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2011 DIPC Highlight
Anharmonic stabilization of
the high-pressure simple cubic
phase of calcium
I. Errea, B. Rousseau, and A. Bergara
Physical Review Letters 106, 165501 (2011)
The understanding of how the atoms vibrate in a solid is crucial to comprehend many of its physical
properties. As an example we can cite thermal transport properties, specific heats, Raman and neutron
diffraction spectra and the very appealing superconductivity. These atomic vibrations are generally explained
in terms of phonons: the quantum of the atomic oscillations. Phonons are calculated in regular
basis applying the harmonic approximation, that is, considering exclusively the behavior of the ionic potential
in the vicinity of the equilibrium position. However, there are cases in which the behavior close to
the equilibrium position does not describe properly the dynamical behavior of the solid and leads to wrong
results. For example, there are experimentally confirmed structures of solids that display imaginary
phonons in ab initio calculations suggesting that the crystal structure is unstable and cannot be realized
experimentally. In order to solve this apparent contradiction the behavior of the atoms farther from equilibrium
needs to be taken into account, that is, anharmonicity cannot be neglected in these systems.
Figure 1: Schematic representation
of how anharmonicity
can stabilize the atomic vibrations
and how the harmonic
approximation can yield to a
wrong interpretation.
The high-pressure simple cubic phase of calcium is a clear example of the latter. Although phonon calculations
based on the harmonic approximation predict that this phase is unstable, many experiments confirm
that this phase is stable and exists in nature. Moreover, calcium starts to superconduct in this simple
cubic phase and becomes the element with the highest superconducting transition temperature (Tc) reaching
29 K above 200 GPa in another phase. Thus, the emergence of superconductivity in Ca cannot be understood
within the harmonic approximation since the electron-phonon interaction cannot be calculated
with imaginary phonons.
Considering that in the harmonic result the ground state energy is not bounded, the anharmonicity in simple
cubic Ca requires a variational approach and suggests the application of the self-consistent harmonic
approximation (SCHA). The SCHA assumes that the best phonons are those that minimize the free energy
of the system. In order to apply the SCHA an arbitrary harmonic term is included and subtracted to the
potential. Then, the free energy is minimized with respect to this arbitrary harmonic term. Finally, the frequencies
that diagonalize the arbitrary harmonic part are the anharmonic frequencies. In order to do so,
the anharmonic coefficients need to be known. In this work performed at the DIPC, this procedure has
been applied for the first time using rigorous ab initio calculations.
The method that has been developed is able to predict that the simple cubic phase of calcium is stabilized
by strong anharmonic effects. Moreover, the anharmonic phonon dispersion that has been obtained has
made possible the calculation of the electron-phonon coupling and predict the superconducting properties
of this system. The calculated value of Tc is in sharp agreement with recent experiments. This strongly
supports the validity of the method used.
The SCHA is a theory that might be used to understand many systems were anharmonicity plays a crucial
role such as ferroelectrics, systems with charge-density waves and hydrides. These results shows that the
SCHA is an appropriate framework to tackle all these type of systems within ab initio calculations, something
not possible so far.
Figure 2: Harmonic phonon spectra and renormalized anharmonic phonon spectra at 0 and 300 K of simple
cubic Ca at 50 GPa. The anharmonic results for the integrated electron-phonon coupling parameter, the Eliashberg
function and the phonon density of states (PDOS) are shown as well.
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