A Dissertation Entitled a Theoretical Study of Bulk and Surface Diffusion

A Dissertation entitled

A Theoretical Study of Bulk and Surface Diﬀusion Processes for Semiconductor Materials Using First Principles Calculations

by Jason L. Roehl

Submitted to the Graduate Faculty as partial fulﬁllment of the requirements for the Doctor of Philosophy Degree in Physics

Dr. Sanjay V. Khare, Committee Chair

Dr. Jacques G. Amar, Committee Member

Dr. Terry Bigioni, Committee Member

Dr. Robert Deck, Committee Member

Dr. Randall Ellingson, Committee Member

Dr. Patricia R. Komuniecki, Dean College of Graduate Studies

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of A Theoretical Study of Bulk and Surface Diﬀusion Processes for Semiconductor Materials Using First Principles Calculations by Jason L. Roehl

Submitted to the Graduate Faculty as partial fulﬁllment of the requirements for the Doctor of Philosophy Degree in Physics The University of Toledo May 2014

Diﬀusion of point defects on crystalline surfaces and in their bulk is an important

and ubiquitous phenomenon aﬀecting ﬁlm quality, electronic properties and device

functionality. A complete understanding of these diﬀusion processes enables one to

predict and then control those processes. Such understanding includes knowledge

of the structural, energetic and electronic properties of these native and non-native

point defect diﬀusion processes. Direct experimental observation of the phenomenon

is diﬃcult and microscopic theories of diﬀusion mechanisms and pathways abound.

Thus, knowing the nature of diﬀusion processes, of speciﬁc point defects in given

materials, has been a challenging task for analytical theory as well as experiment.

The recent advances in computing technology have been a catalyst for the rise of

a third mode of investigation. The advent of tremendous computing power, break-

throughs in algorithmic development in computational applications of electronic den-

sity functional theory now enables direct computation of the diﬀusion process. This

thesis demonstrates such a method applied to several diﬀerent examples of point de-

fect diﬀusion on the (001) surface of gallium arsenide (GaAs) and the bulk of cadmium

telluride (CdTe) and cadmium sulﬁde (CdS).

All results presented in this work are ab initio, total-energy pseudopotential calculations within the local density approximation to density-functional theory. Single

iii particle wavefunctions were expanded in a plane-wave basis and reciprocal space k-

point sampling was achieved by MonkhorstPack generated k-point grids. Both surface

and bulk computations employed a supercell approach using periodic boundary con-

ditions.

Ga adatom adsorption and diﬀusion processes were studied on two reconstruc-

tions of the GaAs(001) surface including the c(4×4) and c(4×4)-heterodimer surface reconstructions. On the GaAs(001)-c(4×4) surface reconstruction, two distinct sets of minima and transition sites were discovered for a Ga adatom relaxing from heights of 3 and 0.5 A˚ from the surface. These two sets show significant differences in the interaction of the Ga adatom with surface As dimers and an electronic signature of the differences in this interaction was identified. The energetic barriers to diffusion were computed between various adsorption sites.

On the GaAs(001)-c(4×4)-heterodimer reconstruction, structural and bonding features of the surface were examined including a comparison with the c(4×4) reconstruction. Minimum energy sites (MES) on the c(4×4)-heterodimer surface were located by mapping the potential energy surface for a Ga adatom. Barriers for diffusion of a Ga adatom between the neighboring MES were calculated by using top hopping- and exchange-diffusion mechanisms. Signature differences between electronic structures of top hopping- and exchange-diffusion mechanisms were studied for relevant atoms. A higher diffusion barrier was observed for the exchange mechanism compared to top hopping.

Diﬀusion proﬁles for native, adatom and vacancy, and non-native interstitial adatoms were investigated along the open [1 1 0] channel in bulk zinc-blende CdTe.

This includes native Cd and S and non-native Cu, Ag, Au, Mo, P, Sb, O, S, and

Cl. High symmetry Wyckoﬀ positions were found to be the global minimum energy location for Cd, Ag, Mo, O and Cl interstitials. Adatoms of Cu, Au, P, Sb, S show an asymmetric shape of the energy diﬀusion barrier with two structurally equivalent

iv minima and two energetically distinct maxima in the pathway. Adatoms of Mo, Ag and Cd interstitial and vacancy, show a symmetric diﬀusion barrier with two structurally unique minima and a maximum. Adatoms of O, Cl, and Te interstitial and vacancy, show a symmetric diﬀusion barrier with a unique maximum and minimum.

Diffusion for Cu, Au, Te and S interstitials proceeds along the [1 1 0] channel in a near straight line path. Diffusion for Cd, Ag, O and Cl proceeds along two nearly straight line paths along [1 1 1] and [1 1 -1]. Diffusion for Mo, P and Sb is along the

[1 1 0] channel deviating slightly from the straight line paths along [1 1 1] and [1 1

-1]. The diffusion barriers range from a low of 0.10 eV for a Ag interstitial to a high of 1.83 eV for a Cd vacancy. The barriers for Cu, Ag, Te, Cl and S are in agreement with the available experimental data. The symmetric or asymmetric nature of the diffusion path as well as the bond length and atomic coordination at the energetic extrema positions were found to influence the size of the diffusion energy barrier. In addition there exist electronic signatures in the local density of states for the bond breaking, difference in the hybridization and energy of occupied states between the global minimum and global maximum energy positions.

Diffusion profiles for native Cd and S, adatom and vacancy, and non-native interstitial adatoms of Te, Cu and Cl were investigated in bulk wurtzite CdS. The interstitial diffusion paths considered in this work were chosen parallel to c-axis as it represents the path encountered by defects diffusing from the CdTe layer. Because of the lattice mismatch between zinc-blende CdTe and hexagonal wurtzite CdS, the c-axis in CdS is normal to the CdTe interface. The global minimum and maximum energy positions in the bulk unit cell vary for different diffusing species. This results in a significant variation, in the bonding configurations and associated strain energies of different extrema positions along the diffusion paths for various defects. The diffusion barriers range from a low of 0.42 eV for an S interstitial to a high of 2.18 eV for a S vacancy. The computed 0.66 eV barrier for a Cu interstitial is in good

v agreement with experimental values in the range of 0.58 - 0.96 eV reported in the literature. There exists an electronic signature in the local density of states for the s- and d-states of the Cu interstitial at the global maximum and global minimum energy position.

The work presented in this thesis is an investigation into diffusion processes for semiconductor bulk and surfaces. The work provides information about these processes at a level of control unavailable experimentally giving an elaborate description into physical and electronic properties associated with diffusion at its most basic level. Not only does this work provide information about GaAs, CdTe and CdS, it is intended to contribute to a foundation of knowledge that can be extended to other systems to expand our overall understanding into the diffusion process.

vi For you, Mom and Dad. Thank you. Acknowledgments

To begin, I would like to thank my research advisor, Prof. Sanjay V. Khare. Not only for his guidance throughout my research and the knowledge he has shared with me during our many informative talks but also for his support and encouragement when it was needed most.

Next, I would like to thank my committee members - Prof. Jacques G. Amar,

Prof. Terry Bigioni, Prof. Robert Deck and Prof. Randall Ellingson for all of their suggestions and time while serving on my committee.

Thank you to all of my professors for sharing their knowledge and helping me to better understand and appreciate the world around us. Thank you to all of the current and former members of Dr. Khare’s research group for their help when I started and their support and friendship throughout. Thank you to all the faculty, past and present, of the Department of Physics and Astronomy for their help and support during my time as a graduate student. A special thanks to Dr. Richard Irving for his computer system support and for taking the time for the many invaluable conversations regarding all that is computing and life.

And ﬁnally I would like to thank God, my family and loved ones for all of their love, support, patience and understanding. It is because of you that all of this has been possible.

viii Contents

Abstract iii

Acknowledgments viii

Contents ix

List of Tables xiii

List of Figures xiv

List of Abbreviations xvii

1 Introduction 1

1.1 Thesis outline ...... 1

2 Materials 3

2.1 GaAs ...... 3

2.1.1 Structure ...... 4

2.1.2 GaAs(001)-c(4×4) Surface Reconstruction ...... 4

2.1.3 GaAs(001)-c(4×4)-hd Surface Reconstruction ...... 4

2.2 CdTe ...... 6

2.2.1 Structure ...... 8

2.3 CdS ...... 9

2.3.1 Structure ...... 11

ix 3 Diﬀusion 13

3.1 Overview ...... 13

3.2 Surface Diﬀusion Mechanisms ...... 13

3.2.1 Top Hopping Diﬀusion ...... 14

3.2.2 Exchange Diﬀusion ...... 15

3.3 Bulk Diﬀusion Mechanisms ...... 17

3.3.1 Interstitial Diﬀusion ...... 17

3.3.2 Bulk Exchange Diﬀusion ...... 18

3.3.3 Vacancy Diﬀusion ...... 19

3.4 Describing Diﬀusion ...... 20

4 Density Functional Theory (DFT) 23

4.1 Introduction ...... 23

4.2 Hohenberg-Kohn Theorems ...... 24

4.3 Kohn-Sham Equations ...... 28

4.4 Exchange-Correlation Approximations ...... 30

4.4.1 The Local Density Approximation (LDA) ...... 30

4.4.2 Generalized Gradient Approximation (GGA) ...... 32

5 Surface diﬀusion on Gallium Arsenide (GaAs) 33

5.1 Computational Method ...... 33

5.2 Modeling Approach ...... 34

5.3 GaAs(001)-c(4×4) ...... 35

5.3.1 Results ...... 35

5.4 GaAs(001)-c(4×4)-heterodimer ...... 42

5.4.1 Results ...... 42

6 Bulk diﬀusion in Cadmium Telluride 51

x 6.1 Computational Method ...... 51

6.2 Modeling Approach ...... 52

6.3 Anion Diﬀusion in CdTe; Cl, Te, O, S, P and Sb ...... 53

6.3.1 Results ...... 53

6.4 Cation Diﬀusion in CdTe; Cd, Cu, Ag, Au and Mo ...... 67

6.4.1 Results ...... 67

7 Bulk diﬀusion in Cadmium Sulﬁde 84

7.1 Computational Method ...... 84

7.2 Cd, S, Te, Cu, and Cl ...... 85

7.2.1 Results ...... 85

8 Conclusions and Future Work 97

8.1 Conclusions ...... 97

8.2 Future Work ...... 100

8.3 Broader Perspective ...... 100

References 102

A Vienna ab initio simulation package (VASP) 113

A.1 Introduction ...... 113

A.2 Approximations ...... 114

A.2.1 DFT ...... 114

A.2.2 Periodicity ...... 114

A.2.2.1 Bloch’s Theorem ...... 114

A.2.2.2 Born-von Karman Boundary Condition ...... 116

A.2.2.3 K point Sampling ...... 119

A.2.2.4 Plane-wave Basis ...... 120

A.2.3 Pseudopotentials ...... 121

xi A.2.4 Minimization Algorithms ...... 124

A.3 Files and Parameters ...... 127

A.3.1 INCAR ...... 127

A.3.1.1 SYSTEM ...... 127

A.3.1.2 NWRITE ...... 127

A.3.1.3 ENCUT ...... 128

A.3.1.4 ISPIN ...... 128

A.3.1.5 ISTART ...... 129

A.3.1.6 IBRION ...... 129

A.3.1.7 POTIM ...... 129

A.3.1.8 EDIFF ...... 130

A.3.1.9 EDIFFG ...... 130

A.3.1.10 LWAVE ...... 130

A.3.1.11 LCHARG ...... 130

A.3.2 POSCAR ...... 131

A.3.3 KPOINTS ...... 132

A.3.4 POTCAR ...... 133

A.3.5 Nudged Elastic Band (NEB) Method ...... 133

xii List of Tables

5.1 Binding energies (eV) of a Ga adatom at various sites on the GaAs(001)-

c(4×4) surface...... 38

5.2 Ga adatom diﬀusion barrier results (eV) on GaAs(001)-c(4×4)...... 39

5.3 Structural diﬀerences between GaAs(001)-c(4×4) and GaAs(001)-c(4×4)-hd 45

5.4 Binding energies (eV) of a Ga adatom relaxed from 3 A˚ above the GaAs(001)-

c(4×4)-hd surface...... 46

5.5 Ga adatom diﬀusion barrier results (eV) on GaAs(001)-c(4×4)-heterodimer. 47

6.1 Extrema positions for anions in CdTe ...... 54

6.2 Average bond length and angle from anion defect to bulk atoms . . . . . 56

6.3 Diﬀusion barriers (eV) for diﬀerent anion point defects...... 57

6.4 Extrema positions for cations in CdTe ...... 71

6.5 Average bond length and angle from cation defect to bulk atoms . . . . . 73

6.6 Diﬀusion barriers (eV) for diﬀerent cation point defects...... 74

7.1 Extrema positions for the diﬀusing adatoms in bulk CdS given in direct

coordinates...... 87

7.2 Average bond length and angle from interstitial defect to nearest neighbor

in CdS ...... 89

7.3 Diﬀusion barriers (eV) for diﬀerent point defects in bulk CdS ...... 92

xiii List of Figures

2-1 GaAs(001)-c(4×4) surface reconstruction...... 5

2-2 GaAs(001)-c(4×4)-heterodimer surface reconstruction...... 6

2-3 CdTe bulk structure ...... 9

2-4 CdS bulk structure ...... 11

3-1 Top-hopping Diﬀusion ...... 14

3-2 Diﬀusion Barrier ...... 15

3-3 Exchange Diﬀusion ...... 16

3-4 Bulk Interstitial Diﬀusion ...... 18

3-5 Bulk Exchange Diﬀusion ...... 19

3-6 Vacancy Diﬀusion ...... 20

3-7 Concentration Gradient ...... 21

4-1 DFT Flowchart...... 30

5-1 GaAs(001)-c(4×4) 3 A˚ minimum energy sites...... 36

5-2 GaAs(001)-c(4×4) 0.5 A˚ minimum energy sites...... 37

5-3 Ga adatom positions having energy minimums or transition sites at 0.5 A˚

distort the surface As dimers...... 40

5-4 Diﬀusion paths on the GaAs(001)-c(4×4) surface at 3 A˚ ...... 41

5-5 Broken As2 dimer LDOS ...... 42 5-6 Diﬀerence between GaAs(001)-c(4×4) and c(4×4)-heterodimer surface re-

constructions ...... 43

xiv 5-7 GaAs(001)-c(4×4)-heterodimer 3 A˚ minimum energy sites ...... 45

5-8 Diﬀusion paths on the GaAs(001)-c(4×4)-heterodimer surface at 3 A˚ . . 48

5-9 Diﬀerence between the GaAs(001)-c(4×4) As2 homodimer and the c(4×4)- heterodimer Ga-As heterodimer ...... 49

5-10 Local density of states for the Ga atom of the surface edge heterodimer

(GaAs) ...... 50

6-1 Bonding conﬁgurations of anion interstitial positions in CdTe ...... 55

6-2 Anion diﬀusion barrier proﬁles in CdTe ...... 60

6-3 Anion diﬀerences in diﬀusing paths down the [110] channel in CdTe . . . 61

6-4 Z-axis deviation from the minimum of anion interstitial atoms while dif-

fusing down the [110] channel in CdTe ...... 62

6-5 Eﬀect of local strain energy and relaxation around diﬀusing interstitial Te

and O atoms in bulk CdTe ...... 63

6-6 LDOS for an interstitial Cl atom, at the minimum and global maximum

positions ...... 64

6-7 LDOS for an interstitial P atom, at the minimum and global maximum

positions ...... 66

6-8 LDOS for an interstitial Sb atom, at the minimum, secondary maximum

and global maximum positions ...... 67

6-9 Bonding conﬁgurations of anion interstitial positions in CdTe ...... 69

6-10 Cation diﬀusion barrier proﬁles in CdTe ...... 75

6-11 Cation diﬀerences in diﬀusing paths down the [110] channel in CdTe . . . 76

6-12 Z-axis deviation from the minimum of the cation interstitial atoms while

diﬀusing between global minimums in adjacent unit cells down the [110]

channel...... 77

xv 6-13 Eﬀect of local strain energy and relaxation around diﬀusing interstitial Cu

and Cd atoms in bulk CdTe ...... 79

6-14 LDOS for an interstitial Ag atom, at the global minimum and secondary

minimum positions ...... 81

6-15 LDOS for an interstitial Cd atom at the global minimum and global max-

imum positions ...... 82

6-16 LDOS for a Cu interstitial at the global minimum and global maximum

positions ...... 83

7-1 Interstitial diﬀusion in bulk CdS ...... 86

7-2 Structural motifs and bonding conﬁgurations in bulk CdS ...... 88

7-3 Energy along the diﬀusion path as a function of the NEB step positions

in bulk CdS ...... 91

7-4 Eﬀect of local strain energy and relaxation around diﬀusing interstitial

atoms Cd, Cu and Te in bulk CdS ...... 94

7-5 LDOS for a Cu interstitial in bulk CdS ...... 96

A-1 Born-von Karmen boundary condition...... 117

A-2 Pseudopotential...... 122

A-3 VASP Flowchart ...... 126

A-4 Plane-wave Convergence ...... 128

A-5 POSCAR File ...... 131

A-6 KPOINTS File ...... 132

xvi List of Abbreviations

CIGS ...... Copper Indium Gallium (di)Selenide

DFT ...... Density Functional Theory

GGA ...... Generalized Gradient Approximation GMax ...... Global Maximum Energy GME ...... Global Minimum Energy

LDA ...... Local Density Approximation (L)DOS ...... (Local) Density of States

MBE ...... Molecular Beam Epitaxy MES ...... Minimum Energy Site

NEB ...... Nudged Elastic Band

RMM-DIIS ...... Residual Minimization Scheme - Direct Inversion in the Iterative Subspace

SMax ...... Secondary Maximum Energy SME ...... Secondary Minimum Energy

VASP ...... Vienna ab initio Simulation Package

xvii Chapter 1

Introduction

1.1 Thesis outline

The thesis presented here is a theoretical study of bulk and surface diffusion in specific semiconductor materials using first principles calculations. Controlling defect concentration and mobility of point defects requires understanding of the structural, energetic and electronic properties of native and non-native point defects and their migration pathways by diffusion. Directly elucidating such pathways experimentally is difficult by current techniques. On the other hand first principles computational methods are aptly suitable for this purpose. However a high requirement of computational resources and complexity has prevented much attention from being given to this problem so far. The following work presented in this thesis is intended to address this problem and partially fill the gap in the literature with computations of (i) Ga atom diffusion on the (001) surface of GaAs and (ii) point defects in bulk CdTe and bulk CdS. A brief outline of the thesis is given below.

Chapter 2 includes an introduction to the materials investigated in this work and the motivation for the work on these materials. This includes GaAs and the

GaAs(001)-c(4×4) and GaAs(001)-c(4×4)-heterodimer surface reconstructions, bulk

CdTe and CdS. In Chapter 3 an overview of diﬀusion and diﬀusion processes for both

1 surfaces and in the bulk is given. This chapter discusses the top-hopping and exchange diﬀusion mechanisms on GaAs(001)-c(4×4) and GaAs(001)-c(4×4)-heterodimer surface reconstructions modeled in this work. Also included in this chapter is the bulk interstitial diﬀusion mechanism for the zinc-blende (CdTe) and hexagonal wurtzite

(CdS) crystal structures presented in this work.

A discussion of density functional theory (DFT) is given in Chapter 4. This chapter includes an introduction to ﬁrst principles and the theories upon which it was built leading to the development of DFT. The discussion on DFT includes sections regarding the theory upon which DFT was built, including Hohenberg-Kohn theorem and Kohn-Sham equations, and a section regarding the implementation of the theory through the use of the local density approximation (LDA).

Chapters 5-7 include the discussion of the results. Chapter 5 begins with a discussion of the computational methods and modeling approach used for the GaAs work.

Then, Chapter 5 follows with a discussion of the results of the surface diffusion of a Ga adatom on GaAs surfaces including the GaAs(001)-c(4×4) and GaAs(001)-c(4×4)- heterodimer surface reconstructions. Chapter 6 includes the results of the work of bulk diffusion in CdTe and Chapter 7 includes the results of the work of bulk diffusion in CdS. Similar to Chapter 5, Chapters 6 and 7 begin with the computational methods and modeling approach used followed by the results. Chapter 8 includes the conclusions of the work presented in chapters 5-7. A look at possibilities for future work along with a broader perspective is presented in closing this thesis.

2 Chapter 2

Materials

2.1 GaAs

The GaAs semiconductor is an important material for today’s mobile devices.

Properties such as high electron mobility and high breakdown voltage have made

GaAs the material of choice. GaAs is used to make devices from solar cells, light emitting diodes and other optoelectronic devices to monolithic microwave integrated circuits used in cellular phones. These applications require precise control during the growth of GaAs thin films. The GaAs(001) surface is the preferred growth surface for this material. The GaAs(001) surface is one of the most frequently studied surfaces during growth by molecular beam epitaxy (MBE). However, recent results show that MBE growth on patterned GaAs(001) under standard conditions leads to instabilities in which the patterned perturbations to a flat surface initially amplify for growth at high temperatures [1–3], and for which multilayer ridges build up around patterned pit-structures at low temperatures [4]. A great deal of work has been done, both theoretically and experimentally, to provide in depth understanding into MBE growth on GaAs(001) [5]. This work has included surface morphology, step-flow layer growth and island nucleation using a number of experimental techniques, such as reflection high-energy electron-diffraction (RHEED) intensity oscillations and scan-

3 ning tunneling microscopy (STM) measurements, and theoretical modeling, including

kinetic Monte Carlo simulations.

2.1.1 Structure

GaAs is a III-V semiconductor that crystallizes in the zinc-blende crystal structure

(space group F43m number 216) under normal conditions. The GaAs zinc-blende

structure has a lattice constant and bulk modulus of 5.653 A˚ and 42 GPa respectively

[6]. GaAs has a direct 1.4 eV band gap.

2.1.2 GaAs(001)-c(4×4) Surface Reconstruction

The GaAs(001) surface exhibits several surface reconstructions [7,8] ranging from

the high temperature β2(2×4) reconstruction to the low temperature c(4×4) reconstruction, both of which are of particular relevance for MBE and other growth

methods. The As-rich GaAs(001)-c(4×4) surface reconstruction, shown in Fig.2-1, is

important because it is the surface typically found during low temperature MBE pro-

cesses under high As/Ga ﬂux conditions [9]. For the As-rich c(4×4) reconstruction, growth requires incorporation of Ga into the surface. Therefore a detailed knowledge of the Ga adatom adsorption and diﬀusion processes on the surface is critical.

2.1.3 GaAs(001)-c(4×4)-hd Surface Reconstruction

Dimerization of surface atoms is a common theme observed in several tetrahedrally coordinated semi-conductor surfaces [10–12]. The GaAs(001) surface is no exception, with the As-rich c(4×4) reconstruction, shown in Fig. 2-1 observed as the dominant surface during low temperature molecular beam epitaxial growth. This structural model with three As2 dimers per surface unit cell was established by Sauvage-Simkin et al. [13] using grazing incidence X-ray diﬀraction and it was found to be stable

4 Figure 2-1: The GaAs(001)-c(4×4) surface reconstruction.

at As-rich conditions by ﬁrst principles calculations [7, 14–16]. Recently Ohtake et

al. [17] proposed a new structure model for c(4×4) reconstruction with three Ga-As heterodimers per unit cell, as shown in Fig. 2-2. They proposed this model based on rocking-curve analysis of RHEED, STM, and reﬂectance diﬀerence spectroscopy.

This new heterodimer (hd) structure was found to be consistent with element speciﬁc

X-ray diﬀraction [18]. This structure referred to henceforth as the GaAs(001)-c(4×4)-

hd reconstruction consists of alternating rows of three Ga-As heterodimers followed

by a missing dimer. The binding energies and diﬀusion barriers for a Ga adatom are

unknown for this recently discovered GaAs(001)-c(4×4)-hd surface reconstruction.

5 Figure 2-2: The GaAs(001)-c(4×4)-heterodimer surface reconstruction.

2.2 CdTe

CdTe based thin film technology has emerged as a leader in the growing market of thin film solar cell module production. The optimal band-gap and high pho- ton absorption coefficient have made CdTe an excellent absorber material in thin

film solar cells. Continuous advancements and the wide variety of fabrication techniques [19–24] have led to improved cell efficiencies [25] at lower costs. The ability to produce high quality CdTe layers is necessary to achieve optimal cell efficiencies.

It is well known that the presence of defects, in these layers, will affect the semiconductor properties [26,27] and hence overall cell efficiencies. Three types of structural defects, planar, linear and point defects, are generally considered influential [28–30].

Speciﬁcally point defects may include native defects of Cd and Te, dopants such as

P or Sb, S from the CdS window layer, other metal atoms such as Cu, Ag, Mo, Au

6 depending on the back contacts used, Cl from CdCl2 treatments and Zn, Sn, O from the transparent conducting oxides [31]. It is well known that Cu plays a number of roles in CdTe. For example, the eﬀect of adding Cu to back contacts dopes the CdTe as p-type and improves the Ohmic contact between the back contact and the p-CdTe.

Increased cell efficiency has also been observed due to diffusion of Cu from the back contact into the CdTe absorber layer [32]. Additionally, the diffusion of Cu into, and its accumulation at, the CdS layer has been the most suspected cause inhibiting long term device stability [33]. For the case of Ag in CdTe, a model has been developed to explain the experimentally observed concentration depth profiles [34]. This model, and similar models, as well as Kinetic Monte Carlo calculations used for growth simulation, require information about the diffusivity of the various defects which depend on the diffusion barriers. Therefore, the ability to control the mobility and concentration of these point defects makes it important to understand their mass transport pathways in CdTe. The variety of diffusion mechanisms of Cu and other defects have to be understood clearly before their control can be achieved. Such understanding will include knowledge of the structural, energetic and electronic properties of these native and non-native point defects. A post-deposition treatment in the presence of

◦ Cl and O2 near 400 C is used by essentially all research groups and manufacturers of CdS/CdTe cells and modules to reach the highest device performance. This ”activation” treatment increases photocurrent and open-circuit voltage and fill factor. It has also been shown that incorporating O into the closed space sublimation growth ambi- ent enhances CdTe device performance [35]. In addition, the activation temperatures that range from 350◦C to 450◦C provide the driving force for bulk interdiffusion of Te and S from the CdTe and CdS layers. The diffusion of CdS into CdTe is a fast process and difficult to control, especially for cell structures with ultrathin, <100 nm, CdS

films [25] and the efficiencies of current devices are believed to be strongly dependent on this interdiffusion at the CdS/CdTe interface [36]. Therefore, controlling defect

7 concentration and mobility of such point defects makes it important to understand the structural, energetic and electronic properties of such native and non-native point defects and their migration pathways by diffusion in CdTe. Directly elucidating such pathways experimentally in the bulk material is difficult by current techniques. On the other hand, first principles computational methods are aptly suitable for this purpose [37].

2.2.1 Structure

CdTe is a group IIB-VIA compound semiconductor, with a direct band gap of

1.5 eV. Under normal conditions, CdTe will crystallize in the zinc-blende crystal structure (space group F43m number 216). Figure 2-3 shows the CdTe zinc-blende structure along the (a) [0 1 0] direction and (b) along the [1 1 0] direction. The CdTe zinc-blende structure has a lattice constant and bulk modulus of 6.46 A˚ and 42 GPa respectively [38]. The simplest mode of diffusion in zinc-blende CdTe is interstitial diffusion down the spacious [1 1 0] channel. Interstitial diffusion down this channel consists of the diffusing adatom passing between two alternating high symmetry sites,

Wyckoff positions 4(b) and 4(d). The 4(b) site is the location of the diffusing adatom when it is tetrahedrally coordinated by Te atoms and the 4(d) site is the location of the diffusing adatom when it is tetrahedrally coordinated by Cd atoms.

8 Figure 2-3: CdTe bulk structure as viewed along the (a) [0 1 0] channel and (b) along the [1 1 0] channel.

2.3 CdS

CdS is known for its applications to optoelectronic devices [39,40]. An important consequence of the large band gap is the high light transmittance of CdS in the visible region. Hence, a prominent use of CdS is as a window layer for thin ﬁlm CdTe and copper indium gallium (di)selenide (CIGS) based solar cells. The recent advances in

CdTe/CdS thin ﬁlm technology and fabrication techniques have allowed CdTe/CdS solar cells to emerge as a leader in the growing market of thin ﬁlm module production.

A number of diﬃculties have slowed the further improvement of CdTe/CdS thin ﬁlm technologies. These include the accumulation of Cu, from the back contacts, at the

CdTe/CdS interface as well as intrinsic and exotic interstitials originating in CdS and diffusing interstitials from CdTe that cross the interface into the CdS layer affecting the cell performance. Diffusion of Cd and Te atoms from the CdTe absorption region

9 into CdS can reduce the light transmission capability of the window in the wavelength region of 500 to 650 nm. The faster process of diffusion of Cd and S atoms into CdTe, in the opposite direction, is more difficult to control, especially for cell structures with ultrathin CdS films [25]. The effect of Cl in CdS is also well known. Secondary ion mass spectrometry measurements suggest that the high Cl concentration in CdS

films yields better solar cell efficiency [41]. Thus, it is well known that semiconductor properties, and hence overall cell efficiencies, are affected by the presence of defects in these layers [27, 42]. To be able to control the defect concentration and mobility of defects in CdS requires understanding of their migration pathways by diffusion in

CdS and the structural and electronic properties of these defects. Revealing these bulk diﬀusion pathways directly is challenging by current experimental techniques alone. Theoretical investigation not only provides results for values of the energetic barriers for diﬀusion of defects but also gives insights into the physical mechanism of adatom of vacancy migration and electronic bonding characteristics [37].

10 Figure 2-4: CdS bulk structure as viewed (a) parallel to the c axis and (b) perpendicular to the c axis.

2.3.1 Structure

CdS is a group IIB-VIA compound semiconductor, with a direct band gap of about 2.42 eV [43]. CdS will crystallize in the both the cubic zinc-blende (space group F43m number 216) and hexagonal wurtzite (space group P63mc number 186) crystal structure. Figure 2-4 shows the CdS hexagonal wurtzite structure as viewed

(a) parallel to the c axis and (b) perpendicular to the c axis. The CdS wurtzite

structure experimental lattice parameters [44] a, c and internal parameter z of 4.1365

A˚ and 6.7160 A˚ and z = 0.3770.

The simplest mode of diﬀusion in wurtzite CdS is interstitial diﬀusion down the

spacious channel along the [0 0 0 1] direction, parallel to the c axis shown in Fig. 2-

4(a). Interstitial diﬀusion down this channel consists of the diﬀusing adatom passing

11 between alternating high symmetry Wyckoﬀ positions 2(a). The 2(a) sites occur symmetrically through the center of hexagons formed by the bulk Cd and S atoms.

12 Chapter 3

Diﬀusion

3.1 Overview

Diffusion is the random process in which mass is transferred by means of atomic movement. There are a number of ways in which diffusion can occur and a number of factors that can influence the diffusion mechanism. These diffusion processes and the equations used to describe them, referred to as Fick’s laws will be discussed in the following sections.

3.2 Surface Diﬀusion Mechanisms

Diﬀusion on solid material surfaces can proceed by a number of diﬀerent methods.

This can include mechanisms in which the diffusing species remains on the surface or mechanisms where the diffusing species is incorporated into the material from the surface. The focus of this work on surface diffusion includes the top-hopping and exchange mechanisms.

13 3.2.1 Top Hopping Diﬀusion

The top-hopping diffusion mechanism occurs when an atom, either native or non- native in origin, moves over the surface by a successive number of jumps between adjacent minimum energy adsorption sites. These diffusing atoms, henceforth referred to as adatoms, are shown diffusing by means of the top-hopping mechanism in Fig.

3-1. The example shown in the ﬁgure depicts top-hopping diﬀusion on a simple cubic surface where the adatom, labeled ’A’ occupies an adsorption site that is characterized by a four-fold coordination with the underlying surface atoms. From this position, adatom ’A’ moves to an adjacent symmetrically equivalent adsorption site.

Figure 3-1: The top-hopping diﬀusion mechanism. The diﬀusing adatom moves from adjacent adsorption sites by ’hopping’ over the underlying surface bulk atoms.

The top-hopping diﬀusion mechanism is the easiest mechanism to relate the associated diﬀusion energy barrier to the motion of the adatom. If we look at one of the four-fold coordinated minimum energy adsorption sites it is clear that the adatom shares four bonds with the four underlying bulk surface atoms. As the adatom moves from one adsorption site to an adjacent adsorption site two of the bonds with the four surface bulk atoms must be broken while the other two bonds remain intact. It is the

14 breaking of the bonds between the adatom and the surface bulk atoms that results in the energy barrier to diﬀusion. This correspondence between bonding, position and energy is illustrated in Figure 3-2.

Figure 3-2: The top-hopping diﬀusion mechanism energy barrier. Panel a) shows the adatom at the minimum energy, four-fold coordinated adsorption position and the corresponding energy versus position is given in panel b). At the minimum energy position the adatom shares four bonds with the underlying bulk surface atoms 1, 2, 3 and 4. In between minimum energy positions, shown in panel c), the adatom stretches and eventually breaks the bonds with surface atoms 1 and 2 while the bonds with surface atoms 3 and 4 remain. It is the breaking of the bonds with the surface atoms that results in the increase in energy shown in d). As the adatom moves back into the adjacent minimum energy adsorption site, panel e), the bonds with surface atoms 3 and 4 remain while the broken bonds with surface atoms 1 and 2 are replaced with similar bonds with surface atoms 5 and 6.

3.2.2 Exchange Diﬀusion

Unlike the top-hopping diffusion mechanism, the exchange diffusion mechanism involves the movement of both the diffusing adatom and the underlying bulk surface atoms. During exchange mediated diffusion the adatom begins at the minimum

15 energy adsorption site, similar to the top-hopping diffusion mechanism, and then proceeds to displace an underlying bulk surface atom. As the adatom takes the place of the bulk surface atom, it promotes the bulk surface atom to an adjacent adsorption site on the surface where it becomes the new adatom. The entire process may then repeat or the new adatom may continue to diffuse by way of the top-hopping method described in section 3.2.1. The process of exchange diffusion is shown in Fig. 3-3.

Figure 3-3: The exchange diﬀusion mechanism involves the simultaneous movement of two atoms. The process begins with the adatom ’A’ at the minimum energy adsorption position, shown from the top in a) and from the side view in b). The next step, c) and d), involves the adatom ’A’ displacing one of the underlying four- fold coordinated bulk surface atoms ’B’. In the ﬁnal step, e) and f), adatom ’A’ takes the place of bulk surface atom ’B’ and is incorporated into the surface and the former bulk surface atom ’B’ becomes the new adatom, on top of the surface, at an adjacent minimum energy adsorption site.

The energy barrier corresponding to the exchange diffusion mechanism is expected to be smaller in magnitude than the energy barrier for the top-hopping diffusion mechanism. This is because during the top-hopping diffusion process, the adatom must break two of the bonds it makes with two of the bulk surface atoms which are reformed with two new surface atoms at the end of the process. During the

16 exchange process, only one bond is broken with the surface atoms which is reformed by the displaced surface atom. There are many factors that will inﬂuence the size of the energy barrier, including but not limited to temperature, the bulk composition, surface reconstruction and the type of adatom.

3.3 Bulk Diﬀusion Mechanisms

Similar to surface diffusion, bulk diffusion can proceed by way of a number of independent mechanisms or a combination of mechanisms. These processes include interstitial diffusion, vacancy diffusion, exchange diffusion and grain boundary diffusion. The diffusion mechanisms investigated in this work include the interstitial and vacancy diffusion mechanisms.

3.3.1 Interstitial Diﬀusion

Interstitial diffusion is characterized by atomic movement through the bulk material where the adatom occupies and moves through the interstitial channels between interstitial positions of a bulk crystalline lattice. This is generally only possible if the adatom is small enough to fit in the interstitial region. Similar to the top-hopping diffusion mechanism, only the adatom moves during interstitial diffusion. The interstitial diffusion process is illustrated for a 2-dimensional example in Fig. 3-4.

17 Figure 3-4: Interstitial diffusion begins with the adatom at one of the minimum energy interstitial sites in the crystal lattice, shown in a). Diffusion can proceed along the interstitial channels of the bulk crystal. Midway through the process the adatom passes between the bulk atoms, b), where the associated diffusion energy barrier is attributed to the broken bonds from the minimum energy interstitial site and the strain energy associated with the close proximity to the bulk atoms. The process is completed when the adatom arrives at an adjacent minimum energy interstitial site.

3.3.2 Bulk Exchange Diﬀusion

Bulk exchange diffusion is similar to the surface exchange diffusion mechanism described in section 3.2.2. Both processes involve the simultaneous motion of both the adatom and one of the bulk atoms. The bulk exchange diffusion mechanism is shown in Fig. 3-5.

18 Figure 3-5: The bulk exchange diﬀusion mechanism involves the simultaneous movement of two atoms. This begins with the adatom ’A’ at the minimum energy interstitial position, shown in a). The next step, b), involves the adatom ’A’ displacing one of the bulk atoms ’B’. In the ﬁnal step, c), adatom ’A’ takes the place of bulk atom ’B’ and bulk atom ’B’ now occupies an adjacent minimum energy interstitial site.

3.3.3 Vacancy Diﬀusion

The absence of a bulk atom in the crystalline lattice is referred to as a vacancy.

The vacancy diffusion mechanism can be regarded as the motion of this vacancy through the bulk crystal. This motion is accomplished by subsequent replacement of the vacancy with another bulk atom from an adjacent bulk position. Therefore vacancy diffusion can be regarded as forward vacancy diffusion or backward bulk atom diffusion. This process is illustrated in Fig. 3-6.

19 Figure 3-6: Vacancy diffusion first requires the presence of a vacancy at the site of a bulk atom in the crystal structure, shown in a). This vacancy be regarded as moving to the right in panel b) or as a bulk atom ’B’ moving to the left. The vacancy diffusion mechanism is complete when the vacancy has occupied another adjacent bulk atom position and the bulk atom that was at that position, atom ’B’ in panel c), now occupies the original vacant bulk position.

3.4 Describing Diﬀusion

As mentioned there are a number of ways in which diffusion can occur. This can include self-diffusion, where all of the atoms involved in diffusion are of the same type, and inter-diffusion, or impurity diffusion, that occurs as the result of the presence of different types of atoms, which creates a concentration gradient. In the following one-dimensional example, the concentration gradient is illustrated by looking at the concentration of type ’A’ atoms as a function of position. Initially, all atoms of type ’A’ atoms are contained to the left of the container, as shown in Fig. 3-6(a), and the graph of the concentration versus position reflects this. After some time the atoms of type ’A’ will mix with the atoms of type ’B’, Fig. 3-6(b). If we look at the concentration of atoms of type ’A’ versus position graph we can define the concentration gradient as the change in concentration of atoms of type ’A’ over a given change in position.

20 Figure 3-7: Initially, (a), all atoms of type ’A’ atoms are on the left of the container and corresponding graph of concentration of atoms of type ’A’ is shown. After some time the atoms of diﬀerent types will mix, (b). We can deﬁne the concentration gradient of atoms of type ’A’ as the change in concentration of atoms of type ’A’ over a given change in position. Figure adapted from Bastawros [45].

This leads us to the definition of Fick’s laws of diffusion. Fick’s first law describes the flux, J, of steady-state diffusion and is given by

∂C J = −D , (3.1) ∂x

∂C where ∂x defines the concentration gradient, the minus sign indicates that diffusion proceeds down the concentration gradient and D is the diffusion coefficient.

Fick’s first law says that the flux is proportional to the concentration gradient and that the flux, or atoms per unit area per unit time, does not change in time. In most situations, however, the flux is not constant and the concentration will change in time. This leads to Fick’s second law of diffusion which describes the change in

21 ﬂux at a particular position by accounting for the change in concentration proﬁle in time. Fick’s second law is given by

∂C ∂J ∂ ∂C ∂2C = − = D = D . (3.2) ∂t ∂x ∂x ∂x ∂x2

One factor that influences diffusion is temperature. As the result of thermal activation, the adatom acquires a sufficient amount of thermal energy to break the bonds with the bulk atoms and proceed via the diffusion mechanisms discussed in the previous sections. The diffusion coefficient D in equations 3.1 and 3.2 describes thermally activated processes and is given by

EA − k T D = D0e b . (3.3)

2 where the pre-exponential D0, with units of m /s, contains information regarding the diffusion length and attempt frequency, kb is Boltzmann’s constant and T is temperature. Here, EA is the activation energy, or the diffusion energy barrier, similar to that shown in Fig. 3-2. Theoretical investigation not only provides results for values of the energetic barriers for diffusion of defects but also gives insights into the physical mechanism of diffusion which is difficult to determine by current experimental techniques alone. It is the investigation of these diffusion barriers and migration pathways that is the focus of this work. This was accomplished by way of total energy calculations made possible by DFT.

22 Chapter 4

Density Functional Theory (DFT)

4.1 Introduction

Understanding of the structural and electronic properties of a quantum many- body system begins with the Schr¨odingerequation. We know that the energy of the system may be found from the solution to the Schr¨odingerequation for a system of

N interacting particles, expressed in the time independent form as

Hˆ Ψ = EΨ , (4.1) where Ψ is the wavefunction for the N particle system and the Hamiltonian Hˆ is given as N 2 N N ˆ X ~ 2 1 X X ZiZj H = − ∇i + . (4.2) 2mi 2 4π0|ri − rj| i=1 i=1 j6=i

th Here ri is the position of the i particle and Zi is its charge. However, an exact analytical solution of the (3N) variable Schr¨odingerequation is not possible for all but the simplest of systems. Numerical solutions, while possible, are limited to equally small systems due to scaling of computational time with system size. DFT provides a means to obtain an exact solution by substituting the many-body wavefunction with a three variable function of the electron density. The following sections will outline

23 the development to DFT.

4.2 Hohenberg-Kohn Theorems

Built upon the theorems of Hohenberg and Kohn [46], DFT owes its development to the approximation methods that preceded it. The idea of using the electron density to calculate the total energy was originally proposed in 1927 by Thomas and

Fermi [47, 48]. They were the ﬁrst to use the electron density to calculate the total energy of the system by using the homogeneous electron gas as an approximation for kinetic energy. However, the Thomas-Fermi method treated the charged particle interaction classically and results were, as stated by Hohenberg and Kohn, to be

”...useful, up to now, for simple though crude descriptions of inhomogeneous systems like atoms and impurities in metals.” Hartree and Fock [49, 50] built upon this idea by treating the particle interaction quantum mechanically and decoupling the system of interacting electrons in a given potential and mapping them to a system of non- interacting electrons in an ’eﬀective’ potential. This was accomplished by replacing the N-body wavefunction with a Slater determinant of N spin-orbitals. The determinant ensures the wavefunction is antisymmetric with respect to exchange of any two electrons, per the Pauli-exclusion principle. The Hartree-Fock method accounts for electron exchange but does not address the correlation between electrons leading to large disagreement with experiment. DFT, however, accounts for both exchange and correlation between electrons.

As mentioned above, the primary purpose of DFT is to reduce the 3N variable

Schr¨odingerequation to a three variable function of the electron density. This reduction is possible based on the two theorems of Pierre Hohenberg and Walter Kohn [46].

Theorem I: For any system of interacting particles, the external potential is a unique functional of the ground state electron density n0(r).

24 Proof I: Assume that the ground state is non-degenerate and there exist two diﬀerent external potentials v(1)(r) and v(2)(r) that diﬀer by more than a constant

(1) (2) and correspond to the same ground state density, n0(r). Clearly v (r) and v (r) belong to distinct Hamiltonians Hˆ (1) and Hˆ (2), and since they diﬀer by more than a

constant they also correspond to distinct wavefunctions Ψ(1) and Ψ(2).

(1) (1) ˆ (1) (1) (1) (1) (1) E = hΨ |H |Ψ i = hΨ |T + Vee + v |Ψ i , (4.3)

Z (1) (1) (1) (1) ⇒ E = n0(r)v (r)dr + hΨ |T + Vee|Ψ i . (4.4)

Similarly, for E(2) we have

Z (2) (2) (2) (2) ⇒ E = n0(r)v (r)dr + hΨ |T + Vee|Ψ i . (4.5)

Since Ψ(2) is not the ground state wavefunction of Hˆ (1) it follows that

Similarly, Ψ(1) is not the ground state wavefunction of Hˆ (2) and it follows that

Z (2) (1) (2) (1) E < E + n0(r)(v (r) − v (r))dr . (4.8)

Adding (4.7) and (4.8) we have

E(1) + E(2) < E(2) + E(1) . (4.9)

This is a contradiction and therefore it has been proved by reductio ad absurdum that

the existence of a second potential v(2)(r) which diﬀers from given potential v(1)(r) 25 by more than a constant and correspond to the same ground state density n0(r) must be wrong. Therefore, the external potential is a unique functional of the ground state electron density.

Theorem II: A universal functional Ev[n(r)] that deﬁnes the energy of a system in a given potential can be deﬁned strictly in terms of the above mentioned electron density n(r). The global minimum value of this energy functional corresponds to the true ground state density, n0(r). Because Ψ is a functional of n(r), then so too is the kinetic and interaction energy,

T and Vee respectively. Here Hohenberg and Kohn deﬁne the functional

F [n(r)] ≡ hΨ|T + Vee|Ψi , (4.10) the second term in (4.4). Here F [n(r)] is a universal functional that is applicable to all ground state, electronic systems regardless of the external potential v(r). With

(4.10), Hohenberg and Kohn deﬁne an energy functional for a given potential v(r)

Z Ev[n(r)] = n(r)v(r)dr + F [n(r)] . (4.11)

Here, Ev[n(r)] will correspond to the ground state energy E0 for the correct ground state density n0(r).

Proof II: It will be shown that for all densities n(r), Ev[n(r)] ≥ E0 where E0 is the ground-state energy for N electrons in the given external potential v(r). From the

ﬁrst theorem, it was shown that any given density n0(r) determines its own external potential v0(r) and wavefunction Ψ0. If this wavefunction, Ψ0, is used as a trial wavefunction for the Hamiltonian corresponding to an external potential, v(r) we

26 have

0 ˆ 0 0 0 hΨ |H|Ψ i = hΨ |T + Vee + v(r)|Ψ i Z = v(r)n0(r)dr + F [n0(r)] (4.12)

0 ˆ = Ev[n (r)] ≥ E0 = hΨ|H|Ψi , by the variational principle. This equality is only true, for non-degenerate ground states, if the trial wavefunction Ψ0 is the true ground state wavefunction for the external potential v(r).

Hohenberg and Kohn proceed to express the functional F [n(r)] as

1 ZZ n(r)n(r0) F [n(r)] = drdr0 + G[n(r)] , (4.13) 2 |r − r0|

Here, Hohenberg and Kohn introduce G[n(r)] as another universal functional similar to F [n(r)]. We can therefore rewrite (4.11) as

Z 1 ZZ n(r)n(r0) E [n(r)] = n(r)v(r)dr + drdr0 + G[n(r)] , (4.14) v 2 |r − r0|

If we knew F [n(r)], the solution would be a straight forward minimization of the energy functional. The exact ground state of the system is the minimum value of this energy functional for a given external potential v(r). Therefore, all the properties of a given system characterized by an external potential v(r) are determined by the electron density n(r). It is important to note that the theorems provided by Hohen- berg and Kohn provide an exact solution to the Schr¨odingerequation for a system of electrons in a given external potential. The problem, however, is that the exact form of F [n(r)], in particular G[n(r)], is unknown. Walter Kohn and Lu Sham provide a solution to this problem by introducing an approximation for the unknown functional

G[n(r)].

27 4.3 Kohn-Sham Equations

Kohn and Sham begin with the energy functional for a given external potential

v(r) in (4.14) and provide an expression for G[n(r)] that contain the both the eﬀects of exchange and correlation [51]. G[n(r)] is deﬁned as

G[n(r)] ≡ Ts[n(r)] + Exc[n(r)] . (4.15)

In the above equation, Ts[n(r)] is the kinetic energy of a system of non-interacting electrons with density n(r), not the kinetic energy of the system, hence the subscript

s. Exc[n(r)] is, by their deﬁnition, the exchange and correlation energy of a system of interacting electrons with density n(r). We can now rewrite the energy functional as

Z 1 ZZ n(r)n(r0) E [n(r)] = n(r)v(r)dr + drdr0 + T [n(r)] + E [n(r)] , (4.16) v 2 |r − r0| s xc

The primary feature of the Kohn-Sham method is the mapping of a system of interact-

ing electrons in a given potential v(r) to a system of non-interacting electrons acting

in an eﬀective potential veff (r). Using the variational approach outlined by Hohen- berg and Kohn, and including a Lagrange multiplier µ as a normalization constraint

on the electron density Z n(r)dr = N, (4.17)

we now have δ Z [E[n(r)] − µ n(r)dr] = 0 , (4.18) δn(r) δE[n(r)] ⇒ = µ . (4.19) δn(r)

28 We can now rewrite (4.19) in terms of an eﬀective potential veff (r).

δT [n(r)] s + V (r) = µ , (4.20) δn(r) eﬀ

where veff (r) is given by

veﬀ (r) = v(r) + vH(r) + vxc(r) . (4.21)

In the above (4.21) vH (r) is the Hartree potential and vxc(r) is the exchange-correlation potential given by Z n(r0) v (r) + dr0 , (4.22) H |r − r0| δE [n(r)] v (r) = XC . (4.23) XC δn(r)

Now, in order to ﬁnd the ground state density and energy we must solve the single particle Schr¨odingerlike Kohn-Sham equations

2 − ~ ∇2 + v (r) ψ (r) = ε ψ (r) , (4.24) 2m eff i i i self-consistently where the density is given by

N X 2 n(r) = |ψi(r)| . (4.25) i=1

A flow chart outlining the method for obtaining the total energy using DFT is given in Fig. 4-1. The first step is to construct the potential v(r) from the atomic positions of the system in question. Next, an initial guess for the electron density n(r) is made. This density is then used to construct the Hartree and the exchange-correlation potential terms in the effective potential, given in (4.21). Once the effective potential is constructed, the next step is to solve the single electron Kohn-Sham equations,

29 (4.24). The solutions to the Kohn-Sham equations are then used to construct a new

density, (4.25).

Figure 4-1: The method for obtaining the total energy from DFT. Figure adapted from Payne et al. [52].

4.4 Exchange-Correlation Approximations

4.4.1 The Local Density Approximation (LDA)

It should be noted that, up to this point, the method presented here provides an exact solution to the many body Schr¨odingerequation. Unfortunately, the exact form

30 of the exchange correlation functional Exc[n(r)] is unknown for any arbitrary density n(r). However, it is possible to approximate Exc[n(r)] and a common approximation is the LDA. If it is assumed the density varies suﬃciently slowly the exchange and correlation energy functional Exc[n(r)] can be expressed as

Z LDA Exc [n(r)] = n(r)xc(n(r))dr , (4.26)

where xc is the exchange and correlation energy per electron. The LDA replaces the exchange correlation energy per electron, in the volume element dr, with that of the exchange correlation energy of an electron in a homogeneous electron gas of the same density. That is

hom xc(n(r)) = xc (n(r)) . (4.27)

The exchange-correlation energies per electron in homogeneous electron gas are well established and have been determined to a great deal of accuracy. This energy is comprised of both the exchange and correlation terms

xc(n(r)) = x(n(r)) + c(n(r)) . (4.28)

The exchange energy term for the homogeneous electron gas is known analytically and is given by 3 3n(r)1/3 hom(n(r)) = − . (4.29) x 4 π

The correlation energy term has been determined through the precise quantum Monte

Carlo simulations of Ceperly and Alder [53] as parameterized by Perdew and Zunger

[54].

31 4.4.2 Generalized Gradient Approximation (GGA)

An obvious shortcoming of the LDA is the assumption of the slowly varying electron density. To address this deﬁciency, and complement the information obtained by the LDA, the GGA exchange-correlation functional was developed to account for the local gradient of the electron density ∇ n(r). The GGA exchange-correlation energy functional is given by

Z GGA hom Exc [n(r)] = n(r)xc (n(r))Fxc(n(r), ∇n(r))dr . (4.30)

The Fxc(n(r), ∇n(r)) term appearing in (4.30) is a correction factor. There are a number of diﬀerent versions of GGA functionals, the diﬀerence being the Fxc(n(r), ∇n(r)) term. Some of the common GGA functionals include those of Perdew-Wang (PW86,

PW91) and Perdew-Burke-Ernzerhof (PBE, Revised PBE). In the following chapter

I will describe the implementation of DFT using the VASP suite of codes.

32 Chapter 5

Surface diﬀusion on Gallium

Arsenide (GaAs)

5.1 Computational Method

All computations were performed by ab initio total energy calculations within the local density approximation to density-functional theory [46] using VASP [55–58].

In this implementation core electrons are implicitly treated by ultrasoft Vanderbilt type pseudopotentials [59] as supplied by Kresse et al. [60] using the Ceperly and

Alder exchange-correlation functional. For each calculation, irreducible k points were

generated according to the Monkhorst-Pack [61] scheme. The single-particle wave

functions were expanded in a planewave basis using a 150 eV energy cutoﬀ. Tests

using a higher plane-wave cutoﬀ and a larger k-point sampling indicated that a nu-

merical convergence better than ± 10 meV was achieved. To obtain the absolute

minimum in total energy, the lattice constant was varied and ﬁt to a parabolic equa-

tion as a function of total energy. The calculated lattice constant of 5.603 A˚ was

found to be within 0.9% of the accepted experimental value of 5.653 A˚ [62]. Full

relaxation of ions was performed to ﬁnd the minimum energy for each conﬁguration.

All atoms were allowed to relax until a force tolerance of 0.03 eV/A˚ was reached for

33 each atom. The calculations for the local density of states (LDOS) were performed with the Methfessel-Paxton scheme [63].

The calculation of the diﬀusion barriers was performed using the NEB [64] method.

The NEB is a method for calculating the diffusion barrier between two known minimum energy sites by optimization of a number of intermediate images or snapshots of the adatom along the diffusing path. To calculate the barrier, the atomic positions in each image are fully relaxed until a force convergence is achieved and the image corresponding to the highest energy is taken to be the top of the diffusion path. The difference between this energy and that of the initial binding site is taken as the diffusion barrier.

5.2 Modeling Approach

The GaAs(001)-c(4×4) surface reconstruction was constructed by ﬁrst isolating a bulk terminated six layer slab which had 16 atoms in one layer normal to [0, 0, 1] direction. Each layer was a square with a side length along the [1, 1, 0] and [1,1,0]¯ directions. An additional layer of As is added to the top As layer to construct the six As dimers. Next, a layer of hydrogen is used to passivate the dangling bonds of the Ga atoms on the bottom of the slab and allowed to relax. Tests for an additional vacuum above the surface normal revealed that a 12 A˚ vacuum was suﬃcient to prevent interaction between supercell images during calculations. The total height of the slab and the vacuum layer was 26.166 A.˚ The top four layers were fully relaxed to obtain a c(4×4) reconstructed surface as shown in Fig. 2-1. This resulting surface is in a series of hills (dimers) and trenches due to the six As dimer pairs in the top layer.

The NEB method requires two stable minimum energy sites for implementation.

The calculation of the minimum energy sites was performed by introducing a Ga

34 adatom, at a position of 3 A,˚ above several prospective minimum energy sites. The z

coordinates of the surface atoms immediately surrounding the prospective minimum

were averaged to determine a reference surface height. The adatom along with the top

four layers of the c(4×4) reconstructed surface were allowed to relax unconstrained

until force convergence was reached. The atomic coordinates of the ﬁnal positions

used as input parameters for the NEB calculations. Finally, the diﬀusion barrier

was calculated using the NEB method. The diﬀusion path between two neighboring

minimum energy sites was modeled using seven images. Two of the images included

the initial and ﬁnal positions (minimum energy sites) and ﬁve linearly interpolated,

intermediate images between the initial and ﬁnal positions. This method was re-

peated for all nearest neighbor minimum energy sites. Similar calculations were also

performed by placing the Ga adatom only 0.5 A˚ above the surface. These latter runs

were motivated by similar distinct minima found on the β2(2×4) surface.

5.3 GaAs(001)-c(4×4)

5.3.1 Results

For the GaAs(001)-c(4×4) reconstructed surface, the minimum energy sites, diffusion paths and corresponding barriers for a Ga adatom have been determined using ab initio total energy calculations. The appearance of two distinct sets of minima and transition sites when the adatom is relaxed from either 3 or 0.5 A˚ above the surface was a unique feature discovered by Kley et al. on the GaAs(001)-β2(2 × 4) surface. [10] The results show similar distinct sets of minima and transition sites for the c(4×4) surface as well. The minimum energy sites and some transition sites for both of these sets are identiﬁed in Fig. 5-1 (3 A)˚ and Fig. 5-2 (0.5 A).˚ Transition sites are labeled with a T and minimum energy sites with a C. I computed a binding energy of 2.915 eV for the C6 (C7) site, which happens to have the lowest total energy. 35 Figure 5-1: Minimum energy and transition sites for a Ga adatom when relaxed from 3 A˚ above the GaAs(001)-c(4×4) surface.

Table 5.1 shows the energies of each site relative to the energy of the Ga adatom at

C6 (C7). You observe that the minimum energy sites in the As dimers at 0.5 A˚ have a higher binding energy than the minimum sites in the dimers at 3 A.˚ All of the 0.5

A˚ minimum sites appear at positions adjacent to transition sites that appear at 3 A.˚

The C1, C5, and C10 sites at 0.5 A˚ are similar in energy to the minimum energy sites at 3 A˚ in the trench and second only to the global minimum occurring at C7 and C6.

The T30 site at 0.5 A˚ is at the same XY position, on the surface, as the T3 site at 3

A˚ but at a diﬀerent Z coordinate. These sites were conﬁrmed by constrained mini- mizations around the minimum. A similar situation arose for the β2(2×4) surface for

the A3 site by Kley et al. [10]. However, a key diﬀerence was that the A3 site was a minimum at 0.5 A˚ and a transition site at 3 A.˚ These two sites (T3 and T30) occur in the middle, or second dimer, of the three consecutive dimers forming the hills on the surface. On the ﬁrst and third such dimers the transition sites T1 and T5 (at 3 A)˚

36 Figure 5-2: Minimum energy and transition sites for a Ga adatom when relaxed from 0.5 A˚ above the GaAs(001)-c(4×4) surface.

turn into C1 and C5 minimum energy sites at 0.5 A.˚ Furthermore the C2 and C4 min-

ima are replaced with different minima at C30 and C300. These Ga adatom positions having energy minimums or transition sites at 0.5 A˚ distort the surface As dimers in a significant manner. These may be considered similar to surface interstitial defects for this surface. Figure 5-3 shows the distortion of the surface from three different views. One view of the As2 dimers without the presence of an adatom, one for the negligibly distorted C2 site and one for the heavily distorted C30 site. The increase in the length of As dimers is significant from 2.46 to 4.57 A,˚ an increase of 86%, going from C2 to C30. Note that the extreme distortion of the surface As dimer, shown in panel (c) of Fig. 5-3, for the C30 site is a general feature of all new binding and transition sites of Fig. 5-2 when relaxing from 0.5 A.˚ For a Ga adatom relaxing from

3 A˚ in the trenches, LePage et al. [16] obtained results comparable with the results

37 Table 5.1: Binding energies (eV) of a Ga adatom on the GaAs(001)-c(4×4) surface. All energies are relative to the lowest energy sites at C6 and C7. The cited values are from [16]. The site C6 (C7) corresponds to their site 1, C8 to their site 2, and C9 to their site 3. Site Energy (eV) C1 (C5, C10) 0.46 T1 (T5, T10) 1.01 C2 (C4) 0.83 C30 (C300) 0.69 T3 1.21 T30 1.08 C6 (C7) 0 (0 [16]) C8 0.26 (0.19 [16]) C9 0.61 (0.50 [16]) presented here. These are also shown in Table 5.1. Slight diﬀerences in numerical values may be attributed to variation between methods. For example, their supercell was of half the area of the supercell used here and rotated by 45◦.

The diffusion paths and energetic barriers between neighboring binding sites were calculated and are shown in Table 5.2. Pairs of neighboring sites consist of hill and hill sites, hill and trench sites, and trench and trench sites. Several observations can be made from Table 5.2. The thermal energies, (kBT)/2, in the experimental temperature range of growth, 500–700 K, are 0.06–0.09 eV. These are comparable to some of the lowest energy barriers, making the corresponding atomic hops likely. The lowest barrier is for diffusion from C8 to C7 and it is only 0.07 eV. In addition, it has been shown, Yildirim et al., [65] that temperature has very little effect on diffusion prefactors for adatom diffusion on metallic surfaces through the experimental temperature range of growth. The return path, from C7-C8 is 0.33 eV. These two barriers may be explained by realizing that only two primary bonds are broken going from

C7 to C8 resulting in an average 0.16 eV energy/bond. From these a bond breaking model can be made which may explain several of these energies when diﬀusion occurs

38 Table 5.2: NEB diffusion barrier results (eV) for neighboring minimum energy sites. The C4-C2 (C2-C4) path occurs over the T3 transition site, the C30-C300 (C300-C30) path occurs over the T30 transition site. Diffusion Pathway Diffusion Barrier (eV) C4-C2 (C2-C4) 0.38 C7-C8 0.33 (0.33 [16]) C8-C7 0.07 (0.15 [16]) C8-C9 0.46 (0.45 [16]) C9-C8 0.11 (0.15 [16]) C4-C9 0.11 C9-C4 0.33 C4-C8 0.09 C8-C4 0.66 C1-C2 (C5-C4) 0.64 C2-C1 (C4-C5) 0.27 C2-C30 (C4-C300) 0.41 C30-C2 (C300-C4) 0.55 C30-C300 (C300-C30) 0.39 C10-C7 (C5-C6) 0.64 C7-C10 (C6-C5) 1.1

39 Figure 5-3: Interaction of the relaxed Ga adatom and the As dimer row. Top (above) and side view (bottom) of the row of As dimers on (a) the GaAs(001)-c(4×4) surface (b) As dimers with a Ga adatom relaxed from 3 A˚ (site C2) and (c) As dimers with a Ga adatom relaxed from 0.5 A˚ (site C3’).

from trench to trench, C7-C8 (C8-C7) or hill to hill, C2-C4 (C4-C2), and C30-C300

(C300-C30). However this model fails to explain diffusion from trench to hill (and vice- versa) such as C10-C7 and C4-C8. This may be due to more complex, second nearest neighbor interactions not captured in a simple bond-counting model. From this we can construct a picture of how a Ga adatom may diffuse on the GaAs(001)-c(4×4) surface. Figure 5-4 shows the lowest energy (C7) site as the initial site for the diffusion process. A C7 site in a neighboring unit cell is taken as the final site. These are shown in Figs. 5-4(b) and 5-4(c) where diffusion occurs through the trench and over the As dimer. In all three paths, diffusion occurs through minimum sites found from

3 A˚ relaxations only. It would be interesting to contrast the electronic properties of sites of the Ga adatom which are at the same surface XY position but at a diﬀerent height from the surface. With this goal in mind, we investigated the local electronic density of states for the T3 and T30 sites, where the Ga adatom sits atop the surface

40 Figure 5-4: The symmetric path in (a) has a rate limiting barrier of 0.46 eV. The diffusion occurs solely in the trench. Diffusion between unit cells requires four jumps shown by red arrows between five minimum sites shown with blue circles. The sequence of jumps is C7-C8-C9-C8-C7. The Ga adatom can diffuse both in the trench and over the As dimers yielding the asymmetric path in (b) with the largest rate limiting barrier of 0.66 eV, requiring five jumps between six sites. The sequence of jumps is C7-C8- C9-C4-C8-C7, and (c) requiring six jumps between seven sites, has a rate limiting barrier of 0.46 eV. The sequence of jumps is C7-C8-C9-C4-C9-C8-C7.

mid-dimer. Figure 5-5 shows the LDOS for the T3 and T30 sites. We observe that the

LDOS for the As dimers shifts away from the Fermi energy for the T30 site compared to the T3 site. We then focus on the peak, with a shoulder, for the Ga LDOS as well as the As2 LDOS for site T3 as seen in panel (a) at 6.25 eV. This peak splits into two distinct peaks in panel (b) for the T30 site. We found this to be a signature of the splitting of the As dimer from 2.46 A˚ as seen in panels (c) and (d) versus 4.57 A˚ in panels (e) and (f). Further investigation by looking at the individual atomic LDOS, split according to diﬀerent orbitals, revealed that the peak at 6.25 eV is formed by s electrons of Ga and p electronic states from As. A closer bond between Ga and As for

T30 of 2.48 A,˚ as opposed to 2.66 A˚ for T3, enhances this sp hybridization causing the lowering of energy. The splitting of the LDOS peak is likely related to the broken As

41 Figure 5-5: LDOS and spatial arrangement of the T3 and T30 transition sites. LDOS for the Ga adatom and As mid-dimer (a) at the T3 transition site and (b) at the T30 transition site. Two side views each are shown: (c) and (d) for the T3 position, and (e) and (f) for the T30 position. The substrate As-As separation is 2.46 A˚ in (c) and (d) and 4.57 A˚ in (e) and (f).

dimer and resulting symmetry-breaking reminiscent of a Jahn-Teller distortion. This lowering of energy, observed in the splitting in the LDOS, may be an intermediate step to the possible growth process outlined by Kunsági-Máté et al. [66]

5.4 GaAs(001)-c(4×4)-heterodimer

5.4.1 Results

Structural diﬀerences between the GaAs(001)-c(4×4) and GaAs(001)-c(4×4)-hd reconstructions are shown in Fig. 5-6. A noticeable diﬀerence between the two reconstructions is the angle the dimers (heterodimers) make with the [0, 0, 1] surface normal. The Ga atoms in Ga-As heterodimers of the c(4×4)-hd reconstruction are

42 Figure 5-6: Top view of rows of As2 dimers on (a) GaAs(001)-c(4×4) and GaAs dimers on (b) GaAs(001)-c(4×4)-hd reconstruction. Side views of the c(4×4) and c(4×4)-hd reconstructions are shown in (c) and (d) respectively to illustrate the heterodimer tilt. The As2 dimers on the c(4×4) surface reconstruction are parallel to each other and lie perpendicular to the surface normal. The GaAs dimers on c(4×4)-hd surface tilt away from the middimer with the Ga atoms moving farther apart.

43 nearly incorporated completely into the surface resulting in an extreme tilt of the

heterodimers. The middle and edge Ga-As heterodimers lie at an angle of 106.5◦

◦ and 104.6 to the [0, 0, 1] surface normal respectively, whereas the As2 middle and edge dimers on c(4×4) reconstruction are perpendicular to the normal. The angle

between the bonds formed by the Ga atom, labeled Ph and the two As sub surface

atoms, labeled Vh and Uh is 122.8◦. The angle between the bonds formed by the same Ga atom with the As sub surface atom, Vh and the As heterodimer atom, Qh is

120.7◦. The angle between the bonds formed by the same Ga atom with the As sub surface atom, Uh and the As heterodimer atom, Qh is 115.2◦. The average of these

three angles is 119.5◦. This is reminiscent of graphite like planar bonding with sp2

hybridized electronic bonds, with bond angles of 120◦ each [67]. This contrasts with

the c(4×4) case, where an As atom replaces the Ga atom and the corresponding angles

are 110.4◦, 103.5◦ and 103.9◦ respectively. The average of these angles is 105.9◦. This

is closer to the ideal value (109.5◦) for sp3 hybridized bonded bulk GaAs structure.

Similar angles for both reconstructions at Ga, As atoms of mid dimers, labeled Rh

and Rc are given in Table 5.3.

Also calculated on the c(4×4)-hd reconstruction is a reduction in the bond lengths

of edge (PQ) and middle (RS) dimers of 3.5% and 3.9% respectively, compared to

those in the c(4×4) reconstruction. These quantitative diﬀerences are summarized in

Table 5.3. The MES for a Ga adatom on the c(4×4)-hd reconstruction are shown in

Fig. 5-7. We obtained a binding energy of 2.95 eV for the global minimum energy

site C2. The binding energies of all MES are shown in Table 5.4. The presence of

heterodimers on the surface forms hills along [1,¯ 1, 0] direction. The surface region

between these hills is referred as the trench. The C7 site, and equivalent C8 and C10

are located between the Ga-As heterodimers, on the hill. And all other MES appear

in the trench region. The computed structures show diﬀerences between the MES

pattern of c(4×4)-hd reconstruction compared to the c(4×4) reconstruction. Four

44 Table 5.3: Structural diﬀerences between GaAs(001)-c(4×4) and GaAs(001)- c(4×4)-hd surfaces are shown. Atom labels are referred from Fig. 5-6. The last two rows give the angle between the surface normal and the surface dimers. Parameter c(4×4) c(4×4)-hd |PQ| 2.56 A˚ 2.47 A˚ |RS| 2.53 A˚ 2.43 A˚ ◦ ◦ ∠VPU 110.4 122.8 ◦ ◦ ∠VPU 103.5 120.7 ◦ ◦ ∠VPQ 103.9 115.2 ◦ ◦ ∠UPQ 103.3 114.3 ◦ ◦ ∠URW 104.2 121.8 ◦ ◦ ∠WRS 104.2 121.8 [0, 0, 1](edge dimers) 90◦ 106.5◦ [0, 0, 1](mid dimers) 90◦ 104.6◦

Figure 5-7: Minimum energy sites (MES), labeled Cn where n varies from 1 to 10, obtained by relaxing a Ga adatom from 3 A˚ above the GaAs(001)-c(4×4)-hd surface reconstruction.

45 Table 5.4: Binding energies of Ga adatom relaxed from 3 A˚ above the GaAs(001)-c(4×4)-hd surface. The binding energies are given relative to the global minimum energy site C2 which has a binding energy of 2.95 eV. Site Energy (eV) C1 0.24 C2 0 C3 0.37 C4 (C9) 0.61 C5 0.63 C6 0.56 C7 (C8, C10) 0.78 diﬀerent MES are present in trench along the [1,¯ 1, 0] direction on the c(4×4)-hd reconstruction compared to only two diﬀerent MES along the same direction on the c(4×4) reconstruction. Furthermore, the three MES present in trench along the [1,

1, 0] direction on the c(4×4)-hd reconstruction have different energies. In contrast, two of the three MES along the same direction on the c(4×4) reconstruction have the same energy. This difference can be explained by considering the symmetry of the two surface reconstructions. The c(4×4) surface reconstruction contains a mirror plane that bisects the As2 dimer bonds along the [1,¯ 1, 0] direction. This mirror plane passes through the global minimum site so the MES on either side of this site have the same energy. The c(4×4)-hd reconstruction does not possess this symmetry. The symmetry is broken by the presence of the Ga-As heterodimer. As a result, the MES labeled C1 and C3, on either side of the global minimum C2, face different type of atoms. MES C1 and C3 are now adjacent to Ga and As atoms respectively from the neighboring rows of heterodimers. Diffusion barriers were calculated using the

NEB method, as explained in section A.2. These barriers for both regular and the exchange mechanism are given in Table 5.5.

The diﬀerences in diﬀusion barriers between neighboring MES (e.g. C1-C2 does

46 Table 5.5: Diffusion barriers for Ga adatom on GaAs(001)-c(4×4)-hd surface. Diffusion barriers for exchange mechanism are labeled with an E. Diffusion Pathway Diffusion Barrier (eV) C1-C2 0.04 C2-C1 0.28 C2-C3 0.42 C3-C2 0.06 C3-C4 0.30 C4-C3 0.06 C4-C5 0.05 C5-C4 0.02 C5-C6 0.02 C6-C5 0.09 C6-C1 0.12 C1-C6 0.45 C1-C8 0.76 C1-C8-E 1.19 C8-C1 0.21 C8-C1-E 0.64 C5-C8 0.51 C5-C8-E 0.80 C8-C5 0.36 C8-C5-E 0.65 C7-C10 (C10-C7) 0.35 C8-C9 0.21 C9-C8 0.37 C8-C2 0.43 C8-C2-E 0.81 C2-C8 1.21 C2-C8-E 1.60

47 Figure 5-8: Two non-collinear diﬀusion paths of a Ga adatom on the GaAs(001)c(4×4)-hd surface. Path (a) occurs between MES C2- C3-C4-C5-C6-C1-C2. The limiting step in this path is the 0.42 eV barrier between C2 and C3 along the [0 1¯ 0] direction and 0.45 eV between C1 and C6 along [0 1 0]. Path (b) occurs between MES C2-C3-C4-C8-C1-C2. The limiting step in this path is the 0.42 eV barrier between C2 and C3 along the [1¯ 1¯ 0] direction and 0.76 eV between C1 and C8 along [1 1 0].

not equal C2-C1) is explained by considering the relative binding energies of the initial and final MES, given in Table 5.4. The difference in energy between the transition point and initial MES is taken as the diffusion barrier for the first path, the difference in energy between the transition point and final MES represents the diffusion barrier for the return path. From the computed diffusion barriers we show two non collinear possible diffusion paths, shown in Fig. 5-8, for a Ga adatom diffusing from the global minimum (C2) to another C2 site in a neighboring unit cell.

The ﬁrst path, (a), shown in Fig. 5-8 occurs only by hops from MES in the trench. For diﬀusion along the [0 1¯ 0] direction, this occurs between MES C2-C3-

C4-C5-C6-C1-C2. For diﬀusion in the opposite direction, along the [0 1 0] direction,

48 Figure 5-9: In the GaAs(001)-c(4×4)-hd reconstruction (a) the Ga atom of the heterodimer, Ph (from Fig. 5-6) bonds with the two sub- surface As atoms, Vh and Uh, and the As atom in the heterodimer, Qh. The structure and angles are similar to those of a planar conﬁguration, as occurs in sp2 hybridization. In the GaAs(001)-c(4×4) reconstruction (b) an As atom replaces the Ga atom and the same bonds form a structure which appears to be a tetrahedral conﬁguration, as occurs in a sp3 hybridized bulk material.

this occurs in the reverse order. The limiting step in the forward path is the 0.42 eV barrier between C2-C3 along the [0 1¯ 0] direction and 0.45 eV between C1-C6 along [0 1 0]. The second diﬀusing path, (b), occurs both in the trench and over the dimer rows. For diﬀusion along the [1¯ 1¯ 0] direction this occurs between MES C2-

C3-C4-C8-C1-C2, for diffusion along [1 1 0], the MES order is reversed. The limiting steps in these paths are the 0.42 eV barrier between C2-C3 along [1¯ 1¯ 0] and 0.76 eV between C1-C8 along [1 1 0]. We report a higher diffusion barrier for the exchange mechanism compared to the regular, top hopping mechanism between MES. This difference may be explained by the bonding configuration of Ga heterodimer atom involved in the exchange mechanism. The Ga atom in the heterodimer, shown in Fig.

5-9(a), bonds with the two As sub surface atoms and the As heterodimer atom. The resulting conﬁguration is close to being planar, similar to that in sp2 hybridization.

The additional energy required for the exchange mechanism, may be explained as the additional energy required by the incoming Ga adatom to break the favorable sp2 like bonding conﬁguration before replacing the Ga heterodimer atom. We explored

49 Figure 5-10: Local density of states for the Ga atom of the surface edge heterodimer (GaAs) at the highest energy point during top hopping and exchange diﬀusion of a Ga adatom between minimum energy sites C8 and C5. We observe a shift in the density of states to the right (in the range 2.5 eV to 1 eV) for the exchange mechanism correlated to a higher diﬀusion barrier compared to that for top hopping.

this feature further by examining the electronic properties of relevant atoms near the exchange process. We computed the LDOS, Fig. 5-10, for the exchange and top hopping mechanisms for diﬀusion between MES C8 and C5. We plotted the LDOS for the combined s and p orbitals of the Ga atom in heterodimer at the transition site

(highest energy point) for both top hopping and exchange diffusion. For exchange diffusion, the Ga atom in the heterodimer has fewer states available in the lower energy (-10.5 eV to -12.5 eV) region and we observe a shift to the right in overall states compared to top hopping, especially near the Fermi energy. The additional states at lower energies for top hopping correspond to the low energy barrier for diffusion.

50 Chapter 6

Bulk diﬀusion in Cadmium

Telluride

6.1 Computational Method

All computations employed ab initio total energy calculations within the local density approximation to DFT [46,51] using VASP [55–58]. Core electrons are treated by ultrasoft Vanderbilt type pseudopotentials [59] as supplied by Kresse et al. [60] using the Ceperly and Alder exchange-correlation functional. A 275 eV energy cutoff was used in expanding the single-particle wave functions in the plane-wave basis. Tests using higher plane-wave energy cutoff indicated that a numerical convergence better than ±1 meV was achieved. The lattice constant was varied and fit to a parabolic equation as a function of total energy to obtain the absolute minimum in total energy.

The calculated lattice constant of 6.43 A˚ is within 0.5% of the experimental lattice constant [38] of 6.46 A.˚ The CdTe bulk structure consists of a two atom primitive cell zinc-blende structure (space group F43m number 216) with the Cd and Te atoms

at Wyckoﬀ positions 4(a) and 4(c) respectively. A cubic unit cell of this zinc-blende

structure consists of 8 atoms. Diﬀusion barrier calculations were computed in a 216

atom supercell, a 3×3×3 repetition of the cubic unit cell. The larger supercell size

51 more closely models an isolated defect by reducing the long-range interactions between defects in neighboring supercells. A Gamma point MonkhorstPack [61] generated k- point grid was used for the Brillouin-zone integrations in all calculations. Tests using larger k-point sampling indicated that a numerical convergence better than ±3 meV was achieved. All atoms were allowed to fully relax to ﬁnd the minimum energy for each conﬁguration. Relaxation was concluded when a force tolerance of 0.01 eV/A˚ was reached for each atom. The calculations for the LDOS were performed with the

Gaussian smearing scheme. Diﬀusion barrier calculations were performed using the

NEB method [64].

6.2 Modeling Approach

The simplest mode of diffusion in zinc-blende CdTe is interstitial diffusion down the spacious [1 1 0] channel. Interstitial diffusion down this channel consists of the diffusing adatom passing between two alternating high symmetry sites, Wyckoff positions 4(b) and 4(d). The 4(b) site is the location of the diffusing adatom when it is tetrahedrally coordinated by Te atoms, thus labeled TT e, and the 4(d) site is the location of the diffusing adatom when it is tetrahedrally coordinated by Cd atoms, thus labeled TCd. To model the interstitial diffusion process, the interstitial adatom was fixed at the TT e position and the surrounding atoms were allowed to relax unconstrained until convergence of forces occurred. A similar run was completed for the adatom at the TCd site. These aforementioned relaxed positions, TT e and TCd, were used as the minimum energy sites for the NEB method, which was used to relax a number of linearly interpolated images along the path between these two positions. If none of the converged intermediate images had an energy lower than either of the initial minimums, the initial minimums were taken as the global minimum energy sites

(GMEs). These GMEs were used to generate a second GME, displaced by lattice

52 vector a[0.5 0.5 0] and allowed to relax until convergence, where a is the computed

lattice constant 6.43 A.˚ Then a second NEB run was used to determine the diﬀusion barrier(s) between these two GMEs. If an image with energy lower than the initial and ﬁnal positions existed between the TT e and TCd positions, it was considered a new presumed GME. With this new presumed GME an unconstrained relaxation was

performed until convergence of forces to conﬁrm its existence. Once conﬁrmed, the

new converged GME was used to generate a second GME, similar to the method

above, down the [1 1 0] channel. Then a second NEB run was used to determine the

diﬀusion barrier(s) between these two new GMEs.

6.3 Anion Diﬀusion in CdTe; Cl, Te, O, S, P and

6.3.1 Results

As mentioned, diﬀusion down the spacious [1 1 0] channel and related symmetry

equivalent directions consists of the diﬀusing adatom passing between two alternat-

ing high symmetry Wyckoﬀ positions 4(b) and 4(d). The 4(b) site is tetrahedrally

coordinated by Te atoms, thus labeled TT e, and the 4(d) site is tetrahedrally coordi-

nated by Cd atoms, thus labeled TCd. The global minimum energy site (GME) for an

interstitial Te atom was found to be slightly oﬀ the TT e site and for O and Cl it was

at the TCd site. The GME was found to be oﬀ the high symmetry site for interstitials of P, S and Sb. For vacancy diﬀusion the GME was found where the bulk Te atom

was removed from a 4(c) site and the surrounding atoms were allowed to relax until

convergence. Using these GME sites, as the initial and ﬁnal positions, NEB runs

were performed for ﬁnding the diﬀusion energy barriers. An odd number of images

were used for symmetric barriers and a distance of 0.5 A˚ was maintained between

53 images to confirm diffusion barriers. As the interstitial adatom diffuses down the [1 1

0] channel we see that different species occupy different extrema positions and atomic configurations as shown in Fig. 6-1.

Table 6.1: Extrema positions given in direct coordinates of a fcc unit cell a1 = a[0, 0.5, 0.5], a2 = a[0.5, 0, 0.5], a3= a[0.5, 0.5, 0], R=c1a1+c2a2+c3a3. For example the S minimum position is given by R[S]=0.45a1+0.45a2+0.91a3.

Global Min Secondary Max Global Max Element c1 c2 c3 c1 c2 c3 c1 c2 c3 0.75 0.75 0.75 Cl 0.75 0.75 1.75 0.49 0.57 1.45 0.79 0.79 1.21 Te 0.79 0.79 2.21 0.75 0.75 1.75 0.75 0.75 0.75 O 0.75 0.75 1.75 0.50 0.50 1.50 0.45 0.45 0.91 S 0.45 0.45 1.20 0.46 0.46 1.04 0.43 0.43 1.58 0.45 0.45 1.91 0.61 0.61 0.61 P 0.61 0.61 1.17 0.43 0.43 1.07 0.70 0.70 1.30 0.61 0.61 1.61 0.60 0.60 0.60 Sb 0.60 0.60 1.21 0.72 0.72 0.78 0.57 0.57 1.43 0.60 0.60 1.60

These diﬀerent minima and maxima positions include the GME and both global maximum energy (GMax) and secondary maximum energy (SMax) positions. Figure

6-1 shows an example of each of the three different bonding scenarios. One which involves both 4(b) and 4(d) sites, for O and Cl, where only Cl is shown. The second scenario involves only one of the high symmetry sites, Te only, and one which involves neither of the high symmetry sites, S, P and Sb, where only S is shown. The extrema positions for the Cl, Te, O, S, P and Sb interstitials are given in Table 6.1. Average bond lengths and angles for the configurations in Fig. 6-1 are given in Table 6.2 and the diffusion barriers for all diffusing species are given in Table 6.3. For a Te

54 Figure 6-1: Structural properties and bonding configuration of interstitial positions corresponding to various minimum and maximum energy sites. The atom labeled x is the diffusing interstitial atom, the surrounding atoms are cadmium (light yellow) and tellurium (dark green). The Te interstitial minimum energy position (a) occupies a square pyramidal site with the nearest neighbor Cd atoms and a seesaw configuration with the second shell Te atoms while the Te interstitial maximum energy position (b) is found to be at a tetrahedrally coordinated position with neighboring Cd atoms and an octahedral site with neighboring Te atoms. The minimum energy position for Cl (c) shares the same coordination as the Te interstitial at the maximum, shown in (b). The maximum position for Cl (d) reverses the coordination observed for the minimum (c). Unlike Te and Cl, the S interstitial atom encounters a second minimum energy position that is not related to the first minimum energy position by a lattice vector and encounters two in-equivalent maximums. The maximum position (f) is a flat sp2 like configuration consisting of two Cd and one Te atom. This is similar to the minimum energy position (e), on either side of this maximum, that breaks this symmetry. The S interstitial global maximum (g) occupies a distorted Te tetrahedral position and a distorted Cd trigonal bipyramidal position.

55 Table 6.2: Average bond length and angle from interstitial defect to the ﬁrst two shells of neighbors which are either only Te atoms or only Cd atoms.

Nearest- Avg. Bond Avg. Bond Element Position Figure 7-1 neighbor Length (A)˚ Angle (◦) Te - - Global Min a Cd - - Te Te 3.39 89.99 Global Max b Cd 2.9 109.58 Te 3.37 89.99 Global Min c Cd 2.7 109.47 Cl Te 3.11 109.27 Global Max d Cd 3.07 89.89 Te 2.38 - Global Min e Cd 2.51 137.2 Te 2.36 - Secondary Max f Cd 2.52 153.3 S Te 3.08 - Global Max g Cd 3.2 - interstitial, the GME, Fig. 6-1(a), appears as a seesaw conﬁguration with the nearest neighbor Te atoms and a square pyramidal shape formed with the closest Cd atoms.

The seesaw conﬁguration and square pyramidal can be viewed as a distorted TT e site where the Te interstitial is displaced from the symmetric 4(b) position. This distortion makes interpreting values of average bond lengths and angles ambiguous.

A similar situation arises for S. Hence some entries in Table 6.2 are not listed. The

Te interstitial GMax, Fig. 6-1(b), occurs at the symmetric TCd position. It appears as a sp3 hybridized tetrahedron of four Cd nearest neighbor atoms with the Te atom in its center as seen from the average Cd-Te-Cd bond angle of 109.58◦. Similarly it is octahedrally coordinated, sp3d2 hybridized, with the next shell of Te atoms with virtually perpendicular bond angles of 89.99◦. Compared to the bulk bond lengths, the average Te-Cd bond length is 4.32% larger and the Te-Te second shell length is

25.5% smaller suggesting considerable expenditure of strain energy and hence a large energy barrier of 1.37 eV as seen in Table 6.3. The computed barrier of 1.37 eV

56 is in excellent agreement with the experimental value of 1.40±0.02 eV reported by

Borsenberger and Stevenson [68] and Woodbury and Hall [69] for a Te interstitial.

The Cl and O interstitial occupy similar GME and GMax positions. The GME

Table 6.3: Diﬀusion barriers (eV) for diﬀerent point defects.

Secondary Global Element Max (eV) Max (eV) Cl - 0.68 (0.63 [70]) O - 1.51 S 0.07 0.65 (0.64 [36]) TeV - 1.42 TeI - 1.37 (1.38 [68], [69]-1.42 [68], [69]) P 0.21 0.68 Sb 0.34 0.49

position for both Cl and O interstitial adatoms occupy the symmetric TCd position, shown for Cl in Fig. 6-1(c). The Cl GME makes an sp3 like average bond length

of 109.47◦ with the four Cd nearest neighbor atoms and an sp3d2 like average bond length of 89.99◦ with the next shell of Te atoms. Similarly, Cl and O both occupy the same TT e GMax position, shown for Cl in Fig. 6-1(d), where the coordination reverses and the Cl interstitial forms an sp3 like average bond angle of 109.27◦ with the four Te nearest neighbor atoms and an sp3d2 like average bond length of 89.89◦ with the next shell of Cd atoms. The Cl and O atoms have a nearest neighbor shell of Cd atoms in the GME compared to the situation in their GMax position where the

ﬁrst shell of atoms is of Te. This must lead to better charge transfer in the former case than the latter thus raising the energy for the GMax site. As the Cl interstitial moves from the GME to the GMax, the Cl-Te bond length decreases 7.72% from 3.37

A˚ to 3.11 A˚ while the Cl-Cd bond length increases 13.7% from 2.70 A˚ to 3.07 A.˚

However, even though the Cl and O interstitial occupy equally coordinated GME and

GMax positions, as the O interstitial moves from the GME to the GMax position, the O-Te bond length decreases 24.9% from 3.42 A˚ to 2.57 A˚ while the O-Cd bond 57 length signiﬁcantly increases 49.1% from 2.22 A˚ to 3.31 A.˚ This substantial disparity

in change of bond length can explain the higher diﬀusion barrier for O of 1.51 eV

to that of Cl of 0.68 eV. The 0.68 eV barrier reported here agrees well with one of

the four values corresponding to diﬀusion proﬁle D04 of 0.63±0.10 eV reported by

Jones et al.. [70]. The D04 profile corresponds to the diffusion profile observed in the greatest CdTe bulk depth where it can be expected to be the most similar the bulk

CdTe modeled in this work.

Unlike the Te, Cl and O interstitials, the GMEs for S, P and Sb are located away from the high symmetry 4(b) and 4(d) sites. The S GMEs, Fig. 6-1(e), is extremely close and on either side of the SMax position, Fig. 6-1(f), and is similar to the sp2 like

SMax conﬁguration, as evidenced by the change in the S-Te and S-Cd bond lengths

and bond angles. As the S interstitial moves from the GME to the SMax position,

the S-Te bond length decreases only 0.84% from 2.38 A˚ to 2.36 A˚ while the S-Cd

bond length increases only 0.40% from 2.51 A˚ to 2.52 A˚ and the Cd-S-Cd bond angle

changes from 137.2◦ to 153.3◦, since there is only one Te atom in this conﬁguration

there is no listing for this value in Table 6.2. This is not surprising as the energy

diﬀerence between these two sites is merely 0.07 eV as seen in Table 6.3. However,

for the S GMEs, the S interstitial no longer lies in the plane with the two Cd and one

Te atom. The S interstitial GMax, Fig. 6-1(g), is located at a distorted TT e position and a distorted trigonal bipyramidal position with the nearest neighbor Cd atoms. At

the S GMax position, although the S-Te bond length increases 29.4% from 2.38 A˚ to

3.08 A˚ from the GME and the S-Cd bond length increases 27.5% from 2.51 A˚ to 3.20

A,˚ the Te coordination changes from 1 to 4 and the Cd coordination changes from 2

to 5. Despite this betterment in coordination the strain energy of the increased bond

lengths causes a signiﬁcant rise in the energy of the GMax site of 0.65 eV. The 0.65

eV barrier for S agrees well with the derived experimental value of 0.64±0.01 eV for

the diﬀusion barrier found deep in bulk CdTe by Lane et al. [36]. The increase in

58 coordination and diﬀerence in symmetry of diﬀusion paths between the other group

VI elements explains why the 0.65 eV barrier for S is considerably lower than the

1.37 eV and 1.51 eV barriers for Te and O respectively. The group V elements, P and

Sb, both display an asymmetric diﬀusion proﬁle as mentioned. As the P interstitial

moves from the GME to the GMax position, the P-Te bond length increases 17.8%

from 2.69 A˚ to 3.17 A˚ while the P-Cd bond length increases 23.1% from 2.86 A˚ to 3.52

A˚ resulting in a rate-limiting global diﬀusion barrier of 0.68 eV. As the Sb interstitial

moves from the GME to the GMax position, the Sb-Te bond length increases 5.78%

from 2.90 A˚ to 3.07 A˚ while the Sb-Cd bond length increases 12.5% from 2.95 A˚ to

3.31 A˚ resulting in a global diﬀusion barrier of 0.49 eV. The larger change in bond length for P again corresponds to a larger diﬀusion barrier compared to that of Sb.

Also, as the Sb interstitial moves from the GME to the SMax position, the Sb-Te bond length decreases 12.5% from 2.90 A˚ to 3.26 A˚ while the Sb-Cd bond length increases 2.37% from 2.95 A˚ to 3.02 A˚ resulting in a barrier of 0.34 eV. We can see that this similar change in bond length for Sb between the GME and the SMax and

GMax position results in a similar diﬀusion barrier at both positions.

The GME for each of the interstitials Cl, O and Te is located at or around either one of the highly symmetric 4(b) or 4(d) sites with only one GMax occurring at the other complementary high symmetry site either 4(d) or 4(b). It leads to a symmetric diﬀusion path down the [1 1 0] channel for these atoms. For P, Sb and S, the GME is not located at either of the 4(b) or 4(d) sites and as a result their diﬀusion path is asymmetric. Furthermore, they have an additional GME, a SMax and a GMax.

The ﬁrst and second GME are related by a lattice vector and hence are equivalent sites, the additional GME is not related to the ﬁrst or second by a lattice vector and as such P, Sb and S posses two distinct GME where as Te, O and Cl posses only one distinct GME. This is because the GME for P, Sb and S do not occur at either of the 4(d) or 4(b) sites and symmetry of the zinc-blende structure along the

59 [1 1 0] channel requires this additional GME. The additional GMEs for P, Sb and S and the diﬀerent symmetries of the diﬀusion paths are clearly seen in Fig. 6-2 for all the interstitials. We observed that both Cl and O pass directly between the two

Figure 6-2: NEB graphs of diffusion barrier profiles as a function of the NEB step positions, (a) the symmetric barriers (i.e. Cl, O and Te) and (b) the asymmetric barriers of P, Sb and S observed in CdTe. The first and last NEB positions on the X axis are displaced through a lattice vector a[0.5 0.5 0] down the [1 1 0] channel, where a is the computed lattice constant 6.43 A,˚ and hence have identical energies.

alternating 4(b) and 4(d) sites resulting in a diﬀusion path down the [1 1 0] channel that involves alternating linear paths down along the [1 1 1]¯ direction then back up along the [1 1 1] direction. This sawtooth diﬀusion character for O is shown in Fig.

6-3. In contrast, the Te interstitial GMax passes through the 4(d) position but the global minimum avoids the 4(b) position resulting in a more linear path down the [1

1 0] channel, also shown in Fig. 6-3.

The z-axis deviation in linearity while diﬀusing down the [1 1 0] channel is shown for all interstitials in Fig. 6-4. As we see from Fig. 6-4, the largest deviation in

60 Figure 6-3: Diﬀerences in diﬀusing paths for (a), (b) O and (c), (d) Te interstitial atoms down the [1 1 0] channel in CdTe. Atoms Cl, P, and Sb have paths similar to O while S has a path similar to Te.

the z-axis is for the symmetric diﬀusing interstitials O, deviating approximately 0.25

A˚ from the O GME z-axis coordinate, followed by Cl that deviates around 0.20

A˚ from the Cl GME. This is expected as the O and Cl interstitials travel in the

sawtooth pattern between the 4(b) and 4(d) sites. We see the previously mentioned

symmetric and linear diﬀusion character of Te interstitial deviates less than 0.05 A˚

from the Te GME z-axis coordinate traveling in an almost straight line path down

the [1 1 0] channel. The S interstitial experiences the least deviation from the z-

axis coordinate of its GME. Although the diﬀusion proﬁle of S is asymmetric, the S

interstitial deviates less than 0.02 A˚ from any GME when passing through the SMax

and GMax position. The asymmetric diffusion profiles of P and Sb are quite different.

The P and Sb interstitial both experience the largest deviation, greater than 0.15 A˚

and 0.10 A˚ respectively, from their GME z-axis coordinates at the SMax position while they experience their smallest z-axis deviation, around 0.10 A˚ and less than

0.05 A˚ respectively, at the GMax position. In addition to their similarity in z-axis

61 Figure 6-4: Z-axis deviation from the minimum of the interstitial atoms while diffusing between global minimums in adjacent unit cells down the [1 1 0] channel. The ordinate axis is defined to be zero at Z- axis position of the diffusing atom in its global minimum energy site.

deviation, both group V elements P and Sb have similar diﬀusion barriers. The SMax energy barriers for P and Sb are 0.21 eV and 0.34 eV with GMax energy barriers of

0.68 eV and 0.49 eV respectively.

To further explore the relative importance of this z-axis deviation during the diffusion path we focused attention on the symmetric diffusing O and Te interstitials which show the highest and lowest z-axis deviation in Fig. 6-4. We performed two additional runs each for a Te and O interstitial to assess the effect of z-axis deviation on local strain energy and relaxation during diffusion. In both runs the interstitial was kept fixed with its x and y coordinates obtained from the fully converged NEB run and all other atoms in the supercell were fixed (i.e. unrelaxed) at their bulk equilibrium positions. The z-coordinate of the interstitial was chosen to be zero in the first, ’straight line’ run, while it was chosen to be the same as the one in the fully converged NEB run in the second, ’bulk’ run. For a Te interstitial, in Fig. 6-5(a), we see that the maximum strain energy occurs at the GME. This can be explained

62 Figure 6-5: Effect of local strain energy and relaxation around diffusing interstitial Te (a) and O (b) atoms. The barrier energy represents the fully relaxed diffusion barrier energy as a function of the NEB position along the diffusing path; the bulk energy is the energy of the interstitial located at the relaxed NEB position in an un- relaxed bulk cell, i.e. the remaining atoms are left unrelaxed in their bulk positions; the straight line energy is the energy of the NEB positions located at the same z-axis position as the minimum energy position in an un-relaxed bulk cell.

by the extreme distortion of the neighboring Te and Cd atoms of the TT e, shown in Fig. 6-1(a). This maximum strain energy occurs at the GME for both the ”straight line” and ”bulk” energy. As the Te interstitial moves towards the GMax position at the TCd position we note there is little difference between ”straight line” and ”bulk” energy. This can be explained by the almost straight line diffusion path of the Te interstitial with little z-axis deviation. At the GMax, the Te interstitial experiences the least ”bulk” and ”straight line” energy of around 3 eV. For the O interstitial the situation is reversed, the maximum strain energy occurs at the GMax TT e position. We also note that there is a substantial difference between the ”bulk” and ”straight line” energies at the maximum position for the O interstitial, around 2 eV, compared to the difference for the Te interstitial case where the difference is negligible. This can again be explained by the character of the diffusion path. For O, the diffusion path is a seesaw pattern that experiences the largest z-axis deviation and hence largest deviation from the straight line path. We also note that the ”bulk” energy for the O

63 interstitial at the GMax is similar to the barrier energy at the same position, which can be explained because the O interstitial occupies a highly symmetric, undistorted

TT e position, unlike the Te interstitial GME. We computed the electronic LDOS of a variety of configurations of all diffusing species studied. We highlight a few observations from these computations here. An example of a symmetric diffusion barrier is illustrated in Fig. 6-6. It shows the LDOS

Figure 6-6: Panel (a) shows the minimum energy position marked ’A’ and global maximum position marked ’B’ for a Cl interstitial. Filled circles show nearest neighbor Te atoms. Position ’A’ is octahedrally coordinated to Te and ’B’ is tetrahedrally coordinated to Te. Nearest Cd atoms not shown. Panel (b) through (d) show LDOS of the p orbitals of Cl interstitial and its nearest neighbor Te atoms. LDOS for Cl atom in position ’A’ are in panels (b) and (c) while for position ’B’ are in panels (d) and (e). LDOS for Te atoms 1, 2 and 3 are in panels (b) and (d) and for atoms 4, 5 and 6, are in panels (c) and (e). At position ’A’, Cl forms similar bonds with all six Te atoms. This similarity is observed in panels (b) and (c). As the Cl interstitial moves from ’A’ to ’B’ there is little change in the LDOS in (d), of Te atoms 1, 2 and 3, while the Cl atom p-states are shifted toward the Fermi energy. This is accompanied by a similar shift for the p-states of Te atoms 4, 5 and 6 seen in (e) that remain bonded to the Cl interstitial throughout the diﬀusion process.

for the diﬀusion of a Cl interstitial as it moves from the GME at the TCd site to the

GMax at the TT e position. The Cl interstitial at the GME position, the white atom

64 marked ’A’ in Fig. 6-6(a), is octahedrally coordinated to Te. The global maximum position, marked ’B’ in Fig. 6-6(a), is tetrahedrally coordinated to Te. The black

ﬁlled circles show nearest neighbor Te atoms and the nearest Cd atoms are not shown.

At the GME ’A’ position the Cl interstitial makes six symmetric bonds with Te atoms

1, 2 and 3 and atoms 4, 5 and 6. As the Cl interstitial moves from the GME ’A’ position to the GMax ’B’ position we clearly see that the three bonds with Te atoms

1, 2 and 3 must be broken while the bonds with Te atoms 4, 5 and 6 remain intact.

Panels (b) and (c) in Fig. 6-6 show the LDOS for the p orbitals of the Cl interstitial and the p orbitals for its nearest neighbor Te atoms 1, 2 and 3 in (b) and for Te atoms 4, 5 and 6 in (c). We see that the substantial p-p bonding peak, around -0.3 eV, is similar in panels (b) and (c) which is expected due to the symmetric nature of the bonding at the 4(d) position. After the Cl interstitial has moved to the GMax position, we see that the p-p bonding between (e) the Cl interstitial and Te atoms 4,

5 and 6 remain and p states of both Cl and the Te atoms have been shifted toward the Fermi energy but the p-p bonding between the Cl interstitial and Te atoms 1,

2 and 3 has been broken and the p states of Te atoms 1, 2 and 3 remain virtually unchanged.

An example of an asymmetric diﬀusion barrier is illustrated in Fig. 6-7. It shows the LDOS for an interstitial P atom at the GME and GMax position. We see that the p-states for the GME are shifted to deeper binding energies than the GMax position suggesting a more energetically favorable position at the GME. When the

P interstitial is at the GMax position we can see that the p-states have split into three distinct peaks, two of which are shifted toward the Fermi to shallower binding energies. In addition to the splitting of the p-states at the GMax position we also observe that s-states and hybridized d-states contribute to the bonding character where as at the GME position only the p-states contribute. A similar situation arises for another asymmetric diﬀusion barrier involving Sb, as shown in Fig. 6-8. In the

65 Figure 6-7: LDOS by atomic orbital type (s, p, d), for an interstitial P atom, at the minimum and global maximum positions. Fermi energy is set to zero in each case. Observe the split from one peak, in the p-states, for the minimum position to three peaks for the global minimum. The overall p-state density is shifted to deeper binding energies for the minimum site. Also observe the contribution to defect states from s, p and hybridized d-states for the global maximum position and only from p-states for the minimum.

ﬁgure we see the LDOS for an interstitial Sb atom at the GME, SMax and GMax positions. Similar to the case for the P interstitial in Fig. 6-7, we see again that the p-states for the GME are shifted to deeper binding energies while the p-states for the GMax are shifted toward the Fermi to shallower binding energies. The less energetically favorable p-state shift toward the Fermi is also observed for the Sb atom at the SMax position. Similar to the case for the P interstitial in Fig. 6-7, we see again that hybridized d-states contribute to the bonding at the GMax position while only p-states contribute at the GME while we do not see any contribution from the s- or d-states for the Sb interstitial at the SMax position.

66 Figure 6-8: LDOS of atomic orbital types, p and d, for an interstitial Sb atom at the minimum, secondary maximum and global maximum positions. Fermi energy set to zero in each case. Observe the shift to deeper binding energies for the minimum energy site and the contribution to defect states at the Fermi level from d-states for the global maximum position only.

6.4 Cation Diﬀusion in CdTe; Cd, Cu, Ag, Au and

6.4.1 Results

TT e site and for Mo it was at TCd. The GME was found to be oﬀ the high symmetry site for interstitials of Cu and Au. For vacancy diﬀusion the GME was found where the bulk Cd atom was removed from its bulk 4(a) site and the surrounding atoms

67 were allowed to relax until convergence. Using these GME sites, as the initial and

final positions, NEB runs were performed for finding the diffusion energy barriers.

An odd number of images were used for symmetric barriers and a distance of 0.5 A˚ was maintained between images to conﬁrm diﬀusion barriers.

As the interstitial adatom diffuses down the [1 1 0] channel we see that different species occupy different extrema positions and atomic configurations. As Cd, Ag and

Mo interstitials diﬀuses between adjacent unit cells down the [1 1 0] channel they encounter a GME and a secondary minimum energy position (SME) and a GMax.

While Cu and Au interstitials diﬀuses between adjacent unit cells they also encounter a GME and a GMax. However, unlike Cd, Ag and Mo, diﬀusion paths for Cu and

Au interstitial possess a SMax and do not possess a SME. These diﬀerent extrema positions and atomic conﬁgurations are shown in Fig. 6-9.

68 Figure 6-9: Structural properties and bonding configuration of interstitial positions corresponding to various minimum and maximum energy sites. The atom labeled x is the diffusing interstitial atom; the surrounding atoms are cadmium (light yellow) and tellurium (dark green). The Cd interstitial global minimum energy position (a) occupies a tetrahedrally coordinated position with neighboring Te atoms and an octahedral site with neighboring Cd atoms. The Cd secondary minimum energy position (b) reverses the coordination observed for the global minimum (a). The Cd global maximum position (c), located between the global minimum and secondary minimum position, forms an sp2 like configuration with both neighboring Cd and Te atoms. Individual Cd and Te sp2 like bonding is also shown in panel (c) for clarity. The Cu interstitial extrema positions are similar to those of Cd but occur at different configurations. The Cu interstitial global minimum (d) occurs at a sp2 like configuration, similar to the global maximum for Cd (c). The Cu secondary maximum configuration is shown in (e) and the global maximum configuration is shown in (f). The Ag interstitial global minimum (g) occupies a tetrahedrally coordinated position with neighboring Te atoms and an octahedral site with neighboring Cd atoms. The Ag secondary minimum (i) reverses the coordination observed for the global minimum (g). The Ag global maximum (h) occurs on either side of the secondary minimum (i) where the symmetry of the secondary minimum position is broken.

69 This ﬁgure shows an example of three diﬀerent bonding scenarios. One scenario involves both 4(b) and 4(d) sites as SME and GME, respectively, and the sp2 like

Cd and Te coordinated site as the GMax, for Cd. The second bonding scenario, for

Ag, where both 4(b) and 4(d) sites reverse roles as the GME and SME, respectively.

The third one involves neither of the high symmetry sites but includes the sp2 like

Cd and Te coordinated site as the GME, for Cu and Au, where only Cu is shown.

The extrema positions for the Cd, Cu, Au, Ag and Mo interstitials are given in Table

6.4. For the configurations in Fig. 6-9, averages for both, bond lengths and angles, are given in Table 6.5 and the diffusion barriers for all diffusing species are given in Table 6.6. The broken symmetry of the Ag GMax position makes interpreting average bond lengths and angles for this position ambiguous and therefore these are omitted in Table 6.5. The Cd interstitial GME position, Fig. 6-9(a), occupies the TT e tetrahedrally coordinated position making an average angle of 109.5◦ with its nearest neighbor Te atoms. The Cd GME also occupies a sp3d2 octahedral site with the closest

Cd atoms making an average angle of 90◦. The Cd SME, Fig. 6-9(b), reverses the coordination observed for the GME in (a) occupying the bulk like TCd position with nearest neighbor Cd atoms and an octahedral site with the closest Te atoms making identical angle of 109.5◦ and 90.0◦ respectively. The Cd SME bonding is comparable to the bulk Cd bonding with a slight increase of Cd-Te and Cd-Cd bond length of

2.52% and 3.60% respectively which explains the small diﬀerence in energy of 0.04 eV between the SME and the GME. The Cd GMax, Fig. 6-9(c) located between the

GME and SME, forms an sp2 like conﬁguration with both nearest neighbor Cd and

Te atoms. We see that the Cd interstitial (c) forms more of a planer sp2 like bond with its three neighboring Cd bulk atoms at distance of 2.89 A˚ making a Cd-Cd-Cd angle of 119◦ with the Cd interstitial, close to the ideal sp2 angle of 120◦. The three neighboring Te atoms are 2.83 A˚ away and make a Te-Cd-Te angle of 114.1◦ with the

Cd interstitial indicating that the Cd interstitial has moved further away from the

70 center of the Te sp2 plane than for the Cd sp2 plane. Individual Cd and Te sp2 like bonding is also shown in panel (c) for clarity. We see this bonding conﬁguration in a number of extrema positions for the other interstitials. This sp2 like conﬁguration is the GME position for a Cu and Au interstitial and it serves as the GMax for a Mo interstitial.

Table 6.4: Extrema positions given in direct coordinates of a fcc unit cell a1 = a[0, 0.5, 0.5], a2 = a[0.5, 0, 0.5], a3= a[0.5, 0.5, 0], R=c1a1+c2a2+c3a3. For example the Mo secondary minimum position is given by R[Mo]=0.53a1+0.53a2+1.48a3.

Global Min Secondary Min Secondary Max Global Max Element c1 c2 c3 c1 c2 c3 c1 c2 c3 c1 c2 c3 0.50 0.50 0.50 Cd 0.75 0.75 0.75 0.65 0.65 0.66 0.50 0.50 1.50 0.60 0.60 0.60 Cu 0.54 0.54 1.46 0.66 0.66 0.83 0.60 0.60 1.60 0.50 0.50 0.50 Ag 0.75 0.75 0.75 0.69 0.69 0.69 0.50 0.50 1.50 0.61 0.61 0.61 Au 0.57 0.57 1.35 0.70 0.70 0.80 0.61 0.61 1.61 0.75 0.75 0.75 Mo 0.53 0.53 1.48 0.62 0.62 1.14 0.75 0.75 1.75

As mentioned above, the Cu interstitial GME position forms a nearly planer sp2 like bond with its neighboring Te bulk atoms, Fig. 6-9(d), with an angle 119.1◦, close to the ideal sp2 angle of 120◦, at a distance of 2.61 A.˚ The Cu interstitial bonding with the neighboring Cd bulk atoms is less planer with a Cd-Cu-Cd bond angle of

112.8◦ at a distance of 2.82 A.˚ We see that the Cu interstitial SMax and GMax, Figs.

6-9(e) and 6-9(f) respectively, deviate from the high symmetry 4(b) and 4(d) sites. At the SMax, Fig. 6-9(e), the Cu interstitial forms a slightly distorted TT e position with a Te-Cu-Te bond angle of 109.2◦ and a distorted octahedral coordination with the neighboring Cd atoms with a less than ideal Cd-Cu-Cd bond angle of 89.7◦. At the

Cu interstitial GMax, Fig 6-9(f), the coordination reverses and the distortion is more

71 pronounced. The Cu interstitial GMax is a distorted TCd position with a Cd-Cu-Cd bond angle of 107.9◦ and a distorted octahedral coordination with the neighboring

Te atoms with a Te-Cu-Te bond angle of 88.1◦. As the Cu interstitial moves from the

GME to the GMax the Cu-Cd bond length only increases 3.55% from 2.82 A˚ to 2.92

A˚ but the Cu-Te bond length changes signiﬁcantly by 23.8% increasing from 2.61 A˚

to 3.23 A.˚ Comparing this to the situation for a Cd interstitial where the Cd-Cd bond

length decreases 1.37% from 2.93 A˚ to 2.89 A˚ and the Cd-Te bond length decreases

12.9% from 3.25 A˚ to 2.83 A,˚ much less than the Cu-Te bond length change, may explain in part why the Cu GMax energy barrier of 0.46 eV is larger than the 0.33 eV GMax energy barrier for a Cd interstitial. The values for the SMax and GMax of 0.12 eV and 0.46 eV, respectively, are in excellent agreement with the theoretical values of 0.12 eV and 0.43 eV reported by Ma and Wei [37]. The rate limiting barrier of 0.46 eV for Cu is also in excellent agreement with the experimental values of 0.33 eV and 0.57 eV reported by Dzhafarov et al. [71] and Jones et al. [72] respectively.

Unlike the Cu interstitial that avoids the high symmetry positions, the Ag inter-

◦ stitial GME, shown in Fig. 6-9(g), occupies the TT e position making the ideal 109.5 angle with its neighboring Te atoms and an octahedral position making the ideal 90.0◦ angle with its neighboring Cd bulk atoms. Figure 6-9(i) shows the SME where the coordination reverses but the ideal symmetries remain, the Ag interstitial occupies

◦ the TCd position making the ideal 109.5 angle with its neighboring Cd atoms and an octahedral position making the ideal 90.0◦ angle with its neighboring Te atoms. The

SME is only 0.02 eV less than the 0.10 eV GMax energy barrier. The Ag GMax, Fig.

6-9(h), lies on either side of the SME where the high symmetry of the TCd position is broken. The broken symmetry of the GMax prevents interpreting average bond

lengths and angles for this position. However, we note as the Ag interstitial moves

from the GME position to the shallow SME we see the Ag-Cd bond length decreases

10.7% from 3.26 A˚ to 2.91 A˚ but the strain of this decrease is compensated by an

72 Table 6.5: Average bond length and angle from interstitial defect to the ﬁrst two shells of neighbors which are either only Te atoms or only Cd atoms.

Nearest- Avg. Bond Avg. Bond Element Position Figure 7-9 neighbor Length (A)˚ Angle (◦) Te 3.25 90.0 Global Min a Cd 2.93 109.5 Te 2.85 109.5 Secondary Min b Cd 3.33 90.0 Cd Te 2.83 114.1 Global Max c Cd 2.89 119.0 Te 2.61 119.1 Global Min d Cd 2.82 112.8 Te 2.74 109.2 Secondary Max e Cd 3.30 89.7 Cu Te 3.23 88.1 Global Max f Cd 2.92 107.9 Te 2.86 109.5 Global Min g Cd 3.26 90.0 Te 3.20 90.0 Secondary Min h Cd 2.91 109.5 Au Te - - Global Max i Cd - - increase in the Ag-Te bond length by 11.9% from 2.86 A˚ to 3.20 A.˚ This balancing of bulk strain results in the small 0.08 eV diﬀerence in energy between the GME and the

SME and may explain the small barrier energy of 0.10 eV at the nearby GMax. The values for the SME and GMax of 0.08 eV and 0.10 eV, respectively, are in agreement with the theoretical values of 0.06 eV and 0.08 eV reported by Ma and Wei [37]. The rate limiting barrier of 0.10 eV for Ag agrees well with the experimental value of 0.15 eV reported by Dzhafarov et al. [71].

The GME for Cd and Ag interstitials is located at the highly symmetric 4(b) site with the SME occurring at the other complementary high symmetry 4(d) site and the GMax located between the two sites on either side of the 4(d) site. A similar situation occurs for Mo. However, the GME for Mo is located at the 4(d) site with

73 Table 6.6: Diﬀusion barriers (eV) for diﬀerent point defects.

Secondary Secondary Global Element Min (eV) Max (eV) Max (eV) CdV - - 1.83 CdI 0.04 - 0.33 Cu - 0.12 (0.12 [37]) 0.46 (0.43 [37], 0.33 [71], 0.57 [72] ) Ag 0.08 (0.06 [37]) - 0.10 (0.08 [37], 0.15 [73]) Au - 0.12 0.27 Mo 0.02 - 0.30 the SME occurring at a slightly distorted 4(b) site and the GMax located between the two sites, immediately on either side of the 4(b) site. In each of these cases,

Cd, Ag and Mo, this leads to a symmetric diﬀusion path down the [1 1 0] channel.

However, for Cu and Au interstitials, the GME is not located at either of the 4(b) or 4(d) sites but rather at the sp2 like position and as a result their diffusion paths are asymmetric. Furthermore, Cu and Au diffusion paths do not possess a SME; instead they possess an SMax and an additional GME. The first and second GME, used for the NEB runs, are related by a lattice vector and hence are equivalent sites, the additional GME is not related to the first or second by a lattice vector and as such Cu and Au posses two distinct GME where as Cd, Ag and Mo posses only one distinct GME. This is because the GME for Cu and Au do not occur at either of the

4(d) or 4(b) sites and symmetry of the zinc-blende structure along the [1 1 0] channel requires this additional GME position. The additional GMEs for Cu and Au, as well as the diﬀerent symmetries of all diﬀusion paths, are clearly seen in Fig. 6-10.

74 Figure 6-10: NEB graphs of diffusion barrier profiles as a function of the NEB step positions, (a) the symmetric barriers (i.e. Cd, Mo and Ag) and (b) the asymmetric barriers of Cu and Au observed in CdTe. The first and last NEB positions on the X axis are displaced through a lattice vector a[0.5 0.5 0] down the [1 1 0] channel, where a is the computed lattice constant 6.43 A,˚ and hence have identical energies. The symmetric Cd vacancy barrier occurs in the middle of the path with an energy of 1.83 eV.

75 As noted earlier both Cd and Ag diﬀusion path minima occur at the 4(b) and

4(d) sites. As a result the diﬀusion path passes directly between the two alternating

4(b) and 4(d) sites resulting in a diﬀusion path down the [1 1 0] channel that involves alternating linear paths down along the [1 1 -1] direction the back up along the [1 1

1] direction. This saw tooth diﬀusion behavior of Cd and Ag is shown in Fig. 6-11(a) for Cd. In contrast, the Cu and Au GME position occurs between the 4(b) and 4(d) sites and the GMax and SMax avoid the 4(d) and 4(b) position resulting in a more linear path down the [1 1 0] channel, shown for Cu in Fig. 6-11(b).

Figure 6-11: Diﬀerences in diﬀusing paths for (a), (b) Cu and (c), (d) Cd interstitial atoms down the [1 1 0] channel in CdTe. Au atoms have paths similar to Cu while Ag has a path similar to Cd. Mo is similar to Cu and Au, with the global minimum occurring at a high symmetry, 4(d), site. However, the path deviates slightly, avoiding the other high symmetry, 4(b), site where as Cu and Au pass between both high symmetry sites.

The different maxima and minima positions for the different interstitials impacts the z-axis deviation of their respective diffusion paths down the [1 1 0] channel. The z-axis deviation in linearity is shown for all interstitials in Fig. 6-12.

76 Figure 6-12: Z-axis deviation from the minimum of the interstitial atoms while diffusing between global minimums in adjacent unit cells down the [1 1 0] channel. The ordinate axis is defined to be zero at Z-axis position of the diffusing atom in its global minimum energy site.

Figure 6-12 quantitatively shows the linearity of the diﬀusion path along the [1

1 0] channel. As we see from Fig. 6-12, the largest deviation in the z-axis is for

the symmetric diﬀusing interstitials Cd and Ag, deviating approximately 0.25 A˚ from

their respective GME z-axis coordinates, followed by Mo that deviates around 0.20

A˚ from the Mo GME. This is expected as the Cd and Ag interstitials travel in the

saw tooth pattern between the 4(b) and 4(d) sites. The asymmetric Cu and Au

interstitials experience the least deviation from the z-axis coordinate of their GME.

We see the linear diﬀusion character of Cu interstitial deviates approximately 0.05

A˚ from the Cu GME z-axis coordinate at both the GMax and SMax, traveling in

an almost straight line path down the [1 1 0] channel. The Au interstitial deviates

less than 0.10 A˚ from its GME when passing through the GMax position but only deviates approximately 0.05 A˚ when passing through its SMax.

77 To further explore the relative importance of this z-axis deviation during the

diﬀusion path we focused attention on the symmetric diﬀusing Cd and asymmetric

diﬀusing Cu interstitials which show the highest and lowest z-axis deviation in Fig

6-12. We performed two additional sets of runs each for a Cd and Cu interstitial

to assess the eﬀect of z-axis deviation on local strain energy and relaxation during

diﬀusion. In both sets the interstitial was kept ﬁxed with its x and y coordinates

obtained from the images of the fully converged NEB run and all other atoms in

the supercell were ﬁxed (i.e. unrelaxed) at their bulk equilibrium positions. For

the ”straight line” set of runs the z-coordinate of the interstitial was ﬁxed in the

entire set at the z-coordinate of the GME. In the ”bulk” set the z-coordinate of the

interstitial was varied. It was chosen to be the z-coordinate of the corresponding

image in the fully converged NEB run. For a Cu interstitial, in Fig. 6-13(a), we see

that the maximum strain energy occurs at the GMax. This can be explained by the

distortion of the symmetric TCd position. This maximum strain energy occurs at the GMax for both the ”straight line” and ”bulk” energy runs. As the Cu interstitial moves towards the SMax position at the TT e position we again see an increase in the ”straight line” and ”bulk” energy. This can be explained again by the distortion of the symmetric TT e tetrahedral and octahedral bonding conﬁgurations. At the GME, the Cu interstitial experiences the least ”bulk” and ”straight line” energy of around

1.2 eV. For the Cd interstitial the situation is diﬀerent, the maximum strain energy

occurs at the SME TT e position. We also note that there is a substantial diﬀerence between the ”bulk” and ”straight line” energies at the maximum strain position for

the Cd interstitial, over 20 eV, compared to the diﬀerence for the Cu interstitial case

where the diﬀerence is around 0.6 eV. This can again be explained by the character

of the diﬀusion path. For Cd, the diﬀusion path is a seesaw pattern that experiences

the largest z-axis deviation and hence largest deviation from the straight line path.

78 Figure 6-13: Effect of local strain energy and relaxation around diffusing interstitial Cu (a) and Cd (b) atoms. The barrier energy represents the fully relaxed diffusion barrier energy as a function of the NEB position along the diffusing path; the bulk energy is the energy of the interstitial located at the relaxed NEB position in an un-relaxed bulk cell, i.e. the remaining atoms are left unrelaxed in their bulk positions; the straight line energy is the energy of the NEB positions located at the same z-axis position as the global minimum energy position in an un-relaxed bulk cell, for Cd (b) the straight line maximum occurs in the middle of the path with an energy of 21.3 eV.

We computed the electronic LDOS of a variety of conﬁgurations of all diﬀusing

species studied. We highlight a few observations from these computations here. An

example of a symmetric diﬀusion barrier is illustrated in Fig. 6-14. It shows the

LDOS for the diﬀusion of a Ag interstitial as it moves from the GME at the TT e site

to the SME at the TCd position. The Ag interstitial at the GME position, the grey atom marked ’A’ in Fig. 6-14(a), is tetrahedrally coordinated to the black Te atoms and octahedrally coordinated to the white Cd atoms. At the SME position, marked

’B’ in Fig. 6-14(a), the coordination is reversed. At the GME ’A’ position the Ag interstitial forms symmetric bonds with Cd atoms 1, 2 and 3 and Te atoms 4, 5 and

6. As the Ag interstitial moves from the GME ’A’ position to the SME ’B’ position we see that the three bonds with Cd atoms 1, 2 and 3 and Te atoms 4, 5 and 6 remain intact. Panels (b) and (c) in Fig. 6-14 show the LDOS for the d orbitals of the Ag interstitial, the s orbitals for Cd atoms 1, 2 and 3 and the p orbitals for Te atoms 4, 5

79 and 6 at (b) the GME and at (c) the SME. At the GME (b) we see two signiﬁcant s-p bonding peaks from the tetrahedrally coordinated nearest neighbor Cd and Te bulk atoms around -5.13 eV and -5.6 eV and two large contributions from the d states of the Ag interstitial again at around -5.13 eV and at -5.25 eV. As the Ag interstitial moves to the SME, in panel (c), we see that the contributions from the Ag p states are shifted to higher energies towards the Fermi level. This is not surprising since the

SME is at a less energetically favorable position than the GME. However, this shift in the Ag p states toward higher energies is accompanied by a considerable shift toward deeper binding energies for the s-p bonding peaks of the nearest neighbor Cd and Te atoms. This compensating shift in the LDOS may in part explain the small 0.08 eV diﬀerence in energies between the Ag interstitial GME and SME positions as well as the small 0.10 eV energy barrier at the GMax position that occurs between the two as the Ag interstitial moves from GME ’A’ position to the SME ’B’ position.

80 Figure 6-14: Panel (a) shows the global minimum energy position marked ’A’ and secondary minimum energy position marked ’B’ for an Ag interstitial. Filled black (white) circles show nearest neighbor Te (Cd) atoms. Position ’A’ is tetrahedrally coordinated to Te and octahedrally coordinated to Cd. At position ’B’ the coordination is reversed. Panels (b) and (c) show the LDOS of the d orbitals of the Ag interstitial and the p orbitals of nearest neighbor Te atoms 1, 2 and 3 and s orbitals of its nearest neighbor Cd atoms 4, 5 and 6. LDOS for the Ag atom in position ’A’ is in panel (b), for position ’B’ in panel (c). At position B, the Ag interstitial d orbitals are shifted toward the Fermi energy, away from the deep binding at position A. However, this shift is accompanied by a shift away from the Fermi, to tighter binding energies for the s and p orbitals of the nearest neighbor Cd and Te atoms respectively. This compensating shift in orbitals may in part explain the low diﬀusion barrier for Ag interstitials.

Another example of a symmetric diﬀusion barrier is illustrated in Fig. 6-15. It shows the LDOS for an interstitial Cd atom at the GME and GMax positions. We see that the s-states for the GME are shifted to deeper binding energies than for the GMax position suggesting a more energetically favorable position at the GME.

When the Cd interstitial is at the GMax position we can see that the Cd s-states have shifted toward the Fermi energy to shallower binding energies. In addition to the shift of the s-states at the GMax position, we observe that the s-states as well as the hybridized p-states contribute to the bonding character where as at the GME

81 position only the s-states contribute.

Figure 6-15: LDOS for an interstitial Cd atom at the global minimum and global maximum positions. Observe the shift to deeper binding energies for global minimum and contribution to defect states at the Fermi level from s, p and d-states for the global maximum position and only from s-states for the global minimum.

An example of an asymmetric diﬀusion barrier is illustrated in Fig. 6-16. It shows

the LDOS for an interstitial Cu atom at the GME and GMax positions. We observe in

Fig. 6-16(a) a shift in the d-states of the Cu interstitial to tighter binding energies for

2 the sp like GME compared to that of the distorted TCd GMax position. In addition we note the three distinct peaks for the GME, at -3.75 eV, -3.25 and around 2.7 eV,

can be attributed to the Cu d-state and Te p-state bonding formed with the three

neighboring Te atoms in the sp2 like bonding conﬁguration. The shift toward deeper binding energies for the GME d-states is also observed in the hybridized s-states, as shown in Fig 6-16(b). In addition to the shift toward shallow binding energies for the

GMax states, we also observe a signiﬁcant increase of the metallic like s-states at the

Fermi energy for the GMax compared to the GME.

82 Figure 6-16: LDOS for a Cu interstitial at the global minimum and global maximum positions. We observe in (a), the shift in the d-states to tighter binding energies for the global minimum position compared to that of the global maximum. This trend is also observed in (b) the s-states. We also observe a signiﬁcant increase of the defect like s-states at the Fermi energy for the global maximum compared to the global minimum.

83 Chapter 7

Bulk diﬀusion in Cadmium Sulﬁde

7.1 Computational Method

All results presented are ab initio total energy calculations within the local density approximation to density-functional theory [46,51] for diffusion energy barrier computations using VASP [55–58]. Core electrons are treated by ultrasoft Vanderbilt type pseudo-potentials [59] as supplied by Kresse et al. [60] using the Ceperly and Alder exchange-correlation functional. The single-particle wave functions were expanded in the plane-wave basis using a 200 eV energy cutoff. A Gamma centered 2×2×2 k-point grid was used for the Brillouin-zone integrations in all calculations. Tests using larger k-point sampling and higher plane-wave energy cutoff indicated that a numerical convergence better than ±10 meV was achieved. The CdS bulk structure consists of a four atom primitive cell wurtzite structure (space group P63mc number √ √ 186) with lattice vectors (a1) = (a/2)[1, - 3, 0], (a2) = (a/2) [1, 3, 0], (a3) = a[0, 0, c/a]. Both Cd and S atoms occupy 2(b) Wyckoff positions (1/3, 2/3, z) and (2/3,

1/3, z+1/2) where z = 0 for Cd and z = the internal parameter for S. The lattice

constants were varied and ﬁt to parabolic equations as a function of total energy to

obtain the absolute minimum in total energy. The calculated lattice constants a and

c were 4.09 A˚ and 6.65 A˚ respectively, with internal parameter z = 0.376, are within

84 1% of the experimental lattice parameters [44] of 4.1365 A˚ and 6.7160 A˚ and z =

0.3770. These computed values give a CdS bond length of 2.50 A.˚ Diﬀusion barrier calculations were computed in a 128 atom supercell, a 4×4×2 repetition of the 4 atom unit cell. The larger supercell size more closely models an isolated defect by reducing the long-range interactions between defects in neighboring supercells. To

ﬁnd the minimum energy for each conﬁguration, all atoms were allowed to fully relax.

Relaxation was completed when a force tolerance of 0.01 eV/A˚ was reached for each atom. The calculations for the LDOS were performed with the tetrahedron method with Bl¨ochl corrections [74]. Diﬀusion barrier calculations were performed using the

NEB method [75].

7.2 Cd, S, Te, Cu, and Cl

7.2.1 Results

We consider here the diffusion of Cd, S, Te, Cu and Cl interstitials and vacancies of Cd and S. Detailed structural information provided here can only be obtained from reliable DFT computations. It is complementary to results of experimental observations where activation energies are measured indirectly leading to deduction of diffusion barriers [36, 68–73, 76–79]. The interstitial diffusion paths considered in this work were chosen parallel to c-axis. An example of this linear diffusion is shown in Fig. 7-1. In the figure, two views of the interstitial diffusion down the c axis are shown. The first one is parallel to the axis and shows that the entire diffusion pathway occurs symmetrically through the center of hexagons formed by the bulk Cd and S atoms. In the second view several positions of the Cu atoms in various NEB images perpendicular to the c axis are shown. Different species of interstitial adatoms were found to occupy different extrema positions. A single GME and GMax position is found for all the diffusing species considered here. This contrasts with the case 85 of Cu, Ag, Au, Mo, S, P and Sb in CdTe where secondary minima and maxima are found. The GME and GMax positions for all interstitials are given in Table 7.1. All positions are given in direct coordinates of the wurtzite unit cell. Of these the GME and GMax positions of Cd, Cl, S and Te show similarities across atomic species in the bonding configuration to neighboring atoms. On the other hand the Cu interstitial behaves differently. We can see from Table 7.1 that the Cd, Cl, S and Te GME occurs between z = 0.17 and z = 0.28, and the GME for Cu occurs at z = 0.37. Similarly, the GMax for Cd, Cl, S and Te occurs between z = 0.43 and z = 0.55, and the GMax for Cu occurs at z = 0.61. To investigate the differences between extrema positions in more detail, we focus on the Cd and Cu interstitials.

Figure 7-1: Interstitial diﬀusion in bulk CdS down the c axis as seen for a Cu adatom (a) parallel and (b) perpendicular to the c axis with several positions of the diﬀusing Cu interstitial shown. The lattice vectors√ for the hexagonal√ wurtzite unit cell are (a1) = (a/2)[1, - 3, 0], (a2) = (a/2) [1, 3, 0], (a3) = a[0, 0, c/a].

In Fig. 7-2, we can see the diﬀerence between the Cd and Cu extrema positions.

The average bond length and bond angle from the interstitial defects, shown in Fig. 7-

2, to nearest neighbors are given in Table 7.2. The Cd interstitial GME position, panel 86 Table 7.1: Extrema positions for the diﬀusing adatoms given in direct coordinates of the hexagonal wurtzite unit cell as, R=c1a1+c2a2+c3a3. For example the S global maximum position is given by R[S]=0a1+0a2+0.53a3. Considering the lattice vectors a1, a2 and a3, the c3 position listed corresponds to the z coordinate.

Global Min Global Max Element c1 c2 c3 c1 c2 c3 0.00 0.00 0.37 Cu 0.00 0.00 0.61 0.00 0.00 0.87 0.00 0.00 0.20 Cd 0.00 0.00 0.43 0.00 0.00 0.70 0.00 0.00 0.25 S 0.00 0.00 0.53 0.00 0.00 0.75 0.00 0.00 0.28 Te 0.00 0.00 0.55 0.00 0.00 0.78 0.00 0.00 0.17 Cl 0.00 0.00 0.44 0.00 0.00 0.67

(a), forms a distorted sp3d2 like octahedral configuration with the nearest neighbor bulk S atoms with an average bond length of 3.03 A˚ and angle of 90.6◦. Due to the distortion of the octahedral configuration, the Cd interstitial forms a bond length and bond angle of 3.29 A˚ and 78.4◦ with three of the S bulk atoms and 2.77 A˚ and 102.8◦ with the other three S bulk atoms. Additionally, the Cd interstitial at the GME position forms a less distorted octahedral configuration with the nearest neighbor bulk Cd atoms with an average bond length and angle of 2.965 A˚ and 90.15◦. The distortion of the Cd bulk atoms is less than that of the S bulk atoms with a bond length and bond angle of 3.10 A˚ and 83.8◦ with three of the Cd bulk atoms and 2.83

A˚ and 96.5◦ with the other three Cd bulk atoms. The Cd interstitial GMax position, panel (b), forms an sp2 like conﬁguration with, both Cd and S, neighboring bulk atoms. The bonding with the bulk S atoms is slightly more sp2 like than the bonding with the bulk Cd atoms. This is indicated by the average bond angle of 118.5◦ with the three neighboring S bulk atoms, closer to the ideal sp2 bond angle of 120.0◦ than the average bond angle of 116.3◦ the Cd interstitial makes with the three Cd bulk

87 atoms.

Figure 7-2: Structural motifs and bonding configuration of interstitial positions along the c axis of minimum and maximum energy. The atom labeled x is the diffusing interstitial atom, the surrounding atoms are of S and Cd. The Cd interstitial begins at the minimum energy position, shown in panel (a), then diffuses down the c axis until it reaches the maximum energy position, shown in (b), and proceeds until it reaches another minimum energy position in a neighboring unit cell (c). The same sequence of diffusion steps is shown for a Cu interstitial in (d)-(f). Atoms of S, Te and Cl diffuse similar to Cd.

Unlike the Cd interstitial, the Cu interstitial, panel (d), forms an sp2 like conﬁg-

uration with both Cd and S neighboring bulk atoms at the GME position. The Cu

interstitial forms an sp2 bond with the three neighboring bulk S atoms with an ideal bond angle of 120.0◦ and a distorted sp2 like conﬁguration with the Cd neighboring

bulk atoms. The distortion of the sp2 like bonds is due to the Cu interstitial being 88 Table 7.2: Average bond length and angle from interstitial defect to nearest neighbor. The Cd interstitial maximum energy position and Cu interstitial minimum energy position each form a sp2 like configuration with their respective Cd and S neighboring bulk atoms. The Cd interstitial minimum energy position and Cu interstitial maximum energy position form a distorted octahedral configuration with their respective nearest neighbor bulk S and Cd atoms. For such data the averages of bond lengths and angles are listed along with their ranges in brackets. For example, the Cu interstitial at the global minimum forms a distorted octahedral bonding configuration with six neighboring S atoms. Three S atoms at 2.87 A˚ making an angle of 87.1◦ and the other three S atoms at 2.74 A˚ making an angle of 92.9◦ for an average bond length of 2.81 A˚ and an average bond angle of 90.0◦.

Nearest Avg. Bond Length Avg. Bond Angle Element Position Figure 1 -neighbor (A)˚ (◦) S 3.03 (2.77-3.29) 90.6 (78.4-102.8) Global Min a Cd 2.97 (2.83-3.10) 90.2 (83.8-96.5) Cd S 2.61 118.5 Global Max b Cd 2.69 116.3 S 2.29 120 Global Min d Cd 2.72 107.5 Cu S 2.81 (2.74-2.87) 90.0 (87.1-92.9) Global Max e Cd 3.12 (2.67-3.57) 91.7 (71.4-111.9)

89 slightly displaced perpendicular to the plane of the Cd bulk atoms resulting in an

average bond angle of 107.5◦, less than the ideal bond angle of 120.0◦. The Cu in-

terstitial GMax position forms a sp3d2 like octahedral conﬁguration with the nearest

neighbor bulk S atoms with an average bond length and angle of 2.805 A˚ and 90.0◦.

The slight distortion of the octahedral conﬁguration creates a Cu interstitial bond length and bond angle of 2.87 A˚ and 87.1◦ with three of the S bulk atoms and 2.74

A˚ and 92.9◦ with the other three S bulk atoms. The Cu interstitial at the GMax position also forms a distorted octahedral conﬁguration with the nearest neighbor bulk Cd atoms with an average bond length and angle of 3.12 A˚ and 91.65◦. The

distortion of the Cd bulk atoms is less than that of the S bulk atoms with a bond

length and bond angle of 2.67 A˚ and 111.9◦ with three of the Cd bulk atoms and 3.57

A˚ and 71.4◦ with the other three Cd bulk atoms.

90 Figure 7-3: Energy along the diffusion path as a function of the NEB step positions. We observe all interstitial diffusion paths contain a single maximum. For diffusion parallel the c axis, Cd and Cu interstitials encounter their respective maximum energy position earlier in the diffusing path than Te, S and Cl interstitials. The relative positions of the interstitials at the extrema positions are given in Table 7.1.

From these GME and GMax positions we have calculated the energetic barriers for diﬀusion along the NEB pathway. The diﬀusion barriers for interstitials of Cd,

Cl, S, Te and Cu are given in Table 7.3 along with the values for Cd and S vacancy diﬀusion.

91 Table 7.3: Diﬀusion barriers (eV) for diﬀerent point defects.

Element Maximum (eV) Cu 0.66 (0.58 [79], 0.72 [76], 0.77 [78], 0.96 [77]) Cd 0.87 CdV 1.09 S 0.42 SV 2.18 Te 0.66 Cl 0.76

The Cd and S vacancy diﬀusion takes place between neighboring bulk positions in √ the plane perpendicular to the c axis along the (a1) = (a/2)[1, - 3, 0] lattice vector direction. Figure 7-4 shows the energy (eV) along the diﬀusion path as a function of the NEB step positions. From Table 7.1 and Figure 7-3 we can see that Cd and

Cu interstitials encounter their respective GMax position earlier in the diffusing path than Te, S and Cl interstitials. The S interstitial has the lowest barrier of 0.42 eV while Cd has the highest barrier of 0.87 eV that is twice that for S. The 0.66 eV value for Cu is in good agreement with experiment. The range of experimental values can be attributed to the different techniques used to determine the barriers including capacitive measurements [76,77], tracer [78] and optical absorption technique [79]. To understand the contribution of strain energy on the total barrier energy we explored the energetics of diffusion by suppressing relaxation of the bulk atoms. To determine the strain energy, an interstitial atom was placed in an un-relaxed bulk cell at the positions determined from the fully converged NEB computations. The energy was computed with the un-relaxed bulk atoms and the interstitial atom fully constrained.

From Table 7.1 we can see three distinct MES position distributions; one for Cl and

Cd with the smallest MES position z coordinates of 0.17 and 0.20, respectively; the second for the next greatest z coordinate S and Te with the MES position z = 0.25 and z = 0.28, respectively; and the third for Cu with the greatest MES position z

92 coordinates of z = 0.37. Figure 7-4 shows the bulk energy along with the diffusion barrier energy for interstitial atoms of each type, Cd (a), Cu (b) and Te (c). The barrier energy represents the fully relaxed diffusion barrier energy as a function of the NEB position along the diffusing path. Fig. 7-4 shows that the highest strain energy is not always associated with the GMax site of the barrier energy. The highest strain energy for Cd, Fig. 7-4(a), does coincide with position of the highest barrier energy. However, for Cu and Te, Fig. 7-4(b) and (c) respectively, the highest strain energy occurs well before the position of maximum barrier energy. Although Cu and

Te share the same diﬀusion barrier energy of 0.66 eV, we see the wide disparity in strain energy; for Cu the maximum strain energy is around 1.2 eV and for Te the maximum strain energy is almost 5 eV, around four times greater than that of Cu.

93 Figure 7-4: Effect of local strain energy and relaxation around diffusing interstitial atoms of Cd (a), Cu (b) and Te (c). The barrier energy represents the fully relaxed diffusion barrier energy as a function of the NEB position along the diffusing path; the bulk energy is the energy of the interstitial located at the relaxed NEB position in an un-relaxed bulk cell, i.e. the remaining atoms are left unrelaxed in their bulk positions.

94 In order to further explore the diﬀerences between the Cd, Cl, S, Te and Cu

interstitials we investigated the electronic properties of the defects in the host CdS.

Since Cu was found to behave diﬀerently from interstitials of Cd, Cl, S and Te we

analyzed the LDOS for Cu at the MES and GMax positions. Figure 7-5 shows

the LDOS for a Cu interstitial diﬀusing parallel to the c axis at the minimum and

maximum energetic positions. As mentioned, the Cu interstitial forms an energetically

favorable sp2 type bond with the S bulk atoms at the MES position and we note that the bonding of the d-states for Cu correspond primarily to the p-states of the neighboring S bulk atoms. We observe the shift toward deeper binding energies for the d-states of the MES and distinct peak for the s-states of the MES around -6.5 eV that is not present at the GMax position. At the sp3d2 like octahedral GMax position,

the bonds between the neighboring S bulk atoms and the Cu interstitial have been

stretched from 2.29 A˚ to an average of 2.81 A˚ and we observe that the d-states of

the Cu interstitial are shifted to higher energies closer to the Fermi energy. We also

note that a sizable contribution to defect states at the Fermi level from s-states for

the GMax position compared to the small contribution from s- and p-states for the

GME conﬁguration, is observed.

95 Figure 7-5: LDOS for a Cu interstitial diﬀusing parallel to the c axis at the minimum and maximum energetic positions. We observe a shift toward deeper binding energies for the d-states of the minimum. Notable contribution to defect states at the Fermi level from s-states for the maximum position compared to the small contribution from s- and d-states for the minimum is also observed.

96 Chapter 8

Conclusions and Future Work

An ab initio study of diﬀusion proﬁles of native and non-native atomic species has

been presented for both surface and bulk diﬀusion mechanisms. In each investigation,

results have been presented that provide insight into the energetics and structural

properties of these mechanisms that will aid experimental measurements of diﬀusion

barriers and provide other complementary information to experimental ﬁndings. A

summary of the conclusions from the diﬀusion investigations are presented below.

8.1 Conclusions

GaAs(001) Surface

GaAs(001)-c(4×4) For the As rich c(4×4) surface reconstruction, growth can

occur only through the incorporation of Ga on the surface. Therefore a detailed

knowledge of the Ga adatom adsorption and diﬀusion on the surface is critical to

understand and control the growth process. In chapter 6, results have been presented

of the energetics of a Ga adatom on the GaAs(001)-c(4×4) surface reconstruction including energies for adsorption of a Ga adatom relaxing from a height of 3 and 0.5

A,˚ and its diﬀusion. When relaxing from 3 A,˚ the calculations reveal two new binding sites in the row of As dimers situated between the dimers. The calculations for the

97 Ga adatom when relaxing from 0.5 A˚ reveal four new binding sites in the row of As

dimers. Some of these sites have lower energy than the ones at 3 A˚ and are comparable

to, and in some cases greater than, the binding energy found in the most stable trench

sites. Such multiple binding sites at varying heights at the same location on a surface

is a unique feature of this particular surface not found in any other system. These

four sites are interesting because they are a result of the separation of the neighboring

As dimer and breaking of the dimer bond. The separation and eventual replacement

of the As dimer with Ga is a necessary step in the growth process and these new sites

provide an energetically favorable starting point for incorporating the Ga atom into

the As rich surface.

GaAs(001)-c(4×4)-heterodimer Also in chapter 6, the energetics of a Ga

adatom on the GaAs(001)-c(4×4)-hd reconstruction were presented which included

the calculated structural differences between the c(4×4) and the c(4×4)-hd reconstructions. MES for a Ga adatom on the c(4×4)-hd surface were identified by computing the potential energy surface of the reconstruction. Energy barriers for diffusion of Ga adatom between neighboring MES were calculated for diffusion by way of top hopping and exchange mechanisms. Two unique paths for diffusion of Ga adatom between the global minimum sites of two neighboring unit cells have been proposed.

Also, there was found to be a correlation between the higher diffusion energy barrier for exchange mechanism compared to top hopping with differences in the LDOS of relevant atoms involved in the diffusion process.

CdTe Bulk In chapter 7, results were presented of the diﬀusion proﬁles in CdTe of native, Te and Cd adatoms and vacancies, anionic non-native interstitial adatoms

P, Sb, O, S, and Cl as well as cationic non-native interstitial adatoms Cu, Ag, Ag and Mo. We have found that both symmetric and asymmetric diﬀusion paths exist.

98 The results of this study show that the rate-limiting diffusion barriers for the anions range from a low of 0.49 eV for the asymmetric diffusion path of an Sb interstitial to a high of 1.51 eV for the symmetric diffusion path of an O interstitial and the rate-limiting diffusion barriers for the cations range from a low of 0.10 eV for the symmetric diffusion path of an Ag interstitial to a high of 1.83 eV for the symmetric diffusion path of a Cd vacancy. Structural motifs were analyzed around the diffusing atom or vacancy through the diffusion process. The intricacies of the process were revealed through a description of the curvature of the diffusion path, relevant bond lengths, bond angles, first and second shell coordination, and local density of states.

It was found that the symmetric or asymmetric nature of the diffusion path as well as the bond length and atomic coordination at the energetic-extrema positions influence the size of the diffusion energy barrier.

For the anions an electronic signature of the bond breaking in the LDOS was found for symmetric diﬀusion barriers, through the p-p bond breaking with neighboring Te atoms, as well as the diﬀerence in hybridization between the GME and

GMax positions for asymmetric diffusion barriers where s- and hybridized d-states are found to exist for only GMax positions. For the cations an electronic signature in the difference in hybridization was found between the GME and GMax positions where d- and hybridized p-states are found to exist for only GMax positions. Also, a compensating shift in the LDOS was found that may explain the small 0.08 eV difference in energies between the Ag interstitial GME and SME positions and the small 0.10 eV energy barrier at the GMax position.

CdS Bulk In chapter 8 diffusion profiles of native, Cd and S adatom and vacancy, and non-native interstitial adatoms Cu, Te and Cl were presented. The rate-limiting diffusion barriers range from a low of 0.42 eV for the diffusion path of an S interstitial to a high of 2.18 eV for the diffusion path of a S vacancy. Differences in structural

99 properties around the diﬀusing atom or vacancy have been explored for the energetic extrema positions along the diﬀusion path. In addition, an electronic signature in the

LDOS of the s- and d-states for a Cu interstitial was found at the MES and GMax positions.

8.2 Future Work

For bulk CdTe and CdS, a number of issues remain that can benefit from the continuation of this work. What role Cu plays in limiting the stability of CdTe/CdS devices and what mechanisms are responsible for the different diffusion profiles Cu displays are but a couple of questions that a comprehensive knowledge of defect diffusion processes, structural and energetic information will undoubtedly assist in answering. Work has already begun on this with preliminary investigations of Cu-

Cd and Cu-Te bulk exchange processes to help explain the Cu diﬀusion proﬁle in

CdTe/CdS.

It is my hope that funding for this work can continue. In order to completely understand and control these diﬀusion mechanisms, and achieve optimal cell eﬃciency, our knowledge should be as complete as possible regarding all of these processes.

This includes information regarding remaining diffusion mechanisms, investigations of different charge states and grain boundary diffusion. With the computing ability available today this is no longer a unrealistic task. The work and findings resulting from these studies have laid the groundwork to enable completion of this task.

8.3 Broader Perspective

The work presented here provides a foundation for a systematic and comprehensive understanding of diﬀusion processes for semiconductor bulk and surfaces. The

100 tremendous computational resources required for examining diffusion mechanisms and the difficulty of the work prohibits such a panoptic study in a single dissertation, therefore a complete understanding will require further investigation. However, with the convergence of the increase in computing power over the last 20 years and the continuous development of DFT codes like VASP, including the algorithms and techniques used, the accomplishment of this goal is certainly attainable. For many more systems of basic scientific and technological interest, beyond the ones investigated here, computational work of this nature can provide insights that analytical theory and experiments cannot provide.

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112 Appendix A

Vienna ab initio simulation package (VASP)

A.1 Introduction

VASP is a set of codes for atomic scale modeling of materials from ab initio

DFT calculations. The code was originally based on a program written by Mike

Payne at MIT and additional contributions were made by J¨urgenHafner, J¨urgen

Furthm¨ullerand Georg Kresse [55, 57, 58, 60]. The VASP codes are used to compute an approximate solution to the many-body Schr¨odinger equation within DFT by solving the Kohn-Sham equations. In the following two sections I will outline the methods and approximations used that enable VASP to compute the solutions to the Kohn-Sham equations and then describe how the code is used in practice. The

first section will include DFT and the Kohn-Sham equations, discussed earlier in chapter 4; periodicity, including Bloch’s theorem; the Born-von Karman boundary condition; plane-wave basis sets and k-point sampling; the use of pseudopotentials to approximate the interactions between the electrons and ions and finally algorithms used to minimize the Kohn-Sham energy functional. The second section will include the files and control parameters used to implement the approximations, outlined in

113 the ﬁrst section, as well as the nudged elastic band method (NEB) used to calculate the diﬀusion barriers presented in this work.

A.2 Approximations

A.2.1 DFT

DFT, as developed by Hohenberg and Kohn and implemented via the Kohn-Sham equations has been described in Chapter 4. The main idea is to transform a system of interacting electrons in a external potential to a system of non-interacting electrons in an eﬀective potential. This is accomplished by solving the Kohn-Sham equations

(4.24) self-consistently so that the computed single electron wavefunction solutions produce the same charge density (4.25) that would create the eﬀective potential (4.21) used to produce the same single electron wavefunction solutions.

A.2.2 Periodicity

Even with the ever increasing computing power of modern computers, the solution to the Kohn-Sham equations would not be possible without the explicit use of periodicity and the approximations assumed therein. This includes the use of periodic boundary conditions made possible by Bloch’s theorem and the use of plane-wave basis sets, the Born-von Karman boundary condition and Brillouin zone k-point sampling methods. The following sections will outline these methods as given, in part, by Ashcroft and Mermin [80].

A.2.2.1 Bloch’s Theorem

We cannot solve the Kohn-Sham equations for an inﬁnite number of non-interacting electrons in an inﬁnitely extended solid but we can make use of the periodicity to

114 reduce the problem to a more tractable form. In a perfect crystalline solid, the atomic ions will produce a periodic potential. The periodicity of the potential U(r) will correspond to the periodicity of the underlying Bravais lattice. This can be expressed as

U(r + R) = U(r) , (A.1) for all Bravais lattice vectors R. Regardless of the form of the potential U(r), as long as the system it describes is a perfect crystalline solid, the potential can be expressed as (A.1). For our purposes we can regard the potential U(r) as the eﬀective potential veff in the single electron Schr¨odinger like Kohn-Sham equations (4.24). Bloch’s theorem extends the idea of using the Bravais lattice periodicity to that of the eigenstates of the single electron Hamiltonian (4.24). The term Bloch electrons is used to describe independent electrons in a periodic potential U(r).

Theorem: Wavefunctions that are eigenstates ψ of the Hamiltonian

2 H = − ~ ∇2 + U(r) , (A.2) 2m in a periodic potential, where U(r) satisﬁes (A.1) for all R in the Bravais lattice, can be expressed as the product of a plane wave and a periodic function that has the periodicity of the periodic potential U(r)

ik·r ψnk(r) = e unk(r) , (A.3) where

unk(r + R) = unk(r) , (A.4) for all R in the Bravais lattice. Here, the quantum number n represents the band index that occurs because of the many independent eigenstates for a given k. From

115 equations (5.3) and (5.4) we can write

ik·(r+R) ik·R ψnk(r + R) = e unk(r + R) = e ψnk(r) , (A.5) or, as it is alternatively stated, the eigenstates of the Hamiltonian are chosen such that with each eigenstate corresponds to a wave vector k such that

ψ(r + R) = eik·Rψ(r) . (A.6)

A.2.2.2 Born-von Karman Boundary Condition

We can again make use of the periodicity of the Bravais lattice by imposing appropriate boundary conditions on the single particle Schrödingerlike Kohn-Sham equations (4.24). If we solve the equations for a given volume V we must account for the surfaces of the volume. Boundary conditions requiring the wavefunction to be zero at the surface produces standing waves which are not convenient for describing charge transport. We can remove this effect by effectively removing the surfaces. This is accomplished by the Born-von Karman boundary condition. For a cubic volume

V =L3, the Born-von Karman boundary condition can be expressed as

ψ(x, y, z + L) = ψ(x, y, z)

ψ(x, y + L, z) = ψ(x, y, z) (A.7)

ψ(x + L, y, z) = ψ(x, y, z) .

It is easy to see how the surface of the given volume can be removed by examining the situation for the 1-dimensional case, Fig. A-1, where we replace the line, from 0 to L, with a circle whose circumference is equal to the line of length L.

116 Figure A-1: The Born-von Karmen boundary condition in one-dimension where we replace the line, of length L, with a circle with circumference of length L.

We extend this idea from a cubic volume V =L3 to a volume corresponding to the primitive cell of the Bravais lattice of the system. We consider a volume V which contains Ni unit cells along the primitive vectors ai. The total number of unit cells in the volume V is then given by N=N1N2N3. We can now express the periodic boundary condition as

ψ(r + Niai) = ψ(r) , i = 1, 2, 3. (A.8)

If we apply Bloch’s theorem to the Born-von Karmen boundary condition we get

iNik·ai ψnk(r + Niai) = e ψnk(r) , i = 1, 2, 3. (A.9)

This requires that

eiNik·ai = 1 , i = 1, 2, 3. (A.10)

117 If k is chosen to have the form

k = x1b1 + x2b2 + x3b3 , (A.11)

where bi are the reciprocal lattice vectors of the Bravais lattice given by

a2 × a3 b1 = 2π a1 · (a2 × a3) a3 × a1 b2 = 2π (A.12) a2 · (a3 × a1) a1 × a2 b3 = 2π , a3 · (a1 × a2)

and satisfy the equation

bi · aj = 2πδij , (A.13)

then it must be that

e2πiNixi = 1 . (A.14)

From (A.10) it follows that

mi xi = , mi ∈ Z, (A.15) Ni

where Z is the set of all integers. Therefore, the general form for the allowed Bloch

wave vectors is 3 X mi k = b , m ∈ Z, (A.16) N i i i=1 i It follows from (A.16) that the volume ∆k of k-space per allowed value of k is just

the volume of the little parallelepiped

b1 b2 b3 1 ∆k = · ( × ) = b1 · (b2 × b3) , (A.17) N1 N2 N3 N

Here, b1·(b2×b3) is the volume of the reciprocal lattice primitive cell, therefore (A.17) 118 states that the number of allowed wave vectors in a primitive cell of the reciprocal

lattice, the Brillouin zone, is equal to the number of sites in the crystal. We see that

as the number of sites in the cell increases, as N → ∞ the number of allowed Bloch

wave vectors becomes inﬁnite. Consider the wave vectors k and k0 that are separated

by a reciprocal wave vector G, that is k0=k+G, according to (A.3) we have

ik0·r ik·r iG·r ik·r ψnk0 (r) = e unk0 (r) = e [unk0 (r)e ] = e u˜(r) = ψn0k(r) , (A.18)

Since the term in brackets on the right hand side has the same periodicity as the

Bravais lattice, the entire product on the right hand side can be expressed as a

Bloch wave function, with a different band index n’. Therefore, with Bloch’s theorem and appropriate boundary conditions we can transform the problem of computing an infinite number of wavefunctions to a problem of computing a finite number of wavefunctions at an infinite number of k-points in the first Brillouin zone.

A.2.2.3 K point Sampling

A number of properties including charge density and density of states require inte- grals over the Brillouin zone. Even though we have reduced the problem significantly, we still cannot handle the infinite number of k points in the first Brillouin zone. To deal with this computationally we use the fact that wavefunctions at k-points that are closely spaced are nearly identical so we can replace the integral over the Brillouin zone with a weighted summation over a few specially chosen k-points, referred to as a k-point mesh. That is

1 Z X f(r) = F (k)d(k) → ω F (k ) , (A.19) Ω j j BZ BZ j

119 where F (k) is the Fourier transform of f(r) and ωj are the weighting factors. One method for selecting a minimal number of special k-points to create a k-point mesh that utilizes the symmetry of the lattice is the Monkhorst and Pack scheme [61]. In the Monkhorst and Pack scheme, the k-points are evenly distributed throughout the reciprocal space and are described using the reciprocal lattice vectors by

kprs = upb1 + urb2 + usb3 , (A.20) where

2r − qr − 1 ur = , r = 1, 2, 3, ..., qr , (A.21) 2qr where qr determines the number of k-points along the b2 lattice vector direction. The k-point mesh must be systematically varied until the total energy is properly converged with respect to the number and location of the k-points.

A.2.2.4 Plane-wave Basis

Bloch’s theorem (A.3) enables us to express the wavefunctions that are eigenstates of the Hamiltonian in a periodic potential as the product of a plane wave basis set expansion and a periodic function. Therefore, we can express the eigenstates ψi of the Kohn-Sham equations (4.24) as

ik·r ψi(r) = e ui(r) , (A.22)

The periodic function can also be expanded in a plane-wave basis, given by the reciprocal lattice vectors G of the crystal as

X iG·r ui(r) = Ci,Ge , (A.23) G

120 Now, combined with (A.22) we can express the wavefunction as

X i(k+G)·r ψi(r) = Ci,k+Ge , (A.24) G

However, because a complete expansion with an infinite number of plane-waves is not computationally feasible we must truncate the expansion, and since the coefficients of the expansion in (A.24) decrease rapidly with increasing kinetic energy, the cut off energy for the plane-wave basis set is chosen based on the kinetic energy of the plane-wave. 2|k + G|2 ~ ≤ E , (A.25) 2m cut

Similar to the k-point mesh, the value for Ecut must be varied and chosen such that the total energy is properly converged. Truncating the plane-wave basis, based on the kinetic energy, eliminates the need for an inﬁnite basis set, however a large basis set is still required to reproduce the rapid oscillations of the wavefunction for the valence electrons near the atomic core region. To further reduce the size of the plane- wave basis set, the pseudopotential method has been developed to approximate the potential near the core region.

A.2.3 Pseudopotentials

The pseudopotential was introduced to reduce the computational cost associated with solving the Kohn-Sham equations. The pseudopotential assumes a softer electron-ion potential felt by the electrons than the larger, true 1/r potential while maintaining the same, important features that correctly describe the valence electrons in the true potential. This softer pseudopotential eliminates the strong electron-ion interaction experienced by the electrons within a predefined radial cutoff. Typically this radius is chosen outside the core, closed shell electrons. Outside this radial cutoff the electrons experience the same potential as the true potential. By eliminating the 121 strong electron-ion interaction, the pseudopotential eliminates the oscillations of the

electron wavefunction within the core radial cutoﬀ. Eliminating this nodal feature

of the electron wavefunction reduces the required planewave basis size needed to ex-

pand the wavefunctions thereby reducing the computational cost. This is shown in

Fig. A-2.

Figure A-2: Representation of a pseudopotential Vpseudo and pseudo- wavefunction Ψpseudo. Beyond the cutoﬀ radius rcutoff , the pseudopotential and pseudo-wavefunction exactly reproduce the all electron potential Vreal and wavefunction Ψreal. Within rcutoff , the ’softer’ pseudopotential removes the rapid oscillations of the real, all electron wavefunction as seen by the smooth pseudo- wavefunction. Figure adapted from Payne et al. [52].

The purpose of the pseudopotential is to reduce the plane-wave basis set by removing the rapid oscillations near the core. This is accomplished by removing the core electrons from the problem and combining them with the atomic core. The core electrons are no longer treated explicitly like the valence electrons and since the core electrons do not contribute signiﬁcantly to the material properties (e.g. bonding

122 chemistry) this approximation is reasonable. Also, by eliminating the core electrons we eliminate the need to compute a number of single electron wavefunctions. The potential experienced by the valence electrons includes the effect of the atomic ion and the core electrons. The long history of pseudopotentials dates back to 1958 and the first attempt at pseudopotential generation is accredited to Phillips [81]. This was the first attempt to separate the potential into two parts, a part for the core electrons and a part for the valence electrons. These early pseudopotentials used the same potential for each angular momentum component of the wave function and are referred to as local pseudopotentials. More recent pseudopotentials, referred to as non-local or semi-local, use a different potential for each angular momentum component of the wave function to better represent the effects of the core on the valance electrons.

To obtain correct exchange-correlation energies it is essential that the pseudo- wavefunction produce the same electron density as the real wavefunction. This is achieved by non-local norm conserving pseudopotentials. The norm conservation condition is given as

Z rc Z rc 2 ps 2 2 ae 2 dr r |Ψnl(r)| = dr r |Ψnl(r)| , (A.26) 0 0 where rc is the cutoﬀ radius rcutoff , ψps and ψae are the pseudo-wavefunction and real, all electron wavefunction shown in Fig. A-2. Even though the elimination of the oscillations of the wavefunction in the core area via the norm conserving pseudopotentials reduces the number of plane-waves required, we can further reduce this number by removing the norm conserving condition. This type of pseudopotential is referred to as an ultrasoft pseudopotential. Ultrasoft pseudopotentials, developed by

Vanderbilt [59], are included in the VASP package. By removing the norm-conserving condition, these pseudopotentials require a much smaller basis size typically two to three times smaller than the norm-conserving pseudopotentials. To account for the

123 relaxation of norm-conservation, the Vanderbilt ultrasoft pseudopotentials require the electron density to be augmented within the rcutoff to account for the lost charge. The Vanderbilt type ultrasoft pseudopotentials were used throughout this work.

A.2.4 Minimization Algorithms

In order to determine the minimum energy configuration of any system, we need to optimize the atomic positions and obtain the ground state density that minimizes the Hamiltonian for that configuration. This process requires a number of steps that can be divided into two parts. The first part involves the refinement of the charge density and wavefunctions and the second part involves the optimization of the atomic positions.

The algorithms used in VASP for the reﬁnement of the wavefunctions and charge density include the conjugate gradient scheme [82,83], block Davidson scheme [84,85], or the residual minimization scheme - direct inversion in the iterative subspace (RMM-

DIIS) [86,87] or a combination thereof. All of the algorithms used in VASP are similar in that the residual vector is the central quantity used in the reﬁnement, the diﬀerence being how each algorithm implements the incorporation of the residual vector. VASP treats these algorithms as ”black boxes” allowing little user control over the behavior of the algorithm. One option includes initial optimization steps using the stable block Davidson scheme followed by RMM-DIIS afterwards. This was the method used for the computations in this work. The charge density mixing is achieved by a

Broyden/Pulay mixing scheme [87,88].

Optimization of the atomic positions can be accomplished through either static or dynamic computations. Static computations involve a number of independent computations where the ionic positions are updated manually by the user before each computation. This method is applicable to determine bulk structural properties for simple structures. To obtain the absolute minimum in total energy, the lattice

124 constant was varied and ﬁt to a parabolic equation as a function of total energy. This

is a simple method for simple structures that possess little or no internal parameters.

The second method relies on a dynamic process where the positions are updated

automatically during the computation process. To accomplish this VASP relies on

minimization of the free energy of the system with respect to the atomic positions

to compute the forces. The force is deﬁned as the derivative of the generalized free

energy functional F , where F depends on the wavefunctions ψ, partial occupancies f and atomic positions R. The variational property of this functional determines the ground state of the given conﬁguration

0 = δF (ψ, f, R) . (A.27)

Variations with respect to ψ and f will be 0 at the ground state of the user deﬁned

initial conﬁguration, determined by the algorithms described above, leaving us to

deﬁne the force as the derivative of the free energy functional F with respect to the

atomic position R dF (ψ, f, R) ∂F force = = . (A.28) dR ∂R

At the electronic ground state of the initial conﬁguration both ψ and f are constant,

so now it becomes simple to compute the force as the partial derivative of the free

energy functional with respect to only atomic position. The problem now reduces to

minimizing the forces due to atomic positions to determine the ﬁnal minimum energy

conﬁguration. There are three algorithms implemented in VASP to ﬁnd this minimum

force including the Quasi-Newton (DIIS) method, the conjugate gradient method and

damped molecular dynamics. The diﬀerence in the methods being the way in which

the algorithm determines the forces to update the atomic positions. In all cases,

the conﬁguration is considered to be in the minimum energy conﬁguration when the

change in energy of the newly updated conﬁguration, with respect to the energy of

125 the previous configuration, has fallen below a certain value previously defined by the user. The conjugate gradient and Quasi-Newton methods were used throughout this work. A flowchart of the dynamic procedure is outlined in the following figure.

Figure A-3: The initial charge density and trial wavefunctions are randomly generated independent quantities. At the beginning of each cycle, the charge density is used to construct the effective potential of the Hamiltonian of the Kohn-Sham equations, then the wavefunctions and charge density are refined for the given atomic configurations Hamiltonian. The optimized wavefunctions are then mixed with the input-charge density to determine the new charge density which is used to compute the total energy from the previous Hamiltonian. If the change in energy is greater than a predefined value, the cycle is repeated. If the computed wavefunctions and charge density minimize the energy for the given Hamiltonian, the force is computed and the atomic positions are updated and the whole process repeats until the minimum energy configuration is determined. Figure adapted from Kresse [89].

126 A.3 Files and Parameters

VASP requires a minimum of four input ﬁles to operate. These include the INCAR,

POSCAR, POTCAR and KPOINTS ﬁles. These ﬁles communicate the computation instructions, atomic positions and approximation parameters described in section A.2.

An example of these ﬁles will be given along with a brief discussion of their use. A complete list of the tags used in VASP is available online [90].

A.3.1 INCAR

The INCAR ﬁle contains all the information VASP requires to know what to compute and how to compute it. This is accomplished with over 50 tags that are read in from the INCAR ﬁle that VASP then uses as parameters in the computations.

Given below is a list of the relevant parameter tags used in this work and their meaning.

A.3.1.1 SYSTEM

The system tag defines a name for the system to be studied. The name given is only used for identification purposes and does not affect the computations. Typically this name will indicate the material and any associated defects.

A.3.1.2 NWRITE

Determines how much information is written to the output ﬁle. This information includes forces, stress, eigenvalues, charge density, total energy and convergence and timing information. A value of NWRITE is chosen from between 0 and 4 to determine how much of the above information is written. Typically for this work the value chosen is 2 which provides the required information while maintaining a reasonably small output ﬁle size, order v1 MB. The default value is NWRITE=2.

127 A.3.1.3 ENCUT

This tag sets the plane-wave basis cutoﬀ energy in eV as described in section

A.2.2.4. This value must be varied and chosen such that the total energy has converged with respect to the plane-wave-basis size. The default value of ENCUT is the enmax value set in the POTCAR, if more than one element is used the default value becomes the largest value of enmax in the concatenated POTCAR. The EN-

CUT value for all the work presented here was determined from convergence tests illustrated in Fig. A-4 for bulk CdS.

Figure A-4: In order to determine the plane-wave basis size, a test must be performed to check for total energy convergence. The ordinate axis is the difference in calculated, self-consistent total energy values for a given ENCUT (abscissa) value between two different lattice constant values. When this difference is less than 1 MeV, as shown in the inset, the plane-wave basis size is considered converged.

A.3.1.4 ISPIN

The ISPIN tag indicates whether or not spin polarized calculations are to be performed. The default value is ISPIN=1 for non spin polarized calculations. 128 A.3.1.5 ISTART

The ISTART tag determines whether a job is to be started from scratch or restarted from information acquired from a previous job (i.e. reading the previous

WAVECAR ﬁle). The default value of ISTART is 0 if no WAVECAR from a previous job is present, this instructs VASP to start a job from scratch with an initial guess for the charge density. The default value of ISTART is 1 if a WAVECAR is present, then VASP will restart, or continue a job by reading the wavefunction solutions to the

Kohn-Sham equations from the WAVECAR ﬁle written after an earlier job, to produce an initial charge density. Producing the charge density from the wavefunction solutions close to the converged wavefunctions will greatly reduce the time required to ﬁnd the converged wavefunction solutions.

A.3.1.6 IBRION

When performing an atomic relaxation job, the IBRION tag instructs VASP how to move and update the ionic positions. This is controlled by choosing the minimization algorithm. For IBRION=0, a molecular dynamics run is performed. For

IBRION=1, the RMM-DIIS algorithm is used. IBRION=2 initializes the conjugate gradient algorithm. IBRION=3 uses damped molecular dynamics. The work presented here used the conjugate gradient algorithm for the initial relaxation and then the RMM-DIIS algorithm to identify the ﬁnal relaxed ionic positions.

A.3.1.7 POTIM

The POTIM tag controls the force scaling for the minimization algorithms IB-

RION=1, 2 and 3 in VASP. For molecular dynamics, IBRION=0, POTIM determines the time step in femtoseconds.

129 A.3.1.8 EDIFF

The EDIFF tag sets the break condition for the electronic self-consistent loop.

This loop is shown as the inner loop in the VASP ﬂowchart in Fig. A-3. The relaxation of the electronic degrees of freedom will stop when the total energy change between two electronic steps are both smaller than EDIFF. The default value of EDIFF=10−4.

A.3.1.9 EDIFFG

The EDIFFG tag sets the break condition for the ionic relaxation loop. This loop is shown as the outer loop in the VASP ﬂowchart in Fig. A-3. If the value of EDIFFG is positive, the relaxation will be stopped when the change in the total energy is smaller than EDIFFG between two ionic steps. If the value of EDIFFG is negative, the relaxation will be stopped when all of the forces are smaller than

EDIFFG. The default value of EDIFFG=EDIFF∗10.

A.3.1.10 LWAVE

The LWAVE tag determines whether (TRUE) or not (FALSE) the wavefunction solutions to the Kohn-Sham equations are written to the output WAVECAR file. As mentioned the WAVECAR file can be used to restart atomic relaxation jobs, however the WAVECAR file can be large, order v10 MB to v1GB, and require large amounts of storage. The default value is LWAVE=.TRUE.

A.3.1.11 LCHARG

The LCHARG determines whether (TRUE) or not (FALSE) the charge densities are written to the output CHGCAR ﬁle. The CHGCAR ﬁle can used to compute the

(local) density of states or (L)DOS.

130 A.3.2 POSCAR

The POSCAR ﬁle contains all the information regarding the atomic positions and lattice geometry of the system. Figure A-5 illustrates a simple example of a POSCAR

ﬁle for bulk zinc-blende GaAs.

Figure A-5: An example of a POSCAR ﬁle for the two atom unit cell of GaAs.

The first line of the POSCAR is a comment line, similar to the SYSTEM tag in the INCAR, this is only for identification and does not affect the calculations. The second line of the POSCAR is the scaling factor, in the example in Fig. A-5 the scaling factor is the lattice constant of bulk silicon. The next three lines, numbers

3-5, are the lattice vectors used deﬁne the unit cell. The sixth line indicates the number of atoms per atomic species. In the example in Fig. A-5, the bulk zinc- blende GaAs is described with only one Ga atom and one As atom. The number of atoms listed and corresponding species is determined by the order in which they appear in the POTCAR. Their respective order is determined by the order in which their respective POTCAR’s were concatenated. The seventh line allows for control in determining which individual atomic coordinates are updated during relaxation jobs. This ’Selective Dynamics’ is shown in the example in Fig. A-5 where the Ga atom, at the origin, is forced to remain at that position. This is because of the three

131 ”F”s (False) to the right of the atom’s coordinates indicate that motion along each of the three lattice vectors is restricted. The As atom however is allowed to have its coordinates updated along each of the three lattice vectors. This is indicated by the three ”T”s (True) next to its coordinates. The eighth line indicates whether the atomic positions speciﬁed in the last lines are given in direct coordinates, fractional coordinates of the unit cell, or standard Cartesian coordinates. The last lines of the

POSCAR ﬁle are the atomic positions of the diﬀerent species given in either direct or Cartesian coordinates. We see that in the example the Ga atom is at the origin

”0 0 0”, the 4(a) Wyckoﬀ position of the space group F43m zinc-blende crystal, and the As atom is at ”0.25 0.25 0.25” which corresponds to the 4(c) Wyckoﬀ position.

A.3.3 KPOINTS

The KPOINTS ﬁle contains the information regarding the k-point mesh as described in section A.2.2.3. An example of a KPOINTS ﬁle is given in Fig. A-6. The

KPOINTS ﬁle example corresponds to the same two atom bulk GaAs zinc-blende structure used in the POSCAR example.

Figure A-6: An example of a KPOINTS ﬁle for the two atom unit cell of GaAs.

The first line of the KPOINTS file is a comment line. The second line in the file states the number of k-points. If the value given in the second line is ’0’, as in the example, the k-points are automatically generated. The third line determines the type of automatically generated mesh. This can either be Gamma centered mesh or, as in

132 the example, a Monkhorst-Pack generated mesh. The fourth line indicates how many k-point divisions are to be taken along each reciprocal lattice vector. Similar to the plane-wave cutoﬀ energy (ENCUT), the number of k-points (i.e. divisions) must also be varied until the total energy has converged. In the example, convergence below

1 MeV corresponds to a k-point mesh with 8 divisions along each of the reciprocal lattice vectors. It should be noted that the 8 division k-point mesh converges for the two atom unit cell. As the unit cell increases in size, we know from (A.12) that the size of the Brillouin zone, or reciprocal lattice unit cell, decreases in size. Therefore, as the real space unit cell gets larger, less k-points are required to sample the smaller reciprocal space.

A.3.4 POTCAR

The POTCAR file is the last file required by VASP. The POTCAR file contains the information regarding the pseudopotentials for each species in the system. VASP supplies the POTCAR files for both the ultra-soft LDA and GGA pseudopotentials.

The POTCAR ﬁles are used as is and do not permit user alteration. The only user requirement involving the POTCAR is when more than one species is involved. If a system is composed of more than one species, as GaAs in our example, the user is required to concatenate the POTCAR for Ga and the POTCAR for As. The order in which this is done determines the order the species numbers are given in the POSCAR as well as the order of the positions listed in the POSCAR.

A.3.5 Nudged Elastic Band (NEB) Method

The diﬀusion barriers presented in this work were calculated using the NEB method [91]. The NEB method is a modiﬁed chain-of-states method used to determine the saddle point for the minimum energy path for a given reaction. To understand

133 how this method works we begin with the principles common to the chain-of-states methods. Chain-of-states methods work by calculating the diﬀusion barrier between two known minimum energy sites by optimization of a number of linearly interpolated images of the adatom along the diﬀusing path. In order to maintain the distances between images and keep them on the minimum energy path the images are connected by springs. The force F describing each of these images i is the sum of the true force and the spring force and is given by

~ ~ ~ ~ s Fi = −∇V (Ri) + Fi , (A.29) where the force associated with the springs in given by

~ s ~ ~ ~ ~ Fi ≡ ki+1(Ri+1 − Ri) − ki(Ri − Ri−1) . (A.30)

This method is referred to as the plain elastic band method. However, the component of the spring force perpendicular to the path will tend to ’cut corners’ and move the chain, or band of images, oﬀ the minimum energy path. In addition, the component of the true force parallel to the direction of the path will pull the images down along the path. To correct for this, the component of the spring force is perpendicular to the path and the component of the real force parallel to the path are projected out.

The projection of these components of the force is referred to as nudging the images and therefore called the NEB method. Here, the force describing the images in the

NEB method is given by

~ ~ ~ ~ s Fi = −∇V (Ri) |⊥ +Fi · τˆiτˆi . (A.31)

To calculate the barrier, a minimization algorithm is used to adjust the atomic positions of each image to minimize the force given in (A.31) then the image corresponding

134 to the highest energy is taken to be the top of the diffusion path. The difference between this energy and that of the initial binding site is taken as the diffusion barrier.

135