Scaling theory of electric-field-assisted tunnelling.

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(-lambda)),where lambda 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.