PCCP View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
PAPER
Cite this: Phys. Chem. Chem. Phys., 2014, 16, 15263
View Journal | View Issue
Vibrational characteristics of graphene sheets elucidated using an elastic network model† Min Hyeok Kim,a Daejoong Kim,b Jae Boong Choic and Moon Ki Kim*ac Recent studies of graphene have demonstrated its great potential for highly sensitive resonators. In order to capture the intrinsic vibrational characteristics of graphene, we propose an atomistic modeling method called the elastic network model (ENM), in which a graphene sheet is modeled as a mass-spring network of adjacent atoms connected by various linear springs with specific bond ratios. Normal mode analysis (NMA) reveals the various vibrational features of bi-layer graphene sheets (BLGSs) clamped at
Received 19th February 2014, Accepted 5th April 2014
two edges. We also propose a coarse-graining (CG) method to extend our graphene study into the
DOI: 10.1039/c4cp00732h
practical. The simulation results show good agreement with experimental observations. Therefore, the
meso- and macroscales, at which experimental measurements and synthesis of graphene become proposed ENM approach will not only shed light on the theoretical study of graphene mechanics, but
www.rsc.org/pccp
also play an important role in the design of highly-sensitive graphene-based resonators.
1 Introduction Since graphene, a two-dimensional crystal of carbon atoms bonded in a hexagonal lattice, was discovered in 2004,1 it has received significant attention due to its extraordinary mechanical, electrical, and thermal properties.2–4 Among these properties, low mass density, ultrahigh aspect ratios, and unusually high stiffness make graphene an ideal candidate for various nanoelectromechanical systems (NEMSs), including graphene-based resonators.5–13 In a recent study, doubly clamped nanomechanical resonators and their array structures have been successfully created from epitaxial graphene in which additional graphene or the inevitable buffer layer is related to stiffness of resonators.14 Furthermore, Bunch and coworkers fabricated electromechanical resonators from graphene sheets and showed that the fundamental resonant frequencies lie in the megahertz range.9 Additionally, it is well known that this range of resonant frequency can increase from the megahertz to the gigahertz regime, as the graphene dimensions decrease.8 These results indicate the possibility of graphene-based resonators, which have great potential
a
SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Suwon, 440-746, Republic of Korea. E-mail: [email protected] b Department of Mechanical Engineering, Sogang University, Seoul, 121-742, Republic of Korea c School of Mechanical Engineering, Sungkyunkwan University, Suwon, 440-746, Republic of Korea † Electronic supplementary information (ESI) available: Simulation details of fundamental frequency versus the aspect ratio, the effect of size on the relationship between frequency and the aspect ratio and atomic structures of few-layer graphenes can be found with 3 supplementary figures. See DOI: 10.1039/ c4cp00732h
This journal is © the Owner Societies 2014
for ultrasensitive mass, force, and charge sensing over a wide frequency range with a high quality factor.8,13,15,16 Following these studies, a rich variety of remarkable research studies related to graphene-based resonators have been reported. However, experimental measurements of the vibrational properties of graphene are very limited because measurement and manufacturing processes at the nanoscale are very difficult and costly.1,17–19 Alternatively, computational methods have been widely used to reveal the vibrational features of graphene sheets. These methods are generally classified into two categories based on the graphene sheet model employed. One model is the atomistic modeling approach, including molecular dynamics (MD)20,21 and molecular mechanics (MM),22 which represent graphene sheets based on a force field among individual carbon atoms. In principle, these atomistic methods can simulate any problem associated with atomic motions, for example the impact of atomic configurations on the mechanical properties of graphene (zigzag or armchair models). However, due to their significant computational burden, their applications are limited to systems containing a small number of atoms or models of relatively short-term phenomena. Recently, researchers have turned their attention to a more efficient method for the modeling of graphene mechanics with continuum structural mechanics. These methods have the merit not only of saving computation time, but also of easy implementation for nanoscale structural components, such as carbon nanotubes (CNTs) and graphene sheets. Some researchers23–26 have modeled graphene sheets as simple plates by balancing the molecular potential energy with the strain energy of representative continuum models. Other groups27–35 have proposed a more sophisticated model that considers the carbon–carbon
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15263
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Paper
sp2 bonds as equivalent beams by matching the harmonic potential of the MM with the mechanical strain energies. From this approach, the detailed characteristics of graphene topology have been obtained, which may extend beyond the abovementioned classical simple plate models. However, the structural mechanics method cannot consider van der Waals interactions just by using simple beams with only bulk material properties. Therefore, distortions between these two different approaches arise, and the accuracy of the obtained results cannot be guaranteed. Moreover, these methods may also require expensive computation costs as the system size increases. In this study, as an alternative and more effective method for graphene modeling, we introduce the elastic network model (ENM) that has been widely used in the study of bio-molecular dynamics.36–39 We built graphene sheets by connecting neighboring carbon atoms with springs of various force constants and then computed their vibrational features in a straightforward manner by Normal mode analysis (NMA). Moreover, coarse-graining (CG) methods are applied to extend the size limitation of graphene analysis to more practical applications without expensive computations. The results provide the accurate and valuable characterization of various types of graphene sheets. The proposed modeling technique will be a useful design tool for NEMSs, including graphene-based resonators.
2 Simulation methodology and model An important factor in determining the dynamics of graphene sheets is the description of the force field among the carbon atoms. In the most general form, the total molecular potential energy is expressed in the form of steric potential energy by the following equation, which is the sum of many individual energy contributions: P P P P P U = Ur + Uy + Uf + Uo + UvdW, (1) where Ur, Uy, Uf, Uo, and UvdW are the energies associated with bond stretching, angle bending, dihedral angle torsion, out-ofplane torsion, and non-bonded van der Waals (vdW) interactions, respectively. Under a specific condition or assumption, various functional forms based on the parameters for a particular force field can be used for these energy terms.40 However, the computational cost required to explore dynamic behaviors, especially for large graphene sheets, is very high. To reduce this computational burden but keep their accuracy, we propose an ENM that represents the interactions among carbon atoms with virtual springs. The ENM retains the geometry of the graphene sheets. The sheets are parallel to each other with an interlayer distance of 3.35 Å. Each layer has the same size, and the carbon atoms are arranged on a hexagonal lattice. The interatomic distance between two adjacent carbon atoms is 1.42 Å. Basically there are two major differences between the force field descriptions in the ENM and those of traditional MM. First, the ENM uses the pairwise Hookean potential instead of a steric potential. Tirion suggested that such a simplified
15264 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
PCCP
potential model would be sufficient to describe the lowfrequency collective motion of macromolecular assemblies.41 Moreover, the recent study42 demonstrated that a more sophisticated lattice structure method using nonlinear interactions can influence only the out-of-plane vibrational characteristics except the fundamental in-plane frequencies focused in this study. This invariance would justify the linear potential-based ENM results. The simplified potential is as follows: U¼
n1 X n 2 1X ki; j xi ðtÞ xj ðtÞ xi ð0Þ xj ð0Þ ; 2 i¼1 j¼iþ1
(2)
where ki, j is a spring constant between the ith and jth atom, xi(0) is the initial position of the ith atom and xi(t) = xi(0) + di. Here, we assume that the di is small displacement of xi(0). Second, the ENM models the covalent and vdW interactions using only two force constants, kcovalent = 9.30 105 dynes per cm and kvdW = 6.99 103 dynes per cm. Our previous study has determined the carbon–carbon single bond as kcovalent = 6.99 105 dynes per cm by matching the computed wavenumber with reported Raman shifts data.43,44 In the modeling of sp2 carbon interaction in graphene, this force constant is adjusted in proportion to the bond order of 1.33. Furthermore, these values are only activated under a certain range of chemical interactions. The following equations define the conditions for determining the type of force constant used: 8 > > kcovalent if xi xj dcovalent > < k ¼ kvdW if dcovalent o xi xj dvdW ; (3) > > > : 0 if xi xj 4 dvdW where dcovalent and dvdW are the cutoff distances of covalent and vdW interactions, which are set to 1.42 and 4 Å, respectively. As the carbon atoms are bonded in a hexagonal lattice, each carbon atom has three covalent bonds. Moreover, interlayer and interatomic vdW interactions are included in the model. Hence, each carbon atom has more than three virtual spring connections to its neighborhood in the BLGS ENM. For a system consisting of N atoms in three-dimensional space, the minimum number of constraints (represented by linear springs in the ENM) needed to achieve mathematical stability is 3N 6.45 Therefore, the proposed BLGS ENM is fully stable. Of course, there might exist more precise and sophisticated potentials, especially in interactions between graphene layers for accurate simulation results.46 However, in this study, we chose the 4 Å cutoff distance as the vdW interaction without loss of generality. The two given force constants also capture the steric potential properly in Cartesian space. The interatomic interactions of the ENM are compared with those of traditional MM in Fig. 1. In the ENM, solid lines with and without springs represent the virtual springs of kcovalent and kvdW, respectively. This simple and intuitive description of potentials not only saves the computational time but also leads to another outstanding advantage of the ENM such as scalability. With the physical insights concerning the graphene sheets, the nodes and their corresponding force fields could be selected at various
This journal is © the Owner Societies 2014
View Article Online
PCCP
Paper
investigated by NMA. Because the standard NMA procedure has been widely reported in the literature, we will not describe it in detail here. A full description is available in ref. 38.
3 Results and discussion
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
3.1 Vibration study of bi-layer graphene sheets (BLGSs) at the atomic scale
Fig. 1 Interatomic interactions in the MM method (a) and ENM (b). The carbon atoms are depicted as red balls that are bonded by four types of force fields, depicted as arrows, in (a) and by only two types of force fields, depicted as springs and lines, in (b).
coarse-grained levels. This feature enables us to investigate the large size of graphene sheets that are beyond the scope of study via molecular simulations. These two great advantages of the ENM, especially the CG-ENM (see Section 3.4), can dramatically reduce the computational burden incurred by using conventional atomistic approaches. Although there could be concern regarding the use of such a simplified force field, Odegard successfully introduced a similar model for CNTs by using two types of pin-jointed truss.33 Furthermore, the huddle caused by the continuum models47 is also solved in the atomic-based ENM. In recent research on the mechanical properties of graphenes,48 the ENM effectively calculated the Young’s modulus of various graphene sheets without size limitation. Finally, in regard to its simulation efficiency, scalability, and accuracy, the proposed ENM is much more appropriate to study the vibrational and mechanical properties of graphene compared to other simulation approaches. Once we constructed an ENM for a typical BLGS, as shown in Fig. 2 the vibrational characteristics of graphene sheets were
Recently, graphene-based resonators have been developed by fixing two edges of graphene sheets on a substrate.8 To mimic this condition, an armchair graphene model clamped at two edges is adopted in this study. Of course, the chirality and boundary condition may have significant impacts on our simulation results at the atomic scale. We investigated the effect of aspect ratios in the BLGSs on normal mode variations in the ESI.† Fig. 2(b) shows a schematic diagram of a doubly clamped armchair BLGS model with width ‘a’ and length ‘b’. The natural frequencies and their corresponding mode shapes are calculated by NMA. The fundamental frequency versus the size of the graphene sheets is presented in Fig. 3. As the overall size of the graphene sheets increases, the fundamental frequency appears to decrease nonlinearly. Similar results, based on MM31 or continuum beam theory,9,49 have been reported elsewhere. Interestingly, the proposed ENM could lead to almost the same results from MM, even though it has used much simpler potentials and less number of interactions than those of MM as shown in Fig. 3. In the case of beam theory, one study treated the graphene sheets as doubly clamped beams and theoretically calculated the fundamental frequency based on the following equation: sffiffiffiffiffi At E f0 ¼ 2 ; (4) L r0 where L is the length of the graphene sheet, t is the thickness of the BLGS (set as 0.67 nm),50 and A is the clamping coefficient, which is 1.03 for doubly-clamped beams.9 E and r0 represent
Fig. 2 Schematic of BLGS simulation concept. (a) The square-shaped BLGS is constructed with an ENM in which the red dots depict carbon atoms and the black solid and dotted lines depict covalent bonds and vdW interactions, respectively. (b) A schematic of a suspended graphene resonator. Two horizontal edges are fixed, and the fixed width and free length of graphene sheet are chosen as ‘a’ and ‘b’, respectively.
This journal is © the Owner Societies 2014
Fig. 3 Fundamental frequencies of a square and doubly clamped BLGS versus its length. The fundamental frequencies from beam theory, MM and our ENM are shown by blue red and black lines, respectively. The number of carbon atom bonds for the ENM and MM is also displayed as the sparse and dense checked bar graph, respectively.
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15265
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Paper
PCCP
Fig. 4 The first five mode shapes of BLGSs for width = 46.73 Å and length = 45.44 Å.
the Young’s modulus and mass density, respectively. The general values for bulk graphite used here yield r0 = 2200 kg m3 and E = 0.872 TPa. Here, the graphene sheet is assumed to have flexural vibrational modes like the bending of doubly clamped beams, which is fundamentally debatable because flexural rigidity of single- or few-layer graphene sheet is negligible. Although this continuum approach could lead to reasonable results on a practical scale, limitations in atomistic details cause large differences in nanoscale graphene vibrations compared to other atomistic approaches. The great disagreement in the small size of BGLS in Fig. 3 represents well the limitation of the continuum approach. The first five mode shapes of the BLGS are shown in Fig. 4. The fundamental vibrational modes of square BLGSs exhibit half sine, inverse half sine, center tip, full sine, and inverse full sine shapes. In addition, these fundamental mode shapes are still observed if the size of the graphene sheet is changed, albeit in a different order. These mode shapes of BLGSs are thought to be appropriate because conventional doubly-clamped systems show similar trends.51 The further studies in ESI† show that these fundamental frequencies and the trend of mode shapes are also insensitive to the aspect ratios when the size of the BLGS varies. The difference in frequency is less than 7% over the entire range of tested lengths from 45.4 to 105.1 Å in Fig. S2 of the ESI.† In conclusion, the non-dependence of graphene vibrations on the size and the aspect ratio gives high versatility for both the design and manufacturing of graphene-based applications. Furthermore, we expect BLGSs to be employed as good candidates for high-fidelity resonators because of their high frequency range as well as their high level of uniformity in mode shapes. 3.2
The layer effect of graphene on vibration frequency
One of the major potentials of graphene for NEMS applications is its remarkable electronic properties, such as the quantum Hall effect and a tunable band gap. Due to intensive progress in research on the geometrical properties of graphene sheets and their electronic properties,52–55 it was found that few-layer graphene sheets (FLGSs) are formed in various stacking orders, which can strongly influence their electronic properties. The following three stable crystallographic configurations are predicted: (1) hexagonal simple graphite with AAA stacking, (2) Bernal graphite with ABA stacking, and (3) rhombohedral graphite with ABC stacking (Fig. S3, ESI†).56,57 The firstprinciples calculations revealed that these stacking sequences
15266 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
Fig. 5 Fundamental frequency versus the layer number for various types of FLGSs. The black, red, and blue lines represent the fundamental frequencies of simple, bernal, and rhombohedral stacking types, respectively.
must be maintained for stability when the number of layers is not a multiple of three.58 For example, we expect only three stable stacking types for tetralayer graphene sheets: AAAA, ABAB, and ABCA. The vibrational characteristics of graphene sheets change with respect to the thickness of the sheets (i.e., the number of layers) and the stacking order. Fig. 5 demonstrates the change in fundamental frequency for three different graphene sheets, each with a different stacking order up to eight layers. The size of each graphene sheet is such that a = 46.73 Å and b = 45.44 Å. In the case of traditional AAA stacking, as the thickness increases, the fundamental frequency also increases but is saturated beyond a discrete layer number. This result can be attributed to the following two factors. The first factor is a geometrical effect in terms of incremental mass and two types of springs, such as the covalent and vdW interactions shown in Table 1. As each new graphene layer is stacked on the FLGS, the increments in mass and spring number are constant. Although the effects of incremental mass and spring number on frequency change are initially dominant, the relative contribution decreases as the number of layers increases. The second factor is a physical effect. Mathematically speaking, the frequency term in NMA is proportional to the square root of the stiffness over the mass. Therefore, the results in Fig. 5 resemble a square-root curve. Due to the combination of these physical and geometrical effects, the fundamental frequency of the FLGSs is eventually saturated. This result seems to be more reasonable than the existing understanding based on beam theory, in which graphene is approximated as a simple beam, where the frequency continuously increases in proportion to the thickness based on eqn (4). In addition, from this result, we can select the best number of graphene sheets to maximize their sensitivity. In the same manner, the other types of FLGSs, Bernal and rhombohedral graphite, were analyzed. For all thicknesses, these forms show higher fundamental frequencies compared to the previous results for simple graphite. In addition, rhombohedral graphite shows an increased frequency according to its specific stacking configuration, while Bernal graphite shows a trend similar to that
This journal is © the Owner Societies 2014
View Article Online
PCCP Table 1
Paper Fundamental frequencies and the number of inter-vdW interactions for three types of FLGSs
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
AAA
ABA
ABC
Layer
f0 (GHz)
vdWinter
vdWinter variation
f0 (GHz)
vdWinter
vdWinter variation
f0 (GHz)
vdWinter
vdWinter variation
2 3 4 5 6 7 8
203 249 271 283 291 297 300
3350 6700 10 050 13 400 16 750 20 100 23 450
3350 3350 3350 3350 3350 3350
233 292 321 337 347 355 361
4109 8218 12 327 16 436 20 545 24 654 28 763
4109 4109 4109 4109 4109 4109
233 298 326 341 354 361 366
4109 9391 15 093 19 202 24 484 30 186 34 295
5282 5702 4109 5282 5702 4109
For all types of FLGSs, the numbers of covalent and intra-vdW interactions are the same. Therefore, these values are not included.
of simple graphite. Such results can also be explained based on the geometrical effect shown in Table 1. When a new graphene layer is stacked on the FLGS, the incremental mass and covalent bond number remain the same for all stacking types, but the incremental vdW interactions between layers are different for different stacking types. In Bernal-type graphite, the incremental spring number for inter-layer vdW interactions is constant at 4109, and this value is greater than that of simple graphite (3350), resulting in a higher frequency in Bernal-type graphite. Likewise, rhombohedral-type graphite has a higher number of incremental springs for inter-layer vdW than simple graphite, but the number increases uniformly according to its unique stacking configuration. These results demonstrate the effects of the three different stacking orders not only on the vibrational features reflected in the fundamental frequency, but also on the electronic properties of FLGSs. 3.3
Highly sensitive mass sensing
Another possible application of graphene-based resonant devices lies in the highly sensitive detection of small bound masses through a frequency shift. In general, the relationship between a frequency shift Df and a change in the absorbed mass Dm is based on the geometry of the resonant device and its fundamental frequency, depicted as Dm ¼
2Meff Df ; f0
the graphene sheets for more precise and efficient calculations at the atomic scale. Substitution of the calculated effective mass into eqn (5) yields the relationship between the frequency shift and the change in added mass, which determines the sensitivity or resolution of a mass sensing device. This estimate implies that it is more favorable to make the graphene sheets as small as possible to obtain the highest sensitivity. Two square BLGSs with lengths of 36.90 and 61.49 Å are tested here. The corresponding fundamental frequencies are 300 and 153 GHz, respectively. Based on these two BLGSs, we calculated the frequency shifts by modulating the adsorbed mass from 1 to 10 yoctograms (1 1024 g). The added mass is placed on the center of the BLGSs, as shown in the inset of Fig. 6, and is connected to the surrounding carbon atoms through vdW interactions. The greater the added mass, the greater the frequency shift of both BLGSs. The slope Df/Dm calculated from the initial approximately linear relationship is in good agreement with the theoretical values for f0/2Meff from eqn (5), 9.17 1033 and 1.71 1033 kg1 s1, until the total mass of inserted atoms reaches that of the graphene sheets. In addition, the sensitivity is greater
(5)
where Meff and f0 are the effective mass and fundamental frequency of the given doubly clamped beam device, respectively, and all inserted atoms are assumed to be located at the center of the graphene sheets.59–61 Graphene is one of the most suitable materials for mass sensing because of its high frequency range and low effective mass.15,16 In this context, ENM-based NMA was performed to verify the sensitivity of a BLGS in detecting a small amount of added mass through a frequency shift. For a doubly clamped BLGS, the effective mass is assumed to be Meff = 0.735ltwr,
(6)
where the constant value of 0.735 reflects the reduction due to double clamping.9 l, t, w, and r are the length, thickness, width, and density of the graphene sheets, respectively. In practice, this effective mass can be calculated by multiplying the mass of a single carbon atom by the total number of atoms contained in
This journal is © the Owner Societies 2014
Fig. 6 Frequency shift versus the change in adsorbed mass located at the center of the BLGS. The red and blue dotted lines represent the relationship between the frequency shift and the added mass based on beam theory (eqn (5)) for small (36.90 Å in length) and large (61.49 Å in length) rectangular graphene sheets, respectively. The red and blue solid lines depict the results from NMA of the ENM for small and large sheets, respectively. The inset shows a schematic of a BLGS sensor with one inserted atom.
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15267
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Paper
for small BLGSs when comparing the Df/Dm slopes of the two graphene sheets from both theoretical and simulation data. Thus, the higher sensitivity of the smaller graphene sheets is primarily due to the small number of atoms composing the graphene sheet, which could exert a relatively strong effect on the frequency shift, even for a small change in mass. The effect of the inserted mass position on the frequency shift was also investigated. It is well known that random positioning of inserted masses causes practical difficulties for the precise operation of resonators.62 As an example, by moving a small mass of 5 yoctograms to all available positions of a BLGS with a length of 36.90 Å, the corresponding changes in fundamental frequency were tested. Fig. 7 shows contours on the unit square graphene sheets. Each contour encloses a region in terms of the frequency shift due to the inserted atoms. In this figure, the highest change in frequency Df is obtained at 159 GHz when the particle is located at the center of the free edge line. This range of frequency shift may occur when a 15-yoctogram particle is placed at the center of the BLGS. In addition, for all available positions, the changes in frequency increase when the particle is located far from the fixed lines while also being located close to the free edge line. These results can be intuitively understood from the general statics equation in which the maximum torque is generated when the weight is far from the fixed pivot position. This result is comparable to a previous experimental result in which the greatest change in frequency is obtained when an absorbed gold bead is positioned at the tip of a cantilever.63 These simulation results show that not only the mass of the adsorbed particle but also the position of the adsorbed particle can influence the change in fundamental frequency. Therefore, the uncertainty in the position of the adsorbed particle must be considered when measuring mass using the frequency shift of a resonator. Our ENM-based NMA will serve as a precise and practical prediction method for this purpose.
Fig. 7 Contours of frequency shift values, e = Df/f0, according to the position of an inserted mass of 5 yoctograms. The square BLGS is 36.90 Å long, and both the top and bottom lines are clamped.
15268 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
PCCP
3.4
CG-ENM for scale-up simulations
In practice, the actual size of graphene-based resonators must be at least on the submicron level due to limitations in both manufacturing and measurement. Although the ENM can dramatically reduce computational burdens, it is also challenging to run a full-atom ENM simulation for graphene sheets that are tens of nanometers long in a timely fashion. To overcome this limitation, a CG-ENM implemented by the fine-trains-coarse approach is proposed here as an attractive alternative for micrometer scale simulations. The main idea for constructing the CG-ENM for graphene is to maintain its structural characteristics (i.e., hexagonal lattice) and force field types with fixed DOFs. For more details, Fig. 8 depicts a two-dimensional hexagonal lattice of carbon atoms with overlaid CG models of different sizes. Three different hexagonal lattices corresponding to three levels of CG are constructed according to their unit of length, rn = 2nr0 where r0 = 1.42 Å and a positive integer, ‘n’, is the level of CG. In the same manner, the unit of mass is mn = 22n m0, where m0 = 2 1026 kg is the mass of one carbon atom. Unlike the in-plane structural change, the distance between each layer remains 3.35 Å. In the case of force field modeling, all spring constants should be modulated by comparing them with those from the original full-atom model. The main modulation factors contributing to the natural frequency that must be considered are the spring connectivity and unit length in each CG-ENM. Basically, a lower number of spring connections cause a lower frequency in CG models. We can compensate for this reduction in the CG model by multiplying the stiffness value by the number ratio of a CG model and a full-atom model. Additionally, the longer unit length leads to lower force constants, and this inverse relationship also contributes to a lower frequency. The unit length in a CG model is defined by the CG level. Therefore, multiplication of the reciprocal of the length ratio and the spring constant is also required in the framework for modeling equivalence. Table 2 gives an overview of the resultant force fields for the full-atom
Fig. 8 The unit honeycomb lattice of the different cg models and their arrangement in the atomic graphene sheet. All CG models have the same DOF and follow the same atomic model. The figure is conceptually adopted from ref. 64.
This journal is © the Owner Societies 2014
View Article Online
PCCP
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Table 2
Paper Resulting variables for given levels of BLGS modeling
Model
Unit length (1.42 Å)
Mass (2 1026 kg)
k_covalent (6.99 105 dynes per cm)
k_intra_vdW (6.99 103 dynes per cm)
k_inter_vdW (6.99 103 dynes per cm)
Full CG1 CG2 CG3
1 2 4 8
1 4 16 64
1 1 (2, 0.5)a 1 (4, 0.25)a 1 (8, 0.125)a
1 2.17 (4.33, 0.5)a 2.172 (18.7, 0.25)a 2.173 (81.0, 0.125)a
1 1.66 (2, 0.83)a 2.21 (4, 0.55)a 2.47 (8, 0.31)a
a The numbers outside of the parentheses in the k_covalent, k_intra_vdW, and k_inter_vdW columns represent specific ratios at each CG level. The left and right numbers in the parentheses are the ratios of bond numbers and inverse unit length, respectively. Unlike the constant or increasing k_covalent and k_intra_vdW cases, k_inter_vdW increases but is eventually saturated.
and three CG models. Based on the concept of force field modeling, only two types of spring constants are arranged in the same configuration as the full-atom graphene model. However, asymmetric growth in the unit length, where the interlayer distance is fixed but the in-plane unit lengths regularly increase, bifurcates the single vdW interaction into inter-vdW and intravdW interactions. For the first CG level, the doubled length in the covalent and intra-vdW interaction reduces the spring constant by one half, whereas the length of the inter-vdW interaction changes very little, because of the constant layer thickness. These effects require the spring constants (i.e. k_covalent and k_intra_vdW cases, k_inter_vdW) in the CG model to be multiplied by 0.5, 0.5, and 0.83, which are inverse values of the unit length ratio between CG and full atomic models. As another critical factor, the number ratios for each connection type are also reflected by multiplying the spring constants in the CG model by 2, 4.33, and 2. For the second, third, and much larger CG levels, these force field ratios are uniform; thus, they can be incorporated into the spring constants according to the CG levels (except the inter-vdW interaction, because of the constant layer thickness). Table 3 compares the fundamental frequencies of the fullatom and CG models. Due to computational limits, only three types of full-atom models (35, 70, and 140 Å in length) were analyzed and compared with appropriate CG models. The models on the diagonal blocks are considered to have the same DOF which also requires the same computational time. At the second and third levels of the CG models, with a length of 70 and 140 Å, the errors in frequency are less than 11%. Although a full-atom simulation was not performed in the case of the fourth level of the CG model, which has a length of 280 Å, the
Table 3 A comparison of fundamental frequencies for four different sizes of BLGSs at each CG level
Fundamental frequency (GHz) Size (Å) 35 70 140 280
DOF 1116 4012 15 180 59 020
Time 6 m 43 s 2 h 39 m 33 s 6 d 6 h 38 s Non
a
Full 256 115 41.3 N/Ab
CG1
CG2
CG3
a
a
N/Aa N/Aa N/Aa 13.0
N/A 120 46.2 14.4
N/A N/Aa 42.4 15.5
We basically use three DOFs: 1116, 4012 and 15 180. The model under 1116 DOFs could be unfeasible due to the side effects. b Due to computational limits, a full model with a size of 280 Å could not be calculated.
This journal is © the Owner Societies 2014
similar ranges of frequency values obtained from the three other CG levels strongly indicate the accuracy of the proposed CG-ENM. Furthermore, a more practical size of the graphene sheet was tested by the proposed CG force fields and also showed good agreement with the existing experimental data. The exfoliated BLGS with dimensions a b t = 1.35 mm 0.85 mm 0.67 mm exhibits a fundamental frequency at 13.33 MHz.65 Allowing for the built-in tension of 1.6 nN which leads to increase in frequency, our CG-ENM successfully predicts the slightly small but almost the same frequency of 10.50 MHz. In short, the use of an atomistic configuration with topologically scalable force fields in the proposed CG-ENM will shed light on mesoscale vibrations by providing both a reasonable computation time and acceptable simulation accuracy.
4 Conclusions In this study, a variety of vibrational characteristics of graphene sheets and their potential as highly sensitive resonators have been thoroughly investigated by an atomistic modeling approach called the ENM. At the atomic scale, BLGSs were systemized as layered carbon atoms connected by two types of linear springs based on chemical information. Then, normal modes for various types of BLGSs were calculated. Fundamental frequencies and their corresponding mode shapes guaranteed the performance of BLGSs as a strong candidate for high-fidelity resonators. Furthermore, the effects of the layer number and the stacking type of the graphene sheets on the fundamental frequency were also investigated. For all stacking types, as the number of layers increased, the fundamental frequencies also increased, but were saturated at a certain number of layers. However, all types of graphene sheets showed unique trends in frequency due to their different topological arrangements. The ENM was also used to explore the potential use of graphene sheets as mass detectors by estimating the frequency shift. Changes in frequency were calculated depending on the quantity of absorbed mass as well as the position of the mass. The linear relationship obtained between the frequency shift and the adsorbed mass matched well with the existing theoretical model. In addition, the simulation results expressing the position dependency of the frequency shift can serve as a practical design factor in high-fidelity resonators for mass sensing. To extend our study to a more practical scale in which both manufacturing and measurements are possible, the CG-ENM was newly proposed. By maintaining the unique hexagonal sub-lattice structure of
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15269
View Article Online
Paper
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
graphene, a reduced set of representative atoms was sampled, and spring constants were appropriately modulated to maintain equivalent force fields. Dramatic reductions in computation time with acceptable accuracy in frequency prediction, even compared to experimental data, can promote our understanding of the vibrational characteristics of graphene based resonators in practical use.
Acknowledgements This research was supported by the Basic Science Research Program (2011-0014584), Pioneer Research Center Program (2012-0009579), and Basic Research Laboratory Program (20110020024) through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology.
Notes and references 1 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubones, I. V. Grigorieva and A. A. Firsov, Science, 2004, 35, 666–669. 2 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature, 2005, 438, 197–200. 3 P. O. Lehtinen, A. S. Foster, Y. Ma, A. V. Krasheninnikov and R. M. Nieminen, Phys. Rev. Lett., 2004, 93, 187202. 4 C. Lee, X. Wei, J. W. Kysar and J. Hone, Science, 2008, 321, 385–388. 5 J. Atalaya, A. Isacsson and J. M. Kinaret, Nano Lett., 2008, 8, 4196–4200. 6 R. B. Karabalin, X. L. Feng and M. L. Roukes, Nano Lett., 2009, 9, 3116–3123. 7 A. M. Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead and P. L. McEuen, Nano Lett., 2010, 10, 4869–4873. 8 C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stromer, T. F. Heinz and J. Hone, Nat. Nanotechnol., 2009, 4, 861–867. 9 J. S. Bunch, A. M. Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead and P. L. McEuen, Science, 2007, 315, 490–493. 10 A. M. Zande, A. S. Paulo, B. Lassagne, P. L. Mceuen and A. Bachtold, Nano Lett., 2008, 8, 1399–1403. 11 M. Poot and H. S. Zant, Phys. Lett., 2008, 92, 063111. 12 J. T. Robinson, M. Zalalutdinov, J. W. Baldwin, E. S. Snow, Z. Wei, P. Sheehan and B. H. Houston, Nano Lett., 2008, 8, 3441–3445. 13 J. S. Bunch, S. S. Verbridge, J. S. Alden, A. M. Zande, J. M. Parpia, H. G. Craighead and P. L. Mceuen, Nano Lett., 2008, 8, 2458–2462. 14 S. Shivaraman, R. A. Barton, X. Yu, J. Alden, L. Herman, M. Chandrashekhar, J. Park, P. L. McEuen, J. M. Parpia, H. G. Craighead and M. G. Spencer, Nano Lett., 2009, 9, 3100–3105.
15270 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
PCCP
15 S. M. Avdoshenko, R. C. Gomes and G. Cuniberti, Nanoscale, 2012, 4, 3168–3174. 16 R. A. Barton, J. Parpia and H. G. Craighead, J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom., 2011, 29, 05081. 17 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson and K. S. Novoselov, Nat. Mater., 2007, 6, 652–655. 18 K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov and A. K. Geim, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 10451–10453. 19 A. J. Van Bommel, J. E. Crombeen and T. Van, Surf. Sci., 1975, 48, 463–472. 20 J. W. Jiang, J. S. Wang and B. Li, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 113405. 21 K. Samadikhah, J. Atalaya, C. Huldt, A. Isacsson and J. Kinaret, MRS Proc., 2007, 1057, 10–20. 22 R. Chowdhury, S. Adhikari, F. Scarpa and M. I. Friswell, J. Phys. D: Appl. Phys., 2011, 44, 205401. 23 K. Behfar and R. Naghdabadi, Compos. Sci. Technol., 2005, 65, 1159–1164. 24 X. Q. He, S. Kitipornchai and K. M. Liew, Nanotechnology, 2005, 16, 2086–2091. 25 M. K. Liew, X. Q. He and S. Kitipornchai, Acta Mater., 2006, 54, 4229–4236. 26 M. R. Lin, Comput. Mater. Sci., 2012, 53, 44–52. 27 F. Scarpa, S. Adhikari and P. A. Srikantha, Nanotechnology, 2009, 20, 065709. 28 A. Sakhaee-Pour, M. T. Ahmadian and A. Vafai, Solid State Commun., 2008, 147, 336–340. 29 S. Arghavan and A. V. Singh, J. Appl. Phys., 2011, 110, 084308. 30 C. Baykasoglu and A. Mugan, Comput. Mater. Sci., 2012, 55, 228–236. 31 Y. Chandra, R. Chowdhury, F. Scarpa and S. Adhikaricor, Thin Solid Films, 2011, 519, 6026–6032. 32 A. Sakhaee-Pour, M. T. Ahmadian and R. Naghdabadi, Nanotechnology, 2009, 19, 085702. 33 G. M. Odegard, T. S. Gates, L. M. Nicholson and K. E. Wise, Compos. Sci. Technol., 2002, 62, 1869–1880. 34 S. K. Georgantzinos, G. I. Giannopoulos and N. K. Anifantis, Mater. Des., 2010, 31, 4646–4654. 35 C. Li and T. W. Chou, Int. J. Solids Struct., 2003, 40, 2487–2499. 36 M. K. Kim, R. L. Jernigan and G. S. Chirikjian, J. Struct. Biol., 2003, 143, 107–117. 37 M. K. Kim, R. L. Jernigan and G. S. Chirikjian, Biophys. J., 2002, 83, 1620–1630. 38 M. K. Kim, G. S. Chirikjian and R. L. Jernigan, J. Mol. Graphics Modell., 2002, 21, 151–160. 39 M. K. Kim, Y. Jeong and J. I. Jeong, Int. J. Control Autom. Syst., 2006, 4, 382–393. 40 C. J. Caseweit, K. S. Colwell and A. K. Rappe, J. Am. Chem. Soc., 1992, 114, 10035–10046. 41 M. Tirion, Phys. Rev. Lett., 1996, 77, 1905–1908. 42 S. Arghaven and A. V. Singh, J. Phys. D: Appl. Phys., 2012, 45, 455305.
This journal is © the Owner Societies 2014
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
PCCP
43 M. Hu, S. Raj, B. Kim, W. K. Liu, S. Baik, T. Kim, B. S. Lim and M. K. Kim, J. Mech. Sci. Technol., 2010, 24, 1771–1780. 44 J. M. Berg, J. L. Tymoczko and L. Stryer, in Biochemistry, W. H. Freeman and Company, New York, 5th edn, 2008. 45 M. H. Kim, S. Seo, J. I. Jeong, B. J. Kim, W. K. Liu, B. S. Lim, J. B. Choi and M. K. Kim, Protein Sci., 2013, 22, 605–613. 46 I. V. Lebedeva, A. A. Knizhnik, A. M. Popov, Y. E. Lozovik and B. V. Potapkin, Phys. Chem. Chem. Phys., 2011, 13, 5687–5695. 47 I. E. Berinskii and F. M. Borodich, Mech. Mater., 2013, 62, 60–68. 48 J. Kang, S. Hwang, J. H. Kim, M. H. Kim, J. Ryu, S. J. Seo, B. H. Hong, M. K. Kim and J. Choi, ACS Nano, 2012, 6, 5360–5365. 49 W. Weaver Jr., S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, Wiley, New York, 5th edn, 1990. 50 Z. H. Ni, H. M. Wang, J. Kasim, H. M. Fan, T. Yu, Y. H. Wu, Y. P. Feng and Z. X. Shen, Nano Lett., 2007, 7, 2758–2763. 51 R. C. Tung, A. Garg, A. Kovacs, D. Peroulis and A. Raman, J. Micromech. Microeng., 2013, 23, 045009. 52 C. L. Lu, C. P. Chang, Y. C. Huang, J. H. Ho, C. C. Hwang and M. F. Hwang, J. Phys. Soc. Jpn., 2007, 76, 024701. 53 F. Guinea, A. H. Castro Neto and N. M. R. Peres, Solid State Commun., 2007, 143, 116–122.
This journal is © the Owner Societies 2014
Paper
54 K. F. Mak, J. Shan and T. F. Heinz, Phys. Rev. Lett., 2010, 104, 1–4. 55 L. M. Malard, M. A. Pimenta, G. Dresselhaus and M. S. Dresselhaus, Phys. Rep., 2009, 473, 51–87. 56 J. C. Charlier, J. P. Michenaud and X. Gonze, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 4531–4539. 57 J. C. Charlier and J. P. Michenaud, Phys. Rev. Lett., 1993, 70, 1858–1861. 58 M. Aoki and H. Amawashi, Solid State Commun., 2010, 150, 458. 59 K. Jensen, K. Kim and A. Zettl, Nat. Nanotechnol., 2008, 3, 533–537. 60 V. Sazonova, PhD thesis, Cornell University, 2006. 61 Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci and M. L. Roukes, Nano Lett., 2006, 6, 583–586. 62 J. Atalaya, J. M. Kinaret and A. Isacsson, Europhys. Lett., 2010, 91, 48001. 63 S. Dohn, R. Sandberg, W. Svendsen and A. Boisen, Appl. Phys. Lett., 2005, 86, 233501. ˜ ol, S. Succi, A. Greiner and 64 D. Kauzlaric´, J. T. Meier, P. Espan J. G. Korvink, J. Chem. Phys., 2011, 134, 064106. 65 J. Lee and P. X. L. Feng, IEEE International Frequency Control Symposium (FCS), Baltimore, 2012.
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15271
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
PAPER
Cite this: Phys. Chem. Chem. Phys., 2014, 16, 15263
View Journal | View Issue
Vibrational characteristics of graphene sheets elucidated using an elastic network model† Min Hyeok Kim,a Daejoong Kim,b Jae Boong Choic and Moon Ki Kim*ac Recent studies of graphene have demonstrated its great potential for highly sensitive resonators. In order to capture the intrinsic vibrational characteristics of graphene, we propose an atomistic modeling method called the elastic network model (ENM), in which a graphene sheet is modeled as a mass-spring network of adjacent atoms connected by various linear springs with specific bond ratios. Normal mode analysis (NMA) reveals the various vibrational features of bi-layer graphene sheets (BLGSs) clamped at
Received 19th February 2014, Accepted 5th April 2014
two edges. We also propose a coarse-graining (CG) method to extend our graphene study into the
DOI: 10.1039/c4cp00732h
practical. The simulation results show good agreement with experimental observations. Therefore, the
meso- and macroscales, at which experimental measurements and synthesis of graphene become proposed ENM approach will not only shed light on the theoretical study of graphene mechanics, but
www.rsc.org/pccp
also play an important role in the design of highly-sensitive graphene-based resonators.
1 Introduction Since graphene, a two-dimensional crystal of carbon atoms bonded in a hexagonal lattice, was discovered in 2004,1 it has received significant attention due to its extraordinary mechanical, electrical, and thermal properties.2–4 Among these properties, low mass density, ultrahigh aspect ratios, and unusually high stiffness make graphene an ideal candidate for various nanoelectromechanical systems (NEMSs), including graphene-based resonators.5–13 In a recent study, doubly clamped nanomechanical resonators and their array structures have been successfully created from epitaxial graphene in which additional graphene or the inevitable buffer layer is related to stiffness of resonators.14 Furthermore, Bunch and coworkers fabricated electromechanical resonators from graphene sheets and showed that the fundamental resonant frequencies lie in the megahertz range.9 Additionally, it is well known that this range of resonant frequency can increase from the megahertz to the gigahertz regime, as the graphene dimensions decrease.8 These results indicate the possibility of graphene-based resonators, which have great potential
a
SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Suwon, 440-746, Republic of Korea. E-mail: [email protected] b Department of Mechanical Engineering, Sogang University, Seoul, 121-742, Republic of Korea c School of Mechanical Engineering, Sungkyunkwan University, Suwon, 440-746, Republic of Korea † Electronic supplementary information (ESI) available: Simulation details of fundamental frequency versus the aspect ratio, the effect of size on the relationship between frequency and the aspect ratio and atomic structures of few-layer graphenes can be found with 3 supplementary figures. See DOI: 10.1039/ c4cp00732h
This journal is © the Owner Societies 2014
for ultrasensitive mass, force, and charge sensing over a wide frequency range with a high quality factor.8,13,15,16 Following these studies, a rich variety of remarkable research studies related to graphene-based resonators have been reported. However, experimental measurements of the vibrational properties of graphene are very limited because measurement and manufacturing processes at the nanoscale are very difficult and costly.1,17–19 Alternatively, computational methods have been widely used to reveal the vibrational features of graphene sheets. These methods are generally classified into two categories based on the graphene sheet model employed. One model is the atomistic modeling approach, including molecular dynamics (MD)20,21 and molecular mechanics (MM),22 which represent graphene sheets based on a force field among individual carbon atoms. In principle, these atomistic methods can simulate any problem associated with atomic motions, for example the impact of atomic configurations on the mechanical properties of graphene (zigzag or armchair models). However, due to their significant computational burden, their applications are limited to systems containing a small number of atoms or models of relatively short-term phenomena. Recently, researchers have turned their attention to a more efficient method for the modeling of graphene mechanics with continuum structural mechanics. These methods have the merit not only of saving computation time, but also of easy implementation for nanoscale structural components, such as carbon nanotubes (CNTs) and graphene sheets. Some researchers23–26 have modeled graphene sheets as simple plates by balancing the molecular potential energy with the strain energy of representative continuum models. Other groups27–35 have proposed a more sophisticated model that considers the carbon–carbon
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15263
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Paper
sp2 bonds as equivalent beams by matching the harmonic potential of the MM with the mechanical strain energies. From this approach, the detailed characteristics of graphene topology have been obtained, which may extend beyond the abovementioned classical simple plate models. However, the structural mechanics method cannot consider van der Waals interactions just by using simple beams with only bulk material properties. Therefore, distortions between these two different approaches arise, and the accuracy of the obtained results cannot be guaranteed. Moreover, these methods may also require expensive computation costs as the system size increases. In this study, as an alternative and more effective method for graphene modeling, we introduce the elastic network model (ENM) that has been widely used in the study of bio-molecular dynamics.36–39 We built graphene sheets by connecting neighboring carbon atoms with springs of various force constants and then computed their vibrational features in a straightforward manner by Normal mode analysis (NMA). Moreover, coarse-graining (CG) methods are applied to extend the size limitation of graphene analysis to more practical applications without expensive computations. The results provide the accurate and valuable characterization of various types of graphene sheets. The proposed modeling technique will be a useful design tool for NEMSs, including graphene-based resonators.
2 Simulation methodology and model An important factor in determining the dynamics of graphene sheets is the description of the force field among the carbon atoms. In the most general form, the total molecular potential energy is expressed in the form of steric potential energy by the following equation, which is the sum of many individual energy contributions: P P P P P U = Ur + Uy + Uf + Uo + UvdW, (1) where Ur, Uy, Uf, Uo, and UvdW are the energies associated with bond stretching, angle bending, dihedral angle torsion, out-ofplane torsion, and non-bonded van der Waals (vdW) interactions, respectively. Under a specific condition or assumption, various functional forms based on the parameters for a particular force field can be used for these energy terms.40 However, the computational cost required to explore dynamic behaviors, especially for large graphene sheets, is very high. To reduce this computational burden but keep their accuracy, we propose an ENM that represents the interactions among carbon atoms with virtual springs. The ENM retains the geometry of the graphene sheets. The sheets are parallel to each other with an interlayer distance of 3.35 Å. Each layer has the same size, and the carbon atoms are arranged on a hexagonal lattice. The interatomic distance between two adjacent carbon atoms is 1.42 Å. Basically there are two major differences between the force field descriptions in the ENM and those of traditional MM. First, the ENM uses the pairwise Hookean potential instead of a steric potential. Tirion suggested that such a simplified
15264 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
PCCP
potential model would be sufficient to describe the lowfrequency collective motion of macromolecular assemblies.41 Moreover, the recent study42 demonstrated that a more sophisticated lattice structure method using nonlinear interactions can influence only the out-of-plane vibrational characteristics except the fundamental in-plane frequencies focused in this study. This invariance would justify the linear potential-based ENM results. The simplified potential is as follows: U¼
n1 X n 2 1X ki; j xi ðtÞ xj ðtÞ xi ð0Þ xj ð0Þ ; 2 i¼1 j¼iþ1
(2)
where ki, j is a spring constant between the ith and jth atom, xi(0) is the initial position of the ith atom and xi(t) = xi(0) + di. Here, we assume that the di is small displacement of xi(0). Second, the ENM models the covalent and vdW interactions using only two force constants, kcovalent = 9.30 105 dynes per cm and kvdW = 6.99 103 dynes per cm. Our previous study has determined the carbon–carbon single bond as kcovalent = 6.99 105 dynes per cm by matching the computed wavenumber with reported Raman shifts data.43,44 In the modeling of sp2 carbon interaction in graphene, this force constant is adjusted in proportion to the bond order of 1.33. Furthermore, these values are only activated under a certain range of chemical interactions. The following equations define the conditions for determining the type of force constant used: 8 > > kcovalent if xi xj dcovalent > < k ¼ kvdW if dcovalent o xi xj dvdW ; (3) > > > : 0 if xi xj 4 dvdW where dcovalent and dvdW are the cutoff distances of covalent and vdW interactions, which are set to 1.42 and 4 Å, respectively. As the carbon atoms are bonded in a hexagonal lattice, each carbon atom has three covalent bonds. Moreover, interlayer and interatomic vdW interactions are included in the model. Hence, each carbon atom has more than three virtual spring connections to its neighborhood in the BLGS ENM. For a system consisting of N atoms in three-dimensional space, the minimum number of constraints (represented by linear springs in the ENM) needed to achieve mathematical stability is 3N 6.45 Therefore, the proposed BLGS ENM is fully stable. Of course, there might exist more precise and sophisticated potentials, especially in interactions between graphene layers for accurate simulation results.46 However, in this study, we chose the 4 Å cutoff distance as the vdW interaction without loss of generality. The two given force constants also capture the steric potential properly in Cartesian space. The interatomic interactions of the ENM are compared with those of traditional MM in Fig. 1. In the ENM, solid lines with and without springs represent the virtual springs of kcovalent and kvdW, respectively. This simple and intuitive description of potentials not only saves the computational time but also leads to another outstanding advantage of the ENM such as scalability. With the physical insights concerning the graphene sheets, the nodes and their corresponding force fields could be selected at various
This journal is © the Owner Societies 2014
View Article Online
PCCP
Paper
investigated by NMA. Because the standard NMA procedure has been widely reported in the literature, we will not describe it in detail here. A full description is available in ref. 38.
3 Results and discussion
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
3.1 Vibration study of bi-layer graphene sheets (BLGSs) at the atomic scale
Fig. 1 Interatomic interactions in the MM method (a) and ENM (b). The carbon atoms are depicted as red balls that are bonded by four types of force fields, depicted as arrows, in (a) and by only two types of force fields, depicted as springs and lines, in (b).
coarse-grained levels. This feature enables us to investigate the large size of graphene sheets that are beyond the scope of study via molecular simulations. These two great advantages of the ENM, especially the CG-ENM (see Section 3.4), can dramatically reduce the computational burden incurred by using conventional atomistic approaches. Although there could be concern regarding the use of such a simplified force field, Odegard successfully introduced a similar model for CNTs by using two types of pin-jointed truss.33 Furthermore, the huddle caused by the continuum models47 is also solved in the atomic-based ENM. In recent research on the mechanical properties of graphenes,48 the ENM effectively calculated the Young’s modulus of various graphene sheets without size limitation. Finally, in regard to its simulation efficiency, scalability, and accuracy, the proposed ENM is much more appropriate to study the vibrational and mechanical properties of graphene compared to other simulation approaches. Once we constructed an ENM for a typical BLGS, as shown in Fig. 2 the vibrational characteristics of graphene sheets were
Recently, graphene-based resonators have been developed by fixing two edges of graphene sheets on a substrate.8 To mimic this condition, an armchair graphene model clamped at two edges is adopted in this study. Of course, the chirality and boundary condition may have significant impacts on our simulation results at the atomic scale. We investigated the effect of aspect ratios in the BLGSs on normal mode variations in the ESI.† Fig. 2(b) shows a schematic diagram of a doubly clamped armchair BLGS model with width ‘a’ and length ‘b’. The natural frequencies and their corresponding mode shapes are calculated by NMA. The fundamental frequency versus the size of the graphene sheets is presented in Fig. 3. As the overall size of the graphene sheets increases, the fundamental frequency appears to decrease nonlinearly. Similar results, based on MM31 or continuum beam theory,9,49 have been reported elsewhere. Interestingly, the proposed ENM could lead to almost the same results from MM, even though it has used much simpler potentials and less number of interactions than those of MM as shown in Fig. 3. In the case of beam theory, one study treated the graphene sheets as doubly clamped beams and theoretically calculated the fundamental frequency based on the following equation: sffiffiffiffiffi At E f0 ¼ 2 ; (4) L r0 where L is the length of the graphene sheet, t is the thickness of the BLGS (set as 0.67 nm),50 and A is the clamping coefficient, which is 1.03 for doubly-clamped beams.9 E and r0 represent
Fig. 2 Schematic of BLGS simulation concept. (a) The square-shaped BLGS is constructed with an ENM in which the red dots depict carbon atoms and the black solid and dotted lines depict covalent bonds and vdW interactions, respectively. (b) A schematic of a suspended graphene resonator. Two horizontal edges are fixed, and the fixed width and free length of graphene sheet are chosen as ‘a’ and ‘b’, respectively.
This journal is © the Owner Societies 2014
Fig. 3 Fundamental frequencies of a square and doubly clamped BLGS versus its length. The fundamental frequencies from beam theory, MM and our ENM are shown by blue red and black lines, respectively. The number of carbon atom bonds for the ENM and MM is also displayed as the sparse and dense checked bar graph, respectively.
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15265
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Paper
PCCP
Fig. 4 The first five mode shapes of BLGSs for width = 46.73 Å and length = 45.44 Å.
the Young’s modulus and mass density, respectively. The general values for bulk graphite used here yield r0 = 2200 kg m3 and E = 0.872 TPa. Here, the graphene sheet is assumed to have flexural vibrational modes like the bending of doubly clamped beams, which is fundamentally debatable because flexural rigidity of single- or few-layer graphene sheet is negligible. Although this continuum approach could lead to reasonable results on a practical scale, limitations in atomistic details cause large differences in nanoscale graphene vibrations compared to other atomistic approaches. The great disagreement in the small size of BGLS in Fig. 3 represents well the limitation of the continuum approach. The first five mode shapes of the BLGS are shown in Fig. 4. The fundamental vibrational modes of square BLGSs exhibit half sine, inverse half sine, center tip, full sine, and inverse full sine shapes. In addition, these fundamental mode shapes are still observed if the size of the graphene sheet is changed, albeit in a different order. These mode shapes of BLGSs are thought to be appropriate because conventional doubly-clamped systems show similar trends.51 The further studies in ESI† show that these fundamental frequencies and the trend of mode shapes are also insensitive to the aspect ratios when the size of the BLGS varies. The difference in frequency is less than 7% over the entire range of tested lengths from 45.4 to 105.1 Å in Fig. S2 of the ESI.† In conclusion, the non-dependence of graphene vibrations on the size and the aspect ratio gives high versatility for both the design and manufacturing of graphene-based applications. Furthermore, we expect BLGSs to be employed as good candidates for high-fidelity resonators because of their high frequency range as well as their high level of uniformity in mode shapes. 3.2
The layer effect of graphene on vibration frequency
One of the major potentials of graphene for NEMS applications is its remarkable electronic properties, such as the quantum Hall effect and a tunable band gap. Due to intensive progress in research on the geometrical properties of graphene sheets and their electronic properties,52–55 it was found that few-layer graphene sheets (FLGSs) are formed in various stacking orders, which can strongly influence their electronic properties. The following three stable crystallographic configurations are predicted: (1) hexagonal simple graphite with AAA stacking, (2) Bernal graphite with ABA stacking, and (3) rhombohedral graphite with ABC stacking (Fig. S3, ESI†).56,57 The firstprinciples calculations revealed that these stacking sequences
15266 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
Fig. 5 Fundamental frequency versus the layer number for various types of FLGSs. The black, red, and blue lines represent the fundamental frequencies of simple, bernal, and rhombohedral stacking types, respectively.
must be maintained for stability when the number of layers is not a multiple of three.58 For example, we expect only three stable stacking types for tetralayer graphene sheets: AAAA, ABAB, and ABCA. The vibrational characteristics of graphene sheets change with respect to the thickness of the sheets (i.e., the number of layers) and the stacking order. Fig. 5 demonstrates the change in fundamental frequency for three different graphene sheets, each with a different stacking order up to eight layers. The size of each graphene sheet is such that a = 46.73 Å and b = 45.44 Å. In the case of traditional AAA stacking, as the thickness increases, the fundamental frequency also increases but is saturated beyond a discrete layer number. This result can be attributed to the following two factors. The first factor is a geometrical effect in terms of incremental mass and two types of springs, such as the covalent and vdW interactions shown in Table 1. As each new graphene layer is stacked on the FLGS, the increments in mass and spring number are constant. Although the effects of incremental mass and spring number on frequency change are initially dominant, the relative contribution decreases as the number of layers increases. The second factor is a physical effect. Mathematically speaking, the frequency term in NMA is proportional to the square root of the stiffness over the mass. Therefore, the results in Fig. 5 resemble a square-root curve. Due to the combination of these physical and geometrical effects, the fundamental frequency of the FLGSs is eventually saturated. This result seems to be more reasonable than the existing understanding based on beam theory, in which graphene is approximated as a simple beam, where the frequency continuously increases in proportion to the thickness based on eqn (4). In addition, from this result, we can select the best number of graphene sheets to maximize their sensitivity. In the same manner, the other types of FLGSs, Bernal and rhombohedral graphite, were analyzed. For all thicknesses, these forms show higher fundamental frequencies compared to the previous results for simple graphite. In addition, rhombohedral graphite shows an increased frequency according to its specific stacking configuration, while Bernal graphite shows a trend similar to that
This journal is © the Owner Societies 2014
View Article Online
PCCP Table 1
Paper Fundamental frequencies and the number of inter-vdW interactions for three types of FLGSs
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
AAA
ABA
ABC
Layer
f0 (GHz)
vdWinter
vdWinter variation
f0 (GHz)
vdWinter
vdWinter variation
f0 (GHz)
vdWinter
vdWinter variation
2 3 4 5 6 7 8
203 249 271 283 291 297 300
3350 6700 10 050 13 400 16 750 20 100 23 450
3350 3350 3350 3350 3350 3350
233 292 321 337 347 355 361
4109 8218 12 327 16 436 20 545 24 654 28 763
4109 4109 4109 4109 4109 4109
233 298 326 341 354 361 366
4109 9391 15 093 19 202 24 484 30 186 34 295
5282 5702 4109 5282 5702 4109
For all types of FLGSs, the numbers of covalent and intra-vdW interactions are the same. Therefore, these values are not included.
of simple graphite. Such results can also be explained based on the geometrical effect shown in Table 1. When a new graphene layer is stacked on the FLGS, the incremental mass and covalent bond number remain the same for all stacking types, but the incremental vdW interactions between layers are different for different stacking types. In Bernal-type graphite, the incremental spring number for inter-layer vdW interactions is constant at 4109, and this value is greater than that of simple graphite (3350), resulting in a higher frequency in Bernal-type graphite. Likewise, rhombohedral-type graphite has a higher number of incremental springs for inter-layer vdW than simple graphite, but the number increases uniformly according to its unique stacking configuration. These results demonstrate the effects of the three different stacking orders not only on the vibrational features reflected in the fundamental frequency, but also on the electronic properties of FLGSs. 3.3
Highly sensitive mass sensing
Another possible application of graphene-based resonant devices lies in the highly sensitive detection of small bound masses through a frequency shift. In general, the relationship between a frequency shift Df and a change in the absorbed mass Dm is based on the geometry of the resonant device and its fundamental frequency, depicted as Dm ¼
2Meff Df ; f0
the graphene sheets for more precise and efficient calculations at the atomic scale. Substitution of the calculated effective mass into eqn (5) yields the relationship between the frequency shift and the change in added mass, which determines the sensitivity or resolution of a mass sensing device. This estimate implies that it is more favorable to make the graphene sheets as small as possible to obtain the highest sensitivity. Two square BLGSs with lengths of 36.90 and 61.49 Å are tested here. The corresponding fundamental frequencies are 300 and 153 GHz, respectively. Based on these two BLGSs, we calculated the frequency shifts by modulating the adsorbed mass from 1 to 10 yoctograms (1 1024 g). The added mass is placed on the center of the BLGSs, as shown in the inset of Fig. 6, and is connected to the surrounding carbon atoms through vdW interactions. The greater the added mass, the greater the frequency shift of both BLGSs. The slope Df/Dm calculated from the initial approximately linear relationship is in good agreement with the theoretical values for f0/2Meff from eqn (5), 9.17 1033 and 1.71 1033 kg1 s1, until the total mass of inserted atoms reaches that of the graphene sheets. In addition, the sensitivity is greater
(5)
where Meff and f0 are the effective mass and fundamental frequency of the given doubly clamped beam device, respectively, and all inserted atoms are assumed to be located at the center of the graphene sheets.59–61 Graphene is one of the most suitable materials for mass sensing because of its high frequency range and low effective mass.15,16 In this context, ENM-based NMA was performed to verify the sensitivity of a BLGS in detecting a small amount of added mass through a frequency shift. For a doubly clamped BLGS, the effective mass is assumed to be Meff = 0.735ltwr,
(6)
where the constant value of 0.735 reflects the reduction due to double clamping.9 l, t, w, and r are the length, thickness, width, and density of the graphene sheets, respectively. In practice, this effective mass can be calculated by multiplying the mass of a single carbon atom by the total number of atoms contained in
This journal is © the Owner Societies 2014
Fig. 6 Frequency shift versus the change in adsorbed mass located at the center of the BLGS. The red and blue dotted lines represent the relationship between the frequency shift and the added mass based on beam theory (eqn (5)) for small (36.90 Å in length) and large (61.49 Å in length) rectangular graphene sheets, respectively. The red and blue solid lines depict the results from NMA of the ENM for small and large sheets, respectively. The inset shows a schematic of a BLGS sensor with one inserted atom.
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15267
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Paper
for small BLGSs when comparing the Df/Dm slopes of the two graphene sheets from both theoretical and simulation data. Thus, the higher sensitivity of the smaller graphene sheets is primarily due to the small number of atoms composing the graphene sheet, which could exert a relatively strong effect on the frequency shift, even for a small change in mass. The effect of the inserted mass position on the frequency shift was also investigated. It is well known that random positioning of inserted masses causes practical difficulties for the precise operation of resonators.62 As an example, by moving a small mass of 5 yoctograms to all available positions of a BLGS with a length of 36.90 Å, the corresponding changes in fundamental frequency were tested. Fig. 7 shows contours on the unit square graphene sheets. Each contour encloses a region in terms of the frequency shift due to the inserted atoms. In this figure, the highest change in frequency Df is obtained at 159 GHz when the particle is located at the center of the free edge line. This range of frequency shift may occur when a 15-yoctogram particle is placed at the center of the BLGS. In addition, for all available positions, the changes in frequency increase when the particle is located far from the fixed lines while also being located close to the free edge line. These results can be intuitively understood from the general statics equation in which the maximum torque is generated when the weight is far from the fixed pivot position. This result is comparable to a previous experimental result in which the greatest change in frequency is obtained when an absorbed gold bead is positioned at the tip of a cantilever.63 These simulation results show that not only the mass of the adsorbed particle but also the position of the adsorbed particle can influence the change in fundamental frequency. Therefore, the uncertainty in the position of the adsorbed particle must be considered when measuring mass using the frequency shift of a resonator. Our ENM-based NMA will serve as a precise and practical prediction method for this purpose.
Fig. 7 Contours of frequency shift values, e = Df/f0, according to the position of an inserted mass of 5 yoctograms. The square BLGS is 36.90 Å long, and both the top and bottom lines are clamped.
15268 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
PCCP
3.4
CG-ENM for scale-up simulations
In practice, the actual size of graphene-based resonators must be at least on the submicron level due to limitations in both manufacturing and measurement. Although the ENM can dramatically reduce computational burdens, it is also challenging to run a full-atom ENM simulation for graphene sheets that are tens of nanometers long in a timely fashion. To overcome this limitation, a CG-ENM implemented by the fine-trains-coarse approach is proposed here as an attractive alternative for micrometer scale simulations. The main idea for constructing the CG-ENM for graphene is to maintain its structural characteristics (i.e., hexagonal lattice) and force field types with fixed DOFs. For more details, Fig. 8 depicts a two-dimensional hexagonal lattice of carbon atoms with overlaid CG models of different sizes. Three different hexagonal lattices corresponding to three levels of CG are constructed according to their unit of length, rn = 2nr0 where r0 = 1.42 Å and a positive integer, ‘n’, is the level of CG. In the same manner, the unit of mass is mn = 22n m0, where m0 = 2 1026 kg is the mass of one carbon atom. Unlike the in-plane structural change, the distance between each layer remains 3.35 Å. In the case of force field modeling, all spring constants should be modulated by comparing them with those from the original full-atom model. The main modulation factors contributing to the natural frequency that must be considered are the spring connectivity and unit length in each CG-ENM. Basically, a lower number of spring connections cause a lower frequency in CG models. We can compensate for this reduction in the CG model by multiplying the stiffness value by the number ratio of a CG model and a full-atom model. Additionally, the longer unit length leads to lower force constants, and this inverse relationship also contributes to a lower frequency. The unit length in a CG model is defined by the CG level. Therefore, multiplication of the reciprocal of the length ratio and the spring constant is also required in the framework for modeling equivalence. Table 2 gives an overview of the resultant force fields for the full-atom
Fig. 8 The unit honeycomb lattice of the different cg models and their arrangement in the atomic graphene sheet. All CG models have the same DOF and follow the same atomic model. The figure is conceptually adopted from ref. 64.
This journal is © the Owner Societies 2014
View Article Online
PCCP
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
Table 2
Paper Resulting variables for given levels of BLGS modeling
Model
Unit length (1.42 Å)
Mass (2 1026 kg)
k_covalent (6.99 105 dynes per cm)
k_intra_vdW (6.99 103 dynes per cm)
k_inter_vdW (6.99 103 dynes per cm)
Full CG1 CG2 CG3
1 2 4 8
1 4 16 64
1 1 (2, 0.5)a 1 (4, 0.25)a 1 (8, 0.125)a
1 2.17 (4.33, 0.5)a 2.172 (18.7, 0.25)a 2.173 (81.0, 0.125)a
1 1.66 (2, 0.83)a 2.21 (4, 0.55)a 2.47 (8, 0.31)a
a The numbers outside of the parentheses in the k_covalent, k_intra_vdW, and k_inter_vdW columns represent specific ratios at each CG level. The left and right numbers in the parentheses are the ratios of bond numbers and inverse unit length, respectively. Unlike the constant or increasing k_covalent and k_intra_vdW cases, k_inter_vdW increases but is eventually saturated.
and three CG models. Based on the concept of force field modeling, only two types of spring constants are arranged in the same configuration as the full-atom graphene model. However, asymmetric growth in the unit length, where the interlayer distance is fixed but the in-plane unit lengths regularly increase, bifurcates the single vdW interaction into inter-vdW and intravdW interactions. For the first CG level, the doubled length in the covalent and intra-vdW interaction reduces the spring constant by one half, whereas the length of the inter-vdW interaction changes very little, because of the constant layer thickness. These effects require the spring constants (i.e. k_covalent and k_intra_vdW cases, k_inter_vdW) in the CG model to be multiplied by 0.5, 0.5, and 0.83, which are inverse values of the unit length ratio between CG and full atomic models. As another critical factor, the number ratios for each connection type are also reflected by multiplying the spring constants in the CG model by 2, 4.33, and 2. For the second, third, and much larger CG levels, these force field ratios are uniform; thus, they can be incorporated into the spring constants according to the CG levels (except the inter-vdW interaction, because of the constant layer thickness). Table 3 compares the fundamental frequencies of the fullatom and CG models. Due to computational limits, only three types of full-atom models (35, 70, and 140 Å in length) were analyzed and compared with appropriate CG models. The models on the diagonal blocks are considered to have the same DOF which also requires the same computational time. At the second and third levels of the CG models, with a length of 70 and 140 Å, the errors in frequency are less than 11%. Although a full-atom simulation was not performed in the case of the fourth level of the CG model, which has a length of 280 Å, the
Table 3 A comparison of fundamental frequencies for four different sizes of BLGSs at each CG level
Fundamental frequency (GHz) Size (Å) 35 70 140 280
DOF 1116 4012 15 180 59 020
Time 6 m 43 s 2 h 39 m 33 s 6 d 6 h 38 s Non
a
Full 256 115 41.3 N/Ab
CG1
CG2
CG3
a
a
N/Aa N/Aa N/Aa 13.0
N/A 120 46.2 14.4
N/A N/Aa 42.4 15.5
We basically use three DOFs: 1116, 4012 and 15 180. The model under 1116 DOFs could be unfeasible due to the side effects. b Due to computational limits, a full model with a size of 280 Å could not be calculated.
This journal is © the Owner Societies 2014
similar ranges of frequency values obtained from the three other CG levels strongly indicate the accuracy of the proposed CG-ENM. Furthermore, a more practical size of the graphene sheet was tested by the proposed CG force fields and also showed good agreement with the existing experimental data. The exfoliated BLGS with dimensions a b t = 1.35 mm 0.85 mm 0.67 mm exhibits a fundamental frequency at 13.33 MHz.65 Allowing for the built-in tension of 1.6 nN which leads to increase in frequency, our CG-ENM successfully predicts the slightly small but almost the same frequency of 10.50 MHz. In short, the use of an atomistic configuration with topologically scalable force fields in the proposed CG-ENM will shed light on mesoscale vibrations by providing both a reasonable computation time and acceptable simulation accuracy.
4 Conclusions In this study, a variety of vibrational characteristics of graphene sheets and their potential as highly sensitive resonators have been thoroughly investigated by an atomistic modeling approach called the ENM. At the atomic scale, BLGSs were systemized as layered carbon atoms connected by two types of linear springs based on chemical information. Then, normal modes for various types of BLGSs were calculated. Fundamental frequencies and their corresponding mode shapes guaranteed the performance of BLGSs as a strong candidate for high-fidelity resonators. Furthermore, the effects of the layer number and the stacking type of the graphene sheets on the fundamental frequency were also investigated. For all stacking types, as the number of layers increased, the fundamental frequencies also increased, but were saturated at a certain number of layers. However, all types of graphene sheets showed unique trends in frequency due to their different topological arrangements. The ENM was also used to explore the potential use of graphene sheets as mass detectors by estimating the frequency shift. Changes in frequency were calculated depending on the quantity of absorbed mass as well as the position of the mass. The linear relationship obtained between the frequency shift and the adsorbed mass matched well with the existing theoretical model. In addition, the simulation results expressing the position dependency of the frequency shift can serve as a practical design factor in high-fidelity resonators for mass sensing. To extend our study to a more practical scale in which both manufacturing and measurements are possible, the CG-ENM was newly proposed. By maintaining the unique hexagonal sub-lattice structure of
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15269
View Article Online
Paper
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
graphene, a reduced set of representative atoms was sampled, and spring constants were appropriately modulated to maintain equivalent force fields. Dramatic reductions in computation time with acceptable accuracy in frequency prediction, even compared to experimental data, can promote our understanding of the vibrational characteristics of graphene based resonators in practical use.
Acknowledgements This research was supported by the Basic Science Research Program (2011-0014584), Pioneer Research Center Program (2012-0009579), and Basic Research Laboratory Program (20110020024) through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology.
Notes and references 1 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubones, I. V. Grigorieva and A. A. Firsov, Science, 2004, 35, 666–669. 2 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature, 2005, 438, 197–200. 3 P. O. Lehtinen, A. S. Foster, Y. Ma, A. V. Krasheninnikov and R. M. Nieminen, Phys. Rev. Lett., 2004, 93, 187202. 4 C. Lee, X. Wei, J. W. Kysar and J. Hone, Science, 2008, 321, 385–388. 5 J. Atalaya, A. Isacsson and J. M. Kinaret, Nano Lett., 2008, 8, 4196–4200. 6 R. B. Karabalin, X. L. Feng and M. L. Roukes, Nano Lett., 2009, 9, 3116–3123. 7 A. M. Zande, R. A. Barton, J. S. Alden, C. S. Ruiz-Vargas, W. S. Whitney, P. H. Q. Pham, J. Park, J. M. Parpia, H. G. Craighead and P. L. McEuen, Nano Lett., 2010, 10, 4869–4873. 8 C. Chen, S. Rosenblatt, K. I. Bolotin, W. Kalb, P. Kim, I. Kymissis, H. L. Stromer, T. F. Heinz and J. Hone, Nat. Nanotechnol., 2009, 4, 861–867. 9 J. S. Bunch, A. M. Zande, S. S. Verbridge, I. W. Frank, D. M. Tanenbaum, J. M. Parpia, H. G. Craighead and P. L. McEuen, Science, 2007, 315, 490–493. 10 A. M. Zande, A. S. Paulo, B. Lassagne, P. L. Mceuen and A. Bachtold, Nano Lett., 2008, 8, 1399–1403. 11 M. Poot and H. S. Zant, Phys. Lett., 2008, 92, 063111. 12 J. T. Robinson, M. Zalalutdinov, J. W. Baldwin, E. S. Snow, Z. Wei, P. Sheehan and B. H. Houston, Nano Lett., 2008, 8, 3441–3445. 13 J. S. Bunch, S. S. Verbridge, J. S. Alden, A. M. Zande, J. M. Parpia, H. G. Craighead and P. L. Mceuen, Nano Lett., 2008, 8, 2458–2462. 14 S. Shivaraman, R. A. Barton, X. Yu, J. Alden, L. Herman, M. Chandrashekhar, J. Park, P. L. McEuen, J. M. Parpia, H. G. Craighead and M. G. Spencer, Nano Lett., 2009, 9, 3100–3105.
15270 | Phys. Chem. Chem. Phys., 2014, 16, 15263--15271
PCCP
15 S. M. Avdoshenko, R. C. Gomes and G. Cuniberti, Nanoscale, 2012, 4, 3168–3174. 16 R. A. Barton, J. Parpia and H. G. Craighead, J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom., 2011, 29, 05081. 17 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson and K. S. Novoselov, Nat. Mater., 2007, 6, 652–655. 18 K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov and A. K. Geim, Proc. Natl. Acad. Sci. U. S. A., 2005, 102, 10451–10453. 19 A. J. Van Bommel, J. E. Crombeen and T. Van, Surf. Sci., 1975, 48, 463–472. 20 J. W. Jiang, J. S. Wang and B. Li, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 113405. 21 K. Samadikhah, J. Atalaya, C. Huldt, A. Isacsson and J. Kinaret, MRS Proc., 2007, 1057, 10–20. 22 R. Chowdhury, S. Adhikari, F. Scarpa and M. I. Friswell, J. Phys. D: Appl. Phys., 2011, 44, 205401. 23 K. Behfar and R. Naghdabadi, Compos. Sci. Technol., 2005, 65, 1159–1164. 24 X. Q. He, S. Kitipornchai and K. M. Liew, Nanotechnology, 2005, 16, 2086–2091. 25 M. K. Liew, X. Q. He and S. Kitipornchai, Acta Mater., 2006, 54, 4229–4236. 26 M. R. Lin, Comput. Mater. Sci., 2012, 53, 44–52. 27 F. Scarpa, S. Adhikari and P. A. Srikantha, Nanotechnology, 2009, 20, 065709. 28 A. Sakhaee-Pour, M. T. Ahmadian and A. Vafai, Solid State Commun., 2008, 147, 336–340. 29 S. Arghavan and A. V. Singh, J. Appl. Phys., 2011, 110, 084308. 30 C. Baykasoglu and A. Mugan, Comput. Mater. Sci., 2012, 55, 228–236. 31 Y. Chandra, R. Chowdhury, F. Scarpa and S. Adhikaricor, Thin Solid Films, 2011, 519, 6026–6032. 32 A. Sakhaee-Pour, M. T. Ahmadian and R. Naghdabadi, Nanotechnology, 2009, 19, 085702. 33 G. M. Odegard, T. S. Gates, L. M. Nicholson and K. E. Wise, Compos. Sci. Technol., 2002, 62, 1869–1880. 34 S. K. Georgantzinos, G. I. Giannopoulos and N. K. Anifantis, Mater. Des., 2010, 31, 4646–4654. 35 C. Li and T. W. Chou, Int. J. Solids Struct., 2003, 40, 2487–2499. 36 M. K. Kim, R. L. Jernigan and G. S. Chirikjian, J. Struct. Biol., 2003, 143, 107–117. 37 M. K. Kim, R. L. Jernigan and G. S. Chirikjian, Biophys. J., 2002, 83, 1620–1630. 38 M. K. Kim, G. S. Chirikjian and R. L. Jernigan, J. Mol. Graphics Modell., 2002, 21, 151–160. 39 M. K. Kim, Y. Jeong and J. I. Jeong, Int. J. Control Autom. Syst., 2006, 4, 382–393. 40 C. J. Caseweit, K. S. Colwell and A. K. Rappe, J. Am. Chem. Soc., 1992, 114, 10035–10046. 41 M. Tirion, Phys. Rev. Lett., 1996, 77, 1905–1908. 42 S. Arghaven and A. V. Singh, J. Phys. D: Appl. Phys., 2012, 45, 455305.
This journal is © the Owner Societies 2014
View Article Online
Published on 13 June 2014. Downloaded by North Dakota State University on 30/10/2014 00:00:33.
PCCP
43 M. Hu, S. Raj, B. Kim, W. K. Liu, S. Baik, T. Kim, B. S. Lim and M. K. Kim, J. Mech. Sci. Technol., 2010, 24, 1771–1780. 44 J. M. Berg, J. L. Tymoczko and L. Stryer, in Biochemistry, W. H. Freeman and Company, New York, 5th edn, 2008. 45 M. H. Kim, S. Seo, J. I. Jeong, B. J. Kim, W. K. Liu, B. S. Lim, J. B. Choi and M. K. Kim, Protein Sci., 2013, 22, 605–613. 46 I. V. Lebedeva, A. A. Knizhnik, A. M. Popov, Y. E. Lozovik and B. V. Potapkin, Phys. Chem. Chem. Phys., 2011, 13, 5687–5695. 47 I. E. Berinskii and F. M. Borodich, Mech. Mater., 2013, 62, 60–68. 48 J. Kang, S. Hwang, J. H. Kim, M. H. Kim, J. Ryu, S. J. Seo, B. H. Hong, M. K. Kim and J. Choi, ACS Nano, 2012, 6, 5360–5365. 49 W. Weaver Jr., S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, Wiley, New York, 5th edn, 1990. 50 Z. H. Ni, H. M. Wang, J. Kasim, H. M. Fan, T. Yu, Y. H. Wu, Y. P. Feng and Z. X. Shen, Nano Lett., 2007, 7, 2758–2763. 51 R. C. Tung, A. Garg, A. Kovacs, D. Peroulis and A. Raman, J. Micromech. Microeng., 2013, 23, 045009. 52 C. L. Lu, C. P. Chang, Y. C. Huang, J. H. Ho, C. C. Hwang and M. F. Hwang, J. Phys. Soc. Jpn., 2007, 76, 024701. 53 F. Guinea, A. H. Castro Neto and N. M. R. Peres, Solid State Commun., 2007, 143, 116–122.
This journal is © the Owner Societies 2014
Paper
54 K. F. Mak, J. Shan and T. F. Heinz, Phys. Rev. Lett., 2010, 104, 1–4. 55 L. M. Malard, M. A. Pimenta, G. Dresselhaus and M. S. Dresselhaus, Phys. Rep., 2009, 473, 51–87. 56 J. C. Charlier, J. P. Michenaud and X. Gonze, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 4531–4539. 57 J. C. Charlier and J. P. Michenaud, Phys. Rev. Lett., 1993, 70, 1858–1861. 58 M. Aoki and H. Amawashi, Solid State Commun., 2010, 150, 458. 59 K. Jensen, K. Kim and A. Zettl, Nat. Nanotechnol., 2008, 3, 533–537. 60 V. Sazonova, PhD thesis, Cornell University, 2006. 61 Y. T. Yang, C. Callegari, X. L. Feng, K. L. Ekinci and M. L. Roukes, Nano Lett., 2006, 6, 583–586. 62 J. Atalaya, J. M. Kinaret and A. Isacsson, Europhys. Lett., 2010, 91, 48001. 63 S. Dohn, R. Sandberg, W. Svendsen and A. Boisen, Appl. Phys. Lett., 2005, 86, 233501. ˜ ol, S. Succi, A. Greiner and 64 D. Kauzlaric´, J. T. Meier, P. Espan J. G. Korvink, J. Chem. Phys., 2011, 134, 064106. 65 J. Lee and P. X. L. Feng, IEEE International Frequency Control Symposium (FCS), Baltimore, 2012.
Phys. Chem. Chem. Phys., 2014, 16, 15263--15271 | 15271
