Abstract
We present systematic ab-initio study on the phonon mode potential as a source of anharmonicity in the crystal. As an example, the transverse optical (TO) mode potential in PbTe has been fitted to density-functional-theory calculated energies of phonons excited with different amplitudes of mode displacements. The corresponding equation of motion has been analytically and numerically solved in 1D and 2D space, respectively. The solution is used for constructing the ensemble of 10,000 systems with potential and kinetic energies selected according to the thermal equilibrium distributions. The velocity auto-correlation function derived from the computed trajectories is then used to calculate the profile of the phonon spectrum for the TO an LA modes at three temperatures of 100, 300, and 600 K. This technique allows for determination of the contribution of non-quadratic potential of the phonon mode to the anharmonicity in the crystal and its effect on the phonon spectrum.
Highlights
We present systematic ab-initio study on the phonon mode potential as a source of anharmonicity in the crystal
We have used an ensemble of 104 oscillators, each followed along 30 ps long trajectory
The final probability density of position (Fig. 6) and velocity (Fig. 7) in the whole ensemble turned out to be adequately close to the target shapes—i.e. Gaussian distribution around equilibrium for position and Rayleigh (i.e. 2D Maxwell–Boltzmann) distribution for velocity—indicating that we have a good approximation of ensemble in thermodynamic equilibrium
Summary
We present systematic ab-initio study on the phonon mode potential as a source of anharmonicity in the crystal. The residuals of the quadratic ( 2nd order) and quartic ( 4th order) polynomial fits to the DFT data (Fig. 3, top panel) clearly indicate that the simple quadratic (harmonic) function is not sufficient to properly reproduce the shape of the mode potential, while the fourth order polynomial provides a much better model of the DFT data points.
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