Electron Transport, Interaction and Spin in Graphene and Graphene Nanoribbons

Linköping Studies in Science and Technology Dissertation No. 1469

Electron transport, interaction and spin in graphene and graphene nanoribbons

Artsem Shylau

Department of Science and Technology Linköping University, SE-601 74 Norrköping, Sweden

Norrköping 2012 Cover illustration: zigzag graphene nanoribbon attached to leads

Electron transport, interaction and spin in graphene and graphene nanoribbons

Department of Science and Technology Linköping University, SE-601 74 Norrköping, Sweden

ISBN 978-91-7519-816-3 ISSN 0345-7524

Printed by LiU-Tryck, Linköping 2012 To my parents, Valentina and Alexander.

Abstract

Since the isolation of graphene in 2004, this novel material has become the major object of modern condensed matter physics. Despite of enor- mous research activity in this field, there are still a number of fundamental phenomena that remain unexplained and challenge researchers for further investigations. Moreover, due to its unique electronic properties, graphene is considered as a promising candidate for future nanoelectronics. Besides experimental and technological issues, utilizing graphene as a fundamental block of electronic devices requires development of new theoretical methods for going deep into understanding of current propagation in graphene constrictions. This thesis is devoted to the investigation of the effects of electron- electron interactions, spin and different types of disorder on electronic and transport properties of graphene and graphene nanoribbons. In paper I we develop an analytical theory for the gate electrostatics of graphene nanoribbons (GNRs). We calculate the classical and quantum capacitance of the GNRs and compare the results with the exact self- consistent numerical model which is based on the tight-binding p-orbital Hamiltonian within the Hartree approximation. It is shown that electron- electron interaction leads to significant modification of the band structure and accumulation of charges near the boundaries of the GNRs. It’s well known that in two-dimensional (2D) bilayer graphene a band gap can be opened by applying a potential difference to its layers. Calcula- tions based on the one-electron model with the Dirac Hamiltonian predict a linear dependence of the energy gap on the potential difference. In paper II we calculate the energy gap in the gated bilayer graphene nanoribbons (bGNRs) taking into account the effect of electron-electron interaction. In contrast to the 2D bilayer systems the energy gap in the bGNRs depends non-linearly on the applied gate voltage. Moreover, at some intermediate gate voltages the energy gap can collapse which is explained by the strong modification of energy spectrum caused by the electron-electron interactions. Paper III reports on conductance quantization in grapehene nanoribbons subjected to a perpendicular magnetic field. We adopt the recursive Green’s function technique to calculate the transmission coefficient which iv is then used to compute the conductance according to the Landauer approach. We find that the conductance quantization is suppressed in the magnetic field. This unexpected behavior results from the interaction- induced modification of the band structure which leads to formation of the compressible strips in the middle of GNRs. We show the existence of the counter-propagating states at the same half of the GNRs. The overlap between these states is significant and can lead to the enhancement of backscattering in realistic (i.e. disordered) GNRs. Magnetotransport in GNRs in the presence of different types of disorder is studied in paper IV. In the regime of the lowest Landau level there are spin polarized states at the Fermi level which propagate in different directions at the same edge. We show that electron interaction leads to the pinning of the Fermi level to the lowest Landau level and subsequent formation of the compressible strips in the middle of the nanoribbon. The states which populate the compressible strips are not spatially localized in contrast to the edge states. They are manifested through the increase of the conductance in the case of the ideal GNRs. However due to their spatial extension these states are very sensitive to different types of disorder and do not significantly contribute to conductance of realistic samples with disorder. In contrast, the edges states are found to be very robust to the disorder. Our calculations show that the edge states can not be easily suppressed and survive even in the case of strong spin-flip scattering. In paper V we study the effect of spatially correlated distribution of impurities on conductivity in 2D graphene sheets. Both short- and long-range impurities are considered. The bulk conductivity is calculated making use of the time-dependent real-space Kubo-Greenwood formalism which allows us to deal with systems consisting of several millions of carbon atoms. Our findings show that correlations in impurities distribution do not significantly influence the conductivity in contrast to the predictions based on the Boltzman equation within the first Born approximation. In paper VI we investigate spin-splitting in graphene in the presence of charged impurities in the substrate and calculate the effective g-factor. We perform self-consistent Thomas-Fermi calculations where the spin effects are included within the Hubbard approximation and show that the effective g-factor in graphene is enhanced in comparison to its one-electron (noninteracting) value. Our findings are in agreement to the recent experimental observations. Populärvetenskaplig sammanfattning

Anda¨ sedan isoleringen av grafen ˚ar 2004, har detta nya material blivit den viktigaste förem˚alet för den moderna kondenserade materiens fysik. Trots enorm forskning inom detta omr˚ade finns det fortfarande ett an- tal grundläggande fenomen som förblir oförklarade och utmanar forskare för vidare undersökningar. Dessutom, p˚agrund av dess unika elektroniska egenskaper, anses grafen vara en lovande kandidat för framtida na- noelektronik. Förutom experimentella och teknologiska fr˚agor, kan grafen användas som ett grundläggande block av elektroniska komponenter som kräver utveckling av nya teoretiska metoder för att fördjupa först˚aelsen av ström utbredning i nanostrukturer av grafen. Denna avhandling tillägnar ˚at utredningen av effekterna av elektron- elektron växelverkan, spin och olika typer av oordning inom elektroniska och transport egenskaper hos grafen och grafen nanoremsor. vi Acknowledgments

Despite of my primary background in applied sciences I always wished to be a theoretical physicist, because I was always impressed by the fact that human mind is able to unravel Nature secrets just with the use of a piece of paper and a pencil (or a computer nowadays). This dream got fulfilled in Sweden where I spent four years as a PhD student at ITN LiU doing my research in theoretical physics. Tack, Sverige! I would like to thank a lot of people who surrounded me during this period. First of all, I would like to thank Prof. Igor Zozoulenko for his great supervision, significant contribution to my scientific development and for motivation when it was really needed. I am thankful to the research administrator Ann-Christin Norén and Elisabeth Andersson for their administrative help during my research work. I had a fruitful collaboration with my colleagues from Germany, Dr. Hengyi Xu and Prof. Thomas Heinzel, and my polish colleague Dr. Jaros law K los. Besides the science there were a lot of other things happening in my life. I met a lot of nice people here, some of them eventually became my friends. First of all, I am grateful to Olga Bubnova for our inspiring philosophical discussions, pleasant lunch time and just for being a good friend. I am also thankful to Lo¨ıg, Anton, Brice, Sergei, Julia and especially Taras for a good company and funny talks. All this time I kept in touch with my belarusian friends, Sergei and Alex. We had a good time together whenever I went home, to my lovely Minsk. Finally, I would like to thank my wife Marina for her love, understanding and patience; my parents and my sister’s family for their love and support which I feel every day no matter where I am.

Artsem Shylau Norrk¨oping, August 2012 viii List of publications

Publications included in the thesis

1. A. A. Shylau, J. W. Klos, and I. V. Zozoulenko, Capacitance of graphene nanoribbons, Phys. Rev. B 80, 205402 (2009). Author’s contribution: Implementation of the self-consistent numerical model and all numerical calculations, partial contribution to development of the analytical model. Preparation of all the ﬁgures. Initial draft of the paper.

2. Hengyi Xu, T. Heinzel, A. A. Shylau, I. V. Zozoulenko, Interactions and screening in gated bilayer graphene nanoribbons, Phys. Rev. B 82, 115311 (2010); selected as ”Editor’s Suggestion”. Author’s contribution: Development of the analytical model. Dis- cussion and analysis of all the obtained results.

3. A. A. Shylau, I. V. Zozoulenko, Hengyi Xu, T. Heinzel, Generic suppression of conductance quantization of interacting electrons in graphene nanoribbons in a perpendicular magnetic ﬁeld, Phys. Rev. B 82, 121410(R) (2010). Author’s contribution: All numerical calculations and preparation of all the ﬁgures. Discussion of the results and writing an initial draft.

4. A. A. Shylau and I. V. Zozoulenko, Interacting electrons in graphene nanoribbons in the lowest Landau level, Phys. Rev. B 84, 075407 (2011). Author’s contribution: All numerical calculations and preparation of all the ﬁgures, discussion of the results and writing an initial draft.

5. T. M. Radchenko, A. A. Shylau, and I. V. Zozoulenko, Inﬂuence of correlated impurities on conductivity of graphene sheets: Time- dependent real-space Kubo approach, Phys. Rev. B 86, 035418 (2012). Author’s contribution: Contribution to implementation of the numerical model, performing initial numerical calculations, discussion of all results. x

6. A. V. Volkov, A. A. Shylau, and I. V. Zozoulenko, Interaction- induced enhancement of g-factor in graphene, arXiv:1208.0522v1 [cond- mat.mes-hall], submitted to PRB. Author’s contribution: Contribution to development of the model, discussion of all results, writing a part of the manuscript, supervision of the numerical calculations.

Relevant publications not included in the thesis

1. J. W. Klos, A. A. Shylau, I. V. Zozoulenko, Hengyi Xu, T. Heinzel, Transition from ballistic to diﬀusive behavior of graphene ribbons in the presence of warping and charged impurities, Phys. Rev. B 80, 245432 (2009).

2. T. Andrijauskas, A. A. Shylau, I. V. Zozoulenko, Thomas-Fermi and Poisson modeling of the gate electrostatics in graphene nanoribbon, Lith. J. of Phys. 52, 63 (2012). Contents

1 Introduction 1

2 Quantum transport 3 2.1 Kuboformalism...... 4 2.2 Kubo-Greenwoodformula ...... 6 2.3 Landauerapproach ...... 7 2.4 S-matrixtechnique ...... 9

3 Electron-electron interactions 11 3.1 Themanybodyproblem ...... 11 3.2 Hartree-Fockapproximation ...... 12 3.3 Density-functionaltheory...... 14 3.4 Kohn-Shamequations ...... 15 3.5 Thomas-Fermi-Dirac approximation ...... 16 3.6 Hubbardmodel ...... 17

4 Electronic structure and transport in graphene 19 4.1 Basic electronic properties ...... 19 4.2 Graphenenanoribbons ...... 24 4.3 Warping...... 27 4.4 Bilayer graphene ...... 28 4.5 Dirac fermions in a magnetic ﬁeld ...... 31

5 Modeling 35 5.1 Tigh-binding Hamiltonian and Green’s function ...... 35 5.2 Recursive Green’s function technique ...... 38 5.2.1 Dysonequation ...... 38 5.2.2 Bloch states ...... 40 5.2.3 Calculation of the Bloch states velocity ...... 42 5.2.4 Surface Green’s function ...... 43 5.2.5 Transmission and reflection ...... 44 5.3 Real-spaceKubomethod...... 45 5.3.1 Diffusion coefficient ...... 45 5.3.2 Transportregimes...... 46 xii Contents

5.3.3 Time evolution ...... 48 5.3.4 Chebyshevmethod ...... 49 5.3.5 Continued fraction technique ...... 51 5.3.6 Tridiagonalization of the Hamiltonian matrix . . . . . 53 5.3.7 Localdensityofstates ...... 54

6 Summary of the papers 55 6.1 PaperI...... 55 6.2 PaperII ...... 56 6.3 PaperIII...... 57 6.4 PaperIV...... 58 6.5 PaperV ...... 59 6.6 PaperVI...... 59 Chapter 1

Introduction

Inside every pencil, there is a neutron star waiting to get out. To release it, just draw a line. (New Scientist, 2006) In 2004 the researchers from Manchester University, Kostya Novoselov, Andre Geim and collaborators, reported on experimental isolation of graphene [1], a pure 2D crystal consisting of carbon atoms arranged in a honey-comb lattice. For a long time before graphene had been considered only by the- oreticians as a basic block used to build theory for graphite [2] and carbon nanotubes [3]. Its existence was doubt since the theory predicted that perfect 2D crystals are not thermodynamically stable [4]. The discovery of graphene triggered a great scientific interest in this field, as a result graphene is one of the most extensively studied object in a modern condensed matter physics [5]. Due to its specific lattice structure graphene posses a number of unique electronic properties which make this material interesting for both theoreti- cians, experimentalists and engineers. One of the most important feature of graphene is its linear energy spectrum. This kind of spectrum is known from high-energy physics where it corresponds to massless particles like neutrino. Relativistic-like dispersion relation is responsible for such effects as Klein tunneling - unimpeded penetration of particle through the in- finitely large potential barrier [6]. The experimental discovery of graphene had led to emergence of a new paradigm of ’relativistic’ condensed-matter physics and provided a way to probe quantum electrodynamics phenomena [4]. That is why graphene is sometimes called ”CERN on a desk”. Being massless fermions electrons in graphene propagate at extremely high velocities, only 300 times smaller than the velocity of light. This makes graphene the best known conductor with mobility up to 200,000 2 1 1 cm V− s− at room temperatures. Graphene subjected to a perpendicular magnetic field exhibits the anomalous quantum Hall effect [7]. This is a result of unusual spectrum quantization and the presence of the 0’th Landau 2 Introduction level, which is equally shared between electrons and holes. Also fractional quantum Hall effect has been experimentally observed in graphene [8]. The range of possible applications of graphene is very broad. With its high mobility graphene is considered as the main candidate for a future post-silicon electronics [9]. It is particularly interesting to use graphene in transistors operating at ultrahigh radio frequencies [10]. Due to its high optical transmittance ( 97.7%) graphene is proposed to be used as a flexi- ble transparent electrode≈ in touchscreen devices [11]. Also graphene posses a broad spectral bandwidth and fast responce times, which makes this material attractive for optoelectronics and, in particular, phototransistors [12]. Chapter 2

Quantum transport

Electrical transport is a non-equilibrium statistical problem. In principle, one can solve the time-dependent Schr¨odinger equation

∂ Ψ(~r, t) i~ | i = Hˆ Ψ(~r, t) (2.1) ∂t | i to ﬁnd a many-body state of the system Ψ(~r, t) at any time and then calculate the expectation value of the current| operatori

Iˆ = ˆj(~r, t) dS.~ (2.2) · ZS In practice, however, the calculation of the many-body state is an unfea- sible task. Moreover, the wave function representing the state provides a detailed information about the system which is often redundant for determination of transport properties. Hence, one needs to introduce a number of approximations in order to simplify the above problem. Prior doing this it is useful to formulate viewpoints underlying quantum transport theories [13]: Viewpoint 1: The electrical current is a consequence of an applied electric field: the field is the cause, the current is the response to this field. Viewpoint 2: The electrical current is determined by the boundary conditions at the surface of the sample. Charge carriers incident to the sample boundaries generate self-consistently an inhomogeneous electric field across the sample. Thus the field is a consequence of the current. In this chapter two approaches for calculation of transport properties are discussed, namely, the Kubo formalism and the Landauer approach. The first one belongs to the viewpoint 1, while the latter belongs to the viewpoint 2. 4 Quantum transport

2.1 Kubo formalism

Kubo formula relates via linear response the conductivity to the equilibrium properties of the system. First we deﬁne the expectation value of the current density operator ~j = Tr ~jρ , (2.3) h i where ρ is a statistical operator h i

H H 1 ρ = Z− e− kB T ,Z = Tr e− kB T . (2.4) h i The Hamiltonian, H, describing the system can be split into two parts

H = H + δH, 0 (2.5) ρ = ρ0 + δρ, where H0 corresponds to the system in a global cannonical equilibrium described by the statistical operator ρ0. External electric field introduces perturbation in the system which is described by the terms δH and δρ. Here we focus on the case of uniform electric field [14]. It’s assumed that the field is applied at time t = and reaches adiabatically its steady value at t = 0 −∞ ( iωt+αt) δH = lim eE~re~ − . (2.6) α 0 → h i Substituting ρ from Eq.(2.5) into Eq.(2.3), we have

~j = Tr ~j(ρ + δρ) = Tr ~jδρ , (2.7) h i 0 h i h i where we took into account that Tr ~jρ0 = 0, since there is no current in the system before applying the externalh i ﬁeld. If we substitute ρ given by Eq.(2.5) into the Liouville equation, i~ρ˙ = [H, ρ], which describes the dynamics of a closed quantum systems, then the change of δρ in time can be found as i~δρ˙ = [H0, δρ] + [δH, ρ0], (2.8) where we used that i~ρ˙0 = [H0, ρ0] and neglected the term [δH, δρ]. Let us now pass to the interaction picture representation of time-dependence of an operator iH0t iH0t δρ = e− ~ ∆ρe ~ . (2.9) Substituting Eq.(2.9) into the left part of Eq.(2.8), we arrive at

iH0t iH0t ~ ~ i~∆ρ = e [δH, ρ0]e− iH0t iH0t (2.10) ( iωt+αt) ~ ~ = limα 0 e − e [e~r, ρ0]e− E~ . → h i Kubo formalism 5

Both δρ and ∆ρ satisfy the following conditions: have the same value at time t = 0, δρ(0) = ∆ρ(0), and equal to zero at t = , δρ( ) = ∆ρ( ) = 0. Thus, integrating Eq.(2.10), we get −∞ −∞ −∞ 0 1 iH0t iH0t ( iωt+αt) ~ ~ δρ(t = 0) = lim dte − e [e~r, ρ0]e− E.~ (2.11) i~ α 0 → Z−∞ Substituting Eq.(2.11) into the expression (2.7) for the expectation value of the current density, we obtain

0 1 iH0t iH0t ( iωt+αt) ~ ~ ~j = Tr ~jδρ = lim dte − Tr ~je [e~r, ρ0]e− E~ . ~ α 0 h i i → h i Z−∞ h i(2.12) The conductivity tensor is deﬁned as

j σ σ E x = xx xy x , (2.13) j σ σ E y yx yy y which allows to deduce from Eq.(2.12) the components of the tensor

0 ( iωt+αt) σµν = lim dte − Kµν, (2.14) α 0 → Z−∞ where 1 iH0t iH0t K = Tr j~ e ~ [er , ρ ]e− ~ . (2.15) µν i~ µ ν 0 Equation (2.15) can be writtenh in another form ifi we use the following relation [14]:

1/kB T λH0 λH0 [rν, ρ0] = ρ0 dλe [H0, rν]e− . (2.16) Z0 Taking into account that [H , r ] = i~r˙ and er˙ = j , we get 0 ν − ν − ν ν

1/kB T λH0 λH0 [erν, ρ0] = i~ρ0 dλe jν e− . (2.17) Z0 Finally, using Heisenberg representation for time-dependence of the current

iH0t iH0t ~j(t) = e ~ ~je− ~ , (2.18) we obtain the Kubo formula for conductivity which is formulated in terms of the current-current response function

1/kB T K = dλ j (0)j (t i~λ) . (2.19) µν h µ ν − i Z0 6 Quantum transport

2.2 Kubo-Greenwood formula

Equations (2.14) and (2.19), which constitute the Kubo formula for conductivity are very important, since they reflect the underlying physics of the linear responce theory. However, these equations are not suitable for a particle calculations and one needs to work out another form of the conductivity formula. As in the previous section we start with splitting the Hamiltonian, H, into two parts H = H0 + δH corresponding to the system in the equilibrium and the perturbation respectively. Hence, if the full Hamiltonian is defined as 1 H = (~p + eA~)2 + eφ (2.20) 2m and the perturbation is incorporated in the vector potential, A~ = A~0 +A~ext, we get for δH: δH = eA~ ~v. (2.21) ext · If we assume the time dependence of the external electric field to be E~ (t) = iωt dA~ Ee~ − , then using the relation E~ = , we arrive at − dt e δH = E~ ~v. (2.22) iω · The expectation value of the current operator, j = Tr ρˆj , can be written h i as h i 2 j = k δρ k′ k′ ˆj k = V dEdE′g(E)g(E′) E δρ E′ E′ ˆj E , h i h | | ih | | i h | | ih | | i kk′ ZZ X (2.23) where g(E) is a density of states and we used Tr[ρ0ˆj] = 0. If we assume δρ(t) = δρ eiωt and use Eq.(2.8), then we get · fFD(E′) fFD(E) E′ δρ E = − E′ δH E , (2.24) h | | i E E ~ω i~αh | | i ′ − − − where α is a small constant and we used that ρ0 E = fFD(E) E [14]. Sub- | i ~ e| i stituting Eq.(2.24) into Eq.(2.23) and recalling that j = V ~v, we obtain the expectation value of the current operator −

2 e e j = V dEdE′g(E)g(E′) h i iω −V ZZ fFD(E′) fFD(E) − E′ E~ ~v E E ~v E′ . (2.25) × E E ~ω i~αh | · | ih | | i ′ − − − This equation allows us to derive the components of the conductivity tensor e2 σ (ω) = Re V dEdE′g(E)g(E′) ij −iω ZZ fFD(E′) fFD(E) − E′ ~v E E ~v E′ , (2.26) × E E ~ω i~αh | i| ih | j| i ′ − − − Landauer approach 7 which can be further simpliﬁed using the relation

1 1 Re lim = πδ(E′ E ~ω) (2.27) α 0 i E E ~ω i~α − − → ′ − − − and performing integration over E′

e2πV σ (ω) = g(E)g(E + ~ω) E + ~ω ~v E E ~v E + ~ω ij − ω h | i| ih | j| i Z [f (E + ~ω) f (E)]dE. (2.28) × FD − FD If one is interested in DC conductivity, Eq.(2.28) should be considered in fFD(E+~ω) fFD(E) ∂fFD(E) a limit ω 0, limω 0 ~ − = , → → ω ∂E ∂f (E) σ (ω) = e2πV ~ g(E) 2 E ~v E E ~v E FD dE. (2.29) ij | | h | i| ih | j| i − ∂E Z Finally, if we consider the case of a very low temperature T 0, the derivative of the Fermi-Dirac function can be substituted by the→ delta function ∂fFD(E) ∂E = δ(E EF ), which simpliﬁes the integration over E. Thus, recalling that g(E−) = Tr[E H], we obtain the relation for the conductivity tensor − e2π~ σ = Tr [v δ(E H)v δ(E H)] . (2.30) ij V i F − j F − This equation is a starting point of the numerical real-space time-dependent Kubo method which will be described in details in the Chapter 5.

2.3 Landauer approach

Landauer approach belongs to the viewpoint 2, i.e. a constant current is forced to flow through a scattering system and the asking question is what the resulting potential distribution will be due to the spatially inhomogeneous distribution of scatters [15]. Calculation of the current in the Landauer approach requires to divide formally the system into three parts, namely, the perfect leads and the scattering region, as depicted on Fig.(2.1). The leads, in turn, are connected to the infinite reservoirs which represent the infinity and contain many electrons in a local equilibrium characterized by the Fermi-Dirac distribution function. The basic idea behind this approach is that the electron has a certain probability to transmit through the scattering region [16]. Hence, the current carrying by an electron in a state with a wave-vector k is

J = ev(k) (k) (2.31) k T 8 Quantum transport

Left Sample Reservoir Right Reservoir lead (scattering region) lead

0 L

Figure 2.1: a) Schematic illustration of the system which consists of the scattering region connected to perfect leads. The leads itself are coupled to microscopic reservoirs. b) Potential proﬁle. The left and right leads have a constant potential µL and µR, respectively, equal to the chemical potential in the reservoirs. with (k) being a transmission probability. Full current supplied by the left leadT is a sum over all states

IL = 2 JkfFD(E(k), µL), (2.32) Xk where the factor 2 is due to spin-degeneracy, µL is a chemical potential in the left lead and 1 fFD(E(k), µ) = (2.33) E(k) µ 1 + exp − kB T is the Fermi-Dirac distribution function. Hence, we have

1 IL = 2e v(k) (k)fFD(E(k), µL) = dk (2.34) T " → 2π # Xk Xk Z 2e ∞ = v(k) (k)f (E(k), µ )dk. (2.35) 2π T FD L Z0 dk Changing integration variables by dk = dE dE and using the expression for 1 dE the group velocity v = ~ dk , we get

2e ∞ I = (E)f (E, µ )dE. (2.36) L h T FD L ZUL Similarly, the current supplied by the right lead is

2e ∞ I = (E)f (E, µ )dE. (2.37) R − h T FD R ZUR S-matrix technique 9

Sum of both contributions gives the net current

2e ∞ I = I + I = (E)[f (E, µ ) f (E, µ )] dE. (2.38) L R h T FD L − FD R ZUL The potential drop between the reservoirs is

eV = µ µ . (2.39) L − R In the case of very low bias, the Fermi-Dirac functions can be expanded in the Taylor series ∂f (E, µ) f (E, µ ) f (E, µ ) eV FD , (2.40) FD L − FD R ≈ − ∂E which results in

2e ∞ ∂f (E, µ) I = (E) eV FD dE. (2.41) h T − ∂E ZUL This equation allows to calculate conductance of the system

2 I 2e ∞ ∂f (E, µ) G = = (E) FD dE. (2.42) V h T − ∂E ZUL At a very low temperature the derivative of the Fermi-Dirac distribution function can be replaced by the Dirac delta function δ(E µ) which reduces − Eq.(2.42) to 2e2 G = (µ). (2.43) h T This equation shows that the conductance of the perfect conductor (i.e. 1 = 1) is ﬁnite and thus the resistance (G− ) is non-zero. The following T explanation can be used [16]: in the contacts (reservoirs) the current is carried by inﬁnitely many transverse modes, however inside the conductor only few modes supply the current. It leads to redistribution of the current among current-carrying modes which results in the interface resistance.

2.4 S-matrix technique

As it was shown in the previous section, the current (or conductance) can be formulated in terms of the transmission function . The powerful method to calculate is the scattering matrix technique.T Scattering matrix (or S- T matrix) relates the outgoing amplitudes b = (b1, b2, ..., bn) to the incident amplitudes a = (a1, a2, ..., an), see Fig.(2.1),

′ r(E) t (E) b = S(E)a,S(E) = ′ . (2.44) t(E) r (E) 10 Quantum transport

S matrix has a 2N 2N dimension, where N is a number of transmission channels. The transmission· probability now equals to

2 tm n(E) = Smn (2.45) ← | | Prior to calculation of the S-matrix, one can determine its general properties. S-matrix must be unitary, that is a consequence of current conser- 2 vation: the incoming electron ﬂux n a must be equal to the outgoing ﬂux b 2 | | n | | P P b+b = a+a, a+(1 S+S)a = 0, (2.46) − S+S = I.

Moreover S-matrix is also a symmetric matrix, S = ST . This fact reflects the time-reversal symmetry of the Schrödinger equation, H = H∗. A non- zero magnetic field breaks the time-reversal symmetry. In this case, we have S = ST . B~ B~ − S-matrix and Green’s function S-matrix can be expressed in terms of Green’s function. Outside the scattering region solution of the Schrödinger equation has the form of plane waves ψ (~r) eiknz, where we assume that n ≈ the system with a cross-section area A is uniform in x and y directions, ~r = (~ρ, z). Each plane wave corresponds to a scattering channel n characterized by a transverse momenta ~qn and longitudinal momenta kn with the energy 2 2 E = (1/2m)(kn + qn). If we define the Green’s function G(E) = (E + iη 1 − H)− with matrix elements between scattering channels m and n as

1 G (z, z′,E) = A− d~ρ dρ~ exp ( i~q ~ρ) exp ( i~q ρ~ ) ~r G(E) r~ . mn ′ − m − n ′ h | | ′i Z Z (2.47) Then, the transmission coeﬃcient, following Fisher and Lee [17], can be calculated as

t = i~√v v G (z, z′,E) exp [ i(k z k z′)], (2.48) mn − m n mn − m − n where z and z′ are taken outside the scattering region, i.e. z > L and z′ < 0, see Fig.(2.1), and vn = kn/m is the velocity in channel n. This relation is very important, since it shows the connection between diﬀerent transport formalisms. Chapter 3

Electron-electron interactions

With advent of quantum mechanics, the physical laws which govern particles motion and interactions between particles became known. However the exact analytical solution is possible only for a system consisting of two particles. A typical piece of solid consists of approximately 1023 particles. Even if it would be possible to write down all the differential equations required to describe this system, the solution of these equation is an un- feasible task in principle. The problem of finding the solution arises from electron-electron interaction which makes the motion of particles correlated and couples corresponding differential equations. Therefore it is of great importance to develop approximated methods which provide a simplified form of the electron-electron interaction and reduces the number of equations needed to be solved.

3.1 The many body problem

The Hamiltonian of a many-body system of interacting particle is written as

Hˆ = Tê + Tˆn + Vê e + Vê n + Vˆn n − − − ~2 ~2 1 e2 = 2 2 + (3.1) − 2m ∇i − 2M ∇I 4πεε ~r ~r i e I I 0 i>j i j X X X | − | 1 Z Z e2 1 Z e2 + I J I , 4πεε0 R~ I R~ J − 4πεε0 ~ri R~ I XI>J | − | Xi,I | − | where the first two terms describe the kinetic energy of electrons and nuclei. The last three terms result from the Coulomb interaction between electrons, electron-nuclei and nuclei-nuclei respectively. The Hamiltonian acts on the wave-function Ψ( ~r , R~ ) which depends on the position of all electrons { i} { I } and nuclei in the system Hˆ Ψ( ~r , R~ ) = EΨ( ~r , R~ ). (3.2) { i} { I } { i} { I } 12 Electron-electron interactions

An essential simplification of Eq.(3.2) can be done with the use of the Born- Oppenheimer approximation which neglects the coupling between the nuclei and electronic motion. In thermodynamic equilibrium electrons move much faster than nuclei, since M m , which allows to treat nuclei as ≫ e stationary particles and neglect the kinetic term Tˆn in the Hamiltonian. Hence one can deal only with the electronic part, Hê, of the full Hamil- tonian which corresponds to the system of interacting electrons moving in the effective potential produced by nuclei

Hê = Tê + Vê e + Vê n + Vˆn n, (3.3) − − − Hˆ Φ( ~r ) = EΦ( ~r ). (3.4) e { i} { i} Even though electronic wave-function Φ( ~r ) still depends on the positions { i} of nuclei, R~ I are just parameters of Eq.(3.3) and the number of differential equations needed to be solved is greatly reduced.

3.2 Hartree-Fock approximation

The Born-Oppenheimer approximation significantly simplifies the problem of interacting particles by eliminating the coupling between electrons and nuclear motion. However determination of the exact solution of the many- particle electronic wave-function is still not feasible. The basic idea of the Hartree-Fock approximation is to substitute the system of interacting electrons by the motion of single electrons in the average self-consistent field generated by all the other electrons in the system. The Hamiltonian of the many-particle system is given by p2 e2 1 Hˆ = + V (~rk)+ = Hˆk + Hˆkk′ , (3.5) ′ 2me 8πεε0 ′ ~rk ~rk ′ Xk Xk Xkk | − | Xk Xkk where the term V (~r ) = V (~r R~ ) describes interaction between k- k I k − I electron with all nuclei in the system located at R~ . The operator Hˆ is a P { I } k one-particle operator, while Hˆkk′ depends on the position of two particles. The simplest way to construct the many-particle wave-function is to write down it in the form of a product of single-particle wave-functions, φk(~r), which have to be determined,

Φ( ~r ) = φ(~r ), φ (~r) φ (~r) = δ . (3.6) { k} k h i | j i ij Yk Let us calculate an expectation value of the energy [14] e2 1 E = Φ Hˆ Φ = φk Hˆk φk + φkφk′ φkφk′ . ′ h | | i h | | i 8πεε0 ′ ~rk ~rk k kk X X − (3.7)

Hartree-Fock approximation 13

According to the variational principle the closer values of φk to the exact solution the smaller value of the energy, thus

δ E Ek ( φk φk 1) = 0, (3.8) − h | i − ! Xk where E are Lagrange parameters. Changing φ φ + δφ and keeping k i → i i terms linear in respect to δφi, we arrive at the Hartree equation 2 2 2 ~ e φk(~rk) ∆ + V (~r) + | | d~rk φi(~r) = Eiφi(~r). (3.9) −2me 4πεε0 ~rk ~ri " k=i # X6 Z | − | The third term in the brackets has a simple interpretation. If we deﬁne the charge density as n(~r) = e φ (~r) 2, then the term i | i | ~ P e n(r′) UH (~r) = dr~′, (3.10) 4πεε0 r~ ~r Z | ′ − | called the Hartree term, describes the Coulomb interaction between the i-th electron located at ~r with all the other electrons in the system. Since electrons are fermions, the many-particle wave-function must change the sign under the interchange of the coordinates of any two particles. The wave-function given by Eq.(3.6) does not satisfy this condition. In order to construct an antisymmetric wave-function one can use Slater determinant

φ1(~q1) . . . φN (~q1) 1 . .. . Φ( ~qk ) = . . . , (3.11) { } √N! φ1(~qN ) . . . φN (~qN )

where ~qi = ~ri, σi denotes both position and spin of the electron and the factor 1 is{ used} for normalization. Following the same way as before, i.e. √N! applying the variational principle to the expectation value of the energy, the new form of the wave-function results in the equation [14] ~2 2 2 e φk(r~′) ∆ + V (~r) φi(~r) + | | dr~′φi(~r) −2me 4πεε0 r~ ~r k=i ′ X6 Z | − | 2 ~ ~ e φk∗(r′)φi(r′) dr~′ φk(~r) = Eiφi(~r), (3.12) −4πεε0 ~ · k=i r′ ~r X6 Z | − | called the Hartree-Fock equation. The additional term in Eq.(3.12) is known as the exchange interaction. It does not have classical analog and results from the Pauli exclusion principle. The exchange interaction term which arises in the Hartree-Fock approximation has a non-local form in contrast to the Coulomb interaction. This makes calculations more com- plicated. In the density-functional theory (discussed in Sec.(3.3)) a number of approximations are used to deduce a local form of the exchange interaction. 14 Electron-electron interactions

3.3 Density-functional theory

Density-functional theory (DFT) is one of the most widely used modeling method applied in physics and chemistry for calculation of electronic properties of complex systems. The basic idea behind DFT is to describe the system in terms of the electronic density instead of operating with a many-body wave function [18].

Hohenberg-Kohn theorems In 1964 Hohenberg and Kohn proved two theorems which made the DFT possible [19]. They state that a knowledge of the ground-state density can, in principle, determine all the ground-state properties of a many-body system [18].

Theorem 1: An external potential Vext(~r) uniquely determines the electronic density for any system of interacting particles.

Proof: Assume that the same electron density n(~r) results from two po- 1 2 tentials Vext(~r) and Vext(~r) differing by more than constant. Obviously, 1 2 ˆ 1 ˆ 2 Vext(~r) and Vext(~r) belong to distinct Hamiltonians H (~r) and H (~r) which produce different wave-functions Ψ1(~r) and Ψ2(~r). The ground-state state energy associated with the Hamiltonian Hˆ 1(~r) is E1 = Ψ1 Hˆ 1 Ψ1 . (3.13) h | | i According to the variational principle no other wave-function can give lower energy, i.e. E1 = Ψ1 Hˆ 1 Ψ1 < Ψ2 Hˆ 1 Ψ2 (3.14) h | | i h | | i Since the Hamiltonians differs by the external potentials only, we can write Hˆ 1 = Hˆ 2 + V 1 V 2 , which gives us for the expectation value ext − ext Ψ2 Hˆ 1 Ψ2 = Ψ2 Hˆ 2 Ψ2 + V 1 V 2 n(~r)d~r. (3.15) h | | i h | | i ext − ext Z Substituting it into Eq.(3.14) and recalling that E 2 = Ψ2 Hˆ 2 Ψ2 , we obtain h | | i E1 < E2 + V 1 V 2 n(~r)d~r. (3.16) ext − ext Z Interchanging labels (1) and (2), we find in the same way that

E2 < E1 + V 2 V 1 n(~r)d~r. (3.17) ext − ext Z Addition of Eq.(3.16) and Eq.(3.17) leads to contradiction E1 + E2 < E1 + E2. (3.18) Hence the theorem is proved by reductio ad absurdum. Kohn-Sham equations 15

Theorem 2: The exact ground-state density n(~r) is the global minimum of the universal functional F [n].

Proof: Since the electron density n(~r) uniquely determines wave-function Ψ, the universal functional F [n] can be deﬁned as

F [n] = Ψ Tˆ + Uˆe e Ψ . (3.19) h | − | i For a given external potential Vext(~r), the energy functional is written as

E[n] = F [n] + Vext(~r)n(~r)d~r. (3.20) Z According to the variational principle, it has a minimum only for the ground-state wave-function Ψ. For any other wave-function Ψ′ which produces density n′(~r), we get

E[Ψ] = F [n]+ Vext(~r)n(~r)d~r< F [n′]+ Vext(~r)n′(~r)d~r = E[Ψ′]. (3.21) Z Z 3.4 Kohn-Sham equations

In the Kohn-Sham method [20] one considers a system of non-interacting electrons moving in some eﬀective potential veff (~r) (which will be deﬁned later) ~2 2 + v (~r) φ (~r) = ǫ φ (~r). (3.22) −2m∇ eff i i i An obtained set of the Kohn-Sham orbitals φi determines the electron density n(~r) = φ (~r) 2. (3.23) | i | i X The energy functional equals to

E[n] = TS[n] + Veff [n] = TS[n] + n(~r)vext(~r)d~r + VH [n] + Exc[n], (3.24) Z where the ﬁrst term,

N ~2 2 T [n] = φ∗(~r) φ (~r)d~r, (3.25) s i −2m∇ i i=1 X Z is a single-electron kinetic energy functional. The second term describes the potential energy acquired by the charged particles in an external electric ﬁeld. The Hartree term is given by

2 e n(~r)n(r~′) VH = d~rdr~′. (3.26) 8πεε0 ~r r~ ZZ | − ′| 16 Electron-electron interactions

The last term, Exc[n], arises from the exchange-correlation interaction. It can be shown [21] that the functional given by Eq.(3.24) corresponds to the the eﬀective potential in the form

e n(r~′) veff (~r) = vext(~r) + dr~′ + vxc(~r). (3.27) 4πεε0 ~r r~ Z | − ′| Equations (3.22), (3.23) and (3.27) constitute the basis of the Kohn-Sham method, and are solved self-consistently for the ground state density and the eﬀective potential. The explicit form of the exchange-correlation potential vxc can be determined using other approximations. The most widely used are the local spin density approximation [21] and the generalized gradient approximation [22].

3.5 Thomas-Fermi-Dirac approximation

According to the Hohenberg-Kohn theorems discussed in Sec.(3.3), the total energy of the system may be written as

E[n(~r)] = T [n(~r)]d~r + V (~r)n(~r)d~r, (3.28) Z Z where T [n(~r)] is a kinetic energy functional of the electron density n(~r) and V (~r) is an external potential. The Thomas-Fermi approximation assumes that the kinetic-energy functional is a local function of the density. This assumption allows to rewrite Eq.(3.28) in the form

µ = T [n(~r)] + V (~r). (3.29)

Let us now derive the relation between the potential and charge density in graphene. Taking into account dispersion relation for graphene, E = ~v ~k , and using n = g g d~k (v = 106 m/s and g = g = 2, see ± F | | v s (2π)2 F v s Chapter 4 for details), we get [28] R

sgn[n(~r)]~vF πn(~r) + V (~r) = µ. (3.30)

An alternative way is to rewrite Eq.(3.28)p in the form which directly relates electron density to the external potential [23, 24]

n(~r) = dEρ(E V (~r))f (E, µ), (3.31) − FD Z where ρ(E) is a single-electron density of states, calculated in the presence of homogeneous potential. Figure (3.1) illustrates application of Eq.(3.31). Hubbard model 17

Figure 3.1: Schematic illustration of Thomas-Fermi model.

Locally the dispersion relation corresponding to the ideal system (with homogeneous external potential) is preserved. Filling up the states lying between the charge neutrality point and the Fermi energy level one obtains local electron density. Even though the Thomas-Fermi model misses quantum mechanical effects (e.g. quantization), it produces quantitatively similar results in comparison to more rigorous models and widely used in graphene physics [25, 26, 27, 28].

3.6 Hubbard model

The Hubbard model, originally proposed by John Hubbard in 1963 [29], is the simplest model of interacting particles in a lattice. The interaction is assumed to take place only between particles located at the same site (or atom). Despite of its simplicity rigorous analytical solution is found only for a one-dimensional problem [30]. The Hubbard Hamiltonian in the tight-binding approximation consists of two terms, namely the kinetic energy term and the on-site potential

ˆ + H = t (ai,σaj,σ + h.c.) + U ni ni , (3.32) − ↑ ↓ i,j ,σ i hXi X + where ni,σ = ai,σai,σ is the occupation number operator. The parameter U = const describes the strength of on-site Coloumb interaction and can be determined using ab-initio calculations or extracted from experimental data. Equation (3.32) can be rewritten within the mean-ﬁeld approach using substitutions

ni = ni + (ni ni ), (3.33) ↑ h ↑i ↑ − h ↑i ni = ni + (ni ni ), (3.34) ↓ h ↓i ↓ − h ↓i 18 Electron-electron interactions where n denotes average occupation of spin σ at site i. Hence, we have h iσi

ni ni = ni ni + ni ni ni ni + (ni ni )(ni ni ), (3.35) ↑ ↓ ↑h ↓i ↓h ↑i − h ↓ih ↑i ↑ − h ↑i ↓ − h ↓i 0 ≈ where the product of two deviations from| the average{z values is} assumed to be small. Substituting the result of Eq.(3.35) into Eq.(3.32) we derive Hubbard Hamiltonian in the mean-ﬁeld approximation

ˆ MF + H = t (ai,σaj,σ + h.c.) + U (ni ni + ni ni ni ni ) . − ↑h ↓i ↓h ↑i − h ↓ih ↑i i,j ,σ i hXi X (3.36) Despite of its simplicity the Hubbard model was applied to investigate the properties of diﬀerent materials. It reproduces a variety of phenomena observed in solid state physics, such as ferromagnetism, metal-insulator transition and superconductivity. Chapter 4

Electronic structure and transport in graphene

4.1 Basic electronic properties y a) b) B A

Figure 4.1: a) Graphene lattice consisting of two interpenetrating triangular sublattices A (red circles) and B (blue circles) with unit vectors ~a1, ~a2 and nearest-neighbours vectors ~δ1, ~δ2, ~δ3. The yellow parallelogram marks unit cell containing two atoms. b) The structure of reciprocal lattice de- ~ ~ ﬁned by unit vectors b1 and b2. The grey hexagon is the ﬁrst Brillouin zone.

Real space and reciprocal lattices Carbon atoms in graphene are arranged in a honeycomb lattice shown on Fig.(4.1). This structure can be described [31, 32] as a triangular lattice with unit vectors a a ~a = cc (3, √3),~a = cc (3, √3), (4.1) 1 2 2 2 − 20 Electronic structure and transport in graphene

where acc = 0.142 nm is a carbon-carbon distance. Unit cell contains two a√3 √ atoms and has an area Scell = 4 , where a = acc 3 = 0.246 nm is a lattice constant. Each point of the sublattice A is connected to its nearest- neighbors by the vectors

√ √ ~ 1 3 ~ 1 3 ~ δ1 = acc , , δ2 = acc , , δ3 = acc ( 1, 0) . (4.2) 2 2 ! 2 − 2 ! −

For a given lattice one can easily build a reciprocal lattice with a unit vectors ~bi deﬁned by the relation ~bi~aj = 2πδij, i.e.

~ 2π ~ 2π b1 = (1, √3), b2 = (1, √3). (4.3) 3acc 3acc − The ﬁrst Brillouin zone, which is the Wigner-Seitz primitive cell of the reciprocal lattice [33], is shown on Fig.(4.1)(b). There are two inequivalent points which are of special interest in graphene physics

~ 2π 1 ~ 2π 1 K = 1, , K′ = 1, . (4.4) 3acc √3 3acc −√3 Dispersion relation Graphene lattice can be considered as two interpenetrating triangular sublattices A and B which are deﬁned by vectors

~ A ~ B ~ Rp,q = p~a1 + q~a2, Rp,q = δ1 + p~a1 + q~a2, (4.5) where p, q are integer numbers. In a single-electron approximation the tight-binding Hamiltonian for electrons in graphene is given by

ˆ + + + Htb = t ap,qbp,q + ap,qbp 1,q + ap,qbp 1,q+1 + h.c., (4.6) − − − p,q X + + where ap,q(ap,q) and bp,q(bp,q) create (annihilate) an electron on sublattices ~A ~B A and B at site Rp,q and Rp,q respectively and t = 2.77 eV is a nearest- neighbor hopping integral. The wave-function for the lattice can be written in the form

Ψ = ζA a+ + ζB b+ 0 , (4.7) | i p,q p,q p,q p,q | i p,q X A(B) A~(B) where ζp,q is a probability amplitude to find the electron at site Rp,q Substituting Eqs.(4.6),(4.7) into the Schrödinger equation, Hˆ Ψ = E Ψ , tb| i | i and calculating the matrix elements 0 a Hˆ a+ 0 and 0 b Hˆ b+ 0 h | p,q tb p,q| i h | p,q tb p,q| i Basic electronic properties 21 with use of the commutation relation, one arrives to the system of difference equations

B B B A t ζp,q + ζp 1,q + ζp 1,q+1 = Eζp,q, (4.8) − − − t ζA + ζA + ζA = EζB . − p,q p+1,q p+1,q+1 p,q A B The states ζp,q and ζp,q can be written in the Bloch form

~ ~A A A ikRp,q A A A ζp,q = ψp,qe , ψp,q = ψp+1,q = ψp+1,q 1, (4.9) − ~ ~B B B ikRp,q B B B ζp,q = ψp,qe , ψp,q = ψp 1,q = ψp 1,q+1. − − Substituting Eq.(4.9) and Eq.(4.2) into Eq.(4.8) and omitting indexes (p, q), one gets

tφ(~k)ψB = EψA, (4.10) − A B tφ∗(~k)ψ = Eψ , − where ~ ~ ~ ~ ~ ~ φ(~k) eikδ1 + eikδ2 + eikδ3 , (4.11) ≡ or in a matrix form

A A ~ ˆ ψ ψ ˆ 0 tφ(k) H B = E B , H − . (4.12) ψ ψ ≡ tφ∗(~k) 0 − ! In order to obtain dispersion relation, one needs to determine the eigenvalues of the matrix Hˆ , which are calculated using the relation det Hˆ IEˆ = 0, | − |

E(~k)2 = t2 φ(~k) 2, | | √3 √3 3 E(~k) = t 1 + 4 cos2 a k + 4 cos a k cos a k . v cc y cc y cc x ± u 2 ! 2 ! 2 u t (4.13)

Dirac equation The spectrum of graphene, given by Eq.(4.13), is symmetric in respect to energy E = 0. If the Fermi energy coincides with this point (EF = 0), i.e the states are occupied only up to zero energy, it corresponds to the case of electrically neutral graphene. One is usually interested in electronic properties close to a charge-neutrality point. There are six points in the k-space where the energy equals to zero. These points are at the corners of the ﬁrst Brillouin zone, see Fig.(4.1). Only two of them, K~ and K~ ′, are inequivalent. 22 Electronic structure and transport in graphene a) 3 b)

Figure 4.2: a) Dispersion relation calculated using Eq.(4.13). The energy is given in units of the hopping integral. b) Close to the charge neutrality point dispersion relation is linear and has a form of a cone determined by the Dirac equation (see Eq.(4.23) below).

Let’s expand the function φ(~k) in the Hamiltonian of Eq.(4.12) near the K~ -point ~k = K~ + ~q, (4.14) where ~q is some small ( ~q < K~ ) vector having origin at K~ . Substituting | | | | Eq.(4.14) in φ(~k), one gets

3 2π 2π iK~ ~δi i~q~δi i i~q~δ1 i0 i~q~δ2 i i~q~δ3 φ(K~ + ~q) = e e = e 3 e + e e + e− 3 e . (4.15) i=1 X Considering the continuum (low energy) limit (a 0) [32], one can ex- cc → pand exponents in Taylor series

i~q~δi lim e 1 + i~q~δi. (4.16) acc 0 → ≃ After performing some straightforward algebra, we arrive at

3acc √3 1 1 √3 φ(~k) = i qx + i qy . (4.17) 2 "− 2 − 2! − 2 2 ! #

π If we rotate the system of coordinates on angle θ = 6 by operator

cos θ sin θ √3 1 R = − = 2 − 2 , (4.18) sin θ cos θ 1 √3 2 2 ! Basic electronic properties 23

Eq.(4.15) is ﬁnally reduced to

3a φ(~q) = cc (q + iq ). (4.19) − 2 x y Substituting Eq.(4.19) into Eq.(4.12), one gets

A A 3 0 qx + iqy ψ ψ acct B = E B . (4.20) 2 qx iqy 0 ψ ψ −

With the use of the Pauli matrices, ~σ = (σx, σy), where

0 1 0 i σx = , σy = − , (4.21) 1 0 i 0 the Hamiltonian can be written in a vector form

HˆK = ~vF ~σ~q, (4.22)

3 acct 6 where vF = 2 ~ 10 m/s is a Fermi velocity. Equation (4.22) is al- gebraically identical≃ to a two-dimensional relativistic Dirac equation with vanishing rest mass known as Weyl’s equation for a neutrino, where the two-component wave function (or spinor) represents pseudo-spin which results from the presence of two sublattices [34, 35]. The eigenenergies of HˆK are E = ~v q. (4.23) ± F In order to ﬁnd eigenfunctions Eq.(4.20) can be rewritten in the following way iθ(~q) A A ~ 0 e ψ ψ vF q iθ(~q) B = E B , (4.24) e− 0 ψ ψ where θ(~q) = arctan(qy/qx). Taking into account Eq.(4.23) for eigenenergies and using normalization condition ψA 2 + ψB 2 = 1, one gets | | | | 1 eiθ(~q)/2 ΨK,s~ (~q) = iθ(~q)/2 , (4.25) | i √2 se− where the sign s = corresponds to the eigenenergies ~v q. ± ± F Besides spin-degeneracy each level is double-degenerated due to valley. One has to operate by a full wave function which includes the contribution from both valleys. The same procedure can be repeated to obtain the eﬀective Hamiltonian and wave function near K′-point

iθ(~q)/2 1 e− HˆK′ = ~vF ~σ∗~q, Ψ ~ ′ (~q) = . (4.26) | K ,s i √2 seiθ(~q)/2 24 Electronic structure and transport in graphene

If the wave-vector ~q rotates once around the Dirac point, i.e. θ θ + 2π, the wave-function acquires an additional phase equals to π, hence→ Ψ (θ 2π) = Ψ (θ) which is a characteristics of fermions. K,s~ ± − K,s~ Electron’s wave function in graphene has a chiral nature. Helicity can be interpreted as a projection of pseudospin vector on direction of motion and deﬁned by the operator hˆ = ~σ~q/ ~q . It can be easily shown using | | Eq.(4.22), that eigenvalues of the helicity operator equal to h = 1. Since hˆ commutes with the Hamiltonian, helicity is a conserved quantity± and responsible for such eﬀects as the Klein tunneling [36].

4.2 Graphene nanoribbons

Electronic properties of graphene nanoribbons (GNR) depend on the type of a edges. One can distinguish two types, namely, zig-zag and armchair GNR’s. a) b) 3

1 0 L 0 E/t

-1

-2

-3-π π kax

Figure 4.3: a) Structure of an armchair graphene nanoribbon lattice. Each edge of the ribbon is terminated by both A and B atoms. b) Dispersion relation of the ribbon with 10 atoms in transverse direction calculated using the tight-binding Hamiltonian.

Armchair graphene nanoribbons Dispersion relation of armchair GNR can be derived solving the Schr¨odinger equation in the following form [37]

0 kx + iky 0 0 ψA ψA kx iky 0 0 0 ψB ψB ~vF  −    = E   , 0 0 0 k + ik ψ′ ψ′ − x y A A  0 0 k ik 0   ψ′   ψ′   − x − y   B   B      (4.27) Graphene nanoribbons 25

where ψA(B) and ψA′ (B) are the probabilty amplitudes on the sublattice A(B) for the state near the K~ and K~ ′ points, respectively. The total wave function has the form iK~r~ iK~ ′~r ′ Ψ = e ΨK~ + e ΨK~ . (4.28) Let us first find a solution near the K point. Substituting in Eq.(4.22) wave vector ~k = i ∂ , i ∂ , one gets − ∂x − ∂y 0 i ∂ + ∂ ψA ψA − ∂x ∂y = ǫ , (4.29) i ∂ ∂ 0 ψB ψB − ∂x − ∂y where ǫ = E . Due to translational invariance in ~x-direction the wave ~vF function can be written in the form φA(y) Ψ (x, y) = eikxx , (4.30) K~ φB(y) which allows us to reduce the problem to a system of two differential equations

∂φB (y) kx 1 ∂φA(y) kxφB(y) + ∂y = ǫφA(y) φB(y) = ǫ φA(y) ǫ ∂y 2 − k φ (y) ∂φA(y) = ǫφ (y) ⇒ ∂ φA(y) + z2φ (y) = 0 ( x A − ∂y B ( ∂y2 A (4.31) 2 2 2 where z = ǫ kx. The general solution of the system of equations is a sum of plane waves− izy izy φA(y) = Ae + Be− kx iz izy kx+iz izy (4.32) φ (y) = − Ae + Be− B ǫ ǫ Similar derivation can be done for the wave functions describing the states near K′-point, which gives izy izy φA′ (y) = Ce + De− kx iz izy kx+iz izy (4.33) φ′ (y) = − − Ce + − De− B ǫ ǫ In order to ﬁnd the unknown coeﬃcients A and B, one can utilize the boundary conditions. The armchair nanoribbon edge consist of atoms be- longing to both sublattices, see Fig.(4.3)(a), therefore one can expect that both ΨA and ΨB should vanish at the edges

ΨA(0) = ΨB(0) = ΨA(L) = ΨB(L) = 0, (4.34) where ΨA(B) is an A(B) component of the total wave function (4.28). Hence, we have

φA(0) + φA′ (0) = 0 φB(0) + φB′ (0) = 0 iKyL iKyL (4.35) e φA(L) + e− φA′ (L) = 0 iKyL iKyL e φB(L) + e− φB′ (L) = 0 26 Electronic structure and transport in graphene

Substituting Eq.(4.32) and Eq.(4.33) into Eq.(4.35) and solving the system of four unknowns, we arrive at

e2izL = ei∆KyL, (4.36)

4π where ∆Ky = 3a . Therefore, the allowed values of z are πn 2π z = + . (4.37) L 3a Finally, the dispersion relation is given by

πn 2π 2 E = ~v k2 + + . (4.38) ± F x L 3a s Note that if the ribbon consists of 3N + 1 atoms, Eq.(4.38) allows zero energy solutions when k 0. This kind of ribbons are called metallic. x → Otherwise the dispersion relation posses an energy gap, i.e. the ribbons are semiconducting. a) b) 3

1 0 L 0 E/t

-1

-2

-3-π π kax

Figure 4.4: a) Structure of a zig-zag graphene nanoribbon lattice. Each edge of the ribbon is terminated by either A or B atoms. b) Dispersion relation of the ribbon with 10 atoms in transverse direction calculated using the tight-binding Hamiltonian.

Zig-zag graphene nanoribbons The Hamiltonian describing zig-zag GNR can be derived from the Hamiltonian used for armchair GNR by rotation of the system on angle π , i.e. changing k k and k k 2 x → y y → − x 0 i ∂ + ∂ ψA ψA ∂y ∂x = ǫ . (4.39) i ∂ ∂ 0 ψB ψB ∂y − ∂x Warping 27

Substituting the wave function in the form given by Eq.(4.30) and following the same procedure as in the case of the armchair GNR, we get

izy izy φA(y) = Ae + Be− ikx z izy ikx+z izy (4.40) φ (y) = − − Ae + − Be− B ǫ ǫ For zig-zag nanoribbons, as illustrated in Fig.(4.4), the boundary conditions are φA(L) = φB(0) = 0. (4.41)

Solving the system of linear equation, we derive the relation between kx and z i2zL ikx + z e− = (4.42) ik z x − In contrast to armchair nanoribbons, the transverse and longitudinal components of the wave vector are coupled in zGNR. Another interesting feature of zGNR is an existence of surface states. Besides the solutions with real z describing propagating states the transcendental equation (4.42) sup- ports also solutions with imaginary values of z. It corresponds to the so- called edge (or surface) states, i.e the states spatially localized near the edges.

4.3 Warping

Figure 4.5: a)Experimental 3D constant current STM image of single layer graphene adopted from Ref.[40] b) Generated surface of a corrugated graphene sheet [39]. Black line corresponds to a characteristic wave-length of the ripples. c) Magniﬁed area marked in a) by black rectangular. M and S stands for maximum (minimum) and saddle points, respectively.

Graphene is a pure two-dimensional crystal. According to the Mermin- Wagner theorem, the long-range order of 2D crystals should be destroyed 28 Electronic structure and transport in graphene by long-wavelength fluctuations and therefore 2D membranes have a ten- dency to get crumpled being in a 3D space [38]. In the case of graphene minimization of energy results in appearance of ripples on graphene sheet. The existence of the ripples was confirmed by a number of experiments [40, 41]. Warping can effect electronic properties. For example, bending the graphene plane changes overlap between p orbitals and in turn hopping integrals in the tight-binding model. Also warping can be related to the formation of electron-hole puddels which cause the spatial modulation of charge density [42]. Another example is a magnetotransport. Electrons propagating through a warped graphene subjected to a magnetic field are influenced by spatially correlated effective magnetic field. The corrugated surface of graphene can be modeled [see Fig.(4.5)] by a superposition of plane waves

h(~r) = C Cqi sin(~qi~ri + δi), (4.43) i X where h(~r) is a out-of-plane displacement at point deﬁned by in-plane position vector ~r = (x, y). The directions ϕ of wave vectors ~q = q (cos ϕ , sin ϕ ) i i | i| i i and the phases δi were chosen randomly. The length of the wave vectors qi covers equidistantly the range 2π/L < qi < 2π/(3acc), where L is a leading linear size of the rectangular area and acc denotes the C-C bond length (we assume that L λ∗). The amplitude of the mode was given ≫ 2 by the harmonic approximation Cq = 2 hq for the wave length λ < λ∗, otherwise it was kept constant and equalq to Cq∗ , where q∗ = 2π/λ∗. We introduced the normalization constant C to keep the averaged amplitude of the out-of-plane displacement h¯ = 2 h2 equal to the experimental values h¯ 1 nm for typical sizes of samples.h i ≈ p 4.4 Bilayer graphene

In a tight-binding approximation bilayer grpahene can be modeled by the following Hamiltonian [31]

ˆ + H = t (al,ibl,i+∆ + h.c.) − i ;Xl=1,2 + γ1 (a1,ia2,i + h.c.) − i X + γ3 (b1,ib2,i+∆ + h.c.), (4.44) − i X + + where al,i (al,i) and bl,i (bl,i) are the creation (annihilation) operators for sublattice A (B), in the layer l = 1, 2, at site R~ i, where i = (p, q). The Bilayer graphene 29

A B1 1 t

γ γ3 1

γ4

B2 A2

Figure 4.6: Structure of bilayer graphene with the illustration of the hopping integrals used in tight-binding model: t is the intralayer nearest- neighbor coupling energy, γ1 is the coupling energy between sublattice A1 and A2 in diﬀerent graphene layer, and γ3 the hopping energy between sublattice B1 and B2 in the upper and lower layers, respectively. meaning of hopping integrals is illustrated in Fig. (4.6): t is the intralayer nearest-neighbor coupling energy, γ1 = 0.39 eV is the coupling energy between sublattice A1 and A2 in diﬀerent graphene layers, and γ3 = 0.315 eV the hopping energy between sublattice B1 and B2 in the upper and lower layers, respectively. The other coupling energy between the nearest- neighboring layers, γ 0.04 eV, is very small compared with γ and 4 ≈ 0 ignored below. Since the unit cell of bilayer graphene consist of four atoms, the wave function can be written in the form

A1 A1 ψp,q c B1 B1 ψp,q c i~kR~ p,q ψp,q =  A2  =  A2  e . (4.45) ψp,q c  ψB2   cB2   p,q        Using this Bloch form of the wave function, we arrive at the eigenvalue problem

~ A1 A1 0 tφ(k) γ1 0 c c ~ ~ B1 B1  tφ∗(k) 0 0 γ3g(k)  c c  A2  = E  A2  , (4.46) γ1 0 0 tφ∗(~k) c − c  ~ ~  cB2 cB2  0 γ3g∗(k) tφ(k) 0                  ~k(~a1+~a1) where g(~k) = e− φ(~k) and φ(~k) is same as in the single layer graphene and deﬁned by Eq.(4.11).

A minimal low-energy model Electronic properties close to the charge- neutrality point can be obtained using a low-energy eﬀective bilayer Hamil- tonian [43, 44]. In this approximation the interaction between layers is described by the interplane hopping term γ1 only (i.e. γ3 is neglected). Also, since we are interested in the properties close to K~ point, one can make 30 Electronic structure and transport in graphene

~ the expansion tφ(k) ~vF k, where k = kx + iky. Taking into account these approximations,− one≈ gets

0 ~vF k γ1 0 ~v k∗ 0 0 0 Hˆ =  F  . (4.47) γ1 0 0 ~vF k∗  0 0 ~v k 0   F    Solution of det(Hˆ EI) = 0 gives a dispersion relation − γ γ2 ~ 1 1 ~2 2 2 E(k) = s1 s2 + + vF k , (4.48) 2 r 4 ! where s1, s2 = 1. One can consider this equation in two limits (below only the conduction± band is discussed): a) low energies, ~vF k < γ1

γ γ2 γ γ ~2v2 k2 ~ 1 1 ~2 2 2 1 1 F E(k) = s2 + + vF k s2 + + 2 , (4.49) 2 r 4 ≈ 2 2 γ1 2 γ 2~2v k2 1 1+ F 2 γ2 1 | {z } ~2v2 k2 F , s = 1 γ1 2 E(k) ~2v2 k2 − (4.50) ≈ γ + F , s = 1 ( 1 γ1 2 Equations (4.50) show that the conduction band of bilayer graphene consists of two parabolic bands separated by the energy interval γ1. In the vicinity of the Dirac point the dispersion relation of the lowest subband can be rewritten in a form which is usually used in a conventional 2DEG systems, i.e. ~2k2 E = , (4.51) 2m∗ γ1 where m∗ 2v2 is an eﬀective mass. ≡ F b) high energies, ~vF k > γ1

γ γ2 γ ~ 1 1 ~2 2 2 1 ~2 2 2 E(k) = s2 + + vF k s2 + vF k , (4.52) 2 r 4 ≈ 2 γ2 1 ~vF k 1+ 8~2v2 k2 F | {z } ~v k, s = 1 E(k) F 2 − (4.53) ≈ ~vF k + γ1, s2 = 1 i.e. the dispersion relation consists of two linear bands (as in the case of the the single layer graphene) separated by the energy interval γ1. Dirac fermions in a magnetic ﬁeld 31

Biased bilayer graphene Electronic properties of bilayer graphene can be strongly modified by applying external electric field, which causes a potential difference between layers. In this case the Hamiltonian (4.47) is modified U1 ~vF k γ1 0 ~v k∗ U 0 0 Hˆ =  F 1  , (4.54) γ1 0 U2 ~vF k∗  0 0 ~v k U   F 2    where U1 and U2 are on-site energies or electrostatic potentials of the 1- st and 2-nd layer respectively. The dispersion relation of biased bilayer graphene is

~ U1+U2 E ,s(k) = ± 2 (4.55) 2 4 ∆U 2 + ~2v2 k2 + γ + ( 1)s (∆U 2 + γ2)~2v2 k2 + γ , ± 4 F 2 − F 4 r q where ∆U = U U and s = 1. 2 − 1 ± Energy gap, ∆Eg, in the vicinity of K~ -point is determined by

∆Eg = E+,1(0) E ,1(0) = U2 U1 = ∆U. (4.56) − − − This equation shows that a band-gap can be tuned by application of an external voltage. This makes bilayer graphene a promising candidate for future electronics.

4.5 Dirac fermions in a magnetic ﬁeld

In a high enough perpendicular magnetic field a continues spectrum of any 2DEG systems is usually modified into a series of Landau levels (LLs). Since the behavior of electrons in graphene described by Dirac rather than Schrödinger equation it should be reflected in the spectrum as well. We start the calculation of the dispersion relation from the Hamiltonian given by Eq.(4.22)

Hˆ = vF ~σ~p. (4.57)

In a magnetic ﬁeld the canonical momentum is changed to ~p ~~k + eA~, where A~ is a vector potential of the magnetic ﬁeld. Using the Landau→ gauge in the form A~ = ( By, 0, 0), one gets − ∂ 0 1 ∂ 0 i Hˆ = vF i~ eBy + i~ − . (4.58) − ∂x − 1 0 − ∂y i 0 32 Electronic structure and transport in graphene

Substituting it in the Schr¨odinger equation, Hˆ Ψ = E Ψ , one arrives at the eigenvalue problem | i | i

0 i~ ∂ eBy ~ ∂ ψA ψA v − ∂x − − ∂y = E . F i~ ∂ eBy + ~ ∂ 0 ψB ψB − ∂x − ∂y (4.59) Due to translational invariance in ~x direction, the solution of the Schr¨odinger equation with the Hamiltonian (4.58) can be written in the Bloch form

ψA φA(y) Ψ(x, y) = = eikx , (4.60) | i ψB φB(y) which gives

∂ y A A 0 + lBk φ (y) φ (y) ~ ∂y lB vF ∂ y − B = lBE B , − + lBk 0 φ (y) φ (y) − ∂y lB − (4.61) ~ where lB = eB is a magnetic length. Introducing dimensionless length y scale ζ = qlBk, one finally gets lB − 0 ∂ + ζ φA(ζ) φA(ζ) ∂ζ = ǫ , (4.62) ∂ + ζ 0 φB(ζ) φB(ζ) − ∂ζ where ǫ = lBE/~vF . This system of the first order differential equations can be reduced− to a second order differential equation

∂2φA(ζ) + (ǫ2 1) ζ2 φA(ζ) = 0. (4.63) ∂ζ2 − − This equation has a form of the harmonic oscillator equation with eigenenergies ǫ = 2(n + 1), where n = 0, 1, 2, .. and eigenfunctions ± 2 p n ζ Φ (ζ) = e− 2 Hn(ζ), (4.64) where Hn is a n-order Hermitian polynomial. Similarly, one can solve an equation for φB(ζ), which gives the eigenenergies ǫ = √2n. Hence, the ﬁnal solution is ±

n 1 y Φ − ( l k) ikx lB B Ψ(x, y) = e n y − , (4.65) | i Φ ( lBk) lB − 1 where we define Φ− 0. Hence, in a magnetic field perpendicular to graphene layer the spectrum≡ is modified into a series of Landau levels with the energies E = ~ω √n, (4.66) ± c Dirac fermions in a magnetic field 33

where ωc = vF 2eB/~ is a cyclotron frequency. There are to distinctive features of graphene spectrum in a magnetic field in comparison to the p spectrum of ordinary 2DEG systems. Firstly, the energy of the ground state (i.e. 0’th LL) of graphene equals to zero and does not depend on a magnetic field value. Secondly, the difference between to successive levels is not constant and decreases with increase of energy. An experimental manifestation of these unusual series is the anomalous quantum Hall effect 4e2 with the Hall conductivity given by σxy = h (n + 1/2) [45, 46]. 34 Electronic structure and transport in graphene Chapter 5

Modeling

In this chapter we describe two widely used techniques for electronic and transport properties calculations, namely, the recursive Green’s function technique (RGFT) and the real-space Kubo method. Both techniques used in the present thesis have their advantages and disadvantages. The recursive Green’s function technique naturally captures all transport regimes: ballistic, diffusive and localization. It allows to take into account any arbitrary shape of a considered structure. Also it is easy to include in the model different types of disorder and a magnetic field. One of the main drawback of the RGFT is that the computational expenses scales as (N 3). In contrast, the real-space Kubo method scales as (N) allowingO to investigate O structures consisting of up to tens of millions of carbon atoms. However, this method can not be properly used to study edge physics. Moreover, even though the real-space Kubo method can be in principle applied to study any transport regime, straightforward application of the method to calculation of conductivity is done only for the diffusive regime. Thus, the recursive Green’s function technique is well suited to study relatively small structures, where edges play a major role; while the real-space Kubo method can be used to investigate bulk properties of large systems.

5.1 Tigh-binding Hamiltonian and Green’s function

In order to model electronic structure and transport properties of graphene we use standard p-orbital tight-binding Hamiltonian [31] (see also Eq.(4.6)),

+ + Hˆ = V (r)ar ar tr,r+∆ar ar+∆, (5.1) r − r r X X, +∆ where the summation runs over all sites in graphene lattice and ∆ includes the nearest-neighbor only. The eﬀect of an external potential as well as 36 Modeling the interaction of an electron with all other particles in the system is incorporated through the change of the on-site energy V (r). In the absence of a magnetic ﬁeld the hopping integral in Hamiltonian is constant and + equals to tr,r+∆ = t0 = 2.77 eV [31]. The operators ar /ar are standard creation/annihilation operators obeying the following anticommutation relations [47],

+ ar, ar′ = δrr′ , (5.2) { } + + ar, ar′ = ar , ar′ = 0. { } { } The Hamiltonian (5.1) acts on the wave-function which is also expressed in terms of the second-quantization operators

+ Ψ = ψrar 0 , 0 = 0,..., 0 (5.3) | i r | i | i | i X with 0 representing a vacuum state and ψr = 0 ar Ψ is a probability amplitude| i to ﬁnd a particle at the site r. h | | i

a) 4 b) 3 4 3 2 2 1 1

Figure 5.1: Depending on the C-C bond orientation the hopping integral acquires diﬀerent phase in a magnetic ﬁeld which is determined by Eq.(5.4).

Pierels substitution If the system is subjected to a magnetic ﬁeld the hopping integral acquires phase,

r+∆ φr,r+∆ tr,r+∆ = t0 exp i2π , φr,r+∆ = A dl, (5.4) φ r · 0 Z where φ0 = h/e is the magnetic-ﬂux quantum and A is the vector potential. In the case of uniform perpendicular magnetic ﬁeld, the convenient choice for the vector potential is the Landau gauge in the form

A = ( By, 0, 0). (5.5) − Using Eqs.(5.4)-(5.5), one can evaluate the hopping integral for geometries depicted on Fig.(5.1). Thus, for zigzag geometry, Fig.(5.1)(a), the phases Tigh-binding Hamiltonian and Green’s function 37 are a √3 1 a √3 1 φ = cc B y + a ; φ = cc B y a ; 12 2 1 4 cc 21 − 2 2 − 4 cc a √3 1 a √3 1 φ = cc B y + a ; φ = cc B y a ; 34 − 2 3 4 cc 43 2 4 − 4 cc φ23 = 0; φ32 = 0; (5.6) and for armchair geometry, Fig.(5.1)(b), we get

acc √3 acc √3 φ12 = B y1 + acc ; φ21 = B y2 acc ; 2 4 ! − 2 − 4 !

acc √3 acc √3 φ23 = B y2 + acc ; φ32 = B y3 acc ; − 2 4 ! 2 − 4 ! φ = By a ; φ = By a ; (5.7) 34 − 3 cc 43 4 cc The Green’s function A standard way to define the Green’s function is [16] 1 [E H iη]G± = 1ˆ G± = [E H iη]− . (5.8) − ± ⇒ − ± where η 0 is an infinitesimal constant. The sign distinguishes between → + ± retarded (G ) and advanced (G−) Green’s functions. In this chapter only retarded Green’s functions are used, therefore we omit the sign thereafter. The Green’s function can also be expressed in terms of eigenvalues and eigenfunctions of the Hamiltonian, Hˆ Ψα = ǫ Ψα , | i α | i α α ψr ψr′∗ G(r, r′; E) = . (5.9) E ǫ + iη α α X − This equation allows to connect the Green’s function to a local density of states (LDOS) which is defined as

α 2 ρ(r,E) = ψr δ(E ǫ ). (5.10) | | − α α X Using the relation 1 lim = πδ(x), (5.11) ℑ η 0 x + iη − → we ﬁnally arrive at the required equation for LDOS 1 ρ(r,E) = lim G(r, r; E) . (5.12) −π ℑ η 0 → Computation of the Green’s function by direct matrix inversion is an extremely time-consuming routine. If the structure consist of M-slices with 38 Modeling

N-sites on each slice, the real-space Hamiltonian matrix has a dimension NM NM, which makes matrix inversion ineﬃcient for structures with a size relevant× for experiments. One way to attack the problem is to switch to an energy (or momentum) representation using the Fourier transforma- tion. However this method introduce approximation in the original exact problem. In the next section we describe the recursive Green’s function technique. This method does not require any simpliﬁcations in comparison with the original problem and allows to avoid the full Hamiltonian matrix inversion.

5.2 Recursive Green’s function technique: application to graphene nanoribbons

In this section we use the technique described in Refs. [48, 49] and applied previously to study properties of graphene nanoribbons and photonic crystals.

5.2.1 Dyson equation

a) b)

0 1 2 i N N+1

Figure 5.2: Schematic illustration showing the application of a) Dyson equation and b) the recursive Green’s function technique.

Imagine that we have two isolated subsystems described by the Hamil- 0 0 tonians H1 and H2 . Then, we put them together and let interact via potential V [see Fig.(5.2) (a)]. Thus, the total Hamiltonian reads

0 0 0 H = H1 + H2 +V = H + V. (5.13) H0

The Green’s function corresponding| {z } to this Hamiltonian equals to

1 0 0 1 G− = E H V = (G )− V. (5.14) − − − (G0)−1 | {z } Recursive Green’s function technique 39

Multiplying the above equation on the left by G0 and on the right by G (or vice versa) and rearranging variables, we derive the Dyson equation

G = G0 + G0V G, (5.15) (G = G0 + GV G0).

The Dyson equation is an extremely useful relation and is a basis of the recursive Green’s function technique. If we deﬁne the matrix element as G = m G n , the Dyson equation reads mn h | | i

0 0 Gmn = Gmn + GmiVijGjn, (5.16) i,j X 0 0 (Gmn = Gmn + GmiVijGjn). i,j X Hence, attaching two slices with the use of the Dyson equation, the Green’s function for the combined system is

0 0 1 0 G11 = (I G11V12G22V21)− G11, 0− 0 0 1 0 G12 = G11V12(I G22V21G11V12)− G22, 0 − 0 0 1 0 G21 = G22V21(I G11V12G22V21)− G11, (5.17) 0 − 0 1 0 G = (I G V G V )− G . 22 − 22 21 11 12 22 Let us now consider a typical structure for transport experiments illustrated in Fig. (5.2)(b). It consists of ideal (non scattering) leads and device (scattering) region. The ﬁrst step is to split the device region into slices i describing by relatively simple Hamiltonians H0 { } i

N 0 H = Hi + U. (5.18) i X By ’simple’ we imply the Hamiltonian of a such size, for which the corre- 0 sponding Green’s function can be found by direct matrix inversion Gii = 0 1 [E Hi ]− . The leads are assumed to be uniform, which often allows to find− the surface Green’s function (Γ) analytically or by using the technique described in Sec.(5.2.4). Thus, having the surface Green’s functions ΓL and ΓR and using Dyson Eqs.(5.17), one can add recursively slice by slice to find the Green’s function of each slice Gii as a part of the whole system, which provide information about LDOS, Eq.(5.12); and the Green’s function of the device GN+1,0, which allows to calculate transmission coefficient (see Eq.(5.45) in Sec.(5.2.5)). 40 Modeling

5.2.2 Bloch states Let us consider an inﬁnitely long ideal graphene nanoribbon consisting of N sites in the transverse j-direction, as illustrated in Fig.(5.3). The structure has a translational symmetry in i-direction with unit cell consisting of M = 2 slices for zigzag GNRs and M = 4 in the case of armchair GNRs.

c) zGNR unit celld) aGNR unit cell a) zigzag graphene nanoribbon 6 6 6 6 6 6 5 5 5 5 5 5 4 4 4 4 4 4 3 3 3 b) armchair graphene nanoribbon 3 3 3 2 2 2 2 2 2 1 1 1 1 1 1 0 1 2 3 0 1 2 3 4 5

Figure 5.3: a) zigzag and b) armchair graphene nanoribbons and corresponding to them unit cells c), d).

The Hamiltonian of the structure can be split into three parts

H = Hcell + Hout + U, (5.19) where the operators H describes the unit cell spanning slices 1 i M, cell ≤ ≤ while Hout describes the region including all other slices < i 0 and M + 1 i < . The coupling between the cell and slices−∞ i = 0≤ and ≤ ∞ i = M +1 is described by the hopping operator U. The total wave function consists of two parts, ψ = ψ + ψ , (5.20) | i | celli | outi corresponding to the unit cell and outside region respectively. Substituting Eqs. (5.20), (5.19) into the Schr¨odinger equation Hˆ ψ = E ψ and making | i | i use of deﬁnition of the Green’s function (5.8), we obtain

ψ = G U ψ , (5.21) | celli cell | outi where Gcell is the Green’s function of the operator Hcell. This equation allows us to relate the wave-function in the cell (slice i = 1 and i = M) with the wave-function in the outside region (slice i = 0 and i = M + 1). Recursive Green’s function technique 41

Hence calculating the matrix element ψ = 0 a ψ , we get i,j h | i,j| i 1,1 1,M + ψ1 = GcellU1,0ψ0 + Gcell U1,0ψM+1, M,1 M,M + ψM = Gcell U1,0ψ0 + Gcell U1,0ψM+1, (5.22) where we used that UM,M+1 = U0,1 due to the periodicity of the ribbon and + U0,1 = U1,0. The vector ψi corresponds to the wave-function on a slice i

ψi,1 . ψi =  .  , (5.23) ψi,N     and the Green’s function on the cell and the coupling matrices are deﬁned by

+ (U ) ′ = 0 a Ua ′ 0 (5.24) 1,0 jj h | 1,j 0,j | i ′ i,i + Gcell = 0 ai,jGcellai′,j′ 0 . jj′ h | | i Equation (5.22) can be rewritten in a compact form

ψ ψ T M+1 = T 1 , (5.25) 1 ψ 2 ψ M 0 where 1,M + 1,1 Gcell U1,0 0 IGcellU1,0 T1 = − ,T2 = − (5.26) GM,M U + I 0 GM,1U − cell 1,0 cell 1,0 and I being the unitary matrix. Since we consider the ideal nanoribbon, the wave-function has the Bloch form with a periodicity equals to the periodicity of the structure,

ikM ψm+M = e Iψm. (5.27)

This allows to rewrite Eq.(5.25) in the form of an eigenvalue problem

1 ψ1 ikM ψ1 T − T = e . (5.28) 1 2 ψ ψ 0 0 For a ﬁxed energy E, solution of (5.28) gives a set of eigenvalues k and { α} eigenfunctions ψα , ψα , 1 α 2N. Among 2N states there are N - { 0 } { 1 } ≤ ≤ prop propagating states and Nevan-evanescent states, which can be separated by the value of imaginary part of kα (i.e. Im[kα] > 0 for evanescent states). The propagating states, in turn, can be| separated| into two equal parts, right- and left-propagating states, by calculating their group velocities. This is described in the next section. 42 Modeling

5.2.3 Calculation of the Bloch states velocity As it is shown in Sec. (5.2.5), in order to calculate the transmission and reflection coefficients, one needs to know the velocities of propagating states. Moreover, the sing of velocity of Bloch state is used to separated the states by its direction of propagation, which is required to construct the surface Green’s functions (see Sec. (5.2.4)). In order to calculate the velocity we use the standard formula for the group velocity 1 ∂E v = . (5.29) ~ ∂k α One can estimate the derivative by direct numerical differentiation (E(k2 ) E(kα))/(kα kα). However it’s not efficient since it requires to solve eigen-− 1 2 − 1 value problem (5.28) twice. Moreover, each time the eigensolver is called, the eigenvalues are given in different order. In this section we derive the explicit form of the derivative. The wave- function of the α-th Bloch state within the unit cell is given by

M ψα = ψα , ψα = eikαxi ϕα . (5.30) | i | i i | i i | i i i=1 X (Thereafter we omit α to simplify notation.) Making use of the relation ψ H ψ = E ψ ψ = E ϕ 2, we obtain h i| | i h i| i | i| 1 ∂E 1 1 M ∂ ψ H ψ v = = h i| | i , (5.31) ~ ∂k ~ M ∂k ϕ 2 i=1 i X | | where the elements of the vector

ϕi,1 . ϕi =  .  (5.32) ϕi,N     equal to ϕ = 0 a ϕ . Finally, calculating the matrix element ψ H ψ i,j h | i,j| i h i| | i with Hamiltonian in the form (5.18), the velocity reads

M T 1 i ϕi∗ ik(xi xi−1) v = (xi xi 1)Ui,i 1ϕi 1e− − −~ M ϕ 2 − − − − i=1 i X | | h ik(xi xi) (x x )U ϕ e +1− . (5.33) − i+1 − i i,i+1 i+1 In the case of zero magnetic ﬁeld, this equation cani be simpliﬁed for zigzag nanoribbons M=2 T acc√3 ϕi∗ 1 v = sin k − Ui 1,iϕi, (5.34) 2~ ϕ 2 − i=1 i 1 X | − | where we used that ϕi = ϕi+M due to periodicity. Recursive Green’s function technique 43

5.2.4 Surface Green’s function Applying the Dyson equation, one can construct the Green’s function of the scattering region of an arbitrary shape. However in calculation of transport properties of the system, one usually assumes that the system is open and attached to a perfect (non-scattering) semi-infinite leads. Hence, one is interested in finding the surface Green’s function of the leads as a starting point for Dyson equation. Most of the methods for calculation of the Green’s function rely on searching for a self-consistent solution for ΓR and ΓL which makes these calculation very time consuming [50]. The method [48, 49] presented in this section does not require self-consistent calculations, and the surface Greens function is expressed in terms of the Bloch states of the graphene lattice. Let us consider a semi-infinite periodic ideal graphene ribbon extended to the right in the region M i < . Suppose that an excitation s is applied to its surface slice− i =≤ M.∞ Introducing the Green’s function| i − of the semi-infinite ribbon, Grib, one can write down the response to the excitation s in a standard form [16] | i ψ = G s . (5.35) | i rib| i The unit cell of a graphene lattice spans slices 1 i M,(M = 2 and ≤ ≤ 4 for the zigzag and armchair lattices, see Fig. 5.3). Applying the Dyson equation between the slices 0 and 1, we get 1, M 0, M Grib− = ΓRU1,0Grib− , (5.36) 1,1 where we defined the right surface Green’s function as ΓR Grib. Evalu- ating the matrix elements 0 a ψ of Eq. (5.35) and making≡ use of Eq. h | 1,j| i (5.36), we obtain for each Bloch state α α α ψ1 = ΓRU1,0ψ0 . (5.37)

This equations can be used to construct ΓR 1 ΓRU1,0 = Ψ1Ψ0− , (5.38) where Ψ1 and Ψ0 are the square matrices composed of the matrix-columns ψα and ψα, (1 α N), Eq. (5.23), i.e. 1 0 ≤ ≤ 1 N 1 N Ψ1 = (ψ1, ..., ψ1 ); Ψ0 = (ψ0, ..., ψ0 ). (5.39)

The expression for the left surface Greens function ΓL can be derived in a similar fashion + 1 ΓLU1,0 = ΨM ΨM− +1, (5.40) where the matrices ΨM and ΨM+1 are deﬁned in a similar way as Ψ1 and Ψ0 above. Note that matrices ΨM and ΨM+1 can be easily obtained from Ψ1 and Ψ0 using the relation (5.25). Note also that in the case of the zero magnetic ﬁeld, the right and left surface Greens functions are identical, ΓL = ΓR. 44 Modeling

5.2.5 Transmission and reﬂection

According to the Landauer formula (2.42), conductance can be expressed in terms of the transmission function. In order to calculate the transmission coefficient (E), the system is divided into three regions, namely, two ideal semi-infiniteT leads of the width N extending in the regions i 0 and i L respectively, and the central scattering region (device),≤ see ≥ Figs.(5.2),(5.3). The left and right leads are assumed to be identical. The i t r incoming, transmitted and reflected states in the leads, ψα , ψα and ψα , have the Bloch form, | i | i | i

N + ψi = eikα xi φα a+ 0 (5.41) | αi i,j i,j| i i 0 j=1 X≤ X N + t ik (xi xL) β + ψ = t e β − φ a 0 (5.42) | αi βα i,j i,j| i i L β j=1 X≥ X X N − ik x β ψr = r e β i φ a+ 0 , (5.43) | αi βα i,j i,j| i i 0 β j=1 X≤ X X where tβα (rβα) are the transmission (reflection) amplitude from the incoming Bloch state α to the transmitted (reflected) Bloch state β (α β), → and we choose x0 = 0. The transmission and reflection coefficients can be expressed through the corresponding amplitudes and the Bloch velocities [16]

vβ 2 vβ 2 = tβα ; = rβα . (5.44) T vα | | R vα | | Xα,β Xα,β The summation includes only propagating states. For the transmission and reﬂection amplitudes we use the equations given in Ref. [49],

L,0 1 Φ1T = G (U0,1Φ1K ΓL− Φ0), (5.45) − 0,0 − 1 Φ R = G (U Φ K Γ − Φ ) Φ , (5.46) 0 − 0,1 1 − L 0 − 0 where the matrices T and R of the dimension N N and have the × prop following meaning, (T )βα = tβα,(R)βα = rβα; (with Nprop being the number of propagating modes in the leads); GL,0 and G0,0 are the Green’s function matrices and ΓL is the left surface Green’s function, Eq. (5.40); U0,1 couples the left lead and the scattering region; K is the diagonal matrix with the + matrix elements Kα,β = exp(ikα x1)δα,β. The square matrices Φ1 and Φ0 α α are composed of the Bloch states φ0 and φ1 , Eq.(5.30), on the slices 0 and 1 N 1 N 1 of the ribbon unit cell, i.e. Φ1 = (φ1, ..., φ1 ) and Φ0 = (φ0, ..., φ0 ). Real-space Kubo method 45

5.3 Real-space Kubo method

The real-space Kubo method is a powerful tool to study transport properties of a system consisting of several millions of atoms. It was developed by S. Roche and D. Mayou [51, 52] and then applied extensively to investigate properties of carbon nanotubes [53, 54] and graphene [55, 56, 57]. The method is based on numerical solution of the time-dependent Schrödinger equation. Having calculated wave-function at different times, one can compute the mean square spreading of a wave packet. This quantity is proportional to the diffusion coefficient, which, in turn, can be related to the conductivity. The derivations provided in this section are primarily based on Refs. [58, 59].

5.3.1 Diﬀusion coeﬃcient The starting point of the model is the Kubo-Greenwood formula for conductivity (see Chapter 2 for details) 2e2π~ σ (E) = Tr [ˆv δ(E H)ˆv δ(E H)] , (5.47) ij V x − x − where V is the system volume,v ˆx is a x-component of the velocity operator and H is the Hamiltonian of the system. The last delta function can be written as a Fourier transform + 1 ∞ i(E H)t/~ δ(E H) = dte − . (5.48) − 2π~ Z−∞ Substituting it in Eq.(5.47), we get

+ 2 ∞ iEt/~ iHt/~ σ(E) = e dtTr vˆ δ(E H)e vˆ e− . (5.49) x − x Z−∞ Using the relation eiEt/~ = eiHt/~ and recalling the Heisenberg form of iHt/~ iHt/~ operators,v ˆx(t) = e vˆxe− , we arrive

+ ∞ σ(E) = e2 dtTr [ˆv (0)δ(E H)ˆv (t)] . (5.50) x − x Z−∞ Introducing the velocity autocorrelation function Tr [ˆv (0)δ(E H)ˆv (t)] vˆ (t)ˆv (0) = x − x , (5.51) h x x iE Tr [δ(E H)] − the equation for conductivity becomes

+ ∞ σ(E) = e2 dtTr [δ(E H)] vˆ (t)ˆv (0) . (5.52) − h x x iE Z−∞ 46 Modeling

It is possible to show that the velocity autocorrelation function vˆx(t)ˆvx(0) can be related to the mean value of the spreading in the hx-direction,i χ2(E, t) = (ˆx(t) xˆ(0))2 as h − iE d2 χ2(E, t) = vˆ (t)ˆv (0) , (5.53) dt2 h x x iE iHt/~ iHt/~ wherex ˆ(t) = e xeˆ − is the x-component of the position operator in the Heisenberg picture. Inserting Eq.(5.53) into Eq.(5.52) and performing integration, we get d σ(E) = e2Tr [δ(E H)] lim χ2(E, t). (5.54) t − →∞ dt Using the definition of χ2(E, t), the conductivity reads as d Tr [δ(E H)(ˆx(t) xˆ(0))2] σ(E) = e2Tr [δ(E H)] lim − − − t dt Tr [δ(E H)] →∞ − d = e2Tr [δ(E H)] lim (tD(E, t)), (5.55) t − →∞ dt where we have introduce a new quantity called diffusion coefficient and defined it as χ2(E, t) 1 Tr [δ(E H)(ˆx(t) xˆ(0))2] 1 n (E, t) D(E, t) = = − − = x , (5.56) t t Tr [δ(E H)] t n(E) − where n(E) is a density of states (DOS). Details of calculation of nx(E, t) are given in Sec.(5.3.3).

5.3.2 Transport regimes The diffusion coefficient, D(E, t), does depend on time. Figure (5.4) illustrates typical time-evolution of the diffusion coefficient. Analyzing its behavior one can distinguish three transport regimes, namely 1. ballistic 2. diffusive 3. localization

Ballistic regime For times t < τball, when the number of scattering events is small, the wave-packet propagates ballistically. This is manifested by linear growth of the diﬀusion coeﬃcient as a function of time such that χ(E, t)2 = v(E)t (5.57) 2 p D(E, t) = v(E) t. (5.58)

If the system is large enough or the impurities concentration ni is high, the diﬀusion coeﬃcient saturates and becomes constant at time τ d/v(E), ball ≈ where d = √ni is an average distance between impurities. Real-space Kubo method 47

Figure 5.4: Diffusion coefficient as a function of time. Three transport regimes (ballistic, diffusion, localization) are separated by dashed lines. The slope of D is proportional to the square of velocity v2. Dotted line denotes maximal value of D which corresponds to a semiclassical limit.

Diffusive regime In this regime the diffusion coefficient is independent of time, D(E, t) = D(E). Hence, the conductivity according to Eq.(5.55) becomes σ(E) = e2ρ(E)D(E), (5.59) where ρ(E) = n(E)/S is a density of states per unit area per spin and S is an area of the system. This equation is also known as Einstein relation for conductivity [16]. In some cases the plateau on the diffusivity plot is small and linear growth of D is immediately changed by decay. This is in contrast to the classical regime where the diffusion coefficient does not change when t . Thus, in order to calculate conductivity in the diffusive regime, Eq.→ (5.59)∞ is modified such that the maximum value of the diffusion coefficient, D(E, t) D (E), is used [55] → max 2 σsc(E) = e ρ(E)Dmax(E). (5.60)

This corresponds to a semi-classical conductivity within the Boltzmann approach where quantum effects leading to localization are not taken into account v2 σ = e2ρ(E)τ F , (5.61) Boltz 2 with τ being a scattering time. Comparing two equations, one can relate the mean free path to the computed diffusion coefficient

2Dmax le = vF τ = . (5.62) vF

Localization regime The eﬀect of localization can be viewed as constructing interference between forward and backward electron trajectories. 48 Modeling

When these effects become dominant, the diffusion coefficient decreases [55, 56]. According to the theory of localization, conductivity decreases with the increase of the system length L. In the Kubo approach we can relate L to time by L = χ(E, t)2. In this case conductivity depends on the size of the system p L(t) σ(E) exp , (5.63) ∼ − ξ with ξ being localization length.

Figure 5.5: Diffusion coefficient as a function of time for different concentration of strong short-range impurities [59]. The graphene sample consists of 6.8 millions of carbon atoms. Figure (5.5) [59] illustrates the normalized diffusion coefficient calculated for graphene sample consisting of 6.8 millions of carbon atoms. De- pending on the concentration of impurities the system appears to be in different transport regimes within plotted time interval. If the impurity concentration is zero, electrons propagate ballistically and the diffusion coefficient (blue line) grows linearly. For concentration n = 2% the diffusion coefficient undergoes crossover from ballistic to diffusive regime (red line) at time t = 20 fs. At n = 5% localization effects becomes dominant already at t = 10 fs which is manifested by the decrease of D (green line) with time.

5.3.3 Time evolution According to Eq.(5.56), in order to calculate the diﬀusion coeﬃcient, we need to compute the quantity n (E, t) = T r δ(E H)(ˆx(t) xˆ(0))2 . (5.64) x − − Time-dependence of the position operator can be rewritten in the Heisen- iHtˆ iHtˆ berg picturex ˆ(t) = e xeˆ − , which yields

iHtˆ iHtˆ iHtˆ iHtˆ n (E, t) = T r (e xeˆ − xˆ)δ(E H)(e xeˆ − xˆ) . (5.65) x − − − h i Real-space Kubo method 49

iHtˆ Using the commutation relation [H,ˆ e− ] = 0, we get

iHtˆ iHtˆ iHtˆ iHtˆ n (E, t) = T r (e xˆ xeˆ )δ(E H)(ˆxe− e− xˆ) x − − − h i = T r [ˆx, Uˆ(t)]+δ(E H)[ˆx, Uˆ(t)] , (5.66) − iHtˆ h i where Uˆ(t) = e− is a time evolution operator. In order to estimate the trace we need to know how the operator [ˆx, Uˆ(t)] acts on the wave-function ψ , i.e | ii [ˆx, Uˆ(t)] ψ =x ˆ Uˆ(t) ψ Uˆ(t) (ˆx ψ ) =x ˆ ψ (t) Uˆ(t) (ˆx ψ ) . (5.67) | ii | ii − | ii | i i− | ii If we denote ψx =x ˆ ψ , then | i i | ii n (E, t) = Ψ (t) δ(E Hˆ ) Ψ (t) , (5.68) x h i | − | i i i X where Ψ (t) = ψ (t) ψx(t) . Mathematically Eq.(5.68) is similar to | i i | i i − | i i equation for ﬁnding of the local density of states. The details of LDOS calculation are given in Sec.(5.3.7).

5.3.4 Exact solution of the time-dependent Schrödinger equation: Chebyshev method As it is seen from Eq.(5.56) and subsequent discussion in Sec. (5.3.2), the diffusion coefficient is a function of time. In order to estimate D(E, t) at different times t one needs to know the wave-function ψ(t) in accordance to Eq.(5.68). In this section we present an efficient method for solution of the time-dependent Schrödinger equation based on the expansion of the time evolution operator in an orthogonal set of Chebyshev polynomials [51, 58, 57]. The starting point is the time-dependent Schrödinger equation ∂ Hˆ ψ(t) = i~ ψ(t) , (5.69) | i ∂t | i with Hˆ being the tight-binding time-independent Hamiltonian. If the initial wave-function at time t0, ψ(t = 0) = ψ0 , is known, the formal solution of Eq.(5.69) can be expressed| via thei time-evolution| i operator (or propagator) Uˆ(t), i Hˆ (t) ψ(t) = Uˆ(t) ψ , Uˆ(t) = e− ~ . (5.70) | i | 0i In order to expand Uˆ(t) in a set of the Chebyshev polynomials Tn(x) (which are defined in the interval x [ 1; 1]), we first renormalize the Hamiltonian such that its spectrum lies in∈ the− above interval, 2Hˆ (E + E )Iˆ Hˆ = − max min , (5.71) norm E E max − min 50 Modeling

where Emax and Emin are the largest and the smallest eigenvalues of the original Hamiltonian Eq. (4.6). (In order to calculate Emax and Emin we use a computational routine that estimates the largest/smallest eigenvalues of the operator without calculation of all the eigenvalues). Expanding Uˆ(t) in Chebyshev polynomials in Eq. (5.70) we obtain for the wave function,

∞ ψ(t) = c (t) Φ , (5.72) | i n | ni n=0 X where the functions Φn = Tn(Hˆnorm) ψ0 are calculated using the recur- rence relations for the| Chebyshevi polynomials,| i

Φ = T (Hˆ ) ψ = ψ (5.73) | 0i 0 norm | 0i | 0i Φ = T (Hˆ ) ψ = Hˆ ψ (5.74) | 1i 1 norm | 0i norm | 0i Φn+1 = Tn+1(Hˆnorm) ψ0 = 2Hˆnorm Φn Φn 1 . (5.75) | i | i | i − | − i

The recursive routine is repeated until the expansion (5.72) converges which is deﬁned by the condition

cn(t) Φn 1 < ǫ, (5.76) k | i k − n=0 X where ǫ is some predeﬁned computational tolerance. The expansion coeﬃ- cients cn(t) are calculated making use of the orthogonality relation for the Chebyshev polynomials

E E t i ( max+ min) n Emax Emin c (t) = 2e− 2~ ( i) J − t . (5.77) n − n 2~

For large t the expansion coeﬃcients cn(t) become exponentially small. This leads to the fast convergence of the expansion series Eq. (5.72), and makes the Chebyshev method very eﬃcient for calculation of the temporal dynamics. Figure (5.6) illustrates temporal dynamics of a wave-packet in a pure (undoped) graphene. The wave-paket, originally (at t = 0) localized at a single site in the middle of the structure, distributes uniformly in all directions. Less than 1000 iterations is required to achieve convergence for 80 fs of elapsed time. Real-space Kubo method 51

Figure 5.6: Temporal dynamics of the wave-packet in 2D graphene. At initial time the wave-packe was localized at a single site in the middle of the structure.

5.3.5 Continued fraction technique Consider a Hamiltonian matrix given in a tridiagonal form,

α1 β1 0 β α β ············0  1 2 2 ·········  0 β α β 0 2 3 3 ······  ......  ˆ  . 0 β3 . . .  Htri =  ···  . (5.78)  ......   . . 0 . . . .   ......   ...... βN 1   −   ......   . . . . . βN 1 αN   −    Our aim is to calculate the ﬁrst diagonal element G11 of the Greens function 1 Gˆ = (EIˆ Hˆ )− without a computing the whole Greens function and all − tri the eigenvalues/eigenfunctions of the Hamiltonian. Let us denote λi = G1i. From the deﬁnition of the Greens function we obtain [60],

(E αi)λi βi 1λi 1 βiλi+1 = 0, 2 i N 1 (5.79) − − − − − ≤ ≤ − with (E αN )λN βN 1λN 1 = 0 and (E α1)λ1 β1λ2 = 1. Expressing − − − − − − sequentially λN by λN 1, λN 1 via λN 2 and λN 3 we express G11 as a − − − − continued fraction [58, 59, 60, 61]: 1 G = . (5.80) 11 β2 E α 1 − 1 − β2 E α 2 2 β2 − − 3 E α3 . − − ..

Even for a huge matrices calculation of G11 does not require a lot of computational time. However if the Hamiltonian matrix is given in a general form, i.e. consist of arbitrary number of diagonals, it must be ﬁrst 52 Modeling

Figure 5.7: LDOS calculated with the use of the continued fraction technique. M is the number of terms included in Eq.(5.81). Energy is in units of the hopping integral. transformed into the tridiagonal form [see Sec.(5.3.6)]. Tridiagonalization procedure is a very time-consuming routine. Therefore it is of great importance to use as less number of terms in Eq.(5.80) as possible. It can be done by truncation of the continues fraction 1 G = . (5.81) 11 β2 E α 1 − 1 − β2 E α 2 2 . − − .. E α β2 Σ(E) − M − M In the above equation we truncated the order-N continued fraction at the fraction M < N by introducing the self-energy Σ(E), 1 1 Σ(E) = = , (5.82) β2 E α β2 Σ(E) E α M − M − M − M − β2 E α M M . − − .. that includes all the remaining terms M + 1 i N. Solving Eq. (5.82) one easily obtains ≤ ≤ E α i 4β2 (E α )2 Σ(E) = − M − M − − M . (5.83) 2β2 p M Real-space Kubo method 53

The number of terms M included in the summation in Eq. (5.81) is determined from the condition for the convergence of G11. As it is seen from Fig.(5.7), the higher energy the more elements should be included in the summation. For this particular case (N = 10000), half of the elements is enough to achieve high accuracy of calculations in a broad energy interval.

5.3.6 Tridiagonalization of the Hamiltonian matrix In the real-space representation the tridiagonal Hamiltonian, which is given by Eq.(5.78), usually corresponds to a one-dimensional chain with nearest- neighbor interaction. In practice, however, Hamiltonian has more com- picated form. For instance, for 2D graphene it can consist of up to 9 diagonals if the periodic boundary conditions are used. In order to utilize the continued fraction technique to calculate G11 the Hamiltonian should be first transformed to the tridiagonalized form H H . This is done → tri by constructing a new orthogonal basis as described below. We start by selecting the first basis vector 1 . Curled brackets i are used to denote the new basis vectors while straight| } brackets i denote| } the old ones. If | i the tridiagolization is performed in order to find the local density if states on the i-th site of the system at hand, the first basis vector is selected as

1 = i , where i = ci† 0 . | } We| requirei that| i the| Hamiltoniani in the new basis be of the tridiagonal form (5.78). By operating Hˆ i we arrive to the following equations [58, | } 59, 60], Hˆ 1 = α 1 + β 2 , (5.84) | } 1| } 1| } Hˆ i = βi 1 i 1 + αi i + βi i + 1 , 2 i N 2, (5.85) | } − | − } | } | } ≤ ≤ − Hˆ N = βN 1 N 1 + αN N . (5.86) | } − | − } | } Using Eq. (5.84) and the orthogonality relation 1 2 = 0 we obtain the { | } second basis vector and the matrix elements α1 and β1, 1 2 = Hˆ 1 α1 1 , (5.87) | } √C2 | } − | } α = 1 Hˆ 1 , β = 2 Hˆ 1 , 1 { | | } 1 { | | } where the normalization coeﬃcient C2 (as well as all other normalization coeﬃcients C , 2 i N) are obtained from the normalization requirement i ≤ ≤ i i = 1. { | }We then proceed to Eq. (5.85) and recursively calculate the basis vectors i , 2 i N 2, and corresponding matrix elements α and β , | } ≤ ≤ − i i 1 i + 1 = Hˆ i βi 1 i 1 αi i , (5.88) | } √Ci+1 | } − − | − } − | } α = i Hˆ i , β = i + 1 Hˆ 1 , 2 i N 2. i { | | } i+1 { | | } ≤ ≤ − 54 Modeling

Finally, from Eq. (5.86) we obtain 1 N = Hˆ N αN N , αN = N Hˆ N , (5.89) | } √CN | } − | } { | | } which concludes the tridiagonalization procedure.

5.3.7 Local density of states The techniques describe in the previous sections, namely tridiagolization and continued fraction technique, provides an eﬃcient way to calculate the local density of states (LDOS). If the system is described by the real-space Hamiltonian H given in the matrix form, LDOS at site i can be calculated by (see Eq.(5.12) in Sec.(5.1))

1 1 ρi(E) = lim Gii(E + iη) ,G(E) = (EIˆ H)− . (5.90) −π ℑ η 0 − →

In order to calculate Gii, the Hamiltonian must be first tridiagonalized. The first basis vector is chosen in a such way that all elements except i-th are equal to zero, hence 1 = i . Then, then the continued fraction technique | } | i is applied to calculate G11(E), which in the new basis corresponds to the required Gii(E). If one needs to compute DOS, it can be done by calculating LDOS on each site and then averaged over them. However this is not an efficient way, since the Hamiltonian must be tridiagonalized for each site separately. Instead we generate a random state, which extends over M-sites in the middle of the structure, 1 ψ = e2iπαi i , (5.91) | rani √ | i M i X and choose the first basis vector as 1 = ψ . Then the Hamiltonian is | } | rani tridiagonalized as described in Sec.(5.3.6) and LDOS computed by ρi(E) = 1 [limη 0 Gii(E + iη)] using the continues fraction technique. In this π → new− ℑ basis one ’site’ corresponds to the chosen domain consisting of M real sites. Therefore calculating LDOS we obtain DOS. Chapter 6

Summary of the papers

6.1 Paper I

In this paper we consider a system consisting of the graphene nanoribbon located on an insulating substrate with a metallic back used to tune charge density. We develop an analytical theory for the gate electrostatics of the GNRs and calculate the capacitance. The total capacitance of the system is a sum of two contributions: the classical capacitance which is determined by the geometry of the system and a dielectric permittivity and the quantum capacitance which arises from a finite density of state of the GNR. To complement our study we also perform numerical calculations which are based on the tight-binding Hamiltonian. The effect of electron-electron interaction is taken into account within the Hartree approximation. For a chosen gate voltage the charge density is calculated self-consistently by integrating LDOS which is computed using the recursive Green’s function technique. We show that the distribution of charge density and potential is not uniform. Due to the electrostatic repulsion electrons tend to accumulate near the boundaries. It is also demonstrated that electron-electron interaction leads to significant modification of the band structure. Our exact numerical calculations show that the density distribution and the potential profile in the GNRs are qualitatively different from those in conventional split-gate quantum wires with a smooth electrostatic confinement where the potential is rather flat and the electron density is constant throughout the wire. At the same time, the electron distribution and the potential profile in the GNR are very similar to those in the cleaved-edge overgrown quantum wires (CEOQW) exhibiting triangular-shaped quantum wells in the vicinity of the wire boundaries. This similarity reflects the fact that both the CEOQWs and the GNRs correspond to the case of the hard-wall confinement at the edges of the structure. 56 Summary of the papers

6.2 Paper II

We study interaction and screening eﬀects in a gated bilayer graphene nanoribbon. We employ the numerical model similar to the model described in paper I in order to study electron distribution and the energy gap in bGNRs. We also derive an analytical expression for a dependence of a potential diﬀerence between graphene layers and the gate voltage. The analytical estimations are shown to be in a good quantitative agreement with the numerical calculations. We demonstrate, see Fig.(6.1), that in

(a) 100 6

80 5

60 g E (meV) 2 40 d=15 nm d=50 nm 20 1 d=100 nm 3 4 0 0 5 10 15 20 25 30 V g (V) (b) E 0.62 E 0.99 F F 1 2

0.58 0.95 0 γ E ( ) 0.54 0.91

0.5 0.87

-0.08 -0.04 0 0.04 0.08 -0.08 -0.04 0 0.04 0.08 1.48 2.12 EF EF 2.08 1.44 3 4 2.04

0 1.4 γ 2 E ( ) 1.36 1.96

1.32 1.92 -0.1 -0.05 0 0.05 0.1 -0.15 -0.1 -0.05 00.05 0.150.1

4.5 E 8.7 F EF

0 5

γ 6 4.4 8.6 E ( )

4.3 8.5

-0.2 -0.1 0 0.1 0.2 -0.2 0-0.1 0.1 0.2 k (1/a) k (1/a)

Figure 6.1: (a) The energy gap as a function of the applied gate voltage for various dielectric thicknesses d. (b) Representative band structures corresponding to the Fermi energies shown in (a). The dashed lines indicate the positions of Fermi energy. contrast to 2D graphene sheets the energy gap of the bGNRs dependence nonlinearly on the applied gate voltage. Also the energy gap can collapse at some intermediate gate voltages which is explained by the strong modiﬁ- cation of the energy spectrum caused by the electron-electron interactions. Paper III 57

6.3 Paper III

Conductance quantization is a hallmark of mesoscopic physics. The first experimental observation of the conductance plateau was done more than 20 years ago [62]. In magnetic field conductance plateau in a quantum point contact becomes more clearly defined, which is attributed to formation of the robust edge states. In this paper we investigate the conductance of gated graphene nanoribbons in a perpendicular magnetic field. We adopt the recursive Green’s function technique to calculate the transmission coefficient which is then used to compute the conductance according to the Landauer approach.

10 20K Hartree one-electron 8 50K Hartree one-electron (d)

(b) 6 (c) 2

(a) w_ G(2e/h) 4 ~ 11 lB

w 2 B nanoribbon d dielectric gate 0 0 2 4 14121086 fillingfactor

Figure 6.2: Conductance of the GNR as a function of ﬁlling factor for interacting and noninteracting electrons at temperatures T = 20 K (red thick lines) and 50 K (blue thin lines) in a magnetic ﬁeld B = 30 T. Inset: sketch of the sample geometry.

Figure (6.2) shows the conductance as a function of filling factor at a fixed magnetic field. We compare both the case of interacting particles calculated within the Hartree approximation (solid lines) and the one-electron case (dashed lines). In the one-electron case the conductance is quantized as it is expected for ideal (without disorder) GNRs. However in the interacting case the conductance quantization is destroyed. This surprising behavior is related to the modification of the band structure of the GNR due to the electron interaction leading, in particular, to the formation of compressible strips in the middle of the ribbon and existence of counter- propagating states in the same half of the GNR. 58 Summary of the papers

6.4 Paper IV

ν =0.0 ν =0.1 ν =0.52 0.04 0.04 0.04 a) b) c) 0.02 0.02 0.02 -2 0.00 0.00 0.00 n(nm)

-0.02 -0.02 -0.02

-0.04 -0.04 -0.04 -10 0.0 10 -10 0.0 10 -10 0.0 10 y(nm) y(nm) y(nm) 0.010 d) 0.105 e) f) 0.505 0.005 0.100 0.500 0 0.000

E(k)/t 0.095 0.495 -0.005 0.090 0.490 -0.010 -0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2

k(1/a)x k(1/a)x k(1/a)x

Figure 6.3: (a)-(c) Charge density distributions, and (d)-(f) band structure of a graphene nanoribbon in a perpendicular magnetic field B = 150T. (a),(d) one-electron approximation, eVg/t0 = 0; (b),(e) self-consistent calculations for eVg/t0 = 0.1 and (c),(f) eVg/t0 = 0.5. Red and blue curves marked by and correspond to the charge densities of spin-up and spin- down electrons↑ respectively,↓ n (y), n (y). Charge densities are averaged over ↑ ↓ three successive sites. Green curves correspond to the total density distribution n(y) = n (y)+n (y). Dashed lines define the Fermi energy position. ↑ ↓ Yellow fields correspond to the energy interval [ 2πkBT, 2πkBT ]; temperature T = 4.2 K. − One of the interesting peculiarity of graphene is the existence of the 0’th Landau level in a magnetic field [63]. This level is located at the Dirac point and equally shared between electrons and holes. In a high enough magnetic fields experiments exhibit the formation of an insulating state at ν = 0 which is manifested by a peak on ρxx plots [64]. The nature of this state is still under debate. It was suggested that backscattering of spin polarized edge states at ν = 0 can be responsible for the increase of ρxx [65]. Figure (6.3) shows the self-consistent distribution of the charge density and the band-structure of the GNR for different values of the filling factor. Our self-consistent calculations demonstrate that, in comparison to the one- Paper V 59 electron picture, electron-electron interaction leads to the drastic changes in the dispersion relation and structure of the propagating states in the regime of the lowest LL such as a formation of the compressible strip and opening of additional conductive channels in the middle of the ribbon. Also we study the effect of different types of disorder (short-range impurities, edge disorder, warping and spin-flipping) on GNRs conductance, focusing on the robustness of, respectively, edge and bulk state transmissions. The latter are shown to be very sensitive to the disorder and get scattered even if the concentration of the disorder is moderate. In contrast, the edge states are very robust and cannot be suppressed even in the presence of the strong spin-flipping.

6.5 Paper V

Recent experiments address the effect of correlation in the spatial distribution of disorder on the conductivity of graphene sheet by doping it with potassium atoms. It has been found that the conductivity of the system at hand increases as the temperature rises, and argued that this was caused by the enhancement of correlation between the potassium ions due to the Coulomb repulsion [66]. In paper V using the efficient time-dependent real-space Kubo formalism, we performed numerical studies of conductivity of large graphene sheets with random and correlated distribution of disorder. In order to describe realistic disorder, we used models of the short- range scattering potential (appropriate for adatoms covalently bound to graphene) and the long-range Gaussian potential (appropriate for screened charged impurities on graphene and/or dielectric surface). The calculations for the uncorrelated potentials are compared to the corresponding predictions based on the semiclassical Boltzmann approach and to exact numerical calculations performed by different methods. We find that for the most important experimentally relevant cases of disorder, namely, the strong short-range potential and the long-range Gaussian potential, the correlation in the distribution of disorder does not affect the conductivity of the graphene sheets as compared to the case when disorder is distributed randomly. Our results strongly indicate that the temperature enhancement of the conductivity reported in the recent study [66] and attributed to the effect of dopant correlations was most likely caused by other factors not related to the correlations in the scattering potential.

6.6 Paper VI

The value of a spin-splitting caused by the Zeeman effect is proportional to the g-factor. For graphene the g-factor equals to its free-electron value 60 Summary of the papers g = 2. If the electron-electron interaction is significant, the spin-splitting is increased. In this case the effective g-factor is introduced to incorporate this effect. Recent experiments shows that effective g-factor in graphene is enhanced in comparison to the free-electron value g∗ = 2.7 0.2 [67], which indicates that electron-electron interaction effects play an± important role and should be taken into account for explanation of the enhanced spin-splitting.

Figure 6.4: The effective g-factor as a function of the filling factor ν for different concentrations of charged impurities, ni = 0%, 0.02%, 0.08%, 0.2%, at the constant perpendicular magnetic field B = 35T. Inset: the dependence of g∗ on the Hubbard constant U for the fixed ν = 2.5. All the calculations are done at the temperature T = 4 K.

In this work we employed the Thomas-Fermi approximation in order to study the effective g-factor in graphene in the presence of a perpendicular magnetic field taking into account the effect of charged impurities in the substrate. We found that electron-electron interaction leads to the enhancement of the spin splitting, see Fig. (6.4), which is characterized by the increase of the effective g-factor. We showed that for a low impurity concentration g∗ oscillates as a function of the filling factor ν in the range from g∗ = 2 to g∗ 4 reaching maxima at even filling factors min max ≈ and minima at odd ones. Also, we outlined the influence of impurities on the spin-splitting and demonstrated that the increase of the impurity concentration leads to the suppression of the oscillation amplitude and to a saturation of the the effective g-factor around a value of g∗ 2.3. ≈ Bibliography

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