A Topological Explanation of the Urbach Tail a Thesis Presented To

A Topological Explanation of the Urbach Tail

A thesis presented to

the faculty of the College of Arts and Sciences of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Dale J. Igram

April 2016

This thesis titled

A Topological Explanation of the Urbach Tail

DALE J. IGRAM

has been approved for

the Department of Physics and Astronomy

and the College of Arts and Sciences by

David A. Drabold

Distinguished Professor of Physics and Astronomy

Robert Frank

Dean, College of Arts and Sciences

ABSTRACT

IGRAM, DALE J., M.S., April 2016, Physics

A Topological Explanation of the Urbach Tail

Director of Thesis: David A. Drabold

The Urbach tail is considered as one of the most significant properties of amorphous structures because of its almost universal characteristics. Several theoretical attempts have been used to explain the existence of the Urbach tail only to fail. Utilizing the best models, it has been shown that Urbach tails are associated with topological filaments [11].

The original work in this thesis is presented in two parts. The first part involves the effects of thermal statistics on eigenvalues and electronic density of states (EDOS) for a 512 atom a-Si model, which revealed a significant variation of the EDOS and energy gap as a function of temperature; thus, showing how a semiconducting state can change to a conducting state. The second part deals with the comparison of two 4096 atom a-Si models (unrelaxed and relaxed), in which physical and electronic structures were compared, revealing that the connectivity of the short and long bonds for both models indicated that short bonds prefer short bonds and similarly for long bonds, and that short bonds of the relaxed model germinated and proliferated faster as compared to the unrelaxed model. The electronic structure calculations for both models were performed using an ab initio code SIESTA.

DEDICATION

To my wife, Esperanza

ACKNOWLEDGEMENTS

First of all, I would like extend my deepest gratitude to Dr. David A Drabold for being my advisor and mentor during my research endeavor, providing patient support and valuable suggestions on my thesis. Second, my appreciation goes to Drs. Charlotte

Elster, David Drabold, and Alexander Neiman for being a committee member and providing helpful suggestions for improving the thesis. Third, a special thank you goes to my student colleagues, Anup Pandey and Bishal Bhattarai, for assistance with Unix and

SIESTA, and Kiran Prasai, for valuable discussions regarding modifications of SIESTA and general support. I would also like to thank all the faculty and staff of the Physics and

Astronomy Department and students that provided support in one form or another. Most of all, I would like to express my deepest appreciation to my wife, Esperanza, who supported me during good times and bad times, and was very patient and understanding.

TABLE OF CONTENTS

Page

Abstract...... ……………………………………………………...... 3

Dedication...... 4 Acknowledgements...... ……………………………………...... 5 List of Figures……………………………………...... ………...... 8

1. Introduction…………………………………………………………...... 14

2. The WWW Model..…………………………………………………...... 21 2.1 Modeling Methodology and Theory...... ……….………...... 22 2.2 Discussion of Model Results …...... …..…...... 31 2.3 Summary...... ……………...... 37

3. Mathematical Description of SIESTA…………………………...... 38

3.1 Basic Assumptions...... ……………………...... 39 3.2 Pseudopotentials...... ……...... 39 3.3 LCAO Basis Sets...... ……………………...... 41 3.4 Matrix Elements of the Hamiltonian Operator…...... ……...... 45 3.5 Total Energy.....………………………………………………...... 48 3.6 Harris Functional...... 49 3.7 Summary...... 50 4. An Explanation of the Urbach Tail………………...... ………...... 51 4.1. Optical Absorption……………………………………..……...... 51 4.2 Electronic Density of States....………………………...... …...... 55 4.2.1 Electronic structure for amorphous materials...... 55 4.2.2 Electronic density of states for amorphous materials...... 58 4.3 Urbach Tails (A Theoretical Review)...... 61 7

4.3.1 Research from other research groups...... 61 4.3.2 Research from our group...... 64 4.4 Summary...... 76 5. Results and Discussion…...... ……………....……………………...... 77 5.1 The 512 Atom a-Si Structure...... 77 5.2 The 4096 Atom a-Si Structure...... 81 5.2.1 Physical structure information...... 81 5.2.2 Electronic density of states...... 84 5.2.3 Bond-center to bond-center correlation distribution...... 93 5.2.4 Connectivity of shortest and longest bonds...... 96 5.3 Summary...... 100 6. Summary and Future Work...... 102 6.1 Summary...... 102 6.2 Future Work...... 104 References……………………………………………...... …………...... 105 Appendix A. Derivation of Equation 2.2...... 111

LIST OF FIGURES

Figure Page

1.1. Disorder Types: (a) topological (no long-range order), (b) spin (on regularlattice), (c) substitutional (on regular lattice), (d) vibrational (about equilibrium positions)[5]...... 15

1.2. A pictorial representation of the structural origin of special featuresin the RDF. An oscillatory behavior is present for the density function ρ(r) as a function of r. The shaded area under a given peak gives the effective coordination number[1]...... 16

1.3 The straight lines represent what is now known as the Urbach tails. These lines were produced from absorption measurement of silver bromide crystals [101]...... 18

2.1. Local bond switching used to create random networks from the diamond cubic structure, where (a) represents the orientation of bonds in the diamond cubic structure and (b) the relaxed orientation of the atoms for a pair bond switch [23]...... 24

2.2. Potential energy variation for two alternative configurations which are related by a bond switch. The dashed line indicates the smooth transition that defines the barrier [34]...... 25

2.3. Procedure for creating continuous random networks...... 27

2.4. The Keating interactions [34]...... 29

2.5. A pictorial of the progression of the spectrum 푛(∆퐸) at different steps during the building process for various bond switch energy, ∆퐸. The bond switch as shown in Fig. 1 is represented by ∆퐸 = 퐸표 [34] ...... 30

2.6. Correlation function for various amounts of randomization. (a) perfect diamond cubic structure; (b) – (f), density of bond-pair switches per atom d = 0.06,0.12, 0.18, 0.24, and 0.30, respectively [34]...... 33

2.7. Correlation function for three different locations in the annealing process for the rms angular deviation of 15.6˚ (top), 13.2˚ (middle), and 12.6˚ (bottom), respectively [34]...... 35

2.8. A comparison of the correlation functions between experimental data for a- Ge and calculated data for a-Si [23]...... 36

3.1. Local pseudopotential and neutral atom pseudopotential for silicon. The dashed line represents −푍⁄푟 and 푉푁퐴 the screened local part of the pseudopotential from an electron charge distribution produced by filling the first-ζ basis orbitals with the free-atom valence occupations [55]...... 41

3.2. A comparison of the convergence of the total energy versus the sizes of LCAO basis sets (from SIESTA) and plane-wave basis sets [55]...... 43

3.3. Total energy per atom versus lattice constant for bulk Si for different basis sets. PW denotes a well converged (50 Ryd cutoff) plane-wave calculation. The minimum for the different curves is represented by the dotted line [55]...... 44

3.4. The dependence of the lattice constant, bulk modulus and cohesive energy for bulk Si versus the cutoff radius of the basis orbitals. To keep the plots simple, the s and p orbital radii are made the same. PW has the same definition as in Fig. 3 [55]...... 45

4.1. Three regions of the optical absorption coefficient for disordered materials [66]...... 52

4.2. A polaron in an energy-configuration diagram with square lattice structures. The open circles and black dots represent atoms and electrons, respectively. The vertical axis represents the total energy of the electron and the horizontal axis a 3N Euclidean space of constituent atoms. A polaron can be considered as a laterally shifted energy minimum of an excited state [67]...... 53

4.3. Optical absorption data for a-Si:H with the open circles representing the experimental data obtained by [9]. All the fits to (3) are for 2.0 푥 102푐푚−1 < 훼 > 5.0 푥 103푐푚−1 [74] ...... 54

4.4. Relationships between atomic and electronic structures for crystals and non- crystals [67]...... 55

4.5. A simple representation of electron distributions of the atoms in solids (middle) and energy levels for isolated atoms and solids (right) [67]...... 56

4.6. Simplified view of three tetrahedrally coordinated lattices and corresponding band structures. (a) represents a crystal lattice, (b) a strained lattice having a dangling bond, and (c) strained fully connected lattice [67]...... 57

4.7. Electronic density of states of crystalline materials [66]...... 59

4.8. Electronic density of states of tetrahedrally bonded amorphous materials [81]...... 60

4.9. Density of states as a function of y displaying the Gaussian tail (푦 ≫ 2), the Urbach tail (0.1 < 푦 > 2), and the H-L tail (0 < 푦 > 0.1) where 휖퐿 ≡ ћ2/2푚퐿2. The essence of the most probable potential fluctuation and wave function in each region is depicted schematically [84]...... 63

4.10. a) Valence band tails for crystalline Si and Crystal Si with two vacancies. (b) Valence band tails for DTW model and a DTW model with randomly distorted bond lengths [70]...... 65

4.11. Electron filaments for different valence and conduction tails. “H” represents the highest occupied molecular orbital state; H1, next lower energy state, etc. “L” represents the lowest unoccupied molecular orbital state; L1, next higher energy state, etc. [70]...... 66

4.12. Number of atoms in short-bond and long-bond cluster (Ncl ) are plotted against the central bond length of the cluster for a-Si. The dotted line indicates the mean bond length [86]...... 67

4.13. Correlation between the two lengths 푅푆푖−푆푖 and 푅푂−푂 . A linear fit to the data is indicated by the red line and the dotted lines represent the mean values of the two lengths [86]...... 68

4.14 Examples of short 푅푂−푂 and long 푅푆푖−푆푖 clusters in a-SiO2 structure. Grey sites indicate the defect nuclei [86]...... 69

4.15. Inverse participation ratio of the density of states for a-Si. Fermi level is represented by dotted line. The valence (conduction) tail is determined by short O-O bonds (long Si-Si bonds) [86]...... 70

4.16. HOMO (purple) and LUMO (cyan) states correspond to short and long bonds. The mean bond length is at 1.36 Å (1.44 Å) for the HOMO (LUMO) [86]...... 70

4.17. Electronic density of states for 105 –atom a-Si model from maxent reconstruction for 107 and 150 moments. The Fermi level is in the middle of the gap [87]...... 72

4.18. A least-square fit to exponentials for valence and conduction tails for maxent reconstruction of the electronic density of states for a 105 –atom a- Si model with 107 moments [87]...... 73

4.19. Comparison of electronic density of states between a 512-atom model and a 105 -atom model [87]...... 73

4.20. Strain recovery in a 512-atom model of a-Si. ∆푟 represents the difference in bond length from the mean [87]...... 74

4.21. Instantaneous snapshots for 512-atom model at 300K for two different times [87]...... 75

4.22. Electronic density of states for 512 atom models without and with filaments [87]...... 76

5.1. A pictorial representation of the electronic density of states for three different temperature with a Fermi level of about -4.110 eV...... 79

5.2. A close up view of the electronic density of states for three different temperatures near the energy gap. The Fermi level for 150K and 300K is about -4.10eV, whereas for 600K around -3.90eV...... 79

5.3. Eigenvalue fluctuations for the HOMO, LUMO, superjacent and subjacent states for a temperature of 150K and 950 MD time steps...... 80

5.4. Eigenvalue fluctuations for the HOMO, LUMO, superjacent and subjacent states for a temperature of 300K and 950 MD time steps...... 80

5.5. Eigenvalue fluctuations for the HOMO, LUMO, superjacent and subjacent states for a temperature of 600K and 950 MD time steps...... 81

5.6. Pair correlation function for the relaxed and unrelaxed 4096 a-Si models..... 82

5.7. Bond length distribution for the unrelaxed and relaxed 4096 a-Si models...... 83

5.8. Bond angel distribution for the unrelaxed and relaxed 4096 a-Si models...... 83

5.9. Dihedral angle distribution for unrelaxed and relaxed 4096 a-Si models...... 84

5.10. Electronic density of states and the Fermi energy levels for three basis sets (SZP, DZ, and SZ) for the unrelaxed structure...... 86

5.11. A closeup of the electronic density of states and the Fermi energy levels for the unrelaxed structure near the energy gap...... 86

5.12. Electronic density of states and the Fermi energy levels for three basis sets (SZP, DZ, and SZ) for the relaxed structure...... 87

5.13. A closeup of the electronic density of states and the Fermi energy levels for the relaxed structure near the energy gap...... 87

5.14. Electronic density of states and associated Fermi energy levels for unrelaxed and relaxed structures using a SZP basis...... 88

5.15. A close-up of the electronic density of states and associated Fermi energy levels near the energy gap for unrelaxed and relaxed structures using a SZP basis...... 89

5.16. Electronic density of states of the valence band with an exponential fit for the unrelaxed model...... 90

5.17. Electronic density of states of the valence band with an exponential fit for the relaxed model...... 90

5.18. Semilog plot of the electronic density of states for the unrelaxed and relaxed structures. The slopes represent the Urbach energies, which are 130 meV (unrelaxed) and 142 meV (relaxed). A SZP basis set was used...... 91

5.19. Electronic density of states of the conduction band with an exponential fit for the unrelaxed model...... 92

5.20. Electronic density of states of the conduction band with an exponential fit for the relaxed model...... 92

5.21. Semilog plot of the electronic density of states for the unrelaxed and relaxed structures. Note: the curves do not represent a good exponential function...... 93

5.22. Bond-center to bond-center correlation distribution for both the 4% shortest and 4% longest bonds for the unrelaxed model...... 94

5.23. Bond-center to bond-center correlation distribution for both the 4% shortest 95 and 4% longest bonds for the relaxed model......

5.24. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) shortest bonds for the unrelaxed model. Note the regions where clustering is occurring...... 96

5.25. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) longest bonds for the unrelaxed model. Note the regions where clustering is occurring...... 97

5.26. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) shortest bonds for the relaxed model. Note the regions where clustering is occurring...... 97

5.27. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) longest bonds for the relaxed model. Note the regions where clustering is occurring...... 98

14 1. INTRODUCTION

Amorphous materials have attracted much attention in the last few years. The first reason is their potential industrial applications as applicable materials for fabricating devices, and the second is the lack of understanding of many properties of these materials, which are very different from those of crystalline materials [1]. Some of their properties are different even from one sample to another of the same material.

Amorphous semiconductors and insulators are used for fabricating many optoelectronic devices. The detection capabilities in some biosensors are based on the electronic and optoelectronic properties of a-Si. As an example, thin film a-Si light sensors can detect emitted light, which is converted into an electrical signal resulting in the optoelectronic detection of DNA molecules [2]. Another example would be ion-sensitive thin film a-Si transistors that produces an induced voltage shift when sensing DNA and proteins [3,4].

The fundamental difference between a crystal and an amorphous material is that a crystal has large-range structural order, due to its periodicity, whereas an amorphous material does not. Amorphous materials consist of what is known as randomness or a lack of long-range order.

By definition, randomness can be described in many forms [5]. The most important ones are: topological, spin, substitutional, and vibrational disorder, which are depicted in Fig. 1.1.

Figure 1.1. Disorder Types: (a) topological (no long-range order), (b) spin (on regular lattice), (c) substitutional (on regular lattice), (d) vibrational (about equilibrium positions), Figure from Ref. [5].

A standard is used to determine the amount of disorder a structure may have.

That standard is the perfect crystal, which is defined as an atomic arrangement that has infinite translational symmetry in three dimensions. Therefore, using this definition, an imperfect crystal can be one that is not infinite but, has surfaces or has a defect, which can be considered as small perturbations to perfect crystallinity. The disorder that is considered in this thesis is topological. This type of disorder has no translational periodicity (no long-range order) and there are various degrees depending on the amount of short-range order a structure may have.

Due to the lack of periodicity, amorphous structures cannot be fabricated very easily when using experimental scattering techniques, unlike a crystal structure. The information from these scattering experiments is mostly restricted to the first two 16 coordination shells for the case of covalent materials [5]. Bond lengths and angles of nearest-neighbor atoms consisting of the basic unit cell can be obtained, however, the relative distribution of these unit cells cannot be ascertained with certainty. An example of this would be in the case of radial distribution functions (RDF) obtained from scattering experiments for monatomic structures, where peaks, besides the first and second, cannot be uniquely associated with a particular interatomic correlation, resulting from many contributions from the higher-lying shells as exemplified by Fig. 1.2.

Figure 1.2. A pictorial representation of the structural origin of special features in the RDF. An oscillatory behavior is present for the density function 휌(푟) as a function of 푟. The oscillatory action is a result of the density function 휌(푟) representing average interatomic seperations. The shaded area under a given peak gives the effective coordination number, Figure from Ref. [1].

If multicomponent structures are considered, things become significantly more complicated where a single diffraction experiment is inadequate for identifying the origin 17 of any peak in the RDF in terms of particular atomic pair correlations. To circumnavigate these problems, constructing models allows the simulation of the structure. Structural parameters, such as RDF, density, etc. can be determined from a model and then compare with experiment. Structural modeling plays a significant role in the understanding of an amorphous structure.

To ascertain electronic structure information, accurate and reliable computer simulations are needed. Some methods used are: 1) empirical or semi-empirical orthogonal tight-binding , 2) ab initio non-orthogonal tight-binding, 3) non-self- consistent Harris-functional, and 4) fully self-consistent density functional theory (DFT).

Despite that fact that items 1) and 3) are computationally inexpensive as compared to ab initio methods, they are inadequate in terms of transferability and reliability [6]. On the other hand, ab initio methods can be applied to several structures for an enormous computational cost with regards to time and memory.

One of the most important characteristics of disordered structures is the Urbach tail. It was not until 1953 that Franz Urbach noted that a semi-log plot of the optical absorption coefficient 훼(휔) versus photon frequency produced a straight line [101], as depicted in Fig. 1.3, where 훼(휔) is associated with the electronic transitions from the valence to conduction band tail in disordered solids. 18

Figure 1.3. The straight lines represent what is now known as the Urbach tails. These lines were produced from absorption measurement of silver bromide crystals, Figure from Ref. [101].

Other experimental studies [7,8,9] on a variety of disordered semiconductors and glasses also revealed an Urbach exponential spectral behavior that lasted over five decades, which firmly suggests that the Urbach absorption edge/tail is almost an universal characteristic of disordered materials. One of the most significant topics of amorphous materials is the connection between structural features and optical or electrical properties of the materials. In the late 1950’s, the work from several researchers indicated that electrons are localized because of disorder [10]. Over several years, many theoretical methods have been developed in an attempt to explain the existence of the Urbach tail associated with disordered structures only to fall short due to inapplicability to all systems or being incomplete in theory. In our group, utilizing the very best models have indicated that Urbach tails are associated with topological filaments, consisting of short and long bonds [11], which will be discussed in more detail later. 19 To better understand the optical and electrical properties of amorphous materials will require an improved comprehension of the Urbach tail [12,13]. The localized states in the Urbach tail for both the valence and conduction bands are responsible for the degradation of electron and hole transport [14,15]. These localized states can act as traps causing the electron and hole mobilities to decrease significantly. The larger the width of the Urbach tail (larger Urbach energy) the greater the degradation of transport.

Perhaps the most significant property of the Urbach tail is its universality.

Urbach tails have been identified in various materials such as, Si, GaAs, InP, chalcogenides [16,17,18], and just recently, carbon nanotubes [19]. So far, the universality of the Urbach tail has not been adequately explained.

The originality of this research, as presented in Chapter 5, is reflected in the outcome of the variation of electronic density of states as a function of temperature and the ab initio calculations performed on an unrelaxed and relaxed 4096 atom a-Si model.

The ab initio calculations for the unrelaxed and relaxed 4096 atom a-Si models revealed good Urbach tails for both the unrelaxed and relaxed models for the valence band, but not well-defined Urbach tails for the conduction band.

The thesis consists of the following chapters. Chapter 2 provides a mathematical description of an ab initio density functional theory (DFT) code named SIESTA [102], which was utilized for the electronic structure calculations. Basic assumptions involving exchange-correlation and norm-conserving pseudopotentials are discussed. A more detail consideration of pseudopotentials and basis sets used are provided, followed by a review of the total energy and the Harris functional. 20 In Chapter 3, a highly successful amorphous structural model, known as the

WWW model, is presented. A detailed description of that model methodology and theory are reviewed. Also, included are correlation function results associated with the randomization and annealing processes for amorphous structures.

A more detail explanation of the Urbach tail is provided in Chapter 4. A review of the optical absorption coefficient of three regions for disordered structures is examined. Since electronic density of states are related to the optical absorption coefficient, electronic density of states is also considered. The remainder of the chapter is a commentary of the work done by other researchers and our group regarding the

Urbach tail.

Computational results are presented and discussed in Chapter 5 for 512 and 4096 atom a-Si models. For the 512 atom a-Si model, the effects of thermal statistics on eigenvalues and electronic density of states are examined. With regards to the 4096 atom a-Si model, two structures (unrelaxed and relaxed) are compared with respect to physical structure, electronic density of states, bond-center to bond-center correlation distribution, and connectivity of 2%, 4% and 8% shortest and longest bonds. 21 2. THE WWW MODEL

An enormous effort has been made in modeling amorphous structures, such as a-

Si[20,21,22,23], a-Ge[22,23], and a-C[27] due to their importance in materials science.

However, because of the lack of long-range order or periodicity, their structures cannot be easily ascertained like a crystal structure when using x-ray diffraction. There does not exists any experimental techniques that produce atomic resolution that is equal to crystallography for these types of structures. In this case, what is typically done is to try to create a model, which will produce a 1-D radial distribution function (RDF) that correlates well with experimentally determined RDF’s [28,29] and has structural distortions (rms bond length and rms bond angle deviations) within experimental estimates. Such a model could be considered as a standard, which has a systematic procedure for constructing a model.

Introduced by Zachariasen in 1932, Continuous Random Network (CRN) models are designed to have short-range order and intermediate-range order with bond lengths and bond angles slightly distorted. Their topology usually involves fivefold and sevenfold rings, when fabricated for a tetrahedral network, as well as a small number of sixfold and eightfold rings [23, 39] that are distinctive of a c-diamond structure. As a result, these models [22] need to have several hundred atoms as a minimum in order to obtain useful information on the properties of the structure and be constrained to periodic boundary conditions to avoid problems associated with free surfaces. The use of periodic boundary conditions allows the surface effects (i.e. dangling bonds) to be avoided.

However, there is a downside to using periodic boundary conditions, which is the introduction of pseudo-crystallinity. Therefore, the model must be large enough so that 22 the correlation function has decayed to its average value for a distance less than l, the unit cell dimension. In other words, the correlation length L needs to be less than l. Also included in the requirements, is that the distortion should not be larger than experimental estimates. Several attempts [20,21,24,25,26] have been made to achieve this goal, only to fall short.

However, one attempt to obtain this goal has proven very successful and is known as the WWW model [22,23] after the developers, Wooten, Winer and Weaire. The model is conceptually simplistic, and yet by utilizing a unique physical process (bond switching) and a numerical scheme (Metropolis algorithm), models of large structures that agree with experiment can be fabricated.

In this chapter, a detailed description of the WWW model, including the modeling methodology, theory and some results are provided.

2.1 Modeling Methodology and Theory

The original WWW scheme begins with a perfect diamond structure with periodic boundary conditions and a cubic supercell, although several years later it was shown [37] that a perfect structure is not required. A process of local bond switching or rearrangement [22] is performed on the diamond cubic structure, as depicted in Fig. 2.1, in a repetitious manner where fourfold coordinates are maintained and fivefold and sevenfold rings are produced. Here a ring is defined as any closed non-returning path of bonds [30]. The bond switches occur at a temperature just above the melting point for the model, where the melting point is defined as the temperature at which the average square displacement of the atoms from their equilibrium positions in the diamond structure increases linearly with time and that the structure factor approaches a value of 23 order 1/푁 [41]. This local bond switching implicates the modification of bonds around two atoms only. More specifically, the bond switching has effects [23] on the topology of the structure in the following way.

1) Introduction of four fivefold rings (due to the reduction of sixfold rings),

2) Removal of twelve sixfold rings (due to the conversion into five- and sevenfold rings),

3) Production of sixteen sevenfold rings, and

4) Elimination of twenty-four eightfold rings.

Bond switching is performed by selecting a pair of nearest-neighbor atoms arbitrarily as shown in Fig. 1 for atoms 2 and 5. The two bonds that are to be switched are selected randomly with the following constraints [22]. 1) To minimize strain energy, the selected bonds must almost be parallel to each other, as illustrated in Fig. 2.1 for bonds 1-2 and 5-6. The parallel requirement can be obtained by requiring that the two bonds are not members of the same 5, 6, or 7-fold ring. Fig. 2.1 also reveals two other possible bond switches associated with bonds 2-4 and 5-8, and bonds 2-3 and 5-7. 2)

Allowing 4-fold rings to exist. The 4-fold rings significantly increase the escape rate from deep metastable states and they are easily removed during the annealing process

[27, 34].

Figure 2.1. Local bond switching used to create random networks from the diamond cubic structure, where (a) represents the orientation of bonds in the diamond cubic structure and (b) the relaxed orientation of the atoms for a pair bond switch, Figure from Ref. [23].

A pictorial representation of the energy barrier [34], represented by the dashed line, that must be conquered in making the bond transposition associated with Fig. 2.1, is illustrated in Fig. 2.2.

Figure 2.2. Potential energy variation for two alternative configurations which are related by a bond switch. The dashed line indicates the smooth transition that defines the barrier, Figure from Ref. [34].

Transitioning from a perfect diamond cubic structure to a randomized structure due to bond switches increases the strain energy of the structure and is represented by moving from a low energy state to a higher energy state (left to right) in Fig. 2.2. If the bond switching process is continued long enough then all identifiable features of the diamond cubic structure will be lost resulting in a highly distorted system with an rms bond angle deviation of about 22˚, which is twice the required value of 12.5˚ [31].

However, this process introduces large strains in the structure, which need to be minimized or eliminated. The elimination of these large strains can be achieved by reducing the temperature in small steps and establish thermal equilibrium at each step, also known as the simulated annealing process. At each step, an accept or reject decision is made according to the relative probability [37]

푃푎 ⁄ = exp (−(퐸푎 − 퐸푏)⁄ 푘퐵푇), (2 .1) 푃푏 26 where 푃푎 and 푃푏 represent the probabilities for the after and before expected bond switches, 푘퐵 the Boltzmann constant, T the temperature, and 퐸푎 and 퐸푏 the total energies of the structure after and before the expected bond switches. If the exponential term is greater than unity, corresponding to 퐸푏 > 퐸푎, the bond switch is accepted. On the other hand, if 퐸푏 < 퐸푎, the bond switch is still accepted, but with a relative probability of

푃푎⁄푃푏 < 1. To accept a bond switch with a probability requires that a uniform random number be selected between 0 and 1, and if the probability is greater than this number, the bond switch is accepted; otherwise, it is rejected. In reference to Fig. 2.2, the simulated annealing process is illustrated by the transition from a high energy state to a lower energy state and at each step a Keating potential (explained later) is applied to assist in partially relaxing the structure; thus, lowering the strain energy. The procedure described above is a Metropolis algorithm.

As this procedure is continued the structure becomes relaxed geometrically (the reduction of the strain energy allowing the stretching and bending of the bonds) and topologically (producing more bond switches). Geometrical relaxation is achieved by calculating the force and force gradient, due to its nearest and next-nearest neighbors, for each atom at a time and relocating that atom at a position of equilibrium associated with the bond-stretching and bond-bending forces, while holding the other atoms fixed. The distance that each atom must travel [34] is determined by taking the gradient of the

Taylor series of the potential energy at the minimum energy location for that atom and then solving for ∆풓 in (2.2).

∇푉 = −(∇2푉)∆풓 (2.2) 27 ∆풓 represents the vector distance that an atom needs to move. The derivation of (2.2) is given in Appendix A. Derivation of Equation 2.2.

Therefore, the total strain energy, rms bond-length deviation, and rms bond-angle deviation are reduced until an optimized amorphous structure is fabricated as the temperature approaches zero. It has been estimated from ring statistics of successful random network models that a concentration of at least 0.1 pair bond switching per atom must be randomly introduced to have a reasonable model.

To summarize the modeling process, a general description of the procedure for creating continuous random networks is shown in Fig. 2.3.

Start with Perfect Diamond Structure

Randomize structure (bond switching) just above melting temperature

Relax structure by simulated annealing at a sequence of successively lower temperatures to T=0

Figure 2.3. Procedure for creating continuous random networks.

As mentioned earlier, the Keating elastic potential came about when attempts were made to fit the elastic and vibrational properties of the column IV elements [32] and was later utilized in the WWW model for describing the bond-bending and -stretching forces of a structure, semi-empirically. The Keating elastic potential involves only short- range interactions. For a pair of atoms, a group of nearest and next-nearest neighbors 28 defines the topology, as well as the structure’s energy. The two atoms interact only if they are a part of the group. By knowing the group of nearest and next-nearest neighbors a relatively simple interaction expression, such as the Keating potential, can be used.

The structure is relaxed by minimizing the Keating elastic potential, equation

(2.3), [32,33,27] after each bond switch.

3 훼 2 2 3 훽 1 2 2 ∑ ∑ ′ ′ 푉 = 2 푙푖(풓풍풊 ∙ 풓풍풊 − 푟표 ) + 2 푙{푖,푖 }(풓풍풊 ∙ 풓풍풊 + 3푟표 ) (2.3) 16 푟표 8 푟표

where α and β represent the bond-stretching and bond-bending force constants, respectively, 푉 the Keating elastic potential, and 푟표 the strain-free equilibrium bond length in the diamond cubic structure. The first sum in (2.3) is for all atoms l and their four nearest neighbors denoted by i and the second sum for all atoms l and pairs of specific neighbors (i,i’ ), and 풓풍풊 is the distance vector from l to its i’th neighbor and

풓풍풊′ the distance vector from l to its next-nearest neighbor i’. It can be seen that the first term in (2.3) represents the bond-stretching energy and the second the bond-bending energy. Utilizing the first term only is not sufficient for stabilizing a tetrahedrally bonded structure. The second term is required since the constraints for maintaining constant bond lengths are less than the degrees of freedom. In other words, the bonds are relaxed to their ideal un-stretched length, which frustates the structure. Thus, the second term provides the means for a structure to minimize the bond-bending energy, resulting in

훽 more tetrahedral order within the constraint of fixed bond lengths. Values for the ratio 훼 of 0.1 to 0.3 are typically used for Si and Ge [34]. The force constants, α and β, can be obtained [38] either by fitting experimental macroscopic elastic constants and bulk moduli or from the microscopic atomic vibrations using temperature-dependent EXAFS 29 (Extended X-ray Absorption Fine Structure) measurements. The Keating interactions are illustrated in Fig. 2.4.

Figure 2.4. The Keating interactions, Figure from Ref. [34].

Another approach [34] to describing the modeling process is to utilize the spectrum of ∆퐸 values (Fig. 2.2), where a single ∆퐸 value is for one bond switch of the type shown in Fig. 2.1, and represented in Fig. 2.5. Using this approach one can create an evolution of the bond switch density of states during the fabrication process. It is important to note that for the final structure, which is annealed at a temperature T=0, the structure has to conform to the condition, 푛(∆퐸) = 0 for ∆퐸 < 0. What is noteworthy is that there exists a finite value of 푛(∆퐸) for ∆퐸 = 0. Understanding the bond switching process for ∆퐸 ≈ 0 could provide deeper comprehension of the abnormal thermal and acoustic properties of amorphous materials at very low temperatures.

Figure 2.5. A pictorial of the progression of the spectrum 푛(∆퐸) at different steps during the building process for various bond switch energy, ∆퐸. The bond switch as shown in Fig. 1 is represented by ∆퐸 = 퐸표, Figure from Ref. [34].

31 2.2 Discussion of Model Results

As was mentioned earlier, unlike crystals, amorphous structures cannot be easily fabricated by conventional means, only by comparing their total correlation function, rms bond length and rms bond angle deviations with experimental total correlation function, rms bond length and rms bond angle deviation estimates, can verification of the structure be done.

From the theory of neutron and X-ray diffraction of amorphous materials, interference effects between the waves scattered from structurally correlated atoms exists, resulting in desirable structural information, such as the total correlation function 푡(푟) defined [25] as

푡 (푟) = 4휋푟푛(푟) (2.4) and related to the differential correlation function 푑(푟) = 4휋푟[푛(푟) − 푛0], by 푡(푟) =

푑(푟) + 푟0(푟), where 푟0(푟) = 4휋푟푛0, 푛0 is an average sample number density, and 푛(푟) the local number density of atoms within a range of 푟 from an atom. The experimental correlation function 푡′(푟) is the convolution of the total correlation function 푡(푟) with a peak function 푃(푟) defined as

1 ∞ 푃(푟) = ∫ 푀(푄) cos 푟푄 푑푄 휋 0 . (2.5) where 푀(푄) is a modification function used to decrease the undesired ripples produced by the termination of a Fourier integral at the maximum scattering vector, 푄푚푎푥. 푄 represents the scattering vector with magnitude of (4휋⁄ 휆) sin 휃, where 휆 is the incident neutron wavelength and 휃 the half scattering angle. 0푀(푄) = for 푄 > 푄푚푎푥. 32 There are basically two reasons why 푡(푟) should be used rather than the radial distribution function

푔(푟) = 푟푡(푟) = 4휋푟2푛(푟). (2.6)

Firstly, using the correlation function allows structural data to be more graphically represented across a larger range of 푟. Secondly, and most importantly, is that experimental broadening by 푃(푟) occurs in 푡(푟), which is symmetric and independent of

푟, resulting in a direct comparison between models and experiment if experimental broadening is incorporated in 푡(푟) of a model by utilizing line shape functions [40], such as a Gaussian function, also known as Gaussian broadening.

Fig. 2.6 illustrates how the total correlation function changes as the structure becomes more randomized. As depicted in Fig. 2.6, the total correlation functions (b) –

(f) are more featureless as the randomization, as denoted by amounts of bond switches per atom d, continues as compared with the perfect diamond cubic structure (a). The first two peaks in (d)-(f) represent tetrahedral bonding.

Figure 2.6. Correlation function for various amounts of randomization. (a) perfect diamond cubic structure; (b) – (f), density of bond-pair switches per atom d = 0.06,0.12, 0.18, 0.24, and 0.30, respectively, Figure from Ref. [34].

Once a well-randomized structure, with very high strain energy, has been created a continuation of bond switches (simulated annealing) are required to further lower the energy. There is always the fear of the well-randomized structure returning to the original diamond cubic structure; thus, a d value greater than 0.6 is considered to avoid this from happening. After a long process of annealing the result should be a well- randomized structure with very low energy.

The correlation function for three locations in the annealing process is depicted in

Fig. 2.7, where the locations are represented by the rms angular deviation of 15.6˚, 13.2˚, 34 and 12.6˚, respectively. Fig. 2.7 illustrates how a long annealing process can result in a good model, which correlates well with experiment that requires a rms angular deviation of 12.5˚. It is well known that for this methodology, if the starting structure is diamond cubic, then the final structure could be the original diamond cubic or a well-randomized structure that agrees well with experiment or something in between.

Figure 2.7. Correlation function for three different locations in the annealing process for the rms angular deviation of 15.6˚ (top), 13.2˚ (middle), and 12.6˚ (bottom), respectively, Figure from Ref. [34].

Fig. 2.8 depicts the comparison between the correlation functions for a-Ge

(experimental [36]) and a-Si, which is scaled to Ge. The scaling of a-Si with respect to a-Ge is proportional to the bond lengths, r, where 푟 = 2.35 Å for a-Si and 2.46 Å for a-

Ge. Gaussian broadening with full width of 0.23 Å at half-maximum was incorporated in 36 the calculations of the correlation functions so that comparison with experiment could be performed.

Using this methodology or algorithm to create models has proven that these models will produce the same RDF and be well-correlated with experiment whether or not the models reveal any diamond cubic memory effects in their structure factor. It is believed that if any diamond cubic memory effects exist in the structure factor it is due to poor initial randomization and not a result of the annealing process.

Figure 2.8. A comparison of the correlation functions between experimental data for a- Ge and calculated data for a-Si, Figure from Ref. [23].

The discrepancies in 푡(푟) at large r were assumed to be due to very large voids of the size of 100 Å, which resulted in the measured macroscopic density of real a-Si films to be around 90% the density of c-Si [36]. For the areas of homogeneity (lacking any voids) the density is very close to that of c-Si.

37 2.3 Summary

Over the years, the WWW approach has proven to be a very successful model.

Despite its simplicity, the WWW model satisfies all the necessary requirements, such as having hundreds of atoms, constraint to boundary conditions, having 4-fold rings, structural distortion not exceeding experimental estimates, producing a structure that correlates well with experiment, and equally important, easy to use.

Many factors contribute to its success. In particular, the specific bond switching scheme that is used for the fabrication process, and the use of the Metropolis algorithm and Keating potential in the simulated annealing process that results in a structure having low angular deviation, a good correlation function, and a low strain energy.

Because of the specific bond switching method considered only the knowledge of the surrounding atoms is needed to implement the semi-empirical Keating elastic potential. This fact alone may explain why the WWW model is so successful with regards to experimental results.

The intent of this chapter was to provide further insight into the WWW model, along with possible reasons for its success. Various concepts pertaining to bond switches, energy variation due to a bond switch, Metropolis algorithm, the general modeling process, Keating interactions, and spectrum of the bond switch density of states were presented. The outcome of the correlation functions as a function of the density of bond switches per atom and rms angular deviation were reported and discussed, as well as a comparison study between calculated and experimental results. Even though other models, like DTW (Djordjevic,Thorpe, Wooten) [27] and BM (Barkema, Mousseau)

[37], exists they are only slight modifications of the WWW model. 38 3. MATHEMATICAL DESCRIPTION OF SIESTA

Accurate and reliable simulations of complex materials containing thousands of atoms are extremely important in the area of materials science. Since the advent of computers, computational power has been steadily increasing, resulting in a computational expense on the order of 푁3, with 푁 representing the number of atoms.

Over the years, several methods have been developed capable of "표푟푑푒푟 − 푁" or 푂(푁) scaling, which results in the computational cost being linearly scaled [42-47]. The past methods consisted, in order of complexity: empirical or semiempirical orthogonal tight- binding methods; ab initio non-orthogonal tight-binding and non-self-consistent Harris- functional methods followed by the fully self-consistent density functional theory (DFT) methods.

SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms)

[102] is a fully developed self-consistent DFT with a flexible linear combination of atomic orbital (LCAO) basis set and 푂(푁) scaling. The LCAO basis set can have multiple zeta and polarized bases, providing flexibility based on required accuracy and available computational power. Other capabilities include spin polarization, generalized gradient approximations (GGA), exchange – correlation (XC), stress tensor and 푘 − sampling integration calculations.

It is for these reasons that SIESTA was used for all the electronic structure calculations in this research. In this chapter, the following topics are presented: Basic assumptions, Pseudopotential, Basis sets, Matrix elements of the Hamiltonian operator,

Total energy, and Harris functional. This chapter is a summary of references [55] and

[56]. 39 3.1 Basic Assumptions

Some of the most fundamental approximations, excluding the Born and

Oppenheimer approximation, involves the XC and the utilization of pseudopotentials, which allows for the elimination of core electrons. The exchange – correlation is considered by utilizing the Kohn-Sham DFT [48]. Both the local (spin) density approximation (LDA/ LSD) [49] and GGA with complete ab initio [50] are incorporated in SIESTA.

Because a real-space grid is utilized, standard norm-conserving pseudopotentials

[51] can be used to avert computations for core electrons as well as make the valence charge density smoother, which will be discussed in more detail in the next section.

Increased simplicity and efficiency can be achieved by converting the pseudopotentials into their separable (Kleinman-Bylander) KB form [52]. In this form, only two-center integrals are needed to determine the nonlocal part of the pseudopotential matrix elements. Finally, for treating of the XC in the core region, nonlinear partial-core correction is employed.

3.2 Pseudopotentials

Even though pseudopotentials are not necessary for atomic basis sets, they are convenient for eliminating core electrons and, more importantly, the expansion of a smooth, pseudo-charge density on a uniform spatial grid. The pseudopotentials are received by SIESTA in a semilocal form, which means a different radial potential 푉푙(푟) for each angular momentum 푙, and parametrized by using the Troullier-Martins parametrization [53]. The semilocal form is converted completely into a total nonlocal form [52] as described below 40 푃푆 퐾퐵 푉 = 푙표푐푎푙(푟) + 푉 (3.1)

퐾퐵 퐾퐵 퐾퐵 푙푚푎푥 푙 푁푙 퐾퐵 퐾퐵 퐾퐵 푉 = ∑푙=0 ∑푚=−1 ∑푛=1 |휒푙푚푛 > 푣푙푛 < 휒푙푚푛| (3.2)

퐾퐵 푣 푙푛 =<휑푙푛|훿푉푙(푟)|휑푙푛 >, (3.3)

where 푟 = |풓| and 훿푉푙(푟) = 푉푙(푟) − 푉푙표푐푎푙(푟) . The KB projection functions are

퐾퐵 퐾퐵 휒푙푚푛(풓) = 휒푙푛 (푟)푌푙푚(풓̂), (3.4)

퐾퐵 where 휒푙푛 (푟) = 훿푉푙(푟)휑푙푛(푟) and 푌푙푚(풓̂) represent a spherical harmonic. 휑푙푛(푟) is defined as

−1 푑2 푙(푙+1) [ 푟 + + 푉 (푟) + 푉퐻(푟) + 푉푋퐶(푟)] 휓 (푟) = 휀 휓 (푟) 2푟 푑푟2 2푟2 푙 푙푛 푙푛 푙푛 . (3.5)

푉퐻(푟) and 푉푋퐶(푟) represent the Hartree and XC potentials for the pseudo-valence charge

ℎ 푒 = = 푚 = 1 density and atomic units ( 2휋 푒 ) are used throughout this chapter. The only pseudopotential part that is represented in the real space grid is the local part of the

퐾퐵 pseudopotential 푉푙표푐푎푙(푟), where the matrix elements of the nonlocal part 푉 can be easily and accurately determined by two-center integrals. 푉푙표푐푎푙(푟) is smoothed through optimization, resulting in it being equal to a potential produced by a positive charge distribution having the form [54]

휌푙표푐푎푙(푟) ∝ exp [−(sinh(푎푏푟) /sinh (푏))2]. (3.6) a and b have been selected to simultaneously optimize real-space localization and reciprocal-space convergence, which after numerical evaluation were given the values of

푎 = 1.82/푟푐표푟푒 and 푏 = 1. 푉푙표푐푎푙(푟) represents the unscreened local part of the pseudopotential that was created as the electrostatic potential made by a localized distribution of positive charge. The local part of the pseudopotential 푉푙표푐푎푙(푟) for silicon is illustrated in Fig. 3.1. 41

Figure 3.1. Local pseudopotential and neutral atom pseudopotential for silicon. The dashed line represents −푍⁄ 푟 and 푉푁퐴 the screened local part of the pseudopotential from an electron charge distribution produced by filling the first-ζ basis orbitals with the free- atom valence occupations, Figure from Ref. [55].

The requirement that 푉푙표푐푎푙(푟) must equal 푉푙(푟) for a distance greater than the pseudopotential core radius 푟푐표푟푒 implies that 훿푉푙(푟) = 0 for 푟 > 푟푐표푟푒 and that

퐾퐵 휒푙푛 (푟) = 0 beyond the radius 푟푐표푟푒 , regardless of the value of 휀푙푛 . The traditional practice of equating 휀푙푛 to the valence atomic eigenvalue 휀푙 and the function 휑푙푛(푟) equivalent to the corresponding eigenstate 휓푙(푟) is incorporated in SIESTA.

3.3 LCAO Basis Sets

As was mentioned in the introduction of this chapter, 푂(푁) scaling is utilized, which depends enormously on the sparsity of the Hamiltonian and overlap matices. The use of compact basis orbitals (orbitals that are zero beyond a certain radius) for handling the sparsity is employed by SIESTA. This approach allows the energy to be only variational. Therefore, for a radius less than the radius of confinement, the atomic basis 42 orbitals turn out to be products of an atomic orbital numerical radial function 휙푖푙푛(푟푙) and a spherical harmonic 푌푙푚(푟̂푙) as defined by (3.7). For a detailed discussion regarding the radius of confinement, refer to [55].

휙푖푙푚푛(풓) = 휙푖푙푛(푟푖)푌푙푚(풓̂풊), (3.7) where 풓풊 = 풓 − 푹풊, and 푹풊 the location of atom i. A multiple zeta basis is defined as a basis with orbitals with the same angular momentum dependence, but different radial dependence. A cubic spline interpolation [16] is used in defining the radial functions, which can have a different cutoff radius and its profile can be user-specific. SIESTA contains several automatic procedures for creating a good minimal SZ basis sets applicable to semi-quantitative simulations and double – zeta plus polarization (DZP) basis sets that provide extremely good results for most systems. The DZP basis is considered as a ‘standard’ basis because it provides an acceptable balance between well converged results and acceptable computational costs. A comparison of the performance between the LCAO atomic basis sets and plane waves basis sets, having the same pseudopotentials and geometries, is depicted in Fig. 3.2. Figure 3.2 reveals that the

LCAO basis sets are more efficient in regards to the number of basis orbitals. The curve in Fig. 3.2 reveals the total energy per Si atom versus the cutoff of a plane-wave basis.

Arrows point out energies obtained with different LCAO basis sets (from SIESTA) and plane-wave cutoff’s which provided the same energies. The basis sizes are represented by numbers in parentheses. Acronyms represented in Fig. 3.2 are: SZ, single ζ (valence s and p orbitals); DZ, double ζ; TZ, triple ζ; DZP, double ζ valence orbitals with SZ polarization d orbitals, TZP, triple ζ valence with SZ polarization; TZDP, triple ζ valence orbitals and double ζ polarization; TZTP, triple ζ valence with triple ζ polarization; 43 TZTPF, same as TZTP with SZ polarization f orbitals. A single ζ (SZ) basis set, also known as a minimum basis set, would consist of only one basis function of valence s and p orbitals per atomic orbital (AO), such as a Slater-type orbital (STO). If a d orbital is added to a single ζ basis set for polarization purposes, the new basis set would be considered a SZP basis set. A basis set with two SZ basis sets per AO is called a double ζ

(DZ) basis set and a DZ basis set with a d orbital included would be known as a DZP basis set.

Figure 3.2. A comparison of the convergence of the total energy versus the sizes of LCAO basis sets (from SIESTA) and plane-wave basis sets, Figure from Ref. [55].

Despite the fact that plane waves have faster algorithms, the LCAO basis was considered for SIESTA because of its suitability for 푂(푁) methodology.

The convergence of total energy for Si versus lattice parameter for different basis sizes is illustrated in Fig. 3.3. From Fig. 3.3, the ‘standard’ DZP basis provides a well converged solution as compared with most plane-wave results. To achieve the same performance as a plane-wave basis, a TZTPF basis would have to be used. 44 The dependence of the lattice constant, bulk modulus and cohesive energy for bulk Si with respect to SZ and DZP basis orbitals is depicted in Fig. 3.4. It reveals that the DZP basis converges well with regards to a well converged plane-wave calculation having the same pseudopotential for lattice constant, bulk modulus and cohesive energy.

Fig. 3.4 also illustrates how well the SZ basis converges with respect to experiment for the bulk modulus and cohesive energy.

Figure 3.3. Total energy per atom versus lattice constant for bulk Si for different basis sets. PW denotes a well converged (50 Ryd cutoff) plane-wave calculation. The minimum for the different curves is represented by the dotted line, Figure from Ref. [55].

Figure 3.4. The dependence of the lattice constant, bulk modulus and cohesive energy for bulk Si versus the cutoff radius of the basis orbitals. To keep the plots simple, the s and p orbital radii are made the same. PW has the same definition as in Fig. 3.3, Figure from Ref. [55].

3.4 Matrix Elements of the Hamiltonian Operator

If the nonlocal-pseudopotential approximation is considered, then the standard

Kohn-Sham Hamiltonian can be expressed as

푙표푐푎푙 퐾퐵 퐻 푋퐶 퐻̂ = 푇̂ + ∑퐼 푉퐼 (풓) + ∑퐼 푉̂퐼 + 푉 (풓) + 푉 (풓), (3.8)

1 푇̂ = − ∇2 푉퐻(푟) 푉푋퐶(푟) where 2 is the kinetic energy operator, I an atom index, and

푙표푐푎푙 퐾퐵 are the total Hartree and XC potentials, and ∑퐼 푉퐼 (푟) and ∑퐼 푉̂퐼 are the local and nonlocal parts of the pseudopotential of atom I. A nonlocal-pseudopotential refers to a pseudopotential that is angular momentum dependent. The nonlocal-pseudopotential

퐾퐵 푉̂퐼 is defined as 46 ̂ 퐾퐵 퐾퐵 퐾퐵 퐾퐵 푉퐼 = ∑푙푚|휙퐼푙푚 > 휖푙푙 < 휙퐼푙푚|, (3.9)

퐾퐵 where lm are angular momentum quantum numbers and 휙퐼푙푚 represent the Kleinman-

Bylander projectors, which are written as

퐾퐵 푙표푐푎푙 휙퐼푙푚 = [푉퐼푙(푟) − 푉퐼 (푟)]휓퐼푙푚(푟), (3.10)

퐾퐵 푙표푐푎푙 −1 휖 푙푙 =<휓퐼푙푚|푉퐼푙 − 푉퐼 |휓퐼푙푚 > . (3.11)

휓퐼푙푚(푟) represent the atomic pseudo-orbitals and are solutions of the radial Schrö dinger

푙표푐푎푙 equation with potentials 푉퐼푙(푟) (with no cutoff radius). Now, 푉퐼푙(푟) − 푉퐼 (푟) is

푐표푟푒 considered zero only outside the pseudopotential core radius 푟퐼 , which implies that

퐾퐵 푐표푟푒 휙퐼푙푚 are also zero beyond 푟퐼 .

푙표푐푎푙 The elimination of the long range of 푉퐼 is accomplished by screening it with

푎푡표푚 푎푡표푚 the potential 푉퐼 , which was produced by an atomic electron density 휌퐼 . The

푎푡표푚 electron density 휌퐼 can be created by populating the basis functions with correct valence atomic charges. It is worth noting that the ‘neutral atom’ (NA) potential,

푁퐴 푙표푐푎푙 푎푡표푚 푐표푟푒 푎푡표푚 푉퐼 = 푉푖 + 푉푖 is zero for 푟 > 푟퐼 . By letting 훿휌(풓) = 휌(풓) − 휌 and

훿푉퐻(풓) be the electrostatic potential produced by 훿휌(풓), equation (3.8) can be expressed as

퐾퐵 푁퐴 퐻 푋퐶 퐻̂ = 푇̂ + ∑퐼 푉̂퐼 + ∑퐼 푉퐼 (풓) + 훿푉 + 푉 (풓). (3.12)

Here 휌(풓) is the self-consistent electron density and 휌푎푡표푚 the sum of atomic densities,

푎푡표푚 푎푡표푚 휌 = ∑퐼 휌퐼 . For the first two terms of (3.12) the matrix elements utilize two-

푁퐴 center integrals of the form < 휙퐼푙푚푛|푉퐼 |휙퐼′푙′푚′푛′ >, which arise in a two-center particle system. These integrals can be evaluated by using an elliptic coordinate system

[58]. The last three terms of (3.12) have potentials that are determined on a three- 47 dimensional real-space grid where the resolution of the grid is manipulated by a ‘grid cutoff’ energy 퐸푐푢푡, which represents the maximum kinetic energy of the plane waves that is characterizable in the grid without any jagged features. The short-range screened

푁퐴 pseudopotentials 푉퐼 (풓) are formulated as a function of the distance to atoms I and easily obtained at any desired grid point. The last two terms require the calculation of the electron density on the grid, which is defined as

∗ 휌(풓) = ∑휇휐 휌휇휐 휙휐(풓)휙휇(풓) (3.13)

∗ where 휌휇휐 = ∑푖 푐휇푖 푛푖푐푖휐, 푛푖 the occupation of state 휓푖 , 푐휇푖 = 〈휑푖|휓푖〉 and 푐푖휐 = 푐휐푖

(from def. of Hermiticity). To determine the density at each grid point, all the atomic basis orbitals need to be known from (3.7), and after interpolating the radial part from numerical tables, (3.13) is used to calculate the density. At any given grid point, there exists a small number of nonzero basis orbitals; thus, allowing the determination of the density to be performed in 푂(푁) operations. However, the calculation of 휌휇휐 does not scale linearly with the system size, requiring the use of special 푂(푁) techniques. Only the matrix elements 휌휇휐 where 휙휇 and 휙휐 overlap are needed for the determination of the density and can be stored as a sparse matrix of 푂(푁) size. After the valence density has been calculated, it is then added to the nonlocal core correction [59], if required. The nonlocal core correction consists of a spherical charge density design to simulate the atomic cores, which is also obtained from a radial grid. Once this all has been done,

푉푋퐶(풓) can be calculated. Now, 훿푉퐻(풓) is determined by finding 휌푎푡표푚 at each grid point and subtracting it from 휌(풓) resulting in 훿휌(풓). Fast Fourier transformations are typically used to solve Poisson’s equation for the determination of 푉퐻(풓) , resulting in finding the total grid potential 48 푉(풓) = 푁퐴(풓) + 훿푉퐻(풓) + 푉푋퐶(풓). (3.14)

∗ 3 Therefore, for each grid point, 푉(풓)휙휐(풓)휙휇(풓)∆풓 is obtained for all nonzero pairs of

∗ 휙휐(풓)휙휇(풓)휇(풓) and then added to the Hamiltonian matrix element 퐻휇휐 .

3.5 Total Energy

The total energy is written in the Kohn-Sham format as a sum of a band-structure

(BS) energy, which is the sum of the energies of the occupied states 휓푖, and some

‘double-count’ corrections terms. The BS term is expressed as

퐵푆 퐸 = ∑ 푛푖푖 < 휓푖|퐻̂|휓푖 >= ∑ 퐻휇휐휇휐 휌휐휇 = 푇푟(퐻휌). (3.15)

The 휓푖 represent the eigenvalues of the Hamiltonian at convergence, whereas, the corrections terms are functionals of the electron density that are obtained from (3.13), and atomic positions. In general, the Kohn-Sham total energy is written as

퐾푆 1 퐻 3 푋퐶 푋퐶 3 푍퐼푍퐽 퐸 = ∑ 퐻휇휐휇휐 휌휐휇 − ∫ 푉 (풓)휌(풓)푑 풓 + ∫(휖 (풓) − 푉 (풓))휌(풓)푑 풓 + ∑퐼<퐽 2 푅퐼퐽

(3.16) where I, J represent atomic indices, 푅퐼퐽 ≡ |푅퐽 − 푅퐼|, 푍퐼, 푍퐽 are the valence ion pseudoatom charges and 휖 푋퐶(풓)휌(풓) is the exchange-correlation energy density.

Because (3.8) is a function of 휌(풓), which in turn is a function of {휙(풓)}, the set of equations defined by 퐻(휌(푟))휙 = 퐸(휌(푟))휙 results in a nonlinear eigenvalue problem.

However, the actual total energy equation utilized by SIESTA is

1 퐸퐾푆 = ∑ (푇 + 푉퐾퐵)휌 + ∑ 푈푁퐴 (푹 ) + ∑ 훿푈푙표푐푎푙(푹 ) − ∑ 푈푙표푐푎푙 휇휐휇휐 휇휐 휇휐 2 퐼퐽 퐼퐽 퐼퐽 퐼<퐽 퐼퐽 퐼퐽 퐼 퐼

1 ∫ 푉푁퐴 (풓)훿휌(풓)푑3풓 + ∫ 훿푉퐻 (풓)훿휌(풓)푑3풓 + ∫ 휖푥푐 (풓)휌(풓)푑3풓 + 2 (3.17) 49 −1 푉푁퐴 = ∑ 푉푁퐴 훿휌 = 휌 − ∑ 휌푎푡표푚 푈푁퐴(푹) = ∫ 푉푁퐴 (풓)∇2푉푁퐴(풓 − 푹)푑3푟 where 퐼 퐼 , 퐼 퐼 , 퐼퐽 4휋 퐼 퐽 and

퐾퐵 퐾퐵 푁퐴 퐾퐵 푉휇휐 = ∑ < 휙휇훼 |휒훼 > 푣훼 < 휒훼|휙휐 > . Both 푈퐼퐽 (푹) and 푉휇휐 can be determined from two-center integrals, whereas, the last three terms are determined using the real- space grid. The main difference between (3.16) and (3.17) is that (3.17) does not have any explicit long-range potentials. In (3.16), the last term represents long-range interactions. Another advantage that (3.17) has, aside from the slowly varying

휖 푋퐶(풓)휌(풓) (exchange-correlation energy density) is that the grid integrals consists of

훿휌(풓), which is much smaller than 휌(풓) resulting in significantly reduced errors linked to the finite grid spacing. By direct differentiation of (3.17) with respect to atomic positions, atomic forces and stresses can be determined.

3.6 Harris Functional

Up to this point, only the self-consistent Kohn-Sham functional has been discussed. A functional that is not self-consistent consumes less computational power and cost, although with less accurate results, is the Harris functional. The Harris functional is more advantageous than the Kohn-Sham functional for large atomic systems

(4096 atoms or more). Results can be obtained in a reasonable amount of time, (only a single diagonalization of 퐻 and 푆 is required) with slightly less accuracy for a very covalent system. The Harris energy functional can be written as [60,61]

1 휌푖푛(풓)휌푖푛(풓′) 퐸퐻푎푟푟푖푠(휌푖푛) = ∑ 푛표푢푡 < 휓표푢푡|퐻̂푖푛|휓표푢푡 > − ∬ 푑3풓푑3풓′ 푖 푖 푖 푖 2 |풓−풓′|

푖푛 푖푛 푖푛 3 푍퐼푍퐽 +∫(휖푥푐 (푟) − 푣푥푐 (푟))휌 (푟)푑 푟 + ∑퐼<퐽 (3.18) 푅퐼퐽 50 ̂푖푛 푖푛 표푢푡 where 퐻 is the KS Hamiltonian created by a trial density 휌 and 휓푖 are its eigenvectors. The first term in (3.18) is represented by 푇푟(퐻푖푛휌표푢푡) and the other terms are considered as the ‘double-count’ corrections. The most significant advantage of

푖푛 (3.18) is that 휌푖 does not have to be determined from a set of orthogonal electron states

푖푛 휓푖 . It only needs to be a superposition of atomic densities. Equation (3.16) complies

표푢푡 표푢푡 푖푛 푖푛 with (3.18) exactly, except that 휓푖 and 푛푖 are replaced with 휓푖 and 푛푖 . By subtracting (3.16) from (3.18) yields

퐻푎푟푟푖푠 푖푛 퐾푆 푖푛 푖푛 표푢푡 푖푛 퐸 [휌 ] − 퐸 [휌 ] = ∑휇휐 퐻휇휐 (휌휇휐 − 휌휇휐) (3.19) or

퐻푎푟푟푖푠 푖푛 퐾푆 푖푛 푖푛 표푢푡 푖푛 퐸 [휌 ] = 퐸 [휌 ] + ∑휇휐 퐻휇휐 (휌휇휐 − 휌휇휐). (3.20)

It has been determined that (3.20) provides no further improvements in the convergence of the atomic forces after two or three iterations.

3.7 Summary

An ab initio code SIESTA was utilized to perform all the electronic structure calculations. A mathematical description of this self-consistent DFT consisted of basic assumptions, pseudopotentials, LCAO basis sets, Hamiltonian matrix elements, total energy and the Harris functional. Basic approximations involving XC and pseudopotentials are considered in SIESTA to avoid dealing with core electrons. LCAO basis sets are employed by SIESTA for manipulating the sparsity of the Hamiltonian and overlap matrices. A Harris functional (non-self-consistent scheme) was utilized in the electronic structure calculations, instead of the SCF method, because it consumes less computational power at a lower cost but, for a price in accuracy. 51 4. AN EXPLANATION OF THE URBACH TAIL

The optical and electrical properties of crystalline and amorphous materials are govern by the electronic density of states in the extended and localized states. Of the two types of properties, the optical properties of amorphous materials are the most important for basically two reasons. Firstly, the optical properties of amorphous materials can easily be related and compared to that of crystalline materials, since the optical properties for c- materials are well known and secondly, optical properties are directly connected to the structural and electronic properties of these materials. Within the optical properties, two very important properties are the absorption coefficient and the electronic density of states (EDOS). Investigations pertaining to the optical properties have provided some important concepts, such as the Tauc tail, mobility edge, mobility gap, Urbach tail, weak absorption tail, and charged defects. Because the optical absorption coefficient and the

EDOS are related [65], the chapter will start with a discussion on optical absorption followed by electronic density of states. The remainder of the chapter will be focused on one of the three band tails, the Urbach tail, which is considered as a universal characteristic of most disordered materials, from crystals with defects to amorphous materials of various chemistry. The universality and exponential behavior of the Urbach tail are two interesting features that require further investigation. The section on the

Urbach tail will contain work performed by others and by our group.

4.1 Optical Absorption

The observed optical absorption coefficient, 훼(퐸푝), for electronic transitions near the mobility edge in disordered materials, can be categorized in three distinct regions as illustrated in Fig. 4.1. Here, 퐸푝 signifies the photon energy. 52

Figure 4.1. Three regions of the optical absorption coefficient for disordered materials, Figure from Ref. [66].

The strong absorption region A is characterized by a power function [65]

푇 푛 훼(퐸푝) ∝ (퐸푝 − 퐸푔 ) , (4.1)

푇 where 퐸푔 represents the Tauc optical gap and 푛 = 2, and is known as the Tauc tail. It is not clear whether the Tauc tail is due to band-to-band (extended-extended states) or band- to-edge (extended-localized states) transitions. However, experimental results [68] have suggested that the Tauc gap is the result of optical transition between localized and extended states. The lower absorption region B, considered as the weakly temperature- dependent Urbach tail [65], is described by an exponential function of the form

훼(퐸푝) ∝ exp (퐸푝⁄퐸푈), (4.2) where 퐸푈 represents the Urbach energy/width of localized tail states. In many amorphous materials, the Urbach tail is almost temperature independent near room temperature, which can be interpreted in a couple of ways: 1) due to a polaron effect 53 (Fig. 4.2) or 2) disorder-induced band tailings [69,70,71]. This is not to say that the disorder is temperature independent.

Figure 4.2. A polaron in an energy-configuration diagram with square lattice structures. The open circles and black dots represent atoms and electrons, respectively. The vertical axis represents the total energy of the electron and the horizontal axis a 3N Euclidean space of constituent atoms. A polaron can be considered as a laterally shifted energy minimum of an excited state, Figure from Ref. [67].

Two puzzling features associated with the Urbach tail are: 1) the existence of a minimal Urbach energy of 퐸푈 ≥ 50 − 60 푚푒푉 in many amorphous materials [72], indicating that some universality exists; and 2) the existence of an Urbach tail focus as exemplified in Fig. 4.3 for a-Si:H [73]. Equation (4.3) is used to fit to the optical absorption data (open circles)

ћ휔−ћ휔표 훼(ћ휔) = 훼표표푒푥푝 ( ), (4.3) 퐸표 where 훼표표 , a pre-exponential constant, and ћ휔표, a characteristic energy, are independent of the amount of disorder. However, the width of the Urbach tail 퐸표 was found to change significantly with the amount of disorder. From experimental analysis of [74], 54 parameters 훼표표 and ћ휔표, known as the Urbach focus parameters, have been determined to have values of 1.5 푥 106푐푚−1 and 2.2 푒푉. As illustrated in Fig.4.3, all the solid lines are extrapolated to a single point known as the Urbach focus. The optical absorptions data was obtained by [74].

Figure 4.3. Optical absorption data for a-Si:H with the open circles representing the experimental data obtained by [9]. All the fits to (3) are for 2.0 푥 102푐푚−1 < 훼 > 5.0 푥 103푐푚−1 , Figure from Ref. [74].

For region C in Fig. 4.1, referred to as the Weak Absorption Tail or residual absorption [65], is characterized by

훼(퐸 푝) ∝ exp (퐸푝⁄퐸푊). (4.4)

The parameter 퐸푊 represents the width of the defect states. This tail appears to be due to several origins, including impurities and the reason for its exponential form has not yet determined [67].

55 4.2 Electronic Density of States

In the previous section, observed optical absorption coefficient results were presented and discussed (Fig. 4.1) for disordered materials. From a theoretical or computational perspective, it is possible to discuss this subject by realizing that the optical absorption coefficient is proportional to the EDOS [67]. Having knowledge of the

EDOS is important for the understanding of the electronic structure of amorphous materials, which will be presented in the first section. A discussion of EDOS will follow.

4.2.1 Electronic structure for amorphous materials

A brief review of the electronic structure of amorphous materials is presented.

The relationships between atomic and electronic structures for crystals and non-crystals are depicted in Fig. 4.4. The electron dispersion curves (electron energy 퐸 vs wavenumber 푘) are calculated by using Bloch functions, assuming one-electron approximations, for periodic structures; thus, allowing the determination of the EDOS

[75]. However, due to the lack of long-range order (lack of periodicity) the wavenumber

푘 cannot be considered a good quantum number because of the localization of the electron wave-functions with respect to a distance ∆푟. Thus, the dispersion curves cannot be determined as well as the EDOS.

Figure 4.4. Relationships between atomic and electronic structures for crystals and non- crystals, Figure from Ref. [67].

56 For disordered materials a different approach must be taken in order to determine the EDOS as illustrated in Fig. 4.4. The EDOS can be described in the following way:

1) Bonds are formed between individual atoms having specific energy levels

(eigenvalues) associated with its atomic orbitals; thus, resulting in a molecular structure

(hybrid structure) with a certain collection of energy levels. 2) As the number of bonds increases the number of energy levels increases resulting in bands due to partial extensions and overlap of atomic orbitals. This is exemplified by Fig. 4.5.

Figure 4.5. A simple representation of electron distributions of the atoms in solids (middle) and energy levels for isolated atoms and solids (right), Figure from Ref. [67].

The density of energy levels (eigenvalues) in the bands are used to determine the EDOS.

Fig. 4.6 depicts a simplistic picture of the relationships between atomic and band structures using only tetrahedrally coordinated 2D lattices. These lattices have similar short-range structures. As illustrated in Fig. 4.6, the band edge is dependent on the structural periodicity. For a perfect crystal (a), the edges of the valence and conduction 57 bands are flat and the electron wave-functions are extend over the whole crystal. In contrast, for the disordered structures, the band edges may be altered by medium-range structures, such as a 5-membered ring (b) or a strained 3-membered ring (c). These modifications may be different in the valence band as compared to the conduction band.

For silicon, bond being a 푠푝3, the angular strain provides a greater effect on the valence band edge since it is governed by the directional 푝 orbital, whereas, the conduction band edge is influenced by the spherical 푠 orbital [76].

Figure 4.6. Simplified view of three tetrahedrally coordinated lattices and corresponding band structures. (a) represents a crystal lattice, (b) a strained lattice having a dangling bond, and (c) strained fully connected lattice, Figure from Ref. [67].

Within the band structure, an electron can experience three different types of movement: 1) energetic transitions, 2) transport, and 3) local motions between gap states.

The first type of movement involves optical processes (absorption/emission) or thermal relaxation (vibrational). The second type is due to drift and diffusion created by dc electric fields and gradients of carrier densities. And thirdly, an electron may hop between gap states by way of tunneling, where the term “hop” refers to the phonon- 58 assisted quantum-mechanical tunneling of an electron from one localized state to another

[77]. Also, it should be noted, that the atomic structure is assumed to be rigid for the conventional band model.

4.2.2 Electronic density of states for amorphous materials

As mentioned earlier, for amorphous materials, the lack of periodicity does not allow the wave vector 풌 to be defined; thus, not allowing the electronic band structure

퐸(풌) to be determined. However, the number of electronic states at energy 퐸 per unit energy 푛(퐸) is considered as a well-defined expression for amorphous materials [78] and can be written as

1 푁 푛(퐸) = ∑ 푓 훿(퐸 − 퐸 ) 푁 푖=1 푖 , (4.5) where 푁푓 represents the total number of occupied states, 퐸푖 the eigenvalue of the electronic Hamiltonian, and 푁 the number of atoms in the material. An important property of crystalline and amorphous materials is the presence of a bandgap, which separates the valence and conduction bands. The existence of perfect periodicity in crystals aids in producing this bandgap. In contrast, the definition of a bandgap for tetrahedrally bonded amorphous materials is not as obvious, according to the pioneering work of [79] in which bands are described by a local arrangement (bonds). The bonds are described using a tight-binding (TB) Hamiltonian defined as

퐻 = 푉1 ∑푖,푘≠푙 |휑푖푘 >< 휑푖푙| + 푉2 ∑푖≠푗,푘 | 휑푖푘 >< 휑푗푘|, (4.6)

3 where 휑푖푘 denotes the 푘th 푠푝 hybrid orbitals associated with the 푖th atom and, 푉1 and 푉2 are matrix elements that are considered as real “banding” and “bonding” parameters .

The first term is a sum of the interactions where the 푘th and 푙th wave functions belong to 59 the same atom (bands), and the second term is a sum associated with the wave functions with the same bonds (bonds). The ratio 푉1⁄푉2 is very important in determining the form of the band structure and for silicon has a value of 1/3. Another important property is the shape of the band tails near the Fermi energy level. It is well known that for crystalline materials, the slope of the band edges (푑푛(퐸)/푑퐸) diverges, known as van Hove singularities, and occurs at values of 퐸 where the constant energy surface contains points at which 훁E(퐤) disappears [80]. This happens at the top of the valence band (HOMO) and bottom of the conduction band (LUMO) as illustrated in Fig. 4.7. However, for amorphous materials smooth edges or band tails exists because of the lack of periodicity

(long-range order), which is exemplified by Fig. 4.8.

Figure 4.7. Electronic density of states of crystalline materials, Figure from Ref. [66].

As shown in Fig. 4.8, delocalized (due to short-range order) and localized regions

(due to no long-range order) are separated by a mobility edge. These localized states do not exist for crystalline materials.

Figure 4.8. Electronic density of states of tetrahedrally bonded amorphous materials, Figure from Ref. [81].

The mobility edge is also known as the Anderson transition, which represent disorder- induced localization. According to the Anderson model of localization, if an electron is placed on an atomic site at time 푡 = 0, then the state, at energy 퐸, is assumed to be localized if as 푡 → ∞ the electron has not diffused away for a temperature 푇 = 0 and has a finite probability 푝 ∝ 푒−2훼푟 of staying at a distance of 푟 within the volume. However, if there exists a probability of diffusion at 푇 = 0, then the state is considered delocalized or extended. The left side (delocalized states) of the Anderson transition, is considered the metal side of the transition and the right side (localized states) the insulator side.

61 4.3 Urbach Tails (A Theoretical Review)

This section is divided in two subsections. The first consisting of a review of the work that has been done by other researchers involving the central idea of fluctuating potentials in explaining the exponential behavior of Urbach tails and the second pertains to the research performed by our group, which utilizes topology to obtain a better understanding of Urbach tails.

4.3.1 Research of other research groups

Many investigators have attempted to explain the existence of the Urbach tail, such as the work done by [82]. The purpose of this paper was to show that 푛 ≈ 1 in

(4.7) was expected for a-Si and for physically reasonable choices of the spatial correlation length 퐿 and photon energy 퐸 .

푛 휌 (퐸) ∝ exp(− 퐸 ⁄퐸표 ), (4.7) with 퐸 being measured away from the band edge, 퐸푏. They start by using an analytical expression derived by [83] (4.8), valid for the band tail region, and show how (4.8)

1 푛 = 푑 = 3 퐿 → 0) 푑 predicts that 2 for within the white-noise limit ( . Here, represents dimensionality.

퐸2 휌(퐸) = (퐸 ⁄푑)푑[푎(푣, 푥)/휁(푑+1)/2)푒푥푝 [− 퐿 푏(푣, 푥)], 퐿 2휁 (4.8) where 푣 = (퐸푏 − 퐸)/퐸 퐿 represents a dimensionless energy variable, 푥 ≡ ћ휔/퐸퐿 a dimensionless variational parameter, 휁 = 〈푉(푟)푉(푟)〉 − 〈푉(푟)〉2 the variance of the

2 ∗ 2 random potential 푉(푟), and 퐸퐿 = ћ /2푚 퐿 the normalizing energy (energy associated with localizing a particle of mass 푚∗ within the correlation length 퐿. 푎(푣, 푥) and 푏(푣, 푥) are dimensionless functions defined as 62 1 푥 푑 푥푑 4 푎(푣, 푥) = [ ][ ] ( + 푣)푑/2 푋 (1 + )푑(푑+1)/4 (4.9) √2휋 2√휋 4 푥

푥푑 4 푏 (푣,푥) = ( + 푣)2(1 + )푑/2. 4 푥 (4.10)

Using (4.8) they were able to obtain the well-known semi-classical result (4.11) of Kane

[83] for the limit → ∞ , which results in 푛 = 2.

퐸 퐸2 − 퐿 푏(푣, 푥) = − . 2휁 2휁 (4.11)

(4.11) is valid in the limit of long-range correlations where the kinetic energy of localization is very small. For 푛 = 1, it was determined that 퐿 is on the order of atomic dimensions and 퐸 resides in a narrow energy range.

Similar to the work of [82], is the work done by [84] where they derive from first principles the density of states in the band tail, resulting in asymptotically exact equations for the Halperin and Lax (H-L) tail (퐿 ≪ 휆) and a Gaussian tail (퐿 ≫ 휆), where 휆

≡ ћ/(2푚|퐸|)1⁄ 2 is the de Broglie wavelength. It was discovered that both regions, (H-

L) tail and Gaussian tail, are experimentally inaccessible for reasonable choices of rms potential fluctuations and spatial correlation length 퐿. However, the tail that is located in the crossover regime (퐿~휆), which exists between the (퐿 ≪ 휆) and (퐿 ≫ 휆) regions, is observable and displays basically linear exponential behavior over many decades.

It is claimed that band tail states are due to potential fluctuations of depth 푉표 that are large compared to the typical fluctuations 푉푟푚푠 and the probability of occurrence is exponentially small. Using the numerical solution of a constraint function 푓(푤) =

2푚푎2|퐸|/ћ2 for a correlated Gaussian potential, they were able to obtain the exponential part of the density of states in the band tail throughout the energy range as illustrated by 63 2 2 Fig. 4.9. Here, 푤 ≡ 2푚푎 푉표/ћ , 푎 represents the range of a Gaussian potential, 푉표 depth of Gaussian potential, 푚 mass of electron, and 퐸 total energy.

Figure 4.9. Density of states as a function of y displaying the Gaussian tail (푦 ≫ 2), the 2 2 Urbach tail (0.1 < 푦 > 2), and the H-L tail (0 < 푦 > 0.1) where 휖퐿 ≡ ћ /2푚퐿 . The essence of the most probable potential fluctuation and wave function in each region is depicted schematically, Figure from Ref. [84].

It is also suggested that the linear behavior of the Urbach tail (see Fig. 4.1) is a result of the influence of the correlation length of the disorder and that the Gaussian potential Ansatz provided a numerically accurate theory by allowing the wave function to tunnel exponentially into the band gap for the H-L states.

To better understand the sensitivity of Urbach tails [85], the authors of [84] performed an extensive study utilizing various correlations functions having a short- range, short-range and long-range, and long-range components depending on the type of disorder. For only short-range correlations, they used the correlation functions of the form 64 2 푚 퐵(푥) = 푉푟푚푠푒푥푝[−(|푥|⁄ 퐿) ] (4.12) for 1 ≤ 푚 ≥ ∞. They have shown that accurate linear behavior in the DOS is characteristic of random potentials which are strongly correlated for lengths less than or equal to the interatomic distance and then rapidly lose correlation on longer length scales.

An example would be for 푚 = 1 which is relevant to heavily doped semiconductors with charged impurities. For amorphous materials having correlations that decay even more rapidly, correlation functions having both a short- and long-range components are used and defined as

2 2 2 2 2 −푥 ⁄퐿1 −푥 ⁄퐿2 퐵(푥) = 푉푟푚푠[훼푒 + (1 − 훼)푒 ]. (4 .13)

For materials with topological disorder or polar semiconductors a set of power law decaying correlation functions can be considered

2 2 2 −푚1⁄2 퐵푉(푥) = 푟푚푠(1 + 푥 ⁄퐿 ) . (4.14)

It is claimed that their results provide a foundation for understanding the universality of

Urbach tails in the one-electron DOS. They discovered that the shape of Urbach tails in disordered materials is a sensitive measure of the microscopic spatial autocorrelations in the random potential and that the linearity of Urbach tails in different materials suggests strong short-range order on the order of interatomic distances, whereas correlations decay more rapidly than exponentially on longer length scales.

4.3.2 Research from our group

A major effort for explaining the origin of Urbach tails has been and continues to be made using the concept of topology, which consists of structural filaments of short and long bonds as revealed in [70]. The purpose of their paper was to reveal the atomistic origin of the Urbach edges by discovering the structural origins of these tails for both 65 defective crystalline and amorphous Si, which was done by utilizing the ab initio density functional theory code SIESTA, for their relaxation studies as well as the spectral properties of their supercell models. They showed that for the relaxed models of crystalline (diamond) Si containing defects (vacancy), and amorphous Si, that exponential (Urbach) tails did appear as exemplified in Fig. 4.10. The relaxed defective crystalline Si model was a 512-atom model of Si with two vacancies, whereas, the relaxed amorphous Si model was a 512-atom Djordjevic,Thorpe and Wooten (DTW) model. Urbach energies of 350 meV and 107 meV were determined for the defective crystalline Si and amorphous Si models, respectively. Both electron filaments (EF) and topological filaments (TF) had been observed.

Figure 4.10. (a) Valence band tails for crystalline Si and Crystal Si with two vacancies. (b) Valence band tails for DTW model and a DTW model with randomly distorted bond lengths, Figure from Ref. [70].

For Fig. 4.10(b), the relaxed DTW model reveals an exponential tail whereas, the distorted DTW model has modified the shape of the tail from an exponential to a more 66 Gaussian form. It has been determined from calculations that valence tail states are associated with short bonds and conduction tail states with long bonds. Their work, associated with amorphous Si, has shown that near the band edge extremes, the states have filament of charge that are strongly associated with the structural filaments. This is illustrated in Fig. 4.11, where electron filaments for five valence tail and four conduction tail states are presented. As can be seen the valence tail contains mostly simple filaments as compared to the conduction tail which consists of filaments and several rings.

Figure 4.11. Electron filaments for different valence and conduction tails. “H” represents the highest occupied molecular orbital state; H1, next lower energy state, etc. “L” represents the lowest unoccupied molecular orbital state; L1, next higher energy state, etc. Figure from Ref. [70].

These filaments, an aftermath of structural relaxation, can be considered as quasiparticles that might govern transport and other electronic properties. This paper has also shown that a simple 1-D structure is the major cause of strong electron-phonon coupling.

Therefore, from the work presented in this paper, it is inferred that the Urbach tail in amorphous Si is the result of the presence of structural filaments. 67 Another effort [86] into better understanding subtle structural correlations and their association with the electronic band tails is considered. The work presented in [86] is very similar to that of [70], but applied to amorphous SiO2 and the organic molecule,

훽-carotene. Also included was a more detail examination for the amorphous Si. In regards to the amorphous Si a more detail analysis was done involving a defect. It was found that there exists a significant difference between the short- and long-bond clusters as depicted in Fig. 4.12, indicating a strong correlation between the relaxation around the short bond defects and the bond length deviation at the nucleus. An implication of this would be that a short bond acts like a catalyst for local volume growth of higher density.

Thus, the shorter the bond is the greater the growth.

Figure 4.12. Number of atoms in short-bond and long-bond cluster (Ncl ) are plotted against the central bond length of the cluster for a-Si. The dotted line indicates the mean bond length, Figure from Ref. [86].

It was suggested that if a short bond was created, independent of method that the system/network would locally densify around that nucleus. For the case of the amorphous SiO2 a 648-atom a-SiO2 model was utilized for studying the network 68 connectivity by considering the methods that were applied to a-Si. It was discovered that no topological correlation exists for the Si-O nearest-neighbor bond lengths resulting in the use of second-neighbor distances 푅푂−푂 (mean length of 2.66 Å) and 푅푆푖−푆푖 (mean length of 3.05 Å). It turns out that the fluctuation in 푅푆푖−푆푖 (FWHM ~0.2 Å) is larger than that of 푅푂−푂 (FWHM ~0.1 Å). The linear dependence between the two lengths is illustrated in Fig. 4.13. Fig. 4.14 exemplifies the connectivity of a substructure which was formed by short and long second neighbors.

Figure 4.13. Correlation between the two lengths 푅푆푖−푆푖 and 푅푂−푂 . A linear fit to the data is indicated by the red line and the dotted lines represent the mean values of the two lengths, Figure from Ref. [86]. 69

Figure 4.14. Examples of short 푅푂−푂 and long 푅푆푖−푆푖 clusters in a-SiO2 structure. Grey sites indicate the defect nuclei, Figure from Ref. [86].

The inverse participation ratio (IPR) of the density of states for a-Si was calculated, and is provided in Fig. 4.15. As shown, an asymmetric profile for the short 푅푂−푂 and long

푅푆푖−푆푖 lengths at the band tails is realized, which means that the localized valence tail states are correlated to the short 푅푂−푂 lengths and the localized conduction tail states with the long 푅푆푖−푆푖 lengths. The fact that the correlation between the denser volumes is similar to that of a-Si may help in the overall understanding of the structure origins of the

Urbach tail.

Figure 4.15. Inverse participation ratio of the density of states for a-Si. Fermi level is represented by dotted line. The valence (conduction) tail is determined by short O-O bonds (long Si-Si bonds), Figure from Ref. [86].

As presented in [86], there are some of the same effects in conjugated organic systems, such as 훽-carotene (C40 H56 ), as in a-Si and a-SiO2 , which is revealed in Fig.

4.16.

Figure 4.16. HOMO (purple) and LUMO (cyan) states correspond to short and long bonds. The mean bond length is at 1.36 Å (1.44 Å) for the HOMO (LUMO), Figure from Ref. [86].

71 The highest occupied molecular orbital (HOMO) state and the lowest unoccupied molecular orbital (LUMO) state can be considered as molecular equivalence to the valence and conduction tails, respectively. Some prior results on 훽-carotene have indicated that the frontier orbitals (HOMO and LUMO) are localized on the conjugation locations, which may result in the lowering of the HOMO energy; therefore, compensating for the increase in energy due to the shortening of the double bonds.

The last paper [87] to be reviewed, investigates the following issues: 1) the role of finite model size on exponential tails , 2) the character of the strain field centered on certain short bonds, and 3) the role of thermal disorder on band tails and filaments. To address the first issue, a high-quality 100,000 atom a-Si model obtained by [88], was used in conjunction with a tight-binding approach. The electronic Hamiltonian matrix used is from [89] and is very sparse making it easily to solve. In addition to the above, the principle of maximum entropy (maxent) was used for reproducing the electronic density of states for the 105 –atom a-Si model with moments of 107 and 150. This is depicted in Fig. 4.17.

Figure 4.17. Electronic density of states for 105 –atom a-Si model from maxent reconstruction for 107 and 150 moments. The Fermi level is in the middle of the gap, Figure from Ref. [87].

A closer view of the band gap region is provided in Fig. 4.18, where the tails have been fitted to an exponential of the form exp(−|퐸 − 퐸푡|⁄ 퐸푈), where 퐸푡 represents the valence or conduction tail, resulting in values of 퐸푈 = 200 meV for the valence tail and

퐸푈 = 96 meV for the conduction tail. It was revealed that exponential tails are not just associated with valence and conduction tails but, also, with “extremal tail” (near -15 and

+8 eV in Fig. 4.17). Because of the unbiased nature of the maxent approach, it has no influence on the exponential form. A comparison between a 512-atom a-Si model and

105 -atom a-Si model is presented in Fig. 4.19, which shows similarities between the two.

Figure 4.18. A least-square fit to exponentials for valence and conduction tails for maxent reconstruction of the electronic density of states for a 105 –atom a-Si model with 107 moments, Figure from Ref. [87].

Figure 4.19. Comparison of electronic density of states between a 512-atom model and a 105 -atom model, Figure from Ref. [87].

For the second issue, it has been shown that a central short bond (aka “defect nucleus”) will tend to connect to other short bonds, where the bond lengths of these short bonds must asymptotically reach the mean bond length of the structure as the distance between the other short bonds and the central short bond increases. As a consequence, a strain field is established by the central short bond. This is exemplified by Fig. 4.20. For 74 a-Si, the valence tail is known to be very broad, largely due to static disorder and not thermal disorder. The pattern is not as clear for the long bonds because of basic asymmetry, resulting in a conduction tail that is not as broad.

Figure 4.20. Strain recovery in a 512-atom model of a-Si. ∆푟 represents the difference in bond length from the mean, Figure from Ref. [87].

The third issue involves thermal effects on band tails and structural filaments. By using the ab initio code SIESTA for temperatures ranging from 20 to 700K, the authors have produced animations of the dynamics of the short bonds. Instantaneous snapshots of the shortest bonds at two different times for 300K are illustrated in Fig. 4.21.

Figure 4.21. Instantaneous snapshots for 512-atom model at 300K for two different times, Figure from Ref. [87].

There exists a significant fluctuation in the identity of the shortest bonds. Also shown in Fig. 4.21, the persistence of the filaments, even at room temperature. Fig. 4.22 compares the electronic density of states of two 512-atom a-Si models with and without filaments. The results were obtained by using a tight-binding approach. A larger band gap exists for the model with filaments than without filaments. 76

Figure 4.22. Electronic density of states for 512 atom models without and with filaments, Figure from Ref. [87].

4.4 Summary

A discussion of the optical absorption characteristics of disordered materials, with a focus on Urbach tails was provided. Realizing that the optical absorption coefficient is proportional to the EDOS, allowed for the understanding of the electronic structure of amorphous materials with regards to the Urbach tail. The main concept, as presented in this chapter, was to interpret the existence of the Urbach tail from two different perspective, one from the point of view of fluctuating potentials and the other from topology. The first approach, although interesting, does not involve EDOS; thus, was not considered for further investigation. However, because the second approach does, made it conducive for further investigation of the Urbach tail, which is presented in the next chapter for an unrelaxed and relaxed 4096 atom a-Si model.

77 5. RESULTS AND DISCUSSION

In this chapter, new results for 512 and 4096 atom amorphous silicon WWW models are presented and discussed. The 512 atom model was used to perform molecular dynamic simulations in order to learn what effect temperature has on eigenvalues and electronic density of states. The utilization of the 4096 atom model was for a comparison study between two sets of atomic coordinates (unrelaxed and relaxed), which was done statically. The topics that were considered for the comparison are: physical structure, electronic density of states, bond-center to bond-center correlation, and connectivity of bonds. The purpose of the comparison study was to investigate what differences, if any, existed between the two sets of atomic coordinates for the selected topics.

These models are defect free. An ab initio code SIESTA [90], consisting of a

SZP basis set and Harris functional, was utilized for the electronic structure calculations for both the 512 and 4096 atom models. Calculations for the exponential curve fitting, bond length – bond length correlation distributions and all the graphical representations were performed using Mathematica. The atomic structure diagrams were created by Jmol

[91] and the physical structure graphical representations by ISAACS [92].

5.1 The 512 Atom a-Si Structure

The structure consisted of a cubic unit cell, containing 512 atoms, with a lattice constant of 21.7 Å, cell volume of 10218.3 Å3 and a mean bond length of 2.35 Å. A MD time step size of 950 steps with a time step duration of 2.0 femtosecond and a Nose’ thermostat algorithm were considered for the thermal statistics study of the electronic density of states and energy gap study. Eigenvalue broadening of 0.05 eV was used. To extract eigenvalues at each MD time step required rewriting three subroutines in 78 SIESTA. The eigenvalues were then averaged over the MD time steps. The EDOS values were calculated using the average eigenvalues.

An ab initio thermal molecular dynamics (MD) simulations of the electronic density of states was conducted for temperatures of 150K, 300K, and 600K, with results illustrated in Fig. 5.1. As shown, the three curves are quite similar, except for the dips in mid-gap. At mid-gap the dip in the curves never reach zero, but increases with increasing temperature, possibly suggesting that the number of mid-gap states are increasing with temperature. This is better elucidated by Fig. 5.2. This is a result of the motion of the eigenstates, in particular, the HOMO and LUMO states. As the temperature increases, the HOMO-LUMO gap decreases as well as the superjacent and subjacent states, resulting in the material going from a semiconducting state to more of a conducting state as depicted in Figs. 5.3-5.5. As a consequence, electrical conductivity increases while electrical resistivity decreases, resulting in requiring lower energy electrons or photons for conductivity to occur. Another interesting feature common in Figs. 5.3-5.5 is that the eigenvalues change very quickly from time step 0 to time step 100. 79

Figure 5.1. A pictorial representation of the electronic density of states for three different temperature with a Fermi level of about -4.110 eV.

Figure 5.2. A close up view of the electronic density of states for three different temperatures near the energy gap. The Fermi level for 150K and 300K is about -4.10eV, whereas for 600K around -3.90eV. 80

Figure 5.3. Eigenvalue fluctuations for the HOMO, LUMO, superjacent and subjacent states for a temperature of 150K and 950 MD time steps.

Figure 5.4. Eigenvalue fluctuations for the HOMO, LUMO, superjacent and subjacent states for a temperature of 300K and 950 MD time steps.

Figure 5.5. Eigenvalue fluctuations for the HOMO, LUMO, superjacent and subjacent states for a temperature of 600K and 950 MD time steps. Note the different vertical axis scale.

5.2 The 4096 Atom a-Si Structure

The structure consisted of a cubic unit cell, containing 4096 atoms, with a lattice constant of 43.4167 Å, cell volume of 81840.913 Å3 and a mean bond length of 2.35 Å.

5.2.1 Physical structure information

In this section, physical structural quantities, such as pair distribution function

푔(푟), bond length distribution, bond angle distribution, and dihedral angle distribution are discussed. The pair distribution function 푔(푟) is depicted in Fig. 5.6 for both the relaxed and unrelaxed models. As can be seen from Fig. 5.6, an oscillatory profile exists for both models, because the peaks of the probability function represent average interatomic distances. Also noteworthy is that 푔(푟) asymptotically approaches unity. 82

Figure 5.6. Pair correlation function for the relaxed and unrelaxed 4096 a-Si models.

The only obvious discrepancy between the two models is with regards to the first peak, where the magnitude of the first peak for the unrelaxed model is considerably less than that of the relaxed model by almost a factor of 0.48. This discrepancy may be the result of the unrelaxed model having fewer bonds at the mean bond length of 2.35Å and a broader bond length distribution than the relaxed model, as illustrated in Fig. 5.7. Fig.

5.7 also reveals that both models have symmetrical curves, but peaks at different bond length values. The unrelaxed model has a bond length peak value of 2.35Å and relaxed of 2.36Å. The curves shown in Fig. 5.7 have been normalized so that the areas under each curve are the same. Figs. 5.6 and 5.7 exemplifies the fact that these two models are quite different. 83

Figure 5.7. Bond length distribution for the unrelaxed and relaxed 4096 a-Si models.

The bond angle characteristics of both models are depicted in Fig. 5.8. For the most part, the two curves are very similar, with the unrelaxed model having a peak value at 108.5˚ and the relaxed at 109.5˚; thus, suggesting that the two models have an equal number of bond angles for the same bond angle range.

Figure 5.8. Bond angle distribution for the unrelaxed and relaxed 4096 a-Si models.

84 With regards to the dihedral angle characteristics the models are almost identical.

A graphical representation of this is provided in Fig. 5.9. Figs. 5.8-5.9 reveals that the two models are essentially identical with respect to angle related quantities

Figure 5.9. Dihedral angle distribution for unrelaxed and relaxed 4096 a-Si models. (bond angle, dihedral angle), but not with bond length.

5.2.2 Electronic density of states

To better understand the electronic structure of a material requires the knowledge of the electronic density of states (EDOS). Both the unrelaxed and relaxed models are considered.

In Fig. 5.10 the effect of different basis sets, with broadening of 0.05eV, on the electronic density of states for the unrelaxed model are elucidated. What really stands out in Fig. 5.10 is that the curves are nearly the same for the valence band but, quite different for the conduction band, where the SZP basis set has the largest values followed by the DZ basis set and the SZ basis set. These values are essentially the same in the valence band because each of the three basis sets (SZ,DZ,SZP) have the same valence 85 basis functions or orbitals (minimal basis), which are fully occupied. In contrast, the three basis sets have different number of conduction basis functions. Therefore, the normalizing factor of the conduction basis functions are different. The SZ basis set has the minimum number of valence basis functions (s- and p-orbitals), the DZ basis set has the minimum number of valence basis functions plus additional basis functions (more s- and p-orbitals), and the SZP basis set has every basis functions that the SZ basis has, in addition to some d-orbitals. As a result, their values vary significantly.

A close-up view of the EDOS and the Fermi energy levels close to the energy gap is illustrated in Fig. 5.11, where there is a significant shift to the left of the SZP curve as compared to the other curves, which was unexpected. What is noteworthy is that the energy gap for the SZP basis set is considerably smaller than the other two basis sets and the gap for DZ basis set slightly narrower than that of the SZ basis set, as anticipated. As the number of terms in a basis set increases there is an increase in the number of eigenstates, resulting in a decrease in the energy gap. Because of the large shift of the

SZP curve there is an equally large shift of the Fermi energy level associated with the

SZP curve as compared to the Fermi levels for the DZ and SZ curves.

Similar curves exist for the relaxed model as exemplified by Figs. 5.12 and 5.13.

However, the energy gaps associated with the relaxed model appear to be smaller than those of the unrelaxed model (Fig. 5.13).

Figure 5.10. Electronic density of states and the Fermi energy levels for three basis sets (SZP, DZ, and SZ) for the unrelaxed structure.

Figure 5.11. A close up of the electronic density of states and the Fermi energy levels for the unrelaxed structure near the energy gap.

Figure 5.12. Electronic density of states and the Fermi energy levels for three basis sets (SZP, DZ, and SZ) for the relaxed structure.

Figure 5.13. A close up of the electronic density of states and the Fermi energy levels for the relaxed structure near the energy gap.

88 The remainder of this section will involve only the SZP basis set for the two models. The SZP basis set was selected over SZ and DZP basis sets because it provides more accurate results as compared to a SZ basis set without a significant increase in computation time for the 4096 atom structure. On the other hand, using a DZP basis set for this size structure would take an enormous amount of computational time. As can be seen from Fig. 5.14 the two models agree fairly well. A closer look of the electronic density of states with corresponding Fermi levels near the energy gap is represented by

Fig. 5.15. A slight shift in the valence band as well as a significant difference in the

Fermi levels are observed.

Figure 5.14. Electronic density of states and associated Fermi energy levels for unrelaxed and relaxed structures using a SZP basis.

Figure 5.15. A close-up of the electronic density of states and associated Fermi energy levels near the energy gap for unrelaxed and relaxed structures using a SZP basis.

The big question is do the curves in Fig. 5.15 exhibit an Urbach tail (exponential function of linear arguments), within a small energy range, for both the valence and conduction bands. The question can be answered by reviewing Figs. 5.16 and 5.17 for the valence band. As illustrated in the figures, an exponential function (Urbach tail) of the form

휌(퐸) ∝ exp (−| 퐸|⁄ 퐸푈) does exists, which has strong agreement with the actual density of states for both the unrelaxed and relaxed models. The Urbach energies 퐸푈 values for unrelaxed and relaxed models are 130 푚푒푉 and 142 푚푒푉, respectively.

Figure 5.16. Electronic density of states of the valence band with an exponential fit for the unrelaxed model.

Figure 5.17. Electronic density of states of the valence band with an exponential fit for the relaxed model.

91 An alternate view is shown in Fig. 5.18, where a semi-log plot reveals essentially the same information as Figs. 5.16 and 5.17, for the valence bands. The Urbach energies are determined by the slopes in Fig. 5.18.

Figure 5.18. Semilog plot of the electronic density of states for the valence bands of the unrelaxed and relaxed structures. The slopes represent the Urbach energies, which are 130 푚푒푉 (unrelaxed) and 142 푚푒푉 (relaxed). A SZP basis set was used.

With respect to the conduction band, Figs. 5.19 and 5.20 illustrates partial correlation between the actual electronic density of states and the exponential fit for the two models. Here, the Urbach energies are 226 푚푒푉 and 246 푚푒푉 for the unrelaxed and relaxed models, respectively. Also noteworthy is that the exponential fit curves diverges at about the same energy of −3.7 푒푉 from their respective electronic density of states curves as a result of the electronic density of states curves decreasing at a higher rate. A semi-log plot of the electronic density of states for the two models is presented in

Fig. 5.21, which reveals that for the conduction band the electronic density of states are 92 not very exponential. Even though, they appear to be in the short energy range of 0.25 eV, which is just short of the experimental range of 0.4eV to 0.8eV.

Figure 5.19. Electronic density of states of the conduction band with an exponential fit for the unrelaxed model.

Figure 5.20. Electronic density of states of the conduction band with an exponential fit for the relaxed model.

Figure 5.21. Semilog plot of the electronic density of states for the unrelaxed and relaxed structures. Note: the curves do not represent a good exponential function.

5.2.3 Bond-center to bond-center correlation distribution

It has been shown [93,94] that there exists self-correlations between short bonds, for valence tail states, and long bonds, for conduction tail states, resulting in clusters and filaments, respectively. The short bond and long bond correlation functions that were computed are defined [1] as

푉 푁1 푁2 훽(푟) = 2 ∑푛 =1 ∑푛 ≠푛 훿( 푟푛1푛2 − 푟) (5.1) 4휋 푁1푁2 1 2 1 where 푁1, 푁2 are the number of short or long bonds required; 푛1 and 푛2 represent

individual bonds that are counted; 푟푛1푛2 is the distance between the bond centers of bond

푛1 and bond 푛2; and 푉 is the volume of the unit cell. Eqn. (1) can be more elucidated by referring to Fig. 5.22, which represents the bond center to bond center correlation distribution for the unrelaxed model. As shown, there appears to be a large correlation at bond lengths of 1.8 Å and magnitude of 0.937 for the 4% shortest bonds and, 2.0 Å and 94 1.474 for the 4% longest bonds, resulting in the 4% longest bonds being 1.57 times more correlated than the 4% shortest bonds. The base of the first peak of the 4% shortest bonds curve is broader with small blips, which was not expected. Also note that the two correlation functions asymptotically approaches a very small value beyond 8.0 Å indicating no correlation between the shortest and longest bonds.

A pictorial representation of bond center to bond center correlation functions for the relaxed model is provided in Fig. 5.23 with some interesting features. One striking feature is that the height of the first peak of the 4% shortest bonds curve is now larger than that of the 4% longest bonds curve. The width of the base associated with the first peak of the 4% shortest bonds is more narrow then that of the 4% longest bonds.

Figure 5.22. Bond-center to bond-center correlation distribution for both the 4% shortest and 4% longest bonds for the unrelaxed model.

95 Another interesting feature is the double peak profile associated with the 4% longest bonds. A possible explanation of this might be that long bonds are much weaker resulting in a broader base and a double peak, whereas relaxation improves the results for the short bonds.

Figure 5.23. Bond-center to bond-center correlation distribution for both the 4% shortest and 4% longest bonds for the relaxed model.

Large correlation exists for bond lengths of 1.9 Å and magnitude of 2.904 for the

4% shortest bonds and, 1.8 Å and 1.148 for the first peak and 2.1 Å and 1.588 for the second peak of the 4% longest bonds. Both of these figures show that, in general, there exists a trend for short bonds to group with other short bonds, and similarly, for long bonds. In comparison with [93], Figs. 5.22 and 5.23 has some similar features with model M2. It appears that Fig. 5.23 may be slightly better, than Fig. 5.22, because the curve for the short-short bonds has a sharp peak similar to [93]. The significant 96 difference between the results presented here and results from [93] is that the two peaks do not have the same magnitude as compared with the peaks for model M2 . Also, as shown in [93], the short-long correlation is much reduced in contrast to the other two curves.

5.2.4 Connectivity of shortest and longest bonds

An intriguing question is: how do the plots in Figs. 5.22 and 5.23 correlate to the bonds spatially for both the unrelaxed and relaxed models. For the unrelaxed model,

Figs. 5.24 and 5.25 exemplify this question by revealing how the shortest and longest bonds germinate to produce clusters, resulting in correlation. A selected percentage of the shortest and longest

Figure 5.24. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) shortest bonds for the unrelaxed model. Note the regions where clustering is occurring.

Figure 5.25. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) longest bonds for the unrelaxed model. Note the regions where clustering is occurring. bonds for the two models were considered.

A feature that is quite pronounced between Figs. 5.24 and 5.25 is the difference in the number of bonds between the 4% shortest and longest bonds. Similarly, Figs. 5.26 and

5.27 demonstrates the percolation of the shortest and longest bonds for the relaxed model, producing correlation between the shortest bonds and longest bonds. The orientation of the last twelve figures is a frontal view.

Figure 5.26. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) shortest bonds for the relaxed model. Note the regions where clustering is occurring.

Figure 5.27. A pictorial representation of 2% (left panel), 4% (center panel), 8% (right panel) longest bonds for the relaxed model. Note the regions where clustering is occurring.

As discussed in the literature [93,94,95,96,97,98] valence tail states (localized eigenstates) are typically associated with short bonds and conduction tail states (localized eigenstates) with long bonds, which may be best explained [93] by realizing that: 1) short bonds decreases the energy of the valence band states, while increasing the energy of the conduction band states and 2) that short bonds also increases the charge in the middle of the bonds (eigenstates near the valence tail). It is understood that if the statements are, indeed true, this would mean that the electronic band energy would be optimized. So how does this relate to the figures above. In general, as the short and long bonds begin to germinate and cluster, the density of the electronic states and charge density of the cluster begin to increase. As an example, a small cluster of short bonds (few bonds) represent a few very localized eigenstates (near the valence tail). As more short bonds percolate and the cluster proliferates, the density of localized eigenstates increases, which may result in the eigenstates becoming less localized. As these clusters continue to grow, and if some of the clusters, of same energy, start to weakly overlap (in real space), due to possible mixing of cluster eigenstates, will form new energy eigenstates for the newly formed 99 group of clusters. Therefore, as this process continues a critical junction is reached where the localized eigenstates become extended eigenstates, which is known as the Anderson transition.

With this information, how do the two models (unrelaxed and relaxed) compare with respect to short and long bonds. A careful review of Figs. 5.24 and 5.26 reveals qualitatively that clusters of short bonds are forming more rapidly, resulting in larger clusters for the relaxed model than the unrelaxed model, implying that the electronic density of states of the valence band edge for the relaxed model should have larger values than the unrelaxed model, in agreement with Fig. 5.15. Comparing Figs. 5.25 and 5.27, the left and right panels appear similar with respect to number and size of clusters for the long bonds, except for the center panels. Qualitatively, this would suggest that the electronic density of states for both models are reasonably close, and in agreement with

Fig. 5.15 for the conduction band edge. Thus, the filament structures appear to be quite similar.

According to [2], states of valence band tail are dominated by contribution of the p-orbitals and less from the s-orbitals, and states in the conduction band tail have fewer contributions from the p-orbitals and more from the s-orbitals. The p-orbital contribution is believed to be due to bond angle distortions, whereas the s-orbital contribution is from the bond length distortions [6,7]. It was also shown that the p-orbital contribution decreases slowly from the valence band tail and across the band gap to the bottom of the conduction band tail. Also noted was that for a-Si the amount of bond angle distortions is greater than that for bond length distortions. It has also been shown, in Fig. 5.13, that the d-orbitals have a significant influence on the both the valence and conduction band tails. 100 It is proposed that by knowing how the atomic p-orbitals are interacting with each other as well as with d-orbitals that the formation of clusters for short and long bonds, exponential behavior of the Urbach tail, and the universality of the Urbach tail might be explained.

5.3 Summary

The electronic density of states for the 512 atom a-Si model varied significantly over temperature and the energy gap decreased as temperature increased, revealing how a semiconductor state can change to a conductor state. With regards to the 4096 atom a-Si models, the pair correlation function and bond length distributions indicated that the unrelaxed and relaxed models are quite different, despite the close similarities among angle distributions. The discrepancy between the two models resulted in only a slight disparity with regards to the electronic density of states for the valence band tail, which revealed excellent agreement between the calculated electronic density of states and the exponential curve fit for both the unrelaxed and relaxed models. However, the conduction band tail did not do so well, the exponential fit was not as close for the two models. A significant disparity existed for bond-center to bond-center correlation distribution for the unrelaxed and relaxed models, which could be due to the apparent difference in the pair correlation function and bond length distributions. The connectivity of the short and long bonds for both models, in relation to bond-center to bond-center correlation and electronic density of states, indicated that short bonds prefer short bonds and long bonds with long bonds and that short bonds of the relaxed model germinated and proliferated faster than that of the unrelaxed model, in contrast to the long bonds. 101 A possible limitation associated with ab initio thermal MD simulations of the 512 atom a-Si model would be the relatively small number of MD time steps used. Increasing this number could possibly improve the results for the electronic density of states.

Finally, the consideration of the p- and d-orbital interactions in the valence and conduction tails for a possible explanation of cluster formation of short and long bonds, exponential characteristic and universality of the Urbach tail was proposed.

102 6. SUMMARY AND FUTURE WORK

6.1 Summary

Despite the enormous amount of research (experimentally and theoretically) that has been conducted over the years there still remains some fundamental questions that needs to be answered. The purpose of this research was to try to address such questions as: Why is the Urbach tail so universal and exponential?

It was shown why the WWW model is considered as the best model to use for computational purposes of amorphous materials. This is a result of the model being able to satisfy all the necessary requirements, such as: containing hundreds of atoms, restricted to boundary conditions, structural distortion not exceeding experimental estimates, and a structure that correlates well with experiment. This was accomplished by utilizing the following: a specific bond switching scheme for the fabrication process, the Metropolis algorithm and the Keating potential in the simulated annealing process. As a consequence, a structure with low angular deviation, a good total correlation function, and low strain energy is realized.

An ab initio code SIESTA was utilized to perform all the electronic structure calculations. A mathematical description of this self-consistent DFT consisted of basic assumptions, pseudopotentials, LCAO basis sets, Hamiltonian matrix elements, total energy and the Harris functional. Basic approximations involving XC and pseudopotentials are considered in SIESTA to avoid dealing with core electrons. LCAO basis sets are employed by SIESTA for manipulating the sparsity of the Hamiltonian and overlap matrices. A Harris functional (non-self-consistent scheme) was utilized in the 103 electronic structure calculations, instead of the SCF method, because it consumes less computational power and cost but, for a price in accuracy.

Also presented was a discussion regarding the optical absorption characteristics of disordered materials as well as the relation between the optical absorption coefficient and electronic density of states. A theoretical review of the Urbach tail was given for work done in explaining the exponential behavior of the Urbach tail, which was associated with researchers utilizing the fluctuating potentials concept and our group that uses topology.

Lastly, a discussion of the results for a 512 and 4096 atom a-Si structures was given. With regards to the 512 atom a-Si structure, the impact on eigenvalues and electronic density of states with respect to temperature revealed how the energy gap changes and the introduction of mid-gap states. As for the 4096 atom a-Si model two sets of atomic coordinates (unrelaxed and relaxed) were compared in regards to physical structure, electronic density of states, correlation of the bond centers and, connectivity of shortest and longest bonds. The physical structure information showed a difference between the unrelaxed and relaxed structures. Despite the disparity, good agreement of the electronic density of states for the valence band tail was shown, along with excellent agreement between the calculated density of states and the exponential fit for both the unrelaxed and relaxed models, in contrast to the conduction band tail; thus, furthering our understanding regarding Urbach tails. There exists a difference for the two models with respect to the correlation distribution of their bond centers. It was shown that through connectivity of short and long bonds for both models that short bonds prefer other short bonds and similarly, for long bonds. Also included was a proposal regarding the explanation of cluster formation of short and long bonds, exponential behavior and 104 universality of the Urbach tail by considering and further understanding the interactions of the p-orbitals associated with the valence and conduction band tails.

6.2 Future Work

It would be good to know why the Urbach tail exists only over a small energy range and how this range varies with different disordered materials. Furthering the comprehension of p-and s-orbital interactions from very localized mid-gap states to extended band states might provide new insight on other tails, such as the Tauc tail.

Investigate p- and s-orbital interactions associated with amorphous carbon to determine differences with respect to amorphous silicon. Another interesting endeavor would be to investigate why Urbach tails exists for carbon nanotubes transistors [100].

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111 APPENDIX A. DERIVATION OF EQUATION 2.2

The derivation of (2.2) is presented and is valid only for the Keating model.

Taking the Taylor series expansion of the potential about the minimum energy position of an atom in the 푥-direction gives

′ 1 ′′ 2 푉(푥) = 푉(푥표) + 푉 (푥표)(푥 − 푥표) + ⁄2 푉 (푥표)(푥 − 푥표) +.... (A.1)

′ We take 푉 (푥표)= 0, then the above equation reduces to

1 ′′ 2 푉(푥) = 푉(푥표) + ⁄2 푉 (푥표)(푥 − 푥표) (A.2)

Similarly,

1 ′′ 2 푉(푦) = 푉(푦표) + ⁄2 푉 (푦표)(푦 − 푦표) (A.3)

1 ′′ 2 푉(푧) = 푉(푧표) + ⁄2 푉 (푧표)(푧 − 푧표) (A.4)

Taking the gradient of (A.2) to (A.4) yields

2 ( ) ′′( )( ) 휕 ⁄ ( ) ∇푥푉 푥 = 푉 푥표 푥 − 푥표 = 휕푥2 푉 푥표 ∆푥 (A.5)

2 ( ) ′′( )( ) 휕 ( ) ∇푦푉 푦 = 푉 푦표 푦 − 푦표 = ⁄휕푦2 푉 푦표 ∆푦 (A.6)

2 ( ) ′′( )( ) 휕 ⁄ ( ) ∇푧푉 푧 = 푉 푧표 푧 − 푧표 = 휕푧2 푉 푧표 ∆푧 (A.7)

where ∆푥 = 푥 − 푥표 , similarly for ∆푦, ∆푧.

The force on the atom can be written as 112 2 2 2 휕 ⁄ ( ) 휕 ( ) 휕 ⁄ ( ) 푭풓 = −훁풓 푉풓 = −( 휕푥2 푉 푥표 ∆풙 + ⁄휕푦2 푉 푦표 ∆풚 + 휕푧2 푉 푧표 ∆풛 (A.8)

Because 푉(푥표) = 푉(푦표) = 푉(푧표), (A8) reduces to

2 2 2 휕 ⁄ 휕 휕 ⁄ ( ) 푭풓 = −[( 휕푥2 + ⁄휕푦2 + 휕푧2)푉 푟표 (∆풙+∆풚+∆풛)] (A.9)

2 2 2 휕 ⁄ 휕 휕 ⁄ ( ) 푭풓 = − [( 휕푥2 + ⁄휕푦2 + 휕푧2) 푉 푟표 ∆풓] (A.10)

In general,

2 푭 풓 = −∇ 푉(푟표)∆풓 (A.11)

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