Graphene's morphology and electronic properties from discrete differential geometry

The geometry of two-dimensional crystalline membranes dictates their mechanical, electronic, and chemical properties. The local geometry of a surface is determined from the two invariants of the metric and the curvature tensors. Here we discuss those invariants directly from atomic positions in terms of angles, areas, and vertex and normal vectors from carbon atoms on the graphene lattice, for arbitrary elastic regimes and atomic conformations, and without recourse to an effective continuum model. The geometrical analysis of graphene membranes under mechanical load is complemented with a study of the local density of states (LDOS), discrete induced gauge potentials, velocity renormalization, and nontrivial electronic effects originating from the scalar deformation potential. The asymmetric LDOS is related to sublattice-specific deformation potential differences, giving rise to the pseudomagnetic field. The results here enable the study of geometrical, mechanical, and electronic properties for arbitrarily shaped graphene membranes in experimentally relevant regimes without recourse to differential geometry and continuum elasticity.

Figure 1

(a) Top view of polyhedra used to determine the four geometrical invariants from the metric and curvature. Circles represent atoms on the A sublattice. Local lattice vectors are ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$; ${\theta }_i$ are internal angles to edges ${\mathbf{e}}_i$ and ${\mathbf{e}}_{i+1}$; and the central shaded hexagon is the Voronoi cell. (b) Side view highlights the differences between continuum and discrete vector fields. (c) (i) $\sqrt{det\left(g\right)}$, (ii) $\text{Tr}\left(g\right)$, (iii) $K$, and (iv) $H$ for a smooth Gaussian bump where discrete and continuum results coincide. Percent differences $\sqrt{det\left(\tilde{g}\right)}-\sqrt{det\left(g\right)}$ and $\text{Tr}\left(\tilde{g}\right)-\text{Tr}\left(g\right)$ are also shown.

Figure 2

(a) Creation of ripples by cutting a square with side $L=0.27\mu \text{m}$ at 1 K: The membrane trades a planar configuration for a rippled one. (b) Geometrical invariants within the dashed square shown in (a).

Figure 3

(a) Graphene under load by a semispherical tip of radius $r_t=15$ Å. (b) Geometrical invariants as the indentation proceeds. Under load, the metric increases are unbounded, yet curvatures can only saturate to $K_t$ and $H_t$ as graphene conforms to the tip (see the flat horizontal lines $K_D=K_t$ and $H_D=H_t$, for $0\le r\lesssim r_t$ at $z_0=-100$ and $-215$ Å). Compare the trends with those in Fig. 1.

Figure 4

(a) The finite-difference curl leading to the pseudomagnetic field $B_s$ [Eq. (4)] is obtained from hoppings among an atom on the A sublattice, and three neighboring atoms on B sublattices. (b) $B_s$ for $z_0=-100$ and $-215$ Å loads. (c) LDOS with screened values of the deformation potential $E_s$ at $r=0$, for $z_0=-100$ and $-215$ Å.

Figure 5

A-sublattice LDOS along the (a) $-{30}^{\circ }$ and (b) $+{30}^{\circ }$ axes ($z_0=-100$ Å). Angular sweeps at $r=75$ Å on the (c) A and (d) B sublattices ($z_0=-100$ Å): Due to time-reversal symmetry, a sign change in $B_s$ in (b) has the same effect as a sublattice exchange (c), (d).