Self-consistency Condition – from s-d Model to Stoner Model

Using functional Keldysh theory [24], one can derive an effective Hamiltonian Hˆ =H0ˆ1−∆_s

2 m~_s~σˆ−∆_d

2 m~_d~σ ,ˆ (2.51)

together with the self-consistency condition for the conduction electrons U

2

Dψ^†(r,t)ˆ~σψ(r,t)E

= U

2Trˆ~σNˆ(r,t) = ∆_sm~_s , (2.52) whereH0 describes the spin-independent part of the Hamiltonian (2.1) for our ferromagnetic conductor. Here, ∆_dm~_d describes the effective field due to the localized d-electrons and like-wise for the itinerants-electrons. U is the exchange coupling between the itinerant electrons and the self-consistency condition (2.52) remains the same in both cases, s-d and Stoner limit. The only difference is the Hamiltonian, to which we have to add an additional term Hˆ_sd = ¹₂∆_dm~_d~σˆ in the s-dcase. This latter term vanishes in the Stoner limit, as opposed to the s-d model, where ∆d is the dominant term as compared to the mean-field contribution from the itinerant electrons described by the ∆_s term, i.e. ∆_d∆_s in thes-dlimit.

If we assume that time scales of external perturbations are much larger than the time scales of the spin dynamics of boths and d-electrons, we havem~s =m~_d incase of a ferromagnetic coupling between the two types of electrons. Then the two magnitudes ∆_s and ∆_d simply

add, and the situation intermediate between s-d and Stoner limit is no different to treat in our mathematical formalism than the two limiting cases. If the separation of timescales is not possible, the intermediate scenario is difficult due to the fact that m~s and m~d are not necessarily parallel and thus, adding up the contributions yields a strong variation of the magnitude of the total exchange field. Consequently, one has to relax the condition of constant ∆ already to lowest order which then would yield terms due to gradients in the magnitude ∆ not treated in the present work.

At any rate, in the present work we assume external perturbations to be much slower than any internal time-scales, so our calculations include also the intermediate situation. However, the two extreme cases ofs-dand Stoner limit are interesting enough, and in the intermediate regime nothing qualitatively new is expected, so in the following we will mainly treat these two limiting cases.

Stoner limit

Let us first treat the Stoner limit, where we can write for the self-consistency condition for the magnetization of the conduction electrons

U

S~₀+S~₂

= (∆−δ∆(r))m~ +δ ~W_⊥(r) , (2.53) whereU is the effective exchange coupling energy between electrons and−δ∆(r)m~ +δ ~W_⊥(r) are anticipated corrections to the self-consistent exchange field in presence of magnetization gradients. The density is obtained from Eq. (2.12) as ˆN = ^N₂^cˆ1 +S~~σ, whereˆ N_c is the charge component and S~ = S~₀+S~₂ the spin density as a series in gradients. To zeroth order, the spin density is S~0 =ρsm~ withρs= ^ν0₂^∆(1 +¹₃P²), and the correction second order in spatial gradients reads explicitly

Sˆ₂ =ν₀

(−P

2δ∆ +δµ)ˆ1 + (−1

2δ∆ +P δµ)m~~σˆ

+1 2ν₀

1 +P²

3

δ ~W_⊥~σˆ +ν₀(∂_rm)~ ²

2me

P²

12ˆ1−P(10 +P²) 60 m~~σˆ

+ ˆS_2,⊥ , (2.54) where ˆS_2,⊥ is the transverse spin component not needed at present. In the Stoner limit of itinerant ferromagnetism, U is fixed by the equilibrium magnetizationU = _ρ^∆_s = _ν ⁶

0(3+P²) and is immediately obtained by solving Eq. (2.53) for the homogeneous case.

Solving (2.53) along with the local charge-neutrality condition (2.49), i.e. Tr ˆS₂= 0 yields δµ^(St)(r) = 15−P²

120

(∂rm)~ ²

2m_e (2.55)

δ∆(r) = 5 + 3P² 20P

(∂rm)~ ²

2m_e , (2.56)

while δ ~W_⊥ exactly drops out from the equation, meaning that in principle it can be any value. However, adding a transverse partδ ~W_⊥would yield modified equations, which however describe the same physics. In fact, we obtain the original equations by simply renormalizing

the magnetization ∆~m → ∆m~ +δ ~W_⊥ (to leading order, since corrections to the magnitude would be of oder δ ~W_⊥² which is of 4th order in gradients and thus beyond our treatment).

Physically speaking, there should be no transverse correctionsδ ~W_⊥, only the magnitude of the magnetization attains correctionsδ∆(r). Thus, settingδ ~W_⊥= 0 and satisfying the transverse part of Eq. (2.53), amounts to ˆS_2,⊥ = 0 which, as we will see in section 2.5.2, corresponds to T~el = 0 and leads to the Landau-Lifshitz-Gilbert equation describing the magnetization dynamics. In fact, this corresponds to the zero-torque theorem [53] which in the Stoner limit emerges from the self-consistency condition for the transverse magnetization, and simply states that the itinerant electron system cannot exert a torque on itself.

Furthermore, in the limit ∆ → 0 viz. P → 0, the correction δ∆ also drops out from the self-consistency equation (2.53) to leading order inP. This means that in the limit of weak itinerant ferromagnetism, a small correction δ∆ to the self-consistent field induces a polar-ization in the electron gas, which in turn corresponds to a small additional effective field and exactly compensates the original perturbationδ∆ in the field. Thus, longitudinal spin-perturbations in the itinerant electron gas cannot be screened efficiently in this limit and therefore, the screening fieldδ∆ from Eq. (2.56) is inversely proportional to P, so that it is greatly enhanced in the limit P → 0. Note that this is true also for additional small trans-verse fields, however for arbitrary strength of the itinerant field ∆, so transtrans-verse perturbations cannot be screened, and instead give rise to the magnetization dynamics mentioned above.

s-d limit

In the s-dlimit, the effective exchange field ∆m(r, t) entering the equation for the itinerant~ electrons is simply given by the magnetization of the localized d-electrons, since we neglect the contribution from the itinerant electrons to the total magnetization, as it is assumed to be much smaller. Thus, we merely have to fulfill the charge neutrality condition for the conduction electrons, which yields

δµ^(sd)(r) =−P² 12

(∂rm)~ ²

2m_e (2.57)

and of course δ ~W_⊥= 0 and δ∆ = 0.

To summarize, we have derived a fully microscopic equation for the spin transport in non-collinear magnetization textures. Our approach takes impurity scattering and spin-flip scat-tering into account on the Hamiltonian level. This paves the way to treat complex magnetic textures and derive microscopic expression for the domain-wall-induced resistance.

2.3. Diffusive Transport in Quasi-1-Dimensional Systems –