Localized Andreev edge states in HgTe quantum wells
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Instability of the topological order and localization of the edge states in HgTequantum wells coupled to s-wave superconductor
I. M. Khaymovich,
N. M. Chtchelkatchev,
1, 3, 4, 5 and V. M. Vinokur Argonne National Laboratory, Argonne, IL 60439, USA Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142190, Moscow Region, Russia L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Moscow, Russia (Dated: September 2, 2018)Using the microscopic tight-binding equations we derive the effective Hamiltonian for the two-layerhybrid structure comprised of the two-dimensional HgTe quantum well-based topological insulator(TI) coupled to the s-wave isotropic superconductor (SC) and show that it contains terms describingmixing of the TI subband branches by the superconducting correlations induced by the proximityeffect. We find that the proximity effect breaks down the rotational symmetry of the TI spectrum.We show that the edge states not only acquire the gap, as follows from the standard theory, butcan also become localized by the Andreev-backscattering mechanism in a small coupling regime. Ina strong coupling regime the edge states merge with the bulk states, and the TI transforms into ananisotropic narrow-gap semiconductor.
I. INTRODUCTION
A topological insulator (TI), a material in which theelectronic spectrum possesses an energy gap in the bulkbut has the special, so-called topologically protected,edge (surface) states falling into this gap, is one of the fo-cal points of current condensed matter studies.
Topo-logical insulators hold high technological promise sincedue to the ability of their topologically protected edgestates to carry nearly dissipationless current they can beutilized in integrated circuits. Thus the question howrobust the edge states are with respect to hybridizationwith the electronic states in the leads has become one ofthe focal points of current TI-related research.Three dimensional (3D) TI have the Dirac spectrumwith the finite mass in the bulk while the spectrum ofthe surface states is massless.
Two-dimensional (2D)TI has the gapless helical edge states.
The surfacestates in 3D TI and the edge states in 2D TI are topolog-ically protected and are robust against all time-reversal-invariant local perturbations. It was shown experimen-tally that 2D TI-state appears in HgTe/CdTe quantumwells (QW).
The TI in three-dimensional (3D) ma-terials was found in Bi − x Sb x , Bi Se and so on. Of special interest are the states that develop at the in-terface between a TI and an s-wave superconductor (SC),like, e.g., in Fig. 1, where the proximity effect gener-ates a superconducting pairing. it was shown that atthe interface of 3D TI coupled to s-wave superconduc-tor p x + ip y -superconducting state appears but withouttime-reversal symmetry breaking. Numerical calcula-tions showed that the proximity of the superconductorleads to a significant renormalization of the original pa-rameters of the effective model describing the surfacestates of a topological insulator. For 2D topologicalinsulator coupled to s-wave superconductor it is knownthat the momentum-independent gap enters in the edge spectrum.
Two-dimensional TIs have a unique property: theirparameters can be tuned over wide ranges of their val-ues by the appropriate choice of the QW width d . Inparticular, the main parameter of 2D TI, the gap in thebulk spectrum, M , can be changed from zero up to roomtemperature energy scale. Phenomenological treatmentof the proximity effect is based on the assumption thatthe gap M in the bulk spectrum of TI is much larger thanthe characteristics energy of the induced superconduct-ing correlations. Unfortunately numerical calculationsdo not give the answer how the spectrum of the TI-SCsystem develops in this regime. In what follows wewill focus on the gaps M of the order of the energy ofsuperconducting correlations.In this Paper we investigate the effect of superconduct-ing correlations on the topologically protected edge statesand the bulk spectrum in the 2D topological insulatorbrought into a contact with the SC layer, see Fig. 1,and show the emergence of the SC correlations in theTI similarly to what was observed in GaAs containinga two-dimensional electron gas. Ordinarily, the ef-fective Hamiltonian of the TI in question is constructedfrom the symmetry considerations, see e.g., Ref. 17–19,and has the trivial structure of the induced supercon-ducting potentials. We demonstrate that the symmetryreasons suggest the additional non-diagonal (in the sub-band space) terms in the Hamiltonian. Moreover, usingthe microscopic tight-binding equations we derive analyt-ically the additional terms in the effective Hamiltonianof the TI describing coupling to the s -wave isotropic su-perconductor (SC) placed on top of it. These terms be-come especially important in the case where the bare gapparameter M of the TI becomes comparable to the char-acteristic energy of the induced superconducting correla-tions. We show, further, that the interplay of the super-conducting and “topological” interactions is essential andresults in several effects, in particular, the collapse of thetopological order in TI. We find that while “topologicalyprotected” edge states can ensure undisturbed propaga-tion of the charge (spin) carriers, superconducting cor-relations can block the edge current causing a peculiarlocalization effect. FIG. 1. (Color online) Sketch of the 2D topological insu-lator coupled to a superconductor. The thick arrows showschematically the edge states.
On the qualitative level our results can be summarizedas follows: in the absence of superconducting correlationsthe edge states spectrum is linear near the Fermi surface,comprising of two counter-propagating electron and twohole branches, respectively. Importantly, the edge spec-trum is isotropic in a sense that it does not depend onthe orientation of the edges with respect to crystallo-graphic axes, although the edge electronic states wavefunctions are orientation dependent. Namely, there is aphase difference between the wave function componentscorresponding to (E) or (H)-subbands in TI. The bulkspectrum is characterized by the gap M . At small SC-TI coupling (the quantitative criteria will be given below)the edge-states spectrum acquire a gap E g . Furthermore,superconducting correlations mix the subband branchesand thus turn the resulting electronic spectrum in TI(edge and bulk), anisotropic, and E g starts to depend onorientation of the edge. This can cause localization ofthe low-lying edge states since an inevitable bending ofthe edge will create the regions along the edge were theedge-particle energy ε < E g thus getting locked betweenthe turning points where ε = E g . At the turning pointselectron and hole excitations undergo Andreev reflectionand form the localized Andreev bound edge states, seeFig. 2a. As the characteristic energy of the induced pair-ing amplitude exceeds M then the gap of the continuumbulk states of TI collapses and TI behaves like the highlyanisotropic narrow-gap semiconductor. II. EFFECTIVE HAMILTONIAN
The low energy Hamiltonian of the two-dimensional(2D) topological insulator formed in the HgTe QW hasthe form: ˇ H = ˆ H
00 ˆ˜ H ! , (1) h -- spin upe -- spin downh -- spin down no edge states on this edge at low energies a)b) ( e xc i t a t i on ene r g y ) / M (longitudinal momentum k) / k e -- spin up FIG. 2. (Color online) a) A sketch of localized Andreevedge states in TI. b) Excitation spectrum in the 2D TI withproximity induced superconducting correlations: the edgemodes at one TI-edge acquired the gap while the edge modeson the opposite edge remain gapless. Solid lines show theedge states and bulk spectrum boundaries in TI without su-perconducting correlations. The parameters are chosen as: mk F t a / (2 π ~ M ) = 1, t b = p | B − | / | B + | t a and the orienta-tion angle ϕ = 0. where ˆ H = ǫ k + d i ˆ σ i , i = { , , } ; ˆ σ i are the Paulimatrices acting in the subband (isospin) space; ǫ k = C − Dk . We choose the frame of reference so that ~d = ( k x A, − k y A, M − Bk ). Here A , B , C , D and M arematerial parameters. The lower block of the Hamilto-nian, ˆ˜ H = ˆ ρ T ˆ H ∗ ˆ ρ = ǫ k − d i ( − k )ˆ σ i , where ˆ ρ = i ˆ σ y is themetric tensor in the spinor space. The chosen represen-tation for ˇ H enables us to employ the machinery of thetensor bispinor algebra developed for Dirac Hamiltonian.To construct the convenient form of the Bogoliubov –de Gennes (BdG) Hamiltonian describing superconduc-tivity, we introduce the time reversal symmetry operator,ˇ T = −
10 0 1 00 − C = − ˆ τ ⊗ i ˆ σ C , (2)where C is the operator of the complex conjugation andˆ τ i , are the Pauli matrices acting in spin space. Then thetime-reverse of the BHZ-Hamiltonian (1) is ˇ T ˇ H ˇ T − =ˇ H . The BdG Hamiltonian is H BDG = (cid:18) ˇ H + ˇ U ˇ∆ TI ˇ∆ + TI − ˇ H − ˇ T ˇ U ˇ T − (cid:19) , (3)where ˇ∆ TI is the effective proximity induced supercon-ducting pairing matrix coupling the spin and subbandspaces. The effective chemical potential shift appearingin the BCS theory has matrix form ˇ U . There is no reason (U +U )/2 a) k=k k=-k ene r g y k b) E g Fermi level e , s p i n do w n h , s p i n upe , s p i n up h , s p i n do w n (U +U )/2s Fermi level E g FIG. 3. (Color online) The edge states at the TI boundary fora weak coupling where the edge states become gapped. a) Un-perturbed edge-spectrum. b) Edge states with the proximity-induced gap. The shift of the zero point reflects the differencein the original chemical potentials. to believe that the matrix structure of ˇ U and ˇ∆ TI is nec-essary trivial like, e.g., in Ref. 17–19; symmetry consid-erations in fact allow nontrivial shape of theses matrices.So both, ˇ∆ TI and ˇ U will be found below microscopically.To proceed further we present the Hamiltonian describ-ing the TI-SC coupling in a form: H = H sc + H D + H int . (4)The superconducting part is H sc = X s = ↑ , ↓ Z d r Ψ + s ( r ) ( ǫ sc − µ ) Ψ s ( r )+ Z d r (cid:16) ∆Ψ + ↑ ( r )Ψ + ↓ ( r ) + ∆ ∗ Ψ ↓ ( r )Ψ ↑ ( r ) (cid:17) (5)where, Ψ ↑ ( ↓ ) (Ψ + ↑ ( ↓ ) ) are the field annihilation (creation)operators for the state with the spin up (down), ∆ isthe superconducting gap, ǫ sc is the single electron kineticenergy, and µ is the Fermi energy.The second quantization representation for the TIHamiltonian is written in the basis of the Wannier func-tions for particles with spin s : H D,s = X RR ′ ,s X σ,σ ′ = a,b c + s R ,σ (cid:0) ǫ D,s ( R σ, R ′ σ ′ )+ Cδ ( R , R ′ ) δ σ,σ ′′ (cid:1) c s R ′ ,σ ′ (6)where ǫ D,s ( R , R ′ ) is the lattice representation of theBHZ-model (1) (see Appendix A). Then H D = P s H D,s .Finally, H int reflects the electronic tunneling betweenthe SC and TI: H int = X R ,s X σ = a,b (cid:16) t σ, R Ψ + s ( R ) c s R ,σ + t ∗ σ, R c + s R ,σ Ψ s ( R ) (cid:17) , -1 1 ( e xc i t a t i on ene r g y ) / M (longitudinal momentum k) / k b) -0.5 k x / k a)
00 -1 110-1 [ c on t i nuou s ( bu l k ) s pe c t r u m ]/ M k y / k FIG. 4. (Color online) Bulk and edge states for the inter-mediate coupling. a) Energy of the bulk states as functionof k x , k y . Without superconductivity, the bulk states dis-persion is isotropic, E ( k ) = ǫ ( k ) ± p A k + ( M − Bk ) , where k = p k x + k y . Superconducting correlations make itanisotropic as follows from the noncommutativity of ˇ∆ and/orˇ U with ˇ H in Eq.(3). b) Gapped edges states. The colors forthe families of the dispersion curves are the same as in Fig. 3.The parameters are chosen as: mk F / (2 π ~ ) t a /M = 1, t b = p | B − | / | B + | t a exp( iπ/
6) and the orientation angle ϕ = π/ where c ↑ ( ↓ ) R ,a is the superposition of (cid:12)(cid:12) Γ , ± (cid:11) , (cid:12)(cid:12) Γ , ± (cid:11) and c ↑ ( ↓ ) R ,b refers to the subband (cid:12)(cid:12) Γ , ± (cid:11) . Integrat-ing out the bulk superconductor variables Ψ s ( R ) usingthe method, developed in Ref. 24 and 25, one obtainsthe effective BdG-Hamiltonian (3) for the homogeneoustunneling amplitudes t σ R = t σ , with the matrix super-conducting order parameter and the effective chemicalpotential shift having the form:ˇ∆ TI = ˆ∆ TI
00 ˆ˜∆ TI ! , ˇ U = ˆ U
00 ˆ˜ U ! . (7)Hereˆ∆ TI = − mk F π ~ (cid:18) t ∗ a t ∗ a t ∗ b t ∗ a t ∗ b t ∗ b (cid:19) , ˆ˜∆ TI = ˆ ρ T ˆ∆ TI ˆ ρ , (8)ˆ U = m π ~ a TI (cid:18) | t a | t ∗ a t b t a t ∗ b | t b | (cid:19) , ˆ˜ U = ˆ ρ T ˆ U ˆ ρ , (9) m and k F are the effective mass and the Fermi momen-tum of the bulk superconductor respectively, a TI is thecharacteristic length scale of the order of the lattice con-stant in TI. Since ∆ TI ≪ ∆, the proximity induced pa-rameters are independent of ∆. In Ref. 17–19 the po-tentials ˆ∆ TI and ˆ U were diagonal (trivial) while the off-diagonal terms were missed.For numerical calculations we take typical parameters: A = 3 . B = − . , D = − . . -2 0 2 c)b) E ( ) g r re turning points ene r g y Boundary orientation angle E ne r g y gap E g a) FIG. 5. (Color online) Energy landscape for the edgestates. a) Sketch of the energy landscape along the edge(parametrized by the coordinate ρ ). The magnitude of thegap may change as a result of spatial fluctuations (like changein a shape of the TI boundary or fluctuations in tunnelingamplitudes) and the state with the energy ε would appeartrapped between the turning points where ε < E g formingAndreev bound edge state like it is shown in Fig. 2. b) Cal-culated E g as function of the edge orientation angle ϕ for theratio t a /t b p | B − /B + | exp( − iπ/
3) equal to 1 (upper curve)and 2 (bottom curve). Such E g behavior can be observed inthe sample shaped into a disc. c) Localized edge states in thesample shaped into a disc for energy ǫ corresponding to thedash-dot line in Fig. 5b. Without a loss of generality we take the energy-shiftparameter C = 0. We do not fix M ( − . M <
0) and use it as the energy unit. Our numer-ical and analytical calculations show the approximatesymmetry relation that satisfies the spectrum of H BDG : χE ( k /χ, M/χ, t a / √ χ, t b / √ χ ) ≈ E ( k , M, t a , t b ), where χ is a dimensionless scaling parameter. The scaling rela-tion appears since M is much smaller than the energyscales one can construct form A , B and D . In addition, M appears to be the most sensitive to the HgTe layerwidth: it changes with it by several orders of magnitudewhile the other parameters change by ∼
20% and theirchanges very slightly modify the spectrum. First we discuss “weak” superconductivity where su-perconducting correlations induced in TI can be treatedperturbatively. In this case matrix elements of ˇ∆ TI andˇ U are smaller than the gap in the continuum spectrum, M , in the bulk of TI. In the absence of superconductingcorrelations ˇ∆ TI = 0 and ˇ U = 0, and there are two elec-tron and two hole edge states at each TI surface. Theedge states have the linear dispersion law with the ve-locity s = A | p B + B − /B | , where B ± = B ± D . Theycross the Fermi energy at k = k = DM/ ( A p B + B − )and k = − k , see Fig. 3a. We denote the wave func-tions of the electron and hole edge states near k = k as ψ (1) = ( ψ edge , ˆ0 , ˆ0 , ˆ0) τ and ψ (2) = (ˆ0 , ˆ0 , ψ edge , ˆ0) τ , respec- tively, where ˆ0 is the zero spinor in the subband space, ψ edge = e − i ˆ σ z ϕ/ p | B | p | B − |− p | B + | ! × (cid:0) e − λ + r · n − e − λ − r · n (cid:1) e ik r · l , (10) k is the momentum component parallel to the edge, r =( x, y ), l , and n are the unit vectors directed along the TIboundary and perpendicular to it correspondingly [ l × n is aligned with the OZ axis], and ϕ is the angle between l and OX axis. The decay length scales of the edge statesinto the bulk of the topological insulators are: λ ± = λ ± s(cid:18) k − DB λ (cid:19) + A B − MB , (11)where λ = A/ (2 p B + B − ). We stress that spinor com-ponents of ψ edge depend on the TI-boundary orientation. ( e xc i t a t i on ene r g y ) / M (longitudinal momentum k) / k FIG. 6. (Color online) Excitation spectrum in 2D TI forthe proximity induced potentials being of the same orderas the gap in TI in the absence of a SC. The gap betweenthe branches of the continuum spectrum collapses and TIacquires metallic conductivity with the relativistic spectrumsimilar to that in graphene. Solid lines correspond to theedge states and the bulk spectrum boundaries in TI with-out superconducting correlations. Parameters are chosen as: mkF/ (2 π ~ ) t a /M = 12, ϕ = 0, and t b = p | B − | / | B + | t a . The dispersion law of the edge states near k = k within the perturbation theory taking in the account thesuperconducting correlations acquires the form: ǫ , ( k ) = ( U + U ± ω ( k )) / , (12)where ω = q (2 s ( k − k ) + U − U ) + 4 E g ; U ii , i =1 , U with respect to thestates ψ (1 , and E g = | ( ˇ∆ TI ) | . They can be parame-terized through T ± = (cid:12)(cid:12)(cid:12) t a p | B − | − t b p | B + | e ± iϕ (cid:12)(cid:12)(cid:12) . So, E g = T + T − , (13)the matrix elements in the see Eq.(9) become ˇ U = α T − ) , and α = ( k F a TI ) − .One now sees that the spectrum of the edge statesbecomes dependent upon the orientation of the boundaryorientation with respect to crystallographic axis. Thisresembles the spectrum that often appear in the 3D TIwhich that are referred to as “strong” TI. There is a wealth of the possible coupling-induced be-haviours of the edge states energy spectrum. If T + = 0or T − = 0, then E g = 0 as well; the situation where E g is very small is also common. A particular picturedepends on the edge orientation angle ϕ and/or on thetunneling amplitudes, t a and t b . Shown in the Fig. 2 isthe situation where at one boundary of the TI-strip theedge states remain gapless ( E g = 0) while at the oppo-site boundary E g = 0 and the edge states have the gap.The stripe within which the edge states are confined hasa finite length (see in Fig. 2), there are points at the edgewhere the TI-boundary changes its direction and, at thesame time, the value of E g changes. At these “turningpoints” electron and hole edge (going in the opposite di-rection) states with the energy smaller than E g undergothe Andreev reflection and form the bound Andreev edgestate, see Figs. 2,5. Illustrated in Figs. 3-4 is the struc-ture of the the edge state energy levels in the case where E g is finite.Now we discuss a general nonperturbative situation.The excitation spectrum in 2D TI accounting for theproximity induced superconducting correlations in thecase where proximity induced potentials in TI are of thesame order as the gap in TI without superconductor ontop is shown in Fig. 6. The gap between the branches ofthe continuum spectrum nearly closes and TI acquires ef-fectively metallic conductivity with the relativistic spec-trum similar to that in graphene. Solid lines correspondto the edge states and bulk spectrum boundaries in TIwithout superconducting correlations. III. CONCLUSIONS
To conclude, we investigated topologically protectededge states in QW of HgTe sandwiched between CdTe and demonstrated that the s -wave isotropic superconduc-tor placed on top of CdTe layer induces superconductingcorrelations in the TI revealing the built in anisotropyof TI which did not affect the spectrum when supercon-ducting correlations were absent. The form of the edgestates spectrum essentially depends on the edge orienta-tion with respect to crystallographic directions of the TI.Depending on the coupling between the superconductorand 2D TI, different scenarios can be realized: (i) theedge states of the topological insulator acquire a gap, (ii)the edge states hybridize into the Andreev localized edgestate and/or (iii) the gap separating the continuum andthe edge modes collapses and TI becomes the narrowgap(anisotropic) semiconductor. Our predictions can be ver-ified by means of, for example, scanning tunnelling spec-troscopy measurements of the spectra showed in Fig. 3where the shift of the zero point U + U can be tunedby the gate placed on top of the CdTe layer. Note : After this work has been completed we becameaware of the recent experiments on InAs/GaSb QW. which revealed the 2D TI state. Since this novel TI isexpected to be well described by the BHZ-model, ourresults apply to InAs/GaSb QW coupled to s-wave su-perconductor as well.
IV. ACKNOWLEDGMENTS
This work was supported by the U.S. Department ofEnergy Office of Science under the Contract No. DE-AC02-06CH11357, the work of IMK and NMC was partlysupported by the Russian president foundation (mk-7674.2010.2) under the Federal program “Scientific andeducational personnel of innovative Russia”.
Appendix A: Lattice model
Microscopic description of the coupling between TI andthe superconductor developed on the base of the standardlattice regularization of the BHZ-model (1) replacing itsparameters by: ǫ k = C − Da − [2 − cos k x a − cos k y a ] , (A1) ~d = (cid:0) Aa − sin k x a, − Aa − sin k y a, M − Ba − [2 − cos k x a − cos k y a ] (cid:1) . (A2)This corresponds to the quadratic Bravais lattice (with the translation vectors a = a x , a = a y ) with two typeof states (corresponding annihilation operators are ˆ c a R and ˆ c b R ) on each site replying to subband states. Thereforethe Hamiltonian describing TI in the second quantization representation in the basis of the Wannier functions forparticles with spin s , takes the form:ˆ H D,s = X RR ′ ,s X σ,σ ′ = a,b ˆ c + s R ,σ (cid:0) ˆ ǫ D,s ( R σ, R ′ σ ′ ) + Cδ ( R , R ′ ) δ σ,σ ′′ (cid:1) ˆ c s R ′ ,σ ′ , (A3)whereˆ ǫ D,s ( R ˜ σ, R ′ ˜ σ ) = δ RR ′ (cid:20)(cid:18) M − Ba (cid:19) ˜ σ − Da (cid:21) + (cid:18) Ba ˜ σ + 4 Da (cid:19) (cid:0) δ R+a , R ′ + δ R-a , R ′ + δ R+a , R ′ + δ R-a , R ′ (cid:1) (A4a)ˆ ǫ D, ↑ ( R a, R ′ b ) = − ˆ ǫ D, ↓ ( R b, R ′ a ) = A a (cid:0) δ R+a , R ′ − δ R-a , R ′ − iδ R+a , R ′ + iδ R-a , R ′ (cid:1) (A4b)ˆ ǫ D, ↑ ( R b, R ′ a ) = − ˆ ǫ D, ↓ ( R a, R ′ b ) = − ˆ ǫ ∗ D, ↑ ( R a, R ′ b ) . (A4c) B. A. Volkov and O. A. Pankratov, JETP Lett. , 178(1985). X. L. Qi, S. C. Zhang, arXiv:1008.2026v1. J. E. Moore, Nature (London) , 194 (2010). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. ,106803 (2007). H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.Zhang, Nat. Phys. , 438 (2009). D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava,and M. Z. Hasan, Nature (London) , 970 (2008). D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier,J. Osterwalder, L. Pattey, J. G. Checkelsky, N. P. Ong, A.V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J.Cava, and M. Z. Hasan, ibid. , 1101 (2009). Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K.Mo,X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C.Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science325, (2009). T. Zhang, P. Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L.Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue,Phys. Rev. Lett. , 266803 (2009). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005); , 226801 (2005). B. A. Bernevig, T. L. Hughes, and S-C. Zhang, Science ,1757 (2006). M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann,L. W. Molenkamp, Xiao-Liang Qi, and S.-C. Zhang, Sci-ence , 766 (2007). A. Roth, C. Br¨une, H. Buhmann, L.W. Molenkamp, J.Maciejko, X.-L. Qi, and S.-C. Zhang, Science , 294(2009). L. Fu and C.L. Kane, Phys. Rev. Lett. , 096407 (2008). T.D. Stanescu, J.D. Sau, R.M. Lutchyn, and S. Das Sarma,Phys. Rev. B , 241310(R) (2010). Liang Fu and C.L. Kane, Phys. Rev. B , 161408(R)(2009). J. Nilsson, A.R. Akhmerov, and C.W.J. Beenakker, Phys.Rev. Lett. , 120403 (2008). P. Adroguer, C. Grenier, D. Carpentier, J. Cayssol, P. De-giovanni, and E. Orignac, Phys. Rev. B , 081303(R)(2010). Hua Jiang, Lei Wang, Qing-feng Sun, and X.C. Xie, Phys.Rev. B , 165316 (2009). Qing-Feng Sun, Yu-Xian Li, Wen Long, and Jian Wang,Phys. Rev. B , 115315 (2011). H. Takayanagi, T. Akazaki, and J. Nitta, Phys. Rev. Lett. , 3533 (1995). I.E. Batov, T. Sch¨apers, N.M. Chtchelkatchev, H. Hardt-degen, and A.V. Ustinov, Phys. Rev. B , 115313 (2007). A.F. Volkov, P.H.C. Magnee, B.J. van Wees, and T.M.Klapwijk, Physica C , 261 (1995). A. S. Mel’nikov, N. B. Kopnin, arXiv:1105.1903. I. Knez, R.R. Du, and G. Sullivan, arXiv:1105.0137; ibidibid