  Based on the dielectric continuum model and transfer-matrix method, the
completing polar optical phonon states in a wurtzite GaN-based superlattice
(SLs) with arbitrary-layer complex bases are investigated. It is proved that
$2^n$ types of phonon modes probably exist in a wurtzite nitride SL with
$n$-layer complex bases. The analytical phonon states of these modes and their
dispersive equations in the wurtzite GaN/Al$_{x}$Ga$_{1-x}$N SL structures are
obtained. Numerical calculations on a three-layer
GaN/Al$_{0.15}$Ga$_{0.85}$N/AlN complex bases SL are performed. Results reveal
that there are interface optical (IO) phonon modes of one type only and four
types of quasi-confined (QC) phonon modes in three-layer
GaN/Al$_{0.15}$Ga$_{0.85}$N/AlN complex bases SLs. The dispersive spectra of
phonon modes in complex bases SLs extend to be a series of frequency bands. The
behaviors of QC modes reducing to IO modes are observed. The present
theoretical scheme and numerical results are quite useful for analyzing the
dispersive spectra of completing phonon modes and their polaronic effect in
wurtzite GaN-based SLs with complex bases.
