Electron-phonon coupling and superconductivity in arsenic under pressure

We perform first-principles calculations of the electronic structure, phonon dispersion, and electron-phonon coupling in elemental As at various pressures above and below the rhombohedral $A7$ to simple cubic ($sc$) structural transition. We find that the electron-phonon coupling constant $\lambda$, and hence the superconducting transition temperature $T_c$, is largest near the structural transition and decreases away from it. Changes in $\lambda$ as a function of pressure are primarily explained by changes in the density of states at the Fermi level for pressures below the transition, and by changes in phonon frequency for pressures above the transition. Although the couplings to the ${\Gamma }_1$ optical phonon mode (for $A7$) and the $R$ phonon mode (for $sc$) are large, the contribution of these modes to $\lambda$ for their respective structures is modest.

Figure 1

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 As in the $A7$ structure with target pressures between 0 and 50 GPa. The transition pressure from $A7$ to $sc$ is found to be between 20 and 25 GPa.

Figure 2

Electronic band structure for $sc$ As at 30, 40, and 50 GPa. Energies are relative to ${\epsilon }_F$.

Figure 3

Electronic density of states $N\left(\epsilon \right)$ for $sc$ As at 30, 40, and 50 GPa. Energies are relative to ${\epsilon }_F$.

Figure 4

Phonon dispersion for $sc$ As at 30, 40, and 50 GPa.

Figure 5

Electron-phonon coupling parameter ${\lambda }_{\mathbf{q}}$ for $sc$ As at 30, 40, and 50 GPa. The values of the peak at $R$ are 35, 9.8, and 6.1 for 30, 40, and 50 GPa, respectively.

Figure 6

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 $sc$ As at 30, 40, and 50 GPa.

Figure 7

Electronic band structure for $A7$ As at 0 GPa. Energies are relative to ${\epsilon }_F$.

Figure 8

Electronic density of states $N\left(\epsilon \right)$ for $A7$ As at 0, 10, and 20 GPa. Energies are relative to ${\epsilon }_F$.

Figure 9

Phonon dispersion ${\omega }_{\mathbf{q}\nu }$ for $A7$ As at 0 GPa (left), 10 GPa (middle), and 20 GPa (right).

Figure 10

Total, acoustic, and optical contributions to the 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 $A7$ As at 0 GPa (left), 10 GPa (middle), and 20 GPa (right).

Figure 11

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 0 denotes the value at 0 GPa.

Figure 12

Superconducting transition temperature $T_c$ as a function of pressure. Blue circles denote experimental results, with error bars, from Ref. 14. For the calculated values, the pressure is determined from the experimental Murnaghan equation of state (EOS) parameters, using the calculated volume as input. For Calculations 1A (green squares) and 1B (red diamonds), ${\mu }^{*}=0.128$ at 41 GPa, and the EOS parameters are from Ref. 5. For Calculation 2A (cyan up-triangles) and 2B (magenta down-triangles), ${\mu }^{*}=0.117$ at 35 GPa, and the EOS parameters are from Ref. 6. For Calculations 1A and 2A, ${\mu }^{*}$ is constant for all pressures, while for Calculations 1B and 2B, ${\mu }^{*}$ varies with pressure according to Eq. (11) in the text. The horizontal line at 1.7 $K$ denotes the lower limit of accessible temperatures in the experiment.