Limitations to the determination of the optical propertiesof a thin film by combined ellipsometric andsurface plasmon resonance measurements

Jay S. Schildkraut

We consider the use of the ellipsometric constant A and the surface plasmon excitation angle 0 of a metalsubstrate covered by a thin transparent isotropic film for the determination of the refractive index n andthickness d of the film. It is proved, using an expression for A(n,d) and 0(n,d) valid to firstorder in d, that it isin general not possible to find a unique n and d from measured values of A and 0. Exact calculation of A(n,d)and 0(n,d) for the case of a silver substrate covered by an organic thin film and 632.8-nm radiation confirms

this result and establishes that the range of d where the proof applies is below -35 nm. For a value of d of -40nm we find that the theoretical limit of uncertainty in n and d when both A and 0 are known to +0.01° is 0.01and 0.05 nm, respectively.

1. Introduction

Metals are commonly used as the substrate on whichthin organic films are formed because: (1) They pro-vide a smooth uniform surface that is often suitable fordeposition of films by the Langmuir-Blodgett tech-nique.1 (2) Chemical reaction between several metalsand organic thiols allows films to be formed by chemi-cal adsorption. 2 3 (3) Most important, use of a metalsubstrate allows the film to be characterized both elec-trically4 and optically by a number of techniques suchas grazing incidence FTIR3 5 and ellipsometry.6 In thecase of a silver or gold substrate surface-enhancedRaman7 and surface plasmon spectroscopy8 are alsopossible. This report is concerned which the com-bined analysis of data obtained by ellipsometry andsurface plasmon spectroscopy.

An ellipsometer determines the relative amplitudeattenuation and phase shift of p- and s-polarized lighton reflection from a surface.6 In the case of a transpar-ent isotropic film on a reflecting substrate with known

The author is with Eastman Kodak Company, Corporate Re-search Laboratories, Rochester, New York 14650.

Received 13 November 1987.0003-6935/88/163329-05$02.00/0.© 1988 Optical Society of America.

refractive index, the two ellipsometric constants A andA, which are related to the relative phase shift andamplitude attenuation, respectively, can both be ex-pressed 'in terms of the refractive index n and thicknessd of the film. When A and ' are measured, this systemof two nonlinear equations can usually be solved toobtain n and d.

A problem arises when the film is extremely thin andthe substrate has a low imaginary component of itsdielectric constant.9 In this case the measured valueof I is essentially the same as for a bare substrate, andthe only useful equation is the one for A(n,d). Thismakes it impossible to determine both n and d. Acommon practice is to assume a value of n and then touse the equation for A to calculated d.

A possible solution to this problem is to make anoth-er measurement which can provide a second equationwith only n and d as unknowns. The measurement ofthe wavenumber of a surface plasmon (SP) excita-tion,1 0 which can be expressed in terms of an excitationangle 0, at a metal surface covered by the thin film mayprovide this second equation. This would be an idealsituation because both SP and ellipsometric experi-ments are sensitive to the presence of <2 nm of anorganic film on the surface of the metal. 8 9

In this paper, we consider the determination of n andd from measured values of A and 0. We prove thatwhen d is small enough so that both A and 0 are a linearfunction of d, it is not possible to determine a unique nand d value. We then use general equations for A(n,d)and 0(n,d) to determine at what d a unique n and d canbe found. In the Appendix we derive an equation thatallows 0 to be precisely calculated for a given n and d.

15 August 1988 / Vol. 27, No. 16 / APPLIED OPTICS 3329

II. Linear Approximation

A configuration in which both reflection ellipso-metry and SP spectroscopy can be performed is shownin Fig. 1. A metal film with refractive index n1 andthickness d, is in contact with the base of a prism withrefractive index no. The metal film is covered by athin film of refractive index n2 and thickness d2. Therefractive index of the ambient is n3. A light beamfrom an ellipsometer is incident on the films from theambient medium at an angle . A p-polarized lightbeam from a SP spectrometer is incident from theprism at the SP excitation angle 0. Both light beamshave the same wavelength X.

When d2 is small compared to X, it has been shownthat both A and Ovary linearly with d2.9"' In this limitwe can write

A = A(n2 )d2 + A, (1)

0 = B(n2 )d2 + 0, (2)

where Al and 00 are the values of A and 0, respectively,in the absence of the thin film. The functions A(n2)and B(n2) are implicit functions of no, nj, and n3 but donot depend on d2.

We now assume that two sets of n2 and d2 values,which we label n, d a and n, d, exist which bothcorrespond to the same A and 0 value. This assump-tion along with Eqs. (1) and (2) leads to the equation

A(nb) B(nb)A n2 2(3)

A(n') B(n')

If nb is close to n we can approximate A(n b) and B(n )by

A(nb) = A(n') + A'(na)(nb -na),

B(nb) = B(na) + B'(na)(n -na),

(4)

(5)

SURFACE PLASMON

y

X

ELUPSOMETERFig. 1. Configuration in which both reflection ellipsometry and SPspectroscopy can be performed. A metal film with refractive indexn, and thickness d is in contact with the base of a prism withrefractive index no. The metal film is covered by a thin film ofrefractive index n2 and thickness d2. The refractive index of theambient is n3 . A light beam from an ellipsometer is incident at anangle I. Alightbeam from aSP spectrometer is incident on the base

of the prism at the SP excitation angle 0.

interface.] The dielectric constant of the metal film Elis in general a complex number, but in the case of ametal that is suitable for SP excitation the imaginarypart is negligible compared with the real part. Withthis condition, the right-hand side of Eq. (7) is animaginary number. Therefore, (tanI) is zero to firstorder in d2, and we obtain the following expression forA(n2):

4rn3 cos e(e 2 - e3)(el - 2 )A (n2 ) = -X e2(el - e3)(el cot 2k - 3) (

When we take the derivative of Eq. (8) with respect ton2 and combine terms, we obtain

A'(n2) = S Xn2 n3 cos-2e)el 3(e 1 cot2

-

e2(el-e)e cot2p-e3 (9)

where the prime indicates the derivative with respectto n2 . Equations (3), (4), and (5) can be combined toobtain the following equation:

The shift in the wave vector of a SP excitation due tothe presence of a thin nonabsorbing film on the surfaceof the metal is given to first order in d2 byll

A'(n') B'(na)

A(n'2) B(na)(6)

Equation (6) is a condition which indicates that aunique n2 and d2 cannot be determined from A and 0values when the linear expansion of A and 0 in terms ofd2, as given by Eqs. (1) and (2), are valid.

We now consider expressions for A(n2 ) and B(n2 )that are available in the literature. The ellipsometricconstants A and T to first order in d2/X can be writtenas9

OtanNF)+ 4irin3 d2cosOtan'i' X

fl(E2 3)(el -e)-2( e 3)(el cot2o - 3)

where ei = n? is the dielectric constant. [Note that Eq.(7) does not take into account reflections at the metalfilm-prism interface. These reflections are negligiblecompared with the reflection at the ambient-metal

6k = w 27rd2 (e2 - 3)(eCe3)2 (E2 - El)c A e2(e1 + e) 2(e3 -e)(-el63) 12

(10)

The shift in wave vector described by Eq. (10) resultsin a shift in SP excitation angle. This shift is given byEq. (2) with the function B(n2) equal to' 2

B(n2) c r kw no cos00

(11)

Using Eqs. (10) and (11) we find that the derivative ofB(n2) with respect to n2 is

B'(n2) = 47rn2 (ele3)2(e2 - )Xno cosoo e2 (el + e3)2(E3 -El)(-elf3)'12

X 1 - (e2 -e 3) [1/e2 - 1/(e2 - e)]-

Using Eqs. (8)-(12) we obtain the ratios

A'(n2 ) B'(n2 ) 2 (e3 el - )A(n2 ) B(n2 ) n2 ( 2 - 3)(e1 -E2)

(12)

(13)

Equation (13) shows that the condition given by Eq.

3330 APPLIED OPTICS / Vol. 27, No. 16 / 15 August 1988

(8)

\ *,

(6) for the nonexistence of a unique n2 and d2 formeasured A and 0 values holds. The above result mayalternatively be stated as follows: When d2 is smallenough so that both A and 0 are linear functions of d2

and if n' and d' correspond to a value of A and 0, for anyn2 that is close to n' there exists a d2 which will lead tothe same A and 0.

Ill. Exact Equations

The proof in Sec. II assumed that d2 is small enoughso that both A(n2,d2) and 0(n2,d2) are linear functionsof d2. In this section we use exact equations forA(n2 ,d2 ) and 0(n2,d2) to establish, for a specific case,the range of d2 where this proof applies.

The ellipsometric constants A and T are defined bythe equation6

(14)P = (tanI) exp[(iA),PS

where pp and p, are the reflectivity coefficient for p-and s-polarized light, respectively. An expression forthe reflectance of a four-layer system can be found inthe literature and will not be given here.13

It is not possible to express 0 explicitly in terms of n2

and d2. However, a nonlinear equation can be derivedwhich can be numerically solved to obtain 0 for givenvalues of n2 and d2. This equation has been given byPockrand.14 In the Appendix we outline the deriva-tion of this equation in more detail and state the equa-tion in matrix form.

The problem of determining values of n2 and d2 fromvalues of A and 0 can be clearly illustrated by plottingcurves of A and 0 for different values of n2 over a rangeof d2. We specialize to the case of 632.8-nm radiation,a fused silica prism (no = 1.457), a silver film (n, =0.115 + 4.18i, d, = 60 nm), air as the ambient medium(n3 = 1.00), 0 = 700, and an organic thin film (n2 =

-1.5). Curves for n2 equal to 1.5 and 1.6 are shown inFig. 2. (A is a negative angle because we use theconvention n = nr + nii, where ni > 0, instead of theconvention commonly used in ellipsometry, n = nr -nii.) We see that for d2 of less than -35 nm bothcurves are identical, which implies that for either valueof n2 a value of d2 exists that will lead to a A and 0 thatlie on the curve. This demonstrates, using exact equa-tions, the result which was proved using linear equa-tions in Sec. II.

For the purpose of comparison, Fig. 3 shows curvesof A and T for the same values of n2 and d2 as in Fig. 2.The relative insensitivity of T compared to 0 to thepresence of a thin film on the metal is evident.

An important question is: At what value of d2 can aunique n2 and d2 be determined from a A and 0 value?This question can only be answered in the context ofthe acceptable uncertainty of n2 and d2 and the accura-cy with which A and 0 can be measured. Figure 4shows curves of A and 0 over a range of d2 close to 40 nmand n2 equal to 1.53, 1,54, and 1.55. Centered at apoint on the curve corresponding to n2 = 1.54 is a boxthat represents an uncertainty in both A and 0 of+0.01°. We see that this uncertainty sets a theoretical

U)-(-

LUiLUiQufC9

-T

ILU

co

00D

I .

40.(

Fig. 2. Curves1.5, 0; 1.6, 0.

U)-

0-coW 9

a-.

1

0

Fig. 3.1.5, 0;

0 45.0 50.0 55.0 60.0THETA (DEGREES)

of A and 0 calculated using exact equations. n2 =

Next to each symbol is written the value of d2 innanometers.

35.0 40.0 45.0 50.0 55.0PSI (DEGREES)

Curves of A and 'IT calculated using exact equations. n2 =

1.6, 0. Next to each symbol is written the value of d 2 innanometers.

limit on how precisely n2 and d2 can be determined of+0.01 and ±0.05 nm, respectively.

IV. Conclusion

Both the ellipsometric constant A and the SP excita-tion angle 0 are extremely sensitive to the presence of athin film on the surface of a metal substrate. We haveproved that, in general, it is not possible to use mea-sured values of A and 0 to determine n and d of the thinfilm for a range of d close to zero. For a specific case,we have determined how large d must be for a unique nand d to be determined.

Despite the above limitation, the measurement of Aand 0 is seen to be superior to the measurement of Aand T for the determination of n and d of a thin film ona metal substrate with a small imaginary component ofits dielectric constant.

15 August 1988 / Vol. 27, No. 16 / APPLIED OPTICS 3331

60

0

1213

2425

36

30 4850 60

To f

ok

9o

9o

00N.

L X

LU L

cflo

C n

N

52.300 52.425 52.550 52.675 52.800THETA ( DEGREES )

Fig. 4. Curves of A and calculated using exact equations. n2 =1.53, d2 = 40.68-41.68 nm, ; n2 1.54, d2 = 40.00-41.00 nm, ; n2 =1.55, d2 = 39.35-40.35 nm, A; indicates an uncertainty in /v and of

0.01°.

associated magnetic fields are found using V X E =i(co/c)B.

Maxwell's equations require that the in-plane com-ponent of the electric and magnetic fields be continu-ous across all interfaces.' 5 Since there are three inter-faces, this requirement results in six equationsinvolving six independent electric field amplitudes.This set of six linear equations with six unknowns canbe written in matrix form as follows:110100

0 01

0where

r, 0 0 0

1 1 1 0

0 r 2 1 1

-ulr, 0 0 0

-01 02 -02 00 92r 2 -02 -03

TOx7

Eo+7

Ej7xE+c2x

E2-

E3x_

00

00

0

0

F = exp(i2ki5 di),

Q = eilkiz,

= [(W/) 21 2-11/2

We acknowledge the help of Anna Cha-Lin withFORTRAN numerical methods subroutines.

Appendix

The configuration for the excitation of a SP wave isshown in Fig. 1. A p-polarized light beam is incidenton the base of the prism. When the in-plane wavevector of light in the prism equals the wave vector ofthe SP wave, the wave will propagate along the silver-thin film interface. The angle 0 at which this occurscan be determined as follows.

There are six electric fields that need to be consid-ered in the case of a SP wave propagating in the four-layer structure. The first is a field in the prism propa-gating away from the films. This field has the form

E = (E0..t + E0.2) exp[i(kxx + ko0 z - wt)], (Al)

(A4)

(A5)

(A6)

(A7)

and ei in Eqs. (A6) and (A7) is the dieletric constant ofmedium i.

In order for Eq. (A4) to have a nontrivial solution (allfields equal to zero), the determinant of the coefficientmatrix must be equal to zero. If we represent this sixby six matrix by the symbol A, this condition can bewritten as

DetA = 0. (AS)

Since A is a complex matrix, and consequently DetA isa complex number, Eq. (A8) expresses two indepen-dent conditions. When the dielectric constants of allfour media and the thickness of the two films are given,Eq. (A8) can be numerically solved to obtain both thereal and imaginary parts of k. The surface plasmonexcitation angle is related to the real part of k, k by

where I and z are unit vectors in the x and z directions,respectively, ko, is the z component of the wave vectorin the prism, and w is the angular frequency; kx, the xcomponent of the wave vector, is the same in all mediabecause of Snell's law. Note that a field propagatingin the prism toward the films, while necessary to excitethe SP wave, is not part of the SP wave itself. Twoelectric fields are present in the silver film because ofreflections. These two fields, E and E, have theform

= (E;1 + EV2) exp[i(kxx kz - t)]. (A2)

The two electric fields in the thin film, E+ and E-, aregiven by an equation analogous to Eq. (A2). The sixthelectric field E3, which propagates in the ambient me-dium away from the films, is given by

E3 = (E3U + E 3, 2 ) exp[i(kxx - k 3 z - ot)]. (A3)

The x and z components of each field are related by therequirement D = 0 in all four media. The six

0=sin- I X .(w/c)no (A9)

References1. I. R. Peterson, G. Veale, and C. M. Montgomery, "The Prepara-

tion of Oleophilic Surfaces for Langmuir-Blodgett Deposition,"J. Colloid Interface Sci. 109, 527 (1986).

2. R. G. Nuzzo, F. A. Fusco, and D. L. Allara, "SpontaneouslyOrganized Molecular Assemblies. 3. Preparation and Proper-ties of Solution Adsorbed Monolayers of Organic Disulfides onGold Surfaces," J. Am. Chem. Soc. 109, 2358 (1987).

3. A. Ulman, N. Tillman, and J. Littman, "Self-assembling Mono-layers of Thiols on Silver and Gold Surfaces: an FTIR Study ofMolecular Orientation," submitted to J. Phys. Chem.

4. P. S. Vincett and G. G. Roberts, "Electrical and PhotoelectricalTransport Properties of Langmuir-Blodgett Films and a Discus-sion of Possible Device Applications," Thin Solid Films 68, 135(1980).

5. J. F. Rabolt, M. Jurich, and J. D. Swalen, "Infrared Reflection-Absorption Studies of Thin Films at Grazing Incidence," Appl.Spectrosc. 39, 269 (1985).

3332 APPLIED OPTICS / Vol. 27, No. 16 / 15 August 1988

6. F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L.

Steinberg, "Measurement of the Thickness and Refractive In-

dex of Very Thin Films and the Optical Properties of Surfaces

by Ellipsometry," J. Res. Natl. Bur. Stand. Sect. A 67, 363

(1963).7. R. M. Corn and M. R. Philpott, "Surface Plasmon-Enhanced

Raman Scattering at Thin Silver Films," J. Chem. Phys. 80,5245(1984).

8. I. Pockrand, J. D. Swalin, J. G. Gordon II, and M. R. Philpott,

"Surface Plasmon Spectroscopy of Organic Monolayer Assem-blies," Surf. Sci. 74, 237 (1977).

9. H. Arwin and D. E. Aspnes, "Determination of Optical Proper-

ties of Thin Organic Films by Spectroellipsometry," Thin Solid

Films 138, 195 (1986).

10. E. Kretschmann and H. Raether, "Radiative Decay of Nonradi-

ative Surface Plasmons Excited by Light," Z. Naturforsch. TeilA 23, 2135 (1968).

11. E. Kretschmann, "The Determination of the Optical Constantsof Metals by Excitation of Surface Plasmons," Z. Phys. 241, 313(1971).

12. I. Pockrand and J. D. Swalin, "Anomalous Dispersion of Surface

Plasma Oscillations," J. Opt. Soc. Am. 68, 1147 (1978).13. 0. S. Heavens, Optical Properties of Thin Solid Films (Butter-

worth, London, 1955), p. 62.14. I. Pockrand, "Surface Plasma Oscillations at Silver Surfaces

with Thin Transparent and Absorbing Coatings," Surf. Sci. 72,577-588 (1978).

15. M. Born and E. Wolf, Principles of Optics (Pergamon, New

York, 1964), p. 6.

Optl Activities (in nd stry/

orted by FRANK COOKE, 59 Summer Street, North Brookfield, Mass. 01535. Mr. Cooke

news and comments for this column which should be sent to him at the above address X

Happy birthday, dear klystron

Donald Smith

Little attention was Daid last year to the fiftieth anniversary of the invention of the klystron, a device thatmade electronic signals oscillate. 'I

Geographic News Service.

Maybe it's because they are so inconspicuous. Peoplecannot see and touch them the way they can most otherbreakthroughs, such as light bulbs and telephones. Forwhatever reason, 1987 is about to slip into history with hard-ly any public notice of the fiftieth anniversary of one of themost important inventions of the electronic era: the kly-stron.

Klystrons are vacuum tubes that transmit and amplifymicrowave signals. They range in size from that of a match-book to 6 feet tall. "It's not really very sexy," said a spokes-man for Stanford University, where the tube was born in aphysics laboratory. "It's sort of like a piece of plumbing."Yet the tube is a basic element of equipment whose purposesrange from TV broadcasting to use in linear atom smashers.Its usefulness in radar systems is credited with helping winthe Battle of Britain in World War II. Today it helps protectU.S. Navy ships patrolling the Persian Gulf.

The klystron's inventors were two brothers, Sigurd andRussell Varian, sons of an itinerant Irish poet-musician whoimmigrated to the United States with his wife in 1894. Rus-sell, the thinker of the duo, by all accounts was a good-hearted, ponderous bear of a man who stood 6-feet 4-inchestall. His eventually diagnosed dyslexia led some of histeachers to believe that he was slow-witted. Despite thishandicap, he earned a master's degree in physics from Stan-ford. The doer was ambitious, driven Sigurd, a dropoutfrom California Polytechnic School in San Luis Obispo, wholater suffered from mental illness. As an adventurousyoung Pan American World Airways pilot, Sigurd was acommercial aviation pioneer in Central America in the early

'his brief historical report is based on a 1987 release from the National

1930s. Sigurd was a cauldron of creative energy, whilingaway long flights dreaming up ideas for things that had neverbeen done before. Later, the brothers would talk and Rus-sell would work on the scientific problems posed by Sigurd'sepiphanies. If all went well, the dexterous Sigurd eventuallywould build working models.

"One out of every forty ideas they had actually pannedout," the Stanford official said. "But those ideas were prettygood." One of their preklystron brainstorms had a notablywild ring. For a while, the brothers worked on developing aninternal combustion engine that required a charge of dyna-mite to get it started. Another project that Sigurd champi-oned was using stencils and metal sprayed from a gun tomake circuits for radio sets. But Russell dismissed thisconcept, obviously a forerunner of printed circuits, as a "bumidea."

The story of the klystron, however, is a technological fairytale. Everything worked out just right. Sigurd's sometimesharrowing aviation experiences had instilled in him an abid-ing interest in improving airplane radar and navigation sys-tems. He worried about the security of the Panama Canal,which he had often overflown, citing its vulnerability asevidence of the need for a better system of spotting enemyairplanes. In 1936, Germany's increasingly accurate bomb-ing of Spanish cities further illustrated how defenseless wereurban populations against air attacks. Sigurd badgeredRussell to help him find a solution. The answer that Russellcame up with involved making electronic signals oscillate.

continued on page 3396

15 August 1988 / Vol. 27, No. 16 / APPLIED OPTICS 3333

6

repc

welcomes letters,