Band convergence and linearization error correction of all-electronGWcalculations: The extreme case of zinc oxide

Abstract

Recently, Shih et al. [Phys. Rev. Lett. 105, 146401 (2010)] published a theoretical band gap for wurtzite ZnO, calculated with the non-self-consistent GW approximation, that agreed surprisingly well with experiment while deviating strongly from previous studies. They showed that a very large number of empty bands is necessary to converge the gap. We reexamine the GW calculation with the full-potential linearized augmented-plane-wave method and find that even with 3000 bands the band gap is not completely converged. A hyperbolical fit is used to extrapolate to infinite bands. Furthermore, we eliminate the linearization error for high-lying states with local orbitals. In fact, our calculated band gap is considerably larger than in previous studies, but somewhat (4)] is constructed from the Kohn-Sham Green function taken from a densityfunctional theory calculation with the local-density approximation (LDA) for the exchange-correlation energy functional. The quasiparticle energy E σ nq with band index n, Bloch vector q, and spin σ is then obtained from the nonlinear equation 5-8 invoking the one-shot LDA + GW approach showed that the band gap of wurtzite ZnO is underestimated with respect to the experimental value by more than 1 eV. They fall in the range 2.12-2.6 eV while the experimental gap amounts to 3.6 eV, 9 after correction for vibrational effects. This large underestimation is untypical for GW calculations of sp-bound systems.The Letter of Shih et al. addressed two issues: first, the erroneous hybridization effects between Zn 3d and O 2p states that results from the self-interaction error within the LDA, 10 and second, the band convergence in the correlation part of the self-energy. The first problem was tackled with the LDA + U approach, 11 in which an orbital-dependent potential corrects the position of the 3d bands and, thus, reduces hybridization effects with the O 2p states. However, the combination LDA + U and GW yields a band gap that is still well below the experimental value. Therefore, the authors investigated the second issue by carefully converging the correlation self-energy and the dielectric matrix with respect to the number of bands. They performed calculations with up to 3000 bands corresponding to a maximal band energy of 67 Ry as well as a cutoff for the dielectric matrix of up to 80 Ry and showed that the resulting GW band gaps, 3.4 eV for LDA + GW and 3.6 eV for LDA + U + GW , turned out to be in very good agreement with experiment. They also demonstrated that a too small energy cutoff for the dielectric matrix can lead to a false convergence behavior: the band gap seems to converge with many fewer bands, but toward a value that is too small.These new results for ZnO are in striking contrast to previous studies. If they are correct, they cast doubt on all GW calculations published so far, not only for ZnO but also for other materials, especially for systems with localized states. In fact, Shih et al. point out at the end of their paper that "many of the previous quasiparticle calculat...

Self-energy calculation of the hydrogen atom: Importance of the unbound states

“…It can be seen that, as l max goes from l = 0 to l = 1, the error of the self-energy or the quasiparticle energy drastically reduces by around 0.6 eV, and by increasing l max , the self-energy decreases monotonically. The self-energy converges slowly with respect to l max , which is reminiscent of a similar problem in GW calculations associated with summing over unoccupied states [11][12][13]. Our estimated value for l max → ∞ by extrapolation is 0.02 ± 0.01 eV.…”

“…One possible reason for this failure could be that, in the G 0 W 0 approximation or RPA, the polarization function contains the unphysical self-screening effect where an electron shields the field produced by itself [10]. In order to improve the theory, careful analysis is required to identify the shortcomings of the GW approximation, but because of the complexity of the GW calculation due to the nonlocality and frequency dependence of the self-energy, it is usually difficult to analyze the self-energy in detail and sometimes it is even more difficult to get converged results [11][12][13]. The hydrogen atom is one of the few real systems where the exact eigenfunctions and energies are known analytically, and it is an ideal system for studying the self-screening problem because, for the 1s state, the error arises entirely from the correlation part of the self-energy.…”

Many-body electronic structure calculations of Eu-doped ZnO

“…It has been shown that a large number of bands is necessary to obtain properly converged results in earlier one-shot G 0 W 0 calculations [51][52][53]. We speculate that the deviations to Ref.…”

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“…Depending on the starting functional and the details of the calculation, values between 2.1 and 3.6 eV are obtained for G 0 W 0 [24,31,39,40,[51][52][53][54], between 2.54-3.6 for GW0 [24,39,[54][55][56] and between 3.2-4.3 for GW [24,37].…”