Reply to "Coupling of electrons to interface phonons in semiconductor quantum wells"

Abstract

We consider the main points raised by Knipp and Reinecke (KR) in their comments on our paper [Phys. Rev. B 43, 9096 (1991)]. We maintain that the interface modes described by KR are interface polaritons and that they couple to electrons via a minimal coupling in terms of a vector potential. We emphasize that guided and bulk polaritons couple weakly to electrons via an A\ensuremath{\cdot}p interaction, in contrast to the interface polaritons, which couple strongly to electrons, also via an A\ensuremath{\cdot}p interaction. We argue that in the radiation (or transverse) gauge, the retarded interface modes have no scalar potential (\ensuremath{\varphi}=0), and their vector potential has only a transverse component ${\mathbf{A}}^{\mathrm{\ensuremath{\perp}}}$ , and not both transverse and longitudinal components. We point out that KR's retarded electric fields are identical to ours, but maintain that our use of only a transverse vector potential is appropriate. For the wave vectors of interest to electron scattering, we agree with KR that the fields are unretarded, but the interaction remains in the minimal coupling form, simply involving the unretarded limit of the same transverse vector potential. We argue that only a unique unitary transformation applied to our total unretarded minimally coupled Hamiltonian can lead to a scalar potential form of the interaction, which we may denote as e\ensuremath{\Phi}, that is identical to that found by Mori and Ando [Phys. Rev. B 40, 6175 (1989)]. The interaction Hamiltonian e${\mathbf{A}}^{\mathrm{\ensuremath{\perp}}}$ \ensuremath{\cdot}p/${\mathit{m}}^{\mathrm{*}}$ and the form e\ensuremath{\Phi} associated with it have different transition matrix elements, but they nontrivially give identical results for all first-order processes that are on the energy shell. We emphasize that the outcome of all first-order transition-rate calculations discussed by KR is indifferent to whether e${\mathbf{A}}^{\mathrm{\ensuremath{\perp}}}$ \ensuremath{\cdot}p/${\mathit{m}}^{\mathrm{*}}$ or e\ensuremath{\Phi} is used.