Open access peer-reviewed chapter

Artificial Neural Networks in Production Scheduling and Yield Prediction of Semiconductor Wafer Fabrication System

By Jie Zhang, Junliang Wang and Wei Qin

Submitted: November 2nd 2015Reviewed: April 4th 2016Published: October 19th 2016

DOI: 10.5772/63444

Downloaded: 1241

Abstract

With the development of artificial intelligence, the artificial neural networks (ANN) are widely used in the control, decision‐making and prediction of complex discrete event manufacturing systems. Wafer fabrication is one of the most complicated and high competence manufacturing phases. The production scheduling and yield prediction are two critical issues in the operation of semiconductor wafer fabrication system (SWFS). This chapter proposed two fuzzy neural networks for the production rescheduling strategy decision and the die yield prediction. Firstly, a fuzzy neural network (FNN)‐based rescheduling decision model is implemented, which can rapidly choose an optimized rescheduling strategy to schedule the semiconductor wafer fabrication lines according to the current system disturbances. The experimental results demonstrate the effectiveness of proposed FNN‐based rescheduling decision mechanism approach over the alternatives (back‐propagation neural network and Multivariate regression). Secondly, a novel fuzzy neural network‐based yield prediction model is proposed to improve prediction accuracy of die yield in which the impact factors of yield and critical electrical test parameters are considered simultaneously and are taken as independent variables. The comparison experiment verifies the proposed yield prediction method improves on three traditional yield prediction methods with respect to prediction accuracy.

Keywords

  • semiconductor wafer fabrication system
  • rescheduling
  • fuzzy neural networks
  • yield prediction
  • decision mechanism

1. The production scheduling and yield prediction of semiconductor wafer fabrication system (SWFS)

The semiconductor wafer fabrication system (SWFS) is one of the most sophisticated manufacturing systems. This kind of manufacture system is characterised by a different type of wafer process (batch and single process), hundreds of process steps, the large and expensive device, production unforeseen circumstances and re‐entrant flow [1]. Semiconductor manufacturing orders are usually global, dynamic and customer driven since the 1990s. As a result, semiconductor manufacturers strive to achieve high‐quality products using advance manufacturing technologies (such as process planning and scheduling and digitized indicators’ prediction technologies) [2]. In recent years, production scheduling and yield prediction are always two issues above all in the complex SWFS.

An organization's competitive advantage is increasingly dependent on its response to market changes and opportunities, and in response to unforeseen circumstances (i.e. Machine breakdown, rush orders), so it is important to reduce inventory and cycle time, and improve resource utilization. Therefore, production scheduling is required to optimize the operation of SWFS and has been reviewed by Uzsoy and his colleagues [3]. SWFS operates in uncertain dynamic environments, facing with a lot of disturbances, such as machine failure, a lot of rework and rush orders [4]. Production rescheduling has been viewed as an efficient approach in responding to these uncertainties raised by the external environment and internal conditions of production [5]. In job shop and flow shop, heuristic algorithms and discrete event simulation methods are mainly applied in production scheduling problems [68]. However, the SWFS is large‐scaled, complicated system with re‐entrant flows, which is different from typical job and flow shop. Many rescheduling strategies improving traditional job shop rescheduling methods have been proposed and applied in SWFS in the recent decade [9, 10]. These methods using a single rescheduling strategy are not enough for the real‐time dynamic manufacturing environment, which is more complex with disruptive events every day. For this reason, a layered rescheduling framework is needed to select rescheduling methodologies in SWFS according to the present system status.

Yield prediction plays an indispensable role in the semiconductor manufacturing factory for its powerful function of reducing cost, increasing production and maintaining a good relationship with customers. Before a malfunction is detected, the accurate prediction model of yield will serve as a warning role and help people take proactive measures to reduce the number of defect's wafers and increase the total yield of SWFS. An accurate prediction of yield plays a useful role in releasing the plan of production and optimizing the process of production, which will make the cycle time shorter and reduce fabrication cost of average units. To offer a reasonable and acceptable price and satisfy the customers, the prediction of manufacturing costs for products is necessary if they are still under development and the accurate prediction of yield can provide some advice for Ref. [11]. To maintain the good relationship with the customers, the order's due data should be guaranteed and the accurate prediction of yield is also useful in this aspect. Some organic problems located on the wafer such as microscopic particles, cluster defects, photo‐resist, critical processing parameters would be the factors which affect the yield of the semiconductor wafer. With the statistic analysis models [12] and traditional artificial neural network (ANN) models [13], the prediction of semiconductor fabrication system's yield is difficult. A fuzzy neural network (FNN)‐based yield model for yield prediction of semiconductor manufacturing systems is proposes in this chapter. In this system, the impacted factors, which are cluster defects, the defect's key attributed parameters, key electrical test parameters, should be considered in the same time. By this way, the precision of the wafer yield's prediction is improved.

2. The application of ANN in production scheduling and yield prediction of the SWFS

For selecting a scheduling strategy, the FNN approach is widely used. FNN is also an effective methodology for prediction of discrete event manufacturing systems, control and decision‐making [14, 15]. For demonstrating the relationship between the monitoring features of a flexible manufacturing system and the conditions of tools, Li et al. [16] presented a fuzzy neural network approach. For controlling manufacturing process, Zhou et al. [17] used a fuzzy neural network approach. Chang et al. [18] created a FNN model of flow time estimation with data, which are generated from a foundry service company. The product design time was estimated with the FNN approach by Xu and Yan [19]. Chang et al. [20] used FNN approach to estimate the influence of the process on the results of the wafer fabrication in SWFS. However, the FNN approach has not been used to solve the problem of SWFS rescheduling problem. This chapter proposes the FNN‐based rescheduling decision mechanism for SWFS. This methodology can solve the uncertainty problem and express the expert knowledge in weighted values. In the neural network, the evaluation of local weight values is the knowledge modelling of control rules. Rescheduling strategies, SWFS state parameters, disturbance parameters can be identified and analysed in this model. In this model, we can build the nonlinear relationship between these three components. With this approach, the layered rescheduling approach will be selected that make the yields rapid responsiveness and high productivity of the SWFS in an environment full of randomness.

To predict the wafer yield, Tong et al. [21] proposed a neural network‐based approach through considering the clustering phenomenon of the defects in integrated circuit manufacturing. It was proved that the proposed approach was effective. For predicting wafer yield for integrated circuit with clustered defects, Tong and Chao [22] used a general regression neural network (GRNN) approach. Defect clustering patterns are simulated from three aspects: the size of chip, percentage of defects and the cluster pattern. A case study demonstrated the effectiveness of the approach of the model. For the lack of reliability and accuracy in the prediction of yield, an approach of a fuzzy set for yield learning was proposed by Chen and Wang [23]. A few of examples enhanced the reliability and precision of the forecasting of the yield. Chen and Lin [24] proposed a fuzzy‐neural system with expert opinions, which can increase the precision of semiconductor yield prediction. The artificial intelligent‐based yield forecasting models demonstrated above have some limitations that it only takes consideration of the physical parameters of wafer and the important attributed parameters of defects in wafer without considering the influence of variation of the key electrical test parameters. With the combining of neural network (NN) and memory‐based reasoning (MBR), an integrated framework for a yield management system with techniques of hybrid machine learning was given by Chung and Sang [25]. In the forecasting model of the yield, some key electrical test parameters have been taken into consideration. With the use of wafer level electrical test data, a parametric neural forecasting model was constructed by Kim et al. [26] and Kim [27]. However, these yield forecasting models have not taken the attributed parameters of defects in wafer into consideration. This chapter proposes a yield forecasting model with the consideration of the wafer electrical test parameters and important attributed parameters of defects in wafer.

3. Artificial neural network for rescheduling decision mechanism in the SWFS

3.1. Layered rescheduling framework of SWFS

A layered rescheduling framework is proposed in order to reschedule the SWFS for the unstable environment which is shown in Figure 1. In the process of rescheduling framework, a three layers of rescheduling strategies are used. a three layers are machine group layer, machine layer and the system layer. The strategies of the rescheduling implement the dynamic scheduling, the global scheduling of SWFS and the machine scheduling. To choose the particular rescheduling strategy, the optimal rescheduling decision mechanism based on FNN approach. The layered rescheduling framework is described in detail in the following paragraph.

Global scheduling of SWFS. If there are some changes in the large‐scale SWFS's condition or there are some disturbances, the rescheduling is needed and the global rescheduling of SWFS is managed for the adjustment of the global scheduling [28]. With the machine group layer's adjusted scheduling objectives, a local dynamic scheduling algorithm is applied for scheduling in the machine group layer [29]. In the end, with the machine group layer's adjusted scheduling objectives, machine scheduling is processed in real‐time and the optimal machine real‐time scheduling solutions are achieved.

Dynamic scheduling of SWFS. If there are some changes in the medium‐scale SWFS's condition or there are some disturbances, the rescheduling in the machine group layer is needed and the local dynamic scheduling of SWFS is managed. In order to adjust the local scheduling of a machine group, a local dynamic scheduling algorithm is applied. With the adjusted scheduling objectives of the machine layer taken into consideration, machine scheduling of SWFS is processed.

Machine scheduling of SWFS. If there are some changes in the large‐scale SWFS's condition or there are some disturbances, the rescheduling is just accomplished and in the same time, the machine scheduling is processed. Though they are same in the operation sequences of the lots, they are different in the operation start times of delayed lots.

FNN‐based optimal rescheduling decision mechanism. With the consideration of the statuses and disturbances to SWFS, the rescheduling layer is chosen by optimal rescheduling decision mechanism. According to the fuzzy neural network, an algorithm for the system is stated in this paper.

Figure 1.

Layered rescheduling framework of SWFS.

3.2. FNN‐based decision mechanism for rescheduling

Fuzzy neural network (FNN) is an ingenious combination of fuzzy logic and neural network, which inherits the advantages from both fuzzy system and neural network. The FNN has the characteristics of processing fuzzy information with fuzzy algorithms and learning with a high‐speed parallel structure. The FNN approach is therefore adaptable and robust, and is well suited for the SMS rescheduling problem.

The FNN‐based rescheduling decision model consists of an input layer, several hidden layers and an output layer. Input parameters connected with disturbances and state parameters are accepted in the input layer. The hidden layers calculate and transform the input parameters using fuzzy logic theory. The output layer produces the decision‐making response of the rescheduling model. More details of this method are described.

3.2.1. Input factors in the proposed FNN model

The SMS's state and disturbance parameters are treated as input of the FNN, which can be detailed as: system disturbances parameter, average queue length, stability of SMS, average relative load and average slack time.

3.2.1.1. System disturbances parameter

Since the operating environments of SMS are uncertain and dynamic, disturbances mainly include: machine failures, lot reworks and rush orders. Once a disturbance has happened, an optimal rescheduling strategy must be selected and carried out to guarantee the stability and efficiency of SMS. Disturbances are converted into machine work times to quantify their effect. The mapping of disturbances to machine work times is defined as follows.

(1) Machine failures. The processing time in SMS increases if machine failures happen. Suppose that tfrefers to the increased process time caused by all machine failures, then,

tf=MjfFfmjiMjftfjiE1

where Mjfis the failed machine group j, mjirefers to the failed machine i of machine group j, tjifrepresents the repair time of machine i of machine group j, and Ffis the set of failed machine group.

(2) Lot reworks. Lot reworks raise the output requirement of SMS. Suppose that tris the additional process time incurred by all lot reworks, then,

tr=MjrFrpjkRjrtjkrE2

where Mjrrefers to the machine group j operating the rework lots, Rjrrefers to the set of the rework lots operated by the machine group j, pjkrefers to the rework lot k operated by the machine group j, tjkrrefers to the process time of the rework lot k operated by the machine group j, Frrefers to the set of the machine group that operate rework lots.

(3) Rush orders. Rush orders also demand more of the production requirement of SMS. Suppose that tois the process time required by all rush orders, then,

to=MjoFoqjkRjotjkoE3

where Mjorepresents the machine group j, that operates the rush orders in current plan time phase, Rjorepresents the set of the lots operated by machine group j in the rush orders; qjkrepresents the lot k in the rush orders operated by machine group j, tjkois the process time of the lot k operated by the machine group j in the rush orders, Forepresents the set of the machine group which are related with rush orders.

(4) System disturbances parameter. Suppose that tdis the system disturbances parameter, denoting the total effect of disturbances on SMS scheduling. The formula to calculate tdis shown as the (4).

td=tf+tr+toE4

3.2.1.2. Average queue length

Average queue length of machine groups reflecting the utility of the machine group is affected by disturbances. L is the average queue length of machine groups affected by disturbances; and the formula is shown in (5).

L=Mj(FrFfFo)LjNE5

where Mjdenotes the machine group j, Ljmeans queue length of machine group Mj, Nrefers to the number of machine group that affected by disturbances.

3.2.1.3. Stability of SMS

The stability of SMS is defined as the deviation in predicted average start time of a rescheduled strategy from the real start time. βdenotes the stability of SMS, which is shown in (6).

β=(i,s)Stcistc|tcistcis|(i,s)StcistcqisE6

where tcisis practical start time of process stage s of product i, tcisis computational start time of process stage s of product i which optimized with a global scheduling algorithm or rescheduling strategy, qisis the number of process stage s of product i, tcis the current time when disturbance happens, Sis set of tasks of all machine group in SMS.

3.2.1.4. Average relative loads

Average relative loads denote the loads of machine groups measured from the current time to the end of the scheduling horizon which can be affected by disturbances. Let ηrepresent the average relative loads, the formula for calculation is shown in (7).

η=(i,s)SdtetistctpisMj(FrFfFo3)nj(tetc)E7

where tpisdenotes process time of process stage s of product i, tedenotes the time point when scheduling is ended, njdenotes the number of machine of machine group Mj, Sdrepresents set of tasks of machine group which affected by disturbances.

3.2.1.5. Average slack time

Average slack time represents the space that the machine groups can be adjusted when disturbances happen. Suppose tsis the average slack time, shown in (8).

ts=(i,s)Sdtetistc(ti(s+1)tistpis)(i,s)SdtetistcqisE8

3.2.2. Output variables

The output variables in the FNN output layer are related to the layered rescheduling strategies, which consists of the rescheduling in system layer, machine group layer, and machine layer. If a particular layered rescheduling strategy is selected, then the corresponding output variable is close to 1, otherwise it equals to 0. In FNN‐based rescheduling decision model, suppose that y1,y2,y3are defined as output variables, then y1,y2,y3correspond to the rescheduling in system layer, rescheduling in machine group layer, and rescheduling in machine layer, respectively.

3.2.3. The structure of FNN

There are five layers in the rescheduling decision model based on FNN, as illustrated in Figure 2.

  1. The input vector is X = [x1, x2, x3, x4, x5]T = [L, β, η, ts, td]T. The function of node input‐output is:

    fi(1)=xi(0)=xi;xi(1)=gi(1)=fi(1);i=1,2,,5E9

  • In the second layer which is the fuzzifer layer, the function of the Gauss membership is adopted.

    uij=e(xicij)2σij2,i=1,2,,5,j=1,2,,kiE10

    In this formula, cij is the centre and σijis width. The node input–output function is:

    fij(2)=(xicij)2σij2;
    xij(2)=uij=gij(2)=efij(2)=e(xicij)2σij2;i=1,2,,5,j=1,2,,kiE11

  • In the third layer as the rule layer, each node in the layer is a fuzzy rule which not only matches the front part of the fuzzy rule but also calculates the adaptive of the rule,

    aj=Πl=15ulil(xlil),j=1,2,,nE12

    In this layer, the input‐–output function is:

    fj(3)=Πl=15xlil(2)=Πl=15ulil(xlil);xj(3)=gj(3)=fj(3);j=1,2,,nE13

  • In the fourth layer which is the normalized layer. In this layer, the node numbers are the same in the third layer. It normalized the adaptive values of these rules. And the input‐output function is:

    fj(4)=xj(3)i=1nxi(3)=aji=1nai;xj(4)=gj(4)=fj(4);j=1,2,,nE14

  • The last layer is the output layer. It defuzzify the output variables. And each node describes a rescheduling strategy. While a rescheduling strategy is chose, the corresponding output is 1 or 0. The input‐output function is:

    fi(5)=j=1nwijxj(4)=j=1nwijbj;xj(5)=gj(5)=fj(5);i=1,2,3E15

  • where wijis the connection weight parameter.

    Figure 2.

    FNN structure.

    3.2.4. The strategy of the fuzzy inference

    The Mamdani‐based fuzzy inference is applied in this FNN‐based rescheduling decision model with a assumption that the fuzzy rule Ri describes the relationship between input and output. Then,

    Ri:

    IF x1 is A1i and x2 is A2i and … and xm is Ami,

    THEN y1 is B1i and y2 is B2i and … and ym is Bki,

    where

    i = 1, 2, …, n.

    n: number of rules.

    m: number of input variables.

    k: number of output variables.

    Aji: value of fuzzy linguistic variable xj.

    Bji: value of fuzzy linguistic variable yj.

    3.3. Result and discussion

    3.3.1. Experiment on the proposed FNN approach

    In this section, the experiments are conducted to evaluate the effectiveness of the proposed FNN rescheduling decision mechanism. A discrete event simulation model is run to gather the experiment data, which is based on a 6‐in. SWFS in Shanghai. This SWFS is composed by eleven machine groups, which add up to thirty‐four machines in total. And three types of wafers are put into the SWFS. The processes of all three types of wafer lots are divided into dozes of stages, which is composed by a key step and several successive normal steps. One hundred and fifty records of rescheduling decision are collected from the simulation model, and shown in Table 1. Ninety records are used in model training, and 60 are taken to evaluate the model. The presented FNN approach is compared with the back propagation network (BPN) approach and the multivariate regression methodology, since the BPN and multivariate regression approaches are widely used in the rescheduling strategy decision and proven to be competitive [30, 31]. Furthermore, the detail numerical comparison of the FNN approach, BPNN approach and multivariate regression are demonstrated as follows.

    Now, it's going to compare the experimental results which are made by these three methods. Figure 3 shows the optimal rescheduling decision value and the model outputs. It shows that the FNN rescheduling method has the best convergence. We also contrast the RMSE and the decision coefficients R2of these three methodologies in Table 2. The FNN has the best performance for the RMSE which is 0.042 and has the largest of the R2values which is 0.9941. Hence, the rescheduling decision based on FNN has the best performance in these three methods.

    Samples noAverage queue length of disturbed machine stations x1 (lot)Stability of scheduling x2 (h)Average load of disturbed machine stations x3 (100%)Average slack time of disturbed machine stations x4 (h)Disturbance x5 (h)Optimal rescheduling decision objective
    Rescheduling in machine layer y1Rescheduling in machine group layer y2Rescheduling in system layer y3
    111.100.596.422.14100
    220.780.526.512.01100
    300.810.465.161.76100
    400.480.15.811.21100
    520.490.144.621.42100
    610.520.125.541.79100
    720.740.175.161.13100
    800.760.074.491.64100
    960.380.294.861.4100
    1050.370.265.172.21100
    ..................
    14164.230.753.819.81001
    14244.150.765.649.18001
    14354.010.764.979.77001
    14420.810.381.879.87001
    14540.870.372.4110.11001
    14620.680.422.189.42001
    14720.720.352.189.76001
    14820.910.232.79.13001
    14910.870.212.979.73001
    15020.870.312.7710.18001

    Table 1.

    One hundred and fifty records for numerical experiments.

    Figure 3.

    The relationship between the rescheduling strategy output and ideal target output for the FNN, BPNN and multivariate regression methods. (a) FNN‐based output value, (b) BPNN‐based output value and (c) multivariate regression‐based output value.

    Rescheduling strategy modelRMSER2
    R2Y1R2Y2R2Y3
    FNN0.00420.98800.97620.9941
    BPNN0.01320.97450.91780.9274
    Multivariate regression0.08970.858870.755660.70813

    Table 2.

    Comparison of RMSE and decision coefficients among the FNN, BPNN and multivariate regression methods.

    3.3.2. Experiment on the proposed rescheduling decision mechanism

    The FNN rescheduling decision mechanism is used in our layered rescheduling method (Method 1). There are two other different rescheduling methods. One is the monolayer‐based rescheduling approach (Method 2). Another one is the first come first served (FCFS) approach (Method 23). In our method, the FNN rescheduling decision mechanism figures out the optimal rescheduling approaches which include the global scheduling of SWFS, the dynamic scheduling and the machine scheduling. By contrast, the Method 2 only considers the rescheduling of the machine group layer. But in practice, the Method 3 is widely used in the Fab. In order to prove the efficiency of our approach, we also compared these three rescheduling methods in terms of the machine utilization and the daily movement, which are the important system targets for SWFS.

    In the case study, the data are collected from a 6‐in. SWFS in Shanghai. It products three kinds of lots which are renamed as A, B and C. The whole process is shown in Table 4. This SWFS has eleven key machine groups (shown in Table 3). which has 34 machines with MTTF and MTTR parameters. They are explained in Section 5. The SWFS simulation model is built by eM‐plant 7.0 software. In the simulation, it took 12 days, including a 5‐day warm‐up. Ten times repeated trials of the same stimulation, in which the initiated loads of machines were different, were performed (3 rescheduling methods 10 replications). The results are shown in Figures 4 and 5, which illustrate:

    1. Method 1 performs well in the rescheduling decision in the SWFS.

    2. Method 1 outperforms method 2 and 3, which indicates the layered rescheduling method is more suitable than the conventional FCFS rescheduling approach and monolayer‐based rescheduling approach in the complex SWFS.

    Machine
    group
    number
    Processing
    type
    Number
    of
    machine
    Batch
    size
    MTBFMTTR
    1Ion implant31701
    2Ion implant41701
    3Diffusion351002
    4Diffusion451102
    5Etching21901
    6Etching41801
    7Etching31601
    8Etching21701
    9Lithography41901
    10Lithography31801
    11Lithography211001

    Table 3.

    Configuration of SWFS.

    Stage
    number
    Number of
    time
    period by
    product A
    Machine
    group
    number of
    product A
    Process
    time of
    product A
    by key
    machine
    t/hour
    Number
    of time
    period by
    product B
    Machine
    group
    number of
    product
    B
    Process
    time of
    product B
    by key
    machine
    t/hour
    Number
    of time
    period by
    product C
    Machine
    group
    number of
    product C
    Process
    time of
    product C
    by key
    machine
    t/hour
    118111011101
    2171181181
    31101451251
    41911101181
    5501111101
    6301161436
    74241101181
    8181191251
    9151601223
    10133101181
    11101141253
    122101181101
    13101211236
    14442323181
    151611101291
    16211181241
    17342252301
    18191131181
    19151251361
    20181191211
    21333111181
    22151251371
    23424191223
    24191301251
    25342323101
    262012101191
    27181271241
    28171111
    29233
    30121

    Table 4.

    Lot products whole process.

    Figure 4.

    The utilization of machine group.

    Figure 5.

    The utilization of machine group.

    4. Artificial neural network approach for die yield prediction in the SWFS

    4.1. FNN‐based yield prediction model

    The yield prediction model based on FNN is composed of three parts, which are an input layer, an output layer and several hidden layers. The three parts do the different jobs respectively. The input layer serves to accept input parameters connected with yield. The output layer does the job to get the yield response of the prediction model. The hidden layers are applied to compute and convert the input parameters which are on the basis of fuzzy logical theory. The following sections show a more detailed yield prediction model based on FNN.

    4.1.1. Variables in FNN input layer

    The input variables in the FNN prediction model include the following parameters: the critical electrical test parameters, wafer physical parameters and key parameters of defects in wafer. Critical process parameters refer to those electrical test parameters which are generally tested at the end of the wafer processing, and they have notable influences on the yield. Wafer physical parameters mainly refer to the size of the chip. Key parameters of defects in wafer Contain a number of defects, clustering parameter, mean number of defects in each chip and mean a number of defects in each unit area. Among these input variables, the critical electrical test parameters and clustering parameters are complex, and we will discuss them in the following sections.

    4.1.1.1. Critical electrical test parameters

    In the process of fabricating complex semiconductor wafer, there are more than one hundred electrical test parameters related to the probed wafer. This paper mainly does the research on establishing the exact relationship of a small number of critical electrical test parameters with yield. These critical electrical test parameters have significant influence on yield, and they have high correlating coefficients or exhibit a ‘cliff’ in the correlation graphs which means they can quickly improve the yield. Wong [32] proposed the hybrid statistical correlation analysis method, and the critical electrical test parameters are identified based on this method. Here, we remove some details of these electrical test parameters for confidentiality.

    4.1.1.2. Clustering parameter

    Clustering parameter displays cluster or clumps degrees of wafer defects in the defect map [33]. Suppose that the clustering parameter is expressed by c, shown in Eq. (1).

    c=min{sv2v¯2,sw2w¯2}E22

    where the sample mean and variance of Viis represented by V¯2and Sv2; and the sample mean and variance of Wiare represented by Sw2. Viand Wiare a series of defect intervals on the x and y axis defined as:

    vi=x(i)x(i1),i=1,2,...,nE23
    wi=y(i)y(i1),i=1,2,...,nE24

    where x(i) refers to the ith smallest defect coordinates on x axis, and similarly, y(i) refers to the ith smallest defect coordinates on y axis, x(0) = y(0) = 0, and n refers to the quantity of defects on one wafer. If the defects are randomly scattered, the value of CI is close to 1, and when clustering of defects appears, the value of CI is likely to be greater than 1.

    4.1.2. FNN structure

    There are five layers in the rescheduling decision model based on FNN, as illustrated in Figure 6.

    Figure 6.

    FNN model structure.

    1. The input vector is X = [x1, x2, x3, …, xm]. The function of node input‐output is:

      fi(1)=xi;xi(1)=gi(1)=fi(1);i=1,2,mE4-4

    2. In the second layer which is the fuzzifer layer, the function of the Gauss membership is adopted.

      uij(xi)=e(xicij)2σij2E25

      In this formula, cij is the centre and σijis width. The node input‐output function is:

      fij(2)=(xi(1)cij)2σij2;xij(2)=uij(xi(1))=gij(2)=efij(2)=e(xicij)2σij2E26

      where i=1,2,mand j=1,2,,li.

    3. In the third layer as the rule layer, each node in the layer is a fuzzy rule which not only matches the front part of the fuzzy rule but also calculates the adaptive of the rule,

      aj=Πi=1muili(xi(1)),j=1,2,,nE27

      In this layer, the input‐output function is:

      fj(3)=Πi=1mxili(2)=Πi=1muili(xi(1));xj(3)=aj=gj(3)=fj(3);j=1,2,,nE28

    4. In the fourth layer which is the normalized layer. In this layer, the node numbers are the same in the third layer. It normalized the adaptive values of these rules. And the input‐output function is:

      bj=aji=1nai,j=1,2,,nE29

      Node input‐output function in this layer is as follows.

      fj(4)=xj(3)i=1nxi(3)=aji=1nai;xj(4)=bj=gj(4)=fj(4);j=1,2,,nE30

    5. The last layer is the output layer. It defuzzify the output variables. And each node describes a rescheduling strategy. While a rescheduling strategy is chose, the corresponding output is 1 or 0. The input‐output function is:

      f(5)=j=1nwjxj(4)=j=1nwjbj;Oout=x(5)=g(5)=f(5)E31

      where Wjis connection weight parameter of output layer, and Ooutis the output of FNN model.

    4.2. Case study

    In this section, the experiments are conducted to evaluate the effectiveness of the proposed FNN method. This section presents a numerical experiment study to demonstrate the effectiveness of the approach proposed. Seven hundred and twenty wafer samples are obtained from a 6 in. SWFS in Shanghai, and each sample includes 360 records of wafer yield. Five hundred and fifty‐two records are used in model training, and 168 are taken to evaluate the model. The attributes contained in each record are, in order, number of defects, clustering parameter, die yield, mean number of defects per unit area, chip size parameter, mean number of defects per chip, and 28 electrical test parameters, which is shown in Table 5. Each feature is acquired by test during the critical manufacturing process. The presented FNN approach is compared with the Poisson model, negative binomial model and BPNN approaches, since the three approaches are widely used in research on yield predicting and have been proved to be competitive [3436]. Furthermore, the detail numerical comparison of the FNN approach, Poisson model, negative binomial model and BPNN approaches are demonstrated as follows.

    Record Number
    of
    defects  
    Mean number
    of
    defects/chip 
    … Chip
    size parameter (cm2) 
    Clustering parameter Process parameter 1 Process parameter 2 … Process parameter 28 Yield (%) 
    1210.1409410.51836322.74980.0608651.1695730.86577
    2450.3020110.70814324.46340.0615731.1724810.73826
    3160.1073810.65597322.19030.0606481.1762230.89262
    4210.1409411.0277313.96590.0650361.1622160.87248
    5460.3087210.59023313.09530.0682861.1834610.73826
    6350.234910.73168323.98320.0618671.1643840.81879
    770.0469810.73807315.90010.0595391.1774360.95302
    8490.3288610.75913310.93560.0608871.1807990.72483
    990.06040310.57871310.65710.064391.1684140.9396
    10330.2214810.83289310.09210.0686951.1799830.80537
    11370.2483210.69348322.25670.0685361.1607160.77181
    12480.3221510.9089323.72770.0699591.1622590.73154
    13120.08053710.6154321.12460.0674031.1763340.91946
    14330.2214810.85056313.05740.0687781.1708390.78523
    15470.3154411.0109313.51330.0632451.1727140.72483
    705310.319591.441.0678310.24840.0658541.1687340.82474
    706350.360821.440.67849321.16130.068831.1660410.83505
    707610.628871.441.00661314.37520.0669451.1849360.75258
    708710.731961.440.95411321.34720.0614561.1646270.7732
    709820.845361.441.5379311.55620.0604571.1603690.73196
    710320.32991.441.2404313.41140.0671971.1602290.80412
    711790.814431.441.5407317.31920.0674541.1759120.7732
    712720.742271.442.4467323.0990.0622851.1719420.76289
    713580.597941.442.32601318.68610.0683441.1748140.76289
    714570.587631.441.0014316.31940.0659811.1739570.79381
    715730.752581.441.3139322.19910.0659621.1808530.73196
    716460.474231.441.6882320.42910.0695171.1662210.79381
    717800.824741.441.57021316.61950.0624611.162930.74227
    718380.391751.442.0034317.06040.066061.1717950.79381
    719180.185571.440.81646323.22130.0649851.1672340.8866
    720720.742271.440.95479319.67680.0686211.1759820.73196

    Table 5.

    Partial wafer measurements parameters and yield.

    4.2.1. Experiment on fuzzy neural network

    The algorithm was programmed in Matlab 6.5, and ten factors were treated as input of the model, which are mean number of defects per chip, chip size, clustering parameter, mean number of defects per unit area, the number of defects per wafer and another five critical electrical test parameters.

    Twenty‐six rules for classification were identified the fuzzifier layer in the model. The 552 samples were utilized in the training of the FNN model with fivefold cross‐validation. The learning process was explored in Figure 7. Afterward, the trained model was assessed by another 168 samples, which is demonstrated in Table 6. Furthermore, the linear regression analysis of the output of the FNN model is detailed in Figure 8.

    Figure 7.

    Fuzzy neural network learning curve.

    SamplesThe actual yieldThe predicted yieldRelative error
    10.724830.725140.000432
    20.691280.691180.000146
    30.82550.83370.009929
    40.751680.743290.011168
    50.758390.7580.000513
    60.731540.730310.001677
    70.899330.898760.000632
    80.751680.752110.000576
    90.744970.744090.001179
    100.751680.750020.002209
    110.76510.76370.001824
    120.852350.856580.004967
    130.899330.904850.006143
    140.93960.939180.000452
    150.812080.815590.00432
    1600.752580.733230.025707
    1610.865980.843150.026365
    1620.742270.743760.002008
    1630.88660.886060.000609
    1640.731960.728170.005175
    1650.77320.768270.006381
    1660.752580.74560.009278
    1670.824740.834130.011391
    1680.835050.822490.01504

    Table 6.

    The predicted yield based on FNN.

    Figure 8.

    The linear regression analysis of the output of the FNN model.

    4.2.2. Experiment of Poisson model

    The Poisson model was built to predict wafer yield as follows.

    Y=eD0AE38

    In the model, Y means the wafer yield, D0denotes the defect density, and Ais the chip size. The yield was forecasted by Poisson model, and the results of 168 samples can be found in Table 7. The lineal correlation analysis between the actual wafer yield and the prediction value is shown in Figure 9.

    SamplesThe actual yieldThe predicted yieldRelative error
    10.724830.576750.20429
    20.691280.481170.30395
    30.82550.795970.035769
    40.751680.65520.12836
    50.758390.668530.11849
    60.731540.580640.20627
    70.899330.892180.007953
    80.751680.650810.13419
    90.744970.616790.17206
    100.751680.612670.18493
    110.76510.596440.22044
    120.852350.839880.014634
    130.899330.874390.027733
    140.93960.928830.011459
    150.812080.749320.077284
    1600.752580.552220.26623
    1610.865980.754220.12905
    1620.742270.490380.33936
    1630.88660.849340.042026
    1640.731960.404310.44763
    1650.77320.483160.37512
    1660.752580.435470.42137
    1670.824740.754220.085502
    1680.835050.743110.1101

    Table 7.

    The predicted yield based on the Poisson model.

    Figure 9.

    The linear regression analysis of the output of the Poisson model.

    4.2.3. Experiment of negative binomial model

    The negative binomial model is built to predict wafer yield as follows.

    Y=11+D0A/aaE38

    In this model, Ymeans the defect‐limited wafer yield, D0denotes the defect density, and Ais the cluster coefficient. The yield was forecasted by negative binomial model and the results of 168 samples can be found in Table 8. The lineal correlation analysis between the actual wafer yield and the prediction value is shown in Figure 10.

    SamplesThe actual yieldThe predicted yieldRelative error
    10.724830.63950.11772
    20.691280.570010.17543
    30.82550.812570.015659
    40.751680.698850.070289
    50.758390.709190.064881
    60.731540.642390.12187
    70.899330.897080.002499
    80.751680.695460.074787
    90.744970.669490.10132
    100.751680.666370.11349
    110.76510.654170.14499
    120.852350.850370.002326
    130.899330.880980.020408
    140.93960.931020.009135
    150.812080.77370.047256
    1600.752580.613460.18486
    1610.865980.77480.10529
    1620.742270.56690.23626
    1630.88660.857460.032863
    1640.731960.503410.31225
    1650.77320.561520.27377
    1660.752580.526260.30072
    1670.824740.77480.06055
    1680.835050.765460.083336

    Table 8.

    The predicted yield based on the negative binomial model.

    Figure 10.

    The linear regression analysis of the output of the negative binomial model.

    4.2.4. Experiment of back‐propagation neural network

    A three layer BPNN is applied to predict wafer yield with ten input factors as same as the proposed FNN. The number of hidden neurons is determined by the empirical formula and selected to be 35. The yield was forecasted by BPNN and the results of 168 samples can be found in Table 9. The lineal correlation analysis between the actual wafer yield and the prediction value is shown in Figure 11.

    SamplesThe actual yieldThe predicted yieldRelative error
    10.724830.734950.013962
    20.691280.72630.050661
    30.82550.823850.001994
    40.751680.763180.015293
    50.758390.759850.001919
    60.731540.750420.025807
    70.899330.898760.000632
    80.751680.761040.012456
    90.744970.759320.019268
    100.751680.739710.015921
    110.76510.766660.002033
    120.852350.830680.025423
    130.899330.873830.028359
    140.93960.934710.005204
    150.812080.794690.021415
    1600.752580.727240.033671
    1610.865980.85810.009105
    1620.742270.70960.044007
    1630.88660.891510.005535
    1640.731960.688050.059996
    1650.77320.727220.059471
    1660.752580.715760.048929
    1670.824740.826830.00253
    1680.835050.828450.007907

    Table 9.

    The predicted yield based on BPNN.

    Figure 11.

    The linear regression analysis of the output of BPNN.

    4.2.5. Results discussion

    Figure 12.

    The relationship between the actual yields and predicted yields based on the FNN, BPNN and Poisson model and negative binomial model approach.

    Aiming to assess the performance the proposed FNN methods, experiment with three contrast method was conducted for comparison. The lineal correlation analyses between the actual wafer yield and the prediction value of four methods are shown in Figure 12, which indicates that the FNN method outperforms other three methods from the view of convergence. The results of four methods in the RMSE and correlation coefficient R is presented in Table 10. The RMSE of the FNN method is 0.017, which is the smallest value above the four methods, and the R of the FNN‐based model is 0.941, which is larger than other three methods. It indicates that the proposed FNN‐based approach is more accurate and effective than other three methods, which are widely used in the yield predicting.

    Yield prediction modelThe actual yieldThe predicted yieldRMSER
    AverageSDAverageSD
    Poisson model0.808640.061680.653940.186940.01690.637
    Negative binomial model0.700470.147890.01230.693
    BPNN0.806910.057360.00240.886
    FNN0.808380.057110.00170.941

    Table 10.

    The comparisons of RMSE and correlation coefficients among the FNN, BPNN, Poisson model and negative binomial model.

    5. Conclusion

    The artificial neural networks (ANN) have a wide range of applications. For example, in complex discrete event manufacturing systems, they can be used to control, make decision and predict. SWFS is exactly such a complex manufacturing system. It has many characteristics, such as a mix of different process types, re‐entrant flows, very expensive equipment and sequence dependent setup times and so on. In order to get more applications of ANN used in quality analysis and production scheduling in the semiconductor wafer fabrication system, this chapter implements two novel fuzzy neural networks that are used in the yield prediction of SWFS and rescheduling decision separately.

    In the respect of rescheduling decision, this chapter puts forward a new method using a FNN model with which a system can make itself adapted to the current states and disturbances. In uncertain dynamic environments, current states and disturbances of the system are mathematically characterized. Rescheduling decision model, which assuming FNN builds the relationship between the inputs (i.e. disturbance, system state parameters) and the outputs (i.e. disturbance, system state parameters) of FNN. According to the current system disturbances, an optimal rescheduling method which can be used to schedule the semiconductor wafer fabrication lines is chosen by the make‐decision model. We do experiment studies in Shanghai, which are based on 6‐in. SWFS. The proposed rescheduling decision mechanism is proved to be effective by the linear regression between ideal targets and output of FNN. The rescheduling decision‐making method which is proposed is demonstrated to be accurate by comparing with regression and traditional BPNN. We also do the comparison between the layered rescheduling method which is on the basis of FNN rescheduling decision mechanism and the two methods that are FCFS approach and the rescheduling approach based on the monolayer. The results indicate that, in respect of machine utilization and daily movement, layered rescheduling method, which is on the basis of FNN rescheduling decision mechanism, is superior to the other two approaches.

    A yield prediction method for semiconductor manufacturing systems which is on the basis of new fuzzy neural networks is proposed for the yield prediction. This method builds the yield prediction model based on FNN by using the following parameters as input variables, which are the number of defects in each wafer, mean number of defects in each chip, mean number of defects in each unit area, clustering parameter, chip size and five critical electrical test parameters.

    According to the data from the experiment studies in Shanghai which are based on 6‐in. SWFS. The proposed rescheduling decision mechanism is proved to be effective by the linear regression between ideal targets and output of FNN. The rescheduling decision‐making method which is proposed is demonstrated to be accurate by comparing with regression and traditional BPNN. The approach proposed in this paper has the advantage that it considers more variables’ influences than other model such as negative binomial yield model, BPNN model and Poisson yield model. The variables here include physical parameters of wafer, key attributed parameters of defects and wafer electrical test parameters on wafer yield and so on. In a word, the model proposed in this paper is more accurate than the other traditional yield prediction approaches.

    Acknowledgments

    This work was supported by the State Key Program of the National Natural Science Foundation of China under Grant No. 51435009.

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    Jie Zhang, Junliang Wang and Wei Qin (October 19th 2016). Artificial Neural Networks in Production Scheduling and Yield Prediction of Semiconductor Wafer Fabrication System, Artificial Neural Networks - Models and Applications, Joao Luis G. Rosa, IntechOpen, DOI: 10.5772/63444. Available from:

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