Quasiparticle energies and excitonic effects in dense solid hydrogen near metallization

We investigate the crucial metallization pressure of the $Cmca\text{−}12$ phase of solid hydrogen (H) using many-body perturbation theory within the $GW$ approximation. We consider the effects of self-consistency, plasmon-pole models, and the vertex correction on the quasiparticle band gap $\left(E_{\mathrm{g}}\right)$. Our calculations show that self-consistency leads to an increase in $E_{\mathrm{g}}$ by 0.33 eV over the one-shot $G_0{W}_0$ approach. Because of error cancellation between the effects of self-consistency and the vertex correction, the simplest $G_0{W}_0$ method underestimates $E_{\mathrm{g}}$ by only 0.16 eV compared with the prediction of the more accurate $GW\Gamma$ approach. Employing the plasmon-pole models underestimates $E_{\mathrm{g}}$ by 0.1–0.2 eV compared to the full-frequency numerical integration results. We thus predict a metallization pressure around 280 GPa, instead of 260 GPa predicted previously. Furthermore, we compute the optical absorption including the electron-hole interaction by solving the Bethe-Salpeter equation (BSE). The resulting absorption spectra demonstrate substantial redshifts and enhancement of absorption peaks compared to the calculated spectra neglecting excitonic effects. We find that the exciton binding energy decreases with increasing pressure from 66 meV at 100 GPa to 12 meV at 200 GPa due to the enhanced electronic screening as solid H approaches metallization. Because optical measurements are so important in identifying the structure of solid H, our BSE results should improve agreement between theory and experiment.

Figure 1

The difference in enthalpy $\left(\Delta H\right)$ relative to the $Cmca$-4 phase for several candidate structures as a function of pressure.

Figure 2

The $Cmca\text{−}12$ crystal structure. (a) A single layer and (b) ABA stacking, with the conventional unit cell shown in red. Blue and green solid spheres represent protons on A and B layers, respectively.

Figure 3

Calculated electronic band structures of the $Cmca\text{−}12$ phase at 250 GPa based on DFT, using (a) a very hard norm-conserving pseudopotential and plane waves, (b) a much softer pseudopotential with plane waves, and (c) the all-electron full potential LAPW method. Overlaps between the VBM and CBM are 0.57, 0.53, and 0.47 eV in panels (a), (b), and (c), respectively.

Figure 4

Band gap of $Cmca\text{−}12$ solid hydrogen as a function of pressure in the $G_0{W}_0$ approximation using 4 PPMs (described in text) and the numerical full-frequency dependent screening. Solid lines are linear fits to their respective data points. The dashed line is a rigid shift of the fit to the PPM 2 data, shifted by the difference between the full-frequency calculation (square data point) and the PPM 2 calculation (red $\times$) at 200 GPa. KS indicates the Kohn-Sham band gap.

Figure 5

$G_0{W}_0$ band gap of $Cmca\text{−}12$ solid hydrogen including off-diagonal matrix elements at 200 GPa, using the PPM of Godby and Needs.

Figure 6

Difference in eigenvector self-consistent $GW$ (PPM of Hybertsen and Louie) and DFT charge densities of solid H at 200 GPa. Red and yellow indicate high and low isosurface values, and the black spheres represent protons.

Figure 7

Imaginary part of the dielectric function for $Cmca\text{−}12$ hydrogen at (a) 100, (b) 150, and (c) 200 Gpa for polarizations along the crystalline axes ${\mathbf{r}}_1$ and ${\mathbf{r}}_3$ with (BSE) and without (GW-RPA) the electron-hole interaction. A thermal broadening of 0.15 eV is applied.