Introduction to Graphene

Home , Landau quantization

Bjarni Hannesson

FacultyFaculty of of Physics Physics UniversityUniversity of of Iceland Iceland 20162016

INTRODUCTION TO GRAPHENE

Bjarni Hannesson

16 ECTS thesis submitted in partial fulﬁllment of a B.Sc. degree in physics

Advisors Lárus Thorlacius Valentina Giangreco M. Puletti

Faculty of Physics School of Engineering and Natural Sciences University of Iceland Reykjavík, June 2016

Abstract

In this report some properties of graphene, a single atomic layer of carbon, are outlined. The structure of graphene is reviewed and its dispersion relation within the tight-binding approximation and the effective mass approximation. The observed half-integer quantum Hall effect in graphene is illustrated and finally bilayer graphene is discussed briefly.

Útdráttur

Í þessari ritgerð verður fjallað um grafín sem er einnar frumeindar þykkt lag af kolefni. Frumeindirnar í grafíni mynda tvívíða sexhyrningagrind og nota má einfalt tvívítt líkan með víxlverkunum milli næstu og þarnæstu nágranna til að lýsa hegðun rafeinda í grafíni. Einnig verður fjallað um skömmtuð Hallhrif í grafíni og að lokum stuttlega um tveggja laga grafín.

Contents

1. Introduction 1

2. Lattice structure 3

3. Energy spectrum 5 3.1. The massless Dirac fermions ...... 6

4. The Quantum Hall effect in Graphene 9 4.1. Classical Hall effect ...... 9 4.2. Landau levels ...... 10 4.3. Quantum Hall Effect ...... 11 4.4. Landau levels in graphene ...... 12 4.5. The anomalous quantum Hall effect in graphene ...... 15 4.6. Symmetry breaking ...... 16 4.7. Shubnikov-de Haas oscillations ...... 18

5. Bilayer Graphene 19 5.1. Fabrication ...... 19 5.2. Identiﬁcation of Layers ...... 20 5.3. Tight-binding model ...... 22

6. Summary 25

A. 2D fermion in an external magnetic ﬁeld 27

B. 2D Dirac fermion in an external magnetic ﬁeld 29

Bibliography 31

vii

1. Introduction

Graphene is a sheet of carbon atoms, ordered in a hexagonal pattern and only one atom thick. We have all come in contact with graphene in our daily lives in its bulk form, named graphite. Graphite is simply a collection of graphene stacks or multiple layers of graphene weakly bound together. Graphite is in your pencil and these weak bonds are the reason the pencil leaves a trace. Although the pencil was invented in the 16-th century, graphene was only ﬁrst observed in 2004 by Novoselov et.al [1]. Novoselov and Geim received the Nobel prize in Physics in 2010 for their groundbreaking studies regarding graphene.

This report will review the structure of graphene in section 2 and its dispersion relation and charge carriers, which behave as massless Dirac fermions, in section 3. We will also discuss the anomalous quantum Hall eﬀect in graphene in section 4 and brieﬂy review bilayer graphene in section 5. This report is mainly based on the review papers by Abergel et.al, Properties of graphene: a theoretical perspective, published in Advances in Physics in 2010 [2] and by Castro Neto et.al, The electronic properties of graphene, published in Reviews of Modern Physics in 2009 [3].

2. Lattice structure

Graphene is a single atomic layer of carbon. The carbon atoms form a hexagonal or honeycomb lattice making the whole sheet resemble chickenwire. The graphene sheet (2D) can be rolled up to form a carbon nanotube (1D) or to a complete sphere to form fullerene (zero-D). To form a sphere some pentagons must replace the hexagons for the pentagons introduce a necessary curvature to the sheet, just like a soccer ball. The sheet of graphene has many qualities such as high electric and thermal conductivity as well as being transparent and robust. Graphene is even more robust than diamond and is in fact carbons most stable form. That is if we excite a diamond with a little energy above a threshold it would break down and form graphene or graphite.

Although fullerenes and nanotubes have a lot of interesting features we will only consider the two dimensional sheet of graphene. Each carbon atom has three nearest neighbours arranged with a 120◦ angle between them, as seen in ﬁgure 2.1. This 2 trigonal plane is caused by sp hybridization of the 2s, 2px and 2py orbitals. These orbitals form strong σ bonds with a separation of 1.42 Å. The σ band forms a full and deep valence band which is responsible for the robustness of graphene. The last valence electron is in the 2pz orbital. A weak π band forms when half ﬁlled 2pz orbitals overlap other 2pz orbitals.

The hexagonal lattice of graphene is bipartite and can be divided into two triangular sublattices, A and B, as shown in ﬁgure 2.1. Each sublattice is a Bravais lattice and the unit cell consist of two atoms. Since the unit cell has two atoms each cell has two 2pz electrons and the π band is completely ﬁlled. The basis vectors of sublattice A with lattice constant a = 2.46Å, are

a √ a √ ~a1 = 3, 1 , ~a2 = 3, −1 . (2.1) 2 2 The location of an A site atom can be expressed as a linear combination of these basis vectors. The location of a B site atom can be expressed as a linear combination of the basis vectors plus an oﬀset,

~ a 1 ~ a 1 ~ 1 δ1 = √ , 1 , δ2 = √ , −1 , δ3 = a −√ , 0 . (2.2) 2 3 2 3 3

3 2. Lattice structure

Figure 2.1: The honeycomb lattice of graphene in real space and reciprocal space. a) The left ﬁgure shows the hexagonal lattice, the A and B sublattices, the basis vectors and the unit cell (dotted lines). b) The corresponding reciprocal lattice and its basis vectors. The shaded area is the ﬁrst Brillouin zone and its high-symmetry points Γ, M and K are illustrated. Figure from [2].

The respective reciprocal vectors are ~ 2π 1 ~ 2π 1 b1 = √ , 1 , b2 = √ , −1 , (2.3) a 3 a 3 √ with a reciprocal lattice constant |~b| = 4π/ 3a. The ﬁrst Brillouin zone is a hexagon with degeneracy points appearing at the corners. These are called K points or Dirac points and, as we shall see below, they are the positions of the apex of so-called Dirac cones in the electron dispersion relation. Only two of the corners are inequivalent of each other and their position in momentum space are 2π 1 2π 1 K~ = √ 1, √ , K~ 0 = √ 1, −√ . (2.4) 3a 3 3a 3

4 3. Energy spectrum

To study graphene we use the tight-binding approximation, where an electron is only allowed to travel between its nearest or next-nearest neighbours. The tight-binding Hamiltonian for the π electrons in graphene was ﬁrst studied by Wallace [4] in 1947 and gave the energy dispersion of the π band. The Hamiltonian is 1

X † 0 X † † Htight-binding = −t (aσ,ibσ,j + H.c.) − t (aσ,iaσ,j + bσ,ibσ,j + H.c.). (3.1) hi,ji,σ hhi,jii,σ

† In the Hamiltonian aσ,i and aσ,i are the annihilation and creation operators for an ~ † electron with spin σ =↑, ↓ at site Ri on sublattice A. Likewise, the bσ,i and bσ,i operators refer to sublattice B. The hopping integral, t, is the energy needed to hop to the nearest neighbour and is t ≈ 2.8eV . The next-nearest neighbour hopping 0 integral, t’, is not well known but calculations in [5] found that 0.02 t . t . 0.2 t. For the tight-binding Hamiltonian the corresponding wavefunction for graphene is a linear combination of Bloch functions for each sublattice.

In the nearest neighbour approximation the energy eigenvalues can be obtained in closed form [6,7] " √ #1/2 3kxa kya 2 kya E(kx, ky) = ±t 1 + 4 cos cos + 4 cos . (3.2) 2 2 2

If we take into account the next-nearest neighbour hopping an electron-hole asymmetry in the K points appears. For a complete picture of the band structure the σ bands should be added since they are the lowest energy bands near the center of the Brillouin zone. The interesting features of graphene however occur in the low energy limit at the K points so we ignore this in this report.

Expanding the energy dispersion (3.2) around the Dirac points such that the wave vector is ~q = K~ +~k and ka 1, where k = |~k|, gives the linear dispersion relation, √ ~ 3 E±(k) ≈ ± tak = ±vF k. (3.3) 2 ~ 1The brackets and double brackets under the sum denote nearest and next-nearest neighbour approximation, respectively, indicating to only sum over neighbouring sites.

5 3. Energy spectrum √ 6 m The Fermi velocity is vF = 3ta/2~ ≈ 10 /s and is independent of charge density and momentum. Note that the energy varies linearly with respect to momentum which indicates that we have relativistic massless particles.

Figure 3.1: a) The energy dispersion of graphene plotted. The π∗ band refers to holes and the ﬁlled π band to electrons. The bands meet at the K and K’ points forming Dirac cones. b) The dispersion along the high-symmetry points ΓMK. Figure from [2].

The π and π∗ bands are shown in ﬁgure 3.1 and they meet at the Dirac points forming the Dirac cones. These cones are a direct result of a linear dispersion relation around the Dirac points. In undoped graphene the Fermi energy level lands exactly at the apex of the cones. Toggling the Fermi level slightly up or down the cones one ﬁnds the Fermi surface to be a circle.

3.1. The massless Dirac fermions

The linear dispersion relation indicates that the charge carriers are massless Dirac fermions. Although the dispersion implies zero effective mass the fermions in graphene do not travel at the speed of light but rather an effective speed of light, which in ∗ graphene is c = vF . Even though the effective mass is zero the cyclotron mass is not. Fermions that satisfy the massless Dirac equation have a cyclotron mass which depends on the charge carrier density while those satisfying the Schrödinger equation have a constant cyclotron mass [3].

Lets convince ourselves that the cyclotron mass is dependent upon the charge density. The cyclotron mass is deﬁned within the semiclassical approximation [8]

2 ∗ ~ ∂A(E) m = . (3.4) 2π ∂E E=EF

2 2 2 2 Here A(E) = πk = πE /vF ~ is the area in momentum space enclosed by the orbit

6 3.1. The massless Dirac fermions traced by the Fermi surface. The cyclotron mass is then √ ∗ EF kF ~ π~√ m = 2 = = n, (3.5) vF vF vF since the Fermi velocity is independent of charge carrier density and the relation 2 between the Fermi momentum and charge density is n = kF /π (with contributions from spin and K points).

Figure 3.2: Experimental data for the ratio of cyclotron mass to rest mass of an electron plotted as a function of charge density. The red circles are for holes and blue circles for electrons. The lines represent a ﬁtted square root behaviour of the cyclotron mass. Figure from [9].

In ﬁgure 3.2 experimental data is plotted against the expected square root dependence. The mass data was collected with the temperature dependence√ of the amplitude of the Shubnikov-de Haas oscillations. From the ﬁgure the n dependence of the cyclotron mass is apparent giving evidence of relativistic massless Dirac fermions in graphene.

4. The Quantum Hall eﬀect in Graphene

4.1. Classical Hall eﬀect

The Hall effect was discovered in 1879 by Edwin Hall, 18 years before the discovery of the electron. The Hall effect can for example be used to measure magnetic field, with a device called a Hall probe. To understand the Hall effect imagine a conducting

Figure 4.1: The classical Hall effect, the figure shows current flowing and a magnetic field perpendicular to the plate. As the charge carriers experience force from the magnetic field they accumulate on one side. Opposite charge will be induced on the other side and an electric field forms which cancels out the magnetic force. Figure from [10]. plate, like the one in figure 4.1, with length L and width W. If a magnetic field is switched on, perpendicular to the plate, the charge carriers will experience the Lorentz-force F~ = −e~v × B~ . Like in figure 4.1 negative charge will accumulate on the left side of the plate which leads to a positive charge build-up on the right side. In steady state the force from the magnetic field and the force due to electric field between the sides of the plates will be equally strong and no net force works on the current. The current passes through in y-direction normally, as if there were no magnetic field. The voltage drop between the sides of the plate is called the Hall R = Vy voltage and the Hall resistance is defined as xy Ix . The resistance in the plate

9 4. The Quantum Hall effect in Graphene is B L/W Rxy = ,Rxx = , (4.1) ne neµ where µ is the conductivity and n is the charge carrier concentration. An interesting feature is that the Hall resistance Rxy is independent of the geometry of the plate. By punching a few holes into the plate we drastically change the resistance, Rxx, while the Hall resistance, Rxy, is unaffected. A similar situation will appear in the quantum Hall effect where there are localized states due to disorder.

4.2. Landau levels

Before we review the quantum Hall effect in graphene we will first look at a non- relativistic two dimensional fermion in an external magnetic field. Let’s put the ~ magnetic field as B = Beˆz perpendicular to the plane in which the fermions lie. A detailed solution to this problem is in appendix A. If we pick the Landau gauge ~ A = (−yB, 0), then the Hamiltonian commutes with the pˆx operator and the fermion behaves as a free particle in x-direction,

ˆ ~ˆ 2 pˆ2 2 ˆ (~p + eA) y mωc 2 H = = + (ˆy − y0) . (4.2) 2m 2m 2

eB px Here ωc = m is the cyclotron frequency and y0 = eB . For the y-component we get 1 a shifted harmonic oscillator with energy levels En = ~ωc n + 2 . These energy levels are also known as Landau levels and are hugely degenerate.

Let’s deﬁne the length Lx so that the fermion can move in a region [0,Lx] in x- direction and assume periodic boundary conditions. Then for the eigenfunctions to be orthogonal the following relationship must be fulﬁlled,

Z Lx 0 = e−iknxeikn+1xdx (4.3) 0

2 This gives kn+1 = kn +2π/Lx. The shift in y-direction is kn = yn/l where the mag- p B netic length is lB = ~/eB. The relationship between two neighboring oscillators 2 is yn+1 = yn + 2πlB/Lx. Now we can calculate the area between two neighbouring oscillators and more importantly the ﬂux between them,

2π 2 2π~ h Lx lBB = B = = Φ0. (4.4) Lx eB e

Here Φ0 is the magnetic flux quantum. Let’s define two other important quantities, B ne nΦ = ν = ne the magnetic degeneracy Φ0 and the filling factor nΦ , where is the electron concentration. The filling factor is a useful parameter since it tells us how

10 4.3. Quantum Hall Effect many of the lowest Landau levels are filled. It becomes relevant when we examine the quantum Hall effect.

The system is also ﬁnite in the y-direction and let’s call that distance Ly. As seen before the wavevector k restricts to kn = 2πn/Lx but now its range is also restricted. If we ignore the boundary eﬀects when the wavefunction overlaps the edges we can estimate the number of available states of one level to be, eB N = LxLy. (4.5) 2π~

4.3. Quantum Hall Eﬀect

The integer quantum Hall effect was first discovered in 1980 by Klauss von Klitzing and in 1985 he won the Nobel Prize in Physics for his discovery. When observing charge-transport in high mobility two-dimensional electron gases at low tempera- tures and high magnetic fields he found that the longitudinal resistance became very small for certain ranges of magnetic field. At those same values for the field, the Hall conductance showed flat regions or plateaus. They seemed independent of e2 samples and appeared at n h where, n, is an integer. Therefore it became apparent that the Hall effect was quantized.

To understand these plateaus let’s look at the current each Landau level carries. We can introduce an electric ﬁeld E in the y-direction by adding a potential term V (y) = −Ey to the Hamiltonian in (4.2), pˆ2 2 ˆ y mωc 2 H = + (ˆy − y0) − eEy.ˆ (4.6) 2m 2

Now the wavefunction of the harmonic oscillator centres around y = kx~/eB + mE/eB2 and the energy levels are [12]

2 1 2 eE m E En,k = ~ωc n + − eE kxlB + 2 + 2 . (4.7) 2 mωc 2 B

We know that the velocity of a particle is m~x˙ = ~p + eA~ and the corresponding current is I~ = −e~x˙. The total current in our quantum mechanical system is the sum over all ﬁlled states, e X I~ = − hψ| − i ∇ + eA~ |ψi . (4.8) m ~ ﬁlled states

11 4. The Quantum Hall eﬀect in Graphene

The sum of all ﬁlled states is then sum of all levels up to the ﬁlling factor and all states in a level. The current in y-direction is then

The later term is the expectation value of y for our harmonic oscillator and is the same as the value they center around,

k~ mE hψn,k| y |ψn,ki = + . (4.11) eB eB2 The momentum term in (4.10) cancels against the ﬁrst term in equation (4.11) giving the total current [12],

X E E eB e2 Ix = −eν = −eν LxLy = −ν ELy. (4.12) BLx BLx 2π h k ~ Here we used that the sum over k gives us the previously calculated number of allowed states of a level, equation (4.5), and the potential diﬀerence is V = ELy.

νe2 The formula for the current gives us the Hall conductance σxy = h and since there is no current in the direction of the electric field the σxx vanishes. In short when the e2 Fermi level lands between Landau levels the Hall conductance is ν h . In a perfect sample the Fermi level never lands between levels but rather jumps from one level to the next. But the reason we see the quantized Hall effect is because of impurities and localized states which give the Fermi level intermediate steps to land in. These localized states however do not affect the Hall resistance just as the classical Hall resistance is independent of the sample geometry.

4.4. Landau levels in graphene

Relativistic fermions in an external magnetic ﬁeld undergo quantized motion, just as the previously discussed non-relativistic fermions. In section 4.2 the Landau levels were equidistant which was due to parabolic dispersion law of free electrons. Close

12 4.4. Landau levels in graphene to the Dirac points the dispersion law is linear and the Landau levels are no longer equidistant. The energy of the n-th level is

Dirac√ En = ±~ωc n, (4.13) where the positive energy applies to electrons and the negative to holes. The energy scale between the first Landau levels for Dirac fermions and conventional electron gas is very different. For conventional electron gas the energy scale is of magnitude E ∼ 10K but the scale between the first, n = 0, and second , n = ±1, level for Dirac fermions is of order E ∼ 1000K. This means that one would expect to see the quantum Hall effect even at room temperature. Furthermore this means that the Zeeman interaction, which is ∼ 5K, is negligible [13].

The Landau quantization has been studied in both the tight binding model [15,16] and in the effective mass relativistic model [17,18]. The tight binding model is valid for a wider range of energy while the effective mass model describes low energy limits. Since we will only review phenomena in the low energy limit, near the Dirac points, the effective mass model will suffice. The effective mass Hamiltonian for a single electron in graphene in a uniform perpendicular magnetic field is   0 πx − iπy 0 0 πx + iπy 0 0 0  HDirac = vF   , (4.14)  0 0 0 πx + iπy 0 0 πx − iπy 0 ~ where ~π = ~p + eA is the canonical momentum and vF is the Fermi velocity. The corresponding wavefunction is [17,18]

 K  ψA K ψB  ψ =  K0  . (4.15) ψA  K0 ψB

K The wavefunction ψA is an envelope wavefunction for A site for the K valley. The same applies to the other terms in equation (4.15). By picking a Landau gauge ~ A = (−yB, 0) the eigenfunctions of HDirac are calculated. A detailed walkthrough is presented in Appendix B. The eigenfunctions of (4.14) are labelled by three indices, the valley index, j = K,K0, the Landau level index, n = 0, ±1, ±2 ... and the k index which is the wavevector along the x-axis. The normalized eigenfunctions are [17,18]

  φ|n|−1,k K Cn ikx ±φ|n|,k  ψ = √ e   (4.16) n,k,± L  0  0

13 4. The Quantum Hall eﬀect in Graphene and  0  K0 Cn ikx  0  ψn,k,± = √ e   , (4.17) L ±φ|n|,k  φ|n|−1,k

2 1/4 mωc(y−y0) 1 mωc − p mωc where φn,k(y) = √ e 2~ Hn (y − y0) and C0 = 1, Cn = √ 2n n! π~ ~ 2 for n 6= 0.

The φn,k are harmonic oscillator wave functions and Hn are Hermite polynomials.

Looking at these wavefunctions we see that Landau levels where n 6= 0 the wavefunctions have non-zero amplitude for both sublattice A and B. However for the n = 0 Landau level we get a zero amplitude at A site for K valley and B site for K’ valley. This, as we will see later, will cause an anomaly in the quantum Hall eﬀect.

Up till now we have disregarded the spin of the electron. If we want to take that into account we must multiply the wavefunction by a two-component spin function. The spin also adds to the degeneracy of Landau levels. The Landau levels are therefore fourfold degenerate, two spin degeneracies and two valley degeneracies, often called pseudospin [19].

For non-relativistic fermions the Kohn theorem applies [20]. It implies that for a non-relativistic fermion in an external magnetic ﬁeld the frequency that corresponds to transitions between neighbouring Landau levels equals the cyclotron frequency. So the frequency of optical transitions is equal to the cyclotron frequency. This is not the case in graphene [20] where the Landau levels are not equidistant. Therefore the frequency of optical transitions depends on electron interactions and the number of electrons. Optical transitions in graphene are of two kinds, intraband transition and interband transitions. The intraband transition is a transition in the valence or conduction band and the interband transition is a transition between the valence and conduction band. In graphene the conduction and valence bands have the same symmetry making the selection rule of the transition the same for both intra- and interband transitions [21],

∆n = |n2| − |n1| = ±1. (4.18)

An example for an intraband transition of an electron state to an electron state could be from level n1 = 2 to n2 = 1 which gives n2 − n1 = 1 − 2 = −1. The interband transition, from the same electron state in n1 = 2 to a hole state would be to Landau level n2 = −3.

14 4.5. The anomalous quantum Hall eﬀect in graphene

4.5. The anomalous quantum Hall eﬀect in graphene

The quantum Hall effect, QHE, occurs at low temperature, large magnetic fields and in high mobility specimens. In graphene however due to large energy scale between Landau levels, or large activation energy, the quantized Hall effect can theoretically be seen at room temperature. This indeed is the case and was observed in [13]. The QHE has also been observed in low-mobility samples [14].

The Hall conductivity favours plateaus at integer values of e2/h. When the Fermi 2 level crosses a Landau level the Hall conductivity increases by an amount of gse /h, where gs is the degeneracy of the Landau level. For graphene the degeneracy is fourfold thus we expect the plateaus of Hall conductance in graphene to appear at e2 4N h . This turns out to be wrong. The QHE in graphene is anomalous and appears at half-integer values, 1 e2 σxy = 4 N + . (4.19) 2 h 1 The filling factor here is ν = 4 N + 2 . The shift in the Hall conductance is because of the special nature of the n = 0 Landau level. This level is robust and its energy is independent of change of magnetic field. It is also at the Dirac point where there is high symmetry between electron and hole levels indicating a twofold degeneracy for electrons and twofold degeneracy for holes [22]. Consequently if the Fermi level lands between Landau levels n = 0 and n = 1 the Hall conductance of the first e2 plateau is 2 h .

The quantum Hall effect for graphene at T = 30mK can be seen in figure 4.2. Obvious plateaus appear in the left figure at high magnetic fields. At low magnetic field the quantization can no longer be observed and we have the classical Hall effect. The right figure shows how the Hall conductivity jumps from plateau to plateau as the Fermi level passes a Landau level.

The activation energies of Hall states have been observed in [24,25] and the excitation gaps of ν = ±2 and ν = 6 have been analyzed. The excitation gaps would be the energy gap between two Landau levels if it were not for broadening of the levels. This broadening introduces a constant offset of excitation gaps from the theoretical value of the sharp levels. The broadening is dependent on magnetic field and disorder. Experimental results show predicted behaviour of the ν = 6 Hall state with Landau level broadening of around 400K. The ν = ±2 Hall state however approaches the bare Landau level separation at high magnetic fields. This is due to the uniqueness of the n = 0 Landau level. As the magnetic field increases the n = 0 Landau level becomes very sharp.

15 4. The Quantum Hall eﬀect in Graphene

Figure 4.2: The figure on the left shows the Hall resistance and magnetoresistance plotted against magnetic field at T=30mK. Plateaus appear for the Hall resistance and the magnetoresistance vanishes at the filling factors. The vertical arrows and number indicates the filling factors and their location. The inset is the quantum Hall effect of a hole gas at 1.6K. The right figure shows the energy plotted against the Hall conductivity and the position of the Fermi level and Landau levels. The Hall conductivity shows plateaus whenever the Fermi level lies between Landau levels and jumps by a constant when the Fermi level passes a Landau level. Figure from [23].

4.6. Symmetry breaking

In high magnetic field, B > 20 T , and high mobility samples, up to 50000 cm2V −1s−1 new plateaus at filling factor ν = 0, ±1 and ± 4 appear. These plateaus have been observed in [26–28]. These plateaus arise from symmetry breaking and lifting of degeneracy. The degeneracy of Landau level n = 0 is completely lifted so when the e2 Fermi level passes through a Landau level the Hall conductance rises by h , giving us plateaus ν = 0, ±1. The fact that the next plateau after filling factor ν = 2 is at 4 but not 3 tells us that the n = ±1 Landau levels degeneracy are only partially lifted. This is shown schematically in figure 4.3. To determine which degeneracy is lifted, the valley or the spin degeneracy, the magnetic field is tilted. The reason being that the spin splitting, Zeeman splitting, depends on the total magnetic field while the valley splitting depends on the projection on the perpendicular axis. The result of a magnetotransport measurement was that the ν = ±4 plateau depends on the total magnetic field and ν = ±1 depends on the perpendicular component [27]. This shows that valley and spin degeneracy of the n = 0 Landau level is lifted whilst only the spin degeneracy of n = 1 Landau level is lifted.

An unusual behaviour of the magnetoresistance, Rxx, arises in high magnetic ﬁeld, when the degeneracy is lifted. The Hall conductance, σxy, behaves normally and plateaus appear at right places for the new ﬁlling factors. The magnetoresistance is

16 4.6. Symmetry breaking

Figure 4.3: An illustration of the lifting of the Landau levels and corresponding filling factors. The number of Landau levels is n and filling factors ν. The arrows show spin orientation of the levels. The degeneracy of n = 0 Landau level is completely lifted while the degeneracy of n = ±1 Landau level is only partially lifted. The spin degeneracy is lifted leaving the valley degeneracy. Figure from [2]. at a minimum and shows an activated behaviour dependent on temperature at the Hall plateaus. But at ν = 0 instead of vanishing the magnetoresistance instead is at a maximum and no activation behaviour. A more detailed analysis of this state shows that it can be in either a metallic or an insulating phase. The difference being finite or infinitely large Rxx. With increasing magnetic field the sample transitions to the insulating phase. This happens at some critical value of magnetic field, measured to be Bc ∼ 30T for all samples in [29].

The lifting of the degeneracy of Landau levels is a result of symmetry breaking. The theory of symmetry breaking in graphene in a magnetic ﬁeld has been studied in detail in [30–44]. The main mechanisms which break the symmetry are the following.

• Disorder can lift the valley degeneracy. Disorder induces intervalley coupling which can be introduced through randomness in hopping integrals between neighbouring sites, t → t + δt [37,38,46]. The randomness in the hopping can be reformulated as an effective random magnetic field, δh [37]. This magnetic field points in z-direction and introduces a Zeeman-like interaction of the valley pseudospin.

• Electron-electron interactions can lift both spin and valley degeneracy. The Zeeman interaction lifts the spin degeneracy of Landau levels. The valley degeneracy is lifted because of lattice symmetry-breaking terms in the Coulomb interaction Hamiltonian in a continuum theory [30, 36]. Backscattering adds an additional term to the Hamiltonian which breaks the valley degeneracy. Another term is added to the Hamiltonian due to shift of the sublattices with respect to each other. This makes the sublattices inequivalent of each other, that is two electrons in a continuous point interact stronger if they belong to

17 4. The Quantum Hall eﬀect in Graphene

the same sublattice [2].

• Lattice distortion can lift the valley degeneracy. An out of plane distortion can occur in a strong magnetic field and is described as a relative shift of the sublattices A and B towards and away from the substrate [45]. Interactions with the substrate give different sublattices different on-site energies. The electrons prefer to occupy the sublattices with the lower energy breaking the valley degeneracy. This effect is only observed if the graphene rests on a substrate. It is not observed in suspended graphene. An analysis in [45] shows that only the degeneracy of Landau level n = 0 is lifted.

4.7. Shubnikov-de Haas oscillations

At low temperature and high magnetic field an effect called the Shubnikov-de Haas oscillations can be observed. They are an oscillation of conductivity with increasing magnetic field. This effect happens because the energy of a Landau level is propor- tional to the strength of the magnetic field and so is the degeneracy of that Landau level. For low temperature all Landau levels below the Fermi level are fully occupied. As the magnetic field is increased, energy of a Landau level increases and eventually passes the Fermi level. Once passed the Landau level empties since the degeneracy of lower Landau levels increases. Each time the Landau level crosses the Fermi level the systems free energy is at a minimum but then increases until the next Landau level crosses. As a consequence the free energy oscillates as a function of magnetic field and so do the linear response coefficients of the material [47]. Therefore the conductivity oscillates as a function of magnetic field.

18 5. Bilayer Graphene

Bilayer graphene is just as interesting as its monolayer counterpart. It shares some of the characteristics of the monolayer graphene and some of the characteristics of a 2D electron gas. The two layers in graphene can be oriented diﬀerently to each other. The most common orientation is the AB, or Bernal, stacking. Bernal stacking is the layout when the top layer is shifted so half of its atoms are directly above the atoms on the lower layer, as seen in ﬁgure 5.1. Other possible orientations include, for example, the top layer being rotated with respect to the lower layer or directly above it.

5.1. Fabrication

Graphene is fabricated mostly in three different ways. The first method is to utilize the weak bond between the graphene planes in graphite and exfoliate the graphene from highly oriented graphite or single graphite crystals [1, 48]. This has proven to be very effective, producing flakes of single-, double- and triple-layer up to 10µm in size. This method is called the Scotch tape method and the flakes are exfoliated from the graphite bulk with adhesive tape [1]. Exfoliated graphene can be used to make suspended graphene, where the monolayer flake does not interact with any substrate and results in a high mobility sample. One way to produce suspended graphene is to place a metallic grid on top of the monolayer and etch away the silicon dioxide substrate [49].

The second method is the epitaxial growth of graphene. Graphene can be grown epitaxially on a metal surface from decomposition of hydrocarbon or carbon oxide [50]. The hydrocarbon or carbon oxide is heated until the hydrogen and oxygen desorb, leaving the carbon atoms which form a monolayer of graphene. This kind of epitaxial graphene could reach up to 1 µm in size with few defects. Another way to make epitaxial graphene is to heat a hexagonal SiC crystal [51–54]. The specimen is heated until the silicon atoms near the boundary desorb and leave only hexagonal carbon. The number of layers can be controlled by limiting the time or temperature of the heating process. The properties of the graphene layers depend on which face of the SiC crystal the graphene is grown. If the SiC(0001)(silicon-terminated) face

19 5. Bilayer Graphene is used a number of high quality graphene layers are made which show the Dirac cone but are n-type doped. Experimental data shows that the graphene layers are covalently bonded to the substrate through a buffer layer [55]. This buffer layer is tightly bound to the substrate and shows no graphene-like π band. If the SiC(0001¯)(carbon-terminated) face is used in the procedure fewer layers are made but of lower quality. Now the graphene does not have a buffer layer but is weakly bound to the substrate.

Yet another method is to grow graphene with chemical vapour [56–58]. A metal foil is placed in a hydrocarbon gas. The hydrocarbon gas can be decomposed and the carbon atoms dissolve into the metal. The foil is cooled at a predetermined rate and graphene layers may form on the foil. These layers can later be transferred to other substrates with chemical etching [58].

[a] [b] Figure 5.1: The ﬁgure on the left shows an overview of the Bernal stacking of bilayer graphene. Half of the atoms of the top layer are positioned directly above the atoms in the lower level. Figure from [59]. The ﬁgure on the right shows the transmittance of white light through a 50 µm aperture with a monolayer and bilayer partially covering it. The transmittance lowers in intervals with increasing number of layers. The inset shows the sample design, a thick metal support structure with apertures of 20, 30 and 50 µm in diameter and graphene crystallites placed over them. Figure from [60].

5.2. Identiﬁcation of Layers

A simple way of determining the number of layers in a graphene ﬂake is its visibility. This problem was tackled in [61–64] and all papers recommended using SiO2 of width ≈ 280 nm as a substrate. The visibility is deﬁned as

(R − R0) V = , (5.1) R0

20 5.2. Identiﬁcation of Layers

where R0 is the reflection coefficient of the substrate and R is the reflection coefficient of the substrate and the graphene flakes. The visibility of a bilayer was reported to be twice that of the monolayer. In figure 5.1 the transmittance through suspended graphene is shown. From the figure we see that the bilayer reflects twice as much white light as the monolayer. Measurements of the visibility of graphene flakes on a glass substrate in [65] also found a linear increase in visibility with increasing number of layers.

An Atomic Force microscope, AFM, can also be used to determine the number of layers. The AFM images the surface by dragging a cantilever with a tip along it and thereby measures the relative height of the specimen. In [66] a statistical analysis of the AFM data of many graphene flakes was made to determine the height dependence of the number of layers. They found that the surface would increase in intervals of 0.35 nm corresponding to the predicted interlayer distance. The height of the first layers however can differ due to difference in strength of interactions with the substrate and one layer can show height difference in measurements due to rippling.

A transmission electron microscope was used in [67] to image exfoliated graphene. They observed stacking faults and relative rotation of the layers could be analysed. The experimental data of rotated layers ﬁtted well with two decoupled monolayers.

A reliable and a fast way to distinguish the number of layers in a flake is to use Raman spectroscopy. The previously mentioned methods are time-consuming with low throughput. Raman spectroscopy is when light interacts with the phonon modes in a system such that the light excites the system in some vibrational energy state to a virtual energy state. The system will only stay in this virtual energy state for a short period of time before scattering a photon. After the scattering the system can be in a different vibrational energy state giving the emitted light higher or lower energy. For characterizing graphene a monochromatic light is used and the Raman shift measured. Two peaks, the G and 2D peaks for two different phonon modes, can be used to distinguish between monolayers and bilayers. The 2D peak broadens for the bilayer and gains a shoulder on its low-energy half [68] and the intensity of the G peak increases.

The broadening of the 2D peak of Bernal stacked bilayer can be seen clearly in ﬁgure 5.2. Two misoriented graphene layers exhibit a shift in the Raman spectrum for the 2D peak and the peak is narrower. In [69] authors suggested that the shift is due to weak coupling of the two monolayers which results in modiﬁcation of the phonon spectrum.

21 5. Bilayer Graphene

Figure 5.2: (a) The G peak of the Raman spectrum for light of wavelength 633 nm on graphene and two layers rotated relative to each other, overlapped. The green line shows the overlapped layers and is oﬀset from the graphene, black line, on the graph for clarity. (b) The 2D peak for graphene, two overlapped layers and Bernal stacked bilayer, yellow line. (c) The 2D peak for graphene and two overlapped layers at wavelength 488 nm and 514.5 nm. Figure from [69].

5.3. Tight-binding model

Earlier in chapter 3 we discussed the tight-binding model of the monolayer graphene. We will now expand this model for the bilayer graphene.

We construct the tight-binding model for the Bernal stacked graphene such that sublattice A is directly above sublattice B. We will from here on denote the labels u, l to describe whether the sublattice is in the upper or lower layer. The hopping integrals can be seen schematically in ﬁgure 5.3. The intralayer hopping integrals t 0 and t are the same as before in the monolayer. The interlayer hop between Au ↔ Bl is the most important and is parametrized by γ1. It is called the dimer bond and since the vector connecting them is in z-direction and there is no projection of momentum on it the momentum matrix term in the Hamiltonian (5.2) vanishes. The hop from Bu ↔ Al is parametrized by γ3 and the hop from Al ↔ Au and Bl ↔ Bu are parametrized by γ4. In each case the momentum dependence are the ~ P3 ~ ~ ~ P6 ~ ~ 0 functions, f(k) = i=1 exp(ik · Ri) and g(k) = i=1 exp(ik · Ri). The dimer bond can add an asymmetry between sublattices which we put as ∆ in our Hamiltonian. The tight-binding Hamiltonian is

 0 ~ ~ ~ ∗ ~ ∗  t g(k) γ3f(k) γ4f(k) tf(k) γ f(~k)∗ t0g(~k) tf(~k) γ f(~k)  H =  3 4  , π  ~ ~ ∗ 0 ~  (5.2)  γ4f(k) tf(k) ∆ + t g(k) γ1  ~ ~ ∗ 0 ~ tf(k) γ4f(k) γ1 ∆ + t g(k) with basis {ψAl , ψBu , ψAu , ψBl }.

22 5.3. Tight-binding model

Figure 5.3: The ﬁgure shows Bernal stacked bilayer graphene. (a) The nearest and next-nearest neighbour hoppings are illustrated. The intralayer hopping integral between sublattices A ↔ B is t and for hops A ↔ A and B ↔ B is t0. The interlayer hopping integrals are γ1 for Au ↔ Bl, γ3 for Bu ↔ Al and γ4 for Al ↔ Au and Bu ↔ Bl. (b) The twelve nearest and next-nearest neighbours are shown here with the the vectors R~ and R~ 0 which come later in the tight-binding formalism. Figure from [2].

The values of the parameters are controversial and seem to diﬀer between exfoliated and epitaxial graphene. Theoretical models also do not agree completely with experimental results. A thorough experimental determination was performed by Kuzmenko [70] and they found that the parameters were t = 3.16 ± 0.03, γ1 = 0.381 ± 0.003, γ3 = 0.38 ± 0.06, γ4 = 0.14 ± 0.03 and ∆ = 0.022 ± 0.003, in eV.

The Hamiltonian in (5.2) describes the whole π band but as for the monolayer we are only interested in the low-energy limit close to the K points, where the electrons are close to the Fermi surface. We expand the Hamiltonian close to the K points and introduce a valley index ξ, where ξ = 1 for K valley and ξ = −1 for K’ valley. Since the K and K’ valley are inequivalent the basis expands to contain eight elements and for convenience we reorder them as such, {ψAl , ψBu , ψAu , ψBl } for K valley and

{ψBu , ψAl , ψBl , ψAu } for K’ valley. The Hamiltonian around the K points with k = |~k| is,

 3 0 2 2 † †  4 t a k ξv3π ξv4π ξvπ † 3 0 2 2  ξv3π 4 t a k ξvπ ξv4π  Hξ =  † 3 0 2 2  . (5.3)  ξv4π ξvπ ∆ + 4 t a k γ1  † 3 0 2 2 ξvπ ξv4π γ1 ∆ + 4 t a k √ √ Here √v = 3at/2~ is the Fermi velocity of monolayer graphene, v3 = 3aγ3/2~ and v4 = 3aγ4/2~ are the velocities of interlayer hops and π = px + ipy. In the nearest

23 5. Bilayer Graphene

0 neighbour approach, we set γ3 = γ4 = ∆ = t = 0 and the energy of Hξ is

r q 2 2 2 2 2 2 E = χ v p + γ1 /2 + αγ1 v p + γ1 /4, (5.4) where χ = ±1 indicates the conduction or valence band and α = ±1 indicates the low energy or split branches, as seen in ﬁgure 5.4. An interesting feature is that this energy dispersion is parabolic near p = 0 and then crosses over to a linear spectrum at p ≈ γ1/2v.

Figure 5.4: (a) The dispersion relation of bilayer graphene around its high-symmetry points. (b) The dispersion close to the K points. The lines are marked (χ, α) indicating the conduction and valence bands and low energy and split branches. Figure from [2].

Lets take a look at the next-nearest approximation with parameters γ4 and ∆ as non-zero, see ﬁgure 5.5. The parameter γ4 gives an electron-hole asymmetry and increases or decreases the curvature of the parabola. The parameter ∆, which was introduced due to the dimer bond giving asymmetry to sublattices in the upper and lower layer, introduces a gap in the dispersion.

Figure 5.5: The dispersion relation of the Hamiltonian in (5.3), kx is measured near the K point with ky = 0. The parameters γ4 and ∆ are set to non-zero. The dashed lines represent the nearest neighbour approximation and the solid line the same calculation with γ4 and ∆ set to 0.15, 0.018 or 0. Figure from [2].

24 6. Summary

In conclusion graphene shows many interesting features. Its two-dimensional hexagonal structure yields a linear dispersion relation close to the Dirac points. Therefore the charge carriers are massless Dirac fermions with an effective speed of light. Due to the relativistic nature of its charge carriers the Landau levels in graphene are not equidistant with a large energy difference between the n = 0 and n = 1 levels. A consequence of this large energy gap is that the quantum Hall effect in graphene can be observed at room temperature. The quantum Hall effect is anomalous and occurs at half-integer values due to the unique nature of the n = 0 Landau level. Bi- layer graphene can also be studied in a tight-binding approximation but the details depend on the stacking of the two layers. Bernal stacking leads to parabolic energy dispersion very close to the Dirac points that transitions to a linear spectrum as one moves away from the Dirac point.

A. 2D fermion in an external magnetic ﬁeld

~ The fermions lie in xy-plane and magnetic ﬁeld is in z-direction so B = Beˆz. Then ~ ~ B = ∇ × A =⇒ B = ∂xAy − ∂yAx. (A.1) Lets choose the Landau gauge A~ = (−yB, 0) and now the Hamiltonian is ˆ (~pˆ + eA~)2 Hˆ = 2m (A.2) (ˆp − eyBˆ )2 pˆ2 = x + y . 2m 2m

This Hamiltonian now commutes with the momentum operator pˆx and can therefore be written as

(p − eyBˆ )2 pˆ2 Hˆ = x + y 2m 2m (A.3) pˆ2 mω2 = y + c (ˆy − y )2 . 2m 2 0

eB px Here ωc = m is the cyclotron frequency and y0 = eB is a shift. Perhaps a more eB y0 important quantity is the wavevector k = y0 = 2 where lB is the magnetic length. ~ lB The Hamiltonian looks like a shifted harmonic oscillator in the y-direction. So the fermions have the eigenstates i px x ψn = e ~ φn(y − y0), (A.4) where φn are the eigenstates of a quantum harmonic oscillator,

1/4 2 r 1 mωc − mωc(y−y0) mωc φn(y) = √ e 2~ Hn (y − y0) , (A.5) 2n n! π~ ~ and Hn are the Hermite polynomials. The energy spectrum of the quantum harmonic oscillator is well known and is 1 En = ωc n + . (A.6) ~ 2

B. 2D Dirac fermion in an external magnetic ﬁeld

In this problem we look at a 2D electron gas in a perpendicular magnetic ﬁeld similar to the previous problem but now we introduce relativistic fermions. Since we are in (2 + 1) dimensions and according to Polchinski [11] we should use a 2D ~ spinor representation. Lets pick the Landau gauge as A = (−By, 0)√and lets use the p Dirac vF magnetic length, lB = ~/eB, to deﬁne the frequency ωc = 2 where vf is √ lB the Fermi velocity. The 2 comes from normalisation.

The 2D Dirac hamiltonian is ~ vF [~σ · (−i∇ + eA/~)]ψ(x, y) = Eψ(x, y), (B.1) and the hamiltonian commutes with the pˆx operator so their eigenfunctions are ψ(x, y) = eikxφ(y). The hamiltonian therefore is 0 ∂y − k + Bey/~ vF φ(y) = Eφ(y), (B.2) −∂y − k + Bey/~ 0 which we rewrite as 0 O ωDirac φ(ξ) = Eφ(ξ). c O† 0 (B.3) y ξ = − lBk Here we have set lB and the 1D harmonic oscillator operators are 1 O = √ (∂ξ + ξ), (B.4) 2

† 1 O = √ (−∂ξ + ξ). (B.5) 2 The operators obey the commutation relation [O,O†] = 1 and n = O†O is the number operator. The previous hamiltonian can be written as

+ † − Dirac (Oσ + O σ )φ = (2E/ωc )φ, (B.6)

± with σ = σx ± iσy. This equation allows for a zero energy solution only if we put

− Oφ0 = 0, σ φ0 = 0. (B.7)

29 B. 2D Dirac fermion in an external magnetic ﬁeld

Since the harmonic oscillator operators are ladder operators there must be a ﬁrst step which gives us the zero energy solution. This step is the ground state of the harmonic oscillator, ψ0(ξ) and we set

φ0(ξ) = ψ0(ξ) ⊗ |⇓i . (B.8)

Here |⇓i denotes a state localized on sublattice A and similar for sublattice B. From the ground state all solutions can be derived from equation (B.6). They are

ψn−1(ξ) φn,±(ξ) = ψn−1(ξ) ⊗ |⇑i ± ψn(ξ) ⊗ |⇓i = . (B.9) ±ψn(ξ) Dirac√ Their energy is E±(n) = ±~ωc n.

30 Bibliography

[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, and A.A. Firosov, Science 306 2004, p. 666.

[2] D. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and T. Chakraborty 2010. Properties of graphene: A theoretical perspective. Advances in Physics, 59(4), 261-482. doi:10.1080/00018732.2010.487978

[3] A. H. Neto, F. Guinea, N.M. Peres, K.S. Novoselov and A.K. Geim 2009. The electronic properties of graphene. Reviews of Modern Physics Rev. Mod. Phys., 81(1), 109-162. doi:10.1103/revmodphys.81.109

[4] P.R. Wallace, Phys. Rev. 71 1947, p. 622

[5] Reich, S., J. Maultzsch, C. Thomsen, and P. Ordejón, 2002, Phys. Rev. B 66, 035412.

[6] R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes,Imperial College Press, London, 1998.

[7] S. Reich, C. Thomsen, and J. Maultzsch, Carbon Nanotubes, Wiley-VCH, Wein- heim,2004.

[8] N. W. Ashcroft, and N. D. Mermin, 1976, Solid State Physics. Saunders College, Philadelphia.

[9] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, 2005, Nature 438, 197.

[10] J. Sun, and J. Kosel, (2012). Finite-Element Modelling and Analysis of Hall Eﬀect and Extraordinary Magnetoresistance Eﬀect. INTECH Open Access Pub- lisher.

[11] J. G. Polchinski, (1998). String theory (Vol. 2, Appendix B). Cambridge, UK

[12] D. Tong, (2016) The Quantum Hall Eﬀect (lecture notes). Cambridge, UK.

31 BIBLIOGRAPHY

[13] K. S. Novoselov, Jiang, Z., Zhang, Y., Morozov, S. V., Stormer, H. L., Zeitler, U., Geim, A. K. (2007). Room-Temperature Quantum Hall Eﬀect in Graphene. Science, 315(5817), 1379-1379. doi:10.1126/science.1137201

[14] S. Masubuchi, K. Suga, M. Ono, K. Kindo, S. Takeyama, and T. Machida, J. Phys. Soc. Japan 77 (2008), p. 113707.

[15] B.A. Bernevig, T.L. Hughes, S.-C. Zhang, H.-D. Chen, and C. Wu, Int. J. Mod. Phys. B 20 (2006), p. 3257.

[16] J.H. Ho, Y.H. Lai, Y.H. Chiu, and M.F. Lin, Physica E 40 (2008), pp. 1722–1725.

[17] T. Ando, in Nano-Physics and Bio-Electronics: A New Odyssey, Chap. 1, T. Chakraborty, F. Peeters, and U. Sivan, eds., Elsevier, Amsterdam, 2002.

[18] Y. Zheng and T. Ando, Phys. Rev. B 65 (2002), p. 245420.

[19] D.A. Abanin, P.A. Lee, and L.S. Levitov, Phys. Rev. Lett. 98 (2007), p. 156801.

[20] W. Kohn, Phys. Rev. 123 (1961), p. 1242.

[21] Z. Jiang, E.A. Henriksen, L.C. Tung, Y.-J. Wang, M.E. Schwartz, M.Y. Han, P. Kim, and H.L. Stormer, Phys. Rev. Lett. 98 (2007), p. 197403.

[22] V.P. Gusynin and S.G. Sharapov, Phys. Rev. Lett. 95 (2005), p. 146801.

[23] Zhang, Y., Tan, Y. W., Stormer, H. L., and Kim, P. (2005). Experimental observation of the quantum Hall eﬀect and Berry’s phase in graphene. Nature, 438(7065), 201-204.

[24] A.J.M. Giesbers, U. Zeitler, M.I. Katsnelson, L.A. Ponomarenko, T.M. Mohi- uddin, and J.C. Maan, Phys. Rev. Lett. 99 (2007), p. 206803.

[25] A.J.M. Giesbers, U. Zeitler, M.I. Katsnelson, L.A. Ponomarenko, T.M.G. Mo- hiuddin, and J.C. Maan, Physica E 40 (2008), pp. 1089–1091.

[26] X. Wu, Y. Hu, M. Ruan, N.K. Madiomanana, J. Hankinson, M. Sprinkle, C. Berger, and W.A. de Heer, Appl. Phys. Lett. 95 (2009), p. 223108.

[27] Z. Jiang, Y. Zhang, H.L. Stormer, and P. Kim, Phys. Rev. Lett. 99 (2007), p. 106802.

[28] Y. Zhang, Z. Jiang, J.P. Small, M.S. Purewal, Y.-W. Tan, M. Fazlollahi, J.D.

32 BIBLIOGRAPHY

Chudow, J.A. Jaszczak, H.L. Stormer, and P. Kim, Phys. Rev. Lett. 96 (2006), p. 136806.

[29] J.G. Checkelsky, L. Li, and N.P. Ong, Phys. Rev. B 79 (2009), p. 115434.

[30] J. Alicea and M.P.A. Fisher, Phys. Rev. B 74 (2006), p. 075422.

[31] P.M. Ostrovsky, I.V. Gornyi, and A.D. Mirlin, Phys. Rev. B 77 (2008), p. 195430.

[32] K. Nomura and A.H. MacDonald, Phys. Rev. Lett. 96 (2006), p. 256602.

[33] D.N. Sheng, L. Sheng, and Z.Y. Weng, Phys. Rev. B 73 (2006), p. 233406.

[34] V.P. Gusynin, V.A. Miransky, S.G. Sharapov, and I.A. Shovkovy, Phys. Rev. B 74 (2006), p. 195429.

[35] K. Yang, S. Das Sarma, and A.H. MacDonald, Phys. Rev. B 74 (2006), p. 075423.

[36] J. Alicea and M.P.A. Fisher, Solid. State. Commun. 143 (2007), pp. 504–509.

[37] D.A. Abanin, P.A. Lee, and L.S. Levitov, Sol. State. Commun. 143 (2007), pp. 77–85.

[38] M. Koshino and T. Ando, Phys. Rev. B 75 (2007), p. 033412.

[39] L. Sheng, D.N. Sheng, F.D.M. Haldane, and L. Balents, Phys. Rev. Lett. 99 (2007), p. 196802.

[40] K. Yang, Solid State Commun. 143 (2007), pp. 27–32.

[41] E.V. Gorbar, V.P. Gusynin, and V.A. Miransky, Low Temp. Phys. 34 (2008), p. 790.

[42] V.P. Gusynin, V.A. Miransky, S.G. Sharapov, and I.A. Shovkovy, Phys. Rev. B 77 (2008), p. 205409.

[43] A.L.C. Pereira and P.A. Schulz, Phys. Rev. B 77 (2008), p. 075416.

[44] V.P. Gusynin, V.A. Miransky, S.G. Sharapov, and I.A. Shovkovy, Low Temp. Phys. 34 (2008), p. 778.

[45] J.-N. Fuchs and P. Lederer, Phys. Rev. Lett. 98 (2007), p. 016803.

33 BIBLIOGRAPHY

[46] T. Kawarabayashi, Y. Hatsugai, and H. Aoki, Phys. Rev. Lett. 103 (2009), p. 156804.

[47] H. Ibach, and H. Lüth, (1995). Solid-state physics: An introduction to principles of materials science. Berlin: Springer-Verlag.

[48] K.S. Novoselov, D. Jiang, F. Schedin, T.J. Booth, V.V. Khotkevich, S.V. Mo- rozov, and A.K. Geim, PNAS 102 (2005), p. 10451.

[49] J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, T.J. Booth, and S. Roth, Nature 446 (2007), p. 60.

[50] J. C. Shelton, H. R. Patil, and J. M. Blakely, 1974, Surf. Sci. 43, 493.

[51] C. Berger, Z. Song, T. Li, X. Li, A.Y. Ogbazghi, R. Feng, Z. Dai, A.N. Marchenkov, E.H. Conrad, P.N. First, and W.A. de Heer, J. Phys. Chem. B 108 (2004), p. 19912.

[52] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A.N. Marchenkov, E.H. Conrad, P.N. First, and W.A. de Heer, Science 312 (2006), p. 397.

[53] J. Hass, W.A. de Heer, and E.H. Conrad, J. Phys.: Condens. Matter 20 (2008), p. 323202.

[54] W.A. de Heer, C. Berger, X. Wu, P.N. First, E.H. Conrad, X. Li, T. Li, M. Sprinkle, J. Hass, M.L. Sadowski, M. Potemski, and G. Martinez, Solid State Commun. 143 (2007), p. 92.

[55] F. Varchon, R. Feng, J. Hass, X. Li, B. Ngoc Nguyen, C. Naud, P. Mallet, J.-Y. Veuillen, C. Berger, E.H. Conrad, and L. Magaud, Phys. Rev. Lett. 99 (2007), p. 126805.

[56] A.N. Obraztsov, E.A. Obraztsova, A.V. Tyurnina, and A.A. Zolotukhin, Car- bon 45 (2007), p. 2017.

[57] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M.S. Dresselhaus, and J. Kong, Nano Lett. 9 (2009), p. 30.

[58] K.S. Kim, Y. Zhao, H. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B.H. Hong, Nature 457 (2009), p. 706.

[59] Yan, X. Z., and Ting, C. S. (2013). Ordered-current state of electrons in bilayer graphene. Physical Review B, 88(4), 045410.

34 BIBLIOGRAPHY

[60] R.R. Nair, P. Blake, A.N. Grigorenko, K.S. Novoselov, T.J. Booth, T. Stauber, N.M.R. Peres, and A.K. Geim, Science 320 (2008), p. 1308.

[61] P. Blake, E.W. Hill, A.H. Castro Neto, K.S. Novoselov, D. Jiang, R. Yang, T.J. Booth, and A.K. Geim, Appl. Phys. Lett. 91 (2007), p. 063124.

[62] D.S.L. Abergel, A. Russell, and V.I. Falko, Appl. Phys. Lett. 91 (2007), p. 063125.

[63] S. Roddaro, P. Pingue, V. Piazza, V. Pellegrini, and F. Beltram, Nano Lett. 7 (2007), p. 2707.

[64] C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K.S. Novoselov, and A.C. Ferrari, Nano Lett. 7 (2007), p. 2711.

[65] P.E. Gaskell, H.S. Skulason, C. Rodenchuk, and T. Szkopek, Appl. Phys. Lett. 94 (2009), p. 143101.

[66] E.A. Obraztsova, A.V. Osadchy, E.D. Obraztsova, S. Lefrant, and I.V. Yamin- sky, Phys. Status Solidi b 245 (2008), p. 2055.

[67] J.H. Warner, M.H. Rümmeli, T. Gemming, B. Büchner, and G.A.D. Briggs, Nano Lett. 9 (2009), p. 102.

[68] A.C. Ferrari, J.C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K.S. Novoselov, S. Roth, and A.K. Geim, Phys. Rev. Lett. 97 (2006), p. 187401.

[69] P. Poncharal, A. Ayari, T. Michel, and J.-L. Sauvajol, Phys. Rev. B 78 (2008), p. 113407.

[70] A.B. Kuzmenko, I. Crassee, D. van der Marel, P. Blake, and K.S. Novoselov, Phys. Rev. B 80 (2009), p. 165406.