TR-PL spectroscopy is a very efficient tool to investigate the processes ruling the relaxation and recombination of carriers and excitons. In this work, firstly we discuss the topic of carrier thermalization.

In Figure 2A, a streak image of the PL time evolution for the bulk sample is shown for the fastest time scale, after nonresonant excitation at 3.35 eV, at a temperature of 10 K. The strong emission at 2.32 eV, which dominates at early times, comes from the free exciton (FE) recombination, as measured from reflectivity (Figure 2D) and in agreement with literature data [15], [18], while the low energy emission below 2.3 eV corresponds to bound exciton (BE) recombination, usually assigned to Br vacancy centers [19].

Figure 2: Bulk sample.

(A) Photoluminescence (PL) time evolution after picosecond excitation at 3.35 eV. (B) PL decay at free carrier recombination (FCR), free exciton (FE) and bound exciton (BE) energies (log scale). (C) Rise of the PL for the FCR, FE and BE energies (linear scale). In green, the experimental time response is reported. (D) Time-integrated PL spectrum (blue) compared with the reflectivity spectrum (red). All the measurements reported in the figures were performed at 10 K.

The Stokes shift value between the PL and the reflectivity spectrum, which resulted in about 2 meV (Figure 2D), is in agreement with literature data [18] and provides evidence of a small inhomogeneous broadening due to disorder.

A significant emission above the FE transition was also detected: from the Arrhenius plot of the PL intensity versus temperature (not shown), we estimate an exciton binding energy of ≈30–35 meV, in agreement with literature data [20]. Therefore, the observation of PL emission at energies above the exciton line has to be ascribed to free carrier recombination (FCR). We want to remark that the polycrystalline nature does not affect the experimental results discussed here and only accounts for the inhomogeneous broadening of the PL spectrum.

PL decays, acquired with the streak camera in a time interval up to 800 ps, are shown in Figure 2B: non-exponential decays are shown at the BE and FE energies, while the FCR decay turns out to be exponential with a decay time of ≈50 ps. An estimate of the initial time decay constant gives values of about 80 and 120 ps for the FE and BE emissions, respectively. Figure 2C shows the risetime of the PL at the same energies as Figure 2B: for the FCR, the risetime is ≈10 ps, while it is ≈20 ps (30 ps) for FE (BE). The time-integrated spectrum is shown for the same experimental conditions in Figure 2D. The PL decay times measured for the different emissions (BE, FE, FCR) agree with literature data at low temperatures [21], [22]. The fast decay of FCR cannot be ascribed to surface recombination as indicated in ref. [23], because in consideration of the absorption length (≈100 nm) in our experimental conditions, it would require a very high mobility, not found in CsPbBr_{3} [24], [25]. Instead it is mainly due to the exciton formation, as also confirmed by its slightly superlinear dependence on the excitation power that is evidence of a prevalence of the exciton formation with respect to the other FCR decay channels.

From the streak images, such as the one shown in Figure 2, we can extract TR spectra at different delays; a set of spectra are shown in Figure 3.

Figure 3: Bulk sample.

Time-resolved photoluminescence (TR-PL) spectra (in log scale) as extracted from streak images in different time scales. For each spectrum, the time delay with respect to the excitation is reported, while Δ*t* corresponds to the time interval between two adjacent spectra. (A) Rise of the TR-PL, (B) early decay times, (C) longer decay times.

From Figures 2C and 3A, it can clearly be seen that the FE emission rises in about 15–20 ps, with a delay of 5–10 ps with respect to the FCR. This corresponds to the transfer of the FE from the high energy non radiative states into the K≤K_{ph} radiative states, where K and K_{ph} indicate the exciton and photon wavevectors, respectively [26]. Given the rise time and the carrier density value we can infer for the bimolecular exciton formation coefficient (C) at 10 K a lower limit C=5×10^{−6} cm^{3} s^{−1}, to be compared, for example with that of GaN where C=1.2×10^{−6} cm^{3} s^{−1} [27]. The emission from FE initially dominates the PL spectra, and successively the excitons localize in shallow states and form bound excitons; Moreover a long living Urbach tail is detected, arising from localized states. A faster rise and decay is observed for the PL signal detected in the high energy side above the FE emission where the contribution comes from the free carriers recombination. From the TR spectra of Figure 3, it is seen that the high energy side of the PL, above the FE state, is characterized by an exponential tail with a slope which becomes steeper, increasing the delay. The presence of the exponential tail is evidence of a thermalization condition for the carriers, where the population can be described, in the non-degenerate case, by a Boltzmann distribution with an effective temperature *T*_{eff}, which is a function of time [26]. Therefore, we can express the PL intensity *I* at a given time *t* by:

$$I\mathrm{(}t\mathrm{)}\propto \text{DOS(}E\text{)}\mathrm{exp}\mathrm{(}-\frac{E}{{K}_{B}{T}_{\text{eff}}\mathrm{(}t\mathrm{)}}\mathrm{)}\text{\hspace{0.17em}}$$(1)

where DOS(*E*) indicates the density of states and the effective temperature *T*_{eff} depends on time. As seen from Figure 3C, after about 0.2 ns, the slope of the high energy tail does not show any remarkable further change, with a very slow decay. From an exponential fit of the high energy tail of the TR-PL spectrum at a time delay of 670 ps where *T*_{eff}=*T*_{fin}, we find that *T*_{fin}=45 K. From Eq. (1), we can extract the DOS(*E*), shown in Figure 4A, where for comparison, a TR-PL spectrum at a time delay of 156 ps is shown. In the DOS(*E*), the excitonic peak at 2.32 eV clearly comes out.

Figure 4: Bulk sample.

(A) Density of states (DOS) extracted from time-resolved photoluminescence (TR-PL) spectra, as discussed in the text, along with a TR-PL spectrum at 10 K. (B) *A*(*t*, *T*) for different time delays (log scale). (C) Effective temperature of the carriers *T*_{eff} as a function of time delay. The red points refer to a thermal distribution, while the blue ones refer to an incomplete thermalization condition. The solid line in (C) is a fit with Eq. (2).

From Eq. (1), we find an exponential dependence on the energy for the quantity

$A\mathrm{(}t,\hspace{0.17em}\text{\hspace{0.17em}}T\mathrm{)}=\frac{I\mathrm{(}t,\text{\hspace{0.17em}}{T}_{\text{eff}}\mathrm{)}}{I\mathrm{(}{t}_{\infty},\text{\hspace{0.17em}}{T}_{\text{fin}}\mathrm{)}}$

as illustrated in Figure 4B for a few TR-PL spectra. From such dependence, we can extract *T*_{eff} which is plotted as a function of the time delay in Figure 4C, for a lattice temperature *T*_{L}=10 K. An evident bottleneck in the relaxation is found and the effective temperature remains around 45 K for hundreds of picoseconds, after a fast initial decay. The blue points in Figure 4C correspond to TR-PL spectra (in the rise of the PL signal up to about 20 ps) where we can roughly estimate a temperature, but the thermalization is incomplete. Given the excitation density and the time resolution of our experiments, the thermalization regime that we investigated is dominated by the interaction of the carriers with acoustic phonons. In this case, the time evolution of *T*_{eff} can be reproduced by [28]:

$${T}_{\text{eff}}\mathrm{(}t\mathrm{)}={T}_{\infty}\text{*}{\mathrm{(}\frac{\text{Aexp}\mathrm{(}\frac{t}{{\tau}_{1}}\mathrm{)}+1}{\text{Aexp}\mathrm{(}\frac{t}{{\tau}_{1}}\mathrm{)}-1\text{\hspace{0.17em}}}\mathrm{)}}^{2}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$$(2)

corresponding to a temperature (energy) relaxation of the free carriers due to the carrier-acoustic phonon interaction via the deformation potential [29]. In Eq. (2), *T*_{∞} indicates the final temperature (that should be the lattice temperature), *τ*_{1} is the time constant describing the fast relaxation and $A=\frac{\sqrt{{T}_{0}}+\sqrt{{T}_{\infty}}}{\sqrt{{T}_{0}}-\sqrt{{T}_{\infty}}}\hspace{0.17em}$ with *T*_{0} the carrier temperature at *t*=0. From the fit (solid line in Figure 4C) we get *τ*_{1}=50 ps, *T*_{0}=67 K and *T*_{∞}=45 K. Such a value for *T*_{∞}, higher than the lattice temperature, does not come from a lattice heating, as confirmed by the temperature dependence of the PL spectra. In fact, at 45 K the bound exciton emission becomes negligible with respect to the free exciton contribution, and by the high energy tail of quasi resonantly excited PL spectra, at 10 K, we extract a carrier temperature lower than 20 K (not shown). As expected from the model discussed in [28], [29], *T*_{∞} should correspond to the lattice value (i.e. 10 K), but TR-PL spectra up to a delay of 1.5 ns show a temperature (*T*≈30 K) still above that of the lattice, indicating the presence of a very slow process.

Similar results were also found for the SC microcrystals: in this case, the PL is characterized by a band the peak energy of which, at 10 K, changes between 2.32 eV and 2.39 eV depending on the position on the sample, possibly due to differences in the local strain induced by the substrate [30]. Results for the band at 2.39 eV are shown in Figure 5, along with some normalized TR-PL spectra, extracted from the image of Figure 5, and the effective temperature of the carriers as a function of time delay.

Figure 5: Spin-coated sample.

(A) Streak image. (B) Normalized time-resolved photoluminescence (TR-PL) spectra where the exponential tail is evidenced in black. (C) Effective temperature of the carriers *T*_{eff} as a function of time delay: the continuous line is a fit with Eq. (2). Measurements refer to a lattice temperature of 10 K.

Therefore, also in the case of the SC sample, a significant bottleneck in the temperature relaxation is found with a faster initial decrease of *T*_{eff} with respect to what is found for the bulk sample. In this case, we get *τ*_{1}=15 ps, *T*_{0}=89 K and *T*_{∞}=58 K. The results previously shown suggest that the same physical mechanism rules the thermalization both in the bulk and the SC sample. The most relevant difference is the relaxation time, which is three times faster for the SC sample, possibly due to a change of the deformation potential coefficient as a consequence of local strain [29]. To our knowledge, there are no results in the literature concerning this specific topic. With regards to the temperature relaxation at longer time delays, and the slow energy dissipation rate of the FCR, similar behavior is found for both samples, possibly related to a long-living free carrier population, especially in the case of the SC sample.

The relaxation bottleneck in H-PVK and I-PVK nanostructures [31], [32], [33], [34], [35], [36], [37], [38], [39] was recently reported and ascribed to several causes such as efficient Auger processes, hot phonons or polaron formations, that keep the carrier temperature higher than the lattice temperature. However, at the excitation density used in our experiment (δ≈10^{16} cm^{−3} carriers per pulse), both hot phonons and Auger heating are negligible, given the Auger coefficient in CsPbBr_{3} bulk, 3×10^{−27} −4×10^{−28} cm^{6} s^{−1} [40]; for the same reason, a screening of the exciton-phonon interaction can be excluded. The large polaron formation has been demonstrated in MAPbBr_{3} and CsPbBr_{3} [32], [36] and strongly affects the cooling in the first few picoseconds, but it is not relevant on the time scale we considered in our experiment. Therefore, the origin of the very slow cooling, in a time scale and an energy range where the interaction between free carriers and acoustic phonons prevails, must be searched for outside the previous causes. It is worth remarking that the persistent non negligible FCR population, living several hundreds of picoseconds (e.g. see Figure 2), may be related to the slowing down of the cooling. Persistent energetic electrons have been already observed in MAPbI_{3} [35] and FAPbI_{3} [41] and a reduced density of states has recently been proposed as the origin of the slow dynamics and cooling in CsPbI_{3} [42]. We recently demonstrated [43], for SC samples, that surface states can efficiently trap the carriers, with a long release time, providing a carrier reservoir for the radiative recombination: therefore, surface states can play a role in the thermalization process and similarly in bulk high energy long-living traps eventually related to small structural defects or inclusions (e.g. CsPbBr_{5}, or PbBr_{2}).

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