previous up next contents Previous: 2.1 The Boltzmann Poisson Up: 2.1 The Boltzmann Poisson Next: 2.1.2 Poisson Equation

2.1.1 The Boltzmann Equation for Parabolic Energy Bands

For a distribution function $ f({\mathbf{x}},{\mathbf{k}},t)$ where $ {\mathbf{x}}$ is the position, $ {\mathbf{k}}$ is the wave vector and $ t$ is the time the Boltzmann equation for parabolic energy bands reads

$\displaystyle \frac{\partial f}{\partial t} + \frac{1}{\hbar} \nabla_{\mathbf{k...
...abla_{{\mathbf{x}}} V({\mathbf{x}},t) \cdot \nabla_{{\mathbf{k}}} f = Q(f)   .$ (2.1)

Here $ \varepsilon$ is the given energy band diagram, $ V$ is the electrostatic potential and $ q$ is the carrier charge (negative for electrons). The independent variables are $ {\mathbf{k}}$ $ \in \mathbb{R}^3$, $ {\mathbf{x}}$ $ \in \mathbb{R}^3$, $ t$ $ \in \mathbb{R}$.
We define the group velocity $ {\mathbf{v}}$

$\displaystyle {\mathbf{v}}({\mathbf{k}}) = \frac{1}{\hbar} \nabla_{{\mathbf{k}}} \varepsilon({\mathbf{k}})   .$ (2.2)

For now we restrict ourselves to parabolic bands. Then

$\displaystyle \varepsilon({\mathbf{k}}) = \frac{\hbar^2}{2m^{*}}\vert{\mathbf{k}}\vert^2   , % + E_c
$ (2.3)

with effective mass $ m^{*}$ and

$\displaystyle {\mathbf{v}}({\mathbf{k}}) = \frac{\hbar {\mathbf{k}}}{m^{*}}   .$ (2.4)

The momentum $ {\mathbf{p}}$ is defined as

$\displaystyle {\mathbf{p}} = \hbar {\mathbf{k}}   .$ (2.5)

Using the group velocity $ {\mathbf{v}}$ the Boltzmann equation becomes

$\displaystyle \frac{\partial f}{\partial t} + {\mathbf{v}} \cdot \nabla_{{\mathbf{x}}} f + \frac{q}{m^{*}} {\mathbf{E}} \cdot \nabla_{{\mathbf{v}}} f = Q(f)   ,$ (2.6)

where we introduced the electrical field $ {\mathbf{E}} =
- \nabla_{{\mathbf{x}}} V({\mathbf{x}},t)$.

previous up next contents Previous: 2.1 The Boltzmann Poisson Up: 2.1 The Boltzmann Poisson Next: 2.1.2 Poisson Equation


R. Kosik: Numerical Challenges on the Road to NanoTCAD