The absolute determination of complex reflectivity

The absolute determination of complex reflectivity

THE ABSOLUTE DETERMINATION REFLECTIVITY OF COMPLEX J. R. BIRCH Division of Electrical Science, National Physical Laboratory. Middlesex TWI 1 OLW...

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THE ABSOLUTE

DETERMINATION REFLECTIVITY

OF COMPLEX

J. R. BIRCH Division

of Electrical

Science, National Physical Laboratory. Middlesex TWI 1 OLW. England

Teddington,

Abstract-An absolute method for the determination of complex reflectivity has been applied to a specimen of high purity silicon between 5 and I20 cm- ’ at 290 K. establishing its amplitude reflectivity with a precision of 10m4 and its phase with a precision of 10m4rad. Using the specimen in a complex reflectivity comparison enabled the complex reflectivity of some thin aluminium films on glass substrates to be precisely measured.

INTRODUCTION

The measurement of the specular power reflectivity of a specimen by broad band techniques at millimetre and submillimetre wavelengths is, conventionally, a relative measurement. The accuracy that can be achieved ultimately rests upon assumptions concerning the reflective properties of a reference surface. In the most usually applied form of the technique the power reflected from a specimen is compared to that reflected from a metal surface which is assumed to have a complex amplitude reflectivity i=

l.Oexpirr

(1)

i.e. is taken to be perfectly reflecting. Thus, the ratio of the two detected powers gives the relative power reflectivity of the specimen. Typically, the metal reflector is an aluminium, silver or gold thin film vacuum evaporated onto a glass substrate. Unless the surface is fresh and unscratched, and sufficiently thick to avoid size and skin depth effects, its reflectivity may depart significantly from unity and, for precise measurements, the above-mentioned procedure becomes invalid. Such problems do not occur in the near-infrared, visible and ultraviolet regions of the spectrum, where techniques that do not rely upon the above assumption exist. (See, for example Refs l-3). These depend upon sampling both the incident and the reflected beams so that the reflectivity can be calculated independently of a reference surface. The sampling is usually accomplished by switching the position of the detector or by beam deflection techniques, with care taken to maintain the equivalence of the switched optical paths and beam geometries. Such methods are not ideally suited to millimetre and submillimetre wavelengths where beams are usually less well-defined and have larger cross sections than at shorter wavelengths. In this work, therefore, a technique is described that has been developed for the determination of reflectivity in which the reflectivity of a suitable specimen is related to the optical constants of vacuum to provide an absolutely determined standard of complex reflectivity against which unknown specimens may be compared. The method is based upon the fact that the specular reflectivity of a solid specimen can be calculated more accurately than it can be measured if its optical constants are known with sufficient precision. The technique relies upon the use of dispersive Fourier transform spectrometry (DFTS) and simply consists of measuring the complex refractive index of a plane-parallel. transparent but fairly reflecting, solid specimen in a transmission DFTS experiment. This can be done with high precision and the complex reflectivity of a plane surface of the solid may then be calculated from the measured complex refractive index using Fresnel’s equations. As the refractive index measurement only required knowledge of the defined refractive index of vacuum fi = 1.0 + i 0.0 613

(2)

J. R. BIRCH

614

and the length standard, this can be taken as providing an absolute determination of complex reflectivity at submillimetre and millimetre wavelengths. Having established this calculated reference material for complex reflectivity it is now possible to perform comparison or calibration experiments in which the complex reflectivity of an unknown specimen is compared to that of the reference in a reflection DFTS meastlrement. The use of a dispersive measurement at this stage is essential as its phase sensitivity ensures that the complex reflectivity scale that is transferred is that of the surface of the reference specimen, and not that of it as a whole. as would result from a comparison using power reflection techniques. Throughout this work the reflectivity spectra that are presented are amplitude spectra as this is the measured quantity in a reflection DFTS experiment. The power reflectivity is the square of this. THE

REFLECTIVITY

STANDARD

The material chosen for these initial investigations of reflectivity standards must satisfy several requirements. It should be sufficiently transparent in the spectral region of interest that its complex refractive index can be measured in a transmission experiment with high accuracy, but it must also be su~cientIy reflecting for its calculated re~ectivity to be accurately transferable to an unknown specimen in a reflection comparison. Additionally, it must be capable of being optically worked to a plane parallel form with its thickness unambiguously determined to better than I part in 103, and must be optically and mechanically stable. These requirements are easily met by pure, elemental semiconductors, and silicon was adopted for this work. Its optical constants have been studied by several workers at millimetre and submillimetre wavelengths(4WhJ and it was chosen in preference to germanium as the presence of phonon bands in germanium in the optical constants, which at wavenumbers as low as 100 cm-’ leads to structure is best avoided for the present application. The corresponding bands in silicon occur not a problem. The particular specimen of silicon above 600 cm- ’ and are consequently used was cut from an undoped single crystal and had a nominal resistivity of lOohm* metre.* It was disc shaped, approximately 40 mm diameter and, after optical polishing. its thickness was determined to be 2.393 mm by averaging micrometer readings taken at different points over its surface. The complex refractive index of this specimen was measured in a transmission dispersive Fourier transform experiment using an interferometer based upon the NPL modular of the modulus and phase spectra computed from interferocube design. (‘) Comparison grams recorded with and without the specimen in the fixed mirror arm of the interferometer gave the attenuation and phase shift caused by the specimen, and the complex refractive index was computed from these using a rapidly converging iterative technique to allow for interface effects. The measurements were performed in two parts. firstly using a liquid helium cooled Rollin detector to cover the spectral region between 5 Golay cell for the region from 30 and 30 cm- ‘, and then with a quartz-windowed to 120cm-‘. The real refractive index, n, and the power absorption coefftcient. 2, of silicon determined in this way are shown in Fig. i for a spectral resolution of 2 cm- ’ and a temperature of 290 K. The spectral variation of both the power absorption coefficient and the refractive index are typical of those expected from free charge carrier absorption, although calculations based on the simple classical model of this effect predict an absorption coefficient which is about an order of magnitude lower than the measured one. This indicates that the quoted resistivity may be too high, but such a discrepancy is not relevant to the present application. The bump on the refractive index spectrum near to 28 cm ’ is not a real feature. It occurs at the high wavenumber end of the measurements made

* The specimen was kindly supptied by Drs W. F. Passchier and D. D. Honijk of fhe Corlaeus Laboratoria. Rijksuniversiteit, Leidcn. and was nominally identical to the specimens studied in their dispersive measurcmerits.‘‘‘’

The absolute

f:

determination

of complex

615

reflectivity

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7 .I 4

3.410

-

3.408

-

E 2

I

I

IO0

I

20.0

I

300

I

400

I

500

I

600

Wavenumber

I

I

70.0

80.0

I

900

I

I

100

I IO

1

( cm-’ I

Fig. 1. The measured values of the real refractive index and power absorption coefficient of the silicon specimen from 5 to 120cm-’ at a spectral resolution of 2 cm-’ and a temperature of 290 K.

the Rollin detector and is due to systematic effects associated with the cut-off of the beamdivider used in those measurements. The level of random noise in these measurements was estimated from the standard deviation of sequential determinations, and for the refractive index this was typically between 0.0001 and 0.0002, while for the absorption spectrum it was about 0.05 Np. cm-‘. At these levels of precision in the refractive index determination it is important to consider possible systematic errors and the most likely cause of systematic error for the refractive index is an incorrect determination of the specimen thickness. By taking the average of many sequential measurements across the specimen surface we believe that the effective thickness for this experiment has been determined to within 0.001 mm. This corresponds to a possible systematic error in the real refractive index of up to 0.001 for this specimen. If this material is to be a suitable reflectivity standard the temperature variation of its reflectivity should, ideally, be very low, or at least known. This will be largely determined by that of the refractive index and so the refractive index was additionally measured over the restricted spectral range between 10 and 40cm-’ at temperatures of 291 K, 292.5 K and 297 K. It was found to increase significantly with increasing temperature in the manner summarised by the three points of Fig. 2 showing the variation at 25 cm-‘, and which was typical of all the measurements. A least-squares straight line fit to this admittedly sparce data yields a temperature coefficient for n of (2.0 + 0.2) x 1O-4 K-l. and the straight line in Fig. 2 represents this best fit to the data. Cardona et u/.(8,9) have measured the temperature dependence of the refractive index for silicon at 3 pm wavelength, but their specimens were of a sufficiently high resistivity that free charge carriers would not have contributed significantly to their result, which cannot, therefore, be compared with the present one. The previous submillimetre wavelength refractive index measurements on this materia1’4-6’ give values that are significantly higher than those found in the present study, with

616

J. R.

18

BIRCH

20

22

Temperature Fig. 2. The temperature

variation

of the real refractive

24 (OC)

index of the silicon specimen

at 25 cm



but only in the paper of Loewenstein et al. CT’ - is the measurement temperature given precisely. Thus, their result at 40 cm - I and 300 K, 3.416 + 0.002, becomes 3.414 f 0.002 when normalised to 290 K by our observed temperature variation, which agrees with our measured value of 3.4125 k 0.0002 within experimental error. Using these measured values of the complex refractive index spectrum, the complex Fresnel reflectivity, ? = I’exp i c$,, for normal incidence from a vacuum was calculated from the usual expressions and the results presented in Fig. 3. Over the spectral range

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I

200

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300

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400

500

600

Wavenumber Fig. 3. The calculated

I

70.0

I

800

I

900

I

I

100

110

I

(cm?1

amplitude and phase of the complex reflectivity of the sihcon for normal incidence from a vacuum at 290 K.

speclmcn

The absolute

Table

determination

of

complex reflectivity

617

1. Estimated errors, Ar and A&,. in the values of the amplitude and phase of the complex of the silicon specimen calculated from its measured complex refractive index

reflectivity

Source

10-S - 2 x lo-5

n, lo-4 - 2 X 10-d z, 0.05 Np. cm _ ’

-

7.4 X lo-s’“’

lO-4 10-s - lo-4

thickness. 10e3 mm temperature change of I-IOK (a) Calculated

A+, @ad)

Ar

Systematic

Random

for i = IOcm~‘.

by these measurements the amplitude reflectivity, r, changes by about 1 part in lo3 as it increases from 0.5462 at 5 cm- * to a virtually constant value just below 0.5469 at 120 cm- ‘. The imaginary part of the complex refractive index was sufficiently small (~0.023) that the amplitude reflectivity was primarily determined by the refractive index and closely followed its spectral variation. The phase, $,, on the other hand is primarily determined by the absorption index and, therefore, for the low values that this takes in this material the phase is very close to rc rad, increasing above this by index is greatest. only 4 mrad close to 5 cm - ‘, where the absorption The curves of Fig. 3 represent the calculated values of the complex reflectivity of an interface between vacuum and the silicon specimen, for normal incidence from the vacuum. As these spectra are going to form the basis of an absolute complex reflectivity it is appropriate to estimate the level of random and systematic error present in them arising from errors in the measurement of the complex refractive index. This is summarised in Table 1. The major uncertainty arises from the possible systematic error of 10m3 mm in the thickness measurement which, through the real refractive index, leads to a possible systematic error in the amplitude reflectivity of 10m4. All other errors in the reflectivity are nearly an order of magnitude below this. The major error in the calculated phase arises from the random error in the determination of the power absorption coefficient, x, and leads to a phase uncertainty of 7.4 x 10e5 rad, which is insignificant compared to the random noise generated in the dispersive measurement of C#J,.“O’

covered

APPLICATION

TO

THIN

METAL

FILMS

The specimen whose reflectivity is to be determined may not always be sufficiently transparent for its complex refractive index to be measured in transmission. Under such circumstances its complex reflectivity must be compared to that of a transparent specimen that has been previously calculated by the above procedure. In order to illustrate the precision with which the calculated complex reflectivity scale of the front surface of the silicon specimen may be transferred to specimens of unknown reflectivity, measurements were made of the complex reflectivity of four thin aluminium films vacuum evaporated onto glass substrates. The film thicknesses were 0.017, 0.040, 0.095 and 0.20 pm and each was deposited onto a 50 mm diameter by 5 mm thick disc of soda-lime-silica glass. The glass blanks had been optically polished to be flat to better than 0.2 pm on their aluminised surfaces. The complex reflectivity of each was measured using the silicon specimen as the reference reflector in an interferometer designed for normal incidence dispersive reflection measurements. (lo) It is important that this comparison of the known and unknown reflectivities be performed by dispersive Fourier transform techniques as only with the phase sensitivity of these techniques can one be sure of transferring the front surface reflectivity of the silicon specimen. Normal power reflection techniques would transfer the reflectivity of the specimen as a whole, which is not the same as that of the front surface. The measurements were made between 10 and 40 cm-’ at a spectral resolution of 2 cm-’ and 290 K using a liquid helium cooled Rollin detector.

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BIRCH

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15

I 20

Wavenumber

I 25

/ 30

I 35

I

(cm-‘)

Fig. 4. The measured amplitude reflectivity of four aluminium films vacuum evaporated onto glass substrates between IO and 40cm. ’ at 290 K. The thickness of the film appropriate to each measurement is indicated. The dashed line shows unity reflectivity.

The results of these measurements are presented in Fig. 4, which shows the amplitude of the complex Fresnel reflectivity, ? = r exp i 4,. for each film-substrate structure. The skin depth at 10 cm-’ for bulk aluminium of resistivity 2.76 x lo-* 0hrn.m is about 0.15 pm so that the observed decrease in reflectivity with decreasing wavenumber for the thinner films is not surprising. Each curve is the average of four independent determinations and over most of the spectral range these independent determinations were reproducible to within _f-0.001. Below 14cm- ’ the reproducibility was somewhat worse than this due to the effects of the beamdivider cut-off, as shown by the measured reflectivity of the 0.20pm film rising above unity. The small periodic structure on all the curves with a period of between 3 and 4 cm _ ’ is not real, but arises from imperfect ratioing out of a channel spectrum present in the background spectrum of the interferometer. The amplitude of this is about 0.001. Thus, the amplitude reflectivity scale of the silicon specimen front surface. which had been calculated with a precision of about 10e4, has been transferred to the thin films with a precision of about 10m3 that was determined by the random and systematic errors in the reflectivity comparison. In a similar manner the phase of the complex reflectivity for all three films was determined from these comparisons to be ILrad within a random scatter of 10 mrad arising from the phase comparison. It is of interest to compare these measurements with calculations based upon simple theory. Carli ‘I” has developed expressions for the reflectivity of a thin metal film using the bulk conductivity model of the complex refractive index in the limit that the film thickness is very much less than the skin depth. As the thinnest of our films is only approaching this condition we have made exact calculations for both the bulk conductivity assumption and a simple size effect model. in both cases treating the reflecting structure as possessing three interfaces, vacuum-metal, metal-glass and glass-vacuum. The infinite summation of the internally reflected rays was dealt with in a manner similar to that described by Rouard. (l’) but using complex arithmetical procedures written into an Algol 60 program to extend the total complex reflectivity at each interface. The values for the optical constants of glass used in these calculations were taken are summarised in from published measurements. (13) The results of these calculations Fig. 5 for the 0.017 and the 0.095 pm films. The crosses are the experimentally determined points replotted from Fig. 4. The lines marked (a) are derived from the bulk conductivity model using the previously mentioned value of 2.76 x 10m8 0hm.m for the resistivity. The measured values are systematically lower than the calculated ones with the discrepancy increasing as the films become thinner, and being particularly pronounced for the thinnest film. The calculated spectra show periodic structure due to multiple beam interference within the layered films, but this occurs below the level of 1 part in lo5 for the amplitude reflectivity and is therefore not apparent on the scale of Fig. 5. In order to improve upon these calculations we have applied Fuchs’ size effect model of the reflectivity for free electrons.“4’ as outlined by Chopra, Cl‘) to the calculation

The absolute

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of complex

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Wavenumber

(cm-t)

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Fig. 5. A comparison between the measured amplitude reflectivities of the 0.017 and 0.095 pm aluminium films and calculations based upon (a) a bulk resistivity model of the complex refractive index, and (b) a simple size-effect model for the resistivity of a thin film.

of the 0.017 pm film. This describes the geometrical limitations imposed by the film boundaries in terms of an effective resistivity in the limit that y, the ratio of film thickness to mean free path, is very much less than 1. The result is derived by assuming diffuse scattering of the electrons at the film boundaries. Application of it to the 0.017pm of film, which most nearly satisfies y < 1, leads to an effective film resistivity 6.7 x 10e8 0hm.m. Using this value in the calculation of the reflectivity of this film gives the curve (b) shown in Fig. 5. Between 20 and 35 cm-’ the measurements and calculations are in good agreement, but as the model gives a frequency independent resistivity it is unable to describe the spectral variation of the reflectivity outside of those limits. In addition to describing size and skin depth effects a more realistic model for the frequency dependence of the resistivity would need to allow for the nature of the substrate surface, the continuity of the metal film and the parameters of the vacuum deposition, as these are all factors known to affect the reflectivity of thin metal films. CONCLUSIONS

An absolute method has been described for the determination of the complex reflectivity of a reasonably transparent solid specimen. It has been applied to a specimen of high quality silicon and its amplitude reflectivity established with a random error of about lo-’ and a possible systematic error of up to 10e4 between 10 and 40cm-‘. The phase of the complex reflectivity was similarly established with a random error of about 7 x IO-’ rad. Using this specimen as a reference reflector, reflectivity comparisons were made with several thin metal films vacuum evaporated onto glass substrates. The transference of the complex reflectivity scale of the reference to these films was limited in accuracy by the errors of the reflectivity comparison at about 10m3 in the amplitude reflectivity and 10mrad in reflectivity phase. Thus, the precision with which the complex reflectivity of the reference specimen may be established. by the absolute method is considerably greater than the precision with which this calculated complex reflectivity may be transferred to an unknown specimen.

620

J. R. BIRCH

.~~kflow/~,~Ul,,,l~,Jlts~The author is extremely grateful to Drs W. F. Passchier and D. D. Honijk at the Gorlaeus Lahoratoria. Rijksuniversiteit. Leiden for the loan of the silicon specimen used in these measurements. Thanks xc also due to Mr M. J. Downs and Miss D. J. Croxford of the Division of Mechanical & Optical Metrology. NPL. for the thin metal films. REFERENCES I. FRAY. S. J.. A. R. GOODWIN. F. A. JOHNSON & J. E. QUAKKINGTON.

J. scirr~r. Iusrr. 40, 387 (1963). 2. FRAY. S. J.. F. A. JOHNSON. J. E. QUARRINGTON & N. WILLIAMS. Proc. Phrs. Sot. (Load.) 85. 153 (1965). 3 JONES. D.. A. H. L~TTINGTON & W. RICHMOND. J. Phys. E. 2. 623 (1969). RANUALL. C. M. & R. D. RAWTLIFFF. Appl. Opr. 6. 1889 (1967). 5 LOIWENSTI:IN. E. V., D. R. SMITH& R. L. MORGAN. Appl. Opt. 12. 39X (1973). PASSCHIEH, W. F.. D. D. HONIJK. M. MANDEL & M. N. AFSAR. J. Phys. D. 10, 509 (1977). 7. CHANTRY. G. W.. H. M. EVANS. J. CHAMBERLAIN & H. A. GEBBI~. Infrared Ph_vs. 9, X5 (1969). X. CAKI)ONA. M.. W. PAN L & H. BR(x)I(s. J. P/IJ,,\. Cl~cm. Solid\ 8. 204 (1959). 9 <‘AHUONA.M.. W. PAI L & H. Bwooks. Solid Sru~c, P/tr.sic,.\ iit E/r,c,/rortic.s uttrl T~,/~,~o,,l,fllr,li~ufi~~~l.\. Vol. I. Part 1 (Edited by DESIRANT.M. & MICHIELS.J. L.). Academic Press. London (1960). IO. BIRCH. J. R. & D. K. Mt KKAY.Itr/rtrrcd P/II,\. 18. 2X3- 191. (1978). I I. CARLI, B.. J.O.S._4. 67. 908 (1977). I’. ROUARU, P.. AIIII. d. Phys. 7. 291 (1937). 13. BIRCH. J. R.. A. F. HARDING.N. R. CROSS& D. W. E. FULLLK. Infrared Php. 16. 421 (1976). Phil. Sot. 34. 100 (1938). 14. FrlcHS. K.. Proc. Camhridgr McGraw-Hill. New York (1969). 15. CHOPKA. K. L.. Thirt Film Phomcwa.

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