Engineering a light–matter strong coupling regime in perovskite-based plasmonic metasurface: quasi-bound state in the continuum and exceptional points

1. INTRODUCTION

Polaritons are eigenstates formed by strong interactions between the excitations of two oscillators: electromagnetic waves and matter. The energy exchange rate between two oscillators is so rapid that it exceeds the decoherent processes of each oscillator, resulting in the creation of polaritons, which is evidenced by an avoided crossing in the energy-momentum curve. These quantum states, that are mixed light–matter quasi-particles, have attracted considerable interest in photonic research. Besides the traditional exciton–photon polaritons [1], the emergence of modern plasmonics has added new categories of polaritons such as exciton-plasmon polaritons [2,3] and exciton-surface plasmon polaritons (ESPPs) [4–8]. The ESPP is a special case when a strong coupling between the excitons of quantum emitters and surface plasmon polaritons (SPPs) occurs. The ESPP consists of electronic energy levels that are distinctive from the emitters and the SPP individually. Despite the original differences on the light side, they all possess a bosonic nature and inherit the best features of their components. Specifically, the photonic/surface plasmon part offers ballistic propagation over macroscopic distances [9,10], as well as optical means to generate and probe polaritons [11]. In parallel, the exciton in the matter plays the role of the active component, not only for light emission but also for nonlinearity interaction, which is orders of magnitude stronger than the pure photonic Kerr effect [12,13]. These special traits make polaritons a fascinating platform to study out-of-equilibrium Bose–Einstein condensation [14,15], quantum fluid of light [16,17], and topological photonics [18,19]. They also enable various applications with novel devices for lasing [20], integrated photonics [21,22], photovoltaics [23], sensing [24], photocatalysis, and chemical kinetics [25,26].

Among active materials for polaritonic devices operating at room temperature, halide perovskites—a type of semiconductor containing organic molecules, lead, and group VII element such as Cl, Br, and I—have recently emerged as an ideal candidate thanks to their excellent properties and low-cost fabrication. Unlike excitons of other room temperature active materials, perovskite excitons manifest strong nonlinear behaviors of Wannier–Mott type [27,28] while exhibiting high binding energy up to a few hundred million electron volts (meV) as strong as that of Frenkel excitons [29–31]. This makes them robust excitons at room temperature. Up to now, making use of the perovskite platform, different groups have successfully demonstrated polariton Bose–Einstein condensation (i.e., polariton lasing) [32–39], laser cooling [40], ballistic propagation [33], ESPPs [6,41], polaritonic microstructures [33,42–44], and lattices [36,45–47]. However, the polaritonic devices reported in these works are optically pumped. In order to realize practical devices, metallic materials are usually introduced to inject them electrically. The use of metals will induce significant challenges, for example, a decreased quality factor of the polariton modes due to unavoidable absorptive losses of the metal. Beside the electrical properties of the metal film, we will exploit its surface plasmons coupled with excitons of perovskite to realize the formation of ESPPs, which have also been observed in several works [6,41]. Lately, it has been demonstrated that perovskite-based metasurfaces of a subwavelength periodic lattice offer an on-demand energy-momentum polaritonic dispersion [47]. While this work only discusses the real part (i.e., energy) of polaritonic dispersion, textured metasurfaces can offer novel strategies to overcome the challenges by engineering the imaginary part (i.e., losses) via concepts such as bound states in the continuum (BIC) [48,49] or exceptional points (EPs) [50–53]. As a result, the metasurface approach would be a promising configuration to minimize the effect of the metallic layer when designing electrically injected polaritonic devices.

In this work, we investigate theoretically the formation of exciton–photon polaritons and ESPPs in a perovskite-based lattice of a subwavelength period on a metallic substrate. It is shown that, depending on the perovskite layer, two different regimes are obtained: solely ESPPs for ultra-thin metasurfaces, and the co-existence of exciton–photon polaritons and ESPPs for metasurfaces of wavelength-scale thickness. In both cases, the lower polariton branches (LPBs) consist of dark and bright modes corresponding to infinite and finite radiative quality factors, respectively. Another salient property of the system is that it allows one to obtain EPs with an enhancement of local density of states (DOS) by simple tuning of geometrical parameters of the perovskite film. In particular, the configuration sustains exciton–photon polaritons, which can be designed in such a way that the non-radiative losses of the dark mode are negligible, and this state becomes a quasi-BIC of high quality factor. Our findings show that the perovskite metasurface is an attractive and rich platform to make polaritonic devices, even with the presence of a metallic layer.

2. SURFACE PLASMON POLARITON AND DIELECTRIC GUIDED MODE OF PEROVSKITE THIN FILM ON METALLIC SUBSTRATE

We first investigate different guided modes formed within a perovskite thin film deposited on a metallic substrate. Two types of guided modes, distinguished by field localization and polarization, are expected: (i) SPP, which is localized at the perovskite-metal interface and only exhibits TM-polarized light; (ii) dielectric guided mode (DGM), which is localized within the perovskite layer and can exhibit both TE and TM-polarized light. These modes will serve as a basis to engineer photonic/plasmonic modes in the perovskite metasurface.

To model the perovskite material, we use the dielectric function of phenethylammonium lead iodide (PEPI) perovskite [i.e., $({\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{C}}_2{\mathrm{H}}_4{\mathrm{NH}}_3{)}_2{\mathrm{PbI}}_4$], one of the most used perovskite materials for polaritonic applications [27,31,37,42,47]. In this model, the excitonic resonance is described by a Lorentz oscillator, with dielectric function given by [54]

The Kretschmann–Raether configuration [57] is used to excite SPPs and DGMs with external light [see Fig. 1(a)]. These guided modes are evidenced as resonant dips in reflectivity spectra obtained by transfer matrix calculations with excitation in TM polarization (i.e., polarized in the $xz$ plane) and TE polarization (i.e., polarized in the $xy$ plane). The angle-resolved reflectivity spectra of passive structures are depicted in Figs. 1(b), 1(e), and 1(f), with corresponding field distributions shown in Figs. 1(d), 1(i), 1(j), and 1(k), and corresponding angle-resolved spectra of active structures depicted in Figs. 1(c), 1(g), and 1(h). These results show that, depending on the perovskite thickness $h$, two different situations are obtained, as discussed in the following.

 

Fig. 1. (a) Sketch of the Kretschmann–Raether configuration to investigate guided modes in a stack consisting of encapsulated perovskite thin film of thickness $h$ on a silver substrate. (b), (c), (d) Results of transfer matrix calculations corresponding to the case $h=30\mathrm{nm}$: (b), (c) angle-resolved reflectivity spectra in TM-polarized light of (b) passive and (c) active structures; (d) field distribution of the SPP mode shown in (b). (e)–(k) Results of transfer matrix calculations corresponding to the case $h=150\mathrm{nm}$: (e)–(h) angle-resolved reflectivity spectra in (e), (g) TM and (f), (h) TE-polarized light of (e), (f) passive and (g), (h) active structures; (i)–(k) field distribution of modes shown in (e) and (f).

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If the thickness of the perovskite layer is ultra-thin, for example $h=30\mathrm{nm}$, the passive structure only sustains the SPP mode [Fig. 1(b)], which is localized at the perovskite-silver interface [Fig. 1(d)]. In the active structure, the strong coupling regime between the SPP mode and exciton is distinctly evidenced by the observation of anti-crossing with Rabi splitting of ${\mathrm{\Omega }}_R=225\mathrm{meV}$ [Fig. 1(c)].

When the optical thickness of the perovskite layer exceeds the half-wavelength scale, for example $h=150\mathrm{nm}$, the passive structure can sustain both SPP mode and DGM modes (${\mathrm{TM}}_1$ in TM polarization and ${\mathrm{TE}}_1$, ${\mathrm{TE}}_2$ in TE polarization) [Figs. 1(e) and 1(f)]. Since the ${\mathrm{TE}}_2$ mode is at much higher energy than the excitonic resonance, we only focus on SPP, ${\mathrm{TM}}_1$, and ${\mathrm{TE}}_1$ modes. Their field distributions are depicted in Figs. 1(i)–1(k). We note that, while mainly localized within the perovskite film, ${\mathrm{TM}}_1$ still has a small amount localized at the perovskite-silver interface. This mode is then not purely DGM but a hybrid of SPP and DGM. In the active structures, SPP, ${\mathrm{TM}}_1$, and ${\mathrm{TE}}_1$ undergo strong coupling with the excitonic resonance. It results in anti-crossings with Rabi splitting amounting to ${\mathrm{\Omega }}_{R1}=241\mathrm{meV}$, ${\mathrm{\Omega }}_{R2}=207\mathrm{meV}$, and ${\mathrm{\Omega }}_{R3}=260\mathrm{meV}$, respectively [Figs. 1(g) and 1(h)].

3. BLOCH RESONANCES OF SPP AND DGM IN WEAKLY PATTERNED STRUCTURES

The design of our perovskite metasurface is a grating of period $a$ in the range of 250–300 nm, with filling fraction $0<f<1$. The total thickness of perovskite is $h=30$ or 150 nm with etching ratio $0<\eta <1$ [see Fig. 2(a)]. From the results of the previous section, we know that the penetration depth of different modes into the metallic substrate can be negligible, and thus the silver substrate can be replaced by 100 nm of silver on the silicon substrate. We note that the patterning of perovskite can be fabricated by thermal nano-imprint [58–60] or electron-beam lithography followed by dry etching [56,61,62].

 

Fig. 2. (a) Design of the perovskite metasurface. (b), (d), (e), (f) Folded dispersion of different guided modes extracted from Figs. 1(b), 1(e), 1(f) with period $a=300\mathrm{nm}$. (c), (g), (h) Angle-resolved reflectivity spectra obtained from RCWA calculations for (c) $h=30\mathrm{nm}$ and (g), (h) $h=150\mathrm{nm}$. All of these calculations are carried out with $a=300\mathrm{nm}$, $\eta =0.3$, and $f=0.9$.

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Periodic patterning of the perovskite layer would out-couple guided modes to free-space via Bragg scattering: SPP and DGM become Bloch resonances via band-folding [63]. When the periodic modulation is strong enough (i.e., high contrast refractive-index grating), Bloch resonances of counter-propagating waves can couple and lead to gap-opening. It is straightforward that the dispersion and losses of these modes, as well as their coupling with excitons, can be strongly controlled via band-gap engineering.

To identify the guided resonances for band-gap engineering, it is important to study the weakly patterned structure ($f\sim 1$ and $\eta \ll 1$). The dispersion of these modes is obtained by folding the dispersion of guided modes extracted from Figs. 1(b), 1(e), and 1(f) into the first Brillouin zone. These folded dispersions are depicted in Figs. 2(b), 2(d), 2(e), and 2(f). The index $m=\pm 1,\pm 2$ attributed to each branch is the diffraction order corresponding to the shift of $2m\pi /a$ in the momentum space with respect to the original guided mode. The dispersion of Bloch resonances is also numerically calculated via angle-resolved reflectivity simulations using the rigorous coupled wave analysis (RCWA) method [64]. The RCWA calculations, shown in Figs. 2(c), 2(g), and 2(h), are in good agreement with the band-folding approach. From these results, we identify two pairs of first-order diffraction: ${\mathrm{SPP}}^{(\pm 1)}$ with $h=30\mathrm{nm}$, and ${\mathrm{TE}}_1^{(\pm 1)}$ with $h=150\mathrm{nm}$, which undergo strong diffraction efficiency and also are well isolated from other modes for band-gap engineering.

4. THEORY OF BAND-GAP ENGINEERING VIA DIFFRACTIVE AND RADIATIVE COUPLING

In this section, we will develop a simple theory to describe band-gap engineering when coupling counter-propagating Bloch resonances. As shown in the previous section, in the vicinity of $k_x=0$, the dispersion of counter-propagating resonances ${\varphi }_{\pm }$ is almost linear, and thus it can be described by

It is noted that a band inversion between bright and dark modes occurs when tuning $U$ from positive to negative values. In particular, a very special phenomenon takes place when $U=0$. In this case, the two complex eigenvalues will degenerate:

5. BAND-GAP ENGINEERING OF SPP RESONANCES IN ULTRA-THIN METASURFACES

We now verify the band-gap engineering theory presented in the previous section with the coupling between ${\mathrm{SPP}}^{(\pm 1)}$ in ultra-thin metasurfaces ($h=30\mathrm{nm}$). The etching factor $\eta$ is fixed at unity, while the filling fraction $f$ is the parameter for tuning the diffracting coupling strength $U$ mentioned in the theory. Figures 3(a)–3(c) present angle-resolved reflectivity spectra obtained via RCWA calculations performed on passive structures with three different values of the filling fraction $f$. In these figures, the dark mode is easily identified via the vanishing of the reflectivity dip at $k_x=0$. The observed dark (bright) nature is well consistent with the odd (even) parity in ${\sigma }_x$ mirror symmetry of the field distribution [Figs. 3(d) and 3(e)]: only the symmetric mode can couple to the radiative continuum. The ratio $Q_r^{\mathrm{dark}}/Q_r^{\mathrm{bright}}$ as a function of $k_x$ for different filling fraction $f$ is depicted in Fig. 3(f). It shows that $Q_r^{\mathrm{dark}}\gg Q_r^{\mathrm{bright}}$ when $k_x$ approaches zero, thus suggesting that $Q_r^{\mathrm{dark}}\underset{k_x\to 0}{\to }\infty$ as expected. Moreover, the results from Figs. 3(a) and 3(c) suggest that $U>0$ for $f<0.29$ and $U<0$ for $f>0.29$. In other words, band inversion takes place when scanning $f$ across 0.29, and EP configuration occurs at $f=0.29$. For further insights into this observation, the quality factor and energy of dark and bright modes at $k_x=0$ as functions of filling fraction $f$ are extracted and presented in Figs. 3(g) and 3(h). These results show plainly the band inversion in the vicinity of $f=0.29$, where EPs take place.

 

Fig. 3. (a)–(c) Angle-resolved reflectivity spectra obtained from RCWA calculations for the passive metasurface of (a) $f=0.16$, (b) $f=0.29$, and (c) $f=0.57$. All of these calculations are carried out with $a=300\mathrm{nm}$, $\eta =1$, and $h=30\mathrm{nm}$. (d), (e) Field distribution of the dark and bright modes. (f) Ratio of radiative quality factors as a function of $k_x$ for different values of the filling fraction $f$. (g), (h) Spectral position and quality factor of the dark and bright modes at $k_x=0$ as functions of the filling fraction $f$.

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Nevertheless, the dispersion at $f=0.29$ [Fig. 3(b)] does not show a double EPs feature ($k_x=\pm k_{EP}$) and looks very similar to a Dirac dispersion, suggesting that $k_{EP}\sim 0$. Otherwise, the enhancement of DOS at EPs, which is accompanied by an enhancement of the resonance dip at $\pm k_{EP}$, is not observed. This can be explained by a dominant role of non-radiative losses: if the absorption overtakes the radiative continuum as the main leaky channel, high quality factor features of the dark state and the DOS enhancement at EPs will be hindered. An evaluation of non-radiative losses versus radiative ones is thus needed. We note that for a given resonance, the quality factor $Q$ is composed of the radiative quality factor $Q_r$ and the non-radiative one $Q_{nr}$ given by

6. BAND-GAP ENGINEERING OF POLARITONIC MODES IN PEROVSKITE METASURFACES

In this section, the coupling between ${\mathrm{TE}}_1^{(\pm 1)}$ in perovskite metasurfaces of thickness $h=150\mathrm{nm}$ will be investigated. Qualitatively, this should be similar to the one of SPP modes discussed in the previous sections: formation of dark, bright bands and EPs at band inversion are expected. Nevertheless, unlike ${\mathrm{SPP}}^{(\pm 1)}$, ${\mathrm{TE}}_1^{(\pm 1)}$ has almost no field at the perovskite-metal interface, thus promising low non-radiative losses. The etching factor $\eta$ is fixed at 0.3, corresponding to a patterning depth of 45 nm. We adopt the same strategy as the one discussed in the previous section: the filling fraction $f$ is the parameter for band-gap engineering. However, for each value of $f$, the period $a$ is slightly varied so that the mid-gap is always pinned at 2.35 eV. This value corresponds to a detuning of $-50\mathrm{meV}$ with respect to the energy of excitonic resonance.

We first discuss the results of passive structures [see Figs. 4(a)–4(c)]. Band inversion is observed when scanning $f$ across 0.51, which corresponds to the EP configuration. The dark and bright statuses are also well confirmed via the parity of field distribution [Figs. 4(d) and 4(e)]. Figure 4(f) depicts the quality factor of the dark and bright modes as a function of $k_x$. It shows that $Q^{\mathrm{dark}}(k_x=0)\sim 1500$, which is one order of magnitude higher than the one of SPP from the previous section. We also highlight that $Q^{\mathrm{dark}}(k_x=0)\gg Q^{\mathrm{bright}}(k_x=0)$. The dark mode of this design is therefore a quasi-BIC.

 

Fig. 4. (a)–(c) Angle-resolved reflectivity spectra obtained from RCWA calculations for the passive metasurface of (a) $f=0.4$ and $a=270\mathrm{nm}$, (b) $f=0.51$ and $a=267\mathrm{nm}$, (c) $f=0.7$ and $a=261\mathrm{nm}$. All of these calculations are carried out with $\eta =0.3$ and $h=150\mathrm{nm}$. (d), (e) Field distribution of the dark and bright modes. (f) Quality factor of the bright and dark modes in (c) as a function of $k_x$. (e), (f) Spectral position and quality factor of the dark and bright modes at $k_x=k_{EP}=0.177{\mathrm{\mu m}}^{-1}$ as functions of the filling fraction $f$. In these calculations, the period $a$ is fixed at 265 nm. (i) Dip amplitude of the reflectivity resonance as a function of the spectral detuning to EP.

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Concerning the EP configuration, its appearance is characterized by an anomaly in the dispersion mapping: the dispersion curve is vertically split into two halves and separated by a small “slit” with a minimum at k_x, corresponding to the gap between both EPs [see Fig. 4(b)] [66]. With this rule, the two EPs can be easily identified at $\pm k_{EP}=\pm 0.177{\mathrm{\mu m}}^{-1}$. Indeed, pinning at $k_x=0.177{\mathrm{\mu m}}^{-1}$, the spectral position and quality factor of the two modes are monitored while scanning $f$. The result [Figs. 4(g) and 4(h)] clearly demonstrates the EP configuration at $f=0.51$ with the simultaneous degeneracy of both energy and quality factors. Strikingly, as shown in Fig. 4(b), the dip resonance is strongly enhanced at EPs. Figure 4(i) presents the dip amplitude as a function of its detuning to the EP. It indicates that there is a four-fold enhancement of resonance amplitude at EPs. Such observation is perfectly in line with theoretical prediction that the DOS is four-fold enhanced at EPs [52].

Since quasi-BIC and EP features are nicely achieved, the main leaky channel in this design should be the radiative continuum. Indeed, as discussed in the previous section, the non-radiative losses are given by the losses of the dark mode at $k_x=0$: $Q_{nr}=Q^{\mathrm{dark}}(k_x=0)\sim 1500$. Moreover, $Q^{EP}\sim 50\ll Q_{nr}$, which leads to $Q_r^{EP}\sim 50\ll Q_{nr}$. The non-radiative losses are thus negligible.

We now study the active structures based on the same designs. As shown in Figs. 5(a)–5(c), four polaritonic modes are observed: two LPBs and two upper polariton branches (UPBs). The polariton states inherit the dark/bright nature of the initial photonic modes with one pair of dark/bright modes on each side of the exciton energy. We highlight that band inversion also takes place at $f=0.51$ as in the case of the passive structure, but occurs for both UPBs and LPBs. To rigorously study these polaritonic modes, a quantum description of photonic crystal polaritons is required [67]. As a matter of fact, we propose here a simple model to explain their formation. Naively, one could suggest a model of three-coupled oscillators involving the two photonic modes and the excitonic resonance. However, the field distribution of dark and bright modes [Figs. 4(d) and 4(e)] indicates that the anti-node position of each mode corresponds to the node position of the other one. Thus, a given exciton in the perovskite metasurface cannot be coupled efficiently to both photonic modes. To a first approximation, we can consider that half of the excitons only couple to the dark mode, and the other half only couple to the bright mode. With two types of excitons and two photonic modes, the system is now described by a four-coupled oscillators Hamiltonian, given by

 

Fig. 5. (a)–(c) Angle-resolved reflectivity spectra obtained from RCWA calculations for the active metasurface of (a) $f=0.4$ and $a=270\mathrm{nm}$, (b) $f=0.51$ and $a=267\mathrm{nm}$, (c) $f=0.7$ and $a=261\mathrm{nm}$. All of these calculations are carried out with $\eta =0.3$ and $h=150\mathrm{nm}$. (d) Quality factor of the dark and bright polaritons in (c) as a function of $k_x$. (g), (h) Spectral position and quality factor of the dark and bright polaritons at $k_x=k_{EP}=0.13{\mathrm{\mu m}}^{-1}$ as functions of the filling fraction $f$. In these calculations, the period $a$ is fixed at 265 nm.

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The quality factors of LPBs for $f=0.7$ are presented in Fig. 5(d). The narrowing of dark polaritons when $k_x$ approaches zero, together with $Q^{\mathrm{dark}-\mathrm{pol}}\gg Q^{\mathrm{bright}-\mathrm{pol}}$, clearly indicates that the dark polariton also inherits the quasi-BIC behavior of the original dark mode. We highlight that a similar observation has been recently reported in the framework of strong coupling between excitons in two-dimensional materials and dielectric Bloch resonance [68].

We now focus on the EP configuration at $f=0.51$. Using the criterion mentioned previously with the passive structure, we obtain $k_{EP}^{\mathrm{LPB}}=0.13{\mathrm{\mu m}}^{-1}$ and $k_{EP}^{\mathrm{UPB}}\sim 0$ for EPs of LPBs and UPBs, respectively. Indeed, Figs. 5(e) and 5(f) depict thespectral position and the quality factor of the dark and bright polaritons of the lower branch while keeping $k_x=0.13{\mathrm{\mu m}}^{-1}$. These data show nicely the creation of polaritonic EPs at $f=0.51$. In addition, with a detuning of −50 meV between the photonic EP and the excitonic resonance [Fig. 4(b)] and a Rabi splitting of 256 meV, the excitonic and photonic fractions of polaritonic EPs are (32% exciton, 68% photon) for EPs of the lower branch and (68% exciton, 32% photon) for EPs of the upper branch. The lower polaritonic EP is therefore more photonic-like, while the upper one is more excitonic-like. Since the excitonic losses (given by ${\mathrm{\Gamma }}_X$) are much more important than the photonic losses (given by ${\mathrm{\Gamma }}_r$ and ${\mathrm{\Gamma }}_{nr}$), the upper EPs exhibit many more losses than the lower ones. This explains a less pronounced EP feature (i.e., $k_{EP}\sim 0$) in the upper branch. Finally, it is important to mention that several groups have reported on polaritonic EPs, but only for localized resonators [69,70]. Our results propose the first polaritonic EPs in band structures of momentum space. This promises the entrance of polaritonic physics in non-Hermitian topology, in which Dirac points are replaced by EPs as singularities [71,72].

7. CONCLUSION

In conclusion, we have successfully introduced a simple design to obtain polaritonic quasi-BIC and EPs with perovskite plasmonic metasurfaces. The implementation of such a design can be achieved with current technology of subwavelength-scale perovskite texturing [41,47,56,58–62,73–76]. For device development, our results propose an elegant solution to integrate a metallic layer into polaritonic devices with minimum photonic degradation. Attractive applications of our design include electrically pumped polariton lasers using high quality factor quasi-BIC or EPs with at least four-fold enhanced DOS [52,65]. In particular, for fundamental photonic research, this finding would pave the way to use the high nonlinearity of perovskite polaritons to demonstrate many intriguing features of nonlinear BICs, such as multistability, self-focusing BIC, and breaking of symmetry protected BIC [77,78]. Finally, the possibility to drive dynamically this non-Hermitian system around the EP via nonlinearity and electrical field would be a new paradigm for exploring non-Hermitian topology [71,72,79].