Temperature-dependent thermal conductivity in Mg-doped and undoped $\beta$-$\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ bulk-crystals

For $\beta$-$\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ only little information exist concerning the thermal properties, especially the thermal conductivity $\lambda$. Here, the thermal conductivity is measured by applying the electrical 3$\omega$-method on Czochralski-grown $\beta$-$\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ bulk crystals, which have a thickness of $200\mu \mathrm{m}$ and $800\mu \mathrm{m}$. At room temperature the thermal conductivity along the [100]-direction in Mg-doped electrical insulating and undoped semiconducting $\beta$-$\mathrm{G}{\mathrm{a}}_2{\mathrm{O}}_3$ is confirmed as $13\pm 1\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ for both crystals. The thermal conductivity increases for decreasing temperature down to $25\mathrm{K}$ to $\lambda (25\mathrm{K})=(5.3\pm 0.6)\cdot {10}^2\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. The phonon contribution of $\lambda$ dominates over the electron contribution below room temperature. The observed function $\lambda (T)$ is in accord with phonon-phonon-Umklapp scattering and the Debye-model for the specific heat at $T\gtrsim 90\mathrm{K}$ which is about $0.1$ fold of the Debye-temperature ${\theta }_{\mathrm{D}}$. Here a detailed discussion of the phonon-phonon-Umklapp scattering for $T<{\theta }_{\mathrm{D}}$ is carried out. The influence of point defect scattering is considered for $T<100\mathrm{K}$.