Electromechanical Properties of Carbon Nanotubes - The Journal of

Article pubs.acs.org/JPCC

Electromechanical Properties of Carbon Nanotubes Rosalba Juarez-Mosqueda, Mahdi Ghorbani-Asl, Agnieszka Kuc, and Thomas Heine* School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany S Supporting Information *

ABSTRACT: Electromechanical properties of carbon nanotubes were studied using Born−Oppenheimer molecular dynamics simulations within the QM/ MM approach. The indentation of nanotubes was simulated using an AFM tip. The electronic structure and transport response to the mechanical deformations were investigated for different deflection points starting from perfect unperturbed systems up to the point where the first bonds break. We found the dependence of the force constant on the diameter size: the smaller the diameter, the larger the k. For the metallic-armchair tubes, with diameters from 8 to 13 Å, the conductance decreases only slightly under radial deformation, and a tiny band gap opening of up to 50 meV was observed.

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INTRODUCTION The research of carbon-based materials has rapidly evolved in the past two decades,1 especially after the discovery of carbon nanotubes (CNTs) by Iijima2 in 1991 and the extraction of single-atom-thick crystallites of graphene in 2004 by Novoselov and Geim.3 Extensive research on graphene and CNTs, especially their extraordinary mechanical and electronic properties, has been widely covered in numerous reviews.4−8 Graphene has been extensively investigated as potential material for the development of nanoelectronic devices due to its remarkably high electron mobility,9 long electron coherence,3 and extreme flexibility and stability.10 Nonetheless, the absence of an intrinsic band gap in graphene makes it difficult to apply this material in transistors or other switching devices.11,12 There have been several strategies proposed toward the band gap opening, among them mechanical deformations. Various theoretical and experimental studies on deformed graphene suggested that tensile strain shifts the Dirac cones and changes the Fermi velocity,13−20 keeping at the same time the metallic characteristics intact.16 Lee et al.21 reported a nonlinear behavior of stress−strain response to indentation of a free-standing graphene. Recent experiments by Huang et al.19 showed that in situ nanoindentation on suspended graphene changes the electrical resistance only slightly, without band gap opening. Moreover, it is already well-known that the bending rigidity of graphene is strongly dependent on the number of layers.22 Neek-Amal et al.23 showed, from classical molecular dynamics simulations, that in order to achieve the same deflection for bilayer graphene a force twice as large as for the monolayer was necessary. On the other hand, carbon nanotubes exhibit a wide range of possible band gap sizes depending on the tube diameter and the chirality. Therefore, the field of CNTs has also been widely explored on the experimental and theoretical bases towards electromechanical properties.24−38 In 1999, Pulson et al.39 investigated the interplay between electronic and mechanical properties of multiwalled CNTs by applying strain to © XXXX American Chemical Society

nanotubes with an AFM (atomic force microscope) tip. The authors reported that the strain in the nanotubes has no measurable effect on the resistance until the material was strained beyond the elastic limit. On the other hand, in early 2000, Tombler et al.24 have reported studies regarding the electronic properties of single-walled CNTs under mechanical deformation triggered by AFM tip indentation. The authors concluded that the conductance decreases by more than 2-fold for bending angles larger than 7°. Several studies based on the tight-binding (TB) method have been carried out to simulate CNTs under tension,33,40,41 torsion,38,41 and bending.24,33−35,38,42,43 Only a few firstprinciples-based calculations on this subject have been reported to date.30,32,44 Most of the theoretical works focus on the standard examples of the (5,5) armchair and the (10,0) zigzag CNTs;30,32,42,44 however, other tube chiralities were investigated as well.31,33−35,38 In 2000, Maiti et al.30 used density functional theory (DFT) to investigate the atomistic deformations in bent (5,5) CNT versus AFM deformed tubes. The authors applied DFT calculations only to the interaction region close to the tip, while the rest of the system and the tip itself were treated classically. They concluded that the bent tubes maintain the allhexagonal network even for large bending angles, while the AFM-deformed CNTs develop an inward conelike structure terminating in an apex. Using a nonorthogonal tight-binding method, Liu et al.42 showed that the drastic reduction in the conductance of the (5,5) CNTs under indentation, as observed by Tombler et al.,24 is a consequence of the change in the bonding nature induced by the local action of the AFM tip. Later, the same authors have shown that for carbon-based nanosystems with delocalized deformations, the change in electronic conductance is relatively Received: March 5, 2014 Revised: June 2, 2014

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dx.doi.org/10.1021/jp502267d | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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small even for large bending angles.43 Moreover, Rochefort et al.28 showed that electromechanical properties of CNTs deformed by an AFM tip strongly dependent on the bending angle as well as the tube diameter. Ivanovskaya et al.32 have used Born−Oppenheimer molecular dynamics simulations within the density functional based tight-binding (DFTB) method to investigate the effect of axial elongation on the electronic and transport properties of (5,5) and the (10,0) CNTs. The authors showed that the conductance of metallic CNTs decreases under strain and reaches a negligible value prior to rupture. The semiconducting character was maintained for any value of strain. In this work, we have studied electromechanical properties and quantum transport of small- and large-diameter CNTs under indentation with a sharp AFM tip. A wide range of chiralities and tube diameters were covered, and as largediameter limit we included graphene mono- and bilayer. Simulations were carried out using the DFTB method.45,46 We put special emphasis on large-diameter CNTs. We have recently shown that properties of large-diameter inorganic nanotubes can be successfully simulated and understood using layered models.47 Therefore, in the present work on CNTs we apply a similar approach. Moreover, we have previously investigated the length and diameter dependence of the band gap in CNTs, 48 showing that the Γ-point approximation can be successfully applied to study electronic properties of honeycomb carbon materials, but the number of unit cells must be carefully selected, such that the Dirac point folds onto the Γ-point.48 Dirac points are the high-symmetry points in the Brillouin zone, where bands cross the Fermi level in a linear manner. At these points electrons and hole have zero effective masses and are called Dirac fermions. In the present work, we have applied the knowledge gained in both earlier works on NTs. Our results show that semiconducting small-diameter SWCNTs under local indentation reduce their band gaps by up to 60%, while the metallic CNTs only slightly open the band gaps up to the maximum of 50 meV. On the other hand, the band gaps of large-diameter CNTs remain essentially robust against the radial deformations. Quantum transport simulations show that the indentation does not significantly change the conductance close to the Fermi level, while away from EF, the transport reduces considerably.

Figure 1. Snapshots of AFM tip indentation into the carbon nanotubes for different stages of deformation. Equilibrium distances (d) between bottom of the tip and the top of the carbon structure are indicated in the left axis. From left: small-diameter SWCNT, largediameter SWCNT, and the large-diameter DWCNT.

NT (∼38 Å). The extension of the orthorhombic cells for the layered models was a = 2.456 Å, b = 4.260 Å. For graphene, we have used three different sizes of the supercells: 14 × 8, 16 × 9, and 18 × 10, whereas the bilayer was takes as a double of the smallest monolayer structure. London dispersion interactions were used following the approach of Zhechkov et al.52 The AFM tip was modeled with 64 atoms of tungsten. We have employed Born−Oppenheimer molecular dynamics (BOMD) simulations within the QM/MM approach. In our simulation setup, CNTs were treated quantum mechanically, while the AFM tip was calculated classically using the nonbonded parameters of the universal force field (UFF),53 an approach that has been successfully applied in an earlier work.54 One ring of the carbon nanotubes, away from the AFM tip, was fixed, serving as model for the support of the tubes. The nanoindentation is simulated in a stepwise procedure,55 where the tip is consecutively moved toward the nanotube. While the tip is kept frozen in order to avoid chemical interactions, the nanotube is described within BOMD. The energy transfer between the tip and the tube is controlled by the NVT ensemble at a temperature of 600 K. According to our experience32,54 at this temperature thermal equilibration is enhanced, but the structures are still stable. In each indentation step, the model has been equilibrated (Berendsen thermostat56 with coupling constant to the bath of τ = 0.2 ps) until the standard deviation of the temperature of the system was within 20 K for 1 ps. The time step was chosen to be 1 fs. A very similar simulation has been used to study the breaking of inorganic nanotubes under axial pressure.54 The transport calculations were performed using the DFTB Hamiltonian in the nonequilibrium Green’s function (NEGF)58 method and the Landauer−Büttiker approach.58−62 Our inhouse DFTB-NEGF software for quantum conductance has already been successfully applied to various 1D and 2D nanomaterials.47,63,64 The simulation setup consists of a finite CNT as a scattering region connected to two semi-infinite perfect CNTs as electrodes (see Figure 2). For the layered models of large-diameter CNTs, the in-plane direction perpendicular to the transport axis is infinite by implementing periodic boundary conditions within the Γ-point approximation.48

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COMPUTATIONAL DETAILS We have studied an extensive set of carbon nanotubes of different chirality and diameter, including armchair (n,n), zigzag (n,0), and chiral (n,m) single-walled carbon nanotubes (SWCNTs). Moreover, we have studied large-diameter SWand double-walled DW-CNTs, approximated by graphene mono- and bilayer models, respectively (see Figure 1). All calculations were carried out using density functional based tight-binding (DFTB) method45,46 as implemented in the deMonNano code.49 For all systems, we have used fixed unit cell parameters with C−C bond lengths of 1.440 Å, reflecting the experimental values.50,51 CNTs and layered models were represented using one- and two-dimensional periodic boundary conditions, respectively. The nonperiodic directions were represented with a 50 Å layer of vacuum. We have used 10 and 17 unit cells for zigzag and armchair NTs, respectively, resulting in the tube lengths of about 42−43 Å. For the chiral NTs we have used 4 unit cells in the case of the (8,4) and the (10,5) NTs (∼46 Å) and 2 unit cells for the (9,6) B



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threshold, the structure was considered to have one broken bond.

Figure 2. Schematic representation of the electronic transport in the CNTs. Left and right electrodes (L and R) that consist of semi-infinite ideal systems are highlighted. The scattering region between the electrodes includes the deformed structures. The transport direction is indicated by the arrow.

Because of the long electron mean-free path in CNTs, the Landauer formalism is expected to be a proper approximation for studying coherent transport properties of our systems.26,65,66 In this approach, and at zero bias, the electrical conductivity through the system can be presented in terms of the Green’s functions of scattering region and the coupling of the device to the electrodes:58 .(E) =

Figure 4. Exemplary bond length evolution in the (9,0) CNT at the tip position of −4.1 Å. The relevant bonds close to the tip are highlighted. The Cx−Cw bond is assumed to be broken as in average it has a value larger than the threshold.

We have calculated the evolution of the band gaps of semiconducting nanotubes using the Γ-point approximation (see Figure 5). For the metallic systems we have calculated the band gaps from the band structure calculations performed through high-symmetry k-points in the Brillouin zone (BZ) along the paths X−Γ−X and Γ−X−R−Γ for NTs and 2D models, respectively (see Figure 6). This is necessary as the Dirac point for those systems may appear away from the Γpoint and spurious band gap opening could be encountered. DeMonNano49 and the DFTB+57 codes were used for the electronic structure calculations. Furthermore, we have calculated the band gap evolution for selected MD trajectories and tip positions either from the Γpoint approximation (see Figure 5) or band structures (see Figure 6) for semiconducting and metallic systems, respectively.

† 2e 2 trace[Ĝ ΓR̂ Ĝ ΓL̂ ] h

where Ĝ is the total Green’s function of the CNT, coupled to the two semi-infinite electrodes and Γ̂α = −2ImΣ̂α is known as the broadening functions. The Σ̂L and Σ̂R are known as selfenergies that can be computed iteratively.62,67 Similar as in Stefanov et al.,54 mechanical properties were determined from force constant (k) calculations. The harmonic approximation was used to calculated k as the second derivative of energy with respect to the position of the AFM tip (see Figure 3). This approach is valid as the relation between the tip position and the deflection of the material is linear. We have analyzed the results from molecular dynamics simulations as follows: We have set the threshold of the broken C−C bond to be at 1.65 Å, which corresponds to about 14.5% elongation. The bond length evolution was analyzed for each MD trajectories at selected tip positions (see Figure 4). If one of the bonds was for longer time at distances larger than our

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RESULTS AND DISCUSSION We have investigated the electromechanical properties of CNTs under AFM tip indentation by calculating the electronic and

Figure 3. Typical potential energy (left) and deflection (right) curves for the indentation of the (6,6) SWCNT (black), large-diameter SWCNT (red), and the large-diameter DWCNT (blue). Dashed lines correspond to the fitted data. Plots until the first bond is broken. d denotes the distance from the bottom of the tip to the top of the carbon structure. C



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Figure 5. Band gap evolution for selected MD trajectories of semiconducting (SC) CNTs calculated from the Γ-point approximation. The tip positions (d) are given in Å.

Figure 6. Band structures of metallic (M) systems calculated for selected tip positions (d), which are given in Å.

We have compared the electronic properties of equilibrated nanotubes with the perturbed ones just before the first bond is broken (see Figures 6 and 7). The results show that for metallic nanotubes the Dirac point shifts and the band gap opens slightly to a maximum value of 50 meV (see Figure 6). The only apparent deviation is the (9,0) CNT, which has an intrinsic band gap of about 120 meV due to the large curvature and resulting σ−π overlap.48 This pronounced curvature breaks the degeneracy at the Γ point and opens a gap. Figure 7 shows the Cx−Cw bond length (cf. Figure 4) and band gap evolution with the tip position. Since BOMD simulations encounter the fluctuations of both properties during trajectories, we have plotted the average numbers from each MD trajectory together with the standard deviation. The results show that with the tip position the fluctuations increase; however, we can draw a few general trends: Semiconducting zigzag nanotubes reduce their band gaps

mechanical properties at selected deformations. The calculated force constants and initial band gaps of small diameter nanotubes are shown in Table 1. The calculated force constants of CNTs decrease when the tube diameter increases. This means that tubes with large curvature need a larger amount of force to be deformed. This could be understood in the way that small-diameter tubes have high strain energy due to the curvature and additional deformations on the surface would require much larger amount of energy than corresponding nanotubes with larger diameters. On the other hand, there is not a clear trend between deflection and tube radius for the values reported in Table 1. The reason is that in this simulation model, which is motivated by a potential experimental setup, the system reacts with deflection and collective motion resulting in a global structure change to the external force. D



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Table 1. Tube Chirality (n,m), Number of Atoms (N) in the Simulation Supercell, Tube Radius (r), Initial (ΔI) Band Gap, the Deflection (δ), and the Force Constant (k) of All Studied Small-Diameter CNTsa (n,m)

N

r (Å)

ΔI (eV)

δ (Å)

k (eV Å−2)

character

(10,10) (8,8) (6,6) (10,5) (9,6) (8,4) (10,0) (9,0) (8,0)

680 544 408 560 456 448 400 360 320

6.876 5.500 4.125 5.251 5.191 4.201 3.969 3.572 3.175

0.00 0.00 0.00 0.62 0.00 0.90 0.92 0.12 1.03

5.368 4.909 5.225 5.170 5.194 5.232 4.902 6.287 5.969

1.405 1.534 1.855 1.478 1.522 1.619 1.662 1.780 2.032

M M M SC M SC SC Mb SC

Table 2. Number of Atoms (N) in the Simulation Supercell, Initial (ΔI) Band Gap, the Deflection (δ), and the Force Constant (k) of SWCNTs and DWCNTs Simulated Using Corresponding 2D Layered Models in the Rectangular Representationa system

N

ΔI (eV)

δ (Å)

k (eV Å−2)

SWCNTs

448 576 720 896

0.00 0.00 0.00 0.00

4.321 4.759 5.171 3.704 3.058

2.054 1.774 1.590 2.742

DWCNTs

a

a The values of k and δ are given for the point right before the rupture of the first C−C bond. For the DWCNT two deflection values correspond to the deformation of the top and the bottom layer, respectively.

significantly under indentation, to values of about 40% of their nondeformed counterparts. We do not observe, however, any semiconductor−metal transition for the elastic deformations. The band gaps of chiral nanotubes change only slightly, reaching for larger diameters a maximum reduction of 50 meV. Furthermore, we have calculated the force constants and the band gaps of graphene and graphene bilayer. These models shall represent very large diameter nanotubes (see Table 2). Here, we have studied the effect of the indentation with respect

to the area of the system. Again, the results show that for larger supercells we obtain smaller force constant (k). The electronic structure changes in the same way as the small-diameter CNTs with band gap opening of no more than 50 meV (see Figure 6). We have simulated the large-diameter limit DWCNT using graphene bilayer with an interlayer distance as found in graphite (3.348 Å).68−71 The force constant in this case is larger than in the corresponding SWNT by about 1 eV Å−2. This is shown in Figure 3, where the slope of the energy curve with respect to the deformation is steeper for the DWCNT (cf. Table 2). For the case of DWCNT, the indentation of the lower layer is of course weaker than in the layer close to the AFM tip. We

The values of k and δ are given for the point right before the rupture of the first C−C bond. M and SC stand for metallic and semiconducting character, respectively. bThe small band gap is due to the large curvature; see text for explanation.

Figure 7. Average bond lengths and band gaps for various tip positions in the semiconducting (SC) NTs. The standard deviations, indicated by error bars, due to the MD fluctuations of the property are given. E



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overlap and consequently in a small band gap opening, in agreement with earlier work.34,48 According to Boungiorno35 and Blase,72 metallic character and transport properties of small diameter nanotubes, such as (9,0) and (6,0), CNTs, are triggered by the tube diameter, leading to the separation of π and π* bands and the increase in the π*−σ* hybridization. On the other hand, our results indicate that semiconducting nanotubes, e.g. the (10,0) or the (8,4) CNT, always keep their semiconducting character under AFM indentation, even though their band gaps drop considerably. These results confirm the same behavior which has been reported earlier.33−35 In the case of SWCNT layered models, we have observed similar changes in the conductance as in the case of metallic CNTs. This indicates that the electronic structure of largediameter CNTs around the Fermi level is almost unaffected by the mechanical deformations. The quantum conductance of DWCNT is exactly twice larger than for the corresponding SWCNT model. This is expected as we have twice as many transport channels available in the energy window. Moreover, the DWCNT is also less sensitive to applied deformation and the changes in the conductance because the tip indentation is less pronounced. Again, the results suggest that multiwalled nanotubes should be more stable against indentation, keeping their transport properties almost unchanged. We have analyzed the conductivity of individual layers in the DWCNT (see Figure 9). We have found that the conductance changes slightly more in the top layer than the bottom layer, due to the larger deflection at given step of the indentation.

have obtained 3.7 and 3.1 Å deflection values for the top and bottom layers, respectively, for the largest deformations before the rupture of the first bond. Moreover, we observe that the band gap opening in the DWCNT is smaller than in the corresponding SWCNT, 10 meV versus 40 meV, due to the less deformed lower layer. This suggests that multiwalled CNTs will be even more stable against the indentation than the SWCNTs presented here. To monitor electromechanical properties of the studied systems, we have calculated the coherent electron transport at different steps of indentation. It is worth mentioning that transport properties of nanotubes are almost independent of the tip shape, in agreement with an earlier investigation by Fa et al.31 Figure 8 shows the quantum conductance of CNTs with respect to different deflection values. The results corresponding

Figure 9. Electronic conductance (. ) for the top and the bottom layers in the DWCNT. Distance (d) is measured from the bottom of the tip to the upper layer. d1 = perfect system, d2 = tip position at 0.0 Å, d3 = tip position at −2.0 Å, and d4 = system with the first C−C broken bond.

Figure 8. Electronic conductance (.) of (a) small- and (b) largediameter CNTs at different indentation steps. d1 = perfect system, d2 = tip position at 0.0 Å, d3 = tip position at −2.0 Å, and d4 = system with the first C−C broken bond.

To better understand the effect of deformation on the electronic properties, we visualized highest-occupied and lowest-unoccupied crystal orbitals (HOCO and LUCO) for the perfect and the deformed systems before the bond breaking (see Figure 10). The effect of indentation on the crystal orbitals is localized to the perturbed region, mostly at the apex formed due to the tip.

to the unperturbed systems are included for comparison. They show that the indentation leads to a slight drop in the conductance of metallic CNTs. This distortion in the transport properties can be attributed to the increase of the σ*−π* coupling induced by applied deformation.31Although the electronic structure of metallic CNTs show a small band gap opening, this is not observed in the conductance results and the nanotubes hold their gapless character, even for high deflection levels (∼5 Å). For the (9,0) CNT, the degeneracy at the Γ point is broken due to the small tube diameter, resulting in σ−π

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CONCLUSIONS In this work, we have studied the electromechanical properties of carbon nanotubes. We have chosen different tube chiralities and diameters in order to account for the metallic and F



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ACKNOWLEDGMENTS This work was partially supported by the European Commission (FP7-PEOPLE-2009-IAPP QUASINANO, GA 251149). We thank Dr. Lyuben Zhechkov for his support programming. Carbon nanostructures were generated using GTK Display Interface for Structures GDIS 0.90.

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Figure 10. Highest-occupied and lowest-unoccupied crystal orbitals (HOCO and LUCO) of selected SWCNTs: (top) small-diameter (6,6) and (bottom) 2D model. The orbitals are plotted for the equilibrated systems and the deflection values before the first bond is broken.

semiconducting systems. We have extended our studies to simulate large diameter single- and double-walled nanotubes using 2D models, as it was earlier successfully applied to inorganic nanotubes.47 We have calculated the electronic structure and transport changes at different steps of deformations under the AFM tip indentation. We found that the force constants of the materials decrease as the diameter is increased. In the case of DWCNTs, the second layer reinforces the resistance of the material and larger forces are required to deform it. Analyzing the electronic transport properties, we have found that the indentation slightly reduces the conductance of metallic nanotubes close to the Fermi level. Although the electronic structure revealed a small band gap opening (of up to 50 meV), there was no gap opening observed in the transport calculations. The indentation of the semiconducting CNTs causes strong reduction in the band gap, however, there are no transport channels opened for the conductance at the Fermi level. Moreover, before the bond breaking, we have not observed any semiconductor-metal transition. The AFM indentation changes the crystal orbitals of CNTs only locally, mostly around the apex formed by the tip.

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ASSOCIATED CONTENT

S Supporting Information *

Hyperlinks to Movies S1−S4. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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