Reverse Monte Carlo Modeling and Simulation of …Reverse Monte Carlo Modeling and Simulation of...

Reverse Monte Carlo Modeling and

Simulation of Disordered Systems

MS-Thesis

Aamir Sha�que

2013-12-0006

Dr. Fakhar-ul-Inam (Supervisor)

Dr. Muhammad Faryad (Thesis committee member)

Department of Physics, SSE

Lahore University of Management Sciences

Reverse Monte Carlo Modeling of Disordered Systems

by

Aamir Sha�que

MS-Thesis

May 2015

Department of Physics

Syed Baber Ali School of Science and Engineering

Lahore University of Management Sciences (LUMS)

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES

Department of Physics

CERTIFICATE

I hereby recommend that the thesis prepared under my supervision by: Aamir

Sha�que on title: "Reverse Monte Carlo Modeling of Disordered Systems" be

accepted in partial ful�llment of the requirements for the MS degree.

Dr. Fakhar ul Inam

��-

Advisor (Chairperson of defense Committee )

Recommendation of Thesis Defense Committee :

Dr. Muhammad Faryad ��-

Name Signature Date

��-

Name Signature Date

Acknowledgement

I would never have been able to �nish my dissertation without the guidance of my

advisor, help from friends, and support of my family. I would like to express my

sincere gratitude to my supervisor Dr. Fakhar-Ul-Inam for his excellent guidance,

encouragement, support and providing me an opportunity to do my research work

at Lahore University of Management Science (LUMS). Also I thank my friends at

LUMS particularly Mr.Yasir Iqbal,Muzamil Shah and Muhammad Arshad Maral for

all kind of their support which was quite helpful for me. I would also like to thank

Mr. Naseem ud Din and Irtaza Hassan for all his valuable assistance in the project

work. My parents and sister were always supporting and encouraging me with their

best wishes.

Aamir Sha�que

Dedication

Dedicated to my loving parents

Abstract

Reverse Monte Carlo (RMC) method is used to generate two models for the amor-

phous graphene. In �rst model crystalline graphene is used as an initial con�guration

and random con�guration of carbon atoms is used in second model. RMC modeling

is based on the experimental data such as structure factor and pair correlation func-

tion. The models of the amorphous graphene are analysed using their ring statistics

and bond angle distributions. The conjugate gradient method is used to make the

models minimum energy con�guration. The electronic structure of these models are

also calculated using density functional theory code SIESTA.

Table of Contents

Declaration

Acknowledgement

Dedication

Abstract

1 Introduction 1

1.1 General introduction and literature review . . . . . . . . . . . . . . . 1

1.2 Disordered materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Why we use RMC method to model disordered materials . . . . . . . 4

2 Structural Analysis 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Pair correlation function . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Derivation of pair correlation function . . . . . . . . . . . . . 7

2.2.2 Thermodynamic properties from pair correlation function . . . 9

2.3 Static structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . 10

TABLE OF CONTENTS

3 Reverse Monte Carlo modeling 14

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 The basic RMC modeling algorithm . . . . . . . . . . . . . . . . . . . 15

3.2.1 Constraints on the RMC method . . . . . . . . . . . . . . . . 17

3.3 Starting con�guration . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 E�ect of move length, number of atoms and beta . . . . . . . . . . . 19

4 Results and Discussions 24

4.1 Reverse Monte Carlo (RMC) modeling of amorphous graphene . . . . 24

4.1.1 RMCmodeling of amorphous graphene using crystalline graphene

con�guration . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2 RMC modeling of amorphous graphene using random con�gu-

ration of carbon atoms . . . . . . . . . . . . . . . . . . . . . . 30

4.1.3 Ring statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.4 Bond angles distribution . . . . . . . . . . . . . . . . . . . . . 33

4.2 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 External potential and electronic density . . . . . . . . . . . . 35

4.2.2 Hohenberg�Kohn theorems . . . . . . . . . . . . . . . . . . . . 37

4.2.3 Kohn�Sham equations . . . . . . . . . . . . . . . . . . . . . . 38

4.2.4 Computational details . . . . . . . . . . . . . . . . . . . . . . 40

4.2.5 Relaxation of amorphous graphene models . . . . . . . . . . . 40

4.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 45

List of Tables

4.1 Ring statistics of amorphous graphene generated by RMC method

using crystalline as an initial con�guration. . . . . . . . . . . . . . . 33

4.2 Ring statistics of amorphous graphene generated by RMC method

Random con�guration of carbon atoms as an initial con�guration. . 34

List of Figures

1.1 The lattice of crystalline (a) and disordered graphene (b) are shown,

the white balls represent carbon atoms and white lines are C-C bonds. 3

2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 A lattice of crystalline silicon atoms is shown in which balls represent

the Si atoms and lines represent Si-Si bonds. All the bond angles and

bond lengths are equal. . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 A supercell of 54 distorted silicon atoms in which the bond angle and

bond length disorder are present. . . . . . . . . . . . . . . . . . . . . 21

3.3 Plots of bond angles distribution vs. Monte Carlo steps (MCS) for

di�erent values of move length (rro) and β and keeping the number of

atoms constant, (a) for lattice of 54 atoms of amorphous silicon using

β = 3 and rro is 0.2 Ao, (b) for lattice 54 atoms of amorphous silicon

using β = 3 and rro is 0.3 Ao, (c) for lattice 54 atoms of amorphous

silicon using β = 4 and rro is 0.2 Ao, (d) for lattice 54 atoms of

amorphous silicon using β = 4 and rro is 0.3 Ao. . . . . . . . . . . . . 22

LIST OF FIGURES

3.4 Plots of bond angles distribution vs. MCS for di�erent number of

atoms in the supercell keeping move length rro and β constant, (a)

for lattice of 54 atoms of amorphous silicon using β = 2 and rro is 0.2

Ao,(b) for lattice of 16 atoms of amorphous silicon using β = 3 and

rro is 0.2 Ao. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 Atomic con�gurations and their PCF's . . . . . . . . . . . . . . . . . 26

4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 RMCmodeling of graphene at di�erent stages using crystalline graphene

as an initial con�guration. . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 RMC modeling of graphene at di�erent stages using random con�gu-

ration of carbon atom as an initial con�guration. . . . . . . . . . . . 31

4.5 RMC modeling of graphene at di�erent stages using random con�gu-

ration of carbon atom as an initial con�guration. . . . . . . . . . . . 32

4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7 Relaxed structures of Amorphous grahene and their PCF's. . . . . . 41

4.8 Amorphous graphene and crystalline graphene densities of states com-

parison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 1

Introduction

1.1 General introduction and literature review

The structure of the materials is key to its properties [8]. Before using the mate-

rials in scienti�c and technological applications, the essential step is the investiga-

tion of materials structure [8, 19]. There are two experimental methods often used

to study materials structure; X-ray di�raction and X-ray absorption spectroscopy

(XAS) [8]. The �rst approach provides information about the equilibrium atomic

structure and the second gives information about the instantaneous local structure

of materials [8, 19].

Several simulation-based approaches can be used to study the materials structure

(specially disordered materials), such as Monte Carlo, classical and ab-initio molec-

ular dynamics (MD) and reverse Monte Carlo(RMC) [8]. Monte Carlo and classi-

cal molecular dynamics were developed based on the inter-atomic pair potential [8].

These techniques calculate a potential energy or force, using local pair potential envi-

1

CHAPTER 1. INTRODUCTION 2

ronment around every simulated atom, and then move the atom randomly through-

out the crystal to minimize the energy [8]. The disadvantage of these techniques

are that they include only the nearest neighbour atomic interaction and neglect the

higher order interactions [20]. A new technique ab-initio is developed to overcome

this problem. The simulated structure here is guided by solving �rst -principle equa-

tions. But due to high computational cost, more than few thousand of atoms cannot

be included. To avoid these problems, one can use RMC-type method [19].

The RMC modeling was �rst introduced by two scientists McGreevy and Pusztai in

1988 [3]. RMC modeling is a structural modeling of materials based on the exper-

imental data [2, 21]. The aim of the RMC modeling is to design a model or series

of models that are consistent with experimental data coupled with a set of appro-

priately chosen constraints [21]. It was primarily developed to study of amorphous

materials and liquids [18]. In RMC, many di�erent types of data can be used such

as neutron di�raction, X-ray di�raction, electron di�raction and nuclear magnetic

resonance etc. and di�erent types of system can be modelled such as liquids, glasses,

polymers, crystals and magnetic materials [6]. RMC modeling emphasizes the ex-

perimental data in order to understand the real materials [2].

1.2 Disordered materials

Disordered materials are those in which there is no long range arrangement like

crystalline materials [12]. It's wrong to say that atoms in disordered material are

randomly distributed in space and there is a short range arrangement in disordered


material [12]. The physical properties of the disordered materials are very similar to

crystalline because most of physical properties depend on the short range arrange-

ment [12]. For example, density of materials change slightly when crystallizes a �uid.

Most of the microscopic properties of material depend on their density and interac-

tion between nearest neighbours,like sound velocity [12]. Disordered materials can

be characterized using pair correlation function, bond angles , bond lengths and ring

statistics.

(a) Crystalline graphene (b) Disordered graphene

Figure 1.1: The lattice of crystalline (a) and disordered graphene (b) are shown, the

white balls represent carbon atoms and white lines are C-C bonds.

The comparison of crystalline and disordered graphene is shown in Figure 1.1. There

is long range arrangement in crystalline graphene but it is not there in disordered

graphene. In crystalline graphene, all bond lengths are about 1.42 Ao and all bond

angles are about 120o in disordered graphene, there is range of bond lengths and

bond angles. In both structures, short range arrangement are present; each carbon


atom is three folded.

1.3 Why we use RMC method to model disordered

materials

RMC method is specially designed to model disordered materials [10]. Numerous

methods have been described in literature for structural modeling of disordered ma-

terials [10]. The RMC method has novel features among other methods because of

the following several advantages [10]:

• The output structural models of this method agree with experimental data [2].

• Inter-atomic potentials are not needed [10].

• Large number of atoms can be used to build a model [2].

Chapter 2

Structural Analysis

2.1 Introduction

After obtaining a model for a disordered material, the most important thing is to

analyse the model to check whether its properties match with experimental data or

not [7]. The pair correlation function and static structure factor are the observables

which are closely related to the experimental data [7]. A brief description of these

observables are given in the next two sections.

2.2 Pair correlation function

In statistical mechanics, the pair correlation function (PCF) tells how density changes

by changing the distance from the reference particle in a material [9]. For homoge-

neous and isotropic system, let a particle be placed at origin and average density of

system of N -particles of volume V is ρ = N/V . Then the local time-averaged den-

5

CHAPTER 2. STRUCTURAL ANALYSIS 6

sity at distance r from origin is ρg(r). Thus, the pair correlation function is simply

probability of �nding particle at distance r from origin [9].

(a) The di�erent atomic

shells are shown. The dif-

ferent size of balls repre-

sent the di�erent types of

atom.

(b) The pair correlation func-

tion (g(r)) of amorphous silicon is

plotted vs distance from the refer-

ence particle (r). PCF is obtained

using position of the atoms.

Figure 2.1

The �rst peak of PCF will give information about three main things. First, the

position of �rst peak provides the information about average bond length between

two atoms [11]. The �rst peak of PCF of amorphous silicon lattice (shown in Figure

2.1(b)), at a distance of around 2.4 Ao, corresponds to the Si-Si bond. Secondly, the

width of peak will tell about the temporal and spatial variations in the bond length.

Thirdly, the integral of the �rst peak tells about neighbors of a single atom [11]. The

integral of amorphous silicon gives an average coordination number of 4.


2.2.1 Derivation of pair correlation function

Suppose we have a system containingN number of particles at position r1, r2, r3, ....rN .

The joint probability distribution for �nding particle 1 at position r1 and particle 2

at r2 can be in the form of following equation [9]

P ( 2N)(r1, r2) =

∫dr3

∫dr4....

∫drNP (rN) (2.1)

where

P (rN) =exp(−βφ(rN))

Z,

where β is equal to 1/kBT and φ(rN) and Z are the potential energy and partition

function respectively. This distribution requires particle 1 must be at r1 and particle 2

at r2 and such requirement are not physical for N-distinguishable particles system [9].

The reduced distribution functions (RDF) are more meaningful quantities such as

the joint distribution function for �nding a particle (any particle) at position r1 and

any other particle(in the N particle system) at r2 can be written in the form of

following equation [9]:

ρ(2N)(r1, r2) = N(N − 1)P ( 2

N)(r1, r2) (2.2)

Note that, there are N number of possible ways of picking �rst particle and N − 1

possible ways of picking second particle. The reduced distribution function for two

particles can also be written in the following form [9]:

ρ(2)(r1, r2) = ρ(r1)ρ(r2)g(2)(r1, r2) (2.3)

in which ρ(r) is one particles density function and g(2)(r1, r2) is the two particle cor-

relation function. For homogeneous system, the two body density function reduced


to following form [7]:

ρ(2)(r1, r2) = ρ2g(2)(r1, r2) (2.4)

in which ρ = N/V . Thus, the two particle pair correlation function becomes:

g(2)(r1, r2) =N(N − 1)

ρ2P (2)(r1, r2)

=N(N − 1)

ρ2Z

∫dr3

∫dr4

∫dr5....

∫drN exp(−βU(r1, r2, ...rN))

=N(N − 1)

ρ2Z

∫dr1δ(r1 − r′1)

∫dr2δ(r2 − r′2)

∫dr3

∫dr4

∫dr5....

∫drN

exp(−βU(r1, r2, ...rN))

=N(N − 1)

ρ2〈δ(r1 − r′1)〉〈δ(r2 − r′2)〉

(2.5)

Using new variables R and r

R =r1 + r2

2,

r = r1 − r2,(2.6)

so that

r1 = R +r

2,

r2 = R− r

2.

So two particle pair correlation can be written as:

g(2)(R, r) =N(N − 1)

ρ2〈δ(R +

1

2r − r′1)〉〈δ(R−

1

2r − r′2)〉 (2.7)

The pair correlation function is de�ned as:

g(2)(r) =1

V

∫dRg(R, r)

=N(N − 1)

V ρ2〈∫dRδ(R +

r

2− r′1)δ(R +

r

2− r′2)〉

=N(N − 1)

ρ2V〈δ(r − r12)

(2.8)


Notice that it is averaging over all possible distances between two atoms.

Now generalized n-particle density function can be written in the following form

ρ(n)(r1, r2, r3...rn) = ρ(r1)ρ(r2)ρ(r3)...ρ(rn)g(r1, r2...rn) (2.9)

n-particle correlation function g(n)(r1, r2, r3...rn) can be write in the following form

g(n)(r1, r2, r3, ..., rn) =1

ρnρ(n)(r1, r2, r3, ...rn)

=V n

ZNNn

N !

(N − n)!

∫e−βUN (r′1,r

′2,...,r

′N )δ(r1 − r′1)...δ(rn − r′n)dr′1, ..., dr

′N

=V n

N

N !

(N − n)!〈n∏i=1

δ(ri − r′i)〉

(2.10)

So the pair correlation function for all particles can be written as

g(r) =N(N − 1)

ρ2V

1

N(N − 1)

∑i6=j

δ(r − rij)

=1

Nρ

∑i6=j

δ(r − rij)(2.11)

The magnitude of the peaks usually decays exponentially with distance as g(r) −→ 1.

When the atomic repulsion is strong enough g(r) becomes zero for pair of atoms by

getting too close [7].

2.2.2 Thermodynamic properties from pair correlation func-

tion

We can directly obtain many thermodynamic properties from the pair correlation

function, such as energy, pressure, compressibility. The relation of these quantities

with g(r) are given below [7]


Energy

E = Eint +3

2NkBT +

1

2NN

V

∫ ∞0

dr4πr2g(r)φ(r) (2.12)

The �rst term in equation 2.12 is the internal energy of the molecules, second term

is translation energy and third term is the interaction of one molecule with all other

molecules [7].

Pressure

The relationship between pressure and g(r) is given as [7]

P

ρkBT= 1− 2πρ

3kBT

∫ ∞0

rdφ

drg(r)dr (2.13)

Compressibility

Compressibility can be written in term of g(r)

ρkBTαT = 1 + ρ

∫V

d3r(g(r)− 1) (2.14)

where αT represents the compressibility.

2.3 Static structure factor

The pair correlation function can be obtained from the X-ray or neutron scattering

experiment. These experiments determine the static structure factor of the material.

The inverse Fourier transform of the static factor gives the pair correlation func-

tion [9].

Figure 2.2 (a) shows the schematic representation of the X-ray scattering experi-


(a) Schematic representation of X-ray scatter-

ing experiment is shown. kin and kout are in-

cident and scattered wave vector, respectively

and RD and and RS are distances from origin

to detector and source, respectively.

(b) Vector addition of in-

cident kinand scattered k0ut

wave vector.

Figure 2.2

ment. A wave, scattered from one atom at RS is:

ΨSc =f(k)ei(kin.RS+kout.(RD−RS))

|RD −RS|(2.15)

If the distance between the detector and scattering center is very large then

|RD −RS| ≈ |RD −RC | (2.16)

condition for elastic scattering is

k′ = |kin| = |kout| =2π

λ(2.17)

and from Figure 2.2 (b)

k = kout − kin (2.18)


|k| =√

2k′2 − 2k′cos(θ)

=

√4k′2sin(

θ)

2

=2k′sin(θ)

2)

(2.19)

The wave function for the scattered wave from one atom becomes:

ΨSc =f(k)eikout.RD .eik.RC

|RD −RC |(2.20)

there are N atoms in the system, and each atom has separate scattered wave. The

superposition of these scattered wave at detector is [9]

ΨT = f(k)eikout.RD

|RC −RD|

N∑j=1

e−ik.rj (2.21)

The intensity observed at the detector of scattered X-ray at the angle θ is:

I(θ) =|f(k)|2

|RD −RC |S(k) (2.22)

where

S(k) = N−1〈N∑

l,j=1

eik.(rl−rj)〉 (2.23)

By expanding the sum into two parts, for l = j and l 6= j, the S(k) becomes:

S(k) =1 +N(N − 1)

N〈exp(ik.(r1 − r2))〉

=1 +N(N − 1)

∫drN exp(ik.(r1 − r2)) exp(−βU)

N∫drN exp(−βU)

(2.24)

From equations 2.1 to 2.4, the second term of the above equation can be written as:

S(k) =1 +N−1∫dr2

∫dr1 exp(ik.(r1 − r2))ρ(2)(r1, r2)

=

∫dr12

∫dr1ρ

2g(r12)

=1 + ρ

∫drg(r) exp(ik.r)

(2.25)


which can be written as:

S(k) = 1 + 4πρ

∫ ∞0

(g(r)− 1)sin(kr)

krr2dr (2.26)

The structure factor for binary system can be written as:

Sαβ(k) = 1 + 4πρ

∫ ∞0

(gαβ(r)− 1)sin(kr)

krr2dr (2.27)

where gαβ is the partial pair correlation function, can be written as:

gαβ =1

4πr2ρNCαCβ

∑i6=j

δ(r − rij) (2.28)

where Cα is the number of α type of atoms divided by total number atoms and Cβ

is the β type of atoms divided by total number of atoms [7].

Chapter 3

Reverse Monte Carlo modeling

3.1 Introduction

The Reverse Monte Carlo modeling (RMC) is a variation in standard Metropolis

Monte Carlo (MMC) method [2]. RMC modeling is simulation method in which the

atoms placed in a con�guration with periodic boundary condition, the position of

atoms can be moved until calculated static structure factor matches the experimental

static structure factor [1]. MMC method adopted same procedure but in this method

minimization of energy occurs instead of static structure factor [2]. The product of

MMC modeling is a con�guration with Boltzman distribution of energies while in

RMC modeling the con�guration is consistent with experimental data within its er-

rors [5, 15].

There are several advantages of RMC modeling, making it more successful method

compared to other related methods. RMC modeling used periodic boundary condi-

tions which avoided edge e�ects [2]. It is a method which is independent of initial

14

CHAPTER 3. REVERSE MONTE CARLO MODELING 15

con�guration because in this method some moves are accepted which increase χ2.

RMC modeling does not use interatomic potential used in the conventional meth-

ods [1]. So in this method, the atomic closest approach can be thought of as a hard

sphere potential [5] therefore, RMC modeling can be applied to any system.

In RMC modeling, one of the most popular misconception related to criticisms is

the lack of a unique solution which means that there many con�gurations that give

good agreement with experimental data [2]. This is one of the main advantages of

the method, it is not disadvantage [2]. If a method produce a single solution, but

that does not means that it is correct [2, 16].

3.2 The basic RMC modeling algorithm

1. Consider N atoms are placed in a cell with periodic boundary condition means

that cell is surrounded by copies of itself. Normally cubic unit cells are used to

make supercell of the crystalline materials. The atomic number density should

be equal to experimental value [2]. The atoms in supercell may be distributed

randomly.

2. Calculate the partial radial distribution functions from the con�guration [2]:

gCoαβ(r) =

ηCoαβ

4πr2drρCα, (3.1)

where ρ represent the atomic number density, Cα denotes the concentration

of the atoms type α and ηCoαβ(r) represents the number of atoms of type β at

distance between r and r + dr from a central atom of type α [2].


3. Take Fourier transforms of partial radial distribution functions to obtain the

partial structure factors [2]:

ACoαβ(Q) = ρ

∫ ∞0

4πr2(gCoαβ(r)− 1)

sinQr

Qrdr, (3.2)

where Q represents the momentum transfer

4. Total structure factor can be obtained from the partial structure factors [2]:

FCo(Q) =∑

CαCβbαbβ(ACoαβ(Q)− 1), (3.3)

where bα and bβ represent the neutron scattering length of type α atoms and

type β atoms, respectively.

5. Calculate the di�erence between the experimentally measured total structure

factor, FE(Q), and that determined from the con�guration, FCo(Q) [2]:

χ2o =

m∑i=1

(FCo(Qi)− FE(Qi))2

σ2(Qi), (3.4)

where m represents the experimental points and σ denotes the experimental

error. Momentum transfer Q should be larger than or equal to 2π/L, where L

represents the dimensions of the con�guration [2].

6. Move an atom randomly in de�ne circle of radius (rro), if any two atoms

approach closer due to the move then reject the move, choose new atom and

new move should be made [2].

7. After the moving the atom, position of the atom is changed [2, 17]. Calculate

the total structure factor of the new con�guration and then the di�erence of


the total structure factors [2]:

χ2n =

m∑i=1

(FCn(Qi)− FE(Qi))2

σ2(Qi)(3.5)

8. Accept the move only if χ2n < χ2

o, if this condition is satisfy new con�guration

becomes the old con�guration. If χ2n > χ2

o, then it is only accepted with certain

probability exp(−(χn − χo)/β), if both conditions are not satisfy, then reject

the move [2].

9. Repeat the process from step 6.

3.2.1 Constraints on the RMC method

It is possible that to �t reference data with calculated data using RMC simulation

without further constraints, but using constraints, we can get more re�ned model [2].

The most commonly used constraints are discussed below.

1. The closest distance constraint: It is most obvious and extremely impor-

tant constraint, it prevents atomic overlap. Reject the move when two atoms

come very close to each other or overlap [14].

2. The atomic coordination number constraint: Atomic coordination num-

ber is de�ned as the number of atoms around a central atom within a certain

�xed distance. Accept only the moves which satisfy the coordination con-

straint [14]. If some fraction of atoms in the con�guration satisfy the coor-

dination constraints as fRMC and the required fraction as freq then following

term [2]


χ2coord =

(freq − fRMC)2

σ2coord

(3.6)

is added to over χ2, σcoord is the weight of coordination constraint [2]. If the

weight is very small then it is called hard constraint and if the weight is larger

then it is called soft constraint. Di�erent types of atoms may be mixed in

a con�guration. For Example in vitreous silica the constraint is all Si to be

coordinate to four O and all O to two Si [2]. In hydrogenated amorphous Si,

one constraint may be all H is coordinated to one silicon and Si to be fourfold

coordinated [2].

3. The bond-angles constraint: To �t the second peak of the pair correlation

function, bond-angles constraint can be used. Accept only the move in which

the angles between the atoms are within a certain range of angles [11].

First constraint is more obvious and extremely important, preventing atomic overlap

[5]. Reject the moves where two atoms come very close to each other or overlap.

The atomic coordination is next commonly used constraint, it is de�ned as number

of atoms around a central atom within the some �xed distance [2].

3.3 Starting con�guration

The crystalline structure or ensemble of atoms which are random in space or previous

models which are generated by molecular dynamics, are used as an initial con�gu-

ration in RMC modeling of disordered systems [10]. The �nal model of RMC did

not depend on the initial con�guration, but initial con�guration can be used as con-

straint [10]. The choice of initial con�guration is di�cult because certain initials


con�guration takes very long time to �t the experimental data, therefore the choice

of initial con�guration is a key step [10].

3.4 E�ect of move length, number of atoms and beta

Before going to RMC modeling, we do some practice to know the e�ect of move

length, number of atoms and beta. First we take a supercell of 54 atoms crystalline

silicon, in which long order arrangement exist and all the angles are about 109.4o, so

the width of bonds angles distribution is very small, approximately zero. The atoms

in cell are well arranged shown in Figure 3.1:

Figure 3.1: A lattice of crystalline silicon atoms is shown in which balls represent

the Si atoms and lines represent Si-Si bonds. All the bond angles and bond lengths

are equal.


Now we want to create broad width, broaden width mean that we want to create

disorderedness in the material. The methodology we apply are quit similar to RMC

modeling. First we calculate the width of the bond angle distribution, we called it

initial width, and then calculate the χinitial:

χinitial = wref − winitial, (3.7)

where wref represents the required width and winitial represents the initial width.

Select an atom randomly from the con�guration and then move the atom randomly

within a certain radius (rro). Now calculate the width of the bond angles distribution

(wfinal) and �nd the χfinal as:

χfinal = wref − wfinal (3.8)

If χfinal < χinitial then accept the move, i. e. new con�guration becomes the old

con�guration. if χfinal > χinitial then we accept the move with certain probability

exp(−β(χfinal − χinitial)), otherwise reject the move. All of the previous steps called

one Monte Carlo Step.

We run a thousand Monte Carlo steps (MCS) simulation for crystalline silicon atoms

and take move length (rro) 0.1 Ao, using reference width of 10o and β = 2 then the

structure of the crystalline silicon atoms becomes completely distorted as shown in

Figure 3.2 below:


Figure 3.2: A supercell of 54 distorted silicon atoms in which the bond angle and

bond length disorder are present.

Now we use di�erent parameters of the method i. e. beta, move length and

number of atoms, and see how these parameters e�ected the bond angles distribution.

In Figure 3.3, width of bond angles distribution are plotted vs. Monte Carlo Steps

(MCS) for di�erent values of beta and move length, here we take reference width

10o, for supercell of 54 silicon atoms:

We compare the plots (a) and (b) in the Figure 3.3 in which the move length (rro)

is changed but β is kept constant. As we increase the move length, the bond angles

distribution width become closer to the reference width of distribution, and it also

increase the variation in width of bond angles distribution. Now we compare the

plot (b) and (d) in �gure 3.3 in which β is changed but move length is kept constant,

the variation in width increases but in this case the distribution width is not close

to reference width. Increase in β means that we increase the accepting probability

therefore, the variation width of distribution increases.


(a) (b)

(c) (d)

Figure 3.3: Plots of bond angles distribution vs. Monte Carlo steps (MCS) for

di�erent values of move length (rro) and β and keeping the number of atoms constant,

(a) for lattice of 54 atoms of amorphous silicon using β = 3 and rro is 0.2 Ao, (b) for

lattice 54 atoms of amorphous silicon using β = 3 and rro is 0.3 Ao, (c) for lattice 54

atoms of amorphous silicon using β = 4 and rro is 0.2 Ao, (d) for lattice 54 atoms of

amorphous silicon using β = 4 and rro is 0.3 Ao.

Now, we changes the number of atoms in the super cell and the value of β and move

length are kept constant. If we decrease the number of atoms in the cell, the variation

in the width of distribution increases.

In the Figure 3.4, we compare the width of distribution for supercell of 16 and 54


(a)

(b)

Figure 3.4: Plots of bond angles distribution vs. MCS for di�erent number of atoms

in the supercell keeping move length rro and β constant, (a) for lattice of 54 atoms

of amorphous silicon using β = 2 and rro is 0.2 Ao,(b) for lattice of 16 atoms of

amorphous silicon using β = 3 and rro is 0.2 Ao.

atoms. The variation in the width distribution is very high in the supercell of the 16

atoms as compared to supercell of 54 atoms.

Chapter 4

Results and Discussions

4.1 Reverse Monte Carlo (RMC) modeling of amor-

phous graphene

Graphene is a closely packed, two dimensional single layer of carbon atoms [26].

Graphene has a honeycomb lattice, with all C-C bond lengths are about 1.42 Ao

and all C-C-C bond angles are about 120o [13]. It is the building block for all other

graphitic materials such as fullerenes, nanotubes and graphite [26].

Graphene has linear dispersion relation, zero e�ective mass and zero band gap [26].

So these properties corresponded that graphene electrons can be treated as spin 1/2

relativistic particles with dirac equation [26]. Due to such unpredictable properties

have led to an array of suggested applications from nano-ribbons to bio-devices [26].

A number of theoretical Scientist has been worked on ideal crystalline graphene.

Many defects (ring disorder, bond length and bond angle disorders) are produced

during the synthesis of graphene [25]. These defects play an important in many

24

CHAPTER 4. RESULTS AND DISCUSSIONS 25

appreciable and specially electronic properties [25]. Many methods were employed

to model amorphous graphene such as Wooten�Weaire�Winer (WWW) method and

molecular dynamics method etc. but here in this chapter we discussed modeling of

amorphous graphene using RMC method.

4.1.1 RMCmodeling of amorphous graphene using crystalline

graphene con�guration

Procedure

In this type of modeling, we are used crystalline graphene as an initial con�guration.

The reference PCF taken from [25]. The method is discussed in the following steps:

1. The �rst step is to break all bonds in the crystalline graphene, allow atoms to

move throughout the crystal to lose the initial memory. For this purpose we

run one lac RMC Steps with maximum allow move of 0.7 Ao to an atom. In

these RMC steps, only closest distance constraint were used to avoid overlap of

atoms and keep the value of β = 1.0. After one lac RMC steps the con�guration

of atoms is shown in Figure 4.1 (c).


(a) An initial con�guration (b) PCF of an initial con�guration, red line

represent the RMC �t PCF and blue line rep-

resent the reference PCF.

(c) Atomic con�guration

after one lac RMC steps

using β = 1.0 and move

length of 0.7 Ao.

(d) PCF after one lac RMC steps using β =

1.0 and move length of 0.7 Ao, red line repre-

sent the RMC �t PCF and blue line represent

the reference PCF.

Figure 4.1: Atomic con�gurations and their PCF's

2. Once all of the bonds in the crystalline graphene were broken and they lost the

previous memory, the next step is to build �rst peak of the Pair correlation

function (g(r)). We run ten thousand RMC steps with maximum allow move of


0.5 Ao to an atom. Here, we used an extra constraint of coordination number.

In these RMC steps, we divided the PCF into three parts, �rst part contains

the PCF from 1.4 Ao to 1.8 Ao (�rst peak), second part contains the PCF from

1.8 Ao to 3.5 Ao (second peak) and the third contains the remaining part of

PCF from 3.5A o to 8 Ao. Now we can write the χ (di�erence between the

reference g(r) and calculated g(r)) as:

χ = w1χ1 + w2χ2 + w3χ3 (4.1)

Where χ1 represents di�erence between �rst peak of reference g(r) and �rst

peak of calculated g(r), χ2 represents di�erence between second peak of ref-

erence g(r) and second peak of calculated g(r) and χ3 represents di�erence

between remaining peaks of reference g(r) and remaining peaks of calculated

g(r). The weighting factors are w1, w2 and w3. In these RMC steps, we were

given the weightage to �rst peak g(r) �ve times greater than other peaks of

g(r), enforce the �rst peak to build. After this step the atomic con�guration

and PCF are shown in Figure 4.2.


(a) Atomic con�guration

after ten thousand RMC

steps using β = 1.0 and

move length of 0.5 Ao.

(b) PCF after ten thousand RMC steps us-

ing β = 1.0 and move length of 0.5 Ao.

Figure 4.2

3. When �rst peak PCF is developed, weightage were given to second peak to

develop. Ten thousand RMC steps were run with allow maximum move to an

atom of 0.5 Ao. All the previous constraints are also applied. In these RMC

steps, the value of β = 5 was used.

4. Angles were �xed to enforce the second peak of PCF to match with reference

PCF. Ten thousand RMC steps were run with maximum allow move 0.2 Ao to

an atom, weightage were also given to �rst peak. In these RMC steps, the value

of β kept higher to avoid the move, which decrease the height of the �rst peak

of g(r). After these RMC steps, the �nal product model is shown in Figure 4.3

(c).




steps using β = 5.0 and

move length of 0.5 Ao.

(b) PCF after ten thousand RMC steps us-

ing β = 5.0 and move length of 0.5 Ao.

(c) Atomic con�guration af-

ter ten thousand RMC steps

using β = 5.0 and move

length of 0.5 Ao and angles

were �xed.

(d) PCF after ten thousand RMC steps us-

ing β = 5.0 and move length of 0.5 Ao and

angles were �xed.

Figure 4.3: RMC modeling of graphene at di�erent stages using crystalline graphene

as an initial con�guration.


4.1.2 RMC modeling of amorphous graphene using random

con�guration of carbon atoms

Procedure

In this type of modeling, random con�guration of carbon atoms are used as an

initial con�guration. The procedure is almost same as discussed above. First step

in above modeling is exempted in type of modeling because the atoms in the system

are already random in space. The remaining procedure is discussed in the following

steps:

1. First step in this type of modeling is to build �rst peak of the PCF. For this

purpose, ten thousand RMC steps were run with maximum allow move of 0.5

Ao to an atom. Closest distance approach and coordination number constraints

were applied. In these RMC steps, the value of β = 2 were used and weightage

is given to �rst peak to build. After �rst step atomic con�guration is shown in

Figure 4.4 (c).

2. When �rst peak of PCF developed, next step is to build the second peak. For

this purpose ten thousand RMC steps were run using β = 3 with maximum

allow move of 0.3 Ao to an atom. Weightage were given to second peak to build

the second peak.

3. When second peak is developed but it is not perfectly match with reference

second peak. For this purpose we applied an extra constraint on the bond

angles, bond angles were �xed within certain range. The value of β kept higher

to avoid the wrong move. The �nal model is shown in Figure 4.5 (c).


(a) Random con�gura-

tion of carbon atoms us-

ing as initial con�gura-

tion.

(b) PCF of the random con�guration of

carbon atoms.

(c) Atomic con�guration


steps using β = 2 with

maximum allow move of

0.5 Ao.

(d) PCF after ten thousand RMC steps

using β = 2 with maximum allow move

of 0.5 Ao.

Figure 4.4: RMCmodeling of graphene at di�erent stages using random con�guration

of carbon atom as an initial con�guration.




steps using β = 3 with

maximum allow move of

0.3 Ao.

(b) PCF of �nal model amorphous

graphene after ten thousand RMC steps

using β = 3 with maximum allow move

of 0.5 Ao.

(c) Final model of

amorphous graphene

after ten thousand

RMC steps with max-

imum allow move of

0.5 Ao and angles were

�xed.

(d) PCF after ten thousand RMC steps

with maximum allow move of 0.5 Ao and

angles were �xed.

Figure 4.5: RMCmodeling of graphene at di�erent stages using random con�guration

of carbon atom as an initial con�guration.


Table 4.1: Ring statistics of amorphous graphene generated by RMC method using

crystalline as an initial con�guration.

Ring Size Percent of Rings

5 32.63

6 32.63

7 32.63

8 2.1

4.1.3 Ring statistics

In analysis of the amorphous materials the ring statistics is important, crystalline

graphene contains only six atoms rings but our amorphous graphene models gener-

ated by RMC method contains di�erent size of rings ranging from �ve atoms rings

to eight. The ring statistics for the two amorphous graphene are given in Table 4.1

and 4.2

4.1.4 Bond angles distribution

The bond angles distribution is the most important tool to analyse the amorphous

structure. The width of distribution tells how the system is disordered. If width of

the distribution is broad its mean that the system is highly disordered and if width

of the distribution is narrow then disorderedness in the system will be small. For

crystalline system width of the distribution is very small or approximately equal to

zero. The bond angle distribution for two amorphous graphene models are show in


Table 4.2: Ring statistics of amorphous graphene generated by RMC method Ran-

dom con�guration of carbon atoms as an initial con�guration.

Ring Size Percent of Rings

5 36.0

6 34 .0

7 24.0

8 6.0

Figure 4.4.

(a) Bond angles distribution of

amorphous graphene generated

by RMC method using crystalline

graphene as an initial con�gura-

tion.

(b) Bond angles distribution of

amorphous graphene generated

by RMC method using random

con�guration of carbon atoms as

an initial con�guration.

Figure 4.6


4.2 Electronic structure

Many properties of the materials can be understood by determining eigenfunctions

and eigenvalues of the many body Hamiltonian [22]. Interacting atoms can be imag-

ined as a piece of matter [22]. The interactions between nuclei and electrons are

Coulombic in nature. [22]. The Hamiltonian for such a system can be written in

following general form [22]:

H = −Q∑J=1

~2

2MJ

∇2J −

n∑j=1

~2

2m∇2j +

e2

2

Q∑J=1

Q∑K 6=J

ZJZK| XI −XK |

+e2

2

n∑j=1

n∑k 6=j

1

| xj − xk |− e2

Q∑J=1

n∑j=1

ZJ| XJ − xj |

, (4.2)

where X = {XI , I = 1, ..., Q} represent the nuclear coordinates, and x = {xi, i =

1, ..., n} represent the electronic coordinates. Where ZJ andMJ represent charge and

mass of the nuclei, respectively and e and m represent charge and mass of electrons,

respectively. The �rst term represents the kinetic energy (K.E) of nuclei, second

term represents the K.E of electrons, third represents the nucleus-nucleus interac-

tion, froth represents the electron-electron interaction and �fth one represents the

nucleus and electron interaction in Equation- 4.2 [22].

4.2.1 External potential and electronic density

Before going to Hohenberg�Kohn theorems, �rst discussed the external potential in

term of electron density. Probability of �nding one electron at r1, another at r2, a


third at r3 etc can be written as [22]

ψ∗({ri})ψ({ri}) = |ψ({ri})|2 (4.3)

Integrating over all variables gives number of electrons∫|ψ({ri})|2dr1dr2...drN = n (4.4)

If we integrate over all electrons except one∫|ψ({ri})|2dr2dr3...drN = n(r1) (4.5)

The external potential contribution to the energy expectation value depends only on

electron density [22]

he =−N∑i=1

~2

2m∇2i +

e2

2

N∑i=1

N∑j 6=i

1

| ri − rj |+

N∑i=1

Vext(ri) (4.6)

〈he〉 =〈ψ|T |ψ〉+ 〈ψ|Uee|ψ〉+N∑i=1

〈ψ|Vext(ri)|ψ〉 (4.7)

The external potential at kth site can be written as

〈ψ|Vext(rk)|ψ〉 =

∫dr1dr2...drNψ

∗({ri})V (rk)ψ({ri}) (4.8)

〈ψ|Vext(rk)|ψ〉 =

∫Vext(rk)drk

∫dr

′|ψ({ri})|2, (4.9)

where dr′ include all the variables except rk

〈ψ|Vext(r)|ψ〉 =

∫Vext(r)n(r)dr, (4.10)

where n(r) represents the electron density.


4.2.2 Hohenberg�Kohn theorems

The �rst theorem of the Hohenberg�Kohn states that "the ground state density

uniquely determine the external potential [27] "

Proof

Suppose there are two potential (di�ering by more than a constant) with the same

ground state density. Potential one and its wave function is given below [27]:

V(1)ext → ψ(1)

o → n(1)e (r) (4.11)

potential two its wave function is given below

V(2)ext → ψ(2)

o → n(2)e (r) (4.12)

and

n(1)e (r) = n(2)

e (r) (4.13)

Expectation value of wave function one in the Hamiltonian one should be lower than

that of wave function two

〈ψ(1)o |H(1)|ψ(1)

o 〉 <〈ψ(2)o |H(1)|ψ(2)

o 〉 (4.14)

〈ψ(1)o |H(1)|ψ(1)

o 〉 =〈ψ(1)o |T |ψ(1)

o 〉+ 〈ψ(1)o |Uee|ψ(1)

o 〉+

∫V

(1)ext (r)n

(1)e (r)dr (4.15)

and

〈ψ(2)o |H(1)|ψ(2)

o 〉 =〈ψ(2)o |T |ψ(2)

o 〉+ 〈ψ(2)o |Uee|ψ(2)

o 〉+

∫V

(1)ext (r)n

(2)e (r)dr (4.16)

put equation 4.15 and 4.16 in 4.14 we get

〈ψ(1)o |T |ψ(1)

o 〉+ 〈ψ(1)o |Uee|ψ(1)

o 〉 < 〈ψ(2)o |T |ψ(2)

o 〉+ 〈ψ(2)o |Uee|ψ(2)

o 〉 (4.17)


Expectation value of wave function two in Hamiltonian two should be lower than

that of wave function one [27]

〈ψ(1)o |H(2)|ψ(1)

o 〉 >〈ψ(2)o |H(2)|ψ(2)

o 〉 (4.18)

We get

〈ψ(1)o |T |ψ(1)

o 〉+ 〈ψ(1)o |Uee|ψ(1)

o 〉 > 〈ψ(2)o |T |ψ(2)

o 〉+ 〈ψ(2)o |Uee|ψ(2)

o 〉 (4.19)

The two equations 4.17 and 4.19 cannot be true at the same time. Two di�erent

potentials cannot lead to identical ground densities which prove that "the ground

state density uniquely determines the external potential" [27].

Ground state wave function ψo(x1, x2, x3...xN) is functional of ground electronic den-

sity no(r). If we know the ground state electronic density as function of position then

we can obtained 3N -dimensional wave function as functional of ground state electron

density and from that we can �nd ground state energy as functional of ground state

electron density, it can be written as [27]

Eo = 〈ψo[no]|H|ψo[no]〉 = E[no] (4.20)

The electron density that minimized the energy functional is the ground state density

[27]

E[no] ≤ E[n] (4.21)

4.2.3 Kohn�Sham equations

Kohn�Sham approach can be used to calculate the energy as functional of electron

density. The total energy can be written in term of K.E, electron -electron interaction


and interaction of electron with external potential [27]:

E[n] = F [n] +

∫vext(r)n(r)d3r (4.22)

where

F [n] = T [n] + Uee (4.23)

Consider a set of non-interacting electrons with density n(r) with orbitals φ(r):

T [n] = − ~2

2m

N∑i

∫φi(r)∇2φi(r)d

3r + Te[n(r)] = TSP [n(r)] + Te[n(r)] (4.24)

The �rst term in above equation 4.24 is K.E of the single particle and second is the

correlation with other particles [27].

Uee[n] = e2∑i6=j

∫|φi(r)|2|φj(r′)|2

r − r′d3rd3r′+UXC [n(r)] = UH [n(r)]+UXC [n(r)] (4.25)

The �rst term is Hartree term and the second is the exchange correlation. The energy

functional can be written as [27]:

E[n] = TSP [n] + UH [n] + Vext[n] + EXC [n] (4.26)

Just like the Hartree equation but with Vext → Vext + EXC and the solution is[− ~2

2m∇2 + vext(r) + vH(r) + vXC(r)

]φi = εiφi (4.27)

With

n(r) =∑|φi(r)|2 (4.28)


4.2.4 Computational details

The electronic structures of the two amorphous graphene models were calculated

using density functional theory (DFT). There are many DFT codes used to calculate

the electronic properties of material but here we used the SIESTA 3.2 code for �nding

the electronic structure. Two approaches can be used in SIESTA 3.2 to �nd electronic

structure, local density approach (LDA) and generalized gradient approach (GGA).

In these calculations, LDA was used. Single zeta orbitals are used as basis set [24].

4.2.5 Relaxation of amorphous graphene models

Once the models of amorphous graphene generated the next step is to make these

con�gurations minimum energy con�guration. Conjugate Gradient (CG) method is

used for energy minimization. In CG, there is no big changes and no bond breaking

occurred but the bond angles and bond length are slightly changed. In Figure 4.7

the relaxed structures and their PCF's are shown


(a) Relaxed structure of

amorphous graphene gen-

erated by RMC method

using crystalline graphene

as a initial con�guration.

(b) PCF of relaxed structure amorphous

graphene generated by RMC method using

crystalline graphene as a initial con�guration.

(c) Relaxed structure of

amorphous graphene gen-

erated by RMCmethod us-

ing random con�guration

of carbon atoms as a initial

con�guration.

(d) PCF of relaxed structure amorphous

graphene generated by RMC method using

random con�guration of carbon atoms as a ini-

tial con�guration.

Figure 4.7: Relaxed structures of Amorphous grahene and their PCF's.


4.2.6 Results

Densities of states of two amorphous graphene and crystalline graphene are compared

in Figure 4.8. The Fermi level is situated at 6.09 eV . The densities of states of

amorphous graphene and crystalline graphene are shown in Figure 4.8 (a) is almost

same. In crystalline graphene there are no state at the Fermi level but density of

states is rise near Fermi level in amorphous graphene. Odd rings are the responsible

for creating states at the Fermi level.


(a) Densities of amorphous graphene generated by

RMC method using random con�guration of car-

bon atoms as a initial con�guration.

(b) Densities of amorphous graphene generated by

RMC method using crystalline graphene as a initial

con�guration.

Figure 4.8: Amorphous graphene and crystalline graphene densities of states com-

parison.


4.3 Conclusions

Reverse Monte Carlo (RMC) basic algorithm are studied, in which minimization

of the di�erence between reference pair correlation function (PCF) and calculated

PCF occurred. Two models of disordered graphene have been generated using RMC

method, in �rst model crystalline graphene used as an initial con�guration and in

second model we used random con�guration of carbon atoms. Electronic struc-

ture of the disordered graphene models are studied using density functional theory

(DFT) code SIESTA. The density of states appear near Fermi level in the disordered

graphene. Next, we will calculate the speci�c heat vs. temperature curve to check

the thermal stability of these con�guration.

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