Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra
with a finite-difference time-domain method
Alexandre Vial,* Anne-Sophie Grimault, Demetrio Macías, Dominique Barchiesi, and Marc Lamy de la Chapelle
Laboratoire de Nanotechnologie et d’Instrumentation Optique - CNRS FRE 2671, Université de Technologie de Troyes,
12 rue Marie Curie, BP-2060 F-10010 Troyes Cedex, France
Received 13 July 2004; revised manuscript received 14 October 2004; published 23 February 2005
We propose an accurate description for the dispersion of gold in the range of 1.24–2.48 eV.We implement
this improved model in an FDTD algorithm and evaluate its efficiency by comparison with an analytical
method. Extinction spectra of gold nanoparticle arrays are then calculated.
DOI: 10.1103/PhysRevB.71.085416 PACS numbers: 41.20.Jb, 42.25.Bs, 77.22.Ch, 78.20.Ci
I. INTRODUCTION
The employment of the finite-difference time-domain
FDTD method in the study of different electromagnetic
phenomena has raised constantly increasing interest over the
past 15 years. Since then, an extensive number of references
describing the principles of the method have been published;
see, e.g., Refs. 1 and 2. Also, a wide variety of software
based on this technique has been developed and is commer-
cially, and noncommercially, available elsewhere.
Due to the fact that accurate results for a full spectrum
can be obtained in a single run of the program, the FDTD has
proven to be well adapted for different kinds of spectro-
scopic studies.
3
Nevertheless, a strong limitation is the re-
quirement of an analytical model of dispersion. Typical
laws used for FDTD simulations are the Debye, Lorentz, or
Drude dispersion models.
3–6
Also, a modified Debye law can
be used.
7
At least in principle, any dispersion law could be
described in terms of a linear combination of Debye and
Lorentz laws.
2
Surprisingly, this property has not been used
in studies using the FDTD method. Rather, it has been suc-
cessfully applied to the description of optical functions for 11
metals over a wide spectrum,
8
or to a calculation of the re-
flectance of single wall nanotubes.
9
In this paper, we will
employ a scheme similar to the one that appears in this last
reference for the study of the optical response of gold nano-
structures.
The structure of this work is as follows. In Sec. II, we
show the results obtained from the implementation of two
classic dispersion models employing the FDTD method. In
order to validate the numerical approaches, we apply them to
the case of a simple structure and compare the results with
those obtained using the analytical method described in Ref.
10. In Sec. III, we employ our FDTD-based implementation
of the Drude-Lorentz model to the calculation of extinction
spectra. In Sec. IV, we present our main conclusions and
final remarks.
II. MODELS OF DISPERSION
A. The Drude model
It is well known that in the near infrared, the relative
permittivity of several metals can be described by means of
the Drude model,
11
D
=
-
D
2
+ i
D
, 1
where
D
is the plasma frequency and
D
is the damping
coefficient. Nevertheless, if frequencies within the range of
the visible spectrum are required for a specific study, the
model in Eq. 1 may not be complete enough to provide
accurate results. To illustrate this fact, we try to fit the rela-
tive permittivity of gold
JC
, tabulated by Johnson and
Christy
12
through the optimization of
,
D
, and
D
for
energies between 1.24 and 2.48 eV wavelengths between
500 and 1000 nm. In order to determine the best set of
parameters, we define a fitness function sometimes called an
objective function as
=
j
Re
JC
j
-
D
j
2
+ Im
JC
j
-
D
j
2
,
2
where
j
are the discrete values of the frequency
=2c / for which the permittivity is calculated. The real
and imaginary parts of a complex value z are, respectively,
Rez and Imz. The minimization of is performed em-
ploying the simulated annealing procedure described in Ref.
13, and results are presented in the first row of Table I.
The real and imaginary parts of the permittivity
D
,
calculated with the Drude model, are, respectively, plotted
with a dotted line in Figs. 1 and 2.
It can be seen that neither Re
JC
nor Im
JC
are well
described for energies above 2.2 and 1.9 eV, respectively.To
emphasize this difference, we also plot in Figs. 1 and 2 the
relative errors on Re
D
and Im
D
. The existence of a
strong discrepancy for energies above 1.9 eV when using the
single Drude model is evident.
Due to the inability of the Drude model to describe the
permittivity of metals over a wide range of frequencies,
some of the authors working with the FDTD restrict their
studies to a zone of the spectrum where the Drude model is
valid Ref. 7 for a gold tip, Refs. 14 and 15 for silver and
aluminum structures; it should be noted that for these two
metals, the Drude model alone works well for the optical
wavelengths. Others try to fit the permittivity in the range of
interest by modifying the values of the parameters
,
D
,
and
D
of the model, as shown in Ref. 5 for the case of silver
PHYSICAL REVIEW B 71, 085416 2005
1098-0121/2005/718/0854167/$23.00 ©2005 The American Physical Society 085416-1