A fast process variation and pattern fidelity aware mask

optimization algorithm

Ahmed AwadAtsushi Takahash Satoshi Tanakay

Chikaaki Kodamay

ICCAD’14

Outline

• Introduction• Problem Formulation And Layout Fragmentation• OPC Algorithm Overview• System Modeling And Solution• OPC Computational Time Reduction Approach• Experimental Results• Conclusion

Introduction

• Sub-100 nm nodes printing suffers from image distortions which badly impact the proper functionality of the circuit. This gave the birth of Optical Proximity Correction(OPC), in which the mask shape is modified to improve the image quality.

• Edge Placement Error (EPE)• Process Variability Band (PV-band) area

EPE

• The geometrical distance between any given point on the target and its corresponding point on the printed image.

PV-band area

• Process variations can be modeled as the area of XOR between two printed images obtained by two extreme(at nominal focus and +2% dose, (ii) at defocus and -2% dose) conditions.

Introduction

• In our algorithm, a mask pattern is optimized using mainly three types of components: core-polygons, hammers, and Sub-Resolution Assist Features (SRAFs).

• we propose a new mask optimization algorithm to minimize EPE and PV-band within a short computational time.

Problem Formulation And Layout Fragmentation

• Given a target layout defined in layout region, the main objective is to find a best mask solution in layout region with the least number of EPE violations (EPEV) under nominal lithographic conditions and least PV-band area between two extreme lithographic conditions within a short optimization time.

Lithographic Model

• In this model , if the intensity is greater than intensity threshold (Ith), the resist will be exposed followed by etching.

transmission cross coefficient function

Lithographic Terminology

• Let R be a layout region which consists of pixels. Let T be a target layout in R

• Let G(M) denotes the printed image of mask M on the wafer

• A mask generated in our algorithm consists of three types of parts; core-polygons, hammers and SRAFs.

Layout Fragmentation

• A segment derived from fragmentation is allowed to shift in orthogonal direction of its length and is used to represent a rectangle which is added into a core-polygon in mask.

Edge Placement Error (EPE) Formulation

Let depe be the allowable distance for EPElet t- be a point in T whose distance from t is depe pixels and which is onthe line that passes t which is perpendicular to the edge of polygon to which t belongs.

Process Variability (PV) Band FormulationTo evaluate the mask robustness against process variations,XOR is applied between two printed images generated usingtwo extreme process variables.

OPC Computational Time

• OPC computational time is the total time required to generate the mask solution. As shown in eq.(1)

• The mask M has to be convoluted with all kernels to obtain an aerial image.

OPC Algorithm Overview

System Modeling And Solution

• Edge Placement Error (EPE) Optimization Model• 1) Segment Shifting• 2) Corners Hammering• Process Variability (PV) Optimization Model• 1) SRAF Insertion

Segment Shifting• In our mask optimization algorithm, a core-polygon is

iteratively modified by shifting two neighboring segments

• h is the distance between the segment current location and the new location after any OPC iteration • The target is to find the solution vector (ha, hb)

Corners Hammering

• When adding a hammer, the intensity in c is defined as a function of the initial intensity and the hammer width w as shown in eq.(10) where P(w) represents the changes in the corner intensity as response of hammer width.

SRAF Insertion and Sizing

• Typically, the distance between contour of Gi(M) and the tap point is closely related to PV-band area.• PV band reduction is achieved either by enlarging the

inner image contour, or by shrinking the outer image contour. However, such shrinking requires shifting segments that could increase EPE value. Therefore, using unprinted SRAFs to increase Ii(M; p) is preferred.

• This is modeled by defining for each tap t as in eq.(11)

• To reduce PV-band area, the number of pixels in the target whose value is positive has to be minimized in case the outer image is exposed with violating EPE.• This can be utilized by minimizing value for each tap point

in the target as long as Io(M; p) >= Ith. Hence, the main objective is formulated as follows:

• Proper SRAFs insertion will reduce for each tap point without violating EPE conditions

OPC Computational Time Reduction Approach• The intensity difference map is defined as the difference

of the intensity map of a large number of kernels and the intensity map of a small number of kernels.• In our fast intensity map estimation, an improved

accuracy intensity map is obtained by combining the intensity map by the first kernel and the intensity difference map as shown in Figure 10• the number of convolutions is given in eq.(14).

Experimental Results

• Implemented in C on a 4 cores 3.6 GHz Linux machine with total memory of 1986912 KB.• Executed on the 10 public benchmarks released in ICCAD contest

2013• To make our results comparable, we used ICCAD cost function

Conclusion

• In this paper, we proposed a new algorithm to minimize EPE and PV-band within a short computational time.• Experimental results on the public benchmarks show

that our algorithm effectiveness can exceed the top 3 teams for ICCAD contest.