The invention of the bipolar transistor at Bell labs in the late 1940s and the subsequent development of field-effect devices and integrated circuits led to many of the important technological advancements of the second half of the 20th century. The widespread adoption of integrated circuits and the continuing efforts to improve the technology has motivated significant efforts to develop detailed models of devices. The semiconductor equations are highly nonlinear, so it is very difficult to develop detailed analytic solutions to them, and consequently numerical modeling has been important in the development of the technology.