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Up: 2.1 The Boltzmann Poisson
Next: 2.1.4 Distribution Function Models
To derive a numerically tractable model from the full
(3+3)-dimensional Boltzmann equation we assume that
the distribution
is
one-dimensional in space
and
has cylinder symmetry in
. That is
where![$\displaystyle \quad k_r^2 = k_1^2 + k_2^2 .$](img38.png) |
(2.11) |
This is known as the ``1+2''-Boltzmann equation.
Note that also the input data has to fulfill this symmetry.
Consequently the cylinder radial components of the
electrical field
vanish.
With this assumption we have
![$\displaystyle \frac{\partial f}{\partial x_1} = \frac{\partial f}{\partial x_2} = 0$](img40.png) |
(2.12) |
and
![$\displaystyle E_{x_1} = E_{x_2} = 0 .$](img41.png) |
(2.13) |
Hence the BTE reduces for parabolic bands to
![$\displaystyle \frac{\partial f}{\partial t} + v_3 \cdot \nabla_{x_3} f + \frac{q}{m^{*}} E_{x_3} \cdot \nabla_{v_3} f = Q(f) .$](img42.png) |
(2.14) |
Previous: 2.1.2 Poisson Equation
Up: 2.1 The Boltzmann Poisson
Next: 2.1.4 Distribution Function Models
R. Kosik: Numerical Challenges on the Road to NanoTCAD