Quantum model of 1D and 2D Child-Langmuir law

Summary form only given, as follows. Summary form only given. In the emerging field of nanotechnology, miniature structures such as nanogaps, nanowires, and nanotubes ranging from <10 to 100s nanometer can be readily fabricated. As a result, vacuum microelectronics (VME) has emerged as a new inter-disciplinary research area, which deals with vacuum devices of micrometer dimensions (or less) that are made by microfabrication techniques developed for semiconductor industry. In VME, interaction of an intense electron beam with its surrounding miniature structures is important in characterizing the device operation However, there is still no fundamental understanding of the interaction of an intense electron beam with its surrounding nanostructures. The classical one dimensional (ID) Child-Langmuir (CL) law gives the maximum current density that can be transported across a planar gap. It is only recently that the 1D CL law is extended to a simple two-dimensional (2D) model. In this paper, we will present a quantum model of the 1D and 2D classical CL law based on a mean-field theory.