3.2.2 Boltzmann's Equation

In this section, Boltzmann's equation is introduced as an important foundation for semi-classical transport description in semiconductor devices. In order to describe the complete device behavior, theoretically every single carrier within the device would have to be described by solving Newton's classical equations of motion

$$\ensuremath{\ensuremath{\frac{\mathrm{d} \ensuremath{\ensuremath{... ...ar} \ensuremath{\ensuremath{\mathitbf{\nabla_{\!k}}}}}\ensuremath{{\cal{H}}}\,,$$ (3.2)

$$\hbar \ensuremath{\ensuremath{\frac{\mathrm{d} \ensuremath{\ensur... ...ath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{{\cal{H}}}\,.$$ (3.3)

Within these, a generalized force $\ensuremath{\ensuremath{\mathitbf{F}}}$ with unique direction comprises the electric field as well as other fields causing a driving force to carriers, such as a thermal gradient (Seebeck-, Peltier-effect) or a magnetic field (Lorentz-force). This force is overlaid by forces with random direction describing the microscopic thermal movement.

The Hamilton function $\ensuremath{{\cal{H}}}$ represents the total carrier energy and can be split into two parts: The potential energy $\ensuremath{\ensuremath{\mathcal{E}}_\mathrm{pot}}$ and the kinetic energy $\ensuremath{\ensuremath{\mathcal{E}}_\mathrm{kin}}$ . The potential energy incorporates the conduction and valence band edge energies $\ensuremath{\ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{c,v}}}}$ for electrons and holes, respectively, $\ensuremath{\Delta \ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{c,v}}}}$ considering band gap narrowing, and the electrostatic potential $\ensuremath{\varphi}$ . Thus, $\ensuremath{{\cal{H}}}$ can be expanded as

$$\ensuremath{{\cal{H}}}= \underbrace{\ensuremath{\ensuremath{\math... ...suremath{\mathcal{E}}}_{\ensuremath{\ensuremath{\mathcal{E}}_\mathrm{kin}}} \,.$$ (3.4)

For the sake of clarity, $\mathrm{q}$ denotes the elementary charge and does not incorporate the different sign of the charges of electrons and holes. The different charge of electrons and holes is treated by $\ensuremath{\mathrm{s}_\nu}$ , which becomes $-1$ for electrons and $1$ for holes.

Instead of keeping the focus on every single carrier, a statistical description is introduced. Neglecting Heisenberg's uncertainty principle, each carrier is exactly described by its momentum and position, so it takes a certain place within 6 dimensional $(\ensuremath{\ensuremath{\mathitbf{r}}}$ , $\ensuremath{\ensuremath{\mathitbf{k}}})$ -space. The normalization of all present carriers within the volume under investigation results in the distribution function, which is the solution variable of Boltzmann's equation. It incorporates the carrier density within 6-dimensional $(\ensuremath{\ensuremath{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}})$ -space.

In the sequel, the Boltzmann transport equation is derived from phenomenological considerations and its range of validity is discussed. Formally, Boltzmann's equation represents a seven-dimensional integro-differential equation within the phase space $(\ensuremath{\ensuremath{\mathitbf{r}}},\ensuremath{\ensuremath{\mathitbf{k}}},t)$ . However, it is accessible using a book-keeping background, which is presented in Section 3.2.2.1. Originally formulated for the description of statistical mechanics of gases, it is the cornerstone for the classical description of transport in semiconductors as well. The transport models derived in this thesis later on are based on the Boltzmann transport equation.

Subsections

• 3.2.2.1 Phenomenological Approach
• 3.2.2.2 Validity

M. Wagner: Simulation of Thermoelectric Devices