- Email: [email protected]
An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates
surface science ELSEVIER
Applied Surface Science 90 (1995) 251-259
An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates T. Easwarakhanthan *, S. Ravelet, P. Renard Laboratoire de Physique des Milieus lords&, CNZLS-URA835, BP 239, Universitk de Nancy Z, 54506
Vandoeuvrelb Nancy Cedex, France
Received 12 January 1995; acceptedfor publication 26 May 1995
Abstract The accurate determination of thicknesses well under 100 A of ultra thin films, such as native, etched or chemically cleaned surface films, from ellipsometric measurements, has always been a difficult problem encountered in the presence of experimental errors, particularly that in the angle of incidence. In this paper, we develop a procedure to determine unambiguously the thickness of such transparent films overlaid on absorbing substrates. From the ellipsometric data measured on two same absorbing substrates overlaid with the same transparent film, but up to different thicknesses, the procedure equally finds out the angle of incidence in addition to the index and the thickness of the two films. Moreover, a very rapid convergence is realized owing to a transformation of the four-parameter problem into a two-dimensional numerical search in which the refractive index and the angle of incidence are found at first independently of the two thicknesses. The validity of the method is also investigated under situations where the assumptions initially made differ slightly. A quantitative error analysis further carried out shows the ability of the method to resist the propagation of experimental angular errors to the parameters sought, in particular to the smaller thickness when the other is chosen relatively thicker. The thicknesses in the order of 10 A such as that of surface films can hence be obtained from an ellipsometer producing angular errors of a few hundredths of a degree and more. The measured angle of incidence can also be verified. The procedure is finally illustrated with data from simulated and real reflecting systems.
1. Introduction
In the fixed wavelength and single angle of incidence ellipsometry, the index of refraction II and the thickness d of transparent films overlying absorbing substrates are generally found by inverting the basic ellipsometric equation [l]. This equation relates the film parameters n and d to the ellipsometric quantities P and A measured at an angle of incidence @J
* Corresponding
author.
0169-4332/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0169-4332(95)00084-4
through the Fresnel’s equations of the two-interface and the optically isotropic three-phase model which represents the air-film-substrate reflecting system. When the substrate optical constants are known, the inversion can be suitably carried out using different numerical procedures accompanied with techniques that transform the basic complex equation into some other convenient form [2-81. The faster convergent algorithms [3,6,7] thus developed, track first the index n iteratively before computing the thickness from the index found. In this way, the multiple critical-point solutions that could have been other-
252
T. Easwarakhanthan
et al. /Applied
wise generated when the two parameters are simultaneously searched, are avoided. However, difficulties arise when one intends to extract thicknesses well under 100 A such as that of native, etched or chemically cleaned thin surface films from the ellipsometric data using the above classical method. Nowadays, ellipsometers inevitably produce experimental errors up to a few hundredths of a degree. These errors, particularly that in the angle of incidence [8-101, propagate to the parameters sought using the inversion methods. The params ters rr and d thereby found out in case where the algorithm tracking them succeeds, could be far away from their real values that one could expect. Taking for example the well-known substrate Si with its surface film SiO, (n = 1.46) having a thickness of 20 A, the sole error in @ of + 0.01” { - 0.01”) drives the index to 1.08 (2.22) and the thickness to 68 w {17 A} at the angle of incidence of 70” and under the wavelength of 6328 A. We present in this paper an inversion procedure in which the angle of incidence, whose angular error modifies largely the parameter solution, is also regarded as an unknown parameter in addition to the index and the thickness of the film. With the angle of incidence and the wavelength fixed, we consider two absorbing substrates overlaid with the same transparent film but having different thicknesses. The assumption made here, that the film’s index of refraction is independent of the thickness, is reasonably valid and verified in many practical situations: polishing and cleaning of substrate surfaces [ll], plasma-assisted [12] or chemical etching [13] of films grown on substrates. However, this assumption may be in its limit of validity at the .initial stages of film growth where its morphology is continuously changing or in the cases of film materials whose microstructure evolves significantly with the thickness. The method as formulated in the following section consists in a two-dimensional numerical search using the Newton-Ralphson technique for a fourparameter problem. Supposing that the substrate optical constants n, and k, are accurately known, the index n and the angle of incidence @ are found at first independently of the two thicknesses d, and d, which are thereafter directly computed. A quantitative error analysis is then carried out by simulating all combinations of experimental errors in ?P, A, n,
Surface Science 90 (1995) 251-259
and k, in order to verify the validity of the parameters thus obtained and to know the conditions under which the uncertainties in the parameters are minimized. The validity of the method is further investigated in situations where the two films have slightly different indexes and are not measured exactly at the same angles of incidence, and where the single uniform layer supposed to represent the films is not enough adequate. The entire procedure is finally illustrated using simulated and experimental data of known reflecting systems. It is thus shown that thicknesses less than 20 A, such as those of surface films, as well as the angle of incidence at which the reflecting system is measured, can be obtained with reasonable accuracy in the presence of angular errors in the order of a few hundredths of a degree.
2. Formulation of the procedure The basic ellipsometric equation pJn, di, ns, ~3) = tan(!Pi:-)exp( j Ai), where pi is the ratio between the “p” and “s” overall complex-amplitude reflection coefficients of the ith film-covered substrate structure {i = 1, 2), ?Pi and A, are the ellipsometric quantities measured on the ith structure and fls (= it, + jk,) is the substrate complex index, can be put in the form: AiqF + Biqi + Ci = 0,
where i = 1,2.
(1)
The complex variables Ai, Bi and Ci are functions of the physical parameters II, rrs, the angle of incidence @ and the ellipsometric angles measured, Pi and Ai. The complex phase shift vi experienced by the light wave as it traverses to and fro between the interfaces, is given by
I,
(2)
where A is the wavelength and Gli the refractive angle at the ambient/film interface of the ith structure. Identifying, respectively, the modulus and phase of Eq. (2) with those of the roots of Eq. cl), once these roots are represented in the polar form with
T. Easwarakhanthan
et al. /Applied
modulus ri and phase pi for the two structures considered, leads to the following equations [3,7]: fii = ln( ri( n, @)) = 0,
for i = 1,2,
(3)
fZi = &( n, CD)+ ei( n, di, @) f 2Xli = 0 for i= 1,2,
(4) where ei(n, di, @I = (4TdJh)n COS(@~~) is the principal value of the phase shift of vi and 1; is an integer constant that takes into account the periodic behaviour of vi described by Eq. (2) and is initially estimated from the approximate values of the two thicknesses. We have now from Eq. (3) a non-linear system of two equations {i = 1 and 2) from which the refractive index n and the angle of incidence @ can be found using the Newton’s iterative procedure: [Xl,=[Xl,-r
-K[/(Xm-l)]-l[F(X,_l)], (5)
where [X]=[n@lT, [Fl=[f,,, fizlT and K is the Levenberg’s parameter which helps in reducing larger Newton steps computed at some subsequent iteration. The partial derivatives intervening in the Jacobean [j] can be found in several references [14,15]. The index n and the angle @ are thus found to a relative accuracy of O.OOl%,converging from a wide range of initial guesses, within a very few seconds in a coprocessed 386SX microcomputer programmed in Turbo Pascal. @ is experimentally known and can be initially well-fed. Now, the two thicknesses di {i = 1, 2) can be directly computed from Eq. (4): d,
=
1
_A
PiCny @)f
2T1i
4-xn cos( Gli)
,
for i= 1,2.
(6)
That multiple solutions are seldom generated is probably due to the reduction of the four-dimensional problem (n, @, d,, d2) to that of two dimensions (n, @>.
Sutface Science 90 (1995) 251-259
253
evaluate the exact uncertainties in the four parameters due to the experimental errors in the angles *r, A,, qz and A,, and to the uncertainties in n, and k,, bearing in mind the highly non-linear nature of the ellipsometric equations. We assume here that the two sets of angles measured {PI, A,) and I!&, AJ are completely independent, which suits besides the situation where the random experimental errors predominate. These errors, which cannot be easily compensated for, occur in the search of minimum intensity, in reading scales and as a result of parasitic beam or beam deviation. On the other hand, the systematic errors, which are in general the apparatus alignment errors, can be more or less compensated for either by physical or numerical alignment [93. These random errors could build up in each of the above six quantities q,, A,, p2, A,, n, and k, in either sense (L-) and consequently there will be 64 (= 26) equally probable solutions for n, @, d, and d, as these six variables could combine in 64 different ways. The maximum uncertainty in each of the four parameters occurring for one or several combinations, is retrieved among these 64 possible solutions. These solutions are found for any given two structures with different film thicknesses: two experimental sets of ly and A corresponding to the two thicknesses are simulated first by generating them at an incidence @ and then adding to them the errors of ArY= f 0.01” (or O.l”), A A = f 0.02” (or 0.2”). The substrate optical constants are also modified according to their uncertainties + An, (0.003) and f Ak, (0.002). As an illustrative example, the following case study is conducted and discussed in the next sections: The standardoreflecting system: Air/SiO,/Si[lOO] at A = 5461 A and @ = 70” with n(Si0,) = 1.460, I(, = 4.086 + 0.003-j 0.0310 f 0.002 [16]. The thickness d, is varied from 20 A to 1200 A while the other thickness d, is fixed at 10 A.
3. Error analysis The uncertainties in the unknown parameters due to the experimental errors in the measured quantities have been generally estimated either by applying the first-order Taylor series approximation of the functions involved at the solution point [5,10,15] or using statistical methods [2,9]. Instead here, we choose to
4. Results and discussion 4.1. Propagation of experimental errors Fig. 1 shows that the maximum possible uncertainty in each of the parameters n, @, d, and d,
254
T. Eawarakhunthan
et al. /Applied
corresponding to the experimental errors of A?@= kO.01” and AA = f 0.02”, Jrecomes less t$an 10% as d, gets thicker than 100 A. Above 200 A for d,, the uncertainty in d,(= 10 A) turns around 10% (* 1 A> even with the errors more important: A?P= f 0.1” and AA = f 0.2”. In particular, the uncertainty in the angle of incidence remains around 0.03% (* 0.02”) and increases to O.ll%(* 0.08”) when the angular errors in W and A are tenfold greater. This result can be used to verify the experimental setting of the angle of incidence. These results can be again seen in the Table 1 which gives the maximum relative uncertainty obtained in each of the parameters with the angular errors of A!P = f0.01” [ f O.l”], A A = f 0.02” [ f 0.2’1 for two different reflecting systems. An, and Ak, are taken f0.003 and f0.002, respectively 1161.Foreeach of these systems, film thicknesses of d, = 10 A and d, = 400 A are considered. One sees
Uncertainty
(%)
‘““!:I=I_
0.0 1 0
200
400
800
800
1000
1200
Thickness d2(A) Fig. 1. Variation of the maximum relative uncertainty in the parameters ?, @, d, and d, with the thickness d, while d, is ped at 10 A, reflecting system considered is SiO, /Si: A = 5461 A, 0 = 70” with n(Si0,) = 1.46, ns = 4.086 kO.003 -j 0.031 f 0.002 [16]; AP = f 0.01” and A A = f 0.02“; the uncertainties An’, Ad’,, Ad, and A@’ correspond to AtY = rtO.1” and AA= f 0.2”.
Surface Science 90 (1995) 251-259
Table 1 The maximum relative uncertainty in the parameters n, @, d, and C$ for two reflecting systems with fifferent film indexes: d, = 400 A and d,=lO A, at A=5461 A, @=70” with n,=4.086f 0.003 -j 0.031~0.002 [16] Structures
Maximum relative uncertainties (%) AP, AA (deg.)
n
An
A@
Ad,
Ad,
SiO, /Si
0.01, 0.02 0.1,0,2
1.46
0.2 2.0
0.03 0.11
3 11
0.4 3.2
Thermal oxides/Si
0.01, 0.02 0.1,0.2
2.8
0.1 0.3
0.03 0.11
3 10
0.2 0.7
1131
that all parameters are found with reasonable uncertainties with the experimental errors specified above. Although An is under 2%, it is found to be more perturbed by the errors than the angle of incidence @. The method generates larger uncertainties in the parameters searched only in cases where d, approaches d, or when these two thicknesses are equally small. The above conclusion can be again percejved in Fig. 2 where the thickness d, is fixed at 200 A and the thickness d, is increased up to 1200 A. The uncertainties in the parametersdncrease substantially when d, passes through 500 A. Figs. 1 yd 2, and Table 1 show that very thin layers (N 10 A) can be detected with an uncertainty of 10 to 20%, depending on the value of the other thickness ( > 100 A) even with ellipsometers producing angular errors in the order of 0.1”. Also, one can easily deduce that the maximum uncertainty in d, is minimized when this thickness lies in the midway neighbourhood of the other Cd,). In the classical method [3-71 generally used, one determines first the refractive index of the film from a set of F and A, measured at an angle of incidence before finding its thickness from the index computed. Now, in order to compare the results obtained here with those found from the classical method, ?P and A are generated at @ = 70” for the reflecting system taken in the above case study for the film thicknesses d, and d, of 20 and 40 A, respectively. The experimental values ?P and A are simulated by adding the angular errors of +O.Ol” and +0.02’, respectively, to the values thus generated.
T. Easwarakhanthan et al. /Applied Surface Science 90 (1995) 251-259
Uncertainty
(46)
loo-
-
Ad2
*
--A@
Adz’
. .
0.01 0
/
I
/
/
200
400
600
600
Thickness
I 1000
1200
d2(A)
Fig. 2. Variation of the maximum relative uncertainty in the para?eters d, and @ with the thickness d, while dIOis fixed at 500 A, for the reflecting system SiO, /Si: h = 5461 A, @ = 70” with t&O,) = 1.46, n, = 4.086 f 0.003 -j 0.031 f 0.002 [16]; Aly = f 0.01” and AA = f 0.02”; the uncertainty Ad; corresponds to AYI = +O.l” and A A = 50.2”.
Table 2 presents the indexes and the thicknesses obtained by the two methods. The method put forward here finds accurate values for the four parameters, which approach their exact values even for thinner films, showing thereby that it is much more resistive to the propagation of errors in the measured quantities to the parameters found. Moreover, it should be noted that the results obtained using the classical method are strongly perturbed by the angle of incidence error especially when the films are
25.5
thinner [lo], whereas the very angle of incidence is also determined by the procedure proposed here. The effect of this angle of incidence error has not been taken into account above in obtaining the parameters by the classical method. The substantial reduction in the uncertainties in the parameters found by the method proposed here may be attributed to the equal determination of the angle of incidence whose error significantly shifts the parameter solutions in the classical method. On the other hand, one could think of the structure with a higher film thickness fixing the uncertainties, knowing that in all cases the effect of the experimental errors diminishes with increasing film thickness. Moreover, the accuracy of the method can be well checked by comparing the measured and the computed angles of incidence. Alternatively, the angle of incidence measured by other means can be verified as the angle of incidence found here is less perturbed by the experimental errors. The other point is that one could evaluate accurately film thicknesses thinner than 10 A, if the thickness of the second film is chosen relatively greater than the thinner one. This may be of much practical interest in situations where one wants to know the thickness of native oxides on substrates or to study the effect of cleaning and etching of substrate surfaces. 4.2. Departure from the assumptions made The first assumption made to formulate the procedure is that the two films of different thicknesses required should have the same refractive index. The ellipsometric characterization carried out on the etching and growth of oxides on the Si substrate [13] shows that the index of thinner oxide films is slightly greater than those of thicker films. In order to per-
Table 2 Parameter solutions obtained by the classical method [3-71 and the method presented here; the errors of AY = + 0.01” and A A = + 0.02” are added to the rY and A values generated at A = 5461 A (n, = 4.086 - j 0.031) and @ = 70” Exact values
Simulated experimental values
Solutions from the classical method [3,7]
Solution from the method here n = 1.4596 @ = 69.996”
n = 1.460
‘J’= 12.0770”
n = 1.1710
d, =2OA n = 1.460
A = 172.8905” Cy= 22.6886”
d, = 37.2 A n = 1.4580
d,=4OOA
A = 100.5166”
d, = 400.8 .fi
d, = 20.0 A d, = 400.4 .&
256
T. Easwarakhanthan
et al/Applied
ceive how the results produced by this method are effected by the above departure from the assumption initially made, !P and ,$ data are now generated at @ = 70” and A = 5461 A (n, = 4.086 - j 0.031) for two reflecting structures having the following film parameters: n = 1.460,
d, = 400 A,
‘# = 22.6786”,
A = 100.4966”,
n = 1.480,
d, = 20 A,
P = 12.0685”,
A = 173.7514”.
Applying our method to the above two structures, we find: n = 1.4599,
@ = 70.0003”,
d, = 20.4 A,
d, = 400.0 A.
d, = 400 A,
‘P = 22.6786”,
A = 100.4966”,
@ = 70.02”, A = 172.8492”.
Using the method, the following parameter values are found: n = 1.4652,
@ = 70.020”,
d,=20.1
A,
d, = 401.4 A.
Contrary to the precedent results, where the index found is closer to that of the film with larger thickness, here the film with the smaller thickness imposes the angle of incidence at which it is measured. Here again, it can be seen that the two thicknesses computed are almost not changed by the difference between the two angles of incidence. Therefore, the method can be also extended to cases where a small difference between the two angles of incidence exists, bearing in mind, however, that the angle of incidence thus obtained will be that at which the thinner film is measured. 4.3. Departure from the model supposed
As ?P and A are generated up to the fourth decimal place, the small deviation of + 0.0003” in @ should be originating from the difference in the index values of the films. The structure that has the larger film thickness (400 A> tends to force its index value of 1.46 as the equivalent index for the two films. If not, the two thicknesses are almost not effected by the above index difference. These conclusions remain valid as long as the two indexes are not that different and one of the film thickness is relatively higher than the other. The other assumption made is that the two structures are measured at the same angle of incidence setting. However, a small shift in the angle of incidence during the replacement of the structures may not be completely discarded. In order to study the effect of a small difference between the two angles of incidence on the parameters obtained, !P and A data are now gener$ted with @ = 70.00” and @ = 70.02“ at h = 5461 A (n, = 4.086 + j 0.031) for two reflecting structures with the same film index 1.460, and having the following film parameters: @ = 70.00”,
Surface Science 90 (1995) 251-259
d, = 20 A,
‘P = 12.0353”,
The two structures needed are represented here by the three-phase model (ambient-film-substrate) of which the film is supposed to be uniform. Several studies [13,17] show that there exists, at the interface between the substrate and the film, another very thin layer having an index of refraction different from that of the bulk oxide layer. In a recent work [17] concerning the native and tkermally grown oxides on Si, a thickness around 12 A has been presumed for this interface layer while assuming a film index of 2.5. In these cases the structures are adequately represented by the double-layer model. In order to know how this thin interface layer modifies the results obtained when applying the method proposed here to the ?P and A data of double film structures, the latter data are generated for the two structures at @= 70.00” and A = 5461 A (n, = 4.086-j 0.031). The index n, of 1.46 and the thicknesses d,, = 20 A and d,, = 400 A are taken for the upper oxide films and the corresponding values for the same interface layer (lower fil$ of the two structures are n, = 2.5 and d, = 12 A, respectively. Table 3 presents values of P and A generated for this double-layer model and the corresponding parameters found by the procedure. The interpretation of the parameters thus obtained is as follows: the “equivalent” refractive index of the simple layer representing the double layer of the
T. Easwarakhanthan
et al. /Applied
Table 3 Parameter solutions found by the method ptesented when applied to Y and A values generated at h = 5461 A and @ = 70” for the double-layer model representing the two structures Exact double-layer values
Generated Cy and A
Parameters obtained
nU = 1.46, d, = 20 A
rY = 12.1351”
n = 1.469
n, = 2.5, d, = 12 A
A = 168.7508”
@ = 70.013”
nU = 1.46, d, = 400 .&
9 = 23.1054”
dC = 33.3 ;i
n, = 2.5, d, = 12 i
A = 99.0197”
dz = 409.3 A
5. An experimental example As an experimental example, some of the ellipsometric measurements presented by Kao et al. [17] are taken to work out the method elaborated. These measurements [17] have been carried out on silicon samples of which the oxides are formed in oxygen at different elevated temperatures. We repeat in Table 4, for the convenience of reading, the quantities ?P and A measured by the same ellipsometer at an angle tf incidence of 70” and under the wavelength 6328 A on these samples which are named here A, B, C and D. The most thick and thin film samples have been chosen here as it is known from the error
solutions
for n, @, d, and d, and their uncertainties
Samples
n
DandA D and B DandC C and A C and B
1.47 1.47 1.42 1.48 1.47
f f f + f
0.03 0.03 0.17 0.03 0.03
257
Table 4 The ellipsometric data ly and A taken from Ref. [17]: data measured on Si samples named here A, B, C and D, at @ = 70” and A = 6328 A (n, = 3.858 -j 0.028); the film thicknesses are obtained from transmission electron microscopy @EM) and ellipsometric data supposing a film index of 1.46 [17] Samples
two structures considered is closer to that of the film with the larger thickness belonging to the upper film of these two structures. This is in agreement with the earlier conclusion where the film with higher thickness tends to force its refractive index as the “equivalent” index when the two films considered have somewhat different indexes. Also, the two “equivalent” thicknesses computed here may be viewed as the sum of the upper and lower film thicknesses of the double-layer model representing the reflecting systems.
Table 5 Parameter
Surface Science 90 (1995) 251-259
A B C D
Experimental data [ 171
From the methods in Ref. [17]
ly
A
(deg.)
(deg.)
d 6) @EM)
d (A) with n=1.46
10.53 10.68 16.43 17.24
168.86 165.90 114.73 111.61
28 37 300 324
33 50 296 325
analysis that a larger difference between the thicknesses of the two films reduces the propagation of experimental errors to the parameters determined. This table also presents the thicknesses of the corresponding samples obtained by the same authors [17] by transmission electron microscopy (TEM) and the ellipsometric data assuming a uniform oxide layer with refractive index 1.46, Table 5 gives the parameter values found by our procedure with their uncertainties estimated from the error analysis supposing A?P = 0.05” and A A = 0.1” [17], for different combinations of the ellipsometric data of the four samples chosen. Only the data combination of the samples D and C brings out inconsistent parameter values deviating from the would-be real values. This result can be foreseen as the nearness between the two thicknesses of the samples D and C should consequently lead to larger uncertainties in the parameters, as shown in the error analysis. For the other data combinations of the samples, the method finds almost the same film thickness for one particular sample, whatever the other sample
found in this work for the ellipsometric
data given in Table 4
d, &
db 61
d, 6)
dd CA)
@ (deg.)
34 f 1 _ 34 f 1 -
45 f 1 -
309 f 59 293 f 9 294* 9
321 f 9 321+ 9 336 f 65 -
70.001 70.009 70.14 70.001 70.008
44*1
+ f f + f
0.03 0.03 0.44 0.03 0.03
258
T. Easwarakhanthan
et al. /Applied
with which it is combined. For example, the same thickness of 34 A has been found for a film of sample A from both the data of the pairs D and A, and C and A. The thicknesses d, and d, of the thinner film samples A and B thereby found are still greater than those deduced from the TEM measurements [17]. The authors proposed [17] a double-layer model, representing the interface region by a thin layer with refractive index 2.5 and thickness 12 A, and the bulk oxide region by a layer with refractive index 1.47, with its thickness deduced from the TRM measurements. They found that the 9 and A computed using this double-layer model agree well with those experimentally measured. Looking again at Table 5, the two thicknesses d, tnd d, are obtained with an uncertainty around + 1 A which originates from the experimental errors in YP and A. Therefore, any difference between the thickness values obtained here and those deduced from the TEM measurements should come from the representation by the single-layer model of a structure which may be better described by a double-layer model with a thin film layer at the interface. It has been noted earlier how the parameters obtained here may be perceived in this situation; the two “equivalent” thicknesses d, and d, computed should be around the sum of the upper and lower film thicknesses of the two structures: the single-layer “equivalent” refractive index of 1.47 f 0.03 obtained here should be closer to that of the thicker bulk oxide layer. Finally, it can be seen that the values obtained for the angle of incidence are in very good agreement with the one measured in Ref. [17].
Surface Science 90 (1995) 251-259
by its very determination. The method is rapidly convergent due to the reduction of the four-parameter problem to a two-dimensional numerical search in which the refractive index and the angle of incidence are found at first independently of the two thicknesses that are directly computed afterwards. The validity of the method is investigated when applied to situations where the conditions differ slightly from the a priori assumptions made to formulate the method. The parameter values found by the procedure in these situations are thus interpreted. Besides, the quantitative error analysis carried out shows the ability of the method to resist the propagation of the experimental errors to all parameters searched, particularly to the smaller thickness d, when d, is chosen sufficiently greater than the other. The method thereby0 allows one to extract thicknesses thinner as 10 A from the ellipsometric angles measured by an automated ellipsometer that induces angular errors in the order of a few hundredths of a degree. This may be of much practical interest in detecting the presence of naturally occurring surface films or in studying the effect of chemical cleaning of the surface films formed on substrates. Furthermore, as the uncertainty in the angle of incidence due to the experimental errors remains relatively small under the above condition, its value thus found may therefore be used to check its measured value in a particular ellipsometric setting.
References
I11R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
6. Conclusions A new procedure has been developed in order to determine unambiguously the thickness of ultra thin films, such as native, etched or chemically cleaned surface films, from the ellipsometric measurements on two reflecting systems having different film thicknesses. The procedure simultaneously finds the angle of incidence whose experimental error modifies largely the parameter solution in the classical method. The propagation of the error in the angle of incidence to the parameters sought is avoided here
121F.L. McCrackin, A FORTRAN Program for Analysis of Ellipsometer Measurements, Technical Note 479 (National Bureau of Standards, US Government Printing Office, Washington, DC, 1969). [31 A.R. Reinberg, Appl. Opt. 11 (1972) 1273. [41 Y. Yoriume, J. Opt. Sot. Am. 73 (1983) 888. iSI D. Charlot and A. Maruani, Appl. Opt. 24 (1985) 3368. [61 E.E. Dagmann, Phys. Chem. Mech. Surf. 3 (1985) 1954. [71 T. Easwarakhanthan, C. Michel and S. Ravelet, Surf. Sci. 197 (1988) 339. k31H.F. Wei, A.K. Henning, J. Slinkman and W.R. Hunter, J. Electrochem. Sot. 139 (1992) 1783. 191 K Riedling, Thin Solid Films 75 (1981) 355. [lOI D. Chandler-Horowitz, Integrated Circuit Metrology, SPIE 342 (1982) 121.
T. Easwarakhanthan et al. /Applied Surface Science 90 (1995) 251-259 [ll]
A.C. Adams and B.R. Pruniaux, J. Electrochem. Sot. 120 (1973) 408. [12] M.C. Plowers, R. Greef, C.M.K. Starbuck, P. Southworth and D.J. Thomas, Vacuum 40 (1990) 483. [13] E. Taft and L. Codes, J. Electrochem. Sot: Solid-State Sci. Technol. 126 (1979) 131.
259
[14] H. Dupoisot and J. Morizet, Appl. Opt. 18 (1979) 2701. [15] T. Easwarakhanthan, P. Mas, M. Renard and S. Ravelet, Surf. Sci. 216 (1989) 198. [16] D.E. Aspnes and A.A. Studna, Phys. Rev. B 27 (1983) 985. [17] Shou-Chen Koo and R.H. Doremus, J. Electrochem. Sot. 114 (1994) 1832.
Applied Surface Science 90 (1995) 251-259
An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates T. Easwarakhanthan *, S. Ravelet, P. Renard Laboratoire de Physique des Milieus lords&, CNZLS-URA835, BP 239, Universitk de Nancy Z, 54506
Vandoeuvrelb Nancy Cedex, France
Received 12 January 1995; acceptedfor publication 26 May 1995
Abstract The accurate determination of thicknesses well under 100 A of ultra thin films, such as native, etched or chemically cleaned surface films, from ellipsometric measurements, has always been a difficult problem encountered in the presence of experimental errors, particularly that in the angle of incidence. In this paper, we develop a procedure to determine unambiguously the thickness of such transparent films overlaid on absorbing substrates. From the ellipsometric data measured on two same absorbing substrates overlaid with the same transparent film, but up to different thicknesses, the procedure equally finds out the angle of incidence in addition to the index and the thickness of the two films. Moreover, a very rapid convergence is realized owing to a transformation of the four-parameter problem into a two-dimensional numerical search in which the refractive index and the angle of incidence are found at first independently of the two thicknesses. The validity of the method is also investigated under situations where the assumptions initially made differ slightly. A quantitative error analysis further carried out shows the ability of the method to resist the propagation of experimental angular errors to the parameters sought, in particular to the smaller thickness when the other is chosen relatively thicker. The thicknesses in the order of 10 A such as that of surface films can hence be obtained from an ellipsometer producing angular errors of a few hundredths of a degree and more. The measured angle of incidence can also be verified. The procedure is finally illustrated with data from simulated and real reflecting systems.
1. Introduction
In the fixed wavelength and single angle of incidence ellipsometry, the index of refraction II and the thickness d of transparent films overlying absorbing substrates are generally found by inverting the basic ellipsometric equation [l]. This equation relates the film parameters n and d to the ellipsometric quantities P and A measured at an angle of incidence @J
* Corresponding
author.
0169-4332/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDZ 0169-4332(95)00084-4
through the Fresnel’s equations of the two-interface and the optically isotropic three-phase model which represents the air-film-substrate reflecting system. When the substrate optical constants are known, the inversion can be suitably carried out using different numerical procedures accompanied with techniques that transform the basic complex equation into some other convenient form [2-81. The faster convergent algorithms [3,6,7] thus developed, track first the index n iteratively before computing the thickness from the index found. In this way, the multiple critical-point solutions that could have been other-
252
T. Easwarakhanthan
et al. /Applied
wise generated when the two parameters are simultaneously searched, are avoided. However, difficulties arise when one intends to extract thicknesses well under 100 A such as that of native, etched or chemically cleaned thin surface films from the ellipsometric data using the above classical method. Nowadays, ellipsometers inevitably produce experimental errors up to a few hundredths of a degree. These errors, particularly that in the angle of incidence [8-101, propagate to the parameters sought using the inversion methods. The params ters rr and d thereby found out in case where the algorithm tracking them succeeds, could be far away from their real values that one could expect. Taking for example the well-known substrate Si with its surface film SiO, (n = 1.46) having a thickness of 20 A, the sole error in @ of + 0.01” { - 0.01”) drives the index to 1.08 (2.22) and the thickness to 68 w {17 A} at the angle of incidence of 70” and under the wavelength of 6328 A. We present in this paper an inversion procedure in which the angle of incidence, whose angular error modifies largely the parameter solution, is also regarded as an unknown parameter in addition to the index and the thickness of the film. With the angle of incidence and the wavelength fixed, we consider two absorbing substrates overlaid with the same transparent film but having different thicknesses. The assumption made here, that the film’s index of refraction is independent of the thickness, is reasonably valid and verified in many practical situations: polishing and cleaning of substrate surfaces [ll], plasma-assisted [12] or chemical etching [13] of films grown on substrates. However, this assumption may be in its limit of validity at the .initial stages of film growth where its morphology is continuously changing or in the cases of film materials whose microstructure evolves significantly with the thickness. The method as formulated in the following section consists in a two-dimensional numerical search using the Newton-Ralphson technique for a fourparameter problem. Supposing that the substrate optical constants n, and k, are accurately known, the index n and the angle of incidence @ are found at first independently of the two thicknesses d, and d, which are thereafter directly computed. A quantitative error analysis is then carried out by simulating all combinations of experimental errors in ?P, A, n,
Surface Science 90 (1995) 251-259
and k, in order to verify the validity of the parameters thus obtained and to know the conditions under which the uncertainties in the parameters are minimized. The validity of the method is further investigated in situations where the two films have slightly different indexes and are not measured exactly at the same angles of incidence, and where the single uniform layer supposed to represent the films is not enough adequate. The entire procedure is finally illustrated using simulated and experimental data of known reflecting systems. It is thus shown that thicknesses less than 20 A, such as those of surface films, as well as the angle of incidence at which the reflecting system is measured, can be obtained with reasonable accuracy in the presence of angular errors in the order of a few hundredths of a degree.
2. Formulation of the procedure The basic ellipsometric equation pJn, di, ns, ~3) = tan(!Pi:-)exp( j Ai), where pi is the ratio between the “p” and “s” overall complex-amplitude reflection coefficients of the ith film-covered substrate structure {i = 1, 2), ?Pi and A, are the ellipsometric quantities measured on the ith structure and fls (= it, + jk,) is the substrate complex index, can be put in the form: AiqF + Biqi + Ci = 0,
where i = 1,2.
(1)
The complex variables Ai, Bi and Ci are functions of the physical parameters II, rrs, the angle of incidence @ and the ellipsometric angles measured, Pi and Ai. The complex phase shift vi experienced by the light wave as it traverses to and fro between the interfaces, is given by
I,
(2)
where A is the wavelength and Gli the refractive angle at the ambient/film interface of the ith structure. Identifying, respectively, the modulus and phase of Eq. (2) with those of the roots of Eq. cl), once these roots are represented in the polar form with
T. Easwarakhanthan
et al. /Applied
modulus ri and phase pi for the two structures considered, leads to the following equations [3,7]: fii = ln( ri( n, @)) = 0,
for i = 1,2,
(3)
fZi = &( n, CD)+ ei( n, di, @) f 2Xli = 0 for i= 1,2,
(4) where ei(n, di, @I = (4TdJh)n COS(@~~) is the principal value of the phase shift of vi and 1; is an integer constant that takes into account the periodic behaviour of vi described by Eq. (2) and is initially estimated from the approximate values of the two thicknesses. We have now from Eq. (3) a non-linear system of two equations {i = 1 and 2) from which the refractive index n and the angle of incidence @ can be found using the Newton’s iterative procedure: [Xl,=[Xl,-r
-K[/(Xm-l)]-l[F(X,_l)], (5)
where [X]=[n@lT, [Fl=[f,,, fizlT and K is the Levenberg’s parameter which helps in reducing larger Newton steps computed at some subsequent iteration. The partial derivatives intervening in the Jacobean [j] can be found in several references [14,15]. The index n and the angle @ are thus found to a relative accuracy of O.OOl%,converging from a wide range of initial guesses, within a very few seconds in a coprocessed 386SX microcomputer programmed in Turbo Pascal. @ is experimentally known and can be initially well-fed. Now, the two thicknesses di {i = 1, 2) can be directly computed from Eq. (4): d,
=
1
_A
PiCny @)f
2T1i
4-xn cos( Gli)
,
for i= 1,2.
(6)
That multiple solutions are seldom generated is probably due to the reduction of the four-dimensional problem (n, @, d,, d2) to that of two dimensions (n, @>.
Sutface Science 90 (1995) 251-259
253
evaluate the exact uncertainties in the four parameters due to the experimental errors in the angles *r, A,, qz and A,, and to the uncertainties in n, and k,, bearing in mind the highly non-linear nature of the ellipsometric equations. We assume here that the two sets of angles measured {PI, A,) and I!&, AJ are completely independent, which suits besides the situation where the random experimental errors predominate. These errors, which cannot be easily compensated for, occur in the search of minimum intensity, in reading scales and as a result of parasitic beam or beam deviation. On the other hand, the systematic errors, which are in general the apparatus alignment errors, can be more or less compensated for either by physical or numerical alignment [93. These random errors could build up in each of the above six quantities q,, A,, p2, A,, n, and k, in either sense (L-) and consequently there will be 64 (= 26) equally probable solutions for n, @, d, and d, as these six variables could combine in 64 different ways. The maximum uncertainty in each of the four parameters occurring for one or several combinations, is retrieved among these 64 possible solutions. These solutions are found for any given two structures with different film thicknesses: two experimental sets of ly and A corresponding to the two thicknesses are simulated first by generating them at an incidence @ and then adding to them the errors of ArY= f 0.01” (or O.l”), A A = f 0.02” (or 0.2”). The substrate optical constants are also modified according to their uncertainties + An, (0.003) and f Ak, (0.002). As an illustrative example, the following case study is conducted and discussed in the next sections: The standardoreflecting system: Air/SiO,/Si[lOO] at A = 5461 A and @ = 70” with n(Si0,) = 1.460, I(, = 4.086 + 0.003-j 0.0310 f 0.002 [16]. The thickness d, is varied from 20 A to 1200 A while the other thickness d, is fixed at 10 A.
3. Error analysis The uncertainties in the unknown parameters due to the experimental errors in the measured quantities have been generally estimated either by applying the first-order Taylor series approximation of the functions involved at the solution point [5,10,15] or using statistical methods [2,9]. Instead here, we choose to
4. Results and discussion 4.1. Propagation of experimental errors Fig. 1 shows that the maximum possible uncertainty in each of the parameters n, @, d, and d,
254
T. Eawarakhunthan
et al. /Applied
corresponding to the experimental errors of A?@= kO.01” and AA = f 0.02”, Jrecomes less t$an 10% as d, gets thicker than 100 A. Above 200 A for d,, the uncertainty in d,(= 10 A) turns around 10% (* 1 A> even with the errors more important: A?P= f 0.1” and AA = f 0.2”. In particular, the uncertainty in the angle of incidence remains around 0.03% (* 0.02”) and increases to O.ll%(* 0.08”) when the angular errors in W and A are tenfold greater. This result can be used to verify the experimental setting of the angle of incidence. These results can be again seen in the Table 1 which gives the maximum relative uncertainty obtained in each of the parameters with the angular errors of A!P = f0.01” [ f O.l”], A A = f 0.02” [ f 0.2’1 for two different reflecting systems. An, and Ak, are taken f0.003 and f0.002, respectively 1161.Foreeach of these systems, film thicknesses of d, = 10 A and d, = 400 A are considered. One sees
Uncertainty
(%)
‘““!:I=I_
0.0 1 0
200
400
800
800
1000
1200
Thickness d2(A) Fig. 1. Variation of the maximum relative uncertainty in the parameters ?, @, d, and d, with the thickness d, while d, is ped at 10 A, reflecting system considered is SiO, /Si: A = 5461 A, 0 = 70” with n(Si0,) = 1.46, ns = 4.086 kO.003 -j 0.031 f 0.002 [16]; AP = f 0.01” and A A = f 0.02“; the uncertainties An’, Ad’,, Ad, and A@’ correspond to AtY = rtO.1” and AA= f 0.2”.
Surface Science 90 (1995) 251-259
Table 1 The maximum relative uncertainty in the parameters n, @, d, and C$ for two reflecting systems with fifferent film indexes: d, = 400 A and d,=lO A, at A=5461 A, @=70” with n,=4.086f 0.003 -j 0.031~0.002 [16] Structures
Maximum relative uncertainties (%) AP, AA (deg.)
n
An
A@
Ad,
Ad,
SiO, /Si
0.01, 0.02 0.1,0,2
1.46
0.2 2.0
0.03 0.11
3 11
0.4 3.2
Thermal oxides/Si
0.01, 0.02 0.1,0.2
2.8
0.1 0.3
0.03 0.11
3 10
0.2 0.7
1131
that all parameters are found with reasonable uncertainties with the experimental errors specified above. Although An is under 2%, it is found to be more perturbed by the errors than the angle of incidence @. The method generates larger uncertainties in the parameters searched only in cases where d, approaches d, or when these two thicknesses are equally small. The above conclusion can be again percejved in Fig. 2 where the thickness d, is fixed at 200 A and the thickness d, is increased up to 1200 A. The uncertainties in the parametersdncrease substantially when d, passes through 500 A. Figs. 1 yd 2, and Table 1 show that very thin layers (N 10 A) can be detected with an uncertainty of 10 to 20%, depending on the value of the other thickness ( > 100 A) even with ellipsometers producing angular errors in the order of 0.1”. Also, one can easily deduce that the maximum uncertainty in d, is minimized when this thickness lies in the midway neighbourhood of the other Cd,). In the classical method [3-71 generally used, one determines first the refractive index of the film from a set of F and A, measured at an angle of incidence before finding its thickness from the index computed. Now, in order to compare the results obtained here with those found from the classical method, ?P and A are generated at @ = 70” for the reflecting system taken in the above case study for the film thicknesses d, and d, of 20 and 40 A, respectively. The experimental values ?P and A are simulated by adding the angular errors of +O.Ol” and +0.02’, respectively, to the values thus generated.
T. Easwarakhanthan et al. /Applied Surface Science 90 (1995) 251-259
Uncertainty
(46)
loo-
-
Ad2
*
--A@
Adz’
. .
0.01 0
/
I
/
/
200
400
600
600
Thickness
I 1000
1200
d2(A)
Fig. 2. Variation of the maximum relative uncertainty in the para?eters d, and @ with the thickness d, while dIOis fixed at 500 A, for the reflecting system SiO, /Si: h = 5461 A, @ = 70” with t&O,) = 1.46, n, = 4.086 f 0.003 -j 0.031 f 0.002 [16]; Aly = f 0.01” and AA = f 0.02”; the uncertainty Ad; corresponds to AYI = +O.l” and A A = 50.2”.
Table 2 presents the indexes and the thicknesses obtained by the two methods. The method put forward here finds accurate values for the four parameters, which approach their exact values even for thinner films, showing thereby that it is much more resistive to the propagation of errors in the measured quantities to the parameters found. Moreover, it should be noted that the results obtained using the classical method are strongly perturbed by the angle of incidence error especially when the films are
25.5
thinner [lo], whereas the very angle of incidence is also determined by the procedure proposed here. The effect of this angle of incidence error has not been taken into account above in obtaining the parameters by the classical method. The substantial reduction in the uncertainties in the parameters found by the method proposed here may be attributed to the equal determination of the angle of incidence whose error significantly shifts the parameter solutions in the classical method. On the other hand, one could think of the structure with a higher film thickness fixing the uncertainties, knowing that in all cases the effect of the experimental errors diminishes with increasing film thickness. Moreover, the accuracy of the method can be well checked by comparing the measured and the computed angles of incidence. Alternatively, the angle of incidence measured by other means can be verified as the angle of incidence found here is less perturbed by the experimental errors. The other point is that one could evaluate accurately film thicknesses thinner than 10 A, if the thickness of the second film is chosen relatively greater than the thinner one. This may be of much practical interest in situations where one wants to know the thickness of native oxides on substrates or to study the effect of cleaning and etching of substrate surfaces. 4.2. Departure from the assumptions made The first assumption made to formulate the procedure is that the two films of different thicknesses required should have the same refractive index. The ellipsometric characterization carried out on the etching and growth of oxides on the Si substrate [13] shows that the index of thinner oxide films is slightly greater than those of thicker films. In order to per-
Table 2 Parameter solutions obtained by the classical method [3-71 and the method presented here; the errors of AY = + 0.01” and A A = + 0.02” are added to the rY and A values generated at A = 5461 A (n, = 4.086 - j 0.031) and @ = 70” Exact values
Simulated experimental values
Solutions from the classical method [3,7]
Solution from the method here n = 1.4596 @ = 69.996”
n = 1.460
‘J’= 12.0770”
n = 1.1710
d, =2OA n = 1.460
A = 172.8905” Cy= 22.6886”
d, = 37.2 A n = 1.4580
d,=4OOA
A = 100.5166”
d, = 400.8 .fi
d, = 20.0 A d, = 400.4 .&
256
T. Easwarakhanthan
et al/Applied
ceive how the results produced by this method are effected by the above departure from the assumption initially made, !P and ,$ data are now generated at @ = 70” and A = 5461 A (n, = 4.086 - j 0.031) for two reflecting structures having the following film parameters: n = 1.460,
d, = 400 A,
‘# = 22.6786”,
A = 100.4966”,
n = 1.480,
d, = 20 A,
P = 12.0685”,
A = 173.7514”.
Applying our method to the above two structures, we find: n = 1.4599,
@ = 70.0003”,
d, = 20.4 A,
d, = 400.0 A.
d, = 400 A,
‘P = 22.6786”,
A = 100.4966”,
@ = 70.02”, A = 172.8492”.
Using the method, the following parameter values are found: n = 1.4652,
@ = 70.020”,
d,=20.1
A,
d, = 401.4 A.
Contrary to the precedent results, where the index found is closer to that of the film with larger thickness, here the film with the smaller thickness imposes the angle of incidence at which it is measured. Here again, it can be seen that the two thicknesses computed are almost not changed by the difference between the two angles of incidence. Therefore, the method can be also extended to cases where a small difference between the two angles of incidence exists, bearing in mind, however, that the angle of incidence thus obtained will be that at which the thinner film is measured. 4.3. Departure from the model supposed
As ?P and A are generated up to the fourth decimal place, the small deviation of + 0.0003” in @ should be originating from the difference in the index values of the films. The structure that has the larger film thickness (400 A> tends to force its index value of 1.46 as the equivalent index for the two films. If not, the two thicknesses are almost not effected by the above index difference. These conclusions remain valid as long as the two indexes are not that different and one of the film thickness is relatively higher than the other. The other assumption made is that the two structures are measured at the same angle of incidence setting. However, a small shift in the angle of incidence during the replacement of the structures may not be completely discarded. In order to study the effect of a small difference between the two angles of incidence on the parameters obtained, !P and A data are now gener$ted with @ = 70.00” and @ = 70.02“ at h = 5461 A (n, = 4.086 + j 0.031) for two reflecting structures with the same film index 1.460, and having the following film parameters: @ = 70.00”,
Surface Science 90 (1995) 251-259
d, = 20 A,
‘P = 12.0353”,
The two structures needed are represented here by the three-phase model (ambient-film-substrate) of which the film is supposed to be uniform. Several studies [13,17] show that there exists, at the interface between the substrate and the film, another very thin layer having an index of refraction different from that of the bulk oxide layer. In a recent work [17] concerning the native and tkermally grown oxides on Si, a thickness around 12 A has been presumed for this interface layer while assuming a film index of 2.5. In these cases the structures are adequately represented by the double-layer model. In order to know how this thin interface layer modifies the results obtained when applying the method proposed here to the ?P and A data of double film structures, the latter data are generated for the two structures at @= 70.00” and A = 5461 A (n, = 4.086-j 0.031). The index n, of 1.46 and the thicknesses d,, = 20 A and d,, = 400 A are taken for the upper oxide films and the corresponding values for the same interface layer (lower fil$ of the two structures are n, = 2.5 and d, = 12 A, respectively. Table 3 presents values of P and A generated for this double-layer model and the corresponding parameters found by the procedure. The interpretation of the parameters thus obtained is as follows: the “equivalent” refractive index of the simple layer representing the double layer of the
T. Easwarakhanthan
et al. /Applied
Table 3 Parameter solutions found by the method ptesented when applied to Y and A values generated at h = 5461 A and @ = 70” for the double-layer model representing the two structures Exact double-layer values
Generated Cy and A
Parameters obtained
nU = 1.46, d, = 20 A
rY = 12.1351”
n = 1.469
n, = 2.5, d, = 12 A
A = 168.7508”
@ = 70.013”
nU = 1.46, d, = 400 .&
9 = 23.1054”
dC = 33.3 ;i
n, = 2.5, d, = 12 i
A = 99.0197”
dz = 409.3 A
5. An experimental example As an experimental example, some of the ellipsometric measurements presented by Kao et al. [17] are taken to work out the method elaborated. These measurements [17] have been carried out on silicon samples of which the oxides are formed in oxygen at different elevated temperatures. We repeat in Table 4, for the convenience of reading, the quantities ?P and A measured by the same ellipsometer at an angle tf incidence of 70” and under the wavelength 6328 A on these samples which are named here A, B, C and D. The most thick and thin film samples have been chosen here as it is known from the error
solutions
for n, @, d, and d, and their uncertainties
Samples
n
DandA D and B DandC C and A C and B
1.47 1.47 1.42 1.48 1.47
f f f + f
0.03 0.03 0.17 0.03 0.03
257
Table 4 The ellipsometric data ly and A taken from Ref. [17]: data measured on Si samples named here A, B, C and D, at @ = 70” and A = 6328 A (n, = 3.858 -j 0.028); the film thicknesses are obtained from transmission electron microscopy @EM) and ellipsometric data supposing a film index of 1.46 [17] Samples
two structures considered is closer to that of the film with the larger thickness belonging to the upper film of these two structures. This is in agreement with the earlier conclusion where the film with higher thickness tends to force its refractive index as the “equivalent” index when the two films considered have somewhat different indexes. Also, the two “equivalent” thicknesses computed here may be viewed as the sum of the upper and lower film thicknesses of the double-layer model representing the reflecting systems.
Table 5 Parameter
Surface Science 90 (1995) 251-259
A B C D
Experimental data [ 171
From the methods in Ref. [17]
ly
A
(deg.)
(deg.)
d 6) @EM)
d (A) with n=1.46
10.53 10.68 16.43 17.24
168.86 165.90 114.73 111.61
28 37 300 324
33 50 296 325
analysis that a larger difference between the thicknesses of the two films reduces the propagation of experimental errors to the parameters determined. This table also presents the thicknesses of the corresponding samples obtained by the same authors [17] by transmission electron microscopy (TEM) and the ellipsometric data assuming a uniform oxide layer with refractive index 1.46, Table 5 gives the parameter values found by our procedure with their uncertainties estimated from the error analysis supposing A?P = 0.05” and A A = 0.1” [17], for different combinations of the ellipsometric data of the four samples chosen. Only the data combination of the samples D and C brings out inconsistent parameter values deviating from the would-be real values. This result can be foreseen as the nearness between the two thicknesses of the samples D and C should consequently lead to larger uncertainties in the parameters, as shown in the error analysis. For the other data combinations of the samples, the method finds almost the same film thickness for one particular sample, whatever the other sample
found in this work for the ellipsometric
data given in Table 4
d, &
db 61
d, 6)
dd CA)
@ (deg.)
34 f 1 _ 34 f 1 -
45 f 1 -
309 f 59 293 f 9 294* 9
321 f 9 321+ 9 336 f 65 -
70.001 70.009 70.14 70.001 70.008
44*1
+ f f + f
0.03 0.03 0.44 0.03 0.03
258
T. Easwarakhanthan
et al. /Applied
with which it is combined. For example, the same thickness of 34 A has been found for a film of sample A from both the data of the pairs D and A, and C and A. The thicknesses d, and d, of the thinner film samples A and B thereby found are still greater than those deduced from the TEM measurements [17]. The authors proposed [17] a double-layer model, representing the interface region by a thin layer with refractive index 2.5 and thickness 12 A, and the bulk oxide region by a layer with refractive index 1.47, with its thickness deduced from the TRM measurements. They found that the 9 and A computed using this double-layer model agree well with those experimentally measured. Looking again at Table 5, the two thicknesses d, tnd d, are obtained with an uncertainty around + 1 A which originates from the experimental errors in YP and A. Therefore, any difference between the thickness values obtained here and those deduced from the TEM measurements should come from the representation by the single-layer model of a structure which may be better described by a double-layer model with a thin film layer at the interface. It has been noted earlier how the parameters obtained here may be perceived in this situation; the two “equivalent” thicknesses d, and d, computed should be around the sum of the upper and lower film thicknesses of the two structures: the single-layer “equivalent” refractive index of 1.47 f 0.03 obtained here should be closer to that of the thicker bulk oxide layer. Finally, it can be seen that the values obtained for the angle of incidence are in very good agreement with the one measured in Ref. [17].
Surface Science 90 (1995) 251-259
by its very determination. The method is rapidly convergent due to the reduction of the four-parameter problem to a two-dimensional numerical search in which the refractive index and the angle of incidence are found at first independently of the two thicknesses that are directly computed afterwards. The validity of the method is investigated when applied to situations where the conditions differ slightly from the a priori assumptions made to formulate the method. The parameter values found by the procedure in these situations are thus interpreted. Besides, the quantitative error analysis carried out shows the ability of the method to resist the propagation of the experimental errors to all parameters searched, particularly to the smaller thickness d, when d, is chosen sufficiently greater than the other. The method thereby0 allows one to extract thicknesses thinner as 10 A from the ellipsometric angles measured by an automated ellipsometer that induces angular errors in the order of a few hundredths of a degree. This may be of much practical interest in detecting the presence of naturally occurring surface films or in studying the effect of chemical cleaning of the surface films formed on substrates. Furthermore, as the uncertainty in the angle of incidence due to the experimental errors remains relatively small under the above condition, its value thus found may therefore be used to check its measured value in a particular ellipsometric setting.
References
I11R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
6. Conclusions A new procedure has been developed in order to determine unambiguously the thickness of ultra thin films, such as native, etched or chemically cleaned surface films, from the ellipsometric measurements on two reflecting systems having different film thicknesses. The procedure simultaneously finds the angle of incidence whose experimental error modifies largely the parameter solution in the classical method. The propagation of the error in the angle of incidence to the parameters sought is avoided here
121F.L. McCrackin, A FORTRAN Program for Analysis of Ellipsometer Measurements, Technical Note 479 (National Bureau of Standards, US Government Printing Office, Washington, DC, 1969). [31 A.R. Reinberg, Appl. Opt. 11 (1972) 1273. [41 Y. Yoriume, J. Opt. Sot. Am. 73 (1983) 888. iSI D. Charlot and A. Maruani, Appl. Opt. 24 (1985) 3368. [61 E.E. Dagmann, Phys. Chem. Mech. Surf. 3 (1985) 1954. [71 T. Easwarakhanthan, C. Michel and S. Ravelet, Surf. Sci. 197 (1988) 339. k31H.F. Wei, A.K. Henning, J. Slinkman and W.R. Hunter, J. Electrochem. Sot. 139 (1992) 1783. 191 K Riedling, Thin Solid Films 75 (1981) 355. [lOI D. Chandler-Horowitz, Integrated Circuit Metrology, SPIE 342 (1982) 121.
T. Easwarakhanthan et al. /Applied Surface Science 90 (1995) 251-259 [ll]
A.C. Adams and B.R. Pruniaux, J. Electrochem. Sot. 120 (1973) 408. [12] M.C. Plowers, R. Greef, C.M.K. Starbuck, P. Southworth and D.J. Thomas, Vacuum 40 (1990) 483. [13] E. Taft and L. Codes, J. Electrochem. Sot: Solid-State Sci. Technol. 126 (1979) 131.
259
[14] H. Dupoisot and J. Morizet, Appl. Opt. 18 (1979) 2701. [15] T. Easwarakhanthan, P. Mas, M. Renard and S. Ravelet, Surf. Sci. 216 (1989) 198. [16] D.E. Aspnes and A.A. Studna, Phys. Rev. B 27 (1983) 985. [17] Shou-Chen Koo and R.H. Doremus, J. Electrochem. Sot. 114 (1994) 1832.