phys. stat. sol. (c) 4, No. 6, 1986–1990 (2007) / DOI 10.1002/pssc.200674420

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Bruggeman effective medium approach for modelling optical

properties of porous silicon: comparison with experiment

M. Khardani, M. Bouaïcha*, and B. Bessaïs

Laboratoire de Photovoltaïque et des Semi-conducteurs, Centre des Recherches et des Technologies de

l’Energie, B. P. 95. 2050 Hammam-Lif, Tunisia

Received 17 March 2006, revised 15 September 2006, accepted 15 November 2006

Published online 9 May 2007

PACS 78.20.Bh, 78.20.Ci, 78.66.Db, 78.67.Hc, 78.67.Lt

While numerous works deal with the properties and applications of porous silicon (PS), some of the re-

lated topics are not complete or could be investigated from different aspects. The main objective of this

paper is to provide novel information associated with the optical properties of nano- and meso-PS by

studying the variation of the effective refractive index (neff) and the relative dielectric constant (εr,eff) as a

function of porosity. For this purpose various PS samples were prepared by electrochemical etching of p

and p+-type silicon wafers in order to form silicon supported nano-PS and free-standing meso-PS layers,

respectively. The experimental effective optical parameters (neff , εr,eff ) of the meso-PS films, were deter-

mined from the transmission spectra and the Bragg law. While, in the case of nano-PS layers, we applied

the Goodman method, and deduced the values of neff and εr,eff from the Bruggeman’s effective medium

approximation (EMA). In the EMA calculation, the PS structure was considered as being a physical com-

bination of three distinct phases formed by silicon, silicon dioxide and voids with a convenient volume

fraction. A good agreement between theory and experiment was found in the case of silicon-supported

nano-PS for all porosities. However, for free-standing meso-PS, the theory does not well fit the experi-

mental results for porosities lower than 50% and higher than 70%.

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The description of optical wave propagation through porous media provides an intriguing and distin-

guished physical problem. Porous silicon (PS) is a material that has awakened interest for a wide variety

of applications including optoelectronics, chemical sensing [1], optical components [2], and as a sub-

strate for carbon nanotubes growth [3]. The use of PS thin films in the above mentioned applications

requires the determination of their refractive index and dielectric constant. According to the IUPAC

guidelines, PS- based media have been classified depending on pore size. Nanoporous and mesoporous

silicon prepared from p and p+-type silicon, consists of randomly interconnected quantum dots (size

dimension ≤ 4 nm), and an arrangement of quantum wires (4 - 50 nm) respectively [4]. It is common

practice to use the Bruggeman effective medium theory [5] to estimate the average ‘’effective’’ optical

and/or electrical parameters [6, 7]. The first part of this work deals with a description of the technique

and the choice of the experimental conditions that allow transmission spectra measurements and estima-

tion of the refractive index and the dielectric constant of nano- and free-standing meso-PS using the

Goodman’s method [8] and the Bragg law, respectively. The experimental values of the refractive index

and the dielectric constant of nano and meso-PS were compared to those calculated from Bruggeman's

effective medium approximation.

* Corresponding author: e-mail: Bouaicha.Mongi@inrst.rnrt.tn, Phone: + 216 71 430 160, Fax: + 216 71 430 934

phys. stat. sol. (c) 4, No. 6 (2007) 1987

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2 Theory

In the present work, we assume that the PS films are an arrangement of interconnected quantum dots

(QDs) in nanoporous p-type silicon and quantum wires (QWs) in mesoporous p+-type silicon. When the

wavelength of the incident light is much larger (Infrared Radiation) than the typical nanocrystallites

(QDs and QWs) size (≈ 4 nm in the case of nanoporous silicon and 10 nm in the case of mesoporous

silicon), it is common practice to use effective medium theory of Bruggeman to estimate average ‘’effec-

tive’’ optical constants. Mesoporous and nanoporous silicon are therefore considered as a physical com-

bination of three distinct phases formed by silicon, silicon dioxide and voids. The effective refractive

index of the studied material ( effn ) is related to the refractive index of each phase (nSi, nOx and nv) as well

as to their volume fractions (νSi, νOx and νV). The Bruggeman equation for the three-component system

looks like:

0( 1) ( 1) ( 1)

Si eff Ox eff V eff

Si Ox V

Si eff Ox eff V eff

n n n n n n

n d n n d n n d nν ν ν

- - -

+ + =

+ - + - + -

(1)

Where d is the system dimensionality [5, 9] (i.e., d = 3 for the nano-PS and d = 2 for the meso-PS). The

volume fraction of each phase satisfies the following equation:

1Si Ox V

ν ν ν+ + = (2)

The effective dielectric constant (εr,eff) is then the contribution of all the material components. It may be

obtained by solving Eq. (3) deduced from the EMA theory:

, , , ,

, , , ,

0( 1) ( 1) ( 1)

r Si r eff Ox r eff V r eff

Si Ox V

r Si r eff Ox r eff V r effd d d

ε ε ε ε ε ε

ν ν ν

ε ε ε ε ε ε

- - -

+ + =

+ - + - + -

(3)

The relative silicon dielectric constant is written as ( )2

rnε = where

in n j n = + , n and

4in

λα

π=

are respectively the real and imaginary parts of the refractive index, and2

1j = - . Applications in IR

are extremely comfortable because in n≺≺ at 800λ nm and then absorption can be neglected.

Hence, we obtain: 2

rnε ª .

Calculation of the effective refractive index for porosities ranging from 30% to 80% has been performed

using the data summarized in Table 1, where only Ox

ν was taken as the fitting parameter.

Table 1 Refractive indexes of the various PS composed phases used to calculate the

effective refractive index of the PS films.

Phase Volume fraction Refractive index

Void νV: porosity 1.00

Oxide νOx 1.43

Silicon νSi 3.44

Equation (1) and Eq. (3) are solved by considering values of nSi from [10] and nOx from [11].

1988 M. Khardani et al.: Bruggeman effective medium approach for modelling optical properties of PS

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

3 Sample preparation

Nano- and meso-PS layers were obtained by electrochemical etching of (100) oriented boron-doped p+-

type (resistivity of about 0.02 Ωcm) and p-type (resistivity of about 1.00 Ωcm) monocrystalline –Si,

respectively. The meso and nanoporous silicon layers were grown in a HF (40%): absolute ethanol (1:1)

electrolyte at different current density values in order to have layers having different thickness and po-

rosities. Preparation of free-standing mesoporous silicon was realized by a subsequent electrochemical

etching at the end of the anodization process. In order to prevent breaking of the nanostructures forming

the PS layers, the latter were kept under ethanol.

4 Optical characterizations

4.1 Mesoporous silicon

For the calculation of the experimental values of the refractive index and the dielectric constant of free-

standing meso-PS films for different porosities, plane-parallel films were investigated by Fourrier Trans-

form Infrared (FT-IR) spectroscopy using a Nicolet spectrometer in transmission mode. The measure-

ments were performed in the 500 cm–1–5000 cm–1 spectral range. The meso-PS refractive index values,

nPS, were deduced from the spectral position of the interferential fringes detected in the 2000 cm–1–5000

cm–1 spectral range, using the simple relation:

2PSn e p λ= (4)

where e denotes the thickness of the porous film and λ the wavelength of the pth fringe. Equation (4) can

be used for the determination of the thickness if the refractive index is known and vice versa. Knowing

that the thickness of the meso-PS films is determined by the gravimetric method, it is easy to determine

the refractive index.

4.2 Nanoporous silicon

Using the Goodman method, the refractive index nPS of the silicon-supported nano-PS may be obtained

independently of the thickness, by calculating the value of the maximum to minimum amplitude ratio

Tmax/Tmin. Figure 1 depicts the variation of the refractive index of silicon-supported nano-PS and meso-

PS layers for porosities ranging from 30% to 80%.

30 40 50 60 70 80

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

Refr

acti

ve i

ndex

Porosity (%)

Fig. 1 Experimental refractive index of the PS medium vs. porosity:

() mesoporous silicon and () nanoporous silicon.

phys. stat. sol. (c) 4, No. 6 (2007) 1989

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It appears obvious that for both nano- and meso-PS layers, nPS decreases as the porosity increases due to

the fact that the volume fraction of voids rises as the porosity increases. At specific porosities the nPS of

the nano-PS is lower than that of the meso-PS film. Such experimental result can be attributed to the

nanocrystallite size dimension of the PS structure. In fact, the doping level can affect the morphological

aspects of the PS structure which in turn affects wavepropagation through nanocrystallites arrangements.

The dielectric constant of silicon-supported nano- and meso-PS is obtained by applying the appropriate

approximation ( 2

rn=ε ).

5 Comparison with the EMA calculations

Figure 2, shows experimental and theoretical values (deduced from the Bruggeman effective medium

approximation) of the refractive index as a function of porosity for nanoporous silicon layers. In this

case, we obtained a good agreement between experiment () and theory (). Figure 3 shows the experi-

mental and theoretical values (deduced from the Bruggeman effective medium approximation) of the

refractive index as a function of porosity for mesoporous silicon films.

30 40 50 60 70 80

1,2

1,4

1,6

1,8

2,0

2,2

2,4

Refr

acti

ve i

ndex

Porosity (%)

Fig. 2 Refractive index as a function of porosity for

nano-PS deduced from experiment () and the prediction

of the EMA theory for nanoporous silicon ()

30 40 50 60 70 80

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

Refr

acti

ve i

ndex

Porosity (%)

Fig. 3 Refractive index as a function of porosity for

meso-PS deduced from experiment () and the predic-

tion of the EMA theory ()

In the case of meso-PS, it was found (Fig. 3) that the calculated values are in good agreement with the

experimental ones in the 50% - 67% porosity range. However, beyond this range of porosities, the calcu-

lated value of the refractive index is lower than the experimental one. This mismatch may be attributed to

be due to the gravimetric method used for the estimation of the thickness of meso-PS films. In fact, the

porosity p is defined as the fraction of voids in the porous structure and satisfies the following equation:

1 2

1 3

m mp

m m

-

=

-

(5)

where, m1, m2 and m3 denote the weight of the original wafer, anodized wafer and stripped wafer (in 1N

NaOH solution), respectively. The small weight differences (m1 - m2) and (m1 - m3) induce some errors in

the p values, especially for high and low porosities. Moreover, the thicknesses of the meso-PS films,

were estimated using the same gravimetric method. Systematically, for high and low porosities, accumu-

lation of errors leads to the observed discrepancy between experimental results and theoretical predic-

tion. Figure 4 depicts the evolution of the experimental and theoretical dielectric constant versus poros-

ity, taking into account the above mentioned approximations.

1990 M. Khardani et al.: Bruggeman effective medium approach for modelling optical properties of PS

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

30 40 50 60 70 80

1

2

3

4

5

6

7

8

Die

lectr

ic c

on

sta

nt

Porosity (%)

Fig. 4 Experimental dielectric constant as a function of porosity for meso () and nano-PS () and theoretical

dielectric constant for meso () and nano-PS ()

6 Conclusions

In this work, PS was assumed to be an optically isotropic medium composed of three components: sili-

con, silicon dioxide and voids. The effective refractive indexes of nano- and meso-PS were evaluated.

Using the Bruggeman effective medium theory, we obtained good agreement between theoretical and

experimental results for nano-PS layers. However, for meso-PS, the Bruggeman effective medium theory

fits experimental results only for porosities ranging between 50% and 70%. The observed disagreement

for large and low porosities was attributed to the fact that for meso-PS, the experimental values of neff

was evaluated using the Bragg law, where the thickness was estimated by the gravimetric method. The

accuracy of the latter was found to induce additional errors especially at high and low porosity ranges.

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