The Child-Langmuir law relates the voltage applied across a planar diode to the saturation value J_CL of current density that can be transmitted through it in case the injection velocity of electrons into the diode is zero. The Child-Langmuir current density J_CL 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 {\em any} value of injected current density above J_CL. Existing generalizations of Child-Langmuir law to non-zero velocities of injection are based on the characteristics (i) and (ii) of J_CL. This paper generalizes the law to non-zero 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.
Comment on "New Scaling of Child-Langmuir Law in the Quantum Regime"
在他们关于量子Child-Langmuir定律的信中，Ang，Kwan和Lau（Phys. Rev. Lett. 91，208303（2003））包括Kohn-Sham密度泛函理论内的交换相关效应，并在数值上探索纳米间隙中的最大传输电流。我们在这里表明，计算是错误的，因为交换相关的化学势的分量已被忽略，而固定的边界条件的Hartree势。
