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Impact ionization coefficients of anisotropic monoclinic $\beta$-Ga$_{2}$O$_{3}$ are estimated along four crystallographic directions and theplot for the $\left[ 010 \right]$ direction is shown. The approximation modelswere fitted to Baraff's universal plot for ionization rate in semiconductorsand the values were obtained for $\beta $-Ga$_{2}$O$_{3}$. The phonon mean freepath 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 oflongitudinal acoustic phonons. The ionization rate has a maximum value of $3.98\times \,10^{6}\,cm^{-1}$ along the $\left[ 010 \right]$ direction over theapplied electric field range $(1.43-4)\times10^{7}\,V\,\cdot\,cm^{-1}$.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 tophonon group velocity ratio. The plots were compared with GaN and 4H-SiC, whichshows 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.

The estimation of impact ionization coefficients for $\beta $-Ga$_{2}$O$_{3}$