Formula (5.2.14) gives an absorption rate. Convert this to an absorption length, that is, show that for a constant fluence of photons hitting a surface at x = 0, the probability of photon absorption at distance x from the surface is given by

where l is the absorption length, which is inversely proportional to the rate 1/τ given by (5.2.14). Estimate this absorption length for the case E_gap = 2 eV, ϵ/ϵ₀ = 10, and effective masses all of the order of the vacuum electron mass, for h̄ω/E_gap = 1.1. The matrix element for a semiconductor is typically given in terms of the unitless oscillator strength

which is of the order of unity for typical semiconductors.

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P(x) = P(0)e-x/1 (5.2.16)