Download Quantum transport and spin effects in lateral semiconductor nanostructures and graphene Martin Evaldsson

Linköping studies in science and technology.
Dissertations, No 1202
Quantum transport and spin effects
in lateral semiconductor
nanostructures and graphene
Martin Evaldsson
Department of Science and Technology
Linköping University, SE-601 74 Norrköping, Sweden
Norrköping, 2008
Quantum transport and spin effects in lateral semiconductor
nanostructures and graphene
c 2008 Martin Evaldsson
Department of Science and Technology
Campus Norrköping, Linköping University
SE-601 74 Norrköping, Sweden
ISBN 978-91-7393-835-8
ISSN 0345-7524
Printed in Sweden by LiU-Tryck, Linköping, 2008
Abstract
This thesis studies electron spin phenomena in lateral semi-conductor quantum dots/anti-dots and electron conductance in graphene nanoribbons by numerical modelling. In paper I we have investigated spin-dependent transport
through open quantum dots, i.e., dots strongly coupled to their leads, within
the Hubbard model. Results in this model were found consistent with experimental data and suggest that spin-degeneracy is lifted inside the dot – even
at zero magnetic field.
Similar systems were also studied with electron-electron effects incorporated via Density Functional Theory (DFT) in the Local Spin Density Approximation (LSDA) in paper II and III. In paper II we found a significant
spin-polarisation in the dot at low electron densities. As the electron density
increases the spin polarisation in the dot gradually diminishes. These findings
are consistent with available experimental observations. Notably, the polarisation is qualitatively different from the one found in the Hubbard model.
Paper III investigates spin polarisation in a quantum wire with a realistic
external potential due to split gates and a random distribution of charged
donors. At low electron densities we recover spin polarisation and a metalinsulator transition when electrons are localised to electron lakes due to ragged
potential profile from the donors.
In paper IV we propose a spin-filter device based on resonant backscattering of edge states against a quantum anti-dot embedded in a quantum wire.
A magnetic field is applied and the spin up/spin down states are separated
through Zeeman splitting. Their respective resonant states may be tuned so
that the device can be used to filter either spin in a controlled way.
Paper V analyses the details of low energy electron transport through a
magnetic barrier in a quantum wire. At sufficiently large magnetisation of the
barrier the conductance is pinched off completely. Furthermore, if the barrier
is sharp we find a resonant reflection close to the pinch off point. This feature
is due to interference between a propagating edge state and quasibond state
inside the magnetic barrier.
Paper VI adapts an efficient numerical method for computing the surface
Green’s function in photonic crystals to graphene nanoribbons (GNR). The
method is used to investigate magnetic barriers in GNR. In contrast to quantum wires, magnetic barriers in GNRs cannot pinch-off the lowest propagating
state. The method is further applied to study edge dislocation defects for realistically sized GNRs in paper VII. In this study we conclude that even modest
edge dislocations are sufficient to explain both the energy gap in narrow GNRs,
and the lack of dependance on the edge structure for electronic properties in
the GNRs.
iii
iv
Preface
This thesis summarises my years as a Ph.D. student at the Department of
Science and Technology (ITN). It consists of two parts, the first serving as a
short introduction both to mesoscopic transport in general and to the papers
included in the second part. There are several persons to whom I would like
to express my gratitude for help and support during these years:
First of all my supervisor Igor Zozoulenko for patiently introducing me to
the field of mesoscopic physics and research in general, I have learnt a lot from
you.
The other members in the Mesoscopic Physics and Photonics group, Aliaksandr Rachachou and Siarhei Ihnatsenka – it is great to have people around
who actually understand what I’m doing.
Torbjörn Blomquist for providing an excellent C++ matrix library which
has significantly facilitated my work.
A lot of people at ITN for various reasons, including Mika Gustafsson (a
lot of things), Michael Hörnquist and Olof Svensson (lunch company and company), Margarita Gonzáles (company and lussebak), Frédéric Cortat (company
and for motivating me to run 5km/year), Steffen Uhlig (company and lussebak), Sixten Nilsson (help with teaching), Sophie Lindesvik and Lise-Lotte
Lönnedahl Ragnar (help with administrative issues).
Also a general thanks to all the past and present members of the ‘Fantastic
Five’, and everyone who has made my coffee breaks more interesting.
I would also like to thank my parents and sister for being there, and of
course my family, Chamilly, Minna and Morris, for keeping my focus where it
matters.
Finally, financial support from The Swedish Research Council (VR) and
the National Graduate School for Scientific Computations (NGSSC) is acknowledged.
v
vi
List of publications
Paper I: M. Evaldsson, I. V. Zozoulenko, M. Ciorga, P. Zawadzki and A. S.
Sachrajda. Spin splitting in open quantum dots. Europhysics Letters
68, 261 (2004)
Author’s contribution: All calculations for the theoretical part. Plotting of all figures. Initial draft of the of the paper (except experimental
part).
Paper II: M. Evaldsson and I. V. Zozoulenko. Spin polarization in open quantum
dots. Physical Review B 73, 035319 (2006)
Author’s contribution: Computations for all figures. Plotting of all
figures. Drafting of paper and contributed to the discussion in the process of writing the paper.
Paper III: M. Evaldsson, S. Ihnatsenka and I. V. Zozoulenko. Spin polarization in
modulation-doped GaAs quantum wires. Physical Review B 77, 165306
(2008)
Author’s contribution: Computations for all figures but figure 4.
Plotting of all figures but figure 4. Drafting of paper and contributed to
the discussion in the process of writing the paper.
Paper IV: I. V. Zozoulenko and M. Evaldsson. Quantum antidot as a controllable
spin injector and spin filter. Applied Physics Letters 85, 3136 (2004)
Author’s contribution: Calculations and plotting of figure 2.
Paper V: Hengyi Xu, T. Heinzel, M. Evaldsson and I. V. Zozoulenko. Resonant
reflection at magnetic barriers in quantum wires. Physical Review B 75,
205301 (2007)
Author’s contribution: Providing source code and technical support
to the calculations. Contributed to the discussion in the process of writing the paper.
Paper VI: Hengyi Xu, T. Heinzel, M. Evaldsson and I. V. Zozoulenko. Magnetic
barriers in graphene nanoribbons: Theoretical study of transport properties Physical Review B 77, 245401 (2008)
Author’s contribution: Part in developing the method and writing
the source code. Contributed to the discussion in the process of writing
the paper.
Paper VII: M. Evaldsson, I. V. Zozoulenko, Hengyi Xu and T. Heinzel. Edge disorder induced Anderson localization and conduction gap in graphene
nanoribbons. Submitted.
Author’s contribution: Calculations for all figures but figure 2. Plotting of all figures. Drafting of paper and contributed to the discussion
in the process of writing the paper.
vii
viii
Contents
Abstract
iii
Preface
v
List of publications
vii
1 Introduction
1
2 Mesoscopic physics
2.1 Heterostructures . . . . . . . . . . . . . . . . . .
2.1.1 Two-dimensional heterostructures . . . .
2.1.2 (Quasi) one-dimensional heterostructures
2.2 Graphene . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Basic properties: experiment . . . . . . .
2.2.2 Basic properties: theory . . . . . . . . . .
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3 Transport in mesoscopic systems
3.1 Landauer formula . . . . . . . . .
3.1.1 Propagating modes . . . .
3.2 Büttiker formalism . . . . . . . .
3.3 Matching wave functions . . . . .
3.3.1 S-matrix formalism . . . .
3.4 Magnetic fields . . . . . . . . . .
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4 Electron-electron interactions
4.0.1 What’s the problem? . .
4.1 The Hubbard model . . . . . .
4.2 The variational principle . . . .
4.3 Thomas-Fermi model . . . . . .
4.4 Hohenberg-Kohn theorems . . .
4.4.1 The first HK-theorem .
4.4.2 The second HK-theorem
4.5 The Kohn-Sham equations . . .
4.6 Local Density Approximation .
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4.7
4.8
Local Spin Density Approximation . . . . . . . . . . . . . . . .
Brief outlook for DFT . . . . . . . . . . . . . . . . . . . . . . .
32
33
5 Modelling
5.1 Tight-binding Hamiltonian . . . . .
5.1.1 Mixed representation . . . . .
5.1.2 Energy dispersion relation . .
5.2 Green’s function . . . . . . . . . . .
5.2.1 Definition of Green’s function
5.2.2 Dyson equation . . . . . . . .
5.2.3 Surface Green’s function . . .
5.2.4 Computational procedure . .
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6 Comments on
6.1 Paper I .
6.2 Paper II .
6.3 Paper III
6.4 Paper IV
6.5 Paper V .
6.6 Paper VI
6.7 Paper VII
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papers
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Chapter 1
Introduction
During the second half of the 20th century, the introduction of semiconductor
materials came to revolutionise modern electronics. The invention of the transistor, followed by the integrated circuit (IC) allowed an increasing number of
components to be put onto a single silicon chip. The efficiency of these ICs
has since then increased several times, partly by straightforward miniaturisation of components. This process was summarised by Gordon E. Moore in
the now famous “Moore’s law”, which states that the number of transistors
on a chip doubles every second year1 . However, as the size of devices continue
to shrink, technology will eventually reach a point when quantum mechanical
effects become a disturbing factor in conventional device design.
From a scientific point of view this miniaturisation is not troubling but,
rather, increasingly interesting. Researchers can manufacture semiconductor
systems, e.g., quantum dots or wires, which are small enough to exhibit pronounced quantum mechanical behaviour and/or mimic some of the physics
seen in atoms. In contrast to working with real atoms or molecules, experimenters can now exercise precise control over external parameters, such as
confining potential, the number of electrons, etc.. In parallel to this novel
research field, a new applied technology is emerging – “spintronics” (spin
electronics). The basic idea of spintronics is to utilise the electron spin as an
additional degree of freedom in order to improve existing devices or innovate
entirely new ones. Existing spintronic devices are built using ferromagnetic
components – the most successful example to date is probably the read head in
modern hard disk drives (see e.g., [94]) based on the Giant Magneto Resistance
effect[10, 15].
Because of the vast knowledge accumulated in semiconductor technology
there is an interest to integrate future spintronic devices into current semiconductor ones. This necessitates a multitude of questions to be answered, e.g.:
1
Moore’s original prediction made in 1965 essentially states (here slightly reformulated),
that “the number of components on chips with the smallest manufacturing costs per component doubles roughly every 12 months”[74]. It has since then been revised and taken on
several different meanings.
2
Introduction
can spin-polarised currents be generated and maintained in semiconductor
materials, does spin-polarisation appear spontaneously in some semiconductor systems?
Experiments, however, are not the only way to approach these new and
interesting phenomena. The continuous improvement of computational power
together with the development of electron many-body theories such as the Density Functional Theory provide a basis for investigating these questions from a
theoretical/computational point of view. Theoretical work may explain phenomena not obvious from experimental results and guide further experimental
work. Modelling of electron transport in lateral semiconductor nanostructures
and graphene is the main topic of this thesis.
Chapter 2
Mesoscopic physics
The terms macroscopic and microscopic traditionally signify the part of the
world that is directly accessible to the naked eye (e.g., a flat wall), and the
part of the world which is to small to see unaided (e.g., the rough and weird
surface of the flat wall in a scanning electron microscope). As the electronic
industry has progressed from the macroscopic world of vacuum tubes towards
the microscopic world of molecular electronics, the need to name an intermediate region has come about. This region is now labelled mesoscopic, where
the prefix derives from the Greek word “mesos”, which means ‘in between’.
Mesoscopic systems are small enough to require a quantum mechanical description but at the same time too big to be described in terms of individual
atoms or molecules, thus ‘in between’ the macroscopic and the microscopic
world.
The mesoscopic length scale is typically of the order of:
• The mean free path of the electrons.
• The phase-relaxation length, the distance after which the original phase
of the electron is lost.
Depending on the material used, the temperature, etc., these lengths and the
actual size of a mesoscopic system could vary from a few nanometres to several
hundred micrometres [22].
This chapter introduces manufacturing techniques, classification and general concepts of low-dimensional and semiconductor systems. The first sections
introduce laterally defined systems in heterostructures; relevant for papers I-V.
In the last section we consider graphene, relevant for papers VI-VII.
2.1
Heterostructures
A heterostructure is a semiconductor composed of more than one material. By
mixing layers of materials with different band gaps, i.e. band-engineering, it
is possible to restrict electron movement to the interface (the heterojunction)
4
Mesoscopic physics
between the materials. This is typically the first step in the fabrication of
low-dimensional devices. Since any defects at the interface will impair electron mobility through surface-roughness scattering, successful heterostructure
fabrication techniques must yield a very fine and smooth interface. Two of the
most common growth methods are molecular beam epitaxy (MBE) and metalorganic chemical vapour deposition (MOCVD). In MBE, a beam of molecules
is directed towards the substrate in an ultra high vacuum chamber, while in
MOCVD a gas mixture of the desired molecules are kept at specific temperatures and pressures in order to promote growth on a substrate. Both these
techniques allow good control of layer thickness and keep impurities low at the
interface.
In addition to the problem with surface-roughness, a mechanical stress due
to the lattice constant mismatch between the heterostructure materials causes
dislocations at the interface. This restricts the number of useful semiconductors to those with close/similar lattice constants. A common and suitable
choice, because of good lattice constant match and band gap alignment (see
figure 2.1), is to grow Alx Ga1−x As (henceforth abbreviated AlGaAs, with the
mixing factor x kept implicit) on top of a GaAs substrate. The position and
relative size of the band gap in a GaAs-AlGaAs heterostructure is schematically shown in figure 2.1. At room temperature the band gap for GaAs is
1.424eV[82], while the band gap in AlGaAs depends on the mixing factor x
and can be approximated by the formula[82]
Eg (x) = 1.424 + 1.429x − 0.14x2 [eV]
0 < x < 0.441 ,
(2.1)
i.e., it varies between 1.424-2.026eV.
2.1.1
Two-dimensional heterostructures
In GaAs-AlGaAs heterostructures, a two-dimensional electron system is typically created by n-doping the AlGaAs (figure 2.2 or the left panel of figure 2.5).
Some of the donor electrons will eventually migrate into the GaAs. These
electrons will still be attracted by the positive donors in the AlGaAs, but be
unable to go back across the heterojunction because of the conduction band
discontinuity. Trapped in a narrow potential well (see figure 2.2), their energy
component in this direction will be quantised. Because the potential well is
very narrow (typically 10nm[24]), the available energy states will be sparsely
spaced, and at sufficiently low temperatures all electrons will be in the same
(the lowest) energy state with respect to motion perpendicular to the interface.
I.e., electrons are free to move in the plane parallel to the heterojunction, but
restricted to the same (lowest) energy state in the third dimension. In this
sense we talk about a two-dimensional electron gas (2DEG).
1
For larger x the smallest band gap in AlGaAs is indirect.
2.1. Heterostructures
5
AlGaAs
GaAs
Conduction band
Band gap
Valence band
Figure 2.1: Band gap for AlGaAs (left) and GaAs (right) schematically. For these
materials their corresponding band gap results in a straddling alignment,
i.e., the smaller band gap in GaAs is entirely enclosed by the larger band
gap in AlGaAs.
Energy
Original n-AlGaAs band
2DEG
Space
Conduction band
donors
n-AlGaAs
Original GaAs band
GaAs
AlGaAs
(spacer-layer)
Figure 2.2: Two-dimensional electron gas seen along the confining dimension.
Remote or modular doping, i.e., to place the donors only in the AlGaAs
layer, prevent the electrons at the heterojunction to scatter against the positive donors. Usually an additional layer of undoped AlGaAs is grown at the
interface as a spacer layer. This will, at the expense of high electron density, further shield the electrons from scattering. Densities in 2DEGs typically
varies between 1 − 5 × 1015 m−1 though values as low as 5 × 1013 m−1 has been
reported[41].
Density of States
A simple but yet powerful characterisation of a system is given by its density of
states (DOS), N (E), where N (E)dE is defined as the number of states in the
energy interval E → E + dE. For free electrons it is possible to determine the
2-dimensional DOS, N2D (E), exactly. This is done by considering electrons
6
Mesoscopic physics
ky
1
0
0
1
1
0
0
1
00
11
0
1
0
1
00
11
0
k 1
0
1
1
0
0
1
0
01
11
k+dk 1
01
000
1
2π
0
1
kx
L
1
0
111111
000000
00
11
0
100
11
n2D (E) =
m
~2 π
n(E)
2π
L
E
(a) 2D k-space
(b) 2D density of states
Figure 2.3: (a) Occupied and unoccupied (filled/empty circles) states in 2D k-space.
(b) The two-dimensional density of states, n2D (E).
situated in a square of area L2 and letting L → ∞. With periodic boundary
conditions the solutions are travelling waves,
φ(r) = eik·r = ei(kx x+ky y) ,
(2.2)
where
k = (kx , ky ) =
2πm 2πn
,
Lx Ly
=
2πm 2πn
,
L
L
m, n = 0, ±1, ±2 . . . .
(2.3)
Plotting these states in the 2-dimensional
k-space,
figure
2.3a,
we
recognise
2
, hence the density of states is
that a unit cell has the area 2π
L
N2D (k) = 2
(2π)2
L2
−1
=
L2
2π 2
(2.4)
where a factor two is added to include spin. Defining the density of states per
unit area,
N2D (k)
1
n2D (k) =
= 2,
(2.5)
L2
2π
we get a quantity that is defined as L → ∞. In order to change variable
from n2D (k) to n2D (E), we look at the annular area described by k and
k+dk in figure 2.3a. This area is approximated by 2πkdk and, thus, contains
n2D (k)2πkdk states. The number of states must of course be the same whether
we express it in terms of wave vector k or energy E, i.e.,
dE
1
dk = 2πkn2D (k)dk = 2πk 2 dk
dk
2π
k dE −1
⇔ n2D (E) =
.
π dk
n2D (E)dE = n2D (E)
(2.6)
(2.7)
2.1. Heterostructures
7
For free electrons, with
E(k) =
~2 k2
,
2m
where k = |k|
(2.8)
n2D (E) =
m
,
~2 π
(2.9)
we finally arrive at
shown in figure 2.3b.
A similar derivation for a one dimensional system yields the density of
states per unit length,
r
1
2m
.
(2.10)
n1D (E) =
~π
E
2.1.2
(Quasi) one-dimensional heterostructures
There are several techniques to further restrict the 2DEG. Two common approaches are chemical etching, or to put metallic gates on top of the sample.
Figures 2.4(a)and (b) show a quantum wire and a quantum dot, respectively,
defined by a potential applied to top gates.
Side gate
Side gate
Top gate
2DEG
(a)
(b)
Figure 2.4: Schematic figure of (a) quantum wire, (b) quantum dot, defined by potentials applied to metalic gates. The layers in the heterostructure are,
from bottom to top, substrate, spacer, donor and cap layer.
Etching
By etching away part of the top dopant layer, the electron gas is located to
the area beneath the remaining dopant as schematically illustrated in figure
2.5.
Metallic gates
A second alternative is to deploy metallic gates on top of the surface as shown
in figure 2.4(a)-(b). By applying a negative voltage to the gates, the 2DEG
will be depleted beneath them. This technique allows experimenters to control
the approximate size of the system by changing the applied potential during
the experiment.
8
Mesoscopic physics
n-AlGaAs
AlGaAs
n-AlGaAs
AlGaAs
11111111111
00000000000
GaAs
2DEG
2DEG
GaAs
Figure 2.5: Left: 2DEG in AlGaAs-GaAs heterostructure. Right: Etching away the
AlGaAs everywhere but in a narrow stripe results in a one-dimensional
quantum wire beneath the remaining AlGaAs.
Subbands in quasi one-dimensional systems
The transversal confinement in a quasi one-dimensional system, such as the
quantum wire in figure 2.4(a), results in a quantisation of energies in this
dimension. Denoting the confining potential U (y) (i.e., electrons are free along
the x-axis), the Schrödinger equation
2
∂
∂2
~2
+
+ U (y) Ψ(x, y) = EΨ(x, y),
(2.11)
−
2m ∂x2 ∂y 2
can, by introducing Ψ(x, y) = ψ(x)φ(y), be separated into
−
~2
2m
−
d2
dy 2
~2 d2
ψ(x) = Ex ψ(x)
2
2m dx
+ U (y) φ(y) = Ey φ(y).
In the simple case of a hard wall potential U (y),
(
0,
0<y<w
U (y) =
∞, y < 0 ∪ w < y,
(2.12)
(2.13)
(2.14)
eq. (2.12) and (2.13) yields the solutions
Ψn (x, y) = ψ(x)φn (y) = eikx x sin
πny w
(2.15)
and eigenenergies
E = Ex + Ey,n
~2
=
2m
kx2
+
πn 2 w
=
~2 2
2
kk + π 2 k⊥
.
2m
(2.16)
The total energy E is the sum of a continuous part Ex , and a discreet part
Ey,n with corresponding continuous (kk ) and and discrete (k⊥ ) wave-vectors.
2 , consists
Hence, the energy dispersion relation, E(k) = E(kk + k⊥ ) ∼ kk2 + k⊥
of subbands, as depicted in figure 2.6.
2.2. Graphene
9
E3
φ1
φ2
φ3
Energy
E2
E1
EF
kk
(a)
(b)
Figure 2.6: (a) The three lowest propagating nodes, φi (y), in a hard wall potential
quantum wire. (b) Corresponding subbands to φ1 (y)–φ3 (y) seen in (a).
The Fermi energy is indicated by EF .
2.2
Graphene
The discovery of the carbon fullerenes in 1985[58] paved the way for extensive
research into the allotropes of carbon. Today, a wide range of structures such
as buckyballs, carbon nanotubes, or a combination thereof (carbon nanobuds)
are known. Basically, these structures consist of a single or several layers of
graphite wrapped up in some configuration. Interestingly, the existence of a
single layer of graphite, not wrapped up, was long thought to be thermodynamically unstable[61, 62, 88, 89]. The initial reports of such structures in
2004[79] was therefore somewhat unexpected. Single-layered graphite is now
known as graphene and has been the focus for intense research during the
past few years. The reason graphene does not become unstable, as theory predicts for 2D structures, seems to be the result of corrugations which stabilizes
the sheet[72]. It is thus more correct to view graphene as a two-dimensional
structure in a three-dimensional space rather than a strictly two-dimensional
structure.
Although graphene hasn’t been experimentally studied for more than a few
years it has been an issue of theoretical interest for a long time. This is partly
because the properties of various carbon based materials such as graphite,
carbon nanotubes, etc., derive from the properties of graphene and partly
because some of the properties of graphene are rather extraordinary, making
graphene an interesting “toy-model” for theoreticians. The honeycomb structure of graphene (figure 2.7) makes the charge carriers in the lattice mimic relativistic particles which can be described by the Dirac equation[29, 103, 105].
This causes a number of effects not expected in non-relativistic systems to be
present; an anomalous quantum hall effect[78, 124], the Klein paradox2 [54],
the presence of a minimum conductivity as charge carrier concentration is
2
Where charge carriers can tunnel through an arbitrary high energy barrier with unity
probability
10
Mesoscopic physics
depleted[78, 113].
A-lattice
B-lattice
acc
{
Figure 2.7: The graphene lattice can be described as two sub-lattices, A and B. Primitive vectors for sub-lattice A are indicated by arrows. The interatomic
distance acc is roughly 1.42Å.
2.2.1
Basic properties: experiment
The discovery of graphene had an illusive air of simplicity – repeated peeling
of a graphite sample with adhesive tape or rubbing it against a surface (i.e.
“drawing”), resulted in flakes a single atom layer thick[78, 79]. Yet the approach is nothing but simple; monolayers are in minority of the flakes found
and conventional techniques for identifying such 2D structures are either too
slow for a random search (atomic force microscopy), lack clear signatures
(transmission electron microscopy) or are invisible under most circumstances
(optical microscopy)[78]. Only by preparing the flakes on a proper substrate,
such as a 300nm thick SiO2 , they become visible in an optical microscope due
to interference[40]. Mechanical extraction of graphene from graphite can produce micrometer sized samples which are sufficient for scientific purposes. For
any commercial use other techniques, such as epitaxial growth[31], need to be
further developed. Mobility measurements of graphene show the extraordinary qualities of its crystal structure. Room temperature mobilities µ of the
order 10,000 cm2 V−1 s−1 [79] has been reported and experiments on suspended
graphene at low temperatures reported a peak mobility at 230,000cm2 V−1 s−1
[17]. The mobilities remain relatively high even when the graphene sheet is
doped[104].
Another intriguing property of graphene is the presence of a minimum
conductivity σmin at the Dirac point. Instead of a metal to insulator transition as the charge density decreases the conductivity stabilizes at σmin ∼
4e2 /h[40]. Although the effect isn’t unexpected for systems governed by the
Dirac equation[37, 54], the theoretically expected value is smaller by a factor
of π, σmin,theor = 4e2 /πh. Occasional measurements approaches the theoretical limit but it is still unclear what physics determines σmin . It has been
suggested that measured σmin depends on the width/length ratio of the measured samples[20, 113] and that wide and short ribbons would give σmin closer
to the theoretical value. Another suggestion is that at low electron density,
2.2. Graphene
11
ky
M K
b2
b1
Γ
kx
E[eV]
5
Dirac point
0
-5
M
(a) Brillouin zone
Γ
Wave vector
K
M
(b) Tight-binding dispersion relation
Figure 2.8: (a) Shaded area shows the Brillouin zone in the reciprocal lattice. b1 , b2
are the reciprocal lattice vectors. (b), Tight-binding dispersion relation
for ǫ2p = γ0 =0, γ1 =-2.7eV along contour in (a). Fermi level is at 0 eV
charges in graphene moves in a poodle like electron-hole landscape where the
underlying physics becomes more complicated than expected[121].
2.2.2
Basic properties: theory
Graphene is a single layer of carbon atoms in a honeycomb-lattice, figure 2.7.
Most of the “exotic” properties of graphene derive from its hexagonal lattice
which results in a linear dispersion relation close to the fermi level instead
of the parabolic dispersion typical for solid state systems. The lattice can
be described as two triangular sub-lattices, figure 2.7, where the interatomic
distance acc roughly equals 1.42Å. Using the tight-binding approximation the
dispersion relation can be computed from the secular equation[48, 98, 102, 117]
HAA − ESAA HAB − ESAB (2.17)
HBA − ESBA HBB − ESBB = 0
Here, HAA = hΦA |H|ΦA i where Φj (k, r) is the Bloch function for site j = A, B,
X
1
Φj √ =
eik·R φj (r − R)
N
R
(2.18)
with φj being the wave function localized at site j and R the position of the
lattice points. Similarly, Sjj ′ = hΦj |Φj ′ i is the overlap matrix and E the
eigenenergies. The solution of eq. (2.17) is of the form[48, 98, 102]
E(k) =
ǫ2p ± γ1 |g(k)|
1 ± γ0 |g(k)|
(2.19)
12
Mesoscopic physics
Figure 2.9: (a) Example of graphene nano ribbon with zigzag edge (Z-GNR). The
GNR extends towards infinity in the x-direction. The width of the Z-GNR
in the y-direction is characterized by the number of zigzag chains, Nz ,
indicated as solid dots across the GNR. Here Nz =6. (b)-(c) Tight-binding
dispersion relation and transmission through a Z-GNR with (b) Nz =7
and (c) Nz =8 transversal sites. Within the tight-binding approximation
Z-GNR:s are metallic for all Nz . (d) Example of graphene nano ribbon
with armchair edge (A-GNR). The width of A-GNR:s is characterized by
the number of dimer lines, Na , indicated as solid dots across the GNR.
Here Na =9. (e)-(f ) Tight-binding dispersion relation and transmission
through an A-GNR with (e) Na =7 and (f) Na =8 transversal sites. AGNR:s are metallic only if Na =3p + 1, p being an integer.
where[1]
v
u
u
|g(k)| = t1 + 4 cos
!
√
3ky a
kx a
kx a
cos
+ 4 cos2
2
2
2
(2.20)
√
and a = 3acc ∼ 2.46nm being the length of the reciprocal lattice vector.
Fitting of the parameters ǫ2p , γ0 and γ1 around the Dirac point, i.e., the
2.2. Graphene
13
K-point in figure 2.8(a),(b)) gives[98]
e2p = 0
−2.5eV < γ1 < −3.0eV
(2.21)
γ0 < 0.1eV.
γ0 is related to the the overlap between nearest neighbours, hφA |φB i, and
is sometimes set to zero to simplify the model further. Figure 2.8(b) shows
the tight-binding dispersion relation along the contour in the Brillouin zone
in fig. 2.8(a). The tight-binding binding approximation generally deviates
significantly from ab initio calculations away from the K-points[98]. Close to
the K-point the dispersion relation is linear as for relativistic particles with
an effective “light” velocity of vF ∼ c/300[86].
Graphene nano ribbons
For application purposes of graphene both theoretical and experimental focus
has been on graphene nano ribbons (GNR:s), i.e., stripes of graphene cut out
of larger sheets (figure 2.9(a),(d)). Hopefully GNR:s will allow the extraordinary properties of graphene, such as high mobility at ambient temperature
and high degree of doping, to be used in conjunction with existing semiconductor components. Graphene nano ribbons has been lithographically patterned
down to widths of ∼20nm[20, 47] and chemically grown with a width less than
10nm[66]. Modelling of GNR:s indicate that their electronic properties are
determined by the width of the wire and the edge structure[39, 98]. Typically,
two kinds of edge structures are considered; zigzag edge graphene nano ribbons (Z-GNR), as in figure 2.9(a), and armchair edge graphene nano ribbons
(A-GNR), as in figure 2.9(d). The width of the ribbon is characterised by
a number Nz (Na ) for Z-GNR (A-GNR) which is defined by the number of
sites across the ribbon along to the paths given by solid dots figure 2.9(a),(d).
Within the tight-binding approximation Z-GNR:s are metallic for all widths,
e.g., figure 2.9(e),(f). For A-GNR only certain widths, Na =3p+1, p being an
integer are metallic (c.f., figure 2.9(e) and (f)). In contrast, Ab initio DFT
calculations of GNR:s yields a band gap for all Z-GNR and A-GNR[109]. The
difference between tight-binding and DFT is due to effects at the edge of
the ribbon, and the discrepancy decreases with increasing ribbon width. The
overall trend of the band gaps is a 1/W dependence, figure 2.10. Because
graphene itself is a gapless semiconductor, very narrow GNR:s has been proposed as a way to engineer a band gap in graphene components. Conductance
measurements of GNR:s show a 1/W scaling of the band gap but surprisingly
no dependence on the crystal direction of the GNR[47]. Although there is
no consensus on the lack of crystal dependence in conductance measurement
yet, possible explanations includes rough edges[20, 47, 66, 96], atomic scale
impurities[20], coulomb blockaded transport[108].
14
Mesoscopic physics
50
Energy gap (units of “t”)
0.5
Width [Na ]
100
150
200
0.4
0.3
0.2
0.1
0
0
5
10
15
Width [nm]
20
25
Figure 2.10: Energy gap of armchair GNR versus ribbon width within the tightbinding approximation. Metallic A-GNR:s, with Na =3p+1 where p ∈ N,
are excluded.
Chapter 3
Transport in mesoscopic
systems
Understanding transport is central to the study of mesoscopic systems. Often transport characteristics work as a flexible tool to probe the electron
states inside a system – well-known examples are the Kondo effect[57] and
the 0.7-anomaly[112]. This chapter introduces some basics of ballistic transport in mesoscopic systems, that is, systems where electrons pass through
the conductor without scattering and remain phase coherent. In brief, we will
start by deriving the Landauer formula which describes transport for a system
with only two leads connected. This is then generalised within the Büttiker
formalism to handle an arbitrary number of leads. A key characteristic for
the quantum conductance in these expressions is the transmission probability
Tn,β←m,α , i.e., the probability for an electron in mode α and lead m to end
up in another mode β and lead n. Finally we will look at effects on transport
when a perpendicular magnetic field is applied.
3.1
Landauer formula
3.1.1
Propagating modes
For simplicity we start with the case of a single propagating mode (see figure
2.6a), i.e., where only the lowest subband in both the left and right lead is
occupied. Figure 3.1 shows electrons surrounding a barrier in a 1-dimensional
system with some bias eV applied. Though in principle, the Fermi energy is
only defined at equilibrium, we will assume this bias to be small enough to
yield a near-equilibrium electron distribution that is characterised by a quasiFermi level, EF l and EF r respectively. To find an explicit expression for the
net current across the barrier due to the bias we first consider the electrons
approaching the barrier from the left. In an infinitesimal momentum interval
16
Transport in mesoscopic systems
EF l
eV
Ul
EF r
Ur
Figure 3.1: Schematic view of a 1D barrier surrounded by electrons. A small bias eV
shifts the (quasi-)Fermi levels in left (EF l ) and right (EF r ) lead.
dk around k, the transmitted current is
Il (k)dk = 2en1D (k)v(k)Tr←l (k)f E(k), EF l dk
(3.1)
where a factor 2 is added to include spin, e the electron charge, n1D (k) is the
one-dimensional density of states, v(k) the velocity of the electrons,
Tr←l (k)
the probability that an electron passes the barrier and f E(k), EF l the FermiDirac distribution. Using eq. (2.10) for the 1D-DOS and integrating both sides
yields
Il =
Z
∞
2e
0
/
/
dk
1
v(k)Tr←l (k)f E(k), EF l dk = dk =
dE
2π
dE
Z ∞
dk
2e
v(E)Tr←l (E)f E, EF l
dE. (3.2)
=
2π 0
dE
Recognising the group velocity as v =
bottom of the left lead brings
Il =
2e
2π
Z
∞
v(E)Tr←l (E)f E, EF l
Ul
dω
dk
=
1 dE
~ dk
and integrating from the
1
dE
~v(E)
Z
2e ∞
=
Tr←l (E)f E, EF l dE. (3.3)
h Ul
There is, except for the different Fermi level and opposite direction of flow, a
similar current Ir from the right to left lead. If the bias is small, the reciprocity
relation Tl←r (E)=Tr←l (E) holds (see e.g. [23]), and we can skip the indices
on the transmission coefficient. The net current becomes
Z
h
i
2e ∞
I = Il + Ir =
T (E) f E, EF l − f E, EF r dE.
(3.4)
h Ul
In the case of very low bias eV , i.e., in the linear response regime, the FermiDirac functions in eq. (3.4) can be Taylor expanded around EF = 21 (EF l + EF r )
according to
1
1
f (E, EF l ) − f (E, EF l ) = f (E, EF + eV ) − f (E, EF − eV )
2
2
3.2. Büttiker formalism
∂f (E, EF )
∂f (E, EF )
+ (eV )2 ≈ −eV
, (3.5)
∂EF
∂E
= eV
resulting in
I=
⇔
17
2e
h
Z
∞
Ul
∂f (E, EF )
T (E) −eV
dE
∂E
Z
I
2e2
G= =
V
h
∞
Ul
∂f (E, EF )
T (E) −
∂E
(3.6)
dE.
(3.7)
∂f
is replaced by
At very low temperatures in the linear response regime, − ∂E
the Dirac delta function δ(E − EF ) and evaluating the integral in eq. (3.7)
gives the famous Landauer formula for quantum conductance
G=
2e2
I
=
T (EF ).
V
h
(3.8)
The result in eq. (3.8) above is readily extended to the case of several
propagating modes in the leads. An electron incoming towards a scatterer in
a specific mode α might be transmitted to some mode β in the opposite lead
or reflected to some mode β in the same lead. By summing over all possible
modes the total conductance is found as
G=
2e2 X
Tβ←α .
h
(3.9)
α,β
3.2
Büttiker formalism
Büttiker formalism describes transport in systems with more than two leads.
A typical example is the three lead system where a net current flows between
two leads (current probes), and the third lead is used to measure the potential
(voltage probe). We will assume fairly low temperature and bias. In this limit
the Fermi-Dirac functions in eq. (3.4) may be replaced by step functions,
2e
I=
h
Z
∞
Ul
h
i
T (E) Θ EF l − E − Θ EF r − E dE
=
2e
h
Z
EF l
T (E)dE. (3.10)
EF r
With low bias we assume T (E) to be constant in the interval EF r < E < EF l ,
thus the integral in eq. (3.10) evaluates to
I=
2e
T × (EF l − EF r ).
h
(3.11)
18
Transport in mesoscopic systems
We now consider a system connected to an arbitrary number of leads (indexed
by m and n) and propagating modes (α and β). Im denotes the total net
current in lead m. It is the sum of the net incident current
P Imm (Iincoming −
Iref lected ) and the current injected from all other leads, n6=m Im←n . With
the lowest quasi-Fermi level in all leads denoted E0 we can write
2e X X
Im←n = −
Tm,α←n,β × (En − E0 ).
(3.12)
h
n6=m α,β
The minus sign indicates a current away from the system and Tm,α←n,β is
the transmission probability from lead n–mode β to lead m–mode α. We will
abbreviate this by the notation
X
(3.13)
T̄m←n =
Tm,α←n,β ,
α,β
and in a similar way
R̄m =
X
(3.14)
Rm,α←m,β
α,β
for the reflection Rm,α←m,β from from mode β to α in lead m. Lead m carries
Nm propagating modes hence the net incident current in lead m is
2e
Imm =
Nm − R̄m (Em − E0 ).
(3.15)
h
Furthermore, conservation of flux requires that
X
X
Nm = R̄m +
T̄n←m = R̄m +
T̄m←n
(3.16)
m6=n
n6=m
where the last equality follows from the sum rule derived in eq. (3.30)–(3.35).
The total current in lead m can be written as
2e X
2e
Im = Imm + Im←n =
Nm − R̄m (Em − E0 ) −
T̄m←n (En − E0 )
h
h
n6=m
(3.17)
X
X
2e
2e
=
T̄m←n (Em − E0 ) −
T̄m←n (En − E0 )
h
h
n6=m
n6=m
(3.18)
2e X
=
T̄m←n Em − T̄m←n En ,
h
(3.19)
n6=m
where eq. (3.16) has been used in the second step. Noting that Em = eVm we
finally get
2e2 X
T̄m←n (Vm − Vn ) .
(3.20)
Im =
h
n6=m
The currents in a multi-lead system are then found from solving the system
of equations defined by equation (3.20).
3.3. Matching wave functions
19
Ce−ik2 x
Aeik1 x
Deik2 x
Be−ik1 x
V0
x=0
Figure 3.2: Incoming and outgoing electrons at a potential step.
3.3
Matching wave functions
We now turn to a simple example for computing the transmission probability T
across a potential step, see figure 3.2. k1 and k2 is the wave vector for incident
and transmitted electrons, respectively. The solution to the Schrödinger eq.
for electrons with energy E > V0 is
(
Aeik1 x + Be−ik1 x x < 0
Ψ(x) =
(3.21)
Ce−ik2 x + Deik2 x x > 0.
The requirement that the wave function and its derivative is continuous at
x = 0 yields the system of equations
A+B = C +D
(3.22)
k1 (A − B) = k2 (D − C).
(3.23)
Solving for the outgoing amplitudes, B and D, results in
1
B
k1 − k2
2k1
A
=
.
D
2k1
k2 − k1
C
k1 + k2
(3.24)
E.g., for a free electron incoming from the left with unit amplitude (A = 1,
C = 0), the transmitted amplitude (D = t2←1 ) becomes
t2←1 =
2k1
,
k1 + k2
(3.25)
and the reflected amplitude (B = r1←1 )
r1←1 =
k1 − k2
.
k1 + k2
(3.26)
Because the velocities in the left and right lead may differ, the flux transmission
coefficient becomes
v2 (k2 )
|t2←1 |2 ,
(3.27)
T2←1 (k1 , k2 ) =
v1 (k1 )
where v2 and v1 are the group velocities for the electron.
20
3.3.1
Transport in mesoscopic systems
S-matrix formalism
By generalizing the notation in the preceding section we can now prove the conservation of flux used in eq. (3.16). The matrix equation (3.24) above relates
the amplitude of the outgoing electrons to the amplitude of the incoming electrons. The current into the system for mode i is given by Ii = −vi (k)|Ai (k)|2
where vi (k) is the group velocity and Ai (k) the amplitude. It is convenient to
define the current amplitude, by
p
ai (k) =Ai (k) −v(k)
(current amplitude for incoming mode i)
p
bi (k) =Bi (k) v(k)
(current amplitude of outgoing mode i),
such that the current carried by mode i simply is Ii = |ai (k)|2 . We now
generalise eq. (3.24) to handle an arbitrary system by defining the S-matrix
(scattering matrix)[22],
b̄ = Sā,
(3.28)
which, for each energy E, relates the incident current amplitudes a = (a1 , a2 ,
. . . , an ) to the outgoing current amplitudes b = (b1 , b2 , . . . , bn ). The transmission probability now equals
Tm←n (E) = |smn |2 ,
(3.29)
where smn is a matrix element in the S-matrix. Now, current conservation
requires that
X
X
Iin = e
|ai |2 = e
|bi |2 = Iout
(3.30)
i
i
or in matrix notation
†
ā† ā = b̄ b̄
⇔
(3.31)
†
†
ā ā = (Sā) Sā
(3.32)
⇔
ā† ā = ā† S† Sā
(3.33)
⇒
S† S = I = SS† ,
(3.34)
eq. (3.28)
i.e., S is unitary and we arrive at
X
X
X
X
Tm←n =
|smn |2 = 1 =
|snm |2 =
Tn←m .
m
3.4
m
m
(3.35)
m
Magnetic fields
Magnetic field is usually incorporated into the Hamiltonian through a vector
potential A defined by the equation
B=∇×A
(3.36)
3.4. Magnetic fields
21
n(E)
(b)
(a)
(c)
1
2 ~ωc
3
2 ~ωc
Energy
5
2 ~ωc
Figure 3.3: (a)–dashed line: 2D density of states with B=0T. (b)–solid line: Ideal
2D density of states in a magnetic field. (c)–dotted line: Broadened 2D
density of states in a magnetic field.
as
1
(3.37)
(−i~∇ + eA)2 + V.
2m
Since ∇ × ∇ξ = 0 for any function ξ, there is a possibility to choose different
gauges, A→A+∇ξ, without changing the physical properties of the system.
Some common choices for a magnetic field Bẑ is Landau gauge A=(−By,0,0)
or A=(0,Bx,0), and the circular symmetric gauge A= 12 (−By, Bx,0). We will
now consider A=(0,Bx,0), and the Schrödinger equation for a 2DEG becomes
H=
HΨ =
~ 2
ie~Bx ∂
(eBx)2
−
∇x,y −
+
Ψ(x, y) = EΨ(x, y).
2m
m ∂y
2m
(3.38)
Trying a solution in the form Ψ(x, y) = u(x)eiky we end up with an equation
only in x,
2 !
~2 d2
mωc2 ~k
−
+
+x
u(x) = ǫu(x)
(3.39)
2m dx2
2
eB
with ωc2 = eB
m being the cyclotron frequency. This is the Schrödinger equation
for a harmonic oscillator so we can write down the eigenenergies as
1
ǫn,k = (n − )~ωc ,
2
n = 1, 2, 3, . . .
(3.40)
The energies in eq. (3.40) are independent of k, hence the density of states –
which for zero magnetic field is constant (eq. (2.9) or fig. 3.3) – now collapses
to very narrow stripes, equidistantly located as shown in figure 3.3. These
collapsed states are called Landau levels and contain all the degenerate states
with the same k. In an actual system they are broadened due to scattering.
Next we consider a hard wall potential electron wave-guide of the width
L, oriented along the y-axis. Trying the same solution as above, Ψ(x, y) =
22
Transport in mesoscopic systems
V (x)
10
(a)
20
(b)
(c)
15
5
10
5
0
-50
0
nm
0
50 -50
0
nm
50
Figure 3.4: (a): Potentials (black) and wave functions (gray) for B=1T (solid lines)
and B = 3T (dashed lines) in a quantum wire, k=0. (b): Potentials (black) and wave functions (gray) for k=0.05nm−1 (solid lines) and
k=0.25nm−1 (dashed lines) in a quantum wire, B = 3T . (c): Classical
skipping orbits in a straight quantum wire.
u(x)eiky , we arrive at
mωc2
~2 d2
+
−
2m dx2
2
~k
+x
eB
2
!
+ V (x) u(x) = ǫu(x)
(3.41)
where the only difference from eq. (3.39) is the confining potential V (x),
(
0
|x| ≤ L2
V (x) =
.
(3.42)
∞ |x| > L2
This equation do not have an analytical solution but we may still draw some
qualitative conclusions from eq. (3.41). The electrons are confined by the hard
wall potential V (x) and the parabolic potential which depends on k and B.
With k=0, the parabolic potential is determined only by the magnetic field
B (through wc ), and an increasing magnetic field will steepen the potential
thereby confining the electrons to the middle of the wire as shown in the leftmost panel of figure 3.4. For |k| >0 the vertex of the parabola is displaced, and
for large |k|’s the wave function is squeezed against the hard wall confinement
(middle panel of figure 3.4). Hence, currents in this system are carried along
the edges of the wire by these “edge states”. The classic analogue to these
states is the so called skipping orbits shown in the rightmost panel of the same
figure. Due to the Lorentz force, an electron in a perpendicular magnetic field
moves in a circle and do not contribute to any net current. However, along
the edges of the wire electrons will bounce off the walls, resulting in a current
in both directions.
Chapter 4
Electron-electron interactions
Some years after the Schrödinger equation and the basics of quantum mechanics had been formulated, Paul Dirac commented that[26]
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus
completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be
soluble.
The results from quantum mechanics were impressive already by 1929, but
there was also an awareness of the huge computational effort needed to solve
some (most) problems. The difficulties comes from electron-electron interactions (e-e interactions) which poses a intractable many-body problem under
most circumstances. Today, a wide range of approximations exists to handle
e-e interactions. This chapter introduces and discusses the methods dealing with e-e interactions relevant for the thesis. First we take a look at the
Hubbard model, where e-e interactions are considered local, i.e., restricted
to the interaction between electrons on the same atom or grid point. Next
we introduce the Thomas-Fermi (TF) model as a precursor to Density Functional Theory (DFT). Both TF and DFT replace the many-body wave function
Ψ(r1 , r2 , . . .) from the Schrödinger equation with the far simpler electron density, n(r). DFT, or any of its extensions, is today one of the most effective
theory for e-e interactions around. It has both a solid theoretical foundation
and can in many cases give excellent predictions on electronic properties.
4.0.1
What’s the problem?
The properties of a stationary N -electron system can be found by solving the
time-independent Schrödinger equation


2
X ~2
X
1
e
ĤΨ = −
∇2 +
+ vext (r) Ψ = EΨ.
(4.1)
2m i
2 ′ |ri − ri′ |
i
i6=i
24
Electron-electron interactions
Ψ=Ψ(r1 , r2 , . . . , rN ) is the N -electron wave function in three dimensions, the
terms in the brackets of eq. (4.1) are the kinetic energy for the i:th electron,
the e-e interaction and finally the external potential vext (r). Because of the
e-e interaction the coordinates r1 , r2 , . . . , rN are coupled and a direct solution
for increasing N is a very difficult many-body problem.
4.1
The Hubbard model
The approach taken by John Hubbard in 1963 was to ignore inter-atomic e-e
interactions completely. In the model, electrons are only interacting with electrons situated on the same atom (or site) in the system. This approximation
is best suited for systems where the electron density is concentrated near the
atom nuclei and sparse between the atoms; for example the d-band of transition metals initially considered by Hubbard. For a brief derivation we will
follow the same procedure as Hubbard[50], and start with Bloch functions Ψk
and energies ǫk calculated in an appropriate spin independent Hartree-Fock
potential. The Hamiltonian can then be approximated by
H=
X
ǫk c†kσ ckσ
kσ
+
1
2
X
X
k1 k2 k′1 k′2 σ1 σ2
−
hk1 k2 |1/r|k′1 k′2 ic†k1 σ1 c†k2 σ2 ck′ σ2 ck′ σ1
2
1
XX
2hkk′ |1/r|kk′ i − hkk′ |1/r|k′ ki νk′ c†kσ ckσ , (4.2)
kk′
σ
where the momentum sum is over the first Brillouin zone, c†kσ and ckσ are the
creation and annihilation operators for the Bloch state k with spin σ (=±1),
the e-e interaction terms are
Z ψ ∗ (x)ψ ′ (x)ψ ∗ (x′ )ψ ′ (x′ )
k1
k2
k1
k2
hk1 k2 |1/r|k′1 k′2 i = e2
dxdx′
(4.3)
′
|x − x |
and νk are the occupation numbers. The first term in eq. (4.2) represents the
band energies of the electrons, the second term their interaction and the third
term counters double counting. Next the Wannier functions are introduced as
a new basis set,
1 X
φ(x) = √
ψk (x),
(4.4)
N k
where N is the number of electrons. The Bloch wave function and creation/annihilation operators can be expressed with the Wannier functions as,
1 X ikRj
e
φ(x − Rj )
ψk (x) = √
N j
(4.5)
4.1. The Hubbard model
and
25
1 X ikRj
ckσ = √
e
cjσ ,
N j
1 X ikRj †
c†kσ = √
e
ciσ ,
N j
(4.6)
Ri being coordinates for lattice site i. The Hamiltonian (eq. 4.2) becomes
H=
XX
i,j
Tij c†iσ cjσ
σ
+
1 XX
hij|1/r|klic†iσ c†jσ′ clσ′ ckσ
2
ijkl σσ′
XX
−
2hij|1/r|kli − hij|1/r|lki νjl c†iσ c†kσ , (4.7)
σ
ijkl
where
Tij = N −1
X
ǫk eik(Ri −Rj ) ,
(4.8)
k
2
hij|1/r|kli = e
Z
φ∗ (x − Ri )φ (x − Rk )φ∗ (x′ − Rj )φ (x′ − Rl )
dxdx′ ,
|x − x′ |
(4.9)
and
νjl = N −1
X
νk eik(Rj −Rl ) .
(4.10)
k
Obviously the main contribution to the e-e interaction comes from the term
hii|1/r|iii = U and Hubbard subsequently suggested to ignore the contribution
from all other terms. The simplified Hubbard Hamiltonian is then,
H=
XX
i,j
σ
X
1 X
Tij c†iσ cjσ + U
niσ ni,−σ − U
νii niσ
2
i,σ
(4.11)
i,σ
where niσ = c†iσ ciσ . A few more simplifications are close at hand, the last term
of eq. (4.11) can be written as
−U
X
i,σ
νii niσ = −U
X
X N −1
νk niσ
i,σ
k
= −U
X
i,σ
1 1
N −1 n niσ = − U N n2 (4.12)
2
2
which is constant and may be ignored. Furthermore, the second term in
eq. (4.11) is
X
1 X
1 X
U
niσ ni,−σ = U
(ni↑ ni↓ + ni↓ ni↑ ) = U
ni↑ ni↓ .
2
2
i,σ
i
i
(4.13)
26
Electron-electron interactions
Finally, by considering only nearest neighbour hopping, i.e. hopping from site
i to site i + ∆, and denoting Tii = E0 , Ti,i+∆ = t we arrive at a common
formulation of the Hubbard Hamiltonian,
X
X †
X
H = E0
niσ + t
ciσ cjσ + U
ni↑ n↓ .
(4.14)
iσ
(i6=j),σ
i
The Hubbard model has been described as a ‘highly oversimplified model’
[5]. The assumption that e-e interactions are local is clearly not satisfactory
for the general case. By Hubbard’s own estimate on 3d transitions metals
the e-e interaction in eq. (4.11) are of the order 10-20 eV while the biggest
neglected term is 2-3 eV. Nevertheless, the model is by no means simple; exact
solutions are only known in one dimension[67] or for two extreme cases
• The ‘Band limit’, U =0 in eq. (4.11)
• The ‘Atomic limit’, t=0 in eq. (4.11)
The model is also sophisticated enough to reproduce a rich variety of phenomena seen in solid state physics such as, metal-insulator transition, antiferromagnetism, ferromagnetism, Tomonaga-Luttinger liquid and superconductivity [111].
4.2
The variational principle
If we are only interested in the ground state of the system, an alternative approach to eq. (4.1) is available through the Rayleigh-Ritz variational method[18].
A trial wave function φ is introduced and an upper bound to the ground state
energy E0 is given by
E = E[φ] =
hφ|Ĥ|φi
≥ E0 .
hφ|φi
(4.15)
Equality holds when φ equals the true ground state wave function (Ψ0 ). In
practice, a number of parameters pi may be introduced in the trial wave function and the ground state energy and wave functions can be found by minimising E = E(p1 , p2 , . . . , pm ) over the parameters pi . The accuracy of the
method depends on how closely the trial wave function φ resembles the actual wave function – the more parameters introduced, the better the result.
However, as pointed out in for example[56], a rough estimate of the number
of parameters needed for anything but very small systems is discouraging. If
we introduce three parameters per spatial variable and consider a system of
N =100 electrons, the total number of parameters (Mp ) over which to perform
the minimisation becomes
Mp = 33N = 3300 ∼ 10143 !
(4.16)
4.3. Thomas-Fermi model
27
Clearly this is not a feasible problem and is sometimes referred to as the
exponential wall, since the size of the problem grows exponentially with the
number of electrons.
4.3
Thomas-Fermi model
Equation (4.1) may be reformulated using the variational principle[85]
δE[Ψ] = 0,
(4.17)
i.e., solutions Ψ to the Schrödinger equation occurs at the extremum for the
functional E[Ψ]. In order to have the wave function normalised, hΨ|Ψi = N ,
a Lagrange multiplier E is introduced and we arrive at
δ hΨ|Ĥ|Ψi − E (hΨ|Ψi − N ) = 0.
(4.18)
This is, however, only a rephrasing of the original Schrödinger equation (with
the constraint hΨ|Ψi = N ) and by no means easier to solve. The idea in the
Thomas-Fermi (TF) model is to assume that the ground state density,
Z
(4.19)
n0 (r) = Ψ0 (r, r2 , . . . , rN )Ψ∗0 (r, r2 , . . . , rN )dr2 dr3 . . . drN
minimises the energy functional
E[n] = T [n] + Uint [n] + Vext [n],
(4.20)
and thereby replace the wave function Ψ(r1 , r2 , . . . , rN ) with the significantly
simpler density n(r). The first term T [n] is the kinetic energy functional, in the
original TF-model this was approximated by the expression for a uniform gas
of non-interacting electrons. For the 2DEG we may find a similar expression
from the 2D-density of states, n2D (ǫ)=m/(~2 π), given in eq. (2.9). The energy
over some area L2 is
Z ǫF
m ǫ2
∆E =
ǫn2D (ǫ)L2 dǫ = 2 F L2 ,
(4.21)
~ π 2
0
where ǫF is the Fermi energy. At the same time the number of electrons in
the area L2 is
Z ǫF
m
N=
n2D (ǫ)L2 dǫ = 2 ǫF L2 .
(4.22)
~
0
Thus,
2
1 ~2 π
mL2
~2 π 2
m ǫ2
N
=
n ,
(4.23)
∆E = 2 F L2 =
2
2
~ π 2
2 mL
~ π
2m
n being the electron density. The kinetic energy functional for a non-uniform
2DEG is now approximated by
Z
~2 π
T [n] ≈ TT F [n] =
n(r)2 dr
(4.24)
2m
28
Electron-electron interactions
where the integration is carried out over two dimensions. The second term in
eq. (4.20) is the e-e interaction functional and was originally approximated by
the classical Hartree energy,
Z Z
n(r1 )n(r2 )
e2
Uint [n] ≈ UH [n] =
dr1 dr2 .
(4.25)
2
|r1 − r2 |
Finally, the last term is the energy due to interaction with some external
potential vext (r)
Z
Vext [n] = e n(r)vext (r)dr.
(4.26)
With the constraint
Z
n(r)dr = N
(4.27)
included through a Lagrange multiplier µ, which may be identified as the
chemical potential, we arrive at
Z
δ E[n] − µ e n(r)dr − N
=0
(4.28)
and consequently get the Euler-Lagrange equation
Z
~2 π
n(r1 )
δE[n]
=
n(r) + e2
dr1 + evext (r),
µ=
δn
m
|r − r1 |
(4.29)
which is the working equation in the Thomas-Fermi model.
4.4
Hohenberg-Kohn theorems
In 1964 Walter Kohn and Pierre Hohenberg proved two theorems[49], essential
to any electronic state theory based on the electron density. The first theorem
justifies the use of the electron density n(r) as a basic variable as it uniquely
defines an external potential and a wave function. The second theorem confirms the use of the energy variational principle for these densities, i.e., for a
trial density ñ(r) > 0, with the condition (4.27) fulfilled and E[n] defined in
eq. (4.20), it is true that
E0 ≤ E[ñ].
(4.30)
The derivations below assume non-degeneracy but can be generalised to include degenerate cases[85].
4.4.1
The first HK-theorem
For an N electron system the Hamiltonian (and thereby the wave function Ψ)
in eq. (4.1) is completely determined by the external potential vext (r). N is
directly obtained from the density through
Z
N = n(r)dr,
(4.31)
4.4. Hohenberg-Kohn theorems
29
while the unique mapping between densities and external potentials are shown
through a proof by contradiction. First assume there exists two different ex′ (r)1 , and thereby two different wave functions
ternal potentials vext (r) and vext
′
Ψ and Ψ , which yield the same electron density n(r). Using eq. (4.15) we can
write
hΨ|Ĥ|Ψi = E0 < hΨ′ |Ĥ|Ψ′ i = hΨ′ |Ĥ ′ |Ψ′ i + hΨ′ |Ĥ − Ĥ ′ |Ψ′ i
Z
′
(r) dr, (4.32)
= E0′ − n(r) vext (r) − vext
and at the same time
hΨ′ |Ĥ ′ |Ψ′ i = E0′ < hΨ|Ĥ ′ |Ψi = hΨ|Ĥ|Ψi + hΨ|Ĥ ′ − Ĥ|Ψi
Z
′
= E0 − n(r) vext (r) − vext
(r) dr, (4.33)
Adding equation (4.32) and (4.33) gives
E0′ + E0 < E0′ + E0 ,
(4.34)
and our assumption that different vext (r) could yield the same density n(r) is
incorrect.
4.4.2
The second HK-theorem
From the first theorem we have that a density n(r) uniquely determines a wave
function Ψ, hence we can we define the functional
F [n(r)] = hΨ|T̂ + Ûint |Ψi,
(4.35)
where T̂ and Ûint signify the kinetic energy and e-e interaction operator. According to the variational principle, the energy functional
Z
E[Ψ] = hΨ| (T + Uint ) |Ψi + Ψ∗ vext (r)Ψdr
(4.36)
has a minimum for the ground state Ψ=Ψ0 . For any other Ψ=Ψ̃
Z
Z
E[Ψ̃] = F [ñ] + vext (r)ñ(r)dr ≥ F [n0 ] + vext (r)n0 (r)dr = E[Ψ0 ] (4.37)
|
{z
} |
{z
}
=E[ñ]
=E[n0 ]
and we arrive at (4.30).
The Hohenberg-Kohn theorems do not help us solve any specific manybody electron problem, however, they do show that there is no principal error
1
′
differing more than vext (r) − vext
(r)=constant
30
Electron-electron interactions
in the Thomas-Fermi approach – only an error due to the approximations
done for T [n] and Uint [n]. With an exact expression for the functional F [n] =
T [n] + Uint [n] we could solve our problem exactly. Furthermore, since there is
no reference to the external potential in F [n], knowing this functional would
allow us to solve any system. For this reason F [n] is referred to as a universal
functional.
4.5
The Kohn-Sham equations
Slightly over a year after the HK-theorems were published Walter Kohn and
Lu Jeu Sham derived a set of equations, the Kohn-Sham equations, that made
density functional calculations feasable[55]. They started by considering a
system of non-interacting electrons moving in some effective potential vef f (r),
which will be defined later. Because the system is non-interacting the ground
state density n(r) can be found by solving the single particle equation
~2 2
∇ + vef f (r) ϕi (r) = εi ϕi (r),
(4.38)
−
2m
and summing
n(r) =
X
i
|ϕi (r)|2 .
(4.39)
The trick applied by Kohn and Sham was to compare the Euler-Lagrange
equation for this non-interacting system with the one in an interacting system.
Using the index s on the kinetic energy functional Ts [n] to remind us that it
refers to the non-interacting (single-electron) system, the energy functional
E[n] equals
Z
E[n] = Ts [n] + Vef f [n] = Ts [n] + n(r)vef f (r)dr,
(4.40)
which, with the constraint from eq. (4.27) included by a Lagrange multiplier
ε, yields the Euler-Lagrange equation
δE[n] δTs [n] δVef f [n]
=
+
−ε
δn
δn
δn
δTs [n]
=
+ vef f (r) − ε = 0.
δn
(4.41)
Meanwhile, the energy functional for the interacting system can, with some
deliberate rearrangements, be written as
E[n] = Ts [n] + UH [n] + Exc [n] + Eext [n],
(4.42)
where UH [n] is defined in eq. (4.25) and Exc [n] are the corrections needed to
make E[n] exact,
Exc [n] = T [n] − Ts [n] + Uint [n] − UH [n].
(4.43)
4.6. Local Density Approximation
31
Once again applying the variational principle with the constraint (4.27), gives
δE[n]
δTs [n] δUH [n] δExc δVext[n]
=
+
+
+
−ε
δn
δn
δn
δn
δn
Z
δTs [n]
n(r1 )
+ e2
dr1 + vxc (r) + vext (r) − ε = 0.
=
δn
|r − r1 |
(4.44)
Comparing eq. (4.41) and eq. (4.44), we realise they are identical if we choose
Z
n(r1 )
vef f (r) =
(4.45)
dr1 + vxc (r) + vext (r),
|r − r1 |
which also was the purpose with the reshuffling made in eq. (4.42). The KohnSham procedure can now be summarised in four steps,
I)
II)
initially guess a density n0 (r)
compute vef f (r) through eq. (4.45)
III)
solve eq. (4.38) and compute a new density n0 (r) through (4.39)
IV)
repeat from step II until n0 (r) is converged
However, the problem to find an explicit expression for the term Exc [n] in
eq. (4.43) remains. This term should include all the corrections needed to
make the energy functional E[n] exact. Part of this correction is due to the
Pauli principle2 – the classical e-e energy UH [n] obviously do not take this
into account. Part of the correction stems from the fact that the actual wave
function can not, in general, be written as some combination of the functions
φi in equation (4.38). These two contributions are often written separate,
namely as an exchange (Pauli principle) and correlation (exact wave function)
contribution,
Exc [n] = Ex [n] + Ec [n].
(4.46)
One of the more successful ways to approximate them is through the Local
Density Approximation (LDA).
4.6
Local Density Approximation
In the case of a uniform electron gas the exchange and correlation energy
per particle, εx (n) and εc (n), can be determined. In LDA it is then assumed
that the total exchange-correlation energy for any system is the sum of these
energies weighted with the local density, i.e.,
Z
(4.47)
Ex [n] = εx [n]n(r)dr,
2
No two electrons in a given system can be in states characterised by the same set of
quantum numbers.
32
Electron-electron interactions
Ec [n] =
Z
εc [n]n(r)dr.
(4.48)
An expression for εx [n] was first proposed by Dirac as an improvement to the
Thomas-Fermi model[27]. For a 2DEG[110],
√ 2
2e p
n(r),
(4.49)
εx [n] = − 3
3π 2 ε0
where e is the electron charge and ε0 the permittivity. A closed expression
for the correlation part is a little bit more troublesome. Using Monte Carlo
methods it may be computed exactly for different densities whereupon these
values are interpolated to fit some analytical expression[92, 110, 115]. E.g.,
with the density parameter rs defined as
rs =
1
√
a0 πn
and a0 being the Bohr radius, Tanatar and Cerperly proposed[110]
√
1 + C 1 rs
C0 e2
εc [n] = −
√
2 ∗ 4πε0 a0 1 + C1 rs + C2 rs + C3 (rs )3/2
(4.50)
(4.51)
where the coefficients Ci were determined from least square fits with their
Monte Carlo simulations.
4.7
Local Spin Density Approximation
Density functional theory within LDA can be extended to include electron-spin
effects using the local spin density approximation (LSDA). Equation (4.38) is
now written as two equations,
~2 2
σ
−
∇ + vef
(r)
ϕσi (r) = εi ϕσi (r),
(4.52)
f
2m
where we differentiate between the two spin species σ =↑, ↓. The spin-up/down
densities are given by
X ↑
n↑ (r) =
|ϕi (r)|2
(4.53)
i
and
n↓ (r) =
X
i
and the total density
|ϕ↓i (r)|2
n(r) = n↑ (r) + n↓ (r).
The effective potential
σ (r)
vef
f
σ
2
vef
f (r) = e
Z
(4.54)
(4.55)
is
n(r1 )
σ
dr1 + vxσ (r) + vcσ (r) + vext
(r).
|r − r1 |
(4.56)
4.8. Brief outlook for DFT
33
The exchange energy per particle now depends on both spin up and spin down,
εx [n↑ , n↓ ]. This is usually rewritten using the polarisation parameter
ζ=
n↑ − n↓
n↑ + n↓
(4.57)
as εx [n, ζ], which may be expressed in terms of the unpolarised εx [n]. For a
2DEG[110]
1
εx [n, ζ] = εx [n] (1 + ζ)3/2 + (1 − ζ)3/2 .
(4.58)
2
The exchange potential vxσ (r) is then given by the functional derivative
vxσ (r) =
where
Ex [n, ζ] =
vcσ (r) is similarly obtained from
Z
vcσ (r) =
δEx [n, ζ]
,
δnσ
(4.59)
n(r)εx [n, ζ]dr.
(4.60)
δEc [n, ζ]
,
δnσ
(4.61)
where a parametrisation for εc [n, ζ] can be found in[7].
4.8
Brief outlook for DFT
Although the local density approximation has performed remarkably well – it
gives systematic errors which often are small (7-10% error in energy)[19, 56] –
it is not sufficient for general quantum chemical computations and occasionally
fail also in solid state calculations. An example of the latter is the (in)famous
case of the ground state in iron not being magnetic within LSDA. Efforts to
improve the accuracy of DFT computations focus on finding better exchangecorrelation energy functionals. This work requires ample of imagination and
mathematical skill as the only guidance are known asymptotic behaviours and
scaling properties of the energy functionals[19, 85].
A natural extension to LSDA is the generalized gradient approximation
(GGA)[12, 63, 90, 93] which makes DFT accurate enough for quantum chemical computations[56]. GGA consider the exchange-correlation energy from
GGA is of the form
both the spin densities and their local gradients, i.e., Exc
(c.f. with LSDA in eq. (4.59)-(4.61))
Z
GGA
(4.62)
Exc [n↑ , n↓ ] = d3 rn(r)ǫGGA
xc (n↑ , n↓ , ∆n↑ , ∆n↓ ).
Some additional extensions to “traditional” DFT are
34
Electron-electron interactions
• Time-dependent DFT (TD-DFT)
“Traditional” DFT solves the time-independent Schrödinger equation. TD-DFT tries to extend DFT-techniques to systems where
the evolution over time is important.
• Hybrid functionals
A hybrid functional is made up part of the exact exchange from
Hartree-Fock theory, part of exchange-correlation from some other
theory (e.g., LDA)[13]. Hybrid functionals has been shown to improve the predictive power of DFT for a number of molecular properties such as atomisation energies, bond lengths and vibration
frequencies[91]
• LDA+U
The LDA+U uses the on-site Coulomb interaction, similar to the
Hubbard term in eq. (4.14), instead of the averaged Coulomb energy
in the energy functional[3, 4, 68]. LDA+U can be more accurate for
localised electron systems, strongly correlated materials, insulators,
etc. . .
Chapter 5
Modelling
To model electron transport in mesoscopic systems, a number of issues need
to be addressed. E.g., how to discretise analytical equations such that they
remain numerically manageable, how, or if, to include electron-electron interactions and how to define boundary conditions without introducing artifacts.
There are numerous ways to handle these issues; this chapter only cover the
techniques relevant for this thesis.
5.1
Tight-binding Hamiltonian
Tight-binding model has been one of the workhorses for problems within semiconductor physics for a long time. To start, we define the notation |m, ni as
the direct product, |mi ⊗ |ni=|mi|ni=|mni, representing a state centred at
site m, n in the discretisation of the 2D semi-infinite waveguide shown in figure
5.1. The |m, ni fulfills the relationship for completeness and orthonormality
X
|m, nihm, n| = 1 and hm, n|m′ , n′ i = δmn,m′ n′ .
(5.1)
m,n
An arbitrary state |Ψi can be expanded in this basis as
X
cmn |m, ni,
|Ψi =
(5.2)
m,n
where |cmn |2 is the probability to find the electron at site m, n. The tightbinding Hamiltonian for an electron moving in a perpendicular magnetic field
may now be written as
Xh
Ĥ =
|m, ni (ε0 + Vmn ) hm, n|
m,n
+|m, nithm, n + 1| + |m, nithm, n − 1|
i
+|m, nite−iqn hm + 1, n| + |m, niteiqn hm − 1, n| .
(5.3)
36
Modelling
n=N
y
x
n=2
n=1
m=1 m=2 m=3 m=4
Figure 5.1: Discretisation of a two-dimensional semi-infinite quantum waveguide with
transversal sites indexed from n=1 . . . N and longitudinal sites m=1 . . . ∞.
Vmn is the on-site potential, ε0 the on-site energy, t the hopping integral
between sites and the phase factor e±iqn comes from inclusion of a perpendicular magnetic field (Landau gauge) via Peierl’s substitution[33, 34]. With the
choice of
1
eBa2
2~2
t = − ε0 ,
q=
,
(5.4)
ε0 = ∗ 2 ,
m a
4
~
m∗ being the effective electron mass, e the electron charge, B the magnetic field
and a the lattice discretisation constant, eq. (5.3) converges to its continuous
counterpart as a → 0.
5.1.1
Mixed representation
Working with quantum wires, it is sometimes convenient to pass from the
real space representation with the states |m, ni to a mixed state representation using the transversal lead eigenfunctions (c.f. with the continuous case in
eq. 2.15),
r
2
πnα
ϕn =
sin
α ∈ N.
(5.5)
N +1
N +1
Denoting these states |m, αi, where m signifies the longitudinal position and
α the transverse mode we can write the transformation between |m, ni and
|m, αi as
|m, αi =
X
m′ ,n′
=
|m′ , n′ ihm′ , n′ | |m, αi
X
m′ ,n′
X
|m′ , n′ i hm′ , n′ |m, αi =
hn′ |αi|m, n′ i
n′
=
X
n′
r
2
sin
N +1
πn′ α
N +1
|m, n′ i. (5.6)
5.1. Tight-binding Hamiltonian
37
Carrying out the transformation gives (see Ref. [127] for further details on the
Hamiltonian)
H=
+
X
αβ
XX
m
α
|α, miǫα hα, m|
αβ
|α, mitβα
r hβ, m + 1| + |α, mitl hβ, m − 1| + |α, mi(Vαβ + ǫ0 )hβ, m|
(5.7)
where[16]
απ
wZ
απy 2t w
βπy
iqy
=
dye sin
sin
w 0
w
w
iqw
α+β
1 − e (−1)
1 − eiqw (−1)α−β
−
= −itqw
w2 q 2 − π 2 (α + β)2 w2 q 2 − π 2 (α − β)2
kα =
tαβ
r
αβ ∗
tαβ
l = (tr )
Z
απy βπy
2 w
dy sin
sin
Vm (y)
Vαβ =
w 0
w
w
ǫ0 = −4t
απ h2 απ 2
ǫα = 2t cos
≈
+ 2t.
w
2m∗ w
w is the width of the wire and
5.1.2
∗
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
denotes complex conjugate.
Energy dispersion relation
We will now consider the energy dispersion relation for the tight-binding
Hamiltonian in zero magnetic field. Expanding the solution |Ψi to eq. (5.3) as
X
|Ψi =
ψmn |m, ni
(5.15)
m,α
and substituting this back into eq. (5.3) yields
ε0 ψmn + tψm+1,n + tψm−1,n + tψm,n+1 + tψm,n−1 = E.
(5.16)
With the discussion of the quantum wire in section 2.1.2 in mind we assume
a separable solution on the form
ψmn = φm ϕn ,
(5.17)
with corresponding longitudinal and transverse energy components,
E = Ek + E⊥ = Ekk + Eα .
(5.18)
38
Modelling
1
Energy
~2
2m∗
Continuous lattice
Discrete lattice
0.5
0
−1
-0.5
0
kk a
0.5
1
Figure 5.2: Continuous (eq. (2.16)) and discrete (eq. (5.23)) energy dispersion relation
for the lowest subband.
After some rearrangements and writing out ε0 and t from eq. (5.4) we get
φm+1 − 2φm + φm−1
~2
= Ekk φm ,
(5.19)
− ∗
2m
a2
~2
ϕn+1 − 2ϕn + ϕn−1
− ∗
= Eα ϕn .
(5.20)
2m
a2
These are discrete derivatives, and we will get travelling wave solutions in the
parallel direction and quantised solutions in the transverse direction,
φm = eikk ma
r
2
πnα
sin
ϕn =
N +1
N +1
kk ∈ R
(5.21)
α ∈ N.
(5.22)
Substituting φm back into eq. (5.19) and ϕn back into eq. (5.22) we get the
dispersion relations,
~2
2 − 2 cos(kk a) ,
∗
2
2m a
~2 π 2 , α2
Eα =
2m∗ w2
transverse : Ekk =
longitudinal :
(5.23)
(5.24)
where w = a(N + 1) is the width
of the channel. A comparison of eq. (5.24)
with eq. (2.16) identifies απ w as k⊥ . If kk a ≪ 1, eq. (5.23) approximates to
the continuous dispersion relation (see figure 5.2),
!!
~2 kk2
(kk a)2
~2
2−2 1−
=
.
(5.25)
Ekk ≈
2m∗ a2
2
2m∗
5.2. Green’s function
39
Knowing the energy dispersion relation allows us to compute the group
velocity, vg , as
1 dE
1 d
dω
vg =
=
=
dkk
~ dkk
~ dkk
5.2
~2
2 − 2 cos(kk a)
2m∗ a2
=
~
sin(kk a).
m∗ a
(5.26)
Green’s function
The S-matrix technique briefly mentioned in section 3.3.1 gives the response
of a system at the boundary due to some excitation at the boundary. This
section introduces the more powerful Green’s function technique which gives
the response of the system at any point due to an excitation at any point.
This will allow us not only to compute the transmission coefficients through
the dot but also the electron density inside the system.
5.2.1
Definition of Green’s function
Given a differential operator Dop which relates an excitation S in some system
to the system response R by
Dop R = S,
(5.27)
the Green’s function is defined as
−1
G = Dop
.
(5.28)
Describing an electron travelling in a lead towards a quantum system as a unit
excitation I at the system boundary, the Green’s function is defined by
HΨ + I =EΨ
(E − H)Ψ =I
⇒
(5.29)
(5.30)
−1
I
(5.31)
−1
.
(5.32)
Ψ =(E − H)
G = Ψ =(E − H)
This definition actually allows two possible Green’s functions, the retarded and
the advanced Green’s function. The former describes the wave function caused
by the excitation while the latter describes the wave function that generates
the excitation. Evidently our interest lies in the retarded Green’s functions
and this may technically be specified by adding a small imaginary part to the
energy[22],
G = (E − H + iδ)−1 .
(5.33)
40
Modelling
Perturbation
|
{z
H01
|
}
{z
H02
}
Figure 5.3: Two systems with respective Hamiltonian H01 and H02 separated by a perturbation.
5.2.2
Dyson equation
The most straightforward way to numerically calculate the Green’s function
is to perform the inverse in equation (5.32). However, matrix inversion is
computationally a very expensive operation and a more manageable approach
is to compute the Green’s function recursively through the Dyson equations.
Considering two adjacent systems H01 and H02 , figure 5.3, and treating hopping
between them as a small perturbation V , we can write the Hamiltonian for
the combined system as
H = H01 + H02 +V = H0 + V.
| {z }
(5.34)
=H0
From equation (5.32) we have
G −1 = E − H = E − H0 − V = G 0
−1
− V,
(5.35)
where G 0 is the Green’s function of H0 . Multiplying equation (5.35) with G 0
from the right and G from the left (or vice versa) yields the Dyson equation(s)
G = G 0 + G 0V G
0
0
(G = G + GV G ).
(5.36)
(5.37)
These equations implicitly define G in terms of the known, unperturbed, G 0 .
For explicit expressions for the Green’s function between two specific points
r,r′ , we introduce the notation
G(r, r′ ) = hr|G|r′ i = hm, n|G|m′ , n′ i = Gmn,m′ n′ .
(5.38)
Similarly Gmm′ is interpreted as hm|G|m′ i. The matrix Gmm′ can then be
written as
X
0
0
Gmi
Vij Gjm′ .
(5.39)
Gmm′ = Gmm
′ +
i,j
For two systems labelled 1 and 2 we have
H = H01 + H02 + V = H0 + V12 + V21
(5.40)
5.2. Green’s function
41
e−
0
m
(b)
(a)
Figure 5.4: (a): Typical system under consideration, two leads are connecting the
quantum dot to an electron reservoirs at infinity. An incoming electron e−
from the right may either be reflected back to the left lead or transmitted
to the right. (b): A straight, homogenous channel with zero magnetic
field.
where
V12 = |1it12 h2| = |1iteiqn h2|
−iqn
V21 = |2it21 h1| = |2ite
h1|.
(5.41)
(5.42)
The Green’s functions for the combined function may now be found through
eq. (5.39). For example,
0
0
+ G11
V12 G21
G11 = G11
G21 =
0
G21
|{z}
0
+G22
V21 G11 ,
(5.43)
(5.44)
=0
substituting back and forth yields, after some rearrangements
0
0
0
G11 = (I − G11
V12 G22
V21 )−1 G11
G21 =
0
G22
V21 (I
−
0
0
0
G11
V12 G22
V21 )−1 G11
.
(5.45)
(5.46)
In a similar way expressions to add entire sections of recursively computed
Green’s functions, such as in figure 5.5 and 5.8, can be derived.
5.2.3
Surface Green’s function
Using eq. (5.39) above, we can find the full Green’s function for a finite system
by recursively adding parts of the system together. However, usually the
system we will consider is infinite, as in figure 5.4a, and the Green’s function
in the leads has to be computed in some other way. For a straight, hard
wall potential channel with zero magnetic field the solutions to the Shrödinger
equation is given by eq. (5.21) and (5.22). An excitation ψ0α |m, αi at slice 0
in figure 5.4b gives the response
|Ψi = Gψ0α |0, αi.
(5.47)
42
Modelling
Because there can be no mixing between different modes α in a homogenous
wire we temporarily suppress index α and write |m, αi → |mi. |Ψi is now
expanded as
X
|Ψi =
ψm |mi = Gψ0 |0i.
(5.48)
m
Multiplying from the right with hm| gives the response at site m
ψm = hm|Gψ0 |0i = Gm0 ψ0 .
(5.49)
From the Dyson equation (5.37) we find Gm+1,0 as
0
0
Gm+1,0 = Gm+1,0
+Gm+1,m+1
Vm+1,m Gm0
| {z }
(5.50)
=0
and thereby
0
ψm+1 = Gm+1,0 ψ0 = Gm+1,m+1
Vm+1,m Gm0 ψ0
=
eq. (5.49)
0
Gm+1,m+1
Vm+1,m ψm . (5.51)
The potential in the wire is periodic and we apply the Bloch theorem,
ikkα ma
ψm = e
um ,
(5.52)
where um is periodic, um = um+1 , and kkα is the wave vector from eq. (2.16)
associated with the transverse mode α. From eq. (5.51) we get
ikkα (m+1)a
e
ikkα a
e
ikkα ma
0
Vm+1,m e
= Gm+1,m+1
0
Vm+1,m .
= Gm+1,m+1
(5.53)
(5.54)
Defining the surface Green’s function as Γ̃ = G0m+1,m+1 , where tilde reminds
us that this is in mixed representation and noting that for zero magnetic field
Vm+1,m = t, we obtain,
ik α a
e k = tΓ̃.
(5.55)
With all N modes α1 , α2 , . . . , αN , the surface Green’s function is a diagonal
matrix
 α1

ik a
e k
0
···


α
ik 2 a

e k
0
1
 0

Γ̃ =  .
(5.56)

.

.
t  ..
.
0


α
ik N a
e k
which may be transformed to real space representation by
Γ(mn, m′ n′ ) = hmn|Γ|m′ n′ i = hn|Γmm′ |n′ i
5.2. Green’s function
43
′
Gm1
′
Gm+1,M
′
G1m
Γ
′
GM,m+1
′
G11
′
GM
M Γ
1
m
′
Gmm
m+1
M
′
Gm+1,m+1
Gmm
Figure 5.5: Recursively adding slices by eq. (5.39) from right and left lead we may
compute the Green’s function
at slice m, Gmm , for the full system. The
diagonal elements of ℑ Gmm is proportional to the density of states
(eq. (5.58)).
=
X
α,α′
hn||αihα|Γmm′ |α′ ihα′ ||n′ i
=
X
α,α′
hα|ni∗ Γ̃(mα, m′ α′ )hα′ |n′ i, (5.57)
where ’∗’ denotes the complex conjugate.
5.2.4
Computational procedure
Figure 5.5 schematically shows the recursive Green’s function technique applied to a two-lead quantum dot. The dot is divided into M slices; the Green’s
function for each slice is described by the matrix Gkk while the Green’s function
between slice k and l by the matrices Gkl and Glk . Starting from the surface
Green’s function Γ, we recursively add slice by slice until we have computed
′
′
the Green’s functions Gmm
and Gm+1,m+1
indicated in figure 5.5. Using the
Dyson equation we may now compute the total Green’s function Gmm for slice
m. The imaginary part of the diagonal elements in this matrix (ℑ[G(r, r)]) are
proportional to the density of states[33],
i
1 h
n(E, r) = − ℑ G(r, r, E) .
π
(5.58)
Integrating the density of states up to the Fermi-energy we find the electron
density inside the dot,
Z E f
1
G(r, r, E)dE .
(5.59)
n(r) = − ℑ
π
−∞
Using DFT in conjunction with the recursive Green’s function technique we
can now write down a procedure to find a self-consistent density to an arbitrary
two-lead quantum dot.
44
Modelling
x 10
20
ℜ(G)
ℑ(G)
G(E, r, r)
0
-5
-10
-15
ℑ
ℑ
E0
-20
EF
0.8
ℜ
0.9
E/EF
ℜ
E0
1
EF
0.8
0.9
ℜ(E)/EF
1
Figure 5.6: Real (solid line) and imaginary (dashed line) part of the Green’s function
at a single point m, n for a typical system. Left panel shows the Green’s
function along the real axis (gray arrow in inset, E0 is the lowest potential
in the dot, EF the Fermi level). Because of the very sharp peaks numerical
integration is difficult along this path. Right panel shows the Green’s
function along the path indicated by the gray arrow in the right inset.
The Green’s function is analytical in the upper half of the complex plane,
hence integration along this path yields the same result as along the real
axis but is numerically easier.
I)
guess an an initial density n(r) (e.g., by Thomas-Fermi)
II)
compute an effective potential vef f (r) by equation (4.45)
III)
update the Hamiltonian (eq. (5.3)) with the new vef f (r)
IV)
update the Green’s function with the new Hamiltonian (eq. (5.32))
V)
VI)
update the density n(r) through eq. (5.59)
repeat from step II until n(r) converges
There are some problems with convergence in this procedure. If the density
between subsequent iterations varies too much, this may cause the solution to
oscillate – this problem is typically addressed by mixing the new density with
the old density by some parameter ǫ,
nnew (r) = ǫnnew (r) + (1 − ǫ)nold (r)
0 < ǫ ≤ 1,
thereby dampening any sudden change in the density.
Green’s function has some very sharp features along the
left panel of figure 5.6), making numerical integration very
poles to the Green’s function are below the real axis this
(5.60)
Furthermore, the
real axis (see the
difficult. Since all
may be remedied
x 10
45
20
2 (a)
G × fF D
0
ℜ(G)
ℑ(G)
-2
fF D (E) ℑ(E)/kB T
5.2. Green’s function
15
10
5
(b)
1
-4
(c)
0
-6
-1
-20
-10
0
(ℜ(x) − EF ) /kB T
-20
-10
0
(ℜ(x) − EF ) /kB T
Figure 5.7: (a) Real and imaginary parts of Green’s function at point m, n multiplied
with the Fermi-Dirac distribution. (b) Integration path in complex plane.
(c) Real and imaginary part of Fermi-Dirac distribution (fF D ) along the
path in (b).
by carrying out the integration in the complex plane (see the right panel of
figure 5.6), where the Green’s function is far smoother.
This may however still not be enough for convergence. With a high density
of states close to the Fermi level the Green’s function may change rapidly as
we return towards the real axis, and the last part of the integration requires a
very fine stepsize ∆E to capture the integral accurately. A simpler approach
is to slightly increase the temperature in the system. With T > 0K some
states above the Fermi level EF will be occupied and the density is given by
Z ∞
(5.61)
n(E, r) fF D (E, EF , T ) dE,
ne (r, EF , T ) =
{z
}
−∞ | {z } |
DOS
F−Ddist.
where fF D is the Fermi-Dirac distribution. We may now continue the integration in the complex plane until the Green’s function is suppressed by the
exponentially decaying Fermi-Dirac distribution, see figure 5.7.
With the self-consistent density nef f (r) we compute the Green’s function
relating the first and last slice as shown in figure 5.8, GM 1 . In mixed representation this matrix gives the transmission amplitudes between mode α in the
left lead to mode β in the opposite lead. At zero magnetic field[33],
r tβ←α = i2|t|
i(kkα m−kkβ n)
sin kkα a sin kkβ a e
G̃M β,1α (E),
(5.62)
where G̃M β,1α (E) = hM β|G̃(E)|1αi, and tilde indicates that it is the Green’s
function in mixed representation. At zero magnetic field the amplitude for the
reflected electron is similarly[33]
r
.
ia(k β +k α )
rβ←α = i sin kkβ a
sin kkα a e k k
46
Modelling
′
′
G1,M
−1 GM −1,1
Γ
′
GM
−1,M −1
′
G11
′
GM
M
1
G11
M −1
Γ
M
GM 1
Figure 5.8: With the self-consistent density n(r), we compute the Green’s functions
GM1 (G11 ) for the full system, which in mixed representation gives the
transmission amplitudes by eq. (5.62) (reflection amplitudes by eq. (5.63)).
i
h
× 2|t| sin kkβ a G̃1β,1α (E) + iδβα , (5.63)
where δαβ is the Dirac delta function.
Chapter 6
Comments on papers
6.1
Paper I
Magnetoconductance through small open quantum dots, i.e., dots with ∼100
electrons or less which are strongly connected to leads, shows a strong correlation between the conductance through the dot and the eigenspectrum of
the corresponding closed dot[126]. A computational example is shown in figure 6.1(a) where conductance is plotted against magnetic field and the number
of propagating modes in the leads. Superimposed on the conductance in the
same figure are the eigenvalues of the corresponding closed dot whose features
roughly match the dips and peaks of the conductance. The correlation is not
perfect, depending on the lifetime broadening some energy levels are missing
or merged in the conductance plot.
In paper I we demonstrate that the magnetoconductance of small lateral
quantum dots in the strongly coupled regime (i.e. when the leads can support
one or more propagating modes) shows a pronounced splitting of the conductance peaks and dips which persists over a wide range of magnetic fields
(from zero field to the edge-state regime) and is virtually independent of the
magnetic field strength. Some of these measurements are shown in figure 6.2
together with their respective top gate layout as insets.
Our numerical analysis of the conductance based on the Hubbard Hamiltonian demonstrates that the splitting is essentially a many-body/spin effect
that can be traced to a splitting of degenerate levels in the corresponding
closed dot – c.f. figure 6.1(b) for the closed dot eigenspectrum with the conductance plot in figure 6.1(c)-(d). The effect in open dots can be regarded as
a counterpart of the Coulomb-blockade effect in weakly coupled dots, with the
difference, however, that the splitting of the peaks originates from interactions
between electrons of opposite spin.
48
Comments on papers
U=0
U=3t
1
1.45 (a)
Closed dot
1.5 (b)
0.8
360nm
k (units π/w)
k (units π/w)
1.4
0.6
1.3
1.25
0.4
1.2
210nm
0.2
kF
1.05
1.1
0
0.2
B (Tesla)
0.4
0
2e2
G( h
)
(c)
k (units π/w)
k (units π/w)
Open dot
0.4
80nm
U=3t
U=3t
1.56
0.2
B (Tesla)
1.52
1.2
(d)
360nm
1.15
1.48
0
0.05
B (Tesla)
0.1
1.1
0.1
0.2
0.3
B (Tesla)
210nm
Figure 6.1: Paper I (a) The conductance G = G(kF , B) for U = 0. Solid lines depict
the eigenspectrum of the corresponding closed dot. (b) The eigenspectrum
of the corresponding closed dot for the same strength of U = 3t. Dotted
line indicates kF .(c),(d) The conductance G = G(B, kF ) for U = 3t. Note
that because of a large computation time the conductance is shown only
for two representative regions.
-0.4
(a)
(b)
-0.62
4
3
VG
VG
-0.5
2
-0.6
-0.66
1
0.2
0.3
0.4
B (Tesla)
0.5
0.6
0
0.2
0.4
0.6
2e
B (Tesla) G( h )
2
Figure 6.2: Paper I Experimental conductance as a function of magnetic field B
and gate voltage VG obtained from different dots. The corresponding
device layouts are shown in the insets. The lithographic size of the dots
is ∼ 450nm.
6.2
Paper II
The existence of a spin polarized state for the two dimensional electron gas
has been the extensively probed for various systems, both experimentally,
e.g., Refs[8, 41, 44, 112], and theoretically, e.g., Refs[114, 118]. It has been
suggested as the explanation for the “0.7”-ananomaly[118] in quantum point
contacts and proposed as a working principle for spin filtering through weakly
6.2. Paper II
49
Figure 6.3: Paper II The density for the spin-up and spin-down electrons n↑ (r),
n↓ (r) (left and right panels respectively) in a (a) low, (b) intermediate and
(c) high density parabolic dot calculated within LSDA. These densities
corresponds to an average electron density of ne =∼ 2 × 1014 m−2 , 8 ×
1014 m−2 and 2 × 1015 m−2 respectively. T=2K.
coupled quantum dots[69, 81].
In open quantum dots, spin polarization has been studied with conflicting
results, c.f. Folk et al.[35, 36] and Evaldsson et al.[32] (Paper I). However,
the dots studied by Folk et al. were significantly larger, with ∼ 1000 − 10000
electrons and an average electron density ne =∼ 1 × 1015 m−2 , compared with
ne =∼ 1 × 1014 m−2 for the dots in Evaldsson et al.. The breaking of spin
degeneracy at sufficiently low electron densities is a well studied phenomena[2,
6, 8, 41, 44, 51, 112, 114, 118], and it is possible that the discrepancy between
Folk et al. and Evaldsson et al. could be explained by the difference in average
electron density in the respective systems.
The focus of paper II is coherent transport through open lateral quantum dots using recursive Greens function technique, incorporating exchangecorrelation effects within the Density Functional Theory (DFT) in the local
spin-density approximation (LSDA). At low electron densities the current is
spin-polarized and the electron density in the dot shows a strong spin polarization. As the electron density increases the spin polarization in the dot
gradually diminishes, see figure 6.3 where the left column show the spin-up
electron density, n↑ (r), and the right column the spin-down density, n↓ (r).
The average density in the plots are n̄(r)/1014 m−2 =, (a) 2, (b) 8, (c) 20.
50
Comments on papers
These findings are consistent with available experimental observations.
6.3
Paper III
As mentioned in the earlier comments to paper II, a spin-polarised state in
low-density 2DEG:s has been studied both experimentally, e.g., Refs. [8, 32,
41, 44, 95, 106, 107, 112] and theoretically, e.g., Refs. [2, 51, 114, 118]. In
experimental studies a difficulty arises because spin polarisation is expected
at very low electron densities, roughly 1 − 3 × 1014 m−2 [42, 107], At such low
Figure 6.4: Paper III Schematic view of the system studied. The heterostructure
consists of (from bottom to top), a GaAs substrate, a 3060 nm AlGaAs
spacer, a 26 nm donor layer, and a 14 nm cap layer. The side gates
define a quantum wire and the top gate controls the electron density in
the section with randomly distributed dopants ρd (r).
electron densities the electrostatic potential due to impurities in the donor
layer can significantly affect the electronic and transport properties of the
2DEG. For example, Nixon et al.[25, 77] showed that a monomode quantum
wire could undergo a metallic-insulator transition MIT[9, 25, 30, 52, 77] due
to the a random potential from unscreened donors.
In paper III we study spin polarization in a split-gate quantum wire focusing on the effect of a realistic smooth potential due to remote donors. A
schematic figure is given in figure 6.4. Electron interaction and spin effects are
included within the density functional theory in the local spin density approximation. We find that depending on the electron density, the spin polarization
exhibits qualitatively different features. For the case of relatively high electron density, when the Fermi energy EF exceeds a characteristic strength of
a long-range impurity potential Vdonors , the density spin polarization inside
6.4. Paper IV
51
eV
700
nm
EF
A)
600
500
−0.02
400
−0.04
300
−0.06
200
−0.08
100
−0.1
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
400
600
800
1000
nm
1200
1400
1600
1800
2000
700
B)
600
nm
500
400
300
200
100
0
0
200
Figure 6.5: Paper III Potential due to split gates and impurities with top gate potential A) Vg =0, B) Vg =-0.04, for the 30nm thick spacer sample. The
first and last segment of ∼400nm show the homogeneous part of the wire.
the wire is practically negligible and the wire conductance is spin-degenerate.
An example of the potential profile of the wire in this regime is given in figure 6.5(A). When the density is decreased such that EF approaches Vdonors , the
electron density and conductance quickly become spin polarized. With further
decrease of the density the electrons are trapped inside the lakes (droplets)
formed by the impurity potential and the wire conductance approaches the
pinch-off regime. An example of the potential profile in this regime is given
in figure 6.5(B). We discuss the limitations of the density functional theory in
the local spin density approximation in this regime and compare the obtained
results with available experimental data.
6.4
Paper IV
The word “Spintronics” has been invented to denote the idea to use the spin
property of the electron as a building block for future devices[120]. With operational sizes of solid state components approaching the meso- and nanometre sized regimes this idea will soon reach its full potential. A remaining
52
Comments on papers
problem within semiconductor spintronics is how to generate and/or inject
spin-currents in semi-conductors. Numerous proposals has been put forward;
transport through spin blockaded lateral quantum dots[69, 80, 81], the use of
spin-orbit coupling effects[100], applying an inhomogeneous in-plane magnetic
field[38], magnetic barriers[46, 83, 84], quantum point contacts[99, 101] and
spin injection[60, 70, 125].
↓
↑, ↓
↑
Figure 6.6: Paper IV Working principle for antidot spin filtering. A perpendicular
magnetic field creates spin-up/down edge states travelling in the direction
indicated by arrows. Edge state may tunnel to the quasi-bound state a
the antidot and then to the edge state at the opposite edge of the quantum
wire. Because of the Zeeman term, ± 21 gµE σB, the maximum tunnelling
rate occurs at different magnetic field strengths for spin-up/down electrons.
Paper IV proposes a device based on an antidot embedded in a narrow
quantum wire in the edge-state regime, figure 6.6, that can be used to inject
and/or control spin-polarized current. The operational principle of the device is based on the effect of resonant backscattering from one edge state into
another through localized quasibound states, combined with the effect of Zeeman splitting of the quasibound states in a sufficiently high magnetic field. We
outline the device geometry, present detailed quantum-mechanical transport
calculations, and suggest a possible scheme to test the device performance and
functionality.
6.5
Paper V
Properties of lateral structures in 2DEGs are often analysed through conductance measurements using a specific device. Additional insights to these
measurement are gained by manipulating external parameter such as the bias
voltage, applying a gate voltage to a top gate or a magnetic field. Paper V
analyse the details of low energy electron transport through a magnetic barrier
(MB), i.e., a static inhomogeneous magnetic field applied locally to the 2DEG.
Typically such a device is prepared by placing a ferromagnetic film on top
of the heterostrucutre[53, 59, 73], figure 6.7. The magnetisation of the film,
µ0 M , induces a perpendicular magnetic field Bz (x) inhomogeneous in the xdirection and homogeneous in the y-direction, see top inset in figure 6.7. The
maximum field strength of Bz (x) and its full width at half maximum (FWHM)
6.5. Paper V
53
Bz (x)
x
side view
z
x
µ0 M
top view
y
x
µ0 M
spacer
magn. film
spacer
2DEG
substrate
magn. film
µ0 M
x=0
Figure 6.7: Paper V Schematic view of a magnetic barrier (MB) structure. A magnetic film is placed on top of a 2DEG heterostructure. Depending on the
magnetization µ0 M , the film thickness and the spacer thickness the shape
of the magnetic field Bz (x) at the 2DEG can be manipulated.
Figure 6.8: Paper V Calculated conductance versus Fermi energy EF for three samples with 15nm, 35nm and 350nm spacer thickness and µ0 M = 1.2T. The
lower inset shows the shapes of the MBs present at the distances considered. Only the most localised MBs, with 15nm and 35nm spacer, displays
the resonance dip.
can be adjusted by changing the magnetisation strength µ0 M , the width of
the spacer and the width of the magnetic film, see inset to figure 6.8. With
increasing magnetic field more propagating modes in the wire are reflected
54
Comments on papers
and quantized conductance steps similar to those appearing at quantum point
contacts (QPC) appear[45, 122, 123]. However, in contrast to QPCs there is
an additional transmission dip for some MB just before the magnetic pinch
off, see conductance curves for 15nm and 35nm samples in figure 6.8. Paper V
explains this resonant feature as an interference effect between an open edge
state in the MB-region and a localised vortex state in the middle of the MB.
6.6
Paper VI
Paper VI adapts a numerical method for efficiently computing the surface
Green’s function in photonic crystals[97] to graphene nanoribbons (GNR).
Present implementations of the Green’s function technique in GNRs, e.g.,
Refs [43, 75], use a self-consistent, typically time-consuming, procedure to
compute the surface Green’s function. The method introduced in paper VI
computes the surface Green’s function directly from the Bloch states by solving
a 2N ×2N eigenequation where N is the number of sites across the nanoribbon.
The paper further reviews steps to compute wave functions, group velocities
and conductance in two-terminal GNRs within the Green’s function formalism.
Next, the method is applied to study the magnetoconductance through
a magnetic barrier (MB), i.e., a static inhomogeneous magnetic field applied
locally to a GNR. MBs are interesting as means to control electron conductance
in 2D-systems[71, 87], and for their predicted spin polarising properties[46, 83,
84]. An example of a MB-setup, for a quantum wire, is given in figure 6.7.
In our study we note that MBs in GNRs locally modify the energy spectrum
and thereby the conductance steps. MBs with steep gradient show pronounced
Fabri-Pérot oscillations whereas smoother MB hardly affect the magnetoconductance at all. For disorderfree zigzag-GNRs, where the lowest propagating
mode is a dispersionless edge state, we find that a MB could not be used as
a conductance “on-off switch” as in the case with MB in quantum wires (Paper V). Analysing the wave function for this mode show that the wave function
remains at the edge of the ribbon as the strength of the MB is varied.
In contrast to simulated (disorderfree) GNRs, experimental measurements
on ribbons of similar widths as studied here always posses an energy gap
independent of the crystal direction of the GNR1 [47]. Theoretical work has
proposed the the presence of edge disorder as an explanation[96] (see also
paper VII) to this phenomena. Because of this we also investigate the effect
of MB in GNRs with edge disorder. For this case we find it is possible that a
MB could delocalise states localised by edge disorder.
1
I.e., their edge structure.
6.7. Paper VII
6.7
55
Paper VII
The high mobility of graphene, even at ambient temperature and high degree of doping, has mad graphene nano ribbons (GNR) a strong candidate for
building blocks in future electronic devices. There are, however, some issues
that need to be addressed regarding graphene electronics. First, graphene is
gapless, making it necessary to engineer a band gap if it is to replace existing
semiconductor devices. Second, modelling indicates that the electronic properties of GNR are expected to depend strongly on the edge structure of the
ribbon[39, 76, 116]. The first issue can be addressed by making sufficiently narrow GNRs as this will induce a band gap. The second issue might be a result of
modelling of ideal GNRs. Measurements by Han et al.[47] found no directional
dependence of the conductance for GNRs of widths 20-100nm. Since these
measurements, a number of explanations have been put forward to address
this discrepancy between experiments and modelling; rough edges[20, 66, 96],
atomic scale impurities[20], coulomb blockaded transport[108] and impurity
scattering[64].
In paper VII we study the effect of the edge disorder on the conductance
of the graphene nanoribbons (GNRs) within the tight-binding model. The
method we use, developed in paper VI, allows us to probe GNRs of similar
size as in the recent experiments by Han et al.. We find that only very modest
edge disorder is sufficient to induce the conduction energy gap in the otherwise metallic GNRs and to lift any difference in the conductance between
nanoribbons of different edge geometry. We relate the formation of the conduction gap to the pronounced edge disorder induced Anderson-type localization
which leads to the strongly enhanced density of states at the edges, formation
of surface-like states and to blocking of conductive paths through the ribbons.
56
Comments on papers
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