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This is to certify that the

dissertation entitled

LOCAL STRUCTURE AND DYNAMICS OF III-V SEMICONDUCTOR ALLOYS
BY HIGH RESOLUTION X-RAY PAIR DISTRIBUTION FUNCTION ANALYSIS

presented by

I L- KYOUNG JEONG

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LOCAL STRUCTURE AND DYNAMICS OF III-V SEMICONDUCTOR ALLOYS
BY HIGH RESOLUTION X-RAY PAIR DISTRIBUTION FUNCTION ANALYSIS

By

IL—KYOUNG JEONG

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

DEPARTMENT OF PHYSICS AND ASTRONOMY

2001

ABSTRACT

LOCAL STRUCTURE AND DYNAMICS OF III-V SEMICONDUCTOR ALLOYS
BY HIGH RESOLUTION X-RAY PAIR DISTRIBUTION FUNCTION ANALYSIS

By

IL—KYOUNG JEONG

In semiconductor alloys such as In1_xGaxAs, the energy band gap as well as
the lattice parameter can be engineered by changing the concentration, :13. Due to
these properties, semiconductor alloys have found wide applications in optoelectronic
devices. In these alloys, local structure information is of fundamental importance in

understanding the physical properties such as band structure.

Using the high real-space resolution atomic Pair Distribution Function, we ob-
tained more complete structural information such as bond length, bond length dis-
tributions, and far-neighbor distances and distributions. From such experimental
information and the Kirkwood model we studied both local static displacements and
correlations in the displacements of atoms. The 3-dimensional As and (In,Ga) atom
iso—probability surfaces were obtained from the supercell relaxed using the Kirkwood
potential. This shows that the As atom displacements are very directional and can
be represented as a combination of (100) and (111) displacements. On the contrary,
the (In,Ga) atom displacements are more or less isotropic. In addition, the single
crystal diffuse scattering calculation of the relaxed supercell shows that the atomic
displacements are correlated over longer range along [110] directions although the

displacements of As atoms are along (100) and (111) directions.

Besides the local static displacements, we studied correlations in thermal atomic

motions of atom pairs from the PDF peak width changes as a function of atom pair

distance. In the PDF the near-neighbor peaks are sharper than those of far-neighbors
due to the correlation in near-neighbor thermal motions. We also determined bond
stretching and bond bending force constants of semiconductor compounds by fitting
the nearest neighbor and far-neighbor peak widths to the lattice dynamic calculations

using the Kirkwood model.

To my family, my Wife Eunhee and daughter Haein

iv

ACKNOWLEDGMENTS

Looking back, so many people have contributed to my life and studies in the
USA. I sincerely express my gratitude to all of you. Especially, I would like to thank
my advisor, Simon Billinge for all the encouragement and guidance he has given me
whenever I was depressed and lost. My sincere gratitude to my guidance committee
members, N. Birge, M. F. Thorpe, J. Linnemann and T. Glasmacher for their time
and comments. I highly appreciate help and advice from J. S. Chung and M. F.
Thorpe and I would like to thank them for making available the Kirkwood model

programs.

I am also grateful to Valeri Petkov for his advice regarding PDF data analysis
programming and experiments we have done together. I must thank Thomas Prof-
fen for helping me to learn DISCUS and diffuse scattering calculations and for many
other things. Thanks to Farida Mohiuddin-Jacobs, K.-S. Choi and Pantelis Trikali—
tis for sample preparations. Thanks to all group members, Peter Peterson, Emil
Bozin, Matthias Gutmann, Xiangyun Qui, and Jeroen Thompson for sharing ideas

and working together in many ways.

Thanks to Korean students in the department for being together at all times.
I wish to thank Sungkyun and Cheongsoo for their encouragement and for being
good friends always. My special thanks to Jeongil for being with me and my family
when we needed help. I am very grateful to all Matthew family members for their
encouragement and prayer all the time. I will never forget the kindness and help from
C. W. Kim and K. Y. Lee families. Finally my gratitude and love to my parents and
parents-in-law. Without their support, love and prayer, I know nothing could have

been done. Praise the Lord, for he is good; his love edures forever.

Contents

LIST OF TABLES viii
LIST OF FIGURES ix
1 Introduction and Motivation 1
1.1 Introduction ................................ 1
1.2 Pr0perties of Semiconductors ...................... 2
1.2.1 Structure of Semiconductors ................... 4

1.2.2 PureSemiconductors...................;... 7

1.2.3 Semiconductor Compounds ................... 7

1.2.4 Semiconductor Alloys ....................... 8

1.2.5 Modeling of Semiconductor Alloy Structures .......... 11

1.3 Measuring Correlated Thermal Motion ................. 12
1.4 PDF Analysis ............................... 14
1.5 Layout of Dissertation .......................... 15

2

Atomic Pair Distribution Function from x-ray Powder Diffraction 16

2.1 Introduction ................................ 16
2.2 Atomic Pair Distribution Function ................... 17
2.2.1 Real space and Q—space representations of local structure . . . 22
2.3 Data Collection and Analysis ...................... 24
2.3.1 X-ray scattering and synchrotron radiation ........... 24
2.3.2 Data Collection and Analysis Procedure ............ 28
2.3.3 PDFgetX ............................. 31
Correlated Thermal Motion in Semiconductor Compounds 32
3.1 Introduction ................................ 32
3.2 Mean-Square Relative Displacements in Crystals ............ 36
3.3 Experiments ................................ 37

vi

3.4 Correlated Thermal Motion in Ni and InAs ............... 38

3.4.1 Modeling and extraction of the PDF peak width ........ 39

3.4.2 Results ............................... 42

3.5 Correlated Thermal Motion: Debye approximation ........... 46

4 Local Structure of InxGa1_xAs Semiconductor Alloys 54
4.1 Introduction ................................ 54
4.2 High Real-Space Resolution PDF measurement of InxGa1_xAs Alloys 55
4.2.1 Sample preparation ........................ 55

4.2.2 PDFs of InxGa1_xAs alloys ................... 56

4.2.3 Bond length & bond length distributions in InxGa1_xAs alloys 58

4.3 Modeling the local structure of InxGa1_xAs semiconductor alloys . . 62
4.3.1 Tetrahedral Cluster Model ................... 62

4.3.2 Supercell model based on the Kirkwood potential ....... 70

4.4 Correlated atomic displacements in InxGa1_xAs alloys ........ 76

5 Discussion and Conclusions 80
APPENDIX 85
PDFgetX User’s Manual 86
Bibliography 128

vii

List of Tables

4.1 Standard deviation of the As and (In,Ga) atom distributions in InxGa1_xAs
alloys obtained from the Kirkwood model ................ 75

A.1 Known Platforms Supporting PDFgetX ................. 89

A2 Summary of structure function refinement ............... 105

viii

List of Figures

1.1
1.2
1.3

1.4
1.5
1.6

2.1
2.2
2.3
2.4

2.5

2.6
2.7
2.8
2.9
2.10

3.1

3.2

3.3

3.4

Diamond and Zincblende structures ...................
Wurtzite and Chalcopyrite structures ..................

A plot of the energy gap as a function of lattice parameter for several
III-V semiconductors ...........................

Composition dependence of the bond lengths in InxGa1_xAs alloys . .
The second neighbor distances in InxGa1_xAs .............

Schematic representation of uncorrelated and correlated thermal mo-
tions of atoms ...............................

The illustration of scattering event in powder sample .........
Dependence of real-space resolution of NN PDF peak on Qnm . . . .
Energy dependence of the anomalous scattering of indium ......

The total and indium differential atomic pair distribution functions of
Ino,5Ga0.5As alloy .............................

The illustration of In0_5Gao.5AS structure and resulting structure func-
tion, and PDF ...............................

Relationship between real-space and momentum-space information . .
Atomic scattering factors of In, Ga, and As atoms ...........
Progress in the brilliance of the X-ray beams during 20“ century . . .

Illustration of the experimental set-up .................
MCA spectrum of InAs at Q=45 A“ ..................

Schematic diagram showing an instantaneous snapshot of atomic po-
sitions in (a) rigid-body model (b) Einstein model (c) Debye model

Thermal diffuse scattering. The TDS intensity is exaggerated in the
picture ...................................

Schematic diagram of determining potential parameters using Kirk-
wood potential and experimental PDF peak widths ..........

Reduced structure functions, F(Q) = Q[S(Q)-1] of (3) Ni, and (b) InAs
measured at T2300 K. ..........................

ix

C3301

10

13

18
20
21

22

23
25
26
27
28
30

33

35

37

38

3.5

3.6

3.7

3.8
3.9

3.10
3.11

3.12
3.13
3.14

4.1
4.2
4.3

4.4
4.5
4.6

4.7
4.8

4.9

4.10

4.11
4.12

4.13
4.14

4.15

Experimental and model PDFs of (a) Ni, and (b) InAs; correlation in

thermal atomic motion is taken into account (refinement B) ..... 40
Experimental PDF (circles) and model PDF (solid line) of InAs; No
correlation in thermal atomic motion is assumed (refinement A). . . . 41
PDF peak width 0,]- and correlation parameter, (I) of Ni and InAs as a
function of atom distance, r ....................... 43
Schematic drawing of a unit cell of InAs ................ 44
Geometrical configurations of atoms involving (a) T3 bond and (b) 7‘5
bond ..................................... 46
Peak width sharpening effects in Correlated Debye model ....... 47

Correlation parameters as a function of atom distance, temperature,
and Debye wavevector .......................... 49

Schematic diagram of phonon modes for different values of wavevector. 50
Correlated Debye model calculation of Ni PDF peak width at 300 K . 51
PDF peak width of GaAs as a function of atom separation ...... 52

The reduced total scattering structure function of InxGa1_.xAs alloys 57

The reduced PDF, G (r), of InxGa1_xAs alloys ............. 58
Composition dependence of Ga-As and In-As bond lengths in InxGa1_xAs
alloys .................................... 59

Mean square relative displacements of atom pairs in InxGa1-xAs alloys 61
Schematic diagram of As displacements in cluster type II, III, and IV 62

Comparison between experimental and cluster model PDFs of Ino,5Gao,5As

alloy .................................... 65
Partial PDF of In and As calculated using the Cluster model. . . . . 66
Comparison of the first peak in experimental and model PDFs of
Ino,5Gao_5As alloy ............................. 68
Z-plot and NN PDF peak shape obtained from the 17-atom Cluster
model ................................... 69
A sketch of the tetrahedron surrounding an impurity atom in the zinc-
blende structure. ............................. 70
Relaxed structure of Ino.5Gao,5As alloy ................. 71

Comparison between experimental and Kirkwood model PDFs of InxGa1_xAs
alloys .................................... 72

ISO-probability surface of the ensemble averaged As atom distribution 73

ISO-probability surface of the ensemble averaged (In,Ga) atom distri-

bution ................................... 74
Single crystal diffuse scattering intensity in the (hk0.0) plane of recip-
rocal space ................................. 77

4.16 Single crystal diffuse scattering intensity in the (hk0.5) plane of recip-

rocal space ................................. 78
A.1 Comparison of normalized elastic scattering .............. 96
A2 MCA spectrum of Ino 33Ga0 67As at Q=40A‘1 ............. 101
A3 Effects of corrections on raw data of Ino 33Gao 67As and comparison

between normalized DATA and ( f 2) .................. 104
A.4 Data corrections in Ino,33Gao,6-;As alloy ................. 105
A5 Reduced Structure Function and PDF of Ino,33Gao.67As semiconductor 106
A6 Data analysis procedure in PDFgetX .................. 112
A.7 Structure function refinement procedure in PDFgetX ......... 114
A8 Dead-time correction in Ino,33Gao.6-,As semiconductor ......... 117
A9 Double scattering ratio for nickel .................... 118
A.10 Absorption factors in transmission and reflection geometry ...... 119
A.11 Compton and elastic scattering intensities in Ino,33Gao,67As alloy . . 120
A.12 MCA file format ............................. 127

xi

Chapter 1

Introduction and Motivation

1 .1 Introduction

The discovery of semiconductors has brought significant advances in modern tech-
nologies. From the elemental semiconductors to the semiconductor compounds, these
materials have been used in computers, telecommunications and Optical devices [1].
Recently, semiconductor alloys, in which electronic structure and material proper-
ties can be controlled, have found wide applications in optoelectronic devices. As
in other interesting materials such as colossal magnetoresistance materials and high
TC superconductors, local structure information is very important in understanding
the physical properties of semiconductor alloys. In pseudobinary semiconductor al-
loys (A1_XBXC), the electronic structure as well as their enthalpies of formation are
strongly affected by the change of concentration, :13. These examples show that the
local structural information is one of the key parameters in understanding physical
properties of these materials. In these disordered semiconductor alloys, however, the
local structure is very different from the average structure, which is determined from

the Rietveld analysis of powder diffraction pattern.

Besides local static displacements of atoms, thermal atomic motions are important

in understanding structural phase transitions [2, 3] and carry information about the

interatomic potential in a crystal. The most detailed information about thermal
atomic motions can be obtained from inelastic neutron scattering [4]. Using suitable
models for the interatomic interactions, the force constants can be determined by

fitting a lattice dynamic model to the experimental phonon dispersion relations.

One of the ways to study the local atomic structure as well as the thermal atomic
motions in a crystal is the total scattering powder diffraction measurement. Here
the total scattering means ‘Bragg peaks + diffuse scattering’. Although directional
information is lost, it is still possible to extract significant information about local
structure [5]. In powder diffraction, diffuse scattering is evident especially in high-
Q space. One way to extract the information stored in the diffuse scattering is to
obtain the atomic pair distribution function (PDF) via a sine Fourier transform of
total scattering data [6]. The resultant PDF is a measure of the probability of finding
atoms at a distance r from a reference atom: PDE peak positions measure pair
separations, the areas under the peaks give the coordination numbers and PDF peak
widths measure bond-length distributions (static and thermal). Therefore, from the
PDF analysis, it is possible to obtain both local structural information as well as

thermal atomic motion.

In this dissertation, we present the application of PDF analysis to the local struc-
ture of InxGa1_xAs semiconductor alloys and correlated atomic motion in semicon-

ductor compounds.

1.2 Properties of Semiconductors

Semiconductors are one of the actively studied subjects and many reviews and use-
ful textbooks are available about semiconductors [7 , 8, 9, 10] and semiconductor

alloys [11, 12]. Here we briefly review some of the semiconductor properties related

to the structure. The energy band gap, E9, between the lowest point of the con-
duction band (conduction band edge) and the highest point of the valence band
(valence band edge) is a useful concept in understanding conductivity in a material.
At T20, a material with an energy gap will be nonconducting unless the applied
field is strong enough to overcome the energy gap. However, at T740, there are some
probabilities that electrons will be thermally excited across the energy gap into the
conduction bands. This probability critically depends on the size of the energy gap

“Ea/”WT where k3 is the Boltzman’s constant. At room

and is roughly of order 6
temperature kBT is around 0.025 eV; therefore, depending on the energy gap, E9,
some electrons can be excited over the energy gap and observable conduction will

occur at room temperature.

Semiconductors are generally classified by their energy band gaps with values
around 0.1~2 eV at room temperature. Depending on the size of the energy gap,
semiconductors can emit from infrared to blue lights. The energy band gap can be
either direct or indirect. In direct gap semiconductors, the conduction band edge '
occurs at the same point in k-space as the valence band edge, whereas in indirect
gap cases the band edges occur at different points in k-space. GaAs is an example
of a direct band gap semiconductor and silicon is an example of an indirect band
gap semiconductor. In indirect gap semiconductors, the efficiency of light emission is
not as good as in direct gap semiconductors since the transition requires emission or

absorption of a phonon to conserve crystal momentum.

One of the most important parameters of semiconductors at temperature T is the
number of carrier densities: electron or hole densities. In intrinsic semiconductors
where impurity contributions are negligible, all carriers come from thermal excitations

and electron density is equal to hole density. In extrinsic semiconductors, impurities

contribute a significant fraction of the carrier density. In this case, electron density
is not equal to the hole density. Thus when impurities provide the major source of
carriers, one or the other type of carrier will be dominant. An extrinsic semiconductor
is called “n-type” or “p-type”, according to whether the dominant carriers type are

electrons or holes, respectively.

Silicon and germanium are the most important semiconductors. These semicon-
ductors have perfect covalent bonding. In addition to the column IV elemental semi-
conductors, compounds made up of elements from columns III and V of the periodic
table form III-V semiconductors. GaAs and InAs are examples. In III-V semicon-
ductors, some electron transfer occurs between the two types of atoms. This uneven
electron distribution increases the Coulomb interaction between the ions. The con-
tribution of ionic character in the bond can be represented using the fractional ionic
character, f, [7]. Most III—V compounds have f,- around 0.2 ~ 0.4. This ionicity
becomes even larger and more important in the II-VI compounds. In the II-VI com-
pounds, the ionicities are in the range of 0.6 ~ 0.8 and the bonding between atoms is
a strong ionic as well as a covalent. For this reason, III-V and II-VI semiconductors

are referred to as polar. They are known as polar semiconductors.

1.2.1 Structure of Semiconductors

Most semiconductors crystallize in diamond, Zincblende, and wurtzite structures. As
shown in Figure 1.1(a), the diamond structure consists of two interpenetrating face-
centered cubic (fcc) sublattices displaced by a/4(i, 3), 2), where a is the lattice con-
stant and :13, 3), 2 are the unit vectors along x, y, z axes. The diamond lattice is not
a Bravais lattice. The underlying Bravais lattice is the fcc lattice and the diamond
structure is obtained with the two-atom basis (0,0,0) and a/4(5c, Q, 2). In the dia-

mond structure, each atom has four nearest neighbor atoms, forming a tetrahedron.

 

Figure 1.1: (a) Diamond structure, (b) Zincblende structure. In the Zincblende struc-
ture, anions occupy one sublattice and cations the other.

Most column IV elements, C, Si, Ge, a-Sn, of the periodic table have this structure.
The Zincblende structure has the same lattice as the diamond structure. However,
in this case, the anions occupy one sublattice and the cation the other. Many im-
portant III-V and II—VI semiconductor compounds such as GaAs and CdTe have this

structure.

The wurtzite structure (Figure 1.2(a)) is a variation of the Zincblende structure.
Both structures are tetrahedrally coordinated and the cohesive energy of the wurtzite
structure is very close to that of the Zincblende structure [8] As a result some III-V
and II—VI compounds crystallize in both the Zincblende and the wurtzite structures. In
the wurtzite structure, the underlying Bravais lattice is a hexagonal. It is interesting
to note that only a few crystals among the III—V compounds crystallize in the wurtzite
structure as a stable phase. This is in clear contrast to the II-VI counterparts where
about half have the wurtzite structure. This is strongly related to the higher ionicity
of II—VI compounds [13]. These diamond, Zincblende and wurtzite structures become

unstable under the high pressure and a phase transition to B-Sn or NaCl structure

 

 

Figure 1.2: (a) Wurtzite structure, (b) Chalcopyrite structure.

occurs depending on the ionicity [14].

Another crystal structure closely related to the semiconductors is a Chalcopyrite
structure [15, 16]. This structure is found in ordered ternary compounds such as
CdGeA82 (ABX2). As shown in Figure 1.2(b), the Chalcopyrite structure is similar to
the zincblende structure. It has nearly the same anion arrangement as the zincblende
structure but has two cations arranged periodically on one of the sublattices. In the
‘ideal chalcopyrite’ structure, the ratio of the unit cell lengths c to a is equal to two.
However, in most real Chalcopyrite crystals, the ratio 6/0 is less than two by as much
as 12%. In general cases, the structure has a tetragonal symmetry and the anions are
displaced from their fcc sites along (100) or (010) directions. The anion displacements
result in two different bond lengths RAX 96 REX. This is very similar to the bond
alternation in InxGa1_xAs alloys. Therefore the Chalcopyrite structure has been used

to explain the local atomic displacements in Ino,5Gao_5As alloy [15].

1.2.2 Pure Semiconductors

Silicon and Germanium are the prototype of a large class of semiconductors. The
crystal structure of Si and Ge is the diamond structure. Both Si and Ge have indirect
band gaps. Thus the optical absorption and emission of Si is much weaker than a
direct band gap material such as GaAs. However, Si is the material of choice for
photodetection because of the high quantum efficiency, high degree of uniformity,

well established technology and low price.

1.2.3 Semiconductor Compounds

Semiconductor compounds made from the elements of the groups III and V (GaAs)
or II and VI (ZnTe) of the periodic table have properties very similar to the elemental
semiconductors. Most III-V compounds crystallize in the cubic zincblende structure
though the wide band gap nitride (GaN, InN) have the hexagonal wurtzite struc-
tures. In II-VI compounds, some crystals like ZnS and CdS have both zincblende and
wurtzite structure as stable phases. Many of the III-V compounds have direct band
gaps that facilitate an optical emission process [17]. Therefore, III-V compounds have
been used in light emitting devices (LED) and lasers [1]. GaAs also has better prop-
erties in electron saturation velocity and electron mobility than those of Si. However,

it is not widely used in computer chips because of the low cost of Si.

The energy band gap of II-VI compounds covers wide ranges from few meV to
a few eV. Wide band gap II—VI compounds such as ZnSe are used in lasers and LEDs,
while narrow band gap compounds such as CdTe are used in solar cells, optical devices,

and infrared detector and emitters [17].

Energy Gap(eV)

 

 

 

L

 

5.4 5.6 5.8 6.0 6.2 6.4 6.6
Lattice Parameter(A)

Figure 1.3: A plot of the energy gap as a function of the lattice parameter for several
III-V semiconductors. The lines that connect points on the graph show how the band
gap and the lattice parameter vary for mixtures of the binary compounds and for the
quaternary compounds to which the points correspond [18].

1.2.4 Semiconductor Alloys

By mixing two binary semiconductor compounds or two elemental semiconductors, it
is possible to make semiconductor alloys. In these alloys, the energy band gap as well
as the lattice parameter can be engineered by changing the concentration as shown
in Figure 1.3. This opens up many new applications [18, 19]. For example, GaAs
electronic integrated circuits are dominant in optoelectronic applications. However,
GaAs generates light near 0.88 pm in the near infrared and the optical loss of optical
fiber is minimum at the wavelength of A215 pm. This wavelength is achievable from
GaAs-InAs or InP-GaAs alloys by choosing an optimum concentration [12]. The Si-
Ge alloy is an another example. By adding Ge to Si, it is possible to improve the

mobilities of the charge carrier and design electronic devices which are faster than

Si [19].

 

2.65 - r . 1 . 1 . ,
Pauling limit

2.6

2.55

2.5

Bond Iength(A)

 

 

 

24 L 1 1 1 1 1 1 1
0 0.2 0.4 0.6 0.8 1
Composition(x in lnxGa,_,/\s)

Figure 1.4: Composition dependence of the bond lengths in InxGa1_xAs alloys [25].
Also shown are bond lengths in Pauling and Vegard limits. See text for details.

In the alloys, however, due to the presence of disorder, the band structure calcula-
tions based on the average crystal structure do not work and the local structure should
be taken into account to obtain accurate electronic structure determinations [20, 21].
In A1_XBXC pseudobinary alloys, two possibilities concerning the composition depen-
dence of the bond lengths dAC(.’II) and (130(33) in an alloy are suggested. First, the
atomic radii are considered as conserved quantities and, hence, are transferable [22, 23]
between materials. Based on this idea (“Pauling limit”), bond lengths dAC(:r) and
dBC(:r) in the alloys are assumed to be composition independent. However, many
x-ray diffraction experiments on semiconductor [24] alloys showed that the average
structure is preserved in the alloys and the lattice parameter is given by the composi-
tion weighted average of the pure end-members (“Vegard law”). In the Pauling limit,
the structure is assumed to be very floppy and complete relaxation occurs. Therefore,

in this limit, two distinct bond lengths exist in an alloy. Conversely, in the Vegard

 

4.32 fi f i r ' 1 ' f

 

 

 

 

 

(a) T vAs-ln-As
4.24 » 4'
4.16 . «
8
g 4.08
a: AAs—Ga—As
271'
‘6 4.00
2
z 4.32
13 Iln-As-ln
g 424 uGa-As—ln
(I) OGa-As-Ga
4.16
4.08
4.00
0 0.2 0.4 0.6 0.8 1

x in Ga In As

1-11 it

Figure 1.5: Symbols are EXAFS experimental data [15] and solid lines [26] are cor-
responding theoretical calculations based on Kirkwood potential. (a) As-As second-
neighbor distances. As-In-As bond (triangle down), As—Ga-As bond (triangle up).
(b) In-As—In (square), Ga-As-In (circle), and Ga-As-Ga (diamond) second-neighbor
distances.

limit, the lattice is rigid and no relaxation occurs, hence the alloy has only a single

average bond length.

The first experimental determination of bond lengths in InxGa1_xAs alloys is
made by Mikkelson and Boyce [25]. Figure 1.4 shows the composition dependence
of bond lengths in InxGa1_xAs alloys obtained using the extended x-ray absorption
fine structure (XAFS) method [25]. As shown, the individual nearest neighbor (N N)
Ga-As and In-As distances in the alloys are rather closer to the pure Ga-As and In-As
distances. In addition, it shows that the local bond lengths differ from the average
bond length by as much as 0.1 A. Figure 1.5 shows the second-neighbor distances in
the InxGa1_xAs alloys. The As-As second-neighbor distances have two distinct bond

lengths; the As—In-As bond (closed squares) is longer than the As-Ga-As bond (closed

10

circles). It also shows that the As-As bond distances are almost completely relaxed.
However, In-In, Ga-Ga, and Ga—In distances are closer to the composition weighted
average value. Further XAFS experiments showed that this is quite general behavior
for many zincblende type alloy systems [27, 28, 29]. However, the XAFS information
is limited to the nearest and next nearest neighbor distances and no bond-length
distribution is available. This limited information makes it difficult to establish in

detail how the alloys accommodate the local distortions due to the alloying.

1.2.5 Modeling of Semiconductor Alloy Structures

One of the simple and useful phenomenological models of semiconductors is the Va-
lence force filed (VFF) model [12]. The most common forms of VFF are the Keating
and Kirkwood models. The Keating and Kirkwood potentials are very similar to each
other. These V FF potentials depend on only two parameters: the bond stretching

and bond bending force constants. The following shows the Kirkwood potential:
. 023' o 2 2 .Bz'jk 1 2

I/ = Z 3-(143' — LU) + Le Z: —8—(6086ijk + 3) . (1.1)
(U) <ij,z'k>
Here Li]- is the bond length between atoms 2' and j, L9]. is the natural bond-length,
and L8 is the average bond length given by Vegard’s law. The potential contains bond
stretching force constants (1,, and bond bending force constant 5131: that couple to the
change in the angle 901: between adjacent nearest neighbor bonds. Under a uniform

expansion, these force constants have a simple relationship to the elastic constants of

cubic crystals [30],

0.. = Bia].+s/3.a]
C12 = §%[a—4/36]
C44 — i[ 4C“? ] (1.2)

 

11

Here C11, C12, and C44 are the elastic constants, B is the bulk modulus and a is the
lattice constant. The force constants in the Kirkwood model can be determined to
fit either the elastic constant C11 and the bulk modulus B 2 1/3(C11 + 2C12) or to
fit C11 and 012. However, the force constants chosen to fit Cu and B result in errors
in C44 and the zone center optic phonon frequency around 15 % ~ 30 ‘70 for most

semiconductor compounds [26] and vice versa.

Using the Kirkwood model, the rigidity of the Al_xBxC semiconductor alloy can
be determined using a topological rigidity parameter, a” [26], which can be fit by the

following function:
H _ 1+ 1.25([J’/a)
_ 1+ 3.6(13/01) + 1.17(13/a)2'

 

(1.3)

The topological rigidity parameter, a" depends on the ratio of the ,3 and a. It can
vary from 0 to 1. a**21 corresponds to the fl0ppy structure (Pauling limit) and a**20
to the rigid structure (Vegard limit). The ratio of ,BAC/aAC is around 0.2 in most III—
V compounds and 0.1 in II-VI compounds. Therefore II-VI compounds (a** ~ 0.8)
are expected to be more floppy than III-V compounds (a** ~ 0.7). And the slopes of

the composition dependence of NN bond lengths are flatter in II-VI compounds.

1.3 Measuring Correlated Thermal Motion

A familiar effect of thermal motions of atoms is the reduction of x-ray scattering
intensities. The decrease in the intensities is represented by e‘2w where W is the
Debye-Waller factor, given as W 2 879/113 sin2 0/A2. Here 17.3 is the mean-square
displacement of a lattice point in a direction perpendicular to the reflection planes,
20 is the scattering angle, A is the incident x-ray wavelength. Therefore from the
decrease of x-ray intensities as a function of scattering angle, the 173 which measures

uncorrelated thermal motion (Figure 1.6(a)), can be determined.

12

 

I
\
I
\

I‘
I
l
\
\
‘-
—>

._>

I.

I

\

\
_>

"-- -\
I \
(a) i = ‘
|
1
\ I
s._ '1
——> ‘—
l
I

(b)

     

Figure 1.6: Schematic representation of (a) uncorrelated and (b) correlated thermal
motions of atoms.

However, atoms in a crystal are coupled together by the interatomic forces and
the atomic motions are correlated (Figure 1.6(b)). In general, near-neighbor atoms
tend to move in-phase with each other and far-neighbors move independently. This
coupling of the atomic motion results in diffuse scattering and provides information
about the interatomic forces. For example, atomic motions in a covalently bonded
material will be more strongly correlated than those in a metallic bond system. In
addition, the degree of correlation will be different depending on atom separation as
well as pair direction [31]. Therefore, from the details of correlated atomic motion, it

is possible to determine empirical interatomic potential parameters.

Since PDF peak widths measure mean-square relative displacements of an atom
pair, their width gives information about the correlation in atomic motions. If the
motion is completely correlated, the peak should be given by a delta-function. And
for the completely independent relative motions, the PDF peak will be broad and
the width is given by the mean-square displacements of atom pairs. Therefore, the
analysis of PDF peak width as a function of atom separation gives information about
the degree of correlation between atoms [32, 33]. For example, the nearest neighbor
PDF peak width depends primarily on the bond stretching force and the far-neighbor

peak width is determined by the bond bending force.

13

1 .4 PDF Analysis

In the pure crystalline material, all the structural information are contained in a unit
cell and can be extracted using Rietveld—type refinement of diffraction patterns. How-
ever, for non-crystalline materials, such an approach does not work due to the lack of
periodicity. The atomic pair distribution function has long been used to characterize
the structure and properties of non-crystalline materials such as liquids, glasses and
amorphous materials. Nice introduction to the PDF method and its application to the
non-crystalline materials are given by many authors [34, 35, 36, 37, 38]. The atomic
pair distribution function measures the probability of finding another atom at a dis-
tance r from the reference atom. Therefore, the PDF provides important information
about structure of materials such as bond length and bond length distributions. Be-
sides the total PDF, differential PDFs have been used to obtain chemical specific
structural information [38]. Application of PDF analysis to crystalline materials has
been relatively recent. The use of the PDF technique to the crystalline materials has
been described in a number of recent publications [39, 40]. In the case of disordered
crystalline materials, obtaining high real-space resolution PDF is very important in
order to obtain local structural information [41, 42, 43]. The real-space resolution of
PDF is mainly determined by the maximum momentum transfer, Qmax. And it is nec-
essary to collect data up to Qmax 2 40A‘1 to obtain reasonable real-space resolution.
However, in conventional x-ray scattering the maximum obtainable Qmar is around
20 A” [37] limiting the real-Space resolution. In addition, the scattering intensity is
very weak in high-Q region due to the atomic scattering factor. These limitations in
the conventional x-ray are overcome with the high energy high intensity synchrotron
x—rays and pulsed neutron sources. In particular, the advent of the third generation

synchrotron radiation make it possible to collect data beyond QmaI240A‘1. In order

14

to use the high energy synchrotron x-rays to the PDF analysis and to study the local
structure of disordered crystalline materials, we developed high energy x—ray PDF
technique and data analysis program. The high real-space resolution PDFs allowed
us to obtain information on the local structure of the InxGa1_xAs semiconductor al-
loys. The details of the high energy x-ray PDF technique are discussed in Chapter 2

and Appendix A.

1.5 Layout of Dissertation

In Chapter 2, we discussed the high energy x-ray PDF technique. The general intro-
duction to the PDF is given in Section 2.2 and the details of data analysis procedures
is described in Section 2.3. The application of the PDF method to correlated atomic
motions in semiconductor compounds is discussed in Chapter 3. In Section 3.4 we
compared correlated thermal motions of atoms in two different interatomic force sys-
tems: Ni and InAs. The experiments are compared to the lattice dynamical calcula-
tions using the Kirkwood potential. Also, we discussed the possibility of obtaining the
interatomic force constants from the PDF. The Correlated Debye model of thermal
motions is given in Section 3.5, The application of PDF method to the InxGa1_xAs
alloys is described in Chapter 4. In Section 4.2 we present the first complete solution
of the distorted local structure of the important III-V semiconductor alloy system
In1_xGa«As using the PDF analysis of powder diffraction data. The modeling of the
local structure of InxGa1_xAs alloys are given in Section 4.3. And the correlations in
static displacements of atoms are described in Section 4.4. Finally, the summary and

discussion is given in Chapter 5.

15

Chapter 2

Atomic Pair Distribution Function
from x—ray Powder Diffraction

2.1 Introduction

X-ray diffraction has long been used to characterize the structure of a crystalline
material [44]. By analyzing the intensities and positions of the Bragg peaks, one can
obtain the average structure information of a crystal using Rietveld-type refinement
method [45]. The success of Bragg law and crystallographic methods is based on
the three dimensional periodicity of a crystalline material. Therefore, for a material
with disorder (local deviation from the average structure), the crystallographic meth-
ods do not reveal the local structural information properly. Any deviation from a
strict long-range order gives rise to diffuse scattering. Therefore, in order to obtain
the local structural information, one has to analyze diffuse scattering [46]. Diffuse
scattering due to local disorder is relatively weak compared to Bragg diffraction and
widely spread in reciprocal space. This often makes the diffuse scattering experiments
on single crystal intensity-limited and very time-consuming [47]. An alternative way
to study diffuse scattering is powder diffraction. Although directional information
is lost, it is still possible to extract significant information about local structure [5].

In conventional crystallographic method, only Bragg peaks are used for the refine-

16

ment and data are usually terminated around Q 2 14 A“ due to the overlap of the
Bragg peaks beyond this value. However, for a disordered material, high-Q space
contains significant information of local structure as a form of diffuse scattering [6].
The information stored in high-Q space cannot be extracted by the crystallographic
method. One way to extract information stored in the diffuse scattering is to ob-
tain the atomic pair distribution function (PDF) via a sine Fourier transform of the

diffraction data [6].

The atomic pair distribution function (PDF) has long been used for the study of
amorphous materials, glasses, and liquids [34]. Its widespread application to crys-
talline materials such as semiconductor alloys is relatively recent [48]. In Section 2.2
we briefly review the formalism of PDF and it’s characteristics. In Section 2.3, we
give a short introduction to synchrotron radiation. And then the data collection and

analysis procedures are described.

2.2 Atomic Pair Distribution Function

In x—ray diffraction experiments, the quantity measured is the scattered intensity of
x-rays. This intensity from a material can be represented in the following way:

2

IMQ) : Ian.ei(k"k)""‘

 

: Z fmei<k' “W": Z fn*e"'“""“"’" (2.1)

2 Z fnzfn*€i(kI _k).rm"
,
m,n

where f,,,, fn are the atomic scattering factors of atom m,n and ‘*’ represents a
. i . .
complex conjugate. k, k are wavevectors of the 1nc1dent and scattered x-rays and

rm, 2 rm —— rn is the vector connecting atoms, m and n. For powder samples, we

17

 

.
u c
o n
.. o
-.-...--.--’

Figure 2.1: Illustration of the scattering event in the powder sample. The dotted
circle shows possible positions of rmn in powder sample.

can assume that the vector l‘mn may have the equal probabilities in all directions as
is shown in Figure 2.1. In this case, we can take a spatial average of the exponential

term in Eq. 2.2 and obtain what is known as the Debye [49] scattering equation,

sin Qrmn
Q)=:4;fmfn—— or... , (2.2)
where Q2 lk— kl— — fsinfl. Here 26’ is the angle between R and R. It is useful

to introduce a density function, pm(rm,,) such that pm(rm,,)dV,, gives the number
of atoms in the volume element an at a distance rmn from atom m. Using the
density function pm(rm,,), for a monoatomic material, Eq. 2.2 can be expressed as the

following:

. sin rmn ,
: 2 f2 + Zf2//)m(rm,,)—Q?———diln. (2.3)

For an atom pair distance, rm” 2 7*, let p('r) 2 (pm(rmn)), where (- - ) means the
average over all atoms, m at a distance r from an atom in the sample. Then, adding

and subtracting the average density, p0, we can rewrite Eq. 2.3 as

Q)=Zf2+§:f2/4m2pr[( —pol S——i22QTdr +Zf2 /4n12po§——i22QTdr- (2-4)

The third term of Eq. 2.4 is the self scattering and is finite only at small angles close

to the primary beam and can be neglected [36]. Therefore we drop this term here,

18

then we have

sin QT
Qr

where N is the total number of scattering atoms in the sample. We can rearrange

 

HQ)“, 2 Nf2 + Nf2 [000 47m"2 [p(r) — [)0] dr, (2.5)

Eq. 2.5 to have the simple form of

Q[S(Q) — 1] 2 A0047rr [p(r) — p0] sin Q7“ dr, (2.6)

where S(Q) is the ‘total scattering structure function’ defined as S(Q) 2 I(Q)e,,/Nf2.
For a polyatomic material, S (Q) can be generalized to have S (Q) 2 [I(Q)e,,/N —
((f2) -— < f)2)]/( 1”)? [34, 35, 36]. In Eq. 2.6, if we define ‘atomic pair distribution
function’, C(r) as G(r) 2 47rr[p(r) — p0], then we can obtain C(r) from the inverse
sine Fourier transform of the experimental reduced total structure function, F (Q) 2
Q[3(Q) - 1],

C(r) 2 g/ooo Q[S(Q) — 1] sin QT dQ. (2.7)
In a real experiment, the upper limit of the integration in Eq. 2.7 is not infinity but
the maximum measured magnitude of momentum transfer, Qmaz. If we consider this

finite Qmal. effect (termination effect), the experimental PDF will be given by the

convolution of the PDF obtained with Qmar 2 00 by the Sinc function [50],

Gemp(7‘) : — _ dT' , (2.8)

w r-W r+f

 

 

1 /°° GUI) [sin Qmm(r — 7") sin Qma$('r + r') ,
0

where, G(r) is the PDF obtained with Qmam 2 00 and Gexp(r) is the experimen-
tally determined PDF. The effect of convolution with finite Qmax is clearly shown in
Figure 2.2. This Figure shows the nearest neighbor peak of Ino.5Gao_5As PDF with
different Qmax. This peak is a doublet made up of two closely separated peaks. With
Qnm higher than 40A”, the two components are well resolved. However, for Qmal
around 30A"1, the splitting is not clear and only a shoulder is noticeable. This shows

that the larger Qmax, the higher is the real-space resolution.

19

 

 

 

 

 

6
4
E

2

O

_2 . 1 . 1 4 9 1 1 7 I
2 2.5 3

«N

Figure 2.2: Real-space resolution of Ino,5Gao_5As NN PDF peak. Symbols show
NN PDF peak obtained with different Qmaz . (plus) Qma$230A‘1, (diamond)
Qmax235A‘1, (square) Qmax240A‘1, and (circle) Qma$245A‘1.

In N-component materials, the total scattering structure function, S(Q) is given

by the weighted sum of the N(N+1) / 2 partial structure functions, Smn(Q):

S(Q) = Z wmnSm..(Q). (2.9)

where Smn(Q) is a partial structure function and wmn 2 cmcn fmfn“ / (f), cm,c,, are
concentration of atom type m and n. Partial structure functions contain chemical
specific information involving element pair (m,n) and differential structure function
yields an element resolved structural information. Therefore, the corresponding PDF
of total structure function is a weighted average of partial pair distribution functions.
When the difference in bond lengths between atom pairs is very small, they cannot
be resolved in the total PDF making determination of bond lengths between specific
atom pairs very difficult. In this case, having partial and/ or differential PDFs helps

to obtain specific bond length information. Chemical specific partial and differential

20

 

 

 

    

 

 

 

 

a 12 - fl.
C
9
‘6
£2
.9,
1 :4 E2: 27.629 keV
_2_* 15,: 27.889 keV. ,. -----
.3.
s— ‘4: ,
-5 « _
, Eedge _27.929 keV
-6-
i
-7 . /
'8 r T ' I r I ' I '
26 27 28 29 30 31
Energy (keV)

Figure 2.3: Energy dependence of the anomalous scattering of indium [51]. Symbols:
theoretical data. The dotted line through the symbols is a guide to the eye. F1111 line
is an experimental data. The two energies below the In edge are marked by arrows.

pair distribution information can be obtained using resonant anomalous x-ray scat-
tering [51, 52, 53]. Resonant x-ray scattering utilizes the energy dependence of the

atomic scattering factor near the absorption edge,

f(Q. E) = me) + HE) + z‘f”(E), (2.10)

where f0(Q) is the usual atomic scattering factor and f '(E ) and f" (E) are the anoma-
lous scattering terms depending on the x-ray photon energy E. Figure 2.3 shows
energy dependence of the anomalous scattering terms of indium [51]. The f'(E)
changes rapidly within 2 100 eV around the absorption edge. By taking the differ-
ence between two data sets measured just below (few eV) the absorption edge and a

few hundred eV from the edge, we obtain a differential structure function which con-

21

 

 

 

 

4..

3- H L
«A 2q
.4
as . \

1- / ln-As

Ga-As
o.
-1'i

 

 

 

1.5 2.0 2.5 ' alo ' 3T5 4:0 ' 4:5 ' 5.0
r (A)

Figure 2.4: The total (full line) and In differential (symbols) atomic pair distribution
functions of Ino.5Gao_5As alloy [51]. The In-differential PDF is significantly broader
than the total-PDF due to the lower Qmax.

tains contributions only from the target atom. The corresponding Fourier transform
of the differential structure function gives pair correlations involving just the target
atom. Figure 2.4 shows the total PDF and In-differential PDF of Ino,5Gao,5As. The

In-differential PDF shows only atom pairs involving indium such as In-As and In—In.

2.2.1 Real space and Q-space representations of local struc-
ture

The PDF, G(r), defined in Eq. 2.7, is a sine Fourier transform of experimentally
measurable ‘reduced structure function’, Q[S (Q — 1]. Therefore it contains equivalent

information to the powder diffraction data. However, the PDF emphasizes the diffuse

22

Structure function : InosGao 5As

 

0(S(Q)— 1]

 

 

rumjsuml Jaimog

 

 

 

 

 

5
r(A)

Figure 2.5: The illustration of Inol5Gao,5As structure and resulting structure func-
tion, and PDF. The high-Q oscillating diffuse scattering contains the local structure
information. This is confirmed by the back-Fourier transform of the PDF NN peak
which reproduces the oscillating diffuse scattering in momentum-space.

scattering through the Fourier transform and reflects the local structure more directly.
Figure 2.5 shows the model structure of InxGa1_xAs at :c 2 0.5. The In and Ga atoms
share one sublattice and As sits on the other sublattice in the zinc-blende structure.
As Ga substitutes for In, arsenic can have two different neighbors: In and Ga. Since
Ga—As and In-As bond lengths are different, the substitution of In by Ga builds
up strain energy. Therefore in order to minimize the strain energy, the As atom
displaces from its average position and the tetrahedron distorts locally. The local
disorder introduced by alloying is evident in powder diffraction as enlarged effective
Debye-Waller factor and Bragg peaks disappear at much lower Q than those of pure

In-As and Ga-As. Instead, diffuse scattering shows up in the high-Q region. The

23

corresponding PDF of the structure function shows clearly resolved nearest neighbor
peak at the position of Ga-As and In-As bond lengths. From this we know that the
alloy sustains two distinct Ga—As and In-As bond lengths. This clearly shows the

advantage of obtaining PDF when a material has disorder.

In order to understand the relationship between real-space and momentum-space
information, peaks in the PDF are Fourier transformed to the momentum-space.
Here, we approximated the PDF peak as a Gaussian. Figure 2.6 shows Gaussian
peaks in real—space and the corresponding representation in the momentum—space.
This shows that the Fourier transform of the Gaussian function is a damped oscillating
function. The damping in momentum-space is determined by the peak width of the
Gaussian function. Broad peaks in the PDF damp out faster in Q-space so high
Q-space contains information about the sharp (nearest neighbor) peak. The position
of nearest neighbor peak can be determined from the wavelength of the oscillation
via rc 2 27r/AQ where rc is a peak position and AQ is the wavelength of oscillation in
Q-space. In Ino,5Gao_5As, the NN peak has two sharp peaks corresponding to Ga-As
and In-As bonds. These sharp peaks generate two oscillating waves with very small
difference in wavelength in Q-space. The oscillating signal in high-Q region shows a

heat from these two waves.

2.3 Data Collection and Analysis

2.3.1 X-ray scattering and synchrotron radiation

As we discussed in section 2.2, the high—Q space stores significant local structural
information. In addition, if Qmax is larger, the real-space resolution becomes better.
And Qmax 2 40 A"1 is required to resolve small bond length difference (2 0.17 A) in

InxGa1_xAs alloys. We can obtain the powder diffraction pattern using either x-ray

24

Gaussian Peaks Sine Fourier transform
4 ' ] ' r r 5 1

 

 

 

 

 

 

 

 

 

0 Muiji'11

o 2 4 6 K 8
r(A)

 

 

 

Figure 2.6: Relationship between real-space and momentum-space information. The
Fourier transform of a Gaussian function is represented as a damped oscillating func-
tion. The damping in momentum-space is determined by the peak width of the
Gaussian function and from the frequency of the oscillation, the Gaussian peak po-
sition is obtained via rC 2 27r//\Q where rc is peak center and AQ is a wavelength of
oscillation in Q-space.

or neutron scattering. In neutron scattering, however, an indium nuclear-absorption
resonance in neutron scattering limits Qnm. Conventional x-ray scattering also has
a serious limitation in obtaining high Qmaz. In x-ray scattering, the incident beam is

scattered by the electron clouds which are spread throughout the atom. This leads

to phase differences in x—rays scattered from different parts of the atom. Therefore,

25

 

50 . fl 2 r . i 2

 

i o 00 In atom
0. 6(1- Ga atom
40 A Q 00 AS atom
*0
3
~ 30
3
§
.2
.2
E 20
2
<
1 0
0

 

 

 

Figure 2.7: Atomic scattering factors of In, Ga, and As atoms

the scattered intensity for the atom decreases as the scattering angle increases [37].
This decrease in scattered intensity is expressed by the atomic scattering factor, f.
Figure 2.7 shows atomic scattering factors of In, Ga, and As atoms as a function
of Q calculated using the analytical formula with nine coefficients developed by D.
Waasmaier & A. Kirfel [54]. It is clear that the scattering factor decreases signif-
icantly as the wavevector, Q, increases and beyond Q220A it is almost 10 times
smaller than the value at Q20. Due to this decreasing atomic scattering factor, x-ray
scattering has not been considered as a suitable choice for obtaining high-Q space
data. In addition, there is another limitation in using conventional x-rays for obtain-
ing high-Q space data. Since the momentum transfer is given by Q 2 47r sin 0//\,
short wavelength (high energy) incident x-rays are required to get high momentum
transfers. However, the highest energy of conventional x-rays is from Ag K0, and

is around 22 keV (z 0.56 A) which corresponds to Qmal. around 20 A‘l. This is

26

 

 

  

(a) 3rd generation . (b)
sources I

1018,_ I 1013 _

P— I t—
, 4
2nd generation . 2nd generation
sources , sources
— ,o —
I
I
lst generation ‘ '
12 - 12 .
10 F sources ,I 10 lst generation
I sources

I .—

   

3rd generation
sources

 
 

Brilliance
Brilliance

 

 

 

 

 

 

 

1' Rotating anode
P—X-r’dy tube Q . _
Q ___________ ' , ’ rotating anode A
9 _
106 P 10 Bremsstrahlung
Continuum
1 1 1 L 1 1 1 1
.2 _ t .,
1900 1940 1980 10 10' 10° 10 10
Year Photon Energy (keV)
l l l l l
1000 100 10 l 0.1
Wavelength (A)

Figure 2.8: (a) Progress in the brilliance of the X-ray beams during 20‘“ century [55],
(b) Brilliance of the x-ray beams as a function of energy.

not enough to get high real-space resolution in the PDF method. These problems,
atomic scattering factor and limited available radiation sources, were addressed us-
ing synchrotron radiation sources and in particular the advent of third generation
synchrotron sources. Figure 2.8 (a) shows the progress in the brilliance (photons )of
x-ray beams during the 20th century. In Figure 2.8 (b), the energy dependence of
the brilliance (photons/sec/mm2/mrad2 in 0.1 ‘70 b.w.) of the x—ray is shown. With
the advent of third generation synchrotron radiation, beams that are 1010 times more
brilliant than conventional x~rays are available. In addition, the third generation syn-
chrotron radiation covers most wavelengths, from infrared to hard x-rays, making it

possible to collect data beyond Qmax=40A"1.

27

A2 Hutch

  
 

Ion chamber monitor

synchrotron source
SI 82 _____
: He tilled tube .'

(”1:5 -------------

 

 

 

 

monochromator and
focusmg system

--------------------------------------------------- ...............................

Figure 2.9: Illustration of the experimental set-up

2.3.2 Data Collection and Analysis Procedure

In this Section, we present brief description of the experimental set-up and data col-
lection strategy. Figure 2.9 shows the experimental set-up at Cornell High Energy
Synchrotron Source (CHESS). The intense white beam is generated using an insertion
device (wiggler) at the A2 beamline. The white beam is dispersed using a Si(111)
double-bounce monochromator. Unlike the conventional tube x-rays, in a synchrotron
experiment the incident x-ray intensity is changing with time. Therefore it is impor-
tant to monitor the incident x-ray intensity. This will be used to normalize for the
incident intensity. Two Ion chamber (IC) monitors with flowing Ar gas were used.
All measurements were carried out in flat plate symmetric transmission geometry. In
order to minimize thermal atomic motion in the samples, and hence increase the sen-
sitivity to static displacements of atoms, the samples were cooled down to 10 K using
a closed cycle helium refrigerator mounted on the Huber 6 circle diffractormeter. The

samples were uniform flat plates of loosely packed fine powder suspended between

28

thin foils of kapton tape. The sample thicknesses were adjusted to achieve sample
absorption at ~ 1 for the incident x—ray energy, where ,u is the linear absorption

coefficient of the sample and t is the sample thickness.

During the whole measurement, the maximum intensity was scaled so that the
count rate across the whole detector energy range in the Ge detector did not exceed
~ 2 x 104 3’1. At these count-rates detector dead-time effects are significant but
can be reliably corrected as we describe below. To reduce the random noise level
below 1%, we repeated runs until the total elastic scattering counts become larger
than 10,000 counts at each value of Q. Also, to obtain a better powder average the
sample was rocked with an amplitude of i0.5° at each Q-position. The scattered x-
rays were detected using an intrinsic Ge solid state detector. The signal from the Ge
detector was processed in two ways. The signal was fed to a multi-channel analyzer
(MCA) so that a complete energy spectrum was recorded at each data-point. The
signals from the elastic and Compton scattered radiation could then be separated
using software after the measurement. In parallel, the data were also fed through
single-channel pulse-height analyzers (SCA) which were preset to collect the elastic
scattering, Compton scattering, and a wider energy window to collect both the elastic

and Compton signals.

For the SCAs, the proper energy channel setting for the elastic scattering is crucial.
Any error in the channel setting could cause an unknown contamination by Compton
scattering and make data corrections very difficult. There’s no such problem in the
MCA method since the entire energy spectrum of the scattered radiation is measured
at each value of Q. In addition, Laaziri et al. [56] developed a fitting procedure for
separating Compton from elastic scattering using an MCA spectrum for the whole

Q-range. The main disadvantage of the MCA method is that it has a larger dead-

29

300

 

 
   
     
 

'§ 200 ~

'3 .

E elastic

7, scattering
$3 100 pulser Compton -

scattering

 

 

 

 

 

 

o 200 400 600 800
MCA channel number

Figure 2.10: MCA spectrum of InAs at Q=45 A‘l. Peaks in the spectrum from the
elastic and Compton scattering are labelled, as is a peak from an electronic pulser
used for dead-time correction. The other peaks in the spectrum come from various
fluorescence and escape peaks.

time, although this can be reliably corrected as we show below. Figure 2.10 shows a
representative MCA spectrum taken from the InAs sample at Q = 45 A. It is clear
that the Compton and elastic scattering are well resolved at this high momentum
transfer. The elastically scattered signal, which contains the structural information,

is obtained by integrating the area under the elastic scattering peak.

In a diffraction experiment, the measured x-ray diffraction data suffers from ex-
perimental effects like sample absorption, x-ray polarization and so on. Therefore we

can express [34] the measured intensity in the following way:
F“%Q)=I%fiNUfih+I$“+fl?UL QJU

where P is the polarization factor, A the absorption factor, N the normalization con-
stant, and IE3“, 1:3”, 1;?“ are the coherent single scattering, incoherent (Compton),

and multiple scattering intensities, respectively, per atom, in electron units. The total

scattering structure function, 5' (Q), is then defined as
5(0) = [12‘3" — (02> — <f>2)1/<f>2, (2.12)

30

where (f) = (f(Q)) is the sample average atomic form factor and (f2) is the sample
average of the square of the atomic form factor. Therefore, to obtain S (Q) from the
measured diffraction data, we have to apply corrections such as multiple scattering,
polarization, absorption, Compton scattering and Laue diffuse corrections on the raw

data [34, 38].
2.3.3 PDFgetX

As we discussed in Section 2.3.2, it is necessary to apply several steps of corrections
to obtain the total scattering structure function, S(Q), from the raw powder diffrac-
tion pattern. In the high intensity high energy synchrotron x-ray experiments, it is
important to correct for detector dead-time effect. In addition, proper corrections for

the multiple scattering and Compton scattering are more important in high—Q region.

For the analysis of the high intensity high energy synchrotron x—ray powder diffrac-
tion data, we developed data analysis program, PDFgetX [57]. PDFgetX is written in
Yorick language. This requires users to install Yorick, a freely available program [58].
PDFgetX applies corrections to “input” data in sequence. During the analysis, it dis-
plays all corrections to raw data and saves all parameters used in the analysis. This
makes the analysis procedure easy to understand and allows reproducible results. The
manual of PDFgetX is attached in Appendix A and the details about the data analysis
procedure and corrections are given in Appendix A.4 and A.5 respectively. The pro-
gram is available from the following website: http://www.pa.msu.edu/cmp/billinge-

group / programs / PDFgetX.html.

31

Chapter 3

Correlated Thermal Motion in
Semiconductor Compounds

3. 1 Introduction

Atoms in crystals are coupled together by the interatomic forces. Therefore, a motion
of one atom influences those of others. Figure 3.1 shows a schematic diagram of
atomic motions in three different interatomic force systems. In a rigid-body system,
the interatomic force is extremely strong and all atoms move in phase. In this case,
we can expect peaks in the PDF to be delta-functions. In another extreme case such
as the Einstein model every atom moves independently. This type of atomic motion
results in broad PDF peaks whose widths are given by the root mean-square atomic
displacement amplitude ( (13)). In real materials, the interatomic forces depend on
the atom separation; it is strong for nearest neighbor interaction and decreases as the
atom separation increases. In this case, near-neighbor atoms tend to move in-phase
with each other and far-neighbors move independently. As a result, the near-neighbor
PDF peaks are sharper than those of far-neighbor pairs. Therefore, from the PDF
peak width as a function of atom separation, we can measure the degree of correlation
in atomic motion. In addition, from the details of correlated atomic motion, we can

obtain information about potential parameters.

32

Rigid-Body Motion

 

(a)

 

 

 

 

Uncorrelated Atomic Motion

 

 

 

 

 

 

 

 

 

1 O Q O 1 1‘:
.9 .2 : w. :1
—> 41- O —> 4"].-1]. 1].. it
1'
Correlated Atomic Motion

G(r) ] l ‘
I l1 ']
(c) 1] 'I i:
'] i~ ‘.

1‘

 

 

 

Figure 3.1: Schematic diagram showing an instantaneous snapshot of atomic positions
in (a) rigid-body model (b) Einstein model (c) Debye model. In (a) and (b) all PDF
peaks have the same width independent of atom separation. In (c) PDF peak width
increases up to the root mean-square displacement as atom separation increases.

In a scattering experiment, the information about correlated atomic motion is
contained in thermal difl'use scattering [32, 33]. Therefore the most detailed infor-
mation can be obtained from phonon dispersion curves determined using inelastic
neutron scattering. More recently very high energy resolution x-ray scattering has
successfully been used to measure phonon dispersion curves [59, 60]. Another ap-
proach that yields this information with lower precision is the study of x-ray thermal
diffuse scattering from single crystals without resolving energy [36, 61]. Other experi-

mental techniques include Raman scattering or IR absorption which can only extract

frequencies of zone center phonons but no information about phonon dispersion or

33

zone edge behavior [4]. The correlated motion of near neighbor atoms also can be
measured in x-ray absorption fine structure (XAF S) experiments [62] which directly
probe their relative motion. In these measurements, however, the information about
the relative motion of far neighbor pairs is very limited and uncorrelated thermal

parameters are not available.

We have taken the approach of extracting correlated thermal motion from powder
diffraction data. This approach may not seem to be favorable since not only is energy
information lost but also the diffuse scattering is isotropically averaged. Nevertheless,
we show that this approach yields extensive information. It has the additional ben-
efit that the experiments are straightforward and do not require single crystals. In
the powder diffraction data, the Bragg peak intensity is attenuated as a function of
scattering angle by the well-known Debye-Waller factor [36]. The ‘remaining’ inten-
sity shows up as diffuse scattering. If the correlated motions are taken into account,
the behavior of the intensities of the Bragg reflections is unchanged but as shown in
Figure 3.2, so-called thermal diffuse scattering (TDS) appears centered at the Bragg
peak positions. The TDS is exaggerated in this picture. By measuring the attenuation
of Bragg-peak intensity it is possible to find the mean—square atomic displacement
amplitude, (112), which corresponds to the uncorrelated value of the thermal motion.
However, in order to learn about the correlated motion of atoms, all diffraction data
including TDS must be used [36, 63]. Although directional information is lost, it
is still possible to extract significant information about the correlation of thermal

motions even in systems with multiple atoms, in the unit cell.

A representation of the powder diffraction data that emphasizes the correlated
atomic motions is the atomic pair distribution function. The PDF peak width gives

information about the correlation of the atomic motion since it is a measure of the

34

. ,1,

ff? ml. ' 3'

 

H

('5
C

"h
N

 

 

 

 

 

 

 

     
 

; ‘ Thermal Diffuse
Scattering

:‘i/
.1
‘1

Figure 3.2: Thermal diffuse scattering. The TDS intensity is exaggerated in the

picture. W is the Debye-Waller factor.

amplitude of relative motions of atom pairs. At low rij where rij is the separation

distance of the pair of atoms, the PDF peaks are relatively sharpened because of the

tendency for near-neighbor atoms to move in-phase with each other. This behavior

was first analyzed by Kaplow in a series of papers [32, 33, 63] and interatomic poten-

tials were determined directly from the PDF for a number of elemental metals. With

the advent of modern synchrotron x-ray and neutron sources and high-speed comput-

ing, the reliability and utility of this technique has dramatically improved from these

early investigations.

35

3.2 Mean-Square Relative Displacements in Crys-
tals

The PDF peak shape can be approximated by a Gaussian function. The peak position
measures atom pair distance and its width is determined by the standard deviation of
atom distance from the average value. Therefore the peak width can be approximated
by the mean-square relative displacement (MSRD) of atom pair [50], projected onto

the vector joining the pair of atoms as is shown in Eq. 3.1

012]- = ([011— 111) ’f'ijl2la (3-1)

where 0,,- is a peak width of atom pair 2', j and ui,uj are thermal displacements of
atom i and j from their average positions. The vector fij is a unit vector parallel
to the vector connecting atoms 1', j, and the angular brackets indicate an ensemble

average. This equation can be rearranged as
01-2} = ([111 ' ful?) + ([uj ' 1‘in - 2((ui ' fij)(uj ' ful)- (3-2)

Equation 3. 2 shows that the MSRD (a, 2) lS composed of the mean- -square displacement
(MSD) of each atom ([u, m]? ), ([uj -r,j]2 ) and the displacement correlation function
(DCF), ((u,~l - fij)(uj - fij)). For a monoatomic crystal, the MSRD can be represented

as the following

 

0;, = —% Zws )(éks'rij)2[n(ws(k))+1/2][1— cos(k - rij)], (3.3)
N‘ k,s

where w,(k) is a phonon frequency with wave vector k in branch 3, n(w,(k)) is the
phonon occupation number, é“ is the polarization vector of the k, s phonon mode.
N is the number of atoms and M is the mass of the atom. If we know the potential
parameters for a system under study, we can obtain phonon frequency (ws(k)) and

polarization (éks) from the eigenvalues and eigenvectors of the dynamical matrix,

36

 

 

 

 

Potential parameters Solve Calculate Compare with

Dynamical . experimental PDF
0" B Matrix PDF peak Width O'ij peak widths: on“ ’00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.3: Schematic diagram of determining potential parameters using Kirkwood
potential and experimental PDF peak widths in InAs compound.

respectively [50]. Then, using Eq. 3.3 we can calculate the MSRD for all atom pairs
and by varying the potential parameters until the theoretical MSRD matches the
experimental values, we can estimate the potential parameters. Figure 3.3 shows a
schematic diagram of the procedure for determining the potential parameters using a
Kirkwood potential in semiconductor compounds with bond stretching (a) and bond

bending ([3) force constants.

3.3 Experiments

All measurements were made using x-ray radiation, at the National Synchrotron
Light Source (NSLS) and at Cornell High Energy Synchrotron Source (CHESS). Ni
and InAs were measured at room temperature at X7A at NSLS using 30 KeV x-rays.
GaAs was measured at the beam line A2 at CHESS using 60 KeV (A = 0.206 A) x-rays.
The data were corrected for absorption, multiple scattering and polarization effects.
Background and Compton scattering were removed and the data were normalized for
flux and number of scatterer to obtain the total scattering structure function, S(Q),
as is described in Section 2.3. The experimental PDF, G (r), is obtained by taking

the Fourier transform of the reduced structure function, F (Q), according to Eq. 2.7.

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50 ’- I I fii I -'
' (a)
t
:7" 30:
g’: ,
$2. .
O 10 -
_\lULJL.llLl L]
_10 A A A A A A L l l I
5°.” . (b) 2
i 30’. j
g , .
g l
O 1oL [J]
will llt t
_10 A A L l A A A A I A A A A A A A A A
o 5 10 15 20
Q(A")

Figure 3.4: Reduced structure functions, F(Q) 2 Q[S(Q)-1] of (a) Ni, and (b) InAs
measured at T2300 K.

3.4 Correlated Thermal Motion in Ni and InAs

In order to study how the atomic correlation depends on the interatomic force, we
first studied correlated atomic motions in Ni and InAs which have very different types
of bonding forces. In Ni, the bonding between atom is isotropic metallic bonding.
On the contrary, InAs has covalent bonding which is very directional. Therefore, we
expect these systems will serve as good examples for the study of the dependence
of atomic correlations on the interatomic forces. Figure 3.4(a) and (b), shows the
reduced structure function, F(Q) 2 Q[S(Q)-1], of the Ni and InAs data measured
at T2300 K. The TDS is evident under the Bragg peaks in the high Q-region. The

Q-range probed is limited by the x—ray energy and flux in these experiments at NSLS.

38

The corresponding PDFs are shown in Figure 3.5 as open circles.

3.4.1 Modeling and extraction of the PDF peak width

In principle one can imagine various ways to extract the widths of the individual peaks
from the experimental PDF. However, the task is not as straightforward as it might
seem. One of the problems is the contribution of termination ripples. These ripples
appear in the PDF from the Fourier transform due to a finite data range. They are
well understood [40], but may cause large errors in conventional fitting or integration
procedures. Here we have used ‘real-space’ Rietveld program PDFFIT [64], which
takes those experimental effects into account, to fit a model PDF, Gm(r), to the

observed PDF. The model PDF, Gm.(r), is defined as [5]

 

GMT) + 47r7‘p0 = 1: Z M? (50 — m). (3.4)
7‘ .- J- <f>‘2

The sums are over all atoms within the sample. The number density of the sample

is given by p0. The value f,- is the atomic scattering factor of atom 2' evaluated at

Q 2 0, in other words the number of electrons Z for atom z'. The sample average

atomic scattering factor at Q 2 0 is (f). Additionally, each atomic pair correlation,

6(7" — rij), is convoluted with a Gaussian to account for thermal motion. Finally, the

Gaussian peaks are convoluted with a Sinc function to account for the experimental

termination effect.

In order to see the correlation effect in the PDF, we first refined the InAs PDF
with the assumption that all PDF peaks have the uncorrelated width (refinement
A) given by the same as the root mean-square displacements of the atoms in the
pair. The result is shown in Figure 3.6. It is immediately obvious that although the
agreement at large values of r is good, the first peak of the model PDF, Gm(r) is too

broad compared to the experimental PDF. This directly shows the effect of correlated

39

 

 

 

 

 

30 ~
‘ .1. ' (a)
20 - .', :
m r -. :2 .. ~ 1; ;,; ;.-.-_
(D 0 -_ ’ 3;. -» : . l
-10 — ' ‘ .'
—20 V v 'A'AVAVV A AVAWAVAVALAVAVAVAVAVAVAV AVA/\VAVAVAVAVAVAVA MAV/
10 l l l I l
.3 (b)
5 ’ if: . .‘
S o ,. '
0 $3
-5 —

 

 

 

_10 1 1 1 1 1
2.5 5 7.5 10 12.5 15

r(A)

 

Figure 3.5: Experimental PDF (circles) and model PDF (solid line) of (3.) Ni, and
(b) InAs; correlation in thermal atomic motion is taken into account (refinement B).
Difference curves are plotted below the data.

motion for the nearest neighbor due to the strong covalent bond between In and As.
In refinement A, the sharpening of the PDF peaks at low values of r was not taken
into account to illustrate this point. Thus, the calculated PDF we see in Figure 3.6

corresponds to the case of completely uncorrelated motion.

The effect of correlated atomic motion is taken into account empirically in our
modeling by describing the r dependence of the PDF peak width, 0,, by the following
equation,

ac 2 00 — —2—. (3.5)

31'

40

 

 

10 ' l ' l ' 1 ' l

 

 

 

 

g .
00
0°
2 5' '. : r d
A 31‘. 5b '1' . :
t, _ lat-gs.” . (.-
(5
_5 _ A -
AAA AAA NVAAA MA A AVA/xv
vvv v \N T] WW ww \«
_14 A I l l l l L l
1 3 5 7 9

r(A)

Figure 3.6: Experimental PDF (circles) and model PDF (solid line) of InAs; No
correlation in thermal atomic motion is assumed (refinement A). All PDF peaks have
the same width as given by the root mean-square displacement. Also shown is a
difference curve between data and fit.

The parameter 00 corresponds to the uncorrelated thermal motion of atom pairs and
is given by 03 2 0,2 + 0?, where a,- and aj are the amplitudes of uncorrelated thermal
motion ( (212)) of atoms 2' and j and 6 is an empirical parameter describing the
sharpening of the PDF peaks. The parameters 00 and 6 are refined in the modeling
process. Figure 3.5 shows the refinement results including the correlated atomic
motion (refinement B) for the Ni and InAs data. We observe a good agreement

between experimental and calculated PDF over the complete r range.

To explicitly extract the peak width, 0c(r), as a function of r, we carried out the
refinement in two steps. In the first step, all parameters are refined using the complete
r range of the experimental PDF. In a second step multiple refinements were carried
out, using only a small region in r around each PDF peak. In this step all parameters
except lattice parameters and the thermal motion parameters were kept fixed. This

approach allowed us to extract individual peak widths reliablely.

41

3.4.2 Results

In order to describe the motional correlations more quantitatively, a correlation pa—

rameter qt can be defined using the following equation [62],

03 = (71-2 + 0'32- — 20',0’j¢. (3.6)

It can be seen from Eq. 3.6 that gt 2 0 corresponds to completely uncorrelated
motion. Positive values of ()5 describe a situation where the atoms move in phase,
thus the resulting value of ac is smaller than for the uncorrelated case. Negative
values of 93 stand for neighboring atoms moving in Opposite directions. Using Eq. 3.6

the correlation parameter 05 can be calculated from the PDF results as

 

2 _ 2
¢> = 00 0.. (3.7)
20,0j

The peak widths and correlation parameters, 95, as a function of the separation dis-

tance r are shown in Figure 3.7 for the Ni and InAs crystals.

A. Ni

The r-dependence of the PDF peak width and the correlation (15 for Ni is shown in
Figure 3.7(a). Even in this close packed compound with isotropic metallic bonding we
observe a weak correlation of the motion of neighboring atoms, e.g. the correlation
parameter for the nearest neighbor is gt 2 0.32 corresponding to a value of cc 2
0.0823(2) A. Fitting the expression given in Eq. 3.5 to the data points in Figure 3.7,
we find the following values: 00 2 010(3) A and 6 2 011(4) A3. Here the width of
the PDF peak is only determined by thermal broadening and zero-point motion and

no static disorder or strain is present in the crystal.

The value for the uncorrelated thermal motion of 00 2 010(3) A determined in this

analysis can directly be compared to the theoretical value for nickel of UN, 2 0.1045 A

42

 

 

0.10-(3) 0.8
‘1
«3:: 0.05
b
0.00
0.15
<3 0.10.
b
0.05
0.00

 

 

 

 

Figure 3.7: (a) Ni: PDF peak width 0,-3- (closed square) and correlation parameter,
cf) (closed circle) as a function of r. The solid line marks the empirical relation in
Eq. 3.5. (b) InAs: Open circles are theoretical values by Chung and Thorpe [50].
The other symbols are the same as in (a).

calculated from the Debye temperature of Ni [65]. The good agreement of the observed

and theoretical value indicates that the PDF analysis allows one to extract thermal

parameters on an absolute scale.

B. InAs

InAs crystallizes in the zincblende structure and shows strong covalent bonding. The
r dependence of the PDF peak width and the correlation parameter 0 are displayed
in Figure 3.7(b). It should be noted that for some separation distances 7' only a
theoretical value is present since no meaningful peak width could be extracted from
the experimental PDF at those points due to PDF peak overlap and weakness of
signal. The correlation parameter for the nearest neighbor (In-As) of gt 2 0.82 is much

larger than in Ni reflecting the differences in the bonding of the nearest neighbors.

43

 

Figure 3.8: Schematic drawing of a unit cell of InAs. The marked distances, r1 to r5
are discussed in the text.

The corresponding value of ac is 0.064(2) A. The resulting values from the empirical
relation given in Eq. 3.5 are: 00 2 0.160(7) A and 6 2 061(9) A3. The overall
agreement between experiment and theory are good. In addition, the empirical Eq. 3.5

shows good agreement with experiment.

In contrast to the Ni results, we can observe a deviation from the empirical behav—
ior given in Eq. 3.5 for low values of rij. These deviation appears in the theory (open
circle) and are accurately determined in the measurement. They are not artifacts but
have a real origin. An explanation of this behavior can be given as follows; First we
assume all bonds to be rigid, thus the first neighboring atoms separated by r1 2 2.6
A would move completely in phase, i.e. ac 2 0. This is obviously an approximation
since we only observe 0., 2 0.064(2) A for the first neighbor. In our simple model

all broadening of subsequent PDF peaks would be caused by bond bending. The

44

structure of InAs is illustrated in Figure 3.8. The first to fifth nearest neighbors are
marked by r1 to r5, respectively. From Figure 3.7 we can see that the PDF peak
width for neighbors separated by r3 is larger whereas for r5 the width is smaller than
given by Eq. 3.5. To understand this behavior we need to consider the bond bending
motion at the two intermediate atoms in both cases. Figure 3.9 shows the geometrical
configuration of atoms involving r3 and r5 bonds. For atoms, 2', j separated by r3,
both bending motions result in almost parallel displacements along the vector r3 and
involve two bending motions resulting in a weaker correlation or broader PDF peak.
For neighbors separated by r5, however, the bending motions lead to displacements
roughly perpendicular to the vector r5. In addition, the thermal motions of atom
i, j involve bond stretching motion of l, k atoms which is very difficult. Due to this
constraint, the combined motion can be considered as a single bond bending. Subse-
quently the correlation should be of the same magnitude as for neighbors separated
by r2 connected by just a single bond bending motion. This is in good agreement
with the observed correlation parameters of 0.27 and 0.31 for neighbors separated
by r2 and r5, respectively, but only 0.10 for neighbors separated by r3. For large
separation distances r,,-, the PDF peak width converges to its uncorrelated limit 00

and the correlation 05 goes to zero.

Chung and Thorpe [50, 66] have calculated the r dependence of the PDF peak
width theoretically using the Kirkwood model [67]. They adjusted the force con-
stants of their model to match the PDF peak width of the first and last peak of
the experimental data presented in this paper [66] and obtained the bond stretching
(a 2 80 N/m) and bond bending ()3 2 10.3 N/m) force constants. These values
Show 10%~30% difference from the literature values [12] determined using the elastic
constants, 011,012,014. However, as we discussed in Section 1.2.5, the force con—

stants determined from the elastic constants also contain errors around 10%~30%.

45

 

 

Figure 3.9: Geometrical configurations of atoms involving (a) r3 bond and (b) r5
bond.

Therefore it is not easy to estimate the accuracy of the force constants obtained from
the PDF. In addition, we used only two data points, nearest and far-neighbor peak
widths, to refine two force constants in our refinement. Therefore this could lead
relatively large errors in the refined force constants. For current purpose, these force
constants are reasonable. The theoretical PDF peak widths are shown as open cir-
cles in Figure 3.7(b). For more details about these calculations see references [50, 66].
The experimental and theoretical values show a good agreement. Even the deviations
of the experimental data from the behavior described by Eq. 3.5 are reproduced by

these theoretical calculations.

3.5 Correlated Thermal Motion: Debye approxi-
mation

We have discussed correlated atomic motion in InAs and showed that the MSRD
obtained from lattice dynamical calculations using the Kirkwood potential are in
good agreement with the experimental PDF peak widths. In the lattice dynamical
calculations, however, the force constants must be known in advance. When the

force constants are not available, the Debye approximation may be used to obtain the

46

 

0004

 

 

 

 

 

(a) .. [1—cos1k0r,,)1/(2k:r.’i
0 1st term of DCF

A 0.002 »~ .
is

a O

.E o

c

8 ° 00000

g 0.000 ‘0 meno °°"--- °°°°°°° 00005....3'" AAAAAAA w." nnnnnn
(D . (b) o

5 >0 a2ndtermotDCFat10K

E 0 010 __ o 0 2nd term Of DCF at 100K

3 ' “Q 0 2nd term of DCF at 300 K

x . .

m °é b/r fitting

0) °o .

O. 00000

0.005 ~
05%!) 00000000000000
% 0000000000000000000
am 00000000000004)
0000 ............ 1 ................... 1 ................... 1.-..i...’ ...........
0 5 10 15 20
«M

Figure 3.10: Peak width sharpening effects. (a) Open circle shows the lst term of

DCF of Eq. 3.10. (b) Symbols are the 2nd term of DCF of Eq. 3.10 at 10 K, 100 K,
and 300 K. Dotted line is a fitting with b/ r. b is a parameter.

PDF peak width changes as a function of atom distance. In Eq. 3.3, if we make no
distinction between longitudinal and transverse phonons and take spherical averaging,

then it will reduce to the following:

of]. 2 <%[n(w) + g] [1 — cos(k . rij):|>, (3.8)

where ( - ) is the average over the 3N modes and N is the number of atoms. This
result is a general expression for all materials and independent of the number of atoms

per unit cell [68]. Using the Debye approximation, w 2 ck, Eq. 3.8 becomes

211 “D p(w) [ 1] [ sin(wr,- /c)
.2. = —— d —— , — 1 — —-——l— , 3.9
0,, 3NM 0 w w "M + 2 wrij/c ( )

where p(w) 2 3N (3w2/wD3) is the phonon density of states, c, the sound velocity

and top 2 ckD is the Debye cut-off frequency. The Debye wave vector is given by

47

[CD 2 (67r2N/V)1/3. Here N / V, the number density of a crystal. After integration

over to, we can rearrange Eq. 3.9 as the following:

02 — —r Him ..
‘7' — Mch 4 es 1 Map

1 — cos(kDrij)
2060713)?

 

 

 

T 2 GD/Tsin(kDr,jTa:/OD)/(kpr'.,jT/OD)
+ H/
91) 0 835—1

d2: , (3.10)
where (1)1 2 foe’)/Tat(ef — 1)"1 dat, and a: is a dimensionless variable and OD is the
Debye temperature. This result is known as the “Correlated Debye model” [69,
70]. In Eq. 3.10, the first term corresponds to the usual uncorrelated mean-square
displacements (MSD) and the second term to the displacement correlation function
(DCF). The MSD shows no 73-, dependence but the DCF depends explicitly on atom
distance, r,,-. Figure 3.10 shows the r,,- dependence of the first and second terms of the
DCF. The first term of the DCF decreases as l/rfj with a cosine modulation and the
second term shows a 1 /r,-,- dependence. In addition, the second term of the DCF shows
a strong temperature dependence. At low temperature, it is negligible compared
to the first term of the DCF. However, as the temperature increases, the second
term becomes dominant. This suggests that the MSRD shows a 1/r,—2j dependence
at low temperature and a 1/r,-,- dependence at room temperature or higher. This Ti,-
dependence of the PDF peak width in Correlated Debye model is a little bit different

from the empirical equation (0C 2 00 — 6/rfj) that we used in Section 3.4. Actually

our approximate expression should be okay:

(If. 2 03 — 2600/7‘; + (5/rfj
~ 03 — 2600/ri2j
2 03 — (SI/r3, (00 ~ (5 << m). (3.11)
What the Correlated Debye model shows is that for higher temperature (on a scale set

by the 190) we might also need a l/rij term. This shows that the empirical equation is

48

 

   

 

 

 

 

0-8 T (a) [f] 1 0A—1
00—010K
0'6 _ a—i-‘i 200K
. «300K

A 0.4 "
3
a5
E 0.2 -
§ ~53.» £41.} 1}}? €3—~fi£i~~~ enete"
o. 0.0 A" v v v "
C
.23 * (b) Temp'=300K
52 0.8 — (3_QI(O=0..8A—1
at) G—Qko=1.0 A-1
o Hko=1.5 A4
O 0.6 “

0.4 r

0.2 r

0.0.11.1...A12“MLLL1L

0 5 10 15 20

r(A)

Figure 3.11: Correlation parameters calculated using the Eq 3.10. (a) Temperature
and atom distance dependence of correlation parameter. Open circle (10 K), open
square (200 K), and open diamond (300 K). Debye wavevector, kg 2 1.0 A‘l, Debye
temperature, GD 2 400 K. (b) Correlation parameter as a function of atom distance
and Debye wavevector. Circle (In; 2 0.8 A“), square (kg 2 1.0 A'l), and diamond
(k0 2 1.5 A“). Debye temperature, 60 2 400 K, temperature T 2 300 K.

a good approximation in representing the rij dependence of PDF peak width although

it appears it could be improved at higher temperatures by including a b/rij term.

The displacement correlation function also depends on the Debye wavevector, k0.
Figure 3.11(a) shows the correlation parameter as a function of atom distance and
temperature. It shows that the correlation parameter increases as the temperature

increases. This is due to the second term of DCF which increases as the temperature

49

9 9 o a 9 9 9 9 9 9 9 Reference

f O k=0
zone center mode

1:1 : «2" o i: : o: o o o 1:1; k=n/2a
‘? k=31d4a
£0 oi o o: 90 o: :o °i .90 01 lo k=1da

zone edge mode

Figure 3.12: Schematic diagram of phonon modes for different values of wavevector.

increases thus results in higher correlation parameter. Figure 3.11(b) shows the kg
dependence of the correlation parameter. It clearly shows that for larger kg, the
correlation becomes smaller. Therefore, in general body-centered cubic (bcc) crystals
have stronger atomic correlation than those of face-centered cubic (fcc) crystals [69]
because the Debye wavevector is larger in the fcc crystals than in the bcc crystals.
Considering phonon modes for different wavevectors may help to understand this
result. Figure 3.12 shows a schematic diagram of longitudinal phonon modes for
different wavevectors, It. For the zone center phonon mode, all atomic displacements
are completely correlated. This mode doesn’t contribute to the PDF peak width
broadening. However, as the phonon wavevector increases, the relative displacement
of neighboring atoms becomes less and less correlated and beyond It 2 7r/2a their
motion becomes more and more anti-correlated until it is completely anti-correlated
at the zone boundary. Thus with increasing Debye wavevector, the NN PDF peak

gets broader.

 

 

 

 

 

 

 

 

 

M T T
~31? (a)
TS 0.10 - .._fi—-—————+1—~-—~———~~ _
.E
.0-0
9 ’- /m .1
3
15 005
a ' ”
LL 0 Experimental PDF peak width of Ni at 300 K
E — Debye model at 300 K, 6:2790 rn/s
0.00 . 1 - 1 . 41 ,
0 5 10 15 20
r(A)
10 . I . ,

’ H Force model (b) 1

3.: 8 ~ ~ Debye model (w,,=49.1 THz) ~
. 6 —
.0
g. ..
(U
:: 4 ~
g .
a: 2 _
O
O

 

(0 (THz)

Figure 3.13: (a) Correlated Debye model calculation of Ni PDF peak width at 300 K.
Circles are experimental PDF peak width and solid line is the calculation. (b)
Solid line: schematic diagram of Ni phonon density of states calculated using force
model [71]. Dotted line: Debye density of states with the same area as the force
model calculation.

Finally, we compared experimental PDF peak widths with the 0,, obtained from
the Correlated Debye model. Figure 3.13(b) shows schematic diagram of Ni phonon
density of states calculated using a force model [71] and corresponding Debye density
of states. In this simple element, the Debye model reasonably approximate the ‘real’
density of states. Figure 3.13(a) shows comparison between experimental PDF peak
widths and Correlated Debye model calculation at 300 K. In this calculation, Debye
wavevector, kD21.757 A"1 and sound velocity, E22790 m/s are used. With these

values given by the atomic geometry and independent measurement, the Correlated

51

 

 

 

 

 

 

 

 

 

 

 

2
‘6 0.06 r
g l
:9
3 0.04 r
x
8
Q. [_ I Experimental PDF peak width at 10 K
LL 0-02 o Kirkwood model (12:95 N/m, 5:10 N/m)
0 Debye model at 10 K. 6:3700 m/s
0- L — Debye model at 10 K. 5:2300 WS
000 1 1 1 l A L 1
0 5 1 0 1 5 20
NA)
1.0 . « , . ,
(b)
H local-density approximation (LDA)
0.3 . Debye model (wo=51.3 THz, $23710 m/s))
3}: — Debye model (1110:2318 THz, 6:2300 m/s) 1
C I
:3 0.6 L
E .
(U
I 0.4 L
g, .
C)
0.2 l-
L.
0.0
0 60

 

00 (THz)

Figure 3.14: (a) PDF peak width of GaAs as a function of atom separation. Symbols:
Experimental PDF peak width (closed square). PDF peak width, 0 obtained from
lattice dynamics calculation (closed diamond). Correlated Debye model peak width
calculation using the average sound velocity, 623710 m/s (dotted line), and 522300
m/s (solid line). (b) Symbol is the GaAs phonon density of states calculated using
local-density approximation [72] and solid (10231.8 THz) and dotted (10251.3 THz)
lines are the corresponding Debye approximation with the same area as the LDA
calculation.

Debye model shows very good agreement with the experiment without any adjustable
parameters. We also compared the Correlated Debye model calculation of PDF peak
width with the experimental peak widths and lattice dynamic calculation using the
Kirkwood potential. Figure 3.14(a) shows the r,,- dependence of the PDF peak widths
of GaAs at 10 K. In the Kirkwood model, the potential parameters are obtained using

the procedure described in Section 3.2. In the Correlated Debye model, the Debye

wavevector, kD21.382 A“ is obtained from the atomic geometry and the sound
velocities, 0 along the [100], [110], and [111] directions are calculated from the elastic
constants. In Figure 3.14(a), the dashed line is obtained using the average sound
velocity, E23710 m/s averaged over three different directions, [100], [110], and [111].
However, the far-neighbor peak widths calculated using this average sound velocity are
off by a factor of 1.2 from the experimental PDF peak widths. This is due to the poor
approximation of density of states by the Debye model as shown in Figure 3.14(b). In
order to match the calculated far-neighbor peak widths to the experimental value, we
used the sound velocity as a parameter. The solid line in Figure 3.14(a) is obtained
using E22300 m/s. This result shows that with just one parameter, the Correlated
Debye model is reasonable in predicting the experimental PDF peak widths even in
more complex systems like GaAs. In addition, the comparison between the Correlated
Debye model and the Kirkwood model calculation shows interesting result. The
difference between these two model calculations are not significant. The general
behavior of PDF peak width changes as a function of atom distance is given by the
Correlated Debye model except small details in PDF peak width changes below ~
10 A. This results suggest that in order to obtain reasonable interatomic potential
parameters, accurate measurements of all the details of the PDF peak width changes

are important.

Chapter 4

Local Structure of InxGa1_xAs
Semiconductor Alloys

4. 1 Introduction

The local structure of InxGa1_xAs was first studied by Mikkelson and Boyce using
extended x-ray absorption fine structure (XAFS) [25]. In this experiment, Mikkel-
son and Boyce showed that the individual nearest neighbor (NN) Ga—As and In-As
distances in the alloys are quite different from the concentration weighted average
value of Ga-As and In-As distances but rather closer to the pure Ga-As and In—As
distances. Further XAFS experiments showed that this is a quite general behavior for
many zinc-blende type alloy systems [27, 28, 29]. Since then a number of theoretical
and model studies have been carried out on the semiconductor alloys to understand

how the alloys accommodate the local displacements [26, 73, 74, 75, 76, 77, 78, 79].

Until now, however, these models and theoretical predictions are tested mainly
by the comparison with XAFS data. The XAFS results give information about the
nearest neighbor and next nearest neighbor distances in the alloys but imprecise in-
formation about bond-length distributions and no information about higher-neighbor
shells. This limited structural data makes it difficult to differentiate between compet-

ing models for the local structure. For example, even a simple radial force model [74]

54

rather accurately predicts the nearest neighbor distances of InxGa1_xAs alloys in the
dilute limit. Therefore, one needs more complete structural information including
nearest neighbor, far-neighbor distances, and bond length distributions to study the

local structure of these alloys more accurately.

In Section 4.2, we present high real—space resolution PDFs of InxGa1_xAs alloys.
In-As and Ga-As bond lengths and distributions are obtained from the InxGa1_xAs
PDFs. In Section 4.3, we describe how the alloys accommodate the local distortion
due to alloying using simple cluster model and more sophisticated Kirkwood potential
based model. Finally, in Section 4.4, we describe correlated static displacements in

InxGa1_xAs alloys.

4.2 High Real-Space Resolution PDF measurement
of InxGa1_xAs Alloys

4.2.1 Sample preparation

The InxGa1_xAs alloy samples with compositions (x 2 0, 0.17, 0.33, 0.5, 0.83, 1)
were prepared by a melt and quench method. An appropriate fraction of InAs and
GaAs crystals were powdered, mixed and sealed under vacuum in quartz ampoules.
The samples were heated beyond the liquidus curve of the respective alloy [24, 80]
to melt them and held in the molten state for 3 hours before quenching them in
cold water. The alloys were powdered, rescaled in vacuum, and annealed just below
the solidus temperature for 72-96 hours to increase the homogeneity of the samples.
After annealing, the sample was cooled down in the furnace by turning off the power.
Depending on the cooling rate, the concentration fluctuation might lead to chemical
clustering [81, 82, 83]. However, because of the slow cooling and the suppression

of mesoscopic chemical concentration fluctuation in the InxGa1_xAs alloys [84] we

55

expect the clustering effects can be neglected in our experiment. The annealing cycle
was repeated until the homogeneity of the samples, as tested by x-ray diffraction, was
satisfactory. After finishing annealing, the sample was ground by hand and sieved
using a 400-mesh sieve. From this, we expect that the particle size distribution is
between a few [1m to 38 pm and the particle size induced strain is negligible. X-ray
diffraction patterns from all the samples showed single, sharp diffraction peaks at the
positions expected for the nominal alloy similar to the results obtained by Mikkelson

and Boyce [25].
4.2.2 PDFs of InxGa1_xAs alloys

Figure 4.1 shows the experimental reduced total scattering structure functions, F (Q) 2
Q[S (Q) — 1], for the InxGa1_xAs alloys measured at 10 K. It is clear that the Bragg
peaks are persistent up to Q ~ 35 A“1 in the end-members, GaAs and InAs. This
reflects both the long range order of the crystalline samples and the small amount
of positional disorder (dynamic or static) on the atomic scale. In the alloy samples,
however, the Bragg peaks disappear at much lower Q-values but still many sharp
Bragg peaks are present in the mid-low Q region. Instead, oscillating diffuse scat-
tering which contains local structural information is evident in high Q region. The
observation of Bragg peaks reflects the presence of long-range crystalline order in
these alloys. The fact that the Bragg peak intensity disappears at lower Q-values in
the alloys than the end-members reflects that there is significant atomic scale disor-
der in the alloys as expected. The oscillating diffuse scattering in the high-Q region

originates from the stiff nearest-neighbor In-As and Ga-As covalent bonds.

Figure 4.2 shows the corresponding reduced PDFs, G (r), obtained using Eq. 2.7.
In the alloys, it is clear that the first peak is split into a doublet corresponding

to shorter Ga—As and longer In-As bonds [51]. The position in r of the left and

56

 

x3 ‘
InAs

M A“ H lfl |n0.83Ga0.17'As

 

 

 

 

 

'"osoGaosoAS
O

:7- 8044111001119me
:63: InoaaGaoe'IAs

& .
l 1 ] moneaocaAs
GaAs

_20 1 1 . 1 , 1 L .

0 1O 20 25 35 45

0(A")

Figure 4.1: The reduced total scattering structure function Q[S (Q) —-1] of InxGa1_xAs
alloys measured at 10K. The data-sets are offset for clarity. The high-Q region is
shown on an expanded scale (x 3) to highlight the presence of diffuse scattering.

right peaks does not disperse significantly on traversing the alloy series. This shows
that the local bond lengths stay close to their end-member values and do not follow
Vegard’s law, in agreement with the earlier XAFS [25] reports. However, already by
10 A the structure is behaving much more like the average structure. For example,
the doublet of PDF peaks around 11 A in GaAs (Figure 4.2) remains a doublet (it
doesn’t become a quadruplet in the alloys) and disperses smoothly across the alloy
series to its position at around 12 A in the pure InAs. This shows that already by
10 A the structure is exhibiting Vegard’s law type behavior. It is also notable that

for the nearest neighbor PDF peak, the peak widths are almost the same in both

57

 

150 ' 1 ' I

 

 

 

InAs
100 Ml;
' InosoGao‘soAs 1
E 50 ’- l”0133(53067As .9
WWW
|“0.17G3083As
O WWW/A
GaAs
_50 1 1
O 5 10 15

NA)

Figure 4.2: The reduced PDF, 0(1), of InxGa1__xAs alloys measured at 10 K. The
data-sets are offset for clarity.

alloys and end-members but for the higher neighbors, the peaks are much broader in

the alloys than in the end-members.

4.2.3 Bond length & bond length distributions in InxGa1_xAs
alloys

We determined the bond lengths of Ga-As and In-As in the alloys from the first
peak of InxGa1_xAs PDFs. Figure 4.3(a) shows the nearest neighbor PDf peaks
of InxGa1_xAs alloys and Figure 4.3(b) shows the evolution of the bond length
across the alloy series. Also shown is the room temperature XAFS data obtained by

Mikkelson and Boyce [25]. We can notice that the individual Ga-As and ln-As bond

58

2.66 T l l T l I T l I T j

 

 

 

 

 

   

 

 

 

 

 

 

I PDF results (10 K) (b) .
o XAFS results (300 K)

2.62 > a

A 2.58 — i e
is s

8 «g ‘

3 C 2 54 ~ -
o- i?
s 5
m

2.50 r d

,

2.46 — a

“'10 ‘ ‘ ‘ ‘ 1 ‘ ‘ ‘ ‘ 2.42 1 1 1 1 1 m 1 1 1 1 1
2 2.5 3 0.0 0.2 0.4 0.6 0.8 1.0
r(A) Composition x

Figure 4.3: (a) NN PDF peaks in InxGa1_xAs alloys. (b) Solid symbols: Compo-
sition dependence of Ga—As and In-As bond lengths in InxGa1_xAs alloys obtained
from PDF measured at 10 K. Open symbols: room-temperature XAF S results from
Ref. [25].

lengths in the alloys are much closer to those of the end-member than the composition
weighted average values. In addition, it shows quite good agreement with the XAFS
results except a constant shift in bond lengths to smaller values by about 0.012 A
in the PDF-based results. This is presumably due to the difference in the measuring

temperature; the PDF data are measured at 10 K whereas the XAFS data were

collected at room temperature.

In AXB1_XC type zinc-blende semiconductor alloys, the angle between A-C and
B-C atom bonds is not orthogonal. Therefore, a displacement of the tetrahedron-
centered C atom leads to a bond length change of all the nearest neighbor atoms. We
obtained the bond length distribution from the PDF peak width. In the alloys, the

PDF peak width (A) has both static and thermal contribution and can be modeled

59

in the following way [35];

A2 2 0% + 0%, (4.1)

where, UT and (ID are peak broadening due to thermal and static disorder, respec—
tively. The thermal contribution to the PDF peak width, (IT, can be determined
from the peak widths of the end-members in which no static disorder exists. In the
alloys, we assume that (IT is given by the linear interpolation of the two end-member
values. Then, we can separate thermal and static contribution. Figure 4.4 shows the
composition dependence of the nearest neighbor (NN) and far-neighbor PDF peak
width. We can see that, for the NN peak, the additional broadening to PDF due to
alloying is very small. It’s less than 15% of the thermal broadening. This makes the
accurate measurements of static bond length distribution very diflicult even with our
high real-space resolution PDF. The difference between Ga—As and In-As peak widths
is less than 2.8% which is within the uncertainty of our measurements. However, the
general trend shows that the Ga—As bond has a larger width than In-As bond. For
a fully relaxed zinc-blende alloy structure, the bond length distributions for A-C and
BC bonds can be modeled as a single peak [76, 85]. In addition, the composition

dependence of the static bond length distribution, 09 can be modeled that
2 _ 2 ,
oD —- 4F0 .1:(1— :13). (4.2)

Schabel et al. [76] calculated F3 using the Keating potential model and obtained
0.000448 A2 and 0.000629 A2 for In-As and Ga—As bonds, respectively. In Fig-
ure 4.4(b) (A-A) and (v-v) show the In-As and Ga—As static peak width calculated
with the F3 given above. This shows reasonable agreement with the experimental
static peak width. For far-neighbor PDF peaks, it is shown that the static contribu-
tion is up to five times larger than the thermal contribution. In addition, As-As pair

peak width is almost three times larger than that of (In, Ga)-(In, Ga) pair. From

60

 

0.025

 

 

 

 

~ (a) Far neighbor E
0.020 - 99999999999 ‘
E As-As
«3:; _, . 1 ..
v 0.015 “' ............... ..
‘9 I" -2 As—(ln Ga) if
a:
as) 0.010 - _________ I,“ _
8 I I ..... E ...........
a , , ’ (In, Ga)—(|n, Ga) _ .
'C 0.005 r xii
E i """""""""""""" .
S
8' 0.000 .1 1 1 1 1 1 1 1 r 1 1.
C (b) Nearest neighbor _ _
E 00040 *’ V,V”V’:: V :‘::v‘~v\ _
37 I , aTA' “I — 7" ‘5 \ .
A ACT’K A i ' 1k 1 V.
00035 E/_“ _E_ _ i _ _ _ 7“ sf
0.0030 ‘ 1 1 1 1 I 1 1 1 1 1
0 0.2 0.4 0.6 0.8 1.0

Composition x

Figure 4.4: Square of the PDF peak widths for far-neighbors (a) and nearest neighbor
(b) vs alloy composition, :17. In (b), o and 0 represent the experimental Ga—As and
In-As peak widths, respectively. The (A-A) and (v-v) are theoretical calculations
for ln-As and Ga—As bonds [76]. In (a), the straight dashed line is the thermal
contribution to PDF peak width. The parabolic dotted curves are the calculations
using the Kirkwood model. Note that for far-neighbor, the static contribution to the
peak width is almost five times larger than that of the nearest neighbor [41].

this result, we can expect that the local distortion on the As sublattice is greater
than the (In, Ga) sublattice distortion. This is understandable considering that (In,

Ga) atoms always have four As neighbors, whereas As atoms have five different local

environments.

61

 

 

Figure 4.5: Schematic diagram of As displacements in cluster (a) type II, (b) type
III, and (c) type IV. Cluster types are discussed in the text. At the corner, a large
dark circle and small grey circle show In and Ga atom positions respectively. At the
center, the grey and dark circles correspond to the As atom position before and after
displacement.

4.3 Modeling the local structure of InxGa1_xAs semi-
conductor alloys

4.3.1 Tetrahedral Cluster Model

The structural disorder in the AxB1_xC type tetrahedral alloys can be intuitively
visualized by considering simple tetrahedral clusters centered about C sites (the un-
alloyed site). In the random alloy this site can have five distinct environments (4
A-neighbors (type-I), 3 A- and 1 B-neighbors (II), 2 A- and 2 B-neighbors (III), 1 A
and 3 B-neighbors (IV) or 4 B neighbors (V)). We assume that the mixed site (A,B)
atoms stay on their ideal crystallographic positions. By considering each cluster type
in turn we can predict the qualitative nature of the atomic displacements present in
the alloy. Let the A atoms be larger than the B atoms. In clusters of type I and V the
C atom will not be displaced away from the center of the tetrahedron. As is shown in
Figure 4.5, in type 11 clusters the C atom will displace away from the center directly
towards the B atom. This is a displacement in a (111) crystallographic direction. In

type III clusters it will displace in a direction between the two B atoms along a (100)

62

crystallographic direction. Finally, in type IV clusters it will again be a (111) type
displacement but this time in a direction directly away from the neighboring A atom.
A simple model of the alloy can then be built up just by determining how many of
each type of cluster are present and applying some simplifying approximations with
regard to the size of the tetrahedron. In the random alloys, the probability of a mixed
sublattice having an A atom is :1: and a B atom is (1 — 3:). Therefore, the probability
of finding type I cluster is 11:4; type II, 4223(1 — as); type III, 63:2(1 — :13)2; type IV,

417(1— 3:)3 and type V, (1 — 2:)4.

Such a cluster model was used to make quantitative comparisons with the nearest
neighbor bond distances observed in XAFS measurements [25] over the whole alloy
series [27]. In this model, it is assumed that no static disorder exists on the mixed
(In,Ga) sublattice. All cluster types have the same lattice parameter that is given
by Vegard’s law [86] for each concentration, :r. Thus, the (In,Ga) sublattice form
a virtual crystal and all the structural relaxation and atomic displacements are ac-
commodated by displacing the As atoms from the centers of the tetrahedra in the
manner described above. Each cluster is independently relaxed according to the pre-
scription of Balzarotti et al. [27] to get the bond-lengths within each cluster type.
In determining the magnitudes of the displacements, it is important to neglect both
bond-bending and relaxation of atoms on mixed sites due to cancellation of errors.
It causes large errors in the average bond length if only one of them is included [27].
Assuming a random alloy the number of each type of cluster that is present can be
estimated using a binomial distribution. This gives the static distribution of bond
lengths predicted by the model. These are then convoluted with the broadening ex-
pected due to thermal motion. This was determined by measuring the width of the

nearest neighbor peaks in the end-member compounds, InAs and GaAs.

63

Starting from this simple cluster model, we studied how the local structure affects
the NN peak shape and far neighbor peak widths in the PDF. The nearest neighbor
PDF peak can be calculated from a linear combination of the different clusters present.
This is shown in Figure 4.8(a). In order to calculate far neighbor peaks in the PDF a
larger model must be constructed. A simple approach is to extend the clusters using
periodic boundary conditions and take a linear combination of these ordered cluster
models. However, this introduces atom—pair correlations which are not there in the
real alloy [87]. In order to avoid this problem, we create a random alloy structure
using program DISCUS [88]. First a 20 x 20 x 20 cubic cell of InAs is generated. Then,
the In atoms were randomly replaced by Ga atoms according to the concentration,
1:. The average lattice parameter of the model was determined using Vegard’s law
for the value of 1' under study. Each resulting cluster was then identified by type
and the As atom was displaced in the appropriate direction by the amount obtained
from calculation carried on the isolated cluster of that size. Now we have static local
displacements in the model structure. However, in order to calculate a model PDF,
we need to take into account one more factor, i.e. thermal atomic motion. As was
discussed in Chapter 3, thermal atomic motion in crystals is correlated [31, 50], e.g.
near-neighbor atoms tend to move in-phase with each other but far-neighbor atoms
move independently. Due to this correlated atomic motion, near-neighbor peaks are
sharper than the far-neighbor peaks in the PDF. The effect of correlated motion is
taken into account in our calculation by describing the r dependence of the PDF peak
width by 00 2 ac — (5/7‘2 (Eq. 3.5). The uncorrelated thermal factors and sharpening
parameter are refined using the program PDF FIT [64] in the pure end-members. To
take into account the thermal broadening in the alloy, the thermal factors of In and
Ga found in the pure end—members were used. For the As atoms, however, the average

of two values found in the pure end-member was used [89]. The thermal factors used

64

 

 

(3(r)

 

 

15 r (C)

. a
5 n "'
0 _

0

 

 

 

1O 15

01%

NA)

Figure 4.6: Experimental PDF (0—0) and cluster model PDF (—) of Ino,5Gao,5As. (a)
Cluster model PDF is calculated using model structure which has chemical disorder
(random distribution of In/ Ga atoms) but no static disorder on both mixed (In,Ga)
and As sublattices. (b) Static disorder on the As site is created using cluster models
(see text) but no disorder on the (In,Ga) sublattice. (c) Disorder on the (In,Ga) sub-
lattice is simulated by randomly displacing (In,Ga) atoms from their ideal sublattice
(see text).

in this model are U 1,, = 0.0445 A, U06 2 0.0417 A, and U ,4, = 0.0558 A, respectively.
For the r-dependent sharpening of the PDF peaks, the parameter 6 = 0.17 A3 is used.
With these information from end-members, the PDFs of the alloys can be calculated

using the cluster model described above with no adjustable parameters.

In order to understand the contribution of the local distortion to the PDF peak
widths, we first compared experimental PDFS with model PDFs calculated using the

average structure (no static distortions on either sub-lattice) of the Ino,5Gao_5As a1-

65

 

5 ~ .. As-As ~
— ln-In

 

 

 

 

 

 

 

 

Figure 4.7: Partial PDF of In and As calculated using the Cluster model.

loy. The comparison is shown in Figure 4.6(a). We notice that all peaks in the model
PDF are too sharp compared with the experimental PDF. Now we added the static
disorder on arsenic sublattice using the method described in the cluster model. F ig-
ure 4.6(b) shows the comparison between the cluster model PDF which has static
distortion on the As sublattice and the experimental PDF. We can see that the near-
est neighbor doublet agrees reasonably well with the experiment. However, still some
peaks in the model-PDF at higher-r are much sharper than those of the experimental
PDF. In order to understand the origin of this difference, we calculated the As and
In partial PDFs from the model. Figure 4.7 shows partial PDFs [90] of As and In
in which only As-As and In-In pairs are shown. In this Figure, it is clear that the
(As-As) correlations are much broader than those of (In-In) correlations. This sug-
gests that the sharp peaks in the model PDF evident in Figure 4.6(b) are coming
mostly from (In,Ga)-(In,Ga) correlations. The mismatch between the model and the
experimental PDFs mainly comes from neglecting mixed (In,Ga) sublattice disorder
in this model. To test this hypothesis we mimic the neglected static disorder on the

mixed (In,Ga) sublattice. All In and Ga atoms in the mixed sublattice have four As

66

nearest neighbors. Since the (In,Ga) sublattice distortion is caused mainly by the
second-neighbor distribution, we expect for the random alloys the second-neighbor
distribution of (In,Ga) atoms is also random. From this, we assume that the static
displacements of (In,Ga) atoms can be approximated as an isotropic distribution.
Based on these arguments, we mimic the static disorder on the mixed (In,Ga) sub-
lattice by randomly displacing (In,Ga) atoms from their ideal sublattice so that the
overall static deviations have a Gaussian distribution. Figure 4.6(c) shows the com—
parison between the model PDF which has both As and (In,Ga) site disorder and the
experimental PDF. Now the higher-r peaks in the model PDF become much broader

and match the experimental PDF better.

We now focus on modeling the first peak of PDF. First of all, as we noted in the
previous section, the additional peak broadening due to the bond length distribution
is very small in the NN peaks. This result suggests that the peaks from different
cluster types should line up with small dispersion. This is illustrated in Figure 4.8(b)
which shows the NN PDF-peak for the a: = 0.5 data—set. The solid line is calculated
in the Pauling limit [23] where all nearest-neighbor bond-lengths are preserved at
their values in the pure end-members. The broadening comes purely from thermal
motion, again taken from the end-members. Clearly this reproduces the data quite
well, though there is a small strain evident in the bonds decreasing the separation
of the peaks in the doublet. The dashed line in this figure shows the peak profile
that we obtain if we make the assumption that the nearest neighbor bond length
changes in the alloy as seen in the “Z-plot” [25, 41], but there is no increase in the
bond length distribution. Again, this gives rather good agreement emphasizing the
fact that there is little additional peak broadening due to the alloying [41]. However,
this effect is clearly exaggerated in the cluster model (Figure 4.8(a)). In the cluster

model, bond-lengths vary significantly depending on the cluster type. These are

67

 

l ' I ' l

6 r (a) Cluster model 000

 

          

I

 

"'2 ‘r l l i
6 _ (c) Kirkwood model

 

 

   

 

2.5 A 2.7 A 2.9
r(A)

2.1 ‘ 2.3

Figure 4.8: Comparison of the first peak in the experimental PDF (open circles)
and model PDF (solid line) of Ino.5Gao_5As. (a) Tetrahedral cluster model with no
disorder present on (In,Ga) sublattice. The sub-peaks represent the contributions
from each type of cluster. Type I (x), type II ([1), type III (0), type IV (A), and
type V(*). (b) The model PDF is calculated in the Pauling limit. The peak positions
were obtained from the InAs and GaAs bond lengths in the end-members (solid line)
and the InAs and GaAs bond lengths in the In0,5Gao,5As PDF (dashed line). See the
text for details. (c) Kirkwood supercell model.

shown under the peak with their respective weights for the :1: = 0.5 sample. The
symbol represents each type of cluster, type I(x), type 11(8), type III(<>), type IV (A),
and type V(*) respectively. The total model PDF is not resolved although it has a
trace of double peaks. The cluster model successfully explains the sloping lines of the

nearest neighbor bonds in the Z-plot [27], as exemplified by the agreement it gets with

68

 

F "_—f—‘——Ti_ ' T” ‘f’ I fir—‘ I 7 7’7

0 Experimental PDF 0
6 ”- 17-alom cluster model PDF

 
   
 
   

2.60 r

G(r)

l l l l l I l l l l l

2.50

Bond Length (A)

l l l l l l l 1

 

 

 

 

 

 

 

0.0 0.2 0.4 0.6 0.8 1.0 2.1 A 2.3 A 2.5 A 2] A 29
X r(A)

Figure 4.9: Left panel: Z-plot obtained using 17-atom cluster model [91]. Right
panel: Comparison between experimental PDF and 17-atom cluster model PDF for
In0.5Gao.5AS-

XAF S, but clearly does a poor job at explaining the nearest neighbor bond length
distribution measured in the PDF. The major discrepancy is that too much intensity
resides at, or close to, the undisplaced position leading to an unresolved broad first
PDF peak in sharp contrast to the measurement. The disagreement is mainly due to
the limited size of the clusters and is somewhat expected. The cluster models using
larger clusters (17 atoms) improve the agreement with the experimental bond length

distribution [78, 91] as is shown in Figure 4.9.

69

 

Figure 4.10: A sketch of the tetrahedron surrounding an impurity atom in the zinc—
blende structure.

4.3.2 Supercell model based on the Kirkwood potential

A. Kirkwood model

A more realistic model for the structure of these alloys [41, 92] is obtained from a

supercell relaxed using a Kirkwood potential given in Eq. 1.1 [67],

5w:

 

1
(0030,17c + —)2.

a,- a
VZZ?J(LiJA-Lij)2+L§ Z 3

<ij> <ij,ik>
The potential contains nearest neighbor bond stretching (an) and bond bending (fly-,6)
force constants. In the relaxed supercell model, the force constants were adjusted to
fit the PDFs of the end-members [26] with aGa_As = 96 N/m, 011,,_A5 = 97 N/m,
flea—As—Ga = fiAs—Ga—As = 10 N/m and flIn—As—In = BAs—In—As = 6 N/m. The addi-
tional angular force constants required in the alloy are taken to be the geometrical
mean, so that ,BGa—As—In = m. The PDFs for the alloys could
then be calculated in a self-consistent way for all the alloys with no adjustable param-
eters [92]. Figure 4.11 shows the relaxed alloy structure for the Ino,5Gao,5As alloy.
The deviations of As atoms from their average positions are evident. In this model,

the lattice dynamics are also included. Starting with the force constants and the

70

@As .Ga OIn

 

Figure 4.11: Relaxed structure of Ino.5Gao_5As alloy. The deviation of As atoms from
their average positions are evident [50]

Kirkwood potential, the thermal broadening of the PDF peaks at any temperature
can be determined directly from the dynamical matrix [41, 92]. The model-PDF is
plotted with the data in Figure 4.12. The excellent agreement with the data over the
entire alloy range suggests that the Kirkwood potential provides an adequate start-
ing point for calculating distorted alloy structures in these III-V alloys. Note that in

comparison with experiment, the theoretical PDF has been convoluted with a Sinc

71

 

15 v r .
10 InonGaoeaAs :

5 _A
0 a
15 Li l l' l

l
- In Ga As <
10 __ 033 067 \ #

    
    

G(r)
En‘

 

. J
' I
- In0 5Gao 5As

 

 

 

 

o < _
15 l i i l i l l '
10 '_ InomGaonAs _

5 E _

30 -l

0‘

_5 4. 1 1 1 1 b
0 2 4 6 8 1O

r(A)

Figure 4.12: Comparison between the experimental PDFs (open circles) and the Kirk—
wood model PDFS (solid lines) of InxGa1_xAs alloys. The model was the Kirkwood
supercell model. The parameters a and [3 are refined from the end—members and the
PDFs for the alloys shown here are then calculated with no adjustable parameters.

function to incorporate the truncation of the-experimental data at Q"m = 45 A.
B. 3-dimensional atomic probability distribution

Now, we analyze the relaxed alloy structures obtained using a Kirkwood potential to
get the average three dimensional atomic probability distribution of As and (In,Ga)
atoms. Figure 4.13 shows iso—probability surfaces for the As site in the InxGa1_xAs
alloy. The probability distributions were created by translating atomic positions of the

displaced arsenic atoms in the supercell (20 x 20 x 20 cubic cell) shown in Figure 4.11

72

      
   

 
 

~e: ..
.S

.5 .i, I

  

”1, j r J‘- ll|\
.5; A . 3":th
tr'--§‘.74l9£f
r ‘k‘ ' ‘

   

Figure 4.13: ISO—probability surface of the ensemble averaged As atom distribu-
tion. The surfaces plotted all enclose the volume where As atoms will be found
With 68 % probabllity. (a) In0.17Gao_83AS (b) Ino_33Ga0_67AS (C) 1110.50Ga050AS (d)
Ino_83Gao.17As. In each case, the probability distribution is viewed down the [001]
axis. Also shown is the cluster types II, III, and IV.

into a single unit cell. To improve statistics, this was done 70 times. The surfaces
shown enclose a volume where the As atom will be found with 68 % probability. The
probability distribution is viewed down the [001] axis. It is clear that the As atom
displacements, though highly symmetric, are far from being isotropic. The same
procedure has been carried out to elucidate the atomic probability distribution on

the (In,Ga) sublattice. The results are shown in Figure 4.14, plotted on the same

scale as in Figure 4.13. In contrast to the As atom static distribution, the (In,Ga)

73

(a) (b)

   

(c) (d)

 
  

x.-."-;-;~:.
#0..» \V
7 1' v v§ \ \
.' w... ‘ .¢‘
7 tame» a“
1A“ ..\"Av."“.'
s sin“. O , e .
==I It: a?
a = _, .~

«'0

     
   

!

            

 

——
qt—
‘—
_.

l J
1 y L
' X

Figure 4.14: ISO-probability surface of the ensemble averaged (In,Ga) atom distri-
bution. The surfaces plotted all enclose the volume where As atoms will be found
with 68 ‘70 probability. (a) Ino,17Gao_83As (b) Ino,33Gao,3-,As (c) Ino_5oGao,5oAs (d)
Ino,83Gao,1-,As. In each case, the probability distribution is viewed down the [001]
axis. These surfaces are plotted on the same scale as those in Figure 4.13.

probability distribution is much more isotropic and sharply peaked in space around

the virtual crystal lattice site.

In all compositions, the As atom distribution is highly anisotropic as evident in
Figure 4.13 with large displacements along (100) and (111) directions. This can be
understood easily within cluster model as we discussed in Section 4.3.1. The (100)
displacements occur in type III clusters and the (111) displacements occur in type
II and IV clusters. This also explains why, in the gallium rich alloy in which the
three and four Ga cluster is dominant, the major As atom displacements are along
[111], [111], [111], and [111] as we observed in Figure 4.13(a). On the contrary, in the

indium rich alloy, the major displacements are along [111], [111], [111], and [111], as

74

Table 4.1: Standard deviation of the As and (In,Ga) atom distributions in InxGa1_xAs
alloys obtained from the Kirkwood model. The numbers in parentheses are the esti-
mated error on the last digit. For both As, and (In,Ga) atoms, 0 = a,U 2 0y 2 02.
See text for details.

 

x=0.17 x=0.33 x=0.50 x=0.83
6(As)(A) 0.072(1) 0.092(1) 0.097(1) 0.074(1)
0(In,Ga)(A) 0.044(1) 0.058(1) 0.060(1) 0.048(1)

553%,“) 0.61 0.63 0.62 0.61

 

 

 

 

 

 

 

 

can be clearly seen in Figure 4.13(d).

The atomic probability distribution obtained from the Kirkwood model for the
(In,Ga) sublattice is shown in Figure 4.14. As we discussed, this is much more
isotropic (though not perfectly so), and more sharply peaked than the As atom distri-
bution. However, contrary to earlier predictions, [27] and borne out quantitatively by
the supercell modeling, there is significant static disorder associated with the (In, Ga)
sublattice. In order to compare the magnitude of the static distortion of the (In,Ga)
sublattice with that of the As sublattice, we calculated the standard deviation, 0,

of the As and (In,Ga) atomic probability distributions. This was calculated using

 

a,- = \/ F171 2le (d,(k))2, (i = :c, y, z), where d,- refers to the displacement from the
undistorted sublattice of atoms in the model supercell in 3:, y, and 2 directions, and
N is the total number of atoms in the supercell. Table 4.1 summarizes the values of
a for the As and (In,Ga) atomic probability distributions in the alloys. It shows that
for all compositions the static disorder on the (In,Ga) sublattice is around 60% of
the disorder on the As sublattice. These static distortions give rise to a broadening
of PDF peaks as is evident in Figure 4.2. To evaluate the static contribution to the

PDF peak broadening, JD, from the 0’s reported in Table 4.1 we used the following

75

expression:

of) = a: + 03, (4.3)

where a, b can be As, or (In,Ga). For example, for a: = 0.5 alloy, we get a?) 2
0.0188(4) A2 for As-As peaks in the PDF, 0.0130(4) A2 for As-(In,Ga) peaks and
0.0072(3) A2 for (In,Ga)-(In,Ga) peaks. These values are in good agreement with
the mean square static PDF peak broadening of As—As, As-(In,Ga) and (In,Ga)-
(In,Ga) peaks, shown in Figure 4.4 of 0.0187(1) A2, 0.0128(1) A2, and 0.0053(1) A?

respectively.

4.4 Correlated atomic displacements in InxGa1_xAs
alloys

We have shown that on the average, atomic displacements of As atoms in InxGa1_xAs
alloy are highly directional. In this section, we would like to address the question
whether these atomic displacements are correlated from site to site. To investigate this
we have calculated theoretically the diffuse scattering intensity which would be ob-
tained from the relaxed Kirkwood supercell model and compare it with the known ex-
perimental diffuse scattering. The Figure 4.15 shows diffuse scattering of In0_5Gao_5As
alloy calculated using the DISCUS program [88]. In this calculation the Bragg-peak
intensities have been removed. Strong diffuse scattering is evident at the Bragg points
in the characteristic butterfly shape pointing towards the origin of reciprocal space.
This is the Huang scattering which is peaked close to Bragg-peak positions and has
already been worked out in detail [93]. In addition to this, clear streaks are appar-
ent running perpendicular to the [110] direction. The diffuse scattering calculations
on (hkl) planes where l 75 0,integer (Figure 4.16), show that these diffuse streaks

are extended along the [hhl] direction consisting of sheets of diffuse scattering per-

76

 

Figure 4.15: Single crystal diffuse scattering intensity obtained from the relaxed super-
cell model for the Ino.5Gao,5As alloy. The cut shown is the diffuse intensity expected
in the (hkO) plane of reciprocal space. Bragg peaks have been removed for clarity.
See text for details.

pendicular to the [110] direction of reciprocal space. Diffuse scattering with exactly
this (110) symmetry was observed in the TEM study of Ino.53Gao‘4-,As [94]. Careful
observation of our calculated diffuse scattering indicates that the diffuse scattering
has a maximum on the low-Q side of the (hh0) planes passing through the Bragg
points, with an intensity minimum on the high-Q side of these planes. This is charac-
teristic size-effect scattering obtained from correlated atomic displacements due to a
mismatch between chemically distinct species as recently observed in a single—crystal
diffuse scattering study on Si1_,¢Gex [95], for example. This asymmetric scattering

was clearly observed in the earlier diffuse scattering study on Ino_53Gao_47As [94].

The single-crystal diffuse scattering intensity which is piled up far from the Bragg-

77

Diffuse scattering Inc'sGao'5As : l= 0.5

 

8
h[r.l.u]

Figure 4.16: Single crystal diffuse scattering intensity obtained from the relaxed su-
percell model of the Ino.5Gao,5As alloy. The cut shown is the diffuse intensity expected
in the (hk0.5) plane of reciprocal space. Bragg peaks have been removed for clarity.
See text for details.

points is giving information about intermediate range ordering of the atomic displace-
ments. It is interesting that it is piled up in planes perpendicular to [110] whereas
the local atomic displacements are predominantly along (100) and (111) directions.
This observation underscores the complementarity of single—crystal diffuse scatter-
ing and real-space measurements such as the PDF. The real-space measurements are
mostly sensitive to the direction and magnitude of local atomic displacements and
less sensitive to how the displacements are correlated over longer-range (though this
information is in the data). On the other hand, the single crystal diffuse scattering
immediately yields the intermediate range correlations of the displacements but one

has to work harder to extract information about the size and nature of the local

78

atomic displacements. Used together these two approaches, together with XAF S,
can reveal a great deal of complementary information about the local structure of

disordered materials.

The single crystal diffuse scattering suggests that atomic displacements are most
strongly correlated (i.e., correlated over the longest range) along [110] directions al-
though the displacements themselves occur along (100) and (111) directions. The
reason may be that the zinc-blende crystal is stiffest along [110] directions because of
the elastic anisotropy in the cubic crystal. This was shown for the case of InAs and
was used to explain why the 5th peak in the PDF (coming from In-As next neighbor
correlations along [110] direction) was anomalously sharp in both experiments and
calculations [31]. If the material is stiffer in this direction, one would expect that
strain fields from displacements will propagate further in these directions than other
directions in the crystal correlating the displacements over longer range. These are
consistent with the displacement pair correlation function calculation by Glas [96]
which shows that the correlation along (110) directions is larger than correlations

along (100) and (111) and extends further.

79

Chapter 5

Discussion and Conclusions

The high real-space resolution PDF measurements of InxGa1_xAs alloys provide more
complete structural information such as bond length, bond length distributions, and
far-neighbor distances and distributions. The bond lengths from the PDF showed
good agreement with those of the XAFS measurements. The high real-space reso-
lution PDF measurement made possible the separation of thermal and static bond
length distributions in the alloys. The static bond length distributions in the alloys
are less than 15 % of thermal broadening at 10 K. This result shows that the accurate
measurements of static distribution becomes more difficult at higher temperatures. In
addition, the composition dependence of NN PDF peak width shows that the In-As
bond has a larger distribution than the Ga-As bond. The Keating [76] and Kirkwood
model [30] also show two different bond length distributions for In-As and Ga-As
bonds in the alloys. Similar results were obtained in InxGa1_xP alloys using Keating
model that Ga—P bond length distributions are larger than those of In-P [89]. In
addition to the NN information, the PDF gives far-neighbor pair distributions which
help to understand a global picture of semiconductor alloys. The far-neighbor dis-
tributions show that static distribution becomes dominant in peak width broadening

although it is less than 15% of thermal broadening for NN.

80

The model structure of III-V InxGa1_xAs alloys was obtained from a supercell
relaxed using the Kirkwood potential. The resultant Kirkwood model PDFs show
good agreement with experimental PDFs in predicting the bond length, bond length
distribution, and far—neighbor pair distributions without adjustable parameters. Re-
cently, it is also shown that the Kirkwood model is successful for more polar II-VI
ZnSe1_xTex alloys [97]. The 3—dimensional As atom iso—probability surface obtained
from the relaxed supercell shows that the As atom displacements are very directional
and symmetric. On the contrary, the (In,Ga) atom displacements are more or less
isotropic. The directional properties of As atom displacements are well understood
with the help of a simple cluster model. The cluster model intuitively explains the

As atom displacements for various nearest neighbor configurations.

In the Kirkwood model, we assumed that (In,Ga) atom distributions are random
according to the concentration, :12. However, in reality the concentration fluctuation
leads to the cluster formation [81]. The Monte-Carlo simulation based on the Keating
potential shows a tendency (7~10 %) of repulsion between first NN Indium pairs and
attraction between second NN pairs [83] in the bulk InxGa1_xAs alloys (58:0.05, 0.2).
Recent simulations on CU3AU show that the PDF can differentiate differences in the
occupational disorder (clustering) in the same sample at different temperatures [98].
In real experiments, it will be more demanding to detect any difference coming from
the clustering effect due to the noises and error in the data. However, these results
open a possibility of using the PDF analysis to extract chemical short range order

information in crystalline materials.

In addition to the atomic displacements and distributions in the alloys, the sin-
gle crystal diffuse scattering calculation of the relaxed supercell reveals a correlation

in static displacements of atoms. The diffuse scattering shows that the atomic dis-

81

placements are correlated over longer range along [110] directions although the dis-
placements of As atoms are along (100) and (111) directions. This result shows the
complementarity of single crystal diffuse scattering and real-space measurements such
as the PDF; the real—space measurements are mostly sensitive to the direction and
magnitude of local atomic displacements but the single crystal diffuse scattering yields
the intermediate range correlations of the displacements. This suggests that if used
together these two approaches can reveal a great deal of complementary information

about the local structure of disordered materials.

Besides local static distortions, the PDF measurements can probe the relative
thermal motions of atoms in semiconductor compounds. In the PDF the near-
neighbor peaks are sharper than those of far-neighbor pairs due to the correlation
in near-neighbor thermal motions. The comparison of correlated atomic motion in Ni
and InAs shows that the details of the correlation in atomic motion depends on the
strength of interatomic interaction and configuration of atom pair. The correlated
thermal motions in simple materials are well explained using the Correlated Debye
model without any adjustable parameters. The Correlated Debye model is simple but

picks up most features of the r),- dependence of the PDF peak width.

The relative motions of nearest neighbor pair atoms are mainly determined by
the bond stretching force constant of a material. Therefore from the NN PDF peak
width, it is possible to estimate the bond stretching force constant. The temperature
dependence of NN peak width, (INN, can be parameterized [99, 100] by the Einstein
frequency, tag as 012v N(T) = EYE—E coth(hwE/2kT), where a is a reduced mass and (12;;
is the Einstein frequency. The Einstein frequencies and corresponding force constants
of several As-crystalline and glass compounds determined from the temperature of

NN peak width are in reasonable agreement with the bond stretching force constants

82

measured from Raman and infrared spectra of same materials [101]. This result shows

the possibility of measuring force constants from PDF the peak width.

We determined bond stretching and bond bending force constants of semiconduc-
tor compounds by fitting the nearest neighbor and far-neighbor peak widths to the
lattice dynamic calculations using the Kirkwood model. The refined bond stretch-
ing and bond bending force constants show 10%~30% difference from the litera-
ture values [12]. However, since the literature values also containes errors around
10%~30% [26], it is not easy to estimate the accuracy of force constants refined us-
ing the PDF. In addition, we used only two data points, nearest and far—neighbor
peak widths, to refine two force constants in our refinement. Therefore this could
lead relatively large errors in the refined force constants. Especially, bond bending
force constant which is mainly determined by the far-neighbor peak width show rela—
tively large variation. Recently, force constants in Ni and CaF2 were determined from
the PDFs of these materials [102]. In this method, one-phonon diffuse scattering is
calculated by varying the force constant until the resultant PDF best matches the
experimental PDF for the whole r-range. The phonon dispersion curves calculated
using these refined force constants show good agreement with the phonon dispersion
curves measured by the inelastic neutron scattering. However, in this study only the
one-phonon contribution to the PDF peak width sharpening is counted. As Thorpe
et al. [103] pointed out, however, the one-phonon contribution recovers only around
90% of the peak sharpening effect and at least two-phonon contributions should be
included to recover 95% of the peak sharpening of the NN PDF peak. In addition, the
accurate correction for the effects of an instrument resolution function to the PDF
peak width broadening [102, 103] should be applied. More fundamentally, in the PDF
the diffuse scattering which contains information about interatomic force constants

are averaged over all directions. This raises questions about how accurately the force

83

constants can be determined using the PDF method. We expect that more system-
atic studies should be conducted both theoretically and experimentally to understand
the limitations and potentials of PDF method in determining the interatomic force

constants.

Finally, we like to mention that most high real-space resolution PDF measure—
ments [41, 97] of semiconductor alloys are carried out for bulk, materials. However,
many semiconductor alloys used in real applications are in the form of thin film.
In addition, a recent XAF S experiment [91] on strained InxGa1_xAs thin-alloy film
grown on GaAs(001) shows very interesting results. The individual Ga—As and In-As
bond lengths in the thin-alloy film are contracted from their bulk-alloy values due
to the strain imposed on the layer by the substrate. As in the bulk material, how-
ever, XAFS gives only nearest neighbor information about the thin-alloy film. This
limits our understanding of the external strain effects on the overall alloy structure.
An important future development would be the ability to study the high real-space

resolution PDFs of the thin film.

84

APPENDIX

User’s Manual

PDFgetX

Version 1.1

 

Ilkyoung Jeong
Jeroen Thompson
Thomas Proffen
Simon Billinge

Department for Physics and Astronomy
Michigan State University
East Lansing, MI, 48824-1116, USA

Contact: billinge@pa.msu.edu

86

Preface

Disclaimer

By downloading the program PDFgetX, you agree to the terms and conditions con-
cerning its use specified in the license agreement that is provided as part of the
distribution. End users wishing to make commercial use of the software must con-
tact Libraries, Computing & Technology, Michigan State University, East Lansing,
MI 48824; (517)353-0722 prior to any commercial distribution to discuss terms. The
Software is provided to End User by MSU on an as is basis. No user support is
provided or implied.

MSU makes no warranty, express or implied to end user or to any other person
or entity. Specifically, MSU makes no warranty of merchantability or fitness for a
particular purpose of the software. MSU will not be liable for special, incidental,
consequential, indirect or other similar damages, even if MSU or its employees have
been advised of the possibility of such damages, regardless of the form of the claim.

Using PDFgetX

Publications of results totally or partially obtained using the program PDFgetX
should state that PDFgetX was used and contain the following reference:

JEONG, I.—K., THOMPSON, J.,PROFFEN, TH., PEREZ, A. AND BILLINGE,
S. J. L. “PDFgetX, a program for obtaining the atomic Pair Distribu-
tion Function from X-ray powder diffraction data” J. Appl. Cryst. (2001),
submitted.

Acknowledgments

The PDFgetX is coded using Yorick language [58]. The atomic scattering factors
are calculated using the analytic formula and coefficients developed by D. Waasmaier
and A. Kirfel [54]. The mass attenuation coefficient data of elements are obtained
from the web at: http: / / physics.nist.gov/ PhysRefData/ FF ast / html/ form.html [104].

Financial support from the National Science Foundation through the grants DMR—
9700966, DMR—0075149, CHE-9633798 and CHE-9903706 as well as the Center for
Fundamental Materials Research (CFMR) is gratefully acknowledged.

87

A. 1 Introduction

A.1.1 What is PDFgetX

PDFgetX is a program to be used to obtain the atomic Pair Distribution Func-
tion (PDF) from a measured X-ray powder diffraction data. PDFgetX is written
using the Yorick, an interpreted language. This will require users to obtain the
Yorick distribution and install it yourself. See Chapter A2 for help in installation.

PDF is the instantaneous atomic number density-density correlation function
which describes the atomic arrangement in materials. A useful characteristic of PDF
method is that it gives both local and average structure information because both
Bragg peaks and diffuse scattering are used in the analysis. And from the PDF
peak width, it’s possible to obtain the information about bond-length distribution
(static, thermal) [41] and correlated atomic thermal motion [31]. By contrast, an
analysis of the Bragg scattered intensities alone, by a Rietveld type analysis for in-
stance, yields the average crystal structure only and the extended x-ray absorption
fine structure(EXAF S) gives nearest-neighbor and next nearest-neighbor distance in-
formation. PDF analysis method has long been used to characterize glasses, liquids
and amorphous materials. Recently, however, it has found more application in the
study of local structural disorder in crystalline materials, where some deviation from
the average structure is expected to take place.

Obtaining total scattering structure function (and PDF) from raw diffraction data
requires many corrections for experimental effects such as absorption, polarization
corrections and removing of Compton and multiple scattering contribution to the
elastic scattering. Also it needs proper error prOpagation to be used in modeling
of PDF using either PDFFIT (real-space Rietveld) [64] or a Reverse Monte Carlo
approach [105] using e.g. DISCUS [88] to yield structural parameters. PDFgetX
allows users to do all these data corrections and error propagation in convenient ways.
During the refinement, PDFgetX displays each correction effect to the raw data and
saves all the parameters used for refinement. This makes the refinement processes
easy to understand and allows reproducible results. PDFgetX supports the following
data formats: multi-column ascii file, SPEC and multi-channel analyzer(MCA) files.

To find out about recent updates of PDFgetX or to get further information visit
the PDFgetX homepage at the following site:

http: / / www.pa.msu.edu / cmp / billinge-group / programs / PDFgetX

88

A.2 Installation

A.2.1 System Requirements

PDFgetX should run on any UNIX/ Linux platform supported by Yorick. This in-
cludes PC / Unix and SGI. It also run on Windows NT and 95. For a list of systems on
which PDFgetX is known to work, see Table A.1. If you successfully install PDFgetX
on a system not included in this list, please contact us and let us know. If you cannot
install PDFgetX on a system, and have studied the documentation thoroughly, please
contact us and ask for help. Without access to a similarly configured system, we may
not be able to help you with the installation, but see Section A.2.2.4 for instructions
on how to report your trouble.

Table A.1: Known Platforms Supporting PDFgetX

 

[ Hardware [ Operating System]

Intel 486 RedHat Linux 6.0
Windows 95 / NT
DEC-ALPHA Digital Unix

SGI Irix

 

 

 

 

 

 

 

 

 

A.2.2 What You Need

A.2.2.1 Yorick

Before you can run PDFgetX, you will need to install Yorick. PDFgetX is written
in the Yorick language, which is an interpreted C-like language (and it’s free). The
distribution of PDFgetX contains only the source code files for PDFgetX; it does not
come with Yorick. The latest version of Yorick can be downloaded from the official
site:

ftp:/ / wuarchive.wustl.edu / languages / yorick / yorick-ad.html
This document provides no information about installing Yorick; see the Yorick

readme files for help with the installation and checking that the installation was
successful. Before installing PDFgetX, be sure that your installation of Yorick works.

89

A.2.2.2 PDFgetX

You may obtain the latest version of PDFgetX from the PDFgetX website:
http: / / www.pa.msu.edu / cmp / billinge—group / programs / PDFgetX

PDFgetX is provided as a compressed file. Use the command

tar -xzvf pdfgetx1.1.tar.gz

which will extract the files into a new directory called “PDF etX/”. And you can
find the following program files under the directory PDFgetX .

pdfgetx.i, pdfgetxdistribution.i, pdfgetx_custom.i
ASF.DAT, PERIODIC_TABLE.DAT, MASS_ABS_COEFF.DAT, LICENSE.TXT

If the -z flag does not work on your system, then use the commands

gzip -d pdfgetx1.1.tar.gz, tar -xvf pdfgetx1.1.tar

to extract the files.

A.2.2.3 Installing and Configuring PDFgetX

To customize the PDFgetX installation, you need to modify two files, “custom.i”
and “pdfgetxdistribution.i”. If you are new user of “Yorick”, you can simply rename
“pdfgetx-custom.i” (included in compressed file) to “customi” and give the proper
path for the “ActuaLPAT H” in the following two lines in “pdfgetx.custom.i” file.

#include "Actua1_PATH/pdfgetx.i"
#include "Actual_PATH/pdfgetxdistribution.i"

And then create a directory “/Yorick” under your home directory and place your
own version of “customi” there. If you already have your own version of “custom.i”,
simply add the above two lines in the “custom.i” file.

For Windows OS, you need to beware of a few things. First, the path should
look like the following: “/c/pdfgetx/..”. Second, in windows OS, users can set the
size of font and graphic window in “customi” file. Therefore it is better to copy the
“custom.i” file coming with the Yorick Window version and add the above two lines
there than just rename “pdfgetx_custom.i” to “customi”. Finally, remember that if
the directory name has a space (“ ”) as in “My Directory”, the Yorick couldn’t find
the directory.

After configuring the “pdfgetx_custom.i” file, if you open the file “pdfgetxdistri-
bution.i” using a text editor, you will find the following code:

asf_file="Actual_PATH/ASF.DAT"

periodic_table="Actual-PATH/PERIODIC_TABLE.DAT"
mabscoeff_file ="Actual_PATH/MASS_ABS_COEFF.DAT"

90

again, give the pr0per path for the “ActuaLPATH”. These code set paths for three
important data files: Atomic scattering factor, Periodic table, and Mass absorption

coefficient. Also, the variable name, e.g. asf_file, should not be changed, otherwise
PDFgetX couldn’t find these data files.

If you want to print graphs directly from PDFgetX, you need to edit the file
“pdfgetxdistribution.i” and change the following line:

printerstring=“lpr -h -Prm31 __temp.ps”.

Modify the printer string to reflect your system. When printing, PDFgetX creates
a temporary postscript file called “..temp.ps” and makes a system call to print the
file. Do not change the name of the file in the printer string or printing will not
work. For windows OS, “lpr” command doesn’t work, instead use “print” command.
This is a DOS command and it seems it sends the graph to a printer connected via
“LPTl” port. If this setting is not working, we would recommend windows OS users
to use another method to print graphs. First, save the graph as a postscript (PS) file
or window meta file (WMF) using [S] option in the main menu, and open it using
ghostview (PS file) or using MS word (WMF file). Then print the graph using print
command in the program.

A.2.2.4 Report problems and suggestions

If you have any problems in installing & running PDFgetX and have any suggestions
about the PDFgetX, please send email to the following address:

billinge@pa. msu. edu
http: / / www.pa.msu.edu / cmp / billinge-group / programs / PDFgetX.html

91

A.3 Tutorial: In0.33Ga0.67As Semi-
conductor Alloy

Now you might have installed PDFgetX and can start it simply by typing pdfgetx
at Yorick prompt. In this tutorial, you’ll get a chance to analyze Ino.33Gao,67As semi-
conductor alloy data collected at Cornell High Energy Synchrotron source (CHESS)
using intense x—rays of 60 KeV (A = 0.206 A). The tutorial files can be downloaded
from the PDFgetX homepage. In this experiment, the incident x-ray energy was
selected using a Si(111) double-bounce monochromator. The data were collected at
10 K to minimize thermal atomic motion in the sample, and hence increase the sen-
sitivity to static displacement of atoms due to alloying using a closed cycle helium
refrigerator mounted on the Huber 6 circle diffractormeter. All the signal measured
was saved to a file using the system controlling software, SPEC.

The signal measured using the intrinsic Ge solid state detector was processed in
two ways. Using single-channel pulse-height analyzer (SCA), the elastic scattering,
Compton scattering, and elastic + Compton scattering were collected separately. In
the measurements using SCA, the proper energy window setting for the elastic scatter-
ing is very important because any error in the window setting could cause an unknown
contamination to the elastic scattering thus make data corrections very difficult. At
the same time, the signal was fed to multi-channel analyzer (MCA) to record the
complete energy spectrum of each value of Q. The elastic and Compton scattered
radiation could then be separated using software after measurement. Collecting data
using MCA has advantages and disadvantages. In the MCA method, since the en—
tire energy spectrum of the scattered radiation is measured at each value of Q, the
error caused by the mis-set of energy window is negligible. The main disadvantage
of the MCA method is that it has a larger dead-time, although this can be reliably
corrected [106].

This tutorial is composed of two subsections, “Preliminary Data Analysis” and
“Refine structure function”. The “Preliminary Data Analysis” section is mainly con-
centrated on how to reduce SPEC and MCA file to build PDFgetX input file for
structure refinement. In “Refine structure function” section, the step by step proce-
dure of structure function refinement is presented. Users can build the input file using
tutorial SPEC file in “Preliminary Data Analysis” section or use a tutorial input file
coming with the program. In this manual, SPEC file refers to the data collected using
SCA and M CA file for the data collected using MCA.

92

A.3.1 Preliminary Data Analysis

A.3.1.1 Reduction of SPEC file

The raw data from x-ray powder diffraction measurements using either a sealed x-ray
tube or synchrotron source could have many different file formats and could contain
multiple scans that ought to be averaged together. Therefore it is very difficult to
use the raw data directly in the structure function refinement. With these things
in mind, we limited the “Preliminary Data Analysis” to support only the SPEC file
format and N-column ascii file format. For details about SPEC file format, please
refer to Appendix App.a This section will show you how to reduce the raw SPEC
data into the input file from which to start analysis. This process includes extracting
scans from SPEC file, comparison of different scans, applying dead-time correction
and combining different scans.

In general, one SPEC file contains many scans. The following shows a scan header
of SPEC file collected at CHESS.

#L me ereal elive Epoch Seconds 1C1 1C3 I_CESR PULSER TOTAL COMPTON IC2 ELASTIC

During the SPEC file reduction process, we will use column number to refer to a
specific variable such as ELASTIC, IC2, PULSER, etc , so you need to remember
which column corresponds to which variable. Follow along with this example terminal
output. It will guide users to learn about how the “Reduction of SPEC file” works.
The comments in /* */ mark are added just for explanation purpose and will not
be shown in the real analysis.

current directory) yorick

Copyright (c) 1996. The Regents of the University of California.
All rights reserved. Yorick 1.4 ready. For help type ’help’

> pdfgetx
Pair Distribution Function from the X-ray powder diffraction (PDFgetX 1.1)

0) Preliminary data reduction
1) Build a setup file
2) Background Substraction
3) Reduction of Structure Function: 8(0)
Input file format: (0, I, d0, d1)
4) PDF calculation: G(r)
Input file format: (Q, S(Q), dQ, dS)
P) Print, S) Save, U) Unzoom, L) Limits windows

Q) Quit
[0-4 qsulp] O /* Enter to the Preliminary data reduction level */
1) Extract Scan(s) from SPEC file

2) Compare N-column(N>=2) ascii files

93

3) Combine N-column(N>=2) ascii files

4) Build PDFgetX input format: (0, I, dQ, dI)
5) Convert MCA file to N-column ascii file

0) Return to Main

[1'50] 1
The Input should be SPEC file format : Continue (y/n)? y

ENTER SPEC FILE NAME TO READ: in33_tutorial.spec
- The following shows scan information in in33_tutorial.spec

#S 1 601 pts --> ascan me 1 13 600 1
#S 2 601 pts --> ascan me 1 13 600 1
#S 3 1401 pts --> ascan me 12 40 1400 1
#S 4 1401 pts --> ascan me 12 40 1400 1
#S 5 1401 pts --> ascan me 12 40 1400 1

EXTRACT SCANS FROM SPEC FILE:
- Each scan will be saved as an ascii file
- Enter all scans to be read: [Ex: 2 4 5] 1 2 3 4 5
/* Extract good scans by entering the scan number */

SAVE SCANS TO ASCII FILE:
- Output file name will be ’samplename_scannumber.asc’

- Enter your ’samplename’ [Ex. InAs] : in33

Return to Preliminary data reduction

Now each scan is saved as ascii file. As it is mentioned during extracting process, the
file name will be “in33-1.asc”, “in33_2.asc”, and so on. Now we can compare these
different scans.

[1-50] 2 /* Compare N-column (N>=2) ascii files */

The Input should be N-column ascii file format :
Plot y(=data/norm) vs. x(=q). Continue (y/n)? y

ENTER FILE NAMES TO COMPARE, TO QUIT READING, ENTER ’0’:
- File name to compare : in33_1.asc

#L me ereal elive Epoch Seconds ICl IC3 LCESR PULSER TOTAL COMPTON IC2 ELASTIC

ASSIGN COLUMN NUMBER TO VARIABLES:

- Column # corresponding to X-axis : 1
- Column # corresponding to DATA : 13

- Normalization of data :
for constant normalization, enter ’0’

94

- Column # corresponding to NORM. : 12

ENTER FILE NAMES T0 COMPARE, TO QUIT READING, ENTER ’0’:
- File name to compare : in33_2.asc

ENTER FILE NAMES TO COMPARE, TO QUIT READING, ENTER ’Q’:
- File name to compare : in33_5.asc

ENTER FILE NAMES TO COMPARE, T0 QUIT READING, ENTER ’Q’:
- File name to compare : q

/* Dead-time correction setting */
READ FILE : in33_1.asc
APPLY DEAD-TIME CORRECTION FOR DATA (y/n)? y
1) Dead-time correction using detector dead-time
2) Dead-time correction using pulser measurement
Q) Exit Dead-time correction
[1-201 2

Enter column # containing pulser : 9

APPLY DEAD-TIME CORRECTION FOR MONITOR (y/n)? n
/* For details, refer to CH. 4 Using PDFgetX */

CHECK VARIABLE AND ASSIGNED COLUMN #

- X-axis : 1
- DATA : 13
- Normalization : 12

Detector dead-time Correction using pulser
- Pulser column : 9

No Monitor dead-time Correction
CHANNEL SETTINGS ARE CORRECT (y/n)? y

READ file: in33_2.asc

READ file: in33_5.asc

Colors in order of reading: magenta, cyan, blue, green, red, and blacks

95

Comparison of scans after dead-time correction

 

(120 1'] .fl, . . .
[ 1 [1[ : —in$1jsmc
i y ‘ [ *inmlzsmc
; : nfiB_aa&:
, in33_4.asc
(115 ” i — mxxfismc ‘

 

 

 

 

 

 

 

 

 

 

 

 

DTC Elastic/Monitor
O
23

 

 

 

(105 ~4.

 

 

 

 

 

(100 ' . 1, . .
0 10 20 30 40
CKA”)

Figure A.1: Comparison of normalized elastic scattering in five scans after dead-time
correction. It shows that the elastic scattering in each scan overlap with each other
quite nicely.

COMPARE OTHER VARIABLE (y/n)? n
EXIT COMPARING FILES:

Return to Preliminary data reduction

Figure A.1 shows dead-time corrected elastic scattering after normalization by mon-
itor. It shows that these scans overlap with each other quite nicely. Maybe it’s good
to check how the comparison looks like if the dead-time correction is not applied.
Since the normalized elastic scattering from different scans overlap with each other,
we can combine all of them.

[1-5Q] 3 /* Combine N-column (N>2=2) ascii file */

The Input should have N-column ascii file format !
The X-axis column should have constant step(dX) !
Combine upto five variables: X-axis(Q, Two-theta)!
DATA, NORM, and Auxl, Aux2 (auxiliary variables) !
Continue (y/n)? y

96

ENTER FILE NAMES
- File name to

T0 COMBINE, TO QUIT READING, ENTER ’0’:
combine : in33_1.asc

#L me ereal elive Epoch Seconds lCl 1C3 I_CESR PULSER TOTAL COMPTON IC2 ELASTIC

ASSIGN COLUMN NUMBER TO VARIABLES:

- If the column # is set to ’0’, the corresponding variable
- will not be propagated !!

- Column # corresponding to X-axis : 1
- Column # corresponding to DATA : 13
- Column # corresponding to NORM. : 12
- Column # corresponding to Auxl : O
- Column # corresponding to Aux2 : O

ENTER FILE NAMES
- File name to

ENTER FILE NAMES
- File name to

ENTER FILE NAMES
- File name to

READ file : in33_

TO COMBINE, TO QUIT READING, ENTER ’0’:
combine : in33_2.asc

TO COMBINE, TO QUIT READING, ENTER ’Q’:
combine : in33_5.asc

T0 COMBINE, T0 QUIT READING, ENTER ’Q’:
combine : q

1.asc

Apply DEAD-TIME CORRECTION FOR DATA (y/n)? y

1) Dead-time

correction using detector dead-time

2) Dead-time correction using pulser measurement
Q) Exit Dead-time correction

[1-2Ql 2

Enter column # containing pulser : 9

APPLY DEAD-TIME CORRECTION FOR MONITOR (y/n)? n

CHECK VARIABLE AND ASSIGNED COLUMN #

- X-axis : 1
- DATA 13
- Normalization : 12
- AUXl : 0
- AUX2 : O

97

Detector dead-time Correction using pulser :
- Pulser column : 9
No Monitor dead-time Correction

CHANNEL SETTINGS ARE CORRECT (y/n)? y

READ file : in33_2.asc

READ file : in33_5.asc

Enter variable name corresponding to X-axis: Q
Enter variable name corresponding to DATA : Elastic
Enter variable name corresponding to NORM. : Monitor

SAVE COMBINED DATA:
- Enter file name for combined data: in33_tutorial.comb

Make sure the X-column and combined variables have a correct value !

Return to Preliminary data reduction

We just obtained combined N-column ascii file. The next step is to convert it to the
PDFgetX input format: (Q, I, dQ, dI). Here Q is the magnitude of scattering vector
and defined as Q = 4 77 sin(6)/A. I is the intensity of elastic scattering normalized by
the monitor, dQ and (11 are errors in Q and I. In this tutorial we’ll not preprocess the
background data. It’ll be given as 4—column (Q, I, dQ, dI) format.

[1-5Q] 4 /* Build PDFgetX input file */

The Input should be N-column(N>=2) ascii file
All columns should have same number of lines
Blanks and commas in columns are not permitted
Continue (y/n)? y

ENTER FILE NAME TO READ: in33_tutorial.comb
- Enter number of comment(and blank) lines in the data header : 2

CHECK FILE FORMAT:

1) Column contains Q/Two-theta?

- Data in Q or Two-theta value?
2) Column contains Intensity?
3) Column contains Monitor/Time? : O

- Normalization by Monitor or Time?: No Normalization
x) Exit.

MDH

[1—3x1 3

98

Column contains Monitor/Time? : 3
- Normalization by Monitor or Time(M/T)? m

1) Column contains Q/Two-theta? . 1

- Data in Q or Two-theta value? : Q
2) Column contains Intensity? . 2
3) Column contains Monitor/Time? : 3

- Normalization by Monitor or Time?: Monitor
x) Exit.
[1-3X] x

SAVE PDFgetX INPUT FILE (Q, I, dQ, dI):
- Enter file name to for input file: in33_tutorial.input

Return to Preliminary data reduction

Now, you obtained the input file for the structure function refinement. If the X-
column is 26, then the program converts it to Q.

A.3.1.2 Reduction of Multi-Channel Analyzer (MCA) data

The MCA file format could be different depending on the instruments. Therefore
here we assume an MCA file format which we used in the data reduction process.
The MCA file format we used is attached in Appendix A.c. In this file, the scattered
intensity at every Q point is distributed to the whole MCA channels. So each block
which is separated by blank corresponds to each Q point. In order to convert this
file to the normal N-column ascii file, we need to know the following information:
Minimum Q or two-theta, number of data points, Q / two-theta step (this should be
constant), total channel number of MCA. Many of these information can be obtained
using “scamsummary” function which is a part of PDFgetX. We will not provide
whole MCA file to do the analysis but just one MCA file to show you how it works.
However we can tell that the structure function obtained using SPEC and MCA data
are basically same.

[1-5Q] 5 /* Convert MCA file to N-column ascii file */
File should have MCA file format: Continue (y/n)? y
ENTER MCA FILE NAME TO READ: in33_tutorial.mca
- Build an X-axis column of corresponding MCA data
- The X-axis should be either ’Q’ or ’Two-theta’

- The X-axis should have constant step (dX)

- Enter variable name corresponding to X-axis, [Q/TT] : q

- Enter minimum Q value : 36
- Enter number of data points : 251
- Enter Q step : 0.02

99

Total channel number of MCA : 1024

READING MCA DATA, BE PATIENT!!

SET

UP THE INTEGRAING REGIONS 0F MCA SPECTRUM:
Propagate upto four variables:
Elastic, Elastic+Compton, Aux1 and Aux2
If the starting and ending channel number of a variable
is same, the corresponding variable will not be propagated

Start of elastic channel : 639
Stop of elastic channel : 657
Start of elastic_compton channel : 470
Stop of elastic_compton channel : 657

Start of MCA auxiliary channel 1
Stop of MCA auxiliary channel 1
Start of MCA auxiliary channel 2
Stop of MCA auxiliary channel 2

0000

read-write binary stream: _mcachanne1
In directory: /home/jeong/analysis/gainas/data/inSO/mcafile/
/* Open a binary file to save MCA channel setting */

CHECK SETTING OF MCA CHANNEL

1)
2)
3)
4)

0)

Start of elastic channel : 639
Stop of elastic channel : 657
Start of elastic_compton channel : 470
Stop of elastic_compton channel : 657
Start of MCA auxiliary channel 1 0
Stop of MCA auxiliary channel 1 : 0
Start of MCA auxiliary channel 2 0
Stop of MCA auxiliary channel 2 0

Exit MCA channel setting

Enter to reset channel setting [1-4Q] : q

INTEGRATE REGIONS OF INTEREST!!

SAVE MCA DATA TO ASCII FILE:

Enter file name for MCA-derived data: in33_mca.dat

READ ANOTHER MCA FILE(y/n)? n

EXIT MCA FILE READING:

Return to Preliminary data reduction

Figure

A.2 shows MCA spectrum of In0.33Gao,67As at Q=40A".

The elastic and

Compton scattering are well separated at this value of Q. It also shows some fluo—

rescence peaks in low channel number side. If you check the output file, you’ll find
that it contains Q, MCA_Elastic, MCA-ElaCompt, and MCA_Det_Tot. When the

100

MCA Spectrum of : InO ”Geo 67A5

 

 

 

 

 

 

 

 

500 . , 1
,
400 r 4
l
S 300 5 Elastic 1
g scattering
§‘
8
93 200 L «
E
Compton
[ scattering
O "Ylwfl HLMALIALAJ k\. .1 i. - A L i A #422ka 1
0 500 1000

MCA Channel number

Figure A.2: MCA spectrum of Ino,33Gao,67As at Q=40A‘1. It shows that the very
sharp elastic scattering around channel number 640 is well separated from the broad,

Compton scattering. The fluorescence of the alloy are shown below channel number
300.

program generates X-column, it assumes constant X-step. The whole MCA files can
be converted to N-column ascii files in this way and then as we did in “Reduction
of SPEC file” these files can be compared and combined. Also it’s possible to apply
dead-time correction for MCA file.

A.3.2 Refine structure function of In0,33Gao,67As

Now we are almost ready to refine structure function of In0,33Gao,67As semiconductor
alloy except two more things; building setup file and background subtraction.

[0-4 qsulpl 1 /* Enter to Build a setup file in Main */

BUILD SETUP FILE:

Note that the setup file is a text file written
using Yorick syntax, and may be modified in emacs without going
through this whole procedure of building a new one.

If PDFgetX crashes, check the (text) input file very closely!

101

ENTER THE TYPE OF INCIDENT X-RAY RADIATION:
- [S]i1ver, [M]olybdenum, [C]opper or enter [EJnergy [smce]? e
- Enter Energy of incident X-ray (KeV) : 59.67

SAMPLE INFORMATION:
- Number of elements in the sample[Ex. InGaAs => 3]? 3

- Element #: 1
- Enter the element (ions not yet supported) : in
- Enter fractional composition: 0.33

- Element #: 2
— Enter the element (ions not yet supported) : ga
- Enter fractional composition: 0.67

- Element #: 3
- Enter the element (ions not yet supported) : as
- Enter fractional composition: 1

- Enter absorption coefficient*thickness of sample, mu*t: 1.11
MONOCHROMATOR INFO.:

1) d-spacing of monochromator: 3.135

2) Position of monochromator : incident_beam
3) Type of monochromator : perfect_cryst
Q) Exit

[1230] Q

SAVE SETUP FILE AS: in33_tutorial.setup

You’ve finished creating a setup file which contains information about the sample
composition and experimental setup. You may take a look at the setup file using a text
editor and check what you have there. Also you can add some more comment using
Yorick syntax if you want. The final step before starting refinement is to subtract
background. Because the sample itself affects magnitude of background, sometimes
instrument background (background measured without sample) over estimate the
real background. So in background correction, the program allows users can change
the magnitude of background by multiplying correction constant in order to make it
match data more nicely in low Q.

[0-4 qsulp] 2 /* Enter to Background substraction in Main */

The Input should be 4-column ascii fileCQ, I, dQ, dI)
Continue (y/n)? y

ENTER DATA FILE NAME TO READ : in33_tutorial.input

ENTER BACKGROUND FILE NAME TO READ : in33_bkg.input

102

BACKGROUND SUBTRACTION:
- Multiply correction constant to background to make
- it match data more nicely in low Q (y/n)? n
- 30 negative intensities set to 0.000208106.
/* negative value are set to minimum intensity */

ENTER FILE NAME FOR BACKGROUND CORRECTED DATA: in33_cfbg.input

[0-4 qsulp] 3 /* start structure function refinement */
ENTER SETUP FILE NAME: in33_tutorial.setup

READ INPUT FILE: It should be 4-column ascii file(Q. I, dQ, dI)
- Enter data file name to read: in33_cfbg.input

DATA REDUCTION :

read-write binary stream: _history.pdb
In directory: /u24/jeong/PDFgetX/manual/
/* Open binary file to save refinement history. Refer to the Appendix A.b */

SMOOTH DATA USING SAVITZKY & GOLAY METHOD (y/n)? n

Flat Symmetric [RJeflection or [T]ransmission geometry (r/t)? t
/* Choose either symmetric flat reflection or transmission geometry */

WINDOW O: CORRECTION EFFETS ON RAW DATA !

APPLY MULTIPLE SCATTERING CORRECTION (y/n)? y
- Does the data contain Compton scattering in high Q (y/n)? n
- Multiple scattering calculation in transmission geometry

WINDOW 3: MULTIPLE SCATTERING RATIO!

APPLY POLARIZATION CORRECTION (y/n)? n
=> Polarization correction NOT applied!

APPLY ABSORPTION CORRECTION (y/n)? y
NORMALIZATION USING MID-HIGH Q PART OF DATA
- Enter a mid-range Q value (roughly 26.4): 25
- 751 points are used for normalization

- Approximate normalization constant: 1904.38

WINDOW 1: CORRECTED DATA vs. TIS!

103

Correction Effects : lnosaGa067As Comparison between Data and <f2>

 

 

 

 

 

 

   
 

 

 

 

 

 

 

 

 

2000 15* .n___
0W7—
6 612-
: owsrTfihsawhnh :
W 1500 r 1
owe 1 ‘ a 1
35 37 39 30 35 40
Q ' Q
g g 1000 9‘-
- Raw data .
. — Data after corrections
~ Alter M‘S' correction Mean—square ave. ASF; <P>
’ NoPoLconmmon 500 ,
‘l-ll Nmer.axmawn
‘ AM I
1 l . '4'— 0 , , —‘ — 1 .
20 30 40 0 1O 20 30 40
-1 —1
0(A ) Q(A )

Figure A.3: (a) Corrections on raw data of Ino_33Gao,67As semiconductor alloy. In
high Q region, after each correction, the change of slope is noticeable. (b) Comparison
between normalized data after corrections and mean-square average atomic scattering
factor, ( f 2). In high Q, those two line up quite nicely.

ENTER NORMALIZATION CONSTANT: 1920
APPLY COMPTON SCATTERING CORRECTION
- Apply Compton correction in MID-LOW Q region using ’Ruland’ method.
— Enter integral width ’b’ (try 0.008): 0.003
- For details, please refer to the MANUAL !!
WINDOW 4: <f‘2>, Compton, and Modified Compton by the Ruland function
WINDOW 2: REDUCED STRUCTURE FUNCTION, 0*(S(Q)-1)!

CHECK IF F(Q) = (S(Q)-1)*Q IS APPROXIMATELY 0 AT HIGH Q
- Is F(Q) approximately 0 at high Q (y/n)? y

SAVE STRUCTURE FUNCTION, 5(0). TO ASCII FILE:
- Enter file name to save data: in33_tutorial.soq

 

Now you’ve obtained structure function. Before we move on, let’s examine the cor-
rections applied during data reduction. First, Figure A.3(a) shows corrections on the
raw data. Each correction effect is not si nificant, however, the changes of slope is
noticeable in high Q region. Figure A.3 b) shows comparison between normalized
data after all correction and total independent scattering(TlS). We can see that TIS
lines up with data in high Q region nicely. Figure A.4 shows absorption factor. dou-
ble scattering, Compton scattering and modified Compton scattering by the Ruland

104

Comparison between Compton and <f2>

 

 

 

 
 
  

  

 

 

 

 

 

   

  
       

 

 

 

 

 

1.0
I— - Rulandlunction
.8 0‘5 - - b = 0.003
. c:
B 0.94 - Absorption factor in 3 0.0 0 + 110 ‘ 20 310 40
< transmission geometry 9 00,-),
0.90 e . 1 1 3
’ b
0.06 ‘17, 200 “' -— Mead-square ave. atomic scattering factor
ac) oo Compton scattering
<1: 0 04 __ E 0‘) Compton modified using Ruland method
"‘ 0 02 Double scattering ratio in —
‘ AA transmission geometry
0000090909on41
0.00 A 1 1 i L . o - ‘ ’ > ... m 3 82A MEATeV—esvc
O 10 20 30 40 0 10 20 30
-1 -1
Q(A ) o(A )

Figure A.4: Data corrections in Ino,33Ga0,6-,As semiconductor alloy: (a) Absorption
factor (,at = 1.11). Absorption effect becomes larger as Q increases. (b) Double

scattering ratio.

(c) Comparison between mean-square average atomic scattering

factor, (f2), Compton (o), and modified Compton (0) using the Ruland function.
Inset shows the Ruland function for the integral width, b=0.003.

function. Finally Figure A.5(a) shows reduced structure function of Ino.33Gao.37As
semiconductor. The oscillating diffuse scattering is clear in high Q region. Table A2
shows all the inputs used in the refinement. Now let’s calculate Pair Distribution
Function(PDF) using the structure function just we obtained.

[0-4 qsulp] 4 /* PDF calculation: G(r) */

Table A2: Summary of structure function refinement

 

 

 

 

 

 

 

 

 

 

Input file in33_tutorial.input

Background file in33_bkg.input

Setup file in33.tutorial.setup

Smoothing No

Geometry Tramsmission

Multiple Scattering Yes

Correction Compton in high Q region is discriminated
Polarization Correction No

Absorption Correction Yes

Normalization Constant 1920

 

Compton Correction

 

 

Remove mid-low Q Compton intensity using
Ruland method. Integral width, b = 0.003

 

 

105

Structure factor: lnoaaGa057As PDF : Iana As

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.67
15 4’ 1 ’
r (a) g" 2 I 1 (b) 1: 5 ~ M
A a t O
10 — 3; -2 ~ ”N lit-sf”
-4 _5 1 1 1
2 2.5 3
E?‘ L [ «A)
e 5 ” l
g l .
0 '— I ‘ A A ' o :; Z/d: :—
\1 i ‘ -
_5 . 1, 1 1 1 1 1 7 _5 1 1 1 1 1 1 1
0 10 20 30 40 0 5 10 15 20
0(1)“) rd)

Figure A.5: (a) Reduced Structure Function of Ino.33Gao_37As semiconductor. The
high Q data shows oscillating diffuse scattering. (b) Pair Distribution Function of
Ino_33Gao_67As semiconductor. The nearest-neighbor peak is split into a doublet cor-
responding to shorter Ga-As and longer In-As bonds

CALCULATE PAIR DISTRIBUTION FUNCTION(PDF): G(r)
- Read structure function: (Q. 8(0). dQ, dS)

ENTER FILE NAME: in33_tutorial.soq
- Enter Qmax at which to cut the data: 40
- Read structure function from Q = 1 to Q = 40

- Enter maxmimum range, r(Angstrom) for PDF calculation: 20
- Enter PDF step size(dr): 0.02

Calculating PDF up to rmax=20 with dr=0.02.

- SAVE PDF: (I, G(r), dr, dG) (y/n)? y
- Enter file name to save data: in33_tutorial.pdf

- Recalculate PDF (y/n)? n

Congratulations! You’ve made a PDF. Figure A.5(b) shows pair distribution function
of Ino,33Gao,6-,As semiconductor alloy. The nearest-neighbor(NN) peak shows well
resolved doublet which corresponds to shorter Ga—As and longer In-As bonds. This
clearly shows the power of high real-space resolution PDF method to study the local
structure of the alloy. It could be instructive to obtain the PDF using different Qmax
to see how it affects the shape of NN peak. The dotted line shows :1: one standard
deviation(o) of PDF; the error propagated to PDF from the raw data. The ripples

106

around sharp peaks are known as the termination ripple. It is caused by the limited
Q value in Sine Fourier transform. And the noise peaks near to r=0 are caused by
noises in the data.

107

A.4 Using PDFgetX

This chapter will teach you how to use PDFgetX. For this purpose, first we’ll give
you overview of PDFgetX. And then'explain how the program works; we will explain
the structure function refinement process.

A.4.1 Overview of PDFgetX

Before learning the specific commands and procedures to control PDFgetX, it is best
to understand how PDFgetX works in a very general way. This section documents
the “broad overview” of PDFgetX while the following sections discuss the specifics
at length.

The function of PDFgetX is to produce PDFs from x-ray powder diffraction data,
whether from an sealed tube x-ray source or from a synchrotron source. Obviously,
to begin the analysis one requires the raw data. The raw data, however, is in general
too “raw” for analytical processing; not only does every facility has a different data
file format, but the data file could contain multiple scans that ought to be averaged
together. PDFgetX can help reduce the raw data into a more convenient format from
which to start the analysis, but ultimately the responsibility for doing so will lie with
the end user.

The input file from which PDFgetX can start the analysis contains the averaged
intensities. Please be aware of the possible name confusion that can occur: the raw
data file refers to the file that is directly output from the computer (like SPEC file)
whereas the input file refers to a input data which will be used for the calculation
of structure function, S(Q). Some information regarding the experiment (such as the
wavelength used) and some information regarding the specimen characteristics (such
as the stoichiometry) are required in order to apply pr0per correction. The experiment
and specimen information are contained in a setup file that is required at every step
of the analysis.

With the setup file and the input file, the analysis can begin. The first stage of
the analysis is to produce S (Q) which is saved as the S(Q) file. However, only the
S(Q) file is used in the second stage of the analysis to produce the PDF. Note that
each stage of the analysis is independent of the others, so long as the necessary input
files are present. That is, to recalculate the PDF of a specimen, you do not have to

108

 

start the analysis from input file; instead, you can specify the correct S(Q) file and
the analysis will immediately create the PDF.

A.4.1.1 Launching PDFgetX

You can start Yorick from any directory by typing yorick at the prompt. At the
Yorick prompt, type pdfgetx and Yorick should begin executing PDFgetX.

current directory: > yorick
Copyright (c) 1996. The Regents of the University of California.
All rights reserved. Yorick 1.4 ready. For help type ’help’

> pdfgetx
Then, this is what you should see:

Pair Distribution Function from the X-ray powder diffraction (PDFgetX 1.1)

0) Preliminary data reduction
1) Build a setup file
2) Background Substraction
3) Reduction of Structure Function: S(Q)
Input file format : (Q. I, dQ, dI)
4) PDF calculation :
Input file format : (Q, S(Q), dQ, dS)
P) Print, S) Save, U) Unzoom, L) Limits windows

Q) Quit

[0-4 hlqpu]

This is the main menu, and Section 5 will explain the menu in detail.

A.4.1.2 Exiting PDFgetX

To quit PDFgetX, type “q” at the main menu prompt. This will exit PDFgetX but
leave you still in Yorick. Type “quit” to exit Yorick.

[0-4 hlqpu] q

Exiting (PDFgetX 1.1):
> quit

current directory: >

109

A.4.1.3. The Main Menu

Pair Distribution Function from the X-ray powder diffraction (PDFgetX 1.1)

0) Preliminary data reduction
1) Build a setup file
2) Background Substraction
3) Reduction of Structure Function: S(Q)
Input file format : (Q, I, dQ, dI)
4) PDF calculation :
Input file format : (Q, S(Q), dQ, dS)
P) Print, S) Save, U) Unzoom, L) Limits windows

Q) Quit

[0-4 hlqpu]

The main menu provides you with several options. Simply type the number or
letter of the option you want and hit “Enter”.

Option 0: This will access an interactive routine that can extract scan(s) from
raw SPEC data and MCA data acquired from x-ray powder diffraction experiments.
Correction for detector and monitor dead-time correction can be applied. You can
also compare variables(e.g. elastic) in each scan and combine scans to get average
value.

Option 1: This will access an interactive routine used to create a setup file de-
scribing the conditions of your experiment. The setup file is needed at several stages
in the analysis; PDFgetX will prompt for the name of the setup file at the appropriate
points.

Option 2: This will access an interactive routine that can subtract a background
from a PDFgetX input file.

Option 3: This will access an interactive routine that applies most of the correc-
tions to the data and produces S (Q) Those corrections requiring feedback from the
user will prompt for the necessary information.

Option 4: This will access an interactive routine that calculates the PDF from

S(Q)-

Option P: When there is a Yorick window present on your screen, you may se-
lect this Option to print the contents of the window. When prompted, specify the
number of the window (this number should be visible in the title bar of the window.
“Yorick 3” would indicate a window number of 3.). This Option is only available from

110

 

the main menu, which means that printing is not possible while doing the analysis.

Option S: This option save the specified window as postscript (PS) file or windows
meta file (WMF) in your directory instead of sending figure to printer.

Option U: This UN-zooms a window. Yorick permits zooming on a data window
(left-button zooms, right-button UN-zooms, and middle—button drags. You may also
click on one axis only to zoom or unzoom that axis.) but sometimes it is diflicult
to make the window look the way it did before the zooming. In that case, select
this option and, when prompted, specify the window number to unzoom the window.
Unzooming only returns the window to the state it was in before the mouse-based
zooms; manually-specified axis limits (option L) supersede the effects of this option.

Option L: This allows you to manually specify the axis limits for a given window.
Enter the window number when prompted.

A.4.2 Data Analysis Procedure in PDFgetX

The Figure A.6 shows schematic diagram of data analysis procedure in PDFgetX. The
procedure is composed of four main blocks, “Preliminary Data Reduction”, “Build
PDFgetX input file”, “Refine Structure Function”, and “PDF Calculation”. Since
most processes in the main blocks are already explained in the tutorial chapter, we
will not repeat the explanation for the whole process. Instead, we’ll give explanation
for the refinement process in detail.

In order to obtain the structure function, we need to apply five major corrections;
dead-time, multiple scattering, polarization, absorption, and Compton scattering cor-
rections. In these corrections, dead-time correction will be applied in the preliminary
data reduction. All other corrections will be applied during structure function anal-
ysis. The Figure A.7 shows flow chart of structure function analysis process. During
the analysis process, the program asks input if it is necessary so please beware of the
messages on the screen.

A.4.3 History File

During data reduction, PDFgetX records all the experimental information, parame-
ters used for corrections, intermediate correction results to a binary file. The default
file name is “_history.pdb”. You can look at the content of this file using Yorick
command.

> o = openb("_history.pdb")
> show, 0

111

 

- Preliminary Data
..Reductio_n

Extract Scan(s)
from SPEC tile

 

 

 

 

 

 

Extract data
from MCA file

 

 

Compare & Combine
Scans

 

 

 

 

Dead-time
Correction

Merged Neolumn
Ascii file

(Q. Elastic. Elastic+ComptOiL Monuor. etc)

 

 

 

 

 

 

 

PDFgetX Data Analysis Procedure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Build PDFgetX . 1' Refine _ r,j.
13.9“! file . Structure Function "
(Q. intensity. dQ, dl) [
_ : 1 p p . YES Multiple Scattering
Background AA A .\?o Correction
-. , menses!) , . . -;
YCS Polarization
x0 Correction
A Build a setup file
’11 ... . .. -.L Yes Absorption
———'—J .\‘0 Correction
— Sample information.
Sample elements and composition Compton Scattering
Correction
—— Experimental setup info:
X-ray wavelength ~ Laue Scattering
Monochromator into Correction

- type. position. d-spaceing
Absorption coefficient

 

-—- File location:

Compton scattering
Atomic scattering factor
Mass absorption coefficient

 

 

Purcmkuuumr'

....-._. . ,.b_,

 

 

 

 

 

G(r)=FrlQ(S(Q)- l )I

 

 

 

 

 

Fourier transform of
reduwd structure
function: Q(S(Q)-l)

l

 

 

 

PDF: G(r) J

 

 

 

 

 

 

 

 

 

 

 

 

Structure Function
_ S(Q)

(Q. S(Q). d0. d5)

 

Figure A.6: Data analysis procedure in PDFgetX

37 non-record variables:

R

Z

absflag

aft

aw

coh_data
compo

compton
compton_hiq
data_cfbg
data_cfbgms
data_cfbgmspf
data_cfbgmspfabs

date
deg_pol
dis_mono
dstran
elementsname
f2ave
fave2
fpara
geometry
lambda
mabscoeff
mscflag
mut

nc
pf

polflag
pos_mono

q

soq_process
rn_data
rn_data_cfcompt
smflag

soq

typ_mono

(r, G(r). dr. dG)

For example, you can simply check your experimental geometry by typing

> o.geometry

"r" /* "r" means reflection geometry */

112

 

For a complete description of history file see Appendix App.b.

A.4.4 Some Yorick Information

This section describes some miscellaneous information regarding the operation of
Yorick, within the context of PDFgetX.

At any time, you may stop the execution of PDFgetX by entering control-C. You
may restart PDFgetX at any time. If PDFgetX, for some reason, crashes, you can
simply restart PDFgetX from within Yorick. It will not usually be necessary to exit
Yorick before restarting PDFgetX. You may zoom any Yorick window using a mouse.
The left mouse button zooms in, the right mouse button zooms out, and the middle
button can be used to drag (if you have a two-button mouse, use both the right and
left buttons at the same time). Click on an individual axis to affect only that axis.

113

Analysis of Structure Function

Start: Read files Input file (Q. l. dQ, dli

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Setup tile
Smooth Data:
- Snvitzky 8r Golay method
No
SMOOTH Transmission or
D AT A Reflection Geometry
T/R
T R
. . Y _ Compton scattering in
Multépcl’e Scanznng c” high 0 region (Y IN)?
No <
Enter degree of polarization . Yes Polarization APPLY MS
of the [Wildcat X-rays Common connecnon
No
APPLY PHI Absorption ' ~ APPLY Ah
CORR ECI'ION C0 ”coho" CORRECTION
No ‘
Noriiializalion using NOR ,
midrhigh 0 part of data MAUZ'ATION

 

 

 

 

 

 

 

l

Compton Scattering
Correction

 

 

 

 

 

      
 

Apply Compton scattering correction only

. ‘ Compton scattering
in mid-low Q region usnig the Ruland method

in high Q region 1'

 

 

Apply Compton scattering conection
" using theoretical Compton prolile

 

 

 

 

 

 

 

APPLY
COMPTON t
CORRECTION

 

 

 

Laue scattering

 

 

Correction

 

    
   

S(Q) = I(Qy<f$

 

 

 

Multiple Scattering Correctiorf]
S(Q) lluctuale around I
O

 
 
      

 

Lblaiization C orrec;[.~

N

   

Nomializanon

 

 

    

Structure Function
S(Q)

(Q. S(Q). dQ, (15)

Q - .
Refinement incomplete

Figure A.7: Structure function analysis procedure in PDFgetX

114

A.5 Data Corrections

The measured X-ray diffraction intensity may be expressed [34] by
Im8“(Q) = PA[N(I§3" + 13:10 + 13%], (A.1)

where P is the polarization factor, A the absorption factor, N normalization constant,
and Igthggalggul are the coherent, incoherent(Compton), and multiple scattering
intensities, respectively, in electron units. We can define the structure function(S(Q))
in the following form.

S(Q) = [1.53" — (<f2> — mat/m? (A2)

where ( f ) is the sample average scattering factor. Therefore to get a structure func-
tion, we have to do the following corrections [38] step by step on raw data.

0) Dead-time correction

1) Multiple scattering correction
2) Polarization correction

3) Absorption correction

4) Normalization

5) Compton scattering correction

6) Lane diffuse correction

A.5.1 Dead-Time Correction

In high-energy, high—intensity synchrotron x-ray diffraction experiments, the detector
and monitor dead-time effect on the measured experimental data is rather lager.
Therefore in these experiments, proper dead-time correction should be applied before
applying standard corrections.

115

In the PDFgetX, the dead-time effect can be corrected by measuring detector
dead-time and using the following Eq. A.3.

N171

(1—7—‘2-312‘)

thc : (A.3)

where 7' is dead-time of detector or monitor, thc the dead time corrected counts,
Nm the measured counts, NM the total counts of detector or monitor and to the
measuring time for each data point. Or dead-time effect can be corrected using the
pulser method [106]. A pulser-train from an electronic pulser of known frequency
can be fed into the detector preamp. The measured counts in the pulser signal in
an SCA window set on the pulser signal is then recorded for each data point. The
data dead-time correction is then obtained by scaling the raw data by the ratio of
the known pulser frequency and the measured pulser counts. The Figure A.8 shows
a comparison of dead-time correction using these two methods.

A.5.2 Multiple Scattering Correction

We’ll consider here only the double scattering process since it represents the major
part of the multiple scattering. To calculate double scattering ratio, we followed the
method suggested by Warren and Mozzi [36]. According to Warren and Mozzi, the
double scattering ratio is given by Eq. A.4

{_(2_) _ B2QIW(26a 0’3 b: #t)
(1) ‘ 1(26) 2 A.u.(m) (M)

 

where, B = Z, Z? and A., p..(m) are the atomic weights and mass absorption coef-
ficients of the atoms. And J (20) is an approximate representation for independent
scattering, 2,- f? or [A f 2 + z'(M)]i depending on whether the measurements include
only the coherent scattering or both the coherent and incoherent scattering and given
in Eq. A.5.

1—a

20 23 ——-———
J( ) (a+1+bsin26

) (A6)
where, a, b are parameters and can be obtained by fitting J (20) to either 2, f? or
ELI f 2 + 2' (M )]1 Q M is a complicated function depending on Q, pt, fitting parameters
a and b and geometry. For details, refer to the papers by Dwiggins Jr. [107, 108]. As
you can see in Figure A.9, the multiple scattering depends on absorption coefficient
and geometry. In transmission geometry it becomes larger as Q increases. In reflection
geometry, however, it increases up to maximum point and decrease a little bit after
that. We can see that smaller the absorption coefficient, smaller double scattering
ratio, in both cases [109].

116

 

 

 
   
 

— DTC: pulser method
a DTC: dead—time(15us)

   

 

 

 

Dead—time correction factor

 

 

 

 

 

 

 

 

I(Q)

After dead-time correction

   

1 x After dead—time correction
W using pulser method

 

 

 

 

 

 

0-1 h i V using dead-time.15ps
y I
0 l | . l l 'N‘“"‘ l l l . l‘
0 10 2O 3O 1O 20 30 40
OM") ow)

Figure A.8: Dead—time correction in Ino_33Gao_6-,As semiconductor alloy: (3.) Com-
parison between the dead-time correction using the pulser method and dead-time
(15 us) measurement. Comparison between low Q and high Q elastic scattering:
(b) before dead-time correction. Low Q data don’t overlap with the high Q data
at Q=12—13 A“, (c) after dead-time correction using the pulser method, (d) after
dead-time correction using the dead-time measurement. After dead-time correction
in both cases, the low Q and high Q data overlaps with each other quite well.

A.5.3 Polarization Correction
Polarization factor P is given by the following Eqs. [34]:
(a) Using a filter

P = (1 + cos2 29)/2 (A.6)

(b) Using a crystal monochromator

P 2 (1+ 2: cos2 26)/(1 + y) (A.7)

117

 

 

 

 

 

 

0.14 t' ' I v I ' t r _
*llt=0.1 .
f ‘ut:0.5 I ‘
— I
009’ .“i-g -...|' 2
I = I
I ..
.0
0.04 r ..-..l:... .0 -
I: .... <<<<<<
'I’: w: «< «<<<<<<<<<<<<::******
000 ::=:!i11113?:*sspeesssssssrs
0203 , - r ‘
. +ut= 0.1 caucuseseaaa
<1Mt=05 Doo
0.02 - Opt=2 Do _,
o <<<<<<<<
o = <4 44
001 GREG
I P a 4
a
figgg+++++++++++++++++++++++++++
u
000" - - . . . . .
0 5 10 15 20
4
Q(A )

Figure A.9: Double Scattering Ratio in Ni, upper panel: transmission geometry,
lower panel: reflection geometry, experimental data includes Compton scattering,
wavelength of x-ray: 0.7107A

where 26 is the scattering angle, :1: = c032 2ozC for a mosaic monochromator crystal or
r = cos 201C for a perfect monochromator crystal where 206 is twice the Bragg angle
of the monochromatic crystal. In Eq. A.7 :1: = y when the monochromator is located
in the incident beam, and y 2: 1 when the monochromator is set in the diffracted
beam.

In the case of the sealed tube X-ray diffractometer, incident beam is unpolarized,
so the full polarization correction should be applied. However, the Synchrotron X-
ray radiation (e.g. CHESS) is almost perpendicularly polarized to the detector plane
therefore only partial polarization correction is necessary, usually less than 5%.

A.5.4 Absorption Correction

Absorption factor A is given by the following Eqs. :

(a) Flat plate reflection geometry

Aref, = [1 - exp(—2ut/sin 6]/2/i (A.8)

118

3.0 . , . ,

 

 

 

 

 

 

+ pi = 0.1 +4(
A > at = 0.3 +++ ‘
520_ th=0.6 +++ ‘
E ' <1 tit = 0.8 ,+++ a
g _ 0 ‘11— _ 1>;++++++>>D>>»
(u
"L D “t - lwvvvvvvvvvv.7
é) 1.0 Wénmmggfigggjggi; ogg°§«::§<<< VvvvvV v‘,
L %UUDDUDDDDO %°°°°0:h
0 0 l l DcinDDUUni-i 0
1,0 pan-nuggnmaggnanugwmm
A 00000 000000000 i
E) 09 ' 000 DUDDDDDDDUD .J
1' ' 0 Mt = 0.5 C’OOO DDDUDDDC
*3 0 Mt : 1 oO 1
w o 8 ~ °°o -
LL - 0 “t = 2 0000
(D. A t = 3 000 i
.o I" 00
< 0.7 *" + “t = 4 0000000 -
r 0 000000“
0.6 i l . 1 2L
0 50 100 150
26

Figure A.10: Absorption Factor in transmission geometry (upper panel) and in re-
flection geometry (lower panel)

(b) Flat plate transmission geometry
Am", = t exp(—p t/cos 6)/cos 6 (A9)

Figure A.10 shows the absorption factor as function of angle and absorption coeffi-
cient. In reflection geometry, if the absorption coefficient is large enough (pt 2 4),
there’s almost no angle dependence of absorption factor. In transmission geometry,
however, when absorption coefficient is around ‘1’, the angle dependence is minimal.

A.5.5 Compton Scattering Correction

Compton scattering correction is very important and difficult in X—ray diffraction
data analysis. Figure A.11 shows elastic and Compton scattering. We can see that
Compton scattering becomes much larger than coherent scattering in high Q. So
even the small error in Compton correction causes big error in determining coher—
ent scattering. Therefore it’s better to discriminate Compton scattering from elastic

119

 

 

0.08

l ' I

 

 

 

 

 

 

A - Elastic scattering 4
E 0 Compton scattering
3
9' .
h
to
V 0.04 r _
.é‘
<0 ' .
C
93 - .
c o
(I)
0.00 m 0 . . . .
0 20 4o

Q(A")

Figure A.11: Comparison between Compton and elastic scattering intensities mea-
sured in Ino_33Gao,37As. Above Q=30A‘1, the Compton becomes larger than the
elastic scattering.

scattering than to correct it theoretically. Compton scattering can be removed ex-
perimentally, particularly at large scattering angles, using an analyzer crystal in the
diffracted beam, or using a solid state detector with a very narrow energy widow
setup. When the Compton scattering is not discriminated, we have to use theoretical
Compton profiles to apply correction. In this case we have to take into account the
‘Breit—Dirac’ recoil factor, R [110]. Formerly, R was usually set equal to unity, which
is still an acceptable approximation for elements of high atomic number. For light
elements and for present-day high-precision diffractometric measurement, however, it
is essential that R be numerically evaluated if the maximum amount of information
inherent in the experimental data is to be extracted. According to Ergun [110], the
following Eq. A.10 should be applied when the number of photons per unit area per
unit time is measured, as with counters.

1

— (1+ r2n_l::sir;20)2

 

(A.10)

where /\ and X are the wavelength of incident and Compton scattered beam. In this
program, we use analytical Compton scattering formula [111] to calculate Compton
profile. One can compare this results with theoretical Compton scattering data from
the ‘International tables for crystallography C’ [112] and find the difference between
these two are very small. Even when the Compton scattering in high Q is discrimi-
nated, the data still contains Compton in mid-low Q region. In order to remove the
Compton in mid-low Q region, we use the method suggested by Ruland [113]. In this
method, the Compton intensity in the data is smoothly attenuated with increasing
Q as is shown in Figure A.4(c).

120

 

A.5.6 Normalization

The measured x-ray intensity is arbitrary value. The intensity should be normalized
properly to get physical meaning. To determine normalization constant, N, we use
high Q part of data. In this method, the normalization constant, N is defined in the
following way.

43,212,102) + 122°(Q)1dQ
fgnf::¢;1[1'cor(Q)] dQ
In Eq. A.11, 1“” corresponds to the data after corrections for background, multiple

scattering, polarization, and absorption. The theoretical atomic scattering factor is
calculated using the analytical formula suggested by D. Waasmaier 8.: A. Kirfel [54].

 

(A.11)

A.5.7 Laue Scattering Correction

Laue term is defined as ( f 2) — ( f )2. The Laue scattering occurs when there is no short-
range order and the atoms are distributed randomly and it decreases monotonically
with increasing scattering angle [36].

A.5.8 Pair Distribution Function

The atomic Pair Distribution Function(PDF), G(r), can be obtained from powder
diffraction data through a sine Fourier transform:

6(7) = 4m {pm - pa] = 2 [0... Q [S(Q) — 1] sin (QT) do (A12)

where p(r) is the microscopic pair density, p0 is the average number density, and Q is
the magnitude of the scattering vector. The PDF is a measure of the probability of
finding an atom at a distance r from another atom and gives information about both
average and the local structure of materials. For more about PDF analysis method,
look up the papers by Egami, Toby and Billinge [40, 48, 114].

A.5.9 Error Propagation

In most diffraction experiments, the measured diffraction intensities are subject to
statistical fluctuations. It is known that the detection process is well represented by
the Poisson distribution. According to Poisson distribution, the standard deviation
of statistical fluctuations is given by x/N for the measured N counts. This error
in measured intensities will be propagated to the error in a function(e.g. PDF)

121

 

determined from these measured intensities. The estimated error in PDF will be
used to test the quality of modeling. In general, an error in function f (121, . . . ,rn)
can be calculated by the following Eq.

 

a '2 a 2

The error in the Structure function, S(Q), is estimated by propagating error in the
measured intensities through each correction step. For the calculation of error in
G(r), the following Eq. is used [114].

2 .
00o) = — E :05(Q,.) Qk AQk SID QM“ (A14)
k

:4

122

 

App.a SPEC file format

In this appendix, the SPEC file format used in the data analysis is presented. The
following shows sample SPEC file.

#F in33_tutorial.spec

#S 1 ascan me 1 13 600 1

#0 Fri Sep 18 16:13:55 1998

#T 1 (Seconds)

#L me ereal elive Epoch Seconds IC1 IC3 I_CESR PULSER TOTAL COMPTON IC2 ELASTIC
1 2.07 1.967 75931 2.11758 556914 396634 394.395 416 2866 233 31718 606

1.02 2.07 1.968 75934 2.11849 558523 396548 394.159 432 3000 217 31791 610

1.04 2.06 1.962 75936 2.10892 555188 394768 392.324 414 3030 253 31569 591

1.06 2.07 1.969 75939 2.11886 558933 396616 394.023 417 3138 240 31776 647

1.08 2.07 1.977 75942 2.1189 559126 396636 393.919 419 2923 246 31839 639

#S 2 ascan me 1 13 600 1

#D Fri Sep 18 16:40:55 1998

#L me ereal elive Epoch Seconds IC1 IC3 I_CESR PULSER TOTAL COMPTON IC2 ELASTIC
1 2.07 1.999 77606 2.11876 490517 396566 353.616 418 2397 186 27129 533

1.02 2.069 1.997 77609 2.11807 490872 396438 353.319 415 2486 194 27167 558

1.04 2.07 1.989 77612 2.11884 489377 396583 353.419 416 2672 177 27045 536
1.06 2.07 1.996 77614 2.11884 492200 396585 353.414 428 2551 195 27218 551
1.08 2.06 1.989 77617 2.10866 488500 394682 351.707 419 2458 199 26993 550

#S 3 ascan me 12 40 1400 1

#L me ereal elive Epoch Seconds IC1 IC3 I_CESR PULSER TOTAL COMPTON IC2 ELASTIC
12 2.07 1.115 88417 2.11861 633382 399154 451.721 317 44519 7243 634757 29557
12.02 2.06 1.137 88419 2.1088 628412 397322 449.505 313 43218 7290 630504 28163
12.04 2.07 1.166 88422 2.11877 625842 399215 451.612 327 42631 7336 630216 27395
12.06 2.07 1.185 88425 2.11884 624286 399227 451.478 299 41732 7166 629057 26469

As shown in the sample SPEC file, all the comments and characters start with #
mark. To specify scan number #S is used and for the scan header, #L and so on. To
separate scans blank line is used. Except these things the SPEC file is the same as
the multi-column ascii file.

123

 

App.b Description of the history

file

In this appendix, the content of the history file (“_history.pdb”) is described. The
history file contains all the experimental information, parameters used for corrections,
intermediate correction results.

Parameter
elementsname
Z

compo

aw

mabscoeff

lambda
mut(pt)
geometry
mscflag

mscParam

polflag

polParam

Description

Name of sample elements ; Ex. [“In”, “Ga”, “As”]

Atomic number of sample elements ; Ex. [49,31,33]
Composition of sample ; Ex. [0.33,0.67,1]

Atomic weight of sample elements ; Ex. [114.82, 69.72, 74.92]
Mass absorption coefficient of sample elements at wavelength /\
Ex. [6.36, 1.88, 2.23] for A=0.2078 A

Wavelength of incident X-ray

Ex. [0.2078] for E = 60 KeV X-ray radiation

Absorption coefficient*sample thickness ;

Configuration of diffractometer

r=Reflection geometry

t=Transmission geometry

0=No multiple scattering correction

1=Multiple scattering correction

mscflag = 1: fpara, dsrefl, dstran

fparazparameters used to approximate scattering

in function J (Warren & Mozzi, 1996)

dsrefl=double scattering ratio, 12/11 in reflection geometry
dstranzdouble scattering ratio, 12 / I1 in transmission geometry
0=No polarization correction

1=Polarization correction

polflag = l: posmono, typ_mon0, dis.mono, deg_pol, pf
p0s_mon : Position of Monochromator

inc-:Primary beam Monochromator

ref=Diffracted beam Monochromator

typ_mono : Type of Monochromator

pczPerfect crystal monochromator

mc=Perfect crystal monochromator

dis_mono : Distance between crystal plane

Ex. Graphite(002), d = 3.3570 A, Si(111), d = 3.135 A
deg_pol : Degree of polarization of incident X—ray beam
Synchrotron source => 1, X-ray tube :> 0

pf = polarization factor

124

 

absflag

absParam

smflag

smoothParam

compton2hiq

comptonParam

nc
R

soq_process

date

f2ave

fave2

compton

q

data2cfbg
data_cfbgms
data_cfbgmspf
data_cfbgmspfabs
rn2data
rn_data_cfcompt
coh_data

soq

0=No absorption correction

1=Absorption correction

absflag = 1: afr, aft, mut(ut)

aft=absorption factor in transmission geometry
afrzabsorption factor in reflection geometry

0=No smoothing of data

1=Smoothing using the Savitzky-Golay filter

smflag = 1: q_s, num_ps

q_s 2 starting point of smoothing

num2ps = number of point used in Savitzky-Golay filter
Y: contain Compton scattering in high Q region of data
N = Compton in high Q is discriminated; no Compton in high Q
compton_hiq = N : integraLwidth, wf

integraLwidth = control parameter for a width of window function
wf = Ruland window function

normalization constant

Breit-Dirac Recoil Factor

0=S(Q) reduction process incomplete, no S(Q) obtained
1=S(Q) reduction process completed, S(Q) obtained

Date of refinement

( f 2), sample average of square of scattering factor

( f )2, square of sample average of scattering factor
Theoretical Compton scattering

Q (=47r sin(0)/A) array

Data after background correction

Data after multiple scattering correction

Data after multiple scattering & polarization correction
Data after multiple scattering, polarization & absorption correction
Normalized data after all necessary corrections

Data corrected for Compton scattering after normalization
Coherent scattering data

Structure function

125

App.c MCA file format

In this appendix, the MCA file format used in the data analysis is presented. The
MCA file format used in this manual is two column ascii file as shown in the following
Figure A.12. The correspond MCA spectrum is shown in Figure A2. The first column
corresponds to the MCA channel number which starts from 0 to 1023 in this case ( so
total MCA channel # = 1024). And the second column is the intensity detected at
each channel. Each 1024 lines corresponds to one Q value and separated by the blank
line. Therefore to convert MCA file to N-column ascii file, the following information
is needed; total MCA channel number and the corresponding “Q” column. You can
get the “Q” column from the corresponding scan (saved in SPEC file) or you can
generate “Q” column if it has constant step. In this case you need Qmin of your scan,
total number of points in your scan, and Q step.

126

MCA channel Intensity

0 0
l 0
0

600 23

601 34

602 15

603 65 Q“)
604 34

605 22
1020 0
1021 0
1023 0

blank line

0 0

1 0

2 0

600 23

601 34

602 15

603 65 -
604 34 Q(i+ l)
605 22

1020 0

102 l 0
1023 0

blank line

0 0

1 0

2 0

600 23

601 34

602 1 5

603 65 .
604 34 Q(t+2)
605 22

1020 0

1021 0

1023 0

Figure A.12: MCA file format

127

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