Entropy Driven Stabilization of Energetically Unstable Crystal Structures Explained from First Principles Theory

Conventional methods to calculate the thermodynamics of crystals evaluate the harmonic phonon spectra and therefore do not work in frequent and important situations where the crystal structure is unstable in the harmonic approximation, such as the body-centered cubic (bcc) crystal structure when it appears as a high-temperature phase of many metals. A method for calculating temperature dependent phonon spectra self-consistently from first principles has been developed to address this issue. The method combines concepts from Born’s interatomic self-consistent phonon approach with first principles calculations of accurate interatomic forces in a supercell. The method has been tested on the high-temperature bcc phase of Ti, Zr, and Hf, as representative examples, and is found to reproduce the observed high-temperature phonon frequencies with good accuracy.

Figure 1

The phonon dispersions of the group IV metals. The solid lines are the first principles self-consistent phonon calculations. In the left column, the finite temperature calculations, and in the right column, the $T=0\mathrm{K}$ calculations. Imaginary frequencies are dashed lines (see text). The filled circles are the experimental data of Refs. 14, 15, 16.

Figure 2

The change in free energy between two consecutive iterations, here plotted as a function of the number of iterations. The inset in the figure shows the same plots but at a smaller energy scale.