Electronic excitations and metallization of dense solid hydrogen

Abstract

Theoretical calculations and an assessment of recent experimental results for dense solid hydrogen lead to a unique scenario for the metallization of hydrogen under pressure. The existence of layered structures based on graphene sheets gives rise to an electronic structure related to unique features found in graphene that are well studied in the carbon phase. The honeycombed layered structure for hydrogen at high density, first predicted in molecular calculations, produces a complex optical response. The metallization of hydrogen is very different from that originally proposed via a phase transition to a close-packed monoatomic structure, and different from simple metallization recently used to interpret recent experimental data. These different mechanisms for metallization have very different experimental signatures. We show that the shift of the main visible absorption edge does not constrain the point of band gap closure, in contrast with recent claims. This conclusion is confirmed by measured optical spectra, including spectra obtained to low photon energies in the infrared region for phases III and IV of hydrogen.

Not only is hydrogen the simplest of all of the elements, it also reflects much of the range of electronic structures observed in materials, ranging from a Mott–Hubbard insulator for an expanded atomic lattice, to a van der Waals bonded condensed phase in the low-pressure molecular crystal, to an atomic metal at low temperatures and extreme compression. At high temperatures hydrogen forms the plasma that is the primary visible component of the universe. Because hydrogen is the most abundant element in the universe, forming most physical bodies, and is subjected to the entire range of pressures in the universe where atomic matter is stable, and because of its fundamental role in the history and structure of quantum mechanics, hydrogen at high pressure has been of intensive theoretical and experimental interest for the last century (1, 2). It has become evident from first-principles quantum-mechanical simulations that hydrogen passes through a set of layered structures as it is compressed. Important new advances in theory and experiment clarify the underlying framework of dense hydrogen physics and chemistry. We show here, from electronic structure calculations and analysis of experimental data, that metallization occurs through an evolution from a molecular solid to an extended crystalline solid in which distorted graphene-like layers provide the primary structural motif.

At low temperatures hydrogen forms a simple hexagonal close-packed structure with freely rotating molecules, called phase I (1). At low temperature, the material transforms to the quantum broken symmetry phase (phase II) at pressures of 10 (50) GPa for H₂ (D₂) and phase III at 150 (165) GPa (1). Vibrational spectroscopy and X-ray diffraction of the lower pressure phases help constrain the structures of the higher pressure phases. The transition to phase III is a sharp phase transition marked by a strong change in the IR vibron absorption spectrum. Recent experimental breakthroughs include the observation of the onset of electrical conductivity at 260 GPa and 300 K, evidence for a new phase (3–5), and bounds on the stability field of phase III (6). With heating above 220 GPa, hydrogen transforms to the recently discovered phase IV (3, 4, 6).

Band-overlap metallization of the molecular solid was predicted in early electronic structure calculations of dense hydrogen based on simple structures found experimentally at low pressures and temperatures (7, 8). It was later realized that there is strong structural control on the predicted band-overlap metallization (9–13). In this connection, Ashcroft (9) pointed out the relevance of graphite to the optical properties of high-pressure hydrogen and its signature for metallization, and proposed that high-pressure hydrogen may consist of honeycomb sheets. Even earlier, calculations by LeSar and Herschbach (14) had predicted the stability of six-membered rings of hydrogen molecules, or honeycomb structures, in the highly compressed solid. Calculations indicated that metallization in the molecular solid is a function of the orientations of the H₂ molecules, with metallization favored by molecules aligned along the c axis for hexagonal close-packed molecular centers, but with low-energy states with molecules canted so as to open a gap (11–13). Calculations showed metallization to occur at small regions in the Brillouin zone in these hexagonal structures (12).

Recent theoretical work has predicted new bounds on stability and has predicted new structures and their stabilities at megabar pressures (15–22). First-principles calculations predict stability of a structure with space group symmetry C2/c in the pressure-temperature (P-T) region found experimentally for phase III (15). The C2/c structure can be considered to be a distortion of planar, graphene-like sheets of hydrogen, with six-membered rings in a honeycomb structure, as we point out below. Several related structures for phase IV, all distinguished by graphene-like planes interlayered by H₂ molecules, are predicted by theory. At high pressures, these calculations predict structures again based on graphene layers with tilted molecules and a more 3D electronic structure. The pairs of hydrogen atoms that form molecules become less obvious with increasing pressure as the intermolecular distances decrease. Simple monoatomic metallic phases related to the original predictions are calculated to form above 400 GPa at low temperatures (23); see refs. 1 and 2 for a review of experiments and recent theoretical work on hydrogen.

Optical measurements using diamond anvil cell techniques have been useful for identifying pressure-induced changes in electronic properties in materials at very high pressures. The predicted onset of metallization of solid hydrogen under pressure, including the predicted band-overlap metallization, has been sought in optical experiments. Direct electrical measurements have also been applied (3, 24) and have provided evidence for transitions, but quantitative interpretation is complicated by technical issues (e.g., potential chemical reactions with the electrodes; ref. 6). Measurements of optical spectra are made difficult by the absorption threshold of the stressed diamond anvil, which moves down through the UV, diffraction effects at long wavelengths, and challenges in determining absorption coefficients. Measurements at lower energies (e.g., the IR) are needed to properly constrain the optical conductivity of the high-pressure phases. Early optical measurements on hydrogen in phase III showed no evidence of metallic conductivity to at least 200 GPa (25, 26). Measurements using synchrotron IR radiation, which is able to probe small samples at the lowest photon energies at megabar pressures, have provided particularly important constraints (26). Recent synchrotron IR measurements on phase III have not only provided additional information on its P-T stability range but also show that the phase could be semimetallic if the effective plasma frequency is <0.1 eV (5, 26).

Here, we present theoretical calculations and an assessment of recent experiments and previous theoretical work on dense hydrogen that leads to a picture of the metallization of hydrogen. What was not realized in early calculations (12) was that bits of Fermi surface were the intersections of Dirac cones from graphene-related structures (near the K point in ideal graphene), and small pockets formed from s–p hybridized states. In solids, electronic states are labeled by k points, which represent single-particle orbitals _, where G is a reciprocal lattice vector and k labels wavevectors in the first Brillouin zone; points and lines of special symmetry have labels, e.g., (0,0,0) is called the Γ-point. In graphene, the band energy E(k) is linear where the bands cross the Fermi level E_F, which separates occupied and unoccupied states, forming cones of energy versus k around the special points labeled K and K′, whereas bands in solids are usually parabolic and not linear. This makes the electron and hole quasiparticles behave as relativistic massless particles; thus the cones of E(k) are known as Dirac cones (see ref. 27).

We first discuss the electronic structure of hypothetical, perfect graphene-type hydrogen, i.e., graphite-structured hydrogen, and then how the proposed high-pressure structures relate to graphene. We show that the experimental optical response and constraints on metallization of hydrogen can be understood from the behavior of stacked honeycomb layers on compression.

Results

Examination of Candidate Structures.

We consider a series of recently proposed structures for dense molecular hydrogen. Only at low pressures or high temperatures, where H₂ molecules are freely rotating, as quantum rotors, is the molecular description ideal. We identify each structure by its international space group symbol. Fig. 1 A and B shows the structure of graphite (space group P6₃mc). The C2/c structure is consistent with most experimental data for the phase III (Fig. 1 C and D). We have drawn the structure so as to emphasize the fact that it consists of distorted graphene-like layers. It is a four-layer structure with stacking ABCD. A linear path in coordinates of the atoms in the layers to perfect graphene sheets (graphite) at 300 GPa shows a monotonic 38 meV/atom (436 K) enthalpy difference; this is appreciable but much lower than zero-point energies and comparable to room temperature, so that quantum and thermal motions will tend to symmetrize the layers, as discussed below. Phase IV has been proposed to be a two-layer structure, with graphene-like sheets separated by molecular H₂ layers related to Pbcn (Fig. 1 E and F) (21). A higher pressure molecular phase is predicted to have the Cmca-4 structure (Fig. 1G), which again is shown in the figure to have a structure related to that of graphene. Note, however, that the sheets in Cmca-4 are cross-linked. These configurations can thus all be thought of as distorted and disordered graphene-based structures. In fact, all of the proposed high-pressure structures below 300 GPa that are consistent with first-principles calculations and experiments (4, 5, 15, 19, 21) can be understood in this way. The H₂ molecules at the vertices of a honeycomb lattice cause the electronic structure of the molecular layers to have similarities to that of the atomic graphene layers, as we now describe.

Fig. 1.

Proposed structures of dense molecular hydrogen: (Left) structures viewed perpendicular to the graphene-like sheets; (Right) a single layer. (A and B) P6₃mc (graphite), (C and D) C2/c, (E and F) Pbcn, (G) Cmca-4, (H) conventional view of C2/c (after ref. 15), which emphasizes the molecular picture.

Pressure-Induced Metallization.

A perfect graphene-structured hydrogen sheet (H graphene) has a similar electronic structure to that of carbon (C) graphene (Fig. 2). Fig. 2 A–C shows the H-graphene band structures as compression is increased. At low pressures it looks very much like C graphene with bands crossing linearly at the K point and with a saddle point at the M point. It is a zero-gap insulator, or zero density-of-states metal, with topologically protected Dirac points at K and K′, as discussed previously (28). One difference from C graphene is that in C graphene the sp² states provide most of the bonding and structural stability, with the Dirac bands being formed from p_z states, whereas in H graphene 1s-electrons provide both the bonding and structural stability and form the linear bands that cross the Fermi level E_F. Nevertheless, the electronic structure around E_F is essentially the same in H graphene as in C graphene. The energies of the 2s states are above the range shown in Fig. 2.

Fig. 2.

Band structures of graphene-structured hydrogen for a single sheet, graphite structure, and high-pressure structures. Hydrogen metallizes through an indirect gap, forming a semimetal over a very small part of the Brillouin zone. (A) H graphene at low pressures, a = 2.328 Å (calculated for graphite c = 21.1 Å). (B) H graphene at moderate pressures, a = 1.90 Å. (C) H graphene at high pressures, a = 1.61 Å. With pressure, the states at Γ move down and cross the Fermi level, leading to a semimetal. However, even at low pressures the graphene structure is a zero-density-of-states metal. (D) The high-pressure (>300 GPa) Cmca-4 structure is 3D. Nevertheless the graphene-like structure leads to a similar electronic structure as the metallized graphene shown in C. (E–G) C2/c, at 150 GPa (E), 300 GPa (F), and 400 GPa (G). The complex band structure is formed from multiple folding of the distorted graphene band structure. At 300 GPa for the static lattice, the C2/c structure has a small gap. (H) Pbcn at 300 GPa. Again, the structure looks like folded graphene, with a large direct gap. Pressures are from the density functional theory computations and all structures are relaxed.

H-graphene sheets tend to distort to open a band gap to reduce the electronic kinetic energy (11). We consider two kinds of such distortions― dimerization, and tilting. The most effective distortions of the first kind are the 2D Peierls-like distortions that induce the energy gap via K and K′ mixing (28, 29). They lead to the formation of superlattices with multiplicities of 3 × 3 or √3×√3 from the primitive graphene cell (30). This occurs, for example, in the proposed phases, Pbcn, Cc, and C2/c, where K and/or K′ are folded to Γ. The second kind of distortion is tilting of molecules or warping of graphene sheets. Whereas the first tends to open a gap at the Dirac points for s states, the second helps prevent metallization through the 2p_z states. In ideal H graphene, the bottom of the 2p_z band is close to the Dirac levels or E_F (it is already within 3 eV when the bond length is as large as 1.25 Å). As the H-graphene sheet is further compressed, these 2p_z bonding states come down in energy at the Γ-point, and pass E_F to form a semimetal. The 2p_z states form π-bonds which are stabilized with compression relative to the more disperse 1s σ-bonds and antibonding states. The system, however, usually avoids the crossing of 1s and 2p_z bands by mixing them, as in the case of diamond-type Si, in which the mixing of 3s and 3p bands forms a gap along the Γ–X direction (31). In contrast with Si, here the s–p mixing can be realized only via special distortions, including the tilting of the H₂ pairs out of the ideal honeycomb plane. Note that such a tilting is found, for example, in the molecular layers of the Pbcn structure and in all layers of the C2/c structure. Further analysis of the predicted electronic structures is presented elsewhere (32).

The band structure for Cmca-4, a phase predicted to appear above 300 GPa (2, 9, 19, 33) (Fig. 2D), is very close to that of graphene, despite the three-dimensional connectivity in Cmca-4. One should not be misled by the change in special k-point labels—the conventional labels change with space group symmetry, but the crossing between Γ and Y in Cmca-4 (Fig. 2D) is the same crossing as at K in graphene (Fig. 2C). However, the segment between Γ and Y is perpendicular to the honeycomb planes, so that there is significant three-dimensional dispersion in Cmca-4.

The band structures of the C2/c, phase, predicted to be stable in the stability range of phase III (from 150 to above 300 GPa at low temperatures (6, 34) are shown in Fig. 2 E–G. All of the special points in graphene get folded to Γ, and the distortions of the graphene layers tend to open an indirect gap, and band extrema off of Γ. Interestingly, the highest occupied states are not at a symmetry point: they are almost degenerate between Γ and Y and Γ and X. The lowest unoccupied states are at C and Γ (Fig. 2 E and F) for 150 and 300 GPa, respectively). With further compression, band overlap between odd points occurs before 400 GPa (Fig. 2G), forming tiny pockets of electrons and holes, reminiscent of the small pockets formed in doped graphene. The candidate for phase IV, Pbcn, is shown in Fig. 2H at 300 GPa. It also has an indirect gap with tiny electron and hole pockets.

In summary, we find that in all of the structures considered here the initial metallization occurs over only a very small part of the Brillouin zone, so that only small patches of the Fermi surfaces first form. Actually, these patches are either the intersections of Dirac cones, as in Cmca-4 structure, or intersections of the bands that are directly involved in the formation of the hybridization gaps via 1s–2p or K–K′ mixings. As pointed out some years ago, accurate simulation of molecular hydrogen metallization requires very large k-point sets due to the initial formation of small pockets of Fermi surface (12). Thus, one can wonder whether small k-point sets in G-W calculations are sufficient (10). Here, we also show that the initial formation of small pockets has great relevance for the signature of metallization expected in experiments.

Optical Properties.

We now examine the optical properties associated with the above classes of predicted structures. Ashcroft (9) discussed the importance of considering both interband and intraband processes, and the expected signatures of metallization and honeycomb structures. Since then much more has been learned about graphene and graphite and their optical properties. A single perfect undoped and zero-potential graphene-like sheet with only Dirac points at E_F for energies has universal low-energy optical properties that do not depend on the atoms involved and lattice constant (27). It has a constant optical conductivity and constant optical absorption in frequency (35–37). Just as in C graphene, there is a saddle point at the M point in H graphene which leads to a large peak in the predicted UV optical response. With increasing frequency, as the saddle point at M is approached in the UV [4 eV in perfect H graphene within Wu-Cohen Generation Gradient Approximation (WC-GGA), ref. 38], there is a large cusp in the optical conductivity and a strong absorption edge (Fig. 3A).

Fig. 3.

Calculated optical spectra of graphene-structured hydrogen, graphite structures, and the high-pressure hydrogen structures. The shape of the peaks depend on the assumed scattering lifetime, as discussed in the text. (A) Optical conductivity of H graphene at a = 2.12 Å (H–H 1.22 Å) is dominated by the interband peak to the saddle point at M, just as in C graphene. (B) Optical conductivity of C graphite shows the signature of the underlying graphene layers. a = 2.46 Å, c = 6.71 Å (C–C 1.42 Å). (C) Optical conductivity of graphite-structured hydrogen at 300 GPa (a = 1.60 Å, c = 2.57 Å, H–H 0.92 Å) shows the same features, with a sharper Drude peak at zero frequency from intraband scattering mainly at the zone boundary. (D) Optical conductivity of C2/c hydrogen at 150 GPa. (E) Reflectivity of C2/c hydrogen against diamond at 150 GPa. (F) Reflectivity of C2/c hydrogen against diamond at 400 GPa. (G) Reflectivity of H graphite against diamond at 300 GPa. (H) Transmission through 1 μm of H graphite in a diamond anvil cell (DAC) at 300 GPa. (I) Reflectivity of Pbcn H at 300 GPa in a DAC. (J) Transmission through 1 μm for Pbcn in a DAC at 300 GPa. (K) Corresponding absorbance of Pbcn at 300 GPa. (L) Reflectivity of Cmca-4 at 300 GPa in a DAC.

A stack of graphene sheets (i.e., graphite) has a slowly varying optical conductivity well below this edge (Fig. 3 B and C), but a linear optical absorption from the additional phase space for scattering. The strong absorption edge is not the band gap as recently claimed (e.g., ref. 4), and its shift does not determine the point of gap closure. The edge results from the scattering from K to the saddle point at M for a graphene sheet, which is folded back to Γ by the tilting and Peierls distortions. So, the edge does not even designate the minimum direct gap. For example, lightly corrugated graphene at zero pressure would have a vanishingly small gap, but still an UV absorption edge. Stronger tilting and distortions push this edge still higher as shown in Fig. 4.

Fig. 4.

Comparison of representative calculated optical absorption spectra for candidate structures of dense hydrogen compared with recent measurements. (Upper) Calculated spectra for C2/c at 150, 300, and 400 GPa, and Pbcn at 300 GPa. The indirect gap is closed for C2/c at 400 GPa and Pbcn at 300 GPa at the current level of theory, giving rise to absorption at low-photon energies (long-wavelength IR). Spectra were calculated for a 1-μm-thick sample measured against diamond, with the reflection losses at the interface included to make direct comparison with experiment. Results were averaged with orientation, to represent randomly oriented fine-grained samples. (Lower) Measurements for phase III at 160 GPa (300 K) (4), phase IV at 310 GPa (300 K) (4), phase III at 360 GPa (17 K) (6), and phase IV at 280 GPa (300 K) (5). The experimental data are relative absorption measurements, and the spectra are offset for clarity (in contrast with the calculated spectra). Sharp features at low energies arise from vibrons, diamond, and hydrogen–diamond interactions; as described in refs. 5 and 6. For comparison, we include optical conductivities calculated assuming the simple Drude models used in previous studies (5, 26) to provide upper bounds on the possible carrier density. The following Drude model parameters were used: ħω_p = 0.2 eV, ħ/τ = 0.1 eV (solid blue line); ħω_p = 0.2 eV, ħ/τ = 0.01 eV (dashed blue line).

With compression of a graphene sheet, as the conduction state bands move through E_F at Γ, a Drude response with very low plasma frequency is added to the interband response described above (Fig. 3A). The C2/c (Fig. 3D) and Pbcn structures behave similarly, with differences being that their zone boundary points corresponding to the graphene K and Ḱ corners are folded into the zone center. This folds the semimetal states to Γ so that there is a predicted absorption edge at 4 eV in C2/c at 150 GPa, which drops to 2 eV at 300 GPa (Fig. 4). The predicted gap (2 eV at 150 GPa, 0.5 eV at 300 GPa) in C2/c is indirect, with a valence band maximum between Γ and Y, and a conduction band minimum at Γ (Fig. 2). The reflectivity of C2/c at 150 GPa in a diamond anvil cell is very small up to over 5 eV. Continued compression of the C2/c structure in the theory closes the band gap, and the reflectivity dramatically increases, below 0.5 eV (Fig. 3F). There is also an increase in optical absorption, but only at very low energies (e.g., initially below 0.1 eV, ref. 6) (Fig. 4).

H graphite is metallic at 300 GPa, with a similar signature in the low-energy reflectivity as in metallic C2/c (Fig. 3G). Even though H graphite is metallic, it has a notch in transmission in the visible (Fig. 3H) for light propagating along rather than perpendicular to the honeycomb layers, even though the reflectivity is more isotropic. Other hydrogen phases are also predicted to be quite anisotropic optically. The Pbcn phase, although metallic, has a very low effective plasma frequency and the reflectivity drops rapidly at low energies, without any indication of it being a metal at higher energies (Fig. 3I). It is also quite transparent in the IR and should appear red in the visible (Fig. 3 J and K). The Cmca-4 phase looks more like an ordinary metal, with 100% reflectivity through the visible (Fig. 3L). The spectra are Drude-like, only at the lowest energies, and are dominated by interband scattering.

The static, ordered Pbcn phase is predicted to be metallic at 300 GPa, with a plasma frequency of 0.8 eV in the plane, and 0.5 eV for light polarized perpendicular to the sheets. Nevertheless, there is a strong absorption edge at 1.8 eV. Despite the metallic character and closing of the band gaps, Pbcn is transparent from 0.18 to 1.8 eV, with an absorptivity of 8 × 10⁴/cm at 0.05 eV (400 cm⁻¹) (Fig. 4). Cmca-4 has a similar optical structure at 1.8 eV and above, but is much more 3D and has much higher predicted plasma frequencies, 1.3, 2.7, and 2.9 eV for xx, yy, and zz, respectively. Interestingly, Cmca-4 is predicted to be a transparent metal, with little absorption below 2 eV, although it has a strong saturation of reflectivity below 3 eV. Note that one cannot simply add together the interband absorption or reflectivity with a Drude intraband component. Although the intraband and interband contributions are linear (additive) in the dielectric constant ε, the reflectivity and absorption are complex functions of ε.

We compare the present calculations with published experimental data (Fig. 4). The form of the absorption edge is largely similar among all structures and contains no information about whether the gap is closed. As mentioned above, this edge is not the band gap, although it has been identified as such in numerous publications (4, 20). Experimentally, there is an absorption feature at 2–3 eV whereas the material remains largely transparent down to 0.1 eV. For the spectrum calculated for the model Pbcn structure, the transparency from 0.18 to 1.8 eV and the absorption of 8 × 10⁴/cm (optical density of 4 for a 1-μm sample) at 0.05 eV ≈ 400 cm⁻¹ are consistent with experimental data for phase IV.

Discussion

We have shown that the seemingly complex high-pressure structures of hydrogen are all related. The analysis reveals that the evolution and the physical properties of solid hydrogen under pressure can be understood in terms of the underlying structural relationship to graphene. Features of the resulting electronic band structures can be traced to the electronic properties of a single graphene-like sheet. Even phases I and II are related in this way, in that the molecules sit on a hexagonal close-packed honeycomb lattice and this dictates the character of their electronic structures. Although the anisotropic and nonclosed packed character of the very high density structures might be considered surprising, numerous other examples of low-symmetry forms are emerging in other elemental systems. We also point out that Wigner and Huntington (39) mentioned the possibility of layered structures forming before an atomic phase, by analogy with other, then-known structures of similar elements.

A series of model structures having perfect lattices with no thermal quantum disorder is assumed in the present analysis. The hybridization gap in the single-particle spectrum due to several types of distortion discussed above will tend to be smeared out by quantum and thermal motions. Another factor contributing to gap formation is hybridization between layers, which is enhanced by the tilting. Stacking disorder will also tend to smear out this tendency to gap formation. So, there are two opposing forces in molecular hydrogen. Gap formation is favored to reduce the kinetic energy (11), as in the Peierls mechanism, and quantum delocalization and thermal disorder tend to decrease the gap through state broadening and symmetrization of the layers.

In many materials disorder can lead to localization and insulating behavior, but here we expect static and dynamic disorder to lead to metallization. This is because the averaged disordered structure is closer to the graphene-like, honeycomb structure than the static ground-state structure. The static relaxations are such as to widen the gap, and disorder will therefore shrink the gap by effectively broadening the electronic states. Thus, both phases III and IV could be conducting in the proposed or closely related structures as a result of q≠0 phonon and defect scattering because the gap is predicted to be indirect.

Conclusions

Theoretical calculations of the electronic structure and optical response for candidate phases of solid molecular hydrogen at multimegabar pressures reveal important features in optical spectra characteristic of the transition from semiconductor to semimetallic and metallic phases. The absorption edge in the visible spectrum (e.g., 2–3 eV) decreases with pressure whereas the material remains largely transparent throughout the near IR for calculated low-energy structures up to ∼300 GPa, in agreement with available experimental data. However, this alone contains no information about whether the gap is closed in these phases because the characteristic features of predicted gap closure are associated with a rise in absorption and reflectivity in the far IR (below 0.2 eV) whereas the material remains transparent in mid IR. This is consistent with synchrotron IR measurements and constraints provided by simple Drude models fit to those data at the lowest energies. The optical response calculated for candidate structures for the full IR and visible spectrum exhibits interband transitions at higher energies. Similar features calculated for the structures arise from the electronic structure of the graphene (or graphene-derived) layers common to the structures. Quantum (40) and thermal disorder and stacking disorder are expected to give rise to gap closure at compressions for which the static lattice calculations would give a small gap. This would in turn result in a semimetallic phase with a signature only in the far IR. The extent to which this occurs in phases III and IV at the conditions of current experiments (>150 GPa and 4 K to 350 K) will require additional experimental and theoretical study. The present analysis provides a unique mechanism for pressure-induced metallization of solid hydrogen.

Methods

We performed density functional computations within the WC-GGA method (38) with the ABINIT (41) pseudopotential and WIEN2K (42) Full-potential Linear Augmented Plane Wave (FLAPW) codes. Structures were based on those described in refs. 15 and 21, which agreed with our relaxed structures in selected tests. All band structures presented here were computed with the WIEN2K code, and optical properties were computed within the random phase approximation using the WIEN2K optics package (43) with 0.01-eV broadening unless stated otherwise. We found that very large k-point sets were required to converge the optical properties. K-point grids ranged from 120 × 120 in the layer plane to 200 × 200 with 4–40 points along the perpendicular direction. The tetrahedron method was used to evaluate Brillouin zone sums.