Quantum Transport and Spin Effects in Lateral Semiconductor

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Link¨oping 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 Link¨oping University, SE-601 74 Norrk¨oping, Sweden

Norrk¨oping, 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, Link¨oping, 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 Ap- proximation (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 metal- insulator 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 real- istically 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, Ali- aksandr 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 ac- knowledged.

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. Plot- ting 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. Plot- ting 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 3 2.1 Heterostructures ...... 3 2.1.1 Two-dimensional heterostructures ...... 4 2.1.2 (Quasi) one-dimensional heterostructures ...... 7 2.2 Graphene ...... 9 2.2.1 Basic properties: experiment ...... 10 2.2.2 Basic properties: theory ...... 11

3 Transport in mesoscopic systems 15 3.1 Landauerformula...... 15 3.1.1 Propagatingmodes...... 15 3.2 B¨uttikerformalism ...... 17 3.3 Matchingwavefunctions...... 19 3.3.1 S-matrixformalism...... 20 3.4 Magneticﬁelds ...... 20

4 Electron-electron interactions 23 4.0.1 What’stheproblem?...... 23 4.1 TheHubbardmodel ...... 24 4.2 Thevariationalprinciple...... 26 4.3 Thomas-Fermimodel...... 27 4.4 Hohenberg-Kohntheorems...... 28 4.4.1 TheﬁrstHK-theorem ...... 28 4.4.2 ThesecondHK-theorem...... 29 4.5 TheKohn-Shamequations...... 30 4.6 Local Density Approximation ...... 31

ix 4.7 Local Spin Density Approximation ...... 32 4.8 BriefoutlookforDFT ...... 33

5 Modelling 35 5.1 Tight-binding Hamiltonian ...... 35 5.1.1 Mixed representation ...... 36 5.1.2 Energy dispersion relation ...... 37 5.2 Green’sfunction ...... 39 5.2.1 Deﬁnition of Green’s function ...... 39 5.2.2 Dysonequation...... 40 5.2.3 SurfaceGreen’sfunction...... 41 5.2.4 Computationalprocedure ...... 43

6 Comments on papers 47 6.1 PaperI ...... 47 6.2 PaperII...... 48 6.3 PaperIII ...... 50 6.4 PaperIV ...... 51 6.5 PaperV...... 52 6.6 PaperVI ...... 54 6.7 PaperVII...... 55

x 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 tran- sistor, 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.:

1Moore’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 Den- sity 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 metal- organic 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 AlxGa1−xAs (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]

E ( x) =1.424 + 1.429x 0.14x2[eV] 0 < x < 0.441 , (2.1) g − 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).

1For 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.

Original n-AlGaAs band Energy

2DEG Space

Conduction band

donors Original GaAs band

GaAs n-AlGaAs AlGaAs (spacer-layer)

Figure 2.2: Two-dimensional electron gas seen along the conﬁning dimension.

Remoter o 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 1015m−1 though values as low as 5 1013m−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 k m n2D(=E) ~2 π k+dk 2π n(E) L kx 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(kxx+kyy), (2.2) where 2πm 2πn 2πm 2πn k = (k , k ) = , = , m, n = 0, 1, 2 .... x y L L L L ± ± x y (2.3) Plotting these states in the 2-dimensional k-space, ﬁgure 2.3a, we recognise 2π 2 that a unit cell has the area L , hence the density of states is

−1 (2π)2 L2 N ( k) =2 = (2.4) 2D L2 2π2 where a factor two is added to include spin. Defining the density of states per unit area, N (k) 1 n (=k) 2D = , (2.5) 2D L2 2π2 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 n2D2(k) πkdk 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 n (EE)d = n (E) dk = 2πkn (kk)d = 2πk dk (2.6) 2D 2D dk 2D 2π2 k dE −1 n (=E) . (2.7) ⇔ 2D π dk 2.1. Heterostructures 7

For free electrons, with ~2k2 E(k) = , where k = k (2.8) 2m | | we finally arrive at m n =(E) , (2.9) 2D ~2π shown in figure 2.3b. A similar derivation for a one dimensional system yields the density of states per unit length, 1 2m n1D=(E) ~ . (2.10) π r 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 n-AlGaAs AlGaAs AlGaAs GaAs 2DEG GaAs 2DEG

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 ∂y2 can, by introducing Ψ(x, y) = ψ(x)φ(y), be separated into

~2 d2 ψ=(x) E ψ(x) (2.12) −2m dx2 x ~2 d2 + U(y) φ(y) = E φ(y). (2.13) −2m dy2 y In the simple case of a hard wall potential U(y),

0, 0 < y < w U(y) = (2.14) , y < 0 w < y, (∞ ∪ eq. (2.12) and (2.13) yields the solutions πny Ψ (x,y) = ψ(x)φ (=y) eikxx sin (2.15) n n w and eigenenergies

~2 πn 2 ~2 E = E + E = k2 + = k2 + π2k2 . (2.16) x y,n 2m x w 2m k ⊥ 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. Hence, the energy dispersion relation, E(k) = E(k + k ) k2 + k2 , consists k ⊥ ∼ k ⊥ of subbands, as depicted in ﬁgure 2.6. 2.2. Graphene 9

Energy E3

φ1 φ2 φ3 E2 E1 EF

(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 thermody- namically 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 pre- dicts 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

2Where charge carriers can tunnel through an arbitrary high energy barrier with unity probability 10 Mesoscopic physics depleted[78, 113].

A-lattice B-lattice a cc {

Figure 2.7: The graphene lattice can be described as two sub-lattices, A and B. Prim- itive vectors for sub-lattice A are indicated by arrows. The interatomic distance acc is roughly 1.42A.˚

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 pro- duce 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 cm2V−1s−1 [79] has been reported and experiments on suspended graphene at low temperatures reported a peak mobility at 230,000cm2V−1s−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 σ min ∼ 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 2 of π, σmin,theor = 4e /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 5 M K 0 Dirac point b2 b1 kx E[eV] Γ -5

M Γ K M Wave vector (a) Brillouin zone (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 1.42A˚. Using the tight-binding approximation the dispersion relation can be computed from the secular equation[48, 98, 102, 117]

H S E H S E AA − AA AB − AB = 0 (2.17) H S E H S E BA − BA BB − BB

Here, HAA = ΦA H Φ A whereΦj( k, r) is the Bloch function for site j = A, B, h | | i

1 ik·R Φj = e φj(r R) (2.18) √N − XR with φj being the wave function localized at site j and R the position of the lattice points. Similarly, S ′ = Φ Φ ′ is the overlap matrix and E the jj h j| j i eigenenergies. The solution of eq. (2.17) is of the form[48, 98, 102]

ǫ γ g(k) E(k) = 2p ± 1 | | (2.19) 1 γ g(k) ± 0 | | 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. A- GNR:s are metallic only if Na=3p + 1, p being an integer.

where[1]

√3k a k a k a g(k) = 1 + 4 cos y cos x + 4 cos2 x (2.20) v | | u 2 ! 2 2 u t and a = √3a 2.46nm being the length of the reciprocal lattice vector. cc ∼ 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 < γ

γ is related to the the overlap between nearest neighbours, φ φ , and 0 h A| Bi 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 v / c 300[86]. F ∼ 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

Width [Na] 50 100 150 200 0.5 ”) t 0.4

0.3

0.2

Energy gap0.1 (units of “

0 0 5 10 15 20 25 Width [nm]

Figure 2.10: Energy gap of armchair GNR versus ribbon width within the tight- binding 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. Of- ten 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 quasi- Fermi 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 EF r Ul 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

Ilk(k)d = 2en1D(k)v(k)Tr←l(Ek)f E(k), F 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 Fermi- Dirac distribution. Using eq. (2.10) for the 1D-DOS and integrating both sides yields

∞ 1 dk I = 2e v(k)T (Ek)f E(k), dk = dk = dE l 2π r←l F l dE Z0 2e ∞ / / dk = v(E)T ,E(E)f E dE. (3.2) 2π r←l F l dE Z0 dω 1 dE Recognising the group velocity as v = dk = ~ dk and integrating from the bottom of the left lead brings

2e ∞ 1 I = v(E)T ,E(E)f E dE l 2π r←l F l ~v(E) ZUl 2e ∞ = T (,EE)f E dE. (3.3) h r←l F l ZUl 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 2e ∞ I = I + I = T,E(E) f E f E,E dE. (3.4) l r h F l − F r ZUl h i In the case of very low bias eV , i.e., in the linear response regime, the Fermi- 1 Dirac functions in eq. (3.4) can be Taylor expanded around EF = 2 (EF l +EF r) according to

1 1 f(E,E ),E f(E ),E = f(E + eV ) f(E,E eV ) F l − F l F 2 − F − 2 3.2. B¨uttiker formalism 17

∂f(E,E ) ∂f(E,E ) = eV F V+ (e )2 eV F , (3.5) ∂EF ≈ − ∂E resulting in

2e ∞ ∂f(E,E ) I = TV (E) e F dE (3.6) h − ∂E ZUl ⇔ I 2e2 ∞ ∂f(E,E ) G = = T (E) F dE. (3.7) V h − ∂E ZUl At very low temperatures in the linear response regime, ∂f is replaced by − ∂E the Dirac delta function δ(E E ) and evaluating the integral in eq. (3.7) − F gives the famous Landauer formula for quantum conductance

I 2e2 G = = T (E ). (3.8) V h F 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 2e2 G = T . (3.9) h β←α Xα,β 3.2 Büttiker formalism

B¨uttiker 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 = T (E) Θ EF l E Θ EF r E E d h U − − − Z l h i 2e EF l = TE.(E)d (3.10) h ZEF 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

2e I = T (E E ). (3.11) h × F l − F r 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 I (I mm incoming − Ireflected) and the current injected from all other leads, n6=m Im←n. With the lowest quasi-Fermi level in all leads denoted E we can write 0 P 2e I = T (E E ). (3.12) m←n − h m,α ←n,β × n − 0 nX6=m Xα,β The minus sign indicates a current away from the system and T m,α←n,β is the transmission probability from lead n–mode β to lead m–mode α. We will abbreviate this by the notation

T¯m←n = Tm,α ←n,β, (3.13) Xα,β and in a similar way R¯m = Rm,α ←m,β (3.14) Xα,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 I = N R¯ (E E ). (3.15) mm h m − m m − 0 Furthermore, conservation of ﬂux requires that

Nm = R¯m + T¯n←m = R¯m + T¯m←n (3.16) mX=6 n nX6=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 2e I = I + I = N R¯ (E E ) T¯ (E E ) m mm m←n h m − m m − 0 − h m←n n − 0 n6=m X (3.17) 2e 2e = T¯ (E E ) T¯ (E E ) h m←n m − 0 − h m←n n − 0 nX6=m nX6=m (3.18) 2e = T¯ E T¯ E , (3.19) h m←n m − m←n n n6=m X where eq. (3.16) has been used in the second step. Noting that Em V= e m we ﬁnally get 2e2 I = T¯ (V V ) . (3.20) m h m←n m − n nX6=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

Aeik1x Ce−k i 2x Deik2x Be−k i 1x V0 x0 =

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 Schr¨odinger eq. for electrons with energy E > V0 is

Aeik1x +e B k−i 1x x < 0 Ψ(x) = (3.21) k−i 2x ik2x (Ce +e D 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) k (=A B) k (D C). (3.23) 1 − 2 − Solving for the outgoing amplitudes, B and D, results in

B 1 k k 2k A = 1 − 2 1 . (3.24) D k + k 2k k k C 1 2 1 2 − 1 E.g., for a free electron incoming from the left with unit amplitude (A = 1, C = 0), the transmitted amplitude (D = t2←1) becomes

2k1 t2←1 = , (3.25) k1 + k2 and the reﬂected amplitude (B = r1←1)

k1 k2 r1←1 = − . (3.26) k1 + k2 Because the velocities in the left and right lead may differ, the flux transmission coefficient becomes v2(k2) 2 T2←1(k1, k2) = t2←1 , (3.27) v1(k1)| | where v2 and v1 are the group velocities for the electron. 20 Transport in mesoscopic systems

3.3.1 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 I = v (k) A (k) 2 i − i | i | where vi (k) is the group velocity and Ai( k) the amplitude. It is convenient to deﬁne the current amplitude, by

a (=k) A (k) v(k) (current amplitude for incoming mode i) i i − bi(=k) Bi(k)pv(k) (current amplitude of outgoing mode i), such that the currentp carried by mode i simply is I = a (k) 2. We now i | i | generalise eq. (3.24) to handle an arbitrary system by deﬁning the S-matrix (scattering matrix)[22], b¯ = Sa¯, (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

T (=E) s 2, (3.29) m←n | mn| where smn is a matrix element in the S-matrix. Now, current conservation requires that I = e a 2 = e b 2 = I (3.30) in | i| | i| out Xi Xi or in matrix notation

a¯†a¯ = b¯†b¯ (3.31) a¯†a¯ = (Sa¯)† Sa¯ (3.32) eq.⇔ (3.28) a¯†a¯ = a¯†S†Sa¯ (3.33) ⇔ S† S = I= SS†, (3.34) ⇒ i.e., S is unitary and we arrive at

T = s 2 = 1 = s 2 = T . (3.35) m←n | mn| | nm| n←m m m m m X X X X 3.4 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

(b) ) E ( n (a) (c) 1 ~ 3 ~ 5 ~ 2 ωc 2 ωc 2 ωc Energy

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 H = ( i~ + eA)2 .+ V (3.37) 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 Bzˆ is Landau gauge A=( By,0,0) − or A=(0,Bx,0), and the circular symmetric gauge A= 1 ( By,Bx,0). We will 2 − now consider A=(0,Bx,0), and the Schrödinger equation for a 2DEG becomes

~ ie~Bx ∂ (eBx)2 HΨ = 2 + Ψ(x, y) = EΨ(x, y). (3.38) −2m∇x,y − m ∂y 2m Trying a solution in the form Ψ(x, y) = u(x)eiky we end up with an equation only in x, ~2 d2 mω2 ~k 2 + c + x u(x) = ǫu(x) (3.39) −2m dx2 2 eB ! 2 eB with ωc = m being the cyclotron frequency. This is the Schr¨odinger equation for a harmonic oscillator so we can write down the eigenenergies as

1 ǫ = (n )~ω ,1 n = , 2, 3,... (3.40) n,k − 2 c 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

10 (a) 20 (b) (c)

15 x ) (

V 5 10

0 0 -50 0 50 -50 0 50 nm nm

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): Poten- tials (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

~2 d2 mω2 ~k 2 + c + x + V (x) u(x) = ǫu(x) (3.41) −2m dx2 2 eB ! where the only diﬀerence from eq. (3.39) is the conﬁning potential V (x),

L 0 x 2 V (x) = | | ≤ L . (3.42) ( x > ∞ | | 2 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 left- most 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 deal- ing 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 Func- tional 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 Schr¨odinger equation

~2 2 ˆ 2 1 e HΨ = i + + vext(=r) Ψ EΨ. (4.1) ′ − 2m∇ 2 ′ ri ri  Xi Xi=6 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 = ǫkckσckσ Xkσ 1 ′ ′ † † + k1k2r1/ k1k2 c c c ′ c ′ 2 h | | i k1σ1 k2σ2 k2σ2 k1σ1 ′ ′ σ1σ2 k1kX2k1k2 X ′ ′ ′ ′ † 2 kk r1/ kk k k r1/ k k ν ′ c c , (4.2) − h | | i − h | | i k kσ kσ ′ σ Xkk X † where the momentum sum is over the ﬁrst Brillouin zone, ckσ and ckσ are the creation and annihilation operators for the Bloch state k with spin σ (= 1), ± the e-e interaction terms are ∗ ∗ ′ ′ ψ (x)ψ ′ (x)ψ (x )ψ ′ (x ) k 1 k k 2 k k k r1/ k′ k′ = e2 1 2 dxdx′ (4.3) h 1 2| | 1 2i x x′ Z | − | 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 φ(x) = ψk(x), (4.4) √N Xk where N is the number of electrons. The Bloch wave function and creation/annihilation operators can be expressed with the Wannier functions as,

1 Rik j ψk(=x) e φ(x Rj) (4.5) √N − Xj 4.1. The Hubbard model 25 and 1 Rik j † 1 iRk j † c = e cjσ, c = e c , (4.6) kσ √N kσ √N iσ Xj Xj Ri being coordinates for lattice site i. The Hamiltonian (eq. 4.2) becomes

† H = Tijciσcjσ σ Xi,j X 1 † † + ji 1/r kl ciσ cjσ′ clσ′ ckσ 2 ′ h | | i Xijkl Xσσ 2 ij 1/r kl ij 1/r lk ν c† c† , (4.7) − h | | i − h | | i jl iσ kσ σ Xijkl X where −1 ik(Ri−Rj ) Tij = N ǫke , (4.8) Xk φ∗(x R )φ (x R )φ∗(x′ R )φ (x′ R ) ij 1/r kl = e2 − i − k − j − l dxdx′, h | | i x x′ Z | − | (4.9) and −1 ik(Rj −Rl) νjl = N νke . (4.10) Xk Obviously the main contribution to the e-e interaction comes from the term ii 1/r ii = U and Hubbard subsequently suggested to ignore the contribution h | | i from all other terms. The simpliﬁed Hubbard Hamiltonian is then,

1 H = T c† c + U n n U ν n (4.11) ij iσ jσ 2 iσ i,−σ − ii iσ σ Xi,j X Xi,σ Xi,σ † where niσ = ciσciσ. A few more simpliﬁcations are close at hand, the last term of eq. (4.11) can be written as

U ν n = U N −1 ν n − ii iσ − k iσ Xi,σ Xi,σ Xk 1 1 = U N −1 n n = UNn2 (4.12) − 2 iσ −2 Xi,σ which is constant and may be ignored. Furthermore, the second term in eq. (4.11) is

1 1 U n n = U (n n + n n ) = U n n . (4.13) 2 iσ i,−σ 2 i↑ i↓ i↓ i↑ i↑ i↓ Xi,σ Xi Xi 26 Electron-electron interactions

Finally, by considering only nearest neighbour hopping, i.e. hopping from site i to site i + ∆, and denoting Tii = E0, T∆i,i+ = t we arrive at a common formulation of the Hubbard Hamiltonian,

† H = E0 niσ + t ciσcjσ + U ni↑n↓. (4.14) Xiσ (iX=6 j),σ Xi 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, antiferro- magnetism, ferromagnetism, Tomonaga-Luttinger liquid and superconductiv- ity [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

φ Hˆ φ E = E[φ] = h | | i E . (4.15) φ φ ≥ 0 h | i 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 min- imising 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

M = 33N = 3300 0 1 143! (4.16) p ∼ 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 Schr¨odinger equation occurs at the extremum for the functional E[Ψ]. In order to have the wave function normalised, Ψ Ψ = N, h | i a Lagrange multiplier E is introduced and we arrive at

δ Ψ Hˆ Ψ E ( Ψ Ψ N) = 0. (4.18) h | | i − h | i − This is, however, only a rephrasing of the original Schr¨odinger equation (with the constraint Ψ Ψ = N) and by no means easier to solve. The idea in the h | i Thomas-Fermi (TF) model is to assume that the ground state density,

∗ n0=(r) Ψ0(r, r2,..., rN)Ψ0(r, r2,..., rN)dr2dr3 . . . drN (4.19) Z 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 2 from the 2D-density of states, n2D(=ǫ) m/(~ π ), given in eq. (2.9). The energy over some area L2 is ǫF m ǫ2 ∆E = ǫn (ǫ)L2dǫ = F L2, (4.21) 2D ~2π 2 Z0 where ǫF is the Fermi energy. At the same time the number of electrons in the area L2 is ǫF m N = n (ǫ)L2dǫ = ǫ L2. (4.22) 2D ~2 F Z0 Thus, m ǫ2 1 ~2π 2 mL2 ~2π ∆E = F L2 = N = n2 , (4.23) ~2π 2 2 mL2 ~2π 2m n being the electron density. The kinetic energy functional for a non-uniform 2DEG is now approximated by ~2π T [n] T [=n] n(r)2dr (4.24) ≈ TF 2m Z 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, e2 n(r )n(r ) U [n] U =[n] 1 2 dr dr . (4.25) int ≈ H 2 r r 1 2 ZZ | 1 − 2| Finally, the last term is the energy due to interaction with some external potential vext(r)

Vext[=n] e n(r)vext(r)dr. (4.26) Z With the constraint n(r)dr = N (4.27) Z included through a Lagrange multiplier µ, which may be identiﬁed as the chemical potential, we arrive at

δ E[n] µ e n(r)dr N = 0 (4.28) − − Z and consequently get the Euler-Lagrange equation δE[n] ~2π n(r ) µ = = n(r) + e2 1 dr +v e ( r), (4.29) δn m r r 1 ext Z | − 1| 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 con- firms the use of the energy variational principle for these densities, i.e., for a trial densityn ˜(r) > 0, with the condition (4.27) fulfilled and E[n] defined in eq. (4.20), it is true that E ˜ E[n]. (4.30) 0 ≤ 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

N = n(r)dr, (4.31) Z 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- ′ 1 ternal potentials vext (r) and vext(r) , and thereby two diﬀerent wave functions Ψ and Ψ′, which yield the same electron density n(r). Using eq. (4.15) we can write

Ψ Hˆ Ψ = E < Ψ′ Hˆ Ψ′ = Ψ′ Hˆ ′ Ψ′ + Ψ′ Hˆ Hˆ ′ Ψ′ h | | i 0 h | | i h | | i h | − | i = E′ n(r) v (r) v′ (r) dr, (4.32) 0 − ext − ext Z and at the same time

Ψ′ Hˆ ′ Ψ′ = E′ < Ψ Hˆ′ Ψ = Ψ Hˆ Ψ + Ψ Hˆ ′ Hˆ Ψ h | | i 0 h | | i h | | i h | − | i = E n(r) v (r) v′ (r) dr, (4.33) 0 − ext − ext Z 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)] = Ψ Tˆ + Uˆ Ψ , (4.35) h | int| i where Tˆ and Uˆint signify the kinetic energy and e-e interaction operator. Ac- cording to the variational principle, the energy functional

E[Ψ] = Ψ (T + U ) Ψ + Ψ∗v Ψ(r) dr (4.36) h | int | i ext Z has a minimum for the ground state Ψ=Ψ0. For any other Ψ=Ψ˜

E[Ψ˜ ] = F [˜n] + v (˜r)n(r)dr F [n ] + v (r)n (r)dr =Ψ E[ ] (4.37) ext ≥ 0 ext 0 0 Z Z =˜ E[n] =E[n0] and we arrive| at (4.30).{z } | {z } The Hohenberg-Kohn theorems do not help us solve any speciﬁc many- body 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 veff ,(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 + v (r) ϕ =(r) ε ϕ (r), (4.38) −2m∇ eff i i i and summing n(r) = ϕ (r) 2. (4.39) | i | Xi 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

E[n] = Ts+[n] Veff [=n] Ts[+n] n(r)veff (r)dr, (4.40) Z which, with the constraint from eq. (4.27) included by a Lagrange multiplier ε, yields the Euler-Lagrange equation δE[n] δT [n] δV [n] = s + eff ε δn δn δn − δT [n] = s + v (0r) ε = . (4.41) δn eff − 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,

E [=n] T [n] T [+n] U [n] U [n]. (4.43) xc − s int − H 4.6. Local Density Approximation 31

Once again applying the variational principle with the constraint (4.27), gives

δE[n] δT [n] δU [n] δE δVext[n] = s + H + xc + ε δn δn δn δn δn − δT [n] n(r ) = s + e2 1 dr + v +(r) v 0(r) ε = . (4.44) δn r r 1 xc ext − Z | − 1| Comparing eq. (4.41) and eq. (4.44), we realise they are identical if we choose n(r ) v (=r) 1 dr + v +(r) v (r), (4.45) eff r r 1 xc ext Z | − 1| which also was the purpose with the reshuﬄing made in eq. (4.42). The Kohn- Sham procedure can now be summarised in four steps,

I) initially guess a density n0(r)

II) compute veff (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 2 Pauli principle – 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.,

Ex=[n] εx[n]n(r)dr, (4.47) Z 2No two electrons in a given system can be in states characterised by the same set of quantum numbers. 32 Electron-electron interactions

Ec[=n] εc[n]n(r)dr. (4.48) Z An expression for εx [n] was ﬁrst proposed by Dirac as an improvement to the Thomas-Fermi model[27]. For a 2DEG[110],

√2e2 εx=[n] 3 n(r), (4.49) −3π 2 ε0 p 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 1 rs = (4.50) a0√πn and a0 being the Bohr radius, Tanatar and Cerperly proposed[110] 2 C0e 1 + C1√rs εc[=n] 3/2 (4.51) −2 4πε0a0 1 + C √r + C r + C (r ) ∗ 1 s 2 s 3 s 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 + vσ (r) ϕσ=(r) ε ϕσ(r), (4.52) −2m∇ eff i i i where we differentiate between the two spin species σ = , . The spin-up/down ↑ ↓ densities are given by n↑=(r) ϕ↑(r) 2 (4.53) | i | Xi and n↓(=r) ϕ↓(r) 2 (4.54) | i | Xi and the total density n(r) = n↑(+r) n↓(r). (4.55) σ The effective potential veff (r) is n(r ) vσ (=r) e2 1 dr + vσ(+r) vσ(+r) vσ ( r). (4.56) eff r r 1 x c ext Z | − 1| 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

n↑ n↓ ζ = − (4.57) n↑ + n↓ as εx[ n,ζ], which may be expressed in terms of the unpolarised εx [n]. For a 2DEG[110] 1 ε [ n,ζ] = ε [ n] (1+ ζ)3/2 + (1 ζ)3/2 . (4.58) x 2 x − σ The exchange potential vx ( r) is then given by the functional derivative δE [n,ζ] vσ(=r) x , (4.59) x δnσ where

Ex[ n,ζ] = n(r)εx [n,ζ]dr. (4.60) Z σ vc ( r) is similarly obtained from δE [n,ζ] vσ=(r) c , (4.61) c δnσ 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 exchange- correlation 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 both the spin densities and their local gradients, i.e., Exc is of the form (c.f. with LSDA in eq. (4.59)-(4.61))

GGA 3 GGA Exc [n↑, n↓] = d rn(r)ǫxc (n↑, n↓, ∆n↑, ∆n↓ ). (4.62) Z Some additional extensions to “traditional” DFT are 34 Electron-electron interactions

Time-dependent DFT (TD-DFT) • “Traditional” DFT solves the time-independent Schr¨odinger 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, n as | i the direct product, m n = m n = mn , representing a state centred at | i ⊗ | i | i| i | i site m, n in the discretisation of the 2D semi-infinite waveguide shown in figure 5.1. The m, n fulfills the relationship for completeness and orthonormality | i ′ ′ m,n m, n = 1 and m, n m , n = δ ′ ′ . (5.1) | ih | h | i mn,m n m,n X An arbitrary state Ψ can be expanded in this basis as | i Ψ = c m,n , (5.2) | i mn| i m,n X where c 2 is the probability to find the electron at site m, n. The tight- | mn| binding Hamiltonian for an electron moving in a perpendicular magnetic field may now be written as

Hˆ = m, n (ε + V ) m,n | i 0 mn h | m,n X h + m, n t m, n + 1 + m, n t m, n 1 | i h | | i h − | + m, n te− iqn1m + , n + m, n teiqnnm 1, . (5.3) | i h | | i h − | i 36 Modelling

n=N

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 2~2 1 eBa2 ε = , t = ε , q = , (5.4) 0 m∗a2 −4 0 ~ 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, n to a mixed state representa- | i tion using the transversal lead eigenfunctions (c.f. with the continuous case in eq. 2.15), 2 πnα ϕ = sin α N. (5.5) n N + 1 N + 1 ∈ r Denoting these states m, α , where m signiﬁes the longitudinal position and | i α the transverse mode we can write the transformation between m, n and | i m, α as | i

m, α = m′, n′ m′, n′ m,α | i ′ ′ | ih | | i mnX, = m′, n′ m′, n′ m,α = n′ α m, n′ ′ ′ | i h | i ′ h | i i mnX, Xn 2 πn′α = sin m, n′ . (5.6) N + 1 N + 1 | i n′ r X 5.1. Tight-binding Hamiltonian 37

Carrying out the transformation gives (see Ref. [127] for further details on the Hamiltonian)

H = α,m ǫ α,m | i αh | m α X X + α, m tβα β,m + 1 + α, m tαβ β,m 1 + α, m (V + ǫ ) β,m | i r h | | i l h − | | i αβ 0 h | Xαβ (5.7) where[16] απ k = (5.8) α w w αβ 2t iqy απy βπy tr = dye sin sin (5.9) w 0 w w Z 1 eiqw)( 1 α+β 1 eiqw() 1 α−β = itqw − − − − (5.10) − w2q2 π2(α + β)2 − w2q2 π2(α β)2 − − − αβ αβ ∗ tl = (tr ) (5.11) 2 w απy βπy Vαβ = dy sin sin Vm(y) (5.12) w 0 w w Z ǫ = 4t (5.13) 0 − απ h2 απ 2 ǫ = 2t cos +. 2t (5.14) α w ≈ 2m∗ w w is the width of the wire and ∗ denotes complex conjugate.

5.1.2 Energy dispersion relation We will now consider the energy dispersion relation for the tight-binding Hamiltonian in zero magnetic ﬁeld. Expanding the solution Ψ to eq. (5.3) as | i Ψ = ψ m,n (5.15) | i mn| i m,α X and substituting this back into eq. (5.3) yields

ε0ψmn +ψ t m1 + ,n +ψ t mn −1, +ψ t 1m,n+ +ψ t m,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 = φmϕn, (5.17) with corresponding longitudinal and transverse energy components,

E = Ek + E⊥ = Ekk + Eα. (5.18) 38 Modelling

1 Continuous lattice Discrete lattice ∗ 2 m ~ 2 0.5 Energy

0 −1 -0.5 0 0.5 1 kka 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

~2 φ 2φ + φ 1m+ − m m−1 = E φ , (5.19) −2m∗ a2 kk m ~2 ϕ 2ϕ + ϕ 1n+ − n n−1 = E ϕ . (5.20) −2m∗ a2 α n These are discrete derivatives, and we will get travelling wave solutions in the parallel direction and quantised solutions in the transverse direction,

φ = eika km k R (5.21) m k ∈ 2 πnα ϕ = sin α N. (5.22) n N + 1 N + 1 ∈ r

Substituting φm back into eq. (5.19) and ϕn back into eq. (5.22) we get the dispersion relations,

~2 transverse : E = 2 2 cos(k a) , (5.23) kk 2m∗a2 − k ~2π2, α2 longitudinal : E = (5.24) α 2m∗w2 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 k a 1, eq. (5.23) approximates to ⊥ k ≪ the continuous dispersion relation (see ﬁgure 5.2), 2 2 ~2 2 ~ (kka) kk Ekk ∗ 2 2 2 1 = ∗ . (5.25) ≈ 2m a − − 2 !! 2m 5.2. Green’s function 39

Knowing the energy dispersion relation allows us to compute the group velocity, vg, as

dω 1 dE 1 d ~2 ~ v = = = 2 2 cos(k a) = sin(k a). g dk ~ dk ~ dk 2m∗a2 − k m∗a k k k k (5.26)

5.2 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 which relates an excitation in some system Dop S to the system response by R = , (5.27) DopR S the Green’s function is deﬁned as

= −1. (5.28) G Dop 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 (5.29) (E H)Ψ =I (5.30) − Ψ =(E H)−1I (5.31) − = Ψ =(E H)−1. (5.32) ⇒ G − 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], = (E H + iδ)−1. (5.33) G − 40 Modelling

Perturbation

H0 H0 1 2

0 0 Figure 5.3: Two systems with| respective{z } Hamiltonian| {z H}1 and H2 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. H0 H0 Considering two adjacent systems 1 and 2, ﬁgure 5.3, and treating hopping between them as a small perturbation V , we can write the Hamiltonian for the combined system as

H H0 H0 H0 = 1 + 2 +V = .+ V (5.34) =H0

From equation (5.32) we have| {z }

−1 −1 = E H = E H0 V = 0 , V (5.35) G − − − G − where 0 is the Green’s function of H0. Multiplying equation (5.35) with 0 G G from the right and from the left (or vice versa) yields the Dyson equation(s) G = 0 + 0V (5.36) G G G G ( = 0 + V 0). (5.37) G G G G These equations implicitly deﬁne in terms of the known, unperturbed, 0. G G For explicit expressions for the Green’s function between two speciﬁc points r,r′, we introduce the notation

′ ′ ′ ′ (r, r ) = r r = m,n m , n = ′ ′ . (5.38) G h |G| i h |G| i Gmn,m n ′ Similarly ′ is interpreted as m m . The matrix ′ can then be Gmm h |G| i Gmm written as 0 0 ′ = ′ + V ′ . (5.39) Gmm Gmm Gmi ijGjm Xi,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 (a) (b)

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

V = 1 t e2 = 1 t iqn 2 (5.41) 12 | i 12h | | i h | V = 2 t e1 = 2 t − iqn 1 . (5.42) 21 | i 21h | | i h | The Green’s functions for the combined function may now be found through eq. (5.39). For example,

= 0 + 0 V (5.43) G11 G11 G11 12G21 = 0 + 0 V , (5.44) G21 G21 G22 21G11 =0 substituting back and forth yields,|{z} after some rearrangements

= (I 0 V 0 V )−1 0 (5.45) G11 G − 11 12G22 21 G11 = 0 V (I 0 V 0 V )−1 0 . (5.46) G21 G22 21G − 11 12G22 21 G11 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 ψ m,α at slice 0 0α| i in figure 5.4b gives the response

Ψ = ψ α0, . (5.47) | i G 0α| i 42 Modelling

Because there can be no mixing between diﬀerent modes α in a homogenous wire we temporarily suppress index α and write m, α m . Ψ is now | i → | i | i expanded as Ψ = ψ m = ψ 0 . (5.48) | i m| i G 0| i m X Multiplying from the right with m gives the response at site m h | ψ = m ψ 0 = ψ . (5.49) m G h | 0| i Gm0 0 From the Dyson equation (5.37) we ﬁnd as Gm1 + ,0 = 0 + 0 V (5.50) G1m+ ,0 Gm+1,0 Gm+1,m+1 m1 + ,mGm0 =0 and thereby | {z }

ψ = ψ = 0 V ψ m1 + Gm1 + ,0 0 Gm+1,m+1 m1 + ,mGm0 0 0 = m+1,m+1Vm1 + ,mψm . (5.51) eq9 . (5.4 ) G

The potential in the wire is periodic and we apply the Bloch theorem,

ika αm ψm = e k um, (5.52)

α where um is periodic, um = u1m+ , and kk is the wave vector from eq. (2.16) associated with the transverse mode α. From eq. (5.51) we get

ikα(m+1)a ika αm e k = 0 V e k (5.53) Gm+1,m+1 m1 + ,m ikαa e k = 0 V . (5.54) Gm+1,m+1 1m+ ,m ˜ 0 Deﬁning the surface Green’s function as Γ = Gm+1,m+1, where tilde reminds us that this is in mixed representation and noting that for zero magnetic ﬁeld Vm1 + ,m = t, we obtain, ikαa e k = tΓ˜. (5.55)

With all N modes α1, α2, . . . , αN , the surface Green’s function is a diagonal matrix α ik 1 a e k 0 α ik··2 a ·  k  ˜ 1 0 e 0 Γ = . . (5.56) t  . 0 ..   α   ik N a  e k    which may be transformed to real space representation by

′ ′ ′ ′ ′ Γ(mn, m n )n = m Γ m n = n Γ ′ n h | | i h | mm | i 5.2. Green’s function 43

G′ ′ m1 Gm+1,M ′ G1m ′ GM,m+1

Γ G′ 11 ′ GMM Γ 1 m m + 1 M

′ ′ Gmm 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, mm, for the full system. The G diagonal elements of mm is proportional to the density of states (eq. (5.58)). ℑ G

′ ′ ′ = n α α Γmm′ α α n ′|h | ih | | ih || i Xα,α = α n ∗Γ˜(mα, m′α′) α′ n′ , (5.57) ′h | i h | i Xα,α 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 while the Green’s function Gkk between slice k and l by the matrices and . Starting from the surface Gkl Glk Green’s function Γ, we recursively add slice by slice until we have computed the Green’s functions ′ and ′ indicated in ﬁgure 5.5. Using the Gmm Gm+1,m+1 Dyson equation we may now compute the total Green’s function for slice Gmm m. The imaginary part of the diagonal elements in this matrix ( [ (r, r)]) are ℑ G proportional to the density of states[33],

1 n(E, r) = (r, r,E) . (5.58) −π ℑ G h i Integrating the density of states up to the Fermi-energy we ﬁnd the electron density inside the dot,

1 Ef n(r) = E(r, r, )dE . (5.59) −π ℑ G Z−∞ 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

20 x 10 ( ) ℜ(G) 0 ℑ G

) -5 r , r E,

( -10

G ℑ

-15 ℑ ℜ ℜ E0 EF E0 EF -20 0.8 0.9 1 0.8 0.9 1 E/E E(E)/ F ℜ F 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 veff (r) by equation (4.45)

III) update the Hamiltonian (eq. (5.3)) with the new veff (r)

IV) update the Green’s function with the new Hamiltonian (eq. (5.32))

V) update the density n(r) through eq. (5.59)

VI) 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,

n (=r) ǫn (r) + (1 ǫ)n (0r) < ǫ 1, (5.60) new new − old ≤ thereby dampening any sudden change in the density. Furthermore, the Green’s function has some very sharp features along the real axis (see the left panel of ﬁgure 5.6), making numerical integration very diﬃcult. Since all poles to the Green’s function are below the real axis this may be remedied 5.2. Green’s function 45

20 x 10 T B 15

(a) /k (b) 2 ( ) ) 10

ℜ G E

( ) ( 0 5 ℑ G ℑ FD

f -2 ) 1 × (c) E ( G -4 0 FD f -6 -1 -20 -10 0 -20 -10 0 ( (x) E )k/ T ( (x) E k) / T ℜ − F B ℜ − F B 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 (fFD) 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 ∞ neE(r, F =,T ) n, (E r) fFD(,EE F ,T ) dE, (5.61) −∞ Z DOS F −Ddist. where fFD is the Fermi-Dirac distribution.| {z } | We may{z 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 neff ( r) we compute the Green’s function relating the first and last slice as shown in figure 5.8, . In mixed represen- GM1 tation 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],

β β i(kαm−k n) t = i2 t sin kαa sin k a e k k ˜ (E), (5.62) β←α | | k k GMβ,1α r where ˜ (=E) Mβ ˜(E) 1α , and tilde indicates that it is the Green’s GMβ,1α h |G | i function in mixed representation. At zero magnetic ﬁeld the amplitude for the reﬂected electron is similarly[33]

β α β α ia(kk +kk ) rβ←α = i sin kk a sin kk a e r . 46 Modelling

′ ′ G1,M−1 GM−1,1

G′ ′ M−1,M−1 Γ G11 ′ GMM Γ 1 M − 1 M

11 G GM1 Figure 5.8: With the self-consistent density n(r), we compute the Green’s functions M1 ( 11) for the full system, which in mixed representation gives the transmissionG G amplitudes by eq. (5.62) (reﬂection amplitudes by eq. (5.63)).

2 t sin kβa ˜ (+E) iδ , (5.63) × | | k G1, β 1α βα h i 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 Hamil- tonian 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 Closed dot 1.45 (a) 1.5 (b)

0.8 1.4 360nm 0.6

1.25 1.3 0.4 k (units π /w) k (units π /w) 1.2 0.2 210nm kF 1.05 1.1 Open dot 2e2 0 0.2 0.4 0 0.2 0.4 G( ) B (Tesla) h B (Tesla) 80nm U=3t U=3t (c) (d) 1.56 1.2 360nm

1.52 1.15

k (units π /w) 1.48 k (units π /w) 1.1 0 0.05 0.1 0.1 0.2 0.3 B (Tesla) 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.

(a) -0.4 (b) -0.62 4

-0.5 3 G G V V

-0.6 2 -0.66 1

0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 2e2 B (Tesla) B (Tesla) G( h )

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 14 −2 corresponds to an average electron density of ne = 2 10 m , 8 1014m−2 and 2 1015m−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 n 0= 1 1 15m−2, compared with e ∼ × n =0 1 1 14m−2 for the dots in Evaldsson et al.. The breaking of spin e ∼ × 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 exchange- correlation 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 aren ¯(r)/1014m−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 1014m−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 fo- cusing 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 E 600 A) F 500 −0.02

400 −0.04

nm 300 −0.06

200 −0.08

100 −0.1 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000

700 600 B) 500

400

nm 300

200

100

0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 nm

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 nanome- tre 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 1 wire. Because of the Zeeman term, 2 gµEσB, 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 Zee- man 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 x- direction 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)

µ0M magn. ﬁlm side view spacer z 2DEG x substrate

top view µ0M y spacer magn. ﬁlm x µ0M 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 µ0M, 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 µ0M = 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 (Pa- per 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.

1I.e., their edge structure. 6.7. Paper VII 55

6.7 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 oth- erwise 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 Bibliography

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