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Photo-induced pure spin currents in quantum wells S.A. Tarasenko and E.L. Ivchenko A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia As is well known the absorption of circularly polarized light in semiconductors results in optical 5 orientation of electron spins and helicity-dependent electric photocurrent, and the absorption of 0 linearly polarized light is accompanied by optical alignment of electron momenta. Here we show 0 thattheabsorption ofunpolarized light inaquantumwell(QW)leadstogeneration ofapurespin 2 current,although both the average electron spin and electric current are vanishing. n a J I. INTRODUCTION pearanceofapurespincurrentunderopticalpumpingis 7 linked with two fundamental properties of semiconduc- 2 Spin andchargeare amongthe basic propertiesof ele- tor QWs, namely, the spin splitting of energy spectrum linear in the wave vector and the spin-sensitive selection mentary particles such as an electron, positron and pro- ] rules for optical transitions. l ton. The perturbation of a system of electrons by an l a electric field or light may lead to a flow of the particles. h Thetypicalexampleisanelectriccurrentthatrepresents - s the directed flow of charge carriers. Usually the electric II. INTERBAND OPTICAL TRANSITIONS e currentsdonotentailaconsiderablespintransferbecause m oftherandomorientationofelectronspins. However,the The effect is most easily conceivable for direct transi- . charge current can be accompanied by a spin current as t tions between the heavy-hole valence subband hh1 and a it happens, e.g., under injection of spin-polarized carri- conduction subband e1 in QWs of the C point symme- m ersfrommagneticmaterialsorintheoptical-orientation- s try, e.g. in (113)- or (110)-grown QWs. In such struc- - inducedcircularphotogalvaniceffect. Furthermore,there d turesthespincomponentalongthe QWnormalz iscou- exists a possibility to create a pure spin currentwhich is n pled with the in-plane electron wave vector. This leads not accompanied by a net charge transfer. This state o to k-linear spin-orbit splitting of the energy spectrum representsa non-equilibriumdistribution when electrons c as sketched in Fig. 1, where heavy hole subband hh1 [ with the spin “up” propagate mainly in one direction is split into two spin branches ±3/2. As a result they and those with the spin “down” propagate in the op- 1 are shifted relative to each other in the k space. In the posite direction. In terms of the kinetic theory, it can v reduced-symmetry structures, the spin splitting of the be illustrated by a spin density matrix with two nonzero 3 conduction subband is usually smaller than that of the 7 components, ρ(s,k;s,k) = ρ(s¯,−k;s¯,−k), where s and valencebandandnotshowninFig.1for simplicity. Due 6 k are the electronspin index and the wave vector,and s¯ to the selection rules the direct optical transitions from 1 means the spin opposite to s. Spin currents in semicon- the valence subband hh1 to the conduction subband e1 0 ductorscanbedrivenbyanelectricfieldactingonunpo- 5 canoccuronlyiftheelectronangularmomentumchanges larizedfreecarrierswhichundergoaspin-dependentscat- 0 by ±1. It follows then that the allowed transitions are tering. This is the so-calledspinHalleffectwherea pure / |+3/2i→|+1/2iand|−3/2i→|−1/2i,asillustratedin t spincurrentappearsinthedirectionperpendiculartothe a Fig.1byverticallines. Underexcitationwithlinearlypo- m electric field. The spin currents can be induced as well larized or unpolarized light the rates of both transitions by optical means as a result of interference of one- and - are equal. In the presence of spin splitting, the optical d two-photon coherent excitation with a two-color electro- n magnetic field or under interband optical transitions in o non-centrosymmetricalsemiconductors [1]. c E Here we show that pure spin currents, accompanied : -1/2 +1/2 v neitherbychargetransfernorbyspinorientation,canbe Xi achieved under absorption of linearly polarized or unpo- j-1/2 j+1/2 larized light in semiconductor low-dimensional systems. r a Particularly, the effect is considered for interband and free-carrier optical transitions in semiconductor QWs. In general, the flux of electron spins can be characterized by a pseudo-tensor Fˆ with the components Fα -3/2 +3/2 β describing the flow in the β direction of spins oriented along α, with α and β being the Cartesian coordinates. k In terms of the kinetic theory such a component of the x spin current is contributed by a non-equilibrium correc- tion ∝ σ k to the electron spin density matrix, where FIG. 1: Microscopic origin of pure spin current induced by α β σ is the Pauli matrix. We demonstrate that the ap- interband photoexcitation. α 2 transitions induced by photons of the fixed energy h¯ω E occur in the opposite points of the k-space for the spin -1/2 +1/2 branches ±1/2. The asymmetry of photoexcitation results in a flow of electrons within each spin branch. The j-1/2 +1/2 -1/2 j+1/2 correspondingfluxesj1/2andj−1/2areofequalstrengths but directed in the opposite directions. Thus, this non- equilibrium electron distribution is characterized by the nonzero spin current jspin = (1/2)(j1/2 −j−1/2) but a vanishing charge current, e(j1/2+j−1/2)=0. 0 k x The directionβ ofthe photo-inducedspin currentand the orientationα oftransmittedspins aredetermined by FIG.2: Microscopicoriginofpurespincurrentinducedunder the formofspin-orbitinteraction. The latteris governed lightabsorptionbyfreeelectrons. Thefree-carrierabsorption bytheQWsymmetryandcanbevaried. ForQWsbased is a combined process involving electron-photon interaction on zinc-blende-lattice semiconductors and grown along (vertical solid lines) and electron scattering (dashed horizon- the crystallographic direction [110]kz, the light absorp- tal lines). tionleadstoaflowalongxk[1¯10]ofspinsorientedalong carrier absorption. The more important contribution z. This component of the electron spin flow can be esti- comes from asymmetry of the electron spin-conserving mated as scattering. InsemiconductorQWsthematrixelementV τ m η Fz =γ(hh1) e h cvI, (1) of electron scattering by static defects or phonons has, x zx 2h¯ me+mh ¯hω in addition to the main contribution V0, an asymmetric spin-dependent term [2] where γ(hh1) is a constant describing the k-linear spin- zx orbit splitting of the hh1 subband, τe is the relaxation V =V0+ Vαβσα(kβ +kβ′), (3) timeofthespincurrent,me andmh aretheelectronand Xαβ hole effective masses in the QW plane, respectively, η cv wherekandk′aretheelectroninitialandscatteredwave is the light absorbance,and I is the light intensity. Note thatthetimeτ candifferfromtheconventionalmomen- vectors, respectively. Microscopically this contribution e is caused by the structural and bulk inversion asymme- tum relaxation time that governs the electron mobility. In (001)-grownQWs the absorptionof linearly- orun- try similar to the Rashba/Dresselhaus spin splitting of the electron subbands. The asymmetry of the electron- polarized light results in a flow of electron spins ori- phonon interaction results in non-equal rates of indirect entedintheQWplane. Incontrasttothelow-symmetry QWs considered above, in (001)-QWs the k-linear spin opticaltransitions for opposite wavevectorsin eachspin subband. This is illustrated in Fig. 2, where the free- splitting of the valence subband hh1 is small and here, carrierabsorptionisshownasacombinedtwo-stagepro- for the sake of simplicity, we assume the parabolic cess involving electron-photon interaction (vertical solid spin-independent dispersion in the hh1 valence subband lines) and electron scattering (dashed horizontal lines). and take into account the spin-dependent contribution γ(e1)σ k to the electron effective Hamiltonian. Then, The scattering asymmetry is shown by thick and thin αβ α β dashedlines: electronswiththespin+1/2arepreferably to the first order in the spin-orbit coupling, the compo- scatteredintothestateswithk >0,whileparticleswith nents of the pure spin current generated in the subband x thespin−1/2arescatteredpredominantlyintothestates e1 are derived to be with k < 0. The asymmetry causes an imbalance in τ m η x Fα =γ(e1) e e cvI . (2) thedistributionofphotoexcitedcarriersineachsubband β αβ 2h¯ me+mh ¯hω s = ±1/2 over the positive and negative kx states and yields oppositely directed electron flows j±1/2 shown by horizontal arrows. Similarly to the interband excitation III. FREE-CARRIER ABSORPTION considered in the previous section, this non-equilibrium distributionischaracterizedbyapurespincurrentwith- Lightabsorptionbyfreecarriers,ortheDrude-likeab- out charge transfer. sorption,occursindopedsemiconductorstructureswhen Let us assume the photon energy h¯ω to exceed the thephotonenergyh¯ωissmallerthanthebandgapaswell typical electron kinetic energy, E for degenerate and F as the intersubband spacing. Because of the energy and k T fornon-degenerateelectrongas. Thenthepurespin B momentum conservation the free-carrier optical transi- current induced by free-carrier light absorption is given tionsbecomepossibleiftheyareaccompaniedbyelectron by scattering by acoustic or optical phonons, static defects etc. Scattering-assisted photoexcitation with unpolar- Fα = τe Vαx 1+ |ex|2−|ey|2 + Vαy e e η I , izedlightalsogivesriseto apurespincurrent. However, x ¯h (cid:20) V (cid:18) 2 (cid:19) V x y(cid:21) e1 0 0 in contrast to the direct transitions considered above, (4) the spin splitting of the energy spectrum leads to no es- where e = (e ,e ) is the light polarization unit vector, x y sential contribution to the spin current induced by free- and η is the light absorbance in this spectral range. e1 3 In addition to the free-carrierabsorption, the spin-de- Acknowledgements pendent asymmetry of electron-phonon interaction can alsogiverisetoapurespincurrentintheprocessofpho- We acknowledgehelpful discussions with V.V. Bel’kov toelectron energy relaxation. In this relaxational mech- andS.D. Ganichev. This workwassupported by RFBR, anism the spin current is generated in a system of hot INTAS,programsoftheRASandFoundation“Dynasty” carriers,independently of heating means. - ICFPM. Besidesthe spin, free chargecarrierscanbe character- References izedbyanotherinternalproperty,e.g.,bythevalleyindex in multi-valley semiconductors. Thus, one can consider [1] R.D.R. Bhat, F. Nastos, A. Najmaie, and J.E. Sipe, pure orbit-valley currents in which case the net electric arXiv:cond-mat/0404066. current vanishes but the partial currents contributed by [2] E.L. Ivchenko and S.A. Tarasenko, Zh. Eksp. Teor. carriers in the particular valleys are nonzero. Fiz. 126, 476 (2004) [JETP 99, 379 (2004)].