Beyond the Child-Langmuir limit

Abstract

This article presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables (i.e., variables that follow particle paths) the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity v(0), electric field E(0), and current J(0). For steady solutions, this approach clarifies the origin of the maximal current J(max), derived by Child and Langmuir for v(0) = 0 and by Jaffe for v(0) > 0. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux J(0) > J(max), which leads to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J(max).

Time evolution of electron flow in a model diode: Non-perturbative analysis

“…Evaluating the derivatives via Eq. (4) one can substitute them into (7), apply (6), and come to the following BC which as Eq. (6) should hold for all t > 0…”

“…Eqs. (6) and (7) should hold for all t ≥ T (t). One can find an explicit expression for the charge density at τ, t ρ(τ, t) = ∂E ∂x = ∂E ∂τ…”

“…We consider the space charge limited one-dimensional (1D) flow, produced by field emission (regime I) or externally (regime II), in conditions which forbid applying linearization techniques [3][4][5][6][7][8]. In particular, we study the transition from no-flow initial state to a stationary current bearing state.…”

Space-charge effect of the time-varying electron injection in a diode: Classical and relativistic regimes

“…For example, a recent work based on the 1D charge sheet model investigated how the temporal interval becomes distorted due to the space-charge effect. 22 Another recent work 13 used a characteristic method to solve the 1D pressureless Euler-Poisson equations and achieved an over-classical limit by varying the anode boundary voltage with time although the relevant characteristics are not solvable in a closed form for arbitrary temporal profiles of injection. Therefore, a general treatment for time-varying flow of electrons is demanded.…”

“…12 There has been renewed interest in investigating whether an injection of time-varying current density can contribute to a larger current density that is transmittable across the diode space. [13][14][15][16][17][18][19][20][21] On the other hand, time-varying current emission through the diode anode arises naturally even when the injection current density is time-invariant, with the diode voltage being fixed. For example, a recent work based on the 1D charge sheet model investigated how the temporal interval becomes distorted due to the space-charge effect.…”