Kinetic-Statistical Theory of Dielectric Breakdown in Nonpolar Crystals

The steady-state Boltzmann kinetic equation for the motion of electrons under an external electric field in an idealized nonpolar insulating crystal has been set up and, after suitable approximations, solved to give the distribution function appropriate to strong fields. A criterion for dielectric breakdown is formulated in terms of crystal parameters (including the ionization probability) and applied to diamond, for which an approximate range of 75,000 to 300,000 volts per centimeter is determined by the existing spread of mobility measurements from 4000 to 900 cm²/volt sec. The reason for the relative insensitivity to impurities of this criterion is shown to be the weak dependence of the result on 〈β〉NAv(F=0)=1/T, where 〈β〉Av=mean trapping cross section of impurities, defects, or any electron acceptors in cm³/sec, N(F=0)=number of such traps vacant at zero field and temperature, and 1/T=mean reciprocal trapping time. It is also necessary and almost always true that Nₑ, the number of electron donors (shallow filled traps) in natural crystals be small relative to the atomic density of the perfect lattice. The new criterion is shown to be more critical than that of Fr\"ohlich or von Hippel, in agreement with the suggestion of Seitz that the fluctuations from mean energy of the electrons govern the breakdown. Since, however, the electron-lattice collision probability decreases with increasing electron energy at energies above h²8mλₘᵢₙ^_², where λₘᵢₙ=shortest lattice vibration wavelength, the ionizing fluctuations governing breakdown are already present in the steady-state solution. Thus the value of the breakdown field is not sensitive to the presence of externally produced electrons, as would be true of an ordinary fluctuation phenomenon, even in uniform fields.