Active strong coupling of exciton and nanocavity based on GSST-WSe<sub>2</sub> hybrid nanostructures

Lijuan Wu; Jing Huang; Jing Huang; Shaojun You; Chenggui Gao; Chenggui Gao; Chaobiao Zhou; Chaobiao Zhou

doi:10.1364/OE.519134

1. Introduction

In recent years, the strong coupling phenomenon between light and matter has attracted significant interest due to its ability to generate exciton-polaritons, which have widespread applications in cavity quantum electrodynamics and quantum information processing [1–5]. A system falls within the strong coupling regime and forms exciton-polaritons when the energy exchange rates between light and matter are larger than their decoherence rates [6–9]. Typically, the strong coupling is characterized by an avoided crossing and Rabi splitting in the spectrum. Exciton-polariton are quasiparticles that combine the advantages of photons and exciton, leading to the emergence of many applications and technologies such as ultra-low-threshold polariton lasers [10,11], ultra-fast all-optical switches [12], and quantum information processing [13].

Transition metal dichalcogenides (TMDCs) are emerging materials with many interesting physical and chemical properties that can be easily integrated with other optoelectronic devices [2,3,4,14–18]. Bulk TMDCs are characterized by large exciton binding energy and high refractive index at room temperature [19–24], which make it possible to overlap the resonance mode and exciton mode in the same structure, and provide an ideal platform for strong coupling due to their unique properties [16,19,25–32]. For instance, in 2021, Zong et al. studied the strong coupling between quasi-bound states in the continuum modes and excitons in a WS$_2$ bulk material, and achieved a Rabi splitting energy of 120 meV [30]. In 2022, Gu et al. increased the Rabi splitting to 235 meV using the similar scheme [31]. In 2023, Xie et al. demonstrated that the coupling strength can be dramatically enhanced with the assistance of a Fabry-Pérot cavity, which represents the highest Rabi splitting value in current strong coupling studies [32]. However, optical cavity realized based on bulk TMDCs are functionally fixed at the fabrication stage. Their actively tunable and reconfigured optical response has the potential to achieve a wide range of applications [33–36]. Reconfigurable photonic devices are a crucial component of next-generation optical technologies, and they will realize utterly new application areas for photonic devices [37–39]. Phase-change materials (PCMs) are an exciting choice for reconfigurable photonic devices [40–44]. The commonly used PCMs, Ge$_{2}$Sb$_{2}$Se$_{4}$Te$_{1}$ (GSST), have a large refractive index change between crystalline and amorphous transitions, as well as a fast response and good stability, making them excellent optical PCMs [45–48]. They are widely used to realize high-performance micro- and nano-optical devices, such as integrated photonic switches [45,49], spatial light modulators [50–52], and nonlinear optics [47,53–55]. For instance, in 2021, Zhang et al. demonstrated an on-chip electrical switching platform enabling both binary switching and quasi-continuous tuning of GSST -based active metasurfaces, and the optical contrast is over 400% [40]. In 2021, the 30-fold enhancement of third-harmonic generation in a highly nonlinear GSST metasurface is demonstrated in the mid-wave infrared spectral region [47]. In 2022, Huang et al. studied the light trapping and manipulation of quasi-BIC in GSST metasurface [36], Tan et al. achieved the polarization-controlled varifocal metalens with GSST materials in mid-infrared [55]. By combining PCMs with bulk TMDCs, it is expected that resonant and exciton mode can be overlapped in the same structure while the coupling strength can be dynamically regulated to realize a wide range of applications. This phenomenon of using PCMs to cause strong coupling and optical switching dynamically has not been reported.

In this work, we propose GSST-WSe$_2$ hybrid nanostructures based on PCMs GSST to realize active modulation of strong coupling of resonance mode with exciton. We first calculate the optical properties of single structures, tetramers, standing array structures, and substrate-array structures. We find that the spectra show significant anti-crossing behavior, indicating the occurrence of a strong coupling phenomenon. To explore the considerable switching effect of PCMs on strong coupling. We compare the effects of crystallinity $m$=0 and $m$=100% of GSST on the interaction of resonance mode and exciton. Furthermore, the same excellent modulation effect of GSST on fluorescence enhancement is demonstrated by the change of crystallinity $m$.

2. Results and discussion

2.1 Active strong coupling base on single GSST-WSe$_2$ hybrid nanostructure

To generate significant strong coupling, we need an emission source with high exciton binding energy and oscillation strength, a nanocavity with a very small mode volume and a high quality factor to generate significant strong coupling. Therefore, we design a hybrid structure consisting of a GSST hollow nanodisk with a bulk WSe$_2$ nanodisk, and the bulk WSe$_2$, as an efficient light emitter, is embedded in the GSST hollow nanodisk, as shown in Fig. 1(a). The advantage of such a design is that the bulk WSe$_2$ has a strong binding energy exciton, which increases the overlap of the resonant mode and exciton to achieve significant strong coupling. At the same time, the transition of PCMs GSST from amorphous (a-GSST) to crystalline (c-GSST) can realize effective active manipulation of light scattering. Therefore, the combination of GSST and bulk WSe$_2$ can realize active modulation of strong coupling of exciton with the nanocavity. Where the optical constant of GSST can be calculated based on the Lorentz-Lorentz relation and the effective dielectric constant theory, and the effective dielectric constants of GSST with different degrees of crystallization are described as [36,40,56]:

(1)$$\frac{\varepsilon_{eff}-1 }{\varepsilon_{eff}+2} =m\times \frac{\varepsilon_{c-GSST}-1 }{\varepsilon_{c-GSST}+2}+\left ( 1-m \right )\frac{\varepsilon_{a-GSST}-1 }{\varepsilon_{a-GSST}+2},$$

where $\varepsilon _{a-GSST}$ and $\varepsilon _{c-GSST}$ are the dielectric constants of a-GSST and c-GSST, respectively, and $m$ is the degree of crystallinity, with $m$ varying from 0 (amorphous) to 100% (crystalline). The calculated refractive indices ($n$, $k$) of the GSST material are shown in Appendix A. It is noted that the transition between states of GSST can be induced by thermal, electrical, and optical means etc [40,52].

Fig. 1. (a) Schematic structure of the hybrid structure GSST-WSe$_2$ with bulk WSe$_2$ embedded inside the hollow GSST nanodisk. (b)-(e) The scattering spectra of GSST nanodisk, GSST annular nanodisk, WSe$_2$ nanodisk, and GSST-WSe$_2$ hybrid structure, respectively. (b) and (d) has the outer and inner radius $R$=120 nm, $r$=0, and height $H$=116 nm, (c) and (e) has the inner radius $r$=60 nm, the other structural parameters are the same as (b) and (d). (f) Multipolar decomposition calculations of the hybrid structure GSST-WSe$_2$. (g) Scattering spectra of hybrid structure with $f$ from 0 to 1, where $m$=0. Points A, B, C, and D represent the positions of the electric field distribution in the $f$= 0, 1 scattering spectra. (h) Scattering spectra of hybrid structure with $m$ from 0 to 100%, where $f$=1. (i) Electric field distribution map of the corresponding four points in (g).

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The artificial dielectric constant of WSe$_2$ can be approximately described as [19,57]:

(2)$$\varepsilon=\varepsilon_{0}+f\frac{\omega^2_{ex} }{\omega^2_{ex}-\omega^2-i\gamma_{ex}\omega} ,$$

where $\varepsilon _{0}$=16 is the background dielectric constant due to higher energy migration, $\omega _{ex}$=1.624 eV is the transition energy, $\gamma _{ex}$=74 meV is the exciton full width [58,59]. The $f$ is the oscillation strength, the change of exciton oscillation intensity can be achieved by changing the transition dipole moment and transition frequency. The transition dipole moment of electrons is an essential property of materials, which can be changed by changing the microstructure of the material, such as doping, alloying, etc [60]. For a background refractive material only, let $f$ be 0.

In this work, the optical response of the above hybrid structures is calculated by the finite difference-time-domain method. The single nanostructures and the tetramers are illuminate using total-field scattered-field light source, with perfectly matched layer is used as boundary condition in the $x$-, $y$, and $z$-directions. For the metasurfaces, a linear light source is employed, periodic boundary conditions are utilized in the $x$- and $y$- directions, and a perfectly matched layer is applied in the $z$-direction. Here, we set the crystallinity $m$ of GSST to 0 and the exciton oscillation strength $f$ to 0. As shown in Figs. 1(b), (c), (d), and (e), we calculate the scattering spectra of GSST nanodisk, GSST annular nanodisk, WSe$_2$ nanodisk, and GSST-WSe$_2$ hybrid nanodisk, respectively. Where the outside and inner radius of the GSST nanodisk is $R$=120 nm, $r$=60 nm. The height of the nanodisk is $H$=116 nm, and the radius of the WSe$_2$ nanodisk is $r$=60 nm, the other structural parameters are the same as GSST and WSe$_2$ nanodisks. By comparing the scattering spectra of the four nanodisks, we find that the GSST annular nanodisk has the widest scattering spectum, GSST nanodisk has the second widest scattering spectum, the GSST-WSe$_2$ hybrid nanodisk has the third widest scattering spectrum, and the WSe$_2$ nanodisk has the narrowest scattering spectral linewidth. This reason is that the non-negligible material loss of GSST with crystallinity $m$=0 in this wavelength range, it exceeds that of WSe$_2$ with $f$=0 (see Appendix A). Moreover, the effective volume of the annular nanodisk is smaller than that other structures. Comparing the resonance wavelengths of the four nanodisks, we find that the wavelengths are shifted, which is mainly due to the change of refractive index. For the hybrid structure, the resonance wavelength is located near 764 nm, corresponding to the exciton wavelength of WSe$_2$. To further analyze the resonance properties, we perform electromagnetic multipole expansion of the resonance mode of the GSST-WSe$_2$ hybrid nanodisk in the Cartesian coordinate system to analyze the multipole moments and their contributions to the far-field radiation [61,62], as shown in Fig. 1(f). The results show that the electric dipole plays a significant role at the resonance wavelength, followed by the magnetic dipole. The response of electric quadrupole, magnetic quadrupole, and toroidal dipole are significantly suppressed.

To explore the effect of $f$ on the coupling strength, the scattering strength of WSe$_2$ at different $f$ is calculated, as shown in Fig. 1(g). The structural parameters are the same as Fig. 1(e), and $m$=0. It can be seen the two splitting peaks appear on the high-energy side and low-energy side of the resonance peak when $f$ is larger than 0. These spectral features indicate the strong coupling between the resonant mode and the excitation state of WSe$_2$. It can also be observed that as the value of $f$ increases from 0 to 1, the splitting phenomenon becomes more obvious. The phenomenon can be explained more clearly by comparing the electric field strengths near the coupling region. As shown in Fig. 1(g), points A, B, C, and D represent the positions of the electric field distribution in the $f=0,1$ scattering spectra. The field distribution at these positions is shown in Fig. 1(i). It is easy to see that the electric field intensity at point C after coupling is significantly lower than at point A at the resonance wavelength before coupling. Furthermore, the electric field intensity at point A larger than at point B and C. It indicates that energy transfer occurs. Similarly, in order to explore the effect of GSST on the active regulation of the spectral splitting, we calculate the scattering spectra under different crystallinity $m$ where $f$=1, as shown in Fig. 1(g). As the value of $m$ gradually increases from 0 to 100%, the scattering spectra splitting gradually becomes smaller until disappears, which is due to the fact that the larger the imaginary part $k$ of the refractive index is, the larger the loss is [36]. By comparing the scattering spectra under the two crystallinity degrees of $m$=0 and $m$=100%, it can be observed that the superior optical switching effect can be realized by the adjustment of GSST crystallinity, which indicates that GSST has an excellent impact of active regulation on the strong coupling. It is worth pointing out that WSe$_2$ can be replaced with other TMDCs, in which case the structural parameters need to be adjusted to obtain the resonance mode of a specific wavelength, such as increasing the thickness and disk radius

To further determine the origin of the spectral features, we analyze the simulated scattering spectra using the coupled oscillator model (COM) [63]:

(3)$$\left( {\begin{array}{cc} {{E_{cav}} - i\hbar \Gamma } & g\\ g & {{E_{ex}} - i\hbar {\Gamma _{ex}}} \end{array}} \right)\left( {\begin{array}{c} \alpha \\ \beta \end{array}} \right) = E\left( {\begin{array}{c} \alpha \\ \beta \end{array}} \right),$$

where $E_{cav}$ and $E_{ex}$ are the energies of the cavity mode and exciton, respectively. $\hbar \Gamma$ and $\hbar \Gamma _{ex}$ are the half-height full widths (HHFW) of the cavity mode and the exciton, and $g$ is the coupling strength. $E$ denotes the hybridization energy of the cavity mode and the exciton. $\alpha$ and $\beta$ are the weights occupied by cavity mode and exciton in the hybridization mode, respectively, and ${\left | \alpha \right |^2} + {\left | \beta \right |^2} = 1$. Therefore, the hybridization mode energy for the coupling of cavity mode and exciton is:

(4)$$\begin{matrix} E_{{\pm}}=\frac{1}{2}[(E_{ex}+E_{cav})-i(\hbar\Gamma_{ex}+\hbar\Gamma_{cav})]\\ \\ \pm\sqrt{g^2-\frac{1}{4}[E_{ex}-E_{cav}+i(\hbar\Gamma_{ex}+\hbar\Gamma_{cav})]^2} \end{matrix}.$$

The energy level of the coupled system is called Rabi splitting when the splitting is:

(5)$$\hbar \Omega = 2\sqrt {{g^2} - 0.25{{\left( {\hbar {\Gamma _{ex}} - \hbar {\Gamma _{cav}}} \right)}^2}}.$$

Existence of strong coupling between cavity mode and exciton mode [63–65]:

(6)$$g > \frac{1}{2}\left| {\hbar {\Gamma _{ex}} - \hbar {\Gamma _{cav}}} \right|,$$

and

(7)$$g > \sqrt {0.5{{(\hbar {\Gamma _{ex}})}^2} + 0.5{{(\hbar {\Gamma _{cav}})}^2}}.$$

The first condition ensures that the Rabi splitting does not disappear, and the second condition guarantees that the Rabi splitting is greater than the HHFW of the cavity mode and exciton.

In hybrid systems consisting of the cavity mode and the exciton mode, an inverse shift behavior in the energy map is usually observed by successfully adjusting the spectral overlap between these two modes. In the current study, the cavity resonance energy can be tuned by adjusting the radius and height of the nanodisk. Figures 2(a) and (b) show the scattering spectra of nanodisks with different heights $H$ and outer radius $R$ when $f$=1, $m$=0. These spectra clearly show that the classical anti-crossing behavior (black dashed line) occurs as the height $H$ of the nanodisk goes from 70 nm to 180 nm, and two branches called the high-energy upper branch (UP) and the low-energy lower branch (LP) are observed. Similarly, when the outer radius $R$ of the hybrid nanostructures is varied from 95 nm to 140 nm, UP and LP modes are formed. Thus, the overlap of the nanocavity mode and WSe$_2$ spectra can be effectively tuned by changing the height and outer radius of the hybrid structure. To observe the modulation effect of GSST, we calculate the scattering spectra at different heights and radii when $m$=100%, as shown in Figs. 2(c) and 2(d). It is clear that the anti-crossing behavior disappears. The reason is that the $m$ of GSST increases, the material loss increases, the cavity mode strength decreases, and the coupling strength decreases to achieve the strong coupling switch.

Fig. 2. Scattering spectra of the hybrid structure. (a),(b) Scattering spectra of the hybrid structure GSST-WSe$_2$ ($f$=1, $m$=0) from 70 nm to 180 nm in height $H$ and 95 nm to 140 nm in outer radius $R$, respectively. (c),(d) The corresponding scattering spectra for $f$=1, $m$=100%.

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2.2 Active strong coupling base on GSST-WSe$_2$ tetramers

In the above study, we consider the active modulation of the crystallinity of GSST by the strong interaction between the single nanostructure cavity and exciton. When multiple nanoparticles are polymerized into particle clusters, strong interactions between the particles and the excitons exist, allowing the tunable degrees of freedom to increase further. Next, the active modulation of the microcavity and exciton coupling strength by a tetramers is analyzed in detail. The GSST-WSe$_2$ tetramers is shown in Fig. 3(a), the distance between two nanodisks is 300 nm in the $x$- direction and 310 nm in the $y$- direction. Here, we calculate the scattering spectra of WSe$_2$ exciton with $f$=0 and 1 when $m$=0, as shown in Fig. 3(b). Evidently, the scattering spectrum changes from one scattering peak to two scattering peaks when the exciton oscillation strength $f$ changes from 0 to 1. The strong coupling occurs when the coherence energy between excitons to photons is larger than their respective decay rates. Thus, the resonance position of the original energy level disappears, and two new energy levels appear. This is an obvious characterization of the strong interaction between excitons and photons. This phenomenon can also be further confirmed from Fig. 3(c), where we observe the dispersion behavior of the exciton polaritons by varying the height of the structure. There is a clear anti-crossing behavior with the formation of UP and LP in the scattering spectrum, and the Rabi splitting $\hbar \Omega$=284 meV. The coupling strength $g$=193.9 meV is calculated via Eq. (1). This result satisfies Eq. (6), which is further evidence that a strong coupling occurs between the exciton and cavity modes. To investigate the effect of GSST on the modulation of strong coupling in the tetramer cavity, the crystallinity of GSST is changed from 0 to 100% for comparison, as shown in Fig. 3(d). At this time, the anti-crossing phenomenon of scattering spectra disappears, and the strong coupling switch in tetrameric cavity is realized. It is noted that the ED radiation is dominant for this mode, which is consistent with multipole response of the single nanodisk, see Appendix B.

Fig. 3. (a) Structural diagram of the tetramers with the hybrid GSST-WSe$_2$. (b) Scattering spectra of the tetramers with $f$=0 (blue line) and 1 (red line), where the blue-gray lines are the exciton positions. (c), (d) Scattering mappings for $m$=0 and 100%, $f$=1, and varying structure heights $H$, respectively.

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2.3 Active strong coupling base on GSST-WSe$_2$ metasurfaces

When nanoparticle are arrayed to form a metasurfaces supported collective resonance, and this resonance mode will greatly suppress the radiation loss in the material, thus improving the coupling of the resonance mode with excitons. Next, we will discuss the active modulation of the strong coupling of the hybrid metasurfaces. The schematic diagram of the structure is shown in Fig. 4(a), the parameters of the unit cell of GSST-WSe$_2$ are the same as those of the single nanostructure mentioned above, with the period of the metasurfaces $P$=710 nm. First, we explore the transmission of the standing array structures without substrate, which is shown in Fig. 4(b). As can be seen from the figure, when $f$=0, resonance occurs at the exciton position (blue line). When $f$=1, resonance splitting at the exciton position is evident, which indicates a strong coupling effect between photons and excitons in the metasurface system. As shown in Fig. 4(c), the transmission spectrum of the coupled system with the change of height shows anti-crossing behavior. The splitting energy is $\hbar \omega$=148 meV, the cavity mode has a HHFW $\hbar \Gamma$=125 meV and a coupling strength $g$=86.09 meV. This result satisfies Eq. (6), which is also a characteristic performance of the strong coupling of the metasurfaces structure. Next, the crystallinity of GSST is changed to modulate the strong coupling effect. The anti-crossing behavior of the transmission spectra disappears and the coupling strength between the resonance mode and the exciton is greatly reduced when the crystallinity is 100%, as shown in Fig. 4(d). The appearance and disappearance of the anti-crossing phenomenon strongly indicate that the crystallinity of GSST effectively modulate the strong coupling.

Fig. 4. (a) Schematic of metasurface without substrate, with the period of the structure $P$=710 nm. (b) Transmission spectra of the array structure $f$=0 and 1, where the gray-blue lines are the exciton positions. (c), (d) Transmission spectra mapping with different heights $H$ for $f$=1 and $m$=0 and 100%. (e), (f) The PL enhancement mapping with different height $H$.

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Optical resonance cavity can trap the optical field, enhance the optical density of states, so that to improve the efficiency of spontaneous radiation and photoluminescence (PL) intensity. In Figs. 4(e) and 4(f), the PL enhancement spectra is calculated, showing an anti-crossing behavior at GSST crystallinity $m$=0, and the anti-crossing spectrum disappears at $m$=100%. Therefore the modulation of the strong coupling in the standing metasurface system by GSST at different crystallinity are further demonstrated. It is noted that the absolute spontaneous emission rate cannot be obtained directly in the simulation. While the quantum emission rate of dipole radiation is correlated with classical power in the same environment. A host of randomly distributed dipole sources are placed within resonant nanostructures as quantum emitters. Finally, the the PL enhancement factor ($F_E$) can be calculated through the radiation power [66]:

(8)$$F_E=\frac{P_{dipole}}{P_{radiated}^0}\times\frac{Q_{E}}{Q_{E_0}},$$

where $P_{dipole}$, $P_{radiated}^0$, $Q_{E}$, and $Q_{E_0}$ are dipole-radiated power in an inhomogeneous environment, dipole-radiated power in a homogeneous environment, quantum efficiency, and intrinsic quantum efficiency, respectively.

In a real device, the substrate needs to be considered. Finally, we explore the modulation of GSST on the strong coupling in metasurface with ${\rm SiO}_{2}$ substrate. The structure schematic is shown in Fig. 5(a), the unit cell of the metasurfaces has the same structural parameters as the single structure above, the period of metasurfaces $P$ is 495 nm. Due to the addition of the substrate, the effective refractive index of the metasurfaces increases leading to a redshift of the resonance mode therefore we reduce the metasurfaces period to bring the resonance mode back to the exciton wavelength. As shown in Fig. 5(b), when $f$=0, a resonance is clearly observed near the exciton position (blue line) in the transmission spectrum, and the mode splitting occurs at the near the exciton position (red line) when $f$=1. These results also fully confirm that the strong coupling between the resonance mode and the exciton occurs at this time. As shown in Fig. 5(c) and 5(f), when $f$=1 and $m$=0, the resonance peaks and PL spectra splitting are still observed, while when $f$=1 and $m$=100%, the splitting phenomenon disappears. This fully confirms the occurrence of strong coupling and its switching characteristics in this system.

Fig. 5. (a) Schematic diagram of the metasurface with substrate (${\rm SiO}_{2}$, $n$=1.445), with structural period $P$=495 nm. (b) Transmission spectra of the $f$=0 and 1 for the substrate-array structures, where the gray-blue line is the exciton positions. (c), (d) Transmission spectra mapping with different heights $H$ for $f$=1 and $m$=0 and 100%. (e), (f) The PL enhancement mapping with different heights $H$ of the array structure.

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3. Conclusion

In conclusion, we investigate the active strong coupling of resonant mode and exciton in various cavities based on GSST. First, the properties of strong interaction between WSe$_2$ excitons and resonant modes supported by nanocavities are demonstrated in the GSST-WSe$_2$ hybrid structure. With the transition of GSST from amorphous to crystalline, the obvious optical switching phenomena are observed in the single nanostructure, tetramers, and metasurfaces systems. Finally, the active appearance and disappearance of PL spectra splitting in metasurfaces further confirm the optical switching phenomenon of these system. Our results may help the emergence of many applications and technologies such as active polariton lasers, tunable all-optical switches, etc.

Appendix

A. Refractive index of GSST

According to Eq. (1), the complex refractive index of GSST material is calculated, as shown in Fig. 6.

Fig. 6. Refractive index of GSST with $m=0-100{\% }$, (a) imaginary part, and (b) real part.

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B. Multipole contributions of nanostructures

For both tetramer and metasurface systems, the electric dipoles still dominate the resonance modes, as illustrated in Fig. 7.

Fig. 7. (a)-(c) Multipole contributions of the tetramer, metasurface without substrate, and metasurface with substrate

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Funding

National Natural Science Foundation of China (12004084, 12164008); Guizhou Provincial Science and Technology Project (ZK[2024]504, ZK[2021]030); Natural Science Foundation of Guizhou Minzu University (GZMUZK[2022]YB04, GZMUZK[2023]CXTD06); Science and Technology Innovation Team Project of Guizhou Colleges and Universities ([2023]060).

Disclosures

The authors declare that they have no conflict of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Active strong coupling of exciton and nanocavity based on GSST-WSe₂ hybrid nanostructures

Abstract

1. Introduction

2. Results and discussion

2.1 Active strong coupling base on single GSST-WSe$_2$ hybrid nanostructure

2.2 Active strong coupling base on GSST-WSe$_2$ tetramers

2.3 Active strong coupling base on GSST-WSe$_2$ metasurfaces

3. Conclusion

Appendix

A. Refractive index of GSST

B. Multipole contributions of nanostructures

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (7)

Equations (8)

Optics Express