Tunable Magnonic Thermal Hall Effect in Skyrmion Crystal Phases of Ferrimagnets

We theoretically study the thermal Hall effect by magnons in skyrmion crystal phases of ferrimagnets in the vicinity of the angular momentum compensation point ($CP$). To this end, we start by deriving the equation of motion for magnons in the background of an arbitrary equilibrium spin texture, which gives rise to the fictitious electromagnetic field for magnons. As the net spin density varies, the resultant equation of motion interpolates between the relativistic Klein-Gordon equation at the $CP$ and the nonrelativistic Schrödinger-like equation away from it. In skyrmion crystal phases, the right- and the left-circularly polarized magnons, with respect to the order parameter, are shown to form the Landau levels separately within the uniform skyrmion-density approximation. For an experimental proposal, we predict that the magnonic thermal Hall conductivity changes its sign when the ferrimagnet is tuned across the $CP$, providing a way to control heat flux in spin-caloritronic devices on the one hand and a feasible way to detect the $CP$ of ferrimagnets on the other hand.

Figure 1

Schematic illustration of the magnonic heat flux ${\mathbf{j}}_Q$ through a ferrimagnet in its skyrmion crystal phase subjected to a temperature gradient $\mathbf{\nabla }T$. The colored small arrows depict a single skyrmion texture of the order parameter $\mathbf{n}$. Magnons can exhibit the thermal Hall effect since the skyrmion crystal gives rise to the fictitious magnetic field, which magnons of left and right circular polarization (with respect to the order parameter) experience as if they carry the positive and the negative charge, respectively. The induced transverse heat flux changes its direction as the net spin density $s$ along $\mathbf{n}$ varies across 0.

Figure 2

The plot of the Landau levels [Eq. (11)] of magnon bands in ferrimagnetic skyrmion crystals in terms of the rescaled energy $\epsilon l/\hslash c$ and the rescaled spin density $sl/2\rho c$. The solid gold and the dashed blue lines represent the right-circularly polarized ($q=+$) and the left-circularly polarized ($q=-$) magnon bands, respectively.

Figure 3

(a) The plot of the rescaled thermal Hall conductivity ${\overline{\kappa }}^{yx}\equiv (2\pi \hslash t/k_B^{2}T){\kappa }^{yx}$, which parametrizes the ratio of the induced transverse heat flux $j_Q^y$ to the applied longitudinal temperature gradient ${\partial }_{x}T$ for the ferrimagnet film of thickness $t$. (b) and (c) the schematic illustrations of the positive-chirality ($+$) and the negative-chirality ($-$) magnon motions subjected to a temperature gradient $\mathbf{\nabla }T=({\partial }_{x}T)\widehat{\mathbf{x}}$ and the resultant transverse energy flux ${\mathbf{j}}_Q=j_Q^y\widehat{\mathbf{y}}$.