Charge and spin transport in edge channels of a ν=0 quantum Hall system on the surface of topological insulators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Charge and spin transport in edge channels of a ν = 0 quantum Hall system on thesurface of topological insulators Takahiro Morimoto, Akira Furusaki,
1, 2 and Naoto Nagaosa Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama, 351-0198, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan Department of Applied Physics, The University of Tokyo, Tokyo, 113-8656, Japan (Dated: August 6, 2018)Three-dimensional topological insulators of finite thickness can show the quantum Hall effect(QHE) at the filling factor ν = 0 under an external magnetic field if there is a finite potential differ-ence between the top and bottom surfaces. We calculate energy spectra of surface Weyl fermions inthe ν = 0 QHE and find that gapped edge states with helical spin structure are formed from Weylfermions on the side surfaces under certain conditions. These edge channels account for the nonlocalcharge transport in the ν = 0 QHE which is observed in a recent experiment on (Bi − x Sb x ) Te films. The edge channels also support spin transport due to the spin-momentum locking. We pro-pose an experimental setup to observe various spintronics functions such as spin transport and spinconversion. Introduction —
The quantum Hall effect (QHE) is arepresentative topological quantum phenomenon whereedge channels play an essential role in low-energy trans-port [1, 2]. The emergence of gapless edge channels isclosely tied to nontrivial topology of gapped electronicstates in the bulk, which is the property called bulk-edgecorrespondence. For topological insulators (TIs) [3, 4],the bulk-edge correspondence dictates that any surfaceof a three-dimensional (3D) TI with a nontrivial Z in-dex has a single (or an odd number of) flavor(s) of Weylfermions with spin-momentum locking. The surface Weylfermions are predicted to give a variety of novel phenom-ena such as the quantized topological magneto-electric(ME) effect [5] and monopole-like magnetic field distri-bution induced by a point charge [6]. Another remarkablephenomenon is the quantized anomalous Hall effect with-out an external magnetic field [7–10]. By contrast, in anexternal magnetic field, a single flavor of Weyl fermionsare expected to show a “half-integer” QHE with the Hallconductance σ xy = ( n + ) e /h ( n ∈ Z ), which has a -shift due to their nontrivial Berry phase, compared withthe quantized Hall conductance of non-relativistic elec-trons, σ xy = ne /h ( n ∈ N ).However, the “half-integer” QHE is forbidden by theNielsen-Ninomiya theorem [11], which dictates that thereshould be an even number of flavors of Weyl fermions inthe first Brillouin zone, implying that the half-integerQHE is not observable in reality. For example, Weylfermions in graphene have four flavors from spin andvalley degrees of freedom, yielding σ xy = 4( n + ) e /h [12]. As for a 3D (strong) TI, excitations at all surfaceshave to be considered together. In a magnetic field ap-plied perpendicular to the top and bottom surfaces of a3D TI, Weyl fermions on the both surfaces contributeto σ xy , yielding ν := σ xy / ( e /h ) = n T + n B + 1 with n T + and n B + coming from the top and bottomsurfaces, respectively. The integers n T and n B are LLindices of the highest occupied LLs at the top and bot- tom surfaces. When the same number of LLs are filledat the top and bottom surfaces ( n T = n B ), the Hall con-ductivity is quantized at an odd integer [13]. However,the QHE of surface Weyl fermions with Hall plateaus at ν = 2 , , , . . . is experimentally observed in 3D HgTe[14]. More recent experiments on (Bi − x Sb x ) Te filmshave reported plateaus at ν = 0 and ± ν = 0 plateau violating the odd-integer rule indicatesthat the degeneracy between the top and bottom sur-faces is lifted in (Bi − x Sb x ) Te films [15, 16]. More-over, a transport measurement on (Bi − x Sb x ) Te filmsrevealed the existence of nonlocal transport in the ν = 0QHE [24]. The nonlocal transport appears to conflictwith the ν = 0 QHE because of the absence of the chiraledge mode supporting the nonlocal transport.In this paper, we study the ν = 0 QHE (i.e., n T + n B + 1 = 0) of Weyl fermions on the surfaces of a 3DTI of finite thickness. Compared with the ν = 0 QHEin graphene, which has been studied actively both ex-perimentally [17–20] and theoretically [21, 22], the QHEin 3D TIs has the following unique features. First, theWeyl fermions on the TI surfaces are two-componentspinors of real spin- degrees of freedom (as opposed topseudo-spins in graphene), and the edge channels sup-port both charge and spin transport. Second, in a mag-netic field the Weyl fermions on the top and bottom sur-faces (which are perpendicular to the field) form Lan-dau levels (LLs), while those on the side surface (parallelto the magnetic field) are not directly affected by thefield. However, one cannot consider independent 2D topand bottom surfaces of σ xy = ± e / h with edge chan-nels. The side surface which connects the top and bot-tom surfaces must be taken into account when discussingthe edge channels. The energy spectrum of the Weylfermions on the side surface is discretized in a thin 3DTI, yielding well-defined edge channels under the con-dition of Eq. (4). These edge channels account for the FIG. 1. Schematic picture of the two-dimensional (2D) modelfor the surface Weyl fermions of a topological insulator ina magnetic field. A 2D plane in the lower panel representsthe top, bottom and side surfaces, where the strength of themagnetic field is ± B for x > d and x < − d and vanishing for − d < x < d . nonlocal charge transport in the ν = 0 QHE observed inthe recent experiment [24]. Furthermore, since these edgechannels are spin-momentum locked, the ν = 0 QHE ofTI thin films offers a new arena for various spintronicsfunctions such as spin transport and spin conversion. Model —
We study quantum Hall states on the surfacesof a TI, treating the top, bottom, and side surfaces asan effective two-dimensional (2D) system as shown inFig. 1. We will ignore the gapped bulk states. The top(bottom) surface is mapped onto the region x > d ( x < − d ) with magnetic fields + B ˆ z ( − B ˆ z ), while the sidesurface is mapped onto the region − d ≤ x ≤ d , where nomagnetic field is applied; see the lower picture in Fig. 1.The effective Hamiltonian for the surface Weyl fermionsis given by H = v F [ − ( p x + eA x ) σ y + ( p y + eA y ) σ x ] + U σ , (1)where v F is the Fermi velocity, p = ( p x , p y ) is the mo-mentum, and Pauli matrices σ = ( σ x , σ y , σ z ) act on thespin degrees of freedom, and σ is a 2 × A = ( A x , A y ) and the scalar potential U are functions of x , A ( x ) = (cid:0)(cid:0)(cid:0) , − B ( x + d ) (cid:1)(cid:1)(cid:1) , U ( x ) = − V , ( x < − d ) , A ( x ) = (0 , , U ( x ) = 0 , ( | x | < d ) , (2) A ( x ) = (cid:0)(cid:0)(cid:0) , B ( x − d ) (cid:1)(cid:1)(cid:1) , U ( x ) = V , ( x > d ) . We have introduced the potential difference between thetop and bottom surfaces, 2 V . Since the 2D modelis translationally invariant along the y direction underthe Landau gauge, the momentum k y is a good quan-tum number. The wave functions with wave number k y - - - - (a) k y (1/nm) E ne r g y ( m e V ) ν =0 N T = − T = − B =1N B =2N B =0N T =0 (b) k y (1/nm) N B =0N T =0 s X s Z - - - - (c) - - - - k y (1/nm) ( n m ) N B =0N T =0 sidesurface -
20 0 200 topsurfacebottomsurface x (nm) sidesurface
FIG. 2. Band structure of the surface Weyl fermions in the2D model. (a) The energy levels of H in the Landau gauge[Eq. (1)] are plotted against the momentum k y for 2 d = 30 nm, B = 15 T, 2 V = 50 meV. Colors encode the expectation valueof the position h x i ; the Landau levels on the top and bottomsurfaces are depicted in red and blue in the region k y < N T ( N B ). (b) Spin textures in the ( s x , s z )-plane and(c) expectation values h x i of the two bands connected to the N T = 0 and N B = 0 LLs. Edge channels ( k y &
0) show thespin-momentum locking, while the N = 0 Landau levels onthe top and bottom surfaces are spin polarized. have a Gaussian form in x , with the expectation value |h x i| ≃ − k y ℓ for k y < h x i ≃ k y >
0, where ℓ = p ~ /eB is the magnetic length; see Fig. 2(c). Edge channels —
We show the band structure of theHamiltonian [Eq. (1)] in Fig. 2(a). The bands are plottedas functions of k y for the following parameters: 2 d =30 nm, B = 15 T, 2 V = 50 meV, and v F = 5 × m / s[23]. The flat bands at k y <
0, shown in red and blue, arethe LLs on the top and bottom surfaces, whose energiesare given by E N = sgn( N ) p | N | ~ ω c + U , (3)with the LL indices N = N T ( N B ) and the scalar poten-tial U = V ( − V ) for the top (bottom) surface, andthe cyclotron frequency ω c = v F p eB/ ~ . Shown ingreen/yellow at k y & N T/B = 0LLs on the top and bottom surfaces are separated bythe energy difference 2 V . Thus, the ν = 0 QHE is real-ized when the Fermi energy E F is in the energy window − V < E F < V , because LLs are filled up to N T = − N B = 0 LLs at top and bottom surfaces, respec-tively, and hence ν = n T + n B + 1 = 0 with n T = − n B = 0. In this case there is no chiral edge mode, asexpected from the bulk-edge correspondence. However,transport through edge channels is still possible in the ν = 0 QHE, under the conditions we discuss below.Three parameters control the existence of edge chan-nels in the ν = 0 QHE: the energy difference of the Weylpoints of the top and bottom surfaces 2 V , the cyclotronenergy ~ ω c , and the energy gap at k y = 0 of the Weylfermions on the side surface ∆ side ≃ ~ v F / d . The edgechannels in the ν = 0 QHE can exist when∆ side < V ≤ ~ ω c . (4)The left inequality implies that the two bands from the N T/B = 0 LLs turn into helical edge channels on the sidesurface with energies in the ν = 0 quantum Hall regime( | E | < V ), whereas the right inequality assures that the N T/B = 0 LLs are located in between the N T = − N B = +1 LLs. The above conditions are satisfied by theparameters used in Fig. 2(a), which shows edge modeswith a gap at k y = 0. Furthermore, we find that, forthe parameters in the experiment of (Bi − x Sb x ) Te inRef. [16], the above inequalities are satisfied with 2 V =70 meV, ~ ω c = 70 meV, and ∆ side ≃
40 meV.In Fig. 2(b) we show the expectation values of( h s x i , h s z i ) for the Bloch wave functions of the valencetop band and the conduction bottom band, connectedto the N B = 0 and N T = 0 LLs, respectively; we find h s y i = 0 for all bands. Note that, since we have rotatedthe side and bottom surfaces around the y axis by 90 ◦ and 180 ◦ , respectively, in our 2D model of TI surfaces(Fig. 1), the spins are also rotated accordingly. Thus thereal spin operators s are written in terms of the Paulimatrices in the Hamiltonian [Eq. (1)] as s ~ / ( σ x , σ y , σ z ) , ( x > d ) , ( − σ z , σ y , σ x ) , ( − d < x < d ) , ( − σ x , σ y , − σ z ) , ( x < − d ) . (5)The spins of the N T/B = 0 LLs on the top and bottomsurfaces are polarized, ( h s x i , h s z i ) = (0 , − ~ /
2) for k y < k y increases, the spin configuration crosses over to ahelical spin structure of the edge channels on the sidesurface for k y &
0, where the spins of the two bandsare polarized to the opposite directions because of thespin-momentum locking. This feature is similar to helical
FIG. 3. A setup for the detection of charge and spin transportthrough edge channels. Applying a voltage between the elec-trodes 1 and 2 in the left induces charge and spin transportthrough the edge channels in the ν = 0 quantum Hall state.The nonlocal charge transport is measured by the voltage V between the electrodes 3 and 4. The spin current is detectedby the voltage V between the two sides of the Pt electrode 3via the inverse spin Hall effect. edge states of quantum spin Hall insulators, except forthe presence of the gap at k y = 0. We note that, when V is negative, all the discussion above applies by replacing V by − V = | V | , except that h s x i becomes −h s x i inthe spin texture. This is because the reflection z → − z changes the sign of V and also ( s x , s y ) → ( − s x , − s y ). Charge and Spin transport by edge channels —
Theedge channels on the side surface are responsible for thecharge and spin transport, when the Fermi energy is inthe range ∆ side < | E F | < V . (Unlike in the quantumHall insulators, however, the charge and spin conduc-tances are not quantized because backscattering is al-lowed in these edge channels.) Here we propose an ex-perimental setup consisting of a TI thin film with fourelectrodes to observe the charge/spin transport throughedge channels; see Fig. 3. In the following we will discussthree characteristic transport phenomena: charge trans-port, spin transport, and spin conversion.First, the edge channels should lead to nonlocal trans-port of charge. We predict that, when a voltage is appliedand electric current flows between the electrodes 1 and 2,a finite voltage V between the electrodes 3 and 4 shouldbe observed. Let us assume V < E F < ∆ side . The ap-plied voltage induces net flow of up-spin electrons to the+ y direction and that of down-spin electrons to the − y direction, according to Fig. 2. This results in the flow ofelectrons to the directions indicated by the thick arrows J and J in Fig. 3 (note that the electric current flowsto the opposite directions), while the bulk transport isvanishing. The thin arrows represent the residual flowof electrons which were not absorbed by electrodes. Thecurrent J flows further along the right and rear sidesurfaces before reaching the electrode 2, yielding a finitevoltage V between the electrodes 3 and 4. The nonlocaltransport in the ν = 0 quantum Hall regime has been ac-tually observed in experiments of (Bi − x Sb x ) Te films[24].Second, the edge channels support spin transport in (a) Energy (meV) S p i n c ondu c t an c e ( e / π ) (b) S p i n c on v e r s i on r a t e E y (kV/cm) - - - - E ne r g y ( m e V ) k y (1/nm) -
20 0 2000.5
FIG. 4. (a) Spin conductance as a function of the Fermi en-ergy. Spin conductance is finite in the ν = 0 quantum Hallstate when the Fermi level is in the edge channels. (b) Spinconversion rate plotted against electric fields E y . Dots rep-resent the spin conversion rate obtained from numerical solu-tions of the time-dependent Schr¨odinger equation, while thesolid curve is given by the Zener tunneling formula [Eq. (9)].Results in both panels are obtained for the same parametersused in Fig. 2(a). addition to the charge transport. As we have discussedabove, the applied voltage V produces the circulatingspin polarized electronic current through the edge chan-nels, as shown in Fig. 3. We predict that the nonvanish-ing spin current should yield a finite voltage V betweenthe two sides of the electrode 3 if it is made of a ma-terial with strong spin-orbit coupling such as Pt. Thespin current J diffusing into the electrode 3 producesa flow of down-spin electrons moving perpendicular tothe side surface. This spin current will generate the volt-age V through the inverse spin Hall effect (ISHE) [25]in the electrode 3. The voltage V is determined by thespin conductance of the edge channels, the diffusion rateacross the interface of the electrode 3, and the efficiencyof the ISHE. The (linear) spin conductance G s z is calcu-lated in the ballistic regime as G s z = X i Z h s z i i,k y | v i,k y | (cid:16) − e (cid:17) df ( ǫ i,k y ) dǫ dk y , (6)where f ( ǫ ) is the Fermi distribution function, and h s z i i,k y is the expectation value of s z for the Bloch wave func-tion ψ i,k y of the i th band with the momentum k y andthe energy ǫ i,k y . The group velocity v i,k y along the y di-rection is given by v i,k y = ~ − dǫ i,k y /dk y . In deriving Eq.(6) we have assumed that the applied voltage V shiftsthe Fermi energy of the right-going (left-going) electronsby V / − V /
2) in the ballistic regime. In Fig. 4(a) weshow the spin conductance at zero temperature which iscalculated as a function of E F with the parameters usedin Fig. 2(a). The spin conductance G s z is non-vanishingwhen the Fermi energy lies in the edge channels. Theasymmetry of the spin conductance in E F reflects thedifferent spin configurations of the two bands connected to the N T/B = 0 LLs shown in Fig. 2(a).Finally, we discuss spin conversion due to interbandtransitions that occur at the side surface in the non-equilibrium regime. We assume that the Fermi energyis within the gap ∆ side . When an electric field is applied,by external gate electrodes, to the sample in the direc-tion from the electrode 2 to the electrode 1, electronsin the N B = 0 band are driven by the electric field andundergo Zener tunneling into the N T = 0 band acrossthe energy gap around k y = 0 as shown in the inset ofFig. 4(b), where k y is along the direction from the elec-trode 1 to the electrode 2. In this process, a pair of aspin-up electron in the N T = 0 band and a spin-downhole in the N B = 0 band is created. Thus the Zenertunneling in the edge channels gives rise to spin conver-sion and spin accumulation at the side surface betweenthe electrodes 1 and 2. In order to quantify the spinconversion, we study the time-evolution of a wave packetdriven by the electric field E y along the y direction forthe time period t f = ~ ( k fy − k iy ) /eE y , from the initialstate ψ ( t = 0) = ψ N B =0 ,k iy in the N B = 0 band with themomentum k iy = − . − to the final state ψ ( t = t f )with the momentum k fy = 0 . − . To this end, wenumerically solve the time-dependent Schr¨odinger equa-tion, i ~ ∂ t ψ ( t ) = ( H + v F eE y tσ x ) ψ ( t ) , (7)and compute the spin conversion rate P ( E y ) P ( E y ) = h s z i t = t f − h s z i N B =0 ,k fy , (8)where h s z i t = t f is the expectation value of s z in the fi-nal state ψ ( t f ). The obtained spin conversion rate inFig. 4(b) shows a nonlinear response to E y , which is con-sistent with the Zener tunneling formula, P ( E y ) ∝ exp (cid:18) − π ~ ∆ v F eE y (cid:19) , (9)for the two-band system of the N T = 0 and N B = 0bands with the energy gap ∆ side ≃ v F ~ / d . Summary —
We have studied charge and spin trans-port in the ν = 0 quantum Hall system of surface Weylfermions of 3D TIs. The charge and spin currents arecarried by the edge channels formed on the side surfacesof TIs. A unique feature of the QHE in TIs is that thetop and bottom surfaces, where LLs are formed, are con-nected by side surfaces, where Weyl fermions form edgechannels, as contrasted to conventional quantum Hall bi-layers and the ν = 0 QHE in graphene. In the proposedspin transport experiment, the voltage V is expected tobe on the order of 10 nV as measured in an experiment ofISHE [26]. The spin conversion rate is typically 0 . ∼ E y ≃ µ rad at the edge of TI thin films[27].The authors are grateful for insightful discussions withM. Kawasaki and Y. Tokura. This work was supportedby Grant-in-Aids for Scientific Research (No. 24224009and No. 24540338) from the Ministry of Education, Cul-ture, Sports, Science and Technology (MEXT) of Japanand from Japan Society for the Promotion of Science. [1] The Quantum Hall Effect , edited by R. E. Prange andS. M. Girvin (Springer-Verlag, New York, 1987).[2]
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Far from the side surface, we have well-defined Landau levels (LLs) at the top and bottom surfaces. In this region,the Hamiltonian [Eq. (1)] reduces to H k y = v F [ − p x σ y + ( ~ k y ± eBx ) σ x ] ± V σ , (S1)where k y is the momentum along the y direction and we choose the sign ± for the top and bottom surfaces. We notethat we replaced the gauge potential with A y = ± Bx for simplicity. We assume B >
0. The energy and wavefunctionof LLs for the above Hamiltonian are written as follows. For the top surface, the LLs are written as E N T ,k y = sgn( N T ) p | N T | ~ ω c + V ,ψ N T ,k y ( x ) = sgn( N T ) φ | N T |− (cid:16) x − x ( k y ) ℓ (cid:17) φ | N T | (cid:16) x − x ( k y ) ℓ (cid:17) ,x ( k y ) = − ℓ k y , (S2)with sgn( n ) = +1 , ( n > , ( n = 0) − , ( n <
0) (S3)where N T is the Landau index at the top surface, φ n is the wavefunction of conventional Landau levels with φ − = 0,the cyclotron energy ~ ω c and the magnetic length ℓ are ~ ω c = v F √ ~ eB, ℓ = r ~ eB . (S4)For the bottom surface, the LLs are written as E N B ,k y = sgn( N B ) p | N B | ~ ω c − V ,ψ N B ,k y ( x ) = φ | N B | (cid:16) x − x ( k y ) ℓ (cid:17) sgn( N B ) φ | N B |− (cid:16) x − x ( k y ) ℓ (cid:17) ,x ( k y ) = ℓ k y . (S5)where N B is the Landau index at the bottom surface. Band structure with parameters in an experiment of TI thin films
We show the band structure of the Hamiltonian [Eq. (1)] using the parameters in experiments by Yoshimi et al.[S1]: 2 d = 8 nm, B = 15 T, 2 V = 70 meV, and v F = 5 × m / s. The band structure is plotted as a function of k y in Fig. S1(a). We find that N B = 0 and N T = − N B = 1 and N T = 0 LLs are almost degenerate. Edgechannels appear in the ν = 0 quantum Hall system (QHS) in the energy window − V < E F < V . The condition forthe presence of edge channels [Eq. (4)] are satisfied with ∆ side ≃ < V . This indicates that nonlocal transportobserved in Ref. [S2] originates from the edge channels in the ν = 0 QHS. The spin expectations values in Fig. S1(b)show that edge channels are spin-momentum locked and the N = 0 LLs are spin-polarized. Thus observations of thespin transport phenomena proposed in this paper are experimentally feasible in available topological thin films withthe experimental setups in Fig. 3. - - - - (a) k y (1/nm) E ne r g y ( m e V ) N T =-1N T =-2N B =1N B =2N B =0N T =0 ν =0(b) k y (1/nm) N B =0N T =0 s X s Z - - - - (c) - - - k y (1/nm) ( n m ) N B =0N T =0 side surface -
20 0 200 topsurfacebottomsurface x (nm) sidesurface
FIG. S1. Band structure of the surface Weyl fermions in the 2D model. (a) The energy levels of H in Eq. (1) are plottedagainst the momentum k y for 2 d = 8 nm, B = 15 T, 2 V = 70 meV. Colors encode the expectation value of the position h x i ;the Landau levels on the top and bottom surfaces are depicted in red and blue in the region k y <
0, while linear bands in greenare edge channels on the side surface. LLs at the top (bottom) surface are labelled by indices N T ( N B ). (b) Spin textures inthe ( s x , s z )-plane and (c) expectation values h x i of the two bands connected to the N T = 0 and N B = 0 LLs. Edge channels( k y &
0) show the spin-momentum locking, while the N = 0 Landau levels on the top and bottom surfaces are spin polarized. Estimates of the parameters controlling presence or absence of the edge channels are given for v F = 5 × m / s asfollows. The cyclotron energy ~ ω c is given by ~ ω c ≃ p B (T) meV . (S6)The gap of the Weyl fermions at the side surface due to the finite size effect can be roughly estimated as∆ side ≃ ~ v F d ≃ d (nm) meV . (S7)(S7)