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 $\pi$ 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.

Figure 1

The uniaxial stress along the $c$ axis applied to A-A and Bernal stacked bilayer graphene and graphite is plotted with the corresponding interlayer distance at which the stress was calculated. The black solid points are for graphite, and the blue open circles are for bilayer graphene. Circles are for Bernal stacking, and squares are for A-A. The data were fitted by Eq. (1), and the fitting results are presented.

Figure 2

The integrated valence charge between the two graphene layers is plotted versus the interlayer distance of the bilayer graphene for A-A (blue open circles) and Bernal stacking (black solid squares). The interlayer distance at equilibrium for each stacking is arrowed. We also plot the calculated equilibrium spacing for Bernal stacked graphite (blue open triangle). The values of $c_{33}$ from the fittings in Fig. 1 at a few key points are presented. The difference between Bernal stacked bilayer graphene and graphite for the elastic constant $c_{33}$ when both have $8\text{−}e/\mathrm{unit}\mathrm{cell}$ valence charge between the graphene bilayers, indicates that the $c_{33}$ is not entirely determined by the amount of interlayer electronic charge.

Figure 3

Hydrostatic pressure was applied to A-A and Bernal stacked graphite. (a) The calculated pressure is plotted versus the ratio of the unit-cell volume to the unstrained volume (A-A stacking in black solid squares and Bernal stacking in blue open circles). Fitting using the Murnaghan equation is shown as dashed lines, black for A-A stacking and blue for Bernal stacking. (b) The frequencies of the LBM for A-A and Bernal stacked graphite are plotted versus hydrostatic pressure (A-A stacking in black solid squares and Bernal stacking in blue open circles). The data are fitted using Eq. (2) as dashed lines, black for A-A stacking and blue for Bernal stacking. (c) The frequencies of the four in-plane phonon modes (as labeled) for A-A stacked graphite are plotted versus pressure. We use the same notation to label the four phonon modes as for Bernal stacked graphite to be consistent, despite the symmetry being different. $L$ is for longitudinal modes, and $T$ is for transverse modes.

Figure 4

Uniaxial stress along the $c$ axis was applied to A-A and Bernal stacked graphite. (a) The frequency of the LBM for A-A stacked graphite (black solid squares) is plotted versus the interlayer distance. The data are fitted by Eq. (3) in dashed lines. The published data and the fit for Bernal stacked graphite is presented for comparison [31]. (b) The frequencies of the four in-plane phonon modes (as labeled) for A-A stacked graphite are plotted versus the interlayer distance under uniaxial stress. We use the same notation to label the phonon modes as those four in Bernal stacked graphite to be consistent, despite the symmetry being different. $L$ is for the longitudinal modes, and $T$ is for the transverse modes.

Figure 5

The total valence charge density of bilayer graphene is plotted along the (110) plane for (a) Bernal and (b) A-A stackings. The color scale is labeled. Positions of atomic cores and $s{p}^2$ bonds are marked. The charge density of the 10% uniaxially compressed bilayer graphene is subtracted from that of the unstrained bilayer when we overlap their bottom layers, and it is then plotted for (c) Bernal and (d) A-A stackings. The contours are separated by $0.0001e/{\AA }^3$. The (110) graphene plane through the carbon atoms is plotted in the middle of each plot. Atomic cores are labeled as black dots. We plot the charge density about 1 Å above and below the graphene plane. Therefore, (c) and (d) are at over twice the magnification of (a) and (b).