3.6.5 Channel Transport

In previous work [50] a mobility model (3.49) [112,178] was used in combination with constant energy relaxation times. An additional exponent $\beta$ was introduced to allow for the modeling of the static overshoot in the mobility model, as given in (3.46). Although being applied successfully for constant ${\it V}_{\mathrm{DS}}$ bias this work uses a modified approach extending the capabilities. In this work an energy dependence of the relaxation times is introduced using the mobility model (3.46).

The results from the high field transport are controlled by one- and two-dimensional MC simulations. Two-dimensional MC simulations are necessary as the high field transport is modified by the existence of interfaces and quantization effects [212] that strongly interact with the high energy carriers. The MC parameters used were obtained from [91,151] and used in a one-dimensional MC device simulator, originally developed by Wang [306]. At instances, results from the two-dimensional MC device simulator DAMOCLES [92] are available for evaluation of two-dimensional high field effects, such as Real Space Transfer.

When discussing the velocity profile in the gate area of a HEMT, as was shown e.g. in [54], one can obtain similar velocity profiles using DD and different HD approaches for one specific bias. In the DD case this is achieved by using the saturation velocity and $\beta$ in (3.43) as a fitting parameter. A fit can also be achieved for HEMTs in the DD approximation. The shortcomings of such an approach can be seen, when regarding the output characteristics. For any multi- ${\it V}_{\mathrm{DS}}$ bias simulation task, e.g. the output characteristics in a HEMT, this approach is not sufficient. To clarify the differences also for velocities, four terms for velocity are carefully distinguished:

The velocity ${\it v}_{{sat}}$ is the measured saturation velocity in bulk material. Manifold references can be found in [226]. They amounts to ${\it v}_{{sat}}$ $\leq 10^7$ cm/s for nearly all semiconductors above E $\geq$ 100 kV/cm except for the GaN based materials. The effective mean carrier velocity ${\it v}_{\mathrm{eff}}$ is extracted, e.g. from S-parameters, assuming a simplified velocity field relation [184]. Typical extracted mean velocities are shown in Fig. 3.27 as a function of gate length ${\it l}_{\mathrm{g}}$ . ${\it v}_{\mathrm{eff}}$ provides an estimate of the speed of the readily processed HEMT. The values of ${\it v}_{\mathrm{eff}}$ clearly exceed the bulk saturation velocities ${\it v}_{{sat}}$ and increase with higher In contents also for ${\it l}_{\mathrm{g}}$ $\leq$ 100 nm.

**Figure 3.27:** Mean carrier velocities $v_{eff}$ as a function of gate length .
$\includegraphics[width=10 cm]{D:/Userquay/Promotion/HtmlDiss/fig-5a.eps}$

**Figure 3.28:** Overshoot effects as a function of background doping.
$\includegraphics[width=10 cm]{D:/Userquay/Promotion/HtmlDiss/fig-15a.eps}$

The third velocity ${\it v}_{\mathrm{eff,het}}$ is the mean velocity ${\it v}_{\mathrm{eff}}$ in the presence of heterointerfaces. As was shown in Fig. 3.28, overshoot effects are visible in the HEMT channel. In the case of carrier transport parallel to interfaces, next to the transitions in k-space, real space transfer modifies the maximum and average carrier velocity, as was shown by Patil et al. in [212]. The fourth velocity is introduced to clarify, whether the effective velocities ${\it v}_{\mathrm{eff}}$ and ${\it v}_{\mathrm{eff,het}}$ have a saturated behaviour for high fields. As we see velocities $\geq$ 3 $\cdot$ 10

cm/s for ${\it l}_{\mathrm{g}}$ = 50 nm in Fig. 3.27, the saturation to bulk levels is not visible.

Fig. 3.28 shows the impact of the background doping concentration on the overshoot in a ${\it l}_{\mathrm{g}}$ = 140 nm AlGaAs/InGaAs device. A maximum overshoot of 3.5 $\times10^7$ cm/s is found, while for higher doping the overshoot significantly drops.