Elsevier

Solid State Communications

Volume 166, July 2013, Pages 70-75
Solid State Communications

Fast-track Communication
Strain-engineering of graphene's electronic structure beyond continuum elasticity

https://doi.org/10.1016/j.ssc.2013.05.002Get rights and content

Highlights

  • New first-order approach to strain-engineering of graphene's electronic structure.

  • No continuous displacement field required.

  • No K-point dependent gauges field on a first-order theory.

  • Method to determine if mechanical strain varies smoothly at each unit cell.

  • Strain-generated fields and local density of electronic states provided.

Abstract

We present a new first-order approach to strain-engineering of graphene's electronic structure where no continuous displacement field u(x,y) is required. The approach is valid for negligible curvature. The theory is directly expressed in terms of atomic displacements under mechanical load, such that one can determine if mechanical strain is varying smoothly at each unit cell, and the extent to which sublattice symmetry holds. Since strain deforms lattice vectors at each unit cell, orthogonality between lattice and reciprocal lattice vectors leads to renormalization of the reciprocal lattice vectors as well, making the K and K points shift in opposite directions. From this observation we conclude that no K-dependent gauges enter on a first-order theory. In this formulation of the theory the deformation potential and pseudo-magnetic field take discrete values at each graphene unit cell. We illustrate the formalism by providing strain-generated fields and local density of electronic states on graphene membranes with large numbers of atoms. The present method complements and goes beyond the prevalent approach, where strain engineering in graphene is based upon first-order continuum elasticity.

Introduction

The interplay between mechanical and electronic effects in carbon nanostructures has been studied for a long time (e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]). The mechanics in those studies invariably enters within the context of continuum elasticity. One of the most interesting predictions of the theory is the creation of large, and roughly uniform pseudo-magnetic fields and deformation potentials under strain configurations having a three-fold symmetry [2]. Those theoretical predictions have been successfully verified experimentally [12], [13].

Nevertheless, different theoretical approaches to strain engineering in graphene possess subtle points and apparent discrepancies [6], [14], which may hinder progress in the field. This motivated us to develop an approach [15] which does not suffer from limitations inherent to continuum elasticity. This new formulation accommodates numerical verifications to determine when arbitrary mechanical deformations preserve sublattice symmetry. Contrary to the conclusions of Ref. [14], with this formulation one can also demonstrate in an explicit manner the absence of K-point dependent gauge fields on a first-order theory (see Refs. [15], [16], [17] as well). The formalism takes as its only direct input raw atomistic data—as the data obtained from molecular dynamics runs. The goal of this paper is to present the method, making the derivation manifest. We illustrate the formalism by computing the gauge fields and the density of states in a graphene membrane under central load.

The theory of strain-engineered electronic effects in graphene is semi-classical. One seeks to determine the effects of mechanical strain across a graphene membrane in terms of spatially modulated pseudospin Hamiltonians Hps; these pseudospin Hamiltonians Hps(q) are low-energy expansions of a Hamiltonian formally defined in reciprocal space. Under “long range” mechanical strain (extending over many unit cells and preserving sublattice symmetry [1], [2], [3]) Hps also become continuous and slowly varying local functions of strain-derived gauges, so that HpsHps(q,r). Within this first-order approach, the salient effect of strain is a local shift of the K and K points in opposite directions, similar to a shift induced by a magnetic field [2], [3]. In the usual formulation of the theory [1], [2], [3], [4], [5], [6], this dependency on position leads to a continuous dependence of strain-induced fields Bs(r) and Es(r). Such continuous fields are customarily superimposed to a discrete lattice, as in Fig. 1 [18].

When expressed in terms of continuous functions, a pseudospin Hamiltonian Hps is defined down to arbitrarily small spatial scales and it spans a zero area. In reality, however, the pseudospin Hamiltonian can only be defined per unit cell, so it should take a single value at an area of order a02 (a0 is the lattice constant in the absence of strain).

This observation tells us already that the scale of the mechanical deformation with respect to a given unit cell is inherently lost in a description based on a continuum model. For this reason, it is important to develop an approach which is directly related to the atomic lattice, as opposed to its idealization as a continuum medium. In the present paper we show that in following this program one gains a deeper understanding of the interrelation between the mechanics and the electronic structure of graphene. Indeed, within this approach we are able to quantitatively analyze whether the proper phase conjugation of the pseudospin Hamiltonian holds at each unit cell. The approach presented here will give (for the first time) the possibility to explicitly check on any given graphene membrane under arbitrary strain if mechanical strain varies smoothly on the scale of interatomic distances. Consistency in the present formalism will also lead to the conclusion that in such scenario strain will not break the sublattice symmetry but the Dirac cones at the K and K points will be shifted in the opposite directions [2], [3].

Clearly, for a reciprocal space to exist one has to preserve crystal symmetry. When crystal symmetry is strongly perturbed, the reciprocal space representation starts to lack physical meaning, which presents a limitation to the semiclassical theory. The lack of sublattice symmetry – observed on actual unit cells on this formulation beyond first-order continuum elasticity – may not allow proper phase conjugation of pseudospin Hamiltonians at unit cells undergoing very large mechanical deformations. Nevertheless this check cannot proceed – and hence has never been discussed – on a description of the theory within a continuum media, because by construction there is no direct reference to actual atoms on a continuum.

As it is well-known, it is also possible to determine the electronic properties directly from a tight-binding Hamiltonian H in real space, without resorting to the semiclassical approximation and without imposing an a priori sublattice symmetry. That is, while the semiclassical Hps(q,r) is defined in reciprocal space (thus assuming some reasonable preservation of crystalline order), the tight-binding Hamiltonian H in real space is more general and can be used for membranes with arbitrary spatial distribution and magnitude of the strain.

In addition, contrary to the claim of Ref. [14], the purported existence of K-point dependent gauge fields does not hold on a first-order formalism [15], [16]. What we find instead, is a shift in opposite directions of the K and K points upon strain [2].

Section snippets

Sublattice symmetry

The continuum theories of strain engineering in graphene, being semiclassical in nature, require sublattice symmetry to hold [1], [2]. One the other hand, no measure exists in the continuum theories [1], [2], [3], [4], [5], [6] to test sublattice symmetry on actual unit cells under a mechanical deformation. For this reason, sublattice symmetry is an implicit assumption embedded in the continuum approach.

To address the problem beyond the continuum approach, let us start by considering the unit

Applying the formalism to rippled graphene membranes

We finish the present contribution by briefly illustrating the formalism on two experimentally relevant case examples. The developments presented here are motivated by recent experiments where freestanding graphene membranes are studied by local probes [22], [23], [24].

Conclusions

We presented a novel framework to study the relation between mechanical strain and the electronic structure of graphene membranes. Gauge fields are expressed directly in terms of changes in atomic positions upon strain. Within this approach, it is possible to determine the extent to which the sublattice symmetry is preserved. In addition, we find that there are no K-dependent gauge fields in the first-order theory. We have illustrated the method by computing the strain-induced gauge fields on a

Acknowledgments

We acknowledge conversations with B. Uchoa, and computer support from HPC at Arkansas (RazorII), and XSEDE (TG-PHY090002, Blacklight, and Stampede). M.V. acknowledges support by the Serbian Ministry of Science, Project no. 171027.

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