Modeling and analysis of cluster tools with chamber revisiting (W.M. Zuberek)

In cluster tools with chamber revisiting, wafers pass through some chambers more than once. Coordinating the flow of wafers is more complicated in this case than for processing without chamber revisiting.

The steady-state, cyclic behavior of a cluster tool can be described by a sequence of configurations that characterize the distributions of wafers in the chambers of the tool. For chamber revisiting, these configurations must take into account that some of the chambers are visited more than once. A convenient approach uses the concept of "virtual chambers" and "extended configurations"; virtual chambers are chambers corresponding to all visits during the processing sequence of a wafer (i.e., chamber revisiting is considered as visiting another chamber), and the fact that more than one virtual chamber corresponds to the same physical chamber is captured by the concept of "coupled chambers" - all virtual chambers which represent the same physical chamber are coupled. An extended configuration is simply a configuration describing virtual chambers.

For example, if the sequence of processing steps in a 4-chamber tool is 1-2-3-4-2-3, which means that each wafer first visits chamber C1, then C2, then C3 and C4, then revisits C2 and finally C3, the configurations are described by 6 variables (i.e., configurations are vectors of 6 components), and variables 2 and 5 as well as 3 and 6 are coupled because they correspond to the first and second visits to C2 and C3, respectively. If any one of coupled variable becomes non-zero, all remaining coupled variables become marked by "x" to indicate that the corresponding (physical) chamber is busy. So, for an implementation of the process 1-2-3-4-2-3, a maximally concurrent initial configuration (i.e., a configuration just before loading a new wafer into the first chamber) can be [0,1,x,1,x,1] or [0,x,x,1,1,1]; [0,1,1,1,x,x] is yet another initial configuration but it is of little interest because, after loading chamber C1, no further continuation is possible.

For the initial configuration [0,1,x,1,x,1], the sequence of consecutive configurations is as follows:

For some configurations there may be more than one possible next operation, which leads to several different schedules with possibly different cycle times. It is also possible that a configuration has no possible continuation, which indicates that the corresponding initial configuration leads to a deadlock. For example, for the processing sequence 1-2-3-4-2-3, the initial configuration [0,0,1,1,0,x] leads to a deadlock:

Sequences of operations leading to a deadlock can easily be identified and eliminated at an early stage of schedule analysis.

As in the case of cluster tools with no chamber revisiting, a net model of a cluster tool is composed of models of all chambers and the model of robot. For revisited chambers, the model of a chamber is slightly more complex because it must allow different temporal characterizations for each visit. Therefore it is in the form of a free-choice structure with the number of choices representing the number of visits of the same wafer to the particular chamber (this number can be different for each chamber). Fig.1 shows a model of a chamber with two visits; each additional visit is represented by an additional cycle on place pi.

Fig.1. Net model of a chamber with two visits.

The complete model is shown in Fig.2. The 4 chamber models are represented by subnets associated with places p1, p2, p3 and p4. The subnets for chambers C2 and C3 are free-choice structures with the upper parts representing the first visits and the lower parts representing the second visits of the wafers. Places p1 and p4 could be removed (together with incident arcs) as they do not contribute to the performance characteristics of the models; they are preserved exclusively for the consistency of the representation. The subnet representing the robot seems to be convoluted but its correspondence to the sequence of operations arising from the sequence of configurations is rather straightforward.

Fig.2. Petri net model of a 4-chamber cluster tool with sequence 1-2-3-4-2-3.

In order to evaluate the cycle time of the model shown in Fig.2, the durations of all operations must be associated with the transitions of the model, and then place invariants can be used to find the cycle time of the model. The net in Fig.2 (after removal of places p1 and p4) has 14 place invariants, so its cycle time, T0, is:

where the cycle times Ti, i=1,...,14 (of subnets implied by place invariants) are obtained by adding the execution times associated with the transitions and dividing this sum by the total count of tokens in the subnet (if it is greater than one).

Research topics include:

• Modeling and analysis of cluster tools with dual-blade robots
• Modeling and analysis of cluster tools processing several types of wafers within the same batch
• Modeling and analysis of cluster tools with additional temporal constraints (e.g., waiting times in chambers, delay between consecutive processing steps, etc.)