The Child-Langmuir law in the quantum domain

It is shown using dimensional analysis that the maximum current density J(QCL) transported on application of a voltage V-g across a gap of size D follows the relation J(QCL) similar to (h) over bar V-3-2 alpha(g)alpha/D5-2 alpha. The classical Child-Langmuir result is recovered at alpha = 3/2 on demanding that the scaling law be independent of (h) over bar. For a nanogap in the deep quantum regime, additional inputs in the form of appropriate boundary conditions and the behaviour of the exchange-correlation potential show that alpha = 5/14. This is verified numerically for several nanogaps. It is also argued that in this regime, the limiting mechanism is quantum reflection from a downhill potential due to a sharp change in slope seen by the electron on emerging through the barrier. Copyright (C) EPLA, 2013