1. Metal Layer Planning for Silicon Interposers with Consideration of Routability and Manufacturing Cost W. Liu, T. Chien and T. Wang Department of CS, NTHU, Taiwan DATE 2014

2. Outline Introduction Preliminaries Problem Formulation The Proposed Variable-Layer Global Router (VGR) Experimental Results Conclusions

3. Introduction Silicon interposer-based 3D IC has been seen as an alternative approach to true 3D ICs. A 2.5D IC places active dies on a silicon interposer, which in turn is placed on a package. Interposer

4. Introduction The number of metal layers in an interposer is one of the critical factors to affect the routability and manufacturing cost of the 2.5D IC. This paper formulate the metal layer planning problem as a variable-layer global routing (VLGR) problem. The VLGR problem asks to find an overflow-free routing result that requires as few metal layer as possible.

5. Preliminaries Background of Global Routing Problem  Given k-layer chip is partitioned into a 3D array of uniform g-cells.  3D grid graph G k (V k, E k ).  The capacity c(e) of a g-edge e indicates the number of routing tracks that can legally cross the abutting boundary of g-cells. 3D grid graph2D grid graph

6. Preliminaries Background of Global Routing Problem  The number of wires that pass through g-edge e is called the demand d(e) of e.  The overflow of e is defined as max(0, d(e)-c(e)).  Most global routers compress the 3D grid graph into a 2D grid graph.  In 2D grid graph, c(e i )= c(e i,1 ) + c(e i,2 ) + c(e i,3 ) 3D grid graph2D grid graph

7. Preliminaries Notations used in Variable-layer Global Routing Problem  (L h, L v ) denotes a layer configuration L h : the number of metal layer with horizontal direction L v : the number of metal layer with vertical direction  t h and t v denote the capacities of each horizontal and vertical g-edge in 3D grid graph.  G(V, E) is a 2D grid graph for the interposer. The capacities of each horizontal and vertical g- edges are t h × L h and t v × L v.

8. Preliminaries Notations used in Variable-layer Global Routing Problem  N is the set of nets to be connected in the interposer. A set of pins corresponding to micro-bumps is located on the top metal layer. A set of pins corresponding to TSVs is located on the bottom metal layer.  Alternate stacking constraint The horizontal layers and vertical layers have to alternately stack to reduce coupling effect between layers, so the difference between L h and L v is at most 1.

9. Problem Formulation The Variable-layer Global Routing Problem  Given G(V, E), t h, t v, and N.  Find a layer configuration (L h, L v ) such that N can be globally routed on G(V, E) without any overflow and |L h -L v |≤1 is satisfied to obey the alternate stacking constraint.  The objective is to minimize the following cost function:

10. The Proposed VGR Flow of VGR

11. Layer Range Identification VGR decomposes each net in N into two-pin subnets by a rectilinear minimum spanning tree algorithm. We want to Identify a lower-bound layer configuration (LB h, LB v ). For each g-cell u, if a two-pin net has a terminal in u and another terminal not in u, this two-pin net is defined to be a global segment of u.

12. Layer Range Identification Let |S(u)| denote the number of global segment of g-cell u. (LB(u) h, LB(u) v ) ensures that |S(u)| is not greater than the total capacity of the g-edges connecting to u and makes |LB(u) h -LB(u) v |≤1.

13. Layer Range Identification

14. The one with the maximum among all g-cells is selected to be (LB h, LB v ). After LB h and LB v are obtained, invoke the initial routing stage to get an initial routing result. To identify the upper-bound layer configuration, VGR analyzes the initial routing results to get a layer configuration (UB h, UB v ) as tight as possible.

15. Layer Range Identification For example, if a horizontal g-edge e i has the maximum demand d(e i ) among all horizontal g- edges, UB h is set to be  d(e i )/t h  to ensure that no horizontal g-edge has overflows. If |UB h -UB v |>1, increase the smaller one of UB h and UB v to make |UB h -UB v |=1.

16. Layer Configuration Selection Explore the layer configurations from the upper to lower bounds in the layer range. Initially, L h =UB h and L v =UB v. VGR reduces either L h or L v by 1. The rip-up and reroute stage iteratively reroutes the global segments in the initial routing results to intend obtaining an overflow-free routing result again.

17. Layer Configuration Selection If an overflow-free result is obtained, the cost of the current routing result is calculated and the current result is treated as a feasible solution to save into a solution pool. R&R stage and metal layer reduction stage are alternatively performed until either an overflow-free result cannot be obtained or (L h, L v ) already reaches the lower bound of the layer range.

18. Metal Layer Reduction When L h is not equal to L v, we always reduce the larger one of them in order to obey the alternate stacking constraint. When L h is equal to L v, reducing which one of them becomes a problem. Given an overflow free result S under (L h, L v ), this stage would judge which one of (L h -1, L v ) and (L h, L v -1) offers better routability.

19. Metal Layer Reduction Construct a congestion map C h from S under (L h -1, L v ). Construct a congestion map C v from S under (L h, L v -1). ChCh CvCv

20. Metal Layer Reduction Panel-based routability evaluation method  Iteratively visit each vertical g-edge from right to left.  Move overflows from overflowed edges to the nearest slack edge to remove overflow.  The cost of moving a unit of overflow to a slack edge is the square of its moving distance.

21. Metal Layer Reduction If a row has not enough slack edges to eliminate all overflows, the remaining overflows are defined to be surplus overflows. The score for a row consists of the total moving cost and the surplus penalty (multiply the amount of surplus overflows by a large constant).

22. Early Termination Scheme Let (L h -1, L v ) and (L h, L v -1) are treated as the children of (L h, L v ). If the children of (L h, L v ) are both unroutable, reducing a layer from (L h, L v ) is meaningless. If the panel-based routability evaluation method reports that the children of (L h, L v ) both have surplus overflows, the layer configuration selection phase can be early terminated.

23. Layer Stacking Arrangement The stacking order of horizontal and vertical layers would impact the via count. When L h -L v =1, the bottom layer has to be the horizontal layer. When L v -L h =1, the bottom layer has to be the vertical layer. When L h =L v, build two possible layer stacking structures.  Use NCTU-GR to map the 2D result to these two 3D grid graphs to get two 3D routing results  Choose the stacking order that offers fewer via count in its 3D result to be the final solution.

24. Experimental Results

27. Conclusions This paper studies a metal layer planning problem for silicon interposers and presents a variable-layer global router called VGR to solve the problem.