Collisional space-charge-limited current with monoenergetic velocity: From Child–Langmuir to Mott–Gurney

We develop a general electrostatic steady-state model to calculate the two-dimensional (2D) space-charge-limited current (SCLC) with an initial velocity v0 (i.e., kinetic energy eKV=mv02/2) injected from a cathode into a vacuum diode with a spacing D and a potential difference of Vg. We consider two types of beam cross section: (a) a sheet beam with a width W and (b) a circular beam with a radius R, where both W and R are larger than D in the calculation. By introducing a parameter ε≡KV/Vg≥0, we can tune the operating regime to be in between the accelerating diode (ε≪1) and drift space (ε≫1). Hence, we verify the 2D SCLC model by converging to some prior analytical findings such as the 2D Child–Langmuir law at ε = 0, 2D SCLC at drift space at ε≫1, and their 1D limits at W/D≫1 or R/D≫1. Our 2D SCLC model indicates that there is a maximal value at ε∼1/60, which is independent of W/D or R/D. Furthermore, the hysteresis behavior is also studied when the injected current density is beyond the SCLC threshold and its dynamics are mainly influenced by ε instead of geometrical effect. This model is useful in high current electron beam emission with a finite emission area and with a non-negligible arbitrary injection kinetic energy.

Two-dimensional space charge limited current in regime between accelerating diode and drift space for sheet and circular beam

Liu,

Ang

2024

Critical current in a two-dimensional non-magnetically insulated crossed-field gap with monoenergetic emission

Zhu,

Wright,

Harsha

et al. 2024

Prior studies have developed theories for the maximum permissible current, or critical current, for one-dimensional planar and cylindrical crossed-field diodes where the magnetic field is below the Hull cutoff, meaning that an electron emitted from the cathode reaches the anode. Here, we develop semi-empirical and analytical models to predict the critical current for a two-dimensional (2D) planar diode with nonzero monoenergetic initial velocity. The semi-empirical method considers the geometry, nonzero initial velocity, and magnetic field as multiplicative corrections to the Child–Langmuir law for space-charge limited current in a one-dimensional planar diode with an initial velocity of zero. These results agree well with 2D particle-in-cell (PIC) simulations using the over-injection method to assess virtual cathode formation for different emission widths, magnetic field strengths, and initial velocities. The analytical solution agrees better with PIC results because it accounts for the coupling of the magnetic field, geometry, and initial velocity that the semi-empirical approach does not.

Limiting current in a collisional crossed-field gap

Breen,

Cartwright,

Loveless

et al. 2024

Crossed-field devices are often used in pulsed power and high-power microwave applications. Previous studies derived closed-form solutions for the limiting current of a vacuum crossed-field system, corresponding to the maximum permissible current for laminar flow, below and above the Hull cutoff BH for magnetic insulation. We extend these studies by introducing collision frequency into the electron force law as a friction term to derive the limiting current in a collisional crossed-field gap. The resulting solution recovers the vacuum crossed-field case in the limit of no collisions and the collisional space-charge limited current with general initial velocity for magnetic field B→0. In the limit of infinite collisions, we obtain a crossed-field equivalent to the Mott–Gurney law for the maximum current permissible in a collisional, nonmagnetic diode. When the collision frequency ν is less than the electron cyclotron frequency Ω, increasing initial velocity makes the critical current nonmonotonic with increasing ν with the critical current higher at B=BH for ν=Ω. As for a misaligned crossed-field gap where a component of the magnetic field was introduced parallel to the electric field across the gap, magnetic insulation is eliminated and the discontinuity at B=BH for limiting current observed in a vacuum crossed-field gap vanishes. As B→∞, the limiting current approaches a constant that depends on the initial velocity and the collision frequency.