Solid State Communications 166 (2013) 70–75

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Solid State Communications

0038-10http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/ssc

Fast-track Communication

Strain-engineering of graphene's electronic structure beyondcontinuum elasticity

Salvador Barraza-Lopez a,n, Alejandro A. Pacheco Sanjuan b, Zhengfei Wang c,Mihajlo Vanević d

a Department of Physics, University of Arkansas, Fayetteville, AR 72701, USAb Departamento de Ingeniería Mecánica, Universidad del Norte, Km. 5 Vía Puerto Colombia, Barranquilla, Colombiac Department of Materials Science and Engineering, University of Utah, Salt Lake City, UT 84112, USAd Department of Physics, University of Belgrade, Studentski trg 12, 11158 Belgrade, Serbia

a r t i c l e i n f o

Article history:Received 16 April 2013Received in revised form23 April 2013Accepted 6 May 2013

by F. Peeters

symmetry holds. Since strain deforms lattice vectors at each unit cell, orthogonality between lattice and

Available online 14 May 2013

Keywords:A. Graphene membranesC. Electronic structureD. Elasticity theory

98/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.ssc.2013.05.002

esponding author.ail address: sbarraza@uark.edu (S. Barraza-Lop

a b s t r a c t

We present a new first-order approach to strain-engineering of graphene's electronic structure where nocontinuous displacement field uðx; yÞ is required. The approach is valid for negligible curvature. Thetheory is directly expressed in terms of atomic displacements under mechanical load, such that one candetermine if mechanical strain is varying smoothly at each unit cell, and the extent to which sublattice

reciprocal lattice vectors leads to renormalization of the reciprocal lattice vectors as well, making the Kand K ′ points shift in opposite directions. From this observation we conclude that no K-dependent gaugesenter 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 providingstrain-generated fields and local density of electronic states on graphene membranes with large numbersof atoms. The present method complements and goes beyond the prevalent approach, where strainengineering in graphene is based upon first-order continuum elasticity.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The interplay between mechanical and electronic effects incarbon nanostructures has been studied for a long time (e.g.,[1–11]). The mechanics in those studies invariably enters withinthe context of continuum elasticity. One of the most interestingpredictions of the theory is the creation of large, and roughlyuniform pseudo-magnetic fields and deformation potentials understrain configurations having a three-fold symmetry [2]. Thosetheoretical predictions have been successfully verified experimen-tally [12,13].

Nevertheless, different theoretical approaches to strain engi-neering in graphene possess subtle points and apparent discre-pancies [6,14], which may hinder progress in the field. Thismotivated us to develop an approach [15] which does not sufferfrom limitations inherent to continuum elasticity. This new for-mulation accommodates numerical verifications to determinewhen arbitrary mechanical deformations preserve sublattice sym-metry. Contrary to the conclusions of Ref. [14], with this formula-tion one can also demonstrate in an explicit manner the absence of

ll rights reserved.

ez).

K-point dependent gauge fields on a first-order theory (see Refs.[15-17] as well). The formalism takes as its only direct input rawatomistic data—as the data obtained from molecular dynamicsruns. The goal of this paper is to present the method, making thederivation manifest. We illustrate the formalism by computing thegauge fields and the density of states in a graphene membraneunder central load.

1.1. Motivation

The theory of strain-engineered electronic effects in grapheneis semi-classical. One seeks to determine the effects of mechanicalstrain across a graphene membrane in terms of spatially modu-lated pseudospin Hamiltonians Hps; these pseudospin Hamilto-nians HpsðqÞ are low-energy expansions of a Hamiltonian formallydefined in reciprocal space. Under “long range” mechanical strain(extending over many unit cells and preserving sublattice sym-metry [1–3]) Hps also become continuous and slowly varying localfunctions of strain-derived gauges, so that Hps-Hpsðq; rÞ. Withinthis first-order approach, the salient effect of strain is a local shiftof the K and K ′ points in opposite directions, similar to a shiftinduced by a magnetic field [2,3]. In the usual formulation of thetheory [1–6], this dependency on position leads to a continuousdependence of strain-induced fields BsðrÞ and EsðrÞ. Such

Fig. 2. (Color online) (a) Definitions of geometrical parameters in a unit cell.(b) Sublattice symmetry relates to how pairs of nearest-neighbor vectors (either inthick or dashed lines) are modified due to strain. These vectors change by Δτj andΔτj ′ upon strain (j¼1,2). Relative displacements of neighboring atoms lead tomodified lattice vectors; the choice of renormalized lattice vectors will be uniqueonly to the extent to which sublattice symmetry is preserved: Δτj ′≃Δτj .

-30 300 0 9 6 918 12Bs (Tesla) Bs (Tesla) Bs (Tesla)

Fig. 1. (Color online) Gauge fields from first-order continuum elasticity are definedregardless of spatial scale. A unit cell is shown in (b) and (c) for comparison. In thiswork, we define the pseudospin Hamiltonian for each unit cell using space-modulated, low-energy expansions of a tight-binding Hamiltonian in reciprocalspace. As a result, in our approach the gauge fields will become discrete.

S. Barraza-Lopez et al. / Solid State Communications 166 (2013) 70–75 71

continuous fields are customarily superimposed to a discretelattice, as in Fig. 1 [18].

When expressed in terms of continuous functions, a pseudos-pin Hamiltonian Hps is defined down to arbitrarily small spatialscales and it spans a zero area. In reality, however, the pseudospinHamiltonian can only be defined per unit cell, so it should take asingle value at an area of order ∼a20 (a0 is the lattice constant in theabsence of strain).

This observation tells us already that the scale of the mechan-ical deformation with respect to a given unit cell is inherently lostin a description based on a continuum model. For this reason, it isimportant to develop an approach which is directly related to theatomic lattice, as opposed to its idealization as a continuummedium. In the present paper we show that in following thisprogram one gains a deeper understanding of the interrelationbetween the mechanics and the electronic structure of graphene.Indeed, within this approach we are able to quantitatively analyzewhether the proper phase conjugation of the pseudospin Hamil-tonian holds at each unit cell. The approach presented here willgive (for the first time) the possibility to explicitly check on anygiven graphene membrane under arbitrary strain if mechanicalstrain varies smoothly on the scale of interatomic distances.Consistency in the present formalism will also lead to the conclu-sion that in such scenario strain will not break the sublatticesymmetry but the Dirac cones at the K and K ′ points will be shiftedin the opposite directions [2,3].

Clearly, for a reciprocal space to exist one has to preserve crystalsymmetry. When crystal symmetry is strongly perturbed, the reci-procal space representation starts to lack physical meaning, whichpresents a limitation to the semiclassical theory. The lack of sublatticesymmetry – observed on actual unit cells on this formulation beyondfirst-order continuum elasticity – may not allow proper phaseconjugation of pseudospin Hamiltonians at unit cells undergoing verylarge mechanical deformations. Nevertheless this check cannot pro-ceed – and hence has never been discussed – on a description of thetheory within a continuum media, because by construction there isno direct reference to actual atoms on a continuum.

As it is well-known, it is also possible to determine theelectronic properties directly from a tight-binding HamiltonianH in real space, without resorting to the semiclassical approxima-tion and without imposing an a priori sublattice symmetry. That is,while the semiclassicalHpsðq; rÞ is defined in reciprocal space (thusassuming some reasonable preservation of crystalline order), thetight-binding Hamiltonian H in real space is more general and canbe used for membranes with arbitrary spatial distribution andmagnitude of the strain.

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

2. Theory

2.1. 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 continuumtheories [1–6] to test sublattice symmetry on actual unit cellsunder a mechanical deformation. For this reason, sublatticesymmetry is an implicit assumption embedded in the continuumapproach.

To address the problem beyond the continuum approach, let usstart by considering the unit cell before (Fig. 2(a)) and afterarbitrary strain has been applied (Fig. 2(b)). For easy comparisonof our results, we make the zigzag direction parallel to the x-axis,which is the choice made in Refs. [2,5]. (Arbitrary choices ofrelative orientation are clearly possible; in Ref. [15] we chose thezigzag direction to be parallel to the y-axis.)

The lattice vectors before the deformation are given by(Fig. 2(a))

a1 ¼ ð1=2;ffiffiffi3

p=2Þa0; a2 ¼ ð−1=2;

ffiffiffi3

p=2Þa0; ð1Þ

τ1 ¼ffiffiffi3

p

2;12

!a0ffiffiffi3

p ; τ2 ¼ −ffiffiffi3

p

2;12

!a0ffiffiffi3

p ; τ3 ¼ 0;−1ð Þ a0ffiffiffi3

p : ð2Þ

(Note that a1 ¼ τ1−τ3, and a2 ¼ τ2−τ3.)After mechanical strain is applied (Fig. 2(b)), each local pseu-

dospin Hamiltonian will only have physical meaning at the unitcells where

Δτj′≃Δτj ðj¼ 1;2Þ: ð3ÞCondition (3) can be re-expressed in terms of changes of anglesΔαj or lengths ΔLj for pairs of nearest-neighbor vectors τj and τ j′[j¼1 is shown in thick solid and j¼2 in thin dashed lines inFig. 2(b)]:

ðτ j þ ΔτjÞ � ðτj þ Δτ′jÞ ¼ jτj þ Δτjjjτj þ Δτ′jjcosðΔαjÞ; ð4Þ

sgnðΔαjÞ ¼ sgnð½ðτ j þ ΔτjÞ � ðτj þ Δτ′jÞ� � kÞ; ð5Þwhere k is a unit vector along the z-axis, sgn is the sign function(sgnðaÞ ¼ þ 1 if a≥0 and sgnðaÞ ¼ −1 if ao0), and

ΔLj≡jτj þ Δτjj−jτj þ Δτ′jj: ð6ÞEven though in the problems of practical interest the deviations

from the sublattice symmetry do tend to be small [15], it is

S. Barraza-Lopez et al. / Solid State Communications 166 (2013) 70–7572

important to bear in mind that the sublattice symmetry does nothold a priori [2]. It is therefore important to have a method toquantify such deviations and check whether the sublattice sym-metry holds at the problem at hand. Forcing the sublatticesymmetry to hold from the start amounts to introducing anartificial mechanical constraint on the membrane which is notjustified on physical grounds [19]. For this reason the method wepropose is discrete and directly related to the actual lattice; it doesnot resort to the approximation of the membrane as a continuummedium [1–6,16]. Being expressed in terms of the actual atomicdisplacements, our formalism holds beyond the linear elasticregime where the first-order continuum elasticity may fail. Thecontinuum formalism is recovered as a special case of the onepresented here in the limit when jΔτjj=a0-0.

2.2. Renormalization of the lattice and reciprocal lattice vectors

In the absence of mechanical strain, the reciprocal latticevectors b1 and b2 are obtained by standard methods: We defineA≡ðaT1; aT2Þ, with a1 and a2 given in Eq. (1) and shown in Fig. 2(a).The reciprocal lattice vectors B≡ðbT

1 ;bT2Þ are related to the lattice

vectors by [20]

BT ¼ 2πA−1: ð7ÞWith the choice we made for a1 and a2 we get

b1 ¼ 1;1ffiffiffi3

p� �

2πa0

and b2 ¼ −1;1ffiffiffi3

p� �

2πa0

: ð8Þ

As seen in Fig. 3(a) the K-points on the first Brillouin zone aredefined by

K1 ¼2b1 þ b2

3; K2 ¼

b1−b2

3and K3 ¼−

b1 þ 2b2

3; ð9Þ

and

K4 ¼ −K1; K5 ¼ −K2 and K6 ¼ −K3: ð10ÞThe relative positions between atoms change when strain is

applied: τj-τj þ Δτj (j¼ 1;2;3Þ, and −τj-−τj−Δτj′ (j¼1,2). Fornegligible curvature, one may assume that Δτj � z ¼Δzj∼0 (andsimilar for the primed displacements Δτj′). We present here aformulation of the theory strictly valid for in-plane strain (it wouldalso be valid for membranes with negligible curvature).

We wish to find out how reciprocal lattice vectors change tofirst order in displacements under mechanical load. In order forreciprocal lattice vectors to make sense at each unit cell, Eq. (3)must hold. In terms of numerical quantities one would need thatΔαj and ΔLj are all close to zero. In that case we set Δτj′-Δτj forj¼1,2, and continue our program.

For this purpose we define

Δa1≡Δτ1−Δτ3 and Δa2≡Δτ2−Δτ3; ð11Þor in terms of (two-dimensional) components:

ΔA≡Δτ1x−Δτ3x Δτ2x−Δτ3xΔτ1y−Δτ3y Δτ2y−Δτ3y

!: ð12Þ

The matrix A changes to A′¼A þ ΔA, and we must modify B sothat Eq. (7) still holds under mechanical load. To first order indisplacements A′−1 becomes

A′−1 ¼ ð1þAΔAÞ−1ðA−1Þ≃A−1−A−1ΔAA−1: ð13ÞBy comparing Eqs. (7) and (13), the reciprocal lattice vectors inFig. 3(b) must be renormalized by

ΔB¼−2πðA−1ΔAA−1ÞT : ð14ÞWe note that the existence of this additional term is quite evidentwhen working directly on the atomic lattice, but it was missed in

Ref. [14], where the theory was expressed on a continuum. Let usnow calculate shifts of the K-points due to strain. For example, K2

(¼ K in Fig. 3(a)) requires an additional contribution, which wefind by explicit calculation to be

ΔK ¼ΔK2 ¼ −4π3a20

Δτ1x−Δτ2x;Δτ1x þ Δτ2x−2Δτ3xffiffiffi

3p

� �;

and using Eq. (10) one immediately sees that ΔK ′¼ −ΔK2, so thatthe K (K2) and K ′ (−K2) points shift in opposite directions, asexpected [2,3].

2.3. Gauge fields

Arbitrary strain breaks down to some extent the periodicity ofthe lattice, and “short-range” strain can be identified to occur atunit cells where Δαj and ΔLj cease to be zero by significantmargins.

This observation provides the rationale for expressing thegauge fields without ever leaving the atomic lattice: WhenΔτj′≃Δτj at each unit cell a mechanical distortion can be consid-ered “long-range,” and the first-order theory is valid. The processto lay down the gauge terms to first order is straightforward. Localgauge fields can be computed as low energy approximations to thefollowing 2�2 pseudospin Hamiltonian:

Es;A gn

g Es;B

!; ð15Þ

with g≡−∑3j ¼ 1ðt þ δtjÞeiðτjþΔτjÞ�ðKnþΔKnþqÞ, and n¼ 1;…;6. We defer

discussion of the diagonal terms for now.Keeping exponents to first order we have

ðτj þ ΔτjÞ � ðKn þ ΔKn þ qÞ≃τj � Kn þ τ j � ΔKn þ Δτj � Kn þ τj � q:The exponent is next expressed to first-order:

eiðτj�Knþτj�ΔKnþΔτj �Knþτj�qÞ

≃ieiτj�Kn τj � qþ eiτj�Kn ½1þ iðτj � ΔKn þ Δτj � KnÞ�: ð16ÞCarrying out explicit calculations, one can see that

∑3

j ¼ 1eiτj�Kn ½1þ iðτj � ΔKn þ Δτ j � KnÞ� ¼ 0: ð17Þ

For example, at K ¼K2 we have

1þ 4iπðΔτ1x þ Δτ2x þ Δτ3xÞ9a0

� �ð1þ e2πi=3−eπi=3Þ;

with phasors adding up to zero. Similar phasor cancelations occurat all other K-point.

The term linear on ΔKn in Eq. (17) cancels out the fictitious K-point dependent gauge fields proposed in Ref. [14], which origi-nated from the term linear on Δτj in the same equation. Thisobservation constitutes yet another reason for the formulation ofthe theory directly on the atomic lattice. With this we havedemonstrated that gauges will not depend explicitly on K-points,so we now continue formulating the theory considering the K2

point only [2,5,3].Eq. (15) takes the following form to first order at K2 in the low-

energy regime:

Hps ¼0 t∑3

j ¼ 1ie−iK2 �τjτj � q

−t∑3j ¼ 1ie

iK2 �τjτ j � q 0

0@

1A

þEs;A −∑3

j ¼ 1δtje−iK2 �τj

−∑3j ¼ 1δtje

iK2 �τj Es;B

0@

1A; ð18Þ

with the first term on the right-hand side reducing to the standardpseudospin Hamiltonian in the absence of strain. β, the change ofthe hopping parameter t, is linearly dependent to variations in

b1b2 K1

K2 (K)

K3-K1

(K’) -K2

-K3 b’1b’2 K3 K3

' KK

K1 K1 K3 K3

K1 K1

KKK2 K2K2 K2

Fig. 3. (Color online) First Brillouin zone (a) before and (b) after mechanical strain is applied. The reciprocal lattice vectors are shown, as well as the changes of the high-symmetry points at the corners of the Brillouin zone. Note that independent K points (K and K ′) move in the opposite directions. The dashed hexagon in (b) represents theboundary of the first Brillouin zone in the absence of strain.

S. Barraza-Lopez et al. / Solid State Communications 166 (2013) 70–75 73

length, as explained in Refs. [1,5]:

δtj ¼−jβjta20

τj � Δτj: ð19Þ

This way Eq. (18) becomes

Hps ¼ ℏvFr � qþEs;A f n1f 1 Es;B

!; ð20Þ

withf n1 ¼ ðjβjt=2a20Þ½2τ3 � Δτ3−τ1 � Δτ1−τ2 � Δτ2 þ

ffiffiffi3

piðτ2 � Δτ2−τ1 � Δτ1Þ�,

and ℏvF≡ffiffiffi3

pa0t=2. f1 relates to the vector potential As the following

way: f 1 ¼ −ℏvFeAs=ℏ. This way

As ¼−jβjϕ0

πa30

2τ3 � Δτ3−τ1 � Δτ1−τ2 � Δτ2ffiffiffi3

p −iðτ2 � Δτ2−τ1 � Δτ1Þ�:

�ð21Þ

We finally analyze the diagonal entries in Eq. (15), which aregiven as follows [15]:

Es;A ¼−0:3 eV0:12

13

∑3

j ¼ 1

jτj−Δτjj−a0=ffiffiffi3

p

a0=ffiffiffi3

p ; ð22Þ

and

Es;B ¼−0:3 eV0:12

13

∑3

j ¼ 1

jτj−Δτ′jj−a0=ffiffiffi3

p

a0=ffiffiffi3

p : ð23Þ

These entries represent the scalar deformation potential which wetake to linear order in the average bond increase [21].

2.4. Relation to the formalism from first-order continuum elasticity

We next establish how the theory based on a continuum relatesto the present formalism. In the absence of significant curvature,the continuum limit is achieved when jΔτjj=a0-0 (for j¼ 1;2;3).We have then (Cauchy–Born rule):

τj � Δτj-τjuxx uxy

uxy uyy

!τTj ;

where uij are the entries of the strain tensor.This way Eq. (21) becomes

As-jβjϕ0

2ffiffiffi3

pπa0

ðuxx−uyy−2iuxyÞ; ð24Þ

as expected [2,5].Eq. (24) confirms that if the zigzag direction is parallel to the

x-axis the vector potential we have obtained is consistent withknown results in the proper limit [2,5]. Besides representing aconsistent first-order formalism, the present approach is excep-tionally suited for the analysis of “raw” atomistic data – obtained,for example, from molecular dynamics simulations – as there is noneed to determine the strain tensor explicitly: the relevant Eqs.

(21)–(23) take as input the changes in atomic positions uponstrain. Within the present approach N=2 space-modulated pseu-dospinor Hamiltonians can be built for a graphene membranehaving N atoms.

3. Applying the formalism to rippled graphene membranes

We finish the present contribution by briefly illustrating theformalism on two experimentally relevant case examples. Thedevelopments presented here are motivated by recent experi-ments where freestanding graphene membranes are studied bylocal probes [22–24].

3.1. Rippled membranes with no external mechanical load

It is an established fact that graphene membranes will benaturally rippled for a number of reasons, including temperature-induced (i.e., dynamic) structural distortions [25], and staticstructural distortions created by the mechanical and electrostaticinteraction with a substrate, a deposition process [26], or linestress at the edges of finite-size membranes [15].

In Ref. [27] it is argued that the rippled texture of freestandinggraphene leads to observable consequences, the strongest being asizeable velocity renormalization. In order to demonstrate suchstatement, one must take a closer look at the underlying mechanicsof the problem. The model [27] assumes that a graphene membrane isoriginally pre-strained (in bringing an analogy, one would say that themembrane would be an “ironed tablecloth”), so that curvature due to asingle wrinkle directly leads to increases in interatomic distances.Those distance increases directly modify the metric on the curvedspace. In practice, an external electrostatic field can be used to realizesuch pre-strained configuration [28].

In improving the consideration of the mechanics beyond first-order continuum elasticity, let us consider what happens if thispre-strained assumption is relaxed (in continuing our analogy, therippled membrane in Fig. 4(a) would then be akin to a “wrinkledtablecloth prior to ironing”): How do the gauge fields look in suchscenario? With our formalism, we can probe the interrelationbetween mechanics and the electronic structure directly. In Fig. 4(a) we display a graphene membrane with three million atoms at1 K after relaxing strain at the edges. The strain relaxationproceeds by the formation of ripples or wrinkles on the mem-brane. This initial configuration is already different to a flat (“pre-strained”) configuration within the continuum formalism, custo-marily enforced prior to the application of strain.

The ripples must be “ironed out” before any significant increase oninteratomic distances can occur: “Isometric deformations” lead tocurvature without any increase on interatomic distances [15] (incontinuing our analogy, this is usually what happens with cloth-ing). We believe that a local determination of the metric tensor

S. Barraza-Lopez et al. / Solid State Communications 166 (2013) 70–7574

from atomic displacements alone will definitely be useful incontinuing making a case for velocity renormalization [6,16,27];this is presently work in progress.

The local density of electronic states is obtained directly fromthe Hamiltonian of the membrane in configuration space H, andshown in Fig. 4(b). When compared to the DOS from a completelyflat membrane, no observable variation on the slope of the DOSappears, and hence, no renormalization of the Fermi velocityeither.

One can determine the extent to which nearest-neighborvectors will preserve sublattice symmetry in terms of Δαj andΔLj, Eqs. (4)–(6). We observe small and apparently randomfluctuations on those measures in Fig. 4(c): ΔLj≲1% and Δαj≲2

○.We display the deformation potential in Fig. 4(d) in terms of

the average (Edef) and difference (Emass) between Es;A and Es;B (Eqs.(22) and (23)) at any given unit cell:

Edef ¼ 12 ðEs;A þ Es;BÞ and Emass ¼ 1

2ðEs;A−Es;BÞ: ð25Þ

Both quantities are of the order of tens of meVs.The ripples lead to the random-looking pseudo-magnetic field

shown in Fig. 4(e), reminiscent of the electron density plotscreated by random charge puddles [29,30]. We next considerhow strain by a sharp probe modifies the results in Fig. 4.

3 million atoms

272 nm

2468

10

-0.4 0.0

Den

sity

of s

tate

s

Energy (eV)

Height(nm)

-2

0

2

-1

0

1%

Deg

-0.2

0.0

0.2

Δα1 Δα2

ΔL1 ΔL2

0.4

Fig. 4. (Color online) A finite-size graphene membrane at 1 K. (a) The membrane formsdiscern changes on the LDOS (which relates to renormalization of the Fermi velocity) on ain angles and lengths at individual unit cells Eqs. (4–6) displaying noise on a small scaleterm and (e) the pseudo-magnetic field are inherently noisy as well.

Δα1 Δα

ΔL1 ΔL

272 nm

−2 20

1.6−1.6 0

Deg

%

71 nm

3 million atoms

Fig. 5. (Color online) Strained membrane: (a) The section in blue (light gray) is kept fixewith a sharp extruder, located at the geometrical center. (b) Deviations from proper subwhere the deformation is the largest and strain is the most inhomogeneous. (c, d) Gau

3.2. Rippled membranes under mechanical load

In what follows we consider a central extruder creating strainon the freestanding membrane. For this, we placed the membraneshown in Fig. 4 on top of a substrate (shown in blue/light gray inFig. 5(a)) with a triangular-shaped hole (in green/dark gray inFig. 5(a)). The membrane is held fixed in position when on thesubstrate, and pushed down by a sharp tip at its geometricalcenter, down to a distance Γ¼10 nm.

As indicated earlier, sublattice symmetry is not exactly satisfiedright underneath the tip, where Δαj and ΔLj take their largestvalues (Fig. 5(c)). While ΔLj still displays some fluctuations, this isnot the case for Δαj (the scale for Δαj is identical to that from Fig. 4(c)). The large white areas indicate that fluctuations on Δαj arewiped out upon load as the extruder removes wrinkles. Thisobservation stems from the lattice-explicit consideration of themechanics.

We have presented a detailed discussion of the problem alongthese lines [15]. We found that for small magnitudes of load arippled membrane will adapt to an extruding tip isometrically.This observation is important in the context of the formulationwith curvature [6,27], because in that formulation there is theassumption that distances between atoms increase as soon asgraphene deviates from a perfect two-dimensional plate.

meV

meV Bs

MassDeformation potential

Tesla-40 400

-20

0

20

0

-10

ripples to relieve mechanical strain originating from its finite size. (b) We could notcompletely flat membrane and after line strain is relieved. (c) Measures for changes, and consistent with the formation of ripples. (d) The deformation potential, mass

Bs

2

2mass

def. potential−0.3 0.0

−150 0 150

−20

0

20meV

Tesla

eV

d, and strain is applied by pushing down the triangular section in green (dark gray)lattice symmetry are concentrated at the section directly underneath the sharp tip,ge fields.

0.01

0.40.20.0-0.2-0.40.00

0.01

0.00

0.01

0.00

0.01

0.00

0.40.20.0-0.2-0.4 0.40.20.0-0.2-0.4

Closestto extruder

Farthestaway

0.40.20.0-0.2-0.4Energy (eV)

Den

sity

of s

tate

s (A

rb. u

nits

)

Fig. 6. (Color online) 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 mostsymmetric 'scan' line patterns are displayed in yellow).

S. Barraza-Lopez et al. / Solid State Communications 166 (2013) 70–75 75

The gauge fields given in Fig. 5(c, d) reflect the circularsymmetry induced by the circular shape of the extruding tip [15].

We finish the discussion by probing the local density of statesat many locations in Fig. 6, which may relate to the discussion ofconfinement by gauge fields [31]. (Although available, Es was notincluded in computing DOS curves shown in Fig. 6.)

Some generic features of DOS are clearly visible: (i) Near theextruder, the deformation is already beyond the linear regime, andthe DOS is indeed anisotropically renormalized (not all DOS curvesat different angular locations overlap) for locations close to themechanical extruder; an observation consistent with an anisotro-pic renormalization of the Fermi velocity [6,16,27]. (ii) A sequenceof features appear on the DOS farther away from the extruder.Because the field is not homogeneous and perhaps due to energybroadening we are unable to tell a central peak. As indicated onthe insets, the plots on Fig. 6(b) and (d) are obtained along high-symmetry lines (the colors on the DOS subplots correspond withthe colored lines on the insets). For this reason they look almostidentical, and the three sets of curves (corresponding to the DOSalong different lines) overlap. Due to lower symmetry, the LDOS inFig. 6(a) and (c) appear symmetric in pairs, with the exception ofthe plots highlighted in gray. (the light ’v’-shaped curve in allsubplots is the reference DOS in the absence of strain).

LDOS curves complement the insight obtained from gauge fieldplots. Hence, they should also be reported in discussing strainengineering of graphene's electronic structure, particularlyin situations where gauge fields are inhomogeneous.

4. Conclusions

We presented a novel framework to study the relation betweenmechanical strain and the electronic structure of graphene mem-branes. Gauge fields are expressed directly in terms of changes inatomic positions upon strain. Within this approach, it is possible todetermine the extent to which the sublattice symmetry is pre-served. In addition, we find that there are no K-dependent gaugefields in the first-order theory. We have illustrated the method bycomputing the strain-induced gauge fields on a rippled graphenemembrane with and without mechanical load. In doing so, wehave initiated a necessary discussion of mechanical effects fallingbeyond a description within first-order continuum elasticity. Suchanalysis is relevant for accurate determination of gauge fields andhas not received proper attention yet.

Acknowledgments

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

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