Energy Gap Closure of Crystalline Molecular Hydrogen with Pressure

We study the gap closure with pressure of crystalline molecular hydrogen. The gaps are obtained from grand-canonical quantum Monte Carlo methods properly extended to quantum and thermal crystals, simulated by coupled electron ion Monte Carlo methods. Nuclear zero point effects cause a large reduction in the gap ($\sim 2\mathrm{eV}$). Depending on the structure, the fundamental indirect gap closes between 380 and 530 GPa for ideal crystals and 330–380 GPa for quantum crystals. Beyond this pressure the system enters into a bad metal phase where the density of states at the Fermi level increases with pressure up to $\sim 450–500\mathrm{GPa}$ when the direct gap closes. Our work partially supports the interpretation of recent experiments in high pressure hydrogen.

Figure 1

Fundamental energy gap for ideal crystals. This work (closed circles): $C2/c\text{−}24$ (blue), $Cmca\text{−}12$ (orange), and $Pc48$ (green), GW results for $C2/c\text{−}24$ (open blue circles [44]). These structures were optimized with the vdW-DF functional. QMC for $C2/c\text{−}24$ optimized with the BLYP from Ref. [51] (closed blues triangles). GW results from Refs. [53, 54, 55] for $C2/c\text{−}24$ (blue) and $Cmca\text{−}12$ (orange) optimized with the PBE functional. Note that pressures from RQMC are 10–15 GPa lower than the nominal optimization pressure.

Figure 2

The fundamental gap of quantum crystals at finite temperature. Closed circles indicate results from this work, for the three structures at various temperature as detailed in the legend. PIMD-DFT results at $T=200\text{K}$ are obtained with two different XC approximations, namely, the HSE (downward open triangles) and the vdW-DF2 (upward open triangles) and the semiclassical averaging are reported for comparison [56].

Figure 3

Direct (closed symbols) and indirect (open symbols) gaps of quantum crystals. GCTABC-RQMC at $T=200\mathrm{K}$: $C2/c\text{−}24$ indirect (blue triangles), direct (blues squares); $Cmca\text{−}12$ indirect (orange triangles), direct (closed squares). Experiments: indirect gap from the Tauc analysis at $T=100\text{K}$ (phase III), (black squares) [64] and at 300 (phase IV), (black triangles) [17, 24]; direct gap at 100 K (black squares) [21, 64].

Figure 4

Integrated density of states for $C2/c\text{−}24$ quantum crystals at 200 K from GCTABC-RQMC (points) and the HSE (smooth lines) at various pressures.

Figure 5

Absorption spectra from the HSE band structure for $C2/c\text{−}24$ quantum crystals (solid lines) and comparison with the available experimental profiles (opened and filled circles). The spectra from the HSE has been shifted in energy by an amount equal to the difference between QMC and the HSE direct gap. The reported pressure are as in Fig. 4 (see the colors). The red dots indicate the location in energy of the direct gap of Fig. 3. Experimental pressures are 296 GPa, open orange circles [64] (corrected by 20 GPa [21]); 386 GPa, magenta filled circles and 406 GPa, red filled circles [21].