Nanoindentation of silicon and structural transformation: Three-dimensional contact theory

Our previous analysis of the stress-pressure phase diagram of silicon [Hebbache, Mattesini, and Szeftel, Phys. Rev. B 63, 205201 (2001)] is extended to study the diamond to β-Sn phase transition that occurs in this material during indentation experiments. We combined the elastic theory of phase transitions and the three-dimensional Hertzian contact theory of anisotropic materials. The stress dependence of the harmonic elastic coefficients, usually omitted in contact theories, is taken into account explicitly. The calculated second-order elastic constants of the stressed cubic and β-Sn phases are given. The shear and the hydrostatic components of the contact pressure beneath the indenter are calculated. Comparison with the experimental results of Williams et al. is made. The spatial variations of the spontaneous tetragonal strain that distorts the cubic phase of Si and the extension of β-Sn phase beneath the indenter are shown.