Density functional theory has been used to investigate the behavior of the $\pi$ 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 $\pi$-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 $\pi$ 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.

中文翻译：

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双层石墨烯的意外柔软性和AA堆叠石墨烯层的软化

密度泛函理论已被用来调查行为 $\pi$ 双层石墨烯和石墨中的电子在压缩时沿着 $C$轴。我们已经研究了石墨烯层的常规Bernal（AB）和AA堆叠。在双层石墨烯中，仅约0.5％$\pi$-电子密度通过 $s{p}^2$网络，无论堆叠顺序如何，压缩率为20％。但是，这会产生很大的影响，导致双层石墨烯的软度约为石墨的六倍。$C$轴。沿压缩$\mathit{C}$ 轴，严重变形的电子轨道（主要是那些 $\pi$电子）增加了石墨烯层之间的层间相互作用，这是预期的，但令人惊讶的是，AA和伯纳尔堆叠的程度相似。另一方面，这种压缩使AA堆叠的石墨烯层的面内声子频率发生了显着变化，与伯纳尔堆叠的层有很大不同。我们将这些结果归因于$s{p}^2$ 在压缩状态下，AA中的电子堆叠逃逸出石墨烯平面并填充较低的电荷密度区域，因此导致电子的非单调变化 $s{p}^2$粘结刚度。