Thin Sohd Films, 125 (1985) 243-250

ELECTRONICS AND OPTICS 243

M U L T I P L E S C ATTERING R E N O R M A L I Z E D T MATRIX T H E O R Y FOR THE DIELECTRIC CONSTANT OF N O N - H O M O G E N E O U S T H IN FILMS*

MANUEL GOMEZ AND LUIS FONSECA

Physics Department, Umverstty of Puerto Rico, Rio Ptedras (Puerto Rico)

(Recewed August 8, 1984, accepted October 19. 1984)

A multiple scattering theory to obtain the effective dielectric constant for a granular metal film is developed considering correlation effects. A renormalization procedure is applied to obtain a field equation within a renormalized first smoothing approximation (RFSA), using T matrix formalism. From the field equation a dispersion relation is obtained that is expressed in terms of the dielectric constant of the metal corrected for size effects, the relative concentration of the constituents, the radius of the inclusions and a correlation length. Using the Maxwell Garnett model for the dielectric constant of the uncorrelated medium, the multiple scattering theory shows that correlation effects move the resonance in the imaginary part of the index of refraction predicted by the Maxwell Garnett model towards lower frequencies and broaden it. Comparison of the RFSA with experiments shows a better fit than do previous theories.

l INTRODUCTION

The inhomogeneous media of granular metal films have been extensively studied because of their importance to technological applications (see for example ref. 1). The optical properties of films composed of random arrays of metallic grains embedded in a dielectric matrix (cermet) have been studied for their spectral selectivity which make them promising selective surfaces ~.

In the range of the visible and near-IR regions of the spectrum where the wavelength of the incident light is large compared with the radii of the metallic grains, the effect of the microstructure is averaged and the film behaves as an effective homogeneous medium. Several theoretical models have been proposed to obtain the effective dielectric constant of cermets in terms of the dielectric constant of the constituents and parameters describing the mlcrostructure, such as the island radii, the relative concentrations of the constituents and the shape of the islands. Two basic and widely used models are the Maxwell Garnett (MG) and the

* Paper presented at the Sixth lnternauonal Conference on Thin Fdms, Stockholm, Sweden, August 13-17, 1984

0040-6090/85/$3 30 © Elsevier Sequoia/Printed m The Netherlands

244 M. GOMEZ, L. FONSECA

Bruggeman models (described elsewhere) 3. The MG model is asymmetric, since it considers the cermet as composed of spheres of the less abundant constituent embedded in a matrix of the more abundant material. The effective dielectric constant is then obtained by performing a volume average over the local fields. While the Bruggeman model is symmetric since it considers the two constituent materials as spheres embedded in an effective medium, a volume average is also performed to obtain a self-consistent expression for the effective dielectric constant. In both theories a resonance in the spectral response of the cermet is obtained, but neither agrees with the experimentally measured complex dielectric constant. For example, comparison between the experimentally obtained imaginary part of the effective index of refraction and the MG model reveals that the theoretical resonance is too high and narrow and in many cases blue shifted with respect to experimental results. Several papers, such as that of Granqvist 4, show that the magnitude and position of the resonance is a strong function of the inclusions' shape. Corrections to the dielectric constant of metallic grains for size effects have been extensively used to reduce the magnitude of the resonance, but they fail to produce sufficient red shift and broadening to explain experimental data.

2. THE THEORY

We consider the cermet as a random array of metallic grains embedded in a dielectric matrix. The spatial fluctuation of the dielectric constant can be envisioned as consisting of a constant dielectric constant 8o (e.g. the dielectric constant of the more abundant materml or a judiciously selected mean value dielectric constant) and a stochastic term 88 that accounts for the spatial fluctuations of the dielectric constant, i.e.

8(r) = 80 q- 88(r)

With this expression the field equations for the propagating electromagnetic field can be separated into a non-stochastic term and a stochastic fluctuating term. The fluctuating term wdl be considered as the perturbing potential responsible for the scattering processes. The resulting field equations cannot be solved exactly and approximation methods must be used.

In the first smoothing approximation (FSA) s'6 only the contribution of two point correlations to the multiply scattered field is considered. This approximation has previously been applied to obtain the effective dielectric constant of a general random medium 7 and more recently to cermets s. Our recent work for cermets has shown that the FSA approximation corrects the magnitude of the resonance but fails to predict a significant spectral shift, suggesting that renormalization corrections to the scattering terms must be included to obtain better agreement with experiments. In order to apply renormalization techniques 6, the T matrix multiple scattering of an individual grain is redefined as the new renormalized potential symbolically represented by ® and responsible for scattering processes in the medium. A new Green's function is then generated by taking into account multiple scattering effects from these new single-scatterer potentials m the absence of correlation. Th~s Green's function associated with ( E ) and represented diagrammatically by - - then

T MATRIX THEORY FOR DIELECTRIC CONSTANT OF FILMS 245

defines an effective medium that, because it is uncorrelated, we shall call the zeroth- order effective medium characterized by a dielectric constant go. In terms of the new renormalized Green's function' and potential, an averaged field equation that takes into account pair correlation functions is written to yield the renormalized first smoothing approximation (RFSA):

( E ) = ( E o ) + ~ ~ ~ ( e ) (1)

where (E0) is the statistically averaged field propagating in the uncorrelated medium and ( E ) the effective total field. The second term on the right-hand side of this self-consistent equation represents scattering processes where the effective total field ( E ) is scattered by pairs of correlated islands ~ and fl located respectively at r, and rj. In order to apply this formalism an explicit mathematical representation of the diagram in the second term of the right-hand side of eqn. (1) must be obtained. Peterson and St r fm 9 developed a formalism to obtain the T matrix from an array of n scattering objects with well-defined spatial coordinates. Their formalism is an extension to a multiplicity of scatterers of that developed by Watermanl o for a single scatterer. In Waterman's formalism the T matrix for a single scatterer is obtained by expanding all the local fields in terms of splaerical harmonics and Bessel or Hankel functions. The term describing this double-scattering process in Peterson and Strfm's formalism is given by

® - - ® --* R(r ~)T~o(r, - r j)TaR(r j - r i )R( - r j ) (2)

where T ~ and T a are the single-scatterer T matrices of islands c~ and fl respectively, expressed with their centre in the origin of the coordinate system. The formalism then translates the particles through matrices R and a to their correct positions r, and r~. Expressions for T ~ are described in several articles 9'1° and expressions for matrices R and a can be found in Peterson and Str6m's article. Using eqn. (2) the field equation can now be expressed as

(E(r)) = (Eo(r)) + V1--- ~ , ~ • ~fg(r , ,r j )R(rj)T~a(r,-r~)x

x TaR(rj - r , )R( - rj)(E> dr, drj (3)

where g(r,, rj) ~s the two-point correlation function and V is the volume of the film. Considering the spheres as impenetrable and the medium as isotropic, exponential correlation functions can be defined as

where 0 is the Heaviside function and L the correlation length. To simplify the calculation we assume all the metallic grains to be spherical in shape and of equal radius which allows us to write T ~ = T p and to make these matrices diagonal. Assuming a microscopic volume we can replace (1/V2)Z by 6 2, where 6 is the density of islands in the cermet. The fields ( E ) and (Eo) will be expanded in terms of the basis functions Re{On(Kor)} that are solutions to the Helmholtz equation,described by Waterman 1°. The 0n are known as the elementary fields and Ko is the

246 M. GOMEZ, L. FONSECA

propagat ion constant of the electromagnetic wave in the zeroth-order effective medium. In terms of this base the fields can be expressed as

<Eo> = ~ d , Re{~.(Kor)} n

<E(r)> = ~ X. Re{~.(Kor)} n

while the total effective field <g> in the second term on the right-hand side ofeqn. (3) is expanded in a base centred at one of the correlated islands:

<E> --- ~ B. Re[~.{Ko(r-rj)}] n'

Substituting these expansions for the fields in eqn. (3) and performing the integration over the relative coordinate between the correlated islands we obtain

X. Re{~.(Kor) } = ~ d. Re{¢.(K0r) } + 6 2 ~ M.tB.~.{Ko(r--rj)}dr j J (4) n n

The matrix elements M. contain all the correlation effects and can be written m terms of the elements of the single-scatterer T matrices. Since the wavelength 2 of the mcident radiation relevant to our problem is greater than 0.3 ~tm and the radii of islands for typical cermets are of the order of 0.01 lam, the long wave hmit can be utilized, thus reducing the expansion of the electric fields to the first few terms. In this long wave limit only the first six terms of these matrices are necessary, which corresponds to a dipolar approximation.

To perform the integral over the rj coordinate we must use the properties of the translation matrix ~r 11 to translate the ~b. from position r~ to the origin:

q,. {Ko(r - r , )} = ~ a.,.(Korj)Re{q,.,(Kor)} (5) n '

All terms in eqn. (4) are now written in the same orthogonal base and are eqmvalent to the following set of scalar equations '

x. = d.+ a2E fM. B. drj

Since we are m the long wave limit where the electromagnetic wave tends to average the microstructure at the scale of island sizes we can assume that the wave propagates in the effective medium as a plane wave. The coefficient d. can be written m terms of Ko, while the coefficmnt representing <E> can also be expressed in terms of the effective propagat ion constant K of the correlated medium:

d. = d. ° exp(iKo'r)

X. = X. ° exp(iK-r)

B. = X, ° exp{1K'(r-rj)}

In terms of these expressions the field equation now becomes

X,° exp(iK'r) = d,° exp(iKo'r)+ 32~ M..X.. ( exp{iK'(r-rj)}~,,,(Korj)drj n' dl, r,r>a

T MATRIX THEORY FOR DIELECTRIC CONSTANT OF FILMS 247

Following the procedure described by Varadan et al) 2 we can use the scalar Helmholtz operator in the zeroth-order approximation to simplify the above equation and to reduce the volume integral to a surface integral:

X. ° = ~M.,X.,°I . . , (6) n '

where

82 l , , , - K 2 _ K o 2 f [ o , , , ( - K o r ) d { e X P ~ r K ' r ) } exp(iK.r)d{o"'(~rK°r)}]ds

I , l>a

Equation (6) represents a system of homogeneous coupled equations with unknown coefficients. Solving the associated secular equation we obtained a new dispersion relationship that takes into account pair correlations in the scattering medium. In terms of the dielectric constants this expression can be expressed as

1 + 2Fk2go e = go- 1 -- Fk2go (7)

where e is the new effective dielectric constant in the correlated medium of the film obtained from the RFSA model, k = o9/c is the propagation constant of the electromagnetic wave in vacuum and F contains all the effects due to correlations between islands. F can be expressed as

22-2 { - 2a'~ 2 + 2 a L ) ( ~ ) 2 F = f f 6 j exPt--- -~J(L

\ - r ' - - i

where a is the radius of the inclusions, f is the relative concentration of metal and e, is the ratio of the dielectric constant of the metallic islands to that of the external medium.

3. RESULTS AND DISCUSSION

The result obtained in eqn. (7) permits the calculation of the complex effective dielectric constant of cermets, with correlation effects taken into account, in terms of basic physical parameters obtainable from experimental data. When the metal concentration is small the correlation length tends to vanish and we recover from eqn. (7) the zeroth-order approximation, i.e. the effective dielectric constant go for the uncorrelated effective medium. To apply the dispersion relationship the MG model is used for the zeroth-order dielectric constant. Conceptually the MG model is compatible with the T matrix formalism; in addition, Varadan et al. have shown that in the limit of no correlation the T matrix calculations yield the MG result 12. For the dielectric constant of the metallic islands both Drude's model with a relaxation time z corrected for the size effect and the quantum size effect (QSE) model 13 will be used; both calculations are corrected for the interband contributions.

Figure 1 shows the real and the imaginary parts of the refractive index for an Ag-80~MgO cermet. In this case the z value used in the Drude model is adjusted to give the best fit to the experimental resonance using the MG model. Then using the same dielectric constant for the metal constituent the RFSA calculation shifts and

248 M. GOMEZ, L. FONSECA

n !

(a)

3 . 0

2 . 0

1 . 0

0 . 6

I I I I

O. 4

112 0 .2

(b) i I i I 0 . 0 1 . 0 2 . 0

Fig 1 (a) The real part nl and (b) the imaginary part n 2 of the refractive index as functions of the wavelength for an Ag-80~oMgO cermet - - , result of the RFSA calculation,---, result obtained using the MG model, ., experimental results of Cralghead ~4 This figure was obtained using the Drude model with z = 2 14 x 10- ~6 s for the &electnc constant of the metalhc grams in both the MG and the RFSA curves The radius of the Islands is taken as 80 ,~ and L = 6 islands.

b r o a d e n s the peak, improv ing the M G ca lcu la t ion and giving a much bet ter fit to exper imenta l d a t a ob t a ined by Craxghead14. F igure 2 shows the same exper imenta l da ta , bu t now the D r u d e mode l has been subs t i tu ted for the Q S E mode l in the ca lcu la t ion for the metal l ic d~electric constant . In this case a more realist ic z value gives the best fit to exper iment by the R F S A mode l bu t the peak is now narrower . The co r r e spond ing mean free pa th for the z value used in this figure is 10 ~,. Values of up to 5/~ had to be used by Cra ighead to ob ta in the best fit of the M G theory to his exper imenta l data . W e also app l i ed the m o d e l to o ther cermets such as N i - A 1 2 0 3 ob ta in ing a very good fit with exper imenta l d a t a for metal l ic concen t ra t ions of up to 54~o.

4. CONCLUSIONS

W e ex tended our F S A mode l calcula t ions , which take mto account cor re la t ion effects a m o n g metal l ic is lands in cermet films, to include renormal ized cor rec t ions and showed tha t cor re la t ion effects are i m p o r t a n t m the b roa de n ing and shifting of the resonance peak to ob t a in a bet ter fit with exper imenta l results. The results of this and o ther recent work a lead us to conc lude tha t cor re la t ion effects and mul t ip le- sca t te r ing cor rec t ions m a k e i m p o r t a n t con t r ibu t ions to the opt ica l p roper t ies of

T MATRIX THEORY FOR DIELECTRIC CONSTANT OF FILMS 249

n I

(a)

nil

3 . 0

2 . 0

1.0

0 . 8

0 . 4 . i " ' ' ' " " -

0 . 2

(b) t i I i 0 . 0 1 . 0 2 . 0

X(~m) Fig. 2 (a) The real part n; and (b) the ]magmary part n2 oftherefractlvemdex for the same cermet as]n Fig 1. Th]s figure lS obtained using the QSE model ~3 with a restricted mean free path of 10/~ for the dielectric constant of the metalhc grams m both the MG and RFSA curves. The island radms is 125 ,~ and L = 6 ]slands

cermet materials and need to be included in theories with predictive value. Equation (7) is restricted to spherical shapes and the long wave limit, but the formalism presented here can be extended to include other island shapes and to permit corrections for higher multipolar contributions to the field equation that may become important when the metal concentration increases and island proximity may make these considerations important. The proposed dispersion relation depends strongly on the model used for the zeroth-order dielectric constant and that used for the dielectric constants of the metallic grains. Other zeroth-order models for the effective medium dielectric constant are being studied as possible starting points for the application of the dispersion relationship presented in this paper. These new calculations might permit the use of a higher • value that corresponds to a more realistic mean free path. Recently other models have been proposed for go; these should help to moderate the height and to broaden the resonance with a higher value 1 ~.

ACKNOWLEDGMENT

This work was supported by the U.S. Army Research Office through Grant DAAG-29-8 l-G0010.

250 M. C~MEZ, L. FONSECA

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10 P. C Waterman, Phys Rev D, 3 (1971) 825 11 V V Varadan, m V. K Varadan and V V Varadan (eds), Acoustw Electromagneuc and Elasttc

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227 14 H Cralghead, Ph.D. Thesis, Cornell Umverslty, 1980. 15 A LlebschandB. N,J Persson, J Phys C, 16(1983) 5375