3.2.5.3 High Field Mobility for the Drift-Diffusion Equation

The dependence of the mobility on the driving force $F_\nu$ is modeled according to (3.43).

$$\mu^{LIF}(F_{\nu}) =  \frac{\mu^{LI}_{\nu}} {\displaystyle \bigg(... ...\cdot F_\nu }{v_{sat \nu}}\bigg)^{\beta_\nu}\bigg)^{1/\beta_\nu}},     \nu= n,p$$ (3.41)

The driving force is calculated using the carrier temperature voltages $U_{T,\nu}$ for the carrier $\nu$. $s_\nu$ is a switch to distinguish electrons and holes which yields: $s_n$= -1 and $s_p$= 1.

$$F_\nu= \bigg\vert \psi+ \frac{s_\nu}{\nu}\cdot (U_{T,\nu}\cdot \nu) \bigg\vert$$ (3.42)

The parameters for this DD high field correction are found in Table 3.16.

Table 3.16: High-field parameters for the DD model.

Material	$\beta_n$	$\beta_p$
III-V	2.0	1.0
GaN	2.0	1.0
Si	2.0	1.0

An alternative DD high field model can be applied for electrons, which is useful for DD simulation of III-V GaAs and also AlGaN/GaN HEMTs. In these devices negative differential mobility occurs for fields, which is considered the extreme high field region for GaAs or InP type semiconductors.

$$\mu^{LIF}(F_{\nu}) =  \frac{\displaystyle \mu^{LI}_{{\it n}_{\mat... ...ta_n-1}}{F_0^{\beta_n}}} { \displaystyle 1+\frac{F_n^{\beta_n}}{F_0^{\beta_n}}}$$ (3.43)

In the model (3.45) $F_0$ is considered a free parameter contrary to (3.43), where $F_0$ is determined by saturation velocity and $\mu^{LI}$. For GaAs and InP, $F_0$ describes the field where negative differential mobility typically occurs. Parameter values for (3.45) are given in Table 3.17. The comparison of $F_0$ shows higher values for InP than for GaAs, while for GaN $F_0$ exceeds by more than an order of magnitude. In comparison to (3.43) the determined $F_0$ for (3.45) exceeds the value of ${\it v}_{{sat}}$/$\mu^{LI}$ by an order of magnitude.

Table 3.17: High-field parameters for the DD solution.

Material	$\beta_n$	$F_0$
		[V/cm]
GaAs	4.0	4.0e3
InP	4.0	10e3
GaN	4.0	2.2e5

The model given in (3.45) is a qualitative estimate of the underlying high field physics and is used only to match the very basic physics of GaN with the overshoot occurring at much higher fields than known from GaAs.