- Email: [email protected]
Neural network modeling of inter-characteristics of silicon nitride film deposited by using a plasma-enhanced chemical vapor deposition
Expert Systems with Applications 38 (2011) 11437–11441
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Neural network modeling of inter-characteristics of silicon nitride film deposited by using a plasma-enhanced chemical vapor deposition Su Jin Lee a, Byungwhan Kim a,⇑, Sung Wook Baik b a b
Department of Electronic Engineering, Sejong University, 98, Goonja-Dong, Kwangjin-Gu, Seoul 143-747, Republic of Korea School of Computer Engineering, Sejong University, Seoul 143-747, Republic of Korea
a r t i c l e
i n f o
Keywords: Neural network Model Silicon nitride Plasma-enhanced chemical vapor deposition Generalized regression neural network Lifetime Solar cell manufacturing
a b s t r a c t Neural network have been widely used to model a relationship between process parameters (or in situ diagnostic variables) and film qualities. A new neural network model relating inter-relationship between the film qualities, not the process parameters is constructed by using a generalized regression neural network and a genetic algorithm. This approach is applied to the lifetime of silicon nitride films deposited by using a plasma-enhanced chemical vapor deposition system. The lifetime is an important quality that determines the efficiency of solar cells. The other film qualities examined are a deposition rate, a refractive index, and a charge density. For a systematic modeling, the deposition process was modeled by using a statistical experiment. Compared to conventional and statistical regression models, the optimized GRNN model demonstrated an improvement of 73% and 81%, respectively. The model predicted important and useful clues to optimizing the lifetime. It is noticeable that higher lifetime was achieved at lower deposition rate. This was also noted as the charge density was decreased. The refractive index played a critical role in improving the lifetime. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction As a passivation or dielectric layer, silicon nitride (SiN) films are widely adopted in manufacturing electronic devices. This stems from the unique properties of a good surface insulation and adherence, and a high resistance to ion migration, moisture and surface oxidation. A plasma-enhanced chemical vapor deposition (PECVD) is widely used to deposit SiN films in a SiH4-based gases. In the context of solar cell manufacturing, film qualities of importance are a deposition rate, a refractive index, a positive charge density, and an effective lifetime. To investigate the effects of the process parameters (e.g., a radio frequency power or pressure), empirical models were constructed by using a neural network in conjunction with a statistical experiment. Many neural network models of SiN film properties have been reported, including the deposition rate (Chen, Lee, & Deng, 2007; Kim, Park, & Lee, 2005), refractive index (Chen et al., 2007; Geisler, George, & May, 2000; Han, Cai, May, & Rohatgi, 1996; Kim & Hong, 2004), charge density (Geisler et al., 2000; Han et al., 1996; Kim & Kwon, 2007), and lifetime (Geisler et al., 2000; Han et al., 1996; Kwon, Kim, & May, 2010). Although these are beneficial to accessing the effect of the parameters under various plasma conditions, they are incapable of providing useful insights into the inter-relationships between the film properties. ⇑ Corresponding author. Tel.: +82 2 3408 3729; fax: +82 2 3408 3329. E-mail address: [email protected] (B. Kim). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.03.016
In fact, most of the film properties are related to each other. For example, the life time determining the cell efficiency is strongly affected on the charge density. High charge density can improve the surface passivation of SiN film, leading to large lifetime. A charge density model-based study revealed that higher charge density is likely to be obtained in N-rich films (Kim & Kim, 2008), which corresponds to low refractive index. Also, a lifetime model implicated its strong dependency on the variation in [N–H] (Kwon et al., 2010). As reported in many studies, the refractive index highly correlates with a ratio of [Si–H]/[N–H] (Classen, Valkenburg, Habraken, & Tamminga, 1983) or with an atomic ratio of Si/N (Boyer, Jalabert, & Masarotto, 2000). Another correlation between the ratios of [Si–H]/[N–H] and SiN was also noted (Tonya, Timothy, Ashfaqul, & Gregory, 1999). Implicated in these reports is that the lifetime is related to the chare density and the refractive index. Moreover, these dependencies might be different depending on the amount of the deposition rate. Unfortunately, up to data, this concern has not been studied systematically and therefore no insight into the inter-relationships between the film properties could have been gained. In this study, a neural network model of lifetime is constructed as a function of other film properties. This type of model is first presented in this study. A generalized regression neural network (GRNN) (Specht, 1991) and a genetic algorithm (GA) (Goldberg, 1989) are used to optimize model performance. A set of experimental data collected by using a statistical experiment is used.
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S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
The optimized model is then used to investigate various inter-relationships between SiN film properties.
Input Layer
Pattern Layer
Summation Layer
Output Layer
2. Experimental data
X1 Using a Plasma-Therm 700 series batch reactor operating at 13.56 MHz, deposition of SiN films was performed. A statistical experiment called a face-centered Box Wilson design was applied to the experimental ranges of the six process parameters reported in the works (Kim & Hong, 2004). The film properties concerned here include a deposition rate, a refractive index, a charge density, and a lifetime. The deposition rate and refractive index were measured by using a Metricon 2010 Prism Coupler and a he–ne laser of 632.8 nm, respectively. The C–V measurement and a laser photoconductive decay tester were used to measure the charge density and the lifetime, respectively. Of four film properties, the lifetime was chosen as a primary film property to model because it a measure of the cell efficiency. In other words, the lifetime becomes the output of the model and the remaining three film properties constitute the input. In this sense, the presented model fundamentally differs from the reported models. A total of 46 experiments were conducted and of them 35 experiments are used to train the GRNN. The training data is composed of 261 factorial experiment and three replicates of a center point. The appropriateness of the trained model is then tested by using the remaining 11 experiments. The experimental ranges of each film characteristic confirmed from the training data are shown in Table 1.
X2
Y
Xn Fig. 1. Schematic of generalized regression neural network.
Dðx; xi Þ ¼
p X xj xij 2 j¼1
f
ð2Þ
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 factor, whose optimal value is often determined experimentally.
3. Generalized regression neural network 4. Results A GRNN was used to construct a model of lifetime. A schematic of GRNN is shown 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 first layer corresponds to individual film quality. The first 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-summation neuron 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 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 set to unity. The output layer merely divides the output of each S-summation neuron by that of each D-summation neuron, yielding a predicted R to an unknown input vector x as
Pn yi exp½Dðx; xi Þ ^i ðxÞ ¼ Pi¼1 y n i¼1 exp½Dðx; xi Þ
4.1. Conventional models For comparison purpose, two types of models were constructed and they are the conventional GRNN and statistical regression models. The former model was developed by experimentally adjusting the spread stated earlier. In this study, the spread was varied from 0.1 to 1.0 with an increment of 0.1. The prediction performance was measured by using the root mean square error, denoted as ‘‘RMSE’’. The RMSE error was measured at each spread and the variation in the error is shown in Fig. 2. As shown in Fig. 2, the smallest RMSE is obtained at 0.1 and the corresponding error is 8.58 ls. Statistical regression models was developed from the general equation
12
where n and xi represents the number of training patterns and the ith training input pattern stored between the first and second layers, respectively. The Gaussian D function in (2) is defined as
Table 1 Experimental ranges of film properties.
Prediction error (µs)
ð1Þ
11 10 9 8 7
Film characteristics
Experimental ranges
Units
Lifetime Deposition rate Positive charge density Refractive index
1.71–2.23 50–234 2.76–8.88 1.71–2.31
ls Å/min /cm2
6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Spread
Fig. 2. Prediction performance of lifetime model as a function of spread.
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S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
y ¼ bo þ
k X
bi xi þ
i¼1
k X
bii x2i þ
XX
i
i
bij xi xj
6
ð3Þ
j
Rather than one single spread value, adopting multi-parameterized spreads (Kim, Kwon, & Kwon, 2009) can improve the prediction performance of GRNN. A genetic algorithm (GA) was once used to search for a set of optimal spreads and the resulting GAGRNN model demonstrated a much improved prediction accuracy in building models of plasma etch or deposition data (Kim & Kwon, 2007; Kim et al., 2005, 2009) and optical resolution of scanning electron microscope (Kim, Kwon, & Kim, 2010). In the GRNN, the number of neurons (e.g., spreads here) in the pattern layer is the same as that of the training input patterns. This indicates that a solution called ‘‘chromosome’’ in genetic terminology consisted of 33 spread parameters. Real-valued, random values generated by using a random generator within a given spread are initially assigned to each parameter and their values are to be evolved to the optimized ones during GA operation. As a parameter for GA operation, the size of initial population was set to 100. The probabilities of crossover and mutation were set to 0.95 and 0.05, respectively. A particular input setting generated by the GA meets a given fitness (F) function expressed as
F¼
1 1 þ RMSE
ð4Þ
Here, the RMSE means the root mean square error measured with the test data. A roulette wheel selection mechanism was employed in generating a subsequent generation of the same size of 100. In GA optimization, the performance of GA-GRNN models was evaluated as a function of the spread range. The spread range was varied from 0.1 to 1.2 with an increment of 0.1. The prediction error of GA-GRNN model is shown in Fig. 3 as a function of the spread range. As shown in Fig. 3, the prediction error seems to increase with increasing the spread. Smaller errors are obtained at two spread ranges of 0.1 and 0.8. Of them, the model corresponding to 0.8 was adopted because the RMSE of about 2.30 is slightly smaller than that of 2.32 at 0.1. Compared to the optimized errors for the conventional and statistical models this shows an
4 3 2 1 0 0.1
0.2 0.3
0.4 0.5 0.6 0.7 Spread Range
Model type
Prediction error (ls)
I II III IV
12.9 13.5 13.2 12.7
0.9
1.0
improvement of 73% and 81%, respectively. A significant improvement of GA-GRNN is therefore demonstrated. The prediction performance of the models is more evaluated by comparing the predictions and actual measurements. This is shown in Fig. 4. As expected from the error comparison, those predictions from GAGRNN the most closely match the actual measurements. 5. Interpretations The optimized model was used to generate three-dimensional plots as a function of the film qualities. Fig. 5 shows the lifetime as a function of the deposition rate and the refractive index. In plotting Fig. 5, the charge density was fixed at 6 1012/cm2. As shown in Fig. 5, the lifetime increases with decreasing the deposition rate. This becomes more drastic at the refractive index of about 2.0. This reveals an important relationship between the life time and the deposition rate. In other words, larger lifetime is achieved at lower deposition rate particularly at the refractive index around 2.0. It should be noted that the largest lifetime is likely to be obtained at fairly high refractive index between 1.9 and 2.1. In contrast, the increase no longer appears at higher refractive index. As shown in Fig. 5, the variation in the refractive index at high
90 80 70 60 50 40
Actual Statistical Regression GRNN GA-GRNN
30
Table 2 Prediction performance of statistical regression models.
0.8
Fig. 3. Prediction performance of lifetime model as a function of the spread range.
Lifetime (µs)
4.2. GA-optimized GRNN model
Prediction Error (µs)
5 where y is the etch attribute, bi and bij are the regression coefficients, and xi is the regressor variable corresponding to the film qualities. An index k denotes the total number of film qualities. For comparison, four types of regression models were constructed. Type I model is composed of the first two terms in (3). Type II model contains all terms comprising (3). Type III or Type IV model corresponds to Type II model with only the third or the fourth term excluded in (3), respectively. Each type of regression models were fitted to the training data and tested with the test data used for GRNN modeling. The prediction errors for two statistical regression models are shown in Table 2 as a function of model type. Table 2 shows that the fourth model yields the smallest prediction error of 12.7 ls and this is much larger than that of the optimized GRNN model mentioned earlier. This indicates that the conventional model is more accurate than the statistical model.
20
1
2
3
4
5 6 7 8 Test Experiment
9
10
11
Fig. 4. Comparison of prediction performance of optimized statistical regression model, GRNN and GA-GRNN models.
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S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
Lifetime (µs)
64
Lifetime (µs)
62 64
60 62
58 60
56 58
54
50 56
0
100 2.4
50
150 Deposition rate
2.2 2.0
( Å /min)
200
1.8 1.6
Refractive index
100 Deposition rate (Å/min)
150 200 250
Fig. 5. Lifetime model as a function of refractive index and deposition rate.
4
6
10
8
Charge density (10¹²/cm²)
Fig. 7. Lifetime model as a function of deposition rate and charge density.
deposition rate gives little impact on the lifetime. In contrast, the impact of the refractive index is quite complicated at lower deposition rate of smaller than 150 Å/min. In consequence, Fig. 5 provides a critical insight into improving the life time, i.e., the fabrication of thinner SiN film at the refractive index of the range stated earlier. Fig. 6 shows the lifetime as a function of charge the density and the refractive index. The deposition rate was fixed at 142 Å/min. As shown in Fig. 6, the lifetime considerably increases with increasing the charge density. This is similarly observed over the entire range of the refractive index. This implies a little interaction between the charge density and the refractive index. In general, the range of refractive index widely employed in manufacturing a reflective SiN film is between 1.9 and 2.2. This means that the largest lifetime predicted at the lowest refractive index has little significance due to the general choice mentioned. Fortunately, a clue to achieving high lifetime without sacrificing the charge density is obtained from the model and this occurs at the refractive index ranging between 2.0 and 2.2. In the context of lifetime optimization, this is a practically useful condition. However, the effect of the refractive index on the lifetime is much different depending on the charge density. Despite the complexity, the lifetime continues to increase with
a decrease in the refractive index especially at higher charge density. Implicated is a higher lifetime for N richer films. This stems from the strong relationship between the refractive index and the ratio of [Si– H/[N–H] (Classen et al., 1983) or Si to N (Boyer et al., 2000). Fig. 7 shows the lifetime as a function of the deposition rate and the charge density. Considering the critical role of the refractive index, two models of the lifetime are generated at two refractive indexes of 2.01 and 2.24 around the range stated earlier. Fig. 7 shows the lifetime generated at a refractive index of 2.01. In Fig. 7, the effect of the deposition rate is very similar to that already observed in Fig. 5. An increase in the charge density at a deposition rates of more than 150 Å/min slightly increases the lifetime. In contrast, this no longer occurs at lower deposition rate. Instead, the lifetime is seen to be improved by controlling the charge density at about 4.0–4.5 1012/cm2 at lower deposition rate. Figs. 5 and 7 emphasize the importance of controlling the refractive index and the charge density in improving the life time particularly at lower deposition rate. Meanwhile, the impact of the deposition rate and the charge density at a higher refractive index of 2.24 is shown in Fig. 8. As shown in Fig. 8, it is clear that two valley peaks exist. The
Lifetime (μs)
60
Lifetime (µs)
59 60
58
59
57
58 10 57 56 1.6
8 6 1.8
56 50
10.0 70
5.0
90 2.0
2.2
Refractive index
4 Charge density 2.4
2.6
2
(10¹²/cm²)
Fig. 6. Lifetime model as a function of refractive index and charge density.
110 Deposition rate (Å/min)
7.5
2.5 Charge density (10¹²/cm²) 130
Fig. 8. Lifetime model as a function of deposition rate and charge density at a refractive index of 2.24.
S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
valleys adjacent at 2.01 each other in Fig. 7 are now clearly separated in Fig. 8. Of the valleys, the one at lower deposition rate is more meaningful because it peaked at much larger lifetime than the other. Fig. 8 implicates the existence of another better condition for improving the lifetime. This corresponds to the condition at lower deposition rate and higher charge density. This condition is preferred because the expected lifetime is likely to be higher than the peak value of the valley. This is supported by the peak value, decreasing as the deposition rate decreases more below about 80 Å/min. 6. Conclusions A new model of SiN films was constructed by using GRNN and GA. Compared to other previous models, the model is distinguished in that it attempted to capture complex inter-relationships between various film qualities. The GA-optimized GRNN model demonstrated a significantly improved prediction over conventional GRNN or statistical regression models. The optimized model showed that the lifetime could be improved by decreasing the deposition rate or by increasing the charge density. The former finding is an important knowledge in that it may prompt a deposition of thinner SiN films in manufacturing Si-based solar cells. Larger lifetime was likely to occur at relatively large refractive index. This is consistent with the current choice of more than 1.90. Controlling the charge density at large refractive index was proven beneficial in improving the lifetime. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20090087476).
11441
References Boyer, P. T., Jalabert, L., & Masarotto, L. (2000). Properties of nitrogen doped silicon films deposited by low-pressure chemical vapor deposition from silence and ammonia. Journal of Vacuum Science and Technology A, 18, 2389–2393. Chen, W. C., Lee, A. H. I., & Deng, W. J. (2007). The implementation of neural network for semiconductor PECVD process. Expert Systems with Applications, 32(4), 1148–1153. Classen, W. A. P., Valkenburg, W. G. J. N., Habraken, F. H. P. M., & Tamminga, Y. (1983). Characterization of plasma silicon nitride layers. Journal of the Electrochemical Society, 130, 2419–2423. Geisler, J. P., George, C. S., & May, G. S. (2000). Neurofuzzy modeling of chemical vapor deposition processes. IEEE Transactions on Semiconductor Manufacturing, 13(1), 46–60. Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Reading, MA: Addison Wesley. Han, S. S., Cai, L., May, G. S., & Rohatgi, A. (1996). Modeling the growth of PECVD silicon nitride films for solar cell applications using neural networks. IEEE Transactions on Semiconductor Manufacturing, 9(3), 303–311. Kim, B., & Hong, W. S. (2004). Use of neural network to characterize a low pressure temperature effect on refractive property of silicon nitride film deposited by PECVD. IEEE Transactions on Plasma Science, 32(1), 84–89. Kim, B., & Kim, S. (2008). Room-temperature high radio-frequency power effects on silicon nitride films deposited by using a plasma-enhanced chemical vapor deposition. Metals and Materials International, 14(5), 637–641. Kim, B., & Kwon, S. (2007). Use of neural network to characterize charge density of PECVD-silicon nitride films. Metals and Materials International, 13(6), 495–499. Kim, B., Kwon, S., & Kim, D. (2010). Optimization of optical lens-controlled scanning electron microscopic resolution using generalized regression neural network and genetic algorithm. Expert Systems and Applications, 37, 182–186. Kim, B., Kwon, M., & Kwon, S. H. (2009). Modeling of plasma process data using a multi-parameterized generalized regression neural network. Microelectronic Engineering, 86, 63–67. Kim, B., Park, K., & Lee, D. (2005). Use of neural network to model a deposition rate of PECVD-silicon nitride films. Plasma Source Science and Technology, 14, 83–88. Kwon, M., Kim, B., & May, G. S. (2010). Use of neural network to model a carrier life time of silicon nitride films deposited by using PECVD. Journal of the Korean Physical Society, 55(5), 1982–1985. Specht, D. F. (1991). A general regression neural network. IEEE Transactions on Neural Network, 2, 568–576. Tonya, M. K., Timothy, M. A., Ashfaqul, I. C., & Gregory, N. P. (1999). Hydrogenated silicon nitride thin films deposited between 50 and 250 °C using nitrogen/silane mixtures with helium dilution. Journal of Vacuum Science and Technology A, 17, 108–112.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Neural network modeling of inter-characteristics of silicon nitride film deposited by using a plasma-enhanced chemical vapor deposition Su Jin Lee a, Byungwhan Kim a,⇑, Sung Wook Baik b a b
Department of Electronic Engineering, Sejong University, 98, Goonja-Dong, Kwangjin-Gu, Seoul 143-747, Republic of Korea School of Computer Engineering, Sejong University, Seoul 143-747, Republic of Korea
a r t i c l e
i n f o
Keywords: Neural network Model Silicon nitride Plasma-enhanced chemical vapor deposition Generalized regression neural network Lifetime Solar cell manufacturing
a b s t r a c t Neural network have been widely used to model a relationship between process parameters (or in situ diagnostic variables) and film qualities. A new neural network model relating inter-relationship between the film qualities, not the process parameters is constructed by using a generalized regression neural network and a genetic algorithm. This approach is applied to the lifetime of silicon nitride films deposited by using a plasma-enhanced chemical vapor deposition system. The lifetime is an important quality that determines the efficiency of solar cells. The other film qualities examined are a deposition rate, a refractive index, and a charge density. For a systematic modeling, the deposition process was modeled by using a statistical experiment. Compared to conventional and statistical regression models, the optimized GRNN model demonstrated an improvement of 73% and 81%, respectively. The model predicted important and useful clues to optimizing the lifetime. It is noticeable that higher lifetime was achieved at lower deposition rate. This was also noted as the charge density was decreased. The refractive index played a critical role in improving the lifetime. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction As a passivation or dielectric layer, silicon nitride (SiN) films are widely adopted in manufacturing electronic devices. This stems from the unique properties of a good surface insulation and adherence, and a high resistance to ion migration, moisture and surface oxidation. A plasma-enhanced chemical vapor deposition (PECVD) is widely used to deposit SiN films in a SiH4-based gases. In the context of solar cell manufacturing, film qualities of importance are a deposition rate, a refractive index, a positive charge density, and an effective lifetime. To investigate the effects of the process parameters (e.g., a radio frequency power or pressure), empirical models were constructed by using a neural network in conjunction with a statistical experiment. Many neural network models of SiN film properties have been reported, including the deposition rate (Chen, Lee, & Deng, 2007; Kim, Park, & Lee, 2005), refractive index (Chen et al., 2007; Geisler, George, & May, 2000; Han, Cai, May, & Rohatgi, 1996; Kim & Hong, 2004), charge density (Geisler et al., 2000; Han et al., 1996; Kim & Kwon, 2007), and lifetime (Geisler et al., 2000; Han et al., 1996; Kwon, Kim, & May, 2010). Although these are beneficial to accessing the effect of the parameters under various plasma conditions, they are incapable of providing useful insights into the inter-relationships between the film properties. ⇑ Corresponding author. Tel.: +82 2 3408 3729; fax: +82 2 3408 3329. E-mail address: [email protected] (B. Kim). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.03.016
In fact, most of the film properties are related to each other. For example, the life time determining the cell efficiency is strongly affected on the charge density. High charge density can improve the surface passivation of SiN film, leading to large lifetime. A charge density model-based study revealed that higher charge density is likely to be obtained in N-rich films (Kim & Kim, 2008), which corresponds to low refractive index. Also, a lifetime model implicated its strong dependency on the variation in [N–H] (Kwon et al., 2010). As reported in many studies, the refractive index highly correlates with a ratio of [Si–H]/[N–H] (Classen, Valkenburg, Habraken, & Tamminga, 1983) or with an atomic ratio of Si/N (Boyer, Jalabert, & Masarotto, 2000). Another correlation between the ratios of [Si–H]/[N–H] and SiN was also noted (Tonya, Timothy, Ashfaqul, & Gregory, 1999). Implicated in these reports is that the lifetime is related to the chare density and the refractive index. Moreover, these dependencies might be different depending on the amount of the deposition rate. Unfortunately, up to data, this concern has not been studied systematically and therefore no insight into the inter-relationships between the film properties could have been gained. In this study, a neural network model of lifetime is constructed as a function of other film properties. This type of model is first presented in this study. A generalized regression neural network (GRNN) (Specht, 1991) and a genetic algorithm (GA) (Goldberg, 1989) are used to optimize model performance. A set of experimental data collected by using a statistical experiment is used.
11438
S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
The optimized model is then used to investigate various inter-relationships between SiN film properties.
Input Layer
Pattern Layer
Summation Layer
Output Layer
2. Experimental data
X1 Using a Plasma-Therm 700 series batch reactor operating at 13.56 MHz, deposition of SiN films was performed. A statistical experiment called a face-centered Box Wilson design was applied to the experimental ranges of the six process parameters reported in the works (Kim & Hong, 2004). The film properties concerned here include a deposition rate, a refractive index, a charge density, and a lifetime. The deposition rate and refractive index were measured by using a Metricon 2010 Prism Coupler and a he–ne laser of 632.8 nm, respectively. The C–V measurement and a laser photoconductive decay tester were used to measure the charge density and the lifetime, respectively. Of four film properties, the lifetime was chosen as a primary film property to model because it a measure of the cell efficiency. In other words, the lifetime becomes the output of the model and the remaining three film properties constitute the input. In this sense, the presented model fundamentally differs from the reported models. A total of 46 experiments were conducted and of them 35 experiments are used to train the GRNN. The training data is composed of 261 factorial experiment and three replicates of a center point. The appropriateness of the trained model is then tested by using the remaining 11 experiments. The experimental ranges of each film characteristic confirmed from the training data are shown in Table 1.
X2
Y
Xn Fig. 1. Schematic of generalized regression neural network.
Dðx; xi Þ ¼
p X xj xij 2 j¼1
f
ð2Þ
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 factor, whose optimal value is often determined experimentally.
3. Generalized regression neural network 4. Results A GRNN was used to construct a model of lifetime. A schematic of GRNN is shown 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 first layer corresponds to individual film quality. The first 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-summation neuron 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 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 set to unity. The output layer merely divides the output of each S-summation neuron by that of each D-summation neuron, yielding a predicted R to an unknown input vector x as
Pn yi exp½Dðx; xi Þ ^i ðxÞ ¼ Pi¼1 y n i¼1 exp½Dðx; xi Þ
4.1. Conventional models For comparison purpose, two types of models were constructed and they are the conventional GRNN and statistical regression models. The former model was developed by experimentally adjusting the spread stated earlier. In this study, the spread was varied from 0.1 to 1.0 with an increment of 0.1. The prediction performance was measured by using the root mean square error, denoted as ‘‘RMSE’’. The RMSE error was measured at each spread and the variation in the error is shown in Fig. 2. As shown in Fig. 2, the smallest RMSE is obtained at 0.1 and the corresponding error is 8.58 ls. Statistical regression models was developed from the general equation
12
where n and xi represents the number of training patterns and the ith training input pattern stored between the first and second layers, respectively. The Gaussian D function in (2) is defined as
Table 1 Experimental ranges of film properties.
Prediction error (µs)
ð1Þ
11 10 9 8 7
Film characteristics
Experimental ranges
Units
Lifetime Deposition rate Positive charge density Refractive index
1.71–2.23 50–234 2.76–8.88 1.71–2.31
ls Å/min /cm2
6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Spread
Fig. 2. Prediction performance of lifetime model as a function of spread.
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S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
y ¼ bo þ
k X
bi xi þ
i¼1
k X
bii x2i þ
XX
i
i
bij xi xj
6
ð3Þ
j
Rather than one single spread value, adopting multi-parameterized spreads (Kim, Kwon, & Kwon, 2009) can improve the prediction performance of GRNN. A genetic algorithm (GA) was once used to search for a set of optimal spreads and the resulting GAGRNN model demonstrated a much improved prediction accuracy in building models of plasma etch or deposition data (Kim & Kwon, 2007; Kim et al., 2005, 2009) and optical resolution of scanning electron microscope (Kim, Kwon, & Kim, 2010). In the GRNN, the number of neurons (e.g., spreads here) in the pattern layer is the same as that of the training input patterns. This indicates that a solution called ‘‘chromosome’’ in genetic terminology consisted of 33 spread parameters. Real-valued, random values generated by using a random generator within a given spread are initially assigned to each parameter and their values are to be evolved to the optimized ones during GA operation. As a parameter for GA operation, the size of initial population was set to 100. The probabilities of crossover and mutation were set to 0.95 and 0.05, respectively. A particular input setting generated by the GA meets a given fitness (F) function expressed as
F¼
1 1 þ RMSE
ð4Þ
Here, the RMSE means the root mean square error measured with the test data. A roulette wheel selection mechanism was employed in generating a subsequent generation of the same size of 100. In GA optimization, the performance of GA-GRNN models was evaluated as a function of the spread range. The spread range was varied from 0.1 to 1.2 with an increment of 0.1. The prediction error of GA-GRNN model is shown in Fig. 3 as a function of the spread range. As shown in Fig. 3, the prediction error seems to increase with increasing the spread. Smaller errors are obtained at two spread ranges of 0.1 and 0.8. Of them, the model corresponding to 0.8 was adopted because the RMSE of about 2.30 is slightly smaller than that of 2.32 at 0.1. Compared to the optimized errors for the conventional and statistical models this shows an
4 3 2 1 0 0.1
0.2 0.3
0.4 0.5 0.6 0.7 Spread Range
Model type
Prediction error (ls)
I II III IV
12.9 13.5 13.2 12.7
0.9
1.0
improvement of 73% and 81%, respectively. A significant improvement of GA-GRNN is therefore demonstrated. The prediction performance of the models is more evaluated by comparing the predictions and actual measurements. This is shown in Fig. 4. As expected from the error comparison, those predictions from GAGRNN the most closely match the actual measurements. 5. Interpretations The optimized model was used to generate three-dimensional plots as a function of the film qualities. Fig. 5 shows the lifetime as a function of the deposition rate and the refractive index. In plotting Fig. 5, the charge density was fixed at 6 1012/cm2. As shown in Fig. 5, the lifetime increases with decreasing the deposition rate. This becomes more drastic at the refractive index of about 2.0. This reveals an important relationship between the life time and the deposition rate. In other words, larger lifetime is achieved at lower deposition rate particularly at the refractive index around 2.0. It should be noted that the largest lifetime is likely to be obtained at fairly high refractive index between 1.9 and 2.1. In contrast, the increase no longer appears at higher refractive index. As shown in Fig. 5, the variation in the refractive index at high
90 80 70 60 50 40
Actual Statistical Regression GRNN GA-GRNN
30
Table 2 Prediction performance of statistical regression models.
0.8
Fig. 3. Prediction performance of lifetime model as a function of the spread range.
Lifetime (µs)
4.2. GA-optimized GRNN model
Prediction Error (µs)
5 where y is the etch attribute, bi and bij are the regression coefficients, and xi is the regressor variable corresponding to the film qualities. An index k denotes the total number of film qualities. For comparison, four types of regression models were constructed. Type I model is composed of the first two terms in (3). Type II model contains all terms comprising (3). Type III or Type IV model corresponds to Type II model with only the third or the fourth term excluded in (3), respectively. Each type of regression models were fitted to the training data and tested with the test data used for GRNN modeling. The prediction errors for two statistical regression models are shown in Table 2 as a function of model type. Table 2 shows that the fourth model yields the smallest prediction error of 12.7 ls and this is much larger than that of the optimized GRNN model mentioned earlier. This indicates that the conventional model is more accurate than the statistical model.
20
1
2
3
4
5 6 7 8 Test Experiment
9
10
11
Fig. 4. Comparison of prediction performance of optimized statistical regression model, GRNN and GA-GRNN models.
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Lifetime (µs)
64
Lifetime (µs)
62 64
60 62
58 60
56 58
54
50 56
0
100 2.4
50
150 Deposition rate
2.2 2.0
( Å /min)
200
1.8 1.6
Refractive index
100 Deposition rate (Å/min)
150 200 250
Fig. 5. Lifetime model as a function of refractive index and deposition rate.
4
6
10
8
Charge density (10¹²/cm²)
Fig. 7. Lifetime model as a function of deposition rate and charge density.
deposition rate gives little impact on the lifetime. In contrast, the impact of the refractive index is quite complicated at lower deposition rate of smaller than 150 Å/min. In consequence, Fig. 5 provides a critical insight into improving the life time, i.e., the fabrication of thinner SiN film at the refractive index of the range stated earlier. Fig. 6 shows the lifetime as a function of charge the density and the refractive index. The deposition rate was fixed at 142 Å/min. As shown in Fig. 6, the lifetime considerably increases with increasing the charge density. This is similarly observed over the entire range of the refractive index. This implies a little interaction between the charge density and the refractive index. In general, the range of refractive index widely employed in manufacturing a reflective SiN film is between 1.9 and 2.2. This means that the largest lifetime predicted at the lowest refractive index has little significance due to the general choice mentioned. Fortunately, a clue to achieving high lifetime without sacrificing the charge density is obtained from the model and this occurs at the refractive index ranging between 2.0 and 2.2. In the context of lifetime optimization, this is a practically useful condition. However, the effect of the refractive index on the lifetime is much different depending on the charge density. Despite the complexity, the lifetime continues to increase with
a decrease in the refractive index especially at higher charge density. Implicated is a higher lifetime for N richer films. This stems from the strong relationship between the refractive index and the ratio of [Si– H/[N–H] (Classen et al., 1983) or Si to N (Boyer et al., 2000). Fig. 7 shows the lifetime as a function of the deposition rate and the charge density. Considering the critical role of the refractive index, two models of the lifetime are generated at two refractive indexes of 2.01 and 2.24 around the range stated earlier. Fig. 7 shows the lifetime generated at a refractive index of 2.01. In Fig. 7, the effect of the deposition rate is very similar to that already observed in Fig. 5. An increase in the charge density at a deposition rates of more than 150 Å/min slightly increases the lifetime. In contrast, this no longer occurs at lower deposition rate. Instead, the lifetime is seen to be improved by controlling the charge density at about 4.0–4.5 1012/cm2 at lower deposition rate. Figs. 5 and 7 emphasize the importance of controlling the refractive index and the charge density in improving the life time particularly at lower deposition rate. Meanwhile, the impact of the deposition rate and the charge density at a higher refractive index of 2.24 is shown in Fig. 8. As shown in Fig. 8, it is clear that two valley peaks exist. The
Lifetime (μs)
60
Lifetime (µs)
59 60
58
59
57
58 10 57 56 1.6
8 6 1.8
56 50
10.0 70
5.0
90 2.0
2.2
Refractive index
4 Charge density 2.4
2.6
2
(10¹²/cm²)
Fig. 6. Lifetime model as a function of refractive index and charge density.
110 Deposition rate (Å/min)
7.5
2.5 Charge density (10¹²/cm²) 130
Fig. 8. Lifetime model as a function of deposition rate and charge density at a refractive index of 2.24.
S.J. Lee et al. / Expert Systems with Applications 38 (2011) 11437–11441
valleys adjacent at 2.01 each other in Fig. 7 are now clearly separated in Fig. 8. Of the valleys, the one at lower deposition rate is more meaningful because it peaked at much larger lifetime than the other. Fig. 8 implicates the existence of another better condition for improving the lifetime. This corresponds to the condition at lower deposition rate and higher charge density. This condition is preferred because the expected lifetime is likely to be higher than the peak value of the valley. This is supported by the peak value, decreasing as the deposition rate decreases more below about 80 Å/min. 6. Conclusions A new model of SiN films was constructed by using GRNN and GA. Compared to other previous models, the model is distinguished in that it attempted to capture complex inter-relationships between various film qualities. The GA-optimized GRNN model demonstrated a significantly improved prediction over conventional GRNN or statistical regression models. The optimized model showed that the lifetime could be improved by decreasing the deposition rate or by increasing the charge density. The former finding is an important knowledge in that it may prompt a deposition of thinner SiN films in manufacturing Si-based solar cells. Larger lifetime was likely to occur at relatively large refractive index. This is consistent with the current choice of more than 1.90. Controlling the charge density at large refractive index was proven beneficial in improving the lifetime. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20090087476).
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