Valley-polarized quantum anomalous Hall phase and disorder induced valley-filtered chiral edge channels
VValley-polarized quantum anomalous Hall phase and disorder induced valley-filteredchiral edge channels
Hui Pan, Xin Li, Hua Jiang, Yugui Yao, and Shengyuan A. Yang Department of Physics, Beihang University, Beijing 100191, China College of Physics, Optoelectronics and Energy, Soochow University, Suzhou 215006, China School of Physics, Beijing Institute of Technology, Beijing 100081, China Engineering Product Development, Singapore University of Technology and Design, Singapore 138682, Singapore
We investigate the topological and transport properties of the recently discovered valley-polarizedquantum anomalous Hall (VQAH) phase. In single layer, the phase is realized through the com-petition between two types of spin-orbit coupling, which breaks the symmetry between the twovalleys. We show that the topological phase transition from conventional quantum anomalous Hallphase to the VQAH phase is due to the change of topological charges with the generation of ad-ditional skyrmions in the real spin texture, when the band gap closes and reopens at one of thevalleys. In the presence of short range disorders, pairs of the gapless edge channels (one from eachvalley in a pair) would be destroyed due to intervalley scattering. However, we discover that inan extended range of moderate scattering strength, the transport through the system is quantizedand fully valley-polarized, i.e. the system is equivalent to a quantum anomalous Hall system withvalley-filtered chiral edge channels. We further show that with additional layer degree of freedom,much richer phase diagram could be realized with multiple VQAH phases. For a bilayer system, wedemonstrate that topological phase transitions could be controlled by the interlayer bias potential.
PACS numbers: 73.43.Cd, 71.70.Ej, 73.22.-f, 73.63.-b
I. INTRODUCTION
Gapless one-dimensional (1D) edge channels are in-triguing physical objects which are usually associatedwith nontrivial topological phases of two-dimensional(2D) systems. For example, the precise quantization ofHall plateaus in quantum Hall effect is tied with thedissipationless chiral edge channels and is related to abulk topological invariant known as the Chern number(or TKNN invariant).
It was later realized that theexistence of such edge channels in fact does not necessar-ily require the orbital effects of external magnetic field.Instead, it could arise from the combined effects of spinpolarization (e.g. due to magnetic ordering) and spin-orbit coupling (SOC).
This topological phase, knownas the quantum anomalous Hall (QAH) phase, has beensought for more than 20 years. Hence its first realizationin magnetically doped topological insulator thin filmshas attracted significant attention and research activitiesrecently.
It is possible that these edge channels may carry addi-tional flavors. For example, in quantum spin Hall effect,the chirality of the edge channel is tied with its spin,hence on each edge, there is a Kramers pair of counter-propagating spin-polarized edge channels, which are pro-tected by time reversal symmetry.
When the bandstructure has multiple energy extremas, carriers couldhave another type of flavor, valley. Similar to spintron-ics, it was proposed that this valley degree of freedommay also be utilized for information processing, leadingto the concept of valleytronics.
It has been shownthat there could be topological charges associated withthe valleys and it is possible to realize 1D channels thatcarry specific valley indices.
However, in these pre- vious studies, the numbers of 1D channels in each valleyare balanced, due to the presence of either time reversalsymmetry or inversion symmetry.In a recent work, we demonstrated that by break-ing both time reversal symmetry and inversion symme-try, it is possible to achieve a novel topological phase,the valley-polarized quantum anomalous Hall (VQAH)phase. The hallmark of this phase is that at systemedges where valleys can be distinguished, there existunbalanced numbers of counter-propagating chiral edgechannels associated with the two valleys. This imbal-ance automatically indicates that the system is in a QAHphase. The additional valley features of the edge chan-nels are manifestations of the unbalanced valley topolog-ical charges in the bulk. Therefore such phase is char-acterized by two bulk topological invariants: the totalChern number C and the valley Chern number C v . Wehave found that such a novel phase could be realized dueto the competition between two types of SOCs in a lowbuckled honeycomb lattice model which may describe 2Dmaterials such as silicene or germanene. Due to length restrictions, several important physicalaspects of the VQAH phase were not exposed in the pre-vious work. In the present paper, we would address thesedetails. More importantly, we greatly extend our previ-ous work by investigating the disorder effects on VQAHphase and the valley-polarized topological phases in a bi-layer system. The valley polarized channels are robustagainst smooth-varying scattering potentials, expectedfrom the large momentum separation between the twovalleys. Through explicit transport calculations, we showthat for short range scatterers, although the interval-ley scattering would destroy pairs of counter-propagatingvalley channels, remarkably, the remaining |C| channels a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec can still retain their valley character in transport. Asa result, each edge could serve as a perfect valley fil-ter with chiral edge channels for one specific valley. Weshow that this happens for an extended window of in-termediate scattering strength. Furthermore, for a bi-layer system formed by stacking two single layers, wefind that the resulting properties are not simply super-position of the two. In fact, the bilayer system exhibitsa much richer phase diagram including several VQAHphases with different ( C , C v ) invariants. The topologicalphase transitions between these phases could be moreeasily controlled by tuning the interlayer bias potential.Our paper is organized as following. In Sec. II, wediscuss the VQAH phase in a single layer lattice model.By decomposing the topological charges into real spinand pseudospin sectors, we show that the topologicalphase transition to VQAH phase is accompanied withthe change of real spin topological charge in one valley.In Sec. III, we study the effects of short range scatteringon the VQAH phase based on a two-terminal transportcalculation and show that in a window of intermediatescattering strength, only one chiral edge channel with K (cid:48) valley character is left. In Sec. IV, we investigate the richtopological phases in a bilayer system and demonstratethe tunability of the phases through interlayer bias andexchange field strength. Finally, we give our conclusionand summarize our results in Sec. V. II. VALLEY-POLARIZED QAH PHASE INSINGLE LAYER SYSTEM
The VQAH phase was first discovered in a latticemodel defined on a low-buckled single layer honeycomblattice. The tight-binding Hamiltonian is written as H = − t (cid:88) (cid:104) ij (cid:105) α c † iα c jα + it SO (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) αβ ν ij c † iα s zαβ c jβ − it R (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) αβ µ ij c † iα ( s × ˆ d ij ) zαβ c jβ + it R (cid:88) (cid:104) ij (cid:105) αβ c † iα ( s × ˆ d ij ) zαβ c jβ + M (cid:88) iαβ c † iα s zαβ c iβ . (1)Here c † iα ( c iα ) is a creation (annihilation) operator for anelectron with spin α at site i . The summation with (cid:104) ... (cid:105) ( (cid:104)(cid:104) ... (cid:105)(cid:105) ) runs over all nearest (next-nearest) neigh-bor sites. The s ’s are the Pauli matrices for real spindegree of freedom. For the right hand side, the first termis the usual nearest neighbor hopping term. The secondterm is the so-called intrinsic SOC term involving thenext-nearest neighbor hopping, ν ij = +1( −
1) if theelectron makes a left (right) turn in going from site j tosite i along the nearest-neighbor bonds. The third andfourth terms are the intrinsic and extrinsic Rashba SOCterms respectively. ˆ d ij is the unit vector pointing fromsite j to i , and µ ij = ± AB -sublattice. FIG. 1. (color online) Upper panel: bulk band structurealong the line of k y = 0. Lower panel: the correspondingenergy spectra for a zigzag edged ribbon with a width of 400atomic sites. (a, d) with only extrinsic Rashba SOC t R =0 .
06 and no intrinsic Rashba SOC; (b, e) with only intrinsicRashba SOC t R = 0 . t R = 0 .
06 and t R = 0 .
1. Other model parameters are set as t = 1, and M = 0 . The last term represents an exchange coupling. The var-ious t ’s and M denote the strengths of the terms.This model was first derived in the study of low en-ergy physics of silicene. The various SOC terms are thesymmetry-allowed terms for the low buckled honeycomblattice structure. The exchange term breaks the timereversal symmetry, which is necessary for the realizationof QAH phase.
As discussed in the previous work, the VQAH phase results from the competition betweenthe intrinsic and extrinsic Rashba SOC terms, i.e. thethird term and the fourth term in Eq.(1). The intrinsicRashba term is due to the mirror symmetry breaking ofthe 2D plane from the lattice buckling. The extrinsicRashba term further breaks the inversion symmetry andit can result from a perpendicular electric field or from asubstrate. To simplify the analysis, in the following weshall focus on these two SOC terms and neglect the in-trinsic SOC term. The effects of the intrinsic SOC termand possible sublattice symmetry breaking term will bediscussed later in the paper.First, we examine the properties of the model when ei-ther intrinsic Rashba or extrinsic Rashba term is present,but not both. Figure 1(a) shows the bulk energy spec-trum near the band gap along k x direction when onlyextrinsic Rashba SOC is present, and Fig. 1(d) shows thecorresponding energy spectrum for a ribbon with zig-zagedge termination. The results for the case with only in-trinsic Rashba SOC are shown in Fig. 1(b) and 1(e). Oneobserves that for both cases, the system is in insulatingstate with a finite band gap. In the results for ribbons,four gapless chiral edge states can be identified in theband gap. From their wave functions, it is easily checkedthat on each edge, there are two edge states propagat-ing in the same direction, indicating a QAH phase with C = 2. The Chern number can be directly calculatedfrom the bulk band structure using the formula C = 12 π (cid:88) n ∈ occ. (cid:90) BZ d k Ω n , (2)where the integration is over the Brillouin zone and thesummation is over all occupied valence bands. Ω n is themomentum-space Berry curvature for the n -th band Ω n ( k ) = − (cid:88) n (cid:48) (cid:54) = n (cid:104) ψ n k | v x | ψ n (cid:48) k (cid:105)(cid:104) ψ n (cid:48) k | v y | ψ n k (cid:105) ( ε n (cid:48) k − ε n k ) , (3)where v x ( y ) is the velocity operator and | ψ n k (cid:105) is theBloch eigenstate with eigen-energy ε n k . The magnitudeof Berry curvature is usually peaked at avoided bandcrossings where the gap is small. For a system withmultiple valleys, such as the case here with two valleys K and K (cid:48) , Berry curvature will be concentrated aroundthe valley centers. This allows us to define a topologi-cal charge associated with each valley by integrating theBerry curvature over the neighborhood of each valley asin Eq.(2).
We denote the results by C K and C K (cid:48) .They represent the contribution to the total Chern num-ber from each valley, and their difference C v = C K − C K (cid:48) is called the valley Chern number. Note that the con-cept of valley as well as the valley topological numbersare well-defined only when the low energy regions arewell separated in the reciprocal space. This condition isensured in our following calculations. In the studied pa-rameter range, the various SOCs are small perturbationscompared with the nearest-neighbor hopping which is thelargest energy scale.Straightforward calculations using the present modelconfirm that for both cases, the system is in the sameQAH phase with C = 2, and C v = 0 showing that contri-bution from the two valleys are equal ( C K = C K (cid:48) = 1).Indeed, on one edge, each valley contributes one chiraledge channel propagating in the same direction.The situation becomes quite different when both SOCsare present. Starting from a fixed intrinsic Rashba SOCas in Fig. 1(b, e), gradually increasing the strength ofextrinsic Rashba SOC, it has been shown that the gapat K valley remains open but the gap at K (cid:48) valley closesand reopens, leading to a topological phase transition tothe VQAH phase with C v (cid:54) = 0. As shown in Fig. 1(c) and1(f), in VQAH phase, the two valleys become asymmet-ric. In K valley, there are still two gapless edge states,but in K (cid:48) valley there are four. These states can be moreclearly seen in the zoom-in images in Fig. 2(a) and 2(b).In Fig. 2(c), we schematically plot the spatial distributionof these edge states in the ribbon geometry. One notesthat on each edge, there are two chiral channels from K (cid:48) valley propagating in opposite direction to only one chan-nel from K valley. Calculation of the topological chargeshows that C K in this case is still 1, but C K (cid:48) changes from1 to −
2, which is consistent with the doubling of the K (cid:48) FIG. 2. (color online) (a) and (b) are the enlarged spectrashowing the gapless edge states (a) in K valley and (b) in K (cid:48) valley, corresponding to the boxes in Fig 1(f). (c) Schematicfigure of the spatial distribution of the edge channels in theribbon. The colors label the edge states at different valleys.(d) Phase diagram as a function of t R and M . The dashedlines are the phase boundaries where the bulk gap closes.Phase I is the conventional QAH phase, while Phase II isthe VQAH phase. valley edge channels and the reversed chirality. In sucha VQAH phase, on the edge, there is an imbalance be-tween channels from different valleys, and in the bulk, itis characterized by both a nonzero C and a nonzero C v .In the present case, we have ( C = − , C v = 3).By looking at the figures in Fig. 1, one notices that forthe cases with either one Rashba SOC, the energy spectraat K and K (cid:48) valleys are symmetric. But when both SOCsare present, the symmetry of the spectra is broken. Tounderstand this, we expand the model around the K and K (cid:48) points to obtain the low energy effective Hamiltonian.The corresponding forms of the kinetic energy term, theextrinsic Rashba term, the intrinsic Rashba term, andthe exchange coupling term are H ( k ) = v ( τ z σ x k x + σ y k y ) , (4a) H R ( k ) = λ R ( τ z σ x s y − σ y s x ) , (4b) H R ( k ) = λ R σ z ( k y s x − k x s y ) , (4c) H M ( k ) = M s z , (4d)where τ z = ± K and K (cid:48) valleys, σ ’s are Paulimatrices representing the AB -sublattice pseudospin de-gree of freedom, the coupling strengths in these termsare related to the parameters in Eq.(1) by v = √ t/ λ R = 3 t R /
2, and λ R = 3 t R /
2. When the extrinsicRashba SOC is absent, i.e. λ R = 0, as in Fig. 1(b, e), theremaining terms all have inversion symmetry, P = σ x ,such that PH ( k ) P − = H ( − k ), meaning that the spec-tra at the two valleys must be symmetric under inver-sion. When extrinsic Rashba term is present, the inver-sion symmetry is broken. However, in the absence ofintrinsic Rashba term, the low energy model has anothersymmetry Q = σ x s z which is an inversion with an ad-ditional spin rotation, such that QH ( k ) Q − = H ( − k ).Therefore the spectra in Fig. 1(a, d) also exhibit simi-lar symmetric feature. We emphasize that Q is not anintrinsic symmetry for the crystal, it is an emergent sym-metry only for the low energy model. Finally when bothRashba terms are present, the two symmetries P and Q are both broken. The spectra at the two valleys becomeasymmetric, as observed in Fig. 1(c, f). The asymme-try between the two valleys is a necessary condition forrealizing the VQAH state.The topological phase transitions from conventionalQAH phase to VQAH phase can be realized by tuningthe model parameters. In Fig. 2(d), we show the phasediagram in the ( t R - M ) plane at a fixed intrinsic Rashbastrength. It can be seen that the VQAH phase has anextended parameter range in the phase diagram (regionII). On each side of VQAH phase, it is the usual QAHphase with ( C = 2 , C v = 0). The color map shows the sizeof the band gap. One observes that the topological phasetransitions are accompanied with the gap closing and re-opening processes, as usually mentioned in the study oftopological insulators. However, in the present model,such gap closing happens only at one valley (the K (cid:48) val-ley), hence the topological charge is only changed at thatvalley, leading to the valley-polarized feature.To gain a better understanding of the change in topo-logical charge between conventional QAH and VQAHphases, we decompose C K and C K (cid:48) at each valley in termsof contributions from real spin ( s ) as well as sublat-tice pseudospin ( σ ) degree of freedom. We calculate thewinding number of the spin (pseudospin) textures usingthe formula n = 14 π (cid:90) (cid:90) dk x dk y ( ∂ k x ˆ h × ∂ k y ˆ h ) · ˆ h , (5)where the unit vector ˆ h ( k ) is the spin (pseudospin) po-larization vector at k .For the conventional QAH phase with either extrinsicor intrinsic Rashba SOC, as in Fig. 1(a, b), we find thatthe band-resolved topological charges carried by the realspin or pseudospin for valleys K and K (cid:48) are n K, s = n K (cid:48) , s ≈ n K, s = n K (cid:48) , s ≈ n K, σ = n K (cid:48) , σ ≈ . n K, σ = n K (cid:48) , σ ≈ − . . (6)Here subscripts 1 and 2 refer to the two valence bandswith band 2 close to the gap. One observes that thetopological charges are symmetric between the two val-leys. The topological charges of pseudospin from the 1stand 2nd valence bands are respectively 0 . − . FIG. 3. (color online) Textures of real spin in the 2nd va-lence band of K (cid:48) valley. (a) Map of (cid:104) s z (cid:105) component for theconventional QAH phase with ( t R = 0 . t R = 0). (b) Mapof (cid:104) s z (cid:105) for the VQAH phase with ( t R = 0 . t R = 0 . φ ofthe in-plane spin component. The white colored loop is the (cid:104) s z (cid:105) = 0 boundary (as in (a) and (b)). The black circledarrows indicate the rotation direction of the angle φ aroundeach vortex. each valley, hence cancelling each other. The net con-tribution is from the real spin in the 2nd valence bandwhich is 1 for each valley, with the texture correspond-ing to one skyrmion. Therefore we have C K = C K (cid:48) = 1and ( C = 2 , C v = 0), which are consistent with previouscalculations.In the VQAH phase, as in Fig. 1(c), straightforwardcalculation shows that n K, s = n K (cid:48) , s ≈ n K, s ≈ n K (cid:48) , s ≈ − n K, σ = n K (cid:48) , σ ≈ . n K, σ = n K (cid:48) , σ ≈ − . . (7)Comparing with the results for the QAH phase in Eq.(6),one can find that the topological charges associated withpseudospin texture remain the same and are still can-celled between the two valence bands. The difference isfrom the real spin in the K (cid:48) valley of the 2nd valenceband. After the gap closing and reopening at the K (cid:48) valley, the topological charge n K (cid:48) , s changes from 1 to −
2. This results in an imbalance of the total topologi-cal charges between the two valleys, leading to C K = 1and C K (cid:48) = − n K (cid:48) , s more clearly,in Fig. 3, we plot the real spin textures in k -space forthe 2nd valence band at K (cid:48) valley in the QAH phaseand in the VQAH phase. Figure 3(a) and 3(b) show the z -component of real spin (cid:104) s z (cid:105) before and after phase tran-sition. One observes that near the valley center, (cid:104) s z (cid:105) isnegative and away from the center it changes to positivevalues. This feature remains the same across the phaseboundary. Figure 3(c) and 3(d) show the azimuthal angleof the in-plane vector ( (cid:104) s x (cid:105) , (cid:104) s y (cid:105) ) in the two phases. Thewinding number n K (cid:48) , s can be visualized by counting thenumber of vortices of the phase winding. For the QAHphase, there is one vortex at the valley center, as seen inFig. 3(c), corresponding to n K (cid:48) , s = 1. In the VQAH,in contrast, close to K (cid:48) point three new vortices appeararound the places where the gap closes during the phasetransition. Their winding directions are opposite to theone at the center, therefore leading to a total windingnumber n K (cid:48) , s = −
2. It is these additional skyrmionsgenerated at K (cid:48) valley in the gap closing and reopen-ing process that are responsible for the realization of theVQAH phase. III. DISORDER INDUCED VALLEY-FILTEREDCHIRAL EDGE CHANNELS
As we discussed in the previous section, in the VQAHphase of the present model, there exist valley-polarizedchiral edge channels as shown in Fig. 2(c). Due to thelarge separation of the two valleys in k -space, the val-ley index is robust against smooth disorder potentials. In the presence of short-range disorder scattering, inter-valley scattering events would be important and wouldtypically destroy the edge channels. Nevertheless, theVQAH phase is first of all a QAH phase characterizedby a Chern number C = −
1. The topological protectionof QAH phase is much stronger than that for the valleyindices. Therefore, we can expect that at moderate short-range scattering strength, a pair of counter-propagatingedge channels (one from K and one from K (cid:48) ) should bedestroyed, leaving only one chiral channel on each edge,as required by the C = − − W/ , W/ W characterizes the disorder strength.We emphasize that due to disorder scattering, valley is FIG. 4. (color online) Schematic figure showing the two-terminal setup for transport calculation. (a) A zigzag edgedribbon is divided into a left lead, a right lead, and a centralscattering region. The propagating modes in the leads withtheir valley characters are indicated. (b) Distribution of theedge modes in the scattering region without scattering or withonly weak long range scattering. (c) With short range scatter-ing, at moderate scattering strength, the transport propertyof the system is equivalent to a QAH system with one chiraledge channel in K (cid:48) valley. The colors label the edge states atdifferent valleys. no long a good quantum number in the central region.However, its valley transport property could be inferredfrom the transmission and reflection amplitudes of valley-polarized carriers from the leads. To this end, the twoleads must have valley well-defined. Our setup resem-bles that of the original proposal of valley filter and weshall demonstrate that our system indeed acts as a per-fect valley filter when moderate short-range scatterers areincorporated.The two-terminal conductance can be calculated basedon the Landauer-B¨uttiker formula G = e h Tr [Γ L G r Γ R G a ] , (8)where G r,a are the retarded and advanced Green’s func-tions of the central scattering region. The quantitiesΓ L/R are the linewidth functions describing the couplingbetween the left/right lead and the central region, andcan be obtained from Γ p = i (Σ rp − Σ ap ). Here, Σ r/ap isthe retarded/advanced self-energy due to the p th semi-infinite lead ( p = L, R ), and can be numerically evaluatedusing a recursive method. Before turning on the disorder potential ( W = 0), weknow that on each edge there are three conducting chan-nels and the propagation directions of the them are tiedto their valley indices. For example, on the upper edge,there are one channel in K valley propagating to theright and two channels in K (cid:48) valley propagating to theleft, as shown schematically in Fig. 4(b). For the loweredge, the directions of the channels are reversed. Ob-viously, the two-terminal conductance for the structureshould be G = 3 (in units of e /h ) due to three ballistictransport channels in each direction.When we increase the disorder strength W , backscat-tering occurs in these edge channels because short-rangescatterer can couple the counter-propagating channels at K and K (cid:48) valleys. This would decrease the conductance.However, at moderate scattering strength, there must beone remaining transport channel as dictated by the to-tal Chern number C = −
1. The chirality requires thatat the upper edge, this channel propagating to the left,while at the lower edge, it propagates to the right. Thisshould lead to a plateau of G = 1 for the two-terminalconductance.In Fig. 5(a) we plot numerical results of the conduc-tance G as a function of the disorder strength W . Thecentral scattering region has a width of 480 atomic sitesand a length of 1200 atomic sites. The Fermi level isset at E F = 0 .
004 in the gap and an ensemble of 100random disorder configurations are taken for each datapoint. Indeed, as we expected, G starts from the valueof 3 at W = 0 and decreases with increasing W . A quan-tized plateau of G = 1 appears around W = 0 . W above ∼ .
75 eventually destroys the QAHchannels by coupling the two chiral channels at the op-posite edges through strong scattering across the bulk.All these are consistent with our previous argument.However, we do not yet know whether the one chiralchannel on the G = 1 plateau still retains a well-definedvalley character, although one may intuitively think thatafter one channel in K valley gets annihilated with onein K (cid:48) valley, the remaining one should be from K (cid:48) val-ley. With finite disorder strength, it is difficult if notimpossible to check the valley feature in energy spec-trum. Instead, we infer the valley character of the chan-nel from its transport properties. More specifically, weconsider the valley resolved transmission probability T vv (cid:48) ( v, v (cid:48) ∈ { K, K (cid:48) } ) which is defined as the transmissionprobability from any incoming mode in valley v (cid:48) of theleft lead to any outgoing mode in valley v of the rightlead. Hence T vv (cid:48) = (cid:80) m ∈ v,n ∈ v (cid:48) T mn , where T mn is thetransmission probability from mode n to mode m , n and m label the propagating modes in the left and the rightlead respectively. In our case, since we consider the Fermilevel in the band gap, the only propagating modes in theleads are the edge modes. The T mn for each pair of in-coming and outgoing propagating modes can be calcu-lated using the technique developed in Ref. 45–47.The result for each T vv (cid:48) as a function of disorderstrength is shown in Fig. 5(b). One observes that when W is small ( < . T KK (cid:48) and T K (cid:48) K are zero. Thisis easily understood by inspecting the configuration ofchannels in Fig. 4(a). In order for an incoming mode in K valley from the left lead to transfer to an outgoing modein K (cid:48) valley at the right lead, it has to cross the insulat-ing bulk hence such probability is negligibly small. For T KK T K’K T KK’ T K’K’ R KK R K’K R KK’ R K’K’ G V Gη V FIG. 5. (color online) (a) The two-terminal conductance asa function of disorder strength W . Each data point is aver-aged over 100 disorder configurations. The error bar showsone standard deviation. (b) The valley resolved transmis-sion probability as functions of W . (c) The valley resolvedreflection probability versus W . (d) The charge ( G ), valley( G v ) conductance, and the valley polarization ( η V ) as func-tions of W . The model parameters used here are M = 0 . t R = 0 .
045 and t R = 0 .
08. Fermi energy is taken as E F = 0 . small W ( < . T K (cid:48) K (cid:48) ≈ T KK ,because the number of channels in K (cid:48) valley doubles thatof the K valley. They both decrease with increasing W .When W reaches the plateau region as in Fig. 5(a), T KK vanishes implying that the transport channel at K val-ley in the central region is totally destroyed. Meanwhile T K (cid:48) K (cid:48) shows a quantized plateau at 1. This indicates thatthe remaining chiral edge channels protected by Chernnumber C = − K (cid:48) valley character. The resultsuggests the physical picture in which the short-rangescattering couples the edge states of the two valleys anddestroys a pair of counter-propagating modes (one fromeach valley), leaving only one edge channel of K (cid:48) valleyin the system. Finally at very large W , scattering cancouple the two edges, and the plateau is destroyed. Thediscussed features are schematically shown in Fig. 4(b)and Fig. 4(c). At moderate disorder strength (in theplateau region), the central region can be viewed effec-tively as having one chiral channel in K (cid:48) valley on eachedge [Fig. 4(c)].The valley resolved reflection probability R vv (cid:48) can bedefined in a similar way as T vv (cid:48) . The results are shownin Fig. 5(c). The key thing to notice is that in theQAH plateau region, R K (cid:48) K (cid:48) remains 0 because such re-flection process requires the electron to transfer acrossthe insulating bulk to the other edge. This condition,combined with T K (cid:48) K (cid:48) = 1 in this region, means thatcarriers in K (cid:48) valley can transmit through the systemwithout reflection while maintaining its valley charac-ter. In addition, the rapid increase of R KK (cid:48) and R K (cid:48) K at small W demonstrates that the short-range scatterersindeed cause strong backscattering between the counter-propagating channels at each edge.Based on the valley resolved transmission probability,we could define a valley resolved conductance by G v = e h (cid:88) m ; n ∈ v T mn , v ∈ { K, K (cid:48) } , (9)which measures the likelihood of transmission of incom-ing carriers in each valley. Then the total conductancecan be written as G = G K + G K (cid:48) . Analogous to quan-tities defined for spin transport, we can define a valleyconductance G V = G K (cid:48) − G K and the valley polarization η V = G V /G . The numerical results for these quantitiesare shown in Fig. 5(d). One observes that in the QAHplateau region, the valley polarization shows a plateauof 1, meaning that the transport through the system isfully valley polarized.From the above results and discussions, we confirmthe intuitive picture that we postulated at the begin-ning of this section. We show that at moderate disorderstrength, the scattering localizes a pair of edge channelson each edge, leaving the system in a C = − K (cid:48) valley character. This impliesthat disorder scattering effectively induces a transitionfrom a VQAH phase with ( C = − , C v = 3) to anotherVQAH phase with ( C = − , C v = 1). Such disorderinduced VQAH phase with |C| = |C v | always has fullyvalley-filtered chiral edge channels in the bulk mobilitygap, and may be termed as a VQAH Anderson insula-tor phase, analogous to the concept introduced in thestudy of transport features of disordered topological in-sulator systems. In this case, whether a carrier canbe transmitted through the system is determined by itsvalley index. Hence this phase could be used to realize aperfect valley filter for valleytronics applications.
IV. VQAH PHASES IN BILAYER SYSTEM
We have shown that VQAH phase can arise in a singlelayer honeycomb lattice model due to the competitionbetween two types of SOCs. In the following, we showthat by combining two such single layers into a bilayersystem, more VQAH phases with different ( C , C v ) couldbe realized. The model we consider is H = H t + H b + t ⊥ (cid:88) i ∈ ( t,A ); j ∈ ( b,B ) ( c † i c j + h.c. )+ U (cid:88) i ξ i c † i c i , (10)where H t and H b each given by Eq.(1) are the Hamilto-nians for the top layer and the bottom layer. The third FIG. 6. (color online) Phase diagram of the bilayer model inthe ( t R , t R ) plane. Five extended insulating phase regionscan be identified. The color indicates the size of the bulkband gap. Other model parameters used here are M = 0 . U = 0, and t ⊥ = 0 . C C v C K C K (cid:48) I 4 0 2 2II 1 3 2 -1III -2 6 2 -4 term on the right hand side is an interlayer coupling.Here we take a bilayer with AB -stacking, hence hoppingbetween the nearest A site in the top layer and B sitein the bottom layer is considered. The last term is aninterlayer bias potential with ξ i = ± U = 0. Figure 6 shows thephase diagram in the ( t R , t R ) plane. One observes thatthere are five phase regions. The color map indicates themagnitude of the bulk band gap. The phase boundariesare the points at which the gap closes. The topologicalinvariants for each phase are listed in Table I. It can beseen that Phase I is the conventional QAH phase with( C = 4 , C v = 0) and Phase III is a VQAH phase with( C = − , C v = 6). These two phases can be understoodas resulting from the corresponding topological phases insingle layer by a direct doubling. Besides these two, in-terestingly there is one additional phase, Phase II. Wefind that this phase is also a VQAH phase. It is char-acterized by ( C = 1 , C v = 3), which differs from thatof Phase III. Therefore, one sees that by combining twosingle layer models, it is possible to generate new VQAHphases. The phase transition between each neighboringphase is accompanied by the gap closing and reopeningprocess, which is a general feature of topological phasetransitions. In this case, the gap closing only occurs inthe K (cid:48) valley, similar to the single layer case. This is con-sistent with the variation of the valley topological charge C K and C K (cid:48) in Table I, i.e. C K is the same for all thethree phases and only C K (cid:48) changes.In the following, let’s have a closer look at the two FIG. 7. (color online) Phase II in Fig. 6. (a) Berry curvaturedistribution in the Brillouin zone. (b) Spectrum of a zigzag-edged ribbon with a width of 400 atomic sites. (c) and (d)The enlarged spectra in the gap region of (b), correspondingto the two valleys K and K (cid:48) . (d) Schematic figure showingthe distribution of edge channels labeled in (c) and (d). Theparameters are set to be t R = 0 . t R = 0 . U = 0, M =0 .
4, and t ⊥ = 0 . VQAH phases, Phase II and Phase III. In Fig. 7(a) andFig. 8(a) we plot the total Berry curvature distribution Ω ( k ) of the valence bands, which sums over the Berrycurvature for each individual valence band. One can findthat the nonzero Berry curvatures are mainly concen-trated around the valley centers and have an overall op-posite sign between the two valleys. This is in contrastto the conventional QAH effect, in which the Berry cur-vature usually has the same sign for different valleys. For both phases, the asymmetry between the two valleyscan be clearly observed. Comparing the two phases, oneobserves that the curvature distributions at K valley arealmost the same, but the curvature at K (cid:48) valley differ alot. Ω ( k ) around K (cid:48) for Phase III has a larger negativemagnitude compared with Phase II. This difference leadsto the different C K (cid:48) between the two phases.In Fig. 7(b), we plot the energy spectra of a zigzag-edged nanoribbon for Phase II. In the zoom-in imagesin Fig. 7(c) and Fig. 7(d), one observes that for valley K , there are two pairs of gapless edge states while forvalley K (cid:48) , there is one pair. On each edge, there aretwo channels from K valley propagating in one directionand another channel from K (cid:48) valley propagating in theopposite direction, as shown schematically in Fig. 7(e),which is consistent with the bulk topological invariants( C = 1 , C v = 3). This net valley polarization of the edgechannels is one signature of VQAH phase.Similarly, for Phase III, we plot its energy spectrum inFig. 8(b-d), in which one identifies two pairs of gaplessedge states in valley K and the other four pairs in val-ley K (cid:48) . As illustrated in Fig. 8(c), on each edge, there FIG. 8. (color online) Phase III in Fig. 6. (a) Berry curvaturedistribution in the Brillouin zone. (b) Spectrum of a zigzag-edged ribbon with a width of 400 atomic sites. (c) and (d)The enlarged spectra in the gap region of (b), correspondingto the two valleys K and K (cid:48) . (d) Schematic figure showingthe distribution of edge channels labeled in (c) and (d). Theparameters are set to be t R = 0 . t R = 0 . U = 0, M = 0 .
4, and t ⊥ = 0 . U , M ) plane. Five extended phase regions can beidentified. The color indicate the size of the bulk band gap.Other model parameters are fixed as t R = 0 . t R = 0 . t ⊥ = 0 . are two edge channels from valley K propagating in onedirection and four edge channels from valley K (cid:48) propa-gating in the opposite direction. Compared with PhaseII, the number of channels in K valley remains the same,while the number in K (cid:48) valley changes from 1 to 4. Thischange reverses the valley polarization of the channels,i.e. now the system has more edge channels in K (cid:48) thanin K .Since bilayer systems provide another layer degree offreedom, which offers additional controllability. In thefollowing, we examine the effect of tuning interlayer biaspotential on the topological phases of the system. InFig. 9, we plot the phase diagram in ( U , M ) plane, withfixed SOC strengths. One can identify five different topo-logical phases in this diagram. The topological invariantsfor each phase are listed in Table II. We take the valuesof t R and t R such that at small U , the system is in theconventional QAH phase with ( C = 4 , C v = 0) (Phase Iin Fig. 9). With increasing U , we see that the systemcan undergo topological phase transitions to a series ofVQAH phases with ( C = 1 , C v = 3) (II), ( C = − , C v = 6)(III), ( C = − , C v = 5) (IV), and ( C = 0 , C v = 2) (V). Byinspecting C K and C K (cid:48) , we can see the sequence of gapclosing and reopening processes at the two valleys duringthese phase transitions. For the transition from I to IIand from II to III, the gap closing is at K (cid:48) valley, whilefor the transition from III to IV, the gap closing hap-pens at K valley. These results are also confirmed by theband structure calculations. Therefore, from above dis-cussion, the transition from conventional QAH to VQAHcan also be controlled by the interlayer bias. This is un-derstandable because finite U breaks the inversion sym-metry connecting the two valleys, hence could drive thesystem towards a valley polarized state. Because inter-layer potential is generally easier to control in practice,e.g. through gating technique, hence this finding offersa potentially convenient route for engineering a VQAHphase in layered structures. Finally at very large U andsmall M , the system’s total Chern number would vanish.The topological charges of the two valleys cancel eachother. This is known as quantum valley Hall phase inprevious studies. TABLE II. Chern number contribution of each valley in Fig. 9Phase
C C v C K C K (cid:48) I 4 0 2 2II 1 3 2 -1III -2 6 2 -4IV -3 5 1 -4V 0 2 1 -1
V. DISCUSSION AND SUMMARY
In the analysis of edge channels, the valley index ofthem is only well-defined provided that the two valleysare separated in momentum when projected to the edge.This depends on the edge orientation. For example, forzigzag edges of a honeycomb lattice, K and K (cid:48) valleysproject to separated points on the edge, which ensuresthe valley index to be defined. In contrast, for arm-chair edges, the two valleys will be projected to the samepoint on the edge. Therefore, the edge channels do nothave a well-defined valley index and usually strong mix-ing between them could gap the edge states. This de-pendence of the edge states on the edge orientation isanalogous to what happens in 3D Dirac and Weyl topo- logical phases.
Nevertheless, the topological invari-ants such as C and C v are defined for the bulk, hence donot depend on the edge orientation.In Sec. III, we used the single layer ( C = − , C v = 3)phase as an example to explicitly demonstrate the ef-fects of short-range disorders. Similar physics also hap-pens for other VQAH phases such as the phases that wediscussed in the bilayer systems. For example, for thephase ( C = − , C v = 6) as shown in Fig. 8, at moder-ate scattering strength, two pairs of counter-propagatingchannels would be localized, leaving only two transportchannels with K (cid:48) valley character. Hence the resultingsystem is equivalent to a C = − K (cid:48) . This implies thatit is possible to achieve full valley polarized transportwith higher conductance values by starting from VQAHstates with higher Chern numbers. From the discussionin Sec. III, one also expects that the valley polarizedtransport plateau could be extended to even higher dis-order strength if one could arrange the disorders to dis-tribute more near the edges than in the bulk, therebysuppressing the coupling between the two edges. Increas-ing the edge roughness may be a possible way to achievethis.The intrinsic SOC as denoted by t SO term is a termthat preserves the inversion symmetry, hence is generallynot helpful for the purpose of inducing valley polariza-tion. Previous studies have shown that this term coulddrive the system to a quantum spin Hall phase in theabsence of time reversal symmetry breaking. For themodels we considered here, with other parameters fixed,the intrinsic SOC tends to drive the system out of theVQAH phase. However, since the topological phase isprotected by the band gap, for small t SO such that thegap is not closed, VQAH phase is still maintained. Inthe single layer model, the inversion symmetry could bebroken by a staggered sublattice potential term. Simi-lar to the effect of interlayer bias potential for the bilayermodel, this term could generate valley polarization, andis capable of driving the system from conventional QAHphase to the VQAH phase in single layer model.Finally, a particular lattice model is adopted here forthe proposition and the study of the novel VQAH phase.We emphasize that the essential features we discuss here,such as the valley-polarization of the gapless edge chan-nels and the disorder effects on the edge channels, aregeneral features of the VQAH phase and are not particu-lar to one specific model. In reality, several 2D materialswith low buckled honeycomb lattice structure have beendiscovered or proposed.
In principle, the strength ofeach individual terms in our model Hamiltonian Eq.(1)could be induced and controlled. For example, the ex-trinsic Rashba SOC could be generated by substrate oradsorbed atoms.
The exchanged coupling could begenerated by defects, magnetic dopants, or in proximityto a magnetic insulator.
Intrinsic SOC and intrinsicRashba SOC could be controlled by structural deforma-tion through applied strains. Furthermore, the candi-0date material is not limited to those with 2D honeycomblattice structure. Any multi-valley systems are possible.Therefore, although fine-tuning the various parametersto achieve the VQAH phase is a challenging task, withthe advance in discovering new 2D materials and in de-veloping new technique to control interactions at submi-cron scale, as demonstrated in the recent realization ofQAH phase, it is promising to also achieve the fascinatingVQAH phase in the future.In summary, we investigated in detail the novel VQAHphase in single layer and bilayer systems. We provide aclear physical picture of the topological phase transitionfrom conventional QAH phase to the VQAH phase. Westudied the transport properties of the edge channels.With short-range disorders, pairs of counter-propagatingedge channels (one from each valley in a pair) could bedestroyed. However, at moderate scattering strength,the transport coefficients exhibit a plateau on whichthe transport is fully valley-filtered, leading to a VQAH Anderson insulator phase. This remarkable effect couldbe used for designing valley filters for valleytronicapplications. Much richer phase diagrams are shownfor the bilayer system with multiple VQAH phases. Wedemonstrate the controllability of the topological phasetransition by tuning the system parameters, especiallythe interlayer bias potential. The study presented hereendows the valley transport with topological protection,which is very important for realizing robust performanceof information processing based on valley degree offreedom.
Acknowledgement.
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