MINISTRY OF EDUCATION
AND TRAINING
VIETNAM ACADEMY OF
SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
……..….***…………
NGUYEN THI THUY NHUNG
ELECTRONIC TRANSPORT PROPERTIES
OF SOME GRAPHENE NANOSTRUCTURES
Major: Theoretical and Mathematical Physics
Code: 9 44 01 03
DISSERTATION SUMMARY
Ha Noi – 2020
Dissertation was completed at: Graduate University of Science and
Technology - Vietnam Academy of Science and Technology
Scientific Supervisor: Prof. Dr. NGUYEN VAN LIEN
1st Reviewer: …
2nd Reviewer: …
3rd Reviewer: ….
The dissertation will be defended at Graduate University of Science and
Technology, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet,
Cau Giay District, Hanoi city.
At … hour … date … month … 2020
The dissertaion can be found in National Library of Vietnam and library of
Graduate University of Science and Technology, Vietnam Academy of Science
and Technology
Introduction
Carbon is a common element in nature and a fundamental element for life. It
is abundant in the crust of the Earth. Diamond and graphite are 3D allotropes
of carbon. In 1985, the 0D allotrope of carbon, fullerence, was found by Kroto,
Smalley and Curl. In 1952, Radushkevich and Lukyanovich reported about
carbon nanotube. In 1991 Lijima and his colleagues successfully fabricated
carbon nanotube.
Wallace was the pioneer who theoretically performed research about a
single layer of carbon in 1947. The term “graphene” was first proposed by
Boehm, Setton and Stumpp in 1994 to indicate a single layer of carbon in
which carbon atoms are arranged at the nodes of a honeycomb lattice. In
2004, Geim and Novoselov successfully separated graphene from graphite.
Graphene became the first ever 2D material created in the laboratory. Other
methods to fabricate graphene have been found gradually.
After being successfully created in the laboratory, graphene has become
a hot subject of research. Researchers expect graphene, with its superior
conductivity and a good heat transfer property, to bring unique and impor-
tant applications. In electronics, graphene is an ideal material for ballistic
transports to be realized. Graphene has an advantage in fabricating p-n-p
junctions, which are the basic components of bipolar devices. Recently, scien-
tists at the Massachusetts Institute of Technology (MIT) have created qubits
in superconducting circuits using graphene. From these facts, we chose the
research topic “The electrical transport properties of some graphene nanos-
tructures”.
The aim of this thesis is to study electrical transport characteristics of
graphene nanostructures. We focus on two research objects associated with
two kinds of graphene nanostructures: graphene bipolar junctions (GBJs) and
graphene quantum dots (GQDs).
GBJs can be created by electrodes that are in contact with a graphene
surface in a configuration that allows to control the transport in 1D. The
electrical transport characteristics of bipolar junctions (BJs) mainly depend
on the potential at the transition region. Previous theoretical studies assumed
that this potential is rectangular or trapezoidal. In this thesis, we propose to
use a Gaussian potential barrier model for studying the transport properties
of GBJs. The advantage of Gaussian potential is that it better describes the
potential profile in real BJ structures. This allows to describe all modes of the
carrier density as well as a smooth transition between the modes. Our study
focuses on calculating the characteristic quantities of electrical transport such
as transmission probability, resistance, Volt-Ampere characteristics and shot
1
noise depending on the model parameters, in order to understand clearly
about the ballistic transport mechanism through GBJs.
Similar to p-njunctions, GQDs can be created by nanoelectrodes. Thanks
to Scanning Tunneling Microscope (STM), it is possible to fabricate nano-
sized electron confinement regions on graphene sheets. In the GQD created by
electrostatic potential, except for certain conditions that allow the existence of
bound states (BSs), the carriers normally exist in quasi-bound states (QBSs)
with a finite lifetime. The determination of the lifetime of charge carriers in
the GQD is paramount for the design of electronic devices based on GQD of
desirable functions. Therefore, in this thesis, we propose a theoretical model
to study the lifetime and local density of states (LDOS) of the carriers in
circular GQD (CGQD) created by electrostatic potentials. The results that
we achieve are compared to experiments.
We use the transfer matrix (T-matrix) method to solve the above prob-
lems. By obtaining the T-matrix, one can calculate electronic properties of the
system such as transmission probability, conductance, current-voltage char-
acteristics and shot noise. From the components of the T-matrix, one can
also determine the energies as well as the level widths of QBSs of electron in
GQD. The local density of states (LDOS) and scattering coefficients can be
expressed accurately in terms of the T-matrix elements. Apart from the T-
matrix method, we propose also a method for calculating LDOS directly from
normalized wave functions for a CGQD with an arbitrary confined potential.
This thesis is divided into 4 chapters, excluding the introduction, the con-
clusion and the references. Chapter 1 presents an overview of the electronic
properties of graphene and the results of the previous research on n-p-njunc-
tion and GQD. Chapter 2 introduces the theoretical and numerical methods
used in the thesis. Chapter 3 presents the results on electronic transport in
n-p-ngraphene junction. Chapter 4 describes the results of energy structure
and related properties of CGQD created by electrostatic potentials as well as a
theoretical development of the T-matrix method for CGQD in a perpendicular
uniform magnetic field.
2
Chapter 1
Electronic properties of
graphene
1.1 Crystal structure and energy band struc-
ture of graphene
Graphene is a monolayer of carbon atoms in which the atoms are on the
vertices of a two-dimensional honeycomb lattice. A graphene lattice can be
considered as a hexagonal Bravais lattice with two atoms in an unit cell. The
energy band structure of graphene can be determined by the tight binding
approximation. When considering only the interaction between the nearest
neighbors in the graphene lattice, the electronic band dispersion is given by
E(k) = ±tq4 cos(πkxa√3) cos(πkya) + 4 cos2(πkya) + 1 ,(1.1)
where t=Rφ∗(r−rA)ˆ
Hφ(r−rB)d3ris the hopping energy between the
nearest neighbors. For graphene, t≈2.7 eV. The energy band structure is
given in terms of the formula (1.1) described in Fig.1.1. The minus sign on
the right hand side of the formula (1.1) corresponds to the lower energy band,
called the πband, whereas the plus sign corresponds to the higher energy
band, called the π∗band. We can see that these two bands are degenerated
at the Kand K′points, called the Dirac points.
At zero Kelvin, the πband is fully filled while the π∗band is empty, and
the Fermi energy is EF= 0. Let k=K+q, where Kis the momentum vector
at the Kor K′point and qis the momentum relative to the momentum at
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