Abstract
The efficient and accurate calculation of how ionic quantum and thermal fluctuations impact the free energy of a crystal, its atomic structure, and phonon spectrum is one of the main challenges of solid state physics, especially when strong anharmonicy invalidates any perturbative approach. To tackle this problem, we present the implementation on a modular Python code of the stochastic self-consistent harmonic approximation (SSCHA) method. This technique rigorously describes the full thermodynamics of crystals accounting for nuclear quantum and thermal anharmonic fluctuations. The approach requires the evaluation of the Born–Oppenheimer energy, as well as its derivatives with respect to ionic positions (forces) and cell parameters (stress tensor) in supercells, which can be provided, for instance, by first principles density-functional-theory codes. The method performs crystal geometry relaxation on the quantum free energy landscape, optimizing the free energy with respect to all degrees of freedom of the crystal structure. It can be used to determine the phase diagram of any crystal at finite temperature. It enables the calculation of phase boundaries for both first-order and second-order phase transitions from the Hessian of the free energy. Finally, the code can also compute the anharmonic phonon spectra, including the phonon linewidths, as well as phonon spectral functions. We review the theoretical framework of the SSCHA and its dynamical extension, making particular emphasis on the physical inter pretation of the variables present in the theory that can enlighten the comparison with any other anharmonic theory. A modular and flexible Python environment is used for the implementation, which allows for a clean interaction with other packages. We briefly present a toy-model calculation to illustrate the potential of the code. Several applications of the method in superconducting hydrides, charge-density-wave materials, and thermoelectric compounds are also reviewed.
Highlights
Ions fluctuate at any temperature in matter, at zero Kelvin due to the quantum zero-point motion
Other methods are based on variational principles [16, 18, 22,23,24, 31], which are mainly inspired on the self-consistent harmonic approximation [32] or vibrational self-consistent field [33] theories, and yield free energies and/or phonon frequencies corrected by anharmonicity non-perturbatively
As a showcase of the stochastic self-consistent harmonic approximation (SSCHA), we provide a simple example in a thermoelectric material in section 6 (SnTe), where we fully characterize the thermodynamics of the phase transition between the highsymmetry and low-symmetry phases
Summary
Ions fluctuate at any temperature in matter, at zero Kelvin due to the quantum zero-point motion. Other methods are based on variational principles [16, 18, 22,23,24, 31], which are mainly inspired on the self-consistent harmonic approximation [32] or vibrational self-consistent field [33] theories, and yield free energies and/or phonon frequencies corrected by anharmonicity non-perturbatively. The SSCHA is defined from a rigorous variational method that directly yields the anharmonic free energy It can optimize completely the crystal structure, including both internal and lattice degrees of freedom, accounting for the quantum nature of the ions at any target pressure or temperature.
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