A Variational Approach of Anharmonicity in Lattice Dynamics

In this work, a variational approach by optimizing the free energy of an anharmonic Hamiltonian with respect to strain tensor, inter-cell atomic displacements and force constants in a lattice system based on first-principle self-consistent phonon theory has been established. The goal is to predict possible phase transitions in crystal structures with the ability to describe phonon spectra at any given finite temperatures. A Fortran code package has been developed by implementing this computational scheme which will be explained. The reliability of the approach is then inspected and validated in toy models and real materials such as Si, PbTe and Bi. This method takes account of anharmonic effects caused by phonon-phonon interactions with the usage of a parametrization of the potential energy as a truncated Taylor expansion. The advantage is that the phase transition can be predicted with a minimal anharmonic model, and computation loads are much lighter compared with other canonical sampling techniques that employs molecular-dynamics simulation. This method also includes time-independent parameters: strain and inter-cell atom displacements to monitor the unit cell shape and volume change at different finite temperatures. Contingent upon this fact, traditionally dropped cubic terms play an important role and are explicitly presented in this formulation. The computational scheme and code package are also constructed in a manner that opens the possibility to incorporate couplings to other order parameters such as orbital-ordering or magnetization.