Lattice dynamics and vibrational spectra of the orthorhombic, tetragonal, and cubic phases of methylammonium lead iodide

The hybrid halide perovskite CH 3 NH 3 PbI 3 exhibits a complex structural behavior, with successive transitions between orthorhombic, tetragonal, and cubic polymorphs around 165 and 327 K. Herein we report ﬁrst-principles lattice dynamics (phonon spectrum) for each phase of CH 3 NH 3 PbI 3 . The equilibrium structures compare well to solutions of temperature-dependent powder neutron diffraction. By following the normal modes, we calculate infrared and Raman intensities of the vibrations, and compare them to the measurement of a single crystal where the Raman laser is controlled to avoid degradation of the sample. Despite a clear separation in energy between low-frequency modes associated with the inorganic (PbI 3 − ) n network and high-frequency modes of the organic CH 3 NH 3 + cation, signiﬁcant coupling between them is found, which emphasizes the interplay between molecular orientation and the corner-sharing octahedral networks in the structural transformations. Soft modes are found at the boundary of the Brillouin zone of the cubic phase, consistent with displacive instabilities and anharmonicity involving tilting of the PbI 6 octahedra around room temperature.


INTRODUCTION
Materials that adopt the perovskite crystal structure are known for their complex structural landscapes, with a large number of accessible polymorphs depending on the temperature, pressure, and/or applied electric field.For ternary ABX 3 perovskites, the A cation is at the centre of a cube formed of corner sharing BX 6 octahedra.Displacement of the A cation is usually associated with a ferroelectric (Brillouin zone centre) instability, while tilting of the BX 6 octahedral network is usually linked to antiferroelectric (Brillouin zone boundary) transitions. 1,2ybrid organic-inorganic perovskites are formed when one of the elemental perovskite building blocks is replaced by a molecular anion or cation. 3,4There exists a large family of such compounds, including the widely studied formate perovskites, which contain both molecular anions and cations. 5,6[9][10][11][12][13][14] Methylammonium lead iodide (MAPbI 3 , where MA represents the CH 3 NH + 3 cation), was first reported by Weber in 1978. 15It is the most relevant hybrid halide perovskite for photovoltaic application.The transition from orthorhombic to tetragonal to cubic perovskite structures as a function of temperature has been studied by techniques including calorimetry and infrared spectroscopy 16 , single-crystal X-ray diffraction 17 , and dia) Electronic mail: a.walsh@bath.ac.uk electric spectroscopy. 18Recently analysis of powder neutron diffraction (PND) measurements has provided more quantitative insights into the temperature dependent behaviour of the MA cation within the anionic (PbI -3 ) n network. 19There is now direct evidence of the degree of order of the MA cation in the different phases, and the average lattice parameters (and thus extent of octahedral tilting) as a function of temperature through the first and second order phase transitions.
In this study, we calculate the phonon dispersion in each phase of MAPbI 3 within the harmonic approximation, computing the force constants with density functional theory (DFT).We use the PBEsol functional, which is a generalised-gradient-approximation (GGA) to the exchange-correlation function, numerically evaluated with Perdew's method, adjusted to give more accurate lattice constants and forces for solids. 20he lattice dynamic calculations allow the atomic origin of each phonon mode to be identified.Changes in lattice polarisation and polarisability for each eigenvector provide the infrared and Raman activity of each mode.Spectral features related to the inorganic and organic components (from 0 to 3000 cm −1 ) are well reproduced in comparison to the Raman spectra of a single crystal of MAPbI 3 .Overlap is found between the vibrations of the CH 3 NH + 3 and PbI - 3 components up to 130 cm −1 , with the modes from 300 to 3000 cm −1 being associated with pure molecular vibrations.The phonon dispersion has implications for developing quantitative models for the generation, transport and recombination of photogenerated electrons and holes in hybrid perovskite solar arXiv:1504.07508v1[cond-mat.mtrl-sci]28 Apr 2015 cells.
Structure Models.The normal modes of a system are defined for an equilibrium configuration.Calculating the vibrations for a non-equilibrium structure will result in imaginary frequencies upon diagonalising the dynamical matrix.Therefore we have generated extremely well optimised structures of MAPbI 3 .One challenge in calculating the phonons of hybrid perovskites is the soft nature and complicated potential energy landscape of some of the restoring potentials, particularly those involving the organic cation.
Orthorhombic Phase.The orthorhombic perovskite structure is the low temperature ground state of MAPbI 3 and maintains its stability up to ca. 165 K. 19,22,23 A comparison of the enthalpy from DFT calculations confirms this ordering in stability.The difference in enthalpy is small, just 2 meV per MAPbI 3 unit compared to the most stable tetragonal phase, yet 90 meV compared to the high-temperature cubic phase.
Initial diffraction pattern solutions assigned the P na2 1 space group. 16,22Recent analysis of higher quality powder neutron diffraction data reassigns it to P nma (a D 2h point group). 19The structure is a √ 2a × √ 2a × 2a supercell expansion of the simple cubic perovskite lattice, i.e. following the lattice transformation matrix In the P na2 1 phase, the PbI 6 octahedra are distorted and tilt as a + b − b − in Glazer notation 1 with respect to the orientation of the conventional cubic cell.In this low-temperature phase, the four molecular cations in the unit cell are static on the diagonals of the ab planes pointing towards the undistorted facets of the cuboctahedral cavity.Correspondingly, molecules belonging to different planes are anti-aligned with a head-tail motif.Such an antiferroelectric alignment is expected from consideration of the molecular dipole-dipole interaction. 24n the low temperature orthorhombic phase the CH 3 NH + 3 sublattice is fully ordered (a low entropy state).The ordering may be sensitive to the material preparation and / or cooling rate into this phase, i.e. the degree of quasi-thermal equilibrium.It is possible that different ordering might be frozen into the low temperature phase by epitaxy or application of external force or electric fields.
Tetragonal Phase.At 165 K MAPbI 3 goes through a first-order phase transition to the tetragonal space group I4/mcm (D 4h point group), which continuously undergoes a second-order phase transition to the cubic phase by ca.327 K 19,22,23 .As with the orthorhombic phase, this can be considered a √ 2a × √ 2a × 2a expansion of the cubic perovskite unit cell.
The molecular cations are no longer in fixed position as in the orthorhombic phase.The molecules are disordered between two non-equivalent positions in each cage. 23,25.The tetragonal distortion parameter, c 2a in the cubic basis is greater than unity (1.01 at 300 K), corresponding to an elongation of the PbI 6 octahedra along the c axis.The associated octahedral tilting pattern is a 0 a 0 c − in Glazer notation.
Atomistic simulations within periodic boundary conditions require an ordered configuration.The solved crystal structure shows that there are several possible configurations for the organic cations within the tetragonal unit cell.These configurations have similar enthalpies within DFT, 26 which is consistent with the observed disorder.We choose to use the most energetically stable structure, which is also consistent with a previous DFT investigation. 27n the tetragonal structure, the MA cations are aligned as in the orthorhombic phase, towards the face of the perovskite cage, i.e. < 100 > in the cubic basis.The MA in different (001) planes are approximately orthogonal to one other.
Cubic Phase.With increasing temperature the tetragonal lattice parameters become more isotropic (i.e.c 2a moves closer to 1), and the molecular disorder increases, to the point where a transition to a cubic phase occurs around 327 K.The transition can be seen clearly from changes in the heat capacity, 16 as well as in temperature dependent neutron diffraction 19 .
The cubic space group P m 3m (O h symmetry) has been assigned to this high-temperature phase.Although the methylammonium ions posses C 3v symmetry, the orien-  I. Equilibrium cell parameters from DFT/PBEsol energy minimisation, including the converged plane-wave cut-off, k -point mesh and force threshold.Z represents the number of formula units of CH 3 NH 3 PbI 3 per cell.The calculated difference in enthalpy (∆H) of each phase is given with respect to the ground-state orthorhombic configuration and per CH 3 NH 3 PbI 3 unit.All equilibrium structures are available in an on-line repository. 21Shown for comparison are the cell parameters from powder neutron diffraction (PND). 19tional disorder gives rise to the effective higher average symmetry.For the bromide and chloride analogues of MAPbI 3 , pair-distribution function analysis of X-ray scattering data indicates a local structure with significant distortion of the lead halide framework at room temperature. 28e previously considered alignment of the molecules along the principal < 100 > (face), < 110 > (edge), and < 111 > (diagonal) directions of the cubic unit cell, and showed that they are of similar DFT enthalpy, with a small barrier for rotation. 29Further ab-initio molecular dynamics showed an average preference for the < 100 > facial configuration at 300 K. 30 Therefore we chose the < 100 > configuration as our reference structure for the lattice vibrations.
Representations of the crystal structure of each phase are shown in Figure 1, the equilibrium structure parameters are listed in Table I, and the structures themselves are available in an on-line repository. 21ETHODS Computational.The total energy and atomic forces were computed from first-principles within density functional theory as implemented in the code VASP. 31,32oise in the lattice vibrations was minimised by rigorous convergence of total Kohn-Sham energy with respect to the basis set (kinetic energy cut-off for plane waves) and sampling of reciprocal space (density of the k -point mesh).The final computational set-up is summarised in Table I.
We performed complete optimisation of the cell volume, shape and atomic positions, with the PBEsol 20 semi-local exchange-correlation functional.The scalarrelativistic projector-augmented wave method 33 was employed, with a pseudo-potential treating the Pb 5d orbitals as valence.All atomic forces were reduced to below a threshold of 1 meV/ Å. Due to the presence of the organic cations, which breaks the ideal lattice symmetry, deviations in the expected parameters can occur, e.g. in the high temperature pseudo-cubic phase, the three equilibrium lattice parameters are not equal.The equilibrium structure parameters (at 0 K and excluding zero-point contributions) are reported in Table I.
The normal modes are calculated within the harmonic approximation, using the Phonopy [34][35][36] package to construct and evaluate the dynamical matrix composed of DFT force constants.Both the finite displacement method (FDM or supercell approach) 37 and density functional perturbation theory (DFPT) 38 approaches to construct the force constants were tested.The results of both approaches produced similar vibrational spectra, with a variance in the mode energies of 6 cm −1 .
Within a primitive cell of N atoms there are 6N possible displacements (±x, ±y, ±z), which can be reduced by the crystal symmetry.For the orthorhomic phase, the 288 possible displacements are reduced to 41, while the tetragonal and cubic phases required 288 and 72 displacements, respectively.The phonon dispersion (for q-points away from the Brillouin zone centre, the Γ point) in the cubic phase was probed in a 2 × 2 × 2 supercell.Due to computational expense, we do not calculate this for the other (larger unit cell) phases, where the phonons are considered at the Γ point only.
Once the normal eigenmodes and eigenvalues are calculated, it is possible to model their associated Raman and infrared (IR) activity by mode following.The two spectroscopic techniques probe different physical responses of the material: the change in polarisation for IR and the change in polarisability for Raman.The IR spectra are simulated with the analytic formula of Gianozzi & Baroni (using the Born effective charge tensor) 38 .Prediction of the Raman spectra required computing the change in macroscopic dielectric tensor with respect to each normal mode of the system, a significant DFT calculation in terms of computational expense. 39xperimental.Methylammonium lead iodide single crystals were grown according to the method of Poglitsch and Weber. 2212.5 g of lead acetate trihydrate (Pb(CH 3 CO 2 ) 2 • 3 H 2 O, Sigma) was dissolved in 10 mL hydroiodic acid (HI aq , 57 wt%, Sigma) in a 50 mL round bottom flask and heated to 100 • C in an oil bath.Separately, 0.597 g of CH 3 NH 2 (aq, 40 wt%, Sigma) was added dropwise to a further 2 mL of HI aq kept at 0 • C in an ice bath under stirring.The methylammonium iodide solution was then added to the lead acetate solution and Raman spectra were collected in backscattering geometry with a high resolution LabRam HR800 spectrometer using a grating with 600 lines per millimetre and equipped with a liquid-nitrogen-cooled charge coupled device (CCD) detector.The 785 nm line of a diodepumped solid state laser was used as excitation beam and focused onto the sample using a long distance 20× microscope objective.Raman measurements were carried out at 100 K using a gas-flow-type cryostat with optical access that fits under the microscope of the Raman setup.
Heating by laser light directly absorbed by CH 3 NH 3 PbI 3 has been shown to lead to rapid degradation of the material resulting in PbI 2 Ra-man signatures. 40Since 785 nm light is only weakly absorbed the heating effect of the laser was low enough to ensure the crystal structure was preserved.The power density incident on the sample was kept at 80 W/cm 2 .Further, samples were kept under vacuum inside the cryostat during the measurements.

RESULTS Harmonic Phonons.
The full phonon density of states (DOS) is shown for the three phases of CH 3 NH 3 PbI 3 in Figure 2. Also plotted is the partial DOS, where assignment to CH 3 NH + 3 or PbI - 3 is performed based on the atomic contribution to each eigenvector.An animation of all 36 eigenmodes of the cubic phase is provided as a WEO.
Due to the large difference in atomic mass of the organic and inorganic components, and to the difference in bonding between the inorganic cage and the covalently bonded molecule, we anticipated that the low frequency modes would comprise entirely of motion by the PbI 6 octahedra, while the high frequency modes will involve the CH 3 NH + 3 cation.While this is qualitatively the case, there is also significant coupling between the two.Taking the example of the cubic phase with 36 modes: the highest-energy 18 modes (forming bands ii and iii) correspond to molecular vibrations, i.e. the 3N -6 modes of the methylammonium ion.For an isolated non-linear molecule, the 6 translational and rotational degrees of freedom do not contribute to the pure vibrational spectrum, but this is not the case for a molecule inside a cuboctahedral cavity.
The 6 additional molecular modes are strongly coupled to the 9 modes (i.e.3N -3 for PbI - 3 ) associated with stretching of the Pb-I bonds and breathing of the PbI 6 octahedra, which results in the spectral overlap observed in the partial DOS of band i. Particularly striking is the low-frequency pivoting motion associated with the libration of the molecular dipole, coupled with a breathing of the octahedral framework (e.g.modes 10 and 15 in the WEO).The final three zero-frequency modes correspond to acoustic translations of the lattice.
Vibrational Spectra.For MAPbI 3 it is not possible to assign the spectral activity directly with group theoretic irreducible representation analysis of the phonon modes due to the anisotropic molecule breaking the crystal symmetry.This allows for simultaneous Raman and IR activity even in the cubic phase.The predicted spectra (the Γ point phonon modes weighted by the computed spectral intensity, convolved with a Lorentizian for experimental comparison) are reported in Figure 2 for each phase.
Raman and IR activity is observed across each of the three phonon bands previously discussed.A notable exception is the lowest-energy purely molecular vibration near 300 cm −1 , which is neither Raman nor IR active.
To understand the effect of embedding the methylammonium in MAPbI 3 , we calculate the normal mode vibrations of an isolated methylammonium ion in vacuum with a similar density functional theory method to our periodic calculations (PBE with an atom centered augmented cc-pVQZ basis set).Thereby we calculate the 18 molecular modes, directly accessing their symmetries and nature.The C 3v symmetry of the molecule separates the vibrational bands into one A symmetric mode and blueshifted two-fold degenerate asymmetric E modes.The six bands we find are in ascending energy: twist around the C-N axis (282, 886 cm −1 ); vibration along the C-N axis (923, 1239 cm −1 ); bending of the C-H bonds (1418, 1451 cm −1 ); bending of the N-H bonds (1478, 1622 cm −1 ); stretching of the C-H bonds (3018, 3119 cm −1 ); stretching of the N-H bonds (3321, 3395 cm −1 ).
Due to the cation charge density being centered towards the N, the motion of the protons associated with N have the strongest affect on the dipole moment and therefore strongest IR activity.Due to the stronger bonds, they are consistently blue shifted relative to the C end.Owing to the large molecular dipole moment (2.2 D) 4 , the two high-frequency asymmetric stretching modes of NH + 3 (band iii) results in the strongest absolute IR intensity.The hydrogen stretching modes (band iii) are responsible for significant Raman activity.The only mode involving C or N motion is the weakly IR and Raman active vibration at 923 cm −1 (vacuum), 1007 cm −1 (cubic perovskite).
The rotation of the CH 3 against the NH 3 unit, while being strongly populated in molecular dynamic simulations, and which forms the main source of quasiinelastic neutron scattering, is entirely IR and Raman inactive in vacuum.This is the mode responsible for the 282 cm −1 (vacuum), 318 cm −1 (cubic) , 300-310 cm −1 (tetragonal), 370-372 cm −1 (orthorhombic) vibration.Progressive confinement of MA from vacuum to the orthorhombic phase blue-shifts the energy of the vibration.
In the solid state, the degeneracies in the molecular modes are typically split by local environment effects, peaks are both redshifted and blueshifted, and the IR and Raman activity varies.As such, it is evident that analysis of the Raman and IR spectra in the experimentally easily accessible molecular frequency range can enable statements to be made about the local structure and configuration of the hybrid perovskite.Our data is collected for a particular representation of the cubic and tetragonal phase; in reality the location of the MA in these phases will be disordered.As such, detailed comparison of theory to experiment will require sampling the thermodynamic ensemble of structures.
The temperature resolved (between 140 and 299 K) IR spectra for tetragonal and orthorhombic phases have been previously reported by Yamamuro 41 .We reproduce the position and the intensity of the peak observed at 900 cm −1 , reliably assigning it to the (also Raman ac-tive) C-N bond stretch.High quality IR measurements, in particular looking at the very low energy transitions, would provide considerable information on the nature of the domains and local structure in a MAPbI 3 film.
To our knowledge, no Raman spectra for the three phases have previously been reported across the full frequency range.Reliable measurements are a challenge due to chemical instability of the material.MAPbI 3 is strongly affected by environmental conditions, such as the presence of ambient moisture 23,42 .Isolated in vacuum the material can still decompose and bleach due to heating, including by that imposed by the (typically high) Raman laser fluence. 40Such degradation leads to the formation of PbI 2 , which overlaps in Raman spectra with MAPbI 3 , and so easily leads to misinterpretation.
The Raman spectra of a high-quality single crystal of MAPbI 3 is shown in Figure 2. The agreement between the predicted and measured spectra is remarkable, with the response across bands i, ii and iii well reproduced.
Anharmonic Effects.The lattice dynamic simulations discussed above were performed within the harmonic approximation.All eigenmodes at the centre of the Brillouin zone were real (positive frequencies) for each phase, i.e. the structures are locally stable.
The phonon dispersion across the first Brillouin zone is shown for the cubic perovskite structure in Figure 3.Here imaginary (negative frequency or 'soft') modes are found at the zone boundaries.Such instabilities are a common feature of the perovskite structure, and correspond to antiferroelectric distortions linked to to rotations and tiling of the octahedra in neighbouring unit cells. 2 The soft modes are centred around the R and M points, which correspond to the < 111 > and < 110 > directions in the cubic lattice.This behaviour is similar to the inorganic perovskite CsPbCl 3 , where neutron scattering was used to probe condensation of these modes, which leads to successive transitions from the cubic to tetragonal to orthorhombic phases. 43The effect of these modes in MAPbI 3 , and the associated high levels of anharmonicity at room temperature can be observed directly in molecular dynamics simulations, where temporal rotations of the CH 3 NH + 3 ions and distortions of the PbI 6 octahedra have been found in several studies. 30,44n approach to including the effects of temperature (thermal expansion) and first-order anharmonicity in lattice dynamic calculations is the quasi-harmonic approximation (QHA). 45,46The computational cost is one order of magnitude higher than the harmonic approximation and thus was considered for the cubic phase only.The volumetric thermal expansion coefficient extracted from the PND data at 300 K is 1.32×10 −4 /K, which compares very well to the value of 1.25 × 10 −4 /K computed within the QHA.The predicted thermal expansion for MAPbI 3 is similar to inorganic semiconductors (e.g. for PbTe the value is 0.7 × 10 −4 /K at 300 K 35 ) and positive over the full temperature range.
The temperature dependence of the phonon modes can be described by the Grüneisen parameter, which has an average of around 1.6 (see Figure 3), slightly below the value of 1.7 found in PbI 2 47 .The imaginary modes at R and M remain at all temperatures, consistent with the cubic lattice being a dynamic average of a locally distorted structure; the same phenomenon is observed in CsSnI 3 . 48The high level of anharmonicity associated with the soft titling modes is consistent with the 'ultra low' (< 1 Wm −1 K −1 at 300 K) lattice thermal conductivity reported for single crystals and polycrystalline MAPbI 3 . 49Hybrid halide perovskites are thus also promising for application in thermoelectric devices. 50ONCLUSIONS The vibrational frequencies of three crystallographic phases of the hybrid perovskite CH 3 NH 3 PbI 3 have been investigated.We identified three main phonon branches present in the three phases.
Two high-frequency branches are associated with the vibration and bond stretching of the molecular cation with frequencies in the range 300 cm −1 to 3300 cm −1 .The lowest energy branch, below 150 cm −1 , arises predominately from the inorganic cage, but with half the modes coupled to the motion of the molecule.The simulated Raman spectra are in good agreement with measurements on a single crystal of MAPbI 3 .Dynamic instabilities occur at the zone boundaries, which requires methods beyond the harmonic approximation, such as self-consistent phonon theory, for an accurate treatment.These results suggest that the room temperature structure of MAPbI 3 is fluctional, owing to the persistent titling and distortion of the octahedral networks and rotations of the molecular cations.
These factors may be important for developing a quantitative understanding and model of how hybrid perovskite solar cells operate.Upon excitation, the relative stability of free carriers and excitons depends intimately on the dielectric screening of the material, which includes a vibrational component (ionic response).The transport and recombination of photo-generated charge carriers will also be influenced by electron-phonon coupling, which can significantly reduce the effective size and distribution of electrons and holes with the perovskite solar cell.

ACKNOWLEDGEMENT
The authors are grateful for helpful discussions with Mariano Campoy-Quiles.The research at Bath has been supported by the EPSRC (Grant No. EP/K016288/1 and EP/M009580/1), the ERC (Grant No. 277757), EU-FP7 (Grant No. 316494), and the Royal Society.AJJ and OJW were funded through the CDT in Sustainable Chemical Technologies (EPSRC Grant No. EP/ G03768X/1).PRFB and AMAL are grateful to the EP-SRC (Grant Nos.EP/J002305/1, EP/M014797/1, and EP/M023532/1).AG acknowledges the Spanish Ministerio de Economia y Competitividad through project number MAT2012-37776.This work benefited from access to both the University of Bath's High Performance Computing Facility and ARCHER, the UK's national highperformance computing service, which is funded by the Office of Science and Technology through EPSRC's High End Computing Programme (Grant No. EP/L000202).

WEB ENHANCED
An animation of the 36 Γ-point phonon modes of the cubic phase of MAPbI 3 in gif format is available http: //people.bath.ac.uk/aw558/temp/mapi_phonon.gif.

FIG. 1 .
FIG. 1.The crystal structures of the (a) orthorhombic, (b) tetragonal and (c) cubic phases of CH 3 NH 3 PbI 3 .The upper and lower panels are oriented through < 100 > and < 001 >, respectively.Lattice parameters and coordinates obtained from powder neutron diffraction were optimised using density functional theory (PBEsol).The PbI 6 octahedra are shaded grey.All structures are available in an on-line repository. 21

FIG. 2 .
FIG. 2. (a-c) Projected phonon density of states (PDOS) for the three phases of CH 3 NH 3 PbI 3 as calculated from DFT/PBEsol, generated by convolution with a 32 cm −1 Lorentzian.(d-f) Simulated infrared (IR) spectra.(g-i) Simulated Raman spectra.(g) The measured Raman spectrum of a single crystal in the othorhombic phase at 100 K is coplotted in orange.Simulated spectra were broadened by convolution of a 10 cm −1 Lorentzian.Insets in the simulated spectra are the same data on a logarithmic scale to show the structure in the low intensity modes.