Unified Drift-Diffusion Theory for Transverse Spin Currents in Spin Valves, Domain Walls, and Other Textured Magnets

Abstract

Spins transverse to the magnetization of a ferromagnet only survive over a short distance. We develop a drift-diffusion approach that captures the main features of transverse spin effects in systems with arbitrary spin textures (e.g., vortices and domain walls) and generalizes the Valet-Fert theory. In addition to the standard characteristic lengths (mean free path for majority and minority electrons, and spin diffusion length), the theory introduces two length scales, the transverse spin coherence length ℓ(⊥) and the (Larmor) spin precession length ℓ(L). We show how ℓ(L) and ℓ(⊥) can be extracted from ab initio calculations or measured with giant magnetoresistance experiments. In long (adiabatic) domain walls, we provide an analytic formula that expresses the so-called "nonadiabatic" (or fieldlike) torque in terms of these length scales. However, this nonadiabatic torque is no longer a simple material parameter but depends on the actual spin texture: in thin (<10 nm) domain walls, we observe very significant deviations from the adiabatic limit.

Thermally driven magnetic precession in spin valves

“…Our starting point is a semi-classical approached for metallic magnetic multilayers that treats the charge degrees of freedom at the drift-diffusion level yet retains all the information about spin degrees of freedom [16,21]. This approach to which we refer as CRMT [16,21,22] (for Continuous Random Matrix Theory) can be seen as a generalization of the Valet Fert theory[23] to systems with non collinear magnetization [24]. It is also equivalent to the so-called (Generalized) Circuit Theory [25].…”

“…(1)- (4) of [24]. The unit vector m is the local direction of the magnetization (bold vectors correspond to spin space while explicit components α = x, y, z are used for real space).…”

“…They can be expressed alternatively in term of * , the average mean free path (1/ * = 1/ ↑ + 1/ ↓ ), and β = ( ↑ − ↓ )/( ↑ + ↓ ), the asymmetry of the spin resolve asymmetry (with a definition identical to the usual Valet-Fert parameter). Two length scales characterize the behavior of a spin perpendicular to the magnetization: the Larmor precession length L and the transverse penetration length ⊥ , see [24]. Finally, H is the heat diffusion length.…”

“…The charge and spin sectors are described by the usual spin dependent interface resistances r b σ , namely Equation (8) and (9) of Ref. [24]. The heat sector is given by (neglecting interface thermoelectric effects),…”

“…(12) and (13) of Ref. [24] for the spin and charge sector while the heat sector reads (n α points towards the system),…”