Coexisting and cooperating light–matter interaction regimes in a polaritonic photovoltaic system

Common quantum frameworks of light–matter coupling demonstrate the interaction between an atom and a cavity occurring through a single feedback channel: an exciton relaxes by emitting a photon that is stored in the cavity for several roundtrips before being re-absorbed to create another exciton, and so on. However, the possibility for the excited system to relax through two different channels belonging to two different regimes has been, until now, neglected. Here, we investigate the case in which the strong coupling regime and the photovoltaic effect cooperate to enhance the wavelength-dependent photocurrent conversion efficiency (defined as the incident photons to converted electrons ratio, namely the external quantum efficiency—EQE) of a photovoltaic cell specifically engineered to behave as an optical cavity tuned to the excitonic transition of the embedded active material (CH3NH3PbI3 perovskite). We exploit the angular dispersion of such photovoltaic cell to show that when the cavity mode approaches the energy of the exciton, the strong coupling regime is achieved and the EQE is significantly enhanced with respect to a classic configuration serving as a benchmark. Our findings do not aim at demonstrating an immediate impact in enhancing the performance of photovoltaic systems but, rather, constitute a proof-of-principle experimental demonstration of how the photovoltaic effect can benefit from the generation of polaritons. Nonetheless, such a peculiar cooperating dual-light–matter interaction could be exploited in future polaritonic photovoltaic architectures.


Introduction
The interaction between excitons and the electromagnetic environment in their proximity can be weak or strong [1,2]. In the so-called 'strong coupling regime', a new quasiparticle called 'polariton' is generated whose nature shares features of both light and matter [3][4][5][6][7]. The insurgence of polaritons can substantially improve the performances of chargetransport related phenomena [8,9]. When a nanometrically delocalized exciton strongly couples with a macroscopically delocalized cavity mode photon, a polariton arises whose mixed-nature wavefunction results coherently delocalized on the length scale of the size of the cavity [10]. Feist and Garcia-Vidal demonstrated that exciton polaritons manifest much longer diffusion lengths than excitons [11]. DelPo et al demonstrated that charge transfer from polaritons can be so fast to compete with the polariton relaxation time, experimentally justifying the use of polaritons for charge-transferrelated applications [12], while Wang et al demonstrated that the rate of polariton-based energy transfer processes is largely increased with respect to the case of simple excitons [13]. In 2022, Zeb et al demonstrated that the so-called incoherent charge hopping process rates occurring in organic materials where Frenkel excitons are mostly present can be largely improved by the presence of polaritons [14]. The same year, leveraging on the aforementioned strongly delocalized character of polaritons, Liu et al proposed the introduction of a polariton-based 'antenna complex' in a system that mimics the natural photosynthesis processes for charge transfer [15]. On a theoretical side, the case for perovskites, where mostly Wannier excitons are expected to be present, is not clear yet but some recent works suggested that even in semiconductors a similar beneficial effect can be obtained. For example, Chevry et al demonstrated that external electric fields could accelerate polaritons by forming new quasiparticles termed polaronpolaritons [16]. It has also been shown that the transport phenomena associated to excitons can be substantially improved by the occurrence of polaritons since, in this last framework, transport is sustained by delocalized polariton modes rather than by tunneling processes [17]. Exceptional propagation lengths of hundreds of microns were demonstrated for polaritons generated in an organic material through the hybridization of Bloch waves and excitons [18].
It seems thus reasonable to expect that photovoltaics could largely benefit from the adoption of a polaritonic environment. For example, it has been recently shown that in photovoltaic systems based on organic polaritons, losses can be substantially lower than in classic exciton-based organic photovoltaic cells [19].
In this paper, we showcase a 'Metal-Insulator-Metal photovoltaic cell' (MIM-cell), that is a system in which strong coupling regime and photovoltaic effect coexist and cooperate to enhance the external quantum efficiency (EQE) of a perovskite-based photovoltaic cell. In the proposed system, the photovoltaic effect involves polaritons, leading to a twofold enhancement of EQE. A possible explanation of the beneficial mechanism lying at the basis of the EQE enhancement due to polaritons could be searched in the enhanced coherent delocalization of the polaritons with respect to excitons. As reported by Schachenmayer et al such a feature shifts the charge transfer from tunneling-based channels, typical of strongly localized excitons, to largely delocalized polaritonic modes that are far less sensitive to defects and associated non-radiative relaxation channels [17]. Our data are far from depicting a technologically mature scenario, but they reveal that this light-matter interaction occurring through the unconventional path of two cooperating regimes coexisting within the same system can be observed also using perovskites.

Results and discussion
The architecture of the MIM-cell is made of a photovoltaic cell embedded within a resonator consisting of a metal/insulator/metal (MIM) cavity. Here we lend the suffix 'meta' and use it with slightly loose hand, to underline that the simple addition of a metallic layer to a classic photovoltaic architecture leads to a resonant response with remarkably different photovoltaic properties. As shown in figure 1, the MIM-cell is fabricated over a glass substrate by depositing different layers.
The proposed MIM-cell was fabricated by stacking from the bottom to the top: Au/ITO/PEDOT:PSS/CH 3 NH 3 PbI 3 perovskite/PC 61 BM/Bathocuproine/Ag. Here, Au and Ag play the role of both the metallic mirrors of the resonator and the photovoltaic cell electrodes. The perovskite represents the active medium that, together with all the other dielectric transport layers, constitutes also the compound dielectric core of the MIM cavity. As such, the MIM-cell is endowed with an angular dispersion stemming from its MIM configuration so that, upon moving the angle of incidence of light radiation away from normal incidence, the wavelength of the cavity mode blue-shifts. Here, it is important to underline that the presence of a top Ag layer, which is usually supposed to be both highly reflective and dissipative, does not prevent the excitation of cavity modes. The peculiar complex refractive index of Ag, indeed, makes it a so-called Hermitian metal, meaning that the imaginary part of its refractive index is far larger than the real one. As a result, photons with frequency within the visible range can tunnel through the Ag layer under the shape of a mostly purely evanescent wave that is not and should not be confused with a surface plasmon [20].
To optimize the photocurrent performances of a photovoltaic system, thick gain material layers are commonly used [21]. In this case, however, due to the high optical density of the perovskite and to its continuous absorption band, it is very difficult to observe the signatures of the strong coupling regime, like mode anti-crossing and Rabi splitting (2 Ω), in scattering parameters like transmittance, absorbance and reflectance [22]. Such features result far more visible if a thin layer of perovskite is used instead of a thick one. Therefore, with the aim of demonstrating the capability of the architecture in figure 1 to sustain strong coupling and polaritons, we carried out scattering matrix method (SMM) simulations by considering a 40 nm thick perovskite layer, thinner than the one we experimentally use. To tune the resonance of the MIM-cell to the wavelength of the exciton, the thickness of the ITO layer is fixed to 250 nm. The simulated structure, made of Ag (20 nm)/PCBM + BCP (50 nm in total), CH 3 NH 3 PbI 3 (40 nm), PEDOT:PSS (50 nm), ITO (250 nm), Au (100 nm) is sketched in figure 2(a). We highlight that the thickness of both the perovskite and the ITO layer used for the simulation are different from the experimental ones. Figures 2(b) and (c) show the simulated absorbance (1 -Reflectance − Transmittance) maps of such MIMcell for p-and s-polarization, respectively, as a function of light incidence angle θ (defined in terms of its deviation from normal incidence). The refractive index of all the involved materials was experimentally measured through spectroscopic ellipsometry (see supporting information). Such an investigation revealed two excitonic transitions, highlighted with dashed white lines in figures 2(b) and (c) and labeled as 'HE-Exciton' and 'LE-Exciton' in figure 2(c). By increasing the incidence angle, the unperturbed cavity mode, occurring for both polarizations at about 830 nm, blue shifts approaching the LE-Exciton wavelength. Such a behavior is expected for a MIM cavity, that follows the well-known Fabry-Pérot . Here, θ i is the angle of incidence, while considering that the refractive index of both the perovskite and the ITO layers are far larger than those of PEDOT:PSS or PCBM + BCP, the quantity n eff can be simply calculated as the thickness weighted average refractive index between ITO and perovskite, n eff = (n ITO ·t ITO + n PSK ·t PSK )/ 2. The quantity t D is the total thickness of the dielectric layer inside the cell, calculated by summing all the thicknesses reported in figure 2(a), so that t D = 390 nm. The quantity λ 0 is the resonant wavelength of the cavity at normal incidence, that can be calculated by leveraging on the analogy between quantum mechanics and classic electromagnetism and, in particular, on the isomorphism between the Helmholtz and Schrödinger equations [20]. For our MIM-cell, the quantity λ 0 , as well as all the additional even harmonics can be calculated by geometrically resolving the following equation: tan(2π/λ·n eff ·t D / 2) = κ M /n eff , where κ M is the imaginary part of the refractive index of Ag. By including all these parameters, the geometric resolution of the previous relation found as the wavelengths where the crossing between the left and the right part of the equation occurs, are λ 1-even = 840 nm, λ 2-even = ∼600 nm and λ 3-even = ∼470 nm. These modes correspond to the first three even harmonics sustained by the MIM-cell. The odd harmonics can be calculated through the related formula-cot(2π/λ·n eff ·t D / 2) = κ M /n eff that provides as solutions λ 1-odd = ∼690 nm and λ 2-odd = ∼530 nm, that correspond to the first two odd harmonics sustained by the cavity. Despite these are approximated wavelengths, such an analysis clarifies the nature of the involved cavity resonances and highlights the presence of additional harmonics lying within the interband transition absorbance region of the perovskite.
A first qualitative investigation of the absorbance maps shown in figures 2(b) and (c) reveals a significant deviation of the absorbance maxima from the original energy of both the two excitonic transitions (LE and HE), evidencing a strong perturbation of the pristine system. We associate this behavior to the occurrence of strong coupling. Since two excitons and one cavity mode are involved, the coupling dynamics is that of three strongly coupled oscillators, with the consequent generation of a low-energy polariton (LP), a middle-energy polariton (MP) and a high-energy polariton (HP). The spectral dispersion for all these three polaritonic branches follows the trend indicated by the colored dots (blue, green and red dots referred to HP, MP and LP, respectively) in figures 2(b) and (c), for both p-and s-polarizations.
To gain insight on the nature of the detected polariton we calculated the Hopfield coefficients for each of them [23][24][25][26][27][28]. Results are shown in figures 2(d) and (e) for the P-and Spolarization, respectively. For both polarizations very similar considerations can be drawn: at small angles, the LP holds mostly a photonic nature and gradually increases its mixing with LE with the angle. On the contrary, the HP holds mainly and excitonic character at small angles and gradually acquires a photonic flavor while increasing the angle of incidence. The MP, in the end, holds mainly an excitonic character and, therefore results lossy and less visible than the others.
To better identify the occurrence of the strong coupling regime in the simulated thin-perovskite system, we fitted the absorbance curves at ten relevant angles (from 0 • to 80 • with 10 • steps) with a tri-Gaussian function: (1) Here, A 0 , λ 0 and σ are the amplitude, central wavelength and standard deviation of the ith oscillator, respectively. To each Gaussian oscillator, a specific mode is associated. In particular, the longer-, middle-and shorter-wavelength oscillator can be related to the low-, middle-and high-energy polaritonic branch labeled as 'LP', 'MP' and 'HP', respectively. The central wavelengths λ 0 calculated through equation (1) can be readily overlapped to both the absorbance maps (see figures 2(b) and (c)) and to the extrapolated absorbance curves, as shown in figures 2(b) and (e). The resulting graph allows to precisely follow the dispersion of the three relevant quantities, HP, MP and LP.
To confirm the achievement of the strong coupling, we analytically modeled the entire system through the interaction Hamiltonian reported in equation (2): Here E CAV , E LE and E HE are the energies of the cavity mode, of the LE-and of the HE-Exciton. The terms g HE-CAV = Ω HE-CAV , g LE-CAV = Ω LE-CAV and g HE-LE are the cavity/HE-Exciton, cavity/LE-Exciton and HE-Exciton/LE-Exciton coupling terms, respectively. Since there is no interaction between the excitons, in our case the term g HE-LE is equal to 0.
The occurrence of the strong coupling between the lowenergy excitonic transition and the cavity mode was evaluated by examining the absorbance curves at the anti-crossing angles θ = 40 • for the p-polarization (figure 2(f)) and θ = 35 • for the s-polarization ( figure 2(g)). The unperturbed cavity dispersion (dashed black line in figures 2(b) and (c)) has been numerically calculated (see supporting information). For both cases, the Rabi Splitting was calculated by deconvoluting the absorbance curves with three Gaussian curves representing the LP, MP and the HP (see figures 2(f) and (g)). Hereafter, we report only the relevant cases of the coupling between the cavity mode and the LE-Exciton. For the p-polarization case, a Rabi Splitting 2 Ω LE-CAV = 82 meV was calculated (see figure 2(f)) while, for the s-polarization, 2 Ω LE-CAV = 100 meV. To assess the occurrence of the strong coupling, the quantity 2 Ω LE-CAV was compared to the linewidth of the cavity mode and of the excitonic transition of the perovskite. In this case, they are equal to ∆E CAV = 41 meV and ∆E EXC ∼ 40 meV, respectively. The full width at half-maximum of the cavity mode was derived as a parameter from equation (1) for the absorbance curve at normal incidence (0 • ), far from the anti-crossing point. At θ = 0 • , indeed, the energy of the LP mode is very far from that of the excitonic transition, ensuring that, at this angle, the mode still holds a marked 'cavity' character, a consideration that is also confirmed by the Hopfield coefficient shown in figure 2(d). The exciton linewidth, on the other hand, was calculated directly from the ellipsometrical measurements (imaginary part of the refractive index of a deposited 40 nm perovskite film). For the p-polarization, Ω LE-CAV ∼ ∆E CAV ∼ ∆E EXC so that the system is barely in the strong coupling regime. On the contrary, for the s-polarization Ω LE-CAV > ∆E CAV and Ω LE-CAV > ∆E EXC , confirming the achievement of the strong coupling condition. In the end, we note that the nature of the polaritonic mode can affect its spectral linewidth. As a consequence, for example, at small angles the LP holds a strong cavity-like behavior (see figure 2(d)) and inherits the Q-factor of the cavity mode from which it is generated. On the contrary, the highly lossy excitonic features of the LP at large angles, confirmed by the increasing mixing with HE as calculated in the Hopfield coefficient of figure 2(d), broaden its linewidth.
As already mentioned, however, a thin active layer is rarely used in experimental frameworks. A thickness of the active layer close to 40 nm would be of the same order as roughness, creating pinholes and increasing shunt currents. In addition, the absorption of the active layer would be very low, inducing low photogenerated currents. Both effects would prevent performance measurements with reasonable accuracy. Nonetheless, the simulations described above can guide the interpretation of experimental results obtained using thicker perovskite layers. Therefore, MIM-cells with the architecture shown in figure 1 were fabricated, with a thickness of the perovskite layer of about 180 nm. To keep the cavity mode and the exciton in tune with each other, the thickness of the ITO layer was decreased to 25 nm. The thickness of the other layers was: Au (100 nm), PEDOT:PSS (50 nm), phenyl-C 61 -butyric acid methyl ester (40 nm), bathocuproine (10 nm), Ag (20 nm). The thickness of the bottom layer ensures optimal conduction properties and suitable quality factor of the cavity mode, while for the top metal, a thickness equal to 20 nm is the best compromise between homogeneity, light tunneling to the active material and an efficient cavity response.
The reflectance of a typical MIM-cell of this kind, measured by an ellipsometric setup is shown in figures 3(a) and (b) as a function of the incidence angle. At θ = 20 • , a cavity mode around 850 nm is generated for both s-and p-polarizations. By increasing the incidence angle, the resonance blue-shifts for both polarizations. Due to the highly lossy nature of the MIM-cell and the high refractive index of the embedded perovskite, the angular dispersion observed for p-polarization does not significantly diverge from what observed in the spolarization case, as would be expected in a standard MIM cavity configuration. Indeed, by recalling the aforementioned relation for the angular blue shift of the cavity mode it turns out that higher values of the real part of the effective refractive index related to the cavity dielectric core (n eff ) induce a smaller angular dispersion [20,29,30]. Exploiting the cavity angular dispersion constitutes also a wise way to sweep the cavity wavevector across the exciton of the active material to investigate the mode anti-crossing typical of the strong coupling regime without losing the optimized thickness of the active layer for the photovoltaic performances.
As we increase the incidence angle, the wavelength of the cavity mode moves towards the optical transition of the perovskite. At an incidence angle of ∼60 • , the resonant wavelength of the hypothetically unperturbed cavity would reach the highest wavelength perovskite absorption. However, a modal splitting is observed, so that a pronounced dip in reflection is found for both p-and s-polarization at about 800 nm (figure 3(e)), and a second, less pronounced dip, is visible at about 730 nm. No reflectance dip is found at the exciton absorption wavelength, at about 758 nm (black curve of figure 3(e)). This is the signature of the occurrence of strong coupling between the cavity mode and the exciton. To confirm this aspect, we evaluated the Rabi splitting 2 Ω as the energy distance between HP and LP in the p-polarization curve and compared it with the energy linewidth of both the bare cavity mode and the excitonic transition of the perovskite. The energy linewidth of the bare cavity has been calculated by fitting with a Gaussian curve the reflectance dip occurring at about 850 nm in the p-polarization curve at an incidence angle of 30 • , extrapolated from the reflectance map of figure 3(b), far from the strong coupling condition, to ensure that the mode holds a strong Fabry-Pérot character. The energy linewidth of the excitonic transition of the perovskite has been calculated directly from the imaginary part of its refractive index. The resulting values are: 2Ω = 140 meV, ∆E CAV = 68 meV and ∆E EXC = 48 meV. Since the condition Ω > ∆E CAV > ∆E EXC is verified, we can confirm the achievement of the strong coupling regime. We can, therefore, associate the former dip at 800 nm to the LP branch and the latter at 730 nm to the HP branch. Moreover, an additional dip in reflectance is detected at ∼575 nm, associated to a higher harmonic cavity mode.
To study how the photovoltaic properties of the MIM-cell are affected by the achievement of the strong coupling regime, angle-resolved EQE measurements were carried out and compared with the same measurements obtained on a benchmark standard photovoltaic cell, i.e. a cell in which the only difference is the absence of the bottom Au layer (see figure 1).

Results are illustrated in figures 4(a)-(d).
A monotonic increase of EQE upon progressively slanting light illumination, from normal incidence (θ = 0 • , black curve) up to θ = 60 • (orange curve) can be observed in figure 4(a) in the spectral region where polariton dispersion occurs, between 650 nm and 800 nm, where a doubling of the EQE values is recorded. In comparison, the EQE of a standard photovoltaic cell identical to the MIM-cell, but without the bottom Au layer, shows little dependence on light incidence angle, with just a slight EQE increase throughout most of the spectral range, as illustrated in figure 4(b). Given the relatively low light extinction achievable with a perovskite thickness of 180 nm, it seems reasonable to attribute this slight increase to the variation of the optical path when θ is increased.
However, in the MIM-cell an increase of EQE of similar entity was also detected in the 500-640 nm range, peaked around 570 nm. This is particularly evident if the normalized EQE values of the standard cell and of the MIM-cell at the same incidence angle are compared, as shown in figures 4(c) and (d) for two incidence angles. It is evident how the EQE increase observed in both the spectral areas centered at 570 nm and at 750 nm is present even for normal incidence. In addition, the high-wavelength EQE improvement becomes quantitatively more important for higher incidence angles.
The refractive index of a single perovskite film was measured by ellipsometry, in order to perform a comparison with the EQE behavior. The measurements, together with the associated Gaussian fit shown in section S6 of the supporting information, allowed us to identify the exact spectral position of the excitons (see figure 4(e)). Among them, the one centered at 782 nm corresponds to the lowest energy perovskite exciton, that can be associated to the direct band-gap transition. It is however true that, as can be evinced from figure 4(e), many other optical transitions are present in the absorption band of the perovskite, to each of which the formation of an exciton can be associated, as in the case for the ones peaked at 484 nm and 590 nm respectively, in figure 4(e). When the excitons related to these optical transitions are brought to interact with the high-energy cavity mode, strong coupling occurs as well. Therefore, the EQE enhancement occurring in the spectral region around 500-640 nm can be associated to the occurrence of polaritons as well. Even though the mechanism behind this effect has not been univocally clarified by theoretical studies, several possibilities were proposed. In fact, strong coupling was already investigated in photovoltaic systems based on organic materials, in which excitons are localized and their diffusion length is typically short. A reduced driving force for charge transfer, as well as a reduction of optical losses, were invoked by Nikolis et al to explain the improved EQE recorded in optical cavities [19]. Elsewhere the improved EQE recorded in optical cavities is attributed to an increased exciton diffusion length due to polaritons [12,13]. Such a polariton-mediated excitation transport, allows a higher percentage of excitation to reach heterojunctions. If the charge transfer from polaritons at the heterojunction is fast enough to compete with polariton decay, the net effect is an improved energy harvesting.
In perovskites like the ones used in this work, excitons are typically more delocalized and their diffusion length much longer than in organics. Therefore, the effect of strong coupling could be expected to be mitigated in perovskite-based MIMs with respect to what observed in their organic counterparts. However, our results show that a relevant effect is also present when perovskites are used and only further studied can clarify whether this is due to similar or different mechanisms.

Conclusions
In conclusion, the experimental observations regarding both the photovoltaic effect and the scattering parameters indicate that two different and cooperating light-matter interaction regimes take place in the proposed system, consisting of a particular kind of photovoltaic cell in which the external electrodes constitute also the mirrors of a MIM resonant cavity. We called such a system 'MIM-cell'. In such a system, the strong-coupling between cavity modes and photo-generated excitons in the active material (CH 3 NH 3 PbI 3 perovskite) gives rise to cavity/exciton polaritons which determine an enhancement of the EQE of the MIM-cell of ∼100% in large areas of the absorption band. In particular, the enhancement of the EQE respects the angular polaritonic dispersion of the system so that, while increasing the angle of incidence of light radiation, the MIM-cell accesses the strong coupling regime, with a consequent gradual enhancement of EQE in the spectral ranges where the cavity acts as a resonator. Our findings, far from showcasing a new photovoltaic cell architecture, represent a remarkable example of coexisting and cooperating lightmatter interaction regimes in a perovskite system, a condition to which a rich and partially unexplored phenomenology can be associated and possibly exploited. Since the strong coupling regime is as more effective as much Ω is larger than both ∆E CAV and ∆E EXC, it is reasonable to expect that minimizing these two quantities, for example by increasing the cavity Qfactor, could potentially increase the described performances.

Chemicals
Poly (3,4- For the MIM-cell, a gold mirror layer of 80 nm was evaporated on the central part of a clean, indium-doped tin oxide (ITO)-coated glass, with a size of 15 × 25 mm 2 . Then a thin layer of ITO was sputtered on the top of the gold layer. In the case of the standard photovoltaic cell, ITO-coated glass was used. The subsequent layer configuration was the same for both kinds of cells.

PEDOT:PSS layer deposition
The PEDOT:PSS colloid was filtered by a 0.22 mm pore PVDF filter and spin coated at 3000 rpm for 60 s on the substrates, followed by annealing at 140 • C for 15 min in air.

MAPbI 3 layer.
The perovskite active layer was deposited onto the above-prepared PEDOT:PSS layer by spincoating at 5000 rpm for 60 s the precursor solution in a N 2 filled glovebox. Immediately after the spin-coating, the prepared film was heated at 100 • C for 30 s.

HTL layers.
The [60] PCBM film was spin-coated onto the MAPbI 3 at 2000 rpm for 60 s. This was followed by the spin-coating of BCP at 6000 rpm for 20 s.
Finally, a very thin silver film, about 20 nm thick, was deposited by thermal evaporation in a high vacuum system (Kurt J Lesker) with a base pressure of 10 −6 mbar.

Angle-resolved EQE
The light of a xenon lamp (300 W) was coupled into a monochromator (Newport 74100) and the resulting monochromatic light was focused onto the MIM or PV cell. The sample was mounted over a rotating holder and the EQE was measured at different angles. The current of the cell at short-circuit conditions was measured by using a source meter (Keithley 2400) between 350 nm and 850 nm, with steps of 5 nm. The EQE was determined by dividing the photocurrent density of the cell by the flux of incoming photons, which was obtained by using a calibrated silicon photodiode Hamamatsu mod. S2386-44 K.

Ellipsometry
The optical constants and thickness of each layer, as well as those of the MIM-cells, were obtained by spectroscopic ellipsometry, carried out by using an M2000 ellipsometer (Woollam). Reflectance (R) and transmittance (T) spectra were ellipsometrically measured. Such an approach gives precise results for multilayers of smooth films.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Acknowledgments
The present work was partially financed by 'Progetto