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with a gap larger that 10 eV are stable in water. Only around 4%
of the materials are stable, when the more strict threshold of
0 eV/atom is used. This may indicate that considering a Δ E
larger than zero can help to identify the materials that are
experimentally observed to be stable in water.
The electronic structures of 20 materials, randomly picked
from the calculated set to cover the full bandgap range and with
a reasonable unit cell size, were also calculated using the nonself
consistent G 0 W 0 and the eigenvalue-self-consistent GW 0
and GW as well as the HSE06 hybrid scheme ( Figure 2 ). This
unconventional set of structures contains ternary and quaternary
materials including oxides, nitrides, sulfi des, phosphates
and chlorides and thus it is a broader set compared to the ones
used elsewhere in the literature. 34
QP gaps were obtained in the G 0 W 0 approximation in a
plane wave representation using LDA wavefunctions and eigenvalues
as starting point. A detailed description of the implementation
of this method in GPAW can be found in ref. 28 .
The initial Kohn-Sham states and energies were calculated in a
plane wave basis with kinetic energies up to 600 eV. The same
value is used for determining the exact exchange contributions.
The G 0 W 0 self-energy was carefully converged with respect to
k points, number of bands and plane wave cutoff energy for
each material individually. Typically, a (7 × 7 × 7) k -point sampling,
100–200 eV energy cutoff and unoccupied bands up to
the same energy (a few hundred bands in total) were found to
be suffi cient in order to converge band gaps within less than
0.1 eV. Both, the plasmon pole approximation (PPA) by Godby
and Needs 35 and the explicit frequency dependence of the dielectric
function, ε (ω ), have been used, yielding almost identical
results.
It is well known that G 0 W 0 underestimates bandgaps compared
to experiments and better results can be obtained using
(partial) self-consistent GW 34 where the LDA wavefunctions
are kept fi xed while the eigenvalues are updated self-consistently.
Recently, 28 it was shown for a set of well known semiconductors
and insulators, that the MAEs for GLLB-SC and
G 0 W 0 with respect to experiments are 0.4 and 0.3 eV, respectively,
with a tendency of the former to overestimate the gaps,
while the latter underestimates them.
Two levels of (partial) self-consistency have been investigated:
i) in the case of GW 0 , the self-consistency in the eigenvalues
is performed for the Green’s functions (G) only, whereas
ii) in the case of GW, the eigenvalues are updated both in G
and in the dielectric matrix of the screened potential ( W ). In
general, for the 20 materials described in this section, three or
four iterations are necessary to converge band gaps within less
than 30 meV and 50 meV for GW 0 and GW, respectively. Due
to the high computational costs, the k-point mesh and energy
cutoff used for GW 0 and GW are coarser than the ones used
for G 0 W 0 . Typically the low convergence criteria of (3 × 3 × 3)
k -point sampling and 100 eV energy cutoff are used for GW 0
and GW. The band gaps are then extrapolated to the dense
k-point grids and high plane wave cut off, using the difference
between the low and high convergence parameters in the G 0 W 0
calculations. For more details about GW 0 and GW, see ref. 34
and references therein.
Hybrid functional based calculations were performed with
the range-separated screened-exchange HSE06 functional. 36,37
The wavefunctions were expanded in a plane-wave basis with
a 700 eV cutoff. We use a Monkhorst-Pack 38 grid of 33 × ( a x −1 ,
a y −1 , a z −1 ) k -points, where a x , a y and a z are the lattice constants
in x , y and z direction, respectively, and the Γ-point is always
included. In the current work, all HSE06 calculations were performed
non-self-consistently from the PBE ground state density
and wavefunctions. Generally, there is good agreement between
the non-self-consistent calculations and the self-consistently
obtained results 24 which indicates that self-consistency will not
be important in the current work.
For all materials in this study, comparison between the different
methods is shown by means of the direct Γ point gap, in
order to avoid the need for interpolation of the band structure
in the case that the minimum of the conduction band is not
located at a high symmetry point in the Brillouin zone.
Figure 2 shows the bandgaps for the 20 selected materials
calculated with LDA, different levels of the GW approximation,
HSE06, and GLLB-SC. Only a few experimental data points are
available, and mainly optical measurements which are therefore
not directly comparable with our values. Ideally photoemission
and inverse photoemission measurements could be used
to compare to our calculated bandgaps, but these are not available
for this set of structures.
It is natural to divide the 20 materials into small and wide
bandgap semiconductors to give a better evaluation of the
signed and mean absolute and relative errors 39 for the different
methods studied here ( Table 1 for the small gap set). Similar
data for the wide gaps is reported in the Supporting Information
together with the comparison of band structures calculated
with different methods for two compounds.
As expected, for both the groups, LDA seriously underestimates
the bandgaps. The mean absolute error (MAE) of GLLBSC
with respect to G 0 W 0 and to HSE06 is larger than 0.5 eV for
the small bandgaps with a clear tendency for GLLB-SC to overestimate
the bandgaps with respect to HSE06 and to G 0 W 0 as
shown by the signed error and Figure 3 a,b. G 0 W 0 and HSE06
are very close, with a MAE of approximately 0.25 eV (G 0 W 0
© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim wileyonlinelibrary.com (3 of 7) 1400915
Adv. Energy Mater. 2015, 5, 1400915
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Figure 2. Bandgaps at Γ-point of 20 structure calculated with LDA (in
black), GW approximations with PPA (G 0 W 0 in red, GW 0 in purple and
GW in orange), GLLB-SC (in blue), and HSE06 (in green). Both the KS
bandgap and the derivative discontinuity are indicated for the GLLBSC
gaps. The materials for which the Γ-point gap corresponds to the
bandgap, are indicated with *.