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

Abstract

Notwithstanding the recent conjecture that the upper bound on the time-averaged current across a space-charge-limited diode is equal to the steady state Child-Langmuir limit (J CL ), Zhu and Ang used a one-dimensional (1D) particle in cell (PIC) code to show that in the regime where space charge effects limit the current to only a few electrons at a time, the time-averaged current can exceed J CL by up to 13% [Y. Zhu and L. K. Ang, Appl. Phys. Lett. 98, 051502 (2011)]. These results are, in fact, verified using our own 1D PIC code. However, the increase in the current is due to special boundary conditions that pertain in this regime and not to the time dependence of the current. To rule out discreteness effects, the conjecture on the upper bound may be reformulated to include only the case when the electric field at the cathode does not fall below zero.The Child-Langmuir Law 1 gives the space-charged limited current in the classical problem of a 1-D diodeMany interesting generalizations of this effect have been considered, particularly with respect to geometry, 2-4 nonzero injection velocities, 5 and relativistic 6,7 and quantum effects. 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. (1), and occurs when space charge in the diode causes the electric field at the cathode to fall to zero. We considered the case where the supply of electrons at the cathode, modelled as a fluid and in the one-dimensional limit, could be controlled as a function of time so that the current emission could be selectively repressed before the space charge limit was reached. The boundary conditions for this situation are qEðx ¼ 0Þ ! 0:We were interested in whether this extra flexibility in the problem allowed for the time-averaged current to exceed the steady state limit. We did provide an upper bound, but not as low an upper bound as the steady state limit, J CL . However, simulations with a 1D particle in cell (PIC) code led us to conjecture that the time-averaged current (for t ! 1) could not exceed J CL . Zhu and Ang 15 considered space charge emission in the "coulomb blockade regime" which is similar to the configuration described above. The difference is that the voltage across the diode is so low that the space charge limit is reached when several electrons at a time pass through the diode. In both cases, the electrons are emitted with zero initial velocity and the voltage across the diode is constant in time. Because the charge can only be emitted in discrete quantities equal to e, the charge of one electron, the current emitted from the cathode is inherently time-dependent. They defined the threshold voltage, V th ...

Over-injection and self-oscillations in an electron vacuum diode

“…Recently, there has been interest in the question of the definition of a steady state SCL current, and whether such a limit consists an upper bound to the SCL current transportable in an unstable diode. [14][15][16] The system discussed in this paper is a very complex unstable dynamical system even though the voltage is kept constant, and no existing analytical steady state bounds would be applicable for comparison. On the other hand, Coulomb forces tend to react as soon as too much charge accumulates in various regions at certain time intervals which means that the dynamics is the result of a SCL upper bound.…”

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

“…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.…”

Modeling of space charge limited current

“…due to the Coulomb blockade effect [110], and it has been confirmed by Griswold et al. [111] in their particle-in-cell simulation.…”