Yijiang Shen

Level-set-based inverse lithography for photomask synthesis

Inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem. We show how the inverse lithography problem can be addressed as an obstacle reconstruction problem or an extended nonlinear image restoration problem, and then solved by a level set time-dependent model with finite difference schemes. We present explicit detailed formulation of the problem together with the first-order temporal and second-order spatial accurate discretization scheme. Experimental results show the superiority of the proposed level set-based ILT over the mainstream gradient methods.

Robust level-set-based inverse lithography

Level-set based inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem, interpreted with a time-dependent model, and then solved as a partial differential equation with finite difference schemes. This paper focuses on developing level-set based ILT for partially coherent systems, and upon that an expectation-orient optimization framework weighting the cost function by random process condition variables. These include defocus and aberration to enhance robustness of layout patterns against process variations. Results demonstrating the benefits of defocus-aberration-aware level-set based ILT are presented.

Hotspot-aware fast source and mask optimization

Source mask optimization (SMO) is a useful technique for printing the integrated circuit (IC) on a wafer with increasingly smaller feature size. However, complex SMO algorithms generally lead to undesirably long runtime resulting from an optimization of largely identical regions over the whole mask pattern. In this work, a weighted SMO scheme incorporating both an awareness of the hotspots and robustness against process variations is proposed. We show how optimal solutions are reached with fewer iterations by applying various degrees of correction in the corresponding regions. The proposed method includes identifying the hotspots and combining a weight matrix to the cost function for adjustment and control. Simulation results are compared with the mask optimization (under a fixed source) and conventional SMO to illustrate the performance improvement in terms of pattern fidelity, convergence rate and process window size.

Binary image restoration by positive semidefinite programming

We report an optimization approach to restore degraded binary images by using positive semidefinite programming when the point spread function (PSF) is known. The approach takes advantage of the combinatorial nature of the problem, considering not only local similarity and spatial context but also the relationship between individual pixel values and the PSF. Numerical experiments confirm the superiority of the approach.

Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step

Inverse mask synthesis is achieved by minimizing a cost function on the difference between the output and desired patterns. Such a minimization problem can be solved by a level-set method where the boundary of the pattern is iteratively evolved. However, this evolution is time-consuming in practice and usually converges to a local minimum. The velocity of the boundary evolution and the size of the evolution step, also known as the descent direction and the step size in optimization theory, have a dramatic influence on the convergence properties. This paper focuses on developing a more efficient algorithm with faster convergence and improved performance such as smaller pattern error, lower mean edge placement error, wider defocus band, and higher normalized image log slope. These improvements are accomplished by employing the conjugate gradient of the cost function as the evolution velocity, and by introducing an optimal time step for each iteration of the boundary evolution. The latter is obtained from an...

Aberration-aware robust mask design with level-set-based inverse lithography

Optical proximity correction (OPC) is one of the most widely used Resolution Enhancement Techniques (RET) in mask designs. Conventional OPC is often designed for a set of nominal imaging parameters without giving sufficient attention to the process variations caused by aspherical wavefront leaving the exit pupil of the lithography system. As a result, the mask designed may deliver poor performance with process variations. In this paper, we first describe how a general point spread function (PSF) with wave aberration can degrade the output pattern quality, and then show how the wave aberration function can be incorporated into an inverse imaging framework for robust input mask pattern design against aberrations. A level-set-based time-dependent model can then be applied to solve it with appropriate finite difference schemes. The optimal mask gives more robust performance against either one specific type of aberration or a combination of different types of aberrations.

A Signomial Programming Approach for Binary Image Restoration by Penalized Least Squares

The authors present a novel optimization approach, using signomial programming (SP), to restore noise-corrupted binary and grayscale images. The approach requires the minimization of a penalized least squares functional over binary variables, which has led to the design of various approximation methods in the past. In this brief, we minimize the functional as a SP problem which is then converted into a reversed geometric programming (GP) problem and solved using standard GP solvers. Numerical experiments show that the proposed approach restores both degraded binary and grayscale images with good accuracy, and is over 20 times faster than the positive semidefinite programming approach.

Restoration of Binary Images Using Positive Semidefinite Programming

We present a novel approach, using positive semidefinite (PSD) programming, to restore blurred and noisy binary images when the point spread function (PSF) is known. The combinatorial nature of the problem is noted: binary image deconvolution requires the minimization of an energy function over binary variables, taking into account not only local similarity and spatial context, but also the relationship between individual pixel values and the PSF. Due to the high computational load the deconvolution process of a large image might face, we segment the binary image into smaller blocks before deconvolving each block. To suppress error propagation, we also process image blocks with different overlapping lines and columns. Superiority of the proposed PSD binary image restoration approach is confirmed by numerical experiments

Finite difference schemes for heat conduction analysis in integrated circuit design and manufacturing

The importance of thermal effects on the reliability and performance of VLSI circuits has grown in recent years. The heat conduction problem is commonly described as a second-order partial differential equation (PDE), and several numerical methods, including simple explicit, simple implicit and Crank–Nicolson methods, all having at most second-order spatial accuracy, have been applied to solve the problem. This paper reviews these methods and further proposes a fourth-order spatial-accurate finite difference scheme to better approximate the PDE solution. Moreover, we devise a fourth-order accurate approximation of the convection boundary condition, and apply it to the proposed finite difference scheme. We use a block cyclic reduction and a recently developed numerically stable algorithm for inversion of block-tridiagonal and banded matrices to solve the PDE-based system efficiently. Despite their higher computation complexity than direct computation in a sequential processor, we make it possible for the very first time to employ a divide-and-conquer algorithm, viable for parallel computation, in heat conduction analysis. Experimental results prove such possibility, suggesting that applying divide-and-conquer algorithms, higher-order finite difference schemes can achieve better simulation accuracy with even faster speed and less memory requirement than conventional methods. Copyright

Interconnect thermal simulation with higher order spatial accuracy

This paper reports on a numerical analysis of interconnect thermal profile with fourth-order accuracy in space. The interconnect thermal simulation is described in a partial differential equation (PDE), and solved by finite difference time domain (FDTD) techniques using a fourth-order approximation of the spatial partial derivative in the PDE. A recently developed numerically stable algorithm for inversion of block tridiagonal and banded matrices is applied when the thermal simulation is conducted using Crank-Nicolson method with fourth-order spatial accuracy. We have promising simulation results, showing that the proposed method can have more accurate temperature profile before reaching the steady state than the traditional menthols and the runtime is linearly proportional to the number of nodes.