Spin-Charge Continuity and Diffusion Equation

In this part, we will investigate the kinetic equations in some more detail, since they are underlying the results to be presented in the remaining part of this chapter. We will explore

some implications of these equations, to which we will refer back in the discussions of the subsequent sections. Also, we will be able to compare our kinetic equation to other theories available in literature.

In this part of our work, high energy terms give important contributions to the results. In an attempt to include the high-energy part into our equations, let us write the equation (2.39) for the spectral density to leading order in time-gradients and electric field,

∂_neqAˆ+v_k(∂_r−eE∂_w) ˆA+ 1 where ∆r ≡∆−δ∆ denotes the magnitude of the exchange field with gradient-corrections included and ˆH is the Hamilton Operator defined in Eq. (2.1). Using v_k =∂_kk=∂_kHˆ and The high energy contribution is simply given by ˆg_h =R _dω

2πAfˆ D(ω), so that the total Greens function is ˆG(k,r, t) = ˆg_h(k,r, t) + ˆg(k,r, t). Therefore, we multiply the equation for ˆA by fD, integrate over ω and partially integrate the derivate in the last term

Z dω

invoking the zero-temperature approximation. We immediately see that the last term exactly corresponds to the source term ˆJt in Eq. (2.46), thus yielding

∂neqˆg_h+vk∂rgˆ_h+ 1 2i

h

∆rm~~σˆ^◦,gˆ_hi

+ (∂rδµ)∂kgˆ_h=−Jˆt . (2.118) Adding the two equations for low- and high-energy dynamics (effectively restoring the ∂_neqˆg term in Eq. (2.44)), the term ˆJt cancels, leaving only the contribution ˆJc. We explicitly write the kinetic equation for the total quantity ˆG(k,r, t) = ˆg_h+ ˆg, where we used the linearity of the collision integral ˆI and we introduced

Jˆeff ≡Jˆc−Iˆ[ˆg_h] =

Here, ˆIω is the fully energy dependent collision integral from Eq. (2.38), while ˆI describes only low-energy collisions and is defined in (2.45). Jˆeff comprises important high-energy contributions to the collision integral and appears as a source term. Essentially, ˆJeff defines a quasi-equilibrium distribution which takes into account situations of external perturbations like time-dependence and external electric field described by∂_neq. We will better understand what it represents, when deriving continuity and diffusion equations in a moment.

Formally, this equation corresponds to the previous equation for the low-energy dynamics, Eq.

(2.44), except that the low-energy Greens function ˆgis replaced by the total Greens function

Gˆ, while instead of the two sources ˆJtand ˆJc, we have ˆJeff. Note that now, we cannot neglect the term ∂_neqG ≈ˆ ∂_neqgˆ_h. So in effect, we have to make the following adjustments to Eqs.

(D.4)-(D.6) in order to obtain the new set of equations for the total quantities: replace the low energy densities by their total density counterparts, i.e. ˆn → N, ˆˆ j → J, ˆˆ T → Tˆ_tot, replace the source terms ˆJt+ ˆJc by ˆJeff and add the term∂_neqNˆ,∂_neqˆj, etc.

Applying this procedure to the spin-charge continuity equation (D.4), we obtain

∂_tNˆ + ˆΓ ˆN +∆r

2i

hm~~σ,ˆ Nˆi

+∂_rJˆ= ˆN_eff = ˆJc⁽⁰⁾+ ˆΓ ˆN_h , (2.121) where the source term explicitly reads

Nˆ_eff,⊥=

βA(∂_r²m)~ _⊥+ (αρs+δαr)m~ ×∂tm~

+βα^nloc₁ m~×∂t∂_r²m~ +βα^nloc₂ (∂rm~ ·∂tm)~ m~ ×∂rm~i

~σ ,ˆ (2.122) which are the correct expressions up to orderO(∂rm)~ ². The so-called non-adiabatic coefficient β ≡ _T2¹_∆ will be recurring throughout the following sections. Furthermore, we will later find thatα= _6ρ^ν0P2

sT₂ corresponds to the Gilbert damping constant andδαr=O(∂rm)~ ²are gradient corrections thereof, which however will be not important in this work, since they turn out to be of higher order in_τ∆¹ , while the dominant term comes from the current and is proportional toσ_c (see result (2.175)). α^nloc_1,2 constitute non-local diffusive type corrections to the Gilbert damping,

The action of the scattering term ˆΓ will be specified explicitly in a moment.

Similarly, for the spin-charge diffusion equation (D.5), we get

∂tJˆ+ ˆΠ ˆJ +∆r

h is the source term which incorporates a high-energy part of the complicated dynamics described by the collision integral. For a definition of the various quantities, see Appendix D.

Corrections to the scattering rates

Now, we want to discuss the relaxation described by ˆΓ and ˆΠ, and in particular, we are interested in gradient corrections to the various scattering times. For illustration, let us first have a look at transverse spin-dephasing which is explicitly given by

Γ ˆˆN_⊥≡ 1

where we can see that only the charge component of ˆa⁽⁰⁾plays a role. T₂ is the spin-dephasing time of the homogeneous system, whileT_2,r=T₂(1+δT₂) incorporates magnetization gradient corrections, which is indicated by the subindex r.

The change in the local density of states is parametrized by the quantity ηr =Er where we introduced the dimensionless energy of magnetization gradients,Er ≡ ^~2_2m^(∂_e^r_E^m)~_F² and η_r=ηE_r.

The corrections to all the remaining scattering times are summarized in the following, and it turns out that all can be reduced toη_r, which seems reasonable since in the end, the scattering rates are essentially given by the density of states. The action of the spin relaxation tensor ˆΓ includes gradient corrections and is defined as ( ˆN = ⁿ₂^cˆ1 + ˆ~σ~s) and likewise, the momentum relaxation tensor ˆΠ performs the following action

ΠˆJˆ= 1−ηr where the corrections to the scattering times for the spin-up/down channels read

δτ↑,↓≡η_r 1∓γ^1+P_2P²

1±P γ . (2.131)

The definitions for the various spin components s_⊥,sk,j↑,↓ are found near Eq. (2.28).

Furthermore, we can see a channel mixing in the presence of magnetization gradients, for example, a transverse spin-currentj_⊥has a finite contribution relaxing into the charge channel.

Current in the presence of non-equilibrium excitations

Let us now derive a diffusion equation in the regime where the elastic mean-free time is much shorter than the precession time, i.e. ∆τ 1. In this limit, gradient corrections to the collision integral and the d-wave contribution ∂rTˆtot become irrelevant and, as it is the case usually in diffusive transport, we need to take into account only s- and p-wave components

for sufficiently weak perturbations so that the distribution function is nearly isotropic. This is contrary to the other sections, where we assumed that the precession time is shorter thanτ which implies that precession due to ∆ induces additional anisotropy in the spin-component of the distribution function, which requires us to also include the d-wave contribution.

Making use of all the simplifications allowed in the regime ∆τ 1, in particular Jeff = 0, the neglect of precession ^∆_2ih

~ m~σ,ˆ Jˆi

and additional terms due to high energy contributions J_h, we can reduce Eq. (2.125) to

∂_tˆj+ ˆΠˆj+ 1 2m_e

2

d∇{Eˆ_F,nˆ} − {∇Eˆ_F,nˆ}

+eE

m_enˆ= 0 , (2.132) where the spin-resolved Fermi-energy is defined as ˆEF ≡ EFˆ1 + ^∆₂m~~σˆ and d = 3 is the dimensionality of the system. Defining d ≡ 1− ^d₂, we can rewrite this equation as (ˆn =

nc

2 ˆ1 + ˆ~σ~s)

∂_tjˆ+ ˆΠˆj+ 1 dme

[E_F∇n_c+ ∆m∇~s~ +_d∆~s∇m] ˆ~ 1 + 1

dm_e

2E_F∇~s+ ∆

2m∇n~ c+∆

2dnc∇m~

~σˆ+ eE

m_eNˆ = 0 . (2.133) Note that for a two-dimensional electron gas d= 2, the gradient terms ∇m~ vanish, which is related to the fact that the density of states is a constant in that case.

We now assume a stationary situation, and by inverting ˆΠ and ignoring gradient corrections to the scattering, we can straightforwardly solve for the charge and spin-currents (note that now, there is an additional factor (−e) included in ˆj and ˆn),

jˆ= 1

2(σ_cE−D₀∇n_c−2P_dD₀m∇~s~ −2Dc~s∇~m) ˆ1 +1

2[P_σσ_cE−2D₀∇~s−P_dD₀∇n_c−2Ds~s∇~m]m~~σˆ−D_⊥(∇~s)_⊥~σˆ− D⊥n_c∇m~~σˆ = 0 . (2.134) Here, we defined the diffusion constants D↑,↓ = _dm²

e(E_F±^∆₂)τ↑,↓ for the spin-up and down channels, its average D0 ≡ ¹₂(D↑+D↓) and the polarization of the diffusion constantsPd=

D↑−D_↓

D↑+D↓. We also introduce the polarization of the conductivity, Pσ ≡ ^σ_σ^↑_↑^−σ_+σ^↓_↓. The transport properties of transverse spin-excitations are summarized in the transverse diffusion constant D_⊥= ^2E_dm^F_eτ. We furthermore defined the off-diagonal diffusion constantsDc/s≡ _2dm^d∆_e(τ↑±τ↓) and D⊥ ≡ _2dm^d∆_eτ that appear as a result of gradient corrections to the spectrum and promote conversion between the spin-degree and the charge-degree of freedom. The latter coupling terms vanish as∇~m= 0.

Finally, in the case of a homogeneous ferromagnet, Zhanget al. phenomenologically derived a set equations describing transport in ferromagnetic conductors in the diffusive limit ∆τ 1 [69]. Thus, for the homogeneous ferromagnet (∇~m = 0), we can compare our result for the current with the expression derived by Zhang et al., and find agreement except for a different transverse diffusion constant D⊥.

Finally, let us stress that in the opposite limit of rapid precession, ∆τ 1, we still obtain an expression for the current like in (2.134), but with the d-wave term included along with some additional terms and renormalized transport parameters due to gradient corrections to the collision integral.

Im Dokument Quantum Transport in Non-Collinear Magnetic Nanostructures (Seite 49-54)