Scaling law for excitons in 2D perovskite quantum wells. — TEI-tagged view

J.-C. BlanconpersName, A. V. StierpersName, H. TsaipersName, W. NiepersName, C. C. StoumpospersName, B. TraorépersName, L. PedesseaupersName, M. KepenekianpersName, F. KatsutanipersName, G. T. NoepersName, J. KonopersName, S. TretiakpersName, S. A. CrookerpersName, C. KatanpersName, M. G. KanatzidispersName, J. J. CrochetpersName, J. EvenpersName, A. D. MohitepersName (2018date). “Scaling law for excitons in 2D perovskite quantum wells..” Nature Communicationstitle·j.

Object of study — excitonsterm·key in Ruddlesden–Popper halide perovskiteterm·dom quantum wellsterm·key of thickness n = 1 to 5

Research question — whether optical resonances in 2D perovskites are excitonsterm·key, and how excitonterm·key mass and binding energy scale with thickness

Contribution — a general scaling lawterm·key for exciton binding energyterm·key in perovskite quantum wellsterm·key of any thickness

Full text — tagged

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Scaling lawterm·key for excitonsterm·key in 2D perovskite quantum wellsterm·key.

Abstract Ruddlesden–Popper halide perovskitesterm·dom are 2D solution-processed quantum wellsterm·key with a general formula A 2 A’ n -1 M n X 3 n +1 , where optoelectronic properties can be tuned by varying the perovskite layer thickness ( n -value), and have recently emerged as efficient semiconductors with technologically relevant stability. However, fundamental questions concerning the nature of optical resonances (excitonsterm·key or free carriers) and the exciton reduced massterm·dom, and their scaling with quantum wellterm·key thickness, which are critical for designing efficient optoelectronic devices, remain unresolved. Here, using optical spectroscopy and 60-Tesla magneto-absorptionterm·key supported by modeling, we unambiguously demonstrate that the optical resonances arise from tightly bound excitonsterm·key with both exciton reduced massesterm·dom and binding energies decreasing, respectively, from 0.221 m 0 to 0.186 m 0 and from 470 meV to 125 meV with increasing thickness from n equals 1 to 5. Based on this study we propose a general scaling lawterm·key to determine the binding energy of excitonsterm·key in perovskite quantum wellsterm·key of any layer thickness.

Introduction Ruddlesden–Popper halide perovskitesterm·dom 1 , 2 (RPPsterm·dom) are solution-processed quantum wellterm·key structures formed by two-dimensional (2D) layers of halide perovskite semiconductors separated by bulky organic spacer layers, whose stoichiometric ratios are defined by the general formula 3 A 2 A’ n -1 M n X 3 n +1 where A, A’ are cations, M is a metal, X is a halide and the integer value n determines the perovskite layer thickness (or quantum wellterm·key thickness). Recent breakthrough in the synthesis of phase-pure (a single n -value) RPPsterm·dom with higher values 3 – 5 of n , up to n equals to 5, has inspired their use as low-cost semiconductors in optoelectronics 5 – 8 as an alternative to three-dimensional (3D) perovskites due to their technologically relevant intrinsic photo- and chemical-stability 5 – 10 . However, key fundamental questions remain unanswered in RPPsterm·dom with n greater than 1, such as the nature of optical transitions, as well as the behavior of Coulomb interactionsterm·dom especially with increasing quantum wellterm·key thickness. In fact there has been an intense ongoing debate 6 – 8 , 11 – 13 regarding the exact nature of the optical transitions (excitonsterm·key versus free carriers) in RPPsterm·dom with large n -values. RPPsterm·dom with n equals to 1 (excitonsterm·key at room temperature), 14 , 15 and 3D perovskites 16 (free carriers at room temperature) are representative of the two limiting regimes at room temperature, but the analysis of the crossover has not been performed. This issue originates from the lack of knowledge of the fundamental quantities such as the exciton reduced massterm·dom, dielectric constant and characteristics like the spatial extension of electron and hole wavefunctions, which play a crucial role in the determination of the exciton binding energyterm·key. In particular, contradictory reports of the value of the exciton reduced massterm·dom have significantly contributed to this uncertainty 15 , 16 making it the most critical experimentally derived parameter required for the quantitative determination of the excitonterm·key characteristics in hybrid (organic–inorganic) perovskite systems. Moreover, unlike bulk inorganic semiconductors, heterostructures 17 and 3D perovskites 18 , 19 , it is non-trivial to determine the exciton reduced massterm·dom in layered hybrid perovskites using a symmetry-based ( k.p ) approach 20 or using many-body ab initio calculations 21 . Furthermore, ambiguity in the determination of the exciton binding energyterm·key in layered 2D perovskites has also been imposed by the limited understanding of the role of dielectric confinementterm·dom versus quantum confinementterm·dom with increasing quantum wellterm·key thickness 22 , 23 . More generally, phase-pure RPPsterm·dom with n greater than 1 present a unique opportunity to explore the physical properties of natural quantum wellterm·key semiconducting crystals intermediate between monolayer 2D materials 24 , 25 and 3D materials 26 , a quasi-dimensional physics only accessible in synthetic inorganic semiconductor quantum wellterm·key structures so far 26 .

Here we present the study on using low-temperature (4 K) magneto-optical spectroscopy to accurately determine the exciton reduced massterm·dom for (BA) 2 (MA) n -1 Pb n I 3 n +1 RPPterm·dom crystals with perovskite layer thickness varying between 0.641 and 3.139 nm, corresponding to n varying between 1 and 5 (Fig. 1a , Supplementary Fig. 1 , and Supplementary Table 1 ), where BA and MA stand for CH 3 (CH 2 ) 3 NH 3 and CH 3 NH 3 , respectively. The reduced mass is used to develop a generalized theoretical model for electron–hole interactions in RPPsterm·dom and determine fundamental characteristics of the excitonterm·key states. In parallel, from low-temperature optical spectroscopy we experimentally determine the exciton binding energyterm·key in the RPPsterm·dom with n varying from 1 to 5. Finally, from these results we produce a general scaling behavior for the binding energy of Wannier–Mott excitonterm·key states in RPPsterm·dom, which allows for prediction of the exciton binding energyterm·key for any given thickness. This study closes a long-standing scientific gap and will lead to the rational design of next-generation layered 2D perovskite-based optoelectronic devices. Fig. 1 Exciton reduced massterm·dom from magneto-absorptionterm·key spectroscopy and theory. a Schematic of the RPPterm·dom structure cut along the direction ĉ of stacking of the 2D layers. b Image of mechanically exfoliated RPPterm·dom crystals. Scale bar is 10 µm. c Magnetic field dependence of the light transmission of an individual RPPterm·dom with n equals to 4 crystal for right- (σ ‒ ) and left-handed (σ + ) circular polarization. d Corresponding shift of the excitonterm·key energy as a function of the magnetic field. Fit of the data using Δ E = ±1/2 g 0 μ B B + c 0 B 2 yields c 0 = 1.04 ± 0.16 µeV T −2 and g 0 = 1.59 ± 0.03. e Derived diamagnetic shift coefficient of the measured RPPsterm·dom (red squares). The value from Tanaka et al. 15 is also reported for RPPsterm·dom with n equals to 1 with a slightly larger organic spacer. f Exciton reduced massterm·dom derived from fitting the diamagnetic shifts with our theoretical model. The gray dotted line is a guide for the eyes. The red dashed line indicates average value of exciton reduced massterm·dom for the 3D perovskite MAPbI 3 . Error bars correspond to s.d. from the fit of the shift of the excitonterm·key energy as a function of the magnetic field Results Exciton reduced massterm·dom from magneto-absorptionterm·key measurements Fig. 1 shows the RPPterm·dom structure and results from magneto-absorptionterm·key spectroscopy, which was employed to probe the strength of the electron–hole interaction in RPPsterm·dom and deduce the exciton reduced massterm·dom. These measurements were performed at 4 K in the Faraday geometry (magnetic field along the stacking direction of the layers, ĉ -axis), and the optical spectra of the right- and left-handed circular polarization components (σ ± ) of the transmitted white light were probed. Application of a high magnetic field on the RPPterm·dom with n equals to 4 results in an energy shift of the optical resonance at about 1.9 eV (Fig. 1c, d ). Similar results were obtained for the other RPPsterm·dom (see Supplementary Fig. 2 ). This optical transition was identified as the excitonterm·key ground state and its energy shift under magnetic field in the Faraday configuration is expressed by 26 Δ E = ±1/2 g 0 μ B B + c 0 B 2 , where the first term describes the Zeeman splitting of the σ + and σ ‒ excitonterm·key transitions ( g 0 is the g -factor in the perovskite plane, μ B the Bohr magneton, B the magnetic field) and the second one the diamagnetic shift ( c 0 is the diamagnetic shift coefficient). Under magnetic field the competition between the Zeeman and diamagnetic effects explains the non-symmetric, opposite-sign energy shift of the σ + and σ ‒ excitonterm·key transitions with respect to the zero field excitonterm·key absorption energy (Fig. 1d and Supplementary Fig. 2 ). We note that this asymmetry becomes more pronounced for thicker perovskite layers ( n approaching 5), which is a consequence of the strong increase of the diamagnetic coefficient with increasing n -value. Fitting the excitonterm·key energy shifts for both σ + and σ ‒ polarizations using the model above yields the diamagnetic coefficients c 0 of the RPPsterm·dom (Fig. 1e ). Here, c 0 increases monotonically with the perovskite layer thickness, and it ranges from ~0.2 to ~1.1 µeV T −2 for the RPPsterm·dom with n varying from 1 to 5, respectively. We note that due to relatively small signal-to-noise ratio in the magneto-absorptionterm·key data for the RPPterm·dom with n equals to 1 (Supplementary Fig. 2a ), the diamagnetic coefficient for this compound was obtained from a combined experimental–theoretical approach which consisted in fitting the c 0 value to be in best agreement with both the magneto-absorptionterm·key data and the experimental exciton binding energyterm·key discussed later in this paper. For comparison, we also included in Fig. 1e the value of c 0 reported by Tanaka et al. 15 ( c 0 = 0.26 µeV T −2 ) for a RPPterm·dom with n equals to 1 and with a slightly larger organic spacer (hexyl-ammonium CH 3 (CH 2 ) 5 NH 3 instead of butylammonium CH 3 (CH 2 ) 3 NH 3 in our case). We note that our value of c 0 for the RPPterm·dom with n equals to 1 with BA as organic spacer is in agreement but slightly smaller than other reports of RPPsterm·dom with n equals to 1 with longer organic spacers 27 – 29 . Concomitant with the diamagnetic shift increase with n -value, the g -factor presents a marked increase from 0.8 to 1.6 for the RPPsterm·dom with n varying from 1 to 5, and a value of about 1.7 was derived from our 3D perovskites magneto-absorptionterm·key data (Supplementary Fig. 3 ) and is consistent with the literature 30 , 31 .

The diamagnetic coefficient is directly connected to both the exciton reduced massterm·dom and the strength of the electron–hole Coulomb interactionterm·dom. Although a simple relation exists for pure 2D systems where quantum confinementterm·dom dominates 22 , 26 , i.e., when confinement effects stem from the confinement of electron and hole wavefunctions in a 2D plane, this model does not apply to the RPPsterm·dom due to the following reasons. First, the perovskite layer thickness is comparable to the spatial extent of the excitonsterm·key 15 , 16 , 27 and the excitonterm·key wavefunction cannot be strictly confined to a 2D plane (see also Supplementary Note 1 to 4 ). Second, the dielectric confinementterm·dom plays a key role in the photo-physics of RPPsterm·dom 15 , 23 , 27 . Dielectric confinementterm·dom (also called image charge effectterm·dom) manifests in a quantum wellterm·key system provided that the thickness of the quantum wellsterm·key is comparable to the excitonterm·key size, and the dielectric constant ratio between the quantum wellsterm·key (perovskite layers with ε w ~4 or higher) and the barriers (organic spacing layers with ε b ~2.2) is larger than unity. This result leads to an enhancement of the Coulomb interactionterm·dom between the electron and hole pair composing each excitonterm·key, which is a consequence of the reduced dielectric screening of the excitonterm·key electric field partially located outside the quantum wellterm·key, i.e., in the surrounding barriers that have a lower dielectric constant.

Therefore, we developed a theoretical model (see details in the next section, Supplementary Note 1 , and Supplementary Fig. 4 ), which describes the electron–hole Coulomb interactionterm·dom in thin semiconductors and includes dielectric confinementterm·dom (Keldysh theoryterm·key 22 ). For this theoretical model to work, it requires an accurate determination of the exciton reduced massterm·dom (labeled µ ). We evaluated the reduced mass for each RPPterm·dom by adjusting the theoretical values of c 0 to those measured experimentally (Fig. 1f and Supplementary Note 5 ). This yielded µ = 0.221 m 0 ( m 0 is the free electron mass) for the RPPterm·dom with n equals to 1, and 0.184 m 0 for the RPPterm·dom with n equals to 1 compound with longer organic spacers reported by Tanaka et al. 15 . Then, we observe a monotonic decrease of the exciton reduced massterm·dom from 0.217 m 0 for the RPPterm·dom with n equals to 2 down to 0.196 m 0 for the RPPterm·dom with n equals to 4. Due to computational limitations, our model was not applied to the RPPterm·dom with n equals to 5; however we estimated, to a first approximation, µ = 0.186 m 0 from the extrapolation of the data for the RPPsterm·dom with n equals 1 to 4.

The deduced values of exciton reduced massterm·dom in the RPPsterm·dom up to n equals to 5 are significantly larger than those reported for 3D perovskites (Supplementary Fig. 3 ), evenpersName though the excitonterm·key states are well confined within the perovskite layer (see next section). This is surprising because a decrease of the exciton reduced massterm·dom is expected for increasing perovskite layer thickness, with limiting value of 0.104 m 0 (3D perovskite), as the vicinity of the excitonterm·key approaches that of the 3D perovskites. However, the equivalent quantum wellterm·key is still very thin evenpersName for the RPPterm·dom with n equals to 5. Moreover, the reduced mass of the excitonterm·key ground state calculated by density functional theoryterm·mth (DFTterm·mth) for each RPPterm·dom (with n equals 1 to 4) yields a similar dependence on the n -value as the experimental ones provided the underestimation of the bandgap computation by DFTterm·mth is taken into account (Supplementary Fig. 5 ). We explain the higher values of exciton reduced massterm·dom in the RPPsterm·dom studied as compared to their 3D counterpart by a progressive reduction of the energy bandgap 26 . The limiting 3D value is thus expected to be reached for larger n -value, i.e., when the RPPterm·dom bandgap equals the 3D perovskite one. Furthermore, electronic band mixing and non-parabolicity effects have been known to induce changes in the exciton reduced massterm·dom in quantum wellterm·key systems 32 – 35 . These effects have also been discussed in classic inorganic semiconductors and agree with the n -value dependence of the experimental bandgap, diamagnetic shift and g -factor (Figs. 1e and 3a , and Supplementary Fig. 6 ). In fact, the bandgap dependence of the exciton reduced massterm·dom was also reported recently in 3D hybrid perovskites 31 . To conclude, the unambiguous determination of the reduced mass allows us to develop a predictive model for excitonsterm·key in solution-processed quantum wellterm·key systems and calculate the excitonterm·key binding energies of the RPPterm·dom compounds.

Calculating the exciton binding energyterm·key in 2D perovskites The binding energy of the excitonterm·key ground state was calculated for each RPPterm·dom (with n equals 1 to 4) by inputting each respective value of the exciton reduced massterm·dom into our theoretical model and semi-empirically solving the Bethe–Salpeter equation based on the effective mass Green’s function approach 23 . This approach includes: calculating the electronic structures of the RPPsterm·dom to extract excitonterm·key wavefunctions and dielectric constant profiles using DFTterm·mth, generalizing the Keldysh theoryterm·key for 2D perovskites and combining both of the above by building a semi-empirical model to simulate the Wannier–Mott excitonterm·key characteristics (Fig. 2 and Supplementary Note 1 , 2 , and 6 ). In our model, the potential function describing the Coulomb interactionterm·dom between the electron and hole forming the excitonterm·key states is based on Keldysh theoryterm·key 22 generalized to a semiconducting dielectric quantum wellterm·key (Fig. 2a ), i.e., a dielectric well (perovskite layer) with thickness d and dielectric constant ε w sandwiched between well barriers (organic spacing layers) with dielectric constant ε b . The non-local, screened electron–hole pair interaction potential is given by: 1 V s q t = - e 2 2 ε w q t ∫ ∫ - d 2 , - d 2 d 2 , d 2 ρ e z e ρ h z h e - q t z e - z h + Δ χ e - q t z e + z h - d + e - q t z e + z h + d + Δ χ 2 e - q t ∣ z e - z h - 2 d ∣ + e - q t ∣ z h - z e - 2 d ∣ d z e d z h , in reciprocal space and in the in-plane a , ^ b ^ direction. In Eq. ( 1 ), the difference of electron and hole transverse (in-plane) wavevectors is q t , Δ = 1 - χ 2 e - 2 q t d - 1 , and χ = ε w - ε b ε w + ε b . Here, the dielectric constants are inputs from our DFTterm·mth results and were taken at optical frequency because the values of binding energy measured in RPPsterm·dom (see next section) are one order of magnitude larger than the highest energy value reported for the lattice optical phonon modes 36 . The electron ( ρ e ) and hole ( ρ h ) probability density profiles along the stacking axis ĉ ( z e,h are real-space coordinates along this direction) were also obtained from our DFTterm·mth results. This method is equivalent to the real-space computation of the screening effect using the image charge method 37 . We also note that the general expression (1) of the electron–hole interaction potential corresponds to the general Keldysh potential in real space 22 . Upon different degrees of approximation (as discussed in Supplementary Note 3 ), the electron–hole interaction potential used in the literature in the limit of monolayer 2D transition metal dichalcogenides 38 – 42 can be retrieved. However, these approximations are not valid in RPPsterm·dom (Supplementary Note 4 ) and the general expression (1) needs to be used. Fig. 2 Semi-empirical model of Wannier–Mott excitonterm·key in RPPsterm·dom. a Schematics of the single quantum wellterm·key system to which our model was applied to. b Computed band structure for the RPPterm·dom with n equals to 4. c Corresponding electron (dashed black lines) and hole (red) probability density profiles, and high-frequency dielectric constant profile (blue) along the stacking axis ĉ . d Calculated binding energy of the excitonterm·key ground state using the exciton reduced massterm·dom in Fig. 1f and the DFTterm·mth results. The values calculated using our model results (red circles) is compared to those obtained in the approximation of a pure 2D system (blue triangles) and of a pure 2D system without dielectric confinementterm·dom (black squares). The binding energy of the excited excitonterm·key states (2 s , 3 s , 4 s ) were also computed and are reported in Supplementary Fig. 7 . For reference, the 3D perovskite MAPbI 3 yields an exciton binding energyterm·key of about 16 meV, from Miyata et al. 16 and in agreement with our results in Supplementary Fig. 3 Overall, theoretical solution to the exciton binding energyterm·key in each RPPterm·dom requires the knowledge of: the electron and hole probability density distributions, the dielectric constant profile along the stacking direction, and the exciton reduced massterm·dom. In RPPsterm·dom, the density profiles and dielectric constants along the stacking direction (Fig. 2c and Supplementary Fig. 4b ), along with the electronic band structure (Fig. 2b ), were derived from the electron and hole wavefunctions calculated by DFTterm·mth (Supplementary Note 6 ). A representative example of the electron and hole probability density profiles is sketched in Fig. 2c for the RPPsterm·dom with n equals to 4. Our calculations reveal that the charge wavefunctions stay confined into the perovskite layers and exhibit little leakage into the organic spacing layers, which is consistent with uncoupled quantum wellterm·key systems. Moreover, the results reveal maximum values of dielectric constant in the range of 4 to 5.2 for the perovskite layers in RPPsterm·dom with n equals 1 to 5, and a minimum value of about 2.2 for the organic spacing layers, consistent with previous estimations 43 , 44 . The contrast of dielectric constants between the perovskite and organic spacers is at the origin of the strong dielectric confinementterm·dom effects observed in RPPsterm·dom, as discussed in the next section.

Gathering all these information, our general model was then applied to the calculation of the exciton binding energyterm·key in RPPsterm·dom, yielding a value of 467 meV for the RPPterm·dom with n equals to 1 (435 meV for the case of longer organic spacers from Tanaka et al. 15 ), and which decreases monotonically to ~135 meV for the RPPterm·dom with n equals to 4 as illustrated in Fig. 2d (see also Supplementary Fig. 7 and Supplementary Table 2 ). We observe a radical difference of the excitonterm·key binding energies calculated using our general approach as compared to those obtained under various degrees of approximation usually suited to 2D materials such as transition metal dichalcogenides 38 – 42 , i.e., in the case the charges are strictly restricted to a 2D plane and including or not dielectric confinementterm·dom (see details in Supplementary Note 3 ). The accuracy of our model to predict the binding energies of excitonsterm·key in 2D perovskites was then verified by directly measuring these values using optical spectroscopy techniques.

Direct measurement of the exciton binding energyterm·key We directly measured the exciton binding energyterm·key in each RPPterm·dom using optical absorption, photoluminescenceterm·mth (PLterm·mth) and photoluminescence excitationterm·mth (PLEterm·mth) spectroscopy (Fig. 3 ). The optical bandgap was tuned over the visible spectral range from 2.540 ± 0.004 eV (488 nm) in the RPPterm·dom with n equals to 1 down to 1.846 ± 0.004 eV (672 nm) in the RPPterm·dom with n equals to 5 (Fig. 3a ). We note that the RPPsterm·dom with n greater than 1 retained their optical bandgap from 290 K down to 4 K. This observation implies that our study conducted at low temperature is directly transposable to room temperature and provides direct insights into the photo-physics of materials used in practical devices. Fig. 3 Optical spectroscopy of the RPPterm·dom crystals with n equals 1 to 5. a Experimental optical bandgap scaling. b Photoluminescenceterm·mth spectra. c Schematics of the Rydberg series of the excitonterm·key ground state (1 s ) and excited excitonterm·key states (2 s , 3 s , etc.) merging with the continuum. (Right) Corresponding absorption or optical density, photoluminescenceterm·mth PLterm·mth, and photoluminescence excitationterm·mth PLEterm·mth spectra typically observed in 2D material systems 26 , 49 . d Optical density OD and e PLEterm·mth spectra of the RPPsterm·dom with n equals 1 to 5, respectively. Stars point to excitonterm·key ground state optical transitions, black squares show the continuum onsets and gray brackets indicate the region of excited excitonterm·key states. Labels A, B and C indicate absorption regions at energies higher than the continuum bandgap E G and which are apparently common to all the RPPsterm·dom given an energy shift proportional to the bandgap Figure 3b shows a single sharp peak in the PLterm·mth spectra, indicating the emission of the excitonterm·key ground state 14 , 45 . The low-energy shoulder observed more than tens of meV below the excitonterm·key peak in all RPPsterm·dom was recently assigned in the RPPterm·dom with n equals to 1 to a phonon replica 46 and self-trapped excitonsterm·key 47 . Based on power dependence measurements of the PLterm·mth (Supplementary Fig. 8 ), we hypothesize a similar origin for the resonances in the RPPsterm·dom with n greater than 1, but we only focus on investigating the ground state excitonterm·key transition in this study.

All the presented absorption spectra show similar features (Fig. 3d ), which includes a single absorption peak at low energy corresponding to the excitonterm·key ground state resonance. Moreover, a steady increase of absorption at higher energies modulated by energy optical transitions is well reproduced in the PLEterm·mth measurements (Fig. 3e ). These spectral features were assigned to the Rydberg series of the Wannier–Mott excitonterm·key (1 s , 2 s , 3 s , 4 s noted as Ns , with N equals to 1, 2, 3 and 4 and having energies E Ns ) and are illustrated in Fig. 3c . The onset of the continuum E G corresponds to electron and hole free carrier states (Fig. 3c ), as previously reported in the RPPterm·dom with n equals to 1 perovskites 15 , 45 , other 2D nanomaterials 39 , 48 and quantum wellterm·key semiconductors 26 , 49 . Absorption transition features, marked as A, B and C, are observed in both the absorption and PLEterm·mth spectra for energy higher than E G , and are apparently common to all the RPPsterm·dom given an energy scaling with the n -value proportional to the bandgap scaling (Fig. 3d, e ). Close to the continuum onset and in our experimental energy range, the optical transitions in RPPsterm·dom at energies above E G have been assigned 11 , 14 , 50 to transitions between the Pb(6 s )-I(5 p ) states in the valence band and Pb(6 p ) states in the conduction band 20 , 51 . The absorption features A, B and C can be understood as transitions between valence and conduction bands overlapping with the continuum of states from the band-edge excitonterm·key. For example, from the calculated RPPterm·dom band structures (Fig. 2b and Supplementary Fig. 4a ), one can assume that the high energy optical transitions involve the series of the mini-bands directly below the highest valence band and above the lowest conduction band at the Γ high symmetry point in the Brillouin zone. Calculation of the broadband optical spectra in the RPPsterm·dom, including dipole allowed transitions from group theory and joint density of states, should be performed in future theoretical studies in order to understand the details of the experimental absorption spectra at energies higher than the continuum E G . 20 , 51 The excitonterm·key binding energies in RPPsterm·dom were derived from the study of the optical transitions corresponding to the excitonterm·key Rydberg series and the continuum. We present a representative example for the RPPterm·dom with n equals to 4 in Fig. 4a, b . Dielectric confinementterm·dom (or image charge effectterm·dom) has been shown 15 , 27 , 44 , 45 to mainly influence the 1 s excitonsterm·key in RPPsterm·dom with n equals 1. Therefore, the 2 s and 3 s excitonterm·key states were fitted using the classic 2D hydrogen Rydberg series with energies 26 E Ns = E G − R y /( N − 1/2) 2 , where R y is the Rydberg energy. This yields E G = 2.078 ± 0.012 eV and R y = 0.11 ± 0.04 eV for the RPPterm·dom with n equals to 4 (Fig. 4b , dashed red line). This procedure is similar to the one used in transition metal dichalcogenides 39 , 52 and 2D perovskites 15 , 45 , 53 . It was also applied to the RPPsterm·dom with n equals to 1, 2, 3 and 5 (Supplementary Fig. 9 ), i.e., the 1 s , 2 s and 3 s excitonterm·key states were identified from the analysis of the experimental spectra and the 2 s , 3 s Rydberg series were fitted with the 2D hydrogen model to derive the continuum energy E G . The binding energy of the 1 s excitonterm·key ground state was derived from the difference | E 1 s − E G |, which ranges from about 470 meV for the RPPterm·dom with n equals to 1 down to 125 meV for the RPPterm·dom with n equals to 5 (Fig. 4c, d , Supplementary Table 1 ). We note that the lower limit of the exciton binding energyterm·key is given by the energy difference between the (1 s ) excitonterm·key ground state and (2 s ) first excited state (Fig. 4d ). We emphasize that the predicted theoretical and experimentally determined values are in excellent agreement (Fig. 4d and Supplementary Fig. 7 ), thus validating the accuracy of our developed theoretical model for Wannier–Mott excitonsterm·key in RPPsterm·dom. These results emphasize the need for an accurate determination of the exciton reduced massterm·dom and taking into account the contributions of quantum and dielectric confinementsterm·dom as well as hole and electron density profiles in order to develop an accurate and robust theoretical model for the 2D perovskite systems. Fig. 4 Direct measurement of the exciton binding energyterm·key and Rydberg states in RPPsterm·dom with n equals 1 to 5. a Optical density OD and photoluminescence excitationterm·mth PLEterm·mth spectra of the RPPterm·dom with n equals to 4 (see the others in Supplementary Fig. 9 ) clearly showing the excitonterm·key ground state 1 s and the excited excitonterm·key states 2 s and 3 s . b Corresponding energy of the excitonterm·key Rydberg states. Dashed line is a fit to the 2 s and 3 s states with the 2D hydrogen model 26 of excitonterm·key Rydberg series using R y = 0.11 ± 0.04 eV and E G = 2.078 ± 0.012 eV in the formula E Ns = E G − R y /( N − 1/2) 2 , with Ns equals to 1 s , 2 s , 3 s . c Evolution of the excitonterm·key ground state, first excited state 2 s , and continuum energies with the 2D perovskite layer thickness (or n ). d Corresponding experimental binding energy of the excitonterm·key ground state (1 s ) and the excited excitonterm·key states (2 s , 3 s , 4 s ), and comparison to theoretical results for the excitonterm·key ground state. The gray open triangle corresponds to the 1 s exciton binding energyterm·key calculated for the diamagnetic shift obtained by Tanaka et al. 15 in a RPPterm·dom with n equals to 1 with larger organic spacers (see Fig. 1d, e ). The values E 1 s − E 2 s provide a lower limit for the exciton binding energyterm·key. Error bars correspond to s.d. from the determination of the energy of the excitonterm·key features After validating the model, we can further extract pertinent details on the specific role of dielectric confinementterm·dom on the excitonterm·key characteristics and the scaling of the exciton binding energyterm·key with perovskite layer thickness. The model elucidates that the overlap of the electron and hole wavefunctions, whose convolution forms the excitonterm·key ground state, becomes more pronounced towards the center of the perovskite layer for the highest n -value (Fig. 2c and Supplementary Fig. 4c ). This suggests that the electric field lines for the excitonterm·key ground state are significantly more localized within the perovskite layer for larger n -values than for the RPPterm·dom with n equals to 1 and 2. This is consistent with the fact that with increasing perovskite layer thickness the strength of dielectric confinementterm·dom decreases with respect to quantum confinementterm·dom (Fig. 2d ). Furthermore, the calculated probability densities of the electron and hole wavefunctions exhibit negligible intensity outside of the perovskite layer (i.e., into the organic spacer layer) as displayed in Fig. 2c and Supplementary Fig. 4c , which is comparable to a technologically relevant multi quantum wellterm·key system. In summary, our theoretical model demonstrates that the strong exciton binding energyterm·key in thin 2D perovskite layers ( n approaching 1) stems from strong dielectric confinementterm·dom effects, which wane progressively for larger perovskite layer thicknesses ( n approaching 5 or greater).

Simple analytical expression of the exciton binding energyterm·key scaling lawterm·key Finally, based on our understanding of the excitonterm·key behavior with varying perovskite layer thickness, we developed a general empirical scaling lawterm·key of the excitonterm·key binding energies based on a classical model for low dimensional systems 54 as described in Eq. ( 1 ): 2 E b , 1 s = E 0 1 + α - 3 2 2 with α = 3 - γ e - L w 2 a 0 .

In this expression for the excitonterm·key ground state binding energy, E 0 (=16 meV) and a 0 (=4.6 nm) are the 3D Rydberg energy and Bohr radius of 3D perovskites 16 , respectively, and L w is the physical width of the quantum wellterm·key (Supplementary Table 1 ) for an infinite quantum wellterm·key potential barrier 55 . In the model (2), the excitonterm·key is considered isotropic in a α -dimensional space ( α strictly greater than 1 and smaller than 3) and we introduce an empirical correction factor γ . The factor γ is a simple way to empirically account for the deviations from the pure quantum confinementterm·dom regime, including electron and holes densities and dielectric confinementterm·dom effects. In a purely quantum confined regime γ equals to 1 in the expression of α , but experimentally derived α −value in RPPsterm·dom can only be fitted with a larger γ value, i.e., γ equals to 1.76 (Fig. 5a ). The corresponding decrease in the value of α highlights additional compression of the excitonterm·key wavefunction in the quantum wellterm·key due to dielectric confinementterm·dom, which results in enhanced values of exciton binding energyterm·key as compared to merely including the quantum confinementterm·dom effect (Fig. 5b ). The surprisingly good agreement of the expression (2) with experimental results, only using a γ correction factor independent of the n -value to take into account dielectric effects, will be understood in future studies involving different RPPterm·dom compositions (Br- and Sn-based 2D perovskites, various organic spacing layers, etc.). We also note that in the simplistic approximation of Hydrogen excitonterm·key model, the changes of excitonterm·key confinement with perovskite layer thickness can be understood as changes of both the effective dielectric constant value and the excitonterm·key Bohr radius as a function of n -value (Supplementary Fig. 10 ). Fig. 5 Scaling lawterm·key of the exciton binding energyterm·key with the perovskite layer thickness. a The dimensionality coefficient α was derived from Eq. ( 2 ), where the exciton binding energyterm·key are the experimental values of Fig. 3d and E 0 = 16 meV, a 0 = 4.6 nm, and L w = 0.6292 × n in nanometers.The black curve indicating γ equals to 1 corresponds to the case of pure quantum confinementterm·dom in quantum wellterm·key systems with infinite potential barriers. The red curve indicating γ equals to 1.76 was derived from the fit to the experimental values of α (red markers) using the expression of α in Eq. ( 2 ). Setting γ greater than 1 leads to a decrease of the value of α which reflects the more pronounced compression of the excitonterm·key ground state wavefunction in the perovskite layer due to dielectric confinementterm·dom, as compared to the case of pure quantum confinementterm·dom. b Corresponding results for the binding energy of the excitonterm·key ground state, showing the enhancement of the binding energy due to dielectric confinementterm·dom. The red curve gives the general scaling lawterm·key of the exciton binding energyterm·key with the perovskite layer thickness based on the Eq. ( 2 ), with E 0 = 16 meV, γ = 1.76, a 0 = 4.6 nm, and L w = 0.6292 × n in nanometers. Error bars correspond to s.d. reported from the analysis in Fig. 4 Applying our model (2) for large n -values, we predict that the exciton binding energyterm·key in RPPsterm·dom is larger than room temperature thermal fluctuations ( k B T ) up to n about 20 (perovskite layer thickness of 12.6 nm). This demonstrates the surprising robustness of excitonterm·key states in RPPsterm·dom, in spite of the small spatial extension of the excitonterm·key as compared to the perovskite layer thickness. In other words, a more abrupt reduction of the exciton binding energyterm·key with increasing perovskite layer thickness was expected given the strong screening of the electron–hole Coulomb interactionterm·dom reported in 3D perovskites 16 , a situation approached by RPPsterm·dom with perovskite layer thickness of a few nanometers. Again, this underlines the unusual nature of photo-excited states in RPPsterm·dom, and to more extent in solution-processed quantum wellterm·key systems as a template for exploring quasi-2D semiconductors.

In summary, we demonstrate the importance of Coulomb interactionsterm·dom in 2D layered perovskites and experimentally elucidate properties of unexpectedly strongly bound excitonsterm·key (>120 meV) in RPPsterm·dom with thickness up to 3.1 nm (corresponding to n equals to 5). We propose a generic formulation of the scaling of the exciton binding energyterm·key with the perovskite layer thickness from the single layer ( n equals to 1) to 3D crystals ( n tends to infinity), further predicting the nature of optical transitions at room temperature to change from excitonic to free carrier like only in RPPsterm·dom with thickness larger than ~12 nm ( n about 20). These results mark a fundamental step towards the design of new 2D perovskite-based semiconductor materials for next-generation optoelectronic and photonic technologies such as solar cells, light-emitting diodes, photodetectors, polariton and electrically driven lasers.

Methods RPPterm·dom crystal synthesis and preparation The crystal structures of the RPPsterm·dom, (BA) 2 (MA) n -1 Pb n I 3 n +1 , is composed of an anionic layer {(MA) n -1 Pb n I 3 n +1 } 2− , derived from bulk methylammonium lead triiodide perovskites (MAPbI 3 ), which is sandwiched between n-butylammonium (BA) spacer cations (Fig. 1a ). BA and MA stand for CH 3 (CH 2 ) 3 NH 3 and CH 3 NH 3 , respectively. RPPsterm·dom with n ranging from 1 to 5, corresponding to perovskite layer thickness between 0.641 and 3.139 nm, were synthetized and purified following previously reported method 3 – 5 . More precisely, the raw crystals were prepared by combining PbO, MACl and BA in appropriate molar ratios in a HI/H 3 PO 2 solvent mixture. The precursor solutions were prepared with 0.225 M of Pb 2+ concentration and stirred at room temperature overnight. Phase purity and crystalline quality of each crystal sample was established by monitoring X-ray diffraction (Supplementary Fig. 1 ). In addition, the small Stokes shifts and relatively sharp linewidths of the excitonterm·key resonances (Fig. 3 ) were another indication of the homogeneity and low disorder in the RPPterm·dom crystals. Thin RPPterm·dom crystals were mechanically exfoliated onto either the 3.5 μm core of a single-mode optical fiber for magneto-absorptionterm·key spectroscopy or transparent quartz substrates for optical absorption, PLterm·mth, and PLEterm·mth spectroscopy experimentsterm·mth.

The 3D perovskite MAPbI 3 ( n tends to infinity) samples for magneto-absorptionterm·key spectroscopy were prepared as thin films on transparent substrates using the hot-casting method previously reported 56 .

Magneto-absorptionterm·key spectroscopy A single RPPterm·dom crystal was affixed over the core of a 3.5 µm diameter single-mode optical fiber to ensure rigid optical alignment of the light path during the magnetic field pulse. The fiber-sample assembly was mounted in a custom optical probe that was fitted in the 4 K bore of a 65 T capacitor-driven pulsed magnet. The sample was in a helium exchange gas environment to ensure thermal anchoring at 4 K. White light from a Xe lamp transmits the sample via the single-mode fiber and was retro-reflected and dispersed in a 300 mm spectrometer with a 300 groove/mm grating. Broadband spectra were recorded every 2.3 ms throughout the magnet pulse. Access to σ + and σ − circular polarization was achieved via a thin-film circular polarizer mounted directly after the sample and by reversing the direction of the magnetic field. Details of the setup can be found elsewhere 57 .

The 3D perovskite MAPbI 3 thin films were measured in the same manner as described above by placing film sample in front of the core of the optical fiber used for illumination.

Optical absorption, PLterm·mth and PLEterm·mth Optical spectroscopies were performed with an in-lab-built confocal microscopy system focusing close to the diffraction limit (about 1 μm resolution), a monochromatic laser tunable over the visible and near-infrared spectral ranges. PLterm·mth spectral responses were obtained through a spectrograph (Spectra-Pro 2300i) and a CCD camera (EMCCD 1024B) yielding an error of less than 2 nm. PLterm·mth data in the main text were measured for light excitation at 440 nm (if not mentioned otherwise) and the excitation intensity was typically of the order of or below 10 3 mW cm −2 . PLEterm·mth spectra were obtained by measuring the PLterm·mth integrated intensity while scanning with a 2 nm step the excitation light wavelength at wavelengths smaller than the one of the PLterm·mth emission peak. Absorption spectra were measured either via a balanced photodiode by tuning the laser excitation wavelength or by detecting the spectral transmission/reflection of the samples exposed to white light. Samples were measured under vacuum (between 10 −5 and 10 −6 Torr) and cooled at 4 K if not mentioned otherwise.

Theory A momentum space representation of the excitonterm·key Green’s function developed previously for classic quantum wellsterm·key 58 was used to solve the Bethe–Salpeter equation in the effective mass approximation. This method was adapted for RPPsterm·dom to include the screening of the electron–hole interaction related to the dielectric confinementterm·dom and the electron and hole wavefunctions overlap. These latter effects were evaluated at the DFTterm·mth level. The Wannier–Mott excitonterm·key Rydberg states appear as the bound states in the absorption spectrum and can also be determined from the corresponding Schrödinger equation for the two particle wave functions (see details in Supplementary Note 1 , 2 , and 6 ).

Data availability The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Electronic supplementary material Supplementary Information