Scale invariance of a diode-like tunnel junction

In Near Field-Emission SEM (NFESEM), electrostatic considerations favor a diode-like tunnel junction consisting of an atomic-sized source mounted at the apex of a thin wire placed at nanometric distances from a collector. The quantum mechanical tunnel process, instead, can provide a barrier toward miniaturization. In the first place, it deteriorates the generation of electrons by introducing non-linearities within the classically forbidden zone that exponentially increase with decreasing sizes. In addition, in the direct tunnelling regime, i.e. when the distance between emitter and collector $d$ approaches the subnanometer range, a characteristic length appears, making the cross-over from the (almost) scale-invariant electric-field assisted regime to the essentially different STM-regime. We have observed that the experimental data relating the current $I$ to the two experimental variables $V$ (bias voltage between tip and collector) and $d$ can be made (almost) collapse onto a ``scaling curve'' relating $I$ to the single variable $V\cdot d^{-\lambda}$, $\lambda$ being some exponent that depends solely on the geometry of the junction. This scaling property can be used to highlight non-linear aspects of the quantum mechanical tunnelling process.