Spin Current and Magnetoelectric Effect in Noncollinear Magnets

A new mechanism of the magnetoelectric effect based on the spin supercurrent is theoretically presented in terms of a microscopic electronic model for noncollinear magnets. The electric polarization ${\overrightarrow{P}}_{ij}$ produced between the two magnetic moments ${\overrightarrow{S}}_i$ and ${\overrightarrow{S}}_j$ is given by $\overrightarrow{P}\propto {\overrightarrow{e}}_{ij}\times ({\overrightarrow{S}}_i\times {\overrightarrow{S}}_j)$ with ${\overrightarrow{e}}_{ij}$ being the unit vector connecting the sites $i$ and $j$. Applications to the spiral spin structure and the gauge theoretical interpretation are discussed.

Figure 1

The cluster model with two transition metal ions M1, M2 with the oxygen atom O between them. With the noncollinear spin directions ${\overrightarrow{e}}_1$ and ${\overrightarrow{e}}_2$, there arises the spin current ${\overrightarrow{j}}_s\propto {\overrightarrow{e}}_1\times {\overrightarrow{e}}_2$ between M1 and M2. Here the direction of the vector ${\overrightarrow{j}}_s$ (denoted by the short arrow near the middle of the diagram) is that of the spin polarization carried by the spin current. The direction of the electric polarization $\overrightarrow{P}$ is given by $\overrightarrow{P}\propto {\overrightarrow{e}}_{12}\times {\overrightarrow{j}}_s$ where ${\overrightarrow{e}}_{12}$ is the unit vector connecting M1 and M2.

Figure 2

(a) Generic spiral spin configuration. (b–d) Some of the specific configurations where the geometrical relation among spins (black arrows), spin current (gray arrows on or below the long right horizontal arrow), and electric polarization [gray arrows above the long right horizontal arrow in (b) and (d)] are shown. The inset shows the comparison of the polarization between the cluster (dotted line) and the periodic array (solid line) at half filling, i.e., superexchange interaction case with the spin configuration in (b).