Optical properties of dense lithium in electride phases by first-principles calculations

The metal-semiconductor-metal transition in dense lithium is considered as an archetype of interplay between interstitial electron localization and delocalization induced by compression, which leads to exotic electride phases. In this work, the dynamic dielectric response and optical properties of the high-pressure electride phases of cI16, oC40 and oC24 in lithium spanning a wide pressure range from 40 to 200 GPa by first-principles calculations are reported. Both interband and intraband contribution to the dielectric function are deliberately treated with the linear response theory. One intraband and two interband plasmons in cI16 at 70 GPa induced by a structural distortion at 2.1, 4.1, and 7.7 eV are discovered, which make the reflectivity of this weak metallic phase abnormally lower than the insulating phase oC40 at the corresponding frequencies. More strikingly, oC24 as a reentrant metallic phase with higher conductivity becomes more transparent than oC40 in infrared and visible light range due to its unique electronic structure around Fermi surface. An intriguing reflectivity anisotropy in both oC40 and oC24 is predicted, with the former being strong enough for experimental detection within the spectrum up to 10 eV. The important role of interstitial localized electrons is highlighted, revealing diversity and rich physics in electrides.


Justification to the computational method
This work (calc.) Rasigni et al. 1977 This work (calc.) Rasigni et al. 1977  The validation and accuracy of the method for optical calculations used in this work has been carefully checked with the bcc structure of lithium at ambient conditions. The plasmon frequency of 7.0 eV for bcc lithium by our calculation is in good agreement with the experimental result of 6.73 eV 2 . The calculated energy loss function also matches the experimental spectrum very well, with a main peak of 7.0 eV against the measured 6.9 eV, as shown in Fig. s2(b). The calculated refractive index compares to the experimental data in Fig.2(a). The peaks in our results are slightly shifted to high frequency, but the overall feature agrees with the experimental spectrum. These lend us the confidence that our results are robust and quantitatively accurate.
Still note that the accuracy of optical calculations is in different levels with calculations of total energy. They should be considered as semi-quantitative, especially for the static electric resistivity. The exactly accurate calculation of static optical resistivity needs considerably dense k-points and huge supercells, so that numerous data in extremely low frequency regime could deduce resistivity for ω → 0. But this extremely high level of exactness is not required in this work. The data in Figure  11 of the main text is to show the metal-semiconductor-metal transition. We cannot determine whether the metallization of oC24 increases at higher pressure only from the subtle difference of resistivity between 115 GPa and 200 GPa.
In addition, VASP Li pseudopotential needs to be carefully checked. That's why we only consider the pressure range only up to 200 GPa. The latest PAW-PBE VASP pseudopotential of sv_GW with a core radius of 1.500 Bohr was used in this work. We checked its accuracy in a structure of P4132 compared to the results obtained by all-electron full potential augment planewave (FP-LAPW) 3 , as Table s1 shows. The pseudopotential of sv_GW shows reliable enough accuracy in calculation of volume of atoms at extreme pressures. The calculation of total energy is certainly affected by the pseudopotential. But the difference is evident only above 200 GPa (see Fig. 1(d) of Reference 3 , where "VASP PAW PBE" means the old pseudopotential). The new sv_GW used in this work has better results. The electronic structure was also checked, in agreement with results by FP-LAPW method. In addition, our electronic structure calculation totally coincides with the results by Marques et al. 4 . Therefore, we believe the pseudopotential we used in this work is accurate enough s3 for our optical research up to 200 GPa.  The absence of the first peak in Reference 5 (see Fig. 2 of the main text) might be due to the premature cutoff of the extremely low-energy excitations (below 0.5 eV) around the Γ point, whose transition probability, however, is much larger than the high frequency ones. An artificial suppression up to 0.5 eV is evident in Reference 5 . The very close bands near the Fermi surfaces might have been treated as fully degenerate in Alonso's calculations, so their contribution to the interband part below 0.5 eV was ignored.     Insulating oC40 and semi-metallic cI16 have totally different symmetries, crystalline and electronic structures, and bonding. It's easy to understand that the electronic bands of oC40 evolve differently with pressure from that of cI16, especially not forming evident degenerate states at Γ point. In addition, as discussions in the manuscript indicate, the cI16 structure is distorted from the bcc structure, where only a few pairs of lithium ions in cI16 are separated by a distance while many others' locations remain unchanged. These unchanged ions also keep relatively unmoved when pressure rises from 40 to 70 GPa. However, the structure of oC40 is optimized more freely and completely at high pressure. All ions will move due to less symmetry constraints when pressure changes, only the distance of very special bonding atoms/quasi-atoms will keep unchanged 7 . Thus, the modification of electronic structure in oC40 is more intuitive, in which the conduction bands not far from the Fermi level are stretched due to different ratios of 2s contribution. In cI16, due to unique structural distortion and the nesting structure, the electronic structure at Γ point alters abnormally, forming a triply-degenerate state at 70 GPa.  oC24 phase inherits a layered structure from oC40, but has a higher symmetry, with a centrosymmetric distribution of atoms within each layer. Layers A' and B' are just another layer s8 of A and B with relative shifts. oC24 has two non-equivalent interstitial electronic localization positions on layer A and B (ELF = 0.921 and 0.908 at 115 GPa), respectively. This structural feature accounts for the observed optical anisotropy.

Extra material for oC40
The interband contribution to DF from electronic transitions with light polarization along axis in the low energy region is much larger than along the other two directions, though this anisotropy disappears gradually under further compression. By the analysis in the main text, the first peak in the imaginary DF occurs within the cone around Γ point and this anisotropy represents an uneven feature of the cone structure. The actual momenta of the second peak differ in the three directions. It is at 3.2 eV along axis, at 2.5 eV along axis, and with two smaller peaks replacing the main one along axis at 2.7 and 3.4 eV, respectively. However, the anisotropy in reflectivity is weak in infrared and visible light regime. But it becomes noticeable from 3 to 12 eV, as can be seen in Fig. s10. The reflectivity at 4 eV along axis is about 0.1 higher than that along and axis, which becomes much more disparate at 7 eV and 10 eV. The local details of electronic structure of oC24 near the Fermi surface actually changes subtly with pressure, as shown in FIG 9 and FIG S11. Its cone feature has more area across the Fermi surface and the corresponding energy bands around Γ point also moves relatively. Although the metallization of oC24 could change subtly, it remains weakly metallic up to 200 GPa.    Table s3 is to aid us understand the electron distribution in the electride phases of lithium. The occupation of interstitial space with high ELF is calculated to represent the degree of the electron localization. The localization is greatly enhanced for cI16 from 40 to 70 GPa, as the occupation volume with ELF greater than 0.9 increases 5 times compared to other two phases. This striking change in the electron distribution results in a structural distortion of cI16, as discussed in the main text. The localization of oC24 is weakened at high pressure, which shows a weak metallic behavior.

Extra discussion
The 2p contribution to the orbitals around ions certainly increases with pressure from 40 to 200 GPa and it plays a significant role in dense lithium. With the increase of 2p contribution, the bonding in cI16 becomes more saturated, allowing more valence electrons to localize at interstitial space. Thus, the structure of cI16 is gradually distorted with the increment of the internal coordinate x, creating more room in the interstitial region. Due to the nesting structure in cI16, there exists an energy gap between the highest not fully occupied valence band and the lowest conduction band. The gap becomes bigger with pressure and finally the structure stabilized by the bonding is not stable anymore. The low-symmetric insulating oC40 has a lower enthalpy above 66 GPa due to the saturated multi-centered chemical bonding and the bonding between interstitial electrons of E1 [7][8][9] . Under further compression the structure alters subtly, and the energy gap becomes larger, because the 2s contribution at Γ-Z, Z-T, and T-Y makes the lowest conduction bands rises faster than the highest valence bands with the 2p contribution. As atomic structure becomes harder, the increasing of 2p contribution becomes slower and it is not the main causes of these structural changes. The repulsion of interstitial electrons is more evident, making the energy reach the value for bond cleavage. Some interstitial electrons tend to be free. As a result, dense lithium transits into the reentrant weak metallic phase oC24. It can be confirmed by that the maximum of ELF in oC40 and oC24 decreases continuously with pressure.
As Figures 1, 6, and 9 in the main text show, the major peaks in the excitation spectra are related to those excitations from the highly occupied 2p states not far below the Fermi surface to the corresponding 2s states in conduction bands. As external pressure rises, though the 2p contribution in valence bands increases, the 2s states contributes more to the changes of excitation s12 spectra, because they rise far faster than the 2p states in band structure, resulting in the movement of the excitation spectra to higher frequencies. Except this, the changes of excitation spectra in cI16 is unique compared to oC40 and oC24, due to its structural distortion as we discussed in the manuscript, which is highly related to the 2p contribution.