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This work characterizes the temperature, channel length, and voltage dependences of substrate current, and presents a local model describing this behavior using Shockley's lucky electron (LE) model as a basis. For n-channel (p-channel) devices, the model is extended using a Maxwell-Boltzmann (MB) distribution of hot-electron (hole) energies above (below) the conduction (valence) band minimum (maximum). The model has been implemented in CADDET, a 2-D device simulator, and is able to explain all of the important features of substrate current which have been reported to date. The model is discussed in the context of works which look at both the local and physical nature of the impact ionization phenomenon. Based on this discussion, the model's parameters are shown to have a solid physical basis, requiring no reliance on curve fitting. The agreement between data and simulations thus enhances physical understanding of substrate current in MOSFET's, and warrants confident design of CMOS technologies for cryogenic operation.