The coupled plasmon--longitudinal-optical-phonon modes of a degenerate inhomogeneous electron or hole gas in a bulk compound semiconductor are studied. The interaction of a point charge with these coupled modes is derived in the random-phase approximation for both the carriers and the phonons in a form which is appropriate for treating inhomogeneous carrier systems. An integral equation governing the interaction of a static point charge with these coupled modes is derived which has the same form as that in the case of a point-charge interacting with plasmons alone but in which the parameters are modified by the coupling to phonons. As a specific example, these results are used to study the binding of the coupled plasmon-phonon modes to point charges in degenerate compound semiconductors, and it is found that such bound states will exist only at very low carrier densities.
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