3D model of Child-Langmuir law

Summary form only given. The study of space-charge limited current or the Child-Langmuir (CL) law began almost a century ago, and continues to be an active field of research. The extension of the 1D classical CL law to a multi-dimensional model or quantum regime is the subject of much interest. In this paper, we present an analytical formulation to calculate the maximum CL current density emitted from a finite emission area consisting of 2 different shapes: ellipse and rectangle. The analytical results of the elliptical shape are more accurate than that postulated by Lau. The analytical results are compared and verified with PIC simulation using MAGIC3D code. The comparison shows good agreement over a wide range of parameters. The 3D classical model is also extended to the relativistic and quantum regime. The implication and the application of this model is discussed.