From Non-equilibrium Spin-excitations to Spin Torque

The treatment will be similar as in section 2.4.3, though, we will be more specific here. In order to obtain the spin-torque, we need to find the transverse non-equilibrium spin-excitation in the system which we denote as~s_⊥. This quantity of interest is obtained by considering just the transverse part of the spin-continuity equation (2.121),

(∂t~s)_⊥+ ∆rm~×~s⊥+ (∇·Js)_⊥=N~_eff,⊥− 1

T_2,r~s⊥ , (2.171) along with N~_eff,⊥ given in Eq. (2.122) and (. . .)_⊥ indicates that we take the vector com-ponent perpendicular to m. The spin-dephasing time~ T2,r = (1 +ηr)T2 has gradient cor-rections included with ηr defined in (2.127). Furthermore, we introduce the spin-current tensor (Js)_ij ≡ ^~₂Tr^σˆ₂^jˆji which contains the spin-currents for the 3 spin-components and (∇·Js)_j ≡∂r_i(Js)_ij is the divergence thereof.² In the preceding sections, we already inves-tigated the spin-currents in presence of time-dependent m~ and external electric field, so the spin-current tensor is explicitly given by Eqs. (2.162), (2.167) and (2.113).

2Here, the spin-current is defined with the usual factor^~₂ in order to obtain the current of angluar momentum.

We can solve this equation for the transverse dynamics iteratively by first solving for~s_⊥,

~s_⊥ = 1

∆_r(β_r−m~×)

−(∂_t~s)_⊥−(∇·J_s)_⊥+N~_eff,⊥

, (2.172)

where here, the non-adiabatic coefficient β_r ≡ _T2,r¹_∆r incorporates gradient-corrections and we drop terms of orderβ². We now start from the zeroth order solution which corresponds to the spin-density of the equilibrium system,~s₀=ρs,rm~ +N~_xc, and consists of the equilibrium spin-density ρ_s,r = ρ_s−δρ_s including gradient corrections, Eq. (2.111), and the transverse spin-excitation due to the exchange current, respectively. Compatible to a linear response treatment, we then plug ~s₀ into the right-hand side of expression (2.172). Finally, the cor-responding spin-torque is given by T~el = −gµ_B∆_rm~ ×~s, where g ≈ 2 is the g-factor and µB ≡ _2m^|e|~_e is the Bohr magneton [66]. Explicitly, we then obtain for the spin-torque

− 1

gµ_BT~el = ∆rm~ ×~s_⊥ = (1 +βrm~×)

−ρs,r∂tm~ −(∂tN~_xc)_⊥−(∇·Js)_⊥+N~_eff,⊥

. (2.173) The time-derivative of the exchange term (2.113) becomes

(∂tN~_xc)_⊥= A

∆ ∂t(∂_r²m)~ _⊥

⊥= A

∆

(∂rm)~ ²∂tm~ + (∂t∂_r²m)~ _⊥ ,

but there are two more contributions related to exchange currents, the obvious one arising from J_s, and the other one a result of the distinction of high- and low-energy scattering, the former expressed in N~_eff,⊥. In fact, we find that it is exactly this term ∝ βA that is responsible for the cancelation of a ”dissipative exchange torque” and that would otherwise appear due to the spin-mistracking originating fromN~_xc. Physically speaking, this is good news, as an intrinsic contribution like the exchange current (2.113) should not be subject to any dissipative mechanism like spin relaxation.

Explicitly writing all the exchange-related terms, we can rewrite the spin-torque in the form³

− 1

gµ_BT~el=−(1 +β ~m×) (∇·J_s)_⊥−ρ_s,r∂_tm~ −αρ_sm~ ×∂_tm~ +α^nloc₁ m~ ×∂_t∂_r²m~ +A

1− β

∆∂_t+ 1

∆m~ ×∂_t

~

m×∂_r²m~

⊥

, (2.174) where now,Js only contains true non-equilibrium contributions and thus, vanishes in absence of time-dependence and an external electric field.

We note that the Gilbert damping constant α emerges from N~_eff,⊥. Furthermore, in this and the following results, we can drop all gradient corrections which would only renormalize the Gilbert damping constant α, since the dominant term arises from ∇·Js and is given by _4e^σc2 (see result (2.175)), while all other corrections would be smaller by a factor _τ2¹∆², and thus negligible in our regime of investigation. This is also the reason why we made the replacements β_r→β and in the prefactor ofα,ρ_s,r →ρ_s.

The last term on the right-hand side resembles the result of (2.138) with the exchange torque given byT~ =A ~m×∂_r²m, except for a factor of 2 whose origin lies in the term~ N~_eff,⊥ and was

3We make use of the vector identities∂tm~×∂_r²m~|⊥=m~×∂_r²m+ (∂rm)~ ²m~×∂tm~ andm~×∂t(m~×∂_r²m) =~ −(∂t∂_r²m)~ ⊥−(∂rm)~ ²∂tm~

discussed above. The term _∆¹m~ ×∂tT~ can be understood as the precession correction and the term −_∆^β(∂tT~)_⊥ as the respective dissipative correction due to spin-dephasing.

Result (2.174) lets us identify the contributions to the non-local exchange (dynamic exchange) which do not directly originate from the non-equilibrium spin-currents Js induced by an external electric field or by the magnetization dynamics. However, in order to study all the terms explicitly appearing in the expression for the spin-torque, we substitute the spin currents using results (2.162), (2.167), and dropping terms of orderβ², we readily obtain⁴

− 1

gµ_BT~el=−(1 +β ~m×)(js∂r)m~ −

ρs,r+A

∆(∂rm)~ ²

∂tm~ −αρsm~ ×∂tm~ +A ~m×∂_r²m~ +β~²σ_c

4e² (∂_rm)~ ²∂_tm~ − ~²σ_c

4e² (∂_r(∂_tm~ ×∂_rm))~ _⊥

+α^(nl)_d m~ ×∂t∂_r²m~ +α^(nl)_r [∂r(∂t∂rm)~ _⊥]_⊥ . (2.175) where the spin-current due to the external electric field is js = σ_sE (the current in units of spin-angular momentum) as introduced in section 2.4.4. The terms of first order in time-gradients and second order in spatial time-gradients include non-local contribution to various terms like Gilbert damping, and will be relevant for magnetization structures varying strongly in space, like short walls.

Taking into account spatial gradient corrections, the effective conduction electron magne-tization density appearing in the LLG equation gets renormalized by the reduction of the spin-density δρ_s determined in (2.111), which is different for Stoner and s-d models. How-ever, in addition to this straightforward term, we also have a renormalization due to the exchange current.

The non-local dissipative (Gilbert damping) and reactive (∝βσ_c) terms proportional to σ_c have also been found by Wong et al. in Ref [74]. Essentially, the reactive term renormalizes the effective electron spin-density by β^~_4e^2σ2^c(∂rm)~ ², in addition to the corrections discussed above. Due to (∂r(∂tm~ ×∂rm))~ _⊥ = (∂rm)~ ²m~ ×∂tm~ −(∂tm~ ·∂rm)~ m~ ×∂rm, the dissipative~ term enhances the Gilbert damping constant αby an isotropic contribution ^~_4e^2σ2^c(∂rm)~ ². The second term essentially modifies the non-adiabatic spin-torque due to the current j_s, giving rise to a fictitious spin-current ∝ ^~_4e^2σ2^c(∂t·∂rm).~

The last two terms are identical for both Stoner and s-d model, namely the dissipative term α^(nl)_d =σd⊥+βσr⊥+βα^nloc₁ −β^A_∆ and the reactive termα^(nl)_r =σr⊥−^A_∆. The dissipative term α^(nl)_d m~ ×∂_t∂_r²m~ has contributions fromN~_eff,⊥but also from the divergence of the spin-current, where the latter give rise toσd⊥+βσr⊥. This term describes additional dissipation mediated by the conduction electrons due to dynamical exchange and has been discussed in Ref. [80], however, there only the link to the transverse spin diffusion coefficientσd⊥was established. In the end, this term can be interpreted as an anisotropic Gilbert damping contribution, which can be seen experimentally using microwave radiation to excite magnetization dynamics, and performing a wave-vector dependent measurement of the Gilbert damping constant α(q).

The Gilbert damping is related to the broadening of the absorption line. Finally, α^(nl)r is an

4We use the identities (∂r(∂tm~ ×∂rm))~ _⊥=−[m~·(∂rm~×∂tm]~ ∂rm~ = (∂rm)~ ²m~×∂tm~−(∂tm~·∂rm)~ m~×∂rm~ and [∂r(∂t∂rm)~ ⊥]_⊥= (∂t∂_r²m)~ ⊥+ (∂tm~·∂rm)~ ∂rm~

additional reactive term due to the transverse conductivityσr⊥and the exchange interaction.

In the experiment just mentioned, it would for example lead to a wave-vector dependent shift in the excitation frequency.

Note that we can identify dissipative terms in the torque by looking for terms which are odd under time-reversal, i.e. ∂t → −∂t, m~ → −m~ and js → −js. The terms even under this transformation are denoted as reactive. It is clear that the dissipative terms vanish in absence of any time-reversal breaking mechanism, but is not true for all of the reactive terms, since they contain intrinsic contributions, like α^(nl)_r . These contributions are reminiscent to the intrinsic contributions of the anomalous Hall effect originating from the spin-orbit interaction. However, we did not include any spin-orbit term in the usual sense in our model, but we have the Zeeman term coupling the electron spin to an inhomogeneous exchange field which actually does couple spin and orbital degrees of freedom, thus acting like a spin-orbit interaction. In fact, for the spatially homogenous case, only the usual Gilbert damping constantα remains, and this term vanishes as T₂ → ∞.

The expression (2.175) is the main result of this section and constitutes the complete torque due to the conduction electrons acting on the magnetization m. It emerges from a treatment~ that is consistent up to order ∂t(∂rm)~ ² and E∂rm~ and is derived in the regime where pre-cession dominates over other time-scales in the system, i.e. ∆τ 1. In this case,β is much smaller than unity and we can neglect terms of order O(β²).