Bipolar Quantum Child-Langmuir Law

Summary form only given. The classical Child-Langmuir (CL) law has been the subject of interest for many years. Since it was first derived in the early twentieth century, many modifications have been made recently to include multi-dimensional, relativistic, and quantum effects. Including the effects of ion flows, bipolar CL law for classical and relativistic theory has been developed . However, the scaling of bipolar flow in a nano-sized diode is still unknown. In this paper, we present a consistent quantum mechanical model of the bipolar CL law based on the mean field theory. In the model, we consider only the quantum effects for electron, but including the space-charge effects of ions consistently. From our calculations, we see that the classical bipolar CL law is no longer valid in quantum regime, when the effects of electron tunneling and exchange-correlation become important. Smooth transition between the classical and quantum regimes is demonstrated