短時間低溫顯影之電子束微影與鄰近效應修正演算法

Short-Time and Low-Temperature Development for Electron-Beam Lithography and the Algorithms of Proximity Effect Correction

粘群(Chun Nien)

指導教授 ： 管傑雄

國立臺灣大學/電機資訊學院/電子工程學研究所/博士(2017年)

https://doi.org/10.6342/NTU201704286

摘要

電子束微影為高解析度奈米結構之無光罩製造技術中最為廣泛使用與重要的一項技術，也提供了電子與光子元件之高速發展的條件。在電子束微影中最關鍵的一項問題是電子散射造成之鄰近效應，使得製造之元件結構失真並嚴重影響元件性能與應用。大多數嘗試修正鄰近效應的研究視顯影圖樣為電子束曝光造成之靜態結果，聚焦於調控電子劑量之空間分佈。事實上，顯影為一動態的過程，並為顯影時間與溫度的函數，而其不斷變動的性質，可能造成顯影圖樣偏離預期。此效應將隨著元件設計圖案尺寸微縮化越為顯著。近來，少數研究開始考慮將顯影的模擬計算加入鄰近效應修正法中，但顯影本為一複雜過程，傳統計算方法需耗用大量計算資源且計算時間相當冗長。

本論文聚焦於改善使用正型電子阻劑之高加速電壓電子束微影中的鄰近效應。我們利用一系列單點曝光實驗探討顯影過程，建立一可用於描述入射電子劑量、電子阻劑、以及顯影時間與溫度等顯影條件之間關係之綜合微分模型。由此模型可發現曝光位置為一具有超高顯影速率之奇異點，並因此可將其視為顯影過程之起始點。此外，我們藉由曝光之電子劑量點散布函數檢視其引致之特徵區域性質，並更進一步發展可有效降低鄰近效應之低溫短時間顯影法，以該法製造線寬僅8奈米之獨立線或週期30奈米線寬9奈米之高密度線陣列。

從單點實驗研究中對於顯影過程的理解，引領我們發明創新之短距離鄰近效應修正方法。我們基於二維顯影模型提出關鍵顯影路徑概念，並以之作為評估電子束微影顯影圖樣適合度之指標。本論文將搜尋關鍵顯影路徑轉化為圖論之最短路徑問題，因此賦予顯影路徑之模擬能有更高效率之演算法之可能性。我們提出基於Dijkstra演算法之關鍵顯影路徑搜尋法，並使用優先佇列資料結構，以確保計算正確性與維持計算效率。此外，我們探討應用於鄰近效應修正問題中的最佳化策略。我們將具群體智慧之最佳化方法引入鄰近效應修正法，並與基於簡單形之最佳化方法比較。從數值分析顯示，在如逐像素式之精細鄰近效應修正法之較複雜適應度函數，選用適當之最佳化策略極為重要。以本論文提出之鄰近效應修正法應用於U形裂環共振器電子束微影圖樣之最佳化，可有效改善鄰近效應並製成與設計具高吻合度之顯影圖樣。我們提出之鄰近效應修正策略，可顯著降低計算成本並適用於具有多重設計限制條件之複雜圖樣。

關鍵字

電子束微影 ； 鄰近效應修正 ； 顯影模型 ； 次十奈米微影 ； 關鍵顯影路徑 ； 群體智能 ； 圖形演算法

並列摘要

Electron-beam lithography (EBL) is one of the most popular and important techniques in manufacturing high-resolution nanopatterns without masks and enabling the fast development of electronic and photonic devices. The proximity effect is one of the most critical issue in EBL, as it can degrade the pattern quality and, thus, impact the performance of the applications greatly. Regarding the development as a static result of electron-beam exposure, most studies solving the proximity effect by focusing on the spatial distribution of electron intensity. In fact, the development is a dynamic process as a function of the development duration and temperature. The continually changing nature of development may lead to pattern deviation. This effect becomes more noticeable as the required feature size continually shrinks. Recently, some of the researches start to consider integrating the development simulation into the proximity effect correction (PEC). However, the EBL development is a complicated process, and the conventional methods are very computationally intensive and lengthy.

This dissertation focuses on solving the proximity effect of the high-voltage EBL using the positive-tone resist. We first use a set of single-spot experiments to categorize the development process and establish a comprehensive differential model of EBL to describe the relation among the incident electrons, resist, and the development conditions such as durations and temperatures. This model identifies the location of exposure point as a singular point of ultra-high development rate, which, thus, can be considered as the beginning point of the development. Further, we verify the characteristic region of each incident spot induced by the point spread function of the electron-beam system. Eliminating the proximity effect effectively, we further achieve the pattern of isolated line with the line width of 8 nm and dense line array with the line width of 9 nm with the pitch size of 30 nm by utilizing the results from single-spot experiments at low development temperatures.

The insights from the study of the single-spot experiments lead to the innovation of a novel short-range PEC method. Based on the 2-D development model, we propose a novel concept of the critical-development path and its usage in the evaluation of the fitness of EBL patterns. For the first time, we also transform the searching of the critical-development path into the shortest-path problem of graph theory, which enables the potential of using a more efficient algorithm in the simulation of the development path. We propose a Dijkstra-based algorithm with the data structure of priority queue to guarantee the correctness and to maintain the efficiency. Also, we investigate the optimization strategies in the PEC problems. The algorithm of swarm intelligence is introduced to the PEC and is compared with the simplex-based method. From the numerical analysis, we demonstrate that choosing of a suitable optimization scheme is important especially in minimizing a complicated function like the fitness function of EBL with pixel-based fine tuning. The PEC algorithm is applied to the fabrication of an U-shaped split-ring resonators and produces an optimized exposure pattern that shows excellent agreement with the targeted design objectives. Our work on the PEC strategy reduces the computational cost significantly and is particularly suitable for the design of complex pattern with various constraints.

並列關鍵字

electron-beam lithography ； proximity effect correction ； development model ； sub-10 nm lithography ； critical-development path ； swarm intelligence ； graph algorithm

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