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 16 zone 2 A. Introduction 16 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]; me- chanical 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 δ det (H Ej S)=0 , (7) H eik. l φ (r R ) φ (r R δ (9)) , − AB ≈ N A − A,i |H| B − A,i − l i=1 where ‘det’ stands for the determinant of the matrix. l=1 In order to model a given system in terms of the gener- where δl are the positions of three nearest B atoms rel- alised eigenvalue problem (5), it is necessary to determine ative to a given A atom, which may be written as δ1 = the matrices H and S. We will proceed by considering 0,a/√3 , δ2 = a/2, a/2√3 , δ3 = a/2, a/2√3 . the relatively simple case of monolayer graphene, before The sum with respect− to the three nearest-neighbour− − generalising the approach to bilayers. In the following B sites is identical for every A site. A hopping parameter sections, we will omit the subscript j on ψj and Ej in may be defined as equation (5), remembering that the number of solutions δ is M, the number of orbitals per unit cell. γ0 = φA (r RA,i) φB (r RA,i l) , (10) − − |H| − − which is positive. Then, the matrix element may be writ- 2. Monolayer graphene ten as 3 ik.δl Here, we will outline how to apply the tight-binding HAB γ0f (k); f (k)= e , (11) ≈− model to graphene, and refer the reader to tutorial-style l=1 reviews [53, 60] for further details. We take into account The other off-diagonal matrix element is given by HBA = one 2pz orbital per atomic site and, as there are two HAB∗ γ0f ∗ (k). The function f (k) describing atoms in the unit cell of monolayer graphene, figure 1(a), nearest-neighbor≈ − hopping, equation (11), is given by we include two orbitals per unit cell labelled as m = A and m = B (the A atoms and the B atoms are each iky a/√3 iky a/2√3 f (k)= e +2e− cos(kxa/2) , (12) arranged on an hexagonal Bravais lattice). 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 Hm and orbital interacting with itself (i.e., in the same unit cell), overlap Sm integral matrices of monolayer graphene may the matrix element may be written as be written as

N ǫA γ0f (k) 1 Hm = − , (13) H φ (r R ) φ (r R ) . (8) γ0f ∗ (k) ǫB AA ≈ N A − A,i |H| A − A,i − i=1 1 s0f (k) Sm = . (14) s0f ∗ (k) 1 This may be regarded as a summation over all unit cells of a parameter ǫ = φ (r R ) φ (r R ) that A A A,i A A,i The corresponding energy may be determined [53] by takes the same value in every− unit|H| cell. Thus,− the matrix solving the secular equation (7). For intrinsic graphene, element may be simply expressed as HAA ǫA. Simi- ≈ i.e., ǫA = ǫB = 0, we have larly, the diagonal element HBB for the B site orbital can be written as HBB = ǫB, while for intrinsic graphene ǫA γ f (k) E = 0 . (15) is equal to ǫB as the two sublattices are identical. The ± | | ± 1 s0 f (k) calculation of the diagonal elements of the overlap inte- ∓ | | 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, equa- tion (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, equa- tion (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 m = γ1/2v (see inset in figure 3). This crossover oc- curs at p γ1/2v. A convenient way to describe the bilayer at≈ low energy and momentum p γ /2v is to To describe the properties of electrons in the vicinity of ≪ 1 the K points, a momentum p = ~k ~K is introduced eliminate the components in the Hamiltonian (30) re- − ξ which is measured from the centre of the Kξ point. Ex- lated to the orbitals on dimer sites B1, A2, resulting in panding in powers of p, the function f(k), equation (12), an effective two-component Hamiltonian describing the orbitals on the non-dimer sites A1, B2, and, thus, the is approximately given by f(k) √3a(ξpx ipy)/2~ ≈ − − two bands that approach each other at zero energy. This which is valid close to the Kξ point, i.e., for pa/~ 1, ≪ is described in the next section, and the solutions of this where p = p = (p2 + p2)1/2. In monolayer graphene, x y Hamiltonian are shown to be massive chiral quasiparti- the Hamiltonian| | matrix (13) is simplified by keeping only cles [8, 9], as opposed to massless chiral ones in monolayer linear terms in momentum p as graphene.

ǫA vπ† Hm = , (29) vπ ǫ B C. Effective two-band Hamiltonian at low energy where π = ξpx + ipy, π† = ξpx ipy, and v = √3aγ0/2~ − In this section we focus on the low-energy electronic is the band velocity. In the intrinsic case, ǫA = ǫB = 0, band structure in the vicinity of the points K+ and K the eigen energy becomes E = v p , which approxi- − mates Eq. (15). In bilayer graphene,± | similarly,| Eq. (16) at the corners of the first Brillouin zone, relevant for en- is reduced to ergies near the Fermi level. A simple model may be ob- tained by eliminating orbitals related to the dimer sites, ǫA1 vπ† v4π† v3π resulting in an effective Hamiltonian for the low-energy − orbitals. First, we outline the procedure in general terms,  vπ ǫB1 γ1 v4π†  Hb = − , (30) because it may be applied to systems other than bilayer v4π γ1 ǫA2 vπ†  −  graphene such as ABC-stacked (rhombohedral) graphene  v π† v π vπ ǫ   3 − 4 B2  multilayers [84, 85], before applying it specifically to bi-   where we introduced the effective velocities, v3 = layer graphene. √3aγ3/2~ and v4 = √3aγ4/2~. 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† pro- portional to γ4, and by considering only interlayer asym- We consider the energy eigenvalue equation, and con- metry U in the on-site energies: ǫA1 = ǫB1 = U/2 and sider separate blocks in the Hamiltonian corresponding to − T T ǫA2 = ǫB2 = U/2. Then, there is electron-hole symme- low-energy θ = (ψA1, ψB2) and dimer χ = (ψA2, ψB1) try, i.e., energies may be written E = εα(p), α =1, 2, components: [9] with ±

2 2 2 hθ u θ θ γ1 U v3 α = E , (33) ε2 = + + v2 + p2 + ( 1) √Γ , (31) u h χ χ α 2 4 2 − † χ 1 2 2 2 2 2 2 2 2 2 2 2 3 Γ= 4 γ1 v3p + v p γ1 + U + v3p + 2ξγ1v3v p cos3Theϕ second , row of (33) allows the dimer components to − be expressed in terms of the low-energy ones: where ϕ is the polar angle of momentum p= (px,py)= 1 p (cos ϕ, sin ϕ). Energy ε2 describes the higher-energy χ = (E hχ)− u†θ , (34) bands split from zero energy by the interlayer coupling − γ1 between the orbitals on the dimer sites B1, A2. Substituting this into the first row of (33) gives an effec- Low-energy bands E = ε1 are related to orbitals on tive eigenvalue equation written solely for the low-energy the non-dimer sites A1, B±2. In an intermediate energy components: 2 range U, (v3/v) γ1 < ε1 < γ1 it is possible to neglect 1 the interlayer asymmetry U and terms proportional to hθ + u (E hχ)− u† θ = Eθ , − γ3 (i.e., set U = v3 = 0), and the low-energy bands may 1 h uh− u† θ E θ , be approximated [9] as θ − χ ≈ S 2 where =1+ uhχ− u†. The second equation is accurate 1 2 2 2 S ε1 γ1 1+4v p /γ 1 , (32) up to linear terms in E. Finally, we perform a transfor- ≈ 2 1 − mation Φ = 1/2θ: S which interpolates between an approximately linear dis- 1 1/2 1/2 hθ uhχ− u† − Φ E Φ , persion ε1 vp at large momentum to a quadratic one − S ≈ S 2 ≈ 1/2 1 1/2 ε1 p /2m at small momentum, where the mass is − hθ uhχ− u† − Φ EΦ . (35) ≈ S − S ≈ 8

This transformation ensures that normalisation of Φ is describes massless 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 val- ley 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 hˆU and hˆAB, 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 mo- mentum, 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∆ + hˆU + hˆAB, (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- hˆAB = , 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 in- terlayer 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 magni- 1. The Landau level spectrum of bilayer graphene tude 2λSO in the spectrum of bilayer graphene, but the two low-energy bands remain spin and valley degenerate (as in a monolayer): E = λ2 + v4p4/γ2. However, We consider the Landau level spectrum of the two- ± SO 1 there are gapless edge excitations and, like monolayer component chiral Hamiltonian hˆ0, 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 , − | | Rashba term hˆR, 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∂x + ∂y ieBx ~ ~ − − a substrate, say [99–101, 105, 110–116]. Taken with the and π† = i ξ∂x ∂y + ieBx, and eigenstates are com- prised of− functions− that are harmonic oscillator states quadratic term hˆ0 in the Hamiltonian (38), hˆR does not produce a gap, but, as in the monolayer, spin-splitting in the x direction and plane waves in the y direction [119, 120], of magnitude 2λR between the bands. That is, there are 2 four valley-degenerate bands at low energy, x pyλB 1 x pyλB pyy φℓ (x, y)= Aℓ ℓ ~ exp ~ + i ~(49) 2 H λB − −2 λB − v4p4 E2 = λ2 1+ 1 . (48) R 2 2 where the magnetic length is λB = ~/eB, ℓ are λRγ1 ± Hermite polynomials of order ℓ for integer ℓ H0, and ℓ ≥ Generally speaking, there is a rich interplay between Aℓ =1/ 2 ℓ!√π is a normalisation constant. The operators π and π appearing in the Hamilto- tuneable interlayer asymmetry U and the influence of the † intrinsic and the Bychkov-Rashba spin-orbit coupling in nian (38) act as raising and lowering operators for the bilayer graphene [105, 110, 111, 115, 116]. For example, harmonic oscillator states (49). At the first valley, K+, the presence of interlayer asymmetry U breaks inversion √ ~ 2i √ symmetry and allows for spin-split levels in the presence K+ : πφℓ = ℓ φℓ 1 , (50) − λB − of intrinsic spin-orbit coupling only (λR = 0) [105], while √2i~ the combination of finite U and very large Rashba cou- K+ : π†φℓ = √ℓ +1 φℓ+1 , (51) pling has been predicted to lead to a topological insulator λB state even with λSO = 0 [115]. and πφ0 = 0. Then, it is possible to show that the Lan- dau level spectrum of the Hamiltonian (38) consists of a series of electron and hole levels with energies and wave J. The integer quantum Hall effect functions [9] given by

Eℓ, = ~ωc ℓ(ℓ 1) , ℓ 2, (52) When a two-dimensional electron gas is placed in a ± ± − ≥ 1 φℓ perpendicular magnetic field, electrons follow cyclotron K+ : ψℓ, = , ℓ 2, (53) orbits and their energies are quantised as Landau levels ± √2 φℓ 2 ≥ ± − [117]. At a high enough magnetic field strength, the dis- where ωc = eB/m and refer to the electron and hole crete nature of the Landau level spectrum is manifest as states, respectively. For± high values of the index, ℓ 1, the integer quantum Hall effect [118–120], whereby the ≫ the levels are approximately equidistant with spacing ~ωc Hall conductivity assumes values that are integer multi- 2 proportional to the magnetic field strength B. However, ples of the quantum of conductivity e /h. this spectrum, equation (52), is only valid for sufficiently The Landau level spectrum of monolayer graphene was small level index and magnetic field ℓ~ωc γ1 because calculated by McClure [121] nearly sixty years ago, and the effective Hamiltonian (38) is only applicable≪ at low there have been a number of related theoretical studies energy. [5, 6, 98, 122–124] considering the consequences of chi- In addition to the field-dependent levels, there are two rality in graphene. The experimental observation of the levels fixed at zero energy E1 = E0 = 0 with eigenfunc- integer quantum Hall effect in monolayer graphene [2, 3] tions: found an unusual sequencing of the quantised plateaus of Hall conductivity, confirming the chiral nature of the φ1 φ0 K+ : ψ1 = , ψ0 = , (54) electrons and prompting an explosion of interest in the 0 0 12

2 2 They may be viewed as arising from the square of the ( a ) s x y ( 4 e / h ) ( b ) s x y ( 4 e / h ) lowering operator in the Hamiltonian (38) which acts on 4 4 both the oscillator ground state and the first excited state 3 3 2 2 to give zero energy π φ0 = π φ1 = 0. The eigenfunctions 2 2 ψ0 and ψ1 have a finite amplitude on the A1 sublattice, 1 1 on the bottom layer, but zero amplitude on the B2 sub- - 4 - 2-3 - 1 1 2 3 4 - 4 - 2-3 - 1 1 2 3 4 lattice. - 1 n ( 4 B / j 0 ) - 1 n ( 4 B / j 0 ) At the second valley, K , the roles of operators π and - 2 - 2 − π† are reversed: - 3 - 3 √2i~ - 4 - 4 K : πφℓ = √ℓ +1 φℓ+1 , (55) − − λB FIG. 5. Schematic of the dependence of the Hall conductiv- √2i~ ity σxy on carrier density n for (a) monolayer graphene and K : π†φℓ = √ℓ φℓ 1 , (56) − λB − (b) bilayer graphene, where ϕ0 = h/e is the flux quantum and B is the magnetic field strength. The dashed line in (b) and π†φ0 = 0. The Landau level spectrum at K is shows the behaviour for a conventional semiconductor or bi- − layer graphene in the presence of large interlayer asymmetry degenerate with that at K+, i.e., Eℓ, = ~ωc ℓ(ℓ 1) for ℓ 2 and E = E = 0, but the± roles± of the A1− and U (section II J 3) with fourfold level degeneracy due to spin 1 0 and valley degrees of freedom. B2 sublattices≥ are reversed:

1 φℓ 2 K : ψℓ, = − , ℓ 2, (57) − ± √2 φ ≥ ± ℓ 0 0 The corresponding Hall conductivity is shown schemati- K : ψ1 = , ψ0 = , (58) 2 − φ1 φ0 cally in figure 5(a). There are steps of height 4e /h be- tween each plateau, as expected by consideration of the The valley structure and electronic spin each contribute a conventional case, but the plateaus occur at half-integer twofold degeneracy to the Landau level spectrum. Thus, values of 4e2/h instead of integer ones, as observed ex- each level in bilayer level graphene is fourfold degenerate, perimentally [2, 3]. This is due to the existence of the except for the zero energy levels which have eightfold fourfold-degenerate level E0 = 0 at zero energy, which degeneracy due to valley, spin and the orbital degeneracy contributes to a step of height 4e2/h at zero density. of ψ0, ψ1. For bilayer graphene, plateaus in the Hall conductiv- 2 ity σxy(n), Fig. 5(b), occur at integer multiples of 4e /h. 2. Three types of integer quantum Hall effect This is similar to a conventional semiconductor with level degeneracy g = 4 arising from the spin and valley de- The form of the Landau level spectrum is manifest in grees of freedom. Deviation from the conventional case a measurement of the integer quantum Hall effect. Here, occurs at low density. In the bilayer there is a step in 2 we will compare the density dependence of the Hall con- σxy of height 8e /h across zero density, accompanied by ductivity σxy(n) for bilayer graphene with that of a con- a plateau separation of 8B/ϕ0 in density [8, 9], arising ventional semiconductor and of monolayer graphene. from the eightfold degeneracy of the zero-energy Lan- The Landau level spectrum of a conventional two- dau levels. This is shown as the solid line in Fig. 5(b), dimensional semiconductor is E = ~ω (ℓ +1/2), ℓ 0, whereas, for a conventional semiconductor, there no step ℓ c ≥ where ωc = eB/m is the cyclotron frequency [119, 120]. across zero density (the dashed line). As density is changed, there is a step in σxy whenever a Landau level is crossed, and the separation of steps on Thus, the chirality of charge carriers in monolayer and the density axis is equal to the maximum carrier den- bilayer graphene give rise to four- and eight-fold degen- sity per Landau level, gB/ϕ0, where ϕ0 = h/e is the erate Landau levels at zero energy and to steps of height flux quantum and g is a degeneracy factor. Each plateau of four and eight times the conductance quantum e2/h in of the Hall conductivity σxy occurs at a quantised value the Hall conductivity at zero density [2, 3, 8]. Here, we of Nge2/h where N is an integer labelling the plateau have assumed that the degeneracy of the Landau levels and g is an integer describing the level degeneracy; steps is preserved, i.e., any splitting of the levels is negligible between adjacent plateaus have height ge2/h. as compared to temperature and level broadening in an The Landau level spectrum of monolayer graphene experiment. The role of electronic interactions in bilayer [98, 121–124] consists of an electron and a hole series of graphene is described in section VIII, while we discuss levels, Eℓ, = √2ℓ~v/λB for ℓ 1, with an additional the influence of interlayer asymmetry on the Landau level ± ± ≥ level at zero energy E0 = 0. All of the levels are fourfold spectrum and integer quantum Hall effect in the next sec- degenerate, due to spin and valley degrees of freedom. tion. 13

3. The role of interlayer asymmetry σb0 σ1 σ2 σt0 V ε ε ε V The Landau level states, equations (53,57), have dif- b b r t t ferent amplitudes on the lower (A1 sublattice) and up- per (B2 sublattice) layers, with the role of the sublattice sites swapped at the two valleys. Thus, interlayer asym- x=-Lb -c 0 c0 x=Lt 2 2 metry U as described by the effective Hamiltonian hˆU , equation (38), leads to a weak splitting of the valley de- generacy of the levels [9]: FIG. 6. Schematic of bilayer graphene in the presence of ex- ternal gates located at x = −Lb and x = +Lt, with po- ξU~ω tentials Vb and Vt, which are separated from the bilayer by E ~ω ℓ(ℓ 1)+ c . (59) ℓ, c media of dielectric constants εb and εt, respectively. The bi- ± ≈± − 2γ1 layer is modelled as two parallel conducting plates positioned Such splitting is prominent for the zero-energy states [9], at x = −c0/2 and +c0/2, separated by a region of dielectric equations (54,58), because they only have non-zero am- constant εr. The layers have change densities σ1 = −en1 and plitude on one of the layers, depending on the valley: σ2 = −en2 corresponding to layer number densities n1 and n2. Charge densities σb0 and σt0 (dashed lines) arise from the 1 presence of additional charged impurities. E = ξU , (60) 0 −2 1 ξU~ωc E1 = ξU + . (61) [140], the presence of disorder [141–143] or disorder and −2 γ1 chiral charge carriers [144]. Broadly speaking, transport When the asymmetry is large enough, then the splitting seems to occur through different mechanisms in different U of the zero energy levels from each valley results in temperature regimes with thermal activation [31, 134– a sequence of quantum Hall plateaus at all integer val- 136, 138, 139] at high temperature (above, roughly, 2 to ues of 4e2/h including a plateau at zero density [17], as 50K) and variable-range [19, 135–137, 139] or nearest- observed experimentally [20]. This behaviour is shown neighbour hopping [135, 138] at low temperature. schematically as the dashed line in figure 5(b). The Lan- dau level spectrum in the presence of large interlayer asymmetry U has been calculated [125–128], including B. Hartree model of screening an analysis of level crossings [128] and a self-consistent calculation of the spectrum in the presence of an exter- External gates are generally used to control the den- nal gate [126, 127], generalising the zero-field procedure sity of electrons n on a graphene device [1], but, in bi- outlined in the next section. layer graphene, external gates will also place the sepa- rate layers of the bilayer at different potential energies resulting in interlayer asymmetry U = ǫ2 ǫ1 (where III. TUNEABLE BAND GAP − ǫ1 = ǫA1 = ǫB1 = U/2 and ǫ2 = ǫA2 = ǫB2 = U/2). Thus, changing the− applied gate voltage(s) will tend to A. Experiments tune both the density n and the interlayer asymmetry U, and, ultimately, the band gap Ug. The dependence of A tuneable band gap in bilayer graphene was first ob- the band gap on the density Ug(n) relies on screening by served with angle-resolved photoemission of epitaxial bi- electrons on the bilayer. In the following, we describe a layer graphene on silicon carbide [16], and the ability to simple model [17, 18, 20, 55, 69, 71, 145, 146] that has control the gap was demonstrated by doping with potas- been developed to take into account screening using the sium. Since then, the majority of experiments probing tight-binding model and Hartree theory. the band gap have used single or dual gate devices based on exfoliated bilayer graphene flakes [19, 20]. The band gap has now been observed in a number of different ex- 1. Electrostatics: asymmetry parameter, layer densities and periments including photoemission [16], magnetotrans- external gates port [20], infrared spectroscopy [55, 78–80, 129, 130], electronic compressibility [131, 132], scanning tunnelling We use the SI system of units, and the electronic charge spectroscopy [133], and transport [19, 31, 134–139]. is e where e> 0. The bilayer graphene device is mod- The gap observed in optics [16, 55, 78–80, 129, 130] is elled− as two parallel conducting plates that are very nar- up to 250 meV [129, 130], the value expected theoretically row in the x-direction, continuous and infinite in the y- (as the gap should saturate at the value of the interlayer z plane, positioned at x = c /2 and +c /2, figure 6. − 0 0 coupling γ1). Transport measurements show insulating Here, c0 is the interlayer spacing and we denote the di- behaviour [19, 31, 134–139], but, generally, not the huge electric constant of the interlayer space as εr (it doesn’t suppression of conductivity expected for a gap of this include the screening by π-band electrons that we are ex- magnitude, and this has been attributed to edge states plicitly modelling). Layer number densities are n1 and 14 n , with corresponding change densities σ = en and integrating over the whole Brillouin zone. The simplified 2 1 − 1 σ2 = en2. two-component Hamiltonian is We− take into account the presence of a back gate and a 2 2 top gate, infinite in the y-z plane, located at x = Lb and ˆ v 0 π† U 1 0 − H2 2 , (67) x =+Lt, with potentials Vb and Vt, which are separated ≈−γ1 π 0 − 2 0 1 from the bilayer by media of dielectric constants εb and − εt, respectively. It is possible to describe the presence Solutions to the energy eigenvalue equation Hˆ2ψ = Eψ of additional charge near the bilayer - due to impurities, are given by say - by introducing density nb0 on the back-gate side and n on the top-gate side. They correspond to charge t0 U 2 v4p4 densities σb0 = enb0 and σt0 = ent0, assuming that nb0 E = + , (68) 4 γ2 and nt0 are positive for positive charge. ± 1 Using Gauss’s Law, it is possible to relate the external E U/2 1 ip.r/~ gate potentials and impurities concentrations to the layer ψ = − 2 2 e . (69) 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) eLb eLt circular Fermi surface 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/(ec0). Equation (62) expresses degeneracy. the total electron density n = n1 + n2 in terms of the ex- For simplicity, we assume the Fermi energy lies within ternal potentials, generalising the relation for monolayer 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 1 E U/2 (n n ) ncb = pdp ∓ , (71) U = U +Λγ 2 − 1 , (64) 1(2) π~2 E ext 1 n ⊥ 1 pF U pF pdp ec0 εb εt (nb0 nt0) = pdp , ~2 ~2 2 4 4 2 Uext = Vb Vt +Λγ1 − , (65) π 0 ∓ 2π 0 U /4+ v p /γ 2εr Lb − Lt n 1 ⊥ 2 where the characteristic density scale n and the dimen- n n U 2 n γ1 2nγ1 ⊥ ⊥ ln | | + 1+ , sionless screening parameter Λ are ≈ 2 ∓ 4γ1 n U n U  ⊥| | ⊥ 2 2 2   γ1 c0e γ1 c0e n ⊥ where the minus (plus) sign is for the first (second) layer, n = 2 2 , Λ= 2 2 . (66) ⊥ π~ v 2π~ v ε0εr ≡ 2γ1ε0εr pF is the Fermi momentum, and the total density is n = 2 ~2 Equation (64) relates the asymmetry parameter U to a pF /π measured with respect to the charge neutrality point, i.e., we assume that the point of zero density is sum of its value, Uext, if screening were negligible plus realised when the Fermi level lies at the crossing point of a term accounting for screening. Parameter values γ1 = 6 1 13 2 the conduction and valence bands. 0.39eV and v =1.0 10 ms− give n =1.1 10 cm− . × ⊥ × In addition, we take into account the contribution of With interlayer spacing c0 = 3.35A˚ and an estimate for the filled valence band nvb to the individual layer den- the dielectric constant of εr 1, then Λ 1, showing 1(2) that screening is relevant. ≈ ∼ sities. Note that, as the asymmetry parameter U varies, the filled valence band does not contribute to any change in the total density n, but it does contribute to the dif- 2. Calculation of individual layer densities ference n1 n2. This may be obtained by integrating with respect− to momentum as in equation (71), but in- troducing an ultraviolet cutoff p = γ /v equivalent Through electrostatics, the asymmetry parameter U max 1 to n = n [18]. Then, the contribution of the filled is related to layer densities n and n , equation (64). max 1 2 valence band⊥ [17, 18, 69, 71, 145, 146] is given by The densities n1 and n2 also depend on U because of the band structure of bilayer graphene. Analytical cal- U pmax pdp culations are possible [17, 18, 145] if only the dominant nvb , 1(2) ~2 2 4 4 2 ≈±2π 0 U /4+ v p /γ inter-layer coupling γ1 is taken into account in the four- 1 band Hamiltonian (30). Here we will use the two-band n U 4γ1 ⊥ ln . (72) model (38) with an explicit ultraviolet cutoff [18] when ≈± 4γ U 1 | | 15

Combining the contributions of equations (71) and (72), 1.0 Ug/γ1 cb vb the individual layer density, n1(2) = n1(2) +n1(2), is given [17, 145] by 0.8 n0/n 0.0 2 2 n n U n 1 n U 0.2 n1(2) ⊥ ln | | + + (73). 0.6 ≈ 2 ∓ 4γ1 2n 2 n 2γ1  0.4 ⊥ ⊥   0.4 Note that some calculations [148] include only the contri- 0.6 bution (71) of the partially-filled conduction band, others [18] include only the filled valence band (72). 0.2

3. Self-consistent screening -1.0-0.5 0 0.5 1.0 n/n Substituting the layer density (73) into the expres- FIG. 7. Density-dependence of the band gap Ug(n) in bi- sion (64) describing screening gives an expression [17, layer graphene as the back-gate voltage Vb is varied for a 60, 145] for the density-dependence of the asymmetry fixed top-gate voltage Vt [17]. The effective offset-density is parameter U: n0 = 2[ε0εtVt/(eLt)+ nt0]. Plots were made for screening 2 2 1 parameter Λ = 1, using Ug = |U|γ1/pU + γ1 and numerical − U(n) Λ n 1 n 2 U 2 solution of equation (74). 1 ln | | + + (74) Uext ≈ − 2 2n 2 n 2γ1  ⊥ ⊥ (n 1-n 2)/n    where Uext is the asymmetry in the absence of screen- 0.4 ing (65). The extent of screening is described by the logarithmic term with argument depending on n and 0.2 U, and a prefactor proportional to the screening pa- rameter Λ 1 (as discussed earlier). A common ex- perimental∼ setup, especially for exfoliated graphene on -1.0 -0.50.5 1.0 a silicon substrate [1–3, 8], includes a single back gate. n/n Figure 7 shows the density-dependence of the band gap -0.2 Ug(n) plotted as the back-gate voltage Vb is varied for a fixed top-gate voltage Vt. In this case, the influence of -0.4 the top-gate voltage Vt may be absorbed into an effec- tive offset-density n0 = 2[ε0εtVt/(eLt)+ nt0] [17] giving Uext = Λγ1(n n0)/n in equation 74. Figure 8 shows − ⊥ FIG. 8. Density-dependence of the difference in layer densities the dependence of the difference in layer densities n1 n2 − n1 − n2 in bilayer graphene as the back-gate voltage Vb is for the case n0 = 0 including both the contribution of varied for a fixed top-gate voltage Vt [17]. Plots were made the partially filled bands as measured with respect to for screening parameter Λ = 1 and the effective offset-density the charge neutrality point (71) (dashed line) and the is n0 = 2[ε0εtVt/(eLt)+ nt0] = 0, corresponding to the data contribution of the full valence band (72) (dotted line). labelled n0 = 0 in Figure 7, obtained by numerical solution The sum of both terms (solid line) shows that n1 n2 of equation (74). The dashed line shows the contribution − is positive (negative) for positive (negative) total density of the partially filled bands as measured with respect to the n. Recalling that layer 1 is closest to the back gate, this charge neutrality point (71) [148], the dotted line shows the shows that the bilayer is polarised along the electric field, contribution of the full valence band (72) [18], the solid line as expected [145]. is their sum (73) [17, 145]. A single back gate in the absence of additional charged dopants may be described by Vt = nb0 = nt0 = 0, re- sulting in simplified expressions n = ε ε V /(eL ) (as in tends to saturate Ug γ1, even if U γ1. At low den- 0 b b b → | | ≫ monolayer graphene [1]) and Uext = Λγ1n/n . Using sity, n n , the logarithmic term describing screening ⊥ | | ≪ ⊥ U γ1, equation (74) simplifies [17, 60, 69, 71] as dominates and Ug U 2γ1( n /n )/ ln(n / n ), in- | | ≪ dependent of the screening≈ | | ≈ parameter| | ⊥ Λ. ⊥ | | 1 Λγ n Λ n − The expressions (74,75) for U(n) take into account U(n) 1 1 ln | | , (75) ≈ n − 2 n screening due to low-energy electrons in pz orbitals using ⊥ ⊥ a simplified Hamiltonian (67) while neglecting other or- At high density n n , the logarithmic term is neg- bitals and the effects of disorder [55, 145, 149–155], crys- | | ∼ ⊥ ligible and U(n) Λγ1n/n is approximately linear in talline inhomogeneity [18] and electron-electron exchange ≈ ⊥ density. Note that the band gap U = U γ / U 2 + γ2 and correlation. Nevertheless, there is generally good g | | 1 1 16 qualitative agreement of the dependence of U(n) on den- L sity n predicted by equations (74,75) with density func- tional theory calculations [18, 73] and experiments in- cluding photoemission [16, 20] and infrared spectroscopy [55, 78–80, 129, 130]. Note that the Hartree screening model has been generalised to describe graphene trilay- ers and multilayers [147, 156–158]. W

IV. TRANSPORT PROPERTIES

A. Introduction x=0 x=L

Bilayer graphene exhibits peculiar transport proper- FIG. 9. Two-probe bilayer graphene device with armchair ties due to its unusual band structure, described in sec- edges, width W and length L. The rectangular shaded regions tion II, where the conduction and valence bands touch represent the ends of semi-infinite leads. with quadratic dispersion. Transport characteristics and the nature of conductivity near the Dirac point were probed experimentally [2, 3, 8, 19, 20, 159–162] and inves- approximation to describe scattering from the disordered potential. Although the two models are quite different, tigated theoretically [163–175]. Neglecting trigonal warp- 2 ing, the minimal conductivity is predicted to be 8e2/(πh) both predict the minimal conductivity to be 8e /(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 prop- erties 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 tween regions of opposite polarity have attracted theo- to that described in section II so the corners of the Bril- retical attention due to the existence of one-dimensional louin zone are located at wavevectors Kξ = ξ(0, 4π/3a). valley-polarised modes along the interface [189–192], a In terms of wavevector measured from the centre of the pseudospintronic analogy of spin-valve devices for trans- valley, i.e., ky ky +ξ4π/3a, then the Hamiltonian (30) → port perpendicular to the interface [193, 194], electronic in basis A1,B1, A2,B2 may be written as confinement [195], and valley-dependent transmission [196]. U vπ† 0 0 In the following we describe two different models of vπ U γ1 0 Hb =   , (76) the conductivity of bilayer graphene at low energy. The 0 γ1 U vπ†   first is for ballistic transport in a clean device of finite  0 0 vπ U    length that is connected to semi-infinite leads, described   using wave matching to calculate the transmission prob- where π = i~(kx + iξky), π† = i~(kx iξky), and U is ability and, then, the conductance. The second model the on-site− energy which describes the− doping of the bi- describes the conductivity of a disordered, infinite sys- layer. For simplicity, we include only the main interlayer tem using the Kubo formula and the self-consistent Born coupling term γ1 of the orbitals on the dimer B1 and A2 17 sites. It is assumed that the transverse wavevector ky as is real and it is conserved at the interfaces between the bilayer and the leads. The Hamiltonian (76) shows there are two values of the longitudinal wavevector kx for given U, ky and energy ε, which we denote as k+ and k : − i i ∓  1  iU∞x/~v  1  iU∞x/~v ψleft, = ∓ e + r+± − e− ± 1 1     ~ 2 ~2 2 2  i   i  vk = (ε U) γ1(ε U) v ky . (77)  −    ± − ± − −   i   − 1 iU∞x/~v iky y + r±   e−  e , (80) − 1 L R    Left- Φ and right- Φ moving wave functions may be  i   written± as ±    i  i  − 1 1 ~      iU∞(x L)/ v+iky y ψright, = t+± − + t± e − . ± 1 − 1 ~      i v( k iξky)   i   i  ∓ − ± −   −   −  L (ε U) ik±x+iky y      Φ = N  ∓ −  e− , (78) ± ± (ε U)  −   i~v( k + iξky)   − − ±    Here, a right-moving wave with unit flux correspond- i~v(k iξky) ∓ ± − ing to a state with wavevector k U /(~v) is in- (ε U) ± ≈ ∞ ΦR = N  ∓ −  eik±x+iky y , (79) jected from the left lead [the first term in equation (80)]. ± ± (ε U) Subsequently, there are two left-moving waves that have  −   i~v(k + iξky)  been reflected with amplitudes r± and r±, and two right-  − ±  +   moving waves are transmitted to the right,− with ampli- tudes t+± and t±. − 1/2 At the charge-neutrality point ε = U = 0 in the central where normalisation N = [4W (ε U)k ]− for unit current. ± − ± bilayer region, the wave functions are evanescent with imaginary longitudinal wavevector, equation (77). Two The aim is to describe a mesoscopic bilayer region of states with kx = iξky have finite amplitude only on the − finite length L connected to macroscopic leads. In order A1, A2 sites and two with kx = iξky have finite amplitude to mimick macroscopic, metallic contacts, there should only on the B1, B2 sites: be many propagating modes in the leads that overlap with the modes in the central bilayer region. If this is the case, then the value of minimal conductance should not depend on the particular model used for the leads, ~ e.g., square lattice or graphene lattice, as demonstrated 1 γ1x/ v 0 0 for monolayer graphene [197]. Note, this will yield quite    ξky y   ξky y ψcentre, = c1± e + c2± e different results from a model with a bilayer lead at the ± 0 1      same level of doping as the central region; then, the sys-   0   0       tem is effectively an infinite system, not a finite, meso-      scopic conductor. 0 0

1 ξky y 0 ξky y iky y +c±   e− + c±   e−  e . Snyman and Beenakker [168] modelled the leads as 3 0 4 0 heavily-doped bilayer graphene, generalising an approach       γ x/~v   1   developed for monolayer graphene [198]. In this way,  − 1          there are many conducting modes present in the leads and it is possible to simply use matching of the bilayer wave functions at the interface between the central re- gion and the leads. In particular, the leads are mod- For each incoming mode from the left lead [the first elled as bilayer graphene with on-site energy U = U term in equation (80)], there± are eight unknown ampli- − ∞ where U > 0 and U ε ,γ1 . Then, in the leads, tudes c1±, c2±, c3±, c4±, r+±, r±, t+±, t±. Continuity of the ∞ ∞ ≫ {| | } − − k+ k U /(~v) and wave functions ψleft, , ψright, , wave functions at the interfaces x = 0 and x = L between in the≈ left− ≈ (x<∞ 0) and right (x>L) leads may± be written± 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 ky L where ℓ1 = ~v/γ1. Solving yields the transmission matrix

+ FIG. 10. The transmission coefficients T± of bilayer t t− + + graphene (89) for L = 50ℓ1 [168] (solid lines). The trans- t = + t t− mission coefficient of monolayer graphene [198] is shown in − − 2i the centre (dashed line). = 2 2 + (L/ℓ1) + 2cosh(2kyL) (L/ℓ 2i)cosh(k L) (L/ℓ ) sinh (ξk L) C. Transport in disordered bilayer graphene 1 − y − 1 y . × (L/ℓ1) sinh (ξkyL) (L/ℓ1 +2i)cosh(kyL) − 1. Conductivity The transmission probability [168] is determined by the eigenvalues T of the product tt†: When the Fermi energy εF is much larger than the level ± broadening caused by the disorder potential, the system 1 is not largely different from a conventional metal, and the T = 2 , (89) ± cosh (ξkyL kcL) conductivity is well described by Boltzmann transport ∓ theory. However, this approximation inevitably breaks L L 2 k L = ln + 1+ , (90) down at the Dirac point, where even the issue of whether c  2ℓ 2ℓ  1 1 the system is metallic or insulating is nontrivial. To model electronic transport at the charge neutrality point,   The transmission coefficients T have the same form we need a refined approximation that properly includes as the transmission in monolayer± graphene T = the finite level broadening. Here, we present a conduc- 2 1/ cosh (kyL) [198] but shifted by the parameter kc, as tivity calculation using the self-consistent Born approxi- shown in figure 10. mation (SCBA) [163]. We define the Green’s function as 1 The conductance G may be determined using the G(ε) = (ε )− . The Green’s function averaged over − H Landauer-B¨uttiker formula [199, 200] the impurity configurations satisfies the Dyson’s equation

′ ′ (0) (0) ′ 2 G (ε) = δ G (ε)+ G (ε) Σ 1 (ε) G 1 ((93)ε) , gvgse α,α α,α α α α,α α ,α G = Tr tt† , (91) α1 h where is an average over configurations of the disorder where the factor of gvgs accounts for valley and spin de- generacy. For a short, wide sample whose width W ex- potential, α is an eigenstate of the ideal Hamiltonian 0, (0) 1 H and Gα = (ε ε )− , with ε being the eigenenergy of ceeds its length L, W L, the transverse wavevector − α α may be assumed to be continuous≫ and the state α in 0. In SCBA, the self-energy is given by [201] H 2 2 gvgse W ∞ 2gvgse W G = (T + T ) dk = (92). ′ ′ ′ + y Σ (ε)= U 1 U ′ G (ε) . (94) h 2π − h πL α,α α,α α1,α α1,α1 −∞ ′ α1,α1 Thus, the minimal conductivity σ = GL/W = 8e2/(πh) The equations (93) and (94) need to be solved self- is twice as large as in the monolayer. In a similar way, it is consistently. The conductivity is calculated using the possible to determine the Fano factor of shot noise which Kubo formula, takes the same value 1/3 [168] as in monolayer graphene [198]. Transmission via evanescent modes in graphene ~e2 σ(ε)= g g ReTr v GR v˜RA GA v GR v˜RR GR , has been described as pseudodiffusive because the Fano v s 2πΩ x x − x x factor takes the same value as in a diffusive metal [198]. (95) 19 where GR = G(ε + i0) and GA = G(ε i0) are retarded 8 − and advanced Green’s functions, v = ∂ /∂p is the (a) Γ / γ x H0 x 1 velocity operator, and gvgs accounts for summation over

] 0.15 RA RR 6 valleys and spins.v ˜x andv ˜x the velocity operators h RA π 0.08 containing the vertex correction, defined byv ˜x =˜vx(ε+ RR 0.03 m*/2 i0,ε i0) andv ˜x =v ˜x(ε + i0,ε + i0) with s g − v 4 0.015

v˜ (ε,ε′)= v + UG(ε)˜v G(ε′)U . (96) DOS x x x

In SCBA,v ˜x should be calculated in the ladder approxi- 2

mation. [in units of g Ideal For the disorder potential, we assume a short-ranged potential within each valley, 0 -1 -0.5 0 0.5 1 E / γ 1000 F 1 0100 25 U = uiδ(r ri)   . (97) − 0010 (b) Γ / γ1 i    0001  20 0.015   h)]

  2 π

and neglect intervalley scattering between K+ and K . /( − 2 15 0.03 e This situation is realized when the length scale of the s g scattering potential is larger than atomic scale but much v shorter than the Fermi wave length. We assume an equal 10 amount of positive and negative scatterers u = u and Conductivity 0.08 i ± a total density of scatterers per unit area nimp.

[in units of g 5 The SCBA formulation is applied to the low-energy 0.15 Hamiltonian equation (38), with the trigonal warping ef- fect due to γ included [163]. Energy broadening due to 0 3 -1 -0.5 0 0.5 1 the disorder potential is characterised by EF / γ1 π m Γ= n u2 ∗ . (98) 2 imp 2π~ FIG. 11. (a) Calculated density of states and (b) conductivity as a function of energy for different disorder strength Γ [202]. When Γ>εL, i.e., the disorder is strong enough to smear In (b), broken curves show the results of the low-energy two- out the∼ fine band structure of the trigonal warping, we band model, and the horizontal line indicates the universal 2 2~ can approximately solve the SCBA equation in an ana- conductivity gvgse /(π ). lytic form. The self-energy, equation (94), becomes diag- onal with respect to index α and it is a constant,

Σ(ε + i0) iΓ. (99) ≈− In the strong disorder regime Γ 2πεL, the correction 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 σ(ε) g g 1+ | | + arctan | | + L . the same approximation. In transport measurements of ≈ 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 The third term in the square bracket arises from the ver- 2 2 gvgse /(π ~). tex correction due to the trigonal warping effect, and ε L The 2 2 (two-band) model works well at low energy, 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- e2 π ε erable. To see this, we numerically solved SCBA equa- σ(ε) g g | |, (101) tion for the original 4 4 (four-band) Hamiltonian. Fig- v s π2~ 4 Γ ≈ ure 11(a) and (b) show× the density of states (DOS) and which increases linearly with energy. The value at zero conductivity, respectively, for several disorder strengths energy becomes [202]. In (b) the results for the 2 2 model in equa- tion (100) are expressed as broken× curves. In (a), we 2 e 2πεL observe that the DOS at zero energy is significantly en- σ(0) = g g 1+ . (102) v s π2~ Γ hanced because states at high energies are shifted toward 20 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 γ1, because the frequency of electron scattering is 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 The electronic structure of bilayer graphene was at zero energy is not universal but depends on the degree probed by spectroscopic measurements in zero magnetic of disorder for scatterers with long-range potential, sim- field [16, 55, 56, 79, 80, 129, 130], and also in high mag- ilar to monolayer graphene [204]. netic fields [78]. The optical absorption for perpendic- ularly incident light is described by the dynamical con- ductivity 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 lo- 2 gvgs e ~ω +2γ1 calisation. In graphene, the presence of spin-like degrees Re σxx(ω)= θ(~ω 2 εF ) 16 ~ ~ω + γ − | | of freedom related to sublattices and valleys, as well as 1 2 real electronic spin itself, creates the possibility of a rich γ1 + [θ(~ω γ1)+ θ(~ω γ1 2 εF )] variety of quantum interference behaviour. Weak local- ~ω − − − | | ~ isation [205, 206] is a particularly useful probe because ω 2γ1 + − θ(~ω 2γ1) it is sensitive to elastic scattering that causes relaxation ~ω γ1 − of the sublattice pseudospin and valley ‘spin’. In the − 2 εF + γ1 ~ absence of symmetry-breaking scattering processes, con- +γ1 log | | δ( ω γ1) , (103) γ1 − servation of pseudospin in monolayer graphene tends to suppress backscattering [88, 207], and the interference of where εF is the Fermi energy and we assumed εF <γ1. chiral electrons would be expected to result in antilocal- We label the four bands in order of descending| energy| ization [208]. However, intravalley symmetry-breaking as 1, 2, 3, 4. The first term in equation (103) represents relaxes the pseudospin and suppresses anti-localisation, absorption from band 2 to 3, the second from 2 to 4 or while intervalley scattering tends to restore conventional from 1 to 3, the third from 1 to 4, and the fourth from 3 weak localisation [209–213], as observed experimentally to 4 or from 1 to 2. Figure 12 (a) shows some examples of [209, 214–219]. Nevertheless, anti-localisation has been calculated dynamical conductivity Re σxx(ω) with several observed at high temperature [218, 220] when the rel- values of the Fermi energy [229]. The curve for εF = 0 ative influence of symmetry-breaking disorder is dimin- has essentially no prominent structure except for a step- ished, and its presence has also been predicted at very like increase corresponding to transitions from 2 to 4. low temperature [221–223] when spin-orbit coupling may With an increase in εF , a delta-function peak appears at influence the spin of the interfering electrons. ~ω = γ1, corresponding to allowed transitions 3 to 4. In bilayer graphene, the pseudospin turns twice as In a magnetic field, an optical excitation by perpen- quickly in the graphene plane as in a monolayer, no dicular incident light is only allowed between the Lan- suppression of backscattering is expected and the quan- dau levels with n and n 1 for arbitrary combinations ± tum correction should be conventional weak localisation of = H,L and s = 1, since the matrix element of ± [212, 213, 224]. However, the relatively strong trigonal the velocity operator vx vanishes otherwise. Figure 12 warping of the Fermi line around each valley (described (b) shows some plots of Re σxx(ω) in magnetic fields at in section II E) can suppress localisation unless interval- εF = 0 and zero temperature [230]. Dotted lines pene- ley scattering is sufficiently strong [224]. Experimental trating panels represent the transition energies between observations confirmed this picture [159], and it was pos- several specific Landau levels as a continuous function sible to determine the temperature and density depen- of ~ωB. Every peak position behaves as a linear func- dence of relevant relaxation lengths by comparing to the tion of B ~ω2 in weak field but it switches over to ∝ B predicted magnetoresistance formula [224]. √B-dependence as the corresponding energy moves out Localisation has also been studied for gapped bilayer of the parabolic band region. In small magnetic fields, graphene in the presence of interlayer potential asymme- the peak structure is smeared out into the zero-field curve try U [225]. It was shown that, as long as the disorder more easily in the bilayer than in the monolayer, because 21

20 [234, 237].

h) (a) π When the spectrum is composed of discrete Landau /2 EF/γ1 2

e 1.20 s levels εn, the thermodynamical potential is generally g v 15 written as 1.00

1 gvgs 0.80 Ω= ϕ(ε ), (104) −β 2πl2 n 10 B n 0.60 β(ε ζ) 0.40 where β =1/kBT , ϕ(ε) = log 1+e− − with ζ being 5 the chemical potential. In weak magnetic field, using the 0.20 Euler-Maclaurin formula, the summation in n in equa- tion (104) can be written as an integral in a continuous Dynamical Conductivity (units of g Dynamical Conductivity (units of 0.00 variable x with a residual term proportional to B2. The 0 magnetization M and the magnetic susceptibility χ can h ω / γ1 be calculated by 2 (b) hωB / γ1 = 0.1 (B ~ 1.1T) ∂Ω ∂M ∂2Ω M = , χ = = (105). − ∂B ζ ∂B B=0 − ∂B2 ζ B=0

/h) 0.2 (4.5T) 2

e s

g v 0.3 (10T) For monolayer graphene, the susceptibility is [121, 239] 1

(units of g 2 2

xx 0.4 (18T) e v ∞ ∂f(ε) σ χ = g g dε. (106) v s 6πc2 ∂ε Re − − −∞ 0.5 (28T) At zero temperature, it becomes a delta function in Fermi 0 energy, 0 1 2 h ω / γ1

(0L, 1L±) (-1L, 0H±) (0H ,± 1H±) ±

(1L ,± 2L±) (0H , 1L±) e2v2

(2L ,± 3L±) (0L, 1H±) χ(εF )= gvgs δ(εF ). (107) ± ± 6πc2 (1H , 2L±) (1L , 2H±) − The delta-function susceptibility of monolayer graphene FIG. 12. Interband part of the dynamical conductivity of bilayer graphene plotted against the frequency ω, in (a) zero is strongly distorted by the interlayer coupling γ1. For the Hamiltonian of the symmetric bilayer graphene, the magnetic field with different εF ’s [229] (b) several magnetic fields with εF = 0 [230]. Dashed curves in (b) indicate the orbital susceptibility is calculated as [234, 237] transition energies between several Landau levels. e2v2 ε 1 χ(ε)= gvgs 2 θ(γ1 ε ) log | | + . (108) 4πc γ1 − | | γ1 3 The susceptibility diverges logarithmically at ε = 0, be- the Landau level spacing is narrower in bilayer due to the F comes slightly paramagnetic near ε = γ , and vanishes finite band mass. | F | 1 for εF > γ1 where the higher subband enters. The in- tegration| | of χ in equation (108) over the Fermi energy 2 2 2 becomes gvgse v /(3πc ) independent of γ1, which is VI. ORBITAL MAGNETISM exactly twice− as large as that of the monolayer graphene, equation (107). Graphite and related materials exhibit a strong or- The susceptibility was also calculated in the presence bital diamagnetism which overcomes the Pauli spin para- of interlayer asymmetry [243]. Figure 13 (a) and (b) show magnetism. Theoretically the diamagnetic susceptibil- the density of states and the susceptibility, respectively, ity was calculated for graphite [121, 231, 232], graphite for bilayer graphene with U = 0, 0.2, and 0.5. The intercalation compounds [233–236], as well as few-layer susceptibility diverges in the paramagnetic direction at graphenes [237, 238]. In particular, monolayer graphene the band edges where the density of states also diverges. has a strong diamagnetic singularity at Dirac point, This huge paramagnetism can be interpreted as the Pauli which is expressed as a Delta function in Fermi energy paramagnetism induced by the valley pseudo-spin split- εF [121, 239–242]. In the bilayer, the singularity is re- ting and diverging density of states [244]. The suscep- laxed by the modification of the band structure caused tibility vanishes in the energy region where the higher by the interlayer coupling as we will see in the following 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 quan- tum 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 effec- tive 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 be- haviour 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 makes an anti-crossing at the intersection. 2 (a) λω symmetric

1 VIII. ELECTRONIC INTERACTIONS

0 Generally speaking, the low-energy behaviour of elec- -1 asymmetric trons in bilayer graphene is well described by the tight-

Frequency Shift (units of binding model without the need to explicitly incorporate )

0 1.0 electron-electron interaction effects. Coulomb screening λω (b) and collective excitations have been described in a num- ber of theoretical papers [152, 292–300] and the impor- 0.5 tance of interaction effects in a bilayer under external gating [293, 301–307] has been stressed. Interaction ef- fects should also be important in the presence of a mag- 0.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 netic field or at very low carrier density, particularly in Broadening (units of 2 2 Electron Density (units of gvgsγ1 /2 πhv ) clean samples. Bilayer graphene has quadratic bands which touch at FIG. 14. (a) Frequency shift and (b) broadening of the Γ- low energy resulting in a non-vanishing density of state point optical phonon in a bilayer graphene [288]. Solid and and it has been predicted to be unstable to electron- dashed lines denote the high- and low-frequency modes, re- electron interactions at half filling. Trigonal warping spectively, and the thin dotted lines in the top panel show tends to cut off infrared singularities and, thus, finite the frequencies for symmetric and antisymmetric modes cal- culated without inclusion of their mixing. coupling strength is generally required to realise corre- lated ground states; if trigonal warping is neglected, then arbitrarily weak interactions are sufficient. Since bilayer graphene possesses pseudospin (i.e., which layer) and val- ley 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,ω0), and the fre- 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,ω0) u . 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 in- troduce 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 i.e., the interband transition excites a valence electron field cumulating in a non-divergent resistance at zero field exactly to the Fermi surface. The coupling between sym- while compressibility measurements of single-gate devices metric and antisymmetric modes arises when n = 0, and [319] found an incompressible region near the charge neu- s 25

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

IX. SUMMARY trality point. These measurements were interpreted as being consistent with either the anomalous quantum Hall state, in which the separate layers of the device are valley This review focused on the single-particle theory polarised, or the gapless nematic phase. of electrons in bilayer graphene, in the shape of the Subsequently, conductivity measurements on single- tight-binding model and the related low-energy effective gate devices [320] observed a weak temperature depen- Hamiltonian. Bilayer graphene has two unique proper- dence of the width of the conductivity minimum near ties: massive chiral quasiparticles in two parabolic bands zero carrier density, suggesting a suppressed density of which touch at zero energy, and the possibility to con- states as compared to that expected for a parabolic dis- trol an infrared gap between these low-energy bands by persion, as well as particularly robust cyclotron gaps at applying an external gate potential. These features have filling factor ν = 4, observations attributed to the pres- a dramatic impact on many physical properties of bi- ence of the nematic± phase. layer graphene including some described here: optical However, other experiments [321–324] observed insu- and transport properties, orbital magnetism, phonons lating states indicating the formation of ordered phases and strain. A number of topics were not covered here in with energy gaps of about 2meV. There was evidence great detail or at all; we refer the reader to relevant de- for the existence of edge states in one of these exper- tailed reviews of graphene including electronic transport iment [321], but not in the others [322–324], and the [176, 177], electronic and photonic devices [344], scan- latter observations including their response to perpen- ning tunnelling microscopy [345], Raman spectroscopy dicular electric field [322, 323] and tilted magnetic field [259, 260], magnetism [346], spintronics and pseudospin- [324] were seen as being consistent with a layer antiferro- tronics [347], Andreev reflection at the interface with a magnetic state, in which the separate layers of the device superconductor and Klein tunnelling [348], growth and are spin polarised, figure 15(d). At present, there appear applications [21], and the properties of graphene in gen- to be contradicting experimental and theoretical results, eral [7, 349]. Finally, although the central features of the but it should be noted that renormalisation group stud- single-particle theory are already established, the same ies [255, 256, 325] have highlighted the sensitivity of the can not be said of the influence of electronic interactions, phase diagram of bilayer graphene to microscopic details. which is likely to remain a subject of intense research, In the absence of interactions, the low-energy Landau both theoretical and experimental, in the near future at level spectrum of bilayer graphene [9] consists of a series least. of fourfold (spin and valley) degenerate levels with an The authors gratefully acknowledge colleagues for eightfold-degenerate level at zero energy, as described in fruitful collaboration in graphene research, in particular section II J. The resulting Hall conductivity consists of T. Ando and V. I. Fal’ko, and we acknowledge funding a series of plateaus at conventional integer positions of by the JST-EPSRC Japan-UK Cooperative Programme 4e2/h, but with a double-sized step of 8e2/h across zero Grant EP/H025804/1. 26

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