Virtual cathode behavior in a crossed-field gap

While the behavior of virtual cathodes (VCs) in a space-charge limited gap in the absence of a magnetic field is well understood, it is less characterized for crossed-field gaps with perpendicular magnetic and electric fields. Thus, this study examines VC formation at steady state in a crossed-field gap with magnetic field B below the Hull cutoff magnetic field B_H for magnetic insulation. VC formation at the critical current density J_c depends strongly on the initial electron velocity u₀ and b = B/B_H. For u₀ ≤ 2ΩD/π, where Ω is the cyclotron frequency and D is the gap distance, there exists a threshold b_VC = B_VC/BH such that a VC forms between the cathode and anode at J_c for b < b_VC, but does not form for b > b_VC. This is unlike the nonmagnetic case, for which a VC forms at J_c whenever u₀ ≠ 0. When no VC forms in a crossed-field gap, the electrons accelerate upon emission from the cathode and decelerate near the anode, causing a peak in electron density. Thus, the limiting current no longer corresponds to the buildup of space-charge at or near the cathode, but instead to buildup at the anode. We also show VC formation for steady-state current densities J < J_c. While a VC also forms at steady state for J < J_c in nonmagnetic diodes with u₀ ≠ 0, VC formation is still necessary at J =J_c. However, for crossed-field gaps, VC formation alone is neither sufficient nor necessary for the limiting current condition.