Active tunability of phonon dispersion and spontaneous emission (SE) are still open problems due to their exciting potential applications. In view of the fact that polaritons are very sensitive to the dielectric environment, in this study, with the help of the differences in optical property between the phase change material vanadium dioxide (VO₂) during the phase transition from the insulating state to metallic state and the tunable surface plasmon polaritons (SPPs) in graphene, a heterostructure composed of hyperbolic material hexagonal boron nitride (hBN) and graphene and VO₂ is proposed to investigate the active tunability of hBN phonon polaritons (PhPs). In order to illustrate the underlying physical mechanism of the above heterostructures, the dispersion distributions of the above heterostructures are calculated and represented by the imaginary part of the p-polarized Fresnel reflection coefficient of the heterostructure, meanwhile the dispersion relation of the hBN/VO₂ heterostructure in hyperbolic region is verified by the quasi-static approximation method.

Results indicate that the active tunability of hBN PhPs inside type-I and type-II hyperbolic bands can be achieved by controlling VO₂ phase transition in hBN/VO₂ heterostructure. The PhP dispersion change of the hBN/VO₂ heterostructure is mainly caused by the change of the VO₂ dielectric function when VO₂ substrate changes from the insulating state into metallic state, which affects the total Fresnel reflection coefficient of the heterostructure, finally resulting in the PhP dispersion change of hBN/VO₂ heterostructure. When graphene is introduced into the hBN/VO₂ heterostructure, coupled hyperbolic plasmon-phonon polaritons (HPPPs) are obtained within type-I and type-II hyperbolic band of hBN, while the surface plasmon-phonon polaritons (SPPPs) are generated outside its hyperbolic bands. Moreover, comparative analysis of SE rates is presented for a quantum emitter positioned with the hBN/VO₂ and graphene/hBN/VO₂ heterostructure, revealing that the SE rates of these heterostructures can be modulated by the passive means including changing the hBN thickness and distance between the dipole emitter and the proposed heterostructure, and also by the active means including tuning VO₂ phase states and graphene chemical potential without changing structural configurations. This study provides a theoretical guidance in realizing active tunability of both phonon dispersion and SE rate of the natural or artificial anisotropic optical materials by using functional materials including phase change materials and graphene.