Film Thickness by X-Ray Emission Spectrography H. A. LIEBHAFSKY and P. D. ZEMANY Research Laboratory, General Electric Co., Schenectady, N. Y.
metal film d cm. thick to excite a characteristic line of intensity I d . The contribution to I d of a volume element of constant area and of thickness dx, located a t depth x , is
When a polychromatic x-ray beam excites the characteristic line of an element in a thin film, the intensity of the line may increase with the thickness of the film, in which case a measurement of line intensity can be used to estimate the thickness. Owing to absorption of x-rays by the filni, however, the relationship of intensity to thickness is not generally simpIe. This absorption filters and attenuates the polychroniatic beam, and attenuates the characteristic line. As a consequence, the number of quanta per second contributed to the emergent beam by a layer of atoms decreases with the distance of the layer beneath the surface. Furtherniore, quanta originating beyond a critical depth will have no measurable effect on the detector because not enough of them will reach it. For the measurement of film thickness by the method under discussion, this critical depth may be regarded as a critical thickness.
+
d I = kIDe- [ ( r d s i n 61) ( r d s i n 8 z ) l ~ xdx (1) where k is a proportionality constant that measures the conversion of incident to characteristic radiant energy p1 and pz are mass absorption coefficients for the incident wave length (A,) and the characteristic (A,) p is the density 81 and 0 2 are the angles made by the incident and emergent beams with the (plane) sample surface (ITigure I) x is the depth (distance perpendicular to surface). The integrated equation may be written
I d = k Io where
a = 7 LOCKER and Schreiber ( 4 ) appreciated the importance of the critical thickness in x-ray emission spectrography. They pointed out that samples of a t least critical thickness, which thciy estimated to be less than 0.01 cm. for their case, could be considered infinitely thick; for such samples, the measured intensity of a characteristic line must be a t the value given by the element in mass. Koh and Caugherty (6) proved experimentally that the intensity of a characteristic line emitted by thin metallic films decreases below that for the massive metal as the thickness of the film falls below the critical value, near 0.003 em. in their cabe. Brissey, Liebhafsky, and Pfeiffer (2)showed by calculation that this value of the critical thickness is reasonable. Koh and Caugherty (6) pointed out that x-ray emission spectrography could be used to measure the thickness of films in the range below the critical value. The present investigation is concerned with the basis and the usefulness of this method (hlethod 11) for measuring thickness, and with its relationship to another, Method 111, in which thickness is established by the extent to which the film attenuates a characteristic line of the substrate. (Method I differs from Method I11 in that no analyzing crystal is used.) The designation of the methods is taken from an earlier paper (11).
ea=
d x = k Io(ead -. l)/a
(2)
+ pz/sin & ) p
(3)
- (pl/sin el
At infinite thickness ( d
(rr
METHOD
Jd =
m),
I m = - kIo/a At the critical thickness dc, the ratio Id/Im = 1
Table I. Region Useless Exponential Linear
(4
- ead
(5)
Capabilities of Method I1
em
Negligible Significant Near unity
d or x Greater t h a n dc Intermediate Smaller t h a n X L
dI/dx Approaches zero Variable Approaches constanc)
The capabilities of Method I1 can be assetised from Equations 1, 3, and 5 with the results shown in Table I, in which three regions are defined according to the value of the exponential term in Equation 1. S o description of the useless region is necessary in light of the discussion that has been given of the critical thickness dc. The linear region is of particular interest. Inasmuch as the relationship of thickness and intensity is made complex by the absorption of x-rays in the film, it seems logical to expect a simplification in the region where the coatings are so thin that absorption is negligible. Intuitively, one is led to surmise that the intensity of the characteristic line in this region will be pioportional to the number of atoms in a coherent film, and hence to film thickness. The correctness of this surmise may be demonstrated by examining Equation 1. -4s x becomes smalIer and smaller, the exponential term in this equation increases to the value unity. Once this increase has gone far enough to make the term indistinguishable experimentally from unity, then the term itself may be regarded as constant; whence
WLYCHROMATIC
Figure 1. Schematic diagram of three x-ray methods for film thickness
A I = kIo A X
(6)
for all values of x below ZL, the value a t mhich the relationship of intensity to thiclrness first becomes linear. The value of X L depends upon the value of a. and upon what variability in the exponential term may be ignored, the latter consideration usually being governed by the precision attained. If 0.99 is chosen as the value of the exponential term for which this consideration holds, then Z L is 10-5 cm. when a is -1000.
The basis of Method I1 may be deduced from Figure 1. T o do this, the ideal case is considered, in which the x-rays involved are monochromatic, all influences of composition are absent, the simplest optics obtain, and excitation of a characteristic line in the film by a characteristic line of the substrate does not occur. Suppose now that a beam of intensity T o falls upon a 455
ANALYTICAL CHEMISTRY
456 Experimental results in accord with Equation G have been obtained by Pfeiffer and Zemany (8) and by Ilhodin (9). The esponential region obviously lies between X L and de. The existence of the three regions of Table I for each value of a is clear from Figuie 3, which contains curves calculated according to Equation 5 for five values of a, one of which was selected to fit exper imental data that mill be discussed Inter.
o P L A T E D SAMPLES 0 M A S S I V E METAL
EXPERIMENTAL
When the counting rate was below about 3000 counts per second, the x-ray tube, which had a tungsten target, mas operated a t 50 kv. and 50 ma. At higher counting rates, the current was reduced to FILM THICKNESS,d.cm 5 ma., but the rates were coirected to 50 ma. and Data for determining background corrections in chroniiuniFigure 2. Iecorded. A flat, reflecting lithium fluoride crystal on-molybdenum system was used throughout. Background Problem. I n this investigation, the background count varies with wave length, and it must be estimated for thewave length of the characteristic line being counted. There is also the special problem of change in the background count from that of the substratemetal to thatof the metal in the film as the thickness increases. The relationship of U this change to atomic number is comples, for it dec pends not only on the way in which s-rays are 5 04 z scattered by a given atom but also on the may in c W z 0 PLATED COATINGS which they are absorbed. 02 0 EVAPORATED COATINGS Over the thickness range for which Method I11 is useful ( I I ) , the coatings are usually thick enough so that the background count may be assumed identical with that of the plating metal in mass a t the 10.6 10-1 lo-' 10-3 10-2 10-1 wave length of the characteristic line of the subFILM THICKNESS.d.cm. strate. I n the case of hlethod 11, the situation Figure 3. Calculated curves showing relationship between intensify is more complex for the reasons given above, ratio and tliicliness for various values of exponent Q and the need for a reliable background count increases as the coatings become thinner. Chromium on Molybdenum. The importance of protecting Figure 2 contains the backgiound data taken on chromiummolybdenum samples for which the thickness of chromium wan> high-temperature molybdenum alloys against ovidation is well known. The present investigation was begun in the hope of estimated both by Method I1 and by Method 111. I n Method demonstrating the usefulness of x-ray methods for measuring the 11, the characteristic line of interest is A f o ( K a ) ,a t a goniometer thicknesses of metallic coatings on niolybdenum alloys. setting of 20.255'; background readings were taken a t 17" I n order to make available a range of thicknesses that would and 23' to bracket h l o ( K a ) . Similarly, corresponding back. test both Method I1 and Method 111, thin films of chromium ground values are plotted for CI ( K a ) , thc characteristic line iii on molybdenum were prepared by evaporation in vacuum and RIethod 111, for which the goniometer setting is G9.25'; thici thicker coatings by electroplating. The molybdenum substrate line is bracketed between 6'7" and i 3 " . The average countiq: was in the form of disks about 2 cm. in diameter. Those to be rates for the massive metals are included in Figure 2; trivial coated by evaporation were about 0.05 cm. thick, and those to d i m epancies aside, these rates may be considered for uppei be plated were about 40 times thicker. Either is, of course, an and lower limits of the curves for the plated samples. "infinite" thickness of molybdenum for present purposes. The background correction a t the wave length of the charactcrI n neither case is the film thickness known so precisely as in istic line was estimated differently in the two cases. For hlethocl the earlier work with iron foil (11). The thickness of the evapo111, in which the background correction is always small relative. to the counting rate for hIo(Zia), the average rate a t the t v a v ~ rated coatings was calculated from the weight of chromium placed on tlie filament from which evaporation occurred, geometrical length of this line was taken as the mean of the averages for 17" factors being' taken into account. In this may, moderately and for 23'. For Method 11, the following average counting: reliable relative values were obtained for the chromium on two rates were obtained on uncoated molybdenum: 113.49 ( G i " ) , pieces of molybdenum a t different distances from the source, 112.23 [69.25"-i.e., Cr(Ka)]-and 95.87 ('73'). Inasmuch at; but the absolute value of the thickness was not well established. Method I1 is useful only for very thin coatings, it seemed best to Furthermore, the thicker chromium coatings made in this way I ely on the background relationships obtained for molybdenum tended to flake off the molybdenum substrate. With the thickest in estimating the background correction. Consequently, the ones, this trouble was so marked that these were not even meas73" values were ignored, and the average background countink; ured. Of those measured, some flaking mas observed a t the rate ( C P S ) B a t Cr(Ka) was taken as two greatest thicknesses, which explains why these points lie ( C ? ) B at K a = at 07'1 112.23/113.49 (7'1 below the curve in Figure 3. The thickness of the electrolytic chromium mas obtained by careful measurement with a miThe foregoing discussion shows that the background correction crometer, but the unavoidable errors here also exceed those is important, and that it can be complcs enough to require attending the use of iron foil. special study.
[(m)~
V O L U M E 2 8 , N O . 4, A P R I L 1 9 5 6
457
The experimental results by hlethod I1 are plotted in Figure 3 for both kinds of coatings. All average counting rates were corrected for background as described above. The rate for infinite thickness, I - , was taken to be the mean of the rates for the four greatest thicknesses measured, which is justifiable inasmuch as these equal or exceed the critical thickness dc (Equation 5). The ratios I d / I m were then computed and plotted, and the bestfitting calculated curve belonging to the family in Figure 3 was passed through the plotted points. Inspection of Figure 3 shows a t once that the esperimental points fit the calculated curve with two exceptions, the points for the two thickest evaporated coatings, which are too low. The critical thickness is near 0.001 cm., and Method I1 is obviously useless for estimating thicknesses greater than this. The curve through the experimental points for chromium-onmolybdenum tvas calculated by use of aexp.= -4410. Substitution of known values in Equation 3 gives (Icslcd. = - (250/0.866 f 110/0.500) G.92 = - 3520 (3a) The mass absorption coefficient 250 is that of chromium for 1.3A., which was taken as the wave length of the incident polychromatic beam (3). According to Equation 5, an arithmetic inrrease in a means an increase in I d ; or, the counting rates measured for the chromium coatings are somewhat higher than one predicts for the simple emission process. This increase could well be due to excitation of C r ( K a ) by the characteristic lines of the molybdenum substrate, notably Mo(Ka). 'To test the correctness of this explanation, the Cr(Ka) line was counted (50 kv., 5 ma.) a t a later date for the thickest plated sample and for massive chromium. The results: plated sample. 2576 counts per second; massive chromium, 2513 counts per second; background, near 4 counts per second in both cases. Escitation does seeni to have occurred as suggested. Metal as a thin plate over the proper substrate can thus yield a more intense characteristic line than the massive metal under the same external excitation. When such strengthening of the characteristic line of the plated metal becomes pronounced, it will have to
'"'F
E
\\
t
1 0
0 EVAPORATE0 CWTINGS
0002
0004
0.006 0008
OD10
FILM T H I C S N E S S , P , cm.
Figure 4. Experimental results for chromium-on-niolybdellllni by RIetbod
IIT
be considered in formulating the expected relationship between thickness and intensity (Figure 3); the esperimental results for chromium on molybdenum are not precise enough to warrant this refinement.
EA lo"o'r
. v)
L
w
2x10' w
t
Ej c w I
I
\
I
\
METHOO I I . A V E R A G E
Figure 5.
CORRECTED COUNTING RATE
lCOUNTSlSECl
Intercomparison of Methods I1 and 111 applied to tin plate Ordinate scale is logarithmic
No exact test of Equation 6 is possible because the thicknesses of the evaporated coatings are not sufficientlywell known. For a = - 4410, ZL is near 2 X cm. under the reasonable conditions postulated when ZL was defined. The four results, two of which could not be plotted in Figure 3, available for this thickness region are in rough agreement with Equation 0. The discrepancies, which are definitely traceable to uncertainties in the method for preparing the films, are great enough to militate against presenting the data. The experiments on tin-plate, discussed later, provide much better support for Equation 6. Fortunately, Rhodin (9, 10) has already presented precise evidence for the applicability of Equation 6 to metallic films a t thicknesses near IOp6 cni. The work of Pfeiffer and Zemany (8),though done on salts deposited on filter paper, also deserves consideration in this connection. Thus, Equation 0 may be regarded as experimentally verified for thicknesses a t which reliable standards can be prepared, and it may lie used with confidence on thinner films. It is clear, of course, that the thicltnesses discussed here are nominal values that are averages over the area of sample viewed by the detector. They are nominal values because they are calculated by use of the bulk density, which may differ considerably from that of thin films T o establish the relationship of Methods I1 and 111,the samples shown in Figure 3 were run by Method 111, which depends upon the attenuation by the chromium coating of the h I o ( K a ) line generated in the substrate. Except for the background correction, which was discussed above, the procedure was so similar to the one already developed (6) that further dexription is unnecessary. The results are plotted in Figure 4, which shows that the
ANALYTICAL CHEMISTRY
458 samples coated by evaporation are crowded so near the ordinate axis as to be indistinguishable. The electroplated samples, on the other hand, which were indistinguishable in Figure 3, are not7 grouped satisfactorily around a straight line as the exponential absorption law requires. The divergences from the line are not greater than possible uncertainties attributable to micrometer measurements. Figure 3 points out that Method I1 is useful up to thicknesses near 0.001 cm. Figure 4 shows that Method I11 is useful from t h a t region up to about 0.01 cm., so that the two methods coinplement each other satisfactorily. Tin on Steel. Electrolytic tin plate is an excellent material for the present investigation, aside from its great industrial importance. Its production is controlled by Method I (1, 7;r, and it is highly uniform. The value of a (Equation 3 ) for tin is near -400, so that X L is about 2.5 X cm., which means t h a t the linear range for tin extends to considerably greater thicknesses than for chromium. An investigation of tin-plate standards might thus make possible an intercomparison of Methods I, 11, and 111, and data bv. Method I1 might provide additional evidence for the validity of Equation 6. (Method I is essentially Method I11 with thi: analyzing crystal omitted.) Such an investigation was carried out on six standards in the form of panels 6 inches square. These panels mere each cut intt2 24 rectangles, 1 X 1.5 inches. For each standard, one of the 24 rectangles was selected a t random and subjected on each face to a measurement of tin thickness by Method I1 and b~ Method 111. The procedure mas generally similar to that for the chromium-on-molybdenum system, with the small difference t h a t the background correction here was calculated for both methods as the mean of appropriate readings-Le., as in blethod I11 for chromium. Average values by Method I for each face were furnished with the standards.
indeed, except a t the greatest thicknesses, where the experimental points lie somewhat below the line. Results by Method I1 are plotted against known thicknesses (obtained by Method I and based ultimately on chcmical determinations) in Figure 6, and results by LIethod I11 similarly in Figure 7. Figure 6 shows that Equation 6 is valid up to about 1.5 X 10-4 cm., or considerably beyond 2.5 X cm., which was estimated above to be the upper limit of the linear range. The points a t the highest thicknesses lie below the line, as mould be espected on the basis of Equation 1; this divergence is probably the cause of the deviation from linearity in Figure 5. Figures 5, 6, and i do not seem to indicate a clearly greater reliability for any one of the three x-ray methods. I os
e
-::
. p z
5110'
$
E IY
r
3110'
z
0 3
0
2x10'
c L
uo
0 w
e Y
>
1x10'
l
2
3
flLM TI'IICKNESS I N C M . X 10.
Figure 7.
Intercomparison of Methods I aiid I11 applied to tin plate
Thickness values based on Method I, ordinate scale logarithmic
F I L M THICKNESS,
a, cm. x io4
Figure 6. Intercoinparison of RIethods I and I1 applied to tin plate Thickness values based on Method I
The results for the 12 thicknesses studied showed that all 12 were in the thickness region in which both methods m r e useful; this is in marked contrast to the chromium-on-molybdenum system, for which the overlapping of the useful regions w s slight. Owing to the complete overlapping for the tin-plate samples, it is possible to compare these methods by plotting the proper counting rates against each other in such a n-ay that a straight line with an intercept a t the counting rate for Fe(Kc!) should result if there is complete agreement between the two methods. Figure 5 is evidence that the agreement is good
Inasmuch as each standard mas subdivided into 24 rectangles, only one of which was run, it seemed desirable to test a t least one of the standards for uniformity by Method 11. This mas done by measuring the time for 214 counts a t the setting for Sn(Rcr) on the same side of each of the 24 rectangles cut from the thinnest tin-plate standard. The standard deviation of these 24 counting periods \vas 2.1%, and placing these times on a map of the standard showed that their distribution was not random; in other words, the plate mas not completely uniform. I n agreement with this conclusion, 12 counting periods for one of the rectangles, withdrawn and reinserted between counts, had a standard deviation of only O.iyO. The predicted standard deviation (6) for 214 counts is 0.870, and the close agreement of these two values shows that operating conditions were good. CONCLUSION
The information given above should make i t possible in general t o predict the usefulness of x-ray methods in problems involving films. I n principle, these methods should be useful occasionally when more than one film is present. The problems in such cases are complex rather than complicated. If both methods ale applicable to a duplex film, for example, there will be three char-
V O L U M E 28, N O . 4, A P R I L 1 9 5 6
459
acteristic lines to be counted, and absorption effects in three regions to consider. The three counts should, however, contain enough information in many cases to permit the drawing of valid conclusions.
(3) (4) (5) (6)
ACKNOWLEDGMENT
CHEW27, 1257 (1955).
Pellissier, G. E., Wicker, E. W., Elec. M j g . 49, 124 ( M a y 1952). Pfeiffer, H. G., Zemany, P. D., dVczture 174, 397 (1954). Rhodin, T. N., ANAL.CHEM.27, 1857 (1!355). Rhodin, T. N., Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Pittsburgh, Pa., March 1955. (11) Zemany, P. D., Liebhafsky, H. A., J . Electrochem. SOC.103, No. 3 (1956).
(7) (8) (9) (10)
The authors wish to thank G. E. Pellissier, Physics and Analytical Chemistry Division, Applied Research Laboratory, United States. Steel Corp., for the set of tin-plate standards; and Virginia Thomas and Gloria Kowaleski for help with the investigation. ,
iMeeting, BSTM, Atlantic City, N. J., June 1953, ASTM Spec. Tech. Publ. 157. Compton, A. H., Allison, S. K., "X-Rays in Theory and Experiment," 2nd ed., Van Nostrand. ivew York, 1935. Glocker, R., Schreiber, H., Ann. P h ~ s i k85, 1089 (1928). Koh, P. K., Caugherty, B., J . A p p l . Phgs. 23, No. 4,427 (1952). Liebhafsky, H. A., Pfeiffer, H. G., Zemany, P. D., ANAL.
LITERATURE CITED RECEIVED for review October 3,1955. Accepted January 17, 1956. Division of Analytical Chemistry, 128th Meeting, ACS, Ninneapolis, Rlinn., September 1955.
(1) Beeghly, H. F., J . Electrochem. SOC.97, 152 (1950). (2) Brissey, R. &I., Liebhafsky, H. d.,Pfeiffer, H. G., 56th Annual
Determination of Iron by Ultraviolet Spectrophotometry ROBERT BASTIAN, RICHARD WEBERLING,
and
Sylvania EIecMc Products Co. he., Kew Gardens,
N. Y.
A n ultraviolet spectrophotometric method for the determination of iron utilizes the absorption of the ferric ion in perchloric acid solution. The system shows excellent stability, good sensitivity, and adherence to the usual absorption laws. The method has been applied to a wide variety of industrial alloys and glass samples containing 0.03 to 0.9% iron. Because no special reagents are needed, iron can often be determined in conjunction with the determination of other elements, and, if desired, the entire sample can be recovered and used for subsequent determinations. Owing to the simplicity and rapidity of the method, it is well suited for routine analyses. The interference of a variety of elements is considered.
FRANK PALILLA
0.7 0.6
0.5 0.40 0 4
=:
0.30
9 0.20 0.10 '200
T
HE ultraviolet absorption spectra of ferric perchlorate
and ferric sulfate solutions have been examined previously for analytical applications (5). The ferric sulfate system was felt to be better because of less interferences encountered from other metals. Further investigation has shown, however, that the ferric perchlorate system can be used to advantage in the analysis of a large number of industrial materials. The precipitation of insoluble sulfates is avoided, and the perchloric acid medium is often more desirable when other elements are to be determined on the same solution. I n addition, the sensitivity is somewhat greater than the ferric sulfate system. APPARATUS AND REAGEKTS
210
220
240 250 260 270 WAVE LENGTH (rnp)
230
2W
290
Figure 1. Absorption spectrum of iron(II1) in perchloric acid solution 0.93 mg. of iron(II1) a n d 10 ml. of 70% perchloric acid per 100 ml.
'
The following were used: Beckman Model DU spectrophotometer with ultraviolet accessories and 1.00-em. silica cells; primary standard iron metal (Hach Chemical Co., Ames, Iowa); 70 to 72% perchloric acid; methanol; 0.03% hydrogen peroxide (freshly prepared I to 1000 dilution of 30% hydrogen peroxide). The above, and other unspecified reagents, were of C.P. quality. CHARACTERISTICS O F FERRIC PERCHLORATE SYSTEM
The absorption spectrum of ferric perchlorate in excess perchloric acid is shown in Figure 1. Although the absorption peak is a t 240 mp, 260 mu is preferred as the wave length for analysis because many other metals interfere less a t that point. The dependence of the absorption on acidity is shown in Figure 2. The absorption rises until about 1 or 2 ml. of perchloric acid are present per 100 ml., then remains nearly constant up t o high
ml. OF 70% PERCHURIC A C I D PER 100 ml.
Figure 2.
Effect of acidity
Solutions contain 0.94 mg. of iron(I[I) per 100 ml.
concentrations of the acid. At high acidity, the iron is presumably present as free ferric ion. A concentration of 10 ml. of perchloric acid per 100 ml. was
