Space-charge-limited flows in the quantum regime^a)

This paper reviews the recent developments of space-charge-limited (SCL) flow or Child-Langmuir (CL) law in the quantum regime. According to the classical CL law for planar diodes, the current density scales as $3∕2$’s power of gap voltage and to the inverse squared power of gap spacing. When the electron de Broglie wavelength is comparable or larger than the gap spacing, the classical SCL current density is enhanced by a large factor due to electron tunneling and exchange-correlation effects, and there is a new quantum scaling for the current density, which is proportional to the $1∕2$’s power of gap voltage, and to the inverse fourth-power of gap spacing. It is also found that the classical concepts of the SCL flow such as bipolar flow, transit time, beam-loaded capacitance, emitted charge density, and magnetic insulation are no longer valid in quantum regime. In the quantum regime, there exists a minimum transit time of the SCL flows, in contrast to the classical solution. By including the surface properties of the emitting surface, there is a threshold voltage that is required to obtain the quantum CL law. The implications of the Fowler-Nordheim-like field emission in the presence of intense space charge over the nanometer scale is discussed.