Rotational motion of magnons and the thermal Hall effect

Due to the Berry curvature in momentum space, the magnon wave packet undergoes two types of orbital motions in analogy with the electron system: the self-rotation motion and a motion along the boundary of the sample (edge current). The magnon edge current causes the thermal Hall effect, and these orbital motions give corrections to the thermal transport coefficients. We also apply our theory to the magnetostatic spin wave in a thin-film ferromagnet and derive expression for the Berry curvature.

Figure 1

Coordinate of the ferromagnet, used for the calculation of the edge current. $U\left(\mathit{r}\right)$ is a confining potential.

Figure 2

Dependence of the thermal Hall conductivity on a magnetic field. The red (broken) curve denotes the result that is calculated from only ${\left(S\right)}_{ij}^{\alpha \beta }$ calculated in Ref. 6; the green (solid) curve denotes the result that is calculated from ${\left(S\right)}_{ij}^{\alpha \beta }+{\left(M\right)}_{ij}^{\alpha \beta }$.

Figure 3

Geometry of the coordinate axes. The magnetization $\mathit{M}$ precesses around ${\mathit{M}}_0$.

Figure 4

(a) Dispersion relation for the MSFVW mode with $n=0,1,\cdots ,5$ and the Berry curvature for the MSFVW mode for (b) $H_0/M_0=0.1$, (c) $H_0/M_0=1.0$, and (d) $H_0/M_0=2.0$.