Theory of long-wavelength optical lattice vibrations in multinary mixed crystals: Application to group-III nitride alloys

The optical lattice vibrations in multinary mixed crystals are investigated in the long-wavelength limit based on the pseudo-unit-cell approach and the Born-Huang procedure. A general equation for calculating the optical phonon frequencies in multinary mixed crystals is presented. The analytical expressions of phonon oscillator strength and dielectric constants of the multinary mixed crystal are obtained as functions of the phonon frequencies. The composition dependence of the oscillator strength shows clearly the phonon-mode behavior of the mixed crystals. As an application of our theory, a group of III-nitride mixed crystals is taken into numerical calculation. The concentration dependences of optical-phonon energy, lattice oscillator strength, and dielectric constants of the crystals are calculated in the whole compound range. It is found that the quaternary AlyInₓGa_1-x-yN mixed crystal shows a two-mode behavior.