An analytical solution for inhomogeneous strain within cylinders of silicon and effect on quantum band structure under the double-punch test

An analytical solution for inhomogeneous strain distributions within a finite cylinder of silicon under the double-punch test is obtained. The stress function method is employed and a new expression for the stress function is proposed so that all of the governing equations and the boundary conditions are satisfied exactly. The solution for isotropic cylinders under the double-punch test is recovered as a special case. Numerical results show that the strain singularities are usually developed near two end surfaces, but the strain distributions are relatively uniform in the central part of the cylinder. The largest tensile strain is always induced along the axis of loading. In addition, based on the envelope-function method of energy-band theory and quantum mechanics, the effect of external loads on the valence-band structure of silicon is analyzed. The spin-orbit interaction is considered. It is found that external load under the double-punch test can alter considerably the quantum behavior of energy-band structure of silicon, which manifests in the change of the constant-energy surfaces of the heavy-hole band, the light-hole band, the split-off band and the corresponding conductivity masses. The present study provides an alternative method to investigate the electro-optic properties of strained silicon.