Youngʹs modulus of silicon nanoplates at finite temperature

Based on the Keating model, a semi-continuum approach is developed in which the strain energy of silicon nanoplates is presented. Using the quasiharmonic approximation, the temperature dependence of the lattice parameter of silicon has been coupled into the semi-continuum approach. By considering (2 × 1) surface reconstruction of the silicon nanoplate, Youngʹs moduli at finite temperature are modeled and the surface effects on the mechanical properties of the silicon nanoplate are predicted. As the nanoplate thickness is scaled down to 100 nm, Youngʹs moduli begin to deviate from that of the bulk silicon. It is interesting to note that Youngʹs moduli exhibits opposite behavior with and without surface reconstruction. Without surface reconstruction, Youngʹs modulus of the nanoplate decreases dramatically as the nanoplate is scaled down to several tens of nanometer, which means that the nanoplate is elastically softer than bulk. The surface reconstruction leads to stronger bonds and hence an increase in the Youngʹs modulus of the material as it is scaled down, which makes the nanoplate stiffer along the [1 0 0] direction. Youngʹs modulus of the nanoplate exhibits a negative temperature coefficient.