On the Throughput of Clustered Photolithography Tools: Wafer - PowerPoint PPT Presentation

On the Throughput of Clustered Photolithography Tools: Wafer Advancement and Intrinsic Equipment Loss Maruthi Kumar Mutnuri James R. Morrison, Ph.D. September 23, 2007

Presentation Outline  Motivation  Model 1: Synchronous photolithography model  System Description  Time between lot completions with diverse lot populations  Time between lot completions with same class of lots  Model 2: Asynchronous photolithography model  System Description  Wafer completion times with a single class of lots  Intrinsic equipment loss:  Reticle change (as a pause in the bottleneck module)  Concluding remarks 2

Motivation  Goal: One generic model for all classes of serial processing cluster tools  Abstract models useful for clustered photolithography tools  Movement of individual wafers can be analyzed  Module level rather than conventional tool level approach  Contribution to flow line literature  High fidelity models with direct application to fabricator simulation  A new class of failures – setup dependent upon state of system  Simplified recursions for system evolution 3

Synchronous Photolithography Model: System Description m 1 m 2 m 3 m 4 m 5 m 6 m 7 Robot M modules m 8 W wafers/lot m 15 m 14 m 13 m 12 m 11 m 10 m 9 Lots may have different deterministic process times in a module  Wafers can only advance at the same instant as all others in the tool – their movement is synchronized  Process time in module m j for family F lots is D F j  May be 0 to model a buffer (only useful for module failure analysis)  Let k(i) denote the number of empty modules in advance of lot l i 4

Synchronous Photolithography Model: Wafer Advancement  Rate of wafer advance is dictated by the maximum module time for all occupied modules m 1 m 2 m 3 m 4 m 5 m 6 m 7 m 8 m 9 m 10 m 11 m 12 m 13 m 14 m 14  For lot l i with family F (i), define the effective module process time as         D F i F i max p , q r     { r : 1 r p or q r M }  The slowest possible effective process time is       D F i F i max j j 5

Synchronous Photolithography Model: Lot Completion for Diverse Lot Populations  There are two families of lots  Time between the departure of the previous lot and departure of lot l i may be calculated as:  M 1    F ( i ) T Time for first wafer to  i j , M 1   reach last module j M k ( i )     F ( i ) ( W M 1 ) Time for last wafer to exit first module  k ( i 1 )    F ( i ) Time until first wafer of 0 , j  lot l i+1 enters tool j 1 M      F ( i ) F ( i 1 ) max( , ) Time until last wafer   0 , j     j 1 k ( i 1 ) , M 1    exits the tool j 2 k ( i 1 ) Here, for notational simplicity, we assume that at most two lots can be on the tool at any instant 6

Synchronous Photolithography Model: Lot Completion for Uniform Lot Population  Lots are of same family  Time between the departure of the previous lot and departure of lot l i may be calculated as:  M 1    Time for first wafer to T  i j , M 1 reach last module   j M k ( i )     ( W M 1 ) Time for last wafer to exit first module  k ( i 1 )    Time until first wafer of 0 , j  lot l i+1 enters tool j 1 M      Time until last wafer    j 1 k ( i 1 ) , j    exits the tool j 2 k ( i 1 ) Here, for notational simplicity, we assume that at most two lots can be on the tool at any instant 7

Synchronous Photolithography Model: Example Time between lot completions, T i 900 850 Parameters: M=11 and W=10 Time between lot completions, T i 800 Process 750 D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 D 11 Times Family F1 20 25 40 35 30 50 15 35 45 20 30 700 Family F2 30 35 50 45 40 60 25 45 55 30 40 650 600 550 1 2 3 4 5 6 7 8 9 10 No. of Empty Modules, K i =K i+1 8

Asynchronous Photolithography Model: System Description  Assume that all wafers in the tool advance at their own rate so long as there are module locations available to do so  Process time in module m j for all lots is D j  May be 0 to model a buffer  Only one class of lots (can readily model many classes but requires full simulation approach) m 1 m 2 m 3 m 4 m 5 m 6 m 7 M modules m 8 W wafers/lot m 15 m 14 m 13 m 12 m 11 m 10 m 9 Lots are of same class, but may have different size 9

Asynchronous Photolithography Model: Parameters m 1 m 2 m 3 m 4 m 5 m 6 m 7 M modules m 8 W wafers/lot m 15 m 14 m 13 m 12 m 11 m 10 m 9 Lots are of same class, but may have different size  Number of modules in the cluster: M Includes buffers D m  Process time for a wafer in module m:   Largest (bottleneck) process time:  Number of wafers per lot W Can easily generalize to depend upon the lot l i  Denote the i-th lot:  Arrival time of lot l i : a i 10

Asynchronous Photolithography Model: Wafer Advancement  The evolution equations for the system may be written  Let x j (w) denote the entry time of wafer w to module m j ,  At the first module: ,         x w max a , x w 1 1 w 2  For the intermediate modules (2 ≤ w ≤ M -1):           D  x w max x w , x w 1    j j 1 j 1 j 1  For the last module:           D   D x w max x w , x w 1   M M 1 M 1 M M 11

Asynchronous Photolithography Model: Towards Completion Time  The completion C i time of lot l i is dictated by two possibilities:  Case i : Lot l i arrives early enough so that its wafers begin to exit immediately after those of lot l i-1    One wafer exits every C C W  i i 1  units of time  Case ii: Lot l i arrives so late that it does not run into lot l i-1 in front of it M       D   Remaining wafers C a W 1 i i j exit every  units  j 1 of time 12 First wafer exits

Asynchronous Photolithography Model: Completion time  The completion C(i) time of lot l i obeys the following recursion:        D     T C max a e , C W 1  i i i 1 with initial condition (for an empty tool)      D   T C a e W 1 i i where   M Number of modules in the track W Number of wafers in a lot D  Processing time of a wafer in module m  a Arrival time of lot i to the system m i     D  D D T  , ,  T  e 1 , , 1 1 M     D   max , 1 i M i Proof: Start with the max-plus algebra representation of the evolution equations and employ an induction within an induction 13

Asynchronous Photolithography Model: Example 1  Example: M = 5, W = 3,  = 50 sec Process Time 15 Module 5 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 30 Module 4 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 0 Module 3 2 2 3 3 3 3 1 2 2 2 3 3 3 3 3 15 Module 2 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 20 Module 1 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 0 25 50 75 100 125 150 175 200 225 L1 enters system at time a1 = 0 L1 exits system at time c1 = 140 L2 enters system at time a2 = 85 L2 exits system at time c2 = 230 No time lost on the     D         bottleneck! T C a e W 1 0 80 ( 3 1 ) 30 140 1 1         D          T C max a e , C W 1 max 85 80 , 140 30 ( 2 )( 30 ) 2 2 1      14 max 165 , 170 60 230

Asynchronous Photolithography Model: Example 2  Example: M = 5, W = 3,  = 50 sec Process Time 15 Module 5 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 30 Module 4 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 0 Module 3 2 2 3 3 3 3 2 2 3 3 3 3 15 Module 2 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 20 Module 1 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 0 25 50 75 100 125 150 175 200 225 L1 enters system at time a1 = 0 L1 exits system at time c1 = 140 L2 enters system at time a2 = 100 L2 exits system at time c2 = 240 10 seconds lost on     D         the bottleneck! T C a e W 1 0 80 ( 3 1 ) 30 140 1 1         D          T C max a e , C W 1 max 100 80 , 140 30 ( 2 )( 30 ) 2 2 1      15 max 180 , 170 60 240