Nonlocal spin-charge conversion via Rashba spin-orbit interaction

We show theoretically that conversion between spin and charge by spin-orbit interaction in metals occurs even in a nonlocal setup where magnetization and spin-orbit interaction are spatially separated if electron diffusion is taken into account. Calculation is carried out for the Rashba spin-orbit interaction treating the coupling with a ferromagnet perturbatively. The results indicate the validity of the concept of effective spin gauge field (spin motive force) in the nonlocal configuration. The inverse Rashba-Edelstein effect observed for a trilayer of a ferromagnet, a normal metal and a heavy metal can be explained in terms of the nonlocal effective spin gauge field.

Figure 1

Schematic figure of a trilayer of a ferromagnet (F), a normal metal (NM), and a heavy metal (HM). The Rashba spin-orbit interaction is localized at the NM-HM interface.

Figure 2

The Feynman diagrams of ${\chi }_{ij}^{(1,n)}$ and ${\chi }_{ij}^{(1,a)}$. The solid lines with two arrows denote the Green functions including the Rashba interaction given by Eq. (6), the filled circle represents the spin vertex, the unfilled triangle describes the normal velocity vertex, and the dashed wavy line indicates the anomalous velocity vertex without the Pauli matrix.

Figure 3

The Feynman diagrams of ${\chi }_{ijk}^{(2,\mathrm{n})}$ and ${\chi }_{ijk}^{(2,\mathrm{a})}$. The lines and symbols are defined in the caption of Fig. 2.

Figure 4

The Feynman diagrams of ${\chi }_{ij}^{\left(1\right)}\left(i{\omega }_{\lambda }\right)$ in the first order of the Rashba spin-orbit interaction; [(a)–(c)] without the ladder type VCs and [(d)–(j)] with the VCs. The solid lines with arrows denote the Green functions without the Rashba interaction, given by Eq. (6); the filled circle represents the spin vertex; the unfilled triangle describes the normal velocity vertex; the dashed wavy line indicates the anomalous velocity vertex; and the solid wavy line depicts the Rashba-interaction vertex without the spin component.

Figure 5

The diagrammatic description for the four-point vertex of the diffusion ladder. The dotted lines denote the impurity potential, and the cross symbol represents the impurity concentration. The solid lines without arrows are for the external momentums.

Figure 6

The Feynman diagrams of ${\chi }_{ijk}^{\left(2\right)}(i{\omega }_{\lambda },i{\omega }_{{\lambda }^{\prime }})$ in the first order of the Rashba interaction without the diffusion ladder VCs. The lines and symbols are defined in the caption of Fig. 4. [(a)–(f)] The contributions of the normal velocity term ${\chi }_{ijk}^{(2,\mathrm{n})}(i{\omega }_{\lambda },i{\omega }_{{\lambda }^{\prime }})$ and [(g) and (h)]: that of the anomalous velocity term ${\chi }_{ijk}^{(2,\mathrm{a})}(i{\omega }_{\lambda },i{\omega }_{{\lambda }^{\prime }})$. Note that (b), (d), (f), and (h) are same contributions as that which are obtained by replacing $j\leftrightarrow k,{\mathit{q}}^{\prime }\leftrightarrow {\mathit{q}}^{″},{\epsilon }_n^{}\to {\epsilon }_n^{-},{\epsilon }_n^{+}\to {\epsilon }_n^{}$, and ${\epsilon }_n^{\prime +}\to {\epsilon }_n^{\prime -}$ in (a), (c), (e), and (g), respectively.

Figure 7

All the Feynman diagrams of ${\chi }_{ijk}^{\left(2\right)}(i{\omega }_{\lambda },i{\omega }_{{\lambda }^{\prime }})$ in the first order of the Rashba interaction; (i), (j), (k), and (n) include the diagrams shown in Figs. 6 and 6, 6 and 6, 6 and 6, and 6 and 6, respectively. The three diagrams surrounded by a thick line include main contributions to the nonlocal effective electric fields. In diagrams (i)–(p), the momentums and Matsubara frequencies are not displayed for readability; they are the same as in the diagrams of Fig. 6 for (i), (j), (k), and (n). The momentums and Matsubara frequencies in (l), (m), (o), and (p) are expected from (i), (j), and (n) by using Fig. 5. The filled triangle and double circle are the full vertexes of the normal velocity and the spin, respectively, given by (q) and (r). The momentums and Matsubara frequencies of the arrowed lines are shared with the corresponding diagrams in Fig. 6.

Figure 8

The path of the integral is described. The solid and dashed line of the path $C_R$ and $C_0$ denote the pathes on the Riemann surfaces $n=0$ and $n=1$, respectively. The contributions from $C_0$ and $C_R$ vanish in the limit of $R\to \infty$.