A.2 Correction for Direct Tunneling

The equation derived above is only valid for triangular barriers, that is the case of high applied voltages. In [19] SCHUEGRAF proposed a correction to the FOWLER-NORDHEIM formula to account for tunneling in the direct tunneling regime. In this case the transmission coefficient is

$$TC({\mathcal{E}})=\exp\left(-\frac{2}{\hbar} \int_0^{\ensuremath{... ...mathcal{E}}_\mathrm{c}}-{\mathcal{E}}_x)}\,\ensuremath {\mathrm{d}}x\right) \ ,$$

where $\ensuremath{t_\mathrm{diel}}$ is the dielectric thickness. The conduction band edge is again approximated by a linear shape

$$\ensuremath {{\mathcal{E}}_\mathrm{c}}(x)=\ensuremath{{\mathcal{E... ... {\mathrm{q}}\Phi_1 - \ensuremath {\mathrm{q}}\ensuremath{E_\mathrm{diel}}x \ .$$

The band edges $\ensuremath {\mathrm{q}}\Phi$ and $\ensuremath {\mathrm{q}}\Phi_0$ are given by (see the right part of Fig. A.1)

$$\renewcommand {\arraystretch }{2.8} \begin{array}{rcl} \ensuremat... ...thrm{q}}\ensuremath{E_\mathrm{diel}}\ensuremath{t_\mathrm{diel}}\ . \end{array}$$

As for the triangular energy barrier, it is assumed that the electrodes have equal work functions: $\Delta \ensuremath{\Phi_\mathrm{W}}= 0$. Using these expressions, the transmission coefficient becomes

$$\renewcommand {\arraystretch }{2.8} \begin{array}{rcl} TC({\mathc... ... {\mathrm{q}}\Phi_0-{\mathcal{E}}_x\right)^{3/2} \right)\right) \ . \end{array}$$ (A.15)

The exponent can be approximated using a first order TAYLOR series expansion around $\ensuremath {\mathrm{q}}\Phi_1$ and $\ensuremath {\mathrm{q}}\Phi_1-\ensuremath {\mathrm{q}}\ensuremath{E_\mathrm{diel}}\ensuremath{t_\mathrm{diel}}$, respectively:

$$\renewcommand {\arraystretch }{2.8} \begin{array}{rcl} \displayst... ...\ensuremath{E_\mathrm{diel}}\ensuremath{t_\mathrm{diel}})^{1/2} \ . \end{array}$$ (A.16)

With the temporary variable $\eta$

$$\displaystyle \eta = \left(\ensuremath {\mathrm{q}}\Phi - {\mathc... ...ensuremath{E_\mathrm{diel}}\ensuremath{t_\mathrm{diel}}\right)^{1/2}\right) \ ,$$

the tunnel current density becomes

$$J=\frac{4\pi \ensuremath {\mathrm{q}}\ensuremath{m_\mathrm{eff}}}... ...}}_\mathrm{f,1}}-{\mathcal{E}}_x) \,\ensuremath {\mathrm{d}}{\mathcal{E}}_x \ .$$ (A.17)

With the abbreviations

$$\renewcommand {\arraystretch }{2.8} \begin{array}{rcl} a &=& \dis... ...ath{E_\mathrm{diel}}\ensuremath{t_\mathrm{diel}})^{1/2} \right) \ , \end{array}$$ (A.18)

the tunnel current density can be written as

$$J = a \exp(b) \int_{\ensuremath{{\mathcal{E}}_\mathrm{f,1}}}^{\en... ...}}_\mathrm{f,1}}- {\mathcal{E}}_x)\,\ensuremath {\mathrm{d}}{\mathcal{E}}_x \ .$$ (A.19)

With $\epsilon = {\mathcal{E}}_x - \ensuremath{{\mathcal{E}}_\mathrm{f,1}}$ this yields

$$J = -a \exp(b) \int_{\ensuremath{{\mathcal{E}}_\mathrm{f,2}}- \en... ...mathrm{f,1}}}^0 \exp(c \epsilon) \epsilon\,\ensuremath {\mathrm{d}}\epsilon \ .$$ (A.20)

Using (A.10) this integral becomes

$$J = \frac{a \exp(b)}{c^2} \left( 1 - \exp\left(-c (\ensuremath{{\... ...thcal{E}}_\mathrm{f,1}}-\ensuremath{{\mathcal{E}}_\mathrm{f,2}})\right) \right)$$ (A.21)

which, for $\ensuremath{{\mathcal{E}}_\mathrm{f,1}} \gg \ensuremath{{\mathcal{E}}_\mathrm{f,2}}$, simplifies to

$$J = \frac{a \exp(b)}{c^2} \ ,$$ (A.22)

or, inserting the expressions for $a$, $b$, and $c$

$$J =\frac{\ensuremath {\mathrm{q}}^3 \ensuremath{m_\mathrm{eff}}}{... ..._1 - \ensuremath {\mathrm{q}}\ensuremath{V_\mathrm{diel}})^{3/2} \right)\right)$$ (A.23)

which is the equation used in [19]. In some publications, the equation is rewritten to make it more similar to the FOWLER-NORDHEIM formula:

$$J= \frac {\ensuremath {\mathrm{q}}^3\ensuremath{m_\mathrm{eff}}} ... ... B_2} {3\hbar \ensuremath {\mathrm{q}}\ensuremath{E_\mathrm{diel}}} \right) \ ,$$ (A.24)

with the additional correction terms $B_1, B_2$ given as

$$\renewedcommand{arraystretch}{2.5}\begin{array}{l} \displaystyle ... ...hrm{diel}}}{\ensuremath {\mathrm{q}}\Phi_1}\right)^{3/2}\right) \ . \end{array}$$ (A.25)

A. Gehring: Simulation of Tunneling in Semiconductor Devices