On the thermal expansion of the tetragonal phase of MAPbI₃ and MAPbBr₃

Apparent shrinkage of [PbX₆] octahedra and Pb-X bond lengths changes in the tetragonal phases of MAPbI₃ and MAPbBr₃

Two previous temperature-dependent crystallographic investigations of MAPbI₃ from Schuck et al.^[6] and of MAPbBr₃ from Weadock et al.^[7] are compared and discussed in relation to the observed volume changes of the [PbX₆] octahedra and the changes in Pb-X bond lengths. In the following, we will focus on the tetragonal phase [Figure 1] and the transition to the cubic structure by looking at the behavior of the [PbX₆] octahedra network in dependence on temperature [Figure 2]. Powder samples of MAPbI₃ were measured by Schuck et al. using synchrotron XRD as a function of temperature, and the resulting diffractograms were refined using the Rietveld method. The MA molecule geometry was taken from the previous density functional theory (DFT) optimization and treated as rigid^[35]. The results of the structure refinements of MAPbI₃^[6] agree with previous temperature-dependent studies^[36-38].

Figure 1. Detail of the tetragonal room-temperature structure of MAPbI₃ with its large iodine anisotropic temperature factors (ADPs) (their flat shape indicates a transverse displacement perpendicular to the Pb-I-Pb bonds and in the direction of the MA molecule), the coordination polyhedra of the MA molecule [MAI₁₂], a cuboctahedron (light yellow), and one [PbI₆] octahedra (light gray)^[37]. The anisotropic shape and orientation of the halide ADPs are partly due to the strongly anharmonic metal-halide bond (see EXAFS section) and not only to the hydrogen bond between the MA and the halide atoms (formed between the jump rotations). In this representation of a single MA molecule, neither the 4-fold rotational axis nor the 2-fold rotational axis of the I4/mcm space group symmetry seems to exist for the molecule. However, since the MA molecule jump rotates^[8] and there are, on average, eight MA molecule orientations in the cage, an average structure with I4/mcm symmetry is observed in diffraction.

Figure 2. (A) The [PbI₆] octahedra network of MAPbI₃ for the cubic high-temperature structure at 350 K^[36] (top) and the tetragonal room-temperature structure at 300 K^[37] (bottom). (B) Pb-Xa-Pb bond angle (octahedra tilt) of MAPbI₃ (red) and MAPbBr₃ (blue) in the tetragonal (solid symbols) and cubic phase (open symbols). Besides the data from Schuck et al.^[6] and Weadock et al.^[7], values for the cubic phase at 350 K of MAPbI₃ from Whitfield et al. are also given^[36]. Adapted with permission from Schuck et al.^[6]. Copyright 2022 American Chemical Society.

The volume of MAPbI₃ increases with temperature over the whole temperature range (volumetric thermal expansion coefficient α_{V, orthorhombic} = 90 × 10^-6 K^-1 and α_{V, tetragonal} = 119 × 10^-6 K^-1). Based on the lattice constants for a pseudo-cubic cell from Schuck et al., a negative linear thermal expansion coefficient of α_{[001], tetragonal} = -23.3(3) × 10^-6 K^-1 in the [001] direction and a positive linear thermal expansion coefficient of α_{[100], tetragonal} = 75(3) × 10^-6 K^-1 in the [100] direction can be determined for MAPbI₃ (temperature range 220-260 K) [Figure 3A]^[6]. The differences between these linear thermal expansion coefficients of Schuck et al. and those of Jacobsson et al. are due to the different temperature ranges used to determine the gradient^[6,27]. Jacobsson et al. used a higher temperature range of 308 to 324 K. Figure 4 shows the percentage changes in Pb-X bond length expansion [Figure 4A] and the percentage changes in volume expansion [Figure 4B] from the onset of the tetragonal phase to higher temperatures^[27]. In the tetragonal phase, the maximum thermal expansion at 300 K is +1.68% (compared to 160 K) for the whole volume and +2.14% for the [MAI₁₂] cuboctahedra. In contrast, the negative thermal expansion (NTE) of the [PbI₆] octahedra is -0.5%, corresponding to the negative change in average bond length, which is -0.17% smaller at 300 K compared to the value at 160 K. Based on the tetragonal lattice constants for a pseudo-cubic cell from Weadock et al., which are available from their temperature-dependent single-crystal structure investigations [crystallographic information files (CIF) were made available]^[7], a NTE of α_{[001], tetragonal} = -66(3) × 10^-6 K^-1 in the [001] direction and a positive thermal expansion of α_{[100], tetragonal} = 95(2) × 10^-6 K^-1 in the [100] direction can be determined for MAPbBr₃ (for the temperature range 200-230 K) [Figure 3B]. As shown in Figure 4, a maximum thermal expansion at 240 K is observed for MAPbBr₃ in the tetragonal phase, analogous to the behavior of MAPbI₃. While the total volume of MAPbBr₃ and the [MABr₁₂] cuboctahedra volume increase by +0.78% and +1.04%, respectively, the Pb-Br bond length decreases by -0.19% with a concomitant NTE of -0.53% for the [PbBr₆] octahedra. In both MAPbI₃ and MAPbBr₃, the temperature-dependent negative changes in the partial volumes of the [PbX₆] octahedra, on the one hand, and positive partial volume changes of the [MAX₁₂] cuboctahedra, on the other hand, differ significantly from the change in the total volume in the tetragonal phase. While the [MAX₁₂] cuboctahedra volume expands almost linearly with increasing temperature (greater expansion than total volume), the [PbX₆] octahedra volume shrinks towards the cubic phase [Figure 4B]. A comparison with pressure-dependent data in the tetragonal phase would certainly also be interesting, but for MAPbBr₃ such data are not yet available^[39] in the corresponding temperature and pressure range, and for MAPbI₃ a clearly different behavior of the tetragonal structure already occurs at 300 K in the range of 0.05 GPa (the [PbI₆] octahedra become 0.15% smaller, and Pb-I shortens by 0.05%)^[40], so that it cannot be excluded that phase transformations also occur at minimal pressures and a direct comparison of the mechanochemical sensitive^[41] MAPbI₃ data is not readily possible.

Figure 3. Temperature-dependent lattice constants for the tetragonal (pseudo-cubic) and cubic unit cell for (A) MAPbI₃ (cubic at 350 K from Whitfield et al.^[36]) and (B) MAPbBr₃ based on the results of Schuck et al.^[6] and Weadock et al.^[7]. In both cases, the linear thermal expansion coefficients in direction [100] [positive thermal expansion (PTE)] and in direction [001] [negative thermal expansion (NTE)] are also given. Adapted with permission from Schuck et al.^[6]. Copyright 2022 American Chemical Society.

Figure 4. (A) Averaged Pb-X bond lengths expansion and (B) volumetric expansion of MAPbI₃ (red) and MAPbBr₃ (blue) in the tetragonal (solid symbols) and cubic phase (open and half-open symbols), the phase transformation temperature is indicated in each case (tetragonal to cubic). Besides the data from Schuck et al.^[6] and Weadock et al.^[7], values for the cubic phase of MAPbI₃ are also given from Whitfield et al.^[36] (left half-open symbols) and Stoumpos et al.^[38] (right half-open symbols). (B) In addition to the volume changes for the [PbX₆] octahedra (squares), the volume changes for the total volume (diamond symbols) and the volume changes for the [MAX₁₂] cuboctahedra (circles) are also given. Note: Both Pb-X bond lengths, i.e., the one along tetragonal c (Pb-Xb in Figure 2A) and the one in the tetragonal a-b plane (Pb-Xa in Figure 2A), become smaller in the tetragonal phase with increasing temperature. Both bond lengths (Pb-Xa and Pb-Xb) for MAPbI₃ and MAPbBr₃ are within the error bars of the averaged value Pb-X. Adapted with permission from Schuck et al.^[6]. Copyright 2022 American Chemical Society.

Temperature-dependent linear thermal expansion coefficients and the Grüneisen tensor

The linear thermal expansion coefficients calculated for MAPbI₃ from the data of Schuck et al. differ significantly from the values of Jacobsson et al.^[6,27]. This is not surprising since one linear fit obviously cannot describe the entire temperature range of the tetragonal phase. For this reason, the temperature dependence of the linear thermal expansion coefficients is investigated using the tetragonal lattice constants of Whitfield et al. [the lattice constants from temperature-dependent synchrotron powder XRD measurements of Whitfield et al. (figure 6C in Ref.^[36]) have been digitized by us and are shown here in Figure 5A]^[36]. The temperature dependence of the linear thermal expansion coefficients [Figure 5B] can be approximately described for the PTE α_[100] with a linear behavior; an increase towards the cubic phase up to 97 × 10^-6 K^-1 is obtained. The NTE α_[001] can be described with a third-order polynomial and reaches a maximum value of -88 × 10^-6 K^-1 towards the cubic phase. Due to the temperature dependence of the two linear thermal expansion coefficients α_[001] and α_[100], the linear thermal expansion coefficient α_[averaged] determined from the averaged tetragonal lattice constant a_averaged = ((2*a_tet)+c_tet)/3 is more suitable for describing the thermal expansion as it has almost no temperature dependence. The value given by Whitfiled et al.^[36] is to be multiplied by 1/L₀ and then results in α_[averaged] = 42.4(4) × 10^-6 K^-1 (dotted line in Figure 5B)^[42]. It is common to describe thermal expansion properties of solids in Grüneisen parameters: NTE results in negative Grüneisen parameter γ.

Figure 5. (A) Temperature-dependent tetragonal (pseudo-cubic) lattice constants according to Whitfield et al.^[36] (black) and from Schuck et al.^[6] (red). (B) Temperature dependence of the linear thermal expansion coefficients α_[100], α_[001], and α_[averaged] based on the lattice constants of Whitfield et al. [displayed in (A)]^[36]. α_[100]: linear behavior; α_[001]: third-order polynomial (solid lines). Adapted with permission from Schuck et al.^[6]. Copyright 2022 American Chemical Society.

Figure 6. (A) Temperature-dependent values of $${\left[2 c_{13}^{S} \alpha_{[100]}+c_{33}^{S} \alpha_{[001]}\right] \sim \gamma_{[001]}}$$ and (B) temperature-dependent values of $${\left[\left(c_{11}^{S}+c_{13}^{S}\right) \alpha_{[100]}+c_{13}^{S} \alpha_{[001]}\right] \sim \gamma_{[100]}}$$. (A and B) for MAPbI₃ using the tabulated c_ij values from Feng et al.^[45].

In the case of the tetragonal crystal structure of hybrid perovskites, two independent Grüneisen parameters, γ_[100] and γ_[001], are related to the principal linear thermal expansion coefficients α_[100] and α_[001] via (according to Schorr et al.^[43]):

where V_m is the molar volume, C_p is the molar specific heat at constant pressure, and the c_ij are the adiabatic elastic stiffness coefficients in this Grüneisen tensor (according to Haussühl and Kitaigorodskii)^[44]. Using the tabulated c_ij values^[45], a negative Grüneisen parameter γ_[001] can only be expected near the phase transition to the cubic phase above 307 K [Figure 6A]. At lower temperatures, γ_[001] is positive and the Grüneisen parameter γ_[100] is always positive for MAPbI₃ [Figure 6B].

Tetragonal to the cubic phase transition

According to Whitfield et al., the phase transition from the tetragonal to the cubic crystal structure in MAPbI₃ and MAPbBr₃ can be described by the temperature-dependent change of three degrees of freedom in the tetragonal structure: the rotational distortion caused by the change of the Pb-Ia-Pb angle (R₄⁺), the strain caused by the thermal expansion (GM₁⁺), and the strain caused by the tetragonal distortion (GM₃⁺)^[36].

The temperature-dependent changes of the three degrees of freedom were analyzed for MAPbI₃ and MAPbBr₃ using ISODISTORT (an internet-server tool for exploring structural phase transitions)^[46] and are shown in Figure 7. The thermal expansion in the tetragonal phase discussed in the previous two sections is related to the progressive phase transformation towards the cubic structure, since the lattice parameters respond to a shear stress coupled to the order parameters (analyzed with ISODISTORT), and this coupling is predicted by a standard Landau theory treatment. The amplitude of the R₄⁺ rotational mode, which is directly proportional to the cubic-tetragonal order parameter, can be fitted by the power law R₄⁺~(T_c-T)^β, where β is the critical exponent [Figure 7]. Both Whitfield et al.^[36] and Liu et al.^[47] conclude in their analysis that the tetragonal/cubic phase transition for MAPbI₃ can be characterized by a single order parameter and that its behavior as a function of temperature and spontaneous strain follows the form of a weakly first-order, nearly tricritical phase transition, which is consistent with the description of a standard Landau free energy function. The standard Landau theory treatment should then also explain the temperature-dependent behavior of the Pb-X bond lengths and the [PbX₆] octahedra in the tetragonal phase towards the cubic structure, although it is worth noting that there is some variance in the description of the order parameter in the case of MAPbI₃ [Figure 7]. However, the differences between the R₄⁺ data of Schuck et al.^[6] and Weadock et al.^[7] seem to be smaller.

Figure 7. ISODISTORT analysis^[46]. The phase transition from tetragonal to cubic, based on the data from Schuck et al.^[6] and Liu et al.^[47] for MAPbI₃ and based on the data from Weadock et al.^[7] for MAPbBr₃ can be described by the temperature-dependent changes of the three degrees of freedom in the tetragonal structure: rotational distortion mode R₄⁺ (blue color, legend on the right) and the two strain mode amplitudes (black color, legend on the left) GM₁⁺ (thermal expansion) and GM₃⁺ (tetragonal distortion (c-a)/c). Fit of the order parameter R₄⁺ for MAPbI₃ based on the data from Schuck et al.^[6] (blue dashed line) and fit for the data from Liu et al.^[47] (blue dotted line), additional also the R₄⁺ fit from Whitfield et al.^[36] (figure 8A in Ref.^[36]) is reproduced (T_c = 329.95 K and β = 0.26, solid blue line). Fit of the order parameter R₄⁺ for MAPbBr₃ based on the data from Weadock et al. (blue dash-dotted line)^[7].

Local structure of MAPbI₃ and MAPbBr₃ with atomic pair distribution function (PDF) analysis

The local atomic structure can be investigated by analyzing the total scattering from diffraction experiments^[48,49], which, in contrast to XRD or neutron powder diffraction methods, use much shorter X-ray or neutron radiation wavelengths. In addition, total scattering measurements are generally made over a much larger scattering vector range (Q_max = 30 Å^-1) than in conventional XRD experiments (Q_max = 5 Å^-1). Besides Bragg scattering, total scattering includes diffusion scattering. The PDF can be used to analyze the total scattering, whereby a Fourier transformation of the measured total scattering must be carried out after extensive data reduction (including a thorough background correction)^[48,49].

In the temperature-dependent pair correlation functions G(r) of MAPbI₃, it is, first of all, noticeable, as already noted by Whitfield et al., that in a range up to about 4 Å, no discontinuous changes occur in the range from 10 to 350 K, but only a gradual broadening of G(r) towards high temperatures (see figure 4 in Ref.^[36]), which is also especially the case in the range of the phase transformation from the orthorhombic to the tetragonal structure (o/t)^[36]. This range up to 4 Å, which does not change discontinuously, corresponds to the Pb-X bond distances. The situation is different for the range from 4 Å to larger distances, where a discontinuous change in G(r) in the temperature range of the o/t phase transformation is observed.

A very similar behavior can also be observed in the data available for MAPbBr₃. Here, there is also no discontinuous change at small distances in the region of the o/t phase transformation (see figure 1 in Ref.^[17]). In the PDF analyses available in the literature, it was observed for both MAPbI₃ and MAPbBr₃ that in the temperature range of the tetragonal and cubic phases, the low range up to 10 Å, can be best described by a short-range octahedra distortion, which essentially corresponds to the orthorhombic low temperature perovskite crystals structure for both MAPbI₃^[14,17] and MAPbBr₃^[15-17]. To illustrate and qualitatively reproduce this effect, PDF fits in the range of 1-10 Å are shown in Figure 8, using the MAPbBr₃ data at 200 K from Bird et al.^[17] and the refinement results from Bernasconi et al.^[16] (here, only the lead and bromine atoms were considered). In the PDF refinement only, the scaling and U_iso were refined. Similar to Bernasconi et al., the use of the tetragonal I4/mcm perovskite structure (Figure 8A, R_w = 0.3449) leads to a poorer refinement than the use of the orthorhombic Pnma perovskite structure (Figure 8B, R_w = 0.1740)^[16]. Although the results of the PDF analysis in Figure 8 show deviations (in the regions of r = 7 Å and r = 9.5 Å, there are peaks in the simulation that are not present in the data, and the peak at r = 4 Å is significantly off in thickness), the same qualitative results can be obtained with the experimental data of Bird et al., and thus, the observations of Bernasconi et al. can be confirmed qualitatively^[16,17]. The temperature-dependent lattice parameters, Pb-Br bond lengths and [PbBr₆] volumes determined by Bernasconi et al. using the orthorhombic perovskite structure in their PDF refinement for the range of 1-10 Å are shown in Figure 9^[16]. Obviously, the thermal expansion of the local structure of MAPbBr₃ is fundamentally different from the thermal expansion of the averaged structure resulting from the analysis of the XRD data. The PDF refinement results from Bernasconi et al. show that all three orthorhombic lattice parameters increase with temperature, as do the Pb-Br bond lengths and [PbBr₆] volumes^[16]. More recent structural PDF studies using symmetry-adapted PDF analysis (SAPA)^[50] have investigated the local structure of ScF₃ and hybrid perovskites utilizing extended degrees of freedom (such as the “scissoring” mode X₅⁺)^[17,51].

Figure 8. Short-range fit (r range = 1-10 Å) by using (A) the tetragonal MAPbBr₃ perovskite structure (I4/mcm) and (B) the orthorhombic MAPbBr₃ perovskite structure (Pnma). For the PDF fit (with PDFgui 1.0), the data from Bird et al.^[17] at 200 K was used together with the fixed refinement results at 200 K from Bernasconi et al.^[16]. In the PDF fit, only the scale and U_iso were refined.

Figure 9. Results of the PDF analysis by Bernasconi et al.^[16] using the G(r) range 1-10 Å with orthorhombic Pnma symmetry. (A) orthorhombic perovskite structure at 200 K; (B) orthorhombic lattice constants (pseudo-cubic); (C) averaged Pb-Br bond lengths expansion and (D) volumetric expansion of MAPbBr₃. Adapted with permission from Bernasconi et al.^[16]. Copyright 2017 American Chemical Society.

Analysis of the anharmonic properties of the lead-halide bond with EXAFS

EXAFS spectroscopy is based on the ionization of an atom when an X-ray quantum is absorbed. This releases an electron whose kinetic energy depends on the energy of the X-rays. The released electron propagates as a matter wave and is scattered by neighboring atoms. Depending on the wavelength of the electron, constructive or destructive interference occurs between the outgoing wave and the backscattered waves. Because the energy of the X-rays is varied, the energy of the released electrons also changes and thus the corresponding wavelength of the electrons. This leads to alternating constructive and destructive interference and, thus, to a change in X-ray absorption depending on the energy, which corresponds to the fine structure of the X-ray absorption spectrum. These X-ray absorption changes can be measured in the energy range from just above the X-ray absorption edge up to a few hundred electron volts above it, i.e., in the upper range of the X-ray absorption edge (hence the term “extended” in EXAFS). From the shape and strength of the X-ray absorption changes, it is possible to deduce at what distance from the ionized atom it is scattered and to what extent, i.e., a so-called radial distribution function is obtained via Fourier transformation of the X-ray absorption “fine structure” data after normalization and background subtraction (for example with the Athena software)^[52]. By comparing the scattering of electrons on and between neighboring atoms with simulations (for example, with the Artemis software)^[6,52,53], the atomic distances of atom pairs R_EXAFS can be determined with high accuracy. However, the radial distribution function not only contains information about the atomic distances but also about the distance variations of the respective atom pairs, which corresponds to the parallel MSRD (EXAFS Debye-Waller factor or C₂). Since the lifetime of the EXAFS photoelectron (~10^-15 s) is very short compared to the atomic vibrations (~10^-12 s), and, at the same time, the limited mean free path of the EXAFS photoelectron is restricted to the local environment of the investigated atom, the anharmonic properties of the investigated bond pair can be studied very well^[34]. The relative ΔC₂ (T) was fitted with an Einstein behavior by Schuck et al.^[6] to determine parallel MSRD (T)^[54] of MAPbI₃:

(µ: reduced mass, Θ_E: Einstein temperature). The resulting parallel MSRD (T) is shown in Figure 10A. This figure also shows the parallel MSRD (T) for MAPbBr₃, which was interpreted by Weadock et al. using both a correlated Debye model and an Einstein behavior [in the Debye model, a distinction was made between a low-temperature (Θ_D,LT = 148 K) and a high-temperature range (Θ_D,LT = 167 K)]^[7]. Weadock et al. noted that the Einstein temperatures for MAPbI₃ and MAPbBr₃ increase with decreasing reduced mass μ = m₁m₂/(m₁ + m₂), which corresponds to the relationship between the frequency and mass of an oscillator^[7]. From Θ_E, a direct derivation of the effective force constant k₀ (bond-stretching effective force constant) could be made since the following applies:

Figure 10. Temperature-dependent MSRDs: (A) parallel, (B) perpendicular and (C) γ_EXAFS. Red: MAPbI₃ (Schuck et al.^[6]); blue: MAPbBr₃ (Weadock et al.^[7]). Solid lines: Einstein fits. In (B) the parallel MSRD values for MAPbI₃ were fitted with two Einstein temperatures, one for the orthorhombic low-temperature range up to 160 K, the other for the tetragonal high-temperature range up to 260 K (here with an additional static component of 0.05). This procedure also leads to a better explanation of the experimental values for the description of the γ_EXAFS dependence in (C). Adapted with permission from Schuck et al.^[6]. Copyright 2022 American Chemical Society.

[1 N/m = 1/16.022 eV/Ų], [1 THz = 47.9924 K], and ν_E: Einstein frequency [THz] [Table 1].

From the temperature-dependent relative interatomic distances ΔR_EXAFS and the averaged relative interatomic distances ΔR_XRD resulting from the synchrotron XRD [Figure 11], Schuck et al. determine the relative “perpendicular MSRD” ($$\Delta < \Delta \mathrm{u}_{perp}^{2} >$$) for MAPbI₃, which correspond to vibrations perpendicular to the lead-halide bonding direction (bond-bending vibrations)^[6].

Figure 11. Relative Pb-X expansion of MAPbI₃ from Schuck et al.^[6] (red) and MAPbBr₃ (Weadock et al.^[7]) (blue). Positive bond expansion ΔR_EXAFS (solid upward triangles) and negative tension effect (solid downward triangles) result in the overall crystallographic expansion ΔR_XRD (open circles). Values for the cubic perovskite phase of MAPbI₃ are also given from Whitfield et al.^[36] (open red star symbol). Adapted with permission from Schuck et al.^[6]. Copyright 2022 American Chemical Society.

The following relationship applied:^[55]

The absolute values of the perpendicular MSRD for MAPbI₃ were then obtained, analogously to the parallel MSRD, through fitting of $$\Delta < \Delta \mathrm{u}_{perp}^{2} >$$ with a correlated Einstein model^[34]. The parallel MSRD values were fitted with two Einstein temperatures: one for the orthorhombic low-temperature range up to 160 K (LT), the other for the tetragonal high-temperature range up to 260 K (HT). This procedure leads to a solid explanation of the experimental values (also for the description of the γ_EXAFS-dependence [Figure 10B and C, Table 1]). The differing dimensionality ($$\Delta \mathrm{u}_{perp}^{2}=\Delta \mathrm{u}_{x}^{2}+\Delta \mathrm{u}_{y}^{2}$$) compared to the parallel MSRD makes it necessary to use the following term:^[33,34]

Analogous to k₀, the corresponding perpendicular effective force constant k_⊥ (bond-bending effective force constant) from the correlated Einstein model could be calculated for MAPbI₃ [Table 1]^[6]. Analogous to the procedure for MAPbI₃, the relative perpendicular MSRD and the bond-bending effective force constant were also determined for MAPbBr₃ from the available data^[7]. For MAPbBr₃, this was possible only for the tetragonal and cubic phases since the values determined for ΔR_EXAFS and ΔR_XRD in the orthorhombic phase [Figure 11], based on the results of Weadock et al., lead only to unphysical values for the relative perpendicular MSRD (here, the digitization of the data from Weadock et al. could not be carried out with sufficient accuracy for the orthorhombic phase)^[7]. However, since the phase transformation from tetragonal to cubic is being discussed here, this is not limiting. The ratio γ_EXAFS = perpendicular MSRD/parallel MSRD describes the degree of anisotropy of the relative vibrations [Figure 10C]^[34]. The uncertainties pointed out by Schuck et al. may be of a similar (if not larger) magnitude in the behavior of MAPbBr₃ shown here, so that the anisotropy of the relative vibrations and the perpendicular MSRD of MAPbI₃ and MAPbBr₃ are unlikely to differ significantly. However, perpendicular MSRD $$< \Delta \mathrm{u}_{perp}^{2} > (T)$$ reveals the negative tension effect in the lead-halide bond [Figure 11]^[6]. Tension is defined as $$-< \Delta \mathrm{u}_{perp}^{2} >$$/2R_XRD^[56]. In the tetragonal/cubic phases of MAPbI₃ and MAPbBr₃, strong negative tension effects result in a shortening of R_XRD, which is accompanied by the observed negative changes in the partial volume of the [PbX₆] octahedra. Tenison effects are often used to explain the structural mechanism of NTE^[57], and these tension effects are also associated with the behavior of optical phonons^[58,59]. Although Dalba et al. point out that “in general the Einstein frequencies cannot be correlated in a simple way to the phonon density of states”^[60], the same authors draw parallels between the relative atomic vibrations of EXAFS and optical phonons^[60,61].

Also, an attempt is made here to establish correlations between the relative atomic vibrations obtained from EXAFS studies on one hand and the behavior of optical phonons on the other. In addition, the Grüneisen parameters of individual phonon modes (also known as phonon deformation potentials) are of interest as a probe of the anharmonicity associated with phonon-phonon scattering and heat transport^[59,62,63]. Two recent publications investigated the temperature dependence of optical phonons using infrared (IR) and terahertz measurements of MAPbI₃ and MAPbBr₃^[64,65]. Two optical phonons at around 0.9 and 1.8 THz have been reported from Boldyrev et al. in MAPbI₃ single crystals using temperature-dependent (figure 5 in Ref.^[64]), high-resolution terahertz reflection spectroscopy^[64]. The authors of the study attribute both modes to the inorganic [PbI₆] framework structure, with the low-frequency mode dominated by bending modes, while stretching processes play a larger role in the higher-frequency mode. Both modes observed from Boldyrev et al. show parallels to the bond-stretching frequencies of ν_E|| = 2 THz and to the bond-bending frequency of $${\nu}_{{E} \perp}$$ = 1 THz for MAPbI₃ both observed with EXAFS^[64]. In a similar temperature-dependent time-domain THz transmission and far IR reflectance investigation on optical phonons in MAPbBr₃ single crystals, the higher frequency mode measured by Železný et al. shows an increase in the frequency in the tetragonal phase from approx^[65]. 2.07 THz to approx. 2.12 THz (figure 5 in Ref.^[65]), which corresponds to a negative mode Grüneisen parameter for this mode. In contrast, the low-frequency mode at 1.4 THz observed by Železný et al. shows hardly any temperature dependence^[65]. The similarities to the EXAFS results of MAPbBr₃ are also present here as the bond-stretching frequencies of ν_E|| = 2.4 THz and the bond-bending frequency of $${\nu}_{{E} \perp}$$ = 1 THz for MAPbBr₃ are observed with EXAFS. Note the frequency increase in the EXAFS results; for MAPbBr₃, the bond-stretching frequencies determined with a Debye model are lower in the low-temperature range (3.1 THz) than in the high-temperature range (3.5 THz). This frequency increase can also be observed, although to a lesser extent, in the bond-stretching frequencies for MAPbI₃.