### 2.3.3.5 Six Moments Transport Model - Closure at

Taking the first six moments, eqns. (2.99) to (2.104), into account give three balance and three flux equations

 (2.168) (2.169) (2.170)

 (2.171) (2.172) (2.173)

By using just a MAXWELL distribution function to close the system one would not obtain any additional information as compared to the energy transport model. A shifted MAXWELL distribution function has only three independent parameters, namely its amplitude, the displacement, and the standard deviation, which correspond to the carrier concentration , the carrier velocity , and the carrier temperature , respectively. By simply increasing the number of considered moments of the distribution function no additional independent variables can be found.

In analogy to statistical mathematics a quantity called kurtosis has been introduced, which is in this work defined as the deviation of the fourth moment of the non-MAXWELL distribution function from the fourth moment of a MAXWELL distribution function with the same standard deviation

 (2.174)

The system is now closed at . Eqn. (2.126) is one possible closure relation obtained from a MAXWELL distribution function. Other empirical closures are also possible (eqn. (2.176)). By introducing an additional temperature 2.11

 (2.175)

the third power of the temperature in eqn. (2.126) is substituted by empirically combining different powers of and

 (2.176)

Simulations have shown, that the combination with fits best to Monte Carlo data [39, G5]. This is depicted in Fig. 2.4 where the different closure relations are compared with the sixth moment obtained from a Monte Carlo simulation of a one-dimensional -- test structure. As can be seen, the closure for the case gives the smallest error within the channel. The convergence behavior of the resulting discretized equation system also appeared most stable when using . Especially for , which corresponds to closing the system with a MAXWELLian distribution function eqn. (2.126) [40] the NEWTON procedure failed to converge in most cases.

Using the closure relation becomes

 (2.177)

and the full six moments transport model reads

 (2.178) (2.179)

 (2.180) (2.181)

 (2.182) (2.183)

In the following the equations for the six moments transport model are rewritten by introducing the charge sign for electrons and the coefficients to . The balance equations become

 (2.184) (2.185) (2.186)

with

 (2.187)

and the following flux equations:

 (2.188) (2.189) (2.190)

The equations for holes are obtained by replacing by and taking into account that :

 (2.191) (2.192)

 (2.193) (2.194)

 (2.195) (2.196)

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF