New Scaling of Child-Langmuir Law in the Quantum Regime

This paper presents a consistent quantum mechanical model of Child-Langmuir (CL) law, including electron exchange-correlation interaction, electrode’s surface curvature, and finite emitter area. The classical value of the CL law is increased by a larger factor due to the electron tunneling through the space-charge potential, and the electron exchange-correlation interaction becomes important when the applied gap voltage $V_g$ and the gap spacing $D$ are, respectively, on the order of Hartree energy level, and nanometer scale. It is found that the classical scaling of $V_g^{3/2}$ and $D^{-2}$ is no longer valid in the quantum regime, and a new scaling of $V_g^{1/2}$ and $D^{-4}$ is established. The smooth transition from the classical regime to the quantum regime is also demonstrated.