1 Introduction

Metal–semiconductor hybrid nanostructures have received increasing attention over the past few years because of their promising applications in optical sensing, light emitter, photovoltaic system, and quantum information [1–6]. The surface plasmons (SPs) in metal nanostructures can interact with the excitons in semiconductor nanostructures when they are in close proximity [5, 7]. The weak couplings between the plasmon and exciton resonances can lead to the induced transparency, enhanced absorption, increased emission, and increased optical nonlinearity [4, 8–13]. The strong plasmon–exciton couplings will modify the responses of the plasmon and exciton modes. J aggregates can strongly couple with the surface plasmon polariton [14–16] or the localized surface plasmon resonance (LSPR) [17, 18] leading to new hybridized plasmon–exciton states (plexcitons). The dispersion curves of two plexcitonic modes show an avoided crossing, which is known as Rabi splitting [5]. Fofang et al. [19, 20] further reported the enhanced and tunable nonlinear optical properties in the Au nanoshell–J aggregate complexes. As a quantum emitter (QE) locates at the center of a dimer nanoantenna, a Rabi splitting has been found in the scattering spectrum [21]. It was also found that the interaction between the quantum emitter and the gap plasmon can result in a Fano resonance in the metallic dimer–QE system [22]. In addition, Marinica et al. [23] found that the electron transfer of the excited electron into the metal can decrease the lifetime of the exciton in the QE.

Recently, Manassah [24] investigated the electrodynamics of CuCl-coated Ag nanoshells. Semiconductor CuCl exhibits a strong Z3 exciton line at about 3.202 eV [7, 25]. The plasmon–exciton couplings in the Ag–CuCl nanoshell lead to a splitting of the excitonic line and a shift in LSPR. In the dimer including a CuCl nanosphere and an Ag nanosphere, the exciton resonances of the CuCl nanosphere interact with the SP resonances of the Ag nanoparticles resulting in the induced transparency and slow light effect [8]. We also found that the plasmon–exciton couplings in the Ag nanoshell with a semiconductor CuCl core can lead to the Rabi splitting and the induced transparency by changing the geometry of the CuCl–Ag nanoshell [26]. To our knowledge, little work has been carried out on the plasmon–exciton couplings in the plexcitonic nanowires [27–29], especially on the associated arrays.

In this paper, we investigate the plasmon–exciton couplings in an Ag nanowire coated with a CuCl outer shell (ACNW) by means of the scattering theory. The strong coupling between the LSPR in the Ag nanowire and the exciton resonance in the CuCl shell leads to two new hybridized plexcitonic modes. The dispersion curves of the plexcitonic modes in the ACNWs have been studied. It is found that the destructive interference between the ω Ag|1 and ω e modes can result in the plasmon–exciton induced transparency. Furthermore, we investigate the transmission properties of the TE wave propagating through the ACNW arrays by using the finite element method (FEM).

2 Electromagnetic scattering model

The TE wave propagating through the ACNW array is shown in Fig. 1. The ACNW comprises an Ag nanowire with radius r 1 and an outer CuCl shell with thickness (r 2 − r 1). The gap between the two ACNWs is defined as d. The ACNW is infinitely long. The propagation and polarization vectors of the TE wave are parallel to y axis and x axis, respectively. The dielectric function of the Ag nanowire ε 1 has real and imaginary frequency-dependent components and can be expressed as [30]

$$\varepsilon_{1} (\omega ) = 1 - \frac{{\omega_{p}^{2} }}{{\omega^{2} + i\omega \gamma_{f} }} + \chi_{\infty }$$
(1)

where the background susceptibility χ ∞ is about 4, the bulk plasma frequency ħω p  = 8.9780 eV, and the bulk collision frequency ħγ f  = 0.0212 eV. The dielectric function of the CuCl shell ε 2 can be expressed as [24, 31]

$$\varepsilon_{2} \left( \omega \right) = \varepsilon_{\infty } + 3\pi \eta \frac{\gamma }{{\omega_{0} - \omega - i\gamma }}$$
(2)

where ε ∞ is the background dielectric function and η is the scaled atomic average density λ 3N/V. The reduced resonance wavelength λ is defined as c/ω 0. The decay rate γ/2π arises from the Z3 exciton line width. For CuCl shell, ε ∞ = 5.59, the exciton resonance frequency ħω 0 = 3.3022 eV, the loss factor ħγ = 4.9206 × 10−5 eV, and 3πη = ε ∞ ħΔLT/ħγ = 632 corresponds to the exciton oscillator strength. Here, ħΔLT = 5.65 meV is the exciton longitudinal transverse splitting. We assume that the ACNWs are embedded in vacuum with ε 3 = 1.

Fig. 1
figure 1

Schematic diagram for TE wave propagating through the ACNW array

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The incident light interacting with an infinite ACNW can be described by the scattering theory [32]. The electromagnetic waves are expanded by using vector cylindrical harmonic functions, and then, Maxwell’s boundary conditions are applied to resolve the unknown expansion coefficients of the scattered and interior waves. For TE wave, the scattering efficiency Q sca,I can be expressed as [32]

$$Q_{{{\text{sca}},I}} = \frac{2}{{kr_{2} }}\left[ {\left| {a_{0I} } \right|^{2} + 2\sum\limits_{n = 1}^{\infty } {\left( {\left| {b_{nI} } \right|^{2} + \left| {a_{nI} } \right|^{2} } \right)} } \right]$$
(3)

where a nI and b nI are the scattering coefficients. a 0I and a 1I correspond to the magnetic (MD) and electric (ED) dipole modes, respectively [33]. The TE wave propagating through the ACNW array can be studied by using the 2D-FEM [34]. The FEM has been proved to be an efficient numerical tool to investigate the optical properties of the metallic nanostructures [35–37]. We calculate the scattering parameters S11 and S21 of the ACNW array. Scattering parameters originate from transmission-line theory. In our model, the incident and transmission ports are defined as ports 1 and 2, respectively. S11 is the reflection coefficient at port 1, and S21 is the transmission coefficient from port 1 to port 2. The time average power reflection and transmission coefficients can be obtained as R = |S11|2 and T = |S21|2, respectively.

3 Results and discussion

We firstly investigate the scattering properties of the ACNWs. In Fig. 2a, the solid and dashed lines show the scattering spectra of an Ag nanowire and a CuCl nanotube, respectively. The radius of the Ag nanowire is fixed at 20 nm. For the CuCl nanotube, r 1 and r 2 are fixed at 20 and 25 nm, respectively, and ε 1 is fixed at 1. A broad peak appears at about 3.2675 eV in the scattering spectrum of the Ag nanowire, which is due to the dipole plasmon resonance. A very sharp exciton peak can be found at about 3.3040 eV in the scattering spectrum of the CuCl nanotube. Figure 2b shows the scattering spectrum of the ACNW. Here, r 1 and r 2 are fixed at 20 and 25 nm, respectively. Figure 2c, d represents the contributions from the ED and MD resonances, respectively. The weak peak at about 3.3015 eV corresponds to the MD resonance in the ACNW, which arises from the high real part of the permittivity of CuCl. The high-permittivity nanowires have been reported to support the artificial magnetic resonances [33]. In the scattering spectrum of the ED mode, the peaks at about 3.3307 and 3.2394 eV correspond to the high- and low-energy plexcitonic modes, respectively. Compared with the plasmon resonance in the Ag nanowire and the exciton resonance in the CuCl nanotube, we can confirm that the strong coupling between the plasmon and exciton resonances occurs in the ACNW. In the ACNW, the strong coupling between the dipole plasmon resonance in the Ag nanowire (ω Ag|1) and the exciton resonance (ω e ) in the CuCl shell results in two new plexcitonic modes. The high-energy mode \((\omega_{\text{Ag}}^{e} |^{h}_{1} )\) and low-energy mode \((\omega_{\text{Ag}}^{e} |^{l}_{1} )\) correspond to the antibonding and bonding between the ω Ag|1 and ω e modes, respectively [18, 19]. A scattering dip can be observed at about 3.2988 eV, which results from the destructive interference between the ω Ag|1 and ω e modes [18, 21]. We also note that a weak peak appears at about 3.3067 eV. It is found with increasing the CuCl shell thickness that this peak always locates at about 3.3067 eV, but the strength of this peak is increased, as shown in Fig. 3. We think that this peak may arise from the interband transition, which corresponds to the bare exciton transition with frequency of ω 0.

Fig. 2
figure 2

a Scattering spectra of an Ag nanowire and a CuCl nanotube. b Scattering spectrum of the ACNW with r 1 = 20 nm and r 2 = 25 nm. Contributions from c the electric dipole mode and d the magnetic dipole mode

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Fig. 3
figure 3

Scattering spectra of the ACNWs with r 2 of a 21, b 23, c 25, d 27, and e 29 nm. Here, r 1 is fixed at 20 nm

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Figure 3 shows the scattering spectra of the ACNWs with r 2 of (a) 21, (b) 23, (c) 25, (d) 27, and (e) 29 nm. Here, r 1 is fixed at 20 nm. In Fig. 3a, a strong peak appears at about 3.5247 eV and a sharp peak appears at about 3.2997 eV. The peaks at about 3.5247 and 3.2997 eV represent the ω Ag|1 mode in the Ag nanowire and the ω e mode in the CuCl shell, respectively. In this case, the line width of the ω e mode is much smaller than that of the ω Ag|1 mode. The sharp ω e peak interferes with the broad ω Ag|1 peak leading to a Fano resonance in the scattering spectrum. The Fano resonances due to the plasmon–exciton couplings have been widely reported in many previous papers [18, 19, 21]. Therefore, an asymmetric Fano peak can be observed at about 3.2997 eV. In Fig. 3b, the increased thickness of the CuCl shell leads to the redshift of the LSPR peak. The increased thickness of the CuCl shell decreases the oscillation strength of the conduction electrons in the Ag nanowire, and hence, the energy of the ω Ag|1 mode decreases. Thus, the ω Ag|1 peak can be tuned through the exciton resonance of the CuCl shell with increasing r 2 values. Meanwhile, the increased thickness of the CuCl shell also will enhance the strength of the exciton peak. However, in Fig. 3b, the line width of the ω e peak is still much smaller than that of ω Ag|1 peak. The interference between the ω Ag|1 and ω e peaks leads to a Fano resonance in the scattering spectrum of the ACNW. As r 2 increases to 25 nm, a true Rabi splitting can be found in Fig. 3c. The peaks at about 3.3307 and 3.2394 eV correspond to the \(\omega_{\text{Ag}}^{e} |^{h}_{1}\) and \(\omega_{\text{Ag}}^{e} |^{l}_{1}\) modes, respectively. A transparency dip also can be observed in the scattering spectrum at about 3.2988 eV, which results from the destructive interference between the ω Ag|1 and ω e modes. In Fig. 3d, e, with further increases in the r 2 value, the \(\omega_{\text{Ag}}^{e} |^{h}_{1}\) and \(\omega_{\text{Ag}}^{e} |^{l}_{1}\) peaks both redshift and the separation between the two peaks increases. As r 2 ≥ 25 nm, the MD mode can be found in the scattering spectra because of the large thickness of the CuCl shell.

It was found with increasing the CuCl shell thickness that the ω Ag|1 peak of the Ag nanowire can be tuned through the ω e peak of the CuCl shell. Therefore, we can carefully study the coherent coupling energy in the ACNW. Figure 4 shows the dispersion curves of two plexcitonic modes in the ACNWs. Here, r 1 is fixed at 20 nm. The dashed-square line shows the resonance energy of the \(\omega_{\text{Ag}}^{e} |^{h}_{1}\) mode, while the dashed-circle line represents the resonance energy of the \(\omega_{\text{Ag}}^{e} |^{l}_{1}\) mode. The solid and dashed lines show the resonance energies of the uncoupled exciton and plasmon modes, respectively. We can find an avoided crossing in the dispersion curves. The Rabi splitting energy is calculated to be about 88 meV, which indicates the strength of the plasmon–exciton coupling in the ACNW. The Rabi splitting energies in different plasmon–exciton complexes can change from a few millielectron volts to several hundreds of millielectron volts [17–19, 21, 38]. The small Rabi splitting energy in the ACNW is due to the narrow exciton line and the large dielectric constant of CuCl. Some previous reports have carefully investigated the transition from Fano resonance to Rabi splitting [18, 21]. It is known that the Rabi splitting occurs when the coupling energy is larger than (γ LSPR − γ 0)/2 [18]. Here, γ LSPR and γ 0 are the relative resonance line widths of the LSPR and exciton modes, respectively. Based on the above calculations, we can confirm that the ACNW can satisfy this criterion, as (γ LSPR − γ 0)/2 ≈ 75 meV.

Fig. 4
figure 4

Dispersion curves of the plexcitonic modes in the ACNWs. Here, r 1 is fixed at 20 nm

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Figure 5a shows the scattering spectrum of the ACNW with r 1 = 20 nm and r 2 = 25 nm, which was calculated by using the 2D-FEM. This scattering spectrum matches well with that obtained by using the scattering theory. Figure 5b–d shows the transmission spectra of the ACNW arrays with d = 2, 10, and 20 nm, respectively. In Fig. 5b, an extraordinary transmittance can be found at about 3.2988 eV, which accords with the location of the scattering dip in the ACNW. In Fig. 5c, d, the extraordinary transmittances of the ACNW arrays always locate at about 3.2988 eV. Thus, the extraordinary transmittance of the ACNW array does not depend on the distance between the two ACNWs. In addition, we also have calculated the transmission spectra of the Ag nanowire arrays. In Fig. 6, the solid, dashed, and dotted lines show the transmission spectra of the arrays of ACNWs, Ag nanowires, and Ag–dielectric nanowires, respectively. Here, the d value for the ACNW array is fixed at 2 nm. In the Ag nanowire array, the radius of the Ag nanowire and the d value are fixed at 20 and 7 nm, respectively. For the Ag–dielectric array, r 1 = 20 nm, r 2 = 25 nm, ε 2 = 5.59, and d = 2 nm. It is obvious that the extraordinary transmittance only happens in the ACNW array, which arises from the plasmon–exciton coupling.

Fig. 5
figure 5

a Scattering spectrum of the ACNW with r 1 = 20 nm and r 2 = 25 nm. Transmission spectra of the ACNW arrays with b d = 2 nm, c d = 10 nm, and d d = 20 nm. Here, r 1 and r 2 are fixed at 20 and 25 nm, respectively

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Fig. 6
figure 6

Transmission spectra of the arrays of ACNWs, Ag nanowires, and Ag–dielectric nanowires

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Figure 7a–c represents the visualizations of the TE wave propagating through the ACNW arrays with d = 2, 10, and 20 nm, respectively. The frequency is fixed at 3.2988 eV. The three ACNW arrays are transparent at this frequency, and the d value does not affect the transmission of the TE wave. Therefore, the plasmon–exciton induced transparency in the ACNW array arises from the single ACNW, which should be a useful meta-atom for designing the metamaterials. Figure 7d, e shows the TE wave propagating through the ACNW array at 3.3307 \((\omega_{\text{Ag}}^{e} |^{h}_{1} )\) and 3.2394 eV \((\omega_{\text{Ag}}^{e} |^{l}_{1} )\), respectively. Here, d is fixed at 20 nm. At the two modes, the ACNW array becomes opaque because of the plasmon–exciton couplings in the ACNWs and the interactions between the adjacent nanowires.

Fig. 7
figure 7

Electric field (E x ) distributions of the TE wave propagating through the ACNW arrays with a d = 2 nm, b d = 10 nm, and c d = 20 nm. Here, the calculation frequency is fixed at 3.2988 eV. Electric field (E x ) distributions of the TE wave propagating through the ACNW array with d = 20 nm at d 3.3307 eV and e 3.2394 eV

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In addition, we should indicate that the above results may have some changes for the finite-length nanowire. For the finite-length ACNW, if the length of the nanowire is much longer than the diameter, the influences of the two cross sections on the plasmon resonances and couplings are weak and can be neglected. However, when the length of the nanowire is comparable to the diameter, the effects of the two cross sections on the plasmon resonances and couplings should be considered. Under this condition, as the incident polarization is parallel to the axis of the nanowire, the collective motions of the conduction electrons in the two cross sections will occur, and hence, another plasmon resonance mode will be found in the Ag nanowire. As the incident polarization is perpendicular to the axis of the nanowire, the localizations from the two cross sections also will affect the plasmon resonances in the Ag nanowire. Therefore, the corresponding plasmon–exciton couplings in the finite-length ACNWs should be different to those in the infinite-length ACNWs. We will further investigate the plasmon–exciton couplings in the finite-length ACNWs in our next work.

4 Conclusions

We have investigated the plasmon–exciton coupling properties of the ACNWs and associated arrays by using the scattering theory and the 2D-FEM. It is found with increasing the CuCl shell thickness that the ω Ag|1 mode can move through the ω e mode. The Rabi splitting can occur in the ACNW. The strong coupling between the ω Ag|1 and ω e modes leads to the hybridized \(\omega_{\text{Ag}}^{e} |^{h}_{1}\) and \(\omega_{\text{Ag}}^{e} |^{l}_{1}\) modes, which correspond to the antibonding and bonding between the ω Ag|1 and ω e modes, respectively. The dispersion curves of the plexcitonic modes in the ACNWs have been studied, and the obtained Rabi splitting energy is about 88 meV. Furthermore, the sharp ω e peak interacts with the broad ω Ag|1 peak resulting in a deep dip in the scattering spectra of the ACNWs. As the TE wave propagates through the ACNW array, an extraordinary transmittance can be observed at the scattering dip for the ACNW. This induced transparency does not depend on the distance between the two ACNWs, which arises from the single ACNW.