Comment on ``Thickness Measurements of Sharp Thin Film Steps''Jean M. Bennett Citation: Review of Scientific Instruments 38, 294 (1967); doi: 10.1063/1.1771392 View online: http://dx.doi.org/10.1063/1.1771392 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/38/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thickness Measurements of Sharp Thin Film Steps Rev. Sci. Instrum. 37, 1260 (1966); 10.1063/1.1720475 Optical Thickness Measurement of Thin Transparent Films on Silicon J. Appl. Phys. 36, 3804 (1965); 10.1063/1.1713951 Automatic Optical Thickness Gauge for Thin Film Measurements Rev. Sci. Instrum. 33, 172 (1962); 10.1063/1.1746529 Measurement of the Thickness of Thin Nylon Films Rev. Sci. Instrum. 20, 457 (1949); 10.1063/1.1741565 The Ellipsometer, an Apparatus to Measure Thicknesses of Thin Surface Films Rev. Sci. Instrum. 16, 26 (1945); 10.1063/1.1770315

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

136.165.238.131 On: Fri, 19 Dec 2014 13:58:31

294 LETTERS TO THE EDITOR

Stieltjes integral which depends only on the properties of the material, and is therefore constant. Observe that (Ro/m)",A-2, where A is the cross section of the wire, and furthermore is independent of length. If the conditions of experiment result in t. remaining nearly constant, as they seem to do in Oktay's cases, then Vo",A necessarily fol­lows. This is Oktay's result.

If the experimental wire is long enough so that arc forma­tion is prevented at vaporization, and if the resistance at melting and beyond exceeds the critical damping resistance, then an exponential current decay with time will be ob­served. Further rise in temperature of the wire or its ex­pansion products would not be large, because heating rate drops off as the square of current, and the decay takes long enough so that expansion and fluid dynamical cooling processes are important.

1 E. Oktay, Rev. Sci. lnstr. 36, 1327 (1965). 2 F. D. Bennett, H. S. Burden, and D. D. Shear, Phys. Fluids 5,

102 (1962). 3 F. D. Bennett, Phys. Fluids 8, 1106 (1965). 4 F. D. Bennett, Phys. Fluids 7, 147 (1964).

Comment on "Thickness Measurements of Sharp Thin Film Steps"

[Glenn C. Bailey, Rev. Sci. lnstr. 37, 1260 (1966)J

JEAN M. BENNETT

Michelson Laboratory, China Lake, California 93555 (Received 31 October 1966)

THE thickness of evaporated films can be accurately determined using interferometric techniques. How­

ever, with the popular Fizeau method in which fringes formed in monochromatic light are used, an involved tech­nique employing two or more monochromatic wavelengths must be employed,t unless the approximate film thickness and the direction of the wedge angle of the interferometer are known. One major advantage to be gained by working with fringes of equal chromatic order is that, as pointed out by Bailey2 and others, there is never any ambiguity in

115,100

... ..< 115,000 ~

114.900

96,200

03: >< 96,100 'c

o

d

1° I I I •

21 i IS,840A

I I I

~pOO~~--------~----------~----------~ 5000 5500 6000 6500

WAVELENGTH A (A)

FIG. 1. Graphical method for determining film thickness using data from Ref. 2. This method completely eliminates dispersion of the phase change.

the order of interference. For thin films (those whose thickness is less than A/2), the method used by Bailey and others is usually adequate, but when working with thicker films (where t>A/2), the dispersion of the phase change on reflection with wavelength cannot be ignored. In the method outlined by Bailey,2 where a constant order of interference is used to compute the film thickness, the effect of phase dispersion is not taken into account, so that erroneous results will occur. For example, the thickness calculated by Bailey for his thick film is in error by more than three times the average deviation of the measured points.

There are three methods of measuring film thickness using fringes of equal chromatic order, all of which nearly or completely compensate for the dispersion of the phase change. The graphical method3 is simple, accurate, can be used for films of any thickness, and completely eliminates the phase dispersion. To illustrate how this method works for Bailey's thick film, see Fig. 1 and Table I. In Table I, the orders of interference nand n', wavelengths A and A', and nA products are listed for the interference fringes in the unfilmed and filmed regions, respectively. The orders

TABLE I. Determination of 1ilm thickness from FeCo fringes.

Un1ilmed region Filmed region n X " nX n' X'

17 6770 A 14771 cm-1 115 090 A 18 6395 15637 115110 19 6055 16515 115045 16 6012 A 20 5748 17397 114960 17 5655 21 5475 18265 114975 18 5335

19 5055

• Thickness calculated from fringes nearest in wavelength: t = (nI2) (X -X') + (X'12) (n -n'l. b Thickness calculated from fringes of same order of interference: t = (nI2) (X -X').

, "

16633 cm-1

17683 18744 19782

n'}..' ta tb

96192 A 9426 A 96135 9412 9477 A 96030 9472 9540 96045 9500

9437±24 9506±23

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

136.165.238.131 On: Fri, 19 Dec 2014 13:58:31

LETTERS TO THE EDITOR 295

of interference have been obtained from the relation" +100

where n1 is the order of interference for the fringe having a wavelength A1, and ni-n1 is the difference in orders of interference (an integer) between fringes having wave­lengths A1 and Ai (where A1>Ai). When the interferometer plates are coated with highly reflecting layers of silver or aluminum, the numbers calculated from Eq. (1) are nearly integers, so that the correct values of the n's are the inte­gers closest to the calculated numbers. In Fig. 1, the nA products are plotted vs A for the fringes in the unfilmed and filmed regions, respectively. Although there is appre­ciable scatter in the points caused by inaccurate measure­ments of the wavelengths of the interference fringes, parallel line graphs can be drawn through the data points. Since both filmed and unfilmed portions of the interferome­ter surface are coated with an opaque layer of the same material,' the dispersion of the phase change is the same for both regions, and hence the lines are strictly parallel, no matter what the curvature of each line is. The difference in the ordinates at any wavelength gives twice the film thickness; in this case t= 9420 A. It is clearly not possible to draw parallel lines through the two sets of data points and obtain Bailey's value of 9506±23 A for t.4

The second method for completely eliminating the effect of phase dispersion from the thickness measurement is Koehler's method of coincidence,5 which is applicable to thick films where there are one or more "crossover points." These points are where the fringe system in one region crosses over from the long wavelength to the short wave­length side of the fringe system in the other region. Bailey's data are not adequate to illustrate this method, since there are no data points on one side of the crossover point, and extrapolation to the coincidence value is un­certain. Nevertheless, the use of the method is illustrated in Fig. 2, where the frequency difference (in cm-1) between fringes in the unfilmed and filmed regions, respectively, is plotted vs the frequency of the fringes in the unfilmed region. At the coincidence frequency V e, where the differ­ence becomes zero, the film thickness is given by the relation

t= (n-n')/2ve• (2)

Even though the extrapolation is uncertain, an approxi­mate value of 15945 cm-1 may be obtained for V e, and, since n-n'=3, t=9407 A, in good agreement with the graphical value of method 1, and about 100 A lower than Bailey's value.

The third method is a calculational one which mini­mizes but does not eliminate the dispersion of the phase change with wavelength. In this method, approximate interference equations6 of the form nA= 2d (where d is the

o

-100

~ ~ -200 I ~

-300

-400

-500

I I I Vc ;15,945 em-I I I I I I

I 16,000 16,500 17,000 17,500

FREOUENCY v (em'l)

18,000 18,500

FIG. 2. Method of coincidence for determining film thickness using data from Ref. 2. This method is applicable to films where t >"10./2.

thickness of the air film separating the two interfering surfaces) are written for the filmed and unfilmed regions, respectively. The difference equation gives

t= (n/2) (A-A')+ (A'/2) (n-n'), (3)

where A and A' are the wavelengths of fringes which are most nearly identical in wavelength in the two regions. The compensation for the phase dispersion is best if A and A' are nearly the same. Table I gives calculated thicknesses for three pairs of fringes between 6055 A and 5475 A. The average value, 9437±24 A, is in good agreement with the values obtained using the other two methods.

In conclusion, Bailey's method of calculating film thick­ness by using fringes having the same order of interference in the unfilmed and filmed'regions (see the last column of Table I) does not eliminate the effect of the dispersion of the phase change on reflection when films thicker than A/2 are measured, and sizeable errors in film thickness may be made. In this case, one of the three methods outlined above should be used instead.

1 S. J. Lins, Trans. 8th Nat!. Vacuum Symp., L. E. Preuss, Ed. (Pergamon Press, Inc., New York, 1961), Vol. 2, p. 846.

2 G. C. Bailey, Rev. Sci. Instr. 37, 1260 (1966). 3 J. M. Bennett, J. Opt. Soc. Am. 54, 612 (1964). 4 There is an arithmetic error in one thickness value in Ref. 2. The

number 9490 A should read 9477 A, as is shown in the last column of Table 1.

6 W. F. Koehler, J. Opt. Soc. Am. 48, 55 (1958). 6 The correct interference equation is given in Eq. (10) of Ref. 3.

Since the phase change on reflection for silver (which is used on most interferometer plates) is not too different from "", {JI and {J2 in Eq. (10) can be set equal to "", and the term (which now equals A) can be trans­ferred to the left hand side of the equation, thus obtaining the approxi­mate equation given here. Note that n in this equation equals N+l in the notation of Ref. 3 [compare Eq. (1) in thip note to Eq. (13) in Ref. 3].

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

136.165.238.131 On: Fri, 19 Dec 2014 13:58:31