10378
J. Phys. Chem. B 1999, 103, 10378-10381
Symmetry Effects on the Conductance of Nanotube Junctions Gabin Treboux* National Institute for AdVanced Interdisciplinary Research, 1-1-4 Higashi, Tsukuba-shi, Ibaraki 305, Japan ReceiVed: April 27, 1999; In Final Form: September 14, 1999
The ballistic conductance of various two-terminal and three-terminal carbon nanotube systems is calculated within the Landauer formalism. Using only metallic nanotubes as terminals, it is shown that only a subset of juctions topologies preserve the metallic function. This behavior is rationalized both in the two-terminal and three-terminal systems through symmetry considerations.
Carbon nanotubes,1 because of their unique electronic properties, are beginning to show promise as nanoelectronic building blocks. Recent experimental observations are consistent with the occurrence of ballistic transport in carbon nanotubes,2 suggesting the use of carbon nanotubes as efficient metallic wires. Rectification has also been experimentally observed in nanotubes,3 providing the first step toward realizing a nanoelectronic diode. The geometrical structure of a homogeneous single-wall nanotube (SWNT) is uniquely determined by the chiral vector C ) na1 + ma2, where a1 and a2 are graphene sheet lattice translation vectors. The (n, n) armchair nanotubes are metallic while the (n, m) tubes are semimetallic if n - m is a nonzero multiple of three, and semiconducting otherwise.4 More specific electronic properties have been predicted for systems which connect nanotubes of different helicity.5-8 In this paper, I study the ballistic conductance of nanotube junctions in relation to their point group symmetry. To predict the conductance of such junctions the algorithm recently proposed by Chico et al.6 has been implemented. Starting with two media L and R representing two perfect nanotubes, a connection is made through a spacer S (see Figure 1). The calculation of the corresponding single particle Green’s function g, defined by
[
E-HR1,1 - HR1,2TR HRS 0 E-HS HSL g ) HSR HLS HL-1,-1-HL-1,-2T hL 0
]
-1
(1)
is performed using the Green’s function matching method (GFM).9 Here HRi,i represents the Hamiltonian matrix corresponding to a one-dimensional unit cell of the perfect nanotube forming the medium R. The index i refers to the position of this unit cell relative to the interface. In the implementation used for this article, the Hamiltonian is restricted to a tightbinding Hamiltonian with one π-electron per atom. Therefore, the matrix HRi,i contains pR × pR elements corresponding to the interaction between the pR π-electrons of the onedimensional unit cell. The Hamiltonian matrix HRi,i+1 of dimension pR × pR represents the coupling between adjacent unit cells of medium R. * E-mail adress:
[email protected].
Figure 1. Schematic view of left (L) and right (R) media connected via a spacer S.
Similarly, the Hamiltonian matrix HS of dimension pS × pS represents the interaction between the pS atoms of the spacer S, and the Hamiltonian matrix HRS of dimension pR × pS represents the coupling between the first unit cell of medium R and the spacer S. The body of the calculation involves obtaining the transfer matrix of each medium, according to
GMn+1,m ) TMGMn,m (n g m)
(2)
h MGMn,m (n e ) GMn-1,m ) T
(3)
GMn,m+1 ) GMn,mSM (m g n)
(4)
GMn,m-1 ) GMn,mShM(m e n) (5)
(5)
Here the Green’s function matrix GMi,j of dimension p × p corresponds to the interaction between the one-dimensional unit cells i and j of the perfect nanotube forming the medium M (M h M, SM, ShM of dimensions p × ) L, R) while the matrices TM, T p are the transfer matrices of the medium M. These transfer matrices are obtained using the algorithm derived by Lopez Sancho et al.10,11 Once the Green’s function for the matched system LSR is obtained, the scattering matrix is defined through
[ ] [ ] φR,out φR,in φL,out ) [S] φL,in
(6)
where φin and φout denote the incoming and outgoing wave function of the corresponding infinite media L or R and where the scattering matrix S is calculated from
[S] )
[
][
]
TRm 0 gRR - gR gRL n g gLL - gL × T hL 0 LR
[
][
SRm 0 GRm,m-1 0 n S hL 0 GLn,n-1 0
10.1021/jp991377y CCC: $18.00 © 1999 American Chemical Society Published on Web 11/06/1999
]
Carbon Nanotube Conductance
J. Phys. Chem. B, Vol. 103, No. 47, 1999 10379 the other.13,14 Calling these atomic sets starred {*} and unstarred {0} respectively, the bonding ψl+ and antibonding ψl- orbitals of an alternant system occur in pairs with opposite energies and can be written as *
ψl ( )
Figure 2. Conductance of the six different carbon nanotubes formed by a linear junction between (12, 0) and (9, 0) metallic zigzag nanotubes together with the conductance of the infinite (9, 0) zigzag (steplike curve). The tight-binding parameter β is set to -2.66 eV. The conductance is unaffected by the value of β. The value of β simply scales the energy axis.
Here gM is the Green’s function of the infinite medium M projected on the p atoms forming the left (unit cell labeled -1 in Figure 1) or right (unit cell labeled 1 in Figure 1) interfaces p using the projector I ) ∑k)1 |k〉〈k| where |k〉 represents the atomic orbital of an atom k of the left or right interface. Here, gRR, gLL, and gLR are obtained from eq 1. The conductance of the system LSR is obtained from the generalization of the Landauer conductance formula corresponding to the two-probe experiment12
Γ(E) )
2e2 h
() VR
∑ji V
|〈φRj|SRL(E)|φLi〉|2
(8)
L
The asymptotic solution of eq 8 is possible far from the interface, i.e., for large values of the indices n and m in eq 7, by setting incident amplitudes as the eingenvectors of the Hamiltonian of the L part and using the conservation of the current in the ballistic regime. I analyze first the possibility of forming a metal-metal junction by connecting two different nanotubes. One way to form such a junction is to insert paired five- and sevenmembered ring defects into the system. Adding a five-seven defect changes the chirality of a metallic (n, m) nanotube to a semiconductor (n ( 1, m ( 1) one. The only way to join two metallic nanotubes by introducing five-seven defects at their junction, without introducing a semiconductor segment between them, is to introduce an exact multiple of three such defects. For instance, there exist six different ways to connect a (12, 0) and a (9, 0) zigzag nanotube by placing three five-seven defects at various places on the circumference of their junction. For all of these systems, the conductance as a function of energy is reported in Figure 2. Whatever the junction considered, an asymmetric conductance curve is obtained. This can be understood in terms of the chemical concept of alternance. In alternant systems, the atoms can be divided into two distinct sets, no atom of one set being adjacent to an atom of
∑i
0
Cliφi (
∑j Cljφj
(9)
so that the electron-hole density of states of the system is symmetric. In terms of chemical properties, a network of sixmembered carbon rings is alternant. In a nonalternant system, these two atomic sets cannot be defined and eq 9 no longer applies. For instance, including a five-membered ring or a sevenmembered ring in a six-membered ring network destroys the electron-hole symmetry, and the resulting system has carbon atoms bearing a net positive or negative charge. In the present systems, the presence of five-seven defects makes the systems nonalternant and so creates an electron-hole asymmetry. A second important feature of the results is the drastic difference between the conductance curves of the junctions around E ) 0 eV. From the curves represented in Figure 2, three different situations are identified and classified as a function of the value of the conductance a E ) 0 eV. The first situation corresponds to a zero conductance, the second to a conductance of approximately 2e2/h and the third to a conductance of approximately 4e2/h. The first situation arises for only one junction. In this junction the three five-seven defects are positioned so that the system conserves a C3 axis. Among all the junction geometries, this is the only one which conserves a C3 axis. In this system the conductance is zero in the energy window within which the corresponding infinite (12, 0) and (9, 0) nanotubes have exactely two ballistic channels. At higher energies, the (12, 0) and (9, 0) nanotubes gain further ballistic channels and the conductance increases. This behavior can be understood with reference to the work of Chico et al.6 For each constituent nanotube, they define an angular momentum for each ballistic channel based on the Cn axis of the nanotube. They show that total reflection of a ballistic electron occurs at the junction if the angular momenta of the matched channels differ. This reflection explains the nonconduction window seen in the first situation. The second situation arises for three junctions. In these junctions the three five-seven defects are positioned so that the system has no overall symmetry. In this case the two ballistic channels of the (12, 0) nanotube and the two ballistic channels of the (9, 0) nanotube have no more angular momentum limitation. Nevertheless, the conductance does not correspond to 4e2/h, the expected value when two channels match, but rather to a value close to 2e2/h, the expected value when one channel matches. This result can be understood from a consideration of symmetry. The wave functions of each atomic site contained within the one-dimensional unit cell of the considered carbon nanotube are combined to form the solutions of symmetry corresponding to the different irreducible representations of the point group symmetry of the considered carbon nanotube. In the context of propagation this procedure defines the ballistic channels. In the vicinity of E ) 0 eV, whatever the index n, the corresponding two channels of a perfect (n, 0) nanotube belong to an irreducible representation of dimension 2 labeled E. At the junction cell, the system has no more symmetry, so the irreducible representation of dimension 2 cannot be formed and degenerates into an irreducible representation of dimension 1. Therefore, at the junction cell, the two channels of the perfect
10380 J. Phys. Chem. B, Vol. 103, No. 47, 1999
Treboux
Figure 4. Side view of a Y-shaped system composed of three identical semi-infinite (12, 0) zigzag nanotubes joined via a triangulene-based spacer. Two seven-membered rings exist at the junction between each terminal (left), leading to nonalternance and electron-hole asymmetry. The removal of a six-membered ring eliminates the two sevenmembered rings and yields an alternant system (right).
Figure 3. Conductance of a carbon nanotube formed by a linear junction between (18, 0) and (9, 0) metallic zigzag nanotubes together with the conductance of the infinite (18, 0) and (9, 0) zigzag nanotubes (open circle and dotted steplike curves respectively). The tight-binding parameter β is set to -2.66 eV.
(n, 0) nanotube should coalesce into a single channel. This consideration of symmetry explains the loss of one of the two conduction channels in the numerical result of Figure 2. The third situation arises for two junctions. In these junctions the three five-seven defects are positioned so that the system has a single plane of symmetry. In this case the conductance is close to 4e2/h, the expected value when two channels match. Here the value of the conductance is explained by the existence of the plane of symmetry, which allows symmetric and antisymmetric solutions to be built from the two channels of the (n, 0) nanotubes. The conductance calculation of a further example of a linear junction obtained by connecting a (18, 0) and a (9, 0) zigzag nanotube is presented in Figure 3. In this example the junction consists of a complete circumference belt of five-seven defects. The system conserves a rotational axis of symmetry but the (18, 0)-(9, 0) zigzag nanotube junction corresponds to a case where the two ballistic channels of each semi-infinite nanotube have an identical angular momentum with respect to the Cn axis. Therefore, in the energy window within which the corresponding infinite (18, 0) and (9, 0) nanotubes have exactely two ballistic channels, the conductance reaches the value of 4e2/h, the expected value, when two channels match. Note that, whatever the energy considered, the value of the conductance is bounded by the conductance of the narrowest element i.e., the (9, 0) nanotube part of the sytem. Having shown how the ability of a linear junction to sustain ballistic transport varies according to its symmetry, I move on to consider branched topologies. A branched junction is a necessary building block for a nanotube network. Three-way (or Y) nanotube junctions have already been observed experimentally15 and their stability has been analyzed theoretically.16,17 Here the conductance characteristics of several such Y-junctions are studied. A Y-junction can be considered as three nanotubes joined via a triangular central spacer. A C3 symmetry is obtained by selecting three identical semi-infinite nanotubes and using a spacer which also conserves C3 symmetry. To obtain a metallic three-terminal junction, semi-infinite metallic nanotube terminals and a metallic spacer are selected. The metallic or semiconductor character of the finite spacer is determined from its energy spectrum, calculated using the
Figure 5. Left: Y-shaped system composed of three identical semiinfinite (12, 0) zigzag nanotubes joined via a triangulene-based spacer. Note that the system has an atom-centered C3 axis. Four 1D unit cells are represented for each semi-infinite terminal. Right: Conductance between two of the semi-infinite terminals.
same Hamiltonian as used for the semi-infinite terminals. The existence of degenerate energy levels at the Fermi level is used here to define the metallic character. Note, however, that the number of such metallic states can be predicted13,14 from the topology of the spacer and corresponds to the difference in cardinality between the starred {*} and unstarred {0} atomic sets discussed earlier. Once the metallic spacer is connected to the semi-infinite metallic nanotube terminals its energy spectrum changes, possibly leading to the disappearance of these metallic states. I have demonstrated in a previous publication18 that whenever the three-terminal system has a C3 axis centered on a sixmembered ring, these metallic states disappear. The chemical basis of this effect lies in the system being able to exploit a local C6 symmetry to stabilize the central region of the Y-junction. Interestingly, whenever the system has a C3 axis centered on a six-membered ring, a characteristic gap appears in the conductance curve, while for the system having an atomcentered C3 axis the metallic character of the junction is preserved. This effect is general enough to be applied for various spacer sizes and for nanotubes of different radii or helicity. As examples, Y-shaped systems composed of three identical semiinfinite (12, 0) zigzag nanotubes built with spacers of different sizes are studied (Figure 5 and Figure 6), together with Y-shaped systems composed of three identical semi-infinite (7, 7) armchair nanotubes (Figure 7) and (6, 6) armchair nanotubes (Figure 8). Note that two seven-membered rings are introduced between each pair of terminals (Figure 4). These seven-membered rings make the system nonalternant and so create an electron-hole asymmetry. I have demonstrated in a previous publication18 that the effect presented here is similar in both the complete junction and the junction where seven-membered rings have been removed. Nevertheless, the removal of the seven-membered rings restores alternance and hence electron-hole symmetry, which allow the energy range under study to be restricted and the effect to be more clearly isolated. Therefore, only the conductance curves for the junctions with the seven-membered rings removed (Figure 4) are presented.
Carbon Nanotube Conductance
Figure 6. Left: Y-shaped system composed of three identical semiinfinite (12, 0) zigzag nanotubes joined via a triangulene-based spacer. Note that the system has a six-membered ring centered C3 axis. Four 1D unit cells are represented for each semi-infinite terminal. Right: Conductance between two of the semi-infinite terminals.
J. Phys. Chem. B, Vol. 103, No. 47, 1999 10381 terminal configuration and C3 axis symmetry of the system, the maximum conductance can only be 2e2/h, i.e., half of the value expected when two channels match. Analyzing the curves, it is seen that every junction produces a significant reflection effect across the full energy range considered. Nevertheless, the Y-junctions which have an atom-centered C3 axis are predicted to allow ballistic transport. Throughout the calculations, the tight-binding parameter β is set to -2.66 eV. The ballistic conductance of two-terminal and three-terminal carbon nanotube junctions is calculated using the Landauer formalism. In the two-terminal configuration, using only metallic nanotube terminals, the symmetry of the junction between the terminals is shown to govern the ballistic transport ability of the system. In the three-terminal Y-junction configuration, again using only metallic nanotube terminals, the system is shown to have a nonconduction energy window whenever it has a C3 axis centered on a six-membered ring. The chemical basis of this effect lies in the system being able to exploit a local C6 symmetry to stabilize the central region of the Y-junction.18 The appearance of a nonconduction energy window associated with the existence of a C3 axis centered on a six-membered ring is shown to be a general phenomenon applicable to both the zigzag and the armchair nanotube cases. Acknowledgment. This work was carried out at Silverbrook Research (Sydney, Australia).
Figure 7. Left: Y-shaped system composed of three identical semiinfinite (7, 7) armchair nanotubes joined via a triangulene-based spacer. Note that the system has an atom-centered C3 axis. Four 1D unit cells are represented for each semi-infinite terminal. Right: Conductance between two of the semi-infinite terminals.
Figure 8. Left: Y-shaped system composed of three identical semiinfinite (6, 6) armchair nanotubes joined via a triangulene-based spacer. Note that the system has a six-membered ring centered C3 axis. Four 1D unit cells are represented for each semi-infinite terminal. Right: Conductance between two of the semi-infinite terminals.
The calculations are confined to the energy window within which the corresponding semi-infinite metallic nanotube has two ballistic channels. In each case, the conductance is calculated from one semi-infinite terminal to another. Due to the three-
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