Bond Polarizability as a Probe of Local Crystal Fields in Hybrid Lead-Halide Perovskites

A rotating organic cation and a dynamically disordered soft inorganic cage are the hallmark features of organic-inorganic lead-halide perovskites. Understanding the interplay between these two subsystems is a challenging problem, but it is this coupling that is widely conjectured to be responsible for the unique behavior of photocarriers in these materials. In this work, we use the fact that the polarizability of the organic cation strongly depends on the ambient electrostatic environment to put the molecule forward as a sensitive probe of the local crystal fields inside the lattice cell. We measure the average polarizability of the C/N–H bond stretching mode by means of infrared spectroscopy, which allows us to deduce the character of the motion of the cation molecule, find the magnitude of the local crystal field, and place an estimate on the strength of the hydrogen bond between the hydrogen and halide atoms. Our results pave the way for understanding electric fields in lead-halide perovskites using infrared bond spectroscopy.

where α H 0 is the static longitudinal polarizability of C(N)-H bonds, Ω H is the resonant frequency of the relevant cluster of modes that correspond to longitudinal bond stretching, and Z is the Born effective charge, which is relatively simple to compute -we refer to previous studies 11 that calculate polarizability in a similar way.
To remind, Born effective charge Z of a given ion is a tensor defined as where p i is the total dipole moment of the molecule and r j is the coordinate of the ion in question. In our case the polarizability of the C(N)-H bond stretching is related to the specific terms of the full Born charge tensor that correspond to the displacements of H ions along the corresponding bonds. In this work we calculate this "longitudinal" Born effective charge by following its definition above. Specifically, we first begin by finding the equilibrium atomic positions in the CH 3 NH + 3 ion by performing an initial geometry optimization and calculating the dipole moment of the resultant configuration. Then we displace the H ion in question by δa = 0.01a 0 in the direction of C(N), where a 0 is the equilibrium C(N)-H bond length. In the new configuration we re-calculate the total dipole moment ⃗ p of the distorted molecule. An important detail here is that since net charge of methylammonium is not zero, we take care to keep the origin unmoved relative to the unperturbed atomic positions when re-calculating ⃗ p. Provided the origin is kept fixed, the change δ⃗ p = ⃗ p − ⃗ p 0 does not depend on it, and the Born effective charge corresponding to the "longitudinal" displacement of H along the C(N)-H bond is found as Z = δ⃗ p/δa.
To confirm that the chosen distortion δa is appropriate for determining Z, we plot in Fig. S1 the change of the dipole moment as a function of δa. We observe a linear dependence within the selected range, which confirms that Z is a well-defined physical property.
S3 Figure S1: Difference in dipole moment of methylammonium molecule upon a slight stretch in one C(N)-H bond from the equilibrium geometry. The bond stretch is presented as a percentage of the equilibrium C(N)-H bond length a 0 and is proportional to δa. The dashed lines are linear fits to the data, where the coefficient of determination R 2 = 1.00 for both N-H and C-H stretch.

Derivation of Eq. (1) of the main text
In an external field, the dipole moment of a bond in a molecule changes as Here, ∥ and ⊥ determine the directions with respect to the C(N)-H axis of the bond. Note that there are two directions perpendicular to the bond -the notation ⊥ is used for both, for convenience. We are interested in the polarizability α ∥∥ , which is denoted as ⟨α H 0 ⟩ in the main text.
To relate ⟨α H 0 ⟩ to the density matrix ρ (cf. Eq. (1) of the main text), we set α ∥⊥ to zero, in agreement with our DFT calculations. [To have a physical picture of this observation, one should consider the C(N)-H bond as a charged ellipsoid, see, e.g., 12 .] Then, we write the 'longitudinal' part of the dipole moment of the molecule that is induced by a weak electric S4 field (assuming that the center of mass is not changed): where the sum is over all C-H and N-H bonds; ⃗ n H is the unit vector that determines the direction of the bond, Z is the associated charge. The change in the position of the H k atom along the bond, ∆X k , is calculated using the harmonic approximation where m H is the mass of a hydrogen atom. Note that we have assumed here that C and N atoms are infinitely heavy. This is done in line with the set of approximations made during the analysis of the data in the main text.
Let us now consider the quantity ∆⃗ p ∥ · ⃗ E. As the HOIP has (on average) a cubic lattice, ∆⃗ p ∥ · ⃗ E = ⟨α H 0 ⟩E 2 , which allows us to write where

Spectroscopy
In the optical Kerr-effect setup, the probe pulses are near-infrared with wavelength λ = 1028nm and pulse duration τ = 270 fs; the pump pulses are variable-wavelength mid-IR, inducing birefringence at the probe sample after being delayed inside the test sample. The The data in Fig. 2B of the main text can also be used to calculate the extinction coefficient, κ: neglecting the Fresnel losses at the perovskite surface in comparison to bulk absorption, κ can be found from the ratio of the intensity transmitted through T to the incident intensity. The ratio of transmission with (I with ) and without (I without ) the test sample (given that the attenuation of light with distance z is exponential, where d is the sample thickness, λ 0 is the wavelength of the pump in vacuum, and we have assumed that the extinction coefficient of air, κ air is 0.  as depicted in Fig. S3B, we achieve an excellent fit with Eq. (2) of the main text, which is derived for a single resonance energy ℏΩ H .

Derivation of Eq. (2) of the main text
We use the Lorentz-Lorenz relation between the phase refractive index n and the polarizabilities of all degrees of freedom of the system {α i } 14 : where N is the number density of atoms and the sum goes over all degrees of freedom including electrons. As can be seen in Fig. 3 of the main text, the molecular contribution to the total phase refractive index n ph = n ph,el + n ph,mol is small. Therefore, n ph,mol and the molecular polarizability α i mol are related as 6n ph,el (n 2 ph,el + 2) 2 n ph,mol ≈ We convert the phase index n ph,mol = n ph −n ph,el into the group index n g,mol ≈ ∂(ωn ph,mol )/∂ω,

Uncorrelated dynamics
If the cage and the molecule are uncorrelated, we write With this, we write where H N (H C ) refers to a hydrogen atom from the N-H (C-H) bond. We assume that the longitudinal polarizability depends only on ⃗ n ⃗ Since the fields in the center of the lattice unit of a cubic lattice are weak, we write where the expansion coefficients C H k do not depend on the electric field. The integral To see this, we expand the vector ⃗ n in components parallel and perpendicular to ⃗ R (for a sketch of the geometry, see Fig. 1 of the main text). The two Gauss's law and Kirchhoff's loop rule, respectively, which allows us to derive the expression presented in the main text:

Correlated dynamics
To estimate the value of the polarizability in case of correlated dynamics, we approximate the density matrix as where ρ mol ( ⃗ X; ⃗ Q 0 ) describes the molecule assuming that the atoms in the cage are located at their average positions, ⃗ Q 0 ; ρ cage ( ⃗ Q) describes the atoms in the cage. Equation (S14) relies on the observation that the time scale for the cage dynamics is sub-ps whereas the dynamics of the C-N reorientation is 'much' longer, on the order of a few ps [18][19][20] .
and that the dynamics of the cage S10 is harmonic, which allows us to write To establish the form of ρ mol ( ⃗ X; ⃗ Q 0 ), we note that there are six degenerate positions of the C-N bond in the three (parallel to the surfaces) planes of symmetry of a cube, see, e.g., 18 .
The interaction between the N-H bonds and the halide atoms in the cage determines the positions of the hydrogen atoms 16,21,22 . For simplicity, we assume that the complex N-H-Br forms a straight line, so that we can directly use our DFT results presented in Fig. 1 of the main text. This assumption is a crude approximation of the molecular position as the C(N)-H bond features a rapid 'wobbling-within-a-cone' motion 18 , for review see 23 . However, with this, we can provide an intuitive illustration of the approach (which is the main purpose of our work), and, furthermore, to put a lower bound on the electric field felt by a molecule.
Finally, we derive where ⃗ E ⃗ Q 0 ( ⃗ X H i ) is the crystal field at the locus of the C(N)-H bond. Using this expression and Fig. 1 of the main text, we estimate ⃗ E ⃗ Q 0 ( ⃗ X H i ) whose value is reported in the main text.

Coulomb fields inside the lattice
In HOIPs, molecules are located close to the centers of lattice units, and experience weak electric fields. To estimate these fields, we calculate the electric potential close to the center of a lattice unit assuming that Br and Pb are point particles with charges -1|e| and +2|e|, S11 respectively. This potential can be calculated from the Taylor series whose first terms read where ⃗ Q Br k ( ⃗ Q Pb k ) is the position of the kth Br (Pb) atom, and the sum goes over all atoms of a lattice unit. Due to the cubic symmetry of the lattice, the sums presented in Eq. (S17) vanish, and one needs to consider higher-order terms in the expansion, which lead to weak electric fields for small values of x.
From the symmetry argument, one can expect that the electric fields behave as | ⃗ E| ∼ ex 3 /(a/2) 5 close to the center of the unit cell, here a = 5.92Å is the lattice constant 15 . [It is worthwhile noting that a fast decay of the electric field, ∼ 1/| ⃗ Q| 5 is the reason why it is enough to consider a single lattice unit for our estimation.] Assuming that x ≃ 1Å, we obtain | ⃗ E| ∼ 0.06 V/Å. [Numerical calculations give even a smaller value, and confirm the order of magnitude of this estimate.] This value is too small to account for the measured value of the polarizability. Therefore, we conclude that one cannot consider ions in the cage as point-like objects. The simplest extension consistent with cubic symmetry of the lattice is a quadruple instead of Pb-Br-Pb complex. This leads to the potential ϕ ∼ D 2d 3 at the locus of the N-H bond, here D is the quadrupolar moment and d = 2.5 Å (see the main text). We put an estimate D ∈ [0.5, 1.6] eÅ 2 -a natural atomic value -by comparing to the electric field deduced from renormalization of the polarizability. In the main text, we report only the central value.

Hydrogen Bond
We estimate H-bonding interactions from the energy required to move the molecule to the center of the cage. Using our estimate of D, we find that this energy per a hydrogen bond S12 is ZD/(2d 3 ) − ZD/(2d 3 1 ) ∼ 0.1 eV, where d 1 ≃ 3.2 Å, and we use the Born charge Z = 0.4e to represent the charge of the N-H hydrogen, see Fig. 1 of the main text. Although, this estimate includes a number of approximations, it agrees well with the values reported in the literature, supporting our conclusion regarding usefulness of bond spectroscopy.