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Contrast inversion on the h -BN nanomesh Masterthesis University of Basel Markus Langer Prof. Ernst Meyer Supervisors: Thilo Glatzel Juli 2010

Masterthesis Markus Langer - Contrast inversion of the h-BN nanomesh

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Page 1: Masterthesis Markus Langer - Contrast inversion of the h-BN nanomesh

Contrast inversion on the h-BN nanomesh

MasterthesisUniversity of Basel

Markus Langer

Prof. Ernst MeyerSupervisors: Thilo Glatzel

Juli 2010

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”Erfolg braucht Leistung””Präzision braucht Leidenschaft”

Markus Langer

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Abstract

In this work an advance analysis of the h-BN nanomesh is given. Theoften seen contrast inversion is discussed in detail and possible expla-nations are shown. After the analysis, a height dependent contrastbecame most probable. In addition, a domain analysis is perfomed,which shows a low domain boundary mobility and no preferred domainorientation. Atomic resolution with the torsional oscillations fTR wasachieved for large scan areas. In all measurements bimodal dynamicforce microscopy was used and the theoretical background is discussed.Further, the torsional oscillation was used for imaging and 2D forcespectroscopy to achieve lateral force information on the nanomesh.

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Contents

1 Introduction 1

2 Theory 22.1 Interaction forces in DFM measurements . . . . . . . . . . . . 2

2.1.1 Electrostatic forces . . . . . . . . . . . . . . . . . . . . 22.1.2 Van der Waals force . . . . . . . . . . . . . . . . . . . 32.1.3 Short range forces . . . . . . . . . . . . . . . . . . . . 4

2.2 Cantilever dynamics . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Bimodal detection . . . . . . . . . . . . . . . . . . . . 72.2.2 Lateral force detection . . . . . . . . . . . . . . . . . . 7

2.3 Theoretical predictions and properties of h-BN nanomesh . . 92.3.1 Theory of the electronic structure and work function . 10

3 Experimental 123.1 Dynamic force microscope . . . . . . . . . . . . . . . . . . . . 123.2 Cantilever preparation . . . . . . . . . . . . . . . . . . . . . . 123.3 Preparation of Rhodium(111) films . . . . . . . . . . . . . . . 133.4 Nanomesh preparation . . . . . . . . . . . . . . . . . . . . . . 14

4 Measurements 164.1 Mesh stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Contrast inversion . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 Tip apex changes . . . . . . . . . . . . . . . . . . . . . 194.2.2 Double tip . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.3 Tip geometry . . . . . . . . . . . . . . . . . . . . . . . 254.2.4 Height dependent contrast inversion . . . . . . . . . . 294.2.5 2D spectroscopy . . . . . . . . . . . . . . . . . . . . . 304.2.6 Detailed spectroscopy analysis . . . . . . . . . . . . . . 334.2.7 2D bimodal spectroscopy . . . . . . . . . . . . . . . . 36

4.3 Domain orientation . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Atomic contrast measured by torsional resonance . . . . . . . 46

5 Conclusion 49

6 Acknowledgment 50

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1 Introduction

Since the technique of feeling atoms by a frequency modulation dynamic forcemicroscope was established in 1995 [1], with a wide range of applications inscience. Applications from human medicine [2], biology [3], applied surfacescience [4] to fundamental surface analysis [5] in physics. The frequencymodulated dynamic force microscopy as it is used for surface characteriza-tion allows insights into the forces with high signal resolution as well as highspacial resolution. For fundamental surface analysis this is a credent fea-ture for revealing unseen surface properties. A major step towards the highresolution data in this work was the use of bimodal excitation of differentresonance frequencies. Beside the fundamental flexural oscillation, a secondoscillation can be superimposed, like the second flexural or the first torsional[20] [21]. Every oscillation is controlled separately, allowing high resolutioninformation acquisition. With the torsional oscillation we gain direct accessto the lateral forces [21].This powerful techniques were used to bring clarity in the analysis of a sp2

monolayer on a metal surface [15]. In recent years, studies on sp2 monolayerhave become significant. One of the well known representatives is graphene[6, 7, 8], a monolayer of carbon. Due to its properties graphene is a fas-cinating material showing an increase of electron mobility, spin transport,increased thermal conductivity and features different mechanical properties.On the other hand, boron nitride forms a sp2 monolayer as well [23]. Evap-oration on a rhodium(111) surface forms a corrugated monolayer with ahoneycomb superstructure, better known as the h-BN nanomesh. Unlikegraphene, the electronic structure makes it an insulator. The topic of thismaster thesis is the study on this corrugated single layer as well as the ex-planation of often observed features, like the contrast inversion.

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2 Theory

By measuring the forces perpendicular to the surface the analysis of fun-damental structures as well as the electronic properties are available. Theinteracting forces between two objects have a large variety of origins andwill be discussed in the order of their decay length. Exact measurementsof forces require high sensitivity as well as precision. Our equipment forsurface characterization is a homebuilt atomic force microscope(AFM) [16].The design and fundamental mode of operation of a nc-AFM is introducedshortly. Cantilever dynamics and the advantage of bimodal dynamic forcemicroscopy (DFM) incorporate with the detected signal will be covered inmore detail.

2.1 Interaction forces in DFM measurements

The forces acting between two separated objects has rarely only one com-ponent. In extreme cases different forces can have a dominant contribution.For example, earth makes his circles around the sun, due to the attractivegravitational force, because every other force has a smaller decay length andis weak compared to gravitation. On the other hand, two electrons repeleach other at very small distances at room temperature, due to repulsivecoulomb forces. Here the strong repulsive force is dominant compared toany other force.Systems in the nanoscale range consist of many different forces. Analyzingand characterizing surfaces with a dynamic force microscope is challenging,due to the sum of interacting forces between tip and sample surface, whichare combined in one signal. Nevertheless it is possible to separate thoseforces and their origins. First of all the decay length is a way for a clas-sification. Two classes are defined, the forces with a long range character,like the electrostatic forces, see section:2.1.1 and the van der Waals forces,see section:2.1.2 and forces with a short range character, like the chemicalinteraction, which is formulated either with the Lennard Jones potential, seesection:2.8 or the Morse potential Eq.:2.10.

2.1.1 Electrostatic forces

A force affects on two electric charges. This force is repulsive, if the chargeshave the same sign. On the other hand the force is attractive for differentlysigned charges. This electrostatic interaction is described by Coulomb’s law,which specifies the strength and the decay length of the force acting on twoobjects:

FCoulomb(z) =1

4πε0

Q1Q2

z2(2.1)

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Here, ε0 is the permittivity of the vacuum and Q1 and Q2 are the sums overall charges in each object with the distance z between them. DFM allowsonly the detection of localized charges within a sample with the tip above.In the most cases localized charges are formed due to surface preparation,like ultra high vacuum (UHV) cleaving of ionic crystals, changing abruptlythe bulk formation. The bulk-vacuum interface forms uncompensated localcharges, due to missing atoms. Ion sputtering as well as plasma depositioncan cause local charges in the surface. Those local charges can last for severalhours up to days under UHV conditions.Electrostatic interaction does not need only defects or adatoms within thesurface. An electrostatic variation can arise due to charge transfer betweendifferent materials, which causes a contact potential difference, resulting inan electrostatic force.In a DFM setup, tip and sample, can be assumed as a distance dependentcapacitor, with a capacity C. The force is given by:

Fel(z) =12

∣∣∣∣∂C

∂z

∣∣∣∣ (VBias − VCPD)2 . (2.2)

The electrostatic force is minimal, when the applied bias voltage Vbias com-pensates the contact potential VCPD. The contact potential arises due tothe different work functions of the materials.The capacity gradient ∂C/∂z can be approximated by assuming the tip asa cone with a half sphere on top defined by radius R. The dominant termcan be [14, 45] reduced to:

Fel(z) = −πε0R

z(VBias − VCPD)2 . (2.3)

2.1.2 Van der Waals force

The van der Waals interaction between two neutral atoms is a quantum me-chanical consequence. The expectation value for an electric dipole momentvanishes for neutral atoms. This expectation value is subject to quantummechanical and thermal fluctuations. Those fluctuations trigger a dipolemoment even in neutral atoms. A spontaneous triggered dipole moment inthe first atom creates an electric field with a decay rate of 1/~r3. A secondatom within the range of the decay rate gets electrically polarized. A dipolemoment is induced by d2 ∼ α2/~r

3. α2 is the polarizability of the secondatom. The interaction of those dipole moments is proportional to −d1d2/~r

3.The electric field of the second atom influences the dipole moment of the firstatom d1 ∼ α1/~r

3. The resulting potential from the van der Waals interactionis described by [11]:

V (~r) ∼ −d1d2

~r3= −α1α2

~r6. (2.4)

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For the case of an integration over a sphere in the distance z above a surfacethe potential adds up to a distance dependence which is proportional toVvdW (z) ∝ −1/z. The force of the interaction is resulting in:

FvdW (z) =HR

6z2(2.5)

with the Hamaker constant H [12], tip radius R and distance z.The inter-action of two dipole moments is fragile to any perturbation. The influenceof a third dipole moment is not additive, hence this theoretical approach isonly qualitative. One reason is the polarizability of an ensemble is not thesum over the polarizabilities of every atom [11].

2.1.3 Short range forces

Short range force or chemical interaction arises, if the wave functions ofatom nuclei overlap. Those atom nuclei experience an attractive force oneach other, if the overlap causes a lower energy state of both systems. Theoverlap can become also repulsive, if the Pauli exclusion principle is violated(pauli repulsion) or due to coulomb repulsion of two nuclei. These forces canbe described via model potentials, which have been fitted on experimentaldata and have no derivations.The Lennard-Jones potential Fig.:1 describes the interaction of two neutralatoms and the Morse potential Fig.:1 describes the interaction of a diatomicmolecule.The Lennard Jones potential is given by:

VLJ(z) = −4ε

[(σ

z

)6−

z

)12]

. (2.6)

ε describes the effective height of the potential wall and σ the effective sep-aration of the atoms. The Lennard Jones potential has a global minimum,which defines the equilibrium bond distance z0 = 21/6σ. The term of theorder of six describes the attractive van der Waals interaction, while theterm of the order of twelve specifies the repulsive interaction [19]. With theequilibrium bond distance z0 we get:

VLJ(z) = −ε

[2

(z0

z

)6−

(z0

z

)12]

. (2.7)

The corresponding force of the Lennard Jones potential is:

FLJ(z) = −12ε

z0

[(z0

z

)7−

(z0

z

)13]

. (2.8)

The Morse potential gives an approximation of a diatomic molecule. Thisempirically derived potential was fitted on measurements, but can also bejustified due to the solution of the Schrödinger’s equation of a H+

2 ions. It is

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fully described by the parameter of the equilibrium bond distance z0, bindingenergy ε and the effective range κ [19]:

VM (z) = ε(1− e−a(z−z0)

)2. (2.9)

a is defined as√

κ/2ε, which defines the potential width and with this thebond dissociation energy. The derivation −∂V/∂z gives:

FM (z) = −∂VM

∂z= −2aε

(e−a(z−z0) − ae−2a(z−z0)

)2. (2.10)

Both potentials can be used to characterize the short range interaction be-tween tip and sample.

Figure 1: Lennard Jones and Morse potential. Parameters: ε = 1eV bindingenergy, z0 = 0.2nm equilibrium bond distance and a = 20eV bond dissocia-tion energy.

Figure 1 shows both potentials. Here we see, that the Morse potential isrepulsive for z ≤ 0.28nm. However, the Lennard Jones potential shows therepulsive interaction at lower equilibrium bond distances z ≤ 0.2nm. On theother hand the latter shows a larger binding energy of ∼ −12nN for interatomic interactions.

2.2 Cantilever dynamics

In non contact dynamic force microscopy, a silicon cantilever prong is excited.This cantilever will oscillate free at the resonance fi for large tip-sample

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separation, for decreasing tip-sample separation the interaction between tipand sample will cause a shift in oscillation frequency ∆fi, in the imagingmode this shift is kept constant to control the tip-sample distance. The freeoscillation becomes perturbated by the interaction, as long as the shift infrequency ∆fi is small compared to the resonance frequency fi, the systemcan be assumed by an slightly perturbed harmonic oscillator. For this systemthe equation of motion is:

miz = −kiz + Aexcicos(ωit + ϕi)− γz + F (z). (2.11)

mi is the effective mass of the oscillating cantilever. Due to the small effectivemass of the cantilever gravitation can be neglected, hence the selection ofthe z axis is free. ki is the effective stiffness of the i-th resonance and F (z)the interaction force. γ defines the damping of the oscillation, Aexci is thedriving amplitude of the i-th resonance with the frequency of ωi = 2πfi.The oscillation amplitude has a phase shift of ϕ = 90◦ in respect to thedriving amplitude Aexc. For a constant oscillation amplitude, one assumesthat the damping and the excitation are compensating each other. Thus,the equation reduces to:

miz = −kiz + F (z). (2.12)

The amplitude Ai is usually larger than the decay length λ of the interactionforces Ai � λ, therefore most of the forces will contribute at the lowerturning point of the oscillation. The cantilever trajectory is assumed tobe harmonic. This justifies the substitution of z = z0 + Aicos(ωit) in theequation of motion 2.12.

−miAiω2i cos(ωit) = −kiz0 − kAicos(ωit) + F (z0 + Aicos(ωit)). (2.13)

The integration over one period of the resonance frequency fi integrates allforces, which contribute to the extension of the period.

Aiki

(1− ωi

ωi0

)=

12π

∫ 2πωi

0F (z0 + Aicos(ωit))cos(ωit)dt (2.14)

Where ωi0 is the angular frequency of the unpertubated system and ωi theeffective angular frequency. If both are comparable, the force interactionaveraged over one period is the experimental determined shift in frequency∆fi divided by the resonance frequency fi. With Θi = ωit

Aiki∆fi

fi=

12π

∫ 1fit

0F (z0 + Aicos(Θi))cos(Θi)dΘi (2.15)

This frequency shift dependence has been shown at first by Giessibl [18] andis valid for any resonance frequency, which is excited alone.

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2.2.1 Bimodal detection

In this mode the cantilever is excited at two different resonance frequen-cies simultaneously [20]. The first resonance frequency is used to controlthe tip-sample distance, while the second resonance detects various surfaceproperties, like surface mechanics, magnetic structure and local contact po-tential of heterogeneous materials. Bimodal DFM imaging was applied forhigh resolution atomic-scale imaging, due to stable distance control of thefundamental frequency and the enhanced sensitivity of higher order oscilla-tion frequencies to short-range forces [20] [21].The two resonance frequencies are incommensurate. Hence, we have to ac-knowledge that the trajectory of one oscillation cycle differs from the nextoscillation cycle, because fi+1/fi is not an integer, therefore we need to av-erage over many oscillation cycles. Due to the limited, system-dependentmeasurement bandwidth of only several Bm ≈ 100Hz and fi � Bm the tipwill perform a large number ni of oscillation cycles on every recording point.Equation 2.15 changes to:

Aiki∆fi

fi=

12πni

∫ 2πni

0F (Tip(Z(t)))cos(Θi)dΘi (2.16)

Tip(Z(t)) denotes the Tip apex position at the time t, while the z positionis adequately described by Z(t) = z0 +Aicos(Θi)+Ajcos(Θj), here z0 is theequilibrium position and Θi = 2πfit the phase of the ith mode.Equation 2.16 is valid for small shifts in frequency ∆fi of any resonance,meaning ∆fi/fi � 1. For i = 1 the cantilever is prevented of jump-to-contacts for large A1st; the condition i = 2 can be fulfilled even for verysmall A2nd, due to the higher effective stiffness k2nd. f1st and f2nd areincommensurate, except for the oscillating force component at fi, whichmakes a finite contribution to the integral equation 2.16. So we expect thesame for A1st = A2nd for the right hand side. For the case, that A1st �λ > A2nd we expect ∆f1 to show the same behaviour as in normal DFM. Inthe limit f2nd � f1st this implies that the integration of Θ2 over successivecycles of the oscillation at f1st is equivalent to a dense sampling over a singlecycle.

k2nd∆f2nd

f2nd= − 1

∫ 2π

0F ′(z0 + A1stcos(Θ1st))dΘ1st (2.17)

F ′ = dF/dz varies quicker than F itself and ∆f2nd with the same A1st has astronger dependence on the short-range forces than ∆f1st in normal DFM.

2.2.2 Lateral force detection

In a usual DFM system, the cantilever oscillates in the vertical direction withrespect to the surface, hence only force variations in the out of plane direc-

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tion FZ are detectable. This gives an access to a large set of information,mainly the topography, local work function and the stiffness of materials,etc.. But, due to the direction of the cantilever oscillation, we do not haveaccess to any information about a lateral force FX,Y .The excitation of a lateral oscillation mode is possible for PPP-NCL at afrequency of around fTR ∼ 1.5MHz. This allows to detect lateral forcecomponents FX [21]. Site-dependent long-range interaction cause no lateralforces, but the strongly distance dependent short-range interactions[21]. So,we expect extremely high spatial resolution in the frequency shift signal ofthe torsional mode ∆fTR. The torsional amplitude of oscillation is verysmall compared to the cantilever dimensions, therefore the tip apex is vi-brating parallel to the surface. This allows an adequate sense of the lateralinteraction forces FX between tip and sample.In bimodal DFM the flexural resonance fi for a stable height control is ex-cited simultaneously to the torsional resonance fTR for lateral resolution.This can be treated in a similar way with respect to the excitation of twoflexural resonances. The trajectory of the tip apex during one oscillationcycle of the flexural resonance differs to the next, due to the incommen-surable resonance frequencies. In every point of the oscillation, the short-range interaction can be split into its components along Z and X. Whilethe site-dependent long range force only contributes to the out of plane forcecomponent FZ . Assuming that all force components weakly perturbate theharmonic oscillation, an expression for the resonance shift can be given as:

Aiki∆fi

fi=

12πni

∫ 2πni

0Fj(Tip(X, Z)))cos(Θi)dΘi (2.18)

Here we denote Fj for the interaction force component in direction j andTip(X, Z) give the tip apex position in the X-Z plane, in the sense of:

Tip(X, Z) = (ATRcos(ΘTR), z0 + AicosΘi) (2.19)

with ATR for the torsional oscillation amplitude and ΘTR = 2πfTRt. Highresolution in lateral force detection is achieved, if the force is not averaged.This means the corresponding amplitude of the lateral oscillation ATR mustbe set smaller compared to the lattice constant. Like in bimodal DFM withtwo flexural resonance modes, the shift ∆fTR is given, for the conditionsthat A1st � λ > ATR:

kTR∆fTR

fTR= − 1

∫ 2π

0F ′X(z0 + A1stcos(Θ1st))dΘ1st (2.20)

here F ′X is the lateral interaction force gradient in the X direction. Theflexural amplitude has to be large compared to the decay length, to neglectany tip apex deformation, due to the lateral interaction force.

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Comparing the detection sensitivities of both bimodal DFM modes, the sec-ond flexural ∆f2nd and the torsional ∆fTR with a large A1st. We notice,that the difference between Equation 2.17 and 2.20 is the direction of theforce gradient. The torsional frequency shift arises from the site depen-dent interactions, hence the macroscopic shape of tip apex should not affectthe lateral force detection, as long as stable imaging remains possible. Thelateral sensitivity is higher and better compared to a flexural mode withultra-small amplitudes. One of the reasons is the better mechanical qualityfactor of the torsional mode QTR, which is typically around ∼ 140000 andso, about 10 times higher than the quality factor of the second flexural modeQ2nd ∼ 14000.

2.3 Theoretical predictions and properties of h-BN nanomesh

The hexagonal boron nitride single layer grows on a transition metal sur-face, due to the good catalytic activity. Transition metals allow a perfectlayer growth by means of vapor deposition [31]. The single layers are sp2

hybridized honeycomb networks and form strong in plane σ and weaker πbonds to the substrate and the adsorbates.A corrugated single layer enhances the binding energy by a lateral bindingcomponent for small molecules. These additional binding energies exceed thethermal energy and makes adsorbates thermally stable up to several 100Kabove RT . The nanomesh is a sp2 hybridized layer, which is very robustto environment conditions so it can be immersed e.g. into liquids withoutstructural changes [44].For developments in nanotechnology it is useful to have single layer systemswhich are inert and remain clean at ambient conditions and are stable up tohigh temperatures.The h-BN nanomesh is an outstanding example for such a sp2 single layersystem, but has a highly reactive surface to bind adsorbates. Compared tographene, which is another sp2 hybridized layer, the nanomesh, grown on atransition metal is an insulator, while graphene is metallic.The atomic structure and the corresponding lattice constant of the sp2 layerplays a key role in the understanding of the corrugated topography, depend-ing on the strong in plane σ bonds of the honeycomb lattice and relatedto these, the weak out of plane π bonds to the substrate. The π bondingdepends on the registry to the substrate atoms, where the layer - substratehybridization causes a tendency for lateral lock-in of the overlayer atoms tothe substrate atoms. If the over layer has the same lattice symmetry as thesubstrate, a lattice mismatch M is defined as:

M =asurface − abulk

abulk

where asurface and abulk are the lattice constant of the surface layers,respectively.

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The sign of M defines a compressive stress of the surface layer (+) or a ten-sile stress (−). If the lattice mismatch M of a laterally rigid sp2 exceeds acritical value, corrugated super structures with a large lattice constant areformed, which lead to a template function.The theory of Greber et al. [15] predicts an importance of occupancy ofthe d-band of the substrate by the nitrogen atoms. Since it is energeticallyunfavorable, if the nitrogen atoms are moved in plane, away form the topsites, it is expected that the energy will scale the adsorption energy. Fora rigid sp2 layer without a lateral lock-in energy available, no corrugationis expected, but instead a moire type pattern. But with a lock-in energygain for preferred bonding sites a dislocation of the surface layer is expected.This involves the formation of commensurate coincidence lattices betweenthe substrate and the surface layer with regular dislocations.For h-BN on rhodium(111) the lattice mismatch is M = −6.7% and the13 × 13 BN units coincide with 12 × 12 Rh units Fig.2a. This leads to aresidual compression of the 13 BN units by 0.9%. The corrugated superstructure Fig.2b, called h-BN nanomesh, has a regular pattern with a peri-odicity of 3.2nm. The pattern consists of a honeycomb mesh with ”wires”and ”holes” or ”pores”.It turned out, that the special super structure is only a corrugated singlelayer of boron nitride. The height difference of the corrugated layer is cal-culated to be 0.05nm. This is sufficient to produce a distinct functionality,like trapping molecules [24]. The reason is, that the super structure has twoelectronically distinct regions, which are related to the topography Fig. 2c[15].

2.3.1 Theory of the electronic structure and work function

The work function is a material dependent constant. It is the minimumenergy to remove a electron from the Fermi to the vacuum level. For a dis-cussion of the electric field near the surface, we have to recall the Helmholtzequation [25] that relates classically the work function Φ of a flat surfacewith vertical electric dipol:

Φ =e

ε0Na · p, (2.21)

e denotes the elementary charge, ε0 the permittivity and Na the areal densityof dipoles p, if the dipoles are assigned to the atoms.If an insulating single layer is placed on top of a metal surface, the verti-cal dipole field of the metal surface will polarize the insulating single layer,this polarization will decrease the work function Φ by ∆Φs = e

ε0Na · pind by

screening Fig.2c. The induced dipole pind is proportional to the out of planeelectric field of the surface E⊥: pind = α · E⊥, with polarizability α. Theelectric field has a strong vertical distance dependence of the surface dipole

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layer, this involves a correlation with the screening induced work functionshift ∆Φs. In the case of the corrugated h-BN surface layer we will expect adifferent local work function for a ”wire” and a ”hole” site, due to the heightdifference. Hence, from the Smoluchowski effect [13], the corrugation alsoinduces a non-uniform surface charge density and thus lateral electric fields.In the case of the h-BN nanomesh the two different electro negativities ofboron and nitride cause a local charge transfer from the boron to the nitrideatoms. If the h-BN is on a metal the screening of the ionic charges increasesslightly a site dependent charge transfer, resulting in a net charge displace-ment of 0.06e− per atom towards the substrate for the hole sites [26]. Ahigher work function is expected for the wires compared to the holes, as wellas lateral electric potential variations Φholes < Φwires.

Figure 2: (a) is a simplified side view of the nanomesh, which show thedislocation of the 13 boron or nitride atoms on top of the 12 rhodium atoms.Figure (b) is a DFT calculation done by Laskowski et al. [27], it shows thetop view of the structure and denotes the corrugation height with respect tothe rhodium surface by the color scale. Figure (c) is a schematic view of theenergy levels above the nanomesh. The levels show a variation, which hasits minimum within the region of the holes. This level variation is expectedto be extended far above ∆z ∼ 10nm the surface of the nanomesh.

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3 Experimental

The atomic force microscope (AFM) [10] can be operated in several modes,depending on the the cantilever dynamic and the imaging mode. The can-tilever dynamic is separated in two classes the static and the dynamic. Staticdeflection AFM or nc-AFM uses the static bending of the cantilever prongfor force detection. In the dynamic mode of operation, the cantilever pronggets actuated in the fundamental flexural resonance f1st, here the force in-teraction is measured by the shift in the resonance frequency ∆f . For themeasurements shown in this work, the non contact dynamic mode is usedand is referred as non contact dynamic force microscopy (DFM).

3.1 Dynamic force microscope

All measurements were made with an in house designed beam deflectionDFM Fig.3 [16] operating at room temperature under ultrahigh vacuum(UHV) conditions (pAnalysis < 1 · 10−10mbar). The advantage of the DFM-system lies in the detection bandwidth of ∼ 3MHz. This allows simul-taneous excitation of multiple cantilever resonances with a large separa-tion in their frequencies. Signal readout is done with two NANONIS PLL(Phase-Lock-Loop) controllers, a Real time-Computer (RT-Computer) anda NANONIS SPM Controller realised in a GUI software.

3.2 Cantilever preparation

Here, highly n+ doped silicon cantilever from Nanosensors PPP-NCL areused. The Cantilever is l ≈ 200µm long, w ≈ 40µm width and t ≈ 7µmthick. On top of the cantilever prong is a conical shaped tip with an heightof h ≈ 20µm. The cantilever support is actuated with an piezoelectriccrystal. The typical fundamental resonance frequency is in the range off1st ≈ 150± 10kHz with a quality factor of Q1st ∼ 30000. The second flex-ural resonance frequency lies at f2nd ≈ 970 ± 20kHz at Q2nd ∼ 15000. Inorder to use these cantilevers under UHV conditions, the cantilevers have tobe glued on Omicron Nanotechnology holders. Therefore, the support of thecantilever is aligned on the holder and glued with a two component Epotekglue and the 4-point-glueing-technique, where the cantilever is glued only atits four edges on the cantilever holder plate. This technique has achievedthe highest Q-values for the first and second resonance frequency.The glue must be baked in an oven at T ≈ 393K for ∆t ≈ 120min to harden.The cantilever with the holder is then introduced into the UHV-chamber forfurther preparation. Inside the vacuum, the cantilever is baked again atT ≈ 393K to evaporate the absorbed water. When the preparation chamberreaches normal pressure values of pprep = (4 − 7) · 10−10mbar the baking isstopped. After the cool down to T = RT the cantilever is sputtered with ar-

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Figure 3: scheme of a ”beam deflection” Atomic Force Microscope

gon ions to remove remaining adsorbates and the silicon dioxide(SiO2) layer.For the argon sputtering a filament current of I ≈ 10mA is set for ionizingthe argon and applying a bias voltage of Vsputter = 680V to accelerate theions to the cantilever. The procedure takes t = 1min, with a longer sputterprocess, double or blunt tips become more likely.

3.3 Preparation of Rhodium(111) films

The rhodium thin film has been grown epitaxial on a sapphire crystal (Al2O3).This crystal was orientated in (0001) direction to have a hexagonal surfacestructure. The lattice mismatch of rhodium(111) can be neglected, hence arhodium film with a thickness of 150nm has no remaining stress, strain andis nearly defect free. Rhodium(111) has a hexagonal surface structure, dueto the growth mode on sapphire.The thin rhodium film has to be cleaned by many cleaning cycles, becauseof the contamination of C, O, H2O. On the other hand, are the rhodiumterraces broadened by these cycles. One cycle consists of:

• Argon sputtering at room temperature T = RT , sputter pressure ofpprep = 3 · 10−6mbar, sputtering voltage Vsputter = 800V for about∆t1 = 45min. This surface bombardment breaks up the surface.

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• The rhodium is heated with a ramp ∆T↗ = 60K/min to T1 = 1023K.

• Temperature T1 is held for ∆t2 = 60min. At this temperature con-taminations diffuse toward the surface and the thin film forms largeterraces.

• The sample is cooled down to T2 = 723K with ∆T↘ = 20K/min.

• At the temperature T2 the surface is exposed to oxygen O2 at a chamberpressure of pprep = 4 · 10−8mbar for ∆t3 = 10min. The temperatureensures a sufficient initial energy, for oxidation the contaminations, toform clean terraces of 100nm width.

For the rhodium around 10 − 15 cycles were performed. A check was donewith a DFM measurement. After the preparation of the rhodium, we haveset up the borazine (B3H6N3) to evaporate it and form the h-BN nanomesh.

3.4 Nanomesh preparation

In this work all measurements were done on a rhodium(111) layer coveredby a monolayer of boron nitride, called the h-BN nanomesh. To form the h-BN nanomesh on the clean Rh(111) surface, borazine is used as a precursormolecule.Borazine is a cyclic inorganic compound with the molecular formula B3H6N3.The units BH and NH alternate in the compound. Borazine is a specialcompound, because it is isoelectronic and isostructural with benzene C6H6.Hence it is called the ”inorganic benzene”. The borazine synthesis was re-ported first in 1926 [28]. It is a reaction with diborane B2H6 and ammoniaNH3 in the ratio 1 : 2 at 520− 570K.

3B2H6 + 6NH3 → 2B3H6N3 + 12H2

Borazine is a colourless liquid with an aromatic smell. It is less toxic, thermalstable, but can decompose to boric acid, ammonia and hydrogen with water.The bond length of the six B −N bonds is 0.1436nm. And it has a partialdelocalisation of the nitrogen lone pair electrons.Applications for borazine and the derivatives are potential precursors toboron nitride ceramics. It is also a starting material for other potentialceramics, as boron nitride ceramics. Borazine is used as a precursor to growthin films on transition metals, to form structures like the nanomesh onrhodium Rh(111) [15].The borazine needs to be clean before it can be evaporated. Therefor it need3− 4 repetitions of a short cleaning cycle:

• The borazine is frozen with liquid nitrogen.

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• It should melt slowly with low pressure pchamber ∼ 10−7mbar. Con-taminations melt and evaporate faster than the borazine itself, due tothe low pressure any dirt is transfered over the load-lock out of thesystem.

• Before the borazine is melted completely, the valve between load-lockand borazine chamber is closed, to avoid pumping the clean borazine.

After the cleaning cycles the borazine is ready to be evaporated on therhodium surface. Here, the frozen borazine is melted and then evaporatedinto a small volume to get a saturated gas phase between the borazine cru-cible and the needle valve leading to the preparation chamber. The borazinevalve should be closed again and the borazine cool back to T = 266K to avoiddecomposing. Meanwhile the rhodium is heated to Tdeposition = 1023K.The borazine is deposited at a pressure of pdeposition = 3 · 10−7mbar for∆tdeposition = 2min. At the deposition rate of 10th of monolayers per sec-onds the whole sample should be covered within the deposition time.The hot rhodium surface has an initial energy of Einit ∼ 0.13eV to splitoff the hydrogen of the borazine, which forms a new bond to a neighboringboron nitride ring. Those boron nitride molecules form the regular, hexago-nal nanomesh.

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4 Measurements

Measurements on the nanomesh are complicated. Stable tip conditions arehard to achieve. Changes in the topography and the pattern of the structureof the acquired data are often seen. These changes are mainly random andnot reproducible. First measurements with low resolution on wide scan areasshow the hexagonal honeycomb structure as the theory predicted see Fig.4.The observed periodicity of the structure is 3.2±0.1nm this is verified by theline scans Fig.4b and 2D FFT Fig.4c. The 2D FFT confirms the hexagonalstructure visible in the 3 fold symmetry of the spots. Line scans show acorrugation amplitude of 45±8pm, here again, we can confirm the theoreticalpredictions, with a calculated corrugation amplitude of 50pm.

Figure 4: Figure (a) shows a zoom of the nanomesh of a larger scan area.With wires ”bright” and holes ”darker”. Even in this small scan area insta-bilities were found. Figure (b) shows a line profile of the topography (see a),the maxima’s of the profile have a periodicity of the predicted 3.2± 0.1nm.Also the corrugation amplitude of 45 ± 8pm is in agreement with the the-ory Fig.:2b. Figure (c) is a 2D Fast Fourier Transformation (FFT) of thetopography. The position, distance and brightness of the spots denote thedirection of the wave vector k, wavelength of the of the k-vector and theamplitude, with corresponds to the regularity of the nanomesh. The wave-length of the k-vector is the inverse of the periodicity, it verifies the peri-odicity measured by the line profile. The FFT spots shows a 3 fold sym-metry, which confirms existence of a hexagonal structure. Image parameter:Scan area 150x150nm2, f1st = 153019Hz, Q1st = 33318, k1st = 25.1N/m,∆f = −149Hz, A1st = 5nm, γ = 8.63fNm0.5.

This measurement verifies the structural prediction, but shows a rather lowresolution at large frequency shifts of ∆f = −149Hz and large gamma fac-tors γ = 8.63fNm0.5.The shown topography Fig.:4a is predicted, but not always seen. We mea-sure three patterns and contrasts. First, the normal, predicted topographywith low resolution. Second, a topography with an inverted contrast but lowresolution. Third, a full inversion of the contrast of the topography, occuring

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mainly with an improvement of the resolution. A detailed discussion on thecontrast inversion is given in chapter 4.2.

4.1 Mesh stability

The borazine precursor molecule is evaporated on the rhodium at Tdeposition =1023K (see Section:3.4) for growing a regular nanomesh. This denotes a highthermal stability of the nanomesh under UHV conditions. Due to a faileddeposition treatment of truxene molecules, the sample had to be cleaned.Starting an annealing cycle at T = 1023K for t = 90min, no changes in thecoverage of the h-BN nanomesh could be observed, also longer annealing cy-cles for t = 150min showed no changes. So annealing above the depositiontemperature and an increasing of the temperature in steps of ∆T = 25Kuntil a change in the coverage can be seen, was the logical step.

Temperature in K Annealing time in min

1023 901023 901023 1501048 751048 901073 1201098 901123 90

Table 1: Table of annealing temperatures and annealing times for cleaningthe nanomesh.

Finally, the nanomesh was annealed at T = 1123K for t = 90min. Afterthe cleaning cycle, small amounts of adsorbates could be registered, whichare not distinguishable, if they remain from the truxene deposition or fromremnant gases in the UHV.Nevertheless, the measurements show still the regular pattern of the honey-comb superstructure. No signs of damages, disordering or evaporation sitesof the h-BN nanomesh were visible. So, the thermal stability for a nanomeshon a rhodium(111) substrate can be assumed to be even higher.Beside the thermal stability the mesh shows also high stability at ambi-ent conditions as well as a distinct longterm stability. Since the prepara-tion of the nanomesh until the most recent measurements, the nanomeshshowed none significant change within on year, which can be clearly leadback to changes in the superstructure. This is quite remarkable, because thenanomesh is only a single layer with no further treatment for stabilizing thestructure.

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4.2 Contrast inversion

The measured data is visualized by a color distribution, the global maximasets the upper end of the color range, while the global minima the lower.This method allows to visualize an acquired data signal, like the frequencyshift, for better recognition. The contrast is then related as the change in thecolor scale between two neighboring points. The corresponding change canbe positive, meaning a change from the lower color towards a higher colorfrom the defined range, or negative vice versa. If the change is large thecorresponding contrast makes these points more distinguishable from eachother. In an acquired data set, which is aligned regular picture like shape, werecognise this set of data as a ”topographical” information, if we consider thesignal of the feedback loop of the z-tube piezo in an atomic force microscope,the acquired data set represents the ”topographical height” information.An inversion of the contrast relation between two neighboring points showsa picture, which has the same shape and pattern, but reflects an inverted”topographical” information. This inversion and its origins are showed andinvestigated.We measure not only the predicted contrast as mentioned before (see Fig.:4),namely wire-hole contrast. Our measurements show over all, three clearlydistinguishable contrasts in the topography. The normal predicted contrastand inverted contrasts, which we call the low resolution inversion and highresolution inversion, shown below.

Figure 5: In the figures (a-c) we see from left to right the normal contrastas expected (a). The inverted contrast with low resolution (b) and the highresolution inversion (c).

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Figure (a) Figure (b) Figure (c)Scan frame in nm2 1502 2502 752

f1st in Hz 153019 153153 157502Q1st 33318 32916 33544k1st in N/m 25.1 25.1 27.3∆f in Hz −149 −8.5 −10.7A1st in nm 5 5 10γ factor in fNm0.5 −8.63 −0.49 −1.86

Table 2: Table of image parameters of Figure 5.

In every image a hexagonal pattern related to the honeycomb structure isshown. With a periodicity being within small errors 3.2±0.2nm. But one ofthe main difference is the corrugation amplitude, which is always enhancedin the inversion by a factor of 2 to 3. The relative corrugation amplitude ofthe low contrast inversion was measured to be 110 ± 15pm, while the highcontrast inversion shows a corrugation of 130 ± 20pm. Another differenceis seen in the high resolution inversion, in the line profile as well as in theimage. Here, it stands out that the z signal has a ”plateau” with a small”dip” in the middle of an hole, which is less elevated. The other contrastsshow no indication of a plateau, neither in the normal contrast, nor in thelow resolution inversion.In comparison with the DFT calculations of Laskowski et al. [27] the nanomeshshould show a flat area inside a hole. But in our case a really faint ”dip” isrecognizable in the middle of a hole.

4.2.1 Tip apex changes

A standard non contact cantilever has a conical tip of ∼ 20µm height. Atthe top of the cone, we consider a half sphere with a radius of ∼ 10nm. Thefront most atom of the tip, which has the closest distance to the surface, hasthe highest contribution in the frequency shift at the lowest turning pointof a cantilever oscillation. A change in the conformation of the front mostatoms, towards a tip shape with more atoms having the same closest distanceto the surface, can change the frequency shift drastically.It has been shown that tip apex changes in combination with polarity changescan produce contrast inversions. This was shown in the studies of Enevoldsonet al. [34] about TiO2 (110) surfaces in 2008. Studies of our measurementsshow often tip apex changes and corresponding height changes of the z piezotube. No additional offset or enhancement of the signal with respect to thenoise level could be measured.

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4.2.2 Double tip

On the other hand a double tip might be induced during tip preparation orscanning.If two tips measure with a single atom at the same distance above the sur-face at different lateral positions, then the acquired data is a superpositionof each topography measured by those tips. In addition, every topographyis a convolution of the surface structure with the tip apex measuring above.Double tips occur most likely with a dominant tip and a side tip. The dif-ference is the distance towards the surface, with the dominant tip having asmaller tip surface distance than the side tip. The side tip influence is seenat step edges, because the same step edge with the same shape is measuredtwice.Considering a double tip measuring above a corrugated surface at the sametip surface distance for both tips. The regular surface pattern is superim-posed with itself, the corrugation amplitude should change, due to inferenceeffects. But interference effects should affect the measured surface patternas well and even an inversion would be possible.

Figure 6: In this figure a scheme of a double tip is shown with the x conditionEq.4.3. Both tips have the same tip-sample distance zc at the lower turningpoint.

To prove the double tip theory and the assumption of a superposition of thesurface structure, a function for a hexagonal structure also used for HOPG[35] was used. Taking into account the periodicity a = 3.2nm and the

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corrugation amplitude ca = 0.05nm, we can write the function:

fnanomesh(x, y) = − ca

4.5

[2 cos

(2π

a· x

)cos

(2π

a√

3· y

)+ cos

(4 ∗ π

a√

3· y

)](4.1)

Figure 7: Figure (a) shows the nanomesh topography measured with a singletip. Figure (b) shows the the nanomesh generated by the function 4.1. Thegenerated nanomesh agrees nicely with the measurement.

The structural simulations were done by Matlab [36], by a small self madeprogram, see the appendix. These simulations are only mathematical func-tions, which fit the structure. They don’t simulate the interaction forces ortake the shape of the tip into account. But Figs. 7 (a,b) are in agreementwith each other. Meaning, that the function is adaptable for the superposi-tion of a second tip.This function is then superimposed with itself, to simulate the influence ofthe second tip, additional variables z1, z2 where inserted to vary the influ-ence of each tip. The range is z1,2 ∈ [0, 1], 0 for no influence, 1 maximalinfluence.

fdoubletip(x1, y1, x2, y2, z1, z2) =

−z1 ·ca

4.5

[2 cos

(2π

a· x1

)cos

(2π

a√

3· y1

)+ cos

(4 ∗ π

a√

3· y1

)]−z2 ·

ca

4.5

[2 cos

(2π

a· x2

)cos

(2π

a√

3· y2

)+ cos

(4 ∗ π

a√

3· y2

)] (4.2)

x1,2, y1,2 denote the positions of tip2 in relation to tip1:

x2 = x1 +(

n± 12

)· ~ax y2 = y1 +

(m± 1

3

)· ~ay (4.3)

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~ax,y denote the displacement vector in x and y direction with a correspondingvector length of 3.2nm, the nanomesh periodicity. The condition in equation4.3 is for a complete inversion of the measured topography. There are moreconditions in the measurements than the inversion condition, two other tipconditions are shown in figure 8.

Figure 8: Figures 1-3 show for column (a) the nanomesh topography mea-surements. Column (b) shows the the nanomesh generated by the function4.1 with the special conditions for the second tip shown in column (c). Thegenerated nanomeshes are in agreement with the different measurements.

The measurements and calculations in Fig.8 give certain hints for doubletip conditions during scanning. The calculated ”topographies” match withthe measurements and explain the seen contrast. But the calculation couldnot reproduce the high resolution inversion. The resolution of the calculated

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Figure 1 Figure 2 Figure 3Scan frame in nm2 502 502 752

f1st in Hz 154797 154797 157502Q1st 33957 33957 33544k1st in N/m 26.0 26.0 27.3∆f in Hz −75 −55 −10.7A1st in nm 6 6 10γ factor in fNm0.5 −5.85 −4.29 −1.86

Table 3: Table of image parameters of Figure 8.

”topographies” is limited by the structural resolution of the function 4.1.Another important feature of the double tip inversion is the position of theholes, when switching contrast. Due to the superposition of two structuralfunctions the holes shift their position during the contrast change. A hintfor this feature, was given by a previously measured 3D spectroscopy of S.Koch.A 3D grid spectroscopy, for a small scan area, represents force distance spec-troscopy curves at every grid point. Composing all spectroscopy curves to-gether a 3 dimensional data cuboid is given, containing 3D frequency shiftinformation. A better visibility of the 3D data set, is provided by cross sec-tion through isodistances of the force distance spectroscopy curves. Withinthese cross sections frequency shift maps are included, allowing comparisonof different sites.In this grid spectroscopy the thermal drift was not compensated and inducedtip changes, visible in the first third of the frequency shift maps Fig. 9.Figs. 9a, c show the 3D spectroscopy cross sections at different tip sampledistances. The upper one has a larger tip sample distance than the lower. Allfigures show measured or calculated frequency shifts with a negative contrastpattern, because the frequency shift is negative proportional to the z-signal.The upper figures show a frequency shift map with bright holes, for largetip sample distances. At one hole, marked with a red dot, a calculation wasdone by using the negative of equation 4.1. The calculated frequency mapwas tilted to match the measurements. Again a bright hole was marked forsite comparison during the contrast switching.The lower figures 9c, d show a contrast inversion in the frequency shift map.The red dot of the measured frequency shift map has the same position, butmarks a site of a wire crossing. The figure 9d was calculated by the negativeof equation 4.2 with a double tip influence. The red dot marks the same spa-tial position a above, but denotes a wire crossing site, too. Calculations arein accordance with the measurements. A comparison inside every column, achange of the hole sites during contrast change is visible.With the upper formalism 4.2 and relation 4.3, the 3D grid spectroscopy

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Figure 9: Figures (a, c) show layers of the frequency shift map of a 3D spec-troscopy at different tip sample distances. Frequency shift map of Fig.:(a)has a larger tip sample distance than the lower. Figures (b, d) show theexpected pattern calculated by the negative of Eg.:4.1. The 3D spectroscopyshowed an inversion of the wire-hole-contrast. A red dot marks the spot of ahole in the frequency shift map of Fig.:(a). In the lower frequency shift mapFig.:(c) the same red dot is on top of a wire. A simulation with the equation4.2 with relation 4.3 showed the same behavior.

measurements as well as the other seen low resolution contrast could beexplained. Therefore, it can be concluded that the low resolution contrastinversion (Fig.5b) is due to a double tip feature: superimposing two topogra-phies, measured at different sites.

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4.2.3 Tip geometry

The double tip induced pattern changes or even the contrast inversion bringsup the question of representativity and reproducibility. We cannot reproduceartificially the patterns of figures 8, which show that, these happen by chance,but they can be observed regularly. These patterns are seen with almostevery tip at some point, so that they should have a common feature.In addition, a consistency with the mentioned theorem Eq.:4.2 is needed. Theproperties for a ”good” AFM tip are firstly a small opening angle θ ∼ 15◦, toseparate the interaction forces, to be more dependent on short range forcesand secondly the atomically sharp tip apex to resolve single atoms or smalldefects. If a tip becomes blunt, meaning the tip apex is not atomically sharpand can be assumed by a tip cone with a tip apex curvature, which is largerthan an atom, then the resolution will be limited by the curvature. A largecurvature results in a large force averaging and features smaller than thecurvature are not resolvable.In our case, the tip cone with a low opening radius must be similar or smallerthan the periodicity of the nanomesh, otherwise it would not be possibleto resolve the wires or holes, meaning rconeopening = periodicity/2. Butthe tip apex curvature should be large, to see the force averaging effect.So a tip apex curvature, which is defined by a hole curvature is assumed.With a periodicity of 3.2nm and a corrugation of 0.05nm, an arc, whichintersects with the left wire (−1.6, 0.05), the center (0, 0) and the right wire(1.6, 0.05) can be defined. This results in an arc radius of rarc = 25.625nm[47], which is also assumed for the tip apex curvature. An atom at thetip apex could image the real topography, but would be accompanied byan averaged background. A second dominant atom at the tip apex withcondition mentioned in equation 4.3 could lead to a stable superposition ofthe topography.The stability during the measurement, is then defined by a stable position ofthe atoms to each other. If an atom change its position due to an influenceof an adsorbate or a step edge, the acquired pattern will change. A possibleway to distinguish a mono atomic tip or a double tip are step edges. Adouble tip influences and broadens the measured path across an step edge.In Scanning Probe Microscopy (SPM) step edges are imaged with the con-volution of the tip geometry [14]. In the acquired data, step edges appearbroaden and smeared out. The transition distance between upper and lowerterraces can determine the tip apex geometry [37]. If the transition distanceis ”small”, then the tip apex has a small curvature and thus has small influ-ence on the convolution between topography and the tip apex in the acquireddata. A single atomic tip resolves the actual topography the best.In our modeled case, two atoms are assumed, each resolves the topographycorrectly. The superposition of the topography will result in a broadeningof the transition distance of a step edge. The minimum transition distance

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Figure 10: The Figure shows a schematic view of the assumed double tip.The tip cone has an opening angle of ∼ 15◦ and at the lower cone end theopening radius is modeled to ∼ 1.6nm, the apex curvature is r = 25nm.Two atoms are supposed at the tip cone end, which measure the dominantforce contribution, with atom position relation 4.3 in respect to each other,a superposition described by the figures 8 can be expected.

will be limited by the separation distance between those two atoms, thetip cone opening radius rcone and the tip apex curvature. Assuming a con-stant tip apex curvature as well as the tip cone opening radius, the tran-sition distance has dominant influence by the measuring atoms, see Fig.10.From the simple schematic point of view the transition distance for a singleatomic tip, which is placed in the center of the tip apex, should be in theregion of ∆strans ≤ 1.6. For a tip with two atoms, the separation in be-tween adds up, but is maximal limited by the cone diameter plus two timesan atomic radius, because we are sensitive to the out of plane force com-ponents. Meaning that the transition distance for two dominant atoms is∼ 1.6nm ≤ ∆stransistion ≤∼ 3.2nm.Figure 11 shows a pattern and contrast change of the topography. The seencontrast will be discussed in the next section 4.2.4. The slow scan directionwas bottom up and the change in the image happened by chance. Thecross sections are perpendicular to the step edges to measure the acquiredtransition distance, without artifacts. Both cross sections are parallel withrespect to each other, allowing a direct comparison of the data, due to thesame relation of slow and fast scan direction components. We suppose,that the upper cross section shows the topography measured by a singletip. We measure the transition distance of a step edge to ∆strans = 1.39 ±0.07nm. This distance is an evidence, that the tip apex has broaden andis decorated by an single dominant atom. The lower cross section shows a

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Figure 11: The Figure 11a shows the topography, which shows a changein the patter as well as in contrast. Two cross sections are taken acrossstep edges. The upper cross section shows the topography measured byone tip. The lower cross section shows the topography measured, we as-sume, by a double tip, see Fig. 11b. The step edge transitions distanceis measured for both cross sections. The transition is broadened for everystep of the lower cross section. Image parameters: Scan area 50x50nm2,f1st = 154793Hz, Q1st = 33000, k1st = 26N/m, ∆f = −75Hz, A1st = 6nm,γ = −5.84fNm0.5.

transition distance, which is enhanced by almost one nanometer, ∆strans =2.33± 0.20nm this holds with our simple assumption of an second dominantatom at the tip apex with a broaden tip apex curvature, which lead to anstable superposition of the acquired data. From the comparison of bothcross sections we can propose, that the topography in the upper part hasless spatial averaging of the forces, while the lower image part gives evidenceto a large spatial averaging, which reduces the corrugation amplitude of thewire hole pattern. Even with a different acquired contrast pattern of thetopography, both cross sections seems to acquire the step height correctly.The height difference between two terraces in both contrast is measured tobe ∆h = 230 ± 20pm, this is in correspondence to the surface relaxation ofrhodium in the (111)-direction, which was measured by low energy electrondiffraction (LEED) dbulk = 219pm [38].The tip apex model we derive is capable to explain the measured pattern,which leads to a low resolution acquisition of the topography. The stabilityof the tip apex configuration we discussed in detail is consistent with thedata. We think the low resolution contrast inversion is a tip apex inducedfeature. The High resolution contrast inversion, from this point of view, is

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not explainable by the mentioned theory.

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4.2.4 Height dependent contrast inversion

The hole wire contrast seen in Fig.5c is characterised by high resolution,reproducibility and stability. Still, it is not clear whether it is a real structuralchange, a surface feature or if it is a tip induced measurement artefact.The investigation was difficult, because a second 3D grid spectroscopy wasno good solution, due to the lack of thermal drift compensation and tipinstabilities. Normal frequency modulated DFM topography measurementsseemed to be the best solution, due to the adequate acquiring time, thuslow thermal drift and high fidelity of the tip conditions. But a topographymeasurement on a clean surface at a fixed frequency shift set point givesno indication on small hole-wire site changes, during a contrast inversion.Therefore, a position marker was needed, for a clear evaluation of hole-wiresites. A simple solution for position markers are random adsorbates alignedin a recognisable way.

Figure 12: Topography maps at the exact same position are shown, visibleby three random adsorbates at the corners. The frequency shift set point wasincreased slowly from left to right dfa,b,c = {−10.5Hz,−12.5Hz,−15.5Hz}.The measurements shows a contrast inversion for a lower frequency set point.Line profiles at the same position, recognisable by the markers, show thecorrugation amplitude behavior and the maxima/minima position for thenanomesh. Figure (b) seems like an ”intermediate” state, here the wires andthe middle of a hole is lower and the rimes of the nanomesh are highlightedeven the corrugation amplitude decreases for the ”intermediate” state. Animportant fact of these topography measurements is the not changing posi-tion of holes, during the high resolution contrast inversion.

Figures 12(a-c) show topography maps of the same position with increasedfrequency shift set point. From left to right a clear contrast inversion ofthe nanomesh is visible. Figure 12 b seems to show an ”intermediate” state

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with enhanced rims and low wires and hole centers. The markers fix theposition of the line profile for a comparison. First visible difference of theseline profiles is the corrugation amplitude. The ”intermediate” state has areduced corrugation amplitude by a factor of ∼ 2 in comparison to normaland inverted contrast. Secondly, the profile structure changed in these threemeasurements. The normal contrast shows a ”sinusoidal” profile curve, whilethe ”intermediate” state as well as the inverted state shows additional peaks.All line profiles were recorded a the same position judged by the three mark-ers. This allows clear comparison of hole-wire positions. In the line profileof Fig. 12a a maxima was marked at a fixed position. In figures 12(b-c) thesame position marker denotes a minima of the measured topography. Thisshows clearly, that holes and wires does not change their position duringa high resolution contrast inversion. In conclusion, the high resolution in-version is a real surface property and is not a measurement artefact due tomultiple tips. Any superposition of multiple nanomeshes should change theposition of holes and wires.If the high resolution contrast inversion is height dependent, a 2D line spec-troscopy should show a height dependent variation in the force fields. Figure12 denote a rather intense dependence ranging within δf = 5Hz betweennormal contrast and inverted contrast. Recalling the nanomesh theory, fig-ure 2c shows a shift in energy levels by virtue of a non uniform electronicdistribution [15]. This enforces the expectation of spatial variation of forcesabove the surface.

4.2.5 2D spectroscopy

The z-spectroscopy correlates one point of the surface (x, y) with a largenumber of measuring points in the surface normal z. Spectroscopy givesinsight in the forces above a certain point, with a high vertical resolution.Here, 2D spectroscopy was used.Theory predicted vertical as well as lateral force deviation within the range ofabout a few nN or in the frequency shift ∆f of some Hertz. These deviationsare supposed to be measurable and extend to several nanometer above thesurface into the vacuum [15].The 2D spectroscopy was performed with the second flexural resonance f2nd,this enhances the short range force sensitivity, due to the measurement ofthe force gradient Eq.:2.17 [20]. The second flexural amplitude was set toA2nd = 400pm to have a enhanced sensitivity and to ensure an appropriatesignal to noise ratio. In the first spectroscopy measurements only the secondflexural resonance was excited. First we wanted to acquire a set of data bymonomodal measurement, to compare this with bimodal spectroscopy data,for additional features and cross talk relations.The 2D spectroscopy was recorded at the path of the cross section seen inFig. 13f. The acquired spectroscopy data was evaluated and further com-

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Figure 13: (a) frequency shift of the second flexural resonance f2nd in de-pendence of z. The average frequency shift of the line was subtracted 13b.The force is calculated by the Sader algorithm 13c and the same spec-troscopy curve is plotted after post processing. A force average of eachline is then subtracted 13d. The acquired excitation was converted to theexcitation per cycle 13e. Figure 13f shows the cross section of the 2D spec-troscopy before the measurements. Spectroscopy parameters: Line distance4.372nm, sweep distance ∆z = 5.0nm, VCPD = 0.304mV , f2nd = 967594Hz,Q2nd = 13099, k2nd = 1701.0N/m, A2nd = 400pm. Image parameter: Scanarea 10nmx10nm, f2nd = 967594Hz, Q2nd = 13099, k2nd = 1701.0N/m,∆f = −230Hz, A2nd = 400pm, γ = −0.29fNm0.5.

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puted by a self written MathWorks Matlab program [36] (see. appendixB). The data is correctly aligned by a force gradient correction of the longrange tail of the spectroscopy curves. A manual drift correction was notapplied, because the gradient correction error is indistinguishable from thecross section error. The fully processed data was smoothed vertically andhorizontally by an Svitzky-Golay filtering just before plotting. This reducesthe measurement and post processing noise in the data.One frequency shift curve after post processing is shown beside figure 13a.A long range spectroscopy curve of a sweep distance of ∆z = 20nm wasadded to each spectroscopy curve to aligned each towards zero frequencyshift, meaning the cantilever is at its free oscillation with no perturbation.The post processed frequency shift data should match the derivative of theLennard Jones potential Fig.1. The spectroscopy curves end before theequilibrium bond distance is reached, to avoid tip instabilities or tip apexchanges.The subtraction of the average frequency shift at distance z in each lineenhances the visibility of small deviations. The resulting figure denotes thechange in frequency shift for a ”quasi” constant height measurement. Figure13b indicates clearly long range frequency shift deviations above wire sites,which extend more than ∼ 3nm above the surface. The periodicity of thefeatures coincident with the nanomesh periodicity. The deviations from themean value increase drastically, if the tip sample distance becomes small,∆ztip ≤ 1nm and the actual surface topography becomes visible and domi-nant in the acquired data set.In addition figure 13b shows some feature at the end of every spectroscopycurve bélow z-distances of z ≤ 80pm. The frequency shift deviation fromthe average line value switch their signs abruptly, within a small distance∆z ∼ 50± 20pm. This feature will be discusses later in detail 4.2.6.Figure 13c shows corresponding to the frequency shift the calculated force.The calculation of the force includes the 20nm long range tail. This wasadded to the spectroscopy to apply the Saders algorithm [39]. A single forcecurve at the same position as the frequency shift curve was picked for clearvisibility. This force distance curve reaches a peak of F2nd = −73.02nN .The force distance curve should be related to the Lennard Jones potential,Fig.:1, which shows in comparison to the frequency shift curve a smootherdecline, due to the dominance of the integral tail.After subtraction of the average force at distance z in a line, the force de-viations at a ”quasi” constant height become enhanced. The force differenceof the wire to hole sites at a fixed distance is clearly visible. The force de-viations show again the periodicity of the nanomesh. The variance in theforce deviation rises presumably from the averaging effect of the integrationand the topography irregularity. In comparison with figure 13b no changein the force deviation is visible close to in the region below z ≤ 80pm. Thisfeature is presumably hidden, due to the dominance in the frequency shift

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integration.Simultaneously to the frequency shift measurements the excitation voltagefor the piezo actuation was recorded. This data set was computed to the ex-citation per a single oscillation cycle by the conversion formula of Giessible,see [18]. The excitation per cycle is related to the dissipation and the energyloss of the cantilever-sample system. On the other hand, a tip apex deforma-tion would be visible in the actuation energy. The excitation data coincidentwith the recorded frequency shift. A larger actuation energy is needed ontop of wire sites. From the excitation data, we have no visible evidence fora tip induced inversion of the frequency shift in the sweep distances belowz ≤ 80pm. Presumably, it can not be seen in the excitation per cycle, dueto the point, that the interaction force acting on the cantilever during oneoscillation, changes the frequency shift and the amplitude. The amplitudechanges in correspondence to the force, while the frequency shift is relatedto the force gradient 2.17 [20]. The force does not show this small variationsclose to the surface, due to that the excitation force exciting the cantileverto a constant amplitude does not show this variations, too.The detailed analysis of the inversion in the frequency shift given below 4.2.6,but from the excitation energy per cycle it becomes unlikely that this inver-sion lasts from tip induced effects, like double tips or any chemical bondingstowards the substrate, because of the relative small changes in the actuationenergy of ∆E = 0.13± 0.01eV .

4.2.6 Detailed spectroscopy analysis

The observed inversion Fig. 13b was neglected in the first place, becauseof uncertainty in data post processing. The data seem to show an smooth-ing error or some artifacts from the applied gradient correction. The postprocessing code was checked and possible errors were tested. It can be con-cluded that the inversion is not introduced by the data post processing ofthe Matlab program [36].In the frequency shift plot Fig. 14, iso lines with the same frequency shifthave been plotted at the frequencies:

∆fisolines = {−230,−200,−170,−140,−110,−80,−50,−20}

The plotted contour lines in the frequency shift data Fig.14a show for ∆f2nd =−20Hz at distance of z = 2.15nm only small deviations, which seem to berelated to the theoretical expected topography. Above this contour line thefrequency shift show a smooth homogeneous decrease to ∆f2nd = 0Hz. Be-low the contour line the frequency shift deviations show the theoretical to-pography more pronounced. For ”tip-sample” distances between z = 0.25nmand z = 0.5nm the contour lines show the largest corrugationamplitude intheir deviations, the maxima and minima of the deviations have the predictednanomesh periodicity and show a surface corrugation within the predicted

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Figure 14: Contour lines of constant frequency shifts were added to theacquired frequency shift data of the spectroscopy, Fig.: 14a. The verticalwhite lines denote the positions of the distance - frequency shift curves.With the left blue curve for a hole and the right red curve for a wire crosssection. Figure 14b is a zoom of the acquired frequency shift map withcontour lines. The contour line for ∆f2nd = −230Hz is marked, becauseit shows a inverted curvature, in spite of the other contour lines. The twocross sections are plotted together on a 5nm scale, the interval between 0and 0.5nm is magnified to show a crossing in the frequency shift curves atthe sweep distance of z ∼ 80pm.

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limits. For ”tip-sample” distances, below z ≤ 0.25nm the amplitude of thedeviations in the frequency shift decreases and show even an inversion of thetheoretical topography for a contour line of ∆f2nd = −230Hz.In figure 14b a detailed view of the closest region with a contour lines aregiven. Here, the contour line of ∆f2nd = −230Hz is marked with an arrow.This contour line shows an inversion of the theoretical predicted topographyin the frequency shift. Meaning, if the tip-sample distance is controlled bya set point below ∆f2nd = −230Hz, the measured z signal or topography isinverted.For a detailed insight, two cross section were extracted from the frequencyshift data. Fig.14c. The blue line is a frequency shift distance curve at ahole site and the red at a wire site. In the upper plot, the full curves areshown. The long range tail of those curves match each other quite well,down to a sweep distance of z = 2.5nm. The red curve, for the wire site,has a less negative frequency shift in comparison to the hole site, blue curve.The difference between both curves become largest for ”tip-sample” distancesof z = 0.16nm. For smaller distances the difference in the frequency shiftbecomes smaller and the curves show an intersection at z ∼ 0.08nm. Fordistances below this intersection the wire sites has a more negative frequencyshift than the hole site.The actual distance at which the crossing occurs is not accurate, becausethe measured frequency shift is strongly tip shape dependent. But the datashow no indication of tip induced features, as it could be estimated by theacquired date in Fig. 13.To distinguish a height dependent contrast inversion clearly from other ef-fects, we can think of probable boundary conditions for such an effect. InFM-DFM mode with small amplitudes the frequency shift is sensitive to theforce gradient. The tip apex is sensitive to a assumed surface potential witha dominance of the Lennard-Jones potential. Any physical property, whichshifts the turning point or the slope of these potentials, would lead to achange of the measured frequency shift on the surface at a certain tip sam-ple distance. If we assume, that wires and holes have independent propertiesand can be described by linear independent potentials with different slopes,we could explain a height dependent contrast inversion by a transition of thepotentials at a certain frequency shift level above the surface. This could re-sult in a distant dependent enhancement of holes compared to wires. In the2D line spectroscopy Fig.13 and Fig.14 such a distant dependent inversionof holes is visible.For the difference in the frequency shift curves Fig.: 14c, a explanation canbe the non uniform electronic properties of the surface [15]. But the com-bination of long and short range forces and the dependence of the tip apexsets a sophisticated task. But, a procedure for measuring and post precess-ing calculations for separation of the influences can be performed.The tip shape or tip apex curvature can be in situ estimated by measuring

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the tip sample distance at a fixed frequency shift for ramping the samplevoltage [40]. This method allows the evaluation, of the tip apex curvatureand the dependence of the conical part. The method should be applied be-fore and after a spectroscopy, if the measured voltage dependence does notshow any difference, it can be assumed, that the tip apex does not change. Incomparison to a frequency shift versus distant curve the advantage of ramp-ing the sample voltage at a larger distance is to avoid touching the surface.For the acquired frequency shift data a analytical fit of the electrostatic forcesbetween the tip and surface can be extracted [41]. This fit takes the tip shapeand the tip apex into account. This values for an initial fit values can beassumed by the bias dependence of the tip position at fix frequency shift[40]. After the subtraction of the long range electrostatic part, the frequencyshift data should contain mainly contributions of the van der Waals forceand the chemical as well as the electrostatic short range force. The chemicalinteraction can be neglected for tip sample separations larger than z ≥ 1nm.For distances below the tip radius the curve can be fitted by the spherical tipmodel Eq.2.5 [17] and the Hamaker constant can be extracted from the data.After subtracting the van der Waals contribution of the frequency shift datathe chemical short range forces remain. For the limit, that the A1st,2nd � zthe data can be fitted by an equation Guggisberg [17] introduced. In themeasurements of Fig.:13 this equation does not apply, because we have anamplitude A2nd, which is comparable to the decay length λ of the chemicalforces.Even though, that the chemical forces cannot be fitted correctly, the contri-bution of the chemical force should be dominant after post processing. Thisseparation method, can be a powerful tool to give a cross check of the fre-quency shift variations of the 2D Spectroscopy curves of the nanomesh dataFig.14.

4.2.7 2D bimodal spectroscopy

To get a detailed insight of the short range interactions a bimodal 2D spec-troscopy was performed exciting a normal and torsional oscillation. Thesecond flexural resonance frequency was used for tip sample distance controlwith an amplitude of A2nd = 600pm. Simultaneously, the torsional reso-nance was excited and measured with an amplitude of ATR = 80pm. Thisspectroscopy had two purposes, first, the analysis of the an expected lateralforce variation of the nanomesh [15]. Second, the influence of the torsionalexcitation on the flexural 2D spectroscopy.Figures 15 with the marked cross sections denote the position of the 2Dspectroscopy. The data was measured simultaneously by bimodal DFM.The height control was warranted by the second frequency shift ∆f2nd, inaddition the torsional frequency shift was acquired with a high sensitivity.

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Figure 15: (a) topography measurement controlled by the second amplitude.(b) the simultaneously measured frequency shift of the torsional resonance.Graph (c) denotes a line profile of the position of the 2D spectroscopy.Image parameters figure (a): Scan area 20x20nm2, f2nd = 967617Hz,Q2nd = 13065, k2nd = 1701.0N/m, ∆f = −46Hz, A2nd = 600pm, γ =−0.11fNm0.5, fTR = 1.486589MHz, QTR = 123253, kTR =∼ 2000N/m,ATR = 80pm

In the recorded frequency shift the wires are visible with a fast decreasingfrequency shift on top of them (at the positions ∼ 1.5nm, ∼ 4nm, ∼ 6.75nmand ∼ 9nm). This spectroscopy data is influenced by uncompensated ther-mal drift in the range of ∆zthermal = 35±7pm calculated by the residual datamismatch after post processing. Compared to the spectroscopy Fig.:13 thebimodal spectroscopy Fig.:16 suffered from a bad tip shape and presumablya large tip apex curvature, visible in the strong averaging of the frequencyshift ∆f2nd.Spatial ∆f2nd variations in the range of δf = 5Hz are visible after subtrac-tion of the mean value in each line, Fig.16b. For large tip sample distancesa spatial variation of the ∆f2nd signal is visible. On top of the wires, morenegative frequencies are recorded compared to the line average as seen beforeFig.:13. Below a tip sample separation z ≤ 0.2nm the wires and holes areclearly distinguishable.The calculated forces Fig.16c show a deviation, which is in correspondenceto the corrugation of the nanomesh below. More detailed information are

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Figure 16: Figures (a-d) show the recorded frequency shift of the secondresonance (a), second frequency shift less the line average (b). From thefrequency shift the distant dependent force was calculated by the Saderformalism (c), while the line average is subtracted in figure (d). The dis-tance dependent excitation is shown in (e). Figure (f) shows the recordedfrequency shift of the torsional resonance. Spectroscopy parameters: Linedistance 9.203nm, sweep distance ∆z = 1.1nm, bias CPD = 0.589mV ,f2nd = 967617Hz, Q2nd = 13065, k2nd = 1701.0N/m, A2nd = 600pm,fTR = 1.486589MHz, QTR = 123253, kTR =∼ 2000N/m, ATR = 80pm

obtained, if the line average of the force is subtracted Fig. 16d. On the leftside of the data plot, holes show a quite extended field with values abovethe line average, marked in blue colors. In comparison to a wire site, thevalues are slightly below the line average, except the case close to the sur-face, here forces have a larger deviation from the mean value. On the right

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side in comparison to the left side the interplay between more attractiveregimes and less attractive regimes seem to change. One possibility could bethe uncompensated thermal x− y − z drift. The Saders formalism is highlydistance dependent. If the distance between tip and sample is varied duringthe spectroscopy measurement, it would show an influence in the force cal-culation. For large tip sample distances the wires are more pronounced. Theforce calculation with the subtracted line average shows this behaviour moreclear. This mentioned feature was also visible in the force calculations of thespectroscopy above, Fig.13b. It seems that this is a systematical error, dueto z drift or large lateral x− y drift, which changes the position to be onlyon top of wires.The excitation signal shows almost no features except close to wires. Incomparison to the spectroscopy of Fig.:13e the excitation energy range hasshifted to larger values.The last figure 16f shows the recorded frequency shift signal of the first tor-sional resonance. Theory has predicted an additional lateral binding forcecomponent close to the surface [15]. This bimodal spectroscopy measurementshows the predicted property for small tip sample distances. The frequencyshift pattern corresponds to the hole wire sites, seen above. Again a smallgradient in the frequency shift is visible, from less negative shifts on the leftto more negative on the right.

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Figure 17: A small intersection of the torsional frequency shift of Fig.16f isshown with a projection of the surface on the x − z-plane. Two wires witha negative frequency shift and three holes with a positive. Far away, smallfrequency shift deviations are measured, which are in accordance with thewire-hole positions. Close to the surface, the lateral variation, due to thecorrugated surface, become more visible. A doubling of the peaks or holesevolves at close distances.

The torsional frequency shift is shown figure 17 with a projection of thesurface on the x − z plane. The sweep distance of the data was stretched,due to the density of data points. The long range part of the torsional fre-quency shift was cut, because of no additional information. The z distanceis arbitrary. For large tip sample distances, the wires have an extended tailin the torsional frequency shift and are showing a more negative shift. Theperiodicity of the extended features are in correspondence with the measuredtopography. The average torsional frequency shift value is for large tip sam-ple separations at fTR = −2Hz. Close to the sample the faint variationsbecome enhanced. First the signal to noise ratio becomes better and thewires and holes are clearly visible. At very small tip sample distances thewires as well as the holes show a doubling of the depressions or peaks. Theholes separate towards two peaks and in the center of a hole the torsionalfrequency shift is almost zero. The wires show a doubling of the depressions,too. But, in comparison with the holes, the frequency shift is almost zero or

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even slightly positive for positions directly above the wires.This torsional frequency shift pattern close to the surface, which relates tothe lateral force variation, becomes comprehensible, if it is assumed, that thelateral force variation is proportional to the second derivative of the topogra-phy. Lateral forces arise, if the surface becomes rough or periodic corrugated[15]. The lateral forces become large, if the lateral gradient of the topogra-phy is large, see Fig.:17 at wires. The measured torsional frequency shift atextreme small amplitudes is related to the derivative of the torsional forceEq. 2.20 [21].If we consider a geometry consisting out of step functions like the Fermi func-tion at non zero temperature, we can model a cross section of a corrugatedtopography, see Fig.18. The shape of the lateral force can be approximatedwith the derivative of the modeled topography in x-direction. The mea-sured torsional frequency shift, is then related to the second derivative inx-direction of the topography. The lateral force gradient should becomezero, if the gradient of the topography is zero.If those Fermi functions are in a small interval, the lateral force gradientwould start overlapping. At a sufficient small interval, the response wouldexpress itself in a doubling of the peak with only one dip or vice versa.The depth of a dip between two peaks should be dependent to the distancebetween the Fermi functions. The double dip feature at wires depend, pre-sumably, on the finite width of the wires itself and the convolution of the tipshape.Figure 18a was modeled in accordance to the periodicity and corrugation ofthe nanomesh for a line cross section through holes. The modeled nanomeshwas derived in x-direction to visualize the shape of the lateral force, Fig.18b.The lateral force is not simulated and does not take real forces into account.Therefore, it is arbitrary in units. The second derivative of the topographydenotes the change in the force and should correspond to the lateral forcegradient 18c, which is measured by the torsional frequency shift Eq.2.20[21]. Figure 18c shows the doubling of the structure, as it was seen in inthe measured frequency shift Fig. 17. Although, the second derivative ofthe topography does neglect any forces and considers only the topographicalshape of the surface.In a detailed comparison between figures 18c and 17, differences can be seen.First, hole peaks are much more broaden and have a less pronounced depres-sion in the real spectroscopy. Second, in the spectroscopy, wire depressionshave a strongly pronounced peak, which has a small peak width. The peakheight reaches the zero frequency shift level and rises also above the zerolevel into positive frequency shifts. Third, the slope between wire depressionand hole peaks in the spectroscopy is not as steep as the change in the mod-eled approach. Forth, due to the fact, that the model has no real physicalproperty the peaks have a arbitrary scale and cannot be compared to thevalues of the torsional frequency shift.

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Figure 18: (a) a topography cross section was modeled with Fermi functionat non zero temperature, with the periodicity and the corrugation of thenanomesh. (b) is the derivative of the topography in x-direction and denotesthe expected lateral forces. (c) shows the second derivative in x-direction,it should show the expected lateral force gradient, which should assumed tobe in accordance with the measured torsional frequency shift.

These differences are mainly due to the fact that lateral forces can corre-spond to the surface topography, but there are more reasons influencing thelateral force, like electrical properties. On the other hand, the measurementsdepend on the tip apex shape.Beside the differences, the agreement between the measured lateral forceand the modeled is good. It can explain the shape and the doubling of thepeaks or depressions. The superpositions of Fermi function at non zero tem-perature was a good choice for modeling the topography. Superimposingsinusoidal functions would not show the features for the second derivative.

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The outlook would be, to link the modeled topography with real forces andcompare these with further spectroscopy data.We can confirm lateral force variations close to the surface by the torsionalmeasurements [15], which should provide additional binding forces. Third, adistant dependent enhancement of holes is seen in the spectroscopy Fig. 14.Further bimodal spectroscopy measurements are needed to affirm these mea-surements, as well as bias spectroscopy for a comparison of the acquired datato the electronic landscape of the nanomesh.

4.3 Domain orientation

Beside the contrast inversions, irregularities in the nanomesh pattern couldbe observed. At bigger scan areas of 150x150nm2 distortions, tilting andbending of the h-BN nanomesh were found. A clear nanomesh symmetrywith the 3 fold relation of the holes can be observed in every domain. On amore detailed investigation, these regions with distinct symmetry are tiltedtowards each other. It was found, that four different symmetry regions, witha difference in the tilt angle of 15◦, are existing, which will be called domains.

Both images 19(a, b) show large range topographical measurements of thenanomesh, in which one finds domains with a 3 fold symmetry. The dif-ference between those measurements are the scan position and the time ofmeasument. Figure 19(a) was measured 3 days after preparation and figure19(b) 6 months later.The topography signal in figure 19a was classified, in a detailed investigation,

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Figure 19: figure (a) shows a large scan area with 131x131nm2 in size. Thismeasurement was done 3 days after the nanomesh preparation. The to-pography signal showed clearly domains with different orientations. Right,every domain was classified and colored, domains with the same orienta-tion have the same color. Beneath, histograms of the four different do-mains with respect to their size are shown. figure (b) is a measurementwith 150x150nm2 in size, done 6 months later. Due to a probable multitipinfluence the nanomesh is not shown properly, but this defect enhances thevisibility of the difference in the domain orientation. Again, the domainswere colored with respect to their orientation.

on its domain orientation Fig. 19. The right overlay picture Fig. 19 was cre-ated afterwards to enhance the visibility of the single domain areas. As wellas allowing a domain size measurement. These were used in the histogram,with respect to their orientation. This was done for both figures 19a,b.After a full classification almost no area is left without a domain affiliation.In figure 19b, were still some unclassified spots left, due to distortions and tipartifacts, which does not allow a proper orientation analysis. By consideringonly the coloring, it is not possible to define a preferred domain orientation.The position and orientation seems to be random, even at step edges thedomains are randomly aligned.Histograms of figure (a) indicate an average domain size between 200nm2−

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Figure (a) Figure (b)Scan frame in nm2 2502 1502

f1st in Hz 153153 158664Q1st 32916 6323k1st in N/m 25.1 28.0∆f in Hz −8.5 −283A1st in nm 5 4.9γ factor in fNm0.5 −0.48 −17.1

Table 4: Table of image parameters of Figure 19.

400nm2 for every orientation. Some satellites at very large values, couldstem from aggregation of smaller domains. From the histogram, there isno evidence of continuous domain sizes, due to the lack of counts in thehistogram, because the overall count of domains is too small for significantstatistic.From these measurements, it can be assumed that the h-BN nanomesh isgrowing on the rhodium(111) substrate in an island growth mode with manynucleation points. We consider two possibilities for the domain creation.First, the rhodium substrate was grown epitactical on sapphire (Al2O3) forsome hundreds of nanometer with a distinguished orientation of (111) withrespect to the Miller notation. By the surface treatment of the prepara-tion cycles large terraces and a plane surface could be achieved. DFM in-vestigations at room temperature of the clean rhodium surface could notresolve rhodium atoms, therefore the substrate surface orientation remainsunknown, whether it has grains with a tilt angle with respect to other grains,or if it is a perfect epitactical layer with a large symmetry. If we consider agrain formation with tilt angle, the borazine would nucleate on such a grainand adept to the orientation. Such a nucleation center would start growingthe nanomesh in the energetic preferred orientation calculated by DFT byLaskowski [27]. If two domains of the nanomesh would intersect the domainstart to form a domain wall region, were an orientation shift will occur.Due to the energetic favoured orientation with respect to the substrate, twodifferent domains would not formate towards an aggregated domain withonly one orientation. This can be deduced by calculations of the rigidityof the sp2-nanomesh layer itself. If we take the second measurement intoaccount, which was done 6 months later, after many annealing cycles withtemperatures up to Tmax = 1123K, we do not observe recognisable changesin the domain sizes and preferred orientation Fig. 19b. In the frame of do-main growth on substrate grains with different tilt angle, it can be assumed,that there is no change in the grain orientation of the rhodium substrate.Therefore, no domain reorganization of the nanomesh was observed. Onepossibility could be the lateral rigidity of the nanomesh and the π-bonding

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towards the substrate, the nanomesh may prevent the substrate atoms ofreorganization.Additional, four different orientations with a tilt angle of 15◦ were measured.If the substrate was grown in grains, the grain orientation in principal canhave every orientation. The relation between the measured nanomesh do-main orientation and the grain orientation is not clear, why four distinctorientations are stable with respect to each other.Second, if the rhodium was grown epitactical with a perfect crystal orienta-tion at the surface, the borazine nucleation centers grow towards differentdomain orientations on the substrate. This would query the DFT calcula-tion [27], which does not predict other possible superstructure alignment onthe rhodium. If a perfect substrate orientation is considered, long annealingcycles at high temperatures should provide enough kinetic energy to enhancethe preferred alignment of the domains. In the histograms of figure 19b, noclear indication of a preferred orientation is recognisable.If the substrate is grown perfect epitactical the question of the orientationstabilization process remains. Considering a different domain orientation ona perfect surface the boron nitride rings are less bounded on the substrate,due to stress and the lack of the lock-in energy of the π orbitals 2.3 [15].From this, a lower surface binding is assumed, which should be indicated bydifferent heights, different spacings between the sp2 layers and the rhodiumsubstrate, which is not observed.A LEED analysis of our sample would solve the task of the crystal composi-tion, whether it has a epitactical or a polycrystalline structure. Nevertheless,it does not explain the stabilization mechanism of the four domain orienta-tions.From the measurements we assume that the nanomesh grows by an islandgrowth with different nucleation centers and is independent of the step edgeinfluence. Also, it is assumed, that the post annealing process does notchange the domain sizes and domain orientations towards a preferred orien-tation and triggers no higher ordering.Recently, the island growth of the h-BN nanomesh was observed in-situ [44].Here, STM measurements of the nanomesh growth on Rh(111) surface weremade. Different domain orientations were observed in addition to domainwall forming. The domain size seems to be related to the deposition rateand the deposition temperature of the surface. For slow deposition rates andlarge temperatures islands become bigger in size.

4.4 Atomic contrast measured by torsional resonance

Atomic resolution on the nanomesh with DFM were demonstrated earlierin my master project work [48]. The resolution was achieved by standardDFM, meaning first resonance for height control and small scan areas. Here,I want to present atomic resolution of the nanomesh recorded by bimodal

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dynamic force microscopy with the second flexural resonance at amplitudesof A2nd = 600pm for height control and the first torsional resonance for ultrasensitive probing of the atomic corrugation by changes of the lateral shortrange forces. The torsional amplitude could be set to ATR = 20pm, whichprovides the drastically increased resolution.

Figure 20: Figure (a) shows the topography signal controlled by the sec-ond flexural resonance. This shows the nanomesh in lower resolution. Infigure (b) the torsional signal clearly indicates the ”atomic” corrugation ofthe nanomesh with high resolution. figure (c) shows a 2D FFT of figure(b). The bright spots in the corners denote the periodicity of the ”atomic”corrugation and the one in the center due to random features in the mea-surement. Additional, the FFT has no information in the y direction whichis predicted by the torsional oscillation mode. Image parameters figure (a):Scan area 18x18nm2, f2nd = 967613Hz, Q2nd = 12236, k2nd = 1881N/m,∆f = −160Hz, A2nd = 600pm, γ = −0.40fNm0.5. figure (b): Scanarea 18x18nm2, fTR = 1.486602MHz, QTR = 124429, kTR ∼ 2200N/m,ATR = 20pm.

The measured topography signal, shown in figure 20a is acquired by the sec-ond flexural resonance. The achieved resolution is rudimental, but denotesthe theoretical nanomesh hole wire pattern. The set point for the heightcontrolling frequency shift was ∆f2nd = −160Hz. A larger frequency shiftset point would drastically increase the probability of a tip apex change orsnap in contact instabilities, which would not allow stable measurement con-

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ditions. No atomic resolution would be possible with a flexural oscillationwith this cantilever. A cantilever change or another cantilever preparationwould be recommendable.In the torsional oscillation measurements of the atomic corrugation by thelateral force variation Fig. 20b is imaged. Although we have a small signalto noise ratio we still achieve high resolution. One of the main goals of thetorsional resonance is the extremely high Q-value of QTR = 124429, whichallows to measure the frequency shift with high fidelity. While the Q-valueprovides a high signal to noise ratio the relatively high spring constant ofkTR ∼ 2200N/m reduces the over all noise level. These conditions in linkwith bimodal dynamic force microscopy are a powerful tool for surface anal-ysis.Figure 20b shows the frequency shift of the torsional signal simultaneouslymeasured as the topography. At wire positions the corrugation of the BN-rings between the atomic sites and the center of a ring is visible. This denotesthe atomic sites of the h-BN nanomesh. In a second approach, atomic res-olution can be achieved by the torsional resonance. The elevations have ahexagonal alignment and have relative distance of ∆a = 220 ± 20pm mea-sured by 2D FFT Fig. 2c. The torsional signal implies not only the moleculecorrugation, additionally an enhancement of the hole rims is measured. Thisis in correspondence with the theoretical predicted lateral force variation ofthe nanomesh, see Fig.2c. The molecule corrugation is clear visible on thewires and faintly visible in holes, this difference arises due to the poor ∆f2nd

signal. Nevertheless, it is still possible to distinguish between them in theacquired signal.In figure 20c the 2D FFT of the torsional signal is shown. The spots onthe corners denote the molecular elevations, while the bright spots at thecenter have information of all other features including the enhancement ofthe rims. Even though we recognise a hexagonal pattern of the moleculecorrugation, which corresponds to a 3-fold symmetry, only four bright spotsat the corners of the FFT can be seen. The angle between the two rightspots is almost 60◦, but the angle of the right between the left upper spotsis about 120◦. For a 3-fold symmetry 2 spots seem to be missing in the FFTof the torsional frequency shift.The torsional resonance oscillates in x direction. This allows a detection offorces with a x-component, y-components are not detectable, due to the os-cillation mode. This is nicely demonstrated in the FFT. A lack informationis seen in the pure y-direction. This is in perfect agreement with the theoryof cantilever dynamics.The torsional frequency shift signal demonstrates the ultra high sensitivity ofthe torsional resonance mode used in bimodal dynamic force microscopy. Forfurther enhanced resolution measurements on the nanomesh bimodal DFMin combination with the torsional resonance is a unremitting tool.

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5 Conclusion

The h-BN nanomesh is a amazing monolayer. A surface characterizationopened up a lot of questions, like a greek hydra it raised two more questions, ifone was solved. Nevertheless, we distinct three different contrast observableon the surface. A normal predicted one, a low resolution inversion, whichlasts from a double tip influence Fig. 8 and a high resolution inversion, whichseems to be a property of the surface. We have seen this high resolutioncontrast in scans and 2D spectroscopy Fig. 13. The spectroscopy showedchanges in the force distance curves Fig. 14, but the origin of this could notbe explained appropriately. Torsional measurements revealed the predictedlateral force variations Fig. 17 [15] close to the surface. The structure ofthe torsional frequency shift was modeled by a simple approach, which takesonly the topographical structure into account and no forces Fig. 18. Thisapproach is nevertheless quite accurate for such simple assumptions.In addition structural domains have been observed. These domains havebeen classified and it was found that there are four different orientationswith a 3 fold symmetry. No preferred domain orientation could be observedand no change in the domain distribution after annealing could be observedFig. 19.Atomic resolution at a large scan area was achieved by bimodal DFM. Theamplitude was controlled by the second flexural oscillation with amplitudesof A2nd = 600pm and the atomic corrugation was measured by the ultrasensitive second torsional resonance with an amplitude of ATR = 20pm,Fig.20.After all, tasks and open questions, I still have the confidence, that thissurface opens a lot of possibilities for room temperature single moleculespectroscopy, due to the high reactivity molecules can be immobilized. If thegrowth mode can be controlled and a preferred orientation is achievable, amagnetic cobalt clusters could be aligned in arrays, controlling the magneticspins of those clusters could lead to a storage of information with a hugedensity. Organic transistor arrays, organic solar cells, even vertical quantumdots would be possible. The high environmental and thermal stability leavean open range of applications in the nano mechanics. This opens a interestingfuture for the h-BN nanomesh.

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6 Acknowledgment

I thank Prof. Ernst Meyer for giving me the possibility to do my masterthesisin his group. Additionally, I am thankful to have access to the UHV systemwith such an amazing microscope. Without this system most of the measure-ments weren’t possible. I really thank Sascha, Shigeki, Marcin and Thilo forthe never ending brain support. Every idea was created or improved by fruit-ful discussions with them. Thank you, Sascha for the the critics, which leadto further improvement of the ideas. Thank you, Shigeki for the amazingknowledge and technical support for the spectroscopy measurements. Thankyou, Thilo for so many initial ideas and so many discussions.I’d like to thank Ernst Meyer and Thilo to open up me the possibility ofpresenting my work at conferences, this was a great opportunity for me andhelped me at my decision of starting a PhD study in physics.

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save(strcat(Folder_path,Spec_folder,backsl,Base_name,Longbase_name,underl,'001')) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% manual drift correction%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear hold_df1_Fclear hold_Exc1_Fclear hold_df2_F drift= (0*100/(Nr_file));for xi = 1:Nr_file drift_corr(xi) = ceil(-xi*(drift/Nr_file)+drift); for zi = 1:drift_corr(xi) hold_df1_F(xi,zi)=df1_F(xi,zi); hold_Exc1_F(xi,zi)=Exc1_F(xi,zi); hold_df2_F(xi,zi)=df2_F(xi,zi); end for zi = 1:(nr_z-drift_corr(xi)-1) zi = (nr_z)-zi+1; df1_F(xi,(zi))= df1_F(xi,(zi-drift_corr(xi))); Exc1_F(xi,(zi))=Exc1_F(xi,(zi-drift_corr(xi))); df2_F(xi,(zi))= df2_F(xi,(zi-drift_corr(xi))); for i = 1:drift_corr(xi) df1_F(xi,i) = hold_df1_F(xi,i); Exc1_F(xi,i) = hold_Exc1_F(xi,i); df2_F(xi,i) = hold_df2_F(xi,i); end end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calculate shift in Longdata%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%average_df1_long = mean(long_df1_F(1:75));df1_correction_long = 0-average_df1_long;long_df1_F = long_df1_F + df1_correction_long; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Shift all the Data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for xi = 1:Nr_file for zi = 1:nr_z df1_F(xi,zi) = df1_F(xi,zi)+(df1_correction_long); endend %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calculate the gradient in Longdata%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% grad_nr=150;Gradient_long_df1 = zeros(1,grad_nr);Gradient_long_x = zeros(1,grad_nr);Gradient_long_y = zeros(1,grad_nr);for zi = nr_z-grad_nr:nr_z Gradient_long_x(zi) = long_Z_F(zi); Gradient_long_y(zi) = long_df1_F(zi);endans = [Gradient_long_x' ones(length(Gradient_long_x),1)]\Gradient_long_y';Gradient_long_df1 = ans(1); Max_Gradient_long = max(Gradient_long_df1); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calculate the gradient%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%gradient_nr=400;Gradient_x = zeros(1,gradient_nr);Gradient_y = zeros(1,gradient_nr); for xi = 1:Nr_file Gradient_df2=zeros(Nr_file); for zi = 1:gradient_nr Gradient_x(zi) = Z_F(zi); Gradient_y(zi) = df1_F(xi,zi); end ans = [Gradient_x' ones(length(Gradient_x),1)]\Gradient_y'; Gradient_df2(xi) = ans(1); endMax_Gradient = max(max(Gradient_df2)); shift_z = zeros(Nr_file,gradient_nr); for xi = 1:Nr_file while(Gradient_df2(xi) < Max_Gradient) for zi = 1:gradient_nr zi = zi+shift_z(xi); Gradient_x(zi) = Z_F(zi);

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Gradient_y(zi) = df1_F(xi,zi); end ans = [Gradient_x' ones(length(Gradient_x),1)]\Gradient_y'; Gradient_df2(xi,:) = ans(1); shift_z(xi) = shift_z(xi) +1; end end for xi = 1:Nr_file for zi = (gradient_nr+1):(nr_z - shift_z(xi)) df1_F(xi,zi)= df1_F(xi,(zi+shift_z(xi))); Exc1_F(xi,zi)=Exc1_F(xi,(zi+shift_z(xi))); df2_F(xi,zi)= df2_F(xi,(zi+shift_z(xi))); end for zi = (nr_z - shift_z(xi))+1 : nr_z df1_F(xi,zi) = df1_F(xi,(nr_z-shift_z(xi))); Exc1_F(xi,zi) = Exc1_F(xi,(nr_z-shift_z(xi))); df2_F(xi,zi) = df2_F(xi,(nr_z-shift_z(xi))); end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Add Long data to the normal set of data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear integ_Aclear integ_Bclear z_stepclear root_amplitudeclear F_Fclear root_distanceall_Z_F; all_A1_F = zeros(1,(2*nr_z-(grad_nr+1)));all_df1_F = zeros((Nr_file),(2*nr_z-(grad_nr+1))); F_F = zeros(Nr_file,(2*nr_z-(grad_nr+1)));for xi = 1:Nr_file for zi = 1:(2*nr_z-grad_nr) if zi <=nr_z-grad_nr all_Z_F(zi)=long_Z_F(zi); all_A1_F(zi)=long_A1_F(zi); all_df1_F(:,zi)=long_df1_F(zi); else all_Z_F(zi)=Z_F(zi-(nr_z-grad_nr)); all_A1_F(zi)=A1_F(zi-(nr_z-grad_nr)); all_df1_F(xi,zi)=df1_F(xi,zi-(nr_z-grad_nr)); end endend %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculating the Force%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for zi = 2:(2*nr_z-grad_nr) integ_A = zeros(Nr_file,1); integ_B = zeros(Nr_file,1); for zzi = 1:(zi-1) z_step = all_Z_F(zzi)-all_Z_F(zzi+1); root_distance = sqrt(abs(all_Z_F(zzi)-all_Z_F(zi))) ; root_amplitude = sqrt(abs(all_A1_F(1,zzi)-all_A1_F(1,zi))); integ_A =integ_A+(1+root_amplitude*z_step/(8*sqrt(pi)*root_distance))*all_df1_F(:,zzi); integ_B =integ_B+ 1/(sqrt(2)*root_distance)*(all_df1_F(:,zzi+1) - all_df1_F(:,zzi)); end F_F(:,zi) = integ_A+integ_B * root_amplitude^3; end all_F_F = 2 * k_1st * F_F / f_1st; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUT OFF the Long data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear F_Fcutoff = zeros(Nr_file,nr_z-grad_nr); for xi = 1:Nr_file for zi = 1:(2*nr_z-grad_nr) if zi <= nr_z-grad_nr cutoff(xi,zi)=all_F_F(xi,zi); else F_F(xi,zi-(nr_z-grad_nr))=all_F_F(xi,zi); end endendsize(F_F);

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Add Long data to the normal set of data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear integ_pot_Aclear integ_pot_Bclear integ_pot_Cclear z_stepclear root_amplitudeclear U_Fclear root_distanceall_Z_F; all_A1_F = zeros(1,(2*nr_z-(grad_nr+1)));all_df1_F = zeros((Nr_file),(2*nr_z-(grad_nr+1))); U_F = zeros(Nr_file,(2*nr_z-(grad_nr+1)));for xi = 1:Nr_file for zi = 1:(2*nr_z-grad_nr) if zi <=nr_z-grad_nr all_Z_F(zi)=long_Z_F(zi); all_A1_F(zi)=long_A1_F(zi); all_df1_F(:,zi)=long_df1_F(zi); else all_Z_F(zi)=Z_F(zi-(nr_z-grad_nr)); all_A1_F(zi)=A1_F(zi-(nr_z-grad_nr)); all_df1_F(xi,zi)=df1_F(xi,zi-(nr_z-grad_nr)); end endend %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Calculating the Potential%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for zi = 2:(2*nr_z-grad_nr) integ_pot_A = zeros(Nr_file,1); integ_pot_B = zeros(Nr_file,1); integ_pot_C = zeros(Nr_file,1); for zzi = 1:(zi-1) z_step = all_Z_F(zzi)-all_Z_F(zzi+1); distance = abs(all_Z_F(zzi)-all_Z_F(zi)); root_distance = sqrt(distance); root_amplitude = sqrt(abs(all_A1_F(1,zzi)-all_A1_F(1,zi))); integ_pot_A = integ_pot_A + (root_amplitude/4)*(root_distance/sqrt(pi))*all_df1_F(:,zzi); integ_pot_B = integ_pot_B + (root_amplitude^3/(sqrt(2)*root_distance))*all_df1_F(:,zzi); integ_pot_C = integ_pot_C + distance*all_df1_F(:,zzi); end U_F(:,zi) = integ_pot_A + integ_pot_B + integ_pot_C; end all_U_F = 2 * k_1st * U_F / f_1st; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUT OFF the Long data%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear U_Fcutoff = zeros(Nr_file,nr_z-grad_nr); for xi = 1:Nr_file for zi = 1:(2*nr_z-grad_nr) if zi <= nr_z-grad_nr cutoff(xi,zi)=all_U_F(xi,zi); else U_F(xi,zi-(nr_z-grad_nr))=all_U_F(xi,zi); end endendsize(U_F); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ddf_1st and dF_F%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%df1_average_F = zeros(1,nr_z);ddf1_F = zeros(Nr_file,nr_z);for xi = 1:Nr_file for zi = 1:nr_z df1_average_F(zi)=mean(df1_F(:,zi)); ddf1_F(xi,zi)=df1_F(xi,zi)-df1_average_F(zi); endend dF_average_F = zeros(1,nr_z);dF_F = zeros(Nr_file-init_nr,nr_z); for zi = 1:(nr_z) dF_average_F(zi)=mean(F_F(:,zi)); for xi = 1:Nr_file dF_F(xi,zi)=F_F(xi,zi)-dF_average_F(zi); end

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end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Get amplitude%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A_1st= mean(mean(A1_F(xi,1:100))); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calculate the average excitation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for xi = 1:Nr_file Average_Exc1_F(xi) = mean(Exc1_F(xi,1:50));end for xi = 1:Nr_file Exc1_F(xi,:)= Exc1_F(xi,:) ./ Average_Exc1_F(xi) - 1 ;end Exc1_F_inst = pi* k_1st * A_1st^2 / Q_1st /eV*1000;Exc1_F = Exc1_F * Exc1_F_inst; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Filter df signal in Z direction%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for xi = (1+1):Nr_file df1_F(xi,:) = sgolayfilt(df1_F(xi,:),2,5); Exc1_F(xi,:) = sgolayfilt(Exc1_F(xi,:),2,5); ddf1_F(xi,:) = sgolayfilt(ddf1_F(xi,:),2,5); df2_F(xi,:) = sgolayfilt(df2_F(xi,:),2,5); %F_F(xi,:) = sgolayfilt(F_F(xi,:),2,5);end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Filter df signal in X direction%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%for zi=2:nr_z df1_F(:,zi) = sgolayfilt(df1_F(:,zi),2,5); Exc1_F(:,zi) = sgolayfilt(Exc1_F(:,zi),2,5); ddf1_F(:,zi) = sgolayfilt(ddf1_F(:,zi),2,5); df2_F(:,zi) = sgolayfilt(df2_F(:,zi),2,5); %F_F(:,zi) = sgolayfilt(F_F(:,zi),2,5);end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Make axis information%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%X = 1:Nr_file;Z = 0:(nr_z-1);Z_F = Z_F * 1.00e+9;X_max = max(Z_F);X_min = min(Z_F);z_nm = Z * (X_max - X_min)/(nr_z-1);x_nm=9.203*(X)/(Nr_file); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plot Range and Contour Spacings%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clear figurescrsz=get(0,'ScreenSize');figure('Position',[1 1 scrsz(3) scrsz(4)],'Name',fname, 'NumberTitle','off');colormap (flipud(jet(96))) clear clevs_df1clear temp_df1temp_df1 = rot90(df1_F,1);%C_max_df1= 55;%C_min_df1= -130;C_max_df1=max(max(temp_df1));C_min_df1=min(min(temp_df1));% for i=1:clev_nr% clevs_df1(i) = C_min_df1+(i*(C_max_df1-C_min_df1))/clev_nr;% end clear clevs_df2clear temp_df2temp_df2 = rot90(df2_F,1);C_max_df2= 2;C_min_df2= -6;% C_max_df2=max(max(temp_df2));% C_min_df2=min(min(temp_df2));% for i=1:clev_nr% clevs_df2(i) = C_min_df1+(i*(C_max_df2-C_min_df2))/clev_nr;% end clear clevs_Exc1

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clear temp_Exc1temp_Exc1 = rot90(Exc1_F,1);%C_max_Exc1= 5;%C_min_Exc1= -5;C_max_Exc1= max(max(temp_Exc1));C_min_Exc1= min(min(temp_Exc1));% C_min_Exc1= 0;%for i=1:clev_nr%clevs_Exc1(i) = C_min_Exc1+(i*(C_max_Exc1-C_min_Exc1))/clev_nr;%end clear clevs_ddf1clear temp_ddf1temp_ddf1 = rot90(ddf1_F,1);C_max_ddf1= 3;C_min_ddf1= -6;% C_max_ddf1=max(max(temp_ddf1));% C_min_ddf1= min(min(temp_ddf1));%for i=1:clev_nr%clevs_ddf1(i) = C_min_ddf1+(i*(C_max_ddf1-C_min_ddf1))/clev_nr;%end clear clevs_dF_Fclear temp_dF_Ftemp_dF = rot90(dF_F,1);C_max_dF= 1.75;C_min_dF= -1.75;% C_max_dF=max(max(temp_dF));% C_min_dF=min(min(temp_dF));% for i=1:clev_nr% clevs_dF(i) = C_min_dF+(i*(C_max_dF-C_min_dF))/clev_nr;% end clear clevs_F_Fclear temp_F_Ftemp_F_F = rot90(F_F,1);% C_max_F_F= 1;% C_min_F_F= -1;C_max_F_F=max(max(temp_F_F));C_min_F_F=min(min(temp_F_F))%for i=1:clev_nr%clevs_F_F(i) = C_min_F_F+(i*(C_max_F_F-C_min_F_F))/clev_nr;%end clear clevs_U_Fclear temp_U_Ftemp_U_F = rot90(U_F,1);%C_max_U_F= 55;%C_min_U_F= -130;C_max_U_F=max(max(temp_U_F));C_min_U_F=min(min(temp_U_F));%for i=1:clev_nr%clevs_U_F(i) = C_min_U_F+(i*(C_max_U_F-C_min_U_F))/clev_nr;%end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Plotting XZ plane%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% subplot(7,1,1) % [c,h] = contour3(x_nm, z_nm, temp_df1,clevs_df1,'k');set(h,'LineWidth',2); hold 'on';surf(x_nm, z_nm, temp_df1,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_df1,C_max_df1]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('\Delta f_{2nd}(z) (Hz)','fontsize', 22)axis equal tight subplot(7,1,2) % [c,h] = contour3(x_nm, z_nm, temp_ddf1,clevs_ddf1,'k');set(h,'LineWidth',2);hold 'on';surf(x_nm, z_nm, temp_ddf1,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_ddf1,C_max_ddf1]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('\Delta f_{2nd}(z) - f_{average}(z) (Hz)','fontsize', 22)

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axis equal tight subplot(7,1,3) % [c,h] = contour3(x_nm, z_nm, temp_df2,clevs_df2,'k');set(h,'LineWidth',2); hold 'on';surf(x_nm, z_nm, temp_df2,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_df2,C_max_df2]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('\Delta f_{TR}(z) (Hz)','fontsize', 22)axis equal tight subplot(7,1,4) %[c,h] = contour3(x_nm, z_nm, temp_F_F,clevs_F_F,'k');set(h,'LineWidth',2);hold 'on';surf(x_nm, z_nm, temp_F_F,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_F_F,C_max_F_F]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('Force_{2nd}(z) in (nN)','fontsize', 22)axis equal tight subplot(7,1,5) % [c,h] = contour3(x_nm, z_nm, temp_dF,clevs_dF,'k');set(h,'LineWidth',2);hold 'on';surf(x_nm, z_nm, temp_dF,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_dF,C_max_dF]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('Force_{2nd}(z) - Force_{average}(z) in (nN)','fontsize', 22)axis equal tight subplot(7,1,6) %[c,h] = contour3(x_nm, z_nm, temp_Exc1,clevs_Exc1,'k');set(h,'LineWidth',2);hold 'on';surf(x_nm, z_nm, temp_Exc1,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_Exc1,C_max_Exc1]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('Excitation_{1st}(mV)','fontsize', 10)axis equal tight subplot(7,1,7) %[c,h] = contour3(x_nm, z_nm, temp_U_F,clevs_U_F,'k');set(h,'LineWidth',2);hold 'on';surf(x_nm, z_nm, temp_U_F,'EdgeColor','none','FaceLighting','phong'),caxis manual,caxis([C_min_U_F,C_max_U_F]), view(0,+90)colorbarxlabel('X(nm)','fontsize', 18);ylabel('Z(nm)','fontsize', 18);set(gca,'XminorTick','on','YminorTick','on','fontsize', 18);title('Potential(z)','fontsize', 10)axis equal tight Plane='XZ';save(strcat(Folder_path,Spec_folder,backsl,Base_name,underl))Outputname=strcat(Base_name,underl,Plane);saveas(gcf,strcat(Folder_path,Spec_folder,backsl,Outputname),'tif')%close(gcf)

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References

[1] F. J. Giessibl Atomic resolution of the Silicon(111)-(7x7) Surface byAtomic Force Microscopy, Science (1995), 267, 68

[2] D. Fotiadis, Y. Liang, S. Filipek, D.A. Sperstein, A. Engel and K.Palczewski Atomic-force-microscopy: Rhodopsin dimers in native discmembranes, Nature (2003), 421, 127-(2)

[3] H. Seelert, A. Poetsch, N.A. Dencher, A. Engel, H. Stahlberg and D.J.Müller Structural biology: Proton-powered turbine of a plant motor, Na-ture (2000), 405, 418-(2)

[4] Y. Sugimoto, P. Pou, M. Abe, P. Jelinek, R. Pérez, S. Morita and O.Custance Chemical identification of individual surface atoms by atomicforce microscopy, Nature (2007), 446, 64-(3)

[5] D. Rugar, R. Budakian, H. J. Mamin and B. W. Chui Single spin de-tection by magnetic resonance force microscopy, Nature (2004), 430,329-(3)

[6] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. vonKlitzing and A. Yacoby Observation of electronhole puddles in grapheneby using a scanning single electron transistor, Nature Physics (2008),4, 144 - 148

[7] K.V. Emtsev, A. Bostwick, K. Horn, J. Jobst, G.L. Kellogg, L. Ley,J.L. McChesney, T. Ohta, S.A. Reshanov, J. Röhrl, E. Rotenberg, A.K.Schmid, D. Waldmann, H.B. Weber and T. Seyller Towards wafer-sizegraphene layers by atmospheric pressure graphitization of silicon carbide,Nature Materials (2009), 8, 203 - 207

[8] K.S. Kim, Y. Zhao, H. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.H. Ahn, P.Kim, J.-Y. Choi and B.H. Hong Large-scale pattern growth of graphenefilms for stretchable transparent electrodes, Nature (2008), 447, 706-(4)

[9] G. Binning, C.F. Quate, Ch. Gerber, Atomic Force Microscope, PRL.(1986), 56, 930-933

[10] G. Binning, h. Rohrer, Ch. Gerber, E. Weibel Surface Studies by Scan-ning Tunneling Microscopy, PRL. (1982), 49, 57-61

[11] Christoph Bruder, Van der Waals- und Casimir-Kräfte, Phys. in unsererZeit (1997), 28, 149-154

[12] D.C. Prieve and W.B. Russel Simplified predictions of Hamaker con-stants from Lifshitz theory, Journal of Colloid and Interface Science,(1987), Vol. 125, No.1

51

Page 64: Masterthesis Markus Langer - Contrast inversion of the h-BN nanomesh

[13] R. Smoluchowski Anisotropy of the electronic work function of metals,Physical Review (1941), 60, 661-674

[14] Ernst Meyer, Hans Josef Hug, Roland Bennewitz Scanning Probe Mi-croscopy: The Lab on a Tip, Springer, Berlin (2003), 1. Auflage

[15] Thomas Greber Chapter:Graphene and Boron Nitride Single Layers,Taylor and Francis Books, (2009) arXiv: 0904.1520v1

[16] Lukas Howald, Raster-Kraftmikroskopie an Silizium und Ionenkristallenim Ultrahochvakuum, Dissertation, Universität Basel (1994)

[17] M.Guggisberg, M. Bammerlin, Ch. Loppacher, O. Pfeiffer, A. Abdurixit,V. Barwich, R. Benewitz, A. Baratoff, E. Meyer, H.-J. Güntherodt Sep-aration of interactions by noncontact force microscopy, PRB (2000),61, 11151-11155

[18] Franz J. Giessibl, Advances in atomic force microscopy, Rev. Mod. Phys.(2003), 75-3, 949-978,

[19] Demtröder, Experimentalphysik 3: Atome, Moleküle und Festkörper,Springer, Berlin (2005), 3. Auflage

[20] S. Kawai, T. Glatzel, S. Koch, B. Such, A. Baratoff and Ernst MeyerSystematic achievement of Improved Atomic-Scale Contrast via BimodalDynamic Force Microscopy, PRL. (2009), 103, 220801 (4)

[21] S. Kawai, T. Glatzel, S. Koch, B. Such, A. Baratoff and ErnstMeyer Ultrasensitive detection of leteral atomic-scale interactions ongraphite(0001) via bimodal dynamic force measurements, PRB. (2010),81, 085420 (7)

[22] A. Nagashima, N Tejima, Y. Gamou, T.Kawai, C. Oshima, ElectronicStructure of Monolayer Hexagonal Boron Nitride Physisorbed on MetalSurfaces, PRL (1995), 75, 3918(4)

[23] Martina Corso, Willi Auwärter, Matthias Muntwiler, Anna Tamai,Thomas Greber, Jürg Osterwalder Boron Nitride Nanomesh, Science(2004), 303, 217-220

[24] Simon Berner, Martina Corso, Roland Widmer, Oliver Groening,Robert Laskowski, Peter Blaha, Karlheinz Schwarz, Andrii Goriachko,Herbert Over, Stefan Gsell, Matthias Schreck, Hermann Sachdev,Thomas Greber, Jürg Osterwalder, Boron Nitride Nanomesh: Function-ality from a Corrugated Monolayer, Angew. Chem. (2007), 46, 5115-5119

[25] J. Hölzl, F.K. Schulte and H. Wagner Work Function of Metals, Springer(1979), Volumen 85 of Springer Tracts in Modern Physics.

52

Page 65: Masterthesis Markus Langer - Contrast inversion of the h-BN nanomesh

[26] G. B. Grad, P. Blaha, K. Schwarz, W. Auwarter and T. Greber Densityfunctional theory investigation of the geometric and spintronic structureof h-bn/ni(111) in view of photoemission and stm experiments, Phys.Rev. B (2003), 68, 085404(8)

[27] Robert Laskowski, Peter Blaha, Thomas Gallauner, Karlheinz Schwarz,Single-Layer Model of the Hexagonal Boron Nitride Nanomesh on theRh(111) Surface, PRL (2007), 98, 106802(4)

[28] A. Stock and E. Pohland Chem. Ber. (1926), 59B, 2215(8)

[29] S. Belaidi, P. Girard, G. Leveque, Electrostatic forces acting on the tipin atomic force microscopy: Modelization and comparison with analyticexpressions, J. Appl. Phys. (1997), 81-3, 1023-1030

[30] Andrii Goriachko, Yunbin He, Marcus Knapp, Herbert Over Self-Assembly of a Hexagonal Boron Nitride Nanomesh on Ru(0001) Lang-muir (2007), 23, 2928-2931

[31] W. Auwaerter, T.J. Kreutz, T. Greber and J. Osterwalder XPD andSTM investigation of hexagonal boron nitride on Ni(111) Surface Sci-ence, (1999) 429, 229-236

[32] , Th. Glatzel, M.Ch. Lux-Steiner, E. Strasbourg, A. Boag, Y. Rosen-waks I.4 Principles of Kelvin Probe Force Microscopy, Springer, Berlin(2007), 113-134

[33] G.H. Enevoldsen, A.S. Foster, M.C. Christensen, J.V. Lauritsen, F.Besenbacher Noncontact atomic force microscopy studies of vacanciesand hydroxyls of TiO2(110): Experiments and atomistic simulationsPhys. Rev. B (2007) 76, 205415(14)

[34] G.H. Enevoldsen, T. Glatzel, M.C. Christensen, J.V. Lauritsen, F. Be-senbacher Atomic Scale Kelvin Probe Force Microscopy Studies of theSurface Potential Variations on the TiO2 (110) Surface PRL, (2008)100, 236104 (4)

[35] P. Steiner, R. Roth, E. Gnecco, A. Baratoff, S. Maier, T. Glatzel andErnst Meyer Two-dimensional simulations of superlubricity on NaCl andhighly oriented pyrolytic graphite PRB, (2009) 79, 045414 (9)

[36] The MathWorks, Matlab R2008a, (2008)

[37] T. Glatzel, L. Zimmerli, S. Koch, B. Such, S. Kawai and Ernst MeyerDetermination of effective tip geometries in Kelvin probe force mi-croscopy on thin insulating films on metals Nanotechnology, (2009) 20,264016 (7)

53

Page 66: Masterthesis Markus Langer - Contrast inversion of the h-BN nanomesh

[38] Van Hove, M. A. Hermann, K. Watson Tables for 4.1.Bonzel SpringerMaterials - The Landolt-Börnstein Database(http://www.springermaterials.com), (06.06.2010) DOI:10.1007/10783464-4

[39] J. E. Sader, S. P. Jarvis Accurate formulas dor interaction force andenergy in frequency modulated force spectroscopy APL, (2004) Vol. 84,No. 10

[40] L. Olsson, N. Lin, V. Yakimov and R. Erlandsson A method for insitucharacterization of tip shape in ac-mode atomic force microscopy usingelectrostatic interaction APL, (1998) Vol. 84, No. 8

[41] S. Hudlet, M. Saint Jean, C. Guthmann and J. Berger Evaulation of thecapacitive force between an atomic force microscopy tip and a metallicsurface Eur. Phys. J. B, (1998) 2, 5-10

[42] S. Hudlet, M.Saint Jean, B. Roulet, J.Berger, C. Guthmann Electro-static forces between metallic tip and semiconductor surfaces J. Appl.Phys. (1995), 77-7, 3308-3314

[43] T. R. Albrecht, P. Grutter, H. K. Horne, D. Rugar Frequency modu-lation detection using high-Q cantilevers for enhanced force microscopesensitivity J. Appl. Phys. (1991), 69-2, 668-673

[44] G. Dong, E.B. Fourre, F. C. Tabak and J.W.M. Frenken How BoronNitride forms a regular Nanomesh on Rh(111) PRL (2010), 104, 096102

[45] Th. Glatzel, S. Sadewasser, M.Ch. Lux-Steiner Amplitude or frequencymodulation-detection in Kelvin probe force microscopy Appl. Surf. Sci.(2003), 210, 84-89

[46] Lars A. Zimmerli, Assemblies of Organic Molecules on Insulating Sur-face Investigated by nc-AFM, Dissertation, Universität Basel (2007)

[47] Arndt Brünner, www.arndt-bruenner.de/mathe/scripts/kreis3p.htm,Homepage, (6.März.2010), 28.Mai.2010

[48] Markus Langer, nc-AFM Analyse des h-BN Nanomesh, MasterprojektWork, Universität Basel (2009)

54