Exciton-Resonance-Enhanced Two-Photon Absorption in Three-Dimensional Hybrid Organic–Inorganic Perovskites

Abstract

Three-dimensional (3D) hybrid organic–inorganic perovskites (HOIPs) have attracted tremendous interest due to strong excitonic effects and large optical nonlinearities. Taking the advantages, 3D HOIPs show great potential for applications in excitonic and nonlinear devices. However, understanding the relevant mechanisms of exciton-associated nonlinear optical phenomena in 3D perovskites is still challenging. Here, we apply the quantum perturbation theory to calculate the exciton-associated degenerate 2PA spectra of 3D HOIPs. The calculated 2PA spectra of twelve 3D HOIPs are predicted to exhibit resonance peaks at both the sub-band and band edges. The exciton-resonance-associated 2PA coefficients are at least one order of magnitude larger than those of band-to-band transitions and are comparable to those of low-dimensional perovskites. To validate our model, we carried out measurements of the static light-intensity-dependent transmission on MAPbBr₃ single crystals. Enhancements of 2PA coefficients are predicted theoretically and observed experimentally with a resonant peak at 1100 nm, indicating intrinsic two-photon transitions to excitonic states in MAPbBr₃ single crystals.

1. Introduction

An exciton is an electrically neutral quasi-particle composed of a Coulombically bound electron–hole pair. The Coulomb interaction between an electron and a hole can be either strong or weak, which refers to two types of excitons: Frenkel and Wannier–Mott excitons [1]. In general, Frenkel excitons are generated in organic semiconductors due to the associated low dielectric constant and have large binding energy (0.1~1 eV), while Wannier–Mott excitons favor inorganic semiconductors with relatively small binding energy (~0.01 eV) [2]. The exciton nature in semiconductors provides them with excellent optical and electrical properties, such as clearly resolved spectrum excitonic absorption [3], high photoluminescence (PL) efficiency [4] and long charge carrier lifetime [5]. Furthermore, excitonic effects have been demonstrated to enhance materials’ optical nonlinearities in a resonance manner and make them great potential for various applications in frequency up-conversion [6], biological imaging [7], microfabrication [8], sub-band photodetection [9], optical data storage [10] and multiphoton-pumped laser [11].

Recently, a renewed interest in hybrid organic–inorganic perovskites (HOIPs) has attracted tremendous research attention due to their excitonic nature at room temperature [12]. The HOIPs are a class of semiconducting materials with the chemical formula of ABX₃, where A interprets a monovalent organic cation (such as CH₃NH₃⁺ = MA⁺ and CH(NH₂)₂ ⁺ = FA⁺), B interprets a metal cation (typically Pb²⁺, Sn²⁺), and X is the anion referring to halide ions (Cl⁻, Br⁻, I⁻). The B-site cation is six-coordinated by the X-site anion to form [BX₆]⁴⁻ octahedrons, which are corner-sharing to constitute three-dimensional frameworks [13,14]. It is commonly demonstrated that the excitons generated in the bulk perovskites are Wannier–Mott excitons arising from the pronounced dielectric constants in the inorganic parts [15]. The motion of organic parts in HOIPs plays a significant role in the exciton–phonon coupling, broadening the PL spectra at room temperature [16]. The Wannier–Mott excitons in HOIPs are hydrogen-like species and the corresponding exciton binding energy generally ranges from tens to hundreds of meV [17,18]. Astonishingly, the exciton binding energies are sensitive to the dielectric environment and can be finely adjusted via tuning chemical components and multiple structures [19]. For example, MAPbCl₃, MAPbBr₃ and MAPbI₃ single crystals have exciton binding energies around 8~69, 15~150 and 2~45 meV, respectively [20,21]. HOIPs in 2D structures, such as layered (C₄H₉NH₃)₂(CH₃NH₃)_n−1Pb_nI_3n+1, (C₉H₁₉NH₃)₂(CH₃NH₃)_n−1Pb_nI_3n+1 (n = 1~4), and (C₆H₁₃NH₃)₂(CH₃NH₃)_n−1Pb_nI_3n+1, are demonstrated to exhibit huge exciton binding energies of several hundred meV due to both quantum and dielectric confinements [22,23,24]. Previous work has shown that the exciton nature in 2D HOIPs enhances the third harmonic generation, as well as multiphoton absorption efficiency by several orders of magnitude [25,26]. A theoretical model based on the quantum perturbation theory has also been proposed for 2PA of (C₄H₉NH₃)_n(CH₃NH₃)_n−1Pb_nI_3n+1 (n = 1~4), where excitonic resonances essentially dominate the nonlinear optical process [27]. Despite the rapid progress in such a research field that has been achieved [28,29,30,31], a systematic theoretical understanding of the exciton-associated 2PA is still lacking in 3D HOIPs, which have been demonstrated to exhibit better thermal stability and higher optical damage threshold for applications in NLO devices [32].

Here, we present a theoretical calculation on degenerate 2PA coefficients of 3D HOIPs by applying a quantum perturbation theory. The calculated 2PA spectra of 3D HOIPs are predicted to be enhanced by at least one order of magnitude over the band edge where the incoming photon is on resonance to the energy of excitonic states. An experimental case study of MAPbBr₃ single crystals to verify our model was conducted. The static light-intensity-dependent transmission measurement is used to measure the 2PA coefficients of MAPbBr₃ single crystals in the near-infrared and visible regions. Excellent agreements between our theory and the experimental findings explicitly confirm a pivotal role of exciton in the 2PA of 3D HOIPs. The excitons in 3D HOIPs provide researchers with an ideal strategy to greatly enhance 2PA for further improving the performance of next-generation photonics and nano-optical devices.

2. Principles and Simulations

Excitons in semiconducting materials are generally characterized via a resonance peak in the optical absorption spectrum, as illustrated in Figure 1a. With a large binding energy, eigenstates of excitons corresponding to the optical resonance are naturally present near the band edge with a Rydberg series. For a simple case with the many-body effects excluded [33], the wavefunction of excitons in bulk crystals is approximated by a 3D hydrogen model as follows [34]:

$$\left|{\psi }_{n,l,m}\left(r\right)\right.〉=\sqrt{{\left({\displaystyle \frac{2}{n{a}_B^{3D}}}\right)}^3{\displaystyle \frac{\left(n-l-1\right)!}{2n{\left[\left(n+l\right)!\right]}^3}}}{\left({\displaystyle \frac{2r}{n{a}_B^{3D}}}\right)}^l{e}^{-{\displaystyle \frac{r}{n{a}_B^{3D}}}}{L}_{n-l-1}^{2l+1}\left({\displaystyle \frac{2r}{n{a}_B^{3D}}}\right)Y_l^m\left(\theta ,\phi \right)$$

(1)

where n, l and m are the principal quantum number, the angular momentum quantum number, and the magnetic quantum number, respectively; $a_B^{3D}$ corresponds to the effective Bohr radius that is much larger than the lattice space. Such excitons are usually called Wannier (Wannier–Mott) excitons, with the motion extending in three dimensions along their crystal structure. L(x) is the generalized Laguerre polynomial and Y(x) is the spherical harmonics. For 2PA, these excitons act as both the intermediate and the final states for electronic transitions when interacting with incoming photons simultaneously.

In the electronic transitions, the 1s exciton acts as a real intermediate state leading to the primary transition from the ground state. The transition dipole moment is given by the following:

$${\mu }_{sg}=〈{\psi }_s(r_e)\left|-e{r}_e\right|{\psi }_g(r_e)〉$$

(2)

where ${\psi }_s(r_e)$ and ${\psi }_g(r_e)$ are wavefunctions of the 1s-exciton state and the ground state, respectively. Here, ${\psi }_g(r_e)$ corresponds to the ground state describing the wavefunction of orbital electrons. ${\psi }_s(r_e)$ is expressed in the formation as $\left|{\psi }_s\left(r_e\right)\right.〉={\displaystyle \int {\psi }_c}\left(r_e\right)\times {\psi }_{n=1,l=0,m=0}\left(r_e-r_h\right)d{r}_h$, with ${\psi }_c(r_e)$ being the Bloch wave functions for the conduction band and ${\psi }_{n=1,l=0,m=0}(r_e-r_h)$ being the hydrogen wave functions of the 1s exciton.

Following the first electronic transition, the 1s exciton makes an intra-excitonic transition to a p exciton as the final state in the 2PA process, the corresponding transition dipole moment is given by the following:

$${\mu }_{ps}=〈{\psi }_p\left(r_e-r_h\right)\left|-e\left(r_e-r_h\right)\right|{\psi }_s\left(r_e-r_h\right)〉$$

(3)

here, ${\psi }_s(r_e-r_h)={\psi }_{n=1,l=0,m=0}(r_e-r_h)$ is calculated based on the 3D hydrogen model for the 1s exciton. ${\psi }_p(r_e-r_h)$ for the p exciton is derived as follows:

$$\begin{array}{l}\left|{\psi }_p\left(r_e-r_h\right)\right.〉=\left|{\psi }_{np,l=1,m=\pm 1}\left(r=r_e-r_h\right)\right.〉\\ =\sqrt{{\left({\displaystyle \frac{2}{n{a}_B^{3D}}}\right)}^3{\displaystyle \frac{\left(n-2\right)!}{2n{\left[\left(n+1\right)!\right]}^3}}}\left({\displaystyle \frac{2r}{n{a}_B^{3D}}}\right)e^{-{\displaystyle \frac{r}{n{a}_B^{3D}}}}{L}_{n-2}^3\left({\displaystyle \frac{2r}{n{a}_B^{3D}}}\right)Y_1^{\pm 1}\left(\theta ,\phi \right)\end{array}$$

(4)

With transition dipole moments calculated from Equations (2) and (3), the coefficients of degenerate two-photon resonances with excitons in 3D semiconducting materials can be derived as follows:

$${\alpha }_2\left(h\nu \right)=CNh\nu {\left(n_0^2+2\right)}^4{\displaystyle \frac{{\left|{\mu }_{sg}\right|}^2}{{\left(E_s-h\nu \right)}^2+{({\Gamma }_s/2)}^2}}{\displaystyle \sum _p\frac{{\left|{\mu }_{ps}\right|}^2{\Gamma }_p/2\pi }{{\left(E_p-2h\nu \right)}^2+{({\Gamma }_p/2)}^2}}$$

(5)

where hν denotes the energy of an incident photon; N is the density of unit cells; ∑ refers to the summation of all final states; μ_sg interprets the transition dipole moment from the ground state to a 1s exciton; μ_ps interprets the transition dipole moment from the 1s exciton to an np exciton; E_s or E_p refers to a certain excitonic state energy level; Γ_s or Γ_p is the linewidth of the corresponding exciton state and is assumed to have the Lorentzian function of density distribution; n₀ is the refractive index; and C is a constant with a value of 4.28 × 10⁴³ in units such that α₂ is in cm/MW, where N is in units of cm⁻³, E_s, E_p, Γ_s, and Γ_p are in units of eV, and μ_sg and μ_ps are in units of esu.

Here, we assume that only 2p exciton is taken into consideration. As derived from Equation (5), there are resonant peaks at both the sub-band and band edges, indicating intrinsic two-photon transitions to excitonic states. At the sub-band edges, individual excitation photon energy is onresonance to the real intermediate states of E_1s and the transition process, as illustrated by Figure 1b. Interestingly, another resonant peak of the ${\alpha }_2$-spectra is predicted near E_2p/2 at the band edge, where the resonance with the 2p-exciton state is the final state for 2PA transitions; see Figure 1c. And the peak values of the ${\alpha }_2$-spectra at the band edge are larger than that at the sub-band edge; this is attributed to the higher density of np-exciton states at the band edge for the two-photon resonance. In our model, only one resonance with the discrete excitonic state is achieved to fulfill the energy conservation requirement, and the calculated ${\alpha }_2$-values are proportional to the density of excitonic states on resonance.

It is noted that the linewidth Γ in our model is a variable parameter that is influenced by acoustic- and longitudinal-optical phonon scatterings. Figure 2a plots the calculated α₂ values of MAPbBr₃ as functions of the excitation wavelength and the linewidth. It is noted that α₂ values exhibit two peaks that correspond to the resonance with the discrete excitonic states as mentioned above. The maximal α₂ value is calculated to be 0.061 cm/MW at 1112 nm. As Γ decreases from 0.4 to 0.05 eV, peak values of 2PA coefficients exhibit a blue shift and increase by at least one order of magnitude. The empirical fitting to the numerical modeling reveals that ${\alpha }_2\propto {\Gamma }^{-0.92}$.

More importantly, α₂ values at the exciton-resonance peak are calculated to be inversely proportional to the excitonic energy level, giving rise to variations in 3D HOIPs with different halide anions (X⁻ = Cl⁻, Br⁻, I⁻); see Figure 2b. Furthermore, α₂ values of APbX₃ are twice that of ASnX₃ at the resonance peak, which may be attributed to larger light–matter interaction.

3. Experimental Results and Discussions

In order to demonstrate the enhancement of 2PA in 3D HOIPs, we conducted the 2PA measurements on MAPbBr₃ single crystals. Single crystals of MAPbBr₃ were synthesized via an antisolvent vapor-assisted growth method (See Appendix A.1) and then prepared in the size of 4 × 4 × 1 mm³; see an image in the inset of Figure 3a. The grew high-quality, millimeter-sized MAPbBr₃ single crystals whose shape conformed to the underlying symmetry of the crystal lattice. The phase purity of the as-grown crystals was confirmed by X-ray diffraction (XRD) from a large batch of crystals (See Appendix A.2). The diffraction peaks of the perovskite are located at 15.45°, 31.06°, and 47.52°, which correspond to the lattice planes (100), (200), and (221), which revealed the characteristics of the perovskite square phase. The XRD pattern in Figure 3a illustrates the well-preferred orientation of MAPbBr₃ single crystals, in good agreement with the previously reported results [35]. This indicates our MAPbBr₃ samples are of high crystalline quality. To characterize the optical properties of these crystals, both linear absorption (See Appendix A.3) and photoluminescence (PL) measurements were carried out. As shown in Figure 3b, a resonance peak near the bandgap rather than a clear band edge cut off in the linear absorption spectrum clearly demonstrates the excitonic effects in MAPbBr₃ single crystals. By fitting the measured linear absorption spectrum with a step function and a Lorentzian function, the optical bandgap of MAPbBr₃ single crystals (E_g) is estimated to be 2.22 eV and the exciton binding energy (E_b) is 40 meV. Under an excitation of 500 nm laser pulses, the one-photon photoluminescence (1PPL) spectrum was measured and it exhibits a peak located at the wavelength of 570 nm and with a full-width half-maximum (FWHM) of 22 nm, indicating homogeneity of excitonic state; see the black dotted line in Figure 3c. It is noted that the same PL peak occurs under the excitation of a pulsed laser at 1100 nm, implying that 2PA takes place where individual photon energy is near the sub-bandgap. The inset shows two-photon-excited photoluminescence (2PPL) versus laser intensity on the log–log scale for the crystals at 1100 nm, and a nearly square dependence of 2PPL on the laser intensity.

A set of static intensity-dependent transmission measurements was conducted [36]. A schematic diagram of the experimental setup is shown in Figure 3d, where a laser system (Carbide-CB5, Light Conversion) is used with a femtosecond laser source, producing 0.1 mJ, 216 fs (τ), 1030 nm pulses at a repetition rate (T) of 60 kHz. The laser source was utilized to pump an optical parametric generator/amplifier (OPA) (Orpheus, Light Conversion, Lithuania) for the generation of laser pulses with the wavelength tuning from 500 nm to 1200 nm by a step of 50 nm. The output power of the laser was continuously tuned via a variable optical attenuator (VOA) (VA-CB-4-CONEX, Newport, America). The detector D1 (919P, Newport, America) after the beam splitter (BS) (BSW29R, Thorlabs, America) was used as a reference channel to measure the incident laser power (P₀), while the detector D2 recorded the laser power transmitted through the sample (P_t). The beam waist (w₀) of the focused laser was measured to be ~3 mm at the focal point. The incident light intensity was limited by 2.0 GW/cm² to avoid thermal effects and laser-induced damage to the sample. To further eliminate the influence of nonlinear refraction, a lens (L2) with a focused length of 10 cm was used to collect signals from the sample. The intensity-dependent transmission of MAPbBr₃ single crystals was recorded as T = P_t/P₀.

Inverse transmissions (1/T) versus light intensity are plotted in Figure 4. At low-intensity light (I < 0.5 GW/cm²), the measured inverse transmissions exhibit linear increase, which manifests a typical characteristic for 2PA, and it is in accordance with the non-saturation model. To extract the 2PA coefficients α₂ of MAPbBr₃ single crystals, we fit the intensity-dependent transmission with the following expression:

$${\displaystyle \frac{1}{T(I)}}={\displaystyle \frac{(1-R){\alpha }_{2}I{L}_{eff}+1}{{(1-R)}^2{e}^{-{\alpha }_{0}L}}}$$

(6)

where R is the reduction factor caused by the reflection and scattering. α₀ is the linear absorption coefficient and L_eff = [1 − exp(−α₀L)]/α₀ is the effective length of the sample, with L = 1 mm being thickness of the sample. Using Equation (7), 2PA coefficients of MAPbBr₃ single crystals are fitted to be in the range of 4.3~12.9 cm/GW, which are on the same order of magnitude as those reported previously at the same wavelength [37,38]. On the other hand, the measured transmission tends to be stable at high light intensity and with excitation wavelengths of 1100, 1150, and 1200 nm; see Figure 4d. This indicates a saturation of 2PA associated with excitonic states that exist below the band edge. These dark excitonic states (or np excitons) provide MAPbBr₃ single crystals with 2PA where individual photon energy is below the sub-bandgap. The saturation process of the MAPbBr₃ single crystals can be well fitted using homogeneously broadened systems:

$${\alpha }_{2-\mathrm{sa}}(I)={\displaystyle \frac{{\alpha }_2}{1+{(I/I_{sa})}^2}}$$

(7)

here, ${\alpha }_{2-sa}$ is the saturated 2PA coefficient, and I_sa is the saturable intensity of 2PA from which the density of excitonic states can be estimated as $\mathrm{N}={\displaystyle \frac{I_{sa}\mathsf{\tau }}{2\mathrm{h}\mathsf{\nu }}}\mathsf{\pi }{\mathsf{\omega }}^2$. With the best fit, the density of excitonic states involved in the 2PA is estimated to be 1.09 × 10¹⁴ cm⁻³, 8.91 × 10¹³ cm⁻³, and 7.27 × 10¹³ cm⁻³ for 1100, 1150, and 1200 nm, respectively. It is consistent with the Lorentz distribution of excitonic states in our model.

The experimental measured 2PA coefficients of MAPbBr₃ single crystals are compared to calculated results from our theoretical model and the two-band model [39]. The 2PA coefficients are plotted as a function of hν/E_g, as shown in Figure 5. It is noted that the experimentally extracted α₂ values exhibit a well-defined peak over the sub-band edge (0.45 < hv/E_g < 0.6), featuring 2PA associated with exciton effects in MAPbBr₃ single crystals. The maximal α₂ value is measured to be 13.9 cm/GW at the excitation wavelength of 1100 nm, corresponding to the resonance with the 2p-excitonic state or the final state in 2PA transitions (2hv = E_2p). As the laser wavelength is tuned from 750 to 600 nm, the experimentally extracted α₂ values show a rising trend, implying another resonance peak near the bandgap (hv~E_1s). The fitting results derived from our model shown by the red solid curve are in good agreement with the experimental data in this wavelength range. However, there is still a discrepancy within one order of magnitude between results from our theoretical model and the experimental data at wavelengths near 1050 nm. It is explained by thermal and scattering effects during measurement, which hinders the intrinsic NLO properties of our samples. More importantly, both experimental and theoretical calculated α₂ values are enhanced by at least one order of magnitude as compared to the results from the two-band model; see the black solid curve in Figure 5. This demonstrates that exciton effects play a significant role in the 2PA in MAPbBr₃ single crystals.

4. Conclusions

In summary, 2PA associated with excitonic effects are theoretically studied in 3D HOIPs based on the quantum perturbation theory. They are at least one order of magnitude larger than the values predicted by the band-to-band transitions and are comparable to those of low-dimensional perovskites. To demonstrate our theory, 2PA spectra of MAPbBr₃ single crystals are calculated in the near-infrared and visible regions. The 2PA coefficients of MAPbBr₃ single crystals were determined via the static intensity-dependent transmission measurement and confirmed the exciton-associated 2PA enhancements. The excitons in 3D HOIPs provide researchers with an ideal strategy to enhance 2PA to further improve the performance of next-generation photonics and nano-optical devices.