IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 59, NO. 10, OCTOBER 2011 2595 A Time-Varying Approach to Circuit Modeling of Plasmonic Nanospheres Using Radial Vector Wave Functions Mehboob Alam, Member, IEEE, Yehia Massoud, Member, IEEE, and George V. Eleftheriades, Fellow, IEEE Abstract—Recent research has demonstrated the use of plas- monic nanoparticles (e.g., a silver or a gold nanosphere) as circuit elements. In these metallic nanoparticles, an electromagnetic wave at optical frequencies excites conduction electrons resulting in a plasmon resonance. The derived values of circuit components are based on the observation that the small size of the particle compared to the wavelength leads to lumped-impedance repre- sentations under the quasi-static approximation. In this paper, we show that circuit representations based on quasi-static approx- imations can often result in large errors for typical nanosphere sizes. To remedy this issue, we present a new approach based on time-varying fields, which uses vector wave functions to explicitly derive accurate resonance frequency and impedance expressions for these metallic nanospheres at and around the plasmon reso- nance. In particular, the proposed approach accurately predicts the dependence of the resonance frequency on the size of the nanoparticle and yields more accurate expressions for the equiva- lent and lumped elements compared to the quasi-static model. The new impedance approach is still compatible with the process of cascading nanoparticles in series and parallel combinations to synthesize more complex nanocircuits. A comparison with Mie and full-wave finite-element simulation results demonstrates that our model provides accurate closed-form expressions, thereby extending the range of the impedance representation to larger radii nanoparticles. Index Terms—Impedance modeling, lumped circuits, Mie theory, modes, nanoparticles, plasmonics, plasmon resonance. I. INTRODUCTION O VER THE last decade there has been great interest in metallic nanostructures and their properties in confining and manipulating light at nanoscale dimensions. The underlying interesting behavior is due to the coupling of electromagnetic waves with the collective oscillation of conduction electrons at the metallic-nanostructure surfaces. Among these metal nanos- tructures, nanoparticles may form the basic building block for the next generation of electronics, opto-electronics, and Manuscript received January 31, 2011; revised May 23, 2011; accepted May 31, 2011. Date of publication August 04, 2011; date of current version October 12, 2011. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). M. Alam and G. V. Eleftheriades are with the Edward S. Rogers Sr. Depart- ment of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 3G4. Y. Massoud is with the Department of Electrical and Computer Engineering, University of Alabama at Birmingham, Birmingham, AL 35294 USA . Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2011.2160872 nanometamaterials [1]–[9]. These are particularly interesting as they exhibit a strong plasmon resonance and the resonance maxima can be easily shifted to hundreds of nanometers by simply synthesizing them in terms of their shape, size, and dielectric environment [10]. The properties of surface plasmon resonance can be described by classical electrodynamics. The same classical theory is often used by many groups to support their ideas for devices and applications based on the control and use of plasmonic reso- nances [3], [9]. In the case of a nanosphere, a plasmon reso- nance at optical frequencies occurs when the negative real part of the nanosphere dielectric constant equals twice the value of the dielectric constant of the surrounding medium [11]. The un- derlying assumption is the quasi-static limit, where the wave- length of light is much longer than the size of the particle. Re- cent research based on this assumption has proposed the inter- esting and useful concept of using nanospheres as circuit el- ements in the optical regime [7]. The theory is based on the observation that the small size of the particle compared to the wavelength leads to lumped-impedance representations under the quasi-static approximation. This approach provides a frame- work of optical nanocircuit theory based on a lumped-element representation of individual nanoparticles. Impedance modeling at microwave and more recently at optical frequencies is well recognized [12]–[15]. This approach yields intuitive analytical tools to the designer by which various types of waveguide dis- continuities and filters can be analyzed and understood. In this paper, we propose a new time-varying approach to circuit mod- eling of plasmonic nanospheres using radial vector wave func- tions. The excited modes in metallic nanoparticles are and with integer values of and varying from . The quasi-static approximation in these nanoparticles works only for very small radii, where phase changes are considered negligible. For larger radii ( nm for gold nanoparticles), the quasi-static approximation does not hold well in predicting the resonant frequency. The problem is also compounded by the fact that the derivation of a voltage and a current for higher order excited modes ( or ) becomes diffi- cult to define well. Therefore, in the case of the excited and modes in a metallic nanoparticle, it would make more sense to derive impedances based on the ratio of . Consequently, for consistency, we adopt in this paper the use of the ratio to define equivalent impedances for the dominant mode. In fact, this is what is customarily done to rep- resent equivalent circuits for non-TEM modes and associated 0018-9480/$26.00 © 2011 IEEE