Microscopic theory of phonon polaritons and long wavelength dielectric response

TL;DR

This work develops a fully first-principles, unified Hamiltonian framework that treats lattice vibrations and the electromagnetic field as dynamical degrees of freedom to describe phonon polaritons. It constructs a quadratic Hamiltonian with coupled phonon and photon sectors, diagonalizes it to obtain polariton modes, and determines all interaction parameters from density functional theory using the temperature-dependent effective potential (TDEP) approach. By including anharmonic terms, it yields a self-energy and spectral function that capture realistic linewidths and non-Lorentzian broadening, enabling predictions of hyperbolic polaritons and anisotropic effects. Demonstrations on GaP, GaN, PbTe, and β-Ga2O3 show agreement with experiment and reveal rich dispersion structures, non-analytical behavior at the zone center being resolved by the polariton picture. The framework provides a predictive, parameter-free tool for studying light-matter coupling in solids with relevance to energy transport, nanophotonic heat control, and potential polariton-enabled chemistry.

Abstract

We present a first-principles approach for calculating phonon-polariton dispersion relations. In this approach, phonon-photon interaction is described by quantization of a Hamiltonian that describes harmonic lattice vibrations coupled with the electromagnetic field inside the material. All Hamiltonian parameters are obtained from first-principles calculations, with diagonalization leading to non-interacting polariton quasiparticles. This method naturally includes retardation effects and resolves non-analytical behavior and ambiguities in phonon frequencies at the Brillouin zone center, especially in non-cubic and optically anisotropic materials. Furthermore, by incorporating higher-order terms in the Hamiltonian, we also account for quasiparticle interactions and spectral broadening. Specifically, we show how anharmonic effects in phonon polaritons lead to a dielectric response that challenges traditional models. The accuracy and consequences of the approach are demonstrated on GaP and GaN as harmonic test systems and PbTe and $β$-Ga$_2$O$_3$ as anharmonic test systems.

Figures (5)

• Figure 1: Phonon-polariton dispersion relation in GaP. Polariton frequencies obtained from eigenvalues of Eq. (\ref{['eq:barepolaritonH']}) are compared with experimental values from Henry.1965, as well as with bare phonon and photon frequencies.
• Figure 2: (a) Phonon dispersion relations for wurtzite GaN. Note the discontinuity at the zone center, $\Gamma$, for the optical modes. (b) Dispersion relations for bare photons (yellow), bare phonons (green), and phonon-polaritons (purple) in a zoomed-in region close to the zone center. No discontinuities arise in the polariton picture and the Hamiltonian is analytical throughout the Brillouin zone.
• Figure 3: Static dielectric constant for NaCl as a function of wavevector $\boldsymbol{q}$, comparing polariton, closed-circuit (constant-E, transverse) and open-circuit (constant-D, longitudinal) limits.
• Figure 4: Left panel: Electromagnetic part of the phonon polariton spectral function in $\beta$-Ga$_2$O$_3$. Panels (i)-(viii) show constant energy cuts of the spectral function, where the energies for the respective slice is marked in the left panel. The $\boldsymbol{q}$-vectors in each slice lie in the plane orthogonal to $(010)$.
• Figure 5: (a) Polariton spectral functions for PbTe at 300K. Panel (b) is only the electromagnetic part of the polaritons, i.e. the photon spectral function. Panel (c) is the anharmonic photon density of states, compared with the one obtained using the ideal Planck's law.