NANO LETTERS
Contact Geometry and Conductance of Crossed Nanotube Junctions under Pressure
2009 Vol. 9, No. 5 1759-1763
Felipe A. Bulat,† Luise Couchman,‡ and Weitao Yang*,† Department of Chemistry, Duke UniVersity, Durham, North Carolina 27708, and Acoustics DiVision, NaVal Research Laboratory, Washington, D.C. 20375 Received November 9, 2008; Revised Manuscript Received January 27, 2009
ABSTRACT We explored the relative stability, structure, and conductance of crossed nanotube junctions with dispersion corrected density functional theory. We found that the most stable junction geometry, not studied before, displays the smallest conductance. While the conductance increases as force is applied, it levels off very rapidly. This behavior contrasts with a less stable junction geometry that show steady increase of the conductance as force is applied. Electromechanical sensing devices based on this effect should exploit the conductance changes close to equilibrium.
Carbon nanotubes have drawn considerable attention for more than a decade.1 Their physical properties have been extensively studied,2 and their application as molecular wires3 or as electromechanical devices4-10 has been extensively discussed in the literature. Crossed nanotube junctions offer a simple of way of contacting nanotubes,9 and the contact quality can be mechanically controlled.11 More recently, field effect transistors have been fabricated by combining semiconducting and metallic nanotubes using the latter as gate electrode.12 Computational modeling has showed that force-measuring biosensors that take advantage of the dispersion interaction between crossing nanotubes would be reliable nanomechanical devices.13 Two- and three-dimensional grids of crossing nanotubes have also been proposed as a possible framework for integrating nanotube devices.14 Because of the limited number of possible junction geometries, crossed nanotube junctions offer the possibility of straightforward comparison of theory and experiment, provided the chirality of the tubes is known. The main challenges are the ab initio simulation of crossed nanotubes with diameters equivalent to those used in experiments and the determination of the chirality of the experimental counterparts. Although the intertube conductance of crossed nanotube junctions built with zigzag nanotubes has been shown to depend on the atomic details14 and the angle at which the tubes cross,8,15 the effect of the contact geometry * Author to whom correspondence should be addressed. E-mail:
[email protected]. † Duke University. ‡ Naval Research Laboratory. 10.1021/nl803388m CCC: $40.75 Published on Web 03/30/2009
2009 American Chemical Society
on the response of the junction to external forces has not been thoroughly studied. More importantly, none of the studies so far has addressed correctly the dispersion forces and the relative stability of the different possible contact geometries. Since the adhesion force of the nanotubes to the substrate is critical and has not been unequivocally determined, it is very important to have accurate profiles for the contact distances as a function of the contact force. In this Letter we consider different contact geometries for a crossed junction built with (5,5) armchair nanotubes, explicitly incorporate the dispersion interactions with correct long-range behavior, determine the most stable geometry, and study the force dependence of the conductance. Imposing the correct long range for dispersion is important in accurately modeling these devices, and considering the (energetically) most favorable junction geometry should improve the agreement with experiment and can qualitatively change the nature of the conductance profiles. We shall be mainly interested in the zero-bias transmission probability of a molecular device (D) T(E) ) Tr{ΓL(E)GD(E)ΓR(E)GD†(E)}
(1)
which represents the permeability of the device to electron flow at energy E.16 In eq 1 GD is the Green’s function of the device and ΓL,R(E) reflects the coupling, at energy E, between the device and the left (L) and right (R) leads. We use density functional theory17 under periodic boundary conditions18,19 combined with the nonequilibrium Green’s functions technique16,20 for the conductance calculations, as described in detail in ref 21. We note here that possible strong
correlations in quasi-one-dimensional systems are not described by current density functional approximations available, especially local and semilocal functionals, like the ones used in the present study. In particular the Luttiner liquid behavior22,23 is not described and could be an interesting topic for further study. A generalized gradient approximation (GGA) exchange-correlation functional24 was used throughout this study, with single-ζ basis for the conductance calculations and double-ζ plus polarization (DZP) basis for the energy and geometries. We use a real space mesh cutoff of 150 meV. The basis set superposition error (BSSE) can be very large for strictly localized basis set, and it increases with the confinement radius (see Supporting Information). The BSSE originates in the fact that the basis set changes with the geometry, and although it can have important intramolecular effects,25,26 it is mostly associated with weakly bound systems (intermolecular interactions). When two weakly bound fragments approach each other, the basis set in one also describes the electron (or orbital) density on the other, effectively increasing the basis set quality, improving the wave function, and hence lowering the energy. This results in an added stabilization of the system, whose origin is the incompleteness of the basis set. To remediate this, we use the counterpoise (CP) method of Boys and Bernardi27 and divide the system in two fragments A and B, each of which corresponds to one of the tubes. The corrected energy ECP(AB) is given by Rβ R β ECP(AB) ) EAB - (EARβ + ERβ B ) + (EA + EB)
(2)
where EXY is the energy of fragment X (X ) A, B, AB) calculated with the basis set Y (Y ) R, β, Rβ, R is the basis set of A, etc). For example, EARβ is the energy of A calculated with its own basis set (R) plus that of B (β), but without the atoms of B. We ran test calculations for the D6d staggered benzene dimer using a DZP basis set with various critical radii ranging from 2.16 to 3.00 Å (energy shifts δε due to confinement from 700 to 100 meV). We found that the uncorrected calculations yield binding energies (equilibrium distances) strongly dependent on the cutoff, ranging from ∼ 27.7 kcal/ mol (∼ 3.0 Å) to ∼ 2.1 kcal/mol (∼ 3.9 Å) for δε ) 700 meV and δε ) 100 meV DZP basis, respectively. High level ab initio wave function calculations predict a binding energy of 1.8 kcal/mol and an equilibrium distance of ∼3.75 Å. The CP correction scheme renders the interaction energy, to an excellent approximation, independent of the energy shift δε, and all the interaction curves strictly repulsive, which is the actual DFT result (see Supporting Information). Of course, dispersion needs to be explicitly accounted for. To properly account for the dispersion interactions, we use a empirical correction to density functional theory.28 This is accomplished by adding a damped pairwise interaction that depends only on the distance Rij between each ion pair 1760
bions}) ) EvdW({R
1 2
∑∑ i
j*i
fd(Rij)
C6ij Rij6
(3)
and where the damping function fd(Rij) is Function I in ref 28 and the pairwise C6 coefficients are computed from the atomic C6 taken also from ref 28. The total energy is then given as the sum of eqs 2 and 3 E ) ECP(AB) + EvdW Rβ R β vdW ) EAB - (EARβ + ERβ B ) + (EA + EB) + E
(4)
In our test calculations for the benzene dimer, we find that the binding energy is 1.9 kcal/mol and the equilibrium distance is 3.75Å, in excellent agreement with very accurate wave function based calculations.29 It also compares favorably with the best uncorrected result (DZP basis set, δε ) 100 meV), which yields 2.1 kcal/mol and 3.9 Å for the binding energy and equilibrium distance, respectively. Most importantly, the uncorrected interaction curve decays faster that the corrected one, which has the correct R-6 asymptotic behavior. Even though metallic wires and surfaces display enhanced dispersion interactions,30-33 this should not affect the configuration considered in the present study. Indeed, while quasi-one-dimensional parallel metallic systems display a much stronger asymptotic law,32 there is no evidence of such behavior for crossing wires. Each of the density functional contributions to the energy in eq 4 and its derivatives with respect to each of the nuclear positions are computed separately18,19 and then combined with the dispersion contribution. The total energy and derivatives are then used to perform the geometry optimization using Gaussian 03.34 For comparison purposes, all the calculations were done with both the fully corrected (DFT+CP+vdW) and uncorrected DFT. In order to simulate the effect of the contact force on the conductance of the junction, geometry optimizations are performed at fixed intertube distance, defined as the distance between the axes of the undeformed nanotube sections (clamped atoms in Figure 1). By systematically bringing the tubes closer while relaxing the junction region on each step, we record the contact distance and contact force as a function of intertube distance. The contact distance is defined as the smallest intertube atom pair distance. The contact force is the sum of the forces applied in the direction perpendicular to the tubes’ axes required to fix the intertube distance at any given value and corresponds to the forces on the clamped atoms in Figure 1 after all the other atoms are fully relaxed. The three junction geometries considered are also shown in Figure 1, and labeled HH (hollow-hollow), HC (hollowcarbon), and CC (carbon-carbon), depending on how one tube is aligned with the other. The contact distance and contact force are shown in Figure 2 as a function of the intertube distance for the three different junction geometries considered, and using both the best uncorrected (100 meV) and corrected DFT (CP+vdW). Note that while in the vicinity of the equilibrium position (zero contact force) the contact distance is a linear function of the Nano Lett., Vol. 9, No. 5, 2009
Figure 1. (Upper panel). Crossed nanotube junction with an HH contact geometry. The geometry optimization cell has a set of clamped (red) and optimized (gray) atoms. Metallic contacts (blue) are added for the transport calculations. (Lower panel) The three junction geometries considered in the present study.
intertube distance (because the tube remains only slightly deformed), for smaller intertube distances the contact distance reaches a minimum value that depends on the specific contact geometry. For all the geometries the minimum contact distance is quite similar, although the CC geometry has a minimum contact distance slightly smaller than the HH and HC geometries (∼0.05 Å). The CP and vdW corrections reduce the minimum contact distance in about ∼0.1 Å and considerably reduce the equilibrium distance of the junction. Our results in Figure 2 indicate that the force grows rapidly when the tubes are brought together from the zero-force position, whereas previous reports11 seem to suggest that from the equilibrium position the tubes could be brought 1.5 Å closer from this same point with a negligible effect on the force. Also, we did not observe the leveling off of the forces for very small intertube distances, and the force grows steadily for all the range of intertube distances studied. Because the force can be very sensitive to the real space mesh cutoff (and basis set confinement), especially around the zero-force region, we used a rather large cutoff (150 meV) as quoted above. We observed that the potential energy surface of the uncorrected DFT is very rugged around the zero-force region because of the basis set confinement (and BSSE), but the DFT+CP+vdW is very smooth as a consequence of the CP correction. It is very interesting that Nano Lett., Vol. 9, No. 5, 2009
Figure 2. Contact distance (upper panel) and contact force (lower panel) as a function of the intertube distance for the various contact geometries. The corrected (black segments line, DFT + CP + vdW) and uncorrected (full red line, DFT) density functional results are shown.
further increasing the mesh did not seem to solve the problem, but the CP correction was very effective in removing this undesirable consequence of the basis set confinement. Because we achieve smooth potential energy surfaces even in the difficult zero-force region, we consider our result well converged. Figure 3 shows the total energy as a function of the contact distance relative to the energy of the HH junction at equilibrium. We observe that the CP and vdW corrections reduce the equilibrium distance of the HH junction by ∼0.25 Å, while the HC and CC junctions appear to be more strongly affected and their equilibrium distances are reduced by ∼0.38 Å. We also observe that the HC junction is more stable than both the HH and CC junctions regardless of the method used (DFT or DFT+CP+vdW). Indeed, the DFT+CP+vdW result predicts that the HC geometry is 1.84 kcal/mol (0.72 kcal/mol) more stable than that of HH. The CC junction is predicted to be 0.82 kcal/mol more stable than the HH junction by the DFT+CP+vdW method, while predicted to be almost degenerate by the uncorrected DFT result (0.05 kcal/mol higher than HH). Although the HC geometry turns 1761
Figure 3. Total energy as a function of contact distance for the various contact geometries. The corrected (black segments line, DFT+CP+vdW) and uncorrected (full red line, DFT) density functional results are shown, and the decrease in the equilibrium contact distance due to the correction is shown with horizontal (green) bars.
out to be the most favorable configuration, it was not considered on previous intertube conductance studies of (5,5)/ (5,5) junctions,11 where results for only the least stable HH junction were reported. Experimental measurements of the conductance of crossed junctions are done with the nanotubes crossing on top of a substrate. The interaction between the substrate and the nanotubes generates a contact force that decreases the distance between the tubes with respect to the equilibrium distances quoted above. The magnitude of that force has been the subject of some controversy, and values ranging from 1 nN8 to 5 nN5 for a 1.4 nm diameter tube have been suggested. The (5,5) nanotubes studied have roughly half that diameter, so that to obtain the same strain energy a much larger contact force (∼3 times) needs to be applied.11 However, even for equivalent strain energies, the contact surface in (5,5)/(5,5) junctions is much smaller than that in the tubes used in the experiments, which have roughly twice the diameter. A much smaller contact surface should mean much smaller intertube transmission probability. This means that the intertube conductance measured in experiments should be an upper bound for the conductances computed for crossed junctions of smaller diameter. We here show that considering the most stable contact geometry is of paramount importance, because the conductance of the HC geometry is smaller and saturates earlier. Figure 4 shows the conductance of the three different junction geometries considered as a function of the contact force. The results for the corrected and uncorrected density functional calculations are shown for comparison. We see that the main effect of the DFT+CP+vdW correction is, because of the added attraction, to displace the profile so that the conductance at given force increases. This is analogous to what was seen for the contact force as a function of intertube distance in Figure 2. In our transport calculations, the conductance of the HH contact geometry seems qualitatively different from the conductance of the HC and CC geometries, in that it increases with applied force in all the range of contact forces considered. The HC and CC 1762
Figure 4. Conductance profiles for the various contact geometries as a function of contact force. The corrected (black segments line, DFT+CP+vdW) and uncorrected (full red line, DFT) density functional results are shown.
junctions, on the other hand, appear to increase their conductance initially, but tend to level off. The CC junction even appears to decrease its conductivity after saturating at around ∼18 nN for the DFT+CP+vdW geometries, and at ∼20 nN for the DFT ones. The HC junction, which is the most stable one according to our analysis above, seems to level off but not to decrease its conductance, for both the DFT and DFT+CP+vdW geometries. The slope of the conductance profiles, however, seems to be slightly different: while for the DFT+CP+vdW geometries the conductance appears almost constant, for the DFT geometries are still growing, albeit very slowly. Previously reported calculations on the conductance of HH junctions,11 though agree in the magnitude of the conductance values, are qualitatively more similar to our results for the CC junction in that saturation is observed at about ∼16 nN, followed by a decrease in the intertube transmission. All these considerations emphasize the sensitivity of the conductance profiles to geometric details. Because the tunneling takes place between a few pair of atoms in the junction, the specific geometric arrangements are of paramount importance, and every effort should be made to obtain accurate geometries, especially when the tunneling is through nonbonded subunits. The intertube conductance measured for 1.4 nm tubes was in the 0.04-0.13 (2e2/h) range.9 If the contact area is taken into consideration, as discussed above, the experimental results should be an upper bound, because the contact surface in the experimental 1.4 nm diameter tubes should be larger than that in the model (5,5)/(5,5) junction. The HH junction, however, overshoots the highest value for all forces >10 nN, which corresponds to the same strain energy on 1.4 nm tubes with contact forces >2.5 nN. We also note that previous calculations on HH junctions11 overshoot the experimental values for all forces >5 nN. While the magnitude of the force exerted by the substrate is not entirely clear, and this obscures any one-to-one comparison, the HH junction clearly shows a conductance that is too large. The HC and CC geometries, on the other hand, show much smaller conductance, in much better agreement with experiment. These contact geometries are also more stable than that of the HH geometry, so it is clear that they are better models for the experiment (they Nano Lett., Vol. 9, No. 5, 2009
are more stable and their conductance is in better agreement with the experimental results). The assessment of the contact surface effects in the tunneling current is thus a critical step forward in enabling direct comparison of theory and experiment. The contact geometry is found to have a significant effect on the conductance, in agreement with previous studies that emphasize the importance of the atomic details. Because of the limited number of possible junction geometries, crossed nanotube junctions enable direct comparison of theory and experiment. This calls for a thorough study of crossed junctions with nanotube diameters comparable to those used in experiments with an emphasis on the region near equilibrium. Transport calculations on the most stable contact geometries should enable direct comparison with experiment and an assessment of the accuracy of computational methods. In summary, density functional theory calculations, corrected for the basis set superposition error inherent to localized basis sets and potentially very large for strictly localized basis set, and coupled with an empirical dispersion correction, have been carried out on crossed nanotube junctions. The corrections significantly reduce the equilibrium distance for all junction geometries and predict that the most stable junction has an HC geometry, which had not been studied before. The HC and CC geometries show a much smaller conductance than the HH geometries and are in excellent agreement with experimental data. We emphasize the importance of considering the most stable geometries, because the atomic details can greatly influence the conductance values and profiles. For designing electromechanical sensing devices, the crossed junctions thus appear to be interesting systems only in the region near equilibrium. This means that the interaction of the tube with the substrate generates a contact force large enough to saturate the conductance of the junction, which would not respond to further pressure. Application of crossed nanotube junctions as electromechanical devices should take this into account to exploit the conductance changes near the equilibrium geometry of the free junction. Acknowledgment. W.Y. gratefully acknowledges financial support from the Office of Naval Research and the National Science Foundation. Supporting Information Available: Discussion of test calculations on the benzene dimer and crossed nanotube junctions. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Iijima, S. Nature 1991, 354, 56. (2) Saito, R.; Dresselhaus, G.; Dresselhauss, M. S. Physical Properties of Carbon Nanotubes; Imperial College Press: London, 1998.
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