Magnetic texture-induced thermal Hall effects

Magnetic excitations in ferromagnetic systems with a noncollinear ground-state magnetization experience a fictitious magnetic field due to the equilibrium magnetic texture. Here, we investigate how such fictitious fields lead to thermal Hall effects in two-dimensional insulating magnets in which the magnetic texture is caused by spin-orbit interaction. Besides the well-known geometric texture contribution to the fictitious magnetic field in such systems, there exists also an equally important contribution due to the original spin-orbit term in the free energy. We consider the different possible ground states in the phase diagram of a two-dimensional ferromagnet with spin-orbit interaction: the spiral state and the skyrmion lattice, and find that thermal Hall effects can occur in certain domain walls as well as the skyrmion lattice.

Figure 1

Pictorial representation of the thermal spin Hall effect. A temperature difference $\Delta T$ applied to a sample leads to a finite heat current. Since the heat current is carried by the magnons in the system, the fictitious magnetic field that magnons experience due to a nontrivial magnetic ground state will lead to a finite thermal Hall conductivity.

Figure 2

Fictitious magnetic field due to the spiral ground state. Parameters are $\zeta =70$ $\mu$m${}^{-1}$, $J/\left(k_B{\epsilon }^2\right)=63$ K, and the interatomic spacing is taken to be $\epsilon =4.5$ Å (see Ref. 22). To make the connection to electromagnetism, we note that a fictitious field $B_z=2\pi \zeta /y_0\left(0\right)\approx 5\times {10}^{15}$ m${}^{-2}$ acting on a spin wave gives rise to the same magnetic length as a $\frac{\hslash }{e}{\zeta }^2\approx 3$ T magnetic field acting on a free electron.

Figure 3

Band structure of the skyrmion lattice with parameters $R=45$ nm, $\zeta =70$ $\mu$m${}^{-1}$, and $2J{B}_0/k_B\approx 50$ mK. The labels on the horizontal axis denote $(k_1,k_2)$, with the wave vectors normalized to $2\pi /a$.

Figure 4

Berry curvature of the two highest magnetic subbands in Fig. 3 in a single Brillouin zone. The subband corresponding to the top figure does not carry a net curvature; the bottom figure carries 2$\pi$.