Scaling law for the space charge oscillation in electron emission diodes and its scale-invariant nature

In this Letter, we derive a full scaling law for describing the space charge oscillation frequency in electron emission diodes based on the Buckingham π-theorem and the similarity method. The scale-invariant nature of the oscillation phenomena is unraveled and demonstrated via first-principles particle-in-cell simulations. By combining the concepts of similarity and scaling laws, two types of scaling networks are established to correlate space charge oscillations at different states, which are essential for designing high-current electron-emitting systems in various fields, particularly for miniaturized electronic devices.