Charge Localization in Cs 2 AgBiBr 6 Double Perovskite: Small Polarons and Self-Trapped Excitons

: Halide double perovskites have gained significant interest as promising materials in optoelectronic applications. Their relatively soft structure allows deformations that can potentially trap charge carriers, leading to the formation of small polarons. In this study, we investigate hole and electron polarons in Cs 2 AgBiBr 6 , a promising double perovskite, using the PBE0( α ) functional. Our results reveal that spin − orbit coupling has a significant effect on the formation energies of small polarons and must be considered for accurate modeling of localized charges in Cs 2 AgBiBr 6 . While electron localization on Bi atoms is favorable, the hole polaron localized at the Ag site is very close in energy to the delocalized state. Simultaneous localization of holes and electrons at neighboring Ag and Bi sites leads to the formation of self-trapped excitons due to the significant attraction between the polarons of opposite charges. Our results provide insights into the essential properties of Cs 2 AgBiBr 6 and have potential implications for further research and development of Cs 2 AgBiBr 6 -based green devices.


■ INTRODUCTION
Halide perovskites have attracted significant attention in the scientific community in recent years due to their exceptional optoelectronic properties. 1,2In particular, their high power conversion efficiencies have led to their use in the design of high-efficiency solar cells, with efficiencies surpassing 25%. 3 Additionally, halide perovskites have been explored for use in a variety of other technologies, such as light-emitting diodes, 4,5 lasers, 6 memories, 7 transistors, 8 batteries, 9 X-ray detectors, 10 and gas sensors. 11However, the most efficient halide perovskites contain lead, which exhibits high toxicity.One possibility to eliminate this issue is to use halide double perovskites, such as Cs 2 AgBiBr 6 , 12−15 where the metal cation in the +2 oxidation state is replaced by the combination of cations in +1 and +3 states.
The nature of excess charges and their mobility are two of the most important factors determining the performance of materials used in optoelectronic devices.Understanding these properties is crucial for the development of new materials with improved performance.Recent experimental works 16−19 reported that excess charges in Cs 2 AgBiBr 6 are subject to self-trapping 16 and in particular showed that the charge carriers in this material undergo an ultrafast, barrier-free self-localization process from an initially mobile delocalized state to a self-trapped, small polaronic state, which diffuses to an emitting color center, revealing an important mechanism for the charge transport and emission properties of the material.−23 While there is a clear need to investigate charge trapping in double perovskites more closely, theoretical investigations on small polarons in this class of materials are lacking.In the present work, we focus on Cs 2 AgBiBr 6 as a representative case of the larger family of double-halide perovskites and computationally assess the nature of self-trapping in these materials.
From a computational point of view, simulation of small polarons poses a challenge since it requires overcoming the self-interaction error related to the use of standard density functional theory (DFT). 24,25To remove the aforementioned error, we use the hybrid functional PBE0(α), satisfying the generalized Koopmans' theorem, 26−28 where α is fraction of the Fock exchange.Hybrid functionals based on the Koopmans' conditions have been shown to well describe the polaronic distortions and to give band gaps with high accuracy without resorting to computationally more demanding GW calculations. 29−33 It has been shown that the perfect cubic structure at 0 K does not represent well the dynamical structure at finite temperatures, which is only cubic on average.One of the ways of overcoming this issue is through the use of the so-called "polymorphous" models proposed by Zhao et al. in ref 31.In the present work, we perform calculations in both monomorphous (perfect cubic) and polymorphous (disordered) structures of Cs 2 AgBiBr 6 to assess the effect of local disorder on polaron formation.Our calculations indicate that charge localization is slightly more favorable in the polymorphous structure.We also examine the effects of spin−orbit coupling (SOC) interactions on the formation energies of the polarons and find that SOC significantly stabilizes the electron polaron.With these two effects considered, the formation of electron polarons in the Cs 2 AgBiBr 6 lattice is favorable, while the hole polarons are marginally unstable.Finally, we determine the formation of the self-trapped exciton (STE) by examining the interaction between the neighboring hole and the electron polarons.The information gained from this study provides insight into the essential properties of Cs 2 AgBiBr 6 , and we anticipate that it will have important implications for future applications in sustainable optoelectronic devices.

■ METHODS
In the present study, we use two DFT packages, CP2K 34,35 to perform fast search and assessment of possible polaronic configurations and the Vienna ab initio simulation package (VASP) 36,37 to achieve more accurate estimates of polaronic formation energies, including SOC.
Within CP2K, Goedecker−Teter−Hutter pseudopotentials are employed to describe the core−valence interactions 38 together with the DZVP-MOLOPT basis sets. 39We set the cutoff energy for the plane-wave basis set to 400 Ry.The PBE0(α) 40 functional with the auxiliary density matrix method 41 is used to perform calculations at the hybrid functional level.We use CP2K to find and explore possible polaronic configurations.
Since the hybrid functional calculations in the CP2K package do not allow for the inclusion of the SOC effect and underestimate the band gap of Cs 2 AgBiBr 6 in the current computational setting (see discussion in Supporting Information), final calculations are performed using VASP. 36,37Within VASP, we run single-shot simulations on top of optimized structures taken from CP2K.We use the plane-wave projectoraugmented wave method with the following valence configurations: Cs [5s5p6s], Ag [5s4d], Bi [6s6p], and Br [4s4p]. 42he plane-wave cutoff energy is set to 300 eV, and the convergence criterion between consequent steps is set to 10 −4 eV.Calculations are run at the PBE0(α) level of theory, and the Brillouin zone is sampled at the Γ point.
To assess the stability of various polaronic states, we calculate the corresponding formation energies with respect to a pristine unit cell containing a delocalized excess charge, represented by the energy of conduction band minimum (CBM) or valence band maximum (VBM).The polaron formation energy, E f , is calculated as where E polaron is the total energy of the charged cell with polaron, E pristine is the total energy of the neutral pristine cell, q is the extra charge in the system (+1 for hole, −1 for electron), ϵ CBM, VBM is the energy of the relevant delocalized state (for q = −1, CBM; q = +1, VBM), and E corr is the finite-size correction term that is calculated based on the scheme proposed by Freysoldt, Neugebauer, and Van de Walle. 43olaron formation energy difference due to SOC effects, Δ SOC , is calculated as where E f withSOC is the polaron formation energy with SOC, and E f w/oSOC is the polaron formation energy without SOC.
We also study the interaction between the hole and electron polarons to assess the possibility of the formation of STEs.Initially, we conduct calculations in a simulation cell that is in a triplet state without considering the effects of SOC.However, in the calculations with SOC, we cannot perform a triplet calculation in the same way as in the spin-polarized case.Therefore, we specify the occupied and unoccupied states that correspond to electron and hole states in the calculations without SOC.We note that the excited state is here described by a single Slater determinant, which in principle approximately represents the first singlet and triplet excited states. 44,45n the case of the singlet state, the spin purification procedure is required to obtain an accurate prediction of its energetic, while the energy of the lowest triplet state is usually not corrected. 46While we focus on the triplet state in this study, we note that an energy difference of about 80 meV was found between the triplet and singlet states, suggesting that the correction for the singlet energy is also of that order.The formation energy of STE, E f STE , where the self-trapped hole and electron are located in neighboring octahedra is calculated by where E exciton is the total energy of the cell with STE and E pristine is the total energy of the neutral pristine structure.The binding energy of the STE, E b STE , is calculated as where E f hole and E f electron are the formation energies of the hole and electron polarons, respectively, as described in eq 1.

■ RESULTS AND DISCUSSION
We first determine α in the hybrid functional, for which the self-interaction error vanishes.This is done through enforcing the Koopmans' condition. 26,27In the calculations, a 2 × 2 × 2 supercell of Cs 2 AgBiBr 6 in the Fm3̅ m phase consisting of 320 atoms is used.The optimal fraction α of Fock exchange in PBE(α) that satisfies Koopmans' theorem is determined by calculating the empty and occupied single-particle energy level of a single Br vacancy.In order to account for electrostatic interactions between periodic images of the charge, finite-size corrections are applied. 47Literature values of high-frequency and static dielectric constants ϵ ∞ (5.42) and ϵ 0 (16.73) are used in the corrections. 48Calculations are carried out for three different fractions, α = 0.40, α = 0.50, and α = 0.60.By choosing relatively high α values, we ensure that the defect charge is localized.Using the linear relation between fraction of Fock exchange and the single-particle energy levels that is reported by Miceli et al., 29 we can determine the optimal α value by extrapolating the value for which the position of the occupied and unoccupied levels coincide.As shown in Figure 1, after applying finite-size corrections, the Fock exchange α is determined to be 0.30 in the CP2K package and 0.28 in the VASP The Journal of Physical Chemistry C package.We note that the uncorrected values of α are 0.23 and 0.22 for CP2K and VASP packages, respectively, highlighting the necessity of including finite-size correction in the procedure. 47,49,50In the subsequent steps of our analysis, we optimized the structures using the CP2K package, with a Fock exchange alpha value set to α = 0.30.We then calculated the total energies of the optimized structures and analyzed the electronic states using the VASP package, with the α value of 0.28.
We note that the fundamental indirect band gap yielded by the PBE0 functional with α = 0.28 (including SOC) is 2.7 eV.This is higher than previously reported values of about 2.0 eV. 16,52However, the extraction of the fundamental indirect band gap from the absorption spectra is rather complicated.Palummo et al. 53 identified indirect excitons in Cs 2 AgBiBr 6 , which would contribute to the absorption signal at about 2 eV.Therefore, this value does not necessarily need to correspond to the fundamental indirect band gap.Considering this complication, we can instead compare the fundamental direct band gap calculated within our hybrid functional (3.5 eV) with experiments.Wright et al. 16 estimated the fundamental direct band gap at low temperatures to be about 3.12 eV; however, the temperature dependence of the band gap exhibits some bending at low temperatures, implying significant effects of zero-point renormalization (ZPR).Linear extrapolation of the band gaps measured at higher temperatures in their study implies a 0 K-band gap without ZPR slightly above 3.2 eV.Kentsch et al. 54 reported a higher value at room temperature of 3.2 eV, which would imply a 0 K-band gap without ZPR of about 3.4 eV.Considering this range of values (3.2−3.4 eV), our fundamental direct band of 3.5 eV is in good agreement with experimental values, which validates the Koopmans functional used here.
Next, we focus on identifying polaronic distortions within the monomorphous model of Cs 2 AgBiBr 6 .Using a high initial alpha value (0.50) to facilitate charge localization, we distort

The Journal of Physical Chemistry C
the Ag−Br and Bi−Br bonds and add, respectively, one excess hole or electron to each system in order to determine possible polaron configurations.The excess hole localizes on the Ag atom, contracting four Ag−Br bonds.The excess electron localizes on the Bi atom, leading to an increase in the length of two Bi−Br bonds.The polaronic geometries are shown in Figure 2.
With the geometries obtained from the previous step, we investigate the formation of the self-trapped hole and electron using the reduced Fock exchange alpha value that satisfies the Koopmans' condition (α = 0.30).Following structural optimization, the hole is found to be delocalized, whereas the electron remains localized on the Bi atom in the monomorphous cubic structure.Without SOC, the formation energy of the electron polaron is +89 meV, and the polaronic level is located at 892 meV below the CBM.
We now focus on the polymorphous model of Cs 2 AgBiBr 6 , since it has been reported that local distortions can significantly affect the electronic structure of cubic halide perovskites 31,55 We first generate several polymorphous structures and compare their energies with those of the monomorphous structure.The details on the structure generation along with energies of different structures are given in Supporting Information.As expected, all polymorphous cells are more stable than the monomorphous one.As a result, to obtain a more accurate description of the energetics of polaronic states in Cs 2 AgBiBr 6 , we carried out our calculations in the polymorphous structure.
We proceeded to investigate the small electron polaron in polymorphous Cs 2 AgBiBr 6 .Similar to the procedure adapted in the monomorphous cell, we add an excess electron to the structure, elongate the Bi−Br bonds surrounding a single Bi atom, and relax the structure with a high initial value of α (0.50), which is then reduced to 0.30.After relaxation, the electron remains localized on the Bi atom.The energy and electronic states of the system are then calculated, and the corresponding charge densities and energy levels are shown in Figure 2b.The formation energy of the electron polaron is determined to be +32 meV without SOC, and the polaron level is found to be located at 1.033 eV below the CBM in the polymorphous cell.Although the formation of the electron polaron remains energetically unfavorable, the polaron is observed to be more stable by 57 meV compared to the electron polaron in the monomorphous cell.
We then investigated the localization of the excess hole in the polymorphous cell.To achieve this, the Ag−Br bond lengths around a single Ag atom in the cell are decreased.The resulting structure is optimized first using α = 0.50, which is then decreased to 0.30.Unlike that in the monomorphous cell, the hole polaron remains localized after the final relaxation.At the same time, we observe that the charge localization induces not only the contraction of Ag−Br bonds but also a noticeable tilting of the whole AgBr 6 unit by 4.5°in comparison to the pristine polymorphous structure (see Figure 2c).We note that to achieve the same polaronic geometry in the monomorphous cell, the AgBr 6 unit would have to tilt by 10.1°compared to the initial state.We therefore conclude that the reason for a more favorable hole localization in the polymorphous cell is the facilitated octahedral tilting in this structure.The corresponding polaron level is located at 0.678 eV.The formation energy of the hole polaron is determined to be +16 meV, indicating that hole localization is not energetically favorable.
The calculations performed thus far have not included SOC effects.However, halide perovskites are known to be strongly influenced by relativistic effects, particularly at the CBM. 30,49,56ence, it is crucial to consider the influence of SOC on the polaronic states.Therefore, we calculate the effects of SOC on electron polaron in monomorphous and polymorphous cells and hole polaron in polymorphous cells.The results are shown in Figure 2. We observe that the stability of the polarons increases with the inclusion of SOC in Cs 2 AgBiBr 6 .This is in contrast to the situation in lead and tin halide perovskites, where SOC makes polaronic states less favorable. 49,57While in the case of the hole polaron, the effect of SOC is weak (12 meV), the formation energy of the electron polaron is strongly reduced (by up to 180 meV).This strong stabilization is related to SOC making the single-particle level of the electron polaron lie deeper in the band gap relative to the CBM (see Table 1 for numerical values).The formation energy of the electron polaron, including SOC, is found to be −91 meV in the monomorphous structure and −141 meV in the polymorphous one.The stability of the electron polaron is in agreement with the experimental reports of barrier-free charge carrier localization in Cs 2 AgBiBr 6 . 16he hole polaron in the polymorphous structure is found to be marginally unstable, as evidenced by its positive formation energy of +4 meV.However, we note that this formation energy is below the expected accuracy of our computational setup, and we cannot rule out the possibility of the formation of hole polarons in Cs 2 AgBiBr 6 .Additionally, we found that the hole polaron is stabilized by facilitated octahedral tilting in the polymorphous cell.Therefore, it could also be stabilized by the thermal effect, which can lead to lattice softening and facilitate octahedral tilting. 49,58Furthermore, we performed additional calculations at the equilibrium lattice constant, which gave a formation energy of −63 meV for the hole polaron (see Supporting Information).This suggests that the current computational setup is unable to rule out the hole localization.
Previous analyses focused on the localization of either holes or electrons separately.We now investigate the interaction between the opposite charges localized nearby.This is done by performing calculations in a cell with distorted neighboring AgBr 6 and BiBr 6 octahedra.After relaxation, we observe hole and electron localization on neighboring sites, as presented in Figure 3.We calculate the resulting formation energy of the STE according to eq 3 and find a value of −293 meV with Change in the formation energy due to SOC is given by Δ SOC .VBM of the pristine cell where SOC effects are not considered is set to 0.
The Journal of Physical Chemistry C SOC (−287 meV without SOC).This is significantly lower than the sum of the formation energies of separate hole and electron polarons (E f hole = +4 meV and E f electron = −141 meV with SOC) within the corresponding computational setup.Therefore, we extract the binding energy E b STE = −156 meV (−335 meV without SOC) from the difference between these energies.Moreover, the electron−hole interaction also results in deeper single-particle states, which are now found at 1.56 eV (1.49 eV without SOC) below the CBM for the electron and 1.06 eV (1.12 eV without SOC) above the VBM for the hole.
We note that all results reported above have been obtained using the experimental lattice constant a = 11.27Å. 59 To verify how our findings depend on this parameter, we also recalculated all formation energies at the equilibrium volume (a = 11.38 Å using PBE0(α) and including SOC).The results, reported in Supporting Information, differ by less than 0.1 eV from the ones calculated using the experimental lattice parameter and lead to the same general conclusions.
We now discuss our results in view of experimental studies on the behavior of excess charges in Cs 2 AgBiBr 6 .Photoluminescence (PL) measurements indicate that an essentially barrier-less self-trapping occurs in this material. 16A PL signal has been detected at the energy of about 1.9 eV and assigned to a spatially localized color center. 60The proposed selftrapping agrees well with the negative formation energy of the electron polaron and the STE found in this study.There are several possible energy transitions that could give rise to the observed PL signal.Initially, upon photoexcitation, an electron−hole pair, likely in the form of a free exciton, is generated.Subsequently, due to atomic relaxation, rapid formation of STEs could occur.Following the STE formation, the first conceivable transition would be the direct electron− hole recombination within the STE.It has been previously shown that this type of STE (related to a non-Jahn−Teller-like octahedral distortion) more likely leads to nonradiative transitions. 61Another possibility would be the trapped electron recombining with a more delocalized hole from the valence band.The corresponding emission energy (based on differences between calculated single-particle levels) would be between 1.20 eV (electron from the STE) and 1.60 eV (electron polaron).This explanation is more plausible; however, the higher value of 1.60 eV is still significantly lower than the experimental value of 1.9 eV.The third option is the recombination of a hole trapped in the STE and a delocalized electron in the conduction band.This transition would correspond to 1.70 eV (see Table S5 in the Supporting Information), which is in better agreement with the experimental signal.We note, however, that the full understanding of photoluminescence from localized charges would require a more detailed study based on methods such as the Bethe-Salpeter equation or time-dependent DFT. 62,63CONCLUSIONS In conclusion, we have studied charge trapping in Cs 2 AgBiBr 6 by employing a Koopmans-compliant hybrid density functional.Possible modes of charge localization were identified by introducing excess holes and electrons and creating initial structural distortions.Excess holes were found to localize on Ag atoms, while excess electrons localized on Bi atoms.We compared charge localization in monomorphous and polymorphous models of Cs 2 AgBiBr 6 and found that facilitated octahedral tilting leads to higher stability of hole polarons in the distorted model.The effects of SOC were also considered to provide a more accurate representation of the CBM and polaronic states.With SOC, electron localization was found to be energetically favorable in both monomorphous and polymorphous cells, with formation energies of −91 and −141 meV, respectively.On the other hand, hole polarons were found to be marginally unstable in the polymorphous structure at 0 K. Finally, the formation of STEs was assessed by considering holes and electrons localized at neighboring sites.We determined the binding energy of the STE to be −156 meV, including the SOC effect.Overall, our results reveal the intricate interplay between charge localization, the effects of SOC, and the stabilizing effect of neighboring electron and hole localization, forming STEs.
Generation and energy assessment of polymorphous structures, polaron formation energies and single-particle levels in optimal polymorphs, comparison between polaron energies and levels between CP2K and VASP, effect of the lattice parameter on polaron energies, and formation and binding energies of STE and corresponding energy levels (PDF)  The Journal of Physical Chemistry C

Figure 1 .
Figure 1.Band edges and single-particle energy levels of the Br vacancy with respect to the fraction of Fock exchange α used in PBE0(α) calculations, determined using (a) CP2K and (b) VASP.Dashed and solid lines represent values excluding and including the finite-size corrections, respectively.Blue and green correspond to the neutral and positive charge states of the defect.

Figure 2 .
Figure 2. Band decomposed charge densities and energy level diagrams at Γ, without and with SOC effects for (a) electron polaron in the monomorphous cell, (b) electron polaron in the polymorphous cell, and (c) hole polaron in the polymorphous cell.In the insets, polaron formation energies including and ignoring SOC are given.Isosurface levels are displayed at 0.003 eV/Å 3 , using VESTA software. 51CBM and VBM levels are taken from the pristine cell.Polaron levels include finite-size corrections.Zero in the energy scale is set to the VBM of the pristine cell without SOC.Electron and hole polaron energy levels are indicated by blue and yellow colors.Arrows in the structures indicate the difference in the corresponding bond lengths.

Figure 3 .
Figure 3. Isodensity of STE with the binding energy calculated including SOC, and the corresponding the energy diagram that shows VBM, CBM, electron (blue), and hole (yellow) levels.Zero in the energy scale is set to the VBM of the pristine cell without SOC.The isosurface level is taken as 0.0015 eV/Å 3 .