Scaling theory of electric-field-assisted tunnelling

Abstract

Recent experiments report the current (I) versus voltage (V) characteristics of a tunnel junction consisting of a metallic tip placed at a distance d from a planar electrode, d varying over six orders of magnitude, from few nanometres to few millimetres. In the 'electric-field-assisted' (or 'field emission') regime, as opposed to the direct tunnelling regime used in conventional scanning tunnelling microscopy, all I-V curves are found to collapse onto one single graph when d is suitably rescaled, suggesting that the current I =I(V, d) is in reality a generalized homogeneous function of one single variable, i.e. I =I(V · d-λ), where λ being some characteristic exponent and I(x) being a scaling function. In this paper, we provide a comprehensive explanation - based on analytical arguments, numerical simulations and further experimental results - for the scaling behaviour that we show to emerge for a variety of tip-plane geometries and thus seems to be a general feature of electric-field-assisted tunnelling. © 2014 The Author(s) Published by the Royal Society. All rights reserved.