Structural phase stability and electron-phonon coupling in lithium Amy Y. Liu Department of Physics, Georgetown University, Washington, DC 20057-0995 Andrew A. Quong Lawrence Livermore National Laboratory, Livermore, California 94550 J. K. Freericks Department of Physics, Georgetown University, Washington, DC 20057-0995 E. J. Nicol Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1 Emily C. Jones Department of Physics, Georgetown University, Washington, DC 20057-0995 Received 14 September 1998 First-principles calculations of the free energy of several structural phases of Li are presented. The density- functional linear-response approach is used to calculate the volume-dependent phonon frequencies needed for computing the vibrational free energy within the quasiharmonic approximation. We show that the transforma- tion from a close-packed structure at low temperatures to the bcc phase upon heating is driven by the large vibrational entropy associated with low-energy phonon modes in bcc Li. In addition, we find that the strength of the electron-phonon interaction in Li is strongly dependent on crystal structure. The coupling strength is significantly reduced in the low-temperature close-packed phases as compared to the bcc phase, and is con- sistent with the observed lack of a superconducting transition in Li. S0163-18299911905-2 I. INTRODUCTION The pressure-temperature structural phase diagram of solid Li, arguably the simplest metal, is not well established. Even along the zero-pressure axis, Li exhibits some complex behavior. At ambient pressure and room temperature, Li crystallizes in the bcc structure, but upon cooling, it under- goes a martensitic transformation around 80 K. The transfor- mation was first observed in 1956, 1 but the crystal structure of the low-temperature phase remained a subject of debate for several decades. In 1984, Overhauser suggested that the neutron scattering data were consistent with the 9R structure, 2 a close-packed phase with a nine-layer stacking sequence. Subsequent investigations confirmed 9R as the pri- mary structure at low temperatures, 3 but also showed that it is accompanied by a large and variable amount of a disor- dered polytype consisting of short-ranged fcc- and hcp-type stacking order, as well as some amount of bcc. 4 Furthermore, upon heating, the 9R phase and disordered polytype appear to transform first to an ordered fcc phase before reverting to bcc Li above about 150 K. 4 Little information is available about the phase diagram of Li at higher pressures. Upon compression, the temperature at which the martensitic transformation begins, M s , rises at a rate of about 20 K/GPa, at least up to pressures of 3 GPa, 5,6 with the structural characteristics of the low-temperature phase remaining the same as at ambient pressure. At room temperature, a structural transition is observed at P =6.9 GPa. 7 The crystal structure of the compressed phase was initially identified as fcc, but the limited diffraction data are also consistent with hcp or 9R. Neutron experiments 3,8,9 show that while there may be some softening of certain phonon modes in bcc Li upon cooling, the transformation occurs well before any phonon frequencies or elastic constants approach zero, ruling out the soft-mode mechanism 10–12 for the transformation. The trans- formation more likely results from a competition between the internal energies and entropies of the different phases. For example, Friedel 13 argued on general grounds that the phonon spectrum should scale roughly with coordination number, leading to an overall lowering of the bcc phonon spectrum compared to that of close-packed structures. The resulting larger vibrational entropy associated with the bcc structure provides a qualitative explanation for the preva- lence of the bcc structure as the high-temperature phase in metals. While these ideas are believed to apply to many sys- tems that exhibit martensitic transformations, it is useful to examine them in the context of detailed materials-specific phonon spectra which are accessible from first-principles cal- culations. Several first-principles theoretical investigations have fo- cused on the relative stability of different crystal structures for Li at zero pressure. 14,15 Conflicting results have emerged, but all the studies agree that the energy differences between structures are small much less than 1 mRy/atom, making it difficult to determine the ordering reliably. Most of the the- oretical work has been based on zero-temperature calcula- tions that do not even include zero-point energies. Given that Li is such a light atom, finite-temperature effects arising from vibrational degrees of freedom are likely to play a role PHYSICAL REVIEW B 1 FEBRUARY 1999-II VOLUME 59, NUMBER 6 PRB 59 0163-1829/99/596/40288/$15.00 4028 ©1999 The American Physical Society