7. Summary and Conclusion

IN SIMULATIONS of partially depleted SOI MOSFETs anomalous output characteristics have been observed which indicate a complete failure of the standard energy transport model. Therefore, the derivation of the energy transport model has been reconsidered. A systematic technique based on the method of moments allows the rigorous derivation of increasingly sophisticated transport models from BOLTZMANN's transport equation. BOLTZMANN's transport equation is multiplied with weight functions of increasing order and is integrated over momentum-space which yields an infinite set of equations. These equations are coupled as the equation for a given moment contains a moment of next higher order. To obtain a tractable equation set this hierarchy has to be truncated. The highest moment has to be modeled as a function of the available moments to close the equation system. The closure has been performed at different orders:

- When only the first two moments are considered the drift-diffusion model is obtained which is still predominantly used in engineering applications. Its advantage is that the numerical methods are robust because only one balance equation needs to be solved. As the drift-diffusion model cannot capture non-local effects, which gain increasing importance for miniaturized devices, its use becomes questionable. Therefore, higher-order equations have been considered.
- The model obtained by using the first three moments is seldomly used. The energy flux equation is often extended in an inconsistent way and the model has no advantage over the four moments transport model, since also two balance equations have to be solved.
- Inclusion of the first four moments of BOLTZMANN's equation results in the full hydrodynamic model which is, however, too complicated for every-day's use. Within the framework of the diffusion approximation the convective terms in the hydrodynamic models are neglected. This results in simpler energy transport models, which are offered nowadays by leading commercial and academic device simulators.
- The derivation of a six moments transport model appeared to be very instructive for the development of the modified energy transport model. However, for every-day's use it cannot be recommended since the third balance equation increases the complexity of the system further and numerical stability seems to be worse than with the energy transport model.

A detailed study reveals some fundamental problems common to transport models using the first three or four moments. Most importantly, the energy distribution function is frequently modeled by assuming a heated MAXWELLian distribution. This distribution function model is then used to derive a closure relation. Monte Carlo simulations show that the energy distribution function is only poorly described by a heated MAXWELLian distribution function, both for bulk and inhomogeneous devices.

In the particular case of partially depleted SOI MOSFETs the error in the closure together with the assumed equipartition of the energy in the directions parallel and normal to the current density cause a complete breakdown of the energy transport model. The number of carriers with sufficient energy to surmount the barrier towards the bulk is significantly overestimated which results in a spurious drop of the body potential with increasing drain voltage instead of the expected rise. Via the body effect the transistor is then virtually turned off, visible as a strong decline of the drain current in the output characteristic.

An improved energy transport model has been developed and implemented in MINIMOS-NT. By using this advanced model it is possible to successfully simulate partially depleted SOI devices. The unphysical current drop in the output characteristics predicted by the standard energy transport model is entirely avoided. The modifications to the standard energy transport model consist of the introduction of an anisotropic carrier temperature and a modified closure relation. The new model appeared to be very stable, especially when compared to simulations of SOI MOSFETs with the standard energy transport model, because it produces a physically sound solution. The spurious diffusion of hot electrons perpendicular to the current direction is sufficiently reduced.

Based on the observations made during the evaluation of transport models including the first four moments of BOLTZMANN's transport equation, an extended model has been proposed which includes the first six moments. The additional even order moment is the kurtosis of the distribution function. While not applicable for SOI simulation due to numerical stability problems, its derivation gave valuable insight in modeling the closure relation used in the modified energy transport model. Furthermore, it has been proven to be highly beneficial for other works performed at the institute. In particular, an analytical model for the energy distribution function [69] has been proposed which accurately captures the features observed in Monte Carlo simulations, notably the thermal tail inside the channel and the contribution of cold carriers inside the drain region. This analytical distribution function model has been used to model impact ionization [75, G6] [76] and the hot-carrier gate currents [77].