[orcid=0000-0002-0669-0984]

Validation, Formal analysis, Investigation, Writing - Review and Editing

[orcid=0000-0001-6339-2986]

Validation, Formal analysis, Investigation, Writing - Review and Editing

[orcid=0000-0002-5566-2656] [1] URL]www.mmj.pl/ aptok/

Conceptualization, Methodology, Validation, Formal analysis, Investigation,
Writing - Original Draft, Writing - Review and Editing, Visualization,
Supervision

[cor1]Corresponding author

# Electronic properties and surface states of RbNi2Se2

Surajit Basak<EMAIL_ADDRESS>Institute of Nuclear Physics, Polish
Academy of Sciences, ul. W. E. Radzikowskiego 152, 31-342 Kraków, Poland
Przemysław Piekarz<EMAIL_ADDRESS>Andrzej Ptok<EMAIL_ADDRESS>
###### Abstract

Iron-based superconductors, with the ThCr2Si2-type tetragonal structure (122
family), due to the iron arsenide/selenide layers exhibit several
characteristic electronic properties. For example, multiband character mosty
associated with the d-orbitals of iron and the quasi-two-dimensional (2D)
cylindrical Fermi surface. Moreover, external hydrostatic pressure leads to
the isostructural phase transition from the tetragonal to collapsed-tetragonal
phase. In this paper, in relation to the iron-based superconductors, we
discuss the electronic properties of novel 122-family member RbNi2Se2 [Liu H.
et al., Phys. Rev. B 106, 094511 (2022)]. We show that the two Fermi pockets
exhibit quasi-2D character. Calculation of the surface spectral function for
the (001) surface shows that the surface states are realized independently on
the surface termination. Additionally, contrary to the iron-based 122
compounds, RbNi2Se2 exhibits extraordinary multiple isostructural phase
transitions under pressure. Moreover, the Lifshizt transition occurs under
external pressure, which results into a strong modification of the shapes of
the Fermi pockets.

###### keywords:

DFT calculations Electronic structure Surface states Hydrostatic pressure
Lifshitz transition

##  1 Introduction

Discovered in 2008, the superconductors based on iron [1] opened a period of
intensive experimental and theoretical studies of this class of compounds [2,
3, 4, 5, 6]. These materials are characterized by the layered structure
containing iron-arsenide/selenide layers – where the iron square lattice is
decorated by anions. As a consequence, the electronic structure exhibits
multiband properties and the characteristic quasi-two-dimensional (2D)
cylindrical Fermi surface.

One of the examples of iron-based superconductors is a group of the ternary
ThCr2Si2-type compounds (called 122 family) [7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20]. The 122-family members exhibit not only
superconductivity, but also a wide range of physical properties, e.g.,
antiferromagnetism [15, 16, 18, 17], spin density waves [7, 8, 9], and
structural phase transitions [11]. Moreover, in the presence of external
hydrostatic pressure, the isostructural transition from the tetragonal to
collapsed-tetragonal phase occurs [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32].

Recently, the RhNi2Se2 compound, showing the multi-gap superconducting
properties below $T_{c}=1.2$ K, was successfully synthesized [33]. Contrary to
a large part of the iron-based materials [34], RhNi2Se2 exhibits the Pauli
paramagnetic behavior. The initial measurements of the specific heat suggest
enhancement of the effective electronic mass $m^{\ast}=6m_{e}$ and a weak role
of correlations.

In this paper, on the base of density functional theory (DFT) calculations, we
discuss the electronic properties of RbNi2Se2 (the band structure and the
Fermi surface). From the DFT band structure, we construct the tight banding
model in the basis of the Wannier orbitals and investigate the surface states.
Mainly we discuss the surface termination dependence of the surface states.
Finally, we demonstrate the existence of the multiple-isostructural phase
transitions occuring under external hydrostatic pressure. The presented
results can also be applicable for other 122-compounds based on nickel, e.g.
KNi2Se2 [35], CsNi2Se2 [36], or TlNi2Se2 [37].

The paper is organized as follows. The computational details are present in
Sec 2. Next, in Sec. 3 we present and discuss our results: electronic
properties (Sec. 1), realized surface states (Sec. 3.2), and properties under
hydrostatic pressure (Sec. 3.3). Finally, a summary is provided in Sect. 4.

##  2 Computational details

The electronic properties were calculated using Quantum ESPRESSO [38, 39, 40].
The calculations were performed within the generalized gradient approximation
(GGA) in the Perdew, Burke, and Ernzerhof (PBE) parameterization [41] within
PsLibrary [42]. In this study we used the experimental lattice parameters
[33].

The DFT results of the electronic band structure calculation, performed for
the primitive unit cell, were used to find the tight binding model in the
basis of the maximally localized Wannier orbitals [43, 44, 45]. It was
performed using the Wannier90 software [46, 47, 48]. In our calculations, we
used the $6\times 6\times 6$ full ${\bm{k}}$-point mesh, starting from the $p$
orbitals of Se, and from the $s$ and $d$ orbitals for Rb and Ni, respectively.
Finally, the resulting $28$-orbital tight binding model of RbNi2Se2 was
applied to investigate the surface Green’s function for a semi-infinite system
[49], using WannierTools [50] software.

Properties of the system under external hydrostatic pressure were investigated
using the projector augmented-wave (PAW) potentials [51] implemented in the
Vienna Ab initio Simulation Package (Vasp) code [52, 53, 54] within the GGA
PBE. The energy cutoff for the plane-wave expansion was set to $400$ eV.
Optimizations of structural parameters (lattice constants and atomic
positions) under pressure were performed in the conventional unit cell
(containing two formula units) using the $12\times 12\times 4$ k–point grid in
the Monkhorst–Pack scheme [55]. The convergence criteria were set to be
$10^{-6}$ eV and $10^{-8}$ eV for ionic and electronic degrees of freedom,
respectively.

##  3 Results and discussion

RbNi2Se2 crystallizes in the ThCr2Si2-type structure (symmetry I4/mmm, space
group No. 139) [33]. In the absence of any external pressure, the lattice
parameters were experimentally measured to be $a=b=3.9272$ Å, and $c=13.8650$.
In this cell three nonequivalent atoms Rb, Ni, and Se are placed at the
crystallographic sites: $2a(0,0,0)$, $4d(0,1/2,1/4)$ and
$4e(0.0,z_{\text{Se}})$, respectively. Experimentally, the free parameter at
the position of Se atom was determined to be $z_{\text{Se}}=0.3498$. The
lattice parameters are comparable to the parameters of the isostructural
KFe2As2 superconductor [30].

Figure 1: The electronic band structure of RbNi2Se2. The results in the
absence and presence of spin–orbit coupling are represented by orange and blue
lines, respectively. Inset presents the Brillouin zone and its high symmetry
points.

###  3.1 Band structure and Fermi surface

The electronic band structure (in the absence and presence of the spin–orbit
coupling) is presented in Fig. 1. In relation to the 122 iron-based compounds,
e.g. KFe2As2 [30], the Fermi level is shifted to the higher energies. This is
associated with the three “extra” electrons in the system (due to the
Fe$\rightarrow$Ni and As$\rightarrow$Se substitutions). Nevertheless, the
Fermi level is still located within the range of energies related to the
d-orbitals [33]. This behaviour is also well visible in the partial density of
states, presented in Fig. 2. For energies from $-6$ to $1$ eV, the Se
p-orbitals and Ni d-orbitals contribute the most. The group of the d-orbitals
are mostly locallized between $-3$ and $1$ eV, and constitute a group of bands
well separated from rest of the spectrum by the gaps (around $-3$ eV and $1$
eV, see Fig. 1). Similar situation was previously reported for FeSe [56],
where bands related to the Fe d-orbitals were well separated. Contrary to
this, in the case of KFe2As2, the bands were not separated due to the strong
mixing with As p-orbitals [56]. In the case of RbNi2Se2, the Se p-orbitals are
mostly localized at energies from $-6$ eV to $-3$ eV, and show only a small
contribution close to the Fermi energy. For bands above $2$ eV, the strong
mixing of orbitals is observed, without dominant role of any of them.

In relation to the 122 iron-based superconductors [30], the Fermi level is
shifted to the higher energies. As a consequence, the strong modification of
the Fermi surface with respect to the 122 iron-based systems is observed (Fig.
3). There are three Fermi pockets with hole-like character. One pocket
exhibits exactly three dimensional character [Fig. 3(c)], while two pockets
along the X–P path exhibit quasi-2D character [Fig. 3(d) and (e)]. It is worth
mentioning that in the case of KFe2As2, the quasi-2D pockets are centered
along the $\Gamma$–Z path, while along the X–P path there are four small
pockets [30].

Figure 2:  The total and partial electronic density of states of RbNi2Se2.
Figure 3: The general view (a) and view from the top (b) of the Fermi surface
of RbNi2Se2. Panels (c)–(e) present the separated Fermi pockets.

The multiband character of this compound can affect its superconducting
properties. Initial investigations of the superconducting states predicted
two-gap BCS-like superconductivity [33]. However, RbNi2Se2 exhibits a weak
ferromagnetic sign from the hysteresis loop, which can be ascribed to a small
amount of Ni impurity [33]. Nevertheless, similarly to the 122 iron-based
superconductors, the magnetic instability can be expected in this system. As a
consequence, the superconducting gap can exhibit non-uniform properties, like
$s_{\pm}$-wave symmetry in iron-based compounds [57]. Unfortunately, the
specific heat study typically does not give unequivocal settlement about the
gap character [58].

Figure 4: (a) Projection of the bulk Brillouin zone on the surface Brillouin
zone for the I4/mmm symmetry. (b) The slab band structure for the (001)
surface of RbNi2Se2 (the bulk states are marked by grey lines, while surface
states for Ni2Se2 and Rb termination, are marked by blue and red lines,
respectively).  Figure 5: Spectral function for the (001) surface of RbNi2Se2
(with different surface terminations, as labeled).

###  3.2 Surface states

Many of the iron-based superconductors exhibit topological properties [59].
Here also, investigation of the electronic band structure indicated the
important role of the spin–orbit coupling (described in earlier paragraphs).

Using the slab type calculations, we studied the termination dependence of the
surface states (Fig. 4). We found that the surface states are realized in the
system, independently on the terminated surface (Rb or Ni2Se2). For both the
terminations, surface states crossing the Fermi level should be visible in the
angle-resolved photoemission spectroscopy (ARPES) measurements. To investigate
it more closely, we calculated the spectral function for the (001) surface
(Fig. 5). The Rb and Ni2Se2 terminations should be distinguishable in high
quality ARPES measurements – the strong differences are visible along the
$\bar{\text{N}}$–$\bar{\text{X}}$ path, i.e. along the edge of 2D Brillouin
zone [cf. Fig. 5(a) and 5(b)]. This is reflected also in the isoenergetic
crossection of the spectral function (Fig. 6). As we can see, the 2D energy
contour at the Fermi level exhibits strong differences, while the surface
states are well visible for the Ni2Se2 termination. Comparing our theoretical
results with the experimental ones (discussed in Ref. [33]), we can conclude
that the experimentally investigated sample had the Rb termination. The Ni2Se2
terminated sample should realize several well visible bands (e.g. along the
$\bar{\Gamma}$–$\bar{\text{N}}$ direction). Contrary to this, for the surface
with the Rb termination, the energy contour are mostly associated with the
bulk-like states (blurred spectrum). Nevertheless, few strong surface states
are also visible around the $\bar{\Gamma}$ point, in the middle of the 2D
Brillouin zone.

Figure 6: The crossection of spectral function for the (001) surface of
RbNi2Se2 at the Fermi level (with different surface terminations, as labeled).
Figure 7: (a) The conventional cell of the I4/mmm structure of RbNi2Se2.
Pressure dependence of the following parameters: (b) lattice constants $a$ and
$c$, (c) volume of the conventional cell $V$ and $c/a$ ratio, (d) distance
$z_{\text{Se}}$ of Se atom from the Ni plane and distance $z_{\text{Se--Se}}$
between Se atoms along the $c$ direction [shown in (a)], and (e) angle between
Ni–Se bonds [shown in (a)]. On panel (b), we compare also lattice constant $a$
and $c$ obtained from calculation with “standard” PAW potentials (solid line),
and PAW potentials containing semicore electrons as valence one (dashed line).

###  3.3 Role of hydrostatic pressure

As we mentioned in the introduction, the external hydrostatic pressure can
lead to the isostructural phase transition from the tetragonal to collapsed-
tetragonal phase (T-cT transition), which was reported in many 122-type
compounds [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. In case of RhNi2Se2
similar properties should be expected. Surprisingly, our study uncover a much
complicated situation (Fig. 7), with multiple isostructural phase transitions.

Now we compare the modification of the structural parameters under the
external hydrostatic pressure, obtained for RbNi2Se2 (Fig. 7) and KFe2As2
[30]:

  * •

Under pressure, we observe simultaneous discontinuous decreasing of $a$ and
$c$ [Fig. 7(b)] – several isostructural phase transitions are visible (marked
by red arrows). In these transitions, the lattice constant $a$ increases by
$\delta a\simeq 0.03$ Å, while $c$ decreases by $\delta c\simeq 0.3$ Å. This
behavior can be compared to the case of KFe2As2, where the lattice constant
modifications were much stronger: $\delta a\simeq 0.1$ Å and $\delta c\simeq
1$ Å [30]. Additionally, in the KFe2As2 compound, $a$ increases slightly just
before the final collapse of the cell at critical pressure. In case of
RbNi2Se2, the lattice constant $a$ increases only at the transition pressures.

  * •

As a natural consequence of the external hydrostatic pressure, the volume $V$
of the unit cell decreases, while the $c/a$ ratio shows discontinuous
decreasing character [Fig. 7(c)]. During the isostructural transitions, the
volume does not exhibit dramatical changes. Additionally, with pressure, the
$c/a$ ratio goes to the saturated value $\sim 2.6$ (marked by red arrow).

  * •

Modification of the lattice constants indicates a change in the distance
between atoms [Fig. 7(d)]. The interlayer distance between two Se atoms
[labeled as $z_{\text{Se--Se}}$ in Fig. 7(a)] decreases slowly. Simultaneously
the distance of Se atoms from the Ni square net plane [labeled as
$z_{\text{Se}}$ in Fig. 7(a)] drops dramatically a few times. The distance
$z_{\text{Se}}$ shows a similar behavior like the distance between As atoms
and Fe square net plane in KFe2As2 [30] – the $z_{\text{Se}}$ drops at the
pressures related to the isostructural transitions.

  * •

The interplanar Se-Ni-Se bond angle [marked in Fig 7(a) as $\alpha$] shows a
nonmonotonous dependence on pressure and changes its value a several times
[Fig. 7(e)]. Between the transitions, $\alpha$ for $p>50$ GPa has
approximately constant value, while for extremely large pressure, its value
saturates to $\alpha\simeq 112^{o}$. For 122 iron-based superconductors,
$\alpha$ increases with pressure to the saturated value $116^{o}$.

Here we would like to mention that the initial lattice constants (in the
absence of pressure), as well as the $c/a$ ration, for RbNi2Se2 and KFe2As2
were comparable. Despite of it, the properties of these systems under external
hydrostatic pressure do not posses many similarities. However, our prediction
of the multiple phase transitions in RbNi2Se2 could be verified experimentally
using the diffraction method.

In context of the superconducting properties of RbNi2Se${2}$ under pressure,
the $z_{\text{Se}}$ parameter can play an important role. In the iron-based
superconductors, the distance between the anion and the Fe plane can be
related to the observed $T_{c}$ [60, 61]. A decrease in the value of
$z_{\text{Se}}$ can suggest a destructive role of pressure on the
superconducting phase.

Figure 8: Electronic band structure (upper panels) and corresponding Fermi
surface (lower panels) of RbNi2Se2 under external hydrostatic pressure (as
labeled). For guide for the eye the bands crossing the Fermi level, are marked
by red, green, and blue color (related to the color of the separated Fermi
pocket on panels a1, a2, a3, etc.).

Recent studies of the system under pressure indicate the important role of the
electrons, which typically are treated as the core ones [62]. Under high
pressure, the electronic structure of a system can change significantly
compared to that at ambient pressure. One effect that has been recognized is
that the transfer of conduction electrons between the s-p states and d states
depends on the degree of compression and varies among crystal structures [63,
64, 65]. This property can be important in context of the system stability,
due to the impact of the conduction electrons. The “standard” PAW potential
can give erroneous thermodynamic properties under high pressure. Therefore,
the PAW potential containing the semicore electrons are also available, and
can improve the obtained results. Indeed, for alkali metal Rb, two versions of
VASP PAW potentials can be used (containing 7 and 9 valence electrons – both
containing the semicore states, s and p, respectively), and similarly for the
transition metal Ni (one PAW potential without semicore electrons, containing
10 valence electrons, and second PAW potential with p semicore electrons,
containing 16 valence electrons). To check quality of obtained results, we
compare the lattice constant of the RbNi2Se2 under pressure (Fig. 3.3(b)) –
results obtained with “standard” PAW potentials (containing 82 valence
electrons within the conventional cell) are presented by dots and solid line,
while results for PAW potentials with semicore electrons (corresponding 106
valence electrons within the conventional cell) by dashed line. As we can see,
both type PAW potentials reproduce the same behavior of $a$ and $c$ under
pressure (also few “jumps” of lattice constant is visible in Fig. 7(a)).
However, under extremely high pressure (above $150$ GPa) the results are
significantly different – jump of $a$ and $c$ is shifted to higher pressure.
Nevertheless, we should have in mind that even if the presented results can
correctly reproduce experimentally observed $a$ and $c$ behavior (like earlier
study of KFe2As2 [30]), it should be used with the uttermost care in the
presence of extremely high-pressure calculations.

#### Electronic properties and Lifshitz transition under pressure.

The external hydrostatic pressure also have an impact on the electronic band
structure. The modified distances between the atoms lead to a stronger overlap
between the atomic orbitals. As a consequence, the electronic band structure
as well as the Fermi surface can be strongly modified.

Firstly, the bandwidth increases – this is well visible when we compare the
electronic band structure at different pressure (Fig. 8). For example at the
$\Gamma$ point, the electron-like parabolic band (marked by blue dashed lines)
is shifted to lower energies, while at the same time, the hole-like band
(marked by red dashed lines) is shifted to higher energies. Similar behavior
was reported previously in context of the transition from the tetragonal to
collapsed-tetragonal phase in KFe2As2 [30], where the most notable
modifications were observed within the $p_{z}$ orbitals of anions.

Secondly, in some situations, specific external conditions can lead to a
change of the Fermi surface topology. Such modification of the Fermi surface,
known as the Lifshitz transition [66], can be induced by doping [67, 68, 69,
70, 71, 72, 73, 74, 75, 76, 77, 78], external pressure [79, 80], or magnetic
field [81, 82, 83, 84, 85]. In the case of 122 iron based superconductors
(e.g., Ba1-xKxFe2As2 [72, 71, 70, 69, 75, 74, 73, 76] or Ba(Fe1-xCox)2As2
[78]), strong dependence of the Fermi surface on doping was observed
experimentally. Also, the pressure dependent Lifshitz transition in the iron-
based systems was investigated theoretically [30, 31, 32, 86, 87, 88]. In our
case, the pressure also leads to a Lifshitz transition (cf. the Fermi surfaces
presented in Fig. 8).

As we mentioned in Sec. 1, at the ambient pressure, the Fermi surface is
composed of three separated pockets (Fig. 8(a1)-(a3)). Number of the packets
does not depend on pressure (in this part, we will call it first, second, and
thirt pocket, what will be corresponding to pockets in Figs. 8(a1), (a2), and
(a3), respectively). The external pressure affects the first Fermi pocket the
most (Fig. 8(a1)). In the ambient pressure, this pocket exhibits the most
three-dimensional character (with the strong $k_{z}$-dependence dispersion).
In the first stage, increasing pressure leads to the disappearance of part of
the pocket along the S0–N–S path. The remnant of this part is a small
ellipsoidal-like shape along S0–$\Gamma$ at $50$ GPa (blue arrow in Fig.
1(b)). Nevertheless, this pocket is mainly realized by a cubic-like shape
centered at the $\Gamma$ point. This modification of the first pocket is
related to the Fermi level by the bands around the N and S0 points (cf. blue
arrows in Fig. 8(a) and (b)). Further increase of pressure leads to a
modification of the cubic-like pocket (cf. Fig. 8(b1), (c1), and (d1)). For
$p>50$ GPa, the band at the $\Gamma$ point is shifted to the lower energy,
while the one at the M point is shifted to the higher energy (cf. blue arrows
in Fig. 8(c) and (d)). During this transition, the first pocket changes shape
from cubic-like (Fig. 8(b1)), through dumbbell-like along $k_{z}$ and centered
at the $\Gamma$ point (Fig. 8(c1)), to bowtie-like along $k_{z}$ and centered
at the M point (Fig. 8(d1)). The second Fermi pocket, with an initially
cylindrical shape ( Fig. 8(a2)), acquires new features – it changes character
from quasi-two-dimensional to directly three-dimensional with the strong
$k_{z}$-dependence dispersion. This is related to crossing the Fermi level by
the bands along the $\Gamma$–S0 path – the bottom of band crosses the Fermi
level for pressure between $50$ GPa and $100$ GPa (cf. red arrows on Fig. 8(b)
and (c)). Further increase of pressure does not introduce any bigger
modification of this pocket shape. The third Fermi pocket does not change its
shape under pressure (cf. Fig. 8(a3), (b3), (c3), and (d3)). The cylindrical
shape of this pocket is related to the quasi-two-dimensional character of the
band, which constitutes it (parabolic band along the X–P path with the weak
$k_{z}$-dependence dispersion).

##  4 Summary

In conclusions, we investigated the electronic properties and surface states
of RbNi2Se2 [89]. The obtained Fermi surface suggests quasi-two-dimensional
character of the electronic states in the two bands crossing the Fermi level,
while the third band has fully three dimensional character. We presented the
electronic band structure in the absence and presence of the spin–orbit
coupling. There are a few places in the band structure, where the spin–orbit
coupling plays an important role. As a consequence of the topological effects,
the surface states can emerge independent of the surface terminations.
However, the spectral function depends on the type of termination (Rb or
Ni2Se2 termination). Indeed, we show that the ARPES results presented in Ref.
[89] suggest that the sample had the Rb termination. Moreover, in case of the
Ni2Se2 termination, the electronic band structure posses much more complex
structure, due to the realization of many surface states crossing the Fermi
level.

Additionally, we discussed the multiple isostructural phase transitions under
external hydrostatic pressure. Contrary to the 122 type iron-based
superconductors, RbNi2Se2 does not exhibit transition from the tetragonal to
collapsed-tetragonal phase. In case of RbNi2Se2, internal chemical pressure
balances the external chemical pressure and the collapse of tetragonal cell
does not occur. Nevertheless, modification of the distance between atoms leads
to a change in the electronic band structure as a consequence of an increased
overlap between the atomic orbitals. This band structure modification is
reflected in the Fermi surface features, which exhibit the Lifshitz transition
– the Fermi pockets show strong pressure dependence.

## Acknowledgments

Some figures in this work were rendered using Vesta [90] and XCrySDen [91]
softwares. S.B. is grateful to IT4Innovations (VŠB-TU Ostrava) for hospitality
during a part of the work on this project. This work was supported by National
Science Centre (NCN, Poland) under Projects No. 2017/25/B/ST3/02586 (S.B.) and
2021/43/B/ST3/02166 (P.P. and A.P.). A.P. appreciates funding in the frame of
scholarships of the Minister of Science and Higher Education (Poland) for
outstanding young scientists (2019 edition, No. 818/STYP/14/2019).

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data Availability Statement

The data presented in this study are available on request from the authors.

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|  Theoretical evaluation of oxynitride, oxyfluoride and nitrofluoride
perovskites with promising photon absorption properties for solar water
splitting†
---|---
|  Manjari Jaina, Deepika Gill, Sanchi Monga and Saswata Bhattacharya∗a
|  Photocatalytic water splitting represents a very promising but at the same
time very challenging contribution to a clean and renewable route to produce
hydrogen fuel. Developing efficient and cost-effective photocatalysts for
water splitting is a growing need. For this purpose, semiconductor
photocatalysts have attracted much more attention due to their stability and
low manufacturing cost. Here, we have systematically applied several state-of-
the-art advanced first-principles-based methodologies, viz., hybrid density
functional theory, many-body perturbation theory (G0W0) and density functional
perturbation theory (DFPT), to understand the electronic structure properties
of ABX2Y perovskites. We have chosen the vast composition space of ABX2Y type
perovskites where A and B are cations and X and Y can be nitrogen, oxygen, or
fluorine. These perovskites exhibit direct band gaps ranging from 1.6 to 3.3
eV. Further, to evaluate the feasibility of the visible light catalytic
performance, we calculate the structural, electronic, and optical properties
of ABX2Y perovskites. In addition, from hydrogen evolution reaction (HER) and
oxygen evolution reaction (OER) mechanism, BaInO2F, InSnO2N, CsPbO2F and
LaNbN2O are found as probable photocatalysts.

##

††footnotetext: aDepartment of Physics, Indian Institute of Technology Delhi,
New Delhi, India. Fax: 91 11 2658 2037; Tel: 91 11 2659 1359; E-mail:
<EMAIL_ADDRESS>
##  1 Introduction

The photocatalytic water splitting has become an intense research area and an
excellent way to capture and store energy from the sun 1, 2, 3, 4. Over recent
decades, the number of applications based on photocatalysis increased acutely
5, 6, 7. Although a wide range of materials has been developed for
photocatalytic performance under visible light, most can only absorb light at
a wavelength of less than 500 nm, so only a small portion of the spectrum can
be utilized 8, 9, 10, 11. Therefore, the design of a semiconductor for the
efficient generation of solar fuel requires a suitable band gap (1.5 $\leq$
E${}_{\textrm{g}}$ $\leq$ 2.8 eV) to efficiently absorb visible light, high
dielectric constant, high charge carrier mobility and suitable band positions
in order to perform the hydrogen and oxygen evolution half-reactions (HER/OER)
12, 13.
So far, most of the existing photocatalysts are oxides 14. However, they have
too large band gap to absorb visible light 14, 15. This is mainly due to a
very low valence band (VB) energy which comes from the 2p orbitals of the
oxygen atoms 15. A requirement for visible light induced photocatalysts is
that the optimum band gap energy should be less than 3 eV 16. In order to
solve this problem, non-oxides such as nitrides and sulfides have been
proposed as their VB position is usually higher in energy 17, 18. Nitrogen is
less electronegative than oxygen that leads to reduction in the band gap.
Hence, the optical gap overlaps with the solar spectrum which makes this class
of materials interesting for application as solar absorbers and visible light-
driven photocatalysts 19. Many useful oxynitrides have been reported, such as:
(i) CaTaO2N and LaTaON2 are non toxic solid solutions, (ii) BaTaO2N has a high
dielectric constant and acts as a photocatalyst in water decomposition, and
(iii) EuNbO2N is ferromagnetic and show colossal magneto resistance 20, 21,
22. Also, it has been reported that LaTaON2, LaTiO2N, SrTaO2N and BaTaO2N
drive the HER and OER half-reactions and can be used in Z-scheme
configurations capturing photons in the 600 to 750 nm range 23, 24, 25, 26,
27. Recently, Zhang et al. reported that CoOx modified LaTiO2N has a high
quantum efficiency of 27% at 440 nm towards water oxidation 28.
LaMgxTa1-xO1+3xN2-3x is found to be the first oxynitride to utilize 600 nm
photons in steady-state overall water splitting 25. Moreover, various
oxyfluoride perovskites such as BaFeO2F, SrFeO2F and PbFeO2F have been
discovered. These perovskites exhibit magnetic ordering until a temperature of
around 645 K, 685 K, and >500 K, respectively 29, 30, 31. Multiferroic
behavior was also shown by iron-based oxyfluoride perovskites 32. Also, Samir
et al. proposed the potential existence of a nitrofluoride (LaZrN2F) from the
first-principles based DFT approach. They showed that LaZrN2F composition
exhibits semiconducting properties with iono-covalent behavior 33. However,
despite several research endeavors in the field of oxynitride, oxyfluoride and
nitrofluoride perovskites, no such in-depth theoretical work is available to
address their electronic, optical, dielectric properties and their application
in photocatalytic water splitting.
In this work, we have used various advanced state-of-the-art first-principles
based methodologies under the framework of density functional theory (DFT) 34,
35, many body perturbation theory (MBPT) 36, 37 and density functional
perturbation theory (DFPT) 38, 39 to provide a comprehensive computational
study of oxynitride, oxyfluoride, and nitrofluoride perovskites. Our
objectives are: (i) to provide a list of compositions that could be likely
experimentally synthesized in the perovskite phase, (ii) to study their
electronic structure, (iii) to study their optical and excitonic properties,
and (iv) to find their application in photocatalytic water splitting. To
achieve this goal, we report calculations for the theoretical electrocatalytic
HER and OER overpotentials using DFT.

##  2 Methodology

We have performed a systematic study to explore the structural, electronic and
optical properties using DFT and beyond approaches under the framework of
MBPT. All calculations are performed with Projector Augmented Wave (PAW)
potentials as implemented in Vienna ab initio simulation package (VASP) 40,
41. All the structures are optimized using generalized gradient approximation
(GGA) as implemented in PBE 42 exchange-correlation ($\epsilon_{\textrm{xc}}$)
functional until the forces are smaller than 0.001 eV/Å. The $\Gamma$-centered
2$\times$2$\times$2 k-mesh sampling is employed for optimization calculations
(optimized structures are shown in Fig. 1). The electronic self-consistency
loop convergence is set to 0.01 meV, and the kinetic energy cutoff is set to
600 eV for plane wave basis set expansion. To explore the electronic
properties, hybrid $\epsilon_{\textrm{xc}}$ functional (HSE06) 43, 44 is used.
For all the energy calculations 4$\times$4$\times$4 k-mesh has been used. The
phonon calculations are performed with 2$\times$2$\times$2 supercells using
the PHONOPY package 45, 46. Note that the surfaces (100 plane) were
constructed by cleaving the fully optimized bulk structure with the lowest
energy. Further, surface geometries are optimized with a force convergence
threshold of 0.001 eV/Å. Reciprocal space is sampled by 4$\times$4$\times$1
k-mesh for the (100) surface. A 20 Å vacuum is added to avoid artificial
interaction between periodic images, and a dipole correction is included. The
two-body Tkatchenko-Scheffler vdW scheme has been used to account for van der
Waals interactions 47, 48. Note that the spin-orbit coupling (SOC) is not
taken into consideration because it has negligible affect on the electronic
structure of ABX2Y perovskites (see section I of supplemental information
(SI)).

##  3 Results and Discussion

###  3.1 Crystal Structure

Here, we focus on the oxynitride, oxyfluoride, and nitrofluoride perovskites
i.e., ABX2Y (A = Ba, Ca, La, Sr, Ag, K, Cs, Pr, In ; B = Nb, Ta, Zr, Ti, In,
Pb, Mg, Sn ; and X, Y = O, N, F) perovskites. We have considered in total 18
perovskite structures, out of which 9 are ABO2N, 3 are ABN2O, 4 are ABO2F, 1
is ABF2O and 1 is ABF2N type of perovskite. The oxynitride perovskite of type
ABO2N e.g., InSnO2N, crystallizes in a hexagonal cell and belongs to the polar
space group P63cm, as shown in Fig. 1(a). The optimized lattice constants are
a = b = 6.16 Å and c = 12.26 Å, which are well in agreement with the previous
findings 18. The perovskite type ABN2O e.g., LaNbN2O, crystallizes in an
orthorhombic cell with lattice constants a = 5.78 Å, b = 5.76 Å and c = 10.02
Å (see Fig. 1(b)). AgTiO2F oxyfluoride perovskite has tetragonal cell edges, a
= b = 5.48 Å and c = 7.45 Å and belongs to the space group I4/mcm as shown in
Fig. 1(c). The system CsPbF2O is an interesting perovskite because of the fact
that the perovskite CsPbI3 is the parent inorganic compound with an
orthorhombic crystal structure. This perovskite has lattice constants a =
13.03 Å, b = 4.79 Å and c= 6.68 Å. Finally, LaMgF2N is the only nitrofluoride
system we predict to have possibilities to be realized experimentally with
space group P121/m1. The lattice constants of this perovskite structure are a
= 6.33 Å, b = 3.69 Å and c = 8.25 Å. Similarly, the lattice parameters of
other perovskite structures are mentioned in SI (see section II of SI).

Fig. 1: Optimized crystal structure of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F,
(d) CsPbF2O, and (e) LaMgF2N. Images produced with VESTA 49.

###  3.2 Structural Stability

The stability of perovskites is often discussed based on the Goldschmidt
tolerance factor 50

$t=\frac{r_{\textrm{A}}+r_{\textrm{anion}}}{\sqrt{2}(r_{\textrm{B}}+r_{\textrm{anion}})}$
(1)

where r${}_{\textrm{A}}$, r${}_{\textrm{B}}$ and r${}_{\textrm{anion}}$ are
the ionic radii of the A and B cations, and of the anion, respectively. A more
recent study based on the novel analytics approach has led to the proposition
of a new factor 51

$\tau=\frac{r_{\textrm{anion}}}{r_{\textrm{B}}}n_{\textrm{A}}\left(n_{\textrm{A}}-\frac{r_{\textrm{A}}/r_{\textrm{B}}}{\textrm{log}(r_{\textrm{A}}/r_{\textrm{B}})}\right)$
(2)

where n${}_{\textrm{A}}$ is the oxidation state of A cation. It has been
reported that for a large experimental dataset of perovskites, 0.825 $<$ t $<$
1.08 gives a classification accuracy of 74$\%$, while $\tau$ $<$ 4.18 has an
accuracy of 92$\%$. Now, to use these formulae for mixed anions, we need to
decide the value of r${}_{\textrm{anion}}$ to be used. In line with the
suggestions provided, we decided to use the arithmetic average of the radii of
two mixed anions, i.e., r${}_{\textrm{anion}}$ =
(2r${}_{\textrm{X}}$+r${}_{\textrm{Y}}$)/3. However, it has been noted that
using r${}_{\textrm{anion}}$, the Goldschmidt tolerance factor fails to
capture the stability trend of mixed anion perovskites 52, 53. For this
purpose, the geometric mean has also been used to approximate the radius of
the mixed anions 54. Also, more complicated factors like the octahedral factor
and atomic packing fraction have been proposed to understand the stability of
these perovskites 54, 55. But in our case, for simplicity, we decided to
consider traditional t and $\tau$.

Table 1: Stability parameters of ABX2Y perovskites. ABX2Y | t | $\tau$
---|---|---
InSnO2N | 0.84 | 3.32
LaNbN2O | 0.84 | 3.74
LaTaN2O | 0.84 | 3.74
PrTaN2O | 0.87 | 2.56
AgTiO2F | 0.90 | 3.81
BaInO2F | 0.89 | 4.20
CsPbO2F | 1.07 | 3.49
KTiO2F | 0.98 | 3.52
LaMgF2N | 0.81 | 5.64

From Table 1, we find that the value of t for all the selected perovskites
lies between 1.07 and 0.81, which is perfectly consistent with the usual range
reported for perovskites 51. Now concerning $\tau$, we find that all selected
perovskites have value $\tau$ $<$ 4.18 except LaMgF2N. For other ABO2N
perovskites also, we have calculated the structural stability parameters
mentioned in section III of SI. In addition, to further analyze the dynamic
stability, we have plotted the phonon band structures for the optimized
perovskite structures. The phonon band structures calculated along the high-
symmetry points of the Brillouin zone for InSnO2N, LaNbN2O, AgTiO2F, CsPbF2O,
and LaMgF2N are shown in Fig. 2. Notably, the absence of negative frequencies
confirms the dynamic stability of these perovskites. Similarly, we have
analyzed the phonon band structures of other ABX2Y perovskites in section IV
of SI.

Fig. 2: Phonon band structures of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F, (d)
CsPbF2O, and (e) LaMgF2N.

Note that we have also checked the structural stability of different ABX2Y
perovskites at a higher temperature using ab initio molecular dynamics (AIMD).
We have obtained the radial distribution function g(r) at T = 0 K and T = 300
K by a 6 ps long MD simulation run with NVT ensemble (Nose-Hoover thermostat)
56. We observe that the nature of the radial distribution function for the
nearest neighbors remains the same at room temperature (see section V of SI).
This confirms the stability of these perovskites at operational temperature of
300 K.

###  3.3 Electronic Structure

We have plotted the electronic band structure and partial density of states
(pDOS) for ABX2Y type of perovskites to better understand the role of A/B
cations and X/Y anions near the valence band maximum (VBM) and the conduction
band minimum (CBm). Fig. 3(a) shows the band structure of InSnO2N with a
direct band gap of 1.60 eV at $\Gamma$ using HSE06 $\epsilon_{\textrm{xc}}$
functional. From Fig. 4(a), we can clearly observe that the valence states are
mainly composed of N and O with a small contribution coming from Sn. While the
conduction states have mostly N contribution. Finally, we see very few states
associated with In in the [-6, 6] eV energy window, which is compatible with
the interpretation that the A atom is fully ionized in the perovskite
structure.

Fig. 3: Electronic band structures of (a) InSnO2N, (b) LaNbN2O, (c) AgTiO2F,
(d) CsPbF2O, and (e) LaMgF2N using HSE06 $\epsilon_{\textrm{xc}}$ functional.

The band structure of LaNbN2O is rather different from that of InSnO2N. It has
an indirect band gap of 2.04 eV with the bottom of the CB at $\Gamma$ and the
top of the VB at M (see Fig. 3(b)). The VB is mainly contributed by N and O
states, while the CB comprises of Nb states. Also, the La states are found
only in the VB, indicating that this atom is ionized in this structure (see
Fig. 4(b)). For the other cases of oxynitride perovskites, all the band
structures and PDOS plots are provided in section VI of SI.

Fig. 4: Partial density of states (pDOS) of (a) InSnO2N, (b) LaNbN2O, (c)
AgTiO2F, (d) CsPbF2O, and (e) LaMgF2N using HSE06 $\epsilon_{\textrm{xc}}$
functional.

From the AgTiO2F band structure (see Fig. 3(c)), we see that it has a direct
band gap of 2.32 eV. Also, from Fig. 4(c), we can clearly observe that the VB
states are mostly composed of Ag and O states, while Ti states mostly
contribute to the CB. The HSE06 band structure of CsPbF2O is depicted in Fig.
3(d). The band gap is 1.87 eV which is indirect, with the top of the valence
and bottom of the conduction band (at $\Gamma$) composed of hybridized F, O,
and Pb states. In this case, the separation of the VB into two manifolds is
incomplete, leading to some overlap between the two sets of bands. The bottom
of the CB, on the other hand, is separated by more than 1 eV from the rest of
the CB (see Fig. 4(d)). Similarly, for the other oxyfluoride perovskites, we
have plotted the band structure and PDOS (see section VI of SI).
In case of nitrofluoride perovskite LaMgF2N, we have obtained a large band gap
of 3.31 eV. The highly dispersive bottom of the CB is mainly constructed from
La states, while top of the VB has mostly N states with smaller La
contribution (see Fig. 3(e) and 4(e)). In LaMgF2N, we also see a clear
splitting of the VB. However, due to a strong electronegativity difference of
N and F, the gap between the two manifolds is considerably larger than for the
oxyfluoride systems.

###  3.4 Optoelectronics and Dielectrics

In order to get an in-depth insight into the suitability of a particular
material in optoelectronic applications, a detailed study of its optical
properties like dielectric function, refractive index, extinction coefficient
and absorption coefficient is indispensable. The absorption coefficient
($\alpha$) of a material is computed from the frequency dependent dielectric
constant using the following formulae 57, where $\varepsilon$1 and
$\varepsilon$2 are the real and imaginary terms of the dielectric constant,
respectively:

$\alpha=\frac{4\pi\kappa(\omega)}{\lambda}$ (3)

where $\kappa(\omega)$ is the extinction coefficient, which is given by:

$\kappa(\omega)=\sqrt{\frac{\sqrt{\varepsilon_{2}^{2}+\varepsilon_{1}^{2}}-\varepsilon_{1}}{2}}$
(4)

The absorption coefficient is one of the most essential properties of a
material in terms of its photovoltaic application as it depicts key
information regarding optimal solar energy conversion efficiency. The typical
$\alpha$ for direct semiconductors is of the order 105 58, 59. The theoretical
results show that all the perovskites exhibit high absorption coefficients
(see Fig. 5 ).

Fig. 5: Absorption coefficient of ABX2Y perovskites.

Further, photovoltaic applications require high charge carrier mobility to
reduce nonradiative recombination losses and efficiently transport the
photogenerated carriers to the electrodes. In order to estimate the carrier
mobility, we have calculated the effective mass of electrons and holes in
ABX2Y perovskites from their band structure obtained by using HSE06
$\epsilon_{\textrm{xc}}$ functional (see Fig. 3 and section VI of SI). All the
perovskites have light electrons, with InSnO2N having me= 0.15m0. However, the
holes are heavier, as is typical for most metal oxides, with InSnO2N having a
hole effective mass of mh= 1.75m0, which is in well agreement with the
previous findings 60. Similarly, BaInO2F has me= 0.41m0 and mh= 0.97m0. In
this case also, the hole effective mass is higher than the electron effective
mass. Here, the low effective mass of the electrons in ABX2Y perovskites can
likely make them promising semiconductors. Also, we have calculated the
electronic and ionic dielectric constant of these perovskites and found that
they have a large dielectric constant value (see Table 2), which are in well
agreement with the previous findings 60, 7. To this point, the relevant
properties for solar energy conversion for some of the ABX2Y perovskites are
compiled and summarized in Table 2 and Table S3 of SI.

Table 2: DFT calculated properties of ABX2Y perovskites. Band gap (Eg), electronic dielectric constant ($\varepsilon_{\infty}$), ionic dielectric constant ($\varepsilon_{r}$), effective mass of electron (m${}_{\textrm{e}}^{*}$) and hole (m${}_{\textrm{h}}^{*}$) in terms of the rest mass of electron (m0). ABX2Y | Eg (eV) | $\varepsilon_{\infty}$ | $\varepsilon_{r}$ | m${}_{\textrm{e}}^{*}$ | m${}_{\textrm{h}}^{*}$
---|---|---|---|---|---
InSnO2N | 1.60 | 4.45 | 10.12 | 0.15 | 1.75
LaNbN2O | 2.04 | 7.27 | 61.06 | 0.33 | 0.41
LaTaN2O | 2.11 | 9.28 | 70.58 | 0.36 | 0.51
AgTiO2F | 2.32 | 2.74 | 39.95 | 0.82 | 1.68
BaInO2F | 2.01 | 4.14 | 15.05 | 0.41 | 0.97
CsPbF2O | 1.87 | 4.66 | 14.97 | 0.15 | 0.17

###  3.5 Photocatalytic Water Splitting

In a photocatalytic process, the perovskite material absorbs light as a result
an electron at the VB is transferred to the CB, and a photogenerated e-h pair
is produced. Subsequently, for the oxidation or reduction reaction, the
photogenerated electrons and holes transfer to the corresponding reaction
sites. The photocatalytic reaction can be described by two half-reactions for
the hydrogen evolution and water oxidation which are summarized as follows:

$2\textrm{H}^{+}+2\textrm{e}^{-}\rightarrow\textrm{H}_{2}$ (5)
$2\textrm{H}_{2}\textrm{O}+4\textrm{h}^{+}\rightarrow\textrm{O}_{2}+4\textrm{H}^{+}$
(6)

For the water splitting reaction, the CBm should be higher than the reduction
potential of H+/H2 (E${}_{\textrm{H}^{+}/\textrm{H}_{2}}$ = –4.44 eV at pH =
0.0). On the other hand, the VBM should be lower than the oxidation potential
of O2/H2O (E${}_{\textrm{O}_{2}/\textrm{H}_{2}\textrm{O}}$ = –5.67 eV at pH =
0.0). From the above discussion, we conclude that in order to evaluate the
feasibility of photocatalytic water splitting, it is crucial to calculate
precisely the edge positions of the materials. The calculation of the band
edges using DFT is quite complicated and computationally expensive because the
surface chemistry and interfacial effects are considered in the process. Galli
et al. have summarized recent progress and open theoretical challenges present
in simulations 61. They use an appropriate method to treat the effect of
interface on the band edge position of the surface structure. However, it is
still complex to predict the absolute energy positions of CBm and VBM for the
bulk structure.
Here, we have used a reliable empirical formula proposed by Xu et al. 62 to
calculate the absolute band edges of ABX2Y bulk perovskites using the
following equation:

$E_{\textrm{VBM}}=\chi-\frac{1}{2}E_{\textrm{g}}$ (7)
$E_{\textrm{CBm}}=E_{\textrm{VBM}}+E_{\textrm{g}}$ (8)

where E${}_{\textrm{VBM}}$ and E${}_{\textrm{CBm}}$ represents the absolute
potentials of VBM and CBm. $\chi$ is the electronegativity of the perovskite
which can be determined by the absolute electronegativities of the constituent
atoms as:

$\chi(\textrm{compound})=\chi_{1}^{a}\chi_{2}^{b}\chi_{3}^{c}....\chi_{n}^{m}$
(9)

where $\chi_{1}$, $\chi_{2}$, $\chi_{3}$ and $\chi_{n}$ represent the
electronegativities of the constituent atoms; a, b, c and m are the molar
fractions of the atoms. The electronegativity of the constituent atoms is
calculated using the Mulliken electronegativity 63:

$\chi_{1}=(I+A)/2$ (10)

where I and A is the ionization energy and the electron affinity of the atom,
respectively.

Fig. 6: Band edge alignment of ABX2Y perovskites w.r.t. water redox potential
levels (H+/H2, O2/H2O)

Fig. 6 visually expresses the absolute energy positions of band edges and the
redox potentials of the water splitting reaction. In case of AgTiO2F, the CBm
is shifted downward by a large amount, and hence, their reduction power is
very low and could not be utilized for hydrogen generation from water. For
KTiO2F and LaMgF2N cases, the band gap is very large thus not inducing the
visible-light response. In view of this, from the applicability in
photocatalytic water splitting, BaInO2F, CsPbO2F, LaNbN2O, LaTaN2O and InSnO2N
are the most desirable ones. For other oxynitride perovskites also, we have
plotted the band edge alignment (see section VIII of SI). Now to better
classify these perovskites for different applications, we have discussed two
applications in detail.

####  3.5.1 One-photon water splitting

The one-photon or overall water splitting process is schematized in Fig. 7(a).
In this process, a single photon creates an e-h pair. Further, the electron
and the hole reach two different regions of the surface to avoid
recombination, and evolve hydrogen and oxidize water, respectively. The
criteria for a material to be used for solar light capture that we have
considered here are: (i) structural stability, (ii) band gap in the visible
light range, i.e., 1.5 $\leq$ E${}_{\textrm{g}}$ $\leq$ 3 eV, and (iii) band
edges straddling with the oxygen and hydrogen evolution potentials i.e., VB
$>$ 1.23 eV and CB $<$ 0 eV. From Fig. 6, we have found various ABX2Y
perovskites (e.g., BaInO2F, CsPbO2F, LaNbN2O, InSnO2N, etc.) which are
suitable for collecting visible light and for the evolution of both hydrogen
and oxygen.

Fig. 7: (a) Overall water splitting scheme, (b) Scheme of a tandem cell with a
transparent protecting shield on the oxygen evolution photocatalyst.

####  3.5.2 Transparent shield

Photocorrosion is one of the major problems related to the use of materials
for oxygen production. In order to overcome this issue, it is required to
develop a highly stable and transparent thin film to cover the oxygen
evolution photocatalyst as a protective shield (see Fig. 7(b)). For this
purpose, the material should have a large band gap of, say, more than 3 eV.
Fig. 7(b) illustrates a device as a photoanode protecting transparent shield
for a tandem cell. Here, the electron with higher energy evolves hydrogen,
while the hole at lower energy moves through the transparent film and evolves
oxygen. The criteria for such material are: (i) a direct band gap in the UV
range in order to avoid absorbing a part of the visible light spectrum, (ii)
VB needs to be placed between the edges of the photocatalyst and the potential
of oxygen evolution so that the hole reaches the reaction region with a small
energy loss i.e., 1.7 $\leq$ VB${}_{\textrm{edge}}$ $\leq$ 2.8 eV. The same
idea can be used to develop a photocathode protecting shield 64. The only
change is the position of the CB i.e., -0.7 $\leq$ CB${}_{\textrm{edge}}$
$\leq$ 0 eV. From Fig. 6, we find that KTiO2F and LaMgF2N can be used as a
protective transparent shield.

###  3.6 HER and OER mechanism

The HER reaction mechanism is schematized in Fig. 8(a). In HER mechanism, the
reaction steps follow: (i) adsorption of H, (ii) hydrogen reduction and (iii)
desorption as H2 65. The Gibbs free energy of hydrogen reaction
($\Delta$G${}_{\textrm{H}}$) is a reasonable descriptor for HER reaction. In
order to calculate the Gibbs free energy changes ($\Delta$G) of the
intermediates in HER, the following expression is used:

$\Delta G_{\textrm{H}}=\Delta E_{\textrm{H}}+\Delta E_{\textrm{ZPE}}-T\Delta
S_{\textrm{H}}$ (11)

where,

$\Delta
E_{\textrm{H}}=E_{\textrm{total}}-E_{\textrm{surface}}-1/2E_{\textrm{H}_{2}}$
(12)

In the above mentioned equations, $\Delta$E${}_{\textrm{H}}$ is the hydrogen
binding energy on the perovskite surface, $\textit{E}_{\textrm{total}}$ is the
total energy for the adsorption state, $\textit{E}_{\textrm{surface}}$ is the
energy of the pure surface, $\textit{E}_{\textrm{H}_{2}}$ is the energy of
hydrogen molecule, $\Delta E_{\textrm{ZPE}}$ is the change in zero-point
energy and $\Delta S_{\textrm{H}}$ is the difference in entropy. At 298 K,
$\Delta E_{\textrm{ZPE}}-T\Delta S_{\textrm{H}}$ = 0.25 eV is well established
in literature 66. The HER diagram along the reaction pathway
$\textrm{H}^{+}+\textrm{e}^{-}\rightarrow\textrm{H}^{*}\rightarrow
1/2\textrm{H}_{2}$ is shown in Fig. 8(c). Under the conditions of pH = 0 and a
standard hydrogen electrode (SHE) potential of 0 V,
$\textrm{H}^{+}+\textrm{e}^{-}$ is equivalent to 1/2H2. From the above
discussion, we found BaInO2F, InSnO2N, CsPbO2F and LaNbN2O as probable
photocatalysts.

Fig. 8: (a) HER steps, (b) OER steps, (c) HER free energy diagram, and (d) OER
free energy diagram for ABX2Y perovskites at an electrode potential
E${}_{\textrm{SHE}}$=0 V and pH=0.

Similarly, the OER mechanism involves four proton-coupled electron transfer
steps on metal ion centers with OH, O, and OOH reaction intermediates, where
$*$ indicates the active metal site (see Fig. 8(b)):

${}^{*}+\textrm{H}_{2}\textrm{O}\rightarrow\textrm{${}^{*}$OH}+\textrm{H}^{+}+\textrm{e}^{-}$
(13)
$\textrm{${}^{*}$OH}\rightarrow\textrm{${}^{*}$O}+\textrm{H}^{+}+\textrm{e}^{-}$
(14)
$\textrm{${}^{*}$O}+\textrm{H}_{2}\textrm{O}\rightarrow\textrm{${}^{*}$OOH}+\textrm{H}^{+}+\textrm{e}^{-}$
(15)
$\textrm{${}^{*}$OOH}\rightarrow^{*}+\textrm{O}_{2}+\textrm{H}^{+}+\textrm{e}^{-}$
(16)

It has already been reported that for ABO3 perovskites, mostly the B site is
considered as the active site for OER reaction steps. Hence, in the case of
ABX2Y perovskites, we have considered the B site as the active site 67, 68.
Now, a computational analysis of the OER is often simplified by only
considering the ∗OH,∗O, and ∗OOH reaction intermediates appearing in the
electrochemical steps, as shown in equations 13\- 16 69, 70, 71, 72.
Following this, we have estimated the catalytic activity of a specific
reaction site by the largest electrochemical step in the OER free energy
profile

$\Delta G^{\textrm{OER}}=\textrm{max}[\Delta G_{i}]$ (17)

where $\Delta G_{i}$ is the change in free energy of the electrochemical OER
step. Then, we derive the thermodynamic overpotential as

$\eta^{\textrm{OER}}=\Delta G^{\textrm{OER}}/e-1.23V$ (18)

where 1.23 V is the equilibrium potential 73. Now, the reaction steps ($\Delta
G_{i}$) are calculated from differences in the adsorption energies of the
various intermediates as given by:

$\Delta
G_{i}=G_{\textrm{total}}-G_{\textrm{surface}}-nG_{\textrm{H}_{2}\textrm{O}}-mG_{\textrm{H}_{2}}$
(19)

where n and m are the stoichiometry coefficients that preserve the number of
atoms on both sides of the respective reaction. Here, we have calculated the
adsorption energies under standard conditions (pH = 0, T= 298 K) and U = 0 V
using the computational hydrogen electrode, as the theoretical overpotential
does not depend on the pH or the potential 74, 75. In this case, the energy of
a proton and an electron equals half the energy of a hydrogen molecule. Fig.
8(d) shows the free energy profile for each elementary step of OER and we have
found that for each case, the overall potential is determined by $\Delta
G_{2}$ step (for details see section IX of SI). From the above results, we
obtain BaInO2F, InSnO2N, CsPbO2F and LaNbN2O as probable photocatalysts for
OER.

##  4 Conclusion

In summary, from our extensive computational study of structural, electronic,
optical and photocatalytic properties of quaternary oxynitride, oxyfluoride,
and nitrofluoride perovskites, we propose them as utmost prospective
candidates for the efficient absorption and conversion of solar energy into
storable fuel. Initially, using different tolerance factor and phonopy
calculations, we have checked the structural and dynamical stability,
respectively. After that, from the band structure and pDOS calculations, we
have found that the varying electronegativity factor affects the electronic
properties. Further, the optoelectronic and dielectric calculations reveal
that these perovskites will enable several promising applications in solar
energy harvesting through efficient solar cells and visible light
optoelectronics. The results show that BaInO2F, CsPbO2F, LaNbN2O, and InSnO2N
have appropriate band edges for the overall water splitting. However, KTiO2F
and LaMgF2N can be used as a protective transparent shield. Following this, we
have calculated the theoretical overpotentials for HER and OER. Low calculated
overpotentials for HER and OER suggest that certain materials may merit
further study as candidates for good photocatalysts. The detailed theoretical
investigation presented in this work will surely help future studies to
improve the stability, engineer the band gap of perovskite absorbers and good
photocatalysts for water splitting.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

MJ acknowledges CSIR, India, for the senior research fellowship [grant no.
[09/086(1344)/2018-EMR-I]]. DG acknowledges UGC, India, for the senior
research fellowship [grant no. [1268/(CSIR-UGC NET JUNE 2018)]]. SM
acknowledges IIT Delhi for the financial support. SB acknowledges financial
support from SERB under a core research grant (grant no. CRG/2019/000647) to
set up his High Performance Computing (HPC) facility “Veena” at IIT Delhi for
computational resources.

## Notes and references

  * 1

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