D.1.4 The Model FrenkelPoole

The FRENKEL-POOLE model can be used to describe trap-assisted tunneling for a highly defective dielectric. The tunneling current is given as a generalization of expression (3.129):

$$J = a \ensuremath{E_\mathrm{diel}}\exp \left( \frac{b \sqrt{\ensu... ...rm{diel}}} -\ensuremath{{\mathcal{E}}_\mathrm{T}}}{{\mathrm{k_B}}T} \right) \ .$$ (D.1)

In this expression ${\mathcal{E}}_\mathrm{T}$ is the trap energy level below the dielectric conduction band, and the values $a$ and $b$ can be used as fitting parameters. Table D.3 summarizes the model keywords.

Table D.3: FrenkelPoole tunneling model keywords.

Symbol	Keyword	Type	Unit
$a$	a	Real
$b$	b	Real
$\ensuremath{{\mathcal{E}}_\mathrm{T}}$	trapNrg	Quantity	eV
	consistent	Boolean

Note that the simple analytic models FNPure, FNLenzlingerSnow, DTSchuegraf, and FrenkelPoole should not be used for the case of a work function difference between the two materials regarded for tunneling. In the case of a work function difference, the electrostatic field in the dielectric does not vanish for zero bias but only at the flat-band voltage. Hence, these models will show the minimum tunneling current if the flat band voltage is applied. The TsuEsaki model, however, takes the work function difference into account and should be used in that case.

A. Gehring: Simulation of Tunneling in Semiconductor Devices