Relativistic Space-Charge-Limited Current in Electrostatic Limit for a Finite-Width Diode with Arbitrary Injection Energy

We develop a weakly relativistic and electrostatic formulation of space-charge-limited current (SCLC) with arbitrary injection energy, covering the accelerating-diode and drift-space regimes. The one-dimensional (1D) relativistic SCLC model is obtained from a stationary-current condition, which is validated against the numerical solutions. Based on this relativistic 1D SCLC model, we investigate two different approaches to account for the effects of finite emission area for a two-dimensional (2D) planar diode of a finite width of $W$, which is larger than the gap spacing $D$. The field-balance (FB) approach evaluates the opposing space-charge fields at the potential minimum, whereas the density-weighted (DW) approach uses the mean absolute distance of electrons from that minimum. The two approaches yield different corrections at finite injection energy, even in the non-relativistic limit, with FB model predicting a larger leading-order enhancement. DW model reproduces our previously published classical coefficient and the classical drift-space limit, whereas the FB model gives a different result. Note both approaches are identical for zero injection velocity. Comparisons with 2D electrostatic particle-in-cell (ES-PIC) simulations, it is found that FB and DW model show better agreement for higher injection energy (= gap voltage) at smaller and larger $W/D$ emission area respectively.