Document [original]

A new X-ray diffractometer

for the online monitoring

of epitaxial processes

Der Fakult¨at f¨ur Naturwissenschaften

der Universit¨at Paderborn

zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften (Dr. rer. nat.)

vorgelegte

Dissertation

von

Alexander Kharchenko

Paderborn, July 2003

Abstract

In this thesis I present an in situ X-ray diffractometer for X-ray diffraction analysis

of epitaxial layers during metal-organic chemical vapor deposition. The conception of

the in situ X-ray diffractometer, the model of data interpretation and the description

of the experimental setup are given in detail. The diffractometer has been constructed

and tested under laboratory conditions with wurtzite type AlGaN- and InGaN-based

materials. The results of tests show that the diffractometer allows to perform X-ray

diffraction measurements under conditions present in a metal-organic chemical vapor

deposition reactor and to obtain accurate information about the composition and relative

thickness (growth rate) of epitaxial layers. The results of measurements have been

compared with the results obtained by conventional high resolution X-ray diffractometer

and perfect agreement has been reached.

The diffractometer has been developed without a goniometer stage. It is not sen-

sitive to precise alignment of the samples before measurements and it has very short

data collection time (up to factor 100 less than by standard diffractometers). These

additional features of the X-ray diffractometer makes it well suited for extremely fast

post-growth diagnostics of multilayer semiconductor structures. This potential of the

diffractometer has been tested with different materials systems which are important

in the semiconductor industry today, namely wurtzite AlGaN single heterostructures,

wurtzite InGaN-based multiple quantum wells and SiGe heterostructures. The measure-

ments of the line scans as well as of reciprocal space maps are presented and analyzed

in terms of composition, periodicity of the multilayers and status of strain. All results

are in good agreement with the results obtained by conventional high resolution X-ray

diffraction.

Contents

1 Introduction 1

2 Some important aspects of high resolution X-ray diffraction 3

2.1 High resolution X-ray diffraction in real and reciprocal space . . . . . . . 3

2.2 Theory of X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Use of reciprocal space mapping for evaluation of strain status . . . . . . 5

2.4 Use of reciprocal space mapping for determination of chemical composition 7

3 Conception of an in situ X-ray diffractometer 11

3.1 Requirements for IXRD and basic principles of an IXRD ........ 11

3.2 IXRD in terms of reciprocal space . . . . . . . . . . . . . . . . . . . . . 13

3.3 Interpretation of measurements with IXRD ................ 18

3.3.1 Choice of suitable diffraction geometry . . . . . . . . . . . . . . . 18

3.3.2 Determination of the vertical lattice parameter of crystalline struc-

tures.................................. 19

3.3.3 Determination of the lateral lattice parameter of crystalline struc-

tures.................................. 21

4 Experimental setup of IXRD 23

4.1 Experimental setup for laboratory tests . . . . . . . . . . . . . . . . . . . 23

4.2 Test of the optical properties of the Johansson monochromator . . . . . . 25

4.3 Performance test of the IXRD ....................... 28

5 Results of experiments under laboratory conditions 31

5.1 Use of the IXRD for fast ex situ characterization . . . . . . . . . . . . . . 31

5.1.1 Characterization of wurtzite AlGaN-, InGaN-based materials . . . 31

5.1.2 Measurements on SiGe . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.3 Fast reciprocal space mapping . . . . . . . . . . . . . . . . . . . . 37

ii Contents

5.2 Measurements under conditions in a MOCVD reactor . . . . . . . . . . . 42

5.2.1 Measurement conditions and principles of data collection . . . . . 42

5.2.2 Results of measurements. Detection and resolution limits of the

IXRD ................................. 44

6 Conclusions 51

A Lattice parameters and stiffness coefficients of relevant materials 53

B Examples of the evaluation of the IXRD spectra 55

Bibliography 58

List of Figures

2.1 Sketch of the diffraction geometry . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The effect of strain on the asymmetrical reciprocal space maps . . . . . . 6

3.1 The principle of Johansson monochromator . . . . . . . . . . . . . . . . . 12

3.2 The geometry of the IXRD setup in real and reciprocal space . . . . . . . 14

3.3 The geometry of the IXRD setup in real and reciprocal space for two

measurementpoints.............................. 15

3.4 Illustration of measurement principles of the IXRD setup . . . . . . . . . 17

3.5 Sketch of reciprocal space for a crystalline structure . . . . . . . . . . . . 19

4.1 Experimental setup of the IXRD for measurements under laboratory con-

ditions..................................... 24

4.2 The arrangement of the setup for the investigation of the optical properties

of the Johansson monochromator . . . . . . . . . . . . . . . . . . . . . . 25

4.3 The distribution of intensity measured at different positions of the narrow

slit in front of the monochromator . . . . . . . . . . . . . . . . . . . . . . 27

4.4 The intensity diffracted from the Johansson monochromator plotted ver-

sus relative detection angle ∆ε........................ 28

4.5 Comparison of spectra measured by the IXRD setup and spectra extracted

from high resolution reciprocal space maps . . . . . . . . . . . . . . . . . 30

5.1 Sketch of the vertical layer structure of AlGaN/GaN single heterostructures 32

5.2 The spectra from AlGaN/GaN single heterostructures collected on the

IXRD and measured with the standard high resolution equipment . . . . 33

5.3 Schematic illustration of the vertical structures of tested InGaN/GaN mul-

tiplequantumwells.............................. 34

5.4 X-ray diffraction patterns of two different InGaN/GaN multiple quantum

wells measured by the IXRD and by the high resolution equipment . . . 35

5.5 Schematic illustration of the vertical structure of SiGe/Si sample . . . . . 36

iii

iv List of Figures

5.6 X-ray diffraction patterns of SiGe/Si measured by the IXRD and by the

high resolution X-ray diffractometer . . . . . . . . . . . . . . . . . . . . . 37

5.7 The procedure of reciprocal space mapping with the IXRD . . . . . . . . 38

5.8 The contours of constant scattered intensity of single heterostructures

measured by the IXRD and collected on the high resolution diffractometer 39

5.9 The reciprocal space maps from multiple quantum wells measured by the

IXRD and by X’Pert MRD diffractometer . . . . . . . . . . . . . . . . . 40

5.10 The (224) reciprocal space maps of SiGe/Si collected by the IXRD and

by the X’Pert MRD diffractometer . . . . . . . . . . . . . . . . . . . . . 42

5.11 The principle of X-ray measurement from rotating and wobbling sample . 44

5.12 X-ray spectra measured by the IXRD from rotating and wobbling In-

GaN/GaN and AlGaN/GaN samples . . . . . . . . . . . . . . . . . . . . 45

5.13 The spectra collected by the IXRD and by the X’Pert MRD from thin

InGaN and AlGaN layers grown on GaN . . . . . . . . . . . . . . . . . . 46

5.14 Dependence of full width at half maximum of the InGaN and AlGaN

peaks on inverse layer thicknesses . . . . . . . . . . . . . . . . . . . . . . 47

5.15 The resolution of the IXRD for the control of the chemical composition

oftheInGaNlayers.............................. 49

5.16 The resolution of the IXRD for the control of the chemical composition

oftheAlGaNlayers.............................. 50

B.1 The diffraction spectrum of AlGaN/GaN single heterostructure taken on

theIXRD ................................... 56

B.2 Measurement result of SiGe/Si sample . . . . . . . . . . . . . . . . . . . 57

List of Tables

3.1 Parameters cos(γ−β)/cos β,tanβ and structure factor Ffor some re-

flectionsofh-GaN............................... 20

3.2 Suitable reflections of some ”substrate crystals” for the determination of

the vertical lattice parameter of multilayer structures . . . . . . . . . . . 21

3.3 Parameters cos(γ−β)/sin β,cotβ and structure factor Ffor some re-

flectionsofh-GaN............................... 21

3.4 Suitable reflections of some ”substrate crystals” for the determination of

the vertical lattice parameter of multilayer structures . . . . . . . . . . . 22

A.1 Lattice a0, c0and stiffness C11, C13 constants for hexagonal GaN, InN and

AlN. ...................................... 53

A.2 Lattice a0, c0and stiffness C11, C12 constants for cubic Si and Ge. . . . . 53

vi List of Tables

1 Introduction

The importance of semiconductor structures for the fabrication of devices in electron-

ics and optoelectronics is obvious. Today, billions of transistors, LEDs and sensors are

manufactured weekly. Generally, the growth of semiconductor multilayer structures are

realized by metal-organic chemical vapor deposition (MOCVD) or molecular-beam epi-

taxy (MBE). The technological process of growth demands methods for characterization

and quality control. X-ray diffraction remains one of the important methods of charac-

terization because it is non-destructive, informative and has clear interpretation. Since

the discovery of X-rays in 1895 by Wilhelm Conrad Roentgen and the discovery of X-ray

diffraction by Max von Laue in 1912, X-ray diffraction has been developed into a powerful

tool for investigation of semiconductor materials. Nowadays, novel X-ray diffractome-

ters equipped with high power sources and high efficiency monochromators, detectors

and analysers are being produced in a large amount. They provide access to impor-

tant parameters of semiconductors such as composition, strain, thickness and structural

quality. There exist a huge amount of experimental and theoretical papers about the

X-ray analysis of different kinds of semiconductor materials, however, most of them are

dealing with post-growth characterization (see for instance reviews on high resolution

X-ray diffraction [1], [2]), i.e. X-ray measurements are performed after the growth of

the samples. Yet in situ characterization techniques are highly demanded especially for

the MOCVD technology, because high ambient pressure preclude the use of electron

diffraction, which is already well established for online monitoring of the MBE growth.

Furthermore, X-ray diffraction is attractive as in situ technique because of the simple

procedure of data interpretation. From the angular positions and shapes of diffraction

peaks the structural information becomes easily available.

Only a few experimental X-ray investigations of semiconductors during the growth

process have been reported ([3], [4], [5], [6], [7], [8], [9], [10], [11]). However, they all

were carried out with synchrotron radiation where a specially designed growth chamber

was brought to synchrotron sources. It is obvious that such studies are not applicable

to most commercial and scientific MOCVD reactors. The intention of this work was to

construct an in situ X-ray diffractometer (IXRD) which is suited for X-ray diffraction

21 Introduction

analysis of semiconductor layers during MOCVD growth. The apparatus has been built

and tested under laboratory conditions. The results of the tests show that our in situ

X-ray diffractometer fulfills all requirements which are important for an analysis of epi-

taxial layers during MOCVD growth: (1) no synchrotron radiation is needed for X-ray

measurements; (2) no parts of the diffractometer are necessary inside the growth reac-

tor; (3) the setup is able to collect X-ray patterns from rotating and wobbling samples;

(4) the data collection time from rotating samples is in the order of one minute and

corresponds to the deposition time of a 10 nm thick layer. Furthermore, the short data

collection time (about 100 times less than with commercially available diffractometers)

and the ability to measure X-ray diffraction without precise adjustment of the sample

makes the equipment also well suited for extremely fast post-growth characterization of

multilayer structures.

The thesis is organized as follows. Chapter 2 describes some important aspects of high

resolution X-ray diffraction. The conception of a new IXRD is introduced in chapter 3.

Chapter 4 describes the details of the experimental setup. Then I will proceed with chap-

ter 5 where I show the results of tests of the IXRD for fast post-growth characterization

as well as the results of measurements under conditions similar to those in a MOCVD

reactor. Finally, the main conclusions are given in chapter 6.

2 Some important aspects of high

resolution X-ray diffraction

2.1 High resolution X-ray diffraction in real and

reciprocal space

Bragg formulated [12] the condition of diffraction of X-rays by a periodic arrangement

of atoms in a solid state crystal:

2·dhkl ·sin(ΘB) = n·λ(2.1)

where dhkl is the spacing between lattice planes with Miller indices (hkl), ΘBis the

Bragg angle and λis the wavelength of X-rays.

It is a convenient and common way to describe X-ray diffraction in reciprocal space.

The reciprocal lattice is formed by the terminal points of reciprocal repetition vectors

b1,~

b2and ~

b3which are related to the primitive vectors of crystal lattice ~a1,~a2and ~a3by:

bi= 2 ·π·~aj×~ak

~a1·(~a2×~a3), i, j, k cycl. (2.2)

In reciprocal space the plane with Miller indices (hkl) is described by the reciprocal

vector which is given by:

Ghkl =h·~

b1+k·~

b2+l·~

b3(2.3)

The condition of diffraction 2.1 by the plane (hkl) can be reformulated in reciprocal

space (see for example [13]) in the form:

Q=~

Ghkl (2.4)

42 Some important aspects of high resolution X-ray diffraction

Qis the scattering vector defined as ~

Q=~

kε−~

kδwhere ~

kδand ~

kεare the wave vectors

of incident and diffracted waves as indicated in figure 2.1.

Figure 2.1: Sketch of the diffraction geometry. The exact Bragg condition for

(hkl) planes is fulfilled, if the end of scattering vector ~

Qends at a reciprocal

lattice point (hkl).

Thus in reciprocal space the diffraction plane is represented as a reciprocal lattice

point and the diffraction geometry defined by the incident and by the detection angles

is represented by the scattering vector. When the scattering vector ends at a reciprocal

lattice point (hkl) the exact Bragg condition is satisfied. Scattered X-ray intensity

around a reciprocal lattice point (RLP) is strongly influenced by the structural properties

of crystalline material. Therefore, the measurement and detailed analysis of diffracted

intensity around reciprocal lattice points is the subject of high resolution diffractometry.

The variation of incident and detection angles allows to scan scattered intensity around

the reciprocal lattice point. Depending on the required information the analysis of one

or two dimensional projections of the intensity distribution around reciprocal lattice

points is performed. Usually these are ω−2Θ, ωscans [1] or two dimensional intensity

distributions of reciprocal lattice points which are called reciprocal space maps.

2.2 Theory of X-ray diffraction

Generally there are two possible ways for the theoretical description of X-ray diffraction

in crystals which depend on the methods for representing a crystalline structure. In the

first method, the calculation of X-ray diffraction is performed by the summation of the

scattered intensity from separate atoms (see for example [13], [14]). In this case the

2.3 Use of reciprocal space mapping for evaluation of strain status 5

scattered intensity is proportional to the square of the structure factor given by:

F=Xfj·exp(i~

Rj) (2.5)

where fjare the atomic scattering factors and summation is performed over all atoms

in the unit cell. The theory of X-ray diffraction in such a form usually completely

neglects the dynamical effects of X-ray diffraction which is the multiple scattering effect

(extinction). For this reason it is called kinematical theory and it is not valid for a highly

perfect type of crystals. The dynamical effects have been taken into account only in the

theory of Darwin developed in 1914 [15].

In the second alternative formalism which has been proposed by Ewald [16] and Laue

[17] the X-ray diffraction theory involves the solution of the Maxwell equations in a

medium with a periodic dielectric function. In this case the electromagnetic field is rep-

resented by the Bloch functions and the dielectric constant is expressed by a Fourier

series. A general description of this theory can be found in [18]. The alternative descrip-

tion was developed by Takagi [19], [20] and Taupin [21] where the solution is represented

by Bloch functions with constant phases and with slowly varying amplitudes. Both for-

malisms are widely used for the calculation of X-ray diffraction spectra. In this work, I

have used the Takagi-Taupin formalism for the simulation of X-ray patterns in multilayer

structures which is described in [22].

2.3 Use of reciprocal space mapping for evaluation of

strain status

The growth of an epitaxial film on a thick substrate causes the modification of the lattice

parameter of the epilayer in growth direction due to the strain. Let aSand cSbe the

lattice parameters of a substrate crystal in direction perpendicular and parallel to the

sample surface. a0,c0and a,care in-plane and perpendicular lattice parameters of bulk

and grown layer respectively. Depending on the lattice mismatch between substrate

and layer crystals, one distinguishes between compressive (a0> aS) and tensile strains

(a0< aS). The lattice parameter cis increased for compressive strain and decreased for

tensile strain. If the layer thickness is below a so-called critical thickness the growth is

pseudomorphic and the layer is fully strained on the substrate. With increasing layer

thickness the elastic energy increases. If the layer thickness is above its critical value,

misfit dislocations are formed and the layer starts to relax.

Reciprocal space mapping offers the unique possibility to perform detailed investiga-

62 Some important aspects of high resolution X-ray diffraction

tion of the relaxation process. In figure 2.2 a) a two dimensional projection of reciprocal

space in the (Qk, Q⊥) plane is shown where Qkand Q⊥are components of the reciprocal

vector which are parallel and perpendicular to the sample surface.

Figure 2.2: a) the effect of strain on the asymmetrical reciprocal space maps is

illustrated. The positions 1, 2, 3 of the reciprocal lattice point correspond to

the strained, partially relaxed and relaxed layer. b) a tilt of the layer causes a

shift of reciprocal lattice points and can lead to a wrong data interpretation.

The information on the layer tilt can be obtained from a symmetrical reciprocal

space map.

We consider (hkl) reflection for which l6= 0 and hor k6= 0, because for this case the

position of reciprocal lattice point depends on both in-plane and perpendicular lattice

parameters. Point S is the reciprocal lattice point of the substrate crystal. The points

1, 2 and 3 correspond to the reciprocal lattice point of layer which has different states of

strain. Below the critical thickness the layer is fully strained. This means that in-plane

lattice parameters of layer and substrate are identical (a=aS) and the vertical lattice

parameter of the layer increases (c > c0). In reciprocal space this corresponds to posi-

tion 1 of the RLP of the layer. Due to the relaxation the strain is relieved therefore the

in-plane constant of layer increases and the perpendicular constant decreases (position

2) until finally the bulk lattice parameters are reached (position 3). The position of the

2.4 Use of reciprocal space mapping for determination of chemical composition 7

reciprocal lattice point of the layer depends on the lattice parameter and therefore recip-

rocal space mapping is an ideal method for strain analysis. However, the measurement

of only one asymmetrical reflection can lead to a wrong interpretation due to a possible

crystallographic tilt of the lattice planes. In figure 2.2 b we demonstrate the situation

in reciprocal space where the positions of layer reflections are shown with and without

tilt. It is not possible to distinguish between effect of strain and tilt of the layer. One

possibility to identify the crystallographic tilt of the layer is to measure a symmetrical

reflection (00l). As shown in figure 2.2 b, in case of a symmetrical reciprocal space map

a tilt of the layer causes the deviation of RLPs from the vertical axis Q⊥. The tilt angle

ωT ILT which can be determined from the symmetrical reflection allows to separate the

effect of strain and tilt of the layer. Thus, measuring both asymmetrical (hkl) and sym-

metrical (00l) reflections the effect of strain and crystallographic tilt can be investigated

in detail.

2.4 Use of reciprocal space mapping for determination

of chemical composition of ternary compounds

Let us assume that a ternary epilayer AxB1−xCis deposited on a substrate S. The epilayer

is an alloy of the binary compounds AC and BC. We consider the general situation when

alloy AxB1−xChas arbitrary status of strain so that AxB1−xCcan be relaxed, partially

relaxed or strained. Compound AC and BC have in-plane and perpendicular lattice

parameters aAC,cAC and aBC,cBC , respectively. Our considerations will be valid for

two type of structures which are of importance in this work: zincblende structures with

tetragonal symmetry and wurtzite type structures with hexagonal symmetry. For the

cubic symmetry the in-plane and perpendicular lattice parameters should be assumed

identical. By measuring the symmetrical and asymmetrical reciprocal space maps we get

reliable information about in-plane and perpendicular lattice parameters of epilayer a

and ctaking into account the effect of a possible crystallographic tilt as described in the

previous section. a0,c0are lattice parameters of the relaxed AxB1−xCalloy, i.e., lattice

parameters of unstrained layer. We assume that there is a linear relationship between

lattice parameters of the relaxed alloy AxB1−xCand the lattice parameters of AC,BC

compounds and the composition x (Vegard’s law):

82 Some important aspects of high resolution X-ray diffraction

a0=x·aAC + (1 −x)·aBC

c0=x·cAC + (1 −x)·cBC

(2.6)

In order to find the relationship between relaxed lattice parameters a0,c0of AxB1−xC

alloy and measured lattice parameters aand cthe effect of strain has to be considered.

It can be described using the elasticity theory [23], [24]. The deformation state of a

crystal is described by the strain tensor εij which is related to the stress tensor σij by

the tensor of stiffness constants. In the case of biaxial strain, the stress normal to the

film vanishes and the following relation between strain in the growth direction and the

in-plane strain is obtained:

c−c0

=−ν(x)·a−a0

(2.7)

where ν(x) is Poisson’s ratio which is:

ν(x) = 2 ·C12(x)

C11(x)for tetragonal symmetry,

ν(x) = 2 ·C13(x)

C33(x)for hexagonal symmetry.

Cij are the elements of the tensor of the elastic moduli (stiffness constants) in Voigt’s

notation which depend on the composition x. The general assumption is a linear de-

pendence of stiffness constants on the composition x. The chemical composition x is

obtained by substitution of expressions 2.6 for a0and c0in 2.7. In case of a linear de-

pendence of stiffness constants on the composition x, this leads to an equation of degree

four for x. The solution is possible only by numerical methods.

In the first-order approximation used in [25] a linear relation for Poisson’s ratio is

applied:

ν(x) = x·νAC + (1 −x)·νBC (2.8)

In this case a cubic equation for x is obtained. As shown in [25], in this case the

analytical solution is straightforward following the formula of Abramowitz and Stegun.

There are three real solutions, but only one physically meaningful value of x exists.

2.4 Use of reciprocal space mapping for determination of chemical composition 9

If the elastic constants of two compounds AC and BC do not differ too much or if x

is close to zero or unity it is a good approximation to assume that the elastic moduli do

not vary with the chemical composition. Then substitution of 2.6 for a0and c0in 2.7

results in a quadratic equation:

Px2+Qx +R= 0 (2.9)

where

P=ν·∆a·∆c+ ∆a·∆c,

Q=ν·∆a·cBC −ν·δa ·∆c+ ∆c·aBC −∆a·δc,

R=−ν·δa ·cBC −δc ·aBC ,

and ∆a=aAC −aBC,∆c=cAC −cBC , δa =a−aBC, δc =c−cBC .

The solution for x is unique and is given by:

x=−Q−√Q2−4·P·R

2·P(2.10)

In this thesis we have measured semiconductor structures with ternary layers with a

mole fraction x of less than 0.25. Therefore approximation 2.9 has been used for the

calculation of the chemical composition. The lattice parameters and stiffness constants

for the relevant materials are given in appendix A.

10 2 Some important aspects of high resolution X-ray diffraction

3 Conception of an in situ X-ray

diffractometer for online monitoring

of the MOCVD growth process

3.1 Requirements for IXRD and basic principles of an

IXRD

As I have mentioned in the introduction, X-ray diffraction is a very promising tool for

online controlling of the MOCVD growth process. However, the application of X-ray

diffraction for in situ monitoring of the MOCVD growth process is a challenging task.

The measurement conditions are complicated due to the construction of the growth re-

actor and the conditions of growth so that they are not ordinary for standard X-ray

diffractometers. There are two ways for the solution of this problem. Previously, spe-

cially designed growth chambers have been developed and brought to the synchrotron

laboratories ([3]-[11]). The drawback of such a solution is obvious, in situ X-ray diffrac-

tometers can not be widely used for commercial as well as for scientific purposes.

The second solution is the development of a specially designed X-ray diffractometer

which can be installed on a MOCVD reactor. Knowing the construction of a MOCVD

reactor and the conditions of growth we formulate below the requirements for such an

in situ X-ray diffractometer:

•no parts of the diffractometer for adjustment and positioning of the samples should

be necessary inside the growth chamber. That allows to avoid any complications to

the construction of the MOCVD reactor and any disturbances of the growth pro-

cess. For the X-ray diffraction setup this means that IXRD should not be sensitive

to the precise alignment of the samples and should not require the positioning of

the samples during measurements;

12 3 Conception of an in situ X-ray diffractometer

•the IXRD has to be able to collect spectra from rotating and wobbling sam-

ples. This requirement is especially important, because samples are rotated during

MOCVD and wobbling of the samples can not be excluded during the growth

process;

•the setup should provide a data collection time which is at least comparable to the

typical deposition time of several nanometer thick layers in a MOCVD reactor.

The design of an IXRD setup is the goal of this thesis. The key features of our IXRD

are: i) the use of focused monochromatic X-ray beams impinging onto the sample instead

of a goniometer system and ii) to collect the diffracted X-rays by a multichannel detector.

To get focused monochromatic beams we use a Johansson monochromator. The princi-

ples of the Johansson monochromator are described in the original paper of T. Johansson

[26]. The diffracting planes of a crystal are bent with a radius 2R and the surface of the

crystal forms a cylinder with a radius R as indicated in figure 3.1.

Figure 3.1: The principle of the Johansson monochromator. The beams coming

from a X-ray source S are monochromized and focused at the point F by the

Johansson monochromator.

When the X-ray source is located at point S on the circle R which is usually called

Rowland circle, all beams strike the lattice planes of the crystal at the same angle. All

3.2 IXRD in terms of reciprocal space 13

reflected beams are focused at point F lying on the Rowland circle. Thus the Johansson

monochromator allows to get monochromatic and focused X-ray beams provided two

conditions are fulfilled. Firstly, X-ray source, monochromator crystal and sample are

placed on the Rowland circle. Secondly, the angle of incidence equals the Bragg angle of

the monochromator crystal. The second condition gives the relation between the radius

of the Rowland circle, Bragg angle and the distance between X-ray source and the center

of the monochromator:

D= 2 ·sin(ΘB)·R(3.1)

The sample to be investigated with the IXRD setup is placed at the focusing point F.

After diffraction the scattered X-rays are collected by a multichannel detector.

3.2 IXRD in terms of reciprocal space

The scattering geometry of IXRD in real space is schematically illustrated in figure 3.2

a. Here we have to distinguish between two important angles: incident angle δand

detection angle ε. Incident angle δis the angle between incident beam and the sample

surface. Detection angle εcorresponds to the angle between direction of the detection of

scattered intensity and the sample surface. The formation of the measurement point in

IXRD is indicated in figure 3.2 (a): the bunch of monochromatic X-rays with incident

angles between δand δ+ Ω strikes the sample. The scattered beams are measured by

one certain channel of a multichannel detector under a detection angle ε.

Figure 3.2 (b) shows the geometry of IXRD in reciprocal space. The beam with the

incident angle δand the beam with the detection angle εare described in reciprocal

space by the wave vectors ~

kδand ~

kε, respectively. Because the incident beams are

monochromatic and we are interested in an elastic scattering, the lengths of wave vectors

kδand ~

kεare the same. In reciprocal space the scattered intensity is plotted as a function

of the scattering vector which is defined as ~

Q=~

kε−~

kδ. In our setup one measurement

point is accomplished by the wave vector ~

kεand the endpoints of the incident wave vectors

between ~

kδand ~

kδ+Ω as indicated in figure 3.2 b. The corresponding scattering vectors

Qdescribe in reciprocal space a line depicted in figure 3.2 b as ”detection window”. The

construction in reciprocal space shows that the intensity measured by one channel of the

detector array is the sum of scattered intensity distributed in reciprocal space within the

”detection window”.

In figure 3.3 (a) we consider two different measurement points 1 and 2. The scattered

14 3 Conception of an in situ X-ray diffractometer

Figure 3.2: The geometry of the IXRD setup in real (a) and reciprocal (b)

space. One measurement point in the IXRD setup is described in angular space

by the certain detection angle εand by the bunch of incident waves with angular

aperture Ω. In reciprocal space this configuration corresponds to the line called

”detection window”.

intensity at two different channels which correspond to the detection angles ε1and ε2.

Figure 3.3 b demonstrates the situation in reciprocal space: the corresponding scatter-

ing vectors ~

Q1and ~

Q2describe lines ”detection window 1” and ”detection window 2”,

respectively. In this case, the intensities of measurement points 1 and 2 in IXRD setup

are the intensities distributed within ”detection window 1” and ”detection window 2”,

respectively. The centers of the ”detection windows” are located on the line shown in

3.2 IXRD in terms of reciprocal space 15

Figure 3.3: The geometry of the IXRD setup in real (a) and reciprocal (b) space

for two measurement points. The different measurement points are described in

angular space by the two different detection angles ε1and ε2. In reciprocal space

each measurement point can be associated with the line ”detection window”.

The direction of measurement in the IXRD setup is described by the line ”scan

direction”.

figure 3.3 b as ”scan direction”. This means that the change in the detection angle ε

results in a shift of the scattering vectors along the ”scan direction”.

In figure 3.4 a, b we show the measurement results of an AlGaN/GaN heterostructure

performed with the IXRD setup and the intensity distribution in reciprocal space mea-

sured by a conventional high resolution equipment. The X-ray intensity measured with

16 3 Conception of an in situ X-ray diffractometer

the IXRD setup is plotted as a function of the relative detection angle ∆ε. The zero

value of the ∆εangle is set to the position of the GaN peak. Each measurement point

in the IXRD setup can be associated with the ”detection window” in reciprocal space as

shown for two arbitrary points 1 and 2 in figure 3.4. The angular distance ∆εbetween

two points 1 and 2 in the IXRD spectrum is related to the distance ∆QIXRD along the

”scan direction” in reciprocal space.

The reciprocal space coordinates are related to the incident angle δand the detection

angle εby the following expressions:

Qx=2·π

λ·(cos(ε)−cos(δ))

Qz=2·π

λ·(sin(ε) + sin(δ)) (3.2)

Distance ∆QIXRD along the ”scan direction” is formed by keeping fixed the incident

angle δ=δoand varying the detection angle from εbr to εbr +∆εwhere εbr is the detection

angle corresponding to the Bragg reflection of GaN. Therefore ∆QIXRD corresponds to

the change of coordinates Qxand Qzby the values ∆Qxand ∆Qzwhich are given by:

∆Qx=2·π

λ·(cos(εbr + ∆ε)−cos(εbr))

∆Qz=2·π

λ·(sin(εbr + ∆ε)−sin(εbr))

If the relative detection angle ∆εis small we can use:

∆Qx=−2·π

λ·sin(εbr)·∆ε

∆Qz=2·π

λ·cos(εbr)·∆ε

The length ∆QIXRD is thus:

∆QIXRD =q∆Q2

x+ ∆Q2

z=2·π

λ·∆ε(3.3)

The expression 3.3 connects the angle between measurement points in the IXRD spec-

trum and the corresponding distance ∆QIXRD in reciprocal space along the ”scan direc-

tion”. It should be emphasized that the orientation of ”scan direction” and ”detection

window” in reciprocal space strongly depends on the diffraction geometry of the IXRD

setup. Therefore the information which can be obtained by the IXRD setup is different

for different scattering geometry. To find out the diffraction geometry which is suitable

for the determination of vertical and lateral lattice parameters, a model for the interpre-

tation of measurements with the IXRD setup will be developed in the next section.

3.2 IXRD in terms of reciprocal space 17

Figure 3.4: Illustration of measurement principles of the IXRD setup. a) inten-

sity measured by the IXRD setup plotted as a function of the relative detection

angle ∆ε, b) the intensity distribution in reciprocal space measured by a stan-

dard high resolution equipment. The relative detection angle ∆εis related to

the distance ∆QIXRD in reciprocal space along the ”scan direction”.

18 3 Conception of an in situ X-ray diffractometer

3.3 Interpretation of measurements with IXRD

3.3.1 Choice of suitable diffraction geometry

Our aim is to investigate a certain multilayer crystal structure with the IXRD setup. Let

us consider the general situation in reciprocal space shown in figure 3.5. We assume that

there is a crystal present in a multilayer structure which lattice parameters are known.

This means that we know the position of the reciprocal lattice point of this crystal in

reciprocal space. We will denote this crystal as a ”substrate crystal” and will use it as

a reference point in reciprocal space. The point ”S” is the center of distribution of the

intensity diffracted from this crystal.

For simplicity reasons we restrict ourselves to the consideration of only one reflection

”M” which is the result of scattering from a multilayer structure. It can be, for example, a

reflection from a crystalline layer included in a multilayer structure or a satellite peak of a

superlattice structure, etc. As full lines we depict the ”detection windows” corresponding

to different measurement points of the detector and the bold dashed line denotes the

”scan direction”.

As described in the previous section the IXRD setup provides information about the

distance ∆QIXRD. However, the distances between reflections along the Qzand Qxaxes

(hereafter vertical ∆Qzand lateral ∆Qxdistances, respectively) are required because

they provide access to the lateral and vertical lattice parameters of crystalline layers. In

case of a single layer, for example, they directly provide the lattice parameters parallel

and perpendicular to the sample surface. In case of a superlattice structure they yield the

average lattice parameter of barrier and well layers. Furthermore, in case of a periodical

structure the distances between reflections in Qzand Qxdirections can additionally

provide information about the periodicity of the structure.

Thus our aim is to find the vertical distance ∆Qzand the lateral distance ∆Qxusing

the distance ∆QIXRD which is measured with the IXRD setup. From geometrical con-

siderations we get the following relations between the vertical distance ∆Qz, the lateral

distance ∆Qx, and the distance ∆QIXRD:

∆Qz= ∆QIXRD ·cos(γ−β)

cos(β)+ ∆Qx·tan(β) (3.4)

3.3 Interpretation of measurements with IXRD 19

∆Qx=−∆QIXRD ·cos(γ−β)

sin(β)+ ∆Qz·cot(β) (3.5)

Figure 3.5: Sketch of reciprocal space for a crystalline structure. Points ”S”

and ”M” are the centers of diffraction from ”substrate crystal” and multilayer

structure, respectively.

where βand γare the angles given by:

β=π

2−δbr, γ =εbr

δbr and εbr are the incident and the detection angles for the exact Bragg diffraction of

the ”substrate crystal”. From equations 3.4 - 3.5 it follows that the relation between

the measured parameters ∆QIXRD and ∆Qz, ∆Qxdepends on the incident and on

the detection Bragg angles of the ”substrate crystal”. Therefore, for the ”substrate

crystal” the suitable reflections must be found depending on the required information.

We analyze equations 3.4 - 3.5 for two important cases: 1) determination of the vertical

lattice parameter and 2) determination of the lateral lattice parameter.

3.3.2 Determination of the vertical lattice parameter of crystalline

structures

From relation 3.4 it follows that we need to consider two parameters. The first one is

parameter cos(γ−β)/cos βwhich describes the resolution of the setup for the vertical

20 3 Conception of an in situ X-ray diffractometer

distance. In order to get optimized measurement conditions this parameter should be

close or less than unity. Otherwise, the measured distance ∆QIXRD will be smaller than

the distance ∆Qzand the setup is not sensitive to small ∆Qz, for example, in case of

a small lattice mismatch between substrate and layer crystals. The second parameter is

tanβ which describes how the reflection depends on the lateral distance ∆Qx. If we have

no information about this distance, i.e. information about the lateral lattice parameter

of the structure, this parameter multiplied by the value of the lateral distance, gives

the systematic error in the determination of the vertical lattice parameter. In order to

decrease the systematic error, this parameter has to be close to zero or a symmetrical

reflection which does not depend on the lateral lattice parameter should be used. Table

3.1 shows these two parameters and structure factor Ffor some reflections of h-GaN.

Table 3.1: Parameters cos(γ−β)/cos β,tanβ and structure factor Ffor some

reflections of h-GaN

cos(γ−β)/cos(β) tan β F

(¯

2024) 4.78 4.96 11.5

(¯

1¯

124) 5.22 5.21 24.5

(0002) 1.91 3.21 48.5

(20¯

24) 0.95 −0.14 11.5

(11¯

24)0.99 0.02 24.5

Table 3.1 is not a full list of all possible reflections, here we intend to illustrate the

most important reflections to understand the criteria for the choice of the reflection.

The table shows that in case of asymmetrical reflections with small incident angles

(reflections (¯

2024) and (¯

1¯

124)) both parameters do not satisfy the requirements for the

vertical mismatch determination. They have poor resolution for the ∆Qzand depend

strongly on the lateral mismatch. The symmetrical reflections for which the sensitivity

to the lateral mismatch is not important (because in this case ∆Qx= 0) have a low

resolution in ∆Qz. After analyzing all possible reflections we conclude that the reflection

with a large incident angle (11¯

24) is most suitable for the determination of the vertical

lattice parameter of structures grown on h-GaN. In table 3.2 we have summarized the

suitable reflections of some ”substrate crystals” for the determination of the vertical

lattice parameter of multilayer structures.

3.3 Interpretation of measurements with IXRD 21

Table 3.2: Suitable reflections of some ”substrate crystals” for the deter-

mination of vertical lattice parameter of multilayer structures. Parameters

cos(γ−β)/cos β,tanβ and structure factors Fare shown.

”substrate crystal” reflection cos(γ−β)/cos(β) tan β F

Si (224) 1.02 0.19 55.8

GaAs (135) 0.96 0.07 82.0

c-GaN (224) 0.92 −0.03 59.7

3.3.3 Determination of the lateral lattice parameter of crystalline

structures

For the determination of the lateral lattice parameter two parameters, namely, cos(γ−

β)/sin βand cot βare of importance. These parameters describe the resolution in Qx-

direction and the dependence of the chosen reflection on the vertical distance, respec-

tively. We illustrate these two parameters and structure factor Ffor some reflections of

h-GaN in table 3.3.

Table 3.3: Parameters cos(γ−β)/sin β,cotβ and structure factors Ffor some

reflections of h-GaN.

cos(γ−β)/sin(β) cot β F

(20¯

24) −6.98 −7.32 11.5

(11¯

24) 61.78 62.72 24.5

(0002) 0.59 0.31 48.5

(¯

2024) 0.96 0.20 11.5

(¯

1¯

124)1.00 0.19 24.5

The asymmetrical reflections with large incident angles which are well suited for the

vertical lattice parameter determination are not appropriate for determination of the

lateral lattice parameter. This means that it is not possible to use only one geometry

of the IXRD for simultaneous determination of lattice parameters in both directions:

parallel and perpendicular to the sample surface. From table 3.3 we see that the (¯

1¯

124)

reflection is the most appropriate for the determination of lateral lattice parameter of

a structure grown on h-GaN. In table 3.4 we have summarized the suitable reflections

22 3 Conception of an in situ X-ray diffractometer

of some ”substrate crystals” for the determination of the lateral lattice parameter of

multilayer structures.

Table 3.4: Suitable reflections of some ”substrate crystals” for the deter-

mination of vertical lattice parameter of multilayer structures. Parameters

cos(γ−β)/sin β,cotβ and structure factors Fare shown.

”substrate crystal” reflection cos(γ−β)/sin(β) cot β F

Si (¯

1¯

13) 0.83 0.05 55.8

GaAs (¯

1¯

13) 0.82 0.03 119.8

c-GaN (¯

1¯

33) 0.99 0.02 58.7

4 Experimental setup of IXRD

4.1 Experimental setup for laboratory tests

The experimental setup of the IXRD for laboratory tests is schematically shown in figure

4.1. The main parts of the setup are X-ray tube, movable slit, Johansson monochromator,

sample stage and multichannel detector.

The X-ray beams are generated by a Cu 2.2 kW ceramic tube which has the possibility

to change between point and line focuses. For our experiments we have used line focus.

The size of line focus is 0.4 mm in the scattering plane and 12 mm in direction perpen-

dicular to the scattering plane. The controller type ”PW3100/00” which consist of a

generator (high tension tank and waterflow control) and electronics rack with CPU as

well as the X-ray tube type ”PW3373/00” have been produced and supplied by ”Philips

Analytical”.

The X-radiation from the X-ray tube is monochromized and focused by the Johansson

monochromator. We have applied a [111]-oriented Si monochromator crystal produced

by INRAD (USA). The (333) reflection is used. The radius of the Rowland circle is

339 mm. Referring to equation 3.1 the distance between X-ray source and center of the

monochromator and between center of the monochromator and center of the sample is

500 mm. The size of the monochromator is 50 mm (curved) x 15 mm (flat). As will be

shown in the next section, the effective angular acceptance range of the monochromator

is about two degrees.

The monochromatic X-beams are focused to the sample position as shown in figure 4.1.

The use of a focused beam as well as of a multichannel detector allows to implement X-ray

measurements without alignment of the samples and without positioning them during

the measurements. Therefore the sample stage was constructed without goniometer.

However, in order to realize conditions present in a MOCVD reactor, the sample holder

has been equipped with a motor which allows to rotate the sample in azimuth direction.

For the collection of diffracted spectra we use the multichannel detector. The multi-

24 4 Experimental setup of IXRD

Figure 4.1: Experimental setup of the IXRD for measurements under laboratory

conditions. The beams from the X-ray tube are monochromized and focused

onto the sample. The diffracted beams are collected by the multichannel de-

tector. The setup is additionally equipped with a movable slit in front of the

monochromator and with a motor for azimuth rotation of the sample during

measurement.

channel detector type ”PW3015/xx” X’Celerator was produced and supplied by ”Philips

Analytical”. Following the product abbreviation of ”Philips Analytical” the detector will

be called X’Celerator.

Figure 4.1 shows that the setup is designed in such a way that it is possible to install

the X’Celerator in two positions. In position A the diffraction measurements can be

performed. Position B is intended for the measurements of X-ray radiation coming

directly from the Johansson monochromator. In the next section we use the position

B of the X’Celerator for the investigation of diffractive and focusing properties of the

Johansson monochromator.

Additionally, we have installed a movable slit in front of the monochromator. The

movable slit equipped with a motion controller allows to perform fast measurements of

4.2 Test of the optical properties of the Johansson monochromator 25

reciprocal space maps as will be shown in chapter 5.1.

4.2 Test of the optical properties of the Johansson

monochromator

The monochromatic and focused X-ray beams are provided by the Johansson monochro-

mator. The efficiency of the whole setup is influenced by the quality of the monochroma-

tor. The production of the Johansson monochromator is complicated due to the bending

radius of the crystal planes and the grinding radius of the crystal surface. Any deviations

from the ideal parameters decrease the quality of the monochromator.

To investigate the properties of the monochromator we used the arrangement of the

setup shown in figure 4.2. Using the narrow slit we pick up a narrow beam which incidents

Figure 4.2: The arrangement of the setup for the investigation of optical prop-

erties of the Johansson monochromator. For each position of the movable slit

the diffracted beams are measured with the X’Celerator at position B.

on the monochromator. By moving the narrow slit we cause an angular deviation of the

26 4 Experimental setup of IXRD

incident beam by the value ∆αas is shown in figure 4.2 for two different beams 1 and 2

corresponding to two different slit positions.

After the reflection by the monochromator the beam is focused at the focusing point.

The change of direction of the incident beam by angle ∆αcauses the angular deviation

∆δof the reflected beam at the focusing point. In case of a monochromator with perfect

optical properties, the relative deviation of the reflected beam ∆δshould be equal to the

relative deviation of the incident beam ∆α.

The deviation of the reflected beam at the focusing point can be detected by the

X’Celerator at position B as detection angle ∆ε. The change in direction of the incident

beam ∆αcan be calculated from the position of the movable slit.

Thus measuring the angular shift ∆εof the diffracted beam for each position of the

movable slit we can check the quality of the monochromator. The results of the measure-

ments can be represented as a two-dimensional contour plot shown in figure 4.3 where

the measured intensity is plotted versus the relative incident angle ∆αand the relative

detection angle ∆ε. Zero value of the relative incident angle corresponds to the position

of the beam at the middle of the monochromator. Zero value of the relative detection

angle ∆εis set to the maximum of the diffraction spectrum measured at ∆α= 0.

For the monochromator with perfect diffractive and focusing properties the scattered

intensity should be uniformly distributed along the black line shown in figure 4.3. Figure

4.3 shows that there is a strong deviation of the direction of the diffracted beams from the

ideal line as well as a decrease of the scattered intensity at both sides of the monochro-

mator. The angular acceptance range of the monochromator is about two degrees which

is less than four degrees specified by the producer of the crystal.

This can be also verified by the measurement of intensity diffracted by the Johansson

monochromator without the slit. Figure 4.4 shows the distribution of intensity coming

directly from the Johansson monochromator. The non-ideality of the monochromator

causes the decrease of the active optical area of the monochromator and the variation of

the reflected intensity by approximately a factor of two.

4.2 Test of the optical properties of the Johansson monochromator 27

Figure 4.3: The distribution of intensity measured at different positions of

narrow slit in front of the monochromator as a function of relative incident

angle ∆αand relative detection angle ∆ε. There are deviations of the direction

of the diffracted beams as well as the diffracted intensity from the ideal form

at both sides of the monochromator.

28 4 Experimental setup of IXRD

Figure 4.4: The intensity diffracted from the Johansson monochromator plotted

versus relative detection angle ∆ε. The imperfection of the monochromator

causes the decrease of the active diffractive area of the monochromator and the

variation of the diffracted intensity by approximately a factor two.

4.3 Performance test of the IXRD

In this section we describe the performance check of the IXRD. One way to do it is to

compare the results of IXRD measurements with the measurements performed with a

conventional high resolution diffractometer.

In section 3.2 we have explained the formation of IXRD spectra in reciprocal space.

Referring to figure 3.4 we see that the measurement point in the IXRD setup can be

obtained from the intensity distribution in reciprocal space (reciprocal space map) by

summation of the intensity located within so-called ”detection windows”. The distance

between different ”detection windows” corresponding to different measurement points

is determined by the distance between pixels of the detector. We have developed a

procedure which allows to obtain a spectrum which is identical to that measured by

the IXRD. The procedure is the following. The measurement point in the IXRD setup

4.3 Performance test of the IXRD 29

is performed at a certain detection angle ε. For this detection angle εand a series of

incident angles δcontinuously varying between δbr −Ω/2 and δbr +Ω/2 (Ω is the angular

aperture of the monochromator, δbr is the incident angle under Bragg condition) we

calculate the positions of points in reciprocal space located in the ”detection windows”

according to the expressions 3.2. For each measurement point we take into account the

width of the ”detection window” which is determined by the size of detector pixels in the

scattering plane. Therefore in our procedure the detection angle εis taken in the range

between ε−εd/2 and ε+εd/2 where εdis the angular width of the detector pixels. After

the summation of the intensities of reciprocal space points located within the ”detection

window” for each detection angle εwe obtain the spectrum which is identical to that

measured by the IXRD setup.

In figure 4.5 the dots show the results of measurements with the IXRD setup from two

different AlGaN/GaN heterostructures. The full lines are the curves extracted from re-

ciprocal space maps using the procedure described above. The high resolution reciprocal

space maps were measured on the Philips X’Pert-MRD diffractometer in triple axis mode

using the hybrid monochromator with a primary beam divergence of 18-25 arc seconds.

Figure 4.5 shows that there is good agreement between measured and extracted spectra

so that we can conclude that the IXRD setup has a good performance.

30 4 Experimental setup of IXRD

Figure 4.5: Comparison of spectra measured by the IXRD setup (dotted lines)

and spectra extracted from reciprocal space maps (full lines). Two different

AlGaN/GaN heterostructures with different Al content have been measured in

a test of the IXRD.

5 Results of experiments under

laboratory conditions

5.1 Use of the IXRD for fast ex situ characterization

5.1.1 Characterization of wurtzite AlGaN-, InGaN-based materials

In this section we focus on the potential of the IXRD for fast ex situ characterization of

semiconductor multilayers. We concentrate mostly on the measurements of III-V nitride

films which are wurtzite type AlGaN- and InGaN-based structures. Enormous progress

in the development of wide-gap III-V nitride semiconductors (see for instance [27],[28])

has recently led to the commercial production of high-brightness light-emitting diodes.

Therefore the structural characterization of these materials is highly demanded.

All tested multilayers are fully strained on the GaN layer so that only the vertical

lattice parameter of the layers is of main interest. In case of strained ternary compounds

the information about the vertical lattice parameter gives access to the chemical com-

position of the layers. Therefore all measurements with the IXRD setup were performed

around the (11¯

24) reflection which is the most appropriate for analysis of the vertical

features of the structures as described in section 3.3.

We have performed the first tests with two different AlGaN/GaN single heterostruc-

tures #1 and #2 which had different Al content. The vertical layer structure of both

samples is the same and is schematically illustrated in figure 5.1.

In figure 5.2 at the left hand side we show the results of measurements performed with

the IXRD setup for both samples. The diffractometer can clearly resolve both GaN and

AlGaN peaks and provides the angular separation ∆εof AlGaN and GaN diffraction

peaks. The detailed description of the interpretation of measurement result is given in

Appendix B. We depict the evaluated compositions of AlGaN layers in figure 5.2.

Figure 5.2 shows that the increase in the Al content of the AlGaN layer of about 4

32 5 Results of experiments under laboratory conditions

Figure 5.1: Sketch of the vertical layer structure of the single heterostructures

#1 and #2.

percent leads to a considerable shifting of the AlGaN peak away from the substrate peak.

The measurement time was 4 seconds for each sample.

For comparison we have performed measurements with the conventional high reso-

lution equipment. The ω- 2Θ measurements are commonly used for the composition

measurements of alloy semiconductors. The high resolution measurements were per-

formed with the Philips X’Pert-MRD diffractometer in triple axis mode using the hybrid

monochromator with a primary beam divergence of 18-25 arc seconds. In figure 5.2 at

the right hand side we show ω- 2Θ scans from samples #1 and #2 around the (0002)

reflection. The relative position of the AlGaN peak with respect to the substrate peak

provide the chemical composition of both samples. The measurement time is consider-

ably longer than for the IXRD setup and is about 30 minutes for each sample.

These results demonstrate that measurements with the IXRD setup give the same

chemical composition of alloy semiconductors as the standard high resolution equipment.

A slight change in Al composition can be clearly resolved by the IXRD setup. As a big

advantage the IXRD setup offers significantly shorter measurement times (more than

two orders of magnitude less than the standard equipment) and absence of alignment of

the samples before measurements.

The next tests are measurements of two different multiple quantum wells (MQWs).

The vertical structures of the investigated MQWs are schematically illustrated in figure

5.3. Samples #3 and #4 consist of 5 and 10 periods of InGaN (well) and GaN (barrier)

layers, respectively. The samples have different periodicity (well thickness + thickness

of barrier). They differ as well in the In content of the active InGaN layers. In figure

5.4 at the left hand side we illustrate the results of measurements with the IXRD. The

X-ray patterns show clearly resolved zero- and first-order superlattice peaks. For our

5.1 Use of the IXRD for fast ex situ characterization 33

Figure 5.2: The spectra of single heterostructures #1 and #2 collected on the

IXRD (at the left hand side) and measured on the standard high resolution

equipment (at the right hand side). The information about the Al content is

identical for both methods within experimental error. The time of measurement

is significantly shorter for the IXRD.

IXRD setup it was not possible to observe higher order superlattice peaks, because the

acceptance angle of the primary beam is only about two degrees. The measurement

time was 40 seconds for each sample. The angular position of the zero-order superlattice

peak corresponds to a lattice parameter determined by (dwcw+dbcb)/(dw+db) where

dw, cwand db, cbare the thicknesses and lattice parameters of the well and the barrier

layers, respectively. Knowing the thicknesses of well and barrier layers from the growth

conditions and the lattice parameter of the GaN layer we get the lattice parameter of

the well layer, i.e., the information about the composition of the well layer. The relative

position of neighboring peaks yield the information about the periodicity D=dw+db

of the structure.

34 5 Results of experiments under laboratory conditions

Figure 5.3: Schematic illustration of the vertical structures of tested multiple

quantum wells.

In figure 5.2 at the right hand side we show the results of (0002) ω-2Θ scans of samples

#3 and #4. Following the procedure of interpretation of such measurements [1] we

determine the In content and the periodicity of the structures for both samples. There

is an perfect agreement between the results obtained by the IXRD and by standard high

resolution equipment.

5.1 Use of the IXRD for fast ex situ characterization 35

Figure 5.4: X-ray diffraction patterns of two different multiple quantum wells

#3 and #4 measured by the IXRD are shown at the left hand side. The chemical

composition of active InGaN layers and the periodicity of the structures were

obtained from the angular position of the zero-order peak and relative position

of neighboring superlattice peaks. For comparison at the right hand side we

show ω−2θscans measured by the high resolution equipment.

5.1.2 Measurements on SiGe

It is likely that SiGe will form the basis of high speed transistor devices. Due to the

successful and rapid development of the growth technology, SiGe structures exhibit very

sharp peaks in X-ray diffraction spectra and X-ray characterization of these materials

usually demand advanced high resolution technique. The question is what is the potential

of the IXRD for fast post-growth characterization of this industrially important material

system. In figure 5.5 we show the sketch of a sample which consists of a Si1−xGexalloy

layer grown on a bulk Si substrate (sample #5).

36 5 Results of experiments under laboratory conditions

Figure 5.5: Schematic illustration of the vertical structure of sample #5.

The left side of figure 5.6 shows the (224) diffraction spectrum measured with the

IXRD. The IXRD allows to resolve the diffraction peak of SiGe alloy which can be

clearly separated from the Si diffraction peak. The angular distance between Si1−xGex

and Si reflection peaks provides the composition x following the procedure described in

Appendix B. Beside the main Si1−xGexand Si diffraction peaks the IXRD diffraction

pattern exhibits interference fringes (or Pendell¨osung fringes). This phenomenon is sim-

ilar to the diffraction pattern of light falling on a narrow slit (details see for instance in

[1] or [24]). The angular distance between fringes is directly connected to the thickness

of Si1−xGexalloy. We show the evaluation of the thickness of the epilayer in Appendix

Both parameters, the composition x and thickness d gathered from X-ray pattern are

shown in figure 5.6. The additional diffraction peak at the right side of the Si peak

results from the diffraction of the Kα2line which is presented in the beams reflected

from the Johansson monochromator.

The right side of figure 5.6 shows the (004) ω−2Θ scan measured by the X’pert MRD

diffractometer. The simulation of X-ray pattern performed by the dynamical theory of

X-ray diffraction gives the composition x and thickness d of the Si1−xGexlayer.

Beside the fact that the results obtained by the IXRD and by the X’Pert MRD diffrac-

tometer are identical, the IXRD offers significantly shorter data collection time.

5.1 Use of the IXRD for fast ex situ characterization 37

Figure 5.6: Left side of the picture shows the (224) diffraction scan of Si1−xGex

alloy grown on the Si substrate measured by the IXRD. The substrate and alloy

peaks as well as interference fringes are clearly resolved and provide composition

x and thickness d of the layer. For comparison the (004) X-ray diffraction

pattern and theoretical curve (full line) are shown at the right hand side of the

picture.

5.1.3 Fast reciprocal space mapping

A more comprehensive analysis of the structural properties can be performed by X-

ray diffraction reciprocal space mapping. Reciprocal space mapping is the collection

of scattered intensity around the reciprocal lattice points of layered structures, and it

is usually performed by series of diffractometer scans. The resulting reciprocal space

map (RSM) represents the contours of constant scattered intensity as a function of

reciprocal space coordinates. The RSMs offer information about the status of strain

and the structural quality of semiconductor multilayers which is not obtainable in ω

- 2Θ measurements [2]. However, the use of the standard X-ray equipment for the

measurement of RSMs demands long data collection times which are in the order of

several hours. Our IXRD allows to measure a RSM in several minutes. The measurement

procedure is explained in figure 5.7.

By moving the narrow slit in front of the monochromator we change the angle of inci-

dence of monochromatic X-rays onto the sample as shown in figure 5.7 for two different

positions 1 and 2 of the slit. Collecting the diffracted beams by the X’Celerator for

different positions of the slit we simultaneously vary both the incident angle δand the

38 5 Results of experiments under laboratory conditions

Figure 5.7: The procedure of reciprocal space mapping with the IXRD. By

moving the narrow slit in front of the monochromator and by collecting the

diffracted spectra by a multichannel detector at different detection angle εwe

collect the distribution of intensity in reciprocal space.

detection angle ε. The intensity of the diffracted X-ray radiation is then represented as a

function of reciprocal space coordinates which are calculated from incident and detection

angles according to equations 3.2.

In figure 5.8 we show (11¯

24) reciprocal space maps measured by the IXRD setup

(at the left hand side) and by the high resolution diffractometer (at the right hand

side) of the single heterostructures #1 and #2. The standard high resolution RSMs

were derived from a series of 2Θ scans performed at different ωangles. The dashed

lines show the calculated locations of fully strained and fully relaxed AlGaN layers of

varying Al content. From figure 5.8 it is obvious what potential is offered for immediate

identification of the strain status of structures by reciprocal space mapping. Both IXRD

and the high resolution diffractometer give the same positions of the Bragg reflexes in

reciprocal space and provide identical compositions of the fully strained layers.

In figure 5.9 we illustrate the reciprocal space maps of the multiple quantum wells #3

and #4 measured by the IXRD setup (at the left right hand) and by the standard high

resolution diffractometer (at the left right hand). The superlattice reflections are clearly

observed with both methods. The dashed lines show the calculated locations of the fully

5.1 Use of the IXRD for fast ex situ characterization 39

Figure 5.8: The (11¯

24) contours of constant scattered intensity of single het-

erostructures #1 and #2 measured by IXRD (at the left side) and collected

on the high resolution diffractometer (at the right side) are shown. The iso-

intensity contours correspond to 0.5, 0.25, 0.1, 0.03 of the maximal diffracted

intensity. The information about strain state and the composition of alloys

are the same whereas the measurement time for IXRD setup is considerably

shorter.

strained and the fully relaxed InGaN layers of varying In content. In this case the strain

status, In content of active layer and the periodicity of the structures are obtained.

40 5 Results of experiments under laboratory conditions

Figure 5.9: The reciprocal space maps from multiple quantum wells #3 and #4

measured by the IXRD (at the left hand side) and by X’Pert MRD diffractome-

ter (at the right hand side) are shown. The iso-intensity contours correspond to

0.5, 0.1, 0.01, 0.001, 0.0005, 0.0002 of the maximal diffracted intensity. The re-

sults of reciprocal space mapping performed by the X’Pert MRD diffractometer

are consistent with those obtained by the IXRD.

In figure 5.10 we illustrate the (224) reciprocal space maps of Si1−xGexalloy #5 mea-

sured by the IXRD (at the left side of the picture) and by the X’Pert MRD diffractometer

5.1 Use of the IXRD for fast ex situ characterization 41

(at the right side of the picture). In contrast to the RSM measured by the X’pert MRD

diffractometer the reciprocal space map collected on the IXRD exhibits a significant

broadening of the main Si and Si1−xGexreflexes as well as of the reflexes due to inter-

ference. There are several reasons for this broadening: i) Due to the use of the narrow

slit in front of the monochromator we can not obtain an ideal parallel beam. For our

experiments we have used a slit of 0.2 mm width. Geometrical estimations show that in

this case the divergence of the primary beam is about 92 arc seconds. The X’Pert MRD

diffractometer has a divergence of the primary beam of only 18-25 arc seconds. This

divergence is responsible for the broadening of reflexes along the Qxaxis. ii) The finite

size of the source of 0.4 mm in the scattering plane reduces the instrumental resolution

∆λ/λ of the IXRD. X-rays emitted from the source which have incidence angles different

from the exact Bragg angle are still reflected by the Johansson monochromator but at

another wavelength. Due to this fact the IXRD has lower instrumental resolution than

the high resolution diffractometer as has been shown in [29]. The presence of a Kα2-line

in IXRD spectrum can not be excluded. iii) The angular resolution of the IXRD in

the ”scan direction” is limited by the size of the detector pixels and by the distance

between the sample and the detector. In the IXRD the angular resolution is about 36

arc seconds which is less than in the case of the high resolution diffractometer which

in triple axis configuration has a resolution of 12 arc seconds. The reduced resolution

causes additional broadening in the ”scan direction” which is almost parallel to the Qz

axis.

This makes the IXRD less suitable for a detailed analysis of the intensity distribution

around reciprocal space lattice points of layered structures with high crystalline perfec-

tion like SiGe/Si or AlGaAs/GaAs. However, the positions of the reflections in IXRD

measurement are consistent with those obtained by the X’Pert MRD diffractometer so

that the structural information on composition, thickness and strain of the epilayers is

identical.

From the series of measurements described in this subsection we can conclude that

the IXRD can perform reciprocal space mapping in significantly shorter times than a

standard X-ray diffractometer. The structural information on composition, thickness

and strain obtained by the IXRD and X’pert diffractometer is identical within the ex-

perimental error.

42 5 Results of experiments under laboratory conditions

Figure 5.10: The reciprocal space maps of SiGe/Si collected by the IXRD and

by X’Pert MRD diffractometer. The iso-intensity contours corresponding to 0.5,

0.1, 0.01, 0.001, 0.0005, 0.0001 of the maximal diffracted intensity are shown.

The required information on the strain status, composition and thickness of the

alloy is identical for both methods.

5.2 Measurements under conditions in a MOCVD

reactor

5.2.1 Measurement conditions and principles of data collection

In this section we describe the abilities of the IXRD for X-ray analysis of crystalline layers

under conditions similar to that in a MOCVD reactor. The MOCVD growth process is

accompanied by a rotation of the sample in a gas flow. Furthermore, the gas flow can

cause a wobble of the sample around the rotation axis. The rotation and wobble during

growth are the most critical points for the in situ X-ray setup, because it can lead to a

deviation of the sample from an angular condition of diffraction and to a reduction of

the diffracted X-ray intensity. In our experiments we have realized the conditions met in

the MOCVD growth reactor Aixtron. In this growth reactor the rotation speed depends

on the flow rate and can be different for different growth modes. However, in average the

rotation speed is about 1 rotation per second, possible fluctuations are not significant

5.2 Measurements under conditions in a MOCVD reactor 43

and can be easily taken into account by the measuring procedure of the IXRD. To clarify

the situation regarding the wobble of the samples, we measured the deflection of a laser

beam during MOCVD growth and found out that the wobble of the sample induces an

angular deviation of the sample rotation axis of about 0.3 degree at maximum.

To simulate these conditions in the laboratory we have equipped the sample stage of

our IXRD setup with an electrical motor which allows to rotate the sample with the

rotation speed of 1 rotation per second. A misalignment of the rotation axis leads to

a wobble of the sample by 0.7 degree at maximum. Thus, we have realized the same

conditions as the conditions present in the real MOCVD growth system.

The use of a focused beam and the multichannel detector in the IXRD setup allows

to get diffraction conditions and to collect the spectra even if a sample deviates from

the exact Bragg condition. We have developed a measurement procedure which allows

to collect X-ray data from rotating and wobbling samples.

In figure 5.11 we demonstrate the principle of data collection from rotating and wob-

bling single heterostructure #1. A multichannel detector collects one X-ray spectrum

within a time interval of 0.15 sec. This time corresponds to the time between two reflec-

tions from different azimuth positions of the wurtzite type crystal ( (hhil) diffractions in

the wurtzite type structures can occur at six different azimuth positions). Taking several

spectra and shifting the maximum of each spectrum to the same position and adding all

spectra together results in the pattern shown in figure 5.11 at the right hand side. The

total measuring time is about 90 sec and exceeds the expected time of 15 sec (equals

100 times 0.15 sec) because of the time needed to transfer the data between the detector

interface board and the computer.

44 5 Results of experiments under laboratory conditions

Figure 5.11: The principle of X-ray measurement from rotating and wobbling

samples. Single shot spectra which are shifted to the maximum peak position

are added. The result is depicted on the right hand side.

5.2.2 Results of measurements. Detection and resolution limits of

the IXRD

In order to analyze the potentials of the IXRD for the X-ray analysis under conditions

in a MOCVD reactor, we have performed X-ray measurements of different InGaN and

AlGaN layers grown on GaN. Our aim is to find out the abilities and restrictions of the

IXRD for X-ray analysis of rotating and wobbling samples.

The left hand side of figure 5.12 shows the (11¯

24) diffraction patterns taken on the

IXRD of samples #6 and #7 which are InGaN and AlGaN layers grown on GaN, re-

spectively. Each X-ray spectrum was obtained by collecting and adding 100 single shot

scans and the total collection time was 90 seconds.

In order to find the accurate positions of the alloy (InGaN or AlGaN) and the GaN

peaks, we have fitted each measured profile with two Pseudo-Voigt functions. The

Pseudo-Voigt functions for GaN and alloy peaks as well as the total fits are plotted

in figure 5.12 as full lines. The fits allow to obtain the relative angular positions ∆εof

alloy peaks in respect to the GaN peak. Using the procedure of composition determina-

tion described in Appendix B for the case of non-rotating layers, we get the In and Al

contents of layers which are indicated in figure 5.12.

At the right side of figure 5.12 we plot the (0002) ω−2Θ scans of samples #6 and

#7 measured with the high resolution diffractometer X’Pert MRD. The full lines are

the calculations performed by means of the dynamical theory of X-ray diffraction. The

5.2 Measurements under conditions in a MOCVD reactor 45

Figure 5.12: At the left hand side the X-ray spectra measured by the IXRD from

rotating and wobbling InGaN/GaN and AlGaN/GaN samples are shown. The

information about the chemical composition of the layers is obtained. (0002)

ω−2Θ measurements of the samples performed by standard high resolution

equipment and dynamical calculations shown at the right hand side provide

accurate thickness d and composition x of the layers.

theoretical simulations provide accurate chemical compositions and thicknesses of InGaN

and AlGaN layers. The collection time for (0002) ω−2Θ was one hour for each sample.

The results of the tests indicated in figure 5.12 demonstrate that the IXRD provides

accurate information about the chemical composition of InGaN and AlGaN layers even

when the samples are rotating and wobbling during measurement. The measurement

time of 90 seconds corresponds approximately to the MOCVD of a 10 nm thick layer

so that under real growth conditions the IXRD averages structural information with a

resolution of about 10 nm.

46 5 Results of experiments under laboratory conditions

Figure 5.13 depicts the spectra measured by the IXRD (at the left hand side) and by

the X’Pert MRD diffractometer (at the right hand side) from samples #8 and #9 which

had the thinnest InGaN and AlGaN layers measured in our tests.

Figure 5.13: The spectra collected by the IXRD (at the right hand side) and

by the X’Pert MRD (at the left hand side) from thin InGaN and AlGaN layers

grown on GaN. The IXRD is able to implement X-ray measurements from 42.5

and 27.5 nm thick InGaN and AlGaN layers.

From the measurements with the IXRD setup we can draw two main conclusions. i)

if we compare the measurement results of samples #8, #9 with the X-ray patterns of

samples #6, #7, we see that the thickness of the alloy strongly influences the width of

the diffraction peak, namely, the decrease of the layer thickness causes the broadening of

the alloy peak. This finite broadening of the diffraction peaks is in a first approximation

proportional to the inverse layer thickness [1], [24].

In figure 5.14 the full width at half maximum (FWHM) of all investigated AlGaN and

5.2 Measurements under conditions in a MOCVD reactor 47

InGaN diffraction peaks is plotted versus the inverse layer thickness. The plot shows a

linear dependence of the FWHMs of the diffraction peaks on the inverse layer thickness.

Thus linear dependence of FWHMs of diffraction peaks on the thickness can be used

for the determination of the relative increase of the thickness during growth, i.e. for the

measurement of the growth rate. The interesting point to note is that the extrapolation

of the curve to the values corresponding to the infinite thick layers results in a FWHM

which equals to that of GaN measured with our IXRD setup. This means that the

broadening of diffraction peaks of alloys due to imperfections is the same as in thick

GaN. On the other hand the quality of GaN buffers predefines the quality of the alloys.

Figure 5.14: Dependence of full width at half maximum of the InGaN and

AlGaN peaks from inverse layer thicknesses.

ii) the broadening of the diffraction peaks and the decrease of the scattered X-ray

intensity from thin layers limit the minimal layer thickness which can be detected by the

IXRD. The X-ray spectra from samples #8 and #9 shown in figure 5.13 indicate that 30

nm thick layers seem to be the detection limit of our IXRD. This means that the IXRD

is able to perform online monitoring starting from 30 nm thick InGaN or AlGaN layers.

The detection limit can be extended to thinner layers by improving the quality of the

Johansson monochromator and increasing the power of the X-ray tube.

As we have shown above, the IXRD offers the potential to obtain information about

chemical composition and relative thickness of layers under conditions present in a

MOCVD reactor. The chemical composition of epilayers is one of the most important

48 5 Results of experiments under laboratory conditions

value, because it defines its optical and electrical properties. It is important to be able

to identify changes in the chemical composition of the layers during growth in order to

influence and correct the growth conditions directly in the growth process. Therefore

we have also investigated minimal differences in the composition of layers which can

be detected by the IXRD. The resolution in composition determination of the IXRD

is limited by the angular resolution of the X’Celerator. The angular resolution of the

X’Celerator depends on the distance between pixels of the detector and on the distance

between the sample and the detector. The angular resolution of the X’Celerator is 0.01

degree and it corresponds to the resolution in composition determination of AlGaN and

InGaN alloys of 0.003 for a chosen reflection. We have measured different InGaN and

AlGaN layers which slightly differ in chemical composition. Figure 5.15 shows the results

of measurements of samples #10 and #11 which are InGaN layers with a difference in In

content of a half percent. We have checked the chemical composition of InGaN alloys by

dynamical simulations of high resolution (0002) ω−2Θ scans shown at the right hand

side of figure 5.15. The shifting of the InGaN peak due to the change in In content is

clearly resolved by the IXRD. Thus these measurement results show that the IXRD has

the resolution in chemical composition even better than half percent.

We have performed the same analysis of the resolution limits for AlGaN layers. The

X-ray pattern of AlGaN/GaN samples #12 and #13 with a difference in Al content

of a half percent are illustrated in figure 5.16. The difference in the position of the

AlGaN diffraction peaks is the limit of capabilities of the IXRD, thus half percent is the

resolution for the determination of chemical composition of AlGaN alloys.

The difference in resolution of the IXRD setup for InGaN and AlGaN layers is due to

the bigger lattice mismatch between InN and GaN lattices than between AlN and GaN.

5.2 Measurements under conditions in a MOCVD reactor 49

Figure 5.15: The resolution of the IXRD for the control of the chemical com-

position of the InGaN layers is analyzed. Two InGaN layers which differ in

only half percent of In content can be distinguished by the IXRD. The results

of IXRD measurements are consistent with the high resolution (0002) ω−2Θ

measurements illustrated on the right hand side of the picture.

50 5 Results of experiments under laboratory conditions

Figure 5.16: The X-ray patterns measured by the IXRD setup (at the left hand

side) and by the X’Pert MRD diffractometer (at the right hand side). The

difference in the Al content of half percent is still resolved by the IXRD setup

and is the limit of IXRD resolution for the control of the chemical composition

of AlGaN alloys.

6 Conclusions

A new X-ray diffractometer which allows to perform in situ control of the MOCVD

growth process using Bragg diffraction of monochromatic X-rays has been realized in

this thesis. The principles of work of the new diffractometer and the description of the

experimental setup are given in detail. We have developed a model for the interpretation

of X-ray spectra taken on the IXRD and have analyzed the possible configuration of

the IXRD for controlling the vertical and in-plane features of layered semiconductor

structures. We show that the IXRD has a good performance for the diagnostics of

epitaxially grown multilayer structures which are used in modern semiconductor industry.

The diffractometer has been tested under conditions typically present in a MOCVD

reactor - with rotation and wobble of the samples during measurements. A procedure

of X-ray measurements of rotating and wobbling samples has been developed. We have

tested our IXRD on different wurtzite type AlGaN and InGaN layers. These materials

were chosen because of their successful use for the fabrication of optoelectronic devices.

Furthermore, today most of these devices are produced by the MOCVD technology. The

tested epilayers were different in composition and thickness allowing to test the IXRD

in different aspects. The results show that the IXRD is able to perform the monitoring

of the layers with a thickness exceeding 30 nm. The IXRD provides composition and

growth rate of the growing layers. We have demonstrated that the IXRD has the same

accuracy and the same resolution for composition monitoring as standard high resolu-

tion X-ray diffractometers which, however, need much longer measurement times and

accurate adjustment of the samples. In the future, the thickness detection limit of 30

nm can be even extended to thinner layers by improving the quality of the Johansson

monochromator and increasing the power of the X-ray tube.

Additionally, the particularities of the IXRD allow to use it for fast post-growth char-

acterization of semiconductor materials. We have tested this potential of the IXRD

with materials which are important for semiconductor industry today and are grown

on a high volume scale. These are InGaN/GaN multiple quantum wells, AlGaN/GaN

single heterostructures and SiGe/Si samples. The structural information obtained from

52 6 Conclusions

the line scans and from reciprocal space maps is in excellent agreement with that gath-

ered from high resolution X-ray diffraction measurements. The IXRD offers advantages

which are decisive for the use of the apparatus for fast post-growth diagnostics: very

short data collection time, insensitivity to alignment of the samples before measurement

and absence of any positioning of the samples during measurement.

We have shown that X-ray diffraction measurements taken on the IXRD provide im-

portant structural information even under conditions in a MOCVD reactor. Since this

information is not obtainable by other methods, IXRD paves the way for an extensive

in situ diagnostics of epitaxial processes.

A Lattice parameters and stiffness

coefficients of relevant materials

The tables A.1 and A.2 contain lattice and elastic constants of materials relevant for

this thesis. For wurtzite type related materials we have used the data reported by O.

Ambacher in his review article on group III - nitrides [30]. For SiGe we have applied the

data published in [31].

Table A.1: Lattice a0, c0and stiffness C11, C13 constants for hexagonal GaN,

InN and AlN.

a0(˚

A) c0(˚

A) C11(GPa)C13(GPa)

GaN 3.189 5.185 374 70

InN 3.54 5.705 190 121

AlN 3.112 4.982 345 120

Table A.2: Lattice a0, c0and stiffness C11, C12 constants for cubic Si and Ge.

a0(˚

A) C11(GPa)C12(GPa)

Si 5.431 165.8 63.9

Ge 5.6575 128.5 48.3

54 A Lattice parameters and stiffness coefficients of relevant materials

B Examples of the evaluation of the

IXRD spectra

In this part of the appendix we describe the procedure of data interpretation of the

measurements performed with the IXRD diffractometer. As examples we consider the

IXRD measurements of the AlGaN/GaN single heterostructure as well as of the SiGe/Si

sample and give the detailed explanation of how we get the structural information from

the spectra measured on the IXRD.

In figure B.1 we depict the (11¯

24) diffraction spectrum of AlGaN/GaN single het-

erostructure #1 collected on the IXRD. The angular separation between the diffraction

peaks of AlGaN and GaN is 0.35 ±0.01 degree. The accuracy in determination of the

angular distance is limited by the resolution of the X’Celerator detector. The distance

between the pixels of the detector and the distance between sample and detector cor-

respond to the angular distance between measurement points of 0.01 degree and we

consider this angle as a systematic error in determination of the angular position. Fol-

lowing equation 3.3 we calculate the distance ∆QIXRD = 0.0249 ±0.0007 ˚

A−1between

reflexes of AlGaN and GaN in reciprocal space. This distance equals the vertical dis-

tance ∆Qzbetween AlGaN and GaN reflexes multiplied by the geometrical factor of

0.99 for the (11¯

24) reflection as shown in section 3.2. Therefore the vertical position of

the AlGaN reflex in reciprocal space is given by QzL=QzS+ (0.99 ·∆QIXRD) where

QzSis the vertical position of the GaN reflex which can be calculated for the (11¯

24)

reflection by QzS= (8 ∗π)/cS= (8 ∗π)/5.185 = 4.8472 ˚

A−1. The vertical position of

the AlGaN reflex is QzL= 4.8719 ˚

A−1and gives access to the vertical lattice parameter

of AlGaN layer cL= (8 ∗π)/QzL= (8 ∗π)/4.8719 = 5.1588 ±0.0008 ˚

A. Thus assuming

that the AlGaN layer is fully strained on GaN we have in-plane aL= 3.189 ˚

A and lattice

parameter cL= 5.1588 ±0.0008 ˚

A perpendicular to the sample surface. Following the

formula 2.10 we find the chemical composition of the layer x= 0.105 ±0.003.

The next example is the evaluation of the (224) diffraction spectrum of SiGe/Si sample

#5 shown in figure B.2 which includes the determination of the composition and the

56 B Examples of the evaluation of the IXRD spectra

Figure B.1: The diffraction spectrum of an AlGaN/GaN single heterostructure

taken on the IXRD. The angular distance between AlGaN and GaN peaks

provides the composition of the layer according to the procedure described in

the text.

thickness of the SiGe layer. The angular distance between SiGe and Si peaks is 0.76±0.01

degree. Referring to equation 3.3 we find the distance ∆QIXRD = 0.0541 ±0.0007 ˚

A−1

between reflexes of SiGe alloy and Si in reciprocal space. From our considerations in

section 3.2 and from table 3.2 we know that for the (224) reflection the vertical distance

∆Qzbetween the SiGe alloy and the Si reflexes equals ∆QIXRD distance multiplied by

the geometrical factor of 1.02. Knowing the vertical position of the Si reflex in reciprocal

space QzS= 4.62765 ˚

A−1we find the vertical position of the reciprocal lattice point of

alloy according to QzL=QzS−(1.02 ·∆QIXRD)=4.5725 ±0.0007 ˚

A−1. The vertical

lattice parameter cLof the SiGe layer is directly connected to the vertical position QzL

and equals cL= (8 ∗π)/QzL= (8 ∗π)/4.5726 = 5.4965 ±0.0008 ˚

A. Assuming that the

layer is fully strained (aL=aS= 5.431 ˚

A) we have both the vertical cL= 5.4965 ±

0.0008 ˚

A and the in-plane aL= 5.431 ˚

A lattice parameters of the alloy. According to

the formula 2.10 we find the chemical composition of the layer to be x= 0.163 ±0.003.

The angular distance between two subsequent thickness oscillations is 0.17±0.01 degree.

This corresponds to the distance in reciprocal space ∆QIXRD = 0.0121 ±0.0007 ˚

A−1

and to the vertical distance ∆Qz= 1.02 ·∆QIXRD = 0.0123 ±0.0007 ˚

A−1. The vertical

distance between thickness oscillations in reciprocal space gives the thickness of the layer

by D= (2 ∗π)/∆Qz= (2 ∗π)/0.0123 = 51 ±3nm.

B Examples of the evaluation of the IXRD spectra 57

Figure B.2: Measurement result of a SiGe/Si sample. The angular distances

between Si, SiGe peaks as well as between two subsequent thickness oscillations

give access to the composition and thickness of the alloy.

We follow the same procedure of data interpretation for the measurements of In-

GaN/GaN multiple quantum wells described in this thesis.

58 B Examples of the evaluation of the IXRD spectra

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ity structure”. Solid State Communications 123, 235 (2002)

A. Tabata, L. K. Teles, L. M. R. Scolfaro, J. R. Leite, A. Khartchenko, T. Frey, D. J.

As, D. Schikora, K. Lischka, J. Furthm¨uller, and F. Bechstedt,

”Phase separation suppression in InGaN epitaxial layers due to biaxial strain”. Appl.

Phys. Lett. 80, 769 (2002)

O. Husberg, A. Khartchenko, H. Vogelsang, D. J. As, K. Lischka, O. C. Noriega, A.

Tabata, L. M. R. Scolfaro, J. R. Leite,

”Photoluminescence associated with quantum dots in cubic GaN/InGaN/GaN double

heterostructures”. Physica E 13, 1090 (2002)

O. C. Noriega, J. R. Leite, E. A. Meneses, J. A. N. T. Soares, S. C. P. Rodrigues, L.

M. R. Scolfaro, G. M. Sipahi, U. K¨ohler, D. J. As, S. Potthast, A. Khartchenko, and K.

Lischka,

”Photoluminescence and Photoreflectance Characterization of Cubic GaN/AlxGa1−xN

Quantum Wells”. phys. stat. sol. (c) 0, 528 (2002)

O. Husberg, A. Khartchenko, D. J. As, K. Lischka, E. Silvera, O. C. Noriega, J. R. L.

Fernandez, and J. R. Leite,

”Thermal Annealing of Cubic InGaN/GaN Double Heterostructures”. phys. stat. sol.

D. J. As, U. K¨ohler, S. Potthast, A. Khartchenko, K. Lischka, V. Potin, and D.

Gerthsen,

”Cathodoluminescence, High-Resolution X-ray Diffraction and Transmission-Electron-

Microscopy Investigations of Cubic AlGaN/GaN Quantum Wells”. phys. stat. sol. (c)

0, 253 (2002)

U. K¨ohler, D. J. As, S. Potthast, A. Khartchenko, K. Lischka, O. C. Noriega, E. A.

Meneses, A. Tabata, S. C. P. Podrigues, L. M. R. Scolfaro, G. M. Sipahi, and J. R. Leite,

”Optical characterization of cubic AlGaN/GaN Quantum Wells”. phys. stat. sol. (a)

192, 129 (2002)

O. Husberg, A. Khartchenko, D. J. As, H. Vogelsang, T. Frey, D. Schikora, and K.

Lischka, O. C. Noriega, A. Tabata, and J. R. Leite,

”Photoluminescence from quantum dots in cubic GaN/InGaN/GaN double heterostruc-

tures”. Appl. Phys. Lett. 79, 1243 (2001)

A. Pawlis, O. Husberg, A. Khartchenko, K. Lischka, and D. Schikora,

”Structural and Optical Investigations of ZnSe Based Semiconductor Microcavities”.

phys. stat. sol. (a) 188, 983 (2001)

A. Kharchenko, U. Englisch, Th. Geue, J. Grenzer, U. Pietsch, R. Siebrecht,

”Investigation of Partially Deuterated Organic Multilayers by Means of X-ray and Po-

larized Neutron Reflectometry”. Neutron News 11, 29 (2000)

V. I. Punegov, A.V. Kharchenko,

”Effect of Multiple Diffuse Scattering on the Dynamical Diffraction of X-rays in Non

uniform Layer Crystals Containing Microdefects”. Crystallography Reports 43, 1020

(1998)

Acknowledgements

First and foremost, I would like to thank sincerely Prof. Dr. Klaus Lischka for the

possibility to work in his group, for valuable advises and discussions during my work on

this PhD thesis.

I thank appl. Prof. Dr. Donat As for helpful discussions.

Special thanks go to S. Igges for his suggestions during the construction of X-ray

diffractometer.

J. Bethke , J. Woitok (PANalytical B.V., The Netherlands) for a good cooperation

during European Project.

K. Schmidegg and Prof. H. Sitter (University of Linz, Austria) for a good cooperation

during the work at University of Linz.

I also want to thank B. Volmer, I. Zimmermann for their help in solving bureaucratical

and technical problems.

I would like to express my gratitude to the PhD students O. Husberg, A. Pawlis, S.

Potthast, Shunfeng Li and to the master degree student J. Sch¨ormann for their helpful

discussions.