Scheduling Single-Armed Cluster Tools With Chamber Cleaning Operations

Scheduling of time-constrained single-arm cluster tools with purge operations in wafer fabrications

In semiconductor manufacturing, wafer residency time constraints make the scheduling problem of cluster tools complicated. A process module (PM) in cluster tools is prone to failure. It is crucial to deal with any such failure in a proper and timely manner. If there are feasible periodic schedules in operating a cluster tool before and after a PM failure, it is desired to make it operate continuously when such a failure occurs. However, due to wafer residency time constraints, it is highly challenging to control a tool such that it can be correctly transferred from a feasible schedule before failure to another after it. To solve this problem, a Petri net model is developed to describe the dynamic behavior of a single-arm cluster tool and failure response policies are proposed. The proposed policies are formulated via simple control laws for their easy implementation. Examples are given to show them.

Cyclic scheduling and operation of a residency time- constrained single-arm cluster tool with failure-prone process modules are highly challenging. In some cases, when a fail- ure occurs, there still exists a feasible cyclic schedule for the performance-degraded tool. In other cases, such a schedule no longer exists. For the latter, it is highly desired to respond to a process module failure properly such that the tool can continue working and the wafers in the tool can be completed in a feasi- ble way. This work is the first one to study this important issue. The idea is to apply Petri nets to describe the dynamic behav- ior of a single-arm cluster tool. With the developed Petri net model, this paper formulates failure response policies to control the cluster tool such that it can keep working without violat- ing any residency time constraint. The failure response policies are implemented via efficient real-time control laws. Illustrative examples are presented to show their usage. Index Terms—Wafer fabrication, Cluster tools, Petri net, Scheduling, Failure response.

For wafer fabrication, a process module (PM) in cluster tools is prone to failure. It is crucial to deal with such failure in a proper and timely manner. With residency time constraints, if there are feasible periodic schedules in operating a cluster tool before and after a PM failure, it is desired to make it operate continuously when such a failure occurs. A Petri net model is developed to describe the dynamic behavior of a single-arm cluster tool and failure response policies are proposed to deal with a failure. The proposed policies are formulated via simple control laws for their real-time and on-line implementation. An example is presented to show their application.

We examine cyclic scheduling of single-armed and dual-armed cluster tools that concurrently process two wafer types by sharing a process module (PM). Because a PM is shared by two different wafers, the backward and swap sequences, which are prevalently used for single-armed and dual-armed tools without such complexity, respectively, are not effective. We therefore propose new sequences, called alternating backward and alternating swap sequences, for steady cycles of single-armed and dual-armed tools, respectively. We then develop optimality conditions for which the proposed sequences achieve the minimum cycle times in a fundamental cycle, and show that the optimality conditions hold for most practical cases. We also develop a condition for which a shared PM becomes the bottleneck and hence the PM sharing increases the cycle time. For general cycles, we propose heuristic scheduling methods that combine both the alternating backward (or swap) sequence and the conventional backward (or swap) sequence. Finally, we experimentally verify the efficiency and effectiveness of the proposed algorithm for dual-armed cluster tools.

Recently, semiconductor manufacturing fabs tend to reduce the wafer lot size, down to just a few. Consequently, the wafer recipe or wafer flow pattern changes frequently. For such problems, it is impossible to apply conventional prevalent cyclic scheduling methods that repeat processing of wafers in an identical cyclic tool operation sequence. We therefore consider the noncyclic scheduling problem of single-armed cluster tools that process wafers with different recipes. Our proposed method is to transforms the problem into a multiobjective problem by considering the ready times of the resources as objectives to minimize. Only feasible states are generated based on the initial state of the system. These feasible states form a multiobjective shortest path problem and give us as an upper bound for the number of states, where , and are the number of different wafer recipes, wafers, and processing chambers. We solve this problem with implicit enumeration by making our scheduling decisions based on the Pareto optimal solutions for each state. The experimental results show that the proposed algorithm can quickly solve large sized problems including ones with arbitrary initial tool states, changing recipes, reentrant wafer Ωows, and parallel chambers. Note to Practitioners-The scheduling method developed in this paper is designed for scheduling robots used in semiconductor production. The method is able to efficiently represent the state of the manufacturing system and uses this to find an optimal schedule to maximize productivity. The same approach can be used for other manufacturing systems with discrete events. The main advantage of this method is that it can fast find an optimal schedule based on the current state of the system. A long-time horizon can be used because of the high efficiency of the algorithm with respect to the number of jobs. However, the method is best suited for manufacturing systems with few buffers and limited degrees of freedom, in other words highly interconnected systems.

Cluster tools have been one of the proposed alternatives to improve operations performance in semiconductor fabrication. The benefits include high yield throughput, less contamination and less human involvement. Perkinson et al. (1994, 1996) developed analytical models to predict the minimum theoretical time required to complete the cycle in a cluster tool. This paper addresses the verification of these analytical models using simulation. Two simulation models were developed -- one with simple configuration and another one that incorporates parallel chambers. The implementation of parallel chambers for the longest process in the cluster tool is tested as the potential area of performance improvement.

Most cluster tool scheduling studies assume identical access times between chambers or do not discuss the impact of access times. However, the optimal scheduling rule and the cycle time can depend on the access times or physical configuration of parallel chambers. Therefore, we examine cyclic scheduling problems for cluster tools that have nonidentical access times. We first develop Petri net models for tool behaviors and analyze the cycle time by identifying the workloads of the process steps. We prove that the conventional backward and swap sequencing strategies are still optimal for single-armed and dual-armed cluster tools, respectively, when a process step is the bottleneck and the tool repeats a minimal cyclic work cycle. We also present a closed-form formula for the cycle time and identify a coprime condition on the number of parallel chambers for which the cycle time is independent of the order of using parallel chambers. Finally, we develop a mixed integer programming model for cases in which the coprime requirement is not satisfied.

Cluster tools, each of which consists of multiple processing modules, one material handling robot, and loadlocks, are widely used for wafer fabrication processes, such as lithography, etching, and deposition. There have been many approaches and algorithms for cyclic scheduling of cluster tools in which the robot repeats a specified sequence for processing identical wafers. However, the lot order size has recently been decreasing due to the larger wafer size and circuit width reductions. In modern fabs, each wafer lot can have different flow patterns and process times for the same process step, and heterogeneous lots are processed consecutively in a tool. Even some tools in a fab have idle time waiting for wafer lots depending on the work-in-process fluctuations. Such different wafer lots and frequent tool state changes cannot be handled with cyclic scheduling methods, and accordingly noncyclic scheduling methods for such cases are required. Therefore, we develop an efficient branch and bound (B&B) procedure for noncyclic scheduling problems of cluster tools to minimize the makespan. Since a timed Petri net (TPN) is known for its powerful modeling ability and analysis capability, the algorithm is developed based on a TPN. We verify the efficiency of the B&B procedure with various cluster tool scheduling problems.

Modeling and implementing a real-time scheduler for dual-armed cluster tools

In semiconductor manufacturing, a process module (PM) failure in cluster tools (CTs) happens from time to time. To effectively operate a CT, such a failure should be handled in a proper and timely manner. This issue becomes much more complicated because wafer residency time constraints (WRTCs) must be met to ensure the quality for some wafer fabrication processes. With such constraints, if a tool is operated under a periodic schedule and a PM fails, it is desired that the tool can still operate under a periodic schedule if it is possible. Nevertheless, the periodic schedule after a PM failure must be different from that before its failure since in this case the tool is degraded. Thus, there must be a transient process between them. It is a great challenge to operate a tool such that it can go through such a transient process with WRTCs being always satisfied. This paper aims to solve this problem by proposing PM failure response policies which can successfully transfer a CT to the feasible schedule after failure from the one before a failure. Then, efficient algorithms are developed to improve these response policies. The proposed policies are composed of simple control laws such that they can be realized in real time and online. Illustrative examples are presented to show their applications.

Scheduling of multicluster tools has received much attention due to its complexity of interaction among cluster tools. A cluster tool comprises several modules such as processing modules, a transfer module with a single or dual-armed handling robot, and loadlock modules. A multicluster tool comprises two, three, or more cluster tools that are interconnected with each other. In this paper, we propose a deadlock prevention policy for noncyclic scheduling of dual-armed multicluster tools. The proposed deadlock prevention policy is effective for generating an efficient schedule for two or three-connected multicluster tools with a single or a dual-path flow. This is proven by analyzing the strict minimal siphon of the Petri net model. The performance of the proposed method is compared with that of a maximal permissible deadlock prevention policy. The results demonstrate that the proposed method is computationally efficient for larger state spaces and more suitable than conventional deadlock prevention policies for generating effective schedules for noncyclic scheduling of multicluster tools. Note to Practitioners —Scheduling of multicluster tools has recently attracted attention in the growing semiconductor manufacturing industry. Most conventional studies on multicluster tools focus on cyclic scheduling to minimize cycle time with swap strategy. The challenge addressed by this paper is to derive a deadlock-free schedule for multicluster tools with noncyclic operations. We propose a simple deadlock prevention policy that can be applied to large-scale timed Petri net (PN) model of general multicluster tools without enumeration of siphon computations or solution of mixed integer programming. A timed PN model is developed for noncyclic operations of dual-armed multicluster tools with a dual ath. The computational results show that the approximately 10% of the total throughput of the proposed method is better than those of the conventional methods. It enables more efficient operations for noncyclic scheduling under transient periods including cleaning operations.

There are wafer fabrication processes in cluster tools that require wafer revisiting. If a swap strategy is applied to dual-arm cluster tools handling wafer revisiting, a three-wafer periodical process is formed with three wafers completed in each period. Such a period contains three cycles in a revisiting process and three cycles in a nonrevisiting one. Hence, analysis and scheduling of such tools become very complicated. In this paper, a Petri net (PN) model is developed to describe their operations. Based on it, it is found that, if a swap strategy is applied, such tools are always in a transient state. A systematic method is then presented to analyze their performance. With the help of the proposed PN model, this work, for the first time, derives the optimality conditions of three-wafer period scheduling. Industrial application examples are given to show the results.

We consider the reentrant flow shop problem F|reentrant, pij = pi |Cmax with identical jobs and makespan criterion. We prove its NP-hardness in the ordinary sense. The proof is performed by polynomial reduction of the problem J3|n = 3|Cmax to the problem F|reentrant, pij = pi |Cmax with three identical jobs. Using the input data of the problem J3|n = 3|Cmax, we have constructed a special type of job for the problem F|reentrant, pij = pi |Cmax. In the proof, we analyze all possible variants of critical paths in the constructed instance. We also propose a dynamic programming algorithm to find the optimal solution of the problem F|reentrant, pij = pi |Cmax. Analysis of the time complexity of the algorithm showed that the problem with fixed number of jobs is pseudopolynomially solvable. Next, we study the use of cyclic schedules to construct approximate solutions. A cyclic schedule with a minimum cycle time can be found in polynomial time. We propose a polynomial algorithm for finding an approximate solution. This algorithm is based on the construction of a cyclic schedule with a minimum cycle time and its subsequent compaction at the beginning and at the end. We construct an upper bound of the makespan of cyclic schedules. This bound depends on the number of jobs processed in one cycle. The paper provides numerous examples characterizing cyclic schedules from both the positive and negative sides to solve the problem F|reentrant, pij = pi |Cmax.

Timed Petri nets are useful in performance evaluation of concurrent systems. The maximum computation rate is achieved for minimal cycle time of timed Petri net. It is known that minimal cycle time problem for P-invariant Petri nets is NP-complete. In this paper we prove that the minimal cycle time problem, for non-P-invariant Petri nets and for a small subclass of P-invariant Petri nets called free-choice nets having live and safe marking, is NP-complete.