ARTICLE

Scaling law for excitons in 2D perovskite quantumwellsJ.-C. Blancon1, A.V. Stier1, H. Tsai1,2, W. Nie 1, C.C. Stoumpos 3, B. Traoré 4, L. Pedesseau5,

M. Kepenekian 4, F. Katsutani6, G.T. Noe6, J. Kono 2,6,7, S. Tretiak 1, S.A. Crooker 1, C. Katan 4,

M.G. Kanatzidis3,8, J.J. Crochet 1, J. Even 5 & A.D. Mohite1,9

Ruddlesden–Popper halide perovskites are 2D solution-processed quantum wells with a

general formula A2A’n-1MnX3n+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 nat-

ure of optical resonances (excitons or free carriers) and the exciton reduced mass, and their

scaling with quantum well thickness, which are critical for designing efficient optoelectronic

devices, remain unresolved. Here, using optical spectroscopy and 60-Tesla magneto-

absorption supported by modeling, we unambiguously demonstrate that the optical

resonances arise from tightly bound excitons with both exciton reduced masses and binding

energies decreasing, respectively, from 0.221m0 to 0.186m0 and from 470meV to 125meV

with increasing thickness from n equals 1 to 5. Based on this study we propose a general

scaling law to determine the binding energy of excitons in perovskite quantum wells of any

layer thickness.

DOI: 10.1038/s41467-018-04659-x OPEN

1 Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2Department of Materials Science and Nanoengineering, Rice University, Houston, TX77005, USA. 3 Department of Chemistry, Northwestern University, Evanston, IL 60208, USA. 4Univ Rennes, ENSCR, INSA Rennes, CNRS, ISCR (Institut desSciences Chimiques de Rennes)–UMR 6226, F-35000 Rennes, France. 5 Univ Rennes, INSA Rennes, CNRS, Institut FOTON–UMR 6082, F-35000 Rennes,France. 6 Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA. 7 Department of Physics and Astronomy, RiceUniversity, Houston, TX 77005, USA. 8Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA. 9 Departmentof Chemical and Biomolecular Engineering, Rice University, Houston, TX 77005, USA. Correspondence and requests for materials should be addressed toJ.-C.B. (email: jblancon@lanl.gov) or to J.E. (email: jacky.even@insa-rennes.fr) or to A.D.M. (email: adm4@rice.edu)

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Ruddlesden–Popper halide perovskites1,2 (RPPs) aresolution-processed quantum well structures formed bytwo-dimensional (2D) layers of halide perovskite

semiconductors separated by bulky organic spacer layers,whose stoichiometric ratios are defined by the general formula3

A2A’n-1MnX3n+1 where A, A’ are cations, M is a metal, X is ahalide and the integer value n determines the perovskite layerthickness (or quantum well thickness). Recent breakthrough inthe synthesis of phase-pure (a single n-value) RPPs with highervalues3–5 of n, up to n equals to 5, has inspired their use as low-cost semiconductors in optoelectronics5–8 as an alternative tothree-dimensional (3D) perovskites due to their technologicallyrelevant intrinsic photo- and chemical-stability5–10. However, keyfundamental questions remain unanswered in RPPs with ngreater than 1, such as the nature of optical transitions, as well asthe behavior of Coulomb interactions especially with increasingquantum well thickness. In fact there has been an intense ongoingdebate6–8,11–13 regarding the exact nature of the optical transi-tions (excitons versus free carriers) in RPPs with large n-values.RPPs with n equals to 1 (excitons at room temperature),14,15 and3D perovskites16 (free carriers at room temperature) are repre-sentative of the two limiting regimes at room temperature, but theanalysis of the crossover has not been performed. This issueoriginates from the lack of knowledge of the fundamentalquantities such as the exciton reduced mass, dielectric constantand characteristics like the spatial extension of electron and holewavefunctions, which play a crucial role in the determination ofthe exciton binding energy. In particular, contradictory reports of

the value of the exciton reduced mass have significantlycontributed to this uncertainty15,16 making it the most criticalexperimentally derived parameter required for the quantitativedetermination of the exciton characteristics in hybrid(organic–inorganic) perovskite systems. Moreover, unlike bulkinorganic semiconductors, heterostructures17 and 3D per-ovskites18,19, it is non-trivial to determine the exciton reducedmass in layered hybrid perovskites using a symmetry-based (k.p)approach20 or using many-body ab initio calculations21. Fur-thermore, ambiguity in the determination of the exciton bindingenergy in layered 2D perovskites has also been imposed by thelimited understanding of the role of dielectric confinement versusquantum confinement with increasing quantum well thick-ness22,23. More generally, phase-pure RPPs with n greater than 1present a unique opportunity to explore the physical properties ofnatural quantum well semiconducting crystals intermediatebetween monolayer 2D materials24,25 and 3D materials26, a quasi-dimensional physics only accessible in synthetic inorganic semi-conductor quantum well structures so far26.

Here we present the study on using low-temperature (4 K)magneto-optical spectroscopy to accurately determine the excitonreduced mass for (BA)2(MA)n-1PbnI3n+1 RPP crystals with per-ovskite 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 BAand MA stand for CH3(CH2)3NH3 and CH3NH3, respectively.The reduced mass is used to develop a generalized theoreticalmodel for electron–hole interactions in RPPs and determine

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Fig. 1 Exciton reduced mass from magneto-absorption spectroscopy and theory. a Schematic of the RPP structure cut along the direction ĉ of stacking ofthe 2D layers. b Image of mechanically exfoliated RPP crystals. Scale bar is 10 µm. c Magnetic field dependence of the light transmission of an individualRPP with n equals to 4 crystal for right- (σ‒) and left-handed (σ+) circular polarization. d Corresponding shift of the exciton energy as a function of themagnetic field. Fit of the data using ΔE= ±1/2 g0μBB+ c0B2 yields c0= 1.04 ± 0.16 µeV T−2 and g0= 1.59 ± 0.03. e Derived diamagnetic shift coefficient ofthe measured RPPs (red squares). The value from Tanaka et al.15 is also reported for RPPs with n equals to 1 with a slightly larger organic spacer. f Excitonreduced mass derived from fitting the diamagnetic shifts with our theoretical model. The gray dotted line is a guide for the eyes. The red dashed lineindicates average value of exciton reduced mass for the 3D perovskite MAPbI3. Error bars correspond to s.d. from the fit of the shift of the exciton energyas a function of the magnetic field

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fundamental characteristics of the exciton states. In parallel, fromlow-temperature optical spectroscopy we experimentally deter-mine the exciton binding energy in the RPPs with n varying from1 to 5. Finally, from these results we produce a general scalingbehavior for the binding energy of Wannier–Mott exciton statesin RPPs, which allows for prediction of the exciton bindingenergy for any given thickness. This study closes a long-standingscientific gap and will lead to the rational design of next-generation layered 2D perovskite-based optoelectronic devices.

ResultsExciton reduced mass from magneto-absorption measure-ments. Fig. 1 shows the RPP structure and results from magneto-absorption spectroscopy, which was employed to probe thestrength of the electron–hole interaction in RPPs and deduce theexciton reduced mass. These measurements were performed at 4K in the Faraday geometry (magnetic field along the stackingdirection of the layers, ĉ-axis), and the optical spectra of the right-and left-handed circular polarization components (σ±) of thetransmitted white light were probed. Application of a highmagnetic field on the RPP with n equals to 4 results in an energyshift of the optical resonance at about 1.9 eV (Fig. 1c, d). Similarresults were obtained for the other RPPs (see SupplementaryFig. 2). This optical transition was identified as the excitonground state and its energy shift under magnetic field in theFaraday configuration is expressed by26 ΔE= ±1/2 g0μBB+ c0B2,where the first term describes the Zeeman splitting of the σ+ andσ‒ exciton transitions (g0 is the g-factor in the perovskite plane, μBthe Bohr magneton, B the magnetic field) and the second one thediamagnetic shift (c0 is the diamagnetic shift coefficient). Undermagnetic field the competition between the Zeeman and dia-magnetic effects explains the non-symmetric, opposite-signenergy shift of the σ+ and σ‒ exciton transitions with respect tothe zero field exciton absorption energy (Fig. 1d and Supple-mentary Fig. 2). We note that this asymmetry becomes morepronounced for thicker perovskite layers (n approaching 5),which is a consequence of the strong increase of the diamagneticcoefficient with increasing n-value. Fitting the exciton energyshifts for both σ+ and σ‒ polarizations using the model aboveyields the diamagnetic coefficients c0 of the RPPs (Fig. 1e). Here,c0 increases monotonically with the perovskite layer thickness,and it ranges from ~0.2 to ~1.1 µeV T−2 for the RPPs with nvarying from 1 to 5, respectively. We note that due to relativelysmall signal-to-noise ratio in the magneto-absorption data for theRPP with n equals to 1 (Supplementary Fig. 2a), the diamagneticcoefficient for this compound was obtained from a combinedexperimental–theoretical approach which consisted in fitting the c0value to be in best agreement with both the magneto-absorptiondata and the experimental exciton binding energy discussed later inthis paper. For comparison, we also included in Fig. 1e the value ofc0 reported by Tanaka et al.15 (c0= 0.26 µeV T−2) for a RPP with nequals to 1 and with a slightly larger organic spacer (hexyl-ammonium CH3(CH2)5NH3 instead of butylammoniumCH3(CH2)3NH3 in our case). We note that our value of c0 for theRPP with n equals to 1 with BA as organic spacer is in agreementbut slightly smaller than other reports of RPPs with n equals to 1with longer organic spacers27–29. Concomitant with the dia-magnetic shift increase with n-value, the g-factor presents amarked increase from 0.8 to 1.6 for the RPPs with n varying from1 to 5, and a value of about 1.7 was derived from our 3D per-ovskites magneto-absorption data (Supplementary Fig. 3) andis consistent with the literature30,31.

The diamagnetic coefficient is directly connected to both theexciton reduced mass and the strength of the electron–holeCoulomb interaction. Although a simple relation exists for pure

2D systems where quantum confinement dominates22,26, i.e.,when confinement effects stem from the confinement of electronand hole wavefunctions in a 2D plane, this model does not applyto the RPPs due to the following reasons. First, the perovskitelayer thickness is comparable to the spatial extent of theexcitons15,16,27 and the exciton wavefunction cannot be strictlyconfined to a 2D plane (see also Supplementary Note 1 to 4).Second, the dielectric confinement plays a key role in the photo-physics of RPPs15,23,27. Dielectric confinement (also called imagecharge effect) manifests in a quantum well system provided thatthe thickness of the quantum wells is comparable to the excitonsize, and the dielectric constant ratio between the quantum wells(perovskite layers with εw~4 or higher) and the barriers (organicspacing layers with εb~2.2) is larger than unity. This result leadsto an enhancement of the Coulomb interaction between theelectron and hole pair composing each exciton, which is aconsequence of the reduced dielectric screening of the excitonelectric field partially located outside the quantum well, i.e., in thesurrounding barriers that have a lower dielectric constant.

Therefore, we developed a theoretical model (see details in thenext section, Supplementary Note 1, and Supplementary Fig. 4),which describes the electron–hole Coulomb interaction in thinsemiconductors and includes dielectric confinement (Keldyshtheory22). For this theoretical model to work, it requires anaccurate determination of the exciton reduced mass (labeled µ).We evaluated the reduced mass for each RPP by adjusting thetheoretical values of c0 to those measured experimentally (Fig. 1fand Supplementary Note 5). This yielded µ= 0.221m0 (m0 is thefree electron mass) for the RPP with n equals to 1, and 0.184m0

for the RPP with n equals to 1 compound with longer organicspacers reported by Tanaka et al.15. Then, we observe amonotonic decrease of the exciton reduced mass from 0.217m0

for the RPP with n equals to 2 down to 0.196m0 for the RPP withn equals to 4. Due to computational limitations, our model wasnot applied to the RPP with n equals to 5; however we estimated,to a first approximation, µ= 0.186m0 from the extrapolation ofthe data for the RPPs with n equals 1 to 4.

The deduced values of exciton reduced mass in the RPPs up ton equals to 5 are significantly larger than those reported for 3Dperovskites (Supplementary Fig. 3), even though the exciton statesare well confined within the perovskite layer (see next section).This is surprising because a decrease of the exciton reduced massis expected for increasing perovskite layer thickness, with limitingvalue of 0.104m0 (3D perovskite), as the vicinity of the excitonapproaches that of the 3D perovskites. However, the equivalentquantum well is still very thin even for the RPP with n equals to 5.Moreover, the reduced mass of the exciton ground statecalculated by density functional theory (DFT) for each RPP(with n equals 1 to 4) yields a similar dependence on the n-valueas the experimental ones provided the underestimation of thebandgap computation by DFT is taken into account (Supple-mentary Fig. 5). We explain the higher values of exciton reducedmass in the RPPs studied as compared to their 3D counterpart bya progressive reduction of the energy bandgap26. The limiting 3Dvalue is thus expected to be reached for larger n-value, i.e., whenthe RPP bandgap equals the 3D perovskite one. Furthermore,electronic band mixing and non-parabolicity effects have beenknown to induce changes in the exciton reduced mass inquantum well systems32–35. These effects have also been discussedin classic inorganic semiconductors and agree with the n-valuedependence of the experimental bandgap, diamagnetic shift andg-factor (Figs. 1e and 3a, and Supplementary Fig. 6). In fact, thebandgap dependence of the exciton reduced mass was alsoreported recently in 3D hybrid perovskites31. To conclude, theunambiguous determination of the reduced mass allows us todevelop a predictive model for excitons in solution-processed

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quantum well systems and calculate the exciton binding energiesof the RPP compounds.

Calculating the exciton binding energy in 2D perovskites. Thebinding energy of the exciton ground state was calculated for eachRPP (with n equals 1 to 4) by inputting each respective value ofthe exciton reduced mass into our theoretical model and semi-empirically solving the Bethe–Salpeter equation based on theeffective mass Green’s function approach23. This approachincludes: calculating the electronic structures of the RPPs toextract exciton wavefunctions and dielectric constant profilesusing DFT, generalizing the Keldysh theory for 2D perovskitesand combining both of the above by building a semi-empiricalmodel to simulate the Wannier–Mott exciton characteristics(Fig. 2 and Supplementary Note 1, 2, and 6). In our model, thepotential function describing the Coulomb interaction betweenthe electron and hole forming the exciton states is based onKeldysh theory22 generalized to a semiconducting dielectricquantum well (Fig. 2a), i.e., a dielectric well (perovskite layer)with thickness d and dielectric constant εw sandwiched betweenwell barriers (organic spacing layers) with dielectric constant εb.The non-local, screened electron–hole pair interaction potential is

given by:

Vs qtð Þ ¼ �e22εwqt

ZZ d2;d2

�d2 ;

�d2

ρe zeð Þρh zhð Þ e�qt ze�zhj j�

þΔχ e�qt zeþzh�dj j þ e�qt zeþzhþdj j� �þΔχ2 e�qtjze�zh�2dj þ e�qtjzh�ze�2dj� ��

dzedzh;

ð1Þ

in reciprocal space and in the in-plane a;b� �

direction. In Eq. (1),

the difference of electron and hole transverse (in-plane) wave-vectors is qt, Δ ¼ 1� χ2e�2qtd

� ��1, and χ ¼ εw�εb

εwþεb. Here, the

dielectric constants are inputs from our DFT results and weretaken at optical frequency because the values of binding energymeasured in RPPs (see next section) are one order of magnitudelarger than the highest energy value reported for the lattice opticalphonon modes36. The electron (ρe) and hole (ρh) probabilitydensity profiles along the stacking axis ĉ (ze,h are real-spacecoordinates along this direction) were also obtained from ourDFT results. This method is equivalent to the real-space com-putation of the screening effect using the image charge method37.

ba

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Fig. 2 Semi-empirical model of Wannier–Mott exciton in RPPs. a Schematics of the single quantum well system to which our model was applied to. bComputed band structure for the RPP with n equals to 4. c Corresponding electron (dashed black lines) and hole (red) probability density profiles, andhigh-frequency dielectric constant profile (blue) along the stacking axis ĉ. d Calculated binding energy of the exciton ground state using the excitonreduced mass in Fig. 1f and the DFT results. The values calculated using our model results (red circles) is compared to those obtained in the approximationof a pure 2D system (blue triangles) and of a pure 2D system without dielectric confinement (black squares). The binding energy of the excited excitonstates (2s, 3s, 4s) were also computed and are reported in Supplementary Fig. 7. For reference, the 3D perovskite MAPbI3 yields an exciton binding energyof about 16 meV, from Miyata et al.16 and in agreement with our results in Supplementary Fig. 3

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We also note that the general expression (1) of the electron–holeinteraction potential corresponds to the general Keldyshpotential in real space22. Upon different degrees of approximation(as discussed in Supplementary Note 3), the electron–holeinteraction potential used in the literature in the limit ofmonolayer 2D transition metal dichalcogenides38–42 can beretrieved. However, these approximations are not valid in RPPs(Supplementary Note 4) and the general expression (1) needs tobe used.

Overall, theoretical solution to the exciton binding energy ineach RPP requires the knowledge of: the electron and holeprobability density distributions, the dielectric constant profilealong the stacking direction, and the exciton reduced mass. InRPPs, the density profiles and dielectric constants along thestacking direction (Fig. 2c and Supplementary Fig. 4b), along withthe electronic band structure (Fig. 2b), were derived from theelectron and hole wavefunctions calculated by DFT (Supplemen-tary Note 6). A representative example of the electron and holeprobability density profiles is sketched in Fig. 2c for the RPPswith n equals to 4. Our calculations reveal that the chargewavefunctions stay confined into the perovskite layers and exhibitlittle leakage into the organic spacing layers, which is consistentwith uncoupled quantum well systems. Moreover, the results

reveal maximum values of dielectric constant in the range of 4 to5.2 for the perovskite layers in RPPs with n equals 1 to 5, and aminimum value of about 2.2 for the organic spacing layers,consistent with previous estimations43,44. The contrast ofdielectric constants between the perovskite and organic spacersis at the origin of the strong dielectric confinement effectsobserved in RPPs, as discussed in the next section.

Gathering all these information, our general model was thenapplied to the calculation of the exciton binding energy in RPPs,yielding a value of 467 meV for the RPP with n equals to 1 (435meV for the case of longer organic spacers from Tanaka et al.15),and which decreases monotonically to ~135 meV for the RPPwith n equals to 4 as illustrated in Fig. 2d (see also SupplementaryFig. 7 and Supplementary Table 2). We observe a radicaldifference of the exciton binding energies calculated using ourgeneral approach as compared to those obtained under variousdegrees of approximation usually suited to 2D materials such astransition metal dichalcogenides38–42, i.e., in the case the chargesare strictly restricted to a 2D plane and including or not dielectricconfinement (see details in Supplementary Note 3). The accuracyof our model to predict the binding energies of excitons in 2Dperovskites was then verified by directly measuring these valuesusing optical spectroscopy techniques.

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Fig. 3 Optical spectroscopy of the RPP crystals with n equals 1 to 5. a Experimental optical bandgap scaling. b Photoluminescence spectra. c Schematics ofthe Rydberg series of the exciton ground state (1s) and excited exciton states (2s, 3s, etc.) merging with the continuum. (Right) Corresponding absorptionor optical density, photoluminescence PL, and photoluminescence excitation PLE spectra typically observed in 2D material systems26,49. d Optical densityOD and e PLE spectra of the RPPs with n equals 1 to 5, respectively. Stars point to exciton ground state optical transitions, black squares show thecontinuum onsets and gray brackets indicate the region of excited exciton states. Labels A, B and C indicate absorption regions at energies higher than thecontinuum bandgap EG and which are apparently common to all the RPPs given an energy shift proportional to the bandgap

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Direct measurement of the exciton binding energy. We directlymeasured the exciton binding energy in each RPP using opticalabsorption, photoluminescence (PL) and photoluminescenceexcitation (PLE) spectroscopy (Fig. 3). The optical bandgap wastuned over the visible spectral range from 2.540 ± 0.004 eV (488nm) in the RPP with n equals to 1 down to 1.846 ± 0.004 eV (672nm) in the RPP with n equals to 5 (Fig. 3a). We note that theRPPs with n greater than 1 retained their optical bandgap from290 K down to 4 K. This observation implies that our studyconducted at low temperature is directly transposable to roomtemperature and provides direct insights into the photo-physicsof materials used in practical devices.

Figure 3b shows a single sharp peak in the PL spectra,indicating the emission of the exciton ground state14,45. The low-energy shoulder observed more than tens of meV below theexciton peak in all RPPs was recently assigned in the RPP with nequals to 1 to a phonon replica46 and self-trapped excitons47.Based on power dependence measurements of the PL (Supple-mentary Fig. 8), we hypothesize a similar origin for theresonances in the RPPs with n greater than 1, but we only focuson investigating the ground state exciton transition in this study.

All the presented absorption spectra show similar features(Fig. 3d), which includes a single absorption peak at low energycorresponding to the exciton ground state resonance. Moreover, a

steady increase of absorption at higher energies modulated byenergy optical transitions is well reproduced in the PLEmeasurements (Fig. 3e). These spectral features were assignedto the Rydberg series of the Wannier–Mott exciton (1s, 2s, 3s, 4snoted as Ns, with N equals to 1, 2, 3 and 4 and having energiesENs) and are illustrated in Fig. 3c. The onset of the continuum EGcorresponds to electron and hole free carrier states (Fig. 3c), aspreviously reported in the RPP with n equals to 1 perovskites15,45,other 2D nanomaterials39,48 and quantum well semiconduc-tors26,49. Absorption transition features, marked as A, B and C,are observed in both the absorption and PLE spectra for energyhigher than EG, and are apparently common to all the RPPs givenan energy scaling with the n-value proportional to the bandgapscaling (Fig. 3d, e). Close to the continuum onset and in ourexperimental energy range, the optical transitions in RPPs atenergies above EG have been assigned11,14,50 to transitionsbetween the Pb(6s)-I(5p) states in the valence band and Pb(6p)states in the conduction band20,51. The absorption features A, Band C can be understood as transitions between valence andconduction bands overlapping with the continuum of states fromthe band-edge exciton. For example, from the calculated RPPband structures (Fig. 2b and Supplementary Fig. 4a), one canassume that the high energy optical transitions involve the seriesof the mini-bands directly below the highest valence band and

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Fig. 4 Direct measurement of the exciton binding energy and Rydberg states in RPPs with n equals 1 to 5. a Optical density OD and photoluminescenceexcitation PLE spectra of the RPP with n equals to 4 (see the others in Supplementary Fig. 9) clearly showing the exciton ground state 1s and the excitedexciton states 2s and 3s. b Corresponding energy of the exciton Rydberg states. Dashed line is a fit to the 2s and 3s states with the 2D hydrogen model26 ofexciton Rydberg series using Ry= 0.11 ± 0.04 eV and EG= 2.078 ± 0.012 eV in the formula ENs= EG− Ry /(N− 1/2)2, with Ns equals to 1s, 2s, 3s. cEvolution of the exciton ground state, first excited state 2s, and continuum energies with the 2D perovskite layer thickness (or n). d Correspondingexperimental binding energy of the exciton ground state (1s) and the excited exciton states (2s, 3s, 4s), and comparison to theoretical results for theexciton ground state. The gray open triangle corresponds to the 1s exciton binding energy calculated for the diamagnetic shift obtained by Tanaka et al.15 ina RPP with n equals to 1 with larger organic spacers (see Fig. 1d, e). The values E1s− E2s provide a lower limit for the exciton binding energy. Error barscorrespond to s.d. from the determination of the energy of the exciton features

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above the lowest conduction band at the Γ high symmetry pointin the Brillouin zone. Calculation of the broadband opticalspectra in the RPPs, including dipole allowed transitions fromgroup theory and joint density of states, should be performed infuture theoretical studies in order to understand the details of theexperimental absorption spectra at energies higher than thecontinuum EG.20,51

The exciton binding energies in RPPs were derived from thestudy of the optical transitions corresponding to the excitonRydberg series and the continuum. We present a representativeexample for the RPP with n equals to 4 in Fig. 4a, b. Dielectricconfinement (or image charge effect) has been shown15,27,44,45 tomainly influence the 1s excitons in RPPs with n equals 1.Therefore, the 2s and 3s exciton states were fitted using the classic2D hydrogen Rydberg series with energies26 ENs= EG− Ry /(N−1/2)2, where Ry is the Rydberg energy. This yields EG= 2.078 ±0.012 eV and Ry= 0.11 ± 0.04 eV for the RPP with n equals to 4(Fig. 4b, dashed red line). This procedure is similar to the oneused in transition metal dichalcogenides39,52 and 2D perovs-kites15,45,53. It was also applied to the RPPs with n equals to 1, 2, 3and 5 (Supplementary Fig. 9), i.e., the 1s, 2s and 3s exciton stateswere identified from the analysis of the experimental spectra andthe 2s, 3s Rydberg series were fitted with the 2D hydrogen modelto derive the continuum energy EG. The binding energy of the 1sexciton ground state was derived from the difference |E1s− EG|,which ranges from about 470 meV for the RPP with n equals to 1down to 125meV for the RPP with n equals to 5 (Fig. 4c, d,Supplementary Table 1). We note that the lower limit of theexciton binding energy is given by the energy difference betweenthe (1s) exciton ground state and (2s) first excited state (Fig. 4d).We emphasize that the predicted theoretical and experimentallydetermined values are in excellent agreement (Fig. 4d andSupplementary Fig. 7), thus validating the accuracy of ourdeveloped theoretical model for Wannier–Mott excitons in RPPs.These results emphasize the need for an accurate determinationof the exciton reduced mass and taking into account thecontributions of quantum and dielectric confinements as well ashole and electron density profiles in order to develop an accurateand robust theoretical model for the 2D perovskite systems.

After validating the model, we can further extract pertinentdetails on the specific role of dielectric confinement on theexciton characteristics and the scaling of the exciton bindingenergy with perovskite layer thickness. The model elucidates thatthe overlap of the electron and hole wavefunctions, whoseconvolution forms the exciton ground state, becomes morepronounced towards the center of the perovskite layer for thehighest n-value (Fig. 2c and Supplementary Fig. 4c). This suggeststhat the electric field lines for the exciton ground state aresignificantly more localized within the perovskite layer for largern-values than for the RPP with n equals to 1 and 2. This isconsistent with the fact that with increasing perovskite layerthickness the strength of dielectric confinement decreases withrespect to quantum confinement (Fig. 2d). Furthermore, thecalculated probability densities of the electron and holewavefunctions exhibit negligible intensity outside of the per-ovskite layer (i.e., into the organic spacer layer) as displayed inFig. 2c and Supplementary Fig. 4c, which is comparable to atechnologically relevant multi quantum well system. In summary,our theoretical model demonstrates that the strong excitonbinding energy in thin 2D perovskite layers (n approaching 1)stems from strong dielectric confinement effects, which waneprogressively for larger perovskite layer thicknesses (n approach-ing 5 or greater).

Simple analytical expression of the exciton binding energyscaling law. Finally, based on our understanding of the excitonbehavior with varying perovskite layer thickness, we developed ageneral empirical scaling law of the exciton binding energiesbased on a classical model for low dimensional systems54 asdescribed in Eq. (1):

Eb;1s ¼E0

1þ α�32

� �2 with α ¼ 3� γe�Lw2a0 : ð2Þ

In this expression for the exciton ground state binding energy,E0 (=16 meV) and a0 (=4.6 nm) are the 3D Rydberg energy andBohr radius of 3D perovskites16, respectively, and Lw is the

500

400

300

200

100

0

Bin

ding

ene

rgy

(meV

)

1 10 100n

Quantumconfinement (� = 1)

� = 1.76

1s excitonexperiment

a b3.0

2.5

2.0

1.5

1 10 100n

Quantumconfinement (� = 1)

1s excitonexperiment

α

� = 1.76

Fig. 5 Scaling law of the exciton binding energy with the perovskite layer thickness. a The dimensionality coefficient α was derived from Eq. (2), where theexciton binding energy are the experimental values of Fig. 3d and E0= 16 meV, a0= 4.6 nm, and Lw= 0.6292 × n in nanometers.The black curve indicatingγ equals to 1 corresponds to the case of pure quantum confinement in quantum well systems with infinite potential barriers. The red curve indicating γequals 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 adecrease of the value of α which reflects the more pronounced compression of the exciton ground state wavefunction in the perovskite layer due todielectric confinement, as compared to the case of pure quantum confinement. b Corresponding results for the binding energy of the exciton ground state,showing the enhancement of the binding energy due to dielectric confinement. The red curve gives the general scaling law of the exciton binding energywith the perovskite layer thickness based on the Eq. (2), with E0= 16 meV, γ= 1.76, a0= 4.6 nm, and Lw= 0.6292 × n in nanometers. Error barscorrespond to s.d. reported from the analysis in Fig. 4

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physical width of the quantum well (Supplementary Table 1) foran infinite quantum well potential barrier55. In the model (2), theexciton is considered isotropic in a α-dimensional space (αstrictly greater than 1 and smaller than 3) and we introduce anempirical correction factor γ. The factor γ is a simple way toempirically account for the deviations from the pure quantumconfinement regime, including electron and holes densities anddielectric confinement effects. In a purely quantum confinedregime γ equals to 1 in the expression of α, but experimentallyderived α−value in RPPs can only be fitted with a larger γ value,i.e., γ equals to 1.76 (Fig. 5a). The corresponding decrease in thevalue of α highlights additional compression of the excitonwavefunction in the quantum well due to dielectric confinement,which results in enhanced values of exciton binding energy ascompared to merely including the quantum confinement effect(Fig. 5b). The surprisingly good agreement of the expression (2)with experimental results, only using a γ correction factorindependent of the n-value to take into account dielectric effects,will be understood in future studies involving different RPPcompositions (Br- and Sn-based 2D perovskites, various organicspacing layers, etc.). We also note that in the simplisticapproximation of Hydrogen exciton model, the changes ofexciton confinement with perovskite layer thickness can beunderstood as changes of both the effective dielectric constantvalue and the exciton Bohr radius as a function of n-value(Supplementary Fig. 10).

Applying our model (2) for large n-values, we predict that theexciton binding energy in RPPs is larger than room temperaturethermal fluctuations (kBT) up to n about 20 (perovskite layerthickness of 12.6 nm). This demonstrates the surprising robust-ness of exciton states in RPPs, in spite of the small spatialextension of the exciton as compared to the perovskite layerthickness. In other words, a more abrupt reduction of the excitonbinding energy with increasing perovskite layer thickness wasexpected given the strong screening of the electron–holeCoulomb interaction reported in 3D perovskites16, a situationapproached by RPPs with perovskite layer thickness of a fewnanometers. Again, this underlines the unusual nature of photo-excited states in RPPs, and to more extent in solution-processedquantum well systems as a template for exploring quasi-2Dsemiconductors.

In summary, we demonstrate the importance of Coulombinteractions in 2D layered perovskites and experimentallyelucidate properties of unexpectedly strongly bound excitons(>120 meV) in RPPs with thickness up to 3.1 nm (correspondingto n equals to 5). We propose a generic formulation of the scalingof the exciton binding energy with the perovskite layer thicknessfrom the single layer (n equals to 1) to 3D crystals (n tends toinfinity), further predicting the nature of optical transitions atroom temperature to change from excitonic to free carrier likeonly in RPPs with thickness larger than ~12 nm (n about 20).These results mark a fundamental step towards the design of new2D perovskite-based semiconductor materials for next-generationoptoelectronic and photonic technologies such as solar cells,light-emitting diodes, photodetectors, polariton and electricallydriven lasers.

MethodsRPP crystal synthesis and preparation. The crystal structures of the RPPs,(BA)2(MA)n-1PbnI3n+1, is composed of an anionic layer {(MA)n-1PbnI3n+1}2−,derived from bulk methylammonium lead triiodide perovskites (MAPbI3), which issandwiched between n-butylammonium (BA) spacer cations (Fig. 1a). BA and MAstand for CH3(CH2)3NH3 and CH3NH3, respectively. RPPs with n ranging from 1to 5, corresponding to perovskite layer thickness between 0.641 and 3.139 nm, weresynthetized and purified following previously reported method3–5. More precisely,the raw crystals were prepared by combining PbO, MACl and BA in appropriatemolar ratios in a HI/H3PO2 solvent mixture. The precursor solutions were

prepared with 0.225 M of Pb2+ concentration and stirred at room temperatureovernight. Phase purity and crystalline quality of each crystal sample was estab-lished by monitoring X-ray diffraction (Supplementary Fig. 1). In addition, thesmall Stokes shifts and relatively sharp linewidths of the exciton resonances (Fig. 3)were another indication of the homogeneity and low disorder in the RPP crystals.Thin RPP crystals were mechanically exfoliated onto either the 3.5 μm core of asingle-mode optical fiber for magneto-absorption spectroscopy or transparentquartz substrates for optical absorption, PL, and PLE spectroscopy experiments.

The 3D perovskite MAPbI3 (n tends to infinity) samples for magneto-absorption spectroscopy were prepared as thin films on transparent substratesusing the hot-casting method previously reported56.

Magneto-absorption spectroscopy. A single RPP crystal was affixed over the coreof a 3.5 µm diameter single-mode optical fiber to ensure rigid optical alignment ofthe light path during the magnetic field pulse. The fiber-sample assembly wasmounted in a custom optical probe that was fitted in the 4 K bore of a 65 Tcapacitor-driven pulsed magnet. The sample was in a helium exchange gasenvironment to ensure thermal anchoring at 4 K. White light from a Xe lamptransmits the sample via the single-mode fiber and was retro-reflected and dis-persed in a 300 mm spectrometer with a 300 groove/mm grating. Broadbandspectra were recorded every 2.3 ms throughout the magnet pulse. Access to σ+ andσ− circular polarization was achieved via a thin-film circular polarizer mounteddirectly after the sample and by reversing the direction of the magnetic field.Details of the setup can be found elsewhere57.

The 3D perovskite MAPbI3 thin films were measured in the same manner asdescribed above by placing film sample in front of the core of the optical fiber usedfor illumination.

Optical absorption, PL and PLE. Optical spectroscopies were performed with anin-lab-built confocal microscopy system focusing close to the diffraction limit(about 1 μm resolution), a monochromatic laser tunable over the visible andnear-infrared spectral ranges. PL spectral responses were obtained through aspectrograph (Spectra-Pro 2300i) and a CCD camera (EMCCD 1024B) yieldingan error of less than 2 nm. PL data in the main text were measured for lightexcitation at 440 nm (if not mentioned otherwise) and the excitation intensity wastypically of the order of or below 103 mW cm−2. PLE spectra were obtained bymeasuring the PL integrated intensity while scanning with a 2 nm step the exci-tation light wavelength at wavelengths smaller than the one of the PL emissionpeak. Absorption spectra were measured either via a balanced photodiode bytuning the laser excitation wavelength or by detecting the spectral transmission/reflection of the samples exposed to white light. Samples were measured undervacuum (between 10−5 and 10−6 Torr) and cooled at 4 K if not mentionedotherwise.

Theory. A momentum space representation of the exciton Green’s functiondeveloped previously for classic quantum wells58 was used to solve theBethe–Salpeter equation in the effective mass approximation. This method wasadapted for RPPs to include the screening of the electron–hole interaction relatedto the dielectric confinement and the electron and hole wavefunctions overlap.These latter effects were evaluated at the DFT level. The Wannier–Mott excitonRydberg states appear as the bound states in the absorption spectrum and can alsobe determined from the corresponding Schrödinger equation for the two particlewave functions (see details in Supplementary Note 1, 2, and 6).

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

Received: 21 December 2017 Accepted: 9 May 2018

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AcknowledgementsThe work at Los Alamos National Laboratory (LANL) was supported by LDRD program(to J.-C.B., W.N., S.T., A.D.M.) and was partially performed at the Center for NonlinearStudies. J.-C.B. and A.D.M. acknowledge support from the DOE (EERE 1647-1544). Thework was conducted, in part, at the Center for Integrated Nanotechnologies (CINT), a U.S. Department of Energy, Office of Science user facility. Part of this work was performedat the National High Magnetic Field Laboratory, which is supported by NSF DMR-1157490, NSF DMR-1644779 and the State of Florida. S.A.C. acknowledges support fromthe DOE Basic Energy Sciences “Science of 100 T” program. Work at NorthwesternUniversity was supported by ONR grant N00014-17-1-2231. The work in France wassupported by Agence Nationale pour la Recherche (TRANSHYPERO project). This workwas granted access to the HPC resources of (TGCC/CINES/IDRIS) under the allocation2017-A0010907682 made by GENCI. The work at Rice University (to F.K., G.T.N., J.K.)

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was supported by the NSF (Grant No. DMR-1310138), the Robert A. Welch Foundation(Grant No. C-1509) and the Air Force Office of Science Research (Grant No. FA9550-14-1-0268).

Author contributionsJ.-C.B. and A.D.M. conceived the idea, designed the experiments and wrote the manu-script. J.-C.B. made the samples, performed the optical spectroscopy measurements,analyzed the data and provided insights into the mechanisms with support from J.J.C. A.V.S. performed the magneto-absorption measurements and analyzed the data under thesupervision of S.A.C. F.K. and G.T.N. performed magneto-absorption measurements onthe RPP with n equals to 1 under the supervision of J.K. J.E. developed the theoreticalmodels. B.T. performed the DFT calculations with support from C.K., L.P. and M.K. M.G.K. and C.C.S. developed the chemistry for the synthesis of phase-pure crystals withsupport from H.T. and W.N. All authors contributed to this work, read the manuscriptand agree to its contents and all data are reported in the main text and supplementalinformation.

Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-018-04659-x.

Competing interests: The authors declare no competing interests.

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1

SUPPLEMENTARY INFORMATION

for

Scaling law for excitons in 2D perovskite quantum wells

J.-C. Blancon1*, A. V. Stier1, H. Tsai1,2, W. Nie1, C. C. Stoumpos3, B. Traoré4, L. Pedesseau5, M.

Kepenekian4, F. Katsutani6, G. T. Noe6, J. Kono2,6,7, S. Tretiak1, S. A. Crooker1, C. Katan4, M.

G. Kanatzidis3,8, J. J. Crochet1, J. Even5* and A. D. Mohite1,9*

1Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.

2Department of Materials Science and Nanoengineering, Rice University, Houston, Texas 77005,

USA.

3Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA.

4Univ Rennes, ENSCR, INSA Rennes, CNRS, ISCR (Institut des Sciences Chimiques de

Rennes) - UMR 6226, F-35000 Rennes, France.

5Univ Rennes, INSA Rennes, CNRS, Institut FOTON - UMR 6082, F-35000 Rennes, France.

6Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005,

USA.

7Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA.

8Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois

60208, USA.

9Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas

77005, USA.

*Correspondence to: jblancon@lanl.gov, jacky.even@insa-rennes.fr, adm4@rice.edu

2

Supplementary Figure 1. Phase purity and crystal structure of RPPs with n=1 to 5. a, X-ray

diffraction spectra in the low diffraction angle region (<16°) demonstrating phase purity of each

RPP compound. b, Corresponding sketch of the crystal structure1–3 along the stacking axis.

3

Supplementary Figure 2. Magneto-absorption spectroscopy of RPPs with n=1, 2, 3, 5. The

results for RPP n=4 compound are presented in Figure 1. The energy shifts of the exciton ground

state (1s) for both right- and left-handed circularly polarization light 𝛔± were fitted using the

classic formula4 𝚫𝑬 = ±𝟏

𝟐𝒈𝟎𝝁𝐁𝑩 + 𝒄𝟎𝑩𝟐, where the first term stand for the Zeeman splitting (𝒈𝟎

the g-factor in the perovskite plane, 𝝁𝐁 the Bohr magneton, B the magnetic field) and the second

one for the diamagnetic shift (𝒄𝟎 the diamagnetic coefficient). OD stands for optical density.

4

Supplementary Figure 3. Magneto-absorption spectroscopy of 3D perovskite MAPbI3. a,

Transmission spectra at 0 T and at the magnetic field extrema 65 T for the two polarization σ±.

Under magnetic field we observe both a shift of the peak associated with the exciton ground state

(1s) at about 1.65 eV and the appearance of new states at high energy at high magnetic field (ripple-

like features above 1.7 eV at ±65 T). The latter features correspond to Landau levels4,5, and their

observation only in 3D perovskite is indicative of a much lower exciton binding energy in 3D

perovskite as compared to the RPPs with n=1 to 5. In the high magnetic field region, the evolution

of the Landau levels are described by Δ𝐸 = (𝑁L +1

2) ℏ𝜔c ±

1

2𝑔0𝜇B𝐵, with 𝜔c = 𝑒𝐵/𝜇 the

cyclotron frequency, NL (=0, 1, 2, etc.) the Landau quantum number, and the other parameters are

defined in the main text. b, Corresponding magnetic field dependence of the 1s exciton and the

two first Landau levels (noted LL1 and LL2). These data are the average of both polarization σ±

so as they do not include the Zeeman effects, i.e., Δ𝐸 = (𝑁L +1

2) ℏ𝜔c . From the linear fit of the

Landau level data we derive the exciton reduced mass of the 3D perovskite: 0.109 m0 for LL1 and

0.099 for LL2. The average exciton mass µ=0.104 m0 is identical to the one reported by Miyata et

al.5. In our experiment we did not observe the 2s exciton state as reported by Miyata et al.5, which

prevented us from obtaining an exact exciton binding energy in 3D perovskite because the

dielectric constant of the system is a priori unknown. Because we obtain experimentally the same

effective mass as in ref.5 in the same 3D perovskite compound, we can assume nearly identical

exciton binding energy (16 meV) and effective dielectric constant (휀eff =9.3). c, Zeeman splitting

for the 1s exciton yielding a g-factor of 𝑔0=1.7 (the data were fitted by 𝑔0𝜇B𝐵).

5

Supplementary Figure 4. DFT computed properties of RPPs with n=1 to 4. a, Computed band

structures. Comparison of the band structure computed for the real crystal structure (blue) with

that obtained by replacing the organic cations BA with Cs+ (red) is shown for n=2. b, Computed

high frequency dielectric constant profiles along the stacking axis ĉ. c, Corresponding probability

densities of valence (VB, red) and conduction (CB, dashed black lines) bands in a single dielectric

quantum well (Figure 2a). For comparison, the dielectric profiles are plotted in blue. Here, the x-

axis stands for the position along the stacking axis ĉ centered at the middle of a perovskite layer

and spanning a single perovskite layer surrounded by an organic spacer layer on each side. We

note that the densities are maximum around the position of the Pb atoms. The dielectric profiles

show a strong contrast between the perovskite layer (position 0) and the organic spacer layers (both

extrema positions on the x-axis).

6

Supplementary Figure 5. DFT computed exciton reduced mass as a function of n, before and

after band gap correction. The results are plotted for the [100] in-plane direction and [001] ([010]

for the RPP n=1) out-of-plane direction. In order to account for the well-known underestimation

of the electronic band gap at the DFT level, the computed single particle effective masses are

multiplied by the ratio between the experimental and computed electronic band gaps.

7

Supplementary Figure 6. Evolution of the g-factor with perovskite layer thickness (n-value).

8

Supplementary Figure 7. a, Calculated and b, measured binding energy of the exciton ground

state (1s) and Rydberg series of the excited exciton states (2s, 3s, 4s).

9

Supplementary Figure 8. Study of the PL spectra features using excitation intensity

dependence of the PL at 4K. a. Photoluminescence spectra showing the main exciton ground

state peak (*) along with low-energy side features. b-f. Excitation power dependence of the main

exciton peak (red circles), the low-energy shoulder of the main PL peak (black squares), and the

broad PL emission observed in n=1 between 1.5 and 2.4 eV (blue triangles). Excitation was

performed at either 440 nm or 480 nm. The integrated signals of all PL features show a nearly

linear dependence on the excitation power in the power range used experimentally. These

observations suggest that the PL side features have an intrinsic origin, as opposed to excitons

bound to defects (sublinear dependence) or bi-excitons (sublinear behavior). Therefore, we

hypothesize that the side features are phonon replica and/or self-trapped excitons as suggested in

previous reports6,7, however further studies will be necessary to understand their properties.

10

Supplementary Figure 9. Experimental analysis of the exciton Rydberg series in RPPs with

n=1 to 5, corresponding to a, to e, respectively. Results follow the method introduced in the

main text and in Figure 4. In the plots of the energy versus exciton state the red dots are

experimental data derived from absorption and PLE, and the blue triangles correspond to the

theoretical results. The dashed lines correspond to the fit of the excited exciton states (2s, 3s, etc.)

using the 2D hydrogen model of exciton Rydberg series. Ry is the Rydberg energy.

11

Supplementary Figure 10. Hydrogen exciton model applied to the 1s exciton state in RPPs.

a, The effective dielectric constant εeff, which describes the effective screening of the electron-hole

Coulomb interaction, was derived in both the 3D and 2D limiting cases considering only quantum

confinement4, i.e. using the hydrogen model of the exciton E1s= EG –Ry for 3D and E1s= EG –4Ry

for 2D, with the Rydberg energy Ry = (13.6 eV)×µ/εeff2. From our interpretation of the experimental

and theoretical results, εeff should adopt a value between the 2D and 3D cases. The RPP n=1 tends

to a pure 2D system but with significant contribution of dielectric confinement, and on the other

hand, the RPP n=5 tends to deviate from the 2D case. b, Corresponding Bohr radius. We emphasize

that these results are inexact in nature because the hydrogen exciton model assumed Dirac

wavefunctions and does not account for dielectric confinement. This is a first level of

approximation to our far more advanced theoretical model, but worth mentioning as a point of

comparison of RPPs to other low dimensional systems.

12

n RPPs Perovskite layer

thickness (nm)a

Exciton reduced

mass µ (units of m0)

Exciton binding

energy (meV)

1 (BA)2Pb1I4 0.641 0.221 467 ± 26

2 (BA)2(MA)Pb2I7 1.255 0.217 251 ± 21

3 (BA)2(MA)2Pb3I10 1.892 0.201 177 ± 19

4 (BA)2(MA)3Pb4I13 2.511 0.196 157 ± 19

5 (BA)2(MA)4Pb5I16 3.139 0.186

(extrapolated)

125 ± 29

∞ MAPbI3 b ∞ 0.104 16 ± 2

Supplementary Table 1. Summary of exciton properties in RPPs derived from the

experimental data. aData from Stoumpos et al.1,3, the organic spacer layer thickness is typically

0.710 nm. bFrom Miyata et al.5.

13

n RPPs

Quantum

well

thickness

(nm)

Quantum

well

dielectric

constant εw

Spacer layer

(barrier)

dielectric

constant εb

Calculated exciton

binding energy

(meV)

1 (BA)2Pb1I4 0.56 4.00 2.10 467

2 (BA)2(MA)Pb2I7 1.13 4.64 2.19 245

3 (BA)2(MA)2Pb3I10 1.70 4.90 2.22 169

4 (BA)2(MA)3Pb4I13 2.28 5.19 2.23 136

Supplementary Table 2. Calculated exciton properties in RPPs and corresponding materials

parameters. Details in Supplementary Text 1 and Supplementary Figure 4.

14

Supplementary Note 1:

Semi-empirical simulation of the Wannier-Mott exciton

The exciton states were modelled using the 3D momentum space representation of the Bethe-

Salpeter equation (BSE) for the polarization function Peh, including a statically screened electron-

hole (e-h) potential interaction 𝑉s, and assuming the effective mass approximation for the

electronic dispersions:8

𝑃eh(𝒌e, 𝒌h, 𝐸) = 𝑃eh0 (𝒌e, 𝒌h, 𝐸) − ∑ ∑ 𝑃eh

0 (𝒌e, 𝒌, 𝐸)

𝒌′𝒌

𝑉s(|𝒌 − 𝒌′|)𝑃eh0 (𝒌′, 𝒌h, 𝐸). (1)

In this expression, ke (kh) is the electron (hole) wavevector in momentum space, E the energy, and

the zero-order polarization function in the low-injection regime reads

𝑃eh0 (𝒌e, 𝒌h, 𝐸) =

1

𝐸 − 𝐸𝐺 −ℏ2|𝒌e|2

2𝜇 + 𝑖Γ𝛿𝒌e,𝒌h

, (2)

with 𝐸𝐺 is the free charge electronic bandgap and Γ a broadening factor.

The potential function describing the Coulomb interaction between the electron and hole forming

the exciton states was expressed for the dielectric quantum well in Figure 2a. Although a real space

approximate expressions can be used to analyse exciton in 2D materials9, our model is based on a

more general approach which described e-h interaction including dielectric confinement effects in

the Fourier space. This method does not require any approximation on the system, and was recently

proposed in a different context10. Therefore, the partial Fourier transform of the non-local,

screened e-h pair interaction potential, in the (��,��) in-plane direction, is given by:11

𝑉s(𝑞t) =−𝑒2

2휀w𝑞t∬ 𝜌e(𝑧e)𝜌h(𝑧h)[𝑒−𝑞t|𝑧e−𝑧h| + Δ𝜒(𝑒−𝑞t|𝑧e+𝑧h−𝑑|+𝑒−𝑞t|𝑧e+𝑧h+𝑑|)

𝑑2

,𝑑2

−𝑑2

,−𝑑2

+ Δ𝜒2(𝑒−𝑞t|𝑧e−𝑧h−2𝑑|+𝑒−𝑞t|𝑧h−𝑧e−2𝑑|)]𝑑𝑧e𝑑𝑧h,

(3)

where the difference of electron and hole transverse (in-plane) wavevectors is 𝑞t = |𝒌te − 𝒌th|,

Δ = (1 − 𝜒2𝑒−2𝑞t𝑑)−1, and χ =𝜀w−𝜀b

𝜀w+𝜀b. Here, 𝜌e and 𝜌h stand for, respectively, the part of the

electron and hole probability density profiles along the stacking axis ĉ located inside the dielectric

well (𝑧e,h are real space coordinates along this direction). The leakage of the density profiles in the

barrier was also considered and complementary expressions of the interaction potential have been

used11. This method is equivalent to the real-space computation of the screening effect using the

image-charge method12. We note that a Gaussian quadrature method was used to integrate the

singularity of the e-h interaction in the reciprocal space13. The general expression (3) of the e-h

interaction potential corresponds to the general Keldysh’s potential in real space14. Upon several

approximations, being 𝑧e = 𝑧h, 𝑞t𝑑 ≪ 1, and 휀w ≫ 휀b, the e-h interaction potential used in the

15

literature in the limit of mono-layer 2D transition metal dichalcogenides9,15 can be retrieved (see

also Supplementary Note 3). However, these approximations are not valid in RPPs and the general

expressions need to be used11.

16

Supplementary Note 2:

Theoretical solution to the exciton binding energy of the Wannier-Mott exciton

As described in the main text, theoretical solution to the exciton binding energy in RPPs requires

the knowledge of (i) the electron and hole probability density profiles, (ii) the dielectric constant

profile along the stacking direction, and (iii) the exciton reduced mass.

(i) Electron and hole probability density profiles. First, the electron and hole wavefunctions, in

RPPs, with in-plane (��,��) isotropic electronic dispersions associated with the transverse

wavevector kt, were calculated from13:

𝜓e,𝒌t(𝒓te, 𝑧e) =

𝑒−𝑖𝒌t.𝒓te

√𝐴𝑢e,𝒌t=0(𝒓te, 𝑧e), and

𝜓h,𝒌t(𝒓th, 𝑧h) =

𝑒−𝑖𝒌t.𝒓th

√𝐴𝑢h,𝒌t=0(𝒓th, 𝑧h),

(4)

where 𝑢e,𝒌t=0(𝒓te, 𝑧e) and 𝑢h,𝒌t=0(𝒓th, 𝑧h) are the ground-state, complex spinor Bloch

functions at the Brillouin zone center, for the electron and hole, respectively, and A is the

surface introduced for normalization purpose. Then the electron and hole probability density

profiles along the stacking axis, corresponding to the envelope functions in classic quantum

wells12,13, were derived by averaging in-plane the Bloch functions:

𝜌e(𝑧e) = ∬|𝑢e,𝒌t=0|2

𝑑2𝒓te and 𝜌h(𝑧h) = ∬|𝑢h,𝒌t=0|2

𝑑2𝒓th.

(5)

In practice, the spinor Bloch functions were calculated only for the highest valence band and

lowest conduction band without loss of information and negligible error on the results.

The charge density profiles (Figure 2c and Supplementary Figure 4c) were obtained from the

expressions (4) and (5) using the wavefunctions calculated by DFT including spin-orbit

coupling. This approach was necessary because methods using effective mass approximation

associated with envelope functions along the stacking axis have shown to be unsuitable for

predicting the electronic state energies and wavefunctions in RPPs16. Precisely, non-physical

superlattice effects are predicted by such methods whereas DFT results exhibit flat dispersions

along the stacking axis (Figure 2b, Supplementary Figure 4a, and refs.3,17). The fundamental

reason explaining this non-conventional behavior is that in RPPs the quantum wells (perovskite

layers) and surrounding well barriers (organic spacing layers) do not share common bulk basis

Bloch functions, unlike classic semiconductor heterostructures16. The density profiles show

that both the electron and hole have maximum probability density at the Pb-atom locations,

and that the conduction and valence band wavefunctions do not overlap perfectly. Convolution

of the electron and hole densities along the stacking axis demonstrated that the distribution of

e-h distances within the exciton ground state has slightly larger spatial expansion for the RPPs

n=3,4 than their n=1,2 counterparts. However, the electron and hole densities (𝜌𝑒(𝑧𝑒) and

17

𝜌ℎ(𝑧ℎ)) have little leakage outside the dielectric quantum well, thereby confirming that the

RPPs can be considered as almost infinite quantum wells16.

(ii) Dielectric constant profiles along the stacking direction. Second, the dielectric constant

profiles of the quantum well systems were obtained from the DFT calculations (Figure 2c,

Supplementary Figure 4b, and Supplementary Table 2) using a methodology developed and

reported previously18–20. Nevertheless, we also checked that an abrupt dielectric interface

approximation does not affect significantly the result for the exciton binding energy, as

compared to the DFT-calculated dielectric profiles or considering a more realistic trapezoidal

shape. In the latter case, the potential response to a point charge, in the linearly graded

dielectric transition region between the perovskite quantum-well and the organic spacer layer,

can be computed analytically with modified Bessel functions (In, Kn) instead of exponential

functions21:

𝑉𝑠(𝑞𝑡) = 𝑎𝐼0 (𝑞𝑡휀(𝑧)

|𝑝|) + 𝑏𝐾0 (

𝑞𝑡휀(𝑧)

|𝑝|),

where 휀(𝑧) is the linear variation of the dielectric constant, 𝑝 its slope and 𝑎, 𝑏 two constants

determined by continuity conditions.

(iii)Exciton reduced mass. Finally, the exciton reduced mass for each RPP was estimated such that

the diamagnetic shift derived from our model matches the experimental values from magneto-

absorption (Figure 1d,e). This approach is justified because it has been shown earlier that

effective masses computed for hybrid perovskites strongly depend on the accuracy of the

theoretical model used22, and that reliable calculations of effective masses can be obtained only

by a self-consistent implementation of both relativistic (SOC) and many-body effects (GW)23.

More precisely, effective masses strongly depend on the accuracy of the DFT approach,

leading to a direct relation between predicted band gaps and effective masses24. From the

computational point of view, such accuracy of theory, which has already been challenging for

small size unit cells in organic-inorganic halide perovskites (as MAPbI3), is out of reach for

crystal structures as large as the ones of RPPs. Moreover, the influence of thermal effects on

the effective masses cannot be captured by DFT.

18

Supplementary Note 3:

Examples of approximated cases of the screened electron-hole interaction potential.

For a strict screened 2D case (electron and hole densities approximated by delta functions

along the direction of layer stacking, i.e. 𝑧e = 𝑧h and 𝜌e(𝑧e) = 𝛿𝑧e and 𝜌h(𝑧h) = 𝛿𝑧h

),

including dielectric confinement effect, the e-h interaction potential reads

𝑉𝑠(𝑞𝑡) =−𝑒2

2휀𝑤𝑞𝑡

1 + 𝜒𝑒−𝑞𝑡𝑑

1 − 𝜒𝑒−𝑞𝑡𝑑

It is equivalent to an effective 2D dielectric function of the form:

휀2𝐷(𝑞𝑡) = 휀𝑤

1 − 𝜒𝑒−𝑞𝑡𝑑

1 + 𝜒𝑒−𝑞𝑡𝑑

This case leads to a strong enhancement of the e-h interaction and was analysed initially by

Keldysh14 yielding approximate expressions of the potential.

In the approximation of long range electron-hole interaction (small in-plane wavevectors) and

quantum wells with small thickness (𝑞t𝑑 ≪ 1), a linear form of the effective 2D dielectric

function is obtained

휀2𝐷(𝑞𝑡) = 휀𝑏 +(휀𝑤

2 − 휀𝑏2)

2휀𝑤𝑞𝑡𝑑,

which can be further approximated14 in the case of 휀w ≫ 휀b by

휀2𝐷(𝑞𝑡) = 휀𝑏 +휀𝑤𝑞𝑡𝑑

2 .

A similar formula can be obtained from ab initio approaches for Van der Waals

heterostructures in the screened 2D limit introducing a 2D polarizability for a monolayer in

vacuum10

휀2𝐷(𝑞𝑡) = 1 +𝛼𝑞𝑡

2

For a pure quantum confinement effect (휀𝑤 = 휀𝑏 , 𝜒 = 0), the screened e-h interaction reduces

to32:

𝑉𝑠(𝑞𝑡) =−𝑒2

2휀𝑤𝑞𝑡∬ 𝜌𝑒(𝑧𝑒)𝜌ℎ(𝑧 ℎ)𝑒−𝑞𝑡|𝑧𝑒−𝑧ℎ|𝑑𝑧𝑒𝑑𝑧ℎ.

19

For a pure quantum confinement effect (휀𝑤 = 휀𝑏), and in the strict 2D case (electron and hole

densities approximated by delta functions along the direction of layer stacking, i.e. 𝑧e = 𝑧h

and 𝜌e(𝑧e) = 𝛿𝑧e and 𝜌h(𝑧h) = 𝛿𝑧h

), the e-h interaction reads:

𝑉𝑠(𝑞𝑡) =−𝑒2

2휀𝑤𝑞𝑡

This 2D limit corresponds to a binding energy of the exciton of 4𝑅𝑦, where 𝑅𝑦 =𝜇𝑒4

2ℏ2(4𝜋𝜀𝑤)2 is

the Rydberg energy of the system.

20

Supplementary Note 4:

Comparison to exciton modelling in Van der Waals heterostructures

Detailed analyses of the exciton binding energy have been proposed recently in the literature for

Van de Waals heterostructures such as hBN and MoS2.25,26 A full ab initio solution of the BSE was

compared to a Mott-Wannier model including a pure 2D screened Coulomb interaction10 or a new

quasi-2D screened Coulomb interaction resulting from an average over step-like representations

or actual distributions of the density. It shows that screened 2D or quasi-2D approaches are both

in good agreement with a full ab initio solution of the BSE for monolayered Van der Waals

heterostructures. By comparison, full ab initio modelling of the BSE for the RPP n=1 case is not

possible with the available computational resources. We nevertheless conclude in the present study

that screened 2D approximations breaks down due the spatial extension of the density profiles.

The method used in the present work is thus more closely related to the quasi-2D approach of

Latini et al.25 where actual density profiles can be used to compute the screened electron-hole

interaction in reciprocal space. However, the effect of the dielectric mismatch is introduced in the

work of Latini et al.25 for Van der Waals heterostructures through a proper 2D transformation of

the 3D bulk dielectric, accounting for local field effects. In the present work due to the absence of

a 3D reference system, a direct computation of the dielectric profile was performed for each RPP

phase.

In Van der Waals heterostructures, the crossover from a 2D monolayer to a 3D bulk crystal can be

studied theoretically by increasing the spacing between the layers10. It allows computing the 2D

polarizability in the 3D bulk reference crystal. For the 3D compound (MAPbI3) corresponding to

the RPP phases, covalent bonds are present along the stacking axis. On the other hand, dangling

bonds are present in the RPP n=1 structure along the same axis. The MAPbI3 crystal can thus not

be regarded as a simple assembly of RPP monolayers, i.e. as a simple reference system to evaluate

dielectric or quantum confinement effects16.

21

Supplementary Note 5:

Computation of the diamagnetic shift.

The diamagnetic shift in the Faraday configuration was computed from 𝑐0 =𝑒2|𝒓𝑡𝑒−𝒓𝑡ℎ|2

8𝜇, and by

numerically averaging the distance between the in-plane (��,��) electron (𝒓𝑡𝑒) and hole (𝒓𝑡ℎ)

positions.

For information, in the case of a strictly 2D system and for only quantum confinement effects, the

analytic expression of the diamagnetic shift 𝑐0 =3

8

𝑒2𝑎B2

8𝜇 matches the numerical result (aB the

exciton Bohr radius).

22

Supplementary Note 6:

DFT computations of 2D RPPs

Electronic band dispersions and spinor Bloch functions were computed within the Density

Functional Theory (DFT)27,28 as implemented in the ABINIT package29. Experimental crystal

structures were used, namely the crystal structure recorded at 293K with Pbca space group for

n=1,30 and those reported in Ref. 3 having Cmcm space group for n = 2 and 4 and Cmca for n = 3.

Band dispersions and the spinor Bloch functions with the organic cations replaced by Cs+ were

computed using the revPBE gradient correction for exchange-correlation31 and PAW datasets32 for

Pb, I, Cs, N, C and H as pseudopotentials. The electronic wavefunctions were expanded onto a

plane-wave basis set with an energy cutoff of 517 eV. The following Monkhorst-Pack grids for

reciprocal space integration were used: 4x4x1 for n=1 and 2x2x4 for n=2, 3 and 4. Spin orbit

coupling interaction was taken into account in our model given its significant role in lead-iodide

perovskites33,34. The computational burden was reduced by replacing the organic cations (BA) by

Cs+ atoms; this method22,33,34 was shown to have minor impact on the electronic bandstructure of

RPPs. DFT results are presented in Figure 2 and Supplementary Figure 4.

High-frequency dielectric constant profiles were computed according to a methodology described

elsewhere18–20, using the DFT implementation of the SIESTA package35 with a basis set of finite-

range of numerical atomic orbitals. We used the generalized gradient approximation with PBE

functional36 to describe the exchange-correlation term. Norm-conserving Troullier-Martins

pseudopotentials were used for each atomic species to account for the core electrons37. Here, 1s1,

2s22p2, 2s22p3, 5s25p5, 5d106s26p2 were used as valence electrons for H, C, N, I, and Pb,

respectively. Polarized Double-Zeta (DZP) basis set with an energy shift of 50 meV were used for

the calculations. For the real space mesh grids, energy cutoffs of 200 Rydberg were used. Slabs

based on the RPPs were constructed and an electric field of 0.01 eV/A was applied along stacking

direction with the relaxation of the sole electron density. The Brillouin zone was sampled with

4x4x1 Monckhorst-Pack grids. To obtain the electronic density of the slabs for the dielectric

constant computations, a nanoscale averaging along the stacking axis (c-axis) was performed38.

23

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