Spin cloud induced by a single impurity

It is well-known that a spin polarization S_y is induced in the bulk by applying an electric field E ˆx to a homogeneous 2DEG in xy-plane with the Rashba SOI [95]. However, the z-polarized spin density is equal to zero due to the SHE in this case. This result can be understood via averaging over impurity positions in a homogeneous electron gas. When the scale of the system is down to microscopic scale, the system becomes non-uniform due to the impurity breaking the homogeneity. Such that the influence of each impurity upon spin polarization can be handled through calculating one single impurity (target impurity) at a fixed position. Under this consideration, other background impurities should be taken average over their positions. Based on this concept, the Landauer electric dipole has been calculated [90, 91]. The target impurity is treated as an elastic scatterer and the electron density can be expressed in terms of the asymptotic expansion of the scattered wave functions of the electron. The wave vectors of an incident electron was weighted with the nonequilibrium part of the Boltzmann distribution function. In this chapter, instead of the Boltzmann equation, the nonequilibrium Green’s function formalism is employed to derive the spin dipole [96]. According to the standard Kubo formula, the response of the spin density is proportional to the linear term of the driving electric field E. The scattering potential of the target impurity should be taken into account in the retarded (advanced) Green’s function G^r(a). Then, assuming a homogeneous electric field is applied to 2DEG.

The electric field can be represented by the vector potential A, where E = iωA/c, as ω → 0 in the dc limit. The spin polarization can be derived similarly to Appendix

to the kinetic term. The velocity operator is obtained by

vj = p_j

m^∗ + ∂h_k· σ

∂p_j . (6.1)

The spin-orbit field h_k depends on the electron wave vector k such that the spin-orbit interaction can be written as

H_so = h_k· σ, (6.2)

where σ = (σ_x, σ_y, σ_z). In the case of Rashba SOI, the spin-orbit field is expressed in the form of

(h^x_k, h^y_k) = (αk_y, −αk_x) (6.3)

with the Rashba spin-orbit coupling constant α. The n-component of the stationary spin polarization is given by

S_n(r) = −e Z

d²r⁰ Z dω

2π

n_F(ω)

dω hT r [σⁿG^r(r, r⁰, ω) (v · E)G^a(r⁰, r, ω)]i, (6.4)

where the angular brackets denotes the averaging over impurity positions and n_F(ω) is the Fermi distribution function, with the trace running over all spin variables. The charge e > 0 such that an electron carries charge −e with its effective mass m^∗ in the semiconductor. The angular momentum is given by ~S_n(r)/2. The electric field E is

The spin cloud induced by a single impurity can be induced by a single impurity, namely, target impurity. The target impurity is located at r_i with a potential V_tg(r − r_i)respecting to an electron position r. We only take into account the Green’s functions in Eq. (6.4) up to the second order of V_tg. Therefore the retarded (advanced) Green’s functions can be expanded in the form of

G^r(a)(r, r⁰) = G^r(a)0(r, r⁰) +R

ds²G^r(a)0(r, s) Vtg(s − ri) G^r(a)0(s, r⁰) +R

ds²ds⁰²G^r(a)0(r, s) V_tg(s − r_i) G^r(a)0(s, s⁰) V_tg(s⁰− r_i) G^r(a)0(s⁰, r⁰) .

(6.5)

Here, G^r(a)0 is the unperturbed Green’s function depending on the scattering of back-ground impurities. The backback-ground impurity with potential Vsc(r) is assumed to be delta potential in the short-range correlations. We have calculated the pair correlation

< V_sc(r)V_sc(r⁰) >= Γδ(r − r⁰)/πN₀ in Chapter3, where Γ = 1/2τ is the scattering rate associated with scattering time τ and the density of state is N₀ at Fermi energy E_F. Actually, the target impurity can be different from background impurities. It could be a special impurity doped into the 2DEG. However, the target impurity and background ones should become identical when all spin dipoles contribute to the spin accumulation near the interface.

One can substitute Eq. (6.5) into Eq. (6.4) to compute the background impurity av-erages in the products of several Green’s functions. If the semiclassical limit E_Fτ À 1 is valid, the perturbation theory can be employed [97]. The building blocks are the ladder perturbation series expressed by the unperturbed averaging Green’s functions

G^r(a)_k = Z

d²(r − r⁰)e^ik·(r−r0)G^r(a)0(r, r⁰) (6.6)

in the momentum space. This Green’s functions are given in the 2×2 matrix form of

G^r(a)_k = (E_F − E_k− h_k· σ ± iΓ)⁻¹, (6.7)

(e) (f) (c) (d)

vE vE vE

vE vE

p-f p k+f p k+q

p p+f p k+q k+f

k k

k+q

k’

k’

k k k+q k’ k

k-q k

σ

_z

σ

_z

σ

_z

σ

_z

σ

_z

k

σ

_z

vE

Figure 6.1: Examples of diagrams for the spin density S_z. Scattering of electrons by the target impurity is shown in the solid circles. Dashed lines denote the ladder series of particles scattered by the background random impurities. p, k, and k⁰ represent the electron momenta.

where signs ± denote the retarded Green’s function in the upper sign and the advanced Green’s function in the lower sign for E_k= k²/(2m^∗). For the ladder approximation, the pairs of retarded and advanced Green’s functions carrying close enough momenta should be chosen to form elements of of the ladder series. We can decouple the mean products of Green’s function into the ladder series and the Fourier transformation of Eq. (6.4) can

Figure 6.2: The constructions of diagram (a) of Fig. 6.1 are decomposed into ladder series.

diagram (a) of Fig. 6.1 by decomposing all ladder series with a target impurity scattering processes in Fig. 6.2 and similar processes can be done for Fig. 6.1 (b)-(e). The vertex Σ_z(q) is related to the qth Fourier component of the induced spin density S_z(r) with the wave vector q. Accordingly, r < l_mean is valid for ballistic regime and r À l_mean is valid for diffusive regime. On the other hand, the vertex T (p) is related to the homogeneous electric field E represented by the ladder at the zeroth wave vector. The vertex Σ_z(q) also contributes to the ballistic results, in which Σ_z(q) has been taken unrenormalized corresponding to q À 1/(vFτ ) in the ballistic regime [60]. Fig. 6.1 (e) and (f) show some diagrams where the diffusion propagator separates two scattering events of the target impurity. These two diagrams give rise to small correlations to the spin density and can be neglected. Therefore, the spin density S_z has to be calculated from contributions of diagrams in Fig. 6.1 (a)-(d). Hence, the spin polarization can be rewritten by

Sz(q) = 1 2π

X

p,k

T r£

G^a_p,kΣz(q) G^r_k+q,pT (p)¤

. (6.8)

to the second order of the scattering potential V_tg.

The vertex Σ_z(q) was calculated in Chapter3 and it can be expressed in terms of propagator

S_z(q) =X

j

D^zjτ^j, j = 0, x, y, z (6.9)

where the 2×2 matrices are τ⁰ = 1 and τⁱ = σⁱ, with i = x, y, z. The matrix ele-ment D^zj(q) of the diffusion propagator satisfying the diffusion equation in Eq. (3.38) of Chapter3. The element D^z0 of the spin-charge mixing vanishes for the case of Rashba SOI [37–40].

The vertex T (q) can be calculated due to the cancellation of diagrams for the case of Rashba SOI , shown in Appendix F. Finally, one obtain the vertex

T (p) = e

m^∗p · E, (6.10)

where the momenta is p = m^∗v. We substitute T (p) and Σ_z(q) into Eq. (6.8) to obtain the spin density in the Fourier q−space

Sz(q) = X

n=x,y,z

D^zn(q)Iⁿ(q), (6.11)

where the source function is

Iⁿ(q) = e 2πm^∗

X(p · E) T r£

G^a_p,kσⁿG^r_k+q,p¤

. (6.12)

tually similar, though different in its context, to the original charge cloud consideration when SOI is not present and the Boltzmann equation is used to describe the subsequent background scattering. For q ¿ l⁻¹_mean ¿ k_F, the source can be expanded in powers of q. Therefore, the wave-vector-independent terms represent the delta source located at r_i, while the terms linear in q are associated with the gradient of the delta function. Below, we will keep only the constant and linear terms for each nth component Iⁿ(q) and as-sume, for simplicity, the short-range scattering potential Vtg(r), such that the kth Fourier transformation is simply V_tgexp(−ik·r_i), where V_tgis a constant. Furthermore, the source can be written by

Iⁿ(q) = I₁ⁿ(q) + I₂ⁿ(q), (6.13)

where I₁ⁿ and I₂ⁿ are the source contributed from the first order and the second order of V_tg. These source terms can be calculated by substituting Eq. (6.5) into Eq. (6.12). The source terms I₁ⁿ can be interpreted by Fig. 6.1 (a) and (b) in the form

I₁ⁿ(q) = eV_tg

Another source term I₂ⁿ can be interpreted by Fig. 6.1 (c) and (d) in the form of

I₂ⁿ(q) = eV_tg²

We assume that the electric field is applied along x axis and z axis is perpendicu-lar to the 2DEG. From Rashba-SOI Hamiltonian α(kyσx − kxσy), there are some use-ful symmetric properties σi → σyσⁱσy by changing momentum (kx, ky) → (kx, −ky) in Rashba-SOI Hamiltonian. Connecting to Eq. (6.12), we have relations of I^x(z)(q_x, q_y) =

−I^x(z)(q_x, −q_y) and I^y(q_x, q_y) = I^y(q_x, −q_y). Also, we can obtain another symmetric

one can easily to see the leading term of expansion of I^z proportional to linear q. The leading term of I^y is a constant and the next order is proportional to quadratic q, which can be neglected. However, the leading term of I^x implies that it is proportional to qxqy

and this source term is too small correlation to be neglected.

Because of energy E_F À Γ À h_kF, the small correlations to band effects h_kF/E_F and Γ/E_F can be ignored. At the same time, q ¿ l⁻¹_mean is valid in the diffusive regime.

Another important length scale is spin-relaxation length l_so which is the distance of spin relaxation due to D’yakonov-Perel’ (DP) mechanism [88]. The spin relaxation length is determined by l_so = √

Dτ_so = v_F/h_kF, where the diffusion constant is D = v²_Fτ /2 and the spin-relaxation time is τ_so = 4(h²_kFτ )⁻¹. In the diffusion approximation, the condition Γ À hkF indicates q ∼ l⁻¹_so ¿ l⁻¹_mean. Hence, we can calculate I₁ⁿ by keeping the leading term h_kF/Γ ¿ 1 in the diffusive regime. From Eq. (6.3), Eq. (6.7) and Eq. (6.14), we can calculate all components of I₁ⁿ in appendix G. Finally, we found the contribution I₁ⁿ = 0.

From appendix G, we can evaluate I₂ⁿ to obtain the total contribution Iⁿ in the forms of













I^x = I₁^x+ I₂^x = 0 I^y = I₁^y+ I₂^y = v_dN₀m^∗αh²_kFΓ^Γ3⁰

I^x = I₁^x+ I₂^x = −iqyvdN0m^∗h²_kF2Γ^Γ03

(6.16)

where Γ⁰ = πN₀V_tg² and v_d= eEτ /m^∗ is the electron drift velocity. If the target impurity is represented by one of the random scatterers, we get Γ⁰ = Γ/ni, where ni is the density of impurities.

In the above calculation, we did not take into account the diagrams shown in Fig. 6.1

diagrams are small by the same reason as I₁ⁿ are, at least, in the most important range of f ¿ l_mean⁻¹ , where f is the small momentum transfer in Fig. 6.1 (e) and (f).

Now, one can combine the source In with the diffusion propagator to find from Eq. (6.11) the shape of the spin cloud around a single scatterer. Taking into account Eq. (6.16), Eq. (6.11) is transformed into

S_z(q) = −v_dN₀h²_kF Γ⁰

2Γ³ (iq_yD^zz(q) − 2m^∗αD^zy(q)) . (6.17)

The matrix elements D^ij satisfy the spin-diffusion equation [42]

X

l

Ã

−δ^ilDq² − Γ^il+ iX

m

R^ilmq_m

!

D^lj(q) = −2Γδ^ij, (6.18)

where the DP relaxation term is given by

Γ^il = 4τ­

δ^ilh²_kF − hⁱ_kFh^l_kF®

(6.19)

with the angular brackets denoting averaging over the Fermi surface. For the case of Rashba SOI, substituting Eq. (6.3) into Eq. (6.19) give us Γ^xx = Γ^yy = 4h²_kFτ and Γ^zz = 2h²_kFτ . The spin precession term associated with SOI field is given by

R^ilm = 4τX

p

ε^ilj­ h^j_kv_F^m®

(6.20)

and nonzero results are iP

m

R^izmq_m= −iP

m

R^zimq_m = 4iDm^∗αq_i for the case of Rashba SOI. We ignored the spin-charge mixing term in 7diffEQ due to the small correlation. This mixing is already taken into account in the source term because Iⁿfor n = x, y, z describes the source of the spin polarization in response to the electric field. From Eq. (6.18),

F

−D^zy = D^yz = 1 2h²_kFτ²

2i˜q_y

(˜q²+ 2) (˜q²+ 1) − 4˜q² D^yy = 1

2h²_kFτ²

˜ q²+ 2

(˜q²+ 2) (˜q²+ 1) − 4˜q², (6.21) where the dimensionless wave vector is defined by ˜q = qlso/2. By substituting Eq. (6.21) into Eq. (6.17), we have spin polarizations

S_z = −2iv_dm^∗α

~ N₀Γ⁰ Γ

˜

q_y(˜q²+ 3)

(˜q²+ 2) (˜q²+ 1) − 4˜q² (6.22)

and

S_y = 2v_dm^∗α

~ N₀Γ⁰ Γ

(3˜q²+ 2)

(˜q²+ 2) (˜q²+ 1) − 4˜q². (6.23)

We have restored the physical unit by putting ~ in the above expressions. The z-component of the spin density in real space is shown in Fig. 6.3. As our expecting, it has the shape of a dipole oriented in y direction perpendicular to the electric field E ˆx.

Its spatial behavior is determined by the single parameter l_so, which gives the range of exponential decay of the spin polarization with increasing distance from an impurity. The S_y component averaged over impurity positions gives the uniform bulk polarization. It is interesting to note that when the target impurities are identical to the background ones (Γ⁰ = Γ), the so obtained uniform polarization Sy|q→0 coincides with the electric spin orientation S_y = ^2vdm_~^∗αN0 agree with the result in Eq. (4.3) of Chapter4.

−4

−2 0 2 4

−4 −2 0 2 4

−4

−2 0 2 4

x y

Spin Density (in arb. unit)

Figure 6.3: Spatial distribution of S_z component of the spin density around a single scatterer. The unit of length is lso.

we will consider a semi-infinite electron gas y > 0 bounded at y = 0 by a boundary parallel to the electric field. Our goal is to calculate a combined effect of spin clouds from random impurities. It is important to note that the summation of spin dipoles from many scatterers does not result in a magnetic potential gradient in the bulk of the sample. This is principally different from the Landauer charge dipoles, which are associated with the macroscopic electric field. The origin of such a distinction can be immediately seen from Eq. (6.22). The magnetic potential µs is proportional to Sz. By taking its gradient, one gets q_yS_z. After averaging over impurity positions q → 0, q_yS_z → 0. It happens due to spin relaxation, which provides at q = 0 a finite value of the denominator in Eq. (6.22).

For the case of the charge cloud, the denominator of the particle diffusion propagator is proportional to q². Hence, the corresponding gradient of the electrochemical potential (electric field) is finite at q = 0. Although the bulk magnetic potential is zero, one cannot expect that it will also be zero near an interface. In order to calculate the spin polarization near the boundary, Eq. (6.18), with q = −i∇ and 2Γδ(r)δ^ij in the right-hand side, has to be solved using appropriate boundary conditions. With the so obtained D^ij(r), the resultant spin density induced by impurities placed at points r_i is given by Eq. (6.9)

S_j(r) = X

n=x,y,z

Z

d²r⁰D^jn(r − r⁰) I_totⁿ (r⁰), (6.24)

where the source term is obtained by the inverse Fourier transform of Eq. (6.16):

I_tot^y (r) = v_dN₀m^∗αh²_kF 1 Γ²n_i

X

i

δ (r − r_i)

where the relation Γ⁰ = Γ/n_i is used because we assumed that the target impurities are identical to the random ones. The macroscopic polarization is obtained by averaging of Eq. (6.24) and Eq. (6.25) over impurity positions. After averaging over x_i and the semi-infinite region y_i > 0, the spin-polarization source Eq. (6.25) transforms to I_avⁿ(y):

I_av^y (y) = vdN0m^∗αh²_kF 1 Γ² I_av^z (y) = −vdN0h²_kFδ¡

y − 0⁺¢ 1

2Γ². (6.26)

It follows from Eq. (6.25) that the corresponding mean value of the spin polarization, S_av(y), satisfies the diffusion equation Eq. (6.18) with the source 2ΓI_avⁿ(y) in its right-hand side. However, this diffusion equation is not complete. We should take into account that the boundary itself can create the interface spin polarization. Most easily, it can be done in the framework of the Boltzmann approach. In terms of the Boltzmann function, the spin density is defined as S_av(y) = P

k

g_k. The equation for the Boltzmann function can be written in the form Ref.[24]

v_y∇_yg_k+ 2 (g_k× h_k) + eE_x∂gk(0)

∂k_x = 1

τ [S_E(y) − g_k] , (6.27)

where S_E(y) = δ (E − E_F)^Sav_N^(y)

0 and g_k⁽⁰⁾ = −h_kδ (E − E_F) is the equilibrium Boltz-mann function. The terms proportional to the charge component of the BoltzBoltz-mann func-tion have been omitted in Eq. (6.27) due to the system local electroneutrality, at least in the scale of the mean free path, which is the smallest characteristic scale of g_k spa-tial variations. The scattering part of Eq. (6.27) is written in the simple relaxation time approximation. Such a scattering term follows from the Keldysh formalism assuming isotropic scattering from impurities, as has been adopted in this work.

The spin-polarization source associated with the boundary is given by a direct interac-tion of the electric field, without taking into account secondary scattering from impurities.

Hence, the term with S_av(y) in the right-hand side of Eq. (6.27) can be ignored. Also, the boundary independent bulk part of g_k has to be subtracted from the general solution

condition is simply

gkx,ky|y=0 = gkx,−ky|y=0 . (6.28)

This condition means that the spin orientation does not change after specular reflection from the interface. The solution of Eq. (6.27) satisfying Eq. (6.30) can be easily found.

By expanding up to the order of α², we obtain

S_if^y (y) = S_if^x (y) = 0 S_if^z (y) = 8vdα²τ m^∗ X

ky>0

kyδ (Ek− EF) exp µ

−m^∗y k_yτ

¶

. (6.29)

Within the diffusion approximation, the second of these equations represents a delta source of the spin polarization with intensity

1 τ

Z _∞

0

dyS_if^z (y) = v_dN₀h²_kF 1

Γ. (6.30)

This source is exactly of the same magnitude, but opposite in sign to the spin polarization emerging from impurities, which is represented by the integral of 2ΓI_av^z (y), with I_av^z (y) given by Eq. (6.26). Taking into account that both sources are located at the interface, so that they cancel each other out, one sees that only the y-component of the source originating from impurity scattering retains in the diffusion equation which acquires the form

∂²S^z ∂S^y

The bulk solutions of this equation are S_av^z = 0 and S_av^y ≡ S^b = 2τ eEN₀α, which coincide with the polarization obtained from Eq. (6.22) and Eq. (6.22) by setting q → 0.

In order to calculate the spin polarization near the boundary (y=0), we employ the hard wall boundary conditions for Eq. (6.31). Such boundary conditions can be easily obtained from Eq. (6.27) by performing its summation over k and integrating from y = 0 to some point y0, placed at a distance much larger than l but still small compared to lso. A simple analysis of Eq. (6.27) shows that up to the order of α², the sum over k of the vector product in the left-hand side of Eq. (6.27) can be neglected, while the right-hand side and the term containing the electric field turn to zero identically. As a result, we get

1 m^∗

X

k

k_yg_kx,ky|_y=y0 = 1 m^∗

X

k

k_yg_kx,ky|_y=0 (6.32)

According to Eq. (6.30), the above sum is zero at y=0. Hence, it is also zero at y=y0.

The latter sum coincides with the spin current within its conventional definition,26 where a contribution associated with the charge density due to the second term of the velocity operator Eq. (6.1) is ignored in an electroneutral system. Using the gradient expansion of Eq. (6.27), this current can easily be expressed through S_av^j |_y = 0, its y derivative, and the last term in the left-hand side of Eq. (6.27). In this way, one arrives at the boundary conditions from Refs. [42, 81]. We generalize these conditions by adding possible surface spin relaxation (see also Ref. [98]). These additional terms are characterized by the two phenomenological parameters rho_y and ρ_z. Finally, we obtain

−D∂S_av^z (y)

∂y |_y=0 + 2Dm^∗α [S_av^y (0) − S_b] = −ρ_zS_av^z (0)

−D∂S_av^y (y)

∂y |_y=0 − 2Dm^∗αS_av^z (0) = −ρ_yS_av^y (0) (6.33) One can easily see from Eq. (6.31) and Eq. (6.33) for ρ_y = ρ_z = 0, the homogeneous bulk solutions S_av^z = 0 and S_av^y = S^b turn out to be the solutions of the diffusion equation everywhere at y > 0. Therefore, the z-components of spin clouds from many impurities

When ρ_i 6= 0 for the soft-wall boundary, the spin density S_av^z is not zero. In the case of weak surface relaxation, ρ_i ¿ D/l_so, Eq. (6.31) and Eq. (6.33) give the finite out-of-plane spin density:

S_av^z (0) = 0.35ρ_yτ eE 1

2π~D, (6.34)

where ~ is restored the conventional units. It is notable that in such a regime of small enough ρi, the surface polarization does not depend on the spin-orbit constant.

6.4 Spin-Hall resistance and energy dissipation

According the above discussions, the finite spin accumulation S_av^z (0) can survive for soft-wall boundary (ρi 6= 0) and we can introduce the spin-Hall resistance due to this spin accumulation. Considering the magnetic potential difference S_av^z (0) = N₀µ_s near the interface y = 0, the spin-Hall resistance is computed by

R_sH = µs

j = S_av^z (0)

jN₀ , (6.35)

where the current density j = σE, with the Drude conductivity σ = ne²τ /m^∗. This spin accumulation is due to the spin-relaxation mechanism and the spin-relaxation mechanism produces the energy dissipation near the interface. We have shown that spin accumulation is associated with the correlation of the electric conductivity of a dc current flowing in the x-direction [42]. For Rashba SOI, the correlation of a current density is given by

The above result is finite within the distance of l_so from the interface. After integration over y, the correlated current has the form of

∆I = e 4m^∗

α²k_F²

Γ² S_av^z (0). (6.37)

The interface dissipation per unit of the interface length can be calculated from Eq. (6.35) and Eq. (6.37)

∆W = E∆I = m^∗

e~³α²τ RsHj². (6.38)

6.5 Summary

We found out that the intrinsic spin-Hall effect induces in 2DEG a nonequilibrium spin density around a spin-independent isotropic elastic scatterer. The z-component of this density has the shape of a dipole directed perpendicular to the external electric field, while the polarization parallel to 2DEG is isotropic. Due to the DP spin relaxation, the spin density decays exponentially at a distance larger than the spin-orbit precession length. It is noteworthy that such a cloud exists even in the case of the Rashba spin-orbit interaction when the macroscopic spin current is absent. We also calculated the macroscopic spin density near an interface by taking the sum of clouds due to many scatterers and independently averaging over their positions. Surprisingly, in the case of the hard wall boundary, the so calculated spin polarization exactly coincides with that found from the drift diffusion or Boltzmann equations. In this case, the out-of-plane spin polarization S_av^z is zero, while the parallel polarization is a constant determined by the electric spin orientation. The spin-Hall resistance of the interface can be calculated by the finite spin accumulation ∆S_av^z (0) for the case of soft boundary.

7.1 Conclusion

In Chapter 2, we studied the characteristics of a spin-dependent pumping in the

In Chapter 2, we studied the characteristics of a spin-dependent pumping in the

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