3.2.4.1 Shift of Conduction Band Minima

In this work the $X$ and $L$ valleys of Si, Ge, and SiGe are considered. The symmetry of these valleys further reduces the number of independent components of the deformation potential tensor to two, namely $\Xi_{u}$ and $\Xi_{d}$. The deformation potential $\Xi_{d}$ relates to pure dilatation while $\Xi_{u}$ is associated with a pure shear involving a uniaxial stretch along the major axis plus a symmetrical compression along the minor axis [59].

Linear deformation-potential theory implies that for conduction band extrema not located in the center of the Brillouin zone the shape of the equal energy surface does not change to the first order in strain. However, a particular extremum of the conduction band shifts under strain. The shift depends on the magnitude of applied forces and their orientation with respect to the quasi-momentum of a given extremum. The degenerate extrema are in general split. This splitting is completely determined by the deformation potentials $\Xi_{d}$ and $\Xi_{u}$.

The general form of the energy shift (3.28) of valley $i$ of type $k=X,L$ for an arbitrary homogeneous deformation can be written in the following form [60]:

$$\Delta\epsilon_{c}^{(i,k)}=\Xi_{d}^{(k)}Tr(\hat{\boldsymbol{\varepsilon}})+\Xi_{u}^{(k)}\vec{a}_{i}^{T}\hat{\boldsymbol{\varepsilon}}\vec{a}_{i},$$ (3.30)

where $\vec{a}_{i}$ is a unit vector parallel to the $\vec{k}$ vector of valley $i$. From (3.30) it follows that degeneracy is reduced by shear strain. S. Smirnov: