# Exploring Exciton and Polaron Dominated Photo-physical Phenomena in
Ruddlesden-Popper Phases of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) from
Many Body Perturbation Theory

Deepika Gill<EMAIL_ADDRESS>Arunima Singh  Manjari Jain
Saswata Bhattacharya<EMAIL_ADDRESS>[

###### Abstract

Ruddlesden-Popper (RP) phases of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) are
evolved as new promising class of chalcogenide perovskites in the field of
optoelectronics, especially in solar cells. However, detailed studies
regarding its optical, excitonic, polaronic and transport properties are
hitherto unknown. Here, we have explored the excitonic and polaronic effect in
RP phases of Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$
(n=[1-3]) using several first-principles based state-of-the-art methodologies
under the framework of Many Body Perturbation Theory. Unlike it’s bulk
counterpart, the optical and excitonic anisotropy are observed in
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases. From Wannier-Mott approach, we show that in the RP phases of this
class of chalcogenide perovskites, capturing the ionic contribution to the
dielectric constant is important. We report significant ionic contribution and
relatively smaller electron-phonon coupling constant for
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ in comparison to
the bulk BaZrS3. The exciton binding energy is found to be dependent on the
presence of large electron-phonon coupling. The charge carrier mobility is
maximum in Ba2ZrS4, computed employing deformation potential of the same. As
per our analysis, the optical phonon modes are observed to dominate the
acoustic phonon modes, leading to decrease in polaron mobility on increasing n
in Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]).

###### keywords:

Ruddlesden-Popper phases, hybrid, DFT, GW-BSE, exciton binding energy, polaron

+91-11-2659 1359 Indian Institute of Technology Delhi] Department of Physics,
Indian Institute of Technology Delhi, New Delhi, India

Perovskites with general chemical formula ABX3 have found great attention in
dielectric, optoelectronics, and solar cell applications due to their superb
ferroelectric, piezoelectric, superconductive and photovoltaic properties 1,
2, 3. During the last decade, in the field of solar cell applications, hybrid
lead-halide perovskites namely, CH3NH3PbX3 and CH(NH2)2PbX3 (X = Cl, Br and I)
have achieved great success owing to their small band gap, long carrier
mobility, low manufacturing cost and high power conversion efficiency 4, 5, 6,
7, 8, 9, 10. However, due to the presence of organic molecules, the stability
of perovskites is affected towards heat, light and moisture, thereby degrading
their efficiency with time in the practical world 11. Moreover, presence of
toxic lead in these materials makes them hazardous for the environment 12.
These shortcomings have hindered their practical applications.

In search of alternative perovskites that can alleviate the limitations of
lead-halide perovskites, chalcogenide perovskites with S or Se-anion have been
proposed for photovoltaic applications 13. Several prototypical chalcogenide
perovskites (viz. SrHfS3 14, AZrS3 (A = Sr, Ca, Ba) 15, 16, 17, along with
their related phases) have been synthesized successfully. Amongst them, BaZrS3
consists of earth-abundant elements and is having moderate band gap
($\sim$1.82 eV 18) ideal for photovoltaics. Moreover, it also exhibits
ambipolar doping 19 and is stable against different environmental conditions
20. In order to optimize the solar cell absorption, doping at Ba/Zr-sites have
been attempted in this material 21, 17, 22. However, such doped/alloyed
configurations seem to lack stability 23. Thin films of BaZrS3 are also
reported, which has directed the research towards its new phases named as
Ruddlesden-Popper (RP) phases 24. Tremendous efforts have been invested to
tune the electrical and optical properties 18. The RP phases as an imitative
of the perovskite structure are evolving as a semiconductor for optoelectronic
applications 25, 26. Their general formula is
A${}_{\textrm{n+1}}$B${}_{\textrm{n}}$X${}_{\textrm{3n+1}}$, where perovskite
structure blocks of unit cell thickness “n” are separated by rock salt layer
AX along [001] direction. Alternate perovskite blocks are displaced in in-
plane direction by half of the unit cell. The RP phases are included in the
broad category of “2D perovskites” owing to their layered structural
arrangement (see Fig. 5).

Note that several studies assigning to the layered perovskites as “2D
perovskites” exist in the literature, where periodic stacking of perovskite
layers result in a bulk structure. Their material properties can be tuned
either by substitution or dimensional reduction 27, 28, 29. Due to quantum
confinement effects 30, considerable change in bulk physical properties (such
as bulk modulus, elastic modulus, charge carrier properties and optical
properties) can be seen on reducing the dimension of material 31, 32. Research
in this field is highly evolving 26, 33, 34, 35.

In optoelectronic materials exciton formation greatly influence the charge
separation properties and hence, excitonic parameter such as exciton binding
energy (E${}_{\textrm{B}}$) acts as an important descriptor for optoelectronic
applications. Solar cell performance depends upon the fraction of thermally
dissociated excitons into electrons and holes, giving rise to the free-charge
carriers. In addition, the concept of polarons has been used to explain
multiple photo-physical phenomena in these materials 36. Polaronic effects
have been suggested to play an important role in the excitation dynamics and
carrier transport. The separation of free charge is also influenced by the
carrier mobility. Hence understanding the effect of electron-phonon coupling
in terms of polaron mobility is important. A systematic study on the excitonic
and polaronic effect in the RP phases of BaZrS3 is hitherto unknown. The
present Letter, therefore, explores the excitonic properties along with
polaronic effect in RP phases of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) under
the framework of Many Body Perturbation Theory. The electron-phonon coupling
is also taken care of using Frohlich model to compute the polaron mobility.

The exciton binding energy is defined as the energy required to decouple the
exciton into individual electron and hole pair. Theoretically, the exciton
binding energy (E${}_{\textrm{B}}$) is calculated by taking the difference of
the energy of bounded electron-hole (e-h) pair (i.e., BSE gap) and unbounded
e-h pair (i.e., GW gap). In order to determine the optical response of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases, we have calculated the imaginary part of dielectric function (Im
($\epsilon$)). Initially, we have benchmarked the exchange-correlation
($\epsilon_{\textrm{xc}}$) functional for our system. As it is already known
that single shot GW (G0W0) calculation strongly depends on the starting point,
we need to validate the suitable starting point for G0W0 calculation. Note
that spin-orbit coupling (SOC) effect is negligible in these systems (see Fig.
S3). Hence, we have excluded SOC in our calculations (see Fig. S5 in SI). The
band gap of Ba2ZrS4, Ba3Zr2S7 and Ba4Zr3S10 are quite underestimated using PBE
and the values are 0.61 eV, 0.42 eV and 0.34 eV, respectively. On the other
hand, the same with default parameters (viz. exact exchange = 25$\%$ and
screening parameter 0.2 Å-1) of HSE06 are 1.39 eV, 1.18 eV and 1.08 eV,
respectively. The HSE06 numbers are in good agreement with the experimental
findings 18. Further, the peak position, which is underestimated by PBE is
improved by G0W0@PBE. Notably, the quasiparticle gaps computed using G0W0@PBE
are overestimated in comparison to experimental band gap, since it does not
take into account the exciton binding energy. However, the gaps are improved
by employing BSE on top of G0W0@PBE. We find, the optical peak position of
Ba2ZrS4, by performing G0W0@PBE and G0W0@HSE06 is 2.11 eV (see Fig. 6(a)) and
2.17 eV (see Fig. S4 in SI), respectively. Since, the results obtained from
the consideration of HSE06 as the starting point of G0W0 deviate more from the
experimental results, we have performed all the calculations using G0W0@PBE.
Although G0W0@PBE gives better result than G0W0@HSE06, it is still far from
the experimental optical band gap i.e., 1.33 eV.

Figure 1: Optimized crystal structure of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3])
Ruddlesden-Popper phases (RP phases).

However, it gets improved when we solve the BSE to obtain optical band gap due
to the fact that BSE takes into account the excitonic effect. The latter is
ignored in G0W0 calculation to obtain accurate optical spectra. Therefore, we
have performed BSE@G0W0@PBE to incorporate e-h interactions. Similarly, we
have performed G0W0@PBE and BSE@G0W0@PBE calculations to capture the optical
and excitonic effect for Ba3Zr2S7 and Ba4Zr3S10, respectively. We find the BSE
peak position for Ba2ZrS4, Ba3Zr2S7 and Ba4Zr3S10 are 1.71 eV, 1.49 eV and
1.43 eV, respectively, whereas G0W0@PBE finds the peak at 2.11 eV, 1.82 eV and
1.69 eV, respectively (see Fig. 6(a-c)).

It should be noted here that these numbers are highly dependent on the k-mesh
and it’s very challenging (even with the fastest supercomputers) to converge
the BSE calculation to obtain the excitonic peak for computing
E${}_{\textrm{B}}$ with absolute accuracy. In Fig. 6, occurrence of red-
shifted peak in BSE@G0W0@PBE than G0W0@PBE signifies the excitonic effect in
the considered RP phases. The computed E${}_{\textrm{B}}$ of first bright
exciton of Ba2ZrS4, Ba3Zr2S7 and Ba4Zr3S10 RP phases are found to be 0.40 eV,
0.33 eV and 0.26 eV, respectively. The obtained E${}_{\textrm{B}}$ values are
somewhat on the larger side. In solar cell, for easy dissociation of exciton
into free charge carriers (viz. e and h) at room temperature, a low
E${}_{\textrm{B}}$ is desirable ($k_{\textrm{B}}$T = 26 meV, T = 300 K). Thus
the discrepancy in the BSE peak position from the experimental value 18 may
lead to incorrect E${}_{\textrm{B}}$ values. Unfortunately, we have already
ensured the highest possible k-mesh to compute the BSE@G0W0@PBE calculations.
Involving a denser k-mesh is not feasible for the superstructures of RP phases
for computing G0W0@PBE and BSE@G0W0@PBE – solely due to computational
limitation. Therefore, this limits us to estimate the accurate
E${}_{\textrm{B}}$ for the given systems. This is why to compute
E${}_{\textrm{B}}$, we have employed a combined state-of-the-art method
comprising of Wannier-Mott and Density Functional Perturbation Theory (DFPT)
approach as explained later. However, one can estimate the error bar in BSE
peak position via a parameterized model for dielectric screening i.e., model-
BSE approach (mBSE). This mBSE approach is computationally cheaper as compared
to BSE but with similar accuracy to estimate the first peak position (see Fig.
S6 in SI). Thus this enables us to sample Brillouin zone with higher number of
k-mesh. As per the mBSE calculation, it is seen that a denser k-mesh sampling
indeed red-shifts the BSE peak by $\sim$ 0.3 eV (see mBSE calculations details
as in section V in SI).

Now, from the above analysis, despite having little discrepancy with the exact
BSE peak position and corresponding E${}_{\textrm{B}}$ values, it’s certain
that the first two excitons are bright excitons in the considered RP phases
and several dark excitons also exist below the second bright exciton of these
systems. From the above studies, it’s also expected to have the correct trend
i.e., the order of E${}_{\textrm{B}}$ for
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ is n=1 $>$ n=2
$>$ n=3. From Fig. 6(a), we have observed the double peak character for
excitonic peak of Ba2ZrS4 that may occur due to two nearby transitions.
Moreover, only Ba2ZrS4 exhibits direct band gap, whereas for other two higher
RP phases in the series band gap becomes indirect in nature. In Fig. 8, we
have shown the broadening of excitonic peak. It is well known that lifetime
($\tau$) of exciton is inversely proportional to the same and therefore, the
qualitative trend of exciton lifetime for
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) is
$\tau_{n=3}$ $>$ $\tau_{n=2}$ $>$ $\tau_{n=1}$. Note that, here broadening is
computed from mBSE approach showing contribution due to the electron-hole
interaction and it does not include the electron-phonon coupling effect.

Figure 2: Imaginary part (Im ($\epsilon$)) of the dielectric function for (a)
Ba2ZrS4, (b) Ba3Zr2S7, (c) Ba4Zr3S10 using single shot GW (G0W0) and BSE.
Imaginary part (Im($\varepsilon$)) of dielectric function for (d) Ba2ZrS4 (e)
Ba3Zr2S7 and (f) Ba4Zr3S10 along E $||$ xy direction and imaginary part
(Im($\varepsilon$)) of dielectric function for (g) Ba2ZrS4 (h) Ba3Zr2S7 and
(i) Ba4Zr3S10 along E $||$ z direction, using G0W0 and BSE. Figure 3: Full
width at half maximum (FWHM) of exciton peak using mBSE approach with dense
k-mesh. Broadening of exciton peak is mainly due to electron-hole interaction.

Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases have tetragonal structure and exhibit optical anisotropy. Hence, it is
required to study their optical and excitonic properties along E$||$xy (i.e.,
in-plane along x- or y-direction) and E$||$z (i.e., out-of-plane along
z-direction) direction. We have observed anisotropy in Ba2ZrS4, Ba3Zr2S7 and
Ba4Zr3S10 which can greatly affect their performance in practical application.
Therefore, it is of paramount importance to understand the anisotropic effect
in the optical and excitonic properties of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases. In Fig. 6 (d-i), we have shown optical and excitonic contribution of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases along different directions viz. x, y and z. Employing Shockley-Queisser
(SQ) criterion 37 for the solar cell and other optoelectronic devices, we can
remark that Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$
(n=[1-3]) RP phases are optically active in in-plane (i.e., along x- and
y-direction) and optically inactive in out-of-plane (i.e., z-direction). These
systems possess similar optical as well as excitonic properties along x- and
y-direction (see Fig. 6(d-f)). However, along z-direction, their optical and
excitonic spectra are not only blue-shifted but also the feature of G0W0 and
BSE peaks are quite different than that in case of in-plane direction (see
Fig. 6(g-i)). It is well known that exciton lifetime is inversely proportional
to the width of the exciton peak. Hence, change in the feature of exciton peak
greatly influences the excitonic parameters as well in different directions.
The d-orbital contribution in valence band maximum (VBM) and conduction band
minimum (CBm) could be responsible for the optical anisotropy in these
systems. Here, Zr 4dxy orbital is contributing maximum in the CBm (see Fig. S7
in SI), whereas in bulk BaZrS3 the CBm is contributed by dyz and dxz. This is
why in the latter, no significant anisotropy is observed 38.

We have used Wannier-Mott model 39 for a simple screened Coulomb potential.
According to this model, the E${}_{\textrm{B}}$ for screened interacting
electron-hole (e-h) pair is given by:

$\begin{split}\textrm{E}_{\textrm{B}}=\left(\frac{\mu}{\epsilon_{\textrm{eff}}^{2}}\right)\textrm{R}_{\infty}\end{split}$
(1)

where, $\mu$ is the reduced mass in term of rest mass of electron,
$\epsilon_{\textrm{eff}}$ is the effective dielectric constant (which includes
electronic as well as ionic contribution to dielectric constant) and R∞ is the
Rydberg constant. The reduced mass of Ba2ZrS4, Ba3Z2S7 and Ba4Zr3S10 in term
of electron rest mass are 0.32, 0.26 and 0.34, respectively (for more
information regarding calculation of reduced mass and effective mass of
electron and hole see section VII in SI). However, in the above expression,
$\epsilon_{\textrm{eff}}$ is still unknown for these systems. It is already
reported that lattice relaxation can influence the exciton binding energy 40.
For example, if $\omega_{\textrm{LO}}$ corresponds to longitudinal optical
phonon frequency and E${}_{\textrm{B}}$ $<<$ $\hbar\omega_{\textrm{LO}}$, one
needs to consider the effect of lattice relaxation to compute
$\epsilon_{\textrm{eff}}$. However, if E${}_{\textrm{B}}$ $>>$
$\hbar\omega_{\textrm{LO}}$, the effect of lattice relaxation can be ignored
as in such cases $\epsilon_{\textrm{eff}}$ $\rightarrow$ $\epsilon_{e}$, where
$\epsilon_{e}$ is the static value of dielectric constant at high frequency
that mainly consists of electronic contribution.

Therefore, for $\epsilon_{\textrm{eff}}$, a value intermediate between the
static electronic dielectric constant at high-frequency i.e., $\epsilon_{e}$
and the static ionic dielectric constant at low frequency i.e., $\epsilon_{i}$
should be considered.

Figure 4: Electronic (Im ($\varepsilon_{e}$) and Re ($\varepsilon_{e}$))
(a)-(c) and ionic (Im ($\varepsilon_{i}$) and Re ($\varepsilon_{i}$)) (d)-(f)
contribution to dielectric function for Ba2ZrS4, Ba3Zr2S7 and Ba4Zr3S10. Red
and black color correspond to real (Re($\varepsilon$)) and imaginary
(Im($\varepsilon$)), respectively.

In Fig. 7(a-c) and Fig. 7(d-f), we have shown the electronic and ionic
contribution to the dielectric function respectively, computed using DFPT
approach. The static real part of ionic dielectric constant for Ba2ZrS4,
Ba3Z2S7 and Ba4Zr3S10 are 39.35, 31.19 and 57.59, respectively. As per Fig. 7,
a considerable increase in the static low frequency of ionic dielectric
constant is attributed to the occurrence of optically active phonon modes
below 10 meV. This shows the ionic nature of the RP phases. Using electronic
and ionic contribution of the dielectric constant and equation 11, we have
calculated the upper and lower bound for the E${}_{\textrm{B}}$ (see Table4).
The effective value of the dielectric constant and hence, the binding energy
lies in between these upper and lower bounds listed in Table4. We have
observed that upper bound and lower bound values of E${}_{\textrm{B}}$ for the
considered RP phases are comparable with that of APbX3 (A = MA, FA; X= I, Br)
perovskites 41, 42. It should be noted that in bulk BaZrS3 ionic contribution
to dielectric constant has been observed to be negligible 38, whereas in RP
phases significant role of ionic contribution to dielectric has been observed.

Table 1: Electronic and ionic contribution to the dielectric constant for Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP phases, where $\epsilon_{e}$ and $\epsilon_{i}$ correspond to the static value of electronic and ionic dielectric constant, respectively. E${}_{\textrm{Bu}}$ and E${}_{\textrm{Bl}}$ correspond to upper and lower bound of exciton binding energy, respectively. Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ | $\epsilon_{e}$ | E${}_{\textrm{Bu}}$ (meV) | $\epsilon_{i}$ | E${}_{\textrm{Bl}}$ (meV)
---|---|---|---|---
Ba2ZrS4 | 8.94 | 56.00 | 39.25 | 2.9
Ba3Zr2S7 | 8.32 | 50.14 | 31.19 | 3.49
Ba4Zr3S10 | 8.51 | 63.45 | 57.59 | 1.39

Recently, Ming et al. have reported the effect of strain on the band gap and
octahedron rotation for Ba2ZrS4. 43. According to their report, a significant
change in the band gap and octahedron rotation is observed with the
application of strain. However, in our case of Ba2ZrS4 RP phase, monotonic
change in the band gap (but not as large as in case of Ming et al.) has been
observed on applying upto $\pm 7\%$ strain along b-axis and c-axis (see Fig.
S9 in SI). The effect on the band gap along b- and a-direction are symmetric.
Further, the effect of strain on the band gap in out-of-plane direction is
more significant than in in-plane direction. We have also noticed slight
octahedral tilt under the application of strain along b-axis. In case of
strain along c-axis, a small rotation of octahedron about c-axis has been
observed. Here, we have calculated the elastic modulus of the RP phases that
is given by

$\begin{split}\textrm{C}_{\textrm{3D}}=\left(\frac{1}{\textrm{V}}\right)\frac{\delta^{2}\textrm{E}}{\delta
s^{2}}\end{split}$ (2)

where V, s and E correspond to the volume of a unit cell, strain and total
energy, respectively. We have observed that elastic modulus of the RP phases
is larger than 2D RP phases 43. In order to compute carrier mobility, we have
used deformation potential model 44, 45, 46, 47. According to this model, the
mobility of charge carrier is defined as:

$\begin{split}\mu_{\textrm{DP}}=\frac{(8\pi)^{\frac{1}{2}}\hbar^{4}e\textrm{C}_{\textrm{3D}}}{3(m^{*})^{5/2}(k_{\textrm{B}}\textrm{T})^{3/2}\textrm{E}_{l}^{2}}\end{split}$
(3)

where T is the temperature, m∗ is the effective mass of charge carrier and e
is the elementary charge of electron. El =
$\Delta\textrm{V}/(\Delta\textrm{l}/\textrm{l}$) is the deformation potential
(for more details regarding calculation of C${}_{\textrm{3D}}$ and El see
section VIII in SI). In Table 5, we have listed the values of
C${}_{\textrm{3D}}$, El and $\mu_{\textrm{DP}}$ for electron and hole of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases. As elastic modulus decreases down the column (see Table 5), we can say
that softness of the RP phases increases on increasing the value of n in
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]).
Mobility of electron also decreases on increasing n in
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]).

Table 2: Elastic modulus, deformation potential and predicted carrier mobility of Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP phases. Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ | C${}_{3\textrm{D}}$ (eV Å-3) | El (eV) |  $\mu_{\textrm{DP}}$ (cm2 V-1 s-1)
---|---|---|---
Ba2ZrS4 (e) | 1.23 | 6.36 | 2.16$\times$104
Ba2ZrS4 (h) | 1.23 | 6.61 | 2.32$\times$103
Ba3Z2S7 (e) | 0.92 | 6.41 | 8.63$\times$103
Ba3Zr2S7 (h) | 0.92 | 6.39 | 4.72$\times$101
Ba4Zr3S10 (e) | 0.67 | 6.72 | 4.22$\times$103
Ba4Zr3S10 (h) | 0.67 | 6.73 | 8.27$\times$101
Table 3: Polaron parameters for
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases.

Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ 1/$\epsilon^{*}$
$\alpha$ $\textrm{m}_{\textrm{P}}$/$\textrm{m}^{*}$ l${}_{\textrm{P}}$ (Å)
$\mu_{\textrm{P}}$ (cm2V-1s-1) Ba2ZrS4 0.09 1.84 1.39 354.30 164.75 Ba3Z2S7
0.08 2.36 1.53 307.12 117.20 Ba4Zr3S10 0.10 1.77 1.37 292.49 76.39

After analyzing the specific free volume (for details see section IX in SI),
we find that study of electron-phonon coupling is important in these
materials. Also, the presence of polarization in the RP phases lays emphasis
on the polaron study. We have examined the electron-phonon interaction in our
system by the mesoscopic model, viz. Frohlich’s model 48, 49, 50 for the
polarons. The dressed “quasiparticles”, formed due to screened interaction of
electron and hole by the lattice, are known as polarons. Frohlich introduced a
parameter to describe theoretically the momentum of electron in the field of
polar lattice vibration. This parameter is known as dimensionless Frohlich
coupling constant 51, 48, 49.

$\begin{split}\alpha=\frac{1}{\epsilon^{*}}\sqrt{\frac{\textrm{R}_{\textrm{y}}}{\textrm{ch}\omega_{\textrm{LO}}}}\sqrt{\frac{\textrm{m}^{*}}{\textrm{m}_{\textrm{e}}}}\end{split}$
(4)

where coupling constant $\alpha$ quantifies the electron-phonon coupling, m∗
is the effective mass of electron, m${}_{\textrm{e}}$ is the rest mass of the
electron, h is Planck’s constant, c is the speed of light,
$\omega_{\textrm{LO}}$ (in [cm-1] units) is the optical phonon frequency,
$1/\epsilon^{*}$ is the ionic screening parameter ($1/\epsilon^{*}$ =
$1/\epsilon_{\infty}$ \- $1/\epsilon_{\textrm{static}}$ were,
$\epsilon_{\textrm{static}}$ and $\epsilon_{\infty}$ are static and high
frequency dielectric constant) and $\textrm{R}_{\textrm{y}}$ is the Rydberg
energy. We have observed that electron-phonon coupling constant of considered
RP phases (Table6), are smaller than that of their bulk BaZrS3 38. Further,
using the extended form of Frohlich’s polaron theory, given by Feynman, the
effective mass of polaron (m${}_{\textrm{P}}$) 49 is defined as:

$\begin{split}\textrm{m}_{\textrm{P}}=\textrm{m}^{*}\left(1+\frac{\alpha}{6}+\frac{\alpha^{2}}{40}+......\right)\end{split}$
(5)

where m∗ is the effective mass calculated from the band structure calculations
(see section VII in SI). The polaron radii 52 can be calculated as follows:

$\begin{split}\textrm{l}_{\textrm{P}}=\sqrt{\frac{h}{2c\textrm{m}^{*}\omega_{\textrm{LO}}}}\end{split}$
(6)

Polaron mobility according to the Hellwarth polaron model is defined as
follows:

$\begin{split}\mu_{\textrm{P}}=\frac{(3\sqrt{\pi}e)}{2\pi\textrm{c}\omega_{\textrm{LO}}\textrm{m}^{*}\alpha}\frac{\textrm{sinh}{(\beta/2)}}{\beta^{5/2}}\frac{w^{3}}{v^{3}}\frac{1}{K}\end{split}$
(7)

where, $\beta$ = hc$\omega_{\textrm{LO}}$/$k_{\textrm{B}}$T, e is the
electronic charge, m∗ is the effective mass of charge carrier, w and v
correspond to temperature dependent variational parameters. K is a function of
v, w, and $\beta$ 50 i.e., defined as follows:

$\begin{split}K(a,b)=\int_{0}^{\infty}du[u^{2}+a^{2}-b\textrm{cos}(vu)]^{-3/2}\textrm{cos}(u)\end{split}$
(8)

Here, a2 and b are calculated as:

$\begin{split}a^{2}=(\beta/2)^{2}+\frac{(v^{2}-w^{2})}{w^{2}v}\beta coth(\beta
v/2)\end{split}$ (9)
$\begin{split}b=\frac{(v^{2}-w^{2})}{w^{2}v}\frac{\beta}{\textrm{sinh}(\beta
v/2)}\end{split}$ (10)

We have used lowest frequency of LO phonon i.e, 1.34, 1.46 and 1.36 THz for
Ba2ZrS4, Ba3Zr2S7 and Ba4Zr3S10, respectively for our calculation.
$\mu_{\textrm{P}}$ gives the upper limit of the charge carrier mobility, under
the assumption that charge carrier (electron) interact only with the optical
phonon. A significant change in the mobility of charge carrier can be seen on
comparing the mobility of electron without including its interaction with the
optical phonon (see Table 5) and with including interaction with the optical
phonon (see Table 6). Here in Table 6, ionic screening is indicative of the
ionicity for a system. On comparing our results with the hybrid inorganic-
organic halide perovskites (ionic screening of MAPbI3, MAPbBr3 and MAPbCl3 are
0.17, 0.18 and 0.22, respectively 52), we can say that Ba2ZrS4, Ba3Zr2S7 and
Ba4Zr3S10 are less ionic than MAPbX3 (X = Cl, Br, I). Also, the obtained
coupling constant is comparable or larger than MAPbX3 (see Table 6). The
lowering of mobility of charge carriers on the inclusion of LO phonon modes
indicate that optical phonon modes are dominating over the acoustical phonon
modes in these materials. Note that, in the absence of experimental data,
these results may help as guideline for further research. Moreover, for
qualitative analysis, our results are very informative to understand the
charge transport properties of these RP phases. From Table6, we can clearly
see that on increasing n in
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) i.e.,
down the column the polaron mobility decreases and for bulk BaZrS3 16 phase it
is very small. In view of this, the considered RP phases are expected to be
better optical material than their bulk phase.

In conclusion, we have reported the electronic and excitonic properties of the
RP phases of Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$
(n=[1-3]) using Many Body Perturbation Theory. The exciton binding energy
decreases on increasing the thickness of the perovskite layer. Double peak
character is observed in the first excitonic peak calculated in in-plane
direction of Ba2ZrS4. Only Ba2ZrS4 possesses direct band gap and with
increasing n in Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$
(n=[1-3]) the band gap becomes more indirect. Using Wannier-Mott approach, we
have obtained the upper and lower bound of E${}_{\textrm{B}}$, from the
electronic and ionic contribution of the dielectric constant, respectively. We
have observed that ionic contribution is more significant in
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ than in bulk
BaZrS3. The charge carrier mobility is maximum in Ba2ZrS4, as computed by
employing deformation potential of the same. Further, amongst
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ and bulk BaZrS3,
the electron-phonon coupling constant is relatively smaller for former RP
phases. From our polaron study, we conclude that the optical phonon modes are
dominating as compared to the acoustical phonon modes for these systems. A
large discrepancy is noticed in the mobility of charge carriers (which
includes the effect of acoustical phonon modes only in electron-phonon
coupling) and polaron mobility (which includes the effect of optical phonon
modes in addition to the acoustic modes in electron-phonon coupling). It shows
the dominating character of optical phonon modes in the electron-phonon
coupling and must be studied to understand charge transport properties of RP
phases.

##  1 Computational Methods

We have executed a systematic study to explore the optical, electronic and
excitonic properties using Density Functional Theory (DFT) 32, 35 and beyond
approaches under the framework of Many Body Perturbation Theory 53, 54, 55.
All calculations are performed with Projected Augmented Wave (PAW) potentials
as implemented in Vienna Ab initio Simulation Package (VASP) 56, 57. The PAW
potential of elements viz., Ba, Zr and S contain ten, twelve and six valence
electrons, respectively.
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases are tetragonal structure having space group I4/mmm [139]. All the
structures are optimized using Generalized Gradient Approximation (GGA) as
implemented in PBE 58 exchange-correlation ($\epsilon_{\textrm{xc}}$)
functional until the forces are smaller than 0.001 eV/Å. The $\Gamma$-centered
2$\times$2$\times$2 k-mesh sampling is employed for optimization calculations
(optimized structures are shown in Fig. 5). The electronic self-consistency
loop convergence is set to 0.01 meV, and the kinetic energy cutoff is set to
600 eV for plane wave basis set expansion. To explore the optical properties
and excitonic effects, Bethe-Salpeter Equation (BSE) is solved. Initially, we
have used light 4$\times$4$\times$1 k-mesh for energy calculation (see Fig.
S1). The convergence criteria for the number of occupied and unoccupied bands
in BSE calculations is given in SI (see Fig. S2). In order to have improved
spectral features with denser k-mesh, we have employed the model-BSE (mBSE) 59
approach. Following this, we have performed Density Functional Perturbation
Theory (DFPT) 60 with k-mesh 12$\times$12$\times$1, to discern the role of
ionic contribution to dielectric function along with electronic contribution.
Note that for GW and BSE calculations, we have used converged NBANDS i.e.,
800. Lastly, by employing Frohlich model approach 61, we have studied polaron
effect in our systems DG acknowledges UGC, India, for the senior research
fellowship [grant no. [1268/(CSIR-UGC NET JUNE 2018)]]. AS acknowledges IIT
Delhi for the financial support. MJ acknowledges CSIR, India, for the senior
research fellowship [grant no. [09/086(1344)/2018-EMR-I]]. SB acknowledges the
financial support from SERB under core research grant (grant no.
CRG/2019/000647). We acknowledge the High Performance Computing (HPC) facility
at IIT Delhi for computational resources. K-mesh convergence for PBE
functional; Number of occupied (NO) and unoccupied (NV) bands convergence in
BSE calculation; Band structure plot with PBE and PBE+SOC; BSE@G0W0@HSE06 for
Ba2ZrS4; model-BSE (mBSE) approach; Projected density of states (PDOS);
Effective mass of electron (e) and hole (h); Calculation of deformation
potential energy and elastic modulus; Strength of electron-phonon coupling.

Supplemental Material

  1. I.

k-mesh convergence for PBE functional

  2. II.

Number of occupied (NO) and unoccupied (NV) bands convergence in BSE
calculation

  3. III.

Band structure plot with PBE and PBE+SOC

  4. IV.

BSE@G0W0@HSE06 for Ba2ZrS4

  5. V.

model-BSE (mBSE) approach

  6. VI.

Projected density of states (PDOS)

  7. VII.

Effective mass of electron (e) and hole (h)

  8. VIII.

Calculation of deformation potential energy and elastic modulus

  9. IX.

Strength of electron-phonon coupling

##  2 k-mesh convergence for PBE functional

Figure 5: Real (Re($\varepsilon$)) and imaginary (Im($\varepsilon$)) part of
dielectric function for
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases using PBE exchange-correlation $\epsilon_{xc}$ functional.

Figure 5(a-c) shows the variation of real part (Re($\varepsilon$)) and Figure
5(d-f) shows the imaginary part (Im($\varepsilon$)) (see Figure 5(d-e)) of
dielectric function for
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases. On increasing k-mesh a significant change in Re($\varepsilon$) can be
seen. However, no shift in first peak position of Im($\varepsilon$) part of
dielectric function is observed on increasing k-mesh. Hence, 4x4x1 light
k-mesh is sufficient to compute quasi particle band gap.

##  3 Number of occupied (NO) and unoccupied (NV) bands convergence in BSE
calculation

Figure 6: Variation of imaginary part (Im($\varepsilon$)) of dielectric
function with number of occupied (NO) and unoccupied (NV) bands using BSE for
Ba2ZrS4.

##  4 Band structure plot with PBE and PBE+SOC

Figure 7: Band structure of (a) Ba2ZrS4, (b) Ba3Zr2S7 using PBE and PBE+SOC
exchange-correlation $\epsilon_{\textrm{xc}}$ functional.

##  5 BSE@G0W0@HSE06 for Ba2ZrS4

Figure 8: Imaginary part (Im ($\epsilon$)) of the dielectric function for
Ba2ZrS4 using G0W0@HSE06 and BSE@G0W0@HSE06.

##  6 model-BSE (mBSE) approach

To compute exciton energy and E${}_{\textrm{B}}$ precisely, one needs to
accurately calculate the optical spectra or optical gap using conventional
BSE@G0W0 approach. However, there is inconsistency observed in BSE exciton
peak, due to insufficient high k-mesh (approx 8$\times$8$\times$1). We can not
use high k-mesh because it is computationally very expensive. This results in
the incorrect E${}_{\textrm{B}}$ value. Therefore, to overcome this issue, a
less expensive but robust mBSE approach was proposed. In this model, the
convergence of the optical spectra as a function of the number of k-mesh is
performed. This method is generally based on two approximations:

(i) Using Eq. 11 the RPA static screening W is replaced by a simple analytical
model. Here, the dielectric function is replaced by the local model function:

$\begin{split}\varepsilon^{-1}_{\textrm{G,G}}(\textrm{$|\textrm{q+G}|$})=1-(1-\varepsilon^{-1}_{\infty})\exp(-\dfrac{\textrm{$|\textrm{q+G}|$}^{2}}{4\lambda^{2}})\end{split}$
(11)

where, $\varepsilon_{\infty}$ is the static ion-clamped dielectric function in
the high frequency limit. $\varepsilon_{\infty}^{-1}$ is calculated either
from DFPT or G0W0. q and G are the wave vector and lattice vector of the
reciprocal cell, respectively. $\lambda$ is the screening length parameter,
calculated by fitting $\varepsilon^{-1}$ at small wave vectors with respect to
$|\textrm{q+G}|$ (see Figure 10 (a-c)). The parameters obtained for the
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP
phases are collected in Table 4.

Table 4: The calculated inverse of static ion-clamped dielectric function $\varepsilon_{\infty}^{-1}$ and the screening length parameter $\lambda$ (Å-1) used in mBSE (Eq. 11) for Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP phases. Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ |  $\varepsilon_{\infty}^{-1}$ (PBE) |  $\lambda$ (PBE)
---|---|---
Ba2ZrS4 | 0.14 | 1.17
Ba3Zr2S7 | 0.15 | 1.19
Ba4Zr3S10 | 0.15 | 1.17
Figure 9: Variation of inverse of the dielectric function $\varepsilon^{-1}$
with respect to $|\textrm{q+G}|$ for (a) Ba2ZrS4, (b) Ba3Zr2S7, and (c)
Ba4Zr3S10, respectively. The red curve is obtained by fitting based on Eq.
(5). The mBSE calculated spectra with different k-mesh for (d) Ba2ZrS4, (e)
Ba3Zr2S7 and (f) Ba4Zr3S10, respectively. Figure 10: Imaginary part (Im
($\varepsilon$)) of the dielectric functional for Ba3Zr2S7 using BSE and mBSE.

Here, in Figure 10, we have shown that the imaginary part of the dielectric
function calculated with BSE@GW@PBE, matches with the one, which is calculated
with the model BSE (mBSE) method, where we have chosen PBE as the starting
point. Therefore, we can say that the excitonic features (i.e first peak)
information are always retained by mBSE approach. Notably, both the
calculations are performed using 4$\times$4$\times$1 k-mesh with the same
starting point.

##  7 Projected density of states (PDOS)

Figure 11: Partial density of states (pDOS) for (a) Ba2ZrS4 and (b) BaZrS3.
Orbital contribution of S and Zr in VBM and CBm, respectively for (c) Ba2ZrS4
and (d) BaZrS3.

##  8 Effective mass of electron (e) and hole (h)

Figure 12: Band structure of
Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]).

To compute effective mass, we have obtained the bandstructure using PBE
$\epsilon_{xc}$ functional (see Figure 14). Here, we have calculated the
effective masses of electron (e) and hole (h) along the symmetric path
$\Gamma$ $\rightarrow$ Z for Ba2ZrS4. In case of Ba3Z2S7 and Ba4Zr3S10, the
effective masses of e and h are calculated along $\Gamma$ $\rightarrow$ Z and
Z $\rightarrow$ $\Gamma$, respectively. All the effective and reduced masses
are tabulated in Table 5. Note that, in Figure 14, the observed flatbands
along high-symmetric path $\Gamma$ $\rightarrow$ X are along z-direction.
While calculating the effective masses, we have excluded the high-symmetric
path along z-direction.

Table 5: Effective masses of electron, hole and reduced mass (in term of rest mass of electron (m${}_{\textrm{e}}$)) for Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]) RP phases. Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ | m${}_{\textrm{e}}^{*}$ | m${}_{\textrm{h}}^{*}$ | $\mu$
---|---|---|---
Ba2ZrS4 | 0.18 | 0.43 | 0.32
Ba3Zr2S7 | 0.23 | 1.85 | 0.26
Ba4Zr3S10 | 0.26 | 1.25 | 0.34

##  9 Calculation of deformation potential energy and elastic modulus

According to deformation potential model the carrier mobility is defined as:

$\begin{split}\mu_{\textrm{DP}}=\frac{(8\pi)^{\frac{1}{2}}\hbar^{4}e\textrm{C}_{\textrm{3D}}}{3(m^{*})^{5/2}(k_{\textrm{B}}\textrm{T})^{3/2}\textrm{E}_{1}^{2}}\end{split}$
(12)

where $k_{\textrm{B}}$, T, and m∗ correspond to the Boltzmann constant,
temperature (i.e., 300 K), and effective mass, respectively. In the above
equation, E1 corresponds to the deformation potential constant along
y-direction for electron and hole, respectively. It is given by:

$\begin{split}\textrm{E}_{1}=\frac{\Delta\textrm{E}_{\textrm{i}}}{\Delta\textrm{l}/\textrm{l}_{0}}\end{split}$
(13)

where, $\Delta\textrm{E}_{\textrm{i}}$ denotes the change in the energy or
shift in the position of VBM or CBm under uniaxial strain along the
y-direction. l0 is the lattice constant along the transport direction.
$\Delta\textrm{l}$ denotes the change or deformation in the lattice constant
l0 on application of uniaxial starin. The elastic modulus C${}_{3\textrm{D}}$
is computed using (E - E0)/V0 = C($\Delta\textrm{l}/\textrm{l}_{0})^{2}/2$,
where E0 and E are the total energy of undeformed system and deformed system.
V0 represents the equilibrium volume of the system. Note, herein, we have
chosen strain range from -1.0% to +1.0% to obtain the fitted values of
C${}_{3\textrm{D}}$ and E1 (see Figure 14).

Figure 13: Band gap variation with strain along b- and c-direction for Ba2ZrS4
using PBE $\epsilon_{\textrm{xc}}$. Figure 14: Deformation potential and
elastic modulus of RP phases.

##  10 Strength of electron-phonon coupling

For the qualitative idea of electron-phonon coupling strength, we have
calculated the specific free volume. It is already reported in the literature
that electron-phonon coupling depends on free volume (unoccupied space) 62.
Lattice free volume is defined as the difference between the unitcell volume
and the constituent ions’ volume. Following this, the ratio between the
unoccupied volume and the total volume is defined as the specific free volume
(see Table 6). From Table 6, we can clearly see that specific free volume is
quite large and hence the electron-phonon coupling could be prompt in these
systems. Hence, electron-phonon coupling is important in these systems.

Table 6: Specific free volume of Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ (n=[1-3]). Ba${}_{\textrm{n+1}}$Zr${}_{\textrm{n}}$S${}_{\textrm{3n+1}}$ | Specific free volume (%)
---|---
Ba2ZrS4 | 94.33
Ba3Zr2S7 | 94.17
Ba4Zr3S10 | 94.10

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