Electron-Phonon Coupling and the Metallization of Solid Helium at Terapascal Pressures

Solid He is studied in the pressure and temperature ranges 1–40 TPa and 0–10 000 K using first-principles methods. Anharmonic vibrational properties are calculated within a self-consistent field framework, including the internal and free energies, density-pressure relation, stress tensor, thermal expansion, and the electron-phonon coupling renormalization of the electronic band gap. We find that an accurate description of electron-phonon coupling requires us to use a nonperturbative approach. The metallization pressure of 32.9 TPa at 0 K is larger than found previously. The vibrational effects are large; for example, at $P=30\mathrm{TPa}$ the band gap is increased by 2.8 eV by electron-phonon coupling and a further 0.1 eV by thermal expansion compared to the static value. The implications of the calculated metallization pressure for the cooling of white dwarfs are discussed.

Figure 1

(a) Enthalpy $\mathrm{\Delta }{H}_{\mathrm{hcp}}$ with respect to the hcp phase of the closed-packed phases described at the static DFT level. (b) Anharmonic energy correction $\mathrm{\Delta }{U}_{\mathrm{anh}}$ to the harmonic energy of the hcp phase at zero temperature.

Figure 2

Electronic band structure (blue, on left-hand side of figure) and density of states (red, on right-hand side of figure) of hcp He at 10 TPa. The dashed black line is the density of states of a free electron gas of the same density as He.

Figure 3

(a) ZP electron-phonon correction to the thermal band gap of hcp solid He as a function of pressure. The inset shows the temperature dependence of the electron-phonon correction at a pressure of $P=25\mathrm{TPa}$. (b) Squared Kohn-Sham eigenstate corresponding to the CBM at 10 and 25 TPa, using a red-green-blue color scale with red corresponding to high density and blue to low density. We have removed the regions of lowest density for clarity.

Figure 4

Density-pressure-temperature phase diagram of solid hcp He. At a given external pressure, the density $\rho$ is renormalized with respect to the static-lattice density ${\rho }_{\text{static}}$ at that pressure. The inset shows the density-pressure relation at $T=0\mathrm{K}$, including the effects of ZP motion.

Figure 5

Thermal band gap $E_{\mathrm{g}}$ of solid hcp He as a function of pressure, calculated by evaluating the difference between the CBM and the VBM (even when the difference is negative). The static gap results correspond to DFT calculations (dash-dotted black line) and DMC calculations (dashed black line) taken from Ref. [6]. The band gap, including the effects of electron-phonon coupling and thermal expansion added to the DMC results, is shown at $T=0\mathrm{K}$ (red solid line).

Figure 6

Phase diagram of He at the pressure-temperature range relevant for cool WDs. The black dashed line is the insulator-metal transition predicted in Ref. [6]. The solid-plasma transition is taken from Ref. [2].