Superconductivity in boron under pressure: A full-potential linear muffin-tin orbitals study

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 (B), body-centered-tetragonal (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 isotropic Eliashberg equation. The fcc phase is found to be stable only at very high pressure $(\text{volume per atom}<21.3{\text{bohrs}}^3)$, estimated to be in excess of 360 GPa. The bct phase $(\text{volume per atom}>21.3{\text{bohrs}}^3)$ is stable at lower pressures in the range 210–360 GPa. In both bct and fcc phases the superconducting transition temperature $T_c$ is found to decrease with increasing pressure, due to the 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. Although it is possible that the method used somewhat overestimates the electron-phonon coupling, its success in studying several other systems, including ${\mathrm{MgB}}_2$, clearly suggests that the experimental work should be reinvestigated. 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 B cannot be ruled out, thus pointing to (substantiating) the need for further experimental investigations of the superconducting properties of high pressure pure phases of B.

Figure 1

Energy vs volume (per atom) curves for the fcc and bct phases of B. A constant energy of 49.0 Ry has been added to the energies per atom for convenience in plotting.

Figure 2

The FP-LMTO energy bands in fcc B for four different lattice parameters. The flat energy band between the symmetry points $X$ and $W$, which provides strong electron-phonon coupling, also renders the structure unstable at larger volumes. The zero of energy has been set at the Fermi level.

Figure 3

Calculated phonon spectra for fcc B for the lattice parameters (a) $a=4.0\mathrm{a}.\mathrm{u}.$, (b) $a=4.4\mathrm{a}.\mathrm{u}.$, and (c) $a=4.6\mathrm{a}.\mathrm{u}.$ The dots represent the calculated phonon frequencies, with solid lines providing only a guide to the eye. The softening of the transverse phonons at the symmetry point $X$ occurs as the system expands to a volume above $3.4{\mathrm{\AA }}^3∕\text{atom}$, in accordance with the earlier results of Mailhiot et al. (Ref. 1).

Figure 4

The Eliashberg spectral function (a) and phonon density of states (b) for fcc B, for three different lattice parameters.

Figure 5

FP-LMTO band structure of bct B for lattice parameter $a=4.35\mathrm{a}.\mathrm{u}.$, $c∕a=0.675$ (a), $a=4.0\mathrm{a}.\mathrm{u}.$, $c∕a=0.675$ (b), and $a=4.35\mathrm{a}.\mathrm{u}.$, $c∕a=0.65$ (c). The zero of energy has been set at the Fermi level.

Figure 6

Calculated phonon spectra for bct B for the lattice parameters (a) $a=4.25\mathrm{a}.\mathrm{u}.$ and (b) $a=4.35\mathrm{a}.\mathrm{u}.$ for the $c∕a$ ratio of 0.675. The dots represent the calculated frequencies, while the solid lines have been drawn through them to guide the eye. For lattice parameters higher than 4.35 a.u. the structure becomes unstable with the $N$-point transverse phonons becoming soft first.

Figure 7

Eliashberg spectral function (a) and phonon density of states (b) for bct B, for three different lattice parameters, and $c∕a=0.675$.

Figure 8

Variation of the superconducting transition temperature $T_c$ with volume per atom. The vertical dashed line is drawn at volume per atom equal to $21.3{\text{bohrs}}^3$, estimated to be the dividing line between the bct and the fcc phases.