Two-dimensional spatially periodic electron flow in various emission regimes

We study the effects of cathode surface curvature on the space charge limited current emitted by a two-dimensional periodic array of field emitters. Each of these emitters has a shape described by a simple analytic function. Linear, quadratic, and Fowler-Nordheim current-field dependences of the cathode emissivity as well as the infinite emissivity Child-Langmuir model are considered. We develop a mathematical anzatz to capture the main features of the potential field structure of this system and supplement it with a set of correction functions with free parameters. A special least square procedure is used for an approximate solution of this nonlinear problem. We find that even a smooth curved cathode can yield significant spatial variations in the current density but for the cases considered it does not change substantially the total current (properly adjusted). When the cathode emissivity and/or the applied voltage are high enough the current density from the top of the cathode bump (where the curvature is maximal) exceeds the current density produced by a flat cathode with infinite emissivity placed at the same distance from the anode. An explanation of this effect is given. The spatial pattern of emission is determined almost solely by the cathode curvature no matter how strong the current is.