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

## TL;DR

This paper investigates the space-charge limited current density in diodes with time-varying electron injection, finding it unlikely to surpass classical or relativistic space-charge limits.

## Contribution

It extends the understanding of space-charge limits to scenarios with time-varying injection in both classical and relativistic regimes.

## Key findings

- Time-varying injection does not typically exceed known space-charge limits.
- Simulations confirm the robustness of classical and relativistic space-charge limits.
- Limits are consistent across different injection profiles.

## Abstract

Conventionally, space-charge (SC) limited current density is defined as the maximal cur- rent density allowed to traverse a diode under a DC voltage when a time-invariant flow is injected from the cathode. In this work, we study the SC limited current density under a time-varying injection for both classical and relativistic regimes and determine the maxi- mal amount of limited current density under certain conditions. Our simulations show that it is unlikely that a time-varying injection emitted from cathode exceeds the known SC limits, in either classical or relativistic regime.