in the direction of the external electric field the potential well height is lowered as much estimated accordingly to the Poole-Frenkel expression. More accurate calculation, performed by Hartke making an average of the electron emission probabilities with respect to any direction, shows that the growth of the free carriers concentration is about an order of magnitude less than that predicted by Poole-Frenkel equation. The Hartke equation is equivalent to where . From a theoretical point of view, Hartke's expression has to be preferred to Poole-Frenkel equation on the grounds that the threedimensionality of the problem of the trap barrier lowering is considered. Additional three-dimensional models have been developed, differing by the treatment they do of the emission process in the upwind direction. Ieda, Sawa, and Kato, for example, have proposed a model that includes the barrier variation in directions both forward and opposite to the electric field. Poole-Frenkel saturation Poole-Frenkel saturation occurs when all the trap sites become ionized, resulting in a maximum of the number of conduction carriers. The corresponding saturation field is obtained from the expression describing the vanishing of the barrier: where is the saturation field. Thus . The trap sites are now necessarily empty, being at the edge of the conduction band. The fact that the Poole–Frenkel effect is described by an expression for the conductivity (and for the current) that diverges with increasing fields and does not experience a saturation, is attributable to the simplifying assumption that the trap population follows the Maxwell-Boltzmann statistics. An enhanced Poole-Frenkel model, comprehensive of a more accurate description of the trap statistics with the Fermi-Dirac formula, and capable to quantitatively represent the saturation, has been devised by Ongaro and Pillonnet. Poole–Frenkel transport in electronic memory devices In charge trap flash memories, charge is stored in a trapping material, typically a silicon-nitride layer, as current flows through a dielectric. In the programming process, electrons are emitted from the substrate towards the trapping layer due to a large positive bias applied to the gate. The current transport is the result of two different conduction mechanisms, to be considered in series: the current through the oxide is by tunneling, the conduction mechanism through the nitride is a Poole–Frenkel transport. The tunneling current is described by a modified Fowler-Nordheim tunneling equation, i.e. a tunneling equation that takes into account that the shape of the tunneling barrier is not triangular (as assumed for the