2.3.2.2 Macroscopic Relaxation Time Approximation

The collision term on the right hand side of eqns. (2.53) and (2.54), which represents the various scattering processes, can be deliberately modeled as

$$\ensuremath{\int \ensuremath{\widetilde{\phi}}_j \, \mathcal{Q} \... ...ath{\widetilde{\phi}}_j \rangle}_0}{\tau_{\ensuremath{\widetilde{\phi}}_j}} \ ,$$ (2.70)

which is commonly termed as relaxation time approximation [14, p.144]. This equation implies that the perturbed distribution function will relax exponentially to the equilibrium function with one time constant $\tau_{\ensuremath{\widetilde{\phi}}_j}$ when the perturbing field is removed. A discussion on the validity of this approximation is given in [20, p.139].

The equilibrium distribution function $f_0(\ensuremath{\boldsymbol{\mathrm{k}}})$ is a symmetric function. Since the even weight functions are symmetric in $\ensuremath{\boldsymbol{\mathrm{k}}}$ and the odd weight functions are anti-symmetric in $\ensuremath{\boldsymbol{\mathrm{k}}}$, only the even moments of the equilibrium distribution function will be non-zero whereas the odd moments will vanish

$$\ensuremath{\langle \phi_j \rangle}_0 = \ensuremath{\int \phi_j(k) \, f_0(k) \,\, \ensuremath{\mathrm{d}}^3 k} \neq 0 \textcolor{lightgrey}{.......}\textrm{for even $j$} \ ,$$ (2.71)

$$\ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{\phi}}}_j \rangle}_0 = \ensuremath{\int \ensuremath{\boldsymbol{\mathrm{\phi}}}_j(\ens... ... \, f_0(\ensuremath{\boldsymbol{\mathrm{k}}}) \,\, \ensuremath{\mathrm{d}}^3 k} = 0 \textcolor{lightgrey}{.......}\textrm{for odd $j$} \ .$$ (2.72)

Applying the relaxation time approximation and inserting the calculated gradients from the previous section into eqns. (2.53) and (2.54) leads to the equation set

$$\phi_0: \textcolor{lightgrey}{.......} \ensuremath{\partial_{t} \, \ensuremath{\langle 1 \rangle}} + \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}... ...remath{\cdot}\ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \rangle}} = - R \ ,$$ (2.73)

$$\phi_2: \textcolor{lightgrey}{.......} \ensuremath{\partial_{t} \, \ensuremath{\langle \mathcal{E} \rangle}} + \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}... ...nsuremath{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \, \mathcal{E} \rangle}} + \mathrm{q}\, \ensuremath{\boldsymbol{\mathrm{E}}}\ensuremath{\cdot}\ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \rangle} = - \frac{\ensuremath{\langle \mathcal{E} \rangle}- \ensuremath{\langle \mathcal{E} \rangle}_0}{\tau_\mathcal{E}} + G_{\mathcal{E}\, n} \ ,$$ (2.74)

$$\phi_4: \textcolor{lightgrey}{.......} \ensuremath{\partial_{t} \, \ensuremath{\langle v^2 \, \mathcal{E} \rangle}} + \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}... ...th{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \, v^2 \, \mathcal{E} \rangle}} + \frac{4 \, \mathrm{q}}{m} \, \ensuremath{\boldsymbol{\mathrm{E}... ...ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \, \mathcal{E} \rangle} = - \frac{\ensuremath{\langle v^2 \, \mathcal{E} \rangle}- \ensuremath{\langle v^2 \, \mathcal{E} \rangle}_0}{\tau_\beta} + G_{\beta \, n} \ ,$$ (2.75)

$$\ensuremath{\boldsymbol{\mathrm{\phi}}}_1: \textcolor{lightgrey}{.......} \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}... ...\mathrm{v}}} \ensuremath{\otimes}\ensuremath{\boldsymbol{\mathrm{p}}} \rangle}} + \mathrm{q}\, \ensuremath{\boldsymbol{\mathrm{E}}}\ensuremath{\cdot}\ensuremath{\langle \ensuremath{\widetilde{\delta}} \rangle} = - \frac{\ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{p}}} \rangle}}{\tau_m} \ ,$$ (2.76)

$$\ensuremath{\boldsymbol{\mathrm{\phi}}}_3: \textcolor{lightgrey}{.......} \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}... ...nsuremath{\otimes}\ensuremath{\boldsymbol{\mathrm{v}}} \, \mathcal{E} \rangle}} + \mathrm{q}\, \ensuremath{\boldsymbol{\mathrm{E}}}\ensuremath{\c... ...{\mathrm{v}}} \ensuremath{\otimes}\ensuremath{\boldsymbol{\mathrm{v}}} \rangle} = - \frac{\ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \, \mathcal{E} \rangle}}{\tau_S} \ ,$$ (2.77)

$$\ensuremath{\boldsymbol{\mathrm{\phi}}}_5: \textcolor{lightgrey}{.......} \ensuremath{\ensuremath{\ensuremath{\boldsymbol{\mathrm{\nabla}}}... ...th{\otimes}\ensuremath{\boldsymbol{\mathrm{v}}} \, v^2 \, \mathcal{E} \rangle}} + \frac{\mathrm{q}}{m} \, \ensuremath{\boldsymbol{\mathrm{E}}}\en... ...\mathrm{v}}} \ensuremath{\otimes}\ensuremath{\boldsymbol{\mathrm{v}}}) \rangle} = - \frac{\ensuremath{\langle \ensuremath{\boldsymbol{\mathrm{v}}} \, v^2 \, \mathcal{E} \rangle}}{\tau_K} \ ,$$ (2.78)

where $\tau_m$, $\tau_\mathcal{E}$, $\tau_S$, $\tau_\beta$, $\tau_K$ are the relaxation times for momentum, energy, energy flux density, kurtosis, and kurtosis flux density, respectively.

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