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2.3.4 Consistency with Bulk Monte Carlo Results

During the development of numerical codes it is very important to find simple test cases for which the analytical solution is known and which can be used to debug and test the simulator.

In the bulk case we assume a constant field $ E$ and constant doping. If all quantities are space independent we get the following system of equations:

$\displaystyle \langle M_1 \rangle$ $\displaystyle = \mu_1 E M_0$ (2.68)
$\displaystyle \langle M_3 \rangle$ $\displaystyle = \mu_3 E \left(1 + \frac{2}{3}\right) M_2$ (2.69)
$\displaystyle \langle M_5 \rangle$ $\displaystyle = \mu_5 E \left(1 + \frac{4}{3}\right) M_4$ (2.70)

and


$\displaystyle M_2$ $\displaystyle = M_{2,{\mathrm{\mathrm{eq}}}} + \frac{2q}{m^*}\tau_2 \frac{M_1}{n}E$ (2.71)
$\displaystyle M_4$ $\displaystyle = M_{4,{\mathrm{\mathrm{eq}}}} + \frac{4q}{m^*}\tau_4 \frac{M_3}{n}E$ (2.72)

The equation for $ M_0$ degenerates and $ M_0$ is fixed by the doping (charge neutrality). The bulk case is used to calibrate the simulator by comparison with Monte Carlo data. The mobilities and relaxation times are extracted to match the Monte Carlo results. Hence the resulting model contains no free fit-parameters.

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