High-Resolution XRD 2 - Stanford University

High-Resolution XRD – 2

Reciprocal Lattice Mapping

Superlattices

(00l)

(hkl)

Symmetricalw - 2q scan

Asymmetricalw - 2q scan

Sample Surface

2q scan

Scan directions

w scanw scan

Reciprocal Space

w-scan is in the direction of an arc centered on the origin2q-scan is an arc along Ewald sphere circumferencew-2q scan is always strait line pointing away from the origin of the reciprocal space

Reciprocal Space

wqw

wqw

2sinsin2

1

2coscos2

1

z

x

Q

Q

Qz

Qx

Q

k = 1/

k0 = 1/

2qw

1/

limiting sphere radius

𝑞𝑥 =4𝜋𝑄𝑥𝜆

𝑞𝑧 =4𝜋𝑄𝑥𝜆

𝑞 =4𝜋𝑠𝑖𝑛𝜃

𝜆

Symmetric and Asymmetric Scans

Symmetric scan

Asymmetric scan. Low angle of incidence.

Asymmetric scan. High angle of incidence.

Reciprocal Space – instrument resolution

• Influence of incident beam divergence

• Influence of detector acceptance angle

• Influence of the wavelength spread

High-Resolution Diffractometry

Schematic of high resolution double-axis instrument

Bartels monochromator

Hybrid monochromator

(000)

(00l)

w direction

w-2q direction

Defined by receiving optics (e.g. slits)

Mosaicity

(000)

(00l)

w direction

w-2q direction

Symmetrical Scan

Open detector

Triple axis

mosaicity

Open detector

Triple axis

d-spacing variation

Triple Axis Geometry

Schematic of high resolution triple-axis instrument

Triple Axis Geometry

Diagram of a high resolution triple axis instrument

Mirror

Sample

F

q = 0.5o

q < 19 arcsec

Ge(220) Crystal

X-ray Tube

Channel Analyzer

Detector

Hybrid Monochromator

High-resolution optics

Separation of Lattice Tilts and Strains

Triple axis measurements: real and reciprocal representations

With receiving slitWith channel analyzer

(002)SrTiO3

(220)SrRuO3

Ge content: 25% 20% 15% 10% 5%

Open detector

Triple axis

Triple-Axis Diffractometry

(hkl)(00l)

(000)

SymmetricalScan

AsymmetricalScan

(h00)

w-scan

w-2q scan(hkl)

(00l)

(000)

(00l) scan

(h00) scan

h-scan

l-scan

Triple-Axis Diffractometry

Relaxed SiGe on Si(001)

Shape of the RLP might provide much more information

Relaxed SiGe on Si(001)(oo4) RLM

Si(004)

SiGe(004)

(004) (113)

(hkl)(00l)

(000)

w-scan

w-2q scan

Real RLP shapes

HeteroepitaxyCompressive stress

L

S

HeteroepitaxyTensile stress

Homoepitaxyd-spacing variation

HeteroepitaxyMosaicity

Finite thickness effect

cL < aScL > aS cL = aS cL < aS

The mosaic spread of the layer is calculated from the angle that the layer peak subtends at the origin of reciprocal space measured perpendicular to the reflecting plane normal.

The lateral correlation length of the layer is calculated from the reciprocal of the FWHM of the peak measured parallel to the interface.

Mosaic Spread and Lateral Correlation Length

LC

MS

To OriginQZ

QX

The Mosaic Spread and Lateral Correlation Length functionality derives information from the shape of a layer peak in a diffraction space map recorded using an asymmetrical reflection

L

qz

L3

L2

L1

Dqx

Dqzx

j

xj

x

cos

cos

2

1

L

L

x

j

cos

sin

2

3 L

L

D

D

DD

z

x

z

x

zx

q

q

q

q

qqL

tan

1

tan

1

22

3

x

j

and

Lateral correlation length

22

2

1

1

zx qq

L

L

Microscopic tilt

qx

(000)

Mosaic Spread and Lateral Correlation Length

Dw

Applications of RLM

Thin epitaxial GaAs film on GaAs(001).Film has more As.

RLM after annealing

Strain in Partially Relaxed Layers

A schematic of the measurements to be extracted from two RLM’s to determine the relaxation of a layer on a substrate.

q

wqw

wqw

sin4

2sinsin1

2

2coscos1

2

q

qs

qs

zz

xx

Superlattices and Multilayers

Transmission electron micrograph depicts one of the earliest multilayer, GeSi / Si

"superlattices." The dots are the actual images of individual columns of atoms.

Substrate

L

t dhkl


General characteristics of large repeat superlattices

The spatial period of the structure

The thickness of the repeating unit

The composition of the layers

The dispersion in the repeating period

The interface roughness

The interface grading


Double-axis rocking curve from an AlGaAs/GaAs superlattice.

Cu Ka radiation, 004 reflection


The rocking curve shows the following features:

A substrate peak from the A substrate

A peak caused by the addition of Bragg reflections from the A and B components of the MQW. This is zero-order or average mismatch peak, from which the average composition of the A+B layers may be obtained by differentiation of Bragg’s law.

A set of subsidiary “satellite” peaks symmetrically surrounding the zero-order peak, with spacing determined by the periodicity (total thickness of the repeating layers) of the MQW.

XRD pattern from multiple quantum well (MQW) with substrate A (GaAs) and stack of AB layers where B is alloy (e.g. Ga1–xAlxAs).


Simulated rocking curves from an epilayer of total thickness 1mm, subdivided into 2, 4, 6 and 10 layers of alternate composition.

2 4

6 8

10

6

(000)

(00l)

10

(000)

(00l)

2

(000)

(00l)

4

(000)

(00l)



Simulated rocking curve from sequence of layers of total thickness 1mm, divided into 2, 4 and 8 repeats of InxGa1–xAs layers of composition x = 0.5 and x = 0.43.


The multilayer periodicity is defined as:

ji

ji nn

ww

sinsin2

L

equation defines satellite peak positions

Structure of SrRuO3

a = 5.567 Å

b = 5.530 Å

c = 7.845 Å

a = 5.578 Å

c = 7.908 Å

a = 3.956 Å

275-550 C 510-702 C

Orthorhombic Tetragonal Cubic

Structure of SrRuO3

SrRuO3

SrTiO3

aobo

co

ao

bo

co

[100]

[001]

[010]

[1-10]

[001]

[110]

SrTiO3

[100]

[001]

[010]

SrRuO3

[001]

[110]

[1-10]

SrRuO3

SrTiO3

(110)

(001)

(1-10)

(001)

(010)

(100)

(2 6 0)(4 4 4)(6 2 0)

(4 4 –4)

(2 2 0)

(0 0 2)(-2 0 4) (2 0 4)

w – 2q scan Reciprocal Space Map

Q scan

SrTiO3

SrRuO3

ab

OrthorhombicSrRuO3

TetragonalSrRuO3

X-ray Diffraction Scan Types

Thickness3100 Å

SrTiO3 (002)

SrRuO3 (220)

SrTiO3 (002)SrRuO3 (220)

Thickness3200 Å

Finite size fringes indicate well ordered films

w – 2q symmetrical scans

f angle0o 90o 180o 270o

(2 2 0)

(0 0 2)

w – 2q scan

SrTiO3

SrRuO3

Reciprocal Lattice Map ofSrRuO3 (220) and SrTiO3 (002)

Substrate

Layer

Distorted perovskite structure:

Films are slightly distorted from orthorhombic, g = 89.1 – 89.4

g

(110)

(110)

(100) (010)

ab

OrthorhombicSrRuO3

(260) (444) (620) (444)

High-Resolution Reciprocal Area Mapping

Substrate

Layer

Orthorombic to Tetragonal Transition

Orthorhombic

Tetragonal

Cubic

Literature: 510-702 C

Transition Orthorhombic to Tetragonal ~ 350 C

Temperature ( oC)

150 200 250 300 350 400

Inte

nsity (

arb

un

its)

0.0

0.2

0.4

0.6

0.8

1.0

Structural Transition, (221) reflection

Orthorhombic

Tetragonal

Cubic

Literature: 510-702 C


O – TTransition

(221) Peak

Orthorhombic Present

Tetragonal Absent


a

Rotation Angle (deg)

0 2 4 6 8 10 12 14 16

Ca

lcu

late

d In

ten

sity (

arb

un

its)

0

10000

20000

30000

40000

50000

60000

(211) peak is absent in cubic SrRuO3


Temperature (oC)

200 300 400 500 600 700

Inte

nsit

y (

a.u

.)

550 600 650 700

Orthorhombic

Tetragonal

Cubic

Attempt forT – C Transition ?

O – T Transition = 310 oC


[0 0 1]

[1 -1 0]

[1 1 0]

NGO(220)

Beam along [1-10]

Beam along [001]

a bc

g

LSMO(444)

LSMO(620)

(110)

(220)

(330)

(440)

Satellite spacing does not change along out-of-plane direction

[001]

LSMO_NGO

[110]

L

The lattice modulations described using kinematical x-ray diffraction.

We used a cubic LSMO unit cell with

LL

HUnit cells have displacements along L-directionwhich are periodic along H-direction withperiodicity, L. Then a one-dimensional complexstructure factor along H-direction can be writtenas:

unit cell structure factor and relative x- and z-positions along H and L directions of a LSMO cubic unit cell

if and then:

H-scans around LSMO(hk0) reflections withh = k = 1,2,3,4. The gray vertical lines areguides to the eye and show that satellite peaksoriginate from periodic atomic displacementsand not from out-of-plane twinning.

L

L

H

(110)

(220)

(330)

(440)

[0 0 1]

[1 -1 0]

[1 1 0]

NGO(220)

L= 23 nmAL = 0.18 ÅL = 5 nm

Reciprocal space for epitaxial thin films is very rich.

Shape and positions of reciprocal lattice points with respect to the substrate reveal information about:

• Mismatch• Strain state• Relaxation• Mosaicity• Composition• Thickness ….

Diffractometer instrumental resolution has to be understood before measurements are performed.

Fun in reciprocal space. © The New Yorker Collection, 1991. John O’Brien, fromwww.cartoonbank.com. All rights reserved.

Things can look very different in reciprocal space than in real space…

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High-Resolution XRD 2 - Stanford University