Theory of the spin Seebeck effect influenced by crystal-field excitations in Tb$_3$Fe$_5$O$_{12}$

TL;DR

This paper models the crystal-field excitations in Tb$_3$Fe$_5$O$_{12}$ and explains how magnetic field dependence of these excitations can enhance the spin Seebeck effect at low temperatures.

Contribution

It introduces a mean field theory calculation of crystal-field excitations in TbIG, linking them to the magnetic field dependence of the spin Seebeck effect.

Findings

CFE calculations show the lowest excitation decreases with magnetic field.

Magnetic field can enhance the SSE at low temperatures in TbIG.

The magnetic structure involves two inequivalent Tb sublattices affecting CFE behavior.

Abstract

The spin Seebeck effect (SSE) is a phenomenon of thermoelectric generation that occurs within a device consisting of a bilayer of a metal and a ferromagnet. When Tb$_3$Fe$_5$O$_{12}$ (TbIG) is substituted for the ferromagnet, the effect goes to zero at low temperatures, yet it increases to positive values with the application of a magnetic field. This is opposite to the expectation that the SSE should be suppressed by a magnetic field due to the increase in the magnon gap. In this paper, the crystal-field excitations (CFE) in TbIG are calculated within a mean field theory exploiting the parameters of Terbium Gallium Garnet Tb$_3$Ga$_5$O$_{12}$ (TGG) from the neutron-scattering experimental literature. Like TGG, the primitive cell of TbIG hosts twelve Tb sites with six inequivalent magnetic sublattices, but due to the net $[111]$-molecular field from the tetrahedral and octahedral Fe ions, these can be classified into two distinct groups, the $\mathbf{C}$ and the $\mathbf{C'}$ sites, which account for the `double umbrella' magnetic structure. We show that when an external magnetic field is applied along the [111] direction of the crystal, the lowest CFE of the $\mathbf{C}$ sublattices decreases. As a consequence of the magnetic field dependence of the lowest CFE, we find that at low temperatures the SSE in TbIG can result enhanced by an applied magnetic field.