As explained previously in Section 2, the main interest of using transient reflectance was to correct the TA spectra of perovskite samples at the bandgap region (∼750 nm), in addition to the carrier mobility studies. Generally speaking, TA spectroscopy provides key information on the excited state dynamics in perovskites and other materials, such as the free carrier dynamics, charge recombination, effective masses, bandgap renormalization, and charge transfer from one material to another[16, 44, 45, 46]. However, the focus of the TA studies on the visible range has left much of the potential information ignored from the NIR part of the spectrum[47]. The few studies that included the NIR were about the interfacial charge transfer between per-ovskite and ETL or HTL[46, 48]. These studies found that the TA response differs greatly if the perovskite is coupled with materials such as TiO2or spiro-OMeTAD, which was attributed to an ultrafast hot-carrier (HC) charge transfer [47, 48]. A carrier is described as "hot" when it resides at a higher energy level than the lowest conduction band: upon excitation, any excess photon energy lifts the electron over the bandgap to higher conduction bands, from where the carrier eventually "cools down" to the lowest conduction band. This cooling usually lasts only few hundred femtoseconds, and the time resolution of many pump-probe systems is only slightly better than that. Therefore it was assumed that the effectively instantaneous differ-ence in the TA spectra between perovskite and perovskite-ETL/HTL samples was caused by an extremely fast charge transfer. On the other hand, alternative methods for charge transfer investigation, such as monitoring the photoluminescence (PL)
de-cay, have too low time resolution to distinguish this fast transfer and could not dis-pute these findings despite witnessing the charge transfer taking place in the nanosec-ond range, thousands of times slower than the transfer claimed by the TA-based stud-ies[49]. Based on these results, the HC charge transfer could be ultrafast while the cooled down carriers transfer much more slowly.
Another aspect of the HCs that is of wide interest is their cooling rate. Novel devices such as HC solar cells would greatly benefit from long-lasting HCs[50], and there are many reports of unusually long cooling times in different perovskite ma-terials. It has been suggested that the so-called hot phonon bottleneck phenomenon extends the HC lifetime[44, 51, 52, 53]: the carriers release their excess energy to the lattice via longitudinal optical phonons, which themselves may take longer to relax and spread the energy to the rest of the perovskite. This maintains a higher carrier temperature because the phonons cannot accept more energy if their energy level is the same as that of the excited carriers’.
In the upcoming Section 6, I will go through theΔñfitting results for CsMAFA perovskite, lead-free perovskite NCs, and GaAs thin-film samples. Based on these fitting results, I will then perform an analysis of the spectral features and carrier lifetimes, including the HCs and other points of interest.
5.1.1 Charge carrier mobility
As mentioned in Section 2, TR is currently the only contact-free technique that can measure carrier diffusion perpendicular to the film surface, and with ultrafast time resolution. The other methods have found the carrier mobility in halide perovskites to range anywhere from less than 1 cm2(Vs)−1(PL studies) to over 100 cm2(Vs)−1 (TRTS and TRMC), and some have even suggested that the hot carriers can travel at "quasi-ballistic" speeds, meaning hundreds of nanometers within few picoseconds before they cool down. Due to the wide discrepancies in the results, it remains un-clear how much the grain boundaries[24, 54, 55], traps [40, 56]and other defects limit the diffusion of charge carriers in polycrystalline perovskite films compared to the ideal single crystals, especially perpendicular to the film surface. Traps refer to variations in energy states caused by defects: deep traps are located close to the middle of the bandgap and facilitate non-radiative recombination, whereas shallow traps are only minor variations and may temporarily prevent the charges from
mov-ing before they receive enough additional energy to leave the trap. In the trappmov-ing- trapping-detrapping model, it is assumed that the charges essentially hop from one trap to another, which is a much slower process than free diffusion[40].
Additionally, excluding the quasi-ballistic diffusion results, a common assump-tion in the literature is that the charge carrier diffusion is ambipolar[22]. Ambipo-lar means that the electric field between the electrons and holes prevents them from moving too far from each other, essentially keeping them together. Ambipolar dif-fusion is therefore dictated by the slower carrier type[57].
Some PL studies have combined the PL measurements with conventional optical microscopy in order to map the carrier concentration across a wide surface area, which enables monitoring how fast the carrier spread across the surface. However, this technique is still often limited to monitoring the diffusion across multiple grain boundaries unless the grain size is very large, and the time resolution is too low to detect very fast phenomena[22]. Similarly, transient absorption has been combined with optical microscopy to follow the spread of charge carriers by using a small pump spot and a wide probe area, which resulted in the aforementioned finding of quasi-ballistic propagation. According to them, the hot carriers diffuse extremely fast, travelling hundreds of nanometers in just a few picoseconds and even across grain boundaries[58, 59, 60, 61]. As an alternative ultrafast technique, TR could either confirm or dispute these findings.
TR has been used previously on polycrystalline MAPbI3and MAPbBr3perovskite films,[14, 15, 21]where the SCC model was used in order to analyse the diffusion constant and the surface recombination velocity. According to them, the carrier mobility was approximately 7 to 11 cm2(Vs)−1, which is between the PL studies and the TRTS results. However, when looking at the original TR data in some of their supporting information, the TR follows the expected signal from Figure 4.11: when theΔnand Δk have opposite signs (as was the case at that probe wavelength), the ΔReither does not immediately respond to changes in distribution or it even be-comes slightly stronger due to the diffusion instead of decaying. Thus it could be that their model significantly underestimated the diffusion constant. Their analysis also ignored the hot carriers, focusing only on what happened well after their cool-ing. Thus so far the TR results have not supported the quasi-ballistic HC diffusion, but there is still room for dispute due to their use of the SCC model and ignoring the ultrafast phenomena.
Our aim is to answer what the transient reflectance can tell us about the carrier diffusion by utilizing the more sophisticated TFI model instead of only relying on the SCC model. The open questions include the impact of traps and grain bound-aries, whether or not the hot carriers travel at quasi-ballistic speeds, and is the diffu-sion ambipolar or not.
5.1.2 GaAs as a reference material
GaAs is one of the most well-known semiconductors, which is why we selected it as a reference sample to validate our new TR model and perovskite results. It has a very long charge carrier lifetime and very high carrier mobility[62, 63], which should make it ideal for TR diffusion studies. However, it was recently published that the TR signal of GaAs is affected by a relaxation process, where theΔñchanges over several picoseconds as the carriers relax to the lowest conduction band [64]. Nonetheless, we set out to validate this finding and see if we can circumvent it to determine the carrier diffusion from the TR response.