Subcell modeling of a Plasmonic Drude Nanosphere A. H. Panaretos1 R. E. Díaz2

1 Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706, USA, e-mail: tassos@asu.edu. 2 Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706, USA, e-mail: rudydiaz@asu.edu, tel.: +1 480 965 4281.

Abstract ! A subcell model is presented that mimics the low frequency response of a Drude nanosphere. The approach is based on the observation that the inclusion of a dielectric sphere into a finite-difference time-domain cell requires to appropriately tailor the cell�’s material properties utilizing an effective medium approach. A circuit based methodology is employed that allows to easily incorporate the effective medium model in the finite-difference time-domain method. The theoretically derived results are supported by numerical experiments.

1 INTRODUCTION

A computational formulation is described for the subcell finite-difference time-domain (FDTD) modeling of a Drude nanosphere, with the ultimate objective being the accurate prediction of its plasmon resonance, the typical low frequency characteristic of such an object. The approach takes advantage of the capacitor-like electric field excited, at the quasi static limit, within a dielectric. It is demonstrated that an FDTD cell can mimic the scattering behavior that would arise from a Drude nanosphere contained within its volume by properly defining the material constitution of the cell. This essentially dictates that the inscription of a Drude sphere into an FDTD cell be treated as a free space-Drude mixture, which effective permittivity needs to be determined. One way to achieve this is to substitute in the appropriate effective medium model the expressions of the host and filler permittivities, carry out laborious algebraic manipulations and derive the effective permittivity expression. Following this methodology, general expressions for the effective permittivity of mixtures consisting of various dispersive dielectric materials are given in [1]. In contrast, our approach stems from the complex analytic properties of dielectrics and effective medium models described in [2] and [3]. Towards this end a thorough effective medium model is derived based on a partially filled capacitor representation of the cell under discussion. Furthermore, a methodology is described that allows to accurately translate the air-Drude mixture into an equivalent RLC circuit, and thus makes the material realization in FDTD considerably straightforward and easy. Finally, the proposed formulation is tested by numerically predicting the plasmon resonance of a Silver nanosphere.

2 AN FDTD COMPATIBLE EQUIVALENT CIRCUIT REPRESENTATION OF A DRUDE MATERIAL

Drude dielectrics are characterized by a complex permittivity function given by

( )"#""

$$!

+= % jp

r

2

(1)

where # is the damping rate, p" is the plasma

frequency and %$ is the material permittivity at infinite frequency. The traditional method to represent a Drude dielectric in a discrete FDTD domain requires the incorporation into Maxwell equations, of an appropriately discretized auxiliary differential equation that relates the electric field E to the polarization current density J [4]. However, we recall that the permittivity function of a dispersive material can be modeled by judiciously choosing the elements of a series RLC circuit as demonstrated by Hippel in [5] and further explored by Diaz in [6]. Accordingly, the circuit equivalent of a Drude material is an RL branch in parallel to a capacitor, as shown in Figure 1.

Figure 1: Circuit model of Drude material.

The main advantage of the circuit representation of material is its inherent compatibility to FDTD since the voltage across the 0C capacitor, that drives the RL branch, can be considered as the voltage due to the electric field component across an FDTD cell. Having said this, and assuming that the RL branch is attached to the z-directed cell edge, then the current flowing along the RL branch satisfies the differential equation:

978-1-61284-978-2/11/$26.00 ©2011 IEEE

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zEVdtdiLiR zCo &!==+ (2)

Consequently the update equations that describe the Drude permittivity along the z-direction are the following:

ztEitRiL nz

nt

nt &&!&!= 22 µ' (3)

( )yx

iEt

nnz

nz

t

&&!()=

&

+++

%

2/12/12/1

0 H'$$ (4)

where t' and tµ are the central difference and averaging operators, respectively. Besides the great physical insight they provide, the practical usefulness of the circuit based update equations will become apparent later in our analysis.

3 INSCRIBING A DRUDE SPHERE IN AN FDTD CELL

In this section the theoretical foundations are set upon which the subcell model is based. First, we point out that since we are looking for a sphere subcell model, we are exploring the properties of spheres with small enough size to be inscribed into an FDTD cell. But given that an FDTD cell, due to resolution requirements, is a subwavelength object, it becomes obvious that the analysis to follow is valid for very small spheres typically with 1<<*k , where * is the sphere radius, and k the wavenumber.

Figure 2: Dielectric sphere in an FDTD cell represented as a partially filled capacitor.

Now the question becomes: why would a dielectric sphere subcell model be feasible? The answer relies on the fact that the electric field distribution inside a dielectric sphere is similar in nature to the electric field across an FDTD cell. In particular, at the quasi-static limit a dielectric sphere exhibits a constant and uniform internal electric field distribution which resembles the field inside a capacitor. Likewise, the electric field across the edge of an FDTD cell can be represented by the field inside a capacitor. The next question to answer is what material properties should we assign across an FDTD cell edge so that it can mimic the low frequency scattering response of a dielectric sphere. The answer

is revealed if we examine this particular subcell problem from its geometric perspective. To this end, we need to approach the derivation of the sphere subcell model from the standpoint of inscribing a sphere into an FDTD cell.

It is straightforward to verify that for a cube of volume 3* the largest sphere that can be inscribed in it, is the one of radius 2/* , which occupies

%526/ +, of the cube�’s volume. Put differently, the largest sphere that can be inscribed into a cube of given volume, fills it, only up to 52%. Consequently, the inscription of the dielectric sphere into a cubical FDTD cell alters its material constitution resulting in an air-Drude mixture rather than a purely Drude material. Therefore, the material properties that should be assigned across the edge of an FDTD cell to correctly mimic a subcell dielectric sphere are those of a mixture composed of free space (empty cell) and Drude dielectric.

4 PARTIALLY FILLED FDTD CELL AS A PARTIALLY FILLED CAPACITOR

The requirement to model the FDTD cell containing the sphere as a cell containing an appropriate material mixture is simply another manifestation of the classic effective medium model problem. As discussed in [3] the correct effective medium model for a mixture is that model that correctly accounts for the flux paths within the medium or, analogously, the one that correctly models all the multipoles contained within a unit cell of the mixture. Since in the problem of the electrically small nanosphere, the dipole term is by far the predominant multipole and since in our problem the adjacent cells need not themselves contain nanospheres, it follows that the appropriate effective medium model is the classic Clausius-Mossoti (CM) [7] model with its well-known dipole-only interaction term. Thus this model is used throughout this paper. In our case the mixture under study consists of free space (empty cell) as the host medium, and has a single spherical inclusion, the Drude sphere, as filler. Now, in [3] it was pointed out that several of the most common and practical effective medium theories, including the CM, can be expressed by the effective admittance of a partially filled capacitor. This statement is graphically illustrated in Figure 2. In other words, if we want to compute the effective permittivity of an FDTD cell which contains a dielectric sphere, we simply need to compute the equivalent capacitance (admittance) of the partially filled capacitor shown in the right subfigure in Figure 2. Returning to [3], it is proved that for the simplest general effective medium model, the effective

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permittivity $e of the mixture, for a host medium with 1=h$ , is related to the filler permittivity as

jue +

=!-

*$ 1 (5)

where -* , are positive real constants, and u is a complex variable that basically isolates the filler�’s frequency dependence defined as

1!

!.f

ju$

(6)

Notice that the left hand side of (5) corresponds to the mixture�’s effective susceptibility. Additionally, notice that the right hand side in (5) resembles the susceptibility expression of a Debye material. Let us now investigate an equivalent geometrical representation of (5). We consider the partially filled capacitor shown in Figure 3. Its equivalent susceptibility, is given by

$e !1 = x

1! x + 1$ f !1

(7)

Substitution of (6) in (5) gives precisely an expression of the form of equation (7). Accordingly, the effective permittivity as provided by a mixing rule can be represented by the capacitance of a partially filled capacitor. The previous outcome, apart from its conceptual beauty, greatly simplifies the treatment of effective media since it translates them to an equivalent capacitor configuration.

4.1 Partially filled capacitor model for the CM model

The effective permittivity that characterizes the resulting material for the CM model is given by

hf

hfhe p

p$$

$$*

**$$

2 ,

121

+!

.!

+= (8)

where p is the volume fraction and * is the polarizability of a dielectric sphere. Now, the objective is to translate the expression in (8) to its equivalent partially filled capacitor form. Therefore, the first step is to determine the functional dependence of w and h , or equivalently how the filler volume varies as p increases. Given that the capacitor in Figure 3 is a unit ideal capacitor such that phw =( , the problem is simply reduced to determining either ( )pww = or ( )phh = . In order to do this we assume that the filler is a perfect electric conductor (PEC), or %/f$ . Then

from Figure 3, the effective permittivity of the partially filled capacitor is given by

( )h

ww hhe !+!=

11 $$$ (9)

In a similar fashion, if we set %/f$ in (8), we get

pp

he !+=

121$$ (10)

Setting the effective permittivity in (10) equal to the one in (9) we can derive the expressions for either w or h as a function of the volume fraction. Specifically, for the CM model we obtain

10 ,33

2 <<+= pph (11)

Figure 3: Unit length cubical capacitor filled with a material parcel of size 1(( hw , and permittivity f$ .

Consequently, given the volume fraction of a filler we can directly derive the geometric characteristics of the partially filled capacitor configuration as shown in Figure 3.

4.2 Equivalent circuit for the partially Drude-filled capacitor

The last task to perform is to transform the partially filled capacitor of Figure 3 into an FDTD compatible form, easily incorporated into the update equations. The partially filled capacitor shown in Figure 3 is equivalent to the circuit topology shown in the left subfigure of Figure 4. Notice within the doted red frame, the RL branch in parallel with the 2C capacitor, which essentially corresponds to the filler Drude dielectric. Additionally notice that all the capacitor elements are scaled according to the volume fraction they occupy as depicted in Figure 3. The scaling of the resistor and the inductor is performed so that the diluted Drude material preserves the characteristics of the original (undiluted) material, which are the values of its damping rate # and of its plasma frequency p" .

213

Figure 4: Equivalent circuit of partially filled capacitor with Drude material.

This circuit can be simplified as shown in the right subfigure of Figure 4. Evidently, the free space-Drude mixture corresponds to an RLC branch in parallel to a capacitance which is a Lorentz material. It can be verified using standard circuit analysis that the elements of the simplified circuit are given by:

12

11

-11 C , !! +==+ CCCCC --* (12)

( )[ ] ( ) 12121

2 L ,!! =+= ** "" CCCL RR (13)

22

21 / CRCR =* (14)

where the values of 1C , 2C , R and L are scaled as shown in Figure 4. At this point the circuit equivalent of the mixture has been fully determined and it is ready to be implemented in FDTD using the set of update equations presented in Section 2.

5 NUMERICAL EXAMPLE

In this section we demonstrate the validity of the proposed methodology. We consider a uniformly discretized FDTD domain with a cell size of 10 nm. We choose the cell at the center of the domain and we assign to its z-directed edge the material properties of a mixture consisting of free space and Silver (Ag) at a volume fraction of 52%. The permittivity properties of Silver at optical frequencies can be modeled as a Drude dielectric with 9.6=%$ ,

184=# THz and 15330=p" THz. According to previous subcell model, this edge upon appropriate illumination will mimic the response of a Silver sphere with radius 5 nm. Therefore, we let a z-polarized plane wave impinge upon the edge and we compute its far field response in the back-scatter direction (RCS). The corresponding results are shown in Figure 5 along with the theoretically expected RCS as provided by Mie theory. Clearly they are in excellent agreement and our model successfully computed the plasmon resonance of the Silver sphere.

6 CONCLUSIONS

We have described a numerical subcell methodology to model a subwavelength dielectric sphere. The methodology was successfully applied for the numerical prediction of a Drude nanosphere�’s plasmon resonance.

Figure 5: Low frequency (plasmon resonance) RCS FDTD predictions of a 5 nm radius Silver sphere

versus Mie theory.

References

[1] G. Kristensson, S. Rikte and A. Shivola, �“Mixing formulas in the time domain�”, J. Opt. Soc. Am. A, vol. 15, no. 5, May, 1998.

[2] R.E. Diaz, and N.G. Alexopoulos,�“An analytic continuation method for the analysis and design of dispersive materials�”, IEEE Trans. Antennas Propag.,vol. 45, no. 11, pp. 1602-1610, Nov. 1997.

[3] R.E. Diaz, W.M. Merrill, and N.G. Alexopoulos, �“Analytic framework for the modeling of effective media�”, Journal of Applied Physics, vol. 84, no. 12, pp. 6815-6826, Dec. 1998.

[4] D.H. Werner and R. Mittra, Eds., �“Frontiers in Electromagnetics�”, IEEE Press, 2000.

[5] A.R.V. Hippel, �“Dielectrics and Waves�”, Artech House, 1995.

[6] R.E. Diaz, and N.G. Alexopoulos, �“The application of the analytic continuation model of dispersive materials to electromagnetic engineering�”, Electromagnetics, vol. 18, no. 4, pp. 395-422, 1998.

[7] A. Ishimaru, �“Electromagnetic Wave Propagation, Radiation, and Scattering�”, Prentice-Hall, 1991.

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