Structural origins of broadband emission from layered Pb–Br hybrid perovskites

We present synthetic design rules for achieving and optimizing broadband emission from layered halide perovskites.

S3 refined using a riding model with an isotropic thermal parameter 1.2 times that of the attached carbon or nitrogen atom. Thermal parameters for all non-hydrogen atoms were refined anisotropically. In some instances, portions of the organic molecules were refined using a disorder model.

Powder X-ray diffraction (PXRD)
PXRD measurements were performed on a PANalytical X'Pert2 powder diffractometer with a Cu anode (Kα 1 = 1.54060 Å, Kα 2 = 1.54443 Å, Kα 2 /Kα 1 = 0.50000), a programmable divergence slit with a nickel filter, and a PIXcel 1D detector. Additional PXRD measurements were performed on a Bruker D8 Advance diffractometer equipped with a Cu anode, fixed divergence slits with a nickel filter, and a LYNXEYE detector. The instrument was operated in a Bragg-Brentano geometry with a step size of 0.01° or 0.02° (2θ). Simulated powder patterns were calculated using the crystallographic information files (CIFs) from single-crystal X-ray experiments.

Optical measurements
Variable-temperature static photoluminescence spectra were collected with a spectrograph (Acton Research SpectraPro 500i) equipped with a silicon CCD (Hamamatsu) detector, using excitation from a 375-nm CW diode laser. Samples were cooled using either liquid nitrogen or liquid helium with a Janis ST-500 cryostat. Single-crystal samples were placed on the cold finger with the inorganic layers oriented perpendicular to the incident beam. Powder samples were prepared by mixing a suspension of the ballmilled perovskites in toluene with a solution of poly(methyl methacrylate) (average M w ≈ 120,000 by GPC) in toluene. This slurry was then allowed to dry at room temperature. For single crystals and powders, the excitation intensity was ca. 0.820 mW·cm −2 or 5.74 mW·cm −2 , respectively, as measured by a Newport 918-UV-L photodiode. For power-dependence measurements on (HIS)PbBr 4 single crystals, the spectrograph entrance slits were closed down to maximize spectral resolution ( Figure S9). The multiple narrow emission PL peaks have been observed in other layered perovskites, and have been variously ascribed to free and bound excitonic emission 6 or phononic sidebands 7 of the free-excitonic emission. Room-temperature absorption measurements were taken using an Agilent Cary 6000i spectrometer in transmission mode.

Calculation of Pb-Br angles and their error
The distortion values D tilt , D out , and D in were calculated using the Matlab script PbBrAngles_witherror.m, which is available as supporting information. The Matlab script contains all mathematical operations necessary for angle and error calculations and comments are included. Atoms were treated as vectors by using their Cartesian coordinates derived from single-crystal X-ray structures. In order to calculate the inand out-of-plane components D in and D out , planes were defined by three Pb atoms rather than crystallographic planes because in certain cases, the Pb atoms do not lie in the (001) plane. Calculating projected angles requires an arccosine function. The uncertainty δf in an arbitrary function f(x) is δf = |df/dx|·δx, hence δ(cos −1 x) = |(1 -x 2 ) −1/2 |·δx. This results in higher error in D values when the θ values are closer to 180°. This does not represent lower precision in a particular X-ray structure, but is a necessary result of correct error propagation.

Structural parameters
In addition to D out , D in , D tilt , and the distance between terminal Br and Pb atoms, we tested many other structural parameters related to the inorganic lattice, which could have potentially yielded a correlation to S4 ln(I BE ·I NE −1 ). Whenever possible, we calculated and tested largest, smallest, average, and distribution values for each parameter. In total we tested 52 structural parameters in the inorganic lattice. The parameters not shown in the manuscript are as follows (terminal Br atoms are denoted Br ax and bridging Br atoms are denoted Br eq ): (1) the set of all unique cis intraoctahedral Br-Pb-Br angles including its subsets (2) cis Br ax -Pb-Br eq intraoctahedral angles and (3) cis Br eq -Pb-Br eq intraoctahedral angles; (4) the set of all unique trans intraoctahedral Br-Pb-Br angles, including its subsets (5) trans Br ax -Pb-Br ax intraoctahedral angles and (6) trans Br eq -Pb-Br eq intraoctahedral angles; (7) all unique intraoctahedral Pb-Br distances, including its subset (8) intraoctahedral distances between Br eq and Pb; intraoctahedral distances between (9) Br ax -Br ax , (10) Br ax -Br eq , and (11) Br eq -Br eq ; (12) a measure of octahedral compression (ratio of distance between Br ax atoms to average of trans Br eq -Br eq distance), (13) SHAPE (see below), (14) octahedral elongation (λ oct , see below) (15) octahedral angle variance (σ 2 oct , see below); (16) the interoctahedral torsion angle between Br ax atoms, and (17) the interoctahedral torsion angle between Br eq atoms.

Calculation of polyhedral distortion
For each crystal structure, Cartesian coordinates were obtained for a lead atom and the six halides that comprise its octahedral coordination environment. These values were supplied to SHAPE, 8 a program that calculates continuous shape measures for atomic positions relative to an idealized polyhedron based on minimal distortion paths, 9 generalized interconversion coordinates, 10 and the following algorithm: 11 Here, S is a dimensionless continuous symmetry measure obtained by assessing the root-mean-square (rms) deviation of N vertices from their idealized positions. Q k is a vector containing the coordinates of the N vertices and P k is the vector for idealized positions. Q 0 is the coordinate vector of the center of mass, and S is normalized by the rms distance from the center of mass to all vertices thus avoiding size effects.
In addition to using S as a measure of polyhedral distortion, two other parameters were calculated, which also provide a measure of polyhedral distortion: octahedral elongation (λ oct ) and octahedral angle variance (σ 2 oct ). Here, λ oct describes the deviation of an octahedron's six Pb-Br bond distances away from the Pb-Br bond distance of a regular octahedron with the same volume. Similarly, σ 2 oct describes deviations of the twelve cis Br-Pb-Br angles within an octahedron away from 90°. These parameters are calculated as follows: 12 Here, d i is the Pb-Br bond length, d 0 is the Pb-Br bond length in a regular octahedron of the same volume, and α i is the Br-Pb-Br angle within the octahedron. Note that in calculating σ 2 oct , Bessel's correction was used to reduce error in population variance due to a finite sample count, hence a prefactor S5 of 1/(n -1) was employed where n is the number of unique Br-Pb-Br angles. This gives a prefactor of 1/11 instead of 1/12.

Calculation of the Arrhenius relation and self-trapping depth.
Following photoexcitation, we assume that the free-exciton (FE) and self-trapped-exciton (STE) states are related by the configuration coordinate diagram shown in the figure. We can use Arrhenius relations to relate the self-trapping depth (−Δ − ; given by the difference in activation energies for trapping and detrapping so that Δ − = , − , < 0) with the relative integrated intensity of the FE and STE emissions, as determined by the following derivation.
where ( ) and ( ) are the populations of the FE and STE states as a function of time, respectively, and , , , , , , , , , and are the rate constants associated with radiative emission from the FE state, non-radiative decay from the FE state, radiative emission from the STE state, non-radiative decay from the STE state, trapping from the FE to the STE state, and detrapping from the STE to FE state, respectively. We define = , + , + and = , + , + for simplicity.
Solving the system of equations yields: For steady-state (continuous-wave, time-integrated) PL experiments, a general expression for PL intensity 13 is: Evaluating the intensities for the free and self-trapped states, we find: We then obtain a general expression for the ratio of steady-state intensities for a two-state system: Assuming that exciton trapping and detrapping are primarily thermally activated processes, the following Arrhenius relations hold for a given temperature: where , and , are the activation energies for exciton trapping and detrapping, respectively, k B is Boltzmann's constant, and and are exponential prefactors, which we assume are temperatureindependent over the range studied. Therefore, Substituting into the expression for the ratio of intensities, we obtain: If we assume that trapping is much faster than radiative and non-radiative decays from the STE state, i.e., that , + , ≪ , we can make the following approximation: Below 80 K, the approximation , + , ≪ may no longer hold, since carrier self-trapping may become decreasingly likely.