An example of a shifted MAXWELLian distribution function together with its symmetric and anti-symmetric part is depicted in Fig. 2.1.
The diffusion approximation now assumes that the displacement of the distribution function is small which means that the anti-symmetric part is much smaller than the symmetric one. Then it is justified to approximate the shifted MAXWELL distribution function by a series expansion with respect to the displacement and to truncate the expansion after the first term:
The interpretation of eqn. (2.111) is that the symmetric part can be approximated by a non-displaced MAXWELL distribution function
For closing the moment equation system at even moments, an assumption about the symmetric part of the distribution function must be introduced since the integrals of even powers of multiplied with the anti-symmetric part vanish. Vice versa, for closing the moment equation system at odd moments, only the anti-symmetric part of the distribution function must be assumed since the integrals of odd powers of multiplied with the symmetric part vanish.
The even moments will be calculated as powers of the energy since for parabolic bands and only differ by a constant prefactor . The same holds true for and where the constant prefactor yields . Starting from
M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF