First-principles electronic structure theory, Graphene ...(LDA). In order to solve the...

First-principles electronic structure theory, Graphenecalculations

Zi Wang0463744

Verslag van Bachelorproject Natuur- en Sterrenkunde,

Omvang 12 EC, uitgevoerd tussen 01-03-2009 en 09-07-2009.

Begeleider: Prof. Dr. A.M.M. Pruisken

Tweede beoordelaar: Prof. Dr. E.P. Verlinde

Inleverdatum: 09-07-2009.

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Instituut voor Theoretische Fysica

c© Zi Wang, 2009

Contents

Abstract iv

Samenvatting v

Acknowledgements vi

1 Introduction 11.1 Physical constants and units . . . . . . . . . . . . . . . . . . . . . . . 11.2 Abbreviations and definitions . . . . . . . . . . . . . . . . . . . . . . 21.3 Electronic structure Schrodinger equation . . . . . . . . . . . . . . . . 21.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Density Functional Theory 52.1 The electron density . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . . 72.3 The self-consistent Kohn-Sham equations . . . . . . . . . . . . . . . . 102.4 The Local Density Approximation . . . . . . . . . . . . . . . . . . . . 14

3 Electronic structure calculations 163.1 Solving the Kohn-Sham equation . . . . . . . . . . . . . . . . . . . . 163.2 Solid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Disordered systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Tight binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 The LMTO method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.7 Electronic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Graphene 31

5 Experiment 345.1 The software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Conclusion 40

A Functionals 41

B Green’s Functions 43B.1 Poisson’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44B.2 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45B.3 Electronic structure Green’s function . . . . . . . . . . . . . . . . . . 46

References 47

ii

List of Figures

2.1 Isosurface of the ground-state density of C60 fullerene as calculatedwith DFT [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1 A Bloch wave in a Si unit cell [18]. . . . . . . . . . . . . . . . . . . . 18

4.1 Energy plot of the graphene BZ as calculated by a tight-binding model[12]. 32

5.1 Graphite band structure calculated by our TB-LMTO-ASA model. . 365.2 Graphite band structure calculated by PWP, a more accurate but much

more expensive method [16]. . . . . . . . . . . . . . . . . . . . . . . . 375.3 Graphite band structure calculated by a TB model. The region around

the K-point is encircled on the left. The encircled point to the rightis the H-point, the corner on the top and bottom of the hexagonalrhomboid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4 DOS of pure graphene as calculated by our TB-LMTO-ASA model. . 395.5 DOS of pure graphene as calculated by a TB model [17]. . . . . . . . 39

iii

Abstract

In this thesis, we will introduce a few ab-initio or first-principles approaches of solvingthe electronic structure many-body Hamiltonian. At first, the density functionaltheory (DFT) is introduced, which is a fundamental technique that maps the many-electron wave function to the electron density. This quantity depends only on the3 spatial variables instead of the 3N variables needed by the wave function, andthus considerably reduces the computational effort required to solve the many-bodySchrodinger equation (SE). It is shown that DFT is an exact transformation of the SEapart from a presumably small exchange-correlation term which can be approximatedby several methods, one of which we will explain is the local density approximation(LDA).

In order to solve the self-consistent Kohn-Sham (KS) equations arising from DFT,the method of tight-binding linear muffin-tin orbitals (TB-LMTO) is used in cooper-ation with the atomic sphere approximation (ASA), and Green’s functions are usedto calculate the necessary parameters. This method is well suited for the electronicstructure problem of solid crystals.

This technique is applied to a newly discovered 2D material called graphene, a sin-gle graphite sheet with many remarkable properties. We study the electronic structureof pure graphene and calculate the local density of states (DOS). The resulting bandstructure is in excellent agreement with consulted literature, as is the calculated DOS.

iv

Samenvatting

In dit bachelorverslag wordt een aantal methodes behandeld die berekenen hoe elek-tronen zich gedragen rondom een samenstelling van atomen. Deze atoomstructuurkan een molecuul zijn, maar ook een blok vaste stof. Deze methodes zijn gebaseerd ophet principe van ab-initio, wat betekent dat de bewegingen van de elektronen in hette bestuderen materiaal bepaald worden met alleen de beginselen van de quantumme-chanica en de atoomstructuur, zonder meetwaarden erbij te raadplegen. Dichtheids-functionaaltheorie (DFT) is een manier om het veel-elektronenprobleem om te zettennaar een elektronendichtheidsprobleem, waar niet de elektronen, maar de kans omeen elektron ergens aan te treffen wordt beschreven. Hier is veel makkelijker mee terekenen dan met de elektronen zelf en geeft dezelfde resultaten. Om de vergelijkingenop te lossen die uit DFT voortkomen wordt een methode toegepast die zeer geschiktis voor vaste kristallen.

Grafiet bestaat uit lagen koolstofatomen gerangschikt in een honinggraatstruc-tuur. De benaming voor 1 zo’n laag is grafeen, een zeer platte laag koolstofatomenmet veel interessante eigenschappen en toekomstige toepassingen. De behandelde the-orie wordt gebruikt om puur grafeen te modelleren en de elektronische eigenschappenervan uit te rekenen. Deze berekeningen blijken goed overeen te komen met de resul-taten uit bestaande literatuur.

v

Acknowledgements

I would like to thank my supervisor, Prof. Dr. Aad Pruisken, for guiding me throughthis process and helping me with many issues. His insight and criticism allowed meto think deeper about the subject and come up with better solutions.

As I did part of this research at McGill University, I would like to thank mysupervisors there, Prof. Hong Guo and Youqi Ke, for providing lots of insight andintricate details of how to set up the software I used.

vi

1

Introduction

1.1 Physical constants and units

Throughout this thesis, atomic units are used, which equates the following constants

to unity:

e = me = h = a0 = 1.

Physical constants.

Planck’s constant h 6.6261× 10−34 J s

Electron mass me 9.1096× 10−31 kg

Proton mass mp 1.6726× 10−27 kg

Bohr radius a0 5.2918× 10−11 m

Electron charge e 1.6022× 10−19 C

Speed of light c 2.9979× 108 m s−1

Fundamental units.

Mass me 9.1094× 10−31 kg

Length a0 5.2918× 10−11 m

Charge e 1.6022× 10−19 C

Angular momentum h 1.0546× 10−34 J s

Energy Eh 4.3597× 10−18 J

Electrostatic force constant 14πε0

8.9876× 109 C−2 N m2

1

1: Introduction 2

1.2 Abbreviations and definitions

List of abbreviations.

SE Schrodinger Equation

DFT Density Functional Theory

HK Hohenberg-Kohn

KS Kohn-Sham

HEG Homogeneous Electron Gas

LMTO Linear Muffin-Tin Orbital

NEGF Non-Equilibrium Green’s Function

List of definitions.

H Molecular Hamiltonian

T N -electron kinetic energy

Vext N -electron external potential energy

Vee Electron-electron interaction energy

Ψ(r1, r2, . . . , rN) Ground state wavefunction

n(r) Ground state electron density

v(r) External potential

1.3 Electronic structure Schrodinger equation

The Hamiltonian of a fully interacting system consisting of many nuclei and electrons,

the molecular Hamiltonian, is the total energy operator of the system. Ignoring

spin, it is given as

H = Te + Tn + Vee + Vnn + Vext, (1.1)

with Te the many-electron kinetic energy operator, Tn the many-nucleus kinetic en-

ergy operator, Vee the electron-electron interaction energy operator, Vnn the nucleus-

nucleus interaction energy operator, and Vext the electron-nucleus interaction energy

1: Introduction 3

operator (external potential). They are given as respectively:

Te = −1

2

∑i

∇2(ri), (1.2)

Tn = −1

2

∑j

1

Mj

∇2(Rj), (1.3)

Vee =1

2

∑i 6=j

1

|ri − rj|, (1.4)

Vnn =1

2

∑i 6=j

ZiZj|Ri −Rj|

, (1.5)

Vext = −∑i,j

Zj|ri −Rj|

, (1.6)

where Mi, Zi, Ri are respectively the mass, atomic number and position of nucleus i.

The Schrodinger eigenvalue equation of this system is

HΨ = EΨ, (1.7)

where the system wavefunction Ψ depends on all configuration variables:

Ψ = Ψ (r1,R1, r2,R2, . . . ) . (1.8)

This is an equation defined in 3M + 3N -parameter space, and is too complex if not

impossible to solve for all but the simplest systems.

Taking into consideration that the mass of a nucleus far exceeds that of an electron,

we can assume that the motion of the nuclei is negligible compared to that of the

electrons and fix their positions. Using this, we can drop (1.3) and (1.5) out of the

equation and rewrite it as a problem of many electrons moving in an external potential

Vext generated by the stationary nuclei:

H = T + Vext + Vee, (1.9)

dropping the electron subscript of T . The SE for this system is

HΨ =

[−1

2

∑i

∇2 −∑i,j

Zj|ri −Rj|

+1

2

∑i 6=j

1

|ri − rj|

]Ψ = EΨ, (1.10)

1: Introduction 4

with Ψ now the many-electron wavefunction, dependent on 3N variables (or 4N

variables when taking spin into consideration),

Ψ = Ψ (r1, r2, . . . ) . (1.11)

This approximation method is called the Born-Oppenheimer approximation and

is used in all systems that are more complex than the hydrogen atom.

1.4 Overview

Starting from the mid-20th century, there has been an increased interest in first-

principles theory, which relied only on fundamental quantum mechanics to calculate

properties as observed in materials. With the emergence of density functional theory

(DFT) and increasingly accurate approximations as well as more powerful computers,

calculations on quantum devices became more and more accessible. Many interesting

phenomena have since then been predicted, alot of them having been verified by

experiment later on.

Graphene, a material that has only been discovered recently (2004), has many

interesting properties and alot of research has already been conducted on it.

This thesis will start with a description of DFT, a modern approach to the many-

body structure problem. Then I will outline the techniques employed in solid state

materials theory, using the Tight-Binding Linear-Muffin-Tin-Orbital with the Atomic

Sphere Approximation (TB-LMTO-ASA) method and the Green’s function method

to solve the configurational Schrodinger equation. The Coherent Potential Approx-

imation is used to conduct calculations on disordered systems, which nearly always

occur in realistic systems. I will also introduce the carbon allotrope Graphene, a

single sheet of Graphite with many interesting properties, and apply the TB-LMTO-

ASA-CPA technique to conduct first-principles calculations on Graphene transport.

2

Density Functional Theory

2.1 The electron density

Figure 2.1: Isosurface of the ground-state density of C60 fullerene as calculated with DFT [18].

The electron density n(r) is the probability to find any electron in a certain vol-

ume element dr. It is defined by square-integrating the wavefunction over all spin

coordinates and all space coordinates except the first one:

n(r) = N∑s1

· · ·∑sN

∫. . .

∫|Ψ(r, s1, r2, s2, . . . , rN , sN)|2 dr2 . . . drN . (2.1)

Since electrons are indistinguishable, the probability of finding any electron in dr is

just N times the probability of finding 1 electron there. Theoretically it is a probability

density, but it is more commonly called the electron density. The boundary conditions

for n(r) thus have to be that it is positive for all r, vanishes at infinity, and integrates

5

2: Density Functional Theory 6

to the total number of electrons N :

n(r) ≥ 0, (2.2)

n(r→∞) = 0, (2.3)∫n(r)dr = N. (2.4)

Using the electron density as a basic variable has several advantages compared to

using the wavefunction. First and foremost of all, it is a function consisting of only

the 3 spatial variables instead of the 3N variables of the wavefunction, thus having a

tremendous computational advantage. Also, it is an observable, so unlike the wave-

function it can be measured experimentally, for example by x-ray diffraction. At all

the atomic sites, it has a finite maximum value due to the positive attraction located

there.

The idea of using the electron density instead of the wavefunction as a method to

obtain information about the electronic structure is almost as old as the theory of

quantum mechanics itself. Thomas and Fermi were amongst the first to independently

develop a theory based on the density, in which they approximated the electron

distribution using a statistical model.

The Thomas-Fermi model (TF) assumes that electrons are distributed uni-

formly in phase space, with two electrons in every h3 of volume. Now, for each

position space volume element d3r we fill out a sphere of momentum space to the

Fermi momentum pf :4

3πp3

f (r). (2.5)

As the number of electrons must be the same in both position space and momentum

space, we find for the electron density n(r):

n(r) =8π

3h3p3f (r). (2.6)

Solving for pf and using this to calculate the classical kinetic energy, we now get a

relation of the kinetic energy as a functional of the electron density:

TTF [n] = CF

∫n5/3(r)d3r, (2.7)


where, now in atomic units,

CF =3

10

(3π2)2/3

. (2.8)

Neglecting the exchange and correlation terms, we can express the total energy of an

atom in just the electron density,

ETF [n] = CF

∫n5/3(r)d3r− Z

∫n(r)

rd3r +

1

2

∫n(r)n(r′)

|r− r′|d3rd3r′. (2.9)

TF theory was thus the first official density functional theory (DFT). It had major

drawbacks however, as the exchange energy and electron correlation was totally ne-

glected, resulting in a large error in the kinetic energy term. It was also proven that

TF theory could not describe molecular bonding. Back then it was also not clear

whether this density functional theory was fundamentally sound.

In 1964, Hohenberg and Kohn published a paper in which they mathematically

proved the exactness and viability of using n(r) as opposed to the much more complex

Ψ. Because of the still unknown exchange and correlation and the major contribution

from those to the kinetic energy, HK-DFT did not achieve a high accuracy in calcu-

lations. It was a year later, in 1965, that Kohn and Sham found a way of approaching

this problem to good accuracy and thus created density functional theory in the form

we know it today.

2.2 The Hohenberg-Kohn theorems

The first Hohenberg-Kohn theorem is stated as:

The external potential v(r) is a unique functional (within a trivial additive constant)

determined by the electron density n(r).

Therefore, as v(r) fixes the Hamiltonian H, n(r) also uniquely determines all other

properties of the system. After 40 years, it has finally been rigorously proved that it

is indeed physically justified to use the electron density n(r) as a basic variable.

The original proof is actually very simple and is done by a reductio ad absurdum.

Let n(r) be the ground-state density for a system of N electrons in an external po-

tential v(r), which has the ground-state wavefunction Ψ and the energy E. Then we


can write for the energy,

E = 〈Ψ|H|Ψ〉

=

∫v(r)n(r)dr + 〈Ψ|T + Vee|Ψ〉. (2.10)

Suppose that another potential v′(r) 6= v(r) + constant with ground state Ψ′ results

in the same density n(r). Ψ′ of course cannot be equal to Ψ as they are the solutions

to different Schrodinger equations. Then

E ′ =

∫v′(r)n(r)dr + 〈Ψ′|T + Vee|Ψ′〉. (2.11)

Now the variational principle states that

E < 〈Ψ′|H|Ψ′〉

=

∫v(r)n(r)dr + 〈Ψ′|T + Vee|Ψ′〉 (2.12)

= E ′ +

∫[v(r)− v′(r)]n(r)dr. (2.13)

If we swap the primed and unprimed quantities, we find in the same way that

E ′ < 〈Ψ|H|Ψ〉

= E +

∫[v′(r)− v(r)]n(r)dr. (2.14)

Adding (2.13) and (2.14) together this implies

E + E ′ < E + E ′. (2.15)

This of course is a contradiction, which establishes the fact that there can only be

one v(r) that produces the ground-state density, and conversely that the ground-state

density n(r) uniquely determines the external potential v(r).

Now, as it is proven that n(r) contains all the necessary information for the entire

system, it follows that all observables are certain functionals of n(r). The total energy

Ev[n] and its constituents, the total kinetic energy T [n], the total external potential

energy Vext[n] and the total electron-electron interaction energy Vee[n] are now written

as

Ev[n] = T [n] + Vext[n] + Vee[n]. (2.16)


Rewriting this in the form of

Ev[n] =

∫v(r)n(r)dr + T [n] + Vee[n]

=

∫v(r)n(r)dr + 〈Ψ|T + Vee|Ψ〉, (2.17)

we can see that one part (∫v(r)n(r)dr) is system dependent. For the other part,

which is universally valid, we define a new Hohenberg-Kohn functional FHK [n] as

FHK [n] ≡ 〈Ψ|T + Vee|Ψ〉 (2.18)

= T [n] + Vee[n],

so that equation (2.17) becomes

Ev[n] =

∫v(r)n(r)dr + FHK [n]. (2.19)

Although this functional FHK [n] appears to be just another formulation, it is actually

very important as it is applicable to each and every system. If it were known exactly,

it would have been possible to solve the Schrodinger equation exactly, and there would

have been no need of approximations. However, it contains the functionals for the

kinetic energy T [n] and the electron-electron interaction energy Vee[n], both of which

the explicit forms are totally unknown. It can be convenient to extract the classical

Coulomb energy from Vee[n] and write

FHK [n] =1

2

∫n(r)n(r′)

|r− r′|drdr′ +G[n], (2.20)

where G[n] is a (unknown) universal functional containing the kinetic energy T [n]

and all the non-classical contributions.

To come back to the total energy Ev[n], it is of course clear that for the correct

n(r) this quantity equals the ground-state energy E0. Using the variational principle,

it is easily shown that insertion of the correct n(r) minimizes Ev[n]. Consider a

trial density n′(r) which satisfies the necessary boundary conditions n′(r) ≥ 0 and∫n′(r)dr = N . As proven, n′(r) has a unique wavefunction Ψ′. Taking this Ψ′ as the


trial wavefunction for the Hamiltonian H corresponding to the real density n0 we can

write

〈Ψ′|H|Ψ′〉 =

∫v(r)n′(r)dr + FHK [n′]

= Ev[n′] ≥ E0[n] = 〈Ψ|H|Ψ〉,

or more compactly

E0[n] ≤ Ev[n′]. (2.21)

This density variational principle is also known as the second Hohenberg-Kohn

theorem:

The Hohenberg-Kohn functional FHK [n] delivers the lowest energy if and only if the

input density is the true ground-state density n0.

For the ground-state density we can minimize the energy functional by using the

constraint that ∫n(r)dr = N. (2.22)

Using the method of Langrangian multipliers we can now write

δ

Ev − µ

[∫n(r)dr−N

]= 0, (2.23)

the stationary principle which the ground-state density has to satisfy. Solving for µ,

we get the Euler-Lagrange equation

µ =δEv[n]

δn(r)

= v(r) +δFHK [n]

δn(r), (2.24)

where µ turns out to be the chemical potential.

2.3 The self-consistent Kohn-Sham equations

The Hohenberg-Kohn theorems allow us to rewrite the energy into two parts, the sys-

tem dependent∫v(r)n(r)dr and the unknown functional FHK [n]. Directly applying

this to real world problems, using explicit approximate forms for T [n] and Vee[n] has


its simplicities, as it creates equations that depend only on n, but the downside is

that this method seemed to be greatly inaccurate. However, a year after the pub-

lication of Hohenberg and Kohn’s paper, Kohn and Sham devised a better way to

handle the unknown FHK [n]. They realized that most of the problems of the direct

density functionals used before arise due to how they determine the kinetic energy

and proposed a new method to indirectly approach T [n]. By doing this they turned

DFT into a practical tool for real world calculations.

Kohn and Sham added orbitals (one electron functions) to create a non-interacting

reference system from which the kinetic energy can be computed to good accuracy,

leaving only a small correction to be calculated separately. To begin with, the exact

formula for the kinetic energy for a set of orbitals is defined as

T = −1

2

N∑i

ni〈φi|∇2|φi〉 (2.25)

with φi and ni the orbitals and their occupation numbers, and 0 ≤ ni ≤ 1. From the

HK theory this energy is a functional of the density,

n(r) =N∑i

ni∑s

|φi(r, s)|2 . (2.26)

However for an interacting system, there will be an infinite number of terms. This

is hard to work with, and Kohn and Sham showed that it is possible to use simpler

formulas to describe a non-interacting reference system, using the non-interacting

Hamiltonian

Hs = −1

2

N∑i

∇2i +

N∑i

vs(r) (2.27)

without the electron-electron interactions and for which the ground-state electron

density is exactly n. This system has kinetic energy

Ts = 〈Ψs| −1

2

N∑i

∇2i |Ψs〉. (2.28)

Hartree-Fock theory already showed that for a non-interacting system like this one,

the ground-state wavefunction can be exactly represented by a determinant:

Ψs =1√N !

det[φ1φ2 . . . φN ], (2.29)


where the orbitals φi are the N lowest eigenstates of the one-electron Hamiltonian

hsφi = [−1

2∇2 + vs(r)]φi = εiφi. (2.30)

Here, vs is the non-interacting external potential. Using this, one can express the

kinetic energy Ts[n] and the electron density n(r) as

Ts[n] = −1

2

N∑i

〈φi|∇2|φi〉, (2.31)

n(r) =N∑i

∑s

|φi(r, s)|2 . (2.32)

This non-interacting kinetic energy Ts is of course still not the exact kinetic energy

T , even if both systems have the same density. Kohn and Sham had the very good

idea of rewriting the problem in such a way that Ts does become the exact kinetic

energy of that problem, simply moving the small residual part of the true kinetic

energy T to the heap of the non-classical contributions. In order to do this, they

rewrote (2.18) as

FHK [n] = Ts[n] + J [n] + Exc[n], (2.33)

with the exchange-correlation energy, containing the small difference between T and

Ts and the nonclassical Vee, defined as

Exc[n] ≡ T [n]− Ts[n] + Vee[n]− J [n]. (2.34)

For this system, we again minimize the energy using the constraint (2.24):

0 = δ

Ev − µ

[∫n(r)dr−N

]= δ

∫v(r)n(r)dr + Ts[n] + J [n] + Exc[n]− µ

[∫n(r)dr−N

]= v(r) +

δTs[n]

δn+δJ [n]

δn+δExc[n]

δn− µ. (2.35)

Defining

veff (r) ≡ v(r) +δJ [n]

δn(r)+δExc[n]

δn(r)(2.36)


and

vxc(r) ≡ δExc[n]

δn(r), (2.37)

we thus get the Euler-Lagrange equation

µ = veff (r) +δTs[n]

δn(r), (2.38)

with the Kohn-Sham effective potential :

veff (r) = v(r) +

∫n(r′)

|r− r′|dr′ + vxc(r). (2.39)

As (2.38) is simply a rearrangement of (2.24), attempting to directly solve it has no

use. Instead, Kohn and Sham proposed an indirect approach.

The KS Euler-Lagrange equation (2.38) with the constraint (2.22) has the exact

same form as the one obtained from conventional DFT, the only difference is that

it applies to a system of non-interacting electrons moving in the external potential

vs(r) = veff (r). Therefore, is it easy to obtain the n(r) that satisfies (2.38), one just

has to solve the N one-electron equations

[−1

2∇2 + veff (r)]φi = εiφi (2.40)

and using the fact that

n(r) =N∑i

∑s

|φi(r, s)|2 . (2.41)

(2.39-2.41) are the canonical Kohn-Sham equations.

As veff depends on n(r), φi depend on veff and finally n(r) depends on φi, these

self-consistent equations need to be solved iteratively. Usually one begins with a

initial guessed n(r), constructs a veff from (2.39), solves for the φi using (2.40) and

finally gets a new n(r) from (2.39). This process continues until the density converges,

at which time self-consistency is achieved.

As of now, all of the theory aforementioned has been exact and no approximations

have been made to arrive here. Unfortunately, an explicit expression for Exc[n] has

not been found, and we will have to resort to some approximations to be able to go

further.


2.4 The Local Density Approximation

The most widely used approximation is the Local Density Approximation (LDA),

which assumes dependency only on the electron density and is based on the uniform

homogeneous electron gas (HEG). To generalize this to spin-polarized systems this

becomes the Local Spin-Density Approximation (LSDA).

The HEG is a system of electrons moving on a positive background charge dis-

tribution such that the system as a whole is electrically neutral. The number of

electrons N and the volume V are considered to approach infinity, while the electron

density n = N/V remains finite and constant everywhere. Physically, this is a model

of an idealized metal, a perfect infinite crystal of valence electrons and positive atoms,

placed so that the positive charge is uniformly distributed. The HEG is actually quite

a good model for simple metals such as Na, but it is far from realistic situations as the

density usually varies rapidly in such systems. The reason for taking the HEG as a

model is that it is the only system for which the form of the exchange and correlation

energy functionals are known exactly or at least to very high accuracy.

The LDA for exchange and correlation energy is defined as:

ELDAxc [n] =

∫n(r)εxc(n)dr (2.42)

where εxc is the exchange and correlation energy per particle of a HEG of density n.

The exchange-correlation potential (2.37) can then be written as

vLDAxc (r) =δELDA

xc

δn(r)= εxc(n(r)) + n(r)

δεxc(n)

δn(2.43)

and the KS equation (2.40) becomes[−1

2∇2 + v(r) +

∫n(r′)

|r− r′|dr′ + vLDAxc (r)

]φi = εiφi. (2.44)

Solving this equation defines the Kohn-Sham LDA (KS-LDA) and is also just called

the LDA method. The exchange-correlation function εxc(n) can be separated into its

constituents, the exchange part and the correlation part:

εxc(n) = εx(n) + εc(n). (2.45)


The exchange part has an exact form of

εx(n) = −3

4

(3

π

)1/3

n(r)1/3. (2.46)

For εc there are numerical forms available, given by very accurate quantum Monte

Carlo calculations.

3

Electronic structure calculations

3.1 Solving the Kohn-Sham equation

In order to solve the Kohn-Sham equation (2.40), the one-electron wavefunction ψi(r)

is expanded in a basis set χν(r):

ψi(r) =N∑ν

ciνχν(r). (3.1)

We now define the Hamiltonian matrix Hνν′ and the overlap or normalization

matrix Sνν′ as respectively:

Hνν′ ≡ 〈χν |hs|χν′〉 = 〈χν | −1

2∇2 + V (r)|χν′〉, (3.2)

Sνν′ ≡ 〈χν |χν′〉. (3.3)

If χν(r) is orthogonal and normalized, then

Sνν′ = δνν′ . (3.4)

Operators are now represented by matrices and functions by vectors, and the KS

one-electron equation can be written in terms of the coefficients ciν′ as an eigenvalue

problem: ∑ν′

(Hνν′ − εiSνν′) ciν′ = 0, (3.5)

with ciν′ the vector of the coefficients ciν′. The eigenvalues can now be obtained

either by diagonalization or by solving the secular equation

det [Hνν′ − εiSνν′ ] = 0. (3.6)

16

3: Electronic structure calculations 17

The basis set or orbital set is chosen according to the characteristics of the problem,

orbitals that are better suited to the problem result in fewer of them needed to describe

the one-electron wave functions ψi(r) and thus less computational effort required.

For practical purposes, a basis set is finite in size and thus incomplete, and does not

need to satisfy orthonormality. It can be shown that this does not complicate the

calculation.

There is a large amount of basis sets available. Amongst them are augmented

plane waves (APW), atomic orbitals (AO), Gaussian orbitals (GO), muffin-tin or-

bitals (MTO), and linear combinations of them (LAPW, LCAO, LCGO, LMTO). The

Korringa-Kohn-Rostoker (KKR) methods uses Green’s functions of the KS equation

(2.40) to solve the problem.

With a chosen basis set, the KS equations (2.39-2.41) are now iterated until self-

consistency has been achieved. The convergence of these calculations depend not

only on the basis, but also on the iterative process. As an initial input, atomic-like

potentials are usually used, and the input and output potentials are mixed together

to some degree and used as the input for the next iteration. In solid state problems,

usually 100 or more iterations are required for a clear convergence.

3.2 Solid crystals

It is much easier to solve the KS one-electron problem for structures that have a

translational symmetry, i.e. the effective potential is periodic:

Veff (r + T) = Veff (r), (3.7)

where T is a translation vector of the crystal lattice. Having this condition we can

now apply Bloch’s theorem, which states that solutions of the one-electron equation

(2.40) can be written as a product of a plane wave and a periodic function:

ψk(r) = eik·ruk(r), (3.8)

with uk having the same periodic property as the crystal lattice,

uk(r + T) = uk(r). (3.9)


Figure 3.1: A Bloch wave in a Si unit cell [18].

It is thus sufficient to find the wave function ψk(r) in the primitive cell.

We now expand the one-electron wave functions ψnk(r) as shown in (3.1):

ψnk(r) =∑i

ci,nkχik(r), (3.10)

with the basis functions χik(r) now satisfying Bloch’s condition (3.8) and forming

a complete set, and n being the band index. Thus we can write the one-electron

equation in terms of the coefficients ci,nk as in (3.5):∑j

[Hij,k − εnkSij,k] cj,nk = 0, (3.11)

where

Sij,k = 〈χik|χjk〉 =

∫Ω

χ∗ik(r)χjk(r)d3r (3.12)

and

Hij,k = 〈χik|hs|χik〉 =

∫Ω

χ∗ik(r)hsχjk(r)d3r, (3.13)

and Ω being the volume of the primitive cell. The energy eigenvalues εnk are given

by the secular equation

det [Hij,k − εnkSij,k] = 0. (3.14)

3.3 Disordered systems

Almost all realistic materials and alloys have slight deficiencies on the atomic scale,

which will reduce their symmetry compared to perfect systems. Bloch’s theorem


does not hold for systems that lack translational invariance, and the standard band-

structure methods cannot be applied.

There are a few kinds of disorders and impurities. Substitutional disorder occurs in

alloys which maintain their periodic lattice, but the atoms occupying these sites are

of a different kind. Topologically disordered materials, such as liquids or amorphous

metals, have no periodic lattice whatsoever. A combination of these 2 disorders occurs

in liquid or amorphous alloys. Lattice defects are another kind, there can be vacant

sites, impurities and dislocations, or the region can belong to the interface. A surface

for example is a large lattice defect of the material.

It is technically impossible to analyse each possible disordered configuration in-

dividually, instead it would be much better if a method averages the system con-

figuration and gives sufficient information to calculate the system properties from

there. Green’s function methods are well suited for these problems, making it pos-

sible to calculate the characteristics of a disordered system directly, without having

to explicitly solve the SE or KS equation inherent of the system configuration. The

coherent potential approximation (CPA) is a powerful tool in this regard, allowing

such disorders to be treated within the KKR and TB-LMTO methods.

For surfaces and interfaces in crystalline materials the periodicity is only lost in the

direction perpendicular to the surface or interface, but in the plane of the material,

two-dimensional translational invariance is still maintained. Here, we can still use

Bloch’s theorem in those two dimensions, and the surface Green’s function is used to

handle this region.

A way of solving defective systems is to insert this defect in a ”supercell” and

extend this supercell in the region, so that when the spacing is large enough, the

defects can be considered isolated and non-interacting. However this is not a realistic

model of a random defective system. Using the CPA to average over the random

defects and surface Green’s functions to handle isolated surfaces and interfaces is a

more powerful way of handling such systems.


3.4 Tight binding

Crystals are periodic solids that are characterized by their periodic lattice. An atom at

a specific site does not contribute a large amount to the potential at sites far away, and

electrons are largely localized near the atomic sites. Also, the electronic interaction

between the atoms themselves is relatively small. It is therefore a good idea to

approximate the electronic wavefunctions as linear combinations of localized orbitals

centered on each site. This approximation is called the Tight-binding approximation

(TB), and most of the band and alloy theory is derived using TB to calculate their

properties, such as s-bands, d -bands, etc.

Decomposing the effective potential Veff (r) into contributions VR(r−R) from the

different sites R, we can write the potential as

Veff (r) =∑R

VR(r−R), (3.15)

and the Hamiltonian becomes

H = −1

2∇2 +

∑R

VR(r−R). (3.16)

These potentials VR(r−R) are now assumed to be not very much different from those

of the atoms when isolated, V atR (|r−R|), which are spherically symmetric. So taking

VR to be spherically symmetric, a set of localized orbitals can be constructed:

χRL(r) ≡ φRl(|r−R|)YL( r−R), (3.17)

satisfying the equation

[−∇2 + VR(r)

]φRl(r)YL(r) = εRlφRl(r)YL(r). (3.18)

Here L stands for the combined angular momentum indices (l,m), φRl(r) are radial

wavefunctions, and YL(r) are spherical harmonics.

Introducing the notation 〈r|RL〉 ≡ χRL(r), we get

H|RL〉 = εRl|RL〉+∑

R′′ 6=R

VR′′|RL〉, (3.19)


and using the fact that 〈RL|RL′〉 = δLL′ we can write for the on-site Hamiltonian

matrix elements:

HRL,RL′ ≡ 〈RL|H|RL′〉

= εRlδLL′ + 〈RL|∑

R′′ 6=R

VR′′ |RL′〉

≡ εRlδLL′ + t(1)R,LL′ , (3.20)

with the one-center integrals t(1)R,LL′ called the crystal field integrals. Defining the

overlap matrix

SRL,R′L′ ≡ 〈RL|R′L′〉, (3.21)

the off-site Hamiltonian matrix elements are given as

HRL,R′L′ ≡ 〈RL|H|R′L′〉

=1

2(εRl + εR′l′)SRL,R′L′ +

1

2〈RL|VR + VR′|R′L′〉

+ 〈RL|∑

R′′ 6=R,R′

VR′′ |R′L′〉

≡ 1

2(εRl + εR′l′) + t

(2)RL,R′L′ + t

(3)RL,R′L′ , (3.22)

where t(2)RL,R′L′ and t

(3)RL,R′L′ are the two-center and sum of the three-center integrals

respectively.

Usually, the three-center terms 〈RL|VR′′|R′L′〉 are much smaller than the two-

center integrals 12〈RL|VR +VR′ |R′L′〉 as R, R′ and R′′ are all different, and the inte-

grals become very small at distances exceeding the lattice constant, and can therefore

be neglected. Also because of this, it is often sufficient to consider the two-center

integrals only for neighbouring sites.

Expanding the electron wave function as the usual

ψn =∑RL

cn,RLχRL(r), (3.23)

we can in principle solve the ES problem by solving the infinite secular equation

det [HRL,R′L′ − εnSRL,R′L′ ] = 0. (3.24)


This approach is still exact within the one-electron approximation, and applies to all

atomic arrangements as Bloch’s theorem has not yet been used. Assuming now that

the solid is a periodic crystal, and for simplicity that there is only one atom per unit

cell, we can then construct a Bloch basis

|kL〉 =1√N

∑R

eik·R|RL〉 (3.25)

and solve the problem in k-space using a secular equation like (3.14).

3.5 Green’s functions

Green’s functions of a certain operator L are called fundamental solutions of L. and

once the Green’s function of the system is found, any physical quantity can in principe

be determined. The Green’s function G(z) of a Hamiltonian H is defined as

(z −H)G(z) = 1,

G(z) = (z −H)−1 , (3.26)

where z is a complex number and 1 is the unity operator. In terms of the Hamiltonian

eigenvectors |ψi〉 and eigenvalues εi, we can also write

G(z) =∑i

|ψi〉1

z − εi〈ψi|. (3.27)

In a particular basis such as (3.1), and assuming ciν forms a complete orthonormal

set, ∑ν

c∗iνcjν = δij,∑i

c∗iνciν′ = δνν′ , (3.28)

the Green’s function can thus be represented as

Gνν′(z) =∑i

ciνc∗iν′

z − εi. (3.29)

We can see that Gνν′(z) has simple poles at z = εi, but is an analytical function of

the complex energy z everywhere else.


Now, in order to determine physical observables, one requires knowledge of G(z)

approaching the real energy axis, i.e. z = E ± iη, where η → 0+ and E are real

numbers, so that we can write

1

z − εi→ P

E − εi∓ iπδ(E − εi), (3.30)

where P is the principal part. In the upper complex half plane near the real axis,

where z = E + iη, η → 0+, and using (3.30), the imaginary part of the diagonal

elements Gνν(z) is given by

limη→0+

[− 1

πIm Gνν(E + iη)

]=∑i

|ciν |2 δ(E − εi). (3.31)

This expression represents the density of states projected on the ν-“orbital”, gν(E).

To write this more compactly,

gν(E) = − 1

πIm Gνν(E + i0). (3.32)

To come back to the Kohn-Sham one-electron Hamiltonian of (2.40):

H = −1

2∇2 + Veff (r), (3.33)

the Green’s function G(r, r′; z) ≡ 〈r| (z −H)−1 |r′〉 of the KS equation is defined by[z +

1

2∇2

r − Veff (r)

]G(r, r′; z) = δ(r− r′). (3.34)

Like above, if we know the eigenvalues εi and the corresponding eigenfunctions ψi(r) ≡

〈r|ψi〉 which satisfy completeness and orthonormality

〈ψi|ψj〉 = δij,∑i

ψ∗i (r)ψi(r′) = δ(r− r′), (3.35)

then accordingly, the Green’s function can be written in the form

G(r, r′; z) =∑i

ψ∗i (r)ψi(r′)

z − εi. (3.36)


Using (3.30) and taking η → 0+, a relation analogous to (3.31) is obtained:

limη→0+

[− 1

πIm G(r, r;E + iη)

]=∑i

|ψi(r)|2 δ(E − εi). (3.37)

The right-hand side represents the energy-resolved single-particle density w(r, E) of

electrons in the (r, E)-space:

w(r, E) = − 1

πIm G(r, r;E + i0). (3.38)

From here one can obtain the single-particle electron density by integrating the energy

over all occupied one-electron states, and the total density of states by integrating

over all space:

n(r) =

∫ EF

−∞w(r, E)dE

= − 1

π

∫ EF

−∞Im G(r, r;E + i0)dE,

g(E) =

∫w(r, E)d3r

= − 1

π

∫Im G(r, r;E + i0)d3r. (3.39)

We have thus found expressions for some observables in terms of the diagonal elements

of the system’s Green’s function.

The Green’s function approach is mathematically equivalent to using wavefunc-

tions, however it is usually much easier to calculate the Green’s function than to

calculate the eigenvalues and eigenfunctions directly. The relations listed above make

it straightforward to calculate the relevant characteristics from the Green’s function.

3.6 The LMTO method

For the crystal problem, the method of muffin-tin-orbitals (MTO) is particularly well

suited. It is a minimal basis set, reducing the size of the Hamiltonian and therefore

the computational effort.

The crystal potential changes rapidly near the atomic sites and varies slowly in the

interstitial region. The muffin-tin (MT) potential VMT approximates the potential by


assuming it is spherically symmetrical near the atomic site and flat in the interstitial

region:

Vext(r) = VMT (r) =

Vext(r), r ≤ s

V0, r > s,

(3.40)

for a certain s, the radius of the MT spheres. This considerably simplifies the problem,

as the wavefunction can now be expressed in terms of the solutions of the SE for each

region, a product of spherical harmonics and radial wave functions inside the sphere

and plane waves in the interstitial region.

The Atomic Sphere Approximation (ASA) chooses s in such way that the total

volume of the MT spheres are equal to the volume of the system, which means that

the spheres have a slight overlap. Also, it takes the kinetic energy outside the MT

sphere equal to zero. For the SE, this translates into:[−1

2∇2 + V (r)− E

]ψ(r) = 0, r ∈ A,

−∇2ψ(r) = 0, r ∈ I, (3.41)

where A is the region inside the atomic spheres and I the interstitial region. Therefore,

as we have set the kinetic energy to 0 in the interstitial region, the wave function

satisfies Laplace’s equation there. Defining rR = r−R and r = |r|, we now write the

potential V (r) as

V (r) = VR(rR), rR ≤ sR, (3.42)

where VR(rR) is the spherically symmetrical potential inside the R-th atomic sphere

of radius sR.

Outside this atomic sphere, where the wave function obeys Laplace’s equation, its

solutions are of the form ψ(r) = ul(r)YL(r), with r = r/r and L standing for (l,m).

YL(r) is a spherical harmonic and ul(r) a radial amplitude, a solution to the radial

equation [− ∂2

∂r2− 2

r

∂

∂r+l(l + 1)

r2

]ul(r) = 0. (3.43)


The regular and irregular solutions to the Laplace equation are respectively:

JL(r) = Jl(r)YL(r), Jl(r) =1

2(2l + 1)

( rw

)l, (3.44)

KL(r) = Kl(r)YL(r), Kl(r) =( rw

)−l−1

(3.45)

where w makes the functions dimensionless. The spherical harmonics used here are

real and satisfy orthonormality:∫YL(r)YL′(r)d2r = δLL′ . (3.46)

Solutions centered at origins of different atomic spheres, namely JL(rR) and KL(rR)

are related to each other, an irregular solution KL(rR) centered at R can be expanded

into regular solutions JL′(rR′) centered at R′ 6= R by

KL(rR) = −∑L′

SRL,R′L′JL′(rR′), (3.47)

where SRL,R′L′ are canonical structure constants, being symmetric (SRL,R′L′ = SR′L′,RL)

and having an inverse power law dependence on the distance |R−R′| as

SRL,R′L′ ∝(

w

|R−R′|

)l+l′+1

. (3.48)

For a single spherically symmetric potential VR(r) centered at R, the SE (3.41)

inside this sphere (r < sR) are given as[−1

2∇2 + VR(r)− E

]φRL(r, E) = 0 (3.49)

and the solutions φRL(r, E) for a given energy E are given as

φRL(r, E) = φRl(r, E)YL(r), (3.50)

with the radial part φRl(r, E) satisfying the radial SE[− ∂2

∂r2− 2

r

∂

∂r+l(l + 1)

r2+ VR(r)− E

]φRl(r, E) = 0. (3.51)

Again, we get regular and irregular solutions when taking into account the asymptotic

behaviour at r → 0, with the regular solutions having rl dependence and the irregular


ones r−l−1. The solutions for inside the atomic sphere and the ones for the interstitial

region must now be combined to form the full solution to the ASA, with the resulting

function and its first derivative being continuous at the sphere boundary (r = sR).

Taking into account the boundary conditions, the full (radial) solution is therefore

the regular solution of (3.51) inside the sphere with φRl(r, E) ∝ rl for r → 0, and the

decaying solution KL(r) of the Laplace equation outside the sphere, with φRl(r, E)→

0 for r →∞.

The MTO have the form of:

ΦL(r, ε) = ilYL(r)

ul(r, ε), r ≤ s[Dl+l+1

2l+1

(rs

)l+ l−Dl

2l+1

(rs

)−l−1]ul(s, ε), r > s,

(3.52)

where L stands for both quantum numbers l and m, ul(r, ε) is a solution of the

radial SE, Ylm(r) a spherical harmonic and Dl(ε) = su′l(s, ε)/ul(s, ε) the logarithmic

derivative of ul(r, ε) at r = s. This solution however is not normalizable because of the(rs

)lfactor outside the atomic sphere which blows up at infinity. But by subtracting

this term from (3.52), we get a form which is normalizable but not a solution to the

SE:

φL(r, ε) = ilYL(r)

2l+1l−Dl

ul(r,ε)ul(s,ε)

− Pl(ε)2(2l+1)

(rs

)l, r ≤ s(

rs

)−l−1, r > s,

(3.53)

with the potential function Pl(ε) defined as:

Pl(ε) = 2(2l + 1)Dl(ε) + l + 1

Dl(ε)− l. (3.54)

In a crystal, there will be an atomic sphere centered on every atom. Inside every

atomic sphere the total wave function is the sum of the wave function from the atomic

sphere itself and the “tails” coming from the other atomic spheres. Knowing that the

solution of the radial equation inside the sphere is ul(r, E), the linear combination of

MTOs centered on different atoms given by

Ψ(r, E) =∑RL

φL(rR, E)CRL (3.55)


will be a solution of the SE of the crystal if all the (r/s)l terms cancel on the central

site. Expanding the tails from sites R′ 6= 0 on the central site as

ilYL(rR)(rRs

)−l−1

= −∑L′

(rR′

s

)l′ 1

2(2l′ + 1)il

′YL′(rR′)SR′L′,RL (3.56)

where rR ≡ r −R, rR ≡ |r −R| and SR′L′,RL the structure constants, we can write

down the tail cancellation condition as:∑R′L′

[PRL(E)δRR′δLL′ − SRL,R′L′ ]CR′L′ = 0. (3.57)

The structure constants SRL,R′L′ contain all information about the crystal structure,

and the potential functions PRL(E) contain all information about the atomic poten-

tials. Thus, this equation can be used to determine the electronic band structure

ε(k), if R′ is being summed over all sites in a crystal and the wave function is a Bloch

state.

MTOs generally have a range extending to infinity. By introducing screening

parameters,

P β(E) = P (E)(1− βP (E))−1,

Sβ = S(1− βS)−1, (3.58)

a tight-binding MTO (TB-MTO) can be constructed which only takes into account

the first and second nearest neighbours when calculating close-packed structures. It

can be shown that this screening transformation does not affect the form of the tail-

cancellation condition.

There is an energy dependence in the potential function, which might prove un-

wieldy when calculating the band structure from (3.57). Therefore, in the case of

self-consistent calculations, Linearized MTOs (LMTO) which remove this energy de-

pendence are used instead.

3.7 Electronic Transport

For devices that obey Ohm’s law, the conductance can be written as

G = R−1 = σA

L, (3.59)


with A the cross section area, L the length of the conductor, σ the conductivity and

R the resistance. However, Ohm’s law breaks down at scales comparable to or shorter

than certain length scales characteristic of the system, when the wave character of the

electrons becomes dominant, and transport has to be treated quantum mechanically

in these regions.

A mesoscopic system is a system of which at least one dimension approaches one of

these characteristic lengths. It lies between the macroscopic and microscopic scale, in

which the quantum properties of a material can be described without having to take

into account the details of the individual atoms. Examples of mesoscopic systems

are 2D electron gases, 1D wires, and 0D quantum dots. In contrast to microscopic

systems (molecules) which are closed, mesoscopic systems (quantum dots) are studied

by calculating the electron transport through it, so they are inherently open systems.

When the system size is decreased to the magnitude of characteristic length scales,

interesting transport phenomena will emerge. Amongst these are quantized conduc-

tance, and the quantum Hall effect (QHE). A few of these length scales are listed

below:

Fermi wavelength λF The Fermi wavelength is the de Broglie wavelength for elec-

trons on the Fermi surface: λF ≡ 2πkF

. In bulk metals, λF ∼ 3.0 A.

Mean free path Lm The mean free path is the average distance travelled by an

electron before its momentum is scattered. It is in the order of Lm ∼ 10µm.

Phase coherence length Lφ The phase coherence length is how long an electron

can travel before it loses its phase information. Lφ is mainly determined by

electron-electron and electron-phonon scattering. An approximate value is Lφ ∼

10µm.

The Landauer approach has proven to be very successful in describing electron

transport of such mesoscopic systems. It expresses the current through a device in

terms of the probability that an electron can transmit through it. An early appli-

cation was calculating the electronic characteristics of tunnelling junctions, where


the transmission probability is usually much less than 1. Buttiker introduces multi-

terminal measurements and magnetic fields into the formalism, and together, the

Landauer-Buttiker formalism has seen widespread use in transport calculations.

The 2-probe Landauer approach considers a device connected to left and right

semi-infinite leads. One can adjust the potentials of either lead independently, which

allows for the study of non-equilibrum systems.

4

Graphene

Carbon is a tetravalent nonmetallic element and its flexible bonding properties allow

it form more compounds than any other element, making it the basis for all life on

earth and the main component in organic chemistry. It occurs in many allotropes

with widely varying physical properties. Diamond and graphite are the most common

allotropes of carbon, the one being amongst the hardest materials known and the other

one being very soft.

The dimensionality of these structures plays a large part in determining its proper-

ties. Graphene, formerly known as single-layer graphite, is an allotrope of carbon, it is

made up of sp2-bonded carbon atoms arranged on a honeycomb lattice, and can been

seen as an infinitely large aromatic molecule. It is a two-dimensional (2D) structure in

the fact that the electrons can move around freely in two dimensions but are severely

restricted in the third. Wrapping up a sheet of graphene by introducing pentagons

causing curvature, creates spherical carbon molecules called fullerenes (C60). Physi-

cally, these buckyballs are zero-dimensional (0D) objects with discrete energy states.

Rolling up a sheet of graphene and reconnecting the bonds at the ends creates carbon

nanotubes, one-dimensional (1D) carbon structures that have remarkable properties.

Stacking graphene in the third dimension (3D) creates the now well-known graphite.

Even though graphene is the basis building block for these allotropes, it was the

latest of them all to be isolated, the first discovery being in 2004 [9]. Before, it

was believed that two-dimensional structures could not be stable in the free state

and that thermal fluctuations would cause dislocations of the material, but since the

discovery of graphene and other 2D crystals such as single-layer hexagonal boron

31

4: Graphene 32

nitride (which has a structure similar to that of graphene) these assumptions have

been proven wrong. Nowadays there are several ways to create graphene, epitaxial

growth being the most promising method of them.

The properties of graphite and graphene have been studied extensively a long

time ago [12, 13], and since the discovery of free graphene there are also experimental

results available.

As said before, the structure of graphene is a honeycomb lattice with carbon atoms

at the hexagonal points. Its primitive unit cell is therefore a rhomb (2 equilateral

triangles) containing 2 carbon atoms, separated by the graphene C-C length of 0.142

nm. The in-plane lattice constant is 0.246 nm, the distance between 2 graphene sheets

is 0.67 nm. The reciprocal lattice or Brillouin zone (BZ) is a hexagon with only 2

points that are not equal by symmetry, the K and K′-points. This degeneracy leads

to 2 separate conduction valleys.

Figure 4.1: Energy plot of the graphene BZ as calculated by a tight-binding model[12].

Graphene is technically a semiconductor, but with a zero band gap, as the con-

4: Graphene 33

duction and valence bands touch each other at the K-point. It exhibits an ambipolar

electric field effect, which means that both electrons and holes can be used as charge

carriers, the one being used depending on the gate voltage. The mobilities of these

charge carriers can exceed 15,000 cm2V−1s−1 at room temperature. The dispersion

relation near the six corners is linear and given by E = ±hν|k|, which means that

charge carriers near these points have zero effective mass and obey the relativistic

Dirac equation. These quasiparticles are called Dirac fermions and the corners of the

BZ are called the Dirac points.

Graphene is also a very suitable material for spintronics, as it has a small spin-

orbit interaction and carbon has almost no nuclear magnetic moment, resulting in

possibilities for spin-current up to room temperature. Spin coherence lengths were

observed to be greater than 1 µm.

It is also remarkable that a quantum Hall effect (QHE) can be observed at room

temperature, in contrast to other instances where usually a very low temperature is

needed to observe such effects. The anomalous QHE in graphene deviates from the

standard sequence by 1/2, so that the Hall plateaus are given by σxy = ±g(N +

1/2)e2/h, where N is the Landau level and g = 4 takes into account the double

valley and double spin degeneracies. It is called a half-integer QHE, as the sequence

is shifted by a fraction but still has an integer scale. This shift is explained due to

QED-quantization of Graphene’s eletronic spectrum in a magnetic field B, which is

given as EN = ±vF√

2ehBN , the sign referring to electrons and holes. Existence of

a quantized level at zero energy implies this shift.

Due to the strong σ-bonds, it is very hard for other atoms to replace the carbon

atoms in the lattice, and graphene sheets exhibit high quality. Despite this however,

even a high quality sample will show some defects and disorders, also the edges

constitute large disorders. Vacancies, adsorbed atoms (adatoms) or molecules, charge

impurities (doping), lattice distortion and substrate interference are amongst the

disorders to be found and each affects the symmetry and transport behaviour in

different ways.

5

Experiment

5.1 The software

The software used in the experiment consists of roughly 4 different parts, namely

atoms, bulk structure, interface, and two probes. I have done calculations on the

bulk structure of graphite in the planar direction, which has the same properties as

graphene. An outline of the procedure is given below.

Establish geometry

The primitive unit cell of the structure must be determined. If the structure is

not close packed, attention must be paid on how to introduce vacuum spheres.

This is dependent on the symmetry of the problem. Then, the radii of the atomic

spheres must be determined, with the total volume of all the spheres equal to

the volume of the unit cell. Again, this depends on the specific geometry.

For graphite, we used the structure provided by Karpan [16]. Graphite consists

of graphene layers stacked in an ABA-configuration. There are thus 4 carbon

atoms in the unit cell, 2 for the first layer and 2 for the stacked second layer. 4

vacuum spheres are placed accordingly to fill up the empty space.

Calculate atomic potential & parameters

This part is called Atomic Electronic Structure (aes). Potentials and parame-

ters are calculated for each atom required from their electronic configurations.

Inputs are the atomic number, radius of the sphere and orbital configuration.

The orbital configuration is futher split into core orbitals which are entirely

34

5: Experiment 35

filled and valence orbitals with their occupancy.

For graphite, the radius for the carbon atom sphere (Z=6) is 1.56 au, the only

core orbital is 1s. The occupancy of the valence orbitals is given as 2s22p23d0,

with the d-orbital also taken into account. Vacuum spheres are of course empty,

with potential zero everywhere in the sphere. Their radius is 2.18 au.

Calculate bulk structure

The program named Bulk Electronic Structure (bes) calculates the potential,

parameters and energies of the given bulk structure. Required inputs are the

geometry with the translation vectors and sites of the atomic spheres, the chem-

ical occupation of those sites, the initial guess for the potential and parameters,

and several parameters used to control self-consistency and accuracy. The ini-

tial potential file is constructed using the atomic potentials calculated by aes.

For the chemical occupation, concentrations less than 1 can be used to indicate

fractional impurities in the material.

Interface, two probes

The goal of all this is to put several bulk structures together and calculate the

electron transport through the created device. The program is also capable of

doing these calculations. However, this part proved to be too hard to do given

the time constraints, so only results of the bulk structure of 2D graphite will be

shown.

5: Experiment 36

5.2 Results

5.2.1 Band structure

Figure 5.1: Graphite band structure calculated by our TB-LMTO-ASA model.

This figure shows the calculated band structure in the Γ-K and K-M directions.

Around the K-point, it can clearly be seen that the valence band and conduction band

touch each other as expected, creating a semiconductor with zero bandgap. Compared

to results obtained by Karpan shown at figure 5.2, our results are qualitatively similar

near the K-point. When looking at figure 5.3 obtained by a theoretical calculation,

we can see that the number of bands at the K-point are equal and follow similar

paths.

5: Experiment 37

Figure 5.2: Graphite band structure calculated by PWP, a more accurate but much more expensivemethod [16].

Figure 5.3: Graphite band structure calculated by a TB model. The region around the K-point isencircled on the left. The encircled point to the right is the H-point, the corner on the top andbottom of the hexagonal rhomboid cell.

5: Experiment 38

5.2.2 Density of States

Figure 5.4 shows the result of the calculation of the density of states (DOS) on pure

graphite calculated by our model. It shows that at the fermi level the DOS drops to 0

as predicted. Comparing this to figure 5.5 of a theoretical tight-binding calculation,

it shows that at the [-5,1] energy interval our model qualitatively resembles the TB

model. For energies higher than the fermi energy, the results are generally inaccurate

due to the fact that DFT is a ground-state theory and it does not handle excited

states very well.

As the DOS drops to zero at the fermi level, this means that there are no free

charge carriers (either electrons or holes) available. However at this level, graphene

still exhibits a minimum conductivity on the order of e2/h [10]. The reasons of this are

still unclear. In transport calculations, where one applies a current to the device, the

DOS will no longer be zero and there will be a tremendous increase in conductivity

as more charge carriers are introduced.

5: Experiment 39

-4 -2 0 2 4 6 80.0

0.5

1.0

1.5

2.0-4 -2 0 2 4 6

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Figure 5.4: DOS of pure graphene as calculated by our TB-LMTO-ASA model.

Figure 5.5: DOS of pure graphene as calculated by a TB model [17].

6

Conclusion

In this thesis, I have outlined some of the contemporary methods used in electronic

structure calculations of solid crystals. Starting with Density Functional Theory

which serves as a fundamental basis for more advanced techniques, I have discussed

some approximation methods used to solve the Kohn-Sham equations, particularly

those that are well suited to the solid crystal problem.

The Tight-Binding Linear Muffin-Tin Orbitals with the Atomic Sphere Approxi-

mation is a minimal orbital approach using only 9 orbitals per atom to solve the KS

equations, the Green’s function (Korringa-Kohn-Rostoker) method is used to obtain

the parameters for LMTO and the Coherent Potential Approximation is used to treat

disordered crystals.

Graphene has many interesting properties allowing future applications such as

highly sensitive particle detection, quantum dots, highly efficient transistors, ultraca-

pacitors and many more. The calculations consisted only of pure graphene, the results

of which were in excellent agreement with current literature. A future prospect could

therefore be a study of impure and disordered graphene, and a study of electron

transport through such a device.

40

A

Functionals

A function maps a set of numbers to another set of numbers. Usually, functions

are defined on the real space R. A functional is a real-valued function on a vector

space V , mapping functions or vectors to scalars. For example, the Hamiltonian

H (qi(t), pi(t), t) is a functional of the functions qi(t) and pi(t). The Thomas-Fermi

kinetic energy functional (ref) is another example.

When it is necessary to differentiate functionals, to look at their rate of change as

a function of the input (functions), one needs to obtain the functional derivative. To

get this, we need to evaluate the functional against all test functions φ, which we can

write as

〈δF [f ], φ〉 =d

dεF [f + εφ]

∣∣∣∣ε=0

, (A.1)

where δF [f ] is short for δF [f(x)]f(x)

.

A definition more used in physics is:

δF [f(x)]

δf(y)= lim

ε→0

F [f(x) + εδ(x− y)]− F [f(x)]

ε. (A.2)

For example, take the electrical potential, which is a functional of the charge

density ρ:

F [ρ(r′)] ≡ V (r) =1

4πε0

∫ρ(r′)

|r− r′|d3r′. (A.3)

If we want to take the functional derivative using (A.2), we get

F [ρ(r′) + εδ3(r′ − r′′)]− F [ρ(r′)]

ε=

1

4πε0

∫δ3(r′ − r′′)

|r− r′|d3r′

=1

4πε0

1

|r− r′′|,

41

A: Functionals 42

or justδF [ρ(r′)]

δρ(r′)=

1

4πε0

1

|r− r′|. (A.4)

The same technique can be applied to the Coulomb potential energy functional

J [n], which was defined as

J [n] =1

2

∫∫n(r)n(r′)

|r− r′|drdr′. (A.5)

The derivative can thus be calculated to be

δJ [n(r)]

δn(r)=

1

2

∫∫∂

∂n

n(r)n(r′)

|r− r′|drdr′ =

1

2

∫n(r′)

|r− r′|dr′. (A.6)

Another example is the dual bra of a given ket |φ〉, 〈φ|, which is a continuous

linear functional that maps functions in the Hilbert space H to the complex numbers

C:

〈φ|(|ψ〉) = (〈φ|, |ψ〉) ≡ 〈φ|ψ〉. (A.7)

In general, any function can be written as a functional, for example

n(r) =

∫n(r′)δ3(r− r′)d3r′. (A.8)

Taking the derivative of this functional results in

δn(r)

δn(r′)=δ∫n(r′)δ3(r− r′)d3r′

δn(r′)=

∂

∂nn(r)δ3(r− r′) = δ3(r− r′). (A.9)

The method of Lagrangian multpliers can also be extended to functionals. If we

want to optimize a functional F [f ] with the constraint G[f ] = 0 we construct a new

functional

Λ[f ] = F [f ]− µG[f ] (A.10)

and extremize it, by setting its functional derivative to 0:

0 =δΛ

δf=δF

δf− µδG

δf. (A.11)

B

Green’s Functions

A Green’s function is a technique used to solve inhomogeneous partial differential

equations (PDE) with boundary conditions. Other such techniques are integral trans-

forms and separation of variables. The Green’s function G(r, r′) of a linear differential

operator L is defined as the solution to

LG(r, r′) = δ(r− r′) (B.1)

and can be used to solve equations of the form

Lφ(r) = f(r). (B.2)

If we multiply both sides of (B.1) by f(r′) and integrate over r′, we obtain∫LG(r, r′)f(r′)d3r′ =

∫δ3(r− r′)f(r′)d3r′ = f(r). (B.3)

By (B.2), the right hand side is equal to Lφ, giving

Lφ(r) =

∫LG(r, r′)f(r′)d3r′. (B.4)

Now, as L is linear and acts only on r, we can rewrite this as

Lφ(r) = L

(∫G(r, r′)f(r′)d3r′

), (B.5)

obtaining a solution of (B.2) in the form of

φ(r) =

∫G(r, r′)f(r′)d3r′. (B.6)

Thus, by linearity of L, it is theoretically possible to obtain φ(r) by knowledge of

the Green’s function and the source term f(r). The problem is now to find the

43

B: Green’s Functions 44

G(r, r′) that satisfies equation (B.1). The Green’s function is therefore also called the

fundamental solution to the operator L.

If it is possible to obtain a complete basis set φn(r) from L, then G(r, r′) can be

expressed in these eigenvectors. Expanding G(r, r′) and δ3(r − r′) in this basis set

gives

G(r, r′) =∞∑n

an(r′)φn(r), (B.7)

δ3(r− r′) =∞∑n

bnφn(r). (B.8)

Multiplying both sides by φm(r) and integrating over r (Fourier’s trick) gives∫φm(r)δ3(r− r′)d3r =

∞∑n

bn

∫φm(r)φn(r)d3r, (B.9)

which, integrated out, gives

φm(r′) =∞∑n

bnδnm = bm. (B.10)

We can thus write the completeness relation for δ3(r− r′):

δ3(r− r′) =∞∑n

φn(r)φn(r′). (B.11)

With (B.7) and (B.11) we now theoretically know enough: plug in the differential

operator L, solve for the ans, substitute back into G(r, r′) and finally solve the in-

homogeneous PDE. A few examples will be given in which this technique will be

applied.

B.1 Poisson’s equation

Poisson’s equation is the inhomogeneous variant of Laplace’s equation, which in gen-

eral form is given as:

∇2φ(r) = f(r). (B.12)

φ(r) is often called a potenial function and f(r) a density function, which for the

electrostatic case is f(r) = − ρε0

. The linear differential operator is ∇2. We are thus


looking for a function G(r, r′) such that

∇2G(r, r′) = δ3(r− r′). (B.13)

But we already know from the laplacian that

∇2

(1

|r− r′|

)= −4πδ3(r− r′), (B.14)

so that

G(r, r′) = − 1

4π|r− r′|. (B.15)

The general solution therefore is

φ(r) =

∫d3r′G(r, r′)f(r′) = −

∫d3r′

f(r′)

4π|r− r′|, (B.16)

which for the electrostatic case is

φ(r) =1

4πε0

∫d3r′

ρ(r′)

|r− r′|. (B.17)

Expansion of G(r, r′) gives

G(r, r′) =1

4π

∞∑l

rl<rl+1>

Pl(cos θ), (B.18)

with r< ≡ min(r, r′), r> ≡ max(r, r′), Pl Legendre polynomials and cos θ ≡ r·r′

rr′.

B.2 Helmholtz equation

The inhomogeneous Helmholtz equation is given as

∇2φ(r) + k2φ(r) = f(r). (B.19)

With the Helmholtz operator L = ∇2 + k2, the Green’s function is defined by

(∇2 + k2

)G(r, r′) = δ3(r− r′). (B.20)

Defining φn as the solutions to the homogeneous equation

∇2φn(r) + k2nφn(r) = 0, (B.21)


we can write G(r, r′) and δ3(r−r′) as in (B.7) and (B.11). Plugging these into (B.20)

gives

∇2

(∞∑n

an(r′)φn(r)

)+ k2

(∞∑n

an(r′)φn(r)

)φn(r) =

∞∑n

φn(r)φn(r′). (B.22)

Using (B.21) we get

−∞∑n

an(r′)k2nφn(r) + k2

∞∑n

an(r′)φn(r) =∞∑n

φn(r)φn(r′)

∞∑n

an(r′)φn(r)(k2 − k2n) =

∞∑n

φn(r)φn(r′). (B.23)

For every n, we can now write

an(r′)φn(r)(k2 − k2n) = φn(r)φn(r′)

an(r′) =φn(r′)

k2 − k2n

, (B.24)

so (B.7) becomes

G(r, r′) =∞∑n

φn(r)φn(r′)

k2 − k2n

. (B.25)

Thus we obtain a general solution:

φ(r) =∞∑n

∫d3r′ρ(r′)

φn(r)φn(r′)

k2 − k2n

. (B.26)

B.3 Electronic structure Green’s function

The Green’s function used in electronic structure calculations has a slightly different

meaning. Consider the single-particle SE with the Hamiltonian H,

Hψi = εiψi, (B.27)

or more explicitly, [−1

2∇2 + V (r)− εi

]ψi(r) = 0. (B.28)

For all realistic applications, ψi has to be expanded in a finite basis set χν(r):

ψi(r) 'N∑ν

ciνχν(r). (B.29)


The finite set means the relation is no longer exact but is now an approximation,

however N can always be increased if higher accuracy is required. As χν is not

complete and not orthonormal, (B.29) is a transformation but it is not unitary. The

expansion coefficients ciν therefore must have the property to diagonalize both the

normalization matrix Sνν′ and the Hamiltonian matrix Hνν′ :∑νν′

c∗iνSνν′cjν′ = δij, (B.30)

where Sνν′ is the normalization matrix for χν(r),

Sνν′ ≡ 〈χν |χν′〉. (B.31)

For the Hamiltonian matrix, this is∑νν′

c∗iνHνν′cjν′ = εiδij, (B.32)

where

Hνν′ ≡ 〈χν | −1

2∇2 + V (r)|χν′〉. (B.33)

We can now define the Green’s function Gνν′(ε):∑ν′′

(εSνν′′ −Hνν′′)Gν′′ν′(ε) ≡ δνν′ . (B.34)

The Green’s function is now not defined in terms of the differential operator −12∇2 +

V (r), but it is just the matrix inverse of εSνν′ −Hνν′ . The matrix approach has the

result that the Green’s function of a system is not specified by the system alone, it

also depends on the choice of the basis set χν(r).

The difference of this compared to the differential operator approach, where the

Green’s function has to be expanded in a basis set of the solution, is that the matrix

approach defines the Green’s function in terms of the matrices Sνν′ andHνν′ , obtaining

an exact solution. The basis-set expansion of the wave function is now the only

approximation that has to be made.

The Green’s function can now be used to calculate quantities such as the total

energy and the electron density.

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Press, Cambridge, 1995).

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[18] Source: Wikipedia

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