Strain gauge fields for rippled graphene membranes under central mechanical load: An approach beyond first-order continuum elasticity

We study the electronic properties of rippled freestanding graphene membranes under central load from a sharp tip. To that end, we develop a gauge field theory on a honeycomb lattice valid beyond the continuum theory. Based on the proper phase conjugation of the tight-binding pseudospin Hamiltonian, we develop a method to determine conditions under which continuum elasticity can be used to extract gauge fields from strain. Along the way, we resolve a recent controversy on the theory of strain engineering in graphene: There are no $K$-point-dependent gauge fields. We combine this lattice gauge field theory with atomistic calculations and find that for moderate load, the rippled graphene membranes conform to the extruding tip without a significant increase in elastic energy. Mechanical strain is created on a membrane only after a certain amount of load is exerted. In addition, we find that the deformation potential—even when partially screened—induces qualitative changes on the electronic spectra, with Landau levels giving way to equally spaced peaks.

Figure 1

(a) Unit cell (shaded); lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$; and nearest-neighbor vectors ${\tau }_1$, ${\tau }_2$, and ${\tau }_3$. (b) Breakdown of CBR: The nearest-neighbor vectors for the atoms $A$ and $B$ under load are not necessarily mirror-symmetric. If such vectors preserve sublattice symmetry to a reasonable extent, then the lattice vectors can be renormalized univocally to become ${\mathbf{a}}_1^{\prime }$ and ${\mathbf{a}}_2^{\prime }$, and a theory for strain engineering can be laid out at each and every unit cell.

Figure 2

(a) The process to produce two-dimensional displacements from three-dimensional ones. (b) Locations from which finite differences are computed.

Figure 3

(a) Triangular graphene membranes under load. (b) $l_1$ is the shortest distance from center to edge. (c) Height profiles and (d) increase of the bond lengths.

Figure 4

(a) The elastic energy vs indentation shows three distinct regimes: (i) isometric, due to the initially rippled conformation, (ii) linear (or harmonic), and (iii) nonlinear (anharmonic). (b) Decomposition of the elastic energy for a load $\Gamma =2.0$ nm (shaded area represents the load time).

Figure 5

Representation of Eqs. (4, 5, 6) for our system. Solid lines highlight antisymmetric patterns.

Figure 6

$\Delta z$ vs in-plane distance to extruding tip $d$. $\Delta z\lesssim 0.3$$\%$ for $d>1$ nm.

Figure 7

(a) $E_s$ and (b) $B_s$ ($\Gamma /l_1=35\%$). (c) Radial dependence of the LDOS ($E_s=0$). (d) Evolution of the LDOS at points 1 and 5 in (c) as $E_s$ is gradually turned on. (e) LDOS for $B_s=0$ and 100$\%$ $E_s$.