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arXiv · 1304.7570

The Child-Langmuir law in the quantum domain

Debabrata Biswas·Raghwendra Kumar
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Abstract

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} ~ \hbar^{3 - 2\alpha} V_g^\alpha /D^{5 - 2\alpha}. The classical Child-Langmuir result is recovered at \alpha = 3/2 on demanding that the scaling law be independent of \hbar. 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.

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Debabrata Biswas, Raghwendra Kumar. 2013-04-29. The Child-Langmuir law in the quantum domain. https://doi.org/10.1209/0295-5075%2F102%2F58002

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