Abstract
In this chapter a short overview will be given regarding the physics of semiconductors and some properties of the Boltzmann equation that is the starting point for describing the semiclassical transport of charge carriers.
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Notes
- 1.
The symbol “∧” denotes the vector product.
- 2.
For any function g(x) belonging to the space \(L^1 (\mathbb {R}^d)\) of the summable functions defined over \(\mathbb {R}^d\), with d integer, we define the Fourier transform as
$$\displaystyle \begin{aligned} \mathscr{F} g (\mathbf{p}) = \int_{\mathbb{R}^d} g(\mathbf{x}) e^{- \frac{i}{\hbar}\mathbf{p} \cdot \mathbf{x}} \, d \mathbf{x} \quad \forall \mathbf{p} \in R^d. \end{aligned}$$Note that p has the dimension of a momentum if x has the dimension of a length.
For any function \(h (\mathbf {p}) \in L^1 (\mathbb {R}^d)\), the inverse Fourier transform is given by
$$\displaystyle \begin{aligned} \mathscr{F}^{-1} h (\mathbf{x}) = \frac{1}{(2 \pi \, \hbar)^d} \int_{\mathbb{R}^d} h (\mathbf{p}) e^{\frac{i}{\hbar}\mathbf{p} \cdot \mathbf{x}} \, d \mathbf{p} \quad \forall \mathbf{x} \in R^d. \end{aligned}$$
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Camiola, V.D., Mascali, G., Romano, V. (2020). Band Structure and Boltzmann Equation. In: Charge Transport in Low Dimensional Semiconductor Structures. Mathematics in Industry(), vol 31. Springer, Cham. https://doi.org/10.1007/978-3-030-35993-5_1
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