First-Principles Studies of the Metallization and the Equation of State of Solid Helium

The insulator-to-metal transition in hcp solid helium at high pressure is studied with first-principles simulations. Diffusion quantum Monte Carlo (DMC) calculations predict that the band gap closes at a density of $21.3\mathrm{g}/{\mathrm{cm}}^3$ and a pressure of 25.7 terapascals, which is 20% higher in density and 40% higher in pressure than predicted by density functional calculations based on the generalized gradient approximation (GGA). The metallization density derived from $GW$ calculations is found to be in very close agreement with DMC predictions. The zero point motion of the nuclei had no effect on the metallization density within the accuracy of the calculation. Finally, fit functions for the equation of state are presented, and the magnitude of the electronic correlation effects left out of the GGA approximation are discussed.

Figure 1

Comparison of band gaps as a function of density. DMC results for a small $2\times 1\times 1$ rectangular supercell lie parallel and above the GGA gaps by 1.4 eV. DMC ($4\times 2\times 2$ rectangular cell) and $GW$ metallization density of $21.3(1)\mathrm{g}/{\mathrm{cm}}^3$ are in agreement. GGA underestimates the band gap by about 4 eV. The inset shows a part of the electronic band structure including the indirect gap. The filled and open circles indicate, respectively, the highest occupied valence state at the $M$ point and the lowest unoccupied conduction state at $\Gamma$. Two densities are shown. The insulating state (black curve) lies below the metallic state (red dashed curve). The DMC error bars are smaller than the DMC symbols.

Figure 2

Panel (a): Difference between DMC and GGA ground state energies. We calculate the DMC energy in two ways, by extrapolations to infinite cell size (circles) and by directly using results from our largest $3\times 3\times 3$ triangular supercell (diamonds). In panel (b), we relate the DFT-DMC energy difference to the amount of mechanical work needed to reach a certain density. We plot the correlation fraction, $(E_{\mathrm{DFT}}-E_{\mathrm{DMC}})/(E_{\mathrm{DFT}}-E_{\mathrm{atom}})$, where $E_{\mathrm{atom}}=-79.0048\mathrm{eV}$ is the exact energy of the isolated helium atom.

Figure 3

Pressure-volume relationship derived from GGA calculations for solid helium. The experimental Vinet fit [26], valid up to 57 GPa, agrees with the low pressure Vinet fit of GGA data. The high pressure (HP) fit to the GGA data approaches the free homogeneous electron gas (HEG). To emphasize differences, $P{V}^3$ was plotted in the main graph.