Strength, elasticity, and equation of state of the nanocrystalline cubic silicon nitrideγ−Si3N4to68GPa

Abstract

Lattice strains in nanocrystalline cubic silicon nitride were measured using an energy-dispersive x-ray diffraction technique under nonhydrostatic stress conditions up to a confining pressure of $68\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. The high-pressure elastic properties of $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ were also investigated theoretically using density-functional theory. The differential stress $t$ between 30 and $68\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ increases from $7\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}23\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and can be described beyond $40\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ as $t=7(4)+0.24(7)P$ where $P$ is the pressure in GPa. The differential stress supported by $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ increases with pressure from 3.5% of the shear modulus at $21\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ to 7.6% at $68\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ is one of the strongest materials yet studied under extreme compression conditions. The elastic anisotropy of $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ is large and only weakly pressure dependent. The elastic anisotropy increases from $A=1.4$ to $A=1.9$ as the parameter $\ensuremath{\alpha}$ that characterizes stress-strain continuity across grain boundaries is decreased from 1 to 0.5. The high elastic anisotropy compares well with our first-principles calculations that lead to $A=1.92--1.93$ at ambient pressure and $A=1.94--1.95$ at $70\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. Using molybdenum as an internal pressure standard, the equation of state depends strongly on $\ensuremath{\psi}$, the direction between the diamond cell axis and the normal of the scattering plane. The bulk modulus increases from $224(3)\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}460(13)\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ as $\ensuremath{\psi}$ varies from 0\ifmmode^\circ\else\textdegree\fi{} to 90\ifmmode^\circ\else\textdegree\fi{}. This large variation highlights the need to account properly for deviatoric stresses in nonhydrostatic x-ray diffraction experiments carried out at angles other than the particular angle of $\ensuremath{\psi}=54.7\ifmmode^\circ\else\textdegree\fi{}$, where deviatoric stress effects on the lattice vanish. At this angle we find a bulk modulus of $339(7)\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ (${K}_{0}^{\ensuremath{'}}=4$, fixed). This result is in general agreement with our local density approximation calculations, ${K}_{0}=321\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, ${K}_{0}^{\ensuremath{'}}=4.0$, and previous shockwave and x-ray diffraction studies. However, our results are significantly lower than the recently reported bulk modulus of ${K}_{0}=685(45)\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ for nanocrystalline $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ below $40\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$.