Quantum correction to the equation of state of an electron gas in a semiconductor

In a recent paper [M. G. Ancona and H. F. Tiersten, Phys. Rev. B 35, 7959 (1987)] a macroscopic description of conduction electrons in a semiconductor was presented in which the equation of state for the electron gas was generalized to include a dependence on the gradient of the density. This generalization led to a new transport equation---often expressible as a generalized diffusion--drift-current equation---which has been shown to accurately describe some important quantum mechanical effects occurring in semiconductor structures. In the present paper sufficient microscopic conditions under which the density-gradient--dependent equation of state does represent lowest-order effects of quantum mechanics are established using methods of quantum statistical mechanics. A microscopic derivation of the transport equation is also given.