Analytic Derivation and Minimization of Line Edge Roughness in Electron-beam Lithography

Abstract

Electron-beam (e-beam) lithography is widely employed in a variety of areas such as fabrication of photomasks, imprint lithography molds, and experimental circuit patterns, etc., because of its ability to transfer ultra-fine features onto the resist and eventually to the substrate. Its main limitations are the low throughput due to the pixel-by-pixel or feature-by-feature writing and the proximity effect caused by electron scattering. The importance of developing effective and efficient schemes for correcting the proximity effect has been well recognized for a long time, and various methods were proposed and implemented by many researchers. As the feature size decreases well below microns into nanoscale, line edge roughness (LER) becomes an increasingly important factor that cannot be ignored since it does not scale with the feature size. It can significantly affect the minimum feature size and maximum feature density realizable in practice, and also the functionality of a device. Therefore, it is unavoidable to minimize the LER in order to maximize the feature density and enhance the yield of fabricated devices.

One important step required in developing an effective method to minimize the LER is to understand the characteristics of LER. That is, it is necessary to be able to estimate the LER accurately. A possible approach is to rely on a simulation. Given a circuit pattern, the stochastic exposure (energy deposited in the resist) distribution is either obtained directly through the Monte Carlo simulation or computed through the convolution with stochastic point spread functions generated via the Monte Carlo simulation. Then, the remaining resist profile (from which the LER is quantified) is derived through a resist development simulation. While the simulation approach is flexible, the main drawback is that it is computationally intensive, since it requires an excessive computation to analyze the quantitative relationship between the LER and e-beam lithographic parameters, e.g., the dose (the amount of electron charge given to a unit area). Once the set-up of e-beam lithographic process is changed, e.g., for a different type of resist, such a process needs to be repeated. An analytic approach to LER estimation is an alternative, which can avoid the repetitive procedure, since the explicit relationship between the LER and e-beam lithographic parameters can be derived.

In this dissertation, an analytic method of deriving and minimizing the LER for a single-line pattern caused by the stochastic variation of developing rate in the resist in the e-beam lithography is described with a realistic 3-D model. The specific objective is to derive an accurate analytic expression of the LER from the moments of developing rate distribution, i.e., the mean and variance. With the developing rate distribution, development paths in the resist development process are derived and used to estimate the variation of edge locations, i.e., LER. The analytic expression of LER is used to minimize the critical dimension (CD) error and LER by optimizing the dose. Then, the analytic method is extended to large-scale uniform patterns exposed with a uniform dose by modeling the global exposure distribution. In addition, to verify the analytic method by using the experimental results, a method of extracting the stochastic information needed by the analytic method from experimental results, i.e., SEM images, is developed. A specific goal of this study is to obtain an explicit analytic expression of LER which can be used in the minimization of LER. In order to achieve the goal, in some steps of derivations, certain assumptions and approximations are made and numerical computations are employed, i.e., the derivations of LER are not completely analytic.