Letter pubs.acs.org/NanoLett
Periodically Modulated Size-Dependent Elastic Properties of Armchair Graphene Nanoribbons X. Li,† Tong-Yi Zhang,‡ and Y. J. Su*,† †
Corrosion and Protection Center, Key Laboratory for Environmental Fracture (MOE), University of Science and Technology Beijing, Beijing 100083, China ‡ Shanghai University Materials Genome Institute and Shanghai Materials Genome Institute, Shanghai University, 99 Shangda Road, Shanghai 200444, China ABSTRACT: First-principles calculations were conducted on armchair graphene nanoribbons (AGNRs) to simulate the elastic behavior of AGNRs with hydrogen-terminated and bare edges. The results show width-dependent elastic properties with a periodicity of three, which depends on the nature of edge. The edge eigenstress and eigendisplacement models are able to predict the widthdependent nominal Young’s modulus and Poisson’s ratio, while the Clar structure explains the crucial role of edges in the periodically modulated size-dependent elastic properties. KEYWORDS: Graphene nanoribbons, size-dependent, Young’s modulus, Poisson’s ratio
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First-principles calculations were performed by using the Vienna ab initio simulation package (VASP)35 with the projectaugmented wave (PAW)36 potentials on the 2p and 2s valence states of C atoms and the Perdew−Burke−Ernzerhof (GGAPBE) exchange correlation functional.37 A plane wave basis set with an energy cutoff of 550 eV, and a Monkhorst−Pack38 kpoint mesh of 39 × 39 × 1 and 21 × 1 × 1 were employed for the bulk graphene and AGNRs, respectively. The energy cutoff value and the number of k-points were tested to ensure the accuracy of 0.1 meV in the total energy. The equilibrium carbon−carbon distance of the stress-free bulk graphene was determined by energy minimization to be 0.1425 nm, which is in good agreement with the experimentally reported value of 0.1419 nm.39 An AGNR with dimensions L0 × w0 × h0 was constructed with the stress-free bulk lattice constant, where L0, w0, and h0 = 0.34 nm denote the length, width and thickness of the unrelaxed ribbon, respectively. The equal-mass method40,41 was used to calculate the unrelaxed AGNR width w0. The calculated structure is periodic along the AGNR length and contains 15 and 20 Å thick vacuum layers, respectively, along the AGNR width and thickness directions. The vacuum layers were sufficient to approximately eliminate the interaction between AGNRs. Hydrogen atoms were added onto each of the edge carbon atoms for the hydrogenterminated edges of AGNRs (HAGNRs), as illustrated in Figure 1. The size of the hydrogen atoms, treated as adsorbates, was not considered in the calculation of the HAGNR width such that its value was equal to the corresponding AGNR width. The width index n, which is the number of carbon atoms
raphene has been widely and intensively studied since its discovery in 2004.1 This is because graphene possesses prominent mechanical, electronic, thermal and optical properties.1 Young’s modulus and intrinsic strength of graphene are extremely high: approximately 1.0 TPa2,3 and 130 GPa,2 respectively. A graphene nanoribbon (GNR) is a graphene strip with an ultranarrow width. The presence of free edges induces the width-dependent Young’s moduli and Poisson’s ratios of GNRs,4−14 as summarized in Table 1. The reported results, however, are inconsistent and some are contradictory one to another, which stimulate the present study. GNRs are classified into armchair graphene nanoribbons (AGNRs) and zigzag graphene nanoribbons (ZGNRs) based on of the edge configuration. The band gap,15−25 crystal structure,23,26 excitonic properties,24 edge stress and edge energy27,28 of AGNRs exhibit three distinct periodicities with the width index n of the number of dimer lines along the width. The three periodicities are interpreted in terms of Clar’s structure26,29−33 and expected to play an important role in the width-dependent nominal Young’s modulus of AGNRs. In atomistic calculations, an AGNR is setup with the lattice constant of stress-free bulk graphene. Before relaxation the free edges of the newly set AGNR have substantially high excess energy and eigenstress34 and thus relaxation will occur unavoidably in the AGNR to reduce the excess energy and edge eigenstress. The relaxation, however, causes initial deformation of the AGNR, which is gauged by initial strain. In the present atomistic calculations, the relaxation was divided into two steps of normal relaxation and parallel relaxation to systematically investigate edge properties. Tensile/compressive tests were then simulated on the relaxed AGNR to determine the nominal Young’s modulus and Poisson’s ratio. © XXXX American Chemical Society
Received: January 30, 2015 Revised: June 29, 2015
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DOI: 10.1021/acs.nanolett.5b00399 Nano Lett. XXXX, XXX, XXX−XXX
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Nano Letters Table 1. Reported Size-Dependent Young’s Modulus and Poisson’s Ratio of AGNRs and HAGNRsa
Symbols “→ pointing up and right”, “→ pointing down and right”, and “∼” represent “increase”, “decrease”, and “complex variation”, respectively, with increasing width of the GNR.
a
When the deformation is linear, the nominal Young’s modulus is given by34
Yn = Yc + 2Ye/w0
(1)
where Yc and Ye are the core and edge moduli, respectively. The applied strain induces a lateral strain, εa⊥, along the width direction (perpendicular to the edge). The perpendicular strain εa⊥ was measured in the simulations, enabling the determination of the nominal Poisson’s ratio, νn = −εa⊥/εa∥, for each AGNR (or HAGNR). According to the surface eigendisplacement model,42 we express the nominal Poisson’s ratio by νn = νc + 2νe/w0
(2)
where vc and ve are the core and edge Poisson’s ratios, respectively. The unrelaxed AGNR possesses a high unrelaxed potential unr energy Uunr = U0 + Uunr exc , where U0 and Uexc denote the reference energy of the stress-free bulk graphene and the unrelaxed excess energy, respectively. The edge is usually treated to be one-dimensional. Thus, the edge energy density at 27 the unrelaxed state is calculated by γunr = Uunr The exc /(2L0). normal relaxation reduces the potential energy from Uunr to Unr nr = U0 + Unr exc, where Uexc denotes the excess energy after normal relaxation, and the edge energy density to γur = Unr exc/(2L0). The parallel relaxation further reduces the potential energy to Uini = ini ini 43 ini ini ini U0 + Uini exc(s+c) = U0 + Wc + Uexc, where Uexc(s+c), Uexc and Wc denotes the excess energy relative to the stress-free bulk counterpart, the excess energy relative to the deformed bulk counterpart, and the strain energy of the bulk counterpart by the relaxation induced initial deformation. For simplicity, the stress-free bulk counterpart is taken as reference here and the edge energy density is calculated from γini = Uini exc(s+c)/(2L0). The two-step-relaxation changes the AGNR dimensions to Lini × wini × h0 with Lini = L0 + ΔL and wini = w0 + 2s0 + Δw, where s0 is the edge eigendisplacement and Δw is caused by the Poisson’s ratio effect. When the initial deformation is small and linear, the initial strain is calculated with εini = (Lini-L0)/L0. Figure 2 shows the edge energy density versus the width index for the AGNRs and HGNRs at the unrelaxed, after normal relaxation, and fully relaxed states, clearly indicating the modulation of the three periodicities on the width-dependent edge density, which is regarded as an quantum manifestation.27 The periodic modulation is caused by three types of dangling bonds on the edges.30 When hydrogen atoms saturate the dangling bonds of the AGNR edges, the edge energy density is greatly reduced. Figure 2 shows that the edge energy density of a HAGNR is
Figure 1. Schematic of the calculated AGNRs (without hydrogen atoms) and HAGNRs (with hydrogen atoms).
across the ribbon width, was adopted to represent the width in this study and the calculated AGNR widths ranged from 0.7 to 2.1 nm. The as-established AGNR was first relaxed along the width direction with fixed length dimension, which is called the normal relaxation. The normal relaxation was followed by the parallel relaxation, in which the atoms were allowed to move in all three directions, which indicates that at equilibrium, zero total force along the length direction must be satisfied along any lateral section perpendicular to the length and the tractionfree boundary condition must be met along the AGNR edges and faces. For HAGNRs, the hydrogen atoms were allowed to move in all three directions during the relaxation. Uniaxial compression/tensile tests were simulated on the relaxed AGNRs and HAGNRs. A uniaxial strain εa∥ was applied along the length direction by adjusting the periodic length L from 0% to 1% (or −1%) with each increment (or decrement) of 0.2%. After each increment (or decrement), the energy was minimized to ensure that the simulated system reached a new equilibrium state. For each of the simulated AGNRs or HAGNRs, the strain energy per unit volume was calculated as a function of applied strain. Differentiating the strain energy density with respect to the applied strain gave the applied stress and further differentiating the stress with respect to the applied strain yielded the nominal Young’s modulus, Yn. In the present study, the analysis is carried out in the Lagrangian coordinates. B
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termination, the curve can be perfectly fitted by eq 1 and the fitting gives the core and edge moduli, which are tableted in Table 2. The bulk Young’s modulus is in good agreement with the experimentally measured value.3 The negative edge Young’s modulus suggests that the nominal uniaxial Young’s modulus is smaller than the bulk Young’s modulus. Figure 3b shows the nominal Poisson’s ratio versus the width, indicating again the obvious periodicity-modulated width-dependent behavior. The AGNR has a higher nominal Poisson’s ratio than the corresponding HAGNR. Fitting the curves in Figure 3b yields the core and edge Poisson’s ratios, which are also listed in Table 2. The edge Poisson’s ratio represents the excess Poisson’s ratio induced by the presence of an edge. After relaxation, the ribbon is at the minimum energy state, ini ini which requires zero total force, i.e., Fini c + Fe = 0, where Fc = ini ini ini ini 0 w = Y ε w and F = 2σ = 2(σ + Y ε ) denote the core σini c 0 c 0 e e e e force and edge force per unit length of the two edges, respectively, with σ0e being the edge eigenstress. The initial strain is then calculated from the force balance equation to be
εini =
Figure 2. Edge energy densities of AGNRs and HAGNRs as a function of the width index.
−2σe0 Ynw0
(3)
Equation 3 indicates that the initial strain depends on the ribbon width. The wider the ribbon is, the smaller the initial deformation will be. Figure 4 shows that the initial strain decreases as the width increases and exhibits also periodic modulation. As expected, the initial strain in a HAGNR was also substantially lower than that in a corresponding AGNR, as shown in Figure 4. In addition, the three-periodicity modulation is more obvious in HAGNRs than in AGNRs. Fitting the initial strain versus ribbon width for each edge termination, we determine the edge eigenstress and list the value in Table 2. All values of the eigenstresses are negative, which suggests that the edges of AGNRs and HAGNRs are compressively stressed when constructed with the bulk graphene lattice constant without any deformation. To release the compressive edge eigenstresses, tensile initial strains must be induced in the relaxation. The absolute value of the edge eigenstress of the HAGNRs is lower than that of the AGNRs because of the saturation of the dangling bonds by hydrogen atoms. The difference in the edge eigenstress also explains the higher initial strain and the more severe width-dependent effect in the AGHRs compared to those in the HAGNRs. The charge density was calculated to understand the difference in the periodically modulated width-dependent behavior between AGNRs and HAGNRs. The charge density of bulk graphene was calculated as a reference (Figure 5a). The C−C bonds on the edges, even after relaxation, possess higher charge density, as shown in Figure 5b for the n = 9 AGNR, which indicates high excess edge energy and edge stress. Hydrogen-terminated edges diminish the obviously high charge density for the n = 9 HAGNR, as indicated in Figure 5c, which leads to low excess edge energy and edge stress. The charge density difference is introduced to investigate the hydrogen effect, which is defined as Δρ = ρHAGNR − ρAGNR − ρH, where ρHAGNR, ρAGNR and ρH are the charge densities of the total HAGNR system and the unperturbed charge densities of the AGNR and H atoms, respectively. Figure 5d shows the charge density difference of the n = 9 HAGNR, indicating that H atoms make electrons move from the C−C bonds to the C−H bonds so that the edge energy density and edge eigenstress of HAGNRs are lower. Consequently, the width-dependence of
one-to-two orders in magnitude smaller than that of an AGNR with the same width, which is consistent with the results of a previously reported DFT study.27,28 Figure 3a shows the nominal uniaxial Young’s modulus Yn versus the ribbon width. In general, the nominal Young’s
Figure 3. Uniaxial Young’s modulus (a) and Poisson’s ratio (b) of AGNRs and HAGNRs vs. the width.
modulus of an AGNR is lower than that of a HAGNR with the same width. The nominal Young’s modulus of AGNRs exhibits more severe width-dependent behavior and less oscillation than that of HAGNRs. A 2D AGNR is considered to be a composite of a geometrical 2D core and two 1D edges. For each edge C
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Nano Letters Table 2. Edge Young’s Modulus, Edge Poisson’s Ratio and Eigenstress of AGNRs and HAGNRs Young’s modulus width index
bulk (GPa)
edge (N m−1)
Poisson’s ratio edgea (N m−1)
3n 3n + 1 3n − 1
976 993 981
−47.17 −54.23 −54.24
−57.80 −57.23 −64.72
3n 3n + 1 3n − 1
1000 1010 1010
−39.45 −40.73 −55.39
−36.21 −35.13 −48.31
bulk AGNRs 0.181 0.161 0.169 HAGNRs 0.146 0.156 0.157
edge (10−11 m)
edgea (10−11 m)
eigenstress (N/m)
4.60 5.38 5.89
4.86 4.58 5.45
−7.08 −6.79 −6.62
1.84 1.23 2.37
0.33 0.05 1.11
−3.19 −2.78 −2.76
a The results are calculated by fitting nominal Young’s modulus (or nominal Poisson’s ratio) vs. width with a fixed bulk Young’s modulus of 997.5 GPa (or bulk Poisson’s ratio of 0.176). The bulk Young’s modulus and Poisson’s ratio were determined from the simulations of compressive and tensile tests on stress-free bulk graphene.
the Clar formula, is the most representative one.26 The Clar sextet might be expressed by its unique area or perimeter, which can be determined from first-principles calculation. Figure 6 shows the perimeter of each hexagon ring in HAGNRs with the widths ranging from 6 to 17. In the case of 3n (6, 9, 12, and 15), there is only one Clar formula and the Clar sextets have a smaller perimeter, corresponding to the bright area in Figure 6. In the case of 3n + 1 (7, 10, 13, and 16), the result shows that the structure is a mixture of two possible Clar structures,26,29 which are represented by dotted circles. The non-Clar sextet appears in dark gray, corresponding to the larger area. In the case of 3n − 1 (8, 11, 14 and 17), there are more than two possible Clar structures,26,29 where the middle hexagons are uniformly distributed except of the edge hexagons. Clearly, Figure 6 shows three Clar structures with distinct bonding distributions and we believe this is the reason for the periodically modulated width-dependent behaviors in AGNRs. Honeycomb-like armchair nanoribbons, such as molybdenum disulfide and silicon nanoribbons, have Clar’s structure exhibit the three periodicities in energy gap.49,50 On the basis of the present study, it is reasonably to predict periodically modulated size-dependent elastic, electrical and excitonic properties in these honeycomb-like armchair nanoribbons.
Figure 4. Initial strain of AGNRs and HAGNRs as a function of the width index.
the elastic properties of HAGNRs is much less obvious compared to that in AGNRs. Clar’s theory44 has been very successful in the understanding of the property and bonding oscillation in sp2-bonded materials such as poly cyclic aromatic hydrocarbons, carbon nanotubes45−48 and graphene nanoribbons.26,29−33 The foundation in Clar’s theory is the Clar sextet, which is defined as six πelectrons localized in a single hexagons ring separated from adjacent rings by C−C single bonds. For a given molecule, the representation with a maximum number of Clar sextets, called
Figure 5. Charge density in bulk graphene (a), the n = 9 AGNR (b), and the n = 9 HAGNR (c), and the charge density difference of the n = 9 HAGNR (d). D
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the width-dependent nominal Young’s modulus and Poisson’s ratio.
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AUTHOR INFORMATION
Corresponding Author
*(Y.J.S.) Telephone and Fax: +86-10-62333884. E-mail: yjsu@ ustb.edu.cn. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This study was supported by the National Basic Research Program of China (Grant No. 2012CB937502).
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REFERENCES
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Figure 6. Clar formulas (left) and hexagon perimeters (right) of each of HAGNRs with width n ranging from 6 to 17, where the solid circle line represents a Clar sextet and the dotted circle represents the superposition of several Clar formulas.
These properties will extend the applications of nanoribbons in nanoscale devices. In summary, first-principle calculations were conducted on AGNRs and HAGNRs by controlled normal relaxation and parallel relaxation and by designed tensile/compressive tests. The relaxation-induced initial in-plane strain and the edge energy density illustrate periodically modulated width-dependent behaviors; i.e., smaller widths correspond to higher initial strain and to smaller edge energy densities of the relaxed AGNRs. The nominal Young’s modulus and Poisson’s ratio of AGNRs and HAGNRs also exhibit periodically modulated width-dependent behaviors. As the width increases, the nominal Young’s modulus increases, whereas the nominal Poisson’s ratio decreases, and both values approach their respective core values. All of the width-dependent properties are periodically modulated by three families. Three sets of values for the Young’s modulus and Poisson’s ratio of the edges can describe E
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