s-d current model

Conduction electrons with energies near the Fermi surface carry spin-dependent transport processes. Localized electrons with energies far below the Fermi level determine the magnetization dynamics. Magnetization of both conduction and localized electrons is considered without orbital moments. The damping mech-anism is based on the electronic scattering of conduction electrons only. Theory presented in this section is based on [67, 75, 76].

The localization ofdelectrons leads to localized magnetic moments, which are now able to polarize conduction electrons. Therefore, bothd (M_d(r, t)) ands electrons (m_s(r, t)) contribute to the magnetizationM:

M(r, t) =M_d(r, t) +m_s(r, t). (6.7) Fig. 6.1 shows the orientation of both magnetizations for a homogeneous FMR mode. With no spin-flip scattering of conduction electrons, the d electrons and selectrons align from thes−dexchange interaction and precess in phase.

6.2 Theoretical modeling of damping

With spin-flip scattering of conduction electrons, part of the magnetization of selectronsδm_swill be orientated perpendicular toM_d. Thesanddmagnetic moments are then not aligned. This induces an extra torque, perpendicular to the precessing dmagnetization, which tries to align the s magnetization with the precessing d moments. The conduction s electrons follow the d electrons with a phase delay. This causes additional damping which increases for larger spin-flip rates.

Figure 6.1: _s−dmodel of a homogeneous FMR mode: orientations of the magnetizations ofsanddelectrons

The Gilbert equation of motion for the magnetic moment of d electrons is a starting point of this model:

M_d

dt =−γ(M_d×H_{ef f,d}) + 1

M_dM_d×αdM_d

dt +T. (6.8)

The H_{ef f,d} is the effective field influencing the d electrons. The second term describes the direct damping of the d electrons, and the last term represents an additional torque due to scattering of conduction electrons. The torque T originates from the non-colinearity ofm_sand M_d and can be expressed as:

T =− 1

τexM_dM_d×m_s , (6.9)

withτexas the period of precession ofm_saroundM_d. To calculate the induced magnetizationm_s of theselectrons, the continuity equation has to be applied:

∂m_s(r, t)

∂t +∇ · =−T −Γ_rel ,

where is the expectation value of the spin-current density tensor operator and Γ_rel represents the spin relaxation term due to spin-flip scattering. The four main contributions to the torqueT are derived from Eq. 6.8. Two of them originate from thedelectron magnetization variation in time and the other two come from the spatial variation ofM_d and the transport current.

Considering these terms, which are not explicitly stated here, Eq. 6.8 can be simplified to an equation of the Gilbert type:

M_d

dt =−γ(M_d×H_{ef f,d}) + 1 Md

M_d×αdM_d dt

with a renormalized gyromagnetic ratioγand the damping parameterα. With-out transport current of electrons, γ and α are expressed as follows:

γ = γ τ_sf represents the spin-flip time of conduction electrons, andm⁰_s represents the adiabatic part of induced magnetization in conduction electrons, m_s. Fig. 6.2

Figure 6.2: Additional damping in metals versus the spin-flip time of the conduction electrons

shows the two main regimes of the damping parameter with respect to the spin-flip scattering time of the conduction electrons. Eq. 6.12 gives the limit values of the damping parameter for infinitely small and infinitely large spin-flip scattering times. For lower spin-flip scattering times, the damping parameter is directly proportional to the spin flip scattering time, τ_sf. For higher spin-flip scattering times (overcritical damping), α is inversely proportional to τsf. Thus,

6.2 Theoretical modeling of damping

A higher spin flip rate (τ_ex>> τ_sf,α∝τ_sf) implies, that the transfer of angular momentum to the lattice will be fast. Further more, the energy dissipation will be faster and the damping parameter will be larger for a higher spin-flip relaxation rate.

In the low spin-flip rate regime (τex<< τsf) the origin of damping is the same as it is in the model for non-local damping by spin currents at the interface of the ferromagnet and normal metal described by Tserkovnyak et al.[77], (See Section 6.2.4). The precessing magnetization in the ferromagnetic layer induces spin current and spins are ”pumped” into the normal metal layer. The spin polarization of the conduction electrons is conserved for a long time and leads to the conduction electrons damping of the magnetization in the normal metal layer. In the s−dmodel, the spins are ”pumped” into their own delocalized states rather than delocalized states outside the ferromagnet.

If the spin-flip scattering time scales monotonically with the temperature, then the dependenceα(τ_sf) can be interpreted as follows: At lower temperatures and smaller spin-flip relaxation times, the damping parameter is proportional to the conductivityσ. At higher temperatures and larger spin-flip relaxation times, α is proportional to the resistivity ρ. Experiments have shown that the temper-ature dependence of α includes contributions proportional to the conductivity and resistivity. Experiments on pure nickel samples with FMAR (ferromagnetic anti-resonance) showed, that the experimentally determined damping parame-ter fits perfectly with the model composed of two damping contributions[78].

One of them is proportional to the conductivity and dominates at lower tem-peratures while the other is proportional to the resistivity and dominates at higher temperatures. The universality of this law could not be confirmed by measurements in other ferromagnetic materials.

For transition metals the s−d model predicts typical values of the damping parameter ∆α ∝10⁻⁵ −10⁻³s⁻¹. Experimental values are more on the order of 10⁻²s⁻¹. The model underestimates the energy dissipation of the precessing magnetization by neglecting thedelectrons contribution to damping and results in a much smaller Gilbert parameterα than experimentally determined.

Thes−dmodel adequately describes the effect of current transport in a metal.

To describe damping quantitatively, it is applied to metallic systems with 3d or 4f impurities, but not to transition metals. d electrons scattering then becomes significant contribution to damping and cannot be neglected. The Breathing Fermi surface model, in contrast, considers the damping mechanism from scattering both s and d electrons on equal footing. This is presented in detail in the following chapter.