Exploring Exciton and Polaron Dominated Photophysical Phenomena in Ruddlesden–Popper Phases of Ban+1Zrns3n+1 (N = 1–3) From

pubs.acs.org/JPCL Letter

Exploring Exciton and Polaron Dominated Photophysical − Phenomena in Ruddlesden Popper Phases of Ban+1ZrnS3n+1 (n =1−3) from Many Body Perturbation Theory Deepika Gill,* Arunima Singh, Manjari Jain, and Saswata Bhattacharya*

Cite This: J. Phys. Chem. Lett. 2021, 12, 6698−6706 Read Online

ACCESS Metrics & More Article Recommendations *sı Supporting Information

− ABSTRACT: Ruddlesden Popper (RP) phases of Ban+1ZrnS3n+1 are an evolving 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 using several first- principles based methodologies under the framework of Many Body Perturbation Theory. Unlike its bulk counterpart, the optical and excitonic anisotropy are observed in Ban+1ZrnS3n+1 (n =1−3) RP phases. As per the Wannier−Mott approach, the ionic contribution to the dielectric constant is important, but it gets decreased on increasing n in Ban+1ZrnS3n+1. The exciton binding energy is found to be dependent on the presence of large electron−phonon coupling. We further observed maximum charge carrier mobility in the Ba2ZrS4 phase. As per our analysis, the optical phonon modes are observed to dominate the acoustic phonon modes, − leading to a decrease in polaron mobility on increasing n in Ban+1ZrnS3n+1 (n =1 3).

− erovskites with the general chemical formula ABX3 have research toward its new phases named as Ruddlesden Popper P found great attention in dielectric, optoelectronic, and (RP) phases.24 Tremendous efforts have been invested to tune solar cell applications due to their superb ferroelectric, the electrical and optical properties.18 The RP phases as an − piezoelectric, superconductive, and photovoltaic properties.1 3 imitation of the perovskite structure are evolving as a During the past decade, in the field of solar cell applications, semiconductor for optoelectronic applications.25,26 Their hybrid lead-halide perovskites, namely, CH3NH3PbX3 and general formula is An+1BnX3n+1, where perovskite structure CH(NH2)2PbX3 (X = Cl, Br, and I) have achieved great blocks of unit cell thickness “n” are separated by rock salt layer success owing to their small band gap, long carrier mobility, AX along the [001] direction. Alternate perovskite blocks are low manufacturing cost, and high power conversion displaced in the in-plane direction by half of the unit cell. The 4−10 efficiency. However, due to the presence of organic RP phases are included in the broad category of “2D molecules, the stability of perovskites is affected by heat, perovskites” owing to their layered structural arrangement light, and moisture, thereby degrading their efficiency with Downloaded via INDIAN INST OF TECH DELHI- IIT on July 15, 2021 at 04:37:19 (UTC). 11 (see Figure 1). time in the practical world. Moreover, the presence of toxic Note that several studies assigned to the layered perovskites See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. lead in these materials makes them hazardous for the “ ” 12 as 2D perovskites exist in the literature, where periodic environment. These shortcomings have hindered their stacking of perovskite layers results in a bulk structure. Their practical applications. material properties can be tuned either by substitution or In search of alternative perovskites that can alleviate the 27−29 fi − dimensional reduction. Due to quantum con nement limitations of lead halide perovskites, chalcogenide perov- effects,30 considerable change in bulk physical properties (such skites with S or Se anion have been proposed for photovoltaic as bulk modulus, elastic modulus, charge carrier properties, and applications.13 Several prototypical chalcogenide perovskites 14 15−17 optical properties) can be seen on reducing the dimension of (viz., SrHfS3 and AZrS3 (A = Sr, Ca, Ba), along with the material.31,32 Research in this field is highly evolv- − their related phases) have been synthesized successfully. ing.26,33 35 Among them, BaZrS3 consists of earth-abundant elements and has a moderate band gap (∼1.82 eV18) ideal for photovoltaics. Moreover, it also exhibits ambipolar doping19 Received: May 9, 2021 and is stable against different environmental conditions.20 In Accepted: July 10, 2021 order to optimize the solar cell absorption, doping at Ba/Zr sites has been attempted in this material.17,21,22 However, such doped/alloyed configurations seem to lack stability.23 Thin fi lms of BaZrS3 are also reported, which has directed the

− Figure 1. Optimized crystal structure of Ban+1ZrnS3n+1 (n =1 3) Ruddlesden−Popper phases (RP phases).

In optoelectronic materials, exciton formation greatly influences the charge separation properties and, hence, the excitonic parameter such as exciton binding energy (EB) 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 photophysical phenomena in 36 these materials. Polaronic effects have been suggested to play Figure 2. Imaginary part (Im(ε)) of the dielectric function for (a) an important role in the excitation dynamics and carrier Ba2ZrS4, (b) Ba3Zr2S7, and (c) Ba4Zr3S10 using single shot GW fl ε transport. The separation of free charge is also in uenced by (G0W0) and BSE. Im( ) of the dielectric function for (d) Ba2ZrS4, (e) ∥ ε the carrier mobility. Hence, understanding the effect of Ba3Zr2S7, and (f) Ba4Zr3S10 along the E xy direction and Im( )of electron−phonon coupling in terms of polaron mobility is dielectric function for (g) Ba2ZrS4, (h) Ba3Zr2S7, and (i) Ba4Zr3S10 ∥ important. A systematic study on the excitonic and polaronic along the E z direction, using G0W0 and BSE. Here, the colored effect in the RP phases of BaZrS is hitherto unknown. The regions indicate the energy window that lies in the visible region of 3 the electromagnetic spectra. present Letter, therefore, explores the excitonic properties ff along with polaronic e ect in RP phases of Ban+1ZrnS3n+1 (n = 1−3) under the framework of Many Body Perturbation HSE06 are overestimated in comparison to the experimental Theory. The electron−phonon coupling is also taken care of band gap, as they do not take into account the exciton binding using the Fröhlich model to compute the polaron mobility. energy. Since BSE takes into account the excitonic effect fi The exciton binding energy is de ned as the energy required (which is ignored in the G0W0 calculation), the results get to decouple the exciton into an individual electron (e) and improved when we solve the BSE to obtain the optical band hole (h) pair. Theoretically, the exciton binding energy (EB)is gap. Therefore, we have performed BSE@G0W0@PBE and ff − calculated by taking the di erence of the energy of the BSE@G0W0@HSE06 to incorporate e h interactions. − − bounded electron hole (e h) pair (i.e., BSE gap) and By performing BSE@G0W0@PBE and BSE@G0W0@ − fi unbounded e h pair (i.e., GW gap). In order to determine HSE06, we nd the optical peak positions of Ba2ZrS4 are, − 18 the optical response of Ban+1ZrnS3n+1 (n =1 3) RP phases, we respectively, 1.71 eV (1.33 eV ) (see Figure 2a) and 1.85 eV have calculated the imaginary part of the dielectric function (see Figure S4 in the SI). Since the former is more close to the (Im(ε)).Initially,wehavebenchmarkedtheexchange- experimental value, we have preferred to compute GW/BSE on ε − correlation ( xc) functional for our system. As it is already top of the PBE Kohn Sham orbitals as the starting point. known that single shot GW (G0W0) calculation strongly Similarly, we have performed G0W0@PBE and BSE@ depends on the starting point, we need to validate the suitable G0W0@PBE calculations to capture the optical and excitonic − ff starting point for G0W0 calculation. Note that the spin orbit e ect for Ba3Zr2S7 and Ba4Zr3S10, respectively. As per the coupling (SOC) effect is negligible in these systems (see previous analysis, we report the BSE peak positions for 18 Figure S3). Hence, we have excluded SOC in our calculations. Ba3Zr2S7 and Ba4Zr3S10 as 1.49 eV (1.28 eV ) and 1.43 eV, The band gaps of Ba2ZrS4,Ba3Zr2S7 and Ba4Zr3S10 are quite respectively, whereas the G0W0@PBE peak is obtained at 1.82 underestimated using PBE, and the values are 0.61 eV, 0.42 eV, and 1.69 eV, respectively (see Figure 2a−c). Note that Figure and 0.34 eV, respectively. On the other hand, the same 2a−c corresponds to the average of the optical responses in the calculations with default parameters (viz., exact exchange = x, y, and z directions. However, Figure 2d−f and Figure 2g−i 25% and screening parameter 0.2 Å−1) of HSE06 are 1.39 eV, show the directional optical response of the RP phases along 1.18 eV, and 1.08 eV, respectively. The HSE06 numbers are in the E∥xy and E∥z directions, respectively (see discussion good agreement with the experimental findings.18 Further, the later). peak position, which is underestimated by PBE, is improved by It should be noted here that these numbers are highly performing G0W0 by taking both PBE and HSE06 as a starting dependent on the k-mesh, and it is very challenging (even with point, which shifts the peak to 2.11 eV (see Figure 2a) and the fastest supercomputers) to converge the BSE calculation to 2.17 eV (see Figure S4 in the SI), respectively. Notably, the obtain the excitonic peak for computing EB with absolute quasiparticle gaps computed using G0W0@PBE and G0W0@ accuracy. In Figure 2, occurrence of the red-shifted peak in

6699 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter fi 1 BSE@G0W0@PBE compared to G0W0@PBE signi es the |ϕ |=2 ff n(0) 33 excitonic e ect in the considered RP phases. The computed EB π()rn (2) fi exc values of the rst bright excitons of the Ba2ZrS4,Ba3Zr2S7, and Ba Zr S RP phases are found to be 0.40 eV, 0.33 eV, and 0.26 The excitonic parameters (computed using the mBSE 4 3 10 fi eV, respectively. However, the discrepancy in the BSE peak approach) are tabulated in Table 1.We nd the probability position from the experimental value18 may lead to some Table 1. Excitonic Parameters for RP Phases (Calculated unprecedented error in the EB values. Unfortunately, we have already ensured the highest possible k-mesh to compute the Using mBSE@G0W0@PBE) BSE@G W @PBE calculations. Involving a denser k-mesh is 0 0 excitonic parameter Ba2ZrS4 Ba3Zr2S7 Ba4Zr3S10 not feasible for the superstructures of the RP phases for  EB (eV) 0.33 0.29 0.23 computing G0W0@PBE and BSE@G0W0@PBE solely due to T (K) 3804 3364 2669 the computational limitation. Therefore, this poses constraints exc rexc (nm) 0.34 0.37 0.55 to estimate the accurate E for the given systems using BSE@ |ϕ |2 27 −3 B n(0) (10 m ) 8.10 6.28 1.91 G0W0@PBE. 1 (10−27 m 3 ) To circumvent this problem, a parametrized model for ϕ(0) 2 0.12 0.20 0.52 dielectric screening i.e., model-BSE approach, (mBSE) is n employed. This mBSE approach is computationally less of the wave function for the e−h pair at zero separation (| expensive as compared to BSE but with similar accuracy to ϕ (0)|2) decreases on increasing n in Ba Zr S . Hence, the estimate the first peak position (see Figure S7 in the SI). Thus, n n+1 n 3n+1 lifetime τ is expected to be increased with increasing n in this enables us to sample the Brillouin zone with a higher τ τ τ Ban+1ZrnS3n+1, i.e., n=3 > n=2 > n=1. number for the k-mesh. As per the mBSE calculation, it is seen Note that the trend in the exciton lifetime can also be further that a denser k-mesh sampling indeed red-shifts the BSE peak verified from the mBSE peaks. This is estimated from the full (see mBSE calculations in Section V in the SI). The EB values width at half maximum (FWHM) of the first peak of Im(ε). In are updated, respectively, to be 0.33 eV, 0.29 eV, and 0.23 eV − Figure 3, we have shown the broadening of the excitonic peak, for n =1 3ofBan+1ZrnS3n+1 (with a maximum possible error of ±0.03 eV as detailed in the SI Section V). These numbers fall well within the range of EB estimated via the state-of-the-art method comprising the Wannier−Mott and Density Func- tional Perturbation Theory (DFPT) approach as explained later. 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 is also noted the order of EB of Ban+1ZrnS3n+1 follows a trend, viz., n =1>n =2>n =3>n = ∞ (i.e., bulk 37 BaZrS3 ). Notably, from Figure 2a, we have observed the double peak character for the excitonic peak of Ba2ZrS4 that may occur due to two nearby transitions. Moreover, for ff Ba2ZrS4 (i.e., n = 1 RP phase) the di erence between the direct and the indirect band gap is ∼0.1 eV, which increases to ∼0.3 eV in the case of n = 2, 3 RP phases. The optimal value of the band gap and relatively smaller difference between direct and indirect values for the n = 1 RP phase may suggest this to be Figure 3. Full width at half-maximum (FWHM) of the exciton peak ideal for solar cell application. using the mBSE@G0W0@PBE approach with a dense k-mesh. Next, we have computed other excitonic parameters. The Broadening of the exciton peak is mainly due to electron−hole excitonic lifetime (τ) is inversely proportional to the interaction. Here, Γ corresponds to the FWHM. probability of a wave function for the electron−hole pair at |ϕ |2 18 zero separation ( n(0) ). The excitonic temperature can be which is well in agreement with previous experiments. calculated by using the relation that E = kBT, where E, kB, and Therefore, the qualitative trend of the exciton lifetime for − τ τ τ  T correspond to the energy, Boltzmann constant, and Ban+1ZrnS3n+1 (n =1 3) is n=3 > n=2 > n=1 which is very temperature, respectively. The excitonic radius (rexc)is much in line with our findings in Table 1. Here broadening has determined as follows: shown the contribution due to the e−h interaction, and it does not include the electron−phonon coupling effect. The latter is m0 2 important, especially for layered systems. But since the trend rexc = εeff nrRy μ (1) matches well with the experiment18 it is therefore expected that it will remain the same even after the inclusion of electron− μ ε where m0, , eff, and n are the free electron mass, reduced phonon coupling. The role of electron−phonon coupling is mass, static effective dielectric constant, and exciton energy discussed in greater detail later but on a different context. − level, respectively. rRy is the Bohr radius for the 2D system, i.e., Ban+1ZrnS3n+1 (n =13) RP phases have a tetragonal half of the Bohr radius for the 3D system. Hence, in our case structure and exhibit optical anisotropy. Hence, it is required rRy = 0.0529/2 = 0.0265 nm. Using the excitonic radius we can to study their optical and excitonic properties along the E∥xy calculate the probability of the wave function for e−h at zero (i.e., in-plane along the x or y directions) and E∥z (i.e., out-of- separation as follows: plane along the z direction) directions. We have observed

6700 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter

ε ε − ε ε − Figure 4. Electronic (Im( e) and Re( e)) (a c) and ionic (Im( i) and Re( i)) (d f) contribution to the dielectric function for Ba2ZrS4,Ba3Zr2S7, ε ε and Ba4Zr3S10. Red and black colors correspond to real (Re( )) and imaginary (Im( )), respectively. anisotropy in Ba2ZrS4,Ba3Zr2S7, and Ba4Zr3S10, which can obtained from the mixing of dyz,dxz, and other orbitals of S and greatly affect their performance in practical applications. Ba are of higher energy. On the other hand, orbitals that Therefore, it is of paramount importance to understand the include dxy are of lower energy. This results in a polarization anisotropic effect in the optical and excitonic properties of the dependent optical gap. As CBm has a contribution from dxy, − − Ban+1ZrnS3n+1 (n =1 3) RP phases. In Figure 2d i, we have the optical transitions are allowed only for in-plane polarized shown the optical and excitonic contributions of the light and not for out-of-plane polarized light. − ff − 39 Ban+1ZrnS3n+1 (n =1 3) RP phases along di erent directions, We have used the Wannier Mott model for a simple viz., x, y, and z. Employing the Shockley−Queisser (SQ) screened Coulomb potential. Recently, it was reported that 38 criterion for the solar cell and other optoelectronic devices, excitons of the RP phases of MAPbX3 are known to be strongly we can remark that Ba Zr S (n =1−3) RP phases are influenced by the quantum confinement effects in the z- n+1 n 3n+1 40,41 optically active in in-plane (i.e., along the x and y directions) direction, exhibiting a quasi-2D character. In the present ff and optically inactive out-of-plane (i.e., z direction). These case as well, we notice the di erence in the xy and z direction fi systems possess similar optical as well as excitonic properties spectra, which signi es overall weak hybridization along the z along the x and y directions (see Figure 2d−f). However, along direction. In ref 41, we also note that with increasing the the z-direction, their optical and excitonic spectra are not only number of layers, there is deviation from the ideal 2D nature. This has been attributed to the screened Coulomb potential. In blue-shifted, but also the features of G0W0 and BSE peaks are ff view of this due to unavailability of experiments, we have also quite di erent than those in case of the in-plane direction (see − Figure 2g−i). It is well-known that the exciton lifetime is provided theoretical analysis considering the 3D Wannier Mott excitons (see the SI). Therefore, for n = 1 and 2, it is inversely proportional to the width of the exciton peak. Hence, − a change in the feature of the exciton peak greatly influences expected more to follow the 2D Wannier Mott model. the excitonic parameters as well in different directions. The d- However, for n = 3 or above, the numbers generated via 3D Wannier−Mott may also seem to be relevant. Therefore, as per orbital contribution in the valence band maximum (VBM) and − conduction band minimum (CBm) could be responsible for Wannier Mott excitons in 2D layered systems, the EB for the screened interacting e−h pair is given by the optical anisotropy in these systems. In bulk BaZrS3,we have observed that Ba(6s), Zr(4d), and S(2p) orbitals are 1 μ hybridized in the VBM. It is basically dominated by 2px,2py, EB = R∞ 1 2 ε2 and 2pz orbitals of the S element. The CBm of bulk BaZrS3 n − eff ()2 (3) consists of hybridized orbitals of S(2p ,2p,2p) and Zr(4d , x y z xy i y 2 μ j z 4dyz,4dxz, 4dz ). Now, in the case of Ba2ZrS4, the VBM where is the reducedj z mass in terms of the rest mass of the ε jff z hybridization is the same as that of bulk BaZrS3. However, in electron, eff is the ek ective{ dielectric constant (which includes CBm, dxy is strongly hybridized with S(2p)/Ba(6s) orbitals, the electronic as well as ionic contribution to the dielectric and dyz and dxz are weakly hybridized. The hybridized orbitals constant), and R∞ is the Rydberg constant. The reduced

6701 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter masses of Ba2ZrS4,Ba3Zr2S7, and Ba4Zr3S10 in terms of the octahedron rotation is observed with the application of the electron rest mass are 0.32, 0.26, and 0.23, respectively (for strain. However, in our case of Ba2ZrS4 RP phases, monotonic more information regarding the calculation of the reduced change in the band gap (but not as large as in the case of Ming mass and effective mass of the electron and hole, see Section et al.45) has been observed on applying up to ±7% strain along ε ff VII in the SI). However, in the above expression, eff is still the b-axis and c-axis (see Figure S10 in the SI). The e ect on unknown for these systems. It is already reported that lattice the band gap along the b and a directions is symmetric. relaxation can influence the exciton binding energy.42 For Further, the effect of the strain on the band gap in the out-of- ω fi example, if LO corresponds to longitudinal optical phonon plane direction is more signi cant than in the in-plane ≪ ℏω ff frequency and EB LO, one needs to consider the e ect of direction. We have also noticed a slight octahedral tilt under ε lattice relaxation. Therefore, for eff, a value intermediate the application of strain along the b-axis. In the case of strain between the static electronic dielectric constant at high along the c-axis, a small rotation of the octahedron about the c- ε frequency, i.e., e, and the static ionic dielectric constant at axis has been observed. Here, we have calculated the elastic ε ≫ low frequency, i.e., i, should be considered. However, if EB modulus of the RP phases that is given by ℏω ff LO, the e ect of the lattice relaxation can be ignored as in 2 such cases ε → ε , where ε is the static value of the dielectric 1 δ E eff e e C2D = constant at high frequency that mainly consists of the A δε2 (4) electronic contribution. ε − − where A, , and E correspond to the area of a unit cell, strain, In Figure 4a c and Figure 4d f, we have shown the and totalji energy,zy respectively. We have observed that the j z electronic and ionic contributions to the dielectric function, elastic modulusk { of the 2D RP phases is larger than those of the respectively, where the ionic contribution is computed using previously reported 2D RP phases.45 In order to compute the the DFPT approach. Notably, the electronic contribution is carrier mobility, we have used the deformation potential − computed using the BSE approach. The static real parts of the model.46 49 According to this model, the mobility of the ionic dielectric constant for Ba2ZrS4,Ba3Zr2S7, and Ba4Zr3S10 charge carrier is defined as are 39.35, 31.19, and 57.59, respectively. As per Figure 3,a 3 considerable increase in the static low frequency of the ionic eCℏ 2D μDP = 2 dielectric constant is attributed to the occurrence of optically kTmmE* (5) active phonon modes below 10 meV. This shows the ionic B dl nature of the RP phases. Using electronic and ionic where T is the temperature, m* is the effective mass of the contribution of the dielectric constant and eq 3, we have charge carrier, and e is the elementary charge of an electron. md calculated the upper and lower bounds for the E (see Table corresponds to the density-of-state effective mass that is B fi 2). The effective values of the dielectric constant and, hence, de ned as md = mmxy. El is the deformation potential (for

more details regarding the calculation of C2D and El see Section Table 2. Electronic and Ionic Contribution to the Dielectric VIII in the SI). In Table 3, we have listed the values of C , El, n − a 2D Constant for Ban+1ZrnS3n+1 ( =1 3) RP Phases ε ε Table 3. Elastic Modulus, Deformation Potential, and Ban+1ZrnS3n+1 e EBu (eV) i EBl (meV) n − Predicted Carrier Mobility of Ban+1ZrnS3n+1 ( =1 3) RP Ba2ZrS4 4.11 1.06 39.25 2.90 Phases Ba3Zr2S7 3.65 1.03 31.19 3.49 −2 μ 2 −1 −1 Ba4Zr3S10 4.89 0.70 57.59 0.95 Ban+1ZrnS3n+1 C2D (eV Å ) El (eV) DP (cm V s ) aε ε e and i correspond to the static values of the electronic and ionic Ba2ZrS4 (e) 20.02 6.36 5217.77 dielectric constant, respectively. EBu and EBl correspond to the upper Ba2ZrS4 (h) 20.02 6.61 862.43 and lower bounds of the exciton binding energy, respectively. Ba3Zr2S7 (e) 23.77 6.41 3731.67

Ba3Zr2S7 (h) 23.77 6.39 57.92 the binding energy lies in between these upper and lower Ba4Zr3S10 (e) 24.11 6.72 2694.99 bounds listed in Table 2. Note that, in Table 2, the static values Ba4Zr3S10 (h) 24.11 6.73 37.06 of the electronic and ionic dielectric constants for the considered RP phases are comparable with those of APbX3 43,44 μ − (A = MA, FA; X = I, Br) perovskites. Hence, we can say and DP for the electron and hole of the Ban+1ZrnS3n+1 (n =1 their optical response resembles that of APbX3 perovskites. 3) RP phases. As the elastic modulus increases down the Despite there being a slight discrepancy in the BSE peak column (see Table 3), we can say that the softness of the RP position, we still can compare the EB obtained after taking the phases decreases on increasing the value of n in Ban+1ZrnS3n+1 ff − di erence of the G0W0 and BSE peaks and the same as (n =1 3). The mobility of the electron also decreases on − − summarized in Table 2 using the 2D Wannier Mott model. increasing n in Ban+1ZrnS3n+1 (n =1 3). After analyzing the We find the order of divergence of the upper bound value specific free volume (for details see Section IX in the SI), we follows n =1>n =2>n = 3. This clearly implies that the find that the study of electron−phonon coupling is important order of significance for the ionic contribution to the dielectric in these materials. Also, the presence of the polarization in the constant is n =1>n =2>n = 3. It means that the bulk BaZrS3 RP phases lays emphasis on the polaron study. We have ionic contribution to the dielectric constant must be negligible, examined the electron−phonon interaction in our system by − which is in good agreement with a recent report regarding bulk the mesoscopic model, viz., Fröhlich’s model,50 52 for the chalcogenide.37 polarons. The dressed “quasiparticles”, formed due to the Recently, Ming et al. have reported the effect of strain on the screened interaction of the electron and hole by the lattice, are 45 ̈ band gap and octahedron rotation for Ba2ZrS4. According to known as polarons. Frohlich introduced a parameter to their report, a significant change in the band gap and describe theoretically the momentum of the electron in the

6702 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter

field of polar lattice vibration. This parameter is known as the the charge carrier can be seen on comparing the mobility of the dimensionless Fröhlich coupling constant.50,51,53 electron without including its interaction with the optical phonon (see Table 3) and with including the interaction with 1 R y m* α = the optical phonon (see Table 4). Here in Table 4, ionic * ε chωLOm e (6) screening is indicative of the ionicity for a system. On comparing our results with the hybrid inorganic−organic where the coupling constant α quantifies the electron−phonon halide perovskites (ionic screenings of MAPbI3, MAPbBr3, and * ff 54 coupling, m is the e ective mass of the electron, me is the rest MAPbCl are 0.17, 0.18, and 0.22, respectively ), we can say ’ 3 mass of the electron, h is Planck s constant, c is the speed of that Ba ZrS ,BaZr S , and Ba Zr S are less ionic than ω −1 2 4 3 2 7 4 3 10 light, LO (in [cm ] units) is the optical phonon frequency, MAPbX (X = Cl, Br, I). Also, the obtained coupling constant ε* ε* ε − ε 3 1/ is the ionic screening parameter (1/ =1/ ∞ 1/ static, is comparable to or larger than MAPbX (see Table 4). The ε ε 3 where, static and ∞ are static and high frequency dielectric lowering of the mobility of the charge carriers on the inclusion constants), and Ry is the Rydberg energy. We have observed of the LO phonon modes indicates that optical phonon modes − that the electron phonon coupling constants of the consid- are dominating over the acoustical phonon modes in these ered RP phases (Table4) are smaller than those of their bulk materials. Note that, in the absence of experimental data, these results may help as a guideline for further research. Moreover, n − Table 4. Polaron Parameters for Ban+1ZrnS3n+1 ( =1 3) RP for qualitative analysis, our results are very informative to Phases understand the charge transport properties of these RP phases. ε* α * μ 2 −1 −1 From Table 4, we can clearly see that, on increasing n in Ban+1ZrnS3n+1 1/ mP/m lP (Å) P (cm V s ) − Ban+1ZrnS3n+1 (n =1 3), i.e., down the column, the polaron Ba2ZrS4 0.09 1.84 1.39 354.30 164.75 16 mobility decreases, and for the bulk BaZrS3 phase it is very Ba3Zr2S7 0.08 2.36 1.53 307.12 117.20 small. In view of this, the considered RP phases are expected to Ba4Zr3S10 0.10 1.77 1.37 292.49 76.39 be a better optical material than their bulk phase. In conclusion, we have reported the electronic and excitonic BaZrS .37 Further, using the extended form of Fröhlich’s 3 properties of the RP phases of Ba Zr S (n =1−3) using polaron theory, given by Feynman, the effective mass of the n+1 n 3n+1 Many Body Perturbation Theory. The exciton binding energy polaron (m )51 is defined as P decreases on increasing the thickness of the perovskite layer. 2 Double peak character is observed in the first excitonic peak * αα mP = m 1 ++ +... calculated in the in-plane direction of Ba ZrS . The difference 640 2 4 (7) between the direct and the indirect band gap increases with − where m* is the effective mass calculated from the band increasing n in Ban+1ZrnS3n+1 (n =1 3), thereby making the ji zy − structure calculationsj (see Sectionz VII in the SI). The polaron band gap more indirect. Using the Wannier Mott approach, 54 j z radii can bek calculated as follows:{ we have obtained the upper and lower bound of EB, from the electronic and ionic contribution of the dielectric constant, h respectively. We have observed that the significance of the l = P * ionic contribution to the dielectric function decreases on 2cm ωLO (8) increasing n in Ban+1ZrnS3n+1. The charge carrier mobility is The polaron mobility according to the Hellwarth polaron maximum in Ba2ZrS4, as computed by employing the model is defined as follows: deformation potential of the same. Further, among Ba Zr S and bulk BaZrS , the electron−phonon coupling (3π e ) sinh(β /2) w3 1 n+1 n 3n+1 3 μ = constant is relatively smaller for former RP phases. From our P * 5/2 3 2cπωLOm α β v K (9) polaron study, we conclude that the optical phonon modes are β ω dominating as compared to the acoustical phonon modes for where = hc LO/kBT and w and v correspond to temperature these systems. A large discrepancy is noticed in the mobility of dependent variational parameters. K is a function of v, w, and ff β 52 fi the charge carriers (which includes the e ect of acoustical , i.e., de ned as follows phonon modes only in electron−phonon coupling) and ∞ ff 22− 3/2 polaron mobility (which includes the e ect of optical phonon Ka(, b )=[+−]∫ d uu a b cos() vu cos() u − 0 modes in addition to the acoustic modes in electron phonon (10) coupling). It shows the dominating character of optical − 2 phonon modes in the electron phonon coupling and must Here, a and b are calculated as be studied to understand the charge transport properties of RP 22 phases. Finally, from the perspective of device applications, 22()vw− a =+(/2)βββcoth(v /2) these RP phases are expected to be promising optoelectronic wv2 (11) materials. ()vw22− β b = ■ COMPUTATIONAL METHODS wv2 sinh(βv /2) (12) We have executed a systematic study to explore the optical, We have used the lowest frequency of the LO phonon, i.e, electronic, and excitonic properties using Density Functional 32,35 1.34, 1.46, and 1.36 THz for Ba2ZrS4,Ba3Zr2S7, and Ba4Zr3S10, Theory (DFT) and beyond approaches under the − μ 55−57 respectively, for our calculation (see Figure 4d f). P gives the framework of Many Body Perturbation Theory. All upper limit of the charge carrier mobility, under the calculations are performed with Projected Augmented Wave assumption that the charge carrier (electron) interacts only (PAW) potentials as implemented in the Vienna Ab initio with the optical phonon. A significant change in the mobility of Simulation Package (VASP).58,59 The PAW potential of the

6703 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter elements, viz., Ba, Zr, and S, contain 10, 12, and 6 valence Notes − electrons, respectively. The Ban+1ZrnS3n+1 (n =1 3) RP phases The authors declare no competing financial interest. are a tetragonal structure having the space group I4/mmm [No. 139]. All the structures are optimized using Generalized ■ ACKNOWLEDGMENTS Gradient Approximation (GGA) as implemented in PBE60 ε D.G. acknowledges UGC, India, for the senior research exchange-correlation ( xc) functional until the forces are smaller than 0.001 eV/Å. The Γ-centered 2 × 2 × 2 k-mesh fellowship [Grant No. 1268/(CSIR-UGC NET JUNE sampling is employed for optimization calculations (optimized 2018)]. A.S. acknowledges IIT Delhi for the financial support. structures are shown in Figure 1). The electronic self- M.J. acknowledges CSIR, India, for the senior research consistency loop convergence is set to 0.01 meV, and the fellowship [Grant No. 09/086(1344)/2018-EMR-I]. S.B. kinetic energy cutoff is set to 600 eV for the plane wave basis acknowledges the financial support from SERB under the set expansion. To explore the optical properties and excitonic core research grant (Grant No. CRG/2019/000647). We effects, the Bethe−Salpeter equation (BSE) is solved. Initially, acknowledge the High Performance Computing (HPC) facility we used a light 4 × 4 × 1 k-mesh for the energy calculation at IIT Delhi for computational resources. (see Figure S1). The convergence criteria for the number of occupied and unoccupied bands in BSE calculations are given ■ REFERENCES in the SI (see Figure S2). In order to have improved spectral (1) Jaffe, B.; Cook, W.; Jaffe, H. Piezoelectric Ceramics; Academic features with a denser k-mesh, we have employed the model- Press: London, 1971; JCPDS 71-2171. BSE (mBSE)61 approach. Following this, we have performed 62 (2) Tombe, S.; Adam, G.; Heilbrunner, H.; Apaydin, D. H.; Density Functional Perturbation Theory (DFPT) with a k- Ulbricht, C.; Sariciftci, N. S.; Arendse, C. J.; Iwuoha, E.; Scharber, M. mesh 12 × 12 × 1, to discern the role of the ionic contribution C. Optical and electronic properties of mixed halide (X = I, Cl, Br) to the dielectric function along with the electronic contribu- methylammonium lead perovskite solar cells. J. Mater. Chem. C 2017, tion. Note that, for GW and BSE calculations, we have used 5, 1714−1723. converged NBANDS, i.e., 800. Lastly, by employing the (3) Zhang, S.; Li, F.; Jiang, X.; Kim, J.; Luo, J.; Geng, X. Advantages 63 Fröhlich model approach, we have studied the polaron effect and challenges of relaxor-PbTiO3 ferroelectric crystals for electro- − − in our systems. All the calculations are very well validated and acoustic transducers A review. Prog. Mater. Sci. 2015, 68,1 66. benchmarked to test the convergence details, which are given (4) Deschler, F.; Price, M.; Pathak, S.; Klintberg, L. E.; Jarausch, D.- in the SI to ensure that the numbers are devoid of numerical D.; Higler, R.; Hu ttner, S.; Leijtens, T.; Stranks, S. D.; Snaith, H. J.; et al. High photoluminescence efficiency and optically pumped lasing error. in solution-processed mixed halide perovskite semiconductors. J. Phys. Chem. Lett. 2014, 5, 1421−1426. ■ ASSOCIATED CONTENT (5) Shi, D.; Adinolfi, V.; Comin, R.; Yuan, M.; Alarousu, E.; Buin, A.; *sı Supporting Information Chen, Y.; Hoogland, S.; Rothenberger, A.; Katsiev, K.; et al. Low trap- The Supporting Information is available free of charge at state density and long carrier diffusion in organolead trihalide − https://pubs.acs.org/doi/10.1021/acs.jpclett.1c01486. perovskite single crystals. Science 2015, 347, 519 522. (6) Stamplecoskie, K. G.; Manser, J. S.; Kamat, P. V. Dual nature of K-mesh convergence for the PBE functional; number of the excited state in organic−inorganic lead halide perovskites. Energy occupied (NO) and unoccupied (NV) band conver- Environ. Sci. 2015, 8, 208−215. gence in the BSE calculation; band structure plot with (7) Bi, Y.; Hutter, E. M.; Fang, Y.; Dong, Q.; Huang, J.; Savenije, T. μ PBE and PBE+SOC; BSE@G0W0@HSE06 for Ba2ZrS4; J. Charge carrier lifetimes exceeding 15 s in methylammonium lead model-BSE (mBSE) approach; projected density of iodide single crystals. J. Phys. Chem. Lett. 2016, 7, 923−928. states (PDOS); effective mass of electron (e) and hole (8) Ramasamy, P.; Lim, D.-H.; Kim, B.; Lee, S.-H.; Lee, M.-S.; Lee, J.-S. All-inorganic cesium lead halide perovskite nanocrystals for (h); calculation of deformation potential energy and − elastic modulus; strength of electron−phonon coupling; photodetector applications. Chem. Commun. 2016, 52, 2067 2070. (9) Leijtens, T.; Bush, K.; Cheacharoen, R.; Beal, R.; Bowring, A.; and calculation using 3D Wannier−Mott formula (PDF) McGehee, M. D. Towards enabling stable lead halide perovskite solar cells; interplay between structural, environmental, and thermal ■ AUTHOR INFORMATION stability. J. Mater. Chem. A 2017, 5, 11483−11500. (10) K, N. N.; Nag, A. Synthesis and luminescence of Mn-doped Corresponding Authors − − Cs2AgInCl6 double perovskites. Chem. Commun. 2018, 54, 5205 Deepika Gill Department of Physics, Indian Institute of 5208. Technology Delhi, New Delhi, Delhi 110016, India; (11) Park, B.-w.; Seok, S. I. Intrinsic instability of inorganic−organic Email: [email protected] hybrid halide perovskite materials. Adv. Mater. 2019, 31, 1805337. Saswata Bhattacharya − Department of Physics, Indian (12) Babayigit, A.; Ethirajan, A.; Muller, M.; Conings, B. Toxicity of Institute of Technology Delhi, New Delhi, Delhi 110016, organometal halide perovskite solar cells. Nat. Mater. 2016, 15, 247− India; orcid.org/0000-0002-4145-4899; Phone: +91- 251. 11-2659 1359; Email: [email protected] (13) Sun, Y.-Y.; Agiorgousis, M. L.; Zhang, P.; Zhang, S. Chalcogenide perovskites for photovoltaics. Nano Lett. 2015, 15, Authors 581−585. Arunima Singh − Department of Physics, Indian Institute of (14) Hanzawa, K.; Iimura, S.; Hiramatsu, H.; Hosono, H. Material Technology Delhi, New Delhi, Delhi 110016, India; design of green-light-emitting semiconductors: Perovskite-type sulfide − orcid.org/0000-0002-7344-9816 SrHfS3. J. Am. Chem. Soc. 2019, 141, 5343 5349. Manjari Jain − Department of Physics, Indian Institute of (15) Lee, C.-S.; Kleinke, K. M.; Kleinke, H. Synthesis, structure, and Technology Delhi, New Delhi, Delhi 110016, India electronic and physical properties of the two SrZrS3 modifications. Solid State Sci. 2005, 7, 1049−1054. Complete contact information is available at: (16) Majumdar, A.; Adeleke, A. A.; Chakraborty, S.; Ahuja, R. https://pubs.acs.org/10.1021/acs.jpclett.1c01486 Emerging piezochromism in lead free alkaline earth chalcogenide

6704 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter perovskite AZrS3 (A= Mg, Ca, Sr and Ba) under pressure. J. Mater. (36) Park, M.; Neukirch, A. J.; Reyes-Lillo, S. E.; Lai, M.; Ellis, S. R.; Chem. C 2020, 8, 16392−16403. Dietze, D.; Neaton, J. B.; Yang, P.; Tretiak, S.; Mathies, R. A. Excited- (17) Perera, S.; Hui, H.; Zhao, C.; Xue, H.; Sun, F.; Deng, C.; Gross, state vibrational dynamics toward the polaron in methylammonium N.; Milleville, C.; Xu, X.; Watson, D. F.; et al. Chalcogenide lead iodide perovskite. Nat. Commun. 2018, 9, 2525. perovskites−an emerging class of ionic semiconductors. Nano Energy (37) Kumar, M.; Singh, A.; Gill, D.; Bhattacharya, S. Optoelectronic 2016, 22, 129−135. Properties of Chalcogenide Perovskites by Many-Body Perturbation (18) Li, W.; Niu, S.; Zhao, B.; Haiges, R.; Zhang, Z.; Ravichandran, Theory. J. Phys. Chem. Lett. 2021, 12, 5301−5307. J.; Janotti, A. Band gap evolution in Ruddlesden-Popper phases. Phys. (38) Shockley, W.; Queisser, H. J. Detailed balance limit of efficiency Rev. Mater. 2019, 3, 101601. of p−n junction solar cells. J. Appl. Phys. 1961, 32, 510−519. (19) Meng, W.; Saparov, B.; Hong, F.; Wang, J.; Mitzi, D. B.; Yan, Y. (39) Waters, M. J.; Hashemi, D.; Kieffer, J. Semiclassical model for Alloying and defect control within chalcogenide perovskites for calculating exciton and polaron pair energetics at interfaces. Mater. Sci. optimized photovoltaic application. Chem. Mater. 2016, 28, 821−829. Eng., B 2020, 261, 114657. (20) Ravi, V. K.; Yu, S. H.; Rajput, P. K.; Nayak, C.; Bhattacharyya, (40) Cho, Y.; Berkelbach, T. C. Optical properties of layered hybrid − D.; Chung, D. S.; Nag, A. Colloidal BaZrS3 chalcogenide perovskite organic inorganic halide perovskites: A tight-binding GW-BSE study. nanocrystals for thin film device fabrication. Nanoscale 2021, 13, J. Phys. Chem. Lett. 2019, 10, 6189−6196. 1616−1623. (41) Wang, Q.; Jiang, C.; Zhang, P.; Hamann, T. W. Overcoming (21) Wei, X.; Hui, H.; Perera, S.; Sheng, A.; Watson, D. F.; Sun, Y.- bulk recombination limits of layered perovskite solar cells with − Y.; Jia, Q.; Zhang, S.; Zeng, H. Ti-alloying of BaZrS3 chalcogenide mesoporous substrates. J. Phys. Chem. C 2018, 122, 14177 14185. perovskite for photovoltaics. ACS omega 2020, 5, 18579−18583. (42) Freysoldt, C.; Grabowski, B.; Hickel, T.; Neugebauer, J.; (22) Kuhar, K.; Crovetto, A.; Pandey, M.; Thygesen, K. S.; Seger, B.; Kresse, G.; Janotti, A.; Van de Walle, C. G. First-principles Vesborg, P. C.; Hansen, O.; Chorkendorff, I.; Jacobsen, K. W. Sulfide calculations for point defects in solids. Rev. Mod. Phys. 2014, 86, 253. perovskites for solar energy conversion applications: computational (43) Basera, P.; Singh, A.; Gill, D.; Bhattacharya, S. Capturing ﬀ screening and synthesis of the selected compound LaYS3. Energy Excitonic E ects in Lead Iodide Perovskites from Many-Body Environ. Sci. 2017, 10, 2579−2593. Perturbation Theory. arXiv, 2020, 2008.03381. (23) Sun, Q.; Chen, H.; Yin, W.-J. Do chalcogenide double (44) Jain, M.; Gill, D.; Bhumla, P.; Basera, P.; Bhattacharya, S. perovskites work as solar cell absorbers: a first-principles study. Chem. Theoretical insights to excitonic effect in lead bromide perovskites. Mater. 2019, 31, 244−250. Appl. Phys. Lett. 2021, 118, 192103. (24) Comparotto, C.; Davydova, A.; Ericson, T.; Riekehr, L.; Moro, (45) Ming, C.; Yang, K.; Zeng, H.; Zhang, S.; Sun, Y.-Y. Octahedron M. V.; Kubart, T.; Scragg, J. Chalcogenide Perovskite BaZrS3: Thin rotation evolution in 2D perovskites and its impact on optoelectronic Film Growth by Sputtering and Rapid Thermal Processing. ACS Appl. properties: the case of Ba−Zr−S chalcogenides. Mater. Horiz. 2020, 7, Energy Mater. 2020, 3, 2762−2770. 2985−2993. (25) Song, J.; Ning, D.; Boukamp, B.; Bassat, J.-M.; Bouwmeester, (46) Szmulowicz, F.; Madarasz, F. L. Deformation-potential-theory H. J. Structure, electrical conductivity and oxygen transport properties calculation of the acoustic-phonon-limited conductivity and Hall − of Ruddlesden Popper phases Lnn+1NinO3n+1 (Ln= La, Pr and Nd; mobilities for p-type silicon. Phys. Rev. B: Condens. Matter Mater. Phys. n= 1, 2 and 3). J. Mater. Chem. A 2020, 8, 22206−22221. 1983, 27, 2605. (26) Ghosh, D.; Acharya, D.; Pedesseau, L.; Katan, C.; Even, J.; (47) Takagi, S.-i.; Toriumi, A.; Iwase, M.; Tango, H. On the Tretiak, S.; Neukirch, A. J. Charge carrier dynamics in two- universality of inversion layer mobility in Si MOSFET’s: Part I-effects dimensional hybrid perovskites: Dion−Jacobson vs. Ruddlesden− of substrate impurity concentration. IEEE Trans. Electron Devices Popper phases. J. Mater. Chem. A 2020, 8, 22009−22022. 1994, 41, 2357−2362. (27) Gill, D.; Kumar, M.; Basera, P.; Bhattacharya, S. Understanding (48) Bruzzone, S.; Fiori, G. Ab-initio simulations of deformation the Ionic Diffusivity in the (Meta) Stable (Un) doped Solid-State potentials and electron mobility in chemically modified graphene and Electrolyte from First-Principles: A Case Study of LISICON. J. Phys. two-dimensional hexagonal boron-nitride. Appl. Phys. Lett. 2011, 99, Chem. C 2020, 124, 17485−17493. 222108. (28) Gill, D.; Bhumla, P.; Kumar, M.; Bhattacharya, S. High- (49) Qiao, J.; Kong, X.; Hu, Z.-X.; Yang, F.; Ji, W. High-mobility throughput screening to modulate electronic and optical properties of transport anisotropy and linear dichroism in few-layer black alloyed Cs2AgBiCl6 for enhanced solar cell efficiency. J. Phys.: Mater. phosphorus. Nat. Commun. 2014, 5, 4475. 2021, 4, 025005. (50) Fröhlich, H. Electrons in lattice fields. Adv. Phys. 1954, 3, 325− (29) Qiu, J.; Xia, Y.; Zheng, Y.; Hui, W.; Gu, H.; Yuan, W.; Yu, H.; 361. Chao, L.; Niu, T.; Yang, Y.; et al. 2D intermediate suppression for (51) Feynman, R. P. Slow electrons in a polar crystal. Phys. Rev. efficient Ruddlesden−Popper (RP) phase lead-free perovskite solar 1955, 97, 660. cells. ACS Energy Lett. 2019, 4, 1513−1520. (52) Hellwarth, R. W.; Biaggio, I. Mobility of an electron in a (30)Stanford,M.G.;Rack,P.D.;Jariwala,D.Emerging multimode polar lattice. Phys. Rev. B: Condens. Matter Mater. Phys. nanofabrication and quantum confinement techniques for 2D 1999, 60, 299. materials beyond graphene. NPJ. 2D Mater. Appl. 2018, 2, 20. (53) Biaggio, I.; Hellwarth, R. W.; Partanen, J. P. Band mobility of (31) Lee, J.-W.; Im, J.-H.; Park, N.-G. Quantum confinement effect photoexcited electrons in Bi12SiO20. Phys. Rev. Lett. 1997, 78, 891. of CdSe induced by nanoscale solvothermal reaction. Nanoscale 2012, (54) Sendner, M.; Nayak, P. K.; Egger, D. A.; Beck, S.; Müller, C.; 4, 6642−6648. Epding, B.; Kowalsky, W.; Kronik, L.; Snaith, H. J.; Pucci, A.; et al. (32) Kohn, W.; Sham, L. J. Self-consistent equations including Optical phonons in methylammonium lead halide perovskites and exchange and correlation effects. Phys. Rev. 1965, 140, A1133. implications for charge transport. Mater. Horiz. 2016, 3, 613−620. (33) Tian, X.; Zhang, Y.; Zheng, R.; Wei, D.; Liu, J. Two- (55) Jiang, H.; Rinke, P.; Scheffler, M. Electronic properties of dimensional organic−inorganic hybrid Ruddlesden−Popper perov- lanthanide oxides from the GW perspective. Phys. Rev. B: Condens. skite materials: preparation, enhanced stability, and applications in Matter Mater. Phys. 2012, 86, 125115. & − photodetection. Sustainable Energy Fuels 2020, 4, 2087 2113. (56) Fuchs, F.; Rödl, C.; Schleife, A.; Bechstedt, F. Efficient O(N2) (34) Yu, Y.; Wang, Z.; Shao, G. Theoretical tuning of Ruddlesden− approach to solve the Bethe-Salpeter equation for excitonic bound Popper type anti-perovskite phases as superb ion conductors and states. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 085103. cathodes for solid sodium ion batteries. J. Mater. Chem. A 2019, 7, (57) Basera, P.; Saini, S.; Bhattacharya, S. Self energy and excitonic − 10483 10493. effect in (un) doped TiO2 anatase: a comparative study of hybrid (35) Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. DFT, GW and BSE to explore optical properties. J. Mater. Chem. C Rev. 1964, 136, B864. 2019, 7, 14284−14293.

6705 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter

(58) Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6,15−50. (59) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758. (60) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865. (61) Bokdam, M.; Sander, T.; Stroppa, A.; Picozzi, S.; Sarma, D.; Franchini, C.; Kresse, G. Role of polar phonons in the photo excited state of metal halide perovskites. Sci. Rep. 2016, 6, 28618. (62) Gajdos,̌ M.; Hummer, K.; Kresse, G.; Furthmüller, J.; Bechstedt, F. Linear optical properties in the projector-augmented wave methodology. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 045112. (63) Frost, J. M. Calculating polaron mobility in halide perovskites. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, 195202.

6706 https://doi.org/10.1021/acs.jpclett.1c01486 J. Phys. Chem. Lett. 2021, 12, 6698−6706