Abstract
Effective–mass equation is very useful to understand the transport and optical properties of semiconductors. In this chapter the effective–mass equation is derived with the help of Wannier function. Using Schrödinger equation based on the effective–mass approximation, we discuss the shallow impurity levels of donors in Ge and Si. Transport properties of electrons and holes are interpreted in terms of the effective mass in the classical mechanics (Newton equation). In this chapter the group velocity (the expectation value of the velocity) is shown to be given by \(\langle \varvec{v}\rangle =(1/\hbar ) \partial \mathcal{E}/\partial \varvec{k}\) in a periodic crystal potential. In the presence of an external force \(\varvec{F}\), an electron is accelerated in \(\varvec{k}\) space in the form of \(\hbar \partial \varvec{k}/\partial t = \varvec{F}\). The electron motion is then expressed in the classical picture of a particle with the effective mass \(m^*\) or \(1/m^* = (1/\hbar ^2)\partial ^2 \mathcal{E}/\partial \varvec{k}^2\) and the momentum \(\varvec{p}=\hbar \varvec{k}= m^* \langle \varvec{v}\rangle \). The results are used to derive transport properties in Chap. 6.
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Notes
- 1.
In a general case (3.91) is defined by replacing \(\alpha (\varvec{k},t)\) with \(\sum _{n}\alpha _{n}(\varvec{k},t)\), where n in the band index.
- 2.
This assumption is based on the fact that the function G will stay un–changed without scattering and thus the maximum value of G is not changed with time t. Scattering induces a change in the wave function and the maximum value will stay constant before the next scattering.
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Hamaguchi, C. (2017). Wannier Function and Effective Mass Approximation. In: Basic Semiconductor Physics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-66860-4_3
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DOI: https://doi.org/10.1007/978-3-319-66860-4_3
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