The electrical conductivity  of dielectrics and semiconductors in presence of high electric fields (more than  for insulators and up to  for semiconductors) increases approximately as described by the Poole's law (eventually leading to electrical breakdown):
where
: is the zero-field electrical conductivity 
: is a constant 
:E is the applied electric field.
In this model the conduction is supposed to be carried by a free electron system moving in a self-consistent periodic potential.
On the contrary, Frenkel derived his formula describing the dielectric (or the semiconductor) as simply composed by neutral atoms acting as positively charged trap states (when empty, i.e. when the atoms are ionized). 
For localized trap states with Coulomb potentials, the barrier height that an electron must cross to move from one atom to another is the 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 SiN, AlO, and SiO, 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 root of the field () is depicted. The idea of such a plot originates from the expression of the Poole-Frenkel current density, which contains this proportionality ( vs ), and would thus result in a straight line in this plot. For a fixed value of the voltage barrier in absence of any applied electric field, the slope is influenced by just one parameter: the dielectric permittivity.
Despite the same functional dependence of the current density upon the electric field intensity, one could differentiate between Poole-Frenkel conduction upon Schottky conduction as they would result in straight lines with different slopes in a Poole-Frenkel plot. The theoretical slopes can be evaluated knowing the high frequency dielectric constant of the material (, where  is the vacuum permittivity), and comparing these with the slopes detected experimentally. As an alternative, one can evaluate the value for  equating the theoretical slopes to the experimental detected ones, provided that it is known if the conductivity is electrode-limited or bulk-limited. Such a value of the high frequency dielectric constant should then conform the relation , where  is the refractive index of the material.