The calculation made for the trap depth reduction is a one-dimensional calculation, resulting in an overestimation of the effective barrier lowering. In fact, only 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