Spin-current in Presence of Inhomogeneous Magnetization

The generation of spin-currents is of fundamental interest in the field of spintronics and the experimentally widespread method is to use spin-pumping due to a dynamic magnetization, especially in non-magnetic materials like semiconductors. Takeuchi et al. [68] calculated charge and spin-currents in presence of s-d exchange coupling and spin-orbit interaction due to an inhomogeneous scalar potential. However, they consider the dirty limit ∆τ 1 and make a perturbation expansion in ∆ and the spin-orbit interaction coupling. The spin-currents in the presence of magnetization dynamics have been investigated in Ref. [79], while they focus on the influence of spin-orbit interaction, which is not part of our study.

In contrast to the charge current, the spin-current can be directly obtained from a given mag-netization profile without having to solve a differential equation. As discussed in the previous sections, the charge current has to be determined together with the electric field by solving the charge continuity equation and the Poisson equation. The reason for this difference is

the following: for the charge current, we have to satisfy the local charge neutrality condition in our case of an incompressible Fermi-liquid. This condition stems from the treatment of Coulomb interaction between electrons in the limiting case of small screening length. How-ever, since we did not take into account any comparable interaction for the spin-degree of freedom, there is no analogue screening effect for the spin, and thus, what matters are the unscreened quantities. With respect to the spin-degree, the Fermi-liquid behaves as being totally compressible.

For the longitudinal spin-current, we obtain to leading order in 1/τ∆ j_s=σ_sE− ~²

4e²σ_c [m~·(∇~m×∂_tm) +~ β∇~m·∂_tm] +~ A

∆∇m~·∂_tm ,~ (2.162) where A is the exchange constant defined in formula (2.112), and for the correction to the spin-conductivity (σ_s)ij =σ_sδij+ (δσ_s)ij, we obtain up to order (∂rm)~ ²

(δσ_s)_ij = ~²σ_s 2meEF

hζ_a^(s)(∂_im~·∂_jm) +~ ζ_i^(s)(∂_rm)~ ²δ_iji

+~²σ↑τ↑+σ↓τ↓

4eme

~

m·(∂_im~×∂_jm)~ , (2.163) with a structure analogue to δσ_c from Eq. (2.140). The prefactor (−_2e^~)² ofσ_c is due to the different units of charge- and spin-currents. Let us note that m~ ·(∇m~ ×∂_tm) constitutes~ a dissipative term while ∇m~ ·∂_tm~ is reactive, in particular, the latter persists even in the absence of dissipative processes due to the intrinsic contribution _∆^A∇m~ ·∂tm. Interestingly,~ the prefactor of this intrinsic term is essentially the same as for the exchange current and spin-density, Eqs. (2.113) and (2.112).

Again, the spin-Hall term and the reactive time-gradientm~ ·(∇~m×∂tm) can be reproduced~ using the notion of the fictitious electric (2.155) and magnetic field (2.154),

js=−~

2e(j↑−j↓) = ~

2e 2

σ_cE− ~²

4m_ee(σ↑τ↑+σ↓τ↓)E×B , (2.164) which was done completely along the same lines as the derivation of (2.157) and in the regime τ∆1.

We can introduce the spin-chemical potential induced by a time-dependent magnetization,

~

µp =m~ ×∂tm~ which gives rise to an effective spin-electric field acting on each spin component.

In fact, we immediately recover the fictitious electric fieldE=−[∇~µ_p]_k≡ −m~·[∇~µ_p]. Taking the gradient of ~µ_p yields a tensor containing the various spin-electric fields as components and which describe effects like spin-pumping due to a time-dependent magnetization. The idea of the spin-chemical potential is analogue to the description of spin-accumulation due to pumping in magnetic multi-layers discussed in Ref. [18]. Then, we can simply write for the pumped charge and spin-currents (ignoring theβ-term for the moment)

jc = −σs[∇µp]_k js = −~²

4e²σc[∇µp]_k .

The difference to the usual electric field is that the electric field yields forces on spin-up/down channels of opposite sign. Therefore, in a non-magnetic medium where transport

quantities do not depend on the spin direction, i.e. σ_s = 0, there is no charge current however, a spin-current ∝σ_c which is a direct consequence of the above relations.

It is also interesting to note that we can incorporate the β-term into the spin-chemical po-tential by defining instead ~µ⁰_p=m~ ×∂_tm~ −β∂_tm, so that~

E⁰ =−

∇~µ⁰_p

k =m~·(∂tm~ ×∇m)~ −β∇~m·∂tm~

completely reproduces all the time-dependent terms in j_c, however, in the longitudinal spin-current js we do not get the intrinsic contribution _∆^A∇~m·∂tm. In summary, the complete~ equations for the charge- and longitudinal spin-current can be compactly written as

jc=σE−σ_s

∇µ⁰_p

k (2.165)

js=σsE− ~² 4e²σc

∇µ⁰_p

k+ A

∆∇~m·∂tm .~ (2.166) Transverse spin currents

As we will find out later, the spin-current gives rise to a spin-torque and thus directly affects the magnetization dynamics. Therefore, it is also interesting to look at the transverse spin-current which is induced by a time-dependent magnetization, and in the limit τ∆ 1 has the form

ˆj_⊥=−σd⊥(∇~µ_p)_⊥~σˆ+σr⊥m~ ×∇~µ_p~σˆ =−σd⊥m~ ×(∂_t∇~m)_⊥~σˆ−σr⊥(∂_t∇m)~ _⊥~σ ,ˆ (2.167) Note that here we are omitting the additional transverse spin-current contribution induced by the electric field. Also, we did not explicitly write the exchange current (2.113), although it has to be added here in order to obtain the complete expression. We defined the transverse spin conductivity with the dissipative part σd⊥ and the reactive part σr⊥,

σd⊥ = ν₀ 12m_eP∆

1−P⁴ τ + 1

T₂ + P²(3 +P²) T₁

, (2.168)

σr⊥ = ν₀ 5 + 5P²+ 2P⁴

60m_eP . (2.169)

The reactive part is due to precession in the exchange field ∆, while the dissipative part is due to momentum and spin-relaxation and vanishes in the absence of scattering mechanisms, τ, T₁, T₂ → ∞, so that in our case ∆τ 1, we have σd⊥ σr⊥. In a fully polarized ferromagnet P = 1, we find that σd⊥ vanishes unless in the presence of spin-relaxation.

Tserkovnyaket al. derived expressions for the dissipative transverse conductivity by calculat-ing the linear spin-density response to a transverse magnetic field [80]. Additionally evaluatcalculat-ing the auto-correlator for the transverse spin-currents (the Kubo formula), they arrive at an ex-pression which exactly corresponds to our result of σd⊥ in the absence of spin-relaxation T₁, T₂ → ∞. They also noted that, formally, there is no difference between Stoner and s-d model, except for the different microscopic origin of the exchange field ∆, a result which we also find.

Finally, using the extended spin-chemical potential~µ⁰_p =m~ ×∂tm~ −β∂tm, we can also rewrite~ the transverse spin-current

ˆj_⊥=−σ_d⊥⁰ ∇~µ⁰_p

⊥

~σˆ+σ⁰_r⊥m~ ×∇~µ⁰_p~σˆ (2.170) so that here, the transverse conductivities are related to the previous ones byσ_d⊥⁰ =σd⊥−βσr⊥

andσ⁰_r⊥=σr⊥, dropping corrections of order β².

This concludes our investigations on the influence of an inhomogeneous and time-dependent magnetization structurem(r, t) on the conduction electrons inside the ferromagnetic metal.~ In particular, we studied the implications on current-flow, also in the presence of an external electric field which gives rise to the domain-wall resistance, and in more complex 2-dimensional magnetization textures like a vortex, it leads to a transverse Hall conductivity.