Quantum extension of Child-Langmuir law

Summary form only given, as follows. In the advent fields of nano-technology, electrode gaps with scales down to nanometer regime can be fabricated. On such a nano scale, quantum effects such as electron tunneling and the electron exclusion principle become important in the study of beam-gap interaction. We extend the classical Child-Langmuir law into the quantum regime including the effects of electron tunneling and the electron exclusion principle for various geometrical electrodes. By using local density approximation, we introduce an exchange potential term in the time-independent Schrodinger equation to account for electron exclusion principle. Our results shows that the limiting current density is higher when exclusion principle is included, and it is more significant when the applied DC gap voltage is smaller than the Hartree energy (EH). Our model shows that limiting current is increased by a large factor when the gap spacing is on order of electron de Broglie wavelength (at energy equals to gap voltage). Geometrical effects of the electrodes are investigated by extending our one-dimensional planar model to cylindrical with one set of parameters used to infer the results expected for other sets of parameters. Further application and implication of this work is discussed.