Image

Tight Binding Descriptions Of Graphene And Its Derivatives

Tight Binding Descriptions Of Graphene And Its Derivatives

ABSTRACT

Graphene is an effectively two dimensional form of carbon atoms arranged in honeycomb lattice. Due to its lightweight, high electron mobility and other special electronic properties it is considered both an academically interesting and industrially promising candidate for various electronics applications. Many investigations focused on graphene require theoretical simulations to be performed over a large number of unit cells of graphene. For simulation scenarios where ab-initio methods are computationally too costly, researchers often refer to the low-cost but still highly accurate tight-binding (TB) model. In TB model, the electron interaction is parametrized, either through the derivation of parameters using rst principles methods, or by fitting to experimental results. The results of TB simulations depend strongly on this parameterization, therefore it is very important to know the level of accuracy and transferability of these parameters. In this research project we will simulate the band structure and density of states of graphene and other derivative of carbon structures such as nano-ribbons using a state-of-the-art parameter set; and compare their performance to the results of ab initio calculations. The resulting comparison will serve as a benchmark for future studies on graphene and derivatives.

TABLE OF CONTENTS

1 Introduction 1
2 Linear Combination of Atomic Orbitals 3
2.1 H-like atom and atomic orbitals . . . . . . . . . . . . . . . . . 3
2.1.1 Asymptotic analysis . . . . . . . . . . . . . . . . . . . 4
2.2 Linear Combination: H2 dimer . . . . . . . . . . . . . . . . . 6
2.3 Linear Combination: H trimer . . . . . . . . . . . . . . . . . . 8
3 Tight Binding Approximation 10
3.1 H chain and Born-von Karman boundary conditions . . . . . . 10
3.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Tight binding model for graphene . . . . . . . . . . . . . . . . 13
4 Density functional theory 16
4.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Graphene 21
5.1 Tight Binding Results . . . . . . . . . . . . . . . . . . . . . . 22
5.1.1 Ab-initio result . . . . . . . . . . . . . . . . . . . . . . 28
6 Nanoribbons 31
6.1 Results for Zigzag Ribbon (zGNR) . . . . . . . . . . . . . . . 31
6.2 Results for Armchair Ribbon (aGNR) . . . . . . . . . . . . . . 33
7 Conclusion and Outlook

Chapter one

Introduction

Carbon is an old but new material. It has been used for centuries going back to antiquity, but yet many new solid forms of carbon have only recently been experimentally obtained in the last few decades. most notably, in 2004, Andre geim and kanstantin novoselov [1] used a scotch tape in remarkably simple technique to extract, for the rst time, a make of carbon with a thickness of just one atom, i.e graphene, from graphite. some other modern crystalline forms of carbon include buckyballs, carbon nanotubes (cnts) as illustrated in fig.1.1.

Since the discovery of geim and novoselov, several studies focused on the electronic structure of graphene and found that graphene has some superior properties, such as very high electron mobility[2], that makes it a promising candidate material for the electronic industry of the future [3]. the superior electronic properties of graphene are mainly attributed to its crystal structure, the 2d honeycomb lattice, and its short-range interactions. Other derivatives of graphene which share these core properties have also been subject to studies, for example 0d fullerenes, or 1d nanotubes or 2d ribbons.

Many of the highly accurate theoretical works on graphene and derivatives use rst principles techniques and their findings support the possibility of using graphene based materials in industrial applications such as batteries [4] solar cells [5] catalysis [6] etc. however more reliable and realistic theoretical simulations of these applications require big scale calculations to be run, which are computationally demanding for ab initio methods.

For this reason theoretical studies also employ numerically more affordable methods such as tight binding model as used by p.r wallace in 1947 [7]. In the tight-binding model electrons are assumed tightly bound to local attraction centers and not interacting with one another but only interacting with the lattice of ions. this allows the use of only a few atomic orbitals to represent the many body wave function of the material. furthermore, in this

Figure 1.1: all dimensionality of carbon

Model the integrals that describe the matrix elements of the hamiltonian, i.e. the kinetic and potential contribution of each pair of independent electronic state of the system, can also be parametrized. the parametrization should be done such that the dimensional tight binding description of the system coincides with truly rst principles results for a wide range of systems. that way, the TB parametrization can be said to be accurate and transferable.

One successful application of TB to graphene is the work of ref.[8] in which they demonstrate that even these simple TB approximations can qualitatively describe the important features of the band structure of graphene
and counts with the correct parametrization.

Their research motivated us to study graphene and its derivatives such as nano-ribbons with tb model using various parameterizations, and explore the relationship between atomistic structure, periodicity and electronic band structure. as in the work of [8], we then compare our results with ab initio calculations in terms of accuracy and computational efficiency so that our results can serve as a benchmark for future studies of large scale Simulations.


Need More or Something Else?

Hire Writer