Topological spin Hall effect resulting from magnetic skyrmions

The intrinsic spin Hall effect originates from the topology of the Bloch bands in momentum space. The duality between real space and momentum space calls for a spin Hall effect induced from a real space topology in analogy to the topological Hall effect of skyrmions. We theoretically demonstrate the topological spin Hall effect in which a pure transverse spin current is generated from a skyrmion spin texture.

Figure 1

The geometry of a $31\times 31$ tight-binding cross bar. The arrows denote the in-plane component of the magnetization texture of a single skyrmion. The color plot demonstrates the $S_z$ component. $S_z$ at the skyrmion center is negative, while the outside values are positive. The four terminals are numbered clockwise.

Figure 2

THE and TSHE for the case of pure spin injection ($t=0.2J_{\mathrm{H}}$). The (a) topological Hall angle ${\theta }_{\mathrm{TH}}$ and the (b) topological spin Hall angle ${\theta }_{\mathrm{TSH}}$ are shown as a function of ${\epsilon }_F$. The surface density of states at terminal 1 is shown in (c). The four scenarios of different carrier-type and spin compositions are illustrated in (d).

Figure 3

THE and TSHE in the case of mixed spin injection ($t=1.5J_{\mathrm{H}}$). (a) and (b) Values of ${\theta }_{\mathrm{TH}}$ and ${\theta }_{\mathrm{TSH}}$ for different positions of ${\epsilon }_F$. The red dashed lines correspond to the case where the central skyrmion is removed. (c) Plot of the surface DOS at terminal 1.

Figure 4

Vector map of ${\vec{J}}_{S_z}$ (arrow plot) and the effective chemical potential distribution (color map) for (a) the THE and (b) the TSHE. A longitudinal applied bias of $\delta {\mu }_1=-\delta {\mu }_3=0.1J_{\mathrm{H}}$ is applied. For the THE (a), the spin current symmetrically circulates on either side of the skyrmion resulting in no net transverse spin current. The electron and hole accumulation induces an imbalanced transverse potential distribution. For the TSHE (b), the transverse chemical potential distribution is symmetric, and a chargeless spin current is established in the transverse direction.