1. Accounting for the TFI will achieve consistent interpretation of the pump-probe data of both reflectance and transmittance modes, and this will in turn 2. allow us to better extract new quantitative information on the key carrier dy-namics that would otherwise be inaccessible, including charge carrier diffusion perpendicular to the sample surface, hot-carrier dynamics, PbI2impurities, and so on.
2 TRANSIENT REFLECTANCE LITERATURE BACKGROUND
The debate around TR and its importance began with attempts to correct the bandgap TA spectra of perovskites in order to better analyse the hot-carrier cooling and other phenomena [15, 16, 17, 18, 19]. However, many of the proposed approaches are either very difficult to implement experimentally or have led to wildly different in-terpretations of the importance and scale of the reflectance effects on the TA spectra [19]. For instance, Ruhman et al[19]. compared the TA spectra of a perovskite thin-film to its nanoparticle counterpart in solvent to reveal any potential effects from reflectance. Obviously, this method only works if one can produce both versions of the material without alterations to the energy states or other properties, which is very unlikely. Meanwhile, Tamming et al. [16]modified a standard TA setup with a white light pulse interferometer to probe the ultrafast refractive index changes but severely limiting the observable timescale. Liu et al.[20]reduced the impact of re-flectance on the TA spectra by applying an oil droplet on the sample, which reduces the reflectance from the film by reducing the gap in the refractive indexes.
Herein are two new methods presented for correcting the TA spectra, by which I mean extracting the real photoinducedΔAbased on the photoinducedΔRandΔT: 1. An efficient linear approximation for calculating theΔnandΔkbased on the
measuredΔRandΔT, and a TFI model.
2. A simple ΔA approximation that only requires the steady-state reflectance spectra in addition to the measuredΔRand ΔT, without any need for TFI modelling.
If the main focus of the study is on the carrier lifetimes or only the changes in absorption are of interest, where information onΔnis not needed, the latter method provides a quick and easy approximation of theΔAfrom theΔRandΔT. I will
also provide an analysis on the accuracy of this approximation, and I believe the methods developed by us are more accurate and universally applicable than the ones previously established in the literature.
The second main application of TR is the determination of charge carrier diffu-sion and mobility. The TR signal, in part, depends on the distribution of charge carriers inside the film. First of all, the reflectance of light at an interface depends on the difference in refractive index between the two media. The strength of the pho-toinducedΔndepends on the number of excited charge carriers, so the more carriers there are at the surface the stronger the TR signal should be. This lead to what I call the surface-carrier concentration (SCC) model[13, 14, 15, 21], which assumes that in the absence of TFI (when the film absorption is high enough to prevent interfer-ence from the film bottom interface), the TR signal strength only depends on the carrier concentration at the surface. Another assumption that was often made was that theΔncaused by a single carrier is constant across time, in other words, there is no change in the state of the carrier apart from recombination. Under these condi-tions, one could easily model the decay in TR signal as a function of recombination and carrier diffusion away from the surface.
However, what was not taken into account was the gradient in the refractive index inside the film: when theΔnis not uniform due to inhomogeneous carrier distribu-tion, the refractive index cannot be uniform either. This gradual change in refractive index generates reflections and interference from inside the film even when the ab-sorption and film thickness are enough to prevent interference from the film bottom interface. The new TFI based model developed by us and presented in Section 4 is the first to account for this effect, which proved to be very significant for modelling carrier diffusion from TR measurements. I will compare the two models, the SCC and TFI models, to validate under which conditions the earlier assumptions made in the literature were valid.
Nonetheless, TR is a powerful method for charge carrier diffusion and mobil-ity studies. Most applications rely on carrier mobilmobil-ity perpendicular to the film surface, which is something the other contact-free methods cannot measure. These other methods, such as combining time-resolved photoluminescence (TRPL)[22] or transient absorption[23, 24]with optical microscopy can only measure the dif-fusion parallel to the film surface. In polycrystalline materials such as perovskites, these methods typically end up measuring diffusion across multiple grain
bound-aries rather than the real diffusion in the devices. Contact-free methods also include time-resolved terahertz spectroscopy (TRTS) and time-resolved microwave conduc-tivity (TRMC), which measure the sum mobility of the carriers across very short dis-tances inside single grains[25, 26, 27]. Even combining these different techniques leaves open major questions: if one technique gives the minimum mobility across grain boundaries and the other the maximum mobility inside single grains, and the two are orders of magnitude apart[28, 29], what is the real carrier mobility inside the devices? PL decay has been employed to study the perpendicular diffusion by adding a charge extraction layer[30], but those results suffer from poor reliability because they require the ETL/HTL to accept the charge carriers as fast as they dif-fuse to the interface. PL decay can therefore measure either the charge transfer rate or the diffusion, but only the slower factor of the two. The contact-based methods for measuring perpendicular diffusion also have their own problems because they require using high voltage across the sample, which may result in ionic movement in addition to the electrode-film interface complications, in addition to other inter-facial problems[31, 32, 33, 34]. Despite its own difficulties, TR remains the only contact-free method for probing the carrier diffusion and mobility perpendicular to the film surface, and it is also one of the few with ultrafast time resolution.
3 STEADY-STATE REFLECTANCE MODELLING
Before we look at modelling the transient reflectance signals, we need to establish how the reflectance is modelled in general. This section goes through these equations and models from the literature that are relevant for the basic modelling of steady-state and therefore transient reflectance signals.