  Hyperbolic phonon polariton is important in precisely controlling photons at
the nanoscale. It was common practice to calculate the dielectric function of
the phonon polariton system with the Drude-Lorenz model. We considered the
impact of LO-TO splitting while applying the Drude-Lorenz model. Then the
dielectric functions become wave vector direction dependent besides electric
polarization direction dependent. Our results show that considering LO-TO
splitting can more accurately predict dielectric functions. Additionally, we
discovered that, besides hexagonal BN, hexagonal AlN exhibits a wide hyperbolic
frequency band range, while the other four materials display it scarcely.
Furthermore, we found that the phonon frequency, lifetime, and the difference
of infrared active transverse optical phonon frequencies with different wave
vector directions are critical factors in determining the width of the
hyperbolic frequency band range. We also found some dumbbell-shaped and
butterfly-shaped isofrequency curves in h-AlN, h-GaP, and especially h-GaN. Our
study provides a fresh perspective on understanding the dielectric properties
of these materials and lays a theoretical foundation for further exploration
and development of new hyperbolic phonon polariton materials.
