Title: Heteroepitaxy and nucleation control for the growth of metal chalcogenides using activated reactant sources
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Title: Heteroepitaxy and nucleation control for the growth of metal chalcogenides using activated reactant sources
Physical Description: Book
Language: English
Creator: Stanbery, B. J ( Billy Jack ), 1952-
Publisher: University of Florida
Place of Publication: Gainesville Fla
Gainesville, Fla
Publication Date: 2001
Copyright Date: 2001
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Subject: Chemical Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )
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Summary: ABSTRACT: A novel rotating-disc reactor has been designed and built to enable modulated flux deposition of CuInSe2 and its related binary compounds. The reactor incorporates both a thermally activated and a novel plasma activated sources of selenium vapor, which have been utilized for the growth of epitaxial and polycrystalline thin-film layers of CuInSe2. A comparison of the different selenium reactant sources has shown evidence of increases in its incorporation when using the plasma source, but no measurable change when the thermally activated source was used. It is concluded that the chemical reactivity of selenium vapor from the plasma source is significantly greater than that provided by the other sources studied. Epitaxially grown CuInSe2 layers on GaAs, ZnTe, and SrF2 demonstrate the importance of nucleation effects on the morphology and crystallographic structure of the resulting materials. These studies have resulted in the first reported growth of the CuAu type-I crystallographic polytype of CuInSe2, and the first reported epitaxial growth of CuInSe2 on ZnTe. Polycrystalline binary (Cu,Se) and (In,Se) thin films have been grown and the molar flux ratio of selenium to metals varied. It is shown that all of the reported binary compounds in each of the corresponding binary phase fields can be synthesized by the modulated flux deposition technique implemented in the reactor by controlling this ratio and the substrate temperature.
Summary: ABSTRACT (cont.): These results were employed to deposit bilayer thin films of specific (Cu,Se) and (In,Se) compounds with low melting point temperatures, which were used to verify the feasibility of synthesizing CuInSe2 by subsequent rapid-thermal processing, a novel approach developed in the course of this research. These studies of the influence of sodium dosing during the initial stages of epitaxy have led to a new model to explain its influences based on the hypothesis that it behaves as a surfactant in the Cu�In�Se material system. This represents the first unified theory of the role of sodium that explains all of its principal effects on the growth and properties of CuInSe2 that have been reported in the prior scientific literature. Finally, statistical mechanical calculations have been combined with published phase diagrams and results of ab-initio quantum mechanical calculations of defect formation enthalpies from the literature to develop the first free energy defect model for CuInSe2 that includes the effects of defect associates (complexes), thereby resolving numerous inadequacies of prior defect models for CuInSe2 that neglected these effects.
Thesis: Thesis (Ph. D.)--University of Florida, 2001.
Bibliography: Includes bibliographical references (p. 380-398).
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Statement of Responsibility: by Billy Jack Stanbery.
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General Note: Vita.
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Bibliographic ID: UF00100801
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: oclc - 50743149
alephbibnum - 002729365
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HETEROEPITAXY AND NUCLEATION CONTROL FOR THE GROWTH OF
METAL CHALCOGENIDES USING ACTIVATED REACTANT SOURCES















By

BILLY JACK STANBERY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2001

























Copyright 2001

by

Billy Jack Stanbery


























To those dedicated teachers whose encouragement enabled me and
whose vision inspired me to persevere:
B. M. Stanberry
M. E. Oakes
W. T. Guy, Jr.
G. C. Hamrick
S. C. Fain
M. P. Gouterman
R. A. Mickelsen
W. S. Chen
L. E. Johns, Jr.
R. Narayanan













ACKNOWLEDGMENTS

The long voyage of discovery I have labored to share in this dissertation

could not have been realized without the contributions of a multitude of others

that have chosen to invest their hope and efforts in mine. My study of this subject

began over two decades ago, and prior to coming to the University of Florida

was conducted mostly while employed by The Boeing Company. Their generous

donation to the University of the key research equipment used to conduct this

research provided irreplaceable physical assets that have made it possible. This

would not have happened without the advocacy within Boeing of Dr. Theodore

L. Johnson, for which I am very grateful.

This research would likewise not have been possible without the financial

support of the United States Department of Energy provided through contract

numbers XCG-4-14194-01, XAF-5-14142-10 and XAK-8-17619-32 from the

National Renewable Energy Laboratory. Although these contracts were won

through competitive procurement processes, the encouragement of K. Zweibel

and J. Benner, providing me hope that I might succeed (but could only if I tried),

was indispensable motivation, and I thank them both.

I would like to thank my advisor, Professor T. J. Anderson, for recruiting

me to the University, providing laboratory space at the University's







MicroFabritech facility, and assembling the interdisciplinary research team

within which I worked over the course of this doctoral research program. Every

member of that team has contributed to this work, but I must particularly thank

Dr. Albert Davydov and Dr. Chih-Hung (Alex) Chang who were always

available, capable, and willing to engage in the intellectual exchanges

that I have found to be the most compelling fount of insight. I am also indebted

to Dr. Weidong Zhuang who provided me an advanced copy of the results of the

critical assessment of the binary Cu-Se system that he and Dr. Chang performed.

Those familiar with laboratory research recognize the enormous value of

those who help a researcher with the essential but unglamorous and tedious

tasks that actually absorb most of the time and effort required to conduct

successful research of this sort. I thank W. P. Axson who helped me turn an

empty space and truck full of unfinished, disassembled equipment into a

productive and safe laboratory filled with operational state-of-the-art research

systems, and taught me plumbing and electrical skills in the process. The control

automation system that finally tamed the reactor and consolidated the data

acquisition process was the work of my research assistant S. Kincal, whom I

thank as well. Without his assistance the quadrapole mass spectrometric

measurements could not have been performed. The EDX composition

measurements vital to calibration and data interpretation were the work of my

other laboratory assistant S. Kim, whom I also thank. I reaped the benefits of the

support of the entire staff of MicroFabritech, especially S. Gapinski and







D. Badylak, who providing a dependable and indispensable laboratory

infrastructure. I also thank the following University staff, faculty, and students

who helped with advice, parts, measurements, and characterization: W. Acree,

M. Davidson, D. Dishman, E. Lambers, and J. Trexler. Outside of this University,

I would like to thank Dr. S. P. Ahrenkiel (NREL), who provided TEM

measurements; Dr. G. Lippold (Universitat Leipzig), who provided Raman

measurements; and Dr. M. Klenk (Universitat Konstanz), who provided XRF

measurements.

I would also like to thank some of those in the scientific community at

large who have shared their time and thoughts with me during the course of this

graduate program: Dr. M. Al-Jassim, Dr. R. Noufi, Dr. K. Ramanathan,

Prof. A. Rockett, Dr. B. von Roedern, Prof. E. Vlieg, and Prof. J. Venables.

Finally, I would like to thank Sue Wagner and my parents, Martha and

Bill Stanberry, without whose faith, hope, support, help, encouragement, and

love I could not have succeeded.














TABLE OF CONTENTS




ACKNOWLEDGMENTS .........................................................................................iv

LIST O F TA BLES ....................................................................................................... x

LIST OF FIGURES ..................................................... ............................................. xi

A BST R A C T ................................................................................................................... xvii

CHAPTERS

1 REVIEW OF PRIOR RESEARCH: CIS MATERIALS FOR
PHOTOVOLTAIC DEVICES......................................... ... ................... 1
Phase Chemistry of Cu-[II-VI Material Systems ..............................................3
The Cu-In-Se (CIS) Material System................................................................
The Cu-Ga-Se (CGs) Material System....................................... ............... 7
The Cu-In-S (cIsu) Material System ........................... .......... .............. 9
Crystallographic Structure of the Ternary cis Compounds............................. 10
a-cis (Chalcopyrite CulnSe2)....................................................................... 11
S (Sphalerite) ......................................................................................... 13
P-cls (Cu2ln4Se7 and Culn3Ses).................................................................... 14
y-cis (C uIn Se ) .................................................................. .........................17
Metastable Crystallographic Structures CuAu-ordering........................ 18
Defect Structure of a-cls .............................................................................. 20
Optical Properties of Ternary Cu-II-VI Materials........................................... 29
Optical Properties of a-cis and P-cs .................................................... 29
Optical Properties of c--CGS ......................................................................... 34
Optical Properties of a-cisu ................................................... .............35
Alloys and Dopants Employed in cas Photovoltaic Devices............................ 35
Gallium Binary Alloy CIGS ...................................................................... 36
Sulfur Binary Alloy ---Css............................................................................ 40
Alkali Impurities in cas and Related Materials ..............................................40
Summary .....................................................................................................44








2 CIS POINT DEFECT CHEMICAL REACTION EQUILIBRIUM MODEL........45
Approach .................................................... ....................................................... 46
Formulation of the Problem............................................................................. 52
R esults................................................................................................................ 63
Interphase Reaction Equilibria ....................................................................... 63
Equilibrium Defect Concentrations in the Cu-In-Se a Phase ....................90
Summary ......................................................................................................... 101

3 REACTOR DESIGN AND CHARACTERIZATION................................................. 103
D esign .................................................................................................................... 103
Operational Characteristics........................................................................... 115
Substrate Temperature Calibration.............................................................. 115
Flux Calibration .......................................................................................... 118

4 ACTIVATED DEPOSITION SOURCES............................................................. 122
Thermally Activated Source and its Molecular Species Distribution........... 125
Plasma Source ................................................................................................. 129
Source Design ............................................................................................. 134
Molecular Species Distribution of the Plasma Source Flux...................... 158
Ion Flux from the Plasma Source................................................................. 161

5 GROWTH OF METAL CHALCOGENIDES..................................................... 164
Binary Chalcogenides .................................................................................... 164
Thermodynamic Phase Control ................................................................... 165
Deposition of RTP Precursor Films ............................................................... 170
Ternary Chalcogenides................................................................................ 174
Deposition of CIS Photovoltaic Absorber Films ......................................... 174
Epitaxial Growth .............................................................................................. 182

6 SUMMARY AND CONCLUSIONS ................................................................... 207

G LO SSA R Y .................................................................................................................... 210

APPENDIX CIS DEFECT AND PHASE EQUILIBRIA CALCULATIONS........ 212
Formula Matrices............................................................................................ 213
Reaction Stoichiometry Matrices................................................................... 228
Boundary Conditions..................................................................................... 232
Thermodynamic Functions ........................................................................... 234
Compounds..................................................................................................... 236
State Vectors ..................................................................................................... 254
Initial Concentration Vector ......................................................................... 255
Reference State Chemical Potential Vector.............................................. 269
Reaction Extents Vector.............................................. .................................. 291
Defect Quasichemical Reaction Equilibria Calculations.................................295








LIST O F REFEREN CES.............................................................................................. 380

BIOGRAPH ICAL SKETCH .................................................... .................................. 399













LIST OF TABLES


Table page

4-1 QMS ion currents generated from the flux of selenium molecules
formed from the predominant mass 80 isotope effusing from the
therm al source. ........................................................................................... 128

4-2 Calculated mode frequencies of semifinal TE011 cavity design at
minimum tuning length limit.a ............................................................... 145

4-3 Calculated mode frequencies of semifinal TE011 cavity design at
mnaximnum tuning length limit. b ................................................................. 146

4-4 Comparison of frequency shifts of the TE011 mode due to dielectric
loading of the cavitya at several different lengths.............................. 150

4-5 Compilation of theoretical calculations and experimental data
demonstrating unloaded semifinal cavity mode assignments.......... 151

4-6 Axial magnetic field strength profiles of the final source assembly......... 154

5-1 Composition of two samples from the cis absorber film deposition
experiments using the three-layer process showing significant
variations in the extent of intermixing between the layers................ 177














LIST OF FIGURES


Figure

1-1 Assessed phase diagram along the Cu2Se [n2Se3 pseudobinary
section of the Cu-In-Se chemical system [26]....................................... 7

1-2 Schematic representation of CuInSe2 chalcopyrite crystal structure:
(a) conventional unit cell of height c, with a square base of width a;
(b) cation-centered first coordination shell; (c) anion-centered first
coordination shell showing bond lengths dc,-se and din-s..................... 12

1-3 Comparison of the crystallographic unit cells of CuInSe2 polytypes:
a) chalcopyrite (cH) structure, and b) CuAu (CA) structure.................. 21

1-4 Predominance diagram for the Cu2Se-In2Se3-Ga2Se3 pseudoternary
phase field at room temperature [113]. In that author's notation,
Ch is the a phase, P1 is the 3 phase, P2 is the y phase, and Zb is the
8 phase..................................................................................................... 37

2-1 Calculated equilibrium phase diagram for the Cu-In-Se system on the
Cu2Se/In2Se3 section where Z= 1....................................................... 64

2-2 Deviation of the Cu2.5Se stoichiometry parameter 6 in hypothetical
equilibrium with stoichiometric CulnSe2 ............................................ 67

2-3 The deviation of the Cu2.-Se stoichiometry parameter 8 from its
minimum allowable value in equilibrium with defective ternary
a-cis in the stoichiometric CulnSe2 mixture............................................ 68

2-4 The Cu2-Se stoichiometry parameter 8 in equilibrium with a-cis in the
stoichiometric CulnSe2 mixture ...................................... ............... 69

2-5 The equilibrium molar extent of binary Cu-Se phase segregation in the
stoichiometric CulnSe2 mixture ...................................... ............... 70







2-6 The negative valency deviation of c-cas in equilibrium with the binary
Cu-Se phase in the stoichiometric CuInSe2 mixture............................. 71

2-7 The negative molecularity deviation of a-cis in equilibrium with the
binary Cu-Se phase in the stoichiometric CulnSe2 mixture ................ 71

2-8 The equilibrium selenium mole fraction of the binary Cui-xSex phase in
the Cu-In-Se mixture with Xa = 0 and Za = +4.5x10-6, and the
temperature dependence of the maximum allowable selenium mole
fraction ...................................................................................................... 73

2-9 The variation of specific Gibbs energy with composition of the binary
Cu-ixSex phase at 393.15K (upper curve) and 398.15K (lower curve) .. 74

2-10 The deviation of equilibrium selenium mole fraction in the binary
Cu.-xSex phase from its minimum constrained value in the Cu-ln-Se
mixture, with Xa= 0 and (left to right) Za= 100, 400, 700, 1000, and
1739 (x10-6).............................................................................................. 75

2-11 The equilibrium molar extent of Cu2.-Se phase segregation in Cu-ln-Se
mixtures, with Xo= 0 and (left to right) Za= 0, 0.11, and 0.22............... 76

2-12 The valency deviation of a-cis in equilibrium with Cu2-aSe, with Xa= 0
and Za= 0.143 or 0.2 (xl0-6) ............................................. ................. 77

2-13 The valency deviation of the two-phase mixture with X=l required to
maintain the valency of the o-cas phase at its STP value...................... 78

2-14 The equilibrium Cu2-sSe/-CIS phase boundaries in the Cu-ln-Se
system for Z = 0 (right) and Za= +0.1% (left) between STP and
the a/p/6-cs eutectoid..................................... ..................... ......... 80

2-15 The composition at the equilibrium Cu2.-Se/a-cis phase boundaries
in the Cu-ln-Se system for Za= 0 (right) and Za= +0.1% (left)
between STP and the a//P6-cIS eutectoid........................ ............... 81

2-16 The variation of the specific Gibbs energy deviation of a-cis from
its value at Z, = 0 on the Cu2..Se/a-cIS two-phase boundary.
Valency deviations between 0 < Z a < 0.1% and temperature
between STP and the oa/P4-cis eutectoid are shown........................... 82

2-17 Temperature variation of the specific Gibbs energy deviation of an
ideal chalcopyrite CuInSe2 crystal from this model's reference value
for the equilibrium stoichiometric mixture............................................ 90








2-18 Temperature variation of the V'cu species mole fraction at the phase
boundaries on the pseudobinary section (left) and with Za= 4x10-
on the Cu2-sSe/a-cIs phase boundary (right) ........................................91

2-19 Temperature variation of the (Vcu E Incu)' species mole fraction at
the phase boundaries on the pseudobinary section (left) and with
Za = 4x104 on the Cu2-sSe/a-CiS phase boundary................................. 93

2-20 Temperature variation of (2Vcu Incu). species mole fraction at the
phase boundaries on the pseudobinary section (left) and with
Z, = 4x104 on the Cu2-.Se/ c-CIS phase boundary............................... 94

2-21 Temperature variation of the V^cu species mole fraction at the phase
boundaries on the pseudobinary section (left) and with Za = 4x104
on the Cu2-aSe/a-cls phase boundary (right) ........................................95

2-22 Temperature variation of the Culn species mole fraction at the phase
boundaries on the pseudobinary section (left) and with Za = 4x10-
on Cu2-~Se/a-cis phase boundary (right)............................................ 96

2-23 Temperature variation of the CuinE Incu species mole fraction at the
phase boundaries on the pseudobinary section (left) and with
Za = 4x10-4 on the Cu2.sSe/a-CiS phase boundary (right) ....................97

2-24 Temperature variation of the h* species mole fraction at the phase
boundaries on the pseudobinary section (left) and with Z = 4x104
on the Cu2-.Se/a-cIs phase boundary (right)........................................ 99

2-25 Contour map of net carrier concentrations in cc-cis in equilibrium with
Cu2-.Se over the temperature range between STP and the OP/6-cis
eutectoid, and the valency deviation range 0= Zoa 50.1%. Contour
intervals are p=2.5x1018 cm-3 and the black region (left) is intrinsic... 100

3-1 Schematic diagram of the MEE reactor showing the source and
shielding configuration. ............................................................................ 110

3-2 Detail of metals deposition shield with chamber removed .......................11

3-3 Detail of the chalcogen (selenium and/or sulfur) deposition zone of
the reactor with the chamber outer walls removed, showing
a) effusion source before the plasma cracker is mounted on the
left and b) radiant heater with power leads and monitoring
therm ocouple at top right ....................................................................... 113








3-4 Detail of reactor viewed from the front load-lock zone with the
chamber walls removed. The NaF Knudsen cell source (a) and QCM
(b) are visible at upper left, in front of the metals deposition shield
(c). The water-cooled selenium sector shield (d) is on the right and
the annular liquid-nitrogen cryoshroud (e) at center......................... 114

3-5 Calibration curve for substrate temperature controller............................ 117

3-6 Absolute selenium molar flux calibration curve for the thermal source. 120

4-1 Ratio of measured ion-currents at high and low thermal source
cracking zone temperature for each selenium molecular species
within the mass detection range of the QMS......................................... 129

4-2 Rendered, cross-sectional CAD drawing of TE011 plasma cracker with
coupled effusion cell ................................................................................ 133

4-3 Calculated resonant frequency contours of TE011 and neighboring
modes as a function of diameter and height of an empty ideal
right circular cylindrical cavity. ............................................... ........ 144

4-4 In-situ impedance measured over a 2GHz range of the final cavity
design at its optimal tuning length for TE011 operation.................... 149

4-5 Final cavity design, tuned and fully loaded, in-situ TE011 mode
im pedance measurement ..................... ..................................................... 152

4-6 QMS ion currents generated by fluxes from the plasma source of
selenium molecules formed from the predominant mass 80 isotope. 159

4-7 QMS ion-current ratio generated from selenium monomer and dimer
fluxes from the plasma source................................................................ 159

5-1 Assessed Cu-Se temperature-composition phase diagram [149]............. 167

5-2 XRD 0-20 scan of desired a-CuSe binary precursor phase for RTP.
Films were grown with up to 54 at.% selenium that showed
sim ilar XRD patterns. ............................................................................... 168

5-3 Assessed In-Se temperature-composition phase diagram [149]............... 169

5-4 Cu-In-Se ternary composition diagram indicating compounds............ 170

5-5 Auger depth profile of Sample 69 showing near surface indium
enrichm ent............................................................................................ 177







5-6 DBOM excess carrier lifetime measured on sample #70 both a) before,
and b) after CBD CdS deposition........................................................... 180

5-7 Illuminated current-voltage curve for the best cis thin-film cell made
by a three-layer codeposition process in the course of this research.. 181

5-8 A comparison of experimental and theoretical TED data.
a) experimental dark-field cross-sections taken with intensities
from the corresponding diffraction spots in the TED pattern along
[010] as shown, and b) theoretical TED patterns of CA and CH
structures in CuInSe2, both along [010]................................................... 185

5-9 Comparison of the XRD spectra of epitaxial chalcopyrite (upper) and
CuAu (lower) crystallographic polytypes of CulnSe2 on (001) GaAs
substrates. .............................................................................................. 186

5-10 Macroscopic Raman scattering spectrum of a CA-CuInSe2 epilayer on
GaAs. Peaks labeled by" are laser plasma lines; the others are
described in the text. .................................................................................. 187

5-11 Spatial distribution and morphology of islands in copper and
indium-rich cases: a) [Cu]/[In] = 1.06 and b) [Cu]/[In] = 0.99............ 189

5-12 AFM images of Cis islands and epilayers. a) islands on Cu-rich films
and b) islands on In-rich film s................................................................ 191

5-13 Cross-sectional TEM on [010]: dark-field using 1/2 (201) spot showing
CH-ordered epitaxial "island" in a sample with [Cu]/[In] = 0.97....... 193

5-14 SE-SEM image of an In-rich cIs film on GaAs dosed with a few
monolayers of NaF. The EMP-measured [Cu]/[In] ratios are
0.94 overall, 0.99 between the islands, and 0.81 within the
island "pools."............................................................................................. 196

5-15 Micro-Raman scattering spectra of islands on two indium-rich cis
films grown on GaAs (100). The uppermost curve is from an
island "pool" on a sodium-dosed film and the lower two are single
and averaged spectra from isolated islands on the sample without
sodium shown in Figure 5-11(b) ............................................................ 197

5-16 XRD 6-20 scan of epitaxial CuInSe2 on (001) ZnTe grown by MEE. The
overall composition of the film was [Cu]=25.5 at.%, [In]=26.3 at.%,
and [Se]=48.2 at.% ................................................................................... 201








5-17 XRD 6-20 scan of epitaxial CuLnSe,:Na on (111) SrF, grown by MEE. The
overall composition of the film was [Cu]=23.4 at.%, [In]=26.3 at.%,
and [Se]=50.3 at.%. The higher curve is a reference SrF2 substrate
w without Cu nSe..................................................................................... 203

5-18 XRD 0-20 scan of epitaxial CuInSe2 on (100) GaAs grown by PMEE.
The overall composition of the film was copper-rich, with
[Cu]=28.1 at.%, [In]=21.1 at.%, and [Se]=50.8 at.%.............................. 204

5-19 XRD 0-20 scan of epitaxial CuInSe2 on (100) GaAs grown by PMEE.
The overall composition of the film was indium-rich, with
[Cu]=23.1 at.%, [In]=26.3 at.%, and [Se]=50.6 at.%.............................. 205

A-1 Temperature dependence of the deviation from one-third of the
minimum stable excess selenium content of Cu2.-Se sufficient to
inhibit metallic copper phase segregation.............................................. 244

A-2 Temperature dependence of the maximum selenium
binary mole fraction of single-phase Cu2.sSe....................................... 247

A-3 Deviation of the Cu2.sSe phase's selenium content in equilibrium with
a-cis at X = Z = 1 from its minimum stable selenium mole fraction.. 328

A-4 Temperature dependence of the valency deviation of a-cis
in equilibrium with Cu2-sSe in the stoichiometric CulnSe2 mixture... 335


















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

HETEROEPITAXY AND NUCLEATION CONTROL FOR THE GROWTH OF
METAL CHALCOGENIDES USING ACTIVATED REACTANT SOURCES



By

Billy Jack Stanbery

May 2001


Chairman: Timothy J. Anderson
Major Department: Chemical Engineering

A novel rotating-disc reactor has been designed and built to enable

modulated flux deposition of CuInSe2 and its related binary compounds. The

reactor incorporates both a thermally activated and a novel plasma activated

sources of selenium vapor, which have been utilized for the growth of epitaxial

and polycrystalline thin-film layers of CuInSe2. A comparison of the different

selenium reactant sources has shown evidence of increases in its incorporation

when using the plasma source, but no measurable change when the thermally

activated source was used. It is concluded that the chemical reactivity of


xvii







selenium vapor from the plasma source is significantly greater than that

provided by the other sources studied.

Epitaxially grown CuInSe2 layers on GaAs, ZnTe, and SrF2 demonstrate

the importance of nucleation effects on the morphology and crystallographic

structure of the resulting materials. These studies have resulted in the first

reported growth of the CuAu type-I crystallographic polytype of CuInSe2, and

the first reported epitaxial growth of CuInSe2 on ZnTe.

Polycrystalline binary (Cu,Se) and (In,Se) thin films have been grown and

the molar flux ratio of selenium to metals varied. It is shown that all of the

reported binary compounds in each of the corresponding binary phase fields can

be synthesized by the modulated flux deposition technique implemented in the

reactor by controlling this ratio and the substrate temperature. These results

were employed to deposit bilayer thin films of specific (Cu,Se) and (In,Se)

compounds with low melting point temperature, which were used to verify the

feasibility of synthesizing CuInSe2 by subsequent rapid-thermal processing, a

novel approach developed in the course of this research.

The studies of the influence of sodium during the initial stages of epitaxy

have led to a new model to explain its influences based on the hypothesis that it

behaves as a surfactant in the Cu-In-Se material system. This represents the first

unified theory on the role of sodium that explains all of sodium's principal

effects on the growth and properties of CuInSe2 that have been reported in the

prior scientific literature.


xviii







Finally, statistical mechanical calculations have been combined with

published phase diagrams and results of ab-initio quantum mechanical

calculations of defect formation enthalpies from the literature to develop the first

free energy defect model for CuInSe2 that includes the effects of defect associates

(complexes). This model correctly predicts the a/1 ternary phase boundary.


















CHAPTER 1
REVIEW OF PRIOR RESEARCH:
CIS MATERIALS FOR PHOTOVOLTAIC DEVICES

Any legitimate review of the prior research in this long-studied field must

of necessity reference a number of excellent reviews already published in the

literature. Nevertheless, the field is rapidly progressing and this critical review

strives to highlight from this author's perspective both some of those research

results that have been previously reviewed and those too recent to have been

available to prior authors. The earliest comprehensive review of chalcopyrite

semiconducting materials [1] by Shay and Wernick is a classic reference in this

field. It focused primarily on the physical and opto-electronic properties of the

general class of I-II-VI2 and II-IV-V2 compound semiconductors. More recent

reviews specifically oriented towards cis materials and electronic properties [2-6]

are also recommended reading for those seeking to familiarize themselves with

key research results in this field.

There are also a number of excellent books and reviews on photovoltaic

device physics [7,8], on the general subject of solar cells and their applications

[9,10], and others specifically oriented towards thin-film solar cells [11,12], the

1







class to which cis solar cells belong. Finally, a non-technical but concise and

current overview of solar cell technology was recently published by Benner and

Kazmerski [13].

The first solid-state photovoltaic (Pv) device was demonstrated in 1876

and consisted of a sheet of selenium mechanically sandwiched between two

metal electrodes [14]. The addition of copper and indium and creation of the first

CIS PV device occurred almost 100 years later in 1973 [15], when a research team

at Salford University annealed a single-crystal of the ternary compound

semiconductor CulnSe2 in indium. Almost all subsequent Cu-In-Se thin-film

deposition process development for PV device applications have sought to make

the compound CuInSe2 or alloys thereof, but in fact generally result in a

multiphase mixture [16], incorporating small amounts of other phases.

Researchers have not always been careful to reserve the use of the compound

designation CuInSe2 for single-phase material of the designated stoichiometry,

an imprecision that is understandable in view of the difficulty in discriminating

CulnSe2 from some other compounds in this material system, as will be

discussed in detail elsewhere in this treatise. The compound designations such as

CulnSe2 will be reserved herein for reference to single-phase material of finite

solid solution extent, and multiphasic or materials of indeterminate structure

composed of copper, indium, and selenium will be referred to by the customary

acronym, in this case CIS.







This review begins with an overview of the physical properties of the

principal copper ternary chalcogenides utilized for Pv devices, including their

thermochemistry, crystallography, and opto-electronic properties. All state-of-

the-art devices rely on alloys of these ternary compounds and employ alkali

impurities, so the physical properties and effects of these additives will be

presented, with an emphasis on their relevance to electronic carrier transport

properties. This foundation will provide a basis from which to address the

additional complexities and variability resulting from the plethora of materials

processing methods and device structures which have been successfully

employed to fabricate high efficiency Pv devices utilizing absorbers belonging to

this class of materials.


Phase Chemistry of Cu-III-VI Material Systems

Significant technological applications exist for Ag-Il-VI2 compounds as

non-linear optical materials [17], but almost all PV devices being developed for

solar energy conversion that utilize ternary chalcogenides are based on the Cu-

III-VI material system. Although the reasons for this may have been initially

historical, this review will demonstrate that fundamental physical properties of

these materials render them uniquely well suited, and underlie the research

community's continuing development of them, for PV applications.







The Cu-In-Se (cs) Material System

The thermochemistry of the Cu-In-Se ternary material system has been

intensely studied, but significant inconsistencies abound and the incompleteness

of the extant scientific literature will become apparent to the reader. One

superficial inconsistency is in the Greek letter designations employed to describe

the various phases, but even today there persist more substantive disagreements,

for example, on the number of phases found in the ternary phase field. To avoid

confusion all discussions herein that employ Greek letter designations to identify

thermodynamic phases will use the identifiers from the work by Boehnke and

Kihn [18].

Experimental studies that require bulk synthesis are extraordinarily

difficult because of the high vapor pressure of selenium and reactivity of copper

with quartz ampoules typically used [19]. It is therefore difficult to insure that

the thermodynamic system remains closed during synthesis and that the

resulting constitution accurately reflects the starting material ratios. Thus it is

difficult to judge whether syntheses intended to lie on the Cu2Se In2Se3

pseudobinary section remain so, hence whether that section is actually an

equilibrium tie-line. Although considerable progress has been made in the bulk

synthesis of these compounds [5], uncertainties such as these persist to this day

in efforts to assess the phase diagram.

The earliest published study of the Cu-In-Se phase diagram [20] was

restricted to a segment of the presumably pseudobinary section between the







compounds Cu2Se and In2Se3, and centered on the equimolar composition

corresponding to CulnSe2. Several key features of Palatnik and Rogacheva's

results have been confirmed in subsequent studies of this system, albeit with

different values of the critical point temperature and compositions. First,

congruent melting of the solid compound with a composition near that of

CuInSe2 at a temperature somewhat less than 1000C (9860C) is observed.

Second, a congruent first-order solid-solid (a -> 8) phase transition at a lower

temperature (8100C) of that high-temperature phase via a crystallographic order-

disorder transition between the sphalerite structure (8 phase) and the

chalcopyrite structure (a phase) is observed. Third, temperature-dependent

extensions of the phase homogeneity range of the chalcopyrite structure to

somewhat indium-rich compositions, but none towards copper-enrichment is

observed. Fourth, peritectoid decomposition of the sphalerite phase at its lowest

stable temperature into the chalcopyrite and a relatively indium-rich defect-

tetragonal structure is observed.

Extension of the characterization of the Cu-In-Se ternary phase field to

compositions off the Cu2Se-ln2Se3 section was finally published in the 1980's by

three groups [18,21,22] although there are significant discrepancies between

them. Boehnke and Kiihn find four phases on the indium-rich side of the

pseudobinary section between the compositions of CuInSe2 and In2Se3, whereas

Fearheiley and coworkers report seven phases based primarily on

crystallographic studies by Folmer et al. [23]. Bachmann and coworkers alone







find a congruently-melting copper-rich compound on this section with a

composition CusInSe4 (analogous to the mineral bornite, CusFeS4), reported to be

unstable at room temperature [24]. Bachmann and coworkers found two critical

point compositions for congruent melting of the solid phases on the indium-rich

side of this section: at 55% In2Se3 mole fraction (corresponding to about 22 at.%

copper) and at 75% In2Se3 mole fraction (corresponding to the compound

CuIn3Ses), whereas the others find only one. More recent study suggests that

there is only one congruently melting composition on this segment of the

liquidus at 52.5 mole% In2Se3 [25]. These and other studies have been assessed by

Chang and coworkers [26] resulting in the T-x section of the phase diagram

shown in Figure 1-1, which will be referenced in further discussions throughout

this treatise.

Another important study has been conducted more recently which

focused on a relatively restricted composition and temperature range directly

relevant to typical cis photovoltaic device materials and processing [27]. Its most

important conclusions were that the composition of the ac- congruent phase

transition occurs at 24.5 at.% Cu (50.8 mole% In2Se3) rather than the

stoichiometric composition of CuInSe2, and that the Cu2Se CuInSe2 phase

boundary at room temperature corresponds to this same composition. Their data

also confirm the retrograde phase boundary between the a-phase and --phase at

temperature below the a+->8 eutectoid transition temperature (which they find

to be 550C, near Rogacheva's but much lower than Boehnke's and Fearheiley's







results), with this boundary at room temperature at 24.0 at.% Cu (51.6 mole%

In2Se3).


1200


1100


1000


900


800


700


600


500 I
0 10 20 30 40 50 60 70 80 90 100
CuzSe X, mol.% InzSea

Figure 1-1 Assessed phase diagram along the Cu2Se In2Se3 pseudobinary
section of the Cu-In-Se chemical system [26].

The Cu-Ga-Se (CGs) Material System

The phase diagram of the Cu-Ga-Se ternary material system remains less

well-characterized and even more controversial than that of Cu-In-Se [28]. The

earliest detailed phase equilibrium study [29], once again restricted to the

presumably pseudobinary CuzSe Ga2Se3 section within the ternary phase field,







reported the existence of one high-temperature disordered phase and 4 room-

temperature stable phases. Two of those latter phases were solid solutions based

on the terminal binary compounds, one was a phase (P3) with the CuGaSe2

composition as its copper-rich boundary, and the last was a relatively indium-

rich phase (8) with a layered structure. The only other comprehensive study of

this ternary phase field [30] failed to confirm the existence of that 8 phase or the

associated compound CuGasSes.

Both studies, however, found that the stoichiometric compound CuGaSe2

has a chalcopyrite structure and does not melt congruently, but instead

undergoes peritectic decomposition at a temperature of 1050-1030C. The earlier

study by Palatnik and Belova [29] characterized the resulting gallium-rich solid,

representing the copper-rich boundary of the high-temperature (y) phase, as the

compound Cu9GanSe21 (55 mole% Ga2Se3) possessing a disordered sphalerite

crystal structure. They found the associated liquid composition at the peritectic

to be 38 mole% Ga2Se3.

A more recent study of CuGaSe2 crystal growth by the gradient freeze

technique [28] provides evidence contradictory to the earlier reports that the

compound decomposes peritectically and suggests instead that it decomposes

congruently and that the earlier studies mistook a solid-phase transformation

which they find at 10450C for peritectic decomposition. Resolution of these

discrepancies will require further scientific inquiry, and a comprehensive

assessment is needed.





9

Perhaps most importantly for photovoltaic-related process development is

the consensus between both of these studies of the phase diagram that the

homogeneity range of the chalcopyrite phase extends significantly to indium-rich

compositions along this section as it does in CulnSe2, but not measurably

towards compositions more copper-rich than that of stoichiometric CuGaSe2.


The Cu-In-S (cisu) Material System

Unlike the other two ternary copper chalcopyrites discussed herein,

CulnS2 occurs naturally, as the mineral roquesite. The earliest comprehensive

study of the Cu2S In2S3 section was conducted by Binsma and coworkers [31].

They found four room-temperature phases, two corresponding to the terminal

binaries and two others containing the compounds CuInS2 (y) and CuInsSs (e).

They did not report the low-temperature homogeneity range of these phases

other than to note that for CuInS2 it was below their detection limits. An earlier

study, however, reported the homogeneity range of -CuInS2 to be 50-52 mole%

In2S3 and that of E-CuInsSs from the stoichometric composition to almost 100%

In2S3 [32]. At higher temperature, but below the chalcopyrite to sphalerite

congruent solid phase order-disorder transition temperature at 9800C, Binsma

found that the homogeneity range of y-CuInS2 extended to copper-rich

compositions, unlike the ternary phases containing CuInSe2 and CuGaSe2. A

third solid-phase transition of the sphalerite structure was detected at 10450C,

just below the congruent melting temperature of 10900C.







Much of the thermochemical data published on the Cu-In-S ternary

system prior to 1993 has been incorporated into an assessment published by

Migge and Grzanna [33]. A more recent experimental study of the CuInS2 In2S3

subsection of the ternary phase field [34] found similar solid phase structures

and transition temperature as those reported by Binsma, including the congruent

melting of the indium-rich phase with a spinel structure and compositions

around that of the compound CuInsSs. They also found, however, an

intermediate phase with a fairly narrow homogeneity range around the

62.5 mole% In2S3 composition of the compound Cu3InsS9, which was reported to

exhibit a monoclinic structure.

Another recent study extended the Cu-In-S ternary phase field

characterization to the CuS InS join [35], and confirmed that the Cu2S In2S3

pseudobinary section appears to be an equilibrium tie-line in this ternary phase

field. They find that the room-temperature homogeneity domain for the

roquesite y-CuInS2 phase is limited to 52 mole% In2S3 but extends towards CuS

enrichment as much as six mole%. They also find that the two indium-rich

ternary phases on the pseudobinary section described in the previous paragraph

do not extend to this join.


Crystallographic Structure of the Ternary cls Compounds

This section is limited to a discussion of those compounds that are stable

at room temperature, with the exception of 8-cis. This is not a particularly







serious restriction for subsequent discussions of thin film growth techniques,

since all of those under development for device applications take place at

temperature well below the solid-phase transition and decomposition

temperature of all of these compounds, with the possible exception of the P to

&cIs transition as discussed in the previous section.


a-cIs (Chalcopyrite CuInSe2)

The crystal structure of a--cs is well established to be chalcopyrite,

corresponding to the space group I42d. It is an adamantine structure, as are -cis

and P-cis, characterized by tetrahedral coordination of every lattice site to its

nearest neighbors. It is distinguished from the zincblende structure of the binary

Grimm-Sommerfeld compounds [36] by ordering of itsfcc cation sublattice into

two distinct sites, one occupied in the ideal structure by copper and the other by

indium (Figure 1-2 (a)), and valency considerations require exactly equal

numbers of each. Single-phase homogeneous crystals will for entropic reasons

always exhibit some degree of disorder at room temperature irrespective of the

deviation of their composition from the stoichiometric compound CuInSe2,

although such deviations will always increase that disorder. The chalcogenide

atoms are located on anotherfcc lattice referred to as the anion sublattice. The

two sublattices interpenetrate such that the four nearest neighboring sites to each

cation site lie on the anion sublattice (Figure 1-2(b)) and conversely the four

nearest neighboring sites to each anion site lie on the cation sublattice (Figure







1-2(c)). Each anion is surrounded by two Cu and two In site types, normally



(a) (b)


O *












shell showing bond lengths d-s and d-s.







The very different chemical nature of the copper and indium atoms result
4 -se *Se







in bonds between each of them and their neighboring selenium atoms with veryoms.

different ionic character and ic represents ation37]. This bond-length alcopyrite rystal structure:
(a) electronventional unieffect of reducing the bandgap energy of tha square compound with the
(b) cation-pyrite sre, relative to th coordinate of the ternary sphalanion-ceritered firstructure with
shell showing bond lengths dcu-se and din-se.


The very different chemical nature of the copper and indium atoms result

in bonds between each of them and their neighboring selenium atoms with very

different ionic character and lengths [37]. This bond-length alternation has the

electronic effect of reducing the bandgap energy of the compound with the

chalcopyrite structure, relative to that of the ternary sphalerite structure with

identical chemical composition, since the latter has a disordered cation sublattice.

This bandgap reduction effect is known as optical bowing.





13

Bond-length alternation also has the effect of making the lattice constants

of the chalcopyrite structure anisotropic in most cases. Binary compounds with

the zincblende structure and the elemental compounds with a diamond structure

require only one lattice constant to quantitatively characterize the crystal

dimensions. The conventional unit cell of the chalcopyrite structure as shown in

Figure 1-2 is equivalent to two cubic zincblende unit cells with sides of length a

stacked in the c-direction and either compressed or dilated along that axis by a

factor i = c/2a, known as the tetragonal distortion.

The lattice constants of CulnSe2 have been widely studied but the early

results by Spiess and coworkers [38] are in excellent agreement with the most

recent measurements of bond lengths by EXAFS [39]. Those values are a = 5.784 A,

c = 11.616 A (and hence r = 1.004), dcu-se = 2.484 A, and din-se = 2.586 A. A more

comprehensive compilation of the various reports of lattice constant

measurements for CuInSe2 may be found in Chang's dissertation [40].


8-cis (Sphalerite)

The &cIs phase is unstable at room temperature, and there is wide

agreement that it forms from either solidification over a wide composition range

of the ternary liquid or a first-order solid-phase transformation from either the

a- or P-cIs phases or mixtures thereof (see Figure 1-1). The &-cIs single-phase

domain exhibits a congruent melting composition, for which the values of

1005C at 52.5 mole% In2Se3 [25] are accepted here. At lower temperature the







domain of -Clis is limited by the eutectoid at 6000C [27] where it decomposes

into a mixture of a- and P-cis. There remains inconsistency between the various

studies over the compositional range of single-phase stability in the relevant

high-temperature regime. Fearheiley's phase diagram [22] posits that this phase

is limited on the copper rich side by a eutectic associated with the putative

compound CusInSe4, and stable to much higher In2Se3 mole fractions than found

by Boehnke and Kihn [18], or than shown in Figure 1-1.

The congruent first-order a-8 solid phase transition at 24.5 at.% Cu (50.8

mole% In2Se3) and 8090C [27] corresponds to the crystallographic order/disorder

transformation from the chalcopyrite to sphalerite structure. The sphalerite

structure is based on the zincblende unit cell (and hence does not exhibit

tetragonal distortion), with no long-range ordering of copper and indium atoms

on the cation sublattice. The persistence of short-range ordering in &8-s,

specifically the dominance of 2 In + 2 Cu tetrahedral clusters around Se anions as

found in a-cIs, has been theoretically predicted [41].


P-cis (Cu2InlSe7 and CuIn3Ses)

It is doubtful that there is any part of the ternary Cu-In-Se phase diagram

that is more controversial and simultaneously more important to understanding

the operation of cIs Pv devices than the indium-rich segment of the pseudobinary

section containing the P-cis phase domain shown in Figure 1-1. There is no

agreement between the many studies of these relatively indium-rich materials on







the phase boundaries' compositions, the number of different phases that lie

between CuInSe2 and CuInsSea (y phase) or their crystallographic structuress.

The situation in this field is very similar to that found in the study of the

metal oxides, wherein there is considerable controversy as to whether

nonstoichiometric phases are single phases with broad ranges of compositional

stability, or a closely spaced series of ordered phases with relatively narrow

ranges of stability [42, 15.2-15.3.].

The existence of the peritectoid decomposition reaction of &cis to the a

phase and another In2Se3-rich solid phase requires that between the compositions

of CuInSe2 and In2Se3 there lies at least one other distinct phase on their tie-line

to satisfy the Gibbs phase rule. A review by Chang [40] finds at least eight

different compounds (Cu2In4Se7, CulIn3Ses, CuInsSes, Cu8InlsSe32, Cu7In9Se32,

Cu14nt16.7Se32, Cu2In3Se5, Cu3ln5Se9), and structures based on eight different space

symmetry groups (I4, 142m, P23, Pm3, P432, P43m, Pm3m, P42c) have been

proposed for 3-Ceis (although not all these compounds lie on the pseudobinary).

Most of these proposed structures are members of the group of adamantine

superstructures derived from the cubic diamond lattice structure [43]. Recently a

twinned structure that does not correspond to any of the 230 regular space

groups [44,45] was also proposed.

Various nomenclatures are used by different researchers to describe the 3-

cis compounds. They are sometimes referred to as P-chalcopyrite, a term coined

by H6nle and coworkers when they concluded that the structure possesses P42c







symmetry [46]. These structures are also sometimes referred to generically as

"Ordered Defect Compounds" (ODc's) but it is important to understand that

"ordering" in the context of this terminology refers to the regular arrangement of

preferred crystallographic sites on which defects are found, which alters the

symmetry properties of the lattice. The defect distributions on those preferred

sites in equilibrium might not have any long-range spatial order, although their

statistical occupation probabilities could nevertheless be well defined.

It is beyond the scope of this review to attempt any resolution of this

continuing controversy. Yet numerous studies of polycrystalline cis [47], Cisu

[48], and CIGS [49] Pv absorber films have shown that the composition at the

surfaces of those films which ultimately yield high efficiency devices exhibits a

[I]/[III] ratio of about 1/3, corresponding to the compound CuIn3Ses (except for

nearly pure CGS where the ratio rises to about 5/6 [49]). Resolution of these

crystallographic and phase boundary uncertainties is essential to testing a recent

theory that this behavior results from copper electromigration limited by the

occurrence of a structural transformation at those compositions [50]. The

existence of such a transformation is consistent with Fearheiley's evidence (which

has not been confirmed) that the compound CuIn3Ses melts congruently [22] and

the crystallographic studies by Folmer [23] that find additional reflections in XRD

spectra for pseudobinary compositions of 77 mole% In2Se3 or greater. The results

of a recent EXAFS study directly prove that the crystallographic structure of





17

CuIn3Ses (75 mole% In2Se3) is defect tetragonal, containing a high concentration

of cation site vacancies [51].


'y-cis (CulnsSes)

Folmer has pointed out [23] that the one common denominator between

all of the structures found along the pseudobinary Cu2Se-In2Se3 section is the

persistence of a close packed lattice of selenium atoms. It is well known that

different stacking sequences of such planes yields different crystallographic

structures, for example the hexagonal dose-packed (...ABAB...) and the face-

centered cubic (...ABCABC...), and that there are an infinite number of possible

stacking arrangements [52, 4]. In cubic notation, these close-packed planes of

thefcc structure are the (1111 family (corresponding to the (221) planes of the

chalcopyrite structure because of the latter's doubled periodicity along the c-

axis).

Although the terminal indium binary compound In2Se3 on the

pseudobinary section has been reported to possess several polymorphic

structures, the low temperature phases are characterized by hexagonal stacking

of the close-packed planes of selenium atoms on the anion sublattice [53]. Hence

the existence of a structural transformation between the cubic stacking

arrangement of thefcc anion sublattice of the chalcopyrite x-cIs structure and the

hexagonal stacking of In2Se3 at some point along that segment of this section is

reasonable. The crystallographic studies by Folmer [23] described previously







find additional reflections in XRD spectra that they index as (114) and (118),

which represents evidence of at least partial hexagonal stacking of the dose-

packed layers of selenium anions, yielding a layered structure, presumably

containing a high density of cation vacancies and antisites.

The segment on the Cu2Se-In2Se3 section containing > 77 mole% In2Se3 is

assigned in Figure 1-1 to a single y-cis phase and a two-phase mixture of

y-cis + In2Se3. Folmer concluded that there are three phases (excluding the

terminal In2Ses) instead of one. Given the diversity of wurtzite-derived ternary

defect adamantine structures with a hexagonal diamond structure [43] the

crystallographic data do not provide clear evidence in favor of either a few

distinct phases in a closely-spaced series or a pseudo-monophasic bivariant

system [54] characterized by coherent intergrowth of two phases.


Metastable Crystallographic Structures CuAu-ordering

Inasmuch as the chalcopyrite structure of a-Cs is itself an ordered variant

of the sphalerite structure of 8-CI, the issue of alternative ordering in the cis

material system has long been an active area of study. Vacancy ordering in

conjunction with the indium-rich |-cIs phase has been described in an earlier

section, but here alternative ordering of materials with a composition within the

equilibrium stability range of c-cis is discussed.

As early as 1992 a theoretical study by Wei and coworkers [41] of the a/-

cIS order-disorder transition calculated that the energy of formation of the CuAu





19

(CA) crystallographic structure (Figure 1-3) differed by only 0.2 meV/atom from

that of the chalcopyrite (CH) at T=0. In 1994 Bode [55], however, reported

evidence of CuPt-ordering (CP) from TEM studies of copper-rich cis films. CuPt-

ordering of HI-V alloys has been widely observed since it was first reported in

the AlGaAs system [56]. In cis the calculated formation energy difference

between the CP and CH structures (at zero Kelvin) was more than 25 times greater

than the difference between that of CA and CH-ordered crystals [41].

The equilibrium CH-CIS crystallographic structure shown in Figure 1-3(a)

consists (in cubic notation) of alternating (201) planes of Cu and In atoms on the

cation sublattice. The CA-CIS structure shown in Figure 1-3(b) consists of

alternating (100) planes and CP-CIS structure consists of alternating (111) planes

[57]. Consequently, each selenium atom in both the CH and CA structures is

surrounded by 2 copper and 2 indium atoms in its first coordination shell

whereas in the CP structure each selenium is surrounded by either (3 Cu + In) or

(3 In + Cu). This variation in local atomic structure is the fundamental reason for

the similar formation energies of the CH and CA structures and their mutual

disparity from that of the CP structure.

The apparent doubling of the periodicity along (111) (cubic notation)

planes that was observed in the study that reported cP-CIs [55] was found in

polycrystalline samples made by codeposition of Cu, In, and Se with an overall

composition in the mixed P-Cu2-&Se + a-CuInSe2 phase domain of the equilibrium

phase diagram (Figure 1-1). Their interpretation has been recently challenged





20

[58] based on the results of a careful study of CIs grown epitaxially on GaAs with

a similar copper-rich composition, where it is shown that coherent intergrowth of

a 3-Cu2-sSe secondary phase can create an apparent doubling of lattice

periodicity and thence of CuPt-ordering in copper-rich cIs. Coherent

intergrowth of 03-Cu2-sSe and CuInSe2 has been suggested by other researchers to

be an energetically favorable strain relief mechanism [59] since these two

compounds share isomorphic, nearly identical Se sublattices.

CuAu-ordering (CA) of the Cu-III-VI2 compounds was first detected

experimentally by TEM in CuInS2 [57] indium-rich MBE-grown epilayers where

the formation of a secondary Cu2-aS phase is unlikely. Recently CA-ordering has

been demonstrated in CuInSe2 in both copper and indium-rich materials grown

by Migration-Enhanced Epitaxy (MEE) [60] using XRD, TEM, and Raman scattering

detection techniques [61]. Further studies of the electronic and optical properties

of CA-CIS are needed to assess their impact on PV device absorber materials,

which very likely contain nanoscale domains of this crystallographic polytype.


Defect Structure of a--cis

The study of the defect structure of a-cis has probably generated more of

the literature on a-cIs than any other fundamental scientific issue. Pure a-cis is

amphoteric: its conductivity type and carrier density varies with composition. It

is incorrect to say, however, that these electronic transport properties in real

materials are determined by composition alone since the defect structures that






must be controlling them are empirically found to vary dramatically between
compositionally indistinguishable materials.


O


O


a) "W W b) w W
Figure 1-3 Comparison of the crystallographic unit cells of CuInSe2 polytypes:
a) chalcopyrite (CH) structure, and b) CuAu (CA) structure.

Conceptually the densities of defect structures found in a single-phase
material system in equilibrium must be determined uniquely by the composition,
temperature, and pressure, else the Gibbs potential, a function of these variables,
is not a legitimate state function for the system. The only intellectually
satisfactory resolution of this conundrum is to conclude that complete
thermodynamic equilibrium is not often found in real cIs materials. As described
in the previous section, recent calculations and experimental results confirm


O







[61,62] that the free energy associated with the formation of some defect

structures is so small that little increase in thermodynamic potential results, and

hence there is insufficient driving force to ensure their elimination under many

synthesis conditions. Furthermore, formation of many atomic defects requires

bond breaking and atom transport processes. At low deposition or synthesis

temperature it is expected that these processes will limit the approach to

equilibrium. Comparison of theory with experiment in this field absolutely

demands constant awareness of the ubiquity of metastable defects in real cis

materials and thus great caution when generalizing limited experimental data.

The starting point for atomistic analyses of the defect chemistry of CuInSe2

is the paper by Groenink and Janse [63] in which they outline a generalized

approach for ternary compounds based on elaboration of an earlier model

developed specifically for spinels by Schmalzried [64]. The number of arbitrary

combinations of possible lattice defects (vacancies, antisites, and interstitials) in a

ternary system is so great that useful insight can only be gained by some

approximation. Antisite defects created by putting anions on cation sites or vice

versa are reasonably neglected because of their extremely high formation energy.

The requirement that the crystal as a whole is electrically neutral also leads

naturally to Schmalzried's assumption that for any given combination of the

thermodynamic variables the concentrations of some pair of defects with

opposite signs will be much higher than the concentrations of all other defects.

Groenink and Janse referred to these as the "majority defect pairs." It is important







to note that their treatment assumes that these pairs behave as non-interacting

point defects, hence in this context these are "pairs" only in the sense that they

occur in roughly equal numbers. It is also significant that this pair dominance

implies those conduction processes in these materials should inevitability be

characterized by significant electrical compensation, deep level ionized impurity

scattering, or both.

The generalized approach by Groenink and Janse was applied specifically

to I-III-VI2 compounds by Rinc6n and Wasim [65] who derived the proper form

for the two parameters most useful for quantifying the deviation of the

composition of these compounds or their alloys from their ideal stoichiometric

values:

Am =- -1 molecularity deviation
[III]

2*[Vii
As = 2 [VI 1 valence stoichiometry deviation
[] + 3 [III ]

Note that in the notation employed in these equations [I], for example, denotes

the Group I atom fraction. Since [I]+[III]+[VI]=1, these two deviation variables

uniquely specify the solid solution composition.

In the same way that a sum rule enables the composition of any ternary

mixture to be specified completely using only two of its three fractional

compositions, the composition can alternatively be specified by the two variables

Am and As. They are coordinates within the ternary I-III-VI composition

triangle of the point corresponding to a compound's actual composition in a







coordinate system whose origin is at the point of I-III-VI 2 stoichiometry and

whose axes are along molecularityy) and transverse (valency) to the I2VI-III2VI

section. Within the composition range where the I-III-VI2 compound or alloy

remains single phase, these variables may be properly viewed as analogous to

the "normal coordinates" of a dynamical system in the Lagrangean formulation

of the physics of motion. This coordinate system divides the ternary composition

triangle into four quadrants and the analysis of Rinc6n and Wasim [65] shows

that the 18 ionized point defects allowed in these approximations yield 81

(= 9* 9) "majority defect pairs," and which might dominate in each of the four

quadrants or at their boundaries.

The merit of molecularity and valency deviations as intrinsically relevant

composition measures in cIs has been empirically demonstrated by careful

studies of conductivity in single crystal CuInSe2 [3]. Neumann and Tomlinson

demonstrated that within the range Am| < 0.08 and |sAs < 0.06, p-type

conductivity occurs whenever As > 0 (electron deficiency) whereas n-type

conductivity occurs for As < 0 (electron surplus). Their Hall effect measurements

also showed that the dominant acceptor changed in p-type cis from shallow (20-

30 meV) whenever Am > 0 (excess copper) to deeper (78-90 meV) when Am < 0

(indium-rich).

The actual predominance of a specific majority defect pair in any given

quadrant of the molecularity vs. valency domain will in equilibrium be

determined by whether its free energy is lower than that of the other probable







pairs. A vast amount of theoretical analysis [66-68] was directed in the 1980's

towards estimation of the enthalpies of formation of the various point defects

since their experimental determination is formidable. There is clear agreement

among those analyses that the energy of formation for an isolated point defect is

lowest for the cation antisite defects Cuin and Incu. There was some disagreement

as to whether the next lowest formation enthalpy values are for the copper

vacancy, Vc. [66,68], or selenium vacancy, Vse [67].

There remained several disturbing issues with those analyses. First is the

lack of the predicted correlation between the composition and net carrier

concentration [3]. Second is the low level of minority carrier recombination in

polycrystalline CIS Pv devices, which are always made with significant negative

molecularity deviation, often in the biphasic a+3 domain. Recalling that the

chalcopyrite unit cell contains 16 atoms, a defect concentration of little more than

6% would yield a statistical probability of one defect per unit cell if they are

randomly distributed.

Defect complexes provide a resolution of these deficiencies, since all the

atomistic models described above exclude defect complexes (associates) which

should be anticipated given the Coulombic attraction between the oppositely

charged members of these "majority defect pairs." The dominant cohesive

bonding force leading to the negative contribution to enthalpy that stabilizes

ionic crystals is the Madelung energy [69] resulting from precisely this

Coulombic attraction, and defect clustering resulting in short-range order has







been shown essential to understanding the defect chemistry of

non-stoichiometric transition metal oxide phases [54].

Theoretical ab-initio quantum-mechanical calculations of cation defect and

defect complex formation enthalpies in CuInSe2 [70] have recently provided

support for these assertions. These results showed that the formation enthalpies

of lattice defects depend on the chemical potential of the constituent atomic

species, and in the case of charged defects, on the chemical potential for electrons

(equal to the Fermi energy at T=0 K). The results showed explicitly that when the

chemical potential of indium sufficiently exceeds that of copper the formation

enthalpy of the (In2 + 2Vc )O neutral defect complex (NDc) actually becomes

negative (energetically favorable). Formation of this defect requires the removal

of three monovalent copper ions and substitution on one of those vacancies of

the trivalent indium; hence it has no net effect on the valence stoichiometry

deviation As. Their calculations were extended to the calculation of the energetic

effects of long-range ordering of the (In2 + 2V ) complex [71]. They show that

the reported compositions of indium-rich compounds (Am < 0) on the

pseudobinary section could be achieved by mathematically rational ratios of the

numbers of this complex to the number of chalcopyrite unit cells, and that

ordering was energetically favorable.

Additional long-range crystallographic ordering possibilities for the

(In"' + 2V )O NDC have been proposed by Rockett [72] and further investigations

are needed to determine the true nature and extent of NDC ordering.







Nevertheless, a recent study of the P-phase compound CuIn3Ses (X=0.75 in

Figure 1-1) [51] has shown that the EXAFS scattering spectrum of selenium in this

compound is best fit by a local structure model having precisely these defect

proportions in the nearest-neighbor tetrahedra surrounding Se atoms in the

lattice (Figure 1-2(c)). This is strong experimental evidence that the "majority

defect pair" found in indium-rich CIS compounds on the pseudobinary section is

in fact this cation NDC.

Deviations from valence stoichiometry off the pseudobinary section

(As 0) cannot be caused by the (Inc + 2V )O NDC. Deviations of As < 0 are

caused by defects which create an excess of electrons compared to those required

to form the "normal valence compound" [73]. As examples, an Incu antisite defect

brings two more valence electrons to that lattice site than when normally

occupied by copper, Cucu; and Vse creation removes two bonding orbitals from

the lattice, which would otherwise be normally occupied, thereby freeing two

electrons to be donated to the conduction band by cations. Conversely,

deviations of As > 0 are caused by defects that create a deficiency of electrons

needed for the normal valence configuration (e.g. Vcu). These considerations lead

to the notation In2, which represents an In ion placed at a cation antisite on

the lattice that is normally occupied by Cu in its +1 oxidation state.

One of the other results from Zhang and coworkers' studies of cation

defect energetic in cIS is their calculation of electronic transitions associated

with the ionization of isolated point defects and busters [70]. Their quantum-







mechanical studies show that the contrast between relative ionicity and

covalency of the copper and indium bonds, respectively, result in an

unexpectedly shallow acceptor level for Vcu (30 meV) and unexpectedly deep

donor levels (Ec-0.24 and Ec-0.59 eV) for the indium cation antisite, Incu. The

shallow donor seen in a-CuInSe2 with deviations of As > 0 had been attributed

in many studies to Incu acting as a donor but these results show that both of its

ionization levels are deeper than that of the (In2 + V&) part of the NDC and all

were too deep to correspond to the very shallow (20-30 meV) donor seen in

numerous studies [3].

One of the limitations of Zhang and coworkers' earlier studies of cation

defect energetic in cis was neglect of defects on the anion sublattice. In

particular the Vse is another widely suggested candidate for this shallow donor

defect [3,68,74]. Investigations of vacancy defects in epitaxial CuInSe2/GaAs via

positron annihilation lifetime studies have been interpreted to suggest that the

most probable defect is the (Vs + Vc,) defect [75-77]. More recent ab-initio

quantum-mechanical calculations of the Vs, V2 electronic transition energy

[78] predict that significant lattice relaxation is associated with the Vse ionization

process, and that the energy level of the indirect (phonon-assisted) transition is

Ec-0.10.05 eV. This represents the most shallow donor level calculated for any of

the point defects investigated theoretically by that group.

The possible role of Vse and cation-anion point-defect complexes in cIs

with deviations from valence stoichiometry (i.e., off the pseudobinary section







with As 0) does not yet appear to have been adequately investigated. Van

Vechten has argued [79] that Vse is unlikely to be stable in indium-rich materials,

proposing a defect annihilation mechanism when both Am < 0 and As < 0 based

on the quasichemical reaction:

2Vc, +In, +2VL Inn +2e -1 crystal unit,

which he suggests would be energetically favorable because of the large cohesive

energy of the lattice compared to the energy of Incu formation.


Optical Properties of Ternary Cu-III-VI Materials

The focus in this section is the fundamental optical bandgaps of the

a-phase compounds CulnSe2, CuInS2, CuGaSe2, and of their associated P-phases.

Discussion of the opto-electronic properties of alloys will be deferred to the

following section.


Optical Properties of a-cis and P-cls

Early measurements of the bandgap energy of single-crystal CuInSe2

exhibited nominal discrepancies [80,81], suggesting a value in the range of 1.02 to

1.04 eV. Subsequent studies [82,83] showed evidence of significant optical

absorption at energies below this fundamental absorption edge. Characterization

of polycrystalline cis absorber films suitable for devices almost always indicate a

significantly lower effective bandgap of -0.90 eV [84], apparently a consequence

of significant collection of carriers generated by absorption in these band-tails. It





30

has been suggested that the widely reported variations in the optical properties

of cIS materials are a direct consequence of variations in composition [85].

The most recent published study of radiative recombination in near-

stoichiometric CuInSe2 epilayers on GaAs yields a value for the fundamental

absorption edge of Eg = 1.046 eV at a temperature of 2 K, with a slight increase to

a value of Eg = 1.048 eV at a temperature of 102 K [86]. Near room temperature,

however, the temperature dependence follows the Varshni relation [87]:

aT2
E,(T) = Eg(O) T+j(

with P= 0 and a = 1.1 xlOeV/ K [85]. Anomolous low-temperature absorption

edge dependency is often observed in of I-IH-VI2 semiconductors [88]. This

phenomenon will be discussed in further detail in the section describing the

optical properties of CuGaSe2, since it has been more thoroughly investigated for

that compound.

This low and high temperature data published by Nakanishi and

coworkers [85] was subsequently fitted over the entire temperature range [89] to

the Manoogian-Lecrerc equation [90]:

E,(T) = E,(0)-UT V[coth ].

The fitting parameters Eg(0), U, V, and s are temperature-independent constants,

although they do have relevant physical significance. For example, the second

and third terms represent the effects of lattice dilation and electron-phonon





31

interactions, respectively. The temperature 0 is the Einstein temperature, related

3
to the Debye temperature by 4 = -_ [89], and the value used in their

calculations was derived from the published value of (o = 225 [K] [91], yeilding

= 170 K. The best fit to that data was found for

Eg(0)= 1.036 [eV],U = -4.238 x 10 [eV K-'] V = 0.875 x 10 [e V K-1I and s = 1.

The corresponding 300 K bandgap energy is 1.01 eV. Note the -10 meV

discrepancy between this value for the bandgap at absolute zero temperature

and that discussed earlier in this section [86].

The spectral dependence of the refractive index of CuInSe2 has been

reported for both bulk and polycrystalline [92] materials as well as epitaxial films

on GaAs [93] Here too, significant discrepancies are found in the reported data.

Analogous discrepancies are found in the reported optical properties of

P-cIs synthesized by different techniques. Polycrystalline films with an overall

composition corresponding to the compound CuIn3Ses are reported to exhibit a

room-temperature fundamental absorption edge at 1.3 eV [47]. Optical

absorption and cathodoluminescence characterization of heteroepitaxial CuInaSes

films on GaAs has been interpreted to indicate a bandgap of Eg 1.18 eV at 8 K

[94]. The most thorough characterization has been conducted on bulk

polycrystalline samples with a nominal composition of Culn3Se5 [95]. The

temperature dependence of the absorption coefficient edge was fitted using the

Manoogian-Lecrerc equation. The best fit to their data was found for







E,(0)=1.25-1.28 [eV], U =2.0 x10- [eV.K-1l V=1.2-1.5x10' [eV.K-'1

0 = 205 213 [K], and s = 1. The corresponding 300 K bandgap energy is in the

range of 1.19 to 1.21 eV. Although there are significant quantitative discrepancies

between the various published data, they all agree without exception that the

bandgap energy of 13-cis is substantially (0.2-0.3 eV) greater than that of a-Cls.

Variation of optical absorption with composition. The fundamental

absorption edge for intrinsic undoped semiconductors can be determined by

extrapolation of the plot of the absorption coefficient a vs. ;iu to a = 0 [96].

Residual absorption at energies below the fundamental absorption edge in

semiconductors which obeys the empirical relationship d(ln a)/d(ho) = 1/kT is

referred to as an Urbach tail [97]. This is known in conventional extrinsically

doped semiconductors to arise via the Franz-Keldysh effect produced by spatial

fluctuations of the internal electrostatic field to give spatial variations in charged

impurity density [98] over distances larger than the Debye screening length.

Photon-assisted tunneling [99] between the resulting exponential bandtails [100]

results in these characteristic exponential optical absorption tails.

The temperature and spectral dependence of the observed sub-bandgap

absorption in single crystal CulnSe2 has been carefully studied by Nakanishi and

coworkers [101]. When they fitted their data to the conventional equation [102] of

the Urbach form:

ro(h E0)]
a = aexpkL kT "







where, with hti representing the optical phonon energy [103]:


a = o Jtan h J),


they found that unphysically large values for the optical phonon energy were

required, and that they depended on composition. However, using the equation:
x(h-E)1
a= a0expE (h,-E0)
L (E,(T,x)"

they separated Ea(T,x) into the sum of two terms, one linearly dependent on

composition and the other a temperature dependent factor that fit the prior two

equations with the reported value for the optical phonon energy. They concluded

that the exponential optical absorption bandtails in CuInSe2 arise both from

phonon and compositional fluctuations, the latter increasing linearly with

negative molecularity deviation.

Further variations in optical absorption and emission of a-cis are

associated with negative valence stoichiometry deviations (As <0). Early

annealing studies [74] showed a significant red-shift of photoluminescence

emission when bulk samples were annealed or synthesized in excess indium

vapor, and a reversible blue-shift after synthesis or annealing in excess selenium

vapor. A more recent study [104] suggests the formation of an impurity (Vse)

subband when As < 0.05.

This phenomenon of strong sub-bandgap absorption in indium-rich cis

giving rise to apparent narrowing of the effective bandgap is also observed in







epitaxial films of cis on GaAs studied by piezoelectric photoacoustic

spectroscopy [105], evidence that it is a consequence of the native defect structure

of these materials, and not an artifact of polycrystallinity, preparation, or

measurement technique. It appears that this effect extends to the biphasic a-03

composition domain, which suggests that the coexistence of these two phases is

accompanied by an interaction between them that results in composition

fluctuations manifested as strong band-tailing in their combined optical

absorption. It is unclear whether this is an equilibrium phenomenon or related to

ubiquitous metastable defect structures common to the materials investigated by

so many researchers.


Optical Properties of a-CGS

The temperature dependence of the bandgap energy of CuGaSe2 has been

well characterized recently [89], with the data also fitted to the Manoogian-

Lecrerc equation. The best fit to the data with s = 1 was found for

E,(0)=1.691 [eV], U =-8.82x 10- [eV. K-'jand V=1.6 x10-' [eV. K-' with

= 189 K, based on the reported Debye temperature for CuGaSe2 of 0~ = 259 K

[91]. The corresponding 300 K bandgap energy is 1.65 eV. Refractive index data

for CuGaSe2 over the range 0.78 to 12.0 pm has been reported by Boyd and

coworkers [106].







Optical Properties of a-cIsu

The most recent determination of the bandgap of a-cIsu was based on

bulk two-phase CuxS + CuInS2 samples with slight negative valence

stoichiometry deviations analyzed by means of photoreflectance spectroscopy,

yielding a value of 1.54 eV at 80 K [107]. Earlier measurements of the bandgap

varied by about 30 meV in the range of 1.52 to 1.55 eV at room temperature [108].

The relationship of the effective bandgap to composition, discussed in the

preceding CIS part of this section, was studied [109], and the variance between

previously published values was attributed to the same effect. In particular, a

decrease in the effective bandgap was observed for negative valence

stoichiometry deviations (As < 0).

The temperature dependence of the CuInS2 bandgap is reported to exhibit

anomalous low-temperature behavior, like that described for all the other Cu

ternary chalcogenides discussed in this section [110,111]. Refractive index data

for CuInS2 over the range 0.9 to 12.0 pm has been reported by Boyd and

coworkers [112].


Alloys and Dopants Employed in cas Photovoltaic Devices

A later section of this review will describe in detail the reasons that most

cis PV devices are not made from the pure ternary compounds, but rather alloys

thereof. Breifly, bandgap engineering is the principal motivation. The

nomenclature might be somewhat confusing in this section unless the reader

keeps clearly in mind the distinction between a compound and an alloy. CulnSe2,







for example, is a ternary compound, as is CuGaSe2. Both of these "compounds:

show a small range of solid solution extent. An alloy of these two ternary

"compounds" is a binary alloy, although it is also a quaternary material (it

contains four elements). One may view this as simple mixing of Cu on the In

sublattice in a-cis. By induction, an alloy of that binary, Cu(In,Ga)Se2, with the

ternary compound CulnS2 yields the ternary alloy Cu(In,Ga)(S,Se)2, which is also

a pentanary material.


Gallium Binary Alloy CIGS

Until the very recent publication of the dissertation of Dr. Cornelia

Beilharz [113] no comprehensive thermochemical study of the quaternary CIGS

phase field was available. This is remarkable in view of the fact that most of the

published world record thin film solar cell efficiencies since 1987 (and all since

1995) have been held by cIGs-based devices. The predominant phase fields in the

pseudoternary CuSe2Sen2Se3-Ga2Se3 composition diagram as reported in that

work are shown Figure 1-4.

The most obviously important aspect of this CIGS pseudoternary

predominance diagram is the monotonic broadening of the ao-CIGS single-phase

domain towards more Group im-rich compositions with increasing Ga.

Practically speaking, this means that synthesis of single a-phase CIGS requires

less precise control over the [I]/[II] ratio molecularityy) than needed for single

phase a-cis synthesis, irrespective of the technique employed. Secondly, the





37
appearance of a domain characterized by both CC- CIGS (designated P1 in Figure

1-4) and p-CIGS (designated P2 in Figure 1-4) plus the disordered zincblende (Zb)

structure, not found at room temperature in either of the pure ternary

compounds. Note that the extent of this domain (designated Ch+P1+Zb in Figure

1-4) along lines of constant [In]/[Ga] molar ratio (i.e., lines emanating from the

Cu2Se corner) is minimal in precisely the composition range around 25% gallium

where the highest efficiency CIGS devices are fabricated [114,115].

Ga2Se3
100
Zb
20 80
Ch+Zb Zb+P1


Ch+P1+Zb ) Zb+P1+P2
60 40 Zb+P2

P2
80 20


100
Cu2Se 20 40 60 80 100 In2Se3

Cu2Se+Ch Ch Ch+P1 P1 P1+P2
Figure 1-4 Predominance diagram for the Cu2Se-In2Se3-Ga2Se3 pseudoternary
phase field at room temperature [113]. In that author's notation, Ch is the a
phase, P1 is the 0 phase, P2 is the y phase, and Zb is the 8 phase.





38

A theoretical study of the effects of gallium addition to CuInSe2 provides

some insight into likely atomic-scale phenomena leading to these effects [116].

First, they calculate that the energy of formation for the isolated group III cation

antisite defect, Gacu, is 0.2 to 0.9 eV greater (depending on its ionization state)

than that of Incu. Second, they calculate that the donor levels for isolated Gacu,

are deeper than those of Incu, hence if present in comparable concentrations Gacu

will not thermally ionize as easily as Incu, and therefore contribute less to

compensation of the acceptors which must dominate for p-type conductivity to

prevail. This is consistent with the experimental observation that hole densities

are higher in CIGS epitaxial films than in CIS epitaxial films with comparable

molecularity and valence stoichiometry [117]. Finally, the (Ga2 + 2Vcj )O Neutral

Defect Complex (NDC) is calculated to require 0.4 eV more energy to form than

the (InC + 2Vcu ) NDC, leading to 0.3 eV higher formation energy per NDC in the

Ordered Defect Compounds (ODC) (i.e., P or P2 phase) containing gallium. This

suggests that in CIGS materials with negative molecularity deviation, under

conditions where NDC aggregation can occur, ODC formation is more

energetically favorable in regions where composition fluctuations have lead to a

lower local gallium concentration.

Bandgap dependence on composition. Alloys of the copper ternary

chalcopyrite compounds, like those of virtually all the zincblende binary alloys,

are found to exhibit a sublinear dependence of their bandgap energy on alloy







composition. Their functional relationship is well approximated by the

expression:

E,(x)= xE,(1)+(1- x)Eg(0)- b(l- x)x,

where the parameter b is referred to as the "bowing parameter." Optical bowing

is now understood to be a consequence of bond alternation in the lattice [37].

Free energy minimization results in a tendency for A and B atoms to avoid each

other as nearest neighbors on the cation sublattice in AxBi-xC alloys, resulting in

short range ordering referred to as anticlustering [118 Chapter 4.].

A very large range of bowing parameters has been reported for CIGS thin

films and bulk Cu(In,Ga)Se2, varying from nearly 0 to 0.025, and data on thin

film CIGS absorber layers strongly supports the contention that this variability is a

consequence of variations in molecularity deviation between the samples

reported by various investigators [119 and reference therein.]. Another study

of combined temperature and composition dependencies of the bandgap in bulk

crystalline Cu(In,Ga)Se2 concluded that the bowing parameter may be

temperature dependent [89]. A theoretical value of 0.21 at absolute zero has also

been calculated [116]. A preponderance of the room temperature data is in the

range of 0.14 [120] to 0.16 [121] so the intermediate value of b = 0.151 from the

original work by Bodnar and coworkers is accepted here [122], leading to the

following expression for Ca-CuIn-.xGaxSe2:

Eac(x) = 1.65x +1.01(1 x) 0.151(1- x)x







Sulfur Binary Alloy --css

Woefully little thermochemical and structural data are available for the

Cu-In-Se-S quaternary system. The bandgap dependence on composition has

been reported by several researchers, with the reported optical bowing

parameters varying from 0 to 0.88 [123-125]. There is substantially better

agreement between a larger number of studies of the mixed-anion alloy

CuGa(SexSl-x)2 that the optical bowing parameter in that system is zero [126, and

references therein]. It has been argued that the bond-alternation which leads to

optical bowing in mixed-cation ternary chalcopyrite alloys does not occur in the

mixed-anion alloys [127], and that the bowing parameter should therefore vanish

in Culn(SexSl-x)2 as reported by Bodnar and coworkers [123]. The substantial

uncertainty and disagreement amongst the published experimental results

suggests that resolution of this question requires further investigation.


Alkali Impurities in cis and Related Materials

The importance of sodium for the optimization of polycrystalline CIS thin-

film solar cell absorber layers has been extensively studied since first suggested

by Hedstr6m and coworkers [128]. Their careful investigation of the

serendipitous sodium "contamination" of cIs absorber films due to exchange

from soda-lime glass substrates contributed to their achievement of the first CIS

device with a reported efficiency exceeding 15%. Subsequent studies have

concluded that whether derived from the substrate [129] or added intentionally

from extrinsic sources [130-132], optimized sodium incorporation is beneficial to







device performance, and excess sodium is detrimental [133-136]. Studies of

sodium's concentration and distribution in the films show it is typically present

at a -0.1 at.% concentration [137], and strongly segregates to the surface [138]

and grain boundaries [139].

A plethora of mechanisms has been suggested in an effort to explain the

beneficial influence of sodium, and an overview of the body of literature taken

together suggests that multiple effects contribute thereto. The primary

phenomenological effects in cIs and CIGS absorber materials may be summarized

as:

1. An increase in p-type conductivity [140] due both to the elimination

of deep hole traps [141], and an increase in net hole concentration resulting

predominately from reduced compensation [142].

2. An increase in the (112) texture and the average grain size in

polycrystalline films [143], with a concomitant reduction in surface roughness.

3. An increased range of compositions (specifically, negative

molecularity deviations) that yield devices with comparable performance

[144-146].

These effects have been attributed to both direct and indirect electronic

effects of sodium in the resulting materials themselves, and to the dynamic

effects of sodium during the synthesis process. These will be each discussed in

turn, beginning with the one model that attributes the improved properties of

absorbers that contain sodium on a bulk defect containing sodium.







Substitution of sodium for indium, creating residual Nain antisite defect

acceptors in the lattice of the resulting material, has been proposed to explain the

observed increase in p-type conductivity [137]. Theoretical calculations predict

[78] that its first ionization level, at 0.20 eV above the valence band edge, is

shallower than that of Cuin, but in typically indium-rich absorbers the formation

of the Cuin defect is less energetically favorable than are Vcu and Incu, the

structural components of the cation NDC. Furthermore, they calculate the

formation enthalpy of the Nain antisite defect is quite large (2.5 eV) when the

compounds CuInSe2 and NaInSe2 are in thermal equilibrium.

The simplest indirect model for the sodium effect on conductivity is that

the Nacu defect is more energetically favorable than the Incu defect, so it

competes effectively for vacant copper sites during growth, thereby reducing the

concentration of the compensating Inc, antisite defect [147] in the resulting

material.

A related model proposes that formation of Nacu substitutional defects in

lieu of Incu is a transition state of the growth reaction in indium-rich materials,

leading to a reduction in the final Incu antisite defect density within the bulk by

inhibiting the incorporation of excess indium into the lattice [148]. In this model,

sodium acts as a surfactant at the boundary between stoichiometric and indium-

rich cIS, forming a two-phase CuInSe2 + NaInSe2 mixture or quaternary

compound if sufficient sodium is available [149,150]. The advantages of this

model are that it predicts a reduction in the concentration of Incu point defects





43

and the NDC defect complexes in the bulk [151]. This model addresses all three of

the primary sodium effects: the morphological changes are a surfactant effect,

and the increased tolerance to negative molecularity deviation a consequence of

enhanced segregation of excess indium. This model has been developed by this

author and will be described in more detail in Chapter 5.

A study of the effects of elemental sodium deposited onto CuInSe2 single

crystals [152] led the authors to conclude that Na atoms at the surface disrupt

Cu-Se bonds, releasing Cu+ ions. These ions subsequently diffuse into the bulk

under the influence of the surface field resulting from band-bending induced by

the sodium itself, thereby increasing the concentration of Vcu acceptors in the

near-surface region. They also suggest that Nac. substitutional defects are

created during this process. For high doses of sodium, they find that this lattice

disruption results in the decomposition of CulnSe2, yielding metallic indium and

Na2Se, and suggest that 13-phase compounds may form at the surface as

intermediate reaction byproducts due to the enhanced Vcu concentrations. It is

difficult to understand how these effects would increase p-type conductivity,

since the excess copper ions released from the surface and driven into the bulk

would most likely recombine with the Vcu shallow acceptors that make it so.

Two other models attribute the influence of sodium on electronic

properties to its effects on the concentration of selenium vacancies. The first of

these [146] suggests that sodium at grain boundaries catalyzes the dissociation of

atmospheric 02, creating atomic oxygen which neutralizes surface Vse by





44

activated chemisorption, leading to the formation of a shallow acceptor [153,154].

Theoretical calculations of the bulk Ose ionization energy level predict very deep

levels [78], however, and studies of the electronic influence of implanted and

annealed sodium in epitaxial Cu(In,Ga)Se2 films provide evidence for

substantially reduced compensation without any evidence of oxygen diffusion

into the bulk [142].

The final published model for the effects of sodium attributes its influence

to increased chemical activity of selenium at the film's surface during growth

[155]. Strong evidence is provided that sodium polyselenides (NazSex) form on

the surface during growth, and they suggest that this acts as a "reservoir" for

selenium on the surface, reducing the formation of compensating Vse donor

defects.


Summary

The various I-III-VI2 material systems described in the foregoing section

show a great deal of similarity in the structure of their phase diagrams. The

common theme among them all is the ubiquity of ordering phenomena

associated with the different phases. Clearly, much more study is needed to

clarify the many unknown properties of each of these material systems and

provide the materials science foundation required to support their successful

application to photovoltaic devices.












CHAPTER 2
CIS POINT DEFECT CHEMICAL REACTION EQUILIBRIUM MODEL


Ternary chalcopyrite I-III-VI, compounds such copper indium diselenide

(CuInSe2) differ at a fundamental level from their binary II-VI zincblende

analogues because of the coexistence in the former of two distinct types of bonds.

Detailed quantum-mechanical calculations [156] show that the I-VI bonds tend

to be far more ionic in character than the III-VI bonds which are predominately

covalent. This heterogeneity leads to extremely strong optical absorption owing

to the resultant high density of unit-cell-scale local dipole fluctuations and to

ionic conduction resulting from the mobility of the relatively weakly-bound

group I atoms.

The point defect chemistry approach expounded by Kr6ger [157] is

employed. The intention is to explore the consequences of the native lattice

disorder (intrinsic point defects and aggregates thereof) caused by finite

temperature and deviations from stoichiometry in the equilibrium a-phase of

ternary Cu-In-Se, usually referred to by its ideal stoichiometric composition

formula CuInSe2.









Approach

An associated solution lattice defect model is developed to calculate the total

Gibbs energy function g(T,P,(Ni ) of a thermodynamic system comprised of a

continuum of electronic states and charge carriers interacting with atoms and ions

which reside on a denumerable lattice of sites. The defect chemical reactions of this

model involve atomic elements and charges within the crystal which is the

thermodynamic system of interest, and atoms, complexes and electrons in an outer

secondary phase which constitutes the reservoir with which the crystal is in

equilibrium. This approach treats specific well-defined point defects and their

complexes embedded in clusters of primitive unit cells on a Bravais lattice as

quasimolecular species and utilizes conventional chemical reaction equilibrium

analysis [158] to calculate their equilibrium concentrations.

An activity-based formulation for the total Gibbs energy of mixing (or mixture

formation) as a function of the temperature T, pressure P, and total number Ni of

each component in the mixture is defined as:
Max(i)
Ag= [T, P, (Ni ] -Ma Ni !j[T, P, (NP)l

where i is the partial molar Gibbs energy of a specie in its reference state,

according to the equations:

g[T,PN] =-Max(i) Ni( [T, P, IN?] + DL [T, P, (NiH + S[T, P, (Nil)


Mai Max(i) Ma
Me rla Ni( s+RT AIn[a) =Z.= Ni Nj ( +RT (ln[x1] +[xi hl))
The relations Ag = 1 x N1 y and a =exp((RT)-1 LX] = y/ x have








been implicitly used, where ai is the activity, yi the activity coefficient, and xi the

concentration of the "jth" component. The separation of the Gibbs energy of mixing

lx into the sum of ideal (random) and excess parts lDL +r/ is particularly

useful when xi -1 in the reference state since yi = 1 if and only if =Y = 0 in that case.

For these computations appropriate but different models for the partial molar total

Gibbs energy of mixing (,Mlx) are used for each component j to solve for their

concentrations, and each activity coefficient yi determined from the solution via the

expression:
0
RT ln[ala, MIX= i- 9 y = x1 exp[_ .

The question of normalization must be addressed carefully in the transition

from an extensive quantity like Ag to the intensive partial molar quantities fSf. This

is a particularly subtle issue in the context of a lattice model where it will sometimes

be necessary to normalize the concentrations xi with respect to the number of lattice

sites. To prevent confusion a number of different concentration notations suitable for

different contexts are introduced and it is simply noted here that the numerical

values for activities and activity coefficients depend explicitly on the choice of

concentration measure [157; 9.4, 158; 6.3].

A building units approach (which is closely related to the more common

structural element approach) and the Kr6ger-Vink notation are used to describe the

crystal lattice and its defects [157; 7.10]. Structure elements are the entities

appearing at particular sites in the lattice such as a vacancy on an interstitial site 'Vi,

or a copper atom on its ideal lattice site Cu', where the superscript 'x' means that it








is in its normal valence state. In addition to this lattice site atom occupancy

information, the change in electronic charge density surrounding a structure element

compared to its normal electronic charge density distribution is of interest. Charge

localization is of course an idealized concept for the fundamental structural elements

of the defect-free crystal whenever covalent bonding and band formation occur. For

many electronically active crystal defects on the other hand, it is reasonable to deal

with the strong electronic/ionic defect interactions by treating them together as a

quasiparticle. The combined defect and perturbed electronic charge density

distribution are represented as a charged structure element. For example In~u

represents a double positively charged indium atom on a lattice site normally

occupied by copper, whereas 'Vu represents a single negatively-charged copper

vacancy. Note that the superscript charge notation represents the deviation of the

defect's local charge distribution from that of the unperturbed lattice site.

Now that the distinction between the physical elements (e.g.: Cu) and

structural elements (e.g.: Cu'u) has been explicitly described, it is appropriate to

introduce the notation for normalization. The notation IICucu II is used to mean the

mole fraction of normal valence copper atoms on copper sites: in other words the

number of moles of CuCu structure elements divided by the total number of moles of

the quasimolecular species comprised of all the elemental species in the system.

Kr6ger used square braces [ ] to denote molar concentration, but that notation cannot

be used with the Mathematica program employed for these calculations since it

identifies and encloses therein the argument sequence of a function. The notation








(Cucu) is used to mean the lattice concentration, or more specifically for this

example, the number of electrically neutral copper atoms on lattice sites normally

occupied in the chalcopyrite lattice by Group I atoms, divided by the total number of

Group I sites in the chalcopyrite lattice. Equivalently, (Cuu ) is the probability that a

Group I lattice site is occupied by a copper atom in its normal charge state. Kr6ger

used curly braces ) but these cannot be used in Mathematica since they are

predefined therein to identify and enclose a list.

One key requirement for the interconsistency between the physical and

structural element thermochemical descriptions of phase and reaction equilibria is

that the difference in their normalization changes reaction equilibrium constants

differently in the two descriptions since exchange of atoms between phases may not

conserve the total number of lattice sites. Strictly speaking, if the species in the model

are structural elements rather than atomic or molecular species, and lattice-

normalized concentrations are used in the equations given above for the Gibbs

energy, the result is instead a quasichemical potential and quasichemical activities

for each of them. These issues must be kept clearly in mind to avoid misapplication

of the results.

This model is similar to the solution defect lattice model developed by

Guggenheim [159]. Guggenheim's model employs his "quasichemical"

approximation (first derived by Bethe [47]) to calculate for point defect associates

(quasimolecular species) the configurational entropy contribution to ASX in the

exact relation AxsG = AgXs TAS'X. The essence of this approximation [160] is that








pairs of nearest-neighbor sites are treated as independent of one another, which

introduces unallowable configurations into the partition function for any species that

occupies more than one lattice site simultaneously. It is nevertheless superior to the

assumption that ASxs vanishes (or equivalently that yi is unity). This is the

assumption used in a regular solution lattice defect model wherein the point defects

are distributed randomly on the lattice despite the existence of interaction enthalpies

between the different point defect species. The introduction of correlated site

distribution probabilities into the theory leads to an associated solution theory and

the implicit possibility of phase segregation or long-range ordering.

The excess entropy ASXS can be partitioned into four components

corresponding to electronic, internal, changes to the lattice vibrational excitations

(phonons) associated with the quasimolecular species, and configurational excess

entropies. These excess entropies are computed for the normal lattice constituents,

point defects, and for defect associates using a cluster expansion method. These

clusters are formally identical to the relative building units used by Schottky [45].

Thus strictly speaking this calculation is based on his building units approach rather

than a structural element approach [157, 7.10]. The overall problem is made

tractable by separating the strong short-range energetic effects due to interactions

between the point defects and the normal lattice components in their immediate

neighborhood into internal interactions within busters which can then be treated as

weakly-interacting. Consequently, the activity coefficients (y) of these dusters in

their mixture corresponding to the actual state of the entire CIS lattice need only be








modified to account for the long-range Coulombic interactions between the charged

species. These corrections are largely compensated by the Fermi degeneracy of the

charge carriers [157, 7.11], so the activity coefficients of the clusters will be

approximated as unity, yielding a simple cluster mixing model.

Prior efforts to identify the structural defects responsible for the electronic

behavior in these materials [65] have relied on estimates of the enthalpy of vacancy

formation by Van Vechten [161] based on a cavity model for vacancy formation

energy. More recently, first principle calculations of these formation enthalpies have

been conducted by Zhang and coworkers [70] which shall be used here. Their

quantum-mechanical calculations provide enthalpies of isolated defect formation

since they allowed for lattice relaxation and hence changes in specific volume

resulting from the formation of a single defect or defect complex within an otherwise

perfect lattice supercell containing 32 atoms. For the dilute point defect and

quasimolecular species in this model which were considered therein, their calculated

formation enthalpy is set equal to WS /NAvo. Those authors, however, estimate the

uncertainty in their calculated defect formation enthalpies to be ~0.2 eV which

represents a potentially significant source of errors in the results of these calculation.

In addition to their formation enthalpy calculations for isolated defects and

complexes, Zhang and coworkers calculated the enthalpies of interaction between

ordered arrays of one specific defect complex, 2 Vcu ( Incu, placed on neighboring

copper sublattice sites along the (110) direction (note the infix notation '(' is used to

denote an associate or defect complex formed from the specified lattice point








defects). This neutral cation defect complex had the lowest formation enthalpy of any

they considered in the dilute limit. The Madelung energy resulting from their

interaction when in a dense array as described above gave an additional reduction in

enthalpy that varied with their concentration. These results will be used when

analyzing the defect model in the case of an overall excess of indium compared to

copper in the isolated thermodynamic system.

Finally, Zhang and coworkers calculated the defect electronic transition

energy levels for isolated cation point defects and complexes. In a defect chemical

model these electronic transition energies correspond to the enthalpy of ionization of

a neutral defect to form an ion or charge localized on a vacant lattice site (an "ionized

vacancy"). Their estimated uncertainty in these electronic transition energy levels is

0.05 eV for isolated point defects and 0.10 eV for defect pairs. This represents

another potential source of errors in the results of these calculation. The entropy of

ionization will be included in an approximation derived by van Vechten [36].

Formulation of the problem

The empirical observation that the compounds which form in the Cu-In-Se

ternary system all exhibit wide compositional ranges of phase homogeneity is proof

of a non-neglible compositional dependence of their partial molar Gibbs energies

,i [T,P,( xi ] on the values of the component atom fractions, xi. It has been proven

that a statistical thermodynamic model can account for this variation by retaining

higher order correction terms to the entropy that are usually neglected, including

lattice vacancy [162] and electronic carrier band-entropy contributions [163].








This model is an adaptation of the ternary alloy model developed by Sha and

Brebrick [163] to the structure of the chalcopyrite lattice, wherein there are three

distinct lattice sites rather than two, as in their model. Unlike their approach, the

statistical mechanics used to compute entropies is based on a cluster configuration

technique. Furthermore, rather than solving the reaction equilibrium problem by the

usual method of Lagrange multipliers, more recently developed matrix techniques

described by Modell and Reid [158] are employed. It is assumed that:

I. The lattice structure of the a and P phases consists of four sublattices,

referred to as Ml, M3, X6 and I (interstitial). Each of the metal-sublattices (Ml and

M3) has N sites and there are 2N X6-sublattice sites for a total of 4N normally-

occupied lattice sites, which comprise an fcc Bravais lattice of N lattice unit cells.

Their are eight normally-occupied lattice sites in each primitive unit cell of the

chalcopyrite crystal structure, which is comprised of four lattice site tetrahedra

distorted along the c-axis. Hence the entire ideal lattice comprises chalcopyrite

primitive unit cells. In anfcc lattice there are a plethora of interstitial sites: eight

tetrahedral, four octahedral, and thirty-two trigonal [164] per fcc unit cell. It is

assumed that the only interstitial species included in this model, the Cu interstitial

(Cui), occupies the tetrahedral interstitial sites only. Note that all of these

tetrahedrally coordinated interstital sites are not equivalent with respect to the

symmetry operations of the I42d point group characteristic of the chalcopyrite

structure (space group 122), but it is assumed nevertheless that they are statistically

equivalent and energetically degenerate. There are therefore 8N total sites available








in the entire lattice including these interstitial sites, and sixteen per primitive unit cell

cluster.

II. Each of the point defect species is distributed randomly on its respective

sublattice within each duster. Defect complexes are defined as short-range (nearest-

neighbor) correlated occupancy on one or more of the sublattices. The correlation is

achieved by restricting each complex to a distinguishable lattice cluster, but the

distribution of those clusters over the available lattice is assumed to be random.

Interactions leading to aggregation of defect complex clusters on this lattice is treated

as a second phase.

III. The excess Gibbs energy of a phase is a first-degree homogeneous linear

function of the numbers of clusters of each kind and the total number of clusters that

comprise the lattice.

The defect structure within the a phase, and phase segregation phenomena

between the a phase and any secondary phase, is analyzed in the context of this

lattice model. The constituent physical elements Cu, In, and Se and charge q are

distributed among the available lattice sites and between phases in accordance with

the principle of minimum total Gibbs energy but the total amounts of these physical

elements are strictly conserved. Hence equilibria are calculated based on the

following basis set:

a = (Cu, In, Se, q, M1, M3, X6, });

The electrochemical state vector with respect to this basis is defined as:








sN := (Ncu, Nn, Nse, Nq, N, N, 2 N, 4 N)


This electrochemical state vector sN can be transformed to express the total

Cu, In, Se and charge q in terms of the reduced set of variables X, y, Z, N, and g

where:


Nln := y N
Ncu := XyN
Z y N (3+ X)
Nse :=
2
Nq :=pN


The reverse transformations are clearly (since N*O):


Ncu
X
Nin
Nin
y N'


3 Nin + Ncu
Nq
=== N'


The dimensionless electrochemical state vector is now defined as:


s := (X, y, Z, 0, N)

Conservation of mass in this chemical context where nuclear transformations

between elements are inadmissable implies the conservation of Ncu, Nse, and Nin

distinctly and therefore of X, Z, and yxN. When the system is closed to charge

transfer, Nq is conserved and therefore exN is conserved. The importance of this

transformation lies in the fact that X and Z are invariant, whereas changes in y and e








can result from a change in N. It is apparent from these observations that the

specification of s uniquely specifies sN. Note that both of these state vectors, s and

sN, are extensive.

The significance of changes in N is apparent in a simple example where

Ncu =Njn =- =N. The state vector sN-(N, N, 2N, Nq, N, N, 2N, 4N). This

represents N formula units of the compound CuInSe2, N primitive unit cells of the

fcc Bravais lattice, and a net electronic charge of Nq. Changes in N therefore

represent the loss or gain of lattice sites resulting from segregation to another phase

in equilibrium which has an incoherent lattice structure. "Incoherent lattice" means a

lattice with a different number of crystallographically distinct sublattices, or different

site ratios, or both. Any such reaction leaves X and Z unchanged but change the ratio

(y) of total indium to lattice sites and the overall charge density on the lattice (e). The

utility of this formulation will become apparent.

The relationship of the variables in the reduced electrochemical state vector s

to prior formulations of this problem [65] is now developed. The chemical

composition of any mixture in this ternary system may be formally written as the

reaction:

x Cu2 Se + (1-x) In Se3+ As Se (CuxInl-)2S3-2. ; 02 x 21 A 2x-32 As.

It is obvious from their definitions that X = = x = x (given that xl1)

and by direct substitution (note that As = 0 4- Z = 1):

S6-4 x+2 As 1+ As (Z-1) (3+X)
2x+6(1-x) T S = (Z-1) (3-2 x) 1+X








If phase segregation of the compound composition on the right hand side of

the reaction does not occur, comparison with [63] shows that this parameter X is the

molecularityy" and Z is the "valence stoichiometry" of that phase. Furthermore,

Z- 1 = is the "valence stoichiometry deviation" of the phase and X- 1= is
3-2x 1-x

Ax, its molecularityy deviation."

The necessary foundation has been laid to address the normalization of atom

fraction and molar quantities in terms of these variables. The atom fraction

corresponding to a number Nk of a given atomic species k is denoted xk, and given by:

Nk= N ___2xNk
xk Ncu+Nu,+Ns. N yx(l+X+ Z ) Ny x(2x(l+X)+Z x (3+X))

The mole fraction corresponding to a number Ni of any given species j is

denoted INi II, or in the code for their computation ci, and has been defined as the

ratio of Nj to the total number of "molecules" of the hypothetical species

(Cux Inl-x)2 Se3-2x+&s. It is stressed that this does not necessarily imply the existence

of any phase within the system with this actual composition, hence the modifier

"hypothetical." Substituting for the stoichiometric coefficients from the foregoing

solutions for x and As in terms of X and Z gives:

(Cux Inl-x)2 Se3-2x+5s = (Cu. InT')2 Se3_ +izX-3.X

= Cu 2x In 2 Se zox .

This form of the quasimolecular species formula can be used to solve for the

atom fractions Xk in terms of the variables X and Z alone:
2X 2 Zx(3+X)
XCu = 2x(l+X) + Zx(3+X) n = 2x(1+X) + Zx(3+X) an x 2x(l+X) + Zx(3+X)

Next the total number of moles, M, of this quasimolecule is sought. By the








definition of y, N1n = y xN and that the number of indium atoms in a mole of the

quasimolecule is XNAvo (Avogadro's number). Hence the equation

M x 2N = N = y xN is solved to give:

(1+X) y xN N1 +Ncu ,andthus:
M 2 NA. 2- and thus:
2 NAvo 2 NAvo
N, 2
Ni11I = MxNAo = Ni x (+) yxN and M= 1 = Ni = 11Ni 1 x NAvo

Since X, Z, and the product yxN are invariants, changes in the atom fraction

or molar fraction of any species, either atomic, molecular, or structural, may be due

only to the change in the number of that species in the entire system.

For completeness the relatively obvious normalizations are given for the

lattice site occupation probability for the species indexed by a given value of j:
N
(Nj)M or M3 = --t for the M1 or M3 (cation) sublattice;

(Nj)N = for the X6 anionn) sublattice;

(Ni). = 4 for the I (interstitial) sublattice.

Unlike the other normalizations, however, note that these species

normalizations may change via their explicit dependence on the unconserved

quantity N.

The equilibrium associated lattice solution theory calculations will be

conducted with respect to the lattice state vector, sL, whose components are lattice

site occupation numbers and which is defined with respect to a subset of all possible

lattice defects: {(Ncu, (Ncu, ) (Nou, (Ncu ), (NLn, {N~g INin1 6 ),

(Nen J,(NseL, I(Nser t, (NsI, g, ( INs }, t(N.e i, IN I m(N e {Ny },

(Ne- i, INh }I }, where the charge q on each of these lattice basis elements assumes all








possible values for each constituent. This complete lattice state ensemble is

unnecessarily large since many configurations which are conceptually possible are so

energetically unfavorable that they may be omitted without significant effects on the

results. The subset chosen for these calculations will be discussed in detail at a later

point. At this juncture it is only necessary to note that sL is constrained by sum rules

that connect it to sN and s. Specifically, the sum of molar concentrations of all

structural elements containing a given physical element must equal the molar

concentration of that physical element in the corresponding thermochemical state

vector. Similarly, the net charge on the lattice calculated from sL must equal the total

charge, Nq,. Finally, the sum of lattice site occupation probabilities must be unity for

each sublattice independently.

Four independent specific variables are required to model the thermochemical

reaction equilibria of a single phase, three component system. The temperature,

pressure, the overall copper to indium molar ratio X, and the anion to cation ratio Z

are chosen: the variable set (T, P, X, Z). The activity of each atomic species is

referenced to its standard state of pure elemental aggregation (Standard Elemental

Reference, SER) at Standard Temperature and Pressure (STP is To =298.15K (250C) and

Po =101.3 kPa = 1 atm) for which its enthalpy of formation, A'HfER, is set to zero by

convention. The absolute scale for entropy where limit S- = 0 for all elemental species
T-*0
is used and the changes in equilibria between phases calculated from mathematical

expressions for G -AH'ER. Furthermore the effects of pressure will not be considered

and all calculations will be conducted at standard pressure, effectively reducing this








to a problem in three variables. Note that the extensive reduced electrochemical state

vector s introduced in the preceding section contains five variables, X, y, Z, Q, and N.

For this initial thermochemical analysis an electroneutrality constraint is imposed,

hence g = 0 (and Nq = 0) in this context. Note that reaction equilibria are intensive

relations and that s may be transformed to an intensive state vector, s, by setting the

total number of moles, M, of the quasimolecules with the formula

(Cux Inl-x)2 Se3-2x+Ay to unity. Using the formula derived in the prior section for M:
(1+X) yxN 2 NA-v
M- 2NAvo y (1+X) N
This transformation from the extensive state vectors sN to s results in no loss

of information regarding the state of the system assuming the ratio of sublattice site

numbers remained fixed with respect to all possible reactions. The variable

transformations therein for y and g, however, are explicitly dependent on N via the

physical requirement that yxN and exN remain constant. Transformation from

either sN or s to the intensive molar state vector s = IX, y, Z, e) places a constraint on

these products, but neither y nor e independently. The choice of the independent

thermodynamic variable set (T, P, X, ZI implies that equilibrium values for the state

variable y (and similarly Q) are dependent variables calculated with the equation

above (or its analog for e) using the equilibrium value of N for one mole of the

quasimolecular species Cu x In 2 Se z x.

The structural element basis of the a and P phases of CIS that will be employed

for these calculations consists of various clusters of the following subset of lattice

species, which comprise the basis set for sL:








LCIS = ((Cu u, Inx, Sex `V, (TeV, x'V, nV, In~, Iu, ~,
(Cuj, Cui,, Cu", eV, n, V, YV", (t', VS's ), (Cu?, Cu;)),
(Cui ( Ycu, Culn ( Incu, "Vcu Incu, (Vcu ( Incu)o'}, (2 c.u ( Incui};

The normal lattice constituents, the isolated point defects on each sublattice,

and defect complexes on different numbers of lattice sites have each been grouped

separately in L.CIS. The various lattice clusters will be labeled by their characteristic

point defect, except for the normal lattice constituents which combine to form the

normal cluster which will be labeled simply CISa. These are grouped in the cluster

basis set afiL by the number of primitive unit cells in the structure element's cluster

of lattice sites, with that corresponding number of primitive unit cells given by the

ordered list ncL.

af3L = (ICISa), Join[Flatten[Take[LCIS, (2, 2111, Take[Take[LCIS, (3, 311]11i, 21,
{(2 Vcu Incu)/3)ll, Join[Drop[Take[LCIS, (3, 3)1~11, 2],
((2 cu a Incu),, (2 Vcu Incu)6ts, (2 'cu E Incu)25 Ill;

afLIll === (CIS,)

apLL2) === (Vu, 'Vu, Inu, In~u, In,, Cuin, Cu'n, Cul, '~n, 'In ,
Vl,'V*, 'Vg, Cu Cu?, Cui e'Vcu, Cuin e Incu, (2 'Vc, Incu),13)

apL31] ===
(YVcu e Incu, ('Vcu Ilncu)', (2 Vcu S Incu),, (2 Vcu E Incu)15s, (2 YCu B Incu)M25)

ncL = (1, 3, 5); Dimensions[ncL] === Dimensions [afL]

To model solid phase equilibria the lattice building unit basis afiL must be

extended to quantify and characterize the transfer of physical constituents from this

lattice to other phases that are in equilibrium with the lattice.








cEgrouped = [afL, [e', h" AN, (Cucu2Se, Cu2.6Sell;

The crystallographically incoherent secondary phase constituents which will

be utilized in this model have been appended, and band-delocalized charge carriers

added to construct a complete basis for the state vector, which will enable the

modeling of phase segregation and electronic carrier concentrations in the

equilibrium system. The basis element AN allows for changes of the sublattice site

proportionality multiplier N independent of phase segregation processes, since

vacancy generation on all sublattices (lattice expansion) is a physical mechanism

whereby the total free energy of the lattice might be reduced even in the absence of a

secondary phase.

The domain of this analysis is limited to the compositional range of greatest

relevance to applications to photovoltaics, with Cu/In molar ratios in the range:

S< X <1. The case of X= 1 and Z= 1, which corresponds to the ideal stoichiometric

compound CuInSe2 is first analyzed. The second case will be for <5 X < 1-e,

corresponding to the a, P, and intermediate a + P two-phase regions [63]. Despite the

present uncertainty regarding the exact crystallographic structure of the P phase, it is

agreed [266, 267] that it must be closely related to the chalcopyrite structure. This

structure persists between the putative a/P phase boundary composition (X= ),

which corresponds to the compound Cu2 In4 Se7 through at least those compositions

corresponding to the compound Culn3Ses (X= 1). Within the restricted limits of this

second case the total number of lattice sites in the system (including interstitial sites)

remains constant, at least in the absence of valency stoichiometry deviations from








zero. This structural coherence between the a and p phases has extremely significant

ramifications which will be addressed at a later point in this treatise.

The temperature domain of these calculations is restricted to below 1048.15K

(750C) in the first case and 873.15K (600C) for the second case. This minimizes the

complications introduced by the high temperature order/disorder phase

transformations of the a phase at 1063K [168] and the P phase at 873K [168].

The details of the interphase and defect equilibria calculations are included in

the appendix to this dissertation, including all the Mathematica code required to

verify the results.



Results

The results of these calculations are divided into two major subsections. The

first details the predicted phase diagrams and the composition of the two different

phases found in equilibrium with a-CIS over the domain of this calculation, Cu2-6 Se

and P-CIS. The second describes the calculated equilibrium defect concentrations

within a-CIS, and their variations with composition and temperature.


Interphase Reaction Equilibria

The predicted equilibrium phase diagram for the Cu-In-Se ternary phase field

along the Cu2Se/In2Se3 tie-line where Z= 1 is shown below as a function of the

atomic fraction of copper and temperature.











800


700


T[K 600
T[KJ

500


400


300
16 18 20 22 24
Cu [at.%]
Figure 2.1 Calculated equilibrium phase diagram for the Cu-In-Se system on the
Cu2Se/In2Se3 tie-line where Z= 1

Two dominant features of this model's predictions are clearly consistent with

the published experimental phase diagrams. The location of the a//3-CIS two-phase

boundary at sTP is predicted to be at 15.35 at.% copper (X= 0.4987), corresponding

almost exactly to the widely reported f--CIS compound formula Cu2 In4 Se7. The

curvature of the copper-rich a/fi-CIS two-phase boundary towards lower copper

content with increasing temperature has also often been reported, although usually

to a much greater extent than found here.

The most striking inconsistency of this diagram with published data are the

narrow width of the predicted single-phase a domain and curvature with increasing

temperature of both the indium-rich a/f-CIS and Cu2Se/a-CIS two-phase

boundaries in the same direction, towards lower copper content.








The detailed discussion of these results in the following subsections will argue

that these inconsistencies are mostly a consequence of two factors. The first is the

inadequacy of the limited, four-species basis used to model the energetic of the

P-CIS phase. The second is that the lowest free energy state of the system is in fact

displaced from this pseudobinary section of the ternary phase field towards a small

selenium enrichment (Z z 1), on a scale below the resolution of current chemical

composition analysis methods.

The effect of such deviations were explicitly modeled for the two-phase

Cu2Se/a-CIS boundary. Those results show a significant increase in the width of the

a-CIS single-phase homogeneity range, and also imply the existence of a kinetic

barrier to Cu2Se/a-CIS equilibration at temperatures below ~100C that would

inhibit the conversion of excess Cu2 Se into a-CIS, creating an apparent shift of this

boundary towards lower copper content.

Stoichiometric CuInSe2 and the Cu2z- Se/a-CIS phase equilibrium

These Cu2-..Se/a-CIS equilibrium calculations have been constrained by an

energy sum rule, which requires that the total Gibbs energy of any Cu-In-Se mixture

with a composition corresponding to CuInSe2 must at every temperature equal a

reference value which has been calculated from three empirical published relations

for the thermodynamic properties of CuInSe2. Their values are given explicitly in the

appendix and include the Gibbs energy at a reference temperature near the a/6-CIS

eutectoid [271], the standard state entropy [2731, and the temperature dependence of

the heat capacity [274].








A mathematical model of the Gibbs energy dependence of Cu2-.6Se with

composition and temperature is used, which was derived as part of a recently

published assessment of the binary Cu-Se phase diagram [270]. It is assumed here

that indium is completely insoluble in Cu2 Se and that a-CIS is in equilibrium with

this phase over the domain of this calculation. Consequently, the constraints on

compositional variation of the non-stoichiometric compound Cu2-_Se imposed by

the other binary Cu-Se phases became implicit constraints within this equilibrium

calculation. This is a direct consequence of the Gibbs phase rule, as the detailed

analysis in the appendix shows, which implies that any unrestricted three-phase

equilibrium in a ternary phase field is confined to a single combination of

temperature and composition. Thus the two-phase boundaries within the Cu-Se

phase field that define limits on the value of the Cu2z-Se stoichiometry deviation

parameter 6 restrict its ability to accommodate stoichiometry variations in a two-phase

mixture that includes indium. Over the range of this equilibrium calculation, these

constraints on 6 are defined by the equilibrium between Cu2-6Se and a number of

different binary Cu-Se phases.

Over the entire temperature range of interest, the Cu2-6 Se binary copper

selenide's copper-rich single-phase domain boundary is determined by its

equilibrium with fcc Cu with a non-vanishing solubility of selenium [270]. Thus

perfectly stoichiometric Cu2 Se is not stable to decomposition in the binary model,

and the Cu2_6Se binary-ternary equilibrium composition is limited by the

corresponding minimum value of 6, or equivalently, this binary compound's









minimum selenium content. This effect is most significant near the

Cu:Se/a-Cu2-6Se/f-Cu2-6 Se peritectoid temperature of 396K (1230C). The

following figure shows the results of the calculated deviation of 6 from that

minimum value if Cu2-6 Se is assumed to be in equilibrium with stoichiometric

CuInSe2.


T [K]


1000

900

800

700




500

400

[ppml
-1200 -1000 -800 -600 -400 -200
Figure 2.2 Calculated deviation of the Cu2- Se stoichiometry parameter 6 in
hypothetical equilibrium with stoichiometric CuInSe2

Figure 2.2 shows that without some mechanism whereby ternary a-CIS could

accommodate stoichiometry variations, Cu2z-Se in equilibrium with CuInSe2 would

not be stable below a calculated temperature of -850K with respect to segregation of

the nearly pure metallic Cu phase found near the Cu vertex in the Cu-In-Se ternary

phase triangle. Since such a three-phase equilibrium over that finite temperature

range would violate the Gibbs phase rule, this cannot occur.








Figure 2.3 shows the results of the equilibrium calculation wherein the

internal defect structure of the a-CIS phase, stoichiometry variation of the Cu2-6 Se

phase, and extent of phase segregation are varied to minimize the total Gibbs energy

of the stoichiometric CuInSe2 mixture. It shows that at high temperatures selenium

will segregate preferentially to the binary phase, increasing 6 above its minimum

value. The temperature at which the equilibrium and constrained minimum values of

6 are equal is lowered by the a-CIS internal defect equilibration to a value of 677K.


6-6min



0.002


0.0015


0.001


0.0005


ST[K]
700 750 800 850 900 950 1000 1050
Figure 2.3 Deviation of the Cu2-6 Se stoichiometry parameter 6 from its minimum
allowable value in equilibrium with defective ternary a-CIS in the stoichiometric
CuInSe2 mixture









Although this segregation of selenium in excess of its constrained minimum to

the Cu2-_Se compound does not continue to lower temperatures in the

stoichiometric mixture, the minimum value of the stoichiometry parameter 6min

is itself positive.



60.

0.01


0.00


0.008


0.006


0.004
0.002


T[K]
300 400 500 600 700 800 900 1000
Figure 2.4 The Cu2-6 Se stoichiometry parameter 6 in equilibrium with a-CIS in the
stoichiometric CuInSe2 mixture

Figure 2.4 shows the total value of 6 over the entire temperature range of this

calculation. Note in particular its rapid increase in equilibrium as the temperature

approaches the peritectoid from above. This also implies that the stoichiometric

composition CuInSe2 is not single phase at equilibrium. Figure 2.5 shows the

calculated extent of phase segregation of this composition over the entire

temperature range of this calculation. Clearly Cu2- Se always segregates to some








extent, thus ideal stoichiometric ternary CuInSe2 always dissociates in equilibrium to

form the two-phase mixture.


IICu2Sell


0.01


0.001


0.0001


0.00001


1.x 10-6


I.x 10-
T[KI
300 400 500 600 700 800 900 1000
Figure 2.5 The equilibrium molar extent of binary Cu-Se phase segregation in the
stoichiometric CuInSe2 mixture

Since the stoichiometry deviation parameter 6 of Cu2-6Se is positive,

the segregation process always removes selenium from the remainder of the mixture

at a rate more than half the rate at which copper is depleted. Hence this segregation

process in the stoichiometric mixture creates negative valency deviation of the

ternary phase in equilibrium, as shown in figure 2.6.











-AZ,


0.00001


1. x 106


1. x 10-


1. x 10-8


1.x 109


I. .. , -. T[KI
300 400 500 600 700 800 900 1000

Figure 2.6 The negative valency deviation of a-CIS in equilibrium with the binary
Cu-Se phase in the stoichiometric CulnSe2 mixture


-AX,,


300 400


500 600 700 800 900 1000


Figure 2.7 The negative molecularity deviation of a-CIS in equilibrium with the
binary Cu-Se phase in the stoichiometric CulnSe2 mixture


0.1




0.001




0.00001




1. x 10-7


T[K]











The segregation of Cu2-a Se does not remove indium from remainder of the

mixture. Hence this segregation process in the stoichiometric mixture also creates

negative molecularity deviation, as well as negative valency deviation of the ternary

phase in equilibrium at this two-phase boundary, as shown in figure 2.7.

Two-phase regions are also present in the binary Cu-Se phase field that define

an upper limit on the single-phase stability range of Cu2_ Se. Over the entire

temperature range, this boundary is defined by the equilibrium between Cu2z-Se

and a number of different phases [270]. The maximum stoichiometry deviation of

equilibrium Cu2-_ Se occurs at a temperature of 650K, where its maximum selenium

binary mole fraction rises to 36.8 at.%. Below that temperature it decreases

monotonically, dropping to 36.0 at.% at the 291K a-Cuz2-sSe/P-Cu2z-Se/Cu3 Se2

eutectoid. The net result of both these upper and lower limits on 6 is a significant

narrowing of the homogeneity range of Cu2-sSe between 396K and 291K.

The equilibrium effects of positive valency deviation in the Cu-In-Se mixture

were also modeled. As previously derived, the relationship between the valency

deviation and excess selenium in the mixture is given by the relation As = (z-) (3+)

All of these calculations were performed for a value of X= 1 in the mixture, so this

relation reduces here to As = 2 AZ. The first issue of concern in these calculations was

to properly include the effects of the constraints on the maximum allowable selenium

content in the binary Cu2-5Se phase. It was found that the secondary phase

composition first exceeds its maximum selenium content at STP when the value of AZ

reaches about +4.5x10-6, as shown in figure 2.8.


















800






700






600
T[K]





500 **






400 ..... ...... *





300

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
Ax



Figure 2.8 The equilibrium selenium mole fraction of the binary Cux Sel-x phase in
the Cu-In-Se mixture with AX = 0 and AZ = +4.5x10-6 (left), and the temperature
dependence of the maximum allowable selenium mole fraction (right)
J00 **













dependence of the maximum allowable selenium mole fraction (right)









Slightly greater selenium enrichment also yields a violation of this limit at the

395K Cu:Se/a-Cu2-6Se/13-Cu2-sSe peritectoid. Figure 2.9 shows the specific Gibbs

energy of the binary phase alone as a function of its selenium mole fraction both a

few degrees above and a few below this peritectoid. The binary's composition

variation with temperature in the ternary equilibrium is clearly a consequence of the

energetic discontinuity at the peritectoid.


g [Joules-mole- ]


-37400

-37600

-37800

-38000

-38200

-38400 0.005 0.01 0.015 0.02 0.025 0.03

-38600
Figure 2.9 The variation of specific Gibbs energy with composition of the binary
Cux Sel-x phase at 393.15K (upper curve) and 398.15K (lower curve)

The constraint on the maximum selenium composition of the binary phase

was incorporated into the calculations for all cases where AZ > 1. The temperature at

which the binary's selenium content saturates is found to increase up to AZ = 0.1739,

at which point the secondary Cu2-aSe phase is found to possess its maximum

selenium content over the entire temperature range from STP to the a/p/6-CIS








eutectoid. Figure 2.10 demonstrates these results for several values of positive

valency deviation.


Ax
0.035

0.03

0.025

0.02

0.015

0.01

0.005

T[K]
300 400 500 600 700 800
Figure 2.10 The deviation of equilibrium selenium mole fraction in the binary
Cux Sel-x phase from its minimum constrained value in the Cu-In-Se mixture, with
AX= 0 and (left to right) AZ= 100, 400, 700, 1000, and 1739 (x10-6)

The results displayed in figure 2.5 for the calculated extent of Cu2-6 Se phase

segregation for the mixture with AZ= 0 were extended to a maximum of AZ= 0.22,

corresponding to a selenium excess of 0.44 at.% in the two-phase mixture with X= 1.

These results are displayed in figure 2.9, and clearly show a significant increase in the

extent of binary segregation with increasing positive valency deviation in the

mixture.

The large decrease in equilibrium solubility of the binary in the mixture with

excess selenium through the final 150 degrees above STP during cool-down after

synthesis may represent a significant kinetic barrier to equilibration. Either a net flux








of copper and selenium into the indium-enriched ternary or a net flux of indium out

of the ternary phase into the binary is needed to effect this transformation. Since the

selenium sublattices of the two phases are nearly identical and selenium interstitials

and antisites so energetically unfavorable, it is unlikely to redistribute. The relative

strength and covalency of the In-Se chemical bond makes indium less mobile than

copper, particularly in this low temperature range. Thus synthesis under conditions

of high selenium fugacity may not fully equilibrate if their composition pass through

the equilibrium two-phase boundary corresponding to its composition during cool-

down. Growth under conditions of indium excess may be more necessary in practice

than the equilibrium phase boundaries suggest, in order to inhibit the formation of

metastable binary copper selenide precipitates.


T[KI


700



600



500


400


II|Cu2Sell
0.002 0.004 0.006 0.008 0.01
Figure 2.9 The equilibrium molar extent of Cu2-6 Se phase segregation in Cu-In-Se
mixtures, with AX= 0 and (left to right) AZ= 0, 0.11, and 0.22









This enhancement of Cu2-6Se phase segregation with increasing positive

valency deviation in the two-phase mixture will exacerbate the consequent negative

molecularity deviation of the a--CIS phase in equilibrium. Since the selenium content

of that binary phase also increases with increasing positive valency deviation, so too

the valency deviation of the ternary must decrease.

These calculations predict that a minimum of about 0.4 ppm excess selenium in

the two-phase mixture is required to insure that the equilibrium valency deviation of

the ternary a-CIS phase remains positive definite over the temperature range

between sTP and the peritectoid. Even more selenium is required at higher

temperatures to inhibit selenium depletion of the ternary, as shown in Figure 2.10.



total AZ=0.143 ppm total AZ=0.2 ppm
1.2 x 10-
1.5 x 10-9 x 10-8
8xl0-9
1 X10-9 6X10-9

5x 10-'o 4x 10-9
2x 10-9

320 340 360 380 400 320 340 360 380 400


7x I- total AZ=0.2 ppm total AZ=0.2 ppm
6x10-8 400 500 600 7 800
5xl0-8 lx106
4 x I0-8 -2 x 10-6
3x 108 -3 x10-6
2 x 10 -4 x 10-6
I1x10-
-5 x 10-6
350 400 450 500 550

Figure 2.10 The valency deviation of a-cis in equilibrium with Cu2-_6Se, with AX= 0
and AZ= 0.143 or 0.2 (x10-6)









Although the relationships between the valency deviation of the two-phase

mixture and those in each of its constituent non-stoichiometric phases are very

complex, they are single-valued. Hence it was possible to invert them and calculate

the valency deviation in the mixture required to yield a specified valency in its

ternary a-cis component. Figure 2.11 shows one example, demonstrating the

temperature dependence of the valency deviation of the mixture with X= 1 that is

required to keep the a-cis phase valency fixed at its equilibrium value in the mixture

at STP. This is equivalent to varying the two-phase mixture's values of X and Z to find

those values at which the extent of Cu2-6 Se phase segregation becomes

infinitesimally small in equilibrium with the ternary at its specified molecularity.


AZ

0.00001



I. x 10-7



1.x 10-9



l.xl0-1



T[K]
300 400 500 600 700 800
Figure 2.11 The valency deviation of the two-phase mixture with X= 1 required to
maintain the valency of the a-cis component at its STP value.








Thus the two-phase boundary value of molecularity can be determined from

the two-phase equilibrium calculations at X= 1. Simplistic application of the "lever-

arm rule" to this situation would give an incorrect answer, without prior knowledge

of the locus of the lever's fulcrum, which effectively varies with temperature and

does not lie in the T-X plane except at the phase boundary itself. This is a

consequence of the non-stoichiometry of both these phases in equilibrium.

The domain over which the two-phase boundary can be calculated by this

method is restricted by the range of the mapping between Z and Z, over the domain

of the two-phase calculation for X= 1. The domain of the two phase calculation

between 0 < AZ < 0.22% maps into the range 0 < AZa s 0.1%, corresponding to a

maximum excess selenium content in the single-phase ternary of about +0.2 at.%. The

calculated phase boundaries both on the pseudobinary section (AZ= 0) and in the

T-X plane where AZ= +0.1% are compared in figure 2.12.

Comparing the two curves in figure 2.12, the increase in valency deviation of

+0.1% has yielded a shift of less than -0.01% in molecularity at srP, but a nominal

shift of -0.4% in the temperature range of -450-600K. Comparison with the extent of

binary phase segregation in figure 2.9 makes it clear that this is a direct consequence

of that process.

The phase boundaries shown in figure 2.12 are more easily compared to the

published literature data when expressed in terms of the atomic fraction of copper, as

in figure 2.13.


















800






700






600
T[K]





500






400






300

0.965 0.97 0.975 0.98 0.985 0.99 0.995 1
AX



Figure 2.12 Calculated equilibrium Cu2.-Se/a-cis phase boundaries in the Cu-In-Se
system for AZ= 0 (right) and AZ= +0.1% (left) between STP and the a/f/6-cis
eutectoid


















800







700







600
T[K]






500







400







300

24.4 24.5 24.6 24.7 24.8 24.9 25
Cu[at.%

Figure 2.13 Copper composition at the equilibrium Cu2-6Se/a--cis phase boundaries
in the Cu-In-Se system for AZ= 0 (right) and AZ= +0.1% (left) between srT and the
a/f//6-cis eutectoid




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