3.6 Linear Deformation Potential Theory

Deformation potential theory was originally developed by Bardeen and Shockley [Bardeen50] and generalized by Herring and Vogt [Herring56]. In an extensive study Bir and Pikus showed later how to calculate strain effects on the band structure of various semiconductors on the basis of deformation potentials by applying laws of group theory [Bir74]. In the following a brief outline of this theory is given before it will be applied to calculate strain-induced energetic shifts of the conduction band valleys of cubic semiconductors.

An additional Hamiltonian attributed to strain, $\mathcal{H}(\hat{{\ensuremath{\varepsilon_{}}}})$, is defined and its effect on the band structure is calculated using first order perturbation theory. The matrix elements of the perturbation Hamiltonian are given by

$$\{ \mathcal{H}(\hat{{\ensuremath{\varepsilon_{}}}}) \}_{ij}=\sum_... ...^{3} \mathcal{D}_{ij}^{\alpha\beta} {\ensuremath{\varepsilon_{\alpha\beta}}}\ ,$$ (3.46)

where ${\ensuremath{\varepsilon_{\alpha\beta}}}$ denotes the $(\alpha\beta)$ component of the strain tensor, and $\mathcal{D}^{\alpha\beta}$ is the deformation potential operator which transforms under symmetry operations as a second rank tensor [Hinckley90]. The additional subscripts $(i,j)$ in $\smash{\mathcal{D}_{ij}^{\alpha\beta}}$ denote the matrix element of the operator $\mathcal{D}^{\alpha\beta}$. From the symmetry of the strain tensor with respect to $\alpha$ and $\beta$ it follows $\mathcal{D}^{\alpha\beta}=\mathcal{D}^{\beta\alpha}$, such that at maximum six independent deformation potential operators can exist.

In cubic semiconductors the edges of the conduction and the valence band are located on symmetry lines which are reflected in symmetries of the energy band structure and of the basis states. Due to the symmetry of the basis states the deformation potential operator of a particular state can be described in terms of two or three deformation potential constants [Herring56,Singh93]. The shift of the band edges can be calculated from these deformation potential constants.

In principle the deformation potential constants can be determined numerically using the empirical pseudopotential method (see Section 3.8) or from ab initio calculations. However, it is common practice to fit deformation potentials to measurements using electrical, optical, microwave techniques or by analyzing stress-induced indirect absorption edges. Although measurements seem to verify theoretical predictions, the deformation potentials obtained by different methods deviate from each other and different values can be found in literature [Fischetti96a].

Subsections

• 3.6.1 Strain-Induced Conduction Band Splitting
• 3.6.2 Strain-Induced Lifting of Degeneracy at X point
• 3.6.3 Strain-Induced Valence Band Splitting

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