The Electronic Properties of Bilayer Graphene

The electronic properties of bilayer graphene

Edward McCann Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK

Mikito Koshino Department of Physics, Tohoku University, Sendai, 980-8578, Japan

We review the electronic properties of bilayer graphene, beginning with a description of the tight- binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.

CONTENTS 1. Electrostatics: asymmetry parameter, layerdensities andexternalgates 13 I. Introduction 1 2. Calculation of individual layer densities 14 3. Self-consistent screening 15 II.Electronicbandstructure 2 A. The crystal structure and the Brillouin IV.Transportproperties 15 zone 2 A. Introduction 15 B. The tight-binding model 3 B. Ballistic transport in a finite system 16 1. Anarbitrarycrystalstructure 3 C. Transport in disordered bilayer graphene 18 2. Monolayer graphene 4 1. Conductivity 18 3. Bilayer graphene 5 2. Localisation effects 20 4. Effective four-band Hamiltonian near the K points 7 V.Opticalproperties 20 C. Effective two-band Hamiltonian at low energy 7 VI.Orbitalmagnetism 21 1.Generalprocedure 7 2. Bilayer graphene 8 VII.Phononsandstrain 22 D. Interlayer coupling γ1: massive chiral A. The influence of strain on electrons in electrons 8 bilayer graphene 22 E. Interlayer coupling γ3: trigonal warping and B. Phonons in bilayer graphene 22 the Lifshitz transition 9 C.Opticalphononanomaly 23 F. Interlayer coupling γ4 and on-site parameter ∆′:electron-holeasymmetry 9 VIII. Electronic interactions 24 G. Asymmetry between on-site energies: band gaps 9 IX. Summary 25 1. Interlayer asymmetry 9 2. Intralayer asymmetry between A and B References 26 sites 10 H. Next-nearest neighbour hopping 10 I. Spin-orbit coupling 10 I. INTRODUCTION J. TheintegerquantumHalleffect 11 1. The Landau level spectrum of bilayer The production by mechanical exfoliation of isolated graphene 11 flakes of graphene with excellent conducting properties 2. Three types of integer quantum Hall [1] was soon followed by the observation of an unusual effect 12 sequence of plateaus in the integer quantum Hall effect in 3. Theroleofinterlayerasymmetry 13 monolayer graphene [2, 3]. This confirmed the fact that charge carriers in monolayer graphene are massless chiral III.Tuneablebandgap 13 quasiparticles with a linear dispersion, as described by a A.Experiments 13 Dirac-like effective Hamiltonian [4–6], and it prompted B.Hartreemodelofscreening 13 an explosion of interest in the field [7]. 2

The integer quantum Hall effect in bilayer graphene [8] and taking into account different parameters that cou- is arguably even more unusual than in monolayer because ple atomic orbitals as well as external factors that may it indicates the presence of massive chiral quasiparticles change the electron bands by, for example, opening a [9] with a parabolic dispersion at low energy. The effec- band gap. We include the Landau level spectrum in the tive Hamiltonian of bilayer graphene may be viewed as presence of a perpendicular magnetic field and the corre- a generalisation of the Dirac-like Hamiltonian of mono- sponding integer quantum Hall effect. In section III we layer graphene and the second (after the monolayer) in a consider the opening of a band gap due to doping or gat- family of chiral Hamiltonians that appear at low energy ing and present a simple analytical model that describes in ABC-stacked (rhombohedral) multilayer graphene [9– the density-dependence of the band gap by taking into 15]. In addition to interesting underlying physics, bilayer account screening by electrons on the bilayer device. The graphene holds potential for electronics applications, not tight-binding model is used to describe transport prop- least because of the possibility to control both carrier erties, section IV, and optical properties, section V. We density and energy band gap through doping or gating also discuss orbital magnetism in section VI, phonons [9, 10, 16–20]. and the influence of strain in section VII. Section VIII Not surprisingly, many of the properties of bilayer concludes with an overview of electronic-interaction ef- graphene are similar to those in monolayer [7, 21]. These fects. Note that this review considers Bernal-stacked include excellent electrical conductivity with room tem- (also known as AB-stacked) bilayer graphene; we do 2 1 1 perature mobility of up to 40, 000 cm V− s− in air [22]; not consider other stacking types such as AA-stacked the possibility to tune electrical properties by changing graphene [40], twisted graphene [41–46] or two graphene the carrier density through gating or doping [1, 8, 16]; sheets separated by a dielectric with, possibly, electronic high thermal conductivity with room temperature ther- interactions between them [47–52]. 1 1 mal conductivity of about 2, 800Wm− K− [23, 24]; mechanical stiffness, strength and flexibility (Young’s mod- ulus is estimated to be about 0.8 TPa [25, 26]); trans- II. ELECTRONIC BAND STRUCTURE parency with transmittance of white light of about 95 % [27]; impermeability to gases [28]; and the ability to be A. The crystal structure and the Brillouin zone chemically functionalised [29]. Thus, as with monolayer graphene, bilayer graphene has potential for future appli- Bilayer graphene consists of two coupled monolayers cations in many areas [21] including transparent, flexible of carbon atoms, each with a honeycomb crystal struc- electrodes for touch screen displays [30]; high-frequency ture. Figure 1 shows the crystal structure of monolayer transistors [31]; thermoelectric devices [32]; photonic de- graphene, figure 2 shows bilayer graphene. In both cases, vices including plasmonic devices [33] and photodetectors primitive lattice vectors a1 and a2 may be defined as [34]; energy applications including batteries [35, 36]; and composite materials [37, 38]. a √3a a √3a a a It should be stressed, however, that bilayer graphene 1 = , , 2 = , , (1) 2 2 2 − 2 has features that make it distinct from monolayer. The low-energy band structure, described in detail in Sec- where a = a1 = a2 is the lattice constant, the distance tion II, is different. Like monolayer, intrinsic bilayer has between adjacent| | | unit| cells, a = 2.46 A˚ [53]. Note that no band gap between its conduction and valence bands, the lattice constant is distinct from the carbon-carbon but the low-energy dispersion is quadratic (rather than bond length aCC = a/√3=1.42 A,˚ which is the distance linear as in monolayer) with massive chiral quasiparti- between adjacent carbon atoms. cles [8, 9] rather than massless ones. As there are two In monolayer graphene, each unit cell contains two layers, bilayer graphene represents the thinnest possible carbon atoms, labelled A and B, figure 1(a). The po- limit of an intercalated material [35, 36]. It is possible sitions of A and B atoms are not equivalent because it to address each layer separately leading to entirely new is not possible to connect them with a lattice vector of functionalities in bilayer graphene including the possibil- the form R = n1a1 + n2a2, where n1 and n2 are inte- ity to control an energy band gap of up to about 300meV gers. Bilayer graphene consists of two coupled monolay- through doping or gating [9, 10, 16–20]. Recently, this ers, with four atoms in the unit cell, labelled A1, B1 band gap has been used to create devices - constrictions on the lower layer and A2, B2 on the upper layer. The and dots - by electrostatic confinement with gates [39]. layers are arranged so that one of the atoms from the Bilayer or multilayer graphene devices may also be prefer- lower layer B1 is directly below an atom, A2, from the able to monolayer ones when there is a need to use more upper layer. We refer to these two atomic sites as ‘dimer’ material for increased electrical or thermal conduction, sites because the electronic orbitals on them are cou- strength [37, 38], or optical signature [33]. pled together by a relatively strong interlayer coupling. In the following we review the electronic properties The other two atoms, A1 and B2, don’t have a counter- of bilayer graphene. Section II is an overview of the part on the other layer that is directly above or below electronic tight-binding Hamiltonian and resulting band them, and are referred to as ‘non-dimer’ sites. Note that structure describing the low-energy chiral Hamiltonian some authors [10, 54–56] employ different definitions of 3

( a ) ( a ) a 1

a 1 A 2 A 1 B 1 B 2 B a A 2 y a 2 a y a x ( b ) g x A 2 0 B 2

( b ) g 1 g b 1 3 g 1 k g 4 y a g 0 A 1 B 1 k K - x K + G FIG. 2. (a) Plan and (b) side view of the crystal structure of bilayer graphene. Atoms A1 and B1 on the lower layer are b 2 shown as white and black circles, A2, B2 on the upper layer M are black and grey, respectively. The shaded rhombus in (a) indicates the conventional unit cell.

B. The tight-binding model FIG. 1. (a) Crystal structure of monolayer graphene with A (B) atoms shown as white (black) circles. The shaded rhombus is the conventional unit cell, a1 and a2 are primitive 1. An arbitrary crystal structure lattice vectors. (b) Reciprocal lattice of monolayer and bilayer graphene with lattice points indicated as crosses, b1 and b2 In the following, we will describe the tight-binding are primitive reciprocal lattice vectors. The shaded hexagon model [53, 59, 60] and its application to bilayer graphene. is the first Brillouin zone with Γ indicating the centre, and We begin by considering an arbitrary crystal with trans- K+ and K− showing two non-equivalent corners. lational invariance and M atomic orbitals φm per unit cell, labelled by index m = 1 ...M. Bloch states Φm(k, r) for a given position vector r and wave vector A and B sites as used here. The point group of the bi- k may be written as layer crystal structure is D3d [12, 57, 58] consisting of N 1 ik.Rm,i elements ( E, 2C3, 3C2′ , i, 2S6, 3σd ), and it may be re- Φ (k, r)= e φ (r R ) , (3) { } m √ m − m,i garded as a direct product of group D3 ( E, 2C3, 3C′ ) N i=1 { 2} with the inversion group Ci ( E,i ). Thus, the lattice is symmetric with respect to spatial{ } inversion symmetry where N is the number of unit cells, i = 1 ...N labels R (x,y,z) ( x, y, z). the unit cell, and m,i is the position vector of the mth → − − − orbital in the ith unit cell. Primitive reciprocal lattice vectors b1 and b2 of mono- The electronic wave function Ψj(k, r) may be ex- layer and bilayer graphene, where a b = a b =2π pressed as a linear superposition of Bloch states 1 1 2 2 and a1 b2 = a2 b1 = 0, are given by M Ψj(k, r)= ψj,m(k)Φm(k, r) , (4) m=1 2π 2π 2π 2π b1 = , , b2 = , . (2) where ψ are expansion coefficients. There are M dif- a √3a a −√3a j,m ferent energy bands, and the energy Ej (k) of the jth band is given by E (k) = Ψ Ψ / Ψ Ψ where j j |H| j j | j H is the Hamiltonian. Minimising the energy Ej with re- As shown in figure 1(b), the reciprocal lattice is an hexag- spect to the expansion coefficients ψj,m [53, 60] leads to onal Bravais lattice, and the first Brillouin zone is an hexagon. Hψj = Ej Sψj , (5) 4

T where ψj is a column vector, ψj = (ψj1, ψj2,...,ψjM ). The off-diagonal element HAB of the transfer integral The transfer integral matrix H and overlap integral ma- matrix H describes the possibility of hopping between trix S are M M matrices with matrix elements defined orbitals on A and B sites. Substituting the Bloch func- by × tion (3) into the matrix element (6) results in a sum over all A sites and a sum over all B sites. We assume that H ′ = Φ Φ ′ , S ′ = Φ Φ ′ . (6) mm m|H| m mm m| m the dominant contribution arises from hopping between adjacent sites. If we consider a given A site, say, then we The band energies E may be determined from the gen- j take into account the possibility of hopping to its three eralised eigenvalue equation (5) by solving the secular nearest-neighbour B sites, labelled by index l =1, 2, 3: equation N 3 1 ik.δ det (H Ej S)=0 , (7) H e l − AB ≈ N i=1 l=1 where ‘det’ stands for the determinant of the matrix. φ (r R ) φ (r R δ ) , (9) In order to model a given system in terms of the gener- × A − A,i |H| B − A,i − l alised eigenvalue problem (5), it is necessary to determine where δl are the positions of three nearest B atoms rel- the matrices H and S. We will proceed by considering ative to a given A atom, which may be written as δ1 = the relatively simple case of monolayer graphene, before 0,a/√3 , δ2 = a/2, a/2√3 , δ3 = a/2, a/2√3 . generalising the approach to bilayers. In the following The sum with respect− to the three nearest-neighbour− − sections, we will omit the subscript j on ψj and Ej in B sites is identical for every A site. A hopping parameter equation (5), remembering that the number of solutions may be defined as is M, the number of orbitals per unit cell. γ = φ (r R ) φ (r R δ ) , (10) 0 − A − A,i |H| B − A,i − l which is positive. Then, the matrix element may be writ- 2. Monolayer graphene ten as 3 Here, we will outline how to apply the tight-binding δ H γ f (k); f (k)= eik. l , (11) model to graphene, and refer the reader to tutorial-style AB ≈− 0 reviews [53, 60] for further details. We take into account l=1 one 2pz orbital per atomic site and, as there are two The other off-diagonal matrix element is given by HBA = atoms in the unit cell of monolayer graphene, figure 1(a), HAB∗ γ0f ∗ (k). The function f (k) describing we include two orbitals per unit cell labelled as m = A nearest-neighbor≈ − hopping, equation (11), is given by and m = B (the A atoms and the B atoms are each iky a/√3 iky a/2√3 arranged on an hexagonal Bravais lattice). f (k)= e +2e− cos(kxa/2) , (12) We begin by considering the diagonal element HAA where k = (kx, ky) is the in-plane wave vector. The cal- of the transfer integral matrix H, equation (6), for the culation of the off-diagonal elements of the overlap in- A site orbital. It may be determined by substituting the tegral matrix S is similar to those of H. A parameter Bloch function (3) for m = A into the matrix element (6), s0 = φA (r RA,i) φB (r RB,l) is introduced to de- which results in a double sum over the positions of the scribe the possibility− | of non-zero− overlap between orbitals unit cells in the crystal. Assuming that the dominant on adjacent sites, giving SAB = SBA∗ = s0f (k). contribution arises from those terms involving a given Gathering the matrix elements, the transfer H and orbital interacting with itself (i.e., in the same unit cell), m overlap Sm integral matrices of monolayer graphene may the matrix element may be written as be written as N ǫ γ f (k) 1 H = A 0 , (13) HAA φA (r RA,i) φA (r RA,i) . (8) m k − ≈ N − |H| − γ0f ∗ ( ) ǫB i=1 − 1 s0f (k) Sm = . (14) This may be regarded as a summation over all unit cells of s0f ∗ (k) 1 a parameter ǫA = φA (r RA,i) φA (r RA,i) that takes the same value in every− unit|H| cell. Thus,− the matrix The corresponding energy may be determined [53] by element may be simply expressed as HAA ǫA. Simi- solving the secular equation (7). For intrinsic graphene, ≈ i.e., ǫ = ǫ = 0, we have larly, the diagonal element HBB for the B site orbital can A B be written as H = ǫ , while for intrinsic graphene ǫ BB B A γ0 f (k) is equal to ǫ as the two sublattices are identical. The E = ± | | . (15) B ± 1 s f (k) calculation of the diagonal elements of the overlap inte- ∓ 0| | gral matrix S, equation (6), proceeds in the same way The parameter values are listed by Saito et al [53] as as that of H, with the overlap of an orbital with itself γ0 =3.033eV and s0 =0.129. equal to unity, φ (r R ) φ (r R ) = 1. Thus, The function f(k), equation (12) is zero at the corners j − j,i | j − j,i SBB = SAA = 1. of the Brillouin zone, two of which are non-equivalent 5

(i.e., they are not connected by a reciprocal lattice vec- The three independent parameters are U to describe in- tor). For example, corners K+ and K with wave vectors terlayer asymmetry between the two layers [9, 10, 16– − K = (4π/3a, 0) are labelled in Figure 1(b). Such po- 20, 68–74], ∆′ for an energy difference between dimer ± ± sitions are called K points or valleys, and we will use a and non-dimer sites [54–56, 67], and δAB for an energy valley index ξ = 1 to distinguish points Kξ. At these difference between A and B sites on each layer [63, 75]. positions, the solutions± (15) are degenerate, marking a These parameters are described in detail in sections II F crossing point and zero band gap between the conduction and II G. and valence bands. The transfer matrix Hm is approxi- The upper-right and lower-left square 2 2 blocks of × mately equal to a Dirac-like Hamiltonian in the vicinity Hb describe inter-layer coupling. Parameter γ1 describes of the K point, describing massless chiral quasiparticles coupling between pairs of orbitals on the dimer sites B1 with a linear dispersion relation. These points are partic- and A2: since this is a vertical coupling, the correspond- ularly important because the Fermi level is located near ing terms in Hb (i.e., HA2,B1 = HB1,A2 = γ1) do not them in pristine graphene. contain f (k) which describes in-plane hopping. Param- eter γ3 describes interlayer coupling between non-dimer orbitals A1 and B2, and γ4 describes interlayer coupling 3. Bilayer graphene between dimer and non-dimer orbitals A1 and A2 or B1 and B2. Both γ3 and γ4 couplings are ‘skew’: they are In the tight-binding description of bilayer graphene, we not strictly vertical, but involve a component of in-plane take into account 2pz orbitals on the four atomic sites in hopping, and each atom on one layer (e.g., A1 for γ3) has the unit cell, labelled as j = A1,B1, A2,B2. Then, the three equidistant nearest-neighbours (e.g., B2 for γ3) on transfer integral matrix of bilayer graphene [9, 10, 54, 61– the other layer. In fact, the in-plane component of this 63] is a 4 4 matrix given by skew hopping is analogous to nearest-neighbour hopping × within a single layer, as parameterised by γ0. Hence, ǫA1 γ0f (k) γ4f (k) γ3f ∗ (k) − − the skew interlayer hopping (e.g., HA1,B2 = γ3f ∗ (k)) γ0f ∗ (k) ǫB1 γ1 γ4f (k) − Hb = − , contains the factor f (k) describing in-plane hopping.  γ f ∗ (k) γ ǫ γ f (k)  4 1 A2 − 0 It is possible to introduce an overlap integral matrix γ f (k) γ f ∗ (k) γ f ∗ (k) ǫ  − 3 4 − 0 B2  for bilayer graphene [63]  (16) where the tight-binding parameters are defined as 1 s0f (k)0 0 s0f ∗ (k) 1 s1 0 γ0 = φA1 φB1 = φA2 φB2 , (17) Sb =   , (28) − |H| − |H| 0 s1 1 s0f (k) γ1 = φA2 φB1 , (18)   |H|  0 0 s0f ∗ (k) 1  γ = φ φ , (19)   3 − A1|H| B2   γ4 = φA1 φA2 = φB1 φB2 . (20) with a form that mirrors Hb. Here, we only include |H| |H| two parameters: s0 = φA1 φB1 = φA2 φB2 describing Here, we use the notation of the Slonczewski-Weiss- non-orthogonality of intra-layer | nearest-neighbours | and McClure (SWM) model [64–67] that describes bulk s1 = φA2 φB1 describing non-orthogonality of orbitals graphite. Note that definitions of the parameters used on dimer | sitesA1 and B2. In principle, it is possible by authors can differ, particularly with respect to signs. to introduce additional parameters analogous to γ3, γ4, The upper-left and lower-right square 2 2 blocks of H × b etc., but generally they will be small and irrelevant. In describe intra-layer terms and are simple generalisations fact, it is common practice to neglect the overlap inte- of the monolayer, equation (13). For bilayer graphene, gral matrix entirely, i.e., replace Sb with a unit matrix, however, we include parameters describing the on-site because the influence of parameters s0 and s1 describ- energies ǫA1, ǫB1, ǫA2, ǫB2 on the four atomic sites, that ing non-orthogonality of adjacent orbitals is small at low are not equal in the most general case. As there are four energy E γ1. Then, the generalised eigenvalue equa- sites, differences between them are described by three | |≤ tion (5) reduces to an eigenvalue equation Hbψ = Eψ parameters [63]: with Hamiltonian Hb, equation (16). 1 ǫ = ( U + δ ) , (21) The energy differences U and δAB are usually at- A1 2 − AB 1 tributed to extrinsic factors such as gates, substrates ǫ = ( U + 2∆′ δ ) , (22) B1 2 − − AB or doping. Thus, there are five independent parame- 1 ǫA2 = 2 (U + 2∆′ + δAB) , (23) ters in the Hamiltonian (16) of intrinsic bilayer graphene, 1 ǫB2 = (U δAB) , (24) namely γ0, γ1, γ3, γ4 and ∆′. The band structure pre- 2 − dicted by the tight-binding model has been compared to where observations from photoemission [16], Raman [76] and U = 1 [(ǫ + ǫ ) (ǫ + ǫ )] , (25) infrared spectroscopy [55, 56, 78–81]. Parameter values 2 A1 B1 − A2 B2 1 determined by fitting to experiments are listed in Table I ∆′ = [(ǫ + ǫ ) (ǫ + ǫ )] , (26) 2 B1 A2 − A1 B2 for bulk graphite [67], for bilayer graphene by Raman δ = 1 [(ǫ + ǫ ) (ǫ + ǫ )] . (27) [76, 77] and infrared [55, 56, 80] spectroscopy, and for AB 2 A1 A2 − B1 B2 6

TABLE I. Values (in eV) of the Slonczewski-Weiss-McClure (SWM) model parameters [64–67] determined experimentally. Numbers in parenthesis indicate estimated accuracy of the ﬁnal digit(s). The energy diﬀerence between dimer and non-dimer ′ sites in the bilayer is ∆ =∆ − γ2 + γ5. Note that next-nearest layer parameters γ2 and γ5 are not present in bilayer graphene. Parameter Graphite [67] Bilayer [76] Bilayer [55] Bilayer [56] Bilayer [80] Trilayer [82] a a γ0 3.16(5) 2.9 3.0 - 3.16(3) 3.1 a γ1 0.39(1) 0.30 0.40(1) 0.404(10) 0.381(3) 0.39 γ2 -0.020(2) - - - - -0.028(4) a a γ3 0.315(15) 0.10 0.3 - 0.38(6) 0.315 γ4 0.044(24) 0.12 0.15(4) - 0.14(3) 0.041(10) γ5 0.038(5) - - - - 0.05(2) ∆ -0.008(2) - 0.018(3) 0.018(2) 0.022(3) -0.03(2) ∆′ 0.050(6) - 0.018(3) 0.018(2) 0.022(3) 0.046(10) a This parameter was not determined by the given experiment, the value quoted was taken from previous literature.

1.5 E(eV) Bernal-stacked trilayer graphene by observation of Lan- 1.0 dau level crossings [82]. Note that there are seven param- E(eV) 0.5

8 px eters in the Slonczewski-Weiss-McClure (SWM) model of -0.5 graphite [64–67] because the next-nearest layer couplings -1.0 6 -1.5 γ2 and γ5, absent in bilayer, are present in graphite (and trilayer graphene, too). Parameter ∆ in the SWM model 4 is related by ∆ = ∆′+γ2 γ5 to the parameter ∆′ describ- 2 ing the energy difference− between dimer and non-dimer sites in bilayer graphene. k -2 x The energy bands are plotted in figure 3 along the K- Γ K+ kx axis in reciprocal space intersecting the corners K , -4 − K+ and the centre Γ of the Brillouin zone [see fig- -6 ure 1(b)]. Plots were made using Hamiltonian Hb, equation (16), with parameter values determined by infrared -8 spectroscopy γ0 = 3.16 eV, γ1 = 0.381 eV, γ3 = 0.38 eV, γ4 = 0.14 eV, ǫB1 = ǫA2 = ∆′ = 0.022 eV, and ǫA1 = ǫB2 = U = δAB = 0 [80]. There are four bands because FIG. 3. Low-energy bands of bilayer graphene arising from the model takes into account one 2pz orbital on each of 2pz orbitals plotted along the kx axis in reciprocal space in- the four atomic sites in the unit cell; a pair of conduc- tersecting the corners K−, K+ and the centre Γ of the Bril- tion bands and a pair of valence bands. Over most of louin zone. The inset shows the bands in the vicinity of the the Brillouin zone, each pair is split by an energy of the K+ point. Plots were made using Hamiltonian Hb, equation (16), with parameter values γ0 = 3.16 eV, γ1 = 0.381 eV, order of the interlayer spacing γ1 0.4 eV [83]. Near ′ the K points, inset of figure 3, one conduction≈ band and γ3 = 0.38 eV, γ4 = 0.14 eV, ǫB1 = ǫA2 = ∆ = 0.022 eV, and ǫA1 = ǫB2 = U = δAB = 0 [80]. one valence band are split away from zero energy by an energy of the order of the interlayer coupling γ1, whereas two bands touch at zero energy [9]. The ‘split’ bands are a bonding and anti-bonding pair arising from the strong coupling (by interlayer coupling γ1) of the orbitals on the dimer B1 and A2 sites, whereas the ‘low-energy’ bands in line with the experimental ones listed in Table I. The arise from hopping between the non-dimer A1 and B2 tight-binding model Hamiltonian Hb, equation (16), used sites. In pristine bilayer graphene, the Fermi level lies at in conjuction with the parameters listed in Table I, is not the point where the two low-energy bands touch (shown accurate over the whole Brillouin zone because the fitting as zero energy in figure 3) and, thus, this region is rele- of tight-binding parameters is generally done in the vicin- vant for the study of electronic properties. It will be the ity of the corners of the Brillouin zone K+ and K (as focus of the following sections. the Fermi level lies near zero energy). For example,− pa- At low energy, the shape of the band structure pre- rameter s0 in equation (28) describing non-orthogonality dicted by the tight-binding model (see inset in figure 3) is of adjacent orbitals has been neglected here, but it con- in good agreement with that calculated by density func- tributes electron-hole asymmetry which is particularly tional theory [18, 57, 68] and it is possible obtain values prevalent near the Γ point at the centre of the Brillouin for the tight-binding parameters in this way, generally zone [53, 60]. 7

2 4. Effective four-band Hamiltonian near the K points ε1 p /2m at small momentum, where the mass is ≈ 2 m = γ1/2v (see inset in figure 3). This crossover occurs at p γ /2v. A convenient way to describe the To describe the properties of electrons in the vicinity of ≈ 1 the K points, a momentum p = ~k ~K is introduced bilayer at low energy and momentum p γ1/2v is to − ξ ≪ which is measured from the centre of the Kξ point. Ex- eliminate the components in the Hamiltonian (30) re- panding in powers of p, the function f(k), equation (12), lated to the orbitals on dimer sites B1, A2, resulting in an effective two-component Hamiltonian describing the is approximately given by f(k) √3a(ξpx ipy)/2~ ≈ − − orbitals on the non-dimer sites A1, B2, and, thus, the which is valid close to the Kξ point, i.e., for pa/~ 1, ≪ two bands that approach each other at zero energy. This where p = p = (p2 + p2)1/2. In monolayer graphene, x y is described in the next section, and the solutions of this the Hamiltonian| | matrix (13) is simplified by keeping only Hamiltonian are shown to be massive chiral quasiparti- linear terms in momentum p as cles [8, 9], as opposed to massless chiral ones in monolayer graphene. ǫA vπ† Hm = , (29) vπ ǫB C. Effective two-band Hamiltonian at low energy where π = ξp + ip , π† = ξp ip , and v = √3aγ /2~ x y x − y 0 is the band velocity. In the intrinsic case, ǫA = ǫB = 0, the eigen energy becomes E = v p , which approxi- In this section we focus on the low-energy electronic ± | | band structure in the vicinity of the points K+ and K mates Eq. (15). In bilayer graphene, similarly, Eq. (16) − is reduced to at the corners of the first Brillouin zone, relevant for energies near the Fermi level. A simple model may be ob- ǫA1 vπ† v4π† v3π tained by eliminating orbitals related to the dimer sites, − resulting in an effective Hamiltonian for the low-energy  vπ ǫB1 γ1 v4π†  Hb = − , (30) orbitals. First, we outline the procedure in general terms, v4π γ1 ǫA2 vπ†  −  because it may be applied to systems other than bilayer  v π† v π vπ ǫ   3 − 4 B2  graphene such as ABC-stacked (rhombohedral) graphene   where we introduced the effective velocities, v3 = multilayers [84, 85], before applying it specifically to bi- √3aγ3/2~ and v4 = √3aγ4/2~. layer graphene. At zero magnetic field Hamiltonian (30) yields four valley-degenerate bands E(p). A simple analytic solution 1. General procedure may be obtained by neglecting the terms v4π, v4π† proportional to γ4, and by considering only interlayer asymmetry U in the on-site energies: ǫ = ǫ = U/2 and We consider the energy eigenvalue equation, and con- A1 B1 − ǫA2 = ǫB2 = U/2. Then, there is electron-hole symme- sider separate blocks in the Hamiltonian corresponding to T T try, i.e., energies may be written E = εα(p), α =1, 2, low-energy θ = (ψA1, ψB2) and dimer χ = (ψA2, ψB1) [9] with ± components:

2 2 2 γ U v α ε2 = 1 + + v2 + 3 p2 + ( 1) √Γ , (31) hθ u θ θ α 2 4 2 − = E , (33) u† hχ χ χ 2 Γ= 1 γ2 v2p2 + v2p2 γ2 + U 2 + v2p2 4 1 − 3 1 3 2 3 The second row of (33) allows the dimer components to +2ξγ1v3v p cos3ϕ , be expressed in terms of the low-energy ones: where ϕ is the polar angle of momentum p = (px,py)= 1 χ = (E hχ)− u†θ , (34) p (cos ϕ, sin ϕ). Energy ε2 describes the higher-energy − bands split from zero energy by the interlayer coupling Substituting this into the ﬁrst row of (33) gives an eﬀec- γ1 between the orbitals on the dimer sites B1, A2. tive eigenvalue equation written solely for the low-energy Low-energy bands E = ε1 are related to orbitals on components: the non-dimer sites A1, B±2. In an intermediate energy 2 1 range U, (v3/v) γ1 < ε1 < γ1 it is possible to neglect hθ + u (E hχ)− u† θ = Eθ , the interlayer asymmetry U and terms proportional to − 1 hθ uh− u† θ E θ , γ3 (i.e., set U = v3 = 0), and the low-energy bands may − χ ≈ S be approximated [9] as 2 where =1+ uhχ− u†. The second equation is accurate up to linearS terms in E. Finally, we perform a transfor- 1 2 2 2 1/2 ε1 γ1 1+4v p /γ1 1 , (32) mation Φ = θ: ≈ 2 − S 1 1/2 1/2 h uh− u† − Φ E Φ , which interpolates between an approximately linear dis- θ − χ S ≈ S 1/2 1 1/2 persion ε1 vp at large momentum to a quadratic one − hθ uhχ− u† − Φ EΦ . (35) ≈ S − S ≈ 8

This transformation ensures that normalisation of Φ is describes massive chiral electrons, section IID. It gener- consistent with that of the original states: ally dominates at low energy E γ1, so that the other ˆ | | ≪ 2 terms in H2 may be considered as perturbations of it. Φ†Φ= θ† θ = θ† 1+ uhχ− u† θ , S The second term hˆw, section IIE, introduces a triangu- θ†θ+ χ†χ , lar distortion of the Fermi circle around each K point ≈ ˆ ˆ 1 known as ‘trigonal warping’. Terms hU and hAB, with where we used equation (34) for small E: χ h− u†θ. ≈− χ 1 on the diagonal, produce a band gap between the Thus, the effective Hamiltonian for low-energy compo- ± ˆ nents is given by equation (35): conduction and valence bands, section IIG, whereas h4 and hˆ∆ introduce electron-hole asymmetry into the band (eff) 1/2 1 1/2 H − h uh− u† − , (36) structure, section IIF. ≈ S θ − χ S 2 The Hamiltonian (38) is written in the vicinity of a =1+ uhχ− u† . (37) S valley with index ξ = 1 distinguishing between K+ and K . In order to briefly± discuss the effect of symmetry op- erations− on it, we introduce Pauli spin matrices σ , σ , σ 2. Bilayer graphene x y z in the A1/B2 sublattice space and Πx, Πy, Πz in the valley space. Then, the first term in the Hamiltonian may be The Hamiltonian (30) is written in basis written as hˆ = (1/2m)[σ (p2 p2)+2Π σ p p ]. The A1,B1, A2,B2. If, instead, it is written in 0 − x x − y z y x y operation of spatial inversion i is represented by Πxσx the basis of low-energy and dimer components because it swaps both valleys and lattice sites, time inver- (θ,χ) A1,B2, A2,B1, equation (33), then ≡ sion is given by complex conjugation and Πx, as it swaps valleys, too. Hamiltonian (38) satisfies time-inversion ǫA1 v3π ǫA2 γ1 hθ = , hχ = , symmetry at zero magnetic field. The intrinsic terms v3π† ǫB2 γ1 ǫB1 hˆ0, hˆw, hˆ4, and hˆ∆ satisfy spatial-inversion symmetry because the bilayer crystal structure is spatial-inversion v4π† vπ† v4π vπ† u = − , u† = − . symmetric, but terms hÛ and hÂB, with 1 on the di- vπ v4π vπ v4π† ± − − agonal, are imposed by external fields and they violate Using the procedure described in the previous section, spatial-inversion symmetry, producing a band gap be- equations (36,37), it is possible to obtain an effective tween the conduction and valence bands. (eff) Hamiltonian H Hˆ2 for components (ψA1, ψB2). An expansion is performed≡ by assuming that the intralayer D. Interlayer coupling γ1: massive chiral electrons hopping γ0 and the interlayer coupling γ1 are larger than other energies: γ ,γ E , vp, γ , γ , U , ∆ , δ . 0 1 3 4 ′ AB ˆ Then, keeping only terms≫ | | that| are| | linear| | in| | the| | small| The Hamiltonian h0 in equation (38) resembles the Dirac-like Hamiltonian of monolayer graphene, but parameters γ3 , γ4 , U , ∆′ , δAB and quadratic in momentum, the| effective| | | | Hamiltonian| | | | | [9, 63] is with a quadratic-in-momentum term on the off-diagonal rather than linear. For example, the term π2/2m ac- Hˆ2 = hˆ0 + hˆw + hˆ4 + hˆ∆ + hÛ + hÂB, (38) counts for an effective hopping between the non-dimer 2 sites A1, B2 via the dimer sites B1, A2 consisting of a 1 0 π† hˆ0 = , hop from A1 to B1 (contributing a factor vπ), followed by −2m π2 0 a transition between B1, A2 dimer sites (giving a ‘mass’ 2 γ1), and a hop from A2 to B2 (a second factor of vπ). 0 π v3a 0 π† ∼ hˆw = v3 , The solutions are massive chiral electrons [8, 9], with π 0 − 4√3~ π2 0 2 2 † parabolic dispersion E = p /2m, m = γ1/2v . The density of states is m/(2π~2±) per spin and per valley, and 2vv4 π†π 0 hˆ4 = , the Fermi velocity vF = pF /m is momentum dependent, γ1 0 ππ† unlike the Fermi velocity v of monolayer graphene. 2 The corresponding wave function is given by ˆ ∆′v π†π 0 h∆ = 2 , γ1 0 ππ† 1 1 ip.r/~ ψ = 2iξϕ e . (39) 2 √2 e ˆ U 1 0 2v π†π 0 ∓ hU = 2 , − 2 0 1 − γ1 0 ππ† The wave function components describe the electronic − − amplitudes on the A1 and B2 sites, and it can be useful δAB 1 0 to introduce the concept of a pseudospin degree of free- hÂB = , 2 0 1 dom [8, 9] that is related to these amplitudes. If all the − electronic density were located on the A1 sites, then the where π = ξpx + ipy, π† = ξpx ipy. In the following pseudospin part of the wave function = (1, 0) could be − |↑ sections, we discuss the terms in Hˆ2. The first term hˆ0 viewed as a pseudospin ‘up’ state, pointing upwards out 9 of the graphene plane. Likewise, a state = (0, 1) with and, thus, it actually only gives a small additional con- |↓ density solely on the B2 sites could be viewed as a pseu- tribution to the mass m. The first term in hˆw causes dospin ‘down’ state. However, density is usually shared trigonal warping of the iso-energetic lines in directions equally between the two layers, so that the pseudospin 2 4 1 5 ϕ = ϕ0, where ϕ0 = 0, 3 π, 3 π at K+, ϕ0 = 3 π, π, 3 π at is a linear combination of up and down, e2iξϕ , K . |↑ ∓ |↓ − equation (39), and it lies in the graphene plane. To analyse the influence of hˆw at low energy, we con- ˆ The Hamiltonian may also be written as h0 = sider just hˆ0 and the first term in hˆw, and the resulting 2 σ σ (p /2m) .ˆn2 where the pseudospin vector is = energy E = ε1 is given by (σx, σy, σz), and ˆn2 = (cos2ϕ, ξ sin 2ϕ, 0) is a unit vec- ± − 3 2 2 tor. This illustrates the chiral nature of the electrons 2 2 ξv3p p σ ε1 = (v3p) cos(3ϕ)+ , (41) [8, 9]: the chiral operator .ˆn2 projects the pseudospin − m 2m onto the direction of quantization ˆn2, which is fixed to lie in the graphene plane, but turns twice as quickly as the in agreement with equation (31) for U = 0, vp/γ1 momentum p. For these chiral quasiparticles, adiabatic 1, and v /v 1. As it is linear in momentum, the≪ 3 ≪ propagation along a closed orbit produces a Berry’s phase influence of hˆw and the resulting triangular distortion [86] change of 2π [8, 9] of the wave function, in contrast of iso-energetic lines tend to increase as the energy and to Berry phase π in monolayer graphene. momentum are decreased until a Lifshitz transition [93] Note that the chiral Hamiltonian hˆ0 may be viewed as a occurs at energy generalisation of the Dirac-like Hamiltonian of monolayer 2 graphene and the second (after the monolayer) in a family γ1 v3 εL = 1 meV. (42) of chiral Hamiltonians HJ , J =1, 2,..., corresponding to 4 v ≈ Berry’s phase Jπ which appear at low energy in ABC- For energies E <ε , iso-energetic lines are broken into stacked (rhombohedral) multilayer graphene [9–15, 84, L four separate| ‘pockets’| consisting of one central pocket 85]: and three ‘leg’ pockets, the latter centred at momentum J p γ v /v2 and angle ϕ , as shown in Figure 4. The 0 π 1 3 0 H = g † , (40) ≈ J J J central pocket is approximately circular for E εL π 0 2 2 | | ≪ with area c πε /(~v3) , while each leg pocket is ap- A ≈ proximately elliptical with area l c/3. Note that where g1 = v for monolayer, g2 = 1/2m for bilayer, and A ≈ A 3 2 − Berry phase 2π is conserved through the Lifshitz transi- g3 = v /γ1 for trilayer graphene. Since the pseudospin is related to the wavefunction amplitude on sites that are tion; the three leg pockets each have Berry phase π while the central pocket has π [12, 94]. located on different layers, pseudospin may be viewed as − a ‘which layer’ degree of freedom [14, 87]. ′ F. Interlayer coupling γ4 and on-site parameter ∆ : electron-hole asymmetry E. Interlayer coupling γ3: trigonal warping and the Lifshitz transition Skew interlayer coupling γ4 between a non-dimer and The Hamiltonian hˆ in equation (38) yields a a dimer site, i.e., between A1 and A2 sites or between 0 ˆ quadratic, isotropic dispersion relation E = p2/2m B1 and B2 sites, is described by h4 in equation (38), ± ~ with circular iso-energetic lines, i.e., there is a circular where the effective velocity is v4 = √3aγ4/2 . This term Fermi line around each K point. This is valid near the produces electron-hole asymmetry in the band struc- K point, pa/~ 1, whereas, at high energy, and mo- ture, as illustrated by considering the energy eigenvalues ≪ 2 mentum p far from the K point, there is a triangular E = (p /2m)(1 2v4/v) of the Hamiltonian hˆ0 + hˆ4. ± ± perturbation of the circular iso-energetic lines known as The energy difference ∆′ between dimer and non-dimer trigonal warping, as in monolayer graphene and graphite. sites, ǫA1 = ǫB2 = 0, ǫB1 = ǫA2 = ∆′, equation (26), It occurs because the band structure follows the symme- also introduces electron-hole asymmetry into the band try of the crystal lattice as described by the full momen- structure: the low-energy bands described by hˆ0 + hˆ∆ 2 tum dependence of the function f(k), equation (12) [88]. are given by E = p /2m(1 ∆′/γ1). In bilayer graphene [9], as in bulk graphite [89–92], a ± ± second source of trigonal warping arises from the skew interlayer coupling γ3 between non-dimer A1 and B2 sites. G. Asymmetry between on-site energies: band The influence of γ3 on the band structure is described gaps by equation (31). In the two-band Hamiltonian, it is described by hˆw in equation (38), the second term of 1. Interlayer asymmetry which arises from a quadratic term in the expansion of ~ 2 2 ~2 f(k) √3a(ξpx ipy)/2 + a (ξpx + ipy) /8 . This Interlayer asymmetry U, equation (25), describes a dif- ≈− − second term has the same momentum dependence as hˆ0, ference in the on-site energies of the orbitals on the two 10

2 25 2. Intralayer asymmetry between A and B sites 20 15

1 10 The energy diﬀerence δAB between A and B sites may 5 be described by the Hamiltonian (30) with ǫ = ǫ = A1 − B1

L ǫA2 = ǫB2 = δAB/2 and v3 = v4 = 0, yielding bands

/ p − y 0 E = εα: p ± 2 2 2 2 δAB γ1 2 2 2 α εα = + 1+4v p /γ1 + ( 1) . (45) -1 4 4 − Thus, δAB creates a band gap, but there is no Mexican -2 hat structure. -2 -1 0 1 2

px / pL H. Next-nearest neighbour hopping

The terms described in Hamiltonians (16,30,38) do not represent an exhaustive list of all possibilities. Addi- tional coupling parameters may be taken into account. For example, next-nearest neighbour hopping within each layer [95–98] results in a term (3 f(k) 2)γ appear- − | | n E / εL ing on every diagonal element of the Hamiltonian (16), where γn is the coupling parameter between next-nearest A (or B sites) on each layer. Ignoring the constant-in- momentum part 3γn produces an additional term in the two-component Hamiltonian (38) py / pL 2 2 γnv p 1 0 p / p ˆ x L hn = 2 , − γ0 0 1 FIG. 4. (a) Trigonal warping of the equi-energy lines in the 2 2 vicinity of each K point, and the Lifshitz transition in bilayer resulting in energies E = p /2m(1 γnγ1/γ0 ). Thus, next-nearest neighbour hopping± represents∓ another graphene. The energy is in units of εL. (b) Corresponding ˆ ˆ three-dimensional plot of the low-energy dispersion. source of electron-hole asymmetry, after h4, h∆, and Sb.

I. Spin-orbit coupling layers ǫA1 = ǫB1 = ǫA2 = ǫB2 = U/2. Its inﬂuence on the bands E = −ε (p) is− described− by equation (31) ± α with v3 = 0: For monolayer graphene, Kane and Mele [99] employed a symmetry analysis to show that there are two distinct γ2 U 2 γ4 types of spin-orbit coupling at the corners K+ and K of 2 1 2 2 α 1 2 2 2 2 − εα = + + v p + ( 1) + v p [γ1 + U ] , the Brillouin zone. These two types of spin-orbit coupling 2 4 − 4 (43) exist in bilayer graphene, too. In both monolayers and bilayers, the magnitude of spin-orbit coupling - although The low-energy bands, α = 1, display a distinctive ‘Mex- the subject of theoretical debate - is generally considered to be very small, with estimates roughly in the range of ican hat’ shape with a band gap Ug between the conduc- 1 to 100 eV [99–109]. tion and valence bands which occurs at momentum pg from the centre of the valley: At the corner of the Brillouin zone Kξ in bilayer graphene, the contribution of spin-orbit coupling to the

2 2 two-component low-energy Hamiltonian (38) may be U γ1 U 2γ1 + U Ug = | | ; pg = | | . (44) written as 2 2 2 2 γ1 + U 2v γ1 + U hˆSO = λSOξσzSz , (46) For small values of the interlayer asymmetry U, the band hˆR = λR (ξσxSy + σySx) , (47) gap is equal to the asymmetry Ug = U , but for large | | where σ , i = x,y,z are Pauli spin matrices in the A1/B2 asymmetry values U γ1 the band gap saturates i U γ . It is possible| | ≫ to induce interlayer asymme- sublattice space, and Sj , j = x,y,z are Pauli spin ma- g → 1 try in bilayer graphene through doping [16] or the use of trices in the spin space. The first term hˆSO is intrin- external gates [17, 19, 20]. This is described in detail in sic to graphene, i.e. it is a full invariant of the sys- section III. tem. Both intra- and inter-layer contributions to hˆSO 11 have been discussed [105–107, 109] with the dominant field [7]. In bilayer graphene, the observation of the in- contribution to its magnitude λSO attributed to skew teger quantum Hall effect [8] and the calculated Landau interlayer coupling between π and σ orbitals [106, 107] level spectrum [9] uncovered additional features related or to the presence of unoccupied d orbitals within each to the chiral nature of the electrons. graphene layer [109]. Taken with the quadratic term hˆ0 in the Hamiltonian (38), hˆSO produces a gap of magnitude 2λSO in the spectrum of bilayer graphene, but the 1. The Landau level spectrum of bilayer graphene two low-energy bands remain spin and valley degenerate (as in a monolayer): E = λ2 + v4p4/γ2. However, ± SO 1 We consider the Landau level spectrum of the two- there are gapless edge excitations and, like monolayer ˆ component chiral Hamiltonian h0, equation (38). The graphene [99], bilayer graphene in the presence of intrin- magnetic field is accounted for by the operator p = sic spin-orbit coupling is a topological insulator with a (px,py) i~ +eA where A is the vector potential and finite spin Hall conductivity [110, 111]. the charge≡− of the∇ electron is e. For a magnetic field per- The second type of spin-orbit coupling is the Bychkov- pendicular to the bilayer, B−= (0, 0, B) where B = B , ˆ − | | Rashba term hR, equation (47), which is permitted only if the vector potential may be written in the Landau gauge mirror reflection symmetry with respect to the graphene A = (0, Bx, 0), which preserves translational invari- plane is broken, by the presence of an electric field or ance in the− y direction. Then, π = i~ξ∂ + ~∂ ieBx − x y − a substrate, say [99–101, 105, 110–116]. Taken with the and π† = i~ξ∂ ~∂ + ieBx, and eigenstates are com- − x − y quadratic term hˆ0 in the Hamiltonian (38), hˆR does not prised of functions that are harmonic oscillator states produce a gap, but, as in the monolayer, spin-splitting in the x direction and plane waves in the y direction of magnitude 2λR between the bands. That is, there are [119, 120], four valley-degenerate bands at low energy, x pyλB 2 φℓ (x, y)= Aℓ ℓ 4 4 H λB − ~ 2 2 v p E = λR 1+ 2 2 1 . (48) 2 λRγ1 ± 1 x pyλB pyy exp ~ + i ~ ,(49) × −2 λB − Generally speaking, there is a rich interplay between tuneable interlayer asymmetry U and the influence of the where the magnetic length is λB = ~/eB, ℓ are intrinsic and the Bychkov-Rashba spin-orbit coupling in Hermite polynomials of order ℓ for integer ℓ H0, and bilayer graphene [105, 110, 111, 115, 116]. For example, ≥ A =1/ 2ℓℓ!√π is a normalisation constant. the presence of interlayer asymmetry U breaks inversion ℓ The operators π and π appearing in the Hamilto- symmetry and allows for spin-split levels in the presence † nian (38) act as raising and lowering operators for the of intrinsic spin-orbit coupling only (λ = 0) [105], while R harmonic oscillator states (49). At the first valley, K , the combination of finite U and very large Rashba cou- + pling has been predicted to lead to a topological insulator √2i~ state even with λSO = 0 [115]. K+ : πφℓ = √ℓ φℓ 1 , (50) − λB − √2i~ K+ : π†φℓ = √ℓ +1 φℓ+1 , (51) J. The integer quantum Hall effect λB

and πφ0 = 0. Then, it is possible to show that the Lan- When a two-dimensional electron gas is placed in a dau level spectrum of the Hamiltonian (38) consists of a perpendicular magnetic field, electrons follow cyclotron series of electron and hole levels with energies and wave orbits and their energies are quantised as Landau levels functions [9] given by [117]. At a high enough magnetic field strength, the discrete nature of the Landau level spectrum is manifest as Eℓ, = ~ωc ℓ(ℓ 1) , ℓ 2, (52) the integer quantum Hall effect [118–120], whereby the ± ± − ≥ Hall conductivity assumes values that are integer multi- 1 φℓ 2 K+ : ψℓ, = , ℓ 2, (53) ples of the quantum of conductivity e /h. ± √2 φℓ 2 ≥ The Landau level spectrum of monolayer graphene was ± − calculated by McClure [121] nearly sixty years ago, and where ωc = eB/m and refer to the electron and hole there have been a number of related theoretical studies states, respectively. For± high values of the index, ℓ 1, ≫ [5, 6, 98, 122–124] considering the consequences of chi- the levels are approximately equidistant with spacing ~ωc rality in graphene. The experimental observation of the proportional to the magnetic field strength B. However, integer quantum Hall effect in monolayer graphene [2, 3] this spectrum, equation (52), is only valid for sufficiently found an unusual sequencing of the quantised plateaus small level index and magnetic field ℓ~ωc γ1 because of Hall conductivity, confirming the chiral nature of the the effective Hamiltonian (38) is only applicable≪ at low electrons and prompting an explosion of interest in the energy. 12

2 2 In addition to the field-dependent levels, there are two ( a ) s x y ( 4 e / h ) ( b ) s x y ( 4 e / h ) 4 4 levels fixed at zero energy E1 = E0 = 0 with eigenfunctions: 3 3 2 2 φ1 φ0 1 1 K+ : ψ1 = , ψ0 = , (54) 0 0 - 4 - 2-3 - 1 1 2 3 4 - 4 - 2-3 - 1 1 2 3 4 - 1 n ( 4 B / j 0 ) - 1 n ( 4 B / j 0 ) They may be viewed as arising from the square of the - 2 - 2 lowering operator in the Hamiltonian (38) which acts on - 3 - 3 both the oscillator ground state and the first excited state - 4 - 4 2 2 to give zero energy π φ0 = π φ1 = 0. The eigenfunctions ψ0 and ψ1 have a finite amplitude on the A1 sublattice, FIG. 5. Schematic of the dependence of the Hall conductiv- on the bottom layer, but zero amplitude on the B2 sub- ity σxy on carrier density n for (a) monolayer graphene and lattice. (b) bilayer graphene, where ϕ0 = h/e is the flux quantum At the second valley, K , the roles of operators π and and B is the magnetic field strength. The dashed line in (b) − π† are reversed: shows the behaviour for a conventional semiconductor or bilayer graphene in the presence of large interlayer asymmetry √2i~ √ U (section II J 3) with fourfold level degeneracy due to spin K : πφℓ = ℓ +1 φℓ+1 , (55) and valley degrees of freedom. − − λB √2i~ K : π†φℓ = √ℓ φℓ 1 , (56) − λ − B and g is an integer describing the level degeneracy; steps 2 and π†φ0 = 0. The Landau level spectrum at K is between adjacent plateaus have height ge /h. − degenerate with that at K+, i.e., Eℓ, = ~ωc ℓ(ℓ 1) The Landau level spectrum of monolayer graphene for ℓ 2 and E = E = 0, but the± roles± of the A1− and [98, 121–124] consists of an electron and a hole series of ≥ 1 0 B2 sublattices are reversed: levels, Eℓ, = √2ℓ~v/λB for ℓ 1, with an additional ± ± ≥ level at zero energy E0 = 0. All of the levels are fourfold 1 φℓ 2 degenerate, due to spin and valley degrees of freedom. K : ψℓ, = − , ℓ 2, (57) − ± √2 φℓ ≥ The corresponding Hall conductivity is shown schemati- ± cally in figure 5(a). There are steps of height 4e2/h be- 0 0 tween each plateau, as expected by consideration of the K : ψ1 = , ψ0 = , (58) − φ1 φ0 conventional case, but the plateaus occur at half-integer values of 4e2/h instead of integer ones, as observed ex- The valley structure and electronic spin each contribute a perimentally [2, 3]. This is due to the existence of the twofold degeneracy to the Landau level spectrum. Thus, fourfold-degenerate level E0 = 0 at zero energy, which each level in bilayer level graphene is fourfold degenerate, contributes to a step of height 4e2/h at zero density. except for the zero energy levels which have eightfold For bilayer graphene, plateaus in the Hall conductiv- 2 degeneracy due to valley, spin and the orbital degeneracy ity σxy(n), Fig. 5(b), occur at integer multiples of 4e /h. of ψ0, ψ1. This is similar to a conventional semiconductor with level degeneracy g = 4 arising from the spin and valley degrees of freedom. Deviation from the conventional case 2. Three types of integer quantum Hall effect occurs at low density. In the bilayer there is a step in 2 σxy of height 8e /h across zero density, accompanied by The form of the Landau level spectrum is manifest in a plateau separation of 8B/ϕ0 in density [8, 9], arising a measurement of the integer quantum Hall effect. Here, from the eightfold degeneracy of the zero-energy Lan- we will compare the density dependence of the Hall con- dau levels. This is shown as the solid line in Fig. 5(b), ductivity σxy(n) for bilayer graphene with that of a con- whereas, for a conventional semiconductor, there no step ventional semiconductor and of monolayer graphene. across zero density (the dashed line). The Landau level spectrum of a conventional two- Thus, the chirality of charge carriers in monolayer and dimensional semiconductor is E = ~ω (ℓ +1/2), ℓ 0, bilayer graphene give rise to four- and eight-fold degen- ℓ c ≥ where ωc = eB/m is the cyclotron frequency [119, 120]. erate Landau levels at zero energy and to steps of height 2 As density is changed, there is a step in σxy whenever a of four and eight times the conductance quantum e /h in Landau level is crossed, and the separation of steps on the Hall conductivity at zero density [2, 3, 8]. Here, we the density axis is equal to the maximum carrier den- have assumed that the degeneracy of the Landau levels sity per Landau level, gB/ϕ0, where ϕ0 = h/e is the is preserved, i.e., any splitting of the levels is negligible flux quantum and g is a degeneracy factor. Each plateau as compared to temperature and level broadening in an of the Hall conductivity σxy occurs at a quantised value experiment. The role of electronic interactions in bilayer of Nge2/h where N is an integer labelling the plateau graphene is described in section VIII, while we discuss 13 the influence of interlayer asymmetry on the Landau level spectrum and integer quantum Hall effect in the next sec- σb0 σ1 σ2 σt0 tion. Vb εb εr εt Vt

3. The role of interlayer asymmetry

x=-Lb -c 0 c0 x=Lt The Landau level states, equations (53,57), have dif- 2 2 ferent amplitudes on the lower (A1 sublattice) and upper (B2 sublattice) layers, with the role of the sublattice FIG. 6. Schematic of bilayer graphene in the presence of ex- sites swapped at the two valleys. Thus, interlayer asym- ternal gates located at x = −Lb and x = +Lt, with potentials Vb and Vt, which are separated from the bilayer by metry U as described by the effective Hamiltonian hÛ , media of dielectric constants εb and εt, respectively. The bi- equation (38), leads to a weak splitting of the valley de- layer is modelled as two parallel conducting plates positioned generacy of the levels [9]: at x = −c0/2 and +c0/2, separated by a region of dielectric constant εr. The layers have charge densities σ1 = −en1 and ξU~ωc Eℓ, ~ωc ℓ(ℓ 1)+ . (59) σ2 = −en2 corresponding to layer number densities n1 and ± ≈± − 2γ1 n2. Charge densities σb0 and σt0 (dashed lines) arise from the presence of additional charged impurities. Such splitting is prominent for the zero-energy states [9], equations (54,58), because they only have non-zero amplitude on one of the layers, depending on the valley: (as the gap should saturate at the value of the interlayer 1 coupling γ1). Transport measurements show insulating E = ξU , (60) 0 −2 behaviour [19, 31, 134–139], but, generally, not the huge suppression of conductivity expected for a gap of this 1 ξU~ωc E1 = ξU + . (61) magnitude, and this has been attributed to edge states −2 γ 1 [140], the presence of disorder [141–143] or disorder and When the asymmetry is large enough, then the splitting chiral charge carriers [144]. Broadly speaking, transport U of the zero energy levels from each valley results in seems to occur through different mechanisms in different a sequence of quantum Hall plateaus at all integer val- temperature regimes with thermal activation [31, 134– ues of 4e2/h including a plateau at zero density [17], as 136, 138, 139] at high temperature (above, roughly, 2 to observed experimentally [20]. This behaviour is shown 50K) and variable-range [19, 135–137, 139] or nearest- schematically as the dashed line in figure 5(b). The Lan- neighbour hopping [135, 138] at low temperature. dau level spectrum in the presence of large interlayer asymmetry U has been calculated [125–128], including an analysis of level crossings [128] and a self-consistent B. Hartree model of screening calculation of the spectrum in the presence of an external gate [126, 127], generalising the zero-field procedure External gates are generally used to control the den- outlined in the next section. sity of electrons n on a graphene device [1], but, in bilayer graphene, external gates will also place the separate layers of the bilayer at different potential energies III. TUNEABLE BAND GAP resulting in interlayer asymmetry U = ǫ ǫ (where 2 − 1 ǫ1 = ǫA1 = ǫB1 = U/2 and ǫ2 = ǫA2 = ǫB2 = U/2). A. Experiments Thus, changing the− applied gate voltage(s) will tend to tune both the density n and the interlayer asymmetry A tuneable band gap in bilayer graphene was first ob- U, and, ultimately, the band gap Ug. The dependence of served with angle-resolved photoemission of epitaxial bi- the band gap on the density Ug(n) relies on screening by layer graphene on silicon carbide [16], and the ability to electrons on the bilayer. In the following, we describe a control the gap was demonstrated by doping with potas- simple model [17, 18, 20, 55, 69, 71, 145, 146] that has sium. Since then, the majority of experiments probing been developed to take into account screening using the the band gap have used single or dual gate devices based tight-binding model and Hartree theory. on exfoliated bilayer graphene flakes [19, 20]. The band gap has now been observed in a number of different experiments including photoemission [16], magnetotrans- 1. Electrostatics: asymmetry parameter, layer densities and port [20], infrared spectroscopy [55, 78–80, 129, 130], external gates electronic compressibility [131, 132], scanning tunnelling spectroscopy [133], and transport [19, 31, 134–139]. We use the SI system of units, and the electronic charge The gap observed in optics [16, 55, 78–80, 129, 130] is is e where e> 0. The bilayer graphene device is mod- up to 250 meV [129, 130], the value expected theoretically elled− as two parallel conducting plates that are very nar- 14 row in the x-direction, continuous and infinite in the y- the band structure of bilayer graphene. Analytical cal- z plane, positioned at x = c /2 and +c /2, figure 6. culations are possible [17, 18, 145] if only the dominant − 0 0 Here, c0 is the interlayer spacing and we denote the di- inter-layer coupling γ1 is taken into account in the four- electric constant of the interlayer space as εr (it doesn’t band Hamiltonian (30). Here we will use the two-band include the screening by π-band electrons that we are ex- model (38) with an explicit ultraviolet cutoff [18] when plicitly modelling). Layer number densities are n1 and integrating over the whole Brillouin zone. The simplified n , with corresponding charge densities σ = en and two-component Hamiltonian is 2 1 − 1 σ2 = en2. − 2 We take into account the presence of a back gate and a v2 0 π U 1 0 Hˆ † , (67) top gate, infinite in the y-z plane, located at x = Lb and 2 2 − ≈−γ1 π 0 − 2 0 1 x =+Lt, with potentials Vb and Vt, which are separated − from the bilayer by media of dielectric constants ε and b ˆ εt, respectively. It is possible to describe the presence Solutions to the energy eigenvalue equation H2ψ = Eψ of additional charge near the bilayer - due to impurities, are given by say - by introducing density nb0 on the back-gate side and nt0 on the top-gate side. They correspond to charge U 2 v4p4 E = + , (68) densities σb0 = enb0 and σt0 = ent0, assuming that nb0 2 ± 4 γ1 and nt0 are positive for positive charge. Using Gauss’s Law, it is possible to relate the external E U/2 1 ip.r/~ ψ = − 2 2 e . (69) gate potentials and impurities concentrations to the layer 2E v p e2iξϕ γ1(E U/2) densities and the interlayer asymmetry [17, 60, 145, 147]: − −

ε0εbVb ε0εtVt Layer densities are determined by integration over the n = n1 + n2 = + + nb0 + nt0 , (62) circular Fermi surface eLb eLt 2 εt c0 e c0 2 2 U = eVt + (n2 nt0) , (63) n1(2) = ψA1(B2)(p) p dp , (70) −εr Lt ε0εr − π~2 | | where the field within the bilayer interlayer space is ap- where a factor of four takes account of spin and valley proximately equal to U/(ec ). Equation (62) expresses 0 degeneracy. the total electron density n = n1 + n2 in terms of the external potentials, generalising the relation for monolayer For simplicity, we assume the Fermi energy lies within the conduction band. Using the solution (69), the contri- graphene [1]. Note that the background densities nb0 and bution of the partially-filled conduction band to the layer nt0 shift the effective values of the gate potentials Vb and densities [17, 145, 148] is given by Vt. The second equation, for U, may be rewritten as

(n2 n1) cb 1 E U/2 U = Uext +Λγ1 − , (64) n = pdp ∓ , (71) n 1(2) π~2 E ⊥ p p ec0 εb εt (nb0 nt0) 1 F U F pdp Uext = Vb Vt +Λγ1 − , (65) = pdp , ~2 ~2 2 4 4 2 2εr Lb − Lt n π 0 ∓ 2π 0 U /4+ v p /γ ⊥ 1 where the characteristic density scale n and the dimen- 2 ⊥ n n U 2 n γ1 2nγ1 sionless screening parameter Λ are ⊥ ln | | + 1+ , ≈ 2 ∓ 4γ1 n U n U  2 2 2 ⊥| | ⊥ γ1 c0e γ1 c0e n n = , Λ= ⊥ . (66) ~2 2 ~2 2   ⊥ π v 2π v ε0εr ≡ 2γ1ε0εr where the minus (plus) sign is for the first (second) layer, Equation (64) relates the asymmetry parameter U to a pF is the Fermi momentum, and the total density is n = 2 ~2 sum of its value, Uext, if screening were negligible plus pF /π measured with respect to the charge neutrality a term accounting for screening. Parameter values γ1 = point, i.e., we assume that the point of zero density is 6 1 13 2 0.39eV and v =1.0 10 ms− give n =1.1 10 cm− . realised when the Fermi level lies at the crossing point of × ⊥ × With interlayer spacing c0 = 3.35A˚ and an estimate for the conduction and valence bands. the dielectric constant of εr 1, then Λ 1, showing In addition, we take into account the contribution of ≈ ∼ vb that screening is relevant. the filled valence band n1(2) to the individual layer densities. Note that, as the asymmetry parameter U varies, the filled valence band does not contribute to any change 2. Calculation of individual layer densities in the total density n, but it does contribute to the difference n1 n2. This may be obtained by integrating Through electrostatics, the asymmetry parameter U with respect− to momentum as in equation (71), but in- is related to layer densities n1 and n2, equation (64). troducing an ultraviolet cutoff pmax = γ1/v equivalent The densities n1 and n2 also depend on U because of to nmax = n [18]. Then, the contribution of the filled ⊥ 15 valence band [17, 18, 69, 71, 145, 146] is given by 1.0 Ug/γ1 U pmax pdp nvb , 1(2) ~2 2 4 4 2 ≈±2π 0 U /4+ v p /γ 0.8 n0/n 1 0.0 n U 4γ1 ⊥ ln . (72) ≈± 4γ U 0.2 1 | | 0.6 0.4 Combining the contributions of equations (71) and (72), the individual layer density, n = ncb +nvb , is given 1(2) 1(2) 1(2) 0.4 0.6 [17, 145] by 0.2 n n U n 1 n 2 U 2 n1(2) ⊥ ln | | + + .(73) ≈ 2 ∓ 4γ1 2n 2 n 2γ1  ⊥ ⊥   -1.0-0.5 0 0.5 1.0 Note that some calculations [148] include only the contri- n/n bution (71) of the partially-filled conduction band, others [18] include only the filled valence band (72). FIG. 7. Density-dependence of the band gap Ug(n) in bilayer graphene as the back-gate voltage Vb is varied for a fixed top-gate voltage Vt [17]. The effective offset-density is 3. Self-consistent screening n0 = 2[ε0εtVt/(eLt)+ nt0]. Plots were made for screening 2 2 parameter Λ = 1, using Ug = |U|γ1/pU + γ1 and numerical Substituting the layer density (73) into the expres- solution of equation (74). sion (64) describing screening gives an expression [17, 60, 145] for the density-dependence of the asymmetry monolayer graphene [1]) and Uext = Λγ1n/n . Using parameter U: ⊥ U γ , equation (74) simplifies [17, 60, 69, 71] as | | ≪ 1 1 1 2 2 − Λγ n Λ n − U(n) Λ n 1 n U U(n) 1 1 ln | | , (75) 1 ln | | + + , ≈ n − 2 n Uext ≈  − 2 2n 2 n 2γ1  ⊥ ⊥ ⊥ ⊥ At high density n n , the logarithmic term is neg-   (74) | | ∼ ⊥ ligible and U(n) Λγ1n/n is approximately linear in ⊥ ≈ 2 2 where Uext is the asymmetry in the absence of screen- density. Note that the band gap Ug = U γ1/ U + γ1 tends to saturate U γ , even if U | γ| . At low den- ing (65). The extent of screening is described by the g → 1 | | ≫ 1 logarithmic term with argument depending on n and sity, n n , the logarithmic term describing screening | | ≪ ⊥ U, and a prefactor proportional to the screening pa- dominates and Ug U 2γ1( n /n )/ ln(n / n ), in- ≈ | | ≈ | | ⊥ ⊥ | | rameter Λ 1 (as discussed earlier). A common ex- dependent of the screening parameter Λ. perimental∼ setup, especially for exfoliated graphene on The expressions (74,75) for U(n) take into account a silicon substrate [1–3, 8], includes a single back gate. screening due to low-energy electrons in pz orbitals using Figure 7 shows the density-dependence of the band gap a simplified Hamiltonian (67) while neglecting other or- Ug(n) plotted as the back-gate voltage Vb is varied for a bitals and the effects of disorder [55, 145, 149–155], crys- fixed top-gate voltage Vt. In this case, the influence of talline inhomogeneity [18] and electron-electron exchange the top-gate voltage Vt may be absorbed into an effec- and correlation. Nevertheless, there is generally good tive offset-density n0 = 2[ε0εtVt/(eLt)+ nt0] [17] giving qualitative agreement of the dependence of U(n) on den- Uext = Λγ1(n n0)/n in equation 74. Figure 8 shows sity n predicted by equations (74,75) with density func- the dependence− of the difference⊥ in layer densities n n tional theory calculations [18, 73] and experiments in- 1 − 2 for the case n0 = 0 including both the contribution of cluding photoemission [16, 20] and infrared spectroscopy the partially filled bands as measured with respect to [55, 78–80, 129, 130]. Note that the Hartree screening the charge neutrality point (71) (dashed line) and the model has been generalised to describe graphene trilay- contribution of the full valence band (72) (dotted line). ers and multilayers [147, 156–158]. The sum of both terms (solid line) shows that n1 n2 is positive (negative) for positive (negative) total density− n. Recalling that layer 1 is closest to the back gate, this IV. TRANSPORT PROPERTIES shows that the bilayer is polarised along the electric field, as expected [145]. A. Introduction A single back gate in the absence of additional charged dopants may be described by Vt = nb0 = nt0 = 0, re- Bilayer graphene exhibits peculiar transport proper- sulting in simplified expressions n = ε0εbVb/(eLb) (as in ties due to its unusual band structure, described in sec- 16

(n 1-n 2)/n L

0.4

0.2 W

-1.0 -0.50.5 1.0 n/n -0.2 x=0 x=L

FIG. 9. Two-probe bilayer graphene device with armchair -0.4 edges, width W and length L. The rectangular shaded regions represent the ends of semi-inﬁnite leads.

FIG. 8. Density-dependence of the difference in layer densities tween regions of opposite polarity have attracted theo- n1 − n2 in bilayer graphene as the back-gate voltage Vb is varied for a fixed top-gate voltage Vt [17]. Plots were made retical attention due to the existence of one-dimensional for screening parameter Λ = 1 and the effective offset-density valley-polarised modes along the interface [189–192], a is n0 = 2[ε0εtVt/(eLt)+ nt0] = 0, corresponding to the data pseudospintronic analogy of spin-valve devices for trans- labelled n0 = 0 in Figure 7, obtained by numerical solution port perpendicular to the interface [193, 194], electronic of equation (74). The dashed line shows the contribution confinement [195], and valley-dependent transmission of the partially filled bands as measured with respect to the [196]. charge neutrality point (71) [148], the dotted line shows the In the following we describe two different models of contribution of the full valence band (72) [18], the solid line the conductivity of bilayer graphene at low energy. The is their sum (73) [17, 145]. first is for ballistic transport in a clean device of finite length that is connected to semi-infinite leads, described using wave matching to calculate the transmission prob- tion II, where the conduction and valence bands touch ability and, then, the conductance. The second model with quadratic dispersion. Transport characteristics and describes the conductivity of a disordered, infinite sys- the nature of conductivity near the Dirac point were tem using the Kubo formula and the self-consistent Born probed experimentally [2, 3, 8, 19, 20, 159–162] and inves- approximation to describe scattering from the disordered tigated theoretically [163–175]. Neglecting trigonal warp- potential. Although the two models are quite different, ing, the minimal conductivity is predicted to be 8e2/(πh) both predict the minimal conductivity to be 8e2/(πh) [163, 167, 168], twice the value in monolayer graphene, [163, 167, 168]. Finally, in section IV C 2, we describe while, in the presence of trigonal warping, it is larger, localisation effects. 24e2/(πh) [163, 169], because of multiple Fermi surface pockets at low energy, section II E. B. Ballistic transport in a finite system For a detailed review of the electronic transport properties of graphene monolayer and bilayers, see Ref. [176, 177]. The characteristics of bilayer graphene in the pres- Ballistic transport in a finite, mesoscopic bilayer ence of short-ranged defects and long-ranged charged- graphene nanostructure has been modelled in a number impurities have been calculated [54, 163, 164, 166, 171– of papers [165, 167–169, 178, 193, 194, 196]. Here, we fol- 175] and it is predicted that the conductivity has an low a wave-matching approach of Snyman and Beenakker approximately linear dependence on density at typical [168]. For bilayer, as compared to monolayer, there is a ~ experimental densities [172]. At interfaces and poten- new length scale ℓ1 = v/γ1 characteristic of the inter- tial barriers, conservation of the pseudospin degree of layer coupling. Here, v = √3aγ0/2~ is the band velocity freedom may influence electronic transmission [178, 179], of monolayer graphene, so ℓ1 = (√3/2)(γ0/γ1)a 18 A˚ as in monolayer graphene [178, 180], including transmis- is several times longer [168] than the lattice constant≈ sion at monolayer-bilayer interfaces [181–184], through a = 2.46 A˚ [53]. For most situations, the sample size multiple electrostatic barriers [185], or magnetic barriers L ℓ1, and the device generally behaves as a (coupled) [186, 187]. Inducing interlayer asymmetry and a band bilayer≫ rather than two separate monolayers [168]. gap using an external gate [9, 17], described in section III, We consider a two-probe geometry with armchair edges may be used to tune transport properties [181, 188]. In- as shown in Fig. 9. There is a central mesoscopic bilayer terlayer asymmetry may also be viewed as creating an region, width W and length L, connected to a left and out-of-plane component of pseudospin and interfaces be- right lead. This orientation is rotated by 90◦ as compared 17 to that described in section II so the corners of the Bril- where U > 0 and U ε ,γ1 . Then, in the leads, ∞ ∞ ≫ {| | } louin zone are located at wavevectors Kξ = ξ(0, 4π/3a). k+ k U /(~v) and wave functions ψleft, , ψright, , In terms of wavevector measured from the centre of the in the≈ left− ≈ (x<∞ 0) and right (x>L) leads may± be written± valley, i.e., ky ky +ξ4π/3a, then the Hamiltonian (30) as in basis A1,B1→, A2,B2 may be written as

U vπ† 0 0 vπ U γ1 0 i i Hb =   , (76) ∓ 0 γ1 U vπ†  1  iU∞x/~v  1  iU∞x/~v   ψleft, = ∓ e + r+± − e−  0 0 vπ U  ± 1 1          i   i  where π = i~(kx + iξky), π† = i~(kx iξky), and U is  −    the on-site− energy which describes the− doping of the bi-   i   − layer. For simplicity, we include only the main interlayer 1 ~   iU∞x/ v iky y coupling term γ of the orbitals on the dimer B1 and A2 + r± e− e , (80) 1 − 1 sites. It is assumed that the transverse wavevector k    y  i   is real and it is conserved at the interfaces between the    bilayer and the leads. The Hamiltonian (76) shows there i  i  − are two values of the longitudinal wavevector kx for given 1 1 ~      iU∞(x L)/ v+iky y U, ky and energy ε, which we denote as k+ and k : ψright, = t+± − + t± e − . − ± 1 − 1      ~ 2 ~2 2 2   i   i  vk = (ε U) γ1(ε U) v ky . (77)   −   −  ± − ± − −      Left- ΦL and right- ΦR moving wave functions may be written± as ± Here, a right-moving wave with unit flux correspond- i~v( k iξky) ing to a state with wavevector k U /(~v) is in- ∓ − ± − ± ≈ ∞ L (ε U) ik±x+iky y jected from the left lead [the first term in equation (80)]. Φ = N  ∓ −  e− , (78) ± ± (ε U) Subsequently, there are two left-moving waves that have −  ~  been reflected with amplitudes r+± and r±, and two right-  i v( k + iξky)  −  − − ±  moving waves are transmitted to the right, with ampli-  ~  i v(k iξky) tudes t+± and t±. ∓ ± − − (ε U) ΦR = N  ∓ −  eik±x+iky y , (79) At the charge-neutrality point ε = U = 0 in the central ± ± (ε U) bilayer region, the wave functions are evanescent with  ~ −   i v(k + iξky)  imaginary longitudinal wavevector, equation (77). Two  − ±    1/2 states with kx = iξky have finite amplitude only on the where normalisation N = [4W (ε U)k ]− for unit − ± ± A1, A2 sites and two with k = iξk have finite amplitude current. − x y only on the B1, B2 sites: The aim is to describe a mesoscopic bilayer region of finite length L connected to macroscopic leads. In order to mimick macroscopic, metallic contacts, there should be many propagating modes in the leads that overlap 1 γ x/~v with the modes in the central bilayer region. If this is 1 0 0 the case, then the value of minimal conductance should    ξky y   ξky y ψcentre, = c1± e + c2± e not depend on the particular model used for the leads, ± 0 1      e.g., square lattice or graphene lattice, as demonstrated   0   0  for monolayer graphene [197]. Note, this will yield quite       0   0  different results from a model with a bilayer lead at the 1 0 same level of doping as the central region; then, the sys-   ξky y   ξky y iky y +c3± e− + c4± e− e . tem is effectively an infinite system, not a finite, meso- 0 0      scopic conductor.  γ x/~v   1    − 1     Snyman and Beenakker [168] modelled the leads as      heavily-doped bilayer graphene, generalising an approach developed for monolayer graphene [198]. In this way, there are many conducting modes present in the leads For each incoming mode from the left lead [the first and it is possible to simply use matching of the bilayer term in equation (80)], there± are eight unknown ampli- wave functions at the interface between the central re- tudes c1±, c2±, c3±, c4±, r+±, r±, t+±, t±. Continuity of the gion and the leads. In particular, the leads are mod- wave functions at the interfaces− x =− 0 and x = L between elled as bilayer graphene with on-site energy U = U the central region and the leads provides eight simulta- − ∞ 18 neous equations: T

i + ir+± ir± = c1± , (81) ∓ − − 1 r+± + r± = c3± , (82) ∓ − − 1+ r+± + r± = c2± , (83) − i + ir+± + ir± = c4± , (84) − − ξky L L ξky L it+± + it± = c1±e + c2± e , (85) − − ℓ1 ξky L t+± + t± = c3±e− , (86) − − ξky L t+± + t± = c2±e , (87) − L ξky L ξky L it+± it± = c3± e− + c4±e− , (88) − − − − ℓ1 where ℓ1 = ~v/γ1. Solving yields the transmission matrix

+ ky L t+ t+− t = + t t− − − FIG. 10. The transmission coefficients T± of bilayer 2i graphene (89) for L = 50ℓ1 [168] (solid lines). The trans- = 2 2 + (L/ℓ1) + 2cosh(2kyL) mission coefficient of monolayer graphene [198] is shown in the centre (dashed line). (L/ℓ 2i)cosh(k L) (L/ℓ ) sinh (ξk L) 1 − y − 1 y . × (L/ℓ1) sinh (ξkyL) (L/ℓ1 +2i)cosh(kyL) − C. Transport in disordered bilayer graphene The transmission probability [168] is determined by the eigenvalues T of the product tt†: 1. Conductivity ± 1 T = , (89) When the Fermi energy ε is much larger than the level ± cosh2 (ξk L k L) F y ∓ c broadening caused by the disorder potential, the system L L 2 is not largely different from a conventional metal, and the kcL = ln + 1+ , (90) conductivity is well described by Boltzmann transport  2ℓ 2ℓ  1 1 theory. However, this approximation inevitably breaks   down at the Dirac point, where even the issue of whether The transmission coefficients T have the same form the system is metallic or insulating is nontrivial. To as the transmission in monolayer± graphene T = 2 model electronic transport at the charge neutrality point, 1/ cosh (kyL) [198] but shifted by the parameter kc, as we need a refined approximation that properly includes shown in figure 10. the finite level broadening. Here, we present a conduc- The conductance G may be determined using the tivity calculation using the self-consistent Born approxi- Landauer-Büttiker formula [199, 200] mation (SCBA) [163]. We define the Green’s function as 1 2 G(ε) = (ε )− . The Green’s function averaged over gvgse − H G = Tr tt† , (91) the impurity configurations satisfies the Dyson’s equation h (0) Gα,α′ (ε) = δα,α′ Gα (ε) where the factor of gvgs accounts for valley and spin de- (0) ′ generacy. For a short, wide sample whose width W ex- + G (ε) Σ 1 (ε) G 1 (ε) , (93) α α,α α ,α ceeds its length L, W L, the transverse wavevector α1 may be assumed to be continuous≫ and where is an average over configurations of the disorder 2 2 gvgse W ∞ 2gvgse W potential, α is an eigenstate of the ideal Hamiltonian 0, G = (T+ + T ) dky = (92). (0) 1 H h 2π − h πL and Gα = (ε εα)− , with εα being the eigenenergy of −∞ − the state α in 0. In SCBA, the self-energy is given by Thus, the minimal conductivity σ = GL/W = 8e2/(πh) [201] H is twice as large as in the monolayer. In a similar way, it is possible to determine the Fano factor of shot noise which ′ ′ ′ ′ Σα,α (ε)= Uα,α1 Uα1,α Gα1,α1 (ε) . (94) ′ takes the same value 1/3 [168] as in monolayer graphene α1,α1 [198]. Transmission via evanescent modes in graphene has been described as pseudodiffusive because the Fano The equations (93) and (94) need to be solved self- factor takes the same value as in a diffusive metal [198]. consistently. The conductivity is calculated using the 19

Kubo formula, 8 (a) Γ / γ ~ 2 1 e R RA A R RR R

] 0.15 σ(ε)= gvgs ReTr vx G v˜x G vx G v˜x G , 6 2πΩ − h π 0.08 (95) R A 0.03 where G = G(ε + i0) and G = G(ε i0) are retarded m*/2 s

− g and advanced Green’s functions, v = ∂ /∂p is the v 4 0.015 x H0 x velocity operator, and gvgs accounts for summation over DOS RA RR valleys and spins.v ˜x andv ˜x the velocity operators RA containing the vertex correction, deﬁned byv ˜x =˜vx(ε+ 2 RR i0,ε i0) andv ˜ =v ˜ (ε + i0,ε + i0) with [in units of g Ideal − x x

v˜x(ε,ε′)= vx + UG(ε)˜vxG(ε′)U . (96) 0 -1 -0.5 0 0.5 1 E / γ In SCBA,v ˜x should be calculated in the ladder approxi- F 1 mation. 25 For the disorder potential, we assume a short-ranged (b) Γ / γ1 potential within each valley, 20 0.015 h)] 2

1000 π /( 2 15 0.03 e

0100 s

  g

U = uiδ(r ri) . (97) v − 0010 i    0001  10

  Conductivity   0.08 and neglect intervalley scattering between K and K . + [in units of g 5 This situation is realized when the length scale of the− 0.15 scattering potential is larger than atomic scale but much 0 shorter than the Fermi wave length. We assume an equal -1 -0.5 0 0.5 1 amount of positive and negative scatterers ui = u and E / γ ± F 1 a total density of scatterers per unit area nimp. The SCBA formulation is applied to the low-energy FIG. 11. (a) Calculated density of states and (b) conductivity Hamiltonian equation (38), with the trigonal warping ef- as a function of energy for diﬀerent disorder strength Γ [202]. fect due to γ3 included [163]. Energy broadening due to In (b), broken curves show the results of the low-energy two- the disorder potential is characterised by band model, and the horizontal line indicates the universal 2 2 conductivity gvgse /(π ~). π m Γ= n u2 ∗ . (98) 2 imp 2π~

When Γ>εL, i.e., the disorder is strong enough to smear out the∼ fine band structure of the trigonal warping, we which increases linearly with energy. The value at zero can approximately solve the SCBA equation in an ana- energy becomes lytic form. The self-energy, equation (94), becomes diag- e2 2πε onal with respect to index α and it is a constant, σ(0) = g g 1+ L . (102) v s π2~ Γ Σ(ε + i0) iΓ. (99) ≈− In the strong disorder regime Γ 2πε , the correction ≫ L Then, the conductivity at the Fermi energy ε is written arising from trigonal warping vanishes and the conductiv- 2 2~ as ity approaches the universal value gvgse /(π ) [163, 167] which is twice as large as that in monolayer graphene in e2 1 ε Γ ε 4πε the same approximation. In transport measurements of σ(ε) g g 1+ | | + arctan | | + L . ≈ v s π2~ 2 Γ ε Γ Γ suspended bilayer graphene [161], the minimum conduc- 4 | | (100) tivity was estimated to be about 10− S, which is close to 2 2~ The third term in the square bracket arises from the ver- gvgse /(π ). The 2 2 (two-band) model works well at low energy, tex correction due to the trigonal warping effect, and εL × is given by equation (42). For high energies ε Γ, σ but it is not expected to be valid in the strong disorder approximates as | | ≫ regime when mixing to higher energy bands is consid- erable. To see this, we numerically solved SCBA equa- e2 π ε tion for the original 4 4 (four-band) Hamiltonian. Fig- σ(ε) g g | |, (101) × ≈ v s π2~ 4 Γ ure 11(a) and (b) show the density of states (DOS) and 20 conductivity, respectively, for several disorder strengths dence of relevant relaxation lengths by comparing to the [202]. In (b) the results for the 2 2 model in equa- predicted magnetoresistance formula [224]. × tion (100) are expressed as broken curves. In (a), we Localisation has also been studied for gapped bilayer observe that the DOS at zero energy is significantly en- graphene in the presence of interlayer potential asymme- hanced because states at high energies are shifted toward try U [225]. It was shown that, as long as the disorder the Dirac point by the disorder potential. However, the potential is long range and does not mix K valleys, gap zero-energy conductivity barely shifts from that of the opening inevitably causes electron delocalisation± some- 2 2 model [equation (102)], even for strong disorder such where between U = 0 and U = , in accordance with × ∞ as Γ/γ1 = 0.15, where the DOS at zero energy becomes the transition of quantum valley Hall conductivity, i.e., nearly twice as large as in the 2 2 model. For higher en- the opposite Hall conductivities associated with two val- × ergy ε > Γ, the conductivity increases linearly with ε leys. This is an analog of quantum Hall physics but can | | | | in qualitative agreement with equation (100) of the 2 2 be controlled purely by an external electric field without × model, while the gradient is generally steeper. The con- any use of magnetic fields. ductivity has a small dip at the higher band edge around ε γ , because the frequency of electron scattering is | | ∼ 1 strongly enhanced by the higher band states. V. OPTICAL PROPERTIES The SCBA calculation was recently extended for long- range scatterers [203]. It was shown that the conductivity at zero energy is not universal but depends on the degree The electronic structure of bilayer graphene was of disorder for scatterers with long-range potential, sim- probed by spectroscopic measurements in zero magnetic ilar to monolayer graphene [204]. field [16, 55, 56, 79, 80, 129, 130], and also in high magnetic fields [78]. The optical absorption for perpendic- ularly incident light is described by the dynamical conductivity in a electric field parallel to the layers, in both 2. Localisation effects symmetric bilayers [164, 226–228] and in the presence of an interlayer-asymmetry gap [227] For symmetric bilayer The SCBA does not take account of some quantum graphene, this is explicitly estimated as [164, 226–228] corrections, such as those included explicitly in weak localisation. In graphene, the presence of spin-like degrees g g e2 ~ω +2γ Re σ (ω)= v s 1 θ(~ω 2 ε ) of freedom related to sublattices and valleys, as well as xx 16 ~ ~ω + γ − | F | 1 real electronic spin itself, creates the possibility of a rich 2 γ1 ~ ~ variety of quantum interference behaviour. Weak local- + ~ [θ( ω γ1)+ θ( ω γ1 2 εF )] isation [205, 206] is a particularly useful probe because ω − − − | | ~ω 2γ it is sensitive to elastic scattering that causes relaxation + − 1 θ(~ω 2γ ) ~ω γ − 1 of the sublattice pseudospin and valley ‘spin’. In the − 1 absence of symmetry-breaking scattering processes, con- 2 εF + γ1 +γ1 log | | δ(~ω γ1) , (103) servation of pseudospin in monolayer graphene tends to γ1 − suppress backscattering [88, 207], and the interference of chiral electrons would be expected to result in antilocal- where εF is the Fermi energy and we assumed εF <γ1. ization [208]. However, intravalley symmetry-breaking We label the four bands in order of descending| energy| relaxes the pseudospin and suppresses anti-localisation, as 1, 2, 3, 4. The first term in equation (103) represents while intervalley scattering tends to restore conventional absorption from band 2 to 3, the second from 2 to 4 or weak localisation [209–213], as observed experimentally from 1 to 3, the third from 1 to 4, and the fourth from 3 [209, 214–219]. Nevertheless, anti-localisation has been to 4 or from 1 to 2. Figure 12 (a) shows some examples of observed at high temperature [218, 220] when the rel- calculated dynamical conductivity Re σxx(ω) with several ative influence of symmetry-breaking disorder is dimin- values of the Fermi energy [229]. The curve for εF = 0 ished, and its presence has also been predicted at very has essentially no prominent structure except for a step- low temperature [221–223] when spin-orbit coupling may like increase corresponding to transitions from 2 to 4. influence the spin of the interfering electrons. With an increase in εF , a delta-function peak appears at In bilayer graphene, the pseudospin turns twice as ~ω = γ1, corresponding to allowed transitions 3 to 4. quickly in the graphene plane as in a monolayer, no In a magnetic field, an optical excitation by perpen- suppression of backscattering is expected and the quan- dicular incident light is only allowed between the Lan- tum correction should be conventional weak localisation dau levels with n and n 1 for arbitrary combinations [212, 213, 224]. However, the relatively strong trigonal of = H,L and s = ±1, since the matrix element of ± warping of the Fermi line around each valley (described the velocity operator vx vanishes otherwise. Figure 12 in section II E) can suppress localisation unless interval- (b) shows some plots of Re σxx(ω) in magnetic fields at ley scattering is sufficiently strong [224]. Experimental εF = 0 and zero temperature [230]. Dotted lines pene- observations confirmed this picture [159], and it was pos- trating panels represent the transition energies between sible to determine the temperature and density depen- several specific Landau levels as a continuous function 21

20 graphenes [237, 238]. In particular, monolayer graphene

h) (a) π has a strong diamagnetic singularity at Dirac point,

/2 EF/γ1 2

e 1.20 which is expressed as a Delta function in Fermi energy s g v 15 εF [121, 239–242]. In the bilayer, the singularity is re- 1.00 laxed by the modiﬁcation of the band structure caused by the interlayer coupling as we will see in the following 0.80 10 [234, 237]. 0.60 When the spectrum is composed of discrete Landau levels εn, the thermodynamical potential is generally 0.40 written as 5 1 g g 0.20 Ω= v s ϕ(ε ), (104) −β 2πl2 n B n

Dynamical Conductivity (units of g Dynamical Conductivity (units of 0.00 0 β(ε ζ) where β =1/kBT , ϕ(ε) = log 1+e− − with ζ being h ω / γ 1 the chemical potential. In weak magnetic ﬁeld, using the 2 Euler-Maclaurin formula, the summation in n in equa- (b) hωB / γ1 = 0.1 (B ~ 1.1T) tion (104) can be written as an integral in a continuous variable x with a residual term proportional to B2. The

/h) 0.2 (4.5T) magnetization M and the magnetic susceptibility χ can 2 e s

g be calculated by v 0.3 (10T) ∂Ω ∂M ∂2Ω 1 M = , χ = = . (105) ζ B=0 2 ζ B=0 (units of g − ∂B ∂B − ∂B

xx 0.4 (18T)

σ For monolayer graphene, the susceptibility is [121, 239] Re 0.5 (28T) 2 2 e v ∞ ∂f(ε) χ = gvgs 2 dε. (106) 0 − 6πc − ∂ε 0 1 2 h ω / γ1 −∞

(0L, 1L±) (-1L, 0H±) (0H ,± 1H±) ± At zero temperature, it becomes a delta function in Fermi

(1L ,± 2L±) (0H , 1L±) ± energy,

(2L , 3L±) (0L, 1H±) ±

(1H , 2L±) (1L ,± 2H±) e2v2 χ(ε )= g g δ(ε ). (107) F − v s 6πc2 F FIG. 12. Interband part of the dynamical conductivity of bilayer graphene plotted against the frequency ω, in (a) zero The delta-function susceptibility of monolayer graphene magnetic field with different εF ’s [229] (b) several magnetic is strongly distorted by the interlayer coupling γ1. For fields with εF = 0 [230]. Dashed curves in (b) indicate the the Hamiltonian of the symmetric bilayer graphene, the transition energies between several Landau levels. orbital susceptibility is calculated as [234, 237] e2v2 ε 1 χ(ε)= gvgs 2 θ(γ1 ε ) log | | + . (108) 4πc γ1 − | | γ1 3 The susceptibility diverges logarithmically at ε = 0, be- of ~ωB. Every peak position behaves as a linear func- F ~ 2 comes slightly paramagnetic near εF = γ1, and vanishes tion of B ωB in weak field but it switches over to ∝ for ε > γ where the higher subband| | enters. The in- √B-dependence as the corresponding energy moves out | F | 1 of the parabolic band region. In small magnetic fields, tegration of χ in equation (108) over the Fermi energy becomes g g e2v2/(3πc2) independent of γ , which is the peak structure is smeared out into the zero-field curve − v s 1 more easily in the bilayer than in the monolayer, because exactly twice as large as that of the monolayer graphene, the Landau level spacing is narrower in bilayer due to the equation (107). finite band mass. The susceptibility was also calculated in the presence of interlayer asymmetry [243]. Figure 13 (a) and (b) show the density of states and the susceptibility, respectively, for bilayer graphene with U = 0, 0.2, and 0.5. The VI. ORBITAL MAGNETISM susceptibility diverges in the paramagnetic direction at the band edges where the density of states also diverges. Graphite and related materials exhibit a strong or- This huge paramagnetism can be interpreted as the Pauli bital diamagnetism which overcomes the Pauli spin para- paramagnetism induced by the valley pseudo-spin split- magnetism. Theoretically the diamagnetic susceptibil- ting and diverging density of states [244]. The suscep- ity was calculated for graphite [121, 231, 232], graphite tibility vanishes in the energy region where the higher intercalation compounds [233–236], as well as few-layer subband enters. 22

(a) DOS ∆ / γ details of the deformation; it is non-zero only when the

) 1 2

v role of skew interlayer coupling γ is taken into account 2 8 0 3 h π [249]. It is estimated that 1% strain would give w

/4 0.2 1

γ | | ∼ s 0.5 ˆ g 6 meV [249]. Taken with the quadratic term h in the v 6 0 Hamiltonian (38), the low-energy bands with energy E = 4 ε are given by ± 1 2 2 2 2 DOS (units of g w p p ε2 = w 2 | | cos(2ϕ + θ)+ . (110) 1 | | − m 2m 0 -1.5 -1 -0.5 0 0.5 1 1.5

Energy (units of γ1) This describes a Lifshitz transition at energy εL = w , below which there are two Dirac points in the vicinity| | ) 1 γ 2 (b) Susceptibility c ∆ / γ of each Brillouin zone corner [249–251], centred at mo- π 1 1 / 2 v

2 0 mentum p 2m w = w γ1/v and angles ϕ = θ/2 e s ≈ | | | | − g

v 0.2 and ϕ = π θ/2. In general, there should be an interplay g − 0.5 0.5 − between terms hˆs, equation (109), and hˆw, equation (38), leading to the possibility of employing strain to annihi- late two Dirac points and, thus, change the low energy 0 topology of the bands from four to two Dirac points [249]. The presence of two Dirac points would cause zero-

Susceptibility (units of -0.5 energy Landau levels to be eightfold degenerate; an ex- -1.5 -1 -0.5 0 0.5 1 1.5 perimental signature of this state is predicted to be the Energy (units of γ ) 1 persistence of filling factor ν = 4 in the low-field quantum Hall effect [249]. This contrasts± with the Lifshitz FIG. 13. (a) Density of states, and (b) susceptibility of bilayer transition that would occur in the presence of parameter graphenes with the asymmetry gap U/γ1 = 0, 0.2, and 0.5 [243]. In (b), the upward direction represents negative (i.e., γ3 without strain when there are four Dirac points, sec- diamagnetic) susceptibility. tion II E, giving a degeneracy of sixteen and ν = 8 at low fields. In both cases, Berry phase 2π is conserved:± two Dirac cones with Berry phase π each [249] or four Dirac cones with three of π and one of π [12, 94]. It has also been predicted that the presence of− the Lifshitz tran- VII. PHONONS AND STRAIN sition will be noticeable in the low-energy conductivity at zero magnetic field [163, 169], and the particular case of A. The influence of strain on electrons in bilayer two Dirac points in the presence of strain has recently graphene been analysed, too [252]. Note that the effect of lateral strain on the low-energy topology of the band structure Deformation of a graphene sheet couples to the elec- is qualitatively similar to that of a gapless nematic phase tronic system and modifies the low-energy Hamiltonian. which possibly arises as the result of electron-electron In monolayer graphene, static changes in distance and interactions in bilayer graphene [253–256]. angles of the atomic bonds can be described as effective scalar or vector potentials in the Dirac Hamiltonian [7, 245]. Bilayer graphene has extra degrees of freedom in B. Phonons in bilayer graphene deformation associated with the presence of two layers. It was shown that a band gap can be opened by giving Raman spectroscopy has been a valuable tool in prob- different distortions to the two layers or by pulling the ing the behaviour of phonons in graphite [257] and it may two layers apart in the perpendicular direction [246–248]. be used to determine the number of layers in multilayer Rather than produce a band gap, it has been predicted graphene [258], differentiating between monolayer and bi- that homogeneous lateral strain in bilayer graphene can layer. For an in-depth review of Raman spectroscopy produce a change in topology of the band structure at low of graphene including bilayer graphene see, for exam- energy [249–251]. This deformation causes tight-binding ple, Refs. [259, 260]. The phonon spectrum of monolayer parameters γ0 and γ3 to become dependent on the hop- graphene has been calculated using a tight-binding force- ping direction and produces an additional term in the constant model with parameters fit to Raman data [261], low-energy two-band Hamiltonian (38) and with density functional theory [262–265]. There are three acoustic (A) and three optical (O) branches consist- 0 w ing of longitudinal (L) and transverse (T) in-plane modes hˆs = , (109) w∗ 0 as well as out-of-plane (Z) modes. At the zone centre (the Γ point), the TA and LA modes display linear dispersion where parameter w = w eiξθ depends on the microscopic ω q but the ZA mode is quadratic ω q2. The ZO | | ∼ ∼ 23

1 mode is at 890 cm− [262, 264], and the LO and TO C. Optical phonon anomaly ∼ 1 modes are degenerate (at 1600 cm− ). At the K point, ∼ 1 the ZA and ZO modes ( 540 cm− ) and the LA and LO In the following, we describe the anomalous optical 1 ∼ modes ( 1240 cm− ) are degenerate, with TA modes phonon spectrum in bilayer graphene taking into account ∼ 1 1 at 1000 cm− and TO at 1300 cm− . For undoped ∼ ∼ electron-phonon coupling [287, 288]. Theoretically, it was graphene, owing to strong electron-phonon coupling, the shown that a continuum model works well in describ- highest optical modes at the Γ and K point (i.e. the LO ing long-wavelength acoustic phonons [245] and optical mode at the Γ point and the TO mode at the K point) phonons [291] in graphene, and this theory was extended display Kohn anomalies [266–268] whereby the phonon to bilayer graphene [287, 288]. An optical phonon on one dispersion ω(q) has an almost linear slope as observed, graphene layer is represented by the relative displacement for example, in inelastic x-ray measurements of graphite of two sub-lattice atoms A and B as [263, 268]. As graphene is a unique system in which the ~ electron or hole concentration can be tuned by an ex- iq r u(r)= (bq, + b† q,)e(q)e , (111) ternal gate voltage, it was realised [269, 270] that the 4NMω − q, 0 change in electron density would also influence the behaviour of the optical phonons through electron-phonon where N is the number of unit cells, M is the mass of a coupling and, in particular, a logarithmic singularity carbon atom, ω0 is the phonon frequency at the Γ point, in their dispersion was predicted [269] when the Fermi q = (qx, qy) is the wave vector, and bq, and bq† , are energy εF is half of the energy of the optical phonon the creation and destruction operators, respectively. The ε = ~ω/2. Subsequently, such tuning of phonon fre- | F | index represents the modes (t for transverse and l for quency and bandwidth by adjusting the electronic den- longitudinal), and corresponding unit vectors are defined sity was observed in monolayer graphene through Raman by el = iq/ q and et = iˆz q/ q . spectroscopy [271, 272]. The Hamiltonian| | of optical× phonons| | is written as 1 The behaviour of phonons in bilayer graphene has been = ~ω b† bq + , (112) Hph 0 q, , 2 observed experimentally through Raman spectroscopy q, [76, 77, 273–279] and infrared spectroscopy [280, 281], with particular focus on optical phonon anomalies and and the interaction with an electron at the K+ point is the influence of gating. Generally, the phonon spectrum ~ K β v of bilayer graphene, which has been calculated using den- = √2 [σxuy(r) σyux(r)] , (113) Hint − a2 − sity functional theory [282, 283] and force-constant mod- CC els [284, 285], is similar to that of monolayer. Near the where the Pauli matrix σi works on the space of Γ point there are additional low-frequency modes. There (φ , φ ) for the phonon on layer 1, and (φ , φ ) for 1 A1 B1 A2 B2 is a doubly-degenerate rigid shear mode at 30cm− ∼ layer 2. The dimensionless parameter β is related to the [284, 285] observed through Raman spectroscopy [278] dependence of the hopping integral on the interatomic 1 and an optical mode at 90cm− which arises from ∼ distance, and is defined by β = d log γ0/d log aCC. We relative motion of the layers in the vertical direction usually expect β 2. The strength− of the electron- (perpendicular to the layer plane), known as a layer- phonon interaction∼ is characterized by a dimensionless breathing mode [286] and observed through Raman spec- parameter troscopy [279]. At the Γ point, interlayer coupling causes 2 the LO/TO modes to split into two doubly-degenerate gvgs 36√3 ~ 1 β branches where the higher (lower) frequency branch cor- λ = 2 . (114) 4 π 2Ma ~ω0 2 responds to symmetric ‘in-phase’ (antisymmetric ‘out-of- 23 phase’) relative motion of atoms on the two layers (in the For M =1.993 10 g and ~ω0 =0.196eV (correspond- ×1 3 2 in-plane direction). Analogously to the monolayer, it was ing to 1583cm− ), we have λ 3 10− (β/2) . For predicted that these optical phonons would be affected the K point, the interaction Hamiltonian≈ × is obtained by electron-phonon coupling, with a logarithmic singu- − by replacing σi with σi∗. larity in the dispersion of the symmetric modes when the The Green’s function− of an optical phonon is given by Fermi energy εF is equal to half of the optical phonon a 2 2 matrix associated with phonons on layers 1 and frequency [287], and hybridisation of the symmetric and 2. This× is written as antisymmetric modes in the presence of interlayer poten- 2~ω tial asymmetry [288, 289]. Experimentally, this anoma- Dˆ(q,ω)= 0 (~ω)2 (~ω )2 2~ω Π(ˆ q,ω) lous phonon dispersion has been observed through Ra- − 0 − 0 man spectroscopy [77, 273–276] including the evolution 1 , (115) of two distinct components in the Raman G band for non- ≈ ~ω ~ω0 Π(ˆ q,ω0) zero interlayer asymmetry [274–276]. The Raman spec- − − trum has also been studied for bilayer graphene in the where Π(ˆ q,ω) is the phonon’s self-energy, and the near- presence of Landau levels in a magnetic field [287, 290]. equality in the second line stands because the self-energy 24 )

0 i.e., the interband transition excites a valence electron 2 (a) λω symmetric exactly to the Fermi surface. The coupling between symmetric and antisymmetric modes arises when n = 0, and s 1 makes an anti-crossing at the intersection.

0 VIII. ELECTRONIC INTERACTIONS

-1 asymmetric Generally speaking, the low-energy behaviour of elec- Frequency Shift (units of

) trons in bilayer graphene is well described by the tight- 0 1.0

λω binding model without the need to explicitly incorporate (b) electron-electron interaction effects. Coulomb screening and collective excitations have been described in a num- 0.5 ber of theoretical papers [152, 292–300] and the impor- tance of interaction effects in a bilayer under external gating [293, 301–307] has been stressed. Interaction ef- 0.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 fects should also be important in the presence of a mag- Broadening (units of 2 2 Electron Density (units of gvgsγ1 /2 πhv ) netic field or at very low carrier density, particularly in clean samples. FIG. 14. (a) Frequency shift and (b) broadening of the Γ- Bilayer graphene has quadratic bands which touch at point optical phonon in a bilayer graphene [288]. Solid and low energy resulting in a non-vanishing density of state dashed lines denote the high- and low-frequency modes, re- and it has been predicted to be unstable to electron- spectively, and the thin dotted lines in the top panel show electron interactions at half filling. Trigonal warping the frequencies for symmetric and antisymmetric modes cal- tends to cut off infrared singularities and, thus, finite culated without inclusion of their mixing. coupling strength is generally required to realise correlated ground states; if trigonal warping is neglected, then arbitrarily weak interactions are sufficient. Since bilayer graphene possesses pseudospin (i.e., which layer) and valley degrees of freedom, in addition to real electron spin, it ~ is much smaller than ω0. Then, the eigenmodes are is possible to imagine a number of different broken sym- given by eigenvectors u of ReΠ(ˆ q,ω ), and the fre- | 0 metry states that could prevail depending on model de- quency shift and broadening are obtained as the real and tails and parameter values. Suggestions include a ferro- imaginary part of u Π(ˆ q,ω ) u . | 0 | magnetic [308], layer antiferromagnetic [14, 87, 309–312], In symmetric bilayer graphene (i.e., no interlayer po- ferroelectric [311, 313] or a charge density wave state tential difference), eigenmodes are always classified into [314]; topologically non-trivial phases with bulk gaps and symmetric and antisymmetric modes in which the dis- gapless edge excitations such as an anomalous quantum placement of the top and bottom layers is given by (u, u) Hall state [87, 311, 315, 316] or a quantum spin Hall state for the former, and (u, u) for the latter. The symmet- [87, 311] (also called a spin flux state [256]); or a gapless − ric mode causes interband transitions between the con- nematic phase [253–256, 317]. duction band and valence band, while the antisymmetric Insulating states contribute a term proportional to σz mode contributes to the transitions between two conduc- in the two-band Hamiltonian (38) with its sign corre- tion bands (or between two valence bands). The potential sponding to the distribution of layer, valley and spin difference between two layers gives rise to hybridisation degrees of freedom, indicated in figure 15, as manifest of symmetric and antisymmetric modes [287]. in their spin- and valley-dependent Hall conductivities Figure 14 (a) and (b) show the calculated frequency [14, 87, 311]. Note that the quantum spin Hall state shift and broadening, respectively, as a function of elec- produces a term equivalent to that of intrinsic spin-orbit tron density ns [288]. Here we take ~ω0 = γ1/2, and in- coupling, equation (46), which describes a topological introduce a phenomenological broadening factor of 0.1~ω0. sulator state [99, 110, 111]. By way of contrast, the gap- We assume that ns is changed by the gate voltage, and less nematic phase has a qualitatively similar effect on appropriately take account of the band deformation due the spectrum of bilayer graphene as lateral strain [249], to the interlayer potential difference when ns = 0, with described in section VII, producing an additional term in the self-consistent screening effect included. The thin the two-band Hamiltonian of the form of equation (109), dotted lines in the top panel in Figure 14 (a) indicate with parameter w taking the role of an order parameter. the frequencies for symmetric and antisymmetric modes Experiments on suspended bilayer graphene devices calculated without inclusion of their mixing. The lower have found evidence for correlated states at very low and higher branches exactly coincide with symmetric and density and zero magnetic field [318–324]. Conductiv- asymmetric modes at ns = 0 where the mixing is absent. ity measurements of double-gate devices [318] observed a The dip at the symmetric mode occurs when ~ω0 =2εF , non-monotonic dependence of the resistance on electric 25

K K K K (a) + - (b) + - section II J. The resulting Hall conductivity consists of a series of plateaus at conventional integer positions of 4e2/h, but with a double-sized step of 8e2/h across zero density [8]. Interaction effects are expected to lift the level degeneracy of quantum Hall ferromagnet states at σz ξσ z integer filling factors [326–331]. Indeed, an insulating quantum valley Hall quantum anomalous Hall state at filling factor ν = 0 and complete splitting of K K K K the eightfold-degenerate level at zero energy have been (c) + - (d) + - observed with quantum states at filling factors ν =0, 1, 2 and 3 in high-mobility suspended bilayer graphene at low fields (with all states resolved at B = 3T) [161, 332] and in samples on silicon substrates at high fields, typically above 20 T [333, 334]. ξsσz sσz quantum spin Hall layer antiferromagnet The fractional quantum Hall effect has been observed recently in monolayer graphene, both in suspended sam- FIG. 15. Schematic of the distribution of spin, valley and ples [335, 336] and graphene on boron nitride [337], and layer degrees of freedom in candidates for spontaneous gapped there is evidence for it in bilayer graphene [338], too. states in bilayer graphene [14, 87, 311]. They each contribute Strongly-correlated states at fractional filling factors and a term proportional to σz in the two-band Hamiltonian (38), the prospect of tuning their properties has been the fo- with its sign depending on the valley ξ = ±1 and spin s = cus of recent theoretical attention [339–343]. Clearly, the ±1 orientation. Arrows indicate spin orientation, located at nature of the electronic properties of bilayer graphene in valley K+ (solid) or K− (dashed) and on different layers (top high-mobility samples is a complicated problem, and it is or bottom) of a state at the top of the valence band. likely to be an area of further intense experimental and theoretical investigation in the following years.

field cumulating in a non-divergent resistance at zero field while compressibility measurements of single-gate devices IX. SUMMARY [319] found an incompressible region near the charge neutrality point. These measurements were interpreted as This review focused on the single-particle theory being consistent with either the anomalous quantum Hall of electrons in bilayer graphene, in the shape of the state, in which the separate layers of the device are valley tight-binding model and the related low-energy effective polarised, or the gapless nematic phase. Hamiltonian. Bilayer graphene has two unique proper- Subsequently, conductivity measurements on single- ties: massive chiral quasiparticles in two parabolic bands gate devices [320] observed a weak temperature depen- which touch at zero energy, and the possibility to con- dence of the width of the conductivity minimum near trol an infrared gap between these low-energy bands by zero carrier density, suggesting a suppressed density of applying an external gate potential. These features have states as compared to that expected for a parabolic dis- a dramatic impact on many physical properties of bi- persion, as well as particularly robust cyclotron gaps at layer graphene including some described here: optical filling factor ν = 4, observations attributed to the pres- and transport properties, orbital magnetism, phonons ence of the nematic± phase. and strain. A number of topics were not covered here in However, other experiments [321–324] observed insu- great detail or at all; we refer the reader to relevant de- lating states indicating the formation of ordered phases tailed reviews of graphene including electronic transport with energy gaps of about 2meV. There was evidence [176, 177], electronic and photonic devices [344], scan- for the existence of edge states in one of these exper- ning tunnelling microscopy [345], Raman spectroscopy iment [321], but not in the others [322–324], and the [259, 260], magnetism [346], spintronics and pseudospin- latter observations including their response to perpen- tronics [347], Andreev reflection at the interface with a dicular electric field [322, 323] and tilted magnetic field superconductor and Klein tunnelling [348], growth and [324] were seen as being consistent with a layer antiferro- applications [21], and the properties of graphene in gen- magnetic state, in which the separate layers of the device eral [7, 349]. Finally, although the central features of the are spin polarised, figure 15(d). At present, there appear single-particle theory are already established, the same to be contradicting experimental and theoretical results, can not be said of the influence of electronic interactions, but it should be noted that renormalisation group stud- which is likely to remain a subject of intense research, ies [255, 256, 325] have highlighted the sensitivity of the both theoretical and experimental, in the near future at phase diagram of bilayer graphene to microscopic details. least. The authors gratefully acknowledge colleagues for In the absence of interactions, the low-energy Landau fruitful collaboration in graphene research, in particular level spectrum of bilayer graphene [9] consists of a series T. Ando and V. I. Fal’ko, and we acknowledge funding of fourfold (spin and valley) degenerate levels with an by the JST-EPSRC Japan-UK Cooperative Programme eightfold-degenerate level at zero energy, as described in Grant EP/H025804/1. 26

[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang W, Katsnelson M I, Geim A K and Novoselov K S 2009 Y, Dubonos S V, Grigorieva I V and Firsov A A 2004 Science 323 610 Science 306 666 [30] Bae S, Kim H, Lee Y, Xu X, Park J-S, Zheng Y, Bal- [2] Novoselov K S, Geim A K, Morozov S V, Jiang D, Kat- akrishnan J, Lei T, Kim H R, Song Y I, Kim Y-J, Kim snelson M I, Grigorieva I V, Dubonos S V and Firsov A K S, Ozyilmaz¨ B, Ahn J-H, Hong B H and Iijima S 2010 A 2005 Nature 438 197 Nature Nanotech. 5 574 [3] Zhang Y B, Tan Y W, Stormer H L and Kim P 2005 [31] Xia F, Farmer D B, Lin Y-M and Avouris P 2010 Nano Nature 438 201 Lett. 10 715 [4] DiVincenzo D P and Mele E J 1984 Phys. Rev. B 29 [32] Wang C-R, Lu W-S, Hao L, Lee W-L, Lee T-K, Lin 1685 F, Cheng I-C and Chen J-Z 2011 Phys. Rev. Lett. 107 [5] Semenoff G W 1984 Phys. Rev. Lett. 53 2449 186602 [6] Haldane F D M 1988 Phys. Rev. Lett. 61 2015 [33] Yan H, Li X, Chandra B, Tulevski G, Wu Y, Freitag M, [7] Castro Neto A H, Guinea F, Peres N M R, Novoselov Zhu W, Avouris P and Xia F 2012 Nature Nanotech. 7 K S and Geim A K 2009 Rev. Mod. Phys. 81 109 330 [8] Novoselov K S, McCann E, Morozov S V, Fal’ko V I, [34] Yan J, Kim M-H, Elle J A, Sushkov A B, Jenkins G S, Katsnelson M I, Zeitler U, Jiang D, Schedin F and Geim Milchberg H M, Fuhrer M S and Drew H D 2012 Nature A K 2006 Nature Phys. 2 177 Nanotech. 7 472 [9] McCann E and Fal’ko V I 2006 Phys. Rev. Lett. 96 [35] Sugawara K, Kanetani K, Sato T and Takahashi T 2011 086805 AIP Adv. 1 022103 [10] Guinea F, Castro Neto A H and Peres N M R 2006 [36] Kanetani K, Sugawara K, Sato T, Shimizu R, Iwaya K, Phys. Rev. B 73 245426 Hitosugi T and Takahashia T 2012 Proc. Nat. Acad. Sci. [11] Koshino M and Ando T 2007 Phys. Rev. B 76 085425 109 19610 [12] Mañes J L, Guinea F and Vozmediano M A H 2007 [37] Gong L, Young R J, Kinloch I A, Riaz I, Jalil R and Phys. Rev. B 75 155424 Novoselov K S 2012 ACS Nano 6 2086 [13] Nakamura M and Hirasawa L 2008 Phys. Rev. B 77 [38] Young R J, Kinloch I A, Gong L and Novoselov K S 045429 2012 Compos. Sci. Technol. 72 1459 [14] Min H, Borghi G, Polini M and MacDonald A H 2008 [39] Goossens A M, Driessen S C M, Baart T A, Watanabe Phys. Rev. B 77 041407 K, Taniguchi T and Vandersypen L M K 2012 Nano [15] Min H and MacDonald A H 2008 Phys. Rev. B 77 letters 12 4656 155416 [40] Liu Z, Suenaga K, Harris P J F and Iijima S 2009 Phys. [16] Ohta T, Bostwick A, Seyller T, Horn K and Rotenberg Rev. Lett. 102, 015501 E 2006 Science 313 951 [41] Lopes dos Santos J M B, Peres N M R and Castro Neto [17] McCann E 2006 Phys. Rev. B 74 161403(R) A H 2007 Phys. Rev. Lett. 99 256802 [18] Min H, Sahu B R, Banerjee S K and MacDonald A H [42] Berger C, Song Z, Li X, Wu X, Brown N, Naud C, 2007 Phys. Rev. B 75 155115 Mayou D, Li T, Hass J, Marchenkov A N, Conrad E [19] Oostinga J B, Heersche H B, Liu X, Morpurgo A F and H, First P N and de Heer W A 2006 Science 312 1191 Vandersypen L M K 2007 Nature Mater. 7 151 [43] Hass J, Varchon F, Millán-Otoya J E, Sprinkle M, [20] Castro E V, Novoselov K S, Morozov S V, Peres N M Sharma N, de Heer W A, Berger C, First P N, Magaud R, Lopes dos Santos J M B, Nilsson J, Guinea F, Geim L and Conrad E H 2008 Phys. Rev. Lett. 100 125504 A K and Castro Neto A H 2007 Phys. Rev. Lett. 99 [44] Mele E J 2010 Phys. Rev. B 81 161405 216802 [45] Li G, Luican A, Lopes dos Santos J M B, Castro Neto A [21] Novoselov K S, Fal’ko V I, Colombo L, Gellert P R, H, Reina A, Kong J and Andrei E Y 2010 Nature Phys. Schwab M G and Kim K 2012 Nature 490 192 6 109 [22] Dean C R, Young A F, Meric I, Lee C, Wang L, Sorgen- [46] Luican A, Li G, Reina A, Kong J, Nair R R, Novoselov frei S, Watanabe K, Taniguchi T, Kim P, Shepard K L K S, Geim A K and Andrei E Y 2011 Phys. Rev. Lett. and Hone J 2010 Nature Nanotech. 5 722 106 126802 [23] Ghosh S, Bao W, Nika D L, Subrina S, Pokatilov E P, [47] Schmidt H, Lüdtke T, Barthold P, McCann E, Fal’ko V Lau C N and Balandin A A 2010 Nature Mater. 9 555 I and Haug R J 2008 Appl. Phys. Lett. 93 172108 [24] Balandin A A 2011 Nature Mater. 10 569 [48] Ni Z, Wang Y, Yu T, You Y and Shen Z 2008 Phys. [25] Neek-Amal M and Peeters F M 2010 Phys. Rev. B 81 Rev. B 77 235403 235421 [49] Min H, Bistritzer R, Su J-J and MacDonald A H 2008 [26] Zhang Y Y, Wang C M, Cheng Y and Xiang Y 2011 Phys. Rev. B 78 121401 Carbon 49 4511 [50] Kharitonov M Y and Efetov K B 2008 Phys. Rev. B 78 [27] Nair R R, Blake P, Grigorenko A N, Novoselov K S, 241401 Booth T J, Stauber T, Peres N M R and Geim A K [51] Schmidt H, Lüdtke T, Barthold P and Haug R J 2010 2008 Science 320 1308 Physica E 42 699 [28] Bunch J S, Verbridge S S, Alden J S, an der Zande A [52] Ponomarenko L A, Geim A K, Zhukov A A, Jalil R, M, Parpia J M, Craighead H G and McEuen P L 2008 Morozov S V, Novoselov K S, Grigorieva I V, Hill E H, Nano Lett. 8 2458 Cheianov V V, Falko V I, Watanabe K, Taniguchi T [29] Elias D C, Nair R R, Mohiuddin T M G, Morozov S and Gorbachev R V 2011 Nature Phys. 7 958 V, Blake P, Halsall M P, Ferrari A C, Boukhvalov D [53] Saito R, Dresselhaus M S and Dresselhaus G 1998 Phys- 27

ical Properties of Carbon Nanotubes (London: Imperial [84] Koshino M and McCann E 2009 Phys. Rev. B 80 165409 College Press) [85] Zhang F, Sahu S, Min H and MacDonald A H 2010 [54] Nilsson J, Castro Neto A H, Guinea F and Peres N M Phys. Rev. B 82 035409 R 2008 Phys. Rev. B 78 045405 [86] Berry M V 1984 Proc. R. Soc. Lond. A 392 45 [55] Zhang L M, Li Z Q, Basov D N, Fogler M M, Hao Z [87] Zhang F, Jung J, Fiete G A, Niu Q and MacDonald A and Martin M C 2008 Phys. Rev. B 78 235408 H 2011 Phys. Rev. Lett. 106 156801 [56] Li Z Q, Henriksen E A, Jiang Z, Hao Z, Martin M C, [88] Ando T, Nakanishi T and Saito R 1998 J. Phys. Soc. Kim P, Stormer H L and Basov D N 2009 Phys. Rev. Jpn. 67 2857 Lett. 102 037403 [89] Dresselhaus G 1974 Phys. Rev. B 10 3602 [57] Latil S and Henrard L 2006 Phys. Rev. Lett. 97 036803 [90] Nakao K 1976 J. Phys. Soc. Japan 40 761 [58] Koshino M and McCann E 2010 Phys. Rev. B 81 115315 [91] Inoue M 1962 J. Phys. Soc. Japan 17 808 [59] Ashcroft N W and Mermin N D 1976 Solid-State Physics [92] Gupta O P and Wallace P R 1972 Phys. Status. Solidi. (Belmont: Brooks/Cole) B 54 53 [60] McCann E 2012 Graphene Nanoelectronics: Metrol- [93] Lifshitz L M 1960 Zh. Exp. Teor. Fiz. 38 1565 [Sov. ogy, Synthesis, Properties and Applications Raza H (ed) Phys. JETP 11 1130] (Berlin: Springer-Verlag) 237-275 [94] Mikitik G P and Sharlai Y 2008 Phys. Rev. B 77 113407 [61] Partoens B and Peeters F M 2006 Phys. Rev. B 74 [95] Wallace P R 1947 Phys. Rev. 71 622 075404 [96] Johnson J G and Dresselhaus G 1973 Phys. Rev. B 7 [62] Mucha-Kruczyński M, Tsyplyatyev O, Grishin A, Mc- 2275 Cann E, Falko V I, Bostwick A and Rotenberg E 2008 [97] Sasaki K, Murakami S and Saito R 2006 Appl. Phys. Phys. Rev. B 77 195403 Lett. 88 113110 [63] Mucha-Kruczyński M, McCann E and Falko V I 2010 [98] Peres N M R, Guinea F and Castro Neto A H 2006 Semicond. Sci. Technol. 25 033001 Phys. Rev. B 73 125411 [64] Slonczewski J C and Weiss P R 1958 Phys. Rev. 109 [99] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801 272 [100] Min H, Hill J E, Sinitsyn N A, Sahu B R, Kleinman L [65] McClure J W 1957 Phys. Rev. 108 612 and MacDonald A H 2006 Phys. Rev. B 74 165310 [66] McClure J W 1960 Phys. Rev. 119 606 [101] Huertas-Hernando D, Guinea F and Brataas A 2006 [67] Dresselhaus M S and Dresselhaus G 2002 Adv. Phys. 51 Phys. Rev. B 74 155426 1 [102] Yao Y, Ye F, Qi X-L, Zhang S-C and Fang Z 2007 Phys. [68] Aoki M and Amawashi H 2007 Solid State Commun. Rev. B 75 041401(R) 142 123 [103] Boettger J C and Trickey S B 2007 Phys. Rev. B 75 [69] McCann E, Abergel D S L and Fal’ko V I 2007 Solid 121402(R) State Commun. 143 110 [104] Gmitra M, Konschuh S, Ertler C, Ambrosch-Draxl C [70] Guinea F, Castro Neto A H and Peres N M R 2007 Solid and Fabian J 2009 Phys. Rev. B 80 235431 State Commun. 143 116 [105] van Gelderen R and Morais Smith C 2010 Phys. Rev. B [71] McCann E, Abergel D S L and Fal’ko V I 2007 Eur. 81 125435 Phys. J. Special Topics 148 15 [106] Guinea F 2010 New J. Phys. 12 083063 [72] Guinea F, Castro Neto A H and Peres N M R 2007 Eur. [107] Liu H-W, Xie X C and Sun Q-F 2010 arXiv:1004.0881 Phys. J. Special Topics 148 117 [108] McCann E and Koshino M 2010 Phys. Rev. B 81 [73] Gava P, Lazzeri M, Saitta A M and Mauri F 2009 Phys. 241409(R) Rev. B 79 165431 [109] Konschuh S, Gmitra M, Kochan D and Fabian J 2012 [74] Boukhvalov D W and Katsnelson M I 2008 Phys. Rev. Phys. Rev. B 85 115423 B 78 085413 [110] Cortijo A, Grushin A G and Vozmediano M A H 2010 [75] Mucha-Kruczyński M, Abergel D S L, McCann E and Phys. Rev. B 82 195438 Falko V I 2009 J. Phys.: Condens. Matter 21 344206 [111] Prada E, San-Jose P, Brey L and Fertig H A 2011 Solid [76] Malard L M, Nilsson J, Elias D C, Brant J C, Plentz State Comm. 151 1075 F, Alves E S, Castro Neto A H and Pimenta M A 2007 [112] Bychkov Y A and Rashba E I 1984 J. Phys. C 17 6039 Phys. Rev. B 76 201401(R) [113] Rashba E I 2009 Phys. Rev. B 79 161409 [77] Das A, Chakraborty B, Piscanec S, Pisana S, Sood A K [114] Rakyta P, Kormanyos A and Cserti J 2010 Phys. Rev. and Ferrari A C 2009 Phys. Rev. B 79 155417 B 82 113405 [78] Henriksen E A, Jiang Z, Tung L-C, Schwartz M E, [115] Qiao Z, Tse W-K, Jiang H, Yao Y and Niu Q 2011 Phys. Takita M, Wang Y-J, Kim P and Stormer H L 2008 Rev. Lett. 107 256801 Phys. Rev. Lett. 100 087403 [116] Mireles F and Schliemann J 2012 New J. Phys. 14 [79] Kuzmenko A B, van Heumen E, van der Marel D, Lerch 093026 P, Blake P, Novoselov K S and Geim A K 2009 Phys. [117] Landau L D 1930 Z. Phys. 64 629 Rev. B 79 115441 [118] von Klitzing K, Dorda G and Pepper M 1980 Phys. Rev. [80] Kuzmenko A B, Crassee I, van der Marel D, Blake P Lett. 45 494 and Novoselov K S 2009 Phys. Rev. B 80 165406 [119] Prange R E and Girvin S M (eds.) 1986 The Quantum [81] Mak K F, Sfeir M Y, Misewich J A and Heinz T F 2010 Hall Effect (New York: Springer-Verlag) Proc. Nat. Acad. Sci. 107 14999 [120] MacDonald A H (ed.) 1989 Quantum Hall Effect: A [82] Taychatanapat T, Watanabe K, Taniguchi T and Perspective (Boston: Kluwer) Jarillo-Herrero P 2011 Nature Phys. 7 621 [121] McClure J W 1956 Phys. Rev. 104 666 [83] Trickey S B, Müller-Plathe F, Diercksen G H F and [122] Zheng Y and Ando T 2002 Phys. Rev. B 65 245420 Boettger J C 1992 Phys. Rev. B 45 4460 [123] Gusynin V P and Sharapov S G 2005 Phys. Rev. Lett. 28

95 146801 Rev. B 79 035421 [124] Herbut I F 2007 Phys. Rev. B 75 165411 [157] Avetisyan A A, Partoens B and Peeters F M 2009 Phys. [125] Milton Pereira Jr. J, Peeters F M and Vasilopoulos P Rev. B 80 195401 2007 Phys. Rev. B 76 115419 [158] Koshino M 2010 Phys. Rev. B 81 125304 [126] Mucha-Kruczyński M, McCann E and Fal’ko V I 2009 [159] Gorbachev R V, Tikhonenko F V, Mayorov A S, Horsell Solid State Comm. 149 1111 D W and Savchenko A K 2007 Phys. Rev. Lett. 98 [127] Nakamura M, Castro E V and Dora B 2009 Phys. Rev. 176805 Lett. 103 266804 [160] Morozov S V, Novoselov K S, Katsnelson M I, Schedin [128] Zhang L M, Fogler M M and Arovas D P 2011 Phys. F, Elias D C, Jaszczak J A and Geim A K 2008 Phys. Rev. B 84 075451 Rev. Lett. 100 016602 [129] Mak K F, Lui C H, Shan J and Heinz T F 2009 Phys. [161] Feldman B E, Martin J and Yacoby A 2009 Nature Phys. Rev. Lett. 102 256405 5 889 [130] Zhang Y, Tang T-T, Girit C, Hao Z, Martin M C, Zettl [162] Xiao S, Chen J-H, Adam S, Williams E D and Fuhrer A, Crommie M F, Shen Y R and Wang F 2009 Nature M S 2010 Phys. Rev. B 82 041406 459 820 [163] Koshino M and Ando T 2006 Phys. Rev. B 73 245403 [131] Henriksen E A and Eisenstein J P 2010 Phys. Rev. B [164] Nilsson J, Castro Neto A H, Guinea F and Peres N M 82 041412 R 2006 Phys. Rev. Lett. 97 266801 [132] Young A F, Dean C R, Meric I, Sorgenfrei S, Ren H, [165] Katsnelson M I 2006 Eur. Phys. J. B 52 151 Watanabe K, Taniguchi T, Hone J, Shepard K L and P [166] Katsnelson M I 2007 Phys. Rev. B 76 073411 Kim 2012 Phys. Rev. B 85 235458 [167] Cserti J 2007 Phys. Rev. B 75 033405 [133] Deshpande A, Bao W, Zhao Z, Lau C N and LeRoy B [168] Snyman I and Beenakker C W J 2007 Phys. Rev. B 75 J 2009 Appl. Phys. Lett. 95 243502 045322 [134] Szafranek B N, Schall D, Otto M, Neumaier D and Kurz [169] Cserti J, Csordas A and David G 2007 Phys. Rev. Lett. H 2010 Appl. Phys. Lett. 96 112103 99 066802 [135] Zou K and Zhu J 2010 Phys. Rev. B 82 081407(R) [170] Kechedzhi K, Fal’ko V I, McCann E and Altshuler B L [136] Miyazaki H, Tsukagoshi K, Kanda A, Otani M and 2007 Phys. Rev. Lett. 98 176806 Okada S 2010 Nano Lett. 10 3888 [171] Adam S and Das Sarma S 2008 Phys. Rev. B 77 115436 [137] Jing L, Velasco Jr. J, Kratz P, Liu G, Bao W Z, Bock- [172] Das Sarma S, Hwang E H and Rossi E 2010 Phys. Rev. rath M and Lau C N 2010 Nano Lett. 10 4000 B 81 161407(R) [138] Taychatanapat T and Jarillo-Herrero P 2010 Phys. Rev. [173] Trushin M, Kailasvuori J, Schliemann J and MacDonald Lett. 105 166601 A H 2010 Phys. Rev. B 82 155308 [139] Yan J and Fuhrer M S 2010 Nano Lett. 10 4521 [174] Hwang E H and Das Sarma S 2010 Phys. Rev. B 82 [140] Li J, Martin I, Büttiker M and Morpurgo A F 2011 081409 Nature Phys. 7 38 [175] Ferreira A, Viana-Gomes J, Nilsson J, Mucciolo E R, [141] Castro E V, Lopez-Sancho M P and Vozmediano M A Peres N M R and Castro Neto A H 2011 Phys. Rev. B H 2010 Phys. Rev. Lett. 104 036802 83 165402 [142] Rossi E and Das Sarma S 2011 Phys. Rev. Lett. 107 [176] Peres N M R 2010 Rev. Mod. Phys. 82 2673 155502 [177] Das Sarma S, Adam S, Hwang E H and Rossi E 2011 [143] Abergel D S L, Min H K, Hwang E H and Das Sarma Rev. Mod. Phys. 83 407 S 2012 Phys. Rev. B 85 045411 [178] Katsnelson M I, Novoselov K S and Geim A K 2006 [144] Trushin M 2012 EPL 98 47007 Nature Phys. 2 620 [145] Fogler M M and McCann E 2010 Phys. Rev. B 82 [179] Poole C J 2010 Solid State Commun. 150 632 197401 [180] Cheianov V V and Falko V I 2006 Phys. Rev. B 74 [146] Castro E V, Novoselov K S, Morozov S V, Peres N M R, 041403(R) 2006 Lopes dos Santos J M B, Nilsson J, Guinea F, Geim A [181] Nilsson J, Castro Neto A H, Guinea F and Peres N M K and Castro Neto A H 2010 J. Phys.: Condens. Matter R 2007 Phys. Rev. B 76 165416 22 175503 [182] Nakanishi T, Koshino M and Ando T 2010 Phys. Rev. [147] Koshino M and McCann E 2009 Phys. Rev. B 79 125443 B 82 125428 [148] Falkovsky L A 2009 Phys. Rev. B 80 113413 [183] Koshino M, Nakanishi T and Ando T 2010 Phys. Rev. [149] Nilsson J and Castro Neto A H 2007 Phys. Rev. Lett. B 82 205436 98 126801 [184] González J W, Santos H, Prada E, Brey L and Chico L [150] Castro E V, Peres N M R and Lopes dos Santos J M B 2011 Phys. Rev. B 83 205402 2007 Phys. Stat. Sol. B 244 2311 [185] Barbier M, Vasilopoulos P, Peeters F M and Milton [151] Mkhitaryan V V and Raikh M E 2008 Phys. Rev. B 78 Pereira Jr J 2009 Phys. Rev. B 79 155402 195409 [186] Ramezani Masir M, Vasilopoulos P and Peeters F M [152] Abergel D S L, Hwang E H and Das Sarma S 2011 Phys. 2009 Phys. Rev. B 79 035409 Rev. B 83 085429 [187] Agrawal (Garg) N, Grover S, Ghosh S and Sharma M [153] Min H, Abergel D S L, Hwang E H and Das Sarma S 2012 J. Phys.: Condens. Matter 24 175003 2011 Phys. Rev. B 84 041406(R) [188] Fiori G and Iannaccone G 2009 IEEE Electr. Device L. [154] Abergel D S L, Min H, Hwang E H and Das Sarma S 30 261 2011 Phys. Rev. B 84 195423 [189] Martin I, Blanter Ya M and Morpurgo A F 2008 Phys. [155] Rutter G M, Jung S, Klimov N N, Newell D B, Zhitenev Rev. Lett. 100 036804 N B and Stroscio J A 2011 Nature Phys. 7 649 [190] Killi M, Wei T-C, Affleck I and Paramekanti A 2010 [156] Avetisyan A A, Partoens B and Peeters F M 2009 Phys. Phys. Rev. Lett. 104 216406 29

[191] Qiao Z, Jung J, Niu Q and MacDonald A H 2011 Nano [226] Abergel D S L and Falko V I 2007 Phys. Rev. B 75 Lett. 11 3453 155430 [192] Zarenia M, Pereira Jr. J M, Farias G A and Peeters F [227] Nicol E J and Carbotte J P 2008 Phys. Rev. B 77 M 2011 Phys. Rev. B 84 125451 155409 [193] San-Jose P, Prada E, McCann E and Schomerus H 2009 [228] Koshino M and Ando T 2009 Solid State Commun. 149 Phys. Rev. Lett. 102 247204 1123 [194] Li X G, Zhang Z Y and Xiao D 2010 Phys. Rev. B 81 [229] Ando T and Koshino M 2009 J. Phys. Soc. Jpn. 78 195402 104716 [195] Xavier L J P, Pereira J M, Chaves A, Farias G A and [230] Koshino M and Ando T 2008 Phys. Rev. B 77 115313 Peeters F M 2010 Appl. Phys. Lett. 96 212108 [231] McClure J W 1960 Phys. Rev. 119 606 [196] Schomerus H 2010 Phys. Rev. B 82 165409 [232] Sharma M P, Johnson L G and McClure J W 1974 Phys. [197] Schomerus H 2007 Phys. Rev. B 76 045433 Rev. B 9 2467 [198] Tworzydlo J, Trauzettel B, Titov M, Rycerz A and [233] Safran S A and DiSalvo F J 1979 Phys. Rev. B 20 4889 Beenakker C W J 2006 Phys. Rev. Lett. 96 246802 [234] Safran S A 1984 Phys. Rev. B 30 421 [199] Landauer R 1970 Phil. Mag. 21 863 [235] Blinowski J and Rigaux C 1984 J. Phys. (Paris) 45 545 [200] Buttiker M 1988 IBM J. Res. Dev. 32 317 [236] Saito R and Kamimura H 1986 Phys. Rev. B 33 7218 [201] Shon N H and Ando T 1998 J. Phys. Soc. Jpn. 67 2421 [237] Koshino M and Ando T 2007 Phys. Rev. B 76 085425 [202] Koshino M 2009 New J. Phys. 11 095010 [238] Nakamura M and Hirasawa L 2008 Phys. Rev. B 77 [203] Ando T 2011 J. Phys. Soc. Jpn. 80 014707 045429 [204] Noro M, Koshino M and Ando T 2010 J. Phys. Soc. [239] Koshino M and Ando T 2007 Phys. Rev. B 75 235333 Jpn. 79 094713 [240] Fukuyama H 2007 J. Phys. Soc. Jpn. 76 043711 [205] Altshuler B L, Khmelnitski D, Larkin A I and Lee P A [241] Nakamura M 2007 Phys. Rev. B 76 113301 1980 Phys. Rev. B 22 5142 [242] Ghosal A, Goswami P and Chakravarty S 2007 Phys. [206] Hikami S, Larkin A I and Nagaoka Y 1980 Prog. Theor Rev. B 75 115123 Phys. 63 707 [243] Koshino M and Ando T 2010 Phys. Rev. B 81 195431 [207] McEuen P L, Bockrath M, Cobden D H, Yoon Y-G and [244] Koshino M 2011 Phys. Rev. B 84 125427 Louie S G 1999 Phys. Rev. Lett. 83 5098 [245] Suzuura H and Ando T 2002 Phys. Rev. B 65 235412 [208] Suzuura H and Ando T 2002 Phys. Rev. Lett. 89 266603 [246] Nanda B R K and Satpathy S 2009 Phys. Rev. B 80 [209] Morozov S V, Novoselov K S, Katsnelson M I, Schedin 165430 F, Ponomarenko L A, Jiang D and Geim A K 2006 Phys. [247] Choi S-M, Jhi S-H and Son Y-W 2010 Nano Lett. 10 Rev. Lett. 97 016801 3486 [210] Morpurgo A F and Guinea F 2006 Phys. Rev. Lett. 97 [248] Verberck B, Partoens B, Peeters F M and Trauzettel B 196804 2012 Phys. Rev. B 85 125403 [211] McCann E, Kechedzhi K, Fal’ko V I, Suzuura H, Ando [249] Mucha-Kruczyński M, Aleiner I L and Falko V I 2011 T and Altshuler B L 2006 Phys. Rev. Lett. 97 146805 Phys. Rev. B 84 041404 [212] Kechedzhi K, McCann E, Fal’ko V I, Suzuura H, Ando [250] Son Y-W, Choi S-M, Hong Y P, Woo S and Jhi S-H T and Altshuler B L 2007 Eur. Phys. J. Special Topics 2011 Phys. Rev. B 84 155410 148 39 [251] Mariani E, Pearce A J and von Oppen F 2012 Phys. [213] Falko V I, Kechedzhi K, McCann E, Altshuler B L, Rev. B 86 165448 Suzuura H and Ando T 2007 Solid State Comm. 143 [252] Gradinar D A, Schomerus H and Fal’ko V I 2012 Phys. 33 Rev. B 85 165429 [214] Heersche H B, Jarillo-Herrero P, Oostinga J B, Vander- [253] Vafek O and Yang K 2010 Phys. Rev. B 81 041401(R) sypen L M K and Morpurgo A F 2007 Nature 446 56 [254] Lemonik Y, Aleiner I L, Toke C and Falko V I 2010 [215] Wu X, Li X, Song Z, Berger C and de Heer W A 2007 Phys. Rev. B 82 201408(R) Phys. Rev. Lett. 98 136801 [255] Throckmorton R E and Vafek O 2012 Phys. Rev. B 86 [216] Tikhonenko F V, Horsell D W, Gorbachev R V and 115447 Savchenko A K 2008 Phys. Rev. Lett. 100 056802 [256] Lemonik Y, Aleiner I L and Fal’ko V I 2012 Phys. Rev. [217] Ki D-K, Jeong D, Choi J-H, Lee H-J and Park K-S 2008 B 85 245451 Phys. Rev. B 78 125409 [257] Reich S and Thomsen C 2004 Phil. Trans. R. Soc. A [218] Tikhonenko F V, Kozikov A A, Savchenko A K and 362 2271 Gorbachev R V 2009 Phys. Rev. Lett. 103 226801 [258] Ferrari A C, Meyer J C, Scardaci V, Casiraghi C, Lazzeri [219] Lara-Avila S, Tzalenchuk A, Kubatkin S, Yakimova R, M, Mauri F, Piscanec S, Jiang D, Novoselov K S, Roth Janssen T J B M, Cedergren K, Bergsten T and Falko S and Geim A K 2006 Phys. Rev. Lett. 97 187401 V 2011 Phys. Rev. Lett. 107 166602 [259] Ferrari A C 2007 Solid State Commun. 143 47 [220] McCann E 2009 Physics 2 98 [260] Malard L M, Pimenta M A, Dresselhaus G and Dressel- [221] Imura K-I, Kuramoto Y and Nomura K 2009 Phys. Rev. haus M S 2009 Phys. Rep. 473 51 B 80 085119 [261] Grüneis A, Saito R, Kimura T, Can¸cado L G, Pimenta [222] Imura K-I, Kuramoto Y and Nomura K 2010 Euro. M A, Jorio A, Souza Filho A G, Dresselhaus G and Phys. Lett. 89 17009 Dresselhaus M S 2002 Phys. Rev. B 65 155405 [223] McCann E and Fal’ko V I 2012 Phys. Rev. Lett. 108 [262] Dubay O and Kresse G 2003 Phys. Rev. B 67 035401 166606 [263] Maultzsch J, Reich S, Thomsen C, Requardt H and Or- [224] Kechedzhi K, Fal’ko V I, McCann E and Altshuler B L dejón P 2004 Phys. Rev. Lett. 92 075501 2007 Phys. Rev. Lett. 98 176806 [264] Wirtz L and Rubio A 2004 Solid State Commun. 131 [225] Koshino M 2008 Phys. Rev. B 78, 155411 141 30

[265] Mounet N and Marzari N 2005 Phys. Rev. B 71 205214 2009 EPL 85 58005 [266] Kohn W 1959 Phys. Rev. Lett. 2 393 [297] Borghi G, Polini M, Asgari R and MacDonald A H 2009 [267] Piscanec S, Lazzeri M, Mauri F, Ferrari A C and Phys. Rev. B 80 241402 Robertson J 2004 Phys. Rev. Lett. 93 185503 [298] Borghi G, Polini M, Asgari R and MacDonald A H 2010 [268] Lazzeri M, Piscanec S, Mauri F, Ferrari A C and Phys. Rev. B 82 155403 Robertson J 2006 Phys. Rev. B 73 155426 [299] Sensarma R, Hwang E H and Das Sarma S 2010 Phys. [269] Ando T 2006 J. Phys. Soc. Jpn. 75 124701 Rev. B 82 195428 [270] Castro Neto A H and Guinea F 2007 Phys. Rev. B 75 [300] Gamayun O V 2011 Phys. Rev. B 84 085112 045404 [301] Castro E V, Peres N M R, Stauber T and Silva N A P [271] Pisana S, Lazzeri M, Casiraghi C, Novoselov K S, Geim 2008 Phys. Rev. Lett. 100 186803 A K, Ferrari A C and Mauri F 2007 Nature Mater. 6 [302] Dillenschneider R and Han J H 2008 Phys. Rev. B 78 198 045401 [272] Yan J, Zhang Y, Kim P and Pinczuk A 2007 Phys. Rev. [303] Sahu B, Min H, MacDonald A H and Banerjee S K 2008 Lett. 98 166802 Phys. Rev. B 78 045404 [273] Yan J, Henriksen E A, Kim P and Pinczuk A 2008 Phys. [304] Park C-H and Louie S G 2010 Nano Lett. 10 426 Rev. Lett. 101, 136804 [305] Wang X-F and Chakraborty T 2010 Phys. Rev. B 81 [274] Malard L M, Elias D C, Alves E S and Pimenta M A 081402 2008 Phys. Rev. Lett. 101 257401 [306] Cortijo A, Oroszlany L and Schomerus H 2010 Phys. [275] Yan J, Villarson T, Henriksen E A, Kim P and Pinczuk Rev. B 81 235422 A 2009 Phys. Rev. B 80 241417 [307] Toke C and Fal’ko V I 2011 Phys. Rev. B 83 115455 [276] Bruna M and Borini S 2010 Phys. Rev. B 81 125421 [308] Nilsson J, Castro Neto A H, Peres N M R and Guinea [277] Garcia J M, He R, Jiang M P, Yan J, Pinczuk A, Zuev Y F 2006 Phys. Rev. B 73 214418 M, Kim K S, Kim P, Baldwin K, West K W and Pfeiffer [309] Vafek O 2010 Phys. Rev. B 82 205106 L N 2010 Solid State Commun. 150 809 [310] Zhang F, Min H, Polini M and MacDonald A H 2010 [278] Tan P H, Han W P, Zhao W J, Wu Z H, Chang K, Wang Phys. Rev. B 81 041402(R) H, Wang Y F, Bonini N, Marzari N, Pugno N, Savini [311] Jung J, Zhang F and MacDonald A H 2011 Phys. Rev. G, Lombardo A and Ferrari A C 2012 Nature Mater. 11 B 83 115408 294 [312] Kharitonov M 2012 Phys. Rev. B 86 195435 [279] Lui C H, Malard L M, Kim S, Lantz G, Laverge F E, [313] Nandkishore R and Levitov L 2010 Phys. Rev. Lett. 104 Saito R and Heinz T F 2012 Nano Lett. 12 5539 156803 [280] Kuzmenko A B, Benfatto L, Cappelluti E, Crassee I, [314] Dahal H P, Wehling T O, Bedell K S, Zhu J-X and van der Marel D, Blake P, Novoselov K S and Geim A Balatsky A V 2010 Physica B 405 2241 K 2009 Phys. Rev. Lett. 103 116804 [315] Nandkishore R and Levitov L arXiv:1002.1966 (unpub- [281] Tang T-T, Zhang Y, Park C-H, Geng B, Girit C, Hao Z, lished). Martin M C, Zettl A, Crommie M F, Louie S G, Shen [316] Nandkishore R and Levitov L 2010 Phys. Rev. B 82 Y R and Wang F 2010 Nature Nano. 5 32 115124 [282] Yan J-A, Ruan W Y and Chou M Y 2008 Phys. Rev. B [317] Sun K, Yao H, Fradkin E and Kivelson S A 2009 Phys. 77 125401 Rev. Lett. 103 046811 [283] Saha S K, Waghmare U V, Krishnamurthy H R and [318] Weitz R T, Allen M T, Feldman B E, Martin J and Sood A K 2008 Phys. Rev. B 78 165421 Yacoby A 2010 Science 330 812 [284] Jiang J-W, Tang H, Wang B-S and Su Z-B 2008 Phys. [319] Martin J, Feldman B E, Weitz R T, Allen M T and Rev. B 77 235421 Yacoby A 2010 Phys. Rev. Lett. 105 256806 [285] Michel K H and Verberck B 2008 Phys. Rev. B 78 [320] Mayorov A S, Elias D C, Mucha-Kruczynski M, Gor- 085424 bachev R V, Tudorovskiy T, Zhukov A, Morozov S V, [286] Kitipornchai S, He X Q and Liew K M 2005 Phys. Rev. Katsnelson M I, Falko V I, Geim A K and Novoselov K B 72 075443 S 2011 Science 333 860 [287] Ando T 2007 J. Phys. Soc. Jpn. 76 104711 [321] Freitag F, Trbovic J, Weiss M and Schönenberger C [288] Ando T and Koshino M 2009 J. Phys. Soc. Jpn. 78 2012 Phys. Rev. Lett. 108 076602 034709 [322] Velasco Jr. J, Jing L, Bao W, Lee Y, Kratz P, Aji V, [289] Gava P, Lazzeri M, Saitta A M and Mauri F 2009 Phys. Bockrath M, Lau C N, Varma C, Stillwell R, Smirnov Rev. B 80 155422 D, Zhang F, Jung J and MacDonald A H 2012 Nature [290] Mucha-Kruczyński M, Kashuba O and Fal’ko V I 2010 Nanotech. 7 156 Phys. Rev. B 82 045405 [323] Bao W, Velasco Jr. J, Zhang F, Jing L, Standley B, [291] Ishikawa K and Ando T 2006 J. Phys. Soc. Jpn. 75 Smirnov D, Bockrath M, MacDonald A H and Lau C N 084713 2012 Proc. Nat. Acad. Sci. 109 10802 [292] Wang X F and Chakraborty T 2007 Phys. Rev. B 75 [324] Veligura A, van Elferen H J, Tombros N, Maan J C, 041404(R) Zeitler U and van Wees B J 2012 Phys. Rev. B 85 [293] Stauber T, Peres N M R, Guinea F and Castro Neto A 155412 H 2007 Phys. Rev. B 75 115425 [325] Zhang F, Min H and MacDonald A H 2012 Phys. Rev. [294] Kusminskiy S V, Nilsson J, Campbell D K and Castro B 86 155128 Neto A H 2008 Phys. Rev. Lett. 100 106805 [326] Nomura K and MacDonald A H 2006 Phys. Rev. Lett. [295] Hwang E H and Das Sarma S 2008 Phys. Rev. Lett. 101 96 256602 156802 [327] Barlas Y, Cote R, Nomura K and MacDonald A H 2008 [296] Kusminskiy S V, Campbell D K and Castro Neto A H Phys. Rev. Lett. 101 097601 31

[328] Shizuya K 2009 Phys. Rev. B 79 165402 lasco Jr J, Smirnov D and Lau C N 2010 Phys. Rev. [329] Abanin D A, Parameswaran S A and Sondhi S L 2009 Lett. 105 246601 Phys. Rev. Lett. 103 076802 [339] Shibata N and Nomura K 2009 J. Phys. Soc. Jpn. 78 [330] Gorbar E V, Gusynin V P and Miransky V A 2010 Phys. 104708 Rev. B 81 155451 [340] Apalkov V M and Chakraborty T 2010 Phys. Rev. Lett. [331] Gorbar E V, Gusynin V P, Jia J and Miransky V A 105 036801 2011 Phys. Rev. B 84 235449 [341] Apalkov V M and Chakraborty T 2011 Phys. Rev. Lett. [332] van Elferen H J, Veligura A, Kurganova E V, Zeitler U, 107 186803 Maan J C, Tombros N, Vera-Marun I J and van Wees [342] Papic Z, Abanin D A, Barlas Y and Bhatt R N 2011 B J 2012 Phys. Rev. B 85 115408 Phys. Rev. B 84 241306(R) [333] Zhao Y, Cadden-Zimansky P, Jiang Z and Kim P 2010 [343] Snizhko K, Cheianov V and Simon S H 2012 Phys. Rev. Phys. Rev. Lett. 104 066801 B 85 201415(R) [334] Kim S, Lee K and Tutuc E 2011 Phys. Rev. Lett. 107 [344] Avouris P 2010 Nano Lett. 10 4285 016803 [345] Andrei E Y, Li G and Du X 2012 Rep. Prog. Phys. 75 [335] Du X, Shachko I, Duerr F, Luican A and Andrei E Y 056501 2009 Nature 462 192 [346] Yazyev O V 2010 Rep. Prog. Phys. 73 056501 [336] Bolotin K I, Ghahari F, Shulman M D, Stormer H L [347] Pesin D and MacDonald A H 2012 Nature Mat. 11 409 and Kim P 2009 Nature 462 196 [348] Beenakker C W J 2008 Rev. Mod. Phys. 80 1337 [337] Dean C R, Young A F, Cadden-Zimansky P, Wang L, [349] Abergel D S L, Apalkov V, Berashevich J, Ziegler K and Ren H, Watanabe K, Taniguchi T, Kim P, Hone J and Chakraborty T 2010 Adv. Phys. 59 261 Shepard K L 2011 Nature Phys. 7 693 [338] Bao W, Zhao Z, Zhang H, Liu G, Kratz P, Jing L, Ve-