Theoretical Linkage of Thermionic, Field, and Space-Charge Limited Emission for a Vacuum Crossed-Field Gap

Crossed-field devices, such as magnetrons and crossed-field amplifiers, where an external magnetic field is orthogonal to the electric field across the gap, are used in radars, communications systems, and material processing. Improving the efficiency and power density of crossed-field devices requires a robust understanding of electron emission, particularly when multiple emission mechanisms may contribute, such as for thermionic and field emission. Previous studies of nonmagnetic diodes have theoretically identified the transitions, or nexuses, between multiple electron emission mechanisms by examining the relevant equations defining each individual mechanism and then derived exact theoretical solutions that combined these mechanisms and approached the standard emission equations under appropriate limits. This work extends those studies by incorporating the orthogonal magnetic field and assessing the transitions between the Richardson-Laue-Dushman (RLD) formulation for thermionic emission, the Fowler-Nordheim (FN) equation for field emission, the space-charge limit given by both the general (for nonzero electron injection velocity) and (zero electron injection velocity) Child-Langmuir (CL) (generalized CL (GCL) and CL, respectively) laws for space-charge limited current, and the space-charge limited current for a noninsulated (magnetic field below the Hull cutoff) and insulated (magnetic field above the Hull cutoff) diode. We derive exact solutions for the current density for these mechanisms and demonstrate their asymptotic behavior under relevant conditions. Furthermore, we plot the exact solution of current density J as a function of applied voltage V by incorporating the general thermo-field emission theory for the current source and demonstrate the transition between each emission mechanisms with increasing V for various temperatures and fixed ratios of magnetic field to the Hull cutoff. We also construct nexus plots to identify transitions between the various regimes for different conditions.