Dynamic emission Stokes shift and liquid-like dielectric solvation of band edge carriers in lead-halide perovskites

Lead-halide perovskites have emerged as promising materials for photovoltaic and optoelectronic applications. Their significantly anharmonic lattice motion, in contrast to conventional harmonic semiconductors, presents a conceptual challenge in understanding the genesis of their exceptional optoelectronic properties. Here we report a strongly temperature dependent luminescence Stokes shift in the electronic spectra of both hybrid and inorganic lead-bromide perovskite single crystals. This behavior stands in stark contrast to that exhibited by more conventional crystalline semiconductors. We correlate the electronic spectra with the anti-Stokes and Stokes Raman vibrational spectra. Dielectric solvation theories, originally developed for excited molecules dissolved in polar liquids, reproduce our experimental observations. Our approach, which invokes a classical Debye-like relaxation process, captures the dielectric response originating from the incipient anharmonicity of the LO phonon at about 20 meV (160 cm−1) in the lead-bromide framework. We reconcile this liquid-like model incorporating thermally-activated dielectric solvation with more standard solid-state theories of the emission Stokes shift in crystalline semiconductors.


Supplementary Figures
Supplementary Figure 1. Optical reflectance and photoluminescence spectra of CsPbBr3 and MAPbBr3 from 4K to 300K. A. CsPbBr3 optical reflectance, B. CsPbBr3 photoluminescence, C. MAPbBr3 optical reflectance, D. MAPbBr3 photoluminescence. Each photoluminescence spectrum is normalized with respect to its peak intensity. Reflectance and photoluminescence spectra are vertically offset for display clarity.  Figure 4A and 4D in the main text.

Supplementary Note 1. Analysis of reflectance and photoluminescence spectra.
Supplementary Figure 1 shows optical reflectance and photoluminescence of CsPbBr3 and MAPbBr3 as a function of temperature.

Supplementary Note 2. Kramers-Kronig analysis of reflection.
We calculated the absorption coefficient from the measured reflectance, using Kramers-Kronig constrained variational analysis. 1-3 The key to the calculation of the absorption coefficient was determining the complex dielectric functions using reflectance data, as expressed in Supplementary Equations 1-3. To extract the complex dielectric function, we used a sum of Lorentz oscillators and a constant background, as expressed in Supplementary Equation 4. The use of Lorentz oscillator functional form guarantees that the Kramers-Kronig relationship is valid.
Numerically, we used N = 256 Lorentz oscillators, half the number of our available data points in each spectrum. The oscillators were placed at fixed positions, evenly spaced out with 3 meV intervals. The oscillators' width were fixed with = 5 meV. The oscillator amplitudes and background level were adjustable parameters. This numerical procedure produced a robust representation of the reflectance data.
In Supplementary Figure  We note that, for accurate determination of the optical dielectric function, one has to take into account the contributions from high-and low-lying transitions beyond the measured spectral range. In the fitting the high energy end of the dielectric function showed an unphysical down turn. The complex refractive indices showed the same behavior accordingly. This is due to the absence of higher energy resonances beyond the spectral range of our data. This apparent lineshape distortion is excluded in subsequent data analysis.
To extract spectral characteristics of the resonance, we fit both the absorption coefficient and photoluminescence resonances with a single Lorentzian function. Spectral background features other than the resonance were modelled with a cubic baseline for absorption and a constant baseline for photoluminescence.

Supplementary Notes 3. Numerical inversion of Laplace transform.
The numerical inversion of a Laplace transform is widely performed. 4 Here we implement the Talbot and the Gaver-Stehfast algorithms. Using simple analytical test function pairs, we test the validity of the inverse transform. As shown in Supplementary Having obtained the dielectric solvation dynamics in the time domain numerically, we examine how the dielectric response affects solvation dynamics. Using the Debye relaxation model expressed in the main text with the point charge solvation, we alter the parameters ε∞, ε0, and τ one at a time and study the effect of each parameter. The results are shown in Supplementary Figure 5. The parameter τ determines the time constant of the relaxation process. These changes do not affect the extent and strength of screening, only the time needed to reach steady state. The parameter ε∞ describes the high frequency, or equivalently short time scale, dielectric response. This affects the screening dynamics shortly after excitation, but not the steady state. As ε∞ increases, |∆E(t = 0)| increases, and |∆E(t = ∞)| remains constant. The parameter ε0 describes the low frequency, or equivalently long time scale dielectric response. This affects the screening dynamics long after excitation. As ε0 increases, |∆E(t = ∞)| increases, and |∆E(t = 0)| remains constant. We note that this examination treats ε0, ε∞, and τ effectively as independent variables. In some physically driven models, ε0 and/or ε∞ are often related to τ. 5 The above examination shows that, ε0 − ε∞ in the numerator is the main factor affecting the steady-state Stokes shift, τ in the denominator has negligible influence on the steady-state Stokes shift. When ε0 − ε∞ is temperature dependent, the observed steady-state Stoke shift will thus change as a function of temperature.

Point charge solvation model for fitting the T dependent Stokes shift
As discussed in the text, the Gaussian solvation model provides an analytical solution of the solvation dynamics in the Laplace domain, as shown in Supplementary Equation 5 (Equation (2) in the main text). This is the equation 5.3 in the Song, Chandler, and Marcus paper. 6 Here a is the solvation cell radius, ε(s) is the dielectric function, and s = iω. The dielectric response function of the solvent can be described by a Debye relaxation model, as shown in Supplementary Equation 6, where is the characteristic time scale of dielectric relaxation.
is sometimes modeled as a function of T and related to 0 − ∞ , 5 but it is not always necessary. 7 as an independent variable is shown to be insignificant in determining steady-state Stokes shift, as discussed in Supplementary Note 3. The temperature dependence of dielectric response can be modelled with Supplementary Equation 7.
Supplementary Figure 6 shows the fitting of the data. In calculation of Supplementary Equation 7 (Equation (5) in the main text) for ε0 − ε∞, ε∞ is taken as 4 from experimental data as discussed in Supplementary Note 5, the prefactor A is taken as 6.5, and Ea is taken as 20 meV. The solvation cell radius a is determined to be about 10 nm.

Point dipole solvation model for fitting the T dependent Stokes shift
The nature of the luminescing band edge species at room temperature has been a topic of ongoing discussion. To account for the stabilization of a possible complex formed by bound electron-hole pairs and colliding free carriers, we also model the dielectric solvation of a point dipole. The procedure is similar to that of the point charge. For a point dipole, the analytical solution of solvation dynamics in the Laplace domain is expressed in Supplementary Equation 8, where p is the dipole moment of excited species, v is the solvation volume, ε(s) is the dielectric function, and s = iω. This is the equation 5.7 in the Song, Chandler, and Marcus paper. 6 The description of solvent dielectric response function and its temperature dependence follow from the previous case of point charge solvation, expressed in Supplementary Equation 6 and 7.
Supplementary Figure 7 shows the results calculated from the above model to match experimental data. In calculation of Supplementary Equation 7 for ε0 − ε∞, ε∞ is taken as 4 from optical data as discussed in Supplementary Note 5, the prefactor A is taken as 6.5, and Ea is taken as 20 meV, identical to the case of point charge solvation. If one takes the e-h separation as 10 nm, then the required solvation volume is about 2.2 × 10 3 nm 3 . The predicted ε0 shown here is identical to Supplementary Figure 6.

Supplementary Note 5. Dielectric function in the optical frequency range.
Optical frequency dielectric functions are obtained from the Kramers-Kronig constrained variational analysis as described above. In Supplementary Figure 8, we plot the dielectric function below the main resonance at the low energy end of our spectral range. The dielectric function in this range is largely constant, independent of temperature evolution. Thus we use ε∞ = 4 for our subsequent analysis of dielectric solvation.

Supplementary Note 6. Prediction of emission Stokes shift based on main text Equation (6)
In the naive prediction of emission Stokes shift based on the solid state Fan model Equation (6) in the main text, we used the following parameters: m0: mass of free electron. ERyd: Rydberg energy of hydrogen atom in vacuum, 13.6 eV. me: band mass of electron, 0.13m0 taken from Miyata et al. 8 mh: band mass of hole, 0.19m0 taken from Miyata et al. 8 ELO: the LO phonon energy, 20 meV as obtained from the fitting of emission Stokes shift as well as the fitting of absorption and emission linewidth. ε∞: dielectric constant at the high frequency limit, 4.0 as obtained from our optical reflectance measurements. ε0: dielectric constant at the low frequency limit, 10.75 as taken from Tilchin et al. 9