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Electrodynamic Modeling of Quantum Dot Luminescence in Plasmonic Metamaterials † ‡ † ‡ ‡ § Ming Fang,*, , Zhixiang Huang,*, Thomas Koschny, and Costas M

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Electrodynamic Modeling of Quantum Dot Luminescence in Plasmonic Metamaterials † ‡ † ‡ ‡ § Ming Fang,*, , Zhixiang Huang,*, Thomas Koschny, and Costas M Article pubs.acs.org/journal/apchd5 Electrodynamic Modeling of Quantum Dot Luminescence in Plasmonic Metamaterials † ‡ † ‡ ‡ § Ming Fang,*, , Zhixiang Huang,*, Thomas Koschny, and Costas M. Soukoulis , † Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University, Hefei 230001, China ‡ Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, United States § Institute of Electronic Structure and Laser, FORTH, 71110 Heraklion, Crete, Greece ABSTRACT: A self-consistent approach is proposed to simulate a coupled system of quantum dots (QDs) and metallic metamaterials. Using a four-level atomic system, an artificial source is introduced to simulate the spontaneous emission process in the QDs. We numerically show that the metamaterials can lead to multifold enhancement and spectral narrowing of photoluminescence from QDs. These results are consistent with recent experimental studies. The proposed method represents an essential step for developing and understanding a metamaterial system with gain medium inclusions. KEYWORDS: photoluminescence, plasmonics, metamaterials, quantum dots, finite-different time-domain, spontaneous emission he fields of plasmonic metamaterials have made device design based on quantum electrodynamics. In an active − T spectacular experimental progress in recent years.1 3 medium, the electromagnetic field is treated classically, whereas The metal-based metamaterial losses at optical frequencies atoms are treated quantum mechanically. According to the are unavoidable. Therefore, control of conductor losses is a key current understanding, the interaction of electromagnetic fields challenge in the development of metamaterial technologies. with an active medium can be modeled by a classical harmonic These losses hamper the development of optical cloaking oscillator model and the rate equations of atomic population devices and negative index media. Recently, several theoreti- densities.24 Thus, simulations of lasing and amplification in − − cal4 7 and experimental8 19 studies demonstrated that one active devices are simple. For example, the finite-difference promising way to reduce or even compensate for the losses is time-domain (FDTD) method is frequently used to study the through combining the metamaterial with a gain medium. The temporal quantum electrodynamics of active systems. In fact, experimentally realized gain materials include organic semi- this method has been used to investigate compensation, 15 18 amplification, and lasing behaviors in various active sys- conductor quantum dots (QDs), quantum wells, and − dyes.20 These studies mainly focused on the coupling between tems.25 27 the gain media and the metamaterial. Without the coupling However, numerical simulation of spontaneous emission system, no loss compensation can occur, and the transmitted cannot be directly implemented using this method because in signal cannot be amplified. In addition, this method can be used the numerical experiments we can simulate the loss to obtain new nanoplasmonic lasers by incorporating gain in compensation and lasing dynamics in active systems using a − the metamaterial.21 23 Improving the laser gain material and probe source, such as a Gaussian pulse. To simulate developing a lasing spaser device constitute a significant target spontaneous emission, we should provide a suitable source of this research. An essential part of this development is the for the coherent FDTD iteration. There are some promising 28−31 study of luminescence of gain media embedded into methods to numerically calculate spontaneous emission. metamaterials. Understanding the mechanism underlying the Here, we present a systematic time-domain simulation of the coupling between the metamaterial and the gain media when quantum electrodynamic behavior of an active metamaterial fi these materials are combined is highly important. system. An arti cial source placed in a gain medium associated Active optical devices, which are used in such applications as with the local population inversion is proposed as the probe spasers, laser cooling of atoms, amplifiers, quantum computing, source in the numerical experiment. We investigate the loss plasmons, and enhanced spontaneous emission in microcavity compensation and emission enhancement in a system of QDs plasmons, require a detailed understanding of the interaction between electromagnetic fields and quantum physics-based gain Received: September 7, 2015 materials. This requirement has generated increased interest in Published: March 18, 2016 © 2016 American Chemical Society 558 DOI: 10.1021/acsphotonics.5b00499 ACS Photonics 2016, 3, 558−563 ACS Photonics Article Figure 1. (a) Schematic of a metamaterial functionalized with QDs. (b) Calculated transmittance T,reflectance R, and absorptance A spectra for the coupled system shown in (a) with a unit cell size of 545 nm. The metamaterial feature sizes are D = 545 nm, lateral slit a = 470 nm, top vertical slit and gap t = g = 170 nm, and slit width w = 65 nm. coupled to a metamaterial, considering both the resonant ∂Nrt(, ) Nrt(, ) 3 =−fNrt(, ) 3 , spectra of the metamaterial and the pumping power. The pump 0 ∂t τ32 numerical simulation results demonstrate that QDs’ photo- ∂Nrt(, ) Nrt(, ) 1 ∂P(,rt ) luminescence properties can be greatly enhanced and regulated 2 =+3 E(,rt )· ∂t τωℏ ∂t by the metamaterial. This result represents an important step in 32 e Nrt2(, ) the development and understanding of active metamaterial − , τ21 devices. ∂Nrt(, )Nrt (, ) 1 ∂P(,rt ) 12=−E(,rt )· ■ THEORETICAL MODEL FOR THE NUMERICAL ∂t τω21ℏ e ∂t EXPERIMENTS Nrt(, ) − 1 , Maxwell’s equations in the time domain for an isotropic media τ10 are given by ∂Nrt(, ) Nrt(, ) 0 =−1 fNrt(, ) ∂t τ pump 0 ∇ ×=−∂∂∇×=∂∂Er(,tttttt ) Br (, )/ , Hr (, ) Dr (, )/ 10 (2) 32 (1) where f pump is the homogeneous pumping rate. In order to solve the response of the active materials in the electromagnetic μμ εε where B(r, t)= 0H(r, t), and D(r, t)= 0E(r, t)+P(r, t). fields numerically, the auxiliary differential equation (ADE) Additionally, P(r, t)=P[E] is the dynamic polarization FDTD method is utilized.33,34 This method enables us describe response, and P(r, t)=P[E] has a nonlinear functional the dynamic electronic responses as induced polarization currents in both linear and nonlinear regimes. For the purpose dependence on the local electric field. In the frequency domain, of saving computing time and storage space, the initial ω ε χ ω ω P( )= 0 ( ) E( ) is associated with a number of condition of the four-level system is set as the saturated state; fundamental electronic response models. We model the that is, the population number changes very slowly with time. metallic metamaterial and active medium using the Drude By the initial condition, we do not need a pump process and a long time decay for pump pulse. The population numbers at and Lorentz models, respectively. each level can be obtained as In the numerical experiments, gain medium is described by a NN=+++/((τττ ) f 1), generic four-level system, and electrons are pumped by a 0 tot 10 21 32 pump homogeneous pumping rate f pump from the ground state level NN=+++ττττ f/(( ) f 1), τ 1tot10pump 10 21 32 pump (N0) to the third level (N3). After a short lifetime 32, electrons NNf=+++ττττ/(( ) f 1), nonradiatively relax into the upper emission level (N2). When 2tot21pump 10 21 32 pump electrons radiatively transfer from upper emission energy to =+++ττττ NN3tot32 fpump /((10 21 32 ) fpump 1) lower emission level, they radiate photons at the central (3) ω − ℏ frequency e =(E2 E1)/ . Lastly, electrons transfer quickly The spontaneous emission in an active medium originates from the radiative process between the upper emission level and nonradiatively from the first level to the ground-state level. and lower emission level. Here an artificial source is introduced The atomic population densities obey the following rate to numerically simulate the spontaneous emission. We assume equations: all electrons in the upper level radiatively transit to lower 559 DOI: 10.1021/acsphotonics.5b00499 ACS Photonics 2016, 3, 558−563 ACS Photonics Article Figure 2. Diagram of the artificial dipoles in the Yee cell. Each dipole has its own direction and random phase. ff Figure 3. (a) Photoluminescence of the QDs without (black line, PL0) and with di erent unit size metamaterials (dotted lines, 545 to 645 nm) with a pumping rate of 17 × 108 s−1. The dashed lines indicate the metamaterial’s absorption spectra, and the solid lines are the Gaussian fits of the dotted lines. (b) Profile of the photoluminescence spectrum without a metamaterial layer. (c) Intensity enhancement and fwhm of the QD−metamaterial coupled device’s photoluminescence spectra. The enhancement is normalized to the photoluminescence peak intensity without the metamaterial (PL0). emission level. The effect on the electric field of a associated with each grid in the gain medium region. The phase fi φ spontaneously emitted photon is accounted for by scaling the of every arti cial dipole has an individual random phase rand. number of excited electrons N2 by the energy per photon at the Within the ADE-FDTD simulation, spontaneous emission is ℏω fi fi fi emission frequency e. Thus, the amplitude of the arti cial introduced as an arti cial dipole source in the electric eld at a η given mesh point: dipole source can be obtained by N2eℏωη0 . Here 0 is the 22 wave impedance in the free space. The artificial dipole −−Δπ((tt0 ) f ) Φ=(,)rtkk Nrt2e (,) ℏωη e sin( ωe t + φ ) generates the probe source with the Lorentzian radiant 0 rand spectrum, which is consistent with that of the gain material. (4) We convert the spontaneous emission into a temporal pulse Here, Δf is the frequency line width of the QDs, and k centered at time t0. The temporal signal is desired because this represents the x, y, and z coordinates. The amplitude of the allows us to perform a Fourier transform to study the spectral artificial source is clearly dependent on the density of emission response. The field components are nodally uncollocated, as level N2.
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