Temperature‐Dependent Excitonic Band Gap in Lead‐Free Bismuth Halide Low‐Dimensional Perovskite Single Crystals

In this study, the optical behavior of lead‐free Bi‐based low‐dimensional perovskite single crystals (Cs3Bi2Cl9, Cs3Bi2Br9, Cs3Bi2I9, and MA3Bi2I9) is investigated by spectroscopic ellipsometry, supported by X‐ray diffraction and density functional theory calculations. All materials exhibit a strong excitonic peak resulting from photogenerated electron–hole Coulomb interactions, whereas the threshold of continuous absorption is found at higher energies. The resonances of the excitonic and continuous bands, along with exciton binding energies, are extracted through Critical Point Analysis of the ellipsometric data over a wide temperature range (from −90 °C to 90 °C), revealing subtle variations in the optical characteristics for each single crystal. These materials can be applied in optoelectronics as photodetectors because of their high stability and lower toxicity compared to their Pb‐based perovskites.

The non-toxic bismuth cation (Bi 3+ ) emerged as one of the most suitable lead counterparts for heterovalent substitution.This is primarily because Bi 3+ shares the same 6s 2 6p 0 electronic structure as the Pb 2+ cation and has a similar effective ionic radius (1.03 Å). [31] Additionally, Bi-based perovskites have shown superior moisture and thermal stability. [32]One of the promising Bi-based perovskite structures is the A 3 Bi 2 X 9 , formed by hexagonal or cubic packing of A and X cations.Whereas the trivalent Bi 3+ cations occupy only two-thirds of the octahedral cavities BiX 6 . [33]Unlike the 3D framework made up of cornersharing PbX 6 octahedra present in APbX 3 , the crystalline lattice of A 3 Bi 2 X 9 Bi-based alternatives is determined by various stackings of trigonal AB 3 layers.
In principle, there are three types of stackings present in these structures: h, hcc, and c. [34] The hexagonal (h) 6 stacking leads to the formation of rhombohedral structures with a 0D framework of face-sharing Bi-X octahedra, i.e., the framework of isolated bi-octahedral B 2 X 9 3− anions.When stacked in cubic (c) mode, they form trigonal structures, where BiX 6 octahedra share cis-vertices with the other three octahedra and thus create the 2D corrugated layers.The (hcc) 2 stacking leads to the formation of hexagonal and orthorhombic structures, where both kinds of octahedron bonding, i.e., both 0D and 2D structures are possible. [33,34]revious reports show that Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 crystallize in hexagonal structure (P6 3 /mmc) [35][36][37] with the 0D motive of bioctahedra, while Cs 3 Bi 2 Br 9 crystallizes in trigonal (P " 3m1) [38,39] and Cs 3 Bi 2 Cl 9 in either orthorhombic (Pnma) [40,41] or trigonal (P " 31c) [42] crystal systems (space groups), with the 2D motive of BiBr 6 and BiCl 6 octahedra, respectively.Due to the reduced dimensionality, A 3 Bi 2 X 9 perovskites have relatively large band gap (1.94-3.02eV) [36,41,42] and extremely low ionic migration.The thin films of Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 have been used as photoactive layers in solar cells while Cs 3 Bi 2 Br 9 and Cs 3 Bi 2 Cl 9 were not used in solar devices due to the too-wide electronic band gap. [43]However, these Bi-based perovskites exhibit low dark current noise (at pA level) and high resistivity (up to 10 12 Ω cm) [36] both very desirable properties for the construction of highly sensitive photodetectors.Additionally, these nontoxic materials showed outstanding thermal stability, fast response speeds (in milliseconds) and high signal-to-noise (onoff) ratio.Li et.al., [44] presented the vertical ITO/Cs 3 Bi 2 I 9 /Au photodetectors with an exceptional on-off ratio of 11 000, measured under −2 V bias and a white LED of 100 mW cm −2 light intensity.Moreover, these devices showed great long-term stability, preserving more than 90% of their initial response after 1000 h of exposure to humid air (50% RH).MA 3 Bi 2 I 9 also proved to be a suitable material for efficient photodetection.Hussain et.al. [45] fabricated Ag/MA 3 Bi 2 I 9 /FTO photodetector with high detectivity (1.3 × 10 12 Jones) and a fast response speed of (26.81/41.98ms) under 0 V bias and low white light intensity of 10 μW cm −2 .Furthermore, Cs 3 Bi 2 I 9 demonstrated great potential for X-ray detection. [46]Zhang et.al. [47] reported Xray detectors based on centimeter-sized Cs 3 Bi 2 I 9 single crystals, which exhibited a high sensitivity of 1652.3 μC Gy air −1 cm −2 and a very low detection limit of 130 nGy air s −1 ≈4 times higher than -Se detectors and ≈40 times lower than required for medical diagnostics.Additionally, these single crystals showed outstanding operational stability, even at a higher temperature of 100 °C.
Unlike their iodide counterparts, Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 , the research on the detection potential of Cs 3 Bi 2 Br 9 and Cs 3 Bi 2 Cl 9 is still in its infancy.Liu et.al. [48] demonstrated the detecting capability of a self-powered FTO/NiO x /Cs 3 Bi 2 Br 9 /Au UV photodetector, with a fast response speed of 3.04/4.65ms, the responsivity of 4.33 mA W −1 and detectivity of 1.3 × 10 11 Jones (measured under week UV light of 15 mW cm −2 , 405 nm).Tailor et al. [41] reported the first Ag/Cs 3 Bi 2 Cl 9 /Ag detectors with a responsivity of 17 mA W −1 and detectivity as high as 6.63 × 10 11 Jones.Although these materials show significant response to incident photons, the relationship between their optical properties and device performance is not fully explained.
Before contemplating their potential application in optoelectronics (photodetectors, scintillators, solar cells etc.), it is crucial to comprehend how the optical properties of these materials evolve concerning temperature, chemical composition, dimensionality, and how such changes could impact the performance of future A 3 Bi 2 X 9 -based devices.In this work, we report an extensive multi-material study of the band gap change for inorganic and hybrid A 3 Bi 2 X 9 perovskite single crystals (Cs 3 Bi 2 Cl 9 , Cs 3 Bi 2 Br 9 , Cs 3 Bi 2 I 9 , and MA 3 Bi 2 I 9 ) using spectroscopic ellipsometry (SE), supported by x-ray diffraction and density functional theory (DFT) calculations.We identified the interband transition energies by applying a Critical Point (CP) Analysis on the absorption spectra, to gain a comprehensive understanding of the excitonic processes in the 1-5 eV energy range, considering temperature variations that span from −90 °C to 90 °C.Within this framework, we discuss exciton and continuous band absorption, phase changes and fine differences in the optical properties arising from the chemical and structural differences of the studied materials.
We point out that, although the Cs 3 Bi 2 Br 9 sample presents more than one set of planes, only the most intense set was selected for SE measurements.On the other hand, we attempted to use a more refined optical model that take into account the anisotropy without obtaining a fitting improvement, very likely due to the very low contribution (i.e., intensity) of other planes to the optical response.
To investigate the optical properties of Bi-based crystals, we performed SE measurements at room temperature (RT) and in air on all the as-prepared samples.The experimental data fit and the extracted dielectric function, as discussed in the experimental section, are shown in Figure S7 (Supporting Information).

Figure 1i
,l shows the absorption coefficient calculated from the real ( 1 ) and the imaginary ( 2 ) parts of the dielectric function using equation: [49] where E is the energy, c is the speed of light in vacuum and h is the Planck constant.The absorption coefficient increases abruptly at a certain energy value that, when excitons are absent, or too weakly bounded, corresponds to the onset of the continuous band and characterizes the band gap (E gap ) of the system.In materials where bound excitons are present, the difference between the exciton energy (E ex ) and the E gap defines the exciton binding energy E B = E gap -E ex . [50]In all Pb-based perovskite materials like MAPbI 3 , [51] MAPbBr 3 , 11 CsPbI 3 , [52] CsPbBr 3 , 11 E ex is hardly distinguishable from E gap due to low exciton binding energies (25-50 meV [53] ).In these materials (bulk or thin layers), the exciton is easily separated in free charges, making them good candidates for solar cell devices. [50]On the contrary, in all studied Bi-based materials, we observe an isolated and narrow excitonic peak, indicated as E ex in Figure 1i,l.This peak is more prominent for the Cs 3 Bi 2 Br 9 and Cs 3 Bi 2 Cl 9 systems with respect to the two iodides (Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 ).We argue that this peak arises from electron-hole interactions and reflects a bound exciton owing to a large binding energy at RT, [54][55][56] while E gap is located at higher energy (Figure 1i,l).For this reason, if the material's band gap is set to the excitonic absorption onset, its value is by far underestimated.This feature was previously reported in the literature for Cs 3 Bi 2 I 9 [57] and MA 3 Bi 2 I 9 [58] single crystals, but only for Cs 3 Bi 2 Cl 9 [41,42,59] and Cs 3 Bi 2 Br 9 [60][61][62][63] nanosystems.For the two latter materials, it is possible to find optical characterizations in the literature that refer to nanocrystals or nanoplatelets, where quantum confinement effects and surface-related phenomena might affect the data with respect to the large single crystals used here. [64]he absorption coefficient provides also information about defects.In particular, the Cs 3 Bi 2 Cl 9 (Figure 1a) and Cs 3 Bi 2 Br 9 (Figure 1b) and to a very lesser extent also MA 3 Bi 2 I 9 (d) have defects inside the gap attested by the broad band at energies just below the excitonic peak.We also calculated the Urbach tail at the band edge according to the following equation  =  0 exp(E/E u ) where  0 is a constant, E is the energy and E u is the Urbach energy.The values of E u are 33 meV for Cs 3 Bi 2 Cl 9 , 20 meV for Cs 3 Bi 2 Br 9 , 96 meV for Cs 3 Bi 2 I 9 , and 34 meV for MA 3 Bi 2 I 9 .[67][68][69][70][71][72] The CPs, which depend on the densities of states in the electronic bands, and their parameters (energy position E, amplitude A, broadening Γ, and phase Φ) are extracted from the simultaneous fit of the real and imaginary parts of the dielectric function () through the following equation: where n is −1/2 for 1D, 0 for 2D, or ½ for 3D critical points and −1 when describing excitonic transitions.The E ex and E gap values calculated for Cs 3 Bi 2 Cl 9 and Cs 3 Bi 2 Br 9 , have not been yet reported in the literature and are respectively: 3.27 and 4.42 eV for Cs 3 Bi 2 Cl 9 , and 2.84 and 4.26 eV for Cs 3 Bi 2 Br 9 .On the other hand, E ex and E gap values for Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 confirm the values reported by Machulin et al. [57] (Cs 3 Bi 2 I 9 , calculated by reflection spectra at RT, 2.578 and 2.857 eV) and by Kawai et al. [58] (MA 3 Bi 2 I 9 , calculated by absorption spectra at RT, 2.49 and 2.9 eV).
Cs 3 Bi 2 Cl 9 and Cs 3 Bi 2 Br 9 exhibit a large E B of 1143 meV and 1424 meV, respectively.The value obtained for Cs 3 Bi 2 Br 9 is higher than those reported by Bass et al. [54] from UV-vis absorption measurements performed on powder (940 meV) and by Wu et al. from similar measurements on nanoplatelets (148 meV), [73] indicating a strong dependence of the optical properties of the material from the characteristics of the crystal structure.
The exciton binding energies for Cs 3 Bi 2 I 9 (334 meV) and MA 3 Bi 2 I 9 (303 meV) are close to those already reported. [57,58]hese high values of E B (not efficient charge separation resulting in low J SC [74] ), along with the wide electronic band gap (low optical absorption) and high Urbach energy (≈50 meV, resulting in high nonradiative recombination losses [75] ), are the main reasons for the low values of photo-conversion efficiency reached using Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 as active materials in solar cells (record of 3.6% [76] and 3.17%, [77] respectively).These results can be compared to those calculated with the Elliot model [78][79][80][81] used to simulate the absorption near the band edge with the equation: where E gap is the electronic band gap, E B is the exciton binding energy (i.e. E B = E gap −E ex where E ex is the energy of the exciton), A c , , ,  are the scaling factors.The first term describes the continuum state absorption and the second term accounts for multiple excitonic states.We neglect the excitonic transition with n>3 because the excitonic peak is well separated from continuous and the oscillator strength decreases as n 3 .The cumulative fit is in good agreement with experimental data as shown in Figure S8 (Supporting Information).The values obtained from fitting are reported in Table 2.
To further investigate the optical behavior of Bi-based crystals as a function of the temperature, being aware that other perovskitic materials are very sensitive to external factors like humidity that affect their properties, we have performed SE measurements in a pure N 2 environment, to avoid possible sample degradation. [82,83]Repeated measurements have demonstrated that all samples are stable in N 2 for several weeks.The measurements were performed in the range of −90 °C to 90 °C, to cover all the possible application fields, from x-ray and gamma detectors, which can work at low temperatures to minimize the dark current to LED and solar cells that can reach high working temperatures under sunlight. [84]he variation of the absorption coefficient as a function of temperature is reported in Figure S9 (Supporting Information).
The E ex and E gap temperature dependence of Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 that have a smaller E B were monitored with a step of 3 °C while for Cs 3 Bi 2 Cl 9 and Cs 3 Bi 2 Br 9 the step was 15 °C.This type of study was previously conducted with a similar approach for the band gap of lead bromide perovskite single crystals. [11,85]n these works, we demonstrated that a lattice phase transition is detectable through SE measurements when a change of slope appears in the energy versus temperature plot of the critical point at the lowest energy.
Figure 2 shows the variation of the second derivative of the real ( 1 ) and imaginary ( 2 ) parts of the dielectric function from which the CPs can be extracted in the range −90 °C to 90 °C (with 30 °C temperature step) for Cs 3 Bi 2 Br 9 (Figure 2a,b) and Cs 3 Bi 2 I 9 (Figure 2c,d).
By lowering the temperature, it is possible to identify a general trend for all Bi-based materials: the E ex peak amplitude (A) increases (y-axis) and the E ex broadening (Γ) decreases (x-axis) (see also Figures S11-S13, Supporting Information).These findings are consistent with expectations, as exciton-phonon interactions are smaller at lower temperatures, leading to a higher energetic localization of the excitons.Furthermore, the E ex position does not vary significantly with the temperature, in contrast with the E gap position which undergoes significant shifts.Quantifying numerically all these observations is possible through the CP analysis.
Figure 3 shows, within the CP analysis, the temperature dependence of the energy position of E ex and E gap with the EB for Cs3Bi2Cl9 (a-c), Cs3Bi2Br9 (d-f), Cs3Bi2I9 (g-i), and MA3Bi2I9 (j-l).The other CP parameters (A, Γ, Φ) trend versus temperature are reported in the Supporting Information (Figures S11-S13, Supporting Information).We observed that Cs3Bi2Cl9 and Cs3Bi2Br9 samples exhibit similar behavior, notwithstanding the differences in their crystal structure.Specifically, E gap and E ex have a linear dependence in the whole temperature range (Figure 3a,d).In agreement with our previous work, [11] we argue that a linear dependence of the energy with the temperature indicates that no phase transitions occur within the studied temperature interval.88] The temperature dependence of E ex and E gap in both I-based crystals (Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 ) differs from above and it is more complex.For Cs 3 Bi 2 I 9 , the E ex curve slope starts to slowly decrease at 45 °C going toward lower temperature and, at ≈−50 °C, there is a net change in the curve slope that we associate with a reversible phase transition (Figure 3g) from the hexagonal to the monoclinic lattice at −53 °C. [57,89]In addition, a second ex-citonic peak (E ex2 in Figure 2e,f) emerges at ≈2.68 eV at the same temperature.The E gap linearly decreases with temperature (Figure 3h), in good agreement with the literature. [57,82]Consequently, E B decreases linearly, differently from the two previous crystals (Figure 3i).This finding suggests that better performances could be achieved for Cs 3 Bi 2 I 9 -based solar cells at the typical operating temperature of 45-85 °C [90] due to the narrowing of the electronic band gap resulting in improved optical absorption and the decrease of E B causing an improved charge extraction.
The behavior of MA 3 Bi 2 I 9 E ex exhibits an even higher degree of complexity.The E ex value shows a linear increase in the range of temperature 90-25 °C (Figure 3j).However, as the sample is further cooled from 25 °C to ≈−50 °C, the slope gradually decreases down to −90 °C.Similarly to the Cs 3 Bi 2 I 9 sample, we could identify at −50 °C a slope change.This is in good agreement with Jakubas et al. [91] who report a second-order phase transition occurring at −50 °C.We note that Kamminga et al. [92] report that the MA 3 Bi 2 I 9 crystal structure gradually evolves from a hexagonal phase to a monoclinic phase by decreasing the temperature from 27 °C to −113 °C through the alignment of the methylammonium cations along the b lattice direction.The trend of E gap with respect to temperature exhibits similarity to the trend of E ex (Figure 3k) and therefore E B increases with temperature (Figure 3l) as for Cs 3 Bi 2 Br 9 , and Cs 3 Bi 2 Cl 9 .We note here that notwithstanding MA 3 Bi 2 I 9 shares similar structural characteristics with Cs 3 Bi 2 I 9 , their optical properties present some fine differences.An E ex2 peak is absent from MA 3 Bi 2 I 9 below the phase transition temperature, whereas the exciton binding energies have different temperature trends.We hypothesize the presence of weakly bound excitons at the conduction band edge that may be responsible for these differences.To better interpret the experimental measurements of the optical constants, we performed density functional theory (DFT) calculations for all the experimentally studied systems in their RT phases and post-processed the obtained electronic structure within the Bethe-Salpeter equation (BSE) [93,94] for the calculation of the real and imaginary parts of the dielectric function.We note that the BSE allows for the calculation of the absorption properties considering the interaction between electrons and holes in the excited electronic spectrum, which is important when strong excitonic features are present.A specific requirement for the studied Bi-based materials was the consideration of spin-orbit coupling effects within the calculation scheme, due to their impact on the description of the conduction band of these systems.(Figure S14, Supporting Information). [95,60]Specifically, in the case of Cs 3 Bi 2 I 9 , spin-orbit interactions split the lower conduction band into two sub-bands (see Cs 3 Bi 2 I 9 Figure S14a, Supporting Information) whereas for Cs 3 Bi 2 Br 9 they "mix" low-dispersed sub-bands in a single band (see Cs 3 Bi 2 Br 9 in Figure S14b, Supporting Information).In both cases, the qualitative differences in the conduction band introduced by spin-orbit interactions cannot be neglected when evaluating the optical transitions of these materials.Moreover, It is interesting to note that all systems share some common characteristics regarding the main orbital projec-tions in their electronic structure (Figure S15, Supporting Information): conduction bands are strongly characterized by Bi p orbitals (with a total angular momentum J = 1/2 for lower energies and J = 3/2 for higher energies), whereas the valence band maxima are mainly shaped by halide p-orbital contributions.The two iodide systems (Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 ) present very similar electronic properties (Figure S16, Supporting Information) arising from the similarity in their structural configuration for a wide range of temperatures.Hereon, only the optical properties of inorganic halides will be discussed.Figure 4 shows the imaginary part of the dielectric function ( 2 ) calculated within the BSE theory for the Cs 3 Bi 2 I 9 , Cs 3 Bi 2 Br 9 and Cs 3 Bi 2 Cl 9 systems and compares the results with experimental measurements at RT.
All curves show strong excitonic characteristics when electron-hole interactions are considered (red lines) with respect to optical calculations that do not account for excitonic effects (blue lines).The Cs 3 Bi 2 I 9 system is characterized by a main excitonic peak having a slightly higher binding energy compared to the experiment.Additionally, the curve is red-shifted with respect to the one calculated without electron-hole interactions.The case of Cs 3 Bi 2 Br 9 is more particular, as between the dominant excitonic peak and the second optical peak of the experimental curve, intermediate peaks (with the most prominent at ≈3.15 eV) appear only in the calculated spectrum.Similar features have been experimentally observed only in Cs 3 Bi 2 Br 9 nanocrystals at certain crystallographic directions, [96] but are absent from crystals of bigger dimensions.This aspect indicates a rather strong influence of the optical characteristics in this system, either from structural characteristics that are not captured in the simple trigonal model used for our calculations, or from local characteristics that are only present at the nanoscale.Nevertheless, our calculations indicate that these intermediate peaks should be intrinsic to the bulk material and independent of surface-related phenomena.It is also interesting to note that the second peak of the experimental optical spectrum in Cs 3 Bi 2 Br 9 practically coincides with the first peak of the  2 curve in the independent particle approximation (i.e., without considering electron-hole interactions in the calculation scheme).Finally, an almost excellent agreement between theoretical and experimental data is obtained in the case of Cs 3 Bi 2 Cl 9 , showing a strongly redshifted spectrum with respect to the independent particle approximation and a main excitonic peak at ∼3.32 eV.Some divergences between the experimental and theoretical data appear only for higher energy values, reflecting the limited number of bands considered for the calculation of the static dielectric matrices (see the Experimental section) with respect to the extremely dense electronic states that are present in the valence band of the material (Figure S12, Supporting Information).Overall, the BSE level of theory appears necessary for the proper estimation of the optical properties of Bi-based halide perovskites.

Conclusion
Our multiparameter analysis provides a comprehensive outlook on the behavior of the excitonic band gap and the continuous absorption onset for A 3 Bi 2 X 9 single crystals depending on the temperature.This is crucial for various optoelectronic applications.In particular, we investigated the structural and optical properties of four Bismuth halide single crystals, namely Cs 3 Bi 2 Cl 9 , Cs 3 Bi 2 Br 9 , Cs 3 Bi 2 I 9 , and MA 3 Bi 2 I 9 .XRD measurements unveiled their crystalline structure, revealing a quasi 1D orthorhombic structure for Cs 3 Bi 2 Cl 9, a quasi 2D trigonal structure for Cs 3 Bi 2 Br 9 and a quasi 0D hexagonal structure for Cs 3 Bi 2 I 9 and MA 3 Bi 2 I 9 .Strong excitonic features were observed for all mate-rials with distinct characteristics, based on the chemical composition of both anions and cations.E B values for Cs 3 Bi 2 Cl 9 , Cs 3 Bi 2 Br 9 and MA 3 Bi 2 I 9 increased with temperature, while for the Cs 3 Bi 2 I 9 the trend was diametrically opposite.We identified a phase transition from the hexagonal to the monoclinic lattice at −53 °C for Cs 3 Bi 2 I 9 , and at −50 °C for MA 3 Bi 2 I 9 .The wide electronic band gap of MA 3 Bi 2 I 9 (2.81 eV) and of Cs 3 Bi 2 I 9 (2.87 eV) and the high exciton binding energies (≈300 meV) are the main reasons for the low-efficiency values of solar cells, suggesting that focused strategies are required to improve the performances. [74]n the other hand, all Bismuth halide single crystals have a great potential for application as highly efficient photodetectors.
Preparation of A 3 Bi 2 X 9 Single Crystals: A 3 Bi 2 X 9 single crystals were prepared using the hydrothermal method.The 0.05 m perovskite solutions were prepared by dissolving the precursors CsX and BiX (molar ratio 3:2) in 20 mL of hydrohalic acids (CsCl and BiCl in HCl, etc.) in hydrothermal autoclave reactor.For detailed precursor masses, see Table S1 and Figure S1 (Supporting Information).The solutions were then heated to 200 °C and kept at constant temperature for 2 h to ensure the complete dissolution of the precursors.In the next step, the solutions were cooled down from 200 °C to 25 °C (temperature gradient 1°C h −1 ) after which millimetre-sized single crystals were obtained.The obtained crystals were then extracted from the solution and separated based on their size and geometry.Samples with the most suitable geometry were used as seeds for further growth.The seeds were placed in the previously filtered perovskite solutions previously filtered (PTFE 0.45 μm) and heated to 50 °C, after which they were slowly cooled down (1°C h −1 ) to 25 °C.The obtained Bi-based perovskite single crystals had exceptionally flat surfaces that were crucial for the optical characterization that they underwent.Microscopic photos including SEM images and a detailed scheme of the synthetic procedure are presented in Figures S2-S5 (Supporting Information).It is important to observe that excitonic bands function as chromophores, ex-

Figure 2 .
Figure 2. a,b) Second derivative of the real ( 1 ) and imaginary ( 2 ) part of the dielectric function across the temperature range from -90 °C to 90 °C (ΔT = 30 °C) for Cs 3 Bi 2 Br 9 and c,d) Cs 3 Bi 2 I 9

Figure 3 .
Figure 3. a-c) Energy position of E ex and E gap with E B =E gap -E ex vs temperature for Cs 3 Bi 2 Cl 9 , d-f) Cs 3 Bi 2 Br 9 , g-i) Cs 3 Bi 2 I 9 , and j-l) MA 3 Bi 2 I 9 .

Figure 4 .
Figure 4.The imaginary part of the dielectric function ( 2 ) calculated within the BSE theory (red lines) and the independent particle approximation (blue lines) for a) Cs 3 Bi 2 I 9 , b) Cs 3 Bi 2 Br 9 , and c) Cs 3 Bi 2 Cl 9 .Experimental data (green dots) correspond to measurements using SE at RT.

Table 1 .
Energy values of E ex , E gap and E 1 at RT extracted through the Critical Points analysis for Cs 3 Bi 2 Cl 9 , Cs 3 Bi 2 Br 9 , Cs 3 Bi 2 I 9 , and MA 3 Bi 2 I 9 .E B (the exciton binding energy) is calculated as E gap -E ex .

Table 2 .
Energy values of E gap, E B at RT extracted through the Elliot Analysis for Cs 3 Bi 2 Cl 9 , Cs 3 Bi 2 Br 9 , Cs 3 Bi 2 I 9 , and MA 3 Bi 2 I 9 .The E ex (the exciton energy) is calculated as E gap -E B .