  Using the full potential linear muffin-tin orbitals (FP-LMTO) method we
examine the pressure-dependence of superconductivity in the two metallic phases
of Boron: bct and fcc. Linear response calculations are carried out to examine
the phonon frequencies and electron-phonon coupling for various lattice
parameters, and superconducting transition temperatures are obtained from the
Eliashberg equation. In both bct and fcc phases the superconducting transition
temperature T$_c$ is found to decrease with increasing pressure, due to
stiffening of phonons with an accompanying decrease in electron-phonon
coupling. This is in contrast to a recent report, where T$_c$ is found to
increase with pressure. Even more drastic is the difference between the
measured T$_c$, in the range 4-11 K, and the calculated values for both bct and
fcc phases, in the range 60-100 K. The calculation reveals that the transition
from the fcc to bct phase, as a result of increasing volume or decreasing
pressure, is caused by the softening of the X-point transverse phonons. This
phonon softening also causes large electron-phonon coupling for high volumes in
the fcc phase, resulting in coupling constants in excess of 2.5 and T$_c$
nearing 100 K. We discuss possible causes as to why the experiment might have
revealed T$_c$'s much lower than what is suggested by the present study. The
main assertion of this paper is that the possibility of high T$_c$, in excess
of 50 K, in high pressure pure metallic phases of boron cannot be ruled out,
thus substantiating the need for further experimental investigations of the
superconducting properties of high pressure pure phases of boron.
