Competing topological phases in few-layer graphene
Pierre Carmier, Oleksii Shevtsov, Christoph Groth, Xavier Waintal
CCompeting topological phases in few-layer graphene
Pierre Carmier, Oleksii Shevtsov, Christoph Groth, and Xavier Waintal
CEA-INAC/UJF Grenoble 1, SPSMS UMR-E 9001, Grenoble F-38054, France (Dated: September 20, 2018)We investigate the effect of spin-orbit coupling on the band structure of graphene-based two-dimensional Dirac fermion gases in the quantum Hall regime. Taking monolayer graphene as ourfirst candidate, we show that a quantum phase transition between two distinct topological states –the quantum Hall and the quantum spin Hall phases – can be driven by simply tuning the Fermilevel with a gate voltage. This transition is characterized by the existence of a chiral spin-polarizededge state propagating along the interface separating the two topological phases. We then applyour analysis to the more difficult case of bilayer graphene. Unlike in monolayer graphene, spin-orbit coupling by itself has indeed been predicted to be unsuccessful in driving bilayer grapheneinto a topological phase, due to the existence of an even number of pairs of spin-polarized edgestates. While we show that this remains the case in the quantum Hall regime, we point out that byadditionally breaking the layer inversion symmetry, a non-trivial quantum spin Hall phase can re-emerge in bilayer graphene at low energy. We consider two different symmetry-breaking mechanisms:inducing spin-orbit coupling only in the upper layer, and applying a perpendicular electric field. Inboth cases, the presence at low energy of an odd number of pairs of edge states can be driven by anexchange field. The related situation in trilayer graphene is also discussed.
PACS numbers: 72.80.Vp, 73.43.Nq, 73.20.At, 73.21.Ac
I. INTRODUCTION
Topological insulators are bulk insulators which pos-sess robust conducting surface states [1, 2]. Paradigmatictwo-dimensional examples of this class are the quantumHall (QH) and the quantum spin Hall (QSH) phases,which are characterized by respectively chiral and he-lical one-dimensional edge states. While the former canbe generated by simply applying a strong perpendicularmagnetic field and has been rather extensively studiedsince the 1980s, the latter requires the presence of spin-orbit coupling and has received very little experimentalevidence. Indeed, despite the wide interest shown in theliterature for the QSH phase (and for topological phasesin general) since the seminal works by Kane and Mele[3, 4], experimental traces of this phase have remainedscarce, with the exception of the remarkable works onHgTe quantum wells [5–7] (see also the experiment in-volving InAs [8]). Recent studies [9–11] have revivedthe possibility of generating a QSH in graphene [12, 13],by showing that low concentrations of suitably chosenadatoms, randomly deposited on graphene, could open alarge non-trivial gap in graphene’s otherwise semimetallicband structure, and yield transport properties showingno trace of the spatially inhomogenous spin-orbit cou-pling. The perspective of successfully turning grapheneinto a QSH insulator is promising, as it would consider-ably enhance the experimental feasibility of engineeringsamples of the latter, which are so far limited to the pre-viously mentioned and experimentally challenging HgTeheterostructures.Although it does not enjoy the conceptual simplicity ofthe monolayer, bilayer graphene is an interesting systemin its own right. It is a gapless semimetal, characterizedby massive chiral excitations carrying a topological Berry phase 2 π [14], with a very rich list of many-body insta-bilities predicted at low density (see [15] for correspond-ing references). One of its most remarkable properties,in contrast to monolayer graphene, is the possibility toopen a gap in its band structure by simply applying a per-pendicular electric field which breaks the layer inversionsymmetry [16]. However, in the presence of a perpen-dicular magnetic field, the electronic properties of bothsystems become qualitatively very similar. In particular,their energy spectrum is characterized by a particle-holesymmetry and by the existence of levels sitting exactlyat zero energy which are at the origin of the anomalousquantization of the Hall conductance in these systems ascompared to other two-dimensional electron gases [17–19]. This is the hallmark property of what we will referto in this article as two-dimensional Dirac fermion gases(2DDFGs).The purpose of this article is to discuss what happensto the band structure of a 2DDFG when the effects ofboth magnetic field and spin-orbit coupling are taken intoaccount simultaneously. Taking graphene as our first ex-ample, we shall review in section II the results alreadypublished elsewhere [20], according to which a topologi-cal phase transition takes place at low energy and can betuned by simply varying the chemical potential. Then,in the following section, we shall investigate the relatedsituation in bilayer graphene and show that the resultsobtained for the monolayer do not extend to it. This canbe traced back to the fact that the topological invariantcharacterizing the QSH phase is non-trivial only if thereare an odd number of pairs of edge states, which trans-lates in multi-layer graphene into the condition of havingan odd number of layers. Nevertheless, we will show insection IV that by adding additional ingredients to ourmodel, namely by breaking the layer inversion symme-try, a non-trivial QSH phase can be generated in bilayer a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov graphene along with a corresponding topological phasetransition. We stress that (almost) all of our results canbe understood by simply looking at the band structuresof the systems we study. Finally, section V discusses thepossible extension of our approach to other systems, andwe conclude in section VI. II. GRAPHENE
Recent investigations of the interplay between QH andQSH phases in some specific examples of 2DDFGs [20–23] have led to surprising results. In these works, it wasshown that the QSH phase can survive the presence ofa perpendicular magnetic field and that the Z topolog-ical invariant [4] remains non-trivial for energies belowthe spin-orbit induced gap, despite the breaking of time-reversal symmetry [59]. As we will exemplify in the caseof graphene, the origin of this intriguing result actuallystems from the existence of zero-energy Landau levels:as soon as the spin-degeneracy of these levels is lifted,spin-polarized edge states characteristic of a QSH phaseemerge [60]. A. Model
Let us start by introducing the model from which ourresults shall be derived. In the vicinity of the zero-energypoints in the Brillouin zone, low-energy excitations canbe described by a Dirac Hamiltonian: H G = v F ( τ ˆ p x σ x + ˆ p y σ y ) , (1)with σ x , σ y the usual set of Pauli matrices acting in thetwo-dimensional space of the inequivalent sublattices Aand B (see Fig. 1), and with v F = √ t ˜ a/ (2 (cid:126) ) the Fermivelocity, expressed as a function of the microscopic latticeparameters t (nearest-neighbor hopping amplitude) and˜ a (lattice constant) which we choose in the following asour working units of energy and length, respectively. τ = ± (cid:15) = ± v F | p | .The presence of a perpendicular magnetic field can bestraightforwardly included by making use of the Peierlssubstitution ˆ p → ˆ Π = ˆ p + e A , which accounts for thepresence of the magnetic vector potential A such that ∇× A = B z . The components of the generalized momen-tum satisfy the Heisenberg algebra [ ˆΠ x , ˆΠ y ] = − i ( (cid:126) /l B ) .By expressing these components in terms of the usualharmonic oscillator ladder operators [19],ˆΠ x = (cid:126) √ l B ( a + a † ) , ˆΠ y = i (cid:126) √ l B ( a − a † ) , (2)and using the standard raising and lowering propertiesof these operators on the eigenstates ( a | n (cid:105) = √ n | n − (cid:105) and a † | n − (cid:105) = √ n | n (cid:105) ), the energy spectrum can then (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) t λ so BA Figure 1: Sketch of the graphene hexagonal lattice, character-ized by nearest-neighbor hopping t and next-nearest-neigborKane-Mele spin-orbit coupling λ so . The unit cell of the latticecontains two inequivalent sites, A and B. straightforwardly be shown to turn into the well-knownLandau levels, (cid:15) n = ± ∆ B (cid:112) | n | (3)with ∆ B = √ (cid:126) v F /l B , and l B = (cid:112) (cid:126) / ( eB ) the magneticlength. As already mentioned before, the main distinc-tive feature of the Landau level spectrum of a 2DDFG ascompared to that of a standard two-dimensional electrongas is the existence of a zero-energy level at n = 0 orig-inating from the pseudo-relativistic nature of the chargecarriers. Also note that all levels enjoy a 4-fold degener-acy arising from spin and valley indices. B. Band structure / edge state correspondence
The appearance of edge states in this context can bebest understood by looking at the band structure of agraphene ribbon. The latter is a system which is trans-lationally invariant in one direction, and confined in theother. In order to derive the band structure numerically,we formulate the above ingredients in terms of a tight-binding model, in which Eq. (1) becomes H G = − t (cid:88) (cid:104) i,j (cid:105) e iφ ij c † i c j . (4)Indices ( i, j ) label lattice sites, while symbol (cid:104) (cid:105) refersto nearest-neighbor coupling (with hopping amplitude t ), as is illustrated in Fig. 1. The Peierls phase φ ij =( e/ (cid:126) ) (cid:82) r i r j A · d r takes into account the contribution fromthe magnetic flux threading the lattice. Numerical cal-culations are performed using kwant, the new quantumtransport software package developed by A. Akhmerov,C. Groth, X. Waintal, and M. Wimmer. In the process,we choose to work with armchair boundary conditions,but our results are qualitatively unaffected by this choice.The band structure associated with the peculiar spec-trum of Eq. (3) in a ribbon geometry (translationally -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.4-0.200.20.4 E v x < >0 = 2= -2 x Figure 2: (Color online): Energy spectrum of a monolayergraphene armchair ribbon in the QH regime ( C (cid:54) = 0). Blackcircles and red triangles respectively stand for spin up andspin down bands, which are here indistinguishable due to spindegeneracy. The horizontal dashed line represents an arbitrar-ily chosen Fermi level. Its intersection with the lowest bandof the system indicates the existence of a (spin-degenerate)edge state which propagates in opposite directions on oppo-site sides. The ribbon width is W = 38 and the magneticlength is l B ≈ invariant in the x -direction, confined in the y -directionwith | y | < W/
2) is shown in Fig. 2. Due to the natureof the classical dynamics, the transverse coordinate ofthe cyclotronic center of motion y c can be identified withthe conserved longitudinal momentum k x via the formula y c = − k x l B . Observing the band structure in terms ofthis real space coordinate, one can see in Fig. 2 that, farfrom the edges of the ribbon, the band structure consistsof flat bands which are none other than the Landau levelsof Eq. (3): electrons in the bulk are classically localizedby the magnetic field along closed cyclotronic orbits. Onthe other hand, electrons in the vicinity of the edges canscatter along them and propagate following skipping or-bits, which translates in the band structure into bulkLandau levels acquiring a finite dispersion as they ap-proach the edges of the ribbon: v ( n ) x = 1 (cid:126) ∂(cid:15) n ∂k x . (5)Because this dispersion is monotonous on a given edge(see Fig. 2), the edge states cannot be backscattered un-less they are coupled to the states living on the oppositeedge, a process the likelihood of which decays exponen-tially with the width of the system. This property ofthe edge states is generally referred to as chirality andis the reason why these states can carry current withoutdissipation: this leads to the celebrated QH effect [24],characterized by a quantized conductance G = C ( e /h ).More formally, the edge states enjoy a topological pro- tection encoded in the Chern number C which is a Z topological invariant characterizing the number of filledbands in the QH regime [25]. It is a topological quantity,in the sense that smooth deformations of the Hamilto-nian (deformations which do not close the gap) cannotchange its value, and shall be defined in the next subsec-tion. We thus see that, for most purposes, the physicsof topological phases such as the QH phase can be verysimply extracted from the corresponding band structure.We now consider the situation where, in addition tothe perpendicular magnetic field, the effect of spin-orbitcoupling as introduced by Kane and Mele [3] is accountedfor in the Hamiltonian as H so = τ s ∆ so σ z , (6)which is characterized by the energy scale ∆ so . τ = ± s = ± H so = iλ so (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ν ij e iφ ij ( c † i,α s αβz c j,β ) , (7)where indices ( i, j ) once more label lattice sites, while( α, β ) label spin indices, symbol (cid:104)(cid:104) (cid:105)(cid:105) refers to next-nearest-neighbor coupling (with SO-induced hopping am-plitude λ so = ∆ so / (3 √
3) [3]), and ν ij = ± and (SO)second nearest-neighbor. The presence of spin-orbit cou-pling modifies the Landau level spectrum according tothe expression (cid:15) n,s = ± (cid:113) ∆ B | n | + ∆ , for n (cid:54) = 0 − s ∆ so , for n = 0 . (8)The latter is characterized by the n = 0 level being liftedfrom zero energy into spin-polarized branches: E = +∆ so features only spin-down states, while E = − ∆ so featuresonly spin-up states [22] (see Fig. 3). While other Lan-dau levels retain their associated chiral edge states, ir-respective of the spin polarization, the lowest Landaulevel now features counter-propagating edge states for | E | < ∆ so , with a spin-dependent direction of propa-gation (see Fig. 3): one has v (0 , ↑ ) x · v (0 , ↓ ) x < Z topological invariant introduced by Kaneand Mele [4], which we do next. C. Topological order
Let us start by recalling the standard topological num-ber characterization of Landau levels when ∆ so = 0. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.4-0.200.20.4 E = 2= -2= 0 = 1 y c x l B x = k- Figure 3: (Color online): Same as in Fig. 2, but with an ad-ditional spin-orbit coupling term λ so = 0 .
02. The latter liftsthe spin-degeneracy of the zero-energy Landau level, yieldinga QSH phase ( ν = 1) with a single pair of counter-propagatingspin-polarized edge states. The sketch above the band struc-ture depicts the edge states in real-space (thick black verticallines on the sides represent the edges of the ribbon). Each Landau level n and its associated eigenfunctionsover the first Brillouin zone are characterized by a topo-logical invariant, the so-called Chern number [25]. Thistopological number takes a value C ( n ) τ,s = +1 for each Lan-dau level, independently of the Landau n , valley τ orspin s indices. For each value of the Fermi energy, wecan characterize the corresponding phase by a topologi-cal number C = (cid:88) τ,s C τ,s , with C τ,s ( E F ) = (cid:88) (cid:15) n
We now switch to the slightly more involved case of(Bernal-stacked) bilayer graphene and start by address-ing the possibility of inducing a QSH phase in bilayer (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
BA BA
112 2 tt t
00 1
Figure 4: Side view of Bernal-stacked bilayer graphene, whichis characterized by intra-layer nearest-neighbor hopping t and inter-layer hopping t . graphene. This is not a trivial endeavour, as the naiveextension of the Kane-Mele model to bilayer grapheneyields a weak Z topological phase, characterized by aneven number (rather than an odd number, as in mono-layer graphene) of pairs of spin-polarized edge states[28, 29]. This doubling of the number of edge states ba-sically arises because a bilayer has twice as many layersas a monolayer. For the same reason, a graphene trilayerwill have an odd number of pairs of edge states and there-fore feature a non-trivial QSH phase. Breaking the layersymmetry by considering the case where spin-orbit cou-pling is present in only one of the layers was shown notto be any more effective [28], the system then remainingsemimetallic.Here we follow a different approach, inspired by themodel presented in the previous section. This seems like anatural idea, as bilayer graphene is also known to featurezero-energy Landau levels [14, 30]. In this section, we willshow that the presence of both spin-orbit coupling and aperpendicular magnetic field in the bilayer yields a bandstructure very similar to that of monolayer graphene, butwith results no different from that of Prada et al. [28]:the obtained QSH phase is topologically trivial due tothe existence of an even number of pairs of spin-polarizededge states. On the other hand, we will show in the nextsection that if spin-orbit coupling is present in only oneof the two layers or if a perpendicular electric field isapplied, then the breaking of layer inversion symmetryopens the door for a non-trivial QSH phase to arise atlow-energy. A. Model
The Hamiltonian for bilayer graphene can be expressedusing two sets of Pauli matrices { σ, η } which respectivelyrefer to sublattice ( A , B ) and layer (1, 2) spaces. We con-sider the usual Bernal stacking, inherited from graphite(see Fig. 4), in which A atoms in the upper layer (2) lieabove B atoms from the lower layer (1). Starting fromthe basis ( A , B , A , B ) T , the Hamiltonian reads [30]: H BG = v F ( τ ˆ p x σ x + ˆ p y σ y ) η + t σ x η x − σ y η y ) . (14) The first term describes the usual low-energy Dirac struc-ture of monolayer graphene. The second term takes intoaccount the coupling between both layers, characterizedby an energy scale t (cid:39) .
15. Corrections to Eq. (14)such as trigonal warping are small effects, typically onlyrelevant below the meV range [15], and will therefore beneglected.The spectrum associated with Eq. (14) is particle-“hole” symmetric, with high-energy bands at (cid:15) high = ± t and low-energy bands touching at two Dirac points char-acterized by a topological Berry phase 2 π . In the vicinityof this point, the energy-dispersion relation is quadratic, (cid:15) low = ± p / (2 m ∗ ), with m ∗ = t / (2 v F ) the effectivemass of the gapless excitations.The presence of a perpendicular magnetic field can bestraightforwardly included by making use of the Peierlssubstitution as before, and the energy spectrum can thenbe shown to turn into the well-known Landau levels [30], (cid:15) n = ± (cid:126) ω c (cid:112) n ( n −
1) (15)with ω c = eB/m ∗ the characteristic cyclotronic fre-quency. The latter can be related to the monolayergraphene energy scale by the simple relation (cid:126) ω c =∆ B /t . Notice how the spectrum in Eq. (15) fea-tures twice as many zero-energy levels as in monolayergraphene, since the n = 1 level also vanishes. Actually,one can prove on general grounds that chirally stacked N -layer graphene should feature a 4 N -fold degeneratezero-energy Landau level [31, 32]. One should also havein mind that the accuracy of expression (15) for n (cid:54) = 0 , B (cid:28) t . B. Quantum Hall regime
To compute the associated band structure numerically,we once more formulate the above ingredients in termsof a tight-binding model, in which Eq. (14) becomes H BG = − t (cid:88) (cid:104) i,j (cid:105) e iφ ij c † i c j + t (cid:88) (cid:104) i ∈ A ,j ∈ B (cid:105) c † i c j , (16)using the same notations as in the previous section. NoPeierls phase appears in the second term, as A - B bondsare oriented along the z -axis. The band structure asso-ciated with the Landau level spectrum of Eq. (15) in aribbon-geometry is displayed in the upper panel of Fig. 5.As expected, it resembles very closely that of monolayergraphene in the QH regime. The main difference betweenthe two lies in the existence of twice as many dispersingbranches arising from the lowest Landau level in bilayergraphene, due to the doubling of the zero-energy Lan-dau level degeneracy. This translates in the languageof topological invariants into Chern numbers per spinspecies C s = − -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.4-0.200.20.4 E = 4= -4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.4-0.200.20.4 E = 4= -4= 0 = 0 Figure 5: (Color online): (Top panel) Energy spectrum of abilayer graphene armchair ribbon in the QH regime. Black cir-cles and red triangles respectively stand for spin up and spindown bands, which are here indistinguishable due to spin de-generacy. Notice that the zero-energy Landau level has twiceas many bands as its counterpart in monolayer graphene. Theribbon width is W = 38 and the magnetic length is l B ≈ λ so = 0 .
02. The latter lifts the spin-degeneracy of the zero-energy Landau level, yielding a weakQSH phase ( ν = 0 (mod 2)) with an even number of pairs ofcounter-propagating spin-polarized edge states. the value of the Chern number as a function of the Fermienergy, yielding: (cid:26) C ↑ = C ↓ = − , for − (cid:126) ω c √ < E F < C ↑ = C ↓ = 2[max(0 , n −
1) + 1] , for E F > n is the index of the highest filled Landau level.Notice that Chern numbers of each spin species are equal(since the spectrum is spin-degenerate) and that C = C ↑ + C ↓ is always non-zero, as expected for a QH phase. C. Effect of layer-symmetric spin-orbit coupling
We now consider the situation where, in addition tothe perpendicular magnetic field, the effect of spin-orbitcoupling as introduced by Kane and Mele for mono-layer graphene is accounted for symmetrically in bothlayers [61]. The layer-degenerate Kane-Mele spin-orbitcoupling term is encoded in the Hamiltonian H so = τ s ∆ so σ z η , (18)which, in terms of a tight-binding model, can be imple-mented as H so = iλ so (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ν ij e iφ ij ( c † i,α s αβz c j,β ) , (19)with similar notations as in the previous section, symbol (cid:104)(cid:104) (cid:105)(cid:105) referring to intra-layer next-nearest-neighbor cou-pling. The presence of spin-orbit coupling modifies theLandau level spectrum according to the expression (cid:15) n,s = (cid:40) ± (cid:112) n ( n − (cid:126) ω c ) + ∆ , for n (cid:54) = 0 , − s ∆ so , for n = 0 , . (20)The latter is characterized by n = 0 and n = 1 levelslifted from zero energy into spin-polarized branches: E =+∆ so features only spin-down states, while E = − ∆ so features only spin-up states (see lower panel of Fig. 5).The topologically trivial nature of the correspondinglow-energy phase can be checked by computing the valueof the Z topological invariant. As a straightforward gen-eralization of the calculations performed in [20], one ob-tains the following results: (cid:26) C ↑ = − C ↓ = +2 , for | E F | < ∆ so C ↑ = C ↓ = 2[max(0 , n −
1) + 1] , for E F > ∆ so (21)where n is once more the index of the highest filled Lan-dau level. This time, one is faced with a QH phasefor | E F | > ∆ so , characterized by the same Chern num-ber as in the absence of spin-orbit coupling, while for | E F | < ∆ so , the total Chern number vanishes C ↑ + C ↓ = 0,indicating that this region is no longer in the QH phase.However, as the Z invariant ν = 0 (mod 2) also vanishes,the phase in this region is not a QSH phase either: rather,it is a weak QSH phase, in the sense that time-reversal-symmetric perturbations can couple the edge states andinduce backscattering. This was not the case in mono-layer graphene, due to the existence in the latter of a sin-gle pair of counter-propagating spin-polarized edge statesat low energy. In the situation discussed in this section,we are thus led to conclude that a similar picture as thatdescribed in Ref. [28] prevails. The way around this in-volves breaking the layer inversion symmetry, as we willsee in the next section.Before moving on, however, we would like to pauseand comment on the fact that the model we consideredin this section could provide a convenient platform fortesting precisely how weak a ν = 0 (mod 2) QSH phasewould be. Indeed, even though theory predicts that pairsof edge states should couple through backscattering pro-cesses, an experimental measure of how strongly edgestate transport would be destroyed by such processes isyet to be done, and one cannot exclude the possibility ofunexpected robustness, similarly to what has recently be-gun to be understood in so-called weak three-dimensionaltopological insulators [33, 34]. Said a little differently,it remains unclear how one could distinguish throughtransport measurements a topological phase from a triv-ial phase which has edge states (such as the one exhibitedin this section). IV. BILAYER GRAPHENE WITH BROKENLAYER INVERSION SYMMETRY
This section is devoted to the study of two layer in-version symmetry breaking mechanisms which enable anexchange-induced QSH phase to arise at low energy: (i)inducing spin-orbit coupling only in one of the layers, and(ii) applying a perpendicular electric field. We provideestimations of the magnitude of the QSH gap, and alsoaddress other possible phases which appear in our set-tings: a spin-polarized QH phase and a quantum valleyHall phase.
A. Mechanism I: layer-asymmetric spin-orbitcoupling
Using the same basis as in Eq. (14), we now replaceEq. (18) by the symmetry-breaking term H asymso = τ s ∆ so σ z (cid:18) η + η z (cid:19) (22)which induces spin-orbit coupling only in the upper layer.This situation is physically relevant if one considers thepossibility of inducing spin-orbit coupling in grapheneby depositing adatoms on the surface [9–11]. The cor-responding tight-binding expression is given by applyingEq. (19) only in the upper layer.In order to obtain the new Landau level spectrum, us-ing the ladder operators introduced in section II, onemust now solve a quartic equation (cid:15) n + α n (cid:15) n + β n (cid:15) n + γ n =0, with a non-vanishing linear term β n (cid:54) = 0: α n = − (cid:0) t + ∆ + (2 n − B (cid:1) β n = − s ∆ so t γ n = ( n − B ( n ∆ B + ∆ ) . (23)Note that the role of opposite spin polarizations is ex-changed when going from the conduction to the valenceband: (cid:15) n ( − s ) = − (cid:15) n ( s ). This leaves 4 (out of the 8)eigenvalues of the lowest Landau levels to be found. Twocan easily be identified: (cid:15) = − s ∆ so (for n = 0) and -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.100.1 E = -2 = -1 = 0 = 1 = 2= = -2= = 2= 0 = 0 = -2 = -2= -1 = 2 = 1 = 2 _ ω c /t E / ∆ s o ∆ so /t E Figure 6: (Color online): (Top panel) Band structure withsame parameters as in Fig. 5, except that spin-orbit cou-pling is only applied in the upper layer. This time, aspin-degenerate zero-energy Landau level survives, while low-energy edge states are characterized by an unbalanced spinpopulation on a given edge ( | C ↑ | (cid:54) = | C ↓ | ). (Bottom panel)Dependence of the eigenvalues (cid:15) ± on the magnetic field for∆ so = 0 .
01. Dashed lines correspond to the analytical pre-dictions (valid in the perturbative limit (cid:126) ω c (cid:28) t ) and thicklines to the numerically obtained values. Inset: Dependenceof (cid:15) ± on the spin-orbit gap for ∆ B = 0 . (cid:15) = 0 (for n = 1) both satisfy the quartic equation. Thelatter implies that a spin-degenerate zero-energy Landaulevel survives in this context. The two remaining eigen-values, (cid:15) − and (cid:15) + , can be estimated perturbatively, inthe limit ∆ so , (cid:126) ω c (cid:28) t , as (cid:15) + ≈ − s ∆ so (1 − (cid:126) ω c /t ) and (cid:15) − ≈ − s ∆ so (cid:126) ω c /t . Their dependence on ∆ so and (cid:126) ω c beyond this perturbative regime is shown in the lowerpanel of Fig. 6. This yields the following ordering ofeigenvalues: 0 = (cid:15) < | (cid:15) − | < | (cid:15) + | < | (cid:15) | = ∆ so .The corresponding band structure is shown in the toppanel of Fig. 6. Its description in terms of spin-polarizedbands is slightly more involved than before, but theChern numbers can nevertheless be computed and shownto evolve as follows, C ↑ = − , C ↓ = − , for − | (cid:15) | < E F < −| (cid:15) + | C ↑ = 0 , C ↓ = − , for − | (cid:15) + | < E F < −| (cid:15) − | C ↑ = 1 , C ↓ = − , for − | (cid:15) − | < E F < C ↑ = 2 , C ↓ = − , for 0 < E F < | (cid:15) − | C ↑ = 2 , C ↓ = 0 , for | (cid:15) − | < E F < | (cid:15) + | C ↑ = 2 , C ↓ = 1 , for | (cid:15) + | < E F < | (cid:15) | (24)indicating that a QH phase is preserved at low energy,since the total Chern number never vanishes. This phaseis peculiar, however, as it is characterized by edge stateswith an unbalanced spin population: | C ↑ | (cid:54) = | C ↓ | . For ex-ample, two spin-up and a single spin-down counterpropa-gating state coexist on the same edge for 0 < E F < | (cid:15) − | .A given spin species can even become fully gapped, astestified by vanishing spin Chern numbers, giving rise tospin-polarized edge state transport over a tunable andquite large energy window | (cid:15) + − (cid:15) − | ≈ ∆ so (1 − (cid:126) ω c /t ).A QSH phase can be generated close to zero energy bylifting the spin-degeneracy of the remaining zero-energyLandau level with an arbitrarily small exchange field, de-riving from H ex = s ∆ ex σ η , (25)where ∆ ex quantifies the magnitude of the effect. Thecorresponding tight-binding expression is given by H ex = ∆ ex (cid:88) i c † i,α s αβz c i,β , (26)using the same notations as before. This yields for thespin Chern numbers at low energy: C ↑ = − C ↓ = 1 , for | E F | < min( | ∆ ex | , | (cid:15) − | − | ∆ ex | ) . (27)The total Chern number is zero and, contrary to the caseof layer-symmetric spin-orbit coupling, this time the Z invariant ν = 1, signaling that the QSH phase is non-trivial. The energy window where this phase can beobserved, i.e. the maximum value of the QSH gap, isbounded by the value | (cid:15) − | /
2, a lower bound of which isgiven by the perturbative limit ∆ maxQSH ≥ ∆ so (cid:126) ω c / (2 t ),as can be checked in the lower panel of Fig. 6. The QSHgap could thus potentially reach several tens of meV, al-though, in graphene-based systems, it will effectively belimited by the highest achievable value of spin splittingwhich should be much smaller [62]. In this respect, anddespite the need for a perpendicular magnetic field, ourproposal offers two advantages with respect to that ofRef. [35], where it was recently shown that gated bilayergraphene could be turned into a Z topological insulatorfor sufficiently strong Rashba spin-orbit coupling: thestrength of spin-orbit coupling need not exceed a criti-cal value, and spin-orbit coupling need not be present inboth layers. The latter condition is particularly conve-nient if one considers that the most promising chance of inducing (intrinsic) spin-orbit coupling in graphene as oftoday is arguably by depositing adatoms on its surface[9–11]. B. Mechanism II: perpendicular electric field
Let us now exhibit another mechanism of symmetry-breaking which can provide a loophole to circumvent theintrinsic difficulty of generating a non-trivial QSH phasein bilayer graphene. Forgetting momentarily about spin-orbit coupling, let us go back to the Hamiltonian ofEq. (14) and consider the effect of an electric field ap-plied perpendicularly to the bilayer, H U = U σ η z . (28)This term opens a gap in the energy-momentum disper-sion relation by breaking the layer symmetry. It can beimplemented in a tight-binding model using the followingexpression: H U = − U (cid:88) i ∈ c † i c i + U (cid:88) i ∈ c † i c i . (29)The derivation of the Landau level spectrum requiressolving once more a quartic equation (cid:15) n + α n (cid:15) n + β n (cid:15) n + γ n = 0, with a non-vanishing linear term β n (cid:54) = 0: α n = − (cid:0) t + 2 U + (2 n − B (cid:1) β n = − τ U ∆ B γ n = U ( U + t ) − (2 n − B U + n ( n − B . (30)Taking into account the spin-degneracy of the spectrumand the additional symmetry (cid:15) n ( − τ ) = − (cid:15) n ( τ ), one isleft with two eigenvalues to compute for the lowest en-ergy Landau levels, one of which can be easily seen tobe (cid:15) + = τ U . The other one, (cid:15) − , must be computed nu-merically. In the limit U, (cid:126) ω c (cid:28) t , it can be estimatedperturbatively [30] as (cid:15) − ≈ τ U (1 − (cid:126) ω c /t ). Its depen-dence on U and (cid:126) ω c beyond this perturbative regime isshown in the lower panel of Fig. 7. Hence, the QH phasehas now been gapped by the perpendicular electric fieldat low energy (see top panel of Fig. 7), yielding the fol-lowing pattern for the Chern number: C ↑ = C ↓ = − , for − | (cid:15) + | < E F < −| (cid:15) − | C ↑ = C ↓ = 0 , for | E F | < | (cid:15) − | C ↑ = C ↓ = 1 , for | (cid:15) − | < E F < | (cid:15) + | (31)However, the layer degeneracy of the former zero-energyLandau levels has now been lifted, which means that anon-trivial QSH phase can once again be generated atlow energy, provided some spin-degeneracy lifting mecha-nism overcomes the gap | (cid:15) − | . This can be achieved eitherby layer-symmetric spin-orbit coupling (18) or by an ex-change term (25). At a critical value of the spin splitting -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.100.1 E = 0= 2= 4= -2= -4 _ ω c /t E / U E Figure 7: (Color online): (Top panel) Effect of an electricfield applied perpendicularly to the plane (yielding a layerpotential asymmetry U = 0 .
1) on the lowest Landau level ofbilayer graphene. A gap is opened arising form the lifting ofthe layer degeneracy for the lowest Landau level. Unspecifiedparameter values are the same as in Fig. 5. (Bottom panel):Dependence of the eigenvalue (cid:15) − on the magnetic field for U = 0 .
01. The dashed line is the analytical prediction (validin the perturbative limit (cid:126) ω c (cid:28) t ) and the thick line is thenumerical calculation. Inset: Dependence of (cid:15) − on the per-pendicular electric field for ∆ B = 0 . | ∆ ex | = | (cid:15) − | , the lowest bands will cross and give rise toa QSH phase C ↑ = − C ↓ = 1 , for | E F | < min( | ∆ ex |−| (cid:15) − | , U −| ∆ ex | ) , (32)characterized by a single pair of counter-propagatingspin-polarized edge states (see Fig. 8). Once againthe total Chern number vanishes while the Z invari-ant ν = 1, indicating the non-trivial character of theQSH phase. Provided the critical spin splitting couldbe achieved, the maximum value of the QSH gap wouldthis time be bounded by the value ( U − | (cid:15) − | ) /
2, an up- -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.100.1 E = 0 = 1 Figure 8: (Color online): Effect of an electric field applied per-pendicularly to the plane (yielding a layer potential asymme-try U = 0 .
1) on the lowest Landau level of bilayer graphene,in the presence of a spin-splitting term ∆ ex = 0 . per bound of which is given by the perturbative limit∆ maxQSH ≤ U (cid:126) ω c /t , as can be checked in the lower panelof Fig. 7.As a closing remark, we note that this second mech-anism of symmetry-breaking shares with that demon-strated in [35] the property of having edge states in thelow-energy region of Eq. (32) that are not only spin-polarized, but that can also be valley-polarized. Thiscan be traced back to the lifting of valley degeneracyby the perpendicular electric field, as is apparent in thevalues of the Landau levels given below Eq. (30). Thisvalley polarization actually translates into an additionaltopological protection, encoded in the valley Chern in-dex ˜ ν = (cid:80) τ τ C τ , with C τ = (cid:80) s C τ,s . In the energyregion of Eq. (32), this index verifies ˜ ν = 1, indicatingthat the low-energy phase is a so-called quantum val-ley Hall phase. The latter is entirely analogous to a QSHphase, if one exchanges spin and valley indices: it is char-acterized by valley-polarized counter-propagating edgestates, which can thus only be backscattered by short-range (valley-coupling) disorder. Hence, the low-energyphase of Eq. (32) should be immune to spin-mixing per-turbations as long as valleys remain uncoupled [63]. C. Discussion
Now that we have identified the regimes in which anon-trivial QSH phase could arise in bilayer graphene,and that we have roughly estimated the order of magni-tude of the associated energy gap ∆
QSH , let us conclude0this section by making a few comments on the experimen-tal relevance of our results. Until now, we have made thenatural assumption of disregarding the effect of disorderin our system, since one of the essential features of a topo-logical phase is its robustness with respect to disorder.The presence of the latter could nevertheless prove prob-lematic if the typical strength of disorder δ dis (cid:29) ∆ QSH .The available experimental data in graphene-like systemsseem to indicate that low-energy disorder is dominated bycharge density fluctuations (electron-hole puddles), butthe use of BN substrates has been shown to significantlyreduce their magnitude [36–38].The main other threat to the QSH phase lies in the var-ious many-body instabilities which have been predictedto occur in bilayer graphene at the Dirac point due to thefinite density of states. This could lead to a spontaneoussymmetry breaking of the spin-valley SU(4) symmetryin undoped bilayer graphene, causing the emergence ofa yet unidentified gapped phase, typically of the orderof a few meV [39–41]. Amusingly, a (many-body driven)QSH phase stands among the list of possible candidates[42, 43].Estimating the importance of disorder and interactioneffects eventually boils down to how big a value of theQSH gap could be achieved. If ∆
QSH lies in the 10 meVrange, then the presence of disorder should be harmlessto the QSH phase, while actually reducing the effect ofthe Coulomb interaction. On the other hand, if ∆
QSH israther in the 1 meV range, then chances are great thatdisorder and/or interactions will wash out the picture wedescribed.
V. EXTENSIONS
Let us now briefly discuss extensions of our model toclosely related systems. We start by considering differenttypes of stacking orders in bilayer graphene, and thenmove on to the case of trilayer graphene.
A. Other stackings
The analysis we performed in this article relied onthe assumption of AB (Bernal) stacking for the bilayer.However, other possibilities may occur. One of them isthe so-called AA-stacking, where both layers are mirror-symmetric: A atoms sit on top of A atoms and B atoms sit on top of B atoms. In this case, follow-ing the exact same steps as described in section II A,the Landau level spectrum can be obtained [44], (cid:15) AAn = ± (cid:112) t ( t ± | n | (cid:126) ω c ). Contrary to the case of Bernal stack-ing, the Landau level with lowest energy is no longernecessarily that corresponding to n = 0, which leads toa peculiar band structure (see Fig. 9) where the low-energy physics is described by counter-propagating spin-degenerate edge states, characterized by a trivial C = 0phase. In a sense, the absence of zero-energy Landau lev- -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.4-0.200.20.4 E = 0 Figure 9: (Color online): Landau level spectrum ofAA-stacked bilayer graphene with parameter values as inFig. 5. Notice how, at low energy, spin-degenerate counter-propagating edge states lead to a trivial topologically trivialphase. els in this system, which we took as our defining criteriumfor a 2DDFG, is directly responsible for the absence of atopological order at zero energy. We additionally checkedthat the symmetry-breaking mechanims investigated inthis work are ineffective for the present system.Besides AB and AA stackings, a whole (continuous)family of bilayers referred to as twisted bilayers can bestudied experimentally. Such bilayers are defined by theangle with which the upper layer is twisted from the lowerlayer. This angle can be probed experimentally by char-acterizing the induced Moir´e patterns. Although suchsystems are also interesting in their own right (and ex-perimentally relevant), they are not well suited to a tight-binding description, especially for small angles, as thelow-energy physics requires potentially very long-rangehoppings to be taken into account. We will thereforenot discuss them any further, and we refer the reader toother approaches developed in the literature to addresstheir properties (see for example [45]).Likewise, so-called double layer graphene [46] – a bi-layer where the coupling between the layers is solelycapacitive (transverse hopping is zero) – crucially re-quires electrostatic screening to be taken into account,and therefore lies beyond the scope of this paper.
B. Trilayer graphene
In the light of our understanding of single layer and bi-layer graphene, we finish by briefly discussing how muchof our previous considerations could find a natural ex-tension in trilayer graphene [64]. One generally distin-guishes two stacking orders (see Fig. 10): ABA stacking,1 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
BA tBA t t BA
01 10 022 3 3
ABA stacking ABC stacking
A B t A B A B t t t t t t Figure 10: Side view of typical stacking sequences of a trilayerof graphite: ABA stacking is mirror-symmetric with respectto the central layer, while ABC stacking can be seen as thenatural extension of Bernal stacking in the bilayer (see Fig. 4). characterized by a combination of linear and quadraticdispersions at low energy, and ABC stacking, character-ized by a cubic dispersion and a corresponding diverg-ing density of states at low energy which favors many-body instabilities. Regardless of the stacking sequence,the odd number of layers implies that, as in monolayergraphene, including spin-orbit coupling in each layer (viaa naive extension of Kane and Mele’s model) will yield anon-trivial QSH phase [65]. The additional presence of aperpendicular magnetic field – which has experimentallybeen shown to give rise to a QH effect [47–50] character-ized by a spectrum with a 12-fold degenerate zero-energyLandau level and described by the following values forthe Chern number, (cid:26) C ↑ = C ↓ = − , for − ∆ LL < E F < C ↑ = C ↓ = +3 , for 0 < E F < ∆ LL (∆ so = 0)(33)where ∆ LL is the energy of the lowest non-zero Landaulevel – will yield a transition from a QH to a non-trivialQSH phase at low-energy (top panel of Fig. 11): (cid:26) C ↑ = − C ↓ = 3 , for | E F | < ∆ so C ↑ = C ↓ = 3 , for ∆ so < E F < (cid:112) ∆ LL + ∆ (34)yielding C = 0 and ν = 1 (mod 2) when | E F | < ∆ so .Exploring further the fate of the Landau level spec-trum, we have checked that applying spin-orbit couplingonly in the upper layer is (QSH-wise) ineffective. How-ever, applying a perpendicular electric field, through thetight-binding expression H U = − U (cid:88) i ∈ c † i c i + U (cid:88) i ∈ c † i c i , (35)has an interesting effect which distinguishes ABA fromABC stacking. In the latter case, it opens a gap, while inthe former it does not (though the QH phase is trivial atlow energy, due to counter-propagating states). When anexchange term is taken into account, a QSH phase withonly a single pair of counter-propagating spin-polarized -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.4-0.200.20.4 E = 6= -6= 0 = 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2k x -0.100.1 E = 0 = 1 Figure 11: (Color online): Landau level spectrum of (ABC-stacked) trilayer graphene with spin-orbit coupling λ so = 0 . U =0 .
1) and spin-splitting ∆ ex = 0 .
05 (bottom panel). The ef-fect of spin-orbit coupling is completely analogous to thatin bilayer graphene (see Fig. 5), causing a lifting of spin-degeneracy in the lowest Landau level. This time, however,the odd number of pairs of spin-polarized counter-propagatingedge states leads to a non-trivial QSH phase at low energy(top panel). Additionally, and also as in bilayer graphene,the simultaneous presence of a layer-degeneracy lifting elec-tric field and a spin-splitting term can also give rise to a non-trivial QSH phase at low energy, with a single pair of counter-propagating spin-polarized edge states (bottom panel). Oncemore, unspecified parameter values are the same as in Fig. 5. states can be accessed (bottom panel of Fig. 11). Thus,our second symmetry-breaking mechanism seems to workequally well in trilayer graphene, although its relevanceis debatable in the present context since, as mentionedabove, a QSH phase could already be obtained in trilayergraphene in the absence of any layer inversion symmetry-breaking. Additionally, the width of the energy window2 N -layer graphene in the QH regime Low-energy topological phase N = 1 (Fig. 2) QH with C = ± N = 1 with ∆ so (Fig. 3) QSH N = 2 (Fig. 5 top) QH with C = ± N = 2 with ∆ so (Fig. 5 bottom) weak QSH with ν = 0 (mod 2) N = 2 with ∆ so only in upper layer (Fig. 6 top) QH with C = ± N = 2 with ∆ so only in upper layer, and ∆ ex QSH N = 2 with U (Fig. 7 top) ∅ N = 2 with U , and | ∆ ex | > | (cid:15) − | (Fig. 8) QSH + QValleyH N = 3 QH with C = ± N = 3 with ∆ so (Fig. 11 top) QSHTable I: Summary of low-energy topological phases in graphene-based 2DDFGs. where our mechanism is effective decreases with the num-ber of layers, which can be qualitatively understood asoriginating from the proliferation of bands (due to theincreasing degeneracy of the lowest Landau level). VI. CONCLUSION
We have considered different examples of graphene-based 2DDFGs and shown that, in the presence of bothspin-orbit coupling and a perpendicular magnetic field,a topological phase transition between a QH and a QSHphase could take place at low energy. An overall sum-mary of the various cases discussed in this article is pro-vided in Table I. While the lifting of spin degeneracy inthe Landau level spectrum was the only requirement toobserve this transition in monolayer graphene, we showedthat a similar prescription proves insufficient in bilayergraphene, yielding a weak QSH phase at low energy.We then proceeded to identify several regimes in whicha non-trivial QSH phase, characterized by a single pairof counter-propagating spin-polarized edge states, can beinduced in bilayer graphene, all of which involved break-ing the layer inversion symmetry. We investigated twopossible ways of achieving this: (i) by considering thepresence of spin-orbit coupling only in the upper layer; (ii) by applying a perpendicular electric field. In bothcases, the resulting low-energy phase can then be tunedinto a non-trivial QSH phase in the presence of an ex-change field: in case (i), an arbitrarily small exchangeterm suffices, while in case (ii), a non-zero critical valueis required. The first of these two cases has the advan-tage of requiring only small spin splitting (which will,however, effectively control the magnitude of the inducedQSH gap) and the presence of spin-orbit coupling onlyin the upper layer. The latter condition is crucial, as themost promising way to induce sizeable (intrinsic) spin-orbit coupling in graphene is arguably by random adatomdeposition [9–11]. Indeed, the effectively weak intrinsicspin-orbit coupling of carbon remains as of today themain obstacle in the attempt of experimentally detectingthe QSH phase in graphene-like systems. In this respect,other recently isolated two-dimensional crystals such assilicene [51–53] or cold atom optical lattices [54] mightoffer an alternative to probe the physics described in thiswork.
Acknowledgments
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