Prediction of Multidimensional Spatial Variation Data via Bayesian Tensor Completion

This paper presents a multidimensional computational method to predict the spatial variation data inside and across multiple dies of a wafer. This technique is based on tensor computation. A tensor is a high-dimensional generalization of a matrix or a vector. By exploiting the hidden low-rank property of a high-dimensional data array, the large amount of unknown variation testing data may be predicted from a few random measurement samples. The tensor rank, which decides the complexity of a tensor representation, is decided by an available variational Bayesian approach. Our approach is validated by a practical chip testing data set, and it can be easily generalized to characterize the process variations of multiple wafers. Our approach is more efficient than the previous virtual probe techniques in terms of memory and computational cost when handling high-dimensional chip testing data.

本文提出了一种多维计算方法，用于预测晶圆多个芯片内部及之间的空间变异数据。该技术基于张量计算。张量是矩阵或向量的高维推广。通过利用高维数据阵列隐藏的低秩特性，可以从少量随机测量样本中预测大量未知的变异测试数据。决定张量表示复杂性的张量秩由一种可用的变分贝叶斯方法确定。我们的方法通过一个实际的芯片测试数据集得到了验证，并且可以很容易地推广到表征多个晶圆的工艺变异。在处理高维芯片测试数据时，我们的方法在内存和计算成本方面比之前的虚拟探针技术更高效。