Performance Evaluation and Schedule Optimization of Multi-Cluster Tools with Process Times Uncertainty

Shengwei Ding Jingang Yi Mike Tao Zhang and Raha Akhavan-TabatabaeiDept of IE & OR Dept of Mechanical Eng AzFSM (Fab 12/22/32) Industrial Eng

University of California Texas A&M University Intel Corporation Berkeley, CA 94720, USA College Station, TX 77843, USA Chandler, AZ 85248, USA

Given the sensitive and proprietary nature of the semiconductor industry, only normalized performance data are used in this paper.

Abstract – Performance evaluation and schedule optimi-zation of cluster tools is challenging, especially when the process times with uncertainty. In this paper, the Critical Path Method (CPM) is used to analyze the throughput of multi-cluster tools with fixed action sequence. Slacks of non-critical path actions are studied and integrated to find the average cycle time of the cluster tool. A thin film tool from Novellus is used as an example to explain this methodology.

Index Terms – Critical Path Method (CPM), Network Model, Event Graph, Cluster Tool, Cycle Time (CT)

I. INTRODUCTION

Modeling, analysis and scheduling of multi-cluster tools is critical to improve the productivity and enhance the design of such equipment. Scheduling of the cluster tool concerns how to manage the relative sequence and timing of all these activities in order to achieve high throughput in steady state. The conventional method to analyze such problems is simulation, but it is time consum-ing to generate good results using these models. Alterna-tively, analytical methods can be more efficient and straightforward.

In the scope of this study, we assume a given and fixed activity sequence and focus on scheduling of activity times and the resulting throughput. Readers can refer to [ 8], [ 7] for guidelines of optimal sequence search and gen-eration. If the process and transfer times are deterministic, a network structure can be generated to model one steady state cycle and CPM (Critical Path Method) can be applied to derive the cyclic schedule throughput as in [ 7]. How-ever, if probabilistic process times or transfer times are involved, the system becomes much more complicated. Probability network analysis, sometimes noted as PERT (Program Evaluation and Review Technique), extends the CPM with uncertainty assumptions and provides means to analyze multi-cluster tools with probabilistic action times.

CPM programming and control methods have been used quite extensively in research and development sched-uling, construction planning and resource allocation mod-

els [5]. When the model assumes that activity durations are generally distributed random variables, the problem of evaluating any one of these measures, though conceptually simple, becomes computationally intractable. The diffi-culty arises because of the statistical dependence intro-duced by arcs that are shared among multiple paths. Customarily, analytical procedures are not computationally efficient for a general PERT problem. Attempts have been made to develop approximation techniques.

In this paper, efforts have been made to build the net-work structure of multi-cluster tools and analyze the throughput with reasonable assumptions of probabilistic action times. The remainder of the paper is as follows. In the Modeling section, we introduce how to construct ac-tion sequences of a cluster tool and model them into a di-rected network. In the Analysis section, using CPM method to calculate cycle time is discussed and the case of probabilistic action timings are studied. In the Implementa-tion section, a thin film tool from Novellus is used as an example to explain the proposed methodology.

II. MODELING

A cluster tool consists of three types of modules: proc-ess modules (PM), transfer modules (TM), and cassette modules (CM). Process modules execute the semiconduc-tor manufacturing processes, cassette modules store wafers for load and unload, and transfer modules move the wafers among process modules as well as between process and cassette modules. In general, a single-cluster tool consists of one transfer module and a few cassette and process modules and a multi-cluster tool consists of several single-clusters that are inter-connected via buffer modules. To simplify the description of modeling and analysis, we con-sider the example of a single-cluster tool with two process modules, P1 and P2, and a single-blade transfer robot R as shown in Figure 1 (C1, C2 denote the cassette modules). Wafer flow is denoted as: C1 P1 P2 C2, where the arrow stands for wafer transfer by robot R. We assume that R is a double blade robot.

Proceeding of the 2006 IEEEInternational Conference on Automation Science and EngineeringShanghai, China, October 7-10, 2006

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Fig. 1. A single-cluster tool.

In order to finish the processing of one single wafer, a series of actions need to be taken by the cluster tool. We denote the action index (ACTi) to symbolize them (see Table 1). In the first step of modeling and analysis, deter-ministic action durations are assumed as denoted by ti.Obviously, if the cluster only needs to finish one wafer, the only action sequence it can take is {ACT1, ACT2,ACT3, ACT4, ACT5, ACT6, ACT7, ACT8}, and the corre-sponding cycle time is the sum of all action durations. However, if there is infinite wafer supply and high throughput is desired, we need to reduce the cycle time and do parallel processing. As described in [ 10], process actions can run in parallel while robot actions should coor-dinate to avoid conflict or deadlock. If running in steady state, the tool scheduling can be divided into many repeat-able cycles and there is one wafer input and one wafer output in each cycle period. As a result, each action takes place only once in one cycle period. Given the assumption that process times are deterministic, we can arrange that the time gap between starting times of two consecutive actions ACTi equals the cycle time. This provides the basis for our analysis that we can generate a sequence of actions ACT1 – ACT8 and repeat them to get steady state schedul-ing.

TABLE 1 ACTION ID AND PROCESSING TIMES

Action Index (ACTi) ti

C1 -> R (pick) 1 5 R -> P1 (place) 2 5 P1 process 3 30 P1 -> R (pick) 4 5 R-> P2 (place) 5 5 P2 process 6 35 P2 -> R 7 5 R -> C2 8 5

The sequence generation is out of the scope of this paper. Existing literature such as [ 8] provides a simulation approach to find all feasible sequence for multi-cluster tools, and [ 7] provides an analytical method to generate the best sequence. For the example in Figure 1, the optimal sequence given by [ 7] is {ACT7, ACT8, ACT4, ACT5,ACT1, ACT6, ACT2, ACT3}, which is known as the

“PULL” method. Under such a sequence, there is no dead-lock or conflict between robot and process modules. We can then transform the cluster tool into a network with the nodes as process or transfer activities and the edges as dependencies between activities.

Given a feasible sequence, we can draw the directed network diagram. In order to differentiate robots and proc-essing units, we represent it in a way that each horizontal line (called group) includes actions in one robot or one process module (as shown in Figure 2). A few representa-tions of the same action can appear in different groups but should be vertically aligned (as shown by dotted lines, e.g. ACT7, ACT4, ACT5, and ACT2). In steady state, each group of actions in the network is repeated with unchanged action orders and relative timings. For example, Figure 2 is the network diagram (as cyclic and we focus on Kth net-work cycle) for the example of the single-cluster tool in the steady state. There are three groups within such a clus-ter tool: robot R (as G1) and those (G2 and G3) for the ac-tion sequences of process modules P1 and P2, respectively.

Fig. 2. A network for the single cluster example shown in Figure 1.

We can then define the network cycle as the span of the structure that includes all actions in the cycle. Such cycles repeat in the steady state with fixed relative action starting time. It is noted that the time span of such a net-work cycle may be larger than a cycle period of the tool. But it is obvious that the repetition of network cycles can overlap in time and the starting time difference of two con-secutive network cycles equals to one cycle period. This initiates the idea to analyze the cluster tool cycle time with the extend CPM method as described in the following sec-tion.

III. APPROACH

Assume there are K groups and I actions in the cluster tool. For each group Gk, we can calculate the minimal time span CTGk. Then the maximum of CTGk is the steady state cycle time of the tool. The goal of the network analysis is to find a set of action times that each group Gk reaches the minimum time span CTGk. We call such a network cycle a compact network cycle. We further assume that this com-pact network cycle runs repeatedly in steady state. For one instance of a compact network cycle, let’s assume the first actions of group Gk starts at time TGST

Gk and the last action of group Gk ends at time TGET

Gk. Then CTGk is calculated as CTGk = TGET

Gk TGETGk , k = 1, · · · , K. For ACTi in the

compact network, denote the earliest starting time as TESTi

and the latest starting time as TLSTi .

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For each action ACTi, i = 1, · · · ,I, we can define the direct preceding actions ACTpre

i as a set of actions that are directly connected with and precede ACTi in the network, and the direct subsequent actions ACTsub

i as a set of ac-tions that are directly connected with and follow ACTi in the network. For example, for ACT5 in the single cluster example, the preceding action set is ACTpre

5 = {ACT4,ACT7} and the subsequent action set is ACTsub

5 = {ACT1,ACT6} (Figure 2). If an action is in the front of all groups that it belongs to within one cycle, its preceding action set is defined as an empty set , e.g., ACT7 in the example. Similarly, an action at the end of all groups within one network cycle will have the empty direct subsequent set, e.g. ACT3 and ACT6 in Figure 2. It is noted that an action with empty ACTpre

i is also preceded by actions in the pre-vious cycle, which we may ignore if considering the cur-rent cycle only, and vice versa for ACTsub

i.We then use iterative calculations to find scheduling

of the compact network cycle. Without loss of generality, we can start with the assumption that TGST

G1=0, TGSTGk,k

0=- , TGETGk= , where the value of – or means the

corresponding time is unknown yet. First, let’s define for-ward calculation and backward calculation as follow.

A. Forward Calculation

TESTFIRST(Gk) = TGST

Gk, k = 1, · · · , K,TEST

i = maxACTj ACTprei {TESTj + tj} , i = 1, · · · , I ,

TGETGk = TEST

LAST(Gk) + t LAST(Gk) , TESTLAST(Gk) - ,

B. Backward Calculation

TLSTLAST(Gk) = TGET

Gk, k = 1, · · · , K,TLST

i = minACTj ACTsubi {TLSTj - tj} , i = 1, · · · , I ,

TGSTGk = TLST

FIRST(Gk) - t FIRST(Gk) , TLSTFIRST(Gk) ,

where tj is the processing time of action ACTj. It is noted that if TGST

Gk are all known, we can find a unique set of TEST

i. With such observations, we can summarize calcula-tion of the compact network cycle schedule as follows,

Algorithm: Computing the compact network cycle GA ={G1,···,GK}; GS = {G1};while GS GA do

Do Forward calculation GE GSfor Gk not GS do

if Gk shares any action in GS thenGE GS + {Gk}

Do Backward calculation GS GEfor Gk GE do

if Gk shares any action in GE thenGS GE + {Gk}

for k = 1 to K do CTGk = TGETGk TGST

Gk

Since the groups within a network are connected (due to the network construction), the algorithm could terminate

eventually within predetermined iterations. With TESTi and

Ti, we can define the slack value TSLKi for action ACTi as

TSLKi = TLST

i TESTi , i = 1, · · · , I.

It is easy to observe that the slack value TSLKi implies

the flexibility of action time ti of ACTi within the compact network structure. Namely, the duration of ACTi can be prolonged to ti + TSLK

i without increasing any CTGk, and thus keeping the cycle time unchanged. On the other hand, if ACTi has a zero slack value, then a small increase in timay cause a comparable time increase on all CTGk that ACTi Gk. By [ 3], there exists a connected path in the network that all actions in the path have zero slack values. Such a path is called the critical path of the network.

Assume that there is only one action that has probabil-istic process time (or transfer time) and the variation of this process time will affect the critical path. First we cal-culate and find all possible optimal schedules by the mean values of all process times. Then we need to pick one from various possible optimal schedules in which the specific action has the least impact on the critical path. To imple-ment it, we can check the critical index of that action for each such schedule. The schedule with the least critical index will be the final optimal schedule. (Critical index of an action denotes the possibility that it will be in the criti-cal path of the network.) This method can be easily im-plemented.

IV. IMPLEMENTATION

A. Verification

Fig. 3, Configuration of “the thin film tool”.

We verified the proposed methodology using “the thin film tool”, which is a thin film tool by Novellus (see Fig-ure 3). The thin film tools are widely used in semiconduc-

PEM-1

AlignXfer

EP-3

PEM-2

PEM-3R2

R1

C 1 C 3 C 2

ANNL1/2

EP-2

EP-1

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tor manufacturing to deposit metals onto the silicon wafer surface using either Chemical Vapor Deposition (CVD), or Physical Vapor Deposition (PVD, or Sputter), or Electro-plate processes. There are many similar cluster tools in fabs which electroplate different metal layers on a wafer during different stages of the process.

“The thin film tool” can be modeled as a two-cluster tool. The Pre-Anneal Cluster includes single-blade Robot

R1, Cassette I, II, III, and redundant Pre-Anneal Chambers 1/2. The Processing Cluster includes single-blade Robot R2, three redundant Electroplating Chambers, EP 1/2/3, and three redundant Post Electroplating Modules, PEM 1/2/3. There are also two interconnection Modules be-tween the two clusters, an Align Module and a Transfer Module. The process flow of this tool is: C 1/2/3 ANNL 1/2 EP 1/2/3 PEM 1/2/3 C 1/2/3.

0 50 100 150 200

Time (second)

3 (Cst to R1 # 1)4 (R1 to ANNL1)5 (R1 to ANNL2)6 (ANNL1 Proc)7 (ANNL2 Proc)

8 (ANNL1 to R1)9 (ANNL2 to R1)

10 (R1 to Align # 1)11 (R1 to Align # 2)

12 (Align Proc #1)13 (Align Proc #2)14 (Align to R2 #

15 (Align to R2 # 2)16 (R2 to EP1)17 (R2 to EP2)18 (R2 to EP3)

19.1 (EP1 Proc-1)20.1 (EP2 Proc-1)21.1 (EP3 Proc-1)

22 (EP1 ro R2)23 (EP2 to R2)24 (EP3 to R2)

25 (R2 to PEM1)26 (R2 to PEM2)27 (R2 to PEM3)28 (PEM1 Proc)

29 (PEM2 Proc-1)30.1 (PEM3 Proc-1)

31 (PEM1to R2)32 (PEM2 to R2)33 (PEM2 to R2)

34 (R2 to Xf er # 1)35 (R2 to Xf er # 2)36 (R2 to Xf er # 3)37 (Xf er to R1 # 1)38 (Xf er to R1 # 2)39 (Xf er to R1 # 3)40 (R1 to Cst # 1)41 (R1 to Cst # 2)42 (R1 to Cst # 3)

3.1 (Cst to R1 # 2)15.1 (Align to R2 #21.2 (EP3 Proc-2)

30.2 (PEM3 Proc-2)20.2 (EP2 Proc-2)

29.2 (PEM2 Proc-2)19.2 (EP1 Proc-2)

Actio

n ID

s an

d De

scri

ptio

ns

Fig. 4, Optimal schedule of “the thin film tool” with the steady state cycle time CT of 65.03 seconds per wafer.

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Given the chamber processing times for metal layer “Y3” and the robot transition times we created the optimal schedule for the layers as illustrated in Figure 4. In this figure, relative time in one steady state cycle is shown on the horizontal axis in seconds. On the vertical axis each action ID represents either a transfer activity or a process-ing activity in the sequence of the optimal schedule. The bars on the graph show the duration of each activity as well as the start and finish times. The resultant steady state cycle time CT for “Y3” is 65.03 seconds per wafer (total of 195.09 sec. for three wafers). This outcome perfectly matches the Intel SPEED model for such a thin film tool and the cycle time falls within the 95% confidence interval of the actual cycle time data from the tool. We have done the same study for other metal layers as well and the re-sults are similarly confirmed with the SPEED Model and the Actual data.

B. Advantages

Although the current schedule by Novellus is very close to optimal in the example above, there is no guaran-tee that it will always do so. The SPEED model is espe-cially incapable of dealing with the following scenarios: (1) different cluster tool configurations; (2) shifting con-straint chambers; (3) processing route changes and (4) process/transition time variations. The proposed method-ology can be applied to effectively address these problems. In this section, we will demonstrate how it can quickly identify the optimal schedule when the processing routes change and how it can help us with process/transition time sensitivity analysis.

C. Processing route changes

Processing routes of cluster tools can change due to various reasons (e.g., chamber down). “The thin film tool” has 3 sets of redundant chambers: Pre-Anneal 1/2, EP Chamber 1/2/3 and PEM 1/2/3. If some of these redundant chambers go down, the tool could still be up for produc-tion with the corresponding changes of the processing routes.

TABLE 2 “THE THIN FILM TOOL” STEADY STATE CYCLE TIMES IN DIFFERENT

CHAMBER DOWN SCENARIOS FOR DIFFERENT METAL LAYERS.

Steady State CT (s) Process Module Down

Layer“Y1”

Layer“Y2”

Layer“Y3”

No Chamber Down 56.26 50.6 65.03

One of the EP Chambers 79.80 75.9 97.55

One of the PEMs 79.80 76.25 97.55

One EP and One PEM 79.80 76.25 97.55

We applied the proposed methodology to quickly gen-

erate the optimal schedules and calculate the steady state cycle times for different scenarios for three metal layers as illustrated in Table 2. In this table CTs for different metal layers are presented under different chamber down scenar-ios. Since each metal layer has different processing times for EP and PEM modules according to the wafer recipe, the cycle time for each layer is different under the same scenario.

D. Process/transition time sensitivity analysis

One of the great advantages of the proposed method-ology is to study the sensitivity of the optimal schedule by leveraging the CPM slack time calculation. This allows us to quickly identify the impacts of the process/transition time variations on throughput and cycle time, provides us with a great tool to quantify the benefits of processing time reduction projects, and also could guide us to choose a robust schedule under such variations.

To illustrate the sensitivity analysis, we choose two optimal schedules SeqA and SeqB of “the thin film tool”, which both produce the CT = 65.03 sec. Table 3 shows all ordered actions in every group of SeqA and SeqB. Here we use table, instead of network to simplify the network presentation. From Table 3, we find that the two networks are the same except the group of Robot 1. In this table sequences of numbers are different Action IDs that follow each other during the optimal schedule and as in Figure 4 each of them corresponds to an activity within the se-quence with a specific duration and description.

Using the CPM slack time calculation, we calculate the slack time for each action. Figure 5 shows the slack time for both schedules SeqA and SeqB.

TABLE 3 TWO OPTIMAL SCHEDULES, SEQA AND SEQB, OF “THE THIN FILM TOOL” IN

GROUP PRESENTATIONGroup Sequence A Sequence B

Robot1

37 40 8 103 4 38

41 9 11 3.1 5 39 42

3.1 5 8 1037 40 34 9 11

38 41 39 42ANNL1 8 4 6ANNL2 9 5 7Align 15.1 10 12 14 11 13

Robot2 31 34 22 25 15.1 16 32 35 2326 14 17 33 36 24 27 15 18

EP1 19.2 22 16 19.1 EP2 20.2 23 17 20EP3 21.2 24 18 21.1 PEM1 31 25 28PEM2 29.2 32 26 29PEM3 30.2 33 27 30.1 Xfer 34 37 35 38 36 39

The processing time variations of different chambers have different effects on the cluster tool performance. For both SeqA and SeqB, the constraint chambers are EP1 and EP3 where the slack times are zero. Any processing time variation of EP1 and EP3 will directly impact the cycle time of “the thin film tool”. However, SeqA and SeqBhave different sensitivity on Pre-Anneal processing time

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variation. The slack time of Pre-Anneal 2 processing is 76.4 sec. in SeqA and 88.5 sec. in SeqB. Hence, the opti-mal schedule SeqB is more robust that SeqA given the Pre-Anneal 2 processing time variation. Similarly, SeqA is more robust than SeqB given the Pre-Anneal 1 processing time variation.

0

20

40

60

80

100

120

140

160

180

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1819.120.121.1 22 23 24 25 26 27 28 2930.1 31 32 33 34 35 36 37 38 39 40 41 423.115.121.230.220.229.219.2

Action ID

Slac

k Ti

me

(sec

Seq A

Seq B

Fig. 5, Sensitivity comparison of two optimal schedules SeqA and SeqB.

E. Software

To automate the process of finding the optimal se-quence we have developed software that can calculate cy-cle time for different chamber down scenarios. This software consists of two modules. The first module takes activity sequences and durations as input and returns the optimal path along with slack times and sensitivity analy-sis in case we compare multiple optimal sequences. The second module takes new processing times, which result from improvement projects, along with chamber down scenarios as input and calculates the cycle time for each scenario. This will especially be useful when improvement projects are to be assessed based on cycle time reduction in different chamber down scenarios. Figure 6 shows a view of user interface for this software.

EP 2 PEM 2 PEM 3

EP 3

R2

Xfer Align

Metal Layer 6

EP Process Time (s) 185.7

PEM Process Time (s) 178.0

Steady State CT 97.55

AN

1 R1

AN

2 C

All Cells

PEM & EP Down

EP Down

Fig. 6, User interface of software for CT calculation under different chamber down scenarios.

V. CONCLUSION

The extended CPM method can quickly estimate the cyclic time of a multi-cluster tool running in steady state with probabilistic processing times and transfer times. Thus it provides an analytical approach to find the optimal schedule and the maximum throughput of such cluster tools. In the future, we will apply this methodology to more complicated equipment like Lithography machines.

ACKNOWLEDGMENT

We would like to thank Intel Fab22 IE Mauricio Gar-cia Giliberti and Cheryl Bell, and Fab22 Thin Films Proc-ess Engineers, who specifically work on the toolset of this case study, for helping us implement this approach on the thin film tool. We also would like to thank Geoffrey Comber, Huang Liu, and Donald G Baker Jr from Intel Fab11x for their help with data collection and implementa-tion. Thanks also go to Intel IE managers Melissa Bowles, Eric Harding, and Mike Gannon for their consistent sup-port on productivity improvement research and applica-tion.

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