Point-Plane and Edge-Plane Space Charge Limited Flows

Numerical calculations and analytic limiting solutions are given for the fields and charge densities obtained from space charge limited current flow in the edge-plane and point-plane geometries. We show rigorously that, at large distances from the symmetry plane (or symmetry axis), the field lines do follow the Laplacian solution, and that the current density on the ground plane follows the experimental Warburg law, j¿ cosn ß, but with exponent 3. We give examples of analytic solutions which are valid everywhere, but unfortunately these require that the permittivity ¿ and mobility ¿ vary rapidly with position, so these solutions are unphysical. We discuss the difficulties associated with direct numerical approaches, and we present a method in which the singularity in charge density at the source electrode is dealt with explicitly, Our numerical solutions give exponents in the range 4.3 to 5.2 close to the symmetry line, in general agreement with the experimental finding.