Effect of Non-homogeneous Strain on the Band Structure of Semi-conductors due to the End Friction under Compression Tests

The exact analytic solution for the inhomogeneous strain field within a finite cubic cylinder of semi-conductors under uniaxial compression test with end constraint is analytically derived. The subsequent inhomogeneous strain-induced valence band structure is studied. More specifically, Lekhnitskii's stress function is employed to uncouple the equations of equilibrium of the material. Numerical results show that the strain is not uniform, and a strain concentration is usually induced by the end constraint. The maximum axial, radial and circumferential strains can be as more than 40%, 65%, 57% respectively, comparing to those without end constraint, but the strain distributions in the central region are relatively uniform, and only 2% extra strain values can be induced by end constraint. The effect of this inhomogeneous strain on the heavy- and light-hole valence band energies are investigated for Sil-xGex alloy (which a popular semi-conductor composed of lattice mismatched materials and x represents the fractional content of germanium). Special focus is paid at the top of the valence band or at the center of Brilliouin zone. Mathematically, it is for zero wave vector k=0 (the wave vector of the solution of the Schrödinger equation describing the relative electron-hole motion), the degeneracy of the eignvalues of the Pikus-Bir strain Hamiltonian, composing of the Luttinger-Kohn Hamiltonian and the strain Hamiltonian, is used to investigate heavy-hole and light-hole energies of the Sil-xGex alloy as a function of x. Physically, the degeneracy corresponding to the existence of different electronic states at the same energy level. In the formulation, the spin-orbit coupling energy is retained. It was found that the presence of the inhomogeneous strains causes significant shifts and splitting of the electronic valence band structure of the semiconducting cylinders.