Quasiparticle GW calculations for solids, molecules, and two-dimensional materials

We present a plane-wave implementation of the G${}_0$W${}_0$ approximation within the projector augmented wave method code gpaw. The computed band gaps of ten bulk semiconductors and insulators deviate on average by $0.2\mathrm{eV}$ ($\sim 5\%$) from the experimental values, the only exception being ZnO where the calculated band gap is around $1\mathrm{eV}$ too low. Similar relative deviations are found for the ionization potentials of a test set of 32 small molecules. The importance of substrate screening for a correct description of quasiparticle energies and Fermi velocities in supported two-dimensional (2D) materials is illustrated by the case of graphene/$h$-BN interfaces. Due to the long-range Coulomb interaction between periodically repeated images, the use of a truncated interaction is found to be essential for obtaining converged results for 2D materials. For all systems studied, a plasmon-pole approximation is found to reproduce the full frequency results to within $0.2\mathrm{eV}$ with a significant gain in computational speed. Throughout, we compare the G${}_0$W${}_0$ results with different exact exchange-based approximations. For completeness, we provide a mathematically rigorous and physically transparent introduction to the notion of quasiparticle states.

Figure 1

Convergence of the band gap of zinc oxide for G${}_0$W${}_0$@LDA with the plasmon-pole approximation. The number of bands is chosen equally to the number of plane waves corresponding to the respective cutoff energy; for example $300\mathrm{eV}$ corresponds to $\sim 1100$ plane waves and bands.

Figure 2

Comparison of calculated and experimental band gaps for the solids listed in Table . The numerical values are listed in Table . A logarithmic scale is used for better visualization. G${}_0$W${}_0$@LDA refers to the fully frequency-dependent nonselfconsistent GW based on LDA. The PBE0 results are obtained nonselfconsistently using LDA orbitals.

Figure 3

Band structure of diamond calculated with LDA (black) and G${}_0$W${}_0$ (red). The bands have been interpolated by splines from a ($15\times 15\times 15$) $k$-point sampling. The band gap is indirect between the $\Gamma$ point and close to the X point with a value of $4.12\mathrm{eV}$ and $5.66\mathrm{eV}$ for LDA and G${}_0$W${}_0$, respectively.

Figure 4

Band structure of fcc gold calculated with LDA (black lines) and G${}_0$W${}_0$@LDA with PPA (red dots). ($45\times 45\times 45$) and ($15\times 15\times 15$) $k$ points have been used for LDA and GW, respectively. The bands are aligned to the respective Fermi level.

Figure 5

Schematic picture of the N-graphene/$h$-BN interface.

Figure 6

Direct G${}_0$W${}_0$ band gap at the $\mathrm{K}$ point for a freestanding $h$-BN sheet as function of the vacuum used to separate layers in neighboring supercell with and without use of the Coulomb truncation method as described in Sec. 3e.

Figure 7

Band structure for a freestanding $h$-BN sheet. The band gap is direct at the $\mathrm{K}$ point with LDA ($4.57\mathrm{eV}$) and GLLBSC ($7.94\mathrm{eV}$) and changes to indirect between the $\mathrm{K}$ and the $\Gamma$ point for G${}_0$W${}_0$ ($7.37\mathrm{eV}$).

Figure 8

LDA and G${}_0$W${}_0$@LDA band structure for a graphene/boron nitride double layer structure. Only the two highest valence bands and the two lowest conduction bands are shown.

Figure 9

The band gap of $h$-BN at the $\mathrm{K}$ point as function of the distance to the graphene sheet (see inset). Dashed horizontal lines indicate the values for the freestanding $h$-BN, corresponding to $d\to \infty$.

Figure 10

$h$-BN gap at the $\mathrm{K}$ point for different number of adsorbed graphene layers. GLLBSC results are plotted without and with the derivative discontinuity ${\Delta }_{xc}$.

Figure 11

Convergence of the Ionization Potential for H${}_2$O with respect to the plane wave cutoff for G${}_0$W${}_0$@LDA. The dashed line shows a linear fit of the points with $E_{\mathrm{cut}}>100\mathrm{eV}$ ($1/E_{\mathrm{cut}}<0.01{\mathrm{eV}}^{-1}$). The IP is given as the negative HOMO energy.

Figure 12

Comparison of theoretical and experimental ionization potentials. The G${}_0$W${}_0$ results are obtained by applying the extrapolation scheme as explained in the text. Corresponding values are listed in Table .

Figure 13

Deviations for the ionization potentials obtained with G${}_0$W${}_0$@LDA compared to (a) Ref. 27 and (b) Ref. 28. The mean deviations are $0.02$ and $0.30\mathrm{eV}$, respectively.