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Giant spin rotation in the normal metal/quantum spin Hall junction Takehito Yokoyama1, Yukio Tanaka1, and Naoto Nagaosa2,3 1Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan 2 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan 3 Cross Correlated Materials Research Group (CMRG), ASI, RIKEN, WAKO 351-0198, Japan (Dated: January 5, 2009) Westudytheoreticallyreflectionprobleminthejunctionbetweenanormalmetalandaninsulator characterizedbyaparameterM,whichisausualinsulatorforM >0oraquantumspinHallsystem 9 forM <0. ThespinrotationangleαatthereflectionisobtainedintheplaneofM andtheincident 0 angleθmeasuredfromthenormaltotheinterface. Theαshowsrichstructuresaroundthequantum 0 critical point M = 0 and θ = 0, i.e., α can be as large as ∼ π at an incident angle in the quatum 2 spin Hall case M < 0 because the helical edge modes resonantly enhance the spin rotation, which n can beusedto map theenergy dispersion of thehelical edgemodes. Asan experimentally relevant a system, we also study spin rotation effect in quantum spin Hall/normal metal/quantum spin Hall J trilayer junction. 5 PACSnumbers: 73.43.Nq,72.25.Dc,85.75.-d ] l l a The ultimate goal of the spintronics is to manipulate tem experiences a quantum phase transition by chang- h the spins without the magnetic field, and the relativis- ing the thickness. Recent experiment has successfully - s tic spin-orbit interaction is the key to realize it. One demonstrated the existence of the helical edge mode for e example is the Datta-Das spin transistor [1] where the thequantumwellofHgTesystembythemeasurementof m spin-orbit interaction modified by the gate voltage con- thequantizedchargeconductance[15]. Alsorecently,one . trols the rotation of the spins of the carriers. However, can obtain the system very close to the quantum critical t a usually the strengthof the spin-orbit interactionis weak point by tuning the thickness of the quantum well. [16] m and its influence is small especially in semiconductors. - Nevertheless, its effects are now clearly observed experi- Most of the previous works on QSH states have fo- d cused on properties of isolated QSH system, in particu- n mentally, e.g., in the Rashba spin splitting [2], and spin lar, edge states of the system. However, in experiments o Hall effect (SHE) [3, 4, 5, 6]. Further enhancement of c the spin-orbiteffects insemiconductorsishighlydesired, to detect some characteristics of the QSH system, some [ probe,e.g. (metallic)electrodeshouldbeattachedtothe andforthatpurposeweproposeinthis Lettertousethe QSH.Therefore,it is animportantissue to clarifytrans- 1 quantum spin Hall (QSH) system[7, 8] and its junction v to the normal metal (N) as the spin rotator with a giant port property in N/QSH junctions. Moreover, the spin 8 transportrelatedtotheQSHsystemisthemostinterest- angle α of the order of π due to the resonance with the 3 ing issue, which has not been well explored and we will helical edge channels. This effect can be used to map 4 address in this paper. Especially, we focus on the spin 0 the energydispersionofthe helicaledge modes, andalso transport properties normal to the edge of the sample, . to produce the spincurrentparallelto the interface with 1 while most of the previous works are interested in the the average over the incident angle. 0 charge transport along the edge channel. 9 Naively,QSHsystemcanberegardedasthetwocopies 0 of the integer quantum Hall systems for up and down In this paper, we study a reflection of the electronic v: spinswiththeoppositechiralities. Hence,thechiraledge wave at the N/QSH interface. It is found that an elec- i modes for up and down spins with the opposite propa- X tron injected from the normal metal shows a spin de- gating directions are expected, which is called the he- pendent reflection at the interface leading to the spin r lical edge modes.[9, 10, 11, 12] Two sets of the helical a rotation. The spin rotation angle α shows rich struc- edge modes can be mixed by e.g., the impurity scatter- ture centered around the quantum critical point M =0, ing, and open the gap, but when the number of these and the normal incident angle θ = 0, i.e., it has a large sets is odd, i.e., Z2 = 1, there always remains a helical value comparable to π and even a winding by 4π in the mode pair which is protected by the Kramers theorem (θ M)-plane in the QSH region (M <0), which stems associated with the time-reversal symmetry. The phase − fromthe helicaledge modes. This isin sharpcontrastto transitionbetweenthetrivialinsulator(Z2 =0),andthe thecaseofnormalmetal/usualinsulatorinterface. Asan topological insulator (Z2 = 1) occurs at the gap closing experimentally relevant system, we also investigate mul- point, wherethe massofthe DiracFermionchangessign tiple reflections in QSH/N/QSH junction. It is clarified [13, 14]. that the multiple reflection strongly enhances the spin The existence of the QSH state has been predicted rotation. in semiconductors with an inverted electronic gap in HgTe/CdTe quantum wells [13]. The quantum well sys- Let us commence with the the effective 4-band model 2 A,B,C,D, and M are material parameters that depend on the quantum well geometry. is equivalent to two H (cid:0) (cid:1)(cid:2)(cid:3) copies of the massive Dirac Hamiltonian but with a k- dependent mass M(k).[17] This model is derived from (cid:4) the Kane model near the Γ point in a quantum well of HgTe/CdTejunction,anditsvalidityislimitedonlynear (cid:8) (cid:7) theΓpoint,namelyforsmallvaluesofk andk . Inthis x y system,thetransitionofelectronicbandstructureoccurs from a normalto an invertedtype when the thickness of (cid:6) (cid:5) the quantum well is varied through a critical thickness. This corresponds to the sign change of the mass M of this system.[13, 18] We consider the interface between normal metal and an insulator, the latter of which is a usual insulator for FIG. 1: (Color online) N/QSH junction and corresponding M > 0 and a QSH system for M < 0, as shown in Fig. band structures (below). Dotted lines represent helical edge 1. For M < 0, helical edge modes are expected to ap- modes. pear at the interface.[13, 18] The interface is parallel to y-axis and located at x = 0, and the energy dispersion proposed for HgTe/CdTe quantum wells [17] of each side is shown schematically in Fig. 1. The insulating side is described by the Hamiltonian given in h(k) 0 Eq.(1). The Hamiltonian in the N side is given by set- = , (1) H (cid:18) 0 h∗( k)(cid:19) tingA=B =M =0inthatofQSH.Thegap M opens − | | on the insulating side. C is the Fermi energy measured with h(k) = ǫ(k)I2×2 + da(k)σa,ǫ(k) = C from the bottom of the|m|etallic band. D(k2 + k2),d (k) = (Ak , Ak ,M(k)), and M(k) −= M x B(ky2 +ak2) wherexw−e hayve used the basis or- Now, let us focus on the spin up state, corresponding − x y der (E1+ , H1+ , E1 , H1 ) (”E” and ”H” repre- tothe upper blockofthe Hamiltonian. Wavefunctionin | i | i | −i | −i sent the electron and hole bands, respectively), and, the N side for E1 state injection is given by 1 1 0 ψ(x 0)= eikFcosθx+r e−ikFcosθx+r′ e−ikFcosθx eikFsinθy (2) ≤ (cid:20)(cid:18)0(cid:19) E(cid:18)0 (cid:19) E(cid:18)1 (cid:19) (cid:21) with k2 =C/D and the incident angle θ. F Wave function in the QSH side reads ψ(x 0)= t Ak+eiθ+ eik+cosθ+x+t′ Ak−eiθ− eik−cosθ−x eikFsinθy (3) ≥ (cid:20) (cid:18)d(k+) M(k+)(cid:19) (cid:18)d(k−) M(k−)(cid:19) (cid:21) − − with d(k)= A2k2+M(k)2 and p 1 k2 = (A2 2MB+2CD) (A2 2MB+2CD)2 4(B2 D2)(M2 C2) (4) ± 2(B2−D2)h− − ±p − − − − i which is obtained by solving E = C Dk2 +d(k) = 0 kF sinθ =k+sinθ+ =k−sinθ−. − for k. Here, θ denote the incident angles for wave- Boundary conditions read ψ(+0) = ψ( 0) and ± functions with wavenumber k . In the following, we v ψ(+0)=v ψ( 0) with v = ∂H. With thes−e bound- ± x x − x ∂kx set C = 0 in the insulating side, when the Fermi en- ary conditions, we can obtain scattering coefficients. ergy is located at the middle of the gap. Notice that we assume insulating state so that k±2 < 0. Due to Thus, the reflection coefficients for E1 state injection the translational symmetry along the y-axis, we have are obtained as follows 3 rE = 1 (Ak+eiθ+ −a+)(d(k−)−M(k−)+b−)−(Ak−eiθ− −a−)(d(k+)−M(k+)+b+) (5) (cid:18)rE′ (cid:19) ∆E (cid:18) 2{−b+(d(k−)−M(k−))+b−(d(k+)−M(k+))} (cid:19) with ∆E = (Ak+eiθ+ + a+)(d(k−) M(k−) + that all these interesting structures occur in the QSH − b−) (Ak−eiθ− + a−)(d(k+) M(k+) + region (M < 0). It should be also noticed that these Mb+()k,aσσ)−)/(2D=kF cos(cid:0)θ1),+bσDB(cid:1)=AλσAk2σkeσiθeσiθσ/−−(2DAk(Fd(ckoσs)θ) −+ cstidruenctcuer,easnadrethseeenrefloervasnmcealltoθ’st,hei.eh.,elaiclmalosetdgneorcmhaanlnine-l 1 B λ (d(k ) M(k )),−λ = k cosθ /(k cosθ), is expected. The angle at which the dispersions of the − D σ σ − σ σ σ σ F (cid:0)and σ =(cid:1) . helical edge modes hit the Fermi energy is given by [18] ± Thecorrespondingreflectioncoefficientsforspindown MD state can be obtained by the substitution A A θ = sin−1 . (6) → − C ± (cid:20)Ak √B2 D2(cid:21) and θ θ. Spin rotation angle α is defined as F − → − α=Imlog(r /r ),whichisthephasedifferenceinthe E↑ E↓ We find that a large magnitude of α in Fig. 2 appears reflectioncoefficientsbetweenupanddownspinelectrons around this angle, which means that the helical edge in the E state. From the definition, we see that α is an modesresonantlyenhancethespinrotation. Thisisplau- odd function of θ since r ( θ) = r (θ). Note that E↑ − E↓ sible since in the helical edge modes, up and down spins when the spin of the incident electron is within the xy- propagateinanoppositedirectionandhenceSU(2)sym- plane, this angle α gives the spin roration angle within metryinspinspaceisstronglybrokenthere. This isalso this plane at the reflection in the same band. Therefore, similar to the Andreev reflection in the superconducting we call it ”spin rotation angle”. analogue of QSH system where the helical edge chan- Now, we show the results for α in the plane of (θ,M) nelproduces a largelyspin-polarizedsupercurrentatthe in Fig. 2 for C = 0.08 eV in (a), C = 0.1 eV in (b), Andreev reflection. [19] − − andC = 1eVin(c)withtheotherparametersfixedas To realize the giant spin rotation experimentally, we A=4 eV−˚A, B = 70eV ˚A2 andD = 50eV ˚A2.[17] propose QSH/N/QSH junction (see the inset in Fig. 3). · − · − · In Figure2 (a), a sharpridge inM <0 andθ >0 region When N layer is sufficiently thin, the electrons in the N and its negative correspondence in M <0 and θ <0 are region are in close contact to the QSH systems and the seen. Thisisinsharpcontrasttotheusualinsulatorcase transport property in this junction is determined by the M > 0 although α is still nonzero there. Note that the reflection at the interfaces. We will also take statistical height of the ridge is as high as π/2. With increasing and incident angular average of the results in order to ∼ C , we find a qualitatively different structure. Near the obtain more experimentally accessible quantity. | | origin in Fig. 2 (b), α reaches π, changes its sign, and To investigate spin rotation effect in this junction, we winds by 4π around the origin, while it does not in the also have to calculate reflection coefficients with H state region far away from the origin. One might wonder that injectionandE orH statereflectioninasimilarway. We the singularity occurs at the origin and there is also an considerthe initial wavefunctionswithdensity matrix as endpoint of the ”branch cut” separating the 4π winding and no winding. (Note that α = π and α = π are a1 − ewqiuseivdailreencttiaonnditshseepcoanratitneudoiunstoinscerveaersaelosfhαeeitns)t.heInclfoacckt-, ψi = aa23 , a∗iaj =(cid:26)01/4othfoerrw{ii,sje} ={1,3},{2,4} (7)   we have checked that sinα, which is physically observ-  a4    able, depends on the direction from which the origin is approached but it does not show any singularity at the where ... denotes statistical average. The density matrix endpoint of the ”branch cut”. As we further increase is chosen so that the initial wavefunctions have a cor- C , the branch cuts extend toward the larger M and relation in the same bands. As seen in Eq. (7), here | | | | θ region, and approaches gradually to the negative M- we consider the initial wavefunctions with spin polariza- | | axis as shown in Fig. 2 (c). Here, one may think that tionalongx-axis. Usingtheseexpressions,wecanobtain whenmagneticfieldisappliedinthexy planeandopens expectation values of σx and σy, Pauli matrices in spin a gap, it will change our results when the Fermi level is space. inside the gap. However, magnitude of the gap is typi- Figure 3 displays expectation values of σ and σ as x y cally smaller than that of M( 10meV), and the Fermi a function of L/w for C = 1 eV and M = 0.01 eV. ∼ − − energyisshiftedfromthecrossingofthehelicaledgedis- Here, L and w are the length and width of the N, re- persions. Hence, our results would be practically robust spectively. An oscillatory dependence on L/w is seen against applied magnetic field in realistic systems. Note because the phase difference between up and down spin 4 (cid:9)(cid:11)(cid:10) (cid:28) (cid:21) (cid:18)(cid:19)(cid:20) ! (cid:17) (cid:22) " (cid:23)(cid:30)(cid:31) (cid:18)(cid:19)(cid:20) (cid:23) (cid:23)(cid:30)(cid:31) (cid:29) (cid:23) (cid:24)(cid:23) (cid:25)(cid:23) (cid:26)(cid:23) (cid:27)(cid:23) (cid:28)(cid:23)(cid:23) (cid:15) (cid:14) (cid:16) (cid:9)(cid:12)(cid:10) FIG. 3: (Color online) expectation values of σx and σy as a function of L/w for C = −1 eV and M = −0.01 eV. Inset shows the model. to rotateelectronspin by π. This may be a greatadvan- tage for application to nanotechnology. In summary, we studied a reflection problem at the N/QSH interface and showed that an electron injected fromthe normalmetalinN/QSHjunctions showsaspin (cid:9) (cid:10) dependent reflectionatthe interface andhence there ap- (cid:13) pearsaphasedifferencebetweenupanddownspinstates inthe reflectioncoefficients. The spin rotationangle can be as large as π in the QSH junction, because the he- ∼ lical edge modes resonantly enhance the spin rotation. In the QSH/N/QSH junction, multiple reflections at the interfaces increasethe phase difference. This also results in a remarkable spin rotation effect in this juction even when the results are averaged over incident angles. The proposed Fabry-Pérotlike heterostructure is experimentally accessible and could be used to unveil another as- pect of the QSH state: spin rotation effect. This work is supported by Grant-in-Aid for Scientific Research(GrantNo. 17071007,17071005)fromtheMin- istryofEducation,Culture,Sports,ScienceandTechnol- FIG. 2: (Color online) spin rotation angle α as a function of ogy of Japan and NTT basic researchlaboratories. T.Y. injection angle and M. (a) C = −0.08 eV, (b) C = −0.1 eV and (c) C = −1 eV. Note that α = π and α = −π are acknowledges support by JSPS. equivalent and the continuous increase of α in the clockwise direction is separated intoseveral sheets in (b) and (c). [1] S. Datta and B. 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