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Spin relaxation in n-type GaAs quantum wells with transient spin grating M. Q. Weng∗ and M. W. Wu† Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui, 230026, China and Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China‡ H. L. Cui Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA 8 (Dated: February 1, 2008) 0 0 By solving the kinetic spin Bloch equations, we study the time evolution of the transient spin 2 grating, whose spin polarization varies periodically in real space, confined in (001) GaAs quantum n wells. With this study we can investigate the properties of both the spin transport and the spin a relaxation atthesametime. TheFouriercomponentofthespinsignaldecaysdoubleexponentially J with two decay rates 1/τ+ and 1/τ−. In high temperature regime, the average of these two rates 9 varies with the grating wave-vector q quadratically, i.e., (1/τ++1/τ−)/2 = Dsq2+1/τ˜s, with Ds 2 andτ˜srepresentingthespindiffusioncoefficientandtheaverageoftheout-of-planeandthein-plane spin relaxation times respectively. τ± calculated from our theory are in good agreement with the ] experimentaldatabyWeberetal. [Phys. Rev. Lett. 98,076604(2007)]. BycomparingDswithand i withouttheelectron-electronCoulombscattering,wecalculatethecontributionofCoulombdragto c s thespin diffusion coefficient. With thetransient spin grating result, we further reveal therelations - amongdifferentcharacteristic parameterssuchasspin diffusioncoefficient D ,spin relaxation time l s r τs, and spin injection length Ls. We show that in the presence of the Dresselhaus and/or Rashba mt spin-orbit coupling, the widely used relation Ls = √Dsτs is generally inaccurate and can even be verywronginsomespecialcases. Wepresentanaccuratewaytoextractthesteady-statetransport . t characteristic parameters from thetransient spin grating signals. a m PACSnumbers: 72.25.-b,72.25.Rb,72.25.Dc,75.40.Gb,71.10.-w - d n I. INTRODUCTION is expected to be 1/τ h2(k)τ∗ . It should be o noted that τ∗ not onlysin∝cluhdes thepciontribution from c p conventional momentum scattering mechanisms such as [ Recently a lot of efforts have been devoted to the electron-impurity and electron-phonon scattering, but study of spin dynamics and spin transport in semicon- 4 also includes the contribution from electron-electron ductor nano-structures in order to realize the spintronic v 5 device.1,2,3,4 In n-type zinc-blende semiconductors, elec- Coulomb scattering which does not directly affect the charge transport properties.14,16,17 As for the steady- 7 tronspinsarerandomizedbytheDresselhausand/orthe state spin injection problem, by assuming that the spin 9 Rashbaspin-orbitcoupling(SOC)whichactsasaneffec- 1 tivemagneticfieldh(k)withitsdirectionandmagnitude dynamics can be separated into two independent pro- 8. depending on the electron momentum k.5,6,7 In spatial cesses,spin diffusion and spin relaxation,the spin polarization is expected to decay exponentially along the in- 0 homogeneous system the spin evolution is characterized jectiondirectionwithdecay“rate”,i.e.,thespininjection 7 by the spin relaxation time τ which describes the decay s 0 rateofspinpolarization;Thesteady-statespintransport length, Ls =√Dsτs, where Ds and τs are two phenome- : nalparameterswhose relationswith other propertiesare v is characterized mainly by the spin injection/diffusion i length L ,8,9,10,11 while the transient spin propagation yet to be determined.8,9,10,11 However in the presence X is charactserizedby spindiffusion coefficient D .12,13 The of the SOC, it has been proved that this assumption is s r relations among these three parameters and other pa- oversimplified. By solving the kinetic spin Bloch equa- a tions, it is shown that the effective magnetic field due rameters such as momentum relaxation time τ , charge p to the SOC alone causes the electron spin to process in diffusioncoefficientD andmobilityµhavebeenactively c real space even in diffusive regime.18,19 The spin polar- discussed. ization varies in the space as e−x/Lscos(x/L ) instead It is understood that spin relaxation/dephasing is in- 0 of simple exponential decay. The spin injection length ducedbytheinhomogeneousbroadeningduetotheSOC L and the spatial oscillation “period” L are obtained together with the (spin conserving) scattering.14,15 The s 0 by solving the kinetic spin Bloch equations,20 the spin scattering provides a channel to speed up the spin de- transportequationswhichincludethecontributionofthe phasingbutalsoslowsdownthespindephasingbyweak- SOC21 or the linear response theory.22 In the diffusive ening the inhomogeneous broadening.16 The competing regime,L andL canbeexpressedbySOCstrengthand effects of scattering have different results on spin de- s 0 momentum scattering time.20,21,22 As for the spin diffu- phasing in different conditions. In weak SOC or strong sioncoefficient, it was widely assumed to be the same as scattering regime, the out-of-plane spin relaxation time 2 the charge diffusion coefficient. Later it was pointed out the TSG problem. Then we show the analytical solution that spin Coulomb drag (SCD), caused by the electron- of simplified equations and the numerical results of the electronCoulombscattering,shouldsuppresstherelative fullkineticspinBlochequations. InSec.IIIwestudythe motion of electrons with different spins and thus reduce relationsamongthespinrelaxationtimeτ ofaspatially s D .13,23,24,25 homogeneous system, the spin injection length L and s s A direct measurement of spin diffusion coefficient the spatial oscillation period L of the steady-state spin 0 can be carried out by transient spin grating (TSG) transport as well as D of the transient spin transport. s experiments.12,13,26,27Withtheassumptionthatspindif- We conclude in Sec. IV. fusionandspinrelaxationareindependentofeachother, the decay rate of TSG was written as Γ =D q2+1/τ , q s s where q is the wave-vectorof the spin grating.12,13 How- ever, from the lesson of steady-state spin injection one II. KINETIC SPIN BLOCH EQUATIONS learns that the spin diffusion and relaxationare not sep- arable even in the diffusive regime. This is further justi- fied by the fact that the decay of TSG can be fitted to The system we study is the electron gas confined in a double-exponential form instead of single exponential a (001) GaAs quantum well (QW) with width a grown one.27 It is therefore still a question of how to get spin along the z-axis. We assume that the well width is nar- diffusion coefficient through TSG experiments. row enough so that only the lowest subband is occu- In this article we study the temporal evolution of the pied. With the help of the nonequilibrium Green func- TSG by solving the kinetic spin Bloch equations. This tionmethod,28 onecanwritedownthekineticspinBloch paper is organizedas following: In Sec. II we first set up equations4,18,19 by using gradient expansion and gener- the kineticspinBlochequationsandapplythemtosolve alized Kandanoff-Baymansatz: ∂ρk(x,t) +eE(x)∂ρk(x,t) kx∂ρk(x,t) i[(gµBB+h(k)) σ/2+εHF(x,k,t),ρk(x,t)] ∂t ∂k − m ∂x − · x ∂ρk(x,t) = . (1) ∂t (cid:12)s (cid:12) (cid:12) (cid:12) Here we assume thatthe transportdirectionis alongthe by the Poisson equation, and the diffusion of electrons, x-axis. ρ(x,k,t) is the density matrix whosediagonalel- respectively. The fourth term describes the spin preces- ementsfkσ(x,t) representthe electrondistributionfunc- sion around the total magnetic field which is composed tions with spin σ(= 1/2) and momentum k = (k ,k ) of the external magnetic field B, the effective magnetic x y at position x. The o±ff-diagonal elements stand for the field h(k) due to the SOC as well as the one from the spin correlations between spin-up and -down electrons. Hartree-Fock term of the electron-electron Coulomb in- The second and third terms of Eq. (1) correspond to teraction εHF(x,k,t). h(k) contains the Dresselhaus and the drift driven by the electric field E(x), determined the Rashba terms:5,6,7 h(k) = β( k cos2θ+k sin2θ,k sin2θ+k cos2θ,0) x y x y − k2 k2 +γ( x− y sin2θ+k k cos2θ)(k , k ,0)+α(k , k ,0) , (2) x y y x y x 2 − − where θ is the angle between x-axis (the spin injec- the electron-impurity, the electron-phononand most im- tion/diffusion direction) and the (100) crystal axis.29 portantly the electron-electron Coulomb scattering. It β =γπ2/a2 withγ beingtheDresselhauscoefficient.5,6 α is noted that in our calculation the electron-electron in- represents the Rashba parameter which depends on the teractionistreatedbeyondthe Hartree-Fockapproxima- electric field along the growth direction of the QW. The tion. The expressionsfor the Hartree-Fockand the scat- scatteringterm ∂ρk(x,t) includes allthe scattering,i.e., tering terms are given in detail in Ref. 30. ∂t s (cid:12) (cid:12) (cid:12) 3 The kinetic spin Blochequationsdescribe the spindy- in the diffusive regime using simplified equations. This namics in the presence of drift, diffusion and spin pre- solution can only provide an intuitive vision of the TSG cession. By choosing some specified initial and bound- dynamics. Wethenpresentthenumericalsolutionofthe ary conditions, one can obtain the evolution of the spin full kinetic spin Bloch equations. signal in time and real space by solving these equations for different systems. In order to study the TSG, the initial spin polarization of the electrons is chosen A. Simplified Solution to be a sinusoidal wave along the x-direction P(x) = (N (x) N (x))/(N (x)+N (x))=P sin(x/L)butuni- ↑ ↓ ↑ ↓ 0 − form along the y-axis, where L = 2π/q is the spatial By neglecting the Hartree-Fock term, the inelastic period. Using periodical boundary condition, one only scattering such as the electron-phonon and the electron- needs to study the dynamics in one period of the space electron Coulomb scatterings and the coupling to the regime. Poisson equation, one is able to rewrite Eq. (1) in dif- Unlessonemakessomesimplifications,thekineticspin fusive regime when the scattering is strong enough, by Blochequationsaretoocomplicatedtobesolvedanalyti- usingasimilarmethodforcalculatingthespinrelaxation cally. Inthispaper,wefirstpresentananalyticalsolution as in Refs. 5,6, ∂S¯(q,t) +Dq2S¯(q,t)+iqh¯ S¯(q,t) ∂t × (α2+βˆ2−2αβˆsin2θ)hk2τ1i 2αβˆ k2τ cos2θ 0 +1 2α+βˆh(γkk23τ)2τ3cio/1s62θ (α2+βˆ2+h2αβˆ1siin2θ)hk2τ1i 0 S¯(q,t)=0 . (3) 2 h 1i +h(γk3)2τ3i/16  0 0 2(α2+βˆ2)hk2τ1i   +h(γk3)2τ3i/8    HereS¯(q,t)= d2kTr ρ(q,k,t)σ dθ istheFouriercom- ifies the coefficient of linear Dresselhaus term β to be ponent of the spin de{nsity functi}on. The second term βˆ = β γk2/4. This relation together with Eq. (3) are R − represents the diffusion term with the diffusion constant briefly derived in the Appendix A. Without the spin ro- D = k2τ /2m2 . 1/τ = 2π 1 cos(lθ)dθ/2π with tation term, the evolution of the TSG is characterized τ(k,θ)hbei1ng theimomelntum0relaτx(ka,θti)on time due to the by the decay rate Dq2 +1/τs. However, the spin rota- R electron-impurity scattering. The third term is the spin tion mixes the in-plane and out-of-plane spin dynamics. rotation caused by the non-vanishing net effective mag- As a result, the evolution is usually described by three netic field relaxation rates. There are two special cases where the evolution can be described by two relaxation rates. The h¯ = k2τ /m ( βˆcos2θ,βˆsin2θ α,0) (4) first one is in the system where only one of the Dres- 1 h i − − selhaus and Rashba terms is important so that the two due to the DP effect and the diffusion. The last in-plane spin dynamics become identical. The other is term is the spin relaxation caused by the DP ef- the spin injection/diffusion along (110) and (1¯10) axises fect and the spin conserving scattering. For a sys- wherethe neteffective magnetic fieldh¯ is parallelto one tem not far away from the equilibrium, = of the characteristic directions of the in-plane spin dy- ∂f(εk)/∂εkd2k/ ∂f(εk)/∂εkd2k with f(hε·)··bieing namics. Thusitcanonlymixtheremainingin-planespin ··· the Fermi distribution function. It is noted that sim- withtheout-of-planespindynamics. TheTSGevolution R R ilar results at low temperature have been obtained by of these cases has a double-exponential form, different approachrecently.21 S (q,t)=S (q,0)(λ e−t/τ+ +λ e−t/τ−) , (5) InthepresenceofboththeDresselhausandtheRashba z z + − terms,thespinrelaxationratesarehighlyanisotropic. In with relaxation rates the weak SOC regime, the in-plane spin relaxation rates tγahr2eehkcc6hhτa3arira/ac3ct2tee,rricizzoeerddrebsdypirotewncdotiiondngesctao(y1t1rha0et)essapnhind(αr(±e1l¯1aβˆ0x))a2tkaio2xτnis1eias/lo2rne+g- Γ± = τ1± =Dq2+ 12(τ1s1 + τ1s)± 2τ1s2s1+ 16Dτsq′12τs2(26), spectively. While the out-of-plane relaxation rate is in which the sum of these two rates 1/τ = (α2 + βˆ2)k2τ + s 1 tγe2mhk,6τth3ie/1p6r.esNenocteetohfatth,eforcuthbeicqDuarseis-sthewlhoa-duismteenrsmionmsioyds-- λ± = 21 1± 1+16D1q2τ2 /τ′ ! . (7) s2 s1 p 4 Here τ (τ ) is the spin relaxation time of the in-plane simplifiedequationsarederivedwithonlytheelasticscat- s1 s2 spin which mixes (does not mix) with the out-plane spin tering. In the equations all of the relaxation times τ 1 due to the net effective magnetic field. For spin injec- that appear in the diffusion coefficient D, net effective tion/diffusion along(110)axis,τ = (α βˆ)2k2τ /2+ magnetic field h¯ as well as the spin relaxation matrix s1 1 h − i γ2 k6τ /32 and τ′ = (α βˆ)2k2τ /2. For spin injec- are the same. For the inelastic electron-phonon scatter- tiohn/di3ffiusion alonsg1(1¯1h0) a−xis,τ =1i(α+βˆ)2k2τ /2+ ing, it is still possible to write down Eq. (3) with same s1 1 h i relaxation time τ by the elastic scattering approxima- γ2 k6τ /32andτ′ = (α+βˆ)2k2τ /2. Inthelongwave 1 lenhgth3liimit (q s11), Γh 1/τ +(11i+4τ /τ′ )Dq2 and tion. This approximation is valid only for the electron- ≪ + ≃ s s2 s1 acoustic phonon scattering at high temperature and is Γ 1/τ +(1 4τ /τ′ )Dq2 become quadratic func- − ≃ s1 − s2 s1 not valid for the electron-LO phonon scattering. When tionsofq,roughlycorrespondtotheout-of-planeandthe the Coulomb scattering is important, it gives different in-plane relaxation rates respectively. In general both contributionstotheserelaxationtimes. Ontheonehand, of these two decay rates [Eq. (6)] are no longer simple therelaxationtimeτ inthespinrelaxationmatrixisaf- quadratic functions of q. If one uses the quadratic fit- 1 fected by the whole Coulomb scattering. On the other ting toyieldthe spindiffusioncoefficient,oneeither gets hand,sincetheCoulombscatteringamongthesamespin larger (for Γ ) or smaller (for Γ ) values than the true + − specie does not change the motion of the center of mass, spin diffusion coefficient. The accurate way to get the it does not directly affect the spin diffusion coefficient. information of spin diffusion coefficient should be from Therefore the relaxationtime τ in the spin diffusion co- the average of these two rates 1 efficient is only affected by part of the Coulomb scattering. For τ in the net effective magnetic field, it is even Γ=(Γ +Γ )/2=Dq2+(1/τ +1/τ )/2 , (8) 1 + − s s1 more complicated to analyze the role of the Coulomb scatteringsince it is the joint result ofdiffusion and spin which differs from the current widely used formula by precession. Thereforetherelaxationtimesτ inthethree replacing the spin decay rate by the average of the out- 1 different terms should be different in the present of the of-plane and in-plane ones. Coulomb scattering. It would be extremely difficult to get the analytical results when the Coulomb scattering 250 is taken into account. In order to study the drag effect, 15 wenumericallysolvethefullkineticspinBlochequations q=0.63 104 cm−1 Eq. (1), with all the scattering explicitly included. The 200 × numerical scheme is laid out in detail in Appendix B. + /τ10 For simplicity, we first consider the spin diffusion along 150 τ− (100) axis in a symmetrical QW in which the Rashba ) termvanishes. In ourcalculation,the SOC strength,the s p ( electron and impurity densities, and the QW width are τ 100 5 chosen to be γ = 11.4 eV˚A3,31 N = 7.8 1011 cm−2, 100 200 300 N = 1 1011 cm−2 and a = 12 nm resp×ectively. The i × T (K) material parameters are listed in detail in Refs. 30. 50 Our numerical results show that the temporal evolution of TSG can not be fitted by a simple exponential function with desirable accuracy. However, if we use 0 0 1 2 3 double-exponential function, the accuracy can be im- proved more than one order of magnitude. This justi- q (104 cm−1) fies that the temporal evolution of the TSG has indeed double-exponential form in high temperature regime. In FIG.1: (Coloronline)Spinrelaxationtimesτ+(open)andτ− Fig. 1 we present the relaxation times of TSG as a (filled) vs. spin grating wavevector q forthreedifferent tem- function of the grating wave-vector q for temperatures peratures T = 120 (red boxes), 150 (blue triangles) and 295 T = 120,150 and 295 K. It is seen that τ decreases K (black circles). The inset is the ratio τ−/τ+ as a function + of temperature for q=0.63 104 cm−1. monotonicallyasq increaseswhileτ− hasapeakatsome × small q . The wave-vector of the peak red-shifts when 0 the temperature increases. In the inset of Fig. 1, we show the ratio τ /τ as the function of temperature for − + fixed q. One finds that the ratio decreases with temper- B. Numerical Results ature. Our results are in contrast to the predictions of earlier theoretical works that the ratio of these two de- The spin diffusion coefficient obtained by the simpli- cay rates and the position of the peak depend only on fied kinetic spin Bloch equations does not include the the SOC and material parameters, but do not vary with contributionofthespinCoulombdragsincetheelectron- thetemperature.22,32 Recentexperimentshowedthatthe electron Coulomb scattering is neglected. Moreover, the ratio of these two decay rates are indeed decreases with 5 the increase of the temperature.27 It is understood that 0.4 0.075 thetemperaturedependenceofthepeakpositionandra- tio between two decay times originate from the cubic k- termintheDresselhauseffectivemagneticfield. Inearlier 0.3 works, it was assumed that only the linear term is im- 0.05 portant. However,inwide quantumwellswithhighelec- ) ) 1 tron density, the cubic term becomes important. Once 1 − − s tqh0e=c√ub1i5cmtβer/m2(iwshciochnsiisdienrdedep,etnhdeenpteaokftqe0mpmeorvaetusrfer)otmo Γ(ps 0.2 ∆Γ(p aboutq =√15mβˆ/2=√15m(β γk2/4)/2. Since k2 0.025 0 − h i increases with temperature, thus q0 decreases. The tem- 0.1 perature dependence of τ /τ also originates from the + − contribution of the cubic Dresselhaus term. 0 0 τ − (b) 103 0 1 2 3 102 τ q (104 cm−1) − ) ) s s p p ( ( τ τ+ 102τ FIG. 3: (Color online) Γ = (Γ+ +Γ−)/2 and ∆Γ = (Γ+− 101 Γ−)/2 vs. q at T = 295 K. Open boxes/triangles are the (a) relaxation rates Γ calculated from the full kinetic spin τ +/− + Bloch equations. Filled/open circles represent Γ and ∆Γ respectively. Noted that the scale for ∆Γ is on the right hand 100 150200250 100 150 200 250 300 side of the frame. The solid curves are the fitting to Γ and T (K) T (K) ∆Γ respectively. The dashed curvesare guide to eyes. FIG. 2: (Color online) Spin relaxation times τ± vs. temperature for (a) high-mobility sample with q=0.58 104 cm−1 cay rates Γ± fit poorly with a quadratic function of q. and (b) low-mobility sample with q =0.69 104×cm−1. The In contrast, the average decay rate Γ fits pretty well by × dots are theexperiment data from Ref. 27. the function Γ =D q2+1/τ′. The resident error of the s s quadraticfittingforΓistwoordersofmagnitudesmaller Due to the natural of the numerical calculation, it is than those of Γ±. Moreover we find that τs′ is very close not possible to show the results with all possible param- to 4τs/3, inverse of the average of the in-plane and out- eter combinations. However our qualitative conclusions of-planespinrelaxationrate. Forexample,atT =295K, arevalidnotjustforthisparticularparametersetbutfor τs′ is about 107.8 ps compares to 4τs/3 = 111.6 ps with a large range of parameters. In order to check the quan- τs calculated by solving the kinetic spin Bloch equations titative accuracy of our numericalcalculation we further for spacial uniform system with the same parameters.16 show the spin relaxation times as functions of temper- Inspired by Eq.(8), the coefficient of the quadratic term ature together with the experimental data from Ref. 27 Ds can be reasonably assumed to be the spin diffusion in Fig. 2.Curves in Fig. 2(a) are the theoretical spin re- coefficient. In this way one can calculate the spin diffu- laxation times of high mobility (µ = 1.5 104 cm2/Vs) sioncoefficientwiththeeffectofCoulombdragincluded. sample with q =0.58 104 cm−1, while t×he dots are the Thedifference ofΓ+ andΓ− fits wellasalinearfunction experimental data fro×m Ref. 27. Figure 2(b) shows the of q. The linear coefficient of ∆Γ is about 2 D /τ′ s s1 resultoflowmobilitysample(µ=3.5 103cm2/Vs)with from Eq. (6). q =0.69 104 cm−1. In the calculatio×nwe use the finite In Fig. 4 we present the spin diffusion coeffipcient cal- × square well assumption.31 All the parameters we use are culated in the above mentioned method as a function of chosen to be the experimental value if available, eg. the temperature. Forcomparison,wealsoincludethecharge grating wave-vector, the electron density, the quantum diffusion coefficient, which is calculated by solving the well width, and the impurity concentration determined kinetic spin Bloch equations with the initial condition from the mobility. The only adjustable parameters are beingthechargegradientinsteadofthespingradient. It the spin-orbit coupling coefficients γ and α. In the cal- is clearly seen from the figure that D < D . For spin- s c culation, γ is chosen to be 11.4 eV˚A3 and 13.8 eV˚A3 for unpolarized charge diffusion, the electrons move along the high and low mobility samples respectively and α is the same direction and the Coulomb scattering does not settobe0.3β,closetothechoiceinRef.27. Onecansee change the center-of-mass motion, therefore it does not from the figure that our theoretical results are in fairly changethe chargediffusioncoefficientdirectly. However, good agreement with the experiment data. in the spin-polarized transport, spin-up and -down elec- In Fig. 3,we plot the decay rates Γ = 1/τ and trons move against each other and the Coulomb scatter- ± ± their average Γ = (Γ + Γ )/2 and difference ∆Γ = ingthereforeslowsdowntherelativemotionofthesetwo + − (Γ Γ )/2 as functions of q at T = 295 K. The de- spin species and reduces spin diffusion coefficient. This + − − 6 700 with φ L = 4 D2τ τ /sin , (9) s s s1 s 2 600 p φ L = 4 D2τ τ /cos . (10) 0 s s1 s 2 s) 500 In these equations p / 2 (cm Dc cosφ=√τs1τs(4/τs′1−1/τs1−1/τs)/2. (11) D 400 It is noted that if one only considers the Rashba term or the linear Dresselhaus term, τ′ =τ , one then recovers s1 s1 the result L = 2√D τ from linear response theory.22 300 D s s s s It is seen that the spin precession actually prolongs the out-of-plane spin injection length by mixing the fast decay of the out-of-plane spin with the slow decay of the 200 in-plane spin. It is further noted from Eq. (9) that the 100 150 200 250 300 spin injection length L is generally larger than √D τ . s s s T (K) The only exception is when the spin injection is along (110)-directionfortheQWwithequallinearDresselhaus FIG. 4: (Color online) Diffusion coefficient as a function of and Rashba spin-orbit couplings. In this case θ = π/4, temperature. Solid circles: Spin diffusion constant D with s therefore the net effective magnetic field [Eq. (4)] van- Coulomb drag; Open circles: Charge diffusion constant D . c ishes. Consequently the in-plane and out-of-plane spin modes do not mix. Formostofthecases,L isinthesameorderof√D τ , s s s is the so-called spin Coulomb drag effect.23,24,25 From althoughtheformerisusuallylarger. However,thereare the figure one can tell that, in the temperature regime some special cases where φ 0, Ls can be orders of → we study, as the temperature increases, both spin and magnitude different from √Dsτs. Specifically, according charge diffusion coefficients decrease and their difference to Cheng et al.,29 when the spin injection/transport di- also decreases. Therefore the Coulomb drag is stronger rection is along (1¯10) in (001) QW, Ls becomes larger in the low temperature regime. However, even at room and larger as α approaches β, regardless of the direction temperature the Coulomb drag is still strong enough to ofspinpolarization. Atthelimitofα=β,thespininjec- reduce the diffusion coefficient by 30 %. These results tionlengthtrendstoinfinity whenthe cubicDresselhaus quantitatively agree with those of Refs. 23,24. term is ignoredand the spin oscillates with a spatial period of 2π/(2mβ).29 As τ is finite and D changes little It is also pointed out that the reduction of spin dif- s s for different grating directions,there is no way to obtain fusion coefficient mostly comes from the Coulomb drag. infinite L from √D τ . The SOC only has slightly effect on the diffusion coeffi- s s s The infinite injection length and the finite oscillation cient since the SOC is verysmall comparedto the Fermi period can also be understood from the TSG point of energy. The numerical result shows that removing the view. Without the cubic Dresselhaus term, 1/τ = SOC only changes spin diffusion coefficient up to one s2 (α β)2 k2τ /2 approaches zero and 1/τ 1/τ = tenth percent for the system we studied. − h 1i s1 → s 2β2 k2τ when α β. Consequently τ = (√Dq 1 ± h i → ± 1/√τ )−2 when α = β. It is then straightforward to see s that τ becomes infinite provided q = q = 1/√Dτ = − 0 s 2mβ. Therefore the steady-state spin injection along III. STEADY-STATE SPIN INJECTION (1¯10) axis is dominated by this non-decay TSG mode which is responsible for the infinite spin injection length In this section we discuss how to obtain the steady- and the spatial oscillation period 2π/q0 =2π/(2mβ). statespininjectioninformationfromtheTSGsignal,i.e., When the electron-phonon and the electron-electron to find out the relation between the steady-state spin Coulombscatteringsare takeninto account,φ should be injection length L and spatial spin oscillation “period” revised accordingly. Unfortunately, there are no analyti- s L0 and the spin diffusion constant Ds together with the calexpressionsforτs,τs1 andτs′1 whenallthescatterings spin relaxation time τ . areincluded. Nevertheless,onecandirectlyusetheTSG s result to calculate the static injection parameters. The We first show from the simplified solution presented numerical result indicates that the two decay rates obey in Sec. IIA that the steady-state spin injection can be extracted from the TSG signal by integrating the TSG Γ =D q2+1/τ′ (cq+d) (12) ± s s± signal Eq. (5) over the time from 0 to and the wave- ∞ where c and d are the fitting parameters to vector from to . Form Eqs. (6) and (7), the in- tegrated TSG−∞reads ∞S (x) = S (0)e−x/Lscos(x/L +ψ) ∆Γ=cq+d . (13) z z 0 7 α Injection D (cm2/s) τ′ (ps) c (µm/ps) d (ps−1) LT (µm) LT (µm) LS (µm) LS (µm) direction s s s 0 s 0 α=0 (100) 246 107.8 0.02 3.5 10−4 2.27 2.44 2.28 2.46 × α=β (110) 243 49.7 9.4 10−3 0.012 0.92 5.1 1 4.9 × α=β (1¯10) 250 28.1 0.058 2.7 10−3 2.7 0.86 2.1 0.9 − × TABLEI:ComparisonofspininjectionlengthLsandspatialoscillationlengthL0alongdifferentinjectiondirectionsatT =295 Kfrom differentapproaches. Thesuperscripts“S”and“T” standforLs andL0 obtained from theinjection calculation inthe steady state and those from TSG parameters by using Eqs. (14) and (15) respectively. Again by integrating the TSG signal over the time and IV. CONCLUSION wave-vectorq with above fitted Γ , one obtains ± In conclusion, we study the evolution of TSG by solv- ingthekineticspinBlochequationswithperiodicbound- Ls = 2Ds/ |c2−4Ds(1/τs′ −d)| , (14) arycondition. Fromthesimplifiedequations,weanalyti- L = 2D /c . (15) callyshowthatduetothespinprecessionthespinsignal 0 s p decays double-exponentially instead of simple exponentially, even in the diffusive regime. Neither of these two We stress that these two equations give the right spin decay rates are simple quadratic functions of the grat- injection length and the spin oscillation period in the ing wave-vector q. However their average value depends presenceoftheSOC.Fromtheexperimentpointofview, quadratically on q and the corresponding coefficient of onecanmonitorthetimeevolutionofTSGwithdifferent the quadratic term is the right diffusion coefficient Ds. wave-vectorsq andobtain the correspondingdecayrates Thereforeitismoreaccuratetoyieldthediffusioncoeffi- Γ . From the q-dependence of the decay rates, one can cient from q dependence of the average of the two decay ± calculate the spin injection length and spin oscillation rates. periodfromEqs.(14) and(15). It is notedthat Eq.(14) We further show that the corresponding solution from naturallygives the infinite injection lengthin the special the full kinetic spin Bloch equations which include all of case29discussedaboveasc=2 D /τ andd=0sothat the scattering mechanisms, especially the Coulombscat- s s the denominator in Eq. (14) tends to zero. In contrast, tering,isalsointheformofthedoubleexponentialdecay. p √D τ always remains finite. Fromtheq-dependenceoftheaveragedecayrate,onecan s s calculate the spin diffusion coefficient with the Coulomb In order to check that the accuracy of this approach drag effect included. It is shown that the Coulomb drag [Eqs. (14) and (15)], we compare L and L obtained s 0 effect is stronger in low temperature and decreases with from directly numerically solving the kinetic spin Bloch the increase of temperature. However, even at room equations [Eq. (1)] for spin injection in the steady state temperature the Coulomb drag is still an important fac- as described in Refs. 20,29, with those from the TSG tor which reduces the spin diffusion coefficient markedly approach. In Table I we list L and L obtained from s 0 compared to the charge diffusion coefficient. the TSGsignalandfromthe steady-statesolutionofthe We also show that by using the TSG result one can kinetic spin Bloch equations for spin injection direction obtain the characteristic steady-state transport parame- along (100) axis in symmetrical (001) QW, as well as ters, such as the injection length and the spatial oscil- (110) and (1¯10) axes in asymmetrical (001) QW with lation length, from diffusion coefficient D and the spin equal Dresselhaus and Rashba coupling. For the sake of s relaxation times. We point out that in a system with clarity, we also list the corresponding fitted parameters the SOC,the wildly adoptedrelationL =√D τ inthe τ′, c and d in the table. It is noted that in the calcula- s s s s literature is generally quantitatively inaccurate and can tionthe cubic Dresselhaustermis included. One cansee be even qualitatively wrong for some special cases. The from the table that even though the injection and the accurate way to extract the spin injection and spatial oscillation lengths at different conditions are quite dif- oscillation lengths directly from the TSG decay rates at ferent, the spin diffusion coefficients D are almost the s different grating wave-vector, which are experimentally same. Their differences are within the numerical fitting measurable, is proposed. We believe this investigation error. ThisconfirmsthattheSOCistoosmallcompared is important in fully understanding the TSG signals in to the Fermi energy to affect D . Moreover, for all the s experiment. caseswestudy,L andL obtainedfromthesetwometh- 0 s odsagreewitheachotherverywell. Althoughwe should point out that the accuracy of L is higher than L due 0 s to the numericalerroranalysis. Sincec2 4D (1/τ′ d) Acknowledgments − s s− is close to 0, its numerical error is much larger than c. Therefore L calculated from TSG is not as accurate as This work was supported by the Natural Science s L . Foundation of China under Grant Nos. 10574120 and 0 8 10725417,theNationalBasicResearchProgramofChina (4) can be derived by expanding Eq. (1) in angular under GrantNo. 2006CB922005,the Knowledge Innova- momentums.5,6 Neglecting the Hartree-Fock term, the tion ProjectofChinese Academy of Sciences andthe US inelasticscatteringandtheelectricfield,theFouriercom- Army Research Office. The authors would like to thank ponent of the l-th order of density matrix obeys the fol- T. Korn for his critical reading of this manuscript. One lowing equation: ofthe authors(MWW) wouldliketo thank J.Fabinand C.Schu¨lleratUniversitätRegensburg,Germanyforhos- pitality where this work was finalized, and the Robert- Bosch Stiftung and GRK 638 for financial support. APPENDIX A: DERIVATION OF SIMPLIFIED SOLUTION By using a similar method for calculating the spin relaxation due to the Dresselhaus effect, Eqs. (3) and ∂ρ (q,k,t) ikq l ρ (q,k,t)+ρ (q,k,t) i h (k) σ,ρ (q,k,t) l+1 l−1 l−m m ∂t − 2m − · (cid:18) (cid:19) m (cid:20) (cid:21) X ρ (q,k,t) l = (1 δ ), (A1) l,0 − τ − l whereρl(q,k,t)= e−ilφρk(q,t)dφ/2π andk=(kcosφ,ksinφ,0). WhenboththeDresselhausandtheRashbaterms are taken into account, R h(k) = γk (π/a)2 k2/4 ( cosφ,sinφ,0)+γk3/4( cos3φ, sin3φ,0) − − − − +αk(sinφ, cosφ,0), (A2) (cid:0) − (cid:1) = βˆk( cosφ,sinφ,0)+αk(sinφ, cosφ,0)+γk3/4( cos3φ, sin3φ,0). (A3) − − − − Therefore there are four effective magnetic field compo- term. The former reads nentswhichdonotvanish,namelyh (k)andh (k). It ±1 ±3 should be noted that, in quasi-two-dimensional system, kx∂f(x) kmx3f(x)−4f(x−2∆∆xx)+f(x−2∆x) kmx >0 . bthyet±he1ccoumbipcotneernmtsβôf=theγ[D(πr/esas)e2lhauks2/te4r]m=arβemoγdki2fi/e4d. m ∂x →( −kmx3f(x)−4f(x+2∆∆xx)+f(x+2∆x) kmx <0 (B1) − − When the scattering is strong, one can drop the terms Theboundaryconditionforxischosentobetheperiodic withτ higherthanthefirstorderandrearrangeEq.(A1) l one when we calculate the TSG problem or fixed when to obtain Eq. (3). The only difference is that in the spa- we calculate the steady-state injection problem.20 In the tialinhomogeneoussystem, we haveadditionalterms re- energy ( ) and angular (φ) space, the drift term reads lying on the wave-vector q. These additional terms give E rise to the second and third terms in Eq. (3). ∂g(k ,k ) eE(x) ∂ eE(x) x y = [2√ cosθg( ,θ)] ∂kx √2m (cid:18)∂E E E ∂ g( ,θ)sinθ E . (B2) −∂θ √ APPENDIX B: NUMERICAL SCHEME E (cid:19) Similar to the diffusion term, one can easily write down In order to solve the kinetic spin Bloch equations nu- the second order up-wind differential schemes for and E merically, one has to discretize the real space, the mo- θ respectively. The numerical schemes for the spin pre- mentum space as well as the time. The real space is cession and the scattering terms are laid out in detail in divided into segments with equal length. The momen- Refs. 20,30. tum space is divided into grids of equal energy and an- We apply the third order semi-implicit Adams- gular differences.30 The second order up-wind differen- Bashforth scheme for the time differential33 to achieve tialschemeisappliedforthediffusiontermandthe drift higher accuracyin temporalevolution. This scheme also 9 saves CPU time. The differential scheme is then given algorithm.34 by The accuracyofthe numericalscheme usedinthis pa- perishigherthantheoneusedinourpreviousworks.19,20 ρk(x,t+∆t) ρk(x,t) 5 Themainnumericalerrorscomefromthedriftanddiffu- − = F[ρk(x,t+∆t)] ∆t 4 sionpartssincethegridsizeofrealspaceandmomentum 23 space is limited by the computing power. We find that F[ρk(x,t)]+3/4F[ρk(x,t ∆t)]+ G[ρk(x,t)] − − 12 the accuracy of the temporal evolution does not change 16 5 the result of the final steady-state spin transport too G[ρk(x,t ∆t)]+ G[ρk(x,t 2∆t)] . (B3) −12 − 12 − much. It is therefore expected that the present numerical scheme and the previous one give very close results Here F[ρk(x,t)] denotes the drift and diffusion terms on the steady-state transport properties. However, the (the second and third terms) in Eq. (1) and G[ρk(x,t)] presentschemeenablesustoalsostudythetimesensitive stands for the spin precession and the scattering terms phenomenons such as the TSG to a sufficient accuracy. 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