Spin generation in a Rashba-type diffusive electron system by nonuniform driving field

Spin generation in a Rashba-type diffusive electron system by nonuniform driving field

L. Y. Wang,1 _{C. S. Chu,}1,2_{and A. G. Mal’shukov}1,2,3

1_{Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan} 2_{National Center for Theoretical Sciences, Physics Division, Hsinchu 30043, Taiwan} 3_{Institute of Spectroscopy, Russian Academy of Science, 142190 Troitsk, Moscow Oblast, Russia}

共Received 11 February 2010; published 10 March 2010兲

We show that the Rashba spin-orbit interaction contributes to edge spin accumulation S_z in a diffusive regime when the driving field is nonuniform. Specifically, we solve the case of nonuniform driving field in the vicinity of a circular void locating in a two-dimensional electron system and we identify the key physical process leading to the edge spin accumulation. The void has radius R₀in the range of spin-relaxation length l_so and is far from both source and drain electrodes. The key physical process we find is originated from the nonuniform in-plane spin polarizations. Their subsequent diffusive contribution to spin current provides the impetus for the edge spin accumulation Szat the void boundary. The edge spin accumulation is proportional to the Rashba coupling constant␣ and is in a spin-dipole form oriented transversely to the driving field. We expect similar spin accumulation to occur if the void is at the sample edge.

DOI:10.1103/PhysRevB.81.115312 PACS number共s兲: 72.25.Dc, 71.70.Ej, 73.40.Lq

I. INTRODUCTION

A major goal for the semiconductor spintronics is to gen-erate and to manipulate spin polarization by mere electrical means. Spin-orbit interaction 共SOI兲 provides the key lever-age and spin-Hall effect共SHE兲 共Refs.1–15兲 provides the key

paradigm, where it is possible for a uniform driving electric field to induce bulk spin polarization and spin current and, in turn, out-of-plane spin accumulations Szat lateral edges. The Rashba SOI共RSOI兲 共Ref.16兲 is of particular interest because

of its gate-tuning capability. However, background scatterers lead to a complete quenching of the RSOI’s contribution to the edge spin accumulation Sz, a direct consequence of its linear dependence on the electron momentum k.17,18 _{It is} le-gitimate then to find ways to restore the RSOI’s contribution to the edge spin accumulation. Our interest here is in the diffusive regime, when the spin-relaxation length l_soⰇl_e, the mean-free path. Even though the spin accumulation is finite in the mesoscopic ballistic regime 共lso⬍le, and L⬍l␾兲,19–21

with L the sample size and l_␾the phase coherent length, it is still important to see whether the RSOI alone can contribute to SHE in the impurity-dominate regime.

Indeed, RSOI was found by Mishchenko et al.18 _{to give} rise to edge spin accumulation Sz near electrodes even though its contribution to bulk spin current vanishes. The edge spin accumulation is concentrated at the two ends of an electrode-sample interface, covering a region of size lso. This

finding was identified to arise from a nonzero spin current Iy z flowing along the sample-electrode interface, in direction yˆ.18 This nonzero spin current was understood from the way the spin current vanishes in the bulk, when an exact cancellation occurs between two terms, one related to the spin polariza-tion and the other related to the driving field.18 _{This exact} cancellation no longer holds at the sample-electrode inter-face, when the driving field has reached its bulk value but the spin polarization has not. Similar result was also obtained by

Raimondi et al.,22 _{where spin-density spatial profiles at the} sample corners were obtained. Yet, it would be more desir-able that we can find schemes and identify physical pro-cesses for the restoring of the RSOI-induced edge spin accu-mulation at locations other than the sample-electrode interfaces and according to our specification.

In this work, we turn to nonuniform driving field for the restoration of the RSOI-induced spin accumulation. The ef-fect of nonuniform driving field on spin accumulation is also interesting in its own right. Earlier study considered nonuni-form driving field in systems in the presence of “extrinsic” SOI, that is, SOI due to SOI impurities.23 _{Here, instead, we} consider nonuniform driving field in the vicinity of a circular void located in a diffusive RSOI-type two-dimensional elec-tron gas共2DEG兲. We obtain spin accumulation in the vicinity of the void. This problem allows us to identify the key physi-cal process for the spin accumulation and also sheds light on the case if the void were to form at a lateral edge. The radius

R0of the void is of the order of lso.

Most important is our finding that the main physical pro-cess is in marked contrast to the conventional one. While the conventional one is associated with the nonvanishing of the out-of-plane spin current I_nz,18 _{the key process we find is} associated with the in-plane spin currents, In

x or In

y

, and with the way they vanishes at the void boundary. Here, nˆ denotes the flow direction normal to the void boundary. The in-plane spin currents consist of two terms, a diffusive term and a term related to the spin accumulation Sz, which are given by

I_␳i = − 2D⳵ ⳵␳Si− R

izi

Sz␳ˆ · iˆ, 共1兲

where␳ is the position vector measured from the center of the void, i苸兵x,y其, and D is the diffusion constant.24 _Rijk denotes the precession of Sj into Si when it flows along kˆ. The factor␳ˆ · iˆ denotes the projection of the flow. The RSOI

governs the symmetry of Rijksuch that Rizj= 0 for i⫽ j.

Our scheme of Szgeneration is made possible by Eq.共1兲, the boundary condition I_␳i= 0, at␳= R0, and the presence of a

radially nonuniform in-plane spin polarization Si. The spin polarization S储= S储

Ed

+⌬S储we obtain in this work has

S储Ed= − N0␣␶e/បzˆ ⫻ E共␳兲, 共2兲

which radial dependence is acquired from the driving field E. Outside the void, E共␳兲=E0xˆ − E0共R0/␳兲2共cos 2␾xˆ + sin 2␾yˆ兲

where E0xˆ is the uniform field far away from the void and

␳· xˆ = cos␾. The driving electric field E = −ⵜ␸共␳兲=␴0j has to

satisfy the steady state condition ⵜ·j=0 and the boundary condition j_␳= 0. Here␴0is the electric conductivity, j is the

electric current density, N0 is the energy density per spin,␣

is the RSOI coupling constant, e⬎0, and␶is the mean-free time. The term S_储Ed is an Edelstein-like spin polarization3 which we have obtained for the case of nonuniform driving fields. That this term of S储alone fails to satisfy the boundary

condition Eq. 共1兲, because of its radial dependence, has

prompted the generation of⌬S储 and Sz.

The aforementioned key physical process associated with the spin current is meant to establish a boundary condition for the spin-diffusion equation. We have used the conven-tional form of the spin current operator J_li=共1/4兲共Vl␴i +␴iVl兲, where spin unit of ប is implied, and the kinetic ve-locity operator Vl=共1/iប兲关xˆl, H兴. This is appropriate for hard wall boundary.24–26_{As the boundary condition is applied to a} region much shorter in distance than l_so from the boundary, the effect of spin torque27,28 here should be of secondary importance. In Sec.IIwe present the spin-diffusion equation for nonuniform driving fields, and the analytical solutions for spin densities around a circular void. In Sec.IIIwe present our numerical results and discussion. Finally, in Sec.IV, we will present our conclusion.

II. THEORY

The derivation of the spin-diffusion equation 共SDE兲 by the Keldysh nonequilibrium Green’s function method13,24_is extended to the case when the driving field is nonuniform. With the RSOI Hamiltonian H_so= hk·␴ and hk= −␣zˆ⫻k, where␴, and hkare, respectively, the Pauli’s matrix vector

and the SOI-effective magnetic field, the SDE is given by

Dⵜ2_S x− ⌫xx ប2Sx+ Rxzx ប ⳵ ⳵xSz− Mx0·⵱ 2ប3 D00= 0, Dⵜ2_S y− ⌫yy ប2Sy+ Ryzy ប ⳵ ⳵ySz− My0·⵱ 2ប3 D00= 0, Dⵜ2Sz− ⌫zz ប2Sz+ Rzxx ប ⳵ ⳵xSx+ Rzyy ប ⳵ ⳵ySy= 0, 共3兲

where the spin density Siis in units ofប, and D=vF

2_␶/2.

Even though the form of the SDE in Eq.共3兲 is essentially

the same as that for the uniform driving field,24 _the spin-charge coupling term, through ⵱D₀0, becomes position de-pendent. To get at this Eq.共3兲, we have performed a

system-atic scrutiny on possible additional terms in it that are up to appropriate orders, as will be detailed in the following. The spin-charge coupling terms, given by −Mi0·⵱D₀0, have Mi0 = 4␶2hk 3 ⳵nki ⳵k= −2␶2hF 2_{␣共iˆ⫻zˆ兲 where D} 0 0 = 2N0e␸共␳兲 is the effec-tive local equilibrium density. The overline denotes angular

average over the Fermi surface, ␸共␳兲=−E0共␳+ R0

2_{/␳兲cos␾}

for ␳ⱖR0 and nk= hk/hk. The Edelstein-like spin polariza-tion S储Ed关Eq. 共2兲兴 is solved directly from Eq. 共3兲.

The D’yakonov-Perel’ 共DP兲 spin-relaxation rates, given by ⌫il= 4␶hk 2_共␦il − nk i nk l_兲,30 have ⌫xx=⌫yy=⌫zz/2=2hF 2_␶ for RSOI. Spin precession arising from diffusive flow is charac-terized by Rilm_{= 4␶兺n⑀}iln_h

k n

v_km, where ⑀iln _{is the Levi-Civita} symbol, and we have Rzii_{= −R}izi_{= −2h}

FvF␶for RSOI and for

i =共x,y兲.24 _{Since k}

FleⰇ1, with le the mean-free path, the

charge neutrality is maintained by the condition of zero charge density throughout due to screening effect. Within the linear response to the driving electric field, the effect of the screening potential on the spin accumulation can be ne-glected.

A brief note on the systematic scrutiny of the possible additional terms in Eq. 共3兲 is in order here. The spin-charge

coupling term in Eq. 共3兲 is resulted from ⌿i0_D

0

0_,29 _which lowest order in RSOI and first order in spatial gradient is given by the expansion of ⌿i0 _{to the order h}

F

3_{q. This is}

ap-propriate for uniform driving field because ⵱␸ would be-come position independent. We take caution here, for the case of nonuniform driving fields, to check for additional terms of higher order in q that could have arisen from⌿i0_D

0 0_. Here,29 ⌿il = ⌫ 2␲N₀

兺

_p ⬘ Tr关␶i Gr共0兲共p

⬘

,␻+␻

⬘

兲␶lGa共0兲共p

⬘

− q,␻

⬘

兲兴, 共4兲

where⌫=1/2␶, Gr/a共0兲_{are retarded共advanced兲 Green’s} func-tions averaged over impurity configuration, ␶i=0_{= 1, and} ␶i=x,y,z_{=␴x,y,z. To identify additional expansion terms in ⌿}i0 for nonuniform driving fields, we note first of all that S_储Edis of order hFq␸. If S储

Ed

is to satisfy the SDE, all the terms in Eq.共3兲 involving Siwill have to be replaced by Si− S储,i

Ed . This implies, according to Eq.共3兲, that terms of order hFq3␸and

hF

2

q2␸will be needed, and thus we should look for terms of the same order in the expansion of the spin-charge coupling ⌿i0_D

0

0_{. The above two orders can also be identified based on}

symmetry argument, that the combined power in hF and q must be even and that they are the lowest RSOI contributions to the respective q orders. Starting from

⌿i0_{共␻= 0,␻}

_⬘

_{,q兲 =} ⌫ 2␲N₀

兺

p Tr

冤

␶i

冉

␻

⬘

−␧p− q · ⳵␧p ⳵p + i⌫

冊

+

冉

hp i + q ·⳵hp i ⳵p

冊

␴ i

冉

␻

⬘

−␧p− q · ⳵␧p ⳵p + i⌫

冊

2

冉

1 ␻

⬘

−␧p− i⌫ + hp i_␴i 共␻

⬘

−␧p− i⌫兲2

冊

冥

, 共5兲

we expand it, for instance, up to the order hFq3, and obtain ⌿i0_共h Fq3兲 = 2N0

冕

d␧ ⫻

冤

共q · vp兲3hp i 共␻

⬘

−␧ + i⌫兲4共␻

⬘

−␧ + i⌫兲2 + 3共q · vp兲2q · ⳵hp i ⳵p 共␻

⬘

−␧ + i⌫兲4共␻

⬘

−␧ − i⌫兲

冥

, 共6兲 where vp= p/mⴱ. The angular averages in Eq. 共6兲 over the

Fermi surface give rise to q dependences of the form q2qj=x,y, which will not contribute to Eq.共3兲 because ⵜ2_{␸= 0.}

Follow-ing similar procedure, ⌿i0_共h F

2_q2_{兲 is found to be identically}

zero. Thus Eq. 共3兲 is the SDE for the case of nonuniform

driving field.

As has been explained in the previous section, S_储Edalone cannot satisfy the boundary condition I_␳i=x,y= 0. Thus in the end we expect to have an additional⌬S so that S=S_储Ed+⌬S. On the other hand, the contribution from S储

Ed

to I_␳z is found to vanish already. The generation of⌬Szthus does not fall into the conventional scheme that spin accumulation Szis caused by In

z

near the sample boundary. Our major task in the fol-lowing is to calculate ⌬S.

Putting the coordinates in units of l_so=

冑

D␶_so, with ␶_so = 2ប2_/共h

F

2_{␶兲, the SDE for ⌬S is given by}

ⵜ2_⌬S x− 4⌬Sx+ 4 ⳵ ⳵x⌬Sz= 0, ⵜ2_⌬S y− 4⌬Sy+ 4 ⳵ ⳵y⌬Sz= 0, ⵜ2_⌬S z− 8⌬Sz− 4 ⳵ ⳵x⌬Sx− 4 ⳵ ⳵y⌬Sy= 0. 共7兲

Modes of solution of Eq. 共7兲 have the form ⌬S共q兲_j

=兺ma共q兲j eim共␦+␾兲Hm共1兲共␥q␳兲, where Hm共1兲共z兲 is the Hankel func-tion of the first kind and the index q denotes the q-th mode. Substituting into Eq.共7兲 we obtain

冤

共−␥2_{− 4兲 0 4i␥sin␦} 0 共−␥2− 4兲 4i␥cos␦ − 4i␥sin␦ − 4i␥cos␦ 共−␥2− 8兲

冥冤

ax共q兲 a共q兲y az共q兲

冥

= 0. 共8兲 The asymptotic behavior required of ⌬S共q兲j leads to Im␥⬎0. Thus ␥₁= 2i, ␥₂=

冑

2 + 2i

冑

7, and ␥₃= −␥₂ⴱ. We

also have 共ax共1兲, ay共1兲, az共1兲兲=a共1兲x 共1,−tan␦, 0兲, 共ax共2兲, ay共2兲, az共2兲兲 = az共2兲共2ig2sin␦, 2ig2cos␦, 1兲, and 共ax共3兲, ay共3兲, az共3兲兲 = az共3兲共2ig2sin␦, 2ig2cos␦, 1兲ⴱ, for q = 1, 2, and 3,

respec-tively. Here g2=␥2/共␥22+ 4兲. As␦takes on continuous values,

there are effective infinite solutions per q-mode. In terms of these modes⌬Sjis expanded in the form

⌬Sj=

冕

0 2␲ d␦

兺

q=1 3

兺

_m aj共q兲共␦兲Hm共1兲共␥q␳兲e im_共␦+␾兲 . 共9兲

The condition that⌬S is real requires ax共1兲to be pure imagi-nary and a_z共2兲= −a_z共3兲ⴱ.

The boundary condition for the nonuniform driving field is established by applying to the spin current expression similar procedure that we have applied to Eq.共3兲. The

spin-current expression is found to resemble the uniform driving field case,24albeit now that⵱␸becomes position dependent. We have Ij i = − 2DⵜjSi− RixjSx− Riy jSy− RizjSz +

兺

l=x,y 4␶2⑀xyiv_Fj

冉

hp⫻ ⳵hp ⳵kl

冊

z eN0ⵜl␸共r兲, 共10兲

where the last term is the explicit contribution from the driv-ing field and is nonzero for Ij

z

only. The boundary condition

I_␳i共␳=␳0兲=0 becomes

−ⵜ_␳⌬Sx− 2 cos␾⌬Sz− 2␣E˜ /␳sin 2␾兩␳=␳₀= 0, −ⵜ_␳⌬Sy− 2 sin␾⌬Sz+ 2␣E˜ /␳cos 2␾兩_␳=␳0= 0,

−ⵜ_␳⌬Sz+ 2 cos␾⌬Sx+ 2 sin␾⌬Sy兩␳=␳₀= 0, 共11兲 whereⵜ_␳⬅⳵/⳵␳,␳0= R0/lso, and E˜ =eE0N0␶/ប. We note that

the E˜ terms in Eq. 共11兲 originate from the spin current due to S_储Ed, which are the driving terms here. We solve Eqs.共9兲 and

共11兲 for aj

q_{共␦兲 by a direct numerical approach and by an} analytical approach. Excellent matching is obtained between the two approaches. The analytical approach is facilitated by the assumed forms ax共1兲= itxsin 2␦and az共2兲= tzcos␦, where tx is real and tzis complex. The former is guided by the obser-vation, from Eq. 共11兲, that ⌬Sx depends on ␾ as sin 2␾. Substituting these forms into Eqs. 共9兲 and 共11兲, and after

−␥1H2共1兲⬘共z1兲tx+ i Im关tzX兴 = i␣E˜ /共␲␳0兲,

− i␥1H1共1兲共z1兲tx+ 2 Im关tzY兴 = 0, 4

z1

H₁共1兲共z1兲tx+ i Im关tzZ兴 = 0, 共12兲

where z1=␥1␳0, f

⬘

共z兲⬅df /dz, X=2H1共1兲共z1兲−2g2␥2H2共1兲⬘共z2兲, Y =共g2␥2− 1兲H1共1兲共z2兲, Z=2共␥2− 4g2兲H1共1兲⬘共z2兲, and z2=␥2␳0.

Equa-tion 共12兲 allows us to solve for txand tzanalytically, which are proportional to␣E˜ . Explicit expressions of txand tzare

tx= − 4␣E˜

␲

Im关YZⴱ_兴

8H₁共1兲共z1兲Im关XYⴱ兴 +␥1z1H1共1兲共z1兲Im关ZXⴱ兴 + 2␥1z1H2共1兲⬘共z1兲Im关ZYⴱ兴

, 共13兲 and tz= ␣E˜ ␲␳0 H₁共1兲共z₁兲Im关8Yⴱ−␥₁z₁Zⴱ兴

8H₁共1兲共z₁兲Im关XYⴱ兴 +␥₁z₁H₁共1兲共z₁兲Im关ZXⴱ兴 + 2␥₁z₁H₂共1兲⬘共z₁兲Im关ZYⴱ兴.

The spin densities⌬Siare then obtained to give ⌬Sx= 2␲兵− itxH2共1兲共␥1␳兲 + 2 Im关tzg2H2共1兲共␥2␳兲兴其sin 2␾,

⌬Sy= 2␲兵− itxH0共1兲共␥1␳兲 − 2 Im关tzg2H0共1兲共␥2␳兲兴其 − ⌬Sxcot 2␾, ⌬Sz= − 4␲Im关tzH共1兲1 共␥2␳兲兴sin␾. 共14兲

This and Eq. 共2兲 together are our main results. In particular,

⌬Sz⫽0 confirms that RSOI’s contribution to spin accumula-tion can be restored in a nonuniform driving field. The parity in␾of⌬Siis consistent with that implied in Eq.共11兲, which is determined by the E˜ terms. The spin accumulation, given in its entirety by⌬Sz, is in a dipole distribution which orients transversely to the driving field E₀xˆ. Furthermore, Eq. 共14兲

shows that ␥₁ and ␥₂ contribute to, respectively, decaying and oscillatory behavior in⌬Si.

III. NUMERICAL RESULTS

Figure1presents the spin accumulation Szin the vicinity of the circular void. We use for our numerical results mate-rial parameters that are consistent with GaAs: effective mass

mⴱ= 0.067m₀with m₀the free-electron mass; electron density

ne= 1⫻1012 cm−2; electron mean free path le= 0.43 ␮m;

ra-dius of the circular hole R0= 0.5lso; and Rashba coupling

constant ␣= 0.3⫻10−12 _{eV m.}31,32 _{The spin-relaxation} length is l_so= 3.76 ␮m and the driving field is E₀ = 40 mV/␮m. As shown in Fig.1, the core of the spin ac-cumulation consists of two spin pockets of opposite spin and of largest spin density magnitude at ␾=⫾␲/2. The spin pockets have radial thickness of about 0.3lso⬃1.1 ␮m. In

the outer region, spin densities of opposite signs and of smaller magnitudes are dispersed to a wider spatial extent, in the form of two curved spin clouds. The spin cloud center is located about one lso from the void boundary at␾=⫾␲/2.

Both the spin pocket thickness and the spin cloud distance from the void boundary are not sensitive to the void radius

R0.

This spin accumulation can be probed optically by Kerr rotation. To simulate the case of an optical probe scanning along the␾=␲/2 direction, we calculate the net number of out-of-plane electron spin within the probe area which center is located at a distance d from the void center. For simplicity, we take the probe area to be the same as that of the void. The result is presented in Fig.2, where we have included several

R0cases, from R0= 0.5lsoup to R0= 1.2lso. Distinct

contribu-tions from the spin pocket and the spin cloud can be identi-fied. The former are negative minima around d⬇0.5l_so and

FIG. 1. 共Color online兲 Spin accumulation Szin the vicinity of a circular void共white circle兲. S_zis in unit of 1/␮m2_{, void radius R}

0 = 0.5lso, and lso= 3.76 ␮m. Dark arrow indicates the driving field direction.

the latter are positive peaks around d⬇R0+ lso. That the

negative minima are essentially unshifted reflects the insen-sitivity of the core radial thickness to the void radius R0.

Additional peak for the R0= 1.2lso curve at d⬇2R0

corre-sponds to the situation when the probe area moves out of the spin pocket.

The spin accumulation in a spin-dipole form oriented transversely to the driving field is a generic feature signify-ing the redistribution of spin rather than the net transport of spin. It has been found in the vicinity of a non-SOI elastic scatterer in a RSOI 2DEG,33,34 _{and in the vicinity of a} me-soscopic cylindrical barrier in a 2DEG with the barrier pro-file providing the SOI.35_{Both objects are of sizes much less} than le. Of course, the physical mechanisms leading to all the

above spin-dipole forms are entirely different. Furthermore, here we demonstrate that such spin-dipole feature can exist in the neighborhood of a much larger object R0⬇lso, is

ro-bust against background scatterers and is within reach of present measurement technology.

Finally, we note that the spin accumulation features we obtain above are relevant to the case when the circular void

is located at a sample edge: an edge-semicircular void共ESV兲 as shown in Fig.3. The nonuniform driving field E共␳兲 for the

circular void satisfies also the additional boundary condition

jy= 0 imposed by the ESV case at the sample edge, ␾ =共0,␲兲. Thus the same E共␳兲 holds in the two cases. How-ever, to satisfy the additional boundary condition for spin current at the sample edge, a further additional spin accumu-lation ⌬SESVis needed, leading to the total spin

accumula-tion S = S储

Ed

+⌬S+SESV. The imposing of the spin current

boundary condition in this case is much more complicated, particularly for the spin accumulation near the two corners of the ESV structure, but we find that the spin pocket and the spin cloud features in Fig.1remains essentially intact except for near corner regions of the ESV structure.36

IV. CONCLUSIONS

In conclusions, we have demonstrated that nonuniform driving field can give rise to spin accumulation in a diffusive Rashba-type 2DEG. The nonuniform driving field can be re-alized by patterning the sample such as with a circular void in the sample or with a semicircular void at the sample edge. The physical process is identified to be associated with spin current for the in-plane spin at the boundary. Our proposed scheme of restoring the RSOI contribution to gate-tunable spin accumulation is relatively simple, and we hope that this will draw experimental effort in the near future.

ACKNOWLEDGMENTS

This work was supported by Taiwan NSC共Contract No. 96-2112-M-009-0038-MY3兲, NCTS Taiwan, and a MOE-ATU grant.

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