Temperature-dependent thermal conductivity in Mg-doped and undoped $\beta$-$\mathrm{Ga_2O_3}$ bulk-crystals

For $\beta$-$\mathrm{Ga_2O_3}$ 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 $\beta$-$\mathrm{Ga_2O_3}$ bulk crystals, which have a thickness of $200~\mathrm{\mu m}$ and $800~\mathrm{\mu m}$. At room temperature the thermal conductivity along the [100]-direction in Mg-doped electrical insulating and undoped semiconducting $\beta$-$\mathrm{Ga_2O_3}$ is confirmed as $13\pm 1~\mathrm{Wm^{-1}K^{-1}}$ for both crystals. The thermal conductivity increases for decreasing temperature down to $25~\mathrm{K}$ to $\lambda(25~\mathrm{K})=(5.3\pm 0.6)\cdot 10^2~\mathrm{Wm^{-1}K^{-1}}$. 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 $T<100~\mathrm{K}$.