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Convergence of lattice approximations and infinite volume limit in the (λφ 4 −σφ 2 −μφ) 3 field theory

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Yong Moon Park; Convergence of lattice approximations and infinite volume limit in the (λφ 4 −σφ 2 −μφ) 3 field theory. J. Math. Phys. 1 March 1977; 18 (3): 354–366. https://doi.org/10.1063/1.523277

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By unified method we prove the convergence of the lattice approximation of the (λφ 4 −σφ 2 −ψφ) 3 field model with periodic, Dirichlet and Neumann boundary conditions in a finite box. This then allows us to take the inifinite volume limit of the Dirichlet states by the Nelson’s monotonicity argument. The model under consideration satisfies all the Wightman axioms except possibly the uniqueness of vacuum for μ=0 and the mass gap.

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• Published: 17 May 2024

Disentangling the effects of structure and lone-pair electrons in the lattice dynamics of halide perovskites

• Sebastián Caicedo-Dávila   ORCID: orcid.org/0000-0001-5135-2979 1 ,
• Adi Cohen 2 ,
• Silvia G. Motti 3 , 4 ,
• Masahiko Isobe   ORCID: orcid.org/0000-0002-9253-7256 5 ,
• Kyle M. McCall   ORCID: orcid.org/0000-0001-8628-3811 6 , 7   nAff10 ,
• Manuel Grumet 1 ,
• Maksym V. Kovalenko   ORCID: orcid.org/0000-0002-6396-8938 6 , 7 ,
• Omer Yaffe 2 ,
• Laura M. Herz   ORCID: orcid.org/0000-0001-9621-334X 3 , 8 ,
• Douglas H. Fabini   ORCID: orcid.org/0000-0002-2391-1507 5 , 9 &
• David A. Egger   ORCID: orcid.org/0000-0001-8424-902X 1  

Nature Communications volume  15 , Article number:  4184 ( 2024 ) Cite this article

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• Materials chemistry
• Materials for energy and catalysis

Halide perovskites show great optoelectronic performance, but their favorable properties are paired with unusually strong anharmonicity. It was proposed that this combination derives from the n s 2 electron configuration of octahedral cations and associated pseudo-Jahn–Teller effect. We show that such cations are not a prerequisite for the strong anharmonicity and low-energy lattice dynamics encountered in these materials. We combine X-ray diffraction, infrared and Raman spectroscopies, and molecular dynamics to contrast the lattice dynamics of CsSrBr 3 with those of CsPbBr 3 , two compounds that are structurally similar but with the former lacking n s 2 cations with the propensity to form electron lone pairs. We exploit low-frequency diffusive Raman scattering, nominally symmetry-forbidden in the cubic phase, as a fingerprint of anharmonicity and reveal that low-frequency tilting occurs irrespective of octahedral cation electron configuration. This highlights the role of structure in perovskite lattice dynamics, providing design rules for the emerging class of soft perovskite semiconductors.

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Introduction.

Halide perovskites (HaPs) with formula AMX 3 generated enormous research interest because of their outstanding performance in optoelectronic devices, most notably in efficient solar cells 1 , 2 , 3 . These compounds are highly unusual among the established semiconductors because they feature an intriguing combination of properties. Strong anharmonic fluctuations 4 , 5 , 6 in these soft materials appear together with optoelectronic characteristics that are favorable for technological applications 7 , 8 . This confluence raised puzzling questions regarding the microscopic characteristics of the materials and the compositional tuning of their properties alike. On the one hand, the soft anharmonic nature of the HaP structure may be beneficial in self-healing mechanisms of the material 9 , 10 , 11 , allowing for low-energy synthesis routes in their fabrication. On the other hand, pairing of anharmonic fluctuations and optoelectronic processes for key quantities of HaPs, e.g ., band gaps 12 , 13 , 14 , 15 , optical absorption profiles 16 , 17 , 18 , and charge-carrier mobilities 8 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , exposed incomplete microscopic rationales for the fundamental physical processes involved in solar-energy conversion. Established materials design rules are now being challenged by these observations, opening a gap in our protocols for making improved compounds.

Significant efforts are now underway to discern the chemical effects giving rise to these remarkable properties of HaPs. Because lattice dynamical and optoelectronic properties appear both to be special and coupled in unusual ways, a common origin in chemical bonding could underlie these phenomena. In this context, an interesting chemical feature is that the octahedral cations in these compounds often bear an n s 2 electron configuration ( e.g ., Pb 2+ with configuration [Xe]6 s 2 ), which is not present in many other semiconductors 26 . This particular aspect of HaPs leads to a strong or weak pseudo-Jahn–Teller (PJT) effect 27 , 28 , 29 , depending on the particulars of cation and anion composition and chemical pressure. The weak case influences local structure 30 , 31 , 32 , lattice dynamics 33 and ionic dielectric responses 26 , 31 , 34 , 35 , while the strong case additionally results in the formation of a stereochemically-expressed electron lone pair and impacts average crystal structures 33 , 36 , 37 , 38 . The weak PJT effect associated with 6 s 2 Pb 2+ coordinated by heavy halides plays a role in optoelectronic properties of these materials: its influence on the dielectric function can modify the Coulomb screening that is relevant for small exciton binding energies, reduced recombination rates and other key properties of HaPs 39 , 40 .

Confluences of the propensity for lone-pair formation with structural and lattice-dynamical properties were investigated in previous work exploring the chemical space of HaPs. Gao et al. 33 found an inverse relationship between the Goldschmidt tolerance factor, t 41 , and anharmonic octahedral tilting motions. Similarly, Huang et al. varied the A-site cation to explore interrelations of chemical, structural, and dynamical effects in HaPs 35 , reporting t -induced modulations of octahedral tiltings and lone-pair stereoactivity. A recent study by several of the present authors found that Cs 2 AgBiBr 6 lacks some expressions of lattice anharmonicity found in other HaP variants 42 . Because every other octahedral cation (Ag + , 4 d 10 ) cannot form a lone pair in this compound, this raised the possibility that changing the electron configuration of the cations may also suppress certain aspects of the lattice dynamics in HaPs. Taken together, previous work assigned a central role of the n s 2 electron configuration and associated PJT effect in the anharmonic lattice dynamics of HaPs in addition to their established effect on the electronic structure and dielectric screening. However, exploring the chemical space of HaPs in this way simultaneously changes their structures. Therefore, isolating the convoluted occurrences of cation lone-pair formation propensity and purely structurally-determined changes in the lattice dynamics of HaPs remained challenging, making an assessment of the precise impact of chemical bonding on anharmonicity in these soft semiconductors largely inaccessible.

Here, we address this issue and show that an n s 2 cation compatible with lone-pair formation is not required for the strong anharmonicity in the low-energy lattice dynamics of soft HaP semiconductors. We disentangle structural and chemical effects in the lattice dynamics of HaPs by comparing the well-known CsPbBr 3 with the far less studied CsSrBr 3 . Both exhibit almost identical geometrical and structural parameters, but CsSrBr 3 exhibits a negligible PJT effect on the octahedral Sr 2+ site, owing to substantially weaker vibronic coupling to degenerate excited states than in the Pb 2+ case (see Supplementary Note  1) , allowing separation of the effects of the n s 2 electron configuration and the geometry on the lattice dynamics in a direct manner. Combining electronic structure and molecular dynamics (MD) calculations with X-ray diffraction (XRD), infrared (IR) and Raman spectroscopies, we assess a key fingerprint of vibrational anharmonicity, i.e ., the Raman central peak, which is a broad peak towards zero frequency in the Raman spectrum resulting from diffusive inelastic scattering 26 , 33 , 34 , 35 , 43 , 44 , 45 . While the electronic structure and dielectric properties of CsPbBr 3 and CsSrBr 3 are very different, their vibrational anharmonicities are found to be remarkably similar. In particular, the crucial dynamic octahedral tiltings giving rise to the Raman central peak are still present even in the absence of n s 2 octahedral cations in CsSrBr 3 . Our results provide microscopic understanding of precisely how the propensity for lone-pair formation influences the anharmonic octahedral tiltings that dynamically break the average cubic symmetry in both compounds, and rule out the weak PJT effect associated with the n s 2 main-group cations as the sole reason for the appearance of such anharmonicity in soft HaPs. These findings are important for chemical tuning of HaPs needed for new materials design.

Electronic structure and bonding

We first investigate the electronic structure and bonding of CsPbBr 3 and CsSrBr 3 using density-functional theory (DFT). Figure  1 shows their band structure, total and projected density of states (DOS), as well as the total and projected crystal-orbital Hamilton population (COHP) of the high-temperature cubic phases of CsPbBr 3 and CsSrBr 3 . The electronic band structure and bonding of CsPbBr 3 were extensively investigated before 46 : the conduction band minimum (CBM) is formed by anti-bonding interactions (positive COHP in Fig.  1 c) between Pb-6 p and Br-4 p /Br-4 s orbitals, while the valence band maximum (VBM) is formed by anti-bonding interactions between Br-4 p and Pb-6 s orbitals.

DFT-computed electronic band structure of cubic CsPbBr 3 a and corresponding total and projected density of states (DOS, b ) and crystal-orbital Hamilton population (COHP, c ). Panels d – f show the same data for CsSrBr 3 .

The electronic structure of CsSrBr 3 exhibits entirely different characteristics 26 , 47 , especially a much larger band gap and weaker covalent interactions. Notably, the magnitude of the COHP is significantly reduced with respect to that of CsPbBr 3 , indicating much greater ionicity, and the COHP is almost entirely recovered by interactions between Cs and Br. Importantly, all bands derived from antibonding interactions between Sr-5 s and Br-4 p /Br-4 s are empty due to the electron configuration of Sr 2+ ([Kr]), and there is no potential for lone pair formation on Sr 2+ . A manifestation of the lack of n s 2 cations in CsSrBr 3 is that there is no cross-gap hybridization of the halide valence orbitals. By contrast, Br-4 p orbitals hybridize with Pb-6 p across the gap of CsPbBr 3 (see the pCOHP in Fig.  1 c). This leads to large Born effective charges, i.e ., large changes in the macroscopic polarization upon ionic displacements 48 , 49 , 50 , 51 reported in Table  1 , which for CsPbBr 3 are more than double the formal charge of Pb (+2) and Br (-1) and much larger than the corresponding values for CsSrBr 3 . Similarly, there is also a larger electronic contribution to the dielectric response in CsPbBr 3 and it features a larger value of the dielectric function at the high-frequency limit ( ε ∞ ) compared to CsSrBr 3 .

Structural properties and phase transitions

In spite of the markedly different electronic structure and bonding characteristics, CsSrBr 3 and CsPbBr 3 exhibit the same high-temperature cubic crystal structure ( \(Pm\bar{3}m\) ) and very similar lattice parameters (see Supplementary Note  3) . One can rationalize this through the nearly identical ionic radii of Pb 2+ and Sr 2+ (119 and 118 pm) and the resulting Goldschmidt factors for the compounds (0.862 and 0.865). Furthermore, both materials exhibit the same sequence of structural phase transitions from the high-temperature cubic to the low-temperature orthorhombic phase (with an intermediate tetragonal phase), as shown by temperature-dependent lattice parameters in Fig.  2 that were determined via XRD. The cubic-to-tetragonal phase transition temperature of CsSrBr 3 (~520 K) is noticeably higher than that of CsPbBr 3 (~400 K) 52 , 53 and slightly higher (~10 K) than that reported for Eu-doped CsSrBr 3 :Eu 5% 54 . The volumetric thermal expansion coefficient ( α V ) of CsSrBr 3 ( ~ 1.32 × 10 −4  K −1 at 300 K) is large and similar to that of CsPbBr 3 ( ~ 1.29 × 10 −4  K −1 , see the Supplementary Note  3) , in good agreement with the one reported for CsSrBr 3 :Eu 54 . Just as for other inorganic HaPs, α V of CsSrBr 3 slightly decreases with temperature 55 , 56 . The similarity of geometric factors and structural phase transitions suggests that the octahedral tilting dynamics in CsSrBr 3 might be similar to those in CsPbBr 3 , which contrasts with their markedly different electronic structure, and prompts a deeper investigation of the impact of the n s 2 cations on structural dynamics.

Temperature-dependent lattice parameters of CsPbBr 3 a and CsSrBr 3 b determined by XRD throughout the orthorhombic—tetragonal—cubic phases. We show reduced lattice parameters \(\tilde{a}\) , \(\tilde{b}\) and \(\tilde{c}\) for better visualization, with the orthorhombic phase expressed in the P b n m setting. Dashed vertical lines indicate phase-transition temperatures. Error bars from Pawley fitting are smaller than the markers and are omitted.

Lower-temperature lattice dynamics

We conduct IR and Raman spectroscopy at different temperatures as well as DFT-based harmonic-phonon calculations. The measured IR spectra show that the dominant CsSrBr 3 features are blue-shifted compared to those of CsPbBr 3 (see Fig.  3 a). Indeed, our DFT calculations of IR activities find a significant softening of the infrared-active TO modes in CsPbBr 3 compared to those in CsSrBr 3 (see Fig.  3 b): the most prominent IR-active TO mode in CsPbBr 3 and CsSrBr 3 appears at ~68 and 146 cm −1 , respectively, corresponding to the same irreducible representation ( B 3u ) with similar eigenvectors (see Supplementary Fig.  9) in each system. This is in line with the theory of weak PJT effects in general 29 and expectations for lone pairs in particular, with significant softening of ungerade modes in CsPbBr 3 that would correspond to lone-pair formation in the strong PJT case relative to those in CsSrBr 3 . Notably, this softening is primarily driven by differences in bonding rather than the difference in the atomic masses (see Supplementary Note  7) .

a IR-reflectivity spectra (dashed curves) and fitted imaginary part of the dielectric function (solid curves, see Supplementary Note  4 for details) of CsPbBr 3 and CsSrBr 3 measured at room temperature. b DFT-calculated IR-absorption spectra within the harmonic approximation for the orthorhombic phases. c Raman spectra of orthorhombic CsPbBr 3 and CsSrBr 3 measured at 80 K. d DFT-calculated Raman spectra of both compounds within the harmonic approximation for the orthorhombic phases.

Moreover, the LO/TO splitting is enhanced in CsPbBr 3 compared to in CsSrBr 3 and the LO phonon modes are hardened. Related to this, the CsPbBr 3 IR spectrum exhibits a broad feature which is known as the Reststrahlen band as has been reported before for MA-based HaPs 57 . This particular effect results in near-zero transmission through the material in a frequency range between the TO and LO modes, represented by high IR intensity values, and occurs in polar materials with larger Born-effective charges. Because the TO modes are softened and the LO modes hardened in CsPbBr 3 compared to CsSrBr 3 , and because the latter is less polar ( cf . Table  1 }, the absence of the n s 2 cations leads to a much less pronounced, blue-shifted Reststrahlen band appearing in a smaller frequency window in CsSrBr 3 (see Fig.  3 a, and Supplementary Note  4) .

Figure  3 c shows the 80 K Raman spectra of CsPbBr 3 and CsSrBr 3 , which are in good agreement with the Raman activities calculated for harmonic phonons (Fig.  3 d). Specifically, the experimental spectrum of CsPbBr 3 in Fig.  3 b finds three broader features at frequencies below and one weaker-intensity feature at frequencies above 100 cm −1 . Conversely, CsSrBr 3 exhibits a structured feature around 50 cm −1 , a pronounced signal close to 100 cm −1 , and then a series of weaker intensities between 100 and 150 cm −1 . While the DFT-computed Raman activities calculated in the harmonic approximation are in broad agreement with these findings (see Fig.  3 d), we note a slightly larger deviation of approximately 20 cm −1 for the higher-frequency peak in CsPbBr 3 . These findings lead us to conclude that unlike in IR, the Raman spectrum of CsSrBr 3 exhibits no substantial energy shifts with respect to CsPbBr 3 . Computing the phonon DOS for the orthorhombic phase of both compounds with DFT (see Supplementary Fig.  8) , we find that they exhibit similar phonon DOS below 100 cm −1 , i.e ., in the region of most of the Raman-active modes. The similar phonon DOS and the contributions of the M-site at low frequencies explain the limited shift of the CsSrBr 3 Raman spectrum, which might be surprising at first sight given the different atomic masses of Sr and Pb. Above this range, CsPbBr 3 exhibits few vibrational states while CsSrBr 3 shows its most pronounced phonon DOS peaks, which correspond well with the strongest IR mode calculated from the harmonic approximation.

High-temperature lattice dynamics

A key signature of vibrational anharmonicity in HaPs at higher temperatures is the Raman central peak 26 , 33 , 34 , 35 , 43 , 44 , 45 . We use this feature that is nominally symmetry-forbidden in the cubic phase as a fingerprint to directly investigate how the propensity for cation lone-pair formation or lack thereof determines anharmonicity in these materials, using Raman spectroscopy and DFT-based MD simulations. Interestingly, a central peak also appears in the high-temperature Raman spectrum of CsSrBr 3 (see Fig.  4 and Supplementary Note  5 for full temperature range). We note that differences in Raman intensity imply that the scattering cross-section of CsSrBr 3 is notably weaker than that of CsPbBr 3 , which is due to its significantly higher bandgap and weaker dielectric response at the Raman excitation wavelength (785 nm) and because a powder sample of CsSrBr 3 has been used for which scattering of light in the back-scattering direction is considerably lower. The presence of a central peak in CsSrBr 3 shows that local fluctuations associated with a cation lone-pair are not required for the low-frequency diffusive Raman scattering and anharmonicity to occur. This result, together with the identical phase-transition sequences of both materials (see Fig.  2 ), led us to investigate the role of tilting instabilities in CsSrBr 3 and CsPbBr 3 .

Raman spectra of CsPbBr 3 a and CsSrBr 3 b in the high-temperature cubic phase measured experimentally and calculated using DFT-MD. The central peak appears for both compounds in the experiments and computations despite significant differences in bonding: [PbBr 6 ] 4− is proximate to lone-pair formation ( i.e ., exhibits a weak PJT effect) 29 , while PJT effects associated with [SrBr 6 ] 4− are negligible.

We first calculate the Raman spectrum for both compounds using MD calculations (see Fig.  4 and Methods section). Remarkably, a central peak appears also in the MD-computed high-temperature Raman spectrum of CsPbBr 3 and CsSrBr 3 . We find good agreement between experiment and theory, both showing a feature between 50 and 100 cm −1 in the Raman spectra of the two materials in addition to the central peak.

Next, we compute harmonic phonon dispersions of both compounds (see Fig.  5 ) and find these to be remarkably similar for cubic CsSrBr 3 and CsPbBr 3 in the low frequency region, in line with the aforementioned similarities in the phonon DOS of the orthorhombic phase. Specifically, both compounds exhibit the same dynamic tilting instabilities at the edge of the Brillouin zone (BZ), governed by in-phase (M point) and three degenerate out-of-phase (R point) rotations. These rotation modes are not only active in the phase transitions, but they also have been discussed to drive the dynamic disorder of halide perovskites 4 , 14 , 58 , 59 , 60 , 61 .

a Harmonic phonon dispersion of cubic CsPbBr 3 and CsSrBr 3 showing the dynamic instabilities in the high-temperature, cubic phase of both compounds. The imaginary modes at the M and R points are the in-phase and out-of-phase tilting depicted in b , c , respectively. The tilting modes are almost identical for CsSrBr 3 and CsPbBr 3 .

Finally, using the MD trajectories of CsPbBr 3 and CsSrBr 3 in the cubic phase, we calculate the frequency-resolved dynamic changes of octahedral rotation angles, Φ α ( ω ) (see Fig.  6 and Eq. ( 1 ) in the Methods Section). Figure  6 b shows Φ α ( ω ) for CsPbBr 3 and CsSrBr 3 and indicates strong low-frequency tilting components in both CsPbBr 3 and CsSrBr 3 . Recently, a phenomenological model for the description of the temperature-dependent Raman spectra of cubic HaPs proposed the inclusion of a low-frequency anharmonic feature, which was associated with transitions between minima of a double-well potential energy surfaces 45 that correspond to different octahedral tiltings 60 , 62 , 63 , 64 . Our results confirm that substantial octahedral dynamics correspond to low-frequency features dynamically breaking the cubic symmetry in CsPbBr 3 and CsSrBr 3 4 , 14 , 44 , 65 , 66 . Interestingly, this low-frequency component appears irrespective of the presence of n s 2 cations and induces the formation of relatively long-lived (tens of ps) structural distortions (see Supplementary Fig.  11) , which strongly deviate from the average cubic symmetry. This suggests that the dynamic deviations from the long-range, crystallographic structure enable the low-frequency Raman response without violating the selection rules.

a Schematic representation of the MBr 6 octahedron aligned along the z Cartesian axis. The octahedral rotation angle around z , ϕ z , is defined as the average of the angles formed by the x / y Cartesian axis and the vector connecting two in-plane Br atoms at opposing edges of the octahedron ( \({\phi }_{z}^{(x)}\) in red and \({\phi }_{z}^{(y)}\) in blue). Note that a clockwise rotation is defined as positive and counter-clockwise as negative. Fourier transform of the octahedral rotation angle, Φ α ( ω ), and cross-correlation between rotation angle and M-site displacement, C α β ( ω ), calculated using DFT-MD trajectories of cubic CsPbBr 3 ( b , c , respectively, T  = 525 K) and CsSrBr 3 ( d , e , respectively, T  = 570 K).

We investigate the impact of the M-site chemistry on octahedral tilting tendencies 33 by computing the Fourier-transform of cross-correlations between rotation angles and M-site displacements, C α β ( ω ) (see Eq. ( 2 ) in the Methods section). Larger values of C α β generally indicate stronger coupling between octahedral rotations and Pb displacements. Absence of the propensity for lone-pair formation becomes evident in the low intensity of C α β ( ω ) for CsSrBr 3 (Fig.  6 c), which is less than half of that of CsPbBr 3 , especially at low-frequencies relevant for the slow, anharmonic, symmetry-breaking rotational features. This suggests that the presence of the n s 2 cations in CsPbBr 3 enhances the low-frequency octahedral tilting, in line with the literature 33 . M-site displacements and octahedral rotations are correlated because the latter is accompanied by changes of the Br-Pb-Br resonant network 17 affecting the charge density in the vicinity of the M-site. While this effect is very weak in CsSrBr 3 (see Supplementary Fig.  12) , the non-zero C α β for this case shows that the presence of n s 2 cations is not necessary to couple octahedral rotations and M-site displacements because the ions are still interacting through other types of interactions, e.g ., electrostatically or due to Pauli repulsion. In CsPbF 3 , the interaction of tilting and M-site displacements is strong enough to drive the adoption of an unusual tilt pattern 37 . We speculate that the lone-pair-enhanced tilting could contribute to the fact that CsPbBr 3 has a lower tetragonal-to-cubic phase transition temperature compared to that of CsSrBr 3 .

We directly disentangled structural and chemical effects in HaPs by comparing CsPbBr 3 and CsSrBr 3 , two compounds with similar ionic radii and structural properties but entirely different orbital interactions that imbue CsPbBr 3 with the weak PJT effect common to technologically-relevant Pb perovskites and CsSrBr 3 with negligible PJT effects. While the n s 2 configuration of the octahedral cations is paramount for the optoelectronic and dielectric properties of these materials, using the Raman central peak at higher temperatures as a fingerprint to detect anharmonicity we found it to appear also for CsSrBr 3 with 5 s 0 cations and to correlate with slow, anharmonic rotations of the octahedra. Notably, the anharmonicity of the tilting motions is different from that of intra-octahedral distortions associated with the PJT effect 29 . Altogether, these findings demonstrate that the perovskite structure allows for anharmonic vibrational dynamics to occur, irrespective of the presence of n s 2 cations with the propensity to form lone pairs, which establishes this somewhat unusual behavior as a generic effect in this material class. We note that recent work by some of the present authors has investigated the commonalities and differences between oxide perovskites and HaPs in this context 45 .

Since octahedral dynamics impact the optoelectronic characteristics of these systems, our results have implications for synthesis of new HaPs with improved properties for technological applications. For instance, Pb-Sr alloying has been proposed as a method to tune the band gap of HaPs for light emission and absorption applications 47 . Our work implies that such Sr alloying for tuning electronic and dielectric properties preserves the strongly anharmonic lattice dynamics. Furthermore, investigating related compounds with distinct electronic configurations on the octahedral cation, such as CsEuBr 3 , may provide further insight about chemical trends in tuning of the HaP properties.

The relevance of these findings for material design strategies of HaP compounds is additionally affirmed when putting our results in the context of previous work discussing anharmonic effects in this class of materials. Specifically, cubic CsPbBr 3 , CsSnBr 3 , CsGeBr 3 , (CH 3 NH 3 ) 0.13 (CH 3 CH 2 NH 3 ) 0.87 PbBr 3 , CH(NH 2 ) 2 PbBr 3 , and, here, CsSrBr 3 are all reported to exhibit dynamic hopping between low symmetry minima on the potential energy surface 33 , 34 , 35 , 67 . By contrast, the high symmetry phase of Cs 2 AgBiBr 6 is anharmonically stabilized and exhibits well-defined normal modes and a soft-mode transition on cooling 42 . Cs 2 SnBr 6 , on the other hand, lacks any phase transitions and similarly exhibits well-defined normal modes 68 . Where previously the strength of the PJT effect associated with n s 2 cations or the density of such cations appeared to be a plausible predictor of broad, nominally symmetry-forbidden Raman scattering resulting in a central peak, our work suggests that instead the differing symmetry in both the structure and the chemical bonding of metal halide perovskites and double-perovskites may be a controlling factor. Notably, CsGeBr 3 , which exhibits no octahedral tilting transitions 69 and a broad Raman central peak in the cubic phase with a mode reflecting persistent pyramidal [GeBr 3 ] − anions 33 , corresponds to the strong PJT 29 case: Stereochemically expressed cation lone pairs are evident in the low temperature average structure 69 and in the local fluctuations of the cubic phase 33 . Dynamic symmetry-breaking giving rise to a broad Raman central peak is thus observed for three distinct bonding regimes with regard to pseudo-Jahn–Teller effects: strong PJT (CsGeBr 3 ) 33 , weak PJT (CsPbBr 3 and others) 34 , and negligible PJT (CsSrBr 3 ).

In conclusion, the n s 2 electron configuration in HaPs that can result in the formation of lone-pairs is crucial to several favorable electronic features 26 , 39 , 46 and gives rise to the elevated ionic dielectric response via enhancement of Born effective charges 39 , 48 . However, we found that presence of a strong or weak PJT effect associated with n s 2 cations is not necessary to produce dynamic symmetry-breaking of the sort that gives rise to broad, intense Raman scattering in the high temperature phases of HaPs and that has been associated with the unique optoelectronic properties in these compounds such as long charge-carrier lifetimes and photoinstabilities. Instead, such dynamic symmetry breaking is common to all cubic bromide and iodide (single-)perovskites thus far studied to the best of our knowledge. These results highlight the key role of structural chemistry in the anharmonic dynamics of halide perovskites, providing an additional criterion for the design of soft optoelectronic semiconductors.

Electronic structure calculations

DFT calculations were performed with Vienna ab-initio simulation package (VASP) code 70 using the projector augmented wave (PAW) method 71 . We employed the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 72 and the Tkatchenko-Scheffler (TS) scheme 73 —using an iterative Hirshfeld partitioning of the charge density 74 , 75 – to account for dispersive interactions. This setup has been shown to accurately describe the structure of HaPs, also in regard to the omission of SOC which impacts electronic-structure properties but does not result in significant changes of quantities related to the total energy 76 , 77 . All static calculations used an energy convergence threshold of 10 −6 eV, a plane-wave cutoff of 500 eV, and a Γ-centered k -grid of 6 × 6 × 6 (6 × 4 × 6) for the \(Pm\bar{3}m\) ( P n m a ) structures. Lattice parameters were optimized by a fitting procedure using the Birch-Murnaghan equation of state 78 , 79 The final structures used in all subsequent calculations were obtained by relaxing the ionic degrees of freedom until the maximum residual force was below 10 −4  eV Å −1 . The total and projected electronic DOS and COHP, were calculated by partitioning the DFT-calculated band structure into bonding and antibonding contributions using the LOBSTER code 80 , 81 . For this task, the DFT-computed electronic wave functions were projected onto Slater-type orbitals (basis set name: pbeVaspFit2015) 80 including Cs 6s, 5p and 5s, Pb 6s and 6p, and Br 4p and 4s states. The maximum charge spilling in this procedure was 1.3%. Spin-orbit coupling was not included in our calculations, since it is currently not supported by the LOBSTER code. We emphasize that our focus is on the orbital contributions to the (anti) bonding interactions, rather than on a quantitative descriptions of the energy.

Phonon calculations

Phonon dispersions and DOSs were obtained via the finite displacements method implemented in the phonopy package 82 . For these calculations, we used 2 × 2 × 2 supercells with 40 (160) atoms of the \(Pm\bar{3}m\) ( P n m a ) CsMBr 3 structures reducing k -space sampling accordingly. IR and Raman spectra were computed with the phonopy-spectroscopy package 83 , using zone-center phonon modes, Born-effective charges and polarizabilities, calculated with density functional perturbation theory (DFPT) 84 .

First-principles molecular dynamics

DFT-based MD calculations were performed for 2 × 2 × 2 supercells of the \(Pm\bar{3}m\) structures using a Nosé-Hoover thermostat within the canonical ensemble (NVT), as implemented in VASP 85 . The simulation temperature was set to T =525 and 570 K for CsPbBr 3 and CsSrBr 3 , respectively. An 8 fs timestep, reduced k -grid of 3 × 3 × 3, and energy convergence threshold of 10 −5  eV were used for the 10 ps equilibration and 115 ps production runs.

Raman spectra from molecular dynamics

DFT-based MD calculations were used to compute the high-temperature Raman spectra of CsPbBr 3 and CsSrBr 3 . We calculated Raman intensities from the autocorrelation function of the polarisability, as detailed elsewhere 86 . The polarizabilities were calculated with DFPT 84 on 400 evenly-spaced snapshots every 0.11 ps for a total of 44.8 ps. The k -grid employed for the DFPT calculations was set to 4 × 4 × 4 after testing convergence of the polarisability tensor for several snapshots.

Octahedral rotation dynamics and cross-correlations

We quantified the octahedral dynamics using the rotation angles, ϕ α , around a given Cartesian axis α (see Fig.  6 a). The frequency-resolved rotational dynamics were calculated as the Fourier transform of ϕ α :

where N steps is the number of snapshots. To compute the angles we selected 1000 equally spaced snapshots. We calculated the frequency-resolved cross-correlation between octahedral rotation angles (around a Cartesian direction α ) and the displacements (along a Cartesian direction β ) of the corresponding M-site, \({d}_{\beta }^{{{{{{{{\rm{M}}}}}}}}}(t)\) , as:

Polycrystalline sample preparation

CsBr (Alfa Aesar, 99.9%), anhydrous SrBr 2 (Alfa Aesar, 99%), Cs 2 CO 3 , PbO, and concentrated aqueous HBr were purchased and used as received. Guided by the reported pseudo-binary phase diagram 87 , polycrystalline CsSrBr 3 for X-ray powder diffraction and Raman spectroscopy was prepared by a solid-state reaction at 600 °C. CsBr (5 mmol, 1064 mg) and SrBr 2 (5 mmol, 1237 mg) were ground and pressed into a 5 mm diameter pellet, placed in an alumina crucible, and flame-sealed under ~ 1/3 atmosphere of argon in a fused silica ampoule. The reaction yields a porous, colorless pellet which is easily separated from the crucible and ground in inert atmosphere. Polycrystalline CsPbBr 3 for X-ray powder diffraction was prepared in ambient atmosphere by precipitation from aqueous hydrobromic acid. PbO (2 mmol, 446.4 mg) was dissolved in 2 mL hot concentrated HBr under stirring. Cs 2 CO 3 (1 mmol, 325.8 mg) was added slowly resulting in an immediate bright orange precipitate. 13 mL additional HBr was added and the mixture left to stir. After an hour, stirring was stopped and the mixture allowed to cool to room temperature. Excess solution was decanted, and the remaining mixture was evaporated to dryness on a hotplate and ground. Phase purity of all prepared compounds was established by powder XRD.

Single crystal preparation

Single crystals of CsSrBr 3 were grown by the Bridgman method from a stoichiometric mixture of the binary metal bromides in a 10 mm diameter quartz ampoule. CsSrBr 3 was pulled at 0.5 mm h −1 through an 800 °C hot zone, yielding a multi-crystalline rod from which several-mm single crystal regions could be cleaved. CsSrBr 3 is hygroscopic and all preparation and handling was performed in an inert atmosphere.

The vertical Bridgman method was used to grow large, high-quality single crystals of CsPbBr 3 . After synthesis and purification (see Supplementary Note  2 for details), the ampoule was reset to the hot zone for the Bridgman Growth. The zone 1 temperature was set to 650 °C with a 150 °C h −1 ramp rate, and held for 12 h to ensure a full melt before sample motion occurred. The zone 2 and 3 temperatures were set to 375 °C. These temperatures were held for 350 h while the ampoule was moved through the furnace at a rate of 0.9 mm h −1 under 0.3 rpm rotation. After the motion had ceased, the zone 1 temperature ramped to 375 °C to make the temperature profile in the furnace uniform. The cooling program was set to slow during the phase transitions occurring near 120 and 90 °C, with a 10 °C h −1 cooling rate from 375 °C to 175 °C, a 2.5 °C h −1 slow cooling rate from 175 °C to 75 °C, and a 10 °C h −1 rate to 30 °C. The resulting CsPbBr 3 ingot was orange-red and had large (≥5 mm) transparent single-crystalline domains, though the edges of some portions exhibited twinning.

X-ray diffraction

Polycrystalline samples were ground with silicon powder (as an internal standard and diluent) and packed in borosilicate glass capillaries. Powder XRD patterns were measured in STOE geometry using a STOE Stadi P diffractometer (Mo K α 1 radiation, Ge-(111) monochromator, Mythen 1K Detector) equipped with a furnace. Data were analyzed by sequential Pawley refinement using GSAS-II 88 .

Infrared reflectivity measurements

IR-reflection spectra in the THz range were measured as a combination of THz spectroscopy (TDS) for the low-frequency end and bolometer detection for the higher frequencies. Bolometer spectra were measured using a Bruker 80v Fourier-transform IR spectrometer with a globar source and a bolometer detector cooled to liquid He temperatures. The crystals were mounted for reflection measurements and the instrument was sealed in vacuum. A gold mirror was used as reflection reference. TDS was performed using a Spectra Physics Mai Tai-Empower-Spitfire Pro Ti:Sapphire regenerative amplifier. The amplifier generates 35 fs pulses centered at 800 nm at a repetition rate of 5 kHz. THz pulses were generated by a spintronic emitter, which was composed of 1.8 nm of Co 4 0 Fe 4 0 B 2 0 sandwiched between 2 nm of tungsten and 2 nm of platinum, all supported by a quartz substrate. The THz pulses were detected using electro-optic sampling in a (100)-ZnTe crystal. A gold mirror was used as reflection reference. The sample crystals, THz emitter and THz detector were held under vacuum during the measurements.

TDS offers better signal at low frequency, while bolometer measurements have an advantage over TDS at higher frequencies. Therefore, the spectra were combined and merged at 100 cm −1 . Owing to scattering losses, the absolute intensity of reflected light can not be taken quantitatively. Therefore, the spectra were scaled to the signal level at 100 cm −1 before merging the data. The final reflectivity spectra are given in arbitrary units. The phonon frequencies and overall spectral shape allows for fitting to the dielectric function.

Raman spectroscopy

All the measurements were taken in a home-built back scattering Raman system. For all measurements, the laser was focused with a 50× objective (Zeiss, USA), and the Rayleigh scattering was then filtered with a notch filter (Ondax Inc., USA). The beam was focused into a spectrometer 1 m long (FHR 1000, Horiba) and then on a CCD detector. To get the unpolarized Raman spectrum for the single crystals (CsSrBr 3 and CsPbBr 3 ), two orthogonal angles were measured in parallel and cross configurations (four measurements overall). The unpolarized spectrum is a summation of all four spectra. The samples were cooled below room temperature by a Janis cryostat ST-500 controlled by Lakeshore model 335 and were heated above room temperature by a closed heating system (Linkam Scientific). Due to the sensitivity of CsSrBr 3 to ambient moisture, CsSrBr 3 powder was flame-sealed in a small quartz capillary for the high-temperature measurements, and a single crystal was loaded into a closed cell under an Ar environment for the low temperatures measurements. CsSrBr 3 low temperature measurements were taken with a 2.5 eV CW diode laser (Toptica Inc.). CsSrBr 3 high-temperature measurement and all the CsPbBr 3 measurements were taken with a 1.57 eV CW diode laser (Toptica Inc.). We note that while Raman spectra on quartz show a contribution towards zero frequency 89 , it is narrower in frequency than what we observe. Results from control experiments (see Supplementary Fig.  7) show that the main signals from quartz do not contribute to the measured Raman spectra of CsSrBr 3 .

Data availability

The computational and experimental data generated in this study have been deposited in the Zenodo repository under accession code 10975217 90 .

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Acknowledgements

Funding provided by the Alexander von Humboldt-Foundation in the framework of the Sofja Kovalevskaja Award, endowed by the German Federal Ministry of Education and Research, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Germany’s Excellence Strategy - EXC 2089/1-390776260, and by TU Munich - IAS, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under Grant Agreement No. 291763, are gratefully acknowledged. Funding was provided by the Engineering and Physical Sciences Research Council (EPSRC), UK. The Gauss Centre for Supercomputing eV is acknowledged for providing computing time through the John von Neumann Institute for Computing on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre. D.H.F. gratefully acknowledges financial support from the Alexander von Humboldt Foundation and the Max Planck Society. D.H.F. thanks Maximilian A. Plass for assistance with flame-sealing the Raman capillaries. K.M.M. and M.V.K. acknowledge financial support by ETH Zürich through the ETH+ Project SynMatLab Laboratory for Multiscale Materials Synthesis.

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Kyle M. McCall

Present address: Department of Materials Science and Engineering, University of Texas at Dallas, Richardson, TX, USA

Authors and Affiliations

Physics Department, TUM School of Natural Sciences, Technical University of Munich, Garching, Germany

Sebastián Caicedo-Dávila, Manuel Grumet & David A. Egger

Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel

Adi Cohen & Omer Yaffe

Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, UK

Silvia G. Motti & Laura M. Herz

School of Physics and Astronomy, Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK

Silvia G. Motti

Max Planck Institute for Solid State Research, Stuttgart, Germany

Masahiko Isobe & Douglas H. Fabini

Laboratory of Inorganic Chemistry, Department of Chemistry and Applied Biosciences, ETH Zurich, Zürich, Switzerland

Kyle M. McCall & Maksym V. Kovalenko

Laboratory for Thin Films and Photovoltaics, EMPA - Swiss National Laboratories for Materials and Technology, Dübendorf, Switzerland

TUM Institute for Advanced Study, Technical University of Munich, Garching, Germany

Laura M. Herz

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, USA

Douglas H. Fabini

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S.C.-D. performed the theoretical calculations, analyzed the data and wrote the initial manuscript under the supervision of D.A.E and with additional support by M.G. A.C. performed Raman measurements and analyzed data under the supervision of O.Y. S.M. performed IR measurements and analyzed data under the supervision of L.M.H. D.H.F. prepared polycrystalline samples, performed XRD measurements, analyzed data, and modeled the PJT effect. M.I. prepared the CsSrBr 3 single crystals. K.M.M. prepared the CsPbBr 3 single crystals under the supervision of M.V.K. D.H.F. and D.A.E. conceived and supervised the project.

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Caicedo-Dávila, S., Cohen, A., Motti, S.G. et al. Disentangling the effects of structure and lone-pair electrons in the lattice dynamics of halide perovskites. Nat Commun 15 , 4184 (2024). https://doi.org/10.1038/s41467-024-48581-x

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DOI : https://doi.org/10.1038/s41467-024-48581-x

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Density Functional Theory and Practice Course

Blog where students post their results, determining the lattice constants for hf.

   For Project 1,  the lattice constants for Hf are calculated. Since Hf is an hcp metal with c/a=1.58, we need to calculate the energy by changing the lattice constant c while keeping c/a fixed. And the optimized a, c can be obtained by minimizing the free energy.

1. The hcp structure of Hf

   Figure 1 shows the unit cell of Hf hcp structure. OA, OB, OC are three lattice vectors. The lengths and angles of the lattices vectors (OA, OB, OC) are a, b, c and α, β, γ, respectively. To keep the symmetry, the constraints for the unit cell are a=b, α=β=90 degrees, γ=120 degrees.

Figure 1 The unit cell of Hf

2. The parameters for the calculations

Some important parameters and inputs for the calculation are listed as following:

Atomic structure for Hf:1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2

The core configuration of the pseudopotential: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10

Functional: GGA Perdew Burke Ernzerhof (PBA) functional

Pseudopotential: OTFG ultrasoft

Smearing width: 0.1

Origin shift of K points: 0, 0, 0

The k points grid: investigated in the next section

The energy cutoff: investigated in the next section

3. The convergence of the calculations

   Before starting the actual calculation for the lattice constant, it is important to explore the convergence of the calculations with respect to the number of k points and the energy cutoff. So the most accurate and efficient results can be obtained. For the following calculation, the lattice constant is fixed at c= 5.0511Å, a= 3.1946Å, while the size of k point grid and the energy cutoff are varied.

   (1) The number of k-points in the irreducible part of the BZ

     In this section 3.1, the energy cutoff is fixed at 400 eV, and the size of k point grid changes and Number of k-points in the irreducible part of the BZ (the number of k points) varies from 1 to 162, shown in Table 1. According to Table 1, the relations between Energy/atom, calculation time and the number of k points can be plotted, which are shown in Figure 2. As shown in Figure 2, the energy/atom becomes stable with the increase of k points. However, the calculation time also increases rapidly when the number of k points increases. By combining the results of these two graphs, 9*9*6 k points with 36 k points is chosen to obtain an accurate result with less calculation time.

   A little discussion here, since my calculation involves unit cells with different volumes, the k points need to be chosen so the density of k points in reciprocal space is comparable for the different supercells. But as shown in section 4, since the volumes of unit cell change just a little during the calculation, fixing the number of k points will not make much difference. So in section 4, the number of k points is chosen as 36 for all calculations.

Table 1 The calculation of different sizes of k point grid

Figure 2 The plots between Energy/atom, calculation time and the number of k points. The solid lines are just used for guide the eye.

(2) The energy cutoff

In this section, the number of k point is fixed at 36, and the energy cutoff varies from 200 to 800, shown in Table 2. And Figure 3 shows the relation between energy/atom, calculation time and energy cutoff. Although the energy/atom keeps increasing with the increase of energy cut off, the differences between each energy/atom are becoming smaller and smaller. On the other hand, the calculation time increases rapidly with the increase of energy cut off. In order to save the time and keep the accuracy, the energy cutoff is chosen as 435.4 eV.

Table 2 The calculation of different energy cutoff

Figure 3 The plots between energy/atom, calculation time and energy cutoff. The solid lines are used to guide the eye.

4 The calculation of the lattice constants

Using the results from section 3, the k points and the energy cutoff are chosen as 9*9*6 and 435.4 eV. Then lattice constants c and a are changed to obtain the minimum energy, shown in Table 3. As shown in Figure 4, the unit cell and the lattice constants can be predicted. As shown in Figure 4a, the quadratic relation between energy/atom and volume/atom only valid for a small range of lattice constants around the equilibrium value. A small range of volume/atom is chosen and the relation is re-plotted in Figure 4b. According to the fitting equation in Figure 4b, the fitting volume/atom is 22.64 Å^3 and the lattice constant are c=5.0673 Å, a=3.2072 Å. And for the experiment data, the lattice constants are a=3.1964 Å, c=5.0511 Å (1). So the calculation is accurate with an error of 0.3%.

Table 3 The calculation of energy with different lattice constants.

Figure 4 The plots between energy/atom and volume/atom. the solid lines are used to guide the eye. The dashed line is the fitting line.

5. Some discussion and future works

Although the calculation result is accurate with an error of 0.3%, there are still some ways to further increase the accuracy. First, the energy cut off can be increased so the energy/atom can be more precise. Second, we can calculate more points around the value c=5.05 Å and maybe a more accurate result can be obtained. Third, instead of using the quadratic fitting, we can use a higher order polynomial fitting.

For some future works: first, it is worthwhile changing the ratio c/a to determine the optimum value of c/a. Second, it should be interesting to analyze the behavior of energy at small or large lattice constants.

6. Reference:

(1) https://www.webelements.com/hafnium/crystal_structure.html

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Identification of nonlinear dynamical system based on adaptive radial basis function neural networks

• Original Article
• Published: 17 May 2024

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• Guo Luo 1 ,
• Hu Min   ORCID: orcid.org/0000-0002-1382-774X 1 &
• Zhi Yang 1  

In this paper, an adaptive radial basis function neural network (RBFNN) is investigated and presented for identifying the nonlinear dynamical system. In traditional RBFNN structure, the selections of the activated radius are decided by the designer’s experience to satisfy the best approximation of nonlinear dynamical function. In order to reduce the experience error, an adaptive RBFNN mechanism that can realize auto-regulation of activated radius is proposed in this paper. Taking the lattice points as the center of activated function, we use Taylor expansion to separate the factor of activated radius in local space. In order to ensure that the identification error has the property of fast convergence, the error conversion function is used for shrinking the gain in the error differential equation. Constructing a Lyapunov function to determine the differential equation of the weights and the activated radius, it is shown that the weights and the activated radius will converge to the neighborhood of its true value and the identification error will converge to neighborhood of zero in a periodic or period-like nonlinear dynamical system. To illustrate the effectiveness of the proposed adaptive RBFNN, Vanderpol and Duffing dynamical system are used as test examples, in comparison with traditional RBFNN and wavelet neural networks (WNN). The results show that the proposed method has best accurate identification and approximating effect in all of test algorithms.

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Luo, G., Min, H. & Yang, Z. Identification of nonlinear dynamical system based on adaptive radial basis function neural networks. Neural Comput & Applic (2024). https://doi.org/10.1007/s00521-024-09794-9

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Received : 17 November 2022

Accepted : 25 March 2024

Published : 17 May 2024

DOI : https://doi.org/10.1007/s00521-024-09794-9

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