Three-dimensional Child–Langmuir law for uniform hot electron emission

This paper presents a three-dimensional (3D) model of Child–Langmuir (CL) law for uniform hot electron emission in planar and cylindrical gap, including the effects of finite emission energy. It is found that the enhancement of 3D CL law (in terms of 1D CL law) can be written in a general form of $JC[3D]∕JC[1D]=1+F×G$⁠, where $F$ is the normalized mean position of 1D electron flow in classical, weakly relativistic, and quantum regime, and $G$ is the geometrical correction factor depending on the geometrical properties of the finite emitting patches on the cathode. In particular, we present the analytical solutions for various emitting patches, such as rectangle, ellipse, square, circle, triangle, and polygon, which agree very well with 3D particle-in-cell simulation. For a cylindrical gap of finite width, it is also found that the convergent flow (cathode outside) has larger enhancement than the divergent flow (cathode inside) at a given aspect ratio of outer radius to inner radius of the gap. Smooth transition from various operating regimes is demonstrated.