The Energy Absorption Rate for Three Metal Nano-ellipsoids in a Three-Dimensional Hybrid System

Plasmonics https://doi.org/10.1007/s11468-024-02408-z RESEARCH The Energy Absorption Rate for Three Metal Nano‑ellipsoids in a Three‑Dimensional Hybrid System Hala M. Hashim1 · Somia Abd‑Elnabi1 Received: 3 May 2024 / Accepted: 24 June 2024 © The Author(s) 2024 Abstract We study the properties of the three-dimensional hybrid system consisting of three metal nano-ellipsoids and semiconductor quantum dots. Our objective is to determine the energy absorption rate of the three metal nano-ellipsoids caused by the indirect contribution of the interaction between the semiconductor quantum dot and the three metal nano-ellipsoids. We compare two situations for the direction of the three external fields and the dipole moment of MNEs. We found that the energy absorption rate depends on the three semi-axes and therefore the polarizability of the three metal nano-ellipsoids. Moreover, the distance between the semiconductor quantum dot and the three metal nano-ellipsoid affects the energy absorption rate. We illustrated that the Rabi frequency of the first external field significantly influences the energy absorption rate. Introduction Combining metallic nanoparticles (MNPs) and semiconductor quantum dots (SQDs) has been a topic of interest in nanoscience and nanoplasmonic technology. This field has become very attractive to scientists and researchers in recent years, particularly those focused on hybrid nanostructure systems [1–9]. A quantum theory of the field-tunable nonlinear Fano effect in the hybrid metal–semiconductor and the Kerr nonlinearity in a four-level quantum system near a plasmonic nanostructure has been developed [10, 11]. Hybrid systems are advantageous due to their optical and nonlinear properties, which can be more easily tuned through the exciton-plasmon interaction [12–16]. A photon Green’s function method based on the exact quantization of electromagnetic field in a dissipative medium, the dependence of the anti-bunching time on the geometrical parameters, has been studied [17]. We found the effect of the core on the absorption in a hybrid nanoshell system, and the absorption can be amplified by using tungsten silicide (WSi) and metallic nano-antenna based on surface plasmons in [18, 19]. The pump-probe response and the nonlinear * Somia Abd‑Elnabi Hala M. Hashim 1 Department of Mathematics, Al-Azhar University, Faculty of Science, Nasr City, Cairo, Egypt four-wave mixing (FWM) effect on a molecule in asymmetric tunneling-controlled double quantum dot moleculemetal nanoparticle hybrids have been studied [20–22]. Also, Fano resonance in the plasmonic structure, localized surface plasmon resonance, splitting of the effective rabi frequencies for the coherent plasmonic fields, and optical susceptibility have been examined in the hybrid Semiconductor quantum dot-metallic nanoparticles [23–26]. Numerous theoretical and experimental studies have recently been conducted on energy absorption rate and energy transfer in hybrid nanostructures based on coupled MNPs and SQDs [27–32]. The nonlinear energy transfer in a quantum dot (QD) and metallic nanorod (MNR) nanocomposite has been investigated, where an intense probe laser field is applied to monitor two-photon energy transfer from the QD to the MNR and a control laser field is applied to control the energy transfer rate [33]. The light-matter interaction in a quantum emitter and metallic graphene flake (MGF) hybrid system deposited on a polar material and energy transfer from an individual silica nanoparticle to graphene quantum dots and resulting enhancement of photodetector responsivity, as well as ultrafast energy transfer in the metal nanoparticles-graphene nanodisks-quantum dots hybrid systems, have been studied [34–36]. A theory for photoluminescence quenching and plasmonic properties in hybrid nanosystems made from three nanosystems such as quantum emitters (QE), metallic nanoparticles (MNP), and graphene (GR) has been developed, where the coupling of resonance energy of surface plasmon polaritons (SPPs) in MNP and GR and exciton Vol.:(0123456789) Plasmonics energy in the QE plays a very importance role. The exciton in the QE interacts with the SPPs via the dipole-dipole interaction, which has been evaluated in the rotating wave approximation [37]. The double quantum dot (DQD)-MNP structure is introduced for a higher energy absorption rate depending on the high linear and nonlinear optical properties in the DQD structure [38]. Also, the energy absorption rate spectrum of an asymmetric double SQD molecule and energy absorption of an exciton-biexciton system in a hybrid structure have been investigated [39, 40]. We theoretically study the energy absorption rate in a three-dimensional hybrid system while the semiconductor quantum dot (SQD) interacts with three metal nano-ellipsoids (MNEs). The SQD is an inverted Y-type four-level system, while the MNEs are located in three dimensions. The hybrid system interacts with three external fields. We will be examining two cases. In the first case, the direction of the three external fields and the dipole moment of MNEs are along the X, Y, and Z directions. In the second case, the direction of the fields and dipole moment of MNEs are along the X direction. This paper is organized as in "Theoretical Model and Solution" section; we describe the theoretical model and solution of the equations for the SQD-MNE hybrid system. In "Result and Discussion" section, we present the results and discussion. Finally, in "Conclusion" section, we provide our conclusion. Theoretical Model and Solution We are discussing a system that combines three metal nanoellipsoids (MNEs) and a small semiconductor quantum dot (SQD). The MNEs are positioned in three dimensions, with MNE1, MNE2 , and MNE3 lying on the X-axis, Y-axis, and Z-axis, respectively. The SQD is located in the center of the axes. Each MNEj has three semi-axis aj,bj and cj where (j = 1, 2 and 3). The dimensions of the semi-axes are such that aj ≻ bj ≻ cj . The distance between SQD and MNEj from the center-to-center is denoted as rj , respectively. The distances between MNE1 and MNE2 , MNE1 and MNE3 and MNE2 and MNE3 from the center-to-center are denoted as r21, r31, and r32 , respectively, as in Fig. 1a. The three MNEs are treated as classical dielectric particles with the dielectric 𝜖j (𝜔) function 𝜖j (𝜔) for MNEj where the dielectric function ( ) is obtained by [41]: 𝜖j (𝜔) = 1 − 𝜔2pj ∕ 𝜔2 + i𝛾pj 𝜔 , where 𝜔pj and 𝛾pj are the plasma frequency and the damping constant for MNEj , respectively. The SQD is a four-level system with an inverted Y shape. The four states in this system are labeled �1⟩, �2⟩, �3⟩, and �4⟩ with corresponding energies of ℏ𝜔1, ℏ𝜔2, ℏ𝜔3 and ℏ𝜔4, respectively, as in Fig. 1. The interband transitions occur at resonance frequencies of 𝜔31 = 𝜔3 − 𝜔1, 𝜔32 = 𝜔3 − 𝜔2, and 𝜔43 = 𝜔4 − 𝜔3. In the hybrid system, there are three external fields denoted as E1, E2, and E3 which derive the excitonic transitions (...truncated)