Super-resonant transport of topological surface states subjected to in-plane magnetic fields
Song-Bo Zhang, Chang-An Li, Francisco Peña-Benitez, Piotr Surówka, Roderich Moessner, Laurens W. Molenkamp, Björn Trauzettel
SSuper-resonant transport of topological surface states subjected to in-plane magneticfields
Song-Bo Zhang, ∗ Chang-An Li, Francisco Pe ˜ na-Benitez,
2, 3
Piotr Surówka,
2, 3, 4
Roderich Moessner,
2, 3
Laurens W. Molenkamp,
5, 6, 3, 7 and Björn Trauzettel
1, 3 Institut für Theoretische Physik und Astrophysik,Universität Würzburg, 97074 Würzburg, Germany Max-Planck-Institut für Physik komplexer Systeme,Nöthnitzer Strasse 38, 01187 Dresden, Germany Würzburg-Dresden Cluster of Excellence ct.qmat, Germany Department of Theoretical Physics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland Physikalisches Institut (EP3), Universität Würzburg, Am Hubland, 97074 Würzburg, Germany Institute for Topological Insulators, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany Max Planck Institute for Chemical Physics of Solids, Dresden D-01187, Germany (Dated: March 1, 2021)Magnetic oscillations of Dirac surface states of topological insulators are expected to be associatedwith the formation of Landau levels or the Aharonov–Bohm effect. We instead study the conductanceof Dirac surface states subjected to an in-plane magnetic field in presence of a barrier potential.Strikingly, we find that, in the case of large barrier potentials, the surface states exhibit pronouncedoscillations in the conductance when varying the magnetic field, in the absence of Landau levels orthe Aharonov–Bohm effect. These novel magnetic oscillations are attributed to the emergence of super-resonant regimes by tuning the magnetic field, in which almost all propagating electrons crossthe barrier with perfect transmission. In the case of small and moderate barrier potentials, we alsoidentify a positive magnetoconductance which is due to the increase of the Fermi surface by tiltingthe surface Dirac cone. Moreover, we show that for weak magnetic fields, the conductance displaysa shifted sinusoidal dependence on the field direction with period π and phase shift determined bythe tilting direction with respect to the field direction. Our predictions can be applied to manytopological insulators, such as HgTe and Bi Se , and provide important insights into exploring andunderstanding exotic magnetotransport properties of topological surface states. Introduction. —Topological insulators host gapless sur-face states which stem from nontrivial bulk topology [1–3]. These surface states can be modeled by a single Diraccone. Over the last two decades, topological insulatorshave been discovered in numerous materials [4–12] in-cluding HgTe [13], Bi − x Sb x [14] and Bi Se [15–18].Magnetotransport on Dirac surface states has been anactive research topic [19–51], theoretically and experi-mentally, since the discovery of topological insulators. Itprovides vital features, which include particularly mag-netic oscillations, to detect and characterize Dirac surfacestates. Magnetic oscillations are usually associated withthe formation of Landau levels or the Aharonov–Bohmeffect [19–27, 33–38]. Thus, a fundamentally intriguingquestion is whether magnetic oscillations of topologicalsurface states can appear in the absence of Landau levelsor the Aharonov–Bohm effect.Notably, in typical topological insulators, electron-holesymmetry in the energy spectrum of surface states is bro-ken by the presence of higher-order momentum correc-tions [52–54]. To fully understand the transport prop-erties of surface states in realistic systems, the consider-ation of this electron-hole asymmetry is important. In-terestingly, the interplay of electron-hole asymmetry andin-plane magnetic fields tilts the surface states at low en-ergies [50].In this Letter, we study the conductance of Dirac sur- Fig. 1. Schematic of the surface states (cyan and magenta)of a topological insulator (gray) with a barrier potential V extending over a length of L (magenta). An in-plane magneticfield B (blue arrows) is applied to the system. face states in presence of a barrier potential and an ex-ternal in-plane magnetic field, taking into account theelectron-hole asymmetry of the energy spectrum. We findthat for small and moderate barrier potentials (compara-ble to the Fermi energy), the surface states exhibit a posi-tive magnetoconductance due to the increase of the Fermisurface by the tilting in any direction. Remarkably, forlarger barrier potentials, super-resonant regimes of sur-face states appear by tuning the magnetic field, whichenable almost all surface propagating electrons to tunnelthrough the barrier without backscattering. These super-resonant regimes result in pronounced oscillations in theconductance as strength or direction of the magnetic fieldare varied. Moreover, we show that for weak magnetic a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b fields, the conductance is a sinusoidal function of fielddirection with period of π and phase shift dependent onthe angle between the tilting and field directions. Ourpredictions are applicable to many topological insulatorsincluding HgTe and Bi Se . Effective Hamiltonian of surface states. —The states ofa topological insulator on a surface can be described bya single Dirac cone [15, 52, 53] H ( k ) = m k + v ( k x s y − k y s x ) , (1)where k = ( k x , k y ) are momenta in the vicinity of the Γ point, v is the Fermi velocity, s x and s y are Pauli matri-ces acting on spin space. Moreover, in certain topologi-cal insulators, for instance, HgTe with zinc-blende crystalstructure, bulk inversion symmetry is broken, leading toextra terms H BIA = v b ( k x s x + k y s y ) + γk x k y [55]. Notethat we have included the quadratic terms in momentum, m k and γk x k y , which preserve time-reversal symmetry.These terms are often ignored in previous studies for sim-plicity. However, they break electron-hole symmetry inthe energy spectrum and can lead to interesting physicsas we show below.Applying an in-plane magnetic field B = B (cos θ, sin θ ) introduces a Zeeman term H Z = gµ B B · s , where g is the g -factor, µ B is the Bohr magneton, B and θ denote thestrength and direction of the magnetic field, respectively.The Zeeman term not only shifts the Dirac cone awayfrom the Γ point in momentum space but also tilts theDirac cone [50]. Considering that v b is typically muchsmaller than v , we can find the position shift of the Diracpoint as k s = k s ( − sin θ, cos θ ) with k s = gµ B B/v [56].It is perpendicular to the magnetic field, i.e., k s · B =0 . Near the Dirac point, the effective model for surfacestates can be written as [55] H ( k ) = v (˜ k x s y − ˜ k y s x ) + t x ˜ k x + t y ˜ k y , (2)where ˜ k = k − k s and the tilting vector t ≡ ( t x , t y ) isgiven by t = k s ( γ cos θ − m sin θ, m cos θ − γ sin θ ) . (3)The eigen-energies are thus tilted as E ± ( k ) = t · ˜ k ± v | ˜ k | . The tilting strength | t | is proportional to the fieldstrength and the tilting direction is controllable by thefield direction. We focus on the realistic case with smalltilting | t | < | v | throughout. Transmission probability. —We consider the surfacestates with a barrier potential V extending over a lengthof L in x -direction, as sketched in Fig. 1. The in-planemagnetic field is applied to the whole system. This setupcan be described by H tot = H ( − i∂ r ) − E F + V ( x ) (4)with E F the Fermi energy and the local electronic po-tential V ( x ) = V for | x | (cid:54) L/ and otherwise [57]. V may be created by local gating [58]. It can be positiveor negative. For simplicity, we assume the system to belarge in y -direction such that the transverse momentum k y is conserved.To study the transport properties of the system, weemploy the scattering approach. In each region, we findtwo eigenstates for given energy E and momentum k y .In the regions away from the barrier, their wavefunctionscan be written as ψ ± ( x, y ) = e i ˜ k y y e i ˜ k ± x (cid:0) e iθ ± , − (cid:1) T / N ± , (5)where e iθ ± ≡ v (˜ k y + i ˜ k ± ) / ( E k y − t x ˜ k ± ) , E k y = E + E F − t y ˜ k y , N ± = (cid:112) | e iθ ± | , and the wave numbers ˜ k ± in x -direction are given by ˜ k ± = [ − t x E k y ± v (cid:113) E k y − ( v − t x )˜ k y ] / ( v − t x ) . (6)In the barrier region, the wavefunctions have the sameform as Eq. (5) but with E k y replaced by E Bk y = E k y − V .Correspondingly, we use the superscript B to indicate theangles θ B ± and wave numbers ˜ k B ± inside the barrier region.The scattering state of injecting an electron from theone lead to the junction can be expanded in terms of thebasis wavefunctions, Eq. (5). Matching the wavefunctionof the scattering state at the interfaces, we derive thetransmission coefficient as t ˜ k y = e − i ˜ k + L e i (˜ k B − +˜ k B + ) L ( e iθ + − e iθ − )( e iθ B + − e iθ B − ) / Z , (7)where Z = e i ˜ k B + L ( e iθ + − e iθ B + )( e iθ − − e iθ B − ) − e i ˜ k B − L ( e iθ + − e iθ B − )( e iθ − − e iθ B + ) . The transmission probability is givenby T ˜ k y = | t ˜ k y | . More details of derivation are presentedin the Supplemental Material (SM) [55]. For the incidentmodes with ˜ k y = 0 , we always have θ ± = θ B ± = ± π/ and hence T ˜ k y =0 = 1 . This perfect transmission resultsfrom spin conservation and is related to Klein tunneling,similar to the case of graphene [59]. Notably, withouttilting, the results are independent of the magnetic field.This indicates that a simple position shift of the Diraccone in momentum space does not change the transportproperties of surface electrons. Positive magnetoconductance. —With the transmissionprobability, the (differential) conductance G (per unitlength) at zero temperature and zero bias voltage can beevaluated as G = e h (cid:90) d ˜ k y π T ˜ k y ( E = 0) , (8)where the sum runs over all modes distinguished by ˜ k y .We first look at the case of small and moderate barrierpotentials, i.e., | V | (cid:46) | E F | , as shown in Fig. 2. Notably, G increases as we increase the tilting strength in any di-rection. Recalling that the tilting strength grows linearlywith increasing the magnetic field B, this indicates a pos-itive magnetoconductance. For small barrier potentials, (a) (b) Fig. 2. (a) Conductance G (in units of e k F /πh with k F = | E F /v | ) as a function of tilt strength t x (for t y = 0 , blue), t y (for t x = 0 , green) and t x = t y = t (magenta) for junctionlength L = 100 v/E F and barrier potential V = 0 (solid), . E F (dotted), and E F (broken), respectively. (b) Number N k of propagating modes (in units of k F /π ) as a function of t x (for t y = 0 ), t y (for t x = 0 ), and t x = t y = t , respectively.Inset shows the Fermi surface for t = (0 . , v , (0 , . v , and (0 . , . v , respectively. | V | (cid:28) | E F | , G increases monotonically with increasing B . A larger barrier potential suppresses G and inducesslight oscillations. However, G increases overall with in-creasing B, see Fig. 2(a). These oscillations are closelyrelated to the super-resonant regimes of tilted surfaceelectrons, which we explain below.The positive magnetoconductance can be attributedto the enhanced Fermi surface of tilted surface states.To understand this, it is instructive to consider the zerobarrier limit V = 0 . In this limit, all propagating modestransmit through the junction without reflection. Thus,the conductance is simply given by the number N k ofpropagating modes, i.e., G = ( e /h ) N k . N k is deter-mined by the size of the Fermi surface in k y -direction,as illustrated in the inset of Fig. 2(b). Tilting the surfaceDirac cone in any direction enlarges the Fermi surfaceand hence the number of propagating modes. As shownby the circles in Fig. 2(b), we calculate N k numerically asa function of the tilting strength in three different direc-tions as considered in Fig. 2(a). Evidently, this depen-dence nicely agrees with the magnetoconductance (solidcurves). When the tilting occurs in x - or y -direction, wecan also obtain N k analytically from the tilted spectrum.Namely, N k = | E F | /π (cid:112) v − t x for tilting in x -directionand N k = v | E F | /π ( v − t y ) for tilting in y -direction. Super-resonant regime and conductance oscilla-tions. —Next, we consider larger barrier potentials, | V | > | E F | , and analyze the pronounced oscillations ofthe conductance. These oscillations can be understoodas the appearance of super-resonant regimes of surfacestates, where almost all propagating modes perfectlytransmit through the barrier at the same magneticfield (i.e., the same tilting). To make this clearer andsimplify the analysis, we focus on the large barrier limit, | V | (cid:29) | E F | . In this limit, we can approximate θ B ± ≈ ± π/ in Eq. (7) and simplify T ˜ k y = 1 − cos( θ + − θ − )1 − sin θ + sin θ − − cos[( k B + − k B − ) L ] cos θ + cos θ − . (9)From this expression, we find that the barrier becomestransparent for the mode with index ˜ k y when the reso-nance condition, sin[( k B + − k B − ) L/
2] = 0 , is fulfilled. Plug-ging the expression for k B ± from Eq. (6) into this equation,the resonance condition becomes ( V − t y ˜ k y ) − ( v − t x )˜ k y = [ nπ ( v − t x ) / ( vL )] (10)with n being an integer.When the tilting is in junction (i.e., x -) direction, wefind the solutions of t x to Eq. (10) as t x = ± (cid:112) | vV | L/ ( πn ) − v (11)for large integers n satisfying v | ˜ k y | (cid:62) | E F | (cid:28) {| V | − ( πnv/L ) , ( πnv/L ) } . Strikingly, these solutions are in-dependent of the mode index ˜ k y . This indicates thesuper-resonant regimes, where all propagating modeswith different ˜ k y exhibit perfect transmission. As aresult, we find resonance peaks in T ˜ k y and hence themaximal conductance G max = ( e /h ) N k at t x ( ∝ B )given by Eq. (11). Moreover, we find that at t x = ± (cid:112) | vV | L/ [ π ( n + 1 / − v , all modes have instead thelowest transmission probabilities given by T ˜ k y = 1 − ( v − t x )˜ k y /E F [55]. Summing over all modes, we hence obtainthe minimal conductance, G min = (2 e / h ) N k . There-fore, we observe pronounced oscillations of G with mag-nitude ∆ G osc as large as one third of the maximal con-ductance ∆ G osc = G max / . (12)Interestingly, the values of G max , G min and ∆ G osc (inunits of N k ) are universal and independent of the poten-tial V and range L of the barrier. Considering the in-crease of N k , when strengthening B , ∆ G osc increases. Incontrast, according to Eq. (11), the separations betweenthe conductance peaks depend strongly on the product V L , whereas they are insensitive to E F . Moreover, theydecrease with increasing B . All these results are in accor-dance with our numerical results displayed in Fig. 3(a),(b) and (g).When the tilting is in transverse (i.e., y -) direction,the solutions to Eq. (10) are given by t y = V / ˜ k y − v (cid:113) πn/ ˜ k y L ) . For n close to n c ≡ [ | V L/πv | ] , thegreatest integer less than | V L/πv | , we find that most ofthe propagating modes exhibit a resonance condition at t y ≈ ± ( | vV | L − πnv ) / | E F L | . (13) (g)(h)(c)(d) (f)(e)(a)(b) Fig. 3. (a) Transmission probability density against ˜ k y and t x for t y = 0 ; (b) Conductance G as a function of t x for t y = 0 ; (c)Transmission probability density against ˜ k y and t y for t x = 0 ; (d) G as a function of t y for t x = 0 . (e) Transmission probabilitydensity against ˜ k y and t x = t y / t ; (f) G as a function of t x = t y / t . In (b), (d) and (e), the peaks (marked by magentaarrows) of G correspond to the super-resonant regimes. (g) Resonance positions in small t x for different propagating modes(with index ˜ k y ) and integers n . The color varies from cyan to red when | ˜ k y | increases from to . k F ; (h) Resonance positionsin small t y for different propagating modes and integers n . The color changes from cyan to red when ˜ k y increases from − . k F to . k F . The parameters are v = 0 . eV · nm, E F = 0 . eV, V = 100 E F and L = 50 /k F . Hence, in this tilting direction, we can also observe theresonance peaks and pronounced oscillations in the con-ductance, as shown in Fig. 4(d). In contrast to the casewith the tilting in junction direction, the separations be-tween the conductance peaks are sensitive not only to L and V individually but also to E F . Moreover, Eq. (13)indicates that the separations between the conductancepeaks are almost constant with respect to B . However, asincreasing B , the resonance positions for different propa-gating modes become more extended [Fig. 3(c) and (h)].Consequently, the magnitude of oscillations is strongestfor small B but suppressed for large B .For the general case with the tilting direction deviat-ing from x - and y -directions, t x = εt y with ε (cid:54) = 0 , we canstill observe magnetoconductance oscillations [Fig. 3(e)and (f)]. These oscillations can be similarly attributedto the super-resonant regimes of surface states as vary-ing B . However, they are less regular, compared to thetwo special cases discussed above. The oscillations areaperiodic in the field strength B and the positions of thepeaks become hard to predict in general.Note that although we focus on the large barrier limitin the above analysis, the conductance oscillations remainpronounced even when the barrier potential is of the sameorder as the Fermi energy, | V | (cid:63) | E F | , see Fig. 2(a) andmore instances in the SM [55]. Dependence on field direction. —As we have discussedpreviously, the conductance depends on the tilting direc- tion which, in turn, is determined periodically by the fielddirection θ , according to Eq. (3). Therefore, the conduc-tance G depends periodically on θ . This field-directiondependence stems from two origins: (i) the anisotropicFermi surface and (ii) the barrier transparency for con-ducting channels. In Fig. 4, we choose typical parametersfor surface states (see the caption), for instance, in HgTe, (b)(a) Fig. 4. (a) Conductance G as a function of field direction θ for m = 0 . eV · nm , γ = 0 and B = 0 . T, T, T and T, respectively. (b) the same as (a) but for B = 2 T, m =0 . − η ) eV · nm and γ = 0 . η eV · nm with η = 0 , / , / , and (from magenta to blue), respectively. We choose g = 20 , E F = 0 . eV and other parameters the same as thosein Fig. 3. and calculate numerically G as a function of θ . Severalinteresting features can be observed.First, G has a period of π in θ . For small field strengths B < B c , G ( θ ) displays approximately a sinusoidal de-pendence, G ( θ ) − G ∝ sin[2( θ − θ )] , where G is a ( θ -independent) constant and B c corresponds to the fieldstrength at which the first conductance peak is located[60]. If the tilting direction is parallel ( m = 0 ) or per-pendicular ( γ = 0 ) to the field direction, the phase shiftbecomes θ = 0 and π/ , respectively. On the otherhand, if the tilting direction is neither parallel nor per-pendicular to the field direction ( mγ (cid:54) = 0 ), then θ isdifferent from 0 and π/ [Fig. 4(b)]. Second, it is evidentthat if we increase the field strength B , the dependenceon θ becomes more pronounced [Fig. 4(a)]. This reflectsthat the anisotropy of surface states is enhanced by in-creasing B via the tilting effect. Finally, for stronger fieldstrengths B > B c , G oscillates with a number of peaksand valleys in each period θ ∈ [0 , π ] (blue curve). Thesedense oscillations with respect to θ can also be relatedto the super-resonant regimes of surface states analyzedbefore. Conclusion and discussion. —We have identified a pos-itive magnetoconductance of Dirac surface states, whichstems from the increase of the Fermi surface by apply-ing in-plane magnetic fields. We have unveiled super-resonant regimes of surface states by tuning the magneticfield, in which almost all propagating electrons transmita barrier potential without backscattering. This super-resonant effect results in pronounced oscillations in themagnetoconductance.We emphasize that the appearance of positive magne-toconductance and conductance oscillations can be di-rectly attributed to the deformation of the surface Diraccone by in-plane magnetic fields. In this work, the crucialrole of deforming is played by tilting the Dirac cone viathe Zeeman effect. Particularly, the anomalous conduc-tance oscillations arising from the super-resonant regimesof surface states are essentially different from conven-tional magnetic oscillations, which typically stem fromthe formation of Landau levels or the Aharonov–Bohmeffect.Our predictions are applicable to various candidatematerials including HgTe and Bi Se where in-planemagnetic fields have been successfully applied to surfacestates [27, 44, 61–64]. The barrier potential can be cre-ated by local gating, as done in Ref. [58]. Appropriatenumbers to estimate the magnitude of our predictions inHgTe-based samples are addressed in the SM [55].This work was supported by the DFG (SPP1666,SFB1170 “ToCoTronics”, and SFB1143 (project-id247310070)), the Würzburg-Dresden Cluster of Excel-lence ct.qmat (EXC2147, project-id 390858490), and theElitenetzwerk Bayern Graduate School on “TopologicalInsulators”. 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