Spin Hall and Edelstein effects in metallic films: from 2D to 3D
SSpin Hall and Edelstein effects in metallic films: from 2D to 3D
J.Borge , C. Gorini , G. Vignale , and R. Raimondi Dipartimento di Matematica e Fisica, Universit`a Roma Tre,Via della Vasca Navale 84, Rome, Italy Service de Physique de l’ ´Etat Condens´e, CNRS URA 2464,CEA Saclay, F-91191 Gif-sur-Yvette, France and Department of Physics and Astronomy,University of Missouri, Columbia MO 65211, USA
Abstract
A normal metallic film sandwiched between two insulators may have strong spin-orbit couplingnear the metal-insulator interfaces, even if spin-orbit coupling is negligible in the bulk of the film.In this paper we study two technologically important and deeply interconnected effects that arisefrom interfacial spin-orbit coupling in metallic films. The first is the spin Hall effect, whereby acharge current in the plane of the film is partially converted into an orthogonal spin current in thesame plane. The second is the Edelstein effect, in which a charge current produces an in-plane,transverse spin polarization. At variance with strictly two-dimensional Rashba systems, we findthat the spin Hall conductivity has a finite value even if spin-orbit interaction with impurities isneglected and “vertex corrections” are properly taken into account. Even more remarkably, suchfinite value becomes “universal” in a certain configuration. This is a direct consequence of thespatial dependence of spin-orbit coupling on the third dimension, perpendicular to the film plane.The non-vanishing spin Hall conductivity has a profound influence on the Edelstein effect, which weshow to consist of two terms, the first with the standard form valid in a strictly two-dimensionalRashba system, and a second arising from the presence of the third dimension. Whereas thestandard term is proportional to the momentum relaxation time, the new one scales with the spinrelaxation time. Our results, although derived in a specific model, should be valid rather generally,whenever a spatially dependent Rashba spin-orbit coupling is present and the electron motion isnot strictly two-dimensional. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r . INTRODUCTION Spin-orbit coupling gives rise to several interesting transport phenomena arising from theinduced correlation between charge and spin degrees of freedom. In particular, it allows oneto manipulate spins without using magnetic electrodes, having as such become one of themost studied topics within the field of spintronics.
Among the many interesting effectsthat arise from spin-orbit coupling, two stand out for their potential technological impor-tance: the spin Hall effect and the Edelstein effect . The spin Hall effect consists in theappearance of a z -polarized spin current flowing in the y -direction produced by an electricfield in the x -direction. The generation of a perpendicular electric field by an injectedspin current, i.e. the inverse spin Hall effect, has been observed in numerous settings andpresently provides the basis for one of the most effective methods to detect spin currents.
The Edelstein effect consists instead in the appearance of a y -spin polarization in re-sponse to an applied electric field in the x -direction. It has been proposed as a promisingway of achieving all-electrical control of magnetic properties in electronic circuits. The two effects are deeply connected, as we will see momentarily.There are, in principle, several possible mechanisms for the spin Hall effect, and it isuseful to divide them in two classes. We call them either extrinsic or intrinsic, depending onwhether their origin is the spin-orbit interaction with impurities or with the regular latticestructure. In this work we will focus exclusively on intrinsic effects. This means that theimpurities (while, of course, needed to give the system a finite electrical conductivity) donot couple to the electron spin.Bychkov and Rashba devised an extremely simple and yet powerful model describingthe intrinsic spin-orbit coupling of the electrons in a 2-Dimensional Electron Gas (2DEG) ina quantum well in the presence of an electric field perpendicular to the plane in which theelectrons move. In spite of its apparent simplicity, this analytically solvable model has severalsubtle features, which arise from the interplay of spin-orbit coupling and impurity scattering.The best-known feature is the vanishing of the Spin Hall Conductivity (SHC) for a uniformand constant in-plane electric field. This would leave spin-orbit coupling with impurities(not included in the original Bychkov-Rashba model) as the only plausible mechanism forthe experimentally observed spin Hall effect in semiconductor-based 2DEGs . However it has been recently pointed out that the vanishing of the SHC need not occur in2 − V + Interfacial spin-orbit(Rashba) z xy
FIG. 1. (color online) Schematic representation of a thin metal film sandwiched between insulatorswith asymmetric interfacial spin-orbit couplings. V + and V − are the heights of the two interfacialpotential barriers. These potentials generate interfacial spin-orbit interactions of the Rashba type,whose strength is controlled by the effective Compton wavelengths λ + and λ − respectively. systems which are not strictly two-dimensional, as explicitly shown in a model schematicallydescribing the interface of the two insulating oxides LaAlO and SrTiO (LAO/STO) . Evenmore recently , it has been suggested that a large SHC could be realized in a thin metal (Cu)film that is sandwiched between two different insulators, such as oxides or the vacuum. Such a system is shown schematically in Fig. 1. The inversion symmetry breaking across theinterfaces produces interfacial Rashba-like spin-orbit couplings, thus allowing metals withoutsubstantial intrinsic bulk spin-orbit to host a non-vanishing SHC. The spin-orbit couplingasymmetry – or, more generally, the fact that the spin-orbit interaction is not homogeneousacross the thickness of the film – is the core issue in this novel approach. In this paper wewill study the influence of the interfacial spin-orbit couplings on the Edelstein and spin Halleffects in this class of heterostructures.Before proceeding to a detailed study of the model depicted in Fig. 1, it is useful torecall the deep connection that exists between the spin Hall and Edelstein effects in theBychkov-Rashba model, described by the Hamiltonian H = p m + α ( σ x p y − σ y p x ) , (1)where m is the effective electron mass and α is the Bychkov-Rashba spin-orbit couplingconstant given by α = λ eE z / (cid:126) , with λ the materials’ effective Compton wavelength, E z e the absolute value of the electroncharge. It is convenient to describe spin-orbit coupling in terms of a non-Abelian gaugefield A = A a σ a /
2, with A xy = 2 mα and A yx = − mα . If not otherwise specified,superscripts indicate spin components, while subscripts stand for spatial components. Thefirst consequence of resorting to this language is the appearance of an SU (2) magnetic field B zz = − (2 mα ) , which arises from the non-commuting components of the Bychkov-Rashbavector potential. Such a spin-magnetic field couples the charge current driven by an electricfield, say along x , to the z -polarized spin current flowing along y . This is very much similarto the standard Hall effect, where two charge currents flowing perpendicular to each other arecoupled by a magnetic field. The drift component of the spin current can thus be describedby a Hall-like term [ J zy ] drift = σ SHEdrift E x . (2)It is however important to appreciate that this is not yet the full spin Hall current, i.e. σ SHEdrift is not the full SHC. In the diffusive regime σ SHEdrift is given by the classic formula σ SHEdrift = ( ω c τ ) σ D /e , where ω c = B /m (cid:126) is the “cyclotron frequency” associated with the SU (2) magnetic field, τ is the elastic momentum scattering time, and σ D is the Drudeconductivity. For a more general formula see Eq. (6) below.In addition to the drift current, there is also a “diffusion current” due to spin precessionaround the Bychkov-Rashba effective spin-orbit field. Within the SU (2) formalism thiscurrent arises from the replacement of the ordinary derivative with the SU (2) covariantderivative in the expression for the diffusion current. The SU (2) covariant derivative, dueto the gauge field, is ∇ j O = ∂ j O + i [ A j , O ] , (3)with O a given quantity being acted upon. The normal derivative, ∂ j , along a given axis j is shifted by the commutator with the gauge field component along that same axis. As aresult of the replacement ∂ → ∇ diffusion-like terms, normally proportional to spin densitygradients, arise even in uniform conditions and the diffusion contribution to the spin currentturns out to be [ J zy ] diff = 2 mα (cid:126) Ds y , (4)where D = v F τ / v F being the Fermi velocity. In the diffusive4egime the full spin current J zy can thus be expressed as J zy = 2 mα (cid:126) Ds y + σ SHEdrift E x . (5)For a detailed justification of Eq. (5) we refer the reader to Refs. 44 and 46. The factor infront of the spin density in the first term of Eq.(5) can also be written as an effective velocity L so /τ s . Here L so = (cid:126) (2 mα ) − is the typical spin length due to the different Fermi momentain the two spin-orbit split bands, whereas τ s = (cid:126) (4 m α D ) − is the Dyakonov-Perel spinrelaxation time. In terms of τ and τ s one has σ SHEdrift = e π (cid:126) ττ s , (6)which is indeed equivalent to the classical surmise given after Eq. (2). If we introduce thetotal SHC and the Edelstein Conductivity (EC) defined by J zy = σ SHE E x , s y = σ EE E x (7)we may rewrite Eq.(5) as σ EE = τ s L so (cid:0) σ SHE − σ SHEdrift (cid:1) . (8)In the standard Bychkov-Rashba model a general constraint from the equation of motiondictates that under steady and uniform conditions J zy = 0. Therefore the EC reads σ EE = − τ s L so σ SHEdrift = − e m π (cid:126) ατ = − eN ατ, (9)which is easily obtained by using the expressions given above and the single particle densityof states in two dimensions, N = m/ π (cid:126) . The remarkable thing is that this expressionremains unchanged for arbitrary ratios between the spin splitting energy and the disorderbroadening of the levels. However, in a more general situation with a non-zero SHC theEC would consist of the two terms appearing in Eq. (8). The latter equation is the “deepconnection” mentioned earlier between the Edelstein and the spin Hall effect. The first termon the r.h.s. is the “regular” contribution to the EC, the only surviving one in the Bychkov-Rashba model where the full SHC vanishes. The second term is “anomalous” in the sensethat it does not appear in the standard Bychkov-Rashba model, but it does appear in moregeneral models such as the one we discuss in this paper. Notice that the “regular” termis proportional to τ (see Eq. (6)), while the “anomalous” term, being proportional to the5yakonov-Perel relaxation time τ s and, in the diffusive regime, is inversely proportional tothe momentum relaxation time.At variance with the Bychkov-Rashba model, the one we choose for our system is notstrictly two-dimensional, and we take into account several states of quantized motion in thedirection perpendicular to the interface ( z ). Another crucial feature of this model is theoccurrence of two different spin-orbit couplings at the two interfaces. The difference arisesbecause (i) the interfacial potential barriers V + and V − are generally different, and (ii) theeffective Compton wavelengths λ + and λ − , characterizing the spin-orbit coupling strengthat the two interfaces, are different.Our central results for the generic asymmetric model are σ SHE = − n c (cid:88) n =1 e π (cid:126) ∆ E (3) nk Fn ∆ E nk Fn , (10)and σ EE = n c (cid:88) n =1 eN k F n (cid:126) (cid:104) ∆ E nk Fn τ + ∆ E (3) nk Fn τ ( n ) DP (cid:105) , (11)the sums running over the n c filled z -subbands of the thin film. To each subband therecorrespond a Fermi wavevector (without spin-orbit) k F n , an intraband spin-orbit energysplitting with a linear- and a cubic-in- k part∆ E nk Fn = (2 E n /d )[ k F n ( λ − λ − ) + 2 mk F n (cid:126) ( λ V + − λ − V − )] (12) ≡ ∆ E (1) nk Fn + ∆ E (3) nk Fn (13)and a Dyakonov-Perel spin relaxation time τ ( n ) DP τ = 2 (cid:20) τ ∆ E nk Fn / (cid:126) ) (2 τ ∆ E nk Fn / (cid:126) ) (cid:21) . (14)In the above formulas d is the film thickness and E = (cid:126) / md . Two particularly interestingregimes are apparent. First, a “quasi-symmetric” configuration, defined by equal spin-orbitstrengths, λ + = λ − ≡ λ , but different barrier heights, V + (cid:54) = V − . In this case ∆ E (1) nk = 0 (dueto Ehrenfest’s theorem ) and a most striking result is obtained: the SHC has a maximalvalue of − e π (cid:126) (independent of λ !) times the number of occupied bands σ SHE = − n c (cid:88) n =1 e π (cid:126) . (15)6t the same time the “anomalous” EC is at its largest. A second very interesting config-uration is a strongly asymmetric insulator-metal-vacuum junction, λ + = 0 , V + → ∞ and λ − ≡ λ, V − ≡ V . In this case the SHC becomes directly proportional to the gap Vσ SHE = − n c (cid:88) n =1 e π (cid:126) mk F n
V λ . (16)Notice however that the SHC cannot be made arbitrarily large simply by engineering a large V , since the above result holds provided 2 mk F n
V λ / (cid:126) < II. THE MODEL AND ITS SOLUTION
Following Ref. 41, we model the normal metallic thin film via the following Hamiltonian H = p m + V C ( z ) + H R + U ( r ) , (17)where the first term represents the kinetic energy associated to the unconstrained motionin the xy plane and p = ( p x , p y ) is the standard two-dimensional momentum operator. Thefinite thickness d of the metallic film is taken into account by a confining potential V C = V + θ ( z − z + ) + V − θ ( z − − z ) , (18)where V ± is the height of the potential barrier at z ± = ± d/ θ ( z ) is the Heavisidefunction. The third term in Eq.(17) describes the Rashba interfacial spin-orbit interactionin the xy plane located at z ± = ± d/ H R = λ − V − δ ( z − z − ) − λ V + δ ( z − z + ) (cid:126) ( p y σ x − p x σ y ) , (19)where λ ± are the effective Compton wavelengths for the two interfaces, σ x , σ y , σ z are thePauli matrices. The last term in Eq.(17) represents the scattering from impurities affectingthe motion in the x − y plane and r = ( x, y ) is the coordinate operator. The impuritypotential is taken in a standard way as a white-noise disorder with variance (cid:104) U ( r ) U ( r (cid:48) ) (cid:105) =72 πN τ ) − δ ( r − r (cid:48) ), where N is the two-dimensional density of states previously introduced.We will assume throughout that the Fermi energy E F n in each subband is much larger thanthe level broadening (cid:126) /τ and use the self-consistent Born approximation.The eigenfunctions of the Hamiltonian (17) have the form ψ n k s ( r , z ) = e i k · r √A √ ise iθ k f n k s ( z ) , (20)where A is the area of the interface, k = ( k x , k y ) is the in-plane wave vector, r is the positionin the interfacial plane and z is the coordinate perpendicular to the plane. θ k is the anglebetween k and the x axis. These states are classified by a subband index n = 1 , .. , whichplays the role of a principal quantum number, an in-plane wave vector k , and an helicityindex , s = +1 or − f n k s ( z ) describing the motion along the z -axis − (cid:126) m f (cid:48)(cid:48) n k s ( z ) + (cid:8) V C ( z ) − ks (cid:2) λ − V − δ ( z + d/ − λ V + δ ( z − d/ (cid:3)(cid:9) f n k s ( z ) = (cid:15) n k s f n k s ( z ) , (21)where the full energy eigenvalues are E n k s = (cid:126) k m + (cid:15) n k s . (22)By taking into account the continuity of the wave function f n k s ( z ) at z = ± d/ (cid:15) n k s the following transcendentalequationarctan √ (cid:15) (cid:114)(cid:16) d d − − (cid:15) (cid:17) − dd − α − sk + arctan √ (cid:15) (cid:114)(cid:16) d d − (cid:15) (cid:17) + dd + α + sk + √ (cid:15) = nπ, (23)where the energy (cid:15) is measured in units of E = (cid:126) / (2 md ) set by the thickness of thefilm. In the absence of spin-orbit coupling ( λ ± = 0) and for infinite heights of the potential( V ± → ∞ ), the solution reduces to the well-known energy levels (cid:15) n k s = E n . In the generalcase with both λ ± and V ± finite we use perturbation theory by assuming d large. There are8wo natural length scales associated with the confining potential d ± = (cid:126) / √ mV ± so thatwe expand in the small parameters d ± /d . Since all the energy scales are set by E , we finduseful to describe the spin-orbit coupling in terms of the parameters α ± = λ ± /d ± in such away that the product E α ± / (cid:126) has the dimensions of a velocity, just as the typical Rashbacoupling parameter. In the following we make an expansion to first order in d ± /d and up tothird order in α ± k .For the eigenvalues of (21) we find (cid:15) n k s = E n (cid:20) − d − + d + d + se k + e k + se k (cid:21) (24)and the eigenfunctions f n k s ( z ) = c n k s sin (cid:34) nπd + d − − α − ks + d + α + ks (cid:18) d z + d − − α − ks (cid:19)(cid:35) , (25)where c n k s = (cid:115) d e [2 − ( se k + e k + se k )] , d e = d + d + + d − ; e = 2 (cid:18) d + d α + − d − d α − (cid:19) , e = − (cid:18) d + d α + d − d α − (cid:19) , e = 2 (cid:18) d + d α − d − d α − (cid:19) . (26)Notice that the sign of the coefficients e and e depends on the relative strength of thespin-orbit coupling λ ± and barrier heights V ± . To avoid troubles with minus signs in thefollowing calculations, we assume that the couplings are labeled in such a way that λ + > λ − ,and V + > V − so that e , e > n = n c is the topmost occupiedsubband. In the following we use units such that (cid:126) = c = 1. III. SPIN HALL CONDUCTIVITY
The SHC is defined as the non-equilibrium spin density response to an applied electricfield. By using a vector gauge with the electric field given by E = − ∂ t A , the Kubo formula,corresponding to the bubble diagram of Fig.2, reads σ SHE = lim ω → Im (cid:104)(cid:104) j zy ; j x (cid:105)(cid:105) ω , (27)9 IG. 2. Feynman bubble diagram for the EC(a+b) or SHC(c). The empty right dot indicates thespin density (EC) or the spin current density (SHC) bare vertex, the left empty one indicates thenormal velocity operator, and the full dot is the dressed charge current density vertex. where we have introduced the spin current operator j zy = σ z k y / m and the charge currentoperator j x = − e ˆ v x . The number current operator, besides the standard velocity component,includes a spin-orbit induced anomalous contribution ˆ v x = k x /m + ˆΓ x . Without vertexcorrections, the anomalous contribution readsˆΓ x = δ ˆ v x = (cid:2) λ V + δ ( z − z + ) − λ − V − δ ( z − z − ) (cid:3) σ y . (28)This expression can be written in terms of the exact Green functions and vertices as σ SHE = − lim ω → Im eω (cid:88) nn (cid:48) kk (cid:48) ss (cid:48) (cid:104) n (cid:48) k (cid:48) s (cid:48) | ˆ v x | n k s (cid:105)(cid:104) n k s | j zy | n (cid:48) k (cid:48) s (cid:48) (cid:105) (cid:90) ∞−∞ d(cid:15) π G ns ( (cid:15) + , k ) G n (cid:48) s (cid:48) ( (cid:15) − , k (cid:48) ) . (29)where e > (cid:15) ± = (cid:15) ± ω/ G ns ( (cid:15), k ) = ( (cid:15) − E n k s + isgn (cid:15)/ τ ) − is theGreen function averaged over disorder in the self-consistent Born approximation with selfenergy Σ ns ( r , r (cid:48) ; (cid:15) ) = δ ( r − r (cid:48) )2 πN τ G ns ( r , r ; (cid:15) ) . (30)After performing the integral over the frequency we obtain σ SHE = − e π (cid:88) nn (cid:48) k ss (cid:48) (cid:104) n (cid:48) k s (cid:48) | ˆ v x | n k s (cid:105)(cid:104) n k s | j zy | n (cid:48) k s (cid:48) (cid:105) G Rn k s G An (cid:48) k s (cid:48) , (31)where we have introduced the retarded and advanced zero-energy Green functions at theFermi level G R,An k s = 1 − E n k s + µ ± i / τ (32)10nd exploited the fact that plane waves at different momentum k are orthogonal.To proceed further we need the expression for the vertices. It is easy to recognize thatthe standard part of the velocity operator k x /m does not contribute since it requires s = s (cid:48) ,whereas the matrix elements of j zy differ from zero only for s (cid:54) = s (cid:48) . Explicitly we have (cid:104) n (cid:48) k s (cid:48) | k x | n k s (cid:105) = k x (cid:104) f n (cid:48) k s (cid:48) | f n k s (cid:105) δ s (cid:48) s = (cid:104) f n (cid:48) k s (cid:48) | f n k s (cid:105) k cos θ k δ s (cid:48) s (33) (cid:104) n k s (cid:48) | δ ˆ v x | n k s (cid:105) = (cos θ k σ z,s (cid:48) s + sin θ k σ y,s (cid:48) s ) ∆ E nk k (cid:104) f n k s (cid:48) | f n k s (cid:105) (34) (cid:104) n k s | j zy | n (cid:48) k s (cid:48) (cid:105) = (cid:104) f n k s | f n (cid:48) k s (cid:48) (cid:105) k m sin θ k σ x,ss (cid:48) , (35)where ∆ E nk = ( E nk + − E nk − ) / E n ( e k + e k ) is half the spin-splitting energy in the n -th band. Eq.(34) is straightforwardly obtained from the eigenvalue equation (21) for thefunctions f n k s ( z ).Let us now discuss the overlaps between the wave functions (cid:104) f n k s | f n (cid:48) k (cid:48) s (cid:48) (cid:105) . If n = n (cid:48) wehave (cid:104) f n k s | f n k (cid:48) s (cid:48) (cid:105) = d e c nks c nk (cid:48) s (cid:48) (cid:20) − e ( ks + k (cid:48) s (cid:48) ) + e ( k + k (cid:48) ) + e ( k s + k (cid:48) s (cid:48) )4 (cid:21) , (36)which is unity plus corrections of order ( d ± /d ) when s, k (cid:54) = s (cid:48) , k (cid:48) . If n (cid:54) = n (cid:48) (cid:104) f n k s | f n (cid:48) k (cid:48) s (cid:48) (cid:105) isat least of order ( d ± /d ). Before continuing our calculation we observe that it is importantto distinguish between the intra-band ( n = n (cid:48) ) and the inter-band ( n (cid:54) = n (cid:48) ) contributions.The inter-band contributions are of second order in d ± /d , because they are proportionalto (cid:104) f n k s | f n (cid:48) k s (cid:48) (cid:105) . Since we limit our expansion to the first order in d ± /d we will from nowon neglect these contributions. Notice, however, that this approximation is no longer validwhen the intra-band splitting controlled by e and e vanishes. In this case one cannot avoidtaking into account the inter-band contributions. In the same spirit, we also approximatethe intra-band overlap (cid:104) f n k s | f n k (cid:48) s (cid:48) (cid:105) (cid:39)
1, because all of our results are at least linear in ( d ± /d )and we neglect higher order terms.The anomalous contribution to the velocity vertex, ˆΓ x , can be computed following theprocedure described in Ref. 37 according to the equations (see Fig.3)ˆΓ x = ˜ γ x + 12 πN τ (cid:88) k (cid:48) G R k (cid:48) ˆΓ x G A k (cid:48) , ˜ γ x = ˆ δv x + 12 πN τ (cid:88) k (cid:48) G R k (cid:48) k (cid:48) x m G A k (cid:48) ≡ ˜ γ (1) + ˜ γ (2) (37)11 IG. 3. Ladder resummation for the spin-dependent part of the dressed charge current densityvertex. The dashed line represents the correlation between propagators scattering off the sameimpurity site.
To extend the treatment to the present case, the projection must be made over the states | n k s (cid:105) . Assuming that the impurity potential does not depend on z , the matrix elements ofthe effective vertex ˜ γ (2) are: γ (2) nnss (cid:48) ( k ) ≡ (cid:104) n k s | ˜ γ (2) | n k s (cid:48) (cid:105) = 12 πN τ (cid:88) n k (cid:48) s (cid:104) n k s | n k (cid:48) s (cid:105) G Rn k (cid:48) s k (cid:48) x m G An k (cid:48) s (cid:104) n k (cid:48) s | n k s (cid:48) (cid:105) , (38)and γ (1) nnss (cid:48) ( k ) ≡ (cid:104) n k s | ˜ γ (1) | n k s (cid:48) (cid:105) is given by Eq.(34). The matrix elements (cid:104) n k s | n k (cid:48) s (cid:105) and (cid:104) n k (cid:48) s | n k s (cid:48) (cid:105) are those of the impurity potential: (cid:104) n k s | n k (cid:48) s (cid:105) = 12 (cid:104) f n k s | f n k (cid:48) s (cid:105) (cid:2) ss e i( θ k (cid:48) − θ k ) (cid:3) (39) (cid:104) n k (cid:48) λ | n k s (cid:48) (cid:105) = 12 (cid:104) f n k (cid:48) s | f n k s (cid:48) (cid:105) (cid:2) s (cid:48) s e − i( θ k (cid:48) − θ k ) (cid:3) . (40)By observing that k (cid:48) x = k (cid:48) cos θ k (cid:48) , one can perform the integration over the direction of k (cid:48) in12he expression of γ (2) nnss (cid:48) ( k )14 (cid:90) π d θ k (cid:48) π (cid:2) ss e i( θ k (cid:48) − θ k ) (cid:3) cos θ k (cid:48) (cid:2) s (cid:48) s e − i( θ k (cid:48) − θ k ) (cid:3) = s (cid:2) se − i θ k + s (cid:48) e i θ k (cid:3) , (41)to get γ (2) nnss (cid:48) ( k ) = (cos θ k σ z,ss (cid:48) + sin θ k σ y,ss (cid:48) )16 πN τ (cid:88) n k (cid:48) s s (cid:104) f n k s | f n k (cid:48) s (cid:105)(cid:104) f n k (cid:48) s | f n k s (cid:48) (cid:105) G Rn k (cid:48) s k (cid:48) m G An k (cid:48) s . (42)Approximating (cid:104) f n k s | f n k (cid:48) s (cid:105) ∼ δ nn , summing over s , and integrating over k with thetechnique shown in the Appendix yields γ (2) nnss (cid:48) ( k ) = − (cos θ k σ z,ss (cid:48) + sin θ k σ y,ss (cid:48) ) E n ( e + 2 e k F n ) , (43)where we have introduced the spin-averaged Fermi momentum in the n -th subband k F n m = µ − E n . (44)On the other hand γ (1) nnss (cid:48) ( k ) is given by γ (1) nnss (cid:48) ( k ) = (cos θ k σ z,ss (cid:48) + sin θ k σ y,ss (cid:48) ) E n ( e + e k F n ) (45)where k has been replaced by k F n at the required level of accuracy. Combining γ (1) nnss (cid:48) ( k )and γ (2) nnss (cid:48) ( k ) as mandated by Eq. (37) we finally obtain γ nnx,ss (cid:48) ( k ) = − (cos θ k σ z,ss (cid:48) + sin θ k σ y,ss (cid:48) ) E n e k F n . (46)Next we project the equation for the vertex corrections in the basis of the eigenstates andget the following integral equation:Γ nnx,ss (cid:48) ( k ) = γ nnx,ss (cid:48) ( k ) + 12 πN τ (cid:88) n n k (cid:48) s s (cid:104) n k s | n k (cid:48) s (cid:105) G Rn k (cid:48) s Γ n n x,s s ( k (cid:48) ) G An k (cid:48) s (cid:104) n k (cid:48) s | n k s (cid:48) (cid:105) , (47)which, by confining to intra-band processes only, can be solved with the ansatz Γ nnx,ss (cid:48) ( k ) =Γ n ( k F n )(cos( θ k )( σ z ) ss (cid:48) + sin( θ k )( σ y ) ss (cid:48) ) yieldingΓ nnx,ss (cid:48) ( k ) = γ nnx,ss (cid:48) ( k ) τ ( n ) DP τ . (48)By performing the integral over momentum and summing over the spin indices in Eq.(31),one obtains the SHC as σ SHE = n c (cid:88) n =1 e π ττ ( n ) DP Γ n ( k F n )∆ E nk Fn /k F n , (49)13here n c is the number of occupied bands.If vertex corrections are ignored, i.e., if we approximate Γ n ( k F n ) = ∆ E nk Fn /k F n (cf.Eq.(34)), Eq.(49) gives us σ SHEdrift = n c (cid:88) n =1 e π ττ ( n ) DP , (50)which, in the weak disorder limit ( τ → ∞ ), reproduces the result of Ref. 41, i.e. σ SHEdrift =( e/ π ) n c .If instead the renormalized vertex (48) is properly taken into account, we obtain σ SHE = − n c (cid:88) n e π e k F n e + e k F n . (51)Notice that, being proportional to λ ± ( e ∝ λ ± , e ∝ λ ± ), this result is consistent withthe result obtained in Ref. 40 for a different but related model. Making use of the explicitexpressions for e and e we finally get the previously reported result of Eq.(10). IV. EDELSTEIN CONDUCTIVITY
In the d.c. limit, i.e., for ω →
0, the Edelstein conductivity (EC) is defined by σ EE = lim ω → Im (cid:104)(cid:104) s y ; j x (cid:105)(cid:105) ω . (52)That can be written as: σ EE = − lim ω → Im eω (cid:88) nn (cid:48) kk (cid:48) ss (cid:48) (cid:104) n (cid:48) k (cid:48) s (cid:48) | ˆ v x | n k s (cid:105)(cid:104) n k s | s y | n (cid:48) k (cid:48) s (cid:48) (cid:105) (cid:90) ∞−∞ d(cid:15) π G ns ( (cid:15) + , k ) G n (cid:48) s (cid:48) ( (cid:15) − , k (cid:48) ) , (53)After performing the integral over frequency we get σ EE = − e π (cid:88) nn (cid:48) k ss (cid:48) (cid:104) n (cid:48) k s (cid:48) | ˆ v x | n k s (cid:105)(cid:104) n k s | s y | n (cid:48) k s (cid:48) (cid:105) G Rn k s G An (cid:48) k s (cid:48) , (54)where we have used again the orthogonality of the eigenvectors with different momentum.As shown in Fig.2, we consider the bare vertex for the spin density s y = σ y / v x = ˆΓ x + k x /m , – ˆΓ x being the renormalizedspin-dependent part of the vertex. Clearly, the two parts of the number current vertex yieldtwo separate contributions to the EC and we are now going to evaluate them separately. We14hen evaluate the (a) diagram in Fig.2 as: σ EE, ( a ) = − e πm (cid:88) nn (cid:48) k ss (cid:48) (cid:104) n (cid:48) k s (cid:48) | k x | n k s (cid:105)(cid:104) n k s | σ y | n (cid:48) k s (cid:48) (cid:105) G Rn k s G An (cid:48) k s (cid:48) , (55)where the matrix elements of the spin vertex is (cid:104) n k s | σ y | n (cid:48) k s (cid:48) (cid:105) = (cid:104) f n k s | f n (cid:48) k s (cid:48) (cid:105) (cos θ k σ z,ss (cid:48) − sin θ k σ y,ss (cid:48) ) . (56)Setting n (cid:48) = n and using Eq.(24) for the energy eigenvalues, we can perform the integra-tion over the momentum in Eq.(55) obtaining for σ EE, ( a ) the expression σ EE, ( a ) = n c (cid:88) n =1 eN τ E n (cid:0) e + 2 e k F n (cid:1) , (57)Next we evaluate the (b) diagram in Fig.2 as: σ EE, ( b ) = − e π (cid:88) nn (cid:48) k ss (cid:48) (cid:104) n (cid:48) k s (cid:48) | ˆΓ x | n k s (cid:105)(cid:104) n k s | σ y | n (cid:48) k s (cid:48) (cid:105) G Rn k s G An (cid:48) k s (cid:48) , (58)We set n = n (cid:48) and insert the result obtained in Eq.(48) for (cid:104) n k s (cid:48) | ˆΓ x | n k s (cid:105) . Since both thematrix elements of ˆΓ x and σ y contain terms proportional to cos( θ k ) and sin( θ k ), we mustdistinguish between s = s (cid:48) (first term in Eq.(46)) and s (cid:54) = s (cid:48) (second term in Eq.(46)). If s = s (cid:48) we have σ EE, ( b )1 = − e π (cid:88) n k s (cid:104) ns | ˜Γ x | n k s (cid:105)(cid:104) n k s | σ y | n k s (cid:105) G Rn k s G An k s (59)The integral over the momentum can be done with the technique shown in the Appendix toyield σ EE, ( b )1 = n c (cid:88) n eN τ E n e k F n τ ( n ) DP τ . (60)If s (cid:54) = s (cid:48) we have instead σ EE, ( b )2 = − e π (cid:88) n k s (cid:104) n k ¯ s | ˜Γ x | n k s (cid:105)(cid:104) n k s | σ y | n k (cid:48) ¯ s (cid:105) G Rn k s G An k ¯ s . (61)So we can conclude that σ EE, ( b )2 = n c (cid:88) n =1 eN τ E n e k F n (2 τ ∆ E nk Fn ) (62)with ∆ E nk Fn defined in Eq.(12). Combining the (a) and (b) contributions, the final resultfor the Edelstein conductivity is found to be: σ EE = n c (cid:88) n =1 eN τ E n (cid:20) e + 3 e k F n + 2 e k F n (2 τ ∆ E nk Fn ) (cid:21) , (63)which is easily seen to be equivalent to Eq. (11).15 . DISCUSSION The two central results (63) and (51) may be interpreted along the lines outlined in theintroduction. We begin by noticing that both conductivities are expressed as simple sumsof independent subband contributions, hence the relation (8) is valid separately within eachsubband. The second step is the identification of the quantity τ s /L so for a given subband.Clearly τ s must be identified with the Dyakonov-Perel relaxation time τ ( n ) DP defined in (14).For the spin-orbit length L so one notices that the quantity 2 αp F in the Rashba modelcorresponds to the band splitting, and hence must here be replaced by − E nk Fn . Thisyields, after restoring (cid:126) in the following, L ( n ) so = (cid:126) v F n E nk Fn , (64)i.e. τ s /L so → τ ( n ) DP /L ( n ) so . With this prescription one can apply Eq. (8) subband-by-subbandand obtain σ EE, ( n ) = τ ( n ) DP L ( n ) so (cid:104) σ SHE, ( n ) − σ SHE, ( n ) drift (cid:105) , (65)where σ SHE, ( n ) , σ SHE, ( n ) drift stand for the n -th band contribution to Eqs. (51) and (50), respec-tively. It is now immediate to see that a sum over the subbands leads to the EC of Eq. (63).We may thus conclude the following: a non vanishing SHC in the presence of Rashba spin-orbit coupling gives rises to an anomalous EC scaling with the inverse scattering time;conversely, an anomalous EC yields a non-vanishing SHC.We now consider two physically interesting limiting cases of the general solution:1. the insulator-metal-vacuum junction, λ + = 0 V + → ∞ , λ − = λ V − = V ;2. films with the same spin orbit constant coupling at the two interfaces, λ − = λ + = λ .In the first case we get σ EE = − n c (cid:88) n eN τ E n λ d (cid:126) (cid:18) (cid:126) π V τ E n (cid:19) , (66) σ SHE = − n c (cid:88) n e π (cid:126) mk F n
V λ . (67)There are some experimental studies of metal-metal-vacuum junctions that shows giant spin-orbit coupling and where one could test the prediction of Eqs.(66-67). Though Eq. (67) is16btained for small values of the parameter 2 mk F n
V λ / (cid:126) (cid:28)
1, the structure of the result isquite interesting: it suggests that this kind of device, the insulator-metal-vacuum junction,could be an efficient spintronic device, its transport properties being proportional to thebarrier height V .In the second case let us first assume a “quasi-symmetric” configuration, i.e. though λ + = λ − ≡ λ , the barrier heights are different, V + (cid:54) = V − . We then obtain that the spinsplitting of the bands vanishes to linear order in k ( e = 0) (see footnote 48) so that σ SHE = − n c (cid:88) n e π (cid:126) , (68)and σ EE = n c (cid:88) n eN τ ∆ E nk Fn k F n (cid:126) (cid:20) (cid:126) τ ∆ E nk Fn ) (cid:21) . (69)The SHC in this limit is independent of λ . This very striking result is reminiscent of theuniversal result e π (cid:126) obtained for a single Bychkov-Rashba band when vertex correctionsare ignored. However vertex corrections are now fully included, yet the SHC is not onlyfinite, but independent of λ and equal to the single band universal result multiplied by afactor −
2! We emphasize that this result has nothing to do with the non-vanishing intrinsicSHC that arises in certain generalized models of spin-orbit coupling with winding numberhigher than 1. Rather, it has everything to do with the k -dependence of the transversesubbands describing the electron wave function in the z - direction. We also find that theanomalous part of the Edelstein effect becomes large, as it is proportional to 1 / ∆ E nk Fn , andthe splitting vanishes with the third power of k at small k .Let us finally discuss the fully inversion-symmetric limit of the model, λ + = λ − and V + = V − . We notice that in this case the limit of Eq. (51) does not exist, because both e and e vanish (the spin splitting is identically zero!) while the value of Eq. (51) dependson the order in which e and e tend to zero, in particular on whether they tend to zerosimultaneously, or e tend to zero before e , as in the “quasi-symmetric” case above. Theorigin of this apparently unphysical non-analytic behavior can be traced back to the singularcharacter of the vertex (48) for vanishing spin splitting. Under these circumstances, theDyakonov-Perel spin relaxation time (14) diverges, apparently implying spin conservation.However, even in the inversion-symmetric limit, interband effects provide spin relaxationprocesses which regularize the vertex. Such effects are typically negligible away from the17nversion-symmetric limit, since they are proportional to the square of the wave-functionoverlap between different bands and therefore scale as ( d ± /d ) . However, in the inversion-symmetric limit they cannot be neglected.A full analysis of interband effects is beyond the scope of the present paper, and we limitourselves to a heuristic discussion of the physical origin of the spin relaxation mechanismdue to interband virtual transitions. In the inversion-symmetric limit, the Hamiltonian isinvariant upon the simultaneous operations of space inversion along the z -direction ( z → − z )and helicity flipping ( s → − s ), i.e., a full mirror reflection in the x − y plane. Hence theeigenfunctions can be classified as even or odd under such a reflection: f n k s ( z ) = P n f n k − s ( − z ) . (70)where P n = ±
1. Furthermore the parity eigenvalue P n is the same as in the absence ofspin-orbit interaction, because the reflection commutes with the spin-orbit interaction.Since states of opposite helicity are degenerate, one can construct, in each band n , statesthat are linear combinations of the helicity eigenstates |±(cid:105) ψ n k ↑ = 12 ( f n k + ( z ) | + (cid:105) + f n k − ( z ) |−(cid:105) ) (71) ψ n k ↓ = 12 ( f n k + ( z ) | + (cid:105) − f n k − ( z ) |−(cid:105) ) . (72)These can be rewritten in terms of the eigenstates | ↑(cid:105) and | ↓(cid:105) of σ z and, after using (70),one obtains ψ n k ↑ = f n k + ( z ) + P n f n k + ( − z )2 | ↑(cid:105) + i e i θ k f n k + ( z ) − P n f n k + ( − z )2 | ↓(cid:105) (73) ψ n k ↓ = f n k + ( z ) − P n f n k + ( − z )2 | ↑(cid:105) + i e i θ k f n k + ( z ) + P n f n k + ( − z )2 | ↓(cid:105) . (74)One sees immediately that, within the first Born approximation, impurity scatteringcannot produce spin flipping within a band because the matrix element of the z -independentdisorder potential between ψ n k ↑ ψ n k (cid:48) ↓ vanishes by symmetry.On the other hand, spin flipping may occur in the second Born approximation by goingthrough an intermediate state in a band of opposite parity. For example, an electron mayfirst jump, under the action of the disorder potential, to a state of opposite spin in anunoccupied band of opposite parity; then in a second step it may return to the original bandwithout flipping its spin. Alternatively the spin may remain unchanged in the transition tothe unoccupied band, and flip on the way back to the original band. As a result of such18econd-order processes, a new mechanism of spin relaxation arises, which we call inter-bandspin relaxation , with rate τ − IB . When this additional relaxation mechanism is taken intoaccount, the diverging DP relaxation time in Eq. (48) for the vertex is replaced by the finitetotal spin relaxation time ( τ − DP + τ − IB ) − . Thus, the non-analyticity is cured.The regime analyzed in this paper corresponds to the situation in which τ − DP (cid:29) τ − IB , andinter-band spin relaxation can be neglected. Clearly, when looking at the fully symmetriclimit, with vanishing spin splitting, inter-band relaxation must be taken into account, to-gether with inter-band contributions to the SHC and EC. Once more, a full-fledged treatmentof this regime is beyond the scope of the present work. VI. CONCLUSIONS
We have developed a simple model for describing spin transport effects and spin-chargeconversion in heterostructures consisting of a metallic film sandwiched between two differ-ent insulators. All the effects we have considered depend crucially on the three-dimensionalnature of the system – in particular, the fact that the transverse wave functions depend onthe in-plane momentum – and on the lack of inversion symmetry caused by the differentproperties of the top and bottom metal-insulator interfaces, each characterized by a differentbarrier height (gap) and spin-orbit coupling strength. After a careful consideration of vertexcorrections we find that the model supports a non-zero intrinsic SHC, in sharp contrastto the 2DEG Rashba case. Strikingly, in a “quasi-symmetric” junction the SHC reachesa maximal and universal value. We have also calculated the Edelstein effect for the samemodel and found that the induced spin polarization is the sum of two different contribu-tions. The first one is analogous to the term found in the 2DEG Rashba case, whereas thesecond “anomalous” one has a completely different nature. Namely, it is inversely propor-tional to the scattering time, indicating that it is caused by the combined action of multipleelectron-impurity scattering and spin-orbit coupling. We have also discussed the generalconnection between the non-vanishing SHC and the anomalous term in the EC. Further-more, by Onsager’s reciprocity relations, our results are immediately relevant to the inverseEdelstein effect , in which a non-equilibrium spin density induces a charge current.The above features, although discussed here for a specific model, are expected to be gen-eral, proper to any non-strictly two-dimensional system in which the spin-orbit interaction19s non-homogeneous across the confining direction. Technical applications of this idea couldlead to a new class of spin-orbit-coupling-based devices.
ACKNOWLEDGMENTS
CG acknowledges support by CEA through the DSM-Energy Program (project E112-7-Meso-Therm-DSM). GV acknowledges support from NSF Grant No. DMR-1104788.
Appendix A: Integrals of Green functions
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