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High-fluence femtosecond laser pulses can induce physical and chemical changes in materials that are unrealizable under standard laboratory conditions. The exact nature of these changes can depend strongly on the gaseous environment in which the material is irradiated since near-surface chemical reactions can occur between the two materials. Surface modifications of silicon are of particular interest due to its significance in semiconductor-based applications. Specifically, the formation of silicon nitride (Si3N4) structures is desirable for multiple applications due to its high stability and low dielectric constant. Herein, we report on femtosecond laser-induced morphological and chemical modifications of silicon in a nitrogen atmosphere. We observed an extremely fast chemical reaction in the silicon-nitrogen system. The presence of crystalline Si3N4 was confirmed using high-resolution transmission electron microscopy, representing the first reported synthesis of Si3N4 nanocrystals through femtosecond laser-based methods. In addition, the surface was found to contain alternating layers of amorphous and crystalline silicon. Provided are plausible mechanisms for the formation of each of these structures. Taken together, these findings on surface modification of silicon using femtosecond laser irradiation may provide new pathways for manufacturing of nanoscale devices. 

Abstract A preceding 2023 study argued that the resistance of a heterogeneous material to the curvature of the displacement field is the most physically realistic localization limiter for softening damage. The curvature was characterized by the second gradient of the displacement vector field, which includes the material rotation gradient, and was named the “sprain” tensor, while the term “spress” is here proposed as the force variable work-conjugate to “sprain.” The partial derivatives of the associated sprain energy density yielded in the preceeding study, sets of curvature resisting self-equilibrated nodal sprain forces. However, the fact that the sprain forces had to be applied on the adjacent nodes of a finite element greatly complicated the programming and extended the simulation time in a commercial code such as abaqus by almost two orders of magnitude. In the present model, Smooth Lagrangian Crack Band Model (slCBM), these computational obstacles are here overcome by using finite elements with linear shape functions for both the displacement vector and for an approximate displacement gradient tensor. A crucial feature is that the nodal values of the approximate gradient tensor are shared by adjacent finite elements. The actual displacement gradient tensor calculated from the nodal displacement vectors is constrained to the approximate displacement gradient tensor by means of a Lagrange multiplier tensor, either one for each element or one for each node. The gradient tensor of the approximate gradient tensor then represents the approximate third-order displacement curvature tensor, or Hessian of the displacement field. Importantly, the Lagrange multiplier behaves as an externally applied generalized moment density that, similar to gravity, does not affect the total strain-plus-sprain energy density of material. The Helmholtz free energy of the finite element and its associated stiffness matrix are formulated and implemented in a user’s element of abaqus. The conditions of stationary values of the total free energy of the structure with respect to the nodal degrees-of-freedom yield the set of equilibrium equations of the structure for each loading step. One- and two-dimensional examples of crack growth in fracture specimens are given. It is demonstrated that the simulation results of the three-point bend test are independent of the orientation of a regular square mesh, capture the width variation of the crack band, the damage strain profile across the band, and converge as the finite element mesh is refined.