ATOMISTIC MATERIALS MODELING OF COMPLEX SYSTEMS: CARBYNES, CARBON NANOTUBE DEVICES AND BULK METALLIC GLASSES

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Weiqi Luo, M.S., B.S.

*****

The Ohio State University

2008

Dissertation Committee: Approved by

Professor Wolfgang Windl, Adviser Professor Katharine M. Flores Adviser Professor John E. Morral Graduate Program in Professor William F. Saam Materials Science and Engineering °c Copyright by

Weiqi Luo

2008

ABSTRACT

The key to understanding and predicting the behavior of materials is the knowledge

of their structures. Many properties of materials samples are not solely determined by

their average chemical compositions which one may easily control. Instead, they are

profoundly influenced by structural features of different characteristic length scales.

Starting in the last century, metallurgical engineering has mostly been microstructure engineering. With the further evolution of materials science, structural features of smaller length scales down to the atomic structure, have become of interest for the purpose of properties engineering and functionalizing materials and are, therefore, subjected to study.

As computer modeling is becoming more powerful due to the dramatic increase of computational resources and software over the recent decades, there is an increasing demand for atomistic simulations with the goal of better understanding materials behavior on the atomic scale. Density functional theory (DFT) is a quantum mechanics based approach to calculate electron distribution, total energy and interatomic forces with high accuracy. From these, atomic structures and thermal effects can be predicted.

However, DFT is mostly applied to relatively simple systems because it is computationally very demanding. In this thesis, the current limits of DFT applications are explored by studying relatively complex systems, namely, carbynes, carbon nanotube

ii (CNT) devices and bulk metallic glasses (BMGs). Special care is taken to overcome the limitations set by small system sizes and time scales that often prohibit DFT from being applied to realistic systems under realistic external conditions.

In the first study, we examine the possible existence of a third solid phase of carbon with linear bonding called carbyne, which has been suggested in the literature and whose formation has been suggested to be detrimental to high-temperature carbon materials. We have suggested potential structures for solid carbynes based on literature data and our calculations and have calculated their free energies by DFT as a function of temperature

(0 – 4000 K) and pressure (0-180 kbar). We propose and verify a simplified approach to calculate the phonon density of states (DOS) to allow a fast calculation of free energies.

We found that all carbyne structures have higher free energies than graphite in the whole temperature and pressure range of this investigation, making pure (carbon-only) carbynes at most meta-stable. The inclusion of impurities was studied as well and may be the key for a stable carbyne phase.

For CNT devices which have been suggested to eventually replace current Si technology, there is currently no equivalent for the highly used Si process modeling methods (“Technology Computer Aided Design” (TCAD)). We suggest accelerated DFT molecular dynamics (MD) simulations as a method for process modeling and apply it to study the contact formation between CNTs and metal contacts consisting of Ti, Pd, Al, and Au. The temperature accelerated dynamics (TAD) technique was adopted to overcome the time limitations of MD simulations in general, which are especially severe for the computationally demanding DFT MD simulations. We found that CNTs undergo a structural transformation when brought into contact with certain metal electrodes (here,

iii Ti and Al). This resulted in a dramatic decrease in electrical conductance of the device.

We also show that the transformation depends on the size of CNTs due to the size- dependent elastic energy and on the electrode materials due to the electronegativity- dependent charge transfer.

In the last study, DFT was used in conjunction with classical MD simulations to predict the electron density of a Cu46Zr54 BMG structure modeled by a 1000-atom cell.

Whereas DFT is capable to calculate the electron distribution in the cell, it is too slow to simulate melting and structural relaxation, which we handle by classical MD within the

Embedded Atom Method. We propose a new model to analyze the open volume distribution based on the electron density and compare it with the traditional hard sphere model. Results from both models agree well, while the former allows a significantly better physical insight into the open volume distribution. As an additional plus, its results can be connected to experimental results by techniques such as Positron Annihilation

Spectroscopy (PAS).

iv

To My Parents

v

ACKNOWLEDGMENTS

I joined the group with limited background knowledge and encountered numerous

problems during my studies. Prof. Windl carried me through all the difficulties since my

first day in the group. My work wouldn’t have been possible without his endless support.

More importantly, as an advisor, he always inspired me with enlightening ideas and always gave me generous help and encouragement. As a teacher, he often explained sophisticated physics to me in simple and precise language. My most sincere thanks go to him first.

I am grateful to Prof. Morral and Prof. Flores, who gave me excellent advice and supported me. Moreover, they introduced me to the experimental world and made it possible for me to connect simulations to experiments.

I wish to thank my fellow students, Tao Liang, Ashwini Bharathula, Naveen Gupta,

Ryan Paul, Ning Zhou, Yuan Zhang, Karthik Ravichandran, Dipanjan Sen, Lanlin Zhang and Anupriya Agrawal who helped me in solving research problems and gave me a lot of laugh in my student life. They gave me very good memories during the past three years.

This work was supported by the National Science Foundation and Semiconductor

Research Cooperation. I am thankful for the computer time awarded by the Ohio

Supercomputer Center, with which most of my simulations were performed.

vi

VITA

2002……………. B.S., Fudan University Shanghai, China

2005……………. M.S., Ohio State University Columbus, OH

2005-present…… Graduate Research Associate Ohio State University Columbus, OH

FIELDS OF STUDY

Major Field: Materials Science and Engineering

vii

TABLE OF CONTENTS

Page

Abstract...... ii

Dedication...... v

Acknowledge ...... vi

Vita...... vii

List of Tables ...... x

List of Figures...... xii

Chapters:

1. Introduction...... 1 1.1 Computational Materials Science ...... 1 1.2 Atomistic Modeling...... 4 1.3 Motivation and Organization of This Thesis ...... 6

2. An Introduction to the Density Functional Theory...... 8 2.1 The Schrödinger Equation and Wave Functions ...... 10 2.2 Born-Oppenheimer Approximation...... 12 2.3 Many-Body Wave Functions ...... 13 2.4 The Variational Method...... 13 2.5 Hartree-Fock Approximation...... 16 2.6 The Exchange and Correlation Hole...... 27 2.7 The Thomas-Fermi Model ...... 30 2.8 The Hohenberg-Kohn Theorem and Kohn-Sham Approach...... 31 2.9 Approximate Exchange-Correlation Functionals...... 35 2.10 Pseudo Potentials...... 38

3. The Structure and Stability of Carbynes...... 41 3.2 Structure and Stability of Carbynes ...... 46 3.2.1 Crystal Structure of Carbyne in Previous Work ...... 46

viii 3.2.2 Computational Method...... 54 3.2.3 Kasatochkin-Type Carbyne...... 55 3.2.4 Kinked Carbyne - Graphite and Lonsdaleite Derived Modifications ...... 59 3.2.5 Graphite-Diamond Intermediate Model...... 62 3.2.6 Kinked Kudryavtsev Type Carbyne and Unified Carbyne Model...... 64 3.2.6.1 Structural Model...... 64 3.2.6.2 Results...... 67 3.3 Free Energy Calculations...... 72 3.3.1 Calculation Model...... 72 3.3.2 Approximations and Accuracy ...... 74 3.3.3 Phase Boundary between Graphite and Diamond ...... 75 3.3.4 Energetics of Carbynes ...... 76 3.4 Discussions ...... 79 3.5 Conclusions...... 83

4. Contact Formation of Carbon Nanotube Devices and Its Effects on Electron Transports ...... 85 4.1 Carbon Nanotube Field Effect Transistors ...... 86 4.1.1 Structure of CNTs...... 87 4.1.2 Electronic Properties of CNTs...... 90 4.1.3 Metal-Oxide-Semiconductor Field Effect Transistors (MOSFETs)...... 93 4.1.4 Structure of CNT-FETs...... 95 4.1.5 The Contact of CNT-FETs...... 97 4.2 Process Modeling of CNT/Metal Contacts...... 100 4.2.1 Temperature-Accelerated Dynamics ...... 101 4.2.2 CNT/Ti Contacts...... 102 4.2.3 Size Effect of the CNT...... 108 4.2.4 Effect of Different Electrode Materials ...... 110 4.3 Conclusions...... 115

5. Open Volume Distrubituion in Cu46Zr54 Bulk Metallic Glasses From Atomistic Simulations ...... 117 5.1 Embedded Atom Method (EAM) ...... 120 5.2 Hard Sphere Model...... 126 5.3 Electron Density Model ...... 130 5.4 Conclusions...... 138

6. Summary of Conclusions...... 139

Bibliography ...... 142

ix

LIST OF TABLES

Table Page

2.1 List of fundamental atomic units from Ref. [7, 12] ...... 10

2.2 Comparison of the lattice constant (a), bulk modulus (B) and cohesive energy (Ec) of silicon obtained by LDA and GGA. The bulk modulus is obtained by fitting energy vs. volume curve with Murnaghan equation. Results are from Refs. [8, 32]...... 38

2.3 Comparison of the lattice constant (a), bulk modulus (B) and cohesive energy (Ec) of copper obtained by LDA and GGA. The bulk modulus is obtained by fitting energy vs. volume curve with the Murnaghan equation. Results are from Refs. [8,33]...... 38

3.1 Crystallographic data of carbyne forms from Ref. [50]. The a0 lattice constant is for a full cell such as shown in Fig. 3.2(a), resulting in an interchain distance d (dashed lines in Fig. 3.2(a)), with the exception of carbon VI, which has a lattice constant a0 smaller by 3 (see Fig. 3.11(b)). The “Heimann” lattice constants (Hei) are calculated within the “kink” model described in the text with n carbon atoms in the chains between two kinks. Bond types are (C) cumulene and (P) polyyne. We have calculated the number of atoms Z in the full unit cell both from the experimental mass density ρ and unit cell volume where available and from multiplying n by 12 (4 in the case of carbon VI), since the structure in Fig. 3.2(a) contains 12 chains per cell (4 in the case of carbon VI)...... 50

3.2 Crystallographic data of optimized kinked-carbyne structures with sp2 (graphite- derived) and sp3 (diamond-derived) bonded kink atoms. Where available, we compare the lattice constants, calculated within LDA to experimental data from Refs. [50] and [67], processed as described in the text. n is the number of C atoms in a chain between two kinks. Several sp3 modifications are unstable and directly relaxed to the sp2 modifications. Type is the bond type (P = polyyne, C = cumulene), determined from the bond lengths...... 61

3.3 Lattice constants and density of optimized kinked Kudryavtsev type carbyne structures with sp2 and sp3 terminated chain lengths of n and m, respectively, sorted by increasing density...... 70

x 4.1 Pauling electronegativities of carbon, titanium, aluminum, palladium and gold [148] ...... 114

5.1 List of EAM parameters used in the present work for Cu and Zr from Ref [173] . 122

5.2 First peak positions of the PRDF of XRD [176] and EXAFS [177] for Cu46Zr54 BMGs in comparison to the results of our simulations and earlier simulations [175, 177] ...... 126

xi

LIST OF FIGURES

Figure Page

2.1 Number of publications with “density functional theory” in title or abstract from 1990 to 2005, identified with the help of Chemical Abstracts...... 9

2.2 Flowchart of self-consistent field (SCF) approach in solving for wave functions within the Hartree-Fock (HF) Approximation...... 25

3.1 Structure and electron density of 5-atom carbyne chains computed within density- functional theory, (a) forced to form alternating single and triple bonds (polyyne) and (b) allowed to relax to double bonds (cumulene) ...... 42

3.2 Structural models suggested for carbyne in (0001) projection by (a) Kasatochkin, (b) Sladkov and Kudryavtsev, and (c) Kudryavtsev et al. [50]...... 47

3.3 Kink-model for carbyne according to Heimann et al. [50] for (a) polyyne and (b) cumulene...... 48

3.4 Graphite bond splitting model for the formation of carbyne proposed by Whittaker [49]. Transition from perfect graphite layer (1) through transition zone (2) with breaking bonds (red dashed lines) to carbyne (3) for two different crystallographic directions...... 52

3.5 Blank's carbyne model of a phase intermediate between graphite and diamond [55]...... 53

3.6 Simulation cell of the simple hexagonal structure of carbyne after Kasatochkin [66] ...... 56

3.7 Typical final structures after MD simulations for 10 ps at different temperatures as indicated. In order to show the graphitic structure from the 3000 K simulation more clearly, the red circle area is replotted separately for a 2× 2 supercell ...... 57

3.8 Final structure (perfect diamond structure) from relaxation of the Kasatochkin-type carbyne from Fig. 3.6 under 50 kB pressure...... 58

xii 3.9 [110] projection of the simulation cell for n=6 carbyne with sp3 bonded kink atoms (a) and sp2 bonded kink atoms (b). Perspective view of the same crystal structures for n=6 sp3 (c) and n=6 sp2 (d) carbyne. The blue (darker) balls in (a) and (c) denote the kink atoms for easier identification...... 60

3.10 [010] projection of the 9R carbon structure that Blank's structure relaxes to for pressures between 669 kbar and just under 700 kbar...... 63

3.11 (a) Structural model of carbyne in (0001) projection suggested by Kudryavtsev et al. [66]. The two different “sublattices” are indicated by grey (brighter) and red (darker) circles, respectively, where the red sublattice has 2/3 occupation. Also shown is the projection of a hexagonal unit cell (solid black line). (b) Supercell of the structure from (a) shown by red lines, encompassing three cells with in-plane lattice constant similar to experimental results. (c) Same structure with fully occupied red sublattice (addition of hatched red circles) and unit cell (dashed red line) for fully occupied case...... 66

3.12 Kudryavtsev model [66] of carbyne with a chain length of six in (001) projection and side view (a) before and (b) after relaxation (different lattice vectors were accidentally chosen, but both structures have the same orientation). The two different “sublattices” are indicated by grey (brighter) and red (darker) spheres, respectively. Also shown is the projection of a hexagonal unit cell (solid black line) ...... 68

3.13 Relaxed Kudryavtsev structure [66] of carbyne in (0001) projection and side view with (a) minimum chain lengths of two (grey, brighter) and four (red, darker), (b) minimum equal chain lengths of four, and (c) 12 atoms in all chains...... 71

3.14 Calculated phonon density of states (DOS) for diamond (top), graphite (middle), and lonsdaleite-derived carbyne with 4 chain atoms, calculated within LDA ...... 73

3.15 Difference in free energy between different carbyne structures and graphite. Solid lines (symbols) are results for zero pressure, dashed lines (open symbols) for a pressure of 60 kbar. Lines are calculated from full phonon density of states within PWSCF, symbols from supercell Γ-point phonons using VASP. Red (top), 6sp3 carbyne (not stable at p = 60 kbar), blue (middle) 4sp2 carbyne, green (bottom) 4sp3 carbyne (n spm: n denotes number of carbon atoms in chains, spm the bonding of the kink atoms) ...... 75

3.16 Phase boundary between diamond (right-lower regime) and graphite (left-upper regime), calculated within LDA as described in the text (dots) in comparisons to experimental results (line) from Ref. [91] ...... 76

xiii 3.17 Difference in free energy between the different proposed carbyne structures that were stable under relaxation and the minimum-energy carbon phase (graphite or diamond, depending on pressure and temperature as shown in Fig. 3.12). Different temperatures are indicated by lines (top-down: 0 K, 1000 K, 2000 K, 3000 K, 4000 K, and 5000 K) as functions of applied pressure. Carbyne structures are labeled by number of carbon atoms in the chains (from 4 to 12) and bonding of kink atoms for the graphite and lonsdaleite-derived modifications, and by number of atoms in the sp2 and sp3 terminated chains, respectively, for the Kudryavtsev-type modifications...... 78

3.18 (001) Projection of the computationally optimized crystal structure of the solid (tert-butyl)C8(tert-butyl) carbynoids suggested in Ref. [45]. The structure consists of alternating layers of molecules with different orientation, indicated by grey C and white H atoms; and blue C and yellow H atoms, respectively ...... 79

3.19 Representative molecule from the (tert-butyl)C8(tert-butyl) structure shown in Fig. 3.18. The numbers next to bonds are the bond lengths after a structural relaxation within LDA in Å, the numbers next to arcs bond angles (experimental values from Ref. [45] in brackets) ...... 80

3.20 Spatial model of the carbyne intercalation compounds, redrawn after the suggestion by Udod et al. [94]. The original caption states that “the length of the rectilinear fragments became the same as a result of the chemical interaction of carbon atoms at kink sites with intercalated metal atoms.” ...... 82

3.21 Relaxed structure of example of cyanopoloyyne, NC − (C ≡ C) x − CN , where the kink atoms are nitrogen, which provide dangling-bond free kinks due to their trivalent bonding as suggested in Ref. [94]...... 82

4.1 (a) Rolling up a single atomic layer of graphite (graphene) forms a SWNT [112]. (b) The unrolled graphene structure, showing basis vectors a1 and a2. The unit cell of a SWNT can be defined by chiral vector, Ch (OA), and translation vector, T (OB). θ is the chiral angle. Rolling up OBB’A along OB and connecting OB with AB’ forms a (4,2) SWNT...... 88

4.2 Schematic drawing of a MWNT. Each cylinder represents a SWNT...... 89

4.3 (a) Unit cell and (b) Brillouin zone of graphene. a1 and a2 (b1 and b2) are (reciprocal) lattice vectors. Special points, Γ, Κ and Μ are defined as the center, the corner and the center of the side of the Brillouin zone, respectively...... 91

xiv 4.4 2D band structure of graphene from tight binding calculation, using ⎛ 3k a ⎞ ⎛ k a ⎞ ⎛ k a ⎞ E k ,k = ±γ 1+ 4cos⎜ x ⎟cos⎜ y ⎟ + 4cos 2 ⎜ y ⎟ with γ = 3 [100]. The ()x y ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ conduction and valence bands touch each other at only six points...... 92

4.5 2D band structure of graphene (top) and the first Brillouin zone (bottom). The allowed states in the nanotubes are cuts of the graphene bands indicated by the white lines. The lines on the bottom are the allowed wave vectors. (b) If the cut passes through a K point, the CNT is metallic; (c) otherwise, the CNT is semiconducting.... 93

4.6 Schematic drawings of (a) P-MOSFET and (b) N-MOSFET...... 94

4.7 Illustration of a CNT-FET device (side view) ...... 95

4.8 I–Vbias characteristic curves of a CNT-FET under various applied gate voltages (Vgate). Inset shows the conductance of the CNT-FET (G) as a function of applied gate voltage. The conductance exhibits a six orders of magnitude difference in the scan range (Vgate= -6V ~ 9V)...... 97

4.9 (a) Energy band diagram of a metal and a semiconductor before contact and (b) after contact, resulting an Schottky barrier ...... 99

4.10 Contact structures between Ti and a (3,3) carbon nanotube, where blue (darker) spheres represent Ti and dark grey (brighter) spheres represent C atoms in (outside of) the overlap region between the CNT and Ti: (a) structure after straightforward relaxation (no MD); (b) side view of the structure in (a); (c) side view of the structure after a 650 K, 1 ps MD run ...... 103

4.11 Opening of an infinitely long (3,3) nanotube on Ti during an MD simulation, where grey (brighter) spheres represent C and blue (darker) spheres Ti atoms. Cross-section of the contact area between the CNT and Ti at various instants during MD: (a) 0 ps, (b) 0.85 ps, (c) 1 ps and (d) 2 ps...... 104

4.12 (a) Predicted realistic contact structure with optimized transition length of ∼ 14 Å between CNT (most stable when no contact between Ti and CNT exists) and flat graphene sheet (most stable for CNT in contact with Ti). (b) Top view of the optimized transition region between tubular CNT (no contact with Ti) and the minimum-energy graphene film for (3,3) CNTs in contact with Ti. Grey (brighter) spheres represent C and blue (darker) spheres Ti atoms...... 105

xv 4.13 Transport curves for the small Ti/CNT/Ti structure after regular relaxation (Fig. 4.10(a), small relaxed) and after a 650 K, 1ps MD run (Fig. 4.10(c), small MD), and for our optimized large contact structure (Fig. 4.12, large realistic) in comparison to those of an ideal CNT to demonstrate the effect of the “electrodes”. (a) Current vs. bias, (b) conductance vs. bias ...... 108

4.14 Snapshots of ab initio TAD results of a infinitely long (4,4) CNT (a) and (5,5) CNT (b) on top of a Ti substrate. Grey (brighter) circles are C atoms and blue (darker) ones are Ti atoms. Both CNTs retain their tubular shape after a ~10 ps MD run at 650 K...... 109

4.15 Grey (brighter) circles represent C atoms, blue (darker) circles Pd atom. (a) CNT on top of Pd after an MD simulation of 3.6 ps at 1000K and relaxation. (b) Manually constructed open structure after relaxation. (c) Initial structure after relaxation...... 111

4.16 The process of CNT disintegration on top of Ti substrate. (a) Breaking C-C bonds; (b) opening up due to strain energy; (c) becoming a flat layer...... 113

4.17 Grey (top) circles represent C atoms, pink (bottom left) circles Al atoms, yellow (bottom right) circles Au atoms. (a) CNT on top of Al after an MD simulation of 1 ps at 800K. (b) CNT on top of Au after an MD simulation of 6.7 ps at 800K...... 115

5.1 Partial radial distribution function (PRDF) of simulated Cu46Zr54 structure with 1000 atoms at 2000K, 1000K and 300K...... 125

5.2 Tangent spheres define the biggest interstitial sphere (center) that can be inserted into the empty spaces between atoms ...... 127

5.3 Volume distribution of the open volume clusters identified in the hard sphere model. Cluster volumes have been corrected for overlapping spheres. Note that for cluster volumes greater than 0.4 Å3, the distribution is well described by the overlaid exponential decay function suggested by Cohen and Turnbull [183]...... 129

5.4 Cross section perpendicular to the z direction of a 1000-atom Cu46Zr54 BMG simulation cell analyzed by the hard sphere model. The open volume calculation was performed in a 29.5 Å × 29.5 Å × 13.5 Å box...... 130

5.5 Radially averaged electron density for a single Cu atom and a single Zr atom truncated at 0.286 e/Å3...... 133

5.6 Contour plot of the electron density distribution in crystalline CuZr B2 structure for a (110) crystallographic plane from (a) radial averaged electron densities and (b) a self- consistent DFT calculation of the charge density. Red corresponds to regions of high electron density (e.g. atom cores), while blue corresponds to regions of low electron density. Periodic boundary conditions are applied to both calculations...... 134

xvi 5.7 Contour plot of the electron density of a cross-section of a 1000-atom Cu46Zr54 BMG structure. The electron density was calculated by our simplified electron density model. Red corresponds to regions of high electron density, blue to regions of low electron density ...... 135

5.8 Size distribution of the open volume regions identified by the electron density model, calculated from ten 1000-atom Cu46Zr54 BMG structures...... 137

5.9 Cross-section of a Cu46Zr54 BMG structure showing the shape of open volume as black from (a) the radially averaged electron density model and (b) a self-consistent DFT calculations...... 137

xvii

CHAPTER 1

INTRODUCTION

1.1 Computational Materials Science

Modeling in computational materials science refers to scientific abstraction of realistic materials systems. The goal of this effort is to produce a description of the system, or part of it, which allows us to obtain a better understanding of physical processes and phenomena and to predict the material’s behavior [1]. This field has been growing rapidly over decades thanks to the overwhelming increase of computational power. As a result, it has become an indispensable part of materials research and has been embraced by academia and industry with great enthusiasm. This is also indicated by the fact that there are now a number of profitable commercial products based on materials modeling [2].

The word ‘model’ usually refers to a set of mathematical equations, but its central idea is to assemble a simplified imitation of the real world while preserving the essential features [3]. In other words, a model must 1) be simple enough to solve and 2) correctly capture the features of interest. The advent of affordable and powerful computers in the

1 last 30 years lead to the development of an array of computational models. These models utilize computers to solve equations numerically, thus providing non-analytic solutions which could not be obtained otherwise. Thanks to the ever increasing speed and memory of computers and the decreasing scale of interest of materials systems, simulations can by now explore the behavior of realistically sized models in reasonable time. As a result, a new range of processes and phenomena are open to study. Also the paradigms of materials theory is changing due to the central role of numerical calculations in today’s modeling – the requirement for a model to be simple is often replaced by the requirement for computational efficiency [1].

According to the characteristic length scale that each materials model deals with, they can be categorized into 3 (or 4) types: continuum, microstructural and atomic scale

(electronic scale) models.

A continuum model is often adopted when solving differential equation problems, such as mechanics, heat transfer and mass transport arising from materials processing such as casting, forging and welding. In this type of model, the macroscopic sample is partitioned into many small volume units and mapped onto a mesh. Materials properties and behavior are assumed to be uniform within each volume unit. The solution to the (set of) differential equation(s) is then approximated by a piecewise smooth function, which can be solved numerically by matrix operations. This approach is also known as the finite element method which is the most dominant model in solving macroscopic mechanical problems. The scale of this model is usually millimeters and larger.

Similarly, a material sample is also partitioned into many small volume units (or cells) in a microstructural model. Physical quantities, such as concentrations in the diffusion

2 problem, assume a value in each cell. The type of equations to be solved depends on the system of interest. For example, Fick’s equations are solved for a diffusion problem.

These equations, also known as governing equations, dictate the interactions between different volume cells and the evolution of each cell. The time is discretized into small pieces called time steps. The physical quantity is updated after each time step for each cell simultaneously according to governing equations. This concept is called cellular automata. For a specific model, usually only one kind of microstructural feature is simulated. This type of model has been applied to study diffusion, grain growth, dislocation motion, phase transformation, precipitate growth, recovery and many other kinds of microstructure evolution. The scale of this model is typically from microns to millimeters.

In atomic scale (atomistic) modeling, materials are visualized as an assembly of many atoms (and possibly electrons). This type of model will be elaborated in the next section.

One may classify models based on different criteria other than the underlying characteristic scale. For example, models can be identified as deterministic, such as molecular dynamics (MD) simulations or stochastic/probabilistic, such as Monte Carlo

(MC) models, according to their predictive character. Or they can be identified as first- principles, such as density functional theory (DFT), or phenomenological, such as the phase field method, according to their descriptive character [3]. With so many choices at hand, it is a really exciting job for materials modelers to analyze a specific problem with appropriate models and try to understand, predict and design materials, which from my understanding is the ultimate goal of computational materials science.

3 1.2 Atomistic Modeling

In an atomistic model, the spatial coordinates of atoms of interest are updated after each time step, usually ranging from femto- (10-15) to pico- (10-12) seconds. The simulation can run for many time steps to represent real physical phenomena on the corresponding time scale. This update of atomic coordinates is governed by Newton’s classical equations of motion. Thus, it is necessary to know the forces acting on atoms r which can be calculated from the gradient of interatomic potentials, F = −∇E .

There are many different choices of potentials depending on the nature of the studied system, the level of accuracy and computational time limitation. Different choices correspond to different types of atomistic models. In general, there are three categories: classical force fields, semi-empirical methods and first principles (ab initio) methods.

In classical MD, the potential of an atom solely depends on its position relative to other atoms. Interactions between atoms are described by analytical potential functions, such as Lennard-Jones, Morse (these two are so-called pair potentials and only depend on the interatomic distances), Embedded-Atom Method (EAM) and Tersoff potentials. For the simplest case of pair potentials, one considers pairs of each individual atom with all other atoms in the system (or just a subset of nearest neighbors for increased computational efficiency), and the energy contribution from each pair is determined by a pair potential (which is a function of the interatomic distance). Then, all pair contributions are summed up to calculated the total energy of the system. This approach allows fast evaluation of energies and provides a quick way to study atomic motion, but

4 with limited accuracy. The typical system size of a classical MD study can currently be up to billions of atoms, but is often in the 1000-100,000 atom range.

Besides some so-called semi-empirical methods which are not used in this thesis, recent developments in first principles modeling allow by now to use quantum mechanical models for the in-depth study of atomic scale features. Treating materials of interest as many-body systems composed of electrons and nuclei on the basis of quantum mechanics without introducing any empirical parameters [4], computationally efficient first principles (or ab initio) methods were developed to determine the electronic structure of solids, surfaces or clusters at a certain acceptable accuracy level with the purpose of predicting properties of materials on the atomic scale. The work of Hohenberg and Kohn [5] gave rise to the development of the density functional theory (DFT) method, which is an important part of the atomic-level materials modeling nowadays. This approach is based on the concept that the properties of the ground state of a many body system can be determined by the electronic ground state electron density, thus allowing one to solve the many-body Schrödinger equation and to simplify the numerical process.

DFT has been very successful; however, it is computationally very demanding which is an inevitable disadvantage. Even with the dramatic improvement in computational power over the recent years, most DFT simulations can only handle less than 1000 atoms (or a few thousand for the most favorable systems) which are on the scale of less than 10 nanometers along each spatial dimension. As a comparison, in a continuum model, there are usually at least 1023 atoms involved.

Because of the ability of DFT to calculate the electron density – something that cannot be done with classical atomistic models – it can also be categorized as electronic

5 scale modeling (or “electronic-structure method”). This ability to also calculate the electronic structure of materials opens a whole new area for materials science studies.

1.3 Motivation and Organization of This Thesis

As mentioned above, the application of DFT is severely limited with respect to the system size and the time scale of dynamic simulations [4]. Traditionally, DFT has been very successful for energy and electronic structure calculations of small and simple systems, such as crystalline Si which can be modeled by a 2-atom primitive unit cell or an 8-atom conventional unit cell. The purpose of this thesis is to examine extending the scope of the application by adopting new approaches and to study complex materials systems with DFT calculations.

Chapter Two provides an introduction to the basic ingredients of DFT. It starts with fundamentals of quantum mechanics including the Schrödinger Equation, many-body wave functions and the variational method. Then the precursor of DFT, the Hartree-Fock approximation, is explained and its application to the uniform electron gas is discussed.

Finally, DFT is introduced through the work of Hohenberg-Kohn [5] and Kohn-Sham [6] with emphasis on the physical meaning. The Born-Oppenheimer approximation and pseudo potential concept are described as well.

Chapter Three is the application of DFT for determining the structure and stability of carbynes which has been proposed as the third allotrope (beside graphite and diamond) of carbon with sp hybridization. In that chapter, the free energy of the different structures is calculated as a function of temperature and pressure within the quasi-harmonic approximation in a simplified way, thus allowing the fast calculation of the free energy of large systems. 6 Chapter Four shows an example of atomic-scale process modeling of carbon nanotube devices with the help of DFT-based accelerated dynamic simulations. The time scale of process modeling is typically nanoseconds or even milliseconds and is far too long for DFT. We tackled this barrier by using ab initio temperature accelerated dynamics, pioneering the use of these methods within the framework of DFT.

Chapter Five discusses the study of the electron density of Cu46Zr54 bulk metallic glasses with DFT in conjunction with classical MD simulations. The system required to model an amorphous structure consists of at least thousand atoms, which is too large for direct DFT simulations. However, the quantity of interest, the electron density, is not available to classical MDs. Thus, we use these two methods complementarily to overcome the size limitation while ensuring accuracy.

Chapter Six is the summary of conclusions and future directions.

7

CHAPTER 2

AN INTRODUCTION TO THE DENSITY FUNCTIONAL THEORY

Density functional theory (DFT) is a quantum mechanics based approach to approximate the electron density and electron wave function of a system of atoms. This approach is widely used for atomic simulations in the field of materials science, solid state physics and quantum chemistry. The number of publications with keywords “DFT” or “density functional theory” has gone up dramatically since the early 1990's. Koch and

Holthausen used Chemical Abstracts for searching DFT related publications and identified that the publication number increased 20 times during the 1990~1998 period [7]. A similar search is performed for the period of 1990~2005 and the results are shown in Fig. 2.1, which indicates the growing trend continued. There are several reasons why DFT became so popular. One of them is that DFT solutions provide information about the electronic structure, which is not available to traditional molecular dynamics

(MD) approaches. It does not require any empirical potentials (but still needs potentials, which will be elaborated in this chapter), which makes it very versatile. It is also quite accurate if conducted carefully. The disadvantage of this approach is that it is

8 computationally expensive and scales badly with the system size in most implementations. However, with the increasing computer power nowadays, DFT becomes a feasible method for larger and larger systems.

Figure 2.1: Number of publications with “density functional theory” in title or abstract from 1990 to 2005, identified with the help of Chemical Abstracts.

In this chapter, the basic ingredients of DFT are surveyed with emphasis on understanding the physics. Detailed derivations can be found in Refs. [7-9]. The presentation of the material and notation follows Refs. [7, 8, 10, 11]. For the sake of conciseness, atomic units are used in all equations in this chapter. In this unit system, six physical constants (see Table 2.1) are set to unity. As a result, some physical constants are avoided in the calculation of electron kinetic energy or Coulomb energy. A list of the fundamental atomic units can be found in Table 2.1. Note that there are two types of

9 atomic units which are Hartree atomic units and Rydberg atomic units. Their difference is in the choice of mass and charge units, and therefore the energy units are different. This chapter uses Hartree atomic units.

Quantity Unit Symbol Value in SI unit −31 Mass electron rest mass me 9.109×10 kg Charge electron charge e 1.602×10−19 C Angular momentum reduced Planck’s constant h/2π 1.055×10−34 J ⋅ s −11 Length Bohr radius a0 5.292×10 m −18 Energy Hartree energy Eh (Ha) 4.360×10 J 9 −2 2 electrostatic constant Coulomb constant 1/4πε0 8.988×10 C Nm

Table 2.1: List of fundamental atomic units from Ref. [7, 12].

2.1 The Schrödinger Equation and Wave Functions

Newtonian mechanics apply well to a broad range of physical processes, ranging from everyday life events such as dropping a ball to sending humans to the moon.

However, as the scales decrease, non-classical effects become prominent. The birth and growth of quantum mechanics is closely related to the insufficiency of Newtonian mechanics for electron orbitals in atoms, where Bohr first proposed new models in

1913 [13].

A particle can be associated with a set of definite coordinates in classical mechanics.

However, in quantum mechanics, it is described in a significantly different way – by a wave function. The Schrödinger equation is the bedrock and centerpiece of non- relativistic quantum mechanics for materials applications. It is applied to derive wave

10 functions and to find out how wave functions evolve, in analogy to Newton's equations of motion in classical mechanics. The goal of density functional theory is to solve

(approximately) the time independent non-relativistic Schrödinger equation

r r r r r r HΨ(r1,r2 ...rN ) = EΨ(r1,r2 ...rN ) (2.1) where H is the Hamilton operator or Hamiltonian, i.e. kinetic energy (T) plus potential energy (V) of an N particle system, Ψ the wave function, and E the energy. The kinetic energy is always

1 N 1 N ⎛ ∂ 2 ∂ 2 ∂ 2 ⎞ T = − ∇ 2 = − ⎜ + + ⎟ , (2.2) ∑ i ∑⎜ 2 2 2 ⎟ 2 i 2 i ⎝ ∂xi ∂yi ∂zi ⎠ while the potential energy is system dependent. Mathematically, the Schrödinger equation defines an eigenvalue problem with eigenvalues E and eigenvectors Ψ.

Given a specific system and its Hamilton operator, the solution to the Schrödinger equation is often not unique. There will be multiple wave functions, each associated with an energy value. The lowest energy is called ground state energy and the associated wave function is called ground state wave function. The wave function itself is not a physically observable quantity but the square of its modulus, Ψ 2 , is the probability to find the particle in a certain state defined by so-called quantum numbers (which can be, e.g., a certain position, momentum, etc.).

The bra-ket notation invented by Dirac will also be used in this chapter. Ψ denotes the wave function Ψ and Ψ denotes the complex conjugate of Ψ, with

Ψ Ψ = ∫ Ψ *Ψdr . The energy of the system can be calculated as

11 Ψ H Ψ = ∫ Ψ * HΨdr . If the wave function is known, other observable quantities can be calculated in the same way.

However, the Schrödinger equation is difficult to solve, especially for the case of many body systems. The equation is solvable for some idealized systems (such as the free particle, the particle in a box and harmonic oscillators) and one simple realistic system, the hydrogen atom. For more complex systems, even for the helium atom with two electrons, no analytically solution can be found anymore. DFT is a method to find the approximate solution for complex systems involving many electrons and many atoms.

2.2 Born-Oppenheimer Approximation

The first approximation that DFT typically uses to tackle the Schrödinger equation is called Born-Oppenheimer approximation, which is taking advantage of the fact that nuclei are significantly heavier than electrons. The lightest nucleus, H+, is approximately

1836 times heavier than an electron, and the second smallest nucleus, He2+, 7357 times heavier. Therefore, all nuclei move very slowly compared to electrons. This leads to a reasonable argument to separate the motion of electrons from that of the nuclei. As a result, one may only consider the wave function of electrons when solving the

Schrödinger equation, and treat the motion of the nuclei by the much simpler formalism of classical mechanics. Therefore, the (electronic-only) Hamiltonian can be written as

N N N N M 1 2 1 1 Z k H = − ∑∇ i + ∑∑ − ∑∑ = T +Vee +VNe , (2.3) 2 i 2 i=≠1 j i rij i==11k rik where N is the number of electrons, M the number of nuclei, Zk the atomic number of atom k, and rij the distance between nuclei i and j. The first term is the kinetic energy T of

12 the electrons, the second term the Coulomb interaction Vee between electrons, and the last term the interactions between electrons and nuclei VNe.

2.3 Many-Body Wave Functions

The Born-Oppenheimer approximation reduces the quantum mechanical problem to that of a system of N identical electrons. In quantum mechanics, identical particles are indistinguishable, which implies that if one swaps any two electrons, the system stays in the same state. Recalling that only the module of the wave function has physical significance, it is required that the module of the wave function doses not change after the

swapping. For a two-electron system, that means ψ (x1 = a, x2 = b) = ψ ()x1 = b, x2 = a , or simply ψ ()a,b = ψ ()b,a . One may conclude that ψ (a,b)()= αψ b,a and α = 1 . It may be further derived that α 2 = 1 [10]. That leads to two possibilities. If α = 1 ,

ψ ()a,b =ψ ()b,a and we say the wave function is symmetric. The corresponding particles are called bosons, such as photons and gravitons. If α = −1, ψ (a,b)()= −ψ b,a and we say the wave function is anti-symmetric. The corresponding particles are called fermions, which include electrons, protons and neutrons. As a result, the wave function for an N- electron system is anti-symmetric because all particles are fermions.

2.4 The Variational Method

It is often impossible to solve for eigenvalues of the Hamiltonian analytically. The variational method is the first approximative approach one may meet in a quantum mechanics textbook.

This method is based on the fact that, for any Hamiltonian H,

13 Ψ H Ψ E[]Ψ ≡ ≥ E , (2.4) Ψ Ψ 0

where Ψ is an arbitrary wave function and E0 is the ground state energy, i.e., the lowest eigenvalue of the Hamiltonian. It can be proved as following. Expand Ψ in terms of eigenfunctions of H, i.e., En : Ψ = ∑α n En . Then, since all En ≥ E0,

2 2 ∑α n En En Ψ E0 ∑α n En Ψ E[]Ψ = 2 ≥ 2 = E0 . (2.5) ∑α n En Ψ ∑α n En Ψ

To find out more properties of the eigenfunctions of the Hamiltonian, let us define a trial function

Ψ = E0 + δΨ . (2.6)

Then

E0 H E0 + δΨ H E0 + E0 H δΨ + δΨ H δΨ 2 E[]Ψ = = E0 + O()δΨ .(2.7) E0 E0 + δΨ E0 + E0 δΨ + δΨ δΨ

This relationship shows that the error in ground state energy is of the second order in trial functions. More generally, this is also true for all other eigenfunctions of H. Let

Ψn = En + δΨn . (2.8)

Then by similar approach

2 E[Ψn ] = En + O(δΨn ) . (2.9)

Therefore, when eigenfunctions of H change by first order, eigenvalues change by second order, i.e., have no first order change. This so-called (n+1)-theorem will be helpful for many-body problems in DFT.

14 It is also necessary to introduce the concept of functionals at this point. Simply speaking, a functional is a function of a function [14]. This can be understood by comparing a functional to a function. For example, let u(x) = x +1. We call u a function

1 of x. Then if we define F[]u = []u()x 2 dx , we call F[u] a functional. Notice that the ∫0 square bracket is often employed for the argument of a functional to distinguish it from a function. A functional takes a function as input and delivers a number. For example,

1 1 F[]u = []u()x 2 dx = 7 / 3. Let v(x) = 2x −1, then F[]v = []v()x 2 dx = 1/ 3 . ∫0 ∫0

The concept of variation for a functional is analogous to the differential of a function.

Let F[y] be a functional, y(x) a function of x, and h(x) a “small” function independent of y(x). Then

δF[h] = F[y + h] − F[y]. (2.10)

The “small” function h(x) represents an increment on the argument. The resulting difference in the functional, δF, is called the variation.

It can be proven that a necessary condition for a functional F[y] to have an extreme value at y = y0 is that δF=0 at y0 [14]. This is again analogous to functions having extreme values when their differential is zero.

The goal is to find the many-body wave function that satisfies the Schrödinger

Equation with Hamiltonian H shown in Eq. (2.3). As pointed out by the variational method, one can find the ground state wave function and the corresponding energy by minimizing the functional E[Ψ] ≡ Ψ H Ψ through all allowed trial wave functions.

The eligible trial function must physically make sense, for example, it must be square

15 integrable and continuous. In other words, if one can search through all eligible N- electron wave functions and indentify the one with lowest energy

E0 = min E(Ψ), (2.11) ∀Ψ then the problem is solved. However, such a search is not possible, because there are infinite numbers of candidates. A feasible approach is to apply the variation method to a subset of all candidates.

2.5 Hartree-Fock Approximation

The Hatree-Fock (HF) approximation is a method to solve for N-electron wave functions. It introduces important concepts such as exchange, self interaction and correlations, which are crucial to DFT. The HF approximation is also of theoretical importance because it is the father of many conventional quantum mechanics methods [7,

8]. It will be discussed as the foundation of DFT.

Although the Born-Oppenheimer approximation significantly simplifies the problem of finding ground states of atoms, the resulting Hamiltonian is still too complex to solve directly due to the electron-electron interactions. To overcome this, the non-interacting or

Hartree approximation [15-17] is introduced. It assumes that all electrons are independent particles, i.e., non-interacting, and that the wave function for this N-electron system can be constructed from the product of single electron wave functions,

H r r r r r r Ψ ()r1 ,r2 Lri = φ1 (r1 )φ2 (r2 )LφN (rN ), (2.12)

r where, φi ()ri is the wave function of a single independent electron placed in the potential of the nuclei. Pictorially speaking, the single-electron wave functions that produce the minimum energy define the molecular or atomic orbitals, for example, 1s, 2s, 2p etc., in

16 the context of electrons in atoms. And they are chosen to be orthonormal states for

computational convenience, i.e, φi φ j = δ ij where δij is the Kronecker delta. This is called Hartree Approximation. The construction would be exact if the non-interaction assumption was true. Then the energy can be calculated from the wave function and

Hamiltonian

2 N N ⎛ ∇ ⎞ 1 1 2 2 E H = Ψ H H Ψ H = drφ * ⎜− +V ⎟φ + dr dr φ φ . (2.13) ∑∫ i ⎜ Ne ⎟ i ∑∑∫∫ i j i j i ⎝ 2 ⎠ 2 i=≠1 j i rij

Once the formulation of the energy has been obtained, the variational method can be applied to solve for wave functions by setting the variation to zero under the constraint of

dr φ r 2 = 1 , which is the normalization condition. This can be done by employing ∫ i ()

Lagrange multipliers εi,

N ⎡ 2 ⎤ δ E H − ε dr φ r −1 = 0 , (2.14) ⎢ ∑ i (∫ i () )⎥ ⎣ i ⎦ which states that when we change the wave function to first order under the constraint,

there is no first-order change in energy. Therefore, when we change φi ()r by an arbitrary

“small” amount, δφi ()r , there should be no change in energy,

2 N ⎡⎛ ∇ 1 2 ⎞ ⎤ δE H = dr δφ * ⎢⎜− +V + dr φ − ε ⎟φ ⎥ = 0 . (2.15) ∫ i i ⎜ 2 Ne ∑∫ j r j i ⎟ i ⎣⎢⎝ j≠i ij ⎠ ⎦⎥

Because δφi ()r is arbitrary, Eq. (2.15) is only possible if the term inside the square bracket is always zero, which is the case for

2 N ⎛ ∇ 1 2 ⎞ ⎜− +V + dr φ ⎟φ = ε φ . (2.16) ⎜ Ne ∑∫ j j ⎟ i i i ⎝ 2 j≠i rij ⎠

17 Equation (2.16) is the Hartree Equation [15-17], which transform the N-electron problem into N independent-electron problems. This new problem can be solved in a self- consistent fashion. However, the total wave function (Eq. (2.12)) is not anti-symmetric under particle exchange. Swapping the positions of any two electrons, Ψ H remains unchanged, which is contradictory to the fact that electrons are fermions.

Another construction of trial wave function can avoid this problem, which can be written in the form of a determinant:

r r r φ1 (r1 ) φ2 (r1 ) L φN (r1 ) r r r HF r 1 φ1 ()2 φ2 ()2 φN ()2 Ψ (){}ri = , (2.17) N! L L L r r r φ1 ()rN φ2 ()rN L φN ()rN

r r where the coordinate ri include the spin state in this case and the φi ()ri are referred to as spin orbitals, for example, 1s ↑ , 2 p ↓ etc. The whole term on the right-hand side is sometimes referred to as a Slater determinant.

This is called the Hatree-Fock approximation, which anti-symmetrizes the total wave function while keeping the spirit of approximating the total wave function by single electron wave functions. Now, if two electrons are swapped (exchange coordinates), it is equivalent to switching two rows in the Slater determinant. This will give a negative sign, which satisfies the anti-symmetry condition for fermions.

As an example, consider two electrons in a helium (He) atom [8]. Let the spin orbitals be

r r ↑ φ1 (r ) =ψ 1s (x)σ (s),

r r ↓ φ2 (r ) =ψ 1s (x)σ (s), (2.18)

18 where x is the spatial coordinate and s is the spin. Then the Slater determinant is

HF 1 r r ↑ ↓ ↑ ↓ Ψ = ψ 1s ()x1 ψ 1s (x2 )[σ ()s1 σ ()s2 −σ ()s2 σ ()s1 ], (2.19) 2 which is anti-symmetric because it is symmetric with respect to the space and is anti- symmetric with respect to the spin states.

Following an approach similar to the Hartree approximation, the energy is first calculated as [7]

N N N E HF = Ψ HF H Ψ HF = ∑()i h i + ∑∑()()ii | jj − ij | ji , (2.20) i i j where

⎛ ∇ 2 M Z ⎞ i h i = drφ * ⎜− − k ⎟φ (2.21) ()∫ i ⎜ ∑ ⎟ i ⎝ 2 k rik ⎠ is the kinetic energy plus nucleus-electron interaction energy,

1 2 2 ii | jj = dr dr φ r φ r (2.22) ()∫∫ i j i ()i j ( j ) rij is the so-called Coulomb integral, and

1 ()ij | ji = dr dr φ ()r φ * ()r φ (r )φ * (r ) (2.23) ∫∫ i j i i j i j j i j rij is the so-called exchange integral.

After the energy is expressed in terms of spin orbitals, the variational method is applied in a similar way as in the Hartree approximation. Spin orbitals are chosen to be orthonormal. Again, Lagrangian multipliers εi are employed to take this constraint into account. The resulting equation is called Hatree-Fock Equation,

fφi = ε iφi (2.24)

19 where f is an operator, which is defined as

2 M ∇ Z k f = − − ∑ +VHF ()i . (2.25) 2 k rik

Through the Hartree-Fock approximation, the N-electron problem is again reduced to

N independent-electron problems. Moreover, the total wave function Ψ HF is anti- symmetric. The Lagrangian multipliers εi in the Hartree-Fock equation (Eq. 2.24) can be interpreted as the orbital energies. The operator f consists of three terms (Eq. 2.25). The first term is kinetic energy, the second term is the nucleus-electron interaction energy, the last term, VHF, is called the Hartree-Fock potential, which is the effective potential from all N electrons acting on the i-th electron. Thus, the complicated electron-electron interactions are taken into account by VHF in an averaged sense. The HF potential VHF is expressed as [7]

N VHF ()ri = ∑[]J j ()ri − K j ()ri . (2.26) j=1

J is the Coulomb operator and defined as

1 2 J r = dr φ r , (2.27) j ()1 ∫ 2 j ()2 r12

2 which corresponds to the Coulomb integral shown in Eq. (2.22). Notice that φ j ()r2 is

the charge density at r2 of an electron belonging to the φ j spin orbital. The integral calculates the Coulomb repulsion experienced by one electron (at r1) due to the charge

distribution of another electron (belonging to φ j spin orbital).

K is an exchange operator and defined as

20 1 K r φ r = dr φ * r φ r φ r , (2.28) j ()()1 i 1 ∫ 2 j ()()()2 i 2 j 1 r12 which corresponds to the exchange integral shown in Eq. (2.23) and takes into account the exchange contribution of the HF potential. As indicated by Eq. (2.28), when K acts on a spin orbital, it causes the spin orbitals to exchange arguments (on the right hand side).

This exchange term arises from the fact that the Slater determinant is anti-symmetric.

Thus, there is no classical counterpart or classical explanation of it. Another property of this term is that the integral will be zero when two spin orbitals are of different spins and non-zero when two spin orbitals are of same (parallel) spin. This is because spin functions themselves are orthonormal. For example, consider the two wave functions in

Eq. (2.18). Substitute them into Eq. (2.28) and the integral will contain a factor

↑ ↓ ↑ ↓ σ ()s2 σ ()s2 . This term is integrated to zero ( σ (s2 ) σ (s2 ) = 0 ) and makes the integral vanish.

Notice that in the expression for the HF potential (Eq. (2.26)), the summation runs over all N electrons, including the i-th electron itself. Thus, there are N Coulomb interactions in J (Eq. (2.27)), one of which is the interaction between the i-th electron and itself. This self-interaction is obviously an artifact of the model. Consider the example of the hydrogen atom which has only one electron. There should be no electron-electron interaction in this case. Nonetheless, the HF potential still has one term

1 2 J r = dr φ r , the self-interaction. The exchange operator K comes to the 1 ()1 ∫ 2 1 ()2 r12 rescue: whenever the self-interaction is counted (when i = j), the exchange term K produces the exactly same term with opposite sign to offset the interaction. It is easy to

21 verify for Eqs. (2.27) and (2.28) that when i = j, K iφi = J iφi . However, K has a minus sign in the HF potential (Eq. 2.26). Thus they cancel out each other and there is no self- interaction in the HF potential.

As an example, the Hartree-Fock approximation can be applied to study electrons in the Jellium model (uniform electron gas) of a solid. In this model, positive charges, i.e., the charges of nuclei are assumed to be uniformly distributed in space. Interacting electrons are moving on top of this uniform “jelly” of positive background charge. This naive model treats solids as if they were jelly, where its name comes from.

To solve the jellium model, consider a solid of cubic shape with edge length L and volume V=L3. Then the free electron wave functions in this cube can be written as

r 1 r r φi ()x = exp(iki ⋅ x) , (2.29) V

r where, 1/ V is the normalization factor and ki is the wave vector. Such a function is called a plane wave. Consider the whole material is made up of many of these cubes. We

may apply periodic boundary condition to it, which requires exp(iki L) = 1 and this leads to [10]

r 2n π 2n π 2n π k = 1 xˆ + 2 xˆ + 3 xˆ n = 0,±1,±2 (2.30) i L 1 L 2 L 3 i L

The ground state of the N-electron system is the state where the N electrons achieve lowest energies. For free electrons, the only energy is the kinetic energy. For an electron

r 1 r 2 with wave vector k , its kinetic energy is k . So if all electrons had zero wave vector, i 2 i their energies would be zero, resulting in a minimum total energy of zero. However,

22 because of the Pauli Exclusion Principle, each wave vector can only accommodate 2 electrons (with different spins). Therefore, the lowest energy configuration is that only 2 electrons fill up the zero wave vector. Another 2 electrons fill up the next smallest wave vector etc., until no electrons are left. Then we reach a wave vector which is the largest among all those occupied by electrons, which is called Fermi wave vector kF. It is a useful concept when dealing with free electrons. It is related to the electron density, n, by

N k 3 n = = F . (2.31) V 3π 2

The total kinetic energy of the N-electron system is

5 / 3 2 (3πn) V T = 2 k = . (2.32) ∑ 2 k ≤kF 5π

Now we can prove that the plane wave function (Eq. (2.29)) is indeed the solution of the HF equation (Eq. (2.24)) in the Jellium model. As mentioned above, the positive charge is uniformly distributed, as well as the electron density, because the electron

2 2 density is calculated as ∑ φi = ∑ 1/ V = N /V , which is a constant throughout space. Moreover, the positive charge is the same as the negative charge (neutrality).

Therefore, the nucleus-electron interaction and electron-electron Coulomb interaction cancel each other, hence the two corresponding terms in the HF potential cancel each other as well. Thus, the HF equation becomes

⎡ ∇ 2 N ⎤ ⎢− − ∑ K j ()ri ⎥φi = ε iφi . (2.33) ⎣ 2 j ⎦

Substituting φi from Eq. (2.29) into Eq. (2.33) results in the kinetic component

23 2 ∇ 2 − φ = k φ (2.34) 2 i i i and the exchange component

N k ⎛ k ⎞ − K r φ = − F F⎜ i ⎟φ , (2.35) ∑ j ()i i ⎜ ⎟ i j π ⎝ k F ⎠ where

1− x 2 ⎛ 1+ x ⎞ F()x = 1+ ln⎜ ⎟ . (2.36) 2x ⎝ 1− x ⎠

Thus, plane waves solve the HF function in the Jellium model. The corresponding energies for individual electrons are

2 k ⎛ k ⎞ ε = k − F F⎜ i ⎟ , (2.37) i i ⎜ ⎟ π ⎝ k F ⎠ and the total exchange energy is

4 / 3 k ⎛ ki ⎞ (3n) V E HF = − F F⎜ ⎟ = − . (2.38) ex ∑ π ⎜ k ⎟ 4π 1/ 3 ki ≤kF ⎝ F ⎠

The total energy is the kinetic energy (Eq. (2.32)) added to the exchange energy (Eq.

(2.38)).

Up to this point, the wave functions and energy of the Jellium model were found within the HF approximation. However, more often than not, no closed-form solution is found for the HF approximation because the exchange term is rather intractable. The self- consistent field (SCF) approach is often employed to deal with those cases. It begins with a trial wave function, which is often a Slater determinant of local orbitals selected based on physical sense. Then the HF operator is generated based on this initial trial, the HF

24 equation is solved, and the new wave function is obtained. The new wave function is compared to the old one, and one determines whether the difference is small enough. If not, the newly obtained wave function is used as the trial function and the whole process is repeated. This is iterated until the difference is smaller than a pre-defined cutoff value.

The process is illustrated in the flowchart shown in Fig. 2.2.

Begin

Input Initial Guess Of Wave Function

Calculate HF operator

No Solve Wave Function Input the obtained Based on HF equation wave function

Converged?

Yes Output End

Figure 2.2: Flowchart of self-consistent field (SCF) approach in solving for wave functions within the Hartree-Fock (HF) Approximation.

25 The Hartree-Fock approximation can be further categorized into restricted HF approximation (RHF) and unrestricted HF (UHF) approximation. The difference is in the construction of the Slater determinant. A lot of systems, such as most molecules, have only paired electrons, where all electrons pair up and each pair share the same spatial orbit but have different spins. For those systems, the assumption can be made that all spatial orbitals enter the formula with double occupancy. If this restriction is imposed, the resulting model is RHF, otherwise UHF. In general, UHF is more accurate in calculating energies, but there are cases where the calculated total spin seriously deviates from the true value [7].

Generally speaking, the HF approximation captures pretty well the physics of N- electron systems. However, the Slater determinant is only an approximation and not the exact wave function. The difference between true energy and the calculated one is called correlation energy [18]

HF HF EC = E0 − E , (2.39) where E0 is the true energy. Recall that the variation principle states that an approximated energy is always greater than the true ground state energy. Therefore, the correlation energy has always a negative value. A large part of the correlation energy, or the error introduced by the HF approximation, is due to the way of calculating the electron- electron Coulomb interaction term (Eq. (2.27)). Although the single-electron wave function may describe the averaged probability of an electron accurately, in real cases, because of the repulsions between electrons, any two electrons tend to stay farther away from each other in any instantaneous moment than the distance predicted by the HF wave function. In other words, the electron-electron distance from the HF approximation is 26 smaller than the real value, which results in an extra energy in the Coulomb term. This part of correlation is called dynamical. Another part of the correlation energy is due to the fact that a single Slater determine may not be a good approximation of the true wave function in some cases. This part is called non-dynamical correlation energy.

2.6 The Exchange and Correlation Hole

To understand better about the correlation effect as mentioned above, as well as the exchange effect (Eq. (2.28)), the concept of the exchange and correlation hole is introduced with the help of electron density and pair density in this section. This concept provides physical pictures of the exchange and correction effect and serves to better understand the HF approximation, as well as DFT.

As has been mentioned above, the probability to find electrons, the electron density, is calculated from the wave function by multi-dimensional integration,

n r = N dr dr Ψ * r r Ψ r r . (2.40) ()1 ∫L∫ 2 L N ( 1 L N ) ( 1 L N )

Unlike the wave function, the electron density is a physically measurable quantity, by experimental methods such as positron annihilation spectroscopy. It is the probability of finding one of the electrons at a specified location, regardless of the whereabouts of the remaining N – 1 electrons. The factor N appears in Eq. (2.40) because each integral gives the probability of finding a specific electron. The probability of finding any one electron is N times that. If the electron density is integrated over the whole space, the result will be the total number of electrons, i.e., dr n r = N . ∫ 1 ( 1 )

27 Closely related to the electron density, the pair density ρ refers to the probability of finding any two electrons at two specified locations simultaneously, regardless of the locations of the remaing N – 2 electrons. It is defined as

ρ r ,r = N N −1 dr dr Ψ * r ,r r Ψ r ,r r , (2.41) ()(1 2 )∫L∫ 3 L N ( 1 2 L N ) ( 1 2 L N ) where the factor N(N – 1) enters because there are that many pairs of electrons that can be randomly picked. The pair density provides information about the exchange and correlation effects, which will be discussed through the hole function later in this section.

It can be proven [7] that, for anti-symmetric wave functions, ρ(r1 ,r1 ) = 0 , which states that the probability of finding two same spin electrons at the same place is zero. This effect corresponds to the Pauli Exclusion Principle and is known as exchange or Fermi correlation. Since the Slater determinant is anti-symmetric, exchange correlation appears in the HF approximation. It is worth noting that the exchange corelation is not related to the HF correlation energy (Eq. (2.39)). As discussed at the end of the last section, the HF correlation energy is mostly due to the error in calculating the electron-electron Coulomb interactions.

Now consider the case of a non-interacting system, where the pair density is related to the electron density by

N −1 ρ()r ,r = n()()r n r . (2.42) 1 2 N 1 2

The (N – 1)/N factor appears because picking any one electron to be placed at r1 will reduce the available number of electrons for the second pick to N – 1. For interacting systems (which is more realistic), the relationship is not that simple. However, a

correlation factor f ()r1;r2 can be defined as 28 ρ(r1 ,r2 ) = [1+ f (r1;r2 )]n(r1 )n(r2 ). (2.43)

Moreover, one may define a quantity called the conditional probability, which specifies the probability of finding any electron at r2 given the information that one electron is at r1. Based on statistics principles, it is

ρ(r1 ,r2 ) Ω()r2 | r1 = . (2.44) n()r1

The difference between this conditional probability and the unconditional probability n(r2) is due to the Pauli exclusion (exchange), Coulomb interactions (correlation) and the self-interaction and defines the exchange-correlation hole function,

hXC ()r1;r2 = Ω(r2 | r1 )− n(r2 ) = f (r1;r2 )n(r2 ), (2.45)

which is negative in the region close to the reference atom (as r2 → r1 ). The hole function is superimposed on top of the electron density to give the conditional probability,

i.e., hXC ()()(r1;r2 + n r2 = Ω r2 | r1 ). This acts like “digging” a hole in the electron density, hence the name.

1 N N 1 Then the electron-electron interaction energy, Eee = Ψ ∑∑ Ψ , can be 2 i=≠1 j i rij expressed by the electron density and the hole function:

1 Ω(r | r ) E = dr dr 2 1 ee 2 ∫∫ 1 2 r 12 , (2.46) 1 n()()r n r 1 n()r h (r ;r ) = dr dr 1 2 + dr dr 1 XC 1 2 ∫∫ 1 2 ∫∫ 1 2 2 r12 2 r12 where the first term is the electrostatic energy of a rigid body of charge distribution n(r) and the second term fixes quantum mechanical effects, as well as the self-interaction.

29 The exchange-correlation hole function can be further divided into Fermi hole and

Coulomb hole:

hXC (r1;r2 ) = hX (r1;r2 )+ hC (r1;r2 ) (2.47)

The former takes care of Pauli exclusion and the latter takes care of the instantaneous

Coulomb interaction leading to greater distances between electrons than the value predicted by the time-averaged electron density n(r).

2.7 The Thomas-Fermi Model

The HF approximation is based on the complex N-electron wave function. However, is it possible to employ a simpler quantity, such as the electron density, to describe the system and predict properties? The first attempts in this regard were by Thomas and

Fermi [19], whose approach was to use a statistical model to find the connection between electron density and momentum of the uniform electron gas, then substitute electron density for momentum in the calculation of the kinetic energy. They obtained this functional [7]:

35 / 3 π 4 / 3 T TF []n()r = n5 / 3 ()r dr . (2.48) 10 ∫

The total energy is obtained by adding nucleus-electron and electron-electron interaction energy to the kinetic energy

35 / 3 π 4 / 3 n(r ) 1 n()()r n r E TF n r = n5 / 3 r dr − Z dr 1 + dr dr 1 2 , (2.49) []() ∫ () ∑ k ∫ 1 ∫∫ 1 2 10 k r1k 2 r12 where the exchange and correlation effects are ignored.

30 2.8 The Hohenberg-Kohn Theorem and Kohn-Sham Approach

The Thomas-Fermi model was the first model in which the electron density was employed as the central quantity. However, the bedrock of modern DFT is work by

Hohenberg and Kohn in 1964 [20], and Kohn and Sham in 1965 [6].

The first paper declares that “the external potential v(r) is (to within a constant) a unique functional of ground state density n(r)” [20]. And because H is determined by the external potential v(r), so is the ground state. Therefore, “the full many particle ground state is a unique functional of n(r)” [20]. The original prove is based upon reductio ad absurdum, which is shown as the following.

Consider there are two external potentials, Vext(r) and V’ext (r), whose difference is more than a trivial additive constant. And assume the corresponding Hamiltonians and ground states are H, Ψ and H’, Ψ’, where H=T+Vee+ Vext and H’=T+Vee+ V’ext. If both external potentials give rise to the same ground state electron density n(r), then T and Vee in H and H’ are the same. The difference between H and H’ is only due to the external potential part. Consider the ground state energy of the first system:

E = Ψ H Ψ < Ψ′ H Ψ′ = Ψ′ H ′ + (Vext −Vext′ ) Ψ′ . (2.50)

Therefore, we find

E < E′ + dr[V r −V ′ r ]n r . (2.51) ∫ ext ( ) ext ( ) ( )

If we start with the ground state energy of the second system and follow the same approach, we get

E′ < E + dr[V ′ r −V r ]n r . (2.52) ∫ ext ( ) ext ( ) ( )

Adding the two inequalities shown above results in

31 E + E′ < E′ + E . (2.53)

This is a clear contradiction. Thus it can be concluded that the external potential Vext is a unique functional of the electron density, and so is the ground state wave function.

Therefore, the kinetic and electron-electron interaction energy (determined by the wave function) are also functionals of the electron density. Then the Hohenberg-Kohn functional is defined as

HK F []n()r ≡ Ψ T +Vee Ψ = T[n]+ Eee [n], (2.54) which is a universal functional, valid for any N-electron system and independent of the number of electrons or external potential.

Once the external potential is known, the energy can be calculated by

E n r = drV r n r + F HK n . (2.55) []( ) ∫ ext ( ) ( ) [ ]

Also in Hohenberg and Kohn’s 1964 paper [20], it was proven that the ground state density gives rise to the minimum value of F HK [n]. This is also proved by Levy with the constrained-search approach [21].

If the form of F HK []n was known, the problem of finding the ground state energy and density would be straightforward, since it could be approached by the minimization of

F HK []n with respect to n under the constraint of ∫ n(r)dr = N . Unfortunately, the form of

F HK []n is unknown.

In order to find an approximate form for F HK [n], Kohn and Sham purposed [6] the non-interacting reference system, in which electrons are only subject to an effective potential VS and the effective Hamiltonian becomes

32 N N 1 2 H S = − ∑∇i + ∑VS . (2.56) 2 i i

Assuming that this non-interacting reference system gave rise to the same electron density as the real system, this system could be used for the study of the real one.

The first advantage of a non-interacting reference system is that its wave function is represented exactly by a Slater determinant. To be distinguishable from its HF counterpart, we write the wave function as

r r r φ1 (r1 ) φ2 (r1 ) L φN (r1 ) r r r r 1 φ1 ()r2 φ2 ()r2 φN ()r2 Θ s (){}ri = , (2.57) N! L L L r r r φ1 ()rN φ2 ()rN L φN ()rN

where the spin orbitals φi , also called Kohn-Sham (KS) orbitals in this context, satisfy the equation

KS f φi = ε iφi , (2.58) where f KS is called the one-electron Kohn-Sham operator and is defined as

∇ 2 f KS = − +V . (2.59) 2 S

The second advantage of the non-interacting reference system is that the total kinetic energy can be calculated exactly by the summation over the KS orbitals,

N 1 2 TS = − ∑ φi ∇ φi . (2.60) 2 i

Of course, it is not equal to the kinetic energy of the real system. Nonetheless, it may represent a large part of the real kinetic energy, since the two systems have the same electron density. So the kinetic energy of the real system can be written as

33 T = TS + TC , (2.61) where TC is the remaining part of the kinetic energy that is not described by TS.

Besides the kinetic energy, F HK [n] (Eq. (2.54)) has the electron-electron interaction.

It has been shown in Eq. (2.46) that

1 n(r )n(r ) E = dr dr 1 2 + E , (2.62) ee ∫∫ 1 2 ncl 2 r12

where Encl is the quantum mechanical correction to the electrostatic energy of a classical charged body, i.e., the correction to the first term on the right hand side.

At this point, it is clear that ground state energy of the real system can be broken into several parts:

E n r = V r n r dr + F HK n []() ∫ ext ()( ) [ ] = V r n r dr + T n + E n ∫ ext ()() []ee [] 1 n()()r n r = V r n r dr + T + T + dr dr 1 2 + E (2.63) ∫ ext ()() S C ∫∫ 1 2 ncl 2 r12 1 n()()r n r = V r n r dr + T + dr dr 1 2 + E ∫ ext ()() S ∫∫ 1 2 XC 2 r12

where EXC is the sum of TC and Encl , i.e., all unknown parts of the energy.

With the help of the reference system, the largest part of the ground state energy (the first three terms on the right hand side of Eq. (2.63)) are calculated accurately, with all

the unknowns being lumped into EXC .

But how can one find a corresponding non-interacting reference system for a real system in the first place? If one applies the variational approach, similar to the one shown in Eqs. (2.13) - (2.16), it can be derived that for KS orbitals [9]

34 ⎧ ∇ 2 ⎡ n(r ) M Z ⎤⎫ − + dr 2 − k +V φ = ε φ , (2.64) ⎨ ⎢∫ 2 ∑ XC ⎥⎬ i i i ⎩ 2 ⎣ ri2 k rik ⎦⎭

where VXC is the exchange-correlation functional and is defined as

δE V ≡ XC . (2.65) XC δn

Comparing Eq. (2.64) to Eqs. (2.58) and (2.59), it is easy to see the effective potential

VS in Eq. (2.59) is the term inside the square bracket on the left hand side of Eq. (2.64):

n(r ) M Z V ≡ dr 2 − k +V . (2.66) S ∫ 2 ∑ XC ri2 k rik

If the form of the exchange-correlation functional VXC is known, one can solve Eq.

(2.64) and derive the ground state properties. A significant difference between HF approximation and the KS approach is that HF introduces the approximation in the very beginning, claiming a Slater determinant as the total wave function, while the KS approach sets up the hypothetical non-interacting system and employs the electron

density as the central quantity, lumping all the unknown factors into VXC . If the exact

form of VXC was known, the solution from the KS approach would be exact.

2.9 Approximate Exchange-Correlation Functionals

Being the only unknown part in the KS formulation, VXC or E XC is crucial to the

accuracy of DFT. The most popular approximations to E XC are local density approximation (LDA), local spin-density approximation (LSD) and generalized gradient approximation (GGA).

35 There is no systemic way of finding or improving approximate functionals. The usual way is to study a model system, derive the functionals from it, and hope that the system of interest, upon which we apply the functionals to do calculations, has a similar behavior.

LDA was historically the first approximation to the exchange-correlation functional.

It is based upon the uniform electron gas model [6] because this model is the simplest one.

The exchange and correlation energy in this case can be calculated to very high accuracy.

E XC in LDA can be written as [7]

E LDA n = n r e n r dr , (2.67) XC [ ] ∫ ( ) XC ( ( ))

where eXC is the exchange-correlation energy per electron. It only depends on the local density of electrons (assuming that eXC(n(r)) is the same as that of a homogeneous electron gas with the same density), hence the name. It is usually split into the exchange part and correlation part,

eXC ()n()r = eX (n(r))+ eC (n(r)). (2.67)

The analytical form of the exchange part is found to be [6, 8]

4 / 3 3 1 ⎛ 3 ⎞ 1/ 3 eX ()n = − k F = − ⎜ ⎟ n . (2.68) 4π π ⎝ 4 ⎠

An analytical form of the correlation part has not been found, but an approximation by Perdew and Wang [8, 22] is

⎡ 1 ⎤ eC ()n = −2c0 (1+ α1rS )ln⎢1+ 1/ 2 3 / 2 2 ⎥ , (2.69) ⎣ 2c0 ()β1rS + β 2 rS + β 3rS + β 4 rS ⎦

3 where rS ≡ 3/ 4πn is the Seitz radius and

36 1 ⎛ c ⎞ ⎜ 1 ⎟ β1 = exp⎜− ⎟ , (2.70) 2c0 ⎝ 2c0 ⎠

2 β 2 = 2c0 β1 (2.71)

with coefficients c0 = 0. 031091, c1 = 0.046644 , α1 = 0.21370 , β 3 = 1.6382 and

β 4 = 0.49294 [8].

LSD, another approximation which is widely used in density functional theory,

↑ ↓ expresses eXC as a function of densities of different spin, n (r) and n ()r ,

E LSD n = n r e n↑ r + n↓ r dr . (2.72) XC [] ∫ ( ) XC ( ( ) ( ))

Since LSD distinguishes different spins, it works better than LDA in general and has been widely used [23]. However, it may not be accurate enough in some studies where the energy needs be determined to high precision. GGA is an extension of LDA and LSD

It takes into account the gradients of the charge density as well. For the example of a spin polarized calculation,

E GGA n = f n↑ r ,n↓ r ,∇n↑ r ,∇n↓ r dr . (2.73) XC [] ∫ ( ( ) ( ) ( ) ( ))

There are many contributions to the development of approximate forms of exchange- correlation functionals [22, 24-30], where the details and formulation can be found. From the point of view of applications, LDA generally predicts reasonably well structural properties such as equilibrium structures (although bond lengths and lattice parameters are often slightly smaller than experiment), vibrational frequencies and elastic constants, but sometimes is said to introduce larger errors for the binding energy. GGA on the other hand, while often improving energy calculations and structural parameters (which are often slightly larger than experiment), may worsen the results for other properties, such 37 as elastic constants for a number of semiconductors. A comparison of the lattice constant, bulk modulus and cohesive energy of silicon predicted by LDA [22] and GGA (PBE96

[31]) are listed in Table 2.2 [8].

LDA GGA experimental a (Å) 5.378 5.463 5.429 B (Mbar) 0.965 0.882 0.978 Ec (eV/atom) 6.00 5.42 4.63

Table 2.2: Comparison of the lattice constant (a), bulk modulus (B) and cohesive energy (Ec) of silicon obtained by LDA and GGA. The bulk modulus is obtained by fitting energy vs. volume curve with Murnaghan equation. Results are from Refs. [8, 32].

A similar comparison of properties predicted by LDA [22] and GGA [31] for copper is listed in Table 2.3 [8].

LDA GGA experimental a (Å) 3.571 3.682 3.61 B (Mbar) 0.902 0.672 1.420 Ec (eV/atom) 4.54 3.58 3.50

Table 2.3: Comparison of the lattice constant (a), bulk modulus (B) and cohesive energy (Ec) of copper obtained by LDA and GGA. The bulk modulus is obtained by fitting energy vs. volume curve with the Murnaghan equation. Results are from Refs. [8, 33].

2.10 Pseudo Potentials

The N-electron problem can be further simplified by dividing electrons into core electrons and valence electrons. Usually core electrons are strongly bound to the nucleus

38 and are hardly affected by the environment of the atom. Therefore, one may combine the nucleus and core electrons together and treat them as one ionic core, while valence electrons experience the potential from this ionic core. This is the pseudo-potential concept, first proposed by Fermi in 1934 [34]. Based on this idea, Hellmann [35] suggested a pseudo-potential for a potassium atom in the form

1 2.74 w()r = − + exp()−1.16r . (2.74) r r

The asymptotic behavior for this potential for r → ∞ is −1/ r , which is the Coulomb interaction of one electron and one positive charge of |e|. The potassium nucleus (Z=19) is mostly shielded by 18 core electrons, thus the ionic core demonstrates itself as a point charge with charge |e|.

Phillips and Kleinman revisited this idea in the 1950’s and provided a more rigorous description [36, 37]. Denoting the wave functions of core electrons and valence electrons

with ψ C and ψ V respectively, ψ V can be written as the sum of a smooth function

ϕV and an oscillating function due to the interaction with the core electrons [8],

ψ V = ϕV + ∑α CV ψ C , (2.75) C where

α CV = − ψ C ϕ V . (2.76)

Substituting Eq. (2.75) into the Schrödinger equation, we obtain

H ϕV = EV ϕV + ∑(EC − EV )ψ C ψ C ϕV . (2.77) C

Then the effective potential experienced by the valence electrons may be defined as

39 PK w ()E = v − ∑(EC − E)ψ C ψ C , (2.78) C where v is the real potential.

Based on this formulation, many modern pseudo-potentials have been proposed [38-

42] and employed in DFT studies.

40

CHAPTER 3

THE STRUCTURE AND STABILITY OF CARBYNES

3.1 Introduction to Carbynes

Many years ago, linear carbon chains with sp hybridization have been proposed as the elemental building blocks of a carbon allotrope called carbine [43]. Such linear carbon chains alone or as parts of molecules are well known in carbon-rich vapor, which was widely studied following the discovery of fullerenes [44, 45]. For solids, the discussion of carbyne started with the discovery of a “new form of carbon” called “white carbon” or

“chaoite”, which is a shock-fused graphite gneiss from the Ries crater in Bavaria [46]. In

Ref. [46], an x-ray powder pattern of the graphite-chaoite structure was best fitted with a

o o hexagonal cell with dimensions a = 8.948 ± 0.09 A and a = 14.078 ± 0.017 A , while no further suggestions were made about the possible crystal structure. Later characterization work suggested that this new phase contained linear carbon [47], as evidenced by, e.g., the appearance of high frequencies > 2000 cm-1 in Raman spectra [48]. A review of the early work suggested a new carbon phase diagram with carbyne as the stable carbon

41 phase for temperatures between ~2600 K and graphite's melting point (~4000 K) and pressures up to ~6 GPa [49].

Linear carbon chains with sp hybridization can contain alternating single and triple bonds (polyyne) or only double bonds (polycumulene) [50], see Fig. 3.1. Past theoretical calculations suggest that polycumulenes are less stable than polyynes [51]. Both species are characterized by an extremely high reactivity against oxygen and a strong tendency to interchain cross linking [43], thus rendering the direct observation of a carbon solid with linear carbon bonds a major challenge.

Figure 3.1: Structure and electron density of 5-atom carbyne chains computed within density-functional theory, (a) forced to form alternating single and triple bonds (polyyne) and (b) allowed to relax to double bonds (cumulene).

While the existence of carbynoids (carbyne stabilized by the addition of other elements besides carbon, see e.g. Ref. [45]) is by now little disputed, the experimental

42 evidence for pure carbyne, mainly based on crystallographic data, has been the object of strong criticism up to a complete rejection of the carbyne concept [43].

This includes work by Smith and Buseck, who argue that the new reflection patterns that supposedly identify the mineral chaoite [52] as a carbyne are due to sheet silicate impurities [53, 54].

Also, Blank et al. [55] claimed that an intermediate structure between graphite and diamond (i.e., graphite strongly compressed along the c-axis) would fit the chaoite x-ray data equally well. Rietmeijer re-interpreted the previous carbyne electron diffraction data and suggested that carbynes that exist in meteorites could be crystalline C-(H-O-N) carbons which are the metastable relics of kinetically-inhibited incomplete pyrolisis rather than carbon allotropes [56, 57]. In that case, the intensities > 2000 cm-1 in the

Raman spectrum could also come from cyanide groups, C ≡ N - , which produce intensities in the same frequency range as sp-bonded C [58].

Synthesis of carbyne has also been reported, for example by high temperature and high-pressure treatment of carbon-based materials. Such an approach seems reasonable since carbyne resembles the known vapor phase which consists of chains of various lengths [59]. However, it seems that some of the early synthesis of solid carbyne (e.g., from graphite by sublimation at 2700~3000 K or by laser irradiation in high vacuum [48]) could not be generally reproduced [60].

A recent success of producing at least carbyne-rich pure carbon material comes from depositing pure carbon clusters from a supersonic beam in ultra-high vacuum [61]. This produces nanostructured carbon films with a strong polyyne and cumulene signal in the

Raman spectrum at frequencies ≥ 2000 cm-1 , which, however, lose their carbyne 43 contribution once taken out of the vacuum. The Raman frequencies of sp bonded carbon

( > 2000 cm-1 ) are considerably higher than those of sp2 bonded carbon (graphite) at

~ 1500 cm-1 and diamond at ~ 1300 cm-1 (see Sec. 3.3.1 and Fig. 3.14 below) and are well known from studying carbynic molecules in the gas phase. A possible drawback for using Raman spectroscopy for identification is that the carbynic frequencies may overlap with the Raman active frequencies of cyanides, C ≡ N - [58].

The authors of Ref. [61] explain the synthesis of the carbyne-rich film with their deposition conditions, where clusters from the gas phase are deposited with an average kinetic energy of 0.3 eV/atom. They state that at such a comparably low energy, the initial clusters were not broken up and maintained their original structure (in case carbon atoms in reactive binding configurations are far enough away from each other). Since it was well known that carbon clusters in carbon vapor contained a considerable fraction of linear carbon chains [44, 59], this explained the linear-carbon signal in their Raman spectrum. Once the system was disturbed (vacuum broken, temperature increased), the carbyne signal quickly disappeared.

Whereas Ravagnan et al. [61] had called their average kinetic-energy value of 0.3 eV too low to cause reactions of the initial clusters, Lamperti and Ossi [62] interpreted this value as the energy required to produce carbyne in a carbon film (equivalent to its heat of formation). This is concluded from an estimate that finds that C clusters with 400-600 atoms with 0.3 eV/atom kinetic energy each ((120 - 180 eV for the cluster) have about the same kinetic energy as an atom in an impacting meteorite (~128 eV), which had been reported to produce the chaoite in the Ries crater. Thus, they call the 0.3 eV/atom the heat of formation of carbyne, although they leave it somewhat unclear why the kinetic energy 44 of a 400-600 atom cluster should be related to the kinetic energy per atom in a meteorite with approximate composition 25% Si, 25% Mg, 10% Fe, and 40% volatile components

(H-2O, COx, SOy).

Carbyne-rich (or carbynoid) solids are reported to have been chemically synthesized, where the polyynes are preserved against cross-linking and chemical decomposition by the presence of metal-based species and molecular groups terminating and separating the chains [45, 63-66]. Some examples will be discussed in Sec. 3.4. On the other hand, the existence of pure carbyne is still a question. Therefore, in the present paper, we will study the thermodynamic stability of pure-carbon carbynes under equilibrium conditions from calculating their free energies at different temperatures and pressures. Since to date there is no unambiguous description of the carbyne crystal structure in the literature (full lattice and basis information), the approach taken in this study was to first construct several candidate structures from the manifold of structural features suggested in the literature, and then examine their stability at zero temperature and finite temperatures elaborated in following sections.

To our best knowledge of the literature, there seem to be three confirmed routes to produce carbynes (or rather carbynoids):

• Low pressure to produce a gas-phase like material with a distance of at least

~3Å and thus minimal interaction between the carbyne chains [see results in

Sec. 3.2].

• Extreme non-equilibrium deposition under vacuum conditions [61], which

results in material unstable at ambient conditions.

45 • Addition of molecular groups [45] and metal ions (e.g. Ag, Fe, Li, Pt [64])

that stabilize the carbyne chains against cross-linking and chemical

decomposition [see Secs. 3.3 and 3.4].

3.2 Structure and Stability of Carbynes

3.2.1 Crystal Structure of Carbyne in Previous Work

A number of ad-hoc models were initially proposed for the crystal structure of carbyne [66]. The first model by Kasatochkin in 1967 assumed a simple parallel, hexagonal arrangement of the carbyne chains as shown in top view in Fig. 3.2(a) with a distance of 2.97 Å between the chains. In 1969, Sladkov and Kudryavtsev revised the model to the structure displayed in Fig. 3.2(b) to match the diffraction data, where, assuming infinite chains, new interchain distances of 1.71 Å appear. Finally, a carbyne chain arrangement as shown in Fig. 3.2 (c) was suggested [66], which, as we will show, can give rise to a sensible carbyne structure when combined with Heimann's kink model

[50] (see below). If all points in this projection would represent infinite carbyne chains in top view, the corresponding c lattice constant should be defined by the period of the

o o carbyne chains, either r1 ()C = C = 1.3 A , or r2 ()C − C ≡ C = 2.6 A . However, measured values for c are 7-15 Å for at least seven modifications and are not whole-numbered

multiples of r1 or r2 .

A possible explanation for this discrepancy has been given by Heimann, Kleiman and

Salansky [50]. They report seven carbyne forms with varying lattice constants, which are

α-carbyne (polyyne type) and β-carbyne (cumulene type); carbon-VI, -VIII, -IX, and -

XIV; and chaoite (a shock-fused natural carbon mineral from the Ries crater). A

46 structural model is suggested based on the postulates that carbynes are formed by a crystallographic arrangement of C − C chains with either conjugated triple or cumulated double bonds; and c-axes lengths are functions of the number of carbon atoms, n, in the chain. Since the measured lattice constants of the acknowledged carbyne modifications

o o are not multiples of r1 ()C = C = 1.282 A , or r2 ()C − C ≡ C = 2.586 A , kinks in the carbon chains as shown in Fig. 3.3 were additionally postulated. Two reasons were given for the addition of the kinks. First, they provide a natural means to arrive at modifications of carbyne with different c-axis lengths by variation of the interkink distance c0. Second,

“kinked chains provide free bonds by splitting double [… and …] triple bonds. […]

These free bonds […] prevent lattice collapse by keeping sufficient distances between the chains.” [50] In the following, we will investigate this postulate with first-principles calculations. One can see from Fig. 3.3 that one needs two kinks in a unit cell to achieve a periodic crystal structure, thus c = 2c0. However, due to the fact that the kinks constitute a 2-fold screw axis, Heimann suggests c0 to be equal to the measured lattice constant.

Figure 3.2: Structural models suggested for carbyne in (0001) projection by (a) Kasatochkin, (b) Sladkov and Kudryavtsev, and (c) Kudryavtsev et al. [50].

47

Figure 3.3: Kink-model for carbyne according to Heimann et al. [50] for (a) polyyne and (b) cumulene.

According to the theory in Ref. [50], the kink spacings define the c0-axis length as

n ⎛ n ⎞ c0 (polyyne) = r()C ≡ C + ⎜ −1+ cosα p ⎟r()C − C 2 ⎝ 2 ⎠ and

c0 (cumulene) = ()()n −1r C = C + cosα c r(C − C)

with on average α p = 60° and α c = 30° . Table 3.1 summarizes the crystallographic data from Ref. [50], where the angles α and number of chain atoms n have been used as fitting parameters. We have also added calculated values for the interchain distance d (which will be relevant for the structures we propose below) and number of atoms, both from the measured density and from the theoretical number of atoms, which in general agree well.

The only exceptions are chaoite and carbon-VI (both identified in the same sample [67]),

48 which however are natural minerals with additional components, possibly clay particles as suggested in Ref. [53]. Heimann also observes that both c0 and a0 increase linearly with the number n of C atoms in the chains, but with different line fits describing polyyne and cumulene. The volume, V, and a0/c0 ratio, on the other hand, were both found to depend linearly on n with one line fit describing both polyynes and cumulenes [50].

49

expt expt Hei expt n a0 (Å) c0 (Å) c0 (Å) α (deg.) d (Å) β−carbyne 6 8.24 7.68 7.60 22.93 2.38 carbon XIV 8 8.7 9.56 9.65 64.44 2.51 carbon VI 10 5.33 12.24 12.24 60.00 2.66 carbon IX 10 9.44 12.5 12.73 45.76 2.73 chaoite 11 8.95 14.08 14.01 23.97 2.58 carbon VIII 12 9.1 14.82 18.83 60.21 2.63 α−carbyne 12 8.94 15.36 15.30 24.18 2.58 Bond type Space group ρ (g/cm3) Z(ρ) Z(n) β−carbyne C P31,2 3.13 71 72 carbon XIV P - - - 96 carbon VI P P3 or P31,2 2.9 44 40 carbon IX C(P?) R3? - - 120 chaoite C P3 or P31,2 3.43 168 132 carbon VIII P - - - 144 α−carbyne C P31,2 2.68 143 144

Table 3.1: Crystallographic data of carbyne forms from Ref. [50]. The a0 lattice constant is for a full cell such as shown in Fig. 3.2(a), resulting in an interchain distance d (dashed lines in Fig. 3.2(a)), with the exception of carbon VI, which has a lattice constant a0 smaller by 3 (see Fig. 3.11(b)). The “Heimann” lattice constants (Hei) are calculated within the “kink” model described in the text with n carbon atoms in the chains between two kinks. Bond types are (C) cumulene and (P) polyyne. We have calculated the number of atoms Z in the full unit cell both from the experimental mass density ρ and unit cell volume where available and from multiplying n by 12 (4 in the case of carbon VI), since the structure in Fig. 3.2(a) contains 12 chains per cell (4 in the case of carbon VI).

50 A problem arises from the bonding suggested in Figs. 3.3 (a) and (b). According to the crystallographic data in Table 3.1, the kink atoms will have the same distance to neighboring atoms in their own chain as to atoms in neighboring chains. In other words, the kink atoms can be expected to form bonds with atoms in neighboring chains. This makes the bonding suggested in Fig. 3.3(a) impossible (since the kink C atoms have already four bonds), but suggests that the kink atoms in Fig. 3.3(b) should be connected with a single bond to a neighboring chain as we will suggest later (Sec. 3.2.4 and

Figs. 3.9(b) and (d)). In that case, the bonding of the kink atoms with now three neighbors will be graphite-like, with three bond angles of 120° , resulting in a cumulene

structure and an angle α c of approximately 60° . However, the resulting structure would have no threefold symmetry along the chain as suggested in Table 3.1 but in the best case rather be orthorhombic. Such a structure has also been suggested within Whittaker's graphite bond splitting model [49] (Fig. 3.4), where breaking of bonds in the 'meta'- position leads to carbyne chains along the [100] direction (Fig. 3.4, lower panel). This structure can also be thought of as graphite with additional carbon atoms added in the bonds along the [100] direction. We will study this type of structure with varying chain lengths in the following.

In order to construct a structure with the experimentally found threefold or sixfold symmetry within the kink model, a kink atom would have to connect to two other, not one other chain, as suggested in Fig. 3.9(a) and (c). Thus, the structure would be similar to hexagonal diamond (lonsdaleite) with additional carbon atoms added in the bonds parallel to the c-axis. In such a structure, the chain bonds next to the kink should be single bonds, forcing a polyyne structure on the chains, but with different bonding than 51 suggested in Fig. 3.3(a). The kink atoms would have four neighbors, similar to the bonding in diamond or lonsdaleite (the resemblance of carbynes to the lonsdaleite structure has already been suggested in Ref. [67]). This would result in kink bond angles

of approximately 109° and suggest an angle α c of approximately 71° . However, these simple structures would have a much smaller unit cell than experimentally found with

a0 = d (Fig. 3.2(a)).

Figure 3.4: Graphite bond splitting model for the formation of carbyne proposed by Whittaker [49]. Transition from perfect graphite layer (1) through transition zone (2) with breaking bonds (red dashed lines) to carbyne (3) for two different crystallographic directions.

Finally, Blank et al. suggested a carbyne structure intermediate between graphite and

o diamond [55]. They measured a triclinic crystal lattice with parameters a = b = 5.1 A ,

o c = 7.45 A , α = β = 82° , and γ = 120° with a density in the range of carbyne densities.

52 o The value of a = 5.1 A was found to be close to twice the value for graphite and to twice the value of d(110) in diamond. They suggested that atoms situated in the nodes of (001)- graphite planes with doubled a-parameter take part in the interlayer connection by sp3 bonds as shown in Fig. 3.5. Then, four interconnected layers form the unit cell of the new phase, which contains 24 atoms.

Figure 3.5: Blank's carbyne model of a phase intermediate between graphite and diamond [55].

Since in the previous literature, no quantitative structural model existed for solid carbyne except for Blank's model [55] (which, however, we find to be unstable), part of the challenge within the present work was to identify and test different structural models.

In the following, we will use density-functional calculations to relax the atomic positions and the lattice parameters of the different suggested structural models. This will examine their stability and determine their lattice parameters in case they are stable. Also, from the relaxed bond lengths, we will be able to identify the nature of the bonds (single, double, triple). The structural models we consider, which encompass the major varieties

53 suggested in the literature, are (a) Kasatochkin type (Sec. 3.2.3); (b) kinked carbyne of the meta bond-splitting type (graphite-derived) with sp2 kink atoms and of the lonsdaleite-derived type with sp3 kink atoms (Sec. 3.2.4); (c) Blank's intermediate graphite-diamond model (Sec. 3.2.5); and (d) kinked Kudryavtsev type and a unified carbyne model derived from it (Sec. 3.2.6).

3.2.2 Computational Method

Density functional theory (DFT) is employed to determine the structure and stability of carbynes. Two possible approximations for the exchange-correlation part of the energy are generally in use, the so-called local density approximation (LDA) and the generalized gradient approximation (GGA). For covalent materials, typical errors for structure predictions are on the order of -1% for LDA, whereas GGA values are most of the time very similar to or slightly larger than experiment. For the example of diamond, we find an LDA (GGA in brackets) value of 3.527 Å (3.568 Å) compared with the experimental value of 3.567 Å, corresponding to a density of 3.63 g/cm3 (experiment: 3.52 g/cm3 [68]).

While GGA often finds lattice constants closer to experiments, this is not the case for graphite, where it is well known that the approximation fails to describe the weak interaction between the graphene layers (see, e.g., Ref. [69]). We calculate graphite

o o o o lattice constants for LDA (GGA) of a = 2.44 A ( a = 2.46 A ) and c = 6.61 A ( c = 8.4 A )

o o (density 2.34 g/cm3), compared to experimental values of a = 2.46 A and c = 6.71 A

(density: 2.09-2.23 g/cm3 [68]). Thus, the GGA c-constant is 25% larger than experiment.

Therefore, LDA is the better choice for calculating lattice constants of carbon structures.

Also, it has been shown that LDA describes the vibrational properties of diamond [70, 71]

54 and graphite [72] excellently, which are important for free-energy calculations as we will show in Sec. 3.3. Thus, we will use in the following LDA. This combination has also been found by others to be very successful to describe the structure, elastic constants, short-range interactions, vibrational frequencies, and energetics for different carbon structures [69, 73-76].

For the following section, we will use DFT-LDA implemented into VASP [77-80] within a constant-pressure algorithm that optimizes atomic positions and lattice vectors simultaneously to find for the different structural modifications the relaxed ground state structure (i.e., the structure where all atoms are force free and the cell is pressure free).

Examining a few test cases, we found negligible differences in the results using ultrasoft [81] or projector-augmented wave (PAW) [82, 83] pseudopotentials respectively.

Since ultrasoft pseudopotentials require for constant-pressure relaxations a kinetic-energy cutoff of only 358 eV vs. 500 eV for PAW potentials, we use ultrasoft potentials to reduce the computational costs. For Brillouin zone integration, we choose Monkhorst-

Pack special k-point meshes [84] such that the product of each lattice constant and the corresponding number of k-points is as close to 20 Å as possible. This corresponds to a

6× 6× 6 mesh for a conventional diamond cell. In order to examine the stability of the unkinked structures, we also use constant-volume, constant-temperature molecular dynamics simulations within VASP with a low energy cutoff of 215 eV and minimal k- point meshes.

3.2.3 Kasatochkin-Type Carbyne

In a first step, we examined Kasatochkin's structure [66] which is just a hexagonal bundle of straight carbon chains as shown for the case of a 48-atom simulation cell in 55 Fig. 3.6. As we will show, the central question that we need to address in this section will be the minimum distance required between carbyne chains (perpendicular to their axis) to prevent them from reacting with each other and forming graphite or diamond.

Figure 3.6: Simulation cell of the simple hexagonal structure of carbyne after Kasatochkin [66].

First principles molecular dynamics simulations for temperatures between 2000 and

4000 K for ~10 ps were performed for this carbyne structure in comparison to graphite, using periodic simulation cells with 24, 36 or 48 atoms. The results presented in the following are from 48-atom cells.

For moderate pressures, we never observed graphite to transform to carbyne during the simulation time. On the other hand, carbyne very often transformed to graphite-like structures. Three sets of lattice constants were applied to the simulation cell shown in

Fig. 3.6 to simulate different interchain distances.

56 For an interchain distance of ~3 Å, the structures resulting from MD runs at different temperatures are shown in Fig. 3.7. It is found that

1. at 2000 K, the carbyne structure is stable;

2. at 3000 K, carbyne transforms into a graphitic structure. Graphitization has

also been observed experimentally under annealing at high temperature [85-

87]. There, XPS was used to probe the initial chain-like structure.

3. at 4000 K, chains begin to form irregular structures (possibly melt).

Figure 3.7: Typical final structures after MD simulations for 10 ps at different temperatures as indicated. In order to show the graphitic structure from the 3000 K simulation more clearly, the red circle area is replotted separately for a 2× 2 supercell.

Kasatochkin-type carbyne is found to be unstable under compressive pressure.

Figure 3.8 shows the resulting structure from a relaxation at 5 GPa pressure. We find that

57 carbyne transform into a perfect diamond structure with simulation cell edge directions along the [110], [110] and [001] directions of the cubic cell.

In summary, we find that during MD runs, the stability of carbyne depends sensitively on the distance between the different chains. When the interchain dimension is increased to more than 3 Å (which can only be achieved for the examined solid composed of straight chains without kinks by applying an external tensile stress of

~40 GPa), we do not observe any structural transformation. This confirms the stability of unkinked linear carbon only for the case of negligible interaction as mentioned previously for the case of vapor [61].

Figure 3.8: Final structure (perfect diamond structure) from relaxation of the Kasatochkin-type carbyne from Fig. 3.6 under 50 kB pressure.

58 3.2.4 Kinked Carbyne - Graphite and Lonsdaleite Derived

Modifications

Following the discussion above of Heimann's kinked-carbyne model [50], we first propose two simple forms of kinked carbyne, one with sp2, the other with sp3 bonded kink atoms. They can be thought of as graphite and hexagonal diamond (lonsdaleite), both with additional carbon atoms added in parallel bonds. For the graphite-derived carbyne, the additional carbon atoms are added in bonds parallel to one of the a-axes. The structure of lonsdaleite is similar to the hexagonal unit cell of cubic diamond (c-axis along the body diagonal of the cubic cell), but with only two stacking layers, (AB-AB-

AB...) instead of three (ABC-ABC-ABC...). Figure 3.9 shows as examples the unit cells of the corresponding structures for carbon chain lengths of n = 6.

As will be shown in following sections, these structures are stable for a variety of chain lengths under most pressure and temperature conditions and have lattice constants in good agreement with the experimentally suggested values. However, the experimental cells have an a-lattice constant several times as long as our hexagonal sp3-carbyne cell

(Fig. 3.9(c) in fact shows a unit cell with doubled a-lattice vectors). We will give a possible explanation for this in Sec. 3.2.6. The sp2-carbyne structure (Fig. 3.9(d)) has orthorhombic symmetry (FMMM) and thus is not compatible with the experimentally suggested point groups with threefold symmetry. Nevertheless, it has been suggested before by Whittaker within his bond-splitting model (see Sec. 3.2.1) and is the simplest structure that would allow a cumulene carbyne structure (i.e., chains that have double bonds between the carbon atoms).

59 In order to allow a comparison to the experimental lattice vectors, we will show in

Sec. 3.2.6 that it is reasonable to multiply the theoretical a-lattice constants by 3 (or 3 in the case of the short lattice constant for carbon VI). We also divide the theoretical c lattice constant (for our cells with two n-atom chains) by two to follow Heimann's 2-fold screw axis argument (Sec. 3.2.1) [50].

Figure 3.9: [110] projection of the simulation cell for n=6 carbyne with sp3 bonded kink atoms (a) and sp2 bonded kink atoms (b). Perspective view of the same crystal structures for n=6 sp3 (c) and n=6 sp2 (d) carbyne. The blue (darker) balls in (a) and (c) denote the kink atoms for easier identification.

For the lonsdaleite derived structure, the minimum chain length has to be four carbon atoms to allow single bonds at the kink atoms and alternating single and triple bonds

60 along the chains. Thus, this is the minimum chain length we consider. For the same reason, we also only consider chains with even numbers of carbon atoms. For the orthorhombic sp2 cells, we report the shortest lattice constant perpendicular (which would be the a lattice constant in graphite) and the c lattice constant parallel to the chains.

Table 3.2 shows compiled results of our calculations in comparison to experimental values.

Expt. (Å) sp3 LDA(Å) sp2 LDA(Å) Type 3 2 n a0 c0 a0=3a c0=c/2 a0=3a c0=c/2 sp sp unknown 4 N/A N/A 8.19 4.60 7.86 4.68 P P β−carbyne 6 8.24 7.68 8.25 7.21 7.95 7.23 P P/C carbon XIV 8 8.70 9.56 → sp2 7.92 9.87 - C carbon VI 10 5.33 12.24 → sp2 8.07 12.42 - C carbon IX 10 9.44 12.50 → sp2 8.07 12.42 - C chaoite 11 8.95 14.08 → sp2 8.22 13.67 - C carbon VIII 12 9.10 14.82 8.67 14.92 8.22 14.97 C C α−carbyne 12 8.94 15.36 8.67 14.92 8.22 14.97 C C

Table 3.2: Crystallographic data of optimized kinked-carbyne structures with sp2 (graphite-derived) and sp3 (diamond-derived) bonded kink atoms. Where available, we compare the lattice constants, calculated within LDA to experimental data from Refs. [50] and [67], processed as described in the text. n is the number of C atoms in a chain between two kinks. Several sp3 modifications are unstable and directly relaxed to the sp2 modifications. Type is the bond type (P = polyyne, C = cumulene), determined from the bond lengths.

Assuming the validity of our assumptions, the agreement between the predicted structures and the experimental values is good, with an average deviation of − 4% . While many c-values are in good agreement, the a-lattice constants for the stable sp3-carbyne

61 modifications are significantly closer to experiment than that of the sp2 modifications.

The sp2 lattice constant is around 10% smaller than experiment.

As mentioned above, LDA predicts the lattice constant of diamond 1.1% smaller than experiment, and the a and c lattice constants for graphite 0.8% and 1.5% smaller than experiment, respectively, which are also typical accuracies for other covalent systems. Also, the C-C bond lengths for the experimentally well characterized (tert- butyl)C8(tert-butyl) carbynoid (see Sec. 3.4 and Figs. 3.18 and3.19) can be predicted with an average deviation of 1.7% . Thus, deviations of 4% average and 10% for a lattice vectors are beyond the expected computational accuracy and suggest that the modeled structures are different from the experimental ones. In nearly all cases the predicted structures are too small.

3.2.5 Graphite-Diamond Intermediate Model

The crystal structure proposed by Blank et al. [55] represents an interesting alternative to the carbyne concept. The paper suggests that an intermediate phase between graphite and diamond (referred to in the following as Blank's model), which is shown in

Fig. 3.5, describes the diffraction pattern of C60- and C70-samples after thermobaric treatment at a temperature of 500 K and a pressure of 9.5 GPa well. The diffraction pattern on the other hand was found to be in accordance with the peaks of chaoite from

Ref. [46]. Since Blank's model can be considered as an interesting link between the traditional carbon structures, diamond and graphite, and carbynes, and furthermore suggests a possible reinterpretation of characterization data of supposedly carbynes, we have included this structure in our investigation.

62 Reference [55] contains besides the crystallographic data also detailed sketches of the atomic arrangement in the cell which can be easily translated into an atomic model

(Fig. 3.5) suitable for ab-initio studies. After prerelaxation with a Tersoff potential with fixed neighbor list (which preserves the structural features), the structure has been relaxed at different ambient pressures up to a value of 80 GPa. However, the intermediate structure was not found to be stable under relaxation for any of the pressures we tried. Up to a pressure of 66.6 GPa, the system relaxes to a graphite-like structure with shifted layers. From 66.9 GPa to just under 70.0 GPa, the structure relaxes into a diamond-like structure with changed stacking sequence. The resulting structure is of the 9R type known from silicon carbide [88], i.e., it has rombohedral symmetry and a stacking sequence of

ABCBCACAB (Fig. 3.10). For pressures between 70 GPa and 80 GPa, the structure keeps fourfold coordination for all carbon atoms, but bond switching begins, similar to amorphous silicon [89].

Figure 3.10: [010] projection of the 9R carbon structure that Blank's structure relaxes to for pressures between 669 kbar and just under 700 kbar.

63 Whereas Blank et al. find that their crystal lattice can be represented as a triclinic one

o o with parameters a = b = 5.1 A , c = 7.45 A , α = β = 82° , and γ = 120° , we find with zero-pressure relaxations for the triclinic representation of the 9R structure

o o a = b = 4.97 A , c = 6.33 A , α = 90° , β = 79° , and γ = 120° ; and for the triclinic

o o representation of the graphite-like structure a = b = 4.88 A , c = 10.39 A , α = 97° ,

β = 76° , and γ = 120° . As expected, Blank's structure is intermediate between these structures.

For none of the examined pressures, Blank's model was stable or metastable under relaxation. Thus, unless there is some dynamic stabilization of Blank's structure, which is not accessible to our methodology here, or stabilization by alloying elements, we cannot confirm evidence for its existence. However, since the structure can relax to both diamond-like and graphite-like carbon, it might be an interesting starting point for studying the transition path from graphite to diamond.

3.2.6 Kinked Kudryavtsev Type Carbyne and Unified Carbyne Model

3.2.6.1 Structural Model

Kudryavtsev's model [66], shown in Fig. 3.11(a) (redrawn from Fig. 3.2(c)), which is derived from x-ray scattering data, suggests an arrangement of carbyne chains in the form of two hexagonal “sublattices”, one in the regular hexagonal positions, which encompass the corners of the hexagon and its center (positions indicated by grey circles in

Fig. 3.11(a)), while the second sublattice (red circles) has an occupation of 2/3 compared to the first one (the equivalent of chain position C is missing) and is shifted by 1.49 Å

64 with respect to the first sublattice in the direction indicated in Fig. 3.11(a). Since we have found in Sec. 3.2.3 that interchain distances of at least ~3 Å are necessary to prevent interchain bonding and formation of graphite or diamond-like material, we propose that chains within the two sublattices cannot be side by side, but have to be in different planes, which we suggest to connect by Heimann-type kinks. We will derive possible models for this type of structure below.

The side length of the hexagon in Fig. 3.11(a) is suggested to be 2.97 Å, making the hexagonal (or maybe trigonal) lattice constants a1 and a2 (indicated in the figure) 5.14 Å.

This is smaller than most of the measured lattice constants summarized in Table 3.1 by a factor of ~3 (except for the short value measured for carbon VI). This suggests that the x-ray measurements find a “supercell” of Kudryavtsev's cell with a-lattice constants along the long diagonal of Kudryavtsev's cell as indicated by the thick red lines in

Fig. 3.11(b).

65

Figure 3.11: (a) Structural model of carbyne in (0001) projection suggested by Kudryavtsev et al [66]. The two different “sublattices” are indicated by grey (brighter) and red (darker) circles, respectively, where the red sublattice has 2/3 occupation. Also shown is the projection of a hexagonal unit cell (solid black line). (b) Supercell of the structure from (a) shown by red lines, encompassing three cells with in-plane lattice constant similar to experimental results. (c) Same structure with fully occupied red sublattice (addition of hatched red circles) and unit cell (dashed red line) for fully occupied case.

The different periodicity can be explained by different occupation of the two sublattices. For example, in case one or more of the red chains (circles) within the boundaries of the red supercell were missing (except for all A-type or all B-type chains,

66 which would preserve the periodicity), this would break the periodicity of the smaller

(black) cell and make the larger cell necessary. Alternatively, the supercell could be explained if the empty trigonal sites between the chains of the grey sublattice (hatched circles in Fig. 3.11(c)) were occupied by atoms of different elements (again, in a pattern that does not have the periodicity of the small, black cell, but that of the large, red cell. A carbyne intercalation compound as an example for this is discussed in Sec. 3.4 and is also shown in Fig. 3.20). Alternatively, varying occupation could explain why sometimes a trigonal (P3), and sometimes a hexagonal cell symmetry [43] have been identified.

Finally, if all sublattice positions were occupied (solid and hatched red circles in

Fig. 3.11(c)), the periodicity (and symmetry) would be increased, resulting in the smallest possible unit cell (dashed line in Fig. 3.11(c)), which is equivalent to our lonsdaleite- derived structure discussed in Sec. 3.2.4. As far as we have seen, all other lattice constants reported in the literature are approximate multiples of one of the cells presented in this section.

3.2.6.2 Results

Kudryavtsev's model (Fig. 3.11(c)) puts the kink atoms of the red chains in sp3 coordination (diamond-like), whereas the kink atoms of the grey chains are in sp2 coordination (graphite like). The sp3 coordination of the red kink atoms forces the first red bond to be a single bond, making the red chain polyyne. The sp2 coordination of the grey kink atoms with single bonds to the red kink atoms makes the first grey chain bond a double bond, resulting in cumulene type for the grey chains. Figure 3.12 shows atomistic models for the perfect (unrelaxed) and for the relaxed cell for the example of a chain length of six carbon atoms in both grey and red sublattices (which we label (62, 63) 67 structure). For the relaxed (62, 63) structure, the sp2 kink atoms try to arrange their three neighbors in a plane as expected from graphite. Since the bonds between gray and red kink atoms at opposite ends of a grey chain cannot be parallel to each other (due to the

109.5° tetrahedral bond angles of sp3 bonding), the grey chains are not straight, but curved in case the chains are long enough. The relaxed cell has lattice vectors of

o o a = 4.97 Å (corresponding to a super-cell lattice constant of ~ 3 × 4.97 A = 8.61A ) and c = 14.09 Å.

Figure 3.12: Kudryavtsev model [66] of carbyne with a chain length of six in (001) projection and side view (a) before and (b) after relaxation (different lattice vectors were accidentally chosen, but both structures have the same orientation). The two different “sublattices” are indicated by grey (brighter) and red (darker) spheres, respectively. Also shown is the projection of a hexagonal unit cell (solid black line).

The cumulene nature of the grey sublattice allows in principle arbitray chain lengths, while the polyyne nature of the red chains with alternating single and triple bonds and

68 single bonds at the ends requires to have (2n + 2) atoms in the red chains with n = 0, 1,

2… . n=0 is for the case without triple bonds in the structure.

To date, we have studied besides the 6-atom chain structure a range of different chain lengths that mostly put the overall c-lattice constant into the experimental range.

Figure 3.13 shows some examples of relaxed structures, while the lattice constants are summarized in Table 3.3.

69

n(sp2) m(sp2) a (Å) 3 a (Å) c (Å) ρ (g/cm3) 12 12 5.11 8.86 29.03 1.82 10 10 5.07 8.79 23.99 1.86 8 8 5.04 8.72 19.05 1.91 6 6 4.97 8.62 14.09 1.98 12 4 5.07 8.79 19.19 2.05 10 4 5.06 8.76 16.63 2.06 2 4 4.83 8.36 6.66 2.08 8 4 4.98 8.62 14.12 2.11 6 4 4.96 8.59 11.54 2.11 4 4 4.86 8.42 9.02 2.16 12 2 5.01 8.69 16.62 2.2 3 4 4.69 8.13 7.97 2.23 10 2 4.98 8.63 14.11 2.23 8 2 4.91 8.51 11.56 2.31 6 2 4.85 8.4 9.02 2.39 5 2 4.75 8.22 7.85 2.47 4 2 4.79 8.29 6.49 2.48

Table 3.3: Lattice constants and density of optimized kinked Kudryavtsev type carbyne structures with sp2 and sp3 terminated chain lengths of n and m, respectively, sorted by increasing density.

70 For the (32, 43) structure, we find that the center atoms in the sp2 terminated chains bind to each other, thus leaving only the sp3 terminated chains in carbyne configurations.

Similar binding of the center atoms happened for (52, 23) chains, but with longer bond lengths. This did not happen for sp2 chains with an even number of chain atoms. No tendency to relax to the layered, graphite-derived kinked carbyne type has been observed for any of the structures as we had found for many chain lengths of the lonsdaleite- derived model.

Figure 3.13: Relaxed Kudryavtsev structure [66] of carbyne in (0001) projection and side view with (a) minimum chain lengths of two (grey, brighter) and four (red, darker), (b) minimum equal chain lengths of four, and (c) 12 atoms in all chains.

71 While the a-lattice constants of the fully-occupied lonsdaleite derived structures agreed well with experimental lattice constants only for the case of β-carbyne, especially the structures with the longer chains (six or higher) have a-lattice constants well within the range of experimental values, making them potential structural candidates for carbynes. However, since there is no more 2-fold screw axis, the longest measured c- lattice constant for α-carbyne, 15.36 Å, limits the chain length to ~6 for the case with equal chain lengths.

3.3 Free Energy Calculations

In order to further evaluate the different structures, we will now examine the free energy of formation for the different carbyne models in comparison to graphite and diamond to examine their potential existence under equilibrium conditions.

3.3.1 Calculation Model

In order to calculate the free energy, we used the quasi-harmonic approximation

(QHA) with VASP input for energies and phonon frequencies, which approximates the free energy by

1 F( p,T) = H ( p) + g[ω( p)]hω( p)dω 4π ∫ (3.1) ⎡ ⎛ hω( p) ⎞⎤ + k T g[ω( p)]ln 1− exp⎜− ⎟ dω B ∫ ⎢ ⎜ ⎟⎥ ⎣ ⎝ 2πk BT ⎠⎦ where g(ω) is the phonon density of states, the second term on the right hand side is the zero-point energy, and the third term comes from the entropy. Thus, we needed to relax the structures for a series of different pressures (typically, 0, 6, 12, and 18 GPa; for diamond and graphite, additional pressures were considered) using a constant-pressure

72 relaxation algorithm and determine their internal energy E(p) as a function of pressure as well as their volume V(p) to calculate H(p) = E(p) + pV(p). Next, the phonon density of states was calculated for each pressure, allowing to calculate the free energy F(p,T) for the full range of temperatures using Eq. (3.1).

Figure 3.14: Calculated phonon density of states (DOS) for diamond (top), graphite (middle), and lonsdaleite-derived carbyne with 4 chain atoms, calculated within LDA.

Although the enthalpy contribution from linear carbon bonds is expected to be larger than that from sp2 and sp3 bonding, the entropy contribution at high temperatures could compensate for that energy difference. For that, we look at the phonon densities of states shown in Fig. 3.14, which is the distribution of all characteristic (thermal) vibrational frequencies from 0 to the maximum possible frequency and thus is a measure of the

73 thermal ability of a system to get from its current configuration into a new one (i.e., its vibrational entropy). The highest characteristic frequency in graphite is the Raman active mode around ~ 1500 cm-1 [72] (1330 cm-1 in the case of diamond [70]), whereas for linear carbon, this frequency is > 2000 cm-1 in agreement with Ref. [61]. Since the argument of the logarithm in the entropy term is always < 1, the whole entropy term has always negative sign. Since the maximum frequency in carbyne is higher than in graphite, this could therefore mean that at increasing temperature, the entropy contribution of carbyne is growing faster than that of graphite and thus make the free energy of carbyne lower at high enough temperature and suitable pressure.

3.3.2 Approximations and Accuracy

For the calculation of the phonon density of states, which is required for the free- energy calculation within the QHA, we used two methods. The first one consists of a full calculation of all phonon frequencies within the complete Brillouin zone within density functional perturbation theory [89] using the PWSCF package [90]. The second approach calculates Γ-point phonons for supercells of the structural unit cell and adds Gaussians on each frequency (with a typical width of 5 meV), which is somewhat more approximate, but much faster (highly desirable, since the computational effort is considerable). For the second approach, we use the VASP program package [77-80]. All calculations are performed within Local Density Approximation (LDA) due to the reasons described in

Sec. 3.2.2.

Figure 3.15 shows a comparison of the results for the difference in free energy between three example carbynes and graphite at two different pressures, calculated from both methods. As can be seen, their agreement is good in general, with an average 74 deviation of 3% ± 8% (0.04 eV ± 0.06 eV) and a maximum error of 0.13 eV (13%). We will show in the next section that the energy penalty for carbyne compared to the lowest stable carbon phase (graphite or diamond) is under all conditions significantly larger than this error, thus justifying a posteriori the use of the approximate method.

Figure 3.15: Difference in free energy between different carbyne structures and graphite. Solid lines (symbols) are results for zero pressure, dashed lines (open symbols) for a pressure of 60 kbar. Lines are calculated from full phonon density of states within PWSCF, symbols from supercell Γ-point phonons using VASP. Red (top), 6sp3 carbyne (not stable at p = 60 kbar), blue (middle) 4sp2 carbyne, green (bottom) 4sp3 carbyne (n spm: n denotes number of carbon atoms in chains, spm the bonding of the kink atoms).

3.3.3 Phase Boundary between Graphite and Diamond

In order to find out if our chosen methodology gives sensible prediction about phase stability, we calculated the free energy of graphite and diamond as a function of pressure and temperature. For a series of different pressures (zero to 12 GPa (where the experimental data stop) in 1.5 GPa increments), we then determined the temperature for

75 which the free energies of graphite and diamond intersect. Figure 3.16 shows our results in comparison to experimental results from Ref. [91]. Our results are in excellent agreement with experiment, as expected from previous work [92] examining the phase boundaries within density-functional theory with similar success. This gives us additional confidence in the validity of our approach, which in general requires increased caution at higher temperatures.

Figure 3.16: Phase boundary between diamond (right-lower regime) and graphite (left- upper regime), calculated within LDA as described in the text (dots) in comparisons to experimental results (line) from Ref. [91].

3.3.4 Energetics of Carbynes

Figure 3.17 shows our results for the free energy difference between all different

(meta-) stable carbyne structures within the lonsdaleite-derived, graphite-derived and

Kudryavtsev type modifications, respectively, and the stable carbon phase (graphite or diamond, depending on temperature and pressure). With the exception of the 6sp3

76 carbyne (which collapses at pressures larger than zero into a graphite-like structure, which is reflected in the energy curves) and to a small degree the hypothetical carbynes with the shortest chain length of 4 (4sp2 and 4sp3, which have not been observed in nature), all energy curves look qualitatively similar, indicating the fact that the energetics are dominated by the linearly bonded carbon atoms. This means that the exact kink structure is of secondary importance, making our exact choice of structure a factor of lesser importance for the energetics.

Clearly, there is a notable free energy penalty for forming linear carbon. Although it decreases with increasing temperature, it never drops below 0.6-0.8 eV per atom (58-77 kJ/mol) at the melting temperature of graphite around 4000 K. A further increase in temperature (we studied the hypothetical case of 5000 K) to decrease the free energy decreased the free energy of formation only marginally. For carbynes with longer carbon chains (n = 10 or larger) and Kudryavtsev-type carbynes, at high temperatures, an increase in pressure decreases the free energy penalty slightly with a minimum around 6

GPa, but fails to drive the free energy into the stable range.

Although this might be interpreted as a hint that the carbyne-like materials previously suggested in the phase diagram of carbon [62] could indeed only be stable under pressure, our results still indicate them to be unstable in the absence of other elements under equilibrium conditions. This of course still means that pure carbynes could potentially form under non-equilibrium conditions, in case they were inhibited by a high enough energy barrier from graphitization. In the present work, we have not studied such barriers.

77

Figure 3.17: Difference in free energy between the different proposed carbyne structures that were stable under relaxation and the minimum-energy carbon phase (graphite or diamond, depending on pressure and temperature as shown in Fig. 3.12). Different temperatures are indicated by lines (top-down: 0 K, 1000 K, 2000 K, 3000 K, 4000 K, and 5000 K) as functions of applied pressure. Carbyne structures are labeled by number of carbon atoms in the chains (from 4 to 12) and bonding of kink atoms for the graphite and lonsdaleite-derived modifications, and by number of atoms in the sp2 and sp3 terminated chains, respectively, for the Kudryavtsev-type modifications.

78 3.4 Discussions

In the previous sections we found that experimentally studied carbyne material

(Table 3.1) is, first, often somewhat larger than our simulated structures (Tables 3.1 and

3.2) and, second, that carbon-only carbyne structures are energetically unfavorable under equilibrium conditions (Fig. 3.17). Combining the larger size of experimental carbynes with the unfavorable energetics of carbon-only carbynes, a possible key to stable carbynes (or better carbynoids) could be the addition of additional impurities which keep their structure from collapsing into graphite or diamond. Possible additions could be organic molecules (such as CH3), which have been shown to produce stable carbynoid solids [45], or metal ions.

Figure 3.18: (001) Projection of the computationally optimized crystal structure of the solid (tert-butyl)C8(tert-butyl) carbynoids suggested in Ref. [45]. The structure consists of alternating layers of molecules with different orientation, indicated by grey C and white H atoms; and blue C and yellow H atoms, respectively.

79 Figure 3.18 shows the unit cell of a solid of (tert-butyl)C8(tert-butyl) (tert-butyl being

3 CH3 groups) that we relaxed from an initial structure suggested in Ref. [45]. The intramolecular distances and angles measured by x-ray diffraction are well reproduced by our ground state structure (Fig. 3.19). Thus, the suggested structure from the characterization results in Ref. [45] seems to be highly reasonable and is one of the few

(if not the only) fully characterized structures of a carbynoid solid.

Figure 3.19: Representative molecule from the (tert-butyl)C8(tert-butyl) structure shown in Fig. 3.18. The numbers next to bonds are the bond lengths after a structural relaxation within LDA in Å, the numbers next to arcs bond angles (experimental values from Ref. [45] in brackets).

Unfortunately, this structure is only stable up to ~130°C. At higher temperatures, polynuclear hydrocarbons and graphitic material begin to form through polymerization

[45]. The reason for this can be understood from Fig. 3.19. The (highly reactive [93]) linear carbon chain in the middle of the molecule bends towards the neighboring layer of crossed chains (blue chain towards grey chain in Fig. 3.18, bending perpendicular to the paper plane, and vice versa). Although the large tert-butyl groups keep them far away from each other and prevent reactions at lower temperatures, the larger vibrational

80 amplitudes at higher temperatures brings the bending C chains into contact. Thus, the tert-butyl groups play the role of the carbon kink-atoms in the carbyne structures studied above. Despite the fact that the energy (or chemical potential) per linear carbon atom should be high again compared to carbon atoms in graphite or diamond, the history of the structure (first formation of the (tert-butyl)C8(tert-butyl) molecules, then formation of the solid) preserves the linear carbon chains.

A similar process should be possible in the presence of metal ions. Carbon intercalation compounds have been proposed [94] as shown in Fig. 3.20. According to

Udod et al. [94], “the destruction of short-chain organometallic compounds under severe conditions (hydrolysis with strong mineral acids) takes place through a reaction which is supposed to bring about the oriented lengthening of the chains with simultaneous accommodation of metal atoms into the existing vacancies of the linear carbon matrix.

The formation of intercalation compounds should affect the C − C bond lengths and kink angles, thus leading to a slight increase in parameter a of the hexagonal carbyne lattice.”

Finally, the reaction occurring in the course of depositing carbon-nitrogen films on hot substrates has been suggested to result in polymerization of cyanopolyynes

NC − (C ≡ C) x − CN [95], as depicted in Fig. 3.21. Nitrogen could naturally play the role of trivalent kink atoms without dangling bonds and thus produce quite stable polymeric molecules.

81

Figure 3.20: Spatial model of the carbyne intercalation compounds, redrawn after the suggestion by Udod et al [94]. The original caption states that “the length of the rectilinear fragments became the same as a result of the chemical interaction of carbon atoms at kink sites with intercalated metal atoms.”

Figure 3.21: Relaxed structure of example of cyanopoloyyne, NC − (,C ≡ C) x − CN where the kink atoms are nitrogen, which provide dangling-bond free kinks due to their trivalent bonding as suggested in Ref. [94].

Thus, although carbon-only carbynes have higher free energies than graphite and diamond, these examples indicate that under the right conditions, carbynes (or better carbynoid structures) can exist and have been identified in the past.

82 3.5 Conclusions

In this study, we have identified potential structures for solid carbynes and have examined their free energy as a function of temperature and pressure for the purpose of identifying possible phase transitions to graphite.

We propose a unified model for carbyne structures, which combines elements from previous models by Kudryavtsev [66, 96] and Heimann [43, 50], where layers of fully or fractionally occupied hexagonal arrangements of carbyne chains of equal length are connected by kink atoms with sp2 and sp3 bonding, respectively. A special case of this model can be derived from hexagonal diamond (lonsdaleite), where additional carbon atoms are added in parallel bonds between existing carbon atoms. The cell parameters of these structures agree well with experimental data, in case they contain “supercells”

(multiples) of the primitive hexagonal unit created by either fractional occupancy of the chain sites, or the presence of elements other than carbon in the free spaces of the carbyne structure. In our kinked-Kudryavtsev model, we find the chains to be forced into curved shapes due to the sp2 and sp3 nature of the kink atoms. The curving of the chains has two effects. (1) it increases the lattice constants, which are then found to be in excellent agreement with experiment and, (2) it can reduce the hexagonal to trigonal symmetry and explain the fact that both symmetries have been measured in the past.

Nevertheless, experimental lattice constants can have the tendency to be slightly larger than the calculated ones, indicating that in experimental carbyne, additional elements may be present that expand the cell and require the space of several primitive carbon-only cells (thus, resulting in measured lattice constants which correspond to supercells).

83 We find that the free energy of carbynes is higher than that of graphite and diamond for all conditions (temperature up to melting temperature, pressures up to 18 GPa). The free energy of formation for carbyne decreases with increasing temperature and is smallest for pressures between 0 and 6 GPa, approximately 0.6-0.8 eV/atom (58-77 kJ/mol or 14-18 kcal/mol) for all examined structures. Unless carbyne can be formed under non-equilibrium conditions and stays metastable afterwards, the free energy calculations indicate that carbyne needs to contain additional components to become stable. In agreement with this, our calculations confirm the stability and structure of the only well characterized carbyne solid we could find, which is stabilized by 3 CH3 end groups on each end of the carbyne chains (but is only stable at temperatures up to

~130°C).

84

CHAPTER 4

CONTACT FORMATION OF CARBON NANOTUBE DEVICES AND ITS EFFECTS ON ELECTRON TRANSPORTS

The continuous minimization of silicon-based electronic devices has been successful for several decades. However, the feature size of those devices is approaching its physical limit and further progress does not seem to be feasible for much longer without changing the traditional device structure. Therefore, molecular devices with carbon nanotubes

(CNTs) as a channel material in field effect transistors are currently studied as a potential replacement for silicon devices.

A lot of effort has been invested in the study of CNTs because of their excellent conductivity arising from their long-range ballistic electron transport. The early research and development in this field was focused on fabricating devices and testing their transport properties. Some desired properties have been realized in CNTs devices, such as small feature size and large on/off current ratio. However, the contact resistance between

CNTs and their metal leads is often much larger compared to that of the CNT itself [97].

Most CNT devices reported possess a Schottky barrier at the CNT/metal contact. It has been shown that the Schottky barrier is crucial for the performance of CNT devices [98].

85 However, it has been recently demonstrated that CNTs contact with Pd electrodes with adsorbed surface hydrogen can lead to barrier-free ballistic transport [99]. These results show that CNT devices with different metal electrodes exhibit very different properties, which is not observed in silicon based field effect transistor devices. Therefore, part of the current research in this area is focused on the contact formation between CNTs and metals, aiming to improve the understanding of the contact structure and investigate its effect on the device performance, which involves both experimental and modeling efforts.

Experimentally, devices with different metal electrodes have been fabricated and the properties were measured and compared. Many of the researchers explained their results by conventional semiconductor-metal junction theory [100]. However, some [101] feel that this theory cannot be applied directly because of the small size of CNTs. On the simulation side, atomic scale modeling has been used to study the contact structure and its effect on device performance, and it has been proved to be a useful tool [102]. The desire for better CNT field effect transistors (CNT-FETs) is the motivation for research in this field.

4.1 Carbon Nanotube Field Effect Transistors

The field effect transistor (FET) is the most important device in modern computers.

The FET has three terminals called gate, source and drain, respectively. Its basic function is that of a switch. By applying different voltages to the gate, one can control the on and off state between the source and drain, i.e., create a conducting or non-conducting connection between them. In current technology, the metal-oxide-semiconductor FET

(MOSFET) is manufactured as the critical element of central processing units (CPUs).

The number of FETs being packaged into one CPU and the feature size of FET, which is 86 the smallest dimension defined by the lithography, are commonly considered as the index of the semiconductor industry. The often quoted Moore’s Law [103] predicts the continued minimization of transistors and the continued increasing number of transistors on an integrated circuit. However, as the feature size of semiconductor devices is shrinking into the nanometer scale, it is becoming increasingly difficult to control product properties. Furthermore, the onset of quantum phenomena, e.g., tunneling, will make the conventional device inoperable. Therefore, molecular devices are studied with the hope of extending Moore’s Law beyond the limit of conventional devices [104-107]. The CNT has been considered as the most promising building block of those devices because of its chemical and mechanical robustness and excellent transport properties [108].

4.1.1 Structure of CNTs

The structure of an ideal CNT can be considered as a tube which has a shell of one or many layers of carbon atoms. This tube can be as long as micrometers, but with diameters as small as several angstroms (Å) to nanometers at the same time. This is where its name is derived from. The first observation of CNTs was reported in 1991 by

Iijima [109]. Only 2 years later, single wall carbon nanotubes (SWNTs) were experimentally discovered by Iijima and Ichihashi [110], and Bethune et al. [111] independently.

The single wall carbon nanotube (SWNT) is the basic building block for all CNT device structures. Rolling up a single atomic layer of graphite and attaching two end edges forms a seamless cylinder which is the SWNT [112], as shown in Fig. 4.1 (a). A single layer of graphite is called graphene by convention.

87

Figure 4.1 (redrawn after Ref.[113]): (a) Rolling up a single atomic layer of graphite (graphene) forms a SWNT [112]. (b) The unrolled graphene structure, showing basis vectors a1 and a2. The unit cell of a SWNT can be defined by chiral vector, Ch (OA), and translation vector, T (OB). θ is the chiral angle. Rolling up OBB’A along OB and connecting OB with AB’ forms a (4,2) SWNT.

The structure of a SWNT can be considered as a periodic repeat of its unit cell along only the axial direction [113]. This unit cell is defined by the chiral vector Ch (OA in Fig.

4.1 (b)) and the translation vector T (OB in Fig. 4.1 (b)). One can construct a SWNT by rolling up rectangle OBB’A along OB direction and attaching edge OB to AB’. The circumference of the SWNT is called the chiral vector, Ch = n a1 + m a2 ≡ (m, n), where, a1 and a2 are basis vectors of the 2D graphene sheet. The chiral vector connects two crystallographically equivalent points. The construction of a SWNT and the subsequent structure uniquely depends on the chiral vector which is specified by the notation (n, m).

According to different type of chiral vectors, SWNTs can be categorized into 3 types: 1) an armchair nanotube when n = m, 2) a zigzag nanotube when m = 0 or n = 0 and 3) a

2 2 chiral nanotube for other (n, m) combinations. The length of Ch is a ()m + mn + n , where a is the length of the basis vectors. The diameter of the SWNT is

88 a ()m 2 + mn + n 2 /π . The chiral angle refers to the angle between the chiral vector and

−1 the basis vector a1, and is given by θ = tan [ 3n /(2m + n)]. Because of the hexagonal symmetry of graphene, the value of θ ranges from 0° to 30°. This chiral angle indicates the degree of deviation of the hexagon edge away from the axial direction. In particular,

θ = 0° for zigzag nanotubes and θ = 30° for armchair nanotubes. As shown in Fig. 4.1(b), the translation vector T defines the other edge of the unit cell, and along which the unit cell is translated and repeated to form a SWNT. The translation vector is normal to the chiral vector and parallel to the axial direction [112].

The first nanotubes discovered [109] were multi-wall carbon nanotubes (MWNTs). A

MWNT is the combination of several coaxial SWNTs with different diameters [114]. The schematic drawing is shown in Fig. 4.2 where each cylinder represents a SWNT [109].

And TEM pictures of MWNTs can be found in Ref. [109]. However, because a MWNT is more complex than a SWNT, the current research and development is focused on

SWNTs.

Figure 4.2 (constructed after Ref. [109]): Schematic drawing of a MWNT. Each cylinder represents a SWNT.

89 4.1.2 Electronic Properties of CNTs

The discovery of CNTs triggered a large body of theoretical development for the properties of CNTs. The most intriguing of these findings is the prediction that CNTs can be metallic or semi-conducting depending on the carbon network structure [115-117].

This prediction was confirmed by later experiments [118, 119]. This exotic property is part of the reason why CNTs have been attracting a lot of attention in the research community, and much effort has been invested into practical application development.

The intriguing electronic properties of CNTs originate from the peculiar electronic structure of graphene, specifically, the band structure of graphene. The band structure is the relationship between energy of electrons and their wave vectors (k) in a solid. It provides a lot of information of electrons such as density of states. Moreover, it describes the allowed and forbidden ranges of energies of an electron in a solid. Figure 4.3 shows the unit cell in real space and the Brillouim zone in reciprocal space of graphene. Special points, Γ, Κ and Μ are defined as the center, a corner and the center of one of the sides of the Brillouin zone, respectively, with coordinates (0,0), (1/2,1/2) and (2/3,1/3) (in the unit of reciprocal lattice vectors), respectively [113]. The reciprocal lattice vectors are

⎛ 2π 2π ⎞ ⎛ 2π 2π ⎞ b1 = ⎜ , ⎟ and b2 = ⎜ ,− ⎟ , where a is the (real-space) lattice constant of ⎝ 3a a ⎠ ⎝ 3a a ⎠ graphene. The conduction and valence band of graphene has been calculated by tight- binding method and can be described by

⎛ 3k a ⎞ ⎛ k a ⎞ ⎛ k a ⎞ E k ,k = ±γ 1+ 4cos⎜ x ⎟cos⎜ y ⎟ + 4cos 2 ⎜ y ⎟ , where the plus sign denotes ()x y ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ the conduction band, the minus sign denotes the valence band, and γ ≈ 3 eV [100]. This

90 2D band structure is plotted in Fig. 4.4. The energy surfaces describing conduction and valence bands touch each other at only six points, namely, six Κ points defined in

Fig. 4.3 (b).

From a conventional point of view, insulators have a large bandgap, i.e., conduction and valence bands do not overlap; semiconductors have a small bandgap (for example, the bandgap of silicon is 1.12 eV); and conductors have a zero bandgap, i.e., conduction and valence band overlap. Graphene is between semiconductors and metals because it has a zero bandgap but conduction and valence bands do not overlap.

Figure 4.3 (redrawn after Ref.[113]): (a) Unit cell and (b) Brillouin zone of graphene. a1 and a2 (b1 and b2) are (reciprocal) lattice vectors. Special points, Γ, Κ and Μ are defined as the center, the corner and the center of the side of the Brillouin zone, respectively.

91

Figure 4.4: 2D band structure of graphene from tight binding calculation, using ⎛ 3k a ⎞ ⎛ k a ⎞ ⎛ k a ⎞ E k ,k = ±γ 1+ 4cos⎜ x ⎟cos⎜ y ⎟ + 4cos 2 ⎜ y ⎟ with γ = 3 [100]. The ()x y ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ conduction and valence bands touch each other at only six points.

The existence of a zero gap at the K points in graphene gives rise to interesting conductance properties of SWNTs. Folding a rectangular graphene forms a SWNT, so one may express a wave vector as K1 in the axial direction (translation vector direction) and K2 in the circumferential direction (chiral vector direction). Due to the tubular structure, there is a periodic boundary condition along the circumferential direction. Thus, it is required that K2·Ch=2π, resulting in a quantized condition for K2. Because the length in axial direction is usually much greater than the diameter, K1 can be considered as continuous [113]. Therefore, the allowed wave vector in SWNT is a set of parallel lines in reciprocal space. Lines are parallel to the axial direction and the separation between lines is 2/diameter [112]. Because of this confinement on electron wave vectors, each two-dimensional energy band of graphene splits into a series of one-dimensional energy

92 bands of the SWNT, i.e., bands described by the allowed energies of electrons along the allowed wave vectors [120]. This relationship is plotted in Fig. 4.5 (a).

(b) (c)

(a)

Figure 4.5: (a) (figure (a) constructed after Ref. [120]) 2D band structure of graphene (top) and the first Brillouin zone (bottom). The allowed states in the nanotubes are cuts of the graphene bands indicated by the dash lines. The lines on the bottom are the allowed wave vectors. (b) If the cut passes through a K point, the CNT is metallic; (c) otherwise, the CNT is semiconducting.

The condition for SWNTs to obtain a zero gap in the energy band structure is that the allowed wave vectors include K points, where graphene has the zero gaps (see Figs. 4.4 and 4.5). Therefore, as illustrated in Figs. 4.5 (b) and (c), if the set of lines indicating allowed wave vectors passes through K points, the SWNT is metallic, otherwise, it is semiconducting [113]. Since the chiral vector dictates the circumference direction and the diameter, it will determine the conductance properties of SWNTs.

4.1.3 Metal-Oxide-Semiconductor Field Effect Transistors (MOSFETs)

CNT FETs are designed in the spirit of current MOSFETs. A MOSFET consists of a metal lead (M), an oxide insulating layer (O) and a semiconductor base (S, traditionally

93 silicon) shown schematically in Fig. 6. The silicon base can be doped with p type dopants

(group III elements, usually B) or n type dopants (group V elements, usually As and P) to enhance the conductivity. MOSFETs can be categorized into two types: p-channel (PNP type, Fig. 4.6(a)) and n-channel (NPN type, Fig. 4.6(b)). They utilize different doping schemes of the silicon base.

Figure 4.6: Schematic drawings of (a) P-MOSFET and (b) N-MOSFET.

Without applied gate voltages, no current can flow between the source and drain because of the two opposite PN-junctions. Upon application of a negative voltage at the p-MOSFET gate electrode (positive voltage for n-MOSFET), the region in the substrate just under the gate oxide is inverted to p type (n type) due to the gate electric field, so that the current can flow through this channel region [121]. The dimension of this channel region (usually is half of the feature size in current technology) is continuously diminished in order to increase the switching speed of MOSFETs. The feature size has been reduced to 45 nano-meters in state-of-the-art processing technology. However, tunneling or leakage current will become prominent if this size gets even smaller. This will post a great challenge for the further down scaling of devices. ITRS (International 94 Technology Roadmap for Semiconductors) has estimated the continuous scaling may be feasible until the year 2022 [122]. But after that, new devices are required to further increase the computer power. The study of molecular devices, such as CNT-FETs, is part of the effort to lay the foundation for next generation electronics.

4.1.4 Structure of CNT-FETs

Most of the proposed CNT-FETs utilize SWNTs instead of MWNTs because

MWNTs are more complex and because of the existence of shell-shell interactions [123,

124]. SWNTs have several advantages [120, 125] for electronics: 1) The strong covalent bonding between carbon atoms of SWNTs assures mechanical toughness, thermal stability and sustainability of high current density [108]; 2) Semiconducting and metallic nanotubes are suitable materials for active devices (such as transistors) and interconnections respectively [120]; 3) Ballistic transport (electrons passing through without scattering) has been observed in SWNTs [126, 127], implying lower power consumption (less heat dissipation) and faster speed of the device; and 4) Unlike the silicon wafer surface, there are no dangling bonds in SWNTs.

Figure 4.7 (redrawn after Ref.[97]): Illustration of a CNT-FET device (side view).

95 CNT-FETs were realized independently by Tans et al. [97] and Martel et al. [128] in

1998 for the first time. Devices from both groups exhibited current-voltage (I-V) curves with FET characteristics. The structure of a CNT-FET bears an analogy with that of a

MOSFET: it is a stack of channel (consisting of semiconducting nanotubes in the case of the CNT-FET), insulating oxide layer and gate contact. An illustration of one of the first

CNT-FETs from Tans’ group [97] is shown in Fig. 4.7, which shows an individual semiconducting carbon nanotube (the tube on the top of Fig. 4.7) lying on top of two Pt electrodes. The next layer shown is the gate oxide (SiO2). Under this oxide layer, there is a layer of doped Si acting as gate electrode. By plotting I-V curves between the source and the drain under different applied gate voltages, Tans et al. [97] found distinguishable on and off states, i.e., high conductance and low conductance states. These I-V curves are shown in Fig. 4.8 [97]. It is obvious that the current is much larger in the case of negative applied gate voltage than that in the case of positive applied gate voltage. As the gate voltage decreases, the current increases for a given value of the CNT end-end bias voltage. In other words, the resistance decreases with decreasing (negative) gate voltage, saturating at around 1MΩ, as shown in the inset of Fig. 4.8 which shows the conductance

(the inverse of the resistance) [97]. A similar resistance has been observed for a metallic

SWNT in a similar configuration [129]. Therefore, it was suggested that the major part of this resistance comes from the contact resistance between SWNTs and electrodes [97].

Moreover, the inset shows that the resistance of the device varies by six orders of magnitude when the gate voltage is changed by 12 V. Thus, Tans et al. suggested that the gate voltage controlled the semiconductor-to-metal transition in this system [97].

96

Figure 4.8 (figures redrawn from data in Ref. [97]): I–Vbias characteristic curves of a CNT-FET under various applied gate voltages (Vgate). Inset shows the conductance of the CNT-FET (G) as a function of applied gate voltage. The conductance exhibits a six orders of magnitude difference in the scan range (Vgate= -6V ~ 9V).

CNT-FETs from Martel’s group exhibited a similar switch behavior [128]. Their device also showed a high-low conductance switching dictated by the gate voltage. The lower bound of resistance was again around 1 MΩ, most of which was argued to be contributed from the contact resistance [128]. Martel’s group also explored the possibility of using MWNTs for FETs, but without much success [128] due to the complexity of

MWNTs. They also found that MWNTs with relatively large diameter (~10 nm) deformed on top of electrodes.

4.1.5 The Contact of CNT-FETs

One of the reasons that make the CNT device so attractive is its excellent transport properties. Nevertheless, the contact resistance is on the order of MΩs as shown in the early work mentioned above [97, 128]. This is a huge disadvantage because contacts of

97 high resistance will consume a large part of the electrical energy and convert it to heat. In a worse situation, a high dissipation of heat results in contact failures.

Some early work has been done to address this problem. Soh et al. reported observing resistance as low as 10 kΩ for one of their CNT/electrode contacts at low temperature

(4.2 K) [130]. In this study, the electrode consisted of double layers of metals: Ti

15nm/Au 60 nm. However, in another plot from the same paper showing I-V curves under different gate voltages at room temperature for an operable device, one can infer that the lowest resistance is again approximately 1 MΩ by calculating the V/I ratio for the steepest I–V curve. These I–Vbias curves of this operable FET are thus very similar to those from Tans et al. as shown in Fig. 4.8.

Other efforts have been invested into decreasing the contact resistance. A very intuitive method is to improve the contact quality and to increase the contact area. In previous studies [97, 128], the CNT was laid on the pre-deposited electrodes and weakly bonded. A different fabrication scheme was used to improve this configuration where semicondcuting SWNTs were dispersed on the silicon oxide substrate first, and then the metal electrodes were deposited on top of them [120, 131, 132]. Thus, the SWNT is sandwiched between the oxide and metal layers in the resulting structure. This indeed yields a better contact with a contact resistance around 250 kΩ derived from the data in

Ref. [131].

It has been initially assumed that the switching of CNT-FETs is due to the effect of the gate voltage on the bulk part of the CNT, similar to conventional silicon devices.

However, some observations suggest an alternative mechanism where the contact plays an important role for switching. It was proposed that there is a Schottky barrier between

98 CNTs and metals, and that this barrier controls the transport properties of CNT-

FETs [120].

Figure 4.9: (a) Energy band diagram of a metal and a semiconductor before contact and (b) after contact, resulting a Schottky barrier.

A Schottky barrier refers to an energy barrier resulting from the contact of a metal with a semiconductor and the subsequent alignment of the Fermi energy levels. Energy band diagrams of the separated metal and semiconductor are shown in Fig. 4.9(a). As they contact, the Fermi energies align at the same level resulting from the equilibration of charges [133]. The band diagram after contact formation is shown in Fig. 4.9(b). Now there exists a barrier ΦB blocking holes going from the metal to the semiconductor.

If Schottky barriers are present at CNT/metal contacts, the current across such contacts will be suppressed. However, if the band bending near the contact is sharp and

99 the barrier thickness is small, the electrons then can tunnel through the thin barrier leading to a much increased conductance [100, 134]. Most CNT-FETs reported to date are believed to possess Schottky barriers at the CNT/metal junctions, which severely limit the on-state conductance and degrade the device performance [99]. A barrier-free contact, also known as Ohmic contact, is rather desired for practical use. The fabrication of Ohmic contact CNT-FETs has been reported recently [99, 135], where transistors exhibit higher conductance. Researchers [100] have tried to explain the results and predict the type of contacts, i.e., Schottky or Ohmic, by conventional semiconductor- metal junction theory. This theory explains a range of experimental observations.

However, it has limitations as well at the same time. For example, it predicts that the contact between gold and CNT is Ohmic [100]. However, this contact was reported to be

Schottky type by experiment [128]. It has been suggested that this dilemma may be due to the small scale of CNTs [101] while semiconductor-metal junction theory is for bulk materials. At the same time, the precise nature of interfaces between nanotubes and metals is still not clear and needs to be further studied, which is the topic of the following sections.

4.2 Process Modeling of CNT/Metal Contacts

In previous work, the CNT was assumed to exhibit unchanged properties when brought into contact with electrodes. This was a straightforward approximation. However, there is also the possibility that CNTs experience a structural change during the formation of contacts. Although those changes may be very small, in the angstrom regime for instance, they are still substantial deformations for CNTs considering the fact that most

CNTs have small diameters (~ 1 nanometer). Thus, such a deformation may well lead to 100 a change of properties. Besides the deformation, other factors, such as the absorption site for a CNT on a metal surface, deserve consideration as well. Foremost, the knowledge of the contact structure is necessary for better understanding of its properties.

The structure of CNT/metal contacts in previous theoretical studies has been predicted by regular geometric relaxation, where temperature effects were ignored. By this approach, the system is usually driven to the closest local-minimum energy configuration. However, real devices are fabricated at finite temperatures, which may have a strong influence on contact formation. This is especially important for low- activation barrier systems, such as molecules on surfaces where even low temperatures will induce large changes in configurations.

4.2.1 Temperature-Accelerated Dynamics

To study the contact formation between CNTs and metal electrodes and its structure under finite temperature over extended timescales, we perform process modeling on the atomic scale for CNT/metal contacts using first principles molecular dynamics. In order to speed up the relatively slow simulations, a temperature-accelerated dynamics

(TAD) [136] type approach is adopted.

In the TAD approach, MD simulations are performed at a considerably higher temperature Thigh than the target temperature Tlow. The effective time at the lower temperature can be calculated by

⎡ ⎛ 1 1 ⎞⎤ t = t exp⎢ΔE⎜ − ⎟⎥ (4.1) low high ⎜ kT kT ⎟ ⎣⎢ ⎝ low high ⎠⎦⎥ by assuming that the transition rate r follows the Arrhenius relationship

101 ⎡ ΔE ⎤ r = v0 exp − (4.2) ⎣⎢ kT ⎦⎥ where, v0 denotes the prefactor, ΔE is the energy barrier, k is Boltzmann’s constant and T is the temperature in Kelvin. Therefore, the effective time step is increased by a factor of exp(ΔE/kTlow – ΔE/kThigh) and the simulation time can be significantly extended.

One extra procedure that needs to be accomplished for TAD is to figure out the most probable events at the target temperature, Tlow. Since the MD simulation is carried out at

Thigh, some events observed in the simulations may be high-temperature activated, i.e., they have high probabilities to occur at Thigh but have low probabilities at Tlow. In order to filter out these events, initial and final structures of the infrequent event are relaxed, and the transition energy barrier is determined with the nudged-elastic band method. Then one can extrapolate to the most-probable event for Tlow. In our study, we run a number of simulations in parallel and stop each one when it produces an infrequent event. This way we could sample a large number of infrequent events and pick the most probable one at low temperature.

4.2.2 CNT/Ti Contacts

This section has been reported previously in Refs. [137, 138]. It is included here for the purpose of completeness of the presentation.

As the first step, the contact structure between CNTs and Ti electrodes has been studied by the TAD-MD method [137, 138]. The structure shown in Fig. 4.10(a) mimics a (3,3) SWNT bridging two pieces of Ti electrodes, where darker spheres represent Ti atoms and lighter ones C atoms. A straightforward relaxation does not lead to any significant change in structure (see Fig. 4.10(b) for a side view). However, after an MD

102 run of 1 pico-second (ps) at 650 K, the CNT starts to disintegrate from its tubular structure (Fig. 4.10(c)) in the region where it contacts the electrode. This deformation is expected to alter the contact properties.

Figure 4.10: Contact structures between Ti and a (3,3) carbon nanotube, where blue (darker) spheres represent Ti and dark grey (brighter) spheres represent C atoms in (outside of) the overlap region between the CNT and Ti: (a) structure after straightforward relaxation (no MD); (b) side view of the structure in (a); (c) side view of the structure after a 650 K, 1 ps MD run.

However, this simulated device is much smaller than a real device and the contact region is only a few atomic layers long. This might lead to unrealistic size effects. Due to the expensive computational cost of first principles methods, the full size device structure, which may well contain tens of thousands of atoms, cannot be modeled directly.

Therefore, we use as feasible alternative approach the other extreme: a periodic cell, shown in Fig. 4.11(a), models an infinitely long CNT on top of an infinite Ti substrate

103 (infinitely long contact region). Snap shots of the structure during simulation are shown in Figs. 4.11(a)~(d) in time sequence. The CNT indeed goes through a huge structural transformation. It opens up and flattens on top of Ti, and adopts a graphene structure. The opening of the CNT results in a significant energy gain of 3.0 eV/Å, which is expected to come from the release of strain energy as the CNT unrolls into a flat graphene ribbon, and from the formation of energetically favored Ti-C bonds, indicated by the Ti-C phase diagram (see Sec. 4.2.4).

Figure 4.11: Opening of an infinitely long (3,3) nanotube on Ti during an MD simulation, where grey (brighter) spheres represent C and blue (darker) spheres Ti atoms. Cross- section of the contact area between the CNT and Ti at various instants during MD: (a) 0 ps, (b) 0.85 ps, (c) 1 ps and (d) 2 ps.

In the configuration of real devices, only part of the CNT lies on top of the electrodes, while the other part is, e.g., suspended between the electrodes. The first part tends to completely open up, as shown above, and the second part tends to retain its tubular shape.

There should be a transition region between them. Following this idea, we can construct 104 this transition region with length a between the open and the tubular parts of the CNT by linear coordinate interpolation, rinterpolated = (x/a) ropen + [(a-x)/a] rtubular, followed by relaxation of the final structure. We then find that, after relaxation, the structure with lowest energy has a transition length between the graphene and tubular part of 10 C layers, approximately ∼ 14 Å (Fig. 4.3).

Figure 4.12: (a) Predicted realistic contact structure with optimized transition length of ∼ 14 Å between CNT (most stable when no contact between Ti and CNT exists) and flat graphene sheet (most stable for CNT in contact with Ti). (b) Top view of the optimized transition region between tubular CNT (no contact with Ti) and the minimum-energy graphene film for (3,3) CNTs in contact with Ti. Grey (brighter) spheres represent C and blue (darker) spheres Ti atoms.

Contact structures, shown in Figs. 4.10, 4.11 and 4.12, are quite different from the ones assumed in previous studies, where the contact region was simply assumed to be a tubular CNT in side contacting with a metal. In some of those studies, it was shown that even some small radial deformations may have profound influence on the device’s

105 performances. Thus, it is very important to know the realistic structure in order to correctly understand the effect of the contact. In our study, the contact region has gone through a major change. We expect that it will lead to dramatic changes in the electron transport properties, which can be calculated based on Landauer’s theory [139, 140]. The current through a molecule under an applied voltage V can be calculated as:

2e EF +V / 2 I ()V = T(E) dE (4.3) p h ∫ EF −V / 2 where e is the charge of an electron, h Planck’s constant, EF the Fermi energy and T(E) the transmission function that can be constructed from matrix elements of a local-orbital

DFT Hamiltonian as:

2 2 T(E) = 4π ∑ Tlr (E) δ (E − El )δ (E − Er ) (4.4) lr where El and Er are energy levels of the left and right electrodes, respectively. Tlr is the matrix element of the scattering operator T, for which the Lippmann-Schwinger equation is used, T = V + VGV, where V is the scattering potential due to the interaction of the conduction electrons with the device, and G is the full Green’s Function of the problem.

We use the approach described in Ref. [140], where the Green’s Functions of the semi- infinite leads is calculated by the block-recursion technique. The matrix elements are evaluated by DFT within a non-orthogonal localized atomic orbital basis using the software package SIESTA [141].

To study the influence of a single CNT/Ti contact on the transport properties in a computationally economical way, we next calculate the transport properties of just one

CNT-metal contact, defining the metal and an infinite non-defective (3,3) CNT, away

106 from the contact region, as the “electrodes” of the device. The resulting current and conductance, for this structure (the left half of the structure depicted in Fig. 12(a) with the right half replaced by an infinite (3,3) non-defective CNT) are found to be less than for the initial structure (Figs. 10(a) and (b)) by a factor of 5. Considering that a device would have a second lead and thus add a second contact (resistance) in series, this results in a decreased conductance of about one order of magnitude, about 30 times worse than the close-to-ballistic transport in an ideal (3,3) CNT without any contacts. Thus, optimizing the structure results in an increased contact resistance. This indicates that one of the most important reasons for disagreement between theoretical and experimental transport results, in the case of molecular devices, should be that the contact models used in simulation work are often too simplistic and do not take into account realistic interactions between the metal surface and the CNT.

107

Figure 4.13: Transport curves for the small Ti/CNT/Ti structure after regular relaxation (Fig. 4.10(a), small relaxed) and after a 650 K, 1ps MD run (Fig. 4.10(c), small MD), and for our optimized large contact structure (Fig. 4.12, large realistic) in comparison to those of an ideal CNT to demonstrate the effect of the “electrodes”. (a) Current vs. bias, (b) conductance vs. bias.

4.2.3 Size Effect of the CNT

Aforementioned structures are used to model the contact between (3,3) CNTs and Ti metal electrodes. However, bigger CNTs are often used in fabrications of CNT devices.

In order to study the dependence of contact formation on the CNT size, we performed ab-

108 inito TAD for (4,4) and (5,5) CNTs on Ti substrate with configurations similar to the structure shown in Fig. 11(a). After TAD at 650K for more than 10 ps, both (4,4) and (5,5)

CNTs keep their tubular shape on top of the Ti substrate. Compared to the case of (3,3)

CNT, this indicates that bigger CNTs might be more stable.

Figure 4.14: Snapshots of ab initio TAD results of a infinitely long (4,4) CNT (a) and (5,5) CNT (b) on top of a Ti substrate. Grey (brighter) circles are C atoms and blue (darker) ones are Ti atoms. Both CNTs retain their tubular shape after a ~10 ps MD run at 650 K.

The effect of CNT size on its stability can be explained from the point of view of strain energy. It is very interesting that the relationship between the strain energy of a

SWNT and its diameter is consistent with the result of continuum elastic modeling in spite of the small dimension [142, 143]. Considering a graphitic tubule as a rolled up elastic graphite film, it has been derived that the strain energy of a graphitic tubule has the general form [144]:

πELt 3 E = (4.5) S 6d

109 where E is the elastic modulus, L and d are the length and diameter of the tubule respectively, and t is the thickness of the wall of the tubule. The strain energy is thus inversely proportional to the diameter of the tubule. This derivation was carried out for carbon filaments with diameters at the order of 0.1 μm. Later, first principles calculations total energy were performed for SWNTs [142, 143]. The results showed that the strain energy per carbon atom has a d-2 dependence. The number of carbon atoms in a SWNT is proportional to the diameter. Therefore, the strain energy of the SWNT in these calculations was found to have a d-1 dependence, which is consistent with the elastic- continuum result for a graphitic tubule mentioned above. Following this relationship, the strain energy decreases with increasing CNT size.. When a CNT disintegrates into a single layer of graphite (as shown in Fig. 4.11(d)), this strain energy is released. Thus, smaller CNTs have more driving force to disintegrate and become less stable.

4.2.4 Effect of Different Electrode Materials

To examine the material dependence of the findings in the previous section, we extended our study to include other electrode materials, foremost Pd, which has been found experimentally to be able to create a system with barrier-free ballistic transport

[99]. For an infinitely long (3,3) CNT on Pd (periodic boundary conditions), even at a very high temperature of 1000 K, the CNT retains its tubular shape for at least 3.6 ps on top of the Pd substrate (Fig. 4.15(a)). Since the expensive computational cost of first principles calculations prevents us from performing significantly longer MD simulations, we have constructed an opened structure and relaxed it (Fig. 4.15(b)). The energy of the open structure is indeed lower than the relaxed initial structure shown in Fig. 4.15(c) by

5.4 eV, and is also lower than the relaxed structure shown in Fig. 4.15(a), implying that 110 the open up structure is energetically favored. Knowing this, the nudged elastic band

(NEB) method can be used to survey an upper limit for the energy barrier and the escape path from the initial to the open-up structure. Calculating the exact value is difficult, since the exact location of the opened relative to the initial CNT has to be chosen in this procedure, and is not found by, e.g., an MD simulation.

Figure 4.15: Grey (brighter) circles represent C atoms, blue (darker) circles Pd atom. (a) CNT on top of Pd after an MD simulation of 3.6 ps at 1000K and relaxation. (b) Manually constructed open structure after relaxation. (c) Initial structure after relaxation.

The result of the NEB search shows that the energy barrier for our constructed case is more than 1 eV between the configurations shown in Figs. 4.15(a) and (b). This is a huge value for surface systems and would explain why the CNT does not disintegrate on top of

Pd in our extended-time MD simulations at high temperatures. Still, this is only an upper limit as only a restricted part of the energy landscape has been searched to find the transition state. Very interestingly, phase diagrams provide some insights that shed light on the different interactions in the CNT/Ti and CNT/Pd systems.

When two materials are brought into contact at moderate temperatures, the bulk parts of them in general remain intact, only forming an interface between them. However, there

111 is the other possibility that they may react to form compounds if these new phases are energetically favored and the temperature is sufficiently high to overcome potential energy barriers between initial system and final compound. From the materials scientist’s point of view, phase diagrams provide rich information on determining the stable phases.

The binary phase diagrams of carbon (C) and electrode materials, namely titanium (Ti), aluminum (Al), palladium (Pd) and gold (Au) can be found in Ref. [145].

These phase diagrams show that Ti and C atoms can form a compound (between

~50% and ~70% Ti for temperatures up to ~3000 oC). In other words, mixing both types of atoms together by a certain ratio (according to the phase diagram, approximately

TiC0.95) will decrease the energy and produce a more stable phase [146]. Similar behavior exists for the Al/C phase diagram, where the Al4C3 line compound (i.e., exact stoichiometry) exists. To the contrary, Pd and Au atoms prefer not to be mixed with C atoms (except for limited C solubility in Pd at high temperatures), and no compounds exist in the phase diagram. In those cases, carbon and the metals stay as separate phases.

Those two distinct tendencies of the atomistic arrangement, i.e., mixing and non-mixing of both types of atoms, will have different influences on the contact formation on the atomic scale.

From the point of view of kinetics, the process of disintegration of a CNT needs to break carbon-carbon bonds, as shown in Figs. 4.16(a)-(c). However, carbon-carbon bonds are usually very strong, with bond energies of 3.60 eV, 6.36 eV and 8.70 eV [147] for single, double and triple bonds, respectively. It is more or less impossible for CNTs to overcome these huge energy barriers by thermal motion even at 1000 K. However, when the CNT is brought into contact with the metal electrode, surface catalysis happens,

112 where some electrons transfer from the metal to the CNT (as will be discussed below), thus saturating some carbon atoms and making carbon-carbon bonds easier to break.

Figure 4.16: The process of CNT disintegration on top of Ti substrate. (a) Breaking C-C bonds; (b) opening up due to strain energy; (c) becoming a flat layer.

The direction and magnitude of the charge transfer depends on the difference in electronegativity of the contacting elements. Generally speaking, elements with higher values of electronegativity tend to obtain electrons from elements with lower value. The greater the difference, the more electrons transfer. Table 4.1 lists the electronegativity of

C, Ti, Al, Pd and Au [148], from which one may conclude that there will be more charge transfer from Ti to C than from Pd to C. This explains the greater energy barrier for CNT to disintegrate on Pd.

113 C Ti Al Pd Au 2.5 1.5 1.5 2.2 2.5

Table 4.1: Pauling electronegativities of carbon, titanium, aluminum, palladium and gold [148].

As pointed out by the Hume-Rothery rule [149], two elements with large difference in electronegativity trend to form compounds instead of solid solutions. Therefore, the abovementioned two explanations of the effect of different metal electrodes are intrinsically related.

To test these central findings of this work, the phase diagrams and/or electronegativity values can now be used to predict the behavior of CNTs brought into contact with other metals. We chose two cases with strongly varying electronegativity,

Al, which has approximately the same electronegativity of ~1.5 as Ti, and Au which has nearly the same electronegativity (2.54) as C (2.55). The Al/C and Au/C phase diagrams and the electronegativities in Table 4.1 both suggest that CNTs should disintegrate on top of Al, similar to the case of Ti, while CNT trends to remain tubular on top of Au, similar to the case of Pd. TAD MD runs using parameters similar to the abovementioned simulations have been carried out. The results are consistent with the predictions. As shown in Fig. 4.17, a (3,3) CNT opens up on top of Al substrate after an MD simulation of 1 ps at 800K while the same CNT stays tubular shape on top of Au substrate after an

MD simulation of 6.7 ps at 800K.

114

Figure 4.17: Grey (top) circles represent C atoms, pink (bottom left) circles Al atoms, yellow (bottom right) circles Au atoms. (a) CNT on top of Al after an MD simulation of 1 ps at 800K. (b) CNT on top of Au after an MD simulation of 6.7 ps at 800K.

4.3 Conclusions

This work shows that the formation of the contact structure between CNTs and metal surfaces is not as simple as putting two parts together. Molecular dynamics simulations indicate that the CNTs can undergo structural transformations on top of metal electrodes when the effect of temperature is taken into account. In this way, we have constructed a realistic contact structure and studied its influence on the electron transport properties of the device. We found that temperature-driven relaxation of the contact structure can change the device conductance by orders of magnitude. This again shows that the precise knowledge of the contact structure is crucial to the understanding of the device performance.

We have also examined the effect of CNT size on the contact structure, which can be explained in terms of the elastic energy as we have shown. We find that CNTs with

115 bigger diameters are more stable on the metal substrate, and that the contact topology depends on the size.

The behavior of CNTs on different metal electrodes has been studied as well. From the point of view of thermodynamics, a CNT is more stable when brought into contact with metals which do not form compound with carbon, such as Pd and Au, and is less stable otherwise, such as for the cases of Ti and Al. We have also shown that metals with lower electronegativity tend to give up electrons to CNTs, which facilitates the opening- up process.

In summary, our results indicate that the contact structure between CNTs and metals is complicated, needs to be carefully studied case by case, and is crucial to device performance.

116

CHAPTER 5

OPEN VOLUME DISTRIBUTION IN Cu46Zr54 BULK METALLIC GLASSES FROM ATOMISTIC SIMULATIONS

Metallic glasses are non-crystalline metallic alloys that have a disordered atomic arrangement. In contrast to crystalline metals, they do not have long range order. Thus, they are also called amorphous metals or glassy metals. Metallic glasses are often produced by a rapid quench of molten metals, during which the crystallization is suppressed.

The first metallic glass was made by Buckel and Hilsch in 1954 [150] by rapid cooling of the vapor of lead and tin. However, this glass was only a few μm thick and crystallized at room temperature. Subsequent development in the fabricating techniques allowed the metallic glass to become thicker and thicker and, more importantly, allowed glass forming at lower cooling rates. In the 1970’s, a Pd-Cu-Si glass with a diameter in the mm range was suction cast at a cooling rate of 1000K/s by Chen [151]. This pushed the size scale from μm to mm and is considered to be the first successful production of a bulk metallic glass (BMG). Later, Turnbull’s group was able to produce a Pd40Ni40P20 metallic glass up to a few mm in diameter by cooling from the liquid with a slow cooling

117 rate of 1.4 K/s in vacuum [152]. After that, new families of BMGs have been discovered, such as La-Al-(Cu-Ni) [153], Mg-Cu-Y [154] and Zr-Ti-Cu-Ni-Be [155].

BMGs often combine an array of excellent mechanical properties, such as high strength, high hardness and large elastic strain (~2%). Usually they have a compressive yield strength as high as 1 GPa and Zr based BMGs have a fracture toughness of 20-55

MPa m [156]. However, the mechanical behavior of BMGs and especially its relationship to the amorphous structure is not well understood. It has been suggested that this behavior might be closely related to the so-called free volume [157, 158], which is very difficult to clearly define, characterize, and extract from modeling. The purpose of this work is employing atomistic modeling to help better understanding the nature, amount and distribution of free volume in BMGs.

The deformation mode of BMGs falls into two distinct regimes: homogenous and inhomogeneous. Homogenous flow often occurs at high temperature and low strain rate, where the entire volume contributes to the strain and deformation uniformly. Fracture occurs after extensive deformation and necking. Inhomogeneous flow often occurs at low temperature and high strain rate, where BMGs fail due to shear localization (shear banding) with a very limited amount of plastic deformation. Obviously, BMGs exhibit strongly different behavior from their crystalline counterparts. Moreover, even when inhomogeneous flow occurs and shear bands appear, they are not in the plane of maximum shear. Rather, tensile shear bands are ~52° and compression shear bands are

~43° to the loading direction [159]. This indicates the dependence of BMG deformation on normal/hydrostatic stress. The deformation of crystalline metals is well known and usually dominated by dislocation motion. On the other hand, the theory of Spaepen [157, 118 158] and Argon [160] states that the flow in BMGs is achieved by cooperative movement of a group of neighboring atoms. This movement is associated with the creation, annihilation and diffusion of the free volume, which is defined as the atomic volume in excess of the ideal densely packed, but still disordered, structure [157]. Generally speaking, the free volume is the region that has lower atomic density. Atoms adjacent to a free volume with comparable size are suggested to jump into this region with diffusion biased by the applied stress. Subsequently, neighboring atoms will rearrange to accommodate this change. This process results in a redistribution of free volume and a plastic deformation.

There are experimental techniques that can characterize the free volume on macroscopic samples, such as differential scanning calorimetry (DSC) [161, 162] and positron annihilation spectroscopy (PAS) [163-165]. In DSC, the average free volume is characterized by measuring the specific heat according to the glass transition model by van den Buekel and Sietsma [161]. In PAS, positrons are injected into the material and subsequently annihilate with the material’s electrons. The energy of this e+-e- pair is emitted as quanta of the electromagnetic field (photons). Therefore, the life time of the positrons can be detected. From this, the electron density at the annihilation site is calculated. As a result, the distribution of open volume, which consists of the areas of low electron density, is characterized. A recent PAS study [164] suggests that there are three types of open volumes with different sizes, which may correspond to inherent interstitial holes (Bernal holes), flow defects and nanoscale voids. The evolution of these open volume regions during inhomogeneous deformation has been studied as well. It was

119 found that the concentration of flow defects increases at the expense of the interstitial holes [164].

Despite these successes in determining free volume distributions, information obtained from both DSC and PAS are the averaged value over the macroscopic sample.

Therefore, they do not provide information about the spatial distribution of free volume.

To overcome this insufficiency of the experimental techniques, we employ in this study atomistic modeling to investigate the open volume distribution in Cu46Zr54 BMG.

5.1 Embedded Atom Method (EAM)

For crystalline materials, a unit cell with a few atoms is sufficient to represent the structure. However, BMGs are amorphous. As a result, it takes much bigger systems and many more atoms to correctly model their structure. In this study, at least 1000 atoms are included in the simulation cell. This makes direct DFT simulations infeasible. As a result, a more economical approach, the embedded atom method (EAM), is adopted in generating BMGs structures. EAM was first proposed by Daw and Baskes [166], who suggested that the total energy of an atom consists of embedding energy and pairwise potentials:

1 Etot = Fi ()ρi + ∑φ(rij ) (5.1) 2 i≠ j

where φ(rij ) is the pair energy of atoms i and j at a distance of rij, and Fi ()ρi is the energy

to embed an atom into a position with electron density ρi . This local electron density is calculated as

ρi = ∑ f (rij ) (5.2) i≠ j

120 where f (rij ) is the electron density contributed from atom j at a distance of rij from atom i.

EAM was further developed and improved by a number of contributors [167-172]. In this study, we adopt the generalized alloy model by Johnson et al. [172-174], in which the pair potential for a specific element is [173]:

⎡ ⎛ r ⎞⎤ ⎡ ⎛ r ⎞⎤ Aexp⎢−α⎜ −1⎟⎥ B exp⎢− β⎜ −1⎟⎥ ⎣ ⎝ re ⎠⎦ ⎣ ⎝ re ⎠⎦ φ()r = 20 − 20 (5.3) ⎛ r ⎞ ⎛ r ⎞ 1+ ⎜ − κ ⎟ 1+ ⎜ − λ ⎟ ⎝ re ⎠ ⎝ re ⎠ where re is the equilibrium distance between nearest neighbors, A, B, α and β are fitting parameters, and κ and λ are two other fitting parameters related to the cut off radius where the interaction energy goes to zero [173]. The first term in the pair potential is repulsive, while the second term is attractive.

The electron density function in Johnson’s model has the same form as the attractive pair potential [173]:

⎡ ⎛ r ⎞⎤ ⎜ ⎟ f e exp⎢− β⎜ −1⎟⎥ ⎣ ⎝ re ⎠⎦ f ()r = 20 (5.4) ⎛ r ⎞ 1+ ⎜ − λ ⎟ ⎝ re ⎠ where fe is a fitting parameters.

Johnson et al. define the cross potential between two different elements a and b as

[173]:

1 ⎛ f b (r) f a (r) ⎞ φ ab r = ⎜ φ aa r + φ bb r ⎟ . (5.5) () ⎜ a () b ()⎟ 2 ⎝ f ()r f ()r ⎠

121 Finally, the embedding energy F is calculated as [173]:

⎧ i 3 ⎛ ρ ⎞ ⎪ ⎜ ⎟ ∑ Fni ⎜ −1⎟ ρ < 0.85ρ e ⎪ i=0 ⎝ 0.85ρ e ⎠ ⎪ i ⎪ 3 ⎛ ρ ⎞ ⎜ ⎟ F()ρ = ⎨∑ Fi ⎜ −1⎟ 0.85ρe ≤ ρ < 1.15ρ e (5.6) ⎪ i=0 ⎝ ρe ⎠ η η ⎪ ⎡ ⎛ ρ ⎞ ⎤⎛ ρ ⎞ ⎪F ⎢1− ln⎜ ⎟ ⎥⎜ ⎟ 1.15ρ ≤ ρ ⎪ e ⎜ ρ ⎟ ⎜ ρ ⎟ e ⎩ ⎣⎢ ⎝ e ⎠ ⎦⎥⎝ e ⎠ where Fni, Fi, Fe, ρe and η are fitting parameters.

Cu Zr re 2.556162 3.199978 fe 1.554485 2.230909 ρe 22.15014 30.87999 α 7.669911 8.55919 β 4.090619 4.564902 A 0.327584 0.424667 B 0.468735 0.640054 κ 0.431307 0.5 λ 0.86214 1 Fn0 -2.17649 -4.48579 Fn1 -0.14004 -0.29313 Fn2 0.285621 0.990148 Fn3 -1.75083 -3.20252 F0 -2.19 -4.51 F1 0 0 F2 0.702991 0.928602 F3 0.683705 -0.98187 η 0.92115 0.597133 Fe -2.19168 -4.50903

Table 5.1: List of EAM parameters used in the present work for Cu and Zr from Ref. [173].

122 In the present work, we use the parameterization of Ref. [173], where all the parameters in the above mentioned formula for elemental Cu and Zr have been determined by fitting basic material properties such as lattice constants, Young’s modulus and bulk modulus. The resulting parameters are listed in Table 5.1 [173].

All Cu46Zr54 BMG structures in this study were generated by standard melt-quench simulations in analogy to real glass forming procedures. We started with a CuZr B2 crystalline structure with 500 Cu and 500 Zr atoms and randomly replaced 40 Cu atoms with Zr atoms to obtain the correct stoichiometry (46:54). Then, we ran an MD simulation at a temperature of 3000K (above the melting temperature) for 100 ps (time step 1 fs), followed by quenching to 0K with a cooling rate of 5×1011 K / s . The resulting structure was a Cu46Zr54 BMG with 1000 atoms and (optimized) dimensions of 33.31 Å ×

33.31 Å × 16.65 Å. This glass structure was then analyzed with the help of the radial distribution function (RDF), which is one of the most useful methods to reveal structural features, especially for noncrystalline materials such as glass. The RDF describes how the number of atoms varies as a function of distance from a given point and is defined as

[175]:

V n(r) g()r = (5.7) N 2 4πr 2 Δr where V is the volume of materials, N is the number of all atoms, r is the distance from a given point and n(r) is the number of atoms found within the spherical shell bounded by r and r+Δr. For binary system such as Cu46Zr54, partial radial distribution functions (PRDF) can be calculated in a similar way:

123 V nAB (r) g AB ()r = 2 (5.8) N A N B 4πr Δr where NA and NB are number of type A atoms and type B atoms respectively, and nAB(r) is the number of B atoms found within the spherical shell bounded by r and r+Δr centered in an A atom. The PRDFs of our simulated 1000-atom BMGs structures at different temperatures are shown in Fig. 5.1. Note that the baseline for the PRDFs for

2000K and 1000K are shifted vertically in the plot for better visibility.

124

Figure 5.1: Partial radial distribution function (PRDF) of simulated Cu46Zr54 structure with 1000 atoms at 2000K, 1000K and 300K.

125 At 2000K (liquid state), all PRDFs are very smooth and smeared out because of the lack of ordering. As the temperature decreases (but still is above the glass transition point), the first (nearest-neighbor) peaks sharpen and second peaks become more pronounced. When the structure is quenched to room temperature, the first peaks are prominent and the second peaks split, which is consistent with experimental data and early simulation results as shown in Table 5.2 [175-177].

Cu-Cu Zr-Zr Cu-Zr XRD 2.54 2.72 3.14 EXAFS 2.53 2.75 3.15 Early Simulation 2.67 2.78 3.22 Current work 2.65 2.86 3.15

Table 5.2: First peak positions of the PRDF of XRD [176] and EXAFS [177] for Cu46Zr54 BMGs in comparison to the results of our simulations and earlier simulations [175, 177].

5.2 Hard Sphere Model

If Cu and Zr atoms are considered to be hard spheres with predefined radii, then one may fit other (interstitial) spheres into the empty spaces between the atoms. A 2D example is illustrated in Fig. 5.2.

126

Figure 5.2: Tangent spheres define the biggest interstitial sphere (center) that can be inserted into the empty spaces between atoms.

These interstitial spheres have been suggested as a representation of the open volume regions in BMG structures in several publications [178-181]. Given the coordinates of

1000 atoms, one feasible approach to find all the interstitial spheres is to partition the space into tetrahedra whose corners lie on the positions (centers in the case of hard- sphere atoms) of the atoms and then, for each tetrahedron, insert a sphere which is tangent to the four atom spheres of this tetrahedron. This method of partitioning space is known as Delaunay tessellation in mathematics and computational geometry (or

Delaunay triangulation in 2D planes). The intrinsic function delaunay3() in MATLAB can perform a Delaunay tessellation and returns the coordinates of the corner points for all tetrahedra. Based on these tetrahedra, interstitial spheres can be obtained by simple geometry calculations as illustrated in Fig. 5.2.

We chose as atomic radii 1.27 Å and 1.58 Å for Cu and Zr respectively, following the suggestion of Ref [182]. The process of Delaunay tessellation and interstitial sphere calculations was carried out for our 1000-atom BMG structure. Surprisingly, some interstitial spheres overlap each other or overlap with atoms. This is due to the fact that

Cu and Zr atoms have different radii while the Delaunay construction assumes that all

127 points are equivalent. Nonetheless, this has previously been the best construction available in the literature for analyzing interstitial space in glasses. To fix the double counting from overlapping spheres, we developed a numerical method to determine the amount of overlapping volume (an analytical method was also coded, but was found to be much slower than the numerical method). In the numerical approach, a mesh with 0.03 Å spatial resolutions is imposed on the BMG structure. Then an algorithm is performed to identify the mesh points that belong to more than one sphere. This way, we found that around 33% of the total volume of interstitial spheres of our structure is overlapping, and the resulting error was corrected. Finally, we identified the interstitial spheres that were intersecting or overlapping, and grouped them into clusters, because they might be thought of as a connected piece of open volume. A histogram describing the volume distribution of these clusters for the CuZr BMG structure, corrected for overlap, is shown in Fig. 5.3 with an overlaid fitted exponential decay function. This exponential decay relationship was suggested for the free volume in BMGs by Cohen and Turnbull [183]. It seems that the distribution of the clusters with volumes greater than 0.4 Å3 agrees well with this function. A cross-section of the simulation cell showing atoms and interstitial spheres is shown in Fig. 5.4. Note that the hard sphere analysis was performed in a

29.5 Å × 29.5 Å × 13.5 Å box, which is smaller than the simulation cell.

128

Figure 5.3: Volume distribution of the open volume clusters identified in the hard sphere model. Cluster volumes have been corrected for overlapping spheres. Note that for cluster volumes greater than 0.4 Å3, the distribution is well described by the overlaid exponential decay function suggested by Cohen and Turnbull [183].

129

Figure 5.4: Cross section perpendicular to the z direction of a 1000-atom Cu46Zr54 BMG simulation cell analyzed by the hard sphere model. The open volume calculation was performed in a 29.5 Å × 29.5 Å × 13.5 Å box.

5.3 Electron Density Model

There are several drawbacks of the hard sphere model. The first and foremost one is that the model needs predefined atomic radii, whose exact values are very difficult to determine because they should depend on the local chemical environment, i.e., bonding, in principle. However, the results of the hard sphere model are very sensitive to the values of atomic radii. Given a different set of values, results will be completely different.

For example, if another set of atomic radii value (RCu = 1.45 Å and RZr = 2.06 Å) from literature [184] is applied to analyzed our structure, one will get a negative value for the unoccupied volume [185] because the total atomic volume exceeds the volume of the simulation cell. The second drawback is that the hard sphere model assumes that all open

130 volumes are spherical. Although we are able to identify the intersecting or overlapping interstitials and group those into clusters, which fixes this issue to a certain degree, this restriction imposed on the shape of open volume is artificial and is not very well justified.

As can be seen from Fig. 5.4, there are unoccupied regions that are not belonging to atoms or interstitials. Part of this unoccupied region should contribute to the open volume.

The third drawback has been already mentioned above: due to the different atomic radii, the Delaunay procedure results in a counter intuitive situation that some calculated interstitial spheres overlap with atoms as can be seen in Fig. 5.4. This situation does not make physical sense because the open volume, represented by interstitial spheres in this model, should correspond to the region with low atomic density. To overcome those drawbacks, we propose a new approach to analyze the open volumes in BMG structures based on the electron density. In reality, atoms are not spatially confined into a sphere with an abrupt surface as described by the hard sphere model, but their electron density decays gradually when moving away from their nuclei. Therefore, the electron density might be a good tool to analyze the open volume in BMGs. Moreover, the electron density is also a physically measurable quantity by techniques such as PAS. This will allow to compare the result of this new model with experiments in the future.

The electron density of any structure can be obtained by self-consistent DFT calculations. However, this method is computationally very expensive for a system with

1000 atoms and becomes formidable for even bigger systems, which are not uncommon for BMG simulations. For example, a DFT calculation of the electron density our 1000- atom Cu46Zr54 BMG structure took more than 3 days to finish on a 32-P4-processor parallel machine. This was performed with the help of the software package VASP.

131 Moreover, the electron density was given on a rather coarse mesh (162×162×80) by

VASP. An alternative approach proposed in this study is to use radially averaged atomic electron densities, which consists of 3 steps.

1. The electron densities of a single Cu atom and a single Zr atom are calculated

separately with DFT. For this, we used VASP with ultra-soft pseudo-

potentials within the generalized gradient approximation.

2. For both Cu and Zr, the radially averaged electron density n(r) is calculated

by

∑ n()xi , yi , zi ΔxΔyΔz n()r = i , r ≤ x 2 + y 2 + z 2 < r + dr (5.9) 4πr 2 dr i i i

3. Superimpose the radially averaged electron densities from each atom in the

BMG structure by

N ⎛ 2 2 2 ⎞ nSuper ()x, y, z = ∑ n⎜ (x − xk )()()+ y − yk + z − zk ⎟ (5.10) k =1 ⎝ ⎠

where N is the total number of atoms, xk, yk, and zk the coordinates of atom k.

The purpose of this work is to study the open volume, which corresponds to low electron density regions. Therefore, the radially averaged electron densities for both Cu and Zr are truncated at 0.286 e/Å3 (which is a high density that only appears in the direct neighborhood of atoms) to allow a better visualization of the details of the low density regions. This specific value is selected for truncation is because it is the value at r ~ 1 Å for both types of atom. Regions for r < 1 Å are very close to an atom and thus of no interest for our purpose. The radial averaged densities for Cu and Zr are shown in Fig. 5.5.

132 This simplified approach enables us to calculate the electron density of a BMG structure much faster and on a much finer mesh. Moreover, this approach is applicable to much larger system because its run time only scales linearly with the system size. The electron density of a crystalline CuZr B2 structure was calculated by this simplified approach and by self-consistent DFT independently and results are compared for the purpose of validation. The contour plot of the electron density for a (110) plane is shown in Fig. 5.6, where high density regions are plotted red and low density regions are plotted blue.

Figure 5.5: Radially averaged electron density for a single Cu atom and a single Zr atom truncated at 0.286 e/Å3.

133

Figure 5.6: Contour plot of the electron density distribution in crystalline CuZr B2 structure for a (110) crystallographic plane from (a) radial averaged electron densities and (b) a self-consistent DFT calculation of the charge density. Red corresponds to regions of high electron density (e.g. atom cores), while blue corresponds to regions of low electron density. Periodic boundary conditions are applied to both calculations.

Both plots in Fig. 5.6 show very similar features, where Cu atoms are bigger and have broader tails as the electron density decays gradually with distance from the nuclei.

Contours of low density regions agree with each other reasonably well, showing similar shapes and identifying the same regions of lowest density. After this validation, this approach was applied in calculating the electron density of several 1000-atom Cu46Zr54

BMG structures. A cross-section plot is shown in Fig. 5.7 with the same color coding as

134 in Fig. 5.6(a), which shows the smooth transition in contrast to the abrupt change seen in the hard sphere model (Fig. 5.4).

Figure 5.7: Contour plot of the electron density of a cross-section of a 1000-atom Cu46Zr54 BMG structure. The electron density was calculated by our simplified electron density model. Red corresponds to regions of high electron density, blue to regions of low electron density.

Based on the result from this simplified approach, regions with electron densities less than 0.11 e/Å3 were identified as open volume. This cutoff value is determined by the fact that the lowest electron density in a B2 CuZr crystal structure is 0.11 e/Å3 identified by our approach. This choice of cutoff value implies that there is no open volume in the crystalline structure. To ensure better statistics, a total of ten Cu46Zr54 BMG structures was generated, each containing 1000 atoms. Open volume analysis was performed for all

135 of them using the simplified electron density model. The results are summarized in the histogram in Fig. 5.8.

Furthermore, a cross-section of one of the structures is shown in Fig. 5.9. The open volume zones are plotted as black regions, and the results from the radially-averaged electron density model are compared to those from a self-consistent DFT calculation.

Again, the comparison shows that the radially averaged electron density method captures all major features of the open volume distribution, while being at the same time highly efficient.

The histograms shown in Fig. 5.3 (hard sphere model) and Fig. 5.8 (electron density model) share similar features. Both may be fitted to an exponential decay function, which is the distribution of free volume suggested in Ref. [183]. However, the decay constant is smaller for the open volumes identified by the electron density model. This is because open volumes can assume any shape in this model while they are restricted to be spherical in the hard sphere model. Therefore, a higher amount of larger-volume regions is found. Although adjacent interstitials in the hard sphere model are combined to form clusters, there is still around 25% of the volume of the simulation cell not accounted for by atoms or interstitials. Thus, the identified size distribution of open volume from the electron density model should be much more realistic than that derived from the hard- sphere model, which captures the basic topology of the volume distribution as well, but imposes restriction on the shape and ignores the unaccounted volume. Thus, our proposed electron density based model represents a significant step toward a more realistic quantification of open volume in BMGs.

136

Figure 5.8: Size distribution of the open volume regions identified by the electron density model, calculated from ten 1000-atom Cu46Zr54 BMG structures.

(a) (b)

Figure 5.9: Cross-section of a Cu46Zr54 BMG structure showing the shape of open volume as black from (a) the radially averaged electron density model and (b) a self-consistent DFT calculations.

137 5.4 Conclusions

Cu46Zr54 BMG structures with 1000 atoms were generated by a melt-quench procedure with classical MD simulations using EAM potentials. PRDFs showed that the simulated structures are sensible as compared to experimental data. We first applied the traditional hard sphere model, with the improvement of correction for overlapping, to analyze the open volume distribution of those BMG structures. It was found that the distribution agrees with the exponential free volume distribution suggested in the literature. However, we also found that the shape restriction in this model limits its capability of better capturing features of free volume regions. Furthermore, we found it very hard to identify the most sensible atomic radii. To overcome these limitations, we proposed an electron density model, in which open volumes are identified as low electron density regions. To make the model fast, we propose to calculate the electron density of the whole structure by superimposing single-atom densities obtained from self-consistent

DFT calculations. We have shown that this approach ensures sufficient accuracy while avoiding the demanding computation efforts required by direct DFT calculations on the whole structure. Defining the cut-off electron density for open volume as the minimum density found in crystalline CuZr, we performed an open volume analysis by the electron density model, which allows a significantly more physical quantification of free volume in BMGs. Also the electron density model allows connections between simulations and experimental techniques such as PAS.

This analysis will be used in conjunction with MD simulations of deformation to investigate the role of free volume in homogeneous and inhomogeneous flow and its evolution during deformation. 138

CHAPTER 6

SUMMARY OF CONCLUSIONS

Materials modeling within the framework of density functional theory (DFT) has demonstrated its capability and accuracy in a plethora of past studies. At the same time, it is computationally very demanding. As a result, DFT is often restricted to the study of small systems and is mostly utilized for static problems or problems involving very small time scales. In this thesis, examples have been shown where the limits of common DFT applications were tested. Different methods have been examined and developed to investigate the temperature and pressure dependent evolution and properties of large systems, ranging from electronic to mechanical applications.

In the carbyne study, potential structures for solid carbynes have been identified and their free energies were calculated by DFT within the quasiharmonic approximation as a function of temperature (0 to melting) and pressure (0-180kBar). More specifically, a unified structural model has been proposed for the first time which combines elements from previous studies and is in agreement with all experimental findings. As one of the important results, we find that for a subset of our structures, carbon chains are forced into curved shapes due to the sp2 and sp3 hybridizations of kink atoms (which we propose to

139 link linear carbon chains into a three-dimensional crystal structure), resulting in a reduction in the crystal symmetry from hexagonal to trigonal, which explains the fact that both symmetries have been reported experimentally. Our free energy calculations show that all carbyne structures have higher energy than graphite in the whole temperature and pressure range. The energy difference is smaller at higher temperatures due to the presence of phonon DOS peaks in carbyne at frequencies significantly higher (> 2000 cm-1) than the highest frequencies present in graphite (~1500 cm-1). This shows that carbyne without any impurity is at most a meta-stable phase. As a first step to include additional elements, we have confirmed that CH3 end groups indeed can stabilize the carbyne chains into a solid structure as previously suggested in the literature. The inclusion of impurities, such as metal ions, may be the key for a stable carbyne phases which have been suggested to be found at elevated temperatures (where the hydrogen- stabilized carbyne structure is not stable anymore) and moderately high pressures. This may also explain the fact that the simulated lattice constants (without impurity) are always slightly smaller than the experimental value (outside of the typical error of DFT values).

In the study of contact formation of CNT devices, process modeling was performed by DFT with temperature accelerated dynamics (TAD) to overcome the time scale problem. We show that finite-temperature process modeling is necessary to successfully predict the contact structure in realistic CNT devices. By this approach, we found that the

CNT went through a significant structural change on top of Ti which resulted in a dramatic decrease in conductance. This phenomenon has not been observed in previous

DFT studies, in which only pure relaxations were performed. It was also found that larger

140 CNTs are more stable on top of Ti because they have less elastic energy. The effect of different electrode materials has been investigated as well. Results show that the charge transfer from electrodes to CNTs assists in the disintegration of CNTs. Future study should include the effect of surface contaminations, which is expected to influence the charge transfer, thus stabilizing CNTs.

DFT has also been applied in the study of the open volume distribution of bulk metallic glasses (BMG), where classical molecular dynamics (MD) were performed to generate Cu46Zr54 BMG structures before the DFT-based calculation of the electron density. For this, we suggested and verified a simplified approach, which can be applied to very large systems orders of magnitude beyond the DFT-limit of ~1000 atoms. In this approach, the total electron density of the system is constructed by superimposing single- atom electron densities calculated by DFT. This makes the information of electron density available in an affordable computation time with reasonable accuracy, as has been benchmarked against self-consistent DFT calculations. Based on this information, a new model has been proposed to capture the free volume distribution in BMGs, along with the traditional hard sphere model. The analysis by this electron density model allows a significantly more physical quantification of free volume and allows the connections with experimental techniques such as PAS. Future work should use this analysis in conjunction with MD simulations of deformation to investigate the role of free volume in homogeneous and inhomogeneous flow and its evolution during deformation.

141

BIBLIOGRAPHY

[1] Bristowe PD, Hasnip PJ. General Aspects of Materials Modelling. In: Barber ZH, editor. Introduction to Materials Modelling. London: Maney Publishing, 2005.

[2] Barber Z. Introduction to Materials Modelling. London: Maney Publishing, 2005.

[3] Raabe D. Computational Materials Science: The simulation of Materials Microstructures and Properties. Weinheim: Wiley-VCH, 1998.

[4] Ohno K, Esfarjani K, Kawzaoe Y. Computational Materials Science, From Ab Initio to Monte Carlo Methods: Springer, 1999.

[5] Hohenberg P, Kohn W. Inhomogeneous Electron Gas. Physical Review 1964;136:B864.

[6] Kohn W, Sham LJ. Self-Consistent Equations Including Exchange and Correlation Effects. Physical Review 1965;140:1133.

[7] Koch W, Holthausen MC. A chemist's guide to density functional theory. Weinheim ; New York: Wiley-VCH, 2001.

[8] Fiolhais C, Nogueira F, Marques M. A primer in density functional theory. Berlin ; New York: Springer, 2003.

[9] Parr RG, Yang W. Density-functional theory of atoms and molecules. New York; Oxford [England]: Oxford University Press ; Clarendon Press, 1989.

[10] Shankar R. Principles of quantum mechanics. New York: Plenum Press, 1994.

[11] Liang Q. Fundamentals of Density Functional Theory. 2005.

[12] Atomic units. http://en.wikipedia.org/wiki/Atomic_units.

[13] Bohr NHD. On the constitution of atoms and molecules; papers of 1913 reprinted from the Philosophical magazine. Copenhagen, New York, W. A. Benjamin: Munksgaard, 1963.

142 [14] Gelfand IM, Fomin SV. Calculus of variations. Englewood Cliffs, N.J.,: Prentice- Hall, 1963.

[15] Hartree DR. Proceedings of the Cambridge Philosophical Society 1928;24:89.

[16] Hartree DR. Proceedings of the Cambridge Philosophical Society 1928;24:111.

[17] Hartree DR. Proceedings of the Cambridge Philosophical Society 1928;24:426.

[18] Lowdin PO. The Historical Development of the Electron Correlation-Problem. International Journal of Quantum Chemistry 1995;55:77.

[19] March NH. Electron density theory of atoms and molecules. London: Academic, 1992.

[20] Hohenberg P, Kohn W. Inhomogeneous Electron Gas. Physical Review 1964;136:B874.

[21] Levy M. Universal Variational Functionals of Electron-Densities, 1st-Order Density-Matrices, and Natural Spin-Orbitals and Solution of the V-Representability Problem. Proceedings of the National Academy of Sciences of the United States of America 1979;76:6062.

[22] Perdew JP, Wang Y. Accurate and Simple Analytic Representation of the Electron-Gas Correlation-Energy. Physical Review B 1992;45:13244.

[23] Jones RO, Gunnarsson O. The Density Functional Formalism, Its Applications and Prospects. Reviews of Modern Physics 1989;61:689.

[24] Becke AD. Density-Functional Exchange-Energy Approximation with Correct Asymptotic-Behavior. Physical Review A 1988;38:3098.

[25] Langreth DC, Mehl MJ. Beyond the Local-Density Approximation in Calculations of Ground-State Electronic-Properties. Physical Review B 1983;28:1809.

[26] Lee CT, Yang WT, Parr RG. Development of the Colle-Salvetti Correlation- Energy Formula into a Functional of the Electron-Density. Physical Review B 1988;37:785.

[27] Perdew JP. Correction. Physical Review B 1986;34:7406.

[28] Perdew JP. Density-Functional Approximation for the Correlation-Energy of the Inhomogeneous Electron-Gas. Physical Review B 1986;33:8822.

143 [29] Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C. Atoms, Molecules, Solids, and Surfaces - Applications of the Generalized Gradient Approximation for Exchange and Correlation. Physical Review B 1992;46:6671.

[30] Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C. Atoms, Molecules, Solids, and Surfaces - Applications of the Generalized Gradient Approximation for Exchange and Correlation (Vol 46, Pg 6671, 1992). Physical Review B 1993;48:4978.

[31] Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Physical Review Letters 1996;77:3865.

[32] Lenosky TJ, Kress JD, Kwon I, Voter AF, Edwards B, Richards DF, Yang S, Adams JB. Highly optimized tight-binding model of silicon. Physical Review B 1997;55:1528.

[33] Rose JH, Smith JR, Guinea F, Ferrante J. Universal Features of the Equation of State of Metals. Physical Review B 1984;29:2963.

[34] Fermi E. Sopra lo spostamento per pressione delle righe elevate delle serie spettrali. Nuovo Cimento 1934;11:157.

[35] Hellmann H. A New Approximation Method in the Problem of Many Electrons. Journal of Chemical Physics 1935;3:61.

[36] Kleinman L, Phillips JC. Crystal Potential and Energy Bands of Semiconductors. III. Self-Consistent Calculations for Silicon. Physical Review 1960;118:1153.

[37] Phillips JC, Kleinman L. New Method for Calculating Wave Functions in Crystals and Molecules. Physical Review 1959;116:287.

[38] Troullier N, Martins JL. Efficient Pseudopotentials for Plane-Wave Calculations. Physical Review B 1991;43:1993.

[39] Troullier N, Martins JL. A Straightforward Method for Generating Soft Transferable Pseudopotentials. Solid State Communications 1990;74:613.

[40] Hamann DR. Generalized Norm-Conserving Pseudopotentials. Physical Review B 1989;40:2980.

[41] Bachelet GB, Hamann DR, Schlüter M. Pseudopotentials That Work - from H to Pu. Physical Review B 1982;26:4199.

[42] Hamann DR, Schlüter M, Chiang C. Norm-Conserving Pseudopotentials. Physical Review Letters 1979;43:1494.

144 [43] Heimann RB, Evsyukov SE, Kavan L. Carbyne and carbynoid structures. Dordrecht ; Boston: Kluwer Academic, 1999.

[44] Heath JR, Zhang Q, Obrien SC, Curl RF, Kroto HW, Smalley RE. The Formation of Long Carbon Chain Molecules During Laser Vaporization of Graphite. Journal of the American Chemical Society 1987;109:359.

[45] Lagow RJ, Kampa JJ, Wei HC, Battle SL, Genge JW, Laude DA, Harper CJ, Bau R, Stevens RC, Haw JF, Munson E. Synthesis of Linear Acetylenic Carbon - the Sp Carbon Allotrope. Science 1995;267:362.

[46] Goresy AE, Donnay G. A New Allotropic Form of Carbon from the Ries Crater. Science 1968;161:363.

[47] Kasatochkin VI, Korshak VV, Kudryavtsev YP, Sladkov AM, Sterenberg IE. Crystalline-Structure of Carbyne. Carbon 1973;11:70.

[48] Nakamizo M, Kammerec.R, Walker PL. Laser Raman Studies on Carbons. Carbon 1974;12:259.

[49] Whittaker AG. Carbon - New View of Its High-Temperature Behavior. Science 1978;200:763.

[50] Heimann RB, Kleiman J, Salansky NM. A Unified Structural Approach to Linear Carbon Polytypes. Nature 1983;306:164.

[51] Rice MJ, Phillpot SR, Bishop AR, Campbell DK. Solitons, Polarons, and Phonons in the Infinite Polyyne Chain. Physical Review B 1986;34:4139.

[52] Whittaker AG, Kintner PL. Carbon - Observations of New Allotropic Form. Science 1969;165:589.

[53] Smith PPK, Buseck PR. Carbyne Forms of Carbon - Do They Exist. Science 1982;216:984.

[54] Whittaker AG. Carbyne Forms of Carbon - Evidence for Their Existence. Science 1985;229:485.

[55] Blank VD, Kulnitskiy BA, Tatyanin YV, Zhigalina OM. A new phase of carbon. Carbon 1999;37:549.

[56] Fitzgerald AG. Structural models of carbyne - electron diffraction and microscopy. In: Heimann RB, Evsyukov SE, Kavan L, editors. Carbyne and carbynoid structures. Dordrecht ; Boston: Kluwer Academic, 1999.

145 [57] Rietmeijer FJM. Are Crystalline C-(H-O-N) Carbons the Elusive Meteoritic Carbynes. Meteoritics 1993;28:242.

[58] Reckeweg O, Simon A. X-ray and Raman investigations on cyanides of mono- and divalent metals and synthesis, crystal structure and Raman spectrum of Tl- 5(CO3)(2)(CN). Zeitschrift Fur Naturforschung Section B-a Journal of Chemical Sciences 2002;57:895.

[59] Grumbach MP, Martin RM. Phase diagram of carbon at high pressures and temperatures. Physical Review B 1996;54:15730.

[60] McKee DW. Carbon and Graphite Science. Annual Review of Materials Science 1973;3:195.

[61] Ravagnan L, Siviero F, Lenardi C, Piseri P, Barborini E, Milani P, Casari CS, Li Bassi A, Bottani CE. Cluster-beam deposition and in situ characterization of carbyne-rich carbon films. Physical Review Letters 2002;89.

[62] Lamperti A, Ossi PM. Energetic condition for carbyne formation. Chemical Physics Letters 2003;376:662.

[63] Kastner J, Kuzmany H, Kavan L, Dousek FP, Kurti J. Reductive Preparation of Carbyne with High-Yield - an in-Situ Raman-Scattering Study. Macromolecules 1995;28:344.

[64] Kavan L. Electrochemical carbon. Chemical Reviews 1997;97:3061.

[65] Kavan L, Hlavaty J, Kastner J, Kuzmany H. Electrochemical Carbyne from Perfluorinated Hydrocarbons - Synthesis and Stability Studied by Raman-Scattering. Carbon 1995;33:1321.

[66] Kudryavtsev YP, Evsyukov S, Guseva M, Babaev V, Khvostov V. Carbyne - A Linear Chainlike Carbon Allotrope. In: Thrower PA, editor. Chemistry and Physics of Carbon. New York: Dekker, 1997.

[67] Whittaker AG, Wolten GM. Carbon - Suggested New Hexagonal Crystal Form. Science 1972;178:54.

[68] Barkema GT, Mousseau N. High-quality continuous random networks. Physical Review B 2000;62:4985.

[69] Ooi N, Rairkar A, Adams JB. Density functional study of graphite bulk and surface properties. Carbon 2006;44:231.

146 [70] Pavone P, Karch K, Schütt O, Windl W, Strauch D, Giannozzi P, Baroni S. Ab- Initio Lattice-Dynamics of Diamond. Physical Review B 1993;48:3156.

[71] Windl W, Pavone P, Karch K, Schütt O, Strauch D, Giannozzi P, Baroni S. 2nd- Order Raman-Spectra of Diamond from Ab-Initio Phonon Calculations. Physical Review B 1993;48:3164.

[72] Pavone P, Bauer R, Karch K, Schütt O, Vent S, Windl W, Strauch D, Baroni S, deGironcoli S. Ab initio phonon calculations in solids. Physica B 1996;220:439.

[73] Boettger JC. All-electron full-potential calculation of the electronic band structure, elastic constants, and equation of state for graphite. Physical Review B 1997;55:11202.

[74] Mounet N, Marzari N. First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives. Physical Review B 2005;71.

[75] Schabel MC, Martins JL. Energetics of Interplanar Binding in Graphite. Physical Review B 1992;46:7185.

[76] Wang YC, Scheerschmidt K, Gösele U. Theoretical investigations of bond properties in graphite and graphitic silicon. Physical Review B 2000;61:12864.

[77] Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B 1996;54:11169.

[78] Kresse G, Furthmüller J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational Materials Science 1996;6:15.

[79] Kresse G, Hafner J. Abinitio Molecular-Dynamics for Liquid-Metals. Physical Review B 1993;47:558.

[80] Kresse G, Hafner J. Ab-Initio Molecular-Dynamics Simulation of the Liquid- Metal Amorphous-Semiconductor Transition in Germanium. Physical Review B 1994;49:14251.

[81] Kresse G, Hafner J. Norm-Conserving and Ultrasoft Pseudopotentials for First- Row and Transition-Elements. Journal of Physics-Condensed Matter 1994;6:8245.

[82] Blöchl PE. Projector Augmented-Wave Method. Physical Review B 1994;50:17953.

[83] Kresse G, Joubert D. From ultrasoft pseudopotentials to the projector augmented- wave method. Physical Review B 1999;59:1758.

147 [84] Monkhorst HJ, Pack JD. Special Points for Brillouin-Zone Integrations. Physical Review B 1976;13:5188.

[85] Evsyukov SE. Pyrolysis of Organic Polymers. In: Heimann RB, Evsyukov SE, Kavan L, editors. Carbyne and carbynoid structures. Dordrecht ; Boston: Kluwer Academic, 1999.

[86] Kugeev FF, Baitinger EM, Teterin YA, Gagarin SG. Khimiya tverdogo topliva 1991;3:120.

[87] Pesin LA, Baitinger EM, Kuznetsov VL, Sokolov OB. ESCA Core Electron- Spectra of Glassy-Carbon. Fizika Tverdogo Tela 1993;35:2262.

[88] Jepps NW, Page TF. 9r-Hrem Observations of a New Silicon-Carbide Polytype. Journal of the American Ceramic Society 1980;63:102.

[89] Baroni S, de Gironcoli S, Dal Corso A, Giannozzi P. Phonons and related crystal properties from density-functional perturbation theory. Reviews of Modern Physics 2001;73:515.

[90] Baroni S, De Gironcoli S, Dal Corso A, Giannozzi P, Cavazzoni C, Ballabio G, Scandolo S, Chiarotti G, Focher P, Pasquarello A, Laasonen K, Trave A, Car R, Marzari N, Kokalj A. Plane-wave self-consistent field (pwscf) computer code, http://www.pwscf.org/.

[91] Bundy FP, Bassett WA, Weathers MS, Hemley RJ, Mao HK, Goncharov AF. The pressure-temperature phase and transformation diagram for carbon; Updated through 1994. Carbon 1996;34:141.

[92] Wang XF, Scandolo S, Car R. Carbon phase diagram from ab initio molecular dynamics. Physical Review Letters 2005;95:185701.

[93] Zhang QL, Obrien SC, Heath JR, Liu Y, Curl RF, Kroto HW, Smalley RE. Reactivity of Large Carbon Clusters - Spheroidal Carbon Shells and Their Possible Relevance to the Formation and Morphology of Soot. Journal of Physical Chemistry 1986;90:525.

[94] Udod IA. Carbyne intercalation compounds. In: Heimann RB, Evsyukov SE, Kavan L, editors. Carbyne and carbynoid structures. Dordrecht ; Boston: Kluwer Academic, 1999. p.269.

[95] Guseva MB, Babaev VG, Babina VM, Khvostov VV. Formation of oriented nitrogen-doped carbon films. JETP Letters 1995;62:715.

148 [96] Kudryavtsev YP, Heimann RB, Evsyukov SE. Carbynes: Advances in the field of linear carbon chain compounds. Journal of Materials Science 1996;31:5557.

[97] Tans SJ, Verschueren ARM, Dekker C. Room-temperature transistor based on a single carbon nanotube. Nature 1998;393:49.

[98] Wind SJ, Radosavljevic M, Appenzeller J, Avouris P. Transistor structures for the study of scaling in carbon nanotubes. Journal of Vacuum Science & Technology B 2003;21:2856.

[99] Javey A, Guo J, Wang Q, Lundstrom M, Dai HJ. Ballistic carbon nanotube field- effect transistors. Nature 2003;424:654.

[100] Anantram MP, Leonard F. Physics of carbon nanotube electronic devices. Reports on Progress in Physics 2006;69:507.

[101] Leonard F, Talin AA. Size-dependent effects on electrical contacts to nanotubes and nanowires. Physical Review Letters 2006;97.

[102] Dag S, Gulseren O, Ciraci S, Yildirim T. Electronic structure of the contact between carbon nanotube and metal electrodes. Applied Physics Letters 2003;83:3180.

[103] Moore GE. Cramming more components onto integrated circuits. Electronics 1965;38.

[104] Petty MC. Molecular electronics: Prospects for instrumentation and measurement science. Measurement Science & Technology 1996;7:725.

[105] Reed MA. Molecular-scale electronics. Proceedings of the Ieee 1999;87:652.

[106] Kouwenhoven L. Applied physics - Single-molecule transistors. Science 1997;275:1896.

[107] Appenzeller T. The Man Who Dared to Think Small. Science 1991;254:1300.

[108] Frank S, Poncharal P, Wang ZL, de Heer WA. Carbon nanotube quantum resistors. Science 1998;280:1744.

[109] Iijima S. Helical Microtubules of Graphitic Carbon. Nature 1991;354:56.

[110] Iijima S, Ichihashi T. Single-Shell Carbon Nanotubes of 1-Nm Diameter. Nature 1993;363:603.

149 [111] Bethune DS, Kiang CH, Devries MS, Gorman G, Savoy R, Vazquez J, Beyers R. Cobalt-Catalyzed Growth of Carbon Nanotubes with Single-Atomic-Layerwalls. Nature 1993;363:605.

[112] Dresselhaus MS, Dresselhaus G, Avouris P. Carbon nanotubes : synthesis, structure, properties, and applications. Berlin: Springer, 2001.

[113] Saito R, Dresselhaus G, Dresselhaus MS. Physical properties of carbon nanotubes. London: Imperial College Press, 1998.

[114] Guo T, Nikolaev P, Rinzler AG, Tomanek D, Colbert DT, Smalley RE. Self- Assembly of Tubular Fullerenes. Journal of Physical Chemistry 1995;99:10694.

[115] Hamada N, Sawada S, Oshiyama A. New One-Dimensional Conductors - Graphitic Microtubules. Physical Review Letters 1992;68:1579.

[116] Mintmire JW, Dunlap BI, White CT. Are Fullerene Tubules Metallic. Physical Review Letters 1992;68:631.

[117] Dresselhaus MS, Dresselhaus G, Saito R. Carbon-Fibers Based on C-60 and Their Symmetry. Physical Review B 1992;45:6234.

[118] Odom TW, Huang JL, Kim P, Lieber CM. Atomic structure and electronic properties of single-walled carbon nanotubes. Nature 1998;391:62.

[119] Wildoer JWG, Venema LC, Rinzler AG, Smalley RE, Dekker C. Electronic structure of atomically resolved carbon nanotubes. Nature 1998;391:59.

[120] Avouris P, Appenzeller J, Martel R, Wind SJ. Carbon nanotube electronics. Proceedings of the Ieee 2003;91:1772.

[121] Sedra A, Smith K. Microelectronic Circuits. New York, Oxford: Oxford University Press, 2004.

[122] ITRS. International Technology Roadmap for Semiconductors, Executive Summary, 2007 Edition. 2007.

[123] Collins PC, Arnold MS, Avouris P. Engineering carbon nanotubes and nanotube circuits using electrical breakdown. Science 2001;292:706.

[124] Collins PG, Avouris P. Multishell conduction in multiwalled carbon nanotubes. Applied Physics A-Materials Science & Processing 2002;74:329.

[125] McEuen PL, Fuhrer MS, Park HK. Single-walled carbon nanotube electronics. Ieee Transactions on Nanotechnology 2002;1:78.

150 [126] Tans SJ, Devoret MH, Dai HJ, Thess A, Smalley RE, Geerligs LJ, Dekker C. Individual single-wall carbon nanotubes as quantum wires. Nature 1997;386:474.

[127] Collins PG, Avouris P. Nanotubes for electronics. Scientific American 2000;283:62.

[128] Martel R, Schmidt T, Shea HR, Hertel T, Avouris P. Single- and multi-wall carbon nanotube field-effect transistors. Applied Physics Letters 1998;73:2447.

[129] Bezryadin A, Verschueren ARM, Tans SJ, Dekker C. Multiprobe transport experiments on individual single-wall carbon nanotubes. Physical Review Letters 1998;80:4036.

[130] Soh HT, Quate CF, Morpurgo AF, Marcus CM, Kong J, Dai HJ. Integrated nanotube circuits: Controlled growth and ohmic contacting of single-walled carbon nanotubes. Applied Physics Letters 1999;75:627.

[131] Martel R, Wong H-SP, KChan K, Avouris P. Carbon nanotube field-effect transistors for logic applications. IEDM 2001, 2001. p.159.

[132] Martel R, Derycke V, Lavoie C, Appenzeller J, Chan KK, Tersoff J, Avouris P. Ambipolar electrical transport in semiconducting single-wall carbon nanotubes. Physical Review Letters 2001;87:256805.

[133] Kasap SO. Principles of Electronic Materials and Devices: Mc-Graw Hill Higher Eduction, 2002.

[134] Appenzeller J, Knoch J, Derycke V, Martel R, Wind S, Avouris P. Field- modulated carrier transport in carbon nanotube transistors. Physical Review Letters 2002;89:126801.

[135] Yaish Y, Park JY, Rosenblatt S, Sazonova V, Brink M, McEuen PL. Electrical nanoprobing of semiconducting carbon nanotubes using an atomic force microscope. Physical Review Letters 2004;92:046401.

[136] Voter AF, Montalenti F, Germann TC. Extending the time scale in atomistic simulation of materials. Annual Review of Materials Research 2002;32:321.

[137] Luo W, Ravichandran K, Windl W, Fonseca L. Contact Structure Formation in Carbon Nanotube Electronic Devices and Its Effect on Electron Transport. Materials Science and Technology 2006. Cincinnati, OH, 2006. p.476.

[138] Ravichandran K. Nano-Scale Process and Device Simulation. Columbus, OH: The Ohio State University, 2005.

151 [139] Datta S. Electronic Transport in Mesoscopic Systems. Cambridge, MA: Cambridge University Press, 1995.

[140] Zhang X, Fonseca L, Demkov AA. The application of density functional, local orbitals, and scattering theory to quantum transport. Physica Status Solidi B-Basic Research 2002;233:70.

[141] Soler JM, Artacho E, Gale JD, Garcia A, Junquera J, Ordejon P, Sanchez-Portal D. The SIESTA method for ab initio order-N materials simulation. Journal of Physics- Condensed Matter 2002;14:2745.

[142] Robertson DH, Brenner DW, Mintmire JW. Energetics of Nanoscale Graphitic Tubules. Physical Review B 1992;45:12592.

[143] Mintmire JW, White CT. Electronic and Structural-Properties of Carbon Nanotubes. Carbon 1995;33:893.

[144] Tibbetts GG. Why Are Carbon Filaments Tubular. Journal of Crystal Growth 1984;66:632.

[145] ASM Alloy Phase Diagrams Center, Villars P, editor-in-chief; Okamoto H, Cenzual K, section editors; http://www.asminternational.org/AsmEnterprise/APD, ASM International, Materials Park, OH, 2006.

[146] Ragone D. Thermodynamics of Materials: John Wiley & Sons, Inc., 1995.

[147] Bond Lengths and Energies. http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html.

[148] Callister WD. Materials science and engineering : an introduction. New York, NY: John Wiley & Sons, 2003.

[149] Reed-Hill RE, Abbaschian R. Physical metallurgy principles. Boston: PWS-Kent Pub., 1992.

[150] Buckel W, Hilsch R. Influence of low-temperature condensation on the electrical resistance and superconductivity of various metals. Zeitschrift fuer Physik 1954;138:109.

[151] Chen HS. Thermodynamic Considerations on Formation and Stability of Metallic Glasses. Acta Metallurgica 1974;22:1505.

[152] Drehman AJ, Greer AL, Turnbull D. Bulk Formation of a Metallic-Glass - Pd40Ni40P20. Applied Physics Letters 1982;41:716.

152 [153] Inoue A, Zhang T, Masumoto T. Zr-Al-Ni Amorphous-Alloys with High Glass- Transition Temperature and Significant Supercooled Liquid Region. Materials Transactions Jim 1990;31:177.

[154] Inoue A, Kato A, Zhang T, Kim SG, Masumoto T. Mg-Cu-Y Amorphous-Alloys with High Mechanical Strengths Produced by a Metallic Mold Casting Method. Materials Transactions Jim 1991;32:609.

[155] Peker A, Johnson WL. A Highly Processable Metallic-Glass - Zr41.2Ti13.8Cu12.5Ni10.0Be22.5. Applied Physics Letters 1993;63:2342.

[156] Lemley B. Glassy metals harder, stronger, and better-the material of the future. Discover 2004;25:46.

[157] Spaepen F. A Microscopic Mechanism for Steady State Inhomogeneous Flow in Metallic Glasses. Acta Metallurgica 1977;25:407.

[158] Spaepen F. Defects in amorphous metals. In: Balian R, editor. Physics of Defects. Les Houches: North Holland Publishing Company, 1980. p.135.

[159] Zhang ZF, Eckert J, Schultz L. Difference in compressive and tensile fracture mechanisms of Zr59Cu20Al10Ni8Ti3 bulk metallic glass. Acta Materialia 2003;51:1167.

[160] Argon AS. Plastic Deformation of Metallic Glasses. Acta Metallurgica 1978;27:12.

[161] de Hey P, Sietsma J, van den Beukel A. Structural disordering in amorphous Pd40Ni40P20 induced by high temperature deformation. Acta Materialia 1998;46:5873.

[162] van den Beukel A, Sietsma J. The Glass-Transition as a Free-Volume Related Kinetic Phenomenon. Acta Metallurgica Et Materialia 1990;38:383.

[163] Flores KM, Kanungo BP, Glade SC, Asoka-Kumar P. Characterization of plasticity-induced structural changes in a Zr-based bulk metallic glass using positron annihilation spectroscopy. Journal of Non-Crystalline solids 2007;353:1201.

[164] Flores KM, Sherer E, Bharathula A, Chen H, Jean YC. Sub-nanometer open volume regions in a bulk metallic glass investigated by positron annihilation. Acta Materialia 2007;55:3403.

[165] Flores KM, Suh D, Asoka-Kumar P, Sterne PA, Howell RH, Dauskardt RH. Characterization of Free Volume in a Bulk Metallic Glass Using Positron Annihilation Spectroscopy. Journal of Materials Research 2002;17:1153.

153 [166] Daw MS, Baskes MI. Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals. Physical Review Letters 1983;50:1285.

[167] Baskes MI. Application of the Embedded-Atom Method to Covalent Materials - a Semiempirical Potential for Silicon. Physical Review Letters 1987;59:2666.

[168] Baskes MI. Modified Embedded-Atom Potentials for Cubic Materials and Impurities. Physical Review B 1992;46:2727.

[169] Baskes MI, Nelson JS, Wright AF. Semiempirical Modified Embedded-Atom Potentials for Silicon and Germanium. Physical Review B 1989;40:6085.

[170] Daw MS, Foiles SM, Baskes MI. The Embedded-Atom Method - a Review of Theory and Applications. Materials Science Reports 1993;9:251.

[171] Foiles SM, Baskes MI, Daw MS. Embedded-Atom-Method Functions for the Fcc Metals Cu, Ag, Au, Ni, Pd, Pt, and Their Alloys. Physical Review B 1986;33:7983.

[172] Johnson RA. Physical Review B 1989;39:12554.

[173] Wadley HNG, Zhou X, Johnson RA. Materials Research Society Symposium Proceedings, vol. 672, O4.1.1, 2001.

[174] Wadley HNG, Zhou X, Johnson RA, Neurock M. Progress in Materials Science 2001;46:329.

[175] Duan G, Xu D, Zhang Q, G.Zhang, T.Cagin, Johnson WL, Goddard WA. Molecular dynamics study of the binary Cu46Zr54 metallic glass motivated by experiments: Glass formation and atomic-level structure. Physical Review B 2005;71:224208.

[176] Chen HS, Waseda Y. Structure of Glassy Zr-Cu and Nb-Ni Alloys. Physica Status Solidi A-Applied Research 1979;51:593.

[177] Sadoc A, Calvayrac Y, Quivy A, Harmelin M, Flank AM. Study of the local structure of Cu-Zr amorphous alloys by EXAFS: Effect of a structural relaxation. Journal of Non-Crystalline Solids 1984;65:109.

[178] Medvedev NN, Voloshin VP. Inter-atomic voids in analysis of computer simulated model structures of liquids and glasses. Journal of Structural Chemistry 2005;46:98.

[179] Medvedev NN, Voloshin VP, Luchnikov VA, Gavrilova ML. An Algorithm for Three Dimensional Voronoi S-network. Wiley InterScience, 2006.

154 [180] Sietsma J, Thijsse J. Characterization of free volume in atomic models of metallic glasses. Physical Review B 1995;52:3248.

[181] Voloshin VP, Naberukhin YI. Empty interatomic space in computer models of simple liquids and amorphous solids. Journal of Physics: Condensed Matter 1993;5:5685.

[182] Egami T, Waseda Y. Atomic size effect on the formability of metallic glasses. Journal of Non-Crystalline Solids 1984;64:113.

[183] Cohen MH, Turnbull D. Molecular transport in Liquids and Glasses. Journal of Chemical Physics 1959;31:1164.

[184] Clementi E, Raimondi DL, Reinhardt WP. Atomic Screening Constants from SCF Functions. Journal of Chemical Physics 1963; 38:2686. Atomic radius. http://www.webelements.com/periodicity/atomic_radius/

[185] Bharathula A, Luo W, Windl W, Flores KM. Characterization of open volume regions in a simulated Cu-Zr metallic glass. Metallurgical and Materials Transactions A- Physical Metallurgy and Materials Science 2008;39A:1779.

155