dataforks = 10' (cm./sec.) exp [(-44 Kcal./mole)/RT] with h, between 5 and 10 cm./sec. Values of ka and h, of the same order of magnitude can be extracted from Theuerer's data shown in Figure 12. While the above magnitude of & is in agreement with an estimate based on mass transfer considerations, the two sets of experimental results contain an intriguing apparent discrepancy. While Theuerer's data, obtained in the vertical reactor, show an increase of film growth rate with increasing flow rate in the maw transfer controlled region, Shepherd found only a very slight dependence of the film growth rate on the gas flow velocity in the horizontal reactor. One pwible explanation for this apparent discrepancy might be through the relative importance of the natural and the forced convection processes in determining the rate of gas phase mass transfer. Evidence of the importance of natural convection in a hot-wall tube reactor is provided by the motion pictures of Smith and Donovan (O), which showed the existence of symmetrical vortices superimposed on the flow by photographing the motion of tracer particlea. The appearance of such vortices, shown in Figure 14, indicates the presence of natural convection. In a cold-wall reactor, the entering gas stream is cold whereas the silicon wafer itself is kept at a very high temperature. As a consequence, temperature gradients of many hundreds of degrees centigrade per centimeter are encountered, further promoting the influence of natural convection. One can estimate that the Grashoff number Gr = (d"gpAT)/v' in an epitaxial reactor is probably of the order of loa to 10'. Thus, it appears reasonable that natural convection should have an important effect in determining the rate of gas phase mass transfer, in which case the gas phase mass transfer c d c i e n t would be independent of the gas flow rate in the reactor. Why natural convection should dominate in the horizontal reactor and not in the vertical, we can only guess. In terms of a simplified picture, the mass transfer rate would depend on the velocity of the gas in the direction normal to the surface of the silicon wafer. In the case of the vertical reactor, the main flow itself is normal to the silicon surface, and the natural convection flow can only m o d i the main flow slightly. In the case of the horizontal reactor, the main flowis parallel to the surface of the wafer, and, therefore, the natural convection flow might have a larger influence in determining the flow velocity in the normal direction.
film is determined by the redistribution of the impurities originally contained within the substrate by solid-state diffusion alone. However, if such precautions are not taken, the impurity distribution may be much more I t has been suggested that the gradual-"sloppy." excess impurities originate from the backside of the heavily doped substrate (7). After escaping from the backside, these impurities are mixed into the gas stream and are then incorporated into the graving film. More recently, experiments with radioactive tracers verified this picture (4). Details of this "autodoping" process must depend very critically on the gas flow pattern. However, this feature of the autodoping phenomenon has not yet been investigated. Conclusions
We have considered the role of gas phase mass transfer in the various process steps of the planar technology. In each case, mass transfer is combined in series with either a chemical reaction or a solid-state diffusion process. On the basis of the available information, it appears that under typical experimental conditions the gas phase maw transfer process has little or no influence on the rate of the oxidation and solid-state diffusion steps but that it may be the rate determining process in epitaxial growth and that it can be involved in determining the impurity distribution in epitaxial films. Little is known about the details ofthe gas phase mass transfer process in any of the above cases. Basic questions, such. as the relative importance of natural us. forced convection or the role of various reactor geometries, are yet to be studied. As a result, gas phase deposition reactors, employed with increasing frequency in the growth of metals and insulators as well as of semiconductors, are designed empirically. Yet, reactions involving semiconductors have great advantages for basic studies of mass transfer processes. The reactions involved are relatively simple, and because the impurities involved are electrically active within the semiconductor, simple electronic measurements can be used to determine concentrations and concentration distributions with extreme accuracy. I t is hoped that chemical engineers will make use of these advantages, with benefits to themselves and to semiconductor technology. REFERENCES
Aulodoplng
Another very interesting phenomenon which is connected intimately. with the gas phase mass transfer process involves the transport of impurities from one place within an epitaxial reactor to another multing in an undesired impurity distribution in the epitaxial film. This process is called "autodoping." In epitaxial film growth it is frequently desired to grow a lightly doped film on a heavily doped substrate. If elaborate precautions are taken to eliminate contamination, it has been shown (3) that the impurity distribution in the
(1) Bausha. H.,Tun& S. K.,M-, R. C.. ThoC. O., Md.Smiendwlor M&. 15, 69 (1962). (2) D a l , B. E.,Dlwc, A. S., J. Appl, Pkn.56,3770 (1965). (3) Drove, A. S., Rods, A,, slh,C. T., Bid., p. 802. (4) J m , B. A,, Wo.vcr, J. C., Made, D. I., J . Ehlncbm.S.c 112,llW (1965). (5) Mwre, 0. E., in "Microelcctr&:' Chap V, McGnr-Hill, New Yak.1969. (6) &hlishtia H "hrndyy Lay- Thcory," Chap. V., 4th d., Md3raw-w.
N w Yak,
fhbso:'
(7) SEhliChtlq, H.,Ibid., Chap. XIV; Bird, R. B., Stewart, W. E., Lightfoot, E. N.,"Tr-prt Phenomena:' p. 802, Why, N w Yak,1960. (8) Shepkd, W., J . Ek1mkm.S.r. 112,989 (1965). (9) Smith, A. M., Doawm, R. P.,*'DinEfDymmnk Oburvationof 1mpV"ly Flow plfah. G & ~ , bDn meuiomof aim,-~*etroch=. s-. F ~ I M A & Washington, D.C., 1964. (IO) Smin,F.M.,Mill~,R.C.,Pklr.fin. IM.1212 (1956). (11) T h e u c r c . , H . C . , J . ~ ~ . S . r . l M , 6 1 9 ( 1 % 1 ) . (lZ&T;yba$ R. E., *'MauTrader Owntioru? Chap. 5, McGraw-Hill, N w
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I THE CHEMISTRY 1 1 I ANDPHYSICS 11 I OFINTERFACES 11 ~
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Liquid-liquid and liqu&gas interfaces Walter Drwt-Hausen: Aqueous IntezfacesMethods of Study and Structural Properties Parts 1and 2 Sydney Ross and
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Keeping it on the Square R. J. DeGray
Data analysis may often be greatly simplified by exploiting the orthogonality of the functions involved. Although most anqlysts formally know this, they occasionally neglect to emptoy the simplifications and unnecessarily complicate their work. The author discusses the virtues of orthogonal relations and illustrates their use with vivid examples and analogies
e begin with a principle which underlies all experiW mental design: balance. Figure 1 is a photograph of an inexpensive type of balance found in many laboratories. With this we introduce the game: keeping one hand behind your back, level this instrument in 30 minutes or less. For leveling, this balance has two adjustable feet, one fixed foot, and a small, bull’s-eye spirit level. Figure 2 shows the arrangement of the two adjustable feet A and B, the fixed foot C, and the spirit level. With only one hand, we can adjust foot A or foot B, but not both simultaneously. Working with A, the bubble may be brought close to the center of the spirit level, but to finish the job with foot B, we upset what had already been achieved with foot A. The best solution to this problem is to make a small adjustment with A, then a small adjustment with B, then back to A, and so on, until the bubble finally is brought into the bull’s-eye. Mathematically, this is known as “successive approximations” and it is usually a tedious procedure. Figure 2 shows what’s wrong. Raising and lowering foot A causes the plane of the balance base to rotate about axis B-C. The bubble in the spirit level will move perpendicular to B-C. Raising
and lowering foot B causes the plane of the balance to rotate about axis ‘ 4 4 and the bubble will move perpendicular to A-C. But A-C is not itself perpendicular IO B-C, so motion perpendicular to one axis has a component in the direction of the other axis. Thus, any adjustment of A upsets, to some extent, the previous adjustment achieved with B, and vice versa; hence, successive approximations and tedium. For contrast, consider Figures 3 and 4: another balance though more expensive. The particular instrument shown here has two spirit Levels a t right angles to each other. This clarifies the discussion, but is not really necessary. For leveling, foot A is raised or lowered until bubble P is between the hairlines, then foot B is raised or lowered until bubble Q is between the hairlines, and the job is complete. The vital point is that angle ACB is 90’. The bubble in P inob’es perpendicular to the axis B-C as before, but since axis d-C is perpendicular to axis B-C, rotation about B-C has no component in the A-C direction; the bubble in P moves with foot A, while the bubble in Q is unaffected. Similarly, the bubble in 2 moves with foot B, while the bubble in Pis unaffected. rhe secret of success is 90’; this balance is on 6ht square, OL
58
NO. 7
JULY 1966
57
., aS&t F&wc 1. A t y p i d bean balance with Icwliig .. . . .. ., ., ban ..., ~
buift inio the
. ..
Figure 2. The lcwl+g axes for the b,4awx of FigUrr 7. A and B are aajustabllafeet in tfreban of the balahc
and. this makes the difference between a tedious, cutand-try procedure and a simple, direct and unequivocal technique. To generake, whenever two fortes, or two vectors, or two variables, or two factors mbve at right angles fo each. other, thelr effects will be mutually independlnt. If fdot A id teGperature, and foot B is pressure, and we keep these moving at right angles to each other, we can move temperature up and down and see what this doeS to the process, and we can move pressure up and down, and see ,what this does to 'the pmess, and each effect wiltbe independent ofthe other.:' We obtain two direct, unequivocal a ~ s w e m :the dfe&of temperature and the &et of pressure. If our experiment is not on the squore, m observechanges .whichare partly due to temperature, and.pattly due to pressure, and we are never sure w M h is which. ' ' ' If evnyihing is perpendicular to everything else, the anglewar&...&.900, so the angles'are all equal. The Grreks had words' for such situations; ortho- is equal, and gokrrl- is angle, 80 orthogonal is equal-angled. 1f:one ~
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+ght ahgles to each other
Kgwe 4. Tfreicmling w p s farhe.0ntJytic.d balance . of-Figurc.3!. . . A &d B ore adjustabli f e d
variable is orthogonal to another, the vectors 'are perpendicular to each other, and independent of each other. When the variables are orthogonal, the &p&< men* is an the square. Orthogonal vari recopnize, the varianci of &i crosP-products"is zero (this is illustrated.below). When this is h e , the experiment is simple to run, simple to calculate, and definite i6 its answers. You can leve vel out" the 'process in a few simple steps. .But how can temperature be perpendicular to p i e sure?; If both variables are rendered dimensionless, and konverted to the proper scale, there is no trouble. Fdr instance, we wish to find 'out what happens to tedile strength when we sliiift the temperature from 500" to 600°* F.,and/or shift the reaction time from 4 hours to 8 hours, and use 5% or 10% of "suppiurn perhapdate.'> ' These.variables can be expressed on dimensionless scales by "orthogonal coding." ORTHOGoNAL CODING Temperature - 550* F. A-
B =
50'
Time
F.
- 6 houm
2 hours 7.5% 2.5%
c = % SP
-
