JP_A Circuit Model for Plasmonic Resonators

A circuit model for plasmonic resonators Di Zhu,1 Michel Bosman,1 and Joel K. W. Yang1,2,*

1Institute of Materials Research and Engineering, A*STAR, 3 Research Link, 117602, Singapore 2 Singapore University of Technology and Design, 20 Dover Drive, 138682, Singapore

*[email protected]

Abstract: 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 OCIS codes: (240.6680) Surface plasmons; (260.5740) Resonance; (350.4990) Particles; (310.6628) Subwavelength structures, nanostructures.

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#207066 - $15.00 USD Received 25 Feb 2014; revised 2 Apr 2014; accepted 3 Apr 2014; published 16 Apr 2014(C) 2014 OSA 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.009809 | OPTICS EXPRESS 9809

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1. Introduction

Localized plasmon resonances are the collective oscillations of electrons in metal nanostruc-tures. 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 [110]. In its first report, a circuit model was constructed for the example of a nanosphere [1]. Although ma-thematically accurate, the determination of the complex impedance of the circuit from the total displacement current and the average voltage across the nanostructure does not necessar-ily produce a physical model. For instance, one would expect that increasing the material re-sistivity would directly cause an increase in the extracted R and a reduction in the quality fac-tor (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 compo-nent 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 po-tential 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 scat-tering (Joule heating) and radiative damping. Intuitively, the electric potential energy is cap-tured by a capacitor (C); the electron kinetic energy by a kinetic inductor (LK) [9]; the magnet-ic 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 [18, 10], where displacement cur-rent 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 reso-nator.

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, iD, can be separated into two parts as follows:

0 0( )i i i i i = = = 0D E E E J D (1)

where J is the conduction current, corresponding to charge flow, and iD0 is the free-space displacement current, caused by charge accumulation (see Fig. 1(a)-1(c)); is the complex permittivity of the material, and 0 is the free-space permittivity; E is the electric field.


This separation leads to a unit-cell circuit with three branches, containing iD, J and iD0 (see Fig. 1(d)). By separating the conduction and the free-space displacement current, we rewrite the standard source-free Maxwells equations into the following form:

0

0 0

0 0

0

0 ( )with

( )i ii

= = = = + = = =

DH E

H D J J E EE H

(2)

where H is the magnetic field; 0 is the free-space permeability; is the charge density within the metal; and is the optical conductivity, equal to i(0).

Fig. 1. The process of forming a lumped circuit model for a metal nanorod. (a) The total dis-placement current density, i(D), can be separated into two parts: (b) the conduction current density, (J), due to free electrons, and (c) free-space displacement current density, i(D)0, that stems from the surface charges accumulated on the nanorod [Fields in (c) were scaled to accentuate the free-space displacement current]. The electron kinetic energy and Joule heating caused by the conduction current in (b) are modeled using an inductor and a resistor in series; while the electric potential energy stored in the displacement current (c) is modeled using a ca-pacitor. (d) Schematic of a unit-cell circuit. (e) Overall RLC lumped circuit formed by defin-ing the lumped current as the net current in each branch in (d) [R, L, and C are per-unit val-ues, while R, L and C are lumped values.]

The insight gained by modeling R and L in series is immediately seen from the Drude model [9]. Here, the permittivity of metal is given by 2 2m 0 P(1 / ( ))i = + , with

2 2P 0/ ( )ne m = (n is the free-electron density, m is electron mass, e is electron charge, and

is the inelastic electron scattering rate (damping parameter)). From Eq. (2), we express the optical resistivity as 1/ = m/(ne2)im/(ne2) = RiL. R and L are identified as the intrin-sic 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 nano-structure, Vmetal, to determine the time-averaged energy stored in L and power dissipated in R. We obtain WL = |J|2Ldv = mv2ndv, which is the expression for the total kinetic energy of electrons; and PR = |J|2Rdv = mv2ndv, 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 neces-sary 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 iD, J, and iD0: 0 US d A ( )dv 0Vi i = = = I D n D (3.1) 1 US d A dv dvV Vi i Q = = = = I J n J (3.2) 2 0 0US ( ) d A ( )dv dvV Vi i i i Q = = = = I D n D (3.3) where US denotes the upper-half surface of the nanorod; Q is the total charge accumulated on the upper half surface. Since I0 = 0 and I1 = I2 = iQ = I, the unit-cell circuit in Fig. 1(d) is transformed into an overall lumped circuit shown in Fig. 1(e) in accordance to Kirchhoffs current law. This source-free circuit models the impulse-response of the RLC circuit that un-dergoes 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 Poyntings Theorem for harmonic fields in phasor notation, in which all fields have a time dependence exp( )i t , we obtain [13]:

2

metal

0

2 20 0

0

1 1 1dv dv2 2

1 ( ) dv21 1dv dv dv2

V V

V

V V V

i

i ii

=

= +

= + +

J E J

H D E

S D H

(4)

where Vmetal is the volume of the nanostructure, and V is the volume of the nanostructure and its surroundings (the entire simulation space in practice); S = E H* is the complex Poynting vector. Practically, the integral is performed for a simulation space that is sufficiently large, where the boundary of V is at least one half wavelength away from the nanostructure.

Separating out the real and imaginary components in Eq. (4) yields the following equa-tions:

MKE

2 2 20 0metal

0

1 1 1 1 1dv Im( ) dv dv 02 2 2V V V

PPP

i i ii

+ =

D J H (5.1)

in radohmic

2in outmetal

1 1 1 1( ) dv Re( ) dv ( ) dv2 2 2V V V

P PP

= + S J S (5.2)

In an RLC circuit, the power delivered at each circuit element can be expressed as Z|I|2, where Z is the impedance and I is the current. As the capacitor and inductor have opposite signs for their impedances (i/C and iL), 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 imped-ance (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/| = 0|E|2dv). (2) Because of the negative imaginary optical resis-tivity for metals, PK is negative (inductive), and the energy stored in the corresponding induc-tor (LK, kinetic inductance) is the electron kinetic energy (|PK/| = Im[1/]|J|2dv/(i)). (3) The last term, PM, is also inductive, and emanates from a Faraday inductance (LF), because it arises from the total magnetic energy (|PM/| = 0|H|2dv).

On the other hand, Eq. (5.2) consists of the balance of three real power terms in accor-dance to the conservation of energy. Note that 12 dvV S in Eq. (4) was separated into two parts: 12 in( ) dvV S and 12 out( ) dvV S , denoting the power flow in and out of V respec-tively. 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 cir-cuit parameters can be expressed as follows based on the simple formula in circuit theory, i.e. P = |I|2Z:

2 2 2ohmic ohmic metal12 / Re dv /

VR P

= = I J I (6.1) 2 2rad out out2 / ( ) dv /VR P= = I S I (6.2) 2 22E 0/ (2 ) | | /( dv)VC i P Q = = I E (6.3) 2 2 2K K metal

1 12 / ( ) Im dv /V

L P i

= =

I J I (6.4) 2 2 2F M 02 / ( ) dv/VL P i = = I H I (6.5)

These expressions provide a means for extracting the RLC components from the corres-ponding 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 [1].

Quasi-static approximation for the electric field inside and outside a nanosphere under an external excitation electric field E0 writes [1, 14]:

int 0 0 0 3ext 0 dip 0 0

3 / ( 2 )

[3 ( ) ] / (4 )r

= +

= + = +

E EE E E E u p u p

(7)

where p = 40a3(0)E0/( + 20); u = r/r; 0 is the free space permittivity; is the permittivi-ty of the nanosphere; a is the radius of the sphere; and r is the position vector, with r = |r|.

Removing the excitation field E0 gives the source-free fields inside and outside the na-nosphere:

i int 0 0 0 0out dip

( ) / ( 2 ) = = +=

E E E EE E

(8)

From Eq. (3), we obtain the current and charge values:

2

i iUS USd A d A= ,

/ ( )

a

Q i

= =

=

I J n E n EI

(9)


where = i(0) is the optical conductivity. From Eq. (6.1), we obtain the resistance value:

23i2 2

ohmic 2metal 2i

41 1 4 13Re dv / Re Re

3V

aR

aa

= = =

EJ I

E (10)

From Eq. (6.4), we obtain the inductance value:

23i2 2

K 2metal 2i

41 1 1 1 4 13Im dv / Im Im

3V

aL

aa

= = =

E

J IE

(11)

From Eq. (6.3), we obtain the capacitance value:

22 22 20 0 i 0 outmetal metal| | /( dv) | | /( dv dv)V V V VC Q Q = = + E E E (12) It is easy to see that 2 230 i 0 imetal

4dv3V

a = E E , but the calculation of 2

0 outmetaldv

V V

E needs to be done in spherical coordinates ( r is the radius, is the polar angle, and is the azimuthal angle) .

Note that,

2 222out dip 3 2

02 2 2

2 2 60

(3 | | cos | | cos ) (| | sin )(4 )

| | 3 | | cos16

r

r

+= =

+=

p p pE E

p p (13)

where p = 40a3(0)E0/( + 20) = 40a3Ei. Take the integration in spherical coordinates, we get

2 2 2 22 2

0 out 0 2 2 6metal00 03 22

230 i0 i3 3

0 0

| | 3 | | cosdv sin d d d16

| 4 || | 86 6 3

V Va

r rr

a aa a

+=

= = =

p pEEp E

(14)

Therefore, the capacitance is

2220 i 0 outmetal metal

2 2 22 3 3i 0 i 0 i

22

0

| | /( dv dv)

4 8/ / ( )3 3

| |4

V V VC Q

a a a

a

= +

= +

=

E EE E E (15)

Hence, the circuit parameters of a nanosphere are extracted as

2ohmic K 20

4 1 4 1Re , Im , and | |3 3 4

aR L Ca a

= = = (16)

As direct comparison, the previous work by Engheta et al. shows the following circuit pa-rameters [1]:


1 2 1sph sph fringe 0( Im[ ]) , ( Re[ ]) , and 2R a L a C a

= = = (17) It is mentioned that the resonance condition for a circuit, LsphCsph = 2, requires the well-

known condition of plasmonic resonance for a nanosphere Re[] = 20 [1, 14]. Similarly, we check whether our circuit satisfies this resonance condition. Assuming =

+ i,

[ ]

2 *K 2 3

0 0

003 2

0 0

4 1 1Im | | Im3 4 3

1 Im ( )3 3

aL Ca

i i

= =

= + =

(18)

It can be seen that, when = 20, LKC = 2, which satisfies the resonance condition. Furthermore, Enghetas model has a parallel connection of R, L and C, whose Q-factor is

[15]

Re[ ] ,Im[ ]

RQL

= = (19)

while our model derives that R and L are in series, whose Q-factor is [15]

Im[1/ ]Re[1/ ]

LQR

= = (20)

We test the Q-factor using the Drude model,

2 2

0 P2 2 2 2 2 2

0 0 P 0 P

(1 / ( ))

/ ( ) / ( ( )),

ii

= +

= + + + (21)

2 20 0 P 0 P1 / 1/ ( ( )) / ( ) / ( )i i = = + (22)

where 2 2P 0/ ( )ne m = (n is the free-electron density, m is electron mass, e is electron charge), and is the inelastic electron scattering rate (damping parameter).

Enghetas model gives Q-factor:

2

2 2P P

Re[ ] / Im[ ] (1 )Q

= = (23)

Our model gives Q-factor:

Im[1/ ] / Re[1/ ] /Q = = (24) In comparison, Wangs derivation gives Q-factor (independent of the plasmon frequency)

[16]:

3 2d / d

2 /Q

= =

+ (25)

Given that 2 2P >> and >> , all three models result in the same prediction, i.e. in-creasing damping factor decreases Q-factor. Our model importantly connects R and L in se-ries, 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 per-mittivity is given by the Drude model Au() = 0(1p2/(2 + i)), with p = 9eV, and =


0.07 eV [17]. 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 (abs), scattering cross-section (scat), extinction cross-section (ext = abs + scat), total electric potential energy (0|E|2dv), total magnetic energy (0|H|2dv), total charge ((||dv), and total current den-sity (|J|2dv) at resonance. Rohmic, C, LK and LF were calculated according to Eq. (6). Equiva-lently, the radiative resistance was estimated using Rrad = Rohmicscat/abs. The resonance fre-quency for the circuit, (LC), and its Q-factor, (L/C)/R, were calculated and compared to simulated values in Fig. 2(c) and 2(d). The resonance energy shows perfect agreement be-tween 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 = (RiL)||1/(iC), 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 Poyntings Theorem for harmonic fields does not consider the dispersion in the metal [13], and therefore slightly overestimates the Q-factor (Fig. 2(d)). According to our model, the Q-factor limit of the material is LK/Rohmic = Im[1/] /Re[1/] = ( 0)/, where and are the real and imaginary parts of the permit-tivity, respectively. Using the Drude model, this value can be further simplified to /. In contrast, by accounting for dispersion, this Q-factor limit is given by (d/d)/(2) = /( + 3/2) [16], which converges to the circuit-model predictions when

use an effective length, leff to account for the non-uniformity. As a result, the impedance of the nanostructure can be expressed as:

rod eff eff eff(1 / ) / Re[1/ ] / Im[1/ ] /Z l A l A i l A R i L = = + = (26)

where leff (l2r)/2, because the fundamental mode approximates a half-wavelength standing wave of current across the nanorod. Figure 3(a) shows a characteristic current-density distri-bution from FDTD simulation along the axial direction in a 100 nm nanorod. The Faraday inductance for a nanorod geometry can be calculated using LF = (0l/2)ln(l/2r) [19]. Figure 3 shows the comparison between the circuit parameters obtained from the full numerical calcu-lation 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 induc-tance in Fig. 3(d) is due to the fact that the equation used to calculate LF from Ref [19]. is for a perfect cylinder, and does not account for the semispherical ends in the simulated structure.

Fig. 3. Using simple formulas to estimate the circuit parameters. (a) Current inside the nanorod forms a standing wave for the fundamental mode. (b) Ohmic resistance, (c) kinetic inductance, (d) Faraday inductance, and (e) resonance energy calculated from full numeric calculation (black solid lines) and simple analytical formula (red dashed lines) are with good agreement.

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) [2023]. 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 na-ture between the dipolar mode in a nanorod and the LC mode in an SRR, an SRR is topologi-cally equivalent to a nanorod, i.e. a nanorod can be reshaped to form an SRR [24]. 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 (||dv) and electric potential energy (0|E|2dv) based on Eq. (6). The gap capacitances were analytically estimated using a formula adapted from Ref [25]:

0gap 0 02 ( ) 8( ) logh whw lC h g w

g g

+= + + + + (27)

where h, w, g and l are the height, width, gap size, and side length of the split ring. From 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 induc-tance 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 nano-rod 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) [24] and Faraday induc-tance. From the circuit model, the resonance frequency is (LC), and the Q-factor is R1(L/C) . 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.

Fig. 4. Extending the circuit model from a nanorod to an SRR. (a) Constructing a circuit model for SRRs from nanorods with equivelent lengths. (b) The total capacitance for SRRs calculated from FDTD simulation (black triangles) is approximately equal to the sum (black dot-dash lines) of the self-capacitance of a nanorod calculated from FDTD simulation (red dashed line) and the gap capacitance estimated using Eq. (27) (blue dotted line).


Fig. 5. Comparison of the circuit parameters between SRRs and nanorods with equivalent lengths. (a) Kinetic inductance and (c) ohmic resistance are comparable for the SRRs and na-norods; while (b) Faraday inductance and (d) radiative resistance are geometry dependent, and their values reduce significantly after bending. (e) Smaller gaps increase gap capacitance, and the resoannce red shifts accordingly. (f) According to circuit theory, the Q-factor is calculated as R1(L/C)1/2. Though SRRs have larger total capacitance, their radiative resistance is signifi-cantly suppressed, giving an even higher Q-factor compared to nanorods. [Black solid lines and dots are for SRRs; red dashed lines and squares are for nanorods; lines are from circuit model, and symbols are from FDTD simulations.]

The extension from nanorods to SRRs demonstrates that our model can be tailored to ac-count 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 nanoan-tennas [11], Fano systems [26], and offer insight to other hybridized plasmonic modes [27].

6. Conclusion

In conclusion, we provide a detailed procedure to extract physical RLC circuit parameters for plasmonic resonators. Based on Poyntings 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 resona-tors, and provides valuable prediction of the spectral response due to geometry and material properties.

Acknowledgments

The authors acknowledge the funding support from Agency for Science, Technology and Re-search (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.


JP_A Circuit Model for Plasmonic Resonators

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