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Physical Review Letters, 104, p. 209602-1, 2010-05-27
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Yao et al. Reply: Comment On “Structural prediction and phase
transformation mechanisms in calcium at high pressure”
Comment On ‘‘Structural Prediction and Phase Transformation Mechanisms in Calcium at High Pressure’’
In an attempt to partially resolve the discrepancy be-tween experiment [1] and theory [2,3] regarding the stabil-ity of the simple cubic (sc) phase of Ca under pressure, Yao et al.[4] have performed metadynamics simulations using a two-electron projector augmented wave (PAW) pseudo-potential (pp). They concluded from their enthalpy analy-sis that the sc structure is thermodynamically stable around 40 GPa and 300 K due to anharmonic free energy contri-butions. Here we show that this conclusion is an artifact of the two-electron pp used in [4]. When semicore states are included as valence electrons, the result is reversed and the sc structure is not preferred at 300 K.
First, we point out that it is not sufficient to use a two-electron pp for Ca in order to explore the asso-ciated anharmonic free energy surface at high-pressure. Fig. 1(a) shows the phonon dispersions of sc Ca at 45 GPa obtained with a ten-electron norm-conserving pp [3]. The frequencies at high-symmetry points are also computed with ten- and two-electron PAW pp’s by diago-nalizing the dynamical matrix of an eight-atom supercell. The two-electron pp results do not yield imaginary fre-quencies at the X and M points, and otherwise differ from the ten-electron pp results overall.
We have investigated the stability of the sc phase at finite-temperature by comparing its Gibbs free energy with that of the previously reported Pnma structure [3] and the I41=amd structure [4], which we also found using a random search method as in [5]. To compute free energies, we performed molecular dynamics (MD) simulations with theVASPcode [6,7] in a canonical ensemble on supercells consisting 32 and 64 atoms. Larger cells with 108 atoms for I41=amd and 125 atoms for sc are also considered to verify the results. The calculations were performed with two- and ten-electron PAW pp’s, 11 Ha plane-wave cutoff, and the PW91-GGA for the exchange-correlation func-tional. The Brillouin zone was sampled at the -point for the 108- and 125-atom cells, and 22 4 and 23 Monkhorst-Pack grids were used for the 32- and 64-atom cells, re-spectively. The equations of motion were integrated with time steps of 1 fs and the ionic temperature was controlled with a Nose´-Hoover thermostat. Initially, the structures were allowed to equilibrate for 2 ps at 300 K. The simu-lations were then carried out for 3 to 5 ps to gather statistical information.
Using the same simulation parameters as in [4], the sc structure appears to have the lowest enthalpy at 300 K, as shown in Fig.1(b). However, when we change the pp to include semicore states, I41=amd becomes more stable. Since entropic effects are essential for describing temperature-driven transitions, we have also computed the Gibbs free energies of the candidate structures. The phonon entropy contribution is calculated from the
vibra-tional density of states, which is the Fourier transform of the velocity autocorrelation function. As shown in Fig.1(c), a two-electron pp leads to sc having the lowest Gibbs free energy, while I41=amd is preferred with the ten-electron pp. We obtained the same result with the larger simulation cells as well. Comparing Figs.1(b)and1(c), it is interesting to note that anharmonic effects appear to play a noticeable effect, but only with the ten-electron pp.
Our conclusion, therefore, is that the apparent stability of sc Ca has not been established theoretically as yet and the discrepancy with experiment continues to remain un-resolved. Further studies, which require the use of more accurate methods and measurements are necessary. Fluctuations and distortions at finite-temperature among several quasidegenerate structures, as proposed in [8], also need to be considered.
A. M. Teweldeberhan1and S. A. Bonev1,2
1
Lawrence Livermore National Laboratory Livermore, California 94550, USA
2Department of Physics
Dalhousie University
Halifax, NS, B3H 3J5, Canada
Received 15 December 2009; published 21 May 2010 DOI:10.1103/PhysRevLett.104.209601
PACS numbers: 61.66.Bi, 61.50.Ks, 63.20.D , 74.70. b
[1] T. Yabuuchi et al.,J. Phys. Soc. Jpn. 74, 2391 (2005).
[2] G. Gao et al.,Solid State Commun. 146, 181 (2008).
[3] A. M. Teweldeberhan and S. A. Bonev,Phys. Rev. B 78,
140101(R) (2008).
[4] Y. Yao et al.,Phys. Rev. Lett. 103, 055503 (2009).
[5] Chris J. Pickard and R. J. Needs, Phys. Rev. Lett. 97,
045504 (2006).
[6] G. Kresse and J. Hafner,Phys. Rev. B 47, 558 (1993).
[7] G. Kresse and J. Furthmu¨ller, Phys. Rev. B 54, 11 169
(1996).
[8] Z. P. Yin, F. Gygi, and W. E. Pickett, Phys. Rev. B 80,
184515 (2009). 2i 0 2 4 6 8 10 Frequency (THz) Γ X M Γ (a) 46 48 50 52 -0.04 -0.02 0 0.02 0.04 H-H sc (eV/atom) I41/amd I41/amd Pnma Pnma 46 48 50 52 -0.04 -0.02 0 0.02 0.04 G-G sc (eV/atom) (c) Pressure (GPa) (b)
FIG. 1 (color online). (a) Phonon dispersion curves from linear
response theory (ten-electron OPIUM pp) and frequencies at X and M calculated with a two-electron PAW pp (diamonds) and a ten-electron PAW pp (circles). (b) Enthalpies and (c) Gibbs free energies relative to sc from 32-atom cell MD at 300 K. Dashed and solid lines indicate relative energies with respect to sc obtained with two- and ten-electron pps, respectively.
PRL 104, 209601 (2010) P H Y S I C A L R E V I E W L E T T E R S 21 MAY 2010week ending
