Exciton-phonon interaction in fractional dimensional space

Explicit analytical expressions of Frohlich-like Hamiltonians for the interaction between an exciton and optical phonons in fractional-dimensional space are derived for two illustrative cases of $1s\to 1s$ and $1s\to 2p$ exciton scattering. The improved Hamiltonians incorporate any modification in the strength of the exciton-optical-phonon interactions due to confinement, by means of a single parameter α, a measure of the dimensionality of the confined system. The Hamiltonians also incorporate the penetration of electron and hole wave functions into the barrier region, an effect which becomes increasingly significant for narrow well widths. The flexibility of the derived Hamiltonians are shown in the ease of systematic study of exciton linewidths in ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ quantum wells. Results show the high sensitivity of the excitonic linewidths to α due to interactions with short-wavelength optical phonons.