Interplay between nanometer-scale strain variations and externally applied strain in graphene

We present a molecular modeling study analyzing nanometer-scale strain variations in graphene as a function of externally applied tensile strain. We consider two different mechanisms that could underlie nanometer-scale strain variations: static perturbations from lattice imperfections of an underlying substrate and thermal fluctuations. For both cases we observe a decrease in the out-of-plane atomic displacements with increasing strain, which is accompanied by an increase in the in-plane displacements. Reflecting the nonlinear elastic properties of graphene, both trends together yield a nonmonotonic variation of the total displacements with increasing tensile strain. This variation allows us to test the role of nanometer-scale strain variations in limiting the carrier mobility of high-quality graphene samples.

Figure 1

Stress-strain curves for a graphene sheet under tensile external strain up to 15%. The linear behavior expected for a material with constant Young's modulus of 1.02 TPa [1] (blue curve) is compared to calculated data for strain applied along the zigzag (red curves) and along the armchair (black curves) directions. The data obtained with the empirical LCBOP potential (solid lines) matches the one obtained by first-principles DFT calculations (dotted lines) within 10%.

Figure 2

Illustration of the simulation cell and the applied Gaussian potential. The Gaussian potential has a width of three interatomic carbon-carbon bonds and a strength of +90 meV. (a) Top view of the central area of the graphene sheet with all individual C atoms overlayed with a color map of the Gaussian potential that is applied in the center of the sheet. (b) Force induced by the Gaussian potential on the C atoms of an ideal sheet and along a line at $y=50\AA$ (red = in-plane force, gray = out-of-plane force, lines are superimposed to guide the eye).

Figure 3

Calculated average displacements under the influence of a static Gaussian potential. Shown as a function of applied equibiaxial tensile strain are the total displacement $d_{\mathrm{tot}}$ (black curves), the in-plane displacement $d_{\mathrm{ip}}$ (red curves), and the out-of-plane displacement $d_{\mathrm{oop}}$ (green curves). The upper panel corresponds to a strength of the Gaussian potential of 10 meV, the middle panel to 50 meV, and the lower panel to 90 meV. In all panels, solid curves correspond to a repulsive potential and dotted curves to an attractive potential. The arrows indicate the minimum in total displacement $d_{\mathrm{tot}}$.

Figure 4

(a) Top view of the graphene sheet depicting the spatial distribution of the atomic displacements under the influence of a Gaussian potential and for equibiaxial tensile strain of 1% (left panels) and 10% (right panels). Compared are the in-plane displacements (upper panels) and the out-of-plane displacements (lower panels). The Gaussian potential of strength 90 meV is centered in the middle of the sheet and the discrete displacement data of the individual C atoms in the sheet has been interpolated to a continuous color map for easier visualization. (b) Relative change in carbon-carbon bond length as a function of the distance to the center where the Gaussian potential is applied. Shown are data for three tensile strain values of 1%, 5%, and 10%. The cyan line is a guide to the eye.

Figure 5

Calculated relative displacements with respect to the atomic positions of the graphene sheet with a Gaussian potential of 90 meV at zero applied strain. The panels show from top to bottom the $x,y$, and $z$ component at 1% and 10% of applied strain. In agreement with Figs. 3 and 4, the in-plane displacements increase with applied strain, whereas the out-of-plane displacement decreases.

Figure 6

Calculated discrete Gaussian curvature $K_{\mathrm{D}}$ and discrete mean curvature $H_{\mathrm{D}}$ for a Gaussian potential of 90 meV at 1% and 10% of applied strain. Both quantities decrease with at least one order of magnitude when increasing the applied strain from 1% to 10%.

Figure 7

(a) Calculated average displacements under the influence of a static Gaussian potential of strength 90 meV. Shown as a function of applied uniaxial tensile strain along the zigzag (left panel) and armchair (right panel) direction are the total displacement $d_{\mathrm{tot}}$ (black curves), the in-plane displacement $d_{\mathrm{ip}}$ (red curves), and the out-of-plane displacement $d_{\mathrm{oop}}$ (green curves). In both panels, solid curves correspond to a repulsive potential and dotted curves to an attractive potential. (b) Top views of the graphene sheet depicting the corresponding spatial distribution of the atomic displacements under 10% strain. Compared are the in-plane displacements (upper panels) and the out-of-plane displacements (lower panels) for uniaxial strain along the armchair (left panels) and zigzag (middle panels) direction, as well as equibiaxial strain (right panels).

Figure 8

Calculated time-average displacements as a function of strain from MD simulations at 300 K. Results are shown for a side length $L$ of 2.5 nm (left panel) and 32 nm (middle panel). In agreement with the MM simulations, $d_{\mathrm{oop}}$ decreases with increasing strain, whereas $d_{\mathrm{ip}}$ increases. The right panel shows how the average out-of-plane displacement scales with the side length $L$ for different values of strain: 0% (black), 0.5% (red), 1% (green), 3% (blue), 5% (cyan), 10% (magenta). The slope of the fitted lines yields the critical exponent of membrane theory, see text.