Semimetallic dense hydrogen above 260 GPa

Abstract

Being the lightest and the most abundant element in the universe, hydrogen is fascinating to physicists. In particular, the conditions of its metallization associated with a possible superconducting state at high temperature have been a matter of much debate in the scientific community, and progress in this field is strongly correlated with the advancements in theoretical methods and experimental techniques. Recently, the existence of hydrogen in a metallic state was reported experimentally at room temperature under a pressure of 260–270 GPa, but was shortly after that disputed in the light of more experiments, finding either a semimetal or a transition to an other phase. With the aim to reconcile the different interpretations proposed, we propose by combining several computational techniques, such as density functional theory and the GW approximation, that phase III at ambient temperature of hydrogen is the Cmca-12 phase, which becomes a semimetal at 260 GPa . From phonon calculations, we demonstrate it to be dynamically stable; calculated electron–phonon coupling is rather weak and therefore this phase is not expected to be a high-temperature superconductor.

The metallization of dense hydrogen has been the most interesting topic since the theoretical prediction of transition to a monoatomic metallic state of hydrogen above 25 GPa by Wigner and Huntington in 1935 (1). Since then, a series of theoretical and experimental studies have been employed to unveil the structural and electronic properties of hydrogen at high pressures (2). Chen et al. (3) and Hemley et al. (4) have reported experimentally that hydrogen remained molecular up to 191 and 216 GPa, respectively, but could be transformed to high-pressure phases. Recently, Pickard and Needs (5) have presented the zero temperature phase diagram of solid hydrogen up to approximately 400 GPa, predicting that the most stable phases remained insulating at very high pressures, which was reported to eliminate the major discrepancy between theoretical (6) and experimental (7) predictions.

However, the metallization of hydrogen at high temperature is a different problem because it cannot directly benefit from theoretical calculations that are usually conducted at T = 0. Recently, Eremets and Troyan (8) have reported the metallization of hydrogen at room temperature and around 260–270 GPa using conductivity measurements, and that it is accompanied by a first-order structural transition. This work has been recently analyzed by Nellis et al. (9) who stated that there was no clear evidence for metallic hydrogen in the experiments of Eremets and Troyan. Eremets and Troyan (8) did not address the structure of the hydrogen crystal, but proposed that it could be the Cmca-12 phase of Pickard and Needs (5). Very recently, Zha et al. (10) studied hydrogen samples above 300 GPa and from 12 to 300 K using synchroton infrared and optical absorption techniques. In contrast to Eremets and Troyan (8), they showed that the properties measured are consistent with a semimetal over a broad range of temperature and pressure. At the same time, Howie et al. (11) performed measurements at 300 K and up to 310 GPa, and proposed the transformation to a new phase (phase IV) at 220 GPa and identified this phase as the Pbcn phase obtained by Pickard and Needs (5). Owing to this particularly complicated situation, it seems necessary to propose a scenario which can reconciliate (at least partially) the different experiments and which can be tested in further theoretical and experimental works.

Using density functional theory (DFT) and a random search method, Pickard and Needs (5) have studied the stability of various phases of solid hydrogen in function of pressure for a temperature equal to 0 K. In our pressure range of interest (from 200 to 300 GPa), they have found that the most stable phases (in Hermann–Mauguin notation) are the C2/c for pressures between 105 and 270 GPa, and the Cmca-12 phase (a phase with the Cmca space group and with 12 atoms per cell noted Cmca-12 by Pickard and Needs, ref. 5, to differentiate it from other Cmca phases) for pressures between 270 and 385 GPa. When zero point effects (ZPE) were included, the transition between the two phases occurred at 240 GPa instead of 270 GPa without ZPE. They have also shown that the C2 and Pbcn phases, although slightly higher in enthalpy, are also relevant for this range of pressure. The C2/c phase has a layered structure with the layers arranged in an ABCDA fashion, and Cmca-12 phase has a layered structure with a stacking of the ABA type. Also, the C2 and Pbcn phases are mixed phases (5) with alternating layers of strongly bonded H₂ molecules and more weakly bonded atoms. The details of the structures such as the H–H distances are reported in the paper of Pickard and Needs (5) and will not be repeated here. However, because these results were obtained for T = 0 K, they cannot be immediately used to interpret the experiments of Eremets and Troyan (8), which were conducted at room temperature.

Therefore to identify the phase seen by Eremets and Troyan (8), we choose the following strategy: We will consider the four phases (C2/c, Cmca-12, C2, and Pbcn) found by Pickard and Needs (5) for the 200–300 GPa range of pressure, but instead of comparing their enthalpy at room temperature, we will rather compare the value of their band gaps and how it closes under pressure. If one of these phases is found to have a pressure of metallization of about the same value as the one reported by Eremets and Troyan, then it will be likely to be the phase observed in ref. 8. To obtain reliable values for the pressure of metallization by band-gap closure of the different phases, DFT with standard approximations for the exchange-correlation potential such as the local density approximation or the generalized gradient approximation (GGA) cannot be used, and therefore we have used the GW approximation (12–14). With the GW approximation, it is possible to overcome the difficulties of standard functionals to obtain reliable values of the band gap by the replacement of the local or semilocal exchange-correlation potential by a nonlocal and energy-dependent self-energy which allows one to describe correctly excited states (see also computational details). Because we use the GW approximation in a perturbative way, density functional theory calculations are a necessary step before the GW calculation. Therefore, we have used the information about the crystal structures of the C2/c, Cmca-12, C2, and Pbcn phases as provided by Pickard and Needs (5) and optimized the geometry for a set of pressures in order to study the metallization with the GW approximation.

Results and Discussion

In Fig. 1, we present our calculated minimum band gap versus pressure for the four phases (C2/c, Cmca-12, Pbcn, and C2) obtained with the GW approximation. When we observe metallization of the four phases under pressure in a range between 200 and 400 GPa, they present different behaviors. The C2/c and Pbcn phases become a metal at pressures of about 350 GPa, the C2 phase is metallic around 300 GPa, whereas the Cmca-12 has the lowest pressure of metallization of all the phases that we have investigated, with a metallization that occurs already for a pressure of about 260 GPa. In the experiments of Eremets and Troyan (8), hydrogen transformed to a metal at 260–270 GPa, and therefore it is likely that the Cmca-12 phase (see Fig. 2 for the picture of the corresponding crystal structure at 260 GPa), as suspected by Eremets and Troyan, is the phase observed during these experiments. Also, they have mentioned (8) that, for pressure above 240 GPa, the band gap should be smaller than 0.7 eV because of the absence of illumination; this is in agreement with our calculated minimum band gap of 0.25 eV at 240 GPa. However, it was shown in these experiments that the transformation to a metal is accompanied by a first-order structural transition, whereas the scenario that we propose is partly different.

Fig. 1.

Band gap calculated with the GW approximation as a function of pressure for the Cmca-12, C2/c, Pbcn, and C2 phases of hydrogen.

Fig. 2.

Crystal structure of the Cmca-12 phase at 260 GPa.

The band structures picturing the metallization of the Cmca-12 phase obtained with either the GGA or with the GW approximation are presented in Fig. 3 (GGA data are presented as a full line; GW results are the red dots, the Fermi level being at 0 eV) for pressures of 210, 240, and 260 GPa. The effect of the GW correction is clearly seen as a shift to higher energy of the GGA conduction bands, which gives a pressure of metallization between 240 and 260 GPa, whereas it would happen at a lower pressure with the GGA. By inspecting in detail the band structures, it is seen that the bottom of the conduction band is at the high-symmetry point Y, whereas the top of the valence bands is along the high-symmetry direction Γ-Y. Therefore the process of metallization is taking place not by a direct overlap of the bands, but rather by the appearance of a semimetal where the electrons at the top of the valence band are transferred to the conduction band at the high-symmetry point Y. The appearance under pressure of a semimetal rather than a full metal is indeed in agreement with the recent work of Zha et al. (10).

Fig. 3.

Calculated band structures for the Cmca-12 phase of hydrogen with GW approximation at (A) 210, (B) 240, and (C) 260 GPa, respectively.

We performed calculations to check the dynamical stability of the Cmca-12 phase. Fig. 4 shows the phonon dispersion curves (Left) and the phonon density of states of this phase at 270 GPa. There is no sign of imaginary modes, so the Cmca-12 phase is stable at this pressure. The dispersion curves consists of two types of phonon bands: bands with a relatively low frequency (0–70 THz) and bands at higher frequency (100–120 THz), which are separated by a gap of approximately 30 THz. The higher six bands correspond to molecular vibron modes and the lower-frequency phonons are dispersive in energy, a softening along the Y-T line being present. Thus, the molecular nature of the Cmca-12 phase still persists at 270 GPa. Using linear response theory, we also checked the electron–phonon coupling for the Cmca-12 phase at 270 GPa, however, the integrated coupling constant is rather small (0.21) and the corresponding superconductivity T_C using the Allen–Dynes equation (15) is estimated to be negligible, which is expected for a semimetal. The calculated averaged phonon momentum, which is the leading parameter to determine T_C, is large (1,483 K, higher than that of MgB₂; ref. 16), but the small number of charge carriers at the Fermi level prevent a strong superconductivity in this phase.

Fig. 4.

Phonon dispersion and density (PhDOS) of states for the Cmca-12 phase of hydrogen at 270 GPa.

A different way to probe the electronic structure of a material is to use optical spectroscopy. In Fig. 5, we present our calculated imaginary part of the dielectric function for the Cmca-12 phase at 300 GPa. It has not been measured experimentally but it is shown here in view of a future comparison. Because the Cmca-12 phase has a different electronic structure than the other phases, its optical properties are also markedly different, and this phase can therefore be identified by measuring its dielectric function. Because it is an orthorhombic phase, it presents three nonequivalent functions depending on the direction of the vanishing incoming wave vector. Each imaginary part of the three dielectric function (Im ϵ_xx, Im ϵ_yy, Im ϵ_zz) has a main peak around 6 eV. Also, Im ϵ_zz has a smaller structure around 2.5 eV and Im ϵ_yy has it at roughly 4 eV, whereas Im ϵ_xx does not seem to have it. However, Im ϵ_xx has a prominent structure around 13 eV which is not present in Im ϵ_yy and Im ϵ_zz.

Fig. 5.

Imaginary dielectric function of the Cmca-12 phase of hydrogen at 300 GPa.

Another point to notice concerns the work of Howie et al. (11), in which it was suggested that a phase appeared with the Pbcn space group. However, they noticed that it has a closing gap above 300 GPa, which is roughly consistent with our calculations (see Fig. 1) on the Pbcn phase.

Conclusions

In summary, we have shown that, among the phases of hydrogen that are relevant in the range of pressure of 200–300 GPa, only the Cmca-12 phase has a pressure of metallization that corresponds to the value reported by Eremets and Troyan (8). A detailed study of the band structure revealed that this phase is a semimetal and that no direct overlap occurs between the valence and the conduction band in the range of pressure that we have investigated. Therefore, the mechanism that we propose for the transition to a metal is different from the one of Eremets and Troyan (8), but the nature of the phase (a semimetal) is in agreement with the work of Zha et al. (10). The Cmca-12 phase is dynamically stable but the corresponding electron–phonon coupling is rather weak, and therefore no strong superconducting behavior is expected to take place. We expect that our work will stimulate further theoretical and experimental works on solid hydrogen.

Methods

The DFT (17, 18) calculations and the GW calculations (12–14) concerning the metallization under pressure of hydrogen have been performed with the Vienna Ab-initio Simulation Package (VASP) (19–21) implementing the projector augmented wave method (22). For the DFT calculations, the generalized gradient approximation of Perdew–Burke–Ernzerhof (23) was used for the exchange-correlation potential. The plane-wave cutoff was set to 1,000 eV and the k-points mesh was adjusted to the size of the different Brillouin zones to ensure convergence. Concerning the GW calculations, a cutoff of 350 eV was used for the polarizability matrix, and 128 bands were used for the summation over the conduction states (see ref. 19 for the details concerning the implementation of the GW approximation in VASP). The optical dielectric function calculations of the Cmca-12 phase were also performed with VASP. To reach convergence, a k-point mesh of 50 × 50 × 50 has been used.

The electronic calculations and ab initio lattice dynamics were performed with density functional perturbation theory implemented in Quantum ESPRESSO (24). The electronic wave function was expanded with a kinetic energy cutoff of 60 Ry. Monkhorst–Pack meshes (25) were used for Brillouin zone (BZ) integrations in the electronic calculations, and BZ sampling in electron–phonon coupling (EPC) calculations (q mesh). EPC matrix elements were computed in the first BZ on an 8 × 8 × 8 q mesh using individual EPC matrices obtained with a 16 × 16 × 16 k mesh.