The Pennsylvania State University

The Graduate School

Eberly College of Science

THEORY OF ELECTRONIC AND

STRUCTURAL PROPERTIES OF MATERIALS:

NOVEL GROUP-IV MATERIALS AND

REAL SPACE METHODS

A Thesis in

Physics

by

Peihong Zhang

c© 2001 Peihong Zhang

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2001

We approve the thesis of Peihong Zhang.

Date of Signature

Vincent H. Crespi

Assistant Professor of PhysicsDownsbrough Professor

Thesis Advisor, Chair of Committee

Renee Diehl

Professor of Physics

Nitin Samarth

Associate Professor of Physics

Thomas E. MalloukProfessor of Chemistry

DuPont Professor of Materials Chemistry

James B. Anderson

Evan Pugh Professor of Chemistry

Jayanth R. Banavar

Professor of Physics

Head of the Department of Physics

Abstract

This dissertation consists of two parts. The first part employs existing computa-

tional techniques to study the electronic and structural properties of novel group-IV

and related materials, namely, carbon nanotubes, boron nitride nanotubes, and crys-

talline group-IV alloys. In the second part, we develop a new electronic structure

calculation technique based on finite element methods with multigrid acceleration.

The computation time of our new technique scales quadratically with the number of

atoms in the system i.e., O(N2), as opposed to the unfavorable cubic scaling (O(N3))

for most existing ab initio methods.

Chapter 1 gives an overview of theoretical methods involved in this dissertation.

Topics covered in this chapter are orthogonal and nonorthogonal tight-binding to-

tal energy models, density functional theory, pesudopotential planewave methods of

electronic structure, and the GW approximation for calculating accurate bandgaps

in semiconductors.

Chapter 2 focuses on the structural properties of carbon and boron nitride nan-

otubes. First, a new nucleation model for carbon nanotubes is proposed. Detailed

atomistic simulations indicate that this model agrees in several aspects with experi-

mental observations. The new model does not resort to the formation of energetically

unfavorable pentagonal rings, as other nucleation models do, to produce the required

curvature for a nanotube nucleus. Second, plastic deformation of carbon nanotube

under high tensile stress is studied using a tight-binding total energy model. The

elastic limits of carbon nanotubes are found to be higher than any other known ma-

terials and very sensitive to the so-called wrapping angle of nanotubes. Elastic limits

of boron nitride nanotubes are also studied and compared against those of carbon

nanotubes.

Chapter 3 is devoted to the electronic and structural properties of novel group-

iii

IV alloys formed from CVD precursor. Group-IV alloys have attracted considerable

research interest recently. Using the newly developed UltraHigh Vacuum Chemical

Vapor Deposition (UHV CVD) technique, group-IV alloys such as Si4C and Ge4C,

which contain 20 atomic % carbon, have been realized. In this chapter, ab initio

results on the electronic and structural properties of these high carbon concentration

group-IV materials are presented. General trends of the effects of substitutional

carbon are understood. More importantly, two molecular precursors are proposed

for synthesizing the group-IV alloys Si2Sn2C and Ge3SnC, which have direct energy

band gaps and lattice match silicon to better than 1%. This result might open a new

way of integrating silicon-based microelectronics with optoelectronics. Finally, a new

class of molecular precursors, X6C2H18 (X=Si, Ge), is proposed for group-IV alloys

containg 25 atomic % carbon.

The computational time of traditional ab initio techniques such as pseudopotential

planewave methods scales at least as O(N3), where N is the number of atoms in the

system. This unfavorable scaling limits the number of atoms one can study using these

methods to several hundreds, even with the most powerful supercomputers available

today. In chapter 4, we develop an O(N2) ab initio electronic structure calculation

technique based on the finite element methods with multigrid acceleration. O(N2)

scaling is achieved by avoiding explicit re-orthogonalization between eigenvectors,

which is made possible by a multigrid algorithm. With these new techniques, we can

perform ab initio calculations for systems containing more than 32 atoms on a single

workstation (Compaq alpha DS10).

iv

Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Chapters

1 Theoretical Methods 1

1.1 Tight-binding total energy models . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Orthogonal tight-binding formalism . . . . . . . . . . . . . . . 2

1.1.2 Nonorthogonal tight-binding models . . . . . . . . . . . . . . 4

1.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 The density functional formalism . . . . . . . . . . . . . . . . 7

1.2.2 Local density approximation . . . . . . . . . . . . . . . . . . . 9

1.3 Pseudopotential plane-wave methods . . . . . . . . . . . . . . . . . . 11

1.3.1 The Bloch theorem and plane-wave expansion . . . . . . . . . 12

1.3.2 Pseudopotential concepts . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 Pseudopotential construction . . . . . . . . . . . . . . . . . . 15

1.3.4 The pseudopotential plane-wave method . . . . . . . . . . . . 18

1.4 Beyond LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.1 The GW approximation . . . . . . . . . . . . . . . . . . . . . 20

v

1.4.2 Generalized Kohn-Sham schemes . . . . . . . . . . . . . . . . 25

2 Carbon and Boron Nitride Nanotubes 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.1 Specification of carbon nanotubes . . . . . . . . . . . . . . . . 28

2.1.2 Novel electronic and structural properties . . . . . . . . . . . . 30

2.2 Nucleation of carbon nanotubes without pentagonal rings . . . . . . . 34

2.3 Plastic deformation of carbon nanotubes . . . . . . . . . . . . . . . . 43

2.4 Plastic deformation of carbon boron-nitride nanotubes: an unexpected

weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Electronic and Structural Properties of Novel Group-IV Compounds

Formed from CVD Precursors 60

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Si4C, Ge4C and Sn4C . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Si2Sn2C and Ge3SnC . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5 Si6C2 and Ge6C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 O(N2) real-space electronic structure methods 87

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1.1 Momentum-space versus real-space formalism . . . . . . . . . 88

4.1.2 O(N) methods . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1.3 O(N2) electronic structure methods: an intermediate step . . 93

4.2 The FE method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2.1 The Ritz-Galerkin Method . . . . . . . . . . . . . . . . . . . . 95

4.2.2 FE triangulation and nodal basis functions . . . . . . . . . . . 96

vi

4.3 FE discretization of the Kohn-Sham equations . . . . . . . . . . . . . 99

4.4 Construction of the ab initio potential matrices . . . . . . . . . . . . 99

4.5 Multigrid Poisson equation solver . . . . . . . . . . . . . . . . . . . . 103

4.5.1 Standard iterative methods for solving linear systems . . . . . 105

4.5.2 Multigrid acceleration . . . . . . . . . . . . . . . . . . . . . . 106

4.5.3 Performance of the MG Poisson equation solver . . . . . . . . 108

4.6 Full multigrid Kohn-Sham equation solver . . . . . . . . . . . . . . . 110

4.6.1 Brief description . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6.2 Coarse grid solution . . . . . . . . . . . . . . . . . . . . . . . 112

4.6.3 O(N2) iterative diagonalization methods . . . . . . . . . . . . 115

4.6.4 Simultaneous convergence of potential and eigenvalues . . . . 120

4.7 Performance of the FMG electronic structure code . . . . . . . . . . . 121

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

vii

List of Figures

1.1 Ionic pseudopotentials of Si. . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Rolling up a graphene sheet into a tube. . . . . . . . . . . . . . . . . 28

2.2 Specification of single-walled carbon nanotubes. The circumferential

vector shown in the figure corresponds to that of a (5,2) nanotube.

The chiral angle θ is also shown. . . . . . . . . . . . . . . . . . . . . . 29

2.3 Near-Fermi surface electronic structure of graphene. The Fermi surface

is reduced to two distinct (Fermi) points. . . . . . . . . . . . . . . . . 30

2.4 Allowed lines of wavevectors in carbon nanotubes: (a) the allowed lines

miss the Fermi points and (b) the allowed lines pass through the Fermi

points. The hexagon is the Brillouin zone of graphene. . . . . . . . . 31

2.5 Electronic properties of carbon nanotubes. Solid circles are metallic;

triangles are small bandgap semiconductors and the rest are semicon-

ductors with relatively large bandgaps. . . . . . . . . . . . . . . . . . 32

2.6 A schematic diagram of the nucleation model in which a nanotube

forms via edge mediated opening of a double-layered graphitic patch. 35

2.7 The relaxed structures from T=0 molecular dynamics for two-layered

graphitic patches of sizes necessary to form nanotube nuclei of diam-

eters (a) 13.7 A and (b) 27.4 A. The smaller nucleus spontaneously

opens into a circular cross-section which defines a strong preferential

growth axis whereas the larger nucleus remains flattened. . . . . . . . 37

viii

2.8 The energy as a function of a reaction coordinate (i.e., the distance

between opposite inner surfaces as the tube nucleus opens) for nucle-

ation patches of varying diameter when opened and of length 1 nm.

The smallest patches have no barrier to opening. For sufficiently large

patches the opened state is energetically disfavored. For intermediate

sizes, thermal excitation can activate opening on experimental time-

scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.9 Two tubes with bond rotation defects. . . . . . . . . . . . . . . . . . 44

2.10 Energy versus length for pure (solid) and defective (dashed) (8,0) nan-

otubes. The cell contains 96 atoms per bond rotation defect with E = 0

for the pure undistorted tube. . . . . . . . . . . . . . . . . . . . . . . 46

2.11 Transition tensions to plastic deformation for a family of (n, 0) and one

of nearly equal-radius (n,m) tubes. The radial dependence is quite

weak except for the smallest tubes, whereas the wrapping angle de-

pendence is very strong. . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.12 Bond rotation defects in (a) (n, 0), (b) (n, n) BN nanotubes, and (c)

(n, 0) and (n, n) carbon nanotubes. There are two distinct bond ro-

tation patterns in a generic (m,n) boron nitride nanotube. For an

(8,0) BN nanotube, the energy difference between the two patterns is

about 0.2 eV/defect at equilibrium (i.e., zero tension), while for (n, n)

tubes the two states are degenerate. Various bond lengths marked (not

shown to scale) correspond to tubes under about 12% strain. . . . . . 53

3.1 Molecular precursors and the corresponding relaxed crystalline phases

of Si4C. Silicon is yellow, carbon is grey, and the terminal group is red.

Ge4C and Sn4C have similar structures. . . . . . . . . . . . . . . . . . 64

3.2 Irreducible Brillouin zone of the bct lattice. . . . . . . . . . . . . . . 66

ix

3.3 LDA band structures of Si4C, Ge4C. . . . . . . . . . . . . . . . . . . 68

3.4 LDA band structures of Sn4C and silicon calculated on a 10-atom unit

cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5 Band structure of Ge4C upon lattice expansion to match that of silicon. 70

3.6 Molecular precursors and the corresponding relaxed crystalline phases

of Si2Sn2C (top) and Ge3SnC (bottom). Silicon is yellow, carbon is

grey, tin is magenta, germanium is green and the terminal group is

red. Ge4C and Sn4C have similar structures. . . . . . . . . . . . . . . 72

3.7 Band structures (in blue) of Si2Sn2C (top) and Ge3SnC (bottom). The

ionicity opens a bandgap deep in the valence band. Symmetry breaking

in Si2Sn2C removes the degeneracy at the X point, pushing one band

up significantly so that it becomes the conduction band maximum and

produces in a direct band gap at X (red arrow). In the case of Ge3SnC,

the conduction band minimum occurs at Γ, which produces a direct

gap at Γ (red arrow). The band structure of silicon (orange) calculated

in the same unit cell is also plotted for comparison. . . . . . . . . . . 77

3.8 Charge density isosurfaces for the top of the valence band at Γ (top)

and X (bottom) in Si2Sn2C. Electrons are localized around Si–Si bonds

(in yellow) for the Γ state, whereas for the X state they are localized

around Sn–Sn bonds (in magenta). . . . . . . . . . . . . . . . . . . . 78

3.9 Proposed molecular precursor for synthesizing Si6C2 and the corre-

sponding relaxed crystalline structure. The molecular precursor and

crystal structure of Ge6C2 are similar to those shown. . . . . . . . . . 80

3.10 LDA band structure of Si6C2 (top) and Ge6C2( bottom). . . . . . . . 83

3.11 Density of states of Si6C2 (top) and Ge6C2( bottom). . . . . . . . . . 84

4.1 n-simplices: 1, 2, and 3-simplices. . . . . . . . . . . . . . . . . . . . . 97

x

4.2 Finite-element triangulation examples in 2D: (a) uniform triangulation

and (b) adaptive (nonuniform) triangulation. . . . . . . . . . . . . . . 97

4.3 A 3D triangulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.4 A linear nodal basis function. The function is defined corresponding

to nodal point Ni. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5 Typical convergence behavior of the GS method . . . . . . . . . . . . 107

4.6 Convergence properties of the MG Poisson equation solver. The solid

and the dotted lines correspond to fine-grid matrix sizes of 32,768×32,768and 131,072×131,072, respectively. . . . . . . . . . . . . . . . . . . . 109

4.7 Comparison of the convergence properties of the standard Davidson

iteration and the MG relaxation scheme we developed. The relaxations

were performed on the finest grid, using the intermediate-grid solution

as a initial guess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xi

List of Tables

2.1 Elastic strength of single-walled carbon nanotubes, (a) assuming the

effective cross-sectional area is πr2, where r is the radius of the tube,

and (b) the cross-section is the area per nanotube in a nanotube bundle

with inter-tube distance 3.0 A. . . . . . . . . . . . . . . . . . . . . . 33

2.2 Comparison of the elastic limits of boron nitride and carbon nanotubes,

showing both full and reduced transition tensions (as defined in the

text). The reduced transition tensions for (n, n) carbon and boron

nitride nanotubes are similar, but these quantities differ greatly for

(n, 0) carbon and boron nitride nanotubes. . . . . . . . . . . . . . . 56

3.1 Structural properties of X4C in the local density approximation. The

bulk moduli in parentheses are for silicon, germanium, and gray tin

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 Structural properties of Si2Sn2C and Ge3SnC. All cited values are LDA

results. The results for silicon calculated with a 10-atom supercell are

also listed for comparison. . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Structural properties of Si6C2 and Ge6C2. The lattice notation of the

bct structure is used since both structures are only slightly distorted

from bct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1 O(N) scaling properties of the MG Poisson equation solver. All calcu-

lations were done on an alpha DS10 workstation. . . . . . . . . . . . 109

xii

4.2 Scaling and convergence properties of the FMG real-space electronic

structure code. All calculations were done on an alpha DS10 worksta-

tion. Note that the errors refer to the differences in the correspond-

ing values between two consecutive iterations, not the absolute errors

with respect to the converged values. CPU time for the corresponding

plane-wave calculation is shown in the parenthesis. . . . . . . . . . . 122

xiii

List of Acronyms

TB Tight-BindingDFT Density Functional TheoryLCAO Linear Combination of Atomic OrbitalsPBC Periodic Boundary ConditionsKS Kohn-ShamLDA Local Density ApproximationSCTB Self-Consistent Tight-BindingHF Hartree-FockCI Configuration InteractionOPW Orthogonalized Plane WaveKB Kleinman-BylanderRPA Random Phase ApproximationGKS Generalized Kohn-ShamBZ Brillouin ZoneDOS Density-of-StatesCVD Chemical Vapor DepositionBN Boron NitrideUHV UltraHigh-VacuumFFT Fast Fourier TransformLNV Li, Nunes, and VanderbiltVSEPR Valence Shell Electron Pair RepulsionFD Finite DifferenceFE Finite ElementMG MultiGridGS Gauss-SiedelCG Conjugate GradientAMG Algebraic MultiGridSOR Successive Over RelaxationFMG Full MultiGridSC Self-Consistent

xiv

Acknowledgments

First, I would like to thank my advisor, Prof. Vincent Crespi, for his constant

support and encouragement. Without his enthusiasm and guidance, the research

projects reported in this dissertation would never have been completed. I would also

like to thank Prof. Steven Louie, Prof. Marvin Cohen and Dr. Eric Chang for the

fruitful collaboration and for hosting me during my visit to UC, Berkeley in Summer,

1999. I thank Dr. Paul Lammert for the enjoyable collaboration and discussion,

and for solving most of my problems with UNIX. Prof. Thomas Mallouk and Prof.

John Kouvetakis are acknowledged for the useful discussion and experimental inputs.

Prof. Jinchao Xu and Prof. Jesse Barlow are acknowledged for their help on numerical

algorithms. Finally, I thank my family for their love and support.

xv

Chapter 1

Theoretical Methods

The work presented in this thesis involves a number of theoretical/computational

techniques, ranging from the simple and fast empirical tight-binding (TB) models to

the fairly sophisticated and computationally intensive GW method. In this chapter,

we give an overview of these methods: orthogonal and nonorthogonal TB total energy

models, the density functional theory (DFT), pseudopotential plane-wave methods,

and the GW approximation.

1.1 Tight-binding total energy models

Empirical classical potential models (known as force-field models in the chem-

istry and biochemistry communities) have been applied extensively to model systems

containing thousands to millions of atoms with great success. However, in situa-

tions where quantum mechanical effects are important, classical potentials often fail

to produce meaningful results. For example, pair potentials fail to stabilize tetra-

hedral structures such as diamond and zincblende, since the directional covalent

bonding in these systems is primarily determined by quantum mechanical effects.

As a result, complicated three-body potentials have been proposed to describe such

systems[1, 2, 3, 4, 5, 6]. DFT-based ab initio techniques (to be discussed in more

detail later), on the other hand, are very accurate and take the quantum mechanical

1

effects into full consideration. However, ab initio methods are also very computation-

ally demanding. Consequently, even with the most powerful supercomputers available

today, the largest system one can handle using ab initio methods is limited to the

order of 1,000 atoms. The TB method[7] takes an intermediate step towards mod-

elling meterials that takes into account quantum mechanical effects without too much

computational effort.

1.1.1 Orthogonal tight-binding formalism

The idea of using a linear combination of atomic orbitals (LCAO) to represent the

wavefunctions of aggregates of atoms (e.g., solids) was due to Bloch[8]. However, the

success of modern orthogonal TB schemes should be credited to Slater and Koster[7],

who presented the first detailed TB formalism for band structure calculations. In the

following, we shall assume periodic boundary conditions (PBC). (Extension of the

TB formalism to finite-size systems is straightforward.)

Denoting an atomic orbital n = (τ, α) at unit cell i as φn(r − rτ −Ri), we have

the Bloch sum

Φnk =1√N

N∑i=1

eik·(rτ+Ri)φn(r− rτ −Ri), (1.1)

where the sum is over the atoms in equivalent positions in all unit cells in the crystal.

τ is the atomic index in the unit cell and α specifies the atomic orbitals of atom τ . For

example, in the diamond structure, τ = 1, 2 and α = s, px, py, pz. We may express

the wavefunctions of the system as linear combinations of Bloch sums. However, the

Bloch sum formed from atomic orbitals is not an ideal basis, since atomic orbitals

at different sites may not be orthogonal to each other. It is more convenient to

work in a orthogonal basis, the so-called Lowdin orbitals ψn(r), which are formed

by the reorthogonalization of atomic orbitals φn(r)[9]. We shall assume that such a

2

reorthogonalization[9] is done and rewrite the Bloch sum as

Ψnk =1√N

N∑i=1

eik·(rτ+Ri)ψn(r− rτ −Ri). (1.2)

The Hamiltonian matrix element between two Bloch sums with the same k is

Hn,n′(k) =N∑i=1

eik·(rτ+Ri−rτ ′)〈ψn(r− rτ )|H|ψn′(r− rτ ′)〉. (1.3)

In the above equation, the sum has to be carried out over all unit cells in principle.

In practice, one always assumes that the matrix element 〈ψn|H|ψn′〉 is short-rangedand decays to zero beyond some cut-off distance[10, 11, 12]. The matrix elements

〈ψn|H|ψn′〉 in the TB formalism are modelled by some empirical parameters which

are either obtained by fitting experimental data or from more accurate ab initio cal-

culations. Eigenvalues and corresponding wavefunctions are then calculated by diag-

onalizing the Hamiltonian matrix H. This is the original Slater-Koster TB model[7].

The Slater-Koster TB formalism computes only the band energies. For atomistic

simulations, the total energy of the system must also be derived. Chadi[10] proposed

that the total energy of a system in the TB framework can be written as band

structure energy plus a sum of pair-wise repulsive potentials

Etot = Ebs + Erep =∑

i∈occupiedεi +

∑i,j

U(|ri − rj|). (1.4)

This is consistent with the local density approximation (LDA) expression of the total

energy, which is as a sum of the band energy and some potential contributions (see

Eqn. (1.27)). The idea that the terms in the total energy which are not included

in the single-electron band energy can be approximated by a sum of pair terms is of

great importance and is almost universally applied[13]. (However, the justification

of doing so is still questionable.) The force acting on an atom in the TB formalism

contains two terms, one arising from the band structure energy Ebs and the other

from the repulsive potential energy[14]. In chapter 2, the TB parameters for carbon

3

systems by Xu, et al.[15] are adopted for studying the structural properties of carbon

nanotubes.

1.1.2 Nonorthogonal tight-binding models

The orthogonal TB model described in section 1.1.1 is conceptually simple and

computationally efficient. However, it shares with other empirical potential methods

a serious drawback, the lack of transferability[13]. The assumption that an orthgo-

nal set of orbitals can be constructed from the nonorthogonal atomic orbitals by the

Lowdin procedure is not universally valid, i.e., an orthogonal transformation is valid

only for a single configuration of the system. Furthermore, the Lowdin orbitals are

usually more extended than the nonorthogonal atomic orbitals, which requires that

the Hamiltonian matrix element 〈ψn|H|ψn′〉 be longer-ranged. The usually short-

ranged TB parametrization is therefore not expected to work well across different en-

vironments. Failure to properly account for the overlap effects is probably the main

reason for the lack of transferability of orthogonal TB models[13], although other

issues such as the two-center approximation of the Hamiltonian matrix element, the

pairwise repulsive potentials, and lack of self-consistency are also important factors.

The transferability can be improved somewhat by including a wider range of empir-

ical data; however, the improvement is limited and not systematic. Dorantes-Davila

et al.[16] and Dorantes-Davila and Pastor[17] have proposed an iterative scheme to

incorporate in an orthogonal TB framework the effects of the overlaps of atomic or-

bitals. A different approach is the so-called nonorthogonal TB model. As its name

tells, this approach eliminates the assumption of orthogonality so that most of the

drawbacks associated with this assumption can be overcome. (The effect of neglecting

the nonorthogonality of the basis has been investigated by Mirabella et al.[18] and

Mckinnon and Choy[19].)

In the nonorthogonal TB model, one has to solve the generalized eigenvalue prob-

4

lem

(H− EiS)Ψi = 0, (1.5)

where S is the overlap matrix between TB orbitals,

Sn,n′ =∫

φn(r)φn′(r)dr, (1.6)

and φ are nonorthogonal atomic TB orbitals as opposed to the orthogonalized Lowdin

orbitals ψ. The repulsive part of the total energy is usually assumed to be pairwise

as in the orthogonal TB models. The on-site (diagonal) Hamiltonian matrix elements

are usually taken as the atomic ionization potentials of the corresponding orbitals, i.e.,

Hn,n = εn and the off-diagonal elements can be determined using Mulliken analysis[20,

21]:

Hn,n′ =1

2K(εn + εn′)Sn,n′, (1.7)

where K is an adjustable parameter, or by other means[22, 23]. We employ a

nonorthogonal TB scheme by Porezag et al.[24] to study the plastic deformation

of boron-nitride nanotubes in Chapter 2.

1.2 Density functional theory

Seeking an accurate yet tractable description of many particle systems has been

an important goal of physicists and chemists during the past few decades. The TB

methods described in the previous section represent the early efforts to incorporate

realistic quantum mechanics effects into a model Hamiltonian. Although TB mod-

els have been successfully applied to a number of systems and have advantages over

empirical potentials, there are limitations. For example, it is very difficult to obtain

TB parameters for systems containing more than two species. Also, there are only a

few accurate and transferrable TB parameters available. TB parameters constructed

by fitting data from one phase might not be readily applied to the study of other

5

phases or the transitions among them, i.e., TB parameters are not very transferable.

Although nonorthogonal TB schemes improve somewhat the transferability of TB by

relaxing the orthogonality assumption, the improvements are limited by their empir-

ical nature. Finally, the assumption that the Hamiltonian matrix does not depend

upon the distribution of electrons brings about severe difficulties when dealing with

systems where charge transfer is important (this problem is partially addressed by

the so-called self-consistent tight binding (SCTB) methods.). Therefore, theoretical

methods that require no empirical parameters, the so-called first principles or ab ini-

tio methods, have attracted intensive research interest since the availability of digital

computers.

The most natural approach to ultimately describe a many-electron system is

to calculate its many-electron wavefunctions. The earliest such attempt is due to

Hartree[25]. In the Hartree method, the many-body wavefunction is approximated

by a product of single-electron wavefunctions.

Ψ(r1, r2, · · · rN) = ψ1(r1)ψ1(r2) · · ·ψN(rN). (1.8)

Each of the wavefunctions ψi satisfies a one-electron Schrodinger equation with the

potential term replaced by an effective self-consistent Coulomb (or Hartree) potential.

Better approximations can be achieved by considering the antisymmetric property of

the many-electron wavefunction as required by the Pauli principle, resulting in a

scheme called Hartree-Fock (HF) approximation[26, 27]. In the HF approximation,

the many-electron wavefunction is represented by an antisymmetrized wavefunction,

i.e., a Slater determinant. The effective potential in the HF approximation consists of

two terms, a local Hartree (or Coulomb) potential and a nonlocal exchange potential

which arises from the antisymmetrization of the many-electron wavefunctions. The

HF approximation has had tremendous success and remains an indispensable tool in

molecular physics and quantum chemistry[28]. However, the nonlocal exchange po-

6

tential introduced by the HF approximation drastically increases the computational

cost. Furthermore, correlation effects are not included in the HF scheme. Systematic

improvement over the HF approximation can be further achieved by the configuration

interaction (CI) scheme, in which the many-electron wavefunction is expanded as a

superposition of many Slater determinants (as opposed to the single Slater determi-

nant used in the HF approximation). CI is a very attractive approach since it yields,

in principle, an exact solution to the many-body problem. However, the explosive

increase in the computer resources required in CI calculations with increasing number

of electrons means that only very small systems can be treated with high accuracy in

this method.

In an effort to avoid the computationally demanding exchange term in the HF ap-

proximation while retaining the overall exchange effects, Slater[29, 30] later replaced

the nonlocal exchange potential with a very simple potential, known as Slater’sXα po-

tential, that depends only on the local charge density, i.e., −α 32π(3π2n(r))1/3. Slater’s

Xα potential can be regarded as an early version of the so-called local density approxi-

mation (LDA) to the exchange-correlation potential in the DFT framework. Actually,

the Thomas-Fermi approximation was the first attempt to describe a many-electron

system using the local charge density as the basic variable[31, 32]. However, it was

not fully realized that the local charge density could indeed serve as the fundamen-

tal variable for solving the many-electron problem until the work of Hohenberg and

Kohn[33] and Kohn and Sham[34].

1.2.1 The density functional formalism

In a paper published in 1964, Hohenberg and Kohn proved that the ground state

properties of a many-electron system can be expressed as functionals of its ground

state charge density[33]. The proof of this remarkable theorem, which was later

generalized by Levy[35], turns out to be rather simple. We follow here the derivation

7

in a review article by Jones and Gunnarsson[28].

Let’s denote n(r) as “N-representable” densities, i.e., those which can be obtained

from some antisymmetric many-electron wavefunction Ψ

n(r) = N∫|Ψ(r, r2, r3, · · · , rN)|2dr2 · · · drN , (1.9)

assuming that the wavefunction Ψ is normalized to one. The Hamiltonian for N

electrons moving in an external potential V ext is

H = T + V ee +N∑i=1

V ext(ri), (1.10)

where T is the kinetic energy operator and V ee is the electron-electron interaction.

Levy defined a functional

F [n] = minΨ→n〈Ψ|T + V ee|Ψ〉, (1.11)

where the minimization is taken over all wavefunctions Ψ that give the charge density

n(r). For all N -representable n(r), the two basic theorems of DFT are

E[n] ≡∫

V ext(r)n(r)dr+ F [n] ≥ E0, (1.12)

where E0 is the ground state energy, and∫V ext(r)n0(r)dr+ F [n0] = E0, (1.13)

where n0 is the ground state charge density.

The variational principle (1.12) is obvious since, for any N -representable density

n, the wavefunction that minimizes the functional F [n], denote it as Ψnmin, is a super-

position of eigenfunctions of the system. Therefore, the expectation value of Ψnmin,

namely, E[n], should be higher or equal to the ground state energy.

If Ψ0 is the ground state wavefunction, n0 is the ground state charge density,

and Ψn0min is the wavefunction that minimizes the functional F [n0], we have, from the

definition of ground state,

E0 ≡ 〈Ψ0|V ext + T + V ee|Ψ0〉 ≤ 〈Ψn0min|V ext + T + V ee|Ψn0min〉. (1.14)

8

Since

〈Ψ0|V ext|Ψ0〉 = 〈Ψn0min|V ext|Ψn0min〉 =∫

V ext(r)n0(r)dr, (1.15)

we have

〈Ψ0|T + V ee|Ψ0〉 ≤ 〈Ψn0min|T + V ee|Ψn0min〉. (1.16)

On the other hand, the definition of functional F [n] yields

〈Ψn0min|T + V ee|Ψn0min〉 ≤ 〈Ψ0|T + V ee|Ψ0〉. (1.17)

This can be true only if

〈Ψ0|T + V ee|Ψ0〉 = 〈Ψn0min|T + V ee|Ψn0min〉. (1.18)

Therefore, we have

E0 ≡ 〈Ψ0|V ext + T + V ee|Ψ0〉=

∫V ext(r)n0(r)dr+ 〈Ψ0|T + V ee|Ψ0〉

=∫V ext(r)n0(r)dr+ 〈Ψn0min|T + V ee|Ψn0min〉

=∫V ext(r)n0(r)dr+ F [n0].

(1.19)

This completes the proof of the two basic theorems of DFT.

1.2.2 Local density approximation

The DFT of Hohenberg and Kohn states that knowledge of the ground state charge

density n0(r) of a system is, in principle, sufficient to calculate any of its ground

state properties, provided that the exact functionals are known. For example, to

calculate the ground state energy, one simply needs to know the functional form of

F [n]. However, obtaining an exact functional F [n] is obviously impossible. Kohn and

Sham[34] proposed the simplest, and surprisingly, the most successful approximation

to the energy functional, which is now known as the local density approximation

(LDA).

Kohn and Sham first partitioned the functional F [n] into three parts:

F [n] = T 0[n] + Eh[n] + Exc[n], (1.20)

9

where T 0[n] is the kinetic energy of a system of noninteracting electrons with density

n(r), Eh[n] is the Hartree energy and Exc[n] is the exchange-correlation energy func-

tional, which remains to be approximated. They assumed a slowly varying charge

density n(r) and approximated the exchange-correlation energy functional as

Exc[n] =∫

n(r)εxc(n(r))dr, (1.21)

where εxc is the exchange-correlation energy per electron of a uniform electron gas

of density n(r)[34]. This expression can be regarded as the leading term of a more

general Taylor expansion of Exc[n]:

Exc[n] =∫

n(r)εxc(n(r))dr+∫

εxc2 (n)|∇n|2dr+ · · · . (1.22)

Therefore, in the slowly varying charge density limit, expression (1.21) becomes exact.

By expanding the charge density n(r) as a sum of single-particle-like densities

n(r) =N∑i=1

|φi(r)|2, (1.23)

and applying the variational principle to the energy functional, they reached a set of

one-particle Schrodinger equations, known as Kohn-Sham equations (in atomic unit)

[−12∇2 + V H(r) + V xc(r) + V ext(r)]φi(r) = εiφi(r), (1.24)

where

V xc(r) =d(nεxc(n))

dn(1.25)

is the exchange-correlation potential and

V H(r) =∫ n(r′)|r− r′|dr

′ (1.26)

is the Hartee potential. The total energy of the system is

E =N∑i=1

εi − 1

2

∫n(r)n(r′)|r− r′| drdr

′ +∫

n(r)[εxc(n(r))− V xc(n(r))]dr. (1.27)

10

The Kohn-Sham equations (1.24) have to be solved self-consistently. Furthermore,

the exchange-correlation energy εxc(n) needs to be approximated.

In their original work, Kohn and Sham did not expect this simple approximation,

namely LDA, to work well for real systems, commenting “we do not expect an accurate

description of the chemical bonding”[34] (by LDA). Therefore, it is a rather surprising

fact that LDA has been applied to a wide variety of systems with great success. In

some cases, the charge densities are not at all slowly varying. Apart from the fact

that LDA (somewhat unexpectedly) produces rather accurate ground state properties

of materials, the Kohn-Sham orbitals φi, which are introduced as a variational basis

and have no obvious connection with the quasiparticle wavefunctions, turn out to be

surprisingly close to the latter[36]. Also, the Kohn-Sham eigenvalues εi are often very

close to quasiparticle energies. Therefore, although DFT was formally established for

computing ground state properties, it is now almost universally applied to calculate

and interpret the electronic structure of materials. However, one should bear in

mind that calculations of the excited state properties are actually out of the scope

of the DFT, and neither LDA nor DFT should be blamed for any inconsistency

between the DFT (LDA) eigenvalues and the experimentally measured quasiparticle

excitation energies. In fact, there are well known problems. For example, LDA

usually underestimates the band gap of semiconductors and insulators. For simple

metals such as lithium and sodium, LDA gives free-electron-like valence band widths

which are typically 20∼30% wider than the experimental ones. We will return to this

point in section 1.4.

1.3 Pseudopotential plane-wave methods

The density functional theory, particularly the Kohn-Sham LDA form, greatly

simplifies the calculation of materials properties and significantly reduces the compu-

tational effort. Many of the modern computational methods in solid state physics are

11

based on DFT. In this section, we give a brief review of pseudopotential plane-wave

methods[37].

1.3.1 The Bloch theorem and plane-wave expansion

The Bloch theorem is one of the most fundamental theorems in solid state physics.

Considering an electron moving in a periodic potential V (r), the Schrodinger equation

is

(−12∇2 + V (r))φi(r) = εiφi(r), (1.28)

where

V (r) = V (r+Rn), (1.29)

and Rn is the lattice vector. Bloch showed that Eqn. (1.28) has solutions

φi,k(r) = eik·rui,k(r), (1.30)

where

ui,k(r) = ui,k(r+Rn). (1.31)

Since uk(r) has the periodicity of the lattice, it can be expanded in a Fourier sum

ui,k(r) =∑G

Ci,G(k)eiG·r, (1.32)

where the sum, in principle, should be carried out over an infinite number of reciprocal

lattice G’s. In practice, beyond a maximum cutoff Gmax, the coefficients can be

assumed to be zero. Substituting the Fourier expansion (1.32) into Eqn. (1.28), we

have the algebraic eigenvalue equations

∑G′

HGG′(k)Ci,G′ = εi,kCi,G, (1.33)

where the Hamiltonian matrix elements are

HGG′(k) ≡ 〈k+G| − 1

2∇2 + V |k+G′〉 = 1

2|k+G|2δG,G′ +VG,G′(k). (1.34)

12

The Hamiltonian matrix can be diagonalized to obtain the band structure and elec-

tronic wavefunctions of the system, provided that the potential matrix elements

VG,G′(k) are known. In the following sections, we shall discuss the concept of pseu-

dopotentials and how to evaluate theVG,G′(k) within the pseudopotential framework.

1.3.2 Pseudopotential concepts

Plane-waves are a very natural basis for describing electronic wavefunctions of pe-

riodic systems. However, if we need to represent highly localized wavefunctions such

as those of core electrons, an enormous number of plane-waves is needed. Fortunately,

the pseudopotential concepts introduced by Fermi[38] and Hellmann[39] make it pos-

sible to accurately describe the properties of materials without dealing explictly with

the core electrons.

The cancellation theorem derived from the orthogonalized plane wave (OPW)

formalism by Phillips and Kleinman is very helpful in understanding the pseudopo-

tential concepts[40]. This theorem demonstrates that the requirement that valence

states should be orthogonal to the core states effectively provides a repulsive potential

for the valence states. If this effective repulsive potential is added to the attractive

potential in the core region, they almost cancel out, leaving a relatively weak net po-

tential or pseudopotential[40]. In the OPW methods, the valence state wavefunctions

are expressed as the sum of a smooth wavefunction φ and a sum over core states φc,

|Ψ〉 = |φ〉+∑c

αc|φc〉, (1.35)

with the requirement that the valence states Ψ are orthogonal to the core states φc,

which yields

|Ψ〉 = |φ〉 −∑c

|φc〉〈φc|φ〉. (1.36)

We now find an effective Schrodinger equation that φ satisfies, assuming that Ψ

13

-16

-14

-12

-10

-8

-6

-4

-2

0

2

0 1 2 3 4 5r (a.u.)

3s

V

3p3d

Figure 1.1: Ionic pseudopotentials of Si.

satisfies the following Schrodinger equation

H|Ψ〉 = (T + Vc)|Ψ〉 = E|Ψ〉, (1.37)

where T is the kinetic energy operator and Vc denotes the attractive core potential.

Substituting (1.36) into the above equation, we have

H|φ〉+∑c

(E −Ec)|φc〉〈φc|φ〉 = E|φ〉, (1.38)

where Ec is the energy of the core state φc, i.e.,

H|φc〉 = Ec|φc〉. (1.39)

Eqn. (1.38) can be rewritten as

(T + V ps)|φ〉 = E|φ〉, (1.40)

where the psuedopotential V ps is

V ps = Vc +∑c

(E − Ec)|φc〉〈φc|. (1.41)

14

It can be shown that the second term of V ps is a strongly repulsive potential

which almost cancels the strongly attractive core potential Vc, leaving a weak net

pseudopotetial. Fig. 1.1 shows the ionic pseudopotentials of silicon. Compared with

the −1/r behavior of the Coulomb potential, the potentials shown in the figure aremuch softer.

1.3.3 Pseudopotential construction

To generate the ionic pseudopotentials, we start by solving the all-electron Kohn-

Sham equations of the atom of interest

[−12∇2 + V H(r) + V xc(r)− Z ion

r]φnlm(r) = εnlφnlm(r). (1.42)

Having obtained the all-electron potential, the valence electron orbital energies and

wavefunctions, there are several proposed schemes to construct the intermediate, so-

called screened pseudopotentials[41, 42, 43, 44, 45]; these will not be discussed in

detail here. We assume that we have constructed the screened pseudopotentials (the

meaning of “screened” will be clear later as we proceed our discussion). The valence

pseudo-wavefunctions satisfy

[−12∇2 + V ps,src

l (r)]φpslm(r) = εpsl φpslm(r). (1.43)

Writing

φpslm(r) =upsl (r)

rYlm(Ωr), (1.44)

we have

[−12

d2

dr2+

l(l + 1)

2r2+ V ps,src

l (r)]upsl (r) = εpsl upsl (r). (1.45)

Note that different angular momentum components have different pseudopotentials.

Most of the modern norm-conserving (see definition below) pseudopotentials share

the following characteristics.

15

(i) The eigenvalues calculated with the pseudopotential are identical to all-electron

eigevalues, i.e.,

εpsl = εl. (1.46)

(ii) The logarithmic derivatives of pseudo-wavefunctions and those of the correspond-

ing all-electron ones agree beyond some cut-off radii rcutl , i.e.,

d

drln upsl (r) =

d

drln ul(r) for r > rcutl . (1.47)

(iii) Pseudo-wavefunctions satisfy the norm conservation condition

upsl (r) = ul(r) for r > rcutl , and∫ ∞

0|upsl (r)|2dr =

∫ ∞

0|ul(r)|2dr = 1. (1.48)

The corresponding pseudopotentials are called norm-conserving pseudopotentials.

(iv) The pseudo-wavefunction contains no radial nodes.

After obtaining the screened pseudopotentials V ps,srcl and the valence pseudo-

wavefunctions upsl , we are ready to generate the so-called ionic, or unscreened pseu-

dopotentials by subtracting the electrostatic and exchange-correlation screening con-

tributions due to the valence electrons from the screened pseudopotentials,

V psl (r) = V ps,src

l (r)− V H [ρpsv ; r]− V xc[ρpsv ; r], (1.49)

where ρpsv is the valence pseudo-charge density. The total ionic pseudopotential is

〈r|V ps|r′〉 = V ps(r, r′) =lmax∑l=0

m=l∑m=−l

Y ∗lm(Ωr)V

psl (r)

δ(r − r′)r2

Ylm(Ωr′), (1.50)

which is obviously nonlocal (or semi-nonlocal since only the angular part is nonlocal).

Applying such nonlocal pseudopotentials on the wavefunctions φ(r) for large systems

is formidable, since the integration

V psφ(r) =lmax∑l=0

m=l∑m=−l

∫Y ∗lm(Ωr)V

psl (r)

δ(r − r′)r2

Ylm(Ωr′)φ(r′)dr′, (1.51)

16

has to be carried out for every distinct point r in the space, which involves O(N3)

operations, where N is the number of atoms in the system. The O(N3) scaling is due

to: first, the number of points in the space scales as O(N); the number of projectors

in Eqn. (1.50) scales as O(N); and finally, a single integration in Eqn. (1.51) requires

O(N) operations.

Note that all V psl (r) at large r (r > rcutl ) reduce to the ionic Coulomb potential,

−Z ion/r, indenpendent of l. Therefore, it is expedient to split the semi-nonlocal

pseudopotential into a multiplicative local term plus some short-ranged corrections

V ps(r, r′) = V pslocδ(r− r′) +

lmax∑l=0

m=l∑m=−l

Y ∗lm(Ωr)δV

psl (r)

δ(r − r′)r2

Ylm(Ωr′), (1.52)

where

V psloc(r) = V ps

l−loc(r), (1.53)

and

δV psl (r) = V ps

l (r)− V psloc(r), (1.54)

which vanishes beyond max(rcutl , rcutloc ). Applying the above nonlocal pseudopotential

on wavefunctions requires O(N3) operations in momentum-space based methods (e.g.,

the pseudopotential plane-wave method, which is discussed in the following section.)

and O(N2) operations in the real-space formalism, since in real-space calculations,

the integration in Eqn. (1.51) can be carried out locally. Real-space methods will be

discussed in Chapter 4.

Kleinman and Bylander (KB)[46] proposed an elegant prescription that transforms

the semi-nonlocal pseudopotential into a fully separable, truly nonlocal pseudopoten-

tial

V psKB =

lmax∑l=0

m=l∑m=−l

|δV psl φpslm〉〈φpslmδV ps

l |〈φpslm|δV ps

l |φpslm〉, (1.55)

where φpslm are the atomic valence pseudo-wavefunctions in Eqn. (1.43) and δV psl is

defined in Eqn. (1.54). We can readily verify that the KB form of the nonlocal

17

pseudopotential reproduces the orbital energy for the reference electronic states, i.e.,

states which we use to construct the pseudopotential:

V psKB|φpsl′m′〉 =

lmax∑l=0

m=l∑m=−l

|δV psl φpslm〉〈φpslmδV ps

l |〈φpslm|δV ps

l |φpslm〉|φpsl′m′〉 = δV ps

l |φpsl′m′〉. (1.56)

Applying the KB nonlocal pseudopotential on the wavefunctions requires O(N) op-

erations in the real-space formalism and O(N2) operation in the momentum-space

formalism. (We will return to this point in Chapter 4.) The KB form of pseudopo-

tential considerably reduces the computational effort associated with the operations

of the semi-nonlocal pseudopotential on the wavefunctions and is now almost univer-

sally used in pseudopotential calculations. However, there is also a minor drawback of

the KB nonlocal pseudopotentials: they sometimes lead to unphysical “ghost” states

at energies below or near those of the physical states. These “ghosts” are artifacts of

the KB potential in which the nodeless pseudo-wavefunctions need not be the lowest

possible eigenstates, unlike in the case of the semi-local form. Usually, ghost states

can be eliminated by properly chosing the local pseudopotential component and/or

varing the cutoff radii rcutl of the pseudopotentials.

1.3.4 The pseudopotential plane-wave method

Most of today’s large-scale electronic structure and total energy calculations for

solids employ pseudopotentials. The plane-wave expansion described in Section (1.3.1),

when combined with pseudopotentials to eliminate the highly localized core electrons,

can be very efficient. In this section, we give a brief review of the pseudopotential

plane-wave formalism[37].

Let’s start with the Kohn-Sham equation (1.24), with the external potential V ext

now replaced by ionic pseudopotential V ps

[−12∇2 + V H(r) + V xc(r) + V ps]φi,k(r) = εi,kφi,k(r). (1.57)

18

We have introduced an additional index k to specify the crystal momentum. V ps is

a nonlocal ionic pseudopotential (in the KB form)

V ps(r, r′) =∑τ,i

V psτ (r− rτ −Ri, r

′ − rτ −Ri), (1.58)

where rτ is the position of τ -atom in unit cell i. The operation of the nonlocal

pseudopotential on the wavefunctions should be understood as

V psφi,k(r) =∫

V ps(r, r′)φi,k(r′)dr′. (1.59)

In a plane-wave basis the wavefunction φ is expanded as

φi,k(r) = eik·r∑G

Ci,G(k)eiG·r, (1.60)

and the Hamiltonian matrix elements

HGG′(k) =1

2|k+G|2δG,G′+V H(G−G′)+V xc(G−G′)+V ps(k+G,k+G′), (1.61)

where the Fourier transform of the pseudopotential in (1.58) can be written

V ps(k+G,k +G′) =∑τ

V psτ (k +G,k+G′)ei(G−G′)·rτ . (1.62)

Note that only the Fourier components of the nonlocal part of the pseudopotential

depend on k.

The eigenvalue problem can then be solved by either traditional direct or newer

iterative diagonalization schemes to obtain the eigenvalues and corresponding wave-

functions. Total energy and other quantities of interest are subsequently evaluated[37].

1.4 Beyond LDA

The Kohn-Sham version of the DFT provides a simple and yet powerful means

to calculate the ground state properties of materials. However, applications of DFT

(particularly in its LDA form) have gone far beyond its original concept. LDA results

19

are now routinely used to interpret electronic band structures of solids with tremen-

dous success. Strictly speaking, there is no clear connection between the Kohn-Sham

eigenvalues and single-particle excitation energies. (In contrast, Koopman’s theorem

provides a formal interpretation of the Hartree-Fock orbitals and their corresponding

eigenvalues for atomic and molecular systems.) Therefore, it is rather surprising that

LDA band structures in general agree well with experimental measurements. De-

spite all of its success, there are serious discrepancies between LDA band structures

and quasiparticle energies measured in, e.g., photoemission experiments: (i) the LDA

bandgaps in semiconductors are usually underestimated by ∼ 30 to 50%; (ii) the

LDA bandwidth in some simple metals such as Li and Na are ∼ 10 to 30% too large.

For the transition metal Ni, the LDA bandwidth is 4.5 eV as compared to 3.3 eV

experimentally; and (iii) the ionization energies of atoms, molecules or clusters are

usually underestimated.

Sham and Schluter[53, 54] and Perdew and Levy[55] have shown formally that

there is an explicit correction to the Kohn-Sham bandgap even when the exact

exchange-correlation functional is used, and therefore LDA is not responsible for

the above-mentioned difficulties. Methods to improve the LDA description of quasi-

particle energies have thus attracted considerable research interest. The GW ap-

proximation, generalized Kohn-Sham schemes (GKS)[49, 50, 51], and time-dependent

LDA[47, 48] represent recent research efforts along this direction. In the following,

we give an brief introduction to the GW approximation.

1.4.1 The GW approximation

As we have mentioned in the previous section, the Kohn-Sham eigenvalues do

not in general agree well with experimental excitation energies, which, in principle,

can be obtained from many-body perturbation theory by solving Dyson’s equation.

Excitations of a system of strongly interacting particles (e.g., electrons) can often be

20

described in terms of weakly interacting quasiparticles, which are composed of bare

particles plus polarization clouds surrounding them. The interaction between quasi-

particles is thus a screened, rather than bare, Coulomb potential. There are funda-

mental differences between a non-interacting single particle picture in the Kohn-Sham

scheme and a quasiparticle description. For example, a non-interacting particle has

infinite lifetime, whereas the lifetime of a quasiparticle is finite. The energy differ-

ence between a non-interacting particle and a quasiparticle is usually described by the

self-energy operator Σ, which must account for all exchange-correlation effects beyond

the Hartree approximation. The GW approximation of Hedin and Lundqvist[52] is

derived systematically from many-body perturbation theory with the self-energy ap-

proximated by a product of the one-particle Green function G and the dynamically

screened interaction W .

Green functions are powerful tools in many-body theory. Quasiparticle properties

such as energies, lifetimes, etc., can be determined by single-particle Green functions.

The one-particle Green function is defined as

iG(x, x′) = 〈N |T [Ψ(x)Ψ(x′)†]|N〉=

〈N |Ψ(x)Ψ(x′)†|N〉 for t > t′ (electron)−〈N |Ψ(x′)†Ψ(x)|N〉 for t′ > t (hole),

(1.63)

where x = (r, t), |N〉 is the N -particle ground state, Ψ(x) is the field operator and Tis the time-order operator. The Fourier transform of the one-particle Green function

G(x, x′), G(r, r′;ω), is related to the spectral function A(r, r′;ω) by

G(r, r′;ω) =∑i

φi(r)φ∗i (r

′)ω − ωi

=∫

A(r, r′;ω′)ω − ω′ dω′, (1.64)

and

A(r, r′;ω) =1

π|ImG(r, r′;ω)|, (1.65)

where φi are quasiparticle wavefunctions. The above expression is also called the

spectral representation of the Green function. The equation of motion of the one-

21

particle Green function is

[i∂

∂t−T−V ext]G(x, x′)+i

∫v(r, r′′)〈N |T [Ψ(r′′, t)†Ψ(r′′, t)Ψ(r, t)Ψ(r′, t′)†]|N〉dr′′ = δ(x, x′).

(1.66)

In the above equation, T is the kinetic energy operator, V ext is the external potential

and v(r, r′) is the interaction between particles, for example, the Coulomb interac-

tion for electrons. The term with four field operators contains a two-particle Green

function and describes the two-body correlations in the system. In principle, Eqn.

(1.66) can be taken as a starting point from which an infinite chain of equations is

generated to introduce successive corrections[52]. Another approach is to introduce

a nonlocal time-dependent self-energy operator Σ, defined by the following equation

[i∂

∂t− T − V (x)]G(x, x′)−

∫Σ(x, x′′)G(x′′, x′)dx′′ = δ(x, x′). (1.67)

V (x) is now the total average potential in the system[52], i.e., the external potential

plus the Hartree potential:

V (x) = V ext(x) + V h(r). (1.68)

The solution of Eqn. (1.67) can be formally written as

G(x, x′) = Go(x, x′) +∫

Go(x, x′′)Σ(x′′, x′′′)G(x′′′, x′)dx′′dx′′′, (1.69)

where G0(x, x′′) is the non-interacting one-particle Green function, which satisfies

[i∂

∂t− T − V (x)]G0(x, x′′) = δ(x, x′′). (1.70)

Eqn. (1.69) is called Dyson equation, which may be written in a more compact and

familiar form

G(k, ω) =Go(k, ω)

1− Σ(k, ω)Go(k, ω)(1.71)

in its Fourier space representation. For a time independent external field, the Fourier

transform of Eqn. (1.67) is

[ω − T − V (r)]G(r, r′;ω)−∫Σ(r, r′′;ω)G(r′′, r′;ω)dr′′ = δ(r, r′), (1.72)

22

and the corresponding quasiparticle wavefunctions satisfy the following equation

[T + V (r)]Ψnk(r) +∫Σ(r, r′;Enk)Ψnk(r′)dr′ = EnkΨnk(r). (1.73)

The complicated many-body character of the above equations is due to the fact that

the self-energy operator Σ is time (i.e., energy) dependent. Hartree and Hartree-Fock

approximations are two examples in which Σ is independent of energy:

ΣH = 0;ΣHF = V ex(r, r′),

(1.74)

where V ex is the exchange potential.

From Eqn. (1.73), our task to calculate the quasiparticle energy Enk and the

corresponding wavefunctions is reduced that of evaluating the self-energy operator Σ

and solving Eqn. (1.73). In general, the Enk are complex; the imaginary part gives

the quasiparticle lifetime. It can be shown that the self-energy Σ and the one-particle

Green function G are related by four integral equations, which are to be solved with

an iterative process[52]:

Σ(1, 2) = i∫

G(1, 3)Γ(3, 2; 4)W (4, 1+)d(34), (1.75)

W (1, 2) = v(1, 2) +∫

v(1, 3)P (3, 4)W (4, 2)d(34), (1.76)

P (1, 2) = −i∫

G(1, 3)G(4, 1+)Γ(3, 4; 2)d(34), (1.77)

and

Γ(1, 2; 3) = δ(1, 2)δ(1, 3) +∫

δΣ(1, 2)

δG(4, 5)G(4, 6)G(7, 5)Γ(6, 7; 3)d(4567). (1.78)

Here we adopt the abbreviated notation (1) ≡ x1 = (r1, t1), etc., and 1+ means that

t1 → t1 + δ where δ is a positive infinitesimal. We have introduced three more quan-

tities, namely the (dynamically) screened Coulomb interaction W , the irreducible

polarization propagator P , and the vertex function Γ. The screened Coulomb inter-

action W is related to the dielectric function ε(x, x′) by

W (1, 2) =∫

v(1, 3)ε−1(3, 2)d(3), (1.79)

23

where the dielectric function ε(x, x′) is

ε(1, 2) = δ(1, 2)−∫

v(1, 3)P (3, 2)d(3). (1.80)

Eqns. (1.75—1.78) can be solved iteratively. For example, we start with Σ = 0 in

Eqn. (1.78), which gives the first approximation of Γ

Γ(1, 2; 3) = δ(1, 2)δ(1, 3), (1.81)

from which we can evaluate Σ

Σ(1, 2) = iG(1, 2)W (2, 1+). (1.82)

The above expression of the self-energy Σ is called the GW approximation. Sometimes

it is more convenient to express the above quantities in their Fourier transforms (for

example, when working with pseudopotential plane-wave methods):

Σ(r, r′, ω) =i

2π

∫G(r, r′, ω + ω′)W (r, r′, ω′)dω′. (1.83)

The one-electron Green function can be calculated by

G(r, r′, ω) =∑n,k

φnk(r)φ∗nk(r

′)ω − εnk − iδnk

=∑k

∑G,G′

ei(k+G)·rGG,G′(k, ω)ei(k+G′)·r′, (1.84)

whereG andG′ are reciprocal lattice vectors, k is the wavevector in the first Brillouin

zone and n is the band index. εnk and φnk are the one-eletron eigenvalues and

the corresponding wavefunctions calculated by, for example, LDA. Similarily, for the

screened interaction W , we have

W (r, r′, ω) =∑k

∑G,G′

ei(k+G)·rWG,G′(k, ω)ei(k+G′)·r′, (1.85)

where

WG,G′(k, ω) = ε−1G,G′(k, ω)v(k+G′). (1.86)

24

The dielectric function ε is related to the irreducible polarizability P by

εG,G′(k, ω) = δG,G′ − v(k+G)PG,G′(k, ω). (1.87)

The real task in GW calculation is to evaluate the dynamic dielectric function

εG,G′(k, ω) or the irreducible polarizability PG,G′(k, ω) in Eqn. (1.87). Note that if

the dielectric function ε (and therefore W ) is calculated using the approximate ir-

reducible polarization propagator P ≈ P 0 = −iG(1, 2)G(2, 1), it corresponds to therandom-phase approximation (RPA)[36]. In most modern GW calculations, the dy-

namic dielectric function is approximated by extending the static dielectric function

calculated from LDA wavefunctions to finite frequencies using the so-called general-

ized plasmon-pole model[36].

Although the GW approximation for computing the quasiparticle energies was

proposed in the 1960’s[56, 52], it was not applied to real materials until the mid

1980s, starting with the work of Hybertsen and Louie[57, 58, 36]. Since then, the

GW approximation has been successfully applied to calculate quasiparticle energies

for various systems. For a review of the GW approximation and its applications, see,

for example, Aulbur, et al.[59].

1.4.2 Generalized Kohn-Sham schemes

The GW approximation has been shown to resolve many of the discrepancies be-

tween LDA band structures and experiments for a variety of materials, ranging from

metals to wide-gap insulators. Unfortunately, GW methods are usually computation-

ally intensive, making it impossible (or very difficult) to perform GW calculations for

large systems. Moreover, most of the GW methods calculate only the quasi-particle

excitation energies. In many cases, it will be very appealing if we can calculate both

the excitation energies and the total energy of a system at the same time with rea-

sonable accuracy. Although it is possible to calculate the total energy within the GW

approximation[60], the accuracy of such calculations has yet to be verified[49].

25

Recently, there has been some interest in the so-called generalized Kohn-Sham

(GKS) schemes[49, 50, 51], which attempt to develop KS-like total energy models

with improved quasi-particle description, while keeping the extra computational effort

to a minimum. For details of recently proposed GKS models, see refs [49, 50, 51].

26

Chapter 2

Carbon and Boron NitrideNanotubes

2.1 Introduction

Carbon is a rather special element which can form very flexible bonding networks,

for example, one dimensional (sp1) carbon chains or rings, two dimensional sp2 lay-

ered structures such as graphite, and three dimensional (sp3) diamond. In contrast,

no sp2 silicon phases are known to exist. It is remarkable that both graphite and dia-

mond are very stable. (Under normal conditions, graphite is just slightly more stable

than diamond). Whereas graphite is a zero-gap semimetal and has the highest in-

plane stiffness, diamond is a wide-gap insulator and is the hardest three-dimensional

material available so far. Diamond and graphite also have the highest thermal con-

ductivity of known materials in 3-D and 2-D. However, the story of carbon seems

to not have ended yet. The discovery of the nearly spherical molecule C60[61] and

related fullerene structures, and the subsequent synthesis of carbon tubules[62], now

referred to as carbon nanotubes, have inspired another wave of research into carbon-

based materials[63]. For an excellent review on various form of carbon materials (for

example, carbon fibers, glassy carbon, carbon black, amorphous carbon, etc.) see

Dresselhaus, et al.[63].

27

Figure 2.1: Rolling up a graphene sheet into a tube.

In a 1991 paper, Iijima reported the synthesis of a new type of carbon structure,

“needle-like tubes”, using an arc-discharge vaporization method similar to that used

for fullerene synthesis[62]. The original carbon tubules consisted of several (ranging

from 2 to about 50) concentric graphitic shells, and had diameters of a few nanome-

ters. These multi-shelled carbon microtubules are now called multi-walled carbon

nanotubes. Single-walled carbon nanotubes were later synthesized by arc-discharge

vaporization [64, 65, 66], laser ablation of carbon targets[67] and chemical vapor de-

position (CVD)[68, 69, 70] in the presence of a catalyst (Fe, Ni, or Co).

2.1.1 Specification of carbon nanotubes

From a purely geometrical perspective, a single-walled carbon nanotube is formed

by rolling up a graphene (single-layer graphite) sheet into a cylinder (see Fig. 2.1).

(Note that it is not likely that carbon nanotubes are formed this way in real ex-

periments. In fact, the nucleation and growth mechanisms of carbon nanotubes are

still an active research topic. Recently, we have proposed a new nucleation model

for carbon nanotubes[71], which is presented in Section 2.2.) A single-walled carbon

28

a1

a2

C

5a1

2a2

θθθθ

Figure 2.2: Specification of single-walled carbon nanotubes. The circumferentialvector shown in the figure corresponds to that of a (5,2) nanotube. The chiral angleθ is also shown.

nanotube is uniquely specified by its circumferential vector C = na1 + ma2[72] as

shown in Fig. 2.2, where a1 and a2 are the unit vectors of graphite, and n and m are

integers. A single-walled carbon nanotube with C = na1 + ma2 is called an (n,m)

nanotube.

Two particular classes of single-walled nanotubes are worth mentioning. One is

(n,0), also called zig-zag nanotubes, and the other is (n,n) or armchair nanotubes.

Generic (n,m) nanotubes are called chiral nanotubes. In contrast, (n,0) and (n,n)

nanotubes are achiral (non-chiral). The diameter d of an (m,n) nanotube can be

calculated by[63]

d =√3aC−C(m2 +mn + n2)1/2/π, (2.1)

where aC−C is the C–C bond length. The chiral angle (or wrapping angle) θ is

θ = tan−1[√3m/(m+ 2n)]. (2.2)

29

Kx

Ky

E

Figure 2.3: Near-Fermi surface electronic structure of graphene. The Fermi surfaceis reduced to two distinct (Fermi) points.

For (n,0) nanotubes, θ = 0o. However, the wrapping angle of (n,n) nanotubes is not

zero but θ = 30o (although they are also non-chiral). A double walled nanotube can

be specified by indexing each of the walls, for example, (5,5)@(10,10).

2.1.2 Novel electronic and structural properties

Carbon nanotubes have been the focus of intensive research for the past decade

due to their unique properties that could have impact on broad areas of science and

technology, ranging from strong and ultra-light composites[73] to nanoelectronics[74,

75, 76]. The novel electronic properties[77, 78, 79] of carbon nanotubes are related

30

K

Γ

K

Γ

(a) (b)

Figure 2.4: Allowed lines of wavevectors in carbon nanotubes: (a) the allowed linesmiss the Fermi points and (b) the allowed lines pass through the Fermi points. Thehexagon is the Brillouin zone of graphene.

to the unique Fermi surface structrure of graphene, combined with the quasi one-

dimensional nature of the tube.

Fig. 2.3 shows the 2-D tight-binding band structure of graphene, considering only

the pz orbital of carbon. The Fermi surface of graphene is reduced to two distinct

points in the Brillouin zone( BZ). Because of the periodic boundary condition of the

nanotube along the circumferential direction, only certain lines of wavevectors in the

BZ are allowed. If the allowed lines pass the Fermi points (corresponding to (n,m)

nanotubes with n−m = 3k, where k is an integer) the nanotube is metallic, otherwise

it is semiconducting, see Fig. 2.4. Therefore, from this purely geometrical argument,

we conclude that, depending on its wrapping indices, a carbon nanotube is either

metallic or semiconducting. Furthermore, since the dispersion of bonding and anti-

bonding pπ bands around the Fermi-points (see Fig. 2.3) is almost linear, the band

31

(8,8)(5,5) (7,7)(6,6)

(6,0)

(9,0)

(12,0)

(4,4)

Figure 2.5: Electronic properties of carbon nanotubes. Solid circles are metallic;triangles are small bandgap semiconductors and the rest are semiconductors withrelatively large bandgaps.

gaps of the semiconducting nanotubes must have an envelope function proportional

to 1/r, where r is the tube diameter.

The discussion thus far does not consider the effect of curvature on the electronic

structure of nanotubes. It turns out that the metallic nature of an (n,m) nanotube

with n−m = 3k and k = 0 is not stable against the perturbation induced by the cur-

vature, which opens up a small bandgap for these nanotubes. As a result, only (n,m)

nanotubes with n = m remain metallic[77] and are free from the Peierls distortion

at room-temperature[78]. (The Peierls distortion is strongly suppressed due to the

rigidity of the sp2 bonding network and the low density-of-states (DOS) of graphene

near the Fermi level, therefore can be easily destroyed by thermal effects.) Therefore,

we have the following general rules for predicting the band gaps of a carbon nanotube:

32

Nanotube Young’s Modulus (TPa)(6,6) 1.20(9,3) 1.15(10,2) 1.11(12,0) 1.03(10,0) 1.18(8,0) 1.52(6,0) 2.12

(a)

Nanotube Young’s Modulus (GPa)(6,6) 554(9,3) 525(10,2) 522(12,0) 500(10,0) 518(8,0) 577(6,0) 628

(b)

Table 2.1: Elastic strength of single-walled carbon nanotubes, (a) assuming the ef-fective cross-sectional area is πr2, where r is the radius of the tube, and (b) thecross-section is the area per nanotube in a nanotube bundle with inter-tube distance3.0 A.

(1) if n = m, metallic;

(2) if n−m = 3k, where k is an integer and k = 0, small band gap semiconductor;

(3) others, semiconducting with band gaps proportional to 1/r.

The exceptionally high elastic strength[80, 81, 82, 83, 84, 85] of carbon nanotubes

is due to the extremely strong sp2 bonding network of carbon. Table 2.1a lists the

Young’s moduli of several carbon nanotubes we calculated with tight-binding param-

eters by Xu, et al.[15], assuming that the effective cross-section is πr2, where r is

the radius of the nanotube (measured from the center of the tube to the carbon nu-

clei). However, a more reasonable estimate of the effective area might be the area per

tube in a nanotube bundle (or nanotube crystal). Rescaled results according to this

estimate are given in Table 2.1b.

In the following sections, we first propose a new nucleation model for carbon

nanotubes[71]. Detailed simulation using a hybrid TB/classical potential technique

shows that this model qualitatively agrees with experimental observations in several

aspects. Next, we address the issue of plastic deformation of carbon nanotubes under

high axial stress[86]. The elastic limits of carbon nanotubes are found to be higher

33

than any other known material and depend strongly on the chiral angle. Finally,

elastic limits of boron nitride nanotubes are studied and compared with those of

carbon nanotubes[87].

2.2 Nucleation of carbon nanotubes without pen-

tagonal rings

The unique one-dimensional structure of carbon nanotubes depends critically upon

a nucleation mechanism which somehow restricts graphitic growth to a single axis.

The transient and extreme growth conditions of carbon nanotubes have obscured

the mechanism of nucleation. Most hypothetical nucleation mechanisms invoke pen-

tagon formation to produce a hemispherical graphitic cap, with the cap’s edges either

open[88, 89, 67], attached to a substrate via a tubular segment[90, 91, 92], or mated

with a second cap into a closed fullerene[93, 94, 95]. These models all require an

abrupt transition from a regime favoring pentagon formation (when the cap is form-

ing) to one that favors exclusively hexagon formation (when the tube is lengthening)

once exactly six pentagons have formed. The last two models are also inconsistent

with experimental evidence favoring open-ended growth[96, 97]. Other models nucle-

ate a nanotube from a ∼40-atom polyyne ring[96, 98]. Unfortunately, such a large

carbon ring is unlikely: carbon clusters change from simple rings to double rings and

cages above ∼20 atoms[99, 100, 101].Very recently, Bandow et al. have discovered that the temperature during synthe-

sis controls the diameter of the nanotubes[102]. Since the diameter of a nanotube is

fixed by its size at nucleation, these results provide rare experimental clues into nu-

cleation itself. Here we propose a new nucleation model for carbon nanotubes which:

(1) contains only hexagonal rings within the tube nucleus and therefore does not re-

quire a transition from pentagon to exclusively hexagon formation, (2) can explain

the temperature dependence of the nanotube diameter distribution, (3) accounts for

34

Figure 2.6: A schematic diagram of the nucleation model in which a nanotube formsvia edge mediated opening of a double-layered graphitic patch.

the narrow diameter distribution of single-walled nanotubes, and (4) can explain the

wider diameter distribution of multi-walled nanotubes.

Carbon nanotube synthesis apparently requires a surface on either an electrode

or a metallic particle. In contrast, smaller closed-shell fullerenes typically form in a

low-density, dangling-bond-dominated gas phase. Surfaces favor the growth of small

flat graphitic patches. (Such graphitic patches can form even in much higher-density

environments[103]). It may be difficult to imagine a means by which a single graphitic

patch could curl into the nucleus of a nanotube without the incorporation of pentagons

to produce the needed curvature, as shown schematically in fig 2.1. One such nucle-

ation mechanism is for a single graphitic sheet to curl around a small particle with

the particle/graphite attraction overcoming the mean curvature energy of graphite

at a characteristic size of ∼ 1 nm. We herein describe a natural kinetic pathway by

which a two-layered graphitic patch can transform into the nucleus of a nanotube.

35

Fig. 2.6 illustrates schematically such a nucleation model. A double-layered

graphitic patch nucleates on a surface, with edges on adjacent layers interconnected

via bridge atoms[104]. When edges are bridged on opposite sides of the patch, such

a two-layered patch looks similar to a very short flattened nanotube[105, 106]. A

small-diameter flattened nanotube is unstable towards popping open, since the at-

traction between the inner surfaces cannot counteract the excess curvature energy on

the edges. If a double-layered patch pops open, then growth in the bridged direction

is arrested and further carbon addition lengthens the nucleus axially to form a nan-

otube. At least one experimental image directly supports this model by capturing a

possible intermediate state of nucleation: High resolution transmission electron mi-

croscopy images show a small-diameter flattened boron nitride nanotube extending

outward in one direction from a metallic particle and simultaneously opening into a

circular cross-section[107]. (However, not too much should be drawn from any single

TEM image of a rare intermediate state.) Since no pentagons are involved, the mech-

anism need not appeal to a fortuitous transition from pentagon formation to exclusive

hexagon formation during growth. The curvature necessary to form a tube nucleus

arises not from pentagons, but from the extreme curvature induced by bridging across

the interface between two adjacent parallel graphitic patches.

Although this model is qualitatively attractive, detailed numerical modelling is

necessary to determine if the energetic balance actually favors this route to nucle-

ation under realistic conditions. We use a tight binding total energy model[15] to

simulate the edge-mediated opening of a double-layered graphitic patch into a single-

walled nanotube nucleus. The tight binding parameterization accounts for the co-

valent electronic energetics and the nearest-neighbor repulsion of atomic cores. The

weak interlayer attraction is incorporated through an empirical Lennard-Jones po-

tential which yields the correct equilibrium interlayer distance and interlayer binding

energy (0.035± 0.015 eV/atom[106]) for graphite.

36

(a) (b)

Figure 2.7: The relaxed structures from T=0 molecular dynamics for two-layeredgraphitic patches of sizes necessary to form nanotube nuclei of diameters (a) 13.7 Aand (b) 27.4 A. The smaller nucleus spontaneously opens into a circular cross-sectionwhich defines a strong preferential growth axis whereas the larger nucleus remainsflattened.

We model two parallel layers of graphite 3.4 A apart with periodic boundary

conditions along one direction and bridge atoms connecting the exposed transverse

edges in the perpendicular dirction. For simplicity we set T = 0 in the simulation; the

primary effects of non-zero temperature are easily incorporated as will be explained

later. Within a few femtoseconds, bulbs form at the bridged edges as the bridge

atoms incorporate themselves into an energetically favorable continuous sp2 bonding

network. If the energetic cost of curvature in the bulbs overcomes the interlayer

attraction between the two layers, then the incipient tube pops open to a circular

cross-section, which then defines a preferential axis for one-dimensional growth. Fig.

2.7a shows this opening process during nucleation of a (10,10) tube. In contrast, the

37

nucleus of a (20,20) tube (Fig. 2.7b) does not pop open, as the interlayer attraction

dominates the energetics.

0

1

2

3

4

5

6

7

8

5 10 15 20 25 30

Energy (eV)

Antipodal Distance (Å)

D=20Å

D=27Å

D=33Å

Figure 2.8: The energy as a function of a reaction coordinate (i.e., the distance be-tween opposite inner surfaces as the tube nucleus opens) for nucleation patches ofvarying diameter when opened and of length 1 nm. The smallest patches have nobarrier to opening. For sufficiently large patches the opened state is energeticallydisfavored. For intermediate sizes, thermal excitation can activate opening on exper-imental time-scales.

Fig. 2.8 shows the total energy during the opening process for incipient nuclei

of (n, n) nanotubes with radii 20, 27, and 33 A (similar results are expected for

other wrapping indices (n,m)). Nuclei smaller than the cross-section of a D = 23 A

diameter tube have no energetic barrier to popping open. The size of the barrier scales

with the length of the tube segment; we choose a length of one nm, which produces

a roughly square patch with a plausible aspect ratio (we return to this point later).

In larger tube nuclei, the energetic barrier against assuming the circular cross-section

38

increases roughly linearly with size as

EB = ε(D − 23A) eV (2.3)

with ε ≈ 0.2 eV/A. The linearity follows from the linear dependence of the interlayer

“glue” area on the diameter D. Above D = 34 A the circular cross-section is en-

ergetically disfavored. Since the curvature energy is more important in multilayered

structures while the interlayer binding energy of the inner surfaces is constant[106],

multiwalled tubes could nucleate at larger diameters; double-walled nanotubes with

inner diameters of ∼45A have been observed[62].

At finite temperature, the system can thermally surmount the barrier towards

popping open. Experimentally, the yield of the smallest-diameter nanotubes (D ≤ 1

nm) is only weakly dependent on temperature[102], in accord with our model, as

these small nanotube nuclei have no energetic barrier to opening. In contrast, the

yield of larger-diameter nanotubes is very sensitive to synthesis temperature with

higher temperatures yielding larger-diameter tubes[102]. Given the energetic barrier

towards opening as a function of the incipient tube diameter, simple kinetic arguments

can then estimate the theoretically expected temperature dependence of the largest

possible nucleation patch produced on experimentally relevant timescales.

These kinetics are dominated by the size dependence of the activation barrier. The

thermally activated opening rate for an incipient nanotube nucleus is approximately

rate = f(R, T )ωvibe−EB/kBT , (2.4)

where f(R, T ) gives the temperature-dependent size distribution of appropriately-

shaped graphitic patches, EB is the calculated energetic barrier and ωvib is the attempt

rate (comparable to the characteristic phonon frequency for transverse vibrations of a

graphitic sheet at a wavevector comparable to the inverse tube radius, i.e., ωvib ∼ 1012

Hz). Several vibrational degrees of freedom might also facilitate opening, as could

39

be revealed by direct finite-temperature molecular dynamics. For small-radius tubes

the exponential factor is absent, hence the number of small-radius tubes produced

reflects directly f(R < R∗, T ) where R∗ is the largest diameter without an activation

barrier to opening. In fact, the number of smallest-radius tubes (R < 1 nm) produced

is nearly independent of temperature from T = 780 K to T = 1000 K[102]. Therefore

f(R < 1nm, T ) is a weak function of T for 780 K < T < 1000 K within the model.

Since the emergence of an energetic barrier at larger radii has no direct influence on

f(R, T ), f(R, T ) should have a relatively weak temperature dependence at somewhat

larger R as well, a regime in which the temperature dependence of the nucleation

rate is then dominated by the rapidly varying exponential term. At sufficiently high

temperatures other physical processes could preemptively cut off f(R, T ) for large R,

regardless of the exponential factor. Recent experiments above 1200 K (well above

the temperatures considered here) show a reduction in typical tube diameter[108].

The number of large-radius tubes falls off rapidly as a function of diameter[102],

consistent with a rapid falloff in the Boltzmann factor. Therefore we focus on the

exponential factor in analyzing the nanotube diameter distribution in the temperature

range from 780 to 1000 K. For example, at 1000 K, the single-site opening rate for

a barrier of 1.5 eV is roughly 104 sec−1. For a barrier of 2.0 eV the opening rate is

∼ 101 sec−1, too slow for significant tube production, while at EB = 1 eV the opening

rate is very fast, ∼ 107 sec−1. We take EB ≈ 1.5 ± 0.3 eV as the maximum barrier

beyond which nuclei do not form on the experimental timescale in the laser ablation

cloud or the dynamic plasma arc discharge.

As tube diameter (and energetic barrier) increases, maintenance of a constant

nucleation rate implies a higher temperature, i.e.,

EB,2/kBT2 = EB,1/kBT1. (2.5)

In other words, at higher temperatures a fixed nucleation rate occurs at a larger

40

radius. The overall diameter distribution at large diameters should shift upwards as

the temperature is increased. Examining the data of Bandow et al.[102] this upward

shift is (1.0−1.3)× 10−2A/K for both the peak and the downslope of the distribution.

Since the maximal surmountable barrier is much larger than kBT , a small change

in temperature produces a large change in the diameter of the largest tube which

can nucleate. As mentioned above, the difference in opening barrier for two tubes

depends linearly on their diameter difference,

∆EB = ε∆D. (2.6)

Where ε ≈ 0.2 eV/A. Therefore a change in temperature ∆T yields a typical diameter

change of∆D

∆T=

EBεT≈ 0.8× 10−2A/K, (2.7)

reasonably consistent with the experimental result of (1.0 − 1.3) × 10−2A/K for the

temperature variation in the most probable nanotube diameter[102]. This pop-open

nucleation mechanism sets a temperature-dependent upper limit on the nanotube

diameter distribution through suppression of thermally activated nucleation; the rel-

atively temperature-independent lower limit at D ≈ 0.8 nm[102, 109, 64, 65] might

then be set by the reduced chemical stability of the highly strained smallest-diameter

tubes[109].

Within this simplified model, the predicted diameter distribution is shifted to-

wards larger diameters than those observed (i.e., ∼ 29 ± 9 A versus the ∼15 A ob-

served at 1000 K; the error bar arises primarily from the uncertainty in the strength

of the interlayer attraction). This discrepancy is expected, for three reasons. First,

an attractive external surface adjacent to one graphitic layer reduces the propensity

for popping open. To quantify this effect, we introduced a rigid external surface with

the same attractive potential as graphite itself; this additional stabilizing force re-

duces the characteristic nucleus diameter by ∼ 3 A. Second, the incipient nuclei of

41

larger-diameter tubes would tend to be longer axially and therefore would have rela-

tively larger barriers to opening than the fixed-length nuclei herein considered. This

effect would also increase ε slightly but would not change the overall consistency with

experiment. Third, the calculations simulate an ideal nanotube nucleus wherein pe-

riodic boundary conditions suppress interlayer bridging along one direction. In more

realistic configurations, the irregular shape and registry of the graphitic patches and

the possible competition between bridging on different pairs of edges reduces the typ-

ical size at which popping open occurs. This competition does not prevent popping

open, since the first edge to bridge induces local bulging, which partially separates

the adjacent edges and leaves the opposite edge as the most likely to bridge next. The

differing reactivities of the zigzag and armchair graphitic edges might favor wrapping

indices around the armchair (n, n) configuration[67, 102, 110, 111]. Finally, patches

must grow from a small size. to pop open at a radius R, one must have failed to

pop open at smaller sizes. This reduces the chance of forming nuclei of large radius

nanotubes.

The nucleation mechanism for carbon nanotubes is one of the most important

open questions in nanotube synthesis. Although definitive determination on this

point awaits further experiments, we have described a plausible and qualitatively new

nucleation mechanism. This mechanism, the spontaneous opening of double-layered

graphitic patches, is quantitatively consistent with the recently discovered tempera-

ture dependence of the nanotube diameter and also produces the required curvature

without the necessity for the formation of exactly six pentagonal rings within the

incipient tube. Independent of the actual nucleation mechanism, the nucleation rate

is likely much higher for single-walled compared to multi-walled tube synthesis. Tube

nucleation and growth occur at temperatures (∼1000 K) where carbon atoms will

likely stick to exposed graphitic surfaces. If the nucleation rate is low, then the con-

centration of tube nuclei is low compared to the concentration of carbon atoms/dimers

42

and outer walls can accrete, particularly when the tubes are long and atoms cannot

quickly diffuse along the outer surface to the reactive ends. If the nucleation rate is

high, then the incipient tube density is high, and the tube nuclei are more likely to

collide with each other rather than with ambient carbon atoms/dimers, so the system

would tend to form bundles of single-walled tubes, as observed.

2.3 Plastic deformation of carbon nanotubes

Carbon nanotubes have remarkable elastic properties. On one hand, they are rel-

atively flexible under twisting[112], bending and squashing[113, 114, 115, 116, 117].

On the other hand, the axial elastic strength (measured by the Young’s modulus)

of carbon nanotubes is exceptionally high[80, 81, 82, 83, 84], of the order of 1 TPa

and is nearly independent of wrapping indices (see Table 2.1). As a comparison,

the Young’s moduli of titanium, stainless steel, and tungsten are 117 GPa, 193 GPa

and 400 GPa, respectively. The extreme elastic strength of carbon nanotubes is not

surprising since graphite is the stiffest planar material known. In fact, most struc-

tural applications proposed for carbon nanotubes, for example, ultra strong fibers

[73] and robust atomic force microscopy tips[118, 119], rely mainly upon their ex-

treme elastic strength. However, elastic response alone provides a limited picture

of mechanical properties. Beyond a certain stress, the elastic limit, a material de-

forms irreversibly via changes in chemical bonding topology. As we have mentioned

in Section 2.1, a nanotube is defined by its radius and chiral angle (or wrapping an-

gle), i.e., the wrapping index. Since the high symmetry of a graphite sheet implies

isotropic elastic properties, a nanotube’s radius alone dominates its elastic response.

Symmetry-breaking plastic distortions, by contrast, are quite sensitive to wrapping

angles.

A π/2 bond rotation in a graphitic network transforms four hexagons into two

pentagons and two heptagons, which is known as a Stone-Wales transformation[120].

43

(9,0) (6,6)

Figure 2.9: Two tubes with bond rotation defects.

The structure elongates along the axis connecting the pentagons and shrinks along

the perpendicular direction, as shown in Fig. 2.9. Thus, in a nanotube, rotation

of a bond from a predominately circumferential direction to a predominately axial

orientation lengthens the tube. If the tube is under tension, such defects relieve

strain and constitute a fundamental mode of plastic deformation. The efficiency of

this strain release depends sensitively on the alignment between the defect and the

tube axis. For example, essentially no strain is relieved by rotation of a bond initially

oriented π/4 from the circumference. One third of the bonds in an (n, n) tube are

aligned precisely with the circumference, the optimal orientation from which to rotate

for the release of strain. In contrast, the most favorable bonds in an (n, 0) nanotube

are misaligned by π/6 from the circumference, as shown in Fig. 2.9.

The occurence of a structural transformation such as plastic deformation is af-

fected by both thermodynamics and kinetics. Whereas the energy barrier controls

the detailed kinetics (i.e., transformation rate, intermediate states, and so on), the

44

phase diagram (i.e., the transition tension) is determined by the relative Gibbs free

energies of the elastically and plastically distorted structures. The energetic barrier

for the Stone-Wales transformation has been discussed extensively in the literature

for the carbon system[121, 122, 123, 124]. As our primary goal is to determine the

phase diagram of carbon nanotubes under tension, we focus on the relative Gibbs free

energies of the elastically and plastically distorted structures.

We use quantum-mechanical atomistic calculations to investigate the plastic de-

formation of carbon nanotubes under strain. A tight binding total energy scheme[15]

is employed. To separate the effects of tube radius from those of wrapping indices,

we study two families of tubes, a series of (n, 0) tubes of different radii and a series

of (n,m) tubes of nearly equal radius. In all cases we first consider transitions at

constant tension, later addressing the transitions at constant length.

Fig. 2.10 shows the energy versus length for the pristine and plastically deformed

(8,0) tube. Since a single defect would produce an inconvenient kinked unit cell,

the deformed tubes we studied contain a pair of well-separated rotated bonds on

opposite sides (but all quoted results are normalized to a single defect). The common

tangent of the energy-versus-length curves gives the critical force (Fc) at which the

Gibbs free energies G = E − FL of the two configurations are equal; Fc = δE/δL,

where δE and δL are the energy difference and elongation across the transition. (This

construction is familiar in three dimensions as yielding the transition pressure between

two structures.) This structural phase transition to a defective phase is eventually

arrested by a repulsive defect-defect interaction at high defect density. In our study

the defect density is low enough that this interaction does not significantly disturb the

onset of plastic deformation (doubling the defect separation changes the transition

tension by less than 1%). Since one expects Fc roughly proportional to the radius,

R, of a tube, we define the reduced transition tension fc ≡ Fc/R, which should vary

weakly amongst nanotubes of identical wrapping angle.

45

0

10

20

30

40

50

60

12 12.5 13 13.5 14 14.5 15

Energy (eV)

Length (A)˚

(8,0)

Figure 2.10: Energy versus length for pure (solid) and defective (dashed) (8,0) nan-otubes. The cell contains 96 atoms per bond rotation defect with E = 0 for the pureundistorted tube.

We define the wrapping angle χ as the angle between the circumferential direction

and the best bonds to rotate. χ ranges from zero for an (n, n) tube to π/6 for an

(n, 0) tube. Note that χ is different from the wrapping angle we defined in Section 2.1

but relates to the latter by χ = 30o− θ. The C2v symmetry of a bond rotation defect

implies that physical quantities are even functions of χ with period π. Hence, to low-

est order in angular dependence, they are well approximated by a(R) + b(R) cos 2χ,

with radius-dependent coefficients a(R) and b(R). Fig. 2.11 reveals a nearly linear

dependence of transition tension on cos 2χ for the family of nearly equal-radius tubes

(12,0), (10,2), (9,3), (8,4), (6,6). The elastic limit varies from about 100 nanoNew-

tons per nanometer for the (6,6) tube to about 180 nN/nm for the (12,0) nanotube.

The radial dependence of the reduced transition tension is signficant only for the

46

6

7

8

9

10

11

12

0.4 0.5 0.6 0.7 0.8 0.9 1

ReducedTransitionTension(eV/A )˚2

Cos(2χ )

(12,0)

(10,2)

(9,3)

(8,4)

(6,6)

(10,0)

(8,0)

(7,0)

(6,0) (nN/nm)

192

176

160

144

128

112

96

Figure 2.11: Transition tensions to plastic deformation for a family of (n, 0) and oneof nearly equal-radius (n,m) tubes. The radial dependence is quite weak except forthe smallest tubes, whereas the wrapping angle dependence is very strong.

smallest tubes.

The radial and angular dependences of the critical tension can be understood by

treating the defect as an extended strain in an elastic continuum with parameters

extracted from the atomistic calculations. For the tension Fc at the transition, the

defect density is vanishingly small and the energy (δE) and length (δL) changes

upon introduction of one defect are close to their values at zero tension. Therefore

the reduced transition tension is well approximated by

FcR=1

R

δE

δL

∣∣∣F=0

. (2.8)

Since the length change δL depends upon only R and the integrated strain of the

defect (with units of L2), δL scales as R−1 with angular dependence in the form

a+ b cos 2χ. Fitting to the atomistic data gives

δL ≈ 3.4A2

2πR(0.05 + cos 2χ), (2.9)

47

which differs by only 15% from a very simple geometrical analysis assuming regular

polygons. The 1/R dependence of the elongation cancels the normalization prefactor

of Eqn. (2.8) and therefore implies that the radial dependence of the critical tension

fc arises only through the defect energy δE. The nanotube at zero tension can freely

relax the defect strain both axially and circumferentially, so this defect energy in

tubes of moderate radius is a weak function of the wrapping angle (4.0 ± 0.1 eV for

the family (12,0), (10,2), (9,3), (8,4), (6,6)). Therefore the angular dependence

of the plastic limit arises mainly from the angular dependence of the defect-induced

elongation and the radial dependence of the plastic limit arises mainly from the radial

dependence of the defect energy.

The detailed radial dependence of the defect energy is a complex combination of

bare defect energy, the elastic energy induced in the surrounding medium and the

curvature-induced rehybridization of the defect. The bare energy is of order 5 eV.

The elastic term is small, of order µ(2πR)−2(∫ε2 d2x) ≈ 0.03 eVnm2/R2, where µ

is the in-plane elastic modulus of graphene and ε is the strain. The rehybridization

energy dominates the radial dependence; narrower nanotubes of the same χ typically

have lower defect formation energies since (i) the additional distortions due to bond

rotation are more easily accomodated into the curved surface of a narrower tube and

(ii) the defective region has a slightly lower mean curvature modulus than a corre-

sponding pristine region. This reduction in defect formation energy with decreasing

radius explains the lower plastic threshold in the smallest (n, 0) tubes. Note that the

radial dependence of the plastic threshold asymptotes (on increasing R) at a relatively

small radius whereas the pronounced χ dependence remains at all sizes.

Beyond the threshold for plastic deformation, the defect density is more easily

controlled by fixing the length rather than the tension. At constant length, the linear

defect density 9 is proportional to the fractional extension beyond the critical length:

9 = (L−Lc)/L δL ≈ (ε−εc)/((1+εc)δL), where ε = (L−L0)/L0 is the axial strain with

48

εc its critical value. Just as the critical force is very sensitive to wrapping angle, so the

critical strain varies from about 6% extension for a (6,6) tube to 12% extension for

a (12,0) tube. These strains greatly exceed those typical of the plastic deformation

in bulk solids, in accord with expectations for the extremely robust bonding in a

defect-free carbon nanotube at T=0. Above this critical extension the defect density

increases rapidly with length, varying as 24(ε − 0.06) nm−1 for the (6,6) tube and

as 43(ε − 0.12) nm−1 for the (12,0) tube. The defect concentration increases most

rapidly in the (n, 0) tubes since in these tubes each individual defect is least efficient

in relieving strain. Since δL varies as 1/R, larger tubes will contain more defects than

smaller tubes of the same wrapping angle at fixed relative strain, although the areal

density of defects remains constant (except for the smallest tubes where the defect

formation energy is a strong function of the radius).

Under an external force above the plastic threshold, proliferation of bond-rotation

defects is eventually halted by repulsive defect-defect interaction. The elastic inter-

action of two defects a distance D apart varies roughly as µ/D2 times the product of

their integrated strains. The strength of this interaction is on the order of 0.08nm2/D2

eV. This simple estimate is consistent with tight binding total energy calculations on

the (6,0) tube, which yield a 0.13 eV increase in the repulsive defect-defect interac-

tion for a reduction in D from 7 to 13 A. A uniform density of defects then has an

interaction per defect quadratic in the linear density λ (neglecting alteration of the

tube’s spring constant by the defects), so that λ ∝ (F − Fc)1/2 at constant force.

These defect densities as a function of strain are interesting not only for the me-

chanical response: the bond rotation defect is also a plausible route towards modu-

lating the electronic properties of carbon nanotubes. An extension of only 1% beyond

the critical length induces enough bond rotation defects to signficantly alter the elec-

tronic structure, increasing or decreasing bandgaps and modulating the density of

states[125]. The repulsive defect-defect interaction may tend to favor periodic defect

49

arrangements and thereby minimize the random component of the defect-induced

potential felt by the electronic states.

The discussion thus far has been in a purely equilibrium context, ignoring the

kinetics of defect formation. We have also calculated the barrier to defect formation

at critical external tension by calculating energy as a function of bond orientation,

considering both in-plane and out-of-plane rotations. Of the kinetic pathways exam-

ined, the lowest barrier for bond rotation in a (6,6) tube occurs when the bond is

tilted roughly 15o out of plane, which yields an upper bound of approximately 4.0

eV for the barrier at the transition, decreasing with increasing strain. This compares

with the 5.4 eV barrier for the related Stone-Wales transformation in C60[123] and

the 10.4 eV barrier for bond rotation in flat graphite[124]. The barrier height could

be further lowered dynamically by thermal fluctuations in the neighborhood (distinct

from thermal excitation of the reaction coordinate itself). One can ascribe the lower

barrier in the strained nanotube to the external tension. The bond rotation barrier

at zero external tension is not symmetric; it is about 8.5eV/defect from the defect

free side and 5.5eV/defect from the defective side. This relatively large barrier pre-

vents both the creation of bond rotation defects and annealing of the defects after

the external strain is released at room temperature. Bond rotation defects comprise

a pair of pentagon-heptagon defects (5-7 and 7-5) which can be separated by ad-

ditional bond rotations. These pairs represent crystal dislocations; the long-range

strain fields favor the gliding of these pairs in opposite directions under an external

stress[126]. However, the local curvature energy of the double pentagon-heptagon

structure is minimized when the oppositely directed curvature dipoles are adjacent.

In a flat unstrained graphitic sheet, tight binding total energy calculations reveal a

very strong short range attraction between a 5-7 and 7-5 defect. Separating them

by one intervening hexagon costs 4.5 eV, while separation by two hexagons costs an

additional 3.5 eV, although in certain situations this barrier may be lower[127]. This

50

tall and wide energetic barrier will help suppress fission of the bond rotation defect.

It will be very interesting to test our predictions experimentally. We here propose

an experimental geometry in which these defects could be created and measured in a

controlled fashion. A nanotube could be securely fixed at each end (e.g., embedded

within blobs of solid material) and then pulled taut and pressed transversely in the

middle of the free section, s s? . This geometry would create a very large mechanical

advantage of order δH/L where L is the suspended length and δH is the transverse

deflection. In addition, the change in tube length upon bond rotation would yield

a greatly amplified transverse deflection. If tube extension decreased the external

force, then the defects would appear one at a time, giving a characteristic signture of

successive quantized defections of the external probe. Note that the defects must be

created via tension, not compression, as a nanotube buckles under compression.

Although the nanotube wrapping angle controls the electronic properties through

the overlay of periodic boundary conditions on the graphitic semimetallic band struc-

ture, the nanotube elastic properties are insensitive to structural details. However,

the wrapping angle reasserts its importance under symmetry breaking plastic dis-

tortions. This strong sensitivity to wrapping angle will affect high-load structural

applications of carbon nanotubes. The electronic consequences of these plastic defor-

mations also open a possible new route towards modulating electronic properties of

carbon nanotubes. Our theoretical prediction on the exceptionally high elastic limits

of carbon nanotubes are consistent with several recent experiments[128, 129, 130].

2.4 Plastic deformation of carbon boron-nitride nan-

otubes: an unexpected weakness

As far as structural properties are concerned, boron nitride systems resemble car-

bon systems in several respects. Both boron nitride and carbon can form cubic and

layered structures. Cubic boron nitride materials are also well-known as hard mate-

51

rials, comparable in hardness to diamond. Therefore, it is not surprising that boron

nitride can also form nanotubes. Unlike carbon nanotubes, which were first observed

in experiment[62], boron nitride (BN) nanotubes were proposed by theorists[131, 132]

and subsequently synthesized in experiments[133]. Indeed, nanotubes based on other

layered compounds have also been successfully synthesized[134].

The elastic strength of BN nanotubes has been shown to be comparable to that of

carbon nanotubes[80, 81, 135, 136]. Therefore, it would be very interesting to compare

the elastic limits of boron nitride nanotubes against those of carbon nanotubes. Since

the boron nitride structure breaks the symmetry of the bipartite graphitic lattice, the

mechanical behavior of BN tubes beyond the elastic limit could differ significantly

from that of their carbon counterparts[86, 121]. In addition, BN tubes have uniformly

larger bandgaps than carbon tubes[131, 132]. This contrast in mechanical response

is potentially important for situations which might require, for example, insulating

behavior coupled with high mechanical toughness. Here we demonstrate that there are

qualitative differences between carbon and boron nitride nanotubes in their response

to high tensile stress. Whereas an (n, n) BN nanotube has an elastic limit against

bond rotation which is comparable to that of an (n, n) carbon nanotube, that of

an (n, 0) BN nanotubes is as much as 40% lower than that of the corresponding

(n, 0) carbon nanotubes In addition, the bond-rotation defects induced by external

tension can induce bond breaking in (n, 0) BN nanotubes even at zero temperature,

whereas carbon nanotubes can sustain greater strains without breaking bonds. These

interesting differences in the structural properties under high tensile stress arise due

to the presence of B–B and N–N bonds during the plastic deformation which we

consider.

The Stone-Wales transformation we discussed in the previous section can be read-

ily applied to BN nanotubes as shown in Fig. 2.12 with subtle differences (discussed in

detail later). (Note that the intermediate states in the rotation of the bond need not

52

1.83(Å)

(Å)1.92

B

N

N

B

tube

axi

s

(a)

1.70 (Å)

1.68 (Å)

B

N

N

B

tube

axi

s

(b)

C

CC

C(Å)1.60

(Å)1.60

tube

axi

s

(c)

Figure 2.12: Bond rotation defects in (a) (n, 0), (b) (n, n) BN nanotubes, and (c)(n, 0) and (n, n) carbon nanotubes. There are two distinct bond rotation patternsin a generic (m,n) boron nitride nanotube. For an (8,0) BN nanotube, the energydifference between the two patterns is about 0.2 eV/defect at equilibrium (i.e., zerotension), while for (n, n) tubes the two states are degenerate. Various bond lengthsmarked (not shown to scale) correspond to tubes under about 12% strain.

53

be in-plane[86, 122], although under tension the intermediate states tend to become

more planar in character). As we have shown in the previous section, such bond rota-

tions in a nanotube can relieve external strain at a moderate formation energy with

an energy barrier that decreases significantly with increasing tensile stress (also see

ref. [86, 121]). For example, under about 10% strain, the energy barrier against bond

rotation in an (n, n) carbon nanotube decreases by 30 – 50% from the value in a C60

fullerene[122], with reasonable agreement between tight binding total energy[86] and

classical Tersoff-Brenner potential[121] models. Since the character of the directional

bonding in planar boron nitride is similar to that in graphitic systems, we expect this

bond rotation deformation mode to be important also for BN nanotubes.

Since bond rotation defects in a BN sp2 network create energetically unfavorable

B–B and N–N bonds, even membered rings (for example, 4- and 8-membered rings,

which require no B–B and N–N bonds) might be more stable[133, 137]. We have ex-

amined this possibility and found that the kinetic route to the formation of such 4884

defects within a hexagonal network is multi-stage and complex, with the final state

involving very large distortions of the sp2 network. An alternative 4875 distortion

is kinetically accessible, but only by passing through the 5775 distortion considered

here as an intermediate state. Therefore, while even-membered rings can be favored

in the nanotube cap, local defects induced by external stress are less likely to be com-

prised exclusively of even membered rings. For (n, 0) nanotubes, due to the particular

arrangement of the bonding network, direct bond breaking along the axial direction

might also be an important deformation mode; we will discuss this briefly later.

We employ a non-orthorgonal tight binding total energy scheme as developed

by Porezag and co-workers[24]. Instead of fitting the tight binding parameters to

functional forms, the repulsive potentials and hopping elements, which are obtained

by fitting ab initio results, are tabulated and interpolated. In relaxed structures of

BN nanotubes, the nitrogen atoms buckle outwards. In other words, a single-walled

54

BN tube has two concentric cylinders, one of solely boron, the other nitrogen. The

difference in the radii of the two cylinders decreases from about 0.2 atomic units (1

atomic unit=0.529 A) for a (6,0) tube to 0.1 atomic units for a (12,0) tube, which is

in good agreement with a previous ab initio calculation[132]. The equilibrium B–N

bond length is about 1.46 A, very close to the bond length of planar hexagonal boron

nitride. This good agreement shows the reliablity of the tight binding parameters

under these conditions.

Fig. 2.12a and 2.12b shows the bond rotation defects in (n, 0) and (n, n) boron

nitride nanotubes. Note that whereas there is only one favorable bond rotation pat-

tern for carbon nanotubes under strain, the situation is a bit more complicated for

BN nanotubes due to the sublattice symmetry breaking. For a generic (n,m) tube

with n=m, clockwise and counter-clockwise bond rotations are distinct. The extremecases are the (n, n) and (n, 0) tubes. For an (n, 0) tube, the difference between the

two choices is maximal: B–B and N–N bonds after the bond rotation align nearly

along either axial or circumferential directions. Tight binding total energy calcula-

tions show that the former is energetically favorable by 0.2 eV/defect. Therefore we

focus on this lowest-energy bond rotation defect. For an (n, n) BN tube, the two

bond rotation patterns are degenerate. The bond rotation defects in (n, 0) and (n, n)

carbon nanotubes are also shown in Fig. 2.12c for comparison. The existence of a

lower-energy bond rotation pattern for (n, 0) tubes is one of the reasons that (n, 0) BN

nanotubes have significantly lower elastic limits than do corresponding (n, 0) carbon

nanotubes, as will be discussed further below.

Table 2.2 compares the elastic limits (i.e., the transition tension) Fc and the

reduced transition tensions fc of carbon and BN nanotubes from the pristine structure

into a plastically deformed bond-rotated phase. (For definitions of Fc and fc, see the

previous section.) Both carbon and BN (n, 0) tubes have higher elastic limits to this

particular failure mode than do (n, n) tubes of similar radii, the reason for which has

55

(8,0) (12,0) (6,6)Fc fc Fc fc Fc fc

(eV/A) (eV/A2) (eV/A) (eV/A2) (eV/A) (eV/A2)BN 20.5 6.4 32.4 6.8 21.0 5.1C 32.7 10.4 51.2 10.9 25.8 6.3

Table 2.2: Comparison of the elastic limits of boron nitride and carbon nanotubes,showing both full and reduced transition tensions (as defined in the text). The reducedtransition tensions for (n, n) carbon and boron nitride nanotubes are similar, but thesequantities differ greatly for (n, 0) carbon and boron nitride nanotubes.

been discussed in the previous section.

As shown in Table 2.2, an (n, n) BN nanotube has an elastic limit only slightly

lower than that of an (n, n) carbon nanotube. This difference can be understood by

considering the approximation for the transition tension[86] we have discussed in the

previous section, namely Eqn. (2.8). The defect formation energy δE in a (6,6) BN

nanotube is smaller than that of a (6,6) carbon nanotube, 3.13 eV versus 3.95 eV

from our tight binding calculations, while the elongation δL is slightly shorter, 0.13

A versus 0.14 A. Taking into account these two factors, this simple model predicts

the elastic limit of a (6,6) BN nanotube to be about 83% of that for a (6,6) carbon

nanotube, which is very close to the actual result (81%) from the detailed simulations.

The elastic limit of an (n, 0) BN nanotube is, however, only about 60% of that for

the corresponding (n, 0) carbon nanotube. The reason that (n, 0) BN nanotubes have

much lower elastic limits is twofold. First, as mentioned above, the lower symmetry

of BN nanotubes results in two distinct bond rotation patterns with different defect

formation energies. If the elastic limit of the (8,0) BN nanotube were calculated

with the higher formation energy defects, the transition tension would have been 22.7

eV/A, as compared to the actual value of 20.5 eV/A. This is, however, still much

smaller than the transition tension of an (8,0) carbon nanotube, which is 32.7 eV/A.

Therefore, the lower defect formation energy alone cannot explain the discrepancy.

56

The remaining discrepancy can be explained by carefully examining the structural

properties in the vicinity of the defect in a BN nanotube. We find that whereas bond

rotation defects in all carbon nanotubes are rather robust and can sustain considerable

tension beyond the elastic limit (at low temperature) without breaking bonds, the

defective area in an (n, 0) BN nanotube is very vulnerable to bond breaking under

tension. Although both carbon and BN nanotubes have an sp2 bonding network, the

BN bonding is partially ionic and less robust under deformation. Rotating a bond

creates unfavorable B–B and N–N neighbors. These homopolar bonds weaken the

entire defect area: The local strain in the defect area is then considerably larger than

in other parts of the tube. Due to the particular arrangement of bonds in an (n, 0)

tube, two bonds align well with the axial direction in the defect area (the shaded

bonds in Fig. 2.12a). These two bonds stretch the most under stress and are then

susceptible to subsequent breakage. For example, for an (8,0) BN nanotube under

12% overall strain, about 30% stretch is observed for those particular bonds (Fig.

2.12a). The most-stretched bond is clearly in the process of breaking. In contrast,

no particularly over-stretched bonds are found in a carbon (8,0) nanotube under the

same amount of strain (Fig. 2.12c). For (n, n) tubes, there are no bonds that align

exactly along axial direction. As a result four bonds (shaded bonds in Fig. 2.12b)

share the stress around the defect area. No bond breaking is observed in an (n, n)

BN nanotube up to 12% strain at zero temperature. Therefore, we conclude that the

weakness of the defect area in a BN nanotube, whose effect is enhanced in an (n, 0)

nanotube, combined with the lower defect formation energy, implies that an (n, 0)

BN nanotube has a much lower elastic limit against this deformation than does a

corresponding (n, 0) carbon nanotube. We show here only the results for (n, 0) and

(n, n) tubes, which represent two extreme cases. For generic (n,m) tubes, the elastic

limit should lie in between that of (n, 0) and (n, n) tubes of similar radius. Note that

kinetic barriers could increase the observed transition tension under certain thermal

57

conditions, whereas preexisting imperfections in the tube (e.g., vacancies) could allow

alternative deformation modes with lower thermodynamic transition tensions.

We now consider one alternative mode of plastic deformation which is relevant

even in ideal systems, namely direct bond breaking. Such a deformation is expected

to be most favorable for (n, 0) tubes wherein certain bonds point directly along the

axial direction. We analyze this possibility by breaking a small fraction of the axially

aligned bonds (i.e., extending them to 2.1 times their normal length) in an (8,0) tube

under the same 9% strain at which bond rotation deformations are thermodynamically

favored. We then relax the structure in two steps, under a constant 9% strain. In

the first step the two atoms comprising the broken bond are fixed. This prevents

the broken bonds from reforming before the neighboring atoms relax into a favorable

configuration. Next, we relax the entire structure without any constraint, i.e., every

atom is allowed to move. After this two-step relaxation, the system spontaneously

heals the broken bonds. These results suggest that the bond-broken state is not a

local minimum of the Gibbs free energy at the strain at which bond rotations are

thermodynamically favored. Therefore direct bond breaking is likely not a favorable

mode of plastic deformation at low temperature thermodynamic under conditions of

constant strain.

In summary, we used a nonorthogonal tight binding total energy model to study

the bond rotation mode of plastic deformation of BN nanotubes. Whereas boron

nitride nanotubes have elastic properties similar to those of their carbon counter-

parts, they have a very different response to plastic deformation. Apart from the

already-understood wrapping angle dependence of this deformation, we found that

BN nanotubes with wrapping indices n m have unusually low-tension failure modes

which contribute to brittle failure via bond breaking. Therefore, although (n, n) boron

nitride nanotubes have comparable plastic limits to those of carbon tubes with sim-

ilar wrapping angle, (n, 0) boron nitride nanotubes have much smaller elastic limits

58

against bond rotations. In (n, 0) tubes, bond rotation defects induced by plastic de-

formation quickly lead to bond breaking and thereby cause tubes in this range of

wrapping angles to be substantially weaker than would be expected from their purely

elastic response. These results could have relevance to the choice of materials for

future high-strength nanomechanical applications of carbon and boron-nitride based

systems.

59

Chapter 3

Electronic and StructuralProperties of Novel Group-IVCompounds Formed from CVDPrecursors

3.1 Introduction

In the previous chapter, we explored some of the interesting properties of car-

bon nanotubes. Actually, most group-IV elements possess fascinating and unique

properties. Moving downward in the periodic table, silicon is undoubtedly the most

technologically important element; germanium has very high hole mobility. Whereas

grey tin (α-tin, diamond structure) is a semimetal, white tin (β-tin, double body

center cubic structure) is a metal. The transition from α-tin to β-tin happens just

slightly below room temperature at atomospheric pressure. Also, the band gaps of

elemental group-IV (C, Si, Ge, and Sn) materials vary from zero (graphite, α-Sn) to

5.5 eV (diamond).

Recently, group-IV alloys have attracted considerable research interest due to their

wide tunability in both structural and electronic properties and the hope that such

alloys might combine the above mentioned properties in new ways[138, 139, 140, 141,

142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152]. Incorporation of substitutional

60

carbon into other group-IV semiconductors is of particular interest. On one hand,

the small radius of carbon might help reduce the lattice mismatch between bandgap

engineered Si/Ge materials and the silicon substrate; on the other hand, the presence

of strongly electronegative carbon atoms, along with the strain introduced by them,

offers a unique way to tailor the electronic structure with a compelling possibility of

obtaining direct gap group-IV semiconductors that lattice match silicon[153].

Although there has long been speculation that alloying carbon with other group-

IV materials might produce interesting properties, little progress was made towards

this goal until recently. The major obstacle for synthesizing group-IV compounds with

high concentrations of carbon is that the solubility of carbon in silicon is very low: less

than 10−4 at. % at the silicon melting point; in germanium the solubility is negligible.

Fortunately, recent advances in synthesis have demonstrated that tetrahedral group-

IV compounds containing high concentrations of carbon are possible[138, 139, 140].

More recently, Kouvetakis et al. reported a synthesis[147, 148] based on UltraHigh-

Vacuum Chemical Vapor Deposition (UHV-CVD) that allows the production of semi-

conductor alloys with well-defined (and thermodynamically inacccessible) 1:4 C:Si and

C:Ge stoichiometries through the use of precursor molecules which build in the re-

quired interatomic bonding, for example, C(SiH3)4. The terminal groups (e.g. H)

are eliminated to produce the pure C/Si (or C/Ge) ordered alloys, such as Si4C or

Ge4C. Such techniques allow one to produce bulk amounts of metastable alloys of con-

trolled stoichiometry which are not accessible by standard near-equilibrium synthesis

methods.

In this chapter, we present systematic ab initio results on the electronic and struc-

tural properties of Si4C, Ge4C, Sn4C and other related group-IV compounds that ei-

ther have already been synthesized or are designed to be accessible to the UHV-CVD

technique using precursor molecules as building blocks. For Si4C and Ge4C, the band

gaps remain indirect while for Sn4C, the the band gap is direct but slightly negative

61

(-0.1 eV). Since the local density approximation (LDA) underestimates the band gap,

we expect Sn4C to have a moderate positive direct band gap (of the order of 0.5 eV)

if quasiparticle corrections are taken into account. In Section 3.3, we will study the

properties of Si4C, Ge4C and Sn4C.

Another subject of interest is silicon-based optically active semiconductors. For all

of their exciting properties, the group-IV (both elemental and compound) semicon-

ductors known so far have indirect energy gaps (not including zero gap materials).

Integration of silicon-based electronics with optical components requires optically

active materials that can be grown on silicon with high-quality interfaces. Unfortu-

nately, no direct-bandgap semiconductors have been produced which lattice-match sil-

icon (and previously suggested structures pose formidable synthetic challenges[154]).

The resulting lattice mismatch generally causes cracks and poor interface properties

whenever the mismatched overlayer exceeds a very thin critical thickness. Much re-

cent work has focused on introducing compliant transition layers between mismatched

components[155, 156, 157]. It might also be possible to make optically active group-

IV materials that are compatible with silicon technologies. Most previous efforts

along this direction have resorted to band folding, surface states and quantum con-

finement. Examples are Si-Ge superlattices[158, 159, 160, 161], porous silicon and

silicon nanowires[162, 163, 164, 165]. Si-Ge superlattices usually have poor optical

properties[159, 160]. The mechanism of light emission of porous silicon is often as-

cribed to the quantum confinement, surface states and/or band folding[166, 167, 168],

although the physical origin is still not fully settled.

We have suggested a more direct solution to this problem[153], proposing two novel

group-IV compounds, namely, Si2Sn2C and Ge3SnC, which have direct energy gaps

and lattice match silicon to within 1% along low-index lattice planes. The bandgaps

of these two group-IV compounds lie within the window of minimal absorption in

current optical fibers. Detailed ab initio studies of Si2Sn2C and Ge3SnC are presented

62

in Section 3.4.

One can build in even higher concentrations of carbon using C2(SiX3)6 and C2(GeX3)6

as precursors for solid Si6C2 or Ge6C2. Although these two compounds contain

C–C bonds, which have long been regarded as energetically unfavorable in group-IV

alloys[142], our detailed ab initio calculations show that they are energetically compa-

rable to the already-synthesized group-IV compounds, e.g., Ge4C. Due to the relative

high strain in these two materials, Si6C2 and Ge6C2 are semi-metallic within LDA.

(Strain induced gap closing has been observed previously in Si-C random alloys[141].)

The highly anisotropic C–C bonds in Si6C2 and Ge6C2 should have significant effects

on the electronic and structural properties, as discussed in Section 3.5.

3.2 Computational details

We use the ab initio pseudopotential plane-wave method[37] in the local density ap-

proximation with norm-conserving pseudopotentials[43] in a nonlocal form suggested

by Kleinman and Bylander[46], as described in chapter 1. The parametrization of

the Ceperley-Alder results[169] by Perdew and Zunger[170] is used for the exchange-

correlation. For germanium and tin, scalar relativistic effects[171] are taken into ac-

count. In all of the calculations, we include the nonlinear partial core correction[172]

to account for the nonlinearity of core-valence exchange-correlation energy. The en-

ergy cutoff of the plane-wave basis is set at 60 Ryd to ensure the convergence of the

calculations, since all of the systems studied include carbon atoms. The k-point set is

generated by the Monkhorst-Pack scheme[173] with a density of 3×3×3, which wouldbe equivalent to a k point density of about 6×6×6 in a more familiar 2-atom diamond

unit cell.

63

3.3 Si4C, Ge4C and Sn4C

Fig. 3.1 shows the precursor molecule C(AX3)4 (A=Si, Ge or Sn) and a correspond-

ing ordered crystalline structure. (Although Sn4C has not been synthesized yet, we

extend our study to this compound for comparison purposes and to gain a better

understanding of the general trends arising from differences in electronegativity and

core radii.) The smallest possible unit cell of these compounds contains 10 atoms and

has a body-centered tetragonal (bct) lattice. In these systems, there are no nearest

or second nearest carbon–carbon pairs, which are normally energetically unfavorable

in group-IV alloys[142].

Figure 3.1: Molecular precursors and the corresponding relaxed crystalline phases ofSi4C. Silicon is yellow, carbon is grey, and the terminal group is red. Ge4C and Sn4Chave similar structures.

Since all of the group-IV compunds studied here are metastable, a measure of their

64

stability is very important. We define the excess energy δE as the negative binding

energy with respect to the constituent elemental materials in the diamond structure.

For group-IV compounds containing Sn, comparing the energy with respect to gray

tin may not be the best way to study their stability, since gray tin is not stable at

room temperature. However, the energy difference between gray and white tin is small

compared to the excess energy. Another quantity of interest regarding the stability of

materials is the energetic barrier against the structural phase transitions. However, a

study of the transition barriers, which are expected to be relatively large for covalent

group-IV compounds, would involve extensive molecular dynamics simulations. Since

all of the structures investigated are locally stable with no indications of soft-mode

behavior, we do not examine the energy barriers in detail.

LatticeConstant(A)

Bond Length(A)

δE(eV/atom)

BulkModulus(GPa)

Si4C a=b=7.82c=4.97

Si–Si 2.30, 2.38Si–C 1.89

0.05 137(100)

Ge4C a=b=8.17c=5.22

Ge–Ge 2.40, 2.48Ge–C 2.00

0.28 108(76)

Sn4C a=b=9.19c=5.93

Sn–Sn 2.77, 2.86Sn–C 2.19

0.42 74(47)

Table 3.1: Structural properties of X4C in the local density approximation. The bulkmoduli in parentheses are for silicon, germanium, and gray tin respectively.

Table 3.1 shows the structural properties of X4C (X=Si, Ge, Sn) compounds. The

excess energy δE for Si4C is very small, 0.05 eV/atom, indicating the relative stability

of this particular compound. The lattice constant of Si4C differs from that of silicon by

about 8.1%. This result is very close to a previous study[142]. The very large lattice

mismatch between Si4C and silicon implies that pseudomorphic growth of Si4C on a

silicon substrates would quickly lead to cracks or other strain relieving defects. The

65

lattice of Ge4C is closer to that of silicon, with a 3.7% mismatch. Another interesting

observation (which will be discussed in more detail later) is that Ge4C becomes a

direct gap semiconductor upon lattice expansion to match that of silicon.

N

Γ

X

P

M

Σ

W

∆

Q

Figure 3.2: Irreducible Brillouin zone of the bct lattice.

Due to the symmetry breaking in these systems, there are two types of X–X (X=Si,

Ge, or Sn) bonds whose bond lengths differ by about 3%, with twice as many shorter

bonds as longer bonds. The weighted average lengths of the X–X bonds are very close

to those in the corresponding elemental bulk materials. For example, the Si–Si bond

length as calculated by LDA is 2.332 A, while the weighted average Si–Si bond length

in Si4C is 2.327 A. Si–C bonds, however, are slightly stretched compared to the bond

length in binary silicon carbide (SiC), 1.89 A vs. 1.87 A. The Ge–C bond length is

2.00 A, which lies between those in C(GeBr3)4 (2.05 A) and C(GeH3)4 (1.97 A)[147].

Although no stable GeC phase exists at ambient conditions, as a privilege for theorists,

66

we can still do the calculation for this idealized compound and we find that compared

to that of GeC, the Ge–C bond length in Ge4C is stretched by 2%, similar to Si–C.

The over-stretched Si–C and Ge–C bonds as compared to their binary counterparts

is expected in these strained systems[147]. The bulk moduli of these materials show

a systematic increase from that of the dominant elemental constituent: 37% for Si4C

and 57% for Sn4C (compared to gray tin).

Fig. 3.3 and 3.4 shows the LDA band structures of Si4C, Ge4C and Sn4C. (A

schematic plot of the irreducible Brillouin zone with high-symmetry k points is shown

in Fig. 3.2.) The LDA band structure of silicon, calculated on the same 10-atom unit

cell, is also plotted for comparison. (The specific points X, etc. mentioned below

refer to this 10-atom bct cell and not the standard 2-atom silicon Brillouin zone).

Although the band gaps of both Si4C and Ge4C remain indirect, the effects of the

substitutional carbon on the band structures of these systems are easily seen. First,

the lowest conduction band at the Γ point is pulled down significantly. The difference

between the direct energy gap at Γ and the minimum indirect gap of silicon is about

2.1 eV, this difference is reduced to about 1.5 eV for Si4C and 0.4 eV for Ge4C. We

believe this trend results from the competition between the overall volume contraction

and charge transfer. Contraction in volume due to the incorporation of carbon tends

to lift the energy of the conduction band at Γ (relatively), while charge transfer tends

to lower it. For Si4C, charge transfer seems dominant whereas for Ge4C, the two effects

are comparable. The LDA band gap of Sn4C is direct and slightly negative (-0.1 eV).

Since LDA calculations usually underestimate the band gap of semiconductors, we

expect the real band gap of Sn4C to be moderately positive.

As we briefly mentioned above, although Ge4C has a indirect energy gap, upon

volume expansion it acquires a direct gap. This high sensitivity arises from the

volume variation in the local potential energy of the conduction band[174]. Such a

lattice expansion could be achieved by deposition of a thin layer of Ge4C on a silicon

67

M Σ Γ ∆ X W P Γ

4Si C

N Q P

ener

gy (e

V)

−15

−10

−5

0

5

M Σ Γ ∆ X W P Γ

4Ge C

N Q P

wavevector K

ener

gy (e

V)

−15

−10

−5

0

5

Figure 3.3: LDA band structures of Si4C, Ge4C.

68

M Σ Γ ∆

4Sn C

X W P Γ N Q P

ener

gy (e

V)

−15

−10

−5

0

5

M Σ Γ ∆ X W P Γ

Si

N Q P

wavevector K

ener

gy (e

V)

−15

−10

−5

0

5

Figure 3.4: LDA band structures of Sn4C and silicon calculated on a 10-atom unitcell.

69

M Σ Γ ∆ X W P Γ N Q P

wavevector K

ener

gy (e

V)

−15

−10

−5

0

5

Figure 3.5: Band structure of Ge4C upon lattice expansion to match that of silicon.

substrate. Fig. 3.5 shows the band structure of Ge4C calculated with the silicon lattice

constant, which shows the direct band gap of Ge4C under expansion. Actually, less

than 2% lattice expansion makes the band gap of Ge4C direct.

3.4 Si2Sn2C and Ge3SnC

Although A4C (A=Si, Ge, Sn) group-IV alloys possess interesting structural and

electronic properties, they might not be good candidates for certain areas such as

optoelectronic applications. Si4C has an indirect gap and the lattice constant differs

from that of silicon by as much as 8%. Ge4C would have a direct band gap upon

lattice expansion to match that of silicon. However, a 3.7% lattice expansion is too

large for most practical applications. Sn4C is expected to have a moderate direct

70

bandgap but again does not lattice match silicon. One may ask, is it possible to

make direct bandgap group-IV alloys that lattice match silicon?

Although bulk silicon is an indirect bandgap semiconductor, it is nearly sur-

rounded by direct-bandgap elemental (unary) and compound semiconductors in the

periodic table. Moving downwards from Si, the unary group-IV materials acquire

larger cores containing d-electrons; these states affect the conduction band states

through orthogonality requirements and changes in the overall volume. On moving

from silicon to germanium to tin, the Γ2′ conduction band (as labelled in silicon) at

k = 0 drops in energy until, in grey tin, the material acquires a direct (and vanishing)

bandgap at k = 0. Moving horizontally on the periodic table, GaAs, a prototypical

direct-bandgap semiconductor, differs from Si or Ge in that the constituents have

different electronegativities. The resulting antisymmetric component in the crystal

potential flattens the bands and opens the bandgap. In addition, the Γ6 point of the

conduction band drops relative to the other points in the band until it becomes the

bottom of the conduction band in most of the well-known direct bandgap III-V and

II-VI semiconductors.

These well-known results suggest that perhaps one can combine these mechanisms:

larger cores and partial ionicity, within alloys containing only group-IV elements

wherein the difference in rows provides the contrast in electronegativity. Such a

system is likely to require the presence of tin to obtain a direct bandgap[175, 176, 177,

178]; the combination of tin (which is substantially larger than silicon) with carbon

(which is substantially smaller than silicon) not only affords the greatest contrast in

electronegativity (which is needed to open the zero-bandgap of elemental Sn), but also

opens the possibility of tuning the stoichiometry to obtain a lattice match to silicon.

Also, the countervailing effects of the larger Sn atoms might compensate the strain

introduced by carbon within a Si/Ge lattice and therefore enhance the solubility of

carbon within a silicon and germanium matrix.

71

Figure 3.6: Molecular precursors and the corresponding relaxed crystalline phasesof Si2Sn2C (top) and Ge3SnC (bottom). Silicon is yellow, carbon is grey, tin ismagenta, germanium is green and the terminal group is red. Ge4C and Sn4C havesimilar structures.

72

To be practical, such an alloy must be accessible to a well-defined synthetic tech-

nique. Ternary group-IV compounds containing Sn, for example, SiySn4−yC and

GeySn4−yC, could be accessible to the aforementioned UHV CVD synthesis techniques

by replacing silicon or germanium atoms with tin in the corresponding molecular

precursors. One interesting previous study[154] in computational design of direct-

bandgap silicon lattice matching materials, in constrast, imposes a formidable syn-

thetic challenge (the material requires the sequential deposition of 8 distinct non-

interdiffusing monolayers of As/Zn/As/Si/As/Zn/P/Si).

Of particular relevance to the current study are several additional synthetic de-

velopments. First, the Sn–D bond has recently been shown to be stable for long

periods (at 0o C) in molecules which incorporate Sn–D3 moieties (the Sn–H3 moiety,

in contrast, is known to be unstable)[179]. Second, techniques have been developed

to synthesize tetrahedral precursors where a central carbon atom is attached to four

tins[180]. Finally, the synthetic yields of, e.g., CSi4H12 and CGe4H12 have improved

significantly since their discovery, to the range where multi-gram quantities can be

regularly prepared[181, 182]. Keeping these important advances in mind, we focus

on novel group-IV alloys which can be built from C(Si,Ge,Sn)4X12 precursors. In

particular, we demonstrate that two alloys, Si2Sn2C and Ge3SnC, have low-energy al-

lotropes which lattice match silicon to better than 0.5-1% and have direct bandgaps.

To obtain the energy gaps more accuratly , we also calculate the quasi-particle energy

gap using the GW approximation[36].

Fig. 3.6 shows the proposed molecular precursors for Si2Sn2C and Ge3SnC, namely,

C(SiX3)2(SnX3)2 and C(GeX3)3(SnX3) (where X is a terminal group), and the cor-

responding ordered crystalline phases. These particular crystal structures are the

lowest-energy (and highest-symmetry) among several allotropes with small unit cells

built from these precursor molecules. The structures shown are fully relaxed locally

and all phonon modes are stable, indicating local structural stability. The absence

73

of strong C-X (e.g. C-H) bonds allows a low growth temperature which is crucial for

synthesizing metastable materials, particularly with tetrahedrally coordinated Sn.

The Si2Sn2C material has a body-centered tetragonal (bct) lattice with 10 atoms

per unit cell like that of the A4C structure. This relatively high-symmetry structure

can minimize the anisotropic strains associated with deposition on a low-index silicon

surface. In this structure, there are four Sn–Sn, four Sn–Si and four Si–Si bonds in a

unit cell, achieving the highest possible rotational symmetry. The fully relaxed den-

sity functional results reveal that the volume per atom in Si2Sn2C is only 0.6% less

than that of bulk silicon (we compare the theoretical LDA volumes to minimize sys-

tematic errors). This is only a slight deviation from Vegard’s law, which predicts that

stoichiometries near Si2.22Sn1.78C lattice-match silicon. The body centered tetragonal

structure of Si2Sn2C provides several lattice planes which can be well-matched to

low-index planes of bulk silicon. For example, the (110) and (111) planes of silicon

match to Si2Sn2C better than 0.5% in all directions.

The 3:1 ratio of noncarbon atomic constituents in Ge3SnC yields a slightly lower-

symmetry structure which is 1.7% lower in volume than bulk silicon. Due to the lower

crystal symmetry, the lattice is slightly distorted from bct. However, the deviation

from bct is very small, about ±1%. Therefore, we use the notation of the special (highsymmetry) k points of bct structure to label the band structure of Ge3SnC. The lower

symmetry of Ge3SnC results in a slightly worse lattice matching to silicon as com-

pared to Si2Sn2C, about 1% in linear dimension (with a combination of compression

and dilation along different axes). Mismatches of this order should yield a critical

thickness of hundreds to thousands of Angstroms[183]. In addition, a synthesis based

on these molecular precursors could allow the addition of a small concentration of a

second precursor that helps to precisely tune the lattice match. Another important

issue regarding lattice compatability is the thermal expansion coefficient. Although

we did not study the thermal expansion compatibility between these two compounds

74

LatticeConstant(A)

Bond Length(A)

δE(eV/atom)

BulkModulus(GPa)

Si2Sn2C a=b=8.43c=5.46

Si–Si 2.36Si–Sn 2.62Sn–Sn 2.72Si–C 1.88Sn–C 2.22

0.30 104

Ge3SnC a=8.50b=8.34c=5.41β=89.6o

Ge–Ge 2.43, 2.49Ge–C 2.00Ge–Sn 2.58, 2.66Sn–C 2.20

0.32 80

Silicon a=b=8.51c=5.38

Si–Si 2.33 0.0 100

Table 3.2: Structural properties of Si2Sn2C and Ge3SnC. All cited values are LDAresults. The results for silicon calculated with a 10-atom supercell are also listed forcomparison.

and silicon, we would expect the difference in thermal expansion to be moderate,

considering that, at room temperature, the thermal expansion coefficient of diamond

(1×10−6/K) is smaller than that of silicon (about 2.5×10−6/K) while both germa-

nium (about 5×10−6/K) and α-tin (about 6×10−6/K) have larger thermal expansion

coefficients.

Table 3.2 shows the structural properties of Si2Sn2C and Ge3SnC. The A–C (A=Si,

Ge, or Sn) bond lengths change little as compared to those in A4C alloys, indicating

the robustness and stiffness of these bonds. However, both Si–Si and Ge–Ge bonds are

longer than those in Si4C and Ge4C, due to the presence of Sn. The bulk modulus of

Si2Sn2C is comparable to that of silicon while the bulk modulus of Ge3SnC is similar

to that of germanium. The excess energies of both Si2Sn2C and Ge3SnC are moderate

and comparable to that of Ge4C, which has been successfully synthesized, indicating

that these two compounds might well be accessible to current CVD techniques so long

75

as the molecular cores of the precursors are stable under the synthesis conditions.

Fig. 3.7. compares the pseudopotential density functional bandstructures of Si2Sn2C

and Ge3SnC to the corresponding Si bandstructure within the same (10-atom) unit

cell. Si2Sn2C has a direct bandgap in an unusual position, the X point. The con-

duction band minimum at the X point is rather close to the folded X point in Si.

Symmetry breaking due to the alloying splits a degeneracy at the X point. In the

uppermost valence band, this splitting raises the higher-energy state substantially

until it passes the Γ point state and thereby becomes the new valence band maxi-

mum. This change in the valence-band maximum compared to Si is facilitated by

the smaller bandwidths in this more ionic semiconductor alloy. Meanwhile, the con-

duction band at Γ follows a well-known pattern: the introduction of Sn pulls down

the lowest conduction band at Γ and produces a local (but in this case not global)

conduction band minimum at Γ.

Although the characteristics of the conduction band of Si2Sn2C are not unex-

pected, having a valence band maximum at X, rarely seen in common semiconductors,

is an intriguing feature. To understand this somewhat abnormal band structure, i.e.,

a direct energy gap at a point other than Γ, we analyze the charge density at the top

valence band for the X and Γ points. Fig. 3.8 shows the three dimensional charge

density isosurfaces for the top valence band at Γ and X for Si2Sn2C. The charge den-

sity at Γ preferentially concentrates around Si–Si bonds, whereas the X-point charge

density is more localized around Sn–Sn bonds. The greater Sn character then ac-

counts for the higher energy of X state. Measuring the bond charge contained within

fixed-width cylinders, we obtain (QΓ(Si–Si)/QΓ(Sn–Sn)):(QX(Si–Si)/QX(Sn–Sn)) ∼4:1. (We define the bond charge as the total charge enclosed by a cylinder connect-

ing the two atoms. Although the absolute value of the bond charge scales with the

volume of the cylinder, the ratio of the bond charges is insensitive to the cylinder

diameter.) Since the valence atomic orbital energies of Sn are higher than those of

76

M Σ Γ ∆ X W P Γ N Q P−15

−10

−5

0

ener

gy (

eV)

M Σ Γ ∆ X W P Γ N Q P

wavevector K

−15

−10

−5

0

ener

gy (

eV)

Figure 3.7: Band structures (in blue) of Si2Sn2C (top) and Ge3SnC (bottom). Theionicity opens a bandgap deep in the valence band. Symmetry breaking in Si2Sn2Cremoves the degeneracy at the X point, pushing one band up significantly so thatit becomes the conduction band maximum and produces in a direct band gap at X(red arrow). In the case of Ge3SnC, the conduction band minimum occurs at Γ,which produces a direct gap at Γ (red arrow). The band structure of silicon (orange)calculated in the same unit cell is also plotted for comparison.

77

Γ

X

Figure 3.8: Charge density isosurfaces for the top of the valence band at Γ (top) andX (bottom) in Si2Sn2C. Electrons are localized around Si–Si bonds (in yellow) for theΓ state, whereas for the X state they are localized around Sn–Sn bonds (in magenta).

78

Si, the greater Sn character of the X point accounts for the unusual valence band

structure of Si2Sn2C. The fundamental LDA band gap of Si2Sn2C at the X point is

0.59 eV. The GW correction increases the band gap to 0.9 eV.

The mechanisms which produce a direct bandgap in Ge3SnC have a more familiar

interpretation. Due to the increased fraction of the lower-row elements Ge and Sn,

the drop in the conduction band at Γ is more pronounced than that in Si2Sn2C. This

Γ-point state becomes the new conduction band minimum, directly above the valence

band maximum, which remains at Γ. Therefore, the direct band gap of Ge3SnC can

be understood as due to a large difference in electronegativity between carbon and

germanium/tin as well as the large ionic core which tends to pull the conduction

band at Γ down more rapidly than at the zone edge. The LDA band gap at Γ is 0.30

eV, increasing to 0.71 eV under the GW correction. The fundamental band gap at

Γ is very sensitive to external pressure, dropping by 0.1 eV for a fractional change

in volume of δV/V = 0.01. This high sensitivity arises from the volume variation

in the local potential energy of the conduction band[174] and provides a potentially

important mechanism for tuning the size of the direct bandgap. For consistency, the

band structures are calculated at the experimental volume per atom of the posited

silicon substrate. Since the volume differences between these materials and bulk

silicon are very small, the band structures are very similar to those calculated at

their equilibrium volumes. Si2Sn2C shows no significant changes in bandstructure

under such a volume change. The high volume sensitivity of Ge3SnC means that

the small expansion to match the experimental volume per atom of silicon actually

drives the bandgap direct. (At the relaxed bulk Ge3SnC volume the bandgap is just

slightly indirect). For Ge3SnC the optical matrix elements at the direct transition are

comparable to those in GaAs. For Si2Sn2C, the optical matrix elements are somewhat

smaller but the transition is still clearly dipole-allowed.

In summary, we have demonstrated in this section how materials design can yield

79

multiple candidate materials which simultaneously possess three desirable materials

properties: a very close lattice match to silicon, a direct bandgap in the 0.7-1.0

electron volt range, and accessibility to well-defined synthetic strategies that can

potentially produce bulk quantities of material.

3.5 Si6C2 and Ge6C2

Figure 3.9: Proposed molecular precursor for synthesizing Si6C2 and the correspond-ing relaxed crystalline structure. The molecular precursor and crystal structure ofGe6C2 are similar to those shown.

In previous sections, we studied group-IV compounds which are isostructural

with already synthesized compounds that use the propotype molecular precursor

80

C(AH3)4, where A = (Si, Ge, Sn). These systems achieve the highest carbon concen-

tration (20 at. %) possible without creating C–C bonds or second nearest neighbor

carbon pairs. An interesting question remains, however. Using specific molecular pre-

cursors, can group-IV compounds with higher carbon concentrations be synthesized?

Such systems would inevitably contain either C–C bonds or second nearest neighbor

carbon pairs. Although C–C bonds are thermodynamically unfavorable in group-IV

alloys, if C–C bonds are built into the molecular cores of precursor molecules and are

stable under film deposition, then we could expect C–C bonds to persist in the result-

ing alloys. We propose here two new molecular precursors, C2(SiH3)6 and C2(GeH3)6,

for synthesizing group-IV compunds with C–C bonds and 25 at. % carbon.

Fig. 3.9 shows the proposed molecular precursor and sample crystalline structure,

fully relaxed from LDA calculations of Si6C2. Both Si6C2 and Ge6C2 have a mono-

clinic lattice which is a slight distortion from bct. For Si6C2, the distortion is very

small, less than 0.1% while for Ge6C2 it is about 1.0%. To a good approximation,

we can still use the more familiar k point notation of the bct structure for these two

compounds.

LatticeConstant(A)

Bond Length(A)

δE(eV/atom)

BulkModulus(GPa)

Si6C2 a=b≈4.91c≈9.46

Si–Si 2.29, 2.322.27, 2.48

C–C 1.55Si–C 1.90, 1.99

0.20 146

Ge6C2 a=b≈5.18c≈9.84

Ge–Ge 2.40, 2.442.37, 2.58

C–C 1.50Ge–C 2.01, 2.12

0.40 111

Table 3.3: Structural properties of Si6C2 and Ge6C2. The lattice notation of the bctstructure is used since both structures are only slightly distorted from bct.

81

Table 3.3 shows the structural properties of these two group-IV compounds. Con-

sidering that the fraction of C–C bonds is large, 1/16, the excess energy of these two

compound is lower than expected. For Si6C2, the excess energy is 0.2 eV/atom, which

is lower than that of Ge4C. One curious result is that the C–C bond in Si6C2 is longer

than the normal bond length in diamond: 1.55 A vs. 1.53 A (both are LDA values),

whereas for Ge6C2, the C–C bond is shorter than the normal one, only 1.52 A. The

strong Si–C bonds, which effectively pull the two carbons apart to minimize the strain

energy, might explain the slightly longer C–C bond length in Si6C2. For Ge6C2, the

relatively weak Ge–C bonds, coupled with stronger C–C bonds due to charge transfer

from Ge atoms to C atoms, imply that shorter C–C bonds are the optimal strain-relief

configuration. Due to the lowering in symmetry, there are two types of A–C (A=Si or

Ge) bonds. The weighted average Si–C and Ge–C bond lengths are 1.93 A and 2.05 A

repectively, substantial longer than the normal Si–C and Ge–C bonds. The existence

of C–C bonds defines a particular direction along which the bond length of both Si–Si

and Ge–Ge bonds vary significantly. For example, the length of Si–Si bonds along

the direction defined by the C–C dimer varies widely from 2.27 to 2.48 A. (However,

the average Si–Si bond length does not deviate significantly from the bulk value.)

The Si–Si bonds more perpendicular to the C–C dimers are closer to the bulk value,

varying from 2.29 to 2.32 A. The presence of over-stretched Si–Si bonds along the

C–C bond direction are consistent with substantial strain on the C–C bonds. Similar

results are seen for Ge6C2.

Fig. 3.10 shows the LDA band structures of Si6C2 and Ge6C2. Both alloys are

metallic within the local density approximation. Strain-induced reduction of the

band gap has been reported before[141]. However, we are not convinced that Si6C2

is indeed metallic since LDA usually underestimates the band gap and in some cases,

semiconductors become metallic in LDA calculations. Since the band overlap for

Si6C2 is small, less than 0.5 eV, Si6C2 might have a small direct gap if quasiparticle

82

M Λ Γ ∆ X W P Q N Γ

ener

gy (e

V)

−15

−10

−5

0

5

M Λ Γ ∆ X W P Q N Γ

−15

wavevector K

ener

gy (e

V)

−10

−5

0

5

Figure 3.10: LDA band structure of Si6C2 (top) and Ge6C2( bottom).

83

energy (eV)

DO

S (

1/eV

)

−16 −12 −8 −4 0 40

1

2

3

4

energy (eV)

DO

S (

1/eV

)

−16 −12 −8 −4 0 40

1

2

3

4

Figure 3.11: Density of states of Si6C2 (top) and Ge6C2( bottom).

84

calculations are carried out. For Ge6C2, the band overlap is large and we do not

expect quasiparticle corrections to fully open the gap. Notice that although the

conduction and valence bands overlap at Γ, we still can see a “quasi-direct” gap,

which is evidence of increasing ionicity, i.e., charge transfer between Si (Ge) and C

in these group-IV compounds. Another consequence of the large ionicity, possibly

along with symmetry breaking, is that the lowest valence band is completely split

off from the rest of valence bands and is very flat, indicating that this band is very

localized (on carbon). Density-of-states plots of Si6C2 and Ge6C2 are shown in Fig.

3.11. The density of states at the Fermi level is very low. However it increases

rapidly below or above the Fermi level. A small amount of doping might change the

electronic properties of these two materials greatly. The anisotropic bonding network

of both materials might produce an anisotropic phonon dispersion, while the covalent

bonding and relatively strong volume sensitivity of the bandstructure in this class

of materials could yield relatively large electron-phonon matrix elements. The large

phonon frequencies of C-associated modes suggests the possibility for a moderately

large superconducting transition temperature under doping heavy enough to climb

out of the pseudogap.

3.6 Conclusion

We have carried out pseudopotential DFT studies on the electronic and structural

properties of novel group-IV alloys that either have been synthesized or are designed

to be synthesizable by UHV-CVD techniques using precursor molecules which build

in the desired bonding. All of the alloys studied are metastable with moderate ex-

cess energies. While the bandgaps of Si4C and Ge4C remain indirect, the effect of

substitional carbon on the band structure is prominent. For Ge4C, a 2% lattice ex-

pansion brings the bandgap direct. Sn4C has a direct and slightly negative LDA gap.

More importantly, we have demonstrated two silicon lattice matching, direct-bandgap

85

group-IV semiconductors, namely, Si2Sn2C and Ge3SnC. Finally, we have proposed a

new class of precursor molecules, X6C2 (X=Si, Ge or Sn), which incorporates 25 at.%

carbon. These high carbon concentration group-IV compounds exhibit interesting

structural and electronic properties which might stimulate experimental search for

this new class of group-IV alloys. The Kouvetakis team at Arizona State University

has recently made some progress in synthesizing X6C2 alloys[184].

The idea of using CVD precursors which build in the desired chemical bonding

to synthesize metastable alloys can be extended to systems beyond group-IV alloys.

This technique opens great possibilities of synthesizing materials that are inaccessible

to conventional thermal equilibrium synthesis techniques. With the help of modern

powerful computers and well-developed ab initio computational techniques, one might

be able to design other specific precursors for novel materials with desired properties.

86

Chapter 4

O(N2) real-space electronicstructure methods

4.1 Introduction

The past few decades have witnessed dramatic progress in our ability to explain

and predict various properties of materials knowing only the identities of their con-

stituent atoms. This progress has been driven mainly by two important factors,

namely, the availability of ever-increasing computer speed and advances in theoret-

ical/computational methods, which have made computer simulation another branch

of scientific research, in addition to the traditional theoretical and experimental ap-

proaches to doing science.

DFT-based ab initio methods, such as pseudopotential plane-wave methods[37] as

we have discussed in Chapter 1, have been very popular and successfully applied to

calculate the electronic, optical, and structural properties of a vast array of materials.

Unfortunately, these traditional electronic structure methods also suffer from some

significant drawbacks which are intrinsic to the formalism, for example, the cubic

scaling of computational time with respect to the number of atoms (or electrons)

in the system, the so-called O(N3) problem. Due to this unfavorable scaling, the

number of atoms one can study using these methods is limited to a few hundred even

87

with the most powerful computers available today. Consequently, an alternative, the

so-called real-space approach, has attracted much research interest recently. (The

plane-wave formalism is also called momentum-space formalism for obvious reasons.)

The real-space methods have some advantages over the momentum-space methods

which we shall discuss in more detail later.

The reason that traditional electronic structure methods, such as pseudopotential

plane-wave methods, must scale at least as O(N3) is the following. The electronic

wavefunctions generally extend over the entire volume of the system, computing an in-

dividual wavefunction takes at least O(N) operations, the number of electrons (hence

the number of wavefunctions needed) grows linearly as the number of atoms, and

finally, most iterative diagonalization algorithms require explicit reorthogonalization

between eigenfunctions which brings in an additional multiplicative O(N) scaling. It

can easily be seen that, if the explicit requirement of wavefunction orthogonalization

is eliminated, an O(N2) scaling can be achieved. Developing such an O(N2) electronic

method is the main subject of this chapter.

Although this chapter involves a considerable amount of mathematical formulae

and numerical algorithms, we shall focus more on the physics, not the completeness

and rigor of the mathematical formalism.

4.1.1 Momentum-space versus real-space formalism

Plane-waves are natural choices for describing the electronic wavefunctions in peri-

odic systems. Plane-wave expansion, when coupled with pseudopotentials to eliminate

the need to explicitly consider the highly localized core electrons (see Chapter 1), can

be very advantageous: (i) the plane-wave basis set is well structured, which greatly

simplifies code development, (ii) the accuracy of the calculation can be improved

systematically by a single parameter, i.e., the highest kinetic energy of the plane-

waves included in the wavefunction expansion, (iii) plane-waves do not depend on the

88

atomic positions and therefore simplify the calculation of forces on atoms and atomic

geometry, (iv) the kinetic energy operator is strictly diagonal in momentum space,

whereas the local pseudopotential is diagonal in real space. Fast Fourier transforms

(FFT) can be carried out between the two spaces.

However, some of the advantages of momentum space formalism may become dis-

advantages in certain circumstances. For example, the fact that plane-waves do not

depend on the atomic positions is problematic when dealing with systems such as

molecules, clusters and surfaces, in which a large number of plane-waves is needed

to decribe the nearly vacuum space. For very localized systems such as the first row

elements or transition metals containing partially filled d or f shells, a plane-wave ex-

pansion becomes very inefficient even if the system contains only one such atom. Also,

the FFT algorithm cannot be efficiently parallelized on modern distributed memory

computers. Furthermore, the computational cost of the traditional momentum-space

methods scales at least as O(N3), where N is the number of atoms in the system, as

we discussed at the beginning of this chapter.

Recently, there has been considerable research interest in real-space methods [185-

205], in which the wavefunctions or other quantities of interest, for example, charge

densities, potentials, etc., are described directly on real space grid(s). In fact, tra-

ditional quantum chemistry calculations are done solely in real space using either

analytical bases (for example, Gaussian-type or Slater-type) or numerical localized

atomic orbitals. These methods usually cannot take advantage of multigrid conver-

gence acceleration[206, 207], so we will not discuss them in detail here. Instead, we

will focus on grid-based real-space methods. Grid-based calculations usually require

large amounts of computer memory (of the order of 10 GB for a system containing a

few hundred atoms) and thus could not be done until recently.

There are a number of reasons for carrying out calculations directly in real space

grid for isolated systems and in some cases, even for periodic systems. First, unlike

89

plane-waves, which extend over the entire system with uniform resolution, the reso-

lution of wavefunctions in real-space can be adjusted to reflect the detailed atomic

arrangement of the system, resulting in an adaptive description of wavefunctions.

Second, the Hamiltonian matrix in real space is usually very sparse. This signifi-

cantly reduces the computation cost for iterative diagonalization algorithms. In con-

trast, the potential matrix V (G,G′) = V (G−G′) = V H(G−G′) + V xc(G−G′) +

V ps(G−G′) in Eqn. (1.61) is a full dense matrix. (Note that we need not to store

the entire potential matrix since V depends only on (G−G′).) Third, parallelization

of real-space based methods is usually more efficient than that of momentum-space

based methods since one can assign specific regions of space to particular proces-

sors and there are no communciation-intensive FFT operations involved. Fourth,

multi-scale[206, 207] (also known as multi-level or multi-grid) convergence accelera-

tion techniques can be readily applied to real space calculations. Finally and very

importantly, a real-space formalism is essential for implementing the so-called O(N)

(or linear scaling) algorithms[208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218,

219, 220, 221, 222, 195], which promise to achieve O(N) scaling with respect to the

number of atoms in the system, as opposed to O(N3) for the traditional techniques.

4.1.2 O(N) methods

The pioneer work of Yang[208],Galli, et al.[209], Li, et al.[210] and Daw[211] in the

early 90’s showed that, by examing the localization properties of quantum systems,

electronic structure calculations are possible with computational cost that scales lin-

early with respect to the number of atoms in the system. Common features of all

proposed O(N) algoritms are (i) the use of a localization approximation, and (ii)

no computation of the individual wavefunctions and eigenenergies. The key idea of

O(N) methods is to evaluate the total energy of the system directly by exploiting the

localization properties of either a Wannier-like orbital basis[209, 212, 213, 214, 195]

90

(the wavefunction approach) or the density matrix ρ(r, r′)[210, 211, 215, 216, 217,

218, 219, 220] (the density matrix approach). There are other O(N) approaches

as well, for example, the projection methods[221, 222] and the divide-and-conquer

methods[208, 224].

It is remarkable that, although the one-particle wavefunctions, which are com-

monly used for describing many particle systems, usually extend over the entire sys-

tem, the physical properties in a given region of a material are affected mainly by its

local environment. This insensitivity of local properties to the long-distance pertur-

bation is called, in Kohn’s word[225], the “nearsightedness” of equilibrium quantum

systems. Actually, locality in quantum systems has been appreciated for a long time

in traditional chemistry. (However, tranditional electronic structure calculations do

not fully exploit this interesting property.) One obvious such example is the cova-

lent bonding in molecules and covalence solids. The bonding nature of these system

is mostly determined by the local atomic arrangement and is described in terms of

electron pairs shared by nearby atoms. The famous Valence Shell Electron Pair Re-

pulsion (VSEPR) theory of Gillespie and Nyholm[226, 227], which is widely used

for determing the molecular geometry, is based on such observations. Surprisingly,

although the conventional bonding concept is no longer valid in metallic systems,

locality still exists. This is supported by the well-known fact that the charge den-

sity in a metal can be reasonably described by the superposition of its consituent

atomic charge densities. The “nearsightedness” of quantum systems is best mani-

fested in the exponentially decaying property of the one-particle charge density ma-

trix ρ(r, r′): ρ(r, r′) ∼ exp(−γ|r− r′|), where γ > 0. It was shown recently that

γ ∝ Eg for semiconductors and γ ∝ T for metals, where Eg is the bandgap and T is

the temperature[228]. The O(N) method of Li, Nunes and Vanderbilt (LNV) is based

on such “nearsightedness” of the density matrix, which is briefly discussed below.

Recall that the traditional way to solve for the band energy Ebs is to diagonalize

91

the Hamiltonian:

HΨi = εiΨi, (4.1)

where Ψi and εi are eigenfunctions and the corresponding eigenvalues. The band

energy is the sum of all occupied eigenvalues:

Ebs =∑i∈occ

εi. (4.2)

As we have discussed in Section 4.1, the above process requires O(N3) operations.

An alternative is to solve for the ground state density-matrix and formulate the band

energy in terms of the density matrix, i.e.,

Ebs = minρtr[ρH ], (4.3)

where ρ satisfies two constraints,

tr[ρ] = Ne, and ρ2 = ρ. (4.4)

The first constraint can be eliminated by working with the grand potential Ω

Ω = Ebs − µNe, (4.5)

where µ is the chemical potential. The second constraint, the idempotency con-

straint, can also be removed by introducing the so-called purification transformation

of Mcweeny[229, 210]: If ρ is a trial density-matrix which is nearly idempotent, then

ρ = 3ρ2 − 2ρ3 (4.6)

is a purified density-matrix which is closer to idempotent than the original ρ.

By taking advantage of the fact that the density-matrix is local in real space, i.e.,

ρ(r, r′) → 0 as |r− r′| → ∞, LNV developed a variational scheme which minimizes

the grand potential Ω in Eqn. (4.5) with the density-matrix ρ replaced by the purified

version ρ. Since the number of non-zero elements of ρ is O(N), the minimization

92

process should take nearly O(N) operations to be completed. The LNV algorithm

is best formulated within the tight-binding (TB) or other empirical models[210, 215,

214, 219], although LNV-like DFT-based O(N) methods using localized Gaussian

orbitals has also been reported[220].

4.1.3 O(N2) electronic structure methods: an intermediatestep

The O(N) methods represent a group of aggressive approaches to ultimately reduce

the computational effort. Whereas the TB O(N) methods[221, 214, 216, 217, 218]

are relatively well developed and widely used, most of the proposed DFT-based

O(N) methods suffer some difficulties[223]. We propose here an intermediate step

towards DFT-based large-scale ab initio calculations by developing an O(N2) real-

space electronic structure method without introducing any localization assumptions.

One important feature of our method is that all occupied electronic wavefunctions and

eigenvalues are computed explicitly, whereas most O(N) methods calculate only the

total energy of the system. Our scheme is based on the finite-element method with

multigrid acceleration, which is one of several proposed grid-based real-space elec-

tronic structure approaches. The O(N2) scaling is achieved by avoiding the explicit

reorthogonalization between eigenfunctions through use of a multigrid algorithm.

Grid-based real-space electronic approaches include finite-difference (FD)[185, 186,

187, 188, 189, 190, 191, 192, 193, 194, 195], finite-element (FE)[196, 197, 198, 199, 200,

201] and wavelet[202, 203, 204, 205] methods. Although the FE method has a long and

successful history in engineering applications, there are only a few quantum mechani-

cal calculations using FE methods. The reasons for this lack of interest in FE methods

among the physics community deserves further discussion, which is beyond the scope

of this thesis. Electronic structure calculations for atoms and diatomic molecules

using FE methods started in the 70’s and 80’s[230, 231, 232, 233, 234, 235, 236].

93

These early calculations used symmetry properties to reduce the dimensionality of

the system. White, et al. carried out the first full 3D FE calculations for atoms and

molecules[197].

Not surprisingly, there have been even fewer solid-state calculations based on FE

methods. Hermansson and Yevick[196] first applied the FE method to 3D solid-state

electronic structure calculations but reached a negative conclusion that “the standard

FEM formalism is far less efficient than standard (plane-wave) techniques”. Recently,

there has been revived interest in FE electronic structure calculations in solids start-

ing from the work of Tsuchida and Tsukada[198, 199, 200] and a recent paper of Pask,

et al.[201]. Unfortunately, none of the aforementioned work implemented the multi-

grid technique[206, 207], which is critical for improving the convergence rate. Here

we present a self-consistent (SC) FE electronic structure calculation with multigrid

acceleration. Furthermore, our method scales nearly as O(N2). In the following, we

give a brief introduction to the FE method.

4.2 The FE method

The mathematical treatment of the FE method is based on the variational for-

mulation of differential equations. Consider the Poisson equation in some domain Ω

with, for example, Dirichlet boundary conditions:

−u(x) = f(x) in Ω,

u(x) = 0 on ∂Ω.(4.7)

Before proceeding with our discussion, we need to introduce some mathematical no-

tations. We use L2(Ω) to denote the space of square-integrable functions over domain

Ω, Hm(Ω) is the Sobolev space of L2(Ω) with square-integrable derivatives up to

order m in Ω and Hm0 (Ω) the subspace of H

m(Ω) of functions with zero boundary

conditions[237]. The above boundary value problem is equivalent to the variational

94

formulation: find u ∈ H10 (Ω) such that

a(u, v) = (f, v) ∀v ∈ H10 (Ω), (4.8)

where

a(u, v) =∫Ω∇u · ∇v, and (f, v) =

∫Ωfv. (4.9)

Actually, the solution of Eqn. (4.8) is the solution of the following variational problem:

find u ∈ H10 (Ω) such that

J(u) =1

2a(u, u)− (f, u) (4.10)

attains its minimum. For the Kohn-Sham equation (1.24), we have the following

variational formulation: find ψi ∈ H1(Ω) such that

1

2

∫Ω∇φ∗ · ∇ψi +

∫Ωφ∗[V eff − εi]ψi = 0, ∀φ ∈ H1(Ω), (4.11)

where V eff = V H + V xc + V ext is the effective potential. Here H1 denotes the space

of functions that satisfies certain boundary conditions. For periodic systems, we may

write ψi,k = exp(ik · r)ui,k and the corresponding variational formulation for ui,k

becomes[201]

1

2

∫Ω∇v∗ ·∇ui,k+

∫Ωv∗[−2ik·∇ui,k+(V eff−k2−εi,k)ui,k] = 0, ∀v ∈ H1(Ω). (4.12)

The function space H1 include all smooth functions satisfying periodic boundary

conditions.

4.2.1 The Ritz-Galerkin Method

Solving the variational form of partial diferential equations, for example, Eqn.

(4.8), (4.11), or (4.12) exactly in the functional space H1 is a formidable challenge

and usually not necessary. Instead, we solve them in some approximated finite-

dimensional subspace Sh, where h measures the discretization parameter. The solu-

tions should converge to the exact ones as h → 0. This is called the Ritz-Galerkin

95

method. The problem is therefore reduced to, in the case of the Poisson equation

(4.8): find uh ∈ Sh such that

a(uh, vh) = (f, vh), ∀vh ∈ Sh(Ω). (4.13)

Suppose φ1, φ2, · · · , φn is a basis of Sh, then Eqn. (4.13) is equivalent to: finduh ∈ Sh such that

a(uh, φi) = (f, φi), i = 1, 2, · · · , n. (4.14)

Furthermore, we can expand uh as a superposition of φi, namely, uh = ∑ciφi, and

reach the following system of algebraic equations

AC = b, (4.15)

where

Aij = a(φi, φj), bj = (f, φj), (4.16)

and C is the column vector of ci, i.e., (c1, c2, · · · , cn)T . The procedure thus far seemsvery familiar. The idea of FE analysis is to approximate the subspace Sh by a piece-

wise polynomial basis with respect to a partition of a domain, the so-called FE space.

The partition of a domain is called the FE triangulation for historical reasons.

4.2.2 FE triangulation and nodal basis functions

In the FE analysis, the domain of the system is divided into subdomains called

“elements”. (The term “element” also refers to the FE space associated with a partic-

ular triangulation, for example, quadratic triangular element, bilinear quadrilateral

element, etc..) For problems in 2D, it is natural to partition the domain into trian-

gles (2-simplices, see Fig. 4.1), as shown in Fig. 4.2. This is probably the reason that

the FE partition is called a triangulation. FE discretization is far more flexible than

FD discretization. For 3D problems, a partition into either tetrahedrons or hexahe-

drons (parallelpipeds, cubes, etc.) are commonly used. Fig. 4.3 shows a tetrahedron

partition example of a cubic domain.

96

(a) (b) (c)

Figure 4.1: n-simplices: 1, 2, and 3-simplices.

(a) (b)

Figure 4.2: Finite-element triangulation examples in 2D: (a) uniform triangulationand (b) adaptive (nonuniform) triangulation.

Figure 4.3: A 3D triangulation.

97

φi

iN

Figure 4.4: A linear nodal basis function. The function is defined corresponding tonodal point Ni.

After partitioning the domain, we need to decide on the polynomial order of the

basis functions. These functions are called nodal basis functions since each is as-

sociated with an individual node. The simplest finite elements are linear elements,

the nodal basis of which consists of linear functions. However, linear elements are

usually not very accurate. In our calculations, we use a quadric tetrahedral element.

Implemention of higher order elements is also possible. Fig. 4.4 shows a linear nodal

basis function φi defined with respect to node i. The nodal basis functions have the

following properties, regardless of their order: (i) φi(xj) = δij, (ii) φi is strictly local-

ized (locally supported), (iii) the nodal basis functions forms a basis φ1, φ2, · · · , φn,over which any function u ∈ Sh can be expanded: u(x) =

∑nj=1 ujφj(x). It can be

easily seen from (i) that the coefficients of expansion are simply the corresponding

functional values, namely, uj = u(xj).

98

4.3 FE discretization of the Kohn-Sham equations

Having given a brief introduction to the FE method, we now return to our orig-

inal goal: to solve the Kohn-Sham equations in real-space with FE approximations.

We will assume periodic boundary conditions. Extension to non-periodic boundary

condition is straightforward. For simplicity, we will also assume k = 0 in Eqn (4.12),

which is then reduced to Eqn. (4.11).

The FE discretization of Eqn (4.11) results in a generalized eigenvalue problem:

HΨi = εiSΨi, (4.17)

where

Hij =∫Ω[1

2∇φi · ∇φj + V effφiφj] (4.18)

is the Hamiltonian matrix and

Sij =∫Ωφiφj (4.19)

is the overlap matrix. Note that the wavefunctions are written as superpositions on

the FE basis, i.e., Ψi(r) =∑

Cijφj(r), and again, φi are the FE basis functions.

Therefore, our task is to compute the relevant matrix elements and solve the general-

ized eigenvalue problem (4.17). Both the kinetic energy matrix Tij =12

∫Ω∇φi · ∇φj

and the overlap matrix Sij can be calculated analytically; we will not discuss this in

detail here. The construction of the potential matrix Vij =∫Ω V effφiφj, however, is

not trivial.

4.4 Construction of the ab initio potential matrices

In the psuedopotential formalism, the effective potential consists of four terms,

the exchange-correlation potential V xc, the Hatree potential V H , the local ionic pseu-

dopotential V ps,loc and the nonlocal pseudopotential, which is usually written in the

Kleinman-Bylander (KB) form[46], V psKB (see Section 1.3.3).

99

First, the charge density ρ is calculated in the usual way:

ρ(r) = 2∑i

Ψi(r)∗Ψ(r), (4.20)

where i sums over all occupied orbitals. (Initially, the charge density is taken as the

superposition of atomic charge densities. We assume no spin polarization.) Since

V xc(r) is a local function of charge density ρ(r) in LDA, its matrix elements are

simple to compute

Vxcij =

∫ΩV xc(r′)φi(r′)φj(r′) =

∑k

V xck

∫Ωφk(r

′)φi(r′)φj(r′). (4.21)

Note that in the above equations, Vxcij are the matrix elements of V

xc(r) while V xcj is

the FE discretization of V xc(r), i.e.,

V xc(r) =∑i

V xci φi(r). (4.22)

We introduce here another notation Fijk =∫Ω φi(r)φj(r)φk(r), which can be computed

analytically in the FE framework. Eqn. (4.21) can be rewritten as

Vxcij =

∑k

V xck Fijk. (4.23)

Similarily, matrix elements of the Hartree and the ionic pseudopotentials are calcu-

lated by

VHij =

∑k

V Hk Fijk, and (4.24)

Vps,locij =

∑k

V ps,lock Fijk, (4.25)

where V Hj and V ps,loc

j are the FE discretizations of the corresponding potentials, i.e.,

V H(r) =∑

V Hj φj(r) and V ps,loc(r) =

∑V ps,locj φj(r).

To calculate the Hartree potential in real space, instead of using the standard

formula

V H(r) =∫Ω

ρ(r′)|r− r′| , (4.26)

100

which requires O(N2) operations, we solve the Poisson equation

−V H(r) = 4πρ(r) (4.27)

with proper boundary conditions. The Poisson equation can be solved in O(N)

operations with multigrid techniques, as will be discussed in detail later.

The local part of the ionic pseudopotential is

V ps,loc(r) =∑i

∑τ

V psτ (r− rτ −Ri), (4.28)

where V psτ (r− rτ −Ri) is the pseudopotential due to the τth atom in the ith unit

cell. Since the ionic pseudopotential is long-ranged, the above expression involves an

infinite sum over all unit cells, making it impossible to carry out the calculation in

real space directly. There are several ways to get around this problem. For example,

we may Fourier transform the pseudopotential to momentum space, do the sum, (see

Eqn. (1.62)) then transform it back to the real space. However, this involves FFT,

which is what we want to avoid (see Section 4.1.1). We could also employ the elegant

procedure, known as the Ewald’s method[238], which would involve O(N2) operations.

Here we describe an alternative scheme that calculates the ionic pseudopotential in

O(N) operations.

First, we notice that the ionic pseudopotential can be regarded as the potential

due to some ficticious charge density, which satisfies

ρfict(r) =1

4πr

d2

dr2(rV ps(r)). (4.29)

This charge density has no obvious connection with the electronic charge density of

the core electrons, except for the fact that the total ficticious charge inside the core

region equals the number of valence electrons, i.e,

∫r<rc

4πρfict(r)r2dr = Zval. (4.30)

101

Since the pseudopotential decays like −Zval/r outside the core cut-off radius rc, theficticious charge density vanishes for r > rc. Therefore, although the pseudopotential

is long-ranged, the ficticious charge density is strictly localized inside the ionic core.

The total ficticious charge density in the unit cell can be evaluate in O(N) operations:

ρfict(r) =∑τ

ρfictτ (|r− rτ |). (4.31)

where τ sum over atoms in the unit cell. The advantage of working with the ficti-

cious charge density is readily seen: there is no infinite sum over all unit cells (see

Eqn. (4.28)). The full ionic pseudopotential is then recovered by solving the Poisson

equation

−V ps,loc(r) = 4πρfict(r) (4.32)

with proper boundary conditions. This is essentially the same procedure as that for

computing the Hartree potential in Eqn. (4.27) and therefore can be done in O(N)

operations.

Finally, we need to evaluate the matrix elements of the nonlocal pseudopotential

(1.55), which can be written as

V ps,nlocKB =

∑τ

lτmax∑l=0

m=l∑m=−l

|χτ,lm(r− rτ )〉〈χτ,lm(r− rτ )|. (4.33)

where the projector function

|χτ,lm〉 = |φps,τlm δV ps,τl 〉√

〈φps,τlm |δV ps,τl |φps,τlm 〉

(4.34)

is due to the contribution from the τth atom in the unit cell. See also Eqn. (1.55).

The corresponding matrix elements of the nonlocal pseudopotential are

Vps,nlocij = 〈φi|V ps|φj〉 =

∑τ

lτmax∑l=0

m=l∑m=−l〈φi|χτ,lm(r− rτ )〉〈χτ,lm(r− rτ )|φj〉. (4.35)

We have suppressed the subscript KB. The advantage of the KB form of nonlocal

pseudopotential is obvious: there is no need to evaluate the full matrix, instead, we

102

only need to calculate the projection vectors

∆τ,lmi = 〈φi|χτ,lm(r− rτ )〉 =

∫Ωφi(r)χ

τ,lm(r− rτ ). (4.36)

The full nonlocal pseudopotential matrix is simply the sum of the direct product of

the projection vectors:

Vps,nloc =∑τ

lτmax∑l=0

m=l∑m=−l

∆τ,lm(∆τ,lm)†. (4.37)

Eqn. (4.36) can be evaluated by a high-order numerical integration

∆lmi =

∫Ωφi(r)χ

lm(r) =∑k

ωkφi(rk)χlm(rk), (4.38)

where ωk is the weight associated with integration point k. We have suppressed

superscript τ for simplicity. An alternative is to discretize χ as

χlm(r) =∑k

χlmk φk(r), (4.39)

and evaluate ∆lmi via

∆lmi =

∑k

Sikχlmk , (4.40)

where Sik is the overlap matrix defined in Eqn. (4.19). We have implemented both

and see no significant differences between the two methods, although the numerical

integration scheme is slightly more accurate.

The speed of applying the nonlocal pseudopotential to the wavefunctions is an-

other advantage of carrying out calculations in real space since it involves only O(N)

operations, compared to O(N2) operations required in the momentum formalism.

4.5 Multigrid Poisson equation solver

As we have discussed in Section 4.2.1, FE discretiation of the The Poisson equation,

e.g., Eqn. (4.27) or Eqn. (4.32), results in a system of linear equations

AV = b, (4.41)

103

where Aij =∫Ω∇φi · ∇φj , V is the FE discretization of the potential, i.e., V (r) =∑

i Viφi(r) and bi = 4π∫Ω ρ(r)φi(r). Again, matrix A can be calculated analytically,

and the right-hand side b can be evaluated by either of the two schemes discussed in

the previous section for calculating ∆lm.

There are subtle issues that need to be addressed when solving the Poisson equa-

tion with periodic boundary conditions. First, solving the equation (4.27) or (4.32)

separately violates the so-called well-posedness condition for periodic systems or sys-

tems with Neumann boundary conditions, which requires∫Ωρ(r) = 0. (4.42)

The physical meaning of this condition is that if the charge in the unit-cell is not zero,

i.e.,∫Ω ρ(r) = 0, then the total charge in the system is infinite. Therefore the potential

diverges. This problem can be easily solved by subtracting the average charge density

ρ from ρ(r), i.e.,

ρ(r)← ρ(r)− ρ. (4.43)

There is another problem, however. Even if the well-posedness condition is satisfied,

the solution V is arbitrary up to a constant since V = 0 if V = const. This

arbitrariness can be avoided by imposing the following constraint on V :∫ΩV (r) = 0. (4.44)

The above two treatments are equivalent to discarding the G = 0 Fourier component

of the potential terms in the plane-wave method[37]. If the system is strictly Coulom-

bic, the G = 0 component of the ionic potential and that of the electronic Hartree

potential cancel out exactly. However, since the ionic pseudopotential (the local part)

is not Coulombic inside the ionic core, the two terms do not cancel each other. There-

fore, the calculated total energy (as well as the band structure) contains a constant

shift. This is a well-known proplem of the pseudopotential plane-wave calculations

and can be remedied by introducing a correction term to the total energy[37].

104

Solving the above linear system by any direct method, e.g., Gaussian elimination

or LU decomposition, is impractical since (i) the dimension of the matrix A is usually

very large, of the order of 106×106, making it impossible to store the entire matrix inthe computer memory, as required for most direct methods, and (ii) direct methods

usually cannot take full advantage of the sparseness of the matrix. Iterative methods

such as Gauss-Siedel (GS) or Conjugate-Gradient (CG)[239], on the other hand, are

very suitable for this type of problem. CG methods are usually stable and very fast,

and are among the best choices for pure algebraic problems. However, the computa-

tional effort of CG methods scales nearly as O(N2) even for sparse matrices. Since

the linear system we are about to solve is not a pure algebraic problem but the result

of the FE discretization of the Poisson equation, numerical mulitgrid methods[207]

can be easily applied. (There are also so-called Algebraic Multigrid (AMG) methods

for pure algebraic problems[207]). MG methods have been the most efficient tech-

niques in solving algebraic systems arising from discretization of partial differential

equations and are well known for their O(N) scaling. Standard iterative methods

such as GS and Jacobi methods, which are briefly discussed below, are often essential

components of MG methods.

4.5.1 Standard iterative methods for solving linear systems

A typical cycle of the standard iterative methods consists of three steps:

(1) form residual r = b−AV old,

(2) solve the residual equation Ae = r approximately, i.e., find B ≈ A−1 and solve

for the error function e = Br, and

(3) update the solution V new = V old + e and go to step (1) until convergence criteria

are met.

Different choices ofB result in different iterative methods. If we writeA = D− L−U,

where D is the diagonal part of A, −L and −U are the lower and upper triangu-

105

lar parts of A, respectively, we have the following classification of standard iterative

methods:

B =

ω, (ω > 0) ,Richardson,D−1 , Jacobi,ωD−1 ,Damped Jacobi,(D− L)−1 ,GS,ω(D− ωL)−1 , Successive Over Relaxation (SOR).

(4.45)

These standard iterative methods usually converge very quickly in the first few itera-

tions, but then the convergence slows down quickly, see Fig. 4.5. The reason for this

behavior is that the standard iterative methods have exceptional smoothing proper-

ties but are very inefficient in correcting the slowly-varying errors. In other words,

most of the high-spatial-frequency errors are smoothed out quickly in the first few

(less than 5) iterations. The low-frequency errors, on the other hand, persist after

many iterations. The quick drop in the error shown in Fig. 4.5 at the very begining re-

flects the fast decrease in high-frequency errors while the tail is due to the long-lasting

low-frequency errors.

4.5.2 Multigrid acceleration

The standard iterative methods are very effective in correcting the high frequency

errors, leaving mainly low frequency errors after a few iterations. The question there-

fore is, how can we speed-up the convergence of the low-frequency errors? Note that

the low-frequency errors are relatively smooth (this is what low-frequency means)

thus can be represented on a coarser grid (e.g., by doubling the grid size). Solving

the problem excatly on a coarser grid might be possible since the dimension of the

matrix A decreases as the grid size increases. Based upon these observations, we can

conceive the following two-grid method:

(1) Smoothing: Do a few (2 or 3, for example) GS iterations on the fine grid h to

smooth out the high-frequency errors, obtaining an approximate solution V h.

(2) Restriction: Form residual rh = bh − AhV h and transfer the residual rh to grid

106

0.01

0.1

1

0 5 10 15 20 25 30 35 40 45 50

Number of iterations

error

||e||

||V||

Figure 4.5: Typical convergence behavior of the GS method

2h by some interpolation scheme, i.e., r2h = I2hh rh.

(3) Coarse grid solution: Formulate the residual equation on grid 2h, namely,A2he2h =

r2h, and solve for the error function e2h.

(4) Prolongation: transfer the error function from grid 2h back to the original fine

grid h and update the solution, i.e., V h ← V h + Ih2he2h.

We have introduced superscript h and 2h to indicate the grid size. Ih2h and I2hh are a

restriction operator and prolongation operator, respectively, which represent interpo-

lation between grid h and grid 2h.

In this two-grid algorithm, we need to solve the residual equation on the coarse

grid 2h. However, if the problem on this grid is still too large, which is often so, exact

solution might still be very costly or even impossible. Fortunately, this problem can be

easily overcome by applying the above two-grid algorithm recursively until a coarsest

possible level is reached. In other words, instead of solving the residual equation on

the coarse grid, we may simply do a few GS iterations again and then transfer the

107

current residual (which now becomes the residual of the residual) equation onto an

even coarser grid. The process is repeated until an exact solution of the residual

equation becomes feasible. This is the basic idea of the multigrid (MG) method.

There are several flavors of MG algoritms[207] which we will not discuss in detail here.

Actually, the MG procedure we have just described is called back-slash MG algorithm.

We adopted a widely used and more powerful MG scheme, the so-called V-cycle MG

algorithm[207] to solve the Poisson equation. The pre-smoothing and post-smoothing

on all grids other than the coarsest one are done with two GS relaxations, while the

residual equation on the coarsest grid is solved nearly exactly.

4.5.3 Performance of the MG Poisson equation solver

Fig. 4.6 show the typical convergence behavior of the MG Poisson equation solver.

Uniform exponential decrease in error can be easily seen, regardless of the matrix

size. Although we only show the 2-norm error, we find the convergence behavior to

be independent of the choice of the functional norm. All calculations start from zero

initial guesses for the potential as indicated by the 100% starting errors. Table 4.1

shows the O(N) scaling properties of this algorithm. The slight deviation from the

exact O(N) scaling might due to other issues that are not related to the algorithm

itself, for example, data transfer latency between the main memory and the cache

tends to increase as the matrix size increases.

108

Number of MG iterations

error

||e||

||V||2

2

1e-14

1e-12

1e-10

1e-8

1e-6

1e-4

1e-2

1

0 1 2 3 4 5 6

N=32,768

N=131,072

Figure 4.6: Convergence properties of the MG Poisson equation solver. The solidand the dotted lines correspond to fine-grid matrix sizes of 32,768×32,768 and131,072×131,072, respectively.

# atoms Matrix size # MG iterations error: ‖e‖2‖V ‖2 CPU time (sec)

2 32,768×32,786 6 1.02E-13 1.654 65,536×65,536 6 7.62E-14 3.598 131,072×131,072 6 7.78E-14 7.7416 262,144×262,144 6 9.46E-14 17.5332 524,288×524,288 6 1.00E-13 34.29

Table 4.1: O(N) scaling properties of the MG Poisson equation solver. All calcula-tions were done on an alpha DS10 workstation.

109

4.6 Full multigrid Kohn-Sham equation solver

Once the Hartree and ionic pseudopotentials are calculated and all necessary ma-

trix elements are evaluated, we are ready to solve the generalized eigenvalue problem

(4.17) arising from discretization of the Kohn-Sham equations. Solving the (general-

ized) eigenvalue problem is, however, far more complicated than solving the Poisson

equation. In this section, we develop a full multigrid (FMG) algorithm for solving the

Kohn-Sham equations self-consistently. (The meaning of FMG will be clear later.)

The FMG algorithm for solving eigenvalue problems was first developed by A. Brandt,

et al.[206]. Costiner, et al. [243, 244] later implemented an FMG algorithm to solve

nonlinear model Schrodinger equations. Our FMG algorithm differs from the previous

implementations in several respects. First, we do not adopt the full approximation

scheme (FAS). (Comments on the drawbacks of applying the FAS to solve the Kohn-

Sham equations for real systems can be found in Ref. [189].) Instead, we employ

the residual correction MG approach similar to that for solving Poisson’s equation

discussed in the previous section, which can be regarded as the MG implementation

of the Davidson method[242] for eigenvalue problems. Second, while the previous

work was on a standard eigenvalue problems, our method deals with the generalized

eigenvalue probem. Third, instead of working on some model problems, we intend

to develop an ab initio electronic structure method for real systems based on the FE

method. Finally, our approach achieves potential self-consistency and convergence in

eigenpairs (eigenvalues and corresponding eigenfunctions) simultaneously in as few as

15 iterations on the finest level.

4.6.1 Brief description

Here we describe a 3-level (i.e., coarse, intermediate and fine levels) FMG scheme

for the Kohm-Sham problem. The FMG eigenvalue solver starts on the coarse level Lc,

110

on which the generalized eigenvalue problem is solved with a CG algorithm[240, 241],

as will be discussed in detail later. After performing a few self-consistent iterations

on level Lc, we obtain a set of initial guesses for the KS eigenvalues and eigenfunc-

tions. The eigenfunctions are then interpolated onto an intermediate level Li and

renormalized. The interpolations are done with the standard FE prolongation pro-

cedure. At the intermediate level, both the self-consistency of the potential and the

accuracy of the eigenpairs are improved simultaneously through an MG algorithm

similar to that for solving the Poisson equations. The results obtained on level Li are

further transferred to the fine level Lf ; subsequent refinement of the eigenpairs and

improvement of the self-consistency are carried out, again, simultaneously, through

an MG algorithm.

The FMG scheme for eigenvalue problems differs from the MG Poisson equation

solver described in the previous section in several respects:(i) Whereas the MG Pois-

son equation solver starts on the fine grid, the FMG eigenvalue solver begin with

the coarse grid solution. The reason is that the MG Poisson equation solver usually

converges very quickly for almost any initial guess satisfying proper boundary condi-

tions (e.g., a zero initial guess). Solving the eigenvalue problem, on the other hand,

is very time consuming and the convergence is usually slow and depends strongly

on the initial guess. Therefore, obtaining a reasonable initial guess is crucial to the

algorithm’s performance. In self-consistent electronic structure calculations, we need

initial guesses for both potentials and eigenpairs. The solutions on the coarse grid,

after being transferred onto the intermediate level Li as initial guesses, could signif-

icantly reduce the computational effort on the level Li. Similarly, results obtained

on level Li serve as good initial guesses for the fine level solutions. (ii) In the FMG

scheme, calculations on all levels other than the coarse one involve an MG loop. This

explains why the algorithm is called full multigrid. Actually, each MG loop in the

FMG eigenvalue solver is similar to the MG algorithm for solving the Poisson equa-

111

tion. (iii) In order to obtain a meaningful initial guess, the coarse grid must be fine

enough that the essential characteristics of the wavefunctions can be represented. In

contrast, the lowest (coarsest) level in the MG Poisson equation solver could be much

coarser than Lc. Since the coarse grid in the FMG scheme can not be too coarse,

we believe that two levels of subsequent refinements are enough for obtaining a fairly

accurate solution, although more intermediate grid levels can be easily implemented.

Note that the coarse grid solution not only provides an initial guess for the finer level

problem but also separates the clusters of eigenvectors so that no explicit reorthogo-

nalization between clusters of eigenvectors is necessary. However, eigenvectors within

a cluster (including degenerate eigenvectors) need to be separated explicitly. We will

return to this point later.

4.6.2 Coarse grid solution

CG algorithms[240, 241] are among the best for finding the lowest eigenpair for

large sparse symmetric (or Hermitian) matrices without any good initial guesses.

Therefore, they are very suitable for solving the coarse grid problem. We adopt

the CG algorithm of Yang[240], which is based on the minimization of the Rayleigh

quotient for generalized eigenvalue problems. Although Yang only proved the con-

vergence properties of the algorithm for the lowest eigenpair[240], we find that the

algorithm is capable of calculating the lowest few eigenpairs with some modifications.

Rayleigh quotient

The Rayleigh Quotient R of the generalized eigenvalue problem

Hx = εSx (4.46)

is defined by

R(x) =x†Hx

x†Sx, (4.47)

112

where x ∈ Cn, x = 0. Obviously, R is bounded by the two extreme eigenvalues of

Eqn. (4.46), i.e., εmin ≤ R(x) ≤ εmax. Since the lower bound of R is the smallest

eigenvalue of the system, minimization of R(x) with respect to x should give λmin.

Minimization of the Rayleigh quotient by the conjugate gradient method

A typical gradient-based minimization algorithm consists of three steps:

(i) Choose a search direction pk,

(ii) Perform one-dimentional minimization along direction pk, i.e.,

minαk

R(xk−1 + αkpk), (4.48)

(iii) Update xk = xk−1 + αkpk.

As we can see, the two basic ingredients of the minimization algorithm are computing

search directions and performing line minimizations.

In the CG method, the search direction p is related to the gradient g by

pk = −gk + βkpk−1, (4.49)

where the gradient g(x) of Rayleigh quotient R(x) is given by

g(x) =2

x†Sx(Hx− R(x)Sx), (4.50)

and has the property

x†g(x) = 0, ∀x = 0. (4.51)

At the kth iteration, βk is given by

βk =

0 if k = 0,

g†kgk/g†k−1gk−1 otherwise.

(4.52)

In practical calculations, βk is usually reset to zero after a fixed number of iterations.

The other parameter (αk) is determined by line minimization of the Rayleigh quotient

113

(4.48). By setting the derivative of R(xk + αkpk) to zero, we have the following

equation for αk:

aα2k + bαk + c = 0, (4.53)

where

a = (x†kSpk)(p

†kHpk)− (x†

kHpk)(p†kSpk), (4.54)

b = (x†kSxk)(p

†kHpk)− (x†

kHxk)(p†kSpk), (4.55)

and

c = (x†kSxk)(x

†kHpk)− (x†

kHxk)(x†kSpk). (4.56)

Solving the Eqn. (4.53) gives

α±k =

−b±√b2 − 4ac2a

. (4.57)

α± corresponds to the minimum and maximum of R along the direction p and we

should choose α+. Numerically more stable formulas for calculating α+ are

α+k =

−2c/(b+√b2 − 4ac) if b > 0,

(−b+√b2 − 4ac)/2a if b ≤ 0 and a = 0.(4.58)

Special care must be taken for the case when a = 0 and b ≤ 0[240]. However, this

almost never happens in numerical calculations.

The CG method for generalized eigenvalue problems described above is rather

stable. In most cases, the minimization converges to the lowest eigenpair with almost

any initial guess. By imposing orthgonalization constrains, we have generalized the

algorithm for computing the lowest few eigenpairs. Unfortunately, this algorithm

scales like O(N3), making it unsuitable for large-scale problems. However, since

the matrix size on the coarse level is relatively small, the computational effort is

insignificant for reasonable system size (say, up to a few hundred). For a system

containing 32 atoms, only about 3% of the total CPU time is spent on this level.

114

4.6.3 O(N2) iterative diagonalization methods

After solving the Kohn-Sham equations on the coarse grid, we obtain a set of

eigenpairs, which, upon being transferred, serves as a reasonable initial guess for the

finer level problem. There are several iterative diagonalization schemes, for example,

Rayleigh-quotient [239] and Davidson iteration[242], which do not require explicit

re-orthogonalization between eigenfunctions if a good initial guess for the eigenvalues

and the eigenvectors is given. As we have discussed at the begining of this chapter,

if no explicit re-orthogonalization is needed, O(N2) scaling can be achieved.

Rayleigh-quotient iteration

Rayleigh-quotient iteration is closely related to the inverse iteration technique,

based on the fact that if λ is close to a distinct eigenvalue, say, λi, of the matrix

H, then the following iteration converges to the eigenvector corresponding to λi with

some initial guess x:

do while (not converged)y = (H− λI)−1xx = y/‖y‖2

end do

(4.59)

The convergence property of the above iteration can be understood by expanding x

as a superposition of eigenvectors xi

x =n∑i

cixi, (4.60)

so that, after one iteration,

y =n∑i

ciλi − λ

xi. (4.61)

It can be easily seen that y has a larger component in the direction of xi than the

original vector x does if λ is close to λi.

The Rayleigh-quotient iteration replaces the fixed parameter λ by the Rayleigh-

115

quotient (x†Hx) (assuming x is normalized):

do while (not converged)λ = x†Hxsolve (H− λI)y = x for yx = y/‖y‖2

end do

(4.62)

For generalized eigenvalue problem, we have

do while (not converged)λ = x†Hxsolve (H− λS)y = Sx for yx = y/‖y‖S

end do

(4.63)

where ‖y‖S = (y†Sy)1/2. The Rayleigh-quotient iteration almost always converges

and when it does, the rate of convergence is cubic[239]. A serious drawback of the

Rayleigh-quotient iteration is that it involves solving the linear equation (H−λS)y =

Sx, which is not as easy as solving the Poisson equation discussed in Section 4.5 due

to the positive-indefiniteness of the matrix (H− λS). (Note that the matrix size of a

typical real-space calculation is of the order of 106× 106). Also, the matrix (H− λS)

gets close to singular as λ approaches to the exact eigenvalue, making it even more

difficult to solve the linear equation. Relaxation techniques, on the other hand, do

not suffer from these drawbacks.

Improving the eigenvectors by relaxation

Recall that the ultimate goal of solving the eigenvalue problem is to find λ ∈ C

and x ∈ Cn such that

Hx = λx, (4.64)

or

(H− λI)x = 0. (4.65)

116

(For simplicity, we return to the standard eigenvalue problem.) This is essentially

a linear equation if λ is known. Given some reasonable initial guess x0 to x, we

may approximate λ by the Rayleigh-quotient λ0 = (x†0Hx0)/(x

†0x0) and improve the

eigenvector x = x0 + δ by relaxing the following equation

(H− λ0I)(δ + x0) = 0, (4.66)

or equivalently,

(H− λ0I)δ = r, (4.67)

where the residual r is

r = −(H− λ0I)x0. (4.68)

Note that we shall not solve Eqn. (4.67) exactly. In fact, the exact solution gives a

trivial result, namely, δ = −x0, or x = δ + x0 = 0. However, we may employ some

relaxation techniques, e.g., GS or Jacobi methods (see Section 4.5.1), to obtain an

approximate solution δ. A Jacobi-type iteration gives

δ = (D− λ0I)−1r, (4.69)

where D is the diagonal of H. This is the so-called Davidson iteration[242]. At this

point, we may update the eigenvector

x0 ← x0 + δ (4.70)

and the Rayleigh-quotient

λ0 ← x†Hx

x†x. (4.71)

The above process, namely, Eqn. (4.69–4.71), is repeated until a certain convergence

criterion is satisfied. Unfortunately, Davidson iteration usually converges very slowly.

This slow convergence rate of Davidson iteration is not unexpected based on our

discussion in Section 4.5, since the Davidson method is similar to the Jacobi method

117

for pure linear systems. The similarity between the two problems, on the other hand,

inspires us to formulate an MG algorithm for the eigenvalue problem.

Multigrid acceleration

1e-12

1e-10

1e-08

1e-06

1e-4

1e-2

1

0 1 2 3 4 5 6 7 8 9

MG relaxation

Davidson relaxation

Eigenvalueerror

Number of iterations

Figure 4.7: Comparison of the convergence properties of the standard Davidson iter-ation and the MG relaxation scheme we developed. The relaxations were performedon the finest grid, using the intermediate-grid solution as a initial guess.

The basic idea of our MG method for the generalized eigenvalue problem (4.17) is

very similar to what we have discussed in Section 4.5.2, which we shall not repeat here.

However, we shall point out several important differences between the two schemes.

First, as we have mentioned previously, we shall not solve Eqn (4.67) exactly. In-

stead, the equation is relaxed (in a MG fashion) and the eigenvalue is updated by

the Rayleigh-quotient (4.71). Second, unlike the Poisson operator, which is positive-

definite, the matrix (H− λI) (or (H− λS) for the generalized eigenvalue problem) is

positive indefinite. Third, whereas the Poisson operator is uniquely defined on each

118

grid level, multigrid representations of the Kohn-Sham Hamiltonian matrix (the po-

tential part) on grids other than the finest grid are somewhat arbitrary. For example,

the ionic potential on the coarse grid may be either calculated by solving the Poisson

equation directly or by interpolating the results from the finer grid. Nevertheless, the

final converged results do not depend on these subtle differences.

Fig. 4.7 compares the typical convergence behavior of traditional Davidson iter-

ation and our MG relaxation algorithm. Both relaxations use the solution on the

coarser grid as a initial guess. Whereas the convergence of the Davidson method

resembles that of standard iteration methods for linear systems (see Fig. 4.5), the

MG relaxation schems for the generalized eigenvalue problem has the similar con-

vergence properties to that of the MG Poisson equation solver (see Fig. 4.6). The

error in eigenvalue is reduced to about 10−12 in less than 10 MG relaxations. (The

absolute value of the eigenvalue is of the order of 1.) Note that there have been

conjectures in the literature that it is not possible to formulate a residual-correction

MG scheme for nonlinear (eigenvalue) problems[245]. Our results demonstrate that

the residual-correction MG scheme is also suitable for solving the eigenvalue problem.

Rayleigh-Ritz Projection

The algorithm described in the above works well if all eigenvalues are distinct, and

since individual eigenpairs can be improved independently, we expect this algorithm

to scale like O(N2). However, if there are clustered or degenerate eigenvalues, this

algorithm is not able to separate the corresponding eigenvectors well. The result is

that the eigenvectors with the cluster might not be orthogonal to each other after

the MG relaxation. Fortunately, this problem can be easily solved by the so-called

Rayleigh-Ritz projection. The Rayleigh-Ritz projection is essentially a subspace diag-

onalization, which can be used to separate the eigenvectors and find the correspond-

ing eigenvalues if only linear combinations of eigenvectors are known. For example,

119

suppose we have m vectors, φ1, φ2, · · ·φm, which are linear combinations of eigenvec-tors corresponding to m-fold degenerate (or clustered) eigenvalues of the generalized

eigenvalue problem (4.46). The eigenvectors and the corresponding eigenvalues can

be obtained by solving the following m×m eigenvalue problem:

HC = εSC, (4.72)

where

Hij = φ†iHφj, and Sij = φ†

iSφj. (4.73)

The separated eigenvector corresponding to eigenvalue εi is

ψi =∑j

φjCji. (4.74)

Therefore, whereas the separation of clusters of eigenvectors is accomplished through

the coarse grid solutions, the eigenvectors within the cluster are separated by Rayleigh-

Ritz projection. We have implemented a simple algorithm to identify clustered eigen-

values and perform the Rayleigh-Ritz projection after every MG relaxation. Although

the Rayleigh-Ritz projection procedure brings about some extra computational effort

to the algorithm, the overhead is reasonably small.

4.6.4 Simultaneous convergence of potential and eigenvalues

In ab initio electronic structure calculations, the electronic potential and wavefunc-

tions have to be calculated in a self-consistent manner. A typical SC loop consists of

4 steps:

120

do while (not converged)

(1) construct the potential V = V (ρold)

(2) solve the Kohn-Sham equations (through MG relaxation),

(3) calculate the new charge density ρnew

(4) mix the old charge density with the new one, e.g.ρnew ← αρold + (1− α)ρnew; ρold = ρnew

end do

Solving the Kohn-Sham equations is obviously the most time consuming part of the

problem if we require the wavefunctions (eigenvalues) to be well converged at each SC

iteration. As Fig. 4.7 shows, it takes about 10 MG iterations to converge the eigen-

value to about 10−12 with a fixed potential. However, before the potential converges,

it might not be neccessary to insist that the wavefunctions be well converged. If we

update the potential after, say, 2 or 3 MG relaxation, we might be able to save 60%

− 80% of the CPU time on this step and hopefully, we will also be able to obtain a

simultaneous convergence in eigenvalues and the potential at the end. In fact, we do

achieve this simultaneous convergence in as few as 15 SC iterations.

4.7 Performance of the FMG electronic structure

code

Table 4.2 shows the scaling and convergence properties of our 3-level FMG code

for silicon systems. (Note that the Poisson equation solver might involve more than

3 levels). The calculations were done on an alpha DS10 workstation equipped with

1GB memory. Although we have only done Γ-point sampling, implementing more

general k-point sampling is straightforward.

The volume of the 2-atom unit cell is uniformly partitioned into 24,576 tetrahedral

elements on a 32×32×32 grid. (Adaptive nonuniform FE triangulation is intended

121

as future work.) The grid sizes of the 4-atom, 8-atom, 16-atom and 32-atom cells are

32×32×64, 32×64×64, 64×64×64 and 64×64×128, respectively.

# atoms CPU time # SC charge density max eigenvalue

(sec) iterations error∫

|δρ|d3r∫ρd3r

error (Ryd)

2 85 (23) 15 1.8E-5 3.3E-74 255 (85) 15 1.4E-5 2.4E-78 895 (349) 15 1.4E-5 2.0E-716 3873 (3676) 15 1.7E-5 6.4E-732 14967 (16987) 15 3.7E-5 3.3E-6

Table 4.2: Scaling and convergence properties of the FMG real-space electronic struc-ture code. All calculations were done on an alpha DS10 workstation. Note that theerrors refer to the differences in the corresponding values between two consecutiveiterations, not the absolute errors with respect to the converged values. CPU timefor the corresponding plane-wave calculation is shown in the parenthesis.

For all of the calculations, we carry out 8 SC iterations on the coarse and inter-

mediate levels before moving onto the fine level, on which we perform an additional

15 SC iterations. The generalized eigenvalue problem on the coarse level is solve with

the CG algorithm described in Section 4.6.2. Actually, the CG algorithm is only used

at the very beginning of the SC iteration. Further improvement of the eigenvectors

and the potential is done with the relaxation methods describe in Section 4.6.3, since

the eigenvalues and eigenvectors from the previous SC step provide reasonable initial

guesses for the relaxation i scheme. The eigenvalue problem is solved by the MG

relaxation algorithm on the intermediate and the fine levels. Within each SC cycle,

we perform only one MG relaxation on the eigenvectors and subsequently update the

eigenvalues and the charge density (potential). The charge density mixing is done

with a linear mixing scheme:

ρnew ← αρold + (1− α)ρnew, (4.75)

122

with α = 0.65 for all calculations. As it is shown in Table 4.2, the CPU time scales

approximately as O(N2), where N is the number of atoms in the system. Simultane-

ous convergence of the charge density (or equivalently, potential) and the eigenvalues

(eigenfunctions) are achieved with as few as 15 iterations on the finest level. CPU

times for the corresponding plane-wave calculation with comparable accuracy are

shown in parentheses for comparison. Our real-space code has potentially better per-

formance than the traditional plane-wave code for system containing more than 32

atoms.

Now we briefly discuss the storage (memory) requirement of the code. Since

we calculate all of the occupied wavefunctions, the memory usage scales as O(N2)

asymptotically for large N . For small systems, the dominant storage requirement is

for the relevant matrices, i.e., the Hamiltonian, kinetic energy, and potential, and

scales linearly with the system size. For large systems, however, the storage for the

wavefunctions becomes more demanding, and scales quadratically with system size.

For example, the grid size of a 32-atom silicon system is 64×64×128=524,288. Sincethe number of occupied bands is 64, the memory required for storing the wavefunc-

tions is 64 × 524, 288 × 8 = 268, 435, 456 Bytes = 256 MB. A supercomputer node

nowadays is usually equipped with 4 − 8 GB of memory (1 GB= 1,024 MB), which

should be capable of handling systems containing up to 150 atoms. One way to get

around the memory bottleneck is to store the wavefunctions temporarily on the hard

disk. However, the performance of the code will be degraded significantly.

4.8 Summary

In summary, we have developed a FMG real-space electronic structure method

based on the FE method. The computational cost of this method scales nearly like

O(N2), as compared to the O(N3) scaling of the traditional methods such as pseu-

dopotential plane-wave methods[37]. The O(N2) scaling is acheived by avoiding the

123

explicit reorthogonalization of wavefunctions. Whereas the separation of clusters

wavefunction is accomplished through the FMG algorithm, wavefunctions within a

cluster are separated by the Rayleigh-Ritz projection.

124

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141

Vita

EducationSummer, 2001 : Ph. D., Condensed Matter Physics

Ph. D. Minor, High Performance ComputingThe Pennsylvania State University

Spring, 1996 : M.S., Atomic & Molecular Physics, Institute of Physics,Chinese Academy of Sciences, China

Spring, 1993 : B.S., Physics, Xiamen (Amoy) University, China

Awards1998— 1999 : Braddocks fellowship, The Pennsylvania State University

2000— 2001 : Braddocks fellowship, The Pennsylvania State University

2001 : The Xerox award for outstanding Ph. D. thesis at PSU in the area ofmaterials

Selected Publications

1 P. Zhang, V. H. Crespi, E. Chang, S. G. Louie, and M. L. Cohen, ComputationalDesign of Direct Bandgap Semiconductors That Lattice-Match Silicon, Nature 409,69 (2001).

2 P. Zhang and V.H. Crespi, Plastic Deformation of Boron Nitride Nanotubes: AnUnexpected Weakness, Phys. Rev. B. 62, 11050 (2000).

3 P. Lammert, P. Zhang and V.H. Crespi, Gapping by Squashing: Metal-InsulatorTransitions in Collapsed Carbon Nanotubes, Phys. Rev. Lett. 84, 2453 (2000).

4 P. Zhang and V.H. Crespi, Nucleation of Carbon Nanotubes without PentagonalRings, Phys. Rev. Lett. 83, 1791 (1999).

5 P. Zhang, P. Lammert, and V.H. Crespi, Plastic Deformations of Carbon Nan-otubes, Phys. Rev. Lett. 81, 5346 (1998).

6 J.-M. Li, P. Zhang, Y. Yang, and L. Lei, Theoretical Study of Adatom Self-diffussion on Metallic fcc001 Surfaces, Chinese Phys. Lett.14, 768 (1997).

7 Y. Zhang, P. Zhang, and J.-M. Li, Near-threshold Structure in Inner-shell Pho-toabsorption Process of N2 and CO, Phys. Rev. A 56, 1819 (1997).

8 P. Zhang and J.-M. Li, Geometry and Electronic Structure of Na3, Acta Phys.Sin. 46, 870 (1997).

9 P. Zhang and J.-M Li, Theoretical Studies of Electronic Excited States for Na3,Phys. Rev. A 54, 665 (1996).

10 J. Yan, P. Zhang, and J.-M Li, Fine Structure Inversion in f Channel of AlkaliAtoms, Acta phys. Sin. (1996).