Graphene physics and insulator-metal transition in compressed hydrogen

Compressed hydrogen passes through a series of layered structures in which the layers can be viewed as distorted graphene sheets. The electronic structures of these layered structures can be understood by studying simple model systems—an ideal single hydrogen graphene sheet and three-dimensional model lattices consisting of such sheets. The energetically stable structures result from structural distortions of model graphene-based systems due to electronic instabilities towards Peierls or other distortions associated with the opening of a band gap. Two factors play crucial roles in the metallization of compressed hydrogen: (i) crossing of conduction and valence bands in hexagonal or graphenelike layers due to topology and (ii) formation of bonding states with $2p_z$ $\pi$ character.

Figure 1

Calculated band structure of H graphene for two different bond lengths: (a) $b=1.23$ and (b) 0.94 Å. The vertical arrow in (a) indicates the bonding conduction band $2p_z$ moving quickly downward as the lattice parameter decreases. At $b=1.10$ Å, this band becomes lower than the Dirac level as seen in (b). The equilibrium $b=1.177$ Å. Note that the bands are colored in order of energy.

Figure 2

Peierls distortions of a honeycomb lattice, leading to a $\sqrt{3}\times \sqrt{3}$ superstructure and described by an amplitude $\delta$. (a) TO distortion. (b) LA distortion; modified after Ref. 22. (c) TO distortions for different $\delta$; $\delta =0$ corresponds to a pristine honeycomb lattice where a nonprimitive unit cell is chosen in the form of hexagon. In the case of $\delta >0$, the initial lattice dimerizes, so that the initial atomic hexagon expands. In the case of $\delta <0$, the initial lattice hexamerizes, and the initial atomic hexagon decreases in size. Modified after Ref. 23.

Figure 3

Total energy vs TO and LA mode amplitude $\delta$ for different initial bond lengths $b$: (a) and (b) 1.40 Å, (c) and (d) 1.177 Å, and (e) and (f) 1.0 Å. The TO and LA modes are actually symmetrized to produce the distortions of the $D_{6h}$ and $D_{3h}$ symmetries, respectively.

Figure 4

Total energy vs $\Gamma$ optical phonon amplitude $\delta$ for different initial bond lengths $b$: (a) 1.177 Å and (b) 1.0 Å. The inserts illustrate that in case of $\delta >0$, two nearest hydrogen atoms in each unit cell approach each other, whereas for $\delta <0$, they move away from each other. The vertical blue arrow indicates the moment when two Dirac points merge to open a band gap.

Figure 5

Calculated band structures at 260 GPa: (a) for $P$6${}_3/$mmc and (b) for Cmca-4. For the sake of comparison, the high-symmetry points in the hexagonal Brillouin zone of the $P$6${}_3/$mmc structure are indicated by double labels with the orthorhombic labels in parentheses.

Figure 6

The band structures of the $C2/c$ phase at 300 GPa (a) and of an artificial phase composed from four ideal honeycomb layers ABCD in such a way that the resulting symmetry is also C2/c (b). The artificial phase, like $C2/c$, has 24 atoms per cell and its lattice parameters are also identical to that for the $C2/c$ structure. The notation of the symmetrical points is borrowed from Ref. 11.

Figure 7

Same as in Fig. 5 but for other directions in the orthorhombic Brillouin zone. In (b), the star * indicates the band gaps that appear due to molecular tilting, $\theta \ne 0$.

Figure 8

Calculated band structures: (a) for Cmca-4 at 260 GPa, (b) for Cmc2${}_1$, and (c) for $C2/m$ phases. In going from Cmca-4 to Cmc2${}_1$, the lattice parameters, intramolecular distances and tilting angles are kept the same. The only difference between these structures is that in the latter the centers of molecules lie almost on an ideal hcp lattice. The $C2/m$ structure differs from Cmc2${}_1$ only by mutual orientation of two molecules in the unit cell, whereas in the Cmc2${}_1$ structure the molecular axes are tilted by the angles $\theta$ and – $\theta$ from the $xy$ plane, in the $C2/m$, the axes are parallel, i.e., molecules tilted by the same angle, $\theta \sim {30}^0$.

Figure 9

Density of states (DOS): (a) for $C2/c$ at 150 GPa and (b) for Cmca-4 at 300 GPa. The calculations have been performed by using the tetrahedron method and sufficiently dense $k$-point meshes: 40$\times$40$\times$40 and 100$\times$100$\times$8 for the $C2/c$ and Cmca-4 structures, respectively.