Application of Variational Mechanics to the Child-Langmuir Law

The Child-Langmuir law yields the space charge limited current density for a one-dimensional, planar vacuum gap for a single charged species with no initial velocity with nonrelativistic mechanics under the electrostatic condition and yields a critical current density J_c ~ V^3/2/D², where V is the applied voltage and D is the gap distance. This theory may be derived from Poisson’s equation, a capacitance model, or a transit time model (P. Zhang, A. Valfells, L.K. Ang, J. W. Luginsland, and Y. Y. Lau, Appl. Phys. Rev. 4, 011304 (2017)). The transit time model may also be used to derive the Langmuir-Blodgett law, which gives J_c ~ E_c^3/2/D^1/2, where E_c is the vacuum electric field on the cathode. One may also derive scaling laws with cathode protrusions to predict an equivalent multiple dimension Child-Langmuir law. Here, we propose the application of variational mechanics to predict space-charge limited current in a manner that is independent of reference frame, and thus applicable to planar, cylindrical, and spherical coordinate systems in 1-D, 2-D, and 3-D geometries. The implications and extension of this technique to other space-charge dominated mechanisms, such as Mott-Gurney for collisions and crossed-field incorporating magnetic field, will be discussed.