Unexpected softness of bilayer graphene and softening of A-A stacked graphene layers

Density functional theory has been used to investigate the behavior of the $\ensuremath{\pi}$ electrons in bilayer graphene and graphite under compression along the $c$ axis. We have studied both conventional Bernal (A-B) and A-A stackings of the graphene layers. In bilayer graphene, only about 0.5% of the $\ensuremath{\pi}$-electron density is squeezed through the $s{p}^{2}$ network for a compression of 20%, regardless of the stacking order. However, this has a major effect, resulting in bilayer graphene being about six times softer than graphite along the $c$ axis. Under compression along the $\mathit{c}$ axis, the heavily deformed electron orbitals (mainly those of the $\ensuremath{\pi}$ electrons) increase the interlayer interaction between the graphene layers as expected, but, surprisingly, to a similar extent for A-A and Bernal stackings. On the other hand, this compression shifts the in-plane phonon frequencies of A-A stacked graphene layers significantly and very differently from the Bernal stacked layers. We attribute these results to some $s{p}^{2}$ electrons in A-A stacking escaping the graphene plane and filling lower charge-density regions when under compression, hence, resulting in a nonmonotonic change in the $s{p}^{2}$-bond stiffness.