Topological Valley Currents in Gapped Dirac Materials
Yuri D. Lensky, Justin C. W. Song, Polnop Samutpraphoot, Leonid S. Levitov
TTopological Valley Currents in Gapped Dirac Materials
Yuri D. Lensky, Justin C. W. Song, Polnop Samutpraphoot, Leonid S. Levitov
Physics Department, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA02139
Gapped 2D Dirac materials, in which inversion symmetry is broken by a gap-opening perturbation,feature a unique valley transport regime. The system ground state hosts dissipationless persistentvalley currents existing even when topologically protected edge modes are absent or when theyare localized due to edge roughness. Topological valley currents in such materials are dominatedby bulk currents produced by electronic states just beneath the gap rather than by edge modes.Dissipationless currents induced by an external bias are characterized by a quantized half-integervalley Hall conductivity. The under-gap currents dominate magnetization and the charge Hall effectin a light-induced valley-polarized state.
Bloch bands in materials with broken inversion sym-metry can feature Berry curvature, an intrinsic physicalfield which dramatically impacts carrier transport[1, 2].The key manifestation of Berry curvature is the anoma-lous Hall effect (AHE), arising in the absence of mag-netic field due to topological currents flowing in systembulk transversely to the applied electric field[3, 4]. Ofhigh current interest are Dirac materials with severalvalleys, such as graphene and transition metal dichalco-genide monolayers[5, 6]. Topological currents in thesesystems have opposite signs in different valleys and, ifintervalley scattering is weak, can give rise to long-rangecharge-neutral valley currents. Such currents have beenobserved recently in graphene/hBN superlattices[7]. Al-ternatively, if valley polarization is induced by light withnonzero helicity, a charge Hall effect is observed[8].Topological effects are particularly striking in gappedsystems where Chern bands support topologically pro-tected edge modes and quantized transport [9–12]. How-ever, existing Valley Hall materials[5–8] lie squarely out-side this paradigm. First, gapless edge states in thesematerials are not enforced by topology or symmetry andmay thus be absent. Second, even when present, thesestates are not protected against backscattering and local-ization. Na¨ıvely, the lack of edge transport would leadone to conclude that topological currents cease to exist.If true, this would imply that the key manifestations,such as the valley Hall conductivity and orbital magne-tization, vanish in the gapped state[6].Here we argue that the opposite is true: the absenceof conducting edge modes does not present an obstaclesince valley currents can be transmitted by bulk statesbeneath the gap. As we will see, rather than being van-ishingly small, valley currents peak in the gapped state.Further, we will argue that such currents are of a per-sistent nature, i.e. they represent a ground state prop-erty, an integral part of thermodynamic equilibrium. Ina valley-polarized state, the under-gap currents dominatemagnetization and the charge Hall effect.The effects due to under-gap states, discussed below,should be contrasted with those due to deep-lying stateswhich are responsible for field-theoretic anomalies[13,
FIG. 1: Persistent valley currents inside and outside pn junc-tion. The currents arise from side jumps of band carriersjust beneath and just above the gap upon reflection from thegapped region, as illustrated by trajectories in Fig.2. Theunder-gap and over-gap currents (red and blue regions) flowin opposite directions and fully cancel deep in the Fermi sea.The two contributions are maximally uncompensated insidethe region − x < x < x , giving a maximum current value of j = e h E per valley, where E is the built-in electric field. He[17]. Importantly, the deep-lying states in our system obey inversion symmetry andthus do not contribute to valley transport. Indeed, aweak gap-opening perturbation which breaks inversionsymmetry for states with energies near the Dirac pointwill have no impact on the deep-lying states. This is quiteunlike the anomaly situation where symmetry is brokenby regularization at the bandwidth scale but remains in-tact at lower energies. The regime studied here, wherevalley currents are dominated by states just beneath thegap, is unique for systems with a weak inversion-breakingperturbation. A similar behavior is expected in systemssuch as graphene bilayers in a transverse E field and a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec twisted graphene bilayers.To gain insight into these delicate issues, we consider amodel edge-free gapped system: a pn junction in gappedgraphene created by a built-in electric field, see Fig.1.This system features an interesting spatial distribution ofvalley currents. As we will see, in contrast to the resultsof Ref.[6], valley currents reach maximum value in thegapped pn region − x < x < x . The origin of such(perhaps counterintuitive) behavior is as follows. Valleycurrents are due to the states just above and just belowthe gap and, crucially, are of opposite sign for the twogroups of states. These states are either both depletedof carriers or both filled away from the gapped region,giving contributions that nearly cancel. This produces anet current decreasing to zero away from the pn region,see Eq.(14). Within the pn region these contributionsare maximally imbalanced, creating a maximum current.Further, the current is quantized to a half-integer valueper valley, j = e h E , where E is the built-in field.Our analysis, which is microscopic and explicit, ap-plies equally well to a spatially uniform gapped systemunder bias (and with no gate-induced built-in field), pre-dicting a quantized Valley Hall effect with σ xy = e h pervalley. In terms of the arrangement shown in Fig.1 thiscorresponds to system sizes L smaller than the gappedregion width 2 x = E g /eE , i.e. weak bias voltage eV = eEL (cid:28) E g . Since valley currents in this case aretransmitted by the under-gap states in the system bulk,they are nondissipative. Below we also discuss valley edgecurrents resulting from the side jumps of the under-gapstates upon reflection from system boundary, see Fig.3.Together with bulk currents, such edge currents ensurethe valley flow continuity. These currents circulate alongthe edge, producing orbital magnetization in the systemground state, see Eq.(15).We model carriers in each valley as 2 + 1 massive Diracparticle in the presence of a static uniform electric fieldwhich defines a pn junction: H = (cid:18) ∆ vp − vp + − ∆ (cid:19) − eEx, p ± = p ± ip , (1)where p , denote momentum components p x,y . The sys-tem ground state is a Fermi sea with a density gradientimposed by the E field, n-doped on one side and p-dopedon the other side of a gapped region, see Fig.1. Simpleas it is, the above Hamiltonian captures all essential ele-ments of interest: tunneling through the gapped region,AHE in surrounding regions, and their interplay.Our approach relies on a mapping onto a fundamentalproblem in quantum dynamics: a pair of quantum levelsdriven through an avoided level crossing. The Landau-Zener (LZ) problem describing these transitions admitsan exact solution[18]. The LZ theory provides a very gen-eral method that accounts for the AHE transport bothoutside and inside the gapped pn region, as well as for FIG. 2: a) The under-gap and over-gap trajectories near thegapped region, Eq.(13). Skewed Hall-like motion gives rise toside jumps. Shown are normally incident trajectories (red forelectrons, blue for holes). The opposite-flowing under-gap andover-gap currents partially cancel when summed over all filledstates, producing net currents flowing in the same directionin the p and n regions, with the maximum current attainedin the middle region − x < x < x , see Eq.(14) and Fig.1.b) Spin-1/2 interpretation of side jumps. Magnetization m ( t )evolves adiabatically in a slowly varying field b ( t ) that sweepsa plane perpendicular to n , see Eq.(7). Magnetization tracksthe field but lags slightly behind, rotating out of the plane andacquiring a component parallel to n , see Eq.(11). So does thevelocity vector which is aligned with m ( t ). tunneling through this region. Below we discuss the rela-tion between our LZ approach and the conventional qua-siclassical approach based on the adiabatic theorem andBerry phase[1, 2]. Since the LZ approach is not restrictedto the adiabatic limit, it provides a full description of non-adiabatic effects, associated with tunneling through thegapped region in our transport problem. Such effects,which are naturally described in the LZ framework, arenot accounted for by the quasiclassical approach.Mapping of Eq.(1) onto the LZ problem is accom-plished in two steps. We first note that in the momen-tum representation εψ = Hψ is a first-order differentialequation, since the only term containing a derivative is − eEx = eEi ¯ h∂ p . We can thus rewrite our equationas a time-dependent Schroedinger equation for a 2 × t = p /eE playing the role of time: i ¯ h∂ t ψ ( t ) = ˜ H ( t ) ψ ( t ) , ˜ H ( t ) = βtσ + vp σ + ∆ σ , (2)where we set ε = 0 without loss of generality and defined β = veE . Next, by interchanging spin components via σ ↔ σ , σ → − σ we bring ˜ H to the canonical LZ form˜ H ( t ) = (cid:18) βt ∆ p ∆ ∗ p − βt (cid:19) , ∆ p = ∆ + ivp, (3)where from now on we use p instead of p for brevity.Time evolution in Eq.(2) defines a unitary S-matrixwhich takes its simplest form in the adiabatic basis of in-stantaneous eigenstates of ˜ H ( t ). These states correspondto a particle moving in a classically allowed region, p orn, without tunneling through the gapped region. Tun-neling is thus described by the LZ transitions betweendifferent adiabatic states. Written in the adiabatic basis,the S-matrix is of the form S = (cid:18) √ q −√ − qe iϕ √ − qe − iϕ √ q (cid:19) , q = e − πδ , (4)where δ = | ∆ p | / β ¯ h . Here the phase ϕ is given by[19] ϕ = π/ − iδ ) + δ (ln δ −
1) + arg ∆ p (5)with Γ( z ) the Gamma function. The non-adiabatic andadiabatic LZ transitions, taking place with the proba-bilities q and 1 − q , correspond to particle transmissionthrough the gapped region and reflection from it. Theevolution is adiabatic at small β , with the system track-ing one of the instantaneous eigenstates of ˜ H ( t ) and non-adiabatic transitions exponentially suppressed, q → y displacement as (cid:104) δy (cid:105) = (cid:104) ψ | i ¯ h∂ p | ψ (cid:105) with the expectation value taken over the left- and right-incident states, | L (cid:105) = S (cid:18) (cid:19) , | R (cid:105) = S (cid:18) (cid:19) . We find (cid:104) δy (cid:105) L,R = ± ∂ϕ/∂p = ± (cid:96) (1 − q ) , (cid:96) = ¯ hv ∆ / | ∆ p | , (6)where only the last term of the phase in Eq.(5) gives acontribution even in p contributing to the net valley cur-rent. Interestingly, the result in Eq.(6) only depends on1 − q that corresponds to reflection, indicating that sidejumps occur only at reflection from the gapped region butnot at transmission through it. The side jump directionreverses upon reversing the sign of ∆, giving values ofopposite sign for valleys K and K (cid:48) as expected for ValleyHall transport.Encouraged by these observations, here we constructindividual one-particle quantum states exhibiting sidejumps. Since Dirac particle velocity is expressed throughits spin, v = i ¯ h [ x , H ] = v ( σ , σ ), it will be convenientto represent LZ dynamics as spin 1 / m ( t ) = (cid:104) ψ ( t ) | s | ψ ( t ) (cid:105) , s i = ¯ h σ i , ∂ t m = b ( t ) × m , b ( t ) = 2¯ h (∆ , − vp, βt ) (7)where the magnetic field b ( t ) orientation changes from − z to + z over −∞ < t < ∞ .We focus on the weak field regime eE (cid:28) ∆ /(cid:96) = ∆ / ¯ hv .In the LZ formulation (3) this corresponds to spin 1 / b ( t ) whichrotates in the plane perpendicular to the vector n = (sin α, cos α, , tan α = vp/ ∆ . (8)Crucially, the adiabatic spin evolution in a rotating field b ( t ) can generate a component of m (and thus of the ve-locity) transverse to the rotation plane and thus pointing along n . This happens because when the field rotates inthe plane perpendicular to n the spin tries to follow it butis left slightly behind. Then, as a result of Bloch preces-sion, the spin rotates out of the plane swept by b ( t ), seeFig.2. This component is proportional to rotation speed,i.e. is not exponentially small in the adiabatic limit.Such a behavior, while somewhat counterintuitive, canbe understood as follows. We usually think of a spinprecessing in a strong but slowly changing magnetic fieldis being “slaved to the field”. This is basically correct,however the spin excursions away from the field directioncan be nonexponential due to the Berry curvature effects.This is precisely the case in our problem.It is convenient to use a (nonuniformly) rotating framein which the field b ( t ) has a frozen orientation. Wewrite | ψ ( t ) (cid:105) = U ( t ) | ψ (cid:48) ( t ) (cid:105) with the unitary transfor-mation U ( t ) chosen so that the field b (cid:48) ( t ) defined by U − ( t ) ( b ( t ) · s ) U ( t ) = b (cid:48) ( t ) · s is directed along a fixedaxis. For the Hamiltonian in Eq.(3) the operator U ( t )with this property can be defined as a spin rotation U ( t ) = e i ¯ h θ ( t ) n · s , tan θ ( t ) = βt/ | ∆ p | , (9)where θ ( t ) is the angle between vectors b ( t ) and b (0) = h (∆ , − vp, i ¯ h∂ t | ψ (cid:48) ( t ) (cid:105) = (cid:16) b (cid:48) ( t ) · s − i ¯ hU − ( t ) ˙ U ( t ) (cid:17) | ψ (cid:48) ( t ) (cid:105) . (10)The last term equals − i ¯ hU − ( t ) ˙ U ( t ) = ∂θ ( t ) ∂t n · s givinga spin Hamiltonian with an effective field b (cid:48) ( t ) + ∂θ ( t ) ∂t n .So far our analysis has been completely general, nowwe specialize to an adiabatic evolution in which the spinorientation tracks the field. In this case, when viewed inour rotated frame, m ( t ) remains aligned with the vector b (cid:48) ( t ) + ∂θ ( t ) ∂t n at all times. Transforming back to the labframe, we conclude that m ( t ) tracks the field˜ b ( t ) = b ( t ) + ∂θ ( t ) ∂t n (11)which, because of the last term, has an additional y com-ponent. Finally, since the velocity operator v = v ( σ , σ )expectation value is aligned with m , the velocity compo-nents are easily evaluated as v x,y = v ˜ b x,y / | ˜ b | giving v x ( t ) = vβtε ( t ) , v y ( t ) = v pε ( t ) + v ∆ β ε ( t ) , (12)where ε ( t ) = ± (cid:112) β t + | ∆ p | with the plus/minus signdescribing p and n states. Here we normalized ˜ b ( t ) ap-proximating | ˜ b ( t ) | ≈ | b ( t ) | . Trajectories are readily ob-tained by integrating velocity, giving x ( t ) = vβ ε ( t ) , y ( t ) = v p ln ε ( t )+ βt | ∆ p | | ∆ p | + v ∆ βt | ∆ p | ε ( t ) (13)(here we have suppressed integration constants). The lastterm in Eqs.(12),(13) originates from Berry curvature,giving rise to side jumps, see Fig.2. The net side jumpvalue is δy = (cid:82) ∞−∞ v y ( t ) dt = v ∆ / | ∆ p | , which matchesthe result found above for p = 0.These results are in accord with the classical equationsof motion augmented with the anomalous velocity termdescribing the nonclassical Berry’s “Lorentz force:”[1, 2]˙ p = e E , ˙ x = ∇ p ε ± ( p ) + Ω( p ) × ˙ p , Ω( p ) = v ∆2 ε ± ( p ) , where ε ± ( p ) = ± ( v p + ∆ ) / is particle dispersion.Current density, found by summing the velocity contri-butions of all states in the Fermi sea, is j ( x ) = (cid:26) j , | x | < x j x / | x | , | x | > x , j = e h E (14)per valley. The current peaks in the gapped region,falling off inversely with distance outside this region, asshown in Fig.1a. An identical result is obtained by in-tegrating the velocity in Eq.(12) over allowed values of p . As discussed above, this behavior originates micro-scopically from the contribution of the under-gap tra-jectories side jumps dominating in the gapped region,however being partially canceled by the over-gap trajec-tories contribution outside this region. Interestingly, thelinear dependence j vs. E translates into a universal, E -independent net current flowing through the gappedregion, I | x |