Oganesson Is a Semiconductor: On the Relativistic Band‐Gap Narrowing in the Heaviest Noble‐Gas Solids

Abstract Oganesson (Og) is the most recent addition to Group 18. Investigations of its atomic electronic structure have unraveled a tremendous impact of relativistic effects, raising the question whether the heaviest noble gas lives up to its position in the periodic table. To address the issue, we explore the electronic structure of bulk Og by means of relativistic Kohn–Sham density functional theory and many‐body perturbation theory in the form of the GW method. Calculating the band structure of the noble‐gas solids from Ne to Og, we demonstrate excellent agreement for the band gaps of the experimentally known solids from Ne to Xe and provide values of 7.1 eV and 1.5 eV for the unknown solids of Rn and Og. While this is in line with periodic trends for Rn, the band gap of Og completely breaks with these trends. The surprisingly small band gap of Og moreover means that, in stark contrast to all other noble‐gas solids, the solid form of Og is a semiconductor.

In this supplementary material more information is provided about the computational details, the generation and testing of PAW potentials for Og, as well as on the calculation of the theoretical best estimates. Input and output files of all shown VASP calculations (including POTCARs) can be provided upon request from one of the authors (JMM). POTCARs are also available for download in the supplementary material of ref. 1.
Details on Employed PAW Potentials -Core electrons are modeled using the projector-augmented wave (PAW) approach of Joubert and Kresse. 2,3 The respective POTCAR files containing the PAW parameters are taken from the VASP library for He−Rn. For He−Xe, the Element GW POTCARS are used, which have 2 valence electrons (VE) for He and 8 VE for Ne. For Xe, Rn and Og, most calculations employ the intermediate Rd d GW POTCARs which additionally include the semi-core d 10 shell (18 VE). Here, we additionally consider the PAWs with the largest valence-space available (Element sv GW POTCARs) which also include the semi-core s and p shells (26 VE, 6s 2 6p 6 6d 10 7s 2 7p 6 valence space). For Og, for which no POTCAR files are available in the library, new POTCARs were devised using the same basic structure as for Rn (PAW approach, PBE all-electron calculation, 18 and 26 VE).
PAW generation and evaluation for Og -This was accomplished using the pseudo-potential (PP) generation package of VASP and the PBE functional for the atomic all-electron calculation. 4,5 Since the POTCAR generation package of VASP only allows for a scalarrelativistic treatment of the valence space, which leads to unacceptable errors for the spin-orbit splitting of the 7p shell in Og, the parameters for POTCAR were further refined to match the one-particle energies of the immediate valence space (7p, 8s) from an atomic calculation with VASP to a four-component 4c-PBE/dyall.d-aug-ae3z reference calculation conducted with DIRAC-17. 6 This was particularly successful for the 18VE PAW, which in turn provides excellent agreement with the 4c-PBE allelectron reference for the one-particle energies (absolute ∆E 7p,8s ≈ 0.1 eV, relative ∆∆E 7p−8s = 0.03 eV, cf. also Table I), as well as a reasonable agreement for the distance in the Og dimer (r AE e = 4.37Å, r 18VE e = 4.34Å). In further tests, both the 18 and 26 VE POTCARs (in combination with the SCAN, PBEsol and PBE-D3BJ functionals) were shown to accurately reproduce the structural parameters, bulk moduli and cohesive energies obtained from a many-body expansion based on relativistic coupled-cluster results. 7 Small vs large valence space PAWs -Although the small-core 26 VE PAW provides slightly improved bulk properties and a better dimer distance (r 26VE e = 4.38Å) for Og, the agreement for the one-particle energies with the all-electron reference is significantly worse. In particular the spin-orbit splitting of the 7p level, which has been found to be critical for the band gap, is too small with the 26 VE PAW, as evident from the too high energy of the 7p 1/2 level (cf.  Table II and Figure 3 main article), which becomes even worse if more bands are included (+0.8 eV for 256 bands), this leads us to the conclusion the smaller valence spaces provides more accurate band gaps for Xe−Og. Whilst this needs to be investigated further, we exclusively focus on the results obtained with the smaller valence space in the main article, and consider the large-core results here for the sake of completeness. Apart from that, the most important outcome of this investigation, namely that Rn is an insulator and Og a semiconductor, is the same for the band gaps obtained with the 26 VE PAWs. Details on GW calculations GW calculations are conducted in the quasi-particle approximation iterating energies as well as wavefunctions and including diagonal and off-diagonal elements as implemented in VASP 5.  II. Calculated and available experimental structural parameters, electronic band gaps Eg (in eV, at the Γ-point) and lowest atomic excitation energies ∆E (in eV) of the Ne−Og. Scalar-relativistic values are set in parentheses (GW and PBE only), and calculations with the larger 26 electron valence-space are marked 26VE. = Note that part of the differences between the SR and spin-orbit relativistic GW calculations for the lighter elements are due to a technical issue, rendering them too large. These differences are more accurate at the DFT/PBE level. Experimental data for electronic band gaps taken from. 8 The value for Ne varies (21.4 − 21.7 eV) depending on the source, 9,10 see ref. 11 for an overview. Atomic excitation energies ∆E (in eV) for Ne to Xe are from experiment, 12 and from FS-CCSD calculations for Rn and Og.

R0
Eg influence is somewhat larger with 0.4 eV.
In general, all calculations are conducted with a k-point grid of 6 3 , and only the final GW/PBE calculations are conducted with a finer 7 3 grid. Exploratory calculations for Og and Ne with an 8 3 grid indicate convergence with respect to the number of k-points (∆E g < 0.02 eV). Already the difference between 6 3 and 7 3 k-points is small (< 0.03 eV) for all elements but Og, for which it amounts to about 0.1 eV. The energy cutoff is set at 300 eV, where exploratory calculations with 400 eV for Rn and Og indicate convergence (∆E g < 0.01 eV). The lowest 128 bands are included for He−Xe, and the lowest 256 bands for Rn and Og. Exploring convergence with respect to the number bands with the reduced 6 3 k-point grid shows that including up to 512 bands has a notable influence of 0.1 − 0.2 eV on the band gaps of Rn and Og. However, since inclusion of this many bands with the highest k points grid leads to prohibitively expensive calculations, we include their influence in the best estimates by extrapolating to an infinite number of included bands. Theoretical best estimates for band gaps -Our theoretical best estimates constitute a compromise between the convergence of the level of theory on one hand, and on the other the observation that for the lighter congeners, not the highest but a specific level provided the best agreement with experimental data, namely GW /PBE with 128 included bands and the smaller valence space. Hence, to obtain balanced theoretical best estimates, we focus on the calculations with the smaller valence space, explicitly include the convergence of k-points and number of included bands, but set the error-bars such that the 1σ-range just includes the results obtained with 128 bands (Rn 6.64 eV, Og 1.00 eV). Doing so leads to the final estimates of 7.1 ± 0.5 eV for Rn and 1.5 ± 0.6 eV for Og.   account for QED effects . 22,23 In both cases QED effects were taken into account from a perturbative treatment at the DHF level using the self-energy effective operator of Flambaum and Ginges and the vacuum polarization from an Uehling potential plus higher order terms (Källèn-Sabry) corrections. 24 We notice that if spin-orbit coupling becomes large, many other transitions from the valence p 3/2 shell into vacant nlj shells become energetically more favorable compared to transitions out of the valence p 1/2 shell. This already happens for Rn, 12 and even more so for Og. 18