Generalization of the Child–Langmuir law for nonzero injection velocities in a planar diode

Abstract

The Child–Langmuir law relates the voltage applied across a planar diode to the saturation value JCL of current density that can be transmitted through it in case the injection velocity of electrons is zero. The Child–Langmuir current density JCL is, at the same time: (i) the maximum current density that can be transmitted through a planar diode, (ii) the current density below which the flow is steady and unidirectional in the long time limit, and (iii) the average transmitted current density for any value of injected current density above JCL. Existing generalizations of Child–Langmuir law to nonzero velocities of injection are based on the characteristics (i) and (ii) of JCL. This paper generalizes the law to nonzero velocities of injection based on the characteristic (iii) by deriving an analytical expression for the saturation value of current density. The analytical expression for the saturation current density is found to be well supported by numerical computations. A reason behind preferring the saturation property of the Child–Langmuir current density as the basis for its generalization is the importance of that property in numerical simulations of high current diode devices.