Effective media equivalent to an asymmetric multilayer and toa rough interface

Tomuo Yamaguchi, Jacques Lafait, Azeddine Bichri, and Kouider Driss-Kodja

A computer fitting program is used to show that not only a symmetric but also an asymmetric multilayer,composed of a periodic structure with small thickness irregularity or with slightly rough interfaces, can beequivalent to an anisotropic single layer within a good approximation. It is also shown that an interface layerequivalent to a rough interface does not exist, in spite of the fact that such an assumption has widely beenused. Key words: Multilayers, effective medium, anisotropy.

1. Introduction

It has been proved that a repeated-symmetric multi-layer is equivalent to an optically anisotropic singlelayer, 1'2 and a repeated-asymmetric multilayer isequivalent to an anisotropic double layer.3 In a previ-ous paper we described various properties of theseequivalent parameters of symmetric multilayers byderiving exact formulas.4

The formulation is not only interesting to under-stand the basic optical features of multilayers but alsouseful to check the validity of anisotropic formulaswithout an experiment. The multilayer can also beused to calibrate such experimental setups as a well-defined anisotropic sample whose principal axes areautomatically adjusted to fit the sample normal to theplane of incidence.

This paper demonstrates that the model of an equiv-alent anisotropic single layer is still valid within a goodapproximation for asymmetric multilayers. We alsoshow how actual unavoidable conditions, such as aninterface roughness and a random thickness error, af-fect the equivalent parameters. A computer fittingprogam is used for that purpose. The optical equiva-lence is obtained when the values of all the observablesare equal. There are eight independent observables,i.e., real and imaginary parts of the complex amplitudereflection and transmission coefficients for s- and p-

The authors are with Pierre et Marie Curie University, CNRSLaboratory of the Optics of Solids, 4 Place Jussieu, F-75252 ParisCEDEX 05, France.

Received 18 October 1989.0003-6935/91/040489-06$05.00/0.© 1991 Optical Society of America.

polarized light r8, tS, rp, and tp, and eight independentparameters in an equivalent anisotropic layer, i.e., realand imaginary parts of the three principal dielectricconstants ex, ey, and e and complex thickness de, the x-,y-, and z-axes being normal to the plane of incidence,normal to the film plane, and normal to the former two,respectively. 4

To check the computer fitting process and to knowwithin which approximation an asymmetric multilayercan be equivalent to a single layer, we first apply thefitting procedure to a symmetric multilayer and theninvestigate the influence of a thickness change in theouter half-layers leading to the asymmetric multilayer.Next we show the effects of interface roughness andrandom thickness distribution.

Finally we discuss the possibility of modeling arough interface using an equivalent layer which hasbeen often assumed for the analysis of, for example,ellipsometric measurements. We show that an inter-face layer equivalent to a rough interface does notexist.

11. Theoretical Formulas

A. Effective Parameters Ex, y, e, and de

The effective parameters of a single anisotropic lay-er equivalent to a symmetric multilayer are derived inthe previous paper4 :effective dielectric tensor (

(1)Z = (m21/m12)s + S2,

i = (1/kde)V(m2 /mi2)p arccos(mll),

Z = S 2 /[1 - y(M2/M11)p];

(2)

(3)

effective thickness of the multilayer periodic struc-ture,

de = (m 12/m2 1)s arccos(mll),/k, (4)

1 February 1991 / Vol. 30, No. 4 / APPLIED OPTICS 489

where mi. is the element of the transfer matrix of theunit symmetric period of the system, S = sinO is theinvariant in all layers, 0 the angle of incidence, and k isthe wave vector in vacuum. Subscripts s and p denotethe states of polarization of light (linear polarization ofthe electric field which is perpendicular and parallel tothe plane of incidence, respectively).

B. Multilayer Formulas 5

The method for calculating the complex amplitudereflection and transmission coefficients r and t of amultilayer stack uses a 2 X 2 matrix which is differentfrom the transfer matrix. The matrix of the L-layerstack including incident layer () and exit layer (L) isgiven by multiplying interlayer matrices Mij and intra-layer matrices Mj as follows [for general use, each layerwas assumed to be itself anisotropic with a dielectrictensor e(e,ey,e,) 5I:

Q2 Q12 = M12 M2 M2 3 . . *ML- * M(L- )L' (5)

where

MJ thj(7i 73}), (6)

Mj= (exPo 6 exp(-i )' 7

= kdV(e - S 2 ), (8)

jp= kdjoV[e - (ejy/ejz)S 21, (9)

ti = itirij fji,(0rijS = (Yi-Yj)/(Yi + Yj), ~~(11)

riM = (Z -Zj)/(Zi + Z) (12)

i= 2Yi/(Y + Yj), (13)

i= (ei/ej)2Z/(Z + Zj) (14)

are amplitude reflection and transmission coefficientsof the ij-interface for s- and p-polarizations, and

Yij= k/(djx-\),: (_~jyl_~j')S2'(15)

Zj= +j-(ejye)S 2I/e y (16)

are admittance and impedance of the jth layer. HereS has the same meaning as in Eq. (1).

A simple but efficient way to introduce the effect ofan interface roughness is to modify the interlayer ma-trix by assuming that the interface is normally distrib-uted around the mean flat interface and to average thephase change resulting from the interface deviationfrom the mean interface. Let f(x) be the deviationfrom the mean interface at point x. The phase devi-ations caused by f(x) on reflection and transmissionare summarized in Table I. These phase deviationshave the following general expressions:

4 = exp[iAf(x)], (17)

where averaging b in the case of a Gaussian distribu-tion results in

Table 1. Phase Change In rand t Resulting From Interface Deviation f(x)From the Mean Smooth Interface

s-polarization p-polarization

rij exp[2ii 8 f(x)] exp[2ibipf(x)]rji exp[-2i6j1 f(x)j exp[-2ijpf(x)]tij(=tji) exp[i(bj, - bj,)f(x)] exp[i(bip-jp)f(x)]

bj = k -(bjx -S 2 ) bjp = kV[bjy - (jy/bjz)S 2]

(18)

where is the standard deviation, in other words, therms roughness.

Equation (18) means that the amplitude of the spec-ular reflection and transmission decreases. For asmooth interface, ,B = 1 in Eqs. (6) and (10), and theinterlayer matrix keeps its symmetry whereas for arough surface, # 5zs 1 and the matrix loses its symmetry,which suggests that an equivalent effective layer nolonger has any reality.

The amplitude reflection and transmission coeffi-cients of the multilayer are given by

r= Q21/Q1,

t = /Q1 .

(19)

(20)

C. Fitting Procedure

Since there are four independent observables ineach polarization, i.e., real and imaginary componentsof r and t, four fitting parameters have been chosen,complex E. and de for s-polarization and complex Ey andE, for p-polarization in the case of absorbing material.For transparent films the number of these parametersis reduced by half. If interface roughness is intro-duced, it is nevertheless necessary to use an equivalentcomplex epsilon even if the materials are transparent.

The fitting procedure was performed by minimizingthe value of the following quantity using a Simplexalgorithm,

error2 = Irmult - requivl2 + Itmult-tequi l 2 (21)

separately for s- and p-polarizations. The subscriptsmult and equiv refer to calculations for the simulatedmultilayer and for the possible equivalent anisotropicsingle layer, respectively.

Ill. Numerical Results and Discussion

Examples of the numerical calculations are shownbelow. Throughout these simulations the wavelengthis put equal to 500 nm and the periodic multilayer isassumed to be composed of two materials a and b ofdielectric constants e, = 6 and eb = 2 close to the squareof the refractive indices of TiO2 and MgF2, respective-ly. Incident and exit media are assumed to be invacuum.

A. Transparent Symmetric System

Two symmetric configurations are simulated: con-figuration A: a/2: b: a: .. .: a: b: a/2 and config-uration B: b/2: a: b: .. .: b: a: b/2, generated,

490 APPLIED OPTICS / Vol. 30, No. 4 / 1 February 1991

((P) = exp[-(o-A)2 /21,

4

W

>4

W

3

2

.1

I .0

. 9

I . 0

0

>I

. I4'

0e

05

00

0.95

1.05

.00

0

>sI

00

( b .o d (m) LvJ

Fig. 1. Unit thickness (d>) dependence of (a) , E,, and ez and (b) deof symmetric A and B systems.

respectively, by the N-time repetition of the basicperiods: case A: a/2: b: a/2 and case B: b/2: a: b/2,where a/2, b/2, a, and b depict, respectively, layers ofmaterials a, b, a, b with thicknesses da/2, db/2, da, db-In both configurations the unit thickness of the periodis du = da + db. For simplicity we assume in thecalculation that da = db-

Figure 1 shows the unit thickness (do = 2da = 2db)dependence of the equivalent parameters E,, Ey, E, andde4 of a symmetric multilayer calculated using Eqs.(1)-(4) at the two angles of incidence, 5 and 850.These equivalent parameters have been normalized tothe bounds calculated in the limit of vanishing thick-nesses and infinite number of layers, 2'4 i.e., Z, = Ey = ,with (da + db)eo = daea + dbeb, Ez e with (da + db) le =dalea + dbleb, and de = d. = da + db. These bounds arealso used as starting values in the fitting procedure.

B. Thickness Change in the Outer Half-Layers

Figure 2 shows the effect of a change d in thicknessdb of the outer half-layers in the B configuration; otherconditions being do = 10 nm, incident angle 0 = 850,and total number of layers = 11 (four elementary peri-ods).

In Fig. 2(a) the value of abscissa d is added to both ofthe outer half-layers, i.e., on the left-hand side of theabscissa. The multilayer structure is the original one,

4.'14

-4

.95

0.90

0.2

0 . I

0 .0

(b) d du

w f + £ X ~~~~~~~r~ / C O

7 .

F _. =/ e

. ~ Iz errorp

_ Io/ / error7

. / ___~~~

d (nm)

Fig. 2. Effect of a thickness change in the outer half layers startingfrom symmetric system B(b/2:a:b:a:b:a:b:a:b:a:b:a:b/2) with ea = 6, eb

= 2, d = 2d = 2db = 10 nm, and the incident angle = 850. In (a) dis added to both the outer layers, finally becoming (b:a:b:a:b:a:b:a-:b:a:b:a:b). Because the system is always symmetric, fitting errorsare always 0. In (b) and (c) d is added to one of the outer layers andis subtracted from the other outer layer, finally leading to an asym-

metric structure (b:a:b:a:b:a:b:a:b:a:b:a).

b/2:a:b:a:b:a:b:a:b:a:b/2.

On the right-hand side of the abscissa, the addition of2.5 nm to both of the outer half-layers results in anequithickness multilayer structure:

b:a:b:a:b:a:b:a:b:a:b.

Since the number of layers is still odd, the system issymmetric and can be replaced by an effective aniso-tropic layer, i.e., the fitting error is still 0.

1 February 1991 / Vol. 30, No. 4 / APPLIED OPTICS 491

(a)de/du

Is, _ £xt/o Iz/re

! ~~~~~~~~~~~~~~~~~~~~~It

60nn On A n

v 1.0

(a)

£C')

(b)

(c) '' ' '

_ _

4

x

2

1 .04

1 .00

0.9 6

0.0 4

0.02

0 2 4 6 8 10

Maximum random thickness error(%)

Fig. 3. Effect of a random thickness distribution on ex, Zy, eZ, and deof symmetric system B with 101 layers; du = 10 nm and 0 = 850.

In Figs. 2(b) and (c) the value of the abscissa is addedto one of the outer half-layers and is subtracted fromthe other outer half-layer, i.e., in this case the systembecomes unsymmetric, except the left-hand side origi-nal one, and finally becomes totally asymmetric on theright-hand side:

b:a:b:a:b:a:b:a:b:a.

Theoretically, this system is no longer equivalent to asingle anisotropic layer. This is confirmed by theincrease of the fitting error calculated using Eq. (21) inFig. 2(c).

Although the fit is not perfect, the fitting error is, atmost, -0.2% even when the system becomes asymmet-ric. A fitting error of 0.2% in amplitude corresponds to

W

t

I .00

0.99

0.980.2

N 0.1

0. 0

r0

0

0

0

0

0 .003

0.002

0.001

0. 00O

rms roughness(nm)

Fig. 4. Effect of an interface rms roughness on Ex, Ey, eZ, and de ofsymmetric system B with thirteen layers; du = 10 nm and = 850.

0.04% in energy reflectance or transmittance, which issmaller than usual measurement errors. We concludethat an asymmetric multilayer system can still be re-garded as an anisotropic single layer within a goodapproximation.

C. Effect of a Random Thickness Error in a SymmetricSystem

Figure 3 shows the influence of a random thicknessdistribution on the equivalent parameters of symmet-ric system B composed of 101 layers. The calculationis made at an incident angle of 850. At each maximumrandom thickness error in the abscissa, ten differentthickness distributions were generated using a randomfunction, and the fit was performed at each distribu-tion.

Figure 3(a) presents the best-fit values of E, Ay, and

492 APPLIED OPTICS / Vol. 30, No. 4 / 1 February 1991

(b) /

/

./ '/'/ /z ''/

/ 7''- - - -

,- -1

I

5.

14

'0P

W

_0N,

.,

0 .00

(c)

error /

.1~~~~.l // / / /

' - -- errorp=0O- - - I I

0 1 2 3 4

5 I .01I

0.95

0 .9 0

0

0

85 h

80 -

0.751 l

0.2 0

(b)

9

0.10 K_ _BT

0.05 | G

300 500 700 900Wavelength(nm)

Fig. 5. Change in T and R spectra induced by an interface rmsroughness between fused quartz and TiO2 calculated with the phaseaveraging model and with the interface layer model using the Brug-

gemann theory (Bt). The thickness is 6 nm.

E, Fig. 3(b) presents the best-fit value for de, and thecorresponding fitting errors for s-polarization areshown in Fig. 3(c).

The scattering range of the equivalent parameters isrelatively small and here too the fitting error is muchsmaller than the usual experimental error. Wechanged the initial values of the fitting procedure byseveral percent, but the result did not depend on thechoice of the initial values.

D. Effect of an Interface Roughness on a SymmetricSystem

We calculated the effect of an interface roughness onthe equivalent parameters of symmetric system Bcomposed of thirteen layers, using the model present-ed in Sec. JI.B and an incident angle of 850.

We can see in Fig. 4(a) that the real part of theeffective epsilon is affected <1% up to a rms roughnessequal to 4 nm at da = db = 5 nm. The remarkable pointis the appearance of an imaginary part in these effec-tive epsilons, as shown in Fig. 4(b). Here, the fittingerror for p-polarization is always 0, and that for s-polarization is much smaller than the usual experi-mental error as seen in Fig. 4(c).

Since the effect of the interface roughness stronglydepends on the angle of incidence, we performed the

same fitting procedure at 5°. The results were quali-tatively similar but the ordinate was multiplied by afactor of 2 for e [Fig. 4(b)] and a factor of 10 for the error[Fig. 4(c)].

E. Effective Interface Layer Equivalent to a RoughInterface

Finally we see whether a single effective interfacelayer can account for the effects predicted by the roughinterface model. This kind of effective interface layerhas often been assumed and its properties have beenextensively discussed for ellipsometric measurementsof a single rough surface 6 7 and for a rough interface in amultilayer.8 9

Chain lines in Fig. 5 show the influence of a roughinterface on normal incidence reflectance]R and trans-mittance T of the interface between fused quartz andTiO2 as calculated by the formula described above.Both R and T decrease with the increase of the rmsroughness.

Qualitatively speaking, it seems difficult to accountfor this variation by any interface layer assumption.The effective dielectric function of the interface layeris some average between those of the two materials.Such an interface layer should act as an antireflectioncoating. As an example, the dashed curves in Fig. 5(marked BT) show R and T calculated, assuming thatthe dielectric constants of the interface layer are givenby the Bruggeman theory 6 with Q = 0.5 and d = 6 nm,where Q is the packing density of the TiO2 particlesembedded in a fused quartz host material. As expect-ed, T becomes greater than the full line characterizingthe smooth interface.

The computer fitting method was used to find valuesof the anisotropic parameters of the interface layerwhich explain simultaneously both r and t calculatedwith a rough interface, but there were no solutions.We tried to fit only with r without t, but we only gotphysically meaningless solutions.

The basic reason for this failure can be explained asfollows: the assumption of the interface layer resultsin a phase change dominant r and t for a small interfaceroughness; whereas the phase averaging method usedin this paper results in amplitude change dominant rand t. The equivalent interface layer model seems,thus, able to explain qualitatively R = r2, but it cannotaccount for phase sensitive measurements such as el-lipsometry.

The ellipsometric parameters and, for a rough sur-face, those calculated by the phase averaging method,are essentially not influenced by a small roughnesswhen the roughness is small enough that such an ap-proximation can be used. On the other hand, theparameters calculated by the equivalent interface lay-er model are so influenced by a small roughness thatthe effective index of refraction of a bulk materialdetermined by ignoring the small roughness deviatesconsiderably from the bulk value. Examples areshown in Table II. Actual surfaces of Ag films as wellas BK7 conventionally polished are believed to have aroughness of several nanometers; however the refrac-

1 February 1991 / Vol. 30, No. 4 / APPLIED OPTICS 493

(a)

9

:I

4 4 - -

Table II. Effective Index of Refraction of BK7 Glass and Ag as a Functionof rms Roughness

BK7 (X = 500 nm) n = 1.521, k = 0.000

a (nm) 0 5 10 20

"60 5.18 5.24 5.41 6.05A6 0 0.00 7.43 14.54 26.84neff 1.521 1.520 1.518 1.507keff 0.000 0.025 0.049 0.098

Ag ( = 500 nm) n = 0.050, k = 3.135

a (nm) 0 1 2 5

*60 44.68 44.25 43.85 42.72A6 0 130.49 129.02 127.60 123.70neff 0.050 0.110 0.161 0.281keff 3.135 3.022 2.917 2.647

a Calculatedon the assumption of an interface layer whose indexof refraction is given by Bruggemann theory with Q = 0.5 and whosethickness is equal to the rms roughness.

tive indices of those materials measured by ellipso-metry ignoring this roughness are not far from the bulkvalues. By ellipsometric measurement, we have neverobtained such large deviations as those suggested inTable II. Deviations caused by the actual polishinglayer or by insufficient annealing are different prob-lems.

The authors are indebted to F. Abeles and J. M.Frigerio for valuable discussions.

T. Yamaguchi is on leave from the Research Insti-tute of Electronics of Shizuoka University, Japan, andK. Driss-Khodja is on leave from the U.R. PhysicsInstitute of Es-Senia University, Algeria.

References

1. F. Abeles, "Quelques proprietes optiques des milieux stratifiessymetriques," Opt Acta 4, 42-44 (1957).

2. 0. Hunderi, "Effective Medium Theory and Nonlocal Effects forSuperlattices," J. Wave-Mater. Interact. 2, 29-39 (1987).

3. F. Abeles, "Recherche sur la propagation des ondes electromag-netiques sinusoidales dans les milieux stratifies. Applicationaux couches minces." Ann. Phys. Paris 5, 637-640 (1950).

4. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri, and K. Driss-Khodja, "Effective Medium Equivalent to a Symmetric Multi-layer at Oblique Incidence," Appl. Opt. 29, 2460-2465 (1990).

5. J. M. Eastman, "Scattering by All-Dielectric Multilayer Band-pass Filters and Mirrors for Lasers," Phys. Thin Films 10, 167(1978).

6. D. E. Aspnes, J. B. Theeten, and H. Hottier, "Investigation ofEffective-Medium Models of Microscopic Surface Roughness bySpectroscopic Ellipsometry," Phys. Rev. B 20,3292-3302 (1979).

7. K. Brudzewski, "Ellipsometric Investigations of a Substrate-Thin Film System With Rough Boundaries Using the EquivalentFilm Theory," Thin Solid Films 61, 183-191 (1979).

8. J. Ebert, H. Pannhorst, H. Kuster, and H. Welling," ScatterLosses of Broadband Interface Coatings," Appl. Opt. 18,818-822(1979).

9. I. Ohlidal, "Reflectance of Multilayer Systems With RandomlyRough Boundaries," Opt. Commun. 71, 323-326 (1989).

494 APPLIED OPTICS / Vol. 30, No. 4 / 1 February 1991