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Electron. Mater. Lett., Vol. 10, No. 1 (2014), pp. 267-269
Optimization of the Optical Properties of Cuprous Oxide and Silicon-Germanium Alloy Using the Lorentz and Debye Models
Md. Ghulam Saber* and Rakibul Hasan Sagor
Department of Electrical and Electronic Engineering, Islamic University of Technology, Board Bazar, Gazipur-1704, Bangladesh
(received date: 24 March 2013 / accepted date: 2 May 2013 / published date: 10 January 2014)
The modeling parameters of cuprous oxide (Cu2O) and silicon-germanium (Si-Ge) alloy for the single-poleLorentz model and the single-pole modified Debye model (MDM) are optimized and presented. A nonlinearoptimization algorithm has been developed in order to optimize the parameters such that they are applicableto a wide frequency range. The obtained parameters have been used to determine the complex relative per-mittivity of the materials and compared with the experimental values. A very good agreement has been observedbetween the experimental values and the optimized parameters in the case of both the material models. Theassociated root mean square (RMS) deviations have been found to be as little as 0.15 and 0.08 for the Lorentzmodel and 0.1638 and 0.3710 for the modified Debye model respectively.
Keywords: Lorentz model, modified debye model, optical properties, material optics
1. INTRODUCTION
The finite-difference time-domain (FDTD)[1] is a widely
used numerical method in the field of computational
electromagnetics. The key advantage of this method is that it
can provide broadband results with a single run which
significantly reduces the memory requirements and time
needed for computation. Problems with arbitrary geometries
can be solved efficiently with this algorithm.
The formulations provided by Yee[1] accounted for the
isotropic materials with static permittivity only. However, in
order to simulate real materials, we also need to incorporate
the frequency dependent properties of the materials. Therefore,
the modeling parameters for the materials should be
available in order to develop accurate simulation models.
Oftentimes researchers use perfect materials due to the lack
of proper modeling parameters of the materials.[2]
In the past few decades, several FDTD based algorithms
have been proposed in order to account for the frequency
dependent properties of anisotropic materials. The properties
of the constituent materials need to be specified as constants
in all these proposed algorithms. Therefore, the need for
appropriate modeling parameters for the materials has
become more prominent in order obtain accurate results
from the simulation. Researchers have been working on
different materials in order to find out the accurate modeling
parameters of their optical properties. Jin et al. have
optimized the parameters for gold applicable in the frequency
range 550 - 950 nm.[3] Rakic et al. have determined the
parameters for 11 metals using the Brendel-Bormann and
Lorentz-Drude models for a wide range of frequencies.[4] Gai
et al. have reported the MDM parameters for five metals.[5]
Alsunaidi et al. determined the parameters for AlGaAs.[6]
In this paper, we present the optical property modeling
parameters for cuprous oxide (Cu2O) and silicon-germanium
(Si-Ge). We have determined the parameters for both the
single-pole Lorentz and the single-pole Debye model. The
equations describing the two models are nonlinear in nature.
Therefore, we have developed a nonlinear algorithm in order
to optimize the material modeling parameters for the two
models. We have determined the complex relative permittivity
using the optimized parameters of the materials for both the
material models. The obtained results have been compared
with the experimental results[7] and an excellent agreement
has been found. The root-mean-square deviations for the
Lorentz model have been found to be 0.15 and 0.08 and for
the Debye model have been found to be 0.1638 and 0.3710
respectively. This study will allow researchers to model the
materials in a more realistic manner and develop accurate
simulation models that will, as a consequence, provide
accurate results.
2. MATERIAL MODELS
The frequency dependent complex permittivity function
for the single-pole Lorentz model[8] is given by,
DOI: 10.1007/s13391-013-3075-5
*Corresponding author: [email protected] ©KIM and Springer
268 M. Ghulam Saber et al.
Electron. Mater. Lett. Vol. 10, No. 1 (2014)
(1)
where, is the infinite frequency relative permittivity, εs is
the zero frequency relative permittivity, j is the imaginary
unit, δ is the damping co-efficient and ωo is the frequency of
the pole pair.
From equation (1), it can be observed that the single-pole
Lorentz model can be described by four parameters which
are , εs, δ and ωo. These four parameters are independent
and need to be optimized if we want to model materials
using the Lorentz model.
The frequency dependent permittivity function of the
modified Debye model[8] is given by,
(2)
where, εr is the complex relative permittivity, ε∞ is the
infinite frequency relative permittivity, εs is the zero frequency
relative permittivity, j is the imaginary unit and τ is the
relaxation time.
From equation (2) we can see that the single-pole modified
Debye model for dielectric material can be described by
three parameters which are ε∞, εs and τ. These three
parameters need to be optimized in order to model dielectric
materials using MDM.
3. OPTIMIZATION METHOD
Numerical solution techniques for solving mathematical
problems with nonlinearity are based on iteration process.
One of the most widely used methods is the trust-region
method.[9] This method has been preferred because of its
capability to handle the singular matrix case as well as its
robustness in determining the initial values of the modeling
parameters. We develop a nonlinear algorithm and utilize the
help of the optimization toolbox of MATLAB to find the
optimum values of the parameters we have chosen. The core
program is the large-scale algorithm[10] which is in fact a sub-
space trust region method and based on interior-reflective
Newton method.[11] The program starts with a set of initial
values and minimizes the object function to find the optimal
values of the modeling parameters.[5]
(3)
. (4)
4. RESULTS AND DISCUSSION
The summary of our optimized parameters for the single-
εr ω( ) ε∞
ωo
2
εs − ε∞
( )
ωo
2j2δω − ω
2+
------------------------------------+=
ε∞
ε∞
εr ω( ) ε∞
εs − ε∞
1 jωτ+( )---------------------+=
min f ε∞εs δ ωo, , ,( )
1
2--- ε̂j ε∞ εs δ ωo, , ,( )− εj
′− iεj″( ) 2
2
j
∑=
min f ε∞εs τ, ,( )
1
2--- ε̂j ε∞ εs τ, ,( )− εj
′−iεj″( ) 2
2
j
∑=
Table 1. Optimized parameters for silicon-germanium alloy andcuprous oxide for single pole pair lorentz model.
Parameters Cuprous OxideSilicon-Germanium
Alloy
(1.41)2 (1.21)2
εS (2.49)2 (3.59)2
δ (rad/sec) 6.1 × 1010 7.1 × 1010
ωo (rad/sec) 0.53 × 1016 5.3 × 1015
Range of Wavelength (nm) 800 - 1500 900 - 1300
RMS Deviation 0.08 0.15
Table 2. Optimized modified debye model parameters for silicon-ger-manium alloy and cuprous oxide.
ParametersCuprous Oxide
(Cu2O)
Silicon-Germanium
Alloy (1.5:1)
ε∞ 6.8396 14.2996
εs 5.819 1.519
τ (sec) 4.261 × 10−15 2.261
Range of Wavelength (nm) 850 - 1500 900 - 1300
RMS Deviation 0.1638 0.3710
ε∞
Fig. 1. Comparison of relative permittivity between our results and experimental values for (i) Cu2O and (ii) Si-Ge, obtained using the single-pole Lorentz model. The red color indicates the imaginary part and the blue color indicates the real part of the complex relative permittivity.
M. Ghulam Saber et al. 269
Electron. Mater. Lett. Vol. 10, No. 1 (2014)
pole Lorentz model and the single-pole modified Debye
model is presented in Table 1 and Table 2 respectively. The
range of applicable wavelength and the RMS deviations are
also stated. From the tables we can see that our parameters
can be applied for a wide range of wavelengths.
The comparisons between our obtained results and the
experimental values have been graphically shown in Fig. 1
and Fig. 2 for the single-pole Lorentz model and the single-
pole modified Debye model respectively. From the figures it
is clearly visible that our optimized parameters agree well
with the experimental results for both the material models.
Therefore, our optimized parameters are valid for the range
of frequency we have mentioned.
5. CONCLUSIONS
In this paper, we have presented the modeling parameters
for cuprous oxide and silicon-germanium alloy for the
single-pole Lorentz model and the single-pole modified
Debye model. The parameters have been optimized using a
nonlinear optimization algorithm. We have validated the
optimized parameters using the experimental results with
relevant data and figures. This analysis will be helpful for the
researchers to develop more accurate simulation models in
order to simulate different nanostructures constructed with
different materials. We also expect that the study presented
here will pave the way for finding new techniques for the
manipulation of light at the nanometer-scale.
REFERENCES
1. K. Yee, IEEE T. Antenn. Propag. 14, 302 (1966).
2. X. Shi and L. Hesselink, J. Opt. Soc. Am. B 21, 1305
(2004).
3. E. X. Jin and X. Xu, Appl. Phys. B-Lasers O. 84, 3 (2006).
4. A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majew-
ski, Appl. Optics 37, 5271 (1998).
5. H. Gai, J. Wang, and Q. Tian, Appl. Optics 46, 2229 (2007).
6. M. A. Alsunaidi and F. S. Al-Hajiri, Proc. Pr. Electromagn.
Res. S., p. 1694, Pr. Electromagn. Res. S, Beijing, China
(2009).
7. E. D. Palik, Handbook of Optical Constants of Solids,
p. 549, Academic Press (1998).
8. K. S. Kunz and R. J. Luebbers, The Finite Difference Time
Domain Method for Electromagnetics, p. 124, CRC (1993).
9. J. J. Moré and D. C. Sorensen, SIAM J. Sci. Stat. Comp. 4,
553 (1983).
10. T. F. Coleman and Y. Li, Math. Program. 67, 189 (1994).
11. R. H. Byrd, R. B. Schnabel, and G. A. Shultz, Math. Pro-
gram. 40, 247 (1988).
Fig. 2. Comparison of relative permittivity between our results and experimental values for (i) Cu2O and (ii) Si-Ge, obtained using the single-pole modified Debye model. The red color indicates the imaginary part and the blue color indicates the real part of the complex relative permit-tivity.