First-Principles Study of Phonon Anharmonicity in Atomically-Thin Black Phosphorus

Black phosphorus (BP) is of interest due to its semiconducting band gap which remains direct independent of the number of layers, as well as its high carrier mobility. It was shown experimentally that the Raman active modes A_g¹, B_2g, and A_g² have frequencies which downshift with increasing temperature. In this work we study this phenomenon for the single-layer using first-principles density functional theory (DFT) calculations. To model the temperature-induced shift of the phonon frequencies, we carry out ab initio molecular dynamics simulations with varied temperature. The normal mode frequencies are located at the peak positions in the power spectrum, which is calculated as the magnitude of the Fourier transform of the total velocity autocorrelation. Anharmonicity induces a frequency shift for each mode individually as well as from phonon-phonon coupling, with the latter interpreted classically as an effective damping of a given mode due to its interaction with all other modes. The effect of thermal expansion is also included by imposing the temperature dependent lattice constant in each simulation as calculated within the quasiharmonic approximation. In general we obtain frequency downshifts with increasing temperature in agreement with experiment.