Theory of the Yield and Fano Factor of Electron-Hole Pairs Generated in Semiconductors by High-Energy Particles

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Abstract
Yield of secondaries and ratio of variance to yield (Fano factor) are associated with a simple statistical model. This model applies almost independently of details of the physics, though introduction in it of refinements from band structure would be straightforward. The model is described in terminology of an equivalent problem which is formulated and called "crazy carpentry." Analytical and Monte Carlo results agree well. The statistics are largely determined at secondary energies of just a few ionization thresholds, above which yield increases linearly with energy, with an apparent threshold the same as the average final energy, or energy of a secondary when first it cannot ionize further. Limiting relative yield, equal to ionization threshold divided by average energy per electron-hole pair, and Fano factor are given in their dependence on losses to optical phonons. The limiting case of given relative energy loss to very many phonons of very small energy generally provides close approximations, so that the results apply also whatever kinds of phonons are produced. Illustrative simple cases are also treated, including the one-dimensional space-filling problem known as the "parking problem." The theoretical yield is used to fit data of Vavilov on the enhanced quantum yield in silicon, and some physical implications are considered.