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:
For the example a series of double-gate MOSFETs were used. They have gate
lengths from
250
nm down to
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
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
[83].
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Figure 6.15 depicts the results of the steady-state simulation of four
double-gate MOSFETs with gate lengths of
250
nm,
100
nm,
50
nm, and
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
is
already significant for the device with
250
nm. The underestimation
continues with smaller gate-lengths, resulting in an error of 50% for
25
nm. The energy-transport model delivers the same results as the six moments model for
250
nm. However, for smaller gate lengths the energy-transport model
systematically overestimates
. Note that in Section 3.3.3, a
comparison of the cut-off frequency results with quasi-static simulations are
shown.
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As derived in Section 2.1.3, the drift-diffusion model is characterized
by a very rough closure of
. 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
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.