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A structural mechanics approach for the phonon dispersion analysis of graphene
Author’s Accepted Manuscript A structural mechanics approach for the phonon dispersion analysis of graphene X.H. Hou, Z.C. Deng, K. Zhang
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To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 19 September 2016 Revised date: 30 November 2016 Accepted date: 7 January 2017 Cite this article as: X.H. Hou, Z.C. Deng and K. Zhang, A structural mechanics approach for the phonon dispersion analysis of graphene, Physica E: Lowdimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2017.01.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A structural mechanics approach for the phonon dispersion analysis of graphene X.H. Hou, Z.C. Deng*, K. Zhang School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072,China
Corresponding author
. E-mail address: [email protected]
Abstract: A molecular structural mechanics model for the numerical simulation of phonon dispersion relations of graphene is developed by relating the C-C bond molecular potential energy to the strain energy of the equivalent beam-truss space frame. With the stiffness matrix known and further based on the periodic structure characteristics, the Bloch theorem is introduced to develop the dispersion relation of graphene sheet. Being different from the existing structural mechanics model, interactions between the fourth-nearest neighbor atoms are further simulated with beam elements to compensate the reduced stretching stiffness, where as a result not only the dispersion relations in the low frequency field are accurately achieved, but results in the high frequency field are also reasonably obtained. This work is expected to provide new opportunities for the dynamic properties analysis of graphene and future application in the engineering sector. Keywords: graphene sheet; energy equivalence; beam-truss space frame; phonon dispersion relation; low-dimensional nanostructure. 1. Introduction
Graphene, a two-dimensional sheet of graphite and the basis to obtain carbon nanotubes (CNTs), which is of great interest for the current researchers, is becoming a crucial area of research now[1]. The phonon dispersion of graphite is a recurrent topic in the literature[2], and the last years have seen a revival of interest due to the successful isolation of a single layer of graphite, i.e. graphene[3]. Several models have been proposed to calculate phonon dispersion of graphite as well as graphene, such as the ab-initio force constant approach[4, 5], first-principles calculations[6], and the molecular dynamics simulations[7]. For the widely-used force constant method, by imposing the symmetry constraints, Falkovsky[8] and Kumar et al[9] calculated the in-plane phonon dispersion in graphene including only the first- and second-nearest neighbor interactions. Kandemir and Aydin[10] included up to the third-nearest neighbor interactions together with a radial bond-bending interaction to calculate the phonon spectra of chiral single-walled carbon nanotubes and graphene within a mass-spring model; Similar mass-spring model was developed by Sahoo and Mishra[1], considering up to the fourth-nearest neighbor interactions, with the force constants taken from[11]. By fitting the force constants to the ab initio dispersion relation[5], the popular fourth-nearest neighbor force-constant approach yields an excellent fit for the low frequency modes and a moderately good fit (with a maximum deviation of 6%) for the high-frequency modes. Wirtz and Rubio[12] reduced the maximum deviation to 4% by taking into account the non-diagonal force-constant for the second-nearest neighbor interaction; Moreover they emphasized the importance of lattice-constant ( a 3ac c ,where ac c denotes
the nearest-neighbor carbon-carbon distance), where the small ( 0.4% ) change in the lattice constant affects strongly the high-frequency modes (up to 2% shift). A fifth-nearest neighbor force constants model, with the force constants fitted to the experimental inelastic x-ray scattering data, was further presented by Mohr et al[13], giving improved force-constants calculations of the phonon dispersion in both graphite and carbon nanotubes. From the structural characteristics of graphene sheet, it is logical to anticipate that there are potential relations between the deformations of graphene and the space-like structures. Different kinds of structural mechanics models have been developed in the past years, such as the beam model suggested by Li and Chou[14], who indicated that a carbon nanotube is a geometrical frame-like structure and the primary bonds between two nearest-neighboring atoms (the first-nearest neighbor interactions) act like load-bearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. Reasonable elastic modulus were obtained. However in the opinion of Leung et al[15], the C-C bond is not allowed to bend and thus a truss model was developed. By relating the bond-angle variation molecular potential energy to the stretching strain energy of the rods in an equivalent-truss model, Odegard et al[16] established an equivalent-continuum model for the graphite sheet and Leung et al[15]introduced the idea of 'spatial periodic strain' according to the laminar substructures for the zigzag graphite sheet and SWCNT. Many mechanical characteristics of graphite sheet, such as the Young's modulus, have been obtained via these approximate structural models. Accordingly the phonon dispersion
relation of graphene sheet can be understood as the vibration properties of the equivalent frame-like nanostructure, i.e. the frequency of the nanostructure. However, according to Zhang et al[17], results in the high-energy region in phonon dispersion analysis is not so satisfactory with the beam model, while for the truss model, although better results in high-energy region can be deduced, this is not true for the case of low frequency. The inter-belt model, which simulated the bond stretch interaction between nearest two atoms with a beam element and placed truss elements between the next-nearest neighbor atoms and the diagonal atoms, is thus introduced. The new proposed inter-belt model was proved to be more efficient that the other structure mechanics models, that is, results obtained by including up to the third-nearest neighbor interactions provide a better approximation for the phonon dispersion than the model considering only the first-neighbor interactions. However, as stated by Sahoo and Mishra[1], the k 2 dependence for the out-of-plane transverse acoustic branch ( k 2 ), obtained in experimental results and other calculations, appears only after including interactions up to the fourth-nearest neighbor atoms. To overcome this deficiency and also inspired by the widely used fourth-nearest neighbor force-constant approach, a structural mechanics model considering up to the fourth nearest-neighbor interactions is proposed in the paper and reasonable results are obtained. The present paper is composed with five sections including Section 1 above. Section 2 briefly describes the structural characteristics of graphene, where the geometrical similarity with the space frame structure is revealed. On the basis, a
structural mechanics model of graphene sheet is developed in Section 3, where the dispersion relation analysis of graphene sheet is further performed based on the equivalent beam-truss space frame structure. Section 4 compares and verifies the results obtained with the structural mechanics model proposed in Section 3. Section 5 finally summarizes the main work of the paper. The results obtained and structural mechanics approach developed in this paper are expected to provide new opportunities for the research of nanostructures, especially the low-dimensional graphene like structure. 2. Structural characteristics of graphene sheet In graphene (a single atomic layer of graphite) the carbon atoms form a honeycomb lattice as shown in Fig.1(a). The unit cell is illustrated as a dashed rhombus containing two atoms Ai , Bi , i 0,1, 2,
n1 , n2
, where a1 and a 2 are the basis vectors. The integer pair
is introduced to label the two-dimensional characteristic of the graphene, and
also to identify the location of the unit cell where with reference to the chosen primitive unit cell
0, 0 , the integer pair n1 , n2 identify any other cell obtained by
n1 translations along the a1 direction and n2 translations along the a 2 direction.
The basis vectors for the hexagonal graphene sheet are expressed in the global coordinate system x, y as
3 1 3 1 a1 , a, a2 , a 2 2 2 2
(1)
where a a1 a2 3ac c is the lattice constant of two-dimensional graphene and ac c is the nearest-neighbor carbon-carbon distance, i.e. acc 0.142nm .
Fig.1 (a) Two-dimensional honeycomb structure of graphene sheet with the unit cell n1 , n2 (containing two types of atoms Ai , Bi , i 0,1, 2, ) illustrated by dashed rhombus along the direction of basic vectors a1 and a 2 . (b) The Brillouin zone of two-dimensional graphene shown as the shaded hexagon with the basis vectors b1 and b 2 of the reciprocal lattice. Energy dispersion relations are obtained along the perimeter of the dashed triangle connecting the high symmetry points, , M ,and K .
The unit vectors b1 and b 2 of the reciprocal lattice are given correspondingly as
2 2 b1 , 3a a
2 2 , , b2 a 3a
(2)
The first Brillouin zone is thus selected as the shaded hexagon shown in Fig.1(b). Here we define the three high symmetry points, , K ,and M as the center, the corner, and the center of the edge, respectively. The energy dispersion relations are calculated along the boundary of the triangle MK shown by the dashed lines in Fig.1(b). 3. Phonon dispersion relation analysis of graphene
3.1 Structural mechanics model to graphene For the case of a graphene lattice subjected to small deformations, only the bond stretching, bond-angle variation energies and the non-bonded van der Waals interaction are considered. The dihedral angle torsion and out-of-plane torsion are negligible. The molecular potential energy of a graphene sheet with carbon-to-carbon bonds is thus given as follows[18]: U steric U r U U vdw
(3)
where U r and U represent the energies due to the bond stretching and the bond angle bending respectively; and U vdw represents that for the non-bonded van der Waals interaction. The main idea of molecular structural mechanics stems from the observation of geometrical
similarities
between
the two-dimensional graphene sheet
and
macroscopic frame structures. By equating the molecular potential energy of nano-structured materials with the mechanical strain energy of a representative structural mechanics model, the stiffness parameters can be obtained, where as a result the mechanical properties can be achieved via finite element method. The equivalent beam-truss structural mechanics model used in the paper (as shown in Fig.2) includes up to the fourth interactions. As listed in Table 1, with reference to the empirical constants of graphene sheet adopted by Li and Chou[14], both the stretching stiffness and the bending stiffness adopted in the structural mechanics model of Zhang et al[17] are greatly reduced, whereas only the reduction of bending stiffness was compensated with the truss
element assumed between the next-nearest atoms. To provide better results, a special beam-truss structural mechanics model is presented in the paper (Fig.2), where the reduction of stretching stiffness is compensated with another beam element IV placed between the fourth-nearest neighbor atoms.
Fig.2 Equivalent beam-truss structural mechanics model considering up to the fourth-nearest neighbor interactions. Table 1 Comparison of the stiffness parameters and force constants adopted in our paper and other structural mechanics models (Zhang et al[17] , Li and Chou[14]) This paper
EA 108 N I
Graphene sheet
6.6
7.0
9.2584
0.23
0.22
0.28
1.2439
EA 108 N
1.37
2.4
2.63
—
EA
III
108 N
0.7
0.668
0.668
—
EA
IV
108 N
0.77
—
—
—
-0.32
—
—
—
231.3
232.39
246.48
326
0.221
0.3470
0.3445
0.438
I
1028 N m2
II
EI IV 1028 N m2
Molecular mechanics force constants
Li and Chou [14]
6.31
EI Structural mechanics stiffness parameters
Numerical values Zhang et al[17] Armchair Zigzag SWCNT SWCNT
N m Kbond anglebending nN nm Kbond stretch
A detailed description of the structural mechanics model is provided as follows. the bond stretching interaction between the nearest two atoms is simulated by beam element I, whereas truss element II is placed between the next-nearest neighbor atoms to compensate the reduction of the bending stiffness. Truss element III is placed between the diagonal atoms to simulate the non-bonded var der Waals interaction. As an innovation point, another beam element IV is assumed between the fourth-nearest atoms to compensate the reduction of the stretching stiffness. The bond stretching energy is thus the sum of the stretch energy of beam element I and beam element IV, and the bending energy of beam element IV, whereas the bond angle bending energy is the sum of the bending energy of beam element I and the stretch energy of truss element II. The non-bonded van der Waals interaction is equivalent with the stretch energy of truss element III. The relation between the molecular potential energy and the mechanical strain energy can thus be expressed as
U r Kbond stretch RI LI
2
EA I EA IV EI IV 2 2 2 RI LI RIV L1V IV IV 2 LI 2 LIV 2 LIV
U Kbond anglebending I I 2
EI EA 2 2 I I RII LII 2 LI 2 LII I
II
(4) where Ri and Li i I , II , IV denote the deformed bond length and undeformed ones between the nearest-neighbor (marked with I), the next-nearest-neighbor (marked with II) and the fourth-nearest-neighbor (marked with IV) atoms respectively.
Corresponding values for Li are appended at the bottom of Fig.2. i and
i I , IV
i
represent the deformed bond angles and undeformed ones.
EA
i I , II , IV
EI
i I , IV
i
i
is the stretch stiffness for beam element or truss element and
is the bending stiffness of a beam element.
Fig.3 Schematic of the deformed geometry of the representative volume element
If it is assumed that the changes in bond angle are small, the following conditions can be achieved from Fig.3: RII LII
I I
LI 2 RIV LIV RI LI cos
IV IV where arctan
(5)
RIV LIV tan LIV
3 5 . Substituting Eq.(5) into Eq.(4) gives
Kbond stretch
1 25 3 I IV IV EA EA EI 2 LI 56 LIV 56 L3IV
Kbond anglebending
L2 1 EI I I EA II 2 LI 8LII
When the force constants Kbond-stretch and Kbond
angle-bending
(6)
are given, the stiffness
parameters can be confirmed. Inspecting Eq.(6), the determination of EI I and II EA have many combinations, similar condition exists for the determination of I IV IV EA , EA and EI .To obtain the optimal fit of the phonon dispersion relation to
the experimental data[6], a optimization procedure is performed to ascertain the stiffness parameters, where reasonable results are listed in Table 1. Compared with the stiffness parameters given by Li and Chou[14], both the stretching stiffness and the bending stiffness of beam element I are greatly reduced. Another beam element IV simulating the interaction between the fourth-nearest neighbor atoms is therefore added to compensate the reduction of the stretching stiffness. And a truss element II simulating the interaction between the next-nearest neighbor atoms is added to compensate the reduction of the bending stiffness. As listed in Table 1, similar stretching stiffness of truss element III with that used in Zhang et al[17]is adopted to account for the var der Waals interaction. Besides, a negative bending stiffness of beam element IV is adopted here to reproduce the
k 2 relation of the optical branch near the point. The corresponding molecular force field constants computed from Eq.(6) are also listed in Table 1.Once the sectional stiffness parameters EA and EI are known, the sectional stiffness matrix
K can be obtained. And then by using the solution procedure of stiffness matrix method for frame structures described in the next section, the phonon dispersion relation of the graphene sheet at the atomic scale can be simulated. 3.2 Frequency calculation of the equivalent beam-truss space frame structure By equivalent the graphene sheet to a special beam-truss space frame, the phonon
dispersion relations of graphene sheet can be understood as the vibration properties of the space frame. Therefore, by calculating the frequency of the equivalent structural mechanics model, the phonon spectra of the graphene structure can be obtained. Once the stiffness and mass matrices are obtained, the Bloch theorem, which allows the reduction of dimensions of the degree of freedom, can be introduced to analyze the wave propagation problem of the two-dimensional graphene sheet. For atoms A0 and B0 of the unit cell n1 , n2 shown in Fig.1(a), the equation of motion is given respectively by
(7) where the sum over n is taken up to the fourth-nearest neighbor interactions relative to the site A0 or B0 . As illustrated in Fig.4, n nI nII nIII nIV 3 6 3 6 18 . According to the Bloch theorem widely used for the wave propagation analysis of periodic structures, the displacement correlation can be given as (Fig.1(a)): u A1 eik1 u A0 , u A2 eik2 u A0 , u A3 ei ( k1 k2 )u A0 ,...... u B1 eik1 u B0 , u B2 eik2 u B0 , u B3 ei ( k1 k2 )u B0 ,......
(8)
where k1 and k 2 represent the components of wave vector k along the boundary of the irreducible Brillouin zone M K as shown in Fig.1(b) [11]. By substituting Eq.(8) into Eq.(7) and assuming the same eigen-frequencies for all u i , the following compact matrix form can be obtained as:
PAA k1 , k2 PAB k1 , k2 0 u A0 2 M 0 M u 0 P k , k P k , k B0 BA 1 2 BB 1 2
(9)
Detailed expressions of the stiffness matrix PAA , PAB , PBA and PBB are given in Appendix A.
Fig.4 Neighbor atoms of a graphene plane up to 4th nearest neighbors for (a) an A atom and (b) a B atom. For ease of viewing, only connections with the reference atoms ( A0 or B0 ) are illustrated.
It is obvious that Eq.(9) represents an eigenvalue problem and it can be solved for obtaining with two propagation constants k k1 , k2 given. The solutions obtained k1 , k2 are termed as the phase constant surfaces or dispersion relation. 4. Results and discussions The phonon dispersion relation of graphene sheet computed using the special beam-truss structural mechanics model is shown in Fig.5(a), where results obtained with the widely used force constant method considering up to the fourth-nearest
neighbor interactions[1] are shown in Fig.5(b) for comparison. Further comparison with the work by Koukaras et al [7] is illustrated in Fig.5(d), where results were obtained by using the molecular dynamics method. The three figures show a good agreement with each other, verifying the effectiveness of the structural mechanics method developed in the paper.
Fig.5 Phonon dispersion relations of graphene: (a) results obtained with the structural mechanics method presented in this paper; (b)results presented by Sahoo and Mishra[1]; (c) results calculated with the stiffness parameters taken from Zhang et al[17]. (d) results presented by Koukaras et al[7]. In contrary to the results of (a) to (c) presented along the boundary of the irreducible Brillouin zone in the direction of M K , results illustrated in (d) are presented in the direction of K M .
The phonon dispersion relation of graphene sheet shown in Fig.5(c) is calculated with stiffness parameters provided by Zhang et al[17], where up to the third-nearest neighbor interactions are considered. Through a comparison of Fig.5(a) and Fig.5(c), it is clear that the k 2 dependence for the out-of-plane transverse acoustic (OTA/ZA) branch[19], which as expected, appears only after including interactions up to the fourth nearest-neighbor. Therefore interactions at least up to the fourth-nearest neighbors are ought to be considered for the calculation of the phonon dispersion of graphene. Further comparison between Fig.5(a) and Fig.5(c) reveals that besides the
k 2 dependence of the OTA/ZA branch, the proposed space frame model not only yields an excellent fit for the low frequency modes, but also gives a good fit for the high-frequency modes, by taking into account the fourth-nearest neighbor interactions. To compare the results quantitatively, we have listed the values at high symmetry points , M , K obtained from different methods in Table 2, including the ab_initio method[5], the force constant method[1, 11, 20], the First Principles Calculations[6, 12, 21] and the experimental values[6, 22]. Results obtained in the paper are in good agreement with existing theoretical and experimental results. 5.Conclusions The two-dimensional graphene is modeled as a beam-truss space frame where the phonon dispersion relation of graphene sheet can be understood as the vibration properties of the space frame. With the stiffness and mass matrices achieved by equating the molecular potential energy of nano-structured materials to the
mechanical strain energy of the equivalent space frame model, the Bloch theorem is introduced for the wave propagation problem of the two-dimensional graphene sheet. Different from existing models, a beam element is added to equivalent interactions between the fourth-nearest neighbor atoms, where as a result the k 2 dependence for the out-of-plane transverse acoustic branch is detected. Besides, the dispersion relations in the low-energy region and high-energy region both show good agreement with those presented in the existing literatures, implying that the inclusion of interactions up to the fourth-nearest neighbor is necessary and also sufficient for reproducing the experimental data for graphene. This work gains insight into the role of structural mechanics method in the dynamic properties analysis of nanostructures. Acknowledgments The authors wish to thank the support from the National Natural Science Foundation of China under grant Nos.11372252 and 11402035
Table 2:Comparison of our results with other theoretical model and experimental results at 1 different high-symmetry points (Frequency, unit: cm ) Meth M K od used to obtai I I L IT OTO L IT OTO L OTA L IT OTO L OTA n the T T O O /ZO O O /ZO A /ZA O O /ZO A /ZA phon A A on dispe rsion Ab initio [5] FCM [1] FCM [11] FCM [20] FPC[ 6] FPC[ 12] FPC[ 21] Exp[ 6] Exp[ 22] SMM (this pape r)
16 90
16 90
860
15 80
14 75
1280
16 05
16 05
874
14 75
13 13
1202
15 88
15 88
863
14 99
13 69
1259
15 81
15 81
893
13 24
13 63
1279
15 81 15 97
15 81 15 97
825
13 50 13 68
14 25 14 28
1315
15 95
15 95
890
13 80
14 42
1339
15 77 15 80
15 65 15 80
13 23 13 90
13 90 13 23
1290
15 86
15 86
15 05
13 72
1256
893
868
837
1346
1290
6 5 1 8 0 0 7 7 3 0 3 3 7 6 7 6 5 6 6 6 5 5 1 5 — —
460
15 20
12 60
1260
89 0
432
13 05
12 22
1222
11 33
460
14 87
12 71
1271
10 11
308
13 49
11 99
1199
10 28
—
—
472
12 20 12 38
1220
6 6 3 2 7 6 6 6 3 1 6 8 — —
13 00 13 26
1238
10 02
475
13 71
12 46
1246
99 4
— 471
11 94 11 84
1194
6 3 0 7 5 1
12 65 13 13 15 13
12 69
6 7 0 6 8 3
483
5 2 0 5 8 0 5 6 7 4 9 5 —
520
535
—
5 3 5 5 3 5 —
1184
—
—
—
1269
96 6
5 9 2
592
580
567
495
—
535
—
*FCM: Force Constant Method; * FPC: First Principles Calculations; *SMM: Structural
Mechanics Method
Appendix A: PAA K
A0 B0
K
A0 B1
K
A0 B9
K
A0 B10
K
A0 A1 ik1
e K
K
A0 B2
K
K
A0 B11
A0 B3
K
A0 A2 ik2
e K
K
A0 A1
A0 B4
K
A0 A3 i k1 k2
e
K
A0 A2
K
A0 B5
K
K
A0 B6
K
A0 A3
K
A0 A4
A0 A4 i k2 k1
K
A0 A6 ik1
e
A0 B7
K e
K
A0 A6
K
A0 B8
K
A0 A7
A0 A7 ik2
e
(A.1)
PAB
K A0 B0 K A0 B1 eik1 K A0 B2 eik2 K A0 B3 ei k1 k2 K A0 B4 ei k1 k2 K A0 B5 ei k1 k2 K A0 B6 eik1 K A0 B7 eik2 K A0 B8 ei 2 k1 k2 K A0 B9 ei k1 2 k2 K A0 B10 e2ik1 K A0 B11 e2ik2 (A.2)
K B0 A0 K B0 A1 eik1 K B0 A2 eik2 K B0 A3 ei k1 k2 K B0 A4 ei k1 k2 K B0 A6 eik1 K B0 A7 eik2 PBA K B0 A12 ei 2 k1 k2 K B0 A13 ei k1 2k2 K B0 A14 e2ik1 K B0 A15 ei k1 k2 K B0 A16 e2ik2 (A.3)
PBB K
B0 A0
K
K
B0 A14
K
B0 B1 ik1
B0 A1
K
K
B0 A15
e K
B0 A2
K
B0 B2 ik2
K
B0 A16
e K
B0 A3
K
K
B0 B1
K
B0 B3 i k1 k2
e
B0 A4
K
B0 B2
K
B0 A6
K
K
B0 A7
B0 B3
K
B0 B4 i k2 k1
K
e
K
B0 B4
B0 A12
K
B0 B6 ik1
e
K
B0 B6
K
B0 A13
K
B0 B7
B0 B7 ik2
e
(A.4)
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Highlights
A structural mechanics method is developed for the dynamic properties analysis of low-dimensional graphene like nanostructure. The phonon dispersion analysis of nanostructures is extended from one-dimensional carbon nanotubes to two-dimensional graphene, with newly proposed theoretical formulas. Accurate results are obtained both in the low frequency field and the high frequency field. The k 2 dependence for the out-of-plane transverse acoustic branch is detected.
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PII: DOI: Reference:
S1386-9477(16)31050-5 http://dx.doi.org/10.1016/j.physe.2017.01.012 PHYSE12700
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 19 September 2016 Revised date: 30 November 2016 Accepted date: 7 January 2017 Cite this article as: X.H. Hou, Z.C. Deng and K. Zhang, A structural mechanics approach for the phonon dispersion analysis of graphene, Physica E: Lowdimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2017.01.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A structural mechanics approach for the phonon dispersion analysis of graphene X.H. Hou, Z.C. Deng*, K. Zhang School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an 710072,China
Corresponding author
. E-mail address: [email protected]
Abstract: A molecular structural mechanics model for the numerical simulation of phonon dispersion relations of graphene is developed by relating the C-C bond molecular potential energy to the strain energy of the equivalent beam-truss space frame. With the stiffness matrix known and further based on the periodic structure characteristics, the Bloch theorem is introduced to develop the dispersion relation of graphene sheet. Being different from the existing structural mechanics model, interactions between the fourth-nearest neighbor atoms are further simulated with beam elements to compensate the reduced stretching stiffness, where as a result not only the dispersion relations in the low frequency field are accurately achieved, but results in the high frequency field are also reasonably obtained. This work is expected to provide new opportunities for the dynamic properties analysis of graphene and future application in the engineering sector. Keywords: graphene sheet; energy equivalence; beam-truss space frame; phonon dispersion relation; low-dimensional nanostructure. 1. Introduction
Graphene, a two-dimensional sheet of graphite and the basis to obtain carbon nanotubes (CNTs), which is of great interest for the current researchers, is becoming a crucial area of research now[1]. The phonon dispersion of graphite is a recurrent topic in the literature[2], and the last years have seen a revival of interest due to the successful isolation of a single layer of graphite, i.e. graphene[3]. Several models have been proposed to calculate phonon dispersion of graphite as well as graphene, such as the ab-initio force constant approach[4, 5], first-principles calculations[6], and the molecular dynamics simulations[7]. For the widely-used force constant method, by imposing the symmetry constraints, Falkovsky[8] and Kumar et al[9] calculated the in-plane phonon dispersion in graphene including only the first- and second-nearest neighbor interactions. Kandemir and Aydin[10] included up to the third-nearest neighbor interactions together with a radial bond-bending interaction to calculate the phonon spectra of chiral single-walled carbon nanotubes and graphene within a mass-spring model; Similar mass-spring model was developed by Sahoo and Mishra[1], considering up to the fourth-nearest neighbor interactions, with the force constants taken from[11]. By fitting the force constants to the ab initio dispersion relation[5], the popular fourth-nearest neighbor force-constant approach yields an excellent fit for the low frequency modes and a moderately good fit (with a maximum deviation of 6%) for the high-frequency modes. Wirtz and Rubio[12] reduced the maximum deviation to 4% by taking into account the non-diagonal force-constant for the second-nearest neighbor interaction; Moreover they emphasized the importance of lattice-constant ( a 3ac c ,where ac c denotes
the nearest-neighbor carbon-carbon distance), where the small ( 0.4% ) change in the lattice constant affects strongly the high-frequency modes (up to 2% shift). A fifth-nearest neighbor force constants model, with the force constants fitted to the experimental inelastic x-ray scattering data, was further presented by Mohr et al[13], giving improved force-constants calculations of the phonon dispersion in both graphite and carbon nanotubes. From the structural characteristics of graphene sheet, it is logical to anticipate that there are potential relations between the deformations of graphene and the space-like structures. Different kinds of structural mechanics models have been developed in the past years, such as the beam model suggested by Li and Chou[14], who indicated that a carbon nanotube is a geometrical frame-like structure and the primary bonds between two nearest-neighboring atoms (the first-nearest neighbor interactions) act like load-bearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. Reasonable elastic modulus were obtained. However in the opinion of Leung et al[15], the C-C bond is not allowed to bend and thus a truss model was developed. By relating the bond-angle variation molecular potential energy to the stretching strain energy of the rods in an equivalent-truss model, Odegard et al[16] established an equivalent-continuum model for the graphite sheet and Leung et al[15]introduced the idea of 'spatial periodic strain' according to the laminar substructures for the zigzag graphite sheet and SWCNT. Many mechanical characteristics of graphite sheet, such as the Young's modulus, have been obtained via these approximate structural models. Accordingly the phonon dispersion
relation of graphene sheet can be understood as the vibration properties of the equivalent frame-like nanostructure, i.e. the frequency of the nanostructure. However, according to Zhang et al[17], results in the high-energy region in phonon dispersion analysis is not so satisfactory with the beam model, while for the truss model, although better results in high-energy region can be deduced, this is not true for the case of low frequency. The inter-belt model, which simulated the bond stretch interaction between nearest two atoms with a beam element and placed truss elements between the next-nearest neighbor atoms and the diagonal atoms, is thus introduced. The new proposed inter-belt model was proved to be more efficient that the other structure mechanics models, that is, results obtained by including up to the third-nearest neighbor interactions provide a better approximation for the phonon dispersion than the model considering only the first-neighbor interactions. However, as stated by Sahoo and Mishra[1], the k 2 dependence for the out-of-plane transverse acoustic branch ( k 2 ), obtained in experimental results and other calculations, appears only after including interactions up to the fourth-nearest neighbor atoms. To overcome this deficiency and also inspired by the widely used fourth-nearest neighbor force-constant approach, a structural mechanics model considering up to the fourth nearest-neighbor interactions is proposed in the paper and reasonable results are obtained. The present paper is composed with five sections including Section 1 above. Section 2 briefly describes the structural characteristics of graphene, where the geometrical similarity with the space frame structure is revealed. On the basis, a
structural mechanics model of graphene sheet is developed in Section 3, where the dispersion relation analysis of graphene sheet is further performed based on the equivalent beam-truss space frame structure. Section 4 compares and verifies the results obtained with the structural mechanics model proposed in Section 3. Section 5 finally summarizes the main work of the paper. The results obtained and structural mechanics approach developed in this paper are expected to provide new opportunities for the research of nanostructures, especially the low-dimensional graphene like structure. 2. Structural characteristics of graphene sheet In graphene (a single atomic layer of graphite) the carbon atoms form a honeycomb lattice as shown in Fig.1(a). The unit cell is illustrated as a dashed rhombus containing two atoms Ai , Bi , i 0,1, 2,
n1 , n2
, where a1 and a 2 are the basis vectors. The integer pair
is introduced to label the two-dimensional characteristic of the graphene, and
also to identify the location of the unit cell where with reference to the chosen primitive unit cell
0, 0 , the integer pair n1 , n2 identify any other cell obtained by
n1 translations along the a1 direction and n2 translations along the a 2 direction.
The basis vectors for the hexagonal graphene sheet are expressed in the global coordinate system x, y as
3 1 3 1 a1 , a, a2 , a 2 2 2 2
(1)
where a a1 a2 3ac c is the lattice constant of two-dimensional graphene and ac c is the nearest-neighbor carbon-carbon distance, i.e. acc 0.142nm .
Fig.1 (a) Two-dimensional honeycomb structure of graphene sheet with the unit cell n1 , n2 (containing two types of atoms Ai , Bi , i 0,1, 2, ) illustrated by dashed rhombus along the direction of basic vectors a1 and a 2 . (b) The Brillouin zone of two-dimensional graphene shown as the shaded hexagon with the basis vectors b1 and b 2 of the reciprocal lattice. Energy dispersion relations are obtained along the perimeter of the dashed triangle connecting the high symmetry points, , M ,and K .
The unit vectors b1 and b 2 of the reciprocal lattice are given correspondingly as
2 2 b1 , 3a a
2 2 , , b2 a 3a
(2)
The first Brillouin zone is thus selected as the shaded hexagon shown in Fig.1(b). Here we define the three high symmetry points, , K ,and M as the center, the corner, and the center of the edge, respectively. The energy dispersion relations are calculated along the boundary of the triangle MK shown by the dashed lines in Fig.1(b). 3. Phonon dispersion relation analysis of graphene
3.1 Structural mechanics model to graphene For the case of a graphene lattice subjected to small deformations, only the bond stretching, bond-angle variation energies and the non-bonded van der Waals interaction are considered. The dihedral angle torsion and out-of-plane torsion are negligible. The molecular potential energy of a graphene sheet with carbon-to-carbon bonds is thus given as follows[18]: U steric U r U U vdw
(3)
where U r and U represent the energies due to the bond stretching and the bond angle bending respectively; and U vdw represents that for the non-bonded van der Waals interaction. The main idea of molecular structural mechanics stems from the observation of geometrical
similarities
between
the two-dimensional graphene sheet
and
macroscopic frame structures. By equating the molecular potential energy of nano-structured materials with the mechanical strain energy of a representative structural mechanics model, the stiffness parameters can be obtained, where as a result the mechanical properties can be achieved via finite element method. The equivalent beam-truss structural mechanics model used in the paper (as shown in Fig.2) includes up to the fourth interactions. As listed in Table 1, with reference to the empirical constants of graphene sheet adopted by Li and Chou[14], both the stretching stiffness and the bending stiffness adopted in the structural mechanics model of Zhang et al[17] are greatly reduced, whereas only the reduction of bending stiffness was compensated with the truss
element assumed between the next-nearest atoms. To provide better results, a special beam-truss structural mechanics model is presented in the paper (Fig.2), where the reduction of stretching stiffness is compensated with another beam element IV placed between the fourth-nearest neighbor atoms.
Fig.2 Equivalent beam-truss structural mechanics model considering up to the fourth-nearest neighbor interactions. Table 1 Comparison of the stiffness parameters and force constants adopted in our paper and other structural mechanics models (Zhang et al[17] , Li and Chou[14]) This paper
EA 108 N I
Graphene sheet
6.6
7.0
9.2584
0.23
0.22
0.28
1.2439
EA 108 N
1.37
2.4
2.63
—
EA
III
108 N
0.7
0.668
0.668
—
EA
IV
108 N
0.77
—
—
—
-0.32
—
—
—
231.3
232.39
246.48
326
0.221
0.3470
0.3445
0.438
I
1028 N m2
II
EI IV 1028 N m2
Molecular mechanics force constants
Li and Chou [14]
6.31
EI Structural mechanics stiffness parameters
Numerical values Zhang et al[17] Armchair Zigzag SWCNT SWCNT
N m Kbond anglebending nN nm Kbond stretch
A detailed description of the structural mechanics model is provided as follows. the bond stretching interaction between the nearest two atoms is simulated by beam element I, whereas truss element II is placed between the next-nearest neighbor atoms to compensate the reduction of the bending stiffness. Truss element III is placed between the diagonal atoms to simulate the non-bonded var der Waals interaction. As an innovation point, another beam element IV is assumed between the fourth-nearest atoms to compensate the reduction of the stretching stiffness. The bond stretching energy is thus the sum of the stretch energy of beam element I and beam element IV, and the bending energy of beam element IV, whereas the bond angle bending energy is the sum of the bending energy of beam element I and the stretch energy of truss element II. The non-bonded van der Waals interaction is equivalent with the stretch energy of truss element III. The relation between the molecular potential energy and the mechanical strain energy can thus be expressed as
U r Kbond stretch RI LI
2
EA I EA IV EI IV 2 2 2 RI LI RIV L1V IV IV 2 LI 2 LIV 2 LIV
U Kbond anglebending I I 2
EI EA 2 2 I I RII LII 2 LI 2 LII I
II
(4) where Ri and Li i I , II , IV denote the deformed bond length and undeformed ones between the nearest-neighbor (marked with I), the next-nearest-neighbor (marked with II) and the fourth-nearest-neighbor (marked with IV) atoms respectively.
Corresponding values for Li are appended at the bottom of Fig.2. i and
i I , IV
i
represent the deformed bond angles and undeformed ones.
EA
i I , II , IV
EI
i I , IV
i
i
is the stretch stiffness for beam element or truss element and
is the bending stiffness of a beam element.
Fig.3 Schematic of the deformed geometry of the representative volume element
If it is assumed that the changes in bond angle are small, the following conditions can be achieved from Fig.3: RII LII
I I
LI 2 RIV LIV RI LI cos
IV IV where arctan
(5)
RIV LIV tan LIV
3 5 . Substituting Eq.(5) into Eq.(4) gives
Kbond stretch
1 25 3 I IV IV EA EA EI 2 LI 56 LIV 56 L3IV
Kbond anglebending
L2 1 EI I I EA II 2 LI 8LII
When the force constants Kbond-stretch and Kbond
angle-bending
(6)
are given, the stiffness
parameters can be confirmed. Inspecting Eq.(6), the determination of EI I and II EA have many combinations, similar condition exists for the determination of I IV IV EA , EA and EI .To obtain the optimal fit of the phonon dispersion relation to
the experimental data[6], a optimization procedure is performed to ascertain the stiffness parameters, where reasonable results are listed in Table 1. Compared with the stiffness parameters given by Li and Chou[14], both the stretching stiffness and the bending stiffness of beam element I are greatly reduced. Another beam element IV simulating the interaction between the fourth-nearest neighbor atoms is therefore added to compensate the reduction of the stretching stiffness. And a truss element II simulating the interaction between the next-nearest neighbor atoms is added to compensate the reduction of the bending stiffness. As listed in Table 1, similar stretching stiffness of truss element III with that used in Zhang et al[17]is adopted to account for the var der Waals interaction. Besides, a negative bending stiffness of beam element IV is adopted here to reproduce the
k 2 relation of the optical branch near the point. The corresponding molecular force field constants computed from Eq.(6) are also listed in Table 1.Once the sectional stiffness parameters EA and EI are known, the sectional stiffness matrix
K can be obtained. And then by using the solution procedure of stiffness matrix method for frame structures described in the next section, the phonon dispersion relation of the graphene sheet at the atomic scale can be simulated. 3.2 Frequency calculation of the equivalent beam-truss space frame structure By equivalent the graphene sheet to a special beam-truss space frame, the phonon
dispersion relations of graphene sheet can be understood as the vibration properties of the space frame. Therefore, by calculating the frequency of the equivalent structural mechanics model, the phonon spectra of the graphene structure can be obtained. Once the stiffness and mass matrices are obtained, the Bloch theorem, which allows the reduction of dimensions of the degree of freedom, can be introduced to analyze the wave propagation problem of the two-dimensional graphene sheet. For atoms A0 and B0 of the unit cell n1 , n2 shown in Fig.1(a), the equation of motion is given respectively by
(7) where the sum over n is taken up to the fourth-nearest neighbor interactions relative to the site A0 or B0 . As illustrated in Fig.4, n nI nII nIII nIV 3 6 3 6 18 . According to the Bloch theorem widely used for the wave propagation analysis of periodic structures, the displacement correlation can be given as (Fig.1(a)): u A1 eik1 u A0 , u A2 eik2 u A0 , u A3 ei ( k1 k2 )u A0 ,...... u B1 eik1 u B0 , u B2 eik2 u B0 , u B3 ei ( k1 k2 )u B0 ,......
(8)
where k1 and k 2 represent the components of wave vector k along the boundary of the irreducible Brillouin zone M K as shown in Fig.1(b) [11]. By substituting Eq.(8) into Eq.(7) and assuming the same eigen-frequencies for all u i , the following compact matrix form can be obtained as:
PAA k1 , k2 PAB k1 , k2 0 u A0 2 M 0 M u 0 P k , k P k , k B0 BA 1 2 BB 1 2
(9)
Detailed expressions of the stiffness matrix PAA , PAB , PBA and PBB are given in Appendix A.
Fig.4 Neighbor atoms of a graphene plane up to 4th nearest neighbors for (a) an A atom and (b) a B atom. For ease of viewing, only connections with the reference atoms ( A0 or B0 ) are illustrated.
It is obvious that Eq.(9) represents an eigenvalue problem and it can be solved for obtaining with two propagation constants k k1 , k2 given. The solutions obtained k1 , k2 are termed as the phase constant surfaces or dispersion relation. 4. Results and discussions The phonon dispersion relation of graphene sheet computed using the special beam-truss structural mechanics model is shown in Fig.5(a), where results obtained with the widely used force constant method considering up to the fourth-nearest
neighbor interactions[1] are shown in Fig.5(b) for comparison. Further comparison with the work by Koukaras et al [7] is illustrated in Fig.5(d), where results were obtained by using the molecular dynamics method. The three figures show a good agreement with each other, verifying the effectiveness of the structural mechanics method developed in the paper.
Fig.5 Phonon dispersion relations of graphene: (a) results obtained with the structural mechanics method presented in this paper; (b)results presented by Sahoo and Mishra[1]; (c) results calculated with the stiffness parameters taken from Zhang et al[17]. (d) results presented by Koukaras et al[7]. In contrary to the results of (a) to (c) presented along the boundary of the irreducible Brillouin zone in the direction of M K , results illustrated in (d) are presented in the direction of K M .
The phonon dispersion relation of graphene sheet shown in Fig.5(c) is calculated with stiffness parameters provided by Zhang et al[17], where up to the third-nearest neighbor interactions are considered. Through a comparison of Fig.5(a) and Fig.5(c), it is clear that the k 2 dependence for the out-of-plane transverse acoustic (OTA/ZA) branch[19], which as expected, appears only after including interactions up to the fourth nearest-neighbor. Therefore interactions at least up to the fourth-nearest neighbors are ought to be considered for the calculation of the phonon dispersion of graphene. Further comparison between Fig.5(a) and Fig.5(c) reveals that besides the
k 2 dependence of the OTA/ZA branch, the proposed space frame model not only yields an excellent fit for the low frequency modes, but also gives a good fit for the high-frequency modes, by taking into account the fourth-nearest neighbor interactions. To compare the results quantitatively, we have listed the values at high symmetry points , M , K obtained from different methods in Table 2, including the ab_initio method[5], the force constant method[1, 11, 20], the First Principles Calculations[6, 12, 21] and the experimental values[6, 22]. Results obtained in the paper are in good agreement with existing theoretical and experimental results. 5.Conclusions The two-dimensional graphene is modeled as a beam-truss space frame where the phonon dispersion relation of graphene sheet can be understood as the vibration properties of the space frame. With the stiffness and mass matrices achieved by equating the molecular potential energy of nano-structured materials to the
mechanical strain energy of the equivalent space frame model, the Bloch theorem is introduced for the wave propagation problem of the two-dimensional graphene sheet. Different from existing models, a beam element is added to equivalent interactions between the fourth-nearest neighbor atoms, where as a result the k 2 dependence for the out-of-plane transverse acoustic branch is detected. Besides, the dispersion relations in the low-energy region and high-energy region both show good agreement with those presented in the existing literatures, implying that the inclusion of interactions up to the fourth-nearest neighbor is necessary and also sufficient for reproducing the experimental data for graphene. This work gains insight into the role of structural mechanics method in the dynamic properties analysis of nanostructures. Acknowledgments The authors wish to thank the support from the National Natural Science Foundation of China under grant Nos.11372252 and 11402035
Table 2:Comparison of our results with other theoretical model and experimental results at 1 different high-symmetry points (Frequency, unit: cm ) Meth M K od used to obtai I I L IT OTO L IT OTO L OTA L IT OTO L OTA n the T T O O /ZO O O /ZO A /ZA O O /ZO A /ZA phon A A on dispe rsion Ab initio [5] FCM [1] FCM [11] FCM [20] FPC[ 6] FPC[ 12] FPC[ 21] Exp[ 6] Exp[ 22] SMM (this pape r)
16 90
16 90
860
15 80
14 75
1280
16 05
16 05
874
14 75
13 13
1202
15 88
15 88
863
14 99
13 69
1259
15 81
15 81
893
13 24
13 63
1279
15 81 15 97
15 81 15 97
825
13 50 13 68
14 25 14 28
1315
15 95
15 95
890
13 80
14 42
1339
15 77 15 80
15 65 15 80
13 23 13 90
13 90 13 23
1290
15 86
15 86
15 05
13 72
1256
893
868
837
1346
1290
6 5 1 8 0 0 7 7 3 0 3 3 7 6 7 6 5 6 6 6 5 5 1 5 — —
460
15 20
12 60
1260
89 0
432
13 05
12 22
1222
11 33
460
14 87
12 71
1271
10 11
308
13 49
11 99
1199
10 28
—
—
472
12 20 12 38
1220
6 6 3 2 7 6 6 6 3 1 6 8 — —
13 00 13 26
1238
10 02
475
13 71
12 46
1246
99 4
— 471
11 94 11 84
1194
6 3 0 7 5 1
12 65 13 13 15 13
12 69
6 7 0 6 8 3
483
5 2 0 5 8 0 5 6 7 4 9 5 —
520
535
—
5 3 5 5 3 5 —
1184
—
—
—
1269
96 6
5 9 2
592
580
567
495
—
535
—
*FCM: Force Constant Method; * FPC: First Principles Calculations; *SMM: Structural
Mechanics Method
Appendix A: PAA K
A0 B0
K
A0 B1
K
A0 B9
K
A0 B10
K
A0 A1 ik1
e K
K
A0 B2
K
K
A0 B11
A0 B3
K
A0 A2 ik2
e K
K
A0 A1
A0 B4
K
A0 A3 i k1 k2
e
K
A0 A2
K
A0 B5
K
K
A0 B6
K
A0 A3
K
A0 A4
A0 A4 i k2 k1
K
A0 A6 ik1
e
A0 B7
K e
K
A0 A6
K
A0 B8
K
A0 A7
A0 A7 ik2
e
(A.1)
PAB
K A0 B0 K A0 B1 eik1 K A0 B2 eik2 K A0 B3 ei k1 k2 K A0 B4 ei k1 k2 K A0 B5 ei k1 k2 K A0 B6 eik1 K A0 B7 eik2 K A0 B8 ei 2 k1 k2 K A0 B9 ei k1 2 k2 K A0 B10 e2ik1 K A0 B11 e2ik2 (A.2)
K B0 A0 K B0 A1 eik1 K B0 A2 eik2 K B0 A3 ei k1 k2 K B0 A4 ei k1 k2 K B0 A6 eik1 K B0 A7 eik2 PBA K B0 A12 ei 2 k1 k2 K B0 A13 ei k1 2k2 K B0 A14 e2ik1 K B0 A15 ei k1 k2 K B0 A16 e2ik2 (A.3)
PBB K
B0 A0
K
K
B0 A14
K
B0 B1 ik1
B0 A1
K
K
B0 A15
e K
B0 A2
K
B0 B2 ik2
K
B0 A16
e K
B0 A3
K
K
B0 B1
K
B0 B3 i k1 k2
e
B0 A4
K
B0 B2
K
B0 A6
K
K
B0 A7
B0 B3
K
B0 B4 i k2 k1
K
e
K
B0 B4
B0 A12
K
B0 B6 ik1
e
K
B0 B6
K
B0 A13
K
B0 B7
B0 B7 ik2
e
(A.4)
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Highlights
A structural mechanics method is developed for the dynamic properties analysis of low-dimensional graphene like nanostructure. The phonon dispersion analysis of nanostructures is extended from one-dimensional carbon nanotubes to two-dimensional graphene, with newly proposed theoretical formulas. Accurate results are obtained both in the low frequency field and the high frequency field. The k 2 dependence for the out-of-plane transverse acoustic branch is detected.