Finite Element Analysis of Child-Langmuir Law in the Quantum Regime

Summary form only given. The Child-Langmuir law gives the maximum electron current, known as the space-charge-limited current, arises because the space charge in the diode presents a potential barrier to the incident electrons. While there are modifications due to geometrical and relativistic effects, the limited current remains a fundamental quantity characterizing the beam-gap interaction. In the research of vacuum electronics on a scale down to a few or tens of nanometers, i.e., nano-scale, the quantum effects should be considered and cannot be neglected. In a previous work of Lau et al. (1991) they extended the classical work of Child and Langmuir to the quantum regime by considering a parallel planar gap. On such a microscopic scale, they used the familiar mean-field theory expressed by the self-consistent, coupled Schrodinger and Poisson´s equation in the Hartree approximation. Thus, they solved the one-dimensional Schrodinger equation and the Poisson´s equation in the gap region by using the Wentzel-Kramers-Brillouin-Jefferys (WKBJ) method. They found that the classical value for the space-charge-limited current in such a diode can be exceeded by a large factor due to the effect of tunnelling. In this work, we quantitatively investigate the space-charge limited current in quantum regime by solving the Schrodinger-Poisson equation based on the finite element method instead of the WKBJ method. Detail analysis and numerical results of the quantitative investigation will be presented.