${\mathbf{A}}^{\mathrm{\perp }}$⋅p versus Φ for coupling electrons to interface optical phonons in quantum wells

The interface optical-phonon modes and their interaction with electrons in layered semiconductor structures are considered. In a canonical theory where retardation effects are retained from the outset, the theory leads naturally to the quantization of the free interface oscillations in the radiation gauge for which the scalar potential is zero and the vector potential is transverse (φ=0, ∇⋅A=0). The description is thus entirely in terms of a transverse vector potential ${\mathbf{A}}^{\mathrm{\perp }}$ which satisfies ∇⋅${\mathbf{A}}^{\mathrm{\perp }}$ =0 everywhere, except at the interfaces where, as usual, only boundary conditions apply. The interaction between the two subsystems (electrons and interface modes) is the well-known minimal-coupling Hamiltonian which is in the form e${\mathbf{A}}^{\mathrm{\perp }}$⋅p/${\mathit{m}}^{\mathrm{*}}$. The main aspects of the quantization of such retarded modes are summarized. It is then shown that for a double heterostructure the nonretarded vector potential can be expressed in terms of the gradient of a unique field operator Λ which enters a unitary transformation ${\mathit{e}}^{\mathit{i}\mathrm{\Lambda }}$. We demonstrate that the result of applying this transformation on the total minimal-coupling Hamiltonian is the unitary-equivalent Hamiltonian in which the coupling to electrons is in the form eΦ. This Φ is identical to that given by Mori and Ando. The matrix elements using the eΦ form of coupling are then compared with those using the e${\mathbf{A}}^{\mathrm{\perp }}$⋅p/${\mathit{m}}^{\mathrm{*}}$ coupling and seen to be clearly different. However, when the emission rates are evaluated using the two coupling Hamiltonians, interesting and nontrivial manipulations are required to prove that the same results emerge for the total emission rate from any given initial electronic state of the double heterostructure. The reasons for the agreement of the two sets of results for first-order transitions are pointed out and discussed.