Poole–Frenkel effect

. Making the dependences from the applied voltage and the temperature explicit, the expression reads: 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 ( \kappa = \epsilon / \epsilon_0 , where \epsilon_0 is the vacuum permittivity), and comparing these with the slopes detected experimentally. As an alternative, one can evaluate the value for \kappa 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 \kappa = n^2, where n is the refractive index of the material. == Improved Poole–Frenkel models ==

Although many progresses have been made on the topic since the classical work of Frenkel, the Poole–Frenkel formula has been spreadly used to interpret several non-ohmic experimental currents observed in dielectrics and also semiconductors. The debate about the underlying assumptions of the classical Poole–Frenkel model has given life to several improved Poole–Frenkel models. These hypotheses are presented in the following. Still, different dependences of the pre-exponential factor from the field can be found: assuming that the carriers could be re-trapped, proportionality to E^{-1/2} or E^{1/2} is obtained, depending on the electron retrapping occurring by the nearest neighbouring trap or after a drift. while dependencies on E^{-3/2} and E^{-3/4} are found to be the result of hopping and diffusion transport processes respectively. In the classical Poole–Frenkel theory a Coulombic trap potential is assumed, but steeper potentials belonging to multipolar defects or screened hydrogenic potentials are considered as well. As a further assumption a single energy level for the traps is assumed. However, the existence of further donor levels is discussed, even if they are supposed to be entirely filled for every field and temperature regime, and thus to not furnish any conduction carrier (this is equivalent to state that the additional donor levels are placed well below the Fermi level). 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. 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: : \phi_B - \sqrt{ \frac{qE_s}{\pi \epsilon} } = 0 where E_s is the saturation field. Thus : E_s = \frac{\pi \epsilon {\phi_B}^2}{q} . 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 Fowler-Nordheim formula derivation), but composed of the series of a trapezoidal barrier in the oxide, and a triangular barrier in the nitride. The Poole–Frenkel process is the limiting mechanism of conduction at the beginning of the memory programming regime due to the higher current provided by tunneling. As the trapped electron charge raises, beginning to screen the field, the modified Fowler-Nordheim tunneling becomes the limiting process. The trapped charge density at the oxide-nitride interface is proportional to the integral of the Poole–Frenkel current flowed across it. With an increasing number of memory write and erase cycles, retention characteristics worsen due to the increasing bulk conductivity in the nitride. == See also ==