Dynamics of electrons accelerated by strong uniform or weakly inhomogeneous electric fields in the presence of phonons

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We have analyzed the dynamics of electrons injected into the region of strong electric field and thermal phonons, as the electron kinetic energy is essential to the initiation of dielectric breakdown. For concreteness we discuss the situation expected in SiO2 which was explored in detail by the IBM group [M. V. Fischetti, Phys. Rev. Lett. 53, 1755 (1984); M. V. Fischetti, et al., Phys. Rev. 31, 8124 (1985)] with Monte Carlo simulation. We have obtained an analytic expression for the electron energy as a function of applied electric field under the collisional equilibrium approximation, which is basically the same for higher energies as that obtained by Fischetti [Phys. Rev. Lett. 53, 1755 (1984)] with a more elaborate Monte Carlo simulation. The difference exists in the lower-energy behavior. Our result shows a sharp energy drop for electric fields below 3.5 MV/cm and a plateau below 2 MV/cm. The former provides a reason for not observing electrons in experiments when the electric field is below about 3.5 MV/cm. We have extended the collisional equilibrium picture to weakly inhomogeneous electric fields to study ‘‘overshooting,’’ which is actually ‘‘undershooting’’ in the sense that the peak energy is less than the collisional equilibrium value for the local electric field.