An upper bound to time-averaged space-charge limited diode currents

Abstract

The Child–Langmuir law limits the steady-state current density across a one-dimensional planar diode. While it is known that the peak current density can surpass this limit when the boundary conditions vary in time, it remains an open question of whether the average current can violate the Child–Langmuir limit under time-dependent conditions. For the case where the applied voltage is constant but the electric field at the cathode is allowed to vary in time, one-dimensional particle-in-cell simulations suggest that such a violation is impossible. Although a formal proof is not given, an upper bound on the time-averaged current density is offered.

Space charge limited current emission for a sharp tip

“…Due to the contemporary needs on the studies of nanogaps and short electron bunches, this 1D classical CL law has been extended to various regimes, including multi-dimensions, [3][4][5][6][7] quantum regime, 8,9 short pulse limit, 10,11 single-electron limit, 12 new scaling in other geometries 13 with applications in THz source 14,15 and time dependent current injection. [16][17][18] Electron emission from a sharp tip is important for many applications, such as vacuum microelectronics, 19 compact high current cathodes for high power microwave sources, [20][21][22] ultrafast laser induced electron emission from sharp tip, [23][24][25][26][27] ultrafast electron imaging, [28][29][30] laser-driven dielectric acceleration, 31,32 and high brightness photo-cathode 33,34 for X-ray free electron laser (FEL). For electron emission from a sharp tip, the localized electron density near to the tip is very high, which induces a space charge electric field, strong enough to modify the external applied electric field significantly, and thus to influence the field emission process.…”

Amended conjecture on an upper bound to time-dependent space-charge limited current

“…8,9 Time-dependent problems have also been studied, for example short current pulses 10,11 and time-varying voltage drops to control startup transients. 12,13 In a previous paper, 14 we considered a diode consisting of a cathode at x ¼ 0 that emits cold electrons with zero initial velocity, an anode at x ¼ d, and a voltage difference V between them. In steady state, the maximum current density that can pass through the diode is given by the ChildLangmuir limit, Eq.…”

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

“…The particular profiles r m ðtÞ are assumed to be polynomial for the same reason as the injection current densities j m ðtÞ in Eqs. (8)- (14). This is the slightly different inflow boundary condition we apply in this subsection.…”

“…Compared with other temporal profiles, these polynomial emission profiles are chosen based on the understanding that a smooth temporal function should be expanded as a series of polynomials according to the Taylor series. We first choose a steady function (8), then monotonic increasing functions (9, 10), monotonic decreasing functions (11,12), and lastly a valley function (13) and a mountain function (14). 26 We sweep magnitudes b m for each m in profile j m ðtÞ and decide that the SC limit is reached when reflection just occurs, i.e., the velocity becomes negative, within the pulse length 0 t s p .…”

“…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.…”