Activation Energy of Organic Cation Rotation in CH3NH3PbI3 and CD3NH3PbI3: Quasi-Elastic Neutron Scattering Measurements and First-Principles Analysis Including Nuclear Quantum Effects

The motion of CH3NH3+ cations in the low-temperature phase of the promising photovoltaic material methylammonium lead triiodide (CH3NH3PbI3) is investigated experimentally as well as theoretically, with a particular focus on the activation energy. Inelastic and quasi-elastic neutron scattering measurements reveal an activation energy of ∼48 meV. Through a combination of experiments and first-principles calculations, we attribute this activation energy to the relative rotation of CH3 against an NH3 group that stays bound to the inorganic cage. The inclusion of nuclear quantum effects through path integral molecular dynamics gives an activation energy of ∼42 meV, in good agreement with the neutron scattering experiments. For deuterated samples (CD3NH3PbI3), both theory and experiment observe a higher activation energy for the rotation of CD3 against NH3, which results from the smaller nuclear quantum effects in CD3. The rotation of the NH3 group, which is bound to the inorganic cage via strong hydrogen bonding, is unlikely to occur at low temperatures due to its high energy barrier of ∼120 meV.


S2. Experimental conditions of neutron scattering measurements
Approximately 1 g of grounded perovskite powder of each compound under investigation was loaded under argon in an aluminum flat can of 3×4×0.05 cm 3 in volume sealed with an indium gasket.
Neutron powder diffraction measurements were carried out at the Institut Laue Langevin (ILL) using the D2B powder diffractometer from 10 to 350 K on CH 3 NH 3 PbI 3 and CD 3 NH 3 PbI 3 samples using a wavelength of 1.594 Å covering the 0.5-4.4 Å −1 Q-range on the available multi-detector.
Incoherent quasi-elastic neutron scattering experiments were performed at ILL on the backscattering spectrometer IN16B. The energy resolution and energy (time) accessible range(s) for this spectrometer are shown in Table S1. More details on the characteristics of these two instruments can be found on the ILL website: http://www.ill.eu/instruments-support/instruments-groups Due to the crystalline character of the investigated materials, we have selected the Q values for which the Bragg diffraction was at its minimum between the peaks as shown in Fig. S2. Accordingly we were obliged to select only 1/3 of scatterring angles for analyzing the quasi-elastic data. Figure S2 illustrates that the much better signal-to-noise ratio obtained for the partially deuterated sample proves that the deuteration brings more coherent scattering to the sample.
Principle of the measurements of elastic fixed window scans (EFWS) obtained on IN16B: EFWS measurements record the intensity at the 0 energy exchange position of the scattering function as a function of temperature. Considering that the total integrated intensity of the scattering function remains constant as long as the quasi-elastic broadening stays measurable in the energy window of the spectrometer, as shown in Fig. S3, the height of this maximum decreases as soon as some quasi-elastic broadening enters the energy window of the spectrometer. In this way, the global evolution of the dynamics visible on this high resolution spectrometer as a function of temperature is recorded. Practically, the Doppler machine of IN16B is stopped and the elastically scattered intensity is directly recorded, where the monochromator and analyzers are constituted of exactly the same single crystals. Figure S2. Powder diffractograms obtained on D2B at ILL at different temperatures. The lines show the Q values at which the incoherent scattering data were analyzed. Figure S3. At 2 K the elastic peak is at a maximum. It can be considered that all local dynamics is frozen. As soon as some quasi-elastic component appears, this maximum decreases as can be seen at 90 K.

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In other words, this experiment measures the scattering function S(Q, 0) which is directly proportional to the Debye-Waller factor: where u 2 is the mean square displacement of the vibrating protons belonging to the molecular species under investigation. The proportionality constant consists of the elastic incoherent structure factor A 0 (Q) and a possibly small part of the inelastic incoherent intensity S IE . The latter can be neglected to good approximation: At very low temperatures, the evolution of the EFWS is mainly driven by the intramolecular vibrational Debye-Waller factor, S3 then it is modulated by the EISF. When the dynamics involved in the system is too rapid compared with the time window of the spectrometer, the EFWS tends towards a constant.
For both all-protonated and methyl-deuterated samples, experimental elastic incoherent structure factors (EISF) recorded at different temperatures are shown in Figs. S4. We can fit the EISF at < 160 K [e.g., Fig. S4a at 100 K as well as Fig. S4b at 90 and 120 K] with the 3-site jump model; while for higher temperatures, e.g., Fig. S4a at 168 and 180 K as well as Fig. S4b at 180 K, we can fit the EISF with the 90°-jump reorientation model. Figure S4. Experimental EISF of (a) CH 3 NH 3 PbI 3 and (b) CD 3 NH 3 PbI 3 . Figure S4 clearly shows that EISF starts to increase again with temperature at > 100 K for CH 3 NH 3 PbI 3 and > 90 K for CD 3 NH 3 PbI 3 . At 150 K, the EISF tends to 1. From the dynamical point of view, we understand that this elastic-intensity increase illustrates the complexity of these compounds when approaching the temperature at which the orthorhombic-tetragonal phase transition occurs. Within this temperature range, quasi-elastic broadening due to rotor dynamics is so large that it becomes more and more difficult to measure accurately. When the temperature is very close to the phase-transition one, this broadening gives a flat background and the elastic-intensity maximum becomes a quasi-constant. In addition, we cannot exclude the possibility that some contribution of the slow dynamics in the tetragonal phase may already exist at 150 K, which will contribute to the increase of the elastic maximum. Thus, the dynamics at 150 K has a very complicated character that cannot be fully contained in the energy window of the spectrometer. In the following discussion, we therefore focus on the dynamics of CH 3 NH 3 PbI 3 at 100 K and CD 3 NH 3 PbI 3 at 90 K.
In Figs shows that the experimental EISFs for both compounds are clearly larger than the simulated ones within nearly the whole investigated Q-range. Thus, the simulation cannot correctly reproduce the experimental EISFs when including the NH 3 rotor in the dynamics. When considering the only-Me rotation, Figure S5b indicates that the simulated EISF agrees relatively well with the experimental one for CH 3 NH 3 PbI 3 . The comparison for the deuterated sample does not produce similar results.
For a better comparison, we weighted the S(Q, 0) functions with the Debye-Waller factor that takes intramolecular and/or lattice vibrational motion into account. Figure S6 shows that, for Me-rotation, a best agreement between the experimental EISF and the simulated one can be obtained by applying a mean-square-displacement (MSD) u 2 = 0.016 Å 2 which is very reasonable for the CH 3 rotor. This convincingly supports our assumption of the Me-rotation dynamics at low temperature. For CD 3 rotors, such a quality of fit requires a larger MSD (0.18 Å 2 ) which is contrary to our physical intuition. To better understand this problem, more measurements with higher resolution are necessary. Here we can only emphasize that, for CD 3 NH 3 PbI 3 , the experimental EISF always lies clearly above the simulated one when including the NH 3 rotors in the motion.
To conclude, considering the analysis of EISF in different dynamical situations, our assumption of Me-rotation is highly credible (even without the activation energies calculated using DFT, which definitely support our model).
Principle of the measurements of inelastic fixed window scans (IFWS) obtained on IN16B:  For IFWS, intensities at a position ω off = 0, i.e., not at the elastic peak (where ω off = 0, were read. The intensity of the scattering profiles is thus recorded as a function of temperature for a given value of energy exchange. Here we present the results of IFWS at ω off = ±2 µeV. For such measurements, the Doppler monochromator moves periodically with a constant velocity ±v D which offsets the incident energy by ω off . Data for the IFWS are acquired during this time frame for obtaining the inelastic incoherent structure factor.  3 . In addition, the Q-dependence of IFWS is also important. A local motion, e.g., rotation or jump between fixed sites, results in a Q-independent T max according to Eq. 2c of the main text, because the relaxation time τ is Q-independent 4 . This is indeed the S5 Figure S7. IFWS measurements on all-proton and partially-deuterated perovskites shown in the whole temperature range. At the phasetransition temperature 160 K, additional protons enter the energy window of the spectrometer.
case for the investigated compounds as shown in Fig. S8.

Method of analysis of the quasi-elastic measurements:
The resolution of the spectrometer for the different compounds was evaluated by measuring the quasi-elastic spectra at 2 K. For the elastic peak, we found a pseudo-Voigt function with FWHM = 0.88 µeV and µ = 0.11 for the all-proton sample, while for the deuterated sample we got a quasi-Gaussian with FWHM = 0.84 µeV and µ = 0.06. For CH 3 NH 3 PbI 3 , we recorded quasi-elastic spectra at 2, 50, 70, 90, 100, 120, 150, 167, 168, 169 and 180 K, while the spectra of CD 3 NH 3 PbI 3 were recorded at 2, 50, 70, 90, 120, 150 and 180 K. Above 180 K the quasi-elastic broadening is too large to measure. Each spectrum was deconvoluted as the sum of resolution peaks plus a Lorentzian component of a given FWHM for all Q values at each temperature. This analysis was performed with setting the flat background at zero.
At different temperatures, the analyzers and detectors was calibrated by measuring the quasi-elastic scattering of a 1 mm thick vanadium sample and an empty can. Figure S9 shows an example of the obtained spectra with the deconvolution. Figure S9. An example of quasi-elastic spectra of (a,b) CH 3 NH 3 PbI 3 and (c,d) CD 3 NH 3 PbI 3 recorded at 90 K and Q = 1.65 Å −1 . We chose these temperature and Q value, at which the EISF is not too high so that we can have a better signal-to-noise ratio. We found that this ratio is always lower for the deuterated sample, and accordingly, the counting times were set as long as possible at the limit of the available beamtime.

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S3. Structure of low-temperature MAPbI 3 with one MA + not taking the most stable geometry Figure S10 shows the relaxed structure of orthorhombic MAPbI 3 (represented in a 2×2×2 right rhombic prism lattice) with one MA + (as highlighted) taking a slightly different orientation as in the fully-relaxed structure (Fig. 4a of the main text). Such a structural change induces a total-energy increase of 108 meV, much larger than the characteristic thermal energy of 160 K, the upper bound for the low-temperature phase. The energy barrier of MA-reorientation from the main-text Fig. 4a structure to the Fig. S10 structure is even larger.
S4. PBE+vdW PES of Me-rotation Figure S11 shows the PBE+vdW PES along the Me-rotation pathway (also shown in Fig. 6 of the main text). DFT data Cosine-potential fit Figure S11. The Me-rotation PES calculated using PBE+vdW (black squares) and the fitted cosine-potential (red curve).
Using Eq. (2) of the main text, the fitter PES can be written as As only coefficient for the 3rd term is nonnegligible, the fitted potential in Fig. S11 exhibits perfect cosine character with period 120°.

S5. A brief analysis of the 1D hindered-rotor model
The time-independent Schrödinger equation of a 1D system with a simple cosine potential given by Eq. (1) (where the potential barrier E b = −2v 3 ) has the form of Mathieu equation It can be solved numerically. We are especially interested in its ground-state energy E 0 . As an appropriate approximation, we can assume its ground-state wave function ψ 0 (θ) is a Gaussian function centered at θ = 0: the expectation value of energy associated withψ 0 (θ) equals The approximate ZPE can be calculated by minimizing Ẽ 0 with an optimal width parameter σ which leads to an equation of σ than can be solved numerically with given I and E b values.
We can further approximate V Me (θ) by a harmonic potential around its minimum at θ = 0: where κ = (V Me ) θ | θ=0 is the curvature of V Me at θ = 0. Within this harmonic approximation (HA), the optimal width parameter for the ground-state Gaussian wavepacket and the ZPE are respectively.
These analysis is not only limited for single-cosine potential given by Eq. (1) but can also be extended to more general cases, such as the potential given by Eq. (3) or (6) of the main text. The ground-state energy calculated by Eq. (5) is a very good approximation to the numerical result. And HA works very well for the investigated problems. For example, the HA groundstate energy for CH 3 -rotation is 10.9 meV, larger than the numerical result 10.3 meV by < 6%. This difference becomes even smaller for deuterated cases, for example HA result 7.7 meV vs. the numerical result 7.4 meV for CD 3 -rotation.
HA allows us to easily analyze the effects of deuteration. When the number of D atoms in the rotating group increases, the moment of inertia I becomes larger, so that the ground-state wave packet becomes narrower and the ground-state energy E 0 decreases. This in turn results in a larger activation energy, which can be calculated by E a = E b − E 0 with only including the 1D zero-point energy (that is, the ground-state energy) on the reactant state.
S6. Theoretical estimation of possibility that CH 3 NH 3 -rotation occurs at low temperature In this section we present two approaches to calculate the possibility that MA-rotation occurs (estimated vs. the possibilities of Me-rotation) using the Arrehnius equation: Both approaches are based on DFT calculations. For E a we adopt the results listed in Table 2 in the main text, 55.5 meV for Me-rotation and 119.3 meV for MA-rotation. In Fig. S12a, we calculate the frequency factor A from the curvature of the potential energy surface at its bottom ( Fig. 6 or 8 in the main text), while in Fig. S12b we use the vibrational frequencies (322 and 80 cm −1 for MA and Me, respectively) obtained from our DFT phonon calculations as A. They both indicate that there would be a contribution from the CH 3 NH 3 -rotation to the dynamics of the system which is very small. For example, at 150 K, the number of MA-rotation events would be ∼500 times less than the number of Me-rotation events.