Phonon dispersion using the ratio of zero-time correlations among conjugate variables: Computing full phonon dispersion surface of graphene

We present a robust family of methods (ZTR) to compute the phonon dispersion curves based on the ratio of zero-time correlations of conjugate variables in the reciprocal space. This general technique extracts the normal mode frequency corresponding to wave vector q and polarization p, from the ratio of the correlation of the nth derivative of displacement to the n-1th derivative in reciprocal space at zero time. A particular version of this method using the ratio of velocity to displacement (n=1) is previously known but seldom employed in atomistic simulations. For n=2, the method involves velocities and accelerations — dynamical variables that are more well-defined than equilibrium displacements in atomistic simulations. We test the ZTR methods and demonstrate that both ZTR methods (n=1,2) can accurately resolve the phonon mode frequencies while offering a significant improvement to the computational speed. We also illustrate the ability of the ZTR methods to handle anharmonicity and phonon softening at high temperatures. Finally, we demonstrate the power of the ZTR approach by computing the full phonon dispersion surface for graphene across the entire Brillouin zone with 3600 wave vectors and six polarizations at finite temperatures — a challenging task for the traditional methods.