Experimental study of Cu-PbC 2 , Cu-NaF, Ag-PbCI 2, andAg-NaF cermet thin films

Alain Chandonnet and Germain Boivin

We present in this paper the experimental spectral transmission curves of four different cermets, Cu-PbC12,Cu-NaF, Ag-PbCl 2 , and Ag-NaF, as well as the calculated curves obtained from the generalized Maxwell-Garnett theory. The results show that the microstructure of the metal particles in the dielectric matrix isindependent of the nature of the dielectric. We also show that this behavior is predicted by the generalizedMaxwell-Garnett theory. However, our measurements also indicate that the microstructure is the same forboth copper and silver cermets. This last experimental evidence cannot be explained within the context ofthe Maxwell-Garnett theory.

1. Introduction

The interest in new composite materials such ascermets has never stopped growing in the last fewyears. The promising optical properties they showhave been mentioned in detail in the AIP ConferenceProceedings on inhomogeneous media.1 However, thediscrepancies between experiment and theory arisingwhen trying to describe these properties show that weneed a better basic comprehension of the physics un-derlying these phenomena.

In this paper we present experimental results show-ing that the generalized Maxwell-Garnett (GMG) the-ory is able to predict the transmission spectra of Cu-PbCl2, Cu-NaF, Ag-PbCl2, and Ag-NaF cermet thinfilms in accordance with the assumption that the mi-crostructure of the particles is independent of the di-electric constant of the matrix in which they are em-bedded.

II. Theory

The complex dielectric constant Kc of a cermet re-sulting from the mixture of a dielectric and a metal insmall volume fraction q can be obtained from the gen-eralized Maxwell-Garnett formula2 3:

KC - Kd _ (KP-Kd) 1KC + (1-F)Kd LFK + (1-F)KdJ

The authors are with Laval University, Physics Department(LROL), Quebec GlK 7P4, Canada.

Received 25 March 1988.0003-6935/89/040717-05$02.00/0.© 1989 Optical Society of America.

where Kd represents the bulk dielectric constant of thedielectric matrix and Kp the dielectric constant of themetal particles. Equation (1) also assumes that me-tallic ellipsoidal particles of cylindrical symmetry areembedded in a perfectly homogeneous dielectric ma-trix. The shape factor F varies from 0 to 1; F = 0corresponds to a disk infinitely thin parallel to thesubstrate, F = 1 to a needle pointing toward the sub-strate, and F = 1/3 to a sphere. The constant Kp isrelated to the bulk dielectric constant Km of the metalby

KP = K - g(L -i) Kp where Kf = (f)K~mgL +iwR w4ig- ) (2This last expression is needed to account for the re-strictions on the mean free path of the electrons in thesmall metal particles of the cermet.4 ,5 The dampingparameter g, the mean free path L, the apparent frac-tion of free electrons f, the plasma frequency of themetal wp, and the light frequency incident on the cer-met , are all known quantities for a given metal. Thefactor R corresponding to the mean radius of the parti-cles and F, the shape factor, can be found from Eq. (1)and measurements of K, from different experimentalsamples. In a previous paper on Cu-PbI2 cermets,3 ithas been shown for the range of volume fraction q ofinterest, that is, from 2% to 12%, that F and R are givenby the empirical formulas

F = -(0.0037 + 0.0001)q + (0.433 + 0.004),

R = (0.031 + 0.007)q + (1.42 + 0.16) (nm),

(3)

(4)

where q is expressed in percent. Since these relationswere obtained using Eq. (1), the uncertainties in theproportionality constants result from the different val-

15 February 1989 / Vol. 28, No. 4/ APPLIED OPTICS 717

(2)

ues available in the literature for the bulk constant Kmof copper.6 -8

Equations (3) and (4) imply that the metal particlesdispersed in the dielectric matrix are on the averageellipsoids that point toward the substrate surface.Obviously, one should not think that the particles inthe real cermet are all exactly the same shape and size.Rather, Eq. (1) says that if all the particles were thesame in shape and size as given by Eqs. (3) and (4) (ahighly idealized cermet), one would measure the samedielectric constant Kc for the real and idealized cer-mets.

The generalized Maxwell-Garnett relation (1) mightguide us in generating new materials where propertiescan be fixed beforehand by correctly choosing the twoconstituents and the proportion in which we mix them.We must also know the size and shape of the particlesin the cermet. However, since relations (3) and (4)were obtained for copper particles embedded in PbI 2 ,nothing can be said about the shape and size of copperparticles in another dielectric matrix. The situation iseven more obscure for another metal embedded in anarbitrarily chosen dielectric. This fundamental limi-tation seems at first to forbid the precise determina-tion of optical constants of a mixture from the mereknowledge of the properties of the constituents.Hopefully, the situation is not as desperate as it looks,at least in the case of a fixed metal in different dielec-tric matrices. As a matter of fact, the answer lies inEq. (1) and in the following argument. Imagine twodifferent dielectrics having the same dielectric con-stant Kd in a small spectral domain and a different Kdoutside this wavelength domain. If we mix them withthe same metal, we see from Eq. (1) that we will obtainthe same microstructures F and R for both cermets inthis limited domain since the Maxwell-Garnett modeldoes not distinguish between the two dielectrics; if thegeneralized Maxwell-Garnett model is applicable toany cermet, the measurements of K, will be the samefor both cermets. The two factors F and R are of ageometrical nature so that they do not change withwavelength. This means that if we now make mea-surements outside the domain where the two dielec-trics have the same Kd, we should measure a differentK, but deduce the same microstructure for both cer-mets since F and R are wavelength independent. Wecan thus conclude that F and R should also be indepen-dent of Kd. In other words, Eqs. (3) and (4) must bevalid for copper particles in any dielectric. Similarequations for silver particles would also be valid what-ever the dielectric in which they are embedded. How-ever, we cannot infer from the above argument that themicrostructure of both copper and silver particlesshould be identical.

II. Experimental Method and Results

The techniques we have used for the fabrication ofthin cermet films have already been presented in detailin Ref. 9. The deposition was made in a vacuum of<10-5 Torr using calibrated film thickness monitors(Sloan MDC-9000 and Temescal FTM-3000). The

thicknesses of the cermet samples were of the order of55 nm. The purity of the constituents was certified tobe better than 99% for the PbCl2 and the NaF andbetter than 99.99% for the silver and copper. Thevolume fraction of metal in the cermets was varied bychanging the deposition rates of the metals while keep-ing the dielectric deposition rates constant. The pre-cise specifications of each cermet film are presented inTable I. The films were deposited on Corning 7059glass substrates.

The transmission measurements were made at nor-mal incidence using a HP 8450A spectrophotometer.The spectral range was from 500 to 650 nm. Thecalculated transmission curves were obtained usingthe optical constants Kc calculated from Eq. (1) foreach experimental sample assuming that Eqs. (3) and(4) are also valid for silver composites. The bulk di-electric constant for PbCl2 was deduced from precisetransmission measurements at normal incidence as-suming no absorption. The value ranges from Kd =5.1 at 500 nm to Kd = 4.9 at 650 nm. The dielectricconstant of NaF is given in Ref. 10. The physicalconstants of bulk silver and copper were taken fromRefs. 11 and 12. The bulk dielectric constant of cop-per was taken from Refs. 6-8, while Km for silver wastaken from Refs.6,7, and 13. The transmission coeffi-cients were determined using coherent light multiplereflections in the thin film and incoherent light multi-ple reflections in the substrate. This is justified by acoherence length of a few microns in the visible spec-trum for the light produced by the spectrophotometer.The basic equations needed to calculate the transmis-sion T were taken from Ref. 14 by replacing the realindex of refraction n of the film by the complex index(n - ik), where n is the index of refraction and k is theextinction coefficient. Equation (1) gives the values nand k of the film since Kc = (n - ik)2. Because theseequations are well known in thin-film physics, we referthe interested readers to Ref. 14. The results for themeasured and calculated intensity transmission of thefour composites are presented in Figs. 1-4. The vol-ume fraction in percent is indicated near each curve.

IV. Discussion

A close examination of Figs. 1-4 reveals that themeasured optical behavior of all the samples includingthe silver cermets matches the general shape of the

Table I. Experimental Parameters of the Samples

Volume fraction q ThicknessComposition (%) (nm)

Cu-PbCl 2 3.7 51.99.7 55.4

Cu-NaF 3.3 51.710.1 55.6

Ag-PbCl 2 3.8 52.07.4 54.0

Ag-NaF 3.4 43.511.2 47.3

718 APPLIED OPTICS / Vol. 28, No. 4 / 15 February 1989

Cu- PbCI,

60

50

40

30

20

8)

Ecc

490 530 570 610 650

WAVELENGTH (nm)

Fig. 1. Transmission vs wavelength for Cu-PbCl 2 cermets.

Cu- NaF

91 -

q3.3%89

87 ,

85 ,

83 / ~~~~~~~~~~~~q=10.1%83

81 /

79 --

77

75 calculatedmeasured

73 I I I490 510 530 550 570 590 610 630 650

WAVELENGTH (nm)

93

91

89

87

85

83

81

79

77

75

73490 510 530 550 570 590 610 630 650

WAVELENGTH (nm)

Fig. 4. Transmission vs wavelength for Ag-NaF cermets.

3.0

2.6 X

21LE

X_jMM

2.0

1.5

Fig. 2. Transmission vs wavelength for Cu-NaF cermets.

Ag- PbCl2

490 510 530 550 570 590 610 630 650WAVELENGTH (nm)

Fig. 3. Transmission vs wavelength for Ag-PbCl 2 cermets.

calculated curves. The discrepancies between themeasured and calculated transmission range from 3%to 5% and are all within the measurement and calcula-tion errors (Sec. II). We also note that, except for asmall portion of Fig. 3, the measured transmissions aresystematically under the calculated curves. This

z 1.00

z 08

06

04

0.2

00

Cu-PbC, cermet

.XMG (a)

~GMG -.s- - Gmfp

.

X measured

calculated

l l l l l l l l l

0 2 4 6 8 10 12

VOLUME FRACTION q (%)

Cu-PbC2 cermet

0 2 4 6 8

VOLUME FRACTION q (%)

10 12

Fig. 5. Bulk optical constants vs volume fraction q for Cu-PbC12cermets: (a) index of refraction n; (b) extinction coefficient k.

leads us to think that a small fraction of the incidentlight is lost in surface diffusion. Surface effects havealready been reported for cermet films in Ref. 9. Inthat paper, the authors propose a method of measure-ment based on transmission interferometry that en-ables the measurement of bulk optical constants at aprecise wavelength. We have used this technique forCu-PbCl2 cermets at 632.8 nm. Figure 5 presents themeasurement of the index of refraction n [Fig. 5(a)]

15 February 1989 / Vol. 28, No. 4/ APPLIED OPTICS 719

z10

zM-

q=3.7% -

__--~~~~ F ~~~~_s

- I q-9.7%

--- calculated- measured

. l l l l I l l I I I I l I l

z

Q

I-of

i:

65

60

55

9 50z°- 45

U)z 40

35

30

25

20

-I

F , q=74%- - - - calculated

- measured

q=3.4%

III

q= 11.2 %

- - - calculated- measured

I . . . I . I . � . . . . . I I .

. . . . . . . . . . . .

Ag-NaF

wWax

4.0

3.0

2.0

1.0

z0

x'Ii

0.2

0.0_

COPPER CERMET

1 2 3 4

INDEX OF THE MATRIX n,

2 3 4

INDEX OF THE MATRIX nd

Fig. 6. Optical constants of copper cermets vs dielectric constant ofthe matrix: (a) index of refraction n; (b) extinction coefficient k.

and of the extinction coefficient k [Fig. 5(b)] for sixsamples of different volume fraction and also the pre-dicted values of these constants calculated from thegeneralized Maxwell-Garnett theory. Moreover, toillustrate the effect of the microstructure on the opti-cal constants, we have calculated and represented inFigs. 5(a) and (b) the values of n and k assumingspherical particles (F = 1/3) as in the original Maxwell-Garnett theory formulation. The curve indicated byMG does not consider the limitation of the mean freepath of the electrons due to the very small size of theparticles, whereas the curve labeled MGmfp does takeit into account. The results seem to confirm our as-sumption that surface effects are responsible for thelower than prediced transmission values since bulkconstants calculated by Eq. (1) agree perfectly with themeasured ones.

The fact that the parameters of the microstructuresF and R are found to be identical for copper and silverindependently of the dielectric in which they are mixedis consistent with the results of Refs. 15 and 16. Thisleads to very interesting behavior in the optical con-stants of cermets as shown in Fig. 6. Substituting Eqs.(3) and (4) into Eq. (1) results in a relation where KCdepends only on Km, Kd, and q. We have plotted theindex of refraction n [Fig. 6(a)] and the extinctioncoefficient k [Fig. 6(b)] of copper cermets vs nd, theindex of refraction of the dielectric matrix knowingthat Kd = n, for three different volume fractions at632.8 nm. Figure 6(a) shows that for nd around 2.7, theindex of the cermet is almost the same whatever theconcentration of copper whereas Fig. 6(b) shows that,for the same value of nd, the absorption coefficientvaries strongly with volume fraction, ranging from 0 to1.4. This represents an important step toward the

fabrication of optical materials with n and k fixedindependently. Investigation of other regions of thespectrum and of other metals can probably reveal do-mains where n varies strongly with q while k remainsrelatively constant.

V. Conclusions

In this paper we have studied the transmission ofCu-PbCl2 , Cu-NaF, Ag-PbCl2, and Ag-NaF cermetsas a function of wavelength. The results agree withthe curves calculated using the generalized Maxwell-Garnett theory, assuming that the microstructures ofthe metallic particles are all the same and identical tothat of Cu-PbI2 cermets.3 This indicates that theshape factor F and the mean radius of the metallicparticles R are, for a given metal, independent of thedielectric constant of the matrix. Implied by this isthe interesting possibility of varying in certain condi-tions the extinction coefficient of the cermet withoutchanging its index of refraction. Moreover, our mea-surements seem to show that the microstructures ofsilver and copper particles are identical: this cannotbe explained by the generalized Maxwell-Garnett the-ory. Further experimental investigations are neededto conclude that the microstructure of metallic parti-cles in cermet composites is the same for all metals inall dielectrics. Finally, measurements of the bulk op-tical constants of the samples also confirm the exis-tence of surface effects and the validity of the general-ized Maxwell-Garnett theory in predicting the bulkoptical constants of cermets.

The authors wish to thank the Quebec Departmentof Education (FCAR) and NSERC of Canada for fi-nancial assistance. We are also grateful to M. D'Au-teuil, M. Lehoux, and Y. Fortier for their technicalassistance.

References1. Special issue on Electrical Transport and Optical Properties of

Inhomogeneous Media, AIP Conf. Proc. 40 (1978).2. J. C. Maxwell-Garnett, "Colours in Metal Glasses and in Metal-

lic Films," Philos. Trans. R. Soc. London 205, 237 (1906).3. J.-M. Theriault and G. Boivin, "Maxwell-Garnett Theory Ex-

tended for Cu-PbCl 2 Cermets," Appl. Opt. 23, 4494 (1984).4. U. Kreibig and C. V. Fragstein, "The Limitation of Electron

Mean Free Path in Small Silver Particles," Z. Phys. 224, 307(1969).

5. P. H. Lissberger and R. G. Nelson, "Optical Properties of ThinFilm Au-MgF 2 Cermets," Thin Solid Films 21, 159 (1974).

6. L. G. Schulz and F. R. Tangherlini, "The Optical Constants ofSilver, Gold, Copper, Aluminum. I: The Absorption Coeffi-cient k. II. The Index of Refraction- n," J. Opt. Soc. Am. 44,357, 362 (1954).

7. P. B. Johnson and R. W. Christy, "Optical Constants of theNoble Metals," Phys. Rev. B 6, 4370 (1972).

8. H.-J. Hagemann, W. Gudat, and C. Kunz, "Optical Constantsfrom the Far Infrared to the X-Ray Region: Mg, Al, Cu, Ag, Au,Bi, C, and A12 0 3," J. Opt. Soc. Am. 65, 742 (1975).

9. G. Boivin and J.-M. Th6riault, "Influence of Surface Effects inthe Determination of the Optical Constants of Cu-PbI2 CermetFilms," Appl. Opt. 23, 4245 (1984).

720 APPLIED OPTICS / Vol. 28, No. 4 / 15 February 1989

(a) 2%

r . I I~~~0

10. E. D. Palik, Ed., Handbook of Optical Constants of Solids(Academic, Orlando, FL, 1985).

11. A. V. Sokolov, Optical Properties of Metals (American Elsevier,New York, 1967).

12. C. Kittel, Introduction a l'6tude de 1'6tat solide (Dunod, Paris,1972).

13. P. Winsemius, F. F. Van Kampen, H. P. Lengkeek, and C. G. VanKent, "Structure Dependence of the Optical Properties of Cop-per, Silver, and Gold," J. Phys. F 6, 1583 (1976).

14. 0. S. Heavens, Optical Properties of Thin Solid Films (Dover,Philadelphia, PA, 1965).

15. R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, "OpticalProperties of Granular Silver and Gold Films," Phys. Rev. B 8,3689 (1973).

16. B. Abeles and J. I. Gittleman, "Composite Material Films: Op-tical Properties and Applications," Appl. Opt. 15, 2328 (1976).

Patents continued from page 644

4,768,851 6 Sept. 1988 (Cl. 350-96.15)Fiber optic modal coupler, interferometer and method of cou-pling spatial modes using same.H. J. SHAW, R. C. YOUNGQUIST, and J. L. BROOKS. Assignedto Board of Trustees of Leland Stanford Junior University. Filed 9July 1986.

Portions of a single-mode fiber are stressed to control coupling light fromone polarization to the other. In some configurations, the stress-causingfixture can act as a polarizer by coupling light to cutoff modes. G.L.M.

4,768,853 6 Sept. 1988 (Cl. 350-96.15)Optical fiber dispersion transformer.V. A. BHAGAVATULA. Assigned to Corning Glass Works. Filed8 Aug. 1986.

The dispersion transformer can be used to compensate for time spreadingelsewhere in a fiber data link. Light is split into a number of modal paths in amultimode fiber. Light from the individual paths is summed. By adjustingthe index profile of the fiber, desired dispersion can be obtained. G.L.M.

4,768,854 6 Sept. 1988 (Cl. 350-96.16)Optical fiber distribution network including nondestructivetaps and method using same.B. D. CAMPBELL and J. ZUCKER. Assigned to Raychem Corp.Filed 11 July 1985.

A familiar fiber optic tap using a fiber bend for coupling to/from radiationmodes is described. Bandwidth advantages are claimed because only highorder modes of the multimode fiber are used. G.L.M.

24

4,770,047 13 Sept. 1988 (Cl. 73-800)Optical fiber sensor.H. ARDITTY, F.-X. DESFORGES, and L. JEUNHOMME. As-signed to Photonetics. Filed 16 Apr' 1987.

Light is coupled into a bent fiber. The radiation from the bent region fallson a retroreflector. Backreflected light is coupled back into the fiber anddetected. The parameter being sensed affects the retroreflector position orfiber bend radius to modulate the reflected light amplitude. G.L.M.

4,770,505 13 Sept. 1988 (Cl. 350-377)Optical isolator.K. OKAZAKI. Assigned to Hoya Corp. Filed 26 Mar. 1985 (inJapan 27 Mar. 1984).

The performance of an optical isolator using a Faraday rotator can bedegraded by reflections from the polarizers used on either end of the rotator.This patent describes a variation on a cubic polarizing beam splitter in theform of a rhombus. This permits the input and output faces to be tilted withrespect to the input beam direction so that the reflected light is deflected outof the beam direction, while maintaining the desired 450 incident angle orangle of incidence on the beam splitter coating. W.J.T.

4,773,063 20 Sept. 1988 (Cl. 370-3)Optical wavelength division multiplexing/demultiplexing sys-tem.R. G. HUNSPERGER and A. J. MALTENFORT. Assigned toUniversity of Delaware. Filed 13 Nov. 1984.

This patent describes a planar integrated optical wavelength division multi-plexer for optical communications systems. Planar waveguide lenses, such asgeodesic lenses, are used to collimate the input signals and focus the outputsignals. Wavelength selection is provided by a thick (Bragg) grating betweenthe lenses. W.J.T.

4,773,753 27 Sept. 1988 (Cl. 356-35.5)Fiber sensor.T. HIROSE and Y. MATSUMURA. Assigned to Daiichi DenshiKogyo K. K. Filed 26 Aug. 1986.

Two polarization preserving fibers are spliced together allowing the mea-surement of temperature or strain. G.L.M.

4,774,405 27 Sept. 1988 (Cl. 250-225)Real time autocollimator device for aligning two surfaces inparallel.M. MALIN. Assigned to U.S.A. as represented by Secretary of theAir Force. Filed 26 May 1987.

This invention describes a real-time autocollimator optical system for mea-suring the relative tilt between two surfaces to adjust them to be parallel to oneanother. This is accomplished through the use of a nonpolarized laser lightsource directed to a series of beam splitters which divide and recombinedissimilarly polarized beams of light. Two quadrant detectors are used tocompare the position of the two focused beams. J.J.J.S.

4,774,406 27 Sept. 1988 (Cl. 250-227)Device for measuring an electrical field by an optical method.P.-A. CHOLLET, F. KAJZAR, and J. MESSIER. Assigned toCommisariat a l'Energie Atomique. Filed 22 May 1987.

This invention describes a device for measuring an electrical field by acompletely optical method. The apparatus comprises a probe, a high powerexciting laser, and a system of optical fibers conducting the light between thelaser and probe and from the probe to an observation chain. This inventioncould be used for the measurement of electric fields in the vicinity of coaxialhigh voltage cables. A.M.

4,776,652 11 Oct. 1988 (Cl. 350-3.72)Optical head having two holographic optical elements, withone element inclined relative to the other.C. S. IH. Assigned to Camber Corp. Filed 1 Oct. 1987.

This patent describes the use of holographic optical elements to replace the

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