考慮製程變異及擁有對功率模型具高度相容性的電熱模擬器

Abstract

本篇論文中提出一個統計型的電熱模擬器，此模擬器考慮了漏電流、晶片與晶片間的製程變異和一個晶片內具有空間相關的製程變異。利用卡洛展開(Karhunen-Loève expansion)，吾人可以將一個具空間相關的製程變異參數，轉換成一組不具相關性的隨機變數做表示，接著在此不具空間相關性的隨機變數，以及代表晶片與晶片間製程變異的隨機變數所共同組成的隨機空間中，使用史摩亞克稀疏網格方法（Smolyak sparse grid method）在此隨機空間中去取樣以求解統計型熱傳方程式。接著透過電熱偶合演算法，可以在每一個取樣點得到一個晶片上的熱分佈。這些計算所得到的熱分佈，會被用來內插在一個晶片上的統計熱分佈，而一個統計上的熱分佈結果可以透過機率的運算所萃取出來。 本篇論文提出的統計型電熱模擬器的準確度，吾人利用蒙地卡羅分析（Monte Carlo analysis）做為比較，而此分析器的效率，是透過蒙地卡羅分析達到同一分析精確度的執行時間做為比較基準。根據實驗結果，本統計型電熱模擬器可以達到比蒙地卡羅分析快一個數量級的速度，且其結果在一個晶片上的溫度期望值最大的誤差在0.36%之內，溫度標準差的誤差小於1.88%。除此之外，本篇論文的電熱模擬器具有對不同功率模型的高度相容性，這個特性對於快速演進的科技是非常重要的。

In this paper, a statistical electro-thermal simulator considering leakage power, inter-die process variations, and intra-die process variations including spatial correlation is developed. With applying Karhunen-Loève expansion, the spatially correlated process parameters can be transformed to a set of uncorrelated random variables. Then, Smolyak sparse grid method is applied to sample the random space expanded by these uncorrelated random variables and inter-die random variables to tackle stochastic heat transfer equations. After that, the thermal profile at each sampling point is built by a constructed electro-thermal coupling algorithm. These calculated thermal profiles are integrated to interpolate the stochastic temperature profile over a chip. Finally, the statistical temperature profile can be extracted. The accuracy and efficiency of the presented statistical electro-thermal simulator are demonstrated by comparing with the Monte Carlo analysis. Experimental results indicate that the developed simulator is orders of magnitude faster than that of the Monte Carlo analysis under the same accuracy level. The maximum error is less than 0.36% and 1.88% in mean and standard deviation of temperature profiles, respectively. The proposed simulator is also highly compatible with different power models and spatial correlation functions. This characteristic is important in such fast innovative technology.