
==== Front
J Phys Chem C Nanomater Interfaces
J Phys Chem C Nanomater Interfaces
jy
jpccck
The Journal of Physical Chemistry. C, Nanomaterials and Interfaces
1932-7447
1932-7455
American Chemical Society

10.1021/acs.jpcc.4c04453
Article
Halide Mixing in Cs2AgBi(IxBr1–x)6 Double Perovskites: A Pathway to Tunable Excitonic Properties
https://orcid.org/0009-0001-2042-1895
Biega Raisa-Ioana †
https://orcid.org/0000-0002-7417-1246
Jöbsis Huygen J. ‡
https://orcid.org/0000-0003-4634-088X
Gijsberg Zamorano ‡
Hüskens Maxim †
https://orcid.org/0000-0002-5537-6545
Hutter Eline M. ‡
https://orcid.org/0000-0002-4361-4382
Leppert Linn *†
† MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands
‡ Debye Institute for Nanomaterials Science, Utrecht University, Princetonlaan 8, 3584 CB Utrecht, The Netherlands
* Email: l.leppert@utwente.nl.
26 08 2024
05 09 2024
128 35 1476714775
03 07 2024
13 08 2024
09 08 2024
© 2024 The Authors. Published by American Chemical Society
2024
The Authors
https://creativecommons.org/licenses/by/4.0/ Permits the broadest form of re-use including for commercial purposes, provided that author attribution and integrity are maintained (https://creativecommons.org/licenses/by/4.0/).

Cs2AgBiBr6 is an emerging double perovskite semiconductor with robust stability. However, its potential for photovoltaics is limited by its indirect band gap and localized electronic structure featuring a resonant exciton with a large binding energy. Cs2AgBi(IxBr1–x)6 nanocrystals with iodide concentrations of up to 100% were recently demonstrated, but an atomistic understanding of how halide mixing affects the electronic and excited-state structure is missing. Here, we use first-principles GW and Bethe–Salpeter Equation calculations to show that halide mixing leads to a pronounced change in the band gap and character of optical excitations. Exciton binding energies are reduced by up to a factor of 5, with significantly more delocalized excitons in I-rich compounds. We further show that phase-pure bulk alloys with x ≤ 0.11 can be fabricated using mechanosynthesis and measure a red-shifted absorption in line with our calculations. Our study highlights that halide mixing in double perovskites can not only lead to significant band gap changes but may also be used for tuning excitonic properties.

Exacte en Natuurwetenschappen 10.13039/501100024870 OCENW.M20.337 Advanced Research Center Chemical Building Blocks Consortium NA NA Exacte en Natuurwetenschappen 10.13039/501100024870 VI.Vidi.223.072 document-id-old-9jp4c04453
document-id-new-14jp4c04453
ccc-price
==== Body
pmcIntroduction

Metal-halide perovskites are energy-converting materials for applications ranging from photovoltaics1,2 to detectors3,4 and photocatalysis.5 Halide perovskites with chemical formula ABX3, in which A is a (molecular) cation such as methylammonium (MA), formamidinium (FA) or Cs+, B is Pb2+ or Sn2+, and X is a halogen ion, typically I–, Br–, or Cl–, have become particularly relevant as absorber materials in single- and multijunction solar cells.6−11

To overcome stability and toxicity limitations of ABX3 perovskites and harness a broader range of compositional tunability, halide double perovskites with chemical formula A2BB′X6 have been widely studied because of their promising optoelectronic properties and stability against thermodynamic degradation and moisture. In particular, the material Cs2AgBiBr612−14 has a suitable band gap for solar and indoor light absorption15,16 and good charge-carrier mobilities.13 Despite these favorable properties, photoconversion efficiencies of solar cells based on Cs2AgBiBr6 have remained ≲6%,17−20 which has been linked to the indirect band gap of ∼2.3 eV in combination with limited charge-carrier diffusion lengths,15,21 large exciton binding energy,22−25 and rapid charge-carrier localization due to strong electron–phonon coupling.26,27

At room temperature, Cs2AgBiBr6 assumes a cubic crystal structure with Fm3̅m space group.15 First-principles electronic structure calculations based on density functional theory (DFT) and the GW approach show that cubic Cs2AgBiBr6 has a highly anisotropic electronic structure, in which the valence band maximum (VBM)—located at k-vector X = (, 0, ), where a is the unit cell length—is primarily derived from Ag dz2, Bi s and Br p orbital contributions, whereas the conduction band minimum (CBM)—at L = (, , )—stems from Bi p and Ag s contributions.14,28,29 This localized and anisotropic electronic structure leads to a resonant, strongly localized exciton with a large binding energy arising from the direct valence band (VB) to conduction band (CB) transition at X and manifests itself as a pronounced particle-like feature in linear optical absorption spectra of this material30,31 and other Ag-pnictogen double perovskites.31,32 The fine structure and spatial extent of this exciton are ill-described by the hydrogenic Wannier–Mott model,33 which assumes isotropic and parabolic band edges and an isotropic and constant dielectric screening on the length scale of the excitonic wave function, a behavior that has also been observed in other double and vacancy-ordered perovskites.34−37

Halide mixing is a robust and well-explored strategy for tuning the band gap and other photophysical properties of metal-halide perovskites. In ABX3 perovskites, halide mixing has been used to achieve band gaps spanning the entire visible spectrum.38 Due to the electronic structure of metal-halide perovskites, which features antibonding halide p states in the valence band maximum, and the size differences between halides with radius rI > rBr > rCl, the band gaps of halide perovskites behave as EIgap < EBrgap < EClgap.39 In ABX3 perovskites, the full range of halide mixing ratios can be accessed38,40 and band gaps change (nearly) linearly as a function of halide mixing ratio.40

In contrast, the synthesis of mixed-halide Cs2AgBiX6 has proven to be less straightforward. Kubicki et al.41 used mechanosynthesis to fabricate mixed-halide Cs2AgBiX6 with X = I, Br, Cl and showed that mixing Br and Cl retains the cubic double perovskite structure up to 100% Cl but deteriorates the photoluminescence of the material. On the other hand, mixing Br and I, while desirable because of the lower band gap predicted for I-featuring compounds,42,43 was reported to produce a range of complex side phases for I ratios exceeding 3 mol %,41 including Cs3Bi2I9–xBrx, Cs3BiBr6–xIx, and other phases that could not be identified in ref (41). However, while bulk Cs2AgBiI6 remains elusive, this material has recently been reported to form at the nanoscale in refs (44) and (45), in which anion exchange was used to fabricate Cs2AgBi(IxBr1–x)6 nanocrystals, including pure Cs2AgBiI6 nanocrystals.

Motivated by these reports, we sought to provide an atomistic understanding of how homogeneous halide mixing affects the electronic and excitonic properties of Cs2AgBi(IxBr1–x)6. We use first-principles Green’s function-based many-body perturbation theory in the GW and Bethe–Salpeter Equation (BSE) approaches to calculate band structures and linear optical absorption spectra of cubic Cs2AgBi(IxBr1–x)6 and demonstrate that increasing the I content in this family of materials not only reduces the band gap like in mixed-halide ABX3 perovskites but also drastically changes the character of excitons. The pronounced excitonic peak, characteristic of the absorption spectrum of Cs2AgBiBr6, is suppressed with increasing I content, corresponding to a significant reduction in the exciton binding energy due to an increase in the macroscopic dielectric constant and a reduction in the charge-carrier effective masses. Furthermore, an additional exciton with a binding energy of ∼50 meV, arising primarily from a Γ-point transition, contributes to the onset of absorption for compounds with x > 0.5. This exciton is more delocalized and described well by the hydrogenic Wannier–Mott model, in contrast to the exciton arising from electronic transitions at the X point. We complement our first-principles predictions with the experimental characterization of mixed-halide Cs2AgBi(IxBr1–x)6 bulk samples fabricated via mechanochemical synthesis. Optical absorption measurements of samples with x ≤ 0.11, which contain no significant amount of side phases, show a red shift of the onset of absorption, in line with our computational predictions. The absorption spectra of samples with x > 0.11 are dominated by nonperovskite side phases, highlighting the previously reported importance of nanostructuring and surface stabilization in obtaining iodide-rich Cs2AgBi(IxBr1–x)6 samples. The combination of lower exciton binding energies and reduced and more balanced effective masses could translate into improved generation and transport of charge carriers in I-rich double perovskites. Thus, we demonstrate that halide mixing may be a feasible strategy not only for band gap tuning but also for tailoring the character of excitons in double perovskites.

Methods

Computational Methods

We used density functional theory (DFT) as implemented in the plane-wave code quantum espresso(46,47) with the exchange–correlation functional by Perdew, Burke, and Ernzerhof (PBE)48 for geometry optimizations and as a starting point for our G0W0 calculations. A plane-wave energy cutoff of 60 Ry and a 10 × 10 × 10 Γ-centered k-mesh were employed. We used norm-conserving fully relativistic pseudopotentials from the PseudoDojo database,49,50 with the following valence electron configurations: Cs 5s2 5p6 6p1, Ag 4s2 4p6 4d10 5s1, Bi 5d10 6s2 6p3, Br 4s2 4p5, and I 5s2 5p5. Spin–orbit coupling (SOC) was taken into account self-consistently for all calculations unless otherwise noted.

We modeled the structure of cubic Cs2AgBiBr6 starting from the experimental room-temperature crystal structure reported in ref (13). The alloyed systems Cs2AgBi(IxBr1–x)6 were represented in the primitive unit cell with Fm3̅m space group using the virtual crystal approximation (VCA)51 and assuming homogeneous mixing of Br and I. The atomic positions and lattice parameters of all structures were fully relaxed using DFT–PBE. The convergence criteria for the geometry optimization are 10–8 Ry for the total energy and 10–6 Ry/Bohr for forces on atoms. A plane-wave energy cutoff of 60 Ry and a 10 × 10 × 10 Γ-centered k-mesh was used. All relaxations were performed without accounting for SOC.

For the calculation of quasiparticle band gaps and band structures, we used the relaxed crystal structures and the one-shot G0W0 approach, in which we constructed the zeroth-order Green’s function G0 and screened Coulomb interaction W0 from DFT eigenvalues and eigenfunctions calculated using the PBE exchange–correlation functional. For the G0W0 + BSE calculations, we used the BerkeleyGW code.52,53 The dielectric screening and quasiparticle band gap of all materials was computed using a polarizability cutoff of 8 Ry, the generalized plasmon-pole model by Godby and Needs,54 an energy cutoff of 48 Ry for the bare Coulomb interaction, and 600 bands. With these settings, we report band gaps converged to within 0.1 eV (see Supporting Information for convergence tests; Figure S7 for Cs2AgBiBr6 and Figure S8 for Cs2AgBiI6, respectively). G0W0@PBE band structures were obtained through Wannier interpolation using the wannier90 code.55 Effective masses were calculated as second derivatives of the energy bands using finite differences as described in the Supporting Information of ref (31).

Optical spectra and excitonic properties were computed by solving the Bethe–Salpeter equation (BSE)56−61 with the Tamm–Dancoff approximation (TDA).61 We obtain linear optical absorption spectra from the imaginary part of the transverse dielectric function using the momentum operator formulation. We constructed the electron–hole interaction kernel Keh on a 4 × 4 × 4 k-point grid, using a set of 22 valence and 22 conduction bands. All absorption spectra were obtained by interpolating the electron–hole kernel on a fine 14 × 14 × 14 k-point grid and applying a constant arbitrary Gaussian smearing of 50 meV. For the fine grid interpolation, we used 4 valence and 6 conduction bands. With these settings, we report exciton binding energies converged to within 5 meV (see convergence tests in Figure S11 of the Supporting Information). Here, and unless otherwise noted, the exciton binding energies are defined as the absolute difference between the computed excitation energy and the lowest-energy direct transition from which the excited state originates.

All computational settings are summarized in Table S1.

Chemicals and Mechanosynthesis

99% cesium bromide (CsBr, Tokyo Chemical Industries), 99.0% cesium iodide (CsI, Tokyo Chemical Industries), 99.998% silver bromide (AgBr, Alfa Aeser), 99.999% silver iodide (AgI, Permion), 99% bismuth bromide (BiBr3, Alfa Aeser), and 99% bismuth iodide (BrI3, Sigma-Aldrich). Prior to the synthesis, BiBr3 was dried at 70 °C under vacuum.

A stochiometric mixture of CsBr, AgBr, BiBr3, CsI, AgI, and BiI3 (ca. 2 g in total) was loaded in a 10 mL stainless steel ball mill jar filled with two stainless steel beads (10 mm in diameter). The double perovskite compositions were synthesized by milling at 30 Hz for 90 min using a Retsch MM500 vario ball mill. All compositions were annealed at 150 °C for 30 min.

X-ray Diffraction

X-ray diffraction experiments were performed using a Bruker D8 Advance equipped with a Cu Kα1,2 (λ = 1.54184 Å) radiation source operating at 40 kV and 40 mA. The patterns were recorded by measuring in Bragg–Brentano geometry at diffraction angles from 5 to 60° 2θ, with a step size of 0.02° and an integration time of 0.5 s.

Rietveld refinements were performed using the FullProf Suite software with the crystallographic information files of AgI (ICSD entry 56552), Cs3Bi2Br9 (ICSD entry 1142), and Cs2AgBiBr6 (ICSD entry 252164) as input files. For the fitting procedure, a pseudo-Voigt function was fit to the experimentally obtained reflections. Moreover, a scaling factor, the lattice parameters, shape factor, and a factor accounting for the reflection width were used as fitting variables.

Diffuse Reflectance Spectra

The absorption profile was approximated by recording the diffuse reflectance spectrum and the Kubelka–Munk relation. The diffuse reflectance was recorded by using a PerkinElmer LAMBDA 950S UV/vis/NIR spectrometer equipped with an integrating sphere. The reflectance was recorded with respect to a poly(tetrafluoroethylene) (PTFE) background from 900 to 350 nm (2 nm step size and 0.4 s integration time).

Results and Discussion

We start by investigating the structural changes in the crystal lattice of cubic Cs2AgBiBr6 (Figure 1(a)) upon incorporation of I. To this end, we perform structural optimizations with DFT using the exchange–correlation functional of Perdew, Burke, and Ernzerhof (PBE)62 as implemented in quantum espresso.63 We represent the mixed halide sites using the virtual crystal approximation,51 i.e., by replacing the X (halide) site with a “virtual atom”, which constitutes an interpolation between the pure halide sites X = Br and X = I. This approximation relies on the assumption of a homogeneous distribution of Br and I in the material that does not lead to pronounced local structural distortions and nonlinear mixing effects such as band gap bowing. It has been used for modeling the structural and optoelectronic properties of halide perovskites with DFT64,65 and in conjunction with the GW + BSE method for understanding optical properties of mixed CsPb(BrxI1–x)3 in excellent agreement with experimental results.40 We note that these calculations by definition do not account for local structural distortions, phase segregation, or the formation of side phases. Rather, we deliberately model homogeneously mixed alloys of Cs2AgBi(IxBr1–x)6 and provide a discussion of the validity of the VCA in the SI (see Figures S1–S5). The DFT–PBE lattice parameters (Table 1) exhibit a nearly linear dependence on I concentration, as shown in Figures 1(c) and S2, in agreement with Vegard’s law66 and the trends reported in ref (45) for Cs2AgBi(IxBr1–x)6 nanocrystals. On experimentally replacing Br with I, we observe a shift of the X-ray diffraction peaks to lower angles (Figure 1(b)), in line with lattice expansion due to incorporation of the larger I (see SI for experimental procedures). From Rietveld refinement of the experimentally obtained X-ray diffraction patterns, we observe successful halide mixing up to x ≤ 0.11 in the mechanochemically synthesized double perovskites. We estimated x using the refined lattice parameters and the sizing curve reported for mixed-halide Cs2AgBi(IxBr1–x)6 nanocyrstals.45 Note that this I/Br ratio is higher than previously reported for bulk compounds,41 but lower than feasible in nanocrystals. The higher degrees of iodide substitution in colloidal nanocrystals, compared with bulk compounds, have been attributed to thermodynamic stabilization. This stabilization is achieved through the increased importance of surface free energy at the nanoscale and the role of stabilizing surface ligands, as discussed in ref (45). Next to the double perovskite phase, Rietveld refinement fits reveal a minor presence of the typically observed silver-free phase Cs3Bi2(Br1–xIx)9 for x = 0.11, which shows a characteristic reflection at 12.76°and becomes the dominating phase for larger iodide concentrations as shown in Figures 1(b) and S6.

Figure 1 (a) Three-dimensional (3D) representation of the cubic crystal structure of Cs2AgBi(IxBr1–x)6, where silver balls represent Ag, orange Bi, white Cs, and red and purple I and Br, respectively. (b) Experimental X-ray diffraction patterns of Cs2AgBi(IxBr1–x)6 with x = 0 and 0.11 and the corresponding Rietveld refinement fit (black) and residual (gray). For x = 0.11, unreacted AgI is still observed in the diffraction pattern and indicated with an asterisk. (c) DFT–PBE (blue, dots) and experimentally (red, star) determined lattice constant, a, as a function of iodide concentration. The black line corresponds to the trendline for Cs2AgBi(IxBr1–x)6 nanocrystals and is reproduced from ref (45). Copyright 2023 American Chemical Society.

Table 1 Computed Lattice Parameters (in Å), Static Dielectric Constant (ε∞), QP Band Gap (in eV) Indirect from XVBM → LCBM, Lowest-Energy Direct Transition (in eV), and the High-Symmetry k-Point at Which It Is Found of Mixed Halide Double Perovskites Cs2AgBi(IxBr1–x)6

 	 	 	QP band gap (eV)	lowest direct transition	
mixing ratio	lattice constant (Å)	dielectric constant (ε∞)	XVBM → LCBM	energy (eV)	k-point	
0.00	11.27	5.41	1.50	2.16	X	
0.16	11.43	5.77	1.34	2.03	X	
0.33	11.55	6.16	1.19	1.91	Γ ≡ X	
0.50	11.70	6.49	1.08	1.81	Γ ≡ X	
0.66	11.82	6.83	0.98	1.70	Γ	
0.83	11.92	7.20	0.89	1.60	Γ	
1.00	11.99	7.61	0.80	1.50	Γ	

Next, we compute the band structure and quasiparticle (QP) band gap of the double perovskite series Cs2AgBi(IxBr1–x)6 within the G0W0 approach,56 in which QP eigenvalues are calculated by perturbatively correcting the DFT–PBE Kohn–Sham eigenvalues. We take spin–orbit coupling into account self-consistently.67Figure 2(a) shows the fundamental (indirect) and lowest-energy direct gap. Note that our calculations consistently underestimate experimental band gaps as reported before and attributed to the dependence of the G0W0 band gap on the DFT “starting point”.31,68,69 Both band gaps decrease linearly with increasing I content as expected and reported previously.43 We further find a linear relationship between these band gaps and the static dielectric constant, which increases with an increasing I content (see Figure S9 and Table 1).

Figure 2 Electronic properties of mixed halide double perovskites Cs2AgBi(IxBr1–x)6. (a) QP band gap (indirect, triangle) and lowest-energy direct transition (circles); (b) QP band structure aligned to the top of the valence band, which is chosen as the zero of the energy scale.

In Figure 2(b), we show the change in the QP band structure upon increasing the I concentration (i.e., from purple to red). I-rich compounds feature a smaller band gap and a reduction in the bandwidth, with a more pronounced effect in the valence manifold. Importantly, with an increase in x, we observe a change in the k-vector of the lowest direct band gap. In Cs2AgBiBr6, the lowest transition is at the X point of the Brillouin zone. At this point, the valence band has Bi s and Ag dz2 character, while the conduction band is derived from Bi p contributions (see Figure S10(a)). Both valence and conduction bands are highly anisotropic at the X point and we have shown previously, that this leads to a localized resonant exciton in Cs2AgBiBr6 and related materials.31 In compounds with x > 0.5, including Cs2AgBiI6, the lowest direct transition is shifted to the Γ point, where the valence band is almost entirely derived from I p orbital contributions, while the conduction band still has contributions predominantly from Bi p, but it is much more disperse (see Figure S10(b)).

In order to quantify this change in band edge anisotropy, we computed effective masses at the k-vector of the lowest-energy direct transition. In Table 2, we report the G0W0 effective masses and the corresponding anisotropy factors , where m and m are transverse and longitudinal effective masses, respectively. The electron effective masses of the I-rich compounds, i.e., with mixing ratios x > 0.5, are at least 3.5 times smaller than that of the pure Cs2AgBiBr6 perovskite. This is a result of the shift in the position of the lowest-energy direct transition to Γ, where the conduction band is more disperse. Another consequence of the shift is the presence of light and heavy holes in the electronic structures of the mixed halide perovskites with x > 0.5. The hole effective mass is the average between the two and, therefore, the I-rich compounds also feature smaller hole effective masses than their Br counterparts. Consequently, the reduced effective mass of the I-rich double perovskites is almost half of that of the pure Br material.

Table 2 G0W0 Hole (mh*), Electron (me*), and Reduced (μ) Effective Masses (in Units of Electron Rest Mass m0) and the Corresponding Anisotropy Factors (λh, λe, and λμ, Respectively) for the Studied Br/I Mixing Ratios

mixing ratio	effective masses	anisotropy factor	
x	mh*	me*	μ	λh	λe	λμ	
0.00	0.202	0.579	0.150	0.54	0.57	0.55	
0.16	0.192	0.489	0.138	0.53	0.54	0.56	
0.33	0.179	0.418	0.126	0.53	0.55	0.54	
0.50	0.181	0.390	0.124	0.53	0.56	0.54	
0.66	0.135	0.162	0.074	0.64	0.65	0.64	
0.83	0.141	0.159	0.075	0.64	0.65	0.65	
1.00	0.144	0.155	0.074	0.64	0.64	0.64	

In Figure 3(a), we show that the observed changes in the electronic band structure lead to pronounced variations in the linear optical absorption spectra of Cs2AgBi(IxBr1–x)6 as x is increased. We calculate these absorption spectra using the BSE approach,70 which includes electron–hole interactions and allows for insights into excitonic effects for a wide range of materials.57,−77 Note that we here consider only direct excitations with zero momentum transfer. In line with the decrease of the QP band gap, we observe a red shift in the absorption spectra with increasing I content. Furthermore, the excitonic peak originating from the direct transition at X24,30,31 is increasingly less pronounced, and the binding energy of the exciton constituting this peak decreases with increasing I concentration (Figure S12). In compounds with x > 0.5, in which the lowest direct transition arises from Γ, we find five dark and three degenerate bright excitations with relatively low oscillator strength below this excitonic peak, which arise from the lowest-energy direct transition at Γ (Table S3 and Figure S13). Our calculated absorption spectra qualitatively agree with those reported for Cs2AgBi(IxBr1–x)6 nanocrystals. These nanocrystals exhibit a broadening of the absorption onset and a suppression of the excitonic peak with increasing iodide substitution.44

Figure 3 Optical properties of mixed halide double perovskites Cs2AgBi(IxBr1–x)6. The color scale denotes the I concentration, i.e., x ·100. (a) Linear optical absorption spectra of Cs2AgBi(IxBr1–x)6 as computed within G0W0@PBE + BSE. (b) Kubelka–Munk transform of the diffuse reflectance spectra of Cs2AgBi(IxBr1–x)6 samples with x = 0 and 0.11. (c, d) Computed exciton binding energies as predicted by the Wannier–Mott model (squares) and BSE binding energy of the exciton arising from direct transitions at (c) X and (d) Γ, respectively.

In line with our predictions, we also observe a red shift in the experimentally determined absorption profiles of Cs2AgBi(IxBr1–x)6 bulk powders with x ≤ 0.11 (Figure 3(b)), indicating a reduction of the band gap energy. Our experimental absorption profiles were obtained from powder samples using diffuse reflectance spectroscopy and approximated with the Kubelka–Munk transform. As a result, they do not show the pronounced excitonic feature seen in previous experiments on thin films.24 Attempts to synthesize thin-film samples with x = 0.11 resulted in low-quality films that were unsuitable for absorption experiments. Therefore, our experimental results cannot confirm excitonic effects but do verify the red shift of the absorption onset predicted by our calculations and observed in nanocrystal experiments.44 The shallow slope at the absorption onset suggests the indirect band gap is maintained on exchanging less than ∼11% of Br with I. For higher I concentrations (x > 0.11), a steeper slope in the absorption profiles is observed, suggesting a change to a more direct band gap transition (Figure S14b). However, control experiments on the silver-free phase, i.e., Cs3Bi2(Br1–xIx)9, confirm that the absorption profile for x = 0.33 is dominated by the silver-free phase (Figure S14c), whose absorption onset is dominated by a pronounced excitonic feature too, as confirmed by previous first-principles calculations.78

In our previous work, we showed that the non-hydrogenic character of excitons in Cs2AgBiBr6 and other double perovskites is intimately linked to the anisotropic electronic structure and dielectric screening in these materials,31,36 as a consequence of the relatively large and anisotropic effective masses at X.29 Here, we show that increasing the I concentration has two distinct effects on excitons in these compounds. For this purpose, we define X- and Γ-excitons, as the lowest-energy excitons arising from transitions at X and Γ, respectively. Their binding energies, EB, are shown in Figure 3(c,d), respectively. First, we find that the binding energy of the X- and Γ-excitons, calculated with respect to the direct QP transition at X and Γ, respectively, decreases with increasing I concentration. The decrease is linear in 1/ε∞2 as expected, albeit with different slopes. In particular, the binding energy of the X-exciton decreases drastically, by a factor of 5, from x = 0 to 1. Second, the character of both excitons also changes, which we demonstrate by comparing our first-principles results with exciton binding energies computed using the Wannier–Mott model,33 i.e., by assuming a hydrogenic series of exciton states with binding energies , where RH is the Rydberg constant, n is the principal quantum number, and values for the reduced effective mass μ and dielectric constant ϵ∞ are obtained from our G0W0 calculations (Tables 1 and 2). The exciton binding energies of the 1s excitons calculated using the Wannier–Mott model are also shown in Figure 3(c,d). We find that the X-exciton of Cs2AgBiBr6 has a non-hydrogenic fine structure and a binding energy ∼200 meV larger than predicted by the Wannier–Mott model. As the I concentration increases, the deviation from the Wannier–Mott model becomes significantly smaller. Additionally, we observe that the Γ-exciton is in much better agreement with the Wannier–Mott model across all I concentrations with an absolute deviation of only ∼30 meV for Cs2AgBiI6. This is in line with the differences in the orbital character at the VBM and CBM at X and Γ, which leads to significantly more disperse and isotropic band edges at Γ (Table S2).

This pronounced change in exciton character is also reflected in the spatial extent of the X- and the Γ-exciton. We investigate this effect by evaluating the probability distribution of the excitonic wave function ΨS(re,rh) = ∑vckAvckSψck(re) ψvk*(rh), where ψv(c) k(rh(e)) are single-particle DFT–PBE wave functions of electrons and holes, and AvckS are coefficients corresponding to the excitonic state S, obtained by solving the BSE. We systematically sample three different hole positions—on Ag, on Bi, and the halide—sum the corresponding probability distributions and plot the resulting excitonic wave function of the pure compounds in Figure 4(a,b) for Cs2AgBiBr6 and Cs2AgBiI6, respectively. This comparison shows that the X-exciton of Cs2AgBiBr6 is significantly more localized than the Γ-exciton of Cs2AgBiI6, which is in line with its less hydrogenic character.

Figure 4 3D representation of the probability density of the exciton wave function in real space, depicted as purple isosurfaces for (a) Cs2AgBiBr6 and red isosurfaces for (b) Cs2AgBiI6, showing 95% of the maximum isovalue. (c, d) Exciton localization as defined in the SI and calculated using G0W0@PBE + BSE (triangles) and using the Wannier–Mott model calculated for the excitons at (c) X and (d) Γ. The gray dashed lines are linear fits denoting the linear relationship between exciton radius and dielectric constant expected from the Wannier–Mott model.

Furthermore, we quantify the spatial extent of the exciton by calculating the average electron–hole separation, σ, using the approach introduced in ref (74) and as used before in ref (31) (see SI for details). In Figure 4(c,d), we show σ and compare it with the Wannier–Mott radius for the X- and Γ-exciton, respectively (see also Table S3). We find that the average electron–hole separation of the X-exciton is significantly smaller than the corresponding Wannier–Mott radii, which is in line with our earlier observations. Furthermore, it increases by a factor of 2 upon increasing the I content from x = 0 to 1. The Γ-exciton shows a much smaller variation as a function of I concentration and reaches a maximum average electron–hole separation of ∼20 Å in Cs2AgBiI6.

Conclusions

In conclusion, we predict a significant change in the optoelectronic properties of Cs2AgBiBr6 upon introduction of I, in particular a reduction of the band gap and the exciton binding energy and a pronounced change of the character of the exciton which becomes more delocalized and “hydrogenic”. We correlate the change in exciton character to a change in the k-vector at which the lowest direct transition, responsible for the onset of absorption, takes place. For x > 0.5, this transition is at the Γ point, at which valence and conduction band edges are more isotropic and disperse than at the X point, from which the lowest direct transition arises in Br-rich compounds. Experimentally, our prediction of a band gap red shift is confirmed in bulk Cs2AgBi(IxBr1–x)6 with x up to 0.11. Our findings show that halide mixing can be a powerful method for tuning not only band gaps of halide perovskites but also their excitonic properties in chemically heterogeneous double perovskites. The reduced band gap, lower exciton binding energies, less confined excitons, and lower effective masses are favorable for photovoltaic performance, and motivate further efforts to experimentally obtain phase-pure Cs2AgBiI6 in bulk form.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcc.4c04453.Additional data on structural and optical characterization, computational methods, information about convergence tests and data supporting the validity of the virtual crystal approximation (PDF)

Supplementary Material

jp4c04453_si_001.pdf

The authors declare no competing financial interest.

Acknowledgments

L.L. acknowledges funding by the Dutch Research Council (NWO) through the grants OCENW.M20.337 and VI.Vidi.223.072. H.J.J. and E.M.H. are supported by the Advanced Research Center Chemical Building Blocks Consortium (ARC CBBC). This work was also supported by the NWO Domain Science for the use of supercomputing facilities.
==== Refs
References

Stranks S. D. ; Snaith H. J. Metal-Halide Perovskites for Photovoltaic and Light-Emitting Devices. Nat. Nanotechnol. 2015, 10 , 391–402. 10.1038/nnano.2015.90.25947963
Nayak P. K. ; Mahesh S. ; Snaith H. J. ; Cahen D. Photovoltaic Solar Cell Technologies: Analysing the State of the Art. Nat. Rev. Mater. 2019, 4 , 269–285. 10.1038/s41578-019-0097-0.
Wei H. ; Huang J. Halide Lead Perovskites for Ionizing Radiation Detection. Nat. Commun. 2019, 10 , 1066 10.1038/s41467-019-08981-w.30842411
Liu F. ; Wu R. ; Wei J. ; Nie W. ; Mohite A. D. ; Brovelli S. ; Manna L. ; Li H. Recent Progress in Halide Perovskite Radiation Detectors for Gamma-Ray Spectroscopy. ACS Energy Lett. 2022, 7 , 1066–1085. 10.1021/acsenergylett.2c00031.
Muscarella L. A. ; Hutter E. M. Halide Double-Perovskite Semiconductors beyond Photovoltaics. ACS Energy Lett. 2022, 7 , 2128–2135. 10.1021/acsenergylett.2c00811.35719270
Kojima A. ; Teshima K. ; Shirai Y. ; Miyasaka T. Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells. J. Am. Chem. Soc. 2009, 131 , 6050–6051. 10.1021/ja809598r.19366264
Eperon G. E. ; Stranks S. D. ; Menelaou C. ; Johnston M. B. ; Herz L. M. ; Snaith H. J. Environmental Science Formamidinium Lead Trihalide: A Broadly Tunable Perovskite for Efficient Planar Heterojunction solar celss. Energy Environ. Sci. 2014, 7 , 982–988. 10.1039/c3ee43822h.
McMeekin D. P. ; Sadoughi G. ; Rehman W. ; Eperon G. E. ; Saliba M. ; Hörantner M. T. ; Haghighirad A. ; Sakai N. ; Korte L. ; Rech B. ; et al. A Mixed-Cation Lead Mixed-Halide Perovskite Absorber for Tandem Solar Cells. Science 2016, 351 , 151–155. 10.1126/science.aad5845.26744401
Bush K. A. ; Palmstrom A. F. ; Yu Z. J. ; Boccard M. ; Cheacharoen R. ; Mailoa J. P. ; McMeekin D. P. ; Hoye R. L. Z. ; Bailie C. D. ; Leijtens T. ; et al. 23.6%-Efficient Monolithic Perovskite/Silicon Tandem Solar Cells with Improved Stability. Nat. Energy 2017, 2 , 1–7. 10.1038/nenergy.2017.9.
Degani M. ; An Q. ; Albaladejo-Siguan M. ; Hofstetter Y. J. ; Cho C. ; Paulus F. ; Grancini G. ; Vaynzof Y. 23.7% Efficient Inverted Perovskite Solar Cells by Dual Interfacial Modification. Sci. Adv. 2021, 7 , eabj7930 10.1126/sciadv.abj7930.34851671
Soto-Montero T. ; Soto-Montero T. ; Kralj S. ; Soltanpoor W. ; Solomon J. S. ; Gómez J. S. ; Zanoni K. P. S. ; Paliwal A. ; Bolink H. J. ; Baeumer C. ; Kentgens A. P. M. Single-Source Vapor-Deposition of MA1–xFAxPbI3 Perovskite Absorbers for Solar Cells. Adv. Funct. Mater. 2023, 2300588 10.1002/adfm.202300588.
McClure E. T. ; Ball M. R. ; Windl W. ; Woodward P. M. Cs2AgBiX6 (X = Br, Cl) – New Visible Light Absorbing, Lead-Free Halide Perovskite Semiconductors. Chem. Mater. 2016, 28 , 1348–1354. 10.1021/acs.chemmater.5b04231.
Slavney A. H. ; Hu T. ; Lindenberg A. M. ; Karunadasa H. I. A Bismuth-Halide Double Perovskite with Long Carrier Recombination Lifetime for Photovoltaic Applications. J. Am. Chem. Soc. 2016, 138 , 2138–2141. 10.1021/jacs.5b13294.26853379
Volonakis G. ; Filip M. R. ; Haghighirad A. A. ; Sakai N. ; Wenger B. ; Snaith H. J. ; Giustino F. Lead-Free Halide Double Perovskites via Heterovalent Substitution of Noble Metals. J. Phys. Chem. Lett. 2016, 7 , 1254–1259. 10.1021/acs.jpclett.6b00376.26982118
Schade L. ; Wright A. D. ; Johnson R. D. ; Dollmann M. ; Wenger B. ; Nayak P. K. ; Prabhakaran D. ; Herz L. M. ; Nicholas R. ; Snaith H. J. ; Radaelli P. G. Structural and Optical Properties of Cs2AgBiBr6 Double Perovskite. ACS Energy Lett. 2019, 4 , 299–305. 10.1021/acsenergylett.8b02090.
Peng Y. ; Huq T. N. ; Mei J. ; Portilla L. ; Jagt R. A. ; Occhipinti L. G. ; MacManus-Driscoll J. L. ; Hoye R. L. Z. ; Pecunia V. Lead-Free Perovskite-Inspired Absorbers for Indoor Photovoltaics. Adv. Energy Mater. 2021, 11 , 2002761 10.1002/aenm.202002761.
Greul E. ; Petrus M. L. ; Binek A. ; Docampo P. ; Bein T. Highly Stable, Phase Pure Cs2AgBiBr6 Double Perovskite Thin Films for Optoelectronic Applications. J. Mater. Chem. A 2017, 5 , 19972–19981. 10.1039/C7TA06816F.
Jin Z. ; Zhang Z. ; Xiu J. ; Song H. ; Gatti T. ; He Z. Critical Review on Bismuth and Antimony Halide Based Perovskites and Their Derivatives for Photovoltaic Applications: Recent Advances and Challenges. J. Mater. Chem. A 2020, 8 , 16166–16188. 10.1039/D0TA05433J.
Sirtl M. T. ; Hooijer R. ; Armer M. ; Ebadi F. G. ; Mohammadi M. ; Maheu C. ; Weis A. ; van Gorkom B. T. ; Häringer S. ; Janssen R. A. J. ; et al. 2D/3D Hybrid Cs2AgBiBr6 Double Perovskite Solar Cells: Improved Energy Level Alignment for Higher Contact-Selectivity and Large Open Circuit Voltage. Adv. Energy Mater. 2022, 12 , 2103215 10.1002/aenm.202103215.
Zhang Z. ; Sun Q. ; Lu Y. ; Lu F. ; Mu X. ; Wei S.-H. ; Sui M. Hydrogenated Cs2AgBiBr6 for Significantly Improved Efficiency of Lead-Free Inorganic Double Perovskite Solar Cell. Nat. Commun. 2022, 13 , 3397 10.1038/s41467-022-31016-w.35697701
Jöbsis H. J. ; Caselli V. M. ; Askes S. H. C. ; Garnett E. C. ; Savenije T. J. ; Rabouw F. T. ; Hutter E. M. Recombination and Localization: Unfolding the Pathways behind Conductivity Losses in Cs2AgBiBr6 Thin Films. Appl. Phys. Lett. 2021, 119 , 131908 10.1063/5.0061899.
Steele J. A. ; Pan W. ; Martin C. ; Keshavarz M. ; Debroye E. ; Yuan H. ; Banerjee S. ; Fron E. ; Jonckheere D. ; Kim C. W. ; et al. Photophysical Pathways in Highly Sensitive Cs2AgBiBr6 Double-Perovskite Single-Crystal X-Ray Detectors. Adv. Mater. 2018, 30 , 1804450 10.1002/adma.201804450.
Zelewski S. J. ; Urban J. M. ; Surrente A. ; Maude D. K. ; Kuc A. ; Schade L. ; Johnson R. D. ; Dollmann M. ; Nayak P. K. ; Snaith H. J. ; et al. Revealing the Nature of Photoluminescence Emission in the Metal-Halide Double Perovskite Cs2AgBiBr6. J. Mater. Chem. C 2019, 7 , 8350–8356. 10.1039/C9TC02402F.
Longo G. ; Mahesh S. ; Buizza L. R. V. ; Wright A. D. ; Ramadan A. J. ; Abdi-Jalebi M. ; Nayak P. K. ; Herz L. M. ; Snaith H. J. Understanding the Performance-Limiting Factors of Cs2AgBiBr6 Double-Perovskite Solar Cells. ACS Energy Lett. 2020, 5 , 2200–2207. 10.1021/acsenergylett.0c01020.
Kentsch R. ; Scholz M. ; Horn J. ; Schlettwein D. ; Oum K. ; Lenzer T. Exciton Dynamics and Electron-Phonon Coupling Affect the Photovoltaic Performance of the Cs2AgBiBr6 Double Perovskite. J. Phys. Chem. C 2018, 122 , 25940–25947. 10.1021/acs.jpcc.8b09911.
Wright A. D. ; Buizza L. R. ; Savill K. J. ; Longo G. ; Snaith H. J. ; Johnston M. B. ; Herz L. M. Ultrafast Excited-State Localization in Cs2AgBiBr6 Double Perovskite. J. Phys. Chem. Lett. 2021, 12 , 3352–3360. 10.1021/acs.jpclett.1c00653.33783218
Baskurt M. ; Wiktor J. Charge Localization in Cs2AgBiBr6 Double Perovskite: Small Polarons and Self-Trapped Excitons. J. Phys. Chem. C 2023, 127 , 23966–23972. 10.1021/acs.jpcc.3c06551.
Filip M. R. ; Hillman S. ; Haghighirad A.-A. ; Snaith H. J. ; Giustino F. Band Gaps of the Lead-Free Halide Double Perovskites Cs2BiAgCl6 and Cs2BiAgBr6 from Theory and Experiment. J. Phys. Chem. Lett. 2016, 7 , 2579–2585. 10.1021/acs.jpclett.6b01041.27322413
Slavney A. H. ; Connor B. ; Leppert L. ; Karunadasa H. A Pencil-and-Paper Method for Elucidating Halide Double Perovskite Band Structures. Chem. Sci. 2019, 10 , 11041–11053. 10.1039/C9SC03219C.32190254
Palummo M. ; Berrios E. ; Varsano D. ; Giorgi G. Optical Properties of Lead-Free Double Perovskites by Ab Initio Excited-State Methods. ACS Energy Lett. 2020, 5 , 457–463. 10.1021/acsenergylett.9b02593.
Biega R. I. ; Filip M. R. ; Leppert L. ; Neaton J. B. Chemically Localized Resonant Excitons in Silver-Pnictogen Halide Double Perovskites. J. Phys. Chem. Lett. 2021, 12 , 2057–2063. 10.1021/acs.jpclett.0c03579.33606534
Righetto M. ; Caicedo-Dávila S. ; Sirtl M. T. ; Lim V. J.-Y. ; Patel J. B. ; Egger D. A. ; Bein T. ; Herz L. M. Alloying Effects on Charge-Carrier Transport in Silver-Bismuth Double Perovskites. J. Phys. Chem. Lett. 2023, 14 , 10340–10347. 10.1021/acs.jpclett.3c02750.37948051
Wannier G. H. The Structure of Electronic Excitation Levels in Insulating Crystals. Phys. Rev. 1937, 52 , 191–197. 10.1103/PhysRev.52.191.
Kavanagh S. R. ; Savory C. N. ; Liga S. M. ; Konstantatos G. ; Walsh A. ; Scanlon D. O. Frenkel Excitons in Vacancy-Ordered Titanium Halide Perovskites (Cs2TiX6). J. Phys. Chem. Lett. 2022, 13 , 10965–10975. 10.1021/acs.jpclett.2c02436.36414263
Cucco B. ; Katan C. ; Even J. ; Kepenekian M. ; Volonakis G. Fine Structure of Excitons in Vacancy-Ordered Halide Double Perovskites. ACS Mater. Lett. 2023, 5 , 52–59. 10.1021/acsmaterialslett.2c01010.
Biega R.-I. ; Chen Y. ; Filip M. R. ; Leppert L. Chemical Mapping of Excitons in Halide Double Perovskites. Nano Lett. 2023, 23 , 8155–8161. 10.1021/acs.nanolett.3c02285.37656044
Leppert L. Excitons in Metal-Halide Perovskites from First-Principles Many-Body Perturbation Theory. J. Chem. Phys. 2024, 160 , 050902 10.1063/5.0187213.38341699
Protesescu L. ; Yakunin S. ; Bodnarchuk M. I. ; Krieg F. ; Caputo R. ; Hendon C. H. ; Yang R. X. ; Walsh A. ; Kovalenko M. V. Nanocrystals of Cesium Lead Halide Perovskites (CsPbX3, X = Cl, Br, and I): Novel Optoelectronic Materials Showing Bright Emission with Wide Color Gamut. Nano Lett. 2015, 15 , 3692–3696. 10.1021/nl5048779.25633588
Tao S. ; Schmidt I. ; Brocks G. ; Jiang J. ; Tranca I. ; Meerholz K. ; Olthof S. Absolute Energy Level Positions in Tin- and Lead-Based Halide Perovskites. Nat. Commun. 2019, 10 , 2560 10.1038/s41467-019-10468-7.31189871
Chen Y. ; Motti S. G. ; Oliver R. D. J. ; Wright A. D. ; Snaith H. J. ; Johnston M. B. ; Herz L. M. ; Filip M. R. Optoelectronic Properties of Mixed Iodide–Bromide Perovskites from First-Principles Computational Modeling and Experiment. J. Phys. Chem. Lett. 2022, 13 , 4184–4192. 10.1021/acs.jpclett.2c00938.35511476
Kubicki D. J. ; Saski M. ; MacPherson S. ; Galkowski K. ; Lewiński J. ; Prochowicz D. ; Titman J. J. ; Stranks S. D. Halide Mixing and Phase Segregation in Cs2AgBiX6 (X = Cl, Br, and I) Double Perovskites from Cesium-133 Solid-State NMR and Optical Spectroscopy. Chem. Mater. 2020, 32 , 8129–8138. 10.1021/acs.chemmater.0c01255.33071455
Silveira J. F. R. V. ; Da Silva J. L. F. Mixed Halide Lead-free Double Perovskite Alloys for Band Gap Engineering. ACS Appl. Energy Mater. 2020, 3 , 7364–7371. 10.1021/acsaem.0c00739.
Wu H. ; Erbing A. ; Johansson M. B. ; Wang J. ; Kamal C. ; Odelius M. ; Johansson E. M. J. Mixed-Halide Double Perovskite Cs2AgBiX6 (X= Br, I) with Tunable Optical Properties via Anion Exchange. ChemSusChem 2021, 14 , 4507–4515. 10.1002/cssc.202101146.34369665
Creutz S. E. ; Crites E. N. ; De Siena M. C. ; Gamelin D. R. Colloidal Nanocrystals of Lead-Free Double-Perovskite (Elpasolite) Semiconductors: Synthesis and Anion Exchange To Access New Materials. Nano Lett. 2018, 18 , 1118–1123. 10.1021/acs.nanolett.7b04659.29376378
Kluherz K. T. ; Mergelsberg S. T. ; De Yoreo J. J. ; Gamelin D. R. Structure and Stability of the Iodide Elpasolite, Cs2AgBiI6. Chem. Mater. 2023, 35 , 5699–5708. 10.1021/acs.chemmater.3c01511.
Giannozzi P. ; Baroni S. ; Bonini N. ; Calandra M. ; Car R. ; Cavazzoni C. ; Ceresoli D. ; Chiarotti G. L. ; Cococcioni M. ; Dabo I. ; et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys.: Condens. Matter 2009, 21 , 395502 10.1088/0953-8984/21/39/395502.21832390
Giannozzi P. ; Andreussi O. ; Brumme T. ; Bunau O. ; Nardelli M. B. ; Calandra M. ; Car R. ; Cavazzoni C. ; Ceresoli D. ; Cococcioni M. ; et al. Advanced capabilities for materials modelling with Quantum ESPRESSO. J. Phys.: Condens. Matter 2017, 29 , 465901 10.1088/1361-648X/aa8f79.29064822
Perdew J. P. ; Burke K. ; Ernzerhof M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77 , 3865–3868. 10.1103/PhysRevLett.77.3865.10062328
Hamann D. R. Optimized norm-conserving Vanderbilt pseudopotentials. Phys. Rev. B 2013, 88 , 085117 10.1103/PhysRevB.88.085117.
van Setten M. ; Giantomassi M. ; Bousquet E. ; Verstraete M. ; Hamann D. ; Gonze X. ; Rignanese G.-M. The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table. Comput. Phys. Commun. 2018, 226 , 39–54. 10.1016/j.cpc.2018.01.012.
Bellaiche L. ; Vanderbilt D. Virtual Crystal Approximation Revisited: Application to Dielectric and Piezoelectric Properties of Perovskites. Phys. Rev. B 2000, 61 , 7877–7882. 10.1103/PhysRevB.61.7877.
Deslippe J. ; Samsonidze G. ; Strubbe D. A. ; Jain M. ; Cohen M. L. ; Louie S. G. BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Comput. Phys. Commun. 2012, 183 , 1269–1289. 10.1016/j.cpc.2011.12.006.
Barker B. A. ; Deslippe J. ; Lischner J. ; Jain M. ; Yazyev O. V. ; Strubbe D. A. ; Louie S. G. ; Spinor G. W. Bethe-Salpeter calculations in BerkeleyGW: Implementation, symmetries, benchmarking, and performance. Phys. Rev. B 2022, 106 , 115127 10.1103/PhysRevB.106.115127.
Godby R. W. ; Needs R. J. Metal-Insulator Transition in Kohn-Sham Theory and Quasiparticle Theory. Phys. Rev. Lett. 1989, 62 , 1169–1172. 10.1103/PhysRevLett.62.1169.10039594
Pizzi G. ; Vitale V. ; Arita R. ; Blügel S. ; Freimuth F. ; Géranton G. ; Gibertini M. ; Gresch D. ; Johnson C. ; Koretsune T. ; et al. Wannier90 as a Community Code: New Features and Applications. J. Phys.: Condens. Matter 2020, 32 , 165902 10.1088/1361-648X/ab51ff.31658458
Hedin L. New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem. Phys. Rev. 1965, 139 , A796–A823. 10.1103/PhysRev.139.A796.
Rohlfing M. ; Louie S. G. Electron-Hole Excitations in Semiconductors and Insulators. Phys. Rev. Lett. 1998, 81 , 2312–2315. 10.1103/PhysRevLett.81.2312.
Albrecht S. ; Reining L. ; Del Sole R. ; Onida G. Ab initio calculation of excitonic effects in the optical spectra of semiconductors. Phys. Rev. Lett. 1998, 80 , 4510 10.1103/PhysRevLett.80.4510.
Rohlfing M. ; Louie S. G. Electron-hole excitations and optical spectra from first principles. Phys. Rev. B 2000, 62 , 4927–4944. 10.1103/PhysRevB.62.4927.
Onida G. ; Reining L. ; Rubio A. Electronic excitations: density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 2002, 74 , 601 10.1103/RevModPhys.74.601.
Kronik L. ; Neaton J. B. Excited-State Properties of Molecular Solids from First Principles. Annu. Rev. Phys. Chem. 2016, 67 , 587–616. 10.1146/annurev-physchem-040214-121351.27090844
Perdew J. P. ; Burke K. ; Ernzerhof M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77 , 3865–3868. 10.1103/PhysRevLett.77.3865.10062328
Giannozzi P. ; Andreussi O. ; Brumme T. ; Bunau O. ; Nardelli M. B. ; Calandra M. ; Car R. ; Cavazzoni C. ; Ceresoli D. ; Cococcioni M. ; et al. Advanced Capabilities for Materials Modelling with Quantum ESPRESSO. J. Phys.: Condens. Matter 2017, 29 , 465901 10.1088/1361-648X/aa8f79.29064822
Jong U.-G. ; Yu C.-J. ; Ri J.-S. ; Kim N.-H. ; Ri G.-C. Influence of Halide Composition on the Structural, Electronic, and Optical Properties of Mixed CH3NH3PbI1–xBrx Perovskites Calculated Using the Virtual Crystal Approximation Method. Phys. Rev. B 2016, 94 , 125139 10.1103/PhysRevB.94.125139.
Ogunniranye I. B. ; Atsue T. ; Oyewande O. E. Structural and Optoelectronic Behavior of the Copper-Doped Cs2AgInCl6 Double Perovskite: A Density Functional Theory Investigation. Phys. Rev. B 2021, 103 , 024102 10.1103/PhysRevB.103.024102.
Vegard L. D. Konstitution Der Mischkristalle Und Die Raumfüllung Der Atome. Z. Phys. 1921, 5 , 17 10.1007/BF01349680.
Barker B. A. ; Deslippe J. ; Lischner J. ; Jain M. ; Yazyev O. V. ; Strubbe D. A. ; Louie S. G. Spinor GW/Bethe-Salpeter Calculations in BerkeleyGW: Implementation, Symmetries, Benchmarking, and Performance. Phys. Rev. B 2022, 106 , 115127 10.1103/PhysRevB.106.115127.
Filip M. R. ; Giustino F. GW Quasiparticle Band Gap of the Hybrid Organic-Inorganic Perovskite CH3NH3PbI3: Effect of Spin-Orbit Interaction, Semicore Electrons, and Self-Consistency. Phys. Rev. B 2014, 90 , 245145 10.1103/PhysRevB.90.245145.
Leppert L. ; Rangel T. ; Neaton J. B. Towards Predictive Band Gaps for Halide Perovskites: Lessons from One-Shot and Eigenvalue Self-Consistent GW. Phys. Rev. Mater. 2019, 3 , 103803 10.1103/PhysRevMaterials.3.103803.
Strinati G. Application of the Green’s Functions Method to the Study of the Optical Properties of Semiconductors. Riv. Nuovo Cimento 1988, 11 , 1–86. 10.1007/BF02725962.
Marini A. ; Hogan C. ; Grüning M. ; Varsano D. Yambo: An Ab Initio Tool for Excited State Calculations. Comput. Phys. Commun. 2009, 180 , 1392–1403. 10.1016/j.cpc.2009.02.003.
Palummo M. ; Hogan C. ; Sottile F. ; Bagalá P. ; Rubio A. Ab Initio Electronic and Optical Spectra of Free-Base Porphyrins: The Role of Electronic Correlation. J. Chem. Phys. 2009, 131 , 084102 10.1063/1.3204938.19725603
Qiu D. Y. ; Da Jornada F. H. ; Louie S. G. Optical Spectrum of MoS2: Many-body Effects and Diversity of Exciton States. Phys. Rev. Lett. 2013, 111 , 216805 10.1103/PhysRevLett.111.216805.24313514
Sharifzadeh S. ; Darancet P. ; Kronik L. ; Neaton J. B. Low-Energy Charge-Transfer Excitons in Organic Solids from First-Principles: The Case of Pentacene. J. Phys. Chem. Lett. 2013, 4 , 2197 10.1021/jz401069f.
Blase X. ; Duchemin I. ; Jacquemin D. The Bethe-Salpeter Equation in Chemistry: Relations with TD-DFT, Applications and Challenges. Chem. Soc. Rev. 2018, 47 , 1022–1043. 10.1039/C7CS00049A.29250615
Filip M. R. ; Haber J. B. ; Neaton J. B. Phonon Screening of Excitons in Semiconductors: Halide Perovskites and Beyond. Phys. Rev. Lett. 2021, 127 , 067401 10.1103/PhysRevLett.127.067401.34420331
Naik M. H. ; Regan E. C. ; Zhang Z. ; Chan Y.-H. ; Li Z. ; Wang D. ; Yoon Y. ; Ong C. S. ; Zhao W. ; Zhao S. ; et al. Intralayer Charge-Transfer Moiré Excitons in van Der Waals Superlattices. Nature 2022, 609 , 52–57. 10.1038/s41586-022-04991-9.36045239
Rieger S. ; Bohn B. J. ; Döblinger M. ; Richter A. F. ; Tong Y. ; Wang K. ; Müller-Buschbaum P. ; Polavarapu L. ; Leppert L. ; Stolarczyk J. K. ; Feldmann J. Excitons and Narrow Bands Determine the Optical Properties of Cesium Bismuth Halides. Phys. Rev. B 2019, 100 , 201404(R) 10.1103/PhysRevB.100.201404.
