Spin edge helices in a perpendicular magnetic field
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Spin edge helices in a perpendicular magnetic field
S. M. Badalyan
1, 2, ∗ and J. Fabian Department of Physics, University of Regensburg, 93040 Regensburg, Germany Department of Radiophysics, Yerevan State University, 1 A. Manoukian St., Yerevan, 375025 Armenia (Dated: November 1, 2018)We present an exact solution to the problem of the spin edge states in the presence of equalBychkov-Rashba and Dresselhaus spin-orbit fields in a two-dimensional electron system, restrictedby a hard-wall confining potential and exposed to a perpendicular magnetic field. We find thatthe spectrum of the spin edge states depends critically on the orientation of the sample edges withrespect to the crystallographic axes. Such a strikingly different spectral behavior generates newmodes of the persistent spin helix—spin edge helices with novel properties, which can be tuned bythe applied electric and magnetic fields.
PACS numbers: 72.25.-b, 72.15. Gd, 85.75.-d
Introduction.
Edge states are a defining factor inmany prominent transport phenomena in condensed mat-ter physics. They emerge and are protected against scat-tering in the quantum Hall systems by applying a perpen-dicular magnetic field to a two-dimensional conductor, orare formed at the interfaces of topological insulators [1]by deforming the bulk band structure in the presence ofstrong spin-orbit coupling. The interplay of the threeeffects: the magnetic field quantization, the spin-orbitcoupling, and the confinement by sample edges, is yetlargely unexplored.Spin-orbit coupling is an important tool to manipu-late electron spins in solids by purely electric means [2–4]. In particular, the spin Hall effect [5] or the spinHall drag [6] allow to create spin accumulation acrossthe transport channels. The Bychkov-Rashba (BR) spin-orbit coupling [7], due to the structure inversion asymme-try, and the Dresselhaus (D) coupling [8], due to the con-finement quantization of the bulk spin-orbit interaction(SOI) dominate in many semiconductor heterostructures.Both BR and D couplings can be tuned by electric gates,asymmetric doping, or strain, allowing for efficient spincontrol [9–12]. If the strengths of the BR and D couplingsare equal, there can be long-lived, persistent spin helices(PSH) formed and protected against spin relaxation [13].This exciting phenomenon was recently observed [14, 15].Here we study the complex interplay of the SOI, cy-clotron effects of an external magnetic field, and thehard wall confinement in a generic zinc-blende two-dimensional electron system (2DES) grown along [001].We present an exact solution to the problem of the spinedge states for the equal BR and D SOI strengths. Thespectrum of the edge states is strikingly different if theedges run along [110] or [110] directions. Depending onthe relative sign of the BR and D couplings, in thosetwo symmetry directions either (a) the spectrum is spindegenerate or (b) the spin splitting of the edge channelsbecomes maximal, i.e. either spin polarized channels andspin current oscillations [16] are possible or not. We findthat in (a) the non-Abelian gauge field via the built-in spin-dependent phase factor generates spin edge helices(SEH) with a precession angle that depends on the trans-verse distance from the edge. In (b) the shifting prop-erty of the spectrum allows the existence of SEH with aprecession angle that depends on the distance along thepropagation direction. In strong magnetic fields the pre-cession angle of the SEH is quantized in case (a), whilea periodic helical structure, extended along the edge, isproduced in case (b). In weak magnetic fields we findinteresting new spin resonances when the cyclotron mo-tion is commensurate with the spin precession [17]. Ex-perimentally, a strong reduction of spin scattering ratetowards the sample edges should be observed.As an important application of our theory we proposeextending the experimental setup in Ref. 14 by exposingthe 2DES additionally to a perpendicular magnetic field B . According to the experimental findings from Ref. 14,the enhancement of the PSH by about two-order of mag-nitude occurs only at intermediate temperatures about T ∼
100 K. Its rapid drop with lowering of T shows thatthe spin Coulomb drag [18–20] via a strong increase ofthe diffusion coefficient destroys the PSH enhancement,for which a finite momentum Q is needed in B = 0. Inthe case of finite B we find that in order to excite SEHin the (b) configuration a finite momentum shift is stillneeded between the spin components but in the (a) con-figuration, where the energy is degenerate, the SEH ex-ists for Q = 0, i.e. without momentum difference alongits propagation direction. This should reduce the roleof spin Coulomb drag in suppressing the enhancementof SEH, which can be useful for spintronic applications.Notice that in three-dimension a nonequivalence in (1¯10)and (¯110) has been already observed by means of theelectron paramagnetic resonance and the electron-dipolespin resonance [17, 21]. Theoretical concept.
We consider electrons in a 2DESexposed to a perpendicular magnetic field B along [001].The electrons are additionally confined by an infinite po-tential V ( x ) = ∞ for x <
0. Then in the presence of theBR and D SOI the Hamiltonian is H = H + H SOI + V ( x ) k y ¢ k x ¢ y x H a L Μ=+ Μ=- @ D @ D - - - - - - k y ¢ k x ¢ yx H b L Μ=+ Μ=- @ D@ D - - - - - - FIG. 1. (Color online) Two physically different configurationsof the 2DES, restricted with a hard-wall confining potentialat x = 0 (filled areas). The spin edge states propagate inthe skipping orbits along [1¯10] and [110] axes, respectively,parallel and perpendicular to the direction of electron spins.The Fermi contours in the absence of the magnetic field areshown in the momentum plane with arrows indicating thedirections of spin. where H = ~π / m ∗ desribes a free particle in a quan-tizing magnetic field, m ∗ denotes the electron effectivemass and ~π = ~p − ( e/c ) ~A the kinetic momentum with ~p = − i ~ ~ ∇ ; the electron charge is − e . There are twopreferential directions in the ( x, y ) plane of 2DES: i) thedirection of the sample boundary along which edge statespropagate, and ii) the direction of the electron spin in thepresence of BR and D SOI of equal strength, determinedby the crystallographic axes. The relative orientation ofthese two directions determines two distinct configura-tions, shown in Fig. 1. In these configurations we choosethe ( x, y ) coordinate system such that the sample bound-ary is always along y . Then in the configuration (a) inFig. 1, in which the axes x and y are along [110] and[110], we have H SOI = ( α R − α D ) π y ˆ σ x − ( α R + α D ) π x ˆ σ y . (1)In the configuration (b), the axes x and y are along [110]and [110] and the Hamiltonian is H SOI = ( α R + α D ) π y ˆ σ x − ( α R − α D ) π x ˆ σ y . (2)Here α R and α D are, respectively, BR and D spin-orbitcoupling constants and ˆ σ x , ˆ σ y are the Pauli matrices.We consider magnetic fields which are strong enough toquantize the electron spectrum but weak enough to causemuch smaller Zeeman splitting than the SOI induced en-ergy splitting [16, 22].We choose the Landau gauge with ~A ( x ) = (0 , xB, ψ ( ~r ) = e ik y y χ k y ( x ) , ~r = ( x, y ) , (3)reduce the two-dimensional Schr¨odinger equation to theeffective one-dimensional problem in the x direction. The transformed Hamiltonian U † HU , generated by the globalunitary transformations U a = 1 √ (cid:18) i − i (cid:19) and U b = 1 √ (cid:18) − (cid:19) , (4)becomes diagonal in the (a) and (b) configurations sothat in the case of α R = α D = α the wave functions χ k y ( x )satisfy the following equation "(cid:18) ddx + iγa ˆ σ z (cid:19) + µ + 12 − ( x − X ( k y ) − γb ˆ σ z ) χ k y ( x ) = 0 . (5)Here the coefficients a and b are given by a = 1 , b = 0 , for the (a) configuration , (6) a = 0 , b = 2 , for the (b) configuration . (7)In Eq. 5 µ = ν + γ and we express the total electron en-ergy E → ( ν + 1 / ~ ω B and the length x → xl B / √ ~ ω B = ~ eB/m ∗ c , and themagnetic length, l B ≡ p ~ c/eB . The dimensionless SOIcoupling constant γ = √ α/v B , where the cyclotron ve-locity v B = ~ /m ∗ l B , while the dimensionless coordinateof the center of orbital rotation X ( k y ) = √ k y l B .In the (a) configuration the SOI with equal BR andD strengths induces a finite non-Abelian gauge field thatdepends on the spin orientation, keeping the cyclotronrotation center unshifted. In contrast, in the (b) configu-ration the SOI induces only a spin-dependent shift of thecyclotron rotation center.With the unitary transformation χ k y ( x ) = exp ( − iγax ˆ σ z ) φ k y ( x ) , (8)we can eliminate the non-Abelian gauge field and mapthe SOI problem of the equal BR and D strengths to theSOI free, shifted edge state problem as follows: h µ ( x − sγb ) φ sk y ( x ) = 0 . (9)Here we introduce the operator h ν ( x ) = d dx + ν + 12 − ( x − X ( k y )) ! (10)and s = ± ↑ and ↓ states in the new spinbasis, created by (4). The general solution of Eq. 9 isgiven in terms of the parabolic cylindric functions D µ ( x )so that the spin edge states are given by ψ sk y ( ~r ) = exp ( − isγax + ik y y ) D µ ( x − X ( k y ) − sγb ) . (11)For sufficiently large positive X ( k y ) the solution (11)recovers the exact spectrum of the dispersionless bulkLandau levels, E sl ( γ ) = (cid:0) l + − γ (cid:1) ~ ω B with the index l = 0 , , . . . [23]; this is valid in both configurations sincethe index µ is the same. The shift of all bulk Landau lev-els due to SOI is independent of spin so they remain spindegenerate. -2 0 2 4 601234 (a) (b) E ne r g y E [ B ] Center of orbital rotation X ( ky ) FIG. 2. The energy spectrum of spin edge states in thepresence of Bychkov-Rashba and Dresselhaus SOI of equalstrengths. The dashed and solid curves correspond to the (a)and (b) configurations in Fig. 1.
Energy spectrum.
The energy spectrum of thespin edge states is obtained by letting the wavefunctions (11) vanish at the sample boundary: D µ ( ν,γ ) ( x − X ( k y ) − sγb ) (cid:12)(cid:12) x =0 = 0. The calculatedenergy branches, E sl ( k y , γ ), are shown in Fig. 2. As seenin the (a) configuration the energy spectrum remainsspin degenerate for all values of k y so that the spinedge states at the Fermi energy, E F , are not separatedin space. The wave functions (11) contain finite spin-dependent phase factors ( a = 1), which describe theprecession of spins with the opposite rotations along theelectron propagation axis y .In contrast, the energy spectrum in (b) develops, fora given principal quantum number l , two spin branches.The degeneracy is lifted already for positive X ( k y ) ∼ | X ( k y ) | fornegative X ( k y ). The spin-dependent phase factor in(11) vanishes identically ( a = 0) and the effect of equalBR and D SOI on the spin edge states is reduced to thespin-dependent shift of the energy branches in the mo-mentum space. At certain values of E F there exist twospatially separated current carrying states with oppositespins, whose directions are locked globally by the geom-etry of the configuration (b).Thus, the spectrum in Fig. 2 shows that the exter-nal magnetic field creates edge states along the [1¯10] and[110]. The linear motion of the edge states via the BRand D SOI of equal strengths induces an effective mag-netic field along [110] and [1¯10], respectively, that is ei-ther perpendicular or parallel to the direction of the spinsthereby keeping the spin degeneracy or resulting in themaximal spin-splitting, respectively, in (a) or (b). Spin edge helices.
The energy bands of the spin edgestates possess a shifting property along y , E ↓ l ( k y , γ ) = E ↑ l ( k y − Q, γ ) , (12) where the shifting wavenumber Q = √ γb/l B is finite in(b) and zero in (a). As in the zero field [13], we introducethe following operators in the transformed spin basis S − Q ( ~r ) = ψ ↓ k y ( ~r ) † ψ ↑ k y − Q ( ~r ) , S + Q ( ~r ) = ψ ↑ k y − Q ( ~r ) † ψ ↓ k y ( ~r ) , (13) S z ( ~r ) = ψ ↑ k y ( ~r ) † ψ ↑ k y ( ~r ) − ψ ↓ k y ( ~r ) † ψ ↓ k y ( ~r ) , which commute with the system Hamiltonian owing tothe shifting property (12) and the SU(2) symmetry ofthe system [13]. The non-diagonal operators S ± Q ( ~r ) = exp ( ± iaγx ± ibγy ) D µ [ x − X ( k y ) + bγ ] (14)represent long-lived spin edge helices of 2DES in the pres-ence of a perpendicular magnetic field. Here the coordi-nate y is also dimensionless, y → yl B / √
2. As seen fromEq. 14, in contrast to the spin edge states, the helicaledge modes have finite spin-dependent phase factors inboth configurations. Going back to the initial spin basiswe see that in the (a) configuration ( a = 1 and b = 0)the factor exp ( ± iγx ), existing also for the edge states,describes the spin precession in ( x, z ) plane with the pre-cession angle ϑ ( x ) = 4 γx , depending on the transverse x coordinate, along which the helices are confined by themagnetic field via the factor D µ [ x − X ( k y ) + bγ ] . Onthe contrary, in the (b) configuration ( a = 0 and b = 2)the factor exp ( ± iγy ) is inherent only to the SEH. Thisfactor arises from the plane wave functions in Eq. 3 due tothe shifting property (12) and describes the spin preces-sion in ( y, z ) plane with the precession angle ϑ ( x ) = 4 γy ,depending on the y coordinate along the free propagationdirection of the SEH. Thus the combined effect of theperpendicular magnetic field and the confining potentialon the SEH critically depends on the orientation of theedges of 2DES relative to the crystallographic axes; as wewill see it depends also on the strength of the magneticfield.In quantizing magnetic fields the spin helices arestrongly localized along the transverse direction x aroundits average position x l ( k y ). In the limit of large nega-tive k y , x l ( k y ) is approximately independent of k y andtakes discrete values x l in the different channels l [16, 27].Therefore, in the (a) configuration the precession angleis quantized around ϑ l = 4 γx l for the spatially separatededge channels l (see bottom of Fig. 3.a). Meanwhile inthe (b) configuration the precession of spin by an angle ϑ l = 4 γy develops a spatially periodic structure alongthe y direction (see bottom of Fig. 3.b), similarly to theusual PSH in an infinite 2DES in the magnetic field freecase [13, 24–26]. Thus, by switching between the (a) and(b) configurations, i.e. by tuning the BR and D couplingstrengths either to the α R = α D or α R = − α D case, onecan realize a selection mechanism between the two alter-natives of the quantized spin edge helices and of the freespin edge helices. FIG. 3. Spin edge helices in a perpendicular magnetic field.Figs. (a) and (c) (Figs. (b) and (d)) correspond to the (a) con-figuration (the (b) configuration). Figs. (a) and (b) (Figs. (c)and (d)) correspond to the limit of strong (weak) magneticfields. The bottom (top) of each figure represents the heli-cal structures in the edge channels (in the quasibulk Landaustates).
In the limit of large positive k y , the electrons are lo-calized in quasibulk Landau states around their rota-tion center X ( k y ) ≫
1, which increases linearly with k y [16, 27]. However, for a given k y one can neglectweak oscillations of the spin precession angle ϑ l sincein strong fields l B is much smaller than the spin-orbitlength, λ SO = 1 / mα , and the variation of ϑ l duringthe period of the cyclotron rotation is of the order of γ = 2 mαl B ≪
1. Thus, the precession angle in the con-figuration (a), ϑ l ≈ γX ( k y ), is independent of y andvaries only along x with k y (see top of Fig. 3a) while inthe (b) configuration, ϑ l ≈ γy is independent of x butvaries along y (see top of Fig. 3b).In weak fields and for l ≫
1, the quasi-classical de-scription is valid and the x coordinate oscillates in time as x = X ( k y )+ √ ν + 1 cos ω B t . For the edge states X ( k y )can be negative, −∞ < X s ( k y ) √ ν + 1 so that in the(a) configuration the spin precesses along x by an an-gle 0 ϑ ϑ where ϑ = 4 γ √ ν + 1. Meanwhile forthe bulk Landau states X ( k y ) > √ ν + 1 and the preces-sion angle in the (a) configuration varies within the range ϑ s − ϑ ϑ ϑ s + ϑ . Therefore, in the magnetic fieldsand electron concentrations such that 4 γ √ ν F + 1 = π/ ν F + 1 = E F / ~ ω B , the spin resonance effect takesplace as shown in the top of Fig. 3c: the spin makesone full precession each time when the electron makes afull cyclotron rotation. Recently, the spin resonance phe-nomenon, driven by the SOI and perpendicular magneticfields, has been experimentally demonstrated in Ref. 28by observing a strong suppression of spin relaxation. Asseen in the top of Fig. 3d and 3c the spin resonance forthe Landau states in the (b) configuration differs fromthat in the (a) by a simple rotation of the cyclotron orbitbecause in the (a) ϑ = 4 γy depends only on y while inthe (b) ϑ = 4 γx depends only on x .For large negative k y the physical picture of the spinresonance in the (a) and (b) configurations differs essen-tially ( cf. the bottom of Figs. 3c and 3d). In (a) ϑ = 4 γx depends only on the x , along which the electron motionoscillates within a finite range. Therefore, the preces-sion angle is independent of the electron motion alongthe edge. In contrast, in (b) ϑ = 4 γy depends on the y coordinate, along which the electron propagates freely.Therefore, in (b) the spin precession generates a spatiallyperiodic structure of the SEH along y and as seen in thebottom of Fig. 3d in the resonance case the spin changesits sign each time the electron makes a half circle in itsskipping orbit.In conclusion, we present a theory of persistent spinedge helices, which exhibit novel features, tuneable byelectric and magnetic fields. We show that either a peri-odic structure of spin edge helices along the sample edgesor a helical structure with a quantized precession anglealong the transverse direction is realized.We thank E. Rashba, G. Vignale, and M. Glazov foruseful discussions and acknowledge support from the EUGrant PIIF-GA-2009-235394 and the DFG SFB 689. ∗ [email protected][1] M. Z. Hasan and C. L. Kane, arXiv:1002.3895 (unpub-lished).[2] D. Awschalom and N. Samarth, Physics , 50 (2009).[3] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I.ˇZutiˇc, Acta Phys. Slov. , 565 (2007).[4] I. ˇZutiˇc, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004).[5] M. I. D’yakonov and Perel’, JRTP Lett. , 144 (1971).[6] S. M. Badalyan and G. Vignale, Phys. Rev. Lett. ,196601 (2009).[7] Yu. Bychkov and E. I. Rashba, JETP Letters , 78(1984).[8] G. Dresselhaus, Phys. Rev. , 580 (1955).[9] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. , 146801 (2003).[10] P. Stano and J. Fabian, Phys. Rev. Lett. , 186602(2006).[11] S. M. Badalyan, A. Matos-Abiague, G. Vignale, and J.Fabian, Phys. Rev. B , 205305 (2009).[12] S. M. Badalyan, A. Matos-Abiague, G. Vignale, and J.Fabian, Phys. Rev. B , 205314 (2010).[13] B. A. Bernevig, J. Orenstein, and S. C. Zhang, Phys.Rev. Lett. , 236601 (2006).[14] J. D. Koralek, C. Weber, J. Orenstein, B. A. Bernevig,S. C. Zhang, S. Mack, and D. D. Awschalom, Nature(London) , 610 (2009).[15] J. Fabian Nature , 580 (2009).[16] V. L. Grigoryan, A. Matos Abiague, and S. M. Badalyan,Phys. Rev. B , 4853 (2000).[19] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein,J. Stephens, and D. D. Awschalom, Nature , 1330(2005).[20] S. M. Badalyan, C. S. Kim, and G. Vignale, Phys. Rev. Lett. , 016603 (2008).[21] M. Dobrowolska et al. , Phys. Rev. Lett. , 134 (1983).[22] S. Debald and B. Kramer, Phys. Rev. B , 085344(2005).[24] Ming-Hao Liu, Kuo-Wei Chen, Son-Hsien Chen, andChing-Ray Chang, Phys. Rev. B , 235322 (2006). [25] Son-Hsien Chen and Ching-Ray Chang, Phys. Rev. B ,045324 (2008).[26] Jyh-Shinn Yang, Xiao-Gang He, Son-Hsien Chen, andChing-Ray Chang, Phys. Rev. B , 085312 (2008).[27] S. M. Badalyan and F. Peeters, Phys. Rev. B , 155303(2001).[28] S. M. Frolov, S. Lscher, W. Yu, Y. Ren, J. A. Folk, andW. Wegscheider, Nature458