Strain and Semiconductor Crystal Symmetry

Abstract

One common question asked about strain on semiconductor band structures is: why is there band splitting with strain? Essentially, the answer to this question can only be sought from the semiconductor crystal symmetry considerations. Band degeneracies are defined by semiconductor crystal symmetry. When strain reduces the original symmetry, some degeneracies are lifted, and thus we see band splitting. Symmetry considerations are intuitively simple yet very complicated when treated in a systematical way. The mathematical tool to treat symmetry is group theory. In this chapter, we are going to investigate the semiconductor crystal symmetry and its effects on the semiconductor band structures, then study strain-altered symmetry-induced band splitting and warping. For simplicity, we will avoid use of the abstract concepts of group theory, but rather wil treat the relation of symmetry and properties of band structures using simple observations and examples. We will divide the discussion in this chapter into two main parts. The first part is for readers who want a simple qualitative picture of symmetry and its effects on band structures. Through the qualitative discussion alone, one can understand how strain changes the band structure by reducing the crystal symmetry. The second part is aimed for readers with an interest in more detailed yet more abstract conceptions and reasonings. Thus, this chapter is organized as follows: in Sect. 3.2, a qualitative discussion is first given for symmetry and its associated band properties; strain-induced symmetry reduction and band property alteration are illustrated using simple diagrams for qualitative understanding. Then in Sect. 3.3 and the following sections, detailed discussions of crystal symmetries are given. Symmetry-determined band properties are studied in a more theoretical point of view. In the last section, a summary is given, which discusses the merits and limitations of symmetry considerations in terms of the current strain research.