Free Carriers versus Self-Trapped Excitons at Different Facets of Ruddlesden–Popper Two-Dimensional Lead Halide Perovskite Single Crystals

The physical origin of sub-band gap photoluminescence in Ruddlesden–Poppers two-dimensional (2D) lead halide perovskites (LHPs) is still under debate. In this paper, we studied the photoluminescence features from two different facets of 2D LHP single crystals: the in-plane facet (IF) containing the 2D inorganic layers and the facet perpendicular to the 2D layers (PF). At the IF, the free carriers (FCs) dominate due to the weak electron–phonon coupling in a symmetric lattice. At the PF, the strain accumulation along the 2D layers enhances the electron–phonon coupling and facilitates self-trapped exciton (STE) formation. The time-resolved PL studies indicate that free carriers (FCs) at the IF can move freely and display the trapping by the intrinsic defects. The STEs at the PF are not likely trapped by the defects due to the reduced mobility. However, with increasing STE density, the STE transport is promoted, enabling the trapping of STE by the intrinsic defects.

glass bottles (20 ml). Afterwards, the glass bottles were sealed and stirred at room temperature for 30 minutes to induce yellow precipitates. Completely clear solutions were obtained after reaction for few minutes at 80 °C as a precursor. Bulk SCs were grown from such a solution at a cooling rate of 0.5 °C/day starting from 50 °C.
Characterizations: Single-crystal XRD measurements were performed on SuperNova Dual Wavelength CCD diffractometer (Agilent Technologies, Mo-Kα with λ = 0.71073 Å) at room temperature (~298 K). Further structural solves and refinements by full-matrix least-squares fitting on F 2 using SHELX-97, the details are similar to our recent work. 1 Crystallographic data and structural refinements for three samples are summarized in Table S1. The Pb-Br bond lengths and Br-Pb-Br angles are listed in Table S2 and S3, respectively. Powder XRD data were collected on a Rigaku MiniFlex 136 II diffractometer using Cu-Kα radiation with λ = 1.5406 Å.

n-PAPB
Br ( When the lone-pair cations (e.g., Sn 2+ , Pb 2+ , Bi 3+ , etc.) and d 0 transition metals (e.g., Nb 5+ , V 5+ , Mo 6+ , etc.) form a polyhedron with oxygen or halide ions, the primary distortive cause can be attributed to second-order Jahn-Teller (SOJT) effects (electronic effects). 2,3 For instance, Figure S3 is a PbBr6 octahedron from the iso-BAPB sample of this work, in which the Pb 2+ deviate from the center position where they should be.

Figure S3. A distorted PbBr6 octahedron
Herein, we can use three parameters, namely the tilting distortion parameter (Δ), the angle distortion parameter (Σ), and the off-center distortion parameter (Δod), to quantitatively compare the PbBr6 octahedral distortions in our iso-BAPB, n-BAPB, and n-PAPB. Δ and Σ represent the distortion degree of the Pb-Br bond lengths and Br-Pb-Br cis-angles, respectively. 4 Δod describes the degree of deviation of the Pb atom from the ideal octahedral S7 center. 2 The calculation equations for Δ and Σ are listed below (Eq. S1 and Eq. S2). And the calculation method of Δod are also given in section.
where the di are the individual Pb-Br bond distances in a the PbBr6 octahedron. <D> is the average Pb-Br bond length, The φi are the twelve cis-angles of Br-Pb-Br around the Pb atom.
For the calculation of Δod, we can first define three trans-bond angles based on Figure S3 are as follows: Then, taking the difference in the associated bond lengths and dividing by the cosine of each angle results in the magnitude of the Δod: 2 The distorted parameters of PbBr6 octahedrons, tilting distortion parameter (Δ) (a), the angle distortion parameter (Σ) (b), and the off-center distortion parameter (Δod) (c).
The calculation results of these three parameters are shown in the Figure S4a-c. It is worth noting that there are two types of Pb 2+ ions in iso-BAPB (i.e., Pb1 and Pb2) since it crystallizes in a low symmetry monoclinic space group. In this work, we took their average parameter value to represent its PbBr6 distortion. The distortion analysis revealed that the Δ and Δod values of n-PAPB are much larger than that of iso-PAPB and n-BAPB. On the other hand, the value of Σ in iso-BAPB is the largest one (19.97°), while the values of n-BAPB (15.95°) and n-PAPB (14.86°) are relatively close. Overall, we can conclude that the PbBr6 octahedron of n-BAPB is the least distorted, whereas that of n-PAPB is distorted the strongest to relax the interface lattice strain.

S3 Calculation of the lattice mismatch between our three 2D RP perovskites.
The lattice mismatches between our 2D RP perovskites and 3D MAPbBr3 were calculated by the following where the α2D is the lattice constant along the 2D layered facets, which were taken from the resolved structures by us (Table S1, Figure S5); α3D is the lattice constant of 3D MAPbBr3 (I4/mcm) ( Figure S5a). 6 The calculation results of lattice mismatches were shown in the Figure S5e. Figure S5, the lattice constants of 3D MAPbBr3 (I4/mcm) (a), and the lattice constants along the 2D layered facets of iso-BAPB (b), n-BAPB (c), and n-PAPB (d); the lattice mismatch between the three 2D RP perovskites and 3D MAPbBr3 (f).

S4 Calculations of the penetration depths and excitation density of the three compounds.
To calculate the penetration depths and excitation density of these samples, we first measured their absorption coefficients using the laser. In order to ensure that the laser spot is completely illuminated on the samples, we prepared a sufficiently large film according to the method in the literature. 7 After the absorption measurements, we used AFM to determine the thickness of the film ( Figure S6). The absorbance (A) can be calculated by the following formula: Here, I0 is incident light intensity and I1 is outgoing light intensity. After measuring the A values at 375 nm of the iso-BAPB, n-BAPB and n-PAPB are about 0.37, 0.39 and 0.33, respectively. The absorption coefficient ε = 2.303A/l, where l is the path length of the light which equals to the thickness of the crystal. So, we obtain ε =1.40 × 10 5 , 1.38 × 10 5 and 1.27 × 10 5 cm -1 for iso-BAPB, n-BAPB and n-PAPB, respectively. The penetration depths of them at 375 nm are about 71, 73 and 79 nm, respectively (δ = 1/ ε). The excitation density n can be calculated as photon flux f in photons/cm 2 multiplied by absorption coefficient ε: n = fε, Table S4 summarizes the corresponding excitation density at different laser photon flux in TRPL measurements: Figure S6. The AFM pictures of iso-BAPB (a), n-BAPB (b) and n-PAPB (c) films with the thickness values.

S10
PL spectra of these three compounds were measured at temperatures ranging 100-280 K in a cryostat ( Figure   S7), liquid nitrogen as the coolant. In order to verify our argumentation, we calculate the electron-phonon coupling strength from the FWHMs of the temperature-dependent PL spectra using the following moldel: 8,9 (T) = 0 + ac Here, Γ0 is a temperature-independent inhomogeneous broadening that arises from scattering due to disorder and  Figure S8).

S6. The fitting analysis of the PL decay.
We can fit the PL kinetics using triexponential decays with one fast component (t1), a medium component (t2), and a slow component (t3) (Table S5). The slow components t3 has a very small amplitude and can be neglected.
Here t1 and t2 can be assigned to the trapping related nonradiative recombination and radiative recombination of the photoexcited charge carriers, respectively.

S7. Detailed model of trap filling and fitting process.
The detailed calculation procedures of the model have been described in the reference. 10 In brief, the equilibrium between free carriers and excitons can be described using the Saha equation. When photodoping (i.e accumulated trap filling) is present, the Saha equation can be generalized as the following equation expressing the concentrations of electrons (ne), holes (nh) and excitons (nx) corresponding to the overall untapped photogenerated species density N: where Rpop and Rdep are the recombination rates of trap population and depopulation, respectively. Taking equations (S9) and (S10) we can obtain the average concentration of electrons as: Here N(0) can be simplified as initial excitation density Nc. γ0 is the total rate of electronic decay not involving traps. Substitution of equation (S11) into (S12) gives: During the fitting of the trapping model, we first assume that both traps exhibit filling but due to different trap population and depopulation rates, the ratio of unoccupied trap densities between two type of traps varies with different excitation intensity ( Figure S9a-b). Therefore we can plot A1/(A1+A2)~Nc data and fit with the expression combination equations S13-S16.
Here nuncT1 and nuncT2 are the concentration of filled traps 1 and 2, respectively. NT1 and NT2 are the concentrations of original concentration of trap 1 and trap 2, respectively. In this work, A of iso-BAPB and n-BAPB were approximately equal to 0.97 × 10 15 cm -3 , respectively, PL recording time t0 was 2.5×10 -7 s, and γ0 was taken as the rate of the charge recombination not contributing to the trap filling process, we first set all the four fitting parameters (NT1, NT2, R1, and R2) while analyzing these two compounds. The best fitting results as shown in Table S6. In this table, R2 of iso-BAPB and n-BAPB are 15.5 and 36.7, respectively. Considering R is the ratio between trap population and depopulation rates, we can then roughly estimate the depopulation time of their traps 2 using the trap population time obtained in PL kinetics (lifetime of the second component 5.05 and ~8.02 ns, respectively) to be ~80 and ~290 ns, respectively. These depopulation time of traps 2 are not much different from the interval between the pulses in our measurement (250 ns). Therefore, we conclude that no considerable trap filling should occur in their traps 2 since the depopulation rate is too fast. 11 The model is then modified for the case that trap filling occurs only in trap1 as follows: (S18) Here the A1/(A1+A2) represents the ratio between unoccupied density of trap 1 and original density of trap 2. We first set all the three fitting parameters (NT1, NT2, and R) free for the values from the equation (S17). Then, we fit the equation (S18) ( Figure S9c) to get the new parameters (Table S6). S14 Figure S9. PL decay kinetics with different excitation fluence for the IFS of iso-BAPB (a) and n-BAPB (b); dependence of the amplitude ratios A1/(A1 + A2) in the multi-exponential fitting of the decays on the initial charge densities Nc for the IFs of iso-BAPB and n-BAPB (c).

S8. Calculation of the free carrier ratio after photo-excitation
After photoexcitation by the laser pulse, actually the free carriers and weak coupled excitons coexist in the crystal under a thermodynamics equilibrium. The ratio between two species is a fixed term resembling the ion-electron balance in a hot plasma, which depends strongly on the exciton binding energy as well as the excitation concentration. Therefore, we can use a classic Saha-Langmuir theory to roughly calculate the ratio of the free carriers X (i.e., fraction of the free carriers among all the excited species) in the system: where Eb refers to the exciton binding energy (We obtained their Eb by fitting the temperature-dependent PL peak intensities of ~59 meV, ~93 meV, ~88 meV for iso-BAPB, n-BAPB, and n-PAPB, respectively), 4 m is the exciton effective mass, T is the temperature and n is the excitation concentration. Previous study has shown that this equation is also applicable to 2D hybrid lead halide perovskites. 12 The following table summarized the calculated x ratio of three samples under the excitation condition in PL decay measurement (Table S7). Apparently, during our measurement, the majority of the excited species in the 2D perovskite crystals is free carriers and the contribution of the exciton can almost be negligible.

S9 Local strain accumulation and phonon coupling analysis of Br-based and I-based perovskites
First, the feature of the 'edge state' should be determined by the stiffness of the lattice. According to the theoretical calculation and the experimental measurement, the Br-based perovskites generally have a higher bulk modulus than I-based perovskites and become less prone to long-range lattice distortions. 13.14 This can lead to two consequences: 1) stronger strain accumulation in Br-based perovskite, 2) small instead of large polaron formation in Br-based perovskites corresponding to the short-range local lattice distortion (e.g. on the surface facet). In addition, the general e-LO phonon coupling strength can be calculated by the following equation :  Here and ϵs and ϵ∞ refers to the dielectric permittivities at static and optical frequencies, respectively. The term within the bracket evaluates the lattice polarity. The lattice polarity made by Pb-Br bonding should be higher than Pb-I due to the larger electronegativity difference between Pb and Br (0.7) compared with Pb and I (0.4).
Therefore, we can expect high phonon coupling in Br-based system, which would facilitate the formation of polaron states. The above two reasons should encounter why in our Br-based 2D perovskites, the STE instead of conventional sub-gap states are formed at PFs.