In this paper, we derive analytically from first principles a generalized Fowler–Nordheim (FN) type equation that takes into account the curvature of a nanoscopic emitter and is generally applicable to any emitter shape provided that the emitter is a good conductor and no field-dependent changes in emitter geometry occur. The traditional FN equation is shown to be a limiting case of our equation in the limit of emitters of large radii of curvature R. Experimental confirmation of the validity of our equation is given by the data of three different groups. Upon applying our equation to experimental FN plots complying with the above limitations, one may deduce (i) R and (ii) standard field emission parameters—e.g. enhancement factor—with better accuracy than by using the FN equation.

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.

The progress in field emission theory from its initial Fowler–Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1–5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system. Significant differences from the usual Cartesian geometry are obtained. (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) The parallel electron velocity vanishes at the apex which may explain the recent spot-size measurements in near-field scanning electron microscopy. (5) Competing effects are found as the tip radius decreases to 1 nm: The electric field increases but the total supply function decreases so that possibly an optimum radius exists.