Giant thermally induced band-gap renormalization in anharmonic silver chalcohalide antiperovskites

This journal is © The Royal Society of Chemistry 2025 J. Mater. Chem. C Cite this: DOI: 10.1039/d5tc00863h Giant thermally induced band-gap renormalization in anharmonic silver chalcohalide antiperovskites† Pol Benı ´tez, * ab Siyu Chen, cd Ruoshi Jiang, c Cibra ´nLo ´pez, ab Josep-Lluı ´s Tamarit, ab Jorge I ´n ˜iguez-Gonza ´lez, ef Edgardo Saucedo, bg Bartomeu Monserrat cd and Claudio Cazorla * ab Silver chalcohalide antiperovskites (CAP), Ag 3 XY (X = S, Se; Y = Br, I), are a family of highly anharmonic inorganic compounds with great potential for energy applications. However, a substantial and unresolved discrepancy exists between the optoelectronic properties predicted by theoretical firstprinciples methods and those measured experimentally at room temperature, hindering the fundamental understanding and rational engineering of CAP. In this work, we employ density functional theory, tightbinding calculations, and anharmonic Fro ¨hlich theory to investigate the optoelectronic properties of CAP at finite temperatures. Near room temperature, we observe a giant band-gap (E g ) reduction of approximately 20–60% relative to the value calculated at T= 0 K, bringing the estimated E g into excellent agreement with experimental measurements. This relative T-induced band-gap renormalization is roughly twice the largest value previously reported in the literature for similar temperature ranges. Lowenergy optical polar phonon modes, which break inversion symmetry and enhance the overlap between silver and chalcogen s electronic orbitals in the conduction band, are identified as the primary drivers of this significant E g reduction. Furthermore, when temperature effects are considered, the optical absorption coefficient of CAP increases by nearly an order of magnitude in the visible light spectrum. These findings not only bridge a critical gap between theory and experiment but also pave the way for future technologies where temperature, electric fields, and light dynamically modulate optoelectronic properties, establishing CAP as a versatile platform for energy and photonic applications. Introduction Electron–phonon coupling (EPC), arising from the interactions between electrons and lattice vibrations, is ubiquitous in materials and is responsible for a wide range of condensed matter physical effects. 1–4 For example, EPC plays a crucial role in the temperature (T) dependence of electrical resistivity in metals, carrier mobility in semiconductors, optical absorption in indirect band gap semiconductors, and the onset of conventional superconductivity. Additionally, EPC enables the thermalization of hot carriers, influences the phonon dispersion in metals, and determines the T-dependence of electronic energy bands in solids. 5,6 Likewise, the band gap (E g ) of semiconducting and dielectric materials can be significantly affected by EPC, typically decreasing with increasing temperature (the so-called Varshni effect 7 ). This common E g behavior can be explained by the Allen–Heine– Cardona perturbative theory, which attributes it to a larger T-induced energy increase in the valence band compared to the conduction band due to a greater sensitivity to phonon a Group of Characterization of Materials, Departament de Fı ´sica, Universitat Polite `cnica de Catalunya, Campus Diagonal Beso `s, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain. E-mail: pol.b[email protected], cla[email protected] b Research Center in Multiscale Science and Engineering, Universitat Polite `cnica de Catalunya, Campus Diagonal-Beso `s, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain c Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, CB30FS, UK d Cavendish Laboratory, University of Cambridge, Cambridge, CB30HE, UK e Materials Research and Technology Department, Luxembourg Institute of Science and Technology (LIST), Avenue des Hauts-Fourneaux 5, L-4362 Esch/Alzette, Luxembourg f Department of Physics and Materials Science, University of Luxembourg, 41 Rue du Brill, L-4422 Belvaux, Luxembourg g Micro and Nanotechnologies Group, Emerging Thin Film Photovoltaics Lab, Departament dEnginyeria Electro `nica, Universitat Polite `cnica de Catalunya, Campus Diagonal Beso `s, Av. Eduard Maristany 10–14, 08019 Barcelona, Spain †Electronic supplementary information (ESI) available. See DOI: https://doi.org/ 10.1039/d5tc00863h Received 27th February 2025, Accepted 11th April 2025 DOI: 10.1039/d5tc00863h rsc.li/materials-c Journal of Materials Chemistry C PAPER Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online View Journal J. Mater. Chem. C This journal is © The Royal Society of Chemistry 2025 population variations (i.e., larger second-order electron–phonon coupling constants). 8–10 Representative examples of this thermal E g dependence include diamond, which exhibits a B5% band-gap reduction at around 1000 K; 11,12 antimony sulfide (Sb 2 S 3 ), which shows a E g reduction of 200 meV in the temperature range of 10 rTr300 K; 13 MgO, which displays a band-gap reduction of B15% in the temperature interval 0 r Tr1500 K; 14 SrTiO 3 , which exhibits a B15% band-gap reduction from 300 to 1000 K; 15 and molecular crystals, which display record band-gap reductions of 15–20% at low temperatures. 16 Anomalous band-gap thermal behaviour, in which E g increases with increasing temperature, has also been observed in a variety of materials such as black phosphorus, 17 halide perovskites, 18 and chalcopyrite 19 and hydride 12 compounds. Highly anharmonic silver chalcohalide antiperovskites (CAP) 20 with chemical formula Ag 3 XY (X = S, Se; Y = Br, I) are structurally similar to lead halide perovskites (e.g., CsPbI 3 ), with the ‘‘anti’’ designation indicating the exchange of anions and cations compared to the typical ionic perovskite arrangement. Analogous to lead halide perovskites, CAP are highly promising materials for energy and optoelectronic applications, 21–26 offering low toxicity due to their lead-free composition. 27,28 The two most extensively studied CAP compounds, Ag 3 SBr and Ag 3 SI, possess experimentally determined band gaps of approximately 1.0 eV, 29,30 making them favorable for photovoltaic applications. These materials have also been recognized as high temperature superionic conductors. 21,22 Additionally, CAP have been investigated as potential thermoelectric materials 25,26 owing to their substantial vibrational anharmonicity and unique charge transport properties. 23,24 Intriguingly, for both Ag 3 SBr and Ag 3 SI, there is an enormous disagreement between the E g predicted by first-principles methods (at T= 0 K, under static lattice conditions) and those measured experimentally at room temperature. In particular, high-level density functional theory (DFT) calculations employing hybrid functionals and including spin–orbit coupling (SOC) effects estimate the band gap of these two archetypal CAP to be 1.8 and 1.4 eV, respectively. 29–31 The E g discrepancies between theory and measurements amount to 60–80% of the experimental values (i.e., differences of 0.5–0.8 eV), which are unusually large and call for a careful inspection of the factors causing them. In this study, we assessed the influence of EPC effects on the E g and optical absorption spectra of CAP using first-principles DFT methods, tight-binding calculations, and anharmonic Fro ¨hlich theory. Near room temperature, our computational investigations revealed a giant E g reduction of 20–60% relative to the value calculated at T= 0 K, bringing the estimated band gap into excellent agreement with the experimental values. Low-energy optical polar phonons, which cause large symmetry-breaking structural distortions and promote the overlap between silver and chalcogen s electronic orbitals in the conduction band, were identified to be the primary mechanism driving this substantial T-induced band-gap reduction. Furthermore, at finite temperatures the optical absorption spectra of CAP were significantly enhanced, in some cases by nearly an order of magnitude. The polar nature of the phonons causing these effects opens up new technological possibilities, where the optoelectronic properties of materials could be effectively manipulated by external electric fields and light. Results The room-temperature phase of both Ag 3 SI and Ag 3 SBr have been experimentally identified as cubic with the space group Pm% 3m. 30,32–34 This phase is characterized by a five-atom unit cell: a chalcogen atom at the center of the cube, halide atoms in each vertex, and silver atoms at the center of each face (Fig. 1a and b). Phonon calculations of this phase within the harmonic approximation (T= 0 K) reveal imaginary phonon branches, thus indicating dynamical instability. However, when phonons are calculated fully accounting for anharmonic effects at finiteTconditions, the resulting phonon spectrum is well-behaved with no signs of instability (Fig. 1c). 20 Thus, the cubic Pm% 3m phase was considered throughout this work for all CAP. As discussed in the Introduction, the discrepancies between the experimentally measured (at T= 300 K) and theoretically determined (at T= 0 K) band gaps of Ag 3 SBr and Ag 3 SI are tremendously large (i.e., 60–80% of the experimental values). Therefore, we investigated their potential causes by assessing the impact of electron–phonon coupling (EPC) and temperature on the band gap of CAP. Band gap renormalization due to electron–phonon interactions is typically estimated using two main approaches: density functional perturbation theory (DFPT) 1 and finite-differences. 2 In DFPT, electron–phonon interactions are treated as a perturbation, with band energy variations (and consequently, band gap shifts) derived from the Fan-Migdal and Debye–Waller selfenergies, as described by Allen and Heine. 36 A key advantage of DFPT is its computational efficiency, as it does not require the use of supercells. This method has been widely employed in band gap renormalization studies 37 and is implemented in popular ab initio codes such as EPW. 38 Finite-differences approaches, on the other hand, are computationally more demanding, requiring supercells to reproduce phonons in real space and dynamical simulations to correctly sample the electronic response to ionic fluctuations. However, they offer distinct advantages over DFPT. One key benefit is their flexibility, as they can be applied with any underlying electronic structure method. Additionally, finitedifferences methods naturally incorporate terms beyond the lowest order in the electron–phonon interaction, making them particularly useful for capturing higher-order effects. 18 Readers seeking a more comprehensive discussion of DFPT and finitedifferences methods are referred to the review articles, 1,2 which extensively cover these techniques. In this study, we employ the finite-differences approach to estimate temperature-renormalized band gaps, as CAP materials exhibit strong anharmonicity. 20 Consequently, renormalizing their electronic band energies at the harmonic level would Paper Journal of Materials Chemistry C Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online This journal is © The Royal Society of Chemistry 2025 J. Mater. Chem. C be inadequate. Moreover, since accurate CAP band gap predictions require the use of hybrid functionals and spin–orbit coupling, the finite-differences approach emerges as the most practical and reliable choice. In particular, we performed first-principles calculations and ab initio molecular dynamics (AIMD) simulations based on DFT (Methods). Additionally, to capture long-range EPC effects, we employed anharmonic Fro ¨hlich theory 39–42 considering longrange dipole–dipole interactions and T-renormalized phonons (Methods). Furthermore, the optical absorption spectra of all CAP were assessed at Ta0 K conditions and the main EPC mechanisms underlying the E g discrepancies were identified with the help of a tight-binding model. EPC band-gap renormalization in CAP The T-renormalized band gap of CAP was calculated as follows: E g (T)=E g (0) + DE g (T), (1) where E g (0) represents the static band gap and the correction term DE g can be expressed as the sum of short-(S) and longwavelength (L) phonon contributions: 40 DE g (T)=DE S g (T)+DE L g (T). (2) The short-wavelength phonon correction was estimated through AIMD simulations using a supercell (Methods), where the band-gap value was averaged over multiple generated configurations, as described in ref. 39: DES gðTÞ¼ 1 NX N k¼1 EgðfRkðTÞgÞ  Egð0Þ;(3) where Nrepresents the total number of considered configurations and {R k } the atomic position of the k-th configuration. To achieve accurate band-gap corrections, it was essential to use hybrid functionals and incorporate spin–orbit coupling effects (Supplementary methods, ESI†), which significantly increase the computational effort. Extensive numerical tests were conducted to optimize the supercell size, k-point mesh, and value of N, ensuring that these critical effects were accurately captured in the calculations (Methods and Supplementary methods, ESI†). In polar materials, there is an additional contribution to the band-gap renormalization stemming from long-range Fro ¨hlich coupling that is not fully captured by the finite size of the supercells employed in the AIMD simulations. 39–43 This longwavelength phonon band-gap correction can be expressed as follows: DEL gTðÞ¼DEL CB TðÞDEL VB TðÞ;(4) where CB and VB refer to the bottom conduction and top valence band levels, respectively. Fig. 1 General physical properties of the archetypal CAP Ag 3 SBr. (a) The cubic Pm % 3mphase experimentally observed at room temperature. (b) Experimental diffractogram of Ag 3 SBr 30 compared with the theoretical one estimated for the cubic Pm % 3mphase. (c) Vibrational phonon spectrum (left) and phonon density of states (right) calculated within the harmonic approximation for the cubic Pm % 3mphase of Ag 3 SBr at T= 0 K (black lines) and at T= 200 K (red lines) fully considering anharmonic effects. (d) Electronic band structure (left) and density of states (right) of Ag 3 SBr calculated with the hybrid functional HSEsol. 35 Red and blue lines (dots) represent valence (top of the valence) and conduction (bottom of the conduction) bands, respectively. Journal of Materials Chemistry C Paper Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online J. Mater. Chem. C This journal is © The Royal Society of Chemistry 2025 For a 3D polar material, the T-induced energy level shifts appearing in eqn (4) can be computed as follows: 39 DEL iðTÞ¼2aP p hoLO tan1qF qLO;i  2nTþ1 ½ ;(5) where a P represents the polaron constant, o LO the phonon frequency averaged over the three longitudinal optical Gphonon modes, 43 and q F a truncation factor. The truncation factor q F can be approximated as the Debye sphere radius and q LO,i is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2moLO þoi ðÞ=h p,m* being the charge carrier effective mass and ho i the state energy. The term n T is the Bose–Einstein occupation number corresponding to the average LO vibrational frequency, and the polaron constant can be computed as follows: 39 aP¼e2 4pE0h 1 e1 1 e0  m 2hoLO  1=2 ;(6) where e N is the high-frequency dielectric constant and e 0 the static permittivity of the system. Quantum nuclear effects have been disregarded throughout this work, hence the T-induced energy level shifts in eqn (5) were offset by their zerotemperature values De L i (0). Fig. 2a presents the anharmonic phonon spectrum calculated for Ag 3 SBr under finite-temperature conditions, accounting for long-range dipole–dipole interactions (i.e., including nonanalytical corrections), which result in LO–TO splitting near the reciprocal space point G. Fig. 2b shows the corresponding shortand long-wavelength phonon band-gap corrections expressed as a function of temperature, which are always negative. Since in this study the DE L g correction term has been calculated using the material’s anharmonic phonon spectrum, we refer to this method as anharmonic Fro ¨hlich theory (Methods). In Fig. 2b, it is observed that near room temperature the DE S g correction is dominant and significantly larger than DE L g , approximately six times greater in the absolute value. Notably, at T= 400 K, the total band-gap correction for Ag 3 SBr amounts to 0.7 eV, which is of giant proportions, representing roughly 40% of the E g value calculated at zero temperature (excluding quantum nuclear effects). Fig. 3 shows the relative band-gap variation, referenced to the value calculated at zero temperature and expressed as a function of temperature, for the four CAP compounds Ag 3 SBr, Ag 3 SI, Ag 3 SeBr and Ag 3 SeI. In all cases, the band gap significantly decreases as the temperature increases (Table 1). The relative T-induced E g reduction is largest for Ag 3 SeBr and smallest for Ag 3 SI. In particular, near room temperature, the band gap of Ag 3 SBr and Ag 3 SI is reduced by 39% and 29% while those of Ag 3 SeBr and Ag 3 SeI decrease by 56% and 38%, respectively (Fig. 3). As shown in Table 1, the agreement between the experimental and theoretical E g values for Ag 3 SBr and Ag 3 SI improves as the temperature increases. In Ag 3 SeBr and Ag 3 SeI, the liquid phase is stabilized over the crystal phase at moderate temperatures (Fig. 3c and d); thus no band gaps were estimated for these two compounds under T4400 K conditions. Notably, our theoretical E g results obtained at T= 400 K are fully consistent with the available experimental data obtained at room temperature. This excellent agreement near ambient conditions strongly suggests that the neglect of EPC effects is the main reason for the huge theoretical–experimental E g discrepancies discussed in the Introduction. The T-induced relative band-gap renormalization found in CAP are of giant proportions, ranging from 20 to 60% near room temperature, setting a new record previously held by molecular crystals, which exhibited a 15 to 20% band-gap renormalization for similar temperature ranges. 16 Table 1 also presents the value of the DE S g and DE L g correction terms estimated for each CAP at three different temperatures. In all cases, both the shortand long-wavelength phonon Fig. 2 Anharmonic phonon spectrum and thermal band-gap corrections estimated for the archetypal CAP Ag 3 SBr. (a) Anharmonic phonon spectrum obtained at T= 200 K neglecting (black solid lines) and considering (red dashed lines) non-analytical corrections (NAC). (b) Shortand long-wavelength phonon band-gap corrections, DE S g and DE L g , respectively, expressed as a function of temperature (excluding quantum nuclear effects). The short-range correction term was evaluated at several temperature points (blue circles and error bars); as a guide to the eye, the DE S g data points were fitted to an arbitrary polynomial function (blue dashed line). Calculations were performed at the HSEsol + SOC level. 35 Paper Journal of Materials Chemistry C Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online This journal is © The Royal Society of Chemistry 2025 J. Mater. Chem. C corrections are negative, with the former term considerably surpassing the latter in absolute value. For example, at T= 400 K, the short-range band-gap corrections are seven and four times larger than the long-range ones calculated for Ag 3 SeBr and Ag 3 SI, respectively. As the temperature is raised, the size of the two band-gap correction terms increases in absolute value, with |DE L g | exhibiting the largest relative enhancement (e.g., approximately a 320% relative increase for Ag 3 SBr from 200 to 600 K). To provide further insights into the impact of thermal effects on the electronic band structure of CAP, we also examined how band morphology evolves with temperature. Since band-gap renormalization in this study is assessed using the finite-differences approach based on AIMD simulations, it is convenient to unfold the energy bands calculated for the supercell into the reciprocal space of the primitive unit cell. 44 This was done using the Easyunfold software, 45 focusing on the representative CAP compound Ag 3 SBr (Fig. S1, ESI†). Specifically, we analyzed five uncorrelated supercell snapshots extracted from a long AIMD trajectory (B100 ps) at T= 200 K. For this particular case, the PBEsol exchange–correlation functional was employed due to the very high computational cost of performing hybrid functional calculations on supercells. Reassuringly, we verified that the band structures obtained using semilocal and hybrid functionals are practically equivalent in morphology (Fig. S2, ESI†). Table 1 Theoretical band gaps of CAP as a function of temperature. E g values were obtained at zero temperature (excluding quantum nuclear effects) at T= 200, 400 and 600 K. Calculations were performed at the HSEsol + SOC level. 35 Numerical uncertainties are provided, which mainly result from the DE S g correction term. Shortand long-wavelength phonon band-gap corrections, DE S g and DE L g , respectively, are provided at each temperature (excluding quantum nuclear effects). The experimental band gaps measured at room temperature for Ag 3 SBr and Ag 3 SI 30 are shown for comparison CAP E 0K g [eV] E 200K g [eV] DE S g [meV] DE L g [meV] E 400K g [eV] DE S g [meV] DE L g [meV] E 600K g [eV] DE S g [meV] DE L g [meV] E exp g [eV] Ag 3 SBr 1.8 1.3 0.1 440 42 1.1 0.1 570 108 0.9 0.2 680 175 1.0 Ag 3 SI 1.4 1.1 0.1 260 29 1.0 0.1 290 75 0.8 0.1 490 122 0.9 Ag 3 SeBr 1.6 0.9 0.1 630 42 0.7 0.1 770 105 Liquid — — — Ag 3 SeI 1.3 0.9 0.1 370 37 0.8 0.2 400 90 Liquid — — — Fig. 3 Temperature-induced relative band-gap variation in CAP. Percentages are referenced to the band gap calculated at T= 0 K conditions (excluding quantum nuclear effects), namely, DE g (T)=E g (T)E g (0), for (a) Ag 3 SBr, (b) Ag 3 SI, (c) Ag 3 SeBr, and (d) Ag 3 SeI. Error bars indicate numerical uncertainties and dashed lines are a guide to the eye. Shaded areas indicate regions of thermodynamic stability of the liquid phase (theory). Calculations were performed at the HSEsol + SOC level. 35 Journal of Materials Chemistry C Paper Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online J. Mater. Chem. C This journal is © The Royal Society of Chemistry 2025 Our results indicate that thermal effects lead to a downward shift of the conduction band minimum, which remains located at the Gpoint, which is consistent with the band-gap trends observed in the corresponding electronic density of states. However, the ionic disorder induced by lattice vibrations causes a noticeable flattening of the valence band near its top. As a result, the valence band maximum becomes poorly defined, unlike in the static case (Fig. S1, ESI†). The consistency of results across the five ionically disordered configurations suggests that this small sampling is sufficient to capture the key temperature-induced changes in band morphology. Thermal effects on the optical absorption coefficient of CAP Following a similar approach to that used for the calculation of T-renormalized band gaps (i.e., performing AIMD simulations with a supercell and averaging the quantity of interest over several of the generated configurations), we determined the frequencydependent complex dielectric tensor of CAP under Ta0K conditions (Methods and Supplementary methods, ESI†), employing linear response theory. From the average dielectric tensor, we computed several macroscopic optical properties like the optical absorption coefficient, a(o), refractive index, and reflectivity. 46 Fig. 4 shows the optical absorption spectra estimated for CAP as a function of incident light wavelength and temperature. It is found that ais significantly enhanced under increasing temperature,insomecasesbyasmuchasanorderofmagnitude. Similarly to the band gap, the T-induced optical absorption variations are the largest for Ag 3 SeBr (for which aB10 3 – 10 5 cm 1 at zero temperature and B10 4 –10 6 cm 1 at 200 K) and smallest for Ag 3 SI (for which aB10 3 –10 5 cm 1 at any temperature). It is also noted that the most significant optical absorption changes generally occur at low temperatures, that is, within the 0 rTr200 K interval. These T-induced atrends align well with the remarkably large influence of the EPC on the band gap, underscoring the critical role of thermal renormalization effects on the optoelectronic properties of CAP. Unfortunately, we cannot directly compare our theoretical a(o) results with experimental data, as such data are not available in the literature. Notably, Can ˜oet al. 30 measured the optical absorption coefficient of CAP films scaled by their layer thickness, d, specifically,  aad. However, since the thickness of the synthesized CAP films was not determined in work, 30 we cannot access the physical quantity of interest. In this regard, performing new optoelectronic experiments on CAP films across a broad range of temperatures, including the low-T regime, would be highly desirable. EPC mechanisms in CAP We have already shown that temperature and EPC effects are essential for understanding the thermal evolution of the Fig. 4 Optical absorption coefficient (a) of CAP calculated at different temperatures as a function of photon energy. (a) Ag 3 SBr, (b) Ag 3 SI, (c) Ag 3 SeBr, and (d) Ag 3 SeI. Solid lines represent the estimated average values and statistical errors are indicated with shaded thick curves. The rainbow-colored region denotes photons with energy in the visible spectrum. Calculations were performed at the HSEsol + SOC level. 35 Paper Journal of Materials Chemistry C Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online This journal is © The Royal Society of Chemistry 2025 J. Mater. Chem. C optoelectronic properties of CAP and for achieving a consistent agreement between first-principles calculations and roomtemperature experiments. Next, we focus on unraveling the primary ionic–electronic mechanisms underlying these key temperature and EPC effects. As shown in Fig. 5a, the influence of each of the fifteen G phonon modes on the band gap of Ag 3 SBr was analysed by monitoring the change in E g driven by frozen-phonon eigenmode distortions of increasing amplitude, u. The Gphonons were classified into acoustic (A), optical polar (P) and optical nonpolar (NP), where the P phonons break the inversion symmetry of the centrosymmetric cubic Pm% 3mphase. It was found that E g is unresponsive to acoustic phonon distortions, as expected, while optical P phonons produce the largest bandgap variations. As the amplitude of the optical phonon distortions increases, E g systematically decreases in both the P and NP cases. Fig. 5b shows the value of the derivative of the band gap with respect to the phonon distortion amplitude, u, expressed as a function of the phonon eigenmode energy (as obtained from T-renormalised phonon calculations, Methods). We found that low-energy polar phonon modes (B10 meV) cause the most significant band-gap reductions, followed by high-energy lattice vibrations of the same type (B200 meV). At room temperature, phonon excitations with the lowest energy host the highest populations and, consequently, represent the most characteristic lattice vibrations in the crystal. Therefore, based on the results shown in Fig. 3 and 5, we conclude that low-energy polar phonon modes are primarily responsible for the substantial temperature-induced E g reduction reported in this study for CAP compounds. The eigenmode of the optical P phonon with the lowest energy is represented in Fig. 5b. As observed therein, this frozen-phonon lattice distortion reduces the distance between the central sulfur atom and one adjacent silver atom (Ag2), while increasing the other two S–Ag1 and S–Ag3 bond lengths, compared to the undistorted cubic unit cell. Fig. 5c summarizes the relative bond length variation, in absolute value, for all pairs of atoms resulting from each of the fifteen Gphonon modes calculated for the cubic Pm% 3mphase. As shown therein, the optical P phonons produce the largest S–Ag distance changes (up to 20%), while the optical NP phonons cause the largest Br–Ag bond length variations (up to 12%). The Br–Ag bond lengths are also appreciably impacted by the optical P phonons (5–10%). This general behaviour is reminiscent of that observed for optical polar phonons in model perovskite oxides like BaTiO 3 (with atomic substitutions Ag 2O, S 2Ti and Br 2Ba). 47,48 After identifying the phonon modes that underpin the giant T-induced band-gap reduction reported in this study for CAP, specifically low-energy optical P modes, we further analyse the induced changes in the electronic band structure. Fig. 6a shows the electronic density of states calculated for the archetypal compound Ag 3 SBr (equilibrium geometry). It is observed that the top of the valence band (VB) is dominated by highly hybridized silver d and chalcohalide p electronic orbitals, while Fig. 5 Phonon-induced band-gap variation estimated for the archetypal CAP Ag 3 SBr. (a) Band gap as a function of the lattice distortion amplitude ufor acoustic, polar optical (P) and non-polar optical (NP) Gphonons. (b) Derivative of the band gap with respect to the phonon distortion amplitude calculated at u 0 = 0.4 Å and expressed as a function of the phonon energy. The eigenmode of the optical polar Gphonon rendering the largest band-gap derivative in absolute value is sketched: Ag, S and Br atoms are represented with grey, yellow and brown spheres, respectively. Calculations were performed at the HSEsol + SOC level. 35 (c) Gphonon-induced relative bond length distortions in the cubic Pm % 3mphase for a distortion amplitude of 0.4 Å. Journal of Materials Chemistry C Paper Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online J. Mater. Chem. C This journal is © The Royal Society of Chemistry 2025 the bottom of the conduction band (CB) is dominated by isotropic and more delocalized S and Ag s orbitals. The electronic band structure in Fig. 6b shows that the VB corresponds to the high-symmetry reciprocal space point M (1/2,1/2,0), while the CB to the center of the Brillouin zone, G(0,0,0); thus the band gap of Ag 3 SBr is indirect (we have checked that the same conclusion applies to the rest of CAP compounds analyzed in this study). The effects on the electronic band structure resulting from a frozen-phonon lattice distortion corresponding to the lowestenergy optical P eigenmode (u 0 = 0.4 Å) are twofold (Fig. 6b). First, due to the breaking of phonon-induced inversion symmetry, the energy band degeneracy at the reciprocal space point M is lifted. However, the band gap of the system is unaffected by this energy degeneracy lifting effect since the VB remains practically invariant. Second, the CB edge experiences a significant decrease in energy and, as a consequence, the band gap of the system is reduced by approximately 30%. Therefore, we may conclude that the giant T-induced E g reduction reported in this study for CAP is primarily caused by lowenergy polar phonon modes that induce a pronounced CB energy decrease. To better understand the electronic origins of the optical P phonon-induced CB energy lowering, we constructed a tightbinding (TB) model based on Wannier functions that accurately reproduces our DFT band structure results (Methods and Fig. S3, ESI†). Specifically, the TB model consists of s, p, and d orbitals for the five atoms in the unit cell, resulting in a total of 45 distinct Wannier orbitals. Consistently, the TB model reproduces the dominant Ag and S s character of the CB and its Fig. 6 Electronic band structure properties of the archetypal CAP Ag 3 SBr. (a) Electronic density of states calculated for the equilibrium cubic Pm % 3m phase. Calculations were performed at the HSEsol + SOC level. 35 (b) Electronic band structure calculated for the equilibrium and phonon distorted (i.e., considering the lowest-energy Goptical P mode with amplitude u 0 = 0.4 Å) cubic Pm % 3mphase; in both cases, the energy bands are referred to a same energy origin, a deep core electronic level that remains unaffected by the distortion (Methods). (c) and (d) The conduction band near its minimum at Gcomputed with a TB model for silver and sulfur atoms (Methods); solid and dashed lines correspond to the equilibrium and distorted structures, respectively. (e) Sketch illustrating the mechanism of band-gap closure induced by low-energy polar soft phonon modes in CAP. Upon phonon distortion, the hybridization of silver and sulfur s electrons in the conduction band is enhanced, lowering (increasing) the energy of the corresponding bonding s(antibonding s*) state. Paper Journal of Materials Chemistry C Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online This journal is © The Royal Society of Chemistry 2025 J. Mater. Chem. C energy lowering under the polar lattice distortion of interest (Fig. 6c and d). According to this TB model, the impact of the frozenphonon distortion on the Ag and S s conduction orbitals is twofold. First, the difference in their kinetic energies, corresponding to the diagonal Hamiltonian matrix elements difference |hAg s|H|Ag sihSs|H|S si|, decreases (Fig. S3, ESI†). And second, the hopping s term involving the Ag2 and S atoms, represented by the off-diagonal Hamiltonian matrix element hAg2 s|H|S si, increases (Fig. S3, ESI†). The general physical interpretation that follows from these TB results is that the polar frozen-phonon distortion enhances the hybridization of Ag2 and S s conduction orbitals, which lowers and increases the energy of the corresponding bonding (s) and antibonding (s*) states, respectively. Consequently, the CB, which is dominated by the Ag2-S sinteraction, is lowered. The revealed EPC mechanism, which overall produces a band-gap reduction, is schematically represented in Fig. 6e. Discussion Thermal expansion is another physical mechanism that can influence the band gap of materials, 40 but it was not explicitly considered in this study due to its high computational cost. However, we performed a series of tests in which we arbitrarily increased and decreased the unit cell volume of Ag 3 SBr and estimated the corresponding relative E g variation (Fig. S4 and Table S1, ESI†). We found that, when the volume of the system increases by a reasonable B1%, as might occur due to thermal expansion near room temperature, the band gap decreases by only B40 meV (i.e., an order of magnitude smaller change than short-wavelength phonon corrections, Table 1). Additionally, thermal expansion likely contributes to the softening of optical P modes, increasing their population and consequently enhancing the E g reduction due to EPC. Therefore, while thermal expansion effects were omitted in our calculations, the reported results likely represent a lower bound of the actual effect, and our conclusions for CAP compounds remain robust and accurate. One may wonder whether, in addition to silver chalcohalide antiperovskites, there exist other families of materials exhibiting similarly large T-renormalization effects on the band gap and optical absorption coefficient. As discussed in previous sections, the polar nature of low-energy optical phonons appears to be essential in this regard. Consequently, a tentative set of necessary conditions for identifying potential materials that display similar T-induced effects on the optoelectronic properties may include dielectric materials exhibiting (1) centrosymmetric crystalline phases, (2) low-energy or even imaginary optical polar phonons, and (3) highly hybridized and delocalized electronic orbitals near the Fermi energy level. The availability of large DFT calculations and phonon databases may enable high-throughput material screening of such a kind. 49,50 Ferroelectric oxide perovskites, exemplified by the archetypal compounds SrTiO 3 (STO) and BaTiO 3 (BTO), appear to satisfy the set of necessary conditions outlined above. Notably, a significant band-gap modulation has been reported for STO under biaxial strain conditions, although this phenomenon arises from different physical mechanisms than those identified in this study for CAP compounds (i.e., energy degeneracy lifting due to symmetry breaking). 51 Moreover, the experimental room-temperature band gap of BTO (E3.2 eV 52 ) shows substantial disagreement with zero-temperature theoretical estimates obtained with hybrid functionals (E4.0 eV 53 ), highlighting an experiment–theory inconsistency similar to that described for Ag 3 SBr and Ag 3 SI in the Introduction. Additionally, thebandgapofthemultiferroicoxideperovskiteBiFeO 3 exhibits a remarkable temperature-dependent shrinkage, decreasing by approximately 50% within the temperature range 300 rTr 1200 K, 54 likely influenced by the magnetic degrees of freedom. 55 These findings suggest that the temperature effects and EPC mechanisms identified in this study for CAP compounds may have broader relevance, potentially extending to other well-known families of functional materials. Theoretical investigations exploring this possibility are currently underway. The polar nature of the optical phonon modes, which cause the significant T-induced reduction in the E g of CAP, opens up exciting technological possibilities. Similar to how an electric field can stabilize a polar phase with ferroelectric polarization over a paraelectric state at constant temperature through a phase transformation, 56 it is likely that polar optical phonons in CAP can also be stimulated using external electric fields. This possibility implies that the optoelectronic properties of CAP could be effectively tuned by applying an electric field rather than altering the temperature, providing a more practical approach for the development of advanced optical devices and other technological applications. Experimental validation of this hypothesis would be highly valuable. Finally, advances in light sources and time-resolved spectroscopy have made it possible to excite specific atomic vibrations in solids and to observe the resulting changes in their electronic and electron–phonon coupling properties. 57–59 These developments also suggest the possibility of tuning the optoelectronic properties of CAP, as well as of similar materials like oxide perovskites, 60–62 through specific phonon excitations using optical means such as lasers. This approach may simplify the design and manufacture of practical setups by eliminating the need for electrode deposition. Therefore, the results presented in this work are significant not only from a fundamental perspective but also for envisioning potential technological applications in which the optical and electronic properties of materials could be effectively tuned by external fields and photoexcitation. Conclusions In this study, we have explored the temperature effects on the band gap of silver chalcohalide antiperovskites (CAP), specifically Ag 3 XY compounds (X = S, Se; Y = Br, I), which are promising for energy and photonic applications due to their Journal of Materials Chemistry C Paper Open Access Article. Published on 14 April 2025. Downloaded on 5/13/2025 10:59:18 AM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online