Macroscopic dielectric response of the metallic particles embedded in host dielectric medium

Microelectronics Reliability 40 (2000) 893±895

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Macroscopic dielectric response of the metallic particles embedded in host dielectric medium L.G. Grechko a,*, K.W. Whites b, V.N. Pustovit a, V.S. Lysenko c b

a Institute of Surface Chemistry, NAS of Ukraine, prospekt Nauki 31, 252022 Kiev, Ukraine Department of Electrical Engineering, University of Kentucky, 453 Anderson Hall, Lexington, KY, USA c Institute of Semiconductor Physics, NAS of Ukraine, prospekt Nauki 29, 252022 Kiev, Ukraine

Abstract A theoretical approach is proposed to calculate an e€ective dielectric constant of a matrix disperse system (MDS) of metallic particles (spheres) randomly distributed and embedded in a uniform dielectric medium. Deviations from the well-known Maxwell±Garnett formula have been observed. The e€ective dielectric constant for di€erent volume fractions of the embedded particles is considered as well as the dependence on pair interaction between particles, and the relation between sizes of each pair of particles. The problem is solved in the electrostatic approximation. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction The problem of the de®nition of an e€ective dielectric constant in a composite medium has been the subject of much interest in this century. A number of di€erent solutions [1±3] have been proposed, but unfortunately, a theory which is able to describe and satisfy the recent experimental data [2] still does not exist until now. In this work, we are concerned with the theoretical description of the optical properties of disordered clusters of particles (spheres) randomly distributed in a host matrix with a dielectric constant e0 . The e€ective dielectric constant ~e of such a system is often given by the Maxwell±Garnett formula at low concentrations of inclusions (metallic particles). A theoretical method, for the calculation of the e€ective dielectric constant of a (MDS) taking into account the pair interaction between inclusions of di€erent radii, is proposed.

2. Dielectric constant of a matrix disperse system Analogous to the method proposed in Refs. [4,5], we have developed an equation for the group expansion of

*

Corresponding author.

the e€ective permittivity, taking into account the pair interaction between two particles [6±8]: P 4p 4p ~e ‡ 2e0 1 a;b na nb ˆP ÿ 3P 2 3 ~e ÿ e0 n a a a … a a na aa †  Z 1  R2ab U…~ Rab † d~ Rab bqab …~ Rab † 0  ~ ‡ 2b? …1† ab …Rab † ; where Rab ˆ j~ Ra ÿ ~ Rb j, ~ Ra and ~ Rb are the origins of the spheres a and b, respectively; na ˆ Na =V ; Na , Nb , Nc ; . . . de®ne the number of particles of each kind; and V is the volume of system. The sum in Eq. (1) is taken for all kinds of particles. The function U…~ Rab † is a two-particle distribution function which was further taken in the form  ea ÿ e0 3 1 Rab > ra ‡ rb ; U…~ Rab † ˆ r aa ˆ 0 Rab 6 ra ‡ rb ; ea ‡ 2e0 a is the usual dipole polarizability of a particle of a given ~ kind a. In Eq. (1), bqab …~ Rab † and b? ab …Rab † de®ne the longitudinal and transverse parts of the tensor polarizability bab …~ Rab † of particle a in the presence of particle b in the external ®eld ~ E~ E0 eÿixt . On taking into account pairs of dipole±dipole interactions between particles, we get

0026-2714/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 9 9 ) 0 0 3 3 6 - 4

894

L.G. Grechko et al. / Microelectronics Reliability 40 (2000) 893±895

aa ab …a† bqab ˆ X10 …~ Rab † ÿ aa ÿ 2 3 ; Rab aa ab …a† ~ ? bab ˆ X11 …Rab † ÿ aa ‡ 3 : Rab

…2†

Rab † and X11 …~ Rab † can be obThe coecients X10 …~ tained from the solution to the electrostatic response for spheres a and b in the external ®eld ~ E0 (see Appendix A). Consider a particular case where the number of particles of each kind a and b is the same, i.e. na ˆ nb ˆ n0 and B ˆ Ba ˆ Bb : We de®ne a ratio of two sphere radii as Dab ˆ rb =ra . From the general Eq. (1), for the case of the problem of two kinds of particles with radii ra ˆ R, rb ˆ r …D ˆ r=R < 1†, we have * + 3f0 …1 ‡ D3 † ~e ˆ e0 1 ‡ 1 ; …3† ÿ f0 …1 ‡ D3 † ÿ 23 f0 D B …a†

…a†

Appendix A Let us de®ne ~ Ra as a radius vector of the origin of the sphere a, and ~ R as an arbitrary point in the medium. In the case of a uniform external ®eld ~ E0 , the potential inside this sphere (which is regular at 1~ Rˆ~ Ra ) has the form

…A:1† where

where Dˆ

able deviations from the experimental data. It should be noted that at D ! 0, the main contribution to ~e gives the particles of large radii.

6

3

1‡D 8‡B D ln ‡ 3 8 ÿ 2B 2…1 ‡ D3 † 1‡D " 2 1 …1 ‡ D†3 ‡ BD3=2  D3=4 ‡ 3=4 ln D …1 ‡ D†3 ÿ 2BD3=2 #  2 1 …1 ‡ D†3 ÿ BD3=2 3=4 ÿ D ÿ 3=4 ln D …1 ‡ D†3 ‡ 2BD3=2

and

is a unit vector along the direction ~ Rÿ~ Ra is the spherical function. Potentials

outside the sphere a can be presented in the form

…4†

and f0 ˆ …4p=3†R3 n0 is the volume fraction of particles of kind a, and n0 , its concentration. Eq. (3) at D ˆ 1 gives the well-known result [5,6] ~e ‡ 2e0 1 ÿ 23 f …e ÿ e0 =e ‡ 2e0 † ln …3e ‡ 5e0 =2e ‡ 6e0 † ˆ : ~e ÿ e0 f …e ÿ e0 =e ‡ 2e0 † …5†

3. Discussion We have presented a theoretical approach to consider the optical properties of an MDS. A method is proposed for the calculation of the e€ective dielectric constant of MDS based on an expanded formulation of the wellknown Maxwell±Garnett law taking into account the pair interaction between particles. This method was applied to describe interactions between two spherical metallic particles of di€erent radii. The obtained results allow us to predict the absorption spectral behavior of MDS in the neighboring infrared and visible regions of the spectrum in the region of the Maxwell±Garnett formula application. The method presented in this article for the calculation of ~e has some restrictions. It is necessary that the volume fraction of inclusions in the matrix should not exceed the threshold of f 6 0:2 [8,9]; otherwise, the calculated values from Eqs. (3) and (4) show consider-

…A:2† where the ®rst term is the potential of the external ®eld, the second term is the potential created by the particle a in the point ~ R, and the third term is the potential from all the remaining particles in this point. In order to reduce the solution of this problem to one center, we shall use expressions for the transformation of the spherical functions from one center to another [10]:

…A:3† where j~ Rÿ~ Ra j < j~ Rb ÿ ~ Ra j. Applying the solution of Eqs. (A.1) and (A.2) to the standard boundary conditions on the surface of inclusions and taking into account Eq. (A.3), we obtain a …a† system of equations for the coecients Bn0 m0 :

L.G. Grechko et al. / Microelectronics Reliability 40 (2000) 893±895 …a†

Bn0 m0 …a† an0

‡

XX b6ˆa n00 m00

00 00 …b† Bn00 m00 Qnn0 mm0 …~ Rb ÿ ~ Ra † ˆ ÿdn0 m0 :

895

…A:4†

In the case of two particles (a and b) with Eq. (A.4), we have

…A:7† where ~ˆ m

~ Rÿ~ Ra ; j~ Rÿ~ Ra j "

…a† X10

ˆ "

…a† X11

ˆ

1

2 ‡ 3 …b† R a1 ab 1

1 ÿ 3 …b† R a1 ab

~ nˆ #" #"

~ E0 ; E0 1

4 ÿ 6 …a† …b† R a1 a1 ab 1

1 ÿ 6 …a† …b† R a1 a1 ab

#ÿ1 ; …A:8†

#ÿ1 :

…A:5† References

Here, we de®ne 00

Knn0 mm0

00



ˆ

…2n0

4p…2n00 ‡ 1†…n0 ‡ n00 ‡ m0 ÿ m00 †!…n0 ‡ n00 ‡ m00 ÿ m0 †! ‡ 1†…2n0 ‡ 2n00 ‡ 1†…n0 ‡ m0 †!…n0 ÿ m0 †!…n00 ‡ m00 †!…n00 ÿ m00 †!

1=2

…A:6† and …a†

an0 ˆ

n0 eea0 ÿ 1

n0 eea0

‡ n0 ‡ 1

0

ra2n ‡1 ;

where Rab ˆ j~ Rb ÿ ~ Ra j, ra is the radius of particle a. Further, we shall restrict ourselves only to the case of pair dipole±dipole interaction (n0 ˆ n00 ˆ 1) and select the axis OZ along the vector …~ Rb ÿ ~ Ra †. Then, from Eqs. (A.2) and (A.5), we obtain expressions for the dipole moment of the particle a, given by

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