Strong nonequilibrium effects in spin-torque systems

We consider a problem of persistent magnetization precession in a single-domain ferromagnetic nanoparticle under the driving by the spin-transfer torque. We find that the adjustment of the electronic distribution function in the particle renders this state unstable. Instead, abrupt switching of the spin orientation is predicted upon increase of the spin-transfer torque current. On the technical level, we derive an effective action of the type of Ambegaokar-Eckern-Schön action for the coupled dynamics of magnetization [gauge group $SU\left(2\right)]$ and voltage [gauge group $U\left(1\right)]$.

Figure 1

Schematic view of the system: A ferromagnetic quantum dot (center) is exposed to an external magnetic field and tunnel coupled to two leads. The left lead is a ferromagnet with a fixed magnetization direction. The right lead is nonmagnetic. A nonequilibrium situation is generated by a voltage $V$ that is applied across the system.

Figure 2

The figure presents schematically the distribution function of Eq. (31). Here $n_d^{\sigma }\left(\epsilon \right)\equiv [1-F_d^{\sigma }\left(\epsilon \right)]/2$.

Figure 3

The stationary solutions and their stability: solid red $\leftrightarrow$ stable, dotted blue $\leftrightarrow$ unstable, dashed purple $\leftrightarrow$ critical solutions. (a) Here $cos{\theta }_0$ is shown for applied $V$. For $V\ne V_{\mathrm{sw}}$ the only stationary solutions are at the poles. For $V<V_{\mathrm{sw}}$ the north pole is stable. For $V>V_{\mathrm{sw}}$ the south pole is stable. (b) In the stationary regime $I_l=I_r=I$. For $V\ne V_{\mathrm{sw}}$ the conductance differs between the steady state solution in the north pole and that in the south poles [cf. Eqs. (48) and (49)]. For either case we obtain Ohm's law, i.e., straight lines (with different slopes) starting at the origin. The scenario shown is for $g_l^{\uparrow \uparrow }+g_l^{\downarrow \downarrow }>g_l^{\uparrow \downarrow }+g_l^{\downarrow \uparrow }$ which results in a higher conductance when the magnetization points at the north pole. In the opposite case, the conductance of the south pole is higher. The stability for voltage biased regime, however, is the same in both cases. In addition, we assume here $B>0$ and ${\mathrm{\Gamma }}_l^{\downarrow }>{\mathrm{\Gamma }}_l^{\uparrow }$, which results in $g_s<0$ and in $V_{\mathrm{sw}}/B>0$.