Contact resistance between carbon nanotubes

w n th 1; f ha ru cc e w o RAPID COMMUNICATIONS PHYSICAL REVIEW B, VOLUME 63, 161403~R! malism the current on terminal i can be written as17 I i5 2e h E2‘ ‘ T¯ i j~E ,V !@ f i~E !2 f j~E !#dE , ~1! where T¯ i j(E ,V) is the transmission coefficient from terminal i to j and f i(E) is the Fermi function for terminal i. In the presence of an applied bias, the energy levels are shifted and T¯ i j(E ,V) is modified. The electronic structure and interac- tions between nanotubes are modeled using the p-orbital tight-binding Hamiltonian.18,19 Our conductance calculations are based on single-particle linear-response theory. Thus, electron-electron interactions are not included. As nanotubes are perfect one dimensional quantum wires, the e-e interac- tion effects may be important. For nanotubes, these were shows the conductance values for armchair-armchair and FIG. 1. ~a! A two-terminal nanotube junction can be formed by bringing two tubes’ ends together in parallel and pointing opposite directions (l is the contact length!. ~b! The transmission coefficient T of the two armchair tube @~10,10!-~10,10!# junction as a function of energy E for l564 Å . Interference of electron waves yields resonances in transport. ~c! Current-voltage characteristics of the ~10,10!-~10,10! junction for l546 Å . Contact resistance bet Alper Buldum a Department of Physics and Astronomy, The University of Nor ~Received 9 February 200 Fascinating properties of nanotubes arise when they cation, we demonstrate that such nanotube junctions tance between the tubes depends strongly on atomic st the optimal electronic transport between nanotubes o contact resistance can vary several orders of magnitud negative differential resistance and nonlinear variation new device applications. DOI: 10.1103/PhysRevB.63.161403 Individual carbon nanotubes are perfect molecular wires with well-known structural, electronic, and transport properties.1–8 Nanoscale contacts can be formed with two or more nanotubes. In this rapid communication, we present fascinating properties of nanotubes when intermolecular nanotube junctions are formed. It is demonstrated here that these nanotube junctions have atomic scale characteristics and the contact resistance depends strongly on the atomic structure in the contact region. The optimal electronic trans- port between the nanotubes occurs when the tubes are in the atomic scale registry. It is also found that the contact resis- tance can vary several orders of magnitude with atomic scale movement. In some configurations the intermolecular con- ductance is comparable to that of perfect nanotubes. Phe- nomena such as the negative differential resistance and non- linear variation of resistance with the contact area are found. The large variation of transport properties found here is simi- lar to the sensitive dependence of mechanical-frictional prop- erties on atomic scale registry.9,10 These unusual properties may lead to new nanoelectronic device applications. Several techniques have been used to calculate the quan- tum conductance of carbon nanotubes.8,12–15 Among these the Green’s-function technique is effective and efficient when localized orbital basis sets are used.8,14,15 In our calcu- lations, the Landauer-Bu¨ttiker formalism is employed to cal- culate the conductance and the I-V characteristics with the surface Green’s-function matching method.15,16 In this for- 0163-1829/2001/63~16!/161403~4!/$20.00 63 1614 een carbon nanotubes d Jian Ping Lu Carolina at Chapel Hill, Chapel Hill, North Carolina 27599 published 5 April 2001! orm intermolecular junctions. In this rapid communi- ve atomic scale characteristics and the contact resis- cture in the contact region. Our calculations show that urs when the tubes are in atomic scale registry. The ith atomic scale movements. Phenomena such as the f resistance are found. These properties may lead to PACS number~s!: 72.80.Rj, 71.15.Ap, 73.61.Wp shown to be low energy effects (,1 meV).20,21 Further in- vestigations may be required for these low energy regimes. Intermolecular nanotube junctions can be formed in many geometrical forms. For example, two tubes can be connected in parallel, perpendicular, or two tubes ends can be brought together. We have studied the quantum conductance and current-voltage characteristics of these junctions for different nanotube positions, orientations and chiralities. The simplest two-terminal nanotube junction is constructed by bringing two tubes’ ends together @see Fig. 1~a!#. This junction con- sists of two semi-infinite tubes in parallel and pointing to opposite directions. The equilibrium positions of these two nanotubes are found using molecular dynamics.9 In equilib- rium positions the tubes are in atomic scale registry and the contact region structure is like the A-B stacking of graphite. As the contact ~or interaction! region is finite this junction shows quantum-interference effects. The interference of waves transmitted and reflected from the ends of the tubes yields resonances in electron transport as shown in Fig. 1~b!. The number of resonances increases with increasing contact length, l. This quantum-interference effect introduces the negative differential resistance ~NDR! in the current-voltage characteristics @shown in Fig. 1~c!#. NDR has many applica- tions including high-speed switching, memory, and amplification.22 An interesting feature of this junction is the sensitive de- pendence of conductance on the contact length, l. Figure 2 ©2001 The American Physical Society03-1 RAPID COMMUNICATIONS ALPER BULDUM AND JIAN PING LU PHYSICAL REVIEW B 63 161403~R! zigzag-zigzag tube junctions. In both cases the dependence of conductance on l is nonlinear and quasiperiodic but the periods are different. In the armchair tubes’ case @Fig. 2~a!# the period is 3az (az52.46 Å , unit cell length of armchair tubes!, which is the Fermi wavelength for armchair tubes. The same periodicity was found in earlier experiments and theoretical calculations on the scanning tunneling micros- copy images of finite nanotubes.23 In zigzag tubes’ case, however, the period is found to be the unit cell length (az 54.26 Å ). As the Fermi wavelength for zigzag tubes is infinite, only atomic corrugation is responsible for the varia- tion of conductance. It is also interesting to note that the conductance values are high and comparable to ideal tubes when the tubes are in-registry. Therefore, this simple end-end contact geometry is an ideal way of connecting multiple tubes in device appli- cations. On the other hand, small displacements of tubes from the in-registry configurations lead to dramatic reduction in the intertube conductance. Thus, rapid switching between high and low conductance states can be achieved and fast atomic scale switches can be constructed by using these end- end junctions. We also have investigated a mixed junction of an armchair ~10,10! and a zigzag ~18,0! tube. In this case, the conductance values are an order of magnitude smaller with no apparent periodic variations. A four-terminal junction can be formed by placing one nanotube perpendicular to another as shown in Fig. 3~a!. Multiprobe measurements can be performed on this junction11 with current passing two terminals and voltage measured using the other two. We find that the conductance between the tubes depends strongly on the atomic structure in the contact region. The conductance is high when two tubes are in-registry where atoms from one tube are placed on top of another like A-B stacking of graphite. Thus, an armchair tube crossing a zigzag tube forms an in-registry junction and the conductance is high. In contrast two perpen- dicular armchair tubes forms an out-of-registry junction, the conductance between the tubes is low. In general, different transport properties can be achieved by manipulating these junctions such as rotating or translat- ing one of the tubes with respect to the other. In Figs. 3~b! and 3~c! the variations of contact resistance with respect to rotation angle Q between the tubes is presented. A large variation of resistance is observed. Lower resistance values FIG. 2. ~a! The variation of conductance at the Fermi energy as a function of contact length, l for the ~10,10!-~10,10! junction. Each pair of peaks form a period with length 3az (az52.46 Å unit-cell length of armchair tubes!. ~b! Variation of conductance with l for the ~18,0!-~18,0! junction. The period is the unit-cell length (az 54.26 Å ) for zigzag tubes. 16140 are found when the junction is in-registry configurations. In the case of the ~18,0!-~10,10! junction the tubes are in- registry at Q530,90,150°. In the ~10,10!-~10,10! junction the tubes are in-registry at Q50,60,120,180°. Even when the tubes are in-registry the contact resistance can be differ- ent at different Q due to change in the contact area. For example, in the ~18,0!-~10,10! junction, the resistance is lower at Q530° than at Q590° as the contact area at Q 530° is larger. In Fig. 3~d!, the variation of resistance with the translation of upper tube is shown for the ~18,0!-~10,10! junction. The variation is small in comparison to the case of rotating the tubes. The lowest resistance is achieved when the contact structure is like A-A stacking of graphite. When the junctions are placed on a substrate, electronic contact can be significantly enhanced by the structural relax- ation of the tubes and adhesion between tubes and the sub- strate. We investigate the effect of relaxation by performing molecular dynamics simulations using empirical potentials.24 The cross junction is relaxed on a rigid surface25 and con- stant forces ~3.0 nN! are applied to the ends of the upper tube ~of length 127 Å ) to simulate the effect of substrate adhe- sion. An example of relaxed junction is shown in Fig. 4~a!. Current-voltage characteristics of rigid and relaxed cross junctions are presented in Figs. 4~b! and 4~c! for two differ- ent nanotube junctions. We found that, when tubes are in- registry, the resistance drops dramatically with relaxation and/or applying forces. In contrast, when the tubes are out of registry, the change in resistance with relaxation and/or ap- plying forces is small. For example, in the case of the ~18,0!- ~10,10! ~in-registry! junction @Fig. 4~b!# the resistance is 2.05 MV for rigid tubes but reduced to 682 KV after re- laxation. When forces are applied the resistance drops to 121 KV . On the other hand, two perpendicular ~10,10! tubes are out of registry @Fig. 4~c!#, the resistance between FIG. 3. ~a! A model of four-terminal junction formed by cross- ing two nanotubes. The terminal labels, rotation angle, Q and the translation directions are shown. The tubes are considered to be rigid. The current is passing between 1 and 4 and voltage is mea- sured between 2 and 3. ~b! Contact resistance of the ~18,0!-~10,10! junction as a function of rotation angle Q . The tubes are in-registry at Q530,90,150°. ~c! Resistance of the ~10,10!-~10,10! junction. The tubes are in-registry at Q50,60,120,180°. ~d! Resistance of the ~18,0!-~10,10! junction as functions of translation of one tube rela- tive to the other in the x and y directions. 3-2 the two conducting channels of the nanotube are extended states around the whole circumference. Increasing the tube size, though increasing the geometrical contact area, in fact reduces the relative weight of conducting channel wave func- RAPID COMMUNICATIONS CONTACT RESISTANCE BETWEEN CARBON NANOTUBES PHYSICAL REVIEW B 63 161403~R! tubes is 3.36 MV for the rigid junction and decreases to 3.21 MV when the junction is relaxed. Applying forces re- duces the resistance to 1.66 MV . The contact resistance we have calculated is in good agreement with recent experiment11 which found 90–360 KV resistances for metal-metal cross junctions on a surface. Our results suggest that modest pressure and/or force can dramatically enhance the intertube transport if the tubes are in-registry. 1 R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Prop- erties of Carbon Nanotubes ~Imperial College Press, London, 1998!. 2 J. W. G. Wildo¨er, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C. Dekker, Nature ~London! 391, 59 ~1998!. 3 T. W. Odom, J. L. Huang, P. Kim, and C. M. Lieber, Nature ~London! 391, 62 ~1998!. 4 P. G. Collins, A. Zettl, H. Bando, A. Thess, and R. E. Smalley, Science 278, 100 ~1997!. 5 M. Bockrath et al., Science 275, 192 ~1997!; S. J. Tans et al., Nature ~London! 386, 474 ~1998!. 6 S. J. Tans, A. R. M. Verschueren, and C. Dekker, Nature ~Lon- don! 393, 49 ~1998!. FIG. 4. ~a! An example of the relaxed cross junction with ap- plied forces. Deformation in the contact region can be clearly seen. The tubes are 127 Å long and only the central part of the junction is shown here. ~b! I-V characteristics of a ~18,0!-~10,10! ~in- registry! cross junction. The resistance value for the rigid case is 2.05 MV , but decreases to 682 KV with relaxation and to 121 KV with force. ~c! I-V characteristics of a ~10,10!-~10,10! ~out-of-registry! cross junction. The resistance values are 3.36, 3.21, and 1.66 MV for a rigid, relaxed without force and relaxed with force cases. 16140 tion around the contact. However, we found the effect of relaxation and forces are more dramatic for larger tubes as they are more susceptible to deformation. In conclusion, quantum transport properties of intermo- lecular nanotube junctions are investigated. We find that nanotube junctions have atomic scale characteristics in their transport properties and the contact resistance strongly de- pends on the atomic structure in the contact region. The op- timal electronic transport between nanotubes occurs when the tubes are in atomic scale registry. The contact resistance can vary several orders of magnitude with atomic scale ma- nipulations. The negative differential resistance is found in nanotube end-end contacts. Similar properties may be found in the contacts of other nanoscale wires and structures. These unusual properties may lead to new atomic scale switches, resistors, amplifiers, and memory devices. Note added: Recent experiments26 performed at UNC- Chapel Hill found that the contact resistance between nano- tube and graphite varied more than an order of magnitude by changing the angular alignment between tube and graphite lattice, similar to the effects predicted in this article. The authors thank R. Superfine, M. R. Falvo, S. Paulson, and S. Washburn for stimulating discussions. This work is supported by U. S. Office of Naval Research ~N00014-98-1- 0593!. 7 Z. Yao, H. W. C. Postma, L. Balents, and C. Dekker, Nature ~London! 402, 273 ~1999!. 8 L. Chico, V. H. Crespi, L. X. Benedict, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 76, 971 ~1996!. 9 A. Buldum and J. P. Lu, Phys. Rev. Lett. 83, 5050 ~1999!. 10 M. R. Falvo, R. M. Taylor, A. Helser, V. Chi, F. P. Brooks, S. Washburn, and R. Superfine, Nature ~London! 397, 236 ~1999!. 11 M. S. Fuhrer et al., Science 288, 494 ~2000!; M. S. Fuhrer et al., Physica E 6, 868 ~2000!. 12 R. Tamura and M. Tsukada, Phys. Rev. B 58, 8120 ~1998!. 13 H. J. Choi and J. Ihm, Phys. Rev. B 59, 2267 ~1999!. 14 M. P. Anantram and T. R. Govindan, Phys. Rev. B 58, 4882 ~1998!. The low contact resistance in the case of in-registry con- figurations can be understood considering the coupling of electronic states between the tubes. Strong coupling occurs when the tubes are in-registry. Although the magnitudes of individual hopping integrals between the atoms on different tubes are similar for in-registry and out-of-registry junctions, phase coherence is achieved in the in-registry junction which enhances the coupling of the electronic states between the tubes. We have also investigated the dependence of the contact resistance on nanotube size and found that the resistance in- crease with tube diameter. 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B 42, 9458 ~1990!. 25 The cross junction is relaxed on a rigid graphite surface in order to have a substrate holding the junction. Periodic boundary con- ditions are applied in lateral directions and all nanotube atoms are allowed to move. Only the nanotubes are considered in con- ductance calculations. 26 S. Paulson, A. Helser, M. B. Nardelli, R. M. Taylor II, M. R. Falvo, R. Superfine, and S. Washburn, Science 290, 1742 ~2000!. RAPID COMMUNICATIONS ALPER BULDUM AND JIAN PING LU PHYSICAL REVIEW B 63 161403~R! 16140 3-4