HAN Xiao-meng (韩晓萌), LI Yan-feng (李彦锋), LIU Yu (刘 宇), HUANG Hong-zhong (黄洪钟)

School of Mechanical Electronic and Industrial Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Reliability Analysis of Lithography Wafer Stage Based on Fuzzy Bayesian Networks

HAN Xiao-meng (韩晓萌)*, LI Yan-feng (李彦锋), LIU Yu (刘 宇), HUANG Hong-zhong (黄洪钟)

SchoolofMechanicalElectronicandIndustrialEngineering,UniversityofElectronicScienceandTechnologyofChina,Chengdu611731,China

Bayesian network (BN) is a powerful tool of uncertainty reasoning. Considering the insufficient information, incorporating fuzzy probability into BN is an effective method. Fuzzy BN was used to solve this problem. In this paper, fuzzy BN was applied in wafer stage system, which was an important part of lithography. BN of wafer stage was transferred from fault tree (FT). The quantitative assessment based on fuzzy BN was carried out. The Birnbaum importance factors of basic events were calculated. Therefore, the system failure probability and the vulnerable components could be gotten.

lithography;waferstage;fuzzyBayesiannetwork(BN);reliabilityanalysis

Introduction

I the production of integrated circuit chips, it is one of the most important procedures to “print” the design configuration of chips to the photoresist on the surface of silicon wafer. Lithography machine is the critical equipment to achieve this process. The work stage of lithography contains two parts, the wafer stage and the mask stage. The exposure of silicon is achieved through the cooperation of these two stages. To a large extent, the accuracy and productivity of wafer stage determine the resolution and exposure efficiency of lithography.

The basic work of the wafer stage is to carry the wafer and move according to the set speed and direction in the process of exposure in order to achieve the accurate transfer of graphics on the mask plate to the fields on silicon wafer. In this paper, the wafer stage adopts the structure of double motion platforms. In the system, the preparatory work of exposure such as alignment is transferred to the second motion platform which moves independently with the motion platform under exposure at the same time. In this way, it greatly shortens the working hours of the wafer stage and improves the productivity[1].

There are many reliability modeling methods such as reliability block diagram, fault tree analysis(FTA), and binary decision diagram. Among them FTA is widely used in the system reliability modeling. However, FTA contains a large amount of complex Boolean calculation that leads to inconvenience in quantitative evaluation process. Bayesian network (BN) offers another effective tool to handle the problem of FTA. BN is a powerful tool for uncertainty reasoning which can be achieved from the fault tree and gets the prior probability of top event and the posterior probability of basic events while avoiding the minimal cut sets and disjoint process[2]. While in practical engineering analysis, the information is always insufficient and rough. To solve this problem, fuzzy probability method is an effective solution to evaluate the occurrence probability of each event.

The reminder of this paper is organized as follows. The structure and function of wafer stage is introduced and the reliability block diagram is established in Section 1. Section 2 briefly reviews the theory of BN. The theory of fuzzy probability and fuzzy BN is presented in Section 3. The detailed process of reliability evaluation of wafer stage based on BN is presented in Section 4. Finally, Section 5 gives some conclusions.

1 Structure of Lithography Wafer Stage

The wafer stage contains drive, cooling, and damper systems. The structure of double motion platforms is the important characteristics of the wafer stage in this system. The double motion platforms with coarse-fine laminated form are used to meet the requirement of nanometer-level positioning accuracy. Each motion platform consists of one macro platform with three degrees of freedom and one micro platform with six degrees of freedom. The cooling system adopts water cooling structure to cool the coil of motors in macro and micro platforms. The damper system consists of air spring and active vibration system. Air spring is used to restrain the vibration transmission between the base seat and balance block. Active vibration system contains balance block, air bearings, and anti-drift motors, which can balance the reaction of double motion platforms.

The relationship diagram of function and structure of this system is shown in Fig.1. And its reliability block diagram is shown in Fig.2. According to Fig.2, any unit’s failure can lead to the entire system failure.

2 The Basic Theory of BN

BN is first proposed by professor Pearl from University of California, which has turned into a popular formalism for uncertainty reasoning and has been applied in a variety of fields. BN includes a directed acyclic graph (DAG) and a conditional probability table (CPT) for each node. DAG is the qualitative part of BN which is composed of a set of nodes, where each node represents a variable. And the aces between nodes represent the relationship between the corresponding variables. CPT is the quantitative part of BN. It describes the probability of each state of the node conditioned on all the combinations of the states of its parent nodes[3].

Given all the nodes’ CPTs, the joint probability distribution contains all the nodes that can be obtained. The conditioned independency assumption simplifies the joint probability distribution. Then, the joint probability distribution of a set of variablesU={U1,U2, …,Un} is

(1)

wherePa(Ui) is the parent set of nodeUi.

3 Fuzzy BN

3.1 Fuzzy number and membership function

In traditional BN, the failure probability of each node is an accurate value which is difficult to obtain. Therefore the fuzzy probability is used to describe the failure probability of each event. There are various forms of membership function for fuzzy numbers. The triangular membership function is widely used because of its simple algebraic operation.

The fuzzy probability ofXcan be expressed by triangular fuzzy number (m-α,m,m+β).

The triangular membership function is

Fig.1 The function-structure relationship diagram of wafer stage

Fig.2 Reliability block diagram of wafer stage

(2)

3.2 Fuzzy BN reliability analysis

Assume thatxi(i=1, 2, …,n) is root node andyj(j=1, 2, …,m) is middle node, andTis leaf node. If the fuzzy probability of each root node is known asP(x1),P(x2), …,P(xn), the failure probability of leaf nodeTis[4]

P(x1)P(x2)…P(xn),

(3)

wherePa(T) is the parent set of nodeT, andPa(y1) is the parent set of nodey1. Andexcept(T)={x1,x2, …,xn,y1,y2, …,ym}.

An important inference task of BN is computing the posterior probability distribution of variables given the evidence (i.e., the state of one or more variables is known with certainty).When the leaf nodeThas failed, the posterior probability of root nodexiis

(4)

4 FTA of Lithography Wafer Stage Based on BN

4.1 BN of wafer stage

According to the structure of wafer stage, we choose the failure mode named “wafer stage doesn’t work” as the top event and build the fault tree. The following assumptions are made:

(1) the bottom events are statistically independent;

(2) each bottom event only has two states: fail and work well;

(3) there aren’t outside interference factors.

The fault tree is shown in Fig.3. The meaning of each symbol is shown in Table 1.

All the events in FT and their logic relationship are mapped to the nodes in BN. Then the corresponding BN is shown in Fig.4.

Fig.3 Fault tree of “wafer stage doesn’t work”

FaultcodeFaultnameFaultcodeFaultnameFaultcodeFaultnameTWaferstagedoesntworkS13AirspringleaksX10SynchronousbeltunitIfailsS1CoolingsystemfailsS14AirbearingunitⅢcantbefullyfloatedX3'AirbearingunitIIisblockedS2DrivesystemfailsS15AirbearingunitIcantbefullyfloatedX4'TheairpressureofairbearingunitIIisinsufficientS3DampersystemfailsS15'AirbearingunitIIcantbefullyfloatedX5'PlanarmotorunitIIfailsS4WatercoolingstructureIIfailsX1CoolingstructureIleaksX6'ScanningmotorunitIIfailsS5WatercoolingstructureIfailsX2ThepipingofcoolingstructureIisblockedX7'SteppingmotorunitIIfailsS6MotionplatformAcantrunX1'CoolingstructureIIleaksX8'VerticalmotorunitIIfailsS7MotionplatformBcantrunX2'ThepipingofcoolingstructureIIisblockedX9'LinearmotorunitIIfailsS8AirspringfailsX3AirbearingunitIisblockedX10'SynchronousbeltunitIIfailsS9ActivevibrationsystemfailsX4TheairpressureofairbearingunitIisinsufficientX11TheairspringagesS10MacroplatformIcantrunX5PlanarmotorunitIfailsX12ThecapsuleofairspringisdamagedS11MicroplatformIcantrunX6ScanningmotorunitIfailsX13ThesealsofairspringisbrokenS12CableplatformIcantrunX7SteppingmotorunitIfailsX3″AirbearingunitⅢisblockedS10'MacroplatformIIcantrunX8VerticalmotorunitIfailsX4″TheairpressureofairbearingunitⅢisinsufficientS11'MicroplatformIIcantrunX9LinearmotorunitIfailsX14Anti-driftmotorunitfailsS12'CableplatformIIcantrun

Fig.4 BN for wafer stage

4.2 Quantitative analysis based on BN

For the sake of brevity, the middle event S6 is taken as an example to be analyzed. The failure probability of S6 is assumed as 10%. And the air bearing unit consists of four air bearings at the bottom of macro platform. Each of the scanning motor unit, stepping motor unit, and vertical motor unit consists of more than one voice coil motor. In the preliminary test, the values of failure probabilities of X3, X4, X5, X6, X7, X8, X9, and X10 are equal to 9%, 9%, 5%, 17%, 17%, 17%, 4%, and 22% respectively. We adopt the 10% deviation as the limit. The fuzzy probability of each root node (i.e., the failure probability of each corresponding basic event in the FT) is shown in Table 2.

Table 2 Prior probability of root nodes

According to Eq.(3), the failure probability of top event S6 can be gotten as follows andU={S6, S10, S11, S12, S15, X3, X4, …, X10}:

P(U) =P(S6, S10, S11, S12, S15, X3, X4, …, X10)=
P(S6|S10, S11, S12)·P(S10|S15, X5)·
P(S11|X6, X7, X8)·
P(S12|X9, X10)·P(S15|X3, X4)·
P(X3)·P(X4)·P(X5)·
P(X6)·P(X7)·P(X8)·P(X9)·P(X10),

The fuzzy failure probability of top event S6 is (0.0866, 0.0958, 0.1049). The minimum probability of S6 is 0.0866 and the maximum probability of S6 is 0.1049.

Given the evidence that the system has failed, the posterior probability of each root node can be calculated and is shown in Table 3. The result shows that X10, X6, X7, and X8 are the most probable causes that induce the system failure.

Table 3 Posterior probability of root nodes

4.3 Sensitivity analysis

Sensitivity analysis can quantify the importance of each basic event and the impact that the reliability improvement of each basic event has on the overall system reliability. This paper uses the Birnbaum importance factor to perform the sensitivity. The Birnbaum importance factor of a basic event Xiis defined as[5]:

I(Xi)=E[R(SYSTEM|Xi=0)-

R(SYSTEM|Xi=1)],

(5)

whereEmeans the gravity values of fuzzy subset.

As an example, the Birnbaum importance factors of basic events of S6 are calculated and are shown in Table 4.

Table 4 Birnbaum importance factors of basic events of S6

So the Birnbaum importance factors of basic events of S6 are sorted by their values as follows:

I(X10)>I(X6)=I(X7)=I(X8)>I(X3)=
I(X4)>I(X5)>I(X9).

5 Conclusions

Fuzzy probability has been incorporated into BN for reliability analysis of lithography wafer stage. The basic theories of Bayesian network and fuzzy Bayesian network are expounded. The function and structure of wafer stage and the reliability block diagram are established. And BN is gotten from FT. Then, the quantitative assessment based on fuzzy BN is carried out. And the Birnbaum importance factors of basic events of S6 are calculated. The result shows that the synchronous belt unit is vulnerable and the scanning motor, stepping motor, and vertical motor units are the key basic events of the middle event S6. In a word, fuzzy BN is a powerful tool for FTA and possesses strong engineering practicability.

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the Fundamental Research Funds for the Central Universities, China (Nos. ZYGX2011J090, ZYGX2011J084)

1672-5220(2014)06-0753-04

Received date: 2014-08-08

* Correspondence should be addressed to HAN Xiao-meng, E-mail: hxmeng2012@163.com

CLC number: TN305 Document code: A