Threshold energies for energy- and momentum-conserving impact ionization of electron-hole pairs in actual semiconductors are determined by differential analysis of the energy-wave-vector relations of the participating charge carriers and phonons. A necessary condition for the initiating carrier to have minimum energy consistent with pair production is that the resultant carriers and all phonons involved have identical real-space velocities. This criterion allows calculations of ionization threshold energies for any semiconductor for which the one-electron-energy-wave-vector relationship is known. A step-by-step graphical procedure is presented for the calculation of threshold energies when the final particles are traveling along a principal axis of a semiconductor. Threshold energies resulting from the application of this procedure are presented for Si, Ge, GaAs, GaP, and InSb. Each of these materials exhibits numerous threshold energies for phononless ionization initiated by either type of carrier. The lowest thresholds for electron-initiated ionization without phonon assistance are 1.1, 0.8, 1.7, 2.6, and 0.2 eV relative to the conduction-band minima in Si, Ge, GaAs, GaP, and InSb, respectively. For ionization initiated by holes, the corresponding results are 1.8, 0.9, 1.4, 2.3, and 0.2 eV relative to the valence-band maximum.

• Received 14 May 1971

DOI:https://doi.org/10.1103/PhysRevB.5.2267