The Child-Langmuir law in the quantum domain
Theoretical Physics Division, Bhabha Atomic Research Centre - Mumbai 400 085, India
Received: 4 February 2013
Accepted: 22 May 2013
It is shown using dimensional analysis that the maximum current density JQCL transported on application of a voltage Vg 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 ℏ. 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.
PACS: 85.45.-w – Vacuum microelectronics / 03.65.Sq – Semiclassical theories and applications / 03.65.Xp – Tunneling, traversal time, quantum Zeno dynamics
© EPLA, 2013