Spin motive force induced by Rashba interaction in the strong $sd$ coupling regime

Spin motive force induced by the Rashba interaction in the presence of strong $sd$ interaction between conduction electron and localized spin is theoretically studied. The motive force is calculated by evaluating the time derivative of the current density on the basis of microscopic formalism. It is shown that there are two motive forces, one proportional to ${\mathit{E}}_{\mathrm{R}}\times \dot{\mathit{n}}$, the other, perpendicular component proportional to ${\mathit{E}}_{\mathrm{R}}\times (\mathit{n}\times \dot{\mathit{n}})$, where ${\mathit{E}}_{\mathrm{R}}$ and $\mathit{n}$ are the Rashba electric field and localized spin direction, respectively. The second type arises in the strong $sd$ coupling regime from the spin relaxation. The appearance of perpendicular component from the spin relaxation is understood from the analogy with the current-induced torques. In the case of domain wall motion, the two contributions to the spin motive force are the same order of magnitude, while the first term dominates in the case of precession of uniform magnetization. Our result explains the appearance of the perpendicular component in the weak $sd$ coupling limit, recently discussed in the context of spin damping monopole. Detection of ac voltage induced by the precession of uniform magnetization serves as a experimental evidence of the Rashba interaction in films and wires.

Figure 1

The Feynman diagrams describing the spin motive force arising from the Rashba interaction at the lowest order. The vertex with ${\mathit{E}}_{\mathrm{R}}$ represents the force induced by the Rashba field [the second term of Eq. (7)], and the interaction with the spin gauge field, ${\mathit{A}}_{\mathrm{s}}$, is included in the linear order.

Figure 2

The Feynman diagrams describing the corrections of the Rashba-induced spin motive force due to the spin relaxation, ${\mathit{F}}^{\left(\mathrm{sr}\right),A}$. These contributions turn out to be higher order in the spin gauge field and thus neglected compared with the contribution of Fig. 3.

Figure 3

The Feynman diagrams describing the spin motive force arising from the Rashba interaction (${\mathit{E}}_{\mathrm{R}}$) and spin relaxation, represented by the random spin-orbit interaction, $v_{\mathrm{so}}$. Although there is no explicit interaction with spin gauge field ($A_{\mathrm{s},}^{}$) in the diagram, the contribution turns out to result in a force containing the spin gauge field, proportional to ${\mathbf{\alpha }}_{\mathrm{R}}\times (\mathit{n}\times \dot{\mathit{n}})$ (Ref. 29).