1. Introduction

Exploiting exciton-plasmon interaction to manipulate optical properties of metal-semiconductor nanostructures is of fundamental importance [1–5]. One of these nanostructures of interest is a nanodimer composed of a semiconductor quantum dot (SQD) and a metal nanoparticle (MNP), which has been subjected to increasing investigation [6–9]. Since the pioneering work of Zhang and coworkers who demonstrated that the exciton-plasmon interaction can lead to some interesting phenomena such as Förster energy transfer, exciton energy shift, and nonlinear Fano resonance [10], a lot of other intriguing optical features arisen from strong exciton-plasmon coupling have also been observed [11–21]. Examples include irreversible ultrafast properties [11], radiative and nonradiative energy transfer [12], tunable emission polarization [13], gain without inversion [14,15], magnetically induced optical transparency [16], controllable Kerr nonlinearity [17], enhanced coherent plasmonic field [11], modified four-wave mixing (FWM) effect [18,19], improved resonance fluorescence properties [20,21], and so on.

Optical bistability (OB) in the SQD/MNP hybrid nanosystem has also been extensively investigated both theoretically and experimentally [22–32]. In a seminal work, Artuso and Bryant found that the bistable effect can be controlled by only changing the sizes of two NPs in a strongly coupled SQD/MNP nanodimer [22]. Malyshev et al. demonstrated that OB can be observed by measuring the optical hysteresis of Rayleigh scattering intensity [23]. Li et al. observed the bistable feature by detecting third-order nonlinear absorption and refraction [24,25]. Nugroho et al. proposed a fresh way to reveal the bistability mechanism and calculated the switching time between two stable branches in the SQD/MNP heterodimer [26]. Also, Asadpour et al. contributed some significant results in this area. They not only studied the role of a two-dimensional array of metal-coated dielectric nanospheres in modulating OB and optical multistability (OM) of a four-level quantum system embedded in a unidirectional ring cavity [27,28], but also demonstrated that the switching between OB and OM is feasible in a unidirectional cavity consisting of a hybrid SQD-MNP molecule [29]. Paspalakis et al. calculated the optical rectification coefficient and found that it is bivalued when the bistable effect arises [30]. In an interesting work of Mari et al., they reported that the fluorescence spectrum is strongly correlated to the initial state of the system when the strong transition and bistable regions appear [31]. In a core-shell NP-QD nanosystem, Miri et al. plotted bistability phase diagrams and found that the borders between Fano, double peaks, weak transition, strong transition, and bistability regions are strongly correlated to the thickness and chemical composition of both core and shell layers [32].

As is well known, surface plasmons of the naked MNP are present at its surface, however, surface plasmons of the metallic nanoshell (MNS) made by a metallic core and a dielectric shell appear at the interface between the core and the shell. Additionally, the dielectric environment of the MNS is obviously different from that of the naked MNP. It is conceivable that these features of the MNS will bring forth some novel optical phenomena in a SQD/MNS heterodimer. Considering this, in this paper we will explore the conditions for producing OB in a heterodimer composed of a SQD and a MNS. We investigate the influence of the shell thickness of the MNS on OB when the pump field is resonant or near-resonant with the exciton transition. We map out bistability phase diagrams in the system’s parameter subspace. We also discuss the role of the multipole polarization in controlling OB.

2. Theoretical model and method

As depicted schematically in Fig. 1, a heterodimer under consideration is composed of a SQD and a MNS. The heterodimer is simultaneously subjected to two applied fields consisting of a strong pump field (E_pu, ω_pu) and a weak probe field (E_pr, ω_pr). The MNS is made by a metallic core and a dielectric shell. The radius of the metallic core is denoted as R₁, the total radius of the MNS is denoted as R₂, and the thickness of the dielectric shell is R₂ − R₁. The SQD can be modeled as a two level system with a ground state |g > and a single exciton state |e > . The dielectric constants for the background, SQD, core and shell of the MNS refer, respectively, to ε_b, ε₁, ε_m and ε_s. The center-to-center distance between the SQD and the MNS is denoted as d.

 

Fig. 1. Schematic of a heterodimer consisting of a SQD and a MNS.

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The total Hamiltonian of the SQD-MNS heterodimer in the rotating frame takes the form [24,33]:

The population inversion (w₀) of the exciton in the SQD can be derived by [24]

3. Results and discussions

We consider a hybrid heterodimer consisting of a CdSe QD and a MNS. The MNS is made of an Au core and the SiO₂ shell. We take ε_m as the bulk dielectric constant of Au [38]. The exciton resonant frequency is chosen to 2.5 eV, which is close to the broad plasmon frequency of gold. The parameters are taken as ε_b = 1, ε_s = 2.16 [39], ε₁ = 6, μ = 40 D, Γ₁ = 1.25 ns⁻¹, Γ₂ = 3.33 ns⁻¹ [10,22], R₁ = 10 nm.

To begin, we study the influence of the shell thickness of the MNS on the FWM signal in the dipole approximation (N = 1). The variation of the exciton-population inversion w₀ versus the exciton-pump field detuning Δ_pu for different shell thicknesses Δr is plotted in Fig. 2(a). As Δr = 0 nm, the spectrum of w₀ exhibits a single-peaked structure. As Δr increases to 3 nm, the spectrum varies from a single-peaked structure to an S-shaped bistable one. However, a fresh bistable curve appears with Δr increasing to 5 nm or more. The physics behind Fig. 2(a) can be partly understood from the hysteresis loops. Take Δr = 3 nm as an example, when the exciton-pump field detuning Δ_pu increases, the system firstly follows the lower stable branch and jumps to the upper branch at a critical of Δ_pu. Upon sweeping Δ_pu back, the system remains on the upper branch and then transitions to the lower one at the other critical value. A hysteresis loop has been completed. The corresponding FWM spectra are plotted in Fig. 2(b). Before the appearance of OB, the FWM spectrum exhibits two asymmetric peaks, which are labeled by P1 and P2 peaks. As Δr = 3 nm, an asymmetric “U-shaped” bistable curve is observed. Such a curve traces out a crossed path ①→②→③→④→⑤. To gain a deeper insight into the physics behind in Fig. 2(b), the variations of the maximum value and position of the P1 peak versus Δr are presented in Fig. 2(c). As Δr increases, the P1 peak value firstly increases, reaching a maximum value ∼ 12.6 × 10⁻³⁰ at Δr = 0.75 nm, then it decreases and reaches a minimum value when Δr = 0.92 nm. With a further increase in Δr, the P1 peak value increases again and then declines during 0.92 nm < Δr < 1.39 nm. Specially, the P1 peak disappears completely as Δr = 1.39 nm. This indicates that a proper Δr can help to modulate the FWM signal. In addition, the P1 peak position keeps almost unchanged. As we expected, the bistable effect occurs as Δr ≥ 1.39 nm. Figure 2(d) shows a charming bistability phase diagram in the system’s parameter subspace [Δr, Δ_puc, d = 20 nm, I_pu = 200 GHz²]. As Δr is enlarged, the width of the bistable region firstly increases, and then decreases, finally increases continuously. As Δr increases from 1.39 nm to 6 nm, the bistable region varies from [-0.002 meV, −0.002 meV] to [-0.001 meV, 0.029 meV], suggesting that a large Δr may make the conditions for generating OB easier to achieve.

 

Fig. 2. Population inversion w₀ (a) and the FWM signal (b), plotted as a function of the exciton-pump field detuning Δ_pu for different Δr. Δr denotes the SiO₂ shell thickness. The inset in Fig. 2(b) shows the corresponding optical hysteresis loop. (c) Variations of the maximum value and position of the P1 peak versus Δr. (d) Bistability phase diagram in the system’s parameter space [Δr, Δ_puc, d = 20 nm, I_pu = 200 GHz²]. I_pu = Ω_pu² denotes the scaled pumping intensity, and Δ_puc refers to the critical value where the bistable effect just happens. The parameters used in Figs. 2(a)-(c) are N = 1, d = 20 nm, and I_pu = 200 GHz².

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Inspired by Fig. 2(b), we find that the pumping frequency (i.e. Δ_puc) and the SiO₂ shell thickness (i.e. Δr) have a great impact on the bistable effect. Considering this, next we will mainly explore the Δr-dependent bistable behaviors in three different examples: (i) Δ_puc = 0.05 meV, (ii) Δ_puc = −0.05 meV, and (iii) Δ_puc = 0 meV. Also, the influence of the multipole effect on OB is also considered. In Fig. 3(a) we study the variation of w₀ as a function of the SQD-MNS distance d in the first example (i.e. Δ_puc = 0.05 meV). Under the conditions of N = 1 and Δr = 0 nm, we observe that w₀ exhibits a wide peak upon increasing d. With a further increase of Δr, this peak moves to a direction with a larger d. Unexpectedly, a standard S-shaped bistable curve appears as Δr = 7 nm. However, the situation for N = 20 is rather different. Herein, the bistable effect only arises as Δr = 0 nm. The corresponding FWM spectra are plotted in Fig. 3(b). We note that the lineshapes of the FWM spectra in the dipole approximation (N = 1) are similar to those of w₀, however, the FWM spectrum evolves from a single-peaked curve into a fish-hook shaped bistable curve at Δr = 7 nm. Herein, the single peak shown in Fig. 3(b) is labeled as the P3 peak. To further clarify the contribution of the shell thickness to the P3 peak, in Fig. 3(c) we explore the variations of the maximum value and position of the P3 peak as a function of Δr. For N = 1, as Δr increases from 0 to 5 nm, the maximum value of the P3 peak increases by 5.6 times, and then it undergoes a sharp increase and reaches to a maximum value at Δr = 6 nm. The position of the P3 peak moves to a direction with a larger d as Δr increases. After eliminate the influence of the dipole polarization, we can draw a conclusion that the multipole polarization (i.e. N > 1) plays an insignificant role in controlling the P3 peak. Figure 3(d) presents the corresponding bistability phase diagram in the system’s parameter space [Δr, d_c, Δ_pu = 0.05 meV, I_pu = 200 GHz²]. In the dipole approximation (N = 1), the bistable effect just appears as Δr_D = 6.14 nm and d_cD = 21.07 nm. As Δr increases to 7.35 nm, the width of the bistable region reaches a maximum value of Δd_c = 2.23 nm. Due to the multipole effect (N = 20), the bistable effect can only be observed in a region of Δr ∈ [0, 0.79] nm. This indicates that the shrinkage of the bistable region can be mainly ascribed to the multipole polarization.

 

Fig. 3. Population inversion w₀ (a) and the FWM signal (b), plotted as a function of the SQD-MNS distance d for N = 1, 20 and different Δr. (c) Variations of the maximum value and position of the P3 peak versus Δr. (d) Bistability phase diagram in the system’s parameter space [Δr, d_c, Δ_pu = 0.05 meV, I_pu = 200 GHz²]. d_c denotes the bistable threshold. The parameters used in Figs. 3(a)-(c) are Δ_pu = 0.05 meV and I_pu = 200 GHz².

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We study how the SQD-MNS distance and the SiO₂ shell thickness affect the FWM signal in the second example (i.e. Δ_puc = −0.05 meV). The curve of w₀ versus d exhibits a wide peak, and w₀ varies slightly (w₀ ≈ −1) for N = 1 and Δr = 0 nm, indicating that there is a lack of a powerful mechanism to promote the population inversion of excitons in the current regime. However, due to the multipole effect (N = 20) and an increase of Δr, the peak of w₀ moves to a direction with a larger d. In Fig. 4(b), the corresponding FWM signal is plotted against d for different Δr. We observe that, in the dipole approximation (N = 1), the contribution of Δr to the FWM signal differs significantly in both sides of the critical value d₀ (i.e. d₀ = 26.8 nm). As d < d₀, the FWM signal is weakened with Δr increasing. However, as d > d₀, the FWM signal is enhanced as Δr increases. Moreover, the magnitude of the FWM signal increases slightly and the FWM peak shifts to the direction with a larger d as Δr increases. This critical value d₀ will be modified by the multipole polarization. Herein, d₀ is equal to 40.5 nm as N = 20. Obviously, the results presented in Figs. 4(a)–4(b) are in sharp contrast to that in Figs. 3(a)–3(b). In the current regime, the bistable effect disappears completely, providing a fresh understanding for the bistability conditions.

 

Fig. 4. Population inversion w₀ (a) and the FWM signal (b), plotted as a function of the SQD-MNS distance d for N = 1, 20 and different Δr. The parameters used are Δ_pu = -0.05 meV and I_pu = 200 GHz².

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Finally, we focus on exploring the FWM behavior in the third example (i.e. Δ_pu = 0 meV). In Fig. 5(a) we observe that, in the dipole approximation (N = 1), the bistable effect occurs and the bistable region is broadened as Δr increases. Compared to the case of N = 1, the multipole polarization (N = 20) induces these bistable regions to move to a direction with a larger d. In Fig. 5(b), we plot the corresponding FWM signal as a function of d by changing the value of Δr. As expected, a serial of “U-shaped” bistable curves appear with increasing Δr (N = 1). The bistable effect arises in the region of d ∈ [12.56 nm, 17.94 nm] when the SiO₂ shell of the MNS is absent (i.e. Δr = 0 nm). When Δr is enlarged to 3 nm, the bistable region becomes d ∈ [12.24 nm, 21.49 nm]. However, the trend for N = 20 is quite different. The FWM spectra assisted by the multipole effect are reshaped and switched from “U-shaped” bistable curves to “fish-hook shaped” ones. Additionally, the FWM signal is weakened at the positions of d_c0 and d_c1 as Δr increases (d_c0 and d_c1 refer to the lower and upper bistable thresholds, respectively). To further reveal the controllability of OB, we map out a bistability phase diagram in the system’s parameter subspace [Δr, d_c, Δ_pu = 0 meV, I_pu = 200 GHz²] for N = 1 and N = 10, as shown in Fig. 5(c). We can see that, as N = 1 and Δr is increased, d_c0 and d_c1 both increase. It is significant to highlight one point that the value of d should take into account the sizes of the SQD and the MNS (i.e. d ≥ R₁ + r_SQD). Considering this, the width of the bistable region keeps almost unchanged in the dipole approximation (N = 1). Additionally, we note that, the FWM spectrum is completely different from that in Fig. 4(b), indicating that the bistable feature in the SQD-MNS heterodimer is strongly influenced by the presence of the resonance (i.e. Δ_pu = 0 meV). In the current regime, the multipole effect becomes more significant. The bistable region for N = 20 becomes narrower and is distributed in an area where d_c1 and d_c0 are both larger in comparison with that for N = 1. Taking the case of N = 1 and Δr = 6 nm for example, a wide bistable region of d ∈ [18.0 nm, 24.47 nm] appears. Unexpectedly, the width of the bistable region for N = 20 is shrunk to nearly 20% of that for N = 1. This indicates that the multipole polarization leads to a significant shrinkage of the bistable region.

 

Fig. 5. Population inversion w₀ (a) and the FWM signal (b), plotted as a function of the SQD-MNS distance d for N = 1, 20 and different Δr. (c) Bistability phase diagram in the system’s parameter space [Δr, d_c, Δ_pu = 0 meV, I_pu = 200 GHz²] for N = 1, 20. The parameters used in Figs. 5(a)-(b) are Δ_pu = 0 meV and I_pu = 200 GHz².

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We proceed to study the variations of w₀ and the FWM signal versus I_pu for different Δr in a strong exciton-plasmon coupling regime. Figure 6(a) shows that the width of the bistable region strongly replies on the value of Δr for N = 1. The larger Δr, the wider the bistable region correlated to the scaled pumping intensity I_pu. Due to the multipole effect (N = 20), the bistable region broadens greatly and moves to a direction with a larger I_pu for a given Δr. It is not difficult to find that the conditions for generating OB in our scheme can be satisfied easily if the coupling between two NPs is sufficiently strong and the pump field is resonant with the exciton transition. The corresponding FWM spectra are plotted in Fig. 6(b). An interesting result seen here is that at the locations of thresholds the FWM signals are weakened greatly with Δr increasing. For a given Δr, the multipole polarization leads to a significant reduction of the FWM signal in comparison to that in the dipole approximation. In addition, the FWM signal will be weakened as Δr increases, indicating that a thicker shell is not conducive to the enhancement of the FWM signal. Analogically, we plot a bistability phase diagram in the parameter subspace [Δr, I_puc, Δ_pu = 0 meV, d = 20 nm] for N = 1 and N = 20 (Fig. 6(c)). As N = 1 and Δr is increased, the lower bistable threshold I_puc0 changes slightly, however, the upper bistable threshold I_puc1 undergoes a sharp rise accompanied by a strong broadening of the bistable region. For Δr = 0 nm, the bistable region is located at I_pu ∈ [19.8 GHz², 49.8 GHz²], while this region is widened more than 30 times as Δr increases to 5 nm. As a comparison, the lower bistable threshold in the SQD-MNS nanodimer is only 20% of that reported in the SQD-MNP molecule [24]. To further uncover the influence of the multipole effect on OB in the strong exciton-plasmon coupling regime, we make a comparison of the bistable region in two cases of N = 20 and N = 1. The results show that, for Δr = 5 nm, the width of the bistable region for N = 20 is broadened by more than five orders of magnitude compared to that for N = 1. From these results, we can draw a conclusion that the multipole polarization can broaden greatly the bistable region as the SQD couples strongly with the MNS.

 

Fig. 6. Population inversion w₀ (a) and the FWM signal (b), plotted as a function of Log[I_pu] for N = 1, 20 and different Δr (I_pu denotes the scaled pumping intensity). (c) Bistability phase diagram in the system’s parameter space [Δr, I_puc, Δ_pu = 0 meV, d = 20 nm]. The parameters used in Figs. 6(a)-(b) are Δ_pu = 0 and d = 20 nm.

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4. Conclusions

We conducted a theoretical study of the conditions for achieving OB in a SQD/MNS heterodimer. Three different examples have been discussed under the dipole and multipole approximations. We find that the frequency of the pump field has an important impact on the generation of OB. We have also calculated bistability phase diagrams in the system’s parameter subspace and found that the dipole-induced bistable region is broadened effectively as the shell thickness Δr increases. We also demonstrated that the multipole polarization not only can narrow the bistable zone but also enlarge d-correlated bistability thresholds for a given intermediate scaled pumping intensity I_pu. For a given shell thickness Δr, the I_pu-related bistable region can be broadened greatly due to the multipole effect in a strong exciton-plasmon coupling regime. This work may open new horizons for fabricating high-performance and low-threshold optical bistable nanodevices.