Pressure

Gas pressure in RIE is typically maintained in a range between a few millitorr and a few hundred millitorr by adjusting gas flow rates or adjusting an exhaust orifice.

Normally, as pressure is decreased below about l00 mTorr, the potential across the discharge characteristically increases. At very low pressure, physical etching mechanisms tend to dominate (figure 3.11) , because of high ion energy, low reactant density, and long mean free paths.

According to the kinetic theory of molecular gases, the mean free path of a gas molecule at constant temperature is inversely proportional with the pressure. So when the pressure decreases the mean free paths of the species increases, and the energetic particles in the plasma can easily transfer their kinetic energy to the atoms at the oxide silicon film surface.

Figure 3.11 Qualitative effect of pressure on ion energy and the etching mechanism [24]

C H A P T E R 4 P ^RINCIPAL C ^OMPONENT A ^NALYSIS

Principal component analysis (PCA) is an important analysis technique in multivariate statistics, it was first suggested in 1901 by Pearson [36], and formally developed by Hotelling [37]. The main idea of principal component analysis (PCA) is to represent number of correlated variables into a smaller number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, the second PC is the linear combination with the second largest variance and orthogonal to the first PC, and so on. There are as many PCs as the number of the original variables. For many datasets, the first several PCs explain most of the variance, so that the rest can be disregarded with minimal loss of information.

The objectives of using PCA are to reduce the dimensionality of a data set, and to identify new underlying variables that are now orthogonal.

To enhance performance of prediction model in this study, PCA is suggested to represent the RIE factors, since simple neural networks with few nodes and connections tend to have better generalization capability. In this chapter, PCA technique

automatically extracts three principle components (PC) from all RIE factors (twenty two factors). Table 4.1 shows the RIE factors and its abbreviation.

Table 4.1 RIE factors and its abbreviation.

x

1 Bias Voltage

x

12 _{Gas 4}

x

2 ESC Clamp Voltage

x

13 _{Gas 7}

x

3 ESC Current1

x

14 He Flow Inner

x

4 ESC Current2

x

15 He Flow Outer

x

5 ESC Temperature

x

16 He Pressure Inner

x

6 Foreline Pressure

x

17 He Pressure Outer

x

7 Forward Power 27MHz

x

18 _Pressure

x

8 Forward Power 2MHz

x

19 Process Time

x

9 _{Gas 1}

x

₂₀ Reflect Power 27MHz

x

10 _{Gas 10}

x

₂₁ Reflect Power 2MHz

x

11 _{Gas 11}

x

₂₂ Top Plate Temperature

It is important to treat each step separately in PCA, because each etching step has different inherent physical/chemical characteristics, and by considering the overall process characteristics and the objective of model simplicity, it was decided that utilizing one PCA for each of the eleven steps might yield a better solution than utilizing a single PCA for the entire process. In this thesis, principal component analysis was utilized for 90 training wafers, the principle components are found by computing the sample covariance* matrix and selecting its eigenvectors (loading vectors) for the k biggest eigenvalues as shown in figure 4.1.

* Covariance matrix Cov(X) is a good choice to capture the dependence between

The covariance matrix Cov(X) can be obtained by:

is data matrix with n samples (rows) and 22 variables (columns), as well as each column represents one of RIE factors.

X

i represents vector i of data matrix

(X −X) stands for subtract the mean value of each column from the corresponding column .

To find the eigenvalues and eigenvector of the covariance matrix, Cov(X) represented by using singular value decomposition (SVD) as shown in the following equation:

( )

^T

Cov X = Λ V V

^(4.2)

where

V =

^⎡_⎣

v v

₁

₂

L v

₂₂^⎤_⎦

is an 22 by 22 unitary matrix of corresponding eigenvector

1

eigenvalues for the eleven steps. Figure 4.2 shows the decrease in eigenvalues λ for each of etching step.

The variance of the ith PC is equal to the ith largest eigenvalue of the covariance matrix [38]. Because of this important property, the

ϕ

percentage (equation 4.3) is used as a guide in choosing an appropriate number of PC. The goal is to choose as small a value of k as possible while achieving a reasonably high percentage of PC variance. The

ϕ

values shown in table 4.3 give the cumulative proportion of the variance explained by

the first k PCs,

ϕ

is larger than 88% when k equals three, thus three principle components are selected to characterise the twenty two RIE factors by using equation 4.4:

1

Score equations (

T = XV

) is tend to identify the three principle components according to RIE factors. Score equations for each etch step is shown in Appendix II

Figure 4.2 Eigenvalues of the covariance matrix for RIE steps

C H A P T E R 5 P ^REDICTION M ^{ODELS FOR} RIE

The computing, telecommunications, aerospace, automotive and consumer electronics industries all rely heavily on integrated circuits (ICs). Next-generation IC manufacturing equipment will require dramatic improvements in cost, quality, throughput, and flexibility. Reducing manufacturing cost involves increasing chip yield, reducing cycle time, maintaining consistent product quality, improving equipment reliability, and maintaining stringent process control. Since IC fabrication consists of hundreds of steps, maintaining product quality requires the control of thousands of variables. Process steps are performed in sequence, and yield loss may occur at every step. However, analyzing wafer defects is the regular method for evaluation semiconductor technologies. Wafer defects carry a lot of wafer status information which can be analyzed in order to characterize the quality of processes and products. If the prediction model accurately predicts the wafer status, the repeated etching failure rate should be prevented, process yield should be greatly enhanced, inspection cost should be reduced and profit should be increased.

The experimental process of this study is as depicted in figure 5.1, and four models are included: offline back-propagation neural network (BPNN), offline principle

component analysis BPNN (PCABPNN), online BPNN and online PCABPNN. These models have the potential to reduce the overall cost of ownership of semiconductor equipment by increasing the wafer yield and throughput of product wafers, and not depend upon monitor wafers or expensive metrology rather it will enable inexpensive real-time wafer-to-wafer control applications in RIE. The capability of the four prediction models to predict the wafer status correctly is discussed in this chapter.

在文檔中 應用類神經網路於蝕刻製程之缺陷分析與預測 (頁 50-62)