The Transition from the Fowler–Nordheim Regime to the Space-Charge-Limited Current Regime in the Case of a Pointed Nanometric Emitter

We examine the transition from the Fowler-Nordheim (FN) field emission regime to the space-charge-limited (SCLC) regime in the case of a pointed nanometric emitter with radius of curvature R ≥ 5 nm, for which the traditional FN equations do not hold. To accomplish this, we use the generalized FN equation for the emission law and the "time of flight" methodology to solve the equations of motion. Taking advantage of the fact that emission from emitters with R = a few nm takes place primarily along the emitter axis, we approximate the paths as linear, provided that the anode is at a far distant position from the cathode. In the approach to the SCLC regime, the calculated currents for emitters with R = a few nm may differ by orders of magnitude, compared to the currents of planar emitters, for the a given fixed anode-cathode separation D. Differences of even greater orders of magnitude are obtained for these currents when R is fixed and D is varied. Furthermore, the variation of currents with D is heavily dependent on R. However, for emitters with R ≥ 25 nm, no appreciable differences in current are observed, compared to the results obtained using the planar theory. An explanation for the observed trends is given.