Computational Characterization of the Dependence of Halide
Perovskite Effective Masses on Chemical Composition and Structure
Negar Ashari-Astani,
†,‡
Simone Meloni,
†,§
Amir Hesam Salavati,
∥,⊥
Giulia Palermo,
†,#
Michael Gra
tzel,
○
and Ursula Rothlisberger*
,†
†
Laboratory of Computational Chemistry and Biochemistry (LCBC),
∥
Audiovisual Communications Laboratory (LCAV), and
○
Laboratory of Photonics and Interfaces (LPI), E
cole Polytechnique Fe
de
rale de Lausanne, Lausanne, Switzerland CH-1015
§
Department of Mechanical and Aerospace Engineering, University of Rome “Sapienza”, via Eudossiana 18, 00184 Roma, Italy
‡
Department of Physics and
⊥
Innovation Center (ICT), Sharif University of Technology, Tehran, Iran
#
Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, California 92093 United States
*
S
Supporting Information
ABSTRACT: Effective masses are calculated for a large variety of perovskites of
the form ABX
3
differing in chemical composition (A= Na, Li, Cs; B = Pb, Sn; X=
Cl, Br, I) and crystal structure. In addition, the effects of some defects and dopants
are assessed. We show that the effective masses are highly correlated with the
energies of the valence-band maximum, conduction-band minimum, and band gap.
Using the k·p theory for the bottom of the conduction band and a tight-binding
model for the top of the valence band, this trend can be rationalized in terms of the
orbital overlap between halide and metal (B cation). Most of the compounds
studied in this work are good charge-carrier transporters, where the effective masses
of the Pb compounds (0 < m
h
* < m
e
* < 1) are systematically larger than those of the
Sn-based compounds (0 < m
h
* ≈ m
e
* < 0.5). The effective masses show anisotropies depending on the crystal symmetry of the
perovskite, whether orthorhombic, tetragonal, or cubic, with the highest anisotropy for the tetragonal phase (ca. 40%). In general,
the effective masses of the perovskites remain low for intrinsic or extrinsic defects, apart from some notable exceptions. Whereas
some dopants, such as Zn(II), flatten the conduction-band edges (m
e
* = 1.7m
0
) and introduce deep defect states, vacancies, more
specifically Pb
2+
vacancies, make the valence-band edge more shallow (m
h
* = 0.9m
0
). From a device-performance point of view,
introducing modifications that increase the orbital overlap [e.g., more cubic structures, larger halides, smaller (larger) monovalent
cations in cubic (tetragonal/orthorhombic) structures] decreases the band gap and, with it, effective masses of the charge carriers.
■
INTRODUCTION
Twenty years after their first discovery as possible transistors,
1
halide organic/inorganic perovskites (HOPs) with the
composition ABX
3
(A = organic or inorganic monovalent
cation, B = bivalent cation, X = halide) have attracted a great
deal of attention because of their breakthrough performance in
third-generation solar cells.
2−4
Efficiencies as high as 22.1%
5
have recently been reported for perovskite solar cells, and
future efficiency increases up to 30%, close to the Shockley−
Queisser limit, seem feasible.
6
Perovskites owe their superb
performance to their high open-circuit voltages (V
oc
≈ 1 V for
iodide perovskites and up to ∼1.5 V for bromide perov-
skites),
7,8
long charge-carrier lifetime (>15 μs),
9,10
and low
nonradiative carrier recombination rates (ca. 8 × 10
−12
cm
3
s
−1
).
11
The appropriate band gap of ∼1.65 eV makes the classic
methylammonium lead iodide perovskite (CH
3
NH
3
PbI
3
)an
excellent light harvester and, performance-wise, puts it in the
class of highly efficient materials for thin-film solar cells, at a
level comparable to that of CdTe and copper indium gallium
(di)selenide (CIGS).
Hand in hand with experimental studies, computational
investigations have been undertaken to shed light on the origin
of the unique electronic properties of halide perovskites. Using
density functional theory (DFT), the effects of halide and
cation variations on the optical band gap and band structure
have been investigated, along with the effects of temperature,
the in fluence of crystal defects, steric effects, and the possible
role of the perovskite/TiO
2
interface.
6,12−17
Because of a
fortuitous cancellation of spin−orbit and many-body effects,
standard DFT calculations within the generalized gradient
approximation (GGA) are able to predict the band gaps of lead
halide perovskites in close agreement with experimental
measurement s. The en ergetics (i.e., the relative energy
differences between various phases) are also well described at
the GGA level.
18
Even more relevant to the present work, using
GW as a reference, Umari et al. showed that spin−orbit
coupling (SOC) is crucial to the accurate determination of the
band structures of halide perovskites and that DFT + SOC can
provide an adequate description of band dispersion close to the
Received: May 22, 2017
Revised: August 3, 2017
Published: August 29, 2017
Article
pubs.acs.org/JPCC
© 2017 American Chemical Society 23886 DOI: 10.1021/acs.jpcc.7b04898
J. Phys. Chem. C 2017, 121, 23886−23895
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valence-band maximum (VBM) and conduction-band mini-
mum (CBM).
19
In our previous work,
16
we demonstrated how one can
rationalize and predict the effects of variations in chemical
composition and symmetry on the VBM and CBM energies of
halide perovskites through their effects on two key parameters,
the overlap between metal and halide orbitals and the effective
charge of the divalent cation, determining the energy of these
orbitals. In this work, we delve deeper into another equally
important property of HOPs that affects the charge-carrier-
transport characteristics: the hole and electron effective masses.
It is worth stressing that charge-carrier transport is also affected
by the scattering of charge carriers by phonons, which is not
discussed here.
In the field of semiconductors, the semiclassical model of
electron dynamics, in which electrons and holes are assigned
effective masses, m
h
and m
e
, respectively, has been very
successful.
20
Retaining much of the simplicity of free-electron
models, an effective-mass picture provides semiquantitative
predictions for one of the terms determining the efficiency of
charge-carrier transport. Recently, this model was applied to
some HOP systems by different groups.
21−31
In these works,
the authors computed effective masses of single halide
perovskite systems
22−24,29
or compared the masses of a limited
number of systems differing in the type of monovalent
27,30,31
or
divalent
26,28
cation or halide.
21,25
Despite the diversity of the
applied approaches for calculating the effective masses, all of the
reported values (except for a few abnormal and inappropriate
values that we discuss in the Theory section) unanimously
agree that the effective masses of both electrons and holes are
in the range of good carrier transporters, in accordance with the
experimental measurements of carrier mobility.
32,33
The
physical/chemical origin of this property, however, has
remained an open question. In this work, we calculate m
h
and m
e
for a wide range of Sn- and Pb-based perovskites that
differ in the chemical nature of the monovalent cation (Na, Li,
Cs, and some organic molecular ions), the halide (I, Br, and
Cl), and the crystal symmetry (cubic, tetragonal, and
orthorhombic). Indeed, the broad range of systems investigated
here made it possible to better understand the dependence of
the effective masses on the chemical and physical properties of
halide perovskites.
The fact that we consider all-inorganic halide perovskites
might appear to be in conflict with the trend of focusing on
hybrid organic−inorganic systems. The reason for our choice is
manifold. Hybrid perovskites, in particular, CH
3
NH
3
PbI
3
, have
a limited stability, which has been attributed to the
decomposition of methylammonium promoted by humidity.
34
Thus, researchers are trying to replace or limit the content of
organic cations by developing all-inorganic or mixed organic/
inorganic-cation halide perovskites. For example, recently, it
was shown that α-CsPbI
3
quantum dots are stable in ambient
air,
35
and this or similar materials are candidates for addressing
the problem of the stability of hybrid perovskites. Other
authors have also reported the synthesis of CsPbI
3
perov-
skites.
36,37
Indeed, the general interest in mixed cation and/or
halide perovskites, which have shown high efficiency and
enhanced stability thanks to the replacement of methylammo-
nium and/or formamidinium by inorganic cations and iodine
by bromine,
18,38
and the addition of small monovalent cations
39
calls for the investigation of a broad set of systems. Moreover,
the research in recent years has shown that halide perovskites
have potential technological applications beyond photovoltaics
(e.g., lasing, light-emitting diodes, photodetection), for which
optical properties different from those needed for solar cells are
sought. Thus, other systems, such as CsPbBr
3
, are of great
interest. Finally, a technical question concerns the modeling of
rotationally highly mobile methylammonium (and formamidi-
nium) ion in static first-principles calculations. Experiments and
simulations (see, e.g., refs 40−44) have shown that the
residence time of methylammonium in metastable orientation
states is on the picosecond time scale. Thus, a single
configuration of methylammonium-based halide perovskites is
not representative of the state of the system. Previous
computational works
13,42
showed that the electronic structures
of the VBM and CBM, which determine the effective masses,
are related to the monovalent cations through the effects of this
ion on the BX
3
−
3D network. In this work, we use Cs-based
perovskites, in particular, CsPbI
3
, also to mimic the average
structure of CH
3
NH
3
PbI
3
. We note that, in simulations, 3D
perovskite CsPbI
3
is metastable and has a band gap similar to
that of CH
3
NH
3
PbI
3
.
16
It is worth mentioning that analogous
approaches have been employed in other works (see, e.g., ref
45; also, the analogies between CH
3
NH
3
+
and Cs
+
are briefly
discussed in ref 46).
Anticipating our results, we found that, consistent with tight-
binding and k·p theories,
47
effective masses are strongly
correlated with the B/ns−X/mp orbital overlap and band
gap.
16
Thus, one can tune the electron and hole effective
masses by acting on those parameters affecting the orbital
overlap. We also investigate the effects of intrinsic (vacancies)
and extrinsic (doping) defects on the effective masses. We
found that, although doping might be beneficial in view of
increasing the concentration of free charge carriers, it turns out
that, for example, Zn(II) doping can have some detrimental
effects on carrier transport in some cases.
■
THEORY
In the semiclassical theory of transport,
48
the effective masses of
holes (m
h
) and electrons ( m
e
) control the response of these
particles to an electric field. Thus, m
h
and m
e
, together with the
scattering of charge carriers by phonons, are two key quantities
determining the transport properties of charge carriers in
perovskites. In 3D crystal systems, the effective mass tensors of
holes and electrons, namely, M
h
and M
e
, respectively, are
related to the Hessian matrix of the energy at the VBM or
CBM: M
h/e
=(ℏ
2
/2)ϵ
h/e
″
−1
, where ϵ
″ is the Hessian matrix of
elements ϵ
i,j
″ = ∂
2
ϵ(k)/∂k
i
∂k
j
, ϵ is the energy of the frontier
orbitals of the valence and conduction bands, and (·)
−1
denotes
an inverse matrix. E ffective masses are more conveniently
computed and analyzed in the reference frame of the
eigenvectors of the Hessian matrix. In this framework, M
h/e
is
a diagonal matrix of elements m
α
= ℏ
2
/(2ϵ
α
″), where ϵ
α
″ is one of
the eigenvalues of the Hessian matrix. Indeed, m
h/e,α
is the
effective mass for hole/electron transport along the direction of
the corresponding eigenvector.
In this work, effective masses are obtained according to the
following algorithm: First, the Hessian matrix is computed from
a parabolic fitting of the energy of the frontier orbitals of the
valence and conduction bands at a set of k-points around k
0
, the
k-point corresponding to the VBM or CBM: ϵ(k)=ϵ(k
0
)+
1
/
2
(k − k
0
)
T
ϵ
″(k − k
0
) [where (·)
T
denotes the transpose of
the vector]. In particular, we use a 3 × 3 × 3 grid of k-points
with a spacing of Δk. The suitable value for Δk is discussed
below. Second, the Hessian matrix is diagonalized to yield the
principal axes of charge-carrier transport (eigenvectors) and
The Journal of Physical Chemistry C Article
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23887
their corresponding eigenvalues. Finally, from the eigenvalues
of the Hessian matrix the effective masses along these
eigenvectors are obtained, as discussed in the previous
paragraph. We note that we focus on a specific sub-band
around its maximum/minimum, that is, a band of same band
index n, and not on the curvature of the overall valence or
conduction band. Thus, we are able to determine the hole/
electron effective masses also when the maximum/minimum is
split because of SOC in locally polarized domains, which has
been shown to be present in some halide perovskites (see, e.g.,
refs 49 and 50).
The above procedure requires suitable values for Δk = |k −
k
0
|. To tune and validate this parameter, we focus on the largely
studied MAPbI
3
system (MA = CH
3
NH
3
+
).
21,23,24,32
In Table
1, we report the average values of the effective masses for three
different values of Δk: |a
|/100, |a
|/400, and |a
|/800, where |a
| is
the length of a reciprocal lattice vector. Thus, although the
spacing of the k-point grid is the same in all directions for cubic
structures, it differs between the axial and equatorial directions
for tetragonal systems and changes in all three directions for
orthorhombic structures. The suitable value of Δk depends on
how broad the energy dispersion curve is around k
0
: Broader
curves are best fitted with coarser grids, and narrower curves
require smoother grids. In fact, if the grid spans a space that is
too wide, the second-order approximation to the energy is
insufficient. On the contrary, if the space spanned is too
narrow, then the change in the energy from the maximum of
the valence band (minimum of the conduction band) is
negligible, and the error in the estimation of the Hessian, and
therefore in the effective masses, is large. The suitability of a
value of Δk is measured by the coefficient of determination R
2
of the parabolic approximation of the dispersion curve k
0
,
which, typically, must be ≥0.9. Table 1 shows that the effects of
the grid spacing on the estimation of the masses can be quite
dramatic, with hole and electron masses reaching values as high
as 18.7m
0
and as low as 0.03m
0
, respectively (where m
0
is the
electron rest mass).
The present analysis suggests that a possible explanation for
the anomalous data reported in refs 25 and 51 for
orthorhombic MAPbI
3
(m
e
= 11.979m
0
) and MASnCl
3
(m
e
=
13.19m
0
) might be due to an improper calculation procedure.
On the contrary, the results reported in Table 1 for Δk = |a
|/
400 (i.e., at the maximum value of R
2
) are in good agreement
with experimental
32
and previous theoretical
19,23
results. It
must be remarked that other explanations are also possible, for
example, the lack of SOC in the calculations of refs 25 and 51.
However, in our simulations, which we performed with and
without SOC, we never observed a change of approximately 2
orders of magnitude associated with spin−orbit effects.
Before closing this section, it is worth mentioning that other
authors have used a different approach to avoid the problem of
determining a suitable value for Δk. Brivio et al.
30
described the
energy dispersion curve close to k
0
by combining the usual
second-order exp ansion wit h an addit ional k-dependent
function. Fitting the energy dispersion curve near k
0
along
one specific direction in reciprocal space with this more
elaborate function, they obtained k-dependent effective masses
that, in the limit of k → k
0
, are consistent with those presented
in this work. R
2
in the table is the coefficient of determination
of the sub-band energy distribution for m
h
* (left) and m
e
*
(right).
■
COMPUTATIONAL SETUP
GGA−DFT in the Perdew−Burke−Ernzerhof (PBE) formula-
tion
52
is used to optimize the atomic configuration and lattice
Table 1. Average Hole and Electron Masses, m
h
and m
e
,
Respectively, for MAPbI
3
Using the Appropriate Grid
Spacing Δk′, Together with Two Inappropriate (Too Small
and Too Large) Grid Spacing Values, Δk and Δk″,
Respectively, Resulting in Abnormal Values for the Effective
Masses
m
h
m
e
R
2
Δk = |a
|/100 0.05 0.35 0.67, 0.87
Δk′ = |a
|/400 0.19 0.32 0.93, 0.9
a
Δk″ = |a
|/800 18.66 0.03 0.80, 0.71
a
Δk′ gives the best R
2
values, showing the significance of the grid
spacing on the fitting procedure.
Figure 1. (Left) Transverse and (right) longitudinal principal axes of hole transport (arrow) for the tetragonal systems. Also for cubic and
orthorhombic systems, the principal axes are oriented along the B−B directions. The gray polygons in the figure represent BX
6
4−
units.
The Journal of Physical Chemistry C Article
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J. Phys. Chem. C 2017, 121, 23886−23895
23888
parameters of all structures. Calculations are performed using
the pw.x code of the Quantum Espresso package.
53
Ultrasoft
pseudopotentials are used to describe the interaction between
the (semi)valence electrons and the nuclei and core electrons
for all of the atoms. The Kohn−Sham orbitals and the total
electronic density are expanded in a plane-wave basis with
energy cutoffs of 40 and 280 Ry, respectively. The Brillouin
zone is sampled with a 3 × 3 × 3or4× 4 × 4 Monkhorst−
Pack k-point grid
54
for cubic and tetragonal/orthorhombic
structures, with supercells containing eight and four stoichio-
metric units, respectively. These values were c hosen by
checking the convergence of the total energy (∼1 × 10
−3
Ry/atom), band gap (0.01 eV), and atomic forces (∼1 × 10
−4
Ry/au).
GGA−SOC calculations were performed on the perovskite
systems to compute their band structures. The effective masses
of holes and electrons were obtained by performing a quadratic
fit of the 3D band structure at k
0
, the k-point corresponding to
the VBM and CBM. This required the calculation of the
energies of frontier states of the valence and conduction bands
on a 3 × 3 × 3 grid of k-points centered at k
0
[see Figure S1 in
the Supporting Information (SI)]. The suitable number of grid
points and grid spacing, Δk, were carefully chosen and tested
system by system. The quadratic least-squares fit of the
dispersion of the valence and conduction bands at k
0
was
performed using the “lsqcurvefit” function of MATLAB 7.12.
55
All of the fits present a value of the norm of the residual lower
than 1.5 × 10
−9
eV, indicating a rather accurate parabolic
approximation of the valence and conduction bands.
■
RESULTS AND DISCUSSION
We first focus on the analysis of the results for defect-free Cs
+
/
Na
+
/Li
+
lead and tin perovskite of I
−
/Br
−
/Cl
−
. All of these
systems present low hole and electron effective masses close to
the supercell Γ point, with minimum values per system ranging
from ∼0.1m
0
to ∼0.6m
0
. The fact that the VBM and CBM
occur near the Γ point is due to the supercell used in the
present work: this point folds back at the proper k
0
point for
the unit cell of the given symmetry, for example, the R point for
cubic systems.
The principal axes of transport, that is, the eigenvectors of
the effective mass tensor, are oriented along the B−B directions
(see Figure 1). These directions are all equivalent in the case of
cubic systems, and thus, charge-carrier transport is isotropic
(i.e., the masses along the three directions are the same). In the
case of tetragonal systems, there are two degenerate principal
axes of transport in the equatorial plane and one along the
tetragonal axis. Finally, in orthorhombic systems, the three
principal axes of transport, and the corresponding effective
masses, are all different.
The dependence of the minimum hole and electron masses
on the chemical composition (monovalent cation and halide
within lead and tin perovskites) and crystal structure presents
interesting trends. For a given chemical composition, the
Figure 2. (Left) Frontier orbital energies and (right) gaps versus minimum effective mass for Sn-based compounds ASnX
3
, A = Li, Na, and Cs
(shown with diamonds, squares, and asterisks, respectively) and X = Cl, Br, and I (shown in green, red, and blue, respectively) for different crystal
structures.
Figure 3. (Left) Frontier orbital energies and (right) gaps versus minimum effective mass for Pb-based compounds APbX
3
, A = Li, Na, and Cs
(shown with diamonds, squares, and asterisks, respectively) and X = Cl, Br, and I (shown in green, red, and blue, respectively) for different crystal
structures.
The Journal of Physical Chemistry C Article
DOI: 10.1021/acs.jpcc.7b04898
J. Phys. Chem. C 2017, 121, 23886−23895
23889
effective masses increase along the series cubic → tetragonal →
orthorhombic. If one considers the dependence on halides,
keeping the types of mono- and bivalent cations and the crystal
structure fixed, the masses grow along the sequence I
−
→ Br
−
→ Cl
−
. The dependence on the type of monovalent cation is
more complex and varies as a function of the crystal symmetry.
For cubic systems, the masses grow along the sequence Li
+
→
Na
+
→ Cs
+
, that is, they grow with the cation size. In tetragonal
and orthorhombic systems, the masses grow in the opposite
order: Cs
+
→ Na
+
→ Li
+
.
An analogous dependence on the chemical and physical
characteristics of the halide perovskites has been observed for
the VBM and CBM energies.
16
This suggests a strong
correlation between the minimum hole and electron masses
(m
h
* and m
e
*) and the energy levels of the frontier orbitals of
the valence and conduction bands. This correlation is shown in
Figures 2 and 3 for lead and tin perovskites, respectively. In the
panels on the left of these figures, the masses of holes and
electrons as function of ΔE
VBM
and ΔE
CBM
are reported, where
ΔE = E − E
min
and E
min
is the minimum of the VBM or CBM
among all of the systems considered. We remark that, in these
and the following figures, the effective masses of holes are
reported with a negative sign, so that the masses of both
carriers can be plotted on the same chart. One can observe that
the trend is almost linear with ΔE: the masses grow with the
value of this parameter. Because E
VBM
and E
CBM
exhibit
opposite trends with the composition and crystal symmetry
(i.e., they concur in the widening or shrinking of the band gap
E
g
= E
CBM
− E
VBM
), m
h
* and m
e
* have a linear trend with the
band gap as well.
The correlation between the hole and electron effective
masses and the band gap in the direction observed in this work
is consistent with the predictions of both the tight-binding
(TB) and k·p theories.
47
Nevertheless, halide perovskite have
some unconventional features that make the relationship
between the properties of the material and its composition
and structure less obvious and intuitive. As explained in ref 16,
the VBM has a (antibonding) covalent character; thus, TB is
well suited for interpreting the properties of this band. On the
contrary, the CBM has a much less covalent character, and k·p
theory works better in describing this case. This suggests that a
single theory will not be adequate for interpreting the
computational results and linking them to the characteristics
of the material, and one has to treat the cases separately.
Concerning m
h
*, the dependence of the effective mass on the
energy of the corresponding band can be explained as follows:
Within TB, the curvature of the VBM grows with the overlap
O
VBM
between the atomic orbitals contributing to the band
(the s orbitals of Sn/Pb and the p orbitals of X atoms
in the present case). [Note that
=
∑∑
*
∈− ∈−
O
ccRe( )
imjm
ijijVBM
Xp Bs
VBM, VBM,
6
,where
∫
=Φ
*
Φrrrd()()
ij i j
6
and c
VBM,i
and c
VBM,j
are the contribu-
tions of atomic orbitals i and j, respectively, to the VBM.] In
practice, the higher the overlap and the higher the curvature of
the band, the lower associated effective mass (see Figure 4). At
the same time, the orbital overlap also determines the energy of
the VBM, which explains the correlation between m
h
* and E
VBM
(Figures 2 and 3).
Concerning m
e
*, as mentioned above, k·p theory is more
suitable for explaining the computational results relative to the
conduction band and, thus, the electron effective masses. k·p
theory is a perturbative approach to the calculation of the
energy of the band at a k point in the neighborhood of the
CBM
∑
=+
ℏ
+
ℏ
|·⟨ || ⟩|
−
EE
k
m
m
uu
EE
kp
2
kk
l
k
l
k
kk
l
CB CB
22 2
2
CB 2
CB
0
00
00
(1)
For the sake of simplicity, we report here the form without
SOC. Nevertheless, this form is sufficient to explain the
dependence of m
e
* on the composition and crystal structure of
the material. m
e
* is related to the second-order perturbative
term, that is, it depends on the transition moment integral, ⟨u
k
0
l
|
p|u
k
0
CB
⟩, and the energy difference between the CB at k
0
and the
other bands at the same point, E
k
0
CB
− E
k
0
m
. In practice, relevant
contributions to E
k
CB
come from bands of suitable symmetry (⟨
u
k
0
l
|p|u
k
0
CB
⟩ ≠ 0) that are close in energy to the CBM (small E
k
0
CB
− E
k
0
m
). Because the transition moment integral vanishes for
most of the conduction bands (see Tables S1 and S2 in the SI),
in the present case, the major contribution comes from the
VBM. Thus, m
e
* depends on the band gap and, as we showed in
ref 16, this is correlated to O
VBM
.
In summary, modifications of the chemical composition and
crystal structure are effective ways of controlling the orbital
overlap and, through it, m
h
* and m
e
*. For example, cubic
perovskites, with linear B−X−B bonds, have maximum overlap,
Figure 4. Antibonding overlap of the VBM orbitals O
VBM
versus the minimum hole effective masses m
h
* for (left) tin-based and (right) lead-based
compounds.
=
∑∑
*
∈− ∈−
O
ccRe( )
imjm
ijijVBM
Xp Bs
VBM, VBM,
6
, where
∫
=Φ
*
Φrrrd()()
ij i j
6
is the overlap between pairs of X and B atomic orbitals
and c
j
and c
i
are projection coefficients of the crystal orbitals onto the atomic orbitals.
The Journal of Physical Chemistry C Article
DOI: 10.1021/acs.jpcc.7b04898
J. Phys. Chem. C 2017, 121, 23886−23895
23890
