Wannier Function and Effective Mass Approximation

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 $$\langle \boldsymbol{v}\rangle =(1/\hbar ) \partial \mathcal{E}/\partial \boldsymbol{k}$$ in a periodic crystal potential. In the presence of an external force $$\boldsymbol{F}$$ , an electron is accelerated in $$\boldsymbol{k}$$ space in the form of $$\hbar \partial \boldsymbol{k}/\partial t = \boldsymbol{F}$$ . The electron motion is then expressed in the classical picture of a particle with the effective mass $$m^*$$ or $$1/m^* = (1/\hbar ^2)\partial ^2 \mathcal{E}/\partial \boldsymbol{k}^2$$ and the momentum $$\boldsymbol{p}=\hbar \boldsymbol{k}= m^* \langle \boldsymbol{v}\rangle $$ . The results are used to derive transport properties in Chap. 6 .