3D Strain in 2D Materials: To What Extent is Monolayer Graphene Graphite?

This work addresses a fundamental question: To what extent is graphene graphite? In particular does 2D graphene have many of the same 3D mechanical properties as graphite, such as the bulk modulus and elastic constant $c_{33}$? We have obtained, for the first time, unambiguous Raman spectra from unsupported monolayer graphene under pressure. We have used these data to quantify the out-of-plane stiffness of monolayer graphene, which is hard to define due to its 2D nature. Our data indicate a first physically meaningful out-of-plane stiffness of monolayer graphene, and find it to be consistent with that of graphite. We also report a shift rate of the in-plane phonon frequency of unsupported monolayer graphene to be $5.4{\mathrm{cm}}^{-1}{\mathrm{GPa}}^{-1}$, very close to that of graphite ($4.7{\mathrm{cm}}^{-1}{\mathrm{GPa}}^{-1}$), contrary to the previous value for supported graphene. Our results imply that monolayer graphene has similar in-plane and out-of-plane stiffnesses, and anharmonicities to graphite.

Figure 1

The GM frequency of graphene in samples $D$ (black squares) and $P$ (orange diamonds) is plotted vs pressure. The data of graphite [9] (blue triangles) are also plotted with the theoretical line of Eq. (4) (blue dashed line) for comparison. The uncertainty in both pressure and frequency is plotted where it exceeds the size of a data point.

Figure 2

The second derivative of the GM frequency to pressure (black dots) of the graphene in sample $D$ is plotted vs pressure, with the optimal fit (blue dashed line) by the second derivative of the Eq. (4) to pressure. Two curves (orange dashed lines) by increasing or decreasing the value of $c_{33}$ by 100 times are plotted to be compared with the optimal fit.

Figure 3

The GM frequency of the graphene in sample $D$ is plotted vs pressure. These data points, taken from Fig. 1, have a small uncertainty in pressure, implying good hydrostaticity. The optimal empirical fit of the first four points (unsupported graphene) is shown in blue dashed line.

Figure 4

The GM frequency of the graphene in sample $P$ is plotted vs pressure. These data points, taken from Fig. 1, have only small uncertainties in pressure, implying good hydrostaticity. The fit of the first four points over the similar range to the fit in Fig. 3 is shown as a blue dashed line.