# Anisotropic hybrid excitation modes in monolayer and double-layer phosphorene on polar substrates

## TL;DR

This study explores the anisotropic hybrid plasmon-phonon modes in monolayer and double-layer phosphorene on various polar substrates, revealing substrate effects and tunability through layer misalignment and separation.

## Contribution

It provides analytical and numerical insights into hybrid mode dispersion relations in phosphorene systems, highlighting substrate influence and tunability mechanisms.

## Key findings

- Strong anisotropy in hybrid modes along light-mass directions
- Substrate type significantly affects dispersion relations
- Layer misalignment and separation enable mode engineering

## Abstract

We investigate the anisotropic hybrid plasmon-SO phonon dispersion relations in monolayer and double-layer phosphorene systems located on the polar substrates, such as SiO2, h-BN and Al2O3. We calculate these hybrid modes with using the dynamical dielectric function in the RPA by considering the electron-electron interaction and long-range electric field generated by the substrate SO phonons via Frohlich interaction. In the long-wavelength limit, we obtain some analytical expressions for the hybrid plasmon-SO phonon dispersion relations which represent the behavior of these modes akin to the modes obtaining from the loss function. Our results indicate a strong anisotropy in plasmon-SO phonon modes, whereas they are stronger along the light-mass direction in our heterostructures. Furthermore, we find that the type of substrate has a significant effect on the dispersion relations of the coupled modes. Also, by tuning the misalignment and separation between layers in double-layer phosphorene on polar substrates, we can engineer the hybrid modes.