2.3.3.2 Discussion of the Diffusion Approximation

Let now be a shifted distribution function

(2.127) |

where is not only symmetric in but also in every component of

(2.128) | ||

(2.129) | ||

(2.130) |

This stronger symmetry property ensures that the resulting tensor quantities are of diagonal shape (see eqn. (2.83)). Functions which satisfy this stronger symmetry criterion are for example an isotropic

As has already been shown in eqns. (2.106) to (2.108), every function can be split into its symmetric and its anti-symmetric part. Since the weight functions and are even functions, only the symmetric part of the distribution function has to be taken into account

Eqn. (2.131) is now used in the evaluation of the statistical average :

Cross terms containing both and vanish because of .

The statistical average of can be evaluated in the same way yielding

For a MAXWELL distribution the first term of the RHS of eqn. (2.133) has already been calculated as eqn. (2.120). Therefore eqn. (2.133) can be written as

As can be seen, without the diffusion approximation the average carrier energy is composed of
a thermal component and a kinetic^{2.8} component.
A consequence of the diffusion approximation is that the kinetic term is neglected
[24, p.71] [25, p.736]. By assuming
,
, and
for electrons
in silicon [9, p.191] the ratio
yields
. However, in reality this ratio is much bigger because in the regions, where the
assumed electron saturation velocity is reached, the electron temperature is much higher than
the lattice temperature [26, p.34]. Neglecting the kinetic term appears
therefore justified. Note that simulations at very low temperatures would have to include
this term. Under dynamic conditions this term can also be significant
[27, p.413].

The first term of the RHS of eqn. (2.132) has also already been calculated as eqn. (2.88). Eqn. (2.132) can therefore be written as

Inserting eqn. (2.135) into eqn. (2.76) yields

By calculating the statistical average

(2.137) |

the current density can be expressed as

(2.138) |

and the second term of the LHS of eqn. (2.136) can be written as

where in the last identity the term containing has been neglected.

Inserting eqn. (2.139) into eqn. (2.136) yields the final form of the current relation

which has compared to eqn. (2.102) an additional second term [28] on the LHS, which is nonlinear in . This example demonstrates that neglection of the term is another consequence of the diffusion approximation. Therefore, if it is justified to neglect the kinetic term in eqn. (2.134) it is equally valid to neglect the additional term in the current equation (2.140).

The importance of this additional term within semiconductor equations is controversial.
Phenomena known from fluid dynamics like super-sonic transport and propagation of electron
shock-waves arise [29] [30]. The resulting transport model is referred to
as (full) hydrodynamic transport model^{2.9}.