Magnetoresistance of edge states of a two-dimensional topological insulator
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Magnetoresistance of edge states of a two-dimensional topological insulator
Leonid Braginsky ∗ and M. V. Entin † Institute of Semiconductor Physics, 630090 Novosibirsk, Russia andNovosibirsk State University, 630090 Novosibirsk, Russia (Dated: January 7, 2021)The theory of magnetoresistance of the edge state of a two-dimensional topological insulator isdeveloped. The magnetic field violates the time reversal invariance. Magnetoresistance arises dueto the energy gap opened by a magnetic field parallel to the sample surface. The combined action ofimpurities and the magnetic field causes backscattering of edge electrons. Although impurities arenecessary for scattering, a sufficiently strong interaction with impurities leads to the suppression ofbackscattering.
I. INTRODUCTION
The edge states of 2D topological insulator is one ofmost inspiring problem of the modern solid state physics.It was shown that these states possess so called topologi-cal protection, preventing the electron from backscatter-ing. At a moment there is a large and rapidly growingnumber of publications on this topic, (see, i.g., [1–4]).The study of the conductance of the edge states of a two-dimensional topological insulator assumes their topologi-cal protection, this makes backscattering prohibited or itis extremely weak. Therefore, the one-dimensional statesare collisionless and the conductance of the electrons onthem is e /h .There were some attempts to implement mechanismsviolating the time-reversibility and, therefore, causingbackscattering. In particular, Ref. [5, 6] ascribe backscat-tering to the transitions between edges with opposite di-rection of travel of the same-spin electrons. The transi-tion between opposite edges of a TI strip were consideredalso in Ref. [7].The present paper was stimulated by the experimentalfinding [8] of the strong magnetoresistence of the edgestate electron conductance. Important experimental factthat has to be explained is the presence of magnetoresis-tance, as well as its gigantic value and fluctuations withthe Fermi level. What is unusual, is the sensitivity of 1Dedge states to the magnetic field, which is absent in other1D systems. The results of [8] can not be explained bymeans of interedge transitions [7] due to large width ofthe TI strip in the experimental conditions.The model is based on the one-dimensional Hamilto-nian H = (cid:18) vp + V ( x ) ∆ ∆ ∗ − vp + V ( x ) (cid:19) . (1)Here p and v are the electron momentum and velocity,and V ( x ) is the impurity potential. The Hamiltonian (1) ∗ [email protected] † [email protected] is the simplest that violates the time reversal invariancein the magnetic field and leads to the gap 2 | ∆ | at p = 0.The gap in the electron spectrum of the 2D topologicalinsulator arises in the magnetic field B directed along thespecimen plane.[9]∆ = µ B ( g xx B x + ig yy B y )for the (0,0,1) orientation. Here µ B is the Bohr mag-neton, and g xx , g yy are the g -factor components in thespecimen plane.The gap is 2 | ∆ | = µ B q g xx B x + g yy B y . At a weak magnetic field the magnetoresistance is dueto the backscattering and localization of previously de-localized states by the magnetic field. Indeed, the pres-ence of off-diagonal terms in the Hamiltonian leads tobackscattering of electrons. The potential itself does notcause transitions between the states. The scattering oc-curs due to the off-diagonal part H nd of the Hamiltonian(1).Two approaches can be utilized. First is the classicalapproach. In the absence of localization the 1D conduc-tivity is of the Drude form: σ = e τ vπ . (2)Since the electron velocity in the linear spectrum is con-stant, the conductivity is determined by the backscatter-ing time τ . The classical approach is valid in the absencethe phase coherence at a finite temperature (similar prob-lem has been considered in connection to the TI strip [7]).This means that during the forward and backward travelat the distance vτ the phase of the electron changes by π due to, e. g., interaction with phonons.Second, the coherence of the electron states resultsin their localization in the infinite volume and, there-fore, exponential size dependence of the conductivity ata zero temperature. In this case the 1D conductivityis not established. Instead, the conductance G of thesystem drops exponentially with the system length L :log G ∝ − L/vτ .In this paper we find the backscattering time τ . Magnetoinduced backscattering
The selfwave functions of the electrons of the Hamil-tonian (1) with energies ǫ ± = ± vp at B = 0 are ψ + = 1 √ L (cid:18) (cid:19) e ipx + i R dxV ( x ) /v ψ − = 1 √ L (cid:18) (cid:19) e ipx − i R dxV ( x ) /v . The amplitude of the backscattering is determined by thefirst order perturbation on the non-diagonal part of theHamiltonian H nd ∝ ∆ , i.e., the matrix element betweenthe | ψ − p, − σ i and | ψ p,σ i states of the same energy ǫ p,σ : A = h ψ − p, − | H nd | ψ p, + i = ∆ L Z L − L exp (cid:18) ipx + 2 iv Z V ( x ) dx (cid:19) dx. II. WEAK IMPURITY POTENTIAL. THEPERTURBATION THEORY
In the first order with regard to the potential V ( x ) wefind A = 2 i h ψ − p, − | H nd | ψ p, + i = 2 i ∆ Lpv ˜ V (2 p ) , ˜ V ( p ) = Z ∞−∞ V ( x ) e ipx dx. Consider the electron scattering at the impurity poten-tial V ( r ) = P n u ( r − r n ), where r n is the position of n-thimpurity and u ( r − r n ) is its potential. The probabilityof backscattering has to be averaged over the randomimpurity positions in 2D layer, so that1 /τ = Lv h| A | i , (3)The 2D Fourier transform of the Coulomb impurity is u ( q ) = 2 πe κq , therefore h| V ( q x ) | i = n i Z dq y π u ( q ) = 2 π n i e q x κ , q = q x + q y . Here n i is the impurity density. For the backscatteringprobability we find1 /τ = 8 π e n i | ∆ | ¯ hκ ( vp F ) = 8 e n i | ∆ | ¯ h κ v πn e . (4)Here n e is the 1D electron density. The probability (4) essentially increases at a small elec-tron density. Therefore, screening of the Coulomb poten-tial by the field electrode has to be taken into account.Then u ( q ) = 2 πe κq (1 − e − qd ) . (5)Here d is the distance to the field electrode. The lastfactor in Eq. (5) restricts the impurity matrix elementsat small q . For 1 /τ value we obtain1 τ = 8 π e n i | ∆ | ¯ hκ ( vp F ) φ ( p F d ) , (6)where φ ( y ) = 2 π Z ∞ (1 − e − y cosh t ) cosh t dt.φ ( y ) ≈ y ln 2 /π at y ≪ φ ( y ) → y → ∞ .Thus, screening reduces the divergence of 1 /τ value atsmall electron density: τ ∝ n e , instead of τ ∝ n e for n e → III. STRONG IMPURITY POTENTIAL
The conductivity of electronic gas with quadratic spec-trum is too complicated, if it considered outside theframes of the perturbation theory. This is not the casefor the electronic 1D gas with the linear spectrum, be-cause this problem has an exact solution.[10] This allowsus to obtain the result at n e → | u ( r − r n | ≪ vp F . Then in Gauss approximation fromEq. (3) we obtain: h| A | i = L − | ∆ | Z Z exp " ip ( x − x ′ ) − v Z Z xx ′ dx dx W ( x − x ) dxdx ′ , where W ( x − x ′ ) ≡ Z ∞−∞ dk (2 π ) − e − ik ( x − x ′ ) ˜ W ( k ) = h V ( x ) V ( x ′ ) i is the potential correlation function. Its Fourier trans-form is ˜ W ( k ) = 4 π e κ n i φ ( kd/ k . / τ s0 5 10 15 20 251234 FIG. 1. (Coloronline) Dependence of 1 /τ on the parameter s .Values of β are shown on the curves. Then using Eq. (3) we obtain1 τ = 2( d/v ) | ∆ | (7) × Z ∞−∞ dz cos( sz ) exp (cid:18) − β Z dyy φ ( y/
2) sin ( yz ) (cid:19) ,s = 4 p F d, β = 2 πe κ v n i d . The behavior of 1 /τ vs β and s is presented in Fig. 1.The dependence on β is linear at small β [or at smallimpurity density, in agreement with Eqs. (4, 6)]] followedby an exponential decrease. The latter is due to therandom potential that results in fluctuating increase ofthe local Fermi momentum reducing the scattering. Conductance of the finit-size electron system at zerotemperature
Without backscattering the zero temperature con-ductance of the edge state G is equal to the conduc- tance quantum G for any edge length. Presence ofthe backscattering leads to the localization of the edgestates and, therefore, to the exponential drop of the con-ductance, if the edge length exceeds the backscatteringlength: ln G/G ∝ − L/vτ . Or ln
G/G ∝ − B in agree-ment with the obtained behavior of τ . DISCUSSIONS
To estimate the edge state conductivity at a low tem-perature, consider the 1D electron gas with the linearspectrum. The electron-phonon interaction affects thephase of the electron wave function, however, this phaseis not important for the scattering time. This means thatthe conductivity obeys the Drude expression Eq. (2) withthe relaxation time found in the present paper.Thus, we found that a magnetic field parallel to theplane of the sample results in elastic backscattering ofthe edge states electrons with the linear spectrum. Thisscattering leads to localization of states in an infinite-sizesystem and, therefore, to finite conductivity at a non-zerotemperature. The localization is due to magnetic field.The transition between the ballistic and localization be-havior of conductance occurs under magnetic field in afinite-size sample. This transition depends on the rela-tion between the magnetic localization length and samplelength. It is shown that the probability of backscatteringincreases with the impurity concentration at a low con-centration followed by decrease at a high concentrationof impurities.Note also that our results correct, if only the Fermilevel is far apart from the gap | E F | ≫ | ∆ | . This isnecessary for the expansion over the magnetic field holds.Another note concerns non-magnetic mechanisms ofbackscattering. The paper [11] showed that in a TI withsmooth edge the overlapping of linear topology-protectededge states with the Dirac gapped branches leads to theelastic backscattering. In this case the magnetoinducedbackscattering complements the latter in the energy do-mains where they coexist.The work is supported by the RFBR grant No.20-02-00622. [1] M.Z. Hasan and C.L. Kane, Colloquium: Topological in-sulators, Rev. Mod. Phys. 80, 3046 (2010);[2] X.-L. Qi and Sh.-Ch. Zhang, Rev. Mod. Phys. 83, 1057(2011); M. Fruchart, D. Carpentier. An Introduction toTopological Insulators. Comptes rendus de l’Academiedes sciences. Serie IV, Physique, astrophysique, Elsevier,2013, 14 (9), pp.779- 815.[3] S.-Q. Shen, Topological Insulators: Dirac Equation inCondensed Matters, Springer Series in Solid-State Sci-ences 174, Springer-Verlag Berlin Heidelberg 2012;[4] B. Andrei Bernevig. Topological Insulators and Topolog-ical Superconductors. Princeton University Press, 2013. [5] J.I. V¨ayrynen, M. Goldstein, and L.I. Glazman, Phys.Rev. Lett. , 216402 (2013).[6] J.I. V¨ayrynen, M. Goldstein, Y. Gefen, and L.I. Glaz-man, Phys. Rev. B , 115309 (2014).[7] L. I. Magarill and M. V. Entin ISSN 00213640, JETPLetters, 2014, Vol. 100, No.9, pp. 541–565 (2014)[8] S. U. Piatrusha, E. S. Tikhonov,Z. D. Kvon, N.N. Mikhailov,S. A. Dvoretsky, and V. S. KhrapaiPhys.Rev.Let. 123, 056801 (2019).[9] M. V. Durnev and S. A. Tarasenko Phys. Rev. B 93,075434 (2016)[10] M. V. Entin and L. Braginsky, EPL 120, 17003 (2017), 115309 (2014).[7] L. I. Magarill and M. V. Entin ISSN 00213640, JETPLetters, 2014, Vol. 100, No.9, pp. 541–565 (2014)[8] S. U. Piatrusha, E. S. Tikhonov,Z. D. Kvon, N.N. Mikhailov,S. A. Dvoretsky, and V. S. KhrapaiPhys.Rev.Let. 123, 056801 (2019).[9] M. V. Durnev and S. A. Tarasenko Phys. Rev. B 93,075434 (2016)[10] M. V. Entin and L. Braginsky, EPL 120, 17003 (2017)