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LOCAL STRUCTURE OF GexSem GLASSES AROUND THE
RIGIDITY PERCOLATION THRESHOLD USING ATOMIC
PAIR DISTRIBUTION FUNCTION
AND X-RAY ABSORPTION FINE STRUCTURE
TECHNIQUES

presented by

MONEEB TAISEER SHATNAWI

has been accepted towards fulfillment
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LOCAL STRUCTURE OF (ErexSe1_zip GLASSES AROUND THE RIGIDITY
PERCOLATION THRESHOLD USING ATOMIC PAIR DISTRIBUTION
FUNCTION AND X—RAY ABSORPTION FINE STRUCTURE TECHNIQUES

By
MONEEB TAISEER SHATNAWI

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

2007

ABSTRACT

LOCAL STRUCTURE OF Ge$Se1_3 GLASSES AROUND THE RIGIDITY
PERCOLATION THRESHOLD USING ATOMIC PAIR DISTRIBUTION
FUNCTION AND X-RAY ABSORPTION FINE STRUCTURE TECHNIQUES

By

Moneeb Taiseer Shatnawi

A search for a structural response to a recently proposed self-organized and stress-
free intermediate phase [1, 2] in semiconductor chalcogenide GexSe1_m glasses has
been performed in this study. These glasses, according to the mean-field approach,
undergo a structural phase transition from floppy to rigid network that occurs at
a mean coordination number of 2.4. Based on thermodynamic and spectroscopic
measurements, these glasses appear to exhibit two transitions instead of one [3].
The region between these transitions has been called the intermediate phase (IP) [3,
4]. The original theoretical work assumed that the network was generic and the
connectivity random [5]. It was therefore suggested [1] that the IP phase is a region
of finite width in composition where the network could self-organize in such a way that
maintains a rigid but unstressed state. However, it has proved difficult to establish
this result experimentally.

High-resolution atomic pair distribution ftmctions (PDF), derived from high en-
ergy synchrotron radiation, coupled with high-resolution X-ray absorption fine struc-
ture (XAFS) measurements on 18 compositions of well-prepared GezSe1_3 glasses that
span the range of the IP have been performed to elucidate aspects of rigidity perco-
lation and the IP. These data sets are the most complete and the highest resolution
data set on this system to date.

Analysis of the structure functions (in reciprocal space) and the PDFs (in real
space) as well as the XAFS data at both Ge and Se edges show no correlations with

the IP. The network evolves smoothly without any break in slope or discontinuity

that might be linked due to the IP. The results obtained in this study contradict
previously published work [6, 7] that claim experimental evidence for a structural
origin of the IP.

The so-called first sharp diffraction peak (FSDP), which is a signature of the
medium range order in these glasses, changes systematically with Ge content. It
develops smoothly from a low background for low Gecontent to a well-defined, sharp
peak at the stoichiometric composition (GeSeg). Its position shifts towards lower Q-
valuas when Ge content is increased. The height of this peak reaches its maximum
at the stoichiometric composition (:1: = 0.33), after which it starts to decrease. This
is interpreted as being due to the change of the role of Ge atoms in the network.
For :1: < 0.33, the Ge atoms work as a network former, so adding Ge results in a
progressive increase in the correlations contributing to this peak. On the contrary,
for a: > 0.33, Ge atoms work as network modifiers. This will weaken the ordering of
the correlations responsible for the FSDP and hence decrease its intensity.

The basic building block in these glasses is the Ge(Se1/2)4 tetrahedron. For low
Ge content, the tetrahedra are immersed in a floppy Se—matrix. The first PDF peak
is mainly due to Ge—Se correlations. Se—Se and Ge—Ge homopolar bonds were found
only in the low-Ge and high-Ge regions, respectively, consistent with the chemically
ordered network (CON) model, in which Ge—Se bonds are always favored over Se—Se
and Ge—Ge bonds.

To my parents, family and friends

iv

Acknowledgments

I would like to express my regards and thanks to my thesis supervisor, Professor
Simon J. L. Billinge, for his continued support and unlimited patience. His kind
treatment, intellectual thinking and his clever way in solving research problems stim-
ulated me to join his research group, which I really enjoyed working with during my
research study. His supervision, recommendations, patience and encouragements are
highly appreciated and will never be forgotten.

I also thank my thesis committee, Prof. Phillip Duxbury, Prof. William Wede—
meyer, Prof. Chong-Yu Ruan and Prof. Carl Schmidt, for their valuable suggestions
and recommendations. They, for sure, directed my research towards better under-
standing, learning and improvements.

I will never forget my friends and colleagues in the Billinge group: Emil Bozin,
Pavol Juhas, Wenduo Zhou, Xiangyun Que, J acques Bloch, Gianluca Paglia, Asel
Sartbaeva, Ahmad Masadeh, Mouath Shatnawi, HyunJeong Kim, Hasan Yavas, Richard
Worhatch, Christopher Farrow and He Lin. I enjoyed working with them. They
helped me in collecting and analyzing the data. Their help and friendship are highly
appreciated.

I also express my thanks and respect to all of the faculty members in the Physics
and Astronomy Department at Michigan State University (MSU). Special thanks
to Prof. Wolfgang Bauer (the department chair) and to Prof. S. D. Mahanti (the
graduate chair) for their continued supervision on my graduate study. I thank Debbie
Simmons (the graduate secretary) and Cathy Cords (the CMP secretary) for their
help and care of all the aspects of my graduate study and research.

I thank Prof. Punit Boolchand from the University of Cincinnati, and his graduate
student Ping Chen for making up the studied samples. Special thanks are also due
to Prof. David Drabold from the University of Ohio and to his graduate student

F. Inam, for their collaboration in modeling the structure of the studied samples. I

thank Prof. M. F. Thorpe from Arizona Sate University for his collaborations and
suggestions in this study. I would like to acknowledge help from Douglas S. Robinson
at the 6—ID beamline of the APS and Qing Ma at the 5-BM beamline of the APS for
help in setting up the experiments.

Many thanks to my family in Jordan for doing everything I asked for. Their
continued encouragements and support will never be forgotten. I also thank my
colleagues in the Physics Department at the University of Jordan. Their continued
support and encouragements are highly appreciated. I thank all of my colleagues and
friends at the MSU. Their friendship facilitated my life and made it possible for me
to achieve my goals. I will never ever forget the years that I lived in East Lansing.
This great place for study and friendship.

This work was supported in part by US Department of Energy grant DE-FG02-
97ER45651. The XRD experiments were performed at the GID-D beamline in the
Midwest Universities Collaborative Access Team (MUCAT) sector at the Advanced
Photon Source (APS). Use of the APS is supported by the US. DOE, Office of Science,
Office of Basic Energy Sciences, under Contract No. W—31—109-Eng—38. The MUCAT
sector at the APS is supported by the US. DOE, Office of Science, Office of Basic
Energy Sciences, through the Ames Laboratory under Contract N o. W-7405-Eng-82.
EXAFS experiments were performed at the DND-CAT at Sector 5 of the Advanced
Photon Source at Argonne National Laboratory. DND-CAT is supported by the El.
DuPont de Nemours & Co., the Dow Chemical Company, the US. National Science
Foundation through Grant No. DMR- 9304725, and the State of Illinois through
the Department of Commerce and the Board of Higher Education Grant No. IBHE
HECA N WU 96.

Contents

1 Introduction 1
1.1 Amorphous materials ........................... 1
1.1.1 Glass transition .......................... 2
1.1.2 Preparation of amorphous materials ............... 4
1.2 Structure and topology of disordered materials ............. 5
1.2.1 Local structural studies ...................... 6
1.2.2 Probes of local structure ..................... 7
1.3 Constraints and rigidity percolation in covalent network glasses 9
1.4 Theoretical prediction of self-organization and the intermediate phase
in glasses .................................. 12
1.5 Experimental evidences about self-organization and the intermediate
phase in glasses .............................. 13
1.5.1 Raman scattering ......................... 14
1.5.2 Heat flow measurements ..................... 14
1.6 GezSe1_$ glasses as a system for studying the intermediate phase . . 21
1.6.1 Introduction ............................ 21
1.6.2 Structural analogs ........................ 22
1.6.3 Structure of glassy GeSez .................... 26
1.6.4 Basic structural characteristics of GexSe1_z glasses ...... 28
1.6.5 Doping dependence of g-GezSe1_x structure .......... 30
1.6.6 Intermediate range order in g-GexSeHE ............. 33
1.6.7 Theoretical studies of short and intermediate range structure
in g—GexSe1_x ........................... 34
1.7 The goal of the Study .......................... 35
1.8 Layout of the dissertation ........................ 36
2 The atomic pair distribution function and X-ray absorption fine
structure techniques 37
2.1 Introduction ................................ 37
2.2 Atomic pair distribution function (PDF) ................ 38
2.2.1 High-resolution X—ray PDF measurements ........... 43
2.2.2 Rapid acquisition PDF (RAPDF) measurements ........ 44
2.3 Structural information obtained from PDF method .......... 46
2.3.1 PDF real space refinement .................... 47
2.4 X—ray absorption fine structure (XAFS) ................. 50

2.4.1 The XAFS phenomenon .....................
2.4.2 XANES versus EXAFS ......................
2.4.3 Basic principles involved in XAFS ................
2.4.4 The EXAFS equation .......................
2.4.5 EXAFS measurements ......................
2.4.6 Structural information obtained from EXAFS experiments . .
2.4.7 Joint EXAFS and PDF refinement of complex materials

3 Search for a structural response to the IP in GexSe1_x glasses
3.1 Introduction ................................
3.2 Experimental procedures and data reduction ..............
3.3 Results and discussion ..........................
3.4 Summary and conclusions ........................

4 Structural modeling of Ge,,Se1_,c glasses around the IP
4.1 Introduction ................................
4.2 Structural insight from crystalline analogs ...............
4.3 First principles molecular dynamics simulations on GexSe1_x system .
4.3.1 Model generation .........................
4.3.2 Assessment of the MD models ..................
4.3.3 Models analysis: Coordination, rings and constraints .....
4.4 Conclusions ................................

5 Summary and conclusions

5.1 Summary and conclusions ........................
5.2 thure work ................................

Bibliography

viii

51
52
55
56
58
60
63

65
65
67
75
93

94
94
95
102
103
103
105
112

113
113
117

119

List of Tables

1.1
1.2

3.1
3.2

4.1

Atomic structural correlations in c-GeSez. ............... 25
Bond lengths and bond angles in Ge—Se system ............. 29
Single bond energy in the GexSe1_z system. .............. 76
xz-values for the different fitting protocols to the Q—space data. . . . 87
Refined fit parameters for a and 6 phases of c-GeSeg. ......... 100

List of Figures

1.1 Schematic illustration of a random network model in 2-dim. .....
1.2 Schematic illustration of the change in volume with temperature as a

supercooled liquid is cooled through the glass-transition temperature,

Tg. .....................................
1.3 Phase diagram of a simple binary system A3B1_m. ...........
1.4 Schematic showing the dihedral angle. .................
1.5 Illustration of Se floppy and SI02 rigid structures. ...........
1.6 Theoretical prediction of the IP based on different models. ......
1.7 Variation of the BSe4/2 (B = Ge, Si) corner-sharing mode frequency

normalized to 1 in Raman spectroscopy with respect to the mean co-

ordination number f. ...........................
1.8 MDSC scan of Geoggseogg glass ......................
1.9 The non-reversing heat flow for GexSe1_m and GexAsxSe1_2x systems.
1.10 The crystal structure of Ge. .......................
1.11 The crystal structures of Se. .......................
1.12 The crystal structure of GeSeg. .....................
1.13 The outrigger raft model .........................

2.1 Illustration for the structural origin of features in the atomic pair dis-

tribution function, G(r), for an amorphous solid .............
2.2 Reduced structure function and PDF of Ni. ..............
2.3 Raw X-ray intensity data from a sample of Ni ..............
2.4 Schematic illustration of calculating the PDF function. ........
2.5 Schematic representation of the outgoing and backscattered X—rays. .
2.6 Schematics of the EXAFS process. ...................
2.7 Typical XAFS spectrum showing the XANES and EXAFS regions . .
2.8 Schematics of the XAFS transmission and fluorescence experimental

setups. ...................................
2.9 XAFS oscillations and their Fourier transform for crystalline-Ge. . . .

3.1 The measured non-reversing heat flow of Ge$Se1_x glasses. ......
3.2 The reduced structure functions of Ge,,,.Se1_,,c glasses. .........
3.3 The reduced pair distribution functions of Ge$Se1_,,. glasses. .....
3.4 kz-weighted XAFS signal (kzx(k)) for GexSe1_,, glasses at both Ge and

Se K—edges. ................................

3.5 Magnitude of the Fourier transform of kz-weighted XAFS signals (k2x(k))

for Gemsel-.. glasses at both Ge and Se K-edges .............

I—l
tor-scams»:

H

16
17
22
23
25
29

40
41
44
49
53
53
54

73

74

3.6
3.7
3.8
3.9
3.10
3.11

3.12
3.13

3.14
3.15

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

Bond counting statistics for the GexSe1_3 networks based on the RCN

and the CON models ............................ 77
Representative plot of the fit quality for both experimental and simu-

lated first PDF peak ............................ 79
Position of the first PDF peak for experimental and simulated data. . 79
Width of the first PDF peak for experimental and simulated data. . . 80

Bond lengths and Debye-Waller factors as obtained from XAFS analysis. 82
Representative plot of the development of the second PDF peak in
GeQBSeH,c glasses. ............................. 84
Representative plot of the low-Q region in S(Q) data for GezSe1_z glasses. 85
Representative plot of the fit quality to the S(Q) data using different

fitting protocols. ............................. 86
FSDP parameters as obtained from different fitting protocols. . . . . 88
Glass transition temperature for GexSeHn glasses ............ 92
The crystal structure of a-GeSe2. .................... 96
The crystal structure of fl-GeSeZ. .................... 97
Comparison between calculated PDFs of the LT and HT phases of
crystalline GeSe2 .............................. 98
Illustration of edge-sharing and corner-sharing of tetrahedra. ..... 99
Fit quality to the PDF of g-GeSez using the LT and HT phases of
c-GeSez. .................................. 101
Representative plot of the structure of Ge0,2OSeg,30 as obtained from
the molecular dynamics simulation. ................... 104
Comparison between calculated PDFs obtained from the MD struc—
tural models with the experimental data ................. 105
Difference curves between successive values of a: in experimental data
and MD models. ............................. 106
Mean coordination number as obtained from MD simulations. . . . . 107
Ge and Se coordination statistics in the MD models. ......... 108
Bond angle distribution and variation of concentration of types of
neighbors for 2-fold Se units ........................ 109
Concentration of corner-sharing tetrahedra and total number of rings
obtained from MD simulations to GemSe1_,, glasses. .......... 109
Comparison between the experimental PDF of g—GeSe2 and the calcu-
lated PDFs from MD model and the crystalline analogues. ...... 111

Chapter 1

Introduction

1. 1 Amorphous materials

NOTE: Images in this dissertation are presented in color.

By definition, amorphous materials are those materials that do not possess the
long-range order (periodicity) that is characteristic of a crystal. Under this definition,
both amorphous and non-crystalline terms are synonymous. On the other hand, the
term glass is more restricted and can be defined as an amorphous solid that exhibits
a glass transition at which a material changes its behavior from being glassy to being
rubbery.

Randomness can occur in several forms, of which topological, spin, substitutional,
and vibrational disorder are the most important. However, this randomness is not
a unique property, as it must be compared to some standard as the ideal crystal in
which atoms (or group of atoms or ‘motifs’) are arranged in a pattern that repeats
periodically in three dimensions to an infinite extent.

Topological (or geometrical) disorder is a form of randomness that lacks transla-
tional periodicity, extended symmetry and long-range order as illustrated schemat-
ically in two dimensions in Figure 1.1. This type of positional disorder forms the

theme of this study.

 

Figure 1.1: Schematic illustration of a random network model in 2 dimensions. In
this model, the structural units are connected ‘randomly’ to give a structure which
lacks periodicity [8]. Images in this figure are presented in color.

1.1.1 Glass transition

The glass transition is the phenomenon in which a solid amorphous phase exhibits
a more or less abrupt change in derivative thermodynamic properties (e.g., heat ca-
pacity or thermal expansivity) from crystal-like to liquid-like values with change of
temperature. This change makes the amorphous phase less thermodynamically stable
than the corresponding crystalline form (i.e., possesses a greater free energy). When
a material in the liquid state is cooled, one of two events may occur. Either crys-
tallization may take place at the melting point Tm, or else, the liquid will become
‘supercooled’ for temperatures below Tm, becoming more viscous with decreasing
temperature, and may ultimately form a glass.

The volume of the material is one of the most important parameters to monitor
during the glass transition. An illustration of the change in volume with temperature
is shown in Figure 1.2. As can be seen in Figure 1.2, the crystallization process is
manifested by an abrupt change in specific volume at Tm, whereas glass formation is
characterized by a gradual break in slope. The point over which the change of slope
occurs is termed the ‘glass—transition temperature’ (T9).

Certain thermodynamic variables (volume, entropy and enthalpy) are continuous

V F Liquid

Supercooled liquid

 

 

 

 

Figure 1.2: Schematic illustration of the change in volume with temperature as a
supercooled liquid is cooled through the glass-transition temperature, T9. The first
order phase transition accompanying crystallization from the melt is also shown. The
vertical arrow illustrates the volume change accompanying the structural relaxation
or stabilization of the glass if held at temperature T1 [9].

through the glass transition, but exhibit a change of slope there. This implies that
at T9, there should be a discontinuity in the derivative (or intensive) variables, such

as the coefficient of thermal expansion (aT = (933%.!)12), the compressibility (5T =

 

—(‘9(’9’,’DV)T) and the heat capacity (0;: = 0%,»).

A convenient way of monitoring glass-transition phenomena is by means of dif-
ferential scanning calorimetry (DSC) or differential thermal analysis (DTA), which
offer an excellent marker of the glass transition. In these experiments, the sample is
heated at a constant rate and the changes in heat (DSC) or temperature (DTA) with
respect to a reference are measured. The T9 of a particular material depends on its
thermal history (heating and cooling rates), and it is not an intrinsic property of the

material.

1.1.2 Preparation of amorphous materials

Preparation of amorphous materials can be regarded as the addition of excess free
energy in some manner to the crystalline polymorph. This comes from the fact that
the entropy of a crystal is less than that of a glass, and according to the third law
of thermodynamics, a perfect crystal at absolute zero has zero entropy; this is true
regardless of its size. On the other hand, a piece of disordered material, such as a
glass, has some finite entropy, So that is greater than zero at absolute zero.

This preparation varies widely depending on the material being prepared as well as
the purpose of the preparation. Thermal evaporation [10], sputtering, glow-discharge
decomposition [11], chemical vapor deposition [12], melt quenching [13, 14], gel desic-
cation, electrolytic deposition [15], chemical reaction, reaction amorphization, irradi-
ation, shock-wave transformation [16], ball-milling [17] and shear amorphization are
the most common methods for preparation of amorphous materials.

The glass-forming tendency (ability to form a glass) for a material depends pri-
marily on its chemical composition. For example, in the case of a binary alloy with
general formula A3B1-x, its glass-forming tendency is maximized at the so-called “eu-
tectic” composition. At this composition, the melting (or liquidus) temperature is a
minimum, as shown in the phase diagram for the A:,,B1_:,c alloy (Figure 1.3). This
deep reduction in the melting point makes the liquid less supercooled at T9, thereby
reducing the possibility of crystallization.

Phillips [18] has proposed a connection between glass-forming ability and the
mean coordination number. His proposal assumed that the glass-forming tendency
is maximized when the number of mechanical constraints experienced by each atom
(here refer to the interatomic forces acting on it) is equal to the number of degrees
of freedom available to it. A system that has a number of constraints greater than
the available degrees of freedom (overconstrained like Si and Ge) can not easily form

a glass, although an amorphous structure may still be obtained by employing rapid

 

Figure 1.3: Phase diagram of a simple binary system A$B1_I. Temperatures a and b
represent the melting points of A and B respectively. At A, :c = 1. 6 represents the
eutectic composition [9]. Images in this figure are presented in color.
cooling techniques (evaporation or sputtering).

It should be emphasized here that one should not expect glasses prepared in

different ways, with different cooling rates, to exhibit a glass transition at the same

temperature.

1.2 Structure and topology of disordered materi-
als

A non-periodic arrangement of atoms could be attained solely as a result of the
incorporation of variations in bond angles (for two dimensions) or bond angles and
‘dihedral’ angles for three dimensions in periodic arrangements. The dihedral angle is
defined as the relative angle of twist between neighboring units as shown in Figure 1.4.
These variations can result in a continuous random network (GEN) [19] which is
considered as a metastable state, since for crystallization to occur, a substantial

topological rearrangement of the structural units must occur to result in one, or at

 
  

\‘, [Atom 3
: ‘ ’

  

Figure 1.4: Schematic showing the dihedral angle. Images in this figure are presented
in color.

most a small number of discrete values of both bond and torsion angles.

1.2.1 Local structural studies

The ultimate goal of structural studies of materials in general, and disordered ma-
terials in particular, is to enhance our understanding of the relationship between
structure and properties as well as to help promote the design of new materials. The
local structure of materials controls their properties as well as their functionality.
Graphite (dark, soft, cheap) and diamond (transparent, hard, precious) are dramat-
ically different materials. However, they are both made up of carbon atoms. The
only difference between them is in their atomic structures. Order in a structure can
itself be classified according to its spatial extent. Short-range order (SRO) describes
the variation between first nearest neighbors, such as the bond length and bond an-
gle distributions. Medium-range order (MRO) describes ordering just beyond SRO,
typically the next two neighbor shells which extend to about 5 A. This kind of order

(MRO) represents the way in which the SRO configurations are brought together to

build up the network structure of the glass. Extended-range order (ERO) describes
the longest spatial order observed in amorphous materials, typically up to 20 A.
Finally, long-range order (LRO) describes spatial order over macroscopic distances,
where such ordering is only present in crystals and quasi-crystals [20].

It is not surprising that some degree of order (SRO and MRO) should exist in
disordered solids, or even in the liquid state, because interatomic forces responsible
for the crystallinity of a solid remain operative even after the solid melts and becomes
a liquid. Heating the solid gives atoms thermal kinetic energy, which prevents atoms
from holding to their regular positions, but the interatomic short range forces remain
strong enough to impart a certain partial order to the disordered or liquid states.

On the other hand, it should be emphasized that in studying the local structure
of glasses, departure from ordinary glass structure can occur in both directions, not
only as a decrease in order, to give conventional defects, but also as an increase in
order, leading to regions of greater perfection.

The typical method of investigating orders beyond short range order is via mod-
eling studies in which an attempt is made to build a ‘characteristic’ region of the
structure that can be used both to compare with experimental data and in the cal-

culation of other properties.

1.2.2 Probes of local structure

One of the oldest experimental approaches for the study of microscopic structure is
the diffraction experiment, where the use of Bragg’s law proved to be very successful
in determining the crystal structure. For normal ordered crystals, periodicity and
symmetry of the crystal lattice are reflected in the diffraction data, and as a result,
it is possible to use the signatures in the diffraction data to identify the size, shape,
and symmetry of the unit cell.

Crystallographic techniques are generally not suited for structural studies of amor-

phous materials, as these materials lack long-range order and produce few, if any,
diffraction peaks, and the diffraction patterns themselves are diffuse. These standard
crystallographic approaches do not capture the aperiodic (disorder) information, as
Bragg diffraction gives information about the “average” structure of a crystalline ma-
terial, namely, the part of the structure that looks the same in each unit cell, whereas
measurements of total scattering, which includes both Bragg peaks and difiuse scat-
tering, can give information about the short-range structure.

A technique called the atomic pair distribution function (PDF) [21], that uti-
lizes the total scattering data averaged isotropically, has proved to be successful in
studying local structures of amorphous materials. Other methods that are suitable
for local structural determination are: Raman spectroscopy, X—ray absorption fine
structure (XAFS) and solid state nuclear magnetic resonance (N MR). As most of our
work concerns the results of PDF and XAFS experiments, chapter 2 will provide an
introductory account of the theory behind them.

The absence of a simple structural formalism, together with the fact that glasses
are normally isotropic on a macroscopic scale, means that the maximum that can
be obtained from a diffraction experiment is a one-dimensional correlation function
from which the regeneration of the underlying three-dimensional structure can never
be unique, and the best result that can be achieved is a structural model that is
consistent with all the known data. However, even if perfect agreement were to be
obtained between a model and experimental data, it is not guaranteed that other

models could not be generated that would fit the data equally well.

1.3 Constraints and rigidity percolation in cova-
lent network glasses

The notion of constraints and their application to classical macroscopic systems such
as the stability of bridges have been introduced and first considered by Lagrange and
Maxwell [22, 23]. On this basis, J. C. Phillips asserted [18, 24] that covalent networks
can be mechanically constrained by interatomic valence forces such as bond-stretching
and bond-bending and optimal glass formation is attained when the network sits at
a mechanically critical point [18, 24]. This happens when the average number of
constraints per atom estimated by Maxwell counting, no, equals the number of degrees
of freedom per atom in three dimensions, i.e., nC = 3.

Phillips [18, 24] considered the mechanical constraints experienced by an atom to
result from the interatomic forces acting on an atom in the “valence-force-field” model,
where the strain potential energy, U,, can be expressed as a sum of contributions from
bond-stretching and from bond-bending forces as given by Equation 1.1:

U, = éozAr2 + éfirgAfiz (1.1)
where a and 6 are the bond-stretching and bond-bending force constants, respectively,
and Ar and A0 represent small deviations in bond length and bond angle from the
equilibrium values for the bond length, 7‘0, and the bond angle, 00, respectively.

In a covalently bonded network, the coordination number, No, for an atom which
has all covalent bonds satisfied, obeys the so-called ‘8-N’ rule (where N is the number
of valence electrons, and NC, is given by the number (8-N)).

The mean coordination number, F, (which should be distinguished from NC), plays
an important role in determining connectivity and rigidity of a network. In the case

of a covalently bonded binary alloy with general formula A3B1_x, the 1" is given by

Equation 1.2:
7" = :rNc(A) + (l — 2:)NC(B) (1.2)

In the mean-field approach, one considers a network of N atoms composed of 72,
atoms that are r-fold coordinated. The enumeration of mechanical constraints in this
system gives r/2 bond-stretching constraints and (2r-3) bond-bending constraints for
an r-fold coordinated atom.

M. F. Thorpe [5] examined such mechanical networks in terms of percolation
theory and showed by a normal mode analysis that the number of zero frequency
solutions (floppy modes ( f )) of the dynamical matrix vanishes when the mean coor-
dination number, 7", of the network reaches the critical value Fe = 2.4. At fa = 2.4,
the glassy network is stable and has a mechanical threshold or critical composition
at which the network changes from an elastically floppy type to a rigid type.

The number of floppy modes, f, in a network of N atoms equals the difference be-
tween the total number of degrees of freedom (3N) and the total number of constraints

present in the network, as given by Equation 1.3:

f = {3N -— 2m; + (2r — 3))}/3N (1.3)

where n, is the number of r-fold coordinated atoms. This reduces to:

“a

II

[\D

|
OBI 01
‘3I
A
y—n
as
v

This number of floppy modes, f, vanishes when 7‘ = 2.4 [5].

According to constraint counting algorithms, a network is considered floppy if the
average number of constraints per atom (no) is less than 3 (the number of degrees
of freedom for an atom in 3 dimensions) as in the case of twofold coordinated single
bond chain networks (shown in Figure 1.5(a)). On the other hand, the network is

rigid if nC is greater than 3 as in networks consisting only of tetrahedral units (such

10

 

Figure 1.5: (a) Se chain structure (floppy) and (b) SiOz tetrahedral network structure
(rigid) [25]. Images in this figure are presented in color.

as the SiOz network shown in Figure 1.5(b)). Thus progressive addition of cross
linking elements (such as As or Ge) to a starting chain network (such as S or Se) will
result in a progressive increase of its connectivity (mean coordination number). At
F = 2.4, the rigidity percolation occurs where the network changes from floppy to a
rigid structure.

The floppy to rigid transition was first theoretically predicted and then numerous
experiments [26, 27, 28] have confirmed it, especially in glass science were chalcogenide
glasses have been used as a benchmark. Various examples about the occurrence of
a rigidity percolation threshold and its application in granular matter, biology and
computational sciences have been reported [27, 29, 30, 31, 32, 33].

 

0.1 -

0.05

Fraction of floppy modes f

.~
Q. .~
.--~.

 

 

 

 

 

2.5
Mean coordination (r)

Figure 1.6: Showing the number of floppy modes plotted against the mean coor-
dination for Maxwell counting (shown as a partly obscured dashed line) with the
associated mean field transition shown by the open square at < r > = 2.4, and for
a randomly diluted diamond lattice using a dot-dash line, where the second-order
transition is indicated by the solid circle at < r > = 2.375. The self—organized model,
where stressed regions are avoided, follows the Maxwell curve, and is shown by a solid
line, and gives a second-order transition at < r > = 2.375, shown by an open circle,
from a floppy to an unstressed rigid state and a first-order transition at < r > =
2.392, shown by an open triangle, to a stressed rigid state. The range of < r > over
which the intermediate phase exists is indicated [1].

1.4 Theoretical prediction of self-organization and

the intermediate phase in glasses

In a mean-field theory, the variation of the number of floppy modes per atom, f,
with the mean coordination number, F, is linear as given by the Equation 1.4. This
result is shown in Figure 1.6, where the transition (from floppy to rigid) at a mean

coordination of 2.4 is indicated by an open square. In applying Maxwell’s counting on

12

randomly diluted diamond lattices, where the network is decomposed into unstressed
(isostatic) rigid regions, and stressed (overconstrained) rigid regions, with flexible
joints, Thorpe et al. [1] found that the Maxwell’s counting is a good guide, and the
transition occurs at a mean coordination of 2.375, as shown in Figure 1.6.

Thorpe et al. [1] proposed that a network can self-organize at it’s fictive tem-
perature (the temperature at which the glass is formed). The term self-organization
here refers to the subtle way by which the structure can incorporate non-random
features to minimize the free energy at the temperature of formation. In studying
a self-organized model of random network in which configurations that are stressed
are avoided if possible, Thorpe et al. [1] found that this avoidance leads to two-phase
transitions and an intermediate phase that is rigid but stress-free (unstressed) as can
be seen in Figure 1.6. By unstressed we mean that each bond length (angle) can
have its natural length (angle) without being forced to change by the surrounding
environment. Otherwise, a bond is stressed.

The transition at fol is assumed to be due to the change of the network from
floppy to an isostatic (stress-free) rigid phase, and the second transition at F02 is
assumed to be due to the change from an iso-static to a stressed rigid phase. These
two transitions at 7‘61 and F62 define an intermediate phase (IP) in which the connected
structure continues to be stress-free (isostatically rigid) [1, 4]. fol and 1762 represent
the lower and upper boundaries of the IP, respectively. The width (A?) of the IP is

given by the difference 7-22 - fol.

1.5 Experimental evidences about self-organization
and the intermediate phase in glasses

Some preliminary experimental evidence supports the picture of the IP. Chalcogenide

glassy systems (materials containing a group VI atom) have been carefully studied

13

and the intermediate phase (IP) defined by two transitions has been discovered by
Boolchand et al. [3, 4] in the context of self-organization. Following is a description of

experimental findings that were interpreted as signatures of the intermediate phase.

1.5.1 Raman scattering

Boolchand et al. [3] suggested from Raman scattering and phase-dependent measure-
ments of the kinetics of the glass transition that two transitions at 1‘01 and 712 appear
when the network stiffens. Their Raman scattering measurements on GexSeH, or
SixSe1_x glasses [3, 34], that probe the elastic thresholds in these binary glasses,
indicated that Ge or Si corner-sharing mode chain frequencies change with mean
coordination number, 7‘, of the glass network. These frequencies exhibit not only a
change in slope at the mean coordination number near 711 = 2.4, but also a first-order
jump at the second transition 732- In germanium systems, the second transition is
located around the mean coordination number of 2.52, whereas it is 2.54 in Si-based
systems. For both systems, a power-law behavior in F — fez is detected for f > Fag
(see Figure 1.7 [35]) and the corresponding measured exponent is very close to the

one obtained in numerical simulations of stressed rigid networks [36].

1.5.2 Heat flow measurements

In conventional differential scanning calorimetry (DSC) measurement, the signature of
softening of a glass is an endothermic (a process or reaction that absorbs energy in the
form of heat) heat flow with respect to an inert reference sample as the temperature
of the glass and reference sample is swept linearly in time at a controlled rate, usually
20 °C / min.

A more sensitive variant of DSC is the so-called temperature-modulated DSC or
MDSC [37]. With MDSC, the endotherm, as one passes through Tg can be decon-

voluted into two contributions; a reversing and a non-reversing heat flow [38]. The

14

 

l'ml'l‘l

.4
_

 

1.03

 

 

 

 

 

 

 

 

 

 

 

0.981.I.1.I.Ir
2.2 2.3 2.4 2.5 2.6 2.7

Mean coordination number —r

Figure 1.7: Variation of the BSe4/2 (B = Ge, Si) corner-sharing mode frequency
normalized to 1 in Raman spectroscopy with respect to the mean coordination number
f. The solid vertical lines define the intermediate phase in Ge-Se [3] while the lower
solid line and the dashed line define it for Si—Se. The intermediate phase in case of
SizSe1_x is larger than the Ge$Se1-x one [35].

deconvolution is made possible by programmm' g a sinusoidal temperature variation
(A sin cat) on the linear ramp (AT / t) [37], and deducing the part of the heat flow that
tracks the sinusoidal temperature variation using fast-Fourier transform. The part
of the heat flow that tracks the temperature oscillations is called the reversing heat
flow [37], and the remainder that does not track the periodic temperature variation
and is thus called the non-reversing heat flow [37]. The difference signal between the
total heat flow and the reversing heat flow, defines the non reversing heat flow. Fig-

ure 1.8 shows MDSC scans of a Geoggseojn glass [39] illustrating the deconvolution

of heat flow into the non-reversing and reversing components.

15

 

  
 
 
  
 

 

 

 

69235972 glass H‘
0-3 t Total
E TQMOC Hf
E __ _
E 0 2 r] endotherm RGVGI'SIHQ
7:5 .— Kw...
I L scan rate: 3°Clmin H",

.0
—l

 

modulation rate:
1 03/1003 Non-Reversing

0.0 r

 

 

1

240 260 280 300 320 340 360

I

 

Temperature (°C)

Figure 1.8: MDSC scan of Geoggseojg glass displaying a T9 = 290 °C from inflexion
point of the reversing heat flow, and a non-reversing enthalpy, AH”, illustrated by
the hash-marked region. The large width ~ 60 °C of the non reversing heat flow
signal is characteristic of networks that are stressed rigid [39].

MDSC experiments on a variety of glasses reveal, in general, a reversing heat flow
with a step-like change, while the non reversing heat flow has a Gaussian-like profile
shown in Figure 1.8 for the case of a Geoggseojz glass [39]. The inflexion point of
the reversing heat flow signal is taken to define the glass transition temperature, T9,
while the shaded area in Figure 1.8 yields the frequency uncorrected non-reversing
enthalpy, AH“, (while heating up), associated with the melting transition. In these
experiments, it is usual to scan up in temperature followed by a scan down in tem-
perature. Hence, the frequency-corrected non-reversing enthalpy, AHM, is obtained
by taking the difference AH", (up) - AH”, (down).

There is no rigorous theory for the decomposition shown in Figure 1.8, but it is

eminently plausible to interpret the reversing component of heat flow as measuring

16

 

.°
a:

AH", (cal/g/sec)
i

 

 

 

 

 

0.2
0.0
2.0 2.2 2.4 2.6 2.8
7’.

Figure 1.9: The non—reversing heat flow for GezSe1_a, (open circles) and GexAsxSe1_2x
(filled circles). Figure is taken from reference [45].

the quasi-equilibrium specific heat of the system as if it were halted at each structural
stage of its transition from glass to melt (or the reverse). On the other hand, the non
reversing heat flow measures the heat absorbed by the system as it passes through the
stages of the transition. Aspects of structural arrest, aging and thermal history that
characterize the non-ergodic character of T9, are all manifested in the non-reversing
enthalpy [38].

Boolchand et al. [4, 38, 40, 41, 42, 43, 44] carried out MDSC experiments on differ-
ent glassy systems, performed as a function of network mean coordination number, f.
They showed regions near 7‘ ~ 2.4, for which the AH," term nearly vanishes as can be
seen in Figure 1.9 which provides the results of the non-reversing heat flows, AH”,

for both Ge$Se1_m and GexAsxSe1_2x glasses. It is clear that in both cases the AH,"

17

has a deep minimum centered near x = 0.23 in case of Ge$Se1_$ glasses and around a:
= 0.19 in case of GexAsmSe1_2x glasses. For under-constrained and over-constrained
glass, the AHm. is sizable, on the other hand, it almost vanishes for an optimally
coordinated glass (glass where the number of constraints per atom is equal to or close
to the total number of degrees of freedom (3)) [46], so the non-reversing heat flow is
considered as an indicator for the optimally coordinated glass sample. In subsequent
studies, the behavior of almost zero non-reversing heat flow in a finite composition
width was recognized as being a universal feature. They called the composition range
for these thermally reversing transitions the intermediate phase (IP) or reversibility
windows.

A more recently realized feature of glass compositions in the IP window is the
absence of aging [47]. There is a purely thermodynamical connection between the
near vanishing of the non reversing heat flow, AHm, and the absence of aging. Since
AS = Smelt — Sglass = f de/T, where AT << T9 is the width of the melting
transition, then AS ~ AH,,,./Tg [39]. Note that the reversing heat flow H,, or specific
heat, is not included; thus the change in the vibrational entropy is not counted and
S here is the configurational entropy only. Outside the window the values of AS are
larger, so 89;“, is smaller there if we assume that Smelt does not change dramatically
in that range of composition. This means that the entropy of melting is small and
that S91”, is a maximum for glasses in the window. N ow, aging of a system requires it
to diffuse over the energy landscape into configurations of higher entropy. These are
absent in the window, so the intermediate phase should not age (at least not much). In
the chalcogenides this means the networks, for compositions in the intermediate phase,
are formed from molecular units, so called isostatic units, which link together without
incurring energy increasing distortions (or stress). For example, in the GezSeH, glass
system, the backbone structures are formed from corner sharing (CS) GeSe4 units and

edge-sharing (ES) GeSe2 units. The number of differently linked clusters of nearly the

18

same energy is large and is comparable with the configuration entropy of the liquid.
So the large scale isostatic network samples a large number of energetically equivalent
configurations over time. This has a direct bearing on the self-organization feature. In
contrast, if the molecular units are made up of stressed units with redundant bonds,
there are energy barriers between different configurations and entropy of the network
decreases.

The connection between thermal properties of GezSe1_m glasses and their elastic
behavior has emerged from Raman scattering measurements [42, 43, 44, 48]. The
Raman determined optical elasticity shows different power laws for glass compositions
in reversibility windows, and for those outside these windows, as discussed above. The
observed power laws are consistent with available numerical simulations [36, 49].

The IP window in binary GezSeH, glasses was found to occur in the 0.20 <
a: < 0.25 range, and has been established by detailed compositional studies [47].
The correlation between thermal and optical behavior of the GezSe1_x glasses places
glass compositions at a: < 0.20 to be in the flexible elastic phase, those in the 0.20 <
a: < 0.25 range to the intermediate phase, while those at x 2 0.26 to the stressed-rigid
elastic phase [47, 48].

The two properties, AH," ~ 0 and non aging have been theoretically connected
with the concept of self-organization of the disordered networks lying in the reversibil-
ity window [1]. Hence, from the above discussion, it is predicted that compositions
in the IP region are stress-free and self-organized.

It should be emphasized that the experimental positions of the IP, as found by
Boolchand et al. (from < r > = 2.40 to 2.45) do not coincidewith the theoretical
prediction (from < r > = 2.375 to 2.392), which is probably due to the simplicity
of the theoretical models. On the other hand, the theory predicts that the IP is
about three times narrower than the experimental finding of Boolchand’s et al. These

two discrepancies between theory and experiment about the IP casts some doubt on

19

making the equivalence between the two.

Despite the fact that covalent bonds are rather stiff and highly directional, it
should be emphasized that there are other interactions in the structure that may act
as constraints, for example, Van Der Waals interactions. These kinds of constraints
are not counted in Maxwell’s constraint counting algorithm, as it just counts bond
stretching and bond-bending constraints. This leakage might be considered as a
reason for the discrepancy between the predicted and experimental finding of the IP.
A methodology that can suppress non-covalent interactions in the network is necessary
to validate Maxwell’s counting algorithm. On the other hand, a new algorithm that
incorporates non-covalent constraints might narrow the gap between the predicted
and the experimentally-found IP.

Micoulaut et al. [35], using size-increasing cluster combinatorics and constraint
counting algorithms, showed that the second transition at f = F62 (the ‘stress tran-
sition’) is a first-order transition while the first transition at F61 is weakly second
order.

Self-organization of the network can be thought of as the creation of a stress free
basic unit that agglomerates to generate new clusters of larger size provided that
stressed rigid regions can be avoided. Upon increasing 7", isostatic rigid regions are
accumulated. However, there will be a limit to this accumulation (which is the upper
boundary of the IP (712)) at which stressed rigid regions can no longer be avoided,

which results in the second “stress” transition.

20

1.6 GezSe1_x glasses as a system for studying the

intermediate phase

1.6.1 Introduction

For studying the IP and rigidity percolation in glasses, a system where the mean
coordination number can be systematically and smoothly changed in the region 2 <
T < 3 to straddle the rigidity percolation threshold of 11:24 is needed. A floppy
network consisting of chains of atoms (i.e., coordination number is 2) with varying
doping ratios of cross linking element (i.e., coordination number is 4) will smoothly
change the mean coordination number of the entire network.

The GexSe1-$ system with 0.15 S a: S 0.40, offers an excellent candidate to
test the concept of rigidity percolation and IP. In this system and according to the
‘8-N’ rule (here N stands for the column number in the periodic table), Ge is 4-
fold coordinated and Se is 2-fold coordinated, which results in a mean coordination
number of 2 + 22:. Increasing, the Ge content in the glass will smoothly increase the
connectivity of the network and the rigidity percolation threshold will be reached at
20 atomic ‘70 Ge (i.e., x=0.20). Coupled with the extent of knowledge on this system,
and the wide compositional range over which it can be made in the glassy state,
GexSe1_z is an ideal model system for IP studies.

On the other hand, GexSe1-x glasses are of special interest, as they have a large
range of transparency from 0.6 to 30 pm, and they have good mechanical properties,
such as hardness, adhesion, low internal stress, and water resistance [50, 51, 52], which
make them very useful in infrared optics, and memory cells. These materials have
long been under development for use as passive optical components in the infrared
and as active electronic device components for photocopying, ultramicrolithography
and electronic switching.

GexSe1_m are canonical binary network glasses, that can be made as glasses over

21

(a)

 

Figure 1.10: The crystal structure of Ge. (a) one unit cell, and (b) multiple unit
cells [56]. Images in this figure are presented in color.

a wide composition range (x = 0.0 to 0.42 of germanium) [50, 51, 53, 52] allowing
systematic composition dependence to be studied. These glasses, when heated above
room temperature, exhibit a glass transition, (T9) [8] with thermodynamic signatures
including an abrupt change in specific heat, compressibility, and thermal expansion.
They have a number of crystalline analogs that facilitate structural comparisons [54,

55] and make it possible to investigate short as well as medium range orders.

1.6.2 Structural analogs

Here we provide a brief summary of the structures of several relevant elements and
compounds that will be used as a reference in this study.

Crystalline Ge

The crystalline Ge structure is described by cubic close-packed (ccp, diamond), with
space group Fm—3m (space group number 225) and a cubic unit cell with dimension

of 5.658 A. A schematic of the crystal structure of Ge is shown in Figure 1.10.

22

 

(a)
sly-f

 

Figure 1.11: The crystal structures of Se. (a) a-Se ring structure [59] and (b) trigonal-
Se chain structure [25].

Crystalline Se

Crystalline selenium has two allotropic forms; the trigonal structure (t-Se) [57], which
is the most stable form, is made up of [Se]n chains, while the a-selenium structure [58]
is built up from Sea rings as is the iso-structural type a-sulfur structure. A schematic
of the structures of c-Se is shown in Figure 1.11.

The t-Se has a hexagonal Bravais lattice, with three atoms in the primitive cell.
The atoms are arranged along helicoidal chains, which winds an angle of 120° between
first neighbors. The atoms in a given chain are covalently bonded to a third atom
(second neighbor) on a parallel chain. Each Se atom in t-Se has a strong covalent

bond with first neighbor and weak bond with second neighbor.

23

Amorphous Ge

The strong preference of the Ge atoms for four-fold coordination gives rise to fully
three-dimensional structures in both crystalline and amorphous forms. Early evidence
for the amorphous phases being actually microcrystalline [60] has been shown to be

inconclusive, so that the random network model is now widely accepted.

Amorphous Se

As might be expected for two-fold coordinated atoms, the SRO of the solid amorphous
phase of Se is based on rings or chains of atoms in varying proportions. Diffraction
studies on amorphous Se suggest that the bulk glass is comprised largely of bent
helical chains, but that evaporated or sputtered thin fihns have varying proportions
of rings and chains [61, 62, 63, 64].

Crystalline GeSez

The structure of crystalline GeSe2 provides a starting point for discussing the struc-
ture of the glassy as well as for the liquidus phases. The short-range structures
are clearly similar, since Ge(Se1/2)4 tetrahedra form the structural basis of all three
phases. The intermediaterange data are clearly different. Figure 1.12 shows the
crystal structure of GeSe2 which consists of Ge(Se1/2)4 tetrahedra that are linked to
each other by both corners and edges.

Atomic structural correlations in crystalline G8882 at room temperature are shown
in Table 1.1 [54, 66], where, for better comparison with the glassy data, the number

of distances of each type of correlation were grouped into coordination shells.

For each shell 3, the mean distance (r,), variance (of) and coordination number

24

 

Figure 1.12: Polyhedral representation of the crystal structure of GeSe2 [65]. Images
in this figure are presented in color.

Table 1.1: Atomic structural correlations in c—GeSe .

 

 

 

Shell no. Type of correlation Mean distance Variance Coordination no.
5 (dd’) r,(A) 03W) 0,.(d')
1 (Ge-Se) 2.355 0.00009 4
2 (Ge-Ge) edge-sharing 3.049 0.0 0.5
3 (Ge-Ge) corner-sharing 3.551 0.00118 3
4 (Se-Se) Within tetrahedra 3.826 0.02511 9.5
4(cont.) (Se-Se) within layers
4(cont.) (Se-Se) between faces

 

 

 

 

 

 

 

(Cd(d‘)) for the crystal are defined by:

1
7“, — TI: ‘6‘, Ti (1.5)
2 _ i , _ 2
as — 713 is (1”, rs) (1.6)
71.3
0.109) = a— (1-7)

for a coordination shell between atoms of types a and b, where n, is the number of

25

atom pairs contributing to the shell 3, and an is the number of atoms of type a in the

unit cell.

1.6.3 Structure of glassy GeSez

GeSe2 glass has received particular attention as a prototypical glass in order to under-
stand the nature of the equilibrium network glass. As a prototypical glass, it works
as a model and as a bench mark for many chalcogenide glasses. Consequently its
structure and properties have been studied extensively.

Nemanich et al. [67] using Raman scattering and neutron scattering identified the
A1 mode to be due to CS tetrahedra and the A1 companion mode to be due to n-fold
rings (n = 4, 5 and 6). On the other hand, Bridenbaugh et al. [68] using Raman
scattering identified the A1-companion mode as a mode of Se—Se cluster edge bonds,
and proposed Ge-rich and Se-rich clusters for GeSez glass. This agrees with the find-

1 was assigned to be due to

ings of Murase et al. [69] where the mode at 246 cm"
bond-stretching and the mode at 145 cm‘1 was assigned to be due to bond-bending.
Sugai [70] using Raman scattering showed that A1 and A1 companion are identified
with CS and ES tetrahedra and the glass structure is described as a stochastic net-
work. This also agrees with the finding of Nemanich et al. [71] which identified the
A1 companion mode as mode of ES tetrahedra and the glass structure is described
as a COCRN. 12"I M6ssbauer emission spectroscopy [72] and 11i’Sn M6ssbauer spec-
troscopy [73] found a finite concentration of Se—Se bonds and Ge—Ge bonds in GeSe2
glass, respectively. Vashishta et al. [7 4] analyzed the structure factor of GeSeg glass
and liquid, measured by Susman et al. [66] using two- and three-body forces for a
648-atom model of GeSe2 and identified the origin of the first sharp diffraction peak
to be due to Ge-Se and Ge—Ge correlations between 4 and 8 A. Cobb and Drabold [75]

and Cobb et al. [76] using ab-initio molecular dynamics for a 216-atom model cal-

culated the vibrational and electronic densities of states, and showed that four- and

26

six-membered ring correlations contribute to the first sharp diffraction peak. In an-
other ab-initio molecular dynamics study for a 120-atom model, Massabrio et al. [77]
analyzed the structure factor and partial distribution functions of GeSe2 liquid, and
showed that the liquid structure is composed of regular Ge(Se1/2)4 tetrahedra coex-
isting with homopolar bonds and threefold centers, but in their study, the origin of
the first sharp diffraction peak is unclear.

Susman et al. [66] using the neutron structure factor of GeSez glass and liquid
showed that the glass structure is composed of CS and ES tetrahedra, similar to that
of a—GeSeg. Penfold and Salmon [78] using the neutron partial distribution function
measured in GeSe2 using isotopic substitution showed experimental evidence for a
finite concentration of homopolar bonds in liquid GeSeg. In a recent neutron partial
distribution function measured in GeSe2 glass using isotopic substitution, Pitri et
al. [7 9] confirmed the existence of a finite concentration of Ge—Ge and Se—Se bonds
in GeSe2 glass with high concentration of distorted CS Ge(Se1/2)4 tetrahedra. In a
recent Raman scattering, 119Sn Mdssbauer spectra and MDSC on Sn—doped GexSe1_x
glasses, Boolchand et al. [34] showed that Tg(x) variation is correlated to Ge—Ge
bond concentration and the ethane units (Ge2(Se1/2)6) are not part of the tetrahedral
backbone.

Raman spectroscopy suggests that the ratio of edge-shared to corner-shared tetra-
hedra is 0.5(3) in g-GeSe2 and the proportion of Ge—Ge bonds is 2% [34]. M6ssbauer
data on lightly 119Sn doped GexSe1_$ samples also detect two sites for Ge; tetra-
hedral and non-tetrahedral, the latter being assigned to the ethane-like Ge2(Se1/2)6
clusters [34]. These results can be compared to isotope substitution partial PDF stud-
ies [80, 79] that find the proportion of edge-shared tetrahedra to be 50% and that
of Ge-Ge bonds to be 3.7(2)% in rather good agreement with the spectroscopic re-
sults. The presence of a small concentration of Ge—Ge bonds in g—GeSe2 implies some

Se-Se bonds and there is experimental evidence that these exist from 1291 Méssbauer

27

emission spectroscopy measurements [72, 81] and Raman scattering [34].

A chemically ordered continuous Random network (COCRN) model was proposed
for the structure of g-GeSe2 [71]. However, measurements suggest that the intrinsic
chemical order of the glass is, however, broken with a maximum of 25(5)% Ge and
20(5)% Se being involved in homopolar bonds at distances of 242(2) and 232(2) A,
respectively [79]. Therefore a defective COCRN model has been suggested [9]. A
model for g—GeSe2 assumes that its structure is made up of stacked chalcogenide-
covered wafers, with an average diameter and stacking thickness of order 10—20 A [68].

These results can be qualitatively reconciled with a model that implies some de-
gree of chemical inhomogeneity, known as the outrigger raft model [68, 24, 82]. Se-rich
regions of the structure resemble the layered structure of the high-temperature crys-
talline phase, with chains of corner-shared Ge(Se1/2)4 bridged by edge-shared tetra-
hedra as shown in Figure 1.13. Stoichiometry is maintained by regions of the material

with Ge2(Se1/2)6 type clusters [68].

1.6.4 Basic structural characteristics of GexSe1_x glasses

The basic structural units in the GexSe1_,, network are the Ge(Se1/2)4 tetrahedra [83,
79], and the Sen chains [84]. The way in which these tetrahedra are linked together
and the existence of long or short Sen chains depend on the Ge content in the network.

The covalent bonding in the GexSeH, network gives rise to well-defined bond
lengths and bond angles as shown in Table 1.2 [85]. Ge is 4-fold and Se is 2-fold
coordinated at all compositions.

Ge-Ge homopolar bonds are absent at small values of :r. Careful structural study
by determining partial structure factors from isotopic substitution neutron PDF anal-
ysis [80] tentatively suggested their existence in g—GeSeg. This seems to have been
confirmed in a more careful recent determination of the PDF partials [79].

Evidence for edge-shared tetrahedra comes from the observation of a split peak

28

 

Figure 1.13: The outrigger raft model [68].

Table 1.2: Bond lengths and bond angles in Ge-Se system

 

 

 

 

 

Parameter Value

Ge—Ge distance 2.44 A

Ge—Se distance 2.38 A

Se—Se distance 2.34 A
Se—Se-Se bond angle 105°

Se—Ge—Se, Ge—Ge-Se, Ge—Ge—Ge bond angles 109°28'
Ge—Se—Se, Ge—Se—Ge bond angle 102°

 

 

29

around 210 - 220 cm‘1 in Raman scattering [68, 71]. Bridenbaugh et al. [68] noticed
the existence of a Raman peak at 219 cm"1 in the high-temperature fl-structural
phase, which includes edge-shared tetrahedra, similar to that seen in the GeSez glass.
The frequency of this anomalous mode can be rationalized from using the Sen-Thorpe
lattice dynamics model [71, 54], applied to edge-shared tetrahedra, also thought of as
“four atom rings” [71]. Also, the intensity of this Raman peak varies in the expected
way with doping [71]. Evidence for the existence of edge-shared tetrahedra also comes
from intensity in the PDF around 7‘ = 3.05 A from Ge-Ge distances in neighboring
tetrahedra [66].

Nemanich et al. [71] also demonstrate a surprisingly good agreement between the
local structure in the crystalline fi-phase and in GeSeg glass, as measured in a low
resolution PDF from neutron scattering data. In fl-GeSeg there exist equal numbers
of edge- and corner-sharing Ge(Se1/2)4 tetrahedra which gives two nearest-neighbor
Ge-Ge distances at 3.05 and 3.55 A, the shortest being the distance between the

centers of edge sharing motifs [79].

1.6.5 Doping dependence of g-GexSe1_x structure

The structure of GexSe1_x glasses depends ultimately on the Ge content. At the low
Ge content (a: < 0.31), no Ge-Ge homopolar bonds are detected [34], but their number
grows rapidly above a: = 0.33 [34]. For the region a: > 0.33, a phase separation into
Ge-rich and Se-rich regions has been postulated [34].

Many structural models have been proposed to describe the structure of Ge$Se1_z
glasses. Some of these models are consistent with some experimental data but fail to
describe other features seen in other experimental data. Here we provide a description
of these various models considered for the structure of the GexSe1_x system:

Random covalent network (RCN) model or Bett’s model:

The structure of glass in this model [86] is essentially three dimensional, where the

30

distribution of bond types is purely statistical [9].

Chemically ordered network (CON) model [71]:
In this model, the glass structure is assumed to be composed of cross-linked structural
units of the stable chemical compounds (of the system) and excess, if any, of the
elements. According to this model, formation of heteropolar bonds is favored over
the formation of homopolar bonds at all compositions. Consequently, Ge—Se bonds
are favored such that Ge—Se and Se—Se bonds are allowed on the Se—rich side of the
GeSe2 composition (:1: < 1 / 3), while only Ge-Se and Ge-Ge bonds are allowed on the
Ge-rich side (a: > 1 / 3). A more complete description of this model will be provided
in Chapter 3.

Both the RCN and the CON models give the same mean coordination number
of f = 2 + 2:1: (a condition that must be satisfied if, in accordance with the ‘8-N’
rule, Ge is fourfold coordinated and Se is twofold coordinated in Ge,,Se1_,c glasses).
The difficulty in distinguishing these two models lies in the characterization of the

intermediate-range atomic arrangements of materials which have no long range order.

Chain crossing model (CCM) [24]:
In this model, the Se chain structure is maintained, but the four-fold, tetrahedrally
coordinated Ge atoms act as chain crossing sites, with Ge—Ge bonds not allowed for
a: S 1 / 3.

For chalcogen-rich glasses, this model (CCM) considers a homogenous glassy struc-
ture made up of short chalcogen chains [Se]n cross-linked by isolated GeSe4 tetrahedra.

Phillips’ outrigger-raft model:
This model was proposed by Phillips for the structure of GeSeg, where the CCM
has been used to interpret the layer structure in glasses. This model consists of an
array of GeSe4/2 tetrahedra, but these units are covalently bonded together in layers

having atomic arrangements which are similar to the layers of crystalline GeSeg. Each

31

layer consists of parallel chains of corner-sharing tetrahedra, cross-linked with pairs
of edge-sharing tetrahedra as shown in Figure 1.13. These layers in the glass, as
well as in the crystal, are held together by Van Der Waals forces, and these layers
are terminated by Se—Se dimers parallel to the chain. Inherent in this model is that
homopolar bonds are an intrinsic part of the structure.

Results from Raman and Méssbauer spectroscopy as well as pressure and optical
measurements and laser recrystallization studies of a-GeSe2 have been used to support
and extend this model.

By contrast, neutron scattering and modeling have shown that layers are not
necessary to describe the scattering and that features of the raft model, such as
three-membered rings of tetrahedra, are not dominant configurations in glassy GeSez.

In this model, the similarity between the glass and the crystal is stressed, and the
medium range order of the network structure is essentially two dimensional (layered)
rather than three dimensional as in SiOz.

Stochastic random network model (stochastic RNM):

A stochastic random network model was proposed for the structure of GexSe1_3
glasses; x3033. This model is based on the interpretation of Raman spectra from the
View point of phonon localization. In this model, the glass structure is characterized
by one parameter, P, that is related to the existing probability of the edge-sharing
bonds between the tetrahedral GeSe4 molecules relative to the corner-sharing bonds.
An advantage of this model is that only one parameter, P, which depends only on the
species of atoms forming the glass and not on x, can characterize the glass structure,
and consistently explains the origin of the A1 and the A‘f peaks in the Raman spectra
which have been a subject of controversy.

Topological model:

The topological model based on constraint theory is successful in accounting for the

features observed in the property-composition dependence of several chalcogenide

32

glass systems.

In this model, the properties are discussed in terms of the mean coordination
number, 7‘, which is indiscriminate to the species of the valence bond. Similarity of
the molecular structure of the glass networks with that of the corresponding crystalline
state is emphasized in this model, but however, reference to specific compositions or

to individual families is not necessary.

1.6.6 Intermediate range order in g-GexSe1_x

Intermediate range order is evident in GemSe1_3 glasses by the observation of a first
sharp diffraction peak (FSDP) in the diffraction pattern at around 1 A‘l. The FSDP’s
are commonly observed in glasses [87, 66, 6, 7]. They appear to have a universal
property that they appear at a position Q1 such that er1 ~ 2.5, where T1 is the
mean position of the nearest-neighbor peak in the PDF. The peaks are sharp (thus
their name) with a full-width at half maximum, AQITI ~ 0.6 [83]. These numbers
indicate that a structural modulation exists in the glass with a wavelength of roughly
27r/Q1 = 2.671 and a correlation length of the order of 10m.

The structural origin of the FSDP’s is still an active area of research. Theoretical
suggestion by Phillips [24] for the FSDP in AS2863 and GeSez based on intercon-
nected rafts of subunits of the crystalline analogs does not seem to be generic enough
to explain the observation of FSDPs in a wide range of systems with different dimen-
sionalities of the network [83]. On the other hand, random packing of rigid structural
units with dimensions of the order of 27r/Q1 also gives rise a sharp peak at around
the right position [60]. FSDP’s also persist into the liquid phase as observed in
GeSez [83]. The FSDP is rapidly destroyed by the addition of silver impurities to
GezSe1_x glasses; and it grows up out of a smooth background when tetrahedrally
coordinated silicon [88], phosphorous [89] or germanium [85] is added to amorphous

selenium. The presence of rigid tetrahedral units therefore appears to be important.

33

Taking a different track, a recent study by Elliott et al. [90, 91] on amorphous sil-
icon and silica showed that the FSDP can be regarded as arising from scattering
from atomic configurations equivalent to a single family of positionally-disordered
local Bragg planes having the furthest separation. A more recent work on FSDP
in Gexselfl, system [6, 7] claimed the observation of structural anomalies in FSDP
parameters from which they characterized the existence of the intermediate phase.

Molecular dynamics (MD) [74, 92] studies suggest that the origin of the FSDP in
GeSeg glass is principally due to the Ge—Ge correlations in the structure. This result
is qualitatively supported by the partial-PDFs measured by Penfold and Salmon [80]
where the FSDP was only significant in the Ge-Ge partial structure factor, SGe_Ge(Q),
though other predictions of the MD studies were less well home out by the measured
partials [80].

The FSDP is clearly an important indicator of intermediate range order in the
glass structure. However, the lack of clear understanding of its structural origin limits
its usefulness as a probe of structural changes that occur with doping, for example,

associated with the IP.

1.6.7 Theoretical studies of short and intermediate range

structure in g-GexSeH,

Vashishta et al. [74, 92] using molecular dynamics simulations, investigated structural
and dynamical correlations in molten and vitreous GeSe2 with an effective interionic
potential. They concluded that the short-range order is dominated by Ge(Se1/2)4
tetrahedra and the Ge-Se, Se—Se, and Ge-Ge bond lengths are 2.35, 3.75, and 4.30
A, respectively, with Ge-Ge and Ge-Se correlations between 4-8 A being responsible
for the FSDP observed in the static-structure factor. They explained the anomalous
decrease in height of the FSDP on cooling by a frustration enhanced by increased

number density.

34

An ab initio molecular-dynamics study of structural, vibrational, and electronic
properties of glassy GeSe2 [76] using a 216 atom model with normal mode analysis
reveals a trend of localization in the band tails of the vibrational density of states
(VDOS) as well as a significant change in the degree of localization for modes above
approximately 15.5 meV. This analysis also shows that chemical disorder has a sig-
nificant effect on the dynamics in g—GeSeg.

Cobb et al. [75] using ab initio molecular dynamics study of liquid GeSe2 showed
many similarities between the topology of the liquid and the glass state, and suggested
that the FSDPs of liquid and glassy GeSe2 are results of the intermediate range
order (IRO) imposed by fourfold and sixfold ring correlations. This study showed
that an increase in Se bond length and bond-angle disorder significantly broadens
the conduction band. The time-dependent behavior of the electronic eigenvalues is
examined and transient events are observed in which an electronic state crosses the

optical gap.

1.7 The goal of the Study

Based on the IP picture, two hypotheses can be inferred: Firstly, if the IP picture is
correct, then samples in the IP region should have zero or minimum strain. Secondly,
samples in the IP window should have maximum intermediate range order due to the
proposed self-organization.

The goal of this study is to search for a structural response to the intermediate
phase (IP) in the Ge3Se1_,, glasses. In particular, our focuss is to examine the local

structural changes in this system looking for evidence of the suggested IP region.

35

1.8 Layout of the dissertation

After the introduction describing the basics of glass networks as well as the onset
of rigidity percolation and the discovery of IP, Chapter 2 describes the experimental
probes used to test the hypotheses of the IP, the PDF and XAFS techniques. Chapter
3 describes the main results of a model-independent analysis of the composition de-
pendence of the peaks in the PDF and of the FSDP. These should indicate underlying
changes in structure that can be correlated with the intermediate phase. Chapter 4
presents a description of modeling of the data using crystalline analogs and results
of molecular dynamics simulations on the Ge$Se1_$ systems. Chapter 5 contains the

summary and the main conclusions of the study.

36

Chapter 2

The atomic pair distribution
function and X-ray absorption fine

structure techniques

2.1 Introduction

One useful way of expressing the structural information is through the use of cor—
relation functions that give information about local structural parameters such as
inter-atomic correlations, coordination numbers and thermal displacements.

In this study we have used the atomic pair distribution function (PDF) and X-
ray absorption fine structure (XAFS) experimental techniques to access structural
information. These structural probes proved to be very successful in determining
the local structure of materials. A brief account of these techniques, together with a

mathematical description of the most relevant relations, will be given in this chapter.

37

2.2 Atomic pair distribution function (PDF)

Characterization of local structure of complex materials is important for a more com-
plete understanding of their ftmctionality. One of the most useful methods to do so
is the atomic pair distribution function (PDF) technique, a total scattering technique
that gives the local structural environment at the atomic scale.

The PDF analysis of powder diffraction data has been used for many years for
studying materials with no long-range order, such as glasses and liquids. More re-
cently, with the advent of high power X-ray and neutron sources and fast computing,
it is making significant impact in the area of locally ordered and crystalline materi-
als [21].

Traditionally, powerful crystallographic methods whereby Bragg-peaks are ana-
lyzed directly in reciprocal space provide extensive information about the underlying
structure. However, as scientific interest shifts to more disordered crystals, and ma:-
terials where the crystallinity is highly compromised, such as nanocrystals or crystal-
lographically challenged materials, more of the important structural information is
contained in the diffuse scattering component

The PDF technique, and closely related total-scattering methods, allow for both
the Bragg and diffuse scattering to be analyzed together on equal terms, revealing
the short and intermediate range order of the material.

The PDF function represents a Fourier transform of scattered X—ray or neutron
total scattering patterns into direct space providing information related to the real-
space arrangement of atom pairs [21]. In contrast to crystallographic methods, no
presumption of periodicity is made, allowing non-periodic structures or aperiodic
modifications to be studied. It is also a highly intuitive function since peaks in the
PDF come directly from pairs of atoms in the solid and are positioned at r-values
corresponding to inter-atomic distances. For example, a PDF peak that is shifted

to lower-r values therefore directly implies that corresponding pair of atoms has a

38

shorter interatomic separation.

Below we briefly define various functions of interest and describe how they are
measured. More details can be found in a recent published book on the subject,
Underneath the Bragg Peaks: Structural Analysis of Complex Materials [21].

The experiments are typical X-ray and neutron powder diffraction measurements. They
are usually carried out at synchrotron X—ray sources and pulsed neutron sources,
rather than usual laboratory sources, since it is important to obtain the data over
a wide range of momentum transfer, Q, for high accuracy and adequate real-space
resolution of the PDF, and the laboratory sources can not meet this requirement.
Since Q is given by, Q = 4r sin(0)//\ (for elastic scattering), and Q > 30 A’1 is
desirable, this implies that short-wavelength, high energy, X-rays or neutrons are re-
quired. X—rays of energy > 45 keV (A = 0.27 A) and up to 100 keV (A = 0.120 A) or
more are typically used. Data from laboratory sources with M0 or Ag tubes can give
acceptable results in some cases, but working at a synchrotron or spallation neutron
source is preferred for the highest resolution measurements.

The atomic PDF, G(r), is defined as:

G(T) = 47”"IP(7‘) - pol (2-1)

where p0 is the average atomic number density, p('r) is the atomic pair-density defined
below, and r is a radial distance.

The function G ('r) is experimentally accessible and gives information about the
number of atoms in a spherical shell of unit thickness at a distance r from a refer-
ence atom. It peaks at characteristic distances separating pairs of atoms, as shown
schematically in Figure 2.1. It is related to the measured X—ray or neutron powder

diffraction pattern through a Fourier transform:

Qmaz
=(2/vr) /=leS(Q-1]sin(Qr)dQ (2.2)

39

   

Radial pair distribution function:

60)

 

 

/ ’ \ '
1“ coordination 2"d coordination
shell shell

Figure 2.1: Illustration for the structural origin of features in the atomic pair distri-
bution function, G(r), for an amorphous solid. Images in this figure are presented in
color.

where S (Q), the total scattering structure function, contains the measured intensities.
An example of S(Q) from fcc nickel, measured over a wide range of Q at beamline
ID-l of the Advanced Photon Source at Argonne National Laboratory, plotted as the
reduced structure function, Q[S(Q) —- 1], and its Fourier transform, G(r), are shown
in Figure 2.2. The structure ftmction is related to the coherent part of the total
scattering intensity of the material, and is given by:

1“”‘(Q) - Zcilfi(Q)I2

S(Q) = Ina-flow

+ 1 (2.3)

where I 6""(Q) is the measured scattering intensity from a powder sample that has
been properly corrected for background and other experimental effects and normalized
by the flux and the amount of the sample in the beam. Here, c,- and f,- are the atomic
concentration and X—ray atomic form factor, respectively, for the atomic species of
typei [93, 21].

In the case of neutron experiments the f,’s are replaced by Q-independent neutron
scattering lengths, b, and the sums run over all isotopes and spin-states as well as over

the atomic species. The choice between carrying out a neutron or X—ray experiment

40

 

'30

no
TN
A ,.
8

([30
V‘—
O

 

 

 

 

 

 

O
0 5 10 15 20 25
out")
In; -
9'. -

G(r)(fi'2)
5

 

 

 

 

—
‘—
_
—E
“
4
.‘
.—
~
.—
‘—
a
‘
1
.-.
£
‘

—5

 

 

l . l s I a I a l

O 5 1 O 1 5 20 25 30
FOR)
Figure 2.2: (a) The reduced structure function, Q[S(Q) — 1], of crystalline Ni. (b)

The PDF obtained by Fourier transforming the data shown in (a). Images in this
figure are presented in color.

 

41

depends on the nature of the sample and nature of the information sought. The
old paradigm that neutrons were intrinsically superior to X—rays for this type of
measurement, as there is no neutron atomic form factor damping the intensity at
high-Q, is no longer valid with the advent of extremely intense high energy X-ray
synchrotron beams that can accurately account for the scattered intensity at very
high Q values.

As can be seen from Equation 2.1 - 3.1, G(r) is simply another representation
of the diffraction data. However, exploring the diffraction data in real space has
advantages especially in the case of materials with significant structural disorder.

The PDF reflects both the long-range atomic structure, manifested in the sharp
Bragg peaks, and the local structural imperfections, manifested in the diffuse com-
ponents of the diffraction pattern. This is because the total scattering, including
Bragg peaks as well as diffuse scattering, contributes to the PDF. The modeling of
the data also does not presume periodicity. Therefore, this technique is particularly
useful for characterizing aperiodic distortions in crystals, analysis of nano structures
and glasses.

The data are corrected for experimental effects such as sample absorption, mul-
tiple scattering, and normalized to get the structure function S (Q) This process is
described in detail elsewhere and various programs are available for carrying out these
corrections. Improper corrections result in distortions to S (Q) but these distortions
vary much more slowly than the signal and are manifested as sharp peaks at very
low-r in the PDF in a region (typically < 1.0 A) where no structural information
exists. A significant advantage of the PDF is that the useful structural data persist
to very long 1“ distances allowing models to be fitted over significant ranges. Pro-
viding they are not over-parameterized, these fits are highly resistant to random and
systematic errors in the data and provide robust structural solutions.

Modeling the experimental PDF is straightforward because it can be calculated

42

directly from a structural model using the following equation:
u 0
G(r) + 47rrp0 = 71.2250) f(f)( >)————“6(r —1',,,,) (2.4)

Here the f (0) ’s are the atomic form factors evaluated at Q = 0 that are, to a good
approximation, given by the number of electrons for a given atomic species, Z. In
the case of neutron diffraction the f (0) ’s should be replaced by neutron scattering
lengths, by. In Equation 2.4, n,“ is the distance separating the nth and m“ atoms
and the sums are over all the atoms in the sample.

For a material whose structure can be described by a small unit cell, which for
the case of disordered crystals will be a supercell of the crystallographic cell, the
first sum in Equation 2.4 runs over atoms in the cell and the second sum runs over
all atoms up to whatever cutoff, rm”, may be of interest. This makes the problem
computationally significantly more tractable. A number of regression programs are
available. A program that uses least-squares in analogy with Rietveld refinement is
PDFFIT2 [94].

Alternative methods include using a reverse Monte-Carlo type approach [95],
where a residuals ftmction is minimized using a simulated annealing algorithm, or

using a potential based modeling scheme.

2.2.1 High-resolution X-ray PDF measurements

The technological advancements that allowed the PDF to be applied to crystalline
materials was the development of spallation neutron sources. Unlike reactor sources,
the spallation sources have high flux of epithermal neutrons with access to short
wavelengths as well.

The real space resolution of the PDF is directly related to the range of Q over

which data are measured; roughly speaking 67‘ = 1r / Qmax where Qmax is the maximum

43

 

y (o.u.)
2x104 5x10‘

 

lntens
10‘

 

 

 

o A AATAAA AAA
0 5 1 O [5 20 25
0(3’ )
Figure 2.3: Raw intensity data from a sample of Ni measured using X-rays of A =

0.1235 A. The effect of the atomic form-factor in suppressing the intensity at high Q
can be noticed. Images in this figure are presented in color.

 

Q-value. Since Q = 47r sin(l9) / A S 47r/ A, to increase Qm it is necessary to decrease
A.

Increasing the real—space resolution of X—ray measurements presents an additional
challenge: the X—ray form-factor, f (Q). The square of this is a measure of the
structural-information containing coherent scattering from the material under study.
The structure factor falls off sharply with increasing Q resulting in a weak signal at
high diffracting angles. This is illustrated in Figure 2.3 that shows the raw scattered
intensity from Ni. The overall drop-off in intensity follows | f (Q)|2 with very little

apparent structure in the scattering in the high—Q region.

2.2.2 Rapid acquisition PDF (RAPDF) measurements

The RAPDF data collection involves straightforward powder diffraction measure-

ments. The main difficulties come from having to use high-energy X-rays, and having

44

a stable and low background setup allowing reliable quantitative measurement of
weak diffuse intensities. This is described in detail in Egami and Billinge [21]. Neu-
tron measurements are generally carried out at spallation sources that have desirable
short-wavelength epithermal neutron fluxes. Data collection times vary, depending
on the source, sample size and composition, from 20 minutes to 20 hours per data-set.
Conventional X-ray measurement protocols use energy resolving solid-state detectors
at a high-energy synchrotron where data collection typically takes 8 to 12 hours per
data—set, depending on the sample properties, flux and required Qmaz. Recently, a
new approach to data collection making use of 2-D image plates as detectors and
using very high energy X-rays (~ 100 keV) to compress the scattering data into a
relatively narrow angular range.

RAPDF measurements are still under development and the data quality and range
of applicability are being extended. However, there appears to be tremendous upside
potential in using this approach, in part because of the fast data collection time,
but also due to the fact that the measurement is static facilitating measurement on
samples in confined geometries and special environments. Potential successful appli-
cations of RAPDF are time-resolved measurements of samples undergoing chemical
reactions or under photo-excitation, for example, materials at high-temperatures and
under pressure and extensive phase diagram studies.

Total scattering and PDF analysis of X—ray and neutron powder diffraction can be
used to solve structural problems that cannot be addressed with traditional crystallo-
graphic methods. In this context the PDF analysis technique goes beyond crystallog-
raphy and captures new structural information. In addition, this type of analysis can
be applied in a complementary fashion to conventional crystallography to check the
accuracy and validity of the crystallographically determined structure. Given that a
crystallographic solution provides only an average structure, one cannot be absolutely

sure whether the local structure is the same as the average one. If they are not, the

45

average structure is then an incorrect representation and does not capture the critical
features that may be responsible for the physical properties of the material under
study.

In summary, the PDF analysis represents an important tool for better under-
standing of structures at the atomic level, coupled with the wide accessibility of high
energy X—ray synchrotrons and neutron sources as well as the short data collection

time, using imaging plates, for example.

2.3 Structural information obtained from PDF method

Structural information obtained from PDF technique can be extracted through the
analysis of the peaks in the PDF data. For well-defined crystals, the peaks are sharp
and very well defined (series of delta functions). The difficulty arises when there are
peaks that involve two or more mixed shells with very little separation between them
as in glass where topological disorder exists.

Generally, extracting peak parameters (position, width and area) is obtained via
fitting the peaks with Gaussian functions convoluted with Sinc functions to account

for termination effects. Following is a description of these parameters:

PDF peak position:

Peak positions in the PDF data give the interatomic correlations in the structure.
The position of the first PDF peak, if it is a single component, gives the bond length
directly, which is very useful in understanding the local atomic structure [96]. The

position of the first PDF peak is also sensitive to homogenous strain in the structure.

PDF peak width:

In real materials atoms are displaced from their perfect positions due to thermal

motion and / or static displacements of the atoms. This gives rise to a distribution of

46

atom-atom distances, which causes the PDF peaks to be broadened. Generally, the
width of a PDF peak gives information about thermal and / or static disorder. For
the first PDF peak, if the number of nearest neighbors is constant, the integrated
area under the peak is invariant, and so the peak height, extracted directly from the
data, gives the inverse peak width. This often gives an accurate determination of
the evolution of peak width with some experimental parameter such as temperature
or composition. Three kinds of information may be obtained by studying the peak
width:

0 The width as a function of temperature yields information about the Debye

temperature of a bond [97].

o The width as a function of atomic separation yields information about correlated

atomic dynamics [98].
o The width as a function of doping gives information about doping-induced dis-
order.
PDF peak intensity

The integrated intensity under the PDF peak yields information about the number
of atoms at a specific distance (the coordination number). This type of analysis is

widely used in studies of glasses [99] and in partially crystalline samples [100].

2.3.1 PDF real space refinement

Conventional structure determination depends on the intensity and position of Bragg
peaks. The most common method for such analysis is the Rietveld method [101]. A
least-squares refinement between the calculated and observed intensities is performed

until the best match with the measured profile is obtained. The calculated intensities

47

are obtained based on the crystal structure, thermal factors, diffraction optics, instru-
mental factors, lattice parameters and other specimen characteristics. The success
of the method can be gauged by the publication of more than a thousand scientific
papers yearly using it [102].

Considering only the Bragg peaks assumes perfect long range periodicity of the
crystal. Such a presumption prevents the studying of non-periodic structures, or
aperiodic modifications of crystalline materials. However many important materials
are disordered and many of these materials owe their important properties to these
deviations from the average structure. These deviations result in the occurrence
of diffuse scattering, which contains information about two-body interactions and
which is disregarded as a background in the Rietveld method. This shows the need
for performing a similar refinement to the PDF data since it considers both kinds of
scattering.

Full profile structural refinement of the PDF can be carried out using the program
PDFfit2 [103]. In this method the model is defined in a small unit cell with atom
positions specified in terms of fractional coordinates. The PDF is then calculated
from the model structure and compared to the experimental PDF.

Following is a description of calculating the PDF data. Figure 2.4 illustrates
successive shells of atoms that surround a given atom. (1) Assume that we have a
sample that consists of N atoms at position 7'” with respect to some origin. Place
the origin of the space randomly at any atom. (2) Systematically find every other
atom in the sample and measure its distance from the origin. (3) Each time we find
an atom we place a tmit of intensity at that position, rm, on the axis of the radial
distribution R(r). When we have cycled over all of the atoms in the sample we move
the origin to another atom and repeat the process, adding the intensity to the R(r)
function. The unit of intensity for each atom-pair is then multiplied by bmbn/ < b >2

where b, is the scattering power of the it” atom. Dividing by the number of atoms to

48

 

 

 

 

123456
rlAl

Figure 2.4: Schematic illustration of calculating the PDF function for a crystalline
material. Images in this figure are presented in color.

keep R(r) an intrinsic function. The R(r) function is then given by Equation 2.5:

em = $2 (3360 — (r. — r...» (2.5)

 

the sum goes over all pairs of atoms m and n within the crystal, (b) is the average
scattering power of the sample.
The calculated PDF is then fitted to the experimental one and the goodness of fit

is determined by the parameter Ru, which is computed according to the relation:

 

 

= 2‘11 w(r‘)[G°bs(Ti) — 0.21147in
Rm \/ 2,11, w(ri)lGobs(Ti)]2 (2.6)

49

Here 10 is the weighting factor, w(r,) = 1/0,2 where a,- is the estimated standard
deviation on the ith data-point at position r,-. The sum goes over all measured data
points 7',- in the experimental PDF.

Both PDF and Rietveld refinements use the same parameters. The main difference
from Rietveld refinement is that PDF refinement allows for different 7' scaling, which
enables one to study the local structure for different r-ranges [21]. The ability to
refine the local structure yields information about disorder and short-range atomic
correlations.

Because of the similarity between the Rietveld method and the PDF modeling a
quantitative comparison between the resulting structures of both refinements may be
made. This is an important first step in revealing the existence of local distortions
beyond the average structure. It should be noted that the Ru, factor used in PDF
modeling is similar to that used in Rietveld analysis but the functions being fitted
are significantly different. Hence, direct comparison of Ru, from PDF and Rietveld
analysis should not be made. Ru, values are useful measures of the goodness-of—fit
when comparing how different models fit to the same PDF data. For well-crystallized
samples, PDF Ru, values greater than 10%, are not uncommon. Obtaining an Rw

value of less than 20%, for nano-crystalline structures is excellent.

2.4 X-ray absorption fine structure (XAFS)

X—ray Absorption Fine Structure (XAFS) spectroscopy has emerged as an incisive
probe of the local structure around selected atomic species in solids, liquids, and
molecular gases. The most important features of XAFS spectroscopy are its applica-
bility to amorphous materials and its tunability (the ability to probe the environments
of different elements in a sample by selecting the incident X—ray energy).

The importance of XAFS for the study of the structure of amorphous materials lies

50

in the fact that the theories describing it are equally valid for ordered or disordered
structures, and perhaps more importantly that the local structure around a given
atom may be ascertained simply by measuring the absorption near the edge of that
particular atom.

There are three fundamental interactions of photons with matter. These are the
photoelectric effect, the Compton scattering and the pair production. The character-
istic energy for XAFS, due to the photdelectric process, ranges from 3 to 40 keV
which is in the X-ray region. In this energy regime pair production is very energet-
ically forbidden since it requires a photon energy of at least two times 511 keV to
create both an electron and a positron. The probability for electron scattering is also
very small since the X—ray wavelength is much larger than the effective electron cross
section. A brief and simplified discussion of the basic ideas behind XAFS is presented

below.

2.4.1 The XAFS phenomenon

The XAFS and the photoelectric processes involve the total absorption of an X-ray by
an atom. The X-ray absorption coefficient of atoms is generally a decreasing function
of energy (because of the increase in energy, high energy X—ray photons can easily
pass through the sample), except at certain discrete energies (absorption edges) at
which there are discontinuous dramatic increases in the absorption. In an atom, every
atomic electron has an associated absorption edge.

When a photon of energy slightly higher than threshold is absorbed by an atom, a
photo-electron is ejected. The photo-electron wave propagates outwards to infinity if
the absorbing atom is isolated (e.g., an inert gas). In this case the absorption coeffi-
cient decreases smoothly from the absorption edge. However, if other atoms surround
the absorbing atom, as in a gaseous molecule or in a condensed phase, the outgoing

photo-electron wave will be back-scattered, and the back-scattered waves will inter-

51

fere with the outgoing waves as shown in Figure 2.5. The interference between the
outgoing and scattered part of the photoelectron at the absorbing atom modifies
the photoelectron. This interference changes the probability matrix elements for the
absorption process itself, for the absorption of an X-ray and is responsible for the
XAFS. Thus, as the X—ray photon energy is increased above threshold, the energy of
the photo-electron also increases and consequently the electron wavelength decreases.
Therefore, a maximum (minimum) in the absorption probability for the X—ray occurs
when the wavelength of the photoelectron, determined by the X-ray energy, and the
path of the photoelectron (the distance to and from the neighboring atoms) corre-
spond to constructive (destructive) interference between the outgoing and scattered
photoelectron waves at the absorbing atom, as shown in Figure 2.6. These periodic
oscillations above the absorption edge are called “fine structure”.

A precise measurement of absorption shows rich XAFS superposed onto the smooth
energy dependence. This has been historically split into two regions. The first con-
tains the fine structure from the absorbing edge to about 50 eV above the edge energy
and is referred to as X—ray absorption near edge structure (XANES). The second re-
gion contains the fine structure from 50 eV to about 1000 eV above the edge energy
and is referred to as extended X—ray absorption fine structure (EXAFS) as shown in

Figure 2.7. which illustrates the two regions.

2.4.2 XANES versus EXAFS

The prevailing behavior of an X-ray absorption spectrum is the monotonic depen-
dence of the X—ray absorption with the photon energy, interspersed with sharp edges.
However, the detailed shape of the edge and of the X—ray absorption spectrum above
it contains useful structural information.

The problem of calculating the outgoing waves in the strong field of adjacent atoms

in a solid or liquid sample is notoriously difficult. It has to be tackled in full for slow

52

 

Figure 2.5: Photoelectron outgoing (solid circles) and backscattered (dashed circles)
waves. The central red circle represents the X-ray absorbing atom. The dark gray
circles represent neighboring atoms. Images in this figure are presented in color.

Constructive interference

 

 

 

 

PXGE)“ ‘ <3

2‘5- Destructive interference M
2.0- fii

‘ M_r
1.5‘ mm—
I .O-I ] >
0 ‘ EXAF S

.. q
I I I I ;
l 7200 l 7400 17600 17800

Figure 2.6: Schematics of the EXAFS process illustrating the origin of EXAFS oscil-
lations due to the interference of outgoing and backscattered photoelectron wave. The
red curve represents a smooth background spline. Images in this figure are presented
in color.

53

 

XANES EXAFS
H

Absorbance

 

\7

E0

 

 

 

Energy

Figure 2.7: Typical XAFS spectrum showing the XANES and EXAFS regions. E0 is
the edge energy. Images in this figure are presented in color.

photoelectrons, i.e., when the incident photon energy is just above the threshold (the
XANES). The XANES region contains valuable information on chemical bonds and
the site symmetry. Further out from the absorption edge, the problem of the photo—
electron wave is considerably simplified. With the shorter photoelectron wavelength,
the adjacent atoms scatter the photoelectrons as point obstacles, each contributing a
tiny wavelet. Each atom scatterer contributes a harmonic oscillatory mode; together
they form a complex quasi-periodic EXAFS signal. Fourier analysis of this signal
resolves the harmonic components into a probability versus distance diagram. Its
peaks occur at rather accurate values of the neighbor atom distances. In addition,
the coordination number and chemical species of the neighbor atoms, as well as the
statistical spread of their distances due to thermal motion or static disorder can be,

in principle, deduced from the size and shape of the peaks.

54

2.4.3 Basic principles involved in XAFS

XAFS happens due to the photoelectric effect, which is described quantum mechani-
cally as the probability for the absorption of an X—ray by an atom. This probability,
as contained in Fermi’s golden rule, is proportional to the square of the matrix ele-
ment between the initial and the final states. The initial state consists of the electron
in an atomic core. The final state consists of a core hole and a photoelectron in the
conduction band, free from the absorbing atom.

The initial state is well localized at the absorbing atom, so it is only necessary
to determine the final state of the photoelectron at the absorbing atom. According
to quantum theory this photoelectron can be visualized as a spherical wave centered
at the excited atom and propagating away with wavelength given by the DeBroglie
relation:

h

where h is Plank’s constant and p is the momentum of the photoelectron. This
momentum is related to the photoelectron’s kinetic energy which equals the difl’erence

between the X-ray energy (E) and the electron binding energy (E0) and is written as:

 

P = \/2me(hV — E0) (2.8)

where me is the mass of the photoelectron, and 1/ is the frequency of the X—ray photon.

In the dipole approximation, where the wavelength of the photon is large compared
with the spatial extent of the excited core state, the absorption can be treated to first
order in perturbation theory and Fermi’s golden rule then yields for the absorption
coeflicient, #51

u. = 4N.w2e2§I<fIzIz'>rp(EF> (2.9)

where [2) and | f) are the initial core state and the final photo-electron wave function,

55

respectively, 02 is the X-ray angular frequency, p(EF) is the density of final states, No
is the number of atoms of one type in the sample, 6 is the charge on the electron and
c is the speed of light.

Since in the dipole approximation, the photon field can be regarded as being
spatially uniform, it can be approximated by a scalar potential proportional to the
distance (2) if the X-ray polarization is in the z-direction. Hence, only the matrix
element of the density of states in Equation 2.9 could give rise to oscillatory behavior
of #3. For photon energies well above threshold, p(EF) varies monotonically since
it is well described by the free electron value. Therefore, the matrix element alone
must be responsible for the oscillatory behavior, and this is because the final state
wave function | f) is made up of contributions from both the outgoing wave and the
back-scattered wave, and interference between the two modulates the matrix element
and hence a...

The phase of the interference is determined by the wavelength and the path of
the photoelectron. The amplitude of the interference is determined, in part, by the
type of neighboring atoms since they determine how strongly the photoelectron will '
be scattered. For the purpose of our study, emphasis is made on the EXAFS part
of the XAFS spectrum, because it is the part that reveals information about bond

lengths, coordination numbers and Debye-Waller factors.

2.4.4 The EXAFS equation

The EXAFS equation is the basis for theoretical models and the resulting fit pa-
rameters are used to extract structural information. The EXAFS equation for single
and multiple scattering events can be generalized from the EXAFS equation for a
photoelectron which has scattered from a single atom.

The oscillatory part of the measured absorption coefficient (p$(E)) is normalized
by the smooth atomic absorption background (no), resulting in the EXAFS signal

56

(x(E)), as shown in the following relation:

9,. E — o E

X(E) = “ ‘ #:(ES‘ ’ (2.10)
where E is the photo-electron energy which, however, is not known exactly since the
threshold energy (i.e., the zero of energy E0) can not be positioned precisely; instead
it is left as an adjustable parameter. The background absorption no is also difficult
to measure, and it is usually fitted by means of a polynomial. Based on this model,
the formula describing the EXAFS can be calculated and in atomic units is given
by [104]:

x(k)= —ZSON:k |————fj(7r xp-( Z)\Eln)exp(—2a§k2)sin(2k:RJ-+25(lc)+7)j(lc)) (2.11)

where a sum over all paths, j, is taken, each with degeneracy N j. 2R,- is the length
of the scattering path. The other parameters are identified in the discussion below.

The magnitude of the EXAFS is proportional to N j, inversely proportional to
R? (since both the outgoing and back-scattered waves are assumed to be spherical,
decreasing in amplitude as 1 / R) and proportional to the back-scattered amplitude
[fj (1r)] from the atoms in the jth shell. The amplitude is attenuated because of the
finite mean free path A6 of the electrons in the material and by the Debye-Waller
term involving root mean square (r.m.s.) displacements a,- (static or thermal) about
the equilibrium position.

The amplitude of the EXAFS is sinusoidally modulated by a function involving
the phase shift of the electron; the additional phase shift 60:) and 17(k) arise because
the photo-electron is emitted and back-scattered, respectively from atomic potentials.

The wavenumber of the photo-electron, k, is given by the relation:

Mil-1) = 27” = (2m??— E°))% (2.12)

 

57

where m. and E, are the mass and threshold energy for the photoelectron respectively,
and E is the energy of the X-ray photon. Note that k here is half the scattering

wavevector Q in Equation 2.13.

_ 41r sin(0)

Q01”) — —/\-— (213)

Thus, from Equation 2.11 we expect the EXAFS to be a complicated oscillatory
function, each path contributing a sinusoidal of a different period which mix together.

An additional amplitude reduction factor (SS) for x09) is introduced in the EXAFS
equation to describe effects of multi-electron excitations accompanying the photo-
effect in the inner shell. Note, however, that it is only the first few shells which
contribute strongly because of the limited mean free path of the photo-electrons, and
the large widths, a, of higher-lying shells.

The radial structure fimction, x(r), is then obtained via Fourier transformation

of the [tn-weighted EXAFS spectra, k”x(k), and is given by Equation 2.14:

1 kmaa:
X0) = W / X(k)M(k)k"exp(2ikr)dk (2.14)

kmin

Here M (k) is a window function, and n represents the k-weight used.

2.4.5 EXAFS measurements

In an EXAFS experiment, the energy of the incident X-ray is increased from approx-
imately 200 eV below to 1000 eV above the absorption edge energy of interest. This
changes the wavelength of the photoelectron (as given by Equation 2.12) and results
in the oscillations in the measured absorption coefficient (#2:)-

Two sources of broad-band X-rays may be used. Either by the Brehmstrahlung
spectrum from conventional X-ray tubes or the radiation emitted by electron syn-

chrotrons. Synchrotron radiation offers the advantage of ~ 104 increase in photon

58

(a) ’0 I [M
as: —0— I —0—I—0

Sample Standard
Fl
0)) "°h"°"'"

x-ray source Sllts I I
Mali—n

Monochromator Sample

Figure 2.8: XAFS experiment setup. (a) Transmission mode and (b) fluorescence
mode. Images in this figure are presented in color.

flux and concomitant reduction in counting time, in both cases the beam is monochro-
matized by a crystal.

EXAFS is conventionally measured using transmission and / or fluorescence geom-
etry. Typical XAFS transmission and fluorescence modes are shown in Figure 2.8.
For the transmission mode, the incident (Io) and transmitted (1,) X-ray intensities
passing through a thin foil of the sample of thickness d are measured as a function of
photon energ. The transmitted and incident intensities are related to the absorption

coefficient, am, by the relation:
It = Io exp(—/ixd) (2.15)

For dilute samples, the desired EXAFS is swamped by background intensity in a
transmission mode, and the more sensitive technique of fluorescence detection must
be employed. In the fluorescence mode, the K-shell hole left by the photo-electron
is filled by a p—electron from the L-shell, emitting an X—ray photon of characteristic

energy less than that of the exciting X-rays. Thus by tuning to the fluorescent

59

wavelength, only those atoms which are excited are monitored, with a consequent
dramatic increase in sensitivity.

With the availability of tunable, high flux, and high energy synchrotron radiation
sources, monochromatic X-ray beams with energy resolution (AE / E) of 10‘4 are
easily obtainable, allowing measurements of high quality absorption spectra in a short
time. These developments made XAFS spectroscopy one of the most widely used
methods for structural research of materials and make it the most unique probe for

identifying the local structure around atoms of a selected type in the sample.

2.4.6 Structural information obtained from EXAFS experi-

ments

In EXAFS, number and species of neighbor atoms, their distance from the selected
atom and the thermal or structural disorder of their positions can be determined from
the oscillatory part of the absorption coefficient above a major absorption edge. The
analysis can be applied to crystalline, nano-structural or amorphous materials, liquids
and molecular gases. EXAFS is often the only practical way to study the arrange-
ment of atoms in materials without long range order, where traditional diffraction
techniques cannot be used.

The structural information available from EXAFS experiments can be seen by
reference to Equation 2.11. In principle, the coordination number, N j, the interatomic
spacing, Rj, and the mean square deviation, of, for each shell, j, of atoms surrounding
the absorbing atom is obtainable.

Two methods of extracting the structural information are commonly employed.
One approach is to treat all the structural parameters in Equation 2.11 as adjustable
variables and to vary these (perhaps with crystalline values as input parameters)
until the best fit between the calculated EXAFS spectrum and the experimental data
is achieved. The major difliculty in this, as in most analyses of EXAFS, is what

60

values to take for the phase shifts 6 and 17. There are two possible remedies to
this problem: either values may be calculated theoretically by solving Schrodinger’s
equation taking account of the perturbations to be potential caused by neighboring
atoms, or else comparison is made with a standard, usually a crystal, for which the
structure is known and by fitting parameters to the EXAFS data of the standard,
values of 6 and n are thereby obtained.

The second method of analysis of EXAFS data is by taking the Fourier transform
(FT) of the measured EXAFS structure as shown in Equation 2.16:

1 kmaa:
¢(7‘)=W f1. . x(k)M(k)k”exp(2ikr)dk (2.16)

where M (k) is a window ftmction, and n can be 1, 2 or 3, the larger weighting the
data more at high k-values.

Via Fourier transform, contributions of individual shells of atoms (first, second
and perhaps higher shells) are separated visually in real space, as can be seen from
Figure 2.9 which represents the EXAFS x(k) for crystalline Ge together with its
Fourier transform. The peaks in the magnitude of the FT spectra appear at the
corresponding positions Rj. To obtain quantitative information on the local environ-
ment, such as number and species of neighboring atoms in a given shell, their distance
from the absorbing atom and their thermal or structural disorder, the peak of interest
is analyzed.

The theoretical basis of the EXAFS method is firmly established and the necessary
electron scattering data known with sufficient accuracy so that ab initio modeling of
the structure is possible.

Several computer programs have been developed for the quantitative analysis,
which take into account single scattering as well as multiple scattering contributions
to the EXAFS signal. In either method of getting structural information (fitting of the

model function to the measured EXAFS spectra in k-space or by Fourier transform to

61

 

k*x(k)
0.25

—O.25 O

 

 

 

 

|x(r)|

0.2 0.3 0.4

0.1

 

 

 

0

 

O 1 2 3

4
r01)

Figure 2.9: (a) The EXAFS x(k) and (b) the magnitude of the Fourier transform of
(a) for crystalline Ge. Images in this figure are presented in color.

R-space), interatomic distances can be determined with very high accuracy (typical
uncertainties below 1%), while for the number of neighbors and the corresponding
Debye-Waller factor lower precision (~ 10%) is only attainable, due to correlations

between the two parameters.

62

2.4.7 Joint EXAFS and PDF refinement of complex materi-
als

EXAFS and PDF are powerful experimental techniques for determining structural
information about materials. Despite the fact that both of them can be applied to
crystalline, nano—structural or amorphous materials, liquids and molecular gases, each
has some strengths and weaknesses that are clearly differentiated one from the other.

On one hand, the PDF method has had some success in characterizing the 10-
cal structure of challenging complex materials. PDF theory is conceptually easy to
understand and to apply. PDF data covers a large spatial range, which makes a
good tool for measuring physical effects that manifest in the medium-range order,
such as charge density waves. However, in unfavorable cases the PDF method cannot
distinguish unambiguously between different structural models.

On the other hand, the chemical specificity of EXAFS makes it an excellent tool
for determining the immediate environment of the absorbing atom. However, due to
the fast decay of the EXAFS signal, such an analysis can seldom be extended past
the second or third coordination shells.

It is certain that there are many problems that are better solved by one technique
over the other. One challenge in determining when to use EXAFS or PDF, or both in
parallel, is comparing the results of an EXAFS and PDF analysis on an equal basis.

One way towards better understanding of the structure of materials is to com-
bine local structural information from different local structural methods. Complex
structural refinement using both XAFS and PDF is a promising approach to achieve
this as two methods complement each other. On one hand, combination of these two
complementary local structural methods in a quantitative and self-consistent manner
holds the promise of getting much more accurate local structural information and is
expected to result in better understanding of the local structure of materials. On

the other hand, refining both XAFS and PDF data for challenging'complex materials

63

will accurately distinguish between two or more structural models that can not be

achieved by a single method.

64

Chapter 3

Search for a structural response to

the IP in GemSe1_z glasses

3.1 Introduction

According to constraint counting theory [18, 24], a network is considered floppy if the
average number of constraints per atom (no) is less than 3 (the number of degrees of
freedom for an atom in 3 dimensions), as in the case of twofold coordinated single
bond chain networks. On the other hand, the network is rigid if n6 is greater than 3
as in networks consisting only of tetrahedral units (such as 8102). Thus, a progressive
addition of cross-linking elements (such as Si, As or Ge) to a starting chain network
(such as S or Se) will result in a progressive increase of its connectivity (mean co-
ordination number (f)). At 7‘ = 2.4, rigidity percolation occurs where the network
changes from floppy to rigid structure [5]. A number of experiments [26, 28, 27] show
responses consistent with a percolation transition.

Canonical systems for studying this phenomenon are the chalcogenide glasses.
Surprisingly, based on thermodynamic and spectroscopic measurements, they appear
to exhibit two transitions instead of one [3]. The region between these transitions has

been called the intermediate phase (IP) [3, 4]. The original theoretical work assumed

65

that the network was generic and the connectivity random [5]. It was therefore
suggested [1] that the IP phase is a region of finite width in composition where the
network could self organize in such a way that maintains a rigid but unstressed state.
However, it has proved very difiicult to establish this result experimentally.

If the interpretation of IP is correct, one expects to see a direct structural response
to the transitions. It should be apparent in structural parameters sensitive to strain,
since it is a transition from unstressed to stressed. Self-organization in the IP may
also be apparent by observing the appearance of intermediate range structural order,
for example, changes in the first sharp diffraction peak (FSDP).

To test the hypothesis of the IP, a careful study has been made of a series of
carefully prepared glasses in the GexSe1_,, system. This system can be made as glasses
over a wide composition range (a: = 0.0 to 0.42 of germanium) [50, 51, 53, 52] allowing
the systematic composition dependence to be studied. Here is described a detailed
systematic composition-dependent study of structural parameters in GexSe1_$ glasses
covering a wide range of composition around the IP (0.15 S :r g 0.40) with a narrow
spacing between points. X-ray diffraction data were measured using advanced high
energy synchrotron radiation with complementary X—ray absorption fine structure
measurements.

The diffraction data have been processed to obtain the total scattering structure
function, S (Q), and the reduced pair distribution function (PDF). The width of the
first peak in the PDF, which contains information about strain in the system, has
been extracted as a function of composition, at. Information about intermediate range
order (IRO) has also been extracted from the FSDP versus :12. A study by Sharma et
al. [7] indicates that the position and width of the FSDP in this system has an anomaly
in the region of the IP, from which they ruled out the existence of three well-resolved
phases consistent with the IP. This result was not reproduced in our work. In fact,

there is no clear evidence for a structural responses from the PDF and XAFS data

66

that correlate with the expected appearance of the IP or the rigidity transition.

3.2 Experimental procedures and data reduction

Bulk glass samples of GezSe1_,,, where a: = 0.15, 0.16, 0.18, 0.19, 0.20, 0.21, 0.22, 0.23,
0.24, 0.26, 0.28, 0.30, 0.33, 0.34, 0.35, 0.36, 0.38 and 0.40 were prepared by a conven-
tional melt-quenching process. The starting ingredients (99.9999% Ge and Se) were
vacuum-sealed (5 x 10‘7 Torr) in quartz tubes, heated to 950 °C for 4 days or more,
and thereafter melt temperatures were slowly lowered to 50 °C above the liquidus (the
temperature above which a substance is completely liquid), where they equilibrated
for 6 hours before quenching in cold water. Samples were allowed to age for 3 weeks
before quartz tubes were opened, and glass transitions were examined in modulated
differential scanning calorimetric measurements. A scan rate of 3 °C / minute and a
modulation rate of 1 °C/ 100 seconds was used to record scans. The frequency cor-
rected non-reversing enthalpies, AHm(a:), for the samples were measured, and showed
a square-well like global minimum as shown in Figure 3.1. As in earlier studies [48],
there is a clear minimum in AHm(a:) which is used as an indicator for the IP and
showing the quality of the samples.

The glasses were gently crushed into fine powder, formed into discs 5 mm in
diameter and 1 mm thick, sealed between thin Kapton foils and subjected to X-ray
diffraction experiments. This approach ensured that the samples in the beam were
of uniform geometry.

The X-ray diffraction measurements were carried out using the rapid acquisition
PDF (RAPDF) technique [105] at the MUCAT 6-ID-D beam line at the Advanced
Photon Source (APS), Argonne National Laboratory at room temperature. A bent
doubleLaue Si <111> crystal [106] was used to monochromatize the white beam
and deliver an intense flux of X—ray photons of energy 87.005 keV (A = 0.14250 A).

A large area image plate detector (Mar345) was placed 108 mm behind the sample.

67

 

 

 

' I [1 I I l I
N I I '
A ”o i
32- I I 4
> - I I o -
3"" GI I'o ‘
"r ' I I "
I‘m... 0 _
<10 + l ]
- o
.. 0 0‘ -
CDI . l . J . I .

 

0.15 0.2 0.25 0.3
Ge content

Figure 3.1: The non-reversing heat flow (AH m) versus composition for the samples
used in this study. The results indicate that these samples exhibit the experimental
signature associated with the presence of an IP [3], which can be located from this
plot in the region between 0.20<x<0.25. The two vertical dashed lines are plotted at
the boundaries of the predicted IP.

The sample-detector distance was calibrated using a silicon standard of known lattice
parameter. The use of X—rays of such high energy allows the access of higher wave
vectors and helps reduce several unwanted experimental effects such as absorption and
multiple scattering. Five diffraction scans, with an irradiation time of 100 seconds
each, were conducted for each sample and the diffracted intensities were averaged to
improve the statistical accuracy and reduce any systematic effects due to instabilities
in the experimental set up. Integration of the MAR images was performed using
program Fit2D [107]. Data reduction to obtain the structure ftmction, S (Q), and the
PDF, G (7‘), were performed using program PDFgetX2 [108].

The structure function, S (Q), is related to the coherent part of the total diffracted

intensity of the material, and is given by:

I°°”(Q) - Zcilfz-(QN2

S(Q) = lZcf.(Q)|2

+ 1 (3.1)

 

68

 

l

I
'(T’T
"»
I

A’,/’:’- I

1 5
2«(

F(II“)

10
e (is,

5

 

 

o 5 10 15 20 25
out“)

Figure 3.2: The measured structure functions, S(Q)’s, for the Ge,,Se1_2 glasses. Data
are offset up for clarity. The compositions (z) are, from the bottom to the top curves,
0.15, 0.16, 0.18, 0.19, 0.20, 0.21, 0.22, 0.23, 0.24, 0.26, 0.28, 0.30, 0.33, 0.34, 0.35,
0.36, 0.38 and 0.40. Images in this figure are presented in color.

 

where I c°"(Q) is the measured scattering intensity from a powder sample that has
been properly corrected for background and other experimental effects and normal-
ized by the flux and number of atoms in the sample. Here, c,- and f,- are the
atomic concentration and X-ray atomic form factor, respectively, for the atomic
species of type i [93, 21]. Figure 3.2 shows the measured reduced structure func-
tions, F(Q) = Q[S(Q) — 1], for all of the studied GexSe1_z glasses.

The overall similarity of the F(Q) curves indicates that there is no appreciable
unaccounted background, or systematic effects interfering with the signal from the
sample. The overall shapes of the curves (including the oscillations about zero at

high-Q) are remarkably similar suggesting that the data reduction and normalization

69

G(K'z)

 

0 2 4

6 8 10
r01)

Figure 3.3: Experimental atomic PDF G(r)’s for GeISe1_,, glasses. Data are offset
up for clarity. The compositions (x) are, from the bottom to the top curves, 0.15,
0.16, 0.18, 0.19, 0.20, 0.21, 0.22, 0.23, 0.24, 0.26, 0.28, 0.30, 0.33, 0.34, 0.35, 0.36,
0.38 and 0.40. Images in this figure are presented in color.

procedures are effective. The curves have not been smoothed and the low level of
noise, even at high-Q, is apparent.

The data were then Fourier transformed to obtain the reduced pair distribution
function, G(r), and these are shown in Figure 3.3. From the experimental PDF’s
shown in Figure 3.3, there are a relatively small number of well defined peaks which is
a signature of such glassy materials since the stronger the disorder in a noncrystalline
material the weaker the correlations between the positions of the atoms in it and,
hence, the lower the number of well-defined peaks in the PDF. The peaks occurring
at short distances (below the first PDF peak at 2.37 A) are unphysical, and are due

to imperfections in the data reduction.

70

The GexSeH, glasses were also subjected to XAFS measurements. Finely pow-
dered glasses were uniformly spread on adhesive scotch tape. Eight to twelve layers
were stacked to get optimized signal-to—noise levels. The XAFS measurements were
performed in the transmission mode at the bending magnet beamline, 5BM-D, at
the Advanced Photon Source, using Si <111> double crystal monochromator. The
energy resolution of the monochromatic beam was determined to be ~ 1.5 eV for
near-edge structure and about 3 eV for the EXAFS. A beam size of about 0.3 x 0.5

2 was used resulting in an incident photon flux of ~ 1010 photons per second.

mm
The synchrotron storage ring operated in the “top-up” mode with the electron beam
current kept around 100 mA.

The monochromator was scanned in energy from 200 eV below to 800 eV above
the Ge and Se K-absorption edges (11103 and 12659 eV, respectively). The inci-
dent, Io, and the transmitted, 1,, X-ray intensities were measured simultaneously at
room temperature using ion-chambers located before and after the sample. The ion-
chambers were filled with a mixture of nitrogen and argon gases. Data were collected
with a step size of 0.20 eV in the respective edge regions. The energy calibrations
were performed using Ge and Se foils between the It and a third, Inf, ion chambers,
respectively.

The EXAFS data reduction was performed using standard procedures using the
Athena software package [109, 110]. The measured absorption spectrum below the
pre-edge region was fit to a straight line. The Autobk algorithm [111] implemented in
Athena was used to determine the background and normalize the X—ray absorption,
ME), data. This algorithm uses a cut off parameter (Rbkg) to define the Fourier
frequency below which the signal is dominated by the background and above which the
signal contains the data. Thus Autobk attempts to remove those Fourier components

that are due to the background while leaving those that contain the data.

Edge-step normalization was also done by the Athena software where the difference

71

between the raw data, 11(E), and the background, 110(E), is divided by an estimation
of the edge step value (110(E0)) resulting in the EXAFS signal, x(E), defined by the

following relation:
x03) = “(2%?” (3.2)

The normalized x(E) spectra were then converted to x(k) in k-space, k = (87r2m(E—
Eo)/h2)1/2. The resulting x(k) functions were then weighted with k2 to account for
the damping of oscillations with increasing k. Based on this, the formula describing
the EXAFS (Equation 2.11) was calculated.

Figure 3.4 shows the XAFS k2x(k) for all the studied samples at both Ge and Se
edges. The pure oscillations in k2x(k) are consistent with such glassy material, in
which they result in a single well-defined shell after being Fourier transformed.

The radial structure function, x(r), was then obtained via Fourier transformation
(Equation 2.14) of the normalized 102-weighted EXAFS spectra, kzx(k), using a k
range of 2.5 to 15.9 A‘1 for both Ge and Se edges. Here M (k) is a window function,
and n represents the k-weight used, which was 2. The Fourier transforms of the
measured EXAFS signals at both Ge and Se edges are shown in Figure 3.5. The
Fourier transforms at both Ge and Se k-edges clearly show a well-defined first shell,
as expected, but the second and higher order shells appear to be observable at a
level only slightly above the noise level. This decreased information in the higher
shells reflects the fact that the considerable disorder (which manifests itself in large
Debye-Waller factors) plays a dominant role in these glasses.

EXAFS data fitting was performed using the Artemis program [110]. A single
scattering path was used to fit each bond type. For Ge—edge data, Ge—Se and Ge-Ge
single scattering paths were used, and for the Se-edge data, Se—Ge and Se—Se single

scattering paths were used.

72

161(k)

15

10

15

 

'ITT'I'I'T‘I'I'I'I

WWW

 

 

W
W

l I I I l I I I L I I I I I I I I

 

2 4 6 81012141618
k(fi")

 

 

 

.l.l.|.l.l.l.l.l.l

 

2 4 6 81012141618
k(fi"’)

Figure 3.4: [1:2 weighted XAF S signal (k2x(k)) for Ge,,Se1_,c glasses at (a) Ge-edge
and (b) Se—edge. Data are shifted up for clarity. The compositions (r) are, from the
bottom to the top curves, 0.15, 0.16, 0.18, 0.19, 0.20, 0.21, 0.22, 0.23, 0.24, 0.26, 0.28,
0.30, 0.33, 0.34, 0.35, 0.36, 0.38 and 0.40. Images in this figure are presented in color.

73

|x(r)l

 
 

 

o
01234567891C

O

|x(r)|

O
012545678916
r01)

Figure 3.5: Magnitude of the Fourier transform of [1:2 weighted XAFS signals (kzx(k))
for GeISeH, glasses at (a) Ge—edge and (b) Se—edge. Data are shifted up for clarity.
The compositions (x) are, from the bottom to the top curves, 0.15, 0.16, 0.18, 0.19,
0.20, 0.21, 0.22, 0.23, 0.24, 0.26, 0.28, 0.30, 0.33, 0.34, 0.35, 0.36, 0.38 and 0.40.
Images in this figure are presented in color.

74

3.3 Results and discussion

The transition from floppy to rigid in GezSe1_,, glasses should be accompanied by the
appearance of strain in the overconstrained region above the transition. One expla-
nation for the appearance of the intermediate phase is that, due to self-organization
of the network, there is a region of finite composition width that is rigid but un-
strained [1]. A measurement of the residual strain-state of the glass could give direct
support to this picture. In the PDF a homogeneous strain can be detected as a shift
in bond—length (PDF peak position) and inhomogeneous strain as a broadening of the
bond-length distribution. The first peak in the PDF yields direct information about
bond-stretching in the nearest neighbor bonds. Changes in peak positions and widths
of higher order peaks yield information about bond-bending relaxations in covalent
systems [112, 113]. Things are complicated in the glass because of peak overlap be—
tween peaks of different structural origin. Nonetheless it is still interesting to look
for evidence of strain in these glasses. Initially, the nearest neighbor PDF peak is
considered.

The first PDF peak in Ge$Se1-3 glasses is a multi-component peak. It has three
unresolved contributions: Tse-se = 2.32 A, Tee—Se = 2.37 A, Tee—Ge = 2.42 A. The
relative intensities of these are determined by the Ge content of the glass. At the low
Ge contents of interest, one expects the peak to be dominated by Se—Se and Ge-Se
correlations. The contribution of each of these sub peaks can be estimated by using
one of two distinctly different models that are consistent with the ‘8-N’ coordination
rule.

The first one is a random covalent network model (RCN) [19], in which the world
random is used to indicate that the distribution of bond-types is statistical and hence
completely determined by the respective local coordinations and the fractional con-
centrations of each atom type. This approach to the bonding neglects factors such as

the relative bond energies. The random covalent bonding description includes Ge-Ge,

75

Table 3.1: Single bond energy in the GexSeH, system.
[Bond Type ] Bond Energy (kcal/moleU

 

 

 

Ge—Ge 37.6
Se-Se 44.0
Ge-Se 49.1

 

 

 

 

Ge-Se and Se—Se bonds at all compositions other than :6 = 0 and a: = 1.

The alternative model for bond statistics is called the chemically ordered network
(CON) model [114]. This model emphasizes the relative bond energies and favored the
hetro-polar (Ge-Se) bonds than the homo-polar (Se-Se or Ge—Ge) bonds. The single
bond energies in the Ge$Se1_z system are given in Table 3.1. The CON model con-
tains a chemically ordered compound phase at a composition me = N53/ (N03 + N33),
where N,- is the coordination number of atom of type i. This phase is the stoichio-
metric composition (GeSeg), a: = 0.33 in this case, which, according to this model,
contains only Ge—Se bonds. For compositions below the stoichiometric composition,
the alloys contain Se—Se and Ge—Se bonds; whereas for compositions defined above
the stoichiometric composition, the alloys contain Ge-Se and Ge—Ge bonds.

Figure 3.6 shows the bond counting statistics for the Ge,.,Se1_,c system based on
the RCN and CON models.

Although neither model is perfectly correct, the existence of a compound phase
at a—GeSe2 suggests that chemical ordering is preferred in the Ge$Se1_$ glasses at
least over the range of bulk-glass formation (x = 0.0 to 0.42 of germanium). This is
supported by other structural measurements, for example PDFs obtained by X—ray
diffraction [115, 116] and by structural interpretations of IR and Raman vibrational
spectroscopy [117, 114], where below x = 1 / 3, the fraction of Ge—Ge homopolar bonds
is less than few percent [79]. We therefore chose to concentrate on the CON model.

In the region of interest, for compositions in the vicinity of the IP, the CON

model predicts only Ge-Se and Se-Se bonds to be present. The width of the bond

76

 

 

 

 

 

        
 

 

 

 

N_ I '. I T I
I» : [’1
w;_ l / ..
416 s | /’ .
\ I
.3’K I ’ '
c _\ I [I d
8 \ "r--/‘~
"Q- \< 1 I ‘\ .-
0 ’ x I \
I. ’ 1“ x ,
I ,’I ~._ \
owl 41 L 4 T‘F-A_g\
0 0.2 0.4 0.6 0.8 1
Ge content
N '. I r I ' I
l
l
IQ l
g.— (3qSe2
«06' I-
\
03'-
U
C
8‘0.
C
o I I
0 0.2 0.4 0.6 0.8 1

Ge content

Figure 3.6: Bond counting statistics for the GexSe1_z networks based on the random
covalent network (RCN) model (upper panel), and the chemically ordered network
(CON) model (lower panel). Red, Blue and Green curves represent Se—Se, GeSe and
Ge-Ge bonds, respectively. The dashed vertical line is plotted at the stoichiometric
composition (GeSeg). Images in this figure are presented in color.

distributions is expected to be ~ 0.1 A at room temperature, whereas the separation of
the centroids of the two peaks is expected to be 133-5,, - r584, = 0.05 A. The intrinsic
width of the peaks is greater than the separation of the peaks and the sub-components
are not resolved. Indeed, the intrinsic width of the sub-components dominates the
observed PDF peak width and results in a peak that is quite Gaussian in its intrinsic
shape, convoluted with the Sinc function coming from the finite termination of the

data in Q-space, as observed in Figure 3.3.

In analyzing the first PDF peak, the parameters have been extracted from both

77

experimental and simulated data. In particular, it has been fit with a single Gaussian
convoluted with a Sinc function to simulate the finite Qmaa, of the measurement.

The width of the first PDF peak contains the desired information about peak
broadening due to strain effects, but because it contains more than one unresolved
bond, its position, width and intensity also evolve due to the changing composition
of the network. To separate these effects, the first PDF peak is simulated taking
into account the changing chemistry. The simulation was done in two ways. In
the first way, the expected behavior of the average position and width was studied
versus doping assuming that the peak consists of three components whose positions
are fixed at the above values and where the widths of the three components are not
changing with doping. In the second way, the simulation was repeated by changing
the positions and widths of the sub-peaks as obtained from the XAFS results. In both
ways, the intensity of each sub—peak is governed by the expected concentration from
the CON model, scaled by the appropriate product of the scattering amplitudes of the
components. A simulated PDF peak was produced from these models by summing
together the three sub—components. Both ways are relevant, as the first way uses
fixed positions and widths, it represents the unstrained model, while the second way
considers the evolution of the experimental positions and width as obtained from
EXAFS measurements, and hence tracks the changes in positions and widths due
to both chemical and strain effects. Similar to the experimental data, the resulting
simulated peak for each composition was then fit using one Gaussian convoluted with
a Sinc function. Figure 3.7 shows a representative plot of the fits to the experimental
and simulated first PDF peak.

The average positions of the first PDF peak, for both experimental and simulated
data, for all the samples, are shown in Figure 3.8. The red circles give the expected
behavior obtained from the simulated data. This shows the behavior with no change

in intrinsic peak width due to strain, but only changes due to the composition of the

78

 

 

 

r01)

 

I n I I I
1.8 2.1 2.4 2.7

IIIJIIJQI

    

 

 

 

 

 

 

2.7

1.8 2.1 2.4
r(A)

Figure 3.7: Representative plot showing the quality of the fit to the first PDF peak of
the Gaussian peak (convoluted with a sine function) in the GeSe2 glass. (8) The fit
to the experimental data and (b) the fit to the simulated data (see text for details).
Experimental or simulated data (blue circles), the fit (red curves). Offset below are
the differences (green curves). Images in this figure are presented in color.

 

 

 

 

 

'91) _I ' I ' | ' I ' I ' I—
o; I
' C
. I .
$300 I _
V").- I § i i .'
C N I i .
.9 _ I O _
'8 ,\ II E o T
D- 10 _ | _
(\i I
I- I -(
l
L l 4 1 L 1 I 1 l l l
0.15 0.2 0.25 0.3 0.35 0.4
Ge content

Figure 3.8: Position of the first PDF peak as obtained from fitting the experimental
PDF data with a single Gaussian (black) and from fitting a simulated PDF peak based
on the XAFS results (green) and CON model (red). The vertical dashed lines are
plotted at the IP boundaries. See text for details. Images in this figure are presented

in color.

79

 

0.7
T

 

0.06
I

0.15 0.2 0.25 0.3 0.35 0.4
Ge content
Figure 3.9: Width of the first PDF peak as obtained from fitting the experimental
PDF data with a single Gaussian (black) and from fitting a simulated PDF peak based
on the XAFS results (green) and CON model (red). The vertical dashed lines are

plotted at the IP boundaries. See text for details. Images in this figure are presented
in color.

Width (21)

 

 

 

peak based on the chemically ordered model and assuming that rGe_5e and 733-5,,
retain their nominal values.

Figure 3.8 shows that the average experimental position of the first PDF peak
(black symbols) agrees well with the expected one from the CON model (red symbols),
with positive deviation that starts to appear after the stoichiometric composition. No
special features are seen in the position of the first PDF peak that may correlate with
the IP.

Of greater interest is to look for evidence of inhomogeneous strain from a peak
broadening. The experimental as well as the simulated width of the first PDF peak
are plotted versus composition in Figure 3.9. There is no apparent change in width,
or even in the slope of the width versus doping, associated with the IP. Any change

in the bond-stretching strain of the sample on passing through the IP must be below

80

the sensitivity of the measurement. This is not surprising as covalent bonds are
rather stiff, and the sensitivity of the measurement is limited because of the multi-
component nature of the first PDF peak. The experimental width of the first PDF
peak is consistent with the simulated one (based on the CON model, where the change
is just due to change in composition of the sample), and shows no correlations with
the IP.

The EXAFS data also contain information about the atomic pair distributions of
the near-neighbor peaks. This information can be extracted by modeling the data.
The relative advantage of EXAFS here is that it allows the positions and widths of the
individual sub-components of the first PDF peak to be separated. The disadvantage
is that parameter correlations arise because of the number of variable parameters in
the fits, which can result in biased results. This was mitigated here because data
from both the Se and Ge edges were collected. In the composition range below the
stoichiometric composition, GeSeg, the number of Ge-Ge homopolar bonds is no more
than a few percent [79]. The analysis of the EXAFS data was concentrated to this
region, neglecting the contribution coming from any Ge—Ge component. The nearest
neighbor coordination shell of the Ge edge data therefore consists of only Ge-Se bonds,
whereas that of the Se edge contains Se—Ge and Se—Se. These edges were fit together,
as described below, which allowed us to extract parameters from these edges with
greater reliability.

In the analysis of the XAFS data, a single scattering path was used for each bond
type. At the Ge—edge, and according to the CON model, only Ge—Se bonds exist,
for a: _<_ 0.33, and so, a single scattering path of Ge tetrahedrally coordinated with
Se atoms was used. For the Se-edge data, two single scattering paths were used to
account for Se-Se and Se-Ge bonds. Ge—Se and Se—Ge path parameters were set to
satisfy the bond consistency relation and the ‘8~N’ rule and sharing the path length

and Debye-Waller factors.

81

 

 

 

l l
I I
#5,"!
try W 1:: 2. l 4:151 .4 I A
‘12. “T “55.13“”? 5;; c) “4:;
AN
5 - : : -
ID I
10.- II I... _
. I
N T I. I
r I I -
o, I, 1 1 : ] : ] :
Qr(b) l | -
05:31 ....t:_:|,fi a» a a-
co | I“
A
50:“ I I -
b0 ll-II' I
to | I
O. I
o I

0.15 0.2 0.25 0.3
Ge content

Figure 3.10: Refined bond lengths (a) and Debye—Waller factors (b) versus .7: (Ge
content). Red circles represent Ge—Se bonds and blue squares represent Se—Se bonds.
The vertical dashed lines are plotted at the lower and upper boundaries of the IP.
Images in this figure are presented in color.

 

 

 

Figure 3.10 shows the refined bond lengths and Debye-Waller factors (a) for both
Ge—Se and Se—Se bonds as a function of Ge content. The bond lengths of the sub-
peaks are close to their nominal values and almost fixed (within their uncertainties)
with Ge content. On the other hand, the Debye-Waller factors for Ge—Se bond are
also fixed (within their uncertainties) with Ge content, but the Se—Se Debye-Waller
factors are decreasing with Ge content, probably due to the decreasing number of

Se-Se bonds when Ge content is increased. No clear correlations exist with the IP,

82

such as a pronounced minimum in the IP compositions range (0.20 to 0.25 in Ge
content).

It is desirable to compare the results of the EXAFS with those from the PDF
to check for consistency. They cannot be compared directly because the PDF data
represent total PDF, and hence information about the partial PDFs (Se-Se, Ge-
Se and Ge—Ge) can not be extracted individually. EXAFS data at both Ge and
Se edges contain structural information about the sub-components (positions and
widths) and so it is therefore possible to simulate the first peak in the diffraction
PDFs directly from the EXAFS data, by summing together the properly weighted
sub-component peaks. This was done and the resulting curves were fit using the same
protocol that was used to fit the PDF data and the simulated peak from the CON
model. The results are shown as the green symbols in Figure 3.8. There is excellent
quantitative agreement with both the diffraction PDF and the result simulated from
the chemically ordered network (CON) model. It is interesting to note that the
EXAFS results indicate that both the Se—Se and Ge-Se bonds are getting slightly
shorter with increasing Ge content, but the position of the compound peak shifts to
higher-r with doping. This is a result of the fact that the contribution of the longer
Ge—Se bond is increasing with 2:.

Bond-bending forces in covalent materials are much weaker than bond-stretch [1 13]
and so one would expect to see a larger response of second and higher neighbor peaks
in the PDF due to a change in the stress state of the sample at the IP. Unfortunately
the second neighbor peak has multiple contributions and interpretation of changes
in peak width, shape and position is somewhat ambiguous. This peak does exhibit
significant changes with composition that can be seen in Figure 3.11. These are
thought to relate to an evolution of the underlying network connectivity, such as the
appearance of edge-sharing GeSe4 tetrahedra [118] as well as effects due to benign

compositional changes. This peak is mainly composed of Se-Se correlations within

83

 

 

 

 

 

Figure 3.11: Second PDF peak plotted on an expanded scale for a selection of the
samples (a: = 0.15 (red), :1: = 0.20 (dark red), :1: = 0.23 (dark magenta), a: = 0.26
(gray), x = 0.33 (dark green), :1: = 0.40 (black). The peak contains multiple contribu-
tions and so changes shape and position with changing composition. Images in this
figure are presented in color.

each tetrahedron and a left-side shoulder that is due to Ge-Ge correlations among
corner and edge shared tetrahedra.

Another important structural indicator that is sensitive to intermediate range
order in a glass is the first sharp diffraction peak (FSDP) [87 , 119]. A response of
this peak to the 1P may indicate an underlying structural ordering consistent with
the idea that the finite width of the IP is a response of the system to remove strain
by introducing structural ordering.

Figure 3.12 shows the S (Q)s for some selected samples in the FSDP region. A
dramatic change in the height and position of the FSDP is clearly observed, in agree-
ment with earlier studies [119]. As evident in Figure 3.12, the FSDP is a strongly

varying function of composition, rising out of a smoothly varying background when

84

 

1.25 1.5

1

 

 

1 l 1 L 1 J 1 I

2.5

 

0 0.25 0.5 0.75

1.5 2
Q (1")

Figure 3.12: Low-Q part of the structure function (S(Q)) for selected samples
(:8 = 0.15 (red), 2: = 0.20 (dark red), :1: = 0.23 (dark magenta), :6 = 0.26 (gray),
:6 = 0.33 (dark green), and r = 0.40 (black)). With increasing Ge content the first
sharp diffraction peak develops in height and shifts to the left until the stoichiometric
composition (a: = 0.33). At higher Ge content it decreases in height. Images in this
figure are presented in color.

Ge is added to selenium glass.

Model independent peak fitting was performed on the S (Q) spectra to extract the
FSDP parameters (position, area, width and height). Due to the uncertainty of the
FSDP profile shape, the S (Q) spectra were analyzed using different fitting protocols.
These include pure Gaussian, pure Lorentzian and pseudo-Voigt which is a linear
combination of Gaussian and Lorentzian line—shapes.

The fits included the first three peaks in the S (Q) curves, with the fitting range
extending from 0.5 A‘1 to 4.5 A“. All the fitting protocols used a zero background.
Figure 3.13 shows the quality of the fits obtained using the above fitting protocols for
two representative sample compositions. The fits of pure Gaussians to all three peaks

were qualitatively worse than the others and so they are not considered further.

85

 

5
00.51 1.5

 

S
0' 0.5 1 1.5

 

 

S
O 0.51 1.5

 

 

 

1 2 5 4 1 2 5 4
Q(A") 0(8")

Figure 3.13: Plots indicating the quality of the fits to S (Q) data for two representative
samples (Geo,2OSeo,30 in the left panels and GeSez in the right panels) using three
different fitting protocols. (a)-(b) Pure Gaussians, (c)-(d) pure Lorentzians and (e)-
(f) pseudo-Voigt type functions. Experimental data are shown as circles, red solid
curves represent the fits with the difference curves offset below. Images in this figure
are presented in color.

86

Table 3.2: X2 values for the different fitting protocols to the Q—space data. The
different fitting protocols are: Pure Lorentzian (protocol 1), pseudo-Voigt but with
the FSDP constrained to be Gaussian (protocol 2) and pseudo-Voigt with the FSDP
line shape being allowed to change between Gaussian and Lorentzian (protocol 3).
x Protocol 1 Protocol 2 Protocol 3
0.15 0.0019 0.0019 0.0016
0.16 0.0018 0.0018 0.0015
0.18 0.0018 0.0017 0.0015
0.19 0.0019 0.0016 0.0016
0.20 0.0020 0.0016 0.0017
0.21 0.0017 0.0016 0.0015
0.22 0.0019 0.0017 0.0016
0.23 0.0019 0.0016 0.0016
0.24 0.0018 0.0016 0.0015
0.26 0.0019 0.0016 0.0017
0.28 0.0022 0.0018 0.0019
0.30 0.0023 0.0018 0.0021
0.33 0.0023 0.0018 0.0022
0.34 0.0042 0.0025 0.0035
0.35 0.0038 0.0021 0.0031
0.36 0.0034 0.0027 0.0027
0.38 0.0032 0.0026 0.0026
0.40 0.0032 0.0027 0.0027

 

 

 

 

 

 

 

 

Table 3.2 lists the x2-values of the different fitting protocols to the S (Q) data for
all the GeISe1_,, glasses:

Figure 3.14 shows the results of the FSDP parameters as obtained from the dif-
ferent fitting protocols. The results from more than one fitting protocol are included
to assess the variability of the results. Greater confidence can be ascribed to results
that are reproduced between the different fitting protocols. The results of the pure
Gaussian fits were not included as these fits did not reproduce the FSDP profile shape.

In each plot in Figure 3.14 the green symbols are the results for Lorentzian line-
shapes and the red and blue are the results of pseudo-Voigt fits. In the case of the red
symbols the FSDP was constrained to be purely Gaussian but the mixing coefficients
in the pseudo—Voigt function were allowed to float for the other peaks, showing the

peak to have a line-shape intermediate between Gaussian and Lorentzian. In the

87

 

 

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1’) 1 1 1 1 1
TA..- 0) I I 1 I' _
5°! | | | _
c . .
fir I I I -
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in!» ' ' i _
1,". l l ,' k: -
vs: 9 I 0
ref ' "' '1
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0.5 o 4
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’44.
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60
:55" H -
.g’ . | I -
Ig— I I I.
- .
g-I . I . I 111111

 

0.15 0.2 0.25 0.3 0.35 0.4
Ge content

Figure 3.14: Fit parameters for the FSDP in binary GexSe1_z glasses as a function
of 2:: (a) position, (b) area, (c) width and ((1) height. The different colors represent
the different fitting protocols. (green) Pure Lorentzian fits, (red) pseudo-Voigt but
with the FSDP constrained to be Gaussian, and (blue) pseudo-Voigt with the FSDP
line shape being allowed to change between Gaussian and Lorentzian as part of the
fit. Square symbols in (a) and (d) represent the position and height of the FSDP,
respectively, obtained directly from the experimental data. The dashed vertical lines
are plotted at the proposed boundary of the IP as well as at the stoichiometric
composition (GeSez). Images in this figure are presented in color.

88

case of the blue symbols, the pseudo-Voigt mixing parameters were allowed to vary
for all three peaks that were fit. The square symbols in Figure 3.14(a) and (d) are
fitting independent parameters for the (a) position and ((1) peak height that were
obtained directly from the data as the a: and y coordinates of the maximum of the
FSDP feature. They track the fits rather well.

Qualitatively, the general behavior of each of the FSDP parameters are well re-
produced by the different fitting protocols. The positions obtained by the different
protocols are within the estimated uncertainties. This is not so for the width and
integrated areas. The differences are largest in the low Ge—content region where the
FSDP is quite indistinct. In this region the different fitting protocols give results
with slightly different slopes. In the case of the purely Lorentzian line-shapes the
widths and areas are offset over the whole range, though detailed x—dependencies
on the refined parameters are reproduced. Below, each parameter is described in
detail. The position of the FSDP tells about characteristic periodicities in the struc-
ture in real space. Figure 3.14(a) shows the behavior of the position of the FSDP in
GexSe1-x glasses as a function of 3:. It starts at about 1.2 A‘1 for low Ge content,
and then shifts towards lower-Q valuas as Ge content is increased, reaching 1.0 A at
:1: = 0.4. This position corresponds to a real space length, d = (21r/ Q FSDP) = 5.2 - 6.3
A, which corresponds well to that of the inter-layer Bragg peaks seen in crystalline
GeSe2 [120], where d is called the inter-layer separation or cluster radius [9, 120].
This is in agreement with more elaborate wavelet analysis of the FSDP by Elliot et
al. [121].

No anomalies in the FSDP position were detected for samples in the vicinity of
the IP or at its upper boundary (a: = 0.25).

The area of the FSDP shows a maximum at the stoichiometric composition (a: = 0.33)
as shown in Figure 3.14(b). This is also apparent in the height of the FSDP max-

imum obtained directly from the data without fitting. These two parameters track

89

each other because the width of the FSDP is nearly constant (Figure 3.14(c)). This
suggests that the proportion of the intermediate range order contributing to this
periodicity in the structure is increasing with Ge content up to the stoichiometric
composition, and then it decreases.

Molecular dynamics [74, 92] calculations indicate the importance of Ge—Ge corre-
lations to this feature in the scattering. It is therefore not a surprise that the peak
intensity scales with the concentration of Ge in the sample being studied.

The dramatic change of the FSDP height as well as area at a: = 0.33, even though
the Ge content is still increasing, is attributed to the change in the role of Ge atoms
in the network. Below 3: = 0.33, Ge atoms work as network formers [119], so adding
Ge will result in a progressive increase in the correlations contributing to this peak,
which are thought to come from well-defined separations of GeSe4 tetrahedra. On the
contrary, above a: = 0.33, Ge atoms work as network modifiers (entities that do not
participate in forming the network structure) [119]. This will weaken the ordering
of the GeSe4 tetrahedra and hence decrease the intensity of this peak. This effect is
also seen when silver ions are added to g—GeSez [83], where they modify the GeSeg
covalent network by bonding to the Se atoms, with the effect of breaking up the larger
ring structure and hence reducing the intensity of the FSDP. The addition of silver
also leads to a softening of the vibrational spectrum.

The IP is thought to be rigid but unstrained where the stress-free state was pro—
posed to be due to self-organization in the network. If this is correct, then one might
expect that the range of the intermediate range order (IRO) is maximal in the IP win-
dow. This would be characterized by a minimum in the FSDP peak width. Actually
the FSDP width is quite composition independent from the fits. There is a broad,
weak minimum that has a global minimum at the GeSe2 composition. This suggests
that the correlation length of the intermediate range order coming from the stacking

of GeSe4 tetrahedra is greatest for the stoichiometric composition and then decreases

90

at higher compositions. There is a suggestion of a change in the slope of FSDP width
versus :0 associated with the rigid to floppy transition at x = 0.20. This is certainly
a suggestion that the range of the IRO starts to increase only after the rigidity of
the network percolates. Apart from offsets due to parameter correlations, this kink
in the curve is reproduced in all the fitting protocols, which builds confidence in its
correctness. However, the FSDP-width continues to decrease (the range of the IRO
continues to increase) with increasing Ge content above the upper limit of the IP, so
there is no convincing evidence that this is a signature for the IP.

The study of the FSDP as an observable for the IP has been pursued in two earlier
studies [6, 7]. The study by Sharma et al. [7] of the FSDP in Ge3Se1_z suggested that
there were three well resolved structural phases consistent with the IP. They found a
plateau-like behavior of the inverse of the FSDP position in the region of the IP. This
is not reproduced in the current work. Wang et al. [6] concluded that around the
stiffness threshold, the area of the FSDP has a plateau-like gradual decrease with :1:
followed by a rapid decrease at cc S 0.18. There is also no evidence for this behavior
in the present study.

Despite that our results of the FSDP parameters reproduced the general trends
in Sharma’s and Wang’s results, as the FSDP position shifts to lower Q-values as Ge
content is increased and its area is increased till a: = 0.33. However, our results show
a smooth and monotonic behavior of the different FSDP parameters when crossing
the IP region. No special features (as constancy of the FSDP area in the IP region
(as in Wang’s paper) and constancy of the FSDP position (as in Sharma’s paper)
happen in FSDP parameters due to the predicted IP, which makes it impossible to
rule out three well-resolved phases from the FSDP.

We think that this discrepancy is due to the fact that the FSDP at the low-Ge
content is a relatively week shoulder that rises out just above the background level,

which makes it difficult to predict its profile shape for certainty. Hence, fitting it

91

 

700
1
_I

600
l
1

T9 (0°)

0
OI— ’ -'
LO

_ Ffl -

l I l l l l

0.15 0.2 0.25 0.3
X( Ge content)

Figure 3.15: Comparison between T9 values of the studied samples (blue circles) and
Sharma samples [7] (red circles). Images in this figure are presented in color.

 

 

 

 

 

with just one fitting protocol (even if that protocol fitted it perfectly) might bias the
results.

Another valid reason of this discrepancy is due to the differences in the methodol—
ogy of samples preparation. Despite that our samples and those of Sharma and Wang
were prepared by conventional melt-quenching technique, any differences in the ther-
mal history of the samples may affect the local structure and hence the intermediate
range order deduced from the FSDP. The glass structure is extremely sensitive to the
method of preparation. Figure 3.15 compares the glass transition temperature of our
samples with those of Sharma. It is clear that there is some discrepancy between the
two values which might be due to the differences in samples preparation.

However, great care was taken in establishing the quality of our samples and char-
acterizing the behavior through the non-reversing heat flow measurements. Multiple
fitting protocols are also reported, that establish the reproducibility and uncertainty
of the results. This is also the only study that combines PDF and EXAFS data to
address this issue. The EXAFS and PDF data are in good agreement, which gives us

confidence that the current results are reliable.

92

3.4 Summary and conclusions

A careful composition-dependent study of the structure of Ge¢Se1_,, glasses, through
the composition range associated with the intermediate phase and using high energy
X—ray total scattering studies coupled with EXAFS measurements at the Ge and Se
edges, do not yield strong evidence for a structural origin for the IP. Structural pa-
rameters associated with strain in the sample (pair distribution function peak widths
from the X-ray and EXAFS data) and intermediate range order (deduced from the
first sharp diffraction peak in S (62)) all evolve smoothly with composition and no
discontinuities or breaks in slope are evidently associated with the boundaries of the

IP.

93

Chapter 4

Structural modeling of GexSe1_$

glasses around the IP

4.1 Introduction

In Chapter 3 we have seen that both PDF and XAFS can give information about
the microscopic structure of amorphous GezSe1_z glasses. However, this information
is limited almost entirely to the first two coordination shells; i.e., the bond lengths
and angles of nearest neighbor atoms comprising the basic structural units (such
as GeSe4 tetrahedra in GeSeg) can be determined but the relative disposition of
such units cannot be ascertained with certainty. For the case of PDFs derived from
scattering experiments on mono-atomic systems, for example, the problem lies in the
fact that peaks other than the first and second cannot be uniquely associated with
a particular interatomic correlation, but are made up from a variety of contributions
from higher-lying shells (see Figure 2.1). Matters are complicated considerably for
multicomponent materials, especially when the constituent atoms are close in their
scattering factors as in Ge$Se1_$ glasses where Ge and Se have atomic numbers of
32 and 34, respectively. In this case, a single diffraction experiment does not identify

the origin of any peak in the PDF in terms of specific atomic pair correlations.

94

One solution of these difficulties is the construction of models which simulate the
structure. Structural parameters such as the PDF, density, etc. may be computed
for the structural model, and the theoretical predictions compared with experiment.
The importance of structural modeling lies in the fact that a detailed quantitative
analysis of the structure may thereby be gained. For instance, the structural origin
of features in the computed PDF may readily be ascertained with the use of a model,
information often impossible to ascertain in any other way. Despite the fact that in
a glass, the structure is very complex in comparison with crystalline materials, a lot
of work has been done towards modeling the structure of glasses [92, 122, 123, 124].

In this chapter, we present the work done towards modeling the structure of
GezSe1_z glasses around the IP. Much emphasis has been done on modeling the
structure of GeSez glass as its structure plays an essential role in the GemSe1-x system

as a prototypical glass.

4.2 Structural insight from crystalline analogs

One key point about the network evolution as a function of Ge content is the evolution
of corner and edge sharing tetrahedra. a—Se has a chain structure where each Se atom
is 2—fold coordinated, adding Ge (cross linking element) to the Se chains brings about
the Ge(Se1/2)4 tetrahedra that are the basic building block in GexSe1_¢ glasses. For
low Ge content, these tetrahedra are immersed in a floppy Se matrix, adding more and
more Ge atoms results in linking these tetrahedra through their corners and edges.
In the following sections we discuss refining the glassy structure of GeSez using the
relaxed crystalline analogues in order to examine the existence of corner and edge
sharing tetrahedra in the structure.

In order to quantify the PDF data, we made a structural comparison of the sto—
ichiometric g—GeSe2 with the known structure of c-GeSeg, in its low temperature

(a-GeSeg) and high temperature (fl-GeSeg) forms.

95

 

Figure 4.1: The crystal structure of the low temperature a-GeSeg. Ball and stick
representation (upper figure) and polyhedral representation (lower figure). Images in
this figure are presented in color.

The structure of the LT a-GeSez is described to form two—dimensional layers
composed of Ge(Se1/2)4 tetrahedra being connected through their corners. This is
shown in Figure 4.1, which shows the unit cell of a—GeSeg. On the other hand, the
structure of the high temperature fi-GeSez phase is composed of Ge(Sel/2)4 tetrahedra
being connected through their corners and edges forming a three-dimensional network,
as shown in Figure 4.2. From the crystalline models, we can calculate total PDFs
to compare with the glassy data, but also partial PDFs which help us to assign
meaning to features in the glassy PDFs. Figure 4.3 shows a comparison between
the LT and HT crystalline phases in their calculated total and partial PDFs. In
the background of each panel in Figure 4.3, the experimental total PDF of g-GeSeg

96

 

Figure 4.2: The crystal structure of the high temperature fi-GeSeg. Ball and stick
representation (upper figure) and polyhedral representation (lower figure). Images in
this figure are presented in color.

97

 

 

 

 

O 5
r ' I
g"

.rIr' I'll
fin-(0) a
d- .
b’o
.m"','f]:]1'r:'
age) .
go
., “1]1'1:]:]a-
m- -
IP.—(C) i
(D
o
0:, :‘1':':'
(1) I ll
0
(I)

0“

 

 

-nI1I.l.l.-

02468
r01)

Figure 4.3: Comparison between the calculated PDFs for both LT (blue curves) and
HT (red curves) phases of crystalline GeSeg. In the background of each panel, the
experimental total PDF of g—GeSeg (green) is plotted for comparison. (a) Total PDFs,
(b) Ge-Ge correlations, (c) Ge—Se correlations, and (d) Se-Se correlations. Images in
this figure are presented in color.

 

is shown for comparison. The excellent match between the first PDF peak in both
crystalline and glassy data is evident, which proves that the basic structural unit in
the glass is the same as in the crystalline phases which is the Ge(Se1/2)4 tetrahedra.
The partial PDFs, Ge-Ge (panel b) and Se—Se (panel d) have almost zero probability
at the location of the first PDF peak, which proves that the crystalline phases have
no homo—polar (wrong) bonds. The relatively broad distribution of the second PDF
peak in the glass is consistent with the fact that the crystalline analogs contain

multiple peaks in this region. Both LT and HT phases contain Ge—Ge, Ge—Se and

98

 

Figure 4.4: (9.) Illustration of edge-sharing tetrahedra (EST) and (b) illustration of
corner-sharing tetrahedra (CST).

Se—Se correlations in this region. However, from their relative intensity, the second
PDF peak is mostly due to the Se-Se correlations, and the shoulder in the leading
edge of this peak is mostly due to Ge—Ge correlations.

Existence of edge-sharing tetrahedra (EST) (see Figure 4.4(a)) and their percent-
age relative to the corner-sharing tetrahedra (CST) (see Figure 4.4(b)) in Ge$Se1_z
glasses plays an important role in determining the intermediate range order (IRO)
and rigidity of the networks. In order to determine if the glassy data have EST or

not, together with determining the extent to which the crystalline models result in

99

Table 4.1: Refined fit parameters for a and 6 phases of c-GeSeg.

 

 

 

 

 

 

 

 

Refined parameter (A2) phase a phase 6
U11 0.006 0.035
U22 0.032 0.022
U33 0.011 0.013

[ R,” | 0.125 | 0.062 ]

 

PDFs that are similar to the glassy data, we have fitted the glassy data of GeSe2 us-
ing PDFgui [94] (a full-profile real-space local-structure refinement program). This is
possible because the a crystalline form of GeSeg has purely corner-shared tetrahedra,
whereas the 6 form has half of the tetrahedra in edge-shared and half in corner-shared
configurations. PDFs were calculated from both the a and [3 crystalline models using
structural parameters from Refs. [65, 54] and [65, 55], respectively. Starting from a
given structural model and given a set of parameters to be refined, PDFgui program
searches for the best structure that is consistent with the experimental PDF data.
The residual function (Rm; defined in Equation 2.6) is used to quantify the agreement
of the calculated PDF from model to the experimental data. The protocol we used
in this approach was fixing the unit cell dimensions and angles to the values of the
crystalline structure, but increasing the thermal parameters (U,,-) to account for the
glass structure.

The PDF peaks were broadened by increasing the atomic displacement parameters
(ADP) in the model; whilst keeping the first PDF peak sharp. The model PDFs were
convoluted with a Sinc fimction to account for the finite Q-cutoff of the image plate
data. The calculated model PDFs in comparison with the experimental data are
shown in Figure 4.5. Table 4.1 lists the resulting refined parameters when using both

the a and 6 phases together with the agreement factor (Rw).

Comparing Figure 4.5(a) and (b) we see that the 6 structural model does a sig-

nificantly better job of reproducing the glass data. Thus, a structure consisting of a

100

 

 

 

 

 

 

 

 

 

3
f(l)
Figure 4.5: The calculated model PDFs (red curve) compared with the experimental

data (blue circles) with the difference (green curve) offset below. (a) Fit to LT a-
GeSe2 and (b) fit to HTfl-GeSeg. Images in this figure are presented in color.

101

mixture of edge and corner-shared tetrahedra is consistent with the g—GeSez PDF and
the shoulder on the leading edge of the second-neighbor peak in the glassy PDF data
is an indicator for the existence of edge sharing of the tetrahedra. The purely corner-
shared structure of a—GeSeg results in a second peak that has the wrong shape and
is significantly shifted compared to the data from the glass. This is strong evidence
for the existence of a mixture of edge- and corner-shared tetrahedra in g—GeSeg.

As the structures of both LT and HT phases are dramatically different in terms
of tetrahedral connectivity, and since the main difference in their calculated PDFs is
in the region of the shoulder in the second PDF peak, it is interesting to investigate
the behavior of this shoulder across the series of the studied samples. Figure 3.11
shows the development of this shoulder, which is the main difference in the PDFs,
when going from low Ge to high Ge content. This supports the suspicion that the
origin of this shoulder is edge-shared tetrahedra that are becoming more frequent
with increasing Ge content.

Structural information about the network connectivity as a function of doping,
percentages of EST and CST, bond-angle distribution, and the ring structure can
not be determined directly from the data, as there are no crystalline analogs at every
concentration, 2:, to be refined similar to the case of g-GeSeg. The accuracy of the
obtained information depends entirely on the accuracy of the obtained models. Fol-
lowing is a description of the efforts done towards modeling the structure of GezSe1_z

glasses around the rigidity percolation threshold.

4.3 First principles molecular dynamics simulations
on Ge$Se1_$ system

In collaboration with Professor David Drabold’s research group at the University of

Ohio, we have obtained a set of structural models for GexSe1_3 glasses using first

102

principles molecular dynamics simulations with compositions that span the region of
the IP. Following is a description of how the models were generated, the statistical

analysis of these models and an assessment of the quality of these models.

4.3.1 Model generation

This work was carried out by F. Inam and D. A. Drabold. The procedure is summa-
rized here for completeness. The ab initio density functional code, FIREBALL, devel-
oped by Sankey and co-workers [125, 126] was used to generate the structural models.
This method has been used very successfully for a variety of covalently bonded sys-
tems, and especially glassy germanium selenide [76, 127, 124]. A set of 500—atom mod-
els for GexSe1_x glasses were generated with a: = 0.10, 0.15, 0.18, 0.22, 0.23, 0.25 and 0.33
using quench from melt technique.

Atoms were randomly placed in a cubic cell with a suitable and fixed volume.
The temperature of the cells in the MD simulation is then set to 4200 K, and then
equilibrated at 1500 K for about 3.5 ps. Following this, they were quenched to 400
K over about 4.5 ps, using velocity re-scaling. Finally, the cells were steepest descent
quenched to 0 K. These model configurations were compared to a series of models
that have been proposed earlier (GeSe4 (x = 0.20), GeSeg (x = 0.10) [124] and GeSez
(x = 0.33) by Cobb et al. [76]. All these models have similar densities. A qualitative

view of one of the generated models is shown in Figure 4.6.

4.3.2 Assessment of the MD models

These models were generated using interatomic potentials that are presumed to be
appropriate, and using MD methods that are presumed to find the equilibrium struc-
ture. To validate the quality of the models we would like to compare them to our
high-quality PDF data. Figure 4.7 shows a comparison between the experimental

and calculated PDFs from the models. From this we see a very good agreement in

103

 

Figure 4.6: Representative plot of the structure of Ge0,2OSeo.8o as obtained from the
molecular dynamics simulations. Black and gold circles represent Ge and Se atoms, re-
spectively. Some of the basic structural units (Ge(Se1/2)4 tetrahedra) are also shown.
Images in this figure are presented in color.

the region of the first shell, but a relatively poor agreement in the second shell. The
relatively large difference between the experimental and calculated PDFs at about
3.5 A is clear.

This indicated that the produced models were able to generate the basic structural
unit( Ge(Se1/2)4 tetrahedra), but were not fully able to reproduce the region between
3 and 4 A. This region, as we showed earlier in this chapter, is of particular interest
as it has important information about the way these tetrahedra are linked to each
other.

In order to test if the MD models are tracking the fine changes that occur in the
experimental data, we have plotted the difference curves between successive values of
:1: (Ge content) for both MD models and the data. Figure 4.8 shows these difference
curves. Despite the fact that the difference curves in the MD data are noisy and do
not depict the fine details, their absolute values are comparable with those in the

experimental data.

104

 

G(R'z)

 

 

Ir01)

Figure 4.7: Comparison between calculated PDFs obtained from the MD structural
models (red curves) with the experimental data (blue circles). Images in this figure
are presented in color.

As the general agreement between the models and the experimental data is rea-
sonable, the experimentally constrained molecular relaxation (ECMR) technique [123]
will be used in subsequent work to tune the structure of these glasses by using the
X—ray diffraction data as a constraint in the model formation process. The starting
point for these calculations will be the set of models that are reported here. Following

is a statistical analysis of the produced MD models.

4.3.3 Models analysis: Coordination, rings and constraints

In the GexSe1_3 system, tetravalent (4-fold) Ge and divalent (2-fold) Se are the fun-

damental building blocks for the network. For :2: small enough, 2-fold Se exist in the

105

 

 

 

 

 

 

 

 

 

Figure 4.8: Difference curves between successive values of :1: (Ge content) in the
experimental data (red curves) and MD models (blue curves). The difference curves
(AG (It-2)) are (a) AG = G(O.18) — G(O.15), (b) AG = G(0.20) — G(0.18) (0)
AG = G(0.22) — G(0.20) ((1) AG = G(0.23) — G(0.22), and (e) AG = G(0.33) —
G (0.23), where here the arguments indicate the composition. Images in this figure
are presented in color.

form of long chains, while 4—fold tetrahedral Ge connect these chains to each other
and form closed rings. Adding Ge atoms in a Se—rich environment increases the mean
coordination per atom. The mean coordination number was defined in Equation 1.2

and can be simplified as in Equation 4.1:

2" m’ (4.1)

 

106

 

f T
[44

V

Mean coordination number
2.2 2.3 2.4 2.5 2.6 2.7

 

 

l n l n I J

0.1 0.2 0.3
Ge content

 

Figure 4.9: Mean coordination number as obtained from MD simulations. The line
between the points is just guide to the eye. Images in this figure are presented in
color.

where 7' runs over all the coordinations present in the system, and n, is the number
of atoms with coordination r. Mean coordination, F, is an important parameter to
describe the network.

Figure 4.9 shows the increase in f with Ge concentration, as obtained from the
MD simulation. It is interesting to note that the increase is not linear, as it is in the
simple-minded modeling that neglects self-organization that was discussed in Chapter
3. Between a: = 0.20 and a: = 0.25, 7‘" briefly saturates suggesting a kind of “resistance”
the system offers to further increase in the number of bonds per atom. This result
depends on a small number of points and needs to be confirmed in greater detail. As
described later, the signature of this behavior is present in the overall evolution of
the network.

Figure 4.10 shows the variation in the coordination of both Ge and Se atoms as

a function of Ge concentration. From a: = 0.10 to a: = 0.18 the concentration of

107

 

 

l
A
O
V

75 100
l
E _
I

I
Concentration (95)
50

0 25
. 1
I .

I'l'lfl'f'

Concentration (Z)
10 20 3O 4O 50 60 7O

 

 

 

 

I I I I I I I I I I I I
0.1 0.2 0.3 0.1 0.2 0.3
Ge content Ge content

0

 

 

Figure 4.10: (a) The concentration of 1 (blue), 2 (red) and 3 (green) fold Se in the
produced MD models and (b) the concentration of 3 (blue), 4 (red) and 5 (green) fold
Ge in the produced MD models. The lines between the points are just guides to the
eye. Images in this figure are presented in color.

2-fold Se increases linearly from 58 to 63%. Between :1: = 0.18 and a: = 0.25 the
concentration first decreases, and then abruptly increases to 72%. These simulations
suggest that a significant proportion of Se is not 2-fold coordinated, contrary to the
assumptions used in Chapter 3.

With increasing Ge concentration, the system gradually eliminates homopolar Se
bonds and forms chemically preferred Se-Ge bonds. A consequence is the transition
which occurs in the neighbors of 2-fold Se (Figure 4.11(b)), that is, a gradual replace-
ment of Se neighbors with Ge neighbors. At lower Ge content, 2-fold Se have more
Se neighbors as compared to Ge neighbors. Increase in Ge content starts to replace
Se with Ge atoms as the neighbors of 2-fold Se. At 3 = 0.20, concentration of 2—fold
Se with neighbors Sel Gel assumes a higher value compared to 2—fold Se with SegGeo
and SeoGe2 neighbors as shown in Figure 4.11(b).

The increase in SelGel units at x = 0.20 affects the evolution of corner sharing

tetrahedra (CST) and the ring structure as shown in Figure 4.12. Due to the high

108

 

 

 

 

 

 

 

 

m _ I I I I I r ‘I T ‘I _ I I F v I I
“I - (a) - g - (b) _
C I A
:9. " N o _ -
3 v co
:9 ’ c
'5 o g o _ _
a S I
_ ID . .
so" §
59- 0g. -
0!
o ’ ‘
- O I- -
o I M I I I
50 75 100 125 0.1 0.2 0.3
Ge content Ge content

Figure 4.11: (a) Bond angle distribution of 2-fold Se units and (b) variation of con-
centration of types of neighbors of 2-fold Se: GeoSe2 (blue), GeISel (red) and Gegseo
(green). Lines are guide to the eye. Note the “flattening” of SelGel concentration
near the IP window. Images in this figure are presented in color.

 

 

 

 

 

 

 

 

l l l r I ' l
" - o
N ' ‘ (I) ' / ‘
V a, m /
0 _ _ c F) ‘ / "'
i7) ‘1' 'c . / .
o . - V— o _ 1/ _1
O O ,_ _ L ‘ / '
n a 10
C I— .—
o «D N
1.3 . ' g ,. q
.51 8 I- —] C 8 f' / "‘
8 * . «g to c " _
S o _ _ I— _ 1
o " o _ _
C I 1 I 1 I 1 ID I 1 I 1 l 1
0.1 0.2 0.3 0.1 0.2 0.3
Ge content Ge content

Figure 4.12: (a) Concentration of corner-sharing tetrahedra and (b) total number of
rings, as obtained from MD simulations to Ge¢Se1_$ glasses. Images in this figure
are presented in color.

109

concentration of SelGel units in the range :6 = 0.20 to 0.25, concentration of CST and
total number of rings tend to saturates in this range (Figure 4.12(a) and (b)). It is
clear that the transition from all Se neighbors around Se sites to Ge neighbors, occurs
roughly through a range a: E (0.20 to 0.25) which may be a signature of the sought-
after “self organization”. Below 2: = 0.18, 4-fold tetrahedral units reveal a slight
increase while it is fluctuating between 3 = 0.20 and x = 0.25. At a: = 0.20, 3-fold
Ge increases to a considerable concentration of 10% of Ge content (Figure 4.10(b)).

Figure 4.11(a) shows the angle distribution for 2-fold Se units averaged over all
as. Interesting feature is the lower angle peak (around 80°) for SeoGe2 units which is
due to the formation of 4-fold rings (edge sharing tetrahedra units). SelGel shows
a broad shoulder around 90° of the main peak centered at about 97°. This shoulder
also comes from 4-fold rings consisting of mainly 3 Se and 1 Ge atoms. SegGeo units
show a high peak around 105° due to open chain like structures or larger size rings
consisting of mainly Se atoms.

Based on the statistical analyses done on the previous molecular dynamics models
for GexSe1_$ glasses, it is clear that some structural anomalies appear to correlate
with the predicted IP. In particular, the network evolves from a—Sez to GeSeg through
a range of Ge composition which roughly coincides with the IP, by keeping a relatively
high concentration of isostatic 2-fold Se units having one Se and one Ge neighbor in
this range. This behavior resists the overall structural evolution of the network,
probably due to the minimization of free energy, F = U — TS, in the IP window.
Increase in the concentration of 2-fold Se bonded with one Ge in the IP range, which
are isostatic in nature, could be the origin of lowering of stress in IP proposed by
Thorpe et al. [1]. This theoretical signature of the IP makes it reasonable to fully
assess the quality of the generated MD models, especially in the region between
3 to 4 A, where dramatic changes occur in the structure in terms of tetrahedral

connectivity. To test the behavior of the MD models in this region (3 - 4 A), we

110

 

4

0

G(r)(fi'2)

 

1

AG(r)(A'2)
0

 

 

 

 

r(fi)

Figure 4.13: (a) Comparison between the MD model calculated PDF and the exper-
imental data for the GeSez glass. (b) The difference curves between experimental
PDF and the calculated PDF from MD models (red curve), LT a phase (green curve)
and the HT 0 phase (blue curve). Images in this figure are presented in color.

have plotted the calculated PDF from the MD model of GeSe2 glass against the
experimental one. Figure 4.13 shows that the MD model is not reproducing the exact
shape of the second PDF peak (see Figure 4.13(a)). In Figure 4.13(b) we show the
difference curves between the experimental PDF of GeSez glass and the calculated
one from the MD model as well as from the LT and HT phases of GeSez glass. It
is clear that the difference curve is large in case of the MD model, with agreement
factor (Rm) of 0.243, compared with 0.125 and 0.062 in case of the LT and HT models,
respectively. From these difference curves, it is clear that even the LT phase (which
has the wrong structure) fitted the data much better than the MD model. This brings
about the importance of validating the quality of the MD models, and shows that
these models are not perfect, and need more improvements in the region between 3

to4A.

111

4.4 Conclusions

As we saw from the above discussion, the basic building block in the glass is similar
to that in the crystalline phase, which is the Ge(Se1/2)4 tetrahedra. The difference is
in how these tetrahedra are linked to each other. Through the structural refinement
using both a and 6 phases of GeSe2 we have proven that the glassy data have both
corner- and edge-sharing tetrahedra.

The connectivity of the tetrahedra affects the average coordination number, and
hence the number of constraints per atom. Edge-sharing tetrahedra (EST) are more
strained than the corner-sharing (CST) ones, and hence, the percentage of EST to
CST plays an essential role in determining strain in the glass. This percentage can
easily be determined from the areas of the peaks responsible for EST and CST.
Unfortunately, these peaks are largely overlapped in GexSeH, glasses which makes it
impossible to get their areas for certainty.

Through the analysis of the MD models, we have seen that these models were
able to reproduce the basic structural unit (Ge(Se1/2)4 tetrahedra), but were unable
to fully reproduce the medium range order. On the other hand, as Ge and Se have
to be 4- and 2-fold coordinated, respectively, any deviation from this (as shown in
Figures ??(b) and (c), where there are a relatively large percentage of 1-fold and 3-fold
Se as well as 3- and 5-fold Ge) is considered as defects in the structure. This draws
into question the validity of the MD models, and shows that even the theoretical
signature of the IP (as these models suggest) is probably unrealistic, and the value of

these models is over estimated.

112

Chapter 5

Summary and conclusions

5.1 Summary and conclusions

Self-organization in network glasses, where the structure can incorporate non-random
features to minimize the free energy at the temperature of formation, has been pro-
posed by Thorpe et al. [1]. Their prediction came from studying a self-organized
model of a network in which configurations that are stressed are avoided where possi-
ble. They found that this avoidance leads to two phase transitions and an intermediate
phase that is rigid but stress-free (unstressed, where all bond lengths and bond angles
have their optimal values).

Chalcogenide glasses exhibit some signs of a first-order transition when the com-
position is varied, as observed by Raman scattering [3, 46], and also some evidence
for the intermediate phase was found using differential scanning calorimetry [3, 46].

Boolchand et al. [3, 34] using Raman scattering studies on Gexxlfi glasses (X=S
or Se) showed a jump in the composition dependence of the frequency of the mode
corresponding to symmetric stretch of Ge(X1/2)4 tetrahedra. For both S and Se,
this change occurs around the composition a: = 0.225, which corresponds to a mean
coordination < r > = 2.45, and may correspond to the transition to the stressed

state. On the other hand, their modulated differential scanning calorimetry (MDSC)

113

measurements on a variety of glasses [4, 40, 41, 38, 42, 43, 44] showed a universal
feature of having the non-reversing heat flow (AH nr) vanish in a finite composition
range. They called the composition range for these thermally reversing transitions
the intermediate phase (IP) or reversibility windows.

However, the experimental positions of the IP, as found by Boolchand et al. [3]
(from < r > = 2.40 to 2.45) do not coincide with the theoretical prediction (from
< r > = 2.375 to 2.392), which may be due to the simplicity of the theoretical models.
On the other hand, the theory predicts that the IP is about three times narrower than
the experimental finding of Boolchand’s et al. [1, 3]. These two discrepancies between
theory and experiment about the IP casts doubt on the equivalence of the two.

In this study, a search for a structural response to the intermediate phase (IP)
and a signature for its predicted self-organization [1, 2] has been performed. Based
on the large number of published papers [2, 46, 48] on the so-called IP and the fact
that many of these papers considered that the existence of the IP had been really well
established from experiments, we were motivated to search for a structural response
to this phase.

High-resolution atomic pair distribution functions, derived from high-energy syn-
chrotron radiation, coupled with high-resolution X—ray absorption fine structure (XAFS)
measurements on 18 compositions of well-prepared GexSe1_x glasses that span the
range of the IP have been performed to elucidate aspects of rigidity percolation and
the IP. These data are the most complete and highest-resolution study of this system
to date.

Analysis of the evolution with composition of the structure functions (in reciprocal
space) and the PDFs (in real space), as well as the XAFS data at both Ge and Se
edges, show no discernable correlations with the IP. Structural parameters evolve
smoothly without any discontinuity or break in slope that might be linked to the IP.
The results obtained in this study contradict previously published works [6, 7] that

114

claim experimental evidence of the existence of the IP.

According to the IP hypothesis, samples in this phase should be in a stress-free
state. Both PDF and XAFS are local structural probes that can detect changes in
the stress state of the network. Analysis of the first PDF peak position (which is
sensitive to homogenous strain) and the first PDF peak width (which is sensitive to
inhomogeneous strain) show no correlation with the IP boundaries. Debye-Waller
factors obtained form the XAFS data for both Ge—Se and Se-Se bonds also indicate
no correlations with the IP. The data show no detected bond strain evident until
approximately a: Z 0.30 as evident from the first PDF peak positions and widths,
surprising given the IP model.

The IP was explained to be in a stress-free state due to self-organization. If the
network is in a self-organized state, one expects this to be reflected in the network’s
medium-range order (MRO). A signature of the MRO is the so-called first sharp
diffraction peak (FSDP). Despite the fact that this peak is a subject of debate in
its structural origin, almost all of the glass community agree that it is a signature
of MRO. Analysis of the different parameters of this peak show no correlation with
the IP. We expect self-organization of the network in the IP to result in a maximum
amplitude and minimum width of this peak for samples in the IP, which is not the
case.

The FSDP changes systematically with Ge content, it develops smoothly from a
relatively small background for low Ge—content to a well-defined and sharp peak at
the stoichiometric composition (GeSeg). Its position shifts towards lower Q-values
when the Ge content is increased. The height of this peak reaches its maximum at
the stoichiometric composition (a: = 0.33) after which it starts to decrease. This is
interpreted as being due to the change in the role of Ge atoms in the network. For a: <
0.33, the Ge atoms work as network formers, so adding Ge will result in a progressive

increase in the correlations contributing to this peak [119]. On the contrary, for a: >

115

0.33, Ge atoms work as network modifiers [119]. This will weaken the ordering of the
correlations responsible for the FSDP and hence decrease its intensity.

Consistency of the PDF and XAFS data was examined by simulating the first PDF
peak using the XAFS data based on the chemically ordered network (CON) model.
The parameters derived from the simulated peak agree well with the experimental
ones, which proves the consistency among the data, and shows that the CON model
is reasonable for describing the structure of Gexselfi glasses.

The basic structural unit in the GezSe1_x glasses is the Ge(Se1/2)4 tetrahedron [80],
where each Ge atoms is tetrahedrally bonded to 4 Se atoms. These tetrahedra are,
for low Ge contents, immersed in a floppy Se-matrix. Adding more Ge results in a
progressive linking of each other through their corners and edges. Results obtained
from the analysis of the first PDF peak are consistent with the CON model, where Ge
and Se atoms maintain the four- and two-fold coordination respectively, consistent
with the ‘8-N’ rule. In the composition range of the studied samples, and according to
the CON model, the first PDF peak is mainly due to Ge—Se bonds, but the Se—Se and
Ge-Ge homopolar bonds are exist in the low Ge and high Ge regions, respectively.

Real-space structure refinement of g-GeSe2 using the two crystalline forms of
GeSez was performed. The LT-a phase is made up of completely corner—sharing
tetrahedra, while the HT-fi phase is made up of both corner- and edge-sharing of
the tetrahedra. The HT—fi phase fit the data perfectly and showed that the basic
structural unit in the glass is similar to that in the crystalline analog and suggests
that the glassy data contains both corner- and edge-sharing of the tetrahedra.

Statistical analysis of structural parameters of a set of structural models of GeISeH,
glasses obtained using ab initio molecular dynamics simulations was performed. Though
the analysis showed some structural features that might be linked to the proposed
IP, the MD models have a relatively large number of structural defects (existence of

1- and 3-fold Se as well as 3- and 5-fold Ge). The MD models did not reproduce the

116

PDF data well enough, nor the composition dependence of changes in the PDF, to
give confidence that they represent the structure of the real glasses.

In summary, our high-resolution PDF and XAFS data do not provide any can-
clusive evidence to support a structural origin for the region of low non-reversing
heat-flow, and no direct evidence for an IP as envisaged theoretically. This could
be due to two reasons: either any structural changes due to the IP are below the

sensitivity of PDF and XAFS methods, or there is no such phase.

5.2 Future work

Through the analysis of the PDF data, it was difficult to identify the structural origins
of the peaks in the MRO with certainty due to the highly overlapped shells. In this
study, we have used the XAFS method to extract the differential PDFs and identify
the origin of the different peaks in the PDF at the short range order (SRO) scale.

However, peaks in the PDF data extend up to about 10 A, and identifying their
origin can not be achieved through XAFS technique, since XAFS is a SRO probe. A
structural probe that is element specific and is able to identify peaks at the medium
range order (MRO) is required to gain full insight about the evolution of the network
and to fully identify the structural origin of the peaks in the PDF.

We propose measuring this set of samples using X-ray anomalous scattering and / or
neutron isotope substitution. These two techniques can fully identify the structural
origins of the different peaks in the PDF, and hence sub-peak parameters can be
analyzed and their evolution versus Ge-doping can be obtained.

The rigidity transition can be expected to manifest directly in the elastic response
of a network. Thus elastic constants probed by ultrasonic echoes [128, 129, 130, 131,
132, 133, 134], Brillouin scattering and Raman scattering [3, 46, 69, 135] appear to
be some of the most direct methods to probe the transition.

It is interesting to look to the behavior of the Boson peak [136] as a function of

117

composition in this set of glasses. This may help both interpret the structural origin
of the Boson peak as well as look for any structural responses to the IP on this peak.

On the other hand, it would be interesting to measure these samples at a relatively
high temperature to suppress non-covalent bonds. This may narrow the gap between
the theoretical prediction of the IP and the experimentally found IP through MDSC
measurement.

For the modeling part, we plan on doing a co-refinement using the PDF and XAFS
data. This is expected to improve the structural models and hence provide better

understanding of the structure of GexSeH, networks.

118

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