Abstract
It is found that the Langmuir-Blodgett solutions for the space charge limited current density, for both cylindrical and spherical diodes, may be approximated by J(app) = (4/9)epsilon(0)root(2e/m)(E-c(3/2)/root D) over a wide range of parameters, where E-c is the surface electric field on the cathode of the vacuum diode and D is the anode-cathode spacing. This dependence is valid whether R-a/R-c is greater than or less than unity, where R-a and R-c are, respectively, the anode and cathode radius. Minor empirical corrections to the above scaling yield fitting formulas that are accurate to within 5% for 3 x 10(-5) < R-c/R-a < 500. An explanation of this scaling is given. An accurate transit time model yields the Langmuir-Blodgett solutions even in the Coulomb blockade regime for a nanogap, where the electron number may be in the single digits, and the transit time frequency is in the THz range.