A Simple Model for Distance-Dependent Mismatch in Semiconductor Devices

We propose a simple model to explain the distance-dependent mismatch in semiconductor devices. The across-chip variations (ACVs) are attributed to multiple long-range imperfections (LRIs), each modeled by a Gaussian spatial function with a random magnitude following a normal distribution. These LRIs are randomly distributed across the wafer according to a uniform distribution, and their effects on a device are combined through linear superposition. The mismatch between the two devices can be analytically derived and is shown to increase and eventually saturate following a Gaussian function. The characteristic length of the mismatch is found to be $\sqrt {{2}}$ times that of an individual LRI. At small distances, the model reproduces the quadratic distance dependence described by the classical Pelgrom equation. Additionally, it effectively describes the measured mismatch data for both common-centroid and noncommon-centroid layouts. This model also provides a first-principles approach for incorporating distance-dependent mismatch into Monte Carlo simulations.