Electron states on the smooth edge of 2D topological insulator: elastic backscattering and light absorption
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electron states on the smooth edge of 2D topological insulator: elastic backscatteringand light absorption
M.M. Mahmoodian ∗ and M.V. Entin † Rzhanov Institute of Semiconductor Physics, Siberian Branch,Russian Academy of Sciences, Novosibirsk, 630090 Russia andNovosibirsk State University, Novosibirsk, 630090 Russia (Dated: January 14, 2021)The 2D TI edge states are considered within the Volkov-Pankratov (VP) Hamiltonian. A smoothtransition between TI and OI is assumed. The edge states are formed in the total gap of homogeneous2D material. A pair of these states are of linear dispersion, others have gapped Dirac spectra. Theoptical selection rules are found. The optical transitions between the neighboring edge states appearin the global 2D gap for the in-plane light electric field directed across the edge.The electrons in linear edge states have no backscattering, that is indicative of the fact of topo-logical protection. However, when linear edge states get to the energy domain of Dirac edge states,the backscattering becomes permitted. The elastic backscattering rate is found. The Drude-likeconductivity is found when the Fermi level gets into the energy domain of the coexistence of linearand Dirac edge states. The localization edge conductance of a finite sample at zero temperature isdetermined.
I. INTRODUCTION
The systems under consideration are 2D topologicalinsulators (TI) (more references can be found in ).The most intriguing property of them is the internal insu-lating part with conducting edges. These systems werebeen first invented by Bernevig-Hughes-Zhang (BHZ) .However, earlier, a similar 3D TI had been studied byVolkov-Pankratov (VP) . The edge states of 2D topo-logical insulators attract attention due to their possibilityof the topological protection of electrons from backscat-tering processes. The difference between VP and BHZmodels is that BHZ is applied to the external edge ofthe TI, while VP can be applicable to the interface be-tween a TI and an ordinary insulator. Many unusualfeatures of the 2D TI edge states were studied. In par-ticular, the non-local conductance due to topologically-protected linear-edge states was observed in , and the-oretically discussed by and; however, the non-localitydoes not obligatorily manifest itself. The edge statelight absorption was theoretically and experimen-tally investigated . Different theoretical approaches tothis intriguing problem were developed. In particular, wefound that the topologically protected edge states possessa unique property: electrons in these states do not inter-act with the external electric field and can not scatter byeach other .If the TI boundary is sharp, the only topologically pro-tected edge states exist. However, the TI boundary isnot necessarily sharp. In this case the additional edgestates are formed , . The gap sign, which determinesif the 2D layer is OI or TI, is controlled by the layerthickness . Due to intentional or non-intentional asmooth transition between TI and OI domains are formednear formal TI boundary.The smooth transition between an OI with a positiveand a TI with a negative gap generates not only the edgestates with linear energy spectrum ± vk x (here and in what follows, ¯ h = 1), where v and k are the velocity andthe momentum of electron on the edge state, but the se-ries of the Dirac-like states ± p v k x + ǫ n . These statescover the 2D energy gap. Unlike linear edge states, Dirac-like edge states have no topological protection. Underillumination they can provide light absorption. The pur-pose of the present paper is to find this absorption andthe elastic backscattering of electrons with linear spectrawith the participation of Dirac edge states.Both planned problems need the knowledge of the ma-trix elements between the edge states due to the interac-tion with light for absorption, or due to the interactionwith impurities for elastic backscattering. Hence, theseproblems can be studied in common.First, we will formulate the problem. Then, the wavefunctions in the model with gap ∝ tanh( z/a ) will be pre-sented. After that, we will present the case of lineartransition between the domains with a negative and pos-itive gap. This is a limiting case of the previous model,which is applicable also to more general models of smoothtransition. Then the elastic backscattering rate will befound. After that the optical absorption, due to transi-tions between linear and the Dirac states, will be found.The results will be summarized in the Discussion section.The calculation details are in the Appendix. Volkov-Pankratov Hamiltonian in 2D
We will base on the VP Hamiltonian for 2D TI . The2D system is located in plane ( x, z ). The transition be-tween a TI and an OI is modeled by the one-dimensionaldependence of gap ∆( z ) = ∆ F ( z/l ), where ∆ is half ofthe bulk gap so that dimensionless function F ( z > > F ( z < <
0. In particular, one can assume that F ( z ) = tanh( z ). If parameter l →
0, the TI-OI transitionis step-like. If l is large, the transition is smooth.The two-dimensional VP Hamiltonian reads H = − ∆( z ) τ y + vτ x ( k x σ x + k z σ z ) , (1)where τ and σ are Pauli matrices, which act on orbitaland spin subspaces, respectively.The energy spectrum of Hamiltonian (7) for the TI-OItransition ∆( z ) = ∆ tanh( z/l ) E λn,σ ( k x ) = λ q v k x + ǫ n,σ , (2) ǫ n,σ = ∆ " − (cid:18) − (cid:18) n + 1 + σ (cid:19) v ∆ l (cid:19) , (3)where n are integers, λ = ± , and 0 ≤ n + (1 + σ ) / ≤ ∆ l/v .For n = 0 , σ = − E λ , − = λvk x . The other Dirac-like brancheswith n, σ = 1 and n + 1 , σ = −
1, have gaps. These statesare double-degenerate (see Fig. 1). E , m e V k x , nm -1 FIG. 1: Edge states for ∆( z ) = ∆ tanh( z/l ) (green,solid) and for ∆( z ) = ∆ ( z/l ) (red, dashed). Thechosen parameters are ∆ = 22 meV and v = 4 . ∗ cm/s, and that corresponds to HgTe layerwidths 55 ˚A and 67 . l = 1750 . Elastic backscattering of edge-state-electrons in atwo-dimensional topological insulator
The presence of Dirac edge states inside the char-acteristic gap of two-dimensional edge states is due tothe smooth transition between the positive and negativegaps. A typical transition scale is described by letter l .The two-dimensional gap 2∆ . The typical distanceto the first Dirac state is p ∆ v/l , where v is the Fermivelocity; the linear branches are of the form ± vk x , whichis less than ∆ for ∆ ≫ v/l . With an abrupt transition, only a linear branch remains inside the two-dimensionalgap. When l is very large, the transition can be replacedby a linear dependence ∆ = ∆ z/l . This dependenceleads to an exactly solvable problem. The spectrum con-sists of a pair of linear branches ± vk x and Dirac branches.In the narrow energy range | E | < p ∆ v/l , there areonly linear topologically protected branches. However,outside the narrow region, these states overlap in energywith Dirac states. At low temperatures, the elastic tran-sition processes between linear and Dirac and betweenDirac states are allowed. This gives the backscatteringof electrons, including those on linear branches, that is,from a state with a velocity v with momentum k x toa state with a velocity – v with momentum − k x . Theexpected process is two-step: vk x → q v k x + ǫ n,σ → ( − v )( − k x ).Consider the probability of transition between states n, k and n ′ , k ′ under the action of exp( iq z z + iq x x ). Ma-trix element J = h n ′ k ′ x | exp( iq z z + iq x x ) | n, k x i is pro-portional to δ ( k ′ x − k x − q x ). The square | J | must bemultiplied by the square of the Fourier transform of theimpurity potential | V ( q ) | and the impurity concentra-tion n i . For Coulomb impurities V ( q ) = 2 πe κq (cid:0) − e − qd (cid:1) , (4)where d is the distance to the metallic gate. As q →
0, theFourier transform of the potential tends to a constant.In a general case, the inverse transition time is1 τ = n i ∞ Z −∞ dq z | V ( q ) | | J | | k ′ x − k x = q x g n ′ ( E λn ′ ,σ,k ′ x ) dE λn ′ ,σ ′ ,k ′ x . (5)The density of states has the form g n ( E ) = Eπv q E − ǫ n,σ . (6)The presence of the transitions indicated here leads toa limitation of the topological security of a linear spec-trum.In Fig. 2 is the energy dependence of the inverse transi-tion time in the region where the linear edge states over-lap the first Dirac subband. II. LIGHT ABSORPTION IN ATWO-DIMENSIONAL TOPOLOGICALINSULATOR
Later on we shall deal with a smooth transition. Whenthe edge state size ξ = v/ ∆ is less than the transi-tion width l , the dependence ∆( z ) may be expanded as∆( z ) = ∆ z/l . In such case the edge states can be ex-pressed via oscillator wave functions.Here it is convenient to use a variant of the Hamilto-nian that was originally proposed by Zhang et al. H = ∆( z ) τ z + v ( k z τ y − k x τ x σ y ) . (7) E, meVt -1 ,10 s -1 FIG. 2: Energy dependence of the inverse transitiontime for transitions E +0 , − = vk x → E + n ′ ,σ ( k x ). Thechosen parameters are ∆ = 22 meV, v = 4 . ∗ cm/s, l = 1750 . d = 1000 ˚A, κ = 10 . n i = 10 cm − ,respectivelyAfter unitary transformation T = exp( iπτ y / z ) = ∆ z/l can be rewrittenas H T = T H T † = v ik x √ l ˆ c − ik x √ l ˆ c √ l ˆ c † − ik x √ l ˆ c † ik x , (8)where the ladder operators areˆ c = l √ (cid:18) zl + ik z (cid:19) and ˆ c † = l √ (cid:18) zl − ik z (cid:19) , (9)which satisfy the usual commutation relations [ˆ c, ˆ c † ] = 1, l = √ lξ . The Hamiltonian (8) has the energy spectrum E λn ( k x ) = λv s k x + 2 nl , (10)where λ = ± . The case n = 0 corresponds to the edgestates with linear dispersion, and n > z -direction is σ zz ( ω ) = ie X m,n ∈ Nλ,λ ′ ∞ Z −∞ dk x π |h ψ λ ′ m | ˆ v z | ψ λn i| E λ ′ m − E λn × f ( E λn ) − f ( E λ ′ m ) E λ ′ m − E λn − ω + iδ . (11)Here, f ( E ) is the Fermi-Dirac distribution function,ˆ v z = vτ y , δ → +0. For the light polarized along the z -direction, according to Eq. (18), only the transitions n → n ± E F >
0, three transitiontypes can be distinguished. The transitions between thestates with λ = − and λ = + are possible at high fre-quencies, starting from ω > √ v/l . At frequencies be-low this value, transitions are possible only between thestates with λ = +. ℜ [ σ zz ( ω )] = e ω X m,n ∈ Nλ,λ ′ ∞ Z −∞ dk x |h ψ λ ′ m | ˆ v z | ψ λn i| × δ (cid:16) ω − ( E λ ′ m − E λn ) (cid:17) (cid:16) f ( E λn ) − f ( E λ ′ m ) (cid:17) . (12)In the Fig. 3 is the optical conductivity dependence ℜ [ σ zz ( ω )] on ω for different values of E F . w/D s zz /(s x) a) b) w/D s zz /(s x) c) w/D s zz /(s x) FIG. 3: Optical conductivity ℜ [ σ zz ] (in units of σ ξ , σ = e /h ), as a function of the excitation energy ¯ hω (in units ∆ ) for E F = 0 (Fig. 3a), E F = 0 . (Fig. 3b) and E F = 0 . (Fig. 3c). III. CONCLUSIONS
We see that, in the absence of scattering, the selec-tion rules allow transitions between the Dirac branchesor between the Dirac and linear branches polarizationof external in-plane microwave electric field across theedge. Besides, they do not allow the transitions betweenthe linear branches for any external microwave electricfield polarization. The transitions are allowed betweenneighboring transversal states. This statement is valid,however, in the assumption of linear dependence of thegap on the transversal coordinate, and that correspondsto the frequency much less than the gap value apart fromedge 2∆ .The selection rules determine the lower threshold forlight absorption p ∆ /vl . This differs the case of smoothedge from the steep one, where the absorption is limitedfrom below by the quantity ∆ due to the processes oftransitions between the edge and two-dimensional states.The light absorption oscillates with the light frequencyand the Fermi level, and has the 1 / √ ω − ω n singularitiesat the thresholds of the pair density of states. Acknowledgments.
This research was supported inpart by the RFBR, grant No. 20-02-00622.
IV. APPENDIX I. ELECTRONS STATES FOR ∆( z ) = ∆ tanh( z/l ) The eigenfunctions of the Hamiltonian (1) for ∆( z ) =∆ tanh( z/l ) are ψ λn, = (cid:18) vk x E λn,σ , iE λn,σ (cid:18) ∆ + v ∂∂z (cid:19) , , (cid:19) Ψ n,σ , n ≥ ,ψ λn, = (cid:18) iE λn,σ (cid:18) ∆ − v ∂∂z (cid:19) , vk x E λn,σ , , (cid:19) Ψ n,σ , n ≥ ,ψ λ = (cid:18) λ, iλvk x (cid:18) ∆ + v ∂∂z (cid:19) , , (cid:19) Ψ n,σ . n = 0 , Here Ψ n,σ ( η ) = A n,σ n !( ε + 1) n (cid:0) − η (cid:1) ε/ P ( ε,ε ) n ( η ) , where P ( ε,ε ) n ( η ) are Jacobi polynomials, ( ε + 1) n is thePochhammer symbol, η = tanh( z/l ) and ε = ∆ l/v − n − (1 + σ ) / A n,σ are normalization constants. V. APPENDIX II. ELECTRONS STATES ANDMATRIX ELEMENTS FOR ∆( z ) = ∆ z/l The eigenstates of Hamiltonian (8) for ∆( z ) = ∆ z/l are | ψ λn i = (cid:0) a λ ,n | n − i , a λ ,n | n − i , a λ ,n | n i , a λ ,n | n i (cid:1) , n ≥ , | ψ λ i = (cid:0) , , a λ , | i , a λ , | i (cid:1) , n = 0 , where | n i is the eigenstate of harmonic oscillator de-termined by ˆ c † and ˆ c . The Hamiltonian written in thisbasis reads: H T ( n ) = v ik x √ nl − ik x √ nl √ nl − ik x √ nl ik x , (13)The corresponding normalized eigenvectors of the Hamil-tonian above are: ψ λn, = 1 √ i cos α n , λ, , sin α n ) , n ≥ , (14) ψ λn, = 1 √ λ, − i cos α n , sin α n , , ) , n ≥ , (15) ψ λ = 1 √ , , i, − λ ) , n = 0 . (16)For n ≥ α n = k x q k x + nl , sin α n = nl q k x + nl . (17)Matrix elements h ψ λ ′ m,d ′ | ˆ v z | ψ λn,d i = iv × h (cid:16) a λ ′ ∗ ,m,d ′ a λ ,n,d + a λ ′ ∗ ,m,d ′ a λ ,n,d (cid:17) δ m,n − − (18) (cid:16) a λ ′ ∗ ,m,d ′ a λ ,n,d + a λ ′ ∗ ,m,d ′ a λ ,n,d (cid:17) δ m − ,n i . ℜ [ σ zz ( ω )] = e v ω ( [1 + cos α ( k )] | ν ′ ( k ) | Θ (cid:18) ω − √ vl (cid:19) (cid:2) f ( E − ( k )) − f ( E +1 ( k )) (cid:3) +[1 − cos α ( k )] | η ′ ( k ) | Θ (cid:18) √ vl − ω (cid:19) (cid:2) f ( E − ( k )) − f ( E +1 ( k )) (cid:3) + X n ≥ [1 − cos α n ( k n )][1 + cos α n +1 ( k n )] | ν ′ n ( k n ) | Θ (cid:18) ω − √ vl (cid:16) √ n + √ n + 1 (cid:17)(cid:19) (cid:2) f ( E − n ( k n )) − f ( E + n +1 ( k n )) (cid:3) +[1 − cos α n ( k n )][1 + cos α n +1 ( k n )]2 | η ′ n ( k n ) | Θ (cid:18) √ vl − ω (cid:19) (cid:2) f ( E + n ( k n )) − f ( E + n +1 ( k n )) (cid:3) ) . Here ν n ( k ) = ω − v (cid:16)q k + n +1) l + q k + nl (cid:17) , η n ( k ) = ω − v (cid:16)q k + n +1) l − q k + nl (cid:17) , k n = q(cid:0) ω v (cid:1) − n +1 l + (cid:0) vωl (cid:1) . ∗ [email protected] † [email protected] M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. , 3045(2010). L. Fu and C.L. Kane, Phys. Rev. B , 045302 (2007). M. K¨onig, S. Wiedmann, C. Br¨une, A. Roth, H. Buhmann,L.W. Molenkamp, X.L. Qi, and S.C. Zhang, Science ,766 (2007). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). B.A. Bernevig, T.L. Hughes, and S.-C. Zhang, Science , no. 5806, 1757 (2006). B.A. Volkov and O.A. Pankratov, Pis’ma Zh. Eksp. Teor.Fiz. , 145 (1985) [JETP Lett. , 178 (1985)]. A. Roth, C. Br¨une, H. Buhmann, L.W. Molenkamp, J.Maciejko, X.-L. Qi, and S.-C. Zhang, Science , 294(2009). G.M. Gusev, Z.D. Kvon, O.A. Shegai, N.N. Mikhailov,S.A. Dvoretsky, and J.C. Portal, Phys. Rev. B ,121302(R) (2011). H. Buhmann, J. Appl. Phys. , 102409 (2011). M.V. Entin, and L.I. Magarill, Pis’ma v ZhETF , 804 (2016) [JETP Lett. , 711 (2016)]. M.M. Mahmoodian, L.I. Magarill, and M.V. Entin, J.Phys.: Condens. Matter , 435303 (2017). M.M. Mahmoodian and M.V. Entin, Phys. Status Solidi B , 1800652 (2019). A. Rahim, A.D. Levin, G.M. Gusev, Z.D. Kvon, E.B. Ol-shanetsky, N.N. Mikhailov, and S.A. Dvoretsky, 2D Mater. , 044015 (2015). M.V. Entin and L. Braginsky, Europhys. Lett. , 17003(2017). S. Tchoumakov, V. Jouffrey, A. Inhofer, E. Bocquillon, B.Placais, D. Carpentier, and M.O. Goerbig, Phys. Rev. B , 201302(R) (2017). X. Lu and M.O. Goerbig, Europhys. Lett. , 67004(2019). M.M. Mahmoodian and M.V. Entin, Phys. Rev. B ,125415 (2020). F. Zhang, C.L. Kane and E.J. Mele, Phys. Rev. B19