⇦Chinese Physics Letters, 2019, Vol. 36, No. 10, Article code 107301 Strong Exciton-Plasmon Coupling and Hybridization of Organic-Inorganic Exciton-Polaritons in Plasmonic Nanocavity * Ping Jiang (江平)1,2, Chao Li (李超)1,2, Yuan-Yuan Chen (陈园园)3, Gang Song (宋钢)1,2, Yi-Lin Wang (王艺霖)1,2, Li Yu (于丽)1,2** Affiliations 1State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876 2School of Science, Beijing University of Posts and Telecommunications, Beijing 100876 3China South Industries Research Academy, Beijing 100089 Received 8 July 2019, online 21 September 2019 *Supported by the National Key Research and Development Program of China under Grant No 2016YFA0301300, the National Natural Science Foundation of China under Grant Nos 11574035 and 11374041, and the State Key Laboratory of Information Photonics and Optical Communications.
**Corresponding author. Email: yuliyuli@bupt.edu.cn Citation Text: Jiang P, Li C, Chen Y Y, Song G and Wang Y L et al 2019 Chin. Phys. Lett. 36 107301    Abstract We investigate strong exciton-plasmon coupling and plasmon-mediated hybridization between the Frenkel (F) and Wannier–Mott (WM) excitons of an organic-inorganic hybrid system consisting of a silver ring separated from a monolayer WS$_{2}$ by J-aggregates. The extinction spectra of the hybrid system calculated by employing the coupled oscillator model are consistent with the results simulated by the finite-difference time-domain method. The calculation results show that strong couplings among F excitons, WM excitons, and localized surface plasmon resonances (LSPRs) lead to the appearance of three plexciton branches in the extinction spectra. The weighting efficiencies of the F exciton, WM exciton and LSPR modes in three plexciton branches are used to analyze the exciton-polaritons in the system. Furthermore, the strong coupling between two different excitons and LSPRs is manipulated by tuning F or WM exciton resonances. DOI:10.1088/0256-307X/36/10/107301 PACS:73.20.Mf, 11.15.Me, 52.25.Tx, 71.36.+c © 2019 Chinese Physics Society Article Text Exciton-surface plasmon polaritons are quasiparticles formed in a plasmonic nanocavity when plasmons couple to excitons such that there is a reversible exchange of energy between the plasmon and exciton. If the rate of the energy exchange between plasmons and excitons becomes faster than the decoherence rates of both individual systems, the system is said to be strongly coupled.[1–6] The strong coupling between the organic Frenkel (F) excitons or inorganic Wannier–Mott (WM) excitons and surface plasmon polaritons (SPPs) has been widely studied.[7–12] SPPs are considered to have inestimable potential application in realization of highly integrated optical circuits.[13,14] F excitons are endowed with large oscillator strength and strong binding energy but smaller scattering cross sections.[8,15] In contrast, WM excitons have smaller oscillator strengths and lower binding energies but large exciton-exciton interaction cross sections.[8] The J-aggregate dye molecular exciton is a kind of F excitons, and two-dimensional transition metal dichalcogenide (TMDs) excitons are WM-type excitons.[16–20] The strong coupling between TMDs and plasmonic nanocavities can be actively controlled via temperature,[2,21,22] electrostatic gating,[16] and femtosecond pumping.[23] Recently, the strong coupling between different excitons and optical microcavities attracts great attention.[12,16,24–28] For example, the simultaneous strong coupling of two different excitons and a photon forms hybrid organic-inorganic polaritons.[12] The strong exciton-plasmon-exciton coupling system consists of a Fabry–Perot hybrid microcavity containing two strongly coupled J-aggregated cyanine dyes is used to study energy transfer processes in light-harvesting.[24] However, strong interactions among localized surface plasmon resonances (LSPRs), organic J-aggregate dye molecular excitons (F excitons), and inorganic WS$_{2}$ excitons (WM excitons) have hardly been studied, which may find application in multimode sensors and multimode lasers.[28] We investigate here the strong couplings among F excitons, WM excitons and LSPRs origin from the Ag-J-aggregates-WS$_{2}$ hybrid system using rigorous electrodynamic simulations based on the finite-difference time-domain (FDTD) method. Using experimental optical properties for J-aggregates and WS$_{2}$, we calculate the extinction spectra of the hybrid system. There are three plexciton branches in the extinction spectra. The extinction spectrum calculated by the FDTD and the corresponding three plexciton branches can be described phenomenologically using a classical coupled-oscillator model. The strong couplings between two different excitons and LSPRs can be manipulated by tuning the temperature or the oscillator strength of J-aggregates. The proposed systems make up a simple platform for the dynamic control of F excitons, WM excitons and LSPRs coupling and have potential applications in inversionless lasing, superfluidity and topologically non-trivial states.[29] Figure 1(a) shows the schematic diagram of the hybrid nanostructure composed of a silver ring separated from a monolayer WS$_{2}$ by J-aggregates. The Ag rings are characterized with the inner diameters in the range of 20–40 nm and outer diameters 80 nm, thickness 20 nm. The solid blue line in Fig. 1(b) shows the normalized extinction spectrum of a single Ag ring, which indicates the resonance energy of the plasmon. The thicknesses of J-aggregates and monolayer WS$_{2}$ are 5 nm and 1 nm, respectively. The simulation accuracy in the FDTD for the Ag ring is 0.5 nm in the $x$, $y$ and $z$ directions. The mesh size for the J-aggregates and monolayer WS$_{2}$ are both 0.5 nm in the $x$ and $y$ directions, and 0.1 nm in the $z$ direction. The permittivities of silver in our simulation are from Johnson and Christy data.[30] The dielectric function of J-aggregates and the monolayer WS$_{2}$ as a function of photon energy can be described by the superposition of several Lorentz models.[19,31–33] We can obtain the J-aggregate exciton resonance at $E_{\rm F} =2.08$ eV, and $\gamma_{\rm F}=50$ meV is the corresponding damping rate as shown in Fig. 1(b). The oscillator strength of J-aggregates varies from 0.02 to 0.1 in different experiments.[34,35] In our FDTD simulations, the oscillator strength of J-aggregates is fixed at 0.05. Furthermore, we can obtain that the A exciton resonance of WS$_{2}$ varies with temperature by the O'Donell model.[2,21,36]
cpl-36-10-107301-fig1.png
Fig. 1. (a) Schematic diagram of the Ag-J-aggregates-WS$_{2}$ hybrid nanostructure. The inset shows the cross section of the hybrid nanostructure with the thickness of Ag ring 20 nm. The thickness of J-aggregates and monolayer WS$_{2}$ are 5 nm and 1 nm, respectively. (b) The solid blue line is the normalized extinction spectrum of a single Ag ring. The solid red line is the normalized absorption spectrum of J-aggregates.
Finite-difference time-domain methods are used to calculate the extinction spectra of the hybrid nanostructures. We further fit the calculated extinction spectra using a phenomenological coupled-oscillator model.[37,38] In this model, F exciton, WM exciton and LSPR are described as damped harmonic oscillators. We can obtain the total extinction cross section of the hybrid system as a function of resonance energies, $$\begin{align} C_{\rm ext} (E)\propto\,&E{\rm Im}\{(ab)/[ab(E^{2}-E_{\rm PL}^{2}+iE\gamma_{\rm PL} )\\ &-E^{2}g_{\rm F}^{2}b-E^{2}g_{\rm WM}^{2}a]\},~~ \tag {1} \end{align} $$ where $a=E^{2}-E_{\rm F}^{2}+iE\gamma_{\rm F}$, $b=E^{2}-E_{\rm WM}^{2}+iE\gamma_{\rm WM}$, and $E_{\rm PL}$, $E_{\rm F}$, and $E_{\rm WM}$ represent the energies of plasmon, F and WM excitons, respectively.
cpl-36-10-107301-fig2.png
Fig. 2. (a) Simulated the transmission spectrum of the monolayer WS$_{2}$ flake. (b) Normalized extinction spectrum of WS$_{2}$ coupled with Ag ring simulated by FDTD solution (red solid line) and calculated by the coupled oscillator model (blue circle). (c) Successive extinction spectra of WS$_{2}$ coupled with Ag rings as a function of LSPR energy. The black scattered-squares and scattered-triangles represent simulated upper and lower energies. The solid white lines are fitted to the coupled oscillator model. The dashed diagonal line and horizontal line correspond to uncoupled LSPR mode and WM exciton resonance, respectively. (d) Weighting efficiencies for LSPR mode and WM exciton contributions to UPB and LPB states as a function of LSPR energy.
For a plasmonic nanocavity with coupling between a plasmon mode and F and WM excitons, the dispersion relation is modeled as a third excitonic oscillator,[7–12] $$ \left[\begin{matrix} {E_{\rm PL} } & {V_{\rm F} } & {V_{\rm WM} } \\ {V_{\rm F} } & {E_{\rm F} } & 0 \\ {V_{\rm WM} } & 0 & {E_{\rm WM} } \\ \end{matrix}\right]\left[ {{ \begin{matrix} \alpha \\ \beta \\ \gamma \\ \end{matrix} }} \right]=E\left[ {{ \begin{matrix} \alpha \\ \beta \\ \gamma \\ \end{matrix} }} \right],~~ \tag {2} $$ where $E$ represents the eigenvalues, $|\alpha|^{2}$, $|\beta|^{2}$ and $|\gamma|^{2}$ represent the weighting efficiencies, $V_{\rm F}$ is the interaction potential between the plasmon mode and F excitons, and $V_{\rm WM}$ is the interaction potential between the plasmon mode and WM excitons. We note that the damping losses are not taken into account, because of their weak effects on the fitting results.[24,27,28] To investigate the strong coupling between the plasmon mode and WM excitons, we first describe the dielectric function of the WS$_{2}$ monolayer by the superposition of several Lorentzian oscillators.[19,31,32] Figure 2(a) gives the simulated transmission spectra of the monolayer WS$_{2}$ flake on quartz, which is consistent with the experimental result in Ref. [31]. We can clearly obtain the A exciton resonance of WS$_{2}$ at $E_{\rm WM} =2.02$ eV and the corresponding damping rate $\gamma_{\rm WM} =28$ meV as shown in Fig. 2(a). Figure 2(b) shows the normalized extinction spectrum of WS$_{2}$ coupled with Ag ring simulated by FDTD solution (red solid line) and calculated by the coupled oscillator model (blue circle),[37] $$\begin{align} C_{\rm ext} (E)\propto\,&E{\rm Im}\{(E^{2}-E_{\rm WM}^{2}+iE\gamma_{\rm WM})/[(E^{2}\\ &-E_{\rm WM}^{2}+iE\gamma_{\rm WM})(E^{2}-E_{\rm PL}^{2}\\ &+iE\gamma_{\rm PL} )-E^{2}g_{\rm WM}^{2}]\}.~~ \tag {3} \end{align} $$ The calculated extinction spectrum by the coupled oscillator model is consistent with the simulated one. Simulated plasmon-dependent successive extinction spectra of WS$_{2}$ coupled with Ag rings are shown in Fig. 2(c). The solid white lines are fitted to the coupled oscillator model, $$ \left[{{\begin{matrix} {E_{\rm PL} -i\frac{\gamma_{\rm PL} }{2}} & {V_{\rm WM} } \\ {V_{\rm WM} } & {E_{\rm WM} -i\frac{\gamma_{\rm WM} }{2}} \\ \end{matrix} }} \right]\left[ {{\begin{matrix} \alpha \\ \beta \\ \end{matrix} }} \right]=E\left[ {{\begin{matrix} \alpha \\ \beta \\ \end{matrix} }} \right],~~ \tag {4} $$ where $E_{\rm PL}$ varies with the inner diameters of Ag ring. In this case, $E_{\rm PL}$ changes from 1.8 eV to 2.3 eV and $V_{\rm WM}=100$ meV. Here $\gamma_{\rm PL} =186$ meV is the average damping loss of LSPRs. The energy of the upper plexciton branches (UPB) and lower plexciton branches (LPB) exhibits a clear anticrossing behavior, which indicates the strong coupling between WM exciton and LSPR mode. The Rabi splitting ($\hslash {\it \Omega} =184$ meV) is extracted as the minimal splitting between the two polariton branches, which is obtained at zero detuning ($\Delta =E_{\rm PL} -E_{\rm WM} =0$). It satisfies the criteria for strong coupling $\hslash {\it \Omega} >(\gamma_{\rm PL} +\gamma_{\rm WM} )/2$. In Fig. 2(d), the weighting efficiencies $|\alpha|^{2}$ and $|\beta|^{2}$ are plotted versus LSPRs for each branch of the hybrid dispersion relation from Fig. 2(c). The two plexciton branches have symmetrically varying amounts of plasmon and WM exciton character.
cpl-36-10-107301-fig3.png
Fig. 3. (a) Top view of electromagnetic energy density distribution within the Ag ring, J-aggregates and WS$_{2}$ at $\lambda=606$ nm. (b) The corresponding charge distributions of top view. (c) Side view of electromagnetic energy density distribution within the Ag ring, J-aggregates and WS$_{2}$ at $\lambda=606$ nm. (d) The corresponding charge distributions of side view.
Let us now turn attention to the strong couplings among F excitons, WM excitons and LSPRs in Ag-J-aggregates-WS$_{2}$ hybrid nanostructure. We first inspect the normalized electric field distributions in the coupled system (Fig. 3). We observe that the mode in the Ag ring is confined mainly to its edges as shown in Figs. 3(a) and 3(c). The charge densities of the hybrid structure can be calculated by solving Poisson's equation. The corresponding charge distributions revealed dipole mode and surface charge distribution of the hybrid system at 606 nm as shown in Figs. 3(b) and 3(d). The large electric field enhancements and the maximizing overlap between the plasmonic field mode and the exciton states are conducive to achieving strong coupling in the hybrid nanostructure.[10,39,40]
cpl-36-10-107301-fig4.png
Fig. 4. (a) Normalized extinction spectrum of J-aggregates and WS$_{2}$ coupled with Ag ring simulated by FDTD solution (red solid line) and calculated by the coupled oscillator model (blue circle). (b) Successive extinction spectra of J-aggregates and WS$_{2}$ coupled with Ag rings as a function of LSPR energy. The black scattered-squares, scattered-circles and scattered-triangles represent simulated upper, middle and lower energies. The solid white lines are fitted to the coupled oscillator model. The dashed diagonal line and horizontal line correspond to uncoupled LSPR mode, F exciton and WM exciton, respectively. (c) Weighting efficiencies for LSPR mode, F exciton and WM exciton contributions to UPB, MPB and LPB states as a function of LSPR energy.
Figure 4(a) shows the normalized extinction spectrum of J-aggregates and WS$_{2}$ coupled with Ag ring simulated by FDTD solution (red solid line) and calculated by the coupled oscillator model based on Eq. (1) (blue circle). The calculated extinction spectrum by the coupled oscillator model matches well with the simulated one, which indicates that strong coupling can be well described by the coupled oscillator model. Simulated plasmon-dependent successive extinction spectra of J-aggregates and WS$_{2}$ coupled with Ag rings are shown in Fig. 4(b). The simulated extinction spectra can be fitted by a three-oscillator model with Eq. (2). The solid white lines in Fig. 4(b) indicate three anticrossed bands corresponding to upper (UPB), middle (MPB), and lower (LPB) plexciton branches. The solid white lines are consistent with the simulated results. The strong couplings among F excitons, WM excitons and LSPRs result in two anticrossings at around $E_{\rm WM} =2.02$ eV and $E_{\rm F} =2.08$ eV, respectively. Here $V_{\rm WM} =60$ meV and $V_{\rm F} =70$ meV are the corresponding interaction potentials, respectively. The interaction potentials can be related to the Rabi splitting energy using $\hslash {\it \Omega} =2$ V. Here $\hslash {\it \Omega}_{\rm MPB-LPB} =2V_{\rm WM} =120$ meV and $\hslash {\it \Omega}_{\rm UPB-MPB} =2V_{\rm F} =140$ meV are the Rabi splitting energies for the WM excitons and F excitons, respectively. Both of them individually satisfy the simplified strong coupling criterion as $\hslash {\it \Omega}_{\rm MPB-LPB} >(\gamma_{\rm PL} +\gamma_{\rm WM} )/2$ and $\hslash {\it \Omega}_{\rm UPB-MPB} >(\gamma_{\rm PL} +\gamma_{\rm F} )/2$. Figure 4(c) shows the weighting efficiencies for LSPR mode, F exciton and WM exciton contributions to UPB, MPB and LPB states as a function of LSPR energy calculated by Eq. (2). The three plexciton branches exhibit significant mixing between plasmon and two different excitons. In the UPB, when $E_{\rm PL} =2.06$ eV, the branch character consists of equal parts F exciton and plasmon ($\sim $42.5%), and $\sim $15% WM exciton character. In the MPB, when $E_{\rm PL} =2.07$ eV, the branch character consists of equal parts WM exciton and F exciton ($\sim $45%), and $\sim $10% plasmon character. In the LPB, when $E_{\rm PL} =2.05$ eV, the branch character consists of equal parts WM exciton and plasmon ($\sim $45%), and $\sim $10% F exciton character. These weighting efficiencies represent the effect and contribution of plasmons, WM exciton, and F exciton to each plexciton branch.
cpl-36-10-107301-fig5.png
Fig. 5. (a) Normalized extinction spectra of the hybrid system with different LSPRs calculated by Eq. (1). (b) Normalized extinction spectra with the damping rate of plasmon ($\gamma_{\rm PL}$) in the range of 80–240 meV. (c) Normalized extinction spectra with the coupling rate ($g_{\rm WM}$) between plasmon and WM exciton in the range of 60–140 meV. (d) Normalized extinction spectra with the coupling rate ($g_{\rm F}$) between plasmon and F exciton in the range of 110–190 meV.
We further use Eq. (1) to calculate extinction spectra of the hybrid system as shown in Fig. 5. Here $E_{\rm F} =2.08$ eV, $\gamma_{\rm F} =50$ meV, $E_{\rm WM} =2.02$ eV and $\gamma_{\rm WM} =28$ meV are consistent with the experimental results.[31] Figure 5(a) shows the normalized extinction spectra of the hybrid system with resonance energies of plasmon in the range of 1.6–2.3 eV and the damping rate of plasmon $\gamma_{\rm PL}=164$ meV. The extinction spectra computed by Eq. (1) are consistent with the results calculated by the FDTD. Figure 5(b) shows the normalized extinction spectra of the hybrid system with resonance energy of plasmon $E_{\rm PL} =2.07$ eV and the damping rates of plasmon $\gamma_{\rm PL}$ in the range of 80–240 meV. It can be seen that, for a given coupling rates $g_{\rm F}$ and $g_{\rm WM}$, depths of the splitting dip in the extinction spectra decrease with the increase of the damping rates. Increasing the coupling rate $g_{\rm WM}$, keeping $g_{\rm F}$ unchanged, fixing the damping rate of plasmon to be $\gamma_{\rm PL} =164$ meV lead to stronger interaction between WM excitons and LSPRs, which in turn increases the splitting and depths of the splitting dip between the left peak and middle peak in the extinction spectra, as shown in Fig. 5(c). Increasing the coupling rate $g_{\rm F}$, keeping $g_{\rm WM}$ unchanged, and fixing the damping rate of plasmon to be $\gamma_{\rm PL} =164$ meV also lead to stronger interaction between F excitons and LSPRs, which in turn increases the splitting and depths of the splitting dip between the middle peak and right peak in the extinction spectra, as shown in Fig. 5(d). We note that the parameters ($\gamma_{\rm PL}$, $g_{\rm WM}$ and $g_{\rm F}$) are interconnected and cannot be optimized independently. However, tuning these parameters individually can help us understand the strong couplings among WM excitons, F excitons and LSPRs intuitively.
cpl-36-10-107301-fig6.png
Fig. 6. (a) Normalized extinction spectra of J-aggregates and WS$_{2}$ coupled with Ag rings for different temperatures. (b) Normalized extinction spectra of J-aggregates and WS$_{2}$ coupled with Ag rings for different oscillator strengths of J-aggregates.
To further understand the strong coupling behavior in the Ag-J-aggregates-WS$_{2}$ hybrid nanostructure, we modulate the strong coupling by tuning the temperature or the oscillator strength of J-aggregates. The normalized extinction spectra of the hybrid system for the temperature range from 200 K to 400 K are shown in Fig. 6(a). The red dashed line and blue dashed line in Fig. 6(a) indicate that low energy peaks and middle energy peaks of extinction spectra have a red shift with the increase of temperature. This phenomenon can be explained by the fact that WM exciton resonance peaks have a red shift with the increase of temperature and the large contributions of WM excitons character to the LPB and MPB. The green dashed line in Fig. 6(a) indicates that high energy peaks of extinction spectra hardly change with the increase of the temperature, which can be attributed to the slight contribution of WM excitons to the UPB in the temperature range from 200 K to 400 K. The normalized extinction spectra of the hybrid system for oscillator strength of F excitons in the range from 0.02 to 0.1 are depicted in Fig. 6(b). The green dashed line and blue dashed line in Fig. 6(b) indicate that the splitting between high energy peaks and middle energy peaks increases with the oscillator strength. However, the splitting between middle energy peaks and low energy peaks decreases with the increase of oscillator strength. These phenomena can be attributed to the large contribution of F excitons to the UPB and MPB as shown in Fig. 4(c), and the strong coupling between plasmon and the WM excitons is suppressed by the strong coupling between plasmon and F excitons with the increase of oscillator strength of F excitons. Therefore, the strong couplings among F excitons, WM excitons and LSPRs could be modulated by tuning the temperature or the oscillator strength of J-aggregates. In summary, we have theoretically studied strong exciton-plasmon coupling between the F exciton of organic J-aggregates and the WM exciton of inorganic WS$_{2}$ through their mutual interaction with a plasmonic nanocavity. The large electric field enhancements and the maximizing overlap between the plasmonic field mode and the exciton states result in three plexciton branches in the extinction spectra of the hybrid system. The extinction spectrum of the hybrid system calculated by the FDTD method is well reproduced by the coupled oscillator model. In addition, the coupled oscillator model is used to analyze the strong coupling behavior and to obtain the weighting efficiencies of the original modes in three plexciton branches. We could further modulate the strong couplings among F excitons, WM excitons and LSPRs by tuning the temperature or the oscillator strength of J-aggregates. 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