2.3.1 Even Moments: Relaxation Times

In the simplest case the relaxation times $\tau_{2i}$ are often modeled as constants.

$$\partial_{x_3} M_{2i+1} - \frac{q}{m^{*}} E_{x_3} (2iM_{2i-1}) = -M_0 \frac{\hat{M}_{2i} - \hat{M}_{2i,\mathrm{eq}}}{\tau_{2i}} .$$ (2.53)

Here the hat denotes the normalized quantities:

$$\hat{M}_{2i} = \frac{\langle O_{2i} \rangle}{\langle 1 \rangle} = \frac{M_{2i}}{M_0} .$$ (2.54)

For the even moments the relaxation time approximation implies that the perturbed moments of the distribution function will relax exponentially to the equilibrium function with time constant $\tau_{2i}$ when the perturbing field is removed. In this case we relax the normalized quantities $\hat{M}_{2i}$ to the values from a cold Maxwellian.

$$\begin{gather*}\begin{split}& \hat{M}_{0,\mathrm{eq}} = 1 , & \hat{M}_{2,\m... ...}{2} \frac{k_{\mathrm{B}}^2}{m} T_{\mathrm{L}}^2 . \end{split}\end{gather*}$$ (2.55)

We combine lattice temperature and local density $M_0(x)$ and hence, in the equation for the second moment, we really relax the temperature.

Variants in the specification of the relaxation time approximation are possible. For the higher order moments relaxation to values from a hot Maxwellian with local temperature $T(x)$ is a valid alternative.

Often the relaxation time is modeled as depending on the normalized $\varepsilon$, that is, local temperature. For a comparison with Monte Carlo results the constant relaxation time approximation can be too simplistic. A better approximation is to tabulate it as a function of temperature and doping extracted from bulk Monte Carlo simulations as discussed in Section 2.3.4.

The modeling of production terms by relaxation type models leads to inconsistencies with the Onsager reciprocity relations as stressed in [ARR00].