Hot-carrier tunable abnormal nonlinear absorption conversion in quasi-2D perovskite

Controlling the high-power laser transmittance is built on the diverse manipulation of multiple nonlinear absorption (NLA) processes in the nonlinear optical (NLO) materials. According to standard saturable absorption (SA) and reverse saturable absorption (RSA) model adapted for traditional semiconductor materials, the coexistence of SA and RSA will result in SA induced transparency at low laser intensity, yet switch to RSA with pump fluence increasing. Here, we observed, in contrast, an unusual RSA to SA conversion in quasi-two-dimensional (2D) perovskite film with a low threshold around 2.6 GW cm−2. With ultrafast transient absorption (TA) spectra measurement, such abnormal NLA is attributed to the competition between excitonic absorption enhancement and non-thermalized carrier induced bleaching. TA singularity from non-thermalized “Fermi Sea” is observed in quasi-2D perovskite film, indicating an ultrafast carrier thermalization within 100 fs. Moreover, the comparative study between the 2D and 3D perovskites uncovers the crucial role of hot-carrier effect to tune the NLA response. The ultrafast carrier cooling of quasi-2D perovskite is pointed out as an important factor to realize such abnormal NLA conversion process. These results provide fresh insights into the NLA mechanisms in low-dimensional perovskites, which may pave a promising way to diversify the NLO material applications.

Here, we consider the overlapping between standard saturable absorption (SA) and reverse-saturable absorption (RSA) stemming from the lowest third-order nonlinear susceptibility. When the RSA and SA coexist, the total absorption coefficient can be expressed as: 1 Where is the linear absorption coefficient, I is the incident laser intensity, is the RSA coefficient, Is is the saturable absorption intensity. Hence, the absorption difference Δα can be calculated by: At low power condition, above equation is simplified as ∆ = ( − ⁄ ) . If < ⁄ , Δα is less than 0 under low excitation intensity, which indicates SA. With the pump power increasing, the SA terms reach saturation, but the RSA is not be limited.
Hence, the RSA will surpass the SA finally to result in a conversion from SA to RSA ( Figure 1a in main text). However, if > ⁄ , the Δα will maintain larger than 0 regardless at low or high excitation density and we can only observe RSA response.
Thus, the combination of SA and RSA only will lead to the conversion from SA to RSA with laser fluence rising.
Additionally, for organic molecules, the RSA is ascribed to strong excited state absorption (ESA) as illustrated in Fig. 1b in main text. the NLA evolution can be described by a set of rate equations for a multi-energy-level: 2 where N is the total population, N0, N1, N2 and N3 are populations in energy level S0, S1, S2 and higher energy level. and are the lifetimes of energy level S1 and S2, respectively. I is the incident intensity and z is the propagation direction. Here, the population N3 is assumed to be zero owing to the rapid relaxation of the higher energy level. According to this model, the NLA response is determined by the absorption cross- Where is the linear absorption coefficient and is the NLA coefficient, I is the incident laser power. The OA Z-scan curves could be fitted by Where I(z) is the incident laser power at z position, which can be expressed by Here, αs is the saturable absorption component, αu is unsaturable absorption component, Is is the saturable absorption intensity defined as the optical intensity when the optical absorbance is reduced to half of its original value. Hence, the normalized transmittance curve of OA Z-scan measurement can be presented as: 10 Where the is known as the modulation depth representing the upper limit of saturable absorption. It is worth noting that, at moderate light intensity ( ≪ ), Eq. 6 can be approximate as Eq. 4, where the linear absorption coefficient under low light field intensity as = + , and the NLA absorption coefficient β equals to − .
Obviously, for SA, the value of NLA absorption coefficient β is negative. Therefore, under moderate light intensity ( ≪ ), Eq. 4 can also be regards as a standard NLA model applying to both RSA and SA process.
Where P is the incident laser pulse energy; R and T are the reflectance and transmittance of the sample for at the excitation wavelength, 1-R-T can be obtained from the absorption spectrum, approximately; ℏ is the energy of the incident photon; S and L is the pump laser beam area on the sample surface and the thickness of the sample. Supplementary Which is known as Elliot model, where is a constant related to the transition matrix element, Eb is the exciton binding energy, Eg is the bandgap, ℏ represents incident photon energy, Θ(∆) is the unit step function, denotes a delta function. The first term in square bracket describes the absorption of discrete excitonic state absorption below the bandgap, while the second term describe the absorption of red side continuum states. To account for the inhomogeneous broadening, we convolve Eq. 9 by a Gaussian function (full-width at half-maximum (FWHM) of 82 meV) to model the actual absorption spectrum. As shown in Supplementary Figure 8a, the theoretical 3D Elliot model consists well with the experimental absorption spectrum of 3D perovskite film.
The contribution form exciton and continuum states absorption are labeled by green and red line, respectively. The obtained bandgap and exciton binding energy is 1.617 eV and 12.5 meV, which is in line with previous reported result. 11,13 When the dimension of semiconductor material reduces to 2D regime, according to 2D hydrogen model, the theoretical absorption coefficient can be modified to: 12 = ℏ 4 The confinement of the carrier enhanced the exciton absorption intensity. However, we note that in the case of hybrid layered perovskites, dielectric confinement effects cause deviations from hydrogenic behaviour. 14 Here, we only can distinguish the 1S absorption peak located at 2.17 eV under room temperature. The higher order excitonic absorption has become sightless. Here, the 1S absorption peak is modeled by an asymmetric gaussian profile given as: Where Γ and Γ are 43 meV and 52 meV. Whereas, even we deduce the 1S exciton absorption, there remains considerable absorption extending below 2.2 eV. This part is ascribed to continuum states absorption which is phenomenologically fitted by a assemble of Gaussian function combined with a Logistic function of: Where k describes the broadening. The fitting result is shown in Supplementary Figure   8b.
For 3D perovskite with weak exciton absorption contribution, the carrier temperature can be obtained more conveniently by fitting the higher energy tails by a Maxwell-Boltzmann distribution. 11,13 However, in low-dimensional system, the confinement leads to remarkable exciton absorption peak.
The continuum band contribution can be calculated by: Where the quasi-equilibrium excited free carriers obey Fermi-Dirac distribution of Under high excitation, large amount temporal populations of created carriers will remain in their original position of energy band which is known as non-thermalized carriers. 16 This fraction of nonequilibrium carriers will block further absorption of incoming photons at the same energy and give rise to a bleaching effect. Normally speaking, the non-thermalized carriers will undergo an ultrafast (within 100 fs) carriercarrier scattering to form a quasi-equilibrium Fermi-Dirac distribution. This rapid intraband relaxation process is known as thermalization. However, if the incident pump photon energy is close to bandgap, non-equilibrium carrier will not totally leave their original energy position. According to Fermi-Dirac distribution, quite amount hot carriers will remain at the excitation energy and make a non-ignorable contribution for bleaching effect. Therefore, in this work, the non-thermalized carrier and quasiequilibrium hot carrier are simplified as two isolated energy states. Given the similar effective mass of electron and hole of perovskite, we only demonstrated the relaxation dynamic of electron in conduction band, the hole in valence band is roughly the same.
For a Gaussian temporal pump pulse with uniform spatial component, the laser intensity can be denoted as: The dynamic for conduction band non-thermalized electron occupation probability can then be described by: Where the first term represents the incident laser excited carrier, Dos( ) is the density of states, is the incident laser FWHM in energy domain, ( ( ), ℏ ) denotes the absorbance of materials; The second term describes the ultrafast non-thermalized carrier thermalization process, is the lifetime of non-thermalized carriers. Herein, the absorption of the incident photon can be expressed as Eq. 4 given in main text. After thermalization, non-thermalized carriers will decay to be quasi-equilibrium hot carriers obeying Fermi-Dirac distribution ( ) = 1/ 1 + exp . Hence, the hot carrier quantity can be given by:

= Dos( ) × ( )
For 2D quantum-well structure with strong quantum confinement, the density of states around band-edge can be approximatively regarded as constant according to 2D Hydrogen model. Therefore, the Dos( ) in Eq. 19 can be replaced by a constant Dos. Subsequently, the equation can be resolved to give the Fermi level of hot carrier as: If plug the calculated into Fermi-Dirac function, we will obtain the occupation possibility of hot carrier, ( ). Then, Eq. 18 ~ 20 and obtained ( ) can be solved by numerical method to give the value of and ( ( ), ℏ ). The absorbed energy per pulse can be given by: The value of the parameters using in theoretical simulation can be found in Supplementary Table 4.
Supplementary Laser beam radius on sample surface 4×10 -4 L (cm) The thickness of the film 8×10 -6 Supplementary Figure 11. NLA simulation with different parameters. The powerdependent transmittance evolution at 540 nm with various hot carrier temperature and thermalization time.
Supplementary Figure 12. Linear absorption spectra of different 2D perovskite materials. Absorption spectra of (PEA)2FAPb2Br7 (n=2), (PEA)2PbI4 (n=1) and (PEA)2FAPb2I7 (n=2) perovskite film normalized at the exciton peak. The absorption form doping states and scattering have been deducted. Dash lines denote the corresponding continuous state absorption shoulder adjacent to the excitonic absorption peak.
Supplementary Figure 14. TA spectrum of (PEA)2FAPb2Br7 (n=2) quasi-2D perovskite film. Supplementary Figure 19. The schematic diagram of the Z-scan system. The femtosecond laser generated from OPA is focused by a lens. The sample is fixed on a one-dimensional moving stage and scans along the laser propagation direction which is known as Z-axis. The incident laser is splitted into two beams with energy ratio of 3:7.
The beam with higher power is focused on the sample and received by detector 1, which reflets the transmittance property of the sample. Another laser beam is collected by detector 2 to monitor the fluctuation of the incident laser intensity. The inset at left bottom illustrates a standard Z-scan curve of a 0.5 mm thick ZnO single crystal at 540 nm, which is fitted well with a standard nonlinear absorption model. The obtain nonlinear absorption coefficient is 7.2 cm GW -1 which is coincident with previous reported result (5 cm GW -1 ). 17