Simulation of the Space-Charge-Limited Current Density for Time-Variant Pulsed Injection

TL;DR

The study addresses how time-varying injection influences space-charge-limited current in short-pulse diode operation, challenging the conventional time-invariant short-pulse limit described by Valfells and the static Child-Langmuir law. It employs one-dimensional PIC simulations with five injected time profiles $j_m(t)$ and the relation $J(t)=\beta_m j_m(t)$ to map the maximum transport density as a function of the normalized pulse length $X_{\rm CL}=\tau_p/\tau_{\rm CL}$, where $\tau_{\rm CL}=3D\sqrt{m_0/(2eV_g)}$. The results show that time-varying injection can raise the time-averaged SCL current by factors of about 2–3 for short pulses, with the largest gains for front-loaded profiles (e.g., $m=4$), while the enhancement diminishes as $X_{\rm CL}\to 1$ due to tail limitations; field evolution $E_{\rm K}(t)-E_{\rm A}(t)$ and end-of-pulse electron distributions corroborate the proposed mechanism. This suggests that tailored time-dependent injection, potentially implemented via ultrafast laser excitation, could enhance high-current diode performance, though further profiling and multidimensional validation are needed.

Abstract

Space-charge-limited (SCL) current density for time-invariant injection (under long-pulse condition) via the diode cathode is the maximum transportable density, while it can be leveraged higher when the injection pulselength becomes shorter than the transmit time for electrons (i.e., under short-pulse condition). However, both limits mentioned above apply for the time-invariant injection condition and the role of time-varying current density for injection remains elusive. In this paper, we numerically investigate the SCL electron flow with time-variant injection. Using particle-in-cell simulation, four different time-variant profiles for electron injection are enforced, and the maximum current densities are determined resulting from the space charge effect for various pulse lengths. We speculate that the time-variant density of injection via the diode cathode will contribute to transport enhancement in the short-pulse condition.

Figures (6)

• Figure 1: (Color online) The ratio of transported charge between the long pulse limit and short pulse $Q_{\rm mod}$ as a function of the normalised pulse length $X_{\rm CL}$ = $\tau_p / \tau_{\rm CL}\leq 1$.
• Figure 2: (Colour online) The five normalized time profiles $j_m(t)\vert_{m = 0, 1,2,3, 4}$ assumed in the model [see Eq. \ref{['Jt']}] to be used in PIC simulation.
• Figure 3: (Color online) Normalised SCL density with respect to $J_{\rm CL}$ for short pulse condition: $\beta_0/J_{\rm CL}$ (symbol) and $J_{\rm S}/J_{\rm CL}$ (line), plotted as a function of normalized pulse length $X_{\rm CL}$. The parameters are $D = 1 {\rm cm}, V_{\rm g}= 30 {\rm MV}$, and SCL transit time $\tau_{\rm p} = 9.24 {\rm ps}$. The reason why a low value occurs near 10 ps is explained near the end of Section \ref{['sec:results']}.
• Figure 4: (Color online) The magnitude of current density $\beta_m$ ($m$ = 0, 1, 2, 3, 4) with repect to $J_{\rm S}$(left ordinate) and $J_{\rm CL}$ (right ordinate) from Eq. \ref{['Valfells']}, as a function of normalized pulse length $X_{\rm CL} < 1$. The same parameters are used as in Fig. \ref{['fig:abs']}. Note that the dashed line indicates the unity level.
• Figure 5: (Color online)The time dependence of the difference between the electric field on the cathode and anode, $E_{\rm K}(t)- E_{\rm A}(t)$ for $m$ =0 case: (black square) and $m$ = 4 case (red dot), which shows, respectively, $t$ and $t^3$ scaling. The used parameters are $D$ = 1 cm, $V_{\rm g}$ = 30 MV, and $\tau_{\rm p} =10 {\rm fs}$.