  We report an analysis on a phonon spectral function of a solid-state plasma
formed in a doped semiconductor. Real and imaginary parts of phonon propagators
are evaluated including carrier screening effects within a random phase
approximation, and finite-temperature spectral behavior of the phonon spectral
function is examined in terms of plasmon--phonon coupled modes and
quasiparticle excitation mode of the plasma. The results are applied to the
case of conduction electrons in a wurtzite GaN considering carrier-phonon
coupling channel via polar optical phonons. We show that the dispersion
relations of the plasmon-LO phonon coupled (`upper' and `lower') modes and the
character of the additional modes via single quasiparticle excitations are
heavily associated with the nonlocal and dynamic behavior of the energy shift
and collisional broadening of the dressed phonon propagator of the plasma.
