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

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

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

For now we restrict ourselves to parabolic bands. Then

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

with effective mass $m^{*}$ and

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

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

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

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

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

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