Scaling properties of a non-Fowler–Nordheim tunnelling junction

Abstract

We have theoretically explained the experimentally observed scaling properties of the current–voltage (I–V) characteristics of a field-emission tunnelling diode with respect to the tip–anode distance d . All the I–V curves for different d -values collapse onto a single curve by a scaling transformation that keeps the electrical field at a given direction constant for different d . Our proof is applicable to more than just the obvious case where the electrostatic potential varies linearly with the distance from the cathode x (and the Fowler–Nordheim plot is also linear). It applies to any general nonlinear potential encountered in emitting tips of small radii of curvature R . Furthermore, we explain why the scaling property is excellent at d ≫ R , but deteriorates when d ∼ R . The scaling property is shown to derive from the simultaneous action of two factors: (i) in a Taylor expansion of the potential in the tunnelling region, the second-order coefficient is found to be proportional to the first-order coefficient and their ratio independent of d . (ii) The angular variation of the electric field is independent of d for d ≫ R . Deviations from scaling at d ∼ R are attributed to both the dependence on d of the angular variation of the electrical field and/or the presence of a third-order term.