(PDF) Linear correlation between binding energy and Young's modulus in graphene nanoribbons

JOURNAL OF APPLIED PHYSICS 106, 054318 共2009兲 Linear correlation between binding energy and Young’s modulus in graphene nanoribbons Constantinos D. Zeinalipour-Yazdi1,a兲 and Constantinos Christofides2 1 Department of Chemistry, University of Cyprus, Nicosia 1678, Cyprus 2 Department of Physics, University of Cyprus, Nicosia 1678, Cyprus 共Received 18 May 2009; accepted 29 July 2009; published online 15 September 2009兲 Graphene nanoribbons 共GNRs兲 have been suggested as a promising material for its use as nanoelectromechanical reasonators for highly sensitive force, mass, and charge detection. Therefore the accurate determination of the size-dependent elastic properties of GNRs is desirable for the design of graphene-based nanoelectromechanical devices. In this study we determine the size-dependent Young’s modulus and carbon-carbon binding energy in a homologous series of GNRs, C4n2+6n+2H6n+4 共n = 2 – 12兲, with the use of all electron first principles computations. An unexpected linearity between the binding energy and Young’s modulus is observed, making possible the prediction of the size-dependent Young’s modulus of GNRs through a single point energy calculation of the GNR ground state. A quantitative-structure-property relationship is derived, which correlates Young’s modulus to the total energy and the number of carbon atoms within the ribbon. In the limit of extended graphene sheets we determine the value of Young’s modulus to be 1.09 TPa, in excellent agreement with experimental estimates derived for graphite and suspended graphene sheets. © 2009 American Institute of Physics. 关doi:10.1063/1.3211944兴 I. INTRODUCTION various ab initio methods employing atom centered basis functions23,24 and planewaves.25 Graphene is an allotropic form of carbon found both in The ideal structure of graphene consists of hexagonally nature and artificially produced by chemical vapor deposition arranged carbon atoms, each covalently bound to three 共CVD兲 of carbonaceous compounds.1 Although stacks of this neighboring carbons, through axial overlap of hybrid sp2 or- two-dimensional 共2D兲 material have been known for more bitals. The remaining 2pz orbitals on each carbon atom over- than a century2 共i.e., graphite兲, recently there has been re- lap in a parallel fashion resulting in a diffuse ␲-cloud located newed interest in graphene in which some of the predictions above and below the graphene layer. Stacks of graphene are of quantum electrodynamics such as the quantum Hall held together by weaker dispersion interactions that result effect,3–5 nonzero Berry’s phase,6 the Klein paradox,7,8 chiral from polarization effects26 of the diffuse ␲-clouds. The an- tunneling,8 and massless Dirac fermions9 are experimentally isotropic intra- and interlayer interactions of the carbon at- testable. Due to the unusual chemical and mechanical prop- oms, evident by the drastically different nearest neighbor dis- erties of nanographene 共i.e., chemical inertness, robustness, tance, known to be 1.42 and 3.35 Å, respectively, clearly and stiffness兲 graphene-based nanoelectromechanical sys- assign nanographene as a 2D solid.27–29 tems 共NEMSs兲 have also been proposed as the most sensitive The elastic properties are evaluated from first principles material for fundamental engineering applications, such as computations of relaxed and deformed GNRs of various mass, force, and charge detection using exfoliated graphene sizes. The stress energy 共⌬U␧兲 of a GNR around equilibrium as an electromechanical resonator.10–12 The elastic properties can be expanded in a Taylor series as a function of strain 共␧兲 of nanosized elements may considerably differ from their given by extended analogs;13 therefore, accurate knowledge of the size-dependent elastic properties of graphene nanoribbon ⌬U␧ = U共␧兲 − U共0兲 共GNR兲 共i.e., Young, shear, and bend moduli, and Poisson’s ratio兲 is desirable not only for the design of graphene-based ⳵ U共0兲 1 ⳵2U共0兲 2 1 ⳵3U共0兲 3 = ␧+ ␧ + ␧ + ¯, 共1兲 NEMSs12,14–16 but also for the future use of graphene in ⳵␧ 2 ⳵ ␧2 6 ⳵ ␧3 carbon-based electronic and magnetoelectronic devices17–19 since the possibility of such deformations may considerably where ␧ is defined as the fractional deformation in the direc- alter the electronic properties of GNRs.20 Experimental de- tion of the applied strain given by termination of Young’s modulus for graphene has only re- ⌬l cently become possible through indentation experiments us- ␧= , 共2兲 l ing atomic force microscopy 共AFM兲.14–16 Therefore, previous attempts have focused in calculating Young’s and U共␧兲 and U共0兲 are the total energies of the strained and modulus using various computational approaches, such as fully relaxed ground-state structure of the GNR, respectively. the membrane theory of shells,21 molecular mechanics,22 and By fitting the stress energy obtained from first principles computations with an nth order polynomial of the form a兲 Electronic mail:

. U共␧兲 = a␧ + b␧2 + c␧3 + ¯ the coefficients a, b, c, etc., have 0021-8979/2009/106共5兲/054318/5/$25.00 106, 054318-1 © 2009 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 054318-2 C. D. Zeinalipour-Yazdi and C. Christofides J. Appl. Phys. 106, 054318 共2009兲 (a) θ (b) 2b⬙ 1 ⳵2U共0兲 ∆ρ B= = , 共9兲 φ V V ⳵ ␤2 w l C650H76 ∆l′ where the bending stress 共␤兲 is given again by C552H70 ∆w C462H64 ⌬l⬘ 2␳共␾ − sin ␾兲 l C380H58 ␤= = , 共10兲 l l C306H52 C240H46 where ␳ is the radius of curvature of the cylinder to which C182H40 the GNR belongs to, and ⌬l⬘ is the in-plane deformation that ~ 6 nm C132H34 (c) C90H38 causes the bending of the GNR by the angle ␾ defined in Fig. C56H22 C30H16 1共b兲. In this report we determine the elastic properties of GNRs and the size-dependent Young’s modulus using all electron first principles computations. The elastic properties are determined in a systematic fashion for GNRs of varying n=2 3 4 5 6 7 8 9 10 11 12 size to the limiting case of an extended graphene sheet, and FIG. 1. 共Color online兲 共a兲 Schematic of GNR 共C240H46兲 under 共a兲 shear and quantitative structure-property relationships 共QSPRs兲 are de- 共b兲 bending stresses. 共c兲 Homologous series of GNRs, C4n2+6n+2H6n+4, ex- rived that can be used to predict the size-dependent Young’s amined for their elastic properties. Anchor atoms are indicated in black in 共a兲 and 共b兲. modulus of GNRs on the basis of their total ground-state energy and the number of carbon atoms within the nanorib- bon. It is noted that a linear correlation between the binding been determined. In the limit of small deformations 共strain/ energy 共BE兲 共sum of bond strengths兲 to Young’s modulus has shear/bend ⬍5%兲 the linear and cubic coefficients were not been previously reported in literature to the best of the found to be negligible 共a , c ⬇ 0兲, simplifying the strain, authors’ knowledge. shear, and bending energy of the GNR to a simple quadratic form, which is given by II. COMPUTATIONAL METHODS U共␧兲 ⬵ b␧2 , 共3兲 Relaxation of the atomic positions was obtained with the use of density functional theory 共DFT兲 computations imple- U共␮兲 ⬵ b⬘␮ , 2 共4兲 mented in the NWCHEM 共Refs. 31 and 32兲 code. The surface and of graphene was modeled by a rectangular H-terminated GNR of varying size shown in Fig. 1 in order to saturate the U共␤兲 ⬵ b⬙␤2 , 共5兲 dangling edge states. The models were either strained, sheared, or bent along the armchair, zigzag, or basal plane respectively. From Eqs. 共1兲 and 共3兲 the in-plane Young’s vectors, respectively, using carbon atoms of fixed nuclear modulus is given by positions 共anchor atoms兲. In order to reduce the computa- tional requirements, the GNR geometries, for straining and 2b 1 ⳵2U共0兲 E= = , 共6兲 shearing deformations, were constrained within the D2h and V V ⳵ ␧2 C2h point group symmetries, whereas for bending deforma- tions the structures were optimized within C2v point group where the GNR volume V = wlh, and the width 共w兲 and the symmetry. The exchange and correlation effects were consid- length 共l兲 are the distances between edge carbon atoms at the ered within the generalized gradient approximation using the middle of the armchair and zigzag edges 共see Fig. 1兲 plus commonly used Gaussian B3LYP hybrid exchange- twice the van der Waals 共vdW兲 radius of carbon in graphite30 correlation functional.33,34 For the calculations standard mo- 共rvdW = 1.78 Å兲. The height 共h兲 was approximated from the lecular basis sets of the STO-3Gⴱ,35,36 6-31Gⴱ,37 and aug-cc- interlayer separation of hexagonal graphite 共3.354 Å兲.28 The pVDZ 共Ref. 38兲 type have been adopted to test the shear modulus 共G兲 given from Eqs. 共1兲 and 共4兲 was evaluated convergence of basis set with respect to the elastic properties using for the smallest GNR in the homologous series of molecules 2b⬘ 1 ⳵2U共0兲 examined. The additional asterisk to the basis set denotes the G= = , 共7兲 inclusion of a shell of d polarization functions. The STO-3Gⴱ V V ⳵ ␮2 basis set was found to yield elastic properties to within 1% of where the shear stress is given as a function of the shear the aug-cc-pVDZ basis for the larger molecular-weight angle ␪ 共where ␪ is small兲 defined in Fig. 1共a兲 and given by GNR, therefore adopted for all subsequent calculations. En- ergetic minima for the smaller molecular-weight GNRs were ⌬w confirmed by vibrational analysis, whereas the larger mol- ␮= = tan ␪ . 共8兲 ecules were converged within the default convergence crite- l ria. This resulted in maximum forces and the root-mean- The bending modulus 共B兲 of the GNR is given by a similar square of the forces to be less than 0.01 and 0.001 eV/Å, relationship respectively. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 054318-3 C. D. Zeinalipour-Yazdi and C. Christofides J. Appl. Phys. 106, 054318 共2009兲 FIG. 2. 共a兲 Strain energy as a function of strain for the various GNRs FIG. 3. 共a兲 BE per carbon atom and 共b兲 Young’s modulus as a function of belonging to the homologous molecular sequence C4n2+6n+2H6n+4, 共b兲 Defor- GNR surface area 共AGNR兲. The curved line is the nonlinear fit to the data mation energy as a function of strain, shear, and bending for C240H46 共n using equations similar to the form of Eq. 共13兲. = 7兲. the electronic structure 共i.e., bandgap opening兲 of GNR III. RESULTS AND DISCUSSION through mechanical deformations.41 For the evaluation of the elastic properties of extended Young’s 共E兲, shear 共G兲, and bend 共B兲 moduli were GNRs 共2D graphene兲 the BE per carbon atom was first con- evaluated by compressing, shearing and bending the GNRs verged with respect to the surface area of the GNR 共see Fig. along the direction of the armchair and zigzag edges and the 3兲. The BE represents the stabilization of the carbon atoms 关0001兴 lattice vector, respectively, by translation of the an- within the GNR framework due to chemical bonding inter- chor atoms. For the evaluation of the elastic properties all actions, covalent in nature 共i.e., ␴ and ␲ bonds兲. The BE was atomic positions, except the anchor atom, were fully relaxed. obtained as the difference between the GNR total energy and Typically the unstrained configurations were calculated first that of the isolated carbon and hydrogen atoms calculated and then strain was applied in steps of 1% in units of strain with the same basis set computed using percentage 共␧%兲 for strains less than 4%. The results of these simulations are presented in Fig. 2共a兲 and clearly demon- BE = 2共EGNR − nCEC − nHEH兲/nC , 共12兲 strate the parabolic form of the strain energy as a function of where EGNR, EC, and EH are the total energies of the GNR, strain, reminiscent to the parabolic potential energy derived and the ground states of atomic carbon 共triplet, from Hook’s law for macroscopic springs. It is interesting to ⫺37.352 397 11Hr兲 and hydrogen 共doublet, note that the same strain can lead to increasingly high strain ⫺0.467 532 59Hr兲, and nC / nH is the number of carbon/ energies in the larger GNR, due to the additivity of the en- hydrogen atoms given by 4n2 + 6n + 2 and 6n + 4 共n = 2 – 12兲, ergy required to compress/elongate a larger number of respectively. carbon-carbon bonds, within the graphene network. In Fig. For the hydrogen terminated GNRs examined, we note 2共b兲 we compare the deformation energies of the largest that the size-dependent BE of the carbon network has the GNR examined for strain, shear, and bending deformations. form On one hand, it is evident that any attempt to strain GNRs would result in bending of the basal plane of the GNR since C BE = BEbulk − 1/2 , 共13兲 for the same deformation the strain energy is one order of AGNR magnitude higher than the shear energy, in agreement with the rippling of suspended graphene sheets previously where BEbulk is the extrapolated BE of an extended graphene reported.39 On the other hand, shear deformations do not sheet, AGNR is the basal plane area, and C is the size- drastically perturb the GNR internal energy, since they only dependent BE stabilization constant 共C = 0.0543⫾ involve bond angle changes, generally accompanied by 0.0001 eV nm/ C atom兲 due to the gradual decrease in the smaller force constants. We note that for supported graphene, edge effects as a function of the GNR surface area. The such as the case of graphite or supported graphene layers, the origin of these edge effects may be caused by various factors strain energy would have to overcome the dispersion forces such as 共a兲 edge states42 as a result of quantum confinement, between the layers of about 0.08 eV/C atom26,40 共i.e., vdW 共b兲 edge stress43,44 due to repulsive or attractive interactions interactions兲 for graphene layers to detach from their neigh- between chemical moieties at the periphery of the GNR, boring layers. Therefore for C240H46 the vdW interactions and/or 共c兲 Peierls45 instability. In any case the edge effect between two adjacent completely overlapping GNRs are causes considerable bond length alternation, especially at the about UvdW = 19.2 eV. From Eqs. 共3兲 and 共6兲 the relationship armchair edge as we observe in our models, which increases between strain and strain energy is given by the total energy of the GNR 共EGNR兲 due to bond compression and elongation at the periphery. Through Eq. 共12兲 increase in ␧= 冑 2U共␧兲 EV . 共11兲 EGNR would cause considerable decrease in the BE 共more exothermic兲, which as we demonstrate in Fig. 3共a兲 dissipates as a function of the lateral dimensions of the GNR. The same So it is expected that bending of the supported GNR will behavior is observed for Young’s modulus and presumably occur when U共␧兲 ⱖ UvdW, which in this particular case hap- for the remaining elastic properties 共i.e., shear and bend pens when the strain percentage exceeds the threshold value moduli兲. It is evident that due to the delocalized electronic of 4.7%. This value should be useful for attempts to modify structure of GNRs there is a considerable expansion of the Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 054318-4 C. D. Zeinalipour-Yazdi and C. Christofides J. Appl. Phys. 106, 054318 共2009兲 Eexc共tot兲 = 兺 Eexc共AB兲 = 兺 兺 P␮␭ P␯␴具␮␭兩␯␴典, 共15兲 where P␮␭ and P␯␴ are the elements of the density matrix that describe the overlap between the atomic orbitals. Since this quantity is essentially nCBE/ 2 in our treatment one can state that the static molecular polarizability of a GNR of arbitrary shape and dimensions 共i.e., PAHs兲 is inversely pro- portional to the BE ¯astatic ⬀ 2 / nCBE. This correlation com- bined with the so called “compressibility sum rule,” from the many body theory of charged systems,51 may provide a physical picture that explains the unexpected linearity we observe between E and BE since the polarizability 共viewed as a response function to a charged field兲 is directly propor- FIG. 4. Graph showing the linear correlation between the BE and Young’s tional to the compressibility, the inverse compressibility be- modulus of the homologous series of GNRs examined. ing equal to Young’s modulus. We argue that when the GNR is uniaxially strained or stressed, the diffuse electron density quantum confinement dimensions reaching widths as large as at the Fermi level 共mostly bound ␴ and ␲ states兲, to which ¯astatic is mostly attributed to,26,52 is polarized in the direction 6 nm in order to reach bulk properties. This is in contrast to other three-dimensional solids such as Si 具001典 nanowires46 perpendicular to the basal plane of the GNR. This polariza- tion decreases the effective polarizability ¯aeff of the GNR in where the elastic properties converge at considerable smaller its deformed state. We note that once the deformation ex- dimensions 共⬃3 – 4 nm兲 due to their semiconducting proper- ceeds the proportional limit and ¯aeff = 0, any further deforma- ties, which are described by less diffuse electron densities at tions reduce the interatomic interactions 共BE兲 resulting in the the Fermi level due to the existence of a bandgap. The find- converging trend commonly observed in stress-strain curves ings here confirm that edge stress may be an important factor of 2D graphene.25 that regulates the size-dependent elastic properties in GNR. Using the extrapolated value of Young’s modulus in the In Fig. 4 we present an interesting linear correlation be- limit of extended graphene sheets in Fig. 3共b兲 we obtain an E tween E and BE in the homologous series of GNR examined, of 1.09 TPa, in good agreement with periodic HF/ 6-31Gⴱ which in combination with Eq. 共12兲 suggests that Young’s and planewave DFT calculations that determined 0.89–1.23 modulus can be predicted on the basis of single point total 共Ref. 24兲 and 1.05 TPa,25 respectively, and with the in-plane energy calculations of the GNR ground state using the fol- E of bulk graphite known to be 1 共Ref. 53兲 and lowing relationship: 1.02⫾ 0.03 TPa,54 respectively; whereas a smaller agree- ment is observed in tip-induced deformation experiments us- ing AFM for chemically derived single graphene sheets,16 ␻ graphene stacks 共less than 5兲,14 and exfoliated graphene E= 关EGNR − 共4n2 + 6n + 2兲EC 共2n + 3n + 1兲 2 monolayers,15 which have determined Young’s moduli of − 共6n + 4兲EH兴 + ␥ , 共14兲 0.25, 0.5, and 1.0 TPa, respectively. Young’s modulus in GNRs obtains the value of extended graphene sheets only when the lateral dimensions are larger than 8 nm 共dimen- where the coefficients ␻ and ␥ are given by 5.2622 共TPa C sions of n = 12 GNR兲. It is therefore suggested that the sus- atom/eV兲 and 3.1824 共TPa兲, respectively. Note that these co- pended portion of the GNR used in graphene-based NEMS efficients are accurate only when the B3LYP/ STO-3Gⴱ exceeds this threshold value; otherwise, Young’s modulus method is used to evaluate the total energy, but in a similar and consequently the fundamental resonating frequency of manner these coefficients can be determined at any level of the NEMS will be size dependent. theory desirable. Such a nearly perfect linear correlation be- tween the elastic properties and the BE energy is reported for IV. CONCLUSIONS the first time, although certain correlations of bond energies and forces’ constants of isolated chemical bonds have previ- We have determined Young’s modulus 共E兲 and carbon- ously been suggested.47–49 An important advantage of this carbon BE in a homologous series of GNRs, C4n2+6n+2H6n+4 simple relationship is that such an assessment of E is com- 共n = 2 – 12兲, with the use of all electron first principles com- putationally less tedious since it requires only a single point putations. An interesting linearity between the BE and energy calculation. Furthermore one can extend the argu- Young’s modulus is observed, which is explained by the de- ments here to the other elastic properties such as the shear crease in the molecular polarizability due to deformations and bend moduli. within the proportional limit of GNRs. The unexpected lin- It has been recently shown50 that for GNR fragments, earity between E and BE makes possible the prediction of such as polycyclic aromatic hydrocarbons 共PAHs兲, the total the size-dependent Young’s modulus of GNRs through a static molecular polarizability 共a ¯ static兲 is inversely propor- single point energy calculation of the GNR ground state. A tional to the total molecular two-center exchange energy, a QSPR is derived that correlates Young’s modulus to the total measure of the interatomic interactions, defined as energy and the number of carbon atoms within the ribbon. In Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 054318-5 C. D. Zeinalipour-Yazdi and C. Christofides J. Appl. Phys. 106, 054318 共2009兲 22 the limit of extended graphene sheets we determine the value A. Sakhaee-Pour, Solid State Commun. 149, 91 共2009兲. 23 K. N. Kudin, G. E. Scusceria, and B. Y. Yakobsen, Phys. Rev. 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