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 It is found that the classical scaling of V-g(3/2) and D-2 is no longer valid in the quantum regime, and a new scaling of V-g(1/2) and D-4 is established.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.