Monte Carlo studies of hot-electron energy distribution in thin insulating films. I. Constant mean free path and a one-dimensional simulation

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Abstract
The hot‐electron energy distribution produced in a high electric field F across a thin insulating film is studied by Monte Carlo calculations on a digital computer. The dominant electron collisions are assumed to be those with the lattice, producing single optical‐phonon emissions of energy εph. The mean free path λ is taken as a constant independent of energy, and both isotropic and anisotropic scattering are studied. A scaling law for the average electron energy Eave,ss in the steady‐state distribution, previously found analytically for isotropic scattering, namely, Eave,ss=k (Fλ)2ε/ph, is found to hold also for anisotropic scattering, k being a numerical constant determined by the detailed nature of the scattering law. Forward scattering produces larger values of k, backward scattering smaller values. No matter how strongly peaked the forward scattering, short of exact (ϑ=0) forward scattering, there is a finite steady‐state distribution. An analogous scaling law is found for the development distance D require to achieve the steady‐state distribution, namely, D=k′ (Fλ2)/εph. Since the study of anisotropic scattering is costly, an extensive computer study was made of a highly simplified simulation, namely, a one‐dimensional random walk (fixed path length between collisions) in a force field. The salient features of the three‐dimensional energy distribution are reproduced by the one‐dimensional simulation, and the computer running time is reduced by a full order of magnitude.