Enhanced lattice perfection by low temperature thermal annealing in photoelectric (CH₃NH₃)PbBr₃

Abstract

The coupling between the organic CH₃NH₃⁺ cations and inorganic perovskite PbBr₃⁻ framework in a large single crystal of (CH₃NH₃)PbBr₃ weighting 13 g was studied using neutron diffraction and inelastic neutron scattering. Two lattice incommensurate (ICM) phases were found, one at higher temperatures, marked ICM_HT, which appeared between 147 and 135 K. The second one, marked ICM_LT, developed below 143 K and remained at 75 K. The transition from the ICM_LT to ICM_HT phase upon warming gave rise to extremely large lattice shrinking, followed by extremely large lattice expansion of the tetragonal basal plane of the PbBr₃ lattice. There was a progressive decrease in the width of the Bragg peaks from the PbBr₃ lattice upon warming, which can be described using a critical exponent for each type of Bragg peak to show complete ordering of the atoms into a (CH₃NH₃)PbBr₃ lattice at 194 K. (CH₃NH₃)PbBr₃ exhibits six definitive acoustic-like phonon branches at 75 K. The six branches renormalizes into two at 200 K, with the frequencies of both the transverse and longitudinal modes greatly enhanced. The asymmetric structure of the CH₃NH₃ ions helps to understand the observed behaviors.

The outstanding photo-to-electric conversion efficiency found in organic-inorganic hybrid lead trihalide perovskites has made this class of material a promising resource for photovoltaic solar cell applications^1-8. Many studies have demonstrated that the orientational degrees-of-freedom of the anisotropic organic cations have decisive impact on the configuration of the electronic properties and the excellent photoelectronic efficiency of hybrid perovskites^9-15. The methylammonium cations (CH₃NH₃⁺, MA) possess a large dipole, which frequently displays orientational disordering at room temperature when incorporated into an inorganic perovskite framework^16-19. High photoelectric performance has been observed in the bromide version of the (CH₃NH₃)PbBr₃ perovskites, where CH₃NH₃Br acts as the light absorbing component^20-26. The crystalline structure of (CH₃NH₃)PbBr₃ may be viewed as consisting of corner-sharing PbBr₆⁻ octahedra that are connected along all three crystallographic axial directions to host the CH₃NH₃⁺ cations in the interstitial spaces of the PbBr₆ octahedra. The CH₃NH₃⁺ cations are relatively weakly bonded to the PbBr₃ framework (Fig. 1a). There are two sublattices of PbBr₃⁻ and CH₃NH₃⁺ that can be distinguished in (CH₃NH₃)PbBr₃. The Pb ions are crystallized in a 6-fold coordination, surrounded by an octahedron of bromide anions together with the CH₃NH₃⁺ cations in a 12-fold cuboctahedral coordination²⁷. (CH₃NH₃)PbBr₃ crystalizes into a cubic Pm3m symmetry at 300 K. It undergoes a series of structural changes due to changes in the thermal orientational order of the CH₃NH₃⁺ cations and tilting of the PrBr₃ host framework upon cooling. (CH₃NH₃)PbBr₃ renormalizes into a tetragonal I4/mcm crystalline symmetry at 237 K. An intermediate structural phase, covering a short temperature range from 146 to 140 K, prior to transformation into the orthorhombic Pnma phase, has been identified by low-frequency Raman scattering^5,28-30. X-ray diffraction has demonstrated the appearance of an incommensurate (ICM) distortion for the intermediate phase, which has been ascribed to be a direct result of organic-inorganic coupling⁵. In addition, the orientational degrees-of-freedom of the organic CH₃NH₃⁺ cations are believed to link directly to the good photoelectric properties of (CH₃NH₃)PbBr₃ (refs 9-15). Understanding the structural coupling between the CH₃NH₃⁺ and the host PbBr₃⁻ framework is essential in understanding the electronic behavior of the materials.

Figure 1 ∣. Crystal structure and the single crystal sample.

(a) Crystal structure of (CH₃NH₃)PbBr₃ in the cubic Pm3m phase at high temperatures. (b) Photo image of the (CH₃NH₃)PbBr₃ single crystal used in this study. The crystal weighed 13 g. The flat surface indicates the basal a-c plane.

To explore the coupling between the CH₃NH₃⁺ cations and the PbBr₃⁻ framework, we carried out detailed neutron diffraction and inelastic neutron scattering measurements on a large single crystal of (CH₃NH₃)PbBr₃ (weighting 13 g, Fig. 1b). Lattice incommensurability which develops below 147 K transforms into a separate ICM phase at 143 K, accompanied by extremely large lattice shrinkage followed by extremely large lattice expansion upon cooling through the transition covering the range from 143 to 135 K. The ICM phase remains at 75 K. Thermal weakening of the CH₃NH₃⁺ sublattice strengthens the atomic ordering of the PbBr₃⁻ framework. The degree of atomic ordering can be described by a critical exponent for each Bragg peak. In addition to the two acoustic phonon branches from the PbBr₃⁻ lattice, the CH₃NH₃⁺ sublattice is solid enough to support another four phonon branches with zone centers at the ICM positions.

Results

Lattice incommensurability.

Neutron diffraction measurements reveal lattice constants of a = c = 8.329(3) Å and b = 11.749(5) Å at 75 K (Fig. 2a), showing a tetragonal structure for the present (CH₃NH₃)PbBr₃ crystal. A neutron diffraction intensity map in the (h0l) scattering plane taken at 75 K reveals satellite components associated with every measured Bragg peak (Fig. 2c). The data were collected by fixing the angular position of the position-sensitive detector, which covers about five degrees, rotating the crystal through a wide angular range, and then combining the scans into a map. The positions of these satellite reflections cannot be expressed as rational numbers of the reciprocal unit cell vectors, demonstrating the appearance of lattice incommensurability. We remark that lattice incommensurability occurs in all three components of the wave vector as discussed in detail below. Four ICM satellite components appear for the Bragg peaks with h≠l. but only two for the Bragg peaks with h=l. The ICM reflections associated with the (200) and (002) Bragg peaks are essentially the same, reflecting the tetragonal nature of the crystal at 75 K (Fig. 2b). Weaker but definitive Bragg intensities appear at the (3/2, 0, 1/2), (5/2, 0, 1/2), (1/2, 0, 3/2) and (1/2, 0, 5/2) positions as well. No ICM component is found to be associated with these half-integer Bragg reflections. The data observed at 75 K can be compared with the observations reported in Ref. 5, where the ICM phase and half-integer Bragg reflections disappear below 140 K. The noticeable differences in intensity for the satellite components of h=k and for the h≠k Bragg peaks reflect the difference in the structure factor of the satellite components.

Figure 2 ∣. Neutron-diffraction intensity map.

(a) (200) (open circles) and (002) (filled triangles) Bragg peaks obtained at 75 K. (b) ICM reflections associated with the (200) (open circles) and (002) (filled triangles) Bragg peaks obtained at 75 K. (c) Neutron diffraction intensity map in the (h0l) scattering plane of the (CH₃NH₃)PbBr₃ single crystal at 75 K. Satellite reflections at incommensurate positions are revealed associated with every Bragg peak. Four satellite reflections associated with the Bragg peaks with h≠l are visible, whereas only two are visible for the Bragg peaks with h=l. The Bragg intensities at the (3/2, 0, 1/2), (5/2, 0, 1/2), (1/2, 0, 3/2) and (1/2, 0, 5/2) positions are also visible.

The satellite component associated with a Bragg peak G(hkl) is specified by the addition of a modulation wavevector q(q_h, q_k, q_l), so that the wave vector of a satellite component Q(Q_hQ_kQ_l) = G(hkl) + q(q_h, q_k, q_l), where G is a reciprocal lattice vector. Detailed investigation of the positions of the ICM reflections reveals that the modulation vectors of the ICM reflections associated with a Bragg peak do not appear in pairs, as viewed from the scattering plane, reflecting the existence of q_k components for the modulation vectors. Interestingly, diffraction measurements performed at a higher q-resolution reveal two additional ICM reflections associated with each ICM reflection, as shown in Fig. 3a for the (101) Bragg peak and in Fig. 3b for the (200) Bragg peak.

Figure 3 ∣. Neutron-diffraction intensity near fundamental Bragg peaks.

(a) Neutron-diffraction intensity map in the (h0l) plane near the (101) Bragg peak at 75 K, revealing two satellite reflections associated with the Bragg peak and two additional satellite reflections associated with each satellite reflection. (b) Neutron-diffraction intensity map in the (h0l) plane near the (200) Bragg peak at 75 K, revealing four satellite reflections associated with the Bragg peak and two additional satellite reflections associated with each satellite reflection.

Thermal evolution of lattice incommensurability.

Four ICM reflections associated with the (200) Bragg reflection appeared in the (hk0) scattering plane at 75 K (Fig. 4a). Two ICM branches appeared along the H-axis direction, marked q₁ and q₃, the other two along the K-axis direction, marked q₂ and q₄. Figures 4b and 4c illustrate the temperature-dependencies of the diffraction intensities across the (200) Bragg peak in longitudinal scans along the H-axis direction (Fig. 4b), marked (200)L, and in transverse scans along the K-axis direction (Fig. 4c), marked (200)T. An ICM phase, marked ICM_LT, appeared at 75 K, the lowest temperature reached in this study. The modulation vectors for the two branches along the H-axis direction are determined to be q_1h = +0.081 and q_3h = −0.063 at 75 K. They appear at each specific q_h, rather than in a ±q_h pair (Fig. 4a). Upon warming, both q_1h, and q_3h shift closer to the (200) Bragg position at the same thermal decrease rate (Fig. 5a). In addition, the intensity of q₁ is much weaker than that of q₃ (the integrated intensity of q₁ is 56% weaker than that of q₃ at 75 K) (Fig. 4b). The differences in position and intensity for q₁ and q₃ detected in the (hk0) scattering plane indicate that q₁ and q₃ have non-zero q_l-components. In addition, there are ICM reflections associated with all q₁, q₂, q₃ and q₄ reflections, as shown in Fig. 3b. Apparently, it was the ICM reflection associated with q₁ that was detected by the instrumental resolution upon scanning in the (hk0) scattering plane, which caused q₁ to appear at a higher q_h with a weaker intensity. On the other hand, the two ICM branches along the K-axis direction appeared in a pair at q_2k = +0.149 and q_4k = −0.149 at 75 K, with the same thermal decrease rate (Fig. 5b).

Figure 4 ∣. Thermal evolution of the diffraction profile near the (200) Bragg peak.

(a) Neutron-diffraction intensity map in the (hk0) plane near the (200) Bragg peak at 75 K. Four satellite reflections, marked q₁, q₂, q₃ and q₄ are clearly visible. (b)-(c) Temperature dependencies of the diffraction pattern near the (200) Bragg peak, taken in (b) longitudinal scans along the a-axis direction and (c) transverse scans along the b-axis direction. Satellite reflections, marked q₁′ and q₃′, in addition to q₁ and q₃, are visible near the transition in (b). Satellite reflections, marked q₂′ and q₄′, separated from q₂ and q₄, are visible near the transition in (c).

Figure 5 ∣. Modulation vectors of the ICM reflections.

(a) Temperature dependencies of the modulation vectors, q_1h, and -q_3h, of the two ICM reflections along the h-axis direction, revealing largely separated values for q_1h, and -q_3h, but essentially the same thermal reduction rate. (b) Temperature dependencies of the modulation vectors, q_2k and -q_4k, of the two ICM reflections on the k-axis direction, revealing that q_2k and q_4k appear in a ±q_k pair.

Another series of ICM reflections, separated from the ICM_LT phase, which developed prior to the CH₃NH₃⁺ sublattice, is thermally disordered (Figs. 4b and 4c). This ICM phase, marked ICM_HT, developed at a higher temperature than the ICM_LT one. Additional intensities developed near the (200) Bragg peak, labelled q₂′ and q₄′, upon warming to 135 K, following a decrease in the intensities of q₂ and q₄ (Fig. 6a). Thus, q₂′ and q₄′ shift away from the (200) Bragg peak on further warming (Fig. 6b), indicating that q₂′ and q₄′ arise from the development of additional peaks rather than the broadening of the (200) Bragg peak. In addition, q₁′ and q₃′, together with q₂′ and q₄′, are all separated from the (200) Bragg peak, as can be clearly seen in the intensity map taken at 139.5 K (Fig. 6c). The ICM_LT and ICM_HT phases coexist between 135 and 143 K, above which ICM _LT is thermally disordered (Fig. 6a). The ICM_HT phase survives up to 147 K, above which the crystal enters an ICM-free phase to form the (CH₃NH₃)PbBr₃ lattice (Fig. 6a). The ICM phase reported in Ref. 5 appears within a short temperature range between 140 and 146 K prior to transfer into the orthorhombic phase below 140 K. In the present crystal, the ICM phase appears over a much broader temperature range, supporting two, rather than one, ICM phases, the one developing at low temperatures remaining at 75 K. Likely, the difference is linked to the amount of CH₃NH₃⁺ that is present in the crystal, since the development of the ICM phase links directly to the orientational order of the CH₃NH₃⁺ ions^3,5. Furthermore, the Bragg peaks become more intense in the ICM-free phase at high temperature (Figs. 4b and 4c). There is a substantial increase in the Bragg intensities upon warming through the transition, reaching maxima upon entering the ICM-free phase at 147 K. The diffraction intensities from the CH₃NH₃⁺ sublattice at low temperatures renormalize into the intensities of the (CH₃NH₃)PbBr₃ lattice after entering the ICM-free phase.

Figure 6 ∣. Diffraction profile in the transition region.

(a) Thermal variations of the intensities near the (200) Bragg peak, covering 130 to 150 K. q₂ and q₄ indicate the ICM reflections in the k-axis direction. They disappeared above 143 K. q₂′ and q₄′ mark the ICM reflections which developed between 135 and 147 K. (b) Diffraction profiles near the (200) Bragg peak taken along the k-axis direction at three representative temperatures, revealing the weakening of q₂ and q₄ together with the development of q₂′ and q₄′. (c) Diffraction intensity map near the (200) Bragg peak in the (hk0) scattering plane at 139.5 K, revealing the appearance of ICM reflections q₁′, q₂′, q₃′ and q₄′ in addition to the ICM reflections q₁, q₂ and q₄.

Enhanced lattice perfection at high temperatures.

Diffraction measurements allow extraction of lattice perfection, where the width of the intrinsic diffraction profile is directly linked to the periodic perfection of the atom arrangement. The widths of both the (020)T (Fig. 7a) and (020)L (Fig. 7b) Bragg peaks become narrower upon warming throughout the whole temperature range studied, from 75 to 185 K, which is likely an extinction effect associated with the depth on penetration of the incident beam and changing perfection of the lattice. For the (020) Bragg peak, the scan taken along the axial b-axis direction (020)L (open triangles in Fig. 8a) is much broader than the scans taken along the in-plane a-axis direction (020)T (open circles in Fig. 8a). Narrowing of the Bragg peak upon warming is also evident, but at a much reduced rate, for the (200)L (Fig. 4b) and (200)T (Fig. 4c) Bragg peaks. Peak widths, in terms of full-width-at-half-maximum (FWHM), W of 0.041 Å⁻¹ for (020)L, 0.023 Å⁻¹ for (020)T, 0.021 Å⁻¹ for (200)T and 0.023 Å⁻¹ for (200)L were obtained at 75 K (Fig. 8). The widths of the (020) and (200) Bragg peaks are much broader than the instrumental resolution of 0.014 Å⁻¹. On the other hand, the ICM reflections are narrower with no significant thermal variation in peak width (Figs. 4c and 7a), as expected since they are much weaker and hence not subject to much if any extinction. The (020)T+q₁ and (020)T+q₃ reflections at 75 K were within the instrumental resolution of W~0.014 Å⁻¹ (filled symbols in Fig. 8a), while those for (200)T+q₂ and (200)T+q₄ were much broader with W~0.02 Å⁻¹ (filled symbols in Fig. 8b). Imperfection in the periodic arrangement of the atoms in the PbBr₃⁻ lattice appeared even at 75 K, especially for the arrangement along the axial b-axis direction. On the other hand, lattice imperfection in the CH₃NH₃⁺ sublattice appeared in the a-c plan, but not along the axial b-axis direction. Unexpectedly, at 75 K the atoms in the CH₃NH₃⁺ sublattice had a higher spatial order than those in the PbBr₃⁻ lattice. Apparently, the CH₃NH₃⁺ sublattice and PbBr₃⁻ lattice are chemically connected internally but must be connected to each other as well, even though they form a sublattice with a different, incommensurate, spatial periodicity.

Figure 7 ∣. Thermal evolutions of the diffraction profile near the (020) Bragg peak.

Thermal variations of the diffraction pattern near the (020) Bragg peaks, taken in (a) transverse scans along the h-axis direction and (b) longitudinal scans along the k-axis direction. The (020) Bragg peak narrows progressively with increasing temperature. Satellite reflections, marked q₁′ and q₃′, are visible in addition to q₁ and q₃ near the transition.

Figure 8 ∣. Thermal variation of the peak widths.

Temperature dependence of the widths of (a) (020) Bragg peak and (b) (200) Bragg peak together with the associated ICM reflections, where T indicates measurements performed in transverse scans and L for longitudinal scans. There is a progressive decrease in the widths of the (020) and (200) Bragg peaks taken in transverse scans (open circles) and in longitudinal scans (open triangles) when the temperature is increased, except in the transition region of 135 to 147 K. The widths of the ICM reflections (filled symbols) are narrower than those of the Bragg peaks (open symbols).

Widths of the Bragg peaks deceased considerably with increasing temperature (open symbols in Figs. 8a and 8b), showing that the atoms in the PbBr₃⁻ lattice became more ordered at high temperatures. Interestingly, the addition of thermal energy improved the periodic arrangement of the atoms. Thermal annealing of lattice perfections can also be seen in the in-plan correlations in the CH₃NH₃⁺ sublattice, but only in the transition from the ICM_LT to ICM_HT phase (filled symbols in Fig. 8b). Improving of the periodic perfection of the atom arrangement by thermal energy in this temperature range is rarely seen. The present observations can be understood assuming that at low temperatures the CH₃NH₃⁺ ions simultaneously participate in the CH₃NH₃⁺ sublattice and in the PbBr₃⁻ lattice, having a different spatial periodicity. The inability of the CH₃NH₃⁺ ions to follow the spatial periodicity of the CH₃NH₃⁺ sublattice or that of the PbBr₃ lattice, gives rise to a CH₃NH₃⁺ arrangement that is imperfect for both lattices. This can happen only if the interaction between two neighboring CH₃NH₃⁺ ions is comparable to that which occurs between CH₃NH₃⁺ and PbBr₃⁻ ions.

Thermal variations of the peak width W(T) display an anomaly in the transition region (135 to 147 K), reflecting the ICM and Bragg reflections are atomically interconnected. The anomaly in W(T) is more pronounced for the (200) Bragg peak than for the (020), signaling that thermal reorientation of the CH₃NH₃⁺ ions is more pronounced in the tetragonal a-c plane than in the axial b-axis direction. Deconvolution of the observed width from the instrumental resolution allows extraction of the intrinsic width W_I of the reflection. Figure 9 illustrates the temperature dependencies of the intrinsic widths excluding the transition region. All four W_I(T) curves observed can be described (dashed curves in Fig. 9) using a critical exponent β as W_I(T) = W₀{1-(T/T_C)}^β, where W₀ indicates the intrinsic width at zero temperature and T_C is the temperature at which thermal reorientation of the CH₃NH₃⁺ ions is completed. All four W_I(T) curves give a T_C≈194 K (Fig. 9).

Figure 9 ∣. Order parameter.

Temperature dependencies of the intrinsic width W_I of the (020) and (020) Bragg peaks. T indicates measurements performed in transverse scans and L for longitudinal scans. The yellow shaded region indicates the transition region. W_I decreases progressively with increasing temperature. The dashed curves indicate the results of fits of the data to the expression listed in the plot. All four curves give a T_C ≈ 194 K.

Two-step transition.

The lattice constants of the PbBr₃₋ unit cell display a two-step change in the transition from the ICM to ICM-free phase, where sharp changes in the in-plane lattice constant a and in the axial lattice constant b appear in separate temperature regimes (Fig. 10). An extremely large negative thermal expansion (shrinking 0.25% from 135 to 143 K) of the lattice constant a is followed by an extremely large positive thermal expansion (expansion of 0.47% from 143 to 147 K) in the transition from the ICM_LT to the ICM_HT phase, whereas the lattice constant b displays no anomalous changes (yellow shaded region in Fig. 10). On the other hand, the transition from the ICM_HT to the ICM-free phase gives rise to an extremely large positive thermal expansion (expansion of 0.23% from 143 to 147 K) of the lattice constant b, while the lattice constant a remains at essentially the same value (blue shaded region in Fig. 10). Remarkably, although the ICM to ICM-free transition occurs within a short temperature range of 12 K, it generates a large lattice expansion of 0.22% in the in-plane a-axis direction and 0.29% in the axial b-axis direction. The extremely large changes of the lattice constants upon warming through the transition link directly to the reorientation of the CH₃NH₃⁺ ions, where the CH₃NH₃⁺ ions renormalize from being ordered to form the CH₃NH₃⁺ sublattice to being attached to the PbBr₃⁻ lattice, with a large change in the spatial periodicity of the CH₃NH₃⁺ ions.

Figure 10 ∣. Thermal variation of the lattice constants.

Temperature dependencies of the lattice constants a (open squares) and b (open triangles). Large negative thermal expansions followed by large positive thermal expansions are seen for a in the regime where ICM_LT and ICM_HT coexist (yellow shaded region). Large positive thermal expansion of b in the ICM_HT-solely regime (blue shaded region) is also visible.

ICM phonons.

Phonon dispersions were measured by inelastic neutron scattering (INS), using the cold neutron triple-axis spectrometer SIKA at ANSTO, with a fixed final energy of 3 meV. The neutron wavevector transfer is denoted by Q(Q_hQ_kQ_l = G(hkl) + p(HKL), where G is a reciprocal lattice vector and p the phonon wavevector. Figure 11 shows several phonon excitation spectra observed in constant-Q scans at Q = (020) + (H00). Two phonon excitations at E = 0.24 and 0.37 meV are detected at H = 0 at 75 K, with peak widths (0.085 meV for the peak at 0.24 meV) that are ~1.6 times the instrumental resolution (at 0.052 meV), reflecting the short lifetimes of the acoustic phonons propagating along the [100] direction (filled squares in Fig. 11a). Deconvolution of the observed width with the instrumental resolution gives an intrinsic full-width-at-half-maximum W_I of 0.067 meV for the E = 0.24 meV phonon at H = 0, corresponding to a lifetime of τ = 19.7 ps calculated using τ = (π·W_I)⁻¹ with τ in units of ps and W_I in THz³¹. These two phonons shift to higher energies at K = 0.02 (open circles in Fig. 11a) but shift to lower energies at K = −0.02 (open triangles in Fig. 11a), which shows that the zone center for these two branches are not at the zone center of the PbBr₃⁻ lattice but belong to the ICM branches. Six phonon lines can be identified in the excitation spectrum taken at H = 0.12 (Fig. 11b). The intensities for these six phonons cannot all originated from the PbBr₃⁻ lattice, but reflect the existence of dynamic structure factor contributions from the CH₃NH₃⁺ sublattice at 75 K.

Figure 11 ∣. Phonon excitation spectra.

Inelastic neutron scattering spectra obtained in constant-Q scans at the representative wavevectors of (a) H = −0.02, 0, 0.02 and (b) H = 0.12, measured in transverse scans along the [H00] direction at 75 K. The inset to (b) shows the high-energy portion of the H = 0.12 spectrum at an enlarged intensity scale. Two phonon excitations at low energy are clearly revealed in the H = −0.02, 0 and 0.02 spectra; whereas six appeared in the H = 0.12 spectrum.

Figure 12a displays the phonon excitations measured at 75 K, covering 30% of the Brillouin zone. These phonon excitations can be grouped into six branches. Each can be satisfactorily described using a harmonic dispersion E(p) = E_hsin{π/2(p+p₀)} (solid lines in Fig. 12a), where p indicates the phonon wavevector in dimensionless reciprocal lattice unit (r.l.u.), p₀ specifies the zone center of the branch based on the PbBr₃⁻ unit cell, and E_h is the harmonic energy at the zone boundary. The six branches fall into three separate groups, with zone centers p₀ at H = +0.08, 0, and −0.08. There are transvers acoustic (TA) and longitudinal acoustic (LA) modes in each group. The two branches at p₀ = 0 are the TA phonons (labeled T_a2) and LA phonons (labeled LA_a2) propagating along the [100] direction of the PbBr₃⁻ lattice, namely the CM phonons; whereas the two at p₀ = +0.08 (labeled TA_a1 and LA_a1) and the two at p₀ = −0.08 (labeled TA_a3 and LA_a3) are from the vibrations of the CH₃NH₃⁺ sublattice, namely the ICM phonons. The two CM branches remain while the four ICM branches disappear at 200 K (Fig. 12b). The energy parameters of the harmonic dispersions obtained from the fits are listed in Table 1. The E_h of the ICM and CM phonons are comparable, revealing that the force constant of the CH₃NH₃⁺ sublattice is weaker as the masses associated with the CH₃NH₃⁺ sublattice are lighter. Unexpectedly, the E_h of the two CM branches become significantly higher (9% increases for the TA mode and 7% for the LA mode) at 200 K, showing that the crystallization of the CH₃NH₃⁺ ions into the PrBr₃⁻ lattice give rise to a stronger force constant for the lattice.

Figure 12 ∣. Phonon dispersions.

Phonon dispersion curves of (CH₃NH₃)PbBr₃ along the [H00] crystallographic direction at (a) 75 K and (b) 200 K. The solid curves indicate the results of fits of the data to harmonic expressions as discussed in the text. Six phonon branches appeared at 75 K, which fall into three groups. The two branches emanating from the zone center at H = 0, marked LA_a2 (solid triangles) and TA_a2 (open triangles). The two emanating from the zone center at H = 0.08, marked LA_a1 (solid squares) and TA_a1 (open squares), and the two emanating from the zone center at H = −0.08, marked LA_a3 (solid circles) and TA_a3 (open circles). The four ICM branches disappeared but the two CM branches remained at 200 K.

Table 1 ∣.

Energy parameters obtained from the fits of the observed phonon energy dispersions at 75 and 200 K. p₀ is the zone center of the branch based on the PbBr₃ unit cell, E_h is the harmonic energy at the zone boundary, and V_0.1 is the phonon group velocity at p = 0.1.

E(p) = Eh sin{π/2(p+p0)}
Temp.		TAa1	LAa1	TAa2	LAa2	TAa3	LAa3
75 K	p0 (r.l.u.)	0.08(1)	0.08(2)	0	0	−0.08(2)	−0.08(1)
	Eh (meV)	2.14(3)	3.16(5)	2.15(3)	3.00(4)	2.15(3)	2.98(3)
	V0.1 (m/s)	685	988	671	942	669	927
200 K	p0 (r.l.u.)	-	-	0	0	-	-
	Eh (meV)	-	-	2.35(3)	3.21(4)	-	-
	V0.1 (m/s)			769	1008

Discussion

The PbBr₃′s lattice in (CH₃NH₃)PbBr₃ constructs a perovskite framework to accommodate CH₃NH₃⁺ in the interstitial spaces of the corner-sharing PbBr₆⁻ octahedra. CH₃NH₃⁺ displays a three-fold symmetry when viewed along the C-N bonds, where rotation about the C-N bonds can result in many different possible molecular conformations for CH₃NH₃⁺. The formation of a structural ICM lattice (Figs. 2 and 3), superimposed on the PbBr₃⁻ lattice arises from the rotational ordering of the CH₃NH₃⁺ ions which bond to the Br⁻ ions, but with a different spatial periodicity from that of the PbBr₃⁻ lattice. There is hence an ICM sublattice from the CH₃NH₃⁺ ions superimposed on the CM lattice from the PrBr₃⁻ ions, each with its own spatial periodicity and generating a series of Bragg reflections. A modulation vector of q = (0.08, 0.15, 0.08) at 75 K (Figs. 5 and 12), reveals the positioning of the CH₃NH₃⁺ ions in the ICM sublattice at an angle of (360°/3)×0.08 = 9.6° relative to the neighboring CH₃NH₃⁺ ions in the in-plane a- and c-axis directions and at (360°/3)×0.15=18° in the axial b-axis direction. The chemical connections of the CH₃NH₃⁺ ions to the PbBr₃⁻ lattice with different orientational angles for the two neighboring CH₃NH₃⁺ ions give rise to an atomic disordering of the CM unit cells, hence the broad CM Bragg peaks (Fig. 9). The formation of an ICM CH₃NH₃⁺ sublattice occurs through the appearance of CH₃NH₃-Br bonding in the next nearest neighboring Br ions along all three crystallographic directions. The coupling strength of the ICM sublattice is comparable to that of the CM lattice and is sufficient to support four acoustic phonon branches with similar dispersions to that of the CM lattice (Fig. 12). This requires strong coupling between two neighboring CH₃NH₃-Br bonds. Apparently, it is the indirect coupling between two neighboring CH₃NH₃-Br ions mediated through the nearest neighbor Br ion that construct the CH₃NH₃⁺ sublattice. One of the Br ions in (CH₃NH₃)PbBr₃ bonds to the CH₃NH₃⁺ ions, the other two mediate the CH₃NH₃-Br bonds that form the solid ICM lattice. We note that in a material that exhibits an ICM structure, the acoustic phonons emanating from the ICM Bragg peaks appear to be distinct from the acoustic phonons emanating from the fundamental Bragg peaks. In the (macroscopic) long wavelength limit they must be the same, but this typically occurs at much smaller wave vectors than investigated in inelastic neutron scattering experiments, as detailed theoretically by Finger and Rice³².

In the low temperature ICM_LT phase, the angle of inclination between the neighboring CH₃NH₃⁺ ions decreases when the temperature is raised (Fig. 5), which reduces the degree of atomic disorder and leads to narrower Bragg peaks than for the CM lattice at higher temperatures (Fig. 9). The ICM^LT phase begins to thermally relax upon warming to 135 K, renormalizing into a new sublattice, characterized by the ICM_HT phase, prior to thermal disordering of the CH₃NH₃⁺ ions (Fig. 6). It is the loss of indirect coupling mediated by the nearest Br ions that melts the ICM lattice, while the CH₃NH₃-Br bonds remain attached. The orientational order of the CH₃NH₃⁺ ions does not switch to follow the periodicity of the CM lattice once the ICM lattice is thermally disordered at 147 K, but joins the CM periodicity at 194 K, above which spatial incomparability between the CH₃NH₃⁺ and PbBr₃⁻ ions disappears to form an atomically ordered (CH₃NH₃)PbBr₃ lattice (Fig. 9).

The thermal evolutions of a CH₃NH₃⁺ ICM sublattice superimposed on the PbBr₃⁻ CM lattice in a large (CH₃NH₃)PbBr₃ single crystal was studied by neutron diffraction and inelastic neutron scattering. Two ICM phases were found. The spatial periodicity of the ICM lattice was characterized by the orientational position of the CH₃NH₃⁺ ions relative to the PbBr₆ octahedra, and is found to be very sensitive to temperature. The low temperature phase, namely ICM_LT, was seen at the lowest temperature reached of 75 K. An ICM_LT lattice formed from the coupling between the nearest neighboring Br ions in the PbBr₆ octahedra, which was strong enough to support lattice vibrations of similar strengths to the CM lattice. The rapid repositioning and reorientation of the CH₃NH₃⁺ ions upon warming through the transition from the ICM_LT phase to the ICM_HT phase, gives rise to extremely large shrinking of the lattice followed by an extremely large expansion of the in-plane lattice. The orientational order of the CH₃NH₃⁺ ions switched to follow the periodicity of the CM lattice at a temperature 1.32 times higher than the temperature at which ICM phase melts.

Methods

Fabrication of (CH₃NH₃)PbBr₃ single crystal.

A single crystal of (CH₃NH₃)PbBr₃ was grown using the solution method. High purity CH₃NH₃Br and PbBr₂ (both 99.999% pure) precursors were dissolved at a molar ratio of 1:1 in dimethylformamide (DMF) to obtain a >1 M solution of (CH₃NH₃)PbBr₃. The solution was filtered by polytetrafluoroethylene (PTFE) filters to remove impurities and then heated to 75 °C for crystal nucleation and growth. One larger (CH₃NH₃)PbBr₃ single crystal (~ 0.5 g) was selected as a seed and soaked in a nearly-saturated (CH₃NH₃)PbBr₃ solution at 75 °C for 2 weeks. The rectangular single crystal used in the present measurements weighed 13 g (Fig. 1b).

Neutron diffraction and inelastic scattering measurements.

Single crystal neutron diffraction measurements performed in the a-c scattering plane were conducted at the NIST Center for Neutron Research, using the BT-7 triple-axis spectrometer and the spin-polarized triple-axis spectrometer SPINS. BT-7 was operated in the diffraction mode, employing an incident wavelength of λ = 2.35 Å (14.7 meV) defined by pyrolytic graphite (PG) (002) crystals at both the monochromator and analyzer positions, with PG filters and a PSD detector system³³. SPINS was operated in the diffraction mode, employing an incident wavelength of λ = 4.04 Å (5 meV) defined by PG(002) crystals with Be filters. The sample temperature was controlled using a liquid He refrigeration system, allowing the sample to be in a helium gas atmosphere.

Single crystal neutron diffraction and inelastic scattering measurements performed in the a-b scattering plane were conducted at the Bragg Institute, ANSTO, using the cold neutron triple-axis spectrometer SIKA with the energy of neutrons defined by PG(002) crystals at both the monochromator and analyzer positions, using a fixed final energy of 3 meV and a Be filter to suppress higher order contaminations. The sample temperature was controlled using a liquid He refrigeration system, allowing the sample to be in a helium gas atmosphere.

Data availability.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.