Effects of loading mode and orientation on deformation mechanismof graphene nano-ribbons

Y. J. Sun,1 F. Ma,1,2,a) Y. H. Huang,3 T. W. Hu,1,2 K. W. Xu,1,4,a) and Paul K. Chu2,a)1State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049,Shaanxi, China2Department of Physics and Materials Science, City University of Hong Kong, Tat Chee Avenue, Kowloon,Hong Kong, China3College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, Shaanxi, China4Department of Physics and Opt-electronic Engineering, Xi’an University of Arts and Science, Xi’an 710065,Shaanxi, China

(Received 8 August 2013; accepted 25 October 2013; published online 7 November 2013)

Molecular dynamics simulation is performed to analyze the deformation mechanism of graphene

nanoribbons. When the load is applied along the zigzag orientation, tensile stress yields brittle

fracture and compressive stress results in lattice shearing and hexagonal-to-orthorhombic phase

transformation. Along the armchair direction, tensile stress produces lattice shearing and phase

transformation, but compressive stress leads to a large bonding force. The phase transformation

induced by lattice shearing is reversible for 17% and 30% strain in compressive loading along the

zigzag direction and tensile loading along the armchair direction. The energy dissipation is less

than 10% and resulting pseudo-elasticity enhances the ductility. VC 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4829480]

Since the successful exfoliation of single-layer gra-

phene in 2004,1 two-dimensional materials have become

one of the research focuses in physics, chemistry, and mate-

rials science.2–5 Thereafter, many approaches have been

developed to fabricate graphene and graphene nanoribbons,

including mechanical exfoliation,6,7 reduction of graphite

oxide,8,9 chemical vapor deposition, decomposition of

SiC,10 and un-zipping of carbon nanotube (CNT).11,12

Generally, residual stress exists in the as-prepared samples

as well as those after substrate transfer,13–15 and it is closely

related to the localized ripples, edges, grain boundaries,

adatom adsorption, and Stone-Wales defects produced in

the synthesis and post thermal processes.16–18 The residual

stress might influence the structural stability of this ultra-

thin system in practice. In addition, graphene possesses me-

chanical strength of about 130 GPa,19 which is almost 100

times larger than that of steels, and so graphene is believed

to be an ideal reinforcing component in composite materi-

als.20,21 Graphene sheets are typically distributed randomly

in the composite host and when the materials are subjected

to an external load along different orientations, the gra-

phene sheets may be activated differently. Hence, it is of

scientific and technological interest to determine the de-

pendence of their deformation behavior on the loading

mode and loading orientation.

It is generally accepted that the competition between

bond rotation and bond rupture determines the deformation

mechanism of graphene. By conducting molecular dynam-

ics (MD) simulation, Grantab et al.22 found that graphenesheets with large-angle tilt boundaries having a high density

of defects were as strong as the pristine one and much

stronger than those with low-angle boundaries having fewer

defects. They ascribed the abnormal behavior to the ability

of the large-angle tilt boundaries to better accommodate the

strained rings via bond rotation.22 Bond-rotation-related

mechanical deformation has been observed from graphene

with Stone-Wales (SW) defects.23–25 Similar to plane slips

in bulk metals, bond rotation depends on the loading direc-

tion and can be described by a physical parameter resem-

bling the Schmidt factor.26 In bulk metals, the deformation

mechanism is determined by the loading modes, namely,

compression and stretching and hence, both the loading

direction and loading mode affect the mechanism of me-

chanical deformation. In this work, MD simulation is per-

formed to determine the effects of compressive/tensile

loading along the zigzag and armchair directions on the de-

formation mechanism.

MD simulation is carried out on the large-scale atomic/

molecular massively parallel simulator (LAMMPS).27 The

interaction between carbon atoms is described by the adaptive

intermolecular reactive bond order (AIREBO) potential,

which can accurately capture the interactions between carbon

atoms as well as bond breaking and re-forming.28,29 The cut-

off parameter describing the short-range C-C interactions is

chosen to be 2.0 Å in order to avoid spuriously high bonding

force and nonphysical results at large deformation.30 A lattice

constant of 1.426 Å is adopted as the initial value and the layer

separation of graphite of 3.4 Å is taken as the effective thick-

ness of the mono-layer graphene.31 A Poisson’s ratio of 0.165

is used.32 Prior to the simulation, the graphene sheets with

periodic conditions in the two in-plane directions are relaxed

to an equilibrium state in the isothermal-isobaric (NPT)

ensembles for 1 000 000 MD steps with a time step of 1 fs.

Graphene nanoribbons (GNRs) along the zigzag/armchair ori-

entations and with a size of 50 � 120 Šwith 2240/2279atoms are created by deleting atoms from the outside part of

the nanoribbons and a vacuum region 15 Å in width is added

a)Authors to whom correspondence should be addressed. Electronic addresses:

mafei@mail.xjtu.edu.cn; kwxu@mail.xjtu.edu.cn; and paul.chu@cityu.edu.hk

0003-6951/2013/103(19)/191906/5/$30.00 VC 2013 AIP Publishing LLC103, 191906-1

APPLIED PHYSICS LETTERS 103, 191906 (2013)

http://dx.doi.org/10.1063/1.4829480http://dx.doi.org/10.1063/1.4829480http://dx.doi.org/10.1063/1.4829480mailto:mafei@mail.xjtu.edu.cnmailto:kwxu@mail.xjtu.edu.cnmailto:paul.chu@cityu.edu.hkhttp://crossmark.crossref.org/dialog/?doi=10.1063/1.4829480&domain=pdf&date_stamp=2013-11-07

perpendicular to the length direction and normal to graphene

sheets, so that the atoms near one edge of the GNRs do not

interact with those near the opposite edge because of the peri-

odic boundary conditions. Simulation is performed under

compressive/tensile loading along the width direction of the

GNRs in the NVT ensemble using the deformation-control

method. A strain increment of about 10�4 corresponding to a

strain rate of 109 s�1 is applied in every time step to the load-

ing and controlled unloading processes. Four combinations

including two orientations (armchair and zigzag) and two

loading modes (tensile and compressive) are considered, that

is, compressive loading along the zigzag direction (CZZ), ten-

sile loading along the zigzag direction (TZZ), compressive

loading along the armchair direction (CAC), and tensile load-

ing along the armchair direction (TAC). The temperature is

kept at 5 K using the Nose-Hoover thermostat. Only in-plane

deformation is allowed so that wrinkles can be avoided thus

enabling the study on deformation behavior of an ideal

two-dimensional system. The strain and stress values are cal-

culated according to Ref. 30. The evolution of atomic configu-

rations and statistical bond lengths are analyzed in different

stages by ATOMEYE.33

Fig. 1 shows the TZZ, TAC, CZZ, and CAC stress-

strain curves of the graphene nanoribbons. The positive and

negative strain values indicate tensile and compressive load-

ing, respectively. During TZZ and CAC, the absolute stress

increases with strain smoothly but abruptly becomes zero at

strain of 30% and 10%, corresponding to the destructive

structural changes in the GNRs. On the other hand, during

CZZ and TAC, three deformation stages are involved. In the

first stage denoted by I, the stress also increases smoothly

with strain, but stress oscillation corresponding to periodic

change in the structure is observed from the second stage

denoted by II (strain of about 12.5% and 20%). In the third

stage denoted by III, the stress increases sharply initially and

then suddenly becomes zero at strain of about 19% and 47%.

In comparison with TZZ and CAC, the ductility in CZZ and

TAC is enhanced by about 111% and 56.7%. The fracture

strain in TAC is the largest (about 47%) whereas that in

CAC is the smallest (10%). In the two compressive loading

processes of CZZ and CAC, the atomic separation is short-

ened and atomic interaction increases thereby producing the

positive second-order elastic constants (second-order deriva-

tive of stress with respect to strain) in stage I. However, in

the two tensile loading processes of TZZ and TAC, the grad-

ually reduced atomic interaction yields negative second-

order elastic constants. The stress terrace in the strain range

between 20% and 45% in TAC resembles elongation of the

yield point induced by Cottrell atmospheres around disloca-

tions and grain boundaries in low-carbon steels.34 It is thus

obvious that the deformation mechanism of graphene sheets

is determined by the combined effect of the loading mode

and orientation.

In order to understand the stress oscillation as well as

enhanced ductility during CZZ and TAC, we systemically

analyze the evolution of the atomic configuration and the

typical snapshots are depicted in Fig. 2. Fig. 3 displays the

FIG. 1. Stress-strain curves of graphene nanoribbons subjected to a strain

rate of 10�4 ps�1 along different orientations (armchair or zigzag) and underdifferent loading modes (tensile or compressive) at 5 K. CZZ, TZZ, CAC

and TAC represent compressive loading along the zigzag direction, tensile

loading along the zigzag direction, compressive loading along the armchair

direction, and tensile loading along the armchair direction, respectively, and

I, II and III denote the elastic, shearing, and hardening deformation stages,

respectively.

FIG. 2. Typical snapshots showing the

local atomic structures of graphene

nanoribbons during loading of differ-

ent modes: (a) TZZ at strain of 0% (I),

10% (II), 20% (III), and 30% (IV); (b)

CAC at strain of 0% (I), 5% (II), 8%

(III), and 9% (IV); (c) TAC at strain of

0% (I), 10% (II), 20% (III), and 47%

(IV); (d) CZZ at strain of 0% (I), 10%

(II), 12.5% (III), and 19% (IV).

191906-2 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)

statistical bond length in local regions with time. During

TZZ, the length of bond 1 and bond 2 increases with strain

but at a strain of 30%, the bond length decreases sharply to

that of the ideal crystal lattice [Fig. 3(a)] as a result of brittle

fracture as illustrated in Fig. 2(a). During CAC, the length of

the two types of bonds decreases with strain and the rate is

larger for bond 1 than bond 2 [Fig. 3(b)]. At about 10%

strain, the length of bond 1 is 1.2 Å and the system is sub-

jected to spuriously high bonding force yielding nonphysical

results as shown in Fig. 2(b). If out-of-plane movement is

allowed, wrinkles should occur in this case. A large lattice

change is also observed during TAC and CZZ. Lattice shear-

ing is activated from the edges of the GNRs along certain

directions at 20% and 12.5% strain and then extends inwards

gradually [Figs. 2(c) and 2(d)]. During lattice shearing, the

bond length in the sheared regions changes suddenly. For

instance, in TAC, the length of bond 1 diminishes from 1.7 Å

to 1.42 Å and it is just as long as the length of bond 2 in the

unsheared region. That of bond 2 increases dramatically

from 1.42 Å to 1.63 Å, which is close to the length of bond 1

in the unsheared region [Fig. 3(b)]. Similar abrupt changes

in the bond length are also observed from the sheared regions

during CZZ. The lattice change corresponds to phase trans-

formation from hexagonal to orthorhombic.26,35 Lattice

shearing proceeds gradually and an energy barrier is over-

come resembling plane slipping in metals. This leads to

stress oscillation in Stage II as shown in Fig. 1. Bond rupture

at local regions occurs after completion of the phase transfor-

mation in the whole system and fracture of GNRs occurs

finally. Our results indicate that the phase transformation

induced by lattice shearing is the origin of stress oscillation

and enhanced ductility during CZZ and TAC.

The orientation-dependent deformation behavior can be

understood by comparing to bulk metals. The Schmidt factor

is suggested to bridge the applied normal stress r and theresolved shear stress s along a given shearing direction in this

two-dimensional system. It can be expressed as sin u � cos u,in which u denotes the angle between the loading and shear-ing directions.26 Since there is the largest atomic line density

and the largest separation between atomic arrays along arm-

chair directions, they are believed to be the possible shearing

directions. In the TAC process, the Schmidt factor has a large

value of about 0.48 and the resolved shear stress exceeds the

critical stress (42.2 GPa (Ref. 26)) required by lattice shear-

ing. Lattice shearing occurs after elastic deformation and

before destructive fracture, as schematically shown in Fig.

4(a). However, in the TZZ process, the Schmidt factor has a

value of 0.44 and decreases with strain gradually. Hence, the

resolved shear stress along the closely packed direction can-

not activate the lattice shearing. Instead, brittle fracture takes

place immediately after elastic deformation, as schematically

shown in Fig. 4(b). Because of the orthorhombic relationship

between the zigzag and armchair directions, CZZ is equiva-

lent to TAC as a result of Poisson’s effect, while CAC is

equivalent to TZZ. Therefore, lattice-shearing-enhanced duc-

tility is found from both the TAC and CZZ processes, but not

in the CAC and TZZ processes as discussed above. Table I

summarizes the main results.

The reversibility of the phase transformation is illus-

trated during both CZZ and TAC when the lattice-sheared

GNRs are unloaded at a controlled strain rate of 109 s�1 from

a strain smaller than the fracture strain. Figs. 5(a) and 5(b)

depict the stress-strain curves of a loading-unloading cycle

during CZZ and TAC for maximum loading strain of 17%

and 30%, respectively. In the low strain range between 0%

and 9% in CZZ and 0% to 12% in TAC, the stress-strain

curves in the unloading process almost coincide with those

in loading process suggesting elastic deformation. However,

at large loading strain, lattice-shearing-induced phase trans-

formation from hexagonal to orthorhombic occurs and

Stone-Wales defects and vacancies are produced sometimes.

In both cases, an energy barrier should be overcome

FIG. 3. Evolution of bond length as a

function of strain during loading along

(a) zigzag and (b) armchair directions

with the positive and negative strain val-

ues indicating tensile and compressive

loading, respectively. When lattice

shearing appears locally, “Sheared”

denotes the bond length evolution in

these regions and “Un-Sheared” in other

regions.

FIG. 4. The schematic models depict-

ing the lattice evolution and the defor-

mation mechanisms in two typical

loading modes: (a) TAC and (b) TZZ.

191906-3 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)

correspondingly and the mechanical energy from the outside

is stored as chemical energy in graphene. Upon unloading,

the stored chemical energy will be dissipated suddenly as

thermal energy but not mechanical energy. This leads to the

difference in the stress-strain curves in the loading and

unloading processes, and stress-strain hysteretic loops are

formed. The magnified loops in grey are shown in the insets

of panels (a) and (b) and the integrated areas reflect the

energy dissipation. The calculated energy dissipation is less

than 10% and quite small. It can be ascribed to the extraordi-

narily low energy barrier of about 50 meV for the inverse

phase transformation.35

Fig. 6 displays the typical snapshots of the atomic config-

uration during loading and unloading and pseudo-color is

employed to indicate the magnitude of the shear strain on

each atom. Lattice shearing occurs mainly along the armchair

directions which are believed to have the highest resolved

shear stress.26 During CZZ, the phase transformation induced

by lattice shearing is nearly completed at 17% strain and the

perfect hexagonal lattice is recovered upon controlled unload-

ing [Fig. 6(a)]. No defects are produced in the circle. If com-

pressive loading is applied again, the phase transformation

takes place. The repeatable behavior endows graphene with

pseudo-elasticity similar to that possessed by metal

nanowires.36–38 During TAC as shown in Fig. 6(b), the bond

length is significantly increased and bond fracture and rotation

in TAC become easier compared to CZZ. Therefore, some

SW defects are produced locally as shown in the inset of Fig.

6(b). The defects are maintained during controlled unloading

and prevent subsequent phase transformation under tensile

loading. Therefore, the pseudo-elasticity at large tensile strain

is degenerated, although the ductility along the armchair

direction is the best. At an elevated temperature, bond fracture

and rotation become easier and so is formation of defects.

Accordingly, the ductility of graphene at high temperature is

not good and it is contrary to the behavior of conventional

bulk metals. This is also the reason why lattice shearing is sel-

dom observed from graphene sheets. Since graphene sheets

are usually distributed randomly in the composite host,

different loading modes along various orientations may be

activated at the same time. The resulting ductility of graphene

sheets can be effectively utilized to improve the mechanical

strength of composites.

In summary, MD simulation is conducted to study the

mechanical behavior of two-dimensional graphene and the

deformation mechanism is determined by both the loading

orientation and loading mode. When the load is applied

along the zigzag orientation, tensile stress gives rise to brittle

fracture, whereas compressive stress results in lattice shear-

ing and phase transformation from hexagonal to orthorhom-

bic. However, when the load is applied along the armchair

direction, tensile stress leads to lattice shearing and phase

transformation, but compressive stress yields large bonding

force. The phase transformation induced by lattice shearing

is reversible even at large strain of 17% and 30% during

CZZ and TAC and the energy dissipation is less than 10%.

As a result, pseudo-elasticity exists along some orientations

under certain loading modes and the ductility is enhanced.

Graphene sheets can be utilized as strengthening materials in

composites to accomplish high elasticity, large strength, and

low energy dissipation.

FIG. 6. Evolution of atomic structures during loading and unloading: (a)

CZZ and (b) TAC. The inset in (b) illustrates nucleation and development of

SW defects.

TABLE I. Deformation mechanisms of GNRs in four loading modes.

Zigzag Armchair

Tensile Brittle fracture Shearing enhanced ductile

Compressive Shearing enhanced ductile Non-physical result

FIG. 5. Loading-unloading stress-strain

curves of graphene nanoribbons at 5 K:

(a) CZZ and (b) TAC. The dissipation

energy can be evaluated from the mag-

nified grey area in the inset.

191906-4 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)

This work was jointly supported by Key Project of

Chinese National Programs for Fundamental Research and

Development (Grant No. 2010CB631002), National Natural

Science Foundation of China (Grant Nos. 51271139,

51171145, and 51302162), New Century Excellent Talents

in University (NCET-10-0679), Fundamental Research

Funds for the Central Universities, and Hong Kong Research

Grants Council (RGC) General Research Funds (GRF) No.

CityU 112212, and City University of Hong Kong Applied

Research Grants (ARG) 9667038 and 9667066.

1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V.

Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).2A. K. Geim, Science 324, 1530 (2009).3A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K.

Geim, Rev. Mod. Phys. 81, 109 (2009).4S. Park and R. S. Ruoff, Nat. Nanotechnol. 4, 217 (2009).5C. N. R. Rao, A. K. Sood, K. S. Subrahmanyam, and A. Govindaraj,

Angew. Chem., Int. Ed. 48, 7752 (2009).6P. Kumar, RSC Adv. 3, 11987 (2013).7J. N. Coleman, M. Lotya, A. O’Neill, and S. D. Bergin, Science 331, 568(2011).

8P. Kumar, K. S. Subrahmanyam, and C. N. R. Rao, Mater. Express 1, 252(2011).

9P. Kumar, B. Das, B. Chitara, and K. S. Subrahmanyam, Macromol.

Chem. Phys. 213, 1146–1163 (2012).10W. Choi, I. Lahiri, R. Seelaboyina, and Y. S. Kang, Crit. Rev. Solid State

Mater. Sci. 35, 52 (2010.).11P. Kumar, L. S. Panchakarla, and C. N. R. Rao, Nanoscale 3, 2127 (2011).12N. Behabtu, J. R. Lomeda, M. J. Green, and A. L. Higginbotham, Nat.

Nanotechnol. 5, 406 (2010).13Y. Liu and B. Yakobson, Nano Lett. 10, 2178 (2010).14S. J. Chae, F. Gǔnes, K. K. Kim, E. S. Kim, G. H. Han, S. M. Kim, H. J.

Shin, S. M. Yoon, J. Y. Choi, M. H. Park, C. W. Yang, D. Pribat, Y. H.

Lee, Adv. Mater. 21, 2328 (2009).

15P. Y. Huang, C. S. Ruiz-Vargas, A. M. Van der Zande, W. S. Whitney, M.

P. Levendorf, J. W. Kevek, S. Garg, J. S. Alden, C. J. Hustedt, Y. Zhu, J.

Park, P. L. McEuen, and D. A. Muller, Nature 469, 389 (2011).16D. A. Schmidt, T. Oha, and T. E. Beechem, Phys. Rev. B 84, 235422

(2011).17Q. Lu and R. Huang, Phys. Rev. B 81, 155410 (2010).18M. Zhou, A. Zhang, Z. Dai, Y. P. Feng, and C. Zhang, J. Phys. Chem. C

114, 16541 (2010).19C. Lee, X. D. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008).20X. Zhao, Q. H. Zhang, D. J. Chen, and P. Lu, Macromolecules 43, 2357

(2010).21B. Das, K. E. Prasad, U. Ramamurty, and C. N. R. Rao, Nanotechnology

20, 125705 (2009).22R. Grantab, B. S. Vivek, and S. R. Rodney, Science 330, 946 (2010).23Y Wei, J. Wu, H. Yin, X. Shi, R. Yang, and M. Dresselhaus, Nature

Mater. 11, 759 (2012).24J. Zhang, J. Zhao, and J. Lu, ACS Nano 6, 2704 (2012).25M. C. Wang, C. Yan, L. Ma, N. Hu, and M. W. Chen, Comput. Mater. Sci.

54, 236 (2012).26F. Ma, Y. J. Sun, D. Y. Ma, K. W. Xu, and P. K. Chu, Acta Mater. 59,

6783 (2011).27S. Plimpton, J. Comput. Phys. 117, 1 (1995).28S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472

(2000).29D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. Ni, and

S. B. Sinnott, J. Phys.: Condens. Matter 14, 783 (2002).30O. A. Shenderova and D. W. Brenner, Phys. Rev. B 61, 3877 (2000).31G. Cocco, E. Cadelano, and L. Colombo, Phys. Rev. B 81, 241412 (2010).32J. Zhou and R. Huang, J. Mech. Phys. Solids 56, 1609 (2008).33J. Li, Modell. Simul. Mater. Sci. Eng. 11, 173 (2003).34M. Calcagnotto, Y. Adachi, D. Ponge, and D. Raabe, Acta Mater. 59, 658

(2011).35P. W. Chung, Phys. Rev. B 73, 075433 (2006).36Q. Cheng, H. A. Wu, Y. Wang, and X. X. Wang, Appl. Phys. Lett. 95,

021911 (2009).37H. S. Park, K. Gall, and J. A. Zimmerman, Phys. Rev. Lett. 95, 255504

(2005).38F. Ma, S. L. Ma, K. W. Xu, and P. K. Chu, Appl. Phys. Lett. 92, 123103

(2008).

191906-5 Sun et al. Appl. Phys. Lett. 103, 191906 (2013)

http://dx.doi.org/10.1126/science.1102896http://dx.doi.org/10.1126/science.1158877http://dx.doi.org/10.1103/RevModPhys.81.109http://dx.doi.org/10.1038/nnano.2009.58http://dx.doi.org/10.1002/anie.200901678http://dx.doi.org/10.1039/c3ra41149dhttp://dx.doi.org/10.1126/science.1194975http://dx.doi.org/10.1166/mex.2011.1024http://dx.doi.org/10.1002/macp.201100451http://dx.doi.org/10.1002/macp.201100451http://dx.doi.org/10.1080/10408430903505036http://dx.doi.org/10.1080/10408430903505036http://dx.doi.org/10.1039/c1nr10137dhttp://dx.doi.org/10.1038/nnano.2010.86http://dx.doi.org/10.1038/nnano.2010.86http://dx.doi.org/10.1021/nl100988rhttp://dx.doi.org/10.1002/adma.200803016http://dx.doi.org/10.1038/nature09718http://dx.doi.org/10.1103/PhysRevB.84.235422http://dx.doi.org/10.1103/PhysRevB.81.155410http://dx.doi.org/10.1021/jp105368jhttp://dx.doi.org/10.1126/science.1157996http://dx.doi.org/10.1021/ma902862uhttp://dx.doi.org/10.1088/0957-4484/20/12/125705http://dx.doi.org/10.1126/science.1196893http://dx.doi.org/10.1038/nmat3370http://dx.doi.org/10.1038/nmat3370http://dx.doi.org/10.1021/nn3001356http://dx.doi.org/10.1016/j.commatsci.2011.10.032http://dx.doi.org/10.1016/j.actamat.2011.07.036http://dx.doi.org/10.1006/jcph.1995.1039http://dx.doi.org/10.1063/1.481208http://dx.doi.org/10.1088/0953-8984/14/4/312http://dx.doi.org/10.1103/PhysRevB.61.3877http://dx.doi.org/10.1103/PhysRevB.81.241412http://dx.doi.org/10.1016/j.jmps.2007.07.013http://dx.doi.org/10.1088/0965-0393/11/2/305http://dx.doi.org/10.1016/j.actamat.2010.10.002http://dx.doi.org/10.1103/PhysRevB.73.075433http://dx.doi.org/10.1063/1.3183584http://dx.doi.org/10.1103/PhysRevLett.95.255504http://dx.doi.org/10.1063/1.2902293