Strong coupling between excitons and magnetic dipole quasi-bound states in the continuum in WS₂–TiO₂ hybrid metasurfaces

To obtain a clearer insight into the physics of magnetic dipole quasi-BIC resonance in the metasurfaces, we analysis the transmission spectrum of the TiO₂ metasurfaces with different asymmetry parameters are shown in Fig. 3(a), which manifests a Fano lineshape resonance as a result of the in-plane symmetry breaking of the unit cell. In our work, when a perturbation is introduced into an in-plane inverse symmetric $\left({x,y}\right)\to\left({-x,-y}\right)$ of a structure, BIC will transform into quasi-BIC and build the radiation channel between a nonradiative bound state and the free space continuum, at the same time, confine part of their electromagnetic field inside the structure is shown in Figs. 3(b) and 3(c). We take the asymmetric parameter $\delta{\rm{=}}0.15$ and $H$=85 nm at the resonance wavelength 616 nm for example, the inverse phase with almost equal amplitude of electric field can be observed in Fig. 3(b), and the circular displacement current in the nanobars generates an out-of-plane magnetic field as shown in Fig. 3(c), which reveal the properties of a magnetic dipole with a strongly localized electrical field inside the nanobars. Then a Fano line will be caught in the transmission spectrum due to the interference between the magnetic dipole and free space continuum. This capture pattern provides a platform for enhancing light-matter interaction at the near-fieldZhang et al. (2013); Koshelev et al. (2018); Wang et al. (2020b).

We then fit the transmission spectrum $T(w)$ by the Fano formulaWu et al. (2014); Yang et al. (2014); Li et al. (2019b)

where $a_{1}$, $a_{2}$, and $b$ are the constant real numbers,${w_{0}}$ is the resonant frequency and $\gamma$ is the dissipation of the quasic-BIC, as depicted in the Fig. 3(b).

Fig. 4(a) describes the absorption spectrum of the hybrid structure of TiO₂ metasurface with monolayer WS₂ on top. Two peaks are located at 612.14 nm and 626.97 nm, respectively. The dip located at 620.06 nm shows that original resonance wavelength (616.2 nm) disappears, and the small red shifts of resonance location due to the large real part of the permittivity of WS₂ monolayer can be shown in Fig. 2(a) blue line. The obvious spectral splitting with two peaks and one dip as a result of the strong coupling between bare monolayer WS₂ and quasi-BIC, which indicates that coherent energy exchange is conducted between excitons and quasi-BIC. This finding can be explained by two-level coupled oscillator modelLi et al. (2019a, b); Qing et al. (2018), as depicted in Fig. 4(b). The incident light can be regarded as ground state with the energ $E_{0}$. When the metasurfaces have proper geometrical parameters, a magnetic dipole with the energy $E_{\rm{MD}}$ can be excited by the incident light. The process can be thought of photon transition from the ground state to one excited state. In the same way, the interaction between photon and exciton in the monolayer WS₂ is considered as the energy transition process from $E_{0}$ to $E$exc. Furthermore, coherent energy exchange will occur between the magnetic dipole and exciton as they share the same energy. When the energy exchange rate is greater than each decay rate, the strong coupling happens, and the original two independent energy levels will be hybridized to form a new hybrid state named polariton with two new energy levels. The electric field distributions of the new hybrid state are shown in Figs. 4(c), 4(d) and 4(e). Comparing the electric field distributions at the absorption peaks and dip, it is found that the local electric field at peaks are much higher than that at the dip, which further proves that original energy state around 2.014 eV (616 nm) disappears and forms two new state are 612.14 nm and 626.97 nm, respectively. Thus, the evident spectral splitting make clear that strong coupling between MD and exciton can be obtained by our design.

It can be seen from the Fig. 5(a) (red line) that the resonant position of WS₂ does not change with thickness $H$, while the resonant wavelength of quasi-BIC shows a linear growth relationship with thickness $H$. Moreover, as the resonant wavelength increases, it will shift across the exciton resonant wavelength, therefore two branches of anti-crossover behavior can be captured and named lower branch (LB) and upper branch (UB), which is depicted from Fig. 5(b) the absorption spectrum of the hybrid structure with different thickness $H$. This can be explained by using coupled-mode theory (CMT). For in-plane vectors the eigenstates can be described asLiu et al. (2014); Deng et al. (2010)

where, ${E_{{\rm{q-}}BIC}}$ and ${\gamma_{{\rm{q-}}BIC}}$ represent the quasi-BIC energy and dissipation, respectively. ${E_{{\rm{exc}}}}$ and ${\gamma_{{\rm{exc}}}}$ represent the energy and nonradiative decay rate of the uncoupled exciton, respectively, and $g$ is the coupling strength. $\alpha$ and $\beta$ are the Hopfield coefficients that describe the weighting of the quasi-BIC and Exciton for LB and UB, which should satisfy ${\left|\alpha\right|^{2}}+{\left|\beta\right|^{2}}=1$. $E_{LB,UB}$ represent the eigvenvalues, which can be obtained from Eq. (2)

When the detuning $\Delta{\rm{=}}{E_{{\rm{q-}}BIC}}{\rm{-}}{E_{{\rm{exc}}}}{\rm{=}}0$, Eq. (3) become

Then we obtain the Rabi splitting energy

which is owning to the strong coupling between quasi-BIC and exciton. Here, we also calculate the ${\gamma_{{\rm{q-}}BIC}}$ =14.93 meV and ${\gamma_{{\rm{exc}}}}$ =15 meV from Fig. 3(a), the Rabi splitting energy $\hbar\Omega$=46 meV can be extracted from FEM simulation results shown in Fig. 5(a) (dashed line), satisfying the condition of strong coupling ( $\hbar\Omega>{{\left({{\gamma_{{\rm{q-}}BIC}}+{\gamma_{{\rm{exc}}}}}\right)}\mathord{\left/{\vphantom{{\left({{\gamma_{{\rm{q-}}BIC}}+{\gamma_{{\rm{exc}}}}}\right)}2}}\right.\kern-1.2pt}2}$ ). We then compare the dissipation rate with the coupling strength $g$. From Eq. (5), we obtained g=23 meV, which indicated $g>{{\left|{{\gamma_{{\rm{exc}}}}-{\gamma_{{\rm{q-}}BIC}}}\right|}\mathord{\left/{\vphantom{{\left|{{\gamma_{{\rm{exc}}}}-{\gamma_{{\rm{q-}}BIC}}}\right|}2}}\right.\kern-1.2pt}2}$ and $g>\sqrt{{{\left({{\gamma_{{\rm{exc}}}}^{2}+{\gamma_{{\rm{q-}}BIC}}^{2}}\right)}\mathord{\left/{\vphantom{{\left({{\gamma_{{\rm{exc}}}}^{2}+{\gamma_{{\rm{q-}}BIC}}^{2}}\right)}2}}\right.\kern-1.2pt}2}}$ . These results are a further proof that we are indeed in the strong coupling regime.

Fig. 6 shows the absorption spectra of quasi-BIC and exciton coupling with the different asymmetric parameters but at the same resonant wavelength by tuning the thickness $H$, which indicates that coupling strength g will reduce with the decrease of the asymmetric parameter. It can be described by CMT that when the dissipation loss of the quasi-BIC resonance gets close to the nonradiative decay rate of the exciton, the Rabi splitting reaches its maximumDeng et al. (2010); Piper and Fan (2014); Xiao et al. (2020b). For the smaller asymmetric parameters, the larger local electric field, accompanied by narrower line width and the dissipation loss of quasi-BIC mode, as a result of limiting the total number of photons related to the interaction with excitons. Therefore, it is important to find a balance between local electric field and spectral line width.

Finally, we also study the absorption curves of two new hybrid sates with a variable thickness $H$, but with fixed asymmetric parameters shown in Fig. 7(a). It is found that the absorption peaks of LB increases while the absorption peaks of UB decreases with the decrease of thickness $H$, which can be explained by the relative weightings of exciton and quasi-BIC in new hybrid state.

The weighting of the quasi-BIC and exciton constituents in the LB and UB can be drived from Eq. (2) and the fractions for the LB/UB polariton are

which are shown in Fig. 7(b). We can found that as the thickness $H$ increase, the exciton (quasi-BIC) fraction increases in UB (LB) and decreases in LB (UB). In other words, with the decrease of thickness in the LB, the weight of excitons decreases, which means that the number of excitons participate in the coupling decreases and the absorption summit of the lower branch decreases.