Carbon nanotubes (CNT) have a very large application potential in the rapid
developing field of molecular electronics. Infinite single-wall metallic CNTs
have theoretically a conductance of 4e2/h because of the two electronic bands
crossing the Fermi level. For finite size CNTs experiments have shown that
other values are also possible, indicating a very strong influence of the
contacts. We study electron transport in single- and double-wall CNTs contacted
to metallic electrodes within the Landauer formalism combined with Green
function techniques. We show that the symmetry of the contact region may lead
to blocking of a transport channel. In the case of double-wall CNTs with both
inner and outer shells being metallic, non-diagonal self energy contributions
from the electrodes may induce channel mixing, precluding a simple addition of
the individual shell conductances

Cuniberti, G.

Gutierrez-Laliga, R.

Krompiewski, S.

Ranjan, N.

English

arXiv

Carbon nanotubes (CNT) have a very large application potential in the rapid developing field of molecular electronics.  Infinite single-wall metallic CNTs have theoretically a conductance of 4e$^{2}$/h because of the two electronic bands crossing the Fermi level. For finite size CNTs experiments have shown that other values are also possible, indicating a very strong influence of the contacts. We study electron transport in single- and double-wall CNTs contacted to metallic electrodes within the Landauer formalism combined with Green function techniques. We show that the symmetry of the contact region may lead to blocking of a transport channel. In the case of double-wall CNTs with both inner and outer shells being metallic, non-diagonal self energy contributions from the electrodes may induce channel mixing, precluding a simple addition of the individual shell conductances

Gutiérrez, Rafael

Krompiewski, Stefan

Cuniberti, Gianaurelio

University of Regensburg Publication Server

arXiv:cond-mat/0507566 v1   25 Jul 2005Electron transport in carbon nanotube-metal systems: contact effectsN. Ranjan,1, 2 R. Gutie´rrez-Laliga,2 S. Krompiewski,3 and G. Cuniberti21Institut fu¨r Physikalische Chemie und Elektrochemie Technische Universita¨t Dresden, Germany2Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany3Institute of Molecular Physics, Polish Academy of Sciences, PL-60179 Poznan´, Poland(Dated: October 10, 2003)Carbon nanotubes (CNT) have a very large application potential in the rapid developing field ofmolecular electronics. Infinite single-wall metallic CNTs have theoretically a conductance of 4e2/hbecause of the two electronic bands crossing the Fermi level. For finite size CNTs experiments haveshown that other values are also possible, indicating a very strong influence of the contacts. Westudy electron transport in single- and double-wall CNTs contacted to metallic electrodes within theLandauer formalism combined with Green function techniques. We show that the symmetry of thecontact region may lead to blocking of a transport channel. In the case of double-wall CNTs withboth inner and outer shells being metallic, non-diagonal self energy contributions from the electrodesmay induce channel mixing, precluding a simple addition of the individual shell conductances.I. INTRODUCTIONCarbon nanotubes belong to one of the most promising candidates in the era of modern nanoelectronics. They can begenerated by wrapping a graphene sheet along different directions as given by the so called chiral vector1. Interestingly,depending on the chiral vector, the tubes show markedly different electronic properties ranging from metallic tosemiconducting. A considerable amount of theoretical and experimental research has been done to explore theirvaried interesting properties, which range from very hard inert materials through good conductors to storage devices2.Concerning the electronic transport properties of metallic single-wall carbon nanotubes (SWCNT) experiments2 havenicely demonstrated ballistic transport and conductance quantization with conductance values equal to 2×G0, G0 =2e2/h being the conductance quantum and the factor 2 in the first expression arising from two spin degenerated bandsat the Fermi level. Similar quantization effects have been recently observed in multi-wall nanotubes (MWCNTs)3,4.However, in contrast to the theoretical expected values conductance steps as low as 0.5G0 or 1×G0 were found3. Evenunder the usual assumption that only the outermost shell is the one contributing to transport, such small conductancevalues suggest that transport channels may be partially or completely closed. Blocking of conductance channels inMWCNTs has been recently addressed in other theoretical works5. We will investigate in this paper conductancequantization in finite size CNTs contacted with metallic electrodes. We will show an example of channel blocking anddemonstrate that the total conductance of muti-wall CNTs cannot be simply obtained by just adding the individualshell conductances.II. THEORETICAL MODELWe investigate electronic transport in a simple model system consisting of carbon nanotubes which are attached tosemi-infinite electrodes having an fcc(111) geometry. A typical configuration is shown in Figure 1.To describe the electronic structure of both subsystems we use a single-orbital nearest-neighbours tight-binding ap-proach. It includes the π-orbitals of the carbon atoms on the tube and s-like orbitals in the electrodes. The metal-CNTcoupling terms are set constant for all neighbours of a given carbon atom. The Hamiltonian isH = HM +Hleads + Vleads,MHM = −tpp∑l,jc†l cj − β∑l, jcos θljea−d l jδ c†jclHleads =∑k∑α∈L,Rǫαkd†kαdkαVleads,M =∑i,k∑α∈L,RVi,αc†idkα +H.c.HM is the CNT Hamiltonian. Its first term describes the intra-shell interaction with a hopping integral tpp whichis set at the constant value of 2.66 eV. The second summand is the inter-shell interaction in the case of MWCNTsTypeset by REVTEX2(β = tpp/k, k > 1). δ = 0.45A˚ is a normalizing factor , a is the difference between the shell radii and θij is the anglebetween the two pz orbitals on different shells. Finally, Hleads is the Hamiltonian of the electrodes and Vleads,M is themutual interaction.The linear conductance G(E) can be related to the electronic transmission probability T (E) according to the Landauerformula (at zero temperature): G = G0T (EF). T (E) can be calculated using Green function techniques6 viaT = TrM[G†MΓRGMΓL]GM is the Green function of the scattering region (in our case the CNT plus two surface layers) which can be calculatedby means of the Dyson equation[E1M −HM − ΣL − ΣR]GM = 1The self-energies Σα = V†αgαVα, α = L,R contain information on the electronic structure of the leads (via the surfaceGreen function gα) as well as on the electrode-scattering region coupling (via Vα). Finally the spectral functions Γαare related to the self-energies by iΓα = (Σα − Σ†α). We do not consider charge transfer effects between the tubesand the metallic electrodes.The use of a single-orbital picture to describe the electrodes allows to write an analyticexpression for the electrodes surface Green function in k-space (assuming L=R)7.g(k, E) =E − ǫ(k)±√(E − ǫ(k))2 − 4|V01(k)|22|V01(k)|2ǫ(k) = 2t0(cos kxa+ 2 coskxa2cos√3kya2)V01(k) = −t0(2 cos(kxa2)eikya2√3 + e−ikya√3 )FIG. 1: View of a (2,2)@(6,6) double walled carbon nanotube sandwiched between two fcc(111) leads (upper panel) and detailsof the contact region (lower panel). The first two layers of the electrodes belong to the scattering region while the third layerextends to infinity and is part of the contact.A. Electronic Transport in single- and double-wall nanotubesWe first consider the diameter dependence of the conductance for SWCNTs. In Fig. 2 we show the transmission as afunction of the energy for (2,2) and (6,6) finite size tubes (10 unit cells). As a reference we also plot the transmissionof an infinite tube. For the latter clear quantization steps are obtained and the conductance around the Fermi levelis 2×G0. The finite size tubes show however a more irregular, spiky behaviour which can be related to finite sizeeffects and to the lifting of some degeneracies as a result of the coupling to the electrodes. More importantly, whilethe (2,2) CNT shows conductance oscillations peaking at 2×G0, the (6,6) tube reaches only on average one quantumof conductance, i.e. a transport channel is apparently closed.We can roughly understand what happens by representing the electronic selfenergy into the eigenstate basis |Φσ〉 ofan isolated CNT, with |Φσ〉 =∑n∈CNT cn,σ |pz,n〉. After some manipulations 8 and assuming a constant coupling ofeach carbon atom to its nearest neighbours metal atoms 9,10, we arrive at the following expression:3Σσσ′ (E) = |V |2∑k||g0,k||(E)Λ†σ(k||)Λσ′(k||) (1)Λσ(k||) =∑m||[n]∑ncn,σeik||m||-3 -2 -1 0 1 2 3E/|tlead|02468T(E)(2,2)(6,6)2G0FIG. 2: Electronic transmission of (2,2) and (6,6) SWCNTs (10 unit cells). The strong oscillations are related to finite sizeeffects. For (2,2) two transport channels at the Fermi level are open, while for the (6,6) CNT only one channel does contribute.Notice that the index n runs now over the CNT atomic slice in direct contact with the metal surface and the indexm||[n] denotes the nearest neighbours on the electrode surface of a given carbon atom n. The function Λσ(k||) containsinformation on the symmetries of the CNT wave functions via cn,σ, and on the electrode surface topology, via theeik||m|| factor. We only need to look at the behaviour of Λσ(k||) for σ = π, π∗, since these are the two eigenstatescrossing the Fermi point in metallic CNTs. Remembering that the expansion coefficients cn,σ of the π and π∗ orbitalsalong the nanotube circumference comprising 2m atoms are proportional to (+1)n and (−1)n, n = 1, · · · , 2m,respectively, the sums in Λσ(k||) can be performed. As a result we find that Λpi∗(k||) identically vanishes for the (6,6)CNT while it is nonzero for the (2,2) tube. Hence the antibonding π∗ orbital does not couple to the electrodes forthe (6,6) CNT and it thus gives no contribution to the conductance.The next issue we have considered is if the conductance of a DWCNT consisting of two armchair SWCNTs can besimply obtained by adding the corresponding conductances of the individual shells. If this holds then, accordning toour previous result, the conductance of a finite size (2,2)@(6,6) DWCNT should yield 3×G0. Two factors can howevermodify this simple picture. One is the inter-shell coupling, the other is the CNT-electrode interface. We have justseen, that the latter can even induce channel blocking.Figure 3 shows the transmission function for different values of the inter-shell coupling parameter β. The maininfluence of β is to introduce mixing of the transport channels which is rather strong at energies far away from theFermi level EF and leads for some energies to a drastic reduction of the conductance when comparing with infinitetubes. The effect near the Fermi level is however less strong. Thus, for β 6= 0 the conductances can not simply beadded since inter-shell interference effects must be considered. More interesting however is the behaviour for zerointer-shell interaction. Even in this case the total conductance near the Fermi level is not simply 3×G0 although it islarger than in the former case (β 6= 0). The imperfect addition of conductances is now related to interference effectscaused by non-diagonal contributions of the electrodes self-energies, Σσ 6=σ′ (E). As a result the transport channelsare mixed in a similar way as for non-zero inter-shell coupling. Although there may be some special cases whereconductances can be added, we can conclude that in general quantum interference effects induced by finite size effects(the existence of the metal-CNT interface) or by the coupling between the nanotube shells will preclude this simpleview.4-3 -2 -1 0 1 2 3E/|tlead|012345T(E)β=0β=tpp/8β=tpp/103G0FIG. 3: Energy dependent transmission of a (2,2)@(6,6) DWCNT (10 unit cells) for different values of the inter-shell couplingβ. The intra-shell hopping tpp = 2.66eV .III. CONCLUSIONSIn this paper we have investigated quantum transport in finite size armchair single- and double-wall carbon nanotubescontacted by metallic electrodes. We have shown that symmetries of the CNT-electrode coupling, hidden in theelectronic self-energies may lead to suppression of transport channels, thus reducing the conductance around theFermi level when comparing with the theoretical ideal case of infinite nanotubes. Moreover, for DWCNTs the simpleapproach of adding the single-shell conductances has been shown to be incorrect, even in the case of no inter-shellinteractions. This can be traced back to interference effects induced by non-diagonal components of the self-energy.These results accentuate the important role played by the interface in determining electronic transport on nanoscalesystems.1 R. Saito, M. S. Dresselhaus, and G. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press (1998).2 S. Reich, C. Thomsen, and J. Maultzsch, Carbon Nanotubes: Basic Concepts and Physical Properties, Wiley-VCH (2004).3 S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer, Carbon Nanotube Quantum Resistors, Science 280, 1744 (1998).4 A. Urbina et al., Quantum Conductance Steps in Solutions of Multiwalled Carbon Nanotubes, Phys. Rev. Lett. 90, 106603(2003).5 S. Sanvito, Y. K. Kwon, D. Tomanek, and C. J. Lambert, Fractional Quantum Conductance in Carbon Nanotubes, Phys.Rev. Lett. 84, 1974 (2000).6 S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge (1995).7 T. N. Todorov, G. A. D. Briggs, and A. P. Sutton, Elastic quantum transport through small structures, J. Phys.-Condens.Matter 5, 2389 (1993).8 , Using |Φσ〉 and the electrode wave functions |k〉 =∑ℓ||eik||ℓ||f(k⊥)∣∣∣χℓ||〉we get: Σσσ′(E) =∑k〈Φσ|V†|k〉g(k, E) 〈k|V |Φσ′〉, which can be put in the form of Eq. (1) by defining Gk|| (E) =∑k⊥ |f(k⊥)|2g(k||,k⊥, E).f(k⊥) is a function of the wave vector perpendicular to the metal surface, only, and the 2d vector ℓ|| runs over the surfacelayer .9 , This approximation can be justified because of the almost perfect fitting of the (2,2) and (6,6) CNTs lattice constants tothe interatomic distances at the electrode surface 10. Thus, each C atom interacts with just 3 atoms on the metal surface .10 S. Krompiewski, Non-equilibrium transport in ferromagnetically contacted metallic carbon nanotubes, Journal of Magnetismand Magnetic Materials In press, (2004).

Electron transport in carbon nanotube-metal systems: contact effects

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