Spinmotive force with static and uniform magnetization induced by a time-varying electric field

A new spinmotive force is predicted in ferromagnets with spin-orbit coupling. By extending the theory of spinmotive force, we show that a time-varying electric field can induce a spinmotive force with static and uniform magnetization. This spinmotive has two advantages; it can be detected free from the inductive voltage owing to the absence of dynamical magnetization and it can be tuned by electric fields. To observe the effect, we propose two experimental setups: electric voltage measurement in a single ferromagnet and spin injection from a ferromagnet into an attached nonmagnetic conductor.

Figure 1

(a) Measurement scheme of the electric voltage $V$ due to the SMF with the static and uniform magnetization $\mathit{m}$. The electric field varying sinusoidally in time is applied by the gate voltage $V_G$. The electric voltage appears perpendicular to both the time derivative of the electric field and the magnetization. (b), (c) Time evolution of a certain applied electric field and the corresponding electric voltage $V$. The ac electric field gives rise to the ac electric voltage with the same frequency and the $\pi /2$ phase shift. The amplitude of $V$ is proportional to the angular frequency of the electric field and the distance between the electrodes $L$.

Figure 2

(a) Schematic of the spin injection from a ferromagnet (F) into a nonmagnet (N). The spin current ${\mathit{j}}_{\mathrm{s}}$ induced in F due to the spin electric field (7) gives rise to a spin accumulation at the F/N interface, which decays in N as the diffusive spin current ${\mathit{j}}_{\mathrm{s}}^N$ within the spin diffusion length. (b) Spatial dependence of the spin accumulation. The spatial gradient of the spin accumulation gives the injected spin current ${\mathit{j}}_{\mathrm{s}}^N$. The coefficients $A_{F_1}$, $A_{F_2}$, and $A_N$ are determined by solving the diffusion equation (11) for F and N layers.