3.6.3 Strain-Induced Valence Band Splitting

The deformation potential theory for the valence bands is different from that for the conduction bands because of the degeneracy of the valence bands at the valence band maximum. The operators $\mathcal{D}^{\alpha\beta}$ are no longer scalars. Instead, they can be expressed as $3\times 3$ matrices. Due to symmetry the six independent operators have only three independent entries, usually labeled $l,m,n$ or $a,b,d$, depending on the used basis for the eigenfunctions [Cardona66].

In the basis $\vert x,s\rangle ,\ \vert y,s\rangle ,\ \vert z,s\rangle$ with $s$ denoting the spin state ($\uparrow$ for $+z$) and ( $\downarrow$ for $-z$), the matrix of the perturbation Hamiltonian is

$$\ensuremath{{\underaccent{\bar}{H}}}_{strain} = \begin{pmatrix}\e... ... \begin{matrix}\vert\uparrow\rangle \\ Vert\downarrow\rangle \\ \end{matrix}\ ,$$ (3.56)

with ${\underaccent{\bar}{H}}$ denoting the $3\times 3$ matrix

$$\ensuremath{{\underaccent{\bar}{H}}} = \begin{pmatrix}l \ensurema... ...egin{matrix}\vert x\rangle \\ Vert y\rangle \\ Vert z\rangle \\ \end{matrix}\ .$$ (3.57)

In Section 3.6.1 and Section 3.6.2 it was shown that from deformation potential theory simple analytical expressions can be derived for the conduction band shifts induced by an arbitrary strain tensor $\varepsilon_{ij}$. For the valence band the expressions for the strain-induced shifts of the heavy-hole, light-hole, and split-off band are more complex, which limits their practical use [Balslev66].

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology