Subsections

• 6.5.1 Simulation Results
• 6.5.2 Conclusions

6.5 Simulations with Higher-Order Transport Models

The final section in this chapter presents results obtained from simulations of advanced MOSFET devices with different transport models. As derived and discussed in Section 2.1.3, the following transport models are compared in this work:

1. The six moments transport model, see Section 2.1.4.
   
2. The energy-transport model, see Section 2.1.5.
   
3. The drift-diffusion transport model, see Section 2.1.6.

For the example a series of double-gate MOSFETs were used. They have gate lengths from $\ensuremath{L_\mathrm{g}}=$250$\,$nm down to $\ensuremath{L_\mathrm{g}}=$25$\,$nm. The main objective of this example is to show that the impact of the higher-order transport models significantly increases with smaller gate lengths and that their application is inevitable for $\ensuremath{L_\mathrm{g}}<$100$\,$nm. This is both demonstrated for IV curves and cut-off frequency extractions. The simulation results are compared with full-band Monte Carlo results [115]. In Figure 6.14, the basic structure of the simulated devices as well as the doping profile including a neutral channel doping are shown depending on the gate length $\ensuremath{L_\mathrm{g}}$ [83].

Figure 6.14: Structure of the simulated double-gate MOSFET devices. The gate length is varied from 250$\$nm down to 25$\$nm [83].

6.5.1 Simulation Results

Figure 6.15 depicts the results of the steady-state simulation of four double-gate MOSFETs with gate lengths of $\ensuremath{L_\mathrm{g}}=$250$\,$nm, $\ensuremath{L_\mathrm{g}}=$100$\,$nm, $\ensuremath{L_\mathrm{g}}=$50$\,$nm, and $\ensuremath{L_\mathrm{g}}=$25$\,$nm. Whereas for the largest device the employment of higher-order transport models does not seem to be necessary, this situation significantly changes for smaller devices. The drift-diffusion model delivers a clear underestimation of the drain current, while the energy-transport model starts to overestimate the current.

For the same devices, small-signal simulations have been performed and the results are presented in Figure 6.16. In contrast to the drain current, the error of the drift-diffusion model regarding the cut-off frequency $f_\textrm {T}$ is already significant for the device with $\ensuremath{L_\mathrm{g}}=$250$\,$nm. The underestimation continues with smaller gate-lengths, resulting in an error of 50% for $\ensuremath{L_\mathrm{g}}=$25$\,$nm. The energy-transport model delivers the same results as the six moments model for $\ensuremath{L_\mathrm{g}}=$250$\,$nm. However, for smaller gate lengths the energy-transport model systematically overestimates $f_\textrm {T}$. Note that in Section 3.3.3, a comparison of the cut-off frequency results with quasi-static simulations are shown.

Figure 6.15: These four figures show the increasing errors of the macroscopic transport models with decreasing gate lengths. Whereas at $L_\textrm {g}=$250$\$nm in the upper left figure the difference of the drain currents is minimal, it can be clearly seen that for $L_\textrm {g}= 50\$nm the six moments transport model delivers the best results. However, for extremely small gate length, it loses its advantages and even more moments would be necessary. Note that the drift-diffusion model results in terminal quantities which underestimates the Monte Carlo results.

Figure 6.16: These four figures show the cut-off frequency versus the drain current and the much higher sensitivity of that small-signal figure of merit is demonstrated.

6.5.2 Conclusions

As derived in Section 2.1.3, the drift-diffusion model is characterized by a very rough closure of $\ensuremath{T_n}=T_{\mathrm{L}}$. Whereas the calculated terminal currents are not severely wrong, the error becomes worse if distributed quantities such as the carrier concentrations or other important quantities such as the cut-off frequency $f_\textrm {T}$ are considered. In fact, the development and underestimating character of the terminal quantity error is used to justify the industrial application of the drift-diffusion model for such devices, which should have already been subject to simulation with higher-order transport models. The main reasons why the drift-diffusion model is still widely applied are its robust convergence behavior and performance.

The energy-transport models do not show an comparable numerical robustness than the drift-diffusion models any more. Due to the additional temperature quantities, the convergence behavior and the performance are generally worse. The simulation setup is more sensitive to the mesh and the heat-flux reduction degrades the condition of the system matrix [83]. However, the benefit of these models are that instead of the cold the heated Maxwell distribution can be used, which allows to take hot-carrier effects into account.

The six moments transport model as applied in the simulations above uses an empirical closure relation calibrated to bulk Monte Carlo data. The six moments models are even more sensitive to the mesh and the condition is more degraded. On an engineering level one can conclude that if the application of energy-transport models has been restrained due to these properties, this will be even more the case for the six moments models. However, they give the best results overall as more details of the distribution function are available. For example, whereas the energy-transport models overestimates the velocity, the six moments models stay closest to the Monte Carlo data.

Furthermore the development of the error of the higher-order transport models with decreasing gate lengths must not be neglected. As already said, the error of the terminal quantities calculated by the drift-diffusion model is not significantly decreasing with smaller gate lengths. In contrast, higher-order transport models indicate that the error is disproportionally increasing with smaller gate lengths. This allows one to conclude that the six moments models should be preferred over the energy-transport models. Although the numerical properties of the assembled equation systems become worse, one can partly counteract on the numerical solver level. The solver evaluation in Section 5.5.5 clearly indicates that some solvers such as the GMRES(M) shows significant advantages over the BICGSTAB in terms of convergence and performance. In addition, as the higher-order transport models are more sensitive to the mesh, advanced generation of adaptive meshes would enable a more convenient and industrial application of that models.