Exciton Fine Structure in 2D Perovskites: The Out‐of‐Plane Excitonic State

2D Ruddlesden‐Popper metal‐halide perovskites feature particularly strong excitonic effects, making them a fascinating playground for studying exciton physics. A complete understanding of the properties of this quasi‐particle is crucial to fully exploit the tremendous potential of 2D perovskites (2DP) in light emission applications. Despite intense investigations, some of the exciton properties remain elusive to date, for example, the energy‐ordering of the exciton states within the so‐called fine structure manifold. Using optical spectroscopy, it demonstrates that in the archetypical 2DP (PEA)2PbI4, in contradiction to theoretical predictions, the energy of the bright out‐of‐plane exciton state is higher than that of two in‐plane states. Having elucidated the order of exciton fine structure, it determines the g‐factor of the dark exciton transition, together with the values of the electron and hole g‐factors in the direction parallel to the c‐axis of the crystal. In this way, it provides for the first time, a complete picture of the exciton fine structure in (PEA)2PbI4 2DP.


Introduction
Optical properties of low-dimensional semiconductor nanostructures are often driven by excitons -a quasi-particle composed of a photo-created electron and hole bound by the Coulomb attraction.Excitonic effects are particularly strong in 2D van der Waals semiconductors, in which simultaneous quantum and DOI: 10.1002/adom.202300877[3] A prominent example is the 2D Ruddlesden-Popper metal-halide perovskites (2DP), in which exciton binding energies reach a few hundred milli-electronvolts. [1,[4][5][6] Additionally, the soft, polar, and low symmetry lattice of metal-halide perovskites creates a unique background for electron-hole interaction [1,2,[7][8][9][10][11] providing a fascinating playground for studying the exciton properties, which challenges our understanding so far.[29][30] In metal halide perovskites, the exchange interaction between the electron and hole spins, together with the crystal field, lifts the degeneracy of the excitonic states.[33][34] Symmetry analysis of the perovskite band structure leads to four Scheme of (PEA) 2 PbI 4 crystal structure. [43]c) Scheme of the "plane" and d) "edge" configurations used in our reflectance measurements.possible exciton states; [17,31,33] one optically inactive i.e., dark state, which we refer to as D, and three optically active bright states (see Figure 1a).In the case of 2D materials, symmetry is naturally broken in the Z direction (out-of-plane), which splits off the state with the dipole moment oriented along the c-axis.We will refer to this state as the Z state.Furthermore, if the inplane symmetry of the crystal is broken, the degeneracy between the two remaining bright states is lifted.In this case, the X and Y states with dipole moments oriented in the plane of the crystal couple to linearly polarized light, and can be observed in the optical spectra. [21,22,26]26] However, even despite the large splitting within the exciton manifold in 2DP, the order of the exciton states remains controversial.[27,29,31] In particular, the position of the Z state [27] as shown schematically in Figure 1a is unclear. Initial thoretical works, based on the multiband effective mass models, together with some experiments, suggested that the exciton state with the out-of-plane dipole moment is the lowest energy state of the bright triplet.[31,[37][38][39] More recent theoretical studies indicate however that this order might be reversed, [27] and strongly depends on the quantum confinement.[24,29] A trace of the Z-state above two in-plane states was recently reported in the high magnetic field measurements in the Voigt configuration.[23] However, its energy at zero magnetic field remains to be elucidated.An experimental determination of the order of the bright exciton states is therefore essential to understand the exciton fine structure, which is crucial for light emission applications and advanced quantum devices.[19,40,41] The detailed quantification of exciton fine structure will also provide a benchmark for band structure modeling, which remains challenging in metal-halide perovskites.[42] Here, we address the problem of the bright exciton fine structure in archetypal 2DP (PEA) 2 PbI 4 .We demonstrate that the exciton with the out-of-plane dipole moment is the highest energy state within the exciton manifold. Our findins challenge the predictions of the effective mass theory while corroborating very recent ab-initio calculations, [27] and magneto-optical studies.[23] The presented results provide a firm base for further studies of the evolution of exciton fine structure splitting with quantum confinement.Moreover, our understanding of the detailed bright exciton fine structure, allows us to determine the previously unknown values of the dark and Z exciton g-factors along the c-axis, and therefore the individual g-factors of the electrons and holes.

Results and Discussion
We have investigated (PEA) 2 PbI 4 , with the crystal structure shown in Figure 1b (the optical images of an investigated crystal can be found in SI.).It consists of inorganic PbI 4 octahedral units slabs separated by phenylethylammonium chains.The planes of the metal-halide octahedra form a quantum well for carriers, while the organic spacers provide quantum and dielectric confinement.We have examined high-quality single crystals grown by two different methods, to exclude any potential influence of the sample quality on the results.The first crystal was grown using a cooling-induced crystallization method [44,45] and the second one was prepared by slow evaporation of a solvent at room temperature [22] (see methods).Our results are both qualitatively and quantitatively the same for both types of crystals.The results presented here correspond to the crystal grown by the first method.The result for the crystal grown by the second method can be found in the Supporting Information (Figures S5-S7, Supporting Information).
For (PEA) 2 PbI 4 a complete lifting of the degeneracy of the exciton states with respect to the angular momentum is expected, [21,22,26] as shown in Figure 2a.Each of the bright states Red and green curves representing the reflectance of spectra measured in the "plane" configuration, with the use of two orthogonal linear polarization of reflected light.The blue curve is the reflectance spectrum measured in the "edge" configuration with the polarization along the out-ofplane direction.The scheme indicates the orientation of the particular exciton dipole moments with respect to the sample.Inset; negative derivative of the reflectance spectrum for the "edge" configuration with polarization oriented perpendicular to the sample plane.b) Dependence of the negative derivative of the reflectance spectrum in the "edge" configuration versus polarization angle.Polarization dependence of spectra in the "plane" configuration can be found in SI, similar data have been also presented in the Ref. [22].c) Polar plot of the intensity of the Z state extracted from the negative derivative of the reflectance spectrum.(PEA) 2 PbI 4 crystal structure is shown to provide a reference for the dipole moment orientation.All measurements were done at T = 4.2 K.
couples selectively to linearly polarized light along the X, Y, or Z directions.Therefore, the simple combination of appropriate sample orientation and polarization-resolved measurements provides direct access to each of the bright exciton states.Here, we have employed polarization-resolved micro-reflectance measurements at T = 4.2 K. Samples were mounted on copper plates in two configurations: "plane" (Figure 1c), where the light was reflected from the crystal plane parallel to the planes of the inorganic sheets.In that configuration, we probe X and Y transitions.In the "edge" configuration (Figure 1d), the light was reflected from the edge of the crystal giving access to the Z-exciton state with a dipole moment oriented perpendicular to both in-plane states.As shown in Figure S1 (Supporting Information), investigated crystals were large and thick enough to easily found flat areas on the plane and edge of the crystal significantly larger than the reflected white light spot ≈2-4 μm.
The results of these measurements are summarized in the Figure 2. The red and green lines in the Figure 2a correspond to the spectrum of two orthogonal polarizations of white light reflected from the plane of the investigated crystal.The two sharp resonances around ≈ 2.353 and ≈2.355 eV are signatures of the X and Y in-plane excitonic states [22] (the direction of polarization were selected to probe selectively X or Y state).The high-energy side of the reflectance spectrum is affected by broad bands with energy spacing around 30-40 meV.][48][49] To probe the Z-state with the out-of-plane dipole orientation (see the sketch in panel (a)), we measure the reflectance spectrum of the light polarized along the c-axis of the crystal (in the "edge" configuration) which is shown as a blue curve.In this case, the spectrum is dominated by a single strong resonance, which we attribute to the Zexciton state.
Interestingly, the high-energy sidebands vanish when the light is polarized perpendicular to the sample plane.52] Assuming based on previous works that the broad sideband visible for in-plane exciton states is a result of phonon-replicas [46,47,49] we conclude that exciton-phonon coupling also exhibits strong anisotropy and vanishes for excitons with the out-of-plane dipole orientation.Further detailed analysis of the broad, high-energy part of the spectrum is beyond the scope of this work, but it certainly characterizes states with an in-plane dipole moment''.When the polarization direction is along the edge (in the "edge" configuration), probing in-plane states, the broad sidebands are recovered (as shown in Figure S3, Supporting Information see also [49,53,54] ).Unfortunately in the "edge" configuration, we cannot clearly resolve any of the in-plane states, probably due to the lower quality of the crystal edge than the surface.
To corroborate the out-of-plane orientation of the Z-state, we measured the full dependence of the reflectance spectrum versus the light polarization angle with respect to the c-axis of the crystal.The energy and oscillator strength of the transition can be more easily seen by taking the negative derivative −dR/dE of the measured reflectance as shown in the inset of Figure2a.The derivation of the reflectance resonance produces a peak at the energies of the transition, while the area under the peak is proportional to the oscillator strength of the transition. [55]As shown in Figure 2b the −dR/dE reaches a maximum when the light The photoluminescence spectrum measured in the magnetic field of 6 T (gray spectrum).The small peak visible below 2.34 eV is a hallmark of the dark exciton state. [22]The colored lines are the negative derivative of the reflectance spectra (without the magnetic field) showing the energy positions of all of the bright exciton states.b) Dependence of the energy of the excitonic state versus the magnetic field applied in the Faraday configuration.Energies of the dark excitonic states are taken after, [22] and lines are calculated according to the Equation 1. c) Summary of electron and hole g-factors for (PEA) 2 PbI 4 (squares with uncertainty) and MAPbI 3 (stars) after. [56]The intermediate values are calculated according to formula Figure 3a shows the derivative of reflectance spectra (-dR/dE) measured with the use of linearly polarized light oriented along particular transitions' dipole moment.The X and Y state spectra were measured in the "plane" configuration, while for the Z state we used the "edge" configuration.As can be seen, the Zstate is located above two in-plane states (this result is consistently observed for both sets of samples as shown in SI), around 3.4 and 1.3 meV above X and Y transitions, respectively.All three bright states are situated around 15 meV above the dark state, revealed in recent magneto-PL studies (grey photoluminescence spectrum), [22] as summarized in the scheme in Figure 3a.Presented results corroborate bright state order emerging from recent first principle calculations and some effective mass models for Lead-Iodide 2D-perovskites and nanocrystals. [27,29,30]However, the theoretical prediction seems to overestimate the value of the Z-state splitting suggesting tens of meV splitting between inplane and out-of-plane states.
In order to exclude the potential impact of the crystal termination at the edge on the energy of excitonic transition we also measure the reflectance spectrum from the plane of the crystal using a high numerical aperture objective (NA=0.82),which provided access to optical transition with the out-of-plane orientation.As it is shown in Figure S4 (Supporting Information) we observed a new feature (with respect to the case with low aperture objective (NA=0.55))1.3-1.5 meV above the highest in-plane states corroborating conclusions derived from "edge" measurements.
Knowing the energy of the Z state allows us not only to complete the picture of excitonic states but also to determine the electron and hole g-factors.[34] The energy shift of the transitions, induced by the magnetic field gives a direct handle to the g-factors, which are described by the following formula (for two pairs of states): [33,34] where B is the magnetic field induction, μ B is the Bohr magneton, g B(D) is the bright (dark) exciton g-factor along the c-axis of the crystal being the sum (difference) of electron and hole g-factors (g-factors along the c axis of the crystal): While the g-factor for the bright in-plane exciton transitions was previously measured to be g B = 1.2 ± 0.1 with the use of the high magnetic field, [6] the value of g-factor of the dark exciton in the direction parallel to the c-axis has been missing for (PEA) 2 PbI 4 .The red-shift of the dark exciton states has been reported previously, [22] however, it is evident from Equation 1that an unambiguous determination of the g-factors of the D and Z states requires the knowledge of the energy difference between these states.The fitting of dark exciton red-shift (taken after [22] ) with Equation 1 is presented as a black line in panel (b) of Figure 3. Taking E Z − E D = 16.9 meV, we obtain g D = 2.7 ± 0.2.Table 1.Summary of exciton states splitting E (with respect to dark state) and the values of the g-factors in the direction parallel (g ∥ ) and perpendicular (g ⊥ ) to the c-axis of the crystal.
16.9 2.7 ± 0.2 4.0 ± 0.3 [23] X Y 15.6 1.2 ± 0.1 [6] , 1.6 [57] 1.9 ± 0.5 [23] X X 13.5 1.2 ± 0.1 [6] , 1.6 [57] 1.9 ± 0.5 [23] X D 0 2 .7 ± 0.2 4.0 ± 0.3 [23] e -1.95 ± 0.3, 2.05 [57] 2.9 ± 0.6 [23] , 2.45 [57] h -−0.75 ± 0.3, −0.45 [57] −1.1 ± 0.6 [23] The dashed curves represent the evolution of the bright states in the magnetic field with previously determined parameters. [6,22]ased on g D = 2.7 and g B = 1.2, [6] we estimate the values of independent electron and hole g-factor parallel to c-axis to be g e∥ = 1.95 ± 0.3 and g h∥ = −0.75 ± 0.3.The determined electron g e∥ = 1.95 is in very good agreement with that obtained from recent Kerr rotation measurements. [57]t is interesting to compare the g-factors of carriers in (PEA) 2 PbI 4 with the bulk ancestor, MAPbI 3 , as such a comparison reveals a deviation from the predictions of effective mass models for the g-factors of (PEA) 2 PbI 4 .Panel (c) in Figure 3 shows combined results of Faraday (Θ = 0°) and Voigt (Θ = 90°) measurements for both compounds. [23,56]Based on an effective mass model, we expect that when the bandgap increases, the electron gfactor should decrease, while the hole g-factor should increase. [56]s can be seen in Figure 3c, these predictions are valid only for the case of g ∥ for the electron.We also notice that the evolution of electron g-factors with Θ are opposite for (PEA) 2 PbI 4 and MAPbI 4 .All those facts indicate that quantum confinement together with octahedral distortion imposed by organic spacers [4][5][6] plays an important role in determining the g-factors.Therefore, conclusions derived for bulk perovskites have to be applied with particular caution in 2DP.For convenience, we summarize the exciton fine structure together with the extracted and previously reported g-factors for (PEA) 2 PbI 4 in the Table 1.

Conclusion
We have provided a full description of the bright exciton fine structure in (PEA) 2 PbI 4 .Using polarization-resolved reflectance measurements from the edge of quantum wells, we have obtained exclusive access to the Z-state with its dipole moment oriented out of the plane of the 2D crystal.We demonstrate that the Z state is energetically located above the two bright in-plane states.Our results contradict the general predictions of effective mass theory [31,37] while corroborating the results of very recent ab-initio calculations. [27,29]Knowing the position of the Z-state, we have for the first time, reported the values of the g-factors of the Z-state and dark excitons (along the c-axis direction).This allows us to determine the individual g-factors of electrons and holes, providing a firm benchmark for band structure models.

Experimental Section
Synthesis and Sample Preparation: Synthesis of two samples was performed using two different methods; i) a cooling-induced crystalliza-tion, described in detail in [44, 45] and ii) a the slow evaporation of a solvent. [22]ptical Measurements: For the reflectance measurements, the samples were mounted in the cold finger He flow optical cryostat.Measurements in the "edge" configuration were done using an additional copper plate with a vertical surface (cube).All these measurements were performed at 4.2 K.For reflectance measurements, the white light was provided by a Tungsten halogen light source.For the excitation and signal collection, it used a long working distance 50× magnification microscope objective with numerical aperture NA=0.55.The signal was dispersed with a 500 mm long monochromator, equipped with a grating of 1200 grooves per mm, and detected using a liquid nitrogen-cooled CCD camera.Polarization optics were mounted in the excitation path of this setup.

Figure 1 .
Figure 1.a) Ladder of exciton fine structure states expected for (PEA) 2 PbI 4 showing the possible positions of the Z-state (indicated by dashed lines).b)Scheme of (PEA) 2 PbI 4 crystal structure.[43]c) Scheme of the "plane" and d) "edge" configurations used in our reflectance measurements.

Figure 2 .
Figure2.a) Red and green curves representing the reflectance of spectra measured in the "plane" configuration, with the use of two orthogonal linear polarization of reflected light.The blue curve is the reflectance spectrum measured in the "edge" configuration with the polarization along the out-ofplane direction.The scheme indicates the orientation of the particular exciton dipole moments with respect to the sample.Inset; negative derivative of the reflectance spectrum for the "edge" configuration with polarization oriented perpendicular to the sample plane.b) Dependence of the negative derivative of the reflectance spectrum in the "edge" configuration versus polarization angle.Polarization dependence of spectra in the "plane" configuration can be found in SI, similar data have been also presented in the Ref.[22].c) Polar plot of the intensity of the Z state extracted from the negative derivative of the reflectance spectrum.(PEA) 2 PbI 4 crystal structure is shown to provide a reference for the dipole moment orientation.All measurements were done at T = 4.2 K.

Figure 3 .
Figure3.a) The photoluminescence spectrum measured in the magnetic field of 6 T (gray spectrum).The small peak visible below 2.34 eV is a hallmark of the dark exciton state.[22]The colored lines are the negative derivative of the reflectance spectra (without the magnetic field) showing the energy positions of all of the bright exciton states.b) Dependence of the energy of the excitonic state versus the magnetic field applied in the Faraday configuration.Energies of the dark excitonic states are taken after,[22] and lines are calculated according to the Equation 1. c) Summary of electron and hole g-factors for (PEA) 2 PbI 4 (squares with uncertainty) and MAPbI 3 (stars) after.[56]The intermediate values are calculated according to formula

√ g 2
e(h)∥ cos 2 () + g 2 e(h)⟂ sin 2 ().All measurements were done in T = 4.2 K.polarization is along the direction of c-axis and the signal attributed to the Z-state vanishes when light is polarized along the crystal edge.The same conclusions can be drawn from the polarization dependence of the transition intensity summarized in the form of the polar plot in Figure2c.The clearly visible double lobe shape, oriented along the c-axis of the crystal, is characteristic for a linearly polarized transition.Having identified all three bright excitonic transitions, we can now elucidate the order of bright exciton states in (PEA) 2 PbI 4 .