The estimation of impact ionization coefficients for $\beta$-Ga${}_2$O${}_3$

Impact ionization coefficients of anisotropic monoclinic $\beta$-Ga${}_2$O${}_3$ are estimated along four crystallographic directions and the plot for the $\left[010\right]$ direction is shown. The approximation models were fitted to Baraff’s universal plot for ionization rate in semiconductors and the values were obtained for $\beta$-Ga${}_2$O${}_3$. The phonon mean free path of $\beta$-Ga${}_2$O${}_3$ was estimated to be 5.2604 \si{\angstrom} using Gray medium approximation. The phonon group velocity takes the value of longitudinal acoustic phonons. The ionization rate has a maximum value of $3.98\times {10}^{6}c{m}^{-1}$ along the $\left[010\right]$ direction over the applied electric field range $(1.43-4)\times {10}^{7}V\cdot c{m}^{-1}$. Contrary to expectations, the phonon mean free path along direction $\left[ \bar{2}01 \right]$ is the lowest since it has a lower thermal conductivity to phonon group velocity ratio. The plots were compared with GaN and 4H-SiC, which shows that as bandgap increases the field required for ionization increases. The critical electric field was estimated to be $ 0.921 \times10^{8},V\cdot,cm^{-1}$along$\left[010 \right]$ direction.