Optical deformation-potential scattering of holes in multiple quantum well structures

The lifetime of excited states in intersubband transitions of holes in nonionic semiconductors, e.g., in ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x,$ is determined predominantly by optical deformation-potential scattering. We present a theory for the calculation of the scattering rates. It includes strain and confinement, the hole-hole Coulomb interaction in Hartree-Fock approximation, and the different possible optical phonon branches. As a consequence of the symmetry of the optical deformation tensor the scattering rate between subbands of different types (e.g., from a heavy-hole subband to a light-hole or split-off subband) is considerably higher than between subbands of the same type. Numerical results are given for ${\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x/\mathrm{Si}$ quantum wells with various Ge concentrations and well widths. We find that the scattering rates decrease both with increasing Ge concentration in the wells and with increasing well width. From this one can derive guidelines how to achieve lifetimes long enough to allow the operation of a silicon-based quantum cascade laser.