Inhomogeneous strain fields within silicon spheres under the point load test and the strain effect on the quantum valence-bands

Abstract

This paper presents an exact solution for inhomogeneous strain and stress distributions within silicon spheres under the point load test. The contact problem between the spherical head of the steel cones and the surfaces of the silicon spheres is considered. Displacement functions are introduced in order to uncouple the governing equations. The Fourier–Legendre expansion technique is employed so that all of the boundary conditions can be satisfied exactly. When the isotropic limit is considered, the classical solution by Hiramatsu and Oka [20] is covered as a special case. It was found that the strain distributions are relatively uniform within central part (say r/R<0.6, where r and R are the distance from the center and the radius of the sphere, respectively) of the silicon spheres under the point load test, but very high tensile strain concentrations are usually developed near r/R=0.9. In addition, based on quantum mechanics and energy band theory, the effect of strain on three quantum valence-bands of silicon is analyzed. Numerical results show that the large strain induced at r/R=0.9 of the sphere under the point load test significantly breaks some symmetric property of quantum valence bands of silicon, and has obvious effect on the constant energy surfaces and the conductivity effective masses of heavy hole (HH) band, the light hole (LH) band and the split-off (SO) band, which is closely related to optical and electric property of silicon.