Uniform description of polar optical phonon states and their Fröhlich electron–phonon interaction Hamiltonians in multi-layer wurtzite nitride low-dimensional quantum structures

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The general features of four types of polar optical phonon modes [including the quasi-confined (QC) modes, the propagating (PR) modes, the half-space (HS) modes and the interface optical (IO) modes] and their electron–phonon coupling properties in wurtzite nitride low-dimensional quantum structures are discussed and concluded. Based on the transfer matrix method and the dielectric continuum (DC) and Loudon’s uniaxial crystal models, the analytical and uniform phonon states of the QC, PR, HS and IO phonon modes in wurtzite quasi-2-dimensional (Q2D) quantum wells (QWs), quasi-1-dimensional (Q1D) quantum wires (QWRs) and quasi-0-dimensional (Q0D) quantum dots (QDs) are given. The Fröhlich electron–phonon interaction Hamiltonian for the four types of dispersive phonon modes in these wurtzite quantum systems is deduced. It is found that the QC, HS and PR phonon modes always appear in GaN/AlGaN or InGaN/GaN low-dimensional quantum systems. But the PR phonon modes only exist in the low-doped x GaN/AlxGa1−xN (x<0.34) and InxGa1−xN/GaN quantum structures (x<0.35). The general features of dispersive phonon spectra in wurtzite QWs, QWRs and QDs including the frequency subranges of phonon modes, the continuous and discrete spectra, the limited frequencies and the “reducing” behaviors are discussed and concluded in detailed. The electron–phonon coupling functions are analyzed. The total tendency of the electron–phonon coupling function decreases with the increase of the free wave-numbers, the orbital and azimuthal quantum numbers. But the electron–phonon coupling functions sometimes take maximum values at some certain wave-(quantum) numbers, which is attributed to the “modulating” effect of dielectric functions ratio (ϵt/ϵz) of wurtzite nitride materials in different directions to the electron–phonon coupling functions. The electron–phonon coupling strength of the high-frequency phonon modes is stronger than that of the low-frequency ones. Moreover, the higher the orders of QC, PR and HS modes are, the weaker the electron–phonon coupling strengths become. The results and conclusions are meaningful and helpful for understanding and analyzing phonon spectra and polaronic effect in wurtzite low-dimensional quantum systems.