Propagating optical-phonon modes and their electron-phonon interactions in wurtzite $\mathrm{GaN}∕{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ quantum wells

The equation of motion for the $p$-polarization field in an arbitrary wurtzite multilayer heterostructure is solved for the propagating optical-phonon (POP) modes in the framework of the dielectric-continuum model and Loudon’s uniaxial crystal model. The polarization eigenvector, the dispersion relation, and the electron-propagating-phonon (EPP) interaction Fröhlich-like Hamiltonian are derived. The analytical formulas can be directly applied to single heterojunctions, single and multiple quantum wells (QW’s), and superlattices. The dispersion relations of the POP modes and the EPP coupling functions are investigated for a given $\mathrm{GaN}∕{\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ single QW with full account of the strains of QW structures and the anisotropy effects of wurtzite crystals. We find that there are infinite POP branches, which can be denoted by a quantum number $n\left(n=1,2,\dots \right)$, with definite symmetry with respect to the center of symmetry of the QW structure. The dispersion of the POP modes with smaller $n$ is more obvious than for larger $n$. Moreover, the modes with smaller $n$ are much more important for the EPP interactions than the modes with larger $n$. In most cases, it is enough to consider the modes with $n=1,2,\dots ,10$ for the EPP interactions in a single QW. The long-wavelength POP modes are much more important for the EPP interactions. Furthermore, the strain effects of the QW structures have a strong influence on the dispersion of the POP modes. The strength of the EPP interactions is markedly increased due to the strains of the QW structures.

Figure 1

Dispersion curves of the POP modes for a symmetric ${\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}∕\mathrm{GaN}∕{\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ single QW with the $\mathrm{GaN}$ well layer thickness of $d=4\mathrm{nm}$. The numbers next to the curves represent the quantum number $n$. Here the strains of the QW structure are included.

Figure 2

Spatial dependence of the coupling function $\Gamma (q_{\perp },z)$ divided by ${(\hslash e^2∕8A{\epsilon }_0)}^{1∕2}$ for the interactions between an electron and POP modes with the quantum number $n=1,2,3,4$ and the wave number $q_{\perp }=0.3{\mathrm{nm}}^{-1}$ for the ${\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}∕\mathrm{GaN}∕{\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ symmetric single QW with the $\mathrm{GaN}$ well layer thickness of $d=4\mathrm{nm}$ and the ${\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ barrier layer thickness of $198\mathrm{nm}$. The heterointerfaces are located at $z=-2$ and $2\mathrm{nm}$, respectively. Here the solid (dashed) lines are for the real (imaginary) part of $\Gamma (q_{\perp },z)$. The strains of the QW are considered.

Figure 3

Absolute values $\mid \Gamma (q_{\perp },z)\mid$ divided by ${(\hslash e^2∕8A{\epsilon }_0)}^{1∕2}$ as a function of $z$ for the same QW structure as in Fig. 2 with different phonon wave number $q_{\perp }$ for the POP modes with the quantum number $n=1,2,3,4$. The electron interaction with the $n=1$ POP mode becomes less pronounced if $q_{\perp }⩾0.6{\mathrm{nm}}^{-1}$ and has been ignored in (b) and (c). The strains of the QW are included.

Figure 4

Dispersion curves of the POP modes for a symmetric ${\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}∕\mathrm{GaN}∕{\mathrm{Al}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ single QW with the $\mathrm{GaN}$ well layer thickness of $d=4\mathrm{nm}$. The numbers next to the curves represent the quantum number $n$. The strains of the QW are ignored.