Pressure dependence of superconductivity in simple cubic phosphorus

The electronic structure and lattice dynamics for simple cubic (sc) P are calculated over the pressure range 0–70 GPa from first principles using the local-density approximation. The $R$ phonon mode is found to be unstable below 20 GPa in the harmonic approximation, but may be stable down to a pressure less than 20 GPa when anharmonicity is considered. The electron-phonon coupling is calculated for pressures above 20 GPa, and the superconducting transition temperature $T_c$ is found to decrease with increasing pressure throughout this pressure range. The result is in agreement with experimental results above 30 GPa. In contrast to experiment, no evidence for a decrease in $T_c$ with decreasing pressure below 30 GPa is found. The structural transition from rhombohedral $A7$ to sc is also investigated. An interesting two-step transition is found to occur theoretically which may have relevance for the pressure dependence of $T_c$. Possible explanations for the discrepancy with experiment are discussed.

Figure 1

Band structure of the simple cubic phase of P for a number of pressures. The Fermi level is located at zero energy.

Figure 2

Phonon dispersion for P in the simple cubic phase for a number of pressures. The dynamical instability at the $R$ point at lower pressures corresponding to the distortion towards the rhombohedral $A7$ structure can be seen.

Figure 3

Phonon frequencies for $R$ phonon mode as a function of pressure. The anharmonic frequencies are estimated for polarizations along the [111] and [100] directions.

Figure 4

Phonon density of states $F\left(\omega \right)$ (top), Eliashberg spectral function ${\alpha }^{2}F\left(\omega \right)$ (bottom, solid), and integrated $\lambda$ (bottom, dashed) for sc P for a number of pressures.

Figure 5

Electron-phonon coupling constant ${\lambda }_{\mathbf{q}}$ (top) and nesting function ${\xi }_{\mathbf{q}}$ (bottom) along high-symmetry lines in the Brillouin zone for sc P at various pressures. Values for ${\lambda }_{\mathbf{q}}$ at $R$ are 57, 6.0, and 2.8 for 20, 40, and 70 GPa, respectively.

Figure 6

Trends in average phonon frequency $\langle {\omega }^2\rangle$, $N\left({\epsilon }_F\right)$, average $e$-$p$ matrix element $\langle g^2\rangle$, and $e$-$p$ coupling $\lambda$ as a function of pressure. The subscript “20” denotes the value at 20 GPa.

Figure 7

Calculated $T_c$ as a function of pressure compared to the experimental results of Karuzawa et al. (Ref. 18). For the calculations, pressures are shifted to match experiment, according to Table .

Figure 8

Lattice parameters (a) $a_{\mathrm{rhom}}$, (b) $\alpha$, and (c) $u$, and (d) nearest-neighbor $d_1$ and next-nearest neighbor $d_2$ distances for variable-cell relaxation calculations of P in the $A7$ structure with target pressures between 0 and 25 GPa.

Figure 9

Simulated x-ray-diffraction spectra for relaxed $A7$ and sc structures at various pressures. The 10.4 and 11.0 ($hkl$) diffraction lines discussed in the text are highlighted for clarity. These lines appear to merge before the transition to the sc phase (with $u=0.25$) is completed.