Adiabatic quantum pumping through surface states in 3D topological insulators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Adiabatic quantum pumping through surface states in 3D topological insulators
M. Alos-Palop, Rakesh P. Tiwari and M. Blaauboer
Delft University of Technology, Kavli Institute of Nanoscience,Department of Quantum Nanoscience, Lorentzweg 1, 2628 CJ Delft, The Netherlands. (Dated: November 9, 2018)We investigate adiabatic quantum pumping of Dirac fermions on the surface of a strong 3Dtopological insulator. Two different geometries are studied in detail, a normal metal – ferromagnetic– normal metal (NFN) junction and a ferromagnetic – normal metal – ferromagnetic (FNF) junction.Using a scattering matrix approach, we first calculate the tunneling conductance and then theadiabatically pumped current using different pumping mechanisms for both types of junctions.We explain the oscillatory behavior of the conductance by studying the condition for resonanttransmission in the junctions and find that each time a new resonant mode appears in the transportwindow, the pumped current diverges. We also predict an experimentally distinguishable differencebetween the pumped current and the rectified current.
PACS numbers: 73.20.-r, 73.40.-c, 73.23.-b, 73.63.-b
I. INTRODUCTION
Recently, surface states in topological insulators haveattracted a lot of attention in the condensed-matter com-munity . Both in two-dimensional (e.g., HgTe) and inthree-dimensional (e.g., Bi Se ) compounds with strongspin-orbit interaction the topological phase has beendemonstrated experimentally . Although these com-pounds are insulating in the bulk (since they have an en-ergy gap between the conduction band and the valanceband), their surface states support topological gaplessexcitations. In the simplest case these low-energy excita-tions of a strong three-dimensional topological insulatorcan be described by a single Dirac cone at the center ofthe two-dimensional Brillouin zone (Γ point) . Thecorresponding Hamiltonian is given by H = ~ v F ~σ · ~k − µI. (1)Here ~σ represents a vector whose three components arethe three Pauli spin matrices, I represents a 2 × v F is the Fermi velocity, and µ is the chemical potential. The low-energy excitationsof H [Eq. (1)] are topologically protected against per-turbations . This has prompted recent research on thetransport properties of surface Dirac fermions. For ex-ample, the conductance and magnetotransport of Diracfermions have been studied in normal metal – ferromag-net (NF), normal metal – ferromagnetic – normal metal(NFN) and arrays of NF junctions on the surface of atopological insulator , suggesting the possibility ofan engineered magnetic switch. An anomalous magne-toresistance effect has been predicted in ferromagnetic-ferromagnetic junctions . Also, electron tunneling andmagnetoresistance have been studied in ferromagnetic –normal metal – ferromagnetic (FNF) junctions , forwhich it has been predicted that the conductance canbe larger in the anti-parallel configuration of the mag-netizations of the two ferromagnetic regions than in theparallel configuration. In addition, a large research efforthas been devoted to studying models which predict the existence of Majorana fermion edge states at the interfacebetween superconductors and ferromagnets deposited ona topological insulator .In this article we investigate adiabatic quantum pump-ing of Dirac fermions through edge states on the sur-face of a strong three-dimensional topological insulator.Quantum pumping refers to a transport mechanism inmeso- and nanoscale devices by which a finite dc cur-rent is generated in the absence of an applied bias byperiodic modulations of at least two system parameters(typically gate voltages or magnetic fields) . In or-der for electrical transport to be adiabatic, the periodof the oscillatory driving signals has to be much longerthan the dwell time τ dwell of the electrons in the system, T = 2 πω − ≫ τ dwell . In the last decade, many differentaspects of quantum pumping have been theoretically in-vestigated in a diverse range of nanodevices, for examplecharge and spin pumping in quantum dots , the roleof electron-electron interactions , quantum pumpingin graphene mono- and bilayers as well as chargeand spin pumping through edge states in quantum Hallsystems and recently a two-dimensional topological in-sulator . On the experimental side, Giazotto et al. have recently reported an experimental demonstration ofcharge pumping in an InAs nanowire embedded in a su-perconducting quantum interference device (SQUID).Our main focus is to study quantum pumping inducedby periodic modulations of gate voltages or exchangefields, which are induced by a ferromagnetic strip in twotopological insulator devices: a NFN and a FNF junc-tion, see Figs. 1 and 2. Using a scattering matrix ap-proach, we obtain analytical expressions for the angle-dependent pumped current in both types of junctions.We find that the adiabatically pumped current in a NFNtopological insulator junction induced by periodic mod-ulations of gate voltages reaches maximum values at spe-cific energy values. In order to explain the position ofthese values, we study in detail the conductance of thejunctions. In particular, we provide an explanation forresonances in the conductance that were predicted but Topological insulator
PSfrag replacements N l N r F~x~y d
FIG. 1. (Color online) Sketch of the N l FN r junction on thesurface of a topological insulator. Pumping is induced byapplying gate voltages (not shown) to the normal leads. Inthe middle region a thin ferromagnetic film induces ferromag-netism on the surface of the topological insulator by meansof the exchange coupling . The arrow in the middle regionindicates the direction of the magnetization M in this region. not explained in detail in the previous works . Weshow that each time a new resonant mode appears in thejunction the conductance increases and the pumped cur-rent reaches a maximum value. For the FNF pump wepredict a non-zero current by periodic modulation of theexchange magnetic coupling in the absence of externalvoltages. We observe and analyze basic similarities anddifferences between the two pumps studied in this paperand highlight an experimentally distinguishable featurebetween the pumped current and the conductance.The remainder of the paper is organized as follows. InSec. II, we describe the NFN and FNF junctions and usea scattering matrix model to calculate the reflection andtransmission coefficients of both junctions. In Sec. III, wereview the conductance of the NFN junction and presenta detailed analysis of the plateau-like steps that appearin the conductance. We also analyze and compare theconductance of the FNF junction with parallel and anti-parallel configuration of the magnetization. In Sec. IV,we calculate the adiabatically pumped current for thetwo different pumps and derive analytical expressions asa function of the angle of incidence of the carriers. Wealso investigate the dependence of the pumped currentand the conductance on the width d of the middle region.Finally, in Sec. V we summarize our main results andpropose possibilities for experimental observation of ourpredictions. II. NFN AND FNF JUNCTIONS
We first describe the NFN junction, see Fig. 1. Thejunction is divided into three regions: region N l (for x < N r (for x > d ) and the ferromagneticregion F in the middle. The left and right-hand side ofthe junction represent the bare topological insulator. Thecharge carriers (surface Dirac fermions) in these regionsare described by the Hamiltonian H [Eq. (1)] whose eigenstates are given by ψ ± N = 1 √ (cid:18) ± e ± iα (cid:19) e ± ik n x e iqy , (2)where +( − ) labels the wavefunctions traveling from theleft (right) to the right (left) of the junction. The angleof incidence α and the momentum k n in the x -directionare given by: sin( α ) = ~ v F q | ǫ + µ | , (3) k n = s(cid:18) ǫ + µ ~ v F (cid:19) − q . (4)Here ǫ represents the energy measured from the Fermi en-ergy ǫ F and q denotes the momentum in the y -direction.In the normal regions N l and N r a dc electrical volt-age can be applied via metallic top gates to tune thechemical potential µ and thereby control the number ofcharge carriers incident on the junction. We assume gatevoltages to be small compared to the bandgap for bulkstates ( eV i ≪ E g ∼ i = l, r ), so that transport iswell described by surface Dirac states . In this case, theeigenstates are given by ψ ± N l = 1 √ (cid:18) ± e ± iα l (cid:19) e ± ik nl x e iqy , (5) ψ ± N r = 1 √ (cid:18) ± e ± iα r (cid:19) e ± ik nr ( x − d ) e iqy , (6)sin( α i ) = ~ v F q | ǫ + µ − eV i | , (7) k n i = s(cid:18) ǫ + µ − eV i ~ v F (cid:19) − q , (8)where the index i = l, r labels the normal sides of thejunction.In the middle region M of the junction (0 < x < d ), thepresence of the ferromagnetic strip modifies the Hamil-tonian by providing an exchange field. The Hamil-tonian that describes the surface states is now H = H + H induced , where the induced exchange Hamiltonianis given by H induced = ~ v F M σ y , (9)with the magnetization ~M = M ˆ y . The magnitude M depends on the strength of the exchange coupling of theferromagnetic film and can be tuned for soft ferromag-netic films by applying an external magnetic field . Theeigenstates of the full Hamiltonian H are then given by: ψ ± F = 1 √ (cid:18) ± e ± iα m (cid:19) e ± ik m x e iqy , (10) Topological insulator
PSfrag replacements F l F r N ~x~y d
FIG. 2. (Color online) Sketch of the F l NF r junction. Ferro-magnetic films are placed on top of the topological insulatoron the left and right providing exchange fields in these re-gions. The arrows indicate the direction of the correspondingmagnetizations M l and M r , see the text for further explana-tion. with sin( α m ) = ~ v F ( q + M ) | ǫ + µ | , (11)and k m = s(cid:18) ǫ + µ ~ v F (cid:19) − ( q + M ) . (12)From Eq. (12) we see that for a given energy there existsa critical magnetization M c = ± | ǫ + µ | / ( ~ v F ) , (13)beyond which for all transverse ( q ) modes the wave-function changes from propagating to spatially decaying(evanescent) along the x -direction . Now we describe the FNF junction, see Fig. 2. Region F l ( x <
0) and region F r ( x > d ) are modeled as ferro-magnetic regions, respectively, with different magnetiza-tions M l , M r along the y -axis and corresponding wave-function ψ F [Eq. (10)]. The Dirac fermions in the middleregion N (0 < x < d ) are described by the wavefunctions ψ N [Eq.(2)]. When calculating transport properties ofthe FNF junction, we focus on two different alignmentsof the magnetizations of the ferromagnetic regions: theparallel configuration (M l k M r ), where the magnetiza-tions in the ferromagnetic regions point in the same di-rection, and the anti-parallel configuration (M l k - M r ),in which the magnetizations are in opposite directions.Using Eqns. (2)-(12) we can calculate the reflection andtransmission coefficients for a Dirac fermion with energy ǫ and transverse momentum q incident from the left on thejunction, for both the NFN and the FNF junctions. Tothis end, we consider a general F l F m F r junction, wherethe wavefunctions in each of the three regions left ( l ),middle ( m ) and right ( r ) are given by: ψ l = ψ + l + r ll ψ − l ,ψ m = p ψ + m + q ψ − m , (14) ψ r = t rl ψ + r . Here ψ ± j ( j = l, m, r ) are the wavefunctions (5), (6) or(10) (depending on the junction considered) and r ll and t rl denote the corresponding reflection and transmissioncoefficients. By requiring continuity of the wavefunctionat the interfaces x = 0 and x = d , we obtain the reflectionand transmission coefficients: r ll = e iα l e ik m d (1 + e i ( α m + α l ) )( e iα m − e iα r ) + ( e iα l − e iα m )(1 + e i ( α m + α r ) ) e ik m d ( e iα m − e iα l )( e iα m − e iα r ) + (1 + e i ( α m + α l ) )(1 + e i ( α m + α r ) ) , (15) t rl = e ik m d (1 + e iα m )(1 + e iα l ) e ik m d ( e iα m − e iα l )( e iα m − e iα r ) + (1 + e i ( α m + α l ) )(1 + e i ( α m + α r ) ) . (16)Here α j denotes the polar angle of the wavevector in re-gion j = l, m, r [Eqns. (7) and (11)]. When consideringan electron incident from the right lead, one can similarlyobtain r rr and t lr . These expressions for the reflectionand transmission coefficients form the basis of our cal-culations of the conductance and the pumped current inSecs. III and IV respectively. III. CONDUCTANCE
The conductance G NFN of a topological insulator NFNjunction has been studied in earlier work by Mondal etal. , who predicted oscillatory behavior of G NFN asa function of the applied bias voltage (see also Fig. 3). In this section we first briefly review their results andthen add a quantitative explanation for the oscillationsof the conductance. This explanation is crucial for under-standing the behavior of the pumped current in the nextsection. We also calculate and analyze the conductancein a FNF junction.The general expression for the conductance G acrossthe junction in terms of the transmission probability T ( α ) ≡ | t rl ( α ) | is given by G = ( G / Z π/ − π/ T ( α ) cos α dα. (17)Here G = e h ρ ( eV ) ~ v F W , ρ ( eV ) = | µ + eV | / (2 π ( ~ v F ) )denotes the density of states, W is the sample width, PSfrag replacements G N F N / G ǫ/µ (1)(2)(3)(4)(1) → ˜ M = 3(2) → ˜ M = 3 . → ˜ M = 4(4) → ˜ M = 4 . FIG. 3. (Color online) The conductance G NFN of the NFNjunction [Eq.(17)] as a function of ǫ/µ for V l = V r = 0and for different values of the effective magnetization ˜ M ≡ ~ v F M/µ = 3 (solid blue line), 3.5 (dashed green line), 4 (dot-dashed red line) and 4.5 (dotted light-blue line). The effectivejunction width ˜ d ≡ µd/ ( ~ v F ) = 5. and the integration is over all the angles of incidence α .For V l = V r = 0 ( α l = α r = α ) the angle-dependenttransmission probability T NFN ( α ) is given by T NFN ( α ) = cos ( α ) cos ( α m ) / (cid:2) cos ( k m d ) cos ( α ) cos ( α m )+ sin ( k m d )(1 − sin( α ) sin( α m )) (cid:3) , (18)where α m is the polar angle of the wave vector in themiddle region as defined in Eq. (11). This angle can beexpressed in terms of α using the fact that the momentumis conserved along y -axis as:sin( α m ) = sin( α ) + M ~ v F | ǫ + µ | . (19)Figure 3 shows the conductance of the NFN junction[obtained from Eqns. (17) and (18)] as a function of theenergy of the incoming carriers ǫ/µ for different values ofthe effective magnetization ˜ M ≡ ~ v F M/µ . For a givenmagnetization M , the conductance is zero for ǫ < ǫ c , i.e., G NFN ( ǫ ) = 0, with the critical energy ǫ c ≡ ~ v F M/ − µ .Below this energy there are no traveling modes inside thebarrier. Our results agree with the previous results in theliterature .From Fig. 3 it can be observed that the conductancechanges from plateau-like to oscillatory as ǫ/µ increases.In order to provide an explanation for this behavior wefirst analyze the plateau like regime in detail. After set-ting α l = α r ≡ α in Eq. (15), we begin by finding the con-ditions when the reflection coefficient is zero, i.e., r ll = 0.The first, trivial, condition α = α m + 2 πn correspondsto the situation of an entirely normal junction (i.e., noferromagnetic region). The second and more interestingcondition is sin( k m d ) = 0. This is the case when trans-mission occurs via a resonant mode of the junction and PSfragreplacements − π/ − π/ π/ π/ T N F N ( α ) α . . . . (a)n=1 → PSfragreplacements − π/ − π/ π/ π/ T N F N ( α ) α . . . . (b)n=2 → PSfragreplacements − π/ − π/ π/ π/ T N F N ( α ) α . . . . (c)n=3 → PSfragreplacements − π/ − π/ π/ π/ T N F N ( α ) α . . . . (d) ← n=4 PSfragreplacements − π/ − π/ π/ π/ T N F N ( α ) α . . . . (e)n=5 → PSfragreplacements − π/ − π/ π/ π/ T N F N ( α ) α . . . . (f)n=6 → FIG. 4. (Color online) The transmission probability T NFN ( α )[Eq. (18)] as a function of the angle of incidence α for differ-ent values of energy ǫ/µ , (a) ǫ/µ = 0 .
7, (b) ǫ/µ = 0 .
9, (c) ǫ/µ = 1 .
2, (d) ǫ/µ = 1 .
6, (e) ǫ/µ = 2 .
4, and (f) ǫ/µ = 2 . d = 5 and ˜ M = 3. can be written as (using Eqns. (3) and (12)) k m d = | ǫ + µ | ~ v F d s − (cid:18) sin α + ~ v F M | ǫ + µ | (cid:19) = nπ. (20)Eq. (20) indicates that for a given M and ǫ there arecertain privileged angles α c for which the barrier becomestransparent:sin( α c ) = ± vuut − nπ ˜ d (1 + ǫµ ) ! − ˜ M | ǫµ | , (21)with ˜ d ≡ dµ/ ( ~ v F ) being the dimensionless barrierlength. These modes are referred to as resonant modesin this article.Figure 4 shows the transmission probability [Eq. (18)]as a function of the angle of incidence α for differentvalues of energy ǫ/µ . The dashed (red) vertical linescorrespond to the angles satisfying Eq. (21) for differ-ent n . It can be seen that as the energy increases moreresonant modes become available for transmission. It isalso worth noting that for energies at which only onemode is present ( n = 1, see Fig. 4(a)) the transmission PSfrag replacements − π/ − π/ π/ α c ǫ/µ n = 1 → n = 2 → ← n = 3 ← n = 4 ← n = 5 n = 6 → FIG. 5. (Color online) The real part of the angle of incidence α c [Eq. (21)] versus energy ǫ/µ for the modes n = 1 , , , , is strongly localized at one particular angle. This prop-erty could be exploited to fabricate single-mode filters.For low energy excitations only negative angles α (i.e., q -momenta anti-parallel to M) contribute to the conduc-tance, see Figs. 4(a)-(d). As the energy increases, theresonant modes move from the left to the right and alsopositive angles α (i.e., q -momenta parallel to M) beginto contribute, see Figs. 4(e) and (f).Now we address the question why a mode becomesresonant in the barrier. Figure 5 shows the angle of inci-dence α c [Eq. (21)] for different values of n as a functionof energy ǫ/µ . For a given n , the resonant mode does notcontribute to the conductance if the energy ǫ satisfies thecondition ǫ c < ǫ < ǫ α n c , because the imaginary part of themomentum k m is nonzero and thus the mode is decayingalong the x -direction. This critical energy ǫ α n c for eachmode is given by: ǫ α n c µ = n π d ˜ M + ˜ M − . (22)When ǫ > ǫ α n c , k m becomes real and the mode be-comes resonant. As we increase the energy, all the modesasymptotically reach their saturation angle α = π/ α = − π/
2. As the energy increases, a new resonantmode appears and the conductance increases in a step-like manner. The plateaus are not sharp due to the factthat each new mode appearing is not sharply peaked,but rather has a certain distribution around a particularangle of incidence, see Fig. 4. Once the energy is largeenough for there to be contributions from both positive and negative angles of incidence, the conductance be-comes oscillatory. For very large energies ( ǫ ≫ ǫ c ), the PSfrag replacements G N F N / G ǫ/µ (1) (2) (3) (4)(1) → eV/µ = 0(2) → eV/µ = 1(3) → eV/µ = 2(4) → eV/µ = 3 FIG. 6. (Color online) The conductance G NFN of the NFNjunction as a function of ǫ/µ for different values of gate volt-ages, eV /µ = 0 (solid blue line), 1 (dashed green line), 2(dashed-dot red line) and 3 (double-dotted light-blue line).As before, ˜ M = 3 and ˜ d = 5. effect of the magnetic barrier disappears and the conduc-tance becomes unity ( G NFN = G ).Figure 6 shows the conductance as a function of energyfor several values of applied bias voltages V l = V r ≡ V .As expected, the features of the conductance remain thesame for finite V . As we increase eV , the critical energy ǫ c = ~ v F M/ eV / − µ for the onset of the conductanceincreases and the spacing between two consecutive reso-nant modes decreases. As a result the plateaus becomenarrower.In the remaining part of this section we study theconductance in a topological insulator FNF junction, asshown in Fig. 2. We consider both the junction withparallel and with anti-parallel magnetization in the fer-romagnetic regions. In the parallel configuration, us-ing α l = α r ≡ α = sin − ( ~ v F ( q + M ) / | ǫ + µ | ) and α m = sin − ( ~ v F q/ | ǫ + µ | ) in Eq. (16), we find that theconductance is similar to the conductance of a NFN junc-tion, as displayed in Fig. 3. However, in the FNF junctionthe first resonant mode becomes resonant for positive α (i.e., transverse q -momentum parallel to M ) and as theenergy increases, the resonances move towards negativevalues of the angle α . This, however, does not affect thetotal conductance, as we sum over all possible angles ofincidence, and the same analysis as for the NFN junctionpresented above can be applied to understand the FNFjunction with parallel magnetization.In the case of anti-parallel alignment of the magne-tization in the two ferromagnetic regions, we substitutesin( α l ) = ~ v F ( q + M ) / | ǫ + µ | , sin( α r ) = ~ v F ( q − M ) / | ǫ + µ | and sin( α m ) = ~ v F q/ | ǫ + µ | in Eq. (16). The conduc-tance of this junction was studied previously in Refs. and the transmission probability T FNF,AP ( α l , α r ) ≡| t rl ( α l , α r ) | is given by: T FNF,AP ( α l , α r ) = cos ( α l ) cos ( α m )cos ( k m d ) cos ( α l + α r ) cos ( α m ) + sin ( k m d ) (cid:2) cos (cid:0) α l − α r (cid:1) − sin( α l + α r ) sin( α m ) (cid:3) . (23) PSfrag replacements G F N F , A P / G ǫ/µ (1)(2)(3)(1) → ˜ M = 3(2) → ˜ M = 3 . → ˜ M = 4 FIG. 7. (Color online) The conductance G FNF,AP of the FNFjunction in the anti-parallel configuration as a function of ǫ/µ ,for ˜ M = 3 (blue solid line), 3 . d = 5. The total conductance is obtained by multiplying T FNF,AP ( α l , α r ) with cos( α r ) / cos( α l ) and then integrat-ing over the allowed angles of incidence , i.e., from α c =sin − (2 ~ v F M/ ( | ǫ + µ | ) −
1) to α c = sin − (2 ~ v F M/ ( | ǫ + µ | ) + 1). Thus we can write G FNF,AP = G / Z π/ α c G FNF,AP ( α l , α r ) cos( α l ) dα l , (24)where G FNF,AP ( α l , α r ) = cos( α r )cos( α l ) T FNF,AP ( α l , α r ) . (25)Fig. 7 shows the conductance of the FNF junction inthe anti-parallel configuration. From the horizontal axiswe see that the critical energy ǫ c for the onset of theconductance is larger than in the corresponding parallelconfiguration. Moreover, as the energy ǫ/µ increases theconductance exhibits no plateau behavior: it increases inan oscillatory fashion. This oscillatory behavior can beunderstood from Fig. 8, which shows G FNF,AP ( α ) for fourdifferent values of ǫ/µ . Note that all the angles ( α l , α m and α r ) can be expressed in terms of one angle, which wechoose to be α l ≡ α . As the energy increases, the areaunder the curve oscillates resulting in oscillations in theconductance.Summarizing, we have obtained a quantitative expla-nation for the behavior of the conductance in topologicalinsulator NFN and FNF-junctions in terms of the num-ber of resonant modes in the junction. This explanationforms the basis for understanding the behavior of thepumped current in the next section. PSfragreplacements G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G α l α l α l α l α l (a)(a)(a)(a)(a)00000 π/ π/ π/ π/ π/ π/ π/ π/ π/ π/ PSfragreplacements G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G α l α l (b)(b)00 π/ π/ π/ π/ PSfragreplacements G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G α l α l α l (c)(c)(c)000 π/ π/ π/ π/ π/ π/ PSfragreplacements G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G G F N F , A P ( α l ) / G α l α l α l α l (d)(d)(d)(d)0000 π/ π/ π/ π/ π/ π/ π/ π/ FIG. 8. The angle-dependent total transmission T FNF,AP ( α l , α r ) for the anti-parallel configuration ofthe FNF junction as a function of the angle of incidence α l for (a) ǫ/µ = 2 .
47, (b) ǫ/µ = 2 .
57, (c) ǫ/µ = 3 .
10 and (d) ǫ/µ = 3 .
40. Parameters used are ˜ M = 3 and ˜ d = 5. IV. ADIABATICALLY PUMPED CURRENT
In this section we investigate adiabatically pumpedcurrents through NFN and FNF junctions in a topo-logical insulator . In general, a pumped current isgenerated by slow variation of two system parameters X and X in the absence of a bias voltage . Forperiodic modulations X ( t ) = X , + δX cos( ωt ) and X ( t ) = X , + δX cos( ωt + φ ), the pumped current I p into the left lead of the junction can be expressed in termsof the area A enclosed by the contour that is traced outin ( X , X )-parameter space during one pumping cycleas : I p = ωe π Z A dX dX X m Π( X , X ) (26a) ≈ ωe π δX δX sin φ X m Π( X , X ) , (26b)withΠ( X , X ) ≡ Im (cid:18) ∂r ∗ ll ∂X ∂r ll ∂X + ∂t ∗ lr ∂X ∂t lr ∂X (cid:19) . (27)Here r ll and t lr represent the reflection and transmissioncoefficients into the left lead and the index m sums overall modes (a similar expression can be obtained for thepumped current into the right lead). Eq. (26b) is validin the bilinear response regime where δX ≪ X , and δX ≪ X , and the integral in Eq. (26a) becomes inde-pendent of the pumping contour.First we analyze the NFN pump, where the pumpedcurrent is generated by adiabatic variation of gate volt-ages V l and V r which change the chemical potential inthe normal leads on the left and right of the junction, re- spectively (see Fig. 1). Calculating the derivatives of thereflection and transmission coefficients r ll [Eq. (15)] and t rl with respect to α l and α r , substituting into Eq. (27)and using ∂α j / ( e∂V j ) = tan( α j ) / | ǫ + µ − eV j | ( j = l, r ),the pumped current for V = V ≡ V and for a specificangle of incidence α is given by: I NFN p ( α ) = − I NFN0 cos ( α m ) sin ( α ) cos( α ) sin(2 k m d )(1 + ǫ/µ − eV /µ ) (cos ( α ) cos ( α m ) cos ( k m d ) + sin ( k m d )(1 − sin( α ) sin( α m )) ) . (28) PSfrag replacements I F N F p / I ǫ/µ (a) FIG. 9. The pumped current I NFN p [Eq. (30)] in the NFNjunction as a function of ǫ/µ for V = 0, ˜ d = 5 and ˜ M = 3. Here I NFN0 ≡ ωe/ (8 π ) sin( φ )( eδV /µ )( eδV /µ ) andsin( α m ) is given by Eq. (19). In the limit M → α m → α ) in an entirely normal junction, we obtain fromEq. (28) the angle-dependent pumped current as: I NFN p | ˜ M =0 = I NFN0 Z π/ − π/ sin(2 k m d ) sin α (1 + ǫ/µ − eV /µ ) cos α dα. (29)On the other hand, the transmission T NFN ( α ) [Eq. (18)]in this limit is given by T NFN | ˜ M → →
1, independentof the angle of incidence α . We notice that even if theprobability for transmission is one, it is possible to pumpa current in the adiabatic driving regime. The totalpumped current I NF Np is then obtained by integratingover α : I NFN p = Z π/ − π/ I NFN p ( α ) cos α dα. (30)In general, this integral cannot be evaluated analytically and we have obtained our results numerically. Figure 9shows the total pumped current I NFN p (in units of I NFN0 )at zero bias V = V = V = 0 for ˜ M = 3. ComparingFigs. 3 and 9 we see that there is a correlation betweenthe pumped current and the conductance for the NFNjunction: for low energies, the pumped current I NF Np iszero as no traveling modes are allowed in the junction.As we increase the energy, each time a resonant modeappears [see Eq. (20)], the pumped current diverges andchanges sign. For energies where both positive and neg-ative angles of incidence contribute to the conductance,the pumped current remains finite but keeps changingits sign. From Figs. 3 and 9 it can also be seen thatthe pumped current vanishes for energies at which sub-sequent resonant modes become fully transmitting. Inorder to gain further insight we plot the analogue of Fig. 4for the pumped current. Figure 10 shows the pumpedcurrent [Eq. 30] as a function of the angle of incidence α for different values of ǫ/µ . The chosen values of ǫ/µ aresame as in Fig. 4. We see that the features in Fig. 10 havea direct correlation with the features in Fig. 4: wheneverthere is a sharp peak in the transmission the pumpedcurrent diverges and changes sign. The key feature thatdistinguishes between the pumped current and the con-ductance is that the pumped current changes sign at par-ticular values of the energy, while the conductance doesnot.Now we analyze the pumped current in the FNF junc-tion with parallel orientation of the magnetizations. Inthis system, the driving parameters are the magneti-zations M l and M r in the left and right contacts, re-spectively, see Fig. 2. After calculating the derivativesof the reflection and transmission coefficients, obtainingthe imaginary part of Eq. (27), and using ∂α j /∂M j = ~ v F / ( | ǫ + µ | cos( α j )) ( j = l, r ), the pumped current I FNF p ( α ) for M = M = M is: I FNF p ( α ) = − I FNF0 cos ( α m ) cos( α ) sin(2 k m d )(1 + ǫ/µ ) (cos ( α ) cos ( α m ) cos ( k m d ) + sin ( k m d )(1 − sin( α ) sin( α m )) ) . (31) −0.3−0.2−0.100.10.20.3−0.3−0.2−0.100.10.20.3 PSfragreplacements I N F N p ( α ) / I I N F N p ( α ) / I αα ( a )( a ) − π/ − π/ − π/ − π/ π/ π/ π/ π/ −0.2−0.100.10.2−0.2−0.100.10.2−0.2−0.100.10.2−0.2−0.100.10.2 PSfragreplacements I N F N p ( α ) / I I N F N p ( α ) / I I N F N p ( α ) / I I N F N p ( α ) / I αααα ( b ) − π/ − π/ − π/ − π/ − π/ − π/ − π/ − π/ π/ π/ π/ π/ π/ π/ π/ π/ −0.2−0.100.10.2−0.2−0.100.10.2 PSfragreplacements I N F N p ( α ) / I I N F N p ( α ) / I αα ( c )( c ) − π/ − π/ − π/ − π/ π/ π/ π/ π/ −0.4−0.3−0.2−0.100.1−0.4−0.3−0.2−0.100.1 PSfragreplacements I N F N p ( α ) / I I N F N p ( α ) / I αα ( d )( d ) − π/ − π/ − π/ − π/ π/ π/ π/ π/ −12−8−404x 10 −3 −12−8−404x 10 −3 PSfragreplacements I N F N p ( α ) / I I N F N p ( α ) / I αα ( e )( e ) − π/ − π/ − π/ − π/ π/ π/ π/ π/ −0.8−0.6−0.4−0.200.2−0.8−0.6−0.4−0.200.2 PSfragreplacements I N F N p ( α ) / I I N F N p ( α ) / I αα ( f )( f ) − π/ − π/ − π/ − π/ π/ π/ π/ π/ FIG. 10. The pumped current I NFN p ( α ) [Eq. (28)] as a func-tion of the angle of incidence α for different values of energy ǫ/µ , (a) ǫ/µ = 0 .
7, (b) ǫ/µ = 0 .
9, (c) ǫ/µ = 1 .
2, (d) ǫ/µ = 1 . ǫ/µ = 2 .
4, and (f) ǫ/µ = 2 .
9. Parameters used are ˜ d = 5and ˜ M = 3. PSfragreplacements G N F N / G ˜ d ( a ) PSfragreplacements I N F N p / I ˜ d ( b ) FIG. 11. (a) The conductance of the NFN junction as afunction of ˜ d . (b) The pumped current for the NFN junctionas a function of ˜ d . Parameters used are ǫ/µ = 2 .
5, ˜ M = 3and V = 0. Here I FNF0 = ωe/ (8 π ) sin( φ )( ~ v F δM l /µ )( ~ v F δM r /µ )and sin( α m ) = sin( α ) − ~ v F M/ ( | ǫ + µ | ).The behavior of the current I FNF p is similar to thatof the pumped current I NFN p in a NFN-junction (shownin Fig. 9). This can also be seen by comparing the de-nominators in Eqns. (28) and (31). Again we observethat the pumped current diverges at exactly the same locations where the conductance changes sharply. Butthere is an important difference between both pumpedcurrents. The pumped current in an NFN-junction atnormal incidence vanishes, I NFN p ( α = 0) = 0, while I FNF p ( α = 0) = 0. This difference arises because thetwo pumps are driven by two different parameters (volt-ages in the NFN pump and magnetizations in the FNFpump).Finally, we briefly analyze the behavior of the pumpedcurrent as a function of the width d of the middle region.For energies below ǫ c , the pumped current of the NFNjunction decays to zero as the width d increases (thereare no resonant modes in the system). For energies largerthan ǫ > ǫ c , the pumped current I NFN p oscillates as afunction of width ˜ d . Fig. 11 shows the conductance andthe pumped current as a function of ˜ d for ǫ/µ = 2 .
5. Thepeaks in the conductance correspond to the resonancecondition Eq. (20). The pumped current I NFN p changesits sign at exactly the same values of ˜ d where the con-ductance has a maximum. This analysis holds as well forthe FNF junction. V. SUMMARY AND DISCUSSION
To summarize, we have analyzed quantum transportby Dirac fermion surface states in NFN and the FNFjunctions in a 3D topological insulator. We have shownthat for low energies the appearance of a new resonantmode results in a plateau-like increment of the conduc-tance and a diverging pumped current in these junctionswhich also changes sign. This is our key result, and rep-resents an experimentally distinguishable signature be-tween conductance and the pumped current. We high-lighted an interesting difference between the two differ-ent pumping mechanisms for the NFN and FNF junc-tions, observing different behaviors for normal incidence( α = 0). Experimentally, the NFN pump could be re-alized using current technology. The FNF pump will bemore difficult to realize since it requires oscillating mag-netizations. A possible way to realize a FNF pump couldbe by moving the two ferromagnetic layers coherentlyusing a nanomechanical oscillator . Experimental veri-fication of our predictions will provide further insight intoquantum transport through these junctions. ACKNOWLEDGMENTS
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