Gold Films ally 1 a z. (3) at x ax(3). whereD, (co)= cE,. (co).E (o) is the electric flux density. At optical and near-infrared frequencies, the permittivity of metals can ...
InternationalJournal of Infrared and Millimeter Waves, Vol. 25, No. 8, August 2004 ( 2004)
MODELING ELECTRICAL PROPERTIES OF GOLD FILMS AT INFRARED FREQUENCY USING FDTD METHOD Rui Qiang, Richard L. Chen, and Ji Chen Department of Electrical and Computer Engineering University of Houston 4800 Calhoun Houston, Texas 77204, USA
Received May 29, 2004
Abstract The finite-difference time-domain (FDTD) algorithm is applied to analyze the electrical properties of gold films, whose relative permittivity is described by the Lorentz-Drude model in infrared and optical frequencies. The skin depth and reflectivity are calculated using the frequency-dependent FDTD method. The results are compared to analytical solutions and an excellent agreement is reported. Key Words: FDTD, Gold Film, Skin depth and Reflectivity
I. Introduction Recently, frequency selective surfaces (FSSs) have been designed in the near-infrared and mid-infrared regions for thermophotovoltaic filter applications around 20 Tera-Hertz (THz) to 300 THz [1][2][3]. In these work, the FSS structures were either studied experimentally or numerically. In the numerical investigations, the finite element method (FEM) or mode matching technique were used to characterize transmission/reflection coefficients for different FSS structures, where infinitely thin metallic surface were used to simplify the modeling. However, the infinitely thin metal assumption may not be valid since these FSS structures are actually electrically thick at 1263 0195 9271/04/0800 1263/0 0 2004 Plenum Publishing Corporation
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near-infrared frequencies [1]. In addition, the conductor loss at near-infrared frequencies is very significant and also frequency dependent [4] [5] [6]. Using a simple surface impedance method to model the loss mechanism appears not sufficient either, since the loss on the walls of apertures is not included [3]. Hence it is very important to develop a tool, which is capable of modelling the dispersive metallic properties with a finite thickness. The FDTD method is one of the effective numerical methods in computational electromagnetics [7]. Recent work has shown that one can use the recursive convolution method, the auxiliary differential equation method, or the Z transform technique to model the frequency-dependent materials [8][9][10]. The advantage of using the FDTD method over the FEM or the mode matching technique is that a wide-band frequency response can be obtained in one time-domain simulation. The goal of this paper is to demonstrate the feasibility of using the FDTD method to analyze metallic structures at optical and infrared frequencies. Through numerical simulations, the reflectivity/absorption and the skin depth of gold at near-infrared frequency are obtained. Analytical results will be used for comparison. The paper is organized as follows. The frequency-dependent FDTD formulation is presented in Section II. In Section III, both the skin depth and the reflectivity of gold are calculated using the FDTD method and compared with the analytical solutions. Finally, conclusions are given in Section IV.
II. FDTD Formulation For simplicity, only the 2D transverse magnetic (TM) polarization is considered. For this case, the field components are E., H and Hy. Hence, the corresponding Maxwell equations are
0D at
at and
OH. ay
(1)
,WE__ 1 p ay
(2)
(OHY ax
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1a z ax(3) x
ally
at whereD, (co)= cE,. (co).E (o)
(3)
is the electric flux density.
At optical and near-infrared frequencies, the permittivity of metals can be described by the Lorentz-Drude (LD) model. With this model, the complex relative permittivity is written as [4][5] Er (co)
where Ef (o) ,b
ef (c)+
r (o),
(4)
describes the intra-band effect with the Drude model and
(o) describes the inter-band effect with the Lorentz model. The two
models are expressed as [6] 2 Ef (W)=1-)
(5)
and r~(G) =
(kipi2_
=l(oo)
2)
+ jri
(6)
where ovp is the plasma frequency, k is the number of oscillators with frequency coi, strength f, and lifetime 1/Fi, while Qp = /Top. To incorporate the frequency-dependent permittivity into the time domain FDTD simulation, one needs to transform (5) and (6) directly into the time domain and perform a time domain convolution procedure to update (1) [93 or use the Z transform method to simplify the time domain update equations [10]. With the Z transform technique, the complicated convolution integrals in the time domain can be avoided. In the following, we shall describe the implementation of the Lorentz-Drude model using the Z transform technique in the FDTD method. For the Drude model, the electric flux density in frequency domain can be written as
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D (o) =s° 1+ (Y,)
-(
io,
+jo) E ().
(7)
W0+j ' )
Transforming (7) into the Z domain yields D, (Z)
=
(8)
co.Ez (Z) +Z-'S, (Z),
where S, (Z) is an auxiliary term, defined as
s,, (z)
At [ [1 (l
0
+e
+e-,,)z-Z
(Z).
(9)
Similarly, for the Lorentz model, the electric flux density in frequency domain can be written as D ()=
where a
=
2
+ +J2 E, (o), az+fi/2+jco2ai _22
F
a
i=l
and /,3
-
4
(10)
. Transforming (10) into the Z domain
yields M(11) O (Z) = Z- S2 (Z), where S2 (Z)=
=
,
( +e 2aZ 2 -E, ( Z). 1-2e-'a1".cos(AAt)Z- Z +e-2,, Z - 2
12eaAi
(12)
Combining (8) and (11) yields the update equation of Ez (Z), written as (13)
E (Z)= D(Z)/o-Z1 [S,(Z)+S2 (Z)]/gO In the time domain, it can be written as E; = D
/o-(S.,
(14)
+ S' )/o,
where
)
eS-(1 +=e-r°'a S 1-e-r° and
E'2Eo
(15)
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2e-n
k
OS (
e2 -A
2 =l
fAA) S2 E ft
L
I,
-AA e
(16)
e-n22
e~~
III. Numerical Results The relative permittivity of gold films associated with the Lorentz-Drude model at the infrared and optical frequencies are obtained from [6]. Figure 1 shows both the real part (-erl) and the imaginary part (er2) of the complex permittivity. As we can see from the figure, the real part of the permittivity is negative in this region and both real and imaginary parts have large variations in the frequency band. 4
10
. >
0 10 °
10 °
10
1 Wavelength [ m]
Figure 1 Gold's real and imaginary parts of permittivity at the optical and infrared frequencies. With this model available, we then calculate the skin depth and the reflection coefficient using the FDTD method. A. Skin depth For
a
medium
with
a
dispersive
relative
permittivity,
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C (O)=
Qiang, Chen, and Chen
(
)-
-,r2 (O)),
where
er (a))
< ,
the
complex propagation
constant is given as
y = + j = jo, ueoer, (Ca) o ))-(
(17)
£r2(0)
Hence the skin depth can be calculated analytically as 1 (18) a Assume a TM wave propagates normally to an interface between a semi-infinite free space and a semi-infinite gold slab. The skin depth of gold slab at optical and infrared frequencies is calculated using the 2D FDTD code developed earlier. The results are compared with the analytical solution, as shown in Figure 2. As we can see from the figure, an excellent agreement is obtained. It is also observed from the figure that the maximum skin depth appears at around 470nm. A ai
0.044
0.039
E C 0.034 0
0.029
0.024
u .u -
1 Wavelength [ m]
10
Figure 2 Skin depth of gold described by the Lorentz-Drude model. B. Reflectivity The FDTD code is then used to calculate the reflectivity between the semi-infinite free space and the semi-infinite gold slab. For this simplified problem, an analytical solution is available. It is given by [6]
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R
(1-n)2 + K2 (l+n) 2 + 2
(19)
'
where n and K are related to the complex frequency-dependent relative permittivity by n+jK =
r
(20)
(J ) -
Figure 3 shows the comparison of the reflectivity obtained using the FDTD method and the analytical solution. As shown in the figure, an excellent agreement is obtained. The figure also indicates that the power absorption, defined as A =1 - R, is very little for the region that the wavelength is greater than ltm. However, in the submicron region, the absorption increases dramatically.
Z. 4, 0 0
1
10
Wavelength [m]
Figure 3 Reflectivity for an interface between a semi-infinte free space and a semi-infinite bulk gold described by the Lorentz-Drude model.
IV.
Conclusions
A frequency-dependent 2D FDTD was developed to study electrical properties, such as the skin depth and the reflectivity, of gold films at the infrared and the optical frequencies. The numerical results were compared with the analytical data. Excellent agreement is achieved. Through this investigation, it is shown that the FDTD method can be applied to analyze
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the FSS structures at the near infrared and the optical frequencies. References [1] Morgan M. D., Home W. E., Sundaram V., Wolfe J. C., Pendharkar S. V. and Tiberion R., "Application of optical filters fabricated by masked ion beam lithography", J. Vac. Sci. Technol. B, Nov/Dec 1996, Vol. 14 No. 6, pp. 3903-3906. [2] Wu T. K., "Infrared filters for high-efficiency thermovoltaic devices", Microwave and Optical Technology Letters, May 1997, Vol. 15 No. 1, pp. 9-12. [3] Raynolds J. E., Munk B. A., Pryor J. B., and Marhefka R. J., "Ohimc loss in frequency-selective surfaces", Journal of Applied Physics, May 2003, Vol. 93 No. 9, pp. 5346-5358. [4] Ehreneich H. and Philipp H. R. and Segall B., "Optical properties of aluminum," Phys, Rev. 132, 1963, pp. 1918-1629. [5] Ehrenreich H. and Philipp H. R., "Optical properties of Ag and Cu," Phys. Rev. 128, 1962, pp. 1622-1629. [6] Rakic A. D., Djurisic A. B., Elazar J. M., and Majewski M. L., "Optical properties of metallic films for vertical-cavity optoelectronic devices", Applied Optics, August 1998, Vol. 37 No. 22,pp. 5271-5283. [7] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas and Propagat.,Vol. 17, 1966, pp. 585-589. [8] Taflove A. Computational Electrodynamics: The Finite Difference Time Domain Method. Norwood, MA: Artech House, 2000. [9] Luebbers R., Hunsberger F., Kunz K., Standler R., and Scheider M., "A frequency-dependent finite-difference time-domain formulation for dispersive materials", IEEE Tran. Electromag. Compat., vol. EMC-32, Aug. 1990, pp. 222-227. [10] Sullivan D. M., "Frequency-dependent FDTD methods using Z transforms', IEEE Trans. Antenna and Propagat., vol. AP-40, Oct. 1992, pp. 12 2 3 -1 2 3 0 .
