Electron Trajectories and Critical Current in a Two-Dimensional Planar Magnetically Insulated Crossed-field Gap

The critical current in a one-dimensional (1D) crossed-field gap is defined by the transition from a cycloidal flow to a near-Brillouin (nB) state characterized by electron flow orthogonal to both the electric and magnetic fields and uniform virtual cathode formation. Motivated by recent studies on space-charge-limited current in non-magnetic diodes, we assess the meaning of critical current in a magentically insulated two-dimensional (2D) planar crossed-field geometry. Particle-in-cell (PIC) simulations demonstrate that binary behavior between a laminar and turbulent state does not occur in 2D because the virtual cathode is nonuniform. Rather than a distinct nB state above the critical current as in 1D, there is an increase in Brillouin contribution with the presence of cycloidal components and noise even at low currents. To evaluate the electron flows in a 2D crossed-field gap in the absence of a binary transition, we developed two metrics to assess the Brillouin and cycloidal components in a 2D planar crossed-field gap for various emission widths and injection current densities by comparing the phase space plots from PIC simulations to analytical solutions for cycloidal and Brillouin flow. For a smaller emission width, less Brillouin contribution occurs for a given injection current, while maximizing the cycloidal noise requires a larger injection current. Once the virtual cathode starts to form and expand with increasing injection current, the cycloidal noise reaches its peak and then decreases while the Brillouin components become significant and increase.