- 3.5.1.1 Strain Induced Conduction Band Splitting
- 3.5.1.2 Strain Induced Degeneracy Lifting at the Point
- 3.5.1.3 Strain Induced Valence Band Splitting

The deformation potential theory introduces an additional Hamiltonian , that is attributed to strain and its effects on the band structure. This Hamiltonian is based on first order perturbation theory and its matrix elements are defined by

denotes the deformation potential operator which transforms under symmetry operations as second rank tensor [167] and describes the strain tensor component. The subscripts in denote the matrix element of the operator . Due to the symmetry of the strain tensor with respect to and , also the deformation potential operator has to obey this symmetry and thus limits the number of independent deformation potential operators to six.

In the case of cubic semiconductors the edges of the conduction band and the valence band are located on symmetry lines. These symmetries are reproduced in the energy band structure and in the basis states. Furthermore, the symmetry of the basis states allows to describe the deformation potential operator of a particular band via two or three deformation potential constants [166].

Although, theoretically the deformation potential constants can be calculated via the empirical pseudo potential method or by ab initio methods, it is more convenient to fit the deformation potentials to experimental results obtained by electrical, optical, microwave techniques, or by analyzing stress induced absorption edges. Even though, theoretical predictions and measurements match quite well, deformation potentials in literature and found by different methods deviate from each other [168].

describes the uniaxial- and the dilatation deformation potential constants for valleys of the type . denotes the unit vector parallel to the vector of valley . The conduction band minimum valley shift can be determined from a single deformation potential constant

(3.17) |

Via the two relations from above the valley splitting from uniaxial stress along arbitrary directions can be calculated.

Bir and Pikus found from **k****.****p** theory, that when the degeneracy at the zone boundary is lifted, a relatively large change in the energy dispersion of the conduction band minimum located close to this point arises [161]. This effect was experimentally proved for by Hensel and Hasegawa [170], who measured the change in effective mass for stress along
, and by Laude [171], who showed the effect via the indirect exciton spectrum.

Therefore, in order to take the lifting of the degeneracy of the two lowest conduction bands and at the points into account, (3.16) has to be adapted [170]

where denotes a new deformation potential,

The solutions of the eigenvalue problem look like:

which shows that at the points the band shifts by an amount of (like before in (3.16)) plus an additional splitting of , which lifts the degeneracy. (3.19) shows the proportional dependence on shear strain for the splitting

(3.21) |

A value of eV has been predicted by Hensel for the shear deformation potential [170]. Laude [171] confirmed this value by his measurement of eV via the indirect exciton spectrum of .

The splitting is already strongly pronounced for shear strain . Due to the lifting of the degeneracy the conduction band is deformed close to the symmetry points (Fig. 3.2).

A non-vanishing shear strain component has the following effects on the energy dispersion of the lowest conduction band:

- The band edge energy of the valley pair along direction shifts down with respect to the other four valleys along and .
- The effective mass of the valley pair along changes with increasing .
- The conduction band minima along move to the zone boundary points at with increasing .

For differing strains ( ), the conduction band minima along the axes are different in their energies, causing a repopulation between the six conduction band valleys. This kind of effect is not covered with (3.16), due to the negligence of possible degeneracy liftings by shear strain and by ignoring a possible repopulation of energy states.

The model presented shows no change in the conduction bands near the zone boundaries and for a shear component (Fig. 3.3). However, shear components like or lift the degeneracy at or .

Applying a degenerate **k****.****p** theory at the zone boundary point [161,170] enables an analytical description for the valley shift along the direction. Shear strain
causes an energy shift between the conduction band valleys along
/
and the valleys along
.
This shift is described by

is a dimensionless parameter and denotes the band separation between the lowest two conduction bands at the conduction band edge

(3.23) |

denotes the position of the band edge in the unstrained lattice.

(3.24) |

denotes the matrix

(3.25) |

In the case of the valence band the description of the strain induced shifts of the heavy-hole, light-hole, and the split-off band are more complex[169].

- ... constants
^{3.1} - neglecting strain induced splitting of the degenerate conduction bands and at the point

T. Windbacher: Engineering Gate Stacks for Field-Effect Transistors