Spin accumulation in the extrinsic spin Hall effect

Spin accumulation in the extrinsic spin Hall effect

Wang-Kong Tse,¹ J. Fabian,²I. Žutić,^1,3and S. Das Sarma¹

1Condensed Matter Theory Center, Department of Physics, University of Maryland at College Park, College Park, Maryland 20742-4111, USA

2Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany

3Center for Computational Materials Science, Naval Research Laboratory, Washington, D.C. 20735, USA 共Received 2 August 2005; revised manuscript received 3 October 2005; published 7 December 2005兲

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 inhomo- geneous spin densities and currents are presented.

DOI:10.1103/PhysRevB.72.241303 PACS number共s兲: 72.25.Dc, 72.25.Hg, 75.80.⫹q

In the presence of spin-orbit共S-O兲coupling, either due to impurities or due to host lattice ions, carriers of opposite spins tend to scatter into opposite directions. With an electric field induced共longitudinal兲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 later revisited by others.³ This effect, which is now called the extrinsic spin Hall effect共SHE兲,⁴ has been recently demon- strated experimentally inn-GaAs and n-InGaAs thin films⁵ and in two-dimensional electron gas confined within 共110兲 AlGaAs quantum wells.⁶The signature of the effect is oppo- site spin accumulation at the edges of the sample, with spin polarization perpendicular to the transport plane.

This paper has two goals. First, we present a formalism for carrier drift and diffusion in inhomogeneous spin- polarized semiconductors in the presence of S-O coupling and spin-dependent band-to-band electron-hole recombina- tion. The formalism, which is a generalization of a previous spin and charge drift-diffusion theory,^7,8applies to both uni- polar and bipolar cases, the former being a subclass of the latter. Second, we apply the formalism to explain the main qualitative features of spin accumulation in the extrinsic SHE in the optical orientation experiment, for two different boundary conditions:共a兲uniform generation of electron-hole pairs, and 共b兲 edge generation of nonequilibrium electrons.

In both cases spin accumulation throughout the sample is calculated analytically. We find that S-O interaction modifies the carrier and spin diffusion lengths and that the spin accu- mulation profile depends, qualitatively, on the specific boundary conditions, implying that interpretation of extrinsic SHE requires detailed case-by-case considerations for spe- cific experimental and sample geometries.

Consider spin-polarized transport in an inhomogeneous nonmagnetic semiconductor in the presence of electric field E. If S-O coupling is present, causing skew scattering and side jump, the phenomenological expression for the carrier 共c=nfor electrons andc=p for holes兲charge current density in the ith direction is readily obtained by generalizing the Dyakonov-Perel’ prediction^1,2

Jc␭,i=q␮c␭c_␭Ei±qDc␭⳵ic_␭+q␭␯c␭c_␭⑀ijzEj+q␭␦c␭⑀ijz⳵jc_␭. 共1兲 Here the upper共lower兲sign is for electrons共holes兲and␭ is the spin index;qis the proton charge. The first two terms are conventional 共longitudinal兲 carrier drift and diffusion, re- spectively, with␮^andDdenoting the spin-dependent mobil- ity and diffusivity. The third共fourth兲term represents the ef- fects of skew spin-orbit scattering and side jump on drift 共diffusion兲. The effects of the scattering are in the transverse direction to E and are opposite for spin-up and spin-down carriers. 共Holes are treated here as spin doublets, which is appropriate in low-dimensional structures with heavy and light hole band splitting; otherwise hole spin does not matter, as we will argue below兲. 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, J_c=J_↑+J_↓, and spin,J_s=J_↑−J_↓, currents. In terms of carrier 共c=c_↑+c_↓兲and spin共sc=c_↑−c_↓兲 densities, the currents are

J_c,i=q共␮cc+␮scsc兲Ei±q共Dc⳵ic+Dsc⳵isc兲

+q⑀ijzEj共␯csc+␯scc兲+q⑀ijz共␦c⳵jsc+␦sc⳵jc兲, 共2兲

J_sc,i=q共␮scc+␮csc兲Ei±q共Dc⳵isc+Dsc⳵ic兲

+q⑀ijzEj共␯cc+␯scsc兲+q⑀ijz共␦c⳵jc+␦sc⳵jsc兲. 共3兲 The transverse carrier charge and spin mobilities are given respectively by␯c=共␯c↑+␯c↓兲/ 2 and␯sc=共␯c↑−␯c↓兲/ 2, while the transverse carrier charge and spin diffusivities are

␦c=共␦c↑+␦c↓兲/ 2 and ␦sc=共␦c↑−␦c↓兲/ 2. The corresponding longitudinal quantities are defined similarly.

Equations共2兲 and共3兲 succinctly describe the appearance of the transverse spin drift and diffusion in the presence of longitudinal charge transport, which is the essence of the extrinsic SHE. If the longitudinal current is spin polarized, the above equations describe the anomalous Hall effect⁹and the appearance of the transverse Hall voltage.

PHYSICAL REVIEW B72, 241303共R兲 共2005兲

RAPID COMMUNICATIONS

1098-0121/2005/72共24兲/241303共4兲/$23.00 241303-1 ©2005 The American Physical Society

To further develop the formalism, we need to include electron-hole recombination and spin relaxation. In the pres- ence of S-O coupling, spin-dependent selection rules¹⁰ for band-to-band transitions need to be considered. In general, the continuity equation reads

±⳵iJc␭,i/q=B₁共c_␭c_␭−c_0␭c_0␭兲+B₂共c_␭c_␭−c_0␭c_0␭兲

+共c_␭−c_␭兲/2T1c. 共4兲

Here¯cisp共n兲ifcisn共p兲,c₀ is the equilibrium carrier den- sity, andT_1cis the T₁ time for spin flipping.

The spin-preserving recombination rate coefficientB₁, as well as the spin-flip coefficientB₂, can be calculated by gen- eralizing the unpolarized case.^11,12The valence band of zinc- blende semiconductors consists of three subbands: heavy hole, light hole, and split-off hole bands. We neglect the split-off band as the energy splitting⌬ⰇkBTat temperatures Tlower than or around room temperature. By explicitly tak- ing into account the angular momentum of the heavy hole and light hole states, and using the optical selection rules for the states,¹⁰we arrive at the following expressions for spin- conserving and spin-flip recombination constants:

B₁=C关mh/共mc+mh兲兴^3/2+²₃关ml/共mc+ml兲兴^3/2

m_h^3/2+m_l^3/2 , 共5兲 B₂=1

3C关mh/共mc+mh兲兴^3/2

m_h^3/2+m_l^3/2 , 共6兲 where C depends on the “maximum” electron energy ប␻max=⑀g+kBT/ 2 共⑀gis the energy gap兲and temperature,

C共ប␻max,T兲= 4e²

ប²m²c³

冉

冊

Here P is the momentum matrix element for optical transi- tions,nris the refractive index, mc共mh andml兲 is the band mass of electrons共heavy holes and light holes兲.

Equations 共1兲 and 共4兲, together with Poisson’s equation form 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 of interest. Our next goal is to introduce qualitative features of spin accumulation in two cases of experimental interest that allow analytical solutions and form a starting point to discuss the concepts and issues to be encountered in more complex situations involving SHE and S-O coupling effects in trans- port. We consider ap-type semiconductor with nondegener- ate electron共minority兲density induced optically. The result- ing spin accumulation via extrinsic SHE can be deduced in a manner similar to spin orientation experiments. We assume that the injected electron density is well below the donor density. Further simplification follows from the fact that mo- bilities and diffusivities are spin independent in the nonde- generate regime.⁷Finally, we assume unpolarized holes since hole 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.共2兲–共4兲, give the drift- diffusion equations for electron spin and carrier density

ⵜ²s+共q/kBT兲共sⵜ ·E+E·ⵜs兲+共␨^q/kBT兲共ⵜn⫻E兲z

+共wp+ 1/T_1n兲s= 0, 共8兲

ⵜ²n+共q/kBT兲共nⵜ ·E+E·ⵜn兲+共␨^q/kBT兲共ⵜs⫻E兲z

+w共np−n₀p₀兲= 0, 共9兲

wherew=共B1+B₂兲/ 2 and␨⁼␯n/␮n=␦n/Dncharacterizes the S-O coupling strength,n₀,p₀are the equilibrium electron and hole densities. The spin quantization axis is taken to bez. We takeE=Eyˆ and considerxto be the transverse direction, the slab boundaries being atx= 0 and x=a, so that all the quan- tities of interest will have xdependence only. Denoting the electron recombination time as ␶n= 1 /共wNa兲, where Na

is the acceptor density, and spin relaxation time as ␶s

= 1 /共␶n

−1+T_1n⁻¹兲, Eqs.共8兲and共9兲become

s¨+共␨^qE/kBT兲n˙−s/L_s²= 0, 共10兲 n¨+共␨^qE/kBT兲s˙−共n−n₀兲/Ln

2= 0, 共11兲

where derivatives with respect to xis denoted by overdots, the longitudinal spin and charge diffusion lengths are defined as Ls=

冑

冑

共10兲and 共11兲 is apparent through the first-order derivatives of spin and charge densities.

We first solve the transverse spin and carrier diffusion for the case of a uniformly illuminated slab共boundary condition BC I兲, with electron-hole spin-unpolarized generation rateG.

We assume that carrier recombination at the edges is not significant so that it is reasonable to impose a uniform共here zero兲 electron transverse current:Jnx共x兲= 0. As for spin cur- rent, we take Js共x= 0兲=Js共x=a兲= 0, implying that spin-flip scattering at the edges is moderate. The solution to the drift- diffusion Eq.共10兲in the presence of S-O scattering is

s共x兲=␨^qELs

kBTG␶nsech共a/2Ls兲sinh共x/L−a/2L兲. 共12兲 The spin-polarization profile is given by ␣共x兲=s共x兲/G␶n, since G␶nⰇn₀ is the average electron density generated in steady-state conditions under illumination. This simple solu- tion demonstrates the essential physics behind spin accumu- lation in extrinsic SHE:共i兲Spin accumulation increases lin- early with E, with possible slight electric field modulation due to the dependence ofLs=Ls共E兲 共not discussed here兲.共ii兲 The magnitude of spin polarization is proportional to the strength of the S-O scattering as well as to the ratio of the voltage drop over min共Ls,a兲 and thermal energy. 共iii兲 For aⰆL_sthe accumulation at the edges is linearly proportional toa, while foraⰇLs, the accumulation is independent ofa.

共iv兲 While␣共x兲⬃x for aⰆLs, spin accumulation is signifi- cant only within the spin diffusion length from the edges whenaⰇLs.

A question now arises: How universal 共i.e., independent of boundary conditions兲is the qualitative behavior discussed

TSEet al. PHYSICAL REVIEW B72, 241303共R兲 共2005兲

RAPID COMMUNICATIONS

241303-2

above? To answer this question we introduce different boundary conditions 共BC II兲, describing the physics of car- rier injection atx= 0 andx=a, while assuming vanishing spin currents at the edges: n共x= 0兲=n₁, n共x=a兲=n₂; J_s,x共x= 0兲

=J_s,x共x=a兲= 0. These conditions mimic the case of ap-doped base in apnpspin-polarized transistor,¹⁵ where the electron injection level can be controlled by the biases to the emitter and collector. In the case discussed here the longitudinal cur- rent would flow perpendicular to the transistor current, which would thus be spin polarized due to the extrinsic SHE.

The vanishing boundary conditions for J_s,x reduce to s˙共x= 0兲= −␨^qEn1/k_BT and s˙共x=a兲= −␨^qEn2/k_BT. Solving Eqs.共10兲and共11兲with the above boundary conditions gives the spin and electron densities inside the slab

s共x兲= ␨^qE

kBT共L₁⁻²−L₂⁻²兲

冋

C₁共x兲− 1

L₂C₂共x兲

册

and

n共x兲−n₀= 1

共L₁⁻²−L₂⁻²兲关共L₁⁻²−L_s⁻²兲S1共x兲

−共L₂⁻²−L_s⁻²兲S2共x兲兴, 共14兲 where for convenience we have defined the functions

C_1,2=

冋

^冉

冊

冉

冊 册 冒

^冉

^冊

共15兲 S_1,2=

冋

^冉

^冊

^冉

^冊 册 冒

^冉

^冊

共16兲 Here we introduce new transverse spin-charge diffusion lengths,L₁ andL₂,

L_1,2⁻² =␥^±

冑

␥⁼共1/2兲关Ls

−2+L_n⁻²+共␨^qE/kBT兲²兴. 共18兲 There is a critical value of the field that separates the regimes of strong and weak spin-charge coupling—Eqs.共10兲and共11兲 reduce to the ordinary spin and charge diffusion equations when EⰆEs, En, where Es=kBT/共␨^qLs兲 and En

=kBT/共␨^qLn兲are the values of the critical fields with respect to spin and charge diffusion. In this case ␥⯝共Ls−2+L_n⁻²兲/ 2 and L_1,2⯝L_s,n. When EⲏEs, En,␥ becomes dependent on the electric field and it is in this regime the spin-field relation deviates from linearity.

For quantitative understanding we take our semiconductor to be GaAs at room temperature,¹⁶ with doping density Na= 3⫻10¹⁵cm⁻³, and transverse size ofa= 6␮^m共which is much greater thanL_s兲. The S-O scattering strength is taken to be ␨^{= 10−4}, reflecting weak S-O coupling in GaAs.

Finally, the boundary conditions for electron density are n共0兲= 2⫻10¹³/ cm³ andn共a兲= 5⫻10¹³/ cm³. Figure 1 shows the profiles of spin and electron densities, as well as spin polarization␣^=s/n, for several values ofE. In contrast to the

purely diffusive behavior exhibited byn, spin density along the slab exhibits weakly oscillatory behavior. Both spin den- sity and spin polarization attain maximum magnitudes at the edges.共We find that spin polarization is enhanced by a de- cade when considering the extrinsic SHE at 77 K.兲What is interesting is, unlike in BC I关conclusion共iv兲兴, that spin ac- cumulation here is significant throughout the sample, not just within Ls from the edges. This oscillatory pattern is due to the existence of spin-charge coupling that couples the spin and charge diffusion lengths together into two spin-charge diffusion lengths Eqs. 共17兲 and 共18兲, so that the net spin- density profile Eq.共13兲can be thought of as the interference between two spin-density profiles, each having a character- istic spin-charge diffusion lengthL₁orL₂. Since, as a result of spin Hall effect, each of these terms changes sign near the boundaries, they interfere destructively and constructively along the slab giving rise to the oscillatory pattern in Fig. 1 共similar arguments apply for the electron density, but since each of these terms is always positive, they add up construc- tively兲. 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.⁶In the presence of magnetic field FIG. 1. Calculated spin densitys, electron densityn, and polar- ization␣for BC II. We have used␨= 10⁻⁴and applied electric field strengths 0.125, 0.250, and 0.500 kV/ cm. While a signature of the extrinsic SHE is the opposite spin accumulation at the edges of the sample, spin accumulation is large also inside.

FIG. 2. Calculated spin current J_s in x 共top兲 and y 共bottom兲 directions for BC II and parameters as in Fig. 1.

SPIN ACCUMULATION IN THE EXTRINSIC SPIN… PHYSICAL REVIEW B72, 241303共R兲 共2005兲

RAPID COMMUNICATIONS

241303-3

共externally applied or effective field due to, e.g., Rashba S-O coupling兲, the oscillatory behavior comes not only from the coupling of spin and charge but also the coupling in between the different components of the spin.¹⁷ Now spin current is not conserved and flows inside the sample, being restricted to zero by our choice of boundary conditions共see Fig. 2兲.

From Eq. 共3兲 we obtain the spin current density in the x direction

J_s,x共x兲= − q␯nE

共L₁⁻²−L₂⁻²兲Ls2关S1共x兲−S₂共x兲兴, 共19兲 and in theydirection

J_s,y共x兲= 1

共L₁⁻²−L₂⁻²兲兵关q␯nE²/kBT+␦n共L₁⁻²−L_s⁻²兲兴C1共x兲/L1

−关q␯nE²/kBT+␦n共L₂⁻²−L_s⁻²兲兴C2共x兲/L2其. 共20兲

Spin current in thex direction flows in the direction of the spin gradient, while theycomponent changes sign inside the slab, reflecting the fact that spin-up and spin-down electrons are deflected into opposite directions due to S-O scattering by impurities. Finally, we wish to see at what value of the electric field spin-charge diffusion lengthsL_1,2will be modi- fied and induce nonlinear behavior in ␣. Figure 3 shows

␣共E兲for different␨. Spin polarization varies linearly withE, except at electric field as large as 10⁵kV/ cm and S-O cou- pling␨⬇1, a clearly unphysical case considered here only to illustrate 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 spin accumulation in the extrinsic spin Hall regime and intro- duced S-O dependent spin-charge diffusion lengths. We have found that spin accumulation is strongly influenced by boundary conditions and thus by specific spintronic device design. Our theory is applicable to any complex device set- ting in which extrinsic spin Hall effect is expected to play a role. 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.

1M. I. D’yakonov and V. I. Perel, JETP Lett. 13, 467共1971兲.

2M. I. D’yakonov and V. I. Perel, Phys. Lett. 35A, 459共1971兲.

3J. E. Hirsch, Phys. Rev. Lett. 83, 1834共1999兲; S. Zhang,ibid. 85, 393共2000兲; H.-A. Engel,et al.,ibid. 95, 166605共2005兲; W.-K.

Tse and S. Das Sarma, cond-mat/0507149共unpublished兲; E. M.

Hankiewicz and G. Vignale, cond-mat/0507228共unpublished兲.

4The term “extrinsic,” to be distinguished from the “intrinsic SHE,” was used in J. Sinovaet al., Phys. Rev. Lett. 92, 126603 共2004兲.

5Y. K. Katoet al.Science 306, 1910共2004兲.

6V. Sihet al., Nat. Phys. 1, 31共2005兲.

7I. Žutićet al., Phys. Rev. Lett. 88, 066603共2002兲.

8J. Fabianet al., Phys. Rev. B 66, 165301共2002兲.

9R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154共1954兲; P.

Noziéres and C. Lewiner, J. Phys.共Paris兲 34, 901共1973兲.

10M. I. D’yakonov and V. I. Perel, inOptical Orientation共Elsevier Science, New York, 1984兲.

11H. Barry Bebb and E. W. Williams, inSemiconductors and Semi-

metals共Academic Press, New York, 1972兲, Vol. 8.

12G. Lasher and F. Stern, Phys. Rev. 133, A553共1964兲.

13I. Žutićet al., Rev. Mod. Phys. 76, 323共2004兲.

14R. A. Smith,Semiconductors共Cambridge University Press, New York, 1978兲.

15J. Fabian, et al.cond-mat/0211639 共unpublished兲; J. Fabian, I.

Žutić, Phys. Rev. B 69, 115314共2004兲.

16ForT= 300 K, the transport parameters for the minority electron are taken as 共Ref. 7兲: diffusivity D_n= 103.6 cm²s⁻¹, mobility

␮n= 4000 cm²V⁻¹s⁻¹, recombination rate w=共1 / 3兲

⫻10⁻⁵cm³s⁻¹and spin-flip timeT₁= 0.2 ns. From these the cal- culated values of electron-hole recombination and spin diffusion lengths are, respectively,L_n= 1␮m andL_s= 0.8␮m. The intrin- sic concentration isn_i= 1.8⫻10⁶cm⁻³.

17A. A. Burkovet al.Phys. Rev. B70155308共2004兲; E. I. Rashba, cond-mat/0507007, Physica E 共to be published兲; I. Adagideli, cond-mat/0506531, Phys. Rev. Lett. 共to be published兲; B. K.

Nikolicet al., Phys. Rev. Lett. 95, 046601共2005兲. FIG. 3. Calculated spin polarization␣ atx= 0 as a function of

electric fieldEfor BC II and␨= 10⁻⁴, 10⁻², and 1. Only in the last 共unphysical兲case␣starts to saturate for largeE, as the dependence ofL₁andL₂onEsets in.

TSEet al. PHYSICAL REVIEW B72, 241303共R兲 共2005兲

RAPID COMMUNICATIONS

241303-4