Short-range ion dynamics in methylammonium lead iodide by multinuclear solid state NMR and ¹²⁷I NQR

Abstract

We explore the short-range ion dynamics in methylammonium lead iodide (MAPbI₃, the archetypal halide perovskite) by means of solid-state NMR (¹H, ¹³C, ¹⁴N, ¹⁵N and ²⁰⁷Pb) and Nuclear Quadrupolar Resonance (¹²⁷I NQR), in combination with molecular dynamics simulations. We focus on the rotational motion of the methylammonium (MA) cation, and on the interaction between MA and the inorganic lattice, since these processes are linked to electronic carrier lifetimes, optical and electronic properties and even structural stability of this promising solar cell material. We show that the motion of the MA cation can be described by a bi-axial rotation, with similar interactions of CH₃ and NH₃⁺ groups with the inorganic framework. This motion becomes nearly isotropic above the cubic phase transition, dominating the spin–lattice relaxation of ¹H, ¹³C and ¹⁵N through spin-rotational interactions. In addition, we observe strong cross-relaxation between ²⁰⁷Pb and ¹²⁷I, which fully controls spin–spin and spin–lattice relaxation in ²⁰⁷Pb.

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Introduction

The outstanding photo-electrochemical properties of hybrid organic–inorganic halide perovskites have motivated in recent years a massive number of studies on these promising solar cell materials.^1–3 Even though these compounds were described nearly 40 years ago,⁴ the characterization of their properties and application in devices is much more recent.^1,5 Despite the momentous effort already spent in characterizing these materials, a significant portion of the physics and chemistry underlying their properties is still missing. As a notable example, the short-range dynamics of the organic cations is still not well understood, despite being of great importance since it is linked to the long carrier lifetimes in these systems.^6–8 In addition, the interaction of the organic cations with the inorganic framework through hydrogen bonding⁹ has been suggested to contribute to the stabilization of the structure,¹⁰ while also influencing the optical and electronic properties of the materials.^11,12 Therefore, the goal of the present study is to probe such short-range ion dynamics in the archetypal halide perovskite MAPbI₃via solid-state NMR (¹H, ¹³C, ¹⁴N, ¹⁵N and ²⁰⁷Pb) and Nuclear Quadrupolar Resonance (¹²⁷I NQR), with particular attention to the rotational motion of the methylammonium (MA) cation, as well as to the interaction between MA and the inorganic lattice. Long-range ion dynamics, that gives rise to a significant iodine conductivity in MAPbI₃, was studied by our group in a previous work.¹³

Owing to its unsurpassed sensitivity to short-range interactions, NMR spectroscopy is very effective to study dynamic processes,^14–18 and its first applications to the investigation of halide perovskites go back as early as 1985.^19–22 In these studies, reorientation of the organic moiety was observed in all crystalline phases of MAPbI₃ (but also MAPbBr₃ and MAPbCl₃) through ¹H, ²H and ¹⁴N spin–lattice relaxation times (T₁) measurements.^20–22 Even though decreasing the temperature naturally hinders MA reorientation, rotation about the C–N axis in a correlated fashion was observed down to 50 K. The phase transitions (orthorhombic–tetragonal at 162 K and tetragonal–cubic at 327 K)²³ are not visible in the T₁ measurements of ²H and ¹⁴N,²¹ but can be detected by ¹H measurements,²² albeit only the lower temperature transition was investigated. More recently, NMR spectroscopy has been applied to halide perovskites to investigate the effects of degradation,²⁴ extended defects and crystal quality²⁵ and details of Pb–X environment and cation dynamics.^13,26–29

In this work, we report on NMR and NQR spectra discussed in light of molecular dynamics simulations and relaxation effects. First we look at the MA cation with ¹H, ¹³C, ¹⁴N and ¹⁵N NMR as a function of temperature and corresponding relaxation time measurements. Combination of these data will show that the motion of the MA cation in MAPbI₃ can be well described by a bi-axial rotation, with similar interactions of CH₃ and NH₃⁺ groups with the inorganic framework. This motion develops into a nearly isotropic rotation above the cubic phase transition, becoming a dominant contribution to the spin–lattice relaxation of ¹H, ¹³C and ¹⁵N through spin-rotational interactions. We follow by presenting ²⁰⁷Pb NMR and ¹²⁷I NQR spectroscopic and relaxation data, where we observe strong cross-relaxation between ²⁰⁷Pb and ¹²⁷I, which fully dominates spin–spin and spin–lattice relaxation in ²⁰⁷Pb.

Experimental part

Synthesis

As a first step, MAI was prepared according to a modified reported procedure.³⁰ Single crystals of MAPbI₃ were then obtained as reported by Saidaminov,³¹ using an equimolar mixture of MAI and commercial PbI₂ powders (99.9995%, Alfa Aesar). The synthesis step was carried out in normal laboratory atmosphere. These single crystals were subsequently crushed under inert atmosphere to obtain crystalline powders and directly transferred into the appropriate NMR sample holders, which were sealed to prevent exposure to oxygen and humidity. In order to synthetize ¹³C- and ¹⁵N-labelled MAPbI₃, we used commercial ¹³C,¹⁵N methylamine (99% enrichment, Sigma-Aldrich) to obtain labelled MAI, that was used in the single crystal synthesis step to yield labelled MAPbI₃ containing 20% ¹³C and ¹⁵N. Powder XRD data on the crushed single crystals are given in the (Fig. S1, ESI†).

NMR measurements

The majority of NMR measurements were performed on a Bruker Avance III 400 MHz instrument at magnetic field of 9.4 T. Measurements with stationary samples used a Bruker PE400 double resonance high power probe with a 5 mm solenoid coil. The spectra of ¹⁴N and ²⁰⁷Pb were obtained using solid-echo (π/2–τ–π/2) and Hahn-echo (π/2–τ–π) pulse sequences with τ = 20 μs, and the relaxation delays set to provide for a complete relaxation (1s for ¹⁴N and 5s for ²⁰⁷Pb at room temperature). Between 100 and 400 scans were accumulated for both nuclei. ¹H spectra were acquired with simple Bloch Decay experiment (i.e., pulse-acquisition), with the π/2 pulse of 2.5 μs, and the relaxation delay set long enough for a complete relaxation (10 s at room temperature). ¹³C and ¹⁵N spectra were obtained using Bloch decay experiment accompanied by a composite proton decoupling. The excitation π/2 pulses were 5 ms and 6 ms, and relaxation delays at room temperature of 150 s and 600 s, respectively. For isotopically enriched samples between 16 and 64 scans were commonly accumulated. Unless stated otherwise, spin–lattice relaxation time measurements were performed in stationary conditions, mainly with the saturation-recovery technique³² in order to minimize the experimental time.

The experiments with Magic Angle Spinning (MAS) used a Bruker BL4 double resonance MAS probe with 4 mm OD ZrO₂ spinners. Dry nitrogen was used as spinning, bearing, and cooling gas. The spinning rate was, when possible, kept between 10 and 14 kHz. The spectra were acquired using ramped Cross Polarization (CP) and also Bloch Decay, both accompanied by composite proton decoupling at the proton RF power of 50 kHz. The temperature in the probes was regulated with a Bruker BVT3000 temperature controller. Actual temperature inside the spinner was calibrated on ²⁰⁷Pb signal of powdered Pb(NO₃)₂.³³ Chemical shifts were referenced externally relative to TMS (¹H and ¹³C, δ_iso = 0.0 ppm), tetramethyl lead (²⁰⁷Pb, δ_iso = 0.0 ppm, set with the absolute referencing to TMS), nitromethane (^14,15N, δ_iso = 0.0 ppm, set with the secondary reference of NH₄Cl, δ_iso = −341.2.0 ppm).³⁴

NQR measurements were performed using the same spectrometer console and probes, which were kept at a sufficient distance from the magnet to avoid the effects of stray magnetic fields. All RF-pulses have been calibrated directly on a sample of MAPbI₃, with π/2 pulses being adjusted to 2 μs. A solid-echo π/2–τ–π/2 pulse sequence with τ = 20 μs and relaxation delay of 2 ms has been used to acquire the spectra, with the number of accumulations being between 128 and 256. Pure NQR spin–lattice relaxation times have been measured using inversion-recovery method.

DFT calculations

All calculations were performed with the Castep code^35,36 integrated within the Biovia Materials Studio 2017 suite. The computations use the generalized gradient approximation (GCA) and Perdew–Burke–Ernzerhof (PBE) functional,³⁷ with the core–valence interactions described by ultra-soft pseudopotentials.²⁸ A Monkhorst–Pack grid³⁸ was used to realize integrations over the Brillouin zone, with k point spacings generally being less than 0.04 Å⁻¹. The convergence of the calculated NMR parameters was tested for both the size of a Monkhorst–Pack k-grid and a basis set cut-off energy, with cut-off energies being in the range of 550–610 eV. To account for relativistic effects in such heavy elements as lead and iodine, all calculations were performed including the zeroth-order regular approximation (ZORA)^39,40 through scalar-relativistic pseudopotentials.⁴¹ Geometry optimization calculations were performed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, with the same functional, k-grid spacings and cut-off energies as in the single-point energy calculations. Convergence tolerance parameters for geometry optimization were as follows: maximum energy 2 × 10⁻⁵ eV per atom, maximum force 0.05 eV Å⁻¹, maximum stress 0.1 GPa and maximum displacement 0.002 Å. Crystallographic data used in the calculations were taken from the literature.⁵¹⁴N NMR parameters in methylammonium cation were calculated in a 3 × 3 × 3 supercell of MAPbI₃ after a molecular dynamic simulation run for 1 ps 270 K. During the DFT MD simulation only the cations move, while the lattice parameters of the inorganic framework composed by heavy atoms of iodine and lead are fixed. Fig. S8 (ESI†) shows the structure before and after the simulations.

Spectral simulations and fitting

Analytical simulations of the experimental spectra were carried out with the DMFit⁴² simulation package and the SOLAGuide module in TopSpin 3.2 processing software from Bruker. ¹⁴N NMR line shapes affected by the motion of the MA-cation were simulated with NMR-WEBLAB,⁴³ and EXPRESS⁴⁴ computational packages.

Results and discussion

¹H solid state NMR

The isotope ¹H offers the highest sensitivity among all NMR active nuclei in MAPbI₃. As shown in Fig. 1a, ¹H spectra obtained with 10–15 kHz Magic Angle Spinning (MAS) show two well-resolved signals at 3.4 ppm (CH₃) and 6.4 ppm (NH₃⁺), the position of which remain nearly constant between 183 and 373 K (Fig. S2, ESI†). On few other occasions, we have observed some minor signals near 0.9 ppm likely due to residual solvent from the synthesis (DMF), while no peaks assignable to water have ever been detected. Although the MAS offers a considerable improvement in spectral resolution, the majority of our measurements have been performed in stationary conditions, since having a consistent spinning of the glass-sealed samples (required to avoid oxygen- and moisture-induced degradation) in the broad range of studied temperatures was rather difficult. A further argument against the MAS measurements was the influence of the spinning on the spin–lattice relaxation time measurements (Fig. S5a, ESI†), which is also found in other materials.^45,46 In contrast to spinning samples, the ¹H stationary spectra show a broadened, unresolved signal with a ∼8 kHz FWHM and a line shape well represented by a Gaussian (Fig. 1a).

Fig. 1 (a) Stationary and MAS ¹H NMR spectra of MAPbI₃ recorded at 297 K. (b) Stationary ¹H spectra as a function of temperature. (c) ¹H T₁ relaxation times obtained from stationary spectra. Extracted activation energy values for the dipole–dipole relaxation mechanism dominating at low temperatures is ΔE_A,DD = 0.13 eV. A value for the spin-rotational contribution of ΔE_A,SR= −0.1 eV is obtained after fitting the entire curve using fixed values initially extracted from the low temperature part. Due to a possibility of sample degradation, only a limited temperature range is available, making the spin-rotational behavior in T₁ challenging to accurately fit and the resulting activation energy intrinsically only indicative.

Before looking further at the experimental results, it can be helpful to discuss the factors affecting the shape and width of NMR signals. The broadening of ¹H NMR signals in a stationary MAPbI₃ sample is primarily due to the homonuclear dipole–dipole (DD) interaction between protons in the MA cation. Additional contributions come from the heteronuclear interaction between protons and ¹⁴N of the amine group and, to a lesser extent, between protons and the atoms of the inorganic lattice. DD interactions in solids can be quantified by the so-called second moment M₂ of the spectrum, which can be calculated for ¹H in MA based on the geometry of the system (Section 3, ESI†). We note a substantial deviation of more than 1 order of magnitude between the M₂ values calculated in a static environment and the experimental values, indicating that the MA cation, along with rotation around the C–N axis, must be involved in another reorientation with a symmetry axis higher than 2-fold (for the detailed discussion, see Section 3, ESI†). Since the methyl and amino groups composing MA have similar masses, such a reorientation is likely a rotational oscillation about an axis passing near the center of the C–N bond, tilted relative to the bond at angle θ (schematics below in Fig. 5). This composite rotation will give a reduction of M₂ by a factor ρ = 1/16·(3 cos² θ − 1)².^20,47 Since 180° flips (θ = 180°) are not the dominant motion reducing M₂ we speculate that a precessional motion of the C–N bond tilted at θ ≤ 60° is responsible for the small observed M₂ value, a hypothesis that is in very good agreement with the ¹⁴N NMR results discussed below. The absence of 180° flips was observed by neutron scattering experiments⁴⁸ and it is also indicated by the quadrupolar splitting visible in our ¹⁴N NMR spectra (given below in Fig. 4a). Note that DD interactions are not completely averaged neither between the protons within the cation, nor to other atoms (¹²⁷I, ²⁰⁷Pb) or to protons from neighboring cells. As a result, both line shape and FWHM of ¹H stationary signals change very little from 503 K down to 183 K (Fig. 1b). Both the temperature independence of ¹H FWHM and the small magnitude of M₂ were previously reported in the tetragonal phases of MAPbCl₃ and MAPbBr₃,²⁰ along with a sharp increase of the second moment after transition into the orthorhombic phase that was assigned to a diminished C₃ reorientation in the MA cation (i.e., around the C–N axis). Similarly, we observe that at T ≤ 173 K the signal undergoes a drastic transformation, morphing into a broad, poorly resolved triplet with separation between the components of ∼18 kHz (Fig. 1b). Such a line shape is expected for a system of three protons at the corners of an equilateral triangle,^49,50 with separation between the peaks being a function of both distance and mobility of the protons.

Additional information about the overall dynamics and specific modes of motion of MA can be obtained from the ¹H spin–lattice relaxation time (T₁) measurements as a function of temperature given in Fig. 1c. Between 183 K and 310 K, the relaxation is characteristic of the fast motion limit of the Bloembergen–Purcell–Pound (BPP) model,⁵¹ and is being governed by DD interactions with other protons (Section 4, ESI†). Assuming an Arrhenius dependence of the motional correlation time (ln(τ_ρ) ∝ 1/T₁), we found an activation barrier (ΔE_A,DD) of 0.13 eV for this temperature range, well within the expected range for rotational motion of the MA cation inside the Pb–I framework and in good agreement with earlier reports.²² We note that the here reported T₁ values are an average of the proton relaxations in CH₃ and NH₃⁺ groups, with the respective contributions likely being equal, or at least very similar, since the observed T₁ is mono-exponential in the whole temperature range, with only a slight deviation near the orthorhombic phase transition (Fig. S3, ESI†). This mono-exponential dependence clearly suggests that CH₃ and NH₃⁺ groups interact similarly with the inorganic framework. In contrast, in a previous report,²⁶¹H T₁ relaxation times measured under MAS conditions were used to claim a preferential interaction of NH₃⁺ groups with the inorganic framework, albeit the difference in T₁ between the two groups was rather small and likely within experimental error. We have also measured ¹H T₁ relaxation times under MAS conditions (Fig. S5a, ESI†), and observed similarly minor differences between the relaxation times of the two groups. We believe that this very minor discrepancy, in conjunction with the strong influence of MAS conditions on T₁ relaxation times,^45,46 makes the above claims of preferential interactions unwarranted.

As expected, the relaxation times observed for ¹H in pure MAPbI₃ and in ¹³C,¹⁵N-labelled samples are identical within experimental error; a similar activation energy is also found for MAPbBr₃ (Fig. S4, ESI†). Interestingly, as seen in Fig. 1c, above room temperature T₁ makes a downward turn, that is characteristic of spin-rotational (SR) interaction. As we will show below, the same behavior is present for ¹³C and ¹⁵N in MAPbI₃, and was reported for ¹H in MAPbBr₃ (also Fig. S4, ESI†) and MAPbCl₃.²⁰ The relaxation rate is therefore defined by a combination of two processes, namely by a dipole–dipole and a spin-rotational mechanism, the latter becoming dominant in the vicinity of the tetragonal to cubic phase transition where the increased free space in the inorganic cage allows for a less hindered (and thus faster) MA rotation. As expected, in the case of ¹H T₁ measurements in MAPbBr₃ (Fig. S4, ESI†), the inversion behavior takes place at lower temperatures, albeit notably not as low as the phase transition into the cubic phase (237 K).²³ This observation is consistent with previous reports.²⁰ Lastly, we note that while we can accurately extract an activation energy for DD relaxation (given above), the experimentally accessible temperature range in which SR contributions dominate is rather narrow, making the estimate of such activation energy inaccurate (see caption of Fig. 1).

¹³C NMR

The ¹³C spectra, as shown in Fig. 2a, give a better spectral resolution than ¹H NMR, though at the expense of substantially longer acquisition times due to the low natural abundance of the isotope (1.1%). To overcome this issue, we use a ¹³C-labelled sample (20% enrichment). Both Cross-Polarization (CP) from protons and direct excitation (Bloch Decay accumulation) combined with the high power proton decoupling (HPDEC) sequence can be successfully used. While in many organic materials CP usually offers a better sensitivity because of the shorter ¹H relaxation times and effective CP dynamics, this is not the case for MAPbI₃. Slow proton magnetization recovery and a very fast relaxation in the rotating frame T_1ρ make CP very ineffective above 250 K (Fig. S5b, ESI†). In contrast, the direct excitation is less affected by the MA dynamics and here the relative intensities of the signals reflect quantitatively the concentration of different sites, making this our approach of choice. MAS has also been applied in order to obtain better resolved spectra (Fig. 2a and Fig. S7a, ESI†). However, since our preliminary results indicated an influence of MAS on ¹³C T₁ relaxation times (Section 5, see also Fig. S6, ESI†), as also reported for other materials,^45,46,52 all T₁ measurements were performed in stationary conditions.

Fig. 2 (a) ¹³C stationary and MAS NMR spectra in (¹³C, ¹⁵N)-labelled MAPbI₃. (b) Spin–lattice relaxation time extracted from stationary ¹³C NMR spectra as a function of temperature. The activation energy (ΔE_A,DD = 0.062 eV) was extracted from the linear low temperature part, and the obtained value was used to generate the complete fit, yielding ΔE_A,SR = −0.38 eV. Again, the narrow range accessible at high temperatures makes the activation energy for SR highly inaccurate.

¹³C spin–lattice relaxation times can be composed of contributions from three primary relaxations mechanisms,^53–55 such as: (i) heteronuclear DD interactions with protons, (ii) relaxation due to the chemical shift anisotropy (CSA) and (iii) relaxation due to SR interactions (as in protons). We leave the full expressions of the relaxation mechanism and of the related contributions to the ESI,† (Section 5); here it suffices to say that the nearly symmetrical signal shape even at very low temperatures suggest a negligible contribution of CSA in ¹³C spectra. This is in agreement with ab initio DFT calculations on the tetragonal phase of MAPbI₃ yielding a very small value (38 ppm) for the shielding anisotropy (Table S1, ESI†). As shown in Fig. 2b, ¹³C T₁ measurements as a function of temperature present a linear trend in the Arrhenius dependence below 340 K, as expected from DD interactions dominating the relaxation (ΔE_A,DD = 0.062 eV). Analogously to ¹H and ¹⁵N (below), above room temperature we observe an inversion of the trend, corresponding to a change in dominant relaxation mechanism from DD to a SR interaction. Again, the trend inverts close to the tetragonal to cubic phase transition, relating it to the increased free space of the inorganic framework giving a reduced obstruction to the rotation of the MA cation.

¹⁵N NMR

Due to very low natural abundance (0.4%), ¹⁵N spectra are barely detectable in non-labelled MAPbI₃. However, since ¹⁵N NMR can produce highly resolved spectra, especially when combined with CP and proton decoupling, we use a ¹⁵N-labelled sample (20% enrichment) to record the spectra (Fig. 3a). MAS conditions have also been applied in order to obtain better resolved spectra (Fig. 3a and Fig. S7b, ESI†). The relaxation mechanisms expecting to contribute into ¹⁵N T₁ are the same as those discussed for ¹³C and described in the previous section (and in Section 5, ESI†). Also here, we can reliably rule out any significant contribution of the CSA to the relaxation process based on our ab initio DFT calculations, which give a low value of 10.2 ppm (Table S1, ESI†) for the ¹⁵N shielding anisotropy, which is by at least two orders of magnitude too small to produce any appreciable contribution to T₁.

Fig. 3 (a) ¹⁵N stationary and MAS spectra (¹³C, ¹⁵N)-enriched MAPbI₃. (b) ¹⁵N Spin–lattice relaxation time T₁ extracted from stationary spectra. The activation energy (ΔE_A,DD = 0.084 eV) was extracted from the linear low temperature part, and the obtained value was subsequently used to generate the complete fit, yielding ΔE_A,SR = −0.17 eV. Also here, the narrow range accessible at high temperatures yields a highly inaccurate activation energy value for the SR contribution.

Also here, T₁ demonstrates a linear trend in the tetragonal phase, as expected for internuclear proton–nitrogen DD relaxation in the fast motion limit (ΔE_A,DD = 0.084 eV), which turns into a negative temperature dependence in the cubic phase due to the increasingly dominating SR contribution. The rather small difference between the activation barriers of the DD relaxation in ¹³C and ¹⁵N is expected for an almost symmetrical rotor such as CH₃–NH₃⁺. In addition, the activation energy extracted from ¹H T₁ measurements is comparable with the ones of ¹³C and ¹⁵N, as expected since they refer to the same motion (and to the same type of relaxation process).

¹⁴N NMR

Fig. 4a shows ¹⁴N NMR spectra of MAPbI₃ in the tetragonal and cubic phases.¹³ In the tetragonal phase, the spectrum is dominated by quadrupolar interactions, demonstrating a typical spin-1 powder pattern with axial symmetry of the electric field gradient (EFG), while in the cubic phase the quadrupolar interactions are averaged, and therefore the splitting absent. Spin–lattice relaxation time T₁ becomes longer with the temperature (Fig. 4b), indicating fast motion limit for the dominant relaxation mechanism in the whole studied range (183 K to 503 K). For a spin-1 nucleus such as ¹⁴N, the dominating mechanism of spin–lattice relaxation is due to quadrupolar interactions. In the fast motion limit, such quadrupolar relaxation can be expressed as:^14,15where χ_Q and η_Q are respectively the nuclear quadrupolar coupling constant and quadrupolar asymmetry parameter, and τ_c is the rotational correlation time. For ¹⁴N relaxation in an axially symmetrical MA cation, τ_c refers to the spatial reorientation of the C–N axis. Due to the fast axial rotation of MA cations, the effective η_Q is reduced to 0 in both tetragonal and cubic phases of MAPbI₃, further simplifying eqn (1). To evaluate τ_c we use χ_Q (¹⁴N) = 0.68 MHz, obtained from our DFT MD simulations (discussed in detail below), resulting in correlation times of 1.0 ps at 303 K (tetragonal phase) and 0.53 ps at 333 K (cubic phase). These values indicate a rapid reorientation of the C–N axis, and are in full agreement with previous reports.^19,21,56 A much higher correlation time of 108 ps has also been reported,²⁸ however this value was obtained by a drastically different method that does not allow for a direct comparison.

Fig. 4 (a) Variable temperature ¹⁴N NMR spectra in MAPbI₃, collected using a solid echo pulse sequence. No difference except in the overall intensity is observed between pure and partially labelled samples. (b) Arrhenius plot of ¹⁴N T₁ relaxation times in MAPbI₃ and (c) ¹⁴N FWHM as a function of temperature in MAPbI₃. For the signals (tetragonal phase) demonstrating the quadrupolar splitting, the value of the splitting has been used in place of the FWHM.

The T₁ temperature dependences (in Arrhenius coordinates, Fig. 4b) are linear in both cubic and tetragonal phases, producing activation barriers of 0.1 eV and 0.16 eV, respectively. These values cannot be directly compared with the one extracted in the ¹⁵N case (Fig. 3b), as the relaxation mechanisms involved are different. Nevertheless, they are fully in agreement with the ones extracted from other ¹⁴N NMR measurements.⁵⁶ The observed decrease of the activation barrier across the phase transition implies a less hindered rotation of MA in the cubic phase, where the ¹⁴N signal displays a Lorentzian line shape with FWHM of only 710 Hz at 333 K, decreasing slowly as the temperature increases to 453 K (Fig. 4c). Such a small and steady change in the signal is consistent with a faster local motion of MA cations.

We can now focus on the analysis of the line shape of ¹⁴N spectra, in order to gain information on the modes of motion involving MA. In general, line shapes in ¹⁴N NMR depend on the interaction between the quadrupole moment of the nuclear spin and the EFG tensor surrounding the nucleus. In the specific case of a MA cation, this quadrupolar coupling is primarily of intramolecular origin, and is therefore strongly affected by the type and the rate of the MA molecular motions. Interestingly, the very large frequency spread of the experimental spectra (Fig. 4a) means that it is possible to probe molecular motion over a very broad frequency range. To help us identify the possible modes of motion involving MA, we performed ¹⁴N spectral simulations. In the general case of non-axial symmetry, the NMR frequencies (ω_Q) in a ¹⁴N spectrum can be expressed as:

where ω_L is the Larmor frequency, a is a combination of constants, F is a function defining the angular dependence of the EFG tensor, χ_Q is the quadrupolar constant and V_zz the largest principal component of the EFG tensor. The asymmetry parameter η describes the deviation from axial symmetry, while the polar angles θ and φ define the tensor orientation with respect to the external magnetic field. The largest principal component of the EFG in MA is directed along the C–N bond. A fast axial motion of a second rank tensor about an axis involving three or more equivalent sites will lead to an axially symmetric averaged tensor with η = 0. Due to the intramolecular origin of quadrupole coupling in MA, one could expect the quadrupolar constant (χ_Q) of the N atom in MAPbI₃ to be similar to that in MA halogenates (values of χ_Q = 1.11 MHz⁵⁷ and 0.91 MHz⁵⁸ have been reported for MACl). We test this hypothesis by performing periodic ab initio DFT MD simulation on a 3 × 3 × 3 super-cell of MAPbI₃. Interestingly, a rather different χ_Q (¹⁴N) of 0.68 MHz with η = 0.52 has been obtained by averaging over all nitrogen sites (Table S1, ESI†), indicating a significant influence of the Pb–I framework on the EFG. These calculated EFG parameters are very close to those recently reported by Franssen et al.,²⁵ and have been used in our ¹⁴N spectral simulations.

As shown in Fig. 5, the ¹⁴N spectra are strongly and diversely influenced by the different internal motions considered, offering a way of discriminating among them. Looking at Fig. 5a, it is clear that a fast (>10⁻⁶ s) continuous rotation around the C–N axis (or equivalently fast ≥3-sites symmetrical jumps) is necessary in order to produce the spectra experimentally observed in the tetragonal phase (Fig. 4a). Also, as shown in Fig. 5b, by adding to a fast rotation around the C–N bond a precessional motion with increasing angle θ (schematized in Fig. 5c), the quadrupolar splitting in the spectra reduces steadily until the angle reaches 54.7°, when it virtually disappears indicating nearly complete averaging of the quadrupolar interactions forming the line shape. This behavior is experimentally observed across the phase transition into the cubic phase (Fig. 4a and 5b), and corresponds to a pseudo-isotropic motion of MA cations within the inorganic framework. We recognize also that, in the cubic phase, identical spectra would be obtained with the presence of a fast enough isotropic motion, for example due to a translational diffusion of MA cations or to the presence of 180° flips involving MA molecules. Both the former and the latter motion were clearly ruled out from previous electrochemical¹³ and neutron scattering experiments⁴⁸ respectively.

Fig. 5 ¹⁴N NMR spectral simulations assuming different MA modes of motion. (a) Simulated spectra in different dynamic conditions. (b) Simulated spectra obtained by modeling fast rotation around C–N axis together with precession in conical fashion at different angles θ. The data are compared with experimental spectra across the cubic phase transition. (c) Sketch of the modeled conical precessional motion.

²⁰⁷Pb NMR

²⁰⁷Pb is a moderately sensitive spin-1/2 nucleus showing a remarkably broad range of chemical shifts (>8000 ppm) depending on its oxidation state and coordination environment, so much so that an unusually large difference (∼1500 ppm) is found between the chemical shifts of MAPbI₃ and PbI₂. In order to rationalize this peculiar chemical shift behavior, we performed extensive DFT calculations of the ²⁰⁷Pb shielding in a series of lead compounds of known environment (results are summarized in Table S2, ESI†). As shown in Fig. 6, the calculations correctly capture the general trend between the experimental chemical shifts and the different chemical environments, confirming the experimental observations. We recognize that the extracted slope in the shielding-shift trend (Fig. 6) has a value of almost 2, while it would commonly not strongly deviate from 1. This semi-quantitative agreement, however, is likely due to the incomplete accounting for relativistic effects in Pb nuclei stemming from the use of pseudopotentials instead of the full electron wave functions and to the lack of spin–orbit effects in the valence wave functions.

Fig. 6 Correlation between calculated isotropic shielding and experimental ²⁰⁷Pb isotropic chemical shifts in various diamagnetic compounds of lead. All experimental data are taken from the literature, with their corresponding ICSD numbers given in Table S2 (ESI†). The chemical shift of lead is referenced to lead tetramethyl (δ_iso = 0 ppm).

The detection of ²⁰⁷Pb NMR in the solid state, particularly in iodine surroundings, presents a substantial challenge. As shown in Fig. 7a, the iodine environment around Pb nuclei causes a very short spin–spin relaxation time T₂ (∼20 μs) and a subsequent formidable broadening of the signals at temperatures above 250 K (Fig. 7b). The ²⁰⁷Pb linewidth remains nearly constant (∼16.5 kHz) in the 433–253 K range, while below 253 K the line shape starts showing distinct asymmetry, consistently with the chemical shift anisotropy, and even significantly broadens at 173 K. Lowering temperature also increases T₂ (Fig. 7b), reaching a peculiar 3-fold enhancement at 183 K, even though the value falls back to 20 μs at 173 K (near the orthorhombic phase transition), in agreement with previous reports.²⁵ This unusual behavior, which is also found in T₁ measurements (Fig. 7b), cannot be ascribed to standard relaxation mechanisms and common spectral density functions, but can be easily understood by taking into account strong spin–spin interactions between the spin-1/2 ²⁰⁷Pb nucleus and the quadrupolar ¹²⁷I nucleus. In this case, the ²⁰⁷Pb T₂ is defined primarily by cross-relaxation to ¹²⁷I, which presents very short relaxation times (Fig. 8c). Reducing the temperature slows down the cross-relaxation rate, yielding an increase in T₂ relaxation times. The transition to the orthorhombic phase appears to alter the dynamics of the spin-exchange, resulting again in a shorter T₂.

Fig. 7 (a) ²⁰⁷Pb NMR spectra of MAPbI₃ as a function of temperature. (b) Temperature dependence of ²⁰⁷Pb spin–lattice and spin–spin relaxation times T₁ and T₂. Dashed lines are only to guide the eye. Vertical lines indicate the phase transitions. Orange () and green () points are T₁ relaxation times collected at 21.1 T in stationary and 8 kHz MAS conditions respectively. Note that the higher field has no influence on T₁ for stationary samples, while spinning greatly reduces it down to 20–30 ms. (c) Isotropic chemical shift of ²⁰⁷Pb signal as a function of temperature, fitted with a quadratic equation δ_iso = 1.04 + 0.96T + 0.001T².

Fig. 8 (a) ¹²⁷I NQR signals in MAPbI₃ at 300 K (tetragonal phase) and at 340 K (cubic phase). (b) Temperature dependence of the ¹²⁷I quadrupolar coupling constant χ_Q and of the asymmetry parameter η. Phase transition is visible at 327 K (blue line). (c) Quadrupolar spin–lattice relaxation time T_1Q as a function of temperature.

Focusing now on spin–lattice relaxation, the magnitude of ²⁰⁷Pb T₁ in MAPbI₃ is similar to that previously reported for PbI₂.⁵⁹ As shown in Fig. 7b, T₁ is practically independent of the magnetic field and depends only marginally on the temperature. Also, T₁ temperature dependences in ¹²⁷I and ²⁰⁷Pb appear rather similar (Fig. 8c and 7b), clearly demonstrating the large influence of cross-relaxation to ¹²⁷I on the ²⁰⁷Pb spin–lattice relaxation process. An additional proof of such coupling is that T₁ is strongly affected by MAS, decreasing by more than 1 order of magnitude under spinning conditions (Fig. 7b, details in the caption). This effect was recently reported and discussed in detail for lead and mercury halides,⁶⁰ where in the specific case of PbI₂ MAS conditions were required to induce the Pb–I coupling; unsurprisingly, thanks to the much stronger quadrupolar interactions, in MAPbI₃ the coupling is present already in stationary conditions (Fig. 7a).

As a last point, we discuss the temperature dependence of the ²⁰⁷Pb isotropic chemical shift (δ_iso). As shown in Fig. 7c, δ_iso exhibits a distinct temperature dependence, as commonly observed in lead compounds^61–64 and also in MAPbCl₃,²⁹ even though our experimental data show a small but detectable deviation from the normally reported linear dependences. For MAPbI₃, this temperature dependence can be fully attributed to changes in the lattice parameters,²⁵ as it is usually the case in Pb compounds.^61–64 We note that this temperature dependence shows no clear sign of the cubic phase transition.

¹²⁷I NQR

¹²⁷I is a quadrupolar spin-5/2 nucleus with 100% natural abundance and reasonably high Larmor frequency, and thus could be considered a high receptivity nucleus. However, due to its very high quadrupole moment (−69.6 fm²),⁶⁵¹²⁷I NMR must be restricted to systems with a relatively high symmetry⁶⁶ (for MAPbI₃, preliminary ¹²⁷I NMR spectra have been reported in ref. 13). In MAPbI₃, however, a significant EFG at the iodine sites (χ_Q > 550 MHz) provides a good sensitivity in NQR spectroscopy, offering an alternative method to probe the I-environment in this material. As shown in Fig. 8a, two pairs of transitions in ¹²⁷I NQR are detected for the two distinct iodine sites present in the tetragonal phase, while in the cubic phase only 2 signals are observed, corresponding to conversion into a single iodine site. The resonance frequencies of the signals, along with the quadrupolar constant χ_Q, show appreciable temperature dependence (Fig. 8b and Fig. S9, ESI†), a behavior that is commonly attributed to a planar-torsional oscillation of a resonant nucleus (I) about another atom to which it is bonded (Pb).

The temperature dependence of the ¹²⁷I quadrupole resonance frequency in MAPbI₃ can be estimated based on the Bayer theory⁶⁷ of torsional oscillations of atoms in the lattice.⁶⁸ In this model the nuclei are agitated by the torsional motion at a rate that is very fast compared to their NQR frequencies, and they are therefore affected by an averaged temperature-dependent EFG that is different from the stationary situation (details in Section 8 and Fig. S9, ESI†). Thanks to this approach,⁶⁸ it is possible to fit the experimental data and extract a torsional frequency ν_t ≈ 1.15 THz, corresponding to 38 cm⁻¹. For a simple comparison, the reported Raman frequencies for Pb–I contributions in MAPbI₃ were calculated to be between 20–50 cm⁻¹, with a perceptible coupling to MA-modes.^69,70 Another work, instead, experimentally observed Pb–I vibrations at higher frequencies (75–125 cm⁻¹), also here with contributions coming from MA-modes.⁷¹ In addition, as shown in Fig. 8c, we record the quadrupolar spin–lattice relaxation time (T_1Q) for ¹²⁷I as a function of temperature. Again according to the Bayer theory,⁶⁷ this temperature dependence can be used to calculate an accurate estimate of the correlation time τ_a of the torsional vibrations, albeit a closed solution for a spin 5/2 nucleus does not exist.⁷² Under certain assumptions, a partial solution can still be found for our system, but only in a very narrow temperature range (290–340 K), making the obtained τ_a values of 2 × 10⁻¹⁰ s (290 K) and 4 × 10⁻¹⁰ s (340 K) only indicative. We refer the reader to the ESI,† (Section 8) for the complete treatment and related assumptions.

Conclusions

In conclusions, we have shown that much information on halide perovskites can be collected by means of NMR spectroscopy. Specifically, we studied the short-range ion dynamics in MAPbI₃ in a broad temperature range using ¹H, ¹³C, ^14,15N, ²⁰⁷Pb multinuclear solid-state NMR and ¹²⁷I NQR. We found that the motion of the MA cation is ascribable to that of a symmetrical rotor, with similar interactions of CH₃ and NH₃⁺ groups with the inorganic lattice. Specifically, the predominant mode of motion for the MA cation is identified as a bi-axial rotation, as supported by ¹⁴N NMR and related spectral simulations. This rotation becomes nearly isotropic above the cubic phase transition, and sufficiently fast to dominate the relaxation mechanism through spin-rotational interactions. Regarding the inorganic lattice, ²⁰⁷Pb NMR shows strong cross-relaxation to quadrupolar ¹²⁷I, which ultimately fully dominates the ²⁰⁷Pb spin–lattice relaxation times T₁ and severely influences the linewidth (T₂). The temperature behavior of ¹²⁷I NQR spectra is indicative of a torsional oscillation of I about Pb atoms, where the frequency of oscillations can be estimated.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors thank V. Terskikh (University of Ottawa, Canada) for help with the UHF NMR experiments. Access to the 900 MHz NMR spectrometer was provided by the National Ultrahigh Field NMR Facility for Solids managed by University of Ottawa, Ottawa, Canada (http://www.nmr900.ca). Dr Helga Hoier is thanked for performing the XRD measurements. Open Access funding provided by the Max Planck Society.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp01535j

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