Numerical solutions to the 1D steady-state Vlasov-Poisson system are used to develop a straightforward empirical formula for the electric current density transmitted through a vacuum diode (voltage gap) as a function of gap distance, gap voltage, the injected current density, and the average velocity and temperature of injected particles, as well as their charge and mass. This formula generalizes the 1D cold beam Child-Langmuir law (which predicts the maximum transmitted current for mono-energetic particles in a planar diode as a function of gap voltage and distance) to the case where particles are injected with a finite velocity spread. Though this case is of practical importance, no analytical solution is known. Found by a best-fit to results from particle-in-cell (PIC) simulations, the empirical formula characterizes the current transmitted across the diode for an injected velocity distribution of a drifting Maxwellian. It is not meant to yield a precise answer, but approximately characterizes the effect of space charge on transmitted current density over a large input space. The formula allows quick quantitative estimation of the effect of space charge in diode-like devices, such as gate-anode gaps in nanoscale vacuum channel transistors.
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。
