depth of the trap potential well. Without any externally applied electric field, the maximum value of the potential is zero and is located at infinite distance from the trap center. When an external electric field is applied, the height of the potential barrier is reduced on one side of the trap by the amount where: q is the elementary charge is the dynamic permittivity. The first contribution is due to the applied electric field, the second is due to the electrostatic attraction between the ionic trap site and the conduction electron. The potential has now a maximum at a distance from the Coulomb trap center, given by . Therefore and . This expression is similar to that obtained for the Schottky effect. The factor 2 in the exponent, that makes the barrier reduction in the Poole–Frenkel effect twice larger than that experienced in the Schottky effect, is due to the interaction of the thermally excited electron with the immobile positive charge of the ion acting as a trap center, rather than with its mobile image charge induced in the metal at the Schottky interface. Now if, without any applied electric field, the number of thermally ionized electrons is proportional to where: is the voltage barrier (in zero applied electric field) that an electron must cross to move from one atom to another in the material is Boltzmann's constant T is the temperature then, in presence of an external electric field the electrical conductivity will be proportional to thus obtaining differing from Poole's law in the dependence on . Taking everything into account (both the frequency with which electrons are excited into the conduction band, and their motion once they're there), and assuming a field-independent mobility of electrons, the standard quantitative expression for the Poole–Frenkel current is: where J is the current density. Making the dependences from the applied voltage and the temperature explicit, the expression reads: where d is the dielectric thickness. For a given dielectric, different conduction processes may dominate in different voltage and temperature ranges. For materials such as Si3N4, Al2O3, and SiO2, at high temperature and high field regimes, the current J is likely due to Poole-Frenkel emission. The detection of Poole-Frenkel emission as the limiting conduction process in a dielectric is usually made studying the slope in the so-called Poole-Frenkel plot, where the logarithm of the current density divided by the field () versus the square