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The spin-Hall effect and inverse spin-Hall effect in a diffusive system L. Y. Wang1, Collaborators: C. S. Chu2 and A. G. Mal’shukov3 1Department of Physics, Fu-Jen Catholic University, New Taipei City, Taiwan 2Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwann 3Institute of Spectroscopy, Russian Academy of Science, Moscow, Russia Highlight Motivation and Highlight The Rashba-type spin-orbit interaction (RSOI) provides the needed tunable capability for spin physics by pure electric means. biased-gate vg InGaAs 2DEG InAlAs Nitta. et al., Phys.Rev.B 60, 7736(1999) Highlight Difficulty confronting RSOI: The effect of RSOI on spin accumulation Sz is quenched in the diffusive regime (lso>>le). lso is the spin relaxation length and le is the electron mean free path. Linear-k SOIs, Rashba or Dresselhaus, do not contribute to spin accumulation when there are background scatterers. J.I. Inoue et al., Phys. Rev. B 70, 041303 (2004) E.I. Rashba, Phys. Rev. B 70, 201309 (2004) O. Chalaev et al., Phys. Rev. B 71, 245318 (2005) E.G. Mishchenko, et al., Phys. Rev. Lett. 93, 226602 (2004) A.A. Burkov, et al, Phys. Rev. B 70, 155308 (2004) O.V. Dimitrova, Phys. Rev. B 71, 245327 (2005) R. Raimondi et al., Phys. Rev. B 71, 033311 (2005) Highlight Efforts to recover the RSOI effects In the mesoscopic ballistic regime: Coherence quantum interference accumulation finite spin B. K. Nikolić et al., Phys. Rev. Lett. 95, 046601 (2005) J. Li and S. Q. Shen, Phys. Rev. B 76, 153302 (2007) P. G. Silvestrov et al., Phys. Rev. Lett. 102, 196802 (2009) Highlight Residual RSOI effects in the diffusive regime : Only along the sample-electrode interfaces was the spin current found to be nonzero. This is understood from the way the spin current vanishes in the bulk, when exact cancellation occurs between two terms, one related to spin polarization and the other related to the driving field. I yz 2 D S z 2hkF vF S y 2 y N 0 e 2 hk2F * m E E. G. Mishchenko et al., PRL, 93, 226602 (2004) Highlight Residual RSOI effects in the diffusive regime : Only along the sample-electrode interfaces was the spin current found to be nonzero. This is understood from the way the spin current vanishes in the bulk, when exact cancellation occurs between two terms, one related to spin polarization and the other related to the driving field. The exact cancellation no longer holds near the electrode interfaces when the driving field has reached its bulk value but the spin polarization has not. E. G. Mishchenko et al., PRL, 93, 226602 (2004) Highlight This work: It is important to see whether the RSOI alone can still contribute to spin accumulation Sz in the diffusive regime. Question we ask in this work: • Is it possible to recover the RSOI effect on the spin accumulation Sz in the diffusive regime ? • And in any location we specify ? Highlight Our finding: The RSOI effect on spin accumulation can be restored in a nonuniform driving field. Spin accumulation around a void Spin accumulation around a void at an edge S ztot (1/ m2 ) 3 1 2.5 0.5 2 0 1.5 1 -0.5 0.5 0 -3 -1 -2 -1 0 1 2 3 Outline 1. Introduction: spin-Hall effect (SHE) 2. Spin diffusion equation for nonuniform driving field 3. Nonuniform driving field around a circular void in the bulk. 4. Nonuniform driving field around a edge semicircular . 5. Inverse spin-Hall effect for Rashba spin-obit interaction Introduction: spin-Hall effect Introduction A driving field results in spin accumulation at the lateral edges of the sample. No magnetic field is needed but SOI is crucial. spin-up -e E z-polarized spins -e spin-down Physical origins of the SOI (1) Extrinsic SOI: arises from SOI impurities. (2) Intrinsic SOI: arises from the lack of structure inversion asymmetry (SIA) or bulk inversion asymmetry (BIA). Introduction Experimental status: extrinsic SHE The “Extrinsic SHE” arises from SOI impurities: Experimental result in n-doped semiconductors. E 77 m Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science, 306, 1910 (2004). Intrinsic spin-Hall effect in the ballistic regime Rashba-type SOI Green arrows: wavevector Red arrows: effective magnetic field py py px px J. Sinova, et al PRL 92, 126603 (2004) However, there is no z-polarized spin polarization induced in the diffusion limit for Rashba SOI. Introduction Intrinsic spin-orbit coupling p2 H h Vimp. r 2m Effective magnetic field: 1. hk (Rashba SOI) k zˆ k y xˆ k x yˆ 2. hk (Dresselhaus SOI) k x k 2y 2 xˆ k y k 2x 2 yˆ for growth direction in [001] Effective magnetic field 1. hk Electron momentum 2. hk Introduction Electric generation of spin polarization in RSOI k dependent effective magnetic field: hk ẑ k where is averaged over the electron distribution given -τehk E by a shifted Fermi sphere f , k = δ F - * m τ : momentum scattering time ky E The Edelstein bulk spin density in unit of , given by e SB N 0 zˆ E N 0 denoting the density of state per spin. kx SBy N 0 e yˆ V. M. Edelstein, Solid State Commun. 73, 233 (1990) Introduction Spin accumulations induced by the spin current z polarized spin current along yˆ I 2 D S z 2hkF vF S y 2 y z y (1) Spin diffusion N 0eE 2 hk2F (2) Spin precession vF2 diffusion constant : D 2 m* (3) Induced by spincharge coupling via a driving field B S For RSOI, the bulk spin polarization y results in exact cancellation between (2) and (3). Outline 1. Introduction: spin-Hall effect (SHE) 2. Spin diffusion equation for nonuniform driving field 3. Nonuniform driving field around a circular void in the bulk. 4. Nonuniform driving field around a edge semicircular . 5. Inverse spin-Hall effect for Rashba spin-obit interaction The diffusive regime k F l 1: Fermi wave length F impurity l F Mean free path l spin relaxation length Lso l impurity Mean free path l Lso Spin diffusion regime The four densities t i Di r , t Tloop [ Dˆ i r , t Sloop down , up ] i G r , r , t , t By using Keldysh Green’s function and consider the linear response approximation to treat the external electric field: G † ˆ ˆ r , r , t , t i Tl r , t r , t i H ' d loop ˆ r'' , r'' , ˆ r'' , H ' d 2 r j imp Keldysh Green’s function is represented by retarded and advanced Green’s functions: dN F r d ' j r'' , G r , r'' , ' Ga r'' , r , ' 2 d dN F r d ' j 2 r d r j r , [G r , r'' , ' G r'', r , ' 2 d a G r , r'' , ' Ga r'' , r , '] G r , r , j d 2 r The densities are obtained by Di r , Di0 r , d 2 r ' ij r , r ', j r ', j Response function is given by d ' dN F ij r , r' , i Tr G a r' , r , i G r r , r' , ' j 2 d ' Di0 q , 2 N 0 ( EF ) i ( q , ) D00 2 N 0 ( EF ) 0 (q , ) Electric potential energy Di0 x , y , z 0 Retarded Green's function: G , k r p i hp Advanced Green's function: G , k a 1 = with is the scattering time 2 i h 2 p 2 p ; p i hp 2 i h p p 2 imp . SHE accumulation Impurity averaging by ladder series p + pq p' p ' p p ' q p , ', q Building block p pq + + +….. pq ci U v 2 r a G p , ' G p q, ' p ci impurity concentration v system volume U impurity strength Diffusion equation in stationary state Di r , Di0 r , d 2 r ' ij r , r ', j r ', j [1 ]il Dl Dl0 i il Dl0 we know that: il 1 i Dq 2 il (...)il [1 ]il Dl Dl0 i il Dl0 i 1 Dl0 [1 ]il D D i D 0 l l stationary state: =0: 0 l D il Dl Dl0 D il Dl Dl0 0 It is convenient to introduce the diffusion propagator D il 1 il F hk in the diffusion regime il , ', q 1 ci U 2V 2 i a l r Tr G p q , ' G p, ' p Finally, we obtain that: il , ', q il 1 i D q 2 ...........h 0 iq 4 2 ilm [v F hPmF ]................qh 4 2 hP2F il nki nkl ..............h 2 i 4 3 q h3p nip p i 0 .........qh3 (Spin-charge coupling) Diffusion equations: density of state at EF for stationary state ( =0): Dil Dl Dl0 0, with Dl0 2 N 0 ( EF ) 0 l 0 homogeneous electric field E along xˆ 0 eEx, (e 0) Dil r il D 2 R ilm m il M i 0 1 2 D vF : diffusion constant 2 R ilm 4 i ln [hFn vFm ] : spin precession term due to SOI field il 4 hF2 il nki nkl : D'yakonov Perel' relaxation M i 0 4 2 hF3 nip hpi / | hp | nip p : spin-charge coupling without the magnetic field Spin precession due to either the SOI field or the external magnetic field: BSOI D’yakonov Perel’ (DP) spin relaxation: Spin relaxation due to spin precession between collisions with no spin-flipping impurities. A change in momentum changes the precession axis of the spin. B’ B spin p p’ Scattering from a normal scatterer : no spin-flipping D xx Dx D xy Dy D xz Dz D x 0 D00 0, yx yy yz y0 0 D Dx D Dy D Dz D D0 0, D zx D D zy D D zz D D z 0 D 0 0, x y z 0 Di 2 Si , D00 2 N 0 e x, y Rashba SOI case: hk k zˆ k y xˆ k x yˆ D xx D 2 xx / 2 D 2 2hF2 / 2 , D yy D 2 yy / 2 D 2 2hF2 / 2 , D zz D 2 zz / 2 D 2 4hF2 / 2 , D xy D yx 0, R xzx 2hF vF R yzy 2hF vF yz D ,D , x x y y xz zxx zyy 2 h v 2h v R R D zx F F , D zy F F , x x y y D x0 2 hF2 2 2 hF2 2 y0 z0 , D , D 0 3 3 y x Spin diffusion equations for Rashba spin-orbit interaction: 2 2 2 2 h 2 h v 2 h 2 F F F F Sz 2 N 0 e x, y 0, D S x 2 S x 3 x y 2 2 2 2 h 2 h v 2 h 2 F F F F Sz 2 N 0 e x, y 0, D S y 2 S y 3 y x 2 4 h 2hF vF 2hF vF 2 F D S z 2 S z Sx S y 0, x y In the bulk, applying a homogeneous electric field Exˆ All spin densities becomes position-independent in the bulk for stationary state. x, y Ex 2 2 2 2 h 2 h v 2 h 2 F F F F D S S S 2 N 0 e x, y 0, x x z 2 3 x y 2 2 2 2 h 2 h v 2 h 2 F F F F D S S S 2 N 0 e x, y 0, y y z 2 3 y x 2 4 h 2h v 2h v D 2 S z F2 S z F F Sx F F S y 0, x y The bulk spin densities are: S x 0; S z 0; Edlestein spin polarization S yED 2 N 0 eE . RSOI field k SDE for nonuniform field Spin diffusion equation for nonuniform driving field RSOI Hamiltonian : hk and hk zˆ k xx R xzx 2 D S x 2 S x yy R yzy 2 D S y 2 S y zz R zxx 2 D S z 2 S z M x0 0 Sz D0 0, x 2 3 M y0 0 Sz D0 0, 3 y 2 R zyy Sx S y 0, x y spin precession arising from diffusive flow: R ilm 4 iln hkn vFm D'yakonov-Perel' spin-relaxation rates: il 4 hk2 il nki nkl spin-charge coupling terms: M i0 nki 4 h k 2 3 k nk hk / | hk | D00 2 N 0 e : effective local equilibrium density (The overlines denote the angular averaging) hk hk2 hk3 q SDE for nonuniform field For uniform driving field, the spin-charge coupling term, given by i0 D 00 , has i0 kept up to the order h 3F q . This is appropriate because is position independent. il 2 N 0 i r 0 l a 0 Tr G p', + ' G p' - q,' , p' G r / a 0 : retarded (advanced) Green's functions averaged over impurity configuration. 1/ 2 , 0 1, x,y,z x , y , z However, for the case of nonuniform driving field, we need to include terms in i0 D 00 that are higher order in q. SDE for nonuniform field xx R xzx 2 D S x 2 S x yy R yzy 2 D S y 2 S y zz zxx R D 2 S z 2 S z M x0 0 Sz D0 0, 3 x 2 M y0 0 Sz D0 0, 3 y 2 R zyy Sx S y 0, x y To identify additional expansion terms in i0 for nonuniform driving field, we note first of all that SEd is of order hF q . If SEd is to satisfy the spin diffusion equations, all the terms involving Si will be replace by Si SEd . For spin diffusion equations, the terms of order hF q 3 and hF2 q 2 will be needed. Outline 1. Introduction: spin-Hall effect (SHE) 2. Spin diffusion equation for nonuniform driving field 3. Nonuniform driving field around a circular void in the bulk. 4. Nonuniform driving field around a edge semicircular . 5. Inverse spin-Hall effect for Rashba spin-obit interaction Nonuniform field with a void Nonuniform driving field around a circular void in the bulk We study the spin accumulation in the vicinity of a circular void, with radius R0~lso, where the driving field becomes nonuniform. Here lso is the spin relaxation length, and lso >> le, the electron mean free path. E 0 j has to satisfy the steady state condition j = 0 and the boundary condition j = 0. E 0 : electric conductivity E E0 xˆ E0 R0 / cos 2 xˆ sin 2 yˆ 2 R0 y x I Nonuniform field with a void In-plane spin currents can recover the RSOI-induced spin accumulation The conventional restoration of RSOI-induced spin accumulation is associated with the nonvanishing of out-ofz plane spin current I n . The key process we find is associated with the in-plane spin currents, I nx or I ny , and with the way they vanishes at the void boundary. (n denotes the flow direction normal to the void boundary) Spin current Spin current operator: 1 1 J il Vl i iVl please note that spin factor have not included 2 2 H p2 velocity operator Vl i i hp p p 2m 1 1 1 Vl i iVl vl i hpi 2 2 2 we consider the spin-Hall current in z-direction:(hpz =0) J il J il vl i Next, we only consider this term contribution in spin-Hall current. I il r , t Tl J il r , t Sl down , up 1 ' {lTr i G r ', r , t , t l' Tr i G r , r ', t , t }r 'r 2m G r , r , j d 2 r d ' j r , G r , r , ' G r , r , ' 2 Useful formula: G r , r , N F G r r , r , G a r , r , G G r G G G G a d ' j j r , N F ' N F ' 2 r G r , r , ' Ga r , r ', ' G r , r , d 2 r d ' j a j r , {N F ' Ga r , r , ' G r , r ', ' 2 r N F ' G r , r , ' Gr r , r ', '} d 2 r Connecting the spin densities to the spin current Si ijy b I y 2 D R S j S j 2 I sH iz y where spin-Hall current: i y I sH hk R S 4 eEN v hk k x z zjy b j 2 y 0 F Nonuniform field with a void In-plane spin currents can recover the RSOI-induced spin accumulation i x or y I 2 D Si Rizi S z ˆ iˆ diffusion term Related to spin accumulation In-plane spin current: no driving field term Rizi denotes the precession of Si into S z when it flows along iˆ. ˆ iˆ denotes the projection of the flow. Rashba SOI governs the symmetry of Rijk such that Rizj =0 for i j. Nonuniform field with a void In-plane spin currents can recover the RSOI-induced spin accumulation In-plane spin polarization is given by S S S Ed Edelstein-like spin polarization for the case of nonuniform driving field S Ed r N0 e / zˆ E N0 : density of state, : electron scattering time, and e>0 In-plane spin currents can recover the RSOI-induced spin accumulation Boundary conditions: E I R0 2 D Si Rizi S z ˆ i ˆi 0 R0 R0 y x If Sz=0, spin current indicates S|| is position-independent.(Wrong!) S Si x or y alone fails to satisfy the boundary condition, because of its radial dependence results in generation of spin accumulation S z and Si . Spin diffusion equation for nonuniform driving field SEd alone cannot satisfy the boundary condition, thus the spin density have an additional S so that S SEd S . Spin diffusion equation for S : 2 S x 4S x 4 x S z 0 2 S y 4S y 4 S z 0 y 2 S z 8S z 4S x 4S y 0 2 2 Dimensionless length in unit of spin relaxation length lso D so , so 2 hF S j q = a jq eim H m1 q m H m1 q is the first kind Hankel function. Spin diffusion equation for nonuniform driving field The asymptotic behavior required of S jq leads to Im >0. The eigen-modes q : 1 2i 2 2 i 2 7 * 3 2 In terms of these modes S j is expanded in the form 2 3 S j d a j 0 q 1 m The coefficient a j q q im 1 H e . m q is determined by the boundary condition! Boundary conditions related to spin currents Spin current hp I 2 D j Si R S x R S y R S z 4 v hp eN 0l r kl z l x, y i j Spin diffusion ixj iyj izj Spin precession 2 xyi j F It is contributed from driving z I field and is nonzero for j only. The boundary conditions are S x 2cos S z 2 E / sin 2 0 0 S y 2sin S z 2 E / cos 2 0 0 I i 0 0 =0 S z 2cos S x 2sin S y 0 0 / , 0 R0 / lso , and E eE0 N 0 / E The analytic approach is assumed ax1 it x sin 2 2 az t z cos 3 * 2 az az t x is real and t z is complex tx tz Substituting into boundary conditions tx , tz Im YZ * 4 E 8H11 z1 Im XY * 1 z1 H11 z1 Im ZX * 2 1 z1 H 21 z1 Im ZY * 1 H1 z1 Im 8Y * 1 z1Z * E 0 8H11 z1 Im XY * 1 z1 H11 z1 Im ZX * 2 1 z1 H 21 z1 Im ZY * zi i 0 , g 2 2 / 22 4 , X 2 H1 z1 2 g 2 2 H 2 z2 , 1 Y g 2 2 1 H11 z2 , Z 2 2 4 g 2 H11 z2 1 Spin density Si We have obtained a j q and from the expression S j d a jq H m1 q eim 2 0 3 q 1 m S 2 it H 1 2 Im t g H 1 sin 2 x 2 1 z 2 2 2 x 1 1 S x cot 2 S 2 it H 2 Im t g H y x 0 1 z 2 0 2 S z 4 Im t z H11 2 sin ΔSx and ΔSz are even parities in φ, ΔSy is odd parity in φ . ΔSz 0 confirms that RSOI’s contribution to spin accumulation can be restored in a nonuniform driving field . Total spin density Si We can calculate the total spin density: Sx SEd x S x Sy SEd y S y S z S z GaAs material parameters: electron density: n 11012 cm 2 effective mass: m* 0.067m0 Rashba SOI constant: 0.3 10-12 eVgm electron mean free path: le 0.43m spin-relaxation length: lso 3.76m Total spin density Si=x,y E E Fig.1: Total spin densities Sx, Sy with radius R0=0.5lso. Total spin density Sz Fig.2: Total spin densities Sz is shown the dipole form normal to a driving field E with radius R0=0.5lso. The spin accumulation is restored due to nonuniform driving field. Number of electron spin Sz d Fig.3: Number of out-of-plane electron spin within a circular area the same size as void. The center is shifted by a distance d from the void center along =/2. probe beam size Spin currents S S I x x 2Sz xˆ x yˆ x y Sy Sy I 2Sz yˆ xˆ x y y Sz Sz 02 02 I 2Sx 2E 2 sin 2 xˆ 2Sy 2E 1 2 cos 2 yˆ x y z Summary I: We have demonstrated that nonuniform driving field can give rise to spin accumulation in a diffusive Rashba-type 2DEG. The nonuniform driving field can be realized by patterning the sample such as with a circular void in the sample or with a semicircular void at the sample edge. The physical process is identified to be associated with spin current for the in-plane spin at the boundary. Outline 1. Introduction: spin-Hall effect (SHE) 2. Spin diffusion equation for nonuniform driving field 3. Nonuniform driving field around a circular void in the bulk. 4. Nonuniform driving field around a edge semicircular . 5. Inverse spin-Hall effect for Rashba spin-obit interaction Spin accumulation for edge-semicircular void (ESV) by nonuniform driving field y R0 x E E0 xˆ E0 R0 / cos 2 xˆ sin 2 yˆ 2 The nonuniform driving field E() for the circular void satisfies also the additional boundary condition jy=0 imposed by the ESV case at the sample edge, =(0, ) . Thus E() holds in the two cases. To satisfy the additional boundary condition for spin current at sample edge, a further additional spin accumulation SESV is needed, leading to Stot SEd S SESV Spin accumulation for ESV y x -R0 R0 For the case of ESV, the void spin density SEd S no longer satisfies the ESV boundary condition. The ESV spin density becomes SEd S SESV . Ed S S will induces new source terms The void spin density Ivoid at y=0 through the spin currents. Spin diffusion equation for ESV ESV 2 ESV ESV S x 4S x 4 S z 0 x ESV 2 ESV ESV S y 4S y 4 S z 0 y 2 ESV ESV ESV ESV S z 8S z 4 S x 4 S y 0 x y y x y=0 First we assume that the sample edge is translation invariant at y=0 such that we can transform the spin diffusion equation into the kspace. The ESV-induced spin density can be expressed in plane wave form with a decay mode in y direction: 3 SiESV dk j 1 i x, y, z ai0 k eikx e y , j k real , complex, Re[ ] 0 Substituting SESV into the spin diffusion equation, one can solve the eigen-modes: k 2 2 4 S xESV 0 4ik ESV 2 2 0 k 4 4 S y 0 2 2 4ik S zESV 4 k 8 S xESV dkeikx ax(1)0 k e 1 y az(2)0 k g 2e 2 y az(3)0 k g 2* e 3 y a k e S yESV dkeikx ax(1)0 k g1e 1 y az(2)0 k g3e 2 y az(3)0 k g3*e 3 y S zESV dkeikx (2) z0 2 y az(3)0 k e 3 y 1 k 2 4, 2 k 2 2 2 7i , g1 = 3 2* k k 2 2 2 7i , g 2 3i 7 ,g 3 3 7i 2 8 8 k 4 ik Procedure to calculate the spin density SESV generate a additional source induce SESV(0) but I ri x, y , z r R0 0 at ESV boundary Applying axiliary source to obtain Saux leading to I ri x, y , z 0 r R 0 The additional source terms induced by SEd S at y=0 for the semi-infinite system ESV (0) Ed add S S S I x x x x y y y 0 y 0 S ESV (0) 2S ESV (0) S S Ed 2S 0 z y z y y y y y 0 y 0 2 ESV (0) ESV (0) Ed add 0 Sz 2S y S 2 S S 2 E 1 cos 2 I z y y z 2 y y y 0 y 0 S ESV (0) satisfies the boundary condition at y 0 Iadd x -R0 Iadd z R0 -R0 I yi x , y , z y 0 0 y R0 x The spin current related to SESV(0) can not satisfy the ESV boundary condition I ri x, y , z -R0 r R0 0 R0 I yi x , y , z y 0 0 New artificial source within (-R0,R0) is needed to satisfy the ESV boundary condition I i x, y , z -R0 R0 ESV 0 The new artificial source can generate a spin density Sart to satisfy the ESV boundary condition Spin currents normal to ESV: art art S I x x y y 0 S art 2S art I art y z y y y 0 S zart 2S yart I zart y y 0 we can obtain spin density S aux such that S ESV S ESV 0 I i x, y , z 0 0 S aux can statisfy the ESV boundary condition Spin density: S z S Ed z lso 3.76m, R 0 =0.5 lso S z S zEd (1/ m 2 ) 3 1 2.5 0.5 2 0 1.5 1 -0.5 0.5 0 -3 -1 -2 -1 0 1 2 3 Fig.4: Spin density S z S zEd are contributed from a circular void. Spin density: S ESV (0) z lso 3.76m, R 0 =0.5 lso 3 SzESV 0 (1/ m2 ) 1 2.5 2 0.5 1.5 0 1 -0.5 0.5 0 -3 -1 -2 -1 ESV (0) 0 1 2 3 Fig.5: Spin density S z is contributed from the additional source induced by S z S zEd . Spin density: S art z lso 3.76m, R 0 =0.5 lso S zaux (1/ m 2 ) 3 1 2.5 0.5 2 1.5 0 1 -0.5 0.5 0 -3 -1 -2 -1 0 1 2 3 art Fig.6: Spin density S z is contributed from the artificial source within (-R0, R0). Spin density: S S tot z S z S Ed z S ESV (0) z S tot z lso 3.76m, R 0 =0.5 lso aux z S ztot (1/ m2 ) 3 1 2.5 0.5 2 0 1.5 1 -0.5 0.5 0 -3 -1 -2 -1 0 1 Fig.7: Total Spin density S ztot . 2 3 Outline 1. Introduction: spin-Hall effect (SHE) 2. Spin diffusion equation for nonuniform driving field 3. Nonuniform driving field around a circular void in the bulk. 4. Nonuniform driving field around a edge semicircular . 5. Inverse spin-Hall effect for Rashba spin-obit interaction Inverse Spin-Hall Effect (ISHE) Inverse Spin-Hall Effect (ISHE) Inverse Spin-Hall Effect (ISHE) Linear response theory H H 0 H so Vi H 0 :unperturbed Hamiltonian of a 2DEG with random elastic spin-independent impurities H so =h k :spin-orbit interaction Vi :the interaction of electrons with the auxiliary fields We consider two kind of spin current sources: (i)Inhomogeneous spin polarization Sz creates the spin flux due to spin diffusion. (ii)Spin current is driving by spatially uniform spin-dependent “electric” field. V1 z B :slowly varying in time nonuniform Zeeman field B V2 z k A :uniform spin-dependent field z A Spin-up driving field Spin-down driving field Spin/charge currents for linear response approach 0: I s/c d , i Tr{Gkr ,k ' Gka,k ' js / cGka'q ,k q V , q N F k , k ' 2 Gkr ,k ' js / c Gkr ' q ,k q Gka' q ,k q Vi , q N F } Vi , q 1 N F Ek Tr s / c , , 2 k k where s z , c e 1 k hk js / c s / c , v , v * 2 m k N F : Fermi distribution For very low temperature, so that N F N F (I)For V1 B z case: 1 2 Dij 1 | U | / 2 ij ij Tr i Gkr q j Gka k ki r K ij * Tr Gk q j Gka k m For Rashba spin-orbit interaction: 2 N 0 I i B K yz Dzz K yx Dxz Dxz 2 c y For Rashba spin-orbit interaction: 2 N 0 K yx ; 2 k F2 N 0 K yz i q 2m*3 2 N 0 c Iy i B K yz Dzz K yx Dxz 2 2 k F2 N 0 i B i q D * 3 zz 2 2m Our goal is to express the charge current through the spin current Ixs. The charge current induces by V1 For Rashba spin-orbit interaction: ki i R jk * Tr j Gkr q k Gka ; m k kx x r a I i B Rzx Dxz Dzz *Tr z Gk q z Gk 2 k m s x Finally, we have 2 * e m s s Ix Ix Inverse spin-Hall effect No ISHE for V2 (II)For V2 z k A case: For Rashba spin-orbit interaction, the charge current is: I i A 2 i c y i | U |2 hk x K D R ii iz 0 yi k y 2 The spin current is: 2 N v s * I x im A 0 F 2 For V2 case, there is no inverse spin-Hall effect. Conclusion The spin accumulation can be restored by nonuniform driving field for RSOI in the diffusive regime. The spin accumulation is shown the dipole form normal to the driving field for the case of a circular void. The spin accumulation is corrected in the corner of edge semicircular void at the sample. The inverse spin-Hall effect depends on the specific form of spin current sources for Rashba SOI case. Thank you! Spin accumulations with Rashba SOI (near the electrode interfaces only) Spin relaxation length N 0 eE 2 hk2F S z 2hkF vF S y 2 y m* (1) (2) (3) When the exact cancellation occurs between (2) and (3), spin current vanishes in the bulk. I yz 2 D The exact cancellation between (2) and (3) no longer holds near the electrode interface when driving field has reached its bulk value but the spin polarization has not. E. G. Mishchenko et al., PRL, 93, 226602 (2004). Summary I: The intrinsic SHE provides a possible way to manipulate electron spins by electric means The RSOI + driving field can generate a bulk spin polarization due to a shifted Fermi sphere. The spin current can be contributed from the spin diffusion, spin precession, and the driving field. The SHE is quenched in the RSOI due to the exact cancellation between the bulk spin polarization and contribution of a driving filed. I ij 2 D j Si Rixj S x Riyj S y R izj S z hp 4 v hp eN 0l r kl z l x, y 2 xyi j F