4.7 Comparison with Monte Carlo

To investigate the accuracy of the six moments (SM) model we consider a series of one-dimensional $n^+$-$n$-$n^+$ test-structures. Although these structures are not of practical relevance, they still display similar features as contemporary MOS and bipolar transistors like velocity overshoot and a mixture of a hot and a cold distribution function in the 'drain' region.

Here we present results of numerical solutions of six moments models (SM) and compare them to self-consistent Monte Carlo data (SCMC). Relaxation times and mobilities are extracted from bulk Monte Carlo data as a function of the carrier temperature and the doping.

In addition to the SM model we consider the corresponding energy-transport (ET) model formulated in moments $V_i$ (see Equations 2.24, 2.25), where the equation for $V_4$ is kept but the equation for the energy-flux $V_3$ is closed with $V_4 = (5/3) V_2^2$, corresponding to a heated Maxwellian distribution. This decouples the equation for $V_4$ from the lower order equations and provides an estimate for $V_4$ and thus $\beta$ [GKGS01], [SYT⁺96].

The doping concentrations were taken to be $5\times 10^{19}$ and $10^{17} \mathrm{cm^{-3}}$. The channel length was varied from $1000 \mathrm{nm}$ down to $50 \mathrm{nm}$ while maintaining a maximum electric field of $300 \mathrm{kV/cm}$.

A comparison of the average velocity $V_0$ and the kurtosis $\beta$ obtained from the SM and ET models with the SCMC simulation is shown in Fig. 4.4 and Fig. 4.5 for the $L_{{c}} =100$nm device. The spurious velocity overshoot is significantly reduced in the SM model, consistent with previous results [GKS01], while the kurtosis produced by the ET model is only a poor approximation to the MC results.

Despite the fact that SM models provide the kurtosis of the distribution function, they do not require a heated Maxwellian closure in the energy flux relation. This has a significant impact on the resulting device currents for channel lengths smaller $100 \mathrm{nm}$ where the ET models show the well-known overestimation of the device currents (see Figure 4.6). The results of the SM model, on the other hand, stay close to the SCMC results which makes the SM model a good choice for TCAD applications.

Figure 4.4: Comparison of the average velocity obtained from the SM and ET models with the self-consistent Monte Carlo simulation for the $L_{{c}} =100$nm device.

Figure 4.5: Comparison of the kurtosis obtained from the SM and ET models with the self-consistent Monte Carlo simulation for the $L_{{c}} =100$nm device.

Figure 4.6: Comparison of the device currents obtained from the SM and ET models with the self-consistent Monte Carlo simulation for varying channel length.