Elastic properties of diamond, silicon, and germanium nanocrystals as functions of their size and shape

The dependence of elastic modulus $B$ on the size (number of atoms $(N)$) and the shape of a nanocrystal of a simple monoatomic substance is studied when the nanocrystal is considered as a rectangular parallelepiped with a varied surface shape. The elastic modulus is shown to decrease during an isomorphic-isothermal decrease of the nanocrystal size. At low temperatures, the $B(N)$ dependence is less pronounced, and the case where the $B(N)$ function increases during an isomorphic-isothermal decrease of the nanocrystal size is possible here. The size dependences of the elastic modulus, Poisson ratio $\mu$, Young’s modulus $Y$, shear modulus $G$, and lattice parameter compression are calculated for diamond, Si, and Ge. It is shown that $B$, $Y$, and $G$ decrease and $\mu$ increases during an isomorphic-isothermal decrease of the nanocrystal size. The surface pressure compresses a nanocrystal at low temperatures and expands it at high temperatures. The larger the deviation of the nanocrystal shape from the most energetically favorable shape, the more pronounced the changes of these functions during an isothermal decrease of the nanocrystal size.