Hsing-Chih Chang Chien Hung-Chih Ou Tung-Chieh Chen Ta-Yu Kuan Yao-Wen Chang Double Patterning Lithography-Aware Analog Placement

Outline Introduction DPL conflict handling The algorithm flow Experimental results Conclusions

Introduction DPL  decompose patterns of a layer into two sub-patterns and then use two masks to manufacture the two sub-patterns

DPL conflict  Two patterns separated by the distance less than S min belong to the same mask.

Analog Placement Problem Previous Works  corner stitching compliant B*-tree (CB-tree) [14] DPL is not considered in previous works.

Analog designers usually reduce mismatches by predefining masks (colors)

Preliminaries For constraint-driven analog placement, CB-tree is the most effective and efficient topological representation with module adjacency information

Review of Corner Stitching Corner stitching is a data structure for representing non-overlapping rectangular modules in a two- dimensional plane (called tile plane).

Analog Constraints Symmetry constraints  are use to place some pairs of modules symmetrically along a vertical or a horizontal symmetry axis. It can reduce mismatches of sensitive modules. Proximity constraints  are use to place modules at closest proximity to reduce process variation.

Problem Formulation Rectangular modules M = {mk|1 ≤ k ≤ |M|} Nets N = {nk|1 ≤ k ≤ |N|} Placement Constraints S = {sk|1 ≤ k ≤ |S|} Pre-coloring Constraints R = {rk|1 ≤ k ≤ |R|} Place all modules in M to minimize the total area, wirelength, and DPL conflict

DPL conflict handling Resolving an analog placement with DPL conflicts. extended conflict graph(ECG) to model the global relationship among patterns/modules.

Module Flipping

Extended Conflict Graph (ECG) The traditional conflict graph considers the information from a fixed layout. Since a poly can only be placed vertically, there are only two choices for module orientations: non-flipped and flipped.

Basic Integer Linear Programming (ILP) Formulation

ILP Problem-Size Reduction Connected Component Computation Two-Edge-Connected Component Computation  identify a bridge

The algorithm flow

Perturbation In stage 1 & 3 In stage 2

The Three-Stage Placement Flow

Tile Plane Updating  to create new tiles and update the corner stitches to record the neighbors for each tile.

Experimental results C++ programming language. Intel Xeon X GHz Linux workstation with 48GB memory. CPLEX12.3 [1] library to solve the ILP problems. Two industrial analog circuits Case1 and Case2.

Conclusions DPL-aware analog placement flow to simultaneously minimize area, wirelength, and DPL conflicts. Propose an extended conflict graph (ECG). Develop an ILP algorithm considering symmetry and pre-coloring constraints to minimize conflicts. A three-stage flow.