Abstract
Anharmonicity and local disorder (polymorphism) are ubiquitous in perovskite physics, inducing various phenomena observed in scattering and spectroscopy experiments. Several of these phenomena still lack interpretation from first principles since, hitherto, no approach is available to account for anharmonicity and disorder in electron–phonon couplings. Here, relying on the special displacement method, we develop a unified treatment of both and demonstrate that electron–phonon coupling is strongly influenced when we employ polymorphous perovskite networks. We uncover that polymorphism in halide perovskites leads to vibrational dynamics far from the ideal noninteracting phonon picture and drives the gradual change in their band gap around phase transition temperatures. We also clarify that combined band gap corrections arising from disorder, spinorbit coupling, exchange–correlation functionals of high accuracy, and electron–phonon coupling are all essential. Our findings agree with experiments, suggesting that polymorphism is the key to address pending questions on perovskites’ technological applications.
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Introduction
Oxide perovskites are fascinating materials with extensive applications owing to their intrinsic ferroelectric, antiferroelectric, and piezoelectric properties^{1}. Halide perovskites are of high interest due to their impressive efficiencies in solar cells^{2,3}, and attractive applications in optoelectronics, electrocatalysis, and thermoelectrics^{4,5,6}. Our understanding of perovskites’ key properties is connected to deviations of the vibrational dynamics and electron–phonon coupling from the standard picture observed in conventional semiconductors^{7,8}. For example, halide perovskites exhibit (i) ultralow thermal conductivities attributed to their lowenergy vibrational densities and peculiar anharmonic characteristics^{9,10} and (ii) limited carrier mobilities which have been discussed in terms of electron–phonon Fröhlich coupling^{11}, dipolar scattering arising from anharmonic halide motion^{12}, and polaronic transport^{13}.
A signature of strong anharmonicity in tetragonal or cubic perovskites (stoichiometry ABX_{3}) is the multiwell potential energy surface (PES), U, described by nuclei displacements, Δτ, away from their staticequilibrium positions (Fig. 1a). Staticequilibrium geometries occur when the net force on each atom vanishes, giving rise to local extrema in the PES. The highsymmetry idealized geometry, also referred to as monomorphous structure^{14}, corresponds to a local maximum; it features perfectly aligned octahedra and can be described by a reference unit cell composed of a few atoms. Local minima are explored when the nuclei move away from their highsymmetry positions, forming a locally disordered (or polymorphous) network characterized by tilted BX_{3} octahedra and a distorted configuration of the A cations (Fig. 1a). Description of this form of static or quasistatic (vide infra) disorder requires supercells that can accommodate symmetrybreaking domains between the repeated unit cells.
Typical density functional theory (DFT) calculations of tetragonal or cubic perovskites rely on the assumption of a highsymmetry network, disregarding the locally disordered ground state configurations. This assumption misses important corrections to the electronic structure^{14,15} and requires enforcing the crystal’s symmetries on anharmonic phonon dynamics^{16}, thus, represented by idealized welldefined dispersions. Such behavior is disconnected from measurements of overdamped optical vibrations, structural disorder, and complex pretransitional dynamics close to structural phase transitions^{17,18,19,20,21,22,23,24,25,26}. Furthermore, direct evidence of local disorder in cubic perovskites is observed in measurements of pair distribution functions (PDFs), Bragg diffraction, and extended diffuse scattering^{23,27,28,29,30,31}.
In this work we demonstrate the important role of anharmonicity and local disorder in the electronic structure, phonon dynamics, and electron–phonon coupling of oxide and halide perovskites (SrTiO_{3}, CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3}). Hybrid halide perovskites undergo additional relaxations related to molecular reorientations, but as a proof of concept we here focus on inorganic compounds. We show from firstprinciples that (i) local disorder and anharmonicity are at the origin of overdamped and strongly coupled phonons; (ii) local disorder and anharmonicity are essential to describe electron–phonon coupling; (iii) lowenergy anharmonic optical vibrations dominate thermal band gap renormalization; (iv) local disorder is the key to explain the smooth evolution of the band gap with temperature around phase transitions; (v) a full description of band gaps and effective masses requires combining disorder with fully relativistic effects. To address points (i)–(iv), we employ a recently developed approach, namely anharmonicity via the special displacement method (ASDM)^{32}, that allows the unified treatment of anharmonic electron–phonon coupling. Our study calls for revisiting open questions related to electron–phonon and anharmonic properties of halide and oxide perovskites.
Results
Lattice dynamics
We start the description of lattice dynamics with the harmonic approximation and take the expansion of a multiwell PES up to second order in atomic displacements to write:
U_{0} is the potential energy with the atoms clamped either at their highsymmetry or locally disordered configuration. This statistically disordered initial configuration can be obtained using a similar procedure (see “Methods”) to the one described in ref. ^{14}. Atomic displacements away from a PES extremum are represented by Δτ_{i} where i is a composite index for the atom, coordinate, and cell. The interatomic force constants (IFCs), defined as \({C}_{i,{i}^{{\prime} }}={\partial }^{2}U/\partial {\tau }_{i}\partial {\tau }_{{i}^{{\prime} }}\), are used to compute the phonons of the highsymmetry or locally disordered structures at 0 K, depending on the stationary point at which the second derivatives are evaluated for.
To incorporate anharmonicity in the lattice dynamics we employ the ASDM that combines the selfconsistent phonon theory developed by Hooton^{33} and the special displacement method (SDM) developed by Zacharias and Giustino (ZG)^{34,35}. In the ASDM, we fixed the nuclei in a supercell at positions determined by ZG displacements and evaluate the IFCs at temperature T as^{32}:
This procedure is performed iteratively until selfconsistency in the phonon spectra is achieved. The merit of the ASDM is that the ZG nuclei coordinates, {τ^{ZG}}, allow to explore automatically an effective temperaturedependent harmonic potential that best captures the solution of the nuclear Schrödinger equation. The anharmonic phonons can then be used to describe the crystal’s vibrational properties. The various schemes used to compute phonon dispersions in this work are described in Supplementary Table 2.
Electron–phonon renormalized observables
Here, we take the ASDM one step beyond and employ the selfconsistent {τ^{ZG}} for the nonperturbative evaluation of electron–phonon coupling in anharmonic systems. Following ref. ^{35}, the renormalization of any temperaturedependent property related to the electronic structure can be expressed as:
where x_{ν} represents the normal coordinate of the phonon, σ_{ν} is the associated meansquare displacement of the atoms, and ν is a composite index for the band and wavevector. The notation \({{{\mathcal{O}}}}({x}_{\nu }^{4})\) represents terms of fourth order and higher in x_{ν}. The ZG displacements derived from the ASDM define the optimum collection of coordinates within a supercell that allows to compute accurately Eq. (3), describing, at the same time, anharmonicity in the PES. In this work, we focus on electron–phonon renormalized band gaps of tetragonal or cubic perovskites, often described^{36} by a harmonic theory introduced by Allen and Heine^{37}. In this case, the derivatives \({\partial }^{2}{O}^{\{\tau \}}/\partial {x}_{\nu }^{2}\) involve linear and second order variations of the PES leading to the Fan–Migdal and Debye–Waller selfenergy corrections^{38}. Computing the renormalization starting from the locally disordered structure yields different results since (i) electron–phonon selfenergy corrections are evaluated for local minima in the PES instead of maxima (c.f. Fig. 1a), (ii) electron wavefunctions are modified, and (iii) phonon frequencies are renormalized.
Potentialwell depth and relation to local disorder
The depth of the potentialwell is, in principle, equivalent to the potential energy lowering obtained for the ground state structure and provides an indicator of the degree of anharmonicity and static disorder. In Table 1 we report the energy lowering (ΔU) and average BXB bond angle (\({\bar{\theta }}_{{{{\rm{BXB}}}}}\)) calculated for locally disordered cubic SrTiO_{3}, CsSnI_{3}, CsPbBr_{3}, and CsPbI_{3}. Our calculations are in good agreement with data reported in Refs. ^{14} and^{15} (see also Supplementary Table 1). In Fig. 1b, we plot the relationship between local disorder, represented by \({\bar{\theta }}_{{{{\rm{BXB}}}}}\), and the potentialwell depth. We find that halide perovskites exhibit a considerably higher degree of anharmonicity than SrTiO_{3} which reflects the larger disorder characterizing their ground state networks (realized schematically in Fig. 1c–f) and PDFs (Fig. 1g, h); this finding is connected to the much softer elastic shear modulus^{39,40} as well as the different ionicity and bonding nature of halide perovskites^{39,41}.
Impact of local disorder on phonons at 0 K
In Fig. 2a, b, we present phonon spectral functions computed for locally disordered SrTiO_{3} and CsPbI_{3}; for CsSnI_{3} and CsPbBr_{3} see Supplementary Fig. 1. We also include phonon dispersions (black) obtained for highsymmetry structures which display large instabilities represented by negative phonon frequencies. Importantly, allowing the systems to explore ground state disorder leads to dynamically stable phonons (color maps). In Fig. 2a, we observe band replicas of the Γ appearing at the R point and vice versa; these features arise from finite size effects and vanish with the supercell size (Supplementary Fig. 2). Local disorder in SrTiO_{3} also induces a large softening of the acoustic branch along RM with the frequency at R reaching as low as 2 meV.
Remarkably, polymorphism induces extensive broadening and nondispersive (flattened) optical bands which are overdamped across the reciprocal space of CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3} (Fig. 2b–d and Supplementary Fig. 1). Focusing in the frequency region below 4 meV (Fig. 2e), only the acoustic phonons around the Γpoint are clearly identified. This behavior is consistent with experiments performed on lead perovskites^{23,42}, suggesting that acoustic phonons emerge from a bath of dispersionless optical vibrations (Fig. 2f). Here, we propose a picture of strongly coupled optical vibrations instead of weaklyinteracting phonon quasiparticles, since momentum information on phonons is smeared out. In fact, local disorder, which is distinct from thermal disorder arising from vibrational fluctuations^{43}, is expected to reduce further the phonon correlation lengths and lifetimes of halide perovskites. Due to its low degree of local disorder, this behavior is not adopted by SrTiO_{3} which exhibits welldefined phonons in the spectral function (Fig. 2a). The extent of vibrational broadening and coupling is also interconnected with the deviation of the PDFs from the archetypal highsymmetry picture (Fig. 1g, h) and lattice softness^{40}. Furthermore, local disorder in halide perovskites causes the decrease in energy of optical vibrations, leading to a narrowing of the phonon dispersion and thereby to enhanced phonon bunching (Fig. 2b).
Figure 3a–c show the phonon spectral functions (color maps) of the three different structural phases of CsPbBr_{3} calculated using locally disordered networks. In each plot, we report the harmonic phonon dispersions of the monomorphous structures (black lines), the potential welldepth (ΔU), and the total energy difference with respect to the energy of the orthorhombic phase (ΔU_{ORH}). As expected, the level of phonon instabilities in the monomorphous networks are related to the depth of ΔU in each phase. As evidenced by the calculated ΔU_{ORH}, the locally disordered tetragonal and cubic CsPbBr_{3} lie higher in energy than their orthorhombic analog [Fig. 3d]. It is also apparent that local disorder in the orthorhombic structure has a negligible effect on the system’s total energy, yielding identical stable phonons with those obtained for the monomorphous phase. On the contrary, local disorder relatively to the cubic and tetragonal highsymmetry networks is much more prominent, increasing the coupling between individual bands and, hence, suggesting a further decrease in the lattice thermal conductivity of these phases^{9}.
Temperaturedependent phonon anharmonicity
Figure 4a, b show temperaturedependent phonon dispersions of the highsymmetry (black) SrTiO_{3} and CsPbBr_{3} calculated using the ASDM^{32}; for CsPbI_{3} and CsSnI_{3} see Supplementary Fig. 3. The phonon spectral functions (color maps) are obtained by combining the atomic positions of locally disordered networks with the IFCs obtained by ASDM (see also discussion around Supplementary Table 2). It can be seen for SrTiO_{3} that the spectral function follows closely the ASDM phonon dispersion. This observation aligns with a picture of noninteracting phonons at low temperatures and it is also related with the minimal level of local disorder in SrTiO_{3} reported in Table 1. On the contrary, symmetrybreaking in halide perovskites induces the coupling of lowenergy optical vibrations and the reduction of their coherence lengths. Moreover, accounting for temperaturedependent anharmonicity in our calculations via the ASDM reproduces the thermal vibrational softening along RM (Fig. 4b and Supplementary Fig. 3), consistent with previous calculations^{25,44}.
Diffuse scattering
In Fig. 4c–h, we present thermal diffuse scattering maps of SrTiO_{3} and CsPbBr_{3} at 300 K and 500 K; for CsPbI_{3} and CsSnI_{3} check Supplementary Fig. 4. We find that using the phonons obtained for the disordered networks (i.e spectral function in Fig. 2a) reproduces better the experimental maps^{45} of SrTiO_{3} in (Q_{x}, Q_{y}, 1/2) (Fig. 4e) and (Q_{x}, Q_{y}, 3/2) (Fig. 4f) reciprocal planes, where Q = (Q_{x}, Q_{y}, Q_{z}) is the scattering wavevector. Importantly, in the calculated maps we can identify the emergence of phononinduced scattering peaks at the Rpoints which correspond to the ultrasoft phonons discussed for Fig. 2a. These features are present in Xray diffuse scattering measurements of ref. ^{45} and attributed to dynamic disorder due to antiphase rotations of the octahedra, mimicked by the static distortions present in our disordered network. Note that diffuse scattering at R is absent when the highsymmetry structure with the ASDM phonons calculated for 2 × 2 × 2 supercells are combined (Supplementary Figs. 5 and 6).
At variance with SrTiO_{3}, the scattering maps computed for the highsymmetry CsPbBr_{3} at 500 K yield better agreement with measurements reported in ref. ^{25}. To illustrate this we perform calculations of scattering maps in the (Q_{x}, Q_{y}, 1/2) plane for two separate frequency ranges (Figs. 4g and 3h) using the ASDM phonons. Focusing on the scattering induced by ultraslow dynamics (<2.5 meV), the acoustic soft branch along RM leads to the formation of vertical and horizontal diffuse rods across several Brillouin zones, as observed in measurements for CsPbBr_{3} (Fig. 4i). We stress that local disorder induces a hardening of the modes along RM (Supplementary Fig. 1), and thus it prevents the formation of diffuse rods (Supplementary Fig. 7). This comes as no surprise since the disordered network should be regarded as a quasistatic approximation that cannot describe the ultraslow dynamical octahedral tilting^{25,26} and, thus, the relaxation of the system between various (deep) minima in the PES. Focusing on the scattering induced by lowenergy excitations (2.5–10 meV), we detect broad diffuse rods along MX (Fig. 4h) which are in close agreement with measurements in the range 2.0–10 meV (Fig. 4j). Now, employing the phonons of the locally disordered network perfectly reproduces the diffuse scattering maps (Supplementary Fig. 4), demonstrating that, unlike the ultraslow octahedra relaxations (<2.5 meV), the lowenergy vibrations (2.5–10 meV) are captured correctly.
Interestingly, in Fig. 4g and Supplementary Fig. 4, we can identify low intensity multiphonon scattering signatures at the Xpoints arising from the combined momenta of two phonons along MR. These fine structures are present in neutron scattering maps of CsPbBr_{3} (Fig. 4i), but not interpreted before. Our findings here suggest that lowenergy multiphonon excitations are another source of manifestation of anharmonicity in halide perovskites, emerging from highly anharmonic zoneedged modes.
Effect of disorder on anharmonic electron–phonon coupling
Figure 5a, b compare the electron spectral functions of locally disordered cubic SrTiO_{3} and CsPbI_{3} (color maps) with the band structures of their highsymmetry counterparts (black); for CsSnI_{3} and CsPbBr_{3} check Supplementary Fig. 8. The effect of symmetrybreaking domains on the electronic structure can be understood intuitively by inspection of the PDFs (Fig. 1g, h). It turns out that local disorder induces slight changes in the electronic structure of SrTiO_{3} with the main impact being on the band edges at Γ and R points (Fig. 5a). In particular, symmetry lowering induces a band gap opening of 0.24 eV which is in agreement with the value reported in ref. ^{15}.
Local disorder causes distinct changes on the electronic structure of cubic halide perovskites (Fig. 5b and Supplementary Fig. 8). Those are large band gap openings, band dispersion renormalization, and band broadening. The quantitative comparison between the band gaps calculated for the highsymmetry and disordered halide perovskites is provided in Table 1. Our calculations reveal a band gap enhancement due to local disorder of more than 0.50 eV for cubic CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3}, similarly to previous calculations^{14}. Owing to a higher degree of local disorder, represented by \({\bar{\theta }}_{{{{\rm{BXB}}}}}\) in Table 1, halide perovskites exhibit a larger band gap opening than SrTiO_{3}. Interestingly, this observation suggests an indirect relationship between anharmonicity and the band gap in perovskite systems. In fact, the connection between \({\bar{\theta }}_{{{{\rm{BXB}}}}}\) with the band gap is related to the changes in the overlap between the metal and halogen states^{41}. Moreover, our calculations show that local disorder in the tetragonal and orthorhombic phases of CsPbBr_{3} yields band gap enhancements of 0.17 and 0.001 eV. These values are much lower than the one obtained for the cubic phase (0.57 eV), in line with the potential welldepth of each phase (Fig. 3d). Our values for the Pbbased compounds in Table 1 show that spinorbit coupling (SOC) induces a giant gap closing of 1.1–1.2 eV, in agreement with ref. ^{46}. We find that SOC has also a strong influence on the effective mass enhancement due to local disorder (Supplementary Table 3). For instance, excluding SOC, local disorder leads to electron and hole mass enhancements λ (see “Methods”) between 0.4–1.2. When SOC is taken into account, the disordered networks yield λ of 1.3–2.3 for CsPbBr_{3} and 3.3–4.7 for CsPbI_{3}. We stress that the calculated effective masses of disordered CsPbBr_{3} and CsPbI_{3}, ranging between 0.13–0.20, compare well with 0.114 and 0.126 measured from interLandau level transitions in CsPbI_{3} and CsPbBr_{3}, respectively^{47}.
The impact of local disorder is clearly manifested in the electronic structure of CsSnI_{3} (Supplementary Fig. 8). The fully relativistic band structure of highsymmetry CsSnI_{3} exhibits an artifact in the conduction band minimum, resulting from the exchange of orbital character between the band edges. This suggests a metalliclike behavior for CsSnI_{3} (hence the value 0.27 eV in Table 1) and leads to unphysical negative electron effective masses at the Rpoint. Instead, accounting for local disorder recovers the standard picture of a parabolic conduction band minimum with positive effective masses of 0.08 and a direct gap of 0.28 eV at the Rpoint.
The fully relativistic DFT band gaps of disordered perovskites still largely underestimate the experimental values^{48,49,50,51}, reported in Table 1, by more than 1 eV, due to the DFT semilocal description of correlation effects. As shown in Table 1, this discrepancy is significantly alleviated when selfenergy corrections through the HSE and PBE0 hybrid functionals are accounted for.
In Fig. 5c, d, we compare the electronic structure around the band extrema of disordered CsPbBr_{3} without and with the effect of phononinduced zeropoint renormalization (ZPR)^{35}. Electron–phonon interactions, incorporated by ZG displacements in 2 × 2 × 2 supercells, induce a band gap opening, yielding a ZPR of 29 meV. Increasing the supercell size to 4 × 4 × 4 reverses the sign of the ZPR and yields a band gap decrease of 35 meV. We also comment that combining disorder with ZG displacements does not lead to an artificial Rashba–Dresselhaus splitting of the doubly degenerate band extrema, reflecting that perovskite crystals should maintain centrosymmetricity at thermal equilibrium^{52}.
Table 1 also reports the phononinduced band gap renormalization, ΔE_{g}(T), of cubic SrTiO_{3}, CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3} at 300, 430, 650, and 500 K, respectively, using 4 × 4 × 4 ZG supercells. It turns out that electron–phonon interactions at finite temperatures result in the closure of the band gap in SrTiO_{3}, whereas in halide perovskites, electron–phonon interactions cause the opening of the band gap. Accounting for local disorder in all compounds reduces ΔE_{g}(T) by 80–210 meV. This is related to the different potential experienced by electrons in the disordered network, affecting the electron–phonon matrix elements (see also discussion around Eq. (3)). Interestingly, for halide perovskites, we observe an almost linear correlation [33 meV/(^{∘})] between the reduction in ΔE_{g}(T) due to disorder and the decrease in \({\bar{\theta }}_{{{{\rm{BXB}}}}}\).
Figure 5e shows that experimental values lie within the range of our electron–phonon corrected HSE and PBE0 band gaps for all disordered compounds. In fact, by adding ΔE_{g}(T) to the average PBE0/HSE gap yields 3.53, 2.08, 1.64, and 1.10 eV for cubic SrTiO_{3}, CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3}, respectively, in good agreement with experiments. Our findings here suggest that accurate electronic structure calculations of cubic perovskites require the combined corrections due to local disorder, SOC, functionals beyond DFT^{53}, and electron–phonon coupling.
In Fig. 5f, g, we show the temperature dependence of the band gap renormalization evaluated for the highsymmetry (red) and locally disordered (blue) cubic SrTiO_{3} and CsPbBr_{3} using the ASDM. Our calculations for highsymmetry SrTiO_{3} underestimate experimental data (black) from ref. ^{48}. This underestimation is reduced when the disordered network is employed. In fact, electron–phonon coupling is strongly modified inducing a correction to the band gap closing of ~30%. This finding together with the computed diffuse scattering patterns further support the presence of local disorder in cubic SrTiO_{3}. As seen in Fig. 5g, using the disordered CsPbBr_{3} also provides an accurate description of the band gap renormalization, explaining the low variation of the experimental data with temperature^{50}. Our analysis (see “Methods”) yields that the lowenergy anharmonic optical vibrations dominate electron–phonon coupling in locally disordered halide perovskites, contributing 88% to the band gap renormalization, but strongly departing from the simplified picture of a Fröhlich interaction related to harmonic modes. This is consistent with photoluminescence spectra measurements of halide perovskite nanocrystals, which suggest a dominant (negligible) contribution of lowenergy optical vibrations (acoustic phonons) to exciton–phonon coupling^{54}. The band gap renormalization calculated using the highsymmetry structure is consistently 300% larger than experiment, fact that further casts doubt on the use of a fully ordered network in firstprinciples calculations of cubic halide perovskites. The remarkable success of the cubic polymorphs in describing electron–phonon coupling is explained by inspecting the electron lifetimes in halide perovskites (~4–6 fs)^{55,56} which are much smaller relatively to the period of atomic vibrations (>200 fs). Hence, anharmonic structural fluctuations look essentially static to the electrons which follow the nuclei in their most probable ground state configuration. We remark that polaronic effects on the band gap renormalization are not included in our calculations. Although it is now possible to combine ab initio polaron distortions^{57} with the ASDM, such calculations are rather challenging and beyond the scope of this work. We also note that corrections to the band gap renormalization coming from hybrid functionals are less than 1 meV (see “Methods”).
In Fig. 6, we show the band gap variation of CsPbBr_{3} with temperature calculated within ASDM using the cubic and tetragonal phases. Remarkably, accounting for local disorder in our anharmonic electron–phonon coupling calculations (blue) yields good agreement with experiment (gray) and captures the smooth variation of the measured band gap around the continuous phase transition temperature at 403 K. Instead, using the highsymmetry structures of the tetragonal and cubic phases (red), the band gap exhibits a spurious abrupt drop of ~0.4 eV, primarily caused by the enforced alignment of the octahedra in the cubic phase. A similar issue for the temperaturedependent band gaps computed for the highsymmetry CsPbI_{3} networks has been observed previously^{58}. The continuous change of the band gap from the disordered cubic to the disordered tetragonal phase, achieved here, is consistent with a secondorder displacive phase transition. We stress that for all computed values we account for the same PBE0 corrections to the band gap; that is we apply an identical shift to the temperaturedependent band gaps of both the tetragonal and cubic phases. In our calculations for the disordered structures we combined ASDM temperaturedependent IFCs with the disordered networks to generate ZG displacements; examples of the resulting phonon spectral functions are shown in Supplementary Fig. 17. Notably, using 0 K ground state phonons obtained for the disordered structures (Fig. 3b, c) yields a similar level of agreement (Supplementary Fig. 18). This demonstrates, essentially, that the vibrational dynamics computed for the locally disordered networks is a reasonable approximation to describe anharmonic electron–phonon effects originating from lowenergy optical vibrations.
Discussion
Taken all together, our work provides insights on the lattice dynamics and electron–phonon couplings in oxide and halide perovskites. Given the agreement with measurements of the vibrationalinduced band gap renormalization of SrTiO_{3} and CsPbBr_{3}, we expect our approach to be widely used for addressing open challenges related to important technological applications of halide and oxide perovskites, such as solar cells, light emitting diodes, and thermoelectric devices, as well as elucidating their unexplored ultrafast spectroscopic properties^{59}. Our results demonstrate that SrTiO_{3} is fully compatible with a static disordered network, while CsPbBr_{3} is better described within a quasistatic picture that captures correctly optical vibrations but not effects arising from ultraslow relaxational rotations of the BX_{3} octahedra. Our findings also confirm that vibrational dynamics in halide perovskites deviate from a textbook noninteracting phonon dispersion; a picture of strongly coupled vibrations should be considered as a precursor to future calculations of perovskites’ peculiar transport properties^{10,11}. In the context of electron–phonon coupling, the description of tetragonal or cubic perovskites with a locally disordered network constitutes the best possible approximation since (i) electron coupling to anharmonic optical vibrations is predominant, (ii) dynamical structural fluctuations look essentially static to the shortlived electrons, and (iii) electrons mostly see the nuclei as fixed in their disordered ground state. This latter point also allows to explain the continuous variation of the band gap around phase transitions in halide perovskites, reflecting the subtle rearrangement of atomic positions between different phases. Some of the important physics of halide perovskites uncovered here are intrinsically related to their extraordinary lattice softness^{40}, and, thus, their remarkable ability to sustain a high degree of disorder. At a fundamental level, our study proposes a radically different way of conceptualize the lattice dynamics in perovskites and sets up a universal framework for accurate simulations of their carrier mobilities, conductivities, excitonic spectra, nonequilibrium dynamics, and polaron physics^{60,61,62}.
Methods
Electronic structure calculations
Electronic structure calculations for SrTiO_{3}, CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3} were performed within density functional theory using plane waves basis sets as implemented in Quantum Espresso (QE)^{63,64}. We employed a kinetic energy cutoff of 120 Ry, the Perdew–Burke–Ernzerhof exchange–correlation functional revised for solids (PBEsol)^{65}, and optimized normconserving Vanderbilt pseudopotentials from the PseudoDojo library^{66,67}. To account for the effect of SOC on the electronic structures we replaced scalar relativistic with fully relativistic pseudopotentials. The uniform sampling of the Brillouin zone of the cubic 1 × 1 × 1, 2 × 2 × 2, 4 × 4 × 4, and 6 × 6 × 6 supercells was performed using 6 × 6 × 6, 3 × 3 × 3, 1 × 1 × 1, and 1 × 1 × 1 kgrids, respectively. The only exception is that for 4 × 4 × 4 supercells of SrTiO_{3} we employed a 2 × 2 × 2 kgrid. Furthermore, for 6 × 6 × 6 supercells we reduced the kinetic energy cutoff to 100 Ry. Initial calculations of the monomorphous structures were performed in the unit cells of the cubic (5 atoms), tetragonal (10 atoms), and orthorhombic (20 atoms) perovskite compounds with the nuclei clamped at their Wyckoff positions (space groups: \(Pm\bar{3}m\) for all cubic perovskites, I4/mcm for tetragonal SrTiO_{3}, P4/mbm for tetragonal CsPbBr_{3}, and Pbnm for orthorhombic CsPbBr_{3}). The lattice constants of cubic SrTiO_{3}, CsPbBr_{3}, CsPbI_{3}, and CsSnI_{3} were fixed to the DFTPBEsol optimized values of 3.889, 5.874, 6.251, and 6.141 Å, respectively. The lattice constants of the tetragonal and orthorhombic CsPbBr_{3} were also fixed to the DFTPBEsol optimized values of (a = b = 5.734, c = 5.963 Å) and (a = 7.971, b = 8.397, and c = 11.640 Å). The lattice constants of the tetragonal SrTiO_{3} were fixed to the experimental lattice constants^{68} of (a = b = 3.896 and c = 3.900 Å) since were found to yield better phonon frequency renormalizations due to anharmonic effects at finite temperatures^{32}. The eigenmodes and eigenfrequencies at each phonon wavevector q were obtained by evaluating the IFCs and corresponding dynamical matrices via the frozenphonon method^{69,70}. Corrections on the phonon dispersions due to longrange dipole–dipole interactions, which vary depending on the degree of staticdisorder^{32}, were included via the linear response approach described in ref. ^{71}.
Electron spectral functions of the disordered cubic structures were calculated using the electron band structure unfolding technique as implemented in the EPW/ZG code^{35,57}. We ran calculations with and without SOC (see Supplementary Figs. 8 and 9) and sampled the Brillouin zone with 417 equallyspaced kpoints along the XRMΓR path. We remark that when SOC is excluded, local distortions in a 2 × 2 × 2 supercell of SrTiO_{3} lead to an artificial degeneracy splitting of 40 and 60 meV of the triply degenerate valence band top and conduction band bottom, respectively. Interestingly, our calculations for a 4 × 4 × 4 supercell show that the splitting in the valence band top is eliminated, while in the conduction band bottom is maintained. This result is consistent with a disordered network that macroscopically might reflect some of the crystal’s symmetries, although local deformations are present. Inclusion of fully relativistic effects in our calculations for SrTiO_{3} induces a small band gap change and spinorbit splitting of the band edges, as shown in Table 1 and Supplementary Fig. 9. Our fully relativistic calculations for the band gaps of the highsymmetry and locally disordered tetragonal CsPbBr_{3} within DFTPBEsol yield 0.69 and 0.86 eV, respectively. Ignoring SOC effects our calculations determine 1.65 and 1.83 eV for the highsymmetry and locally disordered tetragonal CsPbBr_{3}. Data calculated for all cubic compounds are provided in Table 1.
To extract effective masses m^{*} at the band edges we performed parabolic fits to the electron band structure and spectral functions along the specified directions reported in Supplementary Table 3. The mass enhancement due to disorder, λ, was obtained from \({m}_{{{{\rm{d}}}}}^{* }=(1+\lambda )\,{m}_{{{{\rm{hs}}}}}^{* }\), where \({m}_{{{{\rm{d}}}}}^{* }\) and \({m}_{{{{\rm{hs}}}}}^{* }\) are the disordered and highsymmetry structure’s effective masses. We note that at the proximity of the band edges, the electron spectral functions give welldefined bands that do not suffer from band broadening.
All calculations employing the Perdew–Burke–Ernzerhof (PBE0)^{72} and Heyd–Scuseria–Ernzerhof (HSE06)^{73} hybrid functionals were performed using the code VASP^{74}. SOC was taken into account, and a 300 eV cutoff energy was set for the projectoraugmented wave^{75}. For the highsymmetry and 2 × 2 × 2 supercell disordered structures we employed Γcentered kgrids of 4 × 4 × 4 and 2 × 2 × 2, respectively. The HSE06 (PBE0) band gaps with SOC of the highsymmetry and locally disordered tetragonal CsPbBr_{3} are found to be 1.20 eV (1.80 eV) and 1.69 eV (2.31 eV), respectively. The corresponding values for all cubic compounds are reported in Table 1.
The locally disordered (polymorphous) network
To explore, initially, the locally disordered network of cubic SrTiO_{3} we applied three different initial sets of displacements on the atoms of the highsymmetry structure in a 2 × 2 × 2 supercell. Those are: (i) ZG displacements along all phonon modes populated at T = 0 K, (ii) ZG displacements along the soft modes populated at T = 0 K and, (iii) random displacements smaller than 0.1 Å applied to all atomic coordinates. We note that ZG displacements along imaginary soft modes were generated by switching their frequencies to real. Each case was followed by a DFTPBEsol relaxation of the nuclei. Although three different ground state disordered geometries were realized, a consistent energy lowering of 8 meV [f.u.]^{−1} relative to the ordered structure was obtained. Starting from random initial displacements in a 4 × 4 × 4 supercell, the relaxation yields the same energy lowering of 8 meV [f.u.]^{−1} in good agreement with the value of 12 meV [f.u.]^{−1} reported in ref. ^{15}. In Supplementary Fig. 10, we demonstrate that the three disordered structures give identical PDFs. In Supplementary Table 1, we show that accounting for ground state symmetrybreaking domains in 4 × 4 × 4 supercells yields similar band gap openings with 2 × 2 × 2 supercells.
Having demonstrated the equivalence of the three disordered structures, which give the same ground state energy and PDF, now we comment on the best choice of initial displacements [see (i)–(iii) above]. The most computationally inefficient choice is the use of ZG displacements along all phonon modes, bringing the initial configuration well away from its ground state. Instead, the relaxation converges much faster when ZG displacements along the soft modes are used which reproduce the tilting of the octahedra and thus bring the structure closer to the bottom of the potential well^{76}. Using random nudges, the efficiency of generating the ground state network might vary depending on the system and the amplitude of the initial displacement. Therefore, for generating all locally disordered structures, we chose to apply special displacements^{34,35} along the soft modes (computed for the highsymmetry structures) which is the most practical and systematic way to achieve ground state optimization.
The locally disordered (polymorphous) networks of cubic, tetragonal, and orthorhombic phases were explored (unless specified otherwise) by employing 2 × 2 × 2 supercells containing 40, 80, and 160 atoms, respectively. To check whether the locally disordered cubic geometries exhibit any residual symmetries, we perform a symmetry analysis with pymatgen^{77} using a tolerance factor of 0.0001 Å; we confirm that none of the locally disordered cubic structures maintain residual symmetries.
Special displacement method
ZG displacements were generated via the special displacement method^{34,35} (SDM) as implemented in the EPW/ZG code. We used phonons at qpoints commensurate with the supercell size and applied a smooth phase evolution of the phonon eigenvectors in reciprocal space.
Anharmonicity in our calculations was included via the ASDM using 2 × 2 × 2 supercells as described in ref. ^{32}. Selfconsistency in the phonon spectra of each system was achieved using only 34 iterations by means of a linear mixing scheme. To incorporate the effect of anharmonicity in the phononinduced band gap renormalization, we generated ZG displacements in 4 × 4 × 4 (all cubic perovskites), 6 × 6 × 6 (cubic SrTiO_{3}), 4 × 2 × 4 (tetragonal CsPbBr_{3}), and 6 × 4 × 4 (tetragonal SrTiO_{3}) supercells employing the IFCs obtained by ASDM. In all calculations of temperaturedependent band gaps reported in Fig. 5f, g, we allowed the lattice to expand according to the measured expansion coefficient^{51,78}. Symmetry breaking in ZG configurations led to an artificial degeneracy splitting of the band edges of cubic and tetragonal SrTiO_{3}. In this case, the band gap renormalization of SrTiO_{3} at each temperature was evaluated by averaging the energy change of all states participating in the formation of the band edges within an energy window of 20 meV. To ensure high accuracy and limit the errors arising from artificial degeneracy splitting, we also took the average over the band gap renormalization obtained for four different ZG configurations. In Supplementary Fig. 11, we show that the band gap renormalization of cubic SrTiO_{3} remains nearly the same when SDM is combined with 0 K ground state phonons obtained for the disordered structure.
To identify the contribution of ultraslow acoustic (E < 2.5 meV) and lowenergy optical vibrations (3.65 < E < 10 meV) to the band gap renormalization of CsPbBr_{3}, we applied ZG displacements on the nuclei along only the phonons lying within the associated energy windows. We note that using phonons with energies E > 2.5 meV and 2.5 < E < 10 meV we obtain a similar band gap renormalization (within 25 meV) which demonstrates, essentially, that highenergy optical vibrations (E > 10 meV) do not play an important role in the electron–phonon gap renormalization of halide perovskites.
All ASDM calculations for determining the phononinduced band gap renormalization were performed at the DFTPBEsol level. Corrections to the band gap renormalization arising from hybrid functionals were found to be negligible. In particular, our calculations in 2 × 2 × 2 ZG supercells of disordered CsPbBr_{3} yield a ZPR of 29.4, 29.0, and 29.9 meV for DFTPBEsol, HSE, and PBE0 functionals, respectively.
Phonon unfolding
For systems undergoing static symmetry breaking due to lattice distortion coming, e.g., from defects, atomic disorder, or a charge density wave, a supercell is required to compute the phonons. In this case, the crystal’s symmetry operations (translations and rotations) are no longer applicable and all atoms in the supercell need to be displaced for calculating the dynamical matrix and, hence, the renormalized phonon frequencies ω_{Qμ}, where Q and μ are the phonon wavevector and band indices. To illustrate the effect of lattice distortion in the phonons, a common practise is to employ phonon unfolding and evaluate the momentumresolved spectral function given by^{79}:
Here q denotes a wavevector in the Brillouin zone of the unit cell and P_{Qμ,q} represents the spectral weights which are evaluated in the spectral representation of the singleparticle Green’s function as^{80}:
where j is an index for the reciprocal lattice vectors g of the unit cell Brillouin zone, α denotes a Cartesian direction, and κ is the atom index. The symbol ~ indicates quantities calculated using the disordered structure. N_{g} acts as a normalization factor representing the total number of reciprocal lattice vectors entering the summation. The spectral weight can be understood, essentially, as the projection of the phonon eigenvector \({\tilde{e}}_{\alpha \kappa ,\mu }({{{\bf{Q}}}})\) on the phonon eigenvectors e_{ακ,ν}(q) computed in the unit cell, given that Q unfolds into q via Q = q + g_{j} − G, where G is a reciprocal lattice vector of the distorted structure.
To generate vibrational spectral functions we employed Eq. (5) and 417, 382, and 399 equallyspaced qpoints along the XRMΓR (cubic), XAMΓA (tetragonal), and XRSΓR (orthorhombic) paths. Convergence of the spectral weights was ensured by using a 10 × 10 × 10 ggrid of reciprocal lattice vectors. In Supplementary Figs. 12 and 13, we demonstrate our implementation of phonon unfolding by comparing phonon spectral functions computed for 2 × 2 × 2 and 4 × 4 × 4 supercells of SrTiO_{3}. Our implementation of phonon unfolding is available in the EPW/ZG tree. In Supplementary Fig. 14, we also show that the vibrational spectrum of CsPbBr_{3} remains almost identical when two different ground state disordered geometries are considered.
We note that overdamped unfolded phonon spectra of halide perovskites has also been revealed by analysis of velocity autocorrelation functions obtained by molecular dynamics simulations^{43}.
Diffuse scattering
Allphonon diffuse scattering maps were calculated within the LavalBornJames (LBJ) theory using disca.x of the EPW/ZG code^{81,82}. The merit of the LBJ theory is that inelastic scattering arising from onephonon and multiphonon processes is accounted for on the same basis. The DebyeWaller and phononic factors entering LBJ theory [Eq. (1) of ref. ^{81}] were evaluated for a 16 × 16 × 16 qgrid. The phonon eigenmodes and frequencies were obtained by means of Fourier interpolation of the dynamical matrices computed for 2 × 2 × 2 supercells using either the ASDM or the disordered network. A 16 × 16 × 1 uniform Qgrid (scattering wavevectors) per Brillouin zone was used to calculate the phononinduced scattering intensity in the reciprocal lattice planes perpendicular to one Cartesian axis. The atomic scattering amplitudes were determined as a sum of Gaussians with the parameters taken from ref. ^{83}. The diffuse scattering maps of CsPbBr_{3} for ultraslow acoustic and lowenergy phonon dynamics were determined by excluding the modes outside the associated energy windows. In addition to the scattering maps of cubic SrTiO_{3} presented in Fig. 4, we also calculated diffuse scattering in the (Q_{x}, Q_{y}, 0) and (Q_{x}, Q_{y}, 1) planes and found qualitative agreement with measurements of ref. ^{45} (Supplementary Figs. 5 and 6). In Supplementary Figs. 15 and 16, we show the decomposition of the allphonon scattering in the (Q_{x}, Q_{y}, 0) planes of cubic SrTiO_{3} and CsPbBr_{3} into onephonon and multiphonon processes for a large range of scattering wavevectors.
Data availability
The calculations (input and output files) employed for this study are available via the NOMAD archive [https://doi.org/10.17172/NOMAD/2023.07.111 for anharmonic electron–phonon coupling calculations and https://doi.org/10.17172/NOMAD/2023.05.131 for anharmonic phonon calculations], or upon request from the corresponding author.
Code availability
QUANTUM ESPRESSO is available under GNU General Public Licence from the QUANTUM ESPRESSO web site (https://www.quantumespresso.org/). The ZG module of EPW employed for the treatment of local disorder, anharmonicity, and generation of anharmonic selfconsistent special displacements is also available at GitLab (https://gitlab.com/epwcode/qe/tree/ZG).
Change history
04 September 2023
A Correction to this paper has been published: https://doi.org/10.1038/s4152402301119z
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Acknowledgements
We thank M. Kopecký and J. Fábry for kindly sharing experimental data of SrTiO_{3} diffuse scattering maps. We also thank D. R. Ceratti for graciously providing data on temperaturedependent band gaps of CsPbBr_{3} single crystals. M.Z. acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłodowskaCurie Grant Agreement No. 899546. This research was also funded by the European Union (project ULTRA2DPK/HORIZONMSCA2022PF01 / Grant Agreement No. 101106654). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the granting authority can be held responsible for them. J.E. acknowledges financial support from the Institut Universitaire de France. F.G. was supported by the National Science Foundation under CSSI Grant No. 2103991 and DMREF Grant No. 2119555. The work at institute FOTON and ISCR was supported by the European Union’s Horizon 2020 research and innovation program under grant agreement 861985 (PeroCUBE) and grant agreement 899141 (PoLLoC). G.V. acknowledges support from the Agence Nationale pour la Recherche through the CPJ program. We acknowledge that the results of this research have been achieved using the DECI resource Prometheus at CYFRONET in Poland [https://www.cyfronet.pl/] with support from the PRACE aisbl and HPC resources from the Texas Advanced Computing Center (TACC) at The University of Texas at Austin [http://www.tacc.utexas.edu].
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M.Z. performed the computational study and wrote the first draft of the manuscript. G.V. performed the hybrid functional calculations. All authors participated in discussions and the final preparation of the manuscript.
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Zacharias, M., Volonakis, G., Giustino, F. et al. Anharmonic electronphonon coupling in ultrasoft and locally disordered perovskites. npj Comput Mater 9, 153 (2023). https://doi.org/10.1038/s41524023010892
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DOI: https://doi.org/10.1038/s41524023010892
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