Sublattice Magnetization of Yttrium Iron Garnet at Low Temperatures

Abstract

The energies and eigenvectors for long-wavelength acoustic spin waves in yttrium iron garnet are calculated neglecting spin-wave interactions, but including the effects of anisotropy and dipolar interactions. The sublattice magnetization is found to be $S_{\ensuremath{\alpha}}^{}{}_{}{}^{z}(T)=S\ensuremath{-}{\ensuremath{\delta}}_{\ensuremath{\alpha}}\ensuremath{-}cT\ensuremath{-}{A}_{\ensuremath{\alpha}}{T}^{\frac{3}{2}}\ensuremath{-}{B}_{\ensuremath{\alpha}}{T}^{\frac{5}{2}}\ensuremath{-}{C}_{\ensuremath{\alpha}}{T}^{\frac{7}{2}}\ensuremath{\cdots}$. Here $\ensuremath{\alpha}$ labels the sublattice $a$ or $d$, ${\ensuremath{\delta}}_{\ensuremath{\alpha}}$ expresses the effect of zero-point motion, and $\mathrm{cT}$ is the Holstein-Primakoff correction for dipolar interactions. Expressions in terms of the exchange integrals ${J}_{\mathrm{aa}}$, ${J}_{\mathrm{dd}}$, ${J}_{\mathrm{ad}}$, and $J_{\mathrm{ad}}^{}{}_{}{}^{\ensuremath{'}}$, where $J_{\mathrm{ad}}^{}{}_{}{}^{\ensuremath{'}}$ describes interactions between next-nearest-neighbor $a$ and $d$ sites, are given for ${A}_{\ensuremath{\alpha}}$, ${B}_{\ensuremath{\alpha}}$, and, when $J_{\mathrm{ad}}^{}{}_{}{}^{\ensuremath{'}}=0$, for ${C}_{\ensuremath{\alpha}}$. The spin-wave spectrum of some substituted garnets and the effect of spin-wave interactions on the zero-point disordering are treated in appendices.