Modeling of plasma process data using a multi-parameterized generalized regression neural network

Microelectronic Engineering 86 (2009) 63–67

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Modeling of plasma process data using a multi-parameterized generalized regression neural network Byungwhan Kim *, Minji Kwon, Sang Hee Kwon Department of Electronic Engineering, Sejong University, 98 Kunja-Dong, Kwangjin-Ku, Seoul 143-747, Republic of Korea

a r t i c l e

i n f o

Article history: Received 19 November 2007 Received in revised form 4 June 2008 Accepted 15 September 2008 Available online 23 September 2008 Keywords: Generalized regression neural network Genetic algorithm Plasma etching Statistical regression model Training factor Multi-parameterized spreads

a b s t r a c t For characterization or optimization process, a computer prediction model is in demand. A new technique, to improve the prediction performance of conventional generalized regression neural network (GRNN) of plasma process data was presented. Genetic algorithm (GA) was applied to optimize multiparameterized training factors of GRNN. To evaluate the technique, two data sets were collected from the etchings of silica and silicon carbide (SiC) thin films in inductively coupled plasmas. Both data sets called Data I and Data II were statistically characterized by means of 23 and 24 full factorial experiment plus one center point. The GRNN models trained with these data were tested with additional six and 16 experiments. A total of eight etch outputs were modeled and compared with conventional GRNN and statistical regression models. The five etch outputs comprising Data I include silica etch rate, aluminum (Al) etch rate, Al selectivity, profile angle, and DC bias. Data II consisted of three etch outputs, including SiC etch rate, surface roughness, and profile angle. Compared to GRNN models, GA-GRNN models yielded more than 40% and 15% improvements for all etch outputs comprising Data I and Data II, respectively. Similar improvements were also demonstrated with respect to statistical regression models. All these results reveal that a multi-parameterization of training factors and GA optimization is an effective technique to considerably improve the prediction performance of conventional GRNN model. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Plasma processes play a crucial role in depositing thin films or etching fine patterns. In order to manufacture plasma processes in a cost effective way, computer prediction models are in demand. Due to the complexity inherent in plasma systems, it is a complicated task to derive analytical plasma models. To circumvent this difficultly, empirical plasma models based on statistical regression techniques [1], or neural networks [2–9] have been constructed for a variety of manufacturing purposes. The neural network paradigms widely employed for plasma modeling may include the backpropagation neural network (BPNN) [10], radial basis function network (RBFN) [11], or generalized regression neural network (GRNN) [12]. Compared to the BPNN, utilizing the RBFN or GRNN is advantageous in that the training factors to be optimized are very small. RBFN models of plasma etching demonstrated an improved prediction over statistical regression models [8]. In GRNN modeling, it is expected that by adopting multi-parameterized training factors an improved prediction model might be achieved. One related model was once constructed by utilizing a random generator [7]. However, the optimization process was not system* Corresponding author. Fax: +82 2 3408 3329. E-mail address: [email protected] (B. Kim). 0167-9317/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2008.09.015

atic. A low prediction of GRNN is even expected as this is applied to a large size of training patterns, mainly due to a considerable number of training factors to be optimized. To overcome this limitation, a systematic GA-GRNN model was applied to model an etch rate of silicon oxynitride film in a C2F6 plasma [9] as well as a deposition rate of silicon nitride film in a SiH4–NH3–N2 plasma [15]. Here, the GA represents a genetic algorithm [13]. Compared to conventional GRNN (i.e., non-optimized) models, the GA-GRNN models demonstrated of more than 30% improvements [9,15]. However, the previous studies stated earlier are limited in that the improvement of GA-GRNN model was evaluated with only one single film characteristic in plasma chemistry. This limited case study is not enough to ensure the advantage of GA-GRNN model. In other words, the better prediction of GA-GRNN model over conventional GRNN ones should be supported by a large number of comparison results over a wide range of film characteristics. In this study, this concern is examined by extensively evaluating GAGRNN model with two sets of experimental data. The data were collected during the silica etching in CHF3–CF4 plasma and a silicon carbide (SiC) etching in NF3–CF4 plasma. A total of eight etch outputs are modeled. It should be noted that GA-GRNN models for these outputs are first constructed here. For comparison purposes, conventional GRNN or statistical regression models are also constructed.

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2. Experimental details

Table 3 The training data consisted of 24 full factorial experiment and one center point

Two data sets were collected to evaluate the presented technique. The first data set (Data I) was obtained from the etching of silica films during the manufacture of an optical waveguider. The equipment apparatus (Plasma Therm 690 ICP etch system) and test pattern fabrication were detailed in [14]. The process parameters and their experimental ranges are shown in Table 1. In Table 1, the gas flow rate ratio was defined as the flow rate of CHF3 divided by the flow rate of CF4. The total flow rate of gases, CHF3 and CF4, was set at 60 sccm and the flow rate of CHF3 was varied from 10 to 50 sccm. To characterize the etch process, a 23 full factorial experiment [16] was employed. In this modeling works, the resulting eight experiments with one experiment corresponding to the center point in the design were used to train GRNN models. The training data are shown in Table 2. Six experiments were additionally conducted to provide test data for model evaluation. A total of 15 experiments were therefore conducted. The etch outputs to model include silica etch rate, aluminum (Al) etch rate, Al selectivity, silica profile angle, and DC bias. Using scanning electron microscopy (SEM), the etch rates were measured. The Al selectivity was defined as a ratio of silica etch rate to Al etch rate. The profile angle denoted as anisotropy (A) was defined as

Source power (W)

Bias power (W)

Pressure (mTorr)

Gas ratio

Etch rate (nm/min)

Profile angle (°)

Surface roughness (nm)

900 900 900 900 900 900 900 900 800 700 700 700 700 700 700 700 700

150 150 150 150 50 50 50 50 100 150 150 150 150 50 50 50 50

12 12 6 6 12 12 6 6 9 12 12 6 6 12 12 6 6

1.0 0.2 1.0 0.2 1.0 0.2 1.0 0.2 0.6 1.0 0.2 1.0 0.2 1.0 0.2 1.0 0.2

437 20 403 23 202 10 174 14 56 450 33 304 32 184 30 175 19

80 89 83 89 80 83 83 83 84 83 86 88 82 82 80 84 86

0.567 0.431 0.402 0.461 0.561 0.447 0.534 0.404 0.738 0.463 0.531 0.321 0.423 0.556 0.611 0.576 0.527

A ¼ tan1



 2R ð Þ LU

ð1Þ

where U and L represent the widths of the original and etched patterns, respectively. The width of U was 20 lm. The DC bias was measured by using a voltmeter embedded in a radio frequency (rf) match network. Another data set (Data II) was collected during the ICP etching of SiC films in NF3–CH4 chemistry. The equipment and process of test pattern fabrication were detailed in [8]. The etching process was characterized by a 24 full factorial experimental design [16]. The parameters and experimental ranges are contained in Table 1. The gas ratio in Table 1 was defined as a ratio of the NF3 flow rate to the total flow rate, i.e., the sum of NF3 and CH4 flow rates. The total flow rate was set at 30 sccm in every experiment. The result-

Table 1 Experimental ranges of process parameters employed in statistical experimental design Data type

Process parameters

Range

Units

Data I

Source power Bias power Gas ratio Source power Bias power Pressure Gas ratio

100–800 100–400 0.2–5.0 700–900 50–150 6–12 0.2–1.0

W W – W W mTorr –

Data II

ing 16 experiments plus one experiment corresponding to the center point in the design were used to train the GRNN. The training data are shown in Table 3. The trained GRNN was tested with 16 experiments, not pertaining to the training data. The etch outputs modeled include etch rate, profile angle, and surface roughness. The etch rate and surface roughness were measured by using SEM and atomic force microscopy. The angle of profile sidewall was measured in the unit of angle. 3. Generalized regression neural network A schematic of GRNN is depicted in Fig. 1. As shown in Fig. 1, the GRNN consists of four layers, including the input layer, pattern layer, summation layer, and output layer. Each input unit in the input layer corresponds to individual process parameter. The input layer is fully connected to the second, pattern layer, where each unit represents a training pattern and its output is a measure of the distance of the input from the stored patterns. Each pattern layer unit is connected to the two neurons in the summation layer: S- and D-summation neuron. The S-summation neuron computes the sum of the weighted outputs of the pattern layer while the D-summation neuron calculates the unweighted outputs of the pattern neurons. The connection weight between the ith neuron

Input Layer

Pattern Layer

Summation Layer

Output Layer

X1 Table 2 The training data consisted of 23 full factorial experiment and one center point

S

Source power (W)

Bias power (W)

Gas ratio

Silica etch rate (Å/min)

Al etch rate (Å/min)

Al selectivity

Silica profile anisotropy (°)

DC bias (V)

800 800 800 800 100 100 100 100 450

400 400 100 100 400 400 100 100 250

0.2 5.0 0.2 5.0 0.2 5.0 0.2 5.0 1

5550 5812 4370 4380 830 945 280 280 2957

775 1062 180 320 90 110 60 30 457

7 5 24 15 9 9 5 9 6

74.0 76.0 88.7 90.7 59.9 75.6 86.9 67.7 77.1

400 412 128 113 547 533 218 218 333

X2

Y

D X3

Fig. 1. A schematic of generalized regression neural network.

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in the pattern layer and the S-summation neuron is yi, the target output value corresponding to the ith input pattern. For the Dsummation neuron, the connection weight is unity. The output layer merely divides the output of each S-summation neuron by that of each D-summation neuron, yielding the predicted value to an unknown input vector x as

Pn yi exp½Dðx; xi Þ ^i ðxÞ ¼ Pi¼1 y n i i¼1 exp½Dðx; x Þ

ð2Þ

where n indicates the number of training patterns and the Gaussian D function in Eq. (2) is defined as i

Dðx; x Þ ¼

p X xj  xij j¼1

!2 ð3Þ

f

where p indicates the number of elements of an input vector. The xj and xij represent the jth element of x and xi, respectively. The f is generally referred to as the spread, whose optimal value is experimentally determined. It should be noted that in conventional GRNN applications all spreads for the units in the pattern layer are identical. Despite the simplification of the training process, this may limit improving the GRNN prediction performance. The limitation might be circumvented by adopting a multi-parameterization of training factors. 4. Results For each Data I and Data II, three types of models were constructed, including GRNN, statistical regression, and GA-GRNN models. It should be noted that some of GRNN and statistical regression models for both data sets have been reported [8]. For Data I, GRNN and statistical regression models for all etch responses but the Al selectivity was reported [7]. Also, statistical regression models of SiC etch rate and surface roughness were reported [8]. Therefore, new models constructed in this study include GRNN models of Al selectivity and three etch outputs comprising Data II, statistical regression models of Al selectivity and SiC profile angle, and GA-GRNN models for all 8 etch outputs comprising Data I and Data II. Model prediction performance was measured by the root meansquare error (RMSE) defined as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pr 2 j¼1 ðdj  outj Þ RMSE ¼ r

ð4Þ

where dj and outj represent actual etch measurement and model prediction corresponding to the jth experiment. The remaining r indicates the total number of test vectors.

4.1. Models of Data I GA was applied to search one near optimal set of training factors. The operation of GA typically consists of four steps, including the representation, reproduction, recombination, and mutation [13]. During the representation, each possible solution called chromosome is encoded in a binary or real-valued string. In this study, real-valued chromosomes were adopted and each chromosome consisted of multi-parameterized spreads. It should be noted that in GRNN modeling the number of pattern units is identical to that for the training patterns. For example, a chromosome for Data I was composed of nine spreads since the training data consisted of nine experiments. A random generator was used to assign random values to each spread, generated within a given spread range to each spread. The experimental spread range was increased from 0.2 to 1.4 with the increment 0.1. The size of possible chromosomes (i.e., solutions) was set to 200. The first step in GA optimization is to evaluate the fitness of these chromosomes and for this a fitness function was defined as

F¼

1 1 þ RMSETR

ð5Þ

where RMSETR indicates the training error calculated with 9 and 17 training experiments for Data I and II, respectively. On the basis of the fitness of each chromosome, multiple copies of existing chromosomes are produced and this is called the reproduction process. The elitist roulette wheel selection [13] was adopted here to reproduce chromosomes in proportional to the fitness value. The subsequent recombination process is governed by the crossover operator. As the crossover operator, one point crossover was adopted. In other words, only one part of the two chromosomes is swapped at a randomly selected point. Finally, the mutation operator is applied to create new information in the chromosome. The new information was generated by adding the random values generated between 0 and 1. The probabilities for the crossover and mutation operators were 0.95 and 0.05, respectively. As the termination criterion, the maximum generation number was set to 100. As an illustration, GA was applied to optimize a GRNN model of Al etch rate. The spread range was set to 1.0. At each generation, 200 models corresponding to 200 chromosomes were constructed. Two types of models were then chosen. One is the model yielding the smallest RMSETR, and the corresponding RMSETE was referred to as the type I error. Here the RMSETE indicates the testing error measured with the test data prepared earlier. Also, it is possible to choose one model of the smallest RMSETE at each generation. This enables us to choose one model of the smallest RMSETE among 200 smallest RMSETE determined earlier at all generation numbers. This type error was denoted as the type II error. These two types of

Prediction Error ( Å/min)

800 700

Type I

Type II

600 500 400 300 200 100 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100

Generation Number Fig. 2. Prediction performance of two types of GA-GRNN models of Al etch rate. The spread range was set at 1.0.

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errors are plotted in Fig. 2 as a function of the generation number. As shown in Fig. 2, the smallest type I error is obtained at the first generation and is equal to about 105 Å/min. Another smallest type II error of about 103 Å/min is obtained at the tenth generation. Both errors seem to be comparable numerically. In this way, for the other spread ranges, the errors were determined and they are shown in Fig. 3. It is common that for all ranges the type II errors are much smaller than the corresponding type I errors. Fig. 3 reveals that the smallest type I and II errors are obtained at 1.0 and 0.7, and their corresponding RMSETE are about 105 and 27 Å/ min, respectively. It is clear that the type II model demonstrates a considerable improvement over the type I model. This optimization process was repeated for the remaining etch responses. The results and optimized spreads are shown in Table 4. The optimized sets for the multi-parameterized spreads are shown in Table 5 for all etch responses. As shown in Table 4, it is clear that for all etch models the type II errors are much smaller than the type I errors. The type II models were further compared to the conventional GRNN models presented in [7]. The RMSETE reported for the Al etch

Prediction Error ( Å /min)

1400 1200

Type I Type II

1000 800 600 400 200 0

0. 2 0 .3 0. 4 0 .5 0. 6 0 .7 0. 8 0 .9 1. 0 1 .1 1. 2 1 .3 1. 4

DC bias (V) Profile angle (deg) Silica etch rate (Å/min) Al selectivity Al etch rate (Å/min) -20

Table 4 Optimized performance of GA-GRNN models Etch responses

Type I

Al etch rate (Å/min) Al selectivity Silica etch rate (Å/min) Profile angle (°) DC bias (V)

Type II

Spread range

RMSETE

Spread range

RMSETE

1.0 0.8 0.7 1.3 0.3

105 2.12 1126 2.03 87

0.7 0.8 1.0 0.5 0.3

27 1.25 316 0.97 25

Improvement (%)

81.6 40.7 52.0 58.3 72.8

The percent improvement was calculated with respect to GRNN models previously reported in Ref. [5].

Table 5 Multi-parameterized spreads optimized for the etch response models of Data I Silica etch rate (Å/min)

Al etch rate (Å/min)

Al selectivity

Silica profile anisotropy (°)

DC bias (V)

0.355648 0.282891 0.360561 0.10305 0.328091 0.273644 0.3368 0.186092 0.0143882

0.598098 0.376003 0.174787 0.145606 0.475529 0.0138946 0.377013 0.504357 0.00809052

0.198552 0.143024 0.0646143 0.124131 0.199797 0.027986 0.145023 0.17624 0.01113566

0.130039 0.119723 0.0117901 0.178174 0.113616 0.105966 0.138936 0.160704 0.169074

0.0133955 0.315017 0.370084 0.247146 0.0655156 0.355428 0.00391374 0.282153 0.134651

20

40

60

80

100

Improvement (%) Fig. 4. A comparison of GA-GRNN and statistical regression models for the etch outputs comprising Data I.

rate, silica etch rate, profile angle, and DC bias are about 147 Å/min, 659 Å/min, 2.4°, and 93 V, respectively. As stated earlier, the GRNN model for the remaining Al selectivity was newly constructed in this study and the resulting RMSETE is about 2.11. This occurred at the spread of 1.0. The improvements of type II models against the GRNN models are shown in the last column of Table 4. For all etch responses, the GA-GRNN models yield an improvement of more than 40% compared to the GRNN models. This clearly indicates that the type II model can improve considerably the prediction capability of conventional GRNN model. Meanwhile, the type II models were further compared to statistical regression models previously reported [7]. For convenience, the smallest RMSETE reported for the Al etch rate, silica etch rate, profile angle, and DC bias are 106 Å/min, 281 Å/min, and 2.5°, 86 V, respectively. For the remaining Al selectivity, statistical regression models were newly constructed. A typical form of regression model is written as

Spread Range Fig. 3. The prediction performance of GA-GRNN models as a function of spread range.

0

y ¼ bo þ

k X i¼1

bi xi þ

k X i

bii x2i þ

XX i

bij xi xj ;

ð6Þ

j

where y is the etch attribute, bi and bij are the regression coefficients, and xi is the regressor variables corresponding to the process parameters. Index k denotes the total number of process parameters. For comparison, four types of regression models were constructed. Type I model is composed of the first two terms in Eq. (6). Type II model contains all terms comprising Eq. (6). Type III or Type IV model corresponds to Type II model with only the third or the fourth term excluded in Eq. (6), respectively. Each type of regression models were fitted to the training data and tested with the test data used for GRNN modeling. The resulting RMSETE are 2.8, 3.7, 2.4, and 4.5 for the type I, II, III, and IV, respectively. Hence, the smallest RMSETE is achieved at the type III model. The statistical regression models are compared with the type II models in Table 4. The comparison results are shown in Fig. 4. As shown in Fig. 4, for all etch outputs but the silica etch rate, GA-GRNN models produce much smaller RMSETE. The improvements are more than 50%. In consequence, the GA-GRNN models are superior to GRNN or statistical regression models in predicting the etch responses comprising Data I. 4.2. Models of Data II The presented technique is further evaluated with Data II of larger size. As conducted in the case of Data I, three etch outputs comprising Data II were modeled using GA and GRNN. For the two model types, the resulting RMSETE and optimized spread ranges are shown in Table 6. Multi-parameterized spreads for the type II models are shown in Table 7. For comparison, conventional GRNN

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B. Kim et al. / Microelectronic Engineering 86 (2009) 63–67 Table 6 Optimized performance of GA-GRNN models Etch responses

SiC etch rate (nm/min) Profile angle (°) Surface roughness (nm)

Type I

Type II

Spread range

RMSETE

Spread range

RMSETE

0.7 1.1 0.6

43 2.8 0.12

0.5 0.5 0.5

14 2.3 0.09

Silica etch rate (Å /min)

Improvement (%)

Profile angle (deg) 66.6 20.7 18.2

SiC etch rate (nm/min)

The percent improvement was calculated with respect to GRNN models shown in Table 8.

-20

0

20

40

60

80

100

Improvement (%) Table 7 Multi-parameterized spreads optimized for the etch response models of Data II Etch rate (nm/min)

Profile angle (°)

Surface roughness (nm)

0.256625 0.273089 0.109894 0.337776 0.233524 0.342913 0.27046 0.077034 0.198899 0.116319 0.283618 0.298088 0.234797 0.327485 0.411236 0.142177 0.386488

0.0570274 0.03142 0.18688 0.159725 0.0760785 0.0209204 0.101561 0.0802287 0.0638611 0.119285 0.044889 0.154302 0.104338 0.055297 0.0869601 0.0628848 0.0353569

0.0201559 0.0710705 0.105772 0.124547 0.0704323 0.0182535 0.034558 0.00617945 0.148248 0.15229 0.199884 0.15691 0.11273 0.323211 0.114514 0.139974 0.0150195

Fig. 5. A comparison of GA-GRNN and statistical regression models for the etch outputs comprising Data II.

occurs during the plasma etchings. In contrast, for the remaining two etch outputs; GA-GRNN models yield much improved predictions of more than 30% improvements. As a result, similar to Data I, GA-GRNN demonstrated much improved predictions over GRNN and statistical regression models. 5. Conclusions

Table 8 Optimized performance of conventional GRNN model Etch response

Spread

RMSETE

SiC etch rate (nm/min) Profile angle (°) Surface roughness

0.6 1.4 0.8

42 2.9 0.11

models were constructed and the resulting RMSETE and optimized spread are shown in Table 8. The improvements of GA-GRNN models in Table 6 were compared to the corresponding ones in Table 8. The results are shown in the last column of Table 6. For all etch outputs, more than 15% improvements are achieved compared to GRNN models. The improvement is considerable for the SiC etch rate. Meanwhile, the GA-GRNN models are compared to the statistical regression models previously constructed [8]. The smallest RMSETE reported were 49 nm/min and 0.134 nm for the etch rate and surface roughness, respectively. It should be noted that these models were constructed for the purpose of comparisons with RBFN models. In this study, statistical regression models for the remaining SiC profile angle were newly constructed. Among the four types of models, the smallest RMSETE was obtained for the type II regression model and it is about 2.1°. The improvements of GA-GRNN models over statistical models are illustrated in Fig. 5. As seen in Fig. 5, in predicting the profile angle, the regression model demonstrates a RMSETE smaller than that for the GAGRNN model. However, the improvement is only less than 10%. The improvement is not sufficiently large to say that the regression model is better than GA-GRNN model. This is further supported by the measurement error accompanied by simplifying detailed surface variations such as a profile bowing or microtrenching that

In this study, a previous GA-GRNN model was evaluated extensively with a large number of plasma-processed film characteristics. This type of verification work was necessary not only to ensure the advantage of GA-GRNN model as well as for its widespread application. The experimental data consisted of silica and silicon oxynitride films in two different plasma chemistries. Each data set was systematically characterized by means of statistical experimental designs. Compared to conventional GRNN and statistical regression models, GA-GRNN models demonstrated a much improved prediction in all etch responses. By the proven improvement, the GA-GRNN model is expected to be widely applied to other process data. Acknowledgements This work was supported by the Seoul R&BD Program (Grant No.10583), and partly by the Ministry of Knowledge Economy, Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Advancement) (IITA-2008-C109008010030). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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