Multivariate analysis of noise-corrupted PECVD data

E L S E V I E R Thin Solid Films 307 (1997) 65-70

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Multivariate analysis of noise-corrupted PECVD data

A. yon Keudell *, A. Annen, V. Dose Max-Planck-lnstitut fih" Plasmaphysik. Boltzmam~str. 2, 85748 Garching, Germany

Received 6 January i997; accepted 21 May 1997

Abstract

A prominent goal of plasma enhanced chenzical vapor deposition (PECVD) is the description of solid film properties such as, for example, refractive index, carbon content, etc., in terms of input variables such as, for example, gas pressure, substrate temperature and others. This goal is complicated because of the high dimensions of the input and response variable space. In this paper, we analyse data from C:H film deposition with proper account of measurement errors. We show that errors reduce correlations, and that the whole data set can be well described by a multivariate normal distribution. © 1997 Elsevier Science S.A.

Keywords: Deposition process; Hydrocaibons: Plasma processing and deposition

1. Introduct ion

The deposition of thin films by plasma-enhanced chem- ical vapor deposition (PECVD) is an important technique for the preparation of new materials [1]. Different combinations of input variables such as gas pressure, absorbed power in the plasma, substrate temperature and ion energy result in different combinations of response variables such as density, index of refraction, etc. Material tailoring re- quires the knowledge of the influence of an individual input variable on the film properties with the other input variables held fixed. An experimental uniform scan of the input variable space becomes progressively prohibitive as the dimension of the input variable space increases. It may not even be possible in every case. This raises the question of how to describe the relation between input and response variables from a finite set of measurements with arbitrarily scattered input variable vectors. Assuming the simplest possible relation, namely a multilinear one, we show that all the necessary information is condensed in the data covariance matrix [2]. We show that the least informative multivariate distribution, given a data covariance matrix is multivariate normal. We shall derive from this distribution the desired partial correlations between a specific input

* Corresponding author.

0040-6090/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0040-6090(97)00313- 1

and a specific response variable with the other input variables held constant. We then show how the data covariance matrix must be modified when proper account of experimental en'ors is taken. The final result provides a satisfactory description of the data we analyse, and may therefore be used for predictive purposes.

2. Exper imenta l

A radio frequency (RE) plasma is used for the deposition of C:H films from methane [3]. An RF field of 13.56 MHz was applied at the electrode, yielding a DC self-bias (UsB) ranging from - 9 0 to - 3 7 0 V. This DC self-bias determines the "kinetic energy of the impinNng ions during growth of the films. The gas pressure was varied from 0.5 to 8 Pa and the gas flow from 10 to 40 sccm. The silicon substrates were placed on the RE electrode and the substrate temperature was varied from room temperature up to 500 K by heating the electrode and controlling the substrate temperature by a pyrometer. The growth was moni- tored in situ by a He-Ne interferometer and the resulting film thickness was measured with a profilometer. With the "known film thickness, the growth rate can be calculated. The refractive index was determined by the optical modelling of the interference pattern of the He-Ne interfero- gram. The film composition was measured by ion beam analysis, PES (proton enhanced scattering) for the areal

66 A. yon Keudell et al,/Thin Solid Fihns 307 (1997) 65-70

density of carbon and ERD (elastic recoil detection) for the areal density of hydrogen [1]. With the "known film thickness, the total carbon and hydrogen density can be calculated from the areal density. The resulting data set for the measured film properties and process parameters is shown in Table 1. As four input variables--the gas pressure, the gas flow, the subsu'ate temperature and DC self-bias--were varied. These four input variables control the four response variables, namely growth rate, refractive index, carbon density and hydrogen density. The last row in Table 1 contains the experimental errors of each variable. The last column shows a comparison between the measured response variables and the response variables, predicted by multivariate analysis for given input variables.

3. M u l t i v a r i a t e ana lys i s

We denote by 3 = ( X 1 . . . X M) the set of input and response variables for a single measurement. If N such measurements are available, the (i,k)-element of the covariance matrix e is defined by:

1 N = = - - £ % cov(x,,xk) N - l j = t

1 N with 2, = -- £ x,j (1)

N j = t

The dimensionless correlation matrix r is given in terms of e by:

e#k r ,k= $ , ~ (2)

The elements of r are restricted to - 1 < r,~ < 1. The analysis would terminate here if the correlation coefficients between any two input variables would vanish. This is, however, hardly ever the case. We obtain from Eq. (2) the partial correlation rsj, k between x, and xj with the xz. kept fixed:

ru - r'xrJk (3)

ris'k = 7(1 - rS)(1 - r j )

Given the above defined covariance matrix Cik the least informative multivariate distribution function f(2') may be obtained by the principle of maximum entropy [4]. Let S be

s - - - f s ( Z ) l n ( f ( Z ) ) a ' x (4)

We then maximise the entropy S subject to the condition that:

= f x, x y ( 3 ) d"r (5)

Introducing the Lagrange parameters Ask we maximise the functional:

1 4, = - f i ( etln(S(3))d 'x + 7 EA,

¢,,k

the variational solution of this problem gives:

f (x ' ) =f0exp ( - ~£.r4aT) (7)

with A = {A,k}. Since the covariance matrix e is symmet- ric, so is A. Normalization of f (£ ) leads to:

M

= (27r) T(det A) -z /2= Z (8)

given f(.~') the covariances are:

f xix~f( £) d~x 2 c,~= = z = ( A - ' ) ~ , (9)

f f ( 3 ) dMx Z 6A,k

the multivariate distribution f(x-*) is thereby fully speci- fied:

f(K') = (2~ ' ) - Ml2(det C - 1 ) I / 2 e x p ( - l ~ ' r C - ~ ' ) (10)

From Eq. (10) the partial correlation coefficients between any two variables in f(~') with a particular variable kept constant may be derived using the product rule of probabil- ity theory:

f ( xs . . . XM) = f ( . rk ) f (x~ , j . kiXk) (1 1)

f ( x k) is the marginal distribution of x~ defined as

x,) = f / ( ~ ) d x M-' (12) i (

The analysis so far assumed that the data correlation matrix were given free of experimental errors. This is a hypothetical case and we now introduce the modifications necessary for proper account of experimental errors. Let o-ik be the root mean square error of c o v ( x , x k) = c~k. o-ik is obtained from the errors in the specifications of the measurements of xi, x k, A x i and A.rk, respectively. From straightforward error propagation we obtain:

1 N - 1 2 - + (13)

The functional Eq. (6) is then replaced by

1 1 , o~ - - Ea, q9 = o~S 2 . . O'i- k "-2 ik

x(C,~- f x,xkf(Z)d'~'x) (14)

A. ~,on KeudeIl et aI. / Thin Solid Films 307 (1997) 6 5 - 7 0 67

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68 A. yon Keudelt et al./Thin Solid Films 307 (1997) 65-70

V > ,

rxy=O. 8 9 9 4 o o 0

0

0 0 0 ~ 0 0

0 0 0 0 0 0 0

0 £, o~

0 o 0 0 0

cov(x,~ x y x 0.2002 0.1643 y 0.1666

,'-'-= rxy=O'7421 ~ ~

>,

~- ~ l cov(x,y) x y ~ - - - ~ v~"TiZ=~ ' { x 0.2211 0.1499

;. ~ y o.1845

x (a.u.) Fig. i. The correlation coefficients for an arbitrary error-corrupted data set x,y. (a) correlation coefficient according to the normal multivariate analysis. (b) correlation coefficient with proper account of data errors.

these c~.~ are the elements of the data covariance matrix now assumed to be uncertain by cr~ and C~k. the covariance matrix elements calculated from the multivariate distribution function f (£ ) . a is a hyperparameter, which determines the deviations between e~ and C~k, and will finally be adjusted such that

( c i J - C ' j ) 2 = M 2 (15) M E i,: o-q

variation of q5 with respect to f ( £ ) yields:

which is formally identical to Eq. (7), the previous case. Differentiation of • with respect to C~k yields:

o~o',2 hik = 2(Cik -- cd, ) (17)

and differentiation with respect to h,k

c , , = f x,x,f( ~)dx ~' (is)

From the previous analysis we know that

C~k = ( A - ' ),k (19)

and eliminating C from Eq. (17) we finally obtain

A = eik + -~ao ' ,~h~ (20)

which simplifies to the error-free case for oe ~ 0. For a particular data set Eq. (20) is solved for a choice of oe such that Eq. (15) is fulfilled. In the following, the matrix C is then given by A - ~. Since the only difference between the ideal and the error corrupted case is in the actual values of the Lagrange matrix A, the conclusions drawn from f(~'), in particular, the expressions for the partial correlation coefficients remain unaltered.

As an example, the correlation coefficients for arbitrary data vectors x and y with errors A x and A y are calculated with the presented formalism. The data set y(x) is plotted in Fig. 1. In addition, the elements of the covariance matrix C are shown in Fig. 1. The correlation coefficient is calculated from the covariance matrix by Eq. (2). In Fig. la, the correlation between x and y are determined by the normal multivariate analysis yielding r~v = 0.8994. In Fig. lb, the same data set is analysed by the presented algorithm, yielding a correlation coefficient rx:, = 0.7421. It can be seen that the diagonal elements of the covariance matrix are increasing, and the cross elements are decreas- ing due to consideration of measurement errors in the multivariate analysis. Thus, the correlation coefficient has to decrease according to Eq. (2). This result can be illus- trated by the following. The decrease of the correlation coefficient due to the proper account of data errors implies that the linear dependence between x and y is no longer favoured compared to the error-free case.

4. Multivariate analysis of error-corrupted PECVD data

Multivariate analysis without proper account of experimental errors is applied to the PECVD data from Table 1. From Eqs. (2) and (3), the partial correlation coefficients are calculated and shown in Table 2. It can be seen that the deposition rate increases mainly with pressure, because the particle flux towards the substrate increases at higher plasma pressures. It increases with the ion energy due to synergism between the ion bombardment and the incorpo- ration of neutral precursors [5]. The deposition rate decreases with increasing substrate temperature due to reetching with atomic hydrogen [6]. The ion energy con- trols further the other response parameters as carbon/hy-

Table 2 The partial correlation coefficient for the variation of a single input variable with the other input variables held fixed

Growth rate Refractive C-atoms x H-atoms × index n 10 -'2 cm -3 I022 cm -3

p 0.9378 Gas flow UsB 0.9239 0.7187 0.5524 -0.7506

Tsubstrat e - - 0 . 5 0 1 8 -- 0 . 5 8 7 6

Correlation coefficients lower than 0.5 are omitted in the table.

A. yon KeudelI et aI. / Thin Solid Films 307 (1997) 6 5 - 7 0 69

Table 3 The partial correlation coefficient with the proper account of data errors for the variation of a single input variable with the other input variables held fixed

Growth rate Refractive C-atoms × H-atoms × index n 10 22 cm -3 10 22 cm -3

p 0.9202 Gas flow UsB 0.9017 0.465i 0.5108 -0.7499 Z s u b s t r a t e - - 0.2707 - 0.4835

Correlation coefficients lower than 0.5 are omitted in the table.

drogen content and refractive index. The hydrogen content decreases with increasing ion energy, due to the preferen- tial sputtering of bonded hydrogen by ion bombardment [7].

The modified multivariate analysis, as presented in the last section, is also applied to the PECVD data from Table 1. The covariances c,k are calculated by Eq. (1) and the errors of the covariances are detennined by Eq. (13). With this result, the covariance matrix C is calculated by the solution of Eq. (20) and a hyperparameter oe according to Eq. (15). With this covariance matrix C, the correlation coefficients are calculated by Eq. (2), and the partial correlation coefficients according to Eq. (3) are determined. The results are shown in Table 3. In general, all partial correlation coefficients decrease compared to the normal multivariate analysis in Table 2. The largest changes occur for the partial correlation coefficients between substrate temperature and growth rate, and between refractive index and UsB. This shows that an interpretation of the sampled data without the proper account of experimental errors can give misleading correlations between individual variables.

5. Prediction of response variables from input variables by the multivariate analysis

The multivariate analysis is a powerful technique to identify linear dependencies between data vectors. How- ever, this type of analysis is inappropriate for strong nonlinear data sets. This raises the question whether the multivariate analysis describes, in our case, the PECVD data in Table 1 corl"ectly. To verify this, the response variables are calculated from the covariance matrix for a given set of input variables. A difference between the calculated and measured response variables, which is in the error range of the data, implies that the multivariate analysis is capable to describe the data set.

For this calculation, the maximum of the simplest distri- f(~. --', bution f , 3') for a given vector of input variables ~" is

determined by the variation of the vector Y~ne~r of the response variables. For this purpose the matrix C -1 is

divided into the submatrices U 0 ̀ W 0 and V. With this transformation Eq. (10) gives:

1 f(z,y ) =foexp(- 22'rc-I~) = f( z, y )

2-1(5') '1 . . . . . ) I V T W0 V Y'+Ii n e a r 7" ~ = f 0 e x p

(21)

The maximum of the distribution function f(~+, ~) for given input variables g yields for Y'h~ar:

~near = - W o Ivr~ ' (22)

The vector Yi~near represents the prediction of the multivariate analysis on the basis of the covariance matrix and a ~ven input vector ~'. This is compared to the original data ]~(Yl . . . . . Yk) by a least mean square expression:

1 ~ ( Y h n e a r , , - - y , ) 2 X 2 = 7" - - 7 (23) i=1 A ) ' 7

ky, represents the error of the response variable 3;. X e can now be determined for the data set with and without considering experimental errors, yielding X~ and X2, respectively. This is achieved by using the covariance matrix as derived from Eq. (1) for X~ and the en'or corrected covariance matrix from Eq. (20) for X~. The results are also shown in Table 1. A value lower than 1 indicates that the predicted data Yiinear are consistent with the original data in range of the data error A y,. Only very large values indicate that the linear modelling is not capable to describe the measured data. The X: remain almost unaltered, if the input vector ,. is varied in the range of the data errors zXz,. It can also be seen that there is no significant difference between X~ and X~ indicating that both methods can predict the data equivalently. It should also be mentioned that the consideration of experimental errors in the multivariate analysis must not lead to a reduction of Xe since X2 describes only whether the data set can be described by a linear model at all. This type of multivariate analysis is not a method to obtain a better fitting of the data. Instead, the main result of considering experimental errors in the multivariate analysis is the change of the partial correlation coefficients as discussed above.

The largest difference between Y'l,~ea~ and the original data f occur for films with the largest C-density of 8.8 × 102e cm -3 and for films prepared at the lowest DC-self bias of -89 .5 eV. As known from the deposition conditions of C:H films at very low ion energies [8], the dependence of the film structure and composition on ion energy is strongly nonlinear, and thus a linear model cannot describe this measurement.

On the other hand, this observed difference between measurement and predictions by the multivariate analysis

70 A. yon Keudell et al. / Thin Solid Films 307 (1997) 65-70

can serve as a tool for the identification of deposition conditions, which results in strong nonlinear response variables due to physical characteristics of the system under investigation or due to mistakes in the performance of the experiments.

Summarizing, almost all measurements in Table 1 are consistent with the predictions of the multivariate analysis. This justifies the application of this analysis for this pa- rameter space of PECVD data.

6. Conclusion

It has been shown that the multivariate analysis can be a very powerful tool in the quantification of PECVD data, but the consideration of the experimental errors of the data is essential for the analysis. Otherwise, misleading correlations between individual variables are obtained. Further the multivariate analysis can serve as a tool for the identifica-

tion of strong nonlinear dependencies in the deposition system.

References

[1] J.C. Angus, P. Koidl, S. Domitz, in: J. Mort, F. Jansen (Eds.), Plasma Deposited Thin Films, CRC Press, Boca Raton, FL, 1986, p. 89.

[2] W.J. Krzanowski. in: J.B. Copas (Ed.), Principles of Multivariate Analysis, Oxford Science Publications, Clarendon, 1988.

[3] A. Annen, R. Beckmann, W. Jacob, J, Non-Cryst. Solids 209 (1997) 240.

[4] J.N. Kapur, H.K. Kesavan, Entropy Opimization Principles with Applications, Boston Academic Press, 1992.

[5] A. yon Keudell, W. M~ller, R. Hytry, Appl. Phys. Lett. 62 (1993) 937.

[6] A. von Keudell, W. Jacob, J. Appl. Phys. 79 (1996) 1092. t7] W. M~511er, W. Fukarek, K. Lange, A. von Keudell, W. Jacob, Jpn. J.

Appl. Phys. 34 (1995) 2163. [8] W. Jacob, W. MNler, Appl. Phys. Lett. 63 (1993) 177I.

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Multivariate analysis of noise-corrupted PECVD data