Submitted by X-ray Diffraction vs. Dorian Ziss Institute

$Page 1: Submitted by X-ray Diffraction vs. Dorian Ziss Institute$
JOHANNES KEPLER

UNIVERSITY LINZ

Altenberger Str 69

4040 Linz Austria

wwwjkuat

DVR 0093696

Submitted by

Dorian Ziss

Submitted at

Institute of Semiconductor

and Solid State Physics

Supervisor

Julian Stangl

10-2017

X-ray Diffraction vs

Photoluminescence

of

Semiconductor -

Nanostructures

Master Thesis

to obtain the academic degree of

Diplom-Ingenieur

in the Masterrsquos Program

Nanoscience and -Technology

September 22 2017 Dorian 2103


STATUTORY DECLARATION

I hereby declare that the thesis submitted is my own unaided work that I have not used other

than the sources indicated and that all direct and indirect sources are acknowledged as

references

This printed thesis is identical with the electronic version submitted

Linz October 2017



Acknowledgements

The success and outcome of this thesis required a lot of guidance and assistance from many

people and I am extremely privileged to have got this all along the completion of the thesis

Therefore I want to seize the opportunity to thank everyone who contributed and helped me finish

this thesis and lastly also my studies

My first and special thanks go to my supervisor Julian Stangl who always had time for me in all

matters although he had to face turbulent times All that I have done was only possible due to his

enthusiastic supervision and assistance

I highly respect and thank Javier Martiacuten Saacutenchez for providing me with the opportunity to learn

about various fabrication techniques in the cleanroom and for giving me all support and guidance

although he had a busy schedule managing and training many students

I also owe my gratitude to the head of our institute Armando Rastelli as well as to Thomas Lettner

Rinaldo Trotta and Friedrich Schaumlffler who always took keen interest in our concerns and

problems and guided us all along whenever it was needed

Furthermore I should not forget to mention Marc Watzinger Elisabeth Lausecker and Stefanie

Siebeneichler as well as all former and recent colleagues in the office for their encouragement

and moreover for their timely support and guidance until the completion of this thesis

I heartily thank the whole team of the Semiconductor amp Solid State Physics department and

especially Ernst Vorhauer for their support whenever necessary

Finally and most importantly I am very thankful and fortunate to get constant encouragement

support and guidance from my family and specially from my partner in life Bettina Berger who

might have suffered most during the time when I was working on this thesis and related projects


Contents

1 Introduction 9

2 Theory 12

21 Theory of X-ray scattering 12

211 Electromagnetic waves ndash Maxwellrsquos equations 12

212 Generation of X-rays 13

213 Interaction of X-rays with matter 17

214 X-ray scattering on free electrons 18

215 The atomic form factor scattering on atoms 20

216 The structure factor scattering on molecules and crystals 21

217 The reciprocal space Bragg- and Laue-condition 21

218 Refraction and reflection Snellrsquos law for X-rays 23

219 Scanning the reciprocal space by measuring angles and intensities 25

22 Generalized Hookrsquos law the theory of strain and stress 28

221 Introducing the concept of strainstress for isotropic materials 28

222 The elasticity tensor for un-isotropic materials 29

23 Basics of photoluminescence 31

231 Band structure of GaAs 31

232 The effect of stressstrain on the band structure 34

24 Basics of X-ray excited optical luminescence (XEOL) 37

25 Piezoelectric materials 38

251 Properties of PMN-PT 40

3 Device layouts and fabrication 42

31 Investigated samples 42

311 Monolithic device 42

312 Two-leg device 43

32 Fabrication process 43

321 Gold-thermo-compression 45

322 SU8 mediated bonding 45

4 The experiment 46

41 Experimental setups 46

411 XRD-Setup 46

412 PL-Setup 48

413 Synchrotron setup for XRD and XEOL 49

42 Measurements evaluation and data treatment 51


421 Symmetric and asymmetric reflections in two azimuths 51

422 Symmetric reflections used for tilt correction 53

423 Position resolved RSMs 55

425 Footprint strain and tilt distribution 57

426 Track changes with COM calculations 59

427 Tilt varies faster than strain symmetric reflection is the reference 61

428 PL-data ndash Measurements and evaluation 63

5 Results and discussion 65

51 Monolithic devices 65

511 Comparing different bonding techniques ndash Experimental part 65

512 Comparing different bonding techniques - Simulations 67

513 Calibrating the deformation potential XRD vs PL line shift 70


521 Calibrating deformation potentials XRD vs PL line shift 73

522 Discussion of nano-focused beam measurements 76

523 Evaluation of the XEOL measurements 80

6 Summary and outlook 86

7 Appendix 89

71 Python code 89

711 Tilt calculation 89

712 Image processing filters 89

713 Centre-of-mass-calculations routines used for 2D and 3D RSMs 90

72 Reciprocal space maps (RSMs) 93

721 RSMs ndash Monolithic device ndash Gold bonding 93

722 RSMs ndash Monolithic device ndash SU8 bonding 94

723 RSMs ndash Two-leg device 95

724 3D -RSMs ndash Two-leg device - Synchrotron 97

Abbreviations and shortcuts 99

References 100



1 Introduction

Nano-structured semiconductors such as quantum dots (QDs) are very promising for the

realization of light sources used in modern optical applications The QDs offer the possibility to

build single photon sources which can emit single photons on demand The single photon sources

are needed for communication protocols using quantum mechanical properties for encrypted

transfer of information that is intrinsically safe against eavesdropping The most important aspect

in this context is the ability to create single indistinguishable photons which can then be entangled

One problem using QD as the source for single photons is that they have to some extent

ldquostatisticallyrdquo deviating properties after growth via molecular beam epitaxy such as a slightly

different chemical composition or a varying size and strain distribution Thus their ldquoatom-likerdquo

energy states differ slightly from dot to dot and the emitted photons are hence not indistinguishable

anymore To overcome this problem there are generally two possibilities The first one which is

not suitable for integrated devices and rather time-consuming is to simply search for two QDs

with the same intrinsic properties The second possibility is to modify the emission properties of

the QD The first possibility is of course always applicable but there is no guarantee to find two

identical dots Hence the second possibility tuning the energy levels of a QD after growth is the

smarter but more sophisticated solution to overcome this problem Possible ldquotuning-knobsrdquo are

either electricmagnetic fields or strain Several works have been published in the past

demonstrating that the energy levels (band-structure) of QDs can be successfully tuned via strain

electric fields or a combination of both (Trotta et al 2012 Huo et al 2013 Zhang et al 2013

Trotta et al 2015 Martiacuten-Saacutenchez et al 2016 Kremer et al 2014) In this thesis we will focus on

strain which can be used as a ldquotuning knobrdquo for the energy levels The QDs used in this context

are InGaAs QDs embedded in a GaAs matrix which is effectively a 400 nm thick membrane To

get active and reproducible control on the strain state the GaAs membrane is bonded onto a

piezoelectric substrate This configuration allows reversibly inducing strain by applying a voltage

to the piezoelectric substrate with the GaAs membrane on top following the deformation of the

piezo actuator

One particular challenge is that the induced strain in the GaAs cannot be measured precisely from

the optical response of the material (line shift of the GaAs emission) Although the relations

between the strain tensor and the induced changes in the energy levels and band-structure are

theoretically well developed (Chuang 2009) they still rely on material parameters ndash so-called

optical deformation potentials - which are only known to a certain extent (Vurgaftman et al

2001)The colleagues from the institute of semiconductor physics at JKU (Linz) could in principal

determine the effective strain configuration induced in the GaAs just measuring the emission

characteristics extremely accurately (changes in the crystal lattice of about 10-4Å) The most

limiting factors as mentioned are the deformation potentials which can easily deviate by plusmn20

(depending on the particular deformation potential) Hence it is necessary to use a direct method

to get accurate information about the induced strain state without having the problem of intrinsically

high errors Furthermore from the optical measurements alone it is not possible to estimate the

amount of strain that is effectively produced by the piezo actuator which is a benchmark for the

capability of the devices

In this thesis we wanted to find answers to some of the issues which cannot be answered using

only the optical characterization of the devices which have been carried out by our colleagues in

the past Therefore we used X-ray diffraction (XRD) as a technique to get direct information on

the lattice constants in the material and hence information on the induced strain states The first

aim of this work is to investigate how the strain is transferred from the piezo via different bonding

layers to the GaAs and hence to the QDs A second aim is to provide a reliable set of material


parameters (optical deformation potentials) linking the mechanical and optical properties This is

done by comparing the strain measured by X-ray diffraction to the calculated strain form the

changes of the optical emissions and optimizing the deformation potentials to reduce the

differences between calculated and measured strain values

Ultimately this would require measurements by XRD and PL at exactly the same QD under exactly

the same conditions (piezo voltage measurement temperature etc) Such an experiment is

currently not feasible due to a limited resolution of the probing X-rays As a first step we therefore

have performed strain and optical measurements on the GaAs membrane and not on the individual

QDs since the strain state of the QDs follows the induced strain in the membrane With this

approach it was possible to show the feasibility of the experiments and already establish

experimental routines for the final aim which will need to involve X-ray diffraction using nano-

focused synchrotron radiation combined with X-ray excited optical luminescence at cryogenic

temperature On the way we have investigated the strain transfer from the piezo to the membrane

with some unexpected and on first sight counter-intuitive results

The experiments described can be divided into experiments done in our lab and experiments

performed using synchrotron radiation at the ESRF (European Synchrotron Radiation Facility) in

Grenoble France The synchrotron measurements became important when investigating more

complicated structured piezo actuators (Martiacuten-Saacutenchez et al 2016 Trotta et al 2016) which

allow inducing strains with different in-plane orientations Those devices could not properly be

investigated in our X-ray lab due to the limited lateral resolution of the X-ray beam For the optical

experiments in contrast the resolution is just limited by the wavelength of the light-source used

for excitation The problem of having different spot sizes and hence probing different positions on

the same sample makes a valid comparison of PL and XRD measurements complicated To

overcome this discrepancy we employed the focused synchrotron radiation where X-ray spot

sizes in the 500 nm range could be achieved which are even smaller than the spot sizes used for

optical excitation in the lab The synchrotron radiation additionally offers the possibility to measure

XRD and the optical emission in exactly the same sample state This is possible because the

intense and focused high energy X-ray photons induce X-ray excited optical luminescence (XEOL)

in the GaAs membrane The advantage in comparison to the lab measurements is that it is not

necessary to change the setup or move the sample At the synchrotron it is possible to measure

at the same time using only one excitation source for both signals the diffracted X-rays and the

excited photoluminescence Whereas in the lab the sample has to be unmounted from the XRD

setup and re-installed at the optical setup which unavoidably leads to certain changes in the

position that is probed on the sample

For the experiments done in the lab and at the synchrotron many experimental challenges had to

be solved The problems started when investigating the piezo substrates and the GaAs

membranes which did not show a single crystalline behaviour although both were assumed to be

nearly perfect single crystals Distortions and the contribution of different domains to the diffracted

signal lead to a broadening of the Bragg peaks measured via XRD in reciprocal space This made

an accurate determination of the lattice constant rather complicated To be able to resolve and

track differences in the lattice constant which are in the order of 10-3Aring the measurement process

was optimized and a data evaluation procedure which allows a reproducible tracking of the

changes was established

The XRD lab measurements presented in this thesis were used to quantify the effective strain

transfer from the deformed piezo actuator to the GaAs membrane For this purpose using two

different well-established bonding techniques were compared We could successfully explain the

measured differences in the strain transfer by investigating the complicated bonding mechanisms

using finite element method (FEM) simulations for modelling the bonding-interfaces We can say


that the efficiencies we have measured are one of the highest that have been reported for

materials strained in-situ with piezo actuators

In addition a comparison of the XRD data and the measured changes in the PL spectra for various

voltages applied to the actuator allowed for simple and highly symmetric strain configurations to

successfully re-calculate the deformation potential Although a first re-calculation was successful

the error in the measurements was still too high to achieve a valid calibration of the material

parameters This was one of the reasons to perform the same kind of experiments at the

synchrotron For the synchrotron measurements the error was assumed to be much smaller since

the diffracted X-ray beam and the optical luminescence (XEOL) were measured simultaneously

Although the XEOL measurements were very puzzling and not fully successful it could be proven

that XEOL spectra with reasonably high intensities similar to PL measurements in the lab can be

recorded while simultaneously performing XRD measurements which had not been clear at this

point The nano-focused XRD measurements performed at synchrotron additionally showed that

the effective strain- and tilt-distribution of the GaAs membrane is rather complicated and far off a

perfectly modelled system as it was expected to be These results on the other hand were very

helpful to explain the lab measurements where only global changes in strain could be measured

due to the larger beam-spot size

This thesis is divided into six main chapters (including the introduction) It starts with the theoretical

description of the used techniques and mechanisms Within this chapter X-ray diffraction as a

technique to measure interatomic distances in crystalline materials is explained as well as the

theoretical concepts which link the optical properties to the crystal structure using the deformation

potentials Furthermore mechanisms behind XEOL which are relevant when measuring at the

synchrotron and the properties of the piezoelectric material used as actuator are explained In the

next chapter chapter 3 details on the investigated devices (fabrication process propertieshellip) are

presented Chapter 4 is attributed to the experimental details including the XRD and PL

measurements performed in the lab and at the synchrotron Each setup used is explained in detail

together with the data evaluation procedure which was highly relevant for the success of the

experiments Chapter 5 explains and discusses the results for each individual device that was

characterized optically and via XRD By comparing both types of measurements also the

recalculated values for the deformation potentials are discussed in this chapter In the very last

chapter an outlook for further improvements and future projects is given


2 Theory

21 Theory of X-ray scattering

211 Electromagnetic waves ndash Maxwellrsquos equations

This chapter gives an introduction to the basics of X-rays by shortly explaining the most relevant

aspects in terms of electromagnetic radiation their creation and their interaction with matter

X-rays are part of the electromagnetic spectrum and hence can be described as electromagnetic

waves Talking about X-rays the relevant wavelength scale is in the range of 0002 x 10-10m (ultra-

hard X-rays) up to 200 x 10-10m (soft X-rays where the lower end of the UV (ultra-violet) spectrum

starts) Looking at X-ray diffraction the wave-like character of light is the key to understand the

relevant scattering and diffraction processes

In general all electromagnetic waves have to fundamentally obey the electromagnetic wave

equation which can easily be derived starting from Maxwellrsquos equations describing electro-

dynamical processes and assuming that no free charges (120588 = 0) and electrical currents (119895 = 0)

are present The four Maxwellrsquos equations then simplify to

(21) nabla middot = 0 (Gaussrsquos law assuming 120588 = 0)

(22) nabla middot = 0 (Gaussrsquos law for magnetism)

(23) nabla times = minuspart

partt (Farradayrsquos law of induction)

(24) nabla times = 휀01205830part

partt (Amperersquos law assuming 119895 = 0)

Applying the differential operator rot (see appendix) on both sides of Eq (23) and Eq (24) leads

to the so-called wave-equations which must hold for all electromagnetic waves

(25) nabla times (nabla times ) = minuspart

parttnabla times = minus 휀01205830

part2

part1199052 rarr 0 =1

1198882

part2

part1199052 minus ∆

(26) nabla times (nabla times ) = 휀01205830part


part2

part1199052 rarr 0 =1

1198882

part2


The constant c is the speed of light in vacuum and 휀0 the permittivity and 1205830 the permeability in

vacuum They are connected via

(27) 119888 =1

radic12057601205830

Many waves satisfy the electromagnetic wave equations in (25) and (26) but we will focus on

spherical or plane waves When starting with the concepts of scattering and diffraction we will

again refer to these types of waves The electric and magnetic field component of a plane wave

can be described by

(28) = 0ei(119903minusωt) Plane-Wave oscillating of E-field

(29) = 0ei(119903minusωt) Plane-Wave oscillating of B-field


In Eq (28) and (29) is called the wave-vector which points in the direction of propagation The

phase of the plane wave is defined as Φ = 119903 minus ωt with Φ = const ω = 119888|| The length of the

vector for a freely propagating wave is constant and can be written as =2120587

120582 It follows that the

wavelength 120582 and the angular frequency 120596 = 2120587119891 are constant and the energy of each X-ray

photon (basically every photon) can be calculated by

(210) E = ℏ middot ω = h middot f = ℎmiddotc

120582

Eq (210) states that the energy of the photon is proportional to the frequency of the radiation and

indirectly proportional to the wavelength

212 Generation of X-rays

This section will explain in short how X-rays are generated in the context of lab sources and the

differences to the generation of X-rays at a synchrotron radiation facility

Electromagnetic radiation is in the most general description generated if charged particles are

accelerated (indicated by the second derivative of time in Eq (25) and (26)) Having this in mind

and now looking at usual lab sources X-rays are typically generated by accelerating electrons out

of a cathode onto an anode metal-block The cathode is usually a heated filament and the anode

is a solid metal-block mostly copper tungsten or molybdenum When the electrons hit the target

they lose their kinetic energy which can be explained by an acceleration process with negative

sign This process always creates a continuous X-ray spectrum called ldquobremsstrahlungrdquo with a

low-wavelength on-set depending on the maximum electron energy (see Fig 1) The kinetic

energy of the electrons is defined by the acceleration voltage see Eq (211)

(211) 119864119896119894119899 = e middot V = 119898middot1199072

2

If the kinetic energy of the electron exceeds a certain

energy threshold value resonances in the X-ray spectrum

can be observed on-top of the white radiation spectrum

These resonances can be seen as very sharp and intense

lines superimposed on the continuous spectrum They are

called characteristic lines because their position in energy

depends on the anode material That means each anode

material shows a set of characteristic lines unique in

wavelength and energy The appearance of these resonant

lines can be understood by the excitation of K L M

(different electron shells which correspond to the principal

quantum numbers of n=123) electrons which during the

relaxation process emit only radiation in a very narrow

wavelength region corresponding to the energy of the

originally bound electron The short wavelength onset

which has already been mentioned corresponds in terms of energy to the maximum electron

energy which can be converted directly without losses into X-ray radiation This onset can be

calculated using Eq (210) and (211) to

(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881

Figure 1 - Characteristic radiation of a molybdenum target for different electron

energies (Cullity 1978 S 7)


Fig 1 shows the characteristic emission spectrum from a

molybdenum target for different electron energies (measured in

keV)

In Fig 2 one can see the most probable transitions and their

nomination The first letter defines the final shell (K L Mhellip)

and the suffix (α βhellip) the initial shell For instance Kα

means that an electron from the L (n=2) shell undergoes a

radiative transition to the first shell K (n=1) If one zooms

further into the characteristic K or L lines one will see that it

is not one single line in fact they consist of two individual

lines called the doublets for instance Kα1 and Kα2 The

splitting into individual lines can be understood by

considering also different possible orientations for the orbital

angular momentum of the electrons involved in the transition called the fine structure

splitting

In 1913 Henry Moseley discovered an empiric law to calculate the energy of the

characteristic transitions for each element This was historically very important because it

was a justification of Bohrrsquos predicted concept of an atom Moseley originally discovered

his law for the characteristic Kα X-ray emission line which was the most prominent line in

terms of brightness but in general terms this law is valid for each possible (and allowed)

transition see Eq (213) and (214)

(213) 119864119901ℎ119900119905119900119899 = 119891119901ℎ119900119905119900119899 lowast ℎ = 119864119894 minus 119864119891 =119891119877lowast119885119890119891119891

2

ℎ[

1

1198991198912 minus

1

1198991198942] Moseleyrsquos law

(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872

expanded Rydberg-frequency

In Eq (213) Zeff is the effective charge of the nucleus which means it is the atomic number

Z reduced by a factor S which is proportional to the number of electrons shielding the

nucleus at the point of interest Ephoton and fphoton are the energy and the frequency of the

emitted photon and the letters i and f should indicate the initial and the final state of the

transition together with n the principal quantum number For a Kα transition this would

mean that nf = 2 and ni = 1 Eq (214) shows the expanded Rydberg-frequency used in

Eq (213) where M denotes the mass of the nucleus

The process of X-ray generation where the electrons directly interact with the target

atoms is as discussed before widely used for lab sources such as Coolidge-tubes and

Rotating-anode or Metal-jet setups A more detailed discussion on the X-ray source used

for the experiments is given in chapter 4

The important parameter which characterizes the quality of an X-ray source and hence

also sets an intrinsic limit for the resolution obtainable in diffraction experiments is the

brilliance of a light-source (source for X-ray) defined as

(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882

Where mrad2 defines the angular divergence in milli-radiant the source-area is given in

mm2 and the last term (01 BW) denominates the photons falling in relative bandwidth

(BW) of 01 of the chosen wavelength which is a measure for the spectral distribution

Figure 2 - Illustration for the most relevant transitions in terms of X-ray generation


of the emitted photons The given brilliance of a

source already includes the influences of all beam

collimating and shaping elements like mirror-optics

monochromators or apertures Achievable

brilliances for lab sources are in the order of 107-

108 (Eberhardt 2015)

If higher brilliances are needed to successfully

perform the experiments one has to use large

scale facilities dedicated to the production of X-rays

with extremely high brilliances over a large spectral

range known as synchrotron light sources The X-

ray generation on synchrotrons is different to the

generation using lab sources because no direct

interaction of electrons and matter is required Synchrotrons generally are particle

accelerators and use the fact that accelerated electrons emit electromagnetic waves

tangential to the direction of movement Since the speed of the electrons is close to the

speed of light it is a big challenge to keep the electrons on stable trajectories Most

accelerators have a circular shape where magnets and RF (Radio frequency) cavities

form constant electron beams or bunches of electrons which circle in the so-called storage

ring These storage rings could have several hundreds of metres in circumference for

instance the synchrotron located in Grenoble (ldquoEuropean Synchrotron Radiation Facilityrdquo

ESRF) has a circumference of about 844m A sketched synchrotron can be seen in Fig3

The X-ray photons which are finally used for the experiments are produced in bending

magnets or insertion devices Bending magnets are di-pol magnets which ldquobendrdquo the

trajectories of the electrons and hence induce an acceleration proportional to the Lorentz

force

(216) 119865119871 = 119890 + 119890 times

The magnetic field of the bending magnet is given by is the velocity of the electrons

and 119890 their charge The wavelength spectrum (and hence the energy) of the emitted

electromagnetic waves (photons) is directly correlated to the magnetic field inducing the

acceleration (bending radius) and the kinetic energy of the circling electrons The insertion

devices use the same principle but they are built from arrays of magnets to induce an

additional movement or wiggling in the electron

beam which results in higher brilliances There

are two main types of insertion devices which

have differently ordered magnet arrays resulting

in different emission behaviour The first type is

called a wiggler and produces a wavelength

distribution of the emitted radiation very similar

to that of a bending magnet but with a much

higher brilliance The second type is called

undulator In this type the magnet-arrays are

ordered in such a way that the radiation is

coherently added up and hence amplified with

respect to a certain wavelength 120582 This results in

Figure 3 - Simplified sketch of a synchrotron storage ring with an insertion device and beam shaping elements (Als-Nielsen und Des McMorrow 2011)

Figure 4 - Spectral emission characteristics for a bending magnet a wiggler and an undulator are shown on the left side On the right side each of

these devices is sketched


an emission spectrum consisting of sharp lines with much higher brilliance than the peak

brilliances achieved with wigglers but the spectral distribution shows narrow bands in

between these lines The peak brilliances achievable for undulators are in the order of

1020 (Pietsch et al 2004) which is about 12 orders of magnitude higher than for lab

sources The insertion devices and their emission characteristics are depicted in Fig 4

This short introduction of synchrotron radiation should emphasize the advantages of

synchrotrons as sources for the generation of X-rays By simply looking at the brilliances

it becomes clear that experiments which would last for months using lab equipment could

be performed within minutes on a synchrotron (neglecting all other aspects of the

experiment and simply considering the gain in intensity) The details on synchrotron-based

radiation are of course far more complicated than presented in this chapter of the thesis

and I refer for a more detailed discussion to the book ldquoElements of Modern X-ray Physicsrdquo

(Als-Nielsen und Des McMorrow 2011)


213 Interaction of X-rays with matter

When X-rays interact with matter the most prominent effect

observed is known as absorption Absorption describes the loss

of intensity when X-rays are passing through matter which can be

quantitatively explained by the following equation

(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909

I0 is the initial intensity x the penetration length of the X-rays in

the material and μ the absorption coefficient which is directly

proportional to the density of the material The most well-known

picture demonstrating the absorption process of X-rays for

different species of materials is one of the first roentgen-images

recorded by Wilhelm Conrad Roentgen who tried to characterize

the nature of X-rays for the first time in 1895 It shows the right

hand of the anatomist Albert von Koelliker and revealed the power

of X-rays for medical applications see Fig 5 The ldquotruerdquo

absorption process is of course more complicated because in

addition to diffuse scattering the electronic interaction of the X-

ray photons with the atoms of the illuminated material must be

considered The influence of the electronic interaction can be

seen when looking at the mass absorption coefficient μ plotted for different wavelength or

respectively for different energies of the X-ray photons see Fig 6 The resonances which can be

seen as spikes in Fig 6 are specific for each element and are called absorptions edges

Whenever the energy of the X-ray photon is close to the binding energy of an inner bound electron

in the target material the absorption coefficient micro

increases because the photon can then be directly

absorbed by kicking-out an electron The absorption edges

are identified and labelled by the atomic shells where the

absorption occurs This effect is similar to the process of

X-ray generation discussed in the previous section where

resonant peaks in the continuous bremsstrahlung appear

due to the fact that a certain energy threshold which allows

to trigger specific electronic transitions is reached In Fig

6 the mass-absorption coefficient micro for the metal lead (Pb)

is plotted as a function of the wavelength in Aring (equivalent

to 10-10m) and each resonance is labelled according to the

origin (electron shell) of the transition In addition to

absorption effects like diffuse scattering where energy is

transferred to the material also elastic scattering

processes where the energy of X-ray photons is conserved are present

Figure 5 - Contrast differences for the different illuminated materials are clearly visible The higher the density of the material the higher is the absorption

Figure 6 - Atomic mass absorption factor for lead (Pb) at different wavelength The ldquospikesrdquo in the graph are called absorption-edges and are named after the atomic shell where the resonant absorption takes place (Theorie und Praxis der Roumlntgenstrukturanalyse 187)


214 X-ray scattering on free electrons

In case of elastic scattering the X-ray photons

mostly interact with the electrons from the outer

shells which are weakly bound and induce an

oscillation of these electrons with the same

frequency as the incoming photon The

fundamental quantity for this type of scattering

process is called the differential scattering cross-

section (DSC) The DSC describes the flux of

photons scattered in a certain solid angel

element dΩ and is defined as

(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2

The schematic scattering process for a photon scattered on a free electron is shortly discussed in

this paragraph We assume linear polarized photons with an initial intensity I0 and the electron

located in the origin When the photons interact with the electron they induce a force perpendicular

to the direction of the incoming photons The force accelerates the electron and finally results in

an oscillation of the electron around the position of rest which is the origin of the so-called

scattered radiation The scattered radiation has the same frequency as the incoming photon the

energy is conserved

The incoming flux of photons is characterized by the intensity I0 and the electric field vector 119864119894119899

The electron (denoted as e- in Fig 7) gets accelerated along the vector and performs a linear

oscillation which results in linear polarized scattered radiation with the electric field 119864119904119888 and the

intensity Isc measured in point P The vector 119903 has the length R and is tilted by the angle ψ

measured with respect to the incident beam In Fig 7 the scattering process including all

mentioned parameters is sketched

Assuming Thomson scattering (energy of incoming and scattered photon are equal) as dominant

scattering process the scattered electric field can be written as

(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)

The effect of the incoming electromagnetic wave on the electron is described by

(220) 119864119894119899 q = m

This allows re-writing |119864119904119888| and Isc as a function of the angle ψ and the incident electric field |119864119894119899

|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680

For a non-polarized flux of incident photons the electric field vector 119864119894119899 can be decomposed into

an in- and out-of-plane component 119864119894119899 = 119864minus119894119899

+ 119864perpminus119894119899

Figure 7 - Illustration of the scattering process with a free electron On the left-hand side the incoming X-ray photons are characterized by the electric field Ein and the intensity I0 and on the right-hand side the scattered photon flux with Esc and Isc is detected in point P


For minus119894119899 rarr 120595 = 0 119888119900119904 (120595) = 1 and for perpminus119894119899 rarr 120595 =120587

2rarr 119888119900119904(120595) = 0 the intensities can be written

as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2

The total intensity is the sum of the two components The factor 1+cos(120595)2

2 in brackets is called

polarization factor P and the expression 1199022

412058712057601198981198882 is related to the Thomson scattering length or the

classical electron radius 1199030 Summarizing these results the differential scattering cross section

can be re-written as

(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927

The polarization factor is of special interest if synchrotron radiation is used as a source for X-rays

because then the radiation can be strongly polarized


215 The atomic form factor scattering on atoms

From scattering on a single electron the next step is to think about the scattering on atoms with

Z (atomic number) electrons These electrons are quantified by a charge density ρ(119903) which is

defined around the nucleus Since the wavelength of the incident X-ray photons is in the range of

the dimension of the electron cloud one has to consider the phase difference by scattering at

different volume elements in the electron cloud The different scattering contributions from each

point are then super-imposed

The phase difference between the origin and a specific point 119903 in

the electron cloud can be written as the scalar product

(224) ∆120601 = ( minus 119896prime ) 119903 = 119903

denotes the incident wave vector 119896prime the scattered wave vector

and the difference between these vectors the vector is called

scattering vector The length of 119896prime and equals 2120587

120582 considering

an elastic scattering process This scattering process as discussed befor is depicted in Fig 8

Each small volume element dr at position 119903 around the nucleus will contribute to the total scattering

length with a phase-factor of 119890119894119903 hence the total scattering length of an atom can be written as

(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903

The right-hand side of Eq (225) is the fourier--transformed of the charge density and is known as

the atomic form factor In the limit where || = 0 the atomic form factor is equal Z and in the case

where || rarr infin all volume elements scatter out of phase and the atomic form factor becomes

zero

The response of the inner-bound electrons (eg LMhellipshell electrons) to the scattered X-ray

photons is reduced and their contribution to the atomic form factor has to be considered as

frequency-dependent first-order term 119891prime Taking also into account possible absorbtion processes

one has to include an additional imaginary second-order term 119891primeprime The full atomic form factor

includes first- and second-order-dispersion corrections to the original found 1198910and can be written

as

(226) 119891( ℏω) = 1198910() + 119891prime(ℏω) + 119894119891primeprime(ℏω)

For simplicity only 1198910 will be considered as atomic form factor for further explanations but for a

description of absorbtion effects as discussed in 213 within the scattering theory higher orders

are needed

Figure 8 - Scattering on electron clouds


216 The structure factor scattering on molecules and crystals

The explanation of the scattering theory started with the explanation of the scattering process on

a free electron and was then expanded to the scattering on a cloud of electrons surrounding the

nucleus of a single atom the next step is to consider whole molecules as scattering objects This

can be done by writing the sum over all atoms j and their positions 119903 with the corresponding atomic

form factors fj This sum is called structure factor Fmol

(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895

If ony one sort of atoms contibutes to the sum then the structure factor equals the atomic form

factor multiplied by a phase factor 119865119898119900119897 = 119891()119890119894119903 To extend the scattering described for a

molecue to a real crystal as scattering object it is important to define basic properties of perfect

crystals and introduce the concept of the reciprocal space

217 The reciprocal space Bragg- and Laue-condition

The simplest and general description of a crystal is that the atoms are perfectly ordered across

the material This ordering for the simple case where the crystalline material consists of only one

sort of atoms can be qualitatively explained by a regular arrangement of points called lattice The

lattice is defined as an infinite regular pattern of points in a vector space ℝ3 which has a discrete

translational symmetry and can be described by a lattice translation vector to reach every point

in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ

The pre-factors u v w are integers the vectors 119888 are called

primitive lattice vectors These lattice vectors define the edges of

the unit-cell which is the smallest symmetry element still defining

the whole crystal lattice The integer numbers u v w allow an

easy definition of specific direction in the crystal For instance the

space-diagonal is defined by setting u=v=w=1 which is written for

simplicity as [111] and so the face diagonals are defined as [110]

[101] and [011] A crystal lattice which can be characterized this

way is called a Bravais lattice where all lattice points are equal

and the crystal properties remain invariant under translation by a

vector There are 14 different types of Bravais lattices known

which can be constructed but only two of them are of importance

for the investigated materials in this thesis namely tetragonal and the face-cantered-cubic lattice

Therefore only these last two will be discussed in detail For further reading on lattice structures

and crystal types the author refers to (Kittel 2005) In the simple cubic lattice all lattice vectors

have the same length |119886| = |119887| = |119888| and the enclosed angles are equal 90deg = 120573 = 120574 = 90deg The

tetragonal lattice is defined very similarly but not all lattice sides are equal |119886| = |119887| ne |119888| A

sketch of a simple cubic lattice can be seen in Fig 9

For crystals which contain more than one sort of atoms or which have a more complex symmetry

each lattice point can hold a group of atoms or molecules the so-called basis This can eg be

observed in NaCl (Rocksalt) or GaAs (Gallium-arsenide) crystals where one finds two atoms per

lattice point at well-defined positions with respect to the lattice site The simplest basis is of

Figure 9 - Simple cubic crystal lattice


course a basis containing only one atom located at each lattice point Concluding one can say

that a ldquorealrdquo crystal can always be decomposed into a lattice and the basis which defines the sort

and position of atoms sitting on the lattice points

We can now extend the scattering theory deduced for molecules to crystals by introducing another

sum over all lattice points 119877119899 where each lattice point is characterized by the structure factor

which is the contribution of the crystal bases to the scattered intensities So each atom in the

crystal can be accessed with the sum of the lattice vectors and the relative positions of the atoms

119877119899 + 119903 The scattering amplitude for the crystal can be written as the product of two terms the

sum of the unit cell structure factor which is basically the sum of the crystals basis and the second

term is the sum over all lattice points in the crystal as shown in Eq (229)

(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951

Each of these terms is a complex number of the form 119890119894120593 and the sum of all phase factors is in the

order of unity except when all phases fulfil 120593 = 119899 2120587 119899 isin ℕ where the scalar product becomes

(230) = 120784119951120645

In this case the sum in Eq (229) becomes N the number of all lattice points respectively unit

cells To find a unique solution for Eq (230) a vector-space with the dimension of a reciprocal

length [1m] is constructed called reciprocal space The basis vectors lowast lowast lowast fulfil

(231) lowast = lowast = 119888lowast119888 = 2120587 119886119899119889 119891119900119903 119886119897119897 119900119905ℎ119890119903 119901119900119904119904119894119887119890 119904119888119886119897119886119903 119901119903119900119889119906119888119905119904 = 0

Hence a reciprocal space vector can be written in analogy to the real lattice vector 119877 as

(232) = ℎlowast + 119896lowast + 119897119888lowast ℎ 119896 119897 isin ℤ

In Eq (232) the introduced pre-factors h k and l are called Miller-Indices and the vectors lowast lowast lowast

are the basis vectors of the reciprocal space The vector is perpendicular to the set of lattice

planes and is defined by basis vectors and the Miller Indices Since the direction of the basis

vectors is known for most crystal-systems it is sufficient to characterize a certain set of parallel

lattice-planes only by their Miller indices written as [hkl] From the definition of the vector it is

obvious that all reciprocal space vectors satisfy Eq (230) and hence the scattering amplitude for

a crystal given in Eq (229) is non-vanishing when the following equation holds

(233) =

This means when the wave-vector-transfer which is defined

as minus prime equals a reciprocal lattice vector constructive

interference for the scattered intensity in the direction of prime is

observed Eq (233) is also called Laue condition for

constructive interference when talking about X-ray diffraction

The reciprocal basis vectors have to be constructed in such a

way that they are linearly independent but still fulfil Eq (231)

This can be achieved using the following definitions and the

basis vectors in real space to construct them Figure 10 ndash Geometrical explanation of the Laue condition in reciprocal space


(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)

When the basis vectors in real space are known it is rather

easy to construct the corresponding reciprocal space using the

definitions in Eq (234) The spacing between the Bragg peaks

where the Laue condition is fulfilled is given by 2120587119889 see Eq

(235) where d is the lattice spacing in real-space written as

(235) || = || =2120587

119889

In Fig 10 one can see a geometrical explanation of the Laue

condition For an elastic scattering process || = |prime| holds

and one can write the length of the scattering vector as

(236) || =2120587

119889= 2||sin (120579)

The length of the k-vectors is given by 2120587120582 where 120582 is the wavelength of the incoming X-ray

photons which allows re-writing Eq (236) to

(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)

Eq (237) allows an easy explanation for the condition of constructive interference by scattering

on atomic crystal planes in real space Assuming that the angle of the incident X-ray beam equals

the angle of the scattered beam one can calculate the path difference between the reflected beams

on two parallel crystal planes by treating the crystal planes as mirrors which ldquoreflectrdquo the photons

Thereby one obtains the condition where the path difference equals n-times 1205822 (119899 isin ℕ) which

allows constructive interference of the ldquoreflectedrdquo beams This geometrical condition is called

Braggrsquos law and is given in Eq (237) Details on the scattering geometry in real space are

depicted in Fig11

218 Refraction and reflection Snellrsquos law for X-rays

In addition to pure scattering effects of X-rays refraction at sharp interfaces must also be

considered The effect of refraction is characterized by the refraction index n The

refraction index is a complex number which allows considering absorption effects and is

defined as

(238) 119951 = 120783 minus 120633 + 120631119946

The term δ is a function of the first order Taylor-expansion of the structure factor 119891prime and

the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895

Figure 11 - Geometrical interpretation for constructive interference of X-ray photons on atomic layers called Braggs law


The sum over j counts for each different

specimen present in the material with the

corresponding atomic weight Aj and the

corresponding first-order Taylor expansion of

the structure factor 119891119895prime and the atomic

number 119885119895 Na is the Avogadrorsquos number and ρ

the average density of the material

The complex term β is attributed to absorption

effects and is hence a function of 119891119895primeprime (see

chapter 215) fully written as

(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895

If X-rays pass through two different materials with refractive index n1 and n2 with n1gt n2

one part of the photons is reflected into the material with refractive index n1 and the other

part is transmitted into the material with refractive index n2 At the interface between the

two materials a ldquojumprdquo of the electric field vector is observed The momentum component

parallel to the interface-plane is conserved Hence one can write

(241) 1199011 sin(1205791) = 1199012 sin(1205792)

The situation is sketched in Fig12 The energy must be conserved 1198641 = 1198642 which

allows re-writing the refractive index in terms of momentum and energy to

(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642

From these equations (Eq (241) and (242)) one can directly deduce Snellrsquos law for

refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)

For a more detailed explanation of refraction please refer to (Drosdoff und Widom 2005)

Snellrsquos law can be re-written in terms of using the angle 120572 = 90deg minus 120579 as a reference which

is more common in X-ray diffraction where the angles are usually measured against the

materials surface see Eq (244)

(244) 1198991 cos(1205721) = 1198992 cos(1205722)

Like for conventional optics there is also an expression for X-rays for the critical angle

(αc) where all photons are totally reflected Therefore the angle α2 is assumed to equal

zero and one can write

Figure 12 - Snellrsquos law for refraction on a sharp interface Two media with different refractive indices n1 and n2 are assumed where n1gtn2 p1 and p2 are the momentum vectors of the incident and refracted

photons


(245) cos(120572119888) = 1198992

1198991

For X-rays the refractive indices (1198991 1198992) are both close to unity and 120572119888 can be assumed to

be very small which allows the approximation

(246) 120572119888 asymp sin(120572119888) = radic1 minus cos(120572119888)2 = radic1 minus (1198992

1198991

2)

By defining Δn = 1198991 minus 1198992 asymp 0 and hence Δδ = 1205752 minus 1205751 asymp 0 for an interface between

material-1 and the vacuum where 1198991 = 119899119907119886119888119906119906119898 = 1 holds If no absorption effects are

considered the critical angle is defined as

(247) 120572119888 asymp radic2Δn

1198991asymp radic2δ 119908119894119905ℎ 1198992 = 1 minus 120575 (119891119900119903 119907119886119888119906119906119898)

At the angle 120572119888 only an evanescent field which is exponentially damped in the material

can propagate along the surface no direct penetration of the X-ray photons in the material

is possible anymore The calculations for the refractive index and the critical angle when

using X-rays were taken from (Stangl 2014)

219 Scanning the reciprocal space by measuring angles and intensities

In the XRD setup which has been used the sample was mounted on a goniometer which

is a sample stage that allows rotating the sample around four independent axes The

direction of the incident beam was static and the detection system could be rotated around

two axes This configuration allows to ldquoscanrdquo the reciprocal space around certain Q-

positions where constructive interferences for a known set of crystal planes are most

expected This finally allows the determination of the lattice parameter of the investigated

crystalline material with an accuracy of about 10-13m In

this section the methods how to translate angular

measurements into measurements around certain

points in reciprocal space to create so called reciprocal

space maps (RSMs) is discussed

From the six possible axes describing the sample

movement in the incident beam and the movement of

the detection system around the sample two are the

most important ones The translation of the sample

around its axis which is usually defined as angle ω

and the movement of the detection system which is by

convention defined as angle 2120579 The reference or 0deg

position is defined by the direct beam which passes

through the centre-of-rotation with the sample surface

parallel to the beam The geometry is sketched in

Figure 13 - Depiction of the scattering geometry in real space Incident and diffracted beam are indicated by yellow arrows and the sample in dark-blue The possible movement of angle ω is coloured indicated by a light-green circle and the measured deflection as dark-green segment For the angle 2θ the colour scheme is identical the dark red segment

indicates the measured movement


Fig13 The other rotation axes in this experiment are used to

assure that the crystal planes of the sample are in such a

position that the incident and the scattered beam are in the

same plane which is then called coplanar scattering geometry

Looking at the scattering process in reciprocal space the

vectors 119896119894 and 119896119891

indicate the incident and the reflected beam

together with the previously discussed scattering vector see

Fig 14 For the conversion of the measured angles ω and 2θ

to the corresponding components of the scattering vector Qx

and Qz one has to use the geometric relations between the

measured angles 120596 2120579 and the vectors 119896119894 119896119891

and

The length of the incident and diffracted vectors 119896119894 and 119896119891

are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582

From trigonometric relations the angles α β and ϑ (see Fig 14) can be written as

(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587

2minus 120596 119886119899119889 120599 = 120572 minus 120573 = 120596 minus 120579

The length of scattering vector is given by

(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)

This allows finding an expression for the Qx and Qz components using the definition of the

Sine and Cosine functions

(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)

These equations (Eq (251) and (252)) allow

translating each point measured in angular space

defined by 2θ and ω to an equivalent point in the

reciprocal space

If one varies an angle constantly (scans along an

angle) the vector is changed which allows

scanning the reciprocal space along certain

directions Varying the angle ω for instance keeps

the length of preserved but changes its direction

along a circle with the centre located at Qx=0 and

Qz=0 (see Fig 15) Changing the angle 2θ changes

Figure 15 - Illustration of the movement of the

vector along the red or green circles by

changing ω or 2θ If the offset between both angles is kept constant ω-2θ=const only the

length of changes but not the direction ω-

2θ scan

Figure 14 - Scattering process in reciprocal space


the length and the direction of the movement in reciprocal space can again be

described by a circle but the origin is defined by the origin of 119896119894 and 119896119891

If the relation

∆2120579 = 2∆120596 is kept constant by moving both axes simultaneously only the length of is

changed and its direction stays constant This scan is called ω-2θ scan and allows

scanning along a rod in reciprocal space The different angular movements in real space

and their effects in reciprocal space as discussed are schematically sketched in Fig 15


22 Generalized Hookrsquos law the theory of strain and stress

221 Introducing the concept of strainstress for isotropic materials

In this chapter the basic concepts of strain and stress are

introduced since they are of main importance for further

reading and understanding of the thesis

As a first step the physical quantities which describe

strains (ε) and stresses (σ) in materials are introduced

staring with the definition of strain

Strain is a dimensionless quantity which defines the

change in length of an object divided by its original

length written as

(253) 휀 =Δ119871

119871=

119871primeminus119871

119871

Strain is basically a measure for the deformation of the material

The physical quantity stress in contrast to strain is a measure for the force that atoms

exert on each other in a homogenous material and the dimension is that of a pressure

[119873 1198982frasl ] usually given in MPa

Both quantities are expressed by 3x3 tensors the stress-tensor (120590119894119895) and the strain-tensor

(휀119894119895) The basic material properties which are needed to link both quantities in isotropic

materials are

119864119897119886119904119905119894119888 119872119900119889119906119897119906119904 (119884119900119906119899119892prime119904 119872119900119889119906119897119906119904) minus 119916 [119873 1198982frasl ]

119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]

These material properties are not independent they are connected via

(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1

The Youngrsquos modulus describes the effect of normal stress along one of the main axes in the

coordinate system (expressed by the diagonal elements of the stress-tensor 120590119909119909 120590119910119910 120590119911119911) to the

deformation of the material

(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911

Figure 16 - Illustration of the relations between strain stress and the material properties


The deformation of the material along one axis induced by uni-axial stress also forces the

material to deform along the two other axes which are equivalent for isotropic materials The ratio

of deformations for the material along a given axis and perpendicular to this axis is defined by the

Poissonrsquos ratio 120584

(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911

The Shear modulus is related to a deformation of the material in a rotated coordinate system which

means that the strainstress is not given along one of the main axis of the system This deformation

can be expressed by two deformations in a non-rotated coordinate system and is related to the

off-diagonal elements of the strain stress tensors respectively

(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911

The knowledge of either the stress or the strain with the corresponding material properties allows

a full description of the state of the material in terms of internal forces and deformations (see Fig

16 which illustrated these relationships) The strainstress tensor including the diagonal and

shear components is defined as

(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)

222 The elasticity tensor for un-isotropic materials

The concepts developed in chapter 221 for isotropic materials do not hold anymore when one

tries to describe crystalline materials since the elastic properties can vary depending on the

crystalline direction of the material This is even valid for most crystalline materials which consist

of only one sort of atoms (for instance metals) Hence the concept of E G and 120584 has to be modified

to end up with a more general description of the elastic material properties which considers also

the crystalline symmetry

One concept which allows describing un-isotropic materials is the so-called stiffness tensor 119862

The stiffness tensor 119862119894119895119896119897 for a complete un-isotropic material is a four-rank tensor consisting of

36 elements The form of the tensor can be explained by the form and symmetry of the stress

tensor 120590119894119895 having 6 independent components (3 diagonal and 3 off-diagonal) and the stain tensor

휀119894119895 with the same properties

The symmetry of the stressstrain and stiffness tensor allows switching to a more convenient form

of notation called Voigt notation Using Viogtrsquos notation the strain and stress tensors are written

as vectors and the stiffness tensor is written as matrix Voigtrsquos notation is indicated by using the

indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)


The stiffness tensor connects all components of the stress tensor to the strain tensor with the

material parameters 119862120572120573 For symmetry reasons the stiffness tensor in Eq (259) can only have

a maximum of 21 independent elements for a fully un-isotropic material Equation (259) is the

most general form of Hookrsquos law and is applicable for any kind of material

For a perfectly isotropic material for instance the elasticity tensor has only two independent

components and is written as

(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )

The elastic constants of an isotropic material are directly connected to the Youngrsquos modulus and

the Poissonrsquos Ratio via

(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584

For a crystalline material like GaAs which has a cubic crystal structure the number of independent

variables is reduced to 3 and the stiffness tensor can be written as

(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

The stiffness tensor allows calculating the stress-tensor components for a certain deformation in

the material But if one wants to know the deformation caused by certain load on the material one

has to multiply with the inverse stiffness matrix on both sides of Eq (259) Then the 119862120572120573 matrix

becomes a unity matrix and the tensor on the left side of the equation is called compliance matrix

(263) 휀120572 = sum 119878120572120573120590120573120573

Equation (263) connects a load on the material expressed by the stress tensor 120590120573 to a deformation

expressed via 휀120573 Hence by knowing either the stiffness or the compliance matrix one can easily

calculate the other one


23 Basics of photoluminescence

In this chapter the electronic band structure which defines the optical and electrical

properties of the material is introduced and discussed The focus will be on the band

structure properties of GaAs since this material was investigated in detail in this thesis

The theoretical concepts developed will be used to explain the effects of strain and stress

on the measured photoluminescence (PL) for GaAs

231 Band structure of GaAs

GaAs is a single crystalline direct band-gap

semiconductor material For the dispersion relation E(k)

of the crystal this means that the minimum energy state

of the conduction band is at the same crystal momentum

(k-vector) as the maximum energy state of the valence

band Hence the momentum of electrons and holes is

equal and recombination is possible without any

momentum transfer by emitting a photon

A mathematical description of the band structure can be

given if we assume that the electrons in the crystal move

in a periodic potential 119881(119903) The periodicity of the potential

equals the lattice period of the crystal which means that

the potential keeps unchanged if it is translated by a lattice

vector R written as

(264) 119881(119903) = 119881(119903 + )

The Schroumldinger equation for a propagating electron in the lattice periodic potential is

given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]

120595(119903) is the electron wave function which is invariant under translation by a lattice vector

due to the periodicity of the potential 119881(119903)

(266) 120595(119903) = 120595(119903 + )

The general solutions for the Schroumldinger equation in Eq (265) are called Blochrsquos

functions and are given by

(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)

Figure 17 - GaAs band structure calculated in 24-kp model (Zitouni et

al 2005)


119906119899119896(119903) is a periodic function n is the band index and is the wave vector of the electron

with the corresponding energy 119864119899()

An analytical solution which fully solves the band structure model does not exist and

numerical methods are required In Fig 17 a sketch of the calculated band structure for

GaAs is shown

Since we are most interested in the band structure near the direct band-gap (Γ point)

where the radiative transition takes place it is possible to find analytic solutions using the

kp perturbation theory

The next paragraph will explain the most important steps and the results using the

perturbation theory which finally gives an analytical solution for the Γ point of the band

structure

The starting point is to write Eq (265) in terms of (267)

(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)

The full Hamiltonian used in Eq (258) can be written as the sum of

(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898

In Eq (269) 1198670 is the un-perturbated Hamiltonian and 119867119896prime is the perturbation term which

is proportional to the product (kp-perturbation-theory) Solving Eq (268) for the

second order perturbation gives an expression for the eigen-vectors (electron wave

functions) and the eigenvalues (energy bands)

(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩

1198641198990minus119864119899prime0119899primene119899 119906119899prime0

(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2

1198641198990minus119864119899prime0119899primene119899

The term ⟨1199061198990| |119906119899prime0⟩ is called optical matrix element and describes the probability for

a transition from an eigenstate in the valence band to an eigenstate in the conduction

band A matrix-element which equals zero for instance means a forbidden transition

For a complete description of the band structure the Hamiltonian in Eq (269) has to be

modified to take the spin-orbit interaction into account This leads to four discrete bands

conduction heavy-hole (HH) light-hole (LH) and the spin-orbit split-off (SO) bands All

bands are double-degenerated due to two possible spin orientations The new

Hamiltonian is a modification of Eq (268) and in Eq (272) written in terms of the cell

periodic function 119906119899119896


(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2

411989821198882nabla119881 times 120590] times 119906119899119896(119903) = 119864119899

prime119906119899119896(119903)

The operator σ consists of the Pauli spin matrices 120590119909 120590119910 and 120590119911 and acts on the spin-

operator

Another important modification which is necessary for the full description of the band

structure is to consider the degeneracy of the valence bands by using Loumlwdinrsquos

perturbation theory and the correct choice of the basis functions A detailed description of

these band structure calculations using the mentioned methods can be found in (Chuang

2009) The final result of these calculations is the 6x6 Luthering-Kohn Hamilton operator

(given in Eq (273)) with the corresponding eigen-energies and functions which fully

describe the band structure around the direct band-gap

(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877

minus119878lowast 119875 minus 119876 0 119877 minusradic2119876 radic3

2119878

119877lowast 0 119875 minus 119876 119878 radic3

2119878lowast radic2119876

0 119877lowast 119878lowast 119875 + 119876 minusradic2119877lowast minus119878lowast

radic2

minus119878lowast

radic2minusradic2119876lowast radic

3

2119878 minusradic2119877 119875 + Δ 0

radic2119877lowast radic3

2119878lowast radic2119876lowast minus119878

radic20 119875 + Δ

]

The area shaded in red in Eq (273) indicates the matrix without considering the split-orbit

interaction the elements shaded in blue are needed for the full description (the constant

term Δ equals the split-off energy) The matrix coefficients P Q R S and the Hermitian

conjugated which is indicated by the subscription () are given by

(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911

The parameters 1205741 1205742 1205743 are called Luthering-inverse-mass parameters and are

experimentally found correction parameters The eigen-functions will not be discussed

but a detailed description can be found in (Chuang 2009)


232 The effect of stressstrain on the band structure

For a crystalline semiconductor under uniform

deformation it can be assumed that Blochrsquos theorem

still holds because the required periodic properties are

still present they are just ldquomodifiedrdquo This modification

can be expressed by a coordinate transformation from

the strained to the un-strained system schematically

illustrated in Fig 18 for a two- dimensional lattice The

coordinate transformation from an initial position 119903 to the

position 119903prime under uniform strain can be described using the strain-tensor components

휀119894119895 119894 119895 120598 119909 119910 119911

(276) 119903 = 119909 + 119910 + 119911119911 rarr 119903prime = 119909prime + 119910prime + 119911119911prime

(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911

Each strain component can be related to the components of the stress-tensor since the

strain (휀119895) and stress (120590119894) tensors are connected by the materialrsquos stiffness tensor (119862119894119895)

discussed in chapter 22

The coordinate transformation is introduced within the band-structure description by

starting with the Luthering-Kohn Hamiltonian for the un-strained system and then simply

ading a Hamiltonian that is fully attributed to strain which is allowed due to the linearity of

the system

(278) 119867 = 119867119871119870 + 119867120576

Therefore each of the matrix coefficients (P Q R S) in Eq (273) has a corresponding

strain counterpart which is added The new Hamiltonian is called Pikus-Bir and has the

same general arrangement as the Luthering-Kohn Hamiltonian in Eq (273) but the

coefficients are modified and can be decomposed in k-dependent and strain-dependent

parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911 ] + [minus119889(휀119909119911 minus 119894휀119910119911)]

Figure 18 - Transformation from the un-strained to the strained system for a two-

dimensional lattice


The additional factors 119886119907 119887 119889 in the part attributed to strain effects are called the Pikus-Bir

deformation potentials and will be of major interest in this thesis

For the experiments with the monolithic devices which are discussed in detail in the

experimental section the shear components (휀119909119910 휀119909119911 휀119910119911) are negligible and purely

compressive in-plane strain is assumed The experiments will focus only on the region

near the band edge ( = 0) where optical transitions take place Furthermore the split-

off-band can be neglected because for GaAs it is at an energy of 119864119878119874 = Δ = 0341119890119881

(Vurgaftman et al 2001) and the energy differences due to strain in the HH and LH bands

are usually in the range of several tens of milli-electronvolt (Chuang 2009) The Pikus-Bir

Hamiltonian considering all assumptions made for and 휀119894119894 simplifies to

(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]

In this notation it is easy to find an expression for the eigen-energies of the two valence

bands (HH and LH) which are simply the non-vanishing diagonal elements of the matrix

(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

The conduction band energy is given by

(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)

The variable 119864119892 is the gap energy of the un-strained semiconductor (for GaAs 119864119892 = 1519

(Vurgaftman et al 2001)) The eigen-energies of the valence and conduction band allow

calculating the band-gap energy as a function of the applied strain and hence one can

estimate the strain by characterizing the emitted photons in terms of wavelength (and

polarization if the degeneracy of the bands is lifted) These band-gap energies are

(287) 119864119862 minus 119864119867119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

The expression 119886119888 minus 119886119907 in Eq (287) and (288) is simply written as hydrostatic

deformation potential 119886 The parameters 119887 and 119889 in Eq (280 ndash 282) are known as shear

deformation potentials Experimentally found values of the deformation potentials of GaAs

are in the range of

a (hydrostatic) b (shear) d (shear

-814eV ndash -1019eV -166eV ndash -39eV -270eV ndash -60eV

Table 1 - Deformation potential for GaAs (Vurgaftman et al 2001)


In this chapter a mathematical way to describe the band structure which determines the

optical properties is discussed and allows correlating an induced strain to a change in the

optical properties Unfortunately the coefficients which link the strain in the material to the

band gap energies vary within a rather broad range (see Table 1 for GaAs deformation

potentials) which drastically limits the accuracy of calculating strain values from the

measured optical properties


24 Basics of X-ray excited optical luminescence (XEOL)

The XEOL process is very similar to photoluminescence

(PL) both are photon-in photon-out processes where the

light emission from matter in the optical region is

investigated The main differences are the energy ranges

of the incoming photons and hence the excitation paths

For the PL process the incoming photons are usually in the

same energy range as the emitted photons (for visible light

asymp03eV - 1eV) whereas for XEOL the X-ray photons used

for excitation are orders of magnitude higher in energy

(asymp100eV ndash 100keV) than the emitted photons

Looking at the induced PL processes the incoming

photons must have energies equal or higher than the

band-gap of the investigated material When the

photon is absorbed by the material it can create

electron hole pairs in the conduction valence bands

respectively The created electron hole pairs can

undergo non-radiative transitions which are mainly

coulomb scattering or phonon interactions where

momentum can be transferred and the carriers finally

end up in the minimum of the conduction and

maximum of the valence bands The radiative

transition is possible when an electron from the

minimum of the conduction band with a hole from the

maximum of the valence band recombines without

any side-processes by emitting a photon The PL

process is sketched in Fig19

XEOL in direct comparison to induced PL is a more complex process since the energy of

the X-ray photon is sufficient to excite an inner bound electron which leads to de-excitation

cascades and hence many possible side-processes The XEOL process starts when an

X-ray photon is absorbed and an inner bound electron is excited which creates a core-

hole This core-hole is within femtoseconds populated by an electron from a shallower

level via Auger and X-ray fluorescence processes The additionally created shallower

core-holes are filled up by still shallower electrons or valence electrons This cascade

process creates photo- and Auger electrons as well as florescent X-rays which leave an

ionization track in the material until they lose all their energy in thermalization processes

(like in-elastic scattering) or they can escape the sample (Sham und Rosenberg 2007)

When the electron and hole pairs at the very end of these cascade processes recombine

from the conduction valence bands the same type of photon (in terms of energy) is

emitted as in the PL process Concluding one can say that these two techniques are

equivalent and just differ in the energy of the incoming photons which results in a different

relaxation path for the electron hole pairs

Figure 20 - Illustration of the XEOL process the left side shows the excitation of a core level electron the right side the de-excitation cascade with the radiative re-

combination

Figure 19 - Depiction of the PL process The left side shows the excitation of an electron in the valence band to the conduction band The right side shows the radiative recombination process


25 Piezoelectric materials

The piezoelectric effect describes a

material which accumulates an

electric charge proportional to the

pressure applied to the material The

pressure and hence the strain

induced in the material leads to a

change of the intrinsic polarization

which results in the electric charge

accumulation The opposite effect

the converse piezoelectric effect is

observed when a voltage is applied to the material which leads to a mechanical

deformation This effect is generally observed in insulating ferroelectrics and materials

with a permanent dipole moment These materials do not necessarily have to be

crystalline (like rock-salt GaAs and most III-V semiconductors (Fricke 1991 Beya-Wakata

et al 2011)) piezoelectricity is also observed for polymers or materials found in nature

like wood bone or collagen (Ribeiro et al 2015)

The permanent intrinsic dipole moment allows the material to respond to an electric field

by inducing a strain or in the opposite case if a load is applied charges can accumulate

by changing the dipole moment and a voltage can be measured This inherent dipole

moment present in the material requires a certain amount of anisotropy which is

especially important for crystalline materials Looking at centrosymmetric crystal

structures for instance no piezoelectric effect can be observed because the net

polarization chancels out The permanent dipole moments in piezoelectric materials can

be characterized by a dipole-density P which is defined by a vector field Normally these

dipole moments are randomly ordered in ferroelectric domains and have to be re-ordered

(which is possible for most piezoelectric materials) by applying an electric field to the

material The electric field leads to a parallel alignment of the randomly oriented domains

along the field vector and hence to a net-polarization of

the material pointing in the direction of the electric field

The process is called poling and is illustrated in Fig 21

It is crucial to pole the material to get a uniform response

of the material upon an applied electric field in terms of

mechanical deformation The electric field after

successful poling can either be applied parallel to the

net-polarization or anti-parallel If the electric field is

applied parallel this leads to a contraction of the

material along the direction of the electric field vector

For a field applied anti-parallel to the domains the

response of the material is exactly the opposite it

elongates along the field direction see Fig 22 Working

with the electric field anti-parallel to the domains can

Figure 21 - On the left side (a) the un-poled piezoelectric material with randomly oriented domains can be seen The right side (b) shows the poling of the material When a voltage is applied on the top and bottom electrodes an electric field is induced that forced the domains to align parallel to it

Figure 22 - Figure (a) shows the poled piezo without an external field (voltage) applied In figure (b) the applied field is in the opposite direct oriented that the poling axis resulting in a contraction along the poling axis and an expansion perpendicular to it In figure (c) the electric field and the poling axis are aligned parallel which leads to an expansion along the poling axis and a contraction perpendicular to it


easily induce a re-poling of the domains depending on the temperature the piezoelectric

material and the electric field strength

Piezoelectricity can be mathematically described as a combination of the materials

electrical behaviour and Hookrsquos law as discussed in chapter 22 The strain-voltage (strain-

charge) equations describe both effects

(289) 119878 = 119904119864119879 + 119889119905119864 describing the converse piezoelectric effect

(290) 119863 = 휀119879119864 + 119889119879 describing the direct piezoelectric effect

Where D is the electric displacement defined via the permittivity matrix of the material 휀

and the electric field strength 119864 it follows

(291) 119863119894 = 휀119894119895119864119895

S is the strain (in chapter 22 defined as 휀 but re-named to avoid a mixing up with the

permittivity) defined via the compliance matrix 119904 and the stress tensor T (in chapter 22

named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896

The Eq (289) and (290) couple the definitions in Eq (291) and (292) which describe

strainstress and electric effects in the material together with the dielectric constant d

defined in Eq (293) This finally allows calculating the effect of an applied load onto a

material in terms of the measured voltage in Eq (290) or for the converse effect the

structural changes depending on the applied electric field in Eq (291) (assuming all

required material parameters 119889 119904 휀 are known)

In this thesis the converse piezoelectric effect will be used to transfer strain from a

piezoelectric substrate to a semiconductor bonded on top All pre-strains are neglected

and only the changes in strain due to the applied voltage will be considered which

simplifies equation Eq (289) to

(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)


251 Properties of PMN-PT

Here PMN-PT was used as piezoelectric material due to its high

piezoelectric coupling coefficients compared to other commonly

used ferroelectrics such as lead-zirconate-titanate (PZT) or

barium-titanate (BaTiO3) see Tab 2 PMN-PT is a single

crystalline piezoelectric material fully written as lead magnesium

niobate-lead titanate The exact composition that was used for

the experiments was [Pb(Mg13Nb23)O3]071-[PbTiO3]029 In the

given composition the PMN-PT also presents a high

piezoelectric response even at cryogenic temperatures which is

interesting for optical studies which are mostly done at low

temperatures (lt10K) (Herklotz et al 2010 Bukhari et al 2014)

PMN-PT has a perovskite crystal structure and can exist in a

cubic monoclinic triclinic rhombohedral or tetragonal phase

depending on the temperature and exact composition (Ye und Dong 2000 Borges 2011)

In the composition used in our experiments the tetragonal phase was observed

The mineral perovskite has the chemical formula 1198621198861198791198941198743 and in general all perovskite-like

crystal structures can be written in the form 1198601198611198743 where 119860 and 119861 are substituted by the

materialsrsquo specific atoms or molecules For the complex perovskite structure of PMN-PT

the 119860 site is occupied by the 1198751198872+ ion and the site 119861 by either 1198721198922+ 1198731198875+or 1198791198944+ ions

The perovskite crystal structure of PMN-PT is schematically depicted in Fig 23 The

reason for the comparatively high piezoelectric coefficients (119889119894119895) can be understood when

looking at the different possible phases which can be realized in PMN-PT For a specific

chemical composition (similar to the composition which was used in the experiments)

these phases are very close to each other which results in one of the highest reported

piezoelectric responses (Borges 2011)

The response of PMN-PT to an external electric field can be described by Eq (294) taking

into account the crystal structure of the tetragonal phase of PMN-PT (Kholkin et al 2008)

resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915

PMN071-PT029 (Luo et al 2008) 1540 [pCN] -699 [pCN] 164 [pCN]

PZT-5H (Hooker 1998) 585 [pCN] -265 [pCN] -

BaTiO3 (Kholkin et al 2008) 190 [pCN] 038 [pCN]

Table 2 - Strain coupling constants for common piezoelectric materials

Figure 23 - Perovskite structure of PMN-PT the small blue spheres are the oxygen (O) atoms the atom in red is lead centred alternatingly between niobium magnesium and titan atoms The vector P indicates the poling direction along [111] Figure was taken from

(Giorgina Paiella 2014)


The single crystalline PMN-PT was used in the longitudinal extension mode that means

it was poled along the [001] crystalline direction with the corresponding piezoelectric

coefficient 11988933 Along this poling direction the PMN-PT shows the highest response to an

electric field


3 Device layouts and fabrication

This chapter will explain the devices which were used in the experiments A detailed description

of how they are fabricated and an explanation of their working principle will be given for each of

them In general three different devices were fabricated and investigated two of them had a

common device layout but were fabricated using different techniques For the third device the

layout was changed

31 Investigated samples

311 Monolithic device

The first device which were fabricated and investigated were so-

called ldquoMonolithic devicesrdquo Their name comes from the fact that

they had a very simple but effective layout Two devices were

fabricated using different methods to attachbond the GaAs

membrane on top of the PMN-PT substrate

The devices were built from a solid piece of single crystalline

PMN-PT with lateral dimensions of 5 mm x 5 mm and a height

of about 220microm The PMN-PT piezoelectric substrate material

was gold-coated on both sides which was needed for applying

an electrical field On top of the gold-coated PMN-PT a 400 nm

thick GaAs membrane which was gold coated on the side

attached to the PMN-PT was bonded The gold coating was

important for the bonding process (when using gold-bonding

details can be found in chapter 32) and in addition to that it was

very useful for optical investigations of the GaAs because the

gold layer acts as a mirror and enhances the emissions

efficiency Both parts the PMN-PT substrate and the GaAs

membrane on top where glued on an AlN (Aluminium-Nitride)

chip carrier for easier handling and to be able to safely provide

the needed voltage The layout of the assembled device is

sketched in Fig 24

If one applies a voltage parallel to the poling direction of the already poled PMN-PT substrate the

device behaves like an actuator and an in-plane compression and an out-of-plane expansion can

be observed This in-plane compression is transferred via the bondinglayer connecting the PMN-

PT actuator to the GaAs membrane It induces a negative in-plane strain which affects the band

structure as discussed in chapter 23

Figure 24 - Part a) shows a sketch of the monolithic device Small wires connect the top-side of the piezo substrate to a gold-pad on the chip carrier The plus and minus signs indicate the points where the voltage is applied Figure b) shows the working mechanism of the actuator if a voltage is applied parallel to the poling direction This results in an in-plane compression and an out-of-plane expansion


312 Two-leg device

The second type of devices is called ldquoTwo-Legrdquo device

because of the actuator shape The layout differs from the

layout of the monolithic devices since ldquoHrdquo-shaped material is

removed by laser-cutting from the PMN-PT substrate The

bottom side is again fully gold-coated but on the top side only

the two individual legs are coated with gold Between these

two legs which are defined by the ldquoHrdquo-shaped part the GaAs

membrane (gold-coated on one side) is suspended (clamped

on each leg) If now a voltage is applied in poling direction

between the bottom and the top side of the two legs an in-

plane compression of the individual legs can be observed In

Fig 25 the device is schematically depicted Each leg acts as

an individual monolithic device discussed in chapter 311

This bi-axial in-plane compression induced in each of the legs

when a voltage is applied induces a uni-axial in-plane tension

in the suspended GaAs membrane between the legs The

geometrical configuration allows inducing an uni-axial strain-

field in the GaAs membrane which has the opposite sign of

the strain induced in the piezo-actuator Additionally the strain

induced in the suspended part of the GaAs membrane gets

amplified by the ratio of the length of the legs and the length

of the gap between the legs This special strain configuration

is important because it allows studying strain effects on the

band structure for different configurations and not only bi-axial

compressive deformations of the GaAs membrane

32 Fabrication process

The fabrication process is basically identical for both types of devices (monolithic and two-leg)

hence the type of device will not be mentioned explicitly The main differences in the fabrication

are the bonding procedures used to attach the GaAs membrane on the PMN-PT actuator which

could either be done via the gold-thermo-compression technique or by using the polymer SU8

which mediates the bonding For the monolithic devices both techniques were used one device

was fabricated using gold-thermo-compression the other one fabricated using SU8 as bonding

layer This allows a direct comparison of both techniques which is done in chapter 511 For the

two-leg device only SU8 mediated bonding was suitable since the fragile legs can easily break if

a force acts on them which cannot be avoided using gold-thermo-compression Details are

discussed in chapter 321 and 322

The fabrication starts with metallizing the PMN-PT substrate on both sides by depositing a [Cr (10

nm)-Au (100 nm)] bi-layer which is needed for electrically contacting In the next step a multilayer

structure [GaAs(001) substrate- Al07Ga03As(50nm) GaAs (330nm) membrane] which was

separately grown by molecular beam epitaxy (MBE) was coated with the same [Cr (10 nm)-Au

(100 nm)] bi-layer by thermal evaporation The coating on the multilayer structure serves as a

bonding layer and for protection of the membrane during the processing As mentioned before the

metallization of the semiconductor additionally increases the spectral emission efficiency by acting

as a mirror for the optical characterizations

Figure 25 - Figure a) shows a sketch of the gold coated PMN-PT actuator The two individual legs which are gold-coated can be seen the GaAs membrane which is normally suspended between two legs is drawn above the actuator which allows a better view onto the cut PMN-PT actuator Figure b) illustrates what happens if a voltage is applied to the two legs they undergo an in-plane compression which opens the gap between the actuators leading to a tensile strain in the GaAs membrane (not shown in the figure) which is suspended between the legs


The next step is the bonding of the GaAs multilayer structure to the

PMN-PT which is separately discussed in chapter 321 and 322

For further explanations the GaAs structure is assumed to be

bonded on top of the PMN-PT substrate

After the bonding the GaAs membrane is released from the grown

multi-layer structure onto the PMN-PT substrate by selective wet

chemical back-etching of the GaAs substrate This process consists

of three steps (sketched in Fig 26)

i) Starting with a rough non-selective chemical etching of

most of the GaAs substrate with H3PO4 H2O2 (73) (fast

etching rate)

ii) Removal of the remaining GaAs substrate down to the

AlAs sacrificial layer by selectively etching with citric

acid H2O2 (41) (slow etching rate)

iii) Finally etching the Al07Ga03As layer by dipping in HF

(49)

A detailed description of the device fabrication process is given in

(Martiacuten-Saacutenchez et al 2016) The next step is the bonding step

where the membrane is attached to the actuator see chapter 321

and 322

The last processing step is to fix the PMN-PT with the bonded GaAs

membrane on top with silver paint to the AlN chip carrier which had

gold pads for electrical contacts The chip carrier is needed for

easier handling of the devices since they have small dimensions

(05mm x 05mm) and can easily break

To operate the piezoelectric device it is necessary first to pole the

piezoelectric substrate properly which is the very final step This is

done by applying a voltage on top of the substrate progressively in

steps of 1 V up to a total voltage of 150 V This leads to a

ferroelectric ordering of the polarization in the PMN-PT domains

see chapter 23

Figure 26 - The three etching steps are schematically depicted starting with the non-selective etching shown in a) where most of the substrate is removed The next step is the highly selective etching with citric acid and hydrogen-peroxide which stops at the AlGaAs layer depicted in b) and finally the removal of the AlGaAs using HF as shown in c)


321 Gold-thermo-compression

In the bonding process a multi-layer structure containing

the GaAs membrane (part A) is bonded on the PMN-PT

substrate (part B) For gold-thermo-compression

bonding it is crucial that both parts are gold-coated Parts

A and B are then pressed together (with a force of 10

MPa) while they are kept at a temperature of 300degC for

30 minutes to soften the gold-layers in such a way that

the gold can inter-diffuse and form a uniform bonding

layer between both parts Figure 27 shows a sketch of the gold-bonded device

322 SU8 mediated bonding

For the SU8 bonding part B is coated with a 500-nm-thick

SU8 polymer by spin-coating and baked for 5 minutes at

90 degC to evaporate solvents present in as-spinned SU8

Then the SU8-coated part B is pressed against the PMN-

PT substrate part A by applying a mechanical pressure

of about 10 kPa while keeping a temperature of 220 degC

for 15 minutes The 220degC is slightly higher than the

reported glass-transition-temperature of SU8 (Feng und

Farris 2002) which hardens the SU8 and creates an efficient bonding layer It should be mentioned

that a void-less bonding layer is expected when using SU8 by filling all possible gaps between

part A and B during the bonding process Figure 28 shows a sketch of the SU8 bonded device

Figure 27 - Cross section of the gold bonded device

Figure 28 - Cross section of the SU8 bonded device


4 The experiment

In this chapter the experimental details are discussed including explanations of the used

equipment and for data treatment and data evaluation

41 Experimental setups

In the lab two individual setups for the X-ray diffraction measurements and for the PL

measurements were used whereas at the synchrotron a combined setup was available which

allowed the collection of the XEOL data and the data for the diffraction experiments at the same

time Therefore each lab setup will be discussed individually in chapter 411 and 412 For the

synchrotron measurements only the combined XRDXEOL setup will be explained in chapter

413

411 XRD-Setup

The used XRD setup is a semi-commercial rotating anode setup

The X-rays are generated when the electrons are accelerated

from a tungsten cathode (typical acceleration voltage was 40keV

50mA) onto a rotating water-cooled copper cylinder which

was the anode (rotating anode) Due to the rotation of the

anode it is possible to use higher acceleration voltages and

currents resulting in a more intense emission of X-rays

Generator and rotating anode were built by the company Brucker

AXS

The X-rays were horizontally and vertically collimated with a

XENOCS double bent parabolic mirror optic before passing a

monochromator crystal The monochromator used was a

channel-cut monochromator built from a Germanium (220)

crystal which is cut in such a way that the beam is diffracted on two [220] crystal planes When

the beam is diffracted on the first set of crystal planes the Bragg condition (see chapter 217) for

each energy and wavelength is slightly different resulting in an energywavelength separation of

the beam in real space (similar to the colour separation of white light when passing a prism) With

the diffraction on the second set of crystal planes the direction of the beam is changed so that it

is again parallel to the direction of the incident beam

The wavelength separation is achieved by using a slit

system after the monochromator which blocks the

part of the spectrum which is not needed see Fig 29

The monocromator and slit system were aligned to

the CuKα1 line with a wavelength of λ=15406Aring and a

corresponding energy of 8048keV The beam is then

guided through an evacuated tube called flight tube

where a slit system is mounted at the end of the tube

The slit system in addition to the wavelength

separation allows defining the shape of the beam in

the horizontal and vertical direction which was is

necessary to keep the beam-spot size as small as

possible [about 500 microm x 500 microm] The pressure in

Figure 30 ndash Six-circle goniometer with 4 sample axes and 2 detector axes The angle ω (sample movement in direct beam) is named μ and the angle θ (detector movement in scattering plane) is called 120642 (You 1999)

Figure 29 - Channel-cut monochromator built from a Germanium crystal which is cut in such a way that the beam is diffracted on two [220] crystal planes The slit system selects the chosen wavelength by partially blocking the diffracted beam


the flight tube is kept at 10-3 mbar which drastically reduces the loss of intensity The losses with

and without flight tube can be calculated by using Eq 217 describing the absorption The value

of the air absorption coefficient per density for 8kev X-ray photons can be found at NIST data

base and is micro119886119894119903120588119886119894119903 = 99211198881198982119892 (NIST - Database) with a typical air-density of 120588119886119894119903 = 12041 lowast

10minus3 1198921198881198983(119879 = 20deg119862) The length of the flight tube is about one meter 119909 = 1119898 Calculating the

air absorption without the flight tube leads to

(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903

lowast120588119886119894119903lowast119909= 1198909921lowast12041lowast10minus3lowast100 = 03028

This means that if the X-ray beam passes one meter in air 70 of the original intensity gets

absorbed If we now use the flight tube and decrease the density by a factor of 10-4 the absorption

due to air scattering decreases to only 0001 Concluding one can say that using the flight tube

roughly increases the intensity by a factor of 3 for the X-ray setup used in the experiments

Directly after the slit system an adjustable absorber is mounted The absorber is a rotating wheel

basically where aluminium foils of different thickness are embedded Each foil thickness

corresponds to a certain attenuation factor and by rotating the wheel and hence changing the

thickness of the foils the attenuation factor of the beam can be adjusted This allows keeping the

intensity on the detector in a reasonably high range (needed for good statistics) while additionally

ensuring a high dynamic range in the recorded signal

The goniometer where the sample is mounted is a high-resolution 6-circle (4 sample + 2 detector

axes) goniometer built by the company Huber and has an additional piezo-driven x-y-stage

manufactured by Attocube Systems AG in the centre of rotation In Fig 30 the sketched

goniometer can be seen (without the x-y-stage) The x-y-stage allows moving the sample in the

beam with an accuracy of about 1microm and hence allows measuring position-resolved RSMs The

goniometer was additionally equipped with moveable cable connections to a voltage source which

made it possible to safely apply high voltages (up to 250V DC) to the sample while still being able

to move all axes freely (necessary for measuring RSMs)

The detector which was used is a linear detector called Vantec-1 with 1370 channels (50microm width)

manufactured by Brucker AXS From the sample to the detector a second evacuated tube was

mounted

All the circles of the goniometer were equipped with stepper motors and controlled by

programmable logic controllers which are combined in a control-rack developed by

Forschungszentrum Rossendorf The x-y stage and the detector have their own control units

provided by the manufacturing company All components were controlled via a Linux-PC and the

program SPEC SPEC is a software designed especially for instrument control and data

acquisition for X-ray diffraction experiments This software is also widely used at synchrotrons


412 PL-Setup

The PL setup used was originally

designed to provide a collection and

excitation resolution in the microm-range

The samples are normally stored in a He-

cooled cryostat at temperatures below

10K In this thesis however all PL

experiments were performed at room-

temperature and the main focusing

objective which was needed for the microm-

resolution was replaced by a lens

resulting in a collectionemission spot

size on the sample of about asymp300microm to

500microm This adaptation of the original PL

setup was needed to make the measurements comparable to the measurements performed with

the XRD setup

For the excitation a HeNe laser source with a wavelength of 632nm was used For the attenuation

of the excitation power a rotatable 1205822 plate in combination with a fixed polarizer mounted directly

after the laser was used Then the emitted laser-light was filtered by an excitation filter which

supresses the higher harmonics (low-pass filter) The next optical element in the excitation beam

path was a 5050 beam splitter which allows coupling a white light source to the beam path The

white light was needed to optically image the position of the excitation laser-spot on the sample

surface with a CCD Another beam splitter (7030) was needed to send the emission laser and

the white light onto the sample via a focusing lens and collect the emitted light from the sample to

the beam path leading to the spectrometer The emitted light originating from the GaAs

membrane passes again the 7030 beam splitter and a mirror which can be flipped in or out of

the optical path If the mirror is flipped in it allows imaging the sample with a CCD For

measurements with the spectrometer the mirror had to be flipped out Then a second 1205822 plate

and a fixed linear-polarizer were mounted in the beam path This configuration allows performing

polarization-resolved PL measurements through rotation of the 1205822 plate while keeping the linear

polarizer fixed Before the emitted light enters the spectrometer it is filtered through a high-pass

collection filter which blocks the reflected light from the excitation laser The focusing on the

instance slit of the spectrometer was done with a lens In the spectrometer a 600linesmm

diffraction grating was installed and the diffracted signal was measured with a liquid Nitrogen-

cooled Si-based CCD A sketch of the setup containing all mentioned components is shown in

Fig 31 The CCD the spectrometer the rotatable 1205822 plate and the voltage source (to apply high

voltages to the sample) were connected to a control PC which synchronized all measurements

All recorded spectra were stored in the ldquoSPErdquo file format

Figure 31 - Sketch of the PL setup illustrating the excitation (red) the white light (yellow) and emission (violet) beam paths and

indicating the optical elements which are passed


413 Synchrotron setup for XRD and XEOL

The measurements on the

synchrotron offered the great

advantage that it was possible to

measure XRD and the XEOL signal at

the same time This is the only way to

acquire PL and XRD data while the

sample conditions remain absolutely

unchanged which is the preferred way

to characterize these samples The

optical luminescence was excited with

the same X-ray beam as in the

diffraction experiments This combined type of experiment is not possible in the lab due to the fact

that the XEOL process is rather inefficient (see chapter 24) and the excitation intensities are not

sufficient For the lab measurements the sample has to be unmounted and re-installed and the

voltage ramped up and down for each type of experiment which could lead to some possible

changes in the sample conditions

The measurements were performed at the European Synchrotron Radiation Facility (ESRF) in

Grenoble France at the beamline ID1 This beamline is specialized on high resolution micro-

diffraction and imaging X-ray diffraction experiments The minimal beam-spot size that can be

achieved is H 150nm x V 100 nm (ID01 - ESRF) ID01 is a beamline where the X-rays are created

via an undulator insertion device (see chapter 212) which allows to continuously tune the energy

in a range between 60keV ndash 24keV The energy used in the experiments was set to 8keV which

is well below the absorption edges of Ga (103671keV) and As (118667keV) When choosing the

appropriate energy one has to consider the reflections which should be measured and the

possible angular movements of the goniometer axis which are needed to reach these reflections

Furthermore the lenses and focusing optics which might be needed are only optimized for a

certain energy range and one loses resolution (in terms of beam-spot size) by choosing an energy

that is not within this range Experiments at the synchrotron should hence be carefully prepared

in advance since simply varying the energy can lead to many unwanted side-effects

After the undulator the X-ray beam passes some mirror optics which collimate the beam Then a

double crystal monochromator is passed followed by a second mirror system and slits which pre-

define the beam-size and limit the divergence The final focusing for the smallest used beam-spot

sizes (500nm FHW) was done with a Fresnel-zone-plate (FZP) mounted after the slit system If

the FZP was not used for focusing the beam-size was defined only by the slit system and the

dimensions were in the range of about 50microm to 400microm A sketch of the beamline is provided in

Fig 32 The FZP is explained in more detail in Fig33

The sample was mounted on a

6-circle high-resolution

diffractometer (Huber) similar to

the goniometer used in the lab

Also a sample positioning stage

for the x-y-z-directions with

100microm stroke and a resolution

below 1microm was available The

detection system for the X-rays

was a 2D Si-based pixel

Figure 32 - Schematic overview of beamline ID01 with the corresponding distances of the optical elements with respect to the source

Figure 33 - Sketch of the FZP In front of the zone plate a central-beam-stop CBS is mounted to block the 0th-order which would directly pass After the FZP an order-cleaning-aperture OSA is mounted which allows only the 1st order to be in focus on the sample Farther away from the focal spot

(sample) the beam broadens again


detector called Maxipix developed by the ESRF with 512 x 512

pixels (pixel size 55microm) This detector allows measuring RSMs

within one shot no movement of the axes is needed If the

reciprocal space was scanned by moving the axes of the

goniometer the single frames which are measured can be

reconstructed to a 3D reciprocal space map although this was

not necessary for the experiments which were performed

The XEOL emission from the sample was collected using mirror

optics which couples the light into an optical fibre that was

connected to a spectrometer The mirror optics are built from two

parabolic mirrors with fixed focal spots The first focal spot was

aligned to the optical fibre and the second one to the sample

surface Between the two mirrors the light was assumed to be

parallel and a rotating polarizer was mounted to perform

polarization-resolved optical measurements In order to adjust

the focal spot of the second mirror on the sample surface the

whole optical system was mounted on a moveable table with an inclination to the horizon of about

30deg The mirror system as explained is depicted in Fig 34

The spectrometer which was used was borrowed from the company Andor together with the CCD

and a control PC The XRD setup was controlled by the software SPEC and both measurements

were synchronized via a cable connection If the shutter for the X-ray beam was open the CCD

connected to the spectrometer got a signal and started counting for a pre-defined time

Figure 34 - Depiction of the mirror optics The light is collected from the sample using the 1st mirror A rotatable polarizer is mounted in the parallel beam section between the two mirrors The 2nd mirror focuses the light into the optical fibre which is

then connected to a spectrometer


42 Measurements evaluation and data treatment

In this chapter details of the XRD and the PL measurements are discussed Furthermore the

processes which were needed to extract the wanted information from the measured data will be

described Hence the chapter about data evaluation is even more important since most

measurement procedures used are well established state-of-the-art techniques but the important

part was to successfully analyse and interpret the measured data The discussion starts a short

explanation of the measurement procedure for the XRD data the reflections which were measured

and the procedure how the data were evaluated

The software package which has been used to convert the angular measurements in the reciprocal

space is a freely available Python library called Xrayutilities (used versions 14x-15x) A detailed

explanation of the software package can be found in (Kriegner et al 2013) Furthermore self-

written python scripts were used to perform specific evaluation tasks as discussed in chapter 423

426 and 427 all used scripts can be found in the appendix

In chapter 428 the evaluation process for the PL measurements is explained All processing

steps and fitting routines needed for evaluating the PL data which were measured at the institute

were performed with the same software package called XRSP3 The software package as well

as all steps needed to correctly analyse the PL data are discussed in chapter 428

421 Symmetric and asymmetric reflections in two azimuths

The measured GaAs membrane was grown as single

crystalline material with a surface crystallographic

direction of [001] and an [011] in-plane orientation along

the edge of the membrane GaAs in general crystalizes in

a zinc-blend structure which can be described as two

Face-Centred-Cubic (FCC) lattices one containing only

Ga atoms the other one only As atoms which are shifted

along one quarter of the space diagonal into each other

The structure factor for materials crystallizing in the zinc-

blend structure vanishes for certain combinations of hk

and l which means that not all possible reflections are

suitable for diffraction experiments Therefore one has to

choose reflections which have a non-vanishing structure

factor and which can be reached with the setup that is

used The restrictions concerning the setup come from the

fact that the wavelength is usually fixed (for our

experiments CuKα1 λ=15406Aring) and the angel

movement is limited for geometrical reasons hence not all

allowed reflections must necessarily be reachable with

each setup Furthermore the diffraction peak for a certain

hkl combination can be found behind the sample and is

hence not accessible The selection rules for allowed

reflections in the diamond crystal system which shows

basically the same arrangement of atoms as the zinc-

blend structure but consists of only one sort of atoms are

(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889

Figure 35 - In part a) a reciprocal space map of a radial-scan performed for the monolithic device is shown In part b) the intensity is integrated along Qy and the Bragg peaks are assigned to the appropriate materials and reflections


For a zinc-blend system the selection

rules of diamond are applicable but some

forbidden reflections do not completely

vanish In fact all reflections which are

allowed for a FCC lattice still show a small

and non-vanishing diffraction intensity

From the selection rules of the material

and the setup limitations one can choose

the appropriate reflections and calculate

the angles where the Bragg peaks are

expected to be Higher indexed reflections

usually allow a higher resolution in strain

since smaller deviations in the lattice

constant result in larger shifts of the Bragg peaks Unfortunately the scattering intensities become

lower the higher the order of reflection is or the reflection might not be accessible due to the

limitations discussed before

For GaAs the chosen reflections are the [004] symmetric reflection and the [224] asymmetric

reflection both measured in two azimuths (2nd azimuth [-2-24]) For the PMN-PT substrate

underneath the GaAs membrane it was not clear if diffraction peaks could be observed since the

crystalline structure was not fully known and the X-rays had to pass the membrane which

attenuates the X-rays two times To find possible diffraction peaks of the PMN-PT a radial-scan

(120596 minus 2120579 scan see chapter 219) on the monolithic device was performed shown in Fig 35 a) In

this radial scan powder-diffraction streaks from the gold coating covering the whole Qy-range can

be seen The diffraction peaks from the GaAs membrane can be seen as small streaks in Qy and

are a multiple of the [002] reflection which is forbidden according to the diamond selection rules

in Eq (297) but for an FCC lattice still valid The GaAs [004] reflection is also visible as expected

and shows an intensity which is two orders of magnitude higher than the other GaAs reflections

([002] [006]) because it is the first allowed reflection for a diamondzinc-blend structure The bright

and patterned Qy-streaks at equidistant Q-positions correspond to multiples of the [001] PMN-PT

reflection A comparison with database values for X-ray powder diffraction helps to index the

measured diffraction peaks to the different materials and the corresponding sets of crystal planes

see Fig 35 b) From indexing the peaks it is clear that the cubic PMN-PT has similar to the GaAs

a [001] crystalline surface orientation The [002] reflection was the brightest one seen for PMN-PT

and hence it was chosen to be measured The asymmetric reflection which was measured was

the [113] reflection The reciprocal space with all measured reflections [004] and [224] for GaAs

and [002] [113] for PMN-PT is depicted in Fig 36

Figure 36 - Illustration of the reciprocal space Each set of lattice planes is indicated by a coloured dot The yellow dots are the

measured reflections for the GaAs membrane and the blue dots are the reflections measured for the PMN-PT


422 Symmetric reflections used for tilt correction

The symmetric reflections only have a Qz

component which allows calculating the in-

plane lattice constant and furthermore allows

evaluating the tilt caused by an inclination of

the lattice planes to the nominal surface

direction The tilt for the symmetric reflection

can easily be determined by comparing the

nominal q-position (for GaAs [004] and PMN-

PT [002]) to the actual measured q-position

in the RSM and calculating the angle

between the nominal and measured q-vector Fig 37 shows the influence of tilt on the measured

q-vector The measurement has to be corrected by the tilt value which can either be done by

applying a rotation to the recorded reciprocal space map or by simply applying an angular offset

(ωoffset) to each point measured in real space before the conversion to reciprocal space is done

Each point measured in real space is defined by the angles ω and 2θ and the corresponding

intensity value and each point in reciprocal space is defined by a Q-vector and the intensity value

The tilt correction in real space is given by

(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572

Tilt correction in reciprocal space is achieved by an in-plane rotation of the q-vector

(44) 119888119900119903119903 = (cos (120572) minussin(120572) 0sin(120572) cos (120572) 0

0 0 1

)

All measurements were performed in such a way that the incident beam and the diffracted beam

were in the same plane which is known as a coplanar measurement This reduces the Q-vector

to only two non-vanishing components Qx or Qy depending on the azimuth and Qz which is the

same for each azimuth The tilt correction has to be applied for all Bragg peaks measured the

symmetric and the asymmetric ones This is the main reason why measuring the symmetric

reflection is important since a correct tilt calculation is only possible using a symmetric reflection

Figure 37 - The tilt angle is defined as the angle between the nominal q-position of the chosen symmetric reflection and the actual measured q-position of the reflection

Figure 38 - Reciprocal space maps of a GaAs [004] reflection In part a) the RSM as measured p without any tilt correction is shown in b) the tilt corrected RSM is shown


where only the Qz component is not vanishing For the asymmetric reflection the tilt also

influences the lattice constant and the calculation of the correct tilt without knowing the exact lattice

constant (Q-Position of the Bragg peak) is not possible In Fig 38 a tilt correction example for a

measured GaAs [004] symmetric reflection is shown The Python code which was used to

calculate the tilt is given in the appendix chapter 711


423 Position resolved RSMs

For all measurements it was important to know the

exact position where the X-ray beam illuminates the

sample Since the sample dimensions are in the

range of about 5 mm (see Fig 39 a)) and the beam

spot size is about 500 microm x 500 microm positioning the

sample by simply putting it ldquoby eyerdquo in the appropriate

position was not sufficient For the first sample the

monolithic device the wanted position where the

beam should illuminate the sample is in the middle of

the GaAs membrane because at this position the

most uniform strain is expected when a voltage is

applied A position close to a sample edge could

easily lead to an edge dominated strain-distribution

as explained in detail in chapter 512 via finite-

element-method simulations The alignment

procedure to find the middle of the membrane was as

follows first the sample was mounted on the

goniometer in a position where the X-ray beam

somewhere hits the GaAs membrane this was done

by eye Then the GaAs [004] reflection was aligned

properly and x- y-scans along the sample surface

were performed For each position in real space a

short ω-scan of about 15deg was performed This

means a RSM map of a small part in the reciprocal

space was measured for each point in real space

The measured intensities of the RSMs were

integrated and then plotted over the position (see Fig

39) This procedure was necessary since a variation

of the tilt along the sample surface is expected and

without the ω-scan on each x-y position the signal of

the GaAs [004] reflection could be easily lost

although the X-ray beam still illuminates the

membrane

The dimensions of the GaAs membrane could be

perfectly reproduced which is indicated by the rapid

drop of the intensity in Fig 39 b) and c) when the

membrane is not illuminated by the X-ray beam

anymore Finding the middle-position of the

membrane was achieved by performing such x- and

y-scans and aligning the sample to a position between the intensity maxima which indicate the

edge regions of the membrane

For the second measured device the two-leg device the positioning procedure is more elaborate

since the middle of the GaAs membrane is not necessarily exactly the region between the PMN-

PT-legs where the measurement cycles should be performed and the uni-axial strain is expected

The solution was to image the sample while being in diffraction condition Two different images

reproducing the intensitiesrsquo distribution for a certain reflection either the reflection from the GaAs

Figure 39 - In part a) an image of the monolithic device is shown The device is mounted on the sample stage which allows to precisely position the sample in the beam In part b) and c) x and y scans along the samplesrsquo surface are shown the plotted intensity is the integrated intensity of a RSM around the GaAs [004] reflection The intensity enhancement on the edges seen as intensity spikes come from the fact that in the edge regions the etching of the membrane was not uniform and remaining GaAs substrate increases the scattering volume which hence increases the scattered

intensity in these regions


membrane or the PMN-PT substrate were recorded For

the GaAs membrane it was not sufficient to just create

line-scans along the sample surface It was necessary

to create an x-y matrix with the corresponding RSMs for

each combination of x and y similar to the evaluation of

the line-scans for the monolithic device but in two

dimensions

The first image was measured by aligning the PMN-PT

[113] reflection and then raster-scan the sample along

the x and y direction measuring in total 100 x 100

positions and integrating the intensity on the detector for

each of the measured positions The intensity

distribution of the [113] PMN-PT reflection in real space

can be seen in Fig 40 The next step was to align the

GaAs [224] reflection and measure a RSM for each

combination of x and y Looking again at the intensity

distribution this allows the clear identification of the

regions where the GaAs membrane is bonded see Fig

41 Combining both measurements makes an easy

determination of the exact position between the PMN-

PT legs possible where the GaAs is suspended and the

measurement cycles should be performed Although

this procedure for finding the correct measurement

position was very time-consuming (measuring one

PMN-PT map takes about 8 hours) it is the only way to

be sure where to measure on the sample without having

alignment markers which can be identified with X-rays

A combined image of the measured PMN-PT and GaAs

signal in direct comparison with a real sample image is shown in Fig 42 The reason for the choice

of the asymmetric reflections instead of symmetric ones for mapping is the much higher angle of

incidence (ω-angle) which results in a smaller spot-size of the X-ray beam on the sample and

hence sharper contours For

details on the spot-size see

425 The contour plots

shown in Fig 40-42 were

processed by applying

convolution matrix operations

to the raw-data to deduce the

noise and enhance the edge

contrast Details are given in

the appendix see chapter

712

Figure 40 - The integrated intensity of the PMN-PT [113] signal is plotted for each x-y position in real space which allows identifying the position between the legs The intensity fluctuations come from the fact that not each position could be aligned separately and hence the scattering

intensity fluctuates

Figure 41 - The integrated intensity of the GaAs [224] reflection is plotted for each x-y position in

real space

Figure 42 - Fig a) shows a microscope image of the two-leg device and Fig b) the corresponding intensity distribution of the GaAs and PMN-PT reflections


425 Footprint strain and tilt distribution

It was very important to consider the dimensions of the beam-

spot size on the sample (called footprint) since most of the early

measurements did not allow a reproducible correlation of the

applied voltages to a certain strain induced in the GaAs

membrane on top due to the large footprint It was difficult to

accurately determine the changes in strain while simultaneously

correcting the tilt for each voltage step Within these first

approaches the samples were measured with a comparatively

large footprint due to an initial beam diameter of about 1500 microm

(vertically and horizontally) This resulted in a broadening of the

Bragg peak along the Qz and Qy because there were a lot of

inhomogeneities within the illuminated area The measurements with the large footprint were

evaluated in terms of strain and tilt determination by simply manually looking for the global

maximum in the RSMs The additional peak broadening as mentioned in combination with the

manual evaluation of the RSMs does not make an accurate tracking of the strain changes per

applied voltage step possible In this section the reduction of the footprint will shortly be explained

and the effect of peak broadening on the tilt and strain distribution of the measured RSMs will be

discussed

Since the samples are mostly tilted in the direct beam for diffraction experiments the footprint on

the sample can be much larger than the beam diameter An illustration of the effect of the sample

tilt on the footprint is shown in Fig 43 Therefore the first step to reduce the footprint was to reduce

the beam spot size on the sample Two aspects must be considered firstly that the intensity must

still be sufficient for diffraction measurements with reasonable statics and secondly the

collimation of X-ray beam is limited by a certain divergence of the X-ray beam which is the intrinsic

limit of the beam spot size on the sample The divergence effect means that the beam spreads

again after being collimated by a slit system and a further reduction of the spot size cannot be

achieved easily From experimental evaluations of measured RSMs it turned out that a beam-

spot size of 500 microm x 500 microm which is defined by the slit system still allows a reasonably fast

measurement of the RSMs while keeping the illuminated area (or footprint) of the X-ray beam

small enough for a useful strain evaluation of the measured RSMs The effect when the

divergence of the beam limits the minimum spot size was detectable at beam spot sizes smaller

than 250 microm x 250 microm which are not suitable for the measurements due to the low scattering

intensities The calculated footprints for the nominal diffraction angles of the different reflections

are the given in Tab 3

Reflection Incident angle Footprint

PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2

Table 3 - All measured Bragg reflections with the corresponding nominal incident angles and the resulting footprint assuming a beam spot size of 500 microm x 500 microm

Figure 43 - Depiction of the footprint caused by an inclination of the sample in the incident beam


The higher the inclination

and hence the incident

angle (ω) is the smaller is

the footprint The smallest

footprint of 025 mm2 for

the given beam diameter

of 500 microm would obviously

be achieved at an incident

angle of 90deg The

importance of a small

footprint becomes clear

when looking at the strain

and tilt distribution in the

measured RSMs A

smaller footprint leads to

less peak-broadening in

the RSMs because the volume contributing to the diffracted signal is smaller However in the

smaller volume the amount of inhomogeneities (regions with different strain or tilt) which are the

main reason (instrumental resolution is neglected) for the peak broadening is also lower

The inhomogeneities in the illuminated volume can be seen qualitatively by looking at RSMs of

the symmetric reflections where a broadening of the Bragg peak in Qy (or Qx) direction is directly

related to the presence of different tilts and where a broadening along the Qz direction is related

to the presence of different out-of-plane lattice constants (different strains) Examples are given in

Fig 44 for reflections of the GaAs membrane and the PMN-PT In the asymmetric RSM the

broadening is a convolution of effects (the in- and out-of-plane strain distribution as well as the

presence of different tilts) which make an evaluation much more complicated

By decreasing the footprint and hence reducing the broadening induced by inhomogeneities an

evaluation of the RSMs was finally possible

Figure 44 -In part a) the tilt and strain distribution for a symmetric GaAs [004] reflection are shown the marked Qy-range is converted to a tilt distribution of 04deg in total The second peak which appears at higher Qz values can be attributed to parts of the membrane which have a slightly different (0015A) lattice constant than the main peak Part b) shows the PMN-PT [002] reflection and the corresponding tilt range of 072deg for the broadened peak along Qz a broadening is visible but a second peak indicating areas with a different lattice constant cannot be

discriminated


426 Track changes with COM calculations

The evaluation of the RSMs maps by manually determining the global maximum as mentioned in

chapter 425 has a significant drawback since not the whole peak in the RSMs is shifting

continuously Therefore it is not possible to track changes reproducible Applying a voltage and

inducing a strain in the PMN-PT and the membrane leads to a shift of the Bragg peaks but can

additionally induce a strong peak broadening and the emerging of side peaks due to the additional

inhomogeneity and tilt contributions which are induced This is especially relevant for the GaAs

membrane which is in principal a single-crystalline material but inhomogeneities in terms of pre-

strains and different tilts caused by a wrinkling of the membrane are incorporated The wrinkling

is induced due the small thickness of asymp400 nm and the heat treatment during the bonding of the

membrane (see fabrication process in chapter 32) These additional effects make a manual

evaluation of the measured RSMs for different applied voltages too error-prone Therefore the

solution was to use a centre-of-mass (COM) calculation see Eq (45) for each RSM to find the

global maximum and track the changes in strain and tilt effectively

(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894

119868119905119900119905 is the total integrated intensity of the RSM the vector 119894 is a defined position in the RSM with

the corresponding intensity 119868119894 The index 119894 is counts for all measured positions in the q-space The

big advantage of the COM calculation in comparison to the manual evaluation is that it is 100

reproducible However when a voltage is applied and changes in strain and tilt are induced only

the area of the Bragg peaks which causes a shift contributes to the changes in the COM position

Furthermore it should be mentioned that symmetric peak broadening does not affect the COM

position The evaluation procedure using the COM calculation to finally determine the lattice

constant for each voltage set point was the same for all materials and can be decomposed into

three individual steps

- The first step is to take the symmetric RSM as measured and calculate 119862119874119872

- In the second step the 119862119874119872 is used to correct the symmetric and the asymmetric RSMs

for the tilt as discussed in chapter 422

- The third step is to take the tilt-corrected RSMs and calculate for the second time the 119862119874119872

The new COM position is used for the evaluation of the lattice constant

The evaluation process is depicted in Fig 45 for the GaAs [004] and [224] reflections For PMN-

PT the process is the same only the Q-position of the Bragg peaks changes The Python code

used to calculate the COM is given in the appendix see chapter 713


Figure 45 ndash Depiction of the three main evaluation steps starting on the left side by calculating the COM of the symmetric RSM Then the tilt is evaluated and the correction is applied to the symmetric and asymmetric RSM The next step is to calculate again the COM from the tilt-corrected RSMs The new q-position allows a determination of

the the lattice constants


427 Tilt varies faster than strain symmetric reflection is the reference

The data evaluation started with the systematic evaluation

of all measured RSMs in the same way as discussed in the

last chapter 426 For each voltage set point applied to the

piezo four RSMs were measured and evaluated two for

GaAs ([004] [224]) and another two for PMN-PT substrate

([002] [113]) From the symmetric RSMs ([004] and [002])

the tilt and the out-of-plane lattice constants (119876119911 rarr 119886perp)

could be obtained and from the asymmetric RSMs ([224]

[113]) the in-plane (119876119910119909 rarr 119886∥) and out-of-of-plane (119876119911 rarr

119886perp) lattice constants The values obtained for the out-of-

plane lattice constants had to be identical regardless of the

Bragg refection (sym or asym) which was chosen During

the evaluation of the RSMs measured for the two-leg

device a discrepancy in the measured values for 119876119911 and hence out-of-plane lattice constants was

observed depending on chosen GaAs reflections (symmetric or asymmetric) Interestingly this

discrepancy was only significant for the GaAs reflections measured for the two-leg device where

the membrane is suspended between the two PMN-PT legs For the GaAs reflection measured

on the monolithic devices and the PMN-PT reflections no significant discrepancies could be

observed This counter-intuitive behaviour was a bit puzzling and is still not completely

understood but when looking at the tilt distribution of the GaAs [004] reflection (Two-leg device)

one can see that it is twice the tilt distribution of the GaAs [004] reflection from the monolithic

device (compare Fig 44 a) and Fig 46) This is a clear indication that the membrane inbetween

the legs is not perfectly flat and an additional wrinkling can be expected The wrinkling causes a

larger tilt distribution which also leads to higher uncertainties in the evaluation of the asymmetric

RMSs because a broadening of the asymmetric Bragg peak is a result of both the presence of

various tilts and the presence of differently strained areas in the illuminated volume When

applying a voltage to the piezo the changes in tilt vary faster than the induced changes in strain

and the shift of the asymmetric Bragg peak is mainly dominated by the changes in tilt From these

considerations it seems reasonable that the symmetric reflections are more reliable as a

separation of strain and tilt is possible The Qz components obtained from the COM calculations

of the symmetric reflection were

used as a reference for the

asymmetric reflections This

means the length of the Q-

vector obtained from the COM

calculations in the asymmetric

RSMs and the Qz components

from the corresponding

symmetric reflections were used

to calculate the Qy components

via Eq (46) In Fig 47 the

different vector components on

the GaAs [004] and [224]

reflections are illustrated

(46) |119860119904119910119898| = radic119876119910minus1198601199041199101198982 + 119876119911minus119878119910119898

2 rarr 119876119910minus119860119910119904119898 = radic119876119910minus1198601199041199101198982 + 119876119911minus119860119904119910119898

2 minus 119876119911minus1198781199101198982

Figure 46 - Tilt distribution of the suspended GaAs membrane measured on a GaAs [004] reflection for the two-leg device

Figure 47 - RSMs of the GaAs [004] and [224] reflections are shown The components of the Q-vector obtained from COM calculations are sketched the length of the Q-vector in the symmetric reflection has to equal the length of the Qz component in the asymmetric reflection


Avoiding non-physical solutions in terms of

strain configurations that cannot exist by re-

calculating the Qy components according to Eq

(46) one has to proof that Hookrsquos law still holds

(see chapter 22) This was done by measuring

the GaAs reflections in two azimuths one along

the PMN-PT legs and the other one

perpendicular to the legs This allows obtaining

all three strain components 휀119909119909 휀119910119910 휀119911119911 The

two in-plane strain components 휀 119909119909 휀119910119910 were

then re-calculated via Eq (46) Using the elastic

constants of GaAs and Hookrsquos law it is possible

to calculate the 휀119911119911 strain component from the

modified in-plane strain components The

calculated and the measured 휀119911119911 strain values

were finally compared to each other If the

calculated strain values equal the measured

ones (within a certain range defined by the

measurement error) the applied corrections do

not violate Hookrsquos law and non-physical solutions in terms of strain configuration which cannot not

exist can be excluded Therefore we used Eq (259) for an uni-axial strain configuration which

can be expected since the legs of the piezo pull the GaAs membrane only in one direction and

get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )

From Eq (47) an equation for 휀119911119911 can be deduced

(48) 휀119911119911 = minus120576119909119909lowast(11986211+11986212)+120576119910119910lowast(11986211+11986212)

211986212

Using the elastic constants for GaAs (C11 = 1221 [GPa] C12 = 566 [GPa]) taken from (Vurgaftman

et al 2001) the strain component 휀119911119911 can be calculated with the help of the two modified in-plain

strain components (휀119909119909 휀119910119910) In Fig 48 the calculated and the measured out-of-plane strain values

for different voltages applied to the piezo are shown They are in perfect agreement to each other

which proves that the corrections applied via Eq (46) are valid within the elastic regime

The method discussed in this chapter allows a compensation of the measurement error induced

by the rapid changes in tilt while measuring the GaAs membrane suspended between the PMN-

PT legs When applying a voltage at some point the changes in tilt are much more visible in terms

of peak shifting than the changes induced due to different strains For this reason the chapter is

called ldquoTilt varies faster than strain Only the symmetric RSMs allow a separation between strain

and tilt contributions which is hence taken as a reference Furthermore it is shown that the applied

re-calculations are valid within Hookrsquos law and do not lead to forbidden strain configurations All

strain data shown for the two-leg device in chapter 52 are processed according to Eq (46)

Figure 48 ndash Applied voltage vs measured and calculated strain form the GaAs membrane on the two-leg device The black curve shows the calculated strain component

(120634119963119963 = 120656perp) from the two in-plane components (120634119961119961 and 120634119962119962)

and the elastic constants of GaAs The red curve shows the corresponding measured values of 120634119963119963 The differences between both curves are within the estimated error which indicates that the applied corrections are valid


428 PL-data ndash Measurements and evaluation

The recorded PL spectra were evaluated

with the software package XRSP3 (version

33) written in the language Interactive Data

Language (IDL) The software is capable of

displaying and analysing a large number of

recorded spectra and was developed and

continuously improved by Armando

Rastelli head of the Semiconductor and

Solid State Department of the JKU Linz

(2017)

The supported data formats are either

Princeton-files (SPE) or plain ASCII (TXT)

files The displayed data can be fitted with

pre-defined models (Loretz Gauszlig

Bolzman Sine Cosine Mean Min Maxhellip)

which allows an easy extraction of

important information (max energy

polarization intensity) from the acquired PL

spectra Furthermore the program allows

simulating spectra for different zinc-blende

semiconductor materials and the effect of

in-plane-strain on the PL spectra The

simulations are based on the 8-band kp-

theory and are performed by solving the

Luthering-Kohnrsquos Hamiltonian in the tight

binding model near the Gamma point of the

band-structure (details see chapter 23)

The PL spectra can be simulated for most

zinc-blende semiconductors (GaAs

AlGaAs InPhellip) These materials are defined by a set of material parameters including the

deformation potentials and the elastic constants With XRSP3 it is also possible to solve the

inverse problem which means to find the in-plane stressstrain configuration which fits the

measured PL spectrum by providing the set of material parameters In addition to the material

parameters from polarization resolved PL measurements the energies of the two lowest emission

bands E1 E2 (heavy-hole light-hole transitions) the angle of polarization with respect to the [110]

crystal direction and the polarization degree are needed Since the contributions of E1 and E2 are

differently polarized polarization-resolved measurements are required to identify E1 and E2 and

determine their polarization degree The degree of polarization is defined as the ratio between

minimum and maximum intensities (119868119898119894119899119868119898119886119909)11986412 which is measured while rotating the polarizer

and collection a spectrum for each position of the polarizer The values for E1 E2 the polarization

angle and the degree of polarization are used in a chi-square minimization routine which solves

the inverse-problem by finding the strain stress configuration which best fits the measured PL

spectra

For the monolithic device where hydrostatic deformation dominates when a voltage is applied (no

splitting of the light-hole and heavy-hole bands can be measured) it was sufficient to use a

Lorentzian fit-function for each measured spectrum (per applied voltage step) to determine the

Figure 49 - Part a) shows the intensity distribution of the GaAs membrane for each voltage set point applied to the piezo (0V-200V step-size 1V) The red line indicated the peak position in each single spectrum a shift to lower wavelengths for higher voltages can be seen Part b) shows the spectrum measured for 100V applied and the Lorentzian fit in red which was applied to find the exact peak position Note that only a small part of the spectrum indicated by the red line was fitted since the low energy on-set of the spectrum is a temperature dependent broadening and confuses the fitting routine Part c) finally shows the change of the energy peak as a function of the applied voltage


change in the gap-energy Eg The gap-energy is defined by the peak position of the measured PL

spectrum An evaluation example for the monolithic device is given in Fig 49 all data shown in

Fig 49 were processed with XRSP3 and are directly extracted as a screenshot to illustrate the

data processing steps needed to evaluate the PL spectra measured for the monolithic devices

The PL measurements for the two-leg device are more elaborate since for each voltage applied

to the piezo a set of polarization-resolved spectra were recorded Due to uni-axial strain induced

in the GaAs membrane a splitting of the light-hole and heavy-hole bands is expected and hence

two peaks with energies E1 and E2 and different polarizations can be measured For the

polarization resolved measurements for each voltage applied (0V-200V step-size 20V) and each

position of the polarizer (0deg-360deg step-size 8deg) a single spectrum was recorded Each set of

polarization resolved spectra was fitted with a function that considers two individual peaks

contributing to the PL spectrum (combination of

a Lorentzian and a Boltzmann fit function)

From the fitted spectra the parameters E1 E2

and the intensity I1 (corresponding to E1) are

individually plotted as a function of the polarizer

position The position of the high and low

energy components (E1 E2) should not depend

on the polarizer position Therefore the mean

value of E1 and E2 was calculated and finally

used to fit the strain configuration The intensity

distribution I1 allows to calculate the

polarization degree by using a Cosine fit

function the phase-shift of the fitted Cosine

gives the polarization angle Fitting I2 is not

necessary since the polarization degree must

be the same as calculated form the data of I1

and the polarization angle is shifted by 90deg in

respect to the polarization angle of I1 An

evaluation example illustrating all mentioned

steps is given in Fig 50 all data shown in Fig

50 are again processed and evaluated with

XRSP3

The calculation routines in XRSP3 have been

modified by Armando Rastelli to additionally

also vary the deformation potentials a and d if

polarization-resolved spectra and the

corresponding strain configuration are

provided as a set of input parameters This

fitting routine based again on a chi-square

minimization method is not implemented as an

easily accessible function and can only be

activated via the source code requiring a

version of IDL which allows editing and

compiling the source code This fitting routine

is still not fully tested and only works with

specially formatted -TXT files This routine was used in chapter 523 when trying to fit the

deformation potentials by additionally using the strain data obtained via the XRD measurements

Figure 50 ndash Figure a) shows a set of polarization-resolved PL spectra In b) the fitted spectrum for the 80deg polarizer position is shown In c) - e) variation of the fitting parameters E1 and E2 and I1 for the two-peak fitting function vs the polarizer position is plotted These fitting parameters are used to find the mean values and information on the polarization of the high-energy component E1 (shown as inset)


5 Results and discussion

In this chapter all experimental results will be presented and discussed The data processing and

evaluation procedures will not be mentioned explicitly since have already been explained in detail

in chapter 4 Although most problems concerning the lab measurements could be solved we were

not able to determine the absolute strain values with sufficient accuracy however we were able

to track the changes in strain quite precisely The problem is that only distorted membranes and

piezo substrates were measured and no unstrained substrate reflections were available acting as

a reference in reciprocal space Due to the absence of a clear reference the error in strain caused

by a slight mis-alignment of the sample (which cannot be completely avoided) cannot be properly

eliminated This error is in the same order of magnitude range as the pre-strains Instead of

attempting to determine those pre-strains we tracked only the strain changes when a voltage was

applied The measurement for 0V applied is always set as a reference for zero strain

This chapter is divided into two main parts in the first part experimentally measured XRD and PL

data on the monolithic devices are presented and in the second part the XRD PL and XEOL data

for the two-leg device are shown Both types of devices were used to calibrate the deformation

potentials and for each device the results of the calibration calculations are discussed

Furthermore measured strain transfer efficiencies for the differently bonded monolithic devices

with additional simulations on the bonding properties are presented

51 Monolithic devices

The monolithic devices used for investigations are fabricated with two different processing

techniques affecting the bonding procedure discussed in chapter 32 This allows a direct

comparison of the capability of transferring strain from the PMN-PT carrier to the GaAs membrane

for each of the used bonding techniques A detailed discussion of the results will be given in

chapter 511 ndash 512 PL measurements on the bi-axially strained GaAs membrane on top were

performed and the results were compared to the direct strain measurements done with XRD The

comparison between both datasets the induced changes in strain and the PL-line shift allows to

re-calculate the deformation potential a The results on the calibration are shown in chapter 521

511 Comparing different bonding techniques ndash Experimental part

Two differently bonded monolithic devices are investigated each bonding process is discussed in

detail in 32 The data figures and results presented in chapter 511 have already been published

in (Ziss et al 2017) and will not be referenced separately

The measurements started by recoding RSMs around the [004] and [224] Bragg peaks of GaAs

(on-top) and the [002] and [113] Bragg peaks of PMN-PT underneath for the gold and the SU8

bonded devices All reflections were recorded without changing the geometry or moving the

sample in the beam This offered the advantage that a direct comparison of the strain induced in

the PMN-PT and on the corresponding area on top in the GaAs membrane could be measured

simultaneously For each voltage (corresponding to a certain electric field applied across the

piezo) RSMs of the GaAs and PMN-PT reflections were recorded Then the voltage was

increased in a range between 0 V and 200 V with a step size of 25 V After each voltage ramp

(eg 0 V-25 V 1 Vsec) there was a break of about 20 minutes which was needed to limit

drifting effects of the PMN-PT (Ivan et al 2011) that could result in a blurring of the Bragg-peaks

To extract the lattice parameters and hence the strain in each material (GaAs or PMN-PT) the

evaluation procedure for the RSMs was the same as depicted in Fig 45 two COM calculations


were performed the first one was to find

the tilt The second COM calculation was

performed for the asymmetric RSMs [224]

and [113] to find the Q-in-plane

component which was finally used for

evaluation of the in-plane lattice

parameters and the corresponding in-

plain strain values Details on the

reciprocal space RSMs of GaAs and

PMN-PT and the corresponding

calculated 119862119874119872 positions for different

voltages applied can be found in the

appendix (see chapter 721 and 722)

The inhomogeneities in terms of pre-

strains (see (Martiacuten-Saacutenchez et al 2016))

and tilts are in cooperated after bonding

(which can be seen in the RSMs as peak-

broadening distortions or side maxima)

These peak broadening effects are

already discussed for the GaAs

membrane on top in chapter 425 Also in

the RSMs measured for PMN-PT a

significant peak-broadening can be seen

although it is purchased as single

crystalline material and the processing

should not have any influences on the

crystallinity These peak broadening

effects can be attributed to the presence

of multiple domains with a small but finite

angular orientation distribution (ldquomosaicityrdquo) in the order of 03deg (for a single domain) within the

illuminated area (see appendix chapter 721) Atomic force images (AFM) taken from the polished

PMN-PT surface are discussed in detail later in this chapter and confirm the presence of these

domains These peak-width effects are the main contribution to the error of the measured strain

component

Sample PMN-PT - ∆εF (x10-4) GaAs - ∆εF (x10-5) Transfer efficiency -

Gold bonding -1329 plusmn0056 -3360 plusmn028 2528 plusmn 319

SU8 bonding -0950 plusmn0081 -6583 plusmn016 6924 plusmn 783

Table 4 - Strain transfer efficiency ∆εF determined from XRD for the two different bonding techniques gold thermo-compression and bonding mediated by the polymer SU8

Summarizing the results from the XRD measurements in Fig 51 the changes of the in-plane strain

component for both bonding techniques and different electric fields applied can be seen The strain

changes in the bonded GaAs membrane are for both bonding techniques lower than the ones

measured in the PMN-PT actuator which indicates that the strain transfer is not perfect ie the

ratio of the strain changes is smaller than 100 By calculating a linear regression for each set of

data-points a characteristic slope (∆εF) can be obtained which makes it possible to quantify the

Figure 51 - Changes of the in-plane strain component (ε) for the PMN-PT actuator and the GaAs membrane versus electric field for gold bonding (a) and SU8 bonding (b) The areas marked in red represent the loss in strain The error for the given strain values is around 002 and is mainly dominated by the finite peak widths in the RSMs due to mosaicity in the piezo actuator and the bonded GaAs


strain transfer rates The calculated slopes with their corresponding errors and the transfer

efficiencies (strain induced in the PMN-PT actuator equals 100) are given in Tab 4

Interestingly the sample bonded via the SU8 coating shows a much higher transfer efficiency

(69) than for gold thermo-compression bonding (25) although the Youngrsquos modulus of

hardened SU8 (asymp2-4GPa (Gao et al 2010)) is about 20 times lower than the modulus of a thin

gold layer (asymp60-70Gpa (Birleanu et al 2016)) This seems counterintuitive in the first place as one

might think that a harder layer results in a higher strain transfer while the opposite is observed

here The bonding efficiency is actually very sensitive to the interface properties which will be

discussed in chapter 512 and just looking at the material parameters is not sufficient to

understand and model the strain transfer correctly We note that an almost complete strain-

transfer was previously reported for epoxy-glued samples in (Shayegan et al 2003) although the

reached strain levels were about an order of magnitude lower that those achieved here

512 Comparing different bonding techniques - Simulations

FEM simulations for different materials used as bonding

layers and various interface-structures were performed

to allow a deeper understanding of the strain losses The

first step was to transfer the device to an idealized model

by rebuilding each individual layer using the appropriate

elastic material constants and preserving the original

length scales of the device The material parameters

were taken for PMN-PT from (Luo et al 2008) the

polymer SU8 from (Feng und Farris 2002) the thin gold

layers from (Birleanu et al 2016) and the GaAs layer

from (Levinshtein et al 1996) All materials used were

assumed to be linear-elastic

Interestingly the choice of the material mediating the

bonding process (gold or SU8) had no significant

influence on the simulated strain transferred from the

piezo to the semiconductor assuming perfectly bonded

interfaces The strain transfer is always 100 Even for

much softer hypothetic bonding materials such as

rubber-like silicone polymers (Loumltters et al 1997)

(Youngrsquos Moduli of about two orders of magnitude lower

than the Modulus of SU8) no significant strain losses

could be observed in the FEM simulations This on first sight counter-intuitive behaviour can be

understood considering the dimensions of the structure in terms of length scales If the layer-

thickness is much smaller than the lateral dimensions of the structure (which is the case here

since layer thicknesses are in the range of 10-7 m while lateral dimensions are about 10-3 m) the

elastic strain induced and transferred by the individual layers has no possibility to relax except at

the edge regions of the structure Hence only at the edges (on length scales similar to the top

layer thicknesses) significant strain losses are observed due to elastic relaxations whereas in the

sample centre the strain is transferred without any losses from the piezo carrier to the

semiconductor membrane regardless of the materials used for bonding Strain relaxation is thus

relevant only close to structure edges (Zander et al 2009) or for structures with high aspect ratio

(Kremer et al 2014) This is of course true only within the elastic limit ie if no plastic relaxations

Figure 52 - Simulations of the in-plane strain (εyy) transfer efficiency (TE) for the SU8 bonded device It is clearly visible that the only losses in transferred strain from the PMN-PT piezo carrier to the GaAs membrane on-top are observed in the edge regions The effect of strain loss is observed for all edges but only the strain component (εyy) along the y-axis is plotted which explains the asymmetry For a perfectly bonded device as shown the observed losses along the edges are only a few tenth of a percent see colour scale The piezo actuator was set to a compressive strain of -01 which was the maximum measured strain (at 200V applied bias)


or crack formations occur For Au bonding this should be

the case for the material constants and strain ranges in the

order of 01 For SU8 we will discuss this limit below

In Fig 52 simulations of the strain transfer efficiency (TE)

for the in-plane strain component along the surface can be

seen The edge effect is clearly visible whereas the bulk

is strained uniformly The strain transfer efficiency is

quantified by color-coding the relative difference of a strain

component (휀119910119910) in the GaAs and PMN-PT which is given

by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)

휀119910119910(119878119894119898119906119897119886119905119894119900119899) is the simulated strain value and

휀119910119910(119875119872119873 minus 119875119879) the strain induced in the PMN-PT

substrate For the maximum of 100 strain transfer

휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)

In contrast to the simulations the measured strain losses

for the bulk are considerable and can be explained by a more complicated interface structure For

this purpose AFM measurements of the PMN-PT surface before and after gold coating were

performed and revealed a domain-like structure even though the PMN-PT is purchased as single

crystalline material This domain-like structure appears after the surface of the piezo material is

polished to reduce the initial roughness and is also responsible for the mosaicity observed in XRD

The metallization of the piezo crystal does not bury or smoothen the observed domains it

conformably reproduces the PMN-PT surface texture on the surface of the 100-nm-thick gold

layer as shown in Fig 53 The lateral size of a single domain is in the range of 1microm with a peak-

to-valley height of about 4-6 nm This could lead to void areas when the two Au-coated surfaces

are brought into contact during the bonding process resulting in a bonding surface area

significantly below 100 SEM images of cross-sections through gold bonded devices fabricated

in the same way can be found in (Trotta

et al 2012) and clearly confirm the

presence of void areas where no

bonding could be established Thus

qualitative simulations on imperfect

bonding interfaces were performed to

explore the effect of strain losses due to

void areas In Fig 54 simulations of a

rough surface and its effect on the

transfer efficiency can be seen The

rough surfaces have been qualitatively

reproduced replacing the perfect

bonding-layers by a regular pattern of

truncated pyramids on both bonding

faces The top areas of the truncated

pyramids from the gold-coated piezo

and the mirrored ones from the gold-

coated semiconductor are assumed to

be in perfect contact whereas the area

Figure 53 - AFM image of the PMN-PT surface after Au metallization The domain-like structure can be clearly seen The inset shows a height profile along a 5microm line along the surface and allows an estimation of the valley-to-peak distances between the individual domains which is about 4-6 nm

Figure 54 - Simulations of the in-plane strain (εyy) transfer efficiency (TE) for a patterned bonding layer with an array of truncated pyramids mimicking a rough surface In the simulation shown in the left panel only 10 of the total area contributes to the bonding The size of one bonding element in this simulation is about 1microm x 1microm which is in the range of the measured domain-sizes the total lateral model size was reduced due to limited computing power The reduction of the lateral dimensions lead as expected to very prominent edge effects visible along the y-direction for the simulated strain component εyy Nevertheless also in the centre of the structures an overall reduction of the transferred in-plane strain can

be observed


between the truncated pyramids

represents the voids By changing the

size of the top facets of the truncated

pyramids it is possible to continuously

simulate a global bonding ratio between

100 (perfect bonding whole area in

contact) and 0 (no bonding

established) The void parts or

inhomogeneities introduced with the

pyramid-like surface pattern creates

ldquomicro-edgerdquo effects on every imperfect

bonding domain and allows a partial

relaxation of the induced strain That is

why the presence of defects or

inhomogeneities is crucial for the

relaxation of strain and hence to explain

the losses in transferred strain The effect

of bonding inhomogeneities on the

transferred strain can be seen in Fig55

which reveals that the transfer

efficiencies are directly correlated to the bonded area Based on our experimental findings

enhanced gold-bonding quality should be possible by decreasing the surface roughness of the

PMN-PT substrate before bonding ie the height difference between the individual domains In

principle this could be achieved by employing polishing liquids presenting pH factors of ~2 as

demonstrated in (HOPCROFT et al 2005) Other approaches may be viable such as performing

the mechanical polishing of an already poled PMN-PT substrate

Hence in the case of gold-bonding the losses in strain can be very well explained by a reduction

of the effective bonding area due to the intrinsic domain structure of the piezo carrier and a

possible additional roughness induced during gold deposition The efficiency of strain transfer for

a particular sample depends however on the particular details of the bonding surfaces and cannot

easily be predicted quantitatively

For the SU8 bonded samples the domain structure of the piezo substrate should not have any

influence because the liquid SU8 could compensate the

surface roughness by filling up all gaps qualitatively

explaining a higher strain transfer However also for SU8

bonding the strain transfer is significantly below 100 ie

also in this case the bonding layer cannot be

homogeneous and continuous Taking a closer look at the

stress components in the bonding layer for measured

values of strain induced in the piezo material one can see

that the elastic limit of the SU8 (65-100MPa (Spratley et al

2007)) is exceeded at the sample edges or ldquodefectsrdquo in the

interface Simulations shown in Fig 56 illustrate the stress

configuration in the bonding layer We believe that any such

defects or inhomogeneities must trigger plastic deformation

or the formation of small cracks ie lead to a certain

degree of plastic relaxation This most probably occurs

already during the first poling of the device which needs to

Figure 55 - Average strain transfer obtained from a volume-integration in the GaAs layer over the region (12 microm x 12 microm) marked in the inset which is not affected by the edge effect The plot is normalized to 100 strain transfer at 100 bonding area to further eliminate any remainder of edge relaxation effects The plot confirms that a reduction of the bonding area due to domain-like bonding elements leads to substantial losses in transferred

strain

Figure 56 - Simulation of the in-plan strain component for the SU8 bonded device with a given bonding-ratio of 50 The strain differences in the individual bonding-cones are much higher than the elastic limit of SU8


be done after the high temperature bonding step where the Curie-temperature TC asymp 127deg of PMN-

PT (Herklotz et al 2010) is exceeded Afterwards the partly plastically relaxed SU8 layer behaves

elastically Ie during cycling the applied voltage several times over the full range the same strain

state is reached for the same applied bias reproducibly A direct confirmation of this assumption

eg by SEM inspection of cross-section specimen is difficult since the interfaces are hard to

access while the devices are working Further investigations are currently in progress

Concluding one can say that X-ray diffraction measurements clearly show that the devices

fabricated with SU8 as bonding layer show a superior efficiency in terms of strain transfer

compared to the devices fabricated with gold-thermo-compression Considering the different

material constants the measurements seem to be counter-intuitive a ldquosofterrdquo bonding layer leads

to higher transferred strains Simulations on differently modelled bonding-layer-surfaces revealed

that the interface structure is actually more important than the material parameters of the bonding

layer These imperfect interface structures can explain the measured losses in the transferred

strain even if detailed quantitative simulations are not possible due to the interface complexity

Nevertheless these simulations still allow a deeper understanding of processes involved during

the bonding and reveal the reasons for losses in transferred strain

Each of the studied devices is of course individually fabricated and simplified simulation on these

devices cannot fully predict their behaviour Therefore for determining the exact amount of strain

transferred direct strain measurements via independent methods such as X-ray diffraction are

obligatory

513 Calibrating the deformation potential XRD vs PL line shift

For the calibration of the deformation potentials using

the monolithic device where the GaAs membrane is

under bi-axial compression Eq (284) and (285) were

used These equations link the changes in the band

structure to a certain strain induced in the

semiconductor (GaAs membrane) and are deduced and

discussed in detail in chapter 23 (Basics of

photoluminescence) For purely isotropic biaxial strain

the PL emission from the heavy-hole and light-hole

bands cannot be distinguished Therefore their

contributions to the measured PL peak are assumed to

be equal When combining Eq (284) and (285) the

shear deformation potential b cancels out and only the

hydrostatic deformation potential a remains This new equation can be re-written to an expression

were only the changes of the measured band-gap energy the strain components (휀119909119909 휀119910119910 휀119911119911) and

the deformation potential a are contributing

(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)

Equation (52) allows calculating the strain values based on the PL line shifts and the given

deformation potential

Figure 57 ndash Changes in the PL peak position for different electric fields applied to the piezo actuator The maximum electric field applied

corresponds to 200V


For all measurements presented (XRD and PL) the 0V

point is always set as a reference to zero strain This

procedure was necessary since the determination of the

absolute strain values including the pre-strains which

are induced during processing is not possible for the

XRD measurements For calibrating the XRD

measurements in order to obtain quantitative strain

information an unstrained substrate reflection would be

needed as reference in reciprocals space If no

substrate reflection is reachable to correct possible

alignment errors the error is in the range of the

measured strain changes (001 of the lattice

constant)

Equation (52) can be further reduced by considering

only one strain component (휀perp = 휀119911119911 휀∥ = 휀119909119909 = 휀119910119910)

which can be done since the two in-plane strain

components are equal and the out-of-plane strain

component can be expressed via Poissonrsquos ratio and

the in-plane component

(53) Δ119864 = 119938 (2휀∥ + 휀perp) using Poissonrsquos ratio (120584)

Δ119864 = 119938 (2휀∥ + 120584휀∥)

Poissonrsquos ratio 120584 is introduced in chapter 22 and can be

calculated for GaAs using the elastic constants (see

chapter 427) via

(54) 120584 = minus211986212

11986211 (Poissonrsquos ratio)

Equation (53) was used to calculate the changes in

strain from the measured PL shifts In Fig 57 the

changes in the measured band-gap energy (PL line

shift see chapter 428) for the different voltageselectric

fields applied to the gold-bonded monolithic device is

plotted Fig 58 shows the corresponding calculated

strain values using Eq (53) and as an input the

measured PL line shifts from Fig 57 The measured

strain values from the XRD data are shown in Fig 59

Comparing the slopes of the measured in- and out-of-

plane strain values with the calculated ones and

performing a leased square minimization by varying

deformation potential a allows to calibrate this

deformation potential The calibrated value for a is

found where the error between the different slopes is at

a minimum and the highest agreement between the

measured and calculated strain data can be found The calculated strain values using the

calibrated deformation potential together with the measured strain values are sown in Fig 60

Figure 58 ndash The in- and out-of-plane strain values for different voltages applied with the estimated uncertainties The strain values were calculated from the measured changes of the band-gap energy The literature value that was used for deformation potential a is -88eV (Chuang 2009) The dashed lines are linear regressions through each set of data points

Figure 59 - In- and out-of-plane strain obtained from the XRD measurements for different voltages applied to the piezo The dashed lines are linear regressions through each set of data-

points

Figure 60 - Comparison of the strain data obtained from the XRD measurements and the calculated data from the PL measurements using the value for the calibrated deformation potential a shown in red


where it can be seen that both sets of strain data (measuredcalculated) are equal within the

measurement error

The same procedure has been applied for the SU8 bonded device The calibrated values for the

deformation potential of both devices are given in Tab 5

Device Calibrated value deformation potential a

Monolithic device Gold-bonded -104 [eV] plusmn 15 [eV]

Monolithic device SU8-bonded -94 [eV] plusmn 15 [eV]

Table 5 - Calibrated deformation potential for the different devices

The difference between the calibrated values is within the estimated error of 15eV and although

both calibrated values are within the values proposed by literature (8-11eV see Tab1) the main

problem is that the high error (asymp15 eV) does not allow a sufficiently accurate calibration of the

deformation potential to actually set a new reference

The main contribution to the calibration error comes from the error of the XRD measurements

which is caused by the peak broadening The error estimated for the PL data is around one order

of magnitude lower (in finally calculated strain) than the error of the XRD data The PL

measurement error can be easily reduced by cooling the sample which leads to less broadening

of the PL peaks which allows a more accurate determination of the peak position However XRD

measurements at cryogenic temperatures cannot be performed easily since no setup for these

measurements is available and the low temperatures would not improve the XRD measurements

in terms of a smaller error For a reduced error in the XRD measurements the processing of the

devices must be adapted to get rid of the inhomogeneities induced during the processing which

are the main contribution to the broadening of the Bragg peaks


52 Two-leg device

521 Calibrating deformation potentials XRD vs PL line shift

Using the two-leg device to calibrate

deformation potentials has the advantage that

the strain induced in the GaAs membrane

between the two piezo legs is mainly uni-axial

When a voltage is applied to the actuator a bi-

axial compressive stain is induced in the PMN-

PT substrate which leads to a ldquocontractionrdquo of

the legs and hence to a ldquostretchingrdquo of the GaAs

membrane in the direction of the legs The

deformation in the GaAs membrane is hence

not bi-axial anymore (as assumed for the

monolithic devices) which lifts the degeneracy

of the light-hole and heavy-hole bands

Describing the changes in the band structure

deformation potential a is not sufficient anymore

and additionally deformation potential b is

needed This allows re-calculating both

deformation potentials

Systematic XRD measurements were only

performed for the GaAs membrane since the

induced strain in the piezo and the membrane

has an opposite sign and cannot be directly

compared To obtain the information of all three

strain components (휀119909119909 휀119910119910 휀119911119911) XRD

measurements in two azimuths were

performed the first azimuth measured was

parallel to the PMN-PT legs and indexed with

[110] the second azimuth was perpendicular to

the legs and indexed with [1-10] The strain of

the two in-plane strain components (휀119909119909 휀119910119910) is expected to be different since the piezo legs are

pulling in only one direction but the membrane is clamped on all sides onto the piezo substrate

The out-of-plane strain component (휀perp = 휀119911119911) doesnrsquot depend on the chosen azimuth and hence

must be identical in both azimuths In Fig 61 the XRD measurements of the suspended GaAs

membrane for different voltages applied and both azimuths are shown The in-plane strain

components (휀∥minus[110] = 휀119909119909 휀∥minus[1minus10] = 휀119910119910) differ from each other as expected The component

along the legs 휀∥minus[110] in Fig 61 a) shows a tensile strain whereas the strain component

measured perpendicular to the legs 휀∥minus[110] in Fig 61 b) shows a much lower compressive strain

This can be explained since the membrane is stretched in the direction of the legs but clamped

on all edges which leads to a contraction in the direction perpendicular to the legs For the out of

plane strain components (휀perp) no differences in strain are measured which confirms that there is

no dependency on the azimuth

For the optical measurements of the two-leg device for each voltage applied a polarizer was

rotated and every 8deg a spectrum was measured Details of the optical measurements are

Figure 61 ndash In a) the in- and out-of-plane strain components along the legs for different voltages applied are potted b) shows the same measurement but for the azimuth perpendicular to the legs The out-of-plane strain components from the [004] and [224] Bragg peaks are identical sine the [004] measurements were used as a reference


described in chapter 428 The re-

calculation of the deformation potentials

was performed with the software package

XRSP using a chi-square minimization

algorithm to find the best fitting values for

the deformation potentials (a b) The

necessary input parameters were the

measured strain components (휀119909119909 휀119910119910 휀119911119911)

and the energies E1 E2 from the

polarization-resolved spectra the degree

of polarization and the polarization angel

(discussed in chapter 428) However

XRSP calculates the absolute strain

values from the data of the polarization-

resolved spectra and compares them to

the measured ones but the XRD

measurements only allow an accurate

determination of the changes in strain

This means that the calculated and

measured strain values are not

comparable and an offset has to be applied either to the strain values obtained from the XRD

measurements or the calculated values from the PL spectra The chi-squared fitting routine started

by varying both deformation potentials and calculating the strain using the given set of deformation

potentials The strain values calculated for the 0V spectrum were used as an offset for the XRD

measurements which means that both strain values the values obtained via XRD and the

calculated strain values are equal for 0V applied to the piezo This procedure ensured again that

only the changes in strain are relevant and not the absolute values Both sets of strain data for a

chosen combination of a and b are compared to each other and the mean error is calculated For

a certain combination of the deformation potentials the error function will have a minimum and at

this point the deformation potentials are calibrated In Fig 62 a counter plot of the error distribution

for different combinations of both deformation potentials is shown

Deformation potential Calibrated values

a -67 [eV] plusmn 15 [eV]

b -28 [eV] plusmn 15 [eV]

Table 6- Calibrated values for deformation potential a and b

These calibrated values in Tab 6 are far off the values proposed in the literature (refer to Tab 1)

and are unfortunately not consistent with the calibrated values for the monolithic devices

Explanations for these results can be given taking a closer look at the characteristics of the two-

leg device that was used Looking at microscope images (see Fig 63) of the device one can see

that the suspended membrane between the legs seems to be rather flat and not wrinkled at least

on a sub-mm scale (if no voltage is applied) However the bonded area where the membrane is

clamped on the legs of the piezo carrier shows significant inhomogeneities (Fig 63 b)) If a voltage

is applied and the legs start to pull on the membrane the bonding inhomogeneities can induce

highly inhomogeneous strain-fields in the suspended membrane At first thought this effect would

not dramatically influence the reliability of the measurements if both measurements (PL and XRD)

Figure 62 - Colour coded contour plot of the error distribution for the deformation potentials a and b The darker the colour the smaller is the error between measured and calculated strain data The re-calculated values for a and b where the error function has a minimum are given in the inset


probed exactly the same area of the

membrane However since we have a very

limited spatial resolution with the X-ray beam

(see chapter 425) and hence cannot predict

the exact position and shape of the X-ray beam

on the sample (estimated positioning resolution

asymp300 microm) it is very likely that slightly different

areas on the membrane are probed when

performing the XRD and then the PL

measurements Assuming an inhomogeneous

strain distribution on the membrane it seems

reasonable that both measurements are

attributed to different ldquomicroscopicrdquo strain

tensors and hence are not comparable This

could explain that the calibrated deformation

potentials are far from the expected values For

the monolithic device this problem is not

relevant since the bonding is more

homogeneous and the induced strains are

measured to be uniform across the membrane

One solution to avoid the mentioned problems

is to use an intense focused X-ray beam

provided at a synchrotron This offers the advantage that a smaller beam illuminates less

inhomogeneities which can contribute to the measured Bragg peaks Additionally the XEOL

measurements of the exact same position on the membrane would allow a comparable optical

characterization Synchrotron measurements were performed and the results are discussed in the

next chapters 522 and 523

Figure 63 ndash In a) the shape of the two-leg device is schematically sketched and superimposed by a microscope image Part b) shows the zoomed region on one of the PMN-PT legs bubbles indicate that bonding of the membrane on-top of the legs is not perfectly established In c) a photo of the two-leg device mounted on

a chip carrier is shown


522 Discussion of nano-focused beam measurements

All measurements which are presented and discussed in this

chapter were performed at the ESRF (European Synchrotron

Radiation Facility) beamline ID01 with the support of Tobias

Schuumllli beamline responsible who designed the optical setup

and Tao Zhou local contact who helped us performing the

experiments

The samples that were investigated at the synchrotron are leg

devices which are conceptually the same but had different ldquogaprdquo

sizes The ldquogaprdquo size is defined as the distance between the two

tips of the piezo legs where the membrane is suspended We had three different two-leg devices

with 1000 microm (exactly the same device which was used for the lab measurements) 100 microm and

60microm gap size depicted in Fig 64 The advantage of smaller gap sizes is that higher uni-axial

strains along the legs can be induced in the GaAs membranes The ratio between the length of

the legs and gap sizes is a measure for the strain magnification that can be achieved This means

the smaller the gap-size the higher is the strain that can be induced assuming that the length of

the legs is equal Smaller gap sizes could be used since at the synchrotron a nano-focused X-ray

beam was available which fits into the small area between the legs of the piezo

The device with 60 microm gap-size will be discussed in detail We had the possibility to switch

between two different beam configurations in the first configuration the spot size of the X-ray

beam was defined by a slit system which allowed us to use beam spot sizes of 50 microm to 500 microm

The second configuration used a Fresnel zone plate that could be mounted after the slit system

and focused the beam to a spot-size of about 500 nm full width at half maximum The ldquotailrdquo of the

beam could still have several 10 microm The detector that was used was a 2D detector which allows

measuring 3D RSMs For details on the setup see chapter 413

We started measuring the device with 60 microm gap-size and using the

configuration where the beam is defined by the slit system the spot size

was 100 microm x 100 microm (VxH) With this configuration we illuminated the

whole membrane in the region between the piezo legs and could estimate

the average changes of strain induced in the membrane In Fig 65 the

reciprocal space directions with respect to the device position are

depicted For the final measurement the voltage was increased with a

step size of 40V and for each voltage applied two RSMs around the [002]

and [113] reflections were measured The 3D data set of the reciprocal

space measured around each Bragg reflection was integrated along all

three Q-directions and then plotted RSMs of the symmetric GaAs [002]

reflection for 0V and 200V applied to the piezo can be seen in Fig 66 In

the appendix chapter 724 RMSs of all measured reflections symmetric

[002] and asymmetric [113] for different voltages applied are shown All

RSMs show large inhomogeneities contributing to the Bragg peaks

(broadening and side peaks caused by the presence of different tilt and strain contributions) Since

the whole area between the legs was illuminated these results were expected and just confirmed

the lab measurements were qualitatively the same features in the RSM were visible The

evaluation of the RMS to determine the changes of the strain and tilt was identical to the evaluation

of the lab measurements We used a COM algorithm (for three dimensions instead of two used

for the lab measurements) to find the global maximum and tracked the changes when a voltage

was applied

Figure 64 - The gap size is defined as the closest distance between the two legs of the piezo actuator

Figure 65 - Direction in reciprocal space with respect to the position of the device The Qx component is the in-plane component in the direction of the legs and the -Qz the out of plane component


For this device the absolute strain values could

not be determined since the inhomogeneities

dominated the Bragg peak which makes a

quantitative analysis of the absolute strain

values impossible The relative changes of the

induced strain can be tracked rather accurately

and the results are plotted in Fig 67

The changes in the in- and out-of-plane lattice

constants show the same behaviour as

measured for the two-leg device with lab

equipment The out-of-plane component tends

to lower strain values and the in-plane component along the legs to higher ones This behaviour

is perfectly consistent with the changes one would expect see explanations in chapter 512 The

XEOL measurements done with the same device and beam configuration are discussed in chapter

523

This experiment (60microm device and the

configuration with the large beam) was used to get

familiar with the setup and should additionally show

if the lab measurements could be qualitatively

reproduced while simultaneously recording the

XEOL signal

In the next step we changed the beam

configuration by using the Fresnel zone plate

resulting in a spot size of about 500 nm The idea

was to drastically reduce the amount of

inhomogeneities that are illuminated with the beam

to finally end up with RSMs dominated only by a

single Bragg peak This would improve the

precision of the strain determination and would

hence allow a quantitative analysis of the data A RSM around the GaAs [002] reflection is shown

in Fig 68 The beam was already reduced to a size of 500 nm and the Bragg peak was still not

Figure 66 - RSMs of the GaAs [002] reflection are shown by integrating the 3D reciprocal space data along each Q-direction In a) the RMS for the unstrained piezo (0V applied) is shown and in b) the same RSM but for 200V applied is shown

Figure 67 - Relative changes in strain for different voltages applied to the piezo The red triangles show the changes of the out-of-plane lattice constant and the blue hexagons the changes of the in-plane component along

the legs

Figure 68 - RSM of the GaAs [002] reflection with a beam size of 500 nm The piezo was unstrained (0V applied)


homogenous enough to allow an accurate

evaluation of the lattice constant This was very

puzzling since we could not really believe that the

inhomogeneities were still dominating the Bragg

peak at a range below one micrometre that is

illuminated although the membrane is still grown

as single crystalline material

Fortunately the beamline ID01 offered the

possibility to perform high resolution scans along

the sample surface by measuring an individual

RSM for each real space position This technique

is called k-mapping and gave us the change to

investigate the strain and tilt distribution with a

resolution in the microm range of the whole

suspended membrane The idea was to find out

what really happens when a voltage is applied to

the membrane and how the inhomogeneities are

distributed on the different regions and length

scales Each so-called k-map contained 2025

individual 3D RSMs and the challenge was to

extract useful information on strain and tilt For

this purpose the maximum intensity in each

RSM was determined then a cube around this

position was defined and within this cube a COM

calculation was performed to find the position

that was evaluated From these reciprocal space

positions the strains and tilts were calculated

then the voltage was twice increased by 120V

(final voltage applied to the piezo was 240V) and

the same k-map was measured again The

relative changes to the 0V measurement of the

strain and tilt are shown Fig 69 It can be clearly

seen from the strain distribution map in Fig 69-

b) that there is a global trend to lower out-of-

plane strain values when a voltage of 240V is

applied This is in perfect agreement with the

measurements performed with the large beam

where at higher voltages applied the out-of-

plane strain component decreases (see Fig 65)

However there are still ldquoislandsrdquo visible showing

positive strains This is very remarkable since one would expect that the strain induced is rather

homogeneous at least the sign of the strain changes should be constant The areas with the

opposite sign in strain are very prominent at the regions where the membrane is still attached to

the legs of the actuator (grey shaded area in Fig 69 b) and c)) This can be explained since these

regions should follow the strain induced in the piezo like for the monolithic devices but the fact

that even in the middle of the membrane there are areas located next to each other which show

opposite sign in the measured strain changes are an indication that the strains induced are highly

ldquochaoticrdquo This means that although there is a global trend visible the microscopic strain

Figure 69 - In a) the area of the sample that was scanned is highlighted in red The colour-coded map in b) shows the changes of the out-of-plane strain component with respect to the 0V measurements when 240V are applied to the piezo and in c) the same is plotted for the changes in tilt the arrows give the direction and the colour the magnitude of tilt The grey shaded areas in b) and c) are the regions where the membrane is attached to the legs


distribution is mainly dominated by the influence of inhomogeneities that are induced during the

fabrication of the devices Looking at the changes in the lattice-tilt distribution (Fig 69 c)) reflecting

the wrinkling of the membrane after bonding) the same ldquochaoticrdquo behaviour can be observed The

largest changes can be observed in the regions where the membrane is clamped onto the piezo

which can be explained because the forces acting in these regions are strong and the membrane

cannot escape laterally However there are also regions in the middle of the membrane where a

strong wrinkling can be observed indicated by huge changes in tilt which are not expected These

changes are interestingly not correlated with the changes in strain which makes an explanation

even more complicated

With the synchrotron measurements as explained it is clear that it is not possible for the devices

we investigated to determine the exact strain tensor by illuminating the whole area in the gap of

the two-leg device This also explains why the calibration of the deformation potentials using the

two-leg device in the lab was not successful Even small deviations from the measured areas on

the sample could show significantly different strain changes when a voltage was applied

Unfortunately we could not focus the beam small enough to illuminate an area where no

inhomogeneities are visible anymore


523 Evaluation of the XEOL measurements

All XEOL spectra were

recorded with the setup

explained in chapter 413

Fig 34 depicts the setup

schematically and in Fig 70 a

photo of the setup taken at

the synchrotron is shown

The setup is basically built

from two parabolic mirrors a

polarizer and an optical fibre

(All mentioned components

are marked in Fig 70) The

first mirror is used to collect

the light emitted from the

sample surface and the

second one is needed to

focus the collected light into the optical fibre which is connected to a spectrometer In-between the

two mirrors where the beam is parallel a rotatable polarizer is mounted This configuration allows

probing the measured XEOL signal for polarization-dependent components

During our first beam-times it turned out that the most critical issue to detect a reasonably intense

XEOL signal from the GaAs is achieving a good alignment of the whole optical system The setup

is well aligned when the X-ray spot which is the origin of the XEOL emission coincides with the

focus of the collecting mirror (1st mirror) and additionally the focus of the 2nd mirror coincides with

the fibre entrance Ideally also the f-numbers must match to get the optimum transmission of the

signal into the spectrometer Each time the sample is moved during the alignment of the Bragg

peak the optical setup needs to follow in-avoidable small shifts in the range of a few microm The

XEOL emission spot using the highly focused X-ray beam is in the range of 500 nm and the focal

spot of the first mirror (which collects the emitted light) is approximately in the range of 100 microm

hence even a small shift of the setup can lead to significant losses in signal intensity

However an optical table-like setup with individually aligned elements turned out to be

impracticable Instead the whole optical table where the individual optical elements are mounted

(mirrors polarizer fibre holder) is itself mounted on tiltshift stages to decouple the internal

alignment of the individual optical elements from the ldquoglobalrdquo movements of the whole optical

system This configuration is needed to follow the footprint of the X-ray beam For this purpose

the optical table has two in-plane translation axes and one tilt axis which allows adjusting the

inclination The optical table was tilted about 30deg for geometrical reasons which is a compromise

with an acceptable intensity loss in the range of asympCos(30deg) The inclination axis could be adjusted

within about plusmn5deg from the 30deg position

The translation axes are used to bring the focal spot of the parabolic mirror which collects the

XEOL signal from the sample into the centre-of-rotation by moving the whole optical table with

respect to the sample position The mirror used for collecting the XEOL signal is equipped with

two additional tilt axes These axes are needed to align the parallel beam path between the two

mirrors where the polarizer is mounted The second mirror finally focuses the collected light into

the optical fibre The fibre that was used had an opening of 200 microm and was mounted on a fibre-

holder with three translation axes These axes were used to bring the opening of the fibre in the

focus of the second mirror

Figure 70 - Image of the optical table The 1st mirror collects the XEOL signal from the sample and by rotating the polarizer mounted in the parallel beam section between the two mirrors it is possible to differentiate between differently polarized components of the emitted signal The 2nd mirror is used to focus the parallel beam into the optical fiber which is connected to the spectrometer


The alignment procedure started by using a second 10 microm optical fibre mounted at the position

of the sample (centre-of-rotation) which is flooded with white light A CCD was mounted on the

fibre holder instead of the optical fibre which was finally connected to the spectrometer Then a

rough alignment of all optical components was done by checking and optimizing the ldquolight-spotrdquo

emitted from the 10 microm fibre and recorded by the CCD When the light-spot appeared to be sharp

and with high intensity the rough alignment was completed In the next step the CCD was

replaced by the 200 microm fibre which was flooded again with white light to optimize the focal spot

projected on the sample In this configuration the setup is used in the reverse direction light is

sent from the spectrometer side (through the 200 microm fibre) to the sample surface and by changing

the distance and tilt of the optical table the focal spot of the mirror which collects the light can be

adjusted to coincide with the centre of rotation In the last and final alignment step a laser source

was used to illuminate the sample and excite a PL signal from GaAs and by maximizing the

measured intensity of the laser line resolved by the spectrometer the alignment could be refined

The first XEOL signal was measured using a solid piece of GaAs to test the mirror setup and check

if the intensities are high enough to resolve the GaAs emission line which appears at a wavelength

of about 880 nm When looking at the measured spectra it became obvious that a GaAs emission

line was always observed even without exciting the GaAs piece It turned out that the experimental

chamber (experimental hutch) was by far not ldquodarkrdquo as one would expect when the lights were

switched off We found many small ldquolight sourcesrdquo with an emission in the GaAs spectral range

For instance several interferometric position decoders as well as electronic controlling boards

using LEDs contributed to the observed ldquoluminescencerdquo in the measured PL spectra Hence to

avoid measuring the emission from other light sources than the XEOL signal of the GaAs

membrane each possible emission source ndash once it had been identified - was covered with black

tape and in addition to that the optical table was covered with a tentative housing made from

cardboard This procedure allowed us to finally get rid of most of the unexpected emission lines

that could be seen in the recorded PL spectra

Interestingly however there were certain emission lines

appearing only when the X-rays entered the

experimental hutch but they could not be directly

correlated to the excitation of the GaAs Even when the

X-rays did not hit the GaAs these lines were clearly

visible We first thought these lines could be somehow

connected to another LED that indicated the shutter

status (openclose) but we could not find any The

solution for this problem was a different one the X-rays

also ionize the molecules in the air next to the sample

and these ionized air molecules give rise to

characteristic emission lines which are thus always

measured when the X-rays are switched on In Fig 71

the XEOL signal of the GaAs emission and in the

zoomed region the additional gas ionization lines can be clearly resolved Although these

additional lines canrsquot be avoided easily except by using eg an evacuated sample environment

(which is unfortunately incompatible with other experimental requirements like the electrical

connections to the sample) they can be removed during data processing by recording ldquodarkrdquo

spectra with the X-ray beam entering the experimental hutch but without directly illuminating the

GaAs The ldquodarkrdquo spectra have to be measured with exactly the same settings and integration

times as the spectra containing the XEOL signal to avoid intensity differences when correcting for

the measured air ionization peaks

Figure 71 - XEOL signal of GaAs and in the zoomed region shown in the inset the emission lines from the gas molecules of the surrounding

air can be identified


The first XEOL measurements on the two-leg device with different voltages applied were

performed using the X-ray beam with the large-beam configuration (100 microm x 100 microm) The

device measured first was the device with 60 microm gap-size between the actuator legs It turned

out that a re-alignment of the optical setup using the XEOL emission as source (instead of the 10

microm fibre at the sample position) had to be done since otherwise the intensity of the GaAs emission

was extremely low The intensity could be increased by two orders of magnitude doing a re-

alignment For this purpose we used only the translation and tilt axes of the optical table which

define the position and shape of the focal spot from the mirror that collects the signal but we left

the ldquointernal alignmentrdquo of the optical table untouched This additional alignment procedure was

extremely time-consuming since the translation and the tilt axes could only be adjusted by

manually turning a micrometre-screw Simultaneously turning the screw and measuring spectra is

not possible since for radiation safety reasons the experimental hutch is interlocked and canrsquot be

accessed while the shutter of the X-rays is open Hence the only way to align the setup by

tracking the XEOL intensity is to break the interlock open the hutch turn the screw by a small

amount leave the hutch set the safety interlock and open again the shutter for the X-rays This

procedure takes between 2-5 minutes for each tiny movement of each single alignment screw

One can easily imagine that aligning the setup this way is extremely time-consuming but could

not be avoided in our case In future experiments it will thus be mandatory to motorize the optical

alignment stages so that the alignment can be done remotely controlled with the x-ay beam left

on The illuminated area on the sample and the focal spot of the mirror optics have to coincide for

measurements with reasonable intensities

Measured spectra for a fixed polarizer position

and different voltages applied to the two-leg

device (60 microm gap-size) can be seen in Fig 72

Although these first XEOL measurements for the

two-leg device show that the optical setup works

and the intensities are sufficient to track the

changes in the GaAs PL spectra we had

problems with the stability of the optical system

The measurements suffered from intensity

fluctuations when rotating the polarizer or

changing the voltages applied to the actuator

These changes in intensity can either result in a

loss of the whole XEOL signal or it can lead to a

saturation of the detector An example for the

intensity fluctuations measured for the 60 microm

device is given in Fig 73 where polarization-

resolved spectra for three different voltages

applied to the piezo are plotted The higher the applied voltage the higher the intensity which can

be seen in a certain range of polarizer positions where the detector is saturated The intensity

artefacts can also be seen in Fig 72 for the yellow curve (200V applied) which is cut due to the

detector saturation The saturation effects could in principle be compensated if the integration time

per measured spectrum was individually adjusted but these measurement series (10 different

voltages step size 20V and about 72 spectra per voltage step size 5deg in polarizer angle) were

macro-controlled routines lasting for about 3-10 h and checking each individual spectrum was

hence not possible

Figure 72 - Different voltages were applied to the 60 microm two-leg device and XEOL recorded The polarized position was fixed Each individual spectrum is normalized which allows identifying the shift of the PL peak to higher wavelengths due to the uni-axial strain

induced in the membrane


Although not each recorded spectrum could be evaluated the measurements using the polarizer

still reveal the presence of polarization-dependent components The peak maximum position

slightly depends on the polarizer position and shows a periodicity of 180deg See Fig 73 b) and c)

for the sinusoidal intensity distribution of the polarization resolved measurements The observed

sinusoidal intensity distributed can be related to the presence of the differently polarized high- and

low-energy components which contribute to the XEOL signal Corresponding PL lab

measurements are shown in Fig 50 a) for details on polarization resolved PL measurements see

chapter 428

The two-leg devices with 100 microm and 1000 microm gap sizes were investigated with the small X-ray

beam configuration (500 nm beam-spot size) Unfortunately the problems with fluctuating

intensities became worse Rotating the polarizer leads immediately to a complete loss of the

signal In Fig 74 the same measurements as in Fig 73 are shown but with the small beam

configuration and the 100 microm device Individual spectra out of this series (in Fig 74) are plotted

in Fig 75 for all voltages that have been applied and for selected polarizer positions For positions

close to 0deg the spectra independent of the applied voltages show a reasonable intensity and could

be evaluated but the further the polarizer is moved away from the 0deg position the more the

intensity drops until the signal is completely lost When the polarizer is again close to 360deg (360deg

= 0deg) the signal completely recovers Exactly the same behaviour was also observed for the

device with 1000 microm gap size when the small beam configuration was used for the XEOL

measurements The polarization-resolved measurements with the small beam configuration could

not be evaluated (not even qualitatively) since the signal was lost for most of the spectra

Nevertheless we could still demonstrate that as long as the alignment holds the XEOL

Figure 73 - In a) polarization resolved XEOL spectra of the GaAs emission line at 880nm are shown for 0V applied to the actuator b) shows the same measurements for 120V For the measured spectra around 200deg a beginning saturation of the detector is visible In c) 200V are applied to the actuator and the detector is fully saturated which can be seen at the dark blue streaks for a certain wavelength region around the GaAs peak Each individual spectrum is normalized to 1 which allows tracking the polarization dependency of the peak position more easily

Figure 74 - In a) b) and c) polarization-resolved XEOL spectra of the GaAs emission line at 880nm are shown measured on the 100 microm device with a beam-spot size of about 500nm For polarizer positions between 50deg - 300deg the signal got lost and only noise is measured Each individual spectrum is again normalized to 1


measurements with the small X-ray beam configuration are feasible and one can track the GaAs

emission line at 880 nm for different voltages and hence different strain states This was not at all

clear since the smaller X-ray beam also results in a smaller excitation volume which drastically

decreases the XEOL intensity On top of that the X-ray focusing has an efficiency of only a few

percent ie the total flux of the focused beam is much smaller than for the large beam size further

decreasing the XEOL intensity

The intensity drops or saturation effects observed for all XEOL measurements may be attributed

to changes in the beam path which lead to a loss or an enhancement of the signal These changes

can either be caused by moving the sample and hence losing the aligned position of the XEOL

emission spot and focal spot of the collecting mirror or by changing unintendedly the ldquointernal

alignmentrdquo simply by rotating the polarizer This effect on the other hand cannot be easily

explained since the polarizer is mounted in the parallel beam section and rotating it should not

have any influence on the aligned focal spots of the mirrors Most probably some problems are

related to a sort of make-shift mounting of the polarizer which was simply glued on a rotation

stage and not perfectly normal to the parallel beam If the polarizer is mounted with a certain

inclination this could cause a shift of the beam in the parallel beam section which would lower the

intensity collected by the second mirror which focuses the light into the optical fibre Additionally

the transmission function of the grating used in the spectrometer can be highly sensitive to different

polarization directions Hence by rotating the polarizer and changing the polarization direction of

the measured signal the actually measured intensity on the detector can change a lot (depending

on the grating that is used) To overcome the mentioned problems a generally more serious

solution for polarization-resolved measurements should be found in the future which could be a

combination of a rotating λ2-plate and a fixed polarizer This would ensure that the measured

XEOL signal always enters the fibrespectrometer with the same polarization direction which

should make a quantitative evaluation of the spectra possible

As mentioned also changing the applied voltage leads to changes in the measured intensity

which is even more puzzling since the changes cannot be attributed to the induced changes in the

band-structure More likely they are related to changes in the surface morphology of the

Figure 75 - From the left to the right the polarizer position was successively increased Each plot shows spectra for all voltages that were applied The signal intensity drops for polarizer positions between 50deg-300deg and then recovers again close to 360deg


membrane due to the bi-axial strain that is induced The strain could locally lead to changes of the

surface orientation (different tilt) which then leads to a reduced or increased emission collected by

the first mirror

Summarizing the XEOL measurements it

was possible to measure and track the

GaAs emission line at 880nm with both

beam configurations Independent of the

chosen configuration the alignment of the

setup was very critical and unfortunately not

stable within one measurement series In

addition to that the manual alignment

(using micrometre-screws) of the whole

optical setup was extremely time-

consuming and could not be repeated for

each individually measured spectrum The

XEOL measurements for all measured

devices did thus not allow a re-calculation

of the deformation potentials A successful

re-calculation would have required

accurate polarization resolved

measurements with comparable intensities

This could not be achieved with the current

optical setup although the intensity was not

the limiting factor The most limiting factor for the XEOL measurements is up to now the optical

setup that was used We could prove that it works well conceptually and the measurements of

the individual spectra are comparable to the measurements performed in a dedicated optics lab

In Fig 76 the changes of the in-plane strain component and the PL line shift for various voltages

applied are plotted The higher the uni-axial strain along the legs is the higher is the red-shift of

the measure GaAs emission line This proves that the setup works although the error of the

evaluated XEOL spectra is high (the plot in Fig76 indicates the general trend) The most critical

parts are on the one hand the polarizer which needs to be further improved by repaying it with a

rotate-able λ2-plate and a fixed polarizer and on the other hand some of the alignments still need

to be motorized in order to speed up the whole experimental procedure including re-alignments

so that larger datasets can be collected during a beam-time

Figure 76 - The changes of the in-plane strain component and the wavelength of the corresponding PL emission line are plotted for various voltages applied The higher the voltage the more strain is induced and the higher is the observed red-shift of the GaAs emission


6 Summary and outlook

In this very last chapter of this thesis the author wants to summarize and to point out the most

important aspects and highlights which are important for further research and developments

This thesis started with an introduction of the concepts of using piezo-electric actuators to fully-

reversibly transfer strain to direct-band-gap semiconductors (GaAs membranes) which are

attachedbonded onto these actuators The concept of using strain induced via actuators as a

tuning knob to model the band structure has been widely used but the actual magnitude and

direction of the strain which is induced in the semiconductors was not quite clear Our colleagues

at the semiconductor institute (JKU) could characterize the strains in the semiconductor by

measuring the changes in the PL spectrum upon an induced strain and then correlate these

changes in the PL spectra to a certain strain tensor This correlation is mediated by the optical

deformation potentials which are only known with a rather weak accuracy hence a highly accurate

strain measurement from the changes of the PL signal only is not possible Furthermore using

only PL measurements it is not possible to get information on the strain induced in the actuator

itself

This was the starting point for this thesis and with the first set of XRD measurements on the GaAs

we could successfully quantify the amount and crystallographic-orientation of the strain that is

transferred from the actuator to the semiconductor Furthermore we could also measure the

induced strain upon an applied voltage in the actuator itself which allows quantifying the amount

of strain that is effectively transferred to the GaAs membrane Investigating both the strain

induced in the GaAs and the strain induced in the actuator additionally allows characterizing

different bonding techniques that were used to attach the GaAs membrane onto the actuator In

this thesis it could be shown that the bonding mediated with the polymer SU8 is much more

efficient than bonding mediated by using gold as a bonding layer This is a remarkable outcome

since a comparatively ldquosoftrdquo polymer shows a much higher bonding strength than a ldquohardrdquo metal

FEM simulations for different interfaces were performed to find a reasonable explanation for this

strange behaviour In combination with AFM measurements of the rough actuator surface the

FEM simulations show that the interface quality of the different bonding layers is much more

important than the material that mediates the bonding process The simulations show a clear

correlation that an imperfect bonding-interface can significantly reduce the transferred strain

whereas the influence of the materials used for bonding can be neglected This allows explaining

why the SU8 which spreads and covers smoothly across the whole bonding area leads to a better

strain transfer in comparison to the gold-bonding where the membrane is only attached within

domain-like regions which significantly reduces the transferred strain

The monolithic devices which have been used to determine the strain transfer efficiencies were

also used to re-calculate one of the deformation potentials of GaAs For this purpose the devices

were characterized via XRD and PL measurements and by using the equations which describe

the changes in the band-structure induced by bi-axial compressive strain a re-calculation of the

deformation potential a was possible The calibrated values for deformation potential a are very

close to values proposed by the literature although some simplifications in the theoretical

description were applied and all pre-strains were set to zero The successful re-calculation of the

deformation potential shows that the combination of XRD and PL is a powerful approach to

calibrate these fundamental material parameters In the next step we used a more refined actuator

layout where it is possible to induce instead of compressive bi-axial strain mainly uni-axial

tension The different strain configuration gives access to deformation potentials a and b which

are needed to describe the changes in the band structure for uni-axial tension The devices with

the new layout were again characterized via XRD and PL measurements to finally re-calculate


both deformation potentials The re-calculated values were unfortunately not within the range of

values proposed by the literature which can be explained due to the inhomogeneous strain

distribution at the GaAs membrane and the fact that we could only investigate ldquoglobalrdquo strain

changes due to the comparatively large X-ray beam spots we have in the lab By changing the

setup (PL or XRD) also the illuminated area on the membrane as well as the footprint changes

and hence the results from the PL and the XRD measurements are strictly speaking not directly

comparable Looking at the re-calculated values and the corresponding errors for the deformation

potentials we cover the same range as the literature values scatter roughly 20 - 50 depending

on the deformation potential To achieve a reliable calibration of the deformation potentials the

error in the measurements should be decreased by one order of magnitude (strain resolution

should be about 10-4Aring) and additionally allowing to measure absolute changes in strain and not

only relative ones

With the knowledge from the lab measurements (that the induced strains in the GaAs membrane

are not homogeneous within the illuminated area) the synchrotron measurements were performed

The measurements at the synchrotron should offer the possibility to directly measure strain and

the emitted XEOL using only one beam for excitation Furthermore the possibility to drastically

reduce the beam-spot size of the X-ray beam to sizes in the sub microm-range should allow measuring

the strains accurately enough for a successful re-calculation of the deformation potentials These

measurements should provide consistent strain and PL data sets with error-bars that are small

enough to establish the re-calculated potentials as new a reference Unfortunately the

synchrotron measurements did not turn out to be completely successful in terms of achieving

consistent high-quality data sets on the strain and PL changes but could successfully prove that

XEOL can be used as a reliable technique to simultaneously investigate optical properties of the

material in combination with XRD techniques The optical setup that was used caused most of the

troubles by investing a lot of time in the careful alignment of all optical components we performed

XEOL measurements with good statistics although we were using a nano-focused beam

However it turned out that the optical setup was by far not stable enough to perform measurement

series on several Bragg peaks while still successfully tracking the XEOL signal for different

voltages applied The results from the XEOL measurements could only be used qualitatively and

can be seen as a proof of the concept From the nano-focused X-ray measurements we could

show that the strain and tilt distribution of the measured GaAs membrane is indeed highly chaotic

The local distortions in strain and tilt can be an order of magnitude higher and can differ in sign

and orientation in comparison to the global trend These results were quite new although the

induced strains were not expected to be perfectly homogeneous but the local strain and tilt

distribution in detail was not known to this extent

When looking at the synchrotron measurements it is obvious that there are many parameters that

can still be optimized The most puzzling part at the synchrotron was measuring the XEOL signal

and one easy non-scientific optimization approach could be motorizing the microm-screws needed for

the alignment This would easily save hours of synchrotron beam-time which could then be

invested in the measurement process Furthermore the polarizer should be replaced by the

combination of a fixed polarizer and a rotatable λ2 plate to be able to measure independently of

the incoming polarization direction of the XEOL signal In addition to the technical improvements

the devices that have been used can and have to be further optimized to reduce the pre-strains

and inhomogeneities The most critical step where most of the inhomogeneities are created is the

high temperature bonding step where the semiconductor is attached to the actuator in

combination with the final poling of the device Reducing the bonding temperature below the Curie

temperature of the PMN-PT (actuator material) would allow using already poled piezo substrates

and hence drastically limit the amount of inhomogeneities in the final device because no re-poling


would then be necessary Also the thickness of the devices could be increased to reduce the

inhomogeneous since thicker membranes are more resistant against curling and pre-strains

although this would also limit the maximum amount of strain that can be induced Nevertheless if

the error of the measurements became smaller due to the absence of inhomogeneities compared

to the reduction of the transferred strain this would still reduce the error significantly For the PL

measurements the only way to reduce the peak broadening which is the highest contribution to

the error concerning the PL measurements is to reduce the sample temperature ideally to

cryogenic temperatures (in the range of several K) This would imply that also the X-ray (XEOL)

measurements must be performed at cryogenic temperature which is not feasible at the moment

but one could find a compromise and cool the sample in-situ while measuring using a cryojet The

cryojet cools the sample with liquid nitrogen and has the advantage that it could be mounted on

the already existing setup without the need of re-building the whole setup When calibrating the

deformation potentials one should also have in mind that these deformation potentials are only

valid within a linear approximation The equations linking the changes of the band structure to the

changes in the PL spectra only assume small changes in strain (well below 1) If higher strains

are applied more refined theoretical models have to be applied Hence one should always

consider these limitations of the theoretical model used to describe the effect When assuming

also non-linear effects the classical concept of the deformation potentials as introduced in

chapter 23 is not valid anymore

Nevertheless the measurements and methods presented in this thesis could prove that piezo

actuators in combination with XRD and PL (XEOL) measurements can be used to calibrate

deformation potentials The great advantage using piezo actuators to transfer strain is that the

strain transfer is fully reversible Furthermore the semiconductor is attached to the actuator via a

bonding process and not by epitaxial growth (as proposed in some studies) which means that this

process is not limited to GaAs in principle each kind of material can be attached to the actuator

via the bonding process If the measurement procedure for GaAs membranes is well established

and hence a calibration of the potentials with small error-bars is achieved it can be easily applied

to other optically active materials Additionally one can think of investigating the temperature

dependency of the deformation potentials by performing the same measurements at different

temperatures The bottle neck for the investigation of temperature dependent effects are the XRD

measurements where a cryostat (for low temperatures) which can be penetrated by the X-rays

would be necessary Designing a suitable cryostat for this purpose and adapting the goniometer

is already scheduled to be done in the near future

Within this thesis many experimental aspects have been discussed as well as the weaknesses

and strengths of the combination of PL and XRD measurements For further research this

hopefully provides a good starting point and a detailed introduction to this topic There are still

many things that that can be improved but the research presented in this thesis might finally allow

to straightforwardly calibrate the deformation potentials for a wide range of optically active

materials


7 Appendix

71 Python code

711 Tilt calculation

QvecAngSym is the function that was used to calculate the tilt angle A reference vector and the

measured in- and out-of-plane q-component are needed as input parameters The angle in degree

between the two vectors is returned def QvecAngSym(qy_numb qz_numb qref)

q2=qref

q1 = nparray([0qy_numbqz_numb])

dot = npdot(q1 q2)

cos_ang_rad = dot ( npsqrt((q1q1)sum()) npsqrt((q2q2)sum()))

cos_ang_deg = nparccos(cos_ang_rad)360(2nppi)

return cos_ang_deg

712 Image processing filters

The function FilterFunc needs a 2D-intensity array (INT) and the specification of the convolution

matrix (filter) that should be allied to the array (func) as input parameters The image processing

filters which have been applied to the contour plots in Fig 40 41 and 42 were three times the

flattening filter (indicated by the keyword flat) and two times the sharpen filter matrix (indicated by

sharp) The other convolution matrices implemented have not been used to process the data

presented in this thesis def FilterFunc(INT func)

dim_x = INTshape[0]

dim_y = INTshape[1]

INT_ = numpyzeros((dim_xdim_y))

if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])

for var1 in range(0dim_x-1)


for var2 in range(0dim_y-1)

for var3 in range (03)

for var4 in range(03)

x_ind=var1+var3-1

y_ind=var2+var4-1

Avoid index values smaller than 0

if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0

INT_[var1][var2]=INT_[var1][var2]+INT[x_ind][y_ind]ImageMat[var3][var4]

return INT_

713 Centre-of-mass-calculations routines used for 2D and 3D RSMs

The function CenOfMassError calculates the centre of mass for a 2D intensity array The input

parameters are the intensity array INT the two axes qx qz and the variable ROI which can have

values between 0ltROIlt=1 If ROI=1 the whole array will be considered for the calculation of the

COM for values ROIlt1 only a part (1=100 03=30) of the array (around its maximum

intensity value) is considered for the COM calculation If for example ROI=03 and the array is a

2D-matrix with dimensions 100 x 100 then the maximum intensity value of this matrix is taken as

centre for a new 30 x 30 (defined by the value 03) matrix which is used to calculate the COM The

function returns a vector qs which points to the COM position def CenOfMass(qx qz INT roi)

qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00

Find max Int for center Point of roi

temp_arr = npwhere(INT == INTmax())

qz_cen = temp_arr[0][0]

qx_cen = temp_arr[1][0]

if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1

print ROI for center of mass cannot be larger than 1

else

rqx1=int(qx_cen-(roiqxshape[0]2))


rqx2=int(qx_cen+(roiqxshape[0]2))

rqz1=int(qz_cen-(roiqzshape[0]2))

rqz2=int(qz_cen+(roiqzshape[0]2))

for var1 in range(rqx1rqx2)

for var2 in range(rqz1rqz2)

qxs=qxs+qx[var1]INT[var2][var1]

qzs=qzs+qz[var2]INT[var2][var1]

int_ges = int_ges + INT[var2][var1]

qxs = qxsint_ges

qzs = qzsint_ges

qs = nparray([qxsqzs])

return qs

The function CenOfMass3D is a modified version which allows to use 3D intensity arrays and three axes as input The function was needed to evaluate the 3D RSMs measured on the synchrotron

def CenOfMass3D(qx qy qz INT roi)

qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00

Find max Int for Center Point of roi



qy_cen = temp_arr[1][0]


if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]

rqy1=int(qy_cen-(roiqyshape[0]2))

rqy2=int(qy_cen+(roiqyshape[0]2))

if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]


for var2 in range(rqy1rqy2)


qxs=qxs+qx[var1]INT[var3][var2][var1]

qys=qys+qy[var2]INT[var3][var2][var1]

qzs=qzs+qz[var3]INT[var3][var2][var1]

int_ges = int_ges + INT[var3][var2][var1]

qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges

qs = nparray([qxsqzsqys])

return qs


72 Reciprocal space maps (RSMs)

721 RSMs ndash Monolithic device ndash Gold bonding

RSMs for the gold-bonded device are shown In the first row the measured symmetric and

asymmetric reflections of the GaAs membrane and PMN-PT at 0V are plotted The second row

shows the same reflections for the same position but with 50V applied The voltage is then

increased in 50V steps for each row The highest voltage that was applied was 200V and the

corresponding RSMs can be seen in the last row The changes in the COM position which is

indicated by the white cross in each plot can hardly be seen by eye which is also the reason why

the COM calculations were used Furthermore not individual regions of the peaks are shift the

shape remains constant when a voltage is applied and the whole peak shifts


722 RSMs ndash Monolithic device ndash SU8 bonding

RSMs for the SU8 bonded device are shown In the first row the measured symmetric and




corresponding RSMs can be seen in the last row The COM position is indicated by the white cross

in each plot as for the gold-bonded device It is interesting to see that the shape remains constant

when a voltage is applied and that instead of individual regions the whole peak shifts


723 RSMs ndash Two-leg device

RSMs for the two-leg device are shown In the first row the measured symmetric reflections of the

GaAs at 0V are plotted PMN-PT reflections were not measured since the strain configurations in

the membrane and the piezo actuator are not comparable The strain induced in the GaAs

membrane is of uni-axial tension and the strain in the actuator is bi-axial compressive The second

row shows the same reflections for the same position but with 50V applied The voltage is


corresponding RSMs can be seen in the last row The white cross indicates the COM position in

each figure The measured Bragg peaks are shifting as a whole there are no areas which change

individually when a voltage is applied



724 3D -RSMs ndash Two-leg device - Synchrotron

RSMs of the two-leg device with 60microm gap size measured at the synchrotron with a beams-spot

size of approximately 100 microm x 100 microm (VxH) The 3D data set of the reciprocal space measured

around each Bragg reflection is integrated along all three Q-directions and then plotted The

symmetric [002] reflection is shown on the left side and the asymmetric [113] reflection is shown

on the right side for various voltages applied to the piezo The blue dot in each individual figure

indicates the calculated COM position in the 3D RSM



Abbreviations and shortcuts

All variables and functions in bold are vector-valued functions (119813 = F ) all functions which are not

bold are scalar fields

Common used variables

c helliphelliphellip speed of light 299 792 458 mmiddots-1 [meter per second]

휀0 helliphellip vacuum permittivity 8854thinsp187middot10minus12 Fmiddotmminus1 [fardas per meter]

1205830 helliphellip vacuum permeability 4πmiddot10minus7 N middotAminus2 [newton per ampere2]

h helliphelliphellip Planck constant 6626070040times10minus34 Jmiddots [jouls middot seconds]

ℏ helliphelliphellip h2π Jmiddots

e helliphelliphellip Electron charge 16times10minus19 C [coulomb]

119898119890helliphellip Electron mass 9109 384 middot 10minus31 kg

Differential operators

nabla=

(

part

partxpart

partypart

partz

)

nabla2= nabla middot nabla=part2

partx2+

part2

party2+

part2

partz2

119903119900119905119865 (x y z) = 119955119952119957119813(x y z) = nabla times 119813 =

(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part

party119865119911 minus

part

partz119865119910

part

partz119865119909 minus

part

partx119865119911

part

partx119865119910 minus

part

party119865119909

)

119889119894119907119865 (x y z) = 119941119946119959119813(x y z) = nabla middot 119813 =

(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911

119892119903119886119889 F(x y z) = 119944119955119938119941F(x y z) = nabla middot F =

(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References

Als-Nielsen Jens Des McMorrow (2011) Elements of modern X-ray physics 2 ed

Chichester Wiley (A John Wiley amp Sons Ltd publication)

Beya-Wakata Annie Prodhomme Pierre-Yves Bester Gabriel (2011) First- and

second-order piezoelectricity in III-V semiconductors In Phys Rev B 84 (19) DOI

101103PhysRevB84195207

Birleanu C Pustan M Merie V Muumlller R Voicu R Baracu A Craciun S (2016)

Temperature effect on the mechanical properties of gold nano films with different

thickness In IOP Conf Ser Mater Sci Eng 147 S 12021 DOI 1010881757-

899X1471012021

Borges Eudes (2011) Recent Advances in Processing Structural and Dielectric

Properties of PMN-PT Ferroelectric Ceramics at Compositions Around the MPB In

Costas Sikalidis (Hg) Advances in ceramics Electric and magnetic ceramics

bioceramics ceramics and environment InTech InTech

Bukhari Syed Islam Md Haziot Ariel Beamish John (2014) Shear piezoelectric

coefficients of PZT LiNbO 3 and PMN-PT at cryogenic temperatures In J Phys Conf

Ser 568 (3) S 32004 DOI 1010881742-65965683032004

Chuang Shun Lien (2009) Physics of photonic devices 2 ed Hoboken NJ Wiley (Wiley

series in pure and applied optics) Online verfuumlgbar unter httpbvbrbib-

bvbde8991Ffunc=serviceampdoc_library=BVB01ampdoc_number=017057266ampline_numb

er=0002ampfunc_code=DB_RECORDSampservice_type=MEDIA

Drosdoff D Widom A (2005) Snellrsquos law from an elementary particle viewpoint In

American Journal of Physics 73 (10) S 973ndash975 DOI 10111912000974

Eberhardt W (2015) Synchrotron radiation A continuing revolution in X-ray sciencemdash

Diffraction limited storage rings and beyond In Special Anniversary Issue Volume 200

200 S 31ndash39 DOI 101016jelspec201506009

Feng R Farris R J (2002) The characterization of thermal and elastic constants for an

epoxy photoresist SU8 coating In Journal of Materials Science 37 (22) S 4793ndash4799

DOI 101023A1020862129948

Fricke K (1991) Piezoelectric properties of GaAs for application in stress transducers

In J Appl Phys 70 (2) S 914ndash918 DOI 1010631349598

Gao Jiali Guan Le Chu Jinkui (2010) Determining the Youngs modulus of SU-8

negative photoresist through tensile testing for MEMS applications In Sixth International

Symposium on Precision Engineering Measurements and Instrumentation Hangzhou

China Sunday 8 August 2010 SPIE (SPIE Proceedings) S 754464

Herklotz Andreas Plumhof Johannes D Rastelli Armando Schmidt Oliver G Schultz

Ludwig Doumlrr Kathrin (2010) Electrical characterization of PMNndash28PT(001) crystals

used as thin-film substrates In Journal of Applied Physics 108 (9) S 94101 DOI

10106313503209

Hooker Matthew W (1998) Properties of PZT-based Piezoelectric Ceramics Between

minus150 and 250degC Hg v NASA NASA (208708)


HOPCROFT M KRAMER T KIM G TAKASHIMA K HIGO Y MOORE D

BRUGGER J (2005) Micromechanical testing of SU-8 cantilevers In Fat Frac Eng Mat

Struct 28 (8) S 735ndash742 DOI 101111j1460-2695200500873x

Huo Y H Witek B J Kumar S Cardenas J R Zhang J X Akopian N et al (2013)

A light-hole exciton in a quantum dot In Nat Phys 10 (1) S 46ndash51 DOI

101038nphys2799

ID01 - ESRF (Hg) ESRF ID1 - Details on Setup Online verfuumlgbar unter

httpwwwesrfeuUsersAndScienceExperimentsXNPID01 zuletzt gepruumlft am

06042017

Ivan Ioan Alexandru Agnus Joel Rakotondrabe Micky Lutz Philippe Chaillet Nicolas

(2011) Microfabricated PMN-PT on silicon cantilevers with improved static and dynamic

piezoelectric actuation Development characterization and control In 2011 IEEEASME

International Conference on Advanced Intelligent Mechatronics (AIM 2011) Budapest

Hungary 3 - 7 July 2011 2011 IEEEASME International Conference on Advanced

Intelligent Mechatronics (AIM) Budapest Hungary 372011 - 772011 Institute of

Electrical and Electronics Engineers American Society of Mechanical Engineers

Robotics and Automation Society Industrial Electronics Society IEEEASME

International Conference on Advanced Intelligent Mechatronics AIM Piscataway NJ

IEEE S 403ndash408

Kholkin A L Pertsev N A Goltsev A V (2008) Piezoelectricity and Crystal Symmetry

In Ahmad Safari und E Koray Akdoan (Hg) Piezoelectric and Acoustic Materials for

Transducer Applications 1 Aufl sl Springer-Verlag S 17ndash38

Kittel Charles (2005) Introduction to solid state physics 8 ed New York NY ua Wiley

Kremer P E Dada A C Kumar P Ma Y Kumar S Clarke E Gerardot B D

(2014) Strain-tunable quantum dot embedded in a nanowire antenna In Phys Rev B

90 (20) DOI 101103PhysRevB90201408

Kriegner Dominik Wintersberger Eugen Stangl Julian (2013) xrayutilities a versatile

tool for reciprocal space conversion of scattering data recorded with linear and area

detectors In J Appl Crystallogr 46 (Pt 4) S 1162ndash1170 DOI

101107S0021889813017214

Levinshtein M E Rumyantsev Sergey L Shur Michael (1996) Handbook series on

semiconductor parameters Volume 1 Singapore World Scientific Online verfuumlgbar unter

httpsearchebscohostcomloginaspxdirect=trueampscope=siteampdb=e000xatampAN=5656

93

Loumltters J C Olthuis W Veltink P H Bergveld P (1997) The mechanical properties

of the rubber elastic polymer polydimethylsiloxane for sensor applications In J

Micromech Microeng 7 (3) S 145ndash147 DOI 1010880960-131773017

Luo Jun Hackenberger Wesley Zhang Shujun Shrout Tom R (2008) Elastic

piezoelectric and dielectric properties of PIN-PMN-PT crystals grown by Bridgman

method In Elastic piezoelectric and dielectric properties of PIN-PMN-PT crystals grown

by Bridgman method IUS 2008 2 - 5 Nov 2008 Beijing International Convention Center

(BICC) Beijing China 2008 IEEE Ultrasonics Symposium (IUS) Beijing China

2112008 - 5112008 Institute of Electrical and Electronics Engineers Ultrasonics


Ferroelectrics and Frequency Control Society IEEE Ultrasonics Symposium IUS IEEE

International Ultrasonics Symposium Piscataway NJ IEEE S 261ndash264

Martiacuten-Saacutenchez Javier Trotta Rinaldo Piredda Giovanni Schimpf Christian Trevisi

Giovanna Seravalli Luca et al (2016) Reversible Control of In-Plane Elastic Stress

Tensor in Nanomembranes In Advanced Optical Materials 4 (5) S 682ndash687 DOI

101002adom201500779

NIST - Database Absorption coefficient for air Hg v NIST NIST zuletzt gepruumlft am

24042017

Pietsch Ullrich Holyacute Vaacuteclav Baumbach Tilo (2004) High-resolution X-ray scattering

From thin films to lateral nanostructures 2nd ed New York Springer (Advanced texts in

physics)

Ribeiro Clarisse Sencadas Vitor Correia Daniela M Lanceros-Mendez Senentxu

(2015) Piezoelectric polymers as biomaterials for tissue engineering applications In

Colloids and surfaces B Biointerfaces 136 S 46ndash55 DOI

101016jcolsurfb201508043

Sham Tsun-Kong Rosenberg Richard A (2007) Time-resolved synchrotron radiation

excited optical luminescence light-emission properties of silicon-based nanostructures

In Chemphyschem a European journal of chemical physics and physical chemistry 8

(18) S 2557ndash2567 DOI 101002cphc200700226

Shayegan M Karrai K Shkolnikov Y P Vakili K Poortere E P de Manus S

(2003) Low-temperature in situ tunable uniaxial stress measurements in

semiconductors using a piezoelectric actuator In Appl Phys Lett 83 (25) S 5235ndash5237

DOI 10106311635963

Spratley JPF Ward MCL Hall P S (2007) Bending characteristics of SU-8 In

Micro Nano Lett 2 (2) S 20 DOI 101049mnl20070022

Stangl Julian (2014) Nanobeam X-ray scattering Probing matter at the nanoscale

Weinheim Wiley-VCH

Trotta R Atkinson P Plumhof J D Zallo E Rezaev R O Kumar S et al (2012)

Nanomembrane quantum-light-emitting diodes integrated onto piezoelectric actuators In

Advanced materials (Deerfield Beach Fla) 24 (20) S 2668ndash2672 DOI

101002adma201200537

Trotta Rinaldo Martin-Sanchez Javier Daruka Istvan Ortix Carmine Rastelli Armando

(2015) Energy-tunable sources of entangled photons a viable concept for solid-state-

based quantum relays In Physical review letters 114 (15) S 150502 DOI

101103PhysRevLett114150502

Trotta Rinaldo Martin-Sanchez Javier Wildmann Johannes S Piredda Giovanni

Reindl Marcus Schimpf Christian et al (2016) Wavelength-tunable sources of

entangled photons interfaced with atomic vapours In Nature communications 7 S

10375 DOI 101038ncomms10375

Vurgaftman I Meyer J R Ram-Mohan L R (2001) Band parameters for IIIndashV

compound semiconductors and their alloys In J Appl Phys 89 (11) S 5815 DOI

10106311368156


Ye Z-G Dong M (2000) Morphotropic domain structures and phase transitions in

relaxor-based piezo-ferroelectric (1minusx)Pb(Mg13Nb23)O3minusxPbTiO3 single crystals In

J Appl Phys 87 (5) S 2312ndash2319 DOI 1010631372180

Zander Tim Herklotz Andreas Kiravittaya Suwit Benyoucef Mohamed Ding Fei

Atkinson Paola et al (2009) Epitaxial quantum dots in stretchable optical microcavities

In Optics express 17 (25) S 22452ndash22461 DOI 101364OE17022452

Zhang Jiaxiang Ding Fei Zallo Eugenio Trotta Rinaldo Hofer Bianca Han Luyang

et al (2013) A nanomembrane-based wavelength-tunable high-speed single-photon-

emitting diode In Nano letters 13 (12) S 5808ndash5813 DOI 101021nl402307q

Ziss Dorian Martiacuten-Saacutenchez Javier Lettner Thomas Halilovic Alma Trevisi Giovanna

Trotta Rinaldo et al (2017) Comparison of different bonding techniques for efficient strain

transfer using piezoelectric actuators In J Appl Phys 121 (13) S 135303 DOI

10106314979859






references


Linz October 2017



Acknowledgements























Contents

1 Introduction 9

2 Theory 12



























4 The experiment 46


411 XRD-Setup 46

412 PL-Setup 48





















7 Appendix 89

71 Python code 89










References 100



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859





references


Linz October 2017



Acknowledgements























Contents

1 Introduction 9

2 Theory 12



























4 The experiment 46


411 XRD-Setup 46

412 PL-Setup 48





















7 Appendix 89

71 Python code 89










References 100



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859



Acknowledgements























Contents

1 Introduction 9

2 Theory 12



























4 The experiment 46


411 XRD-Setup 46

412 PL-Setup 48





















7 Appendix 89

71 Python code 89










References 100



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859


Acknowledgements























Contents

1 Introduction 9

2 Theory 12



























4 The experiment 46


411 XRD-Setup 46

412 PL-Setup 48





















7 Appendix 89

71 Python code 89










References 100



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859


Contents

1 Introduction 9

2 Theory 12



























4 The experiment 46


411 XRD-Setup 46

412 PL-Setup 48





















7 Appendix 89

71 Python code 89










References 100



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859



















7 Appendix 89

71 Python code 89










References 100



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859



1 Introduction


























piezo actuator







































































































2 Theory
























part2

part1199052 rarr 0 =1

1198882

part2




part2

part1199052 rarr 0 =1

1198882

part2




(27) 119888 =1

radic12057601205830




can be described by











120582















(211) 119864119896119894119899 = e middot V = 119898middot1199072

2



















(212) 120582119900119899119904119890119905 =ℎmiddotc

119890middot119881






keV)












splitting








2

ℎ[

1

1198991198912 minus

1


(214) 119891119877 =119898119890119890

4

812057602ℎ2 lowast

1

1+119898119890119872
















(215) 119887119903119894119897119897119894119886119899119888119890 = 119901ℎ119900119905119900119899119904

119904119890119888119900119899119889lowast

1

1198981199031198861198892 lowast 1198981198982 lowast 01 119861119882































force

(216) 119865119871 = 119890 + 119890 times










































(217) minus119889119868

119868= 120583 119889119909 rarr 119868 = 1198680119890

minus120583119909
















































(218) DSC 119889120590

119889120570=

119901ℎ119900119905119900119899119904 119901119890119903 119904119890119888119900119899119889 119894119899119905119900 ∆120570

119868119899119888119894119889119890119899119905 119865119897119906119909 (∆120570)=

119868119904119888

(11986801198600

)∆120570=

|119904119888|21198772

|119894119899|2







energy is conserved









(219) 119864119904119888 =

119902

412058712057601198882

119886119903

119877 rarr | 119864119904119888

| =119902

412058712057601198981198882

1

119877 || cos(120595)


(220) 119864119894119899 q = m


|

to

(221) |119864119904119888| =

1199022

412058712057601198981198882

1

119877cos(120595) |119864119894119899

| rarr 119868119904119888 = (1199022

412058712057601198981198882)2

1

1198772 cos(120595)2 1198680



+ 119864perpminus119894119899





as

(222) 119868minus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 (121) 119868perpminus119904119888

1198680=

1

2(

1199022

412058712057601198981198882)2

1

1198772 cos(120595)2







(223) 119868119904119888

1198680= (

1199022

412058712057601198981198882)2

1

1198772 (1+cos(120595)2

2) = 1199030

2 1

1198772 (1+cos(120595)2

2) =

11990302119875

1198772 rarr 119941120648

119941120628= 119955120782

120784119927













(224) ∆120601 = ( minus 119896prime ) 119903 = 119903




120582 considering




(225) minus11990301198910() = minus1199030 intρ(119903)119890119894119903119889119903




zero






as




are needed









(227) 119865119898119900119897() = sum 119891119895()119890119894119903119895119895











in the lattice

(228) = 119906 + 119907 + 119908119888 119906 119907 119908 isin ℤ



































(229) 119917119940119955119962119956119957119938119949() = 119880119899119894119905 119888119890119897119897 119904119905119903119906119888119905119906119903119890 119891119886119888119905119900119903 lowast 119871119886119905119905119894119888119890 119904119906119898 = sum 119943119947()119942119946119955119947119947 sum 119942119946119929119951

119951



(230) = 120784119951120645














(233) =











(234) lowast = 2120587 120377 119888

sdot ( 120377 119888) lowast = 2120587

119888 120377

sdot ( 120377 119888) 119888lowast = 2120587

120377

sdot ( 120377 119888)






(235) || = || =2120587

119889




(236) || =2120587

119889= 2||sin (120579)



(237) 2120587

119889= 2|| sin(120579) = 2

2120587

120582sin(120579) rarr 120640 = 120784119941119956119946119951(120637) (119913119955119938119944119944prime119956 119923119938119960)








depicted in Fig11





defined as

(238) 119951 = 120783 minus 120633 + 120631119946


the Z number

(239) 120575 =12058221198902119873119886120588

21205871198981198882 sum 119885119895119891119895

prime119895

sum 119860119895119895













(240) 120573 =12058221198902119873119886120588

21205871198981198882 sum 119891119895

primeprime119895

sum 119860119895119895






(241) 1199011 sin(1205791) = 1199012 sin(1205792)



(242) 1198991 =1198881199011

1198641 119886119899119889 1198992 =

1198881199012

1198642


refraction

(243) 1198991 sin(1205791) = 1198992 sin(1205792)





(244) 1198991 cos(1205721) = 1198992 cos(1205722)





photons


(245) cos(120572119888) = 1198992

1198991




1198991

2)




(247) 120572119888 asymp radic2Δn




































vectors 119896119894 and 119896119891







and


are given by

(248) |119896119894 | = |119896119891

| = | | =2120587

120582


(249) 120587 =2120579

2+ 120572 +

120587

4rarr 120572 =

120587minus2120579

2 119886119899119889 120573 =

120587



(250) sin (2120579

2) =

|119876|2frasl

|119896|rarr || =

4120587

120582119904119894119899(120579)



(251) 119928119961 = || 119852119842119847(120642) =120786120645

120640119956119946119951(120637) 119852119842119847 (120654 minus 120637)

(252) 119928119963 = || 119836119848119852(120642) =120786120645

120640119956119946119951(120637) 119836119848119852 (120654 minus 120637)




reciprocal space












2θ scan





If the relation
















length written as

(253) 휀 =Δ119871

119871=


119871







materials are


119875119900119894119904119904119900119899prime119904 119877119886119905119894119900 minus 120642

119878ℎ119890119886119903 119872119900119889119906119897119906119904 (119872119900119889119906119897119906119904 119900119891 119903119894119892119894119889119894119905119910) minus 119918 [119873 1198982frasl ]


(254) 119864 = 2119866(1 + 120584) 119886119899119889 119866 =119864

2(1+120584) 119886119899119889 120584 =

119864

2119866minus1




(255) 120590119894119894 = 119916120598119894119894 119894 ∊ 119909 119910 119911







(256) 120598119895119895 = minus120584120598119894119894 119894 ne 119895 ∊ 119909 119910 119911





(257) 120590119894119895 = 1198662휀119894119895 119894 ne 119895 ∊ 119909 119910 119911





(258) 휀 = (

휀119909119909 휀119909119910 휀119909119911

휀119910119909 휀119910119910 휀119910119911

휀119911119909 휀119911119910 휀119911119911

) 119886119899119889 = (

120590119909119909 120590119909119910 120590119909119911

120590119910119909 120590119910119910 120590119910119911

120590119911119909 120590119911119910 120590119911119911

)
















indices 120572 and 120573 119862119894119895119896119897 = 119862120572120573 120572 120573 ∊ 119909119909 119910119910 119911119911 119910119911 119911119909 119909119910

(259) 120590120572 = sum 119862120572120573휀120573120573 rarr

(

120590119909119909

120590119910119910

120590119911119911

120590119910119911

120590119911119909

120590119909119910)

=

(

11986211 11986212 11986213 11986214 11986215 11986216

11986221 11986222 11986223 11986224 11986225 11986226

11986231 11986232 11986233 11986234 11986235 11986236

11986241 11986242 11986243 11986244 11986245 11986246

11986251 11986252 11986253 11986254 11986255 11986256

11986261 11986262 11986263 11986264 11986265 11986266)

(

휀119909119909

휀119910119910

120598119911119911

2120598119910119911

2120598119911119909

2120598119909119910)








(260) 119862119868119904119900119905119903119900119901119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 0

0 0 011986211minus 11986212

20 0

0 0 0 011986211minus 11986212

20

0 0 0 0 011986211minus 11986212

2 )



(261) 11986211 =119864

(1+120584)+(1minus2120584) (1 minus 120584) 119886119899119889 11986212 =

119864

(1+120584)+(1minus2120584) 120584



(262) 119862119862119906119887119894119888 =

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)





(263) 휀120572 = sum 119878120572120573120590120573120573

























vector R written as

(264) 119881(119903) = 119881(119903 + )


given by

(265) 1198670120595(119903) = 119864()120595(119903) 119908119894119905ℎ 1198670 = [1199012

2119898nabla2 + 119881(119903)]



(266) 120595(119903) = 120595(119903 + )



(267) 120595119899119896(119903) = 119890minus119894119903119906119899119896(119903) 119908119894119905ℎ 119906119899119896(119903 + ) = 119906119899119896(119903)


al 2005)






GaAs is shown






structure


(268) [1198670 +ℏ2

2119898 +

ℏ2k2

2119898] 119906119899119896(119903) = 119864119899()119906119899119896(119903)


(269) 119867 = 1198670 + 119867119896prime =

1199012

2119898nabla2 + 119881(119903) +

ℏ2

2119898 +

ℏ2k2

2119898





(270) 119906119899119896 = 1199061198990 +ℏ

119898sum

⟨1199061198990| |119906119899prime0⟩


(271) 119864119899119896 = 1198641198990 +ℏ2k2

2119898+

ℏ2

1198982sum

|⟨1199061198990| |119906119899prime0⟩|2












(272) [1198670 +ℏ2

2119898 +

ℏ

41198981198882(nabla119881 times ) 120590 +

ℏ2


prime119906119899119896(119903)


operator








(273) 119867119871119870 = minus

[ 119875 + 119876 minus119878 119877 0

minus119878

radic2radic2119877


2119878




radic2

minus119878lowast


3

2119878 minusradic2119877 119875 + Δ 0



radic20 119875 + Δ

]





(274) 119875 =ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2) 119886119899119889 119876 =ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2)

(275) 119877 =ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] 119886119899119889 119878 =

ℏ21205743

119898radic3(119896119909 minus 119894119896119910)119896119911















휀119894119895 119894 119895 120598 119909 119910 119911


(277) prime = (1 + 휀119909119909) + 휀119909119910 + 휀119909119911119911

prime = 휀119910119909 + (1 + 휀119910119910) + 휀119910119911119911

119911prime = 휀119911119909 + 휀119911119910 + (1 + 휀119911119911)119911







the system

(278) 119867 = 119867119871119870 + 119867120576





parts

(279) 119875 = 119875119896 + 119875120576 = [ℏ21205741

2119898(119896119909

2 + 1198961199102 + 119896119911

2)] + [minus119886119907(휀119909119909 + 휀119910119910 + 휀119911119911)]

(280) 119876 = 119876119896 + 119876120576 = [ℏ21205742

2119898(119896119909

2 + 1198961199102 minus 2119896119911

2) ] + [minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)]

(281) 119877 = 119877119896 + 119877120576 = [ℏ2

2119898[minusradic31205742(119896119909

2 minus 1198961199102) + 1198942radic31205743119896119909119896119910] ] +

[radic3

2119887(휀119909119909 minus 휀119910119910) minus 119894119889휀119909119910]

(282) 119878 = 119878119896 + 119878120576 = [ℏ21205743



dimensional lattice












(283) 119867119875119861 = minus[

119875120576 + 119876120576 0 0 00 119875120576 minus 119876120576 0 00 0 119875120576 minus 119876120576 00 0 0 119875120576 + 119876120576

]



(284) 119864119867119867( = 0) = minus(119875120576 + 119876120576) = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(285) 119864119871119867( = 0) = minus119875120576 + 119876120576 = 119886119907(휀119909119909 + 휀119910119910 + 휀119911119911) minus119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)


(286) 119864119862( = 0) = 119864119892 + 119886119888(휀119909119909 + 휀119910119910 + 휀119911119911)







2(휀119909119909 + 휀119910119910 minus 2휀119911119911)

(288) 119864119862 minus 119864119871119867 = 119864119892 + (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) +119887

2(휀119909119909 + 휀119910119910 minus 2휀119911119911)




are in the range of























































combination

























































(291) 119863119894 = 휀119894119895119864119895



named as 120590)

(292) 119878119894119895 = 119904119894119895119896119897119879119896119897

(293) 119889119894119895119896 =120597119878119894119895

120597119864119896











(294)

(

1198781

1198782

1198783

1198784

1198785

1198786)

= (

11988911 11988912 11988913 11988914 11988915 11988916

11988921 11988922 11988923 11988924 11988925 11988926

11988931 11988932 11988933 11988934 11988935 11988936

)

119905

(1198641

1198642

1198643

) =

(

11988911 11988921 11988931

11988912 11988922 11988932

11988913 11988923 11988933

11988914 11988924 11988934

11988915 11988925 11988935

11988916 11988926 11988936)

(1198641

1198642

1198643

)






























resulting in

(295)

(

1198781

1198782

1198783

1198784

1198785

1198786)

=

(

0 0 11988931

0 0 11988931

0 0 11988933

0 11988915 011988915 0 00 0 0 )

(1198641

1198642

1198643

)

Material 11988933 11988931 11988915











electric field







layout was changed























sketched in Fig 24








312 Two-leg device

























































etching rate)





(49)




and 322











see chapter 23




























4 The experiment









413

411 XRD-Setup









AXS

































(41) 119868

1198680= 119890

micro119886119894119903120588119886119894119903






















mounted








412 PL-Setup















the XRD setup

















































































































































(42) 119863119894119886119898119900119899119889 119903119890119891119897119890119888119905119894119900119899119904 ℎ 119896 119897 119886119897119897 119890119907119890119899 119886119899119889 ℎ + 119896 + 119897 = 4119899 119900119903 ℎ 119896 119897 119886119897119897 119900119889119889




























































(43) 119879119894119897119905 = 120572 119877119890119886119897 119904119901119886119888119890 119901119900119904119894119905119894119900119899 (120596 2120579 119868119899119905119890119899119904119894119905119910) rarr 120596119888119900119903119903 = 120596 + 120572



0 0 1

)

















































membrane























dimensions



































712




real space





















discussed


















PMN-PT [002] 120596 = 2200deg 067 mm2

GaAs [004] 120596 = 3303deg 046 mm2

PMN-PT [113] 120596 = 5500deg 031 mm2

GaAs [224] 120596 = 7714deg 026 mm2












angle of 90deg The





measured RSMs A
















discriminated
















(45) 119862119874119872 =1

119868119905119900119905sum 119868119894119894119894





































































2 minus 119876119911minus1198781199101198982


























get

(47)

(

00

120590119911119911

000 )

=

(

11986211 11986212 11986212 0 0 011986212 11986211 11986212 0 0 011986212 11986212 11986211 0 0 00 0 0 11986244 0 00 0 0 0 11986244 00 0 0 0 0 11986244)

(

휀119909119909

휀119910119910

휀119911119911

000 )



211986212




























(2017)

































spectra






































XRSP3































































































component





































































by

(51) 119879119864 =120576119910119910(119878119894119898119906119897119886119905119894119900119899)

120576119910119910(119875119872119873minus119875119879)




휀119910119910(119878119894119898119906119897119886119905119894119900119899) equals 휀119910119910(119875119872119873 minus 119875119879)


































be observed



















































strain






















obligatory


















(52) Δ119864 = (119886119888 minus 119886119907)(휀119909119909 + 휀119910119910 + 휀119911119911) = 119938 (휀119909119909 + 휀119910119910 + 휀119911119911)




corresponds to 200V














constant)








Δ119864 = 119938 (2휀∥ + 120584휀∥)



chapter 427) via

(54) 120584 = minus211986212






















points




measurement error






















52 Two-leg device


































































































































experiments








































was applied


















523






XEOL signal













the legs























































































































































































































hence not possible











chapter 428


















































the first mirror
















































itself













































only relative ones












































































materials


7 Appendix

71 Python code





q2=qref


dot = npdot(q1 q2)



return cos_ang_deg








dim_x = INTshape[0]

dim_y = INTshape[1]


if (func==flat)

ImageMat=([[011101110111]

[011101110111]

[011101110111]])

elif (func==sharp)

ImageMat=([[0-10]

[-1-5-1]

[0-10]])

elif (func==edge)

ImageMat=([[010]

[1-41]

[010]])

elif (func==relief)

ImageMat=([[-2-10]

[-111]

[012]])






x_ind=var1+var3-1

y_ind=var2+var4-1


if(x_indlt0)

x_ind=0

if(y_indlt0)

y_ind=0


return INT_










qxs = 00

qzs = 00

qx_cen=0

qz_cen=0

var1 = 0

var2 = 0

rqx1=0

rqx2=0

rqz1=0

rqz2=0

int_ges=00





if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

elif roigt1


else











qxs = qxsint_ges

qzs = qzsint_ges


return qs



qxs = 00

qys = 00

qzs = 00

qx_cen=0

qy_cen=0

qz_cen=0

var1 = 0

var2 = 0

var3 = 0

rqx1=0

rqx2=0

rqy1=0

rqy2=0

rqz1=0

rqz2=0

int_ges=00






if roi==1

rqx1=0

rqx2=qxshape[0]

rqz1=0

rqz2=qzshape[0]

rqy1=0

rqy2=qyshape[0]

elif roigt1


else



if rqx1lt0

rqx1=0

if rqx2gtqxshape[0]

rqx2=qxshape[0]




if rqz1lt0

rqz1=0

if rqz2gtqzshape[0]

rqz2=qzshape[0]



if rqy1lt0

rqy1=0

if rqy2gtqyshape[0]

rqy2=qyshape[0]








qxs = qxsint_ges

qys = qysint_ges

qzs = qzsint_ges


return qs























































nabla=

(

part

partxpart

partypart

partz

)


partx2+

part2

party2+

part2

partz2


(

part

partxpart

partypart

partz

)

times (

119865119909119865119910119865119911

) =

(

part


part

partz119865119910

part


part

partx119865119911

part


part

party119865119909

)


(

part

partxpart

partypart

partz

)

middot (

119865119909119865119910119865119911

) =part

partx119865119909 +

part

party119865119910 +

part

partz119865119911


(

part

partxpart

partypart

partz

)

middot F(x y z) =

(

part

partxF(x y z)

part

partyF(x y z)

part

partzF(x y z)

)


References





101103PhysRevB84195207




899X1471012021







Ser 568 (3) S 32004 DOI 1010881742-65965683032004












DOI 101023A1020862129948










10106313503209









101038nphys2799



06042017










IEEE S 403ndash408







90 (20) DOI 101103PhysRevB90201408




101107S0021889813017214




93
















101002adom201500779


24042017



physics)




101016jcolsurfb201508043




(18) S 2557ndash2567 DOI 101002cphc200700226




DOI 10106311635963




Weinheim Wiley-VCH




101002adma201200537








10375 DOI 101038ncomms10375



10106311368156














10106314979859
































































































Documents

Submitted by X-ray Diffraction vs. Dorian Ziss Institute