The drift-diffusion formalism for spin-polarized carrier transport in
semiconductors is generalized to include spin-orbit coupling. The theory is
applied to treat the extrinsic spin Hall effect using realistic boundary
conditions. It is shown that carrier and spin diffusion lengths are modified by
the presence of spin-orbit coupling and that spin accumulation due to the
extrinsic spin Hall effect is strongly and qualitatively influenced by boundary
conditions. Analytical formulas for the spin-dependent carrier recombination
rates and inhomogeneous spin densities and currents are presented.Comment: 5 pages, 3 figure

Fabian, J.

Sarma, S. Das

Tse, Wang-Kong

Zutic, I.

English

arXiv

The drift-diffusion formalism for spin-polarized carrier transport in semiconductors is generalized to include spin-orbit coupling. The theory is applied to treat the extrinsic spin Hall effect using realistic boundary conditions. It is shown that carrier and spin-diffusion lengths are modified by the presence of spin-orbit coupling and that spin accumulation due to the extrinsic spin Hall effect is strongly and qualitatively influenced by boundary conditions. Analytical formulas for the spin-dependent carrier recombination rates and inhomogeneous spin densities and currents are presented

Fabian, Jaroslav

Zutic, Igor

University of Regensburg Publication Server

Spin accumulation in the extrinsic spin Hall effectWang-Kong Tse,1 J. Fabian,2 I. Žutić,1,3 and S. Das Sarma11Condensed Matter Theory Center, Department of Physics, University of Maryland at College Park, College Park,Maryland 20742-4111, USA2Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany3Center for Computational Materials Science, Naval Research Laboratory, Washington, D.C. 20735, USAReceived 2 August 2005; revised manuscript received 3 October 2005; published 7 December 2005The drift-diffusion formalism for spin-polarized carrier transport in semiconductors is generalized to includespin-orbit coupling. The theory is applied to treat the extrinsic spin Hall effect using realistic boundaryconditions. It is shown that carrier and spin-diffusion lengths are modified by the presence of spin-orbitcoupling and that spin accumulation due to the extrinsic spin Hall effect is strongly and qualitatively influencedby boundary conditions. Analytical formulas for the spin-dependent carrier recombination rates and inhomo-geneous spin densities and currents are presented.DOI: 10.1103/PhysRevB.72.241303 PACS numbers: 72.25.Dc, 72.25.Hg, 75.80.qIn the presence of spin-orbit S-O coupling, either due toimpurities or due to host lattice ions, carriers of oppositespins tend to scatter into opposite directions. With an electricfield induced longitudinal motion under bias, the S-O scat-tering results in a transverse spin current and spin accumu-lation, as predicted by D’yakonov and Perel’,1,2 and laterrevisited by others.3 This effect, which is now called theextrinsic spin Hall effect SHE,4 has been recently demon-strated experimentally in n-GaAs and n-InGaAs thin films5and in two-dimensional electron gas confined within 110AlGaAs quantum wells.6 The signature of the effect is oppo-site spin accumulation at the edges of the sample, with spinpolarization perpendicular to the transport plane.This paper has two goals. First, we present a formalismfor carrier drift and diffusion in inhomogeneous spin-polarized semiconductors in the presence of S-O couplingand spin-dependent band-to-band electron-hole recombina-tion. The formalism, which is a generalization of a previousspin and charge drift-diffusion theory,7,8 applies to both uni-polar and bipolar cases, the former being a subclass of thelatter. Second, we apply the formalism to explain the mainqualitative features of spin accumulation in the extrinsicSHE in the optical orientation experiment, for two differentboundary conditions: a uniform generation of electron-holepairs, and b edge generation of nonequilibrium electrons.In both cases spin accumulation throughout the sample iscalculated analytically. We find that S-O interaction modifiesthe carrier and spin diffusion lengths and that the spin accu-mulation profile depends, qualitatively, on the specificboundary conditions, implying that interpretation of extrinsicSHE requires detailed case-by-case considerations for spe-cific experimental and sample geometries.Consider spin-polarized transport in an inhomogeneousnonmagnetic semiconductor in the presence of electric fieldE. If S-O coupling is present, causing skew scattering andside jump, the phenomenological expression for the carrierc=n for electrons and c= p for holes charge current densityin the ith direction is readily obtained by generalizing theDyakonov-Perel’ prediction1,2Jc,i = qccEi ± qDcic + qccijzEj + qcijz jc.1Here the upper lower sign is for electrons holes and  isthe spin index; q is the proton charge. The first two terms areconventional longitudinal carrier drift and diffusion, re-spectively, with  and D denoting the spin-dependent mobil-ity and diffusivity. The third fourth term represents the ef-fects of skew spin-orbit scattering and side jump on driftdiffusion. The effects of the scattering are in the transversedirection to E and are opposite for spin-up and spin-downcarriers. Holes are treated here as spin doublets, which isappropriate in low-dimensional structures with heavy andlight hole band splitting; otherwise hole spin does not matter,as we will argue below. The corresponding transport param-eters are transverse mobility  and transverse diffusivity ;they are proportional to the S-O coupling strength.It is more illuminating to introduce the charge,Jc=J↑+J↓, and spin, Js=J↑−J↓, currents. In terms of carrierc=c↑+c↓ and spin sc=c↑−c↓ densities, the currents areJc,i = qcc + scscEi ± qDcic + Dscisc+ qijzEjcsc + scc + qijzc jsc + sc jc , 2Jsc,i = qscc + cscEi ± qDcisc + Dscic+ qijzEjcc + scsc + qijzc jc + sc jsc . 3The transverse carrier charge and spin mobilities are givenrespectively by c= c↑+c↓ /2 and sc= c↑−c↓ /2, whilethe transverse carrier charge and spin diffusivities arec= c↑+c↓ /2 and sc= c↑−c↓ /2. The correspondinglongitudinal quantities are defined similarly.Equations 2 and 3 succinctly describe the appearanceof the transverse spin drift and diffusion in the presence oflongitudinal charge transport, which is the essence of theextrinsic SHE. If the longitudinal current is spin polarized,the above equations describe the anomalous Hall effect9 andthe appearance of the transverse Hall voltage.PHYSICAL REVIEW B 72, 241303R 2005RAPID COMMUNICATIONS1098-0121/2005/7224/2413034/$23.00 ©2005 The American Physical Society241303-1To further develop the formalism, we need to includeelectron-hole recombination and spin relaxation. In the pres-ence of S-O coupling, spin-dependent selection rules10 forband-to-band transitions need to be considered. In general,the continuity equation reads±iJc,i/q = B1cc − c0c0 + B2cc  − c0c0+ c − c/2T1c. 4Here c¯ is pn if c is np ,c0 is the equilibrium carrier den-sity, and T1c is the T1 time for spin flipping.The spin-preserving recombination rate coefficient B1, aswell as the spin-flip coefficient B2, can be calculated by gen-eralizing the unpolarized case.11,12 The valence band of zinc-blende semiconductors consists of three subbands: heavyhole, light hole, and split-off hole bands. We neglect thesplit-off band as the energy splitting kBT at temperaturesT lower than or around room temperature. By explicitly tak-ing into account the angular momentum of the heavy holeand light hole states, and using the optical selection rules forthe states,10 we arrive at the following expressions for spin-conserving and spin-flip recombination constants:B1 = Cmh/mc + mh3/2 +23 ml/mc + ml3/2mh3/2 + ml3/2 , 5B2 =13Cmh/mc + mh3/2mh3/2 + ml3/2 , 6where C depends on the “maximum” electron energy	max=g+kBT /2 g is the energy gap and temperature,C	max,T =4e2	2m2c32	2kBT 3/2P2nr	 . 7Here P is the momentum matrix element for optical transi-tions, nr is the refractive index, mc mh and ml is the bandmass of electrons heavy holes and light holes.Equations 1 and 4, together with Poisson’s equationform a closed set of nonlinear equations whose solution de-termines charge and spin densities and currents in a semicon-ductor with S-O scattering included. In general, these equa-tions need to be solved numerically for specific cases ofinterest. Our next goal is to introduce qualitative features ofspin accumulation in two cases of experimental interest thatallow analytical solutions and form a starting point to discussthe concepts and issues to be encountered in more complexsituations involving SHE and S-O coupling effects in trans-port. We consider a p-type semiconductor with nondegener-ate electron minority density induced optically. The result-ing spin accumulation via extrinsic SHE can be deduced in amanner similar to spin orientation experiments. We assumethat the injected electron density is well below the donordensity. Further simplification follows from the fact that mo-bilities and diffusivities are spin independent in the nonde-generate regime.7 Finally, we assume unpolarized holes sincehole spin relaxation in zinc-blende semiconductors is ex-tremely fast.13 The only carriers of interest are then spin-polarized electrons.Using the above assumptions, Eqs. 2–4, give the drift-diffusion equations for electron spin and carrier density2s + q/kBTs  · E + E · s + q/kBTn Ez+ wp + 1/T1ns = 0, 82n + q/kBTn  · E + E · n + q/kBTs Ez+ wnp − n0p0 = 0, 9where w= B1+B2 /2 and =n /n=n /Dn characterizes theS-O coupling strength, n0 , p0 are the equilibrium electron andhole densities. The spin quantization axis is taken to be z. Wetake E=Eyˆ and consider x to be the transverse direction, theslab boundaries being at x=0 and x=a, so that all the quan-tities of interest will have x dependence only. Denoting theelectron recombination time as n=1/ wNa, where Nais the acceptor density, and spin relaxation time as s=1/ n−1+T1n−1, Eqs. 8 and 9 becomes¨ + qE/kBTn˙ − s/Ls2= 0, 10n¨ + qE/kBTs˙ − n − n0/Ln2= 0, 11where derivatives with respect to x is denoted by overdots,the longitudinal spin and charge diffusion lengths are definedas Ls=Dss and Ln=Dnn, respectively. In deriving theabove equations we have neglected terms of order s2, sn,and n2 relative to n=n−n0.14 Spin-charge coupling in Eqs.10 and 11 is apparent through the first-order derivativesof spin and charge densities.We first solve the transverse spin and carrier diffusion forthe case of a uniformly illuminated slab boundary conditionBC I, with electron-hole spin-unpolarized generation rate G.We assume that carrier recombination at the edges is notsignificant so that it is reasonable to impose a uniform herezero electron transverse current: Jnxx=0. As for spin cur-rent, we take Jsx=0=Jsx=a=0, implying that spin-flipscattering at the edges is moderate. The solution to the drift-diffusion Eq. 10 in the presence of S-O scattering issx = qELskBTGn secha/2Lssinhx/L − a/2L . 12The spin-polarization profile is given by x=sx /Gn,since Gnn0 is the average electron density generated insteady-state conditions under illumination. This simple solu-tion demonstrates the essential physics behind spin accumu-lation in extrinsic SHE: i Spin accumulation increases lin-early with E, with possible slight electric field modulationdue to the dependence of Ls=LsE not discussed here. iiThe magnitude of spin polarization is proportional to thestrength of the S-O scattering as well as to the ratio of thevoltage drop over min Ls ,a and thermal energy. iii ForaLs the accumulation at the edges is linearly proportionalto a, while for aLs, the accumulation is independent of a.iv While xx for aLs, spin accumulation is signifi-cant only within the spin diffusion length from the edgeswhen aLs.A question now arises: How universal i.e., independentof boundary conditions is the qualitative behavior discussedTSE et al. PHYSICAL REVIEW B 72, 241303R 2005RAPID COMMUNICATIONS241303-2above? To answer this question we introduce differentboundary conditions BC II, describing the physics of car-rier injection at x=0 and x=a, while assuming vanishing spincurrents at the edges: nx=0=n1, nx=a=n2; Js,xx=0=Js,xx=a=0. These conditions mimic the case of a p-dopedbase in a pnp spin-polarized transistor,15 where the electroninjection level can be controlled by the biases to the emitterand collector. In the case discussed here the longitudinal cur-rent would flow perpendicular to the transistor current, whichwould thus be spin polarized due to the extrinsic SHE.The vanishing boundary conditions for Js,x reduce tos˙x=0=−qEn1 /kBT and s˙x=a=−qEn2 /kBT. SolvingEqs. 10 and 11 with the above boundary conditions givesthe spin and electron densities inside the slabsx =qEkBTL1−2− L2−2	 1L1C1x − 1L2C2x , 13andnx − n0 =1L1−2− L2−2L1−2− Ls−2S1x− L2−2− Ls−2S2x , 14where for convenience we have defined the functionsC1,2 = 	n1 cosh a − xL1,2  − n2 cosh  xL1,2 sinh  aL1,2 ,15S1,2 = 	n1 sinh a − xL1,2  + n2 sinh  xL1,2 sinh  aL1,2 .16Here we introduce new transverse spin-charge diffusionlengths, L1 and L2,L1,2−2=  ± 2 − 1/LsLn2, 17where = 1/2Ls−2 + Ln−2 + qE/kBT2 . 18There is a critical value of the field that separates the regimesof strong and weak spin-charge coupling—Eqs. 10 and 11reduce to the ordinary spin and charge diffusion equationswhen EEs, En, where Es=kBT / qLs and En=kBT / qLn are the values of the critical fields with respectto spin and charge diffusion. In this case Ls−2+Ln−2 /2and L1,2Ls,n. When EEs, En , becomes dependent onthe electric field and it is in this regime the spin-field relationdeviates from linearity.For quantitative understanding we take our semiconductorto be GaAs at room temperature,16 with doping densityNa=31015 cm−3, and transverse size of a=6 m which ismuch greater than Ls. The S-O scattering strength is takento be =10−4, reflecting weak S-O coupling in GaAs.Finally, the boundary conditions for electron density aren0=21013/cm3 and na=51013/cm3. Figure 1 showsthe profiles of spin and electron densities, as well as spinpolarization =s /n, for several values of E. In contrast to thepurely diffusive behavior exhibited by n, spin density alongthe slab exhibits weakly oscillatory behavior. Both spin den-sity and spin polarization attain maximum magnitudes at theedges. We find that spin polarization is enhanced by a de-cade when considering the extrinsic SHE at 77 K. What isinteresting is, unlike in BC I conclusion iv, that spin ac-cumulation here is significant throughout the sample, not justwithin Ls from the edges. This oscillatory pattern is due tothe existence of spin-charge coupling that couples the spinand charge diffusion lengths together into two spin-chargediffusion lengths Eqs. 17 and 18, so that the net spin-density profile Eq. 13 can be thought of as the interferencebetween two spin-density profiles, each having a character-istic spin-charge diffusion length L1 or L2. Since, as a resultof spin Hall effect, each of these terms changes sign near theboundaries, they interfere destructively and constructivelyalong the slab giving rise to the oscillatory pattern in Fig. 1similar arguments apply for the electron density, but sinceeach of these terms is always positive, they add up construc-tively. We note that similar oscillatory behavior is also ob-served in a number of recent papers17 on the intrinsic SHE,and in a recent experiment.6 In the presence of magnetic fieldFIG. 1. Calculated spin density s, electron density n, and polar-ization  for BC II. We have used =10−4 and applied electric fieldstrengths 0.125, 0.250, and 0.500 kV/cm. While a signature of theextrinsic SHE is the opposite spin accumulation at the edges of thesample, spin accumulation is large also inside.FIG. 2. Calculated spin current Js in x top and y bottomdirections for BC II and parameters as in Fig. 1.SPIN ACCUMULATION IN THE EXTRINSIC SPIN … PHYSICAL REVIEW B 72, 241303R 2005RAPID COMMUNICATIONS241303-3externally applied or effective field due to, e.g., Rashba S-Ocoupling, the oscillatory behavior comes not only from thecoupling of spin and charge but also the coupling in betweenthe different components of the spin.17 Now spin current isnot conserved and flows inside the sample, being restrictedto zero by our choice of boundary conditions see Fig. 2.From Eq. 3 we obtain the spin current density in the xdirectionJs,xx = −qnEL1−2− L2−2Ls2 S1x − S2x , 19and in the y directionJs,yx =1L1−2− L2−2qnE2/kBT + nL1−2− Ls−2C1x/L1− qnE2/kBT + nL2−2− Ls−2C2x/L2 . 20Spin current in the x direction flows in the direction of thespin gradient, while the y component changes sign inside theslab, reflecting the fact that spin-up and spin-down electronsare deflected into opposite directions due to S-O scatteringby impurities. Finally, we wish to see at what value of theelectric field spin-charge diffusion lengths L1,2 will be modi-fied and induce nonlinear behavior in . Figure 3 showsE for different . Spin polarization varies linearly with E,except at electric field as large as 105 kV/cm and S-O cou-pling 1, a clearly unphysical case considered here only toillustrate the scope of linear behavior.In conclusion, we have presented a drift-diffusion formal-ism which takes into account S-O scattering and spin-dependent carrier recombination. We have calculated spinaccumulation in the extrinsic spin Hall regime and intro-duced S-O dependent spin-charge diffusion lengths. We havefound that spin accumulation is strongly influenced byboundary conditions and thus by specific spintronic devicedesign. Our theory is applicable to any complex device set-ting in which extrinsic spin Hall effect is expected to play arole. We expect similar strong dependence of the “intrinsic”SHE on boundary conditions too, making it difficult to dis-tinguish intrinsic and extrinsic SHE in general.This work was supported by the US-ONR and NSF.1 M. I. D’yakonov and V. I. Perel, JETP Lett. 13, 467 1971.2 M. I. D’yakonov and V. I. Perel, Phys. Lett. 35A, 459 1971.3 J. E. Hirsch, Phys. Rev. Lett. 83, 1834 1999; S. Zhang, ibid. 85,393 2000; H.-A. Engel, et al., ibid. 95, 166605 2005; W.-K.Tse and S. Das Sarma, cond-mat/0507149 unpublished; E. M.Hankiewicz and G. Vignale, cond-mat/0507228 unpublished.4 The term “extrinsic,” to be distinguished from the “intrinsicSHE,” was used in J. Sinova et al., Phys. Rev. Lett. 92, 1266032004.5 Y. K. Kato et al. Science 306, 1910 2004.6 V. Sih et al., Nat. Phys. 1, 31 2005.7 I. Žutić et al., Phys. Rev. Lett. 88, 066603 2002.8 J. Fabian et al., Phys. Rev. B 66, 165301 2002.9 R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 1954; P.Noziéres and C. Lewiner, J. Phys. Paris 34, 901 1973.10 M. I. D’yakonov and V. I. Perel, in Optical Orientation ElsevierScience, New York, 1984.11 H. Barry Bebb and E. W. Williams, in Semiconductors and Semi-metals Academic Press, New York, 1972, Vol. 8.12 G. Lasher and F. Stern, Phys. Rev. 133, A553 1964.13 I. Žutić et al., Rev. Mod. Phys. 76, 323 2004.14 R. A. Smith, Semiconductors Cambridge University Press, NewYork, 1978.15 J. Fabian, et al. cond-mat/0211639 unpublished; J. Fabian, I.Žutić, Phys. Rev. B 69, 115314 2004.16 For T=300 K, the transport parameters for the minority electronare taken as Ref. 7: diffusivity Dn=103.6 cm2 s−1, mobilityn=4000 cm2 V−1 s−1, recombination rate w= 1/310−5 cm3 s−1 and spin-flip time T1=0.2 ns. From these the cal-culated values of electron-hole recombination and spin diffusionlengths are, respectively, Ln=1 m and Ls=0.8 m. The intrin-sic concentration is ni=1.8106 cm−3.17 A. A. Burkov et al. Phys. Rev. B 70 155308 2004; E. I. Rashba,cond-mat/0507007, Physica E to be published; I. Adagideli,cond-mat/0506531, Phys. Rev. Lett. to be published; B. K.Nikolic et al., Phys. Rev. Lett. 95, 046601 2005.FIG. 3. Calculated spin polarization  at x=0 as a function ofelectric field E for BC II and =10−4, 10−2, and 1. Only in the lastunphysical case  starts to saturate for large E, as the dependenceof L1 and L2 on E sets in.TSE et al. PHYSICAL REVIEW B 72, 241303R 2005RAPID COMMUNICATIONS241303-4

Spin Accumulation in the Extrinsic Spin Hall Effect

Abstract

Similar works

Full text

Available Versions

University of Regensburg Publication Server