Finite transverse conductance in topological insulators under an applied in-plane magnetic field
FFinite transverse conductance in topological insulators under an applied in-planemagnetic field
Abhiram Soori ∗ and Dhavala Suri School of Physics, University of Hyderabad, C. R. Rao Road, Gachibowli, Hyderabad-500046, India Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad-500046, India
Recently, in topological insulators (TIs) the phenomenon of planar Hall effect (PHE) wherein acurrent driven in presence an in-plane magnetic field generates a transverse voltage has been exper-imentally witnessed. There have been a couple of theoretical explanations that explain the originof this phenomenon. We suggest an origin of this effect based on scattering theory on a normalmetal-TI-normal metal junction and calculate the conductances in longitudinal and transverse di-rections to the applied bias. The transverse conductance depends on the location and is zero inthe drain electrode when the chemical potentials on the top and the bottom surfaces of the TI areequal. The longitudinal conductance is π -periodic in φ -the angle between the bias direction and thedirection of the in-plane magnetic field. The transverse conductance is π -periodic in φ when thechemical potentials of the top and the bottom TI surfaces are the same whereas it is 2 π -periodicin φ when the chemical potentials of the top and the bottom surfaces are different. As a functionof the magnetic field, the magnitude of transverse conductance increases initially, peaks. At highermagnetic fields, it decays for angles φ closer to 0 , π whereas oscillates for angles φ close to π/
2. Weunderstand the features in the obtained results, which agree with the existing experimental results.
I. INTRODUCTION
In the last few decades, novel materials such as topo-logical insulators (TIs) and Weyl semimetals which ex-hibit nontrivial electrical properties stemming from thetopology of their bandstructures were invented . InWeyl semimetals, a nonzero transverse voltage appears inresponse to current in presence of an external magneticfield . The transverse voltage lies in the same plane ascurrent and applied magnetic field and this phenomenonis called planar Hall effect (PHE). PHE along with neg-ative longitudinal magnetoresistance has been seen as adirect signature of chiral anomaly in Weyl semimetals.PHE has also been observed in TIs and its origin isascribed to spin-flip scattering of surface electrons fromimpurities. Another explanation of PHE comes from thetilting of the Dirac cone that describes the surface statesof the TIs . Also there has been an attempt at explain-ing PHE emanating from the bulk states of the TI . Itis interesting to note that PHE in TIs was predicted byconsidering scattering at junction of TIs with a ferromag-net in proximity to one part of the TI surface , withoutthe need of either the scattering from impurities or thetitling of the Dirac cones due to magnetic field. But aTI has two surfaces- one on top and another at bottom,as a result, it is not clear whether the transverse deflec-tions of the incident electrons will cancel from the twosurfaces. Motivated by these developments, we examinetransport in a system of in-plane magnetic field appliedto top and bottom surfaces of a TI connected to two-dimensional normal metal (NM) leads on either sides.We follow Lanaduer-B¨uttiker approach and calcu-late currents in transverse and longitudinal directions inresponse to a bias applied in the longitudinal direction.This is in contrast to the experiments where a current isdriven in longitudinal direction and voltages developed in transverse and longitudinal directions are measured inHall bar geometry. Finally, we study the effect of unequalchemical potentials on the top and the bottom surfacesof TI which can be achieved in experiments by applyingdifferent gate voltages to the two surfaces. FIG. 1. Schematic diagram of the setup: the topological insu-lator (TI) is connected to normal metal (NM) leads on eithersides. The two NM’s and the TI are taken to be infinitelylong along y . Both the NM leads are semi-infinite along x .A voltage V applied from left NM to the right NM resultsin a current I . Planar Hall effect is when the current I hasa nonzero component along y direction. Such a transversecurrent could be either in the TI region or in the right NMregion. The paper is organized as follows. In sec. II, the sys-tem under consideration and details of the calculationcomprising of the Hamiltonian, the boundary conditionsand the formulae for the longitudinal and the transverseconductances are discussed. In sec. III, the results arepresented and analyzed. In sec. IV, we discuss the impli-cations of our results and conclude. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n II. DETAILS OF CALCULATION
The setup under study is a NM-TI-NM junction, withthe TI in the middle having a top surface and a bottomsurface as shown in the Fig. 1. Both the NMs and theTI are infinitely long along y . The NM lead on the leftextends all the way from x = −∞ to x = 0 and makes a junction with both the surfaces of TI along x = 0. TI ex-tends from x = 0 to x = L and makes a junction with theNM on the right along the line x = L . From now on, weshall denote the coordinates of the top (bottom) surfaceof the TI with a subscript t ( b ). The in-plane magneticfield applied is present only in the TI region. The NMlead on the right extends from x = L to x = ∞ . TheHamiltonian describing the system being investigated is H = (cid:104) − (cid:126) m (cid:16) ∂ ∂x + ∂ ∂y (cid:17) − µ N (cid:105) σ , for x < x > L, = i (cid:126) v F (cid:16) σ y ∂∂x t − σ x ∂∂y t (cid:17) + ( b x · σ x + b y · σ y ) − µ t σ , for 0 < x t < L, = − i (cid:126) v F (cid:16) σ y ∂∂x b − σ x ∂∂y b (cid:17) + ( b x · σ x + b y · σ y ) − µ b σ , for 0 < x b < L. (1)Here, µ N is the chemical potential of the NM leads, µ t/b is the chemical potential on the top/bottom surface of theTI which can be controlled by an applied gate voltage,( b x , b y ) = b (cos φ, sin φ ) where φ is the angle the in-planemagnetic field makes with x -axis (we refer to the Zeemanenergy b as magnetic field), σ is identity matrix and σ x,y are Pauli spin matrices. The in-plane magnetic fieldshifts the Dirac point of the top (bottom) surface to (cid:126)k = ± ( b y , − b x ) / (cid:126) v F respectively. The dispersion relations forthe top and the bottom surfaces are respectively E = − µ t ± (cid:113) ( (cid:126) v F k x − b y ) + ( (cid:126) v F k y + b x ) , (2) E = − µ b ± (cid:113) ( (cid:126) v F k x + b y ) + ( (cid:126) v F k y − b x ) , (3)where k = k x + k y .To solve the scattering problem, boundary conditionsneed to be specified at x = 0 and x = L . Bound-ary conditions at NM-TI junctions have been discussedin literature . The probability current operatorsfor the top and bottom surfaces can be shown to beˆ (cid:126)j t = ( − v F σ y , v F σ x ) and ˆ (cid:126)j b = ( v F σ y , − v F σ x ) respec-tively. So, the conservation of current along x -directionbetween NM and TI surfaces reads (cid:126) Im[ ψ † N ∂ x ψ N ] x = x mv F = − ψ † t σ y ψ t | x t = x + ψ † b σ y ψ b | x b = x , (4)at both the junctions located at x = 0 , L , where ψ N isthe wavefunction on the NM side and ψ t/b is the wave-function on the top/bottom surface of the TI. This leadsto the boundary conditions ψ N = c [ M ( χ t ) ψ t + M ( χ b ) ψ b ] , (cid:126) mv F ∂ x ψ N − χ N ψ N = ic σ y · (cid:104) − M ( χ t ) ψ t + M ( χ b ) ψ b (cid:105) , (5) where all the wavefunctions and ∂ x ψ N are evaluatedat each of the two junctions located at x = 0 and x = L . Here, M ( χ ) = exp[ iχσ y ] and the dimension-less parameters χ N , χ t and χ b quantify the strengthsof the delta-function barriers infinitesimally close to thejunction from the NM-, top TI- and bottom TI- sides re-spectively . We shall set these parameters to zero atboth the junctions to allow maximal transmission. Weshall set c = ( mv F / √ mµ N ) / so that transmission ofnormally incident electron at the junction is perfect atzero energy in absence of a magnetic field .Due to translational invariance of the system in y -direction, the momentum (cid:126) k y along y can be taken tobe equal in all the four regions. The wavefunction of aspin- σ electron incident from the left NM with energy E , making an angle θ with x -axis has the following formin different regions (except for a multiplicative factor of e ik y y ): ψ N ( x ) = ( e ik x x + r σ,σ e − ik x x ) | σ (cid:105) + r σ,σ e − ik x x | σ (cid:105) , for x < ,ψ p ( x p ) = s σ,p, + e ik x,p, + x p | k p, + (cid:105) + s σ,p, − e ik x,p, − x p | k p, − (cid:105) , for 0 < x p < L, and p = t, b,ψ N ( x ) = t ↑ ,σ e ik x x | ↑(cid:105) + t ↓ ,σ e ik x x | ↓(cid:105) , for x > L, (6)where σ = ↑ , ↓ , σ is the spin opposite to σ , | ↑(cid:105) =[1 , T , | ↓(cid:105) = [0 , T , k x = (cid:112) m ( µ N + E ) cos θ/ (cid:126) , k y = (cid:112) m ( µ N + E ) sin θ/ (cid:126) , k x,p,s ’s for s = + , − corre-spond to the two roots for x -wavenumber obtained fromthe dispersion in the p -TI surface ( p = t, b stand for top,bottom surfaces) as a function of E and k y , | k p,s (cid:105) isthe spinor on p -TI surface for electron with wavenum-ber ( k x,p,s , k y ) which can be found from the Hamiltonianfor the TI and the coefficients r σ (cid:48) ,σ , s σ,p,s , t σ (cid:48) ,σ are tobe determined by matching the boundary conditions ineq. (5) at x = 0 , L .The component of the current along x is conservedand is same anywhere. But along y , k y is same in anyregion and the component of current along y need notbe same. If ψ p,σ ( x ) is the wavefunction due to an σ -spinelectron incident at an angle θ at energy E on p -TI sur-face at x p = x in the range 0 ≤ x p ≤ L , the currentalong y at the location x from this wavefunction will be I σ,y ( E, θ )( x ) = ev F (cid:80) p = t,b ψ p,σ ( x ) † σ p σ x ψ p,σ ( x ), where e is electron charge, σ t = 1 and σ b = −
1. If [ I x , I y ( x )] isthe current flowing at x in response to a voltage bias V inthe bias window (0 , eV ), the longitudinal and transversedifferential conductances are defined as G xx = dI x /dV and G yx ( x ) = dI y ( x ) /dV . These are given by the ex-pressions G xx = (cid:112) m ( µ N + eV ) mv F G (cid:88) σ,σ (cid:48) = ↑ , ↓ (cid:90) π/ − π/ dθ cos θ | t σ (cid:48) ,σ | ,G yx ( x ) = G ev F (cid:88) σ = ↑ , ↓ (cid:90) π/ − π/ I σ,y ( eV, θ, x ) dθ , (7)where G = ( e /h ) · ( mv F L y /h ) and L y is the length ofthe system in y -direction. The current deflected in thetransverse direction in the right NM is same at all loca-tions x > L and the transverse differential conductancedue to this current is given by G yx ( x > L )= (cid:112) m ( µ N + eV ) mv F G (cid:88) σ,σ (cid:48) = ↑ , ↓ (cid:90) π/ − π/ dθ sin θ | t σ (cid:48) ,σ | . (8) III. RESULTS AND ANALYSIS
To obtain numerical results, we shall fix µ N and v F ,and choose other parameters as combinations of theseparameters. The mass m decides the size of the Fermiwavenumber. We choose m = 0 . µ N /v F so that thewavenumbers on NM and TI at energy − . µ N are equalwhen µ t = µ b = 0 and b = 0. The length of the TI ischosen to be L = 5 (cid:126) v F /µ N . These are the values of theparameters unless otherwise stated. A. µ t = µ b First, we set µ t = µ b = 0 . µ N and study the depen-dence of G xx and G yx ( L/
2) on the bias at different angles φ when the magnitude of the magnetic field is fixed at b = 0 . µ N in Fig 2(a) and Fig. 2(b) respectively. Theslow increase in G xx with bias is due to the increase indensity of states of incident electrons. For an angle φ be-tween x -axis and the magnetic field, the Dirac cones onthe TI surfaces are displaced in y -direction by an amount | b cos φ/ ( (cid:126) v F ) | thereby making the wavenumbers k x,p,s ’sin the TI region complex (when cos φ (cid:54) = 0) for a range ofangle of incidence θ . This reduces the transmission prob-abilities | t σ,σ (cid:48) | for larger values of | cos φ | and for largervalues of | b | which agrees with the observed features of G xx in Fig. 2(a,c). In Fig. 2(b,d), we find that G yx ( L/ φ = 0 , ± π/ , π . When φ = ± π/
2, theDirac cone is shifted along x -direction and the system issymmetric under y → − y thereby giving zero total cur-rent along y . When φ = 0 , π , the Dirac cones in the topand bottom surfaces are displaced exactly along ± y di-rections, and the currents deflected along y from the topand the bottom surfaces are equal in magnitude and op-posite in sign thus giving zero G yx . Now, we address thequestion why there is a nonzero G yx ( L/
2) for a nonzero b in a direction φ other than 0 , ± π/ , π . Under a magneticfield ( b x , b y ), the Dirac points of the top and the bottomsurfaces are shifted to ± ( b y , − b x ) / ( (cid:126) v F ). The currentsin y -direction carried by the electrons incident at angles θ and − θ on one surface do not cancel due to a finiteshift of the Dirac cone in y -direction. At the same time,the net current in y -direction carried by the top surfaceand the bottom surface do not cancel despite the oppo-site shifts of the two Dirac cones because the wavenum-bers in x -direction of the corresponding surfaces k x,t,s and k x,b,s are different. From Fig. 2(b), it can be seenthat at eV = − µ t = − µ b , the transverse conductance isexactly zero implying that the net current in the trans-verse direction carried by the evanescent waves in the TIregion is zero. The transverse conductance G yx ( x ) is π -periodic in φ . G yx ( x > L ) is exactly zero for the case µ t = µ b .Turning to the dependence of the two conductances on b , in Fig. 2(e), we find monotonic dependence of G xx on | b | for | cos φ | close to 1 and oscillatory dependence of G xx on | b | for | cos φ | small compared to 1. This is because,the displacement of TI Dirac cones on in y direction isby an amount proportional to cos φ . Nearly normal inci-dences with θ close to zero contribute the most to G xx .When | cos φ | is large, for angles of incidences θ close tozero, the transport in TI region is diffusive characterizedby a complex k x,p,s whose imaginary part grows in mag-nitude with b . When | cos φ | is small compared to 1, thedisplacement of the TI Dirac cones along y -direction isminimal. Nearly normal incidences from NM will finda real k x,p,s in the TI and the transport is ballistic ex-cept for scatterings at the interfaces which leads to inter-ference between the forward moving and the backwardmoving waves. This is the reason for oscillatory behaviorof G xx with b . Under the transformation φ → φ + π ,the transmission probabilities | t σ,σ (cid:48) | for angles of inci-dence θ and − θ get interchanged thereby making G xx even in b . The nonzero values of the transverse conduc-tance G yx at certain values of φ increases in magnitudewith | b | for small | b | since increasing value of | b | givesscope for higher asymmetry between scatterings at an-gles of incidence θ and − θ . But beyond a value of | b | ,the displacement of the Dirac cone in y -direction causesthe wavefunction to decay into the TI (which is particu-larly the case for | cos φ | close to 1), resulting in decreasein magnitude of G yx with | b | . For values of φ such that | cos φ | is small compared to 1, the scattering from anglesof incidence away from normal incidence centered around -0.2 -0.1 0 0.1 0.2 eV/ N G xx / G -0.2 -0.1 0 0.1 0.2 eV/ N -0.2-0.100.10.2 G yx ( L / ) / G
0, 0.5 , 0.20.4 0.60.8(b) / G xx / G (c) / -0.1-0.0500.050.1 G yx ( L / ) / G b/ N G xx / G b/ N -0.1-0.0500.050.10.15 G yx ( L / ) / G FIG. 2. (a) G xx and (b) G yx as functions of bias for different angles φ made by the in-plane magnetic field with x -axis with b = 0 . µ N . The values shown in the legend for (a) are the respective values of φ for which G xx is plotted. (c) G xx and (d) G yx at zero bias as functions of φ at different values of magnetic field mentioned within the plot. (e) G xx and (f) G yx at zero biasas functions of magnetic field b for different φ specified in the plot legends. Parameters: L = 5 (cid:126) v F /µ N , µ t = µ b = 0 . µ N and m = 0 . µ N /v F . ± θ b which depend on | b | contribute dominantly to G yx .The Fabry-P´erot type interference of these modes re-sults in oscillations in G yx with | b | . Under φ → φ + π ,the displacement of each of the Dirac cones is oppositeto that before the transformation. This hints at the re-versal of sign of G yx upon b → − b . But, since b y → − b y the displacement of each of the Dirac cones along x isopposite to that before the transformation making thesurface dominantly contributing to G yx switch under thetransformation φ → φ + π . Hence the displacement of theDirac cone along y -direction for the surface dominantlycontributing to G yx is shifted in the same direction forboth choices of magnetic field directions φ and φ + π ,making G yx π -periodic. To study the dependence of thetransverse conductance on the location, we plot G yx ( x )versus x in Fig. 3. We find that the magnitude of thetransverse conductance is peaked at x = L/ B. µ t (cid:54) = µ b To study the conductances in this case, we choose thesame set of parameters as in the Fig. 2 except when men-tioned specifically. We choose µ t = − µ b = 0 . µ N . Thelongitudinal differential conductance shows characteris-tics very similar to the ones in Fig. 2(c) except for achange in the numerical value. In Fig.4, we plot thetransverse differential conductances at x = L/ x/L -0.1-0.0500.050.10.15 G yx ( x ) / G
0, 0.5 , 0.20.4 0.60.8
FIG. 3. Dependence of the transverse differential conduc-tance on the location in the TI region for different angles φ mentioned in the legend with the choice of parameters: L = 5 (cid:126) v F /µ N , µ t = µ b = 0 . µ N and m = 0 . µ N /v F . x > L versus φ . It is interesting to see that G yx ( x ) is2 π -periodic for µ t = − µ b . Also, G yx ( x > L ) is nonzerogenerically except at zero bias for which it is always zero.Also, interestingly both G yx ( L/
2) and G yx ( x > L ) arenonzero at φ = 0 for this case. This is because of the dis-placement of the two Dirac cones in ± y -direction equallybut in opposite directions does not lead to cancellationof transverse currents at nonzero bias due to broken per-fect antisymmetry. For a fixed bias, the values of G yx ( x )for φ and π − φ are equal in magnitude and opposite in / -6-3036912 G yx ( L / ) / G -0.4 N -0.2 N N N (a) / -0.200.20.4 G yx ( x > L ) / G -0.4 N -0.2 N N N (b) FIG. 4. Transverse differential conductance G yx ( x ) at (a) x = L/ x > L/ φ for differentvalues of bias mentioned in the legend. µ t = − µ b = 0 . µ N , L = 5 (cid:126) v F /µ N , and m = 0 . µ N /v F . x/L -4 -2 G yx ( x ) / G FIG. 5. Transverse conductance as a function of the locationfor different choices φ indicated in the legend. Parameterssame as in Fig. 4 and bias 0 . µ N . sign. In Fig. 5, we plot G yx ( x ) versus x for the samechoice of parameters as before, fixing the bias at 0 . µ N for different choices of φ . IV. DISCUSSION AND CONCLUSION
We have essentially studied the phenomenon of PHEin TIs with the scattering theory approach when TI isconnected to NM leads on either sides. We use theboundary condition for the NM-TI junction obtained bydemanding the current conservation. The longitudinaland the transverse conductances are π -periodic in φ when µ t = µ b . When µ t (cid:54) = µ b , the longitudinal conductanceis still π -periodic in φ whereas the transverse conduc-tance becomes 2 π -periodic in φ . For angles φ close to 0or π , the longitudinal conductance decays with magneticfield whereas for angles φ close to ± π/
2, the longitudinalconductance decays with the magnetic field much slowlyshowing a slight periodic behavior with magnetic field at φ = ± π/
2. Magnitude of the transverse conductance firstincreases with magnetic field, peaks and then decreasesfor angles φ close to but not equal to 0 or π whereas oscil-lates after an initial monotonic increase for angles close to ± π/
2. Such oscillations are rooted in Fabry-P´erot typeinterference in the TI region. The transverse conductancedepends on the location and is maximum in magnitudeexactly at the middle between the two TI-NM junctionsand is zero in the right NM lead when µ t = µ b . When µ t = − µ b , the transverse conductance is maximum atthe left NM-TI junction and is nonzero though small inmagnitude in the right NM region. The differential gat-ing of the top and the bottom surfaces of the TI can beexperimentally achieved which means µ t and µ b can beseparately controlled . The angular dependence of thetransverse resistance for the case of differentially gatedtop and bottom surfaces of the TI needs to be probedexperimentally. ACKNOWLEDGMENTS
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