Generalization of the two-dimensional Child-Langmuir law for non-zero injection velocities in a planar diode

A simple analytic expression of the two-dimensional Child-Langmuir law is derived for non-zero injection velocities and Lau's result is obtained in our model by setting the injection velocity equal to zero. The calculation results show that the modify term in our model is larger than Lau's with a non-zero electronic initial energy, and it is twice as large as Lau's when the electronic initial energy is much greater than the potential energy of the gap.