Conditions for the Emergence of Spontaneous Phonon Frequency Combs from Anharmonic Potentials

Abstract

Phononic frequency combs have been demonstrated experimentally and theoretically in the kHz-MHz regime under nonlinear driving; however, their spontaneous formation in the GHz-THz regime remains rare. We investigate the spontaneous formation of frequency combs in van der Waals solid CrGeTe$_3$, where combs of spacing $\sim$2 cm$^{-1}$ were experimentally proposed to occur in a flat phonon mode but are now reported to originate from isotopic distribution. Our spectral energy density calculations using machine-learning-augmented molecular dynamics simulations show comb-like features with reduced spacings of $\sim$0.1 and 0.6 cm$^{-1}$ in the same mode at 50 and 200 K. However, the spacings of the comb-like features are comparable to the phonon linewidth, making precise identification of the frequency comb challenging. From a comprehensive analysis of different modes of CrGeTe$_3$, we identified two conditions to distinctively observe the combs beyond the experimental resolution: (i) an intramolecular or isolated mode with reasonably large third- or higher-order anharmonicity from phonon self-interaction and (ii) limited phonon scattering channels. Our analysis of other van der Waals solids highlighted that these conditions are met for the nearly flat intramolecular $E_{2g}$ and $A_{2u}$ phonon modes of WSe$_2$. The detailed calculations showed the formation of a spontaneous frequency comb with spacings of 1.7 and 0.67 cm$^{-1}$ from coherent superposition of four and five-phonon eigenstates in the $E_{2g}$ and $A_{2u}$ phonon modes, respectively. Available Raman scattering data of the $E_{2g}$ mode in the literature provide preliminary, if not conclusive, evidence of comb formation. Our findings offer a deeper understanding and open avenues for engineering long-lived phononic frequency combs for various practical applications.