Valence and conduction band tuning in halide
perovskites for solar cell applications†
Simone Meloni,‡
a
Giulia Palermo,§
a
Negar Ashari-Astani,{
a
Michael Gr
¨
atzel
b
and Ursula Rothlisberger
*
a
We performed density functional calculations aimed at identifying the atomistic and electronic structure
origin of the valence and conduction band, and band gap tunability of halide perovskites ABX
3
upon
variations of the monovalent and bivalent cations A and B and the halide anion X. We found that the two
key ingredients are the overlap between atomic orbitals of the bivalent cation and the halide anion, and
the electronic charge on the metal center. In particular, lower gaps are associated with higher negative
antibonding overlap of the states at the valence band maximum (VBM), and higher charge on the
bivalent cation in the states at the conduction band minimum (CBM). Both VBM orbital overlap and CBM
charge on the metal ion can be tuned over a wide range by changes in the chemical nature of A, B and
X, as well as by variations of the crystal structure. On the basis of our results, we provide some practical
rules to tune the valence band maximum, respectively the conduction band minimum, and thus the
band gap in this class of materials.
1 Introduction
Mixed organic/inorganic and fully inorganic halide perovskites
of the formula ABX
3
(A ¼ organic or inorganic monovalent
cation, B ¼ bivalent cation, X ¼ halogen anion) (see Fig. 1), have
recently emerged as highly efficient light harvesting and charge
carrier transport materials for solar cells.
1–3
Power Conversion
Efficiencies (PCE), i.e. the percentage of solar light converted
into electric current, as high as a certied 22.1% have been
reported for perovskite based solar cells.
4
Several strategies have been attempted to increase the PCE of
halide perovskites, some targeting the quality of perovskite
lms, e.g. its uniformity
1
and defect passivation,
5
others
focusing on the intrinsic optical and electronic properties, such
as tuning of the band gap, E
g
.
6–10
In the latter case, the objective
is obtaining a compound with a band gap close to the ideal
value of 1.3 eV.
11
Seok and coworkers have shown that in mixed
I/Br perovskites, ABX
3x
Br
x
, E
g
can be tuned by changing the
halide composition, x.
7
In particular, in methyl ammonium lead
iodide/bromide perovskites, MAPbI
3x
Br
x
, (MA ¼ methyl
ammonium), the band gap widens with increasing x. It has also
been shown that
12
employing larger cations, which favor a more
cubic-like structure, the band gap can be reduced. Different
hypotheses have been put forward trying to explain the origin of
band gap variation. Amat et al.
12
suggested the structural vari-
ations of the PbI framework (e.g. the X–B–X angle) as the main
parameter determining the band gap and in particular justied
the effect of larger cations with lower octahedral tilting angles.
Filip et al.
13
further investigated this idea and studied the band
gap dependence as a function of the orientation of PbI
6
octa-
hedra in a Platonic orthorhombic PbI
3
perovskite, i.e. an
orthorhombic lead iodide perovskite in which the monovalent
cations are replaced by a background charge. They have shown
that rotating the octahedra around the equatorial and apical
bond angles in the domain consistent with the other structural
constraints it is possible to change the band gap in the range
1.1–1.9 eV. They have also shown that by changing the mono-
valent cation one can tune the orientation of PbI
6
octahedra,
and through them the band gap. Kanatzidis and coworkers
14
studied the dependence of the valence band maximum (VBM)
and conduction band minimum (CBM) as a function of chem-
ical composition, and concluded that in MASnI
3x
Br
x
the
widening of the band gap with the amount of Br is due to an
increase in CBM rather than a lowering of VBM. Snaith and
coworkers conrmed the ndings of Seok and coworkers,
7
and
showed that a correlation exists between E
g
and the pseudo-
cubic lattice parameter of the crystalline sample.
9
This relation
inspired them to use the size of the monovalent cation, A, as
a
´
Ecole Polytechnique F
´
ed
´
erale de Lausanne, Laboratory of Computational Chemistry
and Biochemistry (LCBC), Lausanne, CH-1015, Switzerland. E-mail: ursula.
roethlisberger@ep.ch; Fax: +41 21 69 30320; Tel: +41 21 69 30321; +41 21 69 30325
b
´
Ecole Polytechnique F
´
ed
´
erale de Lausanne, Laboratory of Photonics and Interfaces
(LPI), Lausanne, CH-1015, Switzerland
† Electronic supplementary information (ESI) available. See DOI:
10.1039/c6ta04949d
‡ Present address: Sapienza University of Rome, Dept. of Mechanical and
Aerospace Engineering, via Eudossiana 18, 00184, Roma, Italy.
§ Present address: Dept. of Chemistry & Biochemistry, Center for Theoretical
Biological Physics, Dept. of Pharmacology, Howard Hughes Medical Institute,
University of California San Diego, La Jolla, CA 92093-0365, USA.
{ Present address: Sharif University of Technology, Dept. of Physics, Tehran, Iran.
Cite this: J. Mater. Chem. A,2016,4,
15997
Received 13th June 2016
Accepted 24th August 2016
DOI: 10.1039/c6ta04949d
www.rsc.org/MaterialsA
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a tuning parameter for changing E
g
. An analogous strategy has
been developed in parallel by Gr
¨
atzel and coworkers,
3,15,16
who
also explored the effect of using a mixture of MA and for-
mamidinium (FA) cations. While each of the aforementioned
models is valid in the scope of the specic system under study,
generalizations and extensions to other compounds are not so
straight forward. For example, the conclusions of Amat et al.,
12
are somewhat counterintuitive in the light of the results of
Snaith and coworkers,
9
where it is shown that FASnBr
3
, which
has a cubic structure and smaller lattice constant, has a larger
band gap than the tetragonal FASnI
3
. This apparently conict-
ing results suggest that there is a signicant interplay between
changes in the chemical composition and variations of the
structure of the perovskite, and the combined effect of these two
parameters on the band gap is non trivial.
The intricacy of the relation between chemical composition
(beyond the substitution of monovalent cations), crystal struc-
ture and the electronic properties of perovskites calls for
a comprehensive theory. Such a theory should aim at rational-
izing experimental and computational results in terms of the
effect of all essential variables, such as chemical composition,
lattice size, as well as distortions in the form of tilting angles, on
the electronic structure, and thus on the optical properties of
perovskites. This is the objective of the present work. We
consider a wide range of perovskites, differing in the chemical
nature of the bivalent (Pb and Sn) and monovalent (Na, Li and
Cs) cations, halogen composition (I, Br, Cl), and crystal
symmetry (cubic, tetragonal and orthorhombic). We also
considered lead-iodide perovskites with A ¼NH
4
+
versus PH
4
+
to
investigate the speciceffects of hydrogen bonding on the
electronic structure.
To this end, we performed density functional theory band
structure calculations and analyzed the nature of VBM and CBM
in terms of atomic orbitals. Interestingly, the character (i.e. the
atomic orbital composition) of the band edges (VBM and CBM)
remains largely unchanged for this wide range of different
chemical and crystallographic variations. This observation
suggested the use of an orbital-based parameter as a central
descriptor to predict band gap variations. In fact, we show here
how the effects of chemical composition and crystal structure
on the bang gap can all be explained using such a parameter.
Finally, as a rule of thumb, we show that strategies for tuning
VBM and CBM energy levels, and the band gap must be crystal
symmetry-dependent since cubic and tetragonal/orthorhombic
structures follow different design guidelines.
2 Results and discussion
2.1 Variations in the crystal structure
We have performed DFT calculations for all chemical compo-
sitions (A ¼ Na, Li and Cs, B ¼ Pb and Sn and X ¼ I, Br, Cl) in
three perovskite phases. The resulting optimized cell parame-
ters show that the chemical nature of halides and bivalent
cations affects mainly the lattice constants (Fig. 1D), as expected
on the basis of the well-established empirical relations between
the ionic radii and the perovskite lattice size.
17
The effect of
substituting the monovalent cation, on the other hand, depends
on the crystalline symmetry of the reference system. If the
system is cubic, the replacement of the original cation results in
a corresponding change of the lattice parameter of the crystal: if
the new cation is larger the lattice expands, if it is smaller the
lattice shrinks (Fig. 1E). While in the cubic case the value of the
tilting angles q
1
, q
2
and q
3
, which measure the relative rotations
of BX
6
octahedra around the three main axes, are xed to zero
(see Fig. 1), in the tetragonal and orthorhombic phases, the
substitution of A can alter both the size of the lattice and the
tilting angles. The effect of substitution on the lattice size is
analogous to the cubic case: larger cations expand the lattice,
and vice versa. The tilting angle(s), instead, increases if the new
cation is smaller, and decreases if it is larger (see Fig. 1G and E).
This affects the linearity of the B–X–B angles: the more the
structure is tilted, the more the B–X–B angles depart from the
180
value of the cubic structures. We observed values of q in the
range 7–19.5
for the non-zero tilting angles in the tetragonal
structures, and 6.5–23
in the orthorhombic ones (where all
tilting angles are non zero). Monovalent cations that form
hydrogen bonds with the halides of the framework are a special
Fig. 1 Configurations of the equilibrium structure of several perov-
skites. Images are ordered in such a way that the corresponding
structures can be thought of as a series of structural or alchemical
alterations starting from the CsPbI
3
cubic structure (A), with a lattice
parameter a ¼6.38
˚
A as a reference. In panels (B) and (C) the tetragonal
and orthorhombic structures of CsPbI
3
are shown, respectively. They
can be obtained from the cubic analogue by tilting the PbI
6
octahedra
along their axis parallel to the tetragonal axis (tetragonal structure) or
along all of their three axes (orthorhombic). Tetragonal CsSnI
3
is
characterized by a tilting angle q
1
¼ 14.3
and a pseudocubic lattice
parameter a* ¼
ffiffiffiffi
V
3
p
¼ 6:12
A (where V is the volume of the unit cell). In
the orthorhombic CsSnI
3
, q
1
q
2
q
3
¼ 8.5
and a* ¼ 6.30
˚
A. (D) is
obtained from the cubic CsPbI
3
by replacing Pb with Sn and I with Cl (q
1
¼ q
3
¼ q
3
¼ 0
and a ¼ 5.85
˚
A). (E–G) are obtained from the cubic,
tetrahedral and orthorhombic CsPbI
3
, respectively, by replacing Cs
with Li. The corresponding tilting angle and pseudocubic lattice
parameters are: (E) a ¼ 6.32
˚
A; (F) q
1
¼ 20.3
, a ¼ 6.12
˚
A; (G) q
1
¼ 20.5
,
q
2
¼ 18.8
, q
3
¼ 10.8
, a ¼ 5.81
˚
A. (H) and (I) are obtained by replacing
Cs
+
in the cubic CsPbI
3
with NH
4
+
and PH
4
+
, respectively. While
PH
4
PbI
3
preserves the original cubic symmetry (a ¼ 6.32
˚
A), NH
4
PbI
3
is
triclinic (a ¼ b ¼85.8
, g ¼ 87.1
, a ¼ 6.38
˚
A) and the PbI
6
octahedra are
highly distorted.
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case. Depending on the strength of the hydrogen bond, the
cubic structure may or may not be a (meta)stable form. In fact,
the structure obtained by replacing Cs
+
with NH
4
+
in the cubic
CsPbI
3
turns out to be triclinic, with crystallographic angles a ¼
b ¼ 85.8
and g ¼ 87.1
. Moreover, strong hydrogen bonds
induce deformations of the BX
6
polyhedra (Fig. 1H). If Cs
+
instead is replaced by a weaker hydrogen bond donor cation,
e.g. PH
4
+
, a stable cubic structure is obtained (Fig. 1I).
2.2 Variations in the electronic structure
To understand the origin of the experimentally observed band
gap variations, we have characterized the inuence of the
chemical nature of A, B and X, and the crystalline symmetry on
the VBM and CBM. It is worth remarking that simple hypoth-
eses to explain the observed phenomenology, e.g. that the
energy of VBM and CBM changes because the energy of the
corresponding atomic orbitals change with the nature of the
halide, turn out to be inadequate. Our calculations show that
the difference between the VBM and CBM energy of perovskites
formed by different halides change in a range of 2 eV (see
Fig. S1†), and sometimes have an opposite sign with respect to
the difference of energy between p atomic orbitals of the
different halides (DE
I/Br
DE
Br/Cl
¼ 0.5–0.6 eV). Thus, it is clear
that other effects, discussed in the following, are responsible for
the high tunability of the valence and conduction band energy,
and, thus, the band gap of halide perovskites.
An interesting feature of all perovskites considered in this
work is the universality of the electronic characteristics of their
VBM and CBM. In fact, in all systems, VBM is formed by an
antibonding combination of B ns and X mp orbitals, namely
Sn-5s or Pb-6p, and Cl-3p, Br-4p or I-5p (see Fig. 2), also in
agreement with previous ndings.
18,19
This orbital has a high
covalent character, with a typical B/X atomic orbital contribu-
tion of 30–40/70–60%, depending on the chemical nature of A,
B and X, and the crystal symmetry. CBM is also characterized by
an antibonding combination, this time by B np and X ms
orbitals (see also ref. 20). In contrast to VBM, the character of
CBM is more ionic, with B orbitals contributing 70–90%.
Therefore it seems that an atomic orbital based model could
provide a universal framework to describe the band gap varia-
tions. In fact concerning the VBM level, the effect of changing
symmetry, from cubic to tetragonal to orthorhombic, is
reducing the overlap between B ns and X mp orbitals (see
Fig. 2A–C). In the cubic structure X mp orbitals are aligned
along the B–X bond. This alignment is worse in the other two
structures; in particular it decreases along the series cubic /
tetragonal / orthorhombic. As for monovalent cation substi-
tution, an effect is observed that depends on the symmetry of
the crystal. In cubic systems, in which the size of A affects only
the size of the lattice, we observe an increase or decrease of the
(negative) B-ns/X-mp overlap with the size of the cation. In
tetragonal and orthorhombic crystals, in which the size of A
affects both the lattice size and the tilting angles, two competing
effects are present: increase or decrease of the B-ns/X-mp overlap
due to (i) size of the lattice and (ii) the extent of tilting. The
overlap increases with the shrinking of the lattice, like in the
case of cubic systems, and decreases with the increase of q.
Calculations show that, typically, the second effect dominates,
and the overlap is reduced in systems with smaller A. The
chemical nature of A can affect VBM orbitals also in a more
direct way, via polarization. This occurs when A can establish
a relative strong bond with the halide of the framework, like for
instance hydrogen bonds (Fig. 2D). The effect is twofold in this
case. First, because the structure is highly distorted, the align-
ment of X-mp orbitals with B–X bonds is very poor. Second, the
electron density on the X ion hydrogen bonded to A is higher,
with corresponding charge depletion on the adjacent B cation
and the remaining X ions. Both effects cooperate in reducing the
antibonding B-ns/X-mp overlap. Finally, concerning the effects
of halide substitution, the change of the chemical nature of this
species affects mainly the geometry of the system by shrinking/
expanding the lattice. However, the size of the p orbitals of X do
not shrink/expand by the same amount, thus producing
a change in the antibonding overlap with the halide composition
of the perovskite.
Fig. 2 VBM (blue frame) and CBM (red frame) orbitals of selected
systems. Panels (A)–(C) shows how the tilting of BX
6
octahedra affects
the overlap: in the cubic structure (A) X mp orbitals are aligned with
B–X bonds and the absolute value of the negative overlap is maximal
(O
VBM
¼0.21). In tetrahedral (B) and orthorhombic (C) structures p
orbitals are no longer well aligned, and the overlap diminishes (tetra-
hedral O
VBM
¼0.17, and DE
VBM
¼0.29 eV is the energy variation
with respect to the cubic case; orthorhombic O
VBM
¼0.11, DE
VBM
¼
0.54 eV). In the case of orthorhombic structures, the tilting along all
the three axes of the octahedra further reduces the overlap, as shown
from the lateral view of the crystal shown in the inset of panel (C). (D)
Monovalent cations forming hydrogen bonds (green dashed lines)
distort the framework reducing the overlap in analogy to the variations
induced by tilting. In addition, hydrogen bonds polarize the VBM
orbitals, resulting in an increase of electronic density on hydrogen
bonded halide ions (red arrow), and a complementary reduction on the
other anions (green arrow) and on bivalent cations. (E) and (F) CBM
orbital in tetragonal CsPbI
3
and CsPbCl
3
. The effect of halide substi-
tution along the series I
,Br
,Cl
is moving the CBM electronic
charge from B to X. For example, q
B
¼ 0.81 and E
CBM
¼ 0.76 eV in
CsPbI
3
, and q
B
¼ 0.86 and E
CBM
¼ 0.34 eV in CsPbI
3
. A similar effect is
produced by the change of crystal structure along the series cubic to
tetragonal to orthorhombic, and with different monovalent cations.
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In summary, for what concerns the tunability of the VMB, all
the changes induced by variations in chemical composition and
crystal structure can be rationalized in terms of the overlap
betweentheBnsandXmpatomicorbitalsformingthisstate.
According to a tight binding formulation, the energy of a crystal
orbital depends on the overlap among the relevant atomic orbitals.
In particular, the larger is the negative (antibonding) overlap the
higher is the energy of the state. This argument, together with the
above analysis, suggests that the key observable correlating with
E
VBM
is the orbital overlap; A, B and X substitutions, and the crystal
symmetry are all effective ways to alter the B ns/X mp overlap.
To validate this idea we computed the overall overlap in the
VBM orbital, O
VBM
¼ Re
X
i˛X-mp
X
j˛B-ms
c
*
VBM;i
c
VBM;j
O
ij
!
,with
O
ij
¼
ð
drF
*
i
ðrÞF
j
ðrÞ being the overlap between pairs of X and B
atomic orbitals, and c
j
and c
j
projection coefficients of crystal
orbitals onto the atomic ones. In Fig. 3, the correlation between
O
VBM
and E
VBM
is shown. The tting of this correlation with
a linear function is very good for Sn-based perovskites (regression
coefficient R
2
0.93), while it is somewhat poorer for Pb-based
systems (R
2
0.52). However, for Pb-perovskites of single halides
the correlation is once again linear (R
2
$ 0.85). The VBM energy of
single halide Pb-perovskites follows the order E
Cl
VBM
< E
Br
VBM
<
E
I
VBM
. This small but nonnegligible effect is due to the energy
variations of X np orbitals, which grows along the series just
introduced. The chemical nature of the halide is more important
in the case of Pb perovskites because the VBM orbital is more ionic
than in corresponding Sn-based systems. We also notice that the
slope of the linear tting is high in both Sn and Pb-based perov-
skites, indicating a high correlation between O
VBM
vs. E
g
.This
conrms our analysis, that all the modications considered in
experiments affect the band gap via the B ms and X np overlap.
The above argument can be also invoked to explain the
dependence of the CBM on chemical composition and crystal
symmetry. Also in this case, O
CBM
vs. E
CBM
data show the ex-
pected trend of a decrease of the energy of the CBM with the
overlap. However, data are more scattered and the slope is
lower, suggesting a poorer correlation between the two observ-
ables. In the case of the CBM, the change of energy is mainly
due to the electron density transfer from B to X (or vice versa)
induced by the change of chemical composition and crystal
structure (see Fig. 2E and F and 4 and S4†). Since the region
around B is positive, and that around X is negative, moving
charge from B to X increases the electrostatic energy of the
orbital. This explanation is proven to be correct by computing
the correlation between the charge on the B ion, q
B
, and E
CBM
.
Indeed, there is a clear linear correlation between these two
observables for the various perovskites of given B and X ions.
The effect of changing halides from I to Br to Cl is to shi this
linear correlation to higher energy values. This is because the Cl
environment, which is more negative, is more repulsive to an
additional electron than Br, which, in turn, is more repulsive
than I.
It is remarkable that the composition and structural
parameters affecting q
B
in the direction of increasing E
BCB
affect
O
VBM
and E
CBM
in the opposite direction (Fig. S5†). Thus,
a single parameter, O
VBM
, can be used to explain the entire
experimental and computational phenomenology. The conse-
quence of this fact is twofold. First, that chemical or structural
alterations affect E
g
by a synergic action on E
VBM
and E
CBM
: they
change simultaneously in the direction of opening or closing
the band gap (Fig. S1†). Second, at variance with other light
harvesting materials, it is therefore not easy to tune the energy
of the valence and conduction band of perovskites indepen-
dently to optimize at the same time light harvesting and charge
injection into the hole and electron transport materials.
However, this does not mean that specic changes cannot affect
more one than the other band. In fact, we found that specic
modications affect the two bands in di fferent ways. Rather, we
must conclude that for halide perovskites, while it is possible to
derive simple design rules for the variations in the energy gap, it
is not straightforward to derive similar guidelines for the
change in the energy of VBM and CBM.
3 Conclusions
In a search for an unied criteria controlling the band gap of
metal halide perovskites, ABX
3
, we performed electronic structure
calculations on a vast number of compounds differing in chem-
ical composition and crystallographic structure. Throughout
these variations the atomic orbital composition of CBM and VBM
Fig. 3 E
VBM
(E
CBM
) vs. O
VBM
(O
CBM
) for Pb-based (right) and Sn-based
(left) perovskites. E
BCB
is shifted as explained in the Methods section.
The relation E
VBM
vs. O
VBM
is linear (R
2
0.93 and 0.53 for Sn-based
and Pb-based perovskites, respectively) and has a large negative slope.
In the case of Pb perovskites, if we fit E
VBM
vs. O
TVB
independently for
each halides (continuous colored lines in the left panel) we obtain
a much higher R
2
, $0.85. The linear fitting of E
CBM
vs. O
CBM
is poorer
(R
2
0.33 and 0.38 for Sn-based and Pb-based perovskites,
respectively), and the slope is lower.
Fig. 4 q
B
and E
CBM
for Sn-based (left panel) and Pb-based (right panel)
systems.
16000
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remained intact, which suggested the use of orbital-overlap based
rationalization scheme. In fact, doing so one can predict the
effects of different chemical and structural variations in a simple
way. In summary, chemical and structural changes increasing the
negative overlap between orbitals of B and X ions result in
a shrinking of the band gap. This principle can be used to suggest
practical rules for tuning (lowering) of the gap. An increase in
overlap can be achieved by choosing B and X so as to reduce the
ratio between the lattice constant and the size of their respective s
and p orbitals. In practice, it is possible to estimate this ratio by
using ionic radii as a proxy for the lattice size,
17
and covalent radii
as a proxy for the size of B and X orbitals. For example, the vari-
ation of E
g
with halide composition follows the trend of the ratio
a ¼ r
I
/r
C
, between their ionic (r
I
) and covalent (r
C
) radii: a
I
¼ 1.47;
a
Br
¼ 1.58; a
Cl
¼ 1.67. As for the substitution of the monovalent
cation, in the case of tetragonal and orthorhombic structures, A
must be large so as to reduce the tilting angles, which makes the
structure as cubic-like as possible. At the same time, A should not
make strong hydrogen bonds with the perovskite framework.
Thus, cations that do not form hydrogen bonds are preferable. For
example, PH
4
+
is better than NH
4
+
: the former is larger than the
latter, and it forms weaker hydrogen bonds.
4 Methods
Density Functional Theory (DFT) calculations are performed
using the Quantum Espresso suite of codes.
21
We use the
Generalized Gradient Approximation (GGA) to density func-
tional theory in the Perdew–Burke–Ernzerhof (PBE) formula-
tion
22
and, for selected systems, the range separated hybrid
exchange and correlation functional of Heyd, Scuseria, and
Ernzerhof (HSE).
23
The interaction between valence electrons
and core electrons and nuclei is described by ultraso pseu-
dopotentials. Norm conserving pseudopotentials are used in
the case of HSE calculations. In GGA calculations, Kohn–Sham
orbitals are expanded in a plane wave basis set with a kinetic
energy cutoff of 40 Ry, and a cutoff of 240 Ry for the density in
the case of ultraso pseudopotentials. HSE calculations are
performed with a cutoff of 60 Ry. The Brillouin zone is sampled
with a 3 3 3 and 4 4 4 Monkhorst– Pack k-points grid
24
for cubic and tetragonal/orthorhombic structures, respectively.
The above values are chosen by checking the convergence of
total energy, band gap and atomic forces.
Sn-Based computational systems are prepared starting from
experimental structures for CsSnBr
3
,
25
replacing Cs
+
with Na
+
,
Li
+
,NH
4
+
and PH
4
+
;andBr
with Cl
and I
, as needed. Struc-
tures (atomic positions and lattice parameters) are then fully
relaxed. Cubic samples consist of a 2 2 2supercellcon-
taining 8 stoichiometric units. For the body-centered tetragonal
structures, we considered the simple tetragonal analogue, which
contains 4 stoichiometric units. Finally, for the orthorhombic
structure we used the experimental unit cell, which already
contains 4 stoichiometric units. Pb-Based samples are created
using the same protocol starting from experimental structures of
ref. 26 and 27.
On the optimized structures, bands calculations are per-
formed including Spin Orbit Coupling effects (SOC) according
to the method proposed by Dal Corso and Mosca Conte.
28
For
this, pseudopotentials obtained from fully relativistic all elec-
trons atomic data are used.
Comparing the energy of Kohn–Sham (KS) orbitals in solid-
state systems is difficult because in periodic boundary calcula-
tions there is no reference value. Several techniques have been
developed to cope with this problem. Two well established
techniques are based on the study of interfaces or surfaces (i.e.
an interface with vacuum). Which one to adopt depends on the
problem at hand, whether one is interested in the interface
between two materials or the effect of a surface on the bands.
Another technique has been used in the case of both bulk and
interface systems. This consists in taking as reference localized
deep impurity states
29,30
or the atomic states of the chemical
species present in the system, e.g. the O-2s states in refs. 31–34,
(i.e. states that are expected to be unaffected by the chemical
and structural nature of the system). The rational behind this
approach is that the energy of these states is not expected to
change from one computational sample to another. Thus, any
change in their energy observed among a set of systems is due to
the arbitrary shi associated to the lack of any absolute refer-
ence. The difference of energy of this state among the systems of
the set can be used for an alignment of their bands. This
procedure is illustrated in Fig. S7.† One drawback of this
approach is that it is not general like the interface/surface based
alignment procedures, as one has to rely on the existence of
such atomic states. Nevertheless, if such states are present, this
procedure provides an accurate and computationally cheap
approach for dening a proper reference.
In the present case, we use as reference nd orbitals of Pb and
Sn. In Fig. S7† the density of states (DOS) of orthorhombic
CsPbI
3
, LiPbI
3
and CsPbBr
3
, and tetragonal CsPbI
3
are shown. It
is evident that while the other bands change among the four
systems the band at 15 eV, corresponding to Pb-5d is only
rigidly shi ed, supporting the hypothesis that the vacuum
levels of these atomic orbitals are independent of the system
considered in the present work. To further conrm the reli-
ability of this approach, we compared the DOS as obtained by
the atomic orbital alignment and the well established vacuum
level alignment (Fig. S7,† panels B and C). The vacuum level
alignment has been obtained on the basis of the electric
potential of a 8 unit cell slab model of the perovskites surfaces,
shown in Fig. S8. † As can be seen from Fig. S8,† the two
alignment procedures are equivalent in the present case, with
a maximum difference of 0.03 eV between the two procedures,
which is within the accuracy of DFT calculations.
Acknowledgements
We thank Alfredo Pasquarello for fruitful discussions. U. R.
acknowledges funding from the Swiss National Science Foun-
dation via individual grant No. 200020-146645, the NCCRs
MUST and MARVEL, and support from the Swiss National
Computing Center (CSCS), the CADMOS project. We also
acknowledge PRACE for awarding us access to resource Super-
MUC based in Germany at Leibniz.
This journal is © The Royal Society of Chemistry 2016 J. Mater. Chem. A,2016,4, 15997–16002 | 16001
Paper Journal of Materials Chemistry A
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