Ultrashort-Pulse Child-Langmuir Law in the Quantum and Relativistic Regimes

This Letter presents a consistent quantum and relativistic model of short-pulse Child-Langmuir (CL) law, of which the pulse length $\tau$ is less than the electron transit time in a gap of spacing $D$ and voltage $V$. The classical value of the short-pulse CL law is enhanced by a large factor due to quantum effects when the pulse length and the size of the beam are, respectively, in femtosecond duration and nanometer scale. At high voltage larger than the electron rest mass, relativistic effects will suppress the enhancement of short-pulse CL law, which is confirmed by particle-in-cell simulation. When the pulse length is much shorter than the gap transit time, the current density is proportional to $V$, and to the inverse power of $D$ and $\tau$.