3.3.1.2 Thermionic Field Emission

For the modeling of heterointerfaces, i.e., with non-negligible band gap discontinuity, a thermionic field emission model is used which can be applied either without or with tunneling over the interface. It constitutes a Neumann interface condition. The relations for the current density $J_\nu$ and energy fluxes $S_\nu$ across the interface read:

$$J_{\nu2} = J_{\nu1}$$ (3.87)

$$J_{\nu2} = q \cdot \bigg(v_{\nu1}\nu_1 - \frac{m_{\nu2}}{m_{\nu1}} \cdot \e... ...(-\frac{\Delta E_\nu - \delta E_{\nu}}{{\it k}_{\mathrm{B}}T_\nu} \bigg) \bigg)$$ (3.88)

$$S_{\nu2} = S_{\nu1}- \frac{1}{q} \cdot (\Delta E_\nu-\delta E_\nu ) \cdot J_{\nu2}$$ (3.89)

$$S_{\nu2} = -\bigg( {\it k}_{\mathrm{B}}\cdot T_{\nu2} \cdot v_{\nu2} \cdot ... ...\frac{\Delta{E}- \delta{E_{\nu}}}{{\it k}_{\mathrm{B}} \cdot T_\nu}\bigg)\bigg)$$ (3.90)

$m_{\nu i}$ with $\nu$ = n,p represents the effective masses at both sides of the interface between the segments i=1 and i=2, respectively. $\Delta{E_{\nu}}$ represents the conduction or valence band discontinuity. The effective barrier reduction $\delta E_{\nu}$ is modeled as function of the electric field $E_2$ orthogonal to the interface.

$$\delta E_\nu= \left\{\begin{array}{r@{\quad:\quad}l} q \cdot E_2 \cdot {\it x}_{\mathrm{tun}}& E_2 > 0, \\ 0 & E_2 \leq 0, \\ \end{array} \right.$$ (3.91)

A fit of the barrier reduction $\delta E_{\nu}$ for the Al$_{0.2}$Ga$_{0.8}$As/In$_{0.25}$Ga$_{0.75}$As interface was performed in [50] and extended to different material systems, such as AlGaAs/GaAs, InAlAs/ InGaAs, and AlGaN/GaN. The tunneling parameters are found in Table 3.33. These are effective fitting parameters, however, they scale with the tunneling probabilities found in Appendix B.

Table 3.33: Tunneling parameters for various materials.

Material	Material Composition	${\it x}_{\mathrm{tun}}$
	$x$	[nm]
Al$_{0.2}$Ga$_{0.8}$As/GaAs	-	3
Al$_{0.2}$Al$_{0.8}$As/In$_{0.25}$Ga$_{0.75}$As	-	7
In$_{0.52}$Al$_{0.48}$As/In$_{0.53}$Ga$_{0.47}$As	-	8
In$_x$Al$_{1-x}$As/In$_x$Ga$_{1-x}$As	0.33 $\leq$x $\leq$ 0.66	7-8
Al$_x$Ga$_{1-x}$N/GaN	0.25	3

The thermionic field emission velocity for the segment i is defined as follows:

$$v_{\nu,i}=\sqrt{\frac{2\cdot {\it k}_{\mathrm{B}}\cdot {\it T}_\mathrm{\nu}}{\pi \cdot m_{\nu,i}}}$$ (3.92)

The thermal boundary condition between semiconductors in general reads as follows:

$$T_{L1} = T_{L2}$$ (3.93)

In the case of heterointerfaces additional entries are necessary in the DD simulation to account for the carriers loosing or gaining energy.

$$\frac{J_n}{q} \cdot \Delta E_C + \frac{J_p}{q}\cdot \Delta E_V = _{A} S_L$$ (3.94)

For the HD case this is not required, since the energy relations yield this condition self-consistently.