# Optical bistability of a plexcitonic system consisting of a quantum dot near a metallic nanorod

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## Abstract

Optical response of a complex nanodimer comprising a semiconductor quantum dot via Coulomb interaction to a metal nanorod is analyzed theoretically. Optical bistability (OB) behavior of a coherently coupled exciton-plasmon (plexcitonic) hybrid system under a laser field is investigated. The bistable exciton population in a hybrid metal–semiconductor nanodimer response is shown, and it could be pointed out that OB behavior is strongly influenced by various parameters such as size of metal nanoparticle, interparticle distance as well as intensity of probe laser field. We show that OB can be observed for a plexcitonic system and depends strongly on the type of the metals which are good candidates for plasmonic applications. The numerical calculations show the gold nanorod exhibits significant optical bistability. The result promises various applications in the field of all-optical information processing at the nanoscale, the most basic of them being the optical switching, optical memory, optical transistor and optical logic.

## Keywords

Optical bistability Plexcitonic Plasmonic nanoparticles## Introduction

The optical properties of composite nanosystems that combine semiconductor quantum dots (SQDs) and plasmonic nanostructures, such as metallic nanoparticles (MNPs), have fascinated substantial interest in recent years [1, 2, 3, 4, 5, 6]. These superstructures have outstanding capability and promising potential in optoelectronic devices [7], quantum computation [8], nanosensors [9], electrical DNA switches [10] and nanodevices [11]. The main phenomenon provoking these applications would be explained that MNPs could alter the optical properties of the semiconductor nanostructures noticeably by their localized surface plasmon resonances (LSPRs). Surface plasmons can increase the electric field experienced by SQDs, which cause the emission enhancement [12]. This process happens by increasing the radiative decay rates of excitons and enhancing the optical excitations of the SQDs. The hybrid systems is coherent coupling between excitons and plasmons so-called plexcitonic which such systems hold potential for tunable nanophotonic devices for sensing and switches [13]. The coupling can happen when these systems interact with coherent light sources, such as laser fields. Recently, several research groups have been studied SQD-MNP systems theoretically and experimentally [14, 15]. It was shown that the quantum coherence has a significant impact on the near fields of the LSPRs and this was done by showing that how the field enhancement factor changes with the center-to-center distance of the SQDs and MNPs for different intensities of the laser field intensities [5]. It was found that the power spectrum of metallic nanorod (MNR) in the SQD-MNR has two transparent states and they can be switched to one transparent state by the control field. Ultrafast switching and sensing nanodevices could be produced using this model; electrodynamic calculation of the near fields for the hybrid system based on the boundary element method was used in this study [16].

It was found that combined hybrid SQD-MNPs system could support a meta-molecular resonance. Such a resonance happens in the space/time domain and is associated with the time delay induced by the plasmonic effects in the response of the quantum dot to a time-dependent optical field. The results demonstrate generation of Rabi oscillation in quantum dots via plasmons and offer alternative ultrasensitive methods for sensing [17]. Four-wave mixing (FWM) response and optical bistability (OB) in a hybrid nanosystem composed of a metal nanoparticle (MNP) and a semiconductor quantum dot (SQD) coupled to a nanomechanical resonator are investigated theoretically [18]. Optical properties of a complex system including a closely spaced spherical SQD and a metal nanosphere were investigated [19]. It was shown that the system can manifest bistability and optical hysteresis [20]. Moreover, the resonant optical bistability condition induced by local-field effects was studied by considering the relationship between the population difference and the excitation field in a two-level system [21].

To the best of our knowledge, OB in coherent coupling of excitons and plasmons in nanoparticle systems consisting of SQDs and metallic nanospheres is recently studied [23], but for metallic nanorods (MNRs) it has not been considered. In this manuscript, we present a semianalytical model to investigate the coupling of an electronic excitation in a SQD to the LSPRs of a metal nanorod. This interaction can lead to resonance splittings in the scattering spectrum of metallic nanostructures and also to enhancement or quenching of the excited state population of SQD. It is shown that nonlinear optical response of this type of hybrid systems can possess more than one output state for a given input state. The modification of field inside SQD leads to some interesting nonlinear phenomena such as Kerr nonlinearity [22] and OB [23]. Besides, we further find that in the presence of quantum coherence SQD-MNR hybrid system, the intensity or polarization of the applied laser field, the distance between the MNR and SQD, the refractive index of the environment and the size of MNR can affect OB behavior dramatically. As an important result, we find that by selecting the better metal plasmonic material such as gold, copper, silver and aluminum, it is possible to achieve maximum or minimum the bistable region for a metal–semiconductor hybrid nanosystem that can be used in the many applications such as optical memory elements and quantum information.

## Theoretical description

In Eq. (3), \(\epsilon _{eff} = {(2\epsilon _{b} + \epsilon _{s})}/{3\epsilon _{b}}\), where \(\epsilon _{b}\) and \(\epsilon _{s}\) are the dielectric constant of the background material and dielectric constant of the SQD. The factor \(S_{a}\) refers to the polarization state of the system and has the value of \(2(-1)\) when the external fields are parallel (perpendicular) to the major axis of the hybrid system.

*E*is parallel to the

*z*(

*x*,

*y*) axis. The

*z*direction corresponds to the axis of the hybrid system. \(\gamma = [\epsilon _{m}(\omega ) - \epsilon _{b}]/{[3\epsilon _{b} + 3\kappa (\epsilon _{m}(\omega ) - \epsilon _{b})]}\) corresponds to polarization of the MNP, which is called the depolarization factor of the MNR. In this study, since the amplitude of the incident electric field lies along the semimajor diameter a, the depolarization factor is considered the same for both polarizations. It is given by the following [24]

## Results and discussion

In this section, we present the results of analytical calculations for the OB of the SQD-MNR hybrid system. It is assumed that the parameters used in our calculation are the transition energy \((\hbar \omega _{0} = 2.36\,\hbox {eV})\), \((\epsilon _{s} = 6.2)\), the SQD transition dipole moment \((\mu = 0.65\times 10^{-9}\,\hbox {e.m})\), the MNP radius \(d = 10\,\hbox {nm}\), the host dielectric constant \(\epsilon _{b} = 1\) and the relaxation constants \(\gamma \) and \(\varGamma \) are defined through \(1/\gamma = 0.8\,\hbox {ns}\).

*q*). It is found that with increasing the aspect ratio parameter, the hysteresis loop of OB can be clearly expanded in comparison with the metal nanosphere (Fig. 2).

From these figures, one can see that the induced field inside the SQD dramatically changes by varying the *q* parameter. This means that the field experienced by the SQD in the presence of the MNP can be considerably different from that of the applied field. It is realized that the increase in the value of the aspect ratio (*q*) tends to widen the hysteresis loop in OB curve. Consequently, OB can happen in a wide range of input power of laser field. This result is unique and important, and no such consequence of hysteresis loops has been shown so far.

It is well known that a drastic reduction in the plasmon dephasing rate in nanorods happens as compared to small nanospheres due to a suppression of interband damping [26]. The rods show very little radiation damping, due to their small volumes. Therefore, it would imply large local-field enhancement factors and relatively high light-scattering efficiencies, making metal nanorods particularly interesting for optical applications. Comparison of the experimental results with theory shows that pure dephasing and interface damping give small contributions to the total plasmon dephasing rate. It should be noted that by increasing *q* very much, the optical response of QD is entirely dependent on metallic nanorod and consequently, the optical bistability of QD would vanished

## Conclusion

We investigate the OB in a SQD modulated by the surface plasmon of a MNR in a hybrid plexcitonic system formed by a SQD placed near a MNR. The SQD is assumed as a two-level atomic system, which is coupled to a classical metal nanorod by the dipole-dipole interaction, and this could modify the relevant decay rates of SQD. It is considered the interaction of the system with a weak probe electromagnetic field and present results from the solution of the relevant density matrix equations. These interaction effects in optical response of the SQD via the MNP, lead to population dependence of the SQD transition frequency and relaxation constant of the SQD dipole moment. This mechanism provides a positive feedback for the nanodimer and induces to the manifestation of OB in the optical response of SQD. Precisely, the total dipole moment of the system can be switched between its two stable states by the incoming field. The OB behavior is strongly dependent on the interparticle distance, and physical parameters of MNR such as aspect ratio (*q*) and the metal substance. It is especially important that the hysterical behaviors of the transmitted light rely on the refractive index of background medium, making this model to be a good candidate for tunable OB device, owing to the possibility of high adjustability of the embedding medium. Therefore, this model suggests such possible applications as optical memories and all-optical switches at the nanoscale.

## Notes

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