Critical current in a two-dimensional non-magnetically insulated crossed-field gap with monoenergetic emission

Abstract

Prior studies have developed theories for the maximum permissible current, or critical current, for one-dimensional planar and cylindrical crossed-field diodes where the magnetic field is below the Hull cutoff, meaning that an electron emitted from the cathode reaches the anode. Here, we develop semi-empirical and analytical models to predict the critical current for a two-dimensional (2D) planar diode with nonzero monoenergetic initial velocity. The semi-empirical method considers the geometry, nonzero initial velocity, and magnetic field as multiplicative corrections to the Child–Langmuir law for space-charge limited current in a one-dimensional planar diode with an initial velocity of zero. These results agree well with 2D particle-in-cell (PIC) simulations using the over-injection method to assess virtual cathode formation for different emission widths, magnetic field strengths, and initial velocities. The analytical solution agrees better with PIC results because it accounts for the coupling of the magnetic field, geometry, and initial velocity that the semi-empirical approach does not.