Uniform space-charge-limited current for a two-dimensional planar emitter with nonzero monoenergetic initial velocity

Abstract

While characterizing space-charge-limited current (SCLC) is important for numerous applications, no analytical solutions for SCLC with monoenergetic initial velocity exist for two-dimensional (2D) geometries. Here, we derive approximate closed-form solutions for uniform SCLC with monoenergetic emission of electrons in a 2D planar diode, where emission is restricted to a long patch of width W for electrodes separated by a distance D. We also derive a semiempirical approach for estimating the SCLC for these cases by treating the geometric and velocity correction factors as multiplicative corrections to the SCLC for a one-dimensional vacuum diode given by the Child–Langmuir (CL) law. We show that the SCLC for a finite patch with nonzero velocity can exceed the CL law by three orders of magnitude. The theoretically calculated SCLCs for various emission widths and initial velocities in the 2D diode agree well with particle-in-cell simulations using the over-injection method in XOOPIC; they agree with the semiempirical relationship for lower initial velocities. In the limit of high initial velocity, the geometry and velocity corrections to the CL law cannot be decoupled, invalidating the assumption of the semiempirical approach and causing it to diverge from the theoretical solution and XOOPIC simulations. These results provide valuable estimates for determining the onset of virtual cathode formation for photocathodes and thermionic cathodes, which operate in the over-injection regime to avoid beam quality degradation.

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

“…Note that applying periodic boundaries would cause the number of macroparticles to increase without bound rather than reach a steady state. The numerical settings, with a square mech size with a length of ∆𝑥 = ∆𝑦 = 5.4 × 10 −5 m and a time step of ∆𝑡 = 9 × 10 −14 s, are identical to our previous work [33] with the numerical stability ensured [45]. We used the analytical Brillouin flow solutions in 1D to evaluate the components of cycloidal, Brillouin, and cycloidal noise from XOOPIC models.…”

“…In our XOOPIC models, the 2D critical current density 𝐽 CR,2D,B z may only be determined by identifying the potential derivative 𝑑𝜙 𝑑𝑥 ⁄ = 0 or the electric field 𝐸 𝑥 = 0 at 𝑥 = 0, which corresponds to the SCLC for zero emission velocity [2]. A more practical way is to identify whether 𝐸 𝑥 > 0 at 𝑥 = 0 by varying 𝐽 in in a binary search manner [33]. The resulting 𝐽 CR,2D,B z from XOOPIC is the minimum 𝐽 in that results in VC formation.…”

“…1). This resembles higher dimensional SCLCD problems where a nonuniform VC appears initially in a limited region [30], [33], [38]. The critical current density 𝐽 CR for 1D and 2D planar CFDs was found analytically by assuming a space-charge-limited condition and examined by electrostatic, particle-in-cell (PIC) simulation in the nonrelativistic regime [3], [23].…”

“…For 𝐵 = 0, the critical current density is the SCLCD, which has been studied for different geometries [26]- [29], multiple dimensions [30]- [33], and nonzero initial velocity [33], [34]. Introducing a magnetic field orthogonal to the electric field in the gap modifies the maximum emitted current density permissible to 𝐽 CR , which may be approximated as a crossed-field SCLCD (CF-SCLCD) [7].…”

“…The SCLCD (or critical current in the case of the crossedfield case) considering specific effects is often represented as a modification of 𝐽 CL . Multiple correction factors may be applied to (2) to incorporate the contributions of other physical phenomena, such as an orthogonal magnetic field [23], multiple dimensions, and nonzero initial velocity [33].…”