Shockley has derived the expression $E_c=\frac{\mathrm{}\left(1.51\right)\mathrm{}{C}_l}{\mu }$ for the critical electric field strength $E_c$ at which the current-voltage relation in a semiconductor departs from Ohm's law. Here $C_l$ is the velocity of longitudinal phonons and $\mu$ the conductivity mobility of the carrier. Experimental values of $E_c$ for Ge and Si are from two to four times larger than those predicted by this formula. We therefore have extended the theory to take account ellipsoidal energy surfaces in the Brillouin Zone and scattering by shear modes of vibration. The effect of the more general effective mass tensor is to raise the theoretical value of $E_c$ by a factor of about 2 for $n$-type Ge and 1.3 for $n$-type Si, whereas shear mode scattering lowers $E_c$ by a factor that is between 1 and the value of the ratio of the velocity of transverse modes to that of longitudinal modes. Moreover, $E_c$ should vary with the direction of the current. The present study still fails to close the gap between theory and experiment: the remaining discrepancy is possibly the result of neglecting intervalley scattering.

• Received 22 April 1955

DOI:https://doi.org/10.1103/PhysRev.99.1189