Spin Pumping and Torque Statistics in the Quantum Noise Limit

We analyze the statistics of charge and energy currents and spin torque in a metallic nanomagnet coupled to a large magnetic metal via a tunnel contact. We derive a Keldysh action for the tunnel barrier, describing the stochastic currents in the presence of a magnetization precessing with the rate $\mathrm{\Omega }$. In contrast to some earlier approaches, our result is valid for an arbitrary ratio of $\hslash \mathrm{\Omega }/k_{B}T$. We illustrate the use of the action by deriving spintronic fluctuation relations, the quantum limit of pumped current noise, and consider the fluctuations in two specific cases: the situation with a stable precession of magnetization driven by spin transfer torque, and the torque-induced switching between the minima of a magnetic anisotropy. The quantum corrections are relevant when the precession rate exceeds the temperature $T$, i.e., for $\hslash \mathrm{\Omega }\gtrsim k_{B}T$.

Figure 1

(a) Tunnel junction between magnetic materials with free ($F^{\prime }$) and fixed ($F$) magnetizations, biased by voltage $V$. The total spin $\mathit{S}=\mathcal{V}\mathit{M}/\gamma$ in $F^{\prime }$ precesses at angular frequency $\mathrm{\Omega }$ around the $z$ axis. As described by Eq. (3), the motion pumps charge, spin, and heat currents through the junction, and the backaction spin transfer torque $\mathit{\tau }$ drives a change in the tilt angle $\theta$. (b)–(c) Schematic of the effective magnetic potential energy, in the presence of an external field $H_{\mathrm{ext}}$ and large spin transfer torque, or, in the presence of a magnetic anisotropy ${\mathrm{\Omega }}_{\mathrm{an}}=\gamma H_{\mathrm{an}}$.

Figure 2

(a) Noise in pumped charge current, for different tilt angles $\theta$ and precession speeds $\mathrm{\Omega }$, for $P_{F^{\prime }}=P_{Fz}=0.9$. (b) Semiclassical trajectories for $V=-{\mathrm{\Omega }}_{\mathrm{ext}}$, $P_{F^{\prime }}=1$, $P_{Fz}=1/2$, $\chi =\xi =0$, $T=0$. Shown are the $H=0$ lines AB and the fixed point $s_{*}=\frac{1}{P_{Fz}{P}_{F^{\prime }}}+\frac{2V}{P_{F^{\prime }}\mathrm{\Omega }}$ (black). Measurement trajectories $C_{\tilde{\chi }}$ for $\tilde{\chi }=1$ (red) and $\tilde{\chi }=-1$ (blue) are also shown. (c) Trajectories with anharmonicity, $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{an}}{s}_z$, $V=1.5{\mathrm{\Omega }}_{\mathrm{an}}$, $P_{Fz}=0.1$, $P_{F^{\prime }}=1$, $T=0$.

Figure 3

(a) Normalized variance of the magnetization $z$ component in the steady state around the fixed point $s_{*}$, as a function of bias voltage and temperature for $P_{F^{\prime }}=1/2$, $P_{Fz}=1/2$. Dotted lines indicate results where the energy shifts in the spin torque noise are neglected. (b) Switching exponent ${\mathrm{\Delta }}_{\mathrm{sw}}$, for $P_{F^{\prime }}=1$, $P_{Fz}=1/4$, and different temperatures and voltages. Dotted lines indicate the range $-{\mathrm{\Omega }}_{\mathrm{an}}/8<V<{\mathrm{\Omega }}_{\mathrm{an}}$ where ${\mathrm{\Delta }}_{sw}=\infty$ at $T=0$.

Figure 4

Current noise $S_I$ as a function of the measurement bandwidth $1/t_0$, for $P_{F^{\prime }}=1$, $P_{Fz}=1/2$, $T=0$.