Spin polarizations and spin Hall currents in a two-dimensional electron gas with magnetic impurities

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Spin polarizations and spin Hall currents in a two-dimensional electron gas with magnetic impurities

C. Gorini, P. Schwab, and M. Dzierzawa

Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany R. Raimondi

CNISM and Dipartimento di Fisica “E. Amaldi,” Università Roma Tre, 00146 Roma, Italy 共Received 16 May 2008; published 29 September 2008兲

We consider a two-dimensional electron gas in the presence of Rashba spin-orbit coupling, and study the effects of magnetics-wave and long-range nonmagnetic impurities on the spin-charge dynamics of the system.

We focus on voltage induced spin polarizations and their relation to spin Hall currents. Our results are obtained using the quasiclassical Green function technique, and hold in the full range of the disorder parameter␣p_F␶. DOI:10.1103/PhysRevB.78.125327 PACS number共s兲: 72.25.⫺b In the field of spintronics, much attention has recently

been paid to spin-orbit related phenomena in semiconduc- tors. One such phenomenon is the spin Hall effect, i.e., a spin current flowing perpendicular to an applied electric field.^1–4 It is now well known that, for linear-in-momentum spin-orbit couplings such as the Rashba or Dresselhaus ones, the spin Hall current vanishes exactly in the bulk of a disordered two-dimensional electron gas共2DEG兲.^5–8This can be under- stood by looking at the peculiar form of the continuity equations for the spin, as derived from its equations of motion in operator form.^9–11 For a magnetically disordered 2DEG things are, however, different, and a nonvanishing spin Hall conductivity is found.^12–14Once more, a look at the continuity equations provides a clear and simple explanation of the effect:¹³a term, whose appearance is due to magnetic impurities, directly relates in-plane spin polarizations, induced by the electric field, to spin currents. As the former, which have been the object of both theoretical and experimental studies,^15–20 are influenced by the type of nonmagnetic scatterers considered, we forgo the simplified assumption that these areswave and take into account the full angle dependence of the scattering potential. Besides going beyond what is currently found in the literature, where, in the presence of magnetic impurities, the nonmagnetic disorder is either neglected or purelyswave, our approach also shows the inter- play between polarizations and spin currents in a 2DEG.²¹ We note that in the correct limits our results agree with what is found in Ref. 14. On the other hand a discrepancy with Ref. 12arises.

For the calculations we rely on the Eilenberger equation for the quasiclassical Green function in the presence of spin- orbit coupling.²² The spin-orbit energy is taken to be small compared to the Fermi energy, i.e., ␣^pFⰆ⑀F—or equiva- lently ␣ⰆvF, and the standard metallic regime condition 1/␶Ⰶ⑀F is also assumed. Here ␣ is the spin-orbit coupling constant,pF共vF兲 the Fermi momentum共velocity兲in the absence of such coupling, and ␶ the elastic quasiparticle lifetime due to nonmagnetic scatterers. Our results hold for a wide range of values of the dimensionless parameter ␣pF␶ since this is not restricted by the above assumptions. Contri- butions of order共␣/vF兲²are neglected throughout. We focus on intrinsic effects in the Rashba model; extrinsic ones,²³

Dresselhaus terms,²⁴ and hole gases²⁵are not taken into account. Finally, weak localization corrections, which could in principle play an important role,¹¹ are beyond the scope of our present work.

The Hamiltonian of the 2DEG, confined to thex−yplane, reads

H= p²

2m−b·␴⁺V共x兲, 共1兲 with b=␣ez⫻p the Rashba internal field, ␴ the vector of Pauli matrices, andV共x兲=V_nm共x兲+V_m共x兲the disorder potential due to randomly distributed impurities.²⁶ Nonmagnetic scatterers give rise to V_nm共x兲,

V_nm共x兲=

兺

_i ^U共x⁻^Rⁱ^兲, ^共2兲

while V_m共x兲 describes magnetics-wave disorder V_m共x兲=

兺

i

B·␴␦共x−Ri兲. 共3兲 Both potentials are treated in the Born approximation, and the standard averaging technique is applied.

To begin with, we look at the continuity equation for the s_yspin polarization,^13,27

⳵tsy+⳵x·js_y= − 2m␣j_s

z

y − 4

3␶sf

sy, 共4兲

where the second term on the right-hand side is due to magnetic impurities. Here␶sfis the spin-flip time that stems from the potential共3兲关cf. Eq.共13兲兴. Under stationary and uniform conditions the above equation implies a vanishing spin current—hence a vanishing spin Hall conductivity—unless magnetic disorder is also present, in which case instead

js_z

y = − 2

3m␣␶sf

sy. 共5兲

Since the out-of-plane polarized spin current is related to the in-plane spin polarization, we now use simple physical arguments to explain how the latter is generated by an applied voltage.^15,28Since the Fermi surface is shifted by an amount

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proportional to the applied electric field 共say along the x direction兲, as shown in Figs. 1共a兲 and 1共b兲, there will be more occupied states with spin up—alongy—than with spin down. In the case of short-range disorder, the total in-plane polarization can be estimated to be proportional to the density of states multiplied by the shift in momentum, sy

⬃N␦^p⬃N兩e兩E␶. Since in the present situation we are dealing with the two Fermi surfaces corresponding to the two helic- ity bands ⑀_⫾=p²/2m⫾␣p, obtained from the Hamiltonian 共1兲, one expectssy⬃共N+−N₋兲␦p, where, for the Rashba in- teraction, one has N_⫾=N₀共1⫿␣/vF兲, N₀=m/2␲^{. Explicit} calculations agree with this simple picture and lead to the result due to Edelstein,¹⁵ s_y= −N₀␣兩e兩E␶. When long-range disorder is considered, a reasonable guess would be to sub- stitute for␶the transport time␶tr

␶→␶tr, 1

␶tr

=

冕

^d^␪^W^共^␪^兲关^{1 − cos}^共^␪^兲兴^, ^共⁶^兲

withW共␪兲 being the angle-dependent scattering probability, so thats_y= −N₀␣兩e兩E␶tr. This was proposed in Ref.29; however, the picture is too simplistic, and therefore the guess is wrong. As discussed in Ref.21, the propersypolarization is given bysy= −N₀␣兩e兩E␶E, with

␶→␶E, 1

␶E

=

冕

^d^␪^W^共^␪^兲关^{1 − cos}^共²^␪^兲兴^. ^共⁷^兲

This particular time ␶E, where “E” stands for Edelstein, arises from the asymmetric shift of the two Fermi surfaces, as depicted in Fig.1共c兲, due to different transport times in the two bands. It shows that contributions from both forward 共␪= 0兲and backward共␪⁼␲兲scatterings are suppressed. The

next step is to consider what happens when magnetic impurities are included. Relying once again on the simple picture of the shifted Fermi surface, one could argue that these have a rather small impact on the spin polarization since the spin- flip scattering time usually makes a small contribution to the total transport time. However, even when this is the case, magnetic disorder does not simply modify the total transport time but has an additional nontrivial effect. In its presence the spins do not align themselves along the internal b field since they acquire nonvanishing components in the plane orthogonal to it 关see Fig. 1共d兲兴. It is these components that give rise to a finite spin Hall conductivity. In this respect, magnetic disorder has an effect similar to that of an in-plane magnetic field: it affects the spin-quantization axis and tilts the spins out of their expected stationary direction. We now make these arguments quantitative.

The starting point is the Eilenberger equation,²²which we write explicitly for a homogeneous Rashba 2DEG in linear response to a constant and homogeneous applied electric field

⳵tg^K=vF·E兩e兩⳵⑀g_eq^K −1

2

再

^p¹^F^⳵^␸^b^·^␴^,e^␸^·^E兩e兩^⳵^⑀^g^eq^K

冎

+i关b·␴^,g^K兴−i关⌺ˇ,gˇ兴^K. 共8兲 The quasiclassical Green function 关gˇ⬅gˇt₁t₂共pˆ;x兲兴is defined as 共␰⁼^p²/2m−␮兲

gˇ= i

␲

冕

^d^␰^G^ˇ^t¹^t²^共^p,x^兲^, ^G^ˇ ⁼

冉

^G⁰^R ^G^G^K^A

冊

^, ^共⁹^兲

where Gˇ

t₁t₂共p,x兲 is the Wigner representation of the Green function, which has a matrix structure in both Keldysh共de- noted by the check symbol兲and spin space. Equation 共8兲 is the equation of motion for the Keldysh component—the one related to physical observables—identified by the superscript

“K,” which will be from now on implicitly assumed and thus dropped. Moreover, g_eq^K= tanh共⑀/2T兲共geq

R−g_eq^A兲, where g_eq^R = −g_eq^A = 1 −⳵_␰^b^·␴, indicates the equilibrium—no electric field—function.²²All objects are evaluated at the Fermi surface in the absence of spin-orbit coupling while ␸ ^{is the} angle defined by the momentum, p=p共cos␸^{, sin}␸兲, ande_␸

=共−sin␸^{, cos}␸兲. From Eqs.共2兲and共3兲one obtains the self- energy contributions

⌺ˇ

_p ^␴^l^G^ˇ^共p兲^␴^l^, ^共11兲

wheren_nmandn_mdenote the concentrations of nonmagnetic and magnetic impurities, respectively. In order to consider long-range nonmagnetic disorder, we first expand the nonmagnetic scattering kernel in spherical harmonics of the scattering angle, and neglect its dependence on the modulus ofp andp

⬘

p_y

p_x p_y

p_x Ex

(a) (b)

p_y

p_x p_x

p_y

(c) (d)

FIG. 1. 共Color online兲关共a兲 and 共b兲兴 The Fermi-surface shift,

␦^p=兩e兩E␶, due to an applied electric field along thexdirection. The white arrows show the direction of the internal fieldb.关共c兲and共d兲兴 Shifted bands and spin polarization in stationary conditions. 共c兲 Asymmetric shift of the two bands when angle-dependent scattering is present. The long dark 共blue兲 arrows show the contributions to the spin polarization arising from a sector d␸ of phase space.共d兲 When magnetic disorder is turned on, additional contributions orthogonal to the internal field b appear, here shown by the short inward and outward pointing共blue兲arrows. Out-of-plane contributions are also present but, for the sake of simplicity, are not shown.

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n_nm兩U兩²= 1

2␲^N0␶^关^{1 + 2K}¹^cos^共␸⁻␸

⬘

兲 + 2K₂cos共2␸^{− 2}␸

⬘

^兲^{+ . . .兴}

⬅ 1

2␲^N0␶^{关1 +}^K共␸⁻␸

⬘

^兲兴^, ^共12兲 with ␶ the nonmagnetic contribution to the elastic lifetime.

Then we write the magnetic scattering kernel in terms of the spin-flip time ␶sf,

n_mB²= 1 2␲^N0␶sf

. 共13兲

The complete disorder self-energy can then be written, sepa- rating itss-wave and higher harmonics contributions,

⌺ˇ=⌺ˇ

m+⌺ˇ

nm 1 +⌺ˇ

nm 2 = − i

6␶sf

兺

l=1 3

␴l具gˇ典␴l− i 2␶^具gˇ典⁻

i 2␶^具Kgˇ典,

共14兲 where具. . .典⬅兰d␸/2␲^{. . ..}

The connection betweengˇand the physical observables is made by integrating over the energy⑀, which is the Fourier conjugate variable of the time differencet₁−t₂. For instance, the spin density is given by the angular average of the Keldysh component,³⁰

s=s^eq−N₀

8

冕

^d^⑀^具Tr共^␴^g兲典. ^共15兲

In order to solve Eq.共8兲, it is convenient to turn it into matrix form, writingg as a four-vector

g=g₀␴0+g·␴^, 共g_␮兲=共g0,g兲. 共16兲 Rather than using the standard共␴x,␴y,␴z兲basis, we choose to rotate to 共␴储,␴⬜,␴z兲, the subscripts ^储 and⬜, indicating, respectively, the directions parallel and perpendicular to the internal field b. Defining the rotation matrixR共␸兲by

冢

^␴^␴^␴^␴⁰^x^y^z

冣

⁼

冢

¹⁰^{0 − cos}⁰ ^sin⁰⁰^␸^␸ ^cos^sin⁰⁰^␸^␸ ⁰⁰⁰¹

冣冢

^␴^␴^␴^␴^⬜⁰^储^z

冣

^, ^共¹⁷^兲

one has

g_␮

⬘

⁼

兺

␮⬘⁼⁰

3

R_␮␮⁻¹_⬘共␸兲g_␮_⬘, 共g_␮

⬘

兲=共g₀,g_储,g_⬜,g_z兲, 共18兲

⬘

兲=K^共^a^兲+ cos共␸⁻␸

⬘

兲K^共^b^兲+ sin共␸⁻␸

⬘

兲K^共^c^兲+ . . . . 共20兲 In the above we have defined

K^共^a^兲=

冢

⁰⁰⁰⁰ ^K⁰⁰⁰¹ ^K⁰⁰⁰¹ ⁰⁰⁰⁰

冣

^, ^K^共^b^兲⁼

冢

^2K⁰⁰⁰¹ ^K⁰⁰⁰² ^K⁰⁰⁰² ^2K⁰⁰⁰¹

冣

^,

共21兲 and

K^共^c^兲=

冢

⁰⁰⁰⁰ ^K⁰⁰⁰² ⁻⁰⁰⁰^K² ⁰⁰⁰⁰

冣

^. ^共22兲

For the purpose of calculating polarizations and spin currents, the higher harmonics play no role and are thus ignored.

By using g_eq^R = −g_eq^A = 1 −⳵␰b·␴ and performing a rotation to the new spin basis, one can write Eq.共8兲as

⳵tg

⬘

⁼_␶¹_ⴱ关−Mg

␶^ⴱ^⬅ 1

␶⁺ 1

␶sf

, 共28兲

which we now use for convenience of notation but will be later incorporated into the proper transport time. Finally, S_E is the source term due to the electric field. We take this to be along thexdirection so that

S_E⬅ 兩e兩vFE⳵⑀关2 tanh共⑀/2T兲兴

冢

^{− cos}^{− sin}^cos⁰^␸^␸^␸^v^v^␣^␣^F^F

冣

^. ^共29兲

Solving for theszspin current flowing alongy, we obtain

j_s

z y = −N₀

4

冕

^d^⑀^v^F^具^p^ˆ^y^g^z^典

= −N₀

4

冕

^d^⑀

冤

⁻ ³⁴^␶^2m^sf⁻^␣ⁱ^␻

冥

^共具p^ˆ^y^g^⬜^典⁻^具p^ˆ^x^g^储^典兲

= −N₀

4

冕

^d^⑀

冤

⁻ ³⁴^␶^2m^sf⁻^␣ⁱ^␻

冥

^具g^y^典

=

冤

⁻ ³⁴^␶^2m^sf⁻^␣ⁱ^␻

冥

^s^y^, ^共³⁰^兲

i.e., the continuity equation result 关Eq. 共4兲兴 under homogeneous conditions. In the third line we have used Eq.共17兲to set具gy典=具pˆ_yg_⬜典−具pˆ_xg_储典. Similarly, one obtains the complete expression for the frequency dependentsyspin polarization

sy= −N₀␣兩e兩E2共␣^pF兲²

冋 ^冉

^␶¹^tr⁻ⁱ^␻

^冊冉

^␶¹^E⁻ⁱ^␻

^冊冉

³⁴^␶^sf⁻ⁱ^␻

^冊

+ 2共␣pF兲²

冉

␶¹E

+ 4 3␶sf

− 2i␻

冊册

⁻¹^. ^共31兲

Besides 1/␶sf, there appear in the above two other different time scales,

1

␶tr

⬅1

␶^{共1 −}^K¹^兲⁺ 1

␶sf

, 1

␶E

⬅1

␶^{共1 −}^K²^兲⁺ 1

␶sf

.

The first,␶tr, is the total transport time. The second,␶E, is the generalization of the characteristic time related to thesyspin polarization introduced in共7兲. By using Eq.共31兲in Eq.共30兲, one obtains the expression for the frequency dependent spin Hall conductivity

␴sH共␻兲= 兩e兩 4␲

冉

3⁴␶sf

−i␻

冊

^2共^␣^p^F^兲²

冋 ^冉

^␶¹^tr⁻ⁱ^␻

^冊冉

^␶¹^E⁻ⁱ^␻

^冊

⫻

冉

³⁴␶sf

−i␻

冊

^{+ 2共}^␣^p^F^兲²

冉

␶¹E

+ 4 3␶sf

− 2i␻

冊册

⁻¹^.

共32兲 Its real part is displayed in Fig. 2for different values of the disorder parameter ␣^pF␶. In the limit ␻→0, the magnitude of the spin Hall conductivity depends on the value of␣pF␶as well as on the ratio␶/␶sf. In the absence of magnetic impurities one has the known result␴sH= 0. As spin-flip scattering grows, the conductivity reaches values of the order of the

“universal”兩e兩/8␲. This was noted already in Ref.12, where however, as pointed out in the beginning, angle-dependent scattering was not considered. Large values of ␣^pF␶^{can be} achieved both in III-V and II-VI semiconducting materials.

Doping the latter with Mn allows controlling of the spin-flip time ␶sf while only weakly affecting the electron mobility^31–33 even though it is not perfectly clear whether these can appropriately be described in terms of the linear Rashba model.³⁴ Additionally, for certain frequencies one can see crossing points关␻␶⬇0.5 and␻␶⬇2 in Fig.2共a兲兴at which magnetic disorder has no effect on the spin Hall response. Such points are well defined only when ␣pF␶⬇1.

For clean共␣^pF␶Ⰷ1兲or dirty共␣^pF␶Ⰶ1兲samples, the different curves cross each other over a progressively wider range of frequencies.

Finally, in the diffusive regime, ␻␶trⰆ1,␣pF␶trⰆ1, and

τ/τsf= 0.4 τ/τsf= 0.3 τ/τsf= 0.2 τ/ττ/τsfsf= 0.1= 0

αpFτ= 1 ωτ ReσsH/(e/8π)

5 4 3 2 1 0 1.2

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

τ/τsf= 0.4 τ/τsf= 0.3 τ/τsf= 0.2 τ/ττ/τsfsf= 0.1= 0

αpFτ= 5

ωτ ReσsH/(e/8π)

20 15

10 5

0 4 3 2 1 0 -1 -2 -3 (a)

(b)

FIG. 2.共Color online兲Real part of the frequency dependent spin Hall conductivity in units of the universal value 兩e兩/8␲ for 共a兲

␣p_F␶= 1 and共b兲 ␣p_F␶= 5. The different curves correspond to different values of the ratio␶/␶sf= 0 , 0.1, 0.2, 0.3, 0.4共from top to bot- tom at the maximum of Re␴sH兲.

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assuming ␶tr/␶sfⰆ1, ␶E/␶sfⰆ1, one obtains the following spin-diffusion equations:

⳵tsx= −

冉

␶¹s

+ 4

3␶sf

冊

^s^x^, ^共33兲

⳵tsy= −

冉

␶¹s

+ 4

3␶sf

冊

^s^y⁻^␣^N⁰^兩e兩E^␶␶^Es

, 共34兲

⳵tsz= −

冉

␶²s

+ 4

3␶sf

冊

^s^z^, ^共35兲

where 共2␣^pF␶tr兲²/2␶tr⬅1/␶s is the D’yakonov-Perel spin- relaxation rate, tied to Rashba spin-orbit coupling. From Eq.

共34兲the sensitivity of the in-plane spin polarization on spin- flip scattering is apparent: in the stationary limit the source

共proportional toE兲is balanced by the spin relaxation. Spin- flip scattering leaves the source unchanged, whereas it en- hances the relaxation rate so that in the end s_yis reduced.

In conclusion, we studied the combined effect of long- range and magnetic disorders on voltage induced spin polarizations and the related spin Hall currents in a Rashba 2DEG.

We investigated homogeneous but nonstatic conditions from the dirty共␣^pF␶Ⰶ1兲to the clean 共␣^pF␶Ⰷ1兲 regime. Care is required when treating long-range disorder because of the two-band structure of the problem while magnetic impurities, even in low concentrations, play a nontrivial role beyond that of a simple redefinition of the time scales.

This work was supported by the Deutsche Forschungsge- meinschaft through SFB 484 and SPP 1285, and by CNISM under Progetti Innesco 2006.

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Spin polarizations and spin Hall currents in a two-dimensional electron gas with magnetic impurities