Voltage drop in mesoscopic systems: A numerical study using a quantum kinetic equation

In this paper, we present a numerical method for evaluating the full Wigner function throughout a device by solving a steady-state quantum kinetic equation in two dimensions, in the linear-response regime. This method has two advantages over conventional treatments of mesoscopic devices. First, dissipative processes can be included within the device, thus allowing a smooth transition from the quantum to the semiclassical regime. Second, the contacts are treated in the same manner as in semiclassical device analysis. A short phase-breaking time can be used in the contact regions so that oscillations in the electron density due to interference effects die out quickly; this is particularly useful when obtaining self-consistent solutions with the Poisson equation. Any quantity of interest, such as electron density or current density per unit energy, can be computed throughout the entire device. We will first show that under low-bias, low-temperature conditions, the diagonal elements of the Wigner function can be used to define a local electrochemical potential (μ) that lends insight into the internal transport physics. We show that separate electrochemical potentials ${\mathrm{\mu }}_{\mathit{L}}$ and ${\mathrm{\mu }}_{\mathit{R}}$ for left- and right-moving electrons show unphysical behavior when defined in a local sense. But sensible results are obtained when these potentials are defined in an average sense over regions the size of a de Broglie wavelength. We then examine the difficulties associated with measuring μ, with numerical examples. Next, we use the local electrochemical potential profile to clarify the nature of the spreading resistance associated with the narrowing of a current lead. Finally, we show that the electrostatic potential (φ) can be viewed as a convolution of μ with a screening function and present example computations of φ.