Electronic and Transport Properties of Graphene Nanostructures.

Poole, Christopher J (2012) Electronic and Transport Properties of Graphene Nanostructures. PhD thesis, Lancaster University.

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Abstract

Graphene --- a crystal of carbon atoms in a two-dimensional (2D) honeycomb lattice --- is a gapless semiconductor, and has attracted great interest since it was fabricated in 2004 [1-3]. This monolayer of graphite has been shown to have remarkable properties, such as a linear energy dispersion relation [4] and massless, chiral fermions [5]. This thesis discusses some of these properties, as well as those found in bilayer graphene [6- 8]. Bilayer graphene is the formation of two coupled layers of graphene, exhibiting Bernal stacking (to be discussed later), and features massive chiral fermions [9]. Chapter 1 discusses the tight binding model, and derives the Hamiltonians for monolayer and bilayer graphene, used in subsequent Chapters. We also review important phenomena that account for the results seen in later Chapters. In addition to the material discussed in this Chapter, several excellent review articles have been written that cover other features and phenomena in few-layer graphene systems [10-17]. Chapter 2 is original published work. We extend the low energy effective two-band Hamiltonian for electrons in bilayer graphene (McCann and Fal'ko [7]) to include a spatially dependent electrostatic potential. We find that this Hamiltonian contains additional terms, as compared to the one used earlier in the analysis of electronic transport in n-p junctions in bilayers (Katsnelson et al. [9]). However, for potential steps |it| < 7 i (where 71 is the interlayer coupling), corrections to the transmission probability due to such terms are small. For the angle-dependent transmission T(0) we find T(6) = sin2(2$) --- (2u/3/ji) sin(4$) sin($) which slightly increases the Fano factor: F = 0.241 for u = 40meV. Chapter 3 is original work which was carried out simultaneously with Barbier et al. [18]. Nevertheless, the method of analysis and parameters considered are different, and the results are reached independently. Agreement is found with [19-21]. The work focuses on the introduction of a periodic potential profile in monolayer and bilayer graphene systems, creating extra Dirac points in the energy dispersion for monolayer graphene. Classical catastrophe theory is then employed to describe caustics and cusps in a system with a periodic potential profile, for cusps forming at regular interfaces. The periodicity of the formation of cusps is matched with the periodicity of the superlattice. The energy at which this occurs is then mapped to the energy spectrum found when analysing monolayer graphene with a periodic potential profile. In Chapter 4 we create a model to characterise the angles and commensurability of a few layers of hexagonal lattice materials, seen in STM images in experiments. The model allows for different lattice constants and rotations for each layer, as well as selectively showing lattice sites that are nearly commensurate with sites in other layers. Two popular cases are turbostratic graphene (bilayer graphene where one layer is rotated relative to the other) and graphene on hBN (hexagonal boron nitride). Finally, appendices with complete source code to reproduce these results are given, with examples of their usage.