Frenkel Excitons in Vacancy-Ordered Titanium Halide Perovskites (Cs2TiX6)

Low-cost, nontoxic, and earth-abundant photovoltaic materials are long-sought targets in the solar cell research community. Perovskite-inspired materials have emerged as promising candidates for this goal, with researchers employing materials design strategies including structural, dimensional, and compositional transformations to avoid the use of rare and toxic elemental constituents, while attempting to maintain high optoelectronic performance. These strategies have recently been invoked to propose Ti-based vacancy-ordered halide perovskites (A2TiX6; A = CH3NH3, Cs, Rb, or K; X = I, Br, or Cl) for photovoltaic operation, following the initial promise of Cs2SnX6 compounds. Theoretical investigations of these materials, however, consistently overestimate their band gaps, a fundamental property for photovoltaic applications. Here, we reveal strong excitonic effects as the origin of this discrepancy between theory and experiment, a consequence of both low structural dimensionality and band localization. These findings have vital implications for the optoelectronic application of these compounds while also highlighting the importance of frontier-orbital character for chemical substitution in materials design strategies.

To purify the NC solutions, 5 ml of acetone was added to 5 ml of solution already diluted in toluene and centrifuged at 5800 rpm for 10 minutes. The supernatant was discarded and the precipitate redispersed in 3 ml of toluene and centrifuged again at 5800 rpm for 10 minutes. The supernatant was discarded again and the precipitate redispersed in toluene and centrifuged again at 2000 rpm for 4 minutes to remove possible aggregates. Finally, the supernatant was filtered using a 0.22 mm PTFE filter. The solutions were stored under inert S2 atmosphere. S1.4 Synthesis and purification of Cs 2 TiX 6 nanocrystals Cs 2 TiBr 6 and Cs 2 TiI 6 NCs were prepared following our previously published method according to ref S2 of the manuscript. Cs 2 TiCl 6 was prepared with the same method, but a higher amount of halide precursor was needed, namely 2.4 ml of TMSCl, while DMOP ligand was not used in the synthesis. The purification of the NC solutions was carried out as for Cs 2 SnX 6 , except that for Cs 2 TiI 6 and Cs 2 TiCl 6 NC solutions no antisolvent was added before the first centrifugation.

S1.5 UV-vis absorption
Optical absorption spectra were collected using a Varian Cary5000 UV-vis-NIR spectrophotometer. When measuring the absorption below 300 nm, the nanocrystals were dispersed in hexane instead of toluene. For the case of Cs 2 SnCl 6 , since the nanocrystals were not stable in hexane, they were deposited on sapphire and the absorption was measured using an internal DRA 2500 integrating sphere.

S1.6 X-Ray Diffraction
The measured x-ray diffraction patterns for the synthesized Cs 2 BX 6 (B = Sn, Ti; X = Cl, Br, I) nanocrystals are shown in Figs. S1 and S2. The cubic lattice constants of the F m3m Cs 2 BX 6 nanocrystals were determined from the measured diffractograms as 11.67Å, 10.86Å and 10.40Å for Cs 2 SnX 6 (X = I, Br, Cl), and 11.50Å, 10.72Å and 10.27Å for Cs 2 TiX 6 (X = I, Br, Cl). The fact that these materials are often synthesized in nanocrystalline form (as opposed to thin films or bulk powders) as in this work, likely contributes to the small differences between reported lattice constants in the literature, due to surface/ligand bonding effects.  Figure S1: X-ray diffraction patterns for synthesized Cs 2 SnX 6 (X = I, Br, Cl) nanocrystals. Measured pattern in blue, expected pattern in orange.  Figure S2: X-ray diffraction patterns for synthesized Cs 2 TiX 6 (X = I, Br, Cl) nanocrystals. Measured pattern in blue, expected pattern in orange.
All computational and experimental data produced during this work is freely available at:  Figure S7: a. Electronic band structure of Cs 2 SnBr 6 calculated with hybrid DFT including spin-orbit coupling (HSE06+SOC), alongside a vertical plot of the orbital-projected electronic density of states. Valence band in blue, conduction band in orange, and VBM set to 0 eV. b. Orbital-projected electronic band structure of Cs 2 SnBr 6 .

S2.6 Cs
Energy (eV) Energy (eV) Cl (p) Sn (p) Sn (s) b a Figure S8: a. Electronic band structure of Cs 2 SnCl 6 calculated with hybrid DFT including spin-orbit coupling (HSE06+SOC), alongside a vertical plot of the orbital-projected electronic density of states. Valence band in blue, conduction band in orange, and VBM set to 0 eV. b. Orbital-projected electronic band structure of Cs 2 SnCl 6 . S2.7 Additional Notes on the Electronic Structure of Cs 2 BX 6 (B = Sn, Ti; X = Cl, Br, I) • The bonding interaction between the metal frontier orbitals with anion p states in the lower valence band (a 1g (σ) and e g (σ) for Ti s and d, t 1u (σ) for Sn p) shows a wider interaction range for Ti than Sn (Figs. S3 to S8), leading to a wider valence bandwidth for Cs 2 TiX 6 as mentioned in the main text (e.g. 4.8 eV vs 4.3 eV for Cs 2 TiI 6 and Cs 2 SnI 6 ).
• Comparing ????c, we witness near-identical VBM charge densities as expected, being comprised almost entirely from non-bonding Iodine p states.
• Much stronger I p contributions to the CBM wavefunctions are witnessed for Cs 2 SnX 6 (Figs. S6 to S8) than for Cs 2 TiX 6 (Figs. S3 to S5), reflecting the greater orbital hybridization and thus stronger dispersion in Cs 2 SnX 6 • Another consequence of mixed ionic-covalent bonding in these systems, similar to other 'perovskite-inspired materials', S13-S15 is that while the formal oxidation state of B in A 2 BX 6 is +4, previous work has shown that the strong hybridization of the (formallyunoccupied) B valence orbitals with X p results in an effective ion charge much less than +4. S16,S17

S3.1 Further GW Calculation Results
Convergence with respect to the number of empty bands (virtual states) and k-point sampling was confirmed in each case, and spin-orbit coupling effects were included in all calculations (which despite a relatively small effect on the band gap are found to significantly affect the absorption shape, particularly for the G 0 W 0 +BSE calculations; Figs. S20 and S21). We note that the quasiparticle band gaps calculated using the GW approach, without explicit electron-hole interactions, are greatly overestimated for all members of the Cs 2 SnX 6 and Cs 2 TiX 6 families (Table S3). This behavior was noted by Cucco et al. S18 for the Cs 2 TiX 6 family, and attributed to poorly-described correlation effects in the initial DFT Ti d orbitals, however we find this overestimation even with the well-described Cs 2 SnX 6 family, and in fact worsening when self-consistently iterating the orbitals within the GW calculation (quasiparticle self-consistent GW ; QSGW ). This failure of the bare GW approach appears to be a consequence of major under-screening within the Coulomb potential W  Figure S9: Real component of the dielectric function (ε real ) for Cs 2 TiI 6 (left) and Cs 2 SnI 6 (right), computed using hybrid DFT (HSE06; blue), G 0 W 0 + BSE (purple) and the Random Phase Approximation (G 0 W 0 (RPA); red).
( Fig. S9 and Table S3), which relies on the dielectric function computed via the Random Phase Approximation (RPA). S19-S22 This error is known to worsen for localized (e.g. d/f -orbital) and lower-dimensional (e.g. vdW-bonded) systems, S18,S21-S23 thus intensifying here due to the combination of structural, orbital and electronic localization ( Fig. 1 and ??). Perturbative inclusion of ladder diagrams in W (i.e. electron-hole screening and thus excitonic effects) via the BSE, rectifies this overestimation for the Sn-based compounds, with absorption onset energies in good agreement with experiment (Table S3). For the Ti-based compounds however, both the absorption onset and exciton binding energies remain apparently overestimated (Tables S2 and S3), consistent with recent studies which have shown this under-screening error in GW (RPA) to cancel under GW +BSE for weakly-correlated systems, but with incomplete cancellation for similar strongly-correlated d-orbital systems, requiring state-of-the-art self-consistent vertex corrections and orbital iteration within the GW calculation to rectify this behavior. S20-S22 S11 Table S2: Difference in the lowest energy bright excitation calculated using G 0 W 0 and G 0 W 0 + BSE a , to estimate the exciton binding energy Cs 2 SnCl 6 Cs 2 SnBr 6 Cs 2 SnI 6 Cs 2 TiCl 6 Cs 2 TiBr 6 Cs 2 TiI 6 1.42 eV 0.79 eV 0.34 eV 2.11 eV 1.60 eV 0.89 eV a Hartree (bubble) diagrams are omitted from the solution of the BSE here, while ladder diagrams are kept included, so that only excitonic effects formally contribute to the energy shifts. We note however that bubble diagrams were found to have a minor effect on the G 0 W 0 + BSE eigenvalues (< 0.1 eV).
From Table S2, the estimated exciton binding energies from G 0 W 0 +BSE agree with the qualitative trends predicted by the Wannier effective mass model (??); namely that

S3.2 Further GW Calculation Details
As mentioned in the main text, despite a relatively modest effect on the bandgap energies (Figs. S17 to S19), SOC was found to have a significant effect on the spectral shape above the absorption onset, as shown in Figs. S20 and S21. The lack of symmetry reduction, requirement for large numbers of virtual states / empty bands (particularly with the use of relativistic wavefunctions) and rapid scaling of computational memory demand with k-point density in the G 0 W 0 +BSE calculations, means that a 3×3×3 mesh (for the 9-atom primitive unit cell, equivalent to a k-point density of 0.33Å −1 in reciprocal space) was the maximum tractable k-point mesh for Cs 2 BX 6 with our computational resources. This reciprocal space sampling density was found to give well-converged absorption spectra for the Br and Cl compounds, however for the iodides (Cs 2 BI 6 ; B = Sn, Ti), the greater band dispersion means the spectra are not well converged for this k-point density. Thus for the iodides, we employ the 'model BSE' (mBSE) approach as described by Varrassi et al. S24 and Tal et al., S25 S12 which has proven to be a reliable method for obtaining approximate GW +BSE spectra at increased k-point densities, with reduced computational cost. This method involves fitting a local model analytic function ε −1 G,G (q) to the screened Coulomb potential W from the explicit G 0 W 0 calculations (with the 3 × 3 × 3 k-point mesh), of the form: where ε ∞ is the high-frequency dielectric constant and λ is the range-separation parameter.
After fitting this model dielectric screening function ε −1 G,G (q), we then calculate the mBSE absorption spectrum with a 3 × 3 × 3 k-mesh to confirm the reproduction of the explicit G 0 W 0 +BSE calculated spectrum with this k-mesh. We then recalculate the absorption spectrum with the fitted ε −1 G,G (q) within the mBSE approach at 8 × 8 × 8 and 4 × 4 × 4 k-point meshes for Cs 2 SnI 6 and Cs 2 TiI 6 respectively. For Cs 2 TiI 6 , a 4 × 4 × 4 k-point mesh is found to give converged results. For Cs 2 SnI 6 , a 8 × 8 × 8 k-point mesh is the maximum tractable k-point mesh for this approach due to the high memory demand, found to be mostly-converged with respect to k-points, though potentially with some small remaining absence of spectral density around ∼2.2 eV (??). Gaussian broadening of the calculated spectra (using σ = 0.1 eV for Cs 2 TiX 6 and σ = 0.15 eV for Cs 2 SnX 6 due to the greater dispersion) and a complex shift of η = 0.1 eV in the Kramers-Krönig transformation of the dielectric function were used in each case.
The GW VASP recommended projector-augmented wave (PAW) pseudopotentials, optimized for excited-state properties, were used for all GW calculations; Cs sv GW, Sn d GW, Ti sv GW, I GW, Br GW and Cl GW; Table S5. In addition, PAW potentials with the largest valence electron configurations available for VASP (PAW2; Table S5) were also trialled, using both semi-local (PBE) and hybrid DFT (HSE06) wavefunctions (Table S4). This pseudopotential choice was found to yield the same trends in G 0 W 0 quasiparticle gaps, G 0 W 0 +BSE excitation energies and binding energies, though with larger absolute values and thus greater S13 overestimation of bandgap and binding energies in each case. Moreover, the lack of available Cl sv GW PAW pseudopotential for VASP means that these parameters give incorrect relative bandgaps for Cs 2 TiBr 6 and Cs 2 TiCl 6 , and thus the PAW1 set was used for all other GW calculations in this work.
Convergence with respect to the number of virtual states / empty bands, imaginary fre-  Table S2, the 'exciton binding energies' are extracted from explicit GW +BSE calculations and then linearly extrapolated to 1/N k → 0 (N k → ∞), though the difference between the N k → ∞ value and the final explicitly calculated datapoint is < 0.06 eV in all cases. S14 Table S3: Calculated electronic bandgaps (E g )(lowest energy vertical excitations) and highfrequency dielectric constants (ε ∞ ) using the G 0 W 0 and G 0 W 0 +BSE approaches for Cs 2 BX 6 (B = Sn, Ti; X = Cl, Br, I), with electronic wavefunctions computed with hybrid DFT including spin-orbit coupling (HSE06+SOC). Bandgaps and dielectric constants calculated using hybrid DFT (HSE06+SOC) and reported experimentally are also included for comparison.  The similarity of the values for ε ∞, G 0 W 0 +BSE between B = Ti/Sn in Table S3, along with the lower hole masses for the contracted Ti compounds (??), provides further evidence for the conduction band effective masses being the dominant factor in the exciton binding strength for A 2 BX 6 .

S3.3 Constrained-Supercell Exciton Calculations
Due to the overestimated quasiparticle band gaps and thus exciton binding energies from GW (+BSE) for both Cs 2 SnX 6 and Cs 2 TiX 6 vacancy-ordered perovskites, a constrainedsupercell approach was employed to estimate the exciton binding energies in these systems.
Here an exciton state is generated in the supercell by constraining the band occupations to place an electron in a conduction band state and a hole in a valence band state, as well as initialising the atomic spins to have an up-spin magnetic moment on a cation (Sn/Ti) site and down-spin magnetic moments on each of the 6 octahedral-coordinating halides, before relaxing the electronic density using hybrid DFT, while keeping the atom positions fixed.
For each X = I, Br, Cl, a localised Frenkel-exciton state is obtained for Cs 2 TiX 6 (Fig. S13), while fully delocalised states are obtained for Cs 2 SnX 6 for supercell sizes up to 23.1Å, as a consequence of delocalised Wannier-Mott exciton behaviour. S18 Cs 2 TiI 6 Cs 2 SnI 6 Figure S13: Charge density isosurfaces of the constrained-occupation excitonic supercells for Cs 2 TiI 6 (left) and Cs 2 SnI 6 (right). Electron states are shown in dark violet, hole states in light green/blue and an isosurface level of 7 × 10 −4 e/Å 3 was used in each case. Caesium atoms in green, titanium/tin in blue/grey, and iodine in purple. Qualitatively similar results were obtained for each X = I, Br, Cl.
The exciton binding energy is then defined as the bandgap minus total energy difference between the constrained-occupation excitonic supercell and the same supercell in the groundstate electronic configuration: For Cs 2 TiI 6 , we compute this this exciton binding energy in supercells of 36, 288 and 972 atoms, corresponding to 1 × 1 × 1, 2 × 2 × 2 and 3 × 3 × 3 expansions of the conventional cubic  Figure S16: Experimentally-reported S4,S16,S26,S37 optical absorption spectra of Cs 2 SnCl 6 (a), and (b) rigidly shifted to align the absorption onsets. As most absorption data is reported in arbitrary units, the spectra have been scaled to best match at the onset peak (E ≃ 4.5 eV).
As noted by Karim et al., S4 a relatively wide distribution of reported bandgaps is seen in the literature for Cs 2 SnCl 6 , mostly in the range 4.4 to 4.9 eV (??). When digitising and plotting the absorption data reported in previous studies (Fig. S16), S4,S16,S26,S37 we find that although the measured absorption onset energy varies within a ∼0.7 eV range, the onset shape is relatively consistent between reported samples. Potential origins include differing morphologies of synthesized samples (thin films vs powders vs nanocrystals), quantum confinement and surface/ligand effects in small nanocrystals (and the effect on excitonic binding), as well as the difficulty in measuring optical absorption in the high energy range (> 4 eV). S4 Figure S18: Electronic band structure of Cs 2 SnI 6 calculated with hybrid DFT (HSE06) excluding ('non-SOC') and including ('SOC') spin-orbit coupling, alongside vertical plots of the orbital-projected electronic density of states. Valence band in blue, conduction band in orange, and VBM set to 0 eV. Figure S19: Band contributions to the brightest exciton state at the absorption onset in Cs 2 TiI 6 (left) and Cs 2 SnI 6 (right), calculated using the G 0 W 0 +BSE approach without ('non-SOC') and with ('SOC') spin-orbit coupling effects. Band eigenvalues are indicated by the black dots, with filled circles weighted by their contributions to the exciton state and gray interpolating bands. The average of the three degenerate brightest states at the absorption onset is used, with the sum area of the filled circles normalized across all compositions. Hole and electron states are shown in blue and orange, respectively, and the VBM is set to 0 eV.  Figure S21: Calculated optical absorption of Cs 2 TiI 6 (left) and Cs 2 SnI 6 (right), computed using the G 0 W 0 +BSE approach without ('non-SOC') and with ('SOC') spin-orbit coupling effects.