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 calculatedwithin the Landauer formalism. Using only metallic nanotubes as terminals, it is shown that only a subset ofjuctions topologies preserve the metallic function. This behavior is rationalized both in the two-terminal andthree-terminal systems through symmetry considerations.

Carbon nanotubes,1 because of their unique electronic proper-ties, are beginning to show promise as nanoelectronic buildingblocks. Recent experimental observations are consistent withthe occurrence of ballistic transport in carbon nanotubes,2

suggesting the use of carbon nanotubes as efficient metallicwires.

Rectification has also been experimentally observed innanotubes,3 providing the first step toward realizing a nano-electronic diode.

The geometrical structure of a homogeneous single-wallnanotube (SWNT) is uniquely determined by the chiral vectorC ) na1 + ma2, wherea1 and a2 are graphene sheet latticetranslation vectors. The (n, n) armchair nanotubes are metallicwhile the (n, m) tubes are semimetallic ifn - m is a nonzeromultiple of three, and semiconducting otherwise.4 More specificelectronic properties have been predicted for systems whichconnect nanotubes of different helicity.5-8

In this paper, I study the ballistic conductance of nanotubejunctions in relation to their point group symmetry. To predictthe conductance of such junctions the algorithm recentlyproposed by Chico et al.6 has been implemented.

Starting with two media L and R representing two perfectnanotubes, a connection is made through a spacer S (see Figure1).

The calculation of the corresponding single particle Green’sfunction g, defined by

is performed using the Green’s function matching method(GFM).9 Here HRi,i represents the Hamiltonian matrix corre-sponding to a one-dimensional unit cell of the perfect nanotubeforming the medium R. The indexi refers to the position ofthis unit cell relative to the interface. In the implementationused for this article, the Hamiltonian is restricted to a tight-binding Hamiltonian with oneπ-electron per atom. Therefore,the matrixHRi,i containspR × pR elements corresponding tothe interaction between thepR π-electrons of the one-dimensional unit cell. The Hamiltonian matrixHRi,i+1 ofdimensionpR × pR represents the coupling between adjacentunit cells of medium R.

Similarly, the Hamiltonian matrixHS of dimensionpS × pS

represents the interaction between thepS atoms of the spacerS, and the Hamiltonian matrixHRS of dimensionpR × pS

represents the coupling between the first unit cell of medium Rand the spacer S. The body of the calculation involves obtainingthe transfer matrix of each medium, according to

Here the Green’s function matrixGMi,j of dimensionp × pcorresponds to the interaction between the one-dimensional unitcells i andj of the perfect nanotube forming the medium M (M) L, R) while the matricesTM, ThM, SM, ShM of dimensionsp ×p are the transfer matrices of the mediumM. These transfermatrices are obtained using the algorithm derived by LopezSancho et al.10,11

Once the Green’s function for the matched system LSR isobtained, the scattering matrix is defined through

where φin and φout denote the incoming and outgoing wavefunction of the corresponding infinite media L or R and wherethe scattering matrix S is calculated from

* E-mail adress: gabin@nair.go.jp.

g ) [E-HR1,1 - HR1,2TR HRS 0HSR E-HS HSL

0 HLS HL-1,-1-HL-1,-2ThL]-1

(1)

Figure 1. Schematic view of left (L) and right (R) media connectedvia a spacer S.

GMn+1,m ) TMGMn,m (n g m) (2)

GMn-1,m ) ThMGMn,m (n e ) (3)

GMn,m+1 ) GMn,mSM (m g n) (4)

GMn,m-1 ) GMn,mShM(m e n) (5) (5)

[φR,out

φL,out ]) [S][φR,in

φL,in ] (6)

[S] ) [TRm 0

0 ThLn][gRR - gR gRL

gLR gLL - gL ] ×

[SRm 0

0 ShLn][GRm,m

-1 0

0 GLn,n-1]

10378 J. Phys. Chem. B1999,103,10378-10381

10.1021/jp991377y CCC: $18.00 © 1999 American Chemical SocietyPublished on Web 11/06/1999

Here gM is the Green’s function of the infinite medium Mprojected on the p atoms forming the left (unit cell labeled-1in Figure 1) or right (unit cell labeled 1 in Figure 1) interfacesusing the projectorI ) ∑k)1

p |k⟩⟨k| where |k⟩ represents theatomic orbital of an atomk of the left or right interface. Here,gRR, gLL, andgLR are obtained from eq 1.

The conductance of the system LSR is obtained from thegeneralization of the Landauer conductance formula correspond-ing to the two-probe experiment12

The asymptotic solution of eq 8 is possible far from theinterface, i.e., for large values of the indicesn andm in eq 7,by setting incident amplitudes as the eingenvectors of theHamiltonian of the L part and using the conservation of thecurrent in the ballistic regime.

I analyze first the possibility of forming a metal-metaljunction by connecting two different nanotubes. One way toform such a junction is to insert paired five- and seven-membered ring defects into the system. Adding a five-sevendefect changes the chirality of a metallic (n, m) nanotube to asemiconductor (n ( 1, m ( 1) one. The only way to join twometallic nanotubes by introducing five-seven defects at theirjunction, without introducing a semiconductor segment betweenthem, 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 defectsat various places on the circumference of their junction. For allof these systems, the conductance as a function of energy isreported in Figure 2.

Whatever the junction considered, an asymmetric conductancecurve is obtained. This can be understood in terms of thechemical concept of alternance.

In alternant systems, the atoms can be divided into twodistinct sets, no atom of one set being adjacent to an atom of

the other.13,14Calling these atomic sets starred{*} and unstarred{0} respectively, the bondingψl+ and antibondingψl- orbitalsof an alternant system occur in pairs with opposite energies andcan be written as

so that the electron-hole density of states of the system issymmetric. In terms of chemical properties, a network of six-membered carbon rings is alternant. In a nonalternant system,these two atomic sets cannot be defined and eq 9 no longerapplies. For instance, including a five-membered ring or a seven-membered ring in a six-membered ring network destroys theelectron-hole symmetry, and the resulting system has carbonatoms bearing a net positive or negative charge. In the presentsystems, the presence of five-seven defects makes the systemsnonalternant and so creates an electron-hole asymmetry.

A second important feature of the results is the drasticdifference between the conductance curves of the junctionsaroundE ) 0 eV. From the curves represented in Figure 2,three different situations are identified and classified as afunction of the value of the conductance a E) 0 eV. The firstsituation corresponds to a zero conductance, the second to aconductance of approximately 2e2/h and the third to a conduc-tance of approximately 4e2/h.

The first situation arises for only one junction. In this junctionthe three five-seven defects are positioned so that the systemconserves aC3 axis. Among all the junction geometries, this isthe only one which conserves aC3 axis. In this system theconductance is zero in the energy window within which thecorresponding infinite (12, 0) and (9, 0) nanotubes have exactelytwo ballistic channels. At higher energies, the (12, 0) and (9,0) nanotubes gain further ballistic channels and the conductanceincreases.

This behavior can be understood with reference to the workof Chico et al.6 For each constituent nanotube, they define anangular momentum for each ballistic channel based on theCn

axis of the nanotube. They show that total reflection of a ballisticelectron occurs at the junction if the angular momenta of thematched channels differ. This reflection explains the noncon-duction window seen in the first situation.

The second situation arises for three junctions. In thesejunctions the three five-seven defects are positioned so thatthe system has no overall symmetry. In this case the two ballisticchannels of the (12, 0) nanotube and the two ballistic channelsof the (9, 0) nanotube have no more angular momentumlimitation. Nevertheless, the conductance does not correspondto 4e2/h, the expected value when two channels match, but ratherto a value close to 2e2/h, the expected value when one channelmatches. This result can be understood from a consideration ofsymmetry. The wave functions of each atomic site containedwithin the one-dimensional unit cell of the considered carbonnanotube are combined to form the solutions of symmetrycorresponding to the different irreducible representations of thepoint group symmetry of the considered carbon nanotube. Inthe context of propagation this procedure defines the ballisticchannels. In the vicinity ofE ) 0 eV, whatever the indexn,the corresponding two channels of a perfect (n, 0) nanotubebelong to an irreducible representation of dimension 2 labeledE. At the junction cell, the system has no more symmetry, sothe irreducible representation of dimension 2 cannot be formedand degenerates into an irreducible representation of dimension1. Therefore, at the junction cell, the two channels of the perfect

Figure 2. Conductance of the six different carbon nanotubes formedby a linear junction between (12, 0) and (9, 0) metallic zigzag nanotubestogether with the conductance of the infinite (9, 0) zigzag (steplikecurve). The tight-binding parameterâ is set to -2.66 eV. Theconductance is unaffected by the value ofâ. The value ofâ simplyscales the energy axis.

Γ(E) )2e2

h∑

ji(VR

VL)|⟨æRj|SRL(E)|æLi⟩|2 (8)

ψl( ) ∑i

*

Cliφi ( ∑j

0

Cljφj (9)

Carbon Nanotube Conductance J. Phys. Chem. B, Vol. 103, No. 47, 199910379

(n, 0) nanotube should coalesce into a single channel. Thisconsideration of symmetry explains the loss of one of the twoconduction channels in the numerical result of Figure 2.

The third situation arises for two junctions. In these junctionsthe three five-seven defects are positioned so that the systemhas a single plane of symmetry. In this case the conductance isclose to 4e2/h, the expected value when two channels match.Here the value of the conductance is explained by the existenceof the plane of symmetry, which allows symmetric andantisymmetric solutions to be built from the two channels ofthe (n, 0) nanotubes.

The conductance calculation of a further example of a linearjunction obtained by connecting a (18, 0) and a (9, 0) zigzagnanotube is presented in Figure 3. In this example the junctionconsists 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 wherethe two ballistic channels of each semi-infinite nanotube havean identical angular momentum with respect to theCn axis.Therefore, in the energy window within which the correspondinginfinite (18, 0) and (9, 0) nanotubes have exactely two ballisticchannels, the conductance reaches the value of 4e2/h, theexpected value, when two channels match. Note that, whateverthe energy considered, the value of the conductance is boundedby 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 sustainballistic transport varies according to its symmetry, I move onto consider branched topologies. A branched junction is anecessary building block for a nanotube network. Three-way(or Y) nanotube junctions have already been observed experi-mentally15 and their stability has been analyzed theoretically.16,17

Here the conductance characteristics of several such Y-junctionsare studied.

A Y-junction can be considered as three nanotubes joinedvia a triangular central spacer. AC3 symmetry is obtained byselecting three identical semi-infinite nanotubes and using aspacer which also conservesC3 symmetry. To obtain a metallicthree-terminal junction, semi-infinite metallic nanotube terminalsand a metallic spacer are selected.

The metallic or semiconductor character of the finite spaceris determined from its energy spectrum, calculated using the

same Hamiltonian as used for the semi-infinite terminals. Theexistence of degenerate energy levels at the Fermi level is usedhere to define the metallic character. Note, however, that thenumber of such metallic states can be predicted13,14 from thetopology of the spacer and corresponds to the difference incardinality between the starred{*} and unstarred{0} atomicsets discussed earlier.

Once the metallic spacer is connected to the semi-infinitemetallic nanotube terminals its energy spectrum changes,possibly leading to the disappearance of these metallic states. Ihave demonstrated in a previous publication18 that wheneverthe three-terminal system has aC3 axis centered on a six-membered ring, these metallic states disappear. The chemicalbasis of this effect lies in the system being able to exploit alocal C6 symmetry to stabilize the central region of theY-junction.

Interestingly, whenever the system has aC3 axis centered ona six-membered ring, a characteristic gap appears in theconductance curve, while for the system having an atom-centeredC3 axis the metallic character of the junction ispreserved.

This effect is general enough to be applied for various spacersizes and for nanotubes of different radii or helicity. Asexamples, Y-shaped systems composed of three identical semi-infinite (12, 0) zigzag nanotubes built with spacers of differentsizes are studied (Figure 5 and Figure 6), together with Y-shapedsystems composed of three identical semi-infinite (7, 7) armchairnanotubes (Figure 7) and (6, 6) armchair nanotubes (Figure 8).

Note that two seven-membered rings are introduced betweeneach pair of terminals (Figure 4). These seven-membered ringsmake the system nonalternant and so create an electron-holeasymmetry. I have demonstrated in a previous publication18 thatthe effect presented here is similar in both the complete junctionand the junction where seven-membered rings have beenremoved. Nevertheless, the removal of the seven-memberedrings restores alternance and hence electron-hole symmetry,which allow the energy range under study to be restricted andthe effect to be more clearly isolated. Therefore, only theconductance curves for the junctions with the seven-memberedrings removed (Figure 4) are presented.

Figure 3. Conductance of a carbon nanotube formed by a linearjunction between (18, 0) and (9, 0) metallic zigzag nanotubes togetherwith the conductance of the infinite (18, 0) and (9, 0) zigzag nanotubes(open circle and dotted steplike curves respectively). The tight-bindingparameterâ is set to-2.66 eV.

Figure 4. Side view of a Y-shaped system composed of three identicalsemi-infinite (12, 0) zigzag nanotubes joined via a triangulene-basedspacer. Two seven-membered rings exist at the junction between eachterminal (left), leading to nonalternance and electron-hole asymmetry.The removal of a six-membered ring eliminates the two seven-membered rings and yields an alternant system (right).

Figure 5. Left: Y-shaped system composed of three identical semi-infinite (12, 0) zigzag nanotubes joined via a triangulene-based spacer.Note that the system has an atom-centeredC3 axis. Four 1D unit cellsare represented for each semi-infinite terminal. Right: Conductancebetween two of the semi-infinite terminals.

10380 J. Phys. Chem. B, Vol. 103, No. 47, 1999 Treboux

The calculations are confined to the energy window withinwhich the corresponding semi-infinite metallic nanotube has twoballistic channels. In each case, the conductance is calculatedfrom one semi-infinite terminal to another. Due to the three-

terminal configuration andC3 axis symmetry of the system, themaximum conductance can only be 2e2/h, i.e., half of the valueexpected when two channels match. Analyzing the curves, it isseen that every junction produces a significant reflection effectacross the full energy range considered. Nevertheless, theY-junctions which have an atom-centeredC3 axis are predictedto allow ballistic transport. Throughout the calculations, thetight-binding parameterâ is set to-2.66 eV.

The ballistic conductance of two-terminal and three-terminalcarbon nanotube junctions is calculated using the Landauerformalism. In the two-terminal configuration, using only metallicnanotube terminals, the symmetry of the junction between theterminals is shown to govern the ballistic transport ability ofthe system. In the three-terminal Y-junction configuration, againusing only metallic nanotube terminals, the system is shown tohave a nonconduction energy window whenever it has aC3 axiscentered on a six-membered ring. The chemical basis of thiseffect lies in the system being able to exploit a localC6

symmetry to stabilize the central region of the Y-junction.18 Theappearance of a nonconduction energy window associated withthe existence of aC3 axis centered on a six-membered ring isshown to be a general phenomenon applicable to both the zigzagand the armchair nanotube cases.

Acknowledgment. This work was carried out at SilverbrookResearch (Sydney, Australia).

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Figure 6. Left: Y-shaped system composed of three identical semi-infinite (12, 0) zigzag nanotubes joined via a triangulene-based spacer.Note that the system has a six-membered ring centeredC3 axis. Four1D unit cells are represented for each semi-infinite terminal. Right:Conductance between two of the semi-infinite terminals.

Figure 7. Left: Y-shaped system composed of three identical semi-infinite (7, 7) armchair nanotubes joined via a triangulene-based spacer.Note that the system has an atom-centeredC3 axis. Four 1D unit cellsare represented for each semi-infinite terminal. Right: Conductancebetween two of the semi-infinite terminals.

Figure 8. Left: Y-shaped system composed of three identical semi-infinite (6, 6) armchair nanotubes joined via a triangulene-based spacer.Note that the system has a six-membered ring centeredC3 axis. Four1D unit cells are represented for each semi-infinite terminal. Right:Conductance between two of the semi-infinite terminals.

Carbon Nanotube Conductance J. Phys. Chem. B, Vol. 103, No. 47, 199910381