Strain-Engineering of Graphene's Electronic Structure Beyond

Strain-engineering of graphene’s electronic structure beyond continuum elasticity

Salvador Barraza-Lopez Department of Physics. University of Arkansas. Fayetteville, AR 72701, USA∗

Alejandro A. Pacheco Sanjuan Departamento de Ingenier´ıaMec´anica. Universidad del Norte. Km. 5 V´ıaPuerto Colombia. Barranquilla, Colombia

Zhengfei Wang Department of Materials Science and Engineering. University of Utah. Salt Lake City, UT 84112, USA

Mihajlo Vanević Department of Physics, University of Belgrade. Studentski trg 12, 11158 Belgrade, Serbia (Dated: Available online: 14 May 2013) 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 K0 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.

I. INTRODUCTION (a) (b) (c)

The interplay between mechanical and electronic effects in carbon nanostructures has been studied for a long time (e.g., [1–11]). The mechanics in those studies invariably enters within the context of continuum elasticity. One of the most interesting predictions of the the- −30 300 0 189 6 129 B (Tesla) B (Tesla) B (Tesla) ory is the creation of large, and roughly uniform pseudo- s s s magnetic fields and deformation potentials under strain FIG. 1: Gauge fields from first-order continuum elasticity are conformations having a three-fold symmetry [2]. Those defined regardless of spatial scale. A unit cell is shown in theoretical predictions have been successfully verified ex- (b) and (c) for comparison. In this work, we define the pseu- perimentally [12, 13]. dospin Hamiltonian for each unit cell using space-modulated, Nevertheless, different theoretical approaches to strain low-energy expansions of a tight-binding Hamiltonian in re- engineering in graphene possess subtle points and ap- ciprocal space. As a result, in our approach the gauge fields parent discrepancies [6, 14], which may hinder progress will become discrete. in the field. This motivated us to develop an approach [15] which does not suffer from limitations inherent to runs. The goal of this paper is to present the method, continuum elasticity. This new formulation accommo- making the derivation manifest. We illustrate the for- dates numerical verifications to determine when arbitrary malism by computing the gauge fields and the density of mechanical deformations preserve sublattice symmetry. arXiv:1310.3622v1 [cond-mat.mes-hall] 14 Oct 2013 states in a graphene membrane under central load. Contrary to the conclusions of Ref. [14], with this formulation one can also demonstrate in an explicit man- ner the absence of K−point dependent gauge fields on a first-order theory (see Refs. [15] and [16, 17] as well). A. Motivation The formalism takes as its only direct input raw atomistic data –as the data obtained from molecular dynamics 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 Hamiltoni- ∗ Electronic address: [email protected] ans Hps; these pseudospin Hamiltonians Hps(q) are low- 2 energy expansions of a Hamiltonian formally defined in semiclassical approximation and without imposing an a reciprocal space. Under “long range” mechanical strain priori sublattice symmetry. That is, while the semiclas- (extending over many unit cells and preserving sublattice sical Hps(q, r) is defined in reciprocal space (thus assum- symmetry [1–3]) Hps also become continuous and slowly- ing some reasonable preservation of crystalline order), varying local functions of strain-derived gauges, so that the tight-binding Hamiltonian H in real space is more Hps → Hps(q, r). Within this first-order approach, the general and can be used for membranes with arbitrary salient effect of strain is a local shift of the K and K0 spatial distribution and magnitude of the strain. points in opposite directions, similar to a shift induced In addition, contrary to the claim of Ref. [14], the pur- by a magnetic field [2, 3]. In the usual formulation of ported existence of K−point dependent gauge fields does the theory [1–6], this dependency on position leads to a not hold on a first-order formalism [15, 16]. What we find 0 continuous dependence of strain-induced fields Bs(r) and instead, is a shift in opposite directions of the K and K Es(r). Such continuous fields are customarily superim- points upon strain [2]. posed to a discrete lattice, as in Figure 1 [18]. When expressed in terms of continuous functions, a pseudospin Hamiltonian Hps is defined down to arbitrar- II. THEORY ily small spatial scales and it spans a zero area. In reality, however, the pseudospin Hamiltonian can only be defined A. Sublattice symmetry per unit cell, so it should take a single value at an area 2 of order ∼ a0 (a0 is the lattice constant in the absence of The continuum theories of strain engineering in strain). graphene, being semiclassical in nature, require sublat- This observation tells us already that the scale of the tice symmetry to hold [1, 2]. One the other hand, no mechanical deformation with respect to a given unit cell measure exists in the continuum theories [1–6] to test is inherently lost in a description based on a continuum sublattice symmetry on actual unit cells under a mechan- model. For this reason, it is important to develop an ical deformation. For this reason, sublattice symmetry is approach which is directly related to the atomic lattice, an implicit assumption embedded in the continuum ap- as opposed to its idealization as a continuum medium. proach. In the present paper we show that in following this program one gains a deeper understanding of the interre- (a) No strain: (b) Under mechanical lation between the mechanics and the electronic struc- strain: ture 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 τ1+∆τ1 τ +∆τ first time) the possibility to explicitly check on any given 2 2 graphene membrane under arbitrary strain if mechanical τ2 τ1 a’2 a’1 strain varies smoothly on the scale of interatomic dis- B 2 tances. Consistency in the present formalism will also a τ3 a1 lead to the conclusion that in such scenario strain will τ3+∆τ3 not break the sublattice symmetry but the Dirac cones −(τ1+∆τ'1) −(τ +∆τ ) 0 2 '2 at the K and K points will be shifted in the opposite A directions [2, 3]. Clearly, for a reciprocal space to exist one has to pre- FIG. 2: (a) Definitions of geometrical parameters in a unit serve crystal symmetry, so that when crystal symmetry cell. (b) Sublattice symmetry relates to how pairs of nearest- is strongly perturbed, the reciprocal space representa- neighbor vectors (either in thick, or dashed lines) are mod- 0 tion starts to lack physical meaning, presenting a limi- ified due to strain. These vectors change by ∆τj and ∆τj tation to the semiclassical theory. The lack of sublattice upon strain (j = 1, 2). Relative displacements of neighboring symmetry –observed on actual unit cells on this formu- atoms lead to modified lattice vectors; the choice of renormal- lation beyond first-order continuum elasticity– may not ized lattice vectors will be unique only to the extent to which 0 allow proper phase conjugation of pseudospin Hamilto- sublattice symmetry is preserved: ∆τj ' ∆τj . nians at unit cells undergoing very large mechanical deformations. Nevertheless this check cannot proceed –and To address the problem beyond the continuum ap- hence has never been discussed– on a description of the proach, let us start by considering the unit cell before theory within a continuum media, because by construc- (Fig. 2(a)) and after arbitrary strain has been applied tion there is no direct reference to actual atoms on a (Fig. 2(b)). For easy comparison of our results, we make continuum. the zigzag direction parallel to the x−axis, which is the As it is well-known, it is also possible to determine choice made in Refs. [2] and [5]. (Arbitrary choices of the electronic properties directly from a tight-binding relative orientation are clearly possible; in Ref. [15] we Hamiltonian H in real space, without resorting to the chose the zigzag direction to be parallel to the y-axis.) 3

The lattice vectors before the deformation are given by and shown in Fig. 2(a). The reciprocal lattice vectors T T (Fig. 2(a)): B ≡ (b1 , b2 ) are related to the lattice vectors by [20]: √ √ BT = 2πA−1. (7) a1 = 1/2, 3/2 a0, a2 = −1/2, 3/2 a0, (1)

With the choice we made for a1 and a2 we get: √ ! √ ! 1 2π 1 2π 3 1 a0 3 1 a0 a0 b1 = 1, √ , and b2 = −1, √ . (8) τ 1 = , √ , τ 2 = − , √ , τ 3 = (0, −1) √ . a a 2 2 3 2 2 3 3 3 0 3 0 (2) As seen in Fig. 3(a) the K−points on the first Brillouin (Note that a1 = τ 1 − τ 3, and a2 = τ 2 − τ 3.) zone are defined by: After mechanical strain is applied (Fig. 2(b)), each local pseudospin Hamiltonian will only have physical mean- 2b1 + b2 b1 − b2 b1 + 2b2 K1 = , K2 = , and K3 = − , ing at the unit cells where: 3 3 3 (9) 0 and: ∆τ j ' ∆τ j (j=1,2). (3) K = −K , K = −K , and K = −K . (10) Condition (3) can be re-expressed in terms of changes of 4 1 5 2 6 3 angles ∆αj or lengths ∆Lj for pairs of nearest-neighbor vectors τ and τ 0 [j = 1 is shown in thick solid and j = 2 j j (a) No strain: (b) Under mechanical in thin dashed lines in Fig. 2(b)]: strain: − −∆ +∆ 0 0 b2 −K K b1 b’2 K3 K3 K1 K1 b’1 (τ j +∆τ j )·(τ j +∆τ j ) = |τ j +∆τ j ||τ j +∆τ j | cos(∆αj ), (4) 3 1 0 ˆ +∆ sgn(∆αj ) = sgn [(τ j + ∆τ j ) × (τ j + ∆τ j )] · k , (5) (Κ') − (Κ) −K −∆K K2 K2 K2 Γ K2 2 2 Γ (Κ'−∆K) (Κ+∆K) where kˆ is a unit vector along the z-axis, sgn is the sign − K1 K3 − −∆ +∆ function (sgn(a) = +1 if a ≥ 0 and sgn(a) = −1 if K1 K1 K3 K3 a < 0), and: FIG. 3: First Brillouin zone (a) before and (b) after mechani- 0 ∆Lj ≡ |τ j + ∆τ j | − |τ j + ∆τ j |. (6) cal strain is applied. The reciprocal lattice vectors are shown, as well as the changes of the high-symmetry points at the cor- Even though in the problems of practical interest the ners of the Brillouin zone. Note that independent K points deviations from the sublattice symmetry do tend to be (K and K0) move in the opposite directions. The dashed small [15], it is important to bear in mind that the sub- hexagon in (b) represents the boundary of the first Brillouin lattice symmetry does not hold a priori [2]. It is there- zone in the absence of strain. fore important to have a method to quantify such deviations and check whether the sublattice symmetry holds The relative positions between atoms change when at the problem at hand. Forcing the sublattice symme- strain is applied: τ j → τ j + ∆τ j (j = 1, 2, 3), and 0 try to hold from the start amounts to introducing an −τ j → −τ j − ∆τ j (j = 1, 2). For negligible curvature, artificial mechanical constraint on the membrane which one may assume that ∆τ j · zˆ = ∆zj ∼ 0 (and similar 0 is not justified on physical grounds [19]. For this rea- for the primed displacements ∆τ j). We present here a son the method we propose is discrete and directly re- formulation of the theory strictly valid for in-plane strain lated to the actual lattice; it does not resort to the ap- (it would also be valid for membranes with negligible cur- proximation of the membrane as a continuum medium vature). [1–6, 16, 17]. Being expressed in terms of the actual We wish to find out how reciprocal lattice vectors atomic displacements, our formalism holds beyond the change to first order in displacements under mechani- linear elastic regime where the first-order continuum elas- cal load. In order for reciprocal lattice vectors to make ticity may fail. The continuum formalism is recovered as sense at each unit cell, Eqn. 3 must hold. In terms of a special case of the one presented here in the limit when numerical quantities one would need that ∆αj and ∆Lj 0 |∆τj|/a0 → 0. are all close to zero. In that case we set ∆τ j → ∆τ j for j=1,2, and continue our program. For this purpose we define: B. Renormalization of the lattice and reciprocal lattice vectors ∆a1 ≡ ∆τ 1 − ∆τ 3, and ∆a2 ≡ ∆τ 2 − ∆τ 3, (11) or in terms of (two-dimensional) components: In the absence of mechanical strain, the reciprocal lattice vectors b1 and b2 are obtained by standard methods: ∆τ1x − ∆τ3x ∆τ2x − ∆τ3x T T ∆A ≡ . (12) We define A ≡ (a1 , a2 ), with a1 and a2 given in Eq. (1) ∆τ1y − ∆τ3y ∆τ2y − ∆τ3y 4

The matrix A changes to A0 = A + ∆A, and we must Carrying out explicit calculations, one can see that: modify B so that Eqn. (7) still holds under mechanical load. To ﬁrst order in displacements A0−1 becomes: 3 X iτ j ·Kn e [1 + i(τ j · ∆Kn + ∆τ j · Kn)] = 0. (17) A0−1 = (1 + A∆A)−1(A−1) 'A−1 − A−1∆AA−1. (13) j=1

By comparing Eqns. (7) and (13), the reciprocal lattice For example, at K = K2 we have: vectors in Fig. 3(b) must be renormalized by: 4iπ(∆τ1x + ∆τ2x + ∆τ3x) 2πi πi 3 3 T 1 + (1 + e − e ), ∆B = −2π A−1∆AA−1 . (14) 9a0

We note that the existence of this additional term is quite with phasors adding up to zero. Similar phasor cancela- evident when working directly on the atomic lattice, but tions occur at every other K−point. it was missed in Ref. [14], where the theory was expressed The term linear on ∆Kn on Eqn. 17 cancels out the on a continuum. Let us now calculate some shifts of fictitious K−point dependent gauge fields proposed in Ref. [14], which originated from the term linear on ∆τ the K−points due to strain. For example, K2 (= K in j Fig. 3(a)) requires an additional contribution, which we on this same equation. This observation constitutes yet find by explicit calculation to be: another reason for the formulation of the theory directly on the atomic lattice. With this we have demonstrated 4π ∆τ + ∆τ − 2∆τ that gauges will not depend explicitly on K−points, so ∆K = ∆K = − ∆τ − ∆τ , 1x √2x 3x , 2 2 1x 2x we now continue formulating the theory considering the 3a0 3 K2 point only [2, 3, 5]. and using Eqn. (10) one immediately sees that ∆K0 = Equation (15) takes the following form to first order at 0 −∆K2, so that the K (K2) and K (−K2) points shift K2 in the low-energy regime: in opposite directions, as expected [2, 3]. P3 −iK2·τ j 0 t j=1 ie τ j ·q Hps = P3 iK2·τ j −t j=1 ie τ j ·q 0

C. Gauge fields P3 −iK2·τ j Es,A − j=1 δtj e + P3 iK2·τ j , (18) − j=1 δtj e Es,B Equation (3) gives a condition for which the mechanical strain that varies smoothly on the scale of interatomic with the first term on the right-hand side reducing to distances does not break the sublattice symmetry [2]. On the standard pseudospin Hamiltonian in the absence of the other hand, arbitrary strain breaks down to some strain. The change of the hopping parameter t is related extent the periodicity of the lattice, and “short-range” to the variation of length, as explained in Refs. [1] and strain can be identified to occur at unit cells where ∆αj [5]: and ∆Lj cease to be zero by significant margins. This observation provides the rationale for expressing |β|t δt = − τ · ∆τ . (19) the gauge fields without ever leaving the atomic lattice: j 2 j j a0 0 When ∆τ j ' ∆τ j at each unit cell a mechanical distor- tion can be considered “long-range,” and the first-order This way Eqn. (18) becomes: theory is valid. The process to lay down the gauge terms ∗ to first order is straightforward. Local gauge fields can be Es,A f1 Hps = ~vF σ · q + , (20) computed as low energy approximations to the following f1 Es,B 2 × 2 pseudospin Hamiltonian: √ ∗ |β|t with f1 = 2a2 [2τ 3 · ∆τ 3 − τ 1 · ∆τ 1 − τ 2 · ∆τ 2 + 3i(τ 2 · ∗ 0 √ Es,A g 3a0t , (15) ∆τ 2 − τ 1 · ∆τ 1)], and ~vF ≡ 2 . The parameter g Es,B f1 can be expressed in terms of a vector potential: As eAs f1 = −~vF . This way: P3 i(τ j +∆τ j )·(Kn+∆Kn+q) ~ with g ≡ − j=1(t + δtj)e , and |β|φ n = 1, ..., 6. We defer discussion of the diagonal terms 0 2τ 3·∆τ 3−τ 1√·∆τ 1−τ 2·∆τ 2 As = − πa3 [ for now. 0 3 Keeping exponents to first order we have: −i(τ 2 · ∆τ 2 − τ 1 · ∆τ 1)]. (21)

(τ j +∆τ j )·(Kn+∆Kn+q) ' τ j ·Kn+τ j ·∆Kn+∆τ j ·Kn+τ j ·q. We finally analyze the diagonal entries in Eqn. (15), which are given as follows [15]: The exponent is next expressed to first-order: 3 √ ei(τ j ·Kn+τ j ·∆Kn+∆τ j ·Kn+τ j ·q) ' 0.3eV 1 X |τ j − ∆τ j| − a0/ 3 Es,A = − √ , (22) iτ j ·Kn iτ j ·Kn 0.12 3 ie τ j · q + e [1 + i(τ j · ∆Kn + ∆τ j · Kn)](16). j=1 a0/ 3 5 and structural distortions [25], and static structural distor- √ tions created by the mechanical and electrostatic inter- 3 0 0.3eV 1 X |τ j − ∆τ j| − a0/ 3 action with a substrate, a deposition process [26], or line E = − √ . (23) s,B 0.12 3 stress at the edges of finite-size membranes [15]. j=1 a0/ 3 In reference [27] it is argued that the rippled texture of These entries represent the scalar deformation potential freestanding graphene leads to observable consequences, which we take to linear order in the average bond increase the strongest being a sizeable velocity renormalization. [21]. In order to demonstrate such statement, one must take a closer look at the underlying mechanics of the problem. The model [27] assumes that a graphene membrane is D. Relation to the formalism from first-order originally pre-strained (in bringing an analogy, one would continuum elasticity say that the membrane would be an “ironed tablecloth”), so that curvature due to a single wrinkle directly leads We next establish how the theory based on a contin- to increases in interatomic distances. Those distance in- uum relates to the present formalism. In the absence creases directly modify the metric on the curved space. of significant curvature, the continuum limit is achieved In practice, an external electrostatic field can be used to when |∆τ j | → 0 (for j = 1, 2, 3). We have then (Cauchy- realize such pre-strained configuration [28]. a0 uxx uxy T In improving the consideration of the mechanics be- Born rule): τ j · ∆τ j → τ j uxy uyy τ j , where uij are the entries of the strain tensor. yond first-order continuum elasticity, let us consider This way Eqn. (21) becomes: what happens if this pre-strained assumption is relaxed (in continuing our analogy, the rippled membrane in |β|φ0 Fig. 4(a) would then be akin to a “wrinkled tablecloth As → √ (uxx − uyy − 2iuxy), (24) 2 3πa prior to ironing”): How do the gauge fields look in such 0 scenario? With our formalism, we can probe the inter- as expected [2, 5]. relation between mechanics and the electronic structure Equation (24) confirms that if the zigzag direction is directly. In Figure 4(a) we display a graphene mem- parallel to the x−axis the vector potential we have ob- brane with three million atoms at 1 Kelvin after relaxing tained is consistent with known results in the proper limit strain at the edges. The strain relaxation proceeds by [2, 5]. Besides representing a consistent first-order for- the formation of ripples or wrinkles on the membrane. malism, the present approach is exceptionally suited for This initial configuration is already different to a flat the analysis of “raw” atomistic data –obtained, for ex- (“pre-strained”) configuration within the continuum for- ample, from molecular dynamics simulations– as there malism, customarily enforced prior to the application of is no need to determine the strain tensor explicitly: the strain. relevant equations (21, 22, 23) take as input the changes The ripples must be “ironed out” before any significant in atomic positions upon strain. Within the present ap- increase on interatomic distances can occur: “Isometric proach N/2 space-modulated pseudospinor Hamiltonians deformations” lead to curvature without any increase on can be built for a graphene membrane having N atoms. interatomic distances [15] (in continuing our analogy, this is usually what happens with clothing). We believe that a local determination of the metric tensor from atomic III. APPLYING THE FORMALISM TO displacements alone will definitely be useful in continuing RIPPLED GRAPHENE MEMBRANES making a case for velocity renormalization [6, 16, 27]; this is presently work in progress [? ]. We finish the present contribution by briefly illustrat- The local density of electronic states is obtained di- ing the formalism on two experimentally relevant case ex- rectly from the Hamiltonian of the membrane in configu- amples. The developments presented here are motivated ration space H, and shown in Fig. 4(b). When compared by recent experiments where freestanding graphene mem- to the DOS from a completely flat membrane, no ob- branes are studied by local probes [22–24]. (One must servable variation on the slope of the DOS appears, and keep in mind, nevertheless, that the theory provided up hence, no renormalization of the Fermi velocity either. to this point is rather general.) One can determine the extent to which nearest- neighbor vectors will preserve sublattice symmetry in A. Rippled membranes with no external terms of ∆αj and ∆Lj, Eqns. (4-6). We observe small mechanical load and apparently random fluctuations on those measures o in Fig. 4(c): ∆Lj . 1% and ∆αj . 2 . It is an established fact that graphene membranes We display the deformation potential in Figure 4(d) will be naturally rippled due to a number of physical in terms of the average (Edef ) and difference (Emass) processes, including temperature-induced (i.e., dynamic) between Es,A and Es,B (Eqns. (22) and (23)) at any given 6

3 million atoms ∆α ∆α − 1 2 meV 40 400 Deg (a) Height (c) (d) (e) Bs Tesla 3 million atoms Tesla 0 (a) ∆α ∆α (c) def. potential (d) (nm) 1 −2 0 2 2 Deg −150 0 150 −0.3 0.0 0.2 (b) eV 0.0 0 2 Bs − 0.2 − 10

− 2 % Deformation potential meV −1.6 0 1.6 mass 272 nm ∆L ∆L 20 ∆ ∆ meV 71 nm 1 2 10 L1 L2 Mass 0 20 8 (b) % 6 −20 0

4 0 1 272 nm 2 − 20 Density of states − 1 −0.4 0.0 0.4 Energy (eV) FIG. 5: Strained membrane: (a) The section in blue (light gray) is kept fixed, and strain is applied by pushing down the FIG. 4: A finite-size graphene membrane at 1 Kelvin. (a) The triangular section in green (dark gray) with a sharp extruder, membrane forms ripples to relieve mechanical strain originat- located at the geometrical center. (b) Deviations from proper ing from its finite size. (b) We could not discern changes on sublattice symmetry are concentrated at the section directly the LDOS (which relates to renormalization of the Fermi ve- underneath the sharp tip, where the deformation is the largest locity) on a completely flat membrane and after line strain and strain is the most inhomogeneous. (c-d) Gauge fields. is relieved. (c) Measures for changes in angles and lengths at individual unit cells (Eqns. 4-6) displaying noise on a small scale, and consistent with the formation of ripples. (d) deviates from a perfect 2-dimensional plate. The deformation potential, mass term and (e) the pseudo- The gauge fields given in Fig. 5(c-d) reflect the circular magnetic field are inherently noisy as well. symmetry induced by the circular shape of the extruding tip [15].

0.01 unit cell: Closest to extruder 1 1 0.00 E = (E + E ), and E = (E − E ). 0.01 def 2 s,A s,B mass 2 s,A s,B 0.00 (25) 0.01 Both quantities are of the order of tens of meVs. 0.00 The ripples lead to the random-looking pseudo- 0.01 (b) (c) (d) Farthest (a) away magnetic ﬁeld shown in Fig. 4(e), reminiscent of the Density of states (Arb. units) 0.00 -0.2-0.4 0.40.20.0 -0.2-0.4 0.40.20.0 -0.2-0.4 0.40.20.0 -0.2-0.4 0.40.20.0 electron density plots created by random charge puddles Energy (eV) [29, 30]. We next consider how strain by a sharp probe modiﬁes the results in Fig. 4. FIG. 6: Local density of states on the membrane under strain shown in Fig. 5. The locations where the DOS is computed are shown in the insets (the most symmetric line patterns are B. Rippled membranes under mechanical load displayed in yellow).

In what follows we consider a central extruder cre- We finish the discussion by probing the local density ating strain on the freestanding membrane. For this, of states at many locations in Fig. 6, which may relate we placed the membrane shown in Fig. 4 on top of a to the discussion of confinement by gauge fields [31]. Es substrate (shown in blue/light gray in Fig. 5(a)) with a was was not included in computing DOS curves. triangular-shaped hole (in green/dark gray in Fig. 5(a)). Some generic features of DOS are clearly visible: (i) The membrane is held fixed in position when on the sub- Near the extruder, the deformation is already beyond the strate, and pushed down by a sharp tip at its geometrical linear regime, and the DOS is indeed renormalized for lo- center, down to a distance Γ=10 nm. As indicated ear- cations close to the mechanical extruder [6, 16, 27]. (ii) lier, sublattice symmetry is not exactly satisfied right un- A sequence of features appear on the DOS farther away derneath the tip, where ∆αj and ∆Lj take their largest from the extruder. Because the field is not homogeneous values (Fig. 5(c)). While ∆Lj still displays some fluctu- and perhaps due to energy broadening we are unable to ations, this is not the case for ∆αj (the scale for ∆αj is tell a central peak. As indicated on the insets, the plots identical to that from Fig. 4(c)). The large white areas on Fig. 6(b) and 6(d) are obtained along high-symmetry tells us that fluctuations on ∆αj are wiped out upon load lines (the colors on the DOS subplots correspond with as the extruder removes wrinkles. This observation stems the colored lines on the insets). For this reason they from the lattice-explicit consideration of the mechanics. look almost identical, and the three sets of curves (corre- We have presented a detailed discussion of the prob- sponding to the DOS along different lines) overlap. Due lem along these lines [15]. We found that for small mag- to lower symmetry, the LDOS in Fig. 6(a) and 6(c) ap- nitudes of load a rippled membrane will adapt to an ex- pear symmetric in pairs, with the exception of the plots truding tip isometrically. This observation is important highlighted in gray. (the light ’v’-shaped curve in all sub- in the context of the formulation with curvature [6, 27], plots is the reference DOS in the absence of strain). because in that formulation there is the assumption that LDOS curves complement the insight obtained from distances between atoms increase as soon as graphene gauge field plots. Hence, they should also be reported 7 in discussing strain engineering of graphene’s electronic the method by computing the strain-induced gauge structure, particularly in situations where gauge fields fields on a rippled graphene membrane with and without are inhomogeneous. mechanical load. In doing so, we have initiated a necessary discussion of mechanical effects falling beyond a description within first-order continuum elasticity. IV. CONCLUSIONS Such analysis is relevant for accurate determination of gauge fields and has not received proper attention yet. We presented a novel framework to study the relation between mechanical strain and the electronic structure of graphene membranes. Gauge fields are expressed Acknowledgments directly in terms of changes in atomic positions upon We acknowledge conversations with B. Uchoa, and com- strain. Within this approach, it is possible to determine puter support from HPC at Arkansas (RazorII ), and the extent to which the sublattice symmetry is preserved. XSEDE (TG-PHY090002, Blacklight, and Stampede). In addition, we find that there are no K−dependent M.V. acknowledges support by the Serbian Ministry of gauge fields in the first-order theory. We have illustrated Science, Project No. 171027.

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