Simple circuit models provide valuable insight into the properties of plasmonic resonators. Yet, it is unclear how the circuit elements can be extracted and connected in the model in an intuitive and accurate manner. Here, we present a detailed treatment for constructing such circuits based on energy and charge oscillation considerations. The accuracy and validity of this approach was demonstrated for a gold nanorod, and extended for a split-ring resonator with varying gap sizes, yielding good intuitive and quantitative agreement with full electromagnetic simulations.
© 2014 Optical Society of America
Localized plasmon resonances are the collective oscillations of electrons in metal nanostructures. They can conveniently be modeled by lumped-circuit elements in the form of a resistor, inductor, and capacitor (RLC). Doing so led to the concept of optical nanocircuits that was first introduced by Engheta et al., and has since generated great interest [1–10]. In its first report, a circuit model was constructed for the example of a nanosphere . Although mathematically accurate, the determination of the complex impedance of the circuit from the total displacement current and the average voltage across the nanostructure does not necessarily produce a physical model. For instance, one would expect that increasing the material resistivity would directly cause an increase in the extracted R and a reduction in the quality factor (Q-factor) of the resonance; or that changing the geometry of the structure to increase the kinetic inductance would increase the extracted L. However, this basic physical intuition is missing in existing treatments of the problem. A detailed treatment is thus needed to explicitly and intuitively link the RLC values of the circuit model to the nanostructure properties, and the physical processes during the resonance.
In this work, we detail the procedure for “circuitizing” plasmonic nanostructures from a thermodynamic perspective. Here, each circuit element is associated with an energy component obtained by considering that a resonator requires the cyclic exchange of different forms of energy. For plasmonic resonators, this cyclic exchange occurs between (1) the electric potential energy (arising from the accumulation of charges on surfaces of the nanostructure), and (2) the sum of kinetic energy of the moving electrons, and the induced magnetic field energy from the current flow. Dissipation or damping of the oscillation occurs through electron scattering (Joule heating) and radiative damping. Intuitively, the electric potential energy is captured by a capacitor (C); the electron kinetic energy by a kinetic inductor (LK) ; the magnetic field energy by a Faraday inductor (LF); and the losses by an ohmic resistor (Rohmic) and a radiative-loss resistor (Rrad). In contrast to previous work [1–8, 10], where displacement current is the only current flow, we explicitly introduce a term for the conduction current due to the moving electrons. Doing so links the currents in the RLC circuit to the charge flow in the nanostructure. We compare our model to the previous work, and test the correctness of the model by observing its ability to reproduce the spectral response for a nanorod plasmonic resonator (including resonance frequency and Q-factor), and its extension to a split-ring resonator.
2. “Circuitizing” a plasmonic resonator
First, we consider the charge oscillation in the metal arising from the free-electron gas that is enclosed by the nanostructure surface. Therefore, the total displacement current, −iωD, can be separated into two parts as follows:Fig. 1(a)-1(c)); is the complex permittivity of the material, and ε0 is the free-space permittivity; E is the electric field.
This separation leads to a “unit-cell” circuit with three branches, containing –iωD, J and −iωD0 (see Fig. 1(d)). By separating the conduction and the free-space displacement current, we rewrite the “standard” source-free Maxwell’s equations into the following form:
The insight gained by modeling R and L in series is immediately seen from the Drude model . Here, the permittivity of metal is given by, with (n is the free-electron density, m is electron mass, e is electron charge, and γ is the inelastic electron scattering rate (damping parameter)). From Eq. (2), we express the optical resistivity as 1/σ = γm/(ne2)−iωm/(ne2) = R′−iωL′. R′ and L′ are identified as the intrinsic resistivity and inductivity of the material. Therefore, a circuit consisting of R in series with L makes good intuitive sense. We then substitute the expression for current density, |J| = nev, where v is the drift velocity of the electron, and integrate over the volume of the metal nanostructure, Vmetal, to determine the time-averaged energy stored in L and power dissipated in R. We obtain WL = ½∫|J|2L′dv = ½∫mv2ndv, which is the expression for the total kinetic energy of electrons; and PR = ½∫|J|2R′dv = ∫½mv2nγdv, which represents the power dissipated due to the inelastic scattering of electrons.
To transform the “unit-cell” circuit to an overall macroscopic lumped circuit, it is necessary to define the lumped currents, I0, I1, and I2. These are the total currents in the three branches shown in Fig. 1(d) obtained via surface integrals of –iωD, J, and –iωD0:Fig. 1(d) is transformed into an overall lumped circuit shown in Fig. 1(e) in accordance to Kirchhoff’s current law. This source-free circuit models the impulse-response of the RLC circuit that undergoes transient oscillation and damping, reminiscent to the response of resonators as excited experimentally by a beam of energetic electrons, using electron energy-loss spectroscopy (EELS) [11, 12], and in pulse-response calculations as done in finite-difference time-domain (FDTD) simulations, e.g. using Lumerical FDTD Solutions.
Next, we consider the power flow in the plasmonic resonator to identify the role and value of each circuit element. Applying Poynting’s Theorem for harmonic fields in phasor notation, in which all fields have a time dependence, we obtain :
Separating out the real and imaginary components in Eq. (4) yields the following equations:
In an RLC circuit, the power delivered at each circuit element can be expressed as ½Z|I|2, where Z is the impedance and I is the current. As the capacitor and inductor have opposite signs for their impedances (i/ωC and −iωL), their reactive powers are 180 Deg out of phase, denoting the cyclic power flow between them. On the other hand, a resistor has a real impedance (R) and power, corresponding to the power dissipation.
Equation (5.1) consists of three reactive power terms that sum to zero, describing the energy oscillation between the inductive and capacitive circuit elements at resonance. (1) PE is positive (capacitive), and the energy stored in the corresponding capacitor (C) is the total electrostatic energy (|PE/ω| = ½ε0∫|E|2dv). (2) Because of the negative imaginary optical resistivity for metals, PK is negative (inductive), and the energy stored in the corresponding inductor (LK, kinetic inductance) is the electron kinetic energy (|PK/ω| = ½Im[1/σ]∫|J|2dv/(−iω)). (3) The last term, PM, is also inductive, and emanates from a Faraday inductance (LF), because it arises from the total magnetic energy (|PM/ω| = ½µ0∫|H|2dv).
On the other hand, Eq. (5.2) consists of the balance of three real power terms in accordance to the conservation of energy. Note that in Eq. (4) was separated into two parts: and , denoting the power flow in and out of V respectively. Thus, the left hand side, Pin, is the power input; and on the right hand side, Pohmic and Prad are the ohmic and radiative losses, respectively.
Finally, linking all the power (Eq. (5)) and current (Eq. (3)) terms, the corresponding circuit parameters can be expressed as follows based on the simple formula in circuit theory, i.e. P = ½|I|2Z:
These expressions provide a means for extracting the RLC components from the corresponding physical quantities.
3. Circuit model for nanospheres
In this section, we analytically derive the circuit parameters of a metal nanosphere using our model, for the ease of direct comparison with the previous work .
Removing the excitation field E0 gives the “source-free” fields inside and outside the nanosphere:
From Eq. (3), we obtain the current and charge values:
From Eq. (6.1), we obtain the resistance value:
From Eq. (6.4), we obtain the inductance value:
From Eq. (6.3), we obtain the capacitance value:
It is easy to see that , but the calculation of needs to be done in spherical coordinates ( is the radius, is the polar angle, and is the azimuthal angle) .
Take the integration in spherical coordinates, we get
Therefore, the capacitance is
Hence, the circuit parameters of a nanosphere are extracted as
As direct comparison, the previous work by Engheta et al. shows the following circuit parameters :
Similarly, we check whether our circuit satisfies this resonance condition. Assuming ε = ε′ + iε″,
It can be seen that, when ε′ = −2ε0, LKC = ω−2, which satisfies the resonance condition.
Furthermore, Engheta’s model has a parallel connection of R, L and C, whose Q-factor is 15]
We test the Q-factor using the Drude model,
Engheta’s model gives Q-factor:
Our model gives Q-factor:
In comparison, Wang’s derivation gives Q-factor (independent of the plasmon frequency) :
Given that and , all three models result in the same prediction, i.e. increasing damping factor decreases Q-factor. Our model importantly connects R and L in series, so that they are directly related to the optical resistivity and conductivity of the metal.
4. Circuit model for nanorods
FDTD simulations were performed using Lumerical FDTD Solutions to evaluate the accuracy of our model. We considered a gold nanorod with hemispherical ends (Fig. 2(a)) whose permittivity is given by the Drude model εAu(ω) = ε0(1−ωp2/(ω2 + iγω)), with ωp = 9eV, and γ = 0.07 eV . The radius (r) was constant at 10 nm, and the length (l) was varied from 100 nm to 260 nm. In the simulations, we determined the absorption cross-section (σabs), scattering cross-section (σscat), extinction cross-section (σext = σabs + σscat), total electric potential energy (½ε0∫|E|2dv), total magnetic energy (½∫µ0|H|2dv), total charge ((½∫|ρ|dv), and total current density (∫|J|2dv) at resonance. Rohmic, C, LK and LF were calculated according to Eq. (6). Equivalently, the radiative resistance was estimated using Rrad = Rohmicσscat/σabs. The resonance frequency for the circuit, (LC)−½, and its Q-factor, (L/C)½/R, were calculated and compared to simulated values in Fig. 2(c) and 2(d). The resonance energy shows perfect agreement between FDTD simulations and the RLC circuit model (see Fig. 2(c)). Assuming that the circuit is driven by an external current source, by considering the power dissipation, ∝Re[Z], where Z = (R–iωL)||1/(–iωC), the extinction spectra of the nanorods were closely reproduced by the circuit model (see Fig. 2(b)). As the circuit model only works for the fundamental mode, the high order mode at ~2.3 eV for the longer nanorod was not accounted for.
It is worth noting that our simple derivation using Poynting’s Theorem for harmonic fields does not consider the dispersion in the metal , and therefore slightly overestimates the Q-factor (Fig. 2(d)). According to our model, the Q-factor limit of the material is ωLK/Rohmic = − Im[1/σ] /Re[1/σ] = (ε′ − ε0)/ε″, where ε′ and ε″ are the real and imaginary parts of the permittivity, respectively. Using the Drude model, this value can be further simplified to ω/γ. In contrast, by accounting for dispersion, this Q-factor limit is given by (ωdε′/dω)/(2ε″) = ω/(γ + γ3/ω2) , which converges to the circuit-model predictions when γ << ω, γ + γ3/ω2 ≈γ. For the sake of simplicity, we excluded the effect of dispersion in this work. If needed, a more accurate expression for ohmic resistance with dispersion can be derived from Poynting’s Theorem for dispersive media [13, 18].
Besides full numerical calculations based on Eq. (6), it is convenient to approximate circuit parameters using simple analytical expressions. More specifically, it can be seen that the ohmic resistance and the kinetic inductance are functions of the optical conductivity (σ) and current density distribution (∫|J|2dv/|I|2). If currents are uniformly distributed in the metal structures, ∫|J|2dv/|I|2 simply reduces to l/A, the ratio between length and cross-sectional area of the nanorod. However, this uniform distribution is generally not satisfied; and therefore we use an effective length, leff to account for the non-uniformity. As a result, the impedance of the nanostructure can be expressed as:Figure 3(a) shows a characteristic current-density distribution from FDTD simulation along the axial direction in a 100 nm nanorod. The Faraday inductance for a nanorod geometry can be calculated using LF = (µ0l/2π)ln(l/2r) . Figure 3 shows the comparison between the circuit parameters obtained from the full numerical calculation based on Eq. (6) and simple analytical approximations. As can be seen in Fig. 3, simple formulas give adequate accuracy, and this is particularly useful for a quick assessment of the resonance behavior for plasmonic resonators. The deviation in the estimate for Faraday inductance in Fig. 3(d) is due to the fact that the equation used to calculate LF from Ref . is for a perfect cylinder, and does not account for the semispherical ends in the simulated structure.
5. Circuit model for split-ring resonators
To demonstrate the applicability of our approach to other structures, we extend our model to split-ring resonators (SRRs) [20–23]. Most applications of SRRs are based on the so called “LC mode”, whose resonance is sensitive to the gap size. Based on the similar resonance nature between the dipolar mode in a nanorod and the LC mode in an SRR, an SRR is topologically equivalent to a nanorod, i.e. a nanorod can be reshaped to form an SRR . Therefore, its equivalent circuit can be intuitively constructed by adding a gap capacitance, as shown in Fig. 4(a). Consequently, the total capacitance of a split ring is the sum of the self-capacitance (Cself) from the nanorod, and the gap capacitance (Cgap).
We simulated 20-nm wide square cross-sectional strips, as analytical expressions for the gap capacitances of these structures are readily available. Figure 4(a) shows the simulated structures. The side length for the SRR was kept constant at 100 nm, and the gap was varied from 10 nm to 50 nm. For comparison, nanorods of equivalent lengths of 270 to 310 nm were simulated (the equivalent total length of an SRR is calculated as 4(side length – width) − gap size). To verify our circuit model, we calculated the total capacitance for the nanorod and SRR using the total charge (½∫|ρ|dv) and electric potential energy (½ε0∫|E|2dv) based on Eq. (6). The gap capacitances were analytically estimated using a formula adapted from Ref :Fig. 4(b) we can see that adding the gap capacitance to the capacitance of the equivalent nanorod gives a good approximation to the total capacitance of the SRR. Furthermore, we show the comparison for other circuit parameters in Fig. 5.Given the same length, width and height, and the similar resonance nature of the fundamental modes, we intuit that the kinetic inductance and ohmic resistance of SRRs are the same for that of the nanorods (Fig. 5(a) and 5(c)). The slight differences of the circuit parameters (C, LK and Rohmic) between the SRR and nanorod may come from the irregular current flow and charge distribution in the SRR caused by the sharp corners. On the other hand, the Faraday inductance and radiative resistance are largely dependent on the shape of the resonators. As shown in Fig. 5(b) and 5(d), bending a nanorod significantly reduces its radiative resistance (i.e. scattering)  and Faraday inductance. From the circuit model, the resonance frequency is (LC)−½, and the Q-factor is R−1(L/C) ½. Because of the significant increase in total capacitance, the resonance of the SRR red shifts relative to the nanorods. Decreasing gap size increases the capacitance at the gap, and reduces the resonance frequency (see Fig. 5(e)). Interestingly, although increased capacitance would reduce the Q-factor, the effect is counteracted by the suppressed radiative resistance of the SRR (Fig. 5(f)). Similar to the case of the nanorods (Section 4), the Q-factor is overestimated because dispersion was not considered.
The extension from nanorods to SRRs demonstrates that our model can be tailored to account for plasmonic nanostructures with different geometries through intuitive manipulation of the equivalent circuit. Moreover, the elementary RLC circuit models for single structures will serve as building blocks for more complex plasmonic structures, such as coupled nanoantennas , Fano systems , and offer insight to other hybridized plasmonic modes .
In conclusion, we provide a detailed procedure to extract physical RLC circuit parameters for plasmonic resonators. Based on Poynting’s Theorem, we identified each energy component and defined its corresponding circuit element. Complete numerical formulas as well as simple analytical approximations were derived and tested, demonstrating the effectiveness of our circuit model. Although we considered only nanorods and split-ring resonator structures, our treatment would enable circuit models to be extracted from more complex plasmonic resonators, and provides valuable prediction of the spectral response due to geometry and material properties.
The authors acknowledge the funding support from Agency for Science, Technology and Research (A*STAR) Young Investigatorship (grant number 0926030138), SERC (grant number 092154099), and National Research Foundation grant award No. NRF-CRP 8-2011-07. The authors thank Huang Shaoying from the Singapore University of Technology and Design for fruitful discussions.
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