Thermally driven spin transfer torque system far from equilibrium: Enhancement of thermoelectric current via pumping current

We consider a small itinerant ferromagnet exposed to an external magnetic field and strongly driven by a thermally induced spin current. For this model, we derive the quasiclassical equations of motion for the magnetization where the effects of a dynamical nonequilibrium distribution function are taken into account self-consistently. We obtain the Landau-Lifshitz-Gilbert equation supplemented by a spin-transfer torque term of Slonczewski form. We identify a regime of persistent precessions in which we find an enhancement of the thermoelectric current by the pumping current.

Figure 1

Schematic view of the system: a small (zero-dimen- sional) itinerant ferromagnet is placed in an external magnetic field and tunnel coupled to two leads. One lead is magnetic with a fixed direction of magnetization (left), while the other lead is a normal metal (right). The system can be driven out of equilibrium by a temperature difference between the leads.

Figure 2

For $d=\frac{{\pi }^2}{3}(T_l^2-T_r^2)$ and ${\mathrm{\Gamma }}_{\mathrm{\Delta }}<0,{\rho }_{\mathrm{\Delta }}^{\prime }<0$ and a symmetric density of states, i.e., ${\rho }_{\mathrm{\Delta }}=0,{\rho }_{\mathrm{\Sigma }}^{\prime }=0$, we show (a) the stationary solutions for $cos{\theta }_0$ with their stability (red solid = stable; blue dotted = unstable) and (b) nonequilibrium distribution functions. The temperature difference tries to drive the magnetization towards the poles for $d>0$ and towards the equator for $d<0$. The Gilbert damping is stronger than thermal driving for $\left|d\right|<d_0$, where $d_0=-{\mathrm{\Gamma }}_{\mathrm{\Sigma }}{\rho }_{\mathrm{\Sigma }}B/\left(\lambda {\mathrm{\Gamma }}_{\mathrm{\Delta }}{\rho }_{\mathrm{\Delta }}^{\prime }\right)$.

Figure 3

For $d=\frac{{\pi }^2}{3}(T_l^2-T_r^2)$ and ${\mathrm{\Gamma }}_{\mathrm{\Delta }}<0,{\rho }_{\mathrm{\Delta }}^{\prime }<0$ and a symmetric density of states, i.e., ${\rho }_{\mathrm{\Delta }}=0,{\rho }_{\mathrm{\Sigma }}^{\prime }=0$, we show the charge current $I_{l\to \mathrm{dot}}$ for the stable (red solid) and unstable (blue dotted) stationary solutions of $cos{\theta }_0$. Furthermore, we show a hypothetical situation (green dashed), in which the magnetization of the dot makes the angle ${\theta }_0$ with the $z$ axis, but does not precess. The value of $cos{\theta }_0$ is the same as in the state of persistent precessions at driving $d$. In the hypothetical situation the pumping and the hybrid currents are absent. At $d<-d_0$, i.e., in the regime of stable persistent precessions, we observe a very interesting effect: while the absolute value of the charge current is reduced in comparison to the stationary solution at the north pole, it is larger than the current for the hypothetical situation without precessions. Thus we conclude that the precession of the magnetization enhances the thermoelectric effect. For $d>d_0$, we observe a regime of double stability and the direction of the thermoelectric charge current depends on the orientation of the magnetization.