Full Two-Dimensional Markov Chain Analysis of Thermal Soft Errors in Subthreshold Nanoscale CMOS Devices

Thermally induced fluctuations in the logic state of a simple flip-flop occur on a timescale that renders them impossible to simulate through Monte Carlo methods. In a previous work, an analytical framework based on Markov chains and queue theory was introduced along with a symbolic solution for a truncated 1-D queue, diagonally connecting the two stable logic states in a two-dimensional (2-D) queue. In this paper, a complete solution for a full 2-D queue is presented, which maps all the possible thermal noise fluctuations of electron populations in flip-flop inverters. The results for the mean time to thermally induced error confirm the estimates given by truncated approximations. This formalism is also capable of computing arbitrary probability moments as well as steady-state distributions and transient behavior of the system. The full 2-D queue can also capture the statistics of other noise sources, like radiation-induced charge generation where the flip-flop can transiently reside in a queue state far from the diagonal connecting the two stable logic states of a flip-flop.