Computational lithography with advanced optimization algorithms

Abstract

Lithography techniques have long been the driving power for the advancement of Moore’s law for the semiconductor industry. In recent years, next generation lithog- raphy techniques, such as extreme ultraviolet lithography, have been extensively de- layed. ArF (Argon Fluoride) immersion lithography with a wavelength of 193 nm is still the primary technique for high volume integrated circuit manufacturing at the 10 nm technology node, and extensive resolution enhancement techniques (RETs) are required to its lifespan. Computational lithography techniques, including source mask optimization (SMO) and inverse lithography technology (ILT), have gained widespread interest due to their strong abilities to enhance the resolution. Yet, it is very challenging to solve computational lithography problems since they are in- trinsically ill-posed, large-scale, and nonlinear, frequently requiring a large amount of computation. At the same time, the solution needs to meet certain requirements, such as the control of source and mask manufacturability and mask shape uncer- tainty. The purpose of this dissertation is to harness the recent development of algorithms, such as convex optimization and sparse optimization, to cope with the challenges arising with computational lithography.

First, to reduce the number of variables in the optimization problems, a repre- sentation method with basis functions is proposed for both source and mask opti- mizations. The source patterns are represented as a superposition of pre-defined Zernike basis functions weighted by a series of coefficients. Similarly, the mask patterns are represented by a linear combination of discrete cosine basis functions. The source optimization problem is then formulated as a convex problem, and can be solved efficiently. The number of iterations in mask optimization can also be significantly reduced, and thus accelerating the optimization computation. Second, a sparse nonlinear inverse imaging method is proposed to tackle mask manufacturability problem in ILT. The mask patterns are represented with pixel- based images to enlarge the solution space in ILT; however, the optimized mask usually contains complex features which are difficult to produce with the current e-beam writers. In order to reduce the complexity of the optimized patterns, a model-based fracturing (MBF), which is formulated as a sparse nonlinear inverse imaging problem, is integrated into the ILT process. It is shown that the integra- tion can effectively reduce the complexity of the optimized pattern and improve the manufacturability. Third, a random field method is proposed to incorporate the mask shape un- certainty into computational lithography. Fabricating the mask after ILT usually introduces shape errors, which are amplified by the imaging system to the wafer especially when the feature sizes are extremely small. In order to reduce the errors propagated to the wafer, the potential mask errors are considered to be a form of mask shape variation, and they are modeled with a random field. Then, an opti- mization technique, which incorporates the random field, is used to derive an opti- mized mask robust against the shape variations. It is demonstrated that the method is effective in reducing the mask error enhancement factor (MEEF) and improving critical dimension (CD) uniformity. (Total words: 493)