FULL PAPER
obtain access to information about the electronic structure
of compounds. The bandgap is a key discriminating property
for a large number of applications, including solar absorbers,
thermoelectrics, transparent conductors, contact and buffer
layers, etc. In recent works, 7,8 the bandgap has been used as
a descriptor for the effi ciency of a light absorber. In this part,
we calculate the electronic bandgaps of experimentally known
compounds. All the structures investigated here are available
in the Materials Project database 9 and have been previously
optimized using the generalized gradient approximation (GGA)
functional by Perdew, Burke, and Ernzerhof (PBE), and GGA
PBE+U for some of the structures. 18 While standard DFT usually
gives good result for the optimization of the crystal structure,
it fails in the calculation of bandgaps. 19 The Kohn-Sham
bandgaps of semiconductors, given by the minimum energy
difference between the bottom of the conduction band and the
top of the valence band, are seriously underestimated because
of the approximate description of the exchange-correlation
functionals, the self-interaction error, 20 and the missing derivative
discontinuity. 21 Many-body methods, such as the GW
approximation, give more reliable results with an increase (at
least one order of magnitude) of the computational cost. Hybrid
functionals, e.g., PBE0 or HSE06, that incorporate a portion of
Hartree-Fock exact exchange, usually give reasonable results
for semiconductors, 22 but fail for metals and wide bandgap
insulators. 23,24 All these methods are expensive to be used in
a screening project of several thousands of materials and, in
particular the GW approximation, can only be effi ciently used
to refi ne the results obtained with computationally cheaper
approximations. 25
Here, the bandgaps are calculated using the GLLB-SC functional,
16 that is an improved description of the original GLLB
functional 14 adapted for solids. The GLLB functional contains
by construction the evaluation of the derivative discontinuity. It
is a further approximation to the KLI approximation to the exact
exchange optimized effective potential (EXX-OEP). 26 The fundamental,
or quasi-particle (QP), bandgap is given as the difference
in the ionization potential (IP) and the electron affi nity
(EA) and thus directly linked to photo-emission and inverse
photo-emission measurements. The Kohn-Sham (KS) bandgap
differs from the QP gap by the derivative discontinuity, Δ xc :
E QP
= IP − EA = E KS +Δ gap gap
xc .
(1)
GW, on the other hand, gives directly QP energies. It is important
to keep in mind that the bandgaps obtained from optical
measurements can be signifi cantly lower than the QP gaps due
to excitonic effects, and one thus speaks of an optical bandgap
instead. 27
The GLLB-SC functional has been recently tested against
other computational methods (mainly non-self-consistent
G 0 W 0 ) and experiments for single metal oxides, 8 for semiconductors,
28 and for perovskite materials for light harvesting. 25
The GLLB-SC results are expected to be within an error of
0.5 eV. We thus expect that this accuracy is good enough for
projects involving thousands of calculations required in a
screening study. In addition, with the GLLB-SC is possible to
calculate larger crystal structures. For example, recently, the
GLLB-SC has been widely used to calculate the bandgaps of 240
organometal halide perovskites 29 which show very interesting
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optical properties for light harvesting and energy conversion. 30
We note that the GLLB-SC has also given good results for the
position of the d -states in noble metals such as silver. 31
The Materials Project database is constantly updated and
so far we have calculated the bandgaps of around 2400 materials.
Those materials have been selected because of their relative
simple structure, their stability and because they show
a bandgap at the GGA level. Despite its low computational
cost, the GLLB-SC functional is at least twice as expensive as
a standard GGA calculation 32 and it is demanding to calculate
the bandgaps of large crystal structures of more than 40/50
atoms. Around 6 months has been the computational time
required for the bandgap calculations for the 2400 materials
using Nifl heim, the supercomputer facility installed at DTU
Physics. On a single core machine, the time required would
have been around 16.5 years. All the calculated quasi-particle
gaps, together with the corresponding ids from the Materials
Project and ICSD databases and the chemical formula, are
listed in the Supporting Information. In addition, this information
is included and freely available in both the Materials Project
database and the Computational Materials Repository.
The distribution of the bandgaps, calculated with GLLBSC,
of the 2400 materials is shown in Figure 1 (in blue). Even
though very large bandgap insulators have been found, the
region with a large number of materials correspond to the visible
light range, between 1.5 and 3.0 eV. When the stability in
water at pH = 7 and at potential 0 V versus normal hydrogen
electrode (NHE) is considered by means of Pourbaix diagrams,
33 the number of materials that might be stable is significantly
reduced. The Pourbaix diagrams give information about
the thermodynamics of the reactions, while other factors, such
as kinetics and surfaces passivation, are not included. For these
and other reasons, here we have considered two energy thresholds
to defi ne if a material is stable (Δ E = 1 and 0 eV/atom,
shown in red and green bars in the fi gure, where Δ E is the total
energy difference between the material and the most stable
phases in which it can separate). Within the energy threshold of
1 eV/atom, more than 50% of the small bandgap semiconductors
are unstable in water, while it seems that all the materials
1400915 (2 of 7) wileyonlinelibrary.com © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Adv. Energy Mater. 2015, 5, 1400915
www.advenergymat.de
Figure 1. Histogram of the GLLB-SC bandgaps for all the 2400 calculated
materials (in blue). We consider the two energy thresholds 1 eV/atom (in
red) and 0 eV/atom (in green) for the stability in water, which is calculated
at zero potential ( U = 0 V vs NHE) and neutral pH.