3. Strain Effects on the Bulk Band Structure

The band structure describes the states of energy in the crystal momentum space that electrons and holes are allowed to have. It presents the electronic dispersion relation under the influence of the potential of the solid. The band structure determines several important characteristics, in particular its electronic and optical properties.

In this chapter, first the basic physical definitions are introduced, such as the strain and stress tensors and how they are related in cubic semiconductors (Section 3.1 to Section 3.3).

The vast majority of semiconductors of interest for electronic and optoelectronic applications have a diamond structure with an underlying face centered cubic (fcc) lattice consisting of two atoms per basis. The basic properties of the diamond structure and the band structure of relaxed Si are presented in Section 3.4.

The symmetry of the crystal in real space is directly reflected in the symmetry of the band structure in momentum space. Using group-theoretic methods many properties of the band structure can be obtained from these symmetries. Section 3.5 is devoted to the effect of strain on the symmetry of the diamond structure. Special focus is put on the consequence of the strain-induced reduction of symmetry for band structure calculations.

Hereafter, three different methods of calculating the effect of strain on the band structure are presented:

• Historically, the first approach was developed by Bardeen and Shockley and is known as the deformation potential theory. The perturbation caused by strain is attributed to an additional Hamiltonian which is linearly proportional to the deformation potential operator and to strain. First order perturbation theory is used to calculate the effect of strain on the band structure and analytical expressions for the strain-induced energy shifts of the conduction- and valence-bands can be obtained. The salient features of this theory and relevant results are summarized in Section 3.6.

• Strain does not only shift the conduction and valence bands, but also changes the curvature of the energy bands. In Section 3.7 the main features of the kp method are briefly outlined, as this method is able to capture the deformation of the shape of the energy bands under strain. It is frequently used to model the influence of strain on the valence bands. On the contrary, the impact of strain on the curvature of the conduction band has often been neglected. In Section 3.7.2 a kp analysis capturing the effect of strain on the lowest conduction band of Si is developed. It shows that shear strain can alter the curvature of the conduction bands and thus can reduce the effective masses of electrons.

• The empirical pseudopotential method (EPM) [Chelikowsky76] including nonlocal effects and spin orbit coupling is frequently used to calculate the band structure of semiconductors, as this method is efficient and requires only a small number of parameters. These are usually calibrated to match energy gaps and effective masses determined from experiments and are available for a large set of materials [Yu03]. The method can very naturally be adapted to incorporate strain effects and has been used to investigate the band structure of biaxially strained Si$_{1-x}$Ge$_{x}$ grown on Si$_{1-y}$Ge$_{y}$ for various surface orientations [Fischetti96a,Van de Walle86,Rieger93]. In Section 3.8 it is shown how a general strain tensor can be taken into account by this method for band structure calculation. Special focus is put on the orthorhombic distortion of the crystal structure resulting from uniaxial stress along the [110] direction. This type of stress is used in practice to enhance the electron mobility in n-channel MOSFETs.

Subsections

• 3.1 Strain
• 3.2 Stress
• 3.3 Stress-Strain Relations
  • 3.3.1 The Miller Index Notation
  • 3.3.2 Strain Resulting from Uniaxial Stress
  • 3.3.3 Strain Resulting from Epitaxy
  
  
• 3.4 Basic Properties of the Diamond Structure
  • 3.4.1 Band Structure of Relaxed Si
  
  
• 3.5 Effect of Strain on Symmetry
  • 3.5.1 Hierarchy of systems
  • 3.5.2 $O_{h}$ symmetry
  • 3.5.3 $D_{4h}$ symmetry
  • 3.5.4 $D_{3d}$ symmetry
  • 3.5.5 $D_{2h}$ symmetry
  • 3.5.6 $C_{2h}$ symmetry
  • 3.5.7 $S_{\mathrm{2}}$ symmetry
  
  
• 3.6 Linear Deformation Potential Theory
  • 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
  
  
• 3.7 The kp method
  • 3.7.1 Effective Electron Mass in Unstrained Si
  • 3.7.2 Strain Effect on the Si Conduction Band Minimum
    • Energy Dispersion of the Conduction Band Minimum of Strained Si: Method 1
    • Energy Dispersion of the Conduction Band Minimum of Strained Si: Method 2
    • Discussion
  
  
• 3.8 EPM for Arbitrary Strain
  • 3.8.1 The Empirical Pseudopotential Method
  • 3.8.2 Inclusion of Strain

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