以幾何規劃方式求解矽鍺異質接面雙極性電晶體摻雜輪廓最佳化之研究

Abstract

隨著高頻電路應用需求之提升，矽鍺(SiGe)異質介面雙極性電晶體(HBTs) 的截止操作頻率(Cut-off frequency)也一直不斷的上升。而藉由調整基極的摻雜 輪廓，可以增加矽鍺異質介面雙極性電晶體的操作速度。然而，設計基極的摻 雜輪廓往往需要經由經驗法則反覆地嘗試而耗費許多時間與金錢成本。 幾何規劃(Geometric programming)為一種數學的最佳化問題，近年來常常 被應用在科學與工程問題。藉由凸函數轉換與對偶定理，以及內點法演算方法 (Interior point method)與電腦近幾十年來電腦計算能力的提升，我們可以迅速 求解具大規模變數(Optimal variable)與限制式(Constraints)的幾何規劃問題，並 且求得全域最佳解(Global solution)。 本論針對基極矽鍺摻雜輪廓設計寫成一幾何規劃問題。首先將矽鍺異質介 面雙極性電晶體的截止頻率數學模型推導成與矽鍺摻雜輪廓有關的連續積分 函數。接著，將此連續積分函數做離散(Discretization)，並將矽鍺摻雜輪廓表 示為與基極位置(Base region)有關之離散的空間變數。透過以上的估計，我們 可以把截止頻率表示成一特殊的函數—正多項式(Posynomial) ，而可將此非線 性的最佳化問題轉成一幾何規劃問題，並以內點法求解。在不失工程準確性的 要求下，此方法有效地提供了快速的矽鍺摻雜輪廓萃取。 為了驗證最佳化模型的準確度，我們以二維度(2-D)元件模擬器對異質介 面雙極性電晶體模型中參數與元件特性做初步校估。結果顯示23%濃度的梯 型形鍺摻雜輪廓可以讓電流增益(Current gain)最大，其值約為1100，比0%濃 度(傳統雙極性電晶體)的電流增益(約為200)大；而12.5%的鍺濃度與三角形的 摻雜輪廓可以讓電晶體達到254GHz 的截止操作頻率，也提升了傳統雙極性電 晶體的截止操作頻率(約為71GHz)。 總之，本研究已運用幾何規劃數學方法來求解矽鍺異質接面雙極性電晶體 最佳摻雜輪廓。此研究有助於電腦模擬器的最佳化功能設計，對於半導體大廠 之技術能力提升有正面之助益。

As the the need of high frequency circuits, the speed of silicon-germanium (SiGe) heterojunction bipolar transistors (HBTs) has been dramatically increased. It is known that the speed of HBTs is dominated by the base transit time, which is influenced by the doping profile in the base region and the Ge concentration. However, to design the doping profile of HBTs requires lots of empirical experiences and time-consuming try-anderror rounds. Geometric programming (GP) is a type of mathematical optimization problem, recently used in applied science and engineering widely. Based on the transformation of the geometric programming into convex form and the duality theory, also benefited from the interior point method and nowadays computing power, we can solve geometric programming problem with large scale optimal variables and constraints efficiently and globally. In this study, the design of the doping profile and Ge-dose concentration for SiGe HBTs are mathematically formulated and solved by the technique of geometric programming. At first, we derive the cut-off frequency model as an integral of Si doping profile and Ge-dose. Then, then discretization of the integral function according to the base region, is applied to obtain the discretized optimal variables of doping profile. Base upon the aforementioned approximation, we could derive the cut-off frequency model as a posynomial function; after that, the interior point method is employed to solve the well-formulated geometric programming. This methodology provides an efficient mechanism to extract the Si doping profile and Ge-dose. The solution calculated by the GP method is guaranteed to be a global optimal. The accuracy of the adopted numerical optimization technique is first confirmed by comparing with a two-dimensional device simulation. The result of this study shows that a 23 % Ge fraction have the maximum current gain, about 1100, which higher than the 0 % Ge fraction (BJT), about 200. Furthermore, a 12.5% Ge may maximize the cut-off frequency for the explored device, where a 254 GHz cut-off frequency is achieved, high than the 0 % Ge fraction case, about 71 GHz. In summary, we have successfully optimized the doping profile of SiGe HBTs using GP method. The results of this study may benefit the technology computer-aided design tool in semiconductor industry.