MICROSCOPIC MODEL FOR THE OPTICAL PROPERTIES OF THIN FILMS

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MICROSCOPIC MODEL FOR THE OPTICAL

PROPERTIES OF THIN FILMS

M. Dignam, J. Fedyk

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C5, supplkment au no 11, Tome 38, Novembre 1977, page C5-57

MICROSCOPIC MODEL FOR THE OPTICAL PROPERTIES OF

THIN FILMS

M. J. DIGNAM and 5. FEDYK

Dept. of Chemistry University of Toronto, Toronto, Ontario, Canada

RBsumB. - Les propri6tCs optiques d'un arrangement planaire de mol6cules, dont l'epaisseur peut atteindre quelques couches mol8culaires, sont examinees en utilisant une reprbentation de dip6les ponctuels pour la mol6cule dans un champ Blectrique. On calcule les constantes optiques en utilisant I'approximation kkctrostatique.

On examine les effets sur les proprietes optiques macroscopiques :

1) des fluctuations de densite et d'orientation moleculaire ;

2) des images de dip6les dans le substrat (trait6 de manikre isotrope) ;

3) du nombre de couches mol6culaires dans le film.

On trouve que les fluctuations de densite n'introduisent genkralement qu'un effet de 2e ordre, tandis que I'image des dip6les et le nombre de couches peuvent dans certaines conditions avoir des effets prononcks, nkcessitant que les constantes optiques effectives soient une fonction de la cons- tante diklectrique du substrat et du nombre de couches.

Abstract. - The optical properties of thin planar arrays of molecules of thickness up to a few molecular layers are examined using a point dipole representation for a molecule in an electric field, and calculating the optical constants using the electrostatic approximation. The effects on the macroscopic or empirical optical constants arising from :

1) fluctuation in number density and molecular orientation ; 2) dipole imaging in the substrate (treated as isotropic) ;

3) the number of molecular layers in the film, are examined.

It is found that fluctuations generally introduce only second order effects, while imaging and number of layers can under certain conditions have pronounced effects, requiring that in general the effective optical constants be a function of substrate dielectric constant and the number of layers.

1. Introduction. -The recent exploitation [ l ] of optical measurements in the study of the electrochemical double-layer acts as an incentive for examining the microscopic basics for the phenomenological equations commonly used in the interpretation of the optical data. By far the most common analytical procedure is to assign a thickness and isotropic complex dielectric constant to each component molecular layer of the double layer, then apply Fresnel's equation to the stratified layer structure, assuming the surface to be flat. In a previous publication [2] we showed that for a stratified layer structure of total optical thickness small compared with the wavelength of light, a more useful procedure is to assign an electric susceptibility per unit area to the entire layer structure. If the structure is taken to be uniaxial in symmetry, then a total of two complex susceptibility components completely characterizes the optical response of the surface. Furthermore, expressions analagous to the Lorentz- Lorenz equation can be derived relating the susceptibility components to the components of the complex polarizability tensor of the species in the double layer and their concentration. It is the purpose of this paper

to examine carefully the approximations implicit in this treatment, to extend the treatment in some areas, and to determine the consequences of the model for data interpretation.

We begin by considering the procedure of assigning non-localized optical constants to a planar array of polarizable species and of relating them to local, microscopic optical properties. We do this in the electrostatic approximation and furthermore neglect qua- dripole and higher order induced moments of the molecules. This is followed by an analysis of substrate and multilayer effects.

2. Dielectric tensor for planar molecular array.

-

We shall consider a disk shaped molecular array oriented normal to the z-axis, with the x, y and z-axes being the optical axes of the array, though not necessa- rily of the individual molecules. We further assume that the array can be considered as optically homo- geneous, which requires that the characteristic length associated with number density fluctuations, and the corresponding length for orientational fluctuations must both be small compared with the wavelength of

5

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C5-58 M. J. DIGNAM AND J. FEDYK

light. The optical properties of the array can then be characterized by a complex valued dielectric tensor which can be calculated using the methods of electro- statics.

We define the mean internal or Maxwell field, E, through the normal boundary conditions on the field,

-+

i. e., E, En = Eon, Et = Eo, where Eo is the external

field, and the subscripts n and t refer to components normal to and tangent to the surface plane respectively, with zbeing the dielectric tensor for the molecular array.

Thus in matrix notation

where

-+

Defining the surface susceptibility tensor, y, as follows

-+

yEo = d 4 nP (3) where P is the mean polarization per unit volume and d the thickness or assumed thickness of the array, we have

+

YEO = 4 n C p j

j

(4) where the sum is over unit area. Furthermore, we have

so that

-+ -+ --f

7 = d ( ~

-

1)

.

(6) The local field polarizing a given molecule, EL, is made up of the external field, E,, which we take to be independent of position, plus the contribution from the induced dipole moments of all of the other molecules, which contribution will be proportional to E, and spe- cific to each molecule. Thus setting

F i s a tensor specific to each molecule in the array. We now have

so that eq. (4) becomes

-f

Proceeding now to evaluate $, the field Eij at mole- cule i due to the induced dipole moment of molecule j is given by :

-+

Eij = fij pj (10) where

3 -+

fij = (3 pij

-

l)/r; for i # j (11) = 0 for i = j (12)

-+ 2

where pij

=

rij rlj/rij and is the projection operator for the subspace corresponding to the internuclear line joining the ith and jth molecules, with the prime denoting the transposed matrix.

The vector rij gives the position of the jth molecule relative to the ith molecule. Note that

+ + 3 + f . . = f!.= f . . = f ! . tJ 1J 31 J i (I3) and also Thus we have -+ --f -+

Eij = fij aj(l

+

Bj)

Eo (15) and -+

C

Eij = Bi EO j (16) so that

Although these equations can be formally solved to yield values for

Kt

the general result is too unwieldly to be useful. We shall proceed instead to examine particular cases.

-+ The simplest case to treat is that for which $ is the same for all molecules, a situation which will be realized for an isolated periodic planar array of identical molecules all oriented with their symmetry axes aligned with the coordinates axes. For such a case

so that

and hence

4

Note that

C

fij must be diagonal with its elements

j

summing to zero, and that for N the number of molecules per unit area, the elements of

?

will have the dimensions of length. For a uniaxial arLay with its optic axis coinciding with the z-axis,

C

fir takes the

i

form

which leads to

(4)

where d, can be regarded as the effective thickness of the layer, and is given by :

For a square planar net, de = 0.935 a where a is the nearest neighbour distance, while it is about 20

%

smaller for an hexagonal net. Choosing d, as the film thickness, the effective dielectric tensor for the array is determined from

-+

E = 1

+

E - ' / d e (25)

4 4

which on substituting for y and

4-'

can be shown to give

which is just the Lorentz-Lorenz expression for the dielectric constant of a macroscopic phase. Only the particular choice for the effective thickness used here leads to this result.

If the arrangement of the molecules in the net is periodic with uniaxial symmetry, but the individual molecules take up a range of orientations, then assuming the fluctuations in to be small, the optical pro- perties will be represented approximately by eqs. (22) and (23) with

2

replaced by

<

2 >,

with diagonal elements

<

a,

>,

<

a,

>, <

a,

>

for overall uniaxial symmetry.

If the fluctuations are not small then we may proceed as follows. We define for a subset of the

3

molecules all having the same $ according to

where

<

>

is the mean polarizability tensor for the subset and Nj their surface number density. The total surface susceptibility tensor

7

is then, according to eq. (9), given by :

-+ -+ Y = C y j .

j

(28) Eqs. (17) and (27) constitute a set of linear equations in$ which can be solved to yield$ as a function of the

<

aj

>

and Njt and hence an expression for

Suppose the molecules are randomly disEibuted in the plane but nevertheless the fluctuations in $ are small formost of them. Once again to a fair approximation

eqs. (22) and (23) result with ;replaced by

<

;>.

If the fluctions in Fare large but to a first approximation

-+ 4 -+ -+ -+

are not correlated witheither fii or a (i. e., f i , a i and $

are nearly uncorrelated) then eq. (17) gives -

To evaluate

< X j

>

the pair distribution function for the molecules must be known.

Finally, if the distribution of molecules can be appro- ximated by close packed groups of molecules, with the number in the groups varying from one on up, and if we neglect local field contributions between groups, then eq. (17) becomes approximately

This then leads to

with

Zk

given by eq. (32).

An important difference arises between the beha- viour predicted by eq. (33) and that predicted by eq. (30) (or eqs. (22) and (23)) at a frequency near that for a moderately sharp absorption band for the molecules in the array. Thus on representing the band by a Lorentzian function, and assuming the same band appears in

<

a,

>

and

<

a,

>,

we have

where a: is the high frequency contribution, s, the oscillator strength, v0 the centre frequency of the absorption band and

I'

the relative band width. Substituting eq. (34) into eqs. (22) and (23) with: replaced by

<

;>,

the results may be represented as follows

r,

= 7,"

+

sul[l

-

( V / V ~ ) ~

+

iru<vlv,">]

,

with y?, S,,

ru

and v: given by :

r:

= 4 n N

<

a," > / [ I

-

q,(4 n/3) (NIde)

<

a:

>I

(363

s,

= 4 nN

<

s, >/[I

-

qu(4 7713) (Nld,)

<

a,"

>]

[1

-

qU(4 ~ 1 3 ) (NIde)

(<

>

+

<

Su

>)I

(37)

r

1

-

qu(4 n/3) (Nld,)

(<

a,"

>

+

<

S, >)'I2

- = Y , =

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C5-60 M. J. DIGNAM AND J. FEDYK

These equations predict a shift in the centre frequency for the band with no significant broadening according to v:

<

v0

<

v:.

Optical measurement will in general produce a result which relates to a linear combination of yn and y , so that a splitting of the band into two bands is predicted. Such a result has been observed for the stretching band of CO adsorbed onto CsCl and accounted for quantitatively in terms of equations analogous to eqs. (34) and (37) [3]. A split of about 12 cm-I was observed and predicted. Without going through the details it is apparent on inspection that eq. (33) will predict a substantial broadening of each of the components of the band, which would result generally in the two components being unresolved.

It is apparent from the above analysis that details concerning the distribution of molecules in a plane can be obtained from optical measurement made in the vicinity of a strong, sharp absorption band. We will not pursue this matter further, but will turn instead to what appears to be an effect of more general importance,

-+

namely the influence on y of placing the planar array of molecules on a semi-infinite substrate. We begin by considering a single planar array and then extending it to a multiple layer for the particular case of a succes- sion of square planar nets.

3. Single molecular layer with imaging. - The system to be analyzed is shown in figure 1. The substrate

/Molecular Layer Electrostatic Image?

FIG. 1.

-

Model of a surface with a monomolecular overlayer.

will be treated as an isotropic dielectric continuum of complex dielectric constant e,. For the particular case of a metallic substrate, this should be a reasonable approximation, as the dielectric properties of metals are determined largely by the Fermi sea of electrons. For other substrates, the results calculated in this way will presumably be at least qualitatively correct.

With this model, we employ the method of images for evaluating the contribution of the substrate to the local field polarizing a given species on the surface. We also take ;and

Sf

to be constant for the array so that eqs. analogous to (18) to (20) but including imaging may be used, and restrict ourselves to arrays of uniaxial symmetry.

The local field acting on a typical molecule in the array, the ith molecule, is given by :

(1

+

$)

E, = Eo

+

where the second term is the contribution due to the images,

7

identifying the image of jth species. Both sums are diagonal tensors of zero trace which can be readily shown to have the form

where

h = z/a is the dimensionless height of the dipoles above the plane of their images, a a characteristic length for the net (e. g). the nearest neighbour distance or lattice parameter), and the second term in eq. (41) is included only if h # 0.

The image of pj is given by

where Xs = (E, - I)/(&,

+

I), SO that on substituting

for pj, replacing pj by 3 1

+

fi

E,,

and making use of eq. (40), eq. (39) becomes

-+

As there are by postulate no fluctuations in

fl,

eqs. (43) and (9) lead to :

Thus the optical constants for the monomolecular net are functions of the substrate optical constant, X,. Furthermore, if we attempt to define an efiective film thickness, d,, such that

<

for the film follows the Lorentz-Lorenz equation, so that 6, = 6, for a, = a,, we obtain

Thus if X, is complex valued, both d, and; will be complex valued even if a, = a, is real valued.

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effect of images is important, we use the following integral approximation for G(h) :

where

G(0) = 2 nNa2 = 2 ₍₄₈₎ since N = l/na2 in this approximation. Expanding h3 G(h) in powers of h-2 for h

>

1 and retaining only the lowest order term gives

Substituting this result and G(0) from eq. (48) into eqs. (44) and (45), the results can be expressed as follows :

..O

*

where y, = 1, y, = 2, and

Thus on approaching the continuum limit, by letting (alz) -+ 0 with z constant and a, varying so that y: remains finite, the image effect is seen to fall off as ( a / ~ ) ~ , i. e. very rapidly.

This result is not surprising as such an image effect has never been included in the continuum treatment of the dielectric properties of matter. For the more realis- tic situation in which z

-

a, however, the image effect is in general not entirely negligible. For this situation, G(h) LY

-

0.65 G(O), while for z/a 2

Jn

as for a

square net of spheres, G(h)

--

- 0.06 G(0). Thus even for a good imaging substrate ( X ,

-

1) the contribution of the images to the local field will usually be small, but not negligible.

Significant effects can nevertheless arise, examples of which will be presented later. Before proceeding to these, however, we extend the treatment of this section to more than one molecular layer.

4. Multiple molecular layers with imaging. - The system to be analyzed is shown in figure 2. Again we treat the substrate as a continuum and use the method of images. We also take ;and $to be constant within any one layer, but differing in general from layer to layer

$

will differ from layer to layer even for; constant). Furthermore, for simplicity, we restrict the analysis to the case in which each layer is a square net of lattice constant a with the molecules in successive layers directly above one another.

Proceeding precisely as for the single layer, except that the local field contributions must now include all the layers and their images, we obtain in place of

Layer

Designation Molecular Layers,

, , . ,

Electrostatic ~ r n o g h s

FIG. 2.

-

Model of a surface with an overlayer of several molecular layers.

eq. (43) which applies to one layer, the following multilayer equations

where [I

+

8>,1

Eo is the local field for any molecule in the kth layer arising from the external field plus all the other molecules, their images and its image, m the number of layers, and

with h for each layer being measured from the plane of images of the first layer (see Fig. 2). Note that G(h), defined by eq. (41) is a function of h only for the model chosen, since rij in eq. (41) is an intra layer distance. Eqs. (52) and (53) are of course equivalent to eq. (43) for the case m = 1.

Defining matrices A, and An with i, j elements (AJij and (An)ij given by :

where

aij

is the Kroneker delta, the solutions to eqs. (52) and (53) take the form

Y u i = 4 na (All)ij , u = n, t (57)

i

where

(7)

C5-62 M. J. DIGNAM AND J. FEDYK

where

I"

r2

-

2 h2 2 xi;

I 5 2 n r d r =

i,,, (r2

+

h2)512 ( i i

+

h2)312 '

In practice, a value of i,,, = 100 is sufficient to obtain G(h) to an accuracy of better than

,

lo-'.

Eq. (57) gives explicit values for the?: in terms of the

+ 3

system parameters, a, hk, a, X,. From these, y can be calculated using eq. (28) which on substituting for y,, from eq. (57) becomes

This represents an exact solution for the model proposed, but the model itself is not, of course, an exact representation of reality. In particular, retardation, multipole moments, and substrate discreteness effects have been neglected. For metal substrates, there are circumstances when retardation effects cannot be neglected even for very thin films [4] since surface plasma waves of wavelength comparable to that of the light can cause large retardation effects. Generally, however, they will be important only near the plasma resonance frequency and will always be unimportant for monomolecular films on dielectric substrates in the U. V., visible and I. R. regions of the spectrum. The importance of multipole moments will depend very much on the nature of the film, as will the importance of discreteness-of-substrate. Actually, eqs. (55) to (60) allow one to treat the substrate as made up of discrete point polarizable species. Such a model is not very satisfactory for metals however.

while for the mixed layers as

Y U = 4 nN(%, 4- a,,) ( 1

+

B,)

(66)

where

pu

is obtained from eq. (43) on setting

X,

= 0

and replacing a, by (a,,

+

au3/2, and N is the number of molecules per unit area in each layer.

In general, the difference between eqs. (65) and (66) is not great, vanishing, of course, for j?,, =

p,,,

and being large only when there is a big disparity between

1

+

pul

and 1

+

P,,,

i. e. between a,, and a,,. The difference will be readily detected when one of the species has an absorption band, and differential optical measuring techniques are being used.

Thus if an absorbing layer is initially present (Iayer 1) and a non-absorbing layer added, then in the event that

+ -+ the layers remain separate, the change in y, Ay, is equal to

T2,

which is independent of in the continuum limit and hence will not lead to any observed optical absorption. (For further explanation see t@ next section.) On the other hand, if the layers mix, AT will be a function of both and

z,

leading to optical absorption. Such an effect has been observed in the infrared following adsorption of CCl, onto an overlayer of formic acid [ 5 ] . The formic acid bands appeared as negative absorbance peaks in the difference spectrum.

6. Optical equations in thin

a m

limit.

-

In order to pursue discreteness effects further, we require equa-

--f

tions relating optical observables to y. Since our inte- rest is in thin films, we restrict attention to the thin film

5. Continuum limit for multilayer structures.

-

The

--

off diagonal elements of A, take the following form on or linearized forms of these. Noting that;= 1

+

y

%

' I d

(eq. (25)) Fresnell's equations for external reflection using the integral approximation for G(h) for h > 1

from a semi-infinite substrate covered by a uniaxial (i. e., using eq. (49))

medium of thickness d and dielectric tensor ;lead to

which goes to zero rapidly as a -t 0. The diagonal

elements, on the other hand, take the following form in the continuum limit

(Au)ii = a31aur

-

3

G(O) (62)

which with eq. (57) gives :

Y u i = 4 ~ ( a u i l a V l [ l

-

3

qu(aui/a3) G(0)] (63)

which is identical to the expression obtained for an isolated single molecular Iayer (i. e. identical to eqs. (22) and (23)). Thus in the continuum limit, y, is simply the sum of the y u j for the isolated layers.

Note that in the continuum thin film limit, the order of the layers in the array has no effect on y,. Inter- mixing the molecular species, however, does. Thus for different molecules in two distinct layers, y, is given in

the continuum approximation by

the following expressions for the change in reflection coefficient upon changing ;by adsorption or desorp- tion 121

i 4 n

AL, =

-

cos ~p

-

[I

+

a,,,

Lo 1 - E,

where

Lo

is the wavelength of the light in vacuuo, cp the angle of incidence, and AL, = In

[EV/Rv]

where

R,

is the complex reflection coefficient for the interface for

v

polarized light, while

xv

corresponds to the reference or initial state of the interface. The base 10 reflectance absorbance, A,, and the ellipsometric para- meters $ and A are then related to AL by

(8)

Note that the observables are independent of film thickness in this approximation, a consequence of a cancellation of d in the expression forTwith that in the first order expansion of Fresnel's equations. For thin films

(Ao

% 4 n

I

E, ( l r 2 d ) therefore, ellipsometry and reflectometry yield no direct information on film thickness, to first order. The first order behaviour is determined soley by

%

the surface susceptibility tensor. Note also that AL, (and hence A , ($ -

F)

and

( A

-

3)

is a linear function of Ay, and Ay, with zero absolute term, which is basically the reason why AL, depends only on A? and not onTitself.

The simplicity of these equations and the relative ease with which AFcan be interpreted in microscopic terms makes this form for the optical equations parti- cularly useful in analyzing the properties of interfacial regions of molecular dimensions in thickness.

The above formulation is most useful when all of the changes in the interfacial region can conveniently be considered to take place in the vacuum phase. If one wishes to relate the changes to the substrate phase, the following equivalent equation can be used

i 4 n A??) 1

-

a,,,

by:)/&, A#)

A t v = - cos q7

-

Lo I/&,

-

1 cot2 (PI

-

l / E s

where?(" is the surface susceptibility referred to the substrate, rather than vacuum phase. Thus y$) is essen- tially 4 n times the excess polarization per unit area, divided by the tangential component of the field in the substrate medium, the excess being with respect to the substrate. A simple application of eq. (70) yields McIntyre7s model [I] for the electroreflectance effect whereby metal electrodes change their reflectance with potential (surface charge). According to this model, the effect is due to a change in the number of conduction electrons near the interface. Using Drude's model for the metal, substrate

where w is the angular frequency of the radiation, w, is the bulk plasma resonance frequency given by

with n the volume concentration of conduction elec- trons of charge -e and mass m, and T the collisional

relaxation time.

This gives, corresponding to a change in n of An the result

where E; is the value of E, for n replaced by n

+

An

and o =

-

And is the surface charge density measured in units of e. Similarly,

so that eq. (70) becomes on setting

EL

.-

E,

i4ncosq1 ( 4 n c e 2 / r n ) AL,

(1

-

) w2

-

iu/T

so that all the observables become proportional to the surface charge, o.

Extension of all the equations in this paper to cover the case of an isotropic continuum ambient phase of dielectric constant differing from unity is straight- forward, and has been done elsewhere [2]. As the main purpose of this paper is t o examine effects arising from the discrete nature of adsorbed layers, we will treat solely the case of a vacuum ambient.

7. Adsorption induced surface pIasma resonance. - For a metal substrate, the presence of the image term

--?-

in the expression for y for a periodic single overlayer, eqs. (44) and (45)' lead to a surface plasma resonance that vanishes in the continuum limit. Thus representing

E, as in eq. (71), X, = (E,

-

I)/(&,

+

1) takes the form :

Xs(o) = [I

-

lo,,)^

+

i ( ~ l ~ , p ) l ~ , p TI-' 1761 where o,, = w , / a and is the surface plasma resonance frequence. Thus at o = a,,, X, is entirely

imaginary and given by :

Xs(osp) =

-

iwsp

.

(77) Since us, Tis typically of the order of lo2 for metals such as silver, copper, and gold, it is clear that pronounced image effects can arise for frequencies near

cusp. Resonance effects will arise primarily from the behaviour of (1

+

PJ

and (1

+

B,).

Thus eq. (67) can be written

which on neglecting the slow variation of 1

-

I/&, and cot2 p

-

I/&, near o,,, leads to resonance effects when either (1

+

P3

or (1

+

8 3

are entirely imaginary. Eqs. (44) and (45) imply

(1 i- 8U) = [l

-

3

qu G(O) (aula3) 4-

(9)

(25-64 M. J. DIGNAM AND J. FEDYK

Substituting for X, from eq. (76) the resonance condition is given by :

where x = o/w,,. For o satisfying eq. (SO),

1

+

BU

= (1

+

Pa),

is given by :

where the approximation involves neglecting the final term in eq. (80) and a small term in G(h). The condition for pronounced resonance effects are satisfied for

I

(1 f Pu),

(

>

1, which for w,, T

-

10' is entirely possible. Note that the resonance conditions are satisfied for o close to a,,, but not equal to it, the exact frequencies being less than w,, and weak functions of h and (q,/a3), and to a lesser extent still, T.

1 . 0 0 -1.0 2.0 3.0 4.0 5.0 LAYER NUMBER (a) 1.001 I I 1.0 2.0 3.0 4.0 5.0 LAYER NUMBER (b)

FIG. 3.

-

Illustration of the adsorption or image induced FIG. 4. -Illustration of the effect of number of layers on the surface plasma resonance for a free electron metallic substrate. dielectric constant for a film consisting of from one to five The calculations are for z/a = 1.0, 1.25 and 1.5 for the curves identical layers of molecules in square planar arrays. (Curves

(10)

MICROSCOPIC MODEL FOR C5-65

8. Numerical calculations. - To obtain an estimate of the possible effects of interlayer local field contributions, preliminary calculations were performed for films of square planar symmetry, the species having isotropic polarizabilities characterized by E,, where a

and eb are related by the Lorentz-Lorenz equation, i. e.

In figure 3, calculated curves are presented showing the adsorption induced surface plasma resonance effect. The calculations were made for a free electron metal having a,, z = 10 and films characterized by 8, = 6, h = 1.0, 1.25 and 1.5 respectively. The magnitude of the resonance effect increases with increasing h, as expected. For h = 1, corresponding to the case of a square planar array of contacting spheres, the resonance effect is small, but by no means negligible. Except for very flat surfaces, one would expect in general that roughness induced plasma resonance [6] would dominate over this effect. More extensive calculations on this effect are currently in progress.

In figure 4 the effect of the number of layers on the dielectric constant of the film is shown. The calculations were performed for E, = 1.5 and es = 1.5 and 6.0 respectively. The film dielectric constant components are defined throught eqs. (6) and (60) with d = ma. The effect is seen to be large for Ah = 1.5, but negligible for Ah = 1, corresponding to a cubic overlayer. 9. Summary and discussion. - A microscopic model for the optical properties of thin layers separating semi-infinite isotropic phases has been developed based on a long wavelength and point dipole approximation, in which the substrate contribution to the local field is handled by the method of images. In general the substrate effect is expected to be small so that to a good approximation in many circumstances the film optical tensor is not an explicit function of the substrate opti- cal constants. A possible exception to this may occur at frequencies just below the surface plasma resonance frequency in the case of a metallic substrate, where adsorption induced resonance effects are predicted. The neglect of the substrate contribution to the local field, and of interlayer local field contributions, is equivalent to treating the layers in the continuum approximation. For this case, the optical tensor for a

[I] For a review see

MCINTYRE, J. D. E., Specular Reflection Spectroscopy, in :

Advances in Electrochemistry and Electrochemical Engineering, Vol. 9, Ed. R. H. Muller (Wiley, New York) (1974) 61-166.

[2] DIGNAM, M. J. and MOSKOVITS, M., J. Chem. Soc. Faraday 2 69 (1973) 56.

[3] RAO, B. and DIGNAM, M. J., J. Chem. Soc. Faraday 2 70 (1974) 492.

stratified layer, represented by;;, is simply the sum (or integral) of the< for the individual stratum.

Furthermore, all of the optical changes accornpany- ing any change in a thin stratified layer are simply

-+-+

proportional to A; the change in y. y is characterized by two complex components for a uniaxial interfacial region, and by three for a biaxial region, such as the (110) face of a ccp crystal.

A similar but not identical treatment to that presented here has been given by Sivukhin [7] in 1951. The author was unaware of this work when the present formulation was developed [2].

Sivukhin defined surface optical constants, also represented by

<

which we will prime to distinguish it from the present He considered only the case of a film separating substrate and vacuum phases, for which +

y' was defined according to one of the following

( p )

-

d ) ~ , =

-

P d .

It is easily shown that these lead to the following relation :

-+ 3

with an identical equation relating y ' f S ) and y f s ) . The two sets of surface optical constants differ, therefore, by a simple function of e,. The interlayer local field contribution which is generally small, has been eva- luated by us by the method of images for the case of a single molecular layer. Sivukhin used an approach in which half the difference between the Lorentz-Lorenz dipole contribution to the local field and the two dimen- sional square planar dipole contribution to the local field, both for the substrate, is taken to be the interlayer contribution to the local field for the adlayer. For metals, the former approach should be superior, while for non-metals it makes little difference which approach is used as the contribution is very small.

The physical meaning of $n our formulation is more transparent. It is simply 4 n: times the excess polarization per unit area of the interfacial region relative to the reference phase (ambient or substrate) divided by the field in the reference phase.

rences

141 NGAI, K. L., Surface Plasma Oscillations and Related Surface

Eflects in Solidr, in : Advances in Chemical Physics, Vol. XXVII, Ed. I. Prigogine and S. A. Rice (Wiley, New York) (1974) 265-354.

[5] RAO, B., STOBIE, R. W. and DXGNAM, M. J., J. Chem. Soc.

Faraday 2 71 (1975) 654.

161 DIGNAM, M. J. and MOSKOVITS, M., J. Chem. Soc. Faraday 2, 69 (1 973) 65.