Thermal and Magnetoelastic Properties of the van der Waals Ferromagnet Fe${}_{3-\delta }$GeTe${}_2$: Anisotropic Spontaneous Magnetostriction and Ferromagnetic Magnon Excitations

By determining the lattice parameters as a function of temperature of the hexagonal van der Waals ferromagnet Fe${}_{2.92(1)}$Ge${}_{1.02(3)}$Te${}_2$ we obtain the temperature dependence of the spontaneous in-plane magnetostriction in the ferromagnetic and the linear thermal expansion coefficients in the paramagnetic state. The spontaneous magnetostriction is clearly seen in the temperature dependence of the in-plane lattice parameter $a(T)$, but less well pronounced perpendicular to the planes along $c$. Below $T_{\mathrm{C}}$ the spontaneous magnetostriction follows the square of the magnetization and leads to an expansion of the hexagonal layers. Extrapolating to $T\to$ 0~K we obtain a spontaneous in-plane saturation magnetostriction of ${\lambda }_{\mathrm{s}\mathrm{p},a}(T\to 0)\approx -220\times {10}^{-6}$. In the paramagnetic state the linear thermal expansion coefficients amount to 13.9(1)$\times$10${}^{-6}$~K${}^{-1}$ and to 23.2(2)$\times$10${}^{-6}$~K${}^{-1}$ for the in-plane and out-of-plane direction, respectively, indicating a linear volume thermal expansion coefficient of 50.8(4)$\times$10${}^{-6}$K${}^{-1}$ which we use to estimate the volume thermal expansion contribution to the heat capacity determined at constant pressure. A Sommerfeld-type linear term in the low-temperature heat capacities can be quantitatively ascribed to 2dim ferromagnetic magnon excitations.