Ideal strength of doped graphene

While the mechanical distortions change the electronic properties of graphene significantly, the effects of electronic manipulation on its mechanical properties have not been known. Using first-principles calculation methods, we show that when graphene expands isotropically under equibiaxial strain, both the electron and hole doping can maintain or improve its ideal strength slightly and enhance the critical breaking strain dramatically. Contrary to the isotropic expansions, the electron doping decreases the ideal strength as well as critical strain of uniaxially strained graphene while the hole doping increases both. Distinct failure mechanisms depending on type of strains are shown to be origins of the different doping induced mechanical stabilities. Our findings may resolve a contradiction between recent experimental and theoretical results on the strength of graphene.

Figure 1

(a) The total energy variation as a function of $A_1^{\prime }$ phonon amplitude under equibiaxial strains of $\epsilon =0.12$, 0.14, 0.15, and 0.16 from top to bottom lines. Inset shows schematic atomic displacements (red arrows) of the $A_1^{\prime }$ phonon mode and the $K$-point supercell (blue arrows). (b) and (c) show the same curves as in (a) at a fixed strain $\epsilon =0.16$, with changing electron and hole doping from top to bottom lines, $n_e\left(n_h\right)=0.04$, 0.02, 0.01, 0.0, respectively.

Figure 2

(a) and (b) show the stress-strain relationship of graphene under equibiaxial strains with different electron and hole doping levels, respectively. The equibiaxial strain is defined by ${\sigma }_{\mathrm{eb}}\equiv \sqrt{{\sigma }_x^2+{\sigma }_y^2}/\sqrt{2}$. The variations (in percentage) of (c) ideal strength ($\delta {\sigma }_{\mathrm{ideal}}$) and (d) corresponding critical breaking strain ($\delta {\epsilon }_{\mathrm{critical}}$) of doped graphene with respect to those of the undoped one as a function of doping.

Figure 3

(a) Phonon dispersion of undoped graphene with equibiaxial strain $\epsilon =0$ (red lines) and 0.148 (blue). (b) Same as in (a) except for doped graphene ($n_e=0.04$) with $\epsilon =0$ (red) and 0.167 (blue). In both panels, the black arrows indicate the Kohn anomaly and the insets show the vectors (black arrows) connecting Fermi surfaces (red lines) related to the Kohn anomaly. Here we do not show irrelevant out-of-plane phonon mode spectrum. (c) $\delta {\sigma }_{\mathrm{ideal}}$ and $\delta {\epsilon }_{\mathrm{critical}}$ (left and right ordinates) obtained from phonon dispersions as a function of electron doping $n_e$ (abscissa).

Figure 4

Stress-strain curves for graphene under (a) zigzag (${\epsilon }_x$) and (b) armchair (${\epsilon }_y$) strain with various doping levels. The variations (in percentage) of (c) ideal strength and (d) critical breaking strain as a function of the doping level for graphene under the ${\epsilon }_x$ (open rectangles) and ${\epsilon }_y$ (solid ones), respectively.

Figure 5

Electronic band structures of graphene under (a) the zigzag strain of ${\epsilon }_x=0.22$ with $n_e=0.05$, (b) ${\epsilon }_x=0.25$ without doping, and (c) ${\epsilon }_x=0.30$ with $n_h=0.05$. The red and black lines are the ${\sigma }^{*}$ and $\pi \left({\pi }^{*}\right)$ bands, respectively, and the blue is the Fermi level. The inset in (a) shows the high symmetry points of BZ of graphene with ${\epsilon }_x$.