Transition from Fowler-Nordheim to Child-Langmuir Law in the quantum regime

摘要

The study of an intense charged particle beam propagating in a nano-gap has been studied in recent years [1,2] and the Child-Langmuir (CL) law or the space-charge-limiting law that predicts the maximum current that can be transported across a planar gap is one of the most important aspect of study in the field of high power vacuum micro- and nano-electronics, accelerator physics and sheath physics. A modified formulation of the Child-Langmuir (CL) law, taking into account the effects of work function and image charge, in the quantum regime has been discussed in a recent review [2]. However, the modified formulation may be inadequate to fully describe the behavior of the high current electron field emitted from metallic cathode according to the Fowler-Nordheim (FN) law. A detailed quantum-mechanical analysis of the current-voltage (I-V) characteristic of a 1D planar nano-diode with gap spacing D (in nm range) and gap voltage V{sub}g, which show the transition from the FN law (in the low current regime) to the quantum CL law in high current regime will be presented. A comparison of the transitions from FN regime to SCL regime for different gap distance D using the FN-quantum CL analysis is shown in the figure for a Barium cathode. It is obvious that for D = 1000 nm (dashed line), the transition from FN to CL regime follows the classical CL law, that is the saturation limit at high electric field E{sub}0 = V{sub}g/D approaches the normalized self-consistent field emission current density γ{sub}C=J{sub}C/J{sub}(CL) (with respect to the 1D CL law J{sub}(CL)) converge to 1 at the space-charge-limiting regime. However, at D = 1 nm and 10 nm, where the quantum effects of tunneling electron inside the gap cannot be neglected, and the transition to quantum CL law with γ{sub}C > 1 is shown. The influence of different cathode materials with different work functions and temperature will be presented.