Determination of the optical constants of thin absorbing films

Thin Solid Films - Elsevier Sequoia S.A., Lausanne - Printed in Switzerland

265

D E T E R M I N A T I O N OF THE OPTICAL CONSTANTS OF T H I N ABSORBING F I L M S *

S. G. TOMLIN University o f Adelaide (Australia)

(Received May 18, 1972)

After a brief reference to the determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence, a method is developed for obtaining dispersion and absorption curves when the transmittance is negligible. This requires one measurement of reflectance from the sample itself, and one from an area of the sample coated with a thin transparent film.

INTRODUCTION

Denton et al.1 have shown how to overcome the difficulties of obtaining the components n and k of the complex refractive index of a film from measurements of reflectance R and transmittance T. The relevant rigorous equations 2 have multiple solutions, the behaviour of which depends critically upon film thickness. Only with this thickness correctly adjusted is it possible to obtain a continuous single-valued dispersion curve (Fig. 1), with elimination of ambiguity

5 Q

lad

O

,x

tY tl. hi ne

3

,, 2

2.0

1.8

1.6

1.~

WAVELENGTH

1.2

1.0

0.8

0 6

IN MICRONS

Fig. 1. Dispersion curve for germanium, with multiple solutions.

* Paper presented at the International Conference on Thin Films, "Application of Thin Films ", Venice, Italy, May 15-19, 1972; Paper 12.5. Thin Solid Films, 13 (1972) 265-268

266

s.G. TOMLIN

about the choice o f solutions, provided there is no systematic error in the measured reflectances and transmittances. The criterion o f requiring an acceptable dispersion curve necessitates measurements over a wide range of wavelengths. Measurements at a single wavelength may be useless. Also it is possible to take account o f the effects of surface layers by using formulae for a double layer on a substrate 3. These methods fail when the transmittance is too small for accurate measurement. A method o f overcoming this difficulty, while retaining the advantages of normal-incidence measurements, is presented. HIGHLY ABSORBING FILMS

Films which are highly absorbing (k > 1.5 roughly) in some part o f the spectral range may transmit so little light that only the reflectance can be measured. The obvious suggestion that reflectances from the air-film and substrate-film interfaces should be used to obtain the optical constants fails because the easily obtained expression for the refractive index depends far too critically upon the measured quantities. Another possibility is to consider the reflectances from the specimen itself, and from an area o f the specimen coated with a thin transparent film. If the specimen does not transmit, its reflectance is R = (n°-n2)2 +k22

(1)

(no + n2)2 + k 2 where no is the refractive index o f air and n 2 - ik2 is the complex refractive index o f the specimen. The reflectance R1 from the transparent film of refractive index nl deposited on the specimen, which may be regarded as a substrate since it does not transmit, is given by the formulae of Tomlin 2 from which I+R 1 1 -

1

R~ - 4non2n--~l [(n2 + n2) (n12+ n2 + k2) + + (n 2 - n 2) ((n 2 - n 2 - k 2) cos 271 + 2nlk2 sin 271 }]

(2)

where 71 = 2nnldl/2; dl is the thickness o f the transparent film and 2 is the wavelength. Equation (1) can be written n

(1 + R)'~2 4n2R 2-no ~ ) +k 2 - (l-R) 2

which is a circle in the n2,

k2

2nox/R/(1 - g). F r o m eqns. (2) and (3) Thin Solid Films, 13 (1972) 265-268

(3)

plane with centre no(1 + R ) / ( 1 - R ) , 0 and radius

267

O P T I C A L CONSTANTS OF T H I N ABSORBING FILMS

k2 -

\I(I+R-R X

n 2 + noz 2 n ~ tan 7x +

2n 0

1

nl(n2_n 2) sin 27~ ~

(n2cos271 +no2sin27t ) _

n 2I+RI" ~ J n 2~

(4)

which is a straight line in the n2, k 2 plane, and the equations may be solved graphically or by computer. Figure 2 shows the solution for a substance with I/2 = k 2 = 4 . 0 , / / 1 - - - 2 . 0 and ~l = rt/4 or 3rc/4. Evidently there may be two

3 kz 2

1

1

I

I

2

3

I

I

I

/,

5

6

N2

Fig. 2. G e o m e t r i c a l s o l u t i o n o f eqns. (3) a n d (4) for n 2 = k 2 = 4.0, nz = 2.0 a n d ?z = ~/4 or 3rr/4.

possible solutions. Choice of the correct one might depend upon continuing the dispersion curve found for the less absorbing part of the wavelength range, or upon making measurements with two different thicknesses of the overlying thin film. There are two special cases to consider. When 7t =pn it follows from eqn. (2) that R1 = R. Thus the product n~dl may be found experimentally by finding the wavelength for which R 1 = R, provided it is within the range for which the specimen is opaque. Secondly, if ?~ = (p +½)n then from eqn. (4), by multiplying through by sin 271 before putting 71 = (P +½)rr, one finds n2 = (n 4 - n 4) 2no

/(n

I+R1 2 1 - R1

I+R~ n°2 ] ~ R )

The accuracy with which It2 and k 2 c a n be found depends upon the accuracy of (1 +RO/(1-RO and (1 +R)/(1-R), which will be poor when R or RI approaches unity. Errors due to inaccuracies in R and Ra are easily calculated. Thin Solid Films, 13 (1972) 265-268

268

s.G. TOMLIN

If R and R~ are about 0.5, IORI = 16Rll = 0.002, nl = 2.0 and 71 = n / 4 , the error in the position of the centre o f the circle is about 0.016 and in the radius 0.02. The error in the slope of the straight line is 0.03 so that under these conditions it appears that n 2 could be found to within 1 ~o. CONCLUSION

A method has been proposed for the measurement of the optical constants o f highly absorbing films, thus making possible the extension o f dispersion and absorption curves obtained by measurements o f reflectance and transmittance. The method is equally applicable to bulk solids. The limitation is that the reflectances must not be too large and the method is therefore unsuitable for highly reflecting metals. REFERENCES 1 R.E. Denton, R. D. Campbell and S. G. Tomlin, J. Phys. D, 5 (1972) 852-863. 2 S.G. Tomtin, J. Phys. D, 1 (1968) 1667-1671. 3 S.G. Tomlin, J. Phys. D, 5 (1972) 847-851.

Thin Solid Filrns, 13 (1972) 265-268