ARTICLE pubs.acs.org/JPCC
Transport Properties of Carbon Nanotubes: Effects of Vacancy Clusters and Disorder Alex Taekyung Lee,† Yong-Ju Kang,‡ and K. J. Chang*,† † ‡
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea CAE TEAM, Memory Division, Samsung El. Company, Hwasung, Republic of Korea
bS Supporting Information ABSTRACT: We investigate the effects of vacancy defects on the electronic and transport properties of carbon nanotubes through density functional calculations. In both cases, where vacancies aggregate into larger clusters and are disordered, conductance changes from metallic to insulating regime, while their origins are different. For small vacancy clusters, the suppression of conductance is led by the defect states associated with π-topological and σ-dangling bond defects, while the local gap opening plays a role for large vacancy clusters. In disordered tubes with various types of vacancy defects, conductance decreases exponentially due to the Anderson localization. The localization length not only depends on the type of vacancy defects but also the tube chirality.
’ INTRODUCTION While pristine single-walled carbon nanotubes (SWNTs) are considered as an extraordinary one-dimensional quantum wire with ballistic conduction,1 3 their electrical properties are strongly affected by defects such as impurities, adatoms, vacancies, and topological defects (five- and seven-membered rings). Among intrinsic defects, vacancies are the most common defects in CNTs, existing even in nearly defect-free nanotubes. Carbon vacancies are easily generated by ion or electron irradiation.4,5 Kinetic Monte Carlo simulations showed that monovacancy (V1) and divacancy (V2) defects are abundant at the early stage of irradiation.6 Upon thermal annealing, monovacancies easily diffuse and aggregate into vacancy clusters,7 9 because V1 is more mobile than V2.10 The formation of vacancy clusters is energetically more favorable than individual V1 and V2 defects. Recent experiments demonstrated that large vacancy clusters migrate and coalesce to form even larger ones at elevated temperatures.9 If the potential range of defects is much larger than the lattice constant, no backscattering occurs, maintaining the perfect transmission.1 On the other hand, point defects, such as vacancies, give rise to a strong potential with a range comparable to the lattice constant and easily destroy the perfect channels. For single V1 and V2 defects, their effects on the conductance have been well studied through tight-binding and first-principles density functional calculations,11 16 which showed that the reduction of conductance near the Fermi level depends on the type of defects. If vacancies are populated in nanotubes, the conductance behavior changes drastically, exhibiting Anderson localization due to the disorder of defects. Recent studies17 19 showed that Anderson localization takes place in the disordered (10,10) tube even with a r 2011 American Chemical Society
small amount of the V1 or V2 defects, using the Green’s function approach within the local-orbital density functional theory, and suggested that divacancies lead to a more dramatic effect in the conductance reduction. Despite the importance of controlling the electrical properties for device applications, the effects of large vacancy clusters and their disorder on the electrical conductance are not well studied. In this work, we perform first-principles density functional calculations to investigate the effects of individual vacancy clusters Vn (1 e n e 36) on the conductance of the (5,5) and (9,0) tubes, where n is the number of missing atoms. As small vacancies aggregate into large vacancy clusters, the conductance near the Fermi level decreases exponentially with increasing n. For small vacancy clusters, the conductance near the Fermi level is strongly affected by the defect states of vacancies, oscillatory decreasing with n, while the local gap opening plays a role in the conductance reduction for large vacancy clusters. We also examine the electrical transport properties of disordered nanotubes, in which a small amount of Vn vacancies with n e 4 or a mixture of different vacancies are randomly distributed. Regardless of the type of vacancy clusters and their distribution, the conductance decreases exponentially with increasing of the number of defects, which is attributed to the strong localization induced by disorder. We discuss the dependences of localization length on the tube chirality and the type of vacancy defects. Received: August 18, 2011 Revised: November 21, 2011 Published: December 01, 2011 1179
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’ CALCULATION METHOD For the electronic properties of vacancy clusters in nanotubes, we performed first-principles density functional calculations by using the generalized gradient approximation (GGA)20 for the exchange-correlation potential and ultrasoft pseudopotentials21 for the ionic potentials, as implemented in the VASP22 and PWSCF23 codes. The wave functions were expanded in plane waves up to an energy cutoff of 400 eV. We employed supercells with the axial length up to 30 Å, including a vacuum region of about 10 Å between adjacent nanotubes. We chose the armchair (5,5) and zigzag (9,0) nanotubes with similar diameters but different extreme tube chiralities to examine the chirality dependence of localization length. For vacancy clusters Vn, in which n missing atoms are serially connected along the tube axis,24 we fully relaxed the lattice constants and the ionic coordinates until the residual forces were less than 0.02 eV/Å. The transport properties were investigated by calculating the transmission function (T) within the matrix Green’s function approach,25,26 which is the matrix version of the nonequilibrium Green’s function theory. For individual vacancy clusters, we set up a device model such that a nanotube containing a single vacancy Vn is sandwiched between two semi-infinite pristine nanotubes, which act as electrodes. In disordered nanotubes, N vacancy defects were randomly distributed in the device region. We generated disordered nanotubes by connecting tube segments containing a single defect. In each tube segment, we fully relax the ionic coordinates for randomly chosen positions and rotational angles of the defect. Two-terminal conductance at zero temperature was obtained from the Landauer B€uttiker formula,27 G = G0T(E), where G0 (=2e2/h) is the conductance quantum. The Hamiltonian and overlap matrices were calculated using the linear combination of the Gaussian orbital basis sets of single-ζ quality for the C atoms within the local-densityfunctional approximation.28 In this case, we performed the selfconsistent calculations by using norm-conserving nonlocal pseudopotentials generated by the scheme of Troullier and Martins.29 ’ RESULTS AND DISCUSSION First, we examine the effects of individual Vn defects on the conductance. The defect states induced by Vn scatter incoming waves and thereby reduce the conductance near the neutrality point, while pristine tubes have the maximum conductance of 2G0, where G0 (=2e2/h) is the conductance quantum. The defect states are determined by reconstructions, which allow for the saturation of dangling bonds and the creation of topological defects.24 When n is even, dangling bonds are completely eliminated by forming reconstructed bonds as well as pentagon and heptagon defects, while, for odd n, a large hole such as 8-, 9-, or 10-membered rings is generated, leaving one dangling bond. Because of the unsaturated dangling bond, the energies of Vn are generally higher for odd n. Theoretical calculations showed that the formation energies per missing atom of Vn defects decrease with increasing of n; thus, large vacancy clusters are energetically more favorable.24 As pentagon, heptagon, and dangling bond defects have different defect levels, the conductance is strongly affected by the type of Vn. The symmetry of Vn is described by two mirror planes, M1 and M2, which contain the tube circumference and the tube axis, respectively (Supporting Information, Figure S1). In the (5,5) tube, Vn defects do not preserve the M1 and M2 mirror planes because the orientation of topological defects is slightly tilted from the tube axis, while, in the (9,0) tube,
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Figure 1. (a f) The electrical conductance is plotted as a function of energy in the (5,5) nanotube with a single Vn defect (1 e n e 6). The atomic structure of the defect and the distribution of the wave function for the defect state marked by an arrow are shown in each panel. Red and blue clouds represent different signs of the isosurfaces at (0.001 electrons per Å3. Left (right) panels represent Vn defects with odd (even) numbers n. The Fermi level is set to zero.
most of Vn defects have the M2 symmetry due to the parallel alignment of topological defects to the tube axis.24 In the (5,5) tube with a single V1(5 9) defect, which consists of one five-membered ring and one nine-membered ring, we observe two dips in conductance at 1.48 and 1.1 eV below the Fermi level (EF) (Figure 1a). According to H€uckel’s rule, the pentagon and heptagon defects act as acceptor- and donor-like defects, respectively. The lower-energy dip is attributed to the pentagon defect, whereas the upper dip is due to the pz-dangling bond in the nine-membered ring. Because the mirror planes are not preserved, the defect states have both the π and the π* characters, which are even and odd under mirror symmetry operations, respectively. As one-half of each channel is reflected, the conductance is reduced to 1G0 at each dip. The conductance exhibits an additional dip at 0.34 eV above the Fermi level. The conductance reduction of 1G0 results from scattering by the σ-dangling bond in the nine-membered ring. We find that the σ-dangling bond hybridizes with the reconstructed bond and the unsaturated C atom is protruded from the tangential plane, with the bond distance of 2.5 Å. However, as the dip is slightly broadened, the conductance at the Fermi level is not significantly affected, with G(EF) = 1.75G0, similar to other calculations.13,16 The V2(5 8 5) defect is composed of two pentagons and one octagon, without dangling bonds. Thus, all conductance dips are associated with the topological defects (Figure 1b). Because of the lack of mirror symmetries, the defect states do not preserve the channel parity, with a mixture of the π and π* characters. The dip at 0.32 eV above the Fermi level is related to the octagon 1180
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The Journal of Physical Chemistry C defect. This dip is broadened because of several defect levels; thus, the conductance at EF is greatly affected, with G(EF) = 1.09G0, in good agreement with the previous result.14,16 Similar to the V1 defect, the dip at the low energy of 1.36 eV is caused by two pentagons, which are nearly equivalent to each other and thus give similar defect levels (Supporting Information, Figure S2). The proximity of these defect levels reduces the conductance to nearly zero. In the case of V3(5 10 5), the characteristics of conductance dips are similar to those of V1 (Figure 1c). The dip at 0.06 eV near the Fermi level is associated with the σ-dangling bond state. The distance between the reconstructed bond and the dangling bond is about 2.9 Å, larger than that for the V1 defect. Because the dangling bond does not interact with the reconstructed bond, the conductance dip becomes sharp. Thus, the conductance at EF is slightly affected, with G(EF) = 1.42G0, which is larger than that for the V2 defect. Two pentagon defects give rise to the broad conductance dip around 0.94 eV. Although the 10-membered ring breaks globally the M2 mirror symmetry, this symmetry is locally preserved for two pentagons. Thus, two defect states, which are close in energy, maintain the π and π* characters, which block the π and π* channels, respectively. As a result, the conductance is significantly reduced to 0.45G0. It is interesting to note that two dips appear at 1.38 and +1.38 eV. Similar to the V1 defect, the low-energy dip results from the pz-dangling bond in the 10-membered ring, whereas the sharp dip at the higher energy corresponds to the antibonding counterpart of the former defect state. The V4(5 7 7 5) defect consists of two pentagons and two heptagons, which are connected along the tube axis. In the Stone Wales defect, which has two pentagon heptagon pairs formed by rotating a single C C bond, two conductance dips occur at energies far from the Fermi level due to acceptor- and donor-like defect states.11 Similarly, for the V4(5 7 7 5) defect, we find two dips at 1.32 and +0.90 eV, which are associated with the pentagon and heptagon defects, respectively (Figure 1d). As the Stone Wales defect preserves both the M1 and the M2 symmetries, one of the conducting channels is blocked, giving the conductance close to 1G0. However, in the V4(5 7 7 5) defect, the mirror symmetries are broken due to the spiral alignment of the topological defects, inducing a mixture of the π and π* characters for an eigenchannel. Two pentagon-related defect states give rise to a large suppression of the conductance around 1.32 eV. There are also two heptagon-related states, the bonding and antibonding states of the reconstructed bond, which is formed between two heptagons. Similar to the pentagonrelated dip, the heptagon-related defect states block two eigenchannels, resulting in a very small conductance at +0.90 eV. Because of the acceptor- and donor-like defect states, the conductance near EF is significantly reduced, with G(EF) = 0.60G0. In large vacancy clusters with n g 5, as the missing atoms are serially connected along the tube, the chirality of the vacancy region becomes close to (5,4). In this case, pentagon heptagon defects are positioned at the junction boundaries between the (5,5) and (5,4)-like tubes, and their separation increases with n. In the (5,5) tube, the V5(5 7 6 8) defect with n = 5 was shown to be less stable by 0.22 eV than the V5(5 6 10 5) defect, while this defect is the ground state in the (9,0) tube.24 Here, we consider the V5(5 7 6 8) defect for the consistency in comparison with the (9,0) tube, because the alignment of the pentagon heptagon pair and the eight-membered ring with a dangling bond is the same as that in the (9,0) tube. The V5(5 7 6 8)
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Figure 2. (a) The variation of conductance at the Fermi energy with n is drawn for the (5,5) nanotube with a single Vn defect (1 e n e 36). The inset shows the exponential decrease of G(EF) with increasing of x, where x is the length of the vacancy region of the Vn defect. Dotted line represents the fitted function, exp( x/x0), where x0 is 3.56 Å. (b) The conductance and (c) the local density of states (LDOS) are compared for different Vn defects. As n increases, the development of the local gap is shown. (d) The LDOS is drawn along the tube with a single V36 defect and exhibits the local gap in the vacancy region. The Fermi level is set to zero.
defect can be easily obtained by removing a C atom from the V4(5 7 7 5) defect along the axis. With the V5(5 7 6 8) defect, the conductance is suppressed over a wide range of energies around EF (Figure 1e). The σ- and pz-dangling bond states lie around 0.5 eV and just above the Fermi level, respectively, whereas the broad dips associated with the pentagon and heptagon defects are located around 0.5 and +0.74 eV, respectively. Thus, the conductance at EF is significantly reduced to G(EF) = 0.79G0, as compared to the V1 and V3 defects. In the V6(5 7 6 7 5) defect, the conductance at EF is further reduced to G(EF) = 0.58G0 due to the enlarged (5,4)-like tube region. Similar to the V4 defect, the dip at +0.54 eV results from two heptagons (Figure 1f), while the broad dip around 0.82 eV is related to two pentagons. The variation of G(EF) with n is plotted up to 36 missing atoms in Figure 2a. The conductance at the Fermi level has the tendency to decrease with n. For small odd numbers of n, G(EF) is less affected by sharp dips associated with the σ-dangling bond, while broad dips caused by the topological defects reduce more significantly G(EF) for even numbers of n. Thus, G(EF) decreases in an oscillatory manner for n e 6. If n increases continuously, the (5,4)-like tube region plays a role in the conductance reduction. For n larger than 16, G(EF) drops quickly to nearly zero conductance due to the gap opening in the (5,4)-like region. As the local gap is developed by increasing n, the conductance near EF is suppressed, as shown in Figure 2b and c. For n = 36, the conductance and the local density of states (LDOS) show a large local gap of 1.4 eV in the deformed tube region (Figure 2d), which is comparable to the band gap 1.2 eV of the (5,4) tube. The variation of G(EF) with n is well described by the exponential 1181
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Figure 4. (a,b) In the disordered (5,5) nanotubes with the V2 defects, the conductance and the local density of states are compared for different numbers of the defects. (c,d) The same plots are shown for the disordered (9,0) nanotubes with the V2 defects. The Fermi energy and the neutrality level are set to zero for the (5,5) and (9,0) tubes, respectively.
Figure 3. (a f) The electrical conductance is plotted as a function of energy in the (9,0) nanotube with a single Vn defect (1 e n e 6). The atomic structure of the defect and the distribution of the wave function for the defect state marked by an arrow are shown in each panel. Red and blue clouds represent different signs of the isosurfaces at (0.001 electrons per Å3. Left (right) panels represent Vn defects with odd (even) numbers n. The neutrality level is set to zero.
function, exp( x/x0), where x is the tube length of the (5,4)-like region (Figure 2a). For n g 6, x0 is estimated to be 3.56 Å. We point out that x0 is very similar to the decay length of the metalinduced gap states at the junction between the (5,5) and (5,4)like tubes. In fact, the decay length of the metal induced gap states is estimated to be 3.64 Å for the V36 defect (Supporting Information, Figure S3). In the zigzag (9,0) tube, the alignments of topological defects in Vn defects are similar to those in the (5,5) tube (Figure 3). Thus, conductance dips have the same origins, except for zero conductance at the neutrality point, which is attributed to the small gap of about 110 meV. As topological defects are aligned parallel to the tube axis, the M1 or M2 mirror symmetries are mostly preserved in Vn defects. Because the π and π* states do not mix, the conductance dips become sharper in the (9,0) tube, as compared to the (5,5) tube. With the V1(5 9) defect, the (9,0) tube exhibits two conductance dips at 0.26 and 1.36 eV from the neutrality point (Figure 3a). The former is associated with the σ-dangling bond in the nine-membered ring, as in the (5,5) tube. Because of the M2 plane, the π channel is reflected by the defect state with the same parity, resulting in the conductance reduction of 1G0. Note that the conductance near the valence band maximum (VBM) is not affected by the σ-dangling bond, close to 2G0. The dip at 1.36 eV results from scattering by two defect states formed by the pentagon and pz-dangling bond defects, which have odd and even parities, respectively, with respect to the M2 mirror plane. As the π and π* channels are blocked by these defect states, which are very close in energy, the conductance becomes nearly zero.
The V2(5 8 5) defect has both the M1 and the M2 mirror symmetries; thus, the π and π* states do not mix. The octagon defect level with the π bonding character is located at 0.20 eV and reduces the conductance to about 1G0, as shown in Figure 3b. This level tends to shift upward as the tube diameter increases, because the bond length of the reconstructed bonds at the boundaries between the octagon and pentagon defects increases. For example, in the (18,0) tube, the octagon defect level lies at 15 meV below the neutrality point. Thus, the octagon defect more strongly affects the conductance near the band edge. Similar to the (5,5) tube, the sharp dip at 1.34 eV is associated with two pentagon defects, which give rise to two defect states with different parities (Supporting Information, Figure S2). For the V3(5 10 5) defect, the σ-dangling bond state is lowered toward the band gap (Figure 3c). As a sharp resonant peak appears in the gap, pretty high conductance is maintained near the VBM. The pentagon and the 10-membered ring containing the pz-dangling bond do not preserve the mirror symmetries. Thus, their defect states are characterized by a mixture of the π and π* states. As these states are located around 1.25 eV, the conductance is significantly reduced, with the broad dip. With the V4(5 7 7 5) defect, we find the pentagon and heptagon defect levels at 1.29 and 0.13 eV, respectively (Figure 3d). In fact, these defects suppress the conductance over a wide range of energies between 1.29 and 0 eV. For the V5(5 7 6 8) defect, the octagon and pentagon defect levels have the π and π* characters, respectively. The broad dip near 1.1 eV is also associated with these topological defects. On the other hand, the σ-dangling bond state lies just below the VBM, reducing the conductance near the band edge (Figure 3e). For the V6(5 7 6 7 5) defect, we also observe a broad dip around 0.5 eV, which is attributed to the pentagon and heptagon defects levels (Figure 3f). For n > 6, the chirality of the vacancy region changes from (9,0) to (8,0). Thus, the conductance near the band edge is greatly affected by the development of the local band gap, similar to the (5,5) tube. Next, we study the Anderson localization induced by defects in nanotubes, in which N vacancies are randomly distributed, with 1182
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Figure 5. (a) The conductance at the Fermi energy is plotted as a function of the number of defects (N) in the disordered (5,5) nanotubes with different types of Vn defects (V1, V2, V3, and V4). (b) The variations of conductance with N are compared for the disordered (5,5) and (9,0) nanotubes with the V2 defects. The Fermi energy is considered for the (5,5) tube, whereas the energy at 0.15 eV from the neutrality point is chosen for the (9,0) tube.
the defect concentrations ranging from 0.5% to 0.63%. In the device region with the tube length L, the mean distance between defects is d, satisfying L = Nd. In one-dimensional disordered systems, the localization theory suggested that the electronic states become localized as L goes to infinite, independent of the strength of disorder.30 The conductance decreases with increase of the system size L, following the relation G = G0 exp( L/L0), if the sample size is larger than the localization length L0. For a given d, we examine the variation of conductance near the Fermi level with increasing N. To consider the randomness of defects, we choose more than 10 configurations. For V2 and V4 defects, we find that the localization indeed takes place, satisfying the criterion, ΔT/T > 1 and Δ ln T/ln T < 1, where T and ΔT are the mean value and the standard deviation of T, respectively.31 Figure 4 shows the conductance and the local density of states in the disordered (5,5) and (9,0) tubes with the V2 defects. As N increases, both the conductance and the local density of states strongly fluctuate, indicating the localization of the electronic states. The conductance near EF is plotted as a function of N for different types of defects in Figure 5a. We take the conductance average in the energy window within (25 meV from the Fermi level, which corresponds to room temperature.17 19 Thus, the fluctuation of conductance is suppressed, following the ergodic hypothesis.30 For the V1 and V3 defects, the localization condition, L > L0, is not satisfied for the device lengths considered. However, the conductance well exhibits the exponentially decreasing behavior for N g 3, similar to previous calculations.18,19 In previous kinetic Monte Carlo simulations,6 the V1 and V2 defects were shown to be abundant vacancies in single-walled nanotubes under ion irradiation. With only the V1 defects in the (5,5) tube, we estimate the localization lengths to be L0 = 10.4d and 25.6d for d = 14.76 and 19.68 Å, respectively. Our calculated localization lengths are much smaller than the tight-binding result of L0 = 600d for the (10,10) tube with d g 45 nm.18 The difference between the two calculations comes from the use of smaller distances d in the present work, whereas large distances d were used in the local-orbital density functional calculations.18 We point out that L0 is sensitive to the mean distance d, increasing rapidly with increasing of d. The localization length also depends on the tube diameter. For various disordered systems, previous calculations showed that L0 is generally proportional to the tube diameter,3,32,33 while it is independent of the tube chirality in some cases.32,33 This behavior was attributed to the reduced mean-square amplitude of the effective
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disorder3 or the increased number of diffusing paths in largediameter tubes.32 However, if realistic vacancy defects are considered, the situation will change because the conductance is more affected by the positions of defect levels. In large-diameter tubes, the bond length of the reconstructed bond increases due to the reduced curvature. Thus, the conductance dip just above EF, which is produced by a single V1 defect, becomes sharper and less affects the value of G(EF). In fact, in the (10,10) tube with a V1 defect, the conductance at EF was shown to be 1.98G0, which is larger than that in the (5,5) tube. Thus, the localization length is expected to increase in tubes with large diameters. As discussed earlier, a single V2 defect affects significantly the conductance due to the broad dip around 0.32 eV. In the disordered (5,5) tube with the V2 defects, the reduction of G near EF becomes significant with increasing N, as shown in Figures 4a and 5a. The localization lengths are calculated to be L0 = 4.65d and 5.88d for d = 19.68 and 24.6 Å, respectively, in good agreement with previous calculations, where L0 = 5.2d for d = 16.3, 37.6, and 75.5 nm.19 As compared to the V1 defects, the sensitivity of L0 to d is suppressed. As the localization lengths are much smaller than those for the V1 defects, the V2 defects induce a more significant effect on the localization. The concentration of the V2 defects grows upon thermal annealing because of the coalescence of mobile V1 defects. As the localization can be induced with a small amount of the V2 defects, annealing is very important to control the electrical properties of defected tubes. Without heat treatment, nanotubes are likely to be disordered by a mixture of the V1 and V2 defects.6 In the disordered (5,5) tubes with the equal numbers of V1 and V2, we obtain the localization length of L0 = 11.11d for d = 19.68 Å, which lies between L0(V1) and L0(V2), where L0(Vn) denotes the localization length for the Vn defects. When d decreases to 16.65 Å, L0 is reduced to 8.20d. In this case, the mean distance between the V2 defects increases from 43.3 to 54.1 Å. As L0 is not sensitive to the mean distance between the V2 defects, the reduction of L0 by decreasing d mainly results from the sensitivity to the mean distance between the V1 defects. On the other hand, as d increases, the localization length will be governed by the V2 defects because the localization by the V1 defects is suppressed. The formation of large vacancy clusters was shown to be energetically more favorable due to their low formation energies.24 To study the effect of large vacancy clusters on the localization, we consider the V3 and V4 defects in the (5,5) tube. We find that the localization length is also affected by the presence of conductance dip near EF, which is initially formed by a single defect. In the disordered (5,5) tubes with the V3 and V4 defects, the localization lengths are estimated to be L0 = 17.54d and 1.97d for d = 24.6 Å, respectively (Figure 5a and Supporting Information, Figure S4). The localization lengths follow the trend, L0(V4) < L0(V2) < L0(V3) < L0(V1), similar to the variation of G(EF) with a single Vn defect. This trend indicates that the Vn defects with even integer n induce more easily the localization than those with odd integer n. In the disordered (9,0) tube, the localization can be more easily induced by the V2 defects because the octagon π state by a single V2 defect suppresses significantly the conductance near EF. Similar to the (5,5) tube, the conductance and the local density of states strongly fluctuate (Figure 4c and d), and the conductance near the VBM decays exponentially with increasing of N. To calculate the localization length, we consider the conductance at 0.15 eV from the neutrality point, because the localization lengths at lower energies right next to the gap are affected by the 1183
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The Journal of Physical Chemistry C intrinsic gap. For the V2 defects with d = 21.30 Å, we estimate the localization length to be L0 = 2.56d, which is smaller by a factor of 2 than that for the (5,5) tube (Figure 5b). This result indicates that the localization length depends not only on the type of defects but also the tube chirality.
’ CONCLUSIONS We have investigated the electronic and transport properties of the (5,5) and (9,0) carbon nanotubes with vacancy clusters as well as disordered vacancies. For individual vacancy clusters, in which the missing atoms are serially connected along the tube axis, the conductance near the Fermi level has the tendency of decreasing exponentially with increasing of cluster size. We find that the defect levels formed by a single defect strongly affect the conductance for small vacancy clusters, whereas the conductance reduction in large vacancy clusters is dominated by the local gap opening. With a small amount of monovacancies, divacancies, or a mixture of both defects, disorder induces the Anderson localization in conductance. The localization is found to be stronger for the V2 and V4 defects with even numbers of the missing atoms due to the π-topological defects than for the V1 and V3 defects, which have the defect levels characterized by the σ-dangling bond. The exponential decrease of conductance is attributed to the localized states induced by disorder, not to the development of the local energy gap. In the case of divacancies, the zigzag (9,0) tube exhibits stronger localization than does the armchair (5,5) tube. ’ ASSOCIATED CONTENT
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Supporting Information. Mirror planes in the nanotube with a single vacancy defect, pentagon-related defect states in the nanotubes with and without the mirror symmetries, metalinduced gap states in the (5,5) tube with a single V36 defect, and electronic and transport properties of the disordered (5,5) nanotube with the V4 defects. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION
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’ ACKNOWLEDGMENT This work was supported by the Korea Research Foundation Grant (KRF-2010-0093845). A.T.L. thanks Ja-Eun Lee and Prof. Yong-Hoon Kim for their computational help. ’ REFERENCES (1) Ando, T. J. Phys. Soc. Jpn. 2005, 74, 777. (2) Tans, S. J.; Devoret, M. H.; Dai, H.; Thess, A.; Smalley, R. E.; Geerligs, L. J.; Dekker, C. Nature (London) 1997, 386, 474. (3) White, C. T.; Todorov, T. N. Nature (London) 1998, 393, 240. (4) Sun, L.; Banhart, L.; Krasheninnikov, A. V.; Rodríguez-Manzo, J. A.; Terrones, M.; Ajayan, P. M. Science 2006, 312, 1199. (5) Rodriguez-Manzo, J. A.; Banhart, F. Nano Lett. 2009, 9, 2285. (6) Tolvanen, A.; Kotakoski, J.; Krasheninnikov, A. V.; Nordlund, K. Appl. Phys. Lett. 2007, 91, 173109. (7) Yuzvinsky, T. D.; Michelson, W.; Aloni, S.; Begtrup, G. E.; Kis, A.; Zettl, A. Nano Lett. 2006, 6, 2718. 1184
dx.doi.org/10.1021/jp207943u |J. Phys. Chem. C 2012, 116, 1179–1184
