Spectral Shape and Attenuation Length for Hot Electrons in the Presence of Finite Absorption

Abstract

The space and energy distribution of electrons released into a field-free semiconductor provides a means of studying the interaction of the electrons with the material, as experiments by Bartelink, Moll, and Meyer have shown. In the presence of finite absorption, or large energy losses per collision, the conventional methods of solving the Boltzmann equation governing the distribution function fail to give reliable results. We have found that a considerable amount of information may be obtained from the Boltzmann equation itself without a spherical harmonic expansion by studying its Laplace transform with respect to energy. In particular, we obtain an analytic expression for the exponential attenuation length which reduces to the BMM expression in the limit of small absorption and small energy loss. We obtain expressions for the average energy loss and average spread of the distribution, both of which increase linearly with distance, and for the total intensity of the distribution, which decreases exponentially with the distance through which the electrons must diffuse. This form of variation is independent of the energy distribution of the source of hot electrons. Therefore, if the energy distribution of electrons is measured at two different distances from the same source, the rate at which the average energy, for example, decreases with distance may be determined. This rate, being independent of the source distribution, is a characteristic of the medium, as is the BMM attenuation length. These two quantities together provide sufficient information to determine the mean free path for optical phonon emission and the mean free path for impact ionization. Measuring the energy distribution of the electrons in the medium can be done by measuring the distribution of electrons emitted into the vacuum provided that the angular distribution of the particles is known. This angular distribution may be approximated from knowledge of the Laplace transform. The effect of this analysis on the interpretation of the BMM experiments is to suggest that the mean free path for impact ionization in silicon may be closer to 300 Å than to 200 Å.