D.1.5 The Model TsuEsaki

The tunneling current density is calculated by expression (3.13) which involves a numerical integration in the energy domain. The values ${\mathcal{E}}_\mathrm{min}$ and ${\mathcal{E}}_\mathrm{max}$ are found automatically for the ECB, HVB, and EVB processes. If the keyword dfType is set to fermi, the supply function is calculated using (3.14). Alternatively, the supply function can be calculated by numerical integration of the distribution function as described in Section 3.3 if the keyword dfType is set to general. With this model it is possible to simulate electron tunneling from the conduction band and hole tunneling from the valence band. The transmission coefficient can be calculated using numerical integration of the WKB expression (3.55) by setting tcType to numericalWKB, by the analytical expression for a linear energy barrier (see Section 3.5.1) by setting the keyword tcType to analyticalWKB, or using a SCHRÖDINGER solver based on the quantum transmitting-boundary method qtbm, see Section 3.5.4. If the keyword imageForce is set to yes, the energy barrier is corrected using the image force correction term described in Section 3.4.2. For the numerical integration, the step width can be given in the keyword dNrg. If the keywords tat and direct are set to yes, both direct and trap-assisted tunneling is calculated. The model keywords are summarized in Table D.4.

Table D.4: TsuEsaki tunneling model keywords.

Symbol	Keyword	Type	Unit
	direct	Boolean
	tat	Boolean
	consistent	Boolean
	imageForce	Boolean
	dNrg	Quantity	eV
$h\omega$	phononNrg	Quantity	eV
$S$	huangRhys	Real
$\ensuremath{m_\mathrm{diel}}/\mathrm{m}_0$	mOx	Real

The keywords which are related to trap-assisted tunneling are only relevant if tat=yes and are described in Section D.4. The possible values of the keywords tcType and dfType are given in Table D.5.

Table D.5: TsuEsaki transmission coefficient and supply function keywords.

Keyword	Type	Description
tcType	String	analyticalWKB, numericalWKB, qtbm
dfType	String	fermi, general

A. Gehring: Simulation of Tunneling in Semiconductor Devices