It is shown using dimensional analysis that the maximum current density J_(QCL) transported on application of a voltage h ~(3-2α)V~α _g/D~(5-2α) across a gap of size D follows the relation. The classical Child-Langmuir result is recovered at α = 3/2 on demanding that the scaling law be independent of ?. 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 α = 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.
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