Show that the spontaneous-emission lifetime is a few hundred femtoseconds for a conduction electron initially in a state with kinetic energy 30 meV falling into an empty valence-band state. Assume the unitless oscillator strength defined in Exercise 5.2.1 is equal to unity, the gap energy E_gap = 1.5 eV, the effective masses of both bands are about one-tenth the free-electron mass, and the dielectric constant of the medium is ϵ = 10ϵ₀. (These are values typical for a III–V semiconductor such as GaAs.)

Data from Exercise 5.2.1

Show that the interaction Hamiltonian (5.2.8) is Hermitian, even if 〈n′|p(vector)|n〉 is complex. The fact that both n and n′are summed over is important.

Transcribed Image Text:

He-phot m X h ΣΣΣ, Ξενω ἢκλ · {n'\p\n} 2€ nn' k₁ ka κατα bt .b = + a²b² kλ n',ki+k nk1 -b κλη',k -k nky (5.2.8)