Spin heat accumulation and its relaxation in spin valves

We study the concept of spin heat accumulation in excited spin valves, more precisely the effective electron temperature that may become spin dependent, both in linear response and far from equilibrium. A temperature or voltage gradient create nonequilibrium energy distributions of the two spin ensembles in the normal-metal spacer, which approach Fermi-Dirac functions through energy relaxation mediated by electron-electron and electron-phonon coupling. Both mechanisms also exchange energy between the spin subsystems. This interspin energy exchange may strongly affect thermoelectric properties of spin valves, leading, e.g., to violations of the Wiedemann-Franz law.

Figure 1

(Color online) Schematic spin valve biased with a voltage and/or temperature difference. Spin-flip and inelastic electron-electron and electron-phonon scattering in the normal-metal spacer lead to interspin energy exchange and change the thermoelectric characteristics. $I$ and $\dot{Q}$ stand for the charge and heat currents flowing into the island. $T_{\text{ph}}$ is the temperature of the phonon bath.

Figure 2

(Color online) Temperature dependence of the heat conductance $K$ (solid lines, left axis) and thermopower $S$ (dashed line, right axis) of a structurally left-right symmetric spin valve with $P=0.9$, $P^{\prime }=0.5$ and when the electron-phonon relaxation dominates the interspin energy exchange. The lines are plots of Eqs. (3, 4, 5) and the symbols have been calculated from the full nonequilibrium distribution function [Eqs. (10, 11)]. The results have been calculated for P configuration with $r=0$ (circles) and $r=1/2$ (squares) and AP configuration with $r=0$ (stars) and $r=1/2$ (triangles).

Figure 3

Electron-electron scattering vertices. (a) Equal-spin scattering, which equilibrates the electrons but does not thermalize the spins. (b) Spin-conserving scattering and (c) spin exchange scattering, which do thermalize the spins.

Figure 4

(Color online) Spin-dependent effective temperature vs voltage in an asymmetric spin valve with $P=0.9$, $P^{\prime }=0.5$, and $G_R=0.1G_L$. The lines are calculated from Eqs. (2, 13) and the symbols from Eq. (12) for numerical solutions of the kinetic equations. The upper curves are for majority, the lower for minority spins, and different strengths of $e\text{−}e$ scattering with $F=0$: no scattering (solid line and circles), weak scattering with $E_T=0.05k_{B}T$ and $G_L=100e^2/h$ (dashed line and squares), and strong scattering with $E_T=0.001k_{B}T$ and $G_L=100e^2/h$ (dash-dotted line and stars). Here $T$ denotes the temperature of the reservoirs. Left inset: behavior at low bias with thermoelectric effects $c_L=10c_R=0.02/\left(kT\right)$. Right inset: distribution function at $e\Delta V=10E_T$ with different strengths of $e\text{−}e$ scattering.