On the Infrared Thickness Measurement of Epitaxially Grown Silicon Layers P. J. Severin The thickness of epitaxially grown layers is commonly measured by infrared multiple reflection. The phase shift at the layer-substrate interface is very often neglected. It has been computed as a function of the wavelength, and for accurate measurement it is said to be advisable to apply this correction. By direct calculation from measured spectra, a phase shift independent of wavelength is found. The layer- substrate reflection coefficient,generally assumed to be constant, has been verified to vary exponentially with the wavenumber, yielding another constant phase shift. The measured constant phase shift can- not be explained from this dependence. An alternative procedure for measuring the thickness of epitaxial layers with a precision to within 0.5% is suggested. Introduction The thickness of a silicon layer epitaxially grown on a silicon substrate of higher conductivity is generally measured by interference between the directly reflected wave and the wave reflected at the interface. The wavelength X of the incident radiation is varied nom- inally between 2.5 g and 50 u for an epitaxial layer a few microns thick, and usually one or more fringes are recorded. For these wavelengths the complex refrac- tive index can be calculated, treating the free charge carrier gas as a classical plasma with losses. In this way the phase change upon reflection at the high-low interface can be calculated and computed. The main motive for this communication is the author's experimental observation that this phase change does not show the theoretical wavelength dependence, but that an arbitrary wavelength inde- pendent value 3o is found, which is in general different for each slice and over a slice. The value is uniform only when the slices are produced as a perfectly flat batch. In the next section the usual procedure, common for several years, is critically reviewed. In the third section experiments are presented, and for the reader's convenience of reference, similar but published spectra are analyzed and interpreted, yielding a wavelength independent value of the phase shift. It is shown that an exponential decrease of the interface reflection coefficient 12, with the wavenumber k = 1/X, is The author is with Philips Research Laboratories, N.V. Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands. Received 5 February 1970. formally identical with a constant phase shift. Experi- ments are presented which show that the reflection coefficient 12 in fact decreases in accordance with this law, but that the constant phase shift is not due to this effect only. It is surprising that in the literature, whereas the phase shift is supposed to be wavelength dependent-which in fact it is not-the reflection co- efficient is assumed to be constant, which is not true either. This interpretation of multiple interference spectra is shown to be applicable to GaAs too. In the last section, the nature of the high-low junction is qualitatively discussed, and it is argued that the com- monly assumed wavelength dependence cannot possibly be expected. In this paper an alternative procedure for measuring the thickness of epitaxial layers with a precision to within 0.5% is suggested, and the evidence is produced on which this recommendation is based. By way of example, the thickness has been calculated from pub- lished spectra, using the method of least squares with a desk calculator. A high degree of correlation is found for the straight line approximation. Theory of Multiple Reflection for Thickness Measurement Weighing a silicon sample before and after growth of an epitaxial layer, and angle lapping followed by stain- ing, were the only methods for measuring the thickness of the layer when Spitzer and Tanenbaum' introduced multiple reflection. They stated that the layer ought to be transparent, and layer and substrate ought to have sufficiently different refractive indices n, and n 2 . The complex dielectric constant as a function of carrier con- centrations had been studied before by Spitzer and Fan. 2 Several of the experimental curves of intensity October 1970/ Vol. 9, No. 10 / APPLIED OPTICS 2381
Transcript
On the Infrared Thickness Measurement of
Epitaxially Grown Silicon Layers
P. J. Severin
The thickness of epitaxially grown layers is commonly measured by infrared multiple reflection. Thephase shift at the layer-substrate interface is very often neglected. It has been computed as a functionof the wavelength, and for accurate measurement it is said to be advisable to apply this correction. Bydirect calculation from measured spectra, a phase shift independent of wavelength is found. The layer-substrate reflection coefficient, generally assumed to be constant, has been verified to vary exponentiallywith the wavenumber, yielding another constant phase shift. The measured constant phase shift can-not be explained from this dependence. An alternative procedure for measuring the thickness of epitaxiallayers with a precision to within 0.5% is suggested.
Introduction
The thickness of a silicon layer epitaxially grown on asilicon substrate of higher conductivity is generallymeasured by interference between the directly reflectedwave and the wave reflected at the interface. Thewavelength X of the incident radiation is varied nom-inally between 2.5 g and 50 u for an epitaxial layer a fewmicrons thick, and usually one or more fringes arerecorded. For these wavelengths the complex refrac-tive index can be calculated, treating the free chargecarrier gas as a classical plasma with losses. In thisway the phase change upon reflection at the high-lowinterface can be calculated and computed.
The main motive for this communication is theauthor's experimental observation that this phasechange does not show the theoretical wavelengthdependence, but that an arbitrary wavelength inde-pendent value 3o is found, which is in general differentfor each slice and over a slice. The value is uniformonly when the slices are produced as a perfectly flatbatch.
In the next section the usual procedure, common forseveral years, is critically reviewed. In the thirdsection experiments are presented, and for the reader'sconvenience of reference, similar but published spectraare analyzed and interpreted, yielding a wavelengthindependent value of the phase shift. It is shown thatan exponential decrease of the interface reflectioncoefficient 12, with the wavenumber k = 1/X, is
The author is with Philips Research Laboratories, N.V.Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands.
Received 5 February 1970.
formally identical with a constant phase shift. Experi-ments are presented which show that the reflectioncoefficient 12 in fact decreases in accordance with thislaw, but that the constant phase shift is not due to thiseffect only. It is surprising that in the literature,whereas the phase shift is supposed to be wavelengthdependent-which in fact it is not-the reflection co-efficient is assumed to be constant, which is not trueeither. This interpretation of multiple interferencespectra is shown to be applicable to GaAs too.
In the last section, the nature of the high-low junctionis qualitatively discussed, and it is argued that the com-monly assumed wavelength dependence cannot possiblybe expected.
In this paper an alternative procedure for measuringthe thickness of epitaxial layers with a precision towithin 0.5% is suggested, and the evidence is producedon which this recommendation is based. By way ofexample, the thickness has been calculated from pub-lished spectra, using the method of least squares with adesk calculator. A high degree of correlation is foundfor the straight line approximation.
Theory of Multiple Reflection forThickness Measurement
Weighing a silicon sample before and after growth ofan epitaxial layer, and angle lapping followed by stain-ing, were the only methods for measuring the thicknessof the layer when Spitzer and Tanenbaum' introducedmultiple reflection. They stated that the layer oughtto be transparent, and layer and substrate ought to havesufficiently different refractive indices n, and n2. Thecomplex dielectric constant as a function of carrier con-centrations had been studied before by Spitzer andFan.2 Several of the experimental curves of intensity
reflection coefficient R vs X show pronounced minima.In this approach, the semiconductor charge carrier gas istreated as a plasma with losses. The characteristicparameters are the charge concentration N and the colli-sion time T. MV/any authors extended this theory andconfirmed it experimentally, in particular by studyingthe reflection minima.3 -9
The infrared interference method was introduced forindustrial application by Albert and Combs. 1 It isshown in any textbook on optics that the rays directlyreflected at the (01) interface (o) combine with the raystransmitted through layer (1) and reflected at the (12)interface ( 12 ), to an effective amplitude reflection coeffi-cient r given by
rOl + 7'12 exp-jo1 + rlr 12 exp-jo
Fig. 1. Definition of the geometry used.
(1)
where 0 = 4rdnlc cos 61, and the geometry is defined inFig. 1. The intensity of the reflected light, as detectedby a parabolic detector, is given by rr*. Writing for thecomplex reflection coefficients r = r expj8, the intensityreflection coefficient R is found to be
From electromagnetic theory it can be shown that,when the layer (1) is lossless, and the substrate (2)shows absorption, the reflection coefficient 12 and thephase shift 12 upon reflection are given by
treme values are different. Determining again theextreme values of the denominator in Eq. (2a), therelation
where the substrate refractive index is complex n2 =n2 ,+jn2'-
For a silicon epitaxial layer, with typically 1- cmresistivity, on a silicon substrate, the refractive indexni is 3.42 with a negligible imaginary part. Hence, fromEqs. (3) and (4), the phase shift Bo is found to be r, *oi2
This expression can be approximated with 122 << 1 and*O1*2 << 1 to give
1 = ol - 2(1 - o2) ol*2 cos(O - 5). (2b)Extreme values for both Eqs. (2a) and (2b) are foundwhen
- = 127r for minima, and- 5 = (1 - )2ir for maxima,
where the order is indicated by an integer 1.However, when it is assumed that the reflection
coefficient 12 depends on the wavenumber k, the ex-
* The suffix 12 will henceforth be dropped for S.
and assuming *01*12 << 1, as found from Eq. (2b),
cos - ( - *) = 0. (6b)
The procedure for measuring the thickness of anepitaxial layer recommended by Albert and Combs isbased on Eq. (5), where, in addition to a wavelengthindependent 12, it is assumed that = 0 or, since onlyperpendicular incidence is considered,
2dnk = l for minima, and2dnk = (1 - 1) for maxima.
(5a)
For easy measurement based on Eq. (5a) they sug-gested a fringe chart consisting of straight lines d vs Xwith as a parameter, and d can be read when allextrema fit the lines. However, from Eq. (4) it isnumerically obvious that for reflection from a substrateof 10-2-Q-cm or 10-3-Q-cm N-type silicon, the phase shift8 should have a finite c-dependent value. Schumann,Phillips, and Olshefskill2 present their expressions forn2 ' and n2", from which as a function of X is computedfor N on N+ and P on P+ silicon, with substrate re-sistivities between 0.001 Q-cm and 0.2 -cm.
The method they adopt for the analysis of infraredspectra is as follows. From the key formula
2dnk = I - + /27r, (5b)
at two different and not too close maxima, X1,11, and2,12 = m + 1, the order is found
Using then the appropriate computed (k) relation inEq. (5), for every maximum a value of d is found which,after averaging, is supposed to give the true thickness.This procedure has been accepted by the AmericanSociety for Testing and Materials 3 as a tentativemethod pending adoption as a standard. A similarprocedure is sketched by Basnett,' 4 and a most ad-vanced instrument for thickness measurement isdescribed by Reichardt."1 An interesting spectrum hasbeen published by Frieser,'0 who was able to grow anepitaxial layer at 700'C from an Si2Cl6 source in thepresence of uv radiation. In contrast to normallygrown epitaxial layers, which show interference withincreasing fringe amplitude from 10 /.z onwards, inter-ference starts here at 2 A. He adduces this as evidencethat the low temperature grown layers have a sharperinterface.
It is an essential characteristic of all calculationsreferred to above to assume that the high-low junctionis abrupt, which is not true, of course. Sato, Ishikawa,and Sugawara' 7 found that slices of identical thicknessdo not produce identical interference fringes, and thatupon heat treatment, the fringes shift toward higher kand decrease in amplitude but reveal the same thick-ness. They introduced a wavelength independent re-duced refractive index, which cannot explain all observa-tions. Abe and Kato"5 treat the diffusion layer as astack of thin layers of constant n' and n" and computethe characteristic matrix. They assume that the diffu-sion coefficient in the epitaxial layer has three times thebulk value but cannot explain all observations in thisway. The infrared thickness measurement has alsobeen used by Abe, Nishi, Goto, and Konakal 9 formeasuring the depth of an N+ diffusion in an N typelayer from the back.
Experimental Results on the Phase Shift a andthe Reflection Coefficient rl,
If the simple theory discussed in the preceding sec-tion, and the ASTM procedure based on this theoryhold true, it should be possible to find the relation (k)experimentally. According to Eq. (5), maxima plottedin an vs k diagram should be on a straight line whichintersects the axis at = , if 8(k) = 0. Since, accord-ing to theory, should increase from zero to r with in-creasing wavelength, (k) = 0 holds only at high valuesof k. The difference between this straight line and theexperimental 1 vs k plot yields 8 - (k). In all caseswhere it cannot be found owing to lack of experimentalaccuracy, it is irrelevant as a correction; in all caseswhere the stated value is not found, it is not correct as acorrection.
The procedure described above has been applied tohundreds of spectra obtained from slices grown under avariety of circumstances by many different manufac-turers. It has always been found that vs k for the ex-treme values forms a straight line, which intersects theI axis at any value +0.5 > > -0.5; in other words,0 < bo < 27r and 6(k) = o in Eq. (5). Since this result isin contrast to what other workers have reported, possi-
ble causes of experimental errors have been carefullychecked. The type of spectrophotometer is irrelevant,the magnitude of the exposed spot is immaterial, andtilting the slice does not influence the spectrum. Itreproduces well when the slice is removed and carefullyreplaced. The k scale is well calibrated, and the scan-ning speed does not influence the result.
The value 3o, independent of k, is generally found tobe different from slice to slice, even when grown in thesame batch and at different spots on the slice. How-ever, on the basis of a theory recently developed byEversteyn et al.,2 0 the growth of epitaxial layers of out-standingly constant thickness along the reactor has beenachieved. In slices produced in this way, 85 has beenfound to be fairly uniform over the slice and onlyslightly different from slice to slice. The results on onebatch are presented in Table I, and more conspicuouslyin Fig. 2. The constant phase shift 6o has also beenfound with less conventional layered systems, like 10-/.&thick Si layers sputtered onto Si substrates of a varietyof conductivities, an Si layer grown on a lappedsubstrate, or a (100) Si layer grown on (0112) A1203 or
Table I. The Thickness d, the Characteristic Length do,Measured Phase Shift 80, and the Angle * For a Number of
Fig. 2. Thickness d and wavelength independent phase shift50, as a function of position along the susceptor (top and lowercurve). Middle curve yields thickness as measured by bevel andstain. Each of the eight N on N+ slices has been measured on
GaAs epitaxial layers. On the other hand, single N-type epitaxial layers produced by electrolytic etchingwith the van Dijk2 ' process show the proper wavelength-dependence with no phase shift at all, both in trans-mission and reflection.
For the reader's convenience of reference, we haveanalyzed a number of published spectra, and the resultsare presented in Table II. The k values of maxima andminima, as read from a spectrum or quoted from a table,are given. Then it turns out to be possible to draw astraight line through the extrema plotted on vs kpaper. (In fact, the maxima are plotted and theminima are inserted.) The thickness can be determinedby combining two, not too close, values kl, and k 2,12 =
14 + m to give
d = ?n/2n(k2 - k). (9)
All extrema can be combined with one of them, and thevalues of d obtained in this way are averaged. Inorder to obtain the highest accuracy available, thethickness d as presented in Table II has been calculatedusing the method of least squares with a Hewlett-Packard 9100 calculator. The high correlation of thedata, typically 0.9998, shows that the straight line ap-proximation is correct. The value of 2dnk yields formaxima 1 - + /2 7r, and for minima + /27r, fromwhich can be found. From Table II it is clear that ,is in each case remarkably constant. If a mistake hadbeen made in assessing d, this should be manifested by amonotonous increase or decrease in with k, as can beseen from Eq. (5). Since always decreases with in-creasing k, with a = 7r at k = 0, an underestimate of dalways yields a monotonous decrease of d with k, and anoverestimate may yield a minimum in . There is noindication of either possibility in Table II. The aver-age phase shift 6o obtained from the same machine cal-culation is also presented in Table II.
This is the procedure suggested for epitaxial layerthickness evaluation and the experimental evidence onwhich this recombination is based. A fringe chartbased on Eq. (5b) can easily be made by plotting k vs with d as a parameter and floating scale. For mostaccurate thickness evaluation the use of a calculatorprogrammed for the method of least squares is stronglyrecommended.
In order to judge correctly the noted values of 0 it isof interest to discuss the accuracy. Most infraredspectrophotometers may safely claim a maximumreading accuracy on the k scale between 400 cm-' and1200 cm-l of about i 3 cm-l. Since two values of k aresubstrated, there is always a loss of relative accuracyamounting to about 0.5% in d. i/lultiplying the valueof 2dn by k, one finds the lowest value of and 1 + /27rwith an accuracy to within 0.5%. The final accuracy in5/ 2ir depends on the value of 1 and of / 2 7r itself.Roughly it can be said that for order 4 to 8 at X = 10 ,the value d/27r is known to within 0.05. It can be seenfrom Table II that the values measured at differentorders are in line with this sort of accuracy. For an N-type epitaxial layer on an 10-2 -Q-cm N silicon sub-
strate, the phase shift /2 7r calculated by Schumannet al."1 decreases from 0.25 at 25 1A to 0.1 at 10 A; there isno indication of such influence in Table II. In otherwords, if such correction should be applied, the truethickness d could not be determined as accurately.
It is surprising that although much attention has beenpaid in the literature to 8(k), the more conspicuousP12(k) dependence has been neglected. In all inter-ference spectra of epitaxial layer systems produced inthe usual way, the fringes disappear at wavelengthsbelow about 10 . Since p012 = 0.30, this implies thatr12 gets below about 0.01 as can be seen from Eq.(2b). With this equation rl2 has been calculatedas a function of the wavenumber k from spectra of alarge number of slices produced under a variety ofconditions. They all show an exponential dependenceon the wavenumber according to
12 = ?120 exp-kdo, (10)
where the characteristic length do assumes any valuebetween 40 A and 100 A and the long wavelength valueP,20 between 0.1 and 0.3. Just like (k), this depen-dence cannot be explained by the classical expressions(3) and (4). Their understanding should be based onspecific properties of the epitaxial layer system. FromEq. (10) it follows that the procedure suggested byAST/I, as based on Eqs. (5) and (8), with constantr12, is not realistic.
It can easily be shown that relations (7) and (10)can be combined to yield a constant * such that with6(k) 8,
tg* = 4dn/do. (11)
It would be tempting to identify in Eq. (6b) theangle * with the wavelength independent phase shift
= . However, 3o/2 7r has been found to cover therange 0 to 1, whereas */27r can cover the range 0 to0.25 only. In other words, * should be added to themeasured 8, in order to get . Summarizing, it shouldbe admitted that the observations reported are incon-clusive as to the true origin of the constant phase shift.
Discussion
It was assumed in the theories dealing with the prac-tical thickness measurement and with the wavelengthdependent phase shift that the high-low junction isdiscontinuous, that is to say, that the charge carrierconcentration changes abruptly. This may be math-ematically convenient, but it is evident that even withan abrupt impurity concentration change an accumula-tion layer extends into the lightly doped region over alarge distance, related to the Debye length. For a1-Q-cm N-type epitaxial layer on a 10-2 -Q-cm N+ typesilicon substrate excess electrons penetrate over about0.3 into the epitaxial layer. It should be realizedthat electrical and optical measurements allow con-clusions to be drawn only as to the available chargecarrier distribution. This applies most strongly toconsiderations involving the phase change upon reflec-tion of light at the interface. Moreover, the above
Note: The phase shift corrections 5(k) have not been applied and apparently are not relevant.because the horizontal scale is not calibrated in most published spectra.
assumption of a discontinuous impurity concentrationcannot be justified. In epitaxy, charge carrier profilesare related both to the impurity diffusion length and tothe Debye length. This also implies that the thick-ness measured can be at variance with the mechanicalthickness. The results of different methods of thick-ness measurement show a fairly systematic discrepancy.An example of this is shown in Fig. 2.
It is clear that for electromagnetic waves reflectingfrom an interface diffuse on a scale comparable to thewavelength, the reflection coefficient 12 is smallerthan for electromagnetic waves of so long wavelengththat the interface looks sharp. This must be, qual-itatively speaking, the origin of the decrease of 12 withk, as expressed in Eq. (11). A quantitative explanationfor the exponential nature of this decrease or for theactual values of P12' and do is not yet available. Apartfrom giving a formal contribution to the experiencedphase shift incorporated in 6*, it is clear that the realphase shift may be different for wavelengths short orlong with respect to the length of the transition layer.Reflection in stratified or graded ionized media hasbeen considered in monographs by Wait2 2 and Budden.2 3
No evidence has been found for a constant phase shift.
The angle 8* could not be calculated
Though the substrate surface is polished and etchedbefore growing, it may be that, as well known in crystalgrowth, the transition can become rough. No reason-ing can be given for a constant phase shift. As athird possibility it is worth noticing that the transitionlayer in an epitaxially grown high-low junction mustbe highly stressed. Seeking for an explanation herewould be mere speculation.
The interest in multiple reflection measurements onepitaxial layers will remain as long as no alternativemethod is available for equally accurate thicknessmeasurement. Although alternative methods do lookpromising and growth parameters can be controlled tosuch an extent that a whole batch consists of slicesvarying less than 1% in thickness,20 the infrared methodwill be of practical interest for some time to come. Inaddition the values ?1 2
0, do, and 5o must have a physicalmeaning from which practical information can be ex-tracted. That is why, within the framework of con-centrated work on epitaxial layer characterization inthis laboratory an ellipsometer has been built for the3 .3 9-.t and 10.6-,4 wavelengths. Results of this workwill be reported in due course.
