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: gsaber@iut-dhaka.edu ©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.

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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.