Finite element modeling of high-pressure deformation and phase transformation of silicon beneath a sharp indenter

Abstract

Modeling of the mechanical response of single crystalline silicon to a sharp indenter is an essential step for the optimization of wafer manufacturing processes. In this paper, deformation of silicon during indenter loading and unloading was analyzed by the finite element method, and the changes of stress field and high-pressure phase distribution were dynamically simulated. We found that the deformation of silicon in nanoindentation can be simply characterized by two factors: one is the elastic strain of each high-pressure phase and the other is the equivalent elastic strain of each phase transformation. In loading, indentation energy is absorbed mostly by phase transformation, and accumulated as the elastic strain of the high-pressure phases. The distribution pattern of the high-pressure phases beneath the indenter is independent of the indentation load, and the depth of the phase-transformed region is approximately twice the indentation depth. In unloading, high-pressure phases except the β-Sn phase undergo reverse phase transformations. The β-Sn phase does not transform back to the diamond phase but changes to other non-equilibrium phases, which becomes the dominant reason for residual strain. During unloading, the non-equilibrium phase expands from the diamond phase region toward the indenter tip, while the boundary between the non-equilibrium phase and the diamond phase remains unchanged. The unloading mechanism is independent of the change in the maximum indentation load and the presence/absence of pop-out events.