Bias-driven local density of states alterations and transport in ballistic molecular devices
BBias-driven local density of states alterations and transport in ballistic moleculardevices
Ioannis Deretzis ∗ and Antonino La Magna † Istituto per la Microelettronica e Microsistemi (CNR-IMM)Stradale Primosole 50, Catania 95121, Italy (Dated: October 31, 2018)We study dynamic nonequilibrium electron charging phenomena in ballistic molecular devicesat room temperature that compromise their response to bias, and whose nature is evidently dis-tinguishable from static Schottky-type potential barriers. Using various metallic/semiconductingcarbon nanotubes and alkane dithiol molecules as active parts of a molecular bridge, we performself-consistent quantum transport calculations under the non-equilibrium Green’s function formalismcoupled to a three-dimensional Poisson solver for a mutual description of chemistry and electrostat-ics. Our results sketch a particular tracking relationship between the device’s local density of statesand the contact electrochemical potentials that can effectively condition the conduction processby altering the electronic structure of the molecular system. Such change is unassociated to elec-tronic/phononic scattering effects while its extent is highly correlated to the conducting character ofthe system, giving rise to an increase of the intrinsic resistance of molecules with a semiconductingcharacter and a symmetric mass-center disposition.
PACS numbers: 73.63.Fg; 31.15.bu
I. INTRODUCTION
It is a common perception nowadays that both industry and academic research have largely focused their attentionon molecular structures as active part candidates of nanoscale electronic devices. Such trend is other than unfounded,as reduced dimensionality could significantly advance miniaturization and improve performance of plausible logicand storage devices. Aside the increasing electronic transport measurement paradigms of synthetic organic/inorganicmolecular structures[1], it is undoubtable that carbon nanotubes (CNTs) have constituted the central point of molec-ular research during the last decade[2]. Moreover, recent zeolite template growth techniques[3] have clearly demon-strated that the era of thin ( d ≈ . nm diameter) CNTs cannot be considered too distant. Nonetheless advances innanoscale engineering have brought knowledge of molecular electronics up to a level of integrated circuit assembling[4],the comprehension of the quantum effects that rule the conduction mechanism has not been fully mastered, in partic-ular when it comes to interface interaction issues between device and electrode atoms. Freshman electronics teachesthat when two surfaces with different work functions come to contact, charge transfer takes place that tends to mini-mize this difference. Such effect has been repeatedly noted in the case of CNTs embedded to metallic electrodes[5, 6],with Schottky-type potential barriers that vary on the basis of the metallic contact element and the CNT diameter.Although conceptually convenient, the categorization of the quality of metallic contacts considering only metal-CNTwork function differences finds experimental inconsistency, e.g. in the case of group 10 transition metals palladiumand platinum[7, 8]. Theoretically, the electronic configuration regime of a molecule changes with respect to its iso-lated equilibrium picture when a) the device is attached to metallic contacts (appearance of metal-induced states[9],broadening of energy levels[10], charge transfer due to work function differences[11]) and b) when the device goes outof equilibrium (bias-induced charge transfer[12]). It is therefore reasonable that a complete theoretical investigationof quantum transport effects in nanodevices accounts also for dynamic nonequilibrium processes[9, 13] apart fromthe static equilibrium ones. In such context we have preliminarily evidenced the presence of transmission probabilityalterations in ballistic semiconducting carbon nanotube molecular bridges[12], purely related to the application of abias on the two electrodes of the system. Moreover such effect manifests in a strong coupling regime at room temper-ature, excluding Kondo or Coulomb blockade implications. By presenting equilibrium/nonequilibrium local densityof states (LDOS) distributions and electronic density modifications, this article attempts to intensely examine thephysical processes that lead to the aforementioned behavior for various metallic/semiconducting CNTs, and generalizefindings by demonstrating similar conduction aspects also in alkane dithiol molecules. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Modeling of carbon-nanotube-based devices can often result puzzling, since too short CNTs manifest contact-induced states that can significantly change the density of states spectrum with respect to the respective bulk device[9],while long ‘real-size’ systems are computationally prohibited out of the single-orbital tight-binding model. This workimplements a mathematical formalism that can capture geometrical, chemical and charging interactions without atthe same time resulting computationally weighty. In this sense the non-equilibrium Green’s function formalism hasbeen based on an all-valence-electron extended H¨uckel Hamiltonian[14], iteratively coupled to a three-dimensionalPoisson solver in the self-consistent field regime. Such approximation can combine a) the flexibility of semiempiricalmethods, b) the qualitative aspects of the Extended H¨uckel method in the calculation of CNT band structures[15], c)the inclusion of charging effects under bias[13] and d) the uniform description of both device and contacts, allowingat the same time multiple hundreds-of-atoms simulations with a relatively low CPU load. Based on this model, thisstudy is focused on the correlation between the electrochemical potentials of the metallic electrodes and the localdensity of states of a biased molecular system, the change in the electronic density that such interaction provokes andthe consequences of these processes on the conduction mechanism.This paper is organized as follows: Section II describes the details of the mathematical/computational formalism,section III shows transmission probability and LDOS spectra for CNT and alkane dithiol systems out of equilibrium,section IV demonstrates changes in electronic density and potential profiles for a metallic and a semiconducting CNT,and finally in section V we discuss the presented results.
II. METHODOLOGY
The theoretical foundation of our computational model is based on the method developed by Zahid et al. [13],which couples the Green’s function formalism based on an extended H¨uckel Hamiltonian with a complete neglectof differential overlap (CNDO) mean field theory for the inclusion of Coulomb interactions, with appropriate in-house mathematical enhancements that render computations more affordable. The basic points of such formalismare briefly described in the following: a non-equilibrium Green’s function[16] is used for the quantum calculationof transport for various finite size CNTs and alkane dithiol molecules embedded between two semi-infinite metallicplanes (source and drain contacts). Such approach is based on the single particle retarded Green’s function matrix G = [ ES − H − Σ L − Σ R ] − , where E is the scalar energy, H is the device Hamiltonian matrix in an appropriate basisset, S is the overlap matrix in that basis set and Σ L,R is the self energy, which includes the effect of scattering due to theleft ( L ) and right ( R ) contacts. Self energy Σ L,R terms can be expressed as Σ = ( τ − ES ) g s ( τ † − ES ), where g s is thesurface Green function specific to the contact type and τ is the Hamiltonian relative to the mutual interaction betweenthe device and the contact[16]. Device and contacts Hamiltonian matrices are obtained using an all-valence-orbitalH¨uckel method[14], where orbitals are approximated by Slater-type functions and the parameterization of diagonalelements is based on theoretical and experimental data of ionization energies for the respective atoms[10, 17]. Non-diagonal Hamiltonian elements are calculated on the basis of the overlap matrices between different atomic orbitals[17].Considering that the computational framework takes place in the coherent limit, a Landauer-type expression can beused for the current calculation: I = 2 eh (cid:90) + ∞−∞ dET ( E )[ f ( E, µ L ) − f ( E, µ R )] . (1)In this expression T ( E ) = T r [Γ L G Γ R G † ] represents the transmission probability for the device current as a functionof energy and Γ L,R = i [Σ L,R − Σ † L,R ] are the anti-Hermitian parts of the self-energy terms. The statistical Fermi-Diracdistribution of electrons f ( E, µ
L,R ) in the contact at chemical potential µ L,R has a particular importance here sincesimulation temperatures have been set to 300 K. Charging effects can be introduced in the model with the inclusionof a self-consistent potential U SC (∆ ρ ) that is a functional of the device density matrix difference between equilibriumand non-equilibrium states (∆ ρ = ρ − ρ eq ), and which is added to the bare device’s Hamiltonian ( H = H + U SC (∆ ρ )).The U SC (∆ ρ ) is calculated self-consistently by the addition of three separate constituent terms[13]: U SC (∆ ρ ) = U Laplace + U P oisson (∆ ρ ) + U Image (∆ ρ ) , (2)whereas ρ is given by ρ = 12 π (cid:90) + ∞−∞ dE [ f ( E, µ L ) G Γ L G † + f ( E, µ R ) G Γ R G † ] . (3) U Laplace and U Image parts of the potential have been numerically calculated with a three-dimensional finite elementmethod solving the ∇ U = 0 equation with appropriate box boundary conditions[26]. The calculation of the Poissonterm can derive in the framework of the CNDO theory, using only the Hartree potential for the Coulomb interaction[13].CNDO parameters have been determined for device atoms using ionization potential and electron affinity data fromref.[18]. All potential components have been evaluated on the atoms (a linear interpolation between grid and atomicsites has been implemented in the Laplace and Image term case).For the integration of the density matrix at each step of the self-consistent process a contour integration techniquehas been applied[19] up to energies where the Fermi-Dirac coefficient is still a unity ( f ( E, µ
L,R ) ≈ III. LOCAL DENSITY OF STATES AND BIAS CORRELATION IN CARBON NANOTUBE ANDALKANE DITHIOL MOLECULES
One of the most appealing aspects of semiempirical methods is their flexibility in the determination of various systemparameters without necessarily compromising on the description of the qualitative characteristics of the physicalsystem. In this sense the implemented method allows for a separate study of the effects of electron charging on theconduction mechanism under equilibrium and non-equilibrium conditions. Although there are plenty of theoreticalworks that affront the problem of CNTs and static Schottky potential barriers in a multidisciplinary level[11, 22,23], to the best of our knowledge there has been no systematic attempt to evidence charging manifestation undernonequilibrium conditions and its role on the propagation of current in CNT systems. In the current study, thenonequilibrium transport features have been separated from the electrostatic effects of equilibrium (Schottky barrierformations) by setting the Fermi level of the whole system (device and contacts) to the Fermi energy level of therespective bulk CNTs. Such condition imposes the alignment between the metallic contact’s work function andthe charge neutrality level of the CNT by shifting the energy bands of the device with respect to the ones of thecontact, affecting only equilibrium charge transfer features while leaving all chemical and nonequilibrium interactionsunchanged. In the laboratory, an analogous phenomenon can be realized by using a gate electrode strongly coupled tothe device levels, to unidirectionally shift the CNT bands towards higher or lower energies, depending on the spectralposition of the electrodes’ work function.Strong interface coupling conditions have been explicitly imposed at this study by bringing Au(111) metallic elec-trodes at an L = 0 . nm distance from the two open edges of the CNTs[17]. Simulations have been undertakenfor various metallic and semiconducting nanotubes in equilibrium and under bias, and figure 1 shows transmissionfunction outcomes for CNTs with basis vector indices (3,2), (3,3), (8,0) and (5,5). The diameters of these CNTs are d (3 , ≈ . nm , d (3 , ≈ . nm , d (8 , ≈ . nm , d (5 , ≈ . nm , their lengths L (3 , ≈ . nm , L (3 , ≈ . nm , L (8 , ≈ . nm , L (5 , ≈ . nm , while their conducting character is fully determined by their helicity and (in the caseof narrow CNTs) σ − π hybrid orbital mixing[17]. The point of differentiation between semiconducting and metalliccarbon nanotubes lies in the response of the intrinsic conducting capability of each system to bias. Semiconductortubes, apart from shifting their transmission peaks corresponding to a shift of energy eigenvalues[13], demonstrate alsoa clear diminishment of their total transmission probability, which also affects the energy zone close to the Fermi levelof the system. In terms of the Landauer formalism this can be translated in a decrease of the device’s current-carryingcapacity[12] and could be seen as an increase of the nanotube resistance compared to the one estimated from theequilibrium transmission spectrum(see figure 2), which has no relation with phononic or electronic scattering interac-tions, static Shottky-type potential barriers or low-temperature/weak coupling effects. Rather, such process impliesan implicit alteration of the propagation modes of current under non-equilibrium. On the other hand, although thephenomenon is present also for the metallic CNTs, its strength is evidently limited and the transmission coefficientdoes not diverge significantly from the equilibrium case. Such non-uniform situation cannot be simply attributed toenergy level shifts and needs to be better evaluated.The flow of current in molecules strongly depends on their electronic structure and its correlation with the metallicelectrodes, in terms of interface chemical bonding and electrochemical potential positioning[10]. Both qualitative andquantitative aspects of the latter are impressed on the device’s density of states distribution. Hence, an LDOS ringanalysis[24] has been performed for the (8,0) and (5,5) tubes in the cases of 0V and 2V applied bias, in order to evidencethe processes that render the aforementioned metallic/semiconducting behaviors divergent. Figures 3 and 4 show therespective results. The observation of the distribution of states in the designated rings under equilibrium can evidencea high electronic density near the Fermi energy level for both CNTs close to the contacts that can be attributed to metalinduced states, which arise from the tails of metallic electron wavefunctions that decay exponentially into the devicebody[11]. Moving towards the inner part of the CNT, such concentration tends to diminish to a finite value for the (5,5)tube and almost vanishes for the (8,0) tube, revealing respectively the metallic and semiconducting character of thestudied CNTs. The application of bias has a qualitatively similar influence on the electronic bands of the two systems;nonetheless their different conducting character, the LDOS’s of the various rings of the two systems tend to follow themovement of the electrochemical potential of the contact to which they are better correlated, both topologically andchemically. Thus, rings closer to the positively biased electrode (bottom electrode in figures 3,4) tend to shift theirLDOS’s towards lower energies, rings that are near to the negatively biased electrode (upper one in figures 3,4) tend tomove LDOS’s towards higher energies, while middle rings of both systems have LDOS’s with minor differences that areattributed to geometrically -and therefore chemically- unequal couplings between the two electrodes and the device.Note that such minor mid-device alterations would not have been evidenced with a tight-binding representation of themetallic contacts that neglects geometrical differentiations in the reconstruction of interface chemical bonds[17]. Thetransmission probability of the semiconducting CNT is affected by the aforementioned LDOS redistribution in a biggerextent with respect to that of the metallic one, since this bidirectional LDOS motion at various parts of the CNTinserts and pulls out states from the conduction gap. These newly inserted gap states cannot always energeticallycorrelate with states of neighboring rings. Such lack of correlation leads to a spatial localization of the electronicwavefunctions related to the aforementioned states (e.g. see the peaks appearing near the Fermi energy level in thenon-equilibrium LDOS spectra of figure 3) that cannot give any contribution to the globally evaluated transmissioncoefficient, and thus, the conductivity of the system. The implicit increase of the device resistance can therefore beseen as a bias-induced change of its electronic structure that provokes a decrease of availability of non-localized stateswithin the conduction window. On the other hand, the magnitude of this effect is minor in the metallic CNT sinceshifted states can almost always correlate with neighboring ring states throughout the energy spectrum of interest forthe conduction of current. In this case, the respective bias-induced change of the device’s electronic structure doesnot lead to an extended localization of electronic wavefunctions.The previous concepts can be better visualized in the analysis of few level molecular systems. In this sense alkanedithiols are theoretically ideal case study objects due to their small size and large conduction gap[1]. As such, thesame type of investigation has been carried out for C8, C10 and C12 alkane dithiol molecules whereas representativeresults shown in figure 5 are for the C12 molecule. Here, in accordance with ab initio reconstructions that calculateequilibrium Au-S distance between Au and SCH complexes at L Au − S = 0 . nm [25], the vertical distance betweenthe device and contacts has been set to L = 0 . nm . The local density of states has been calculated on the atomicsites and fig. 5 demonstrates LDOS’s for the two external sulfur atoms that spectrally give the most importantdensity contribution with respect to C and H atoms. The LDOS trend that tracks electrochemical potentials is clearlyconfirmed also here, whereas states from the right sulfur atom that enter the conduction gap become localized and donot contribute to the transmission probability of the system. Attempting therefore a generalization of the presentedresults, it would be appropriate to state that the increase of the intrinsic resistance due to bias-induced factors shouldbe expected in all molecules with a semiconducting/isolating character that exhibit a spatial symmetry with regardto their center of mass. The prerequisite of symmetry has a sense of isotropic contribution of the various parts ofthe molecule to the total electronic density. Asymmetry could have an impact similar to disorder on the electronicstructure of a molecular system whereas the application of bias could affect singularly each distinct structure. Inthis sense, even if asymmetric molecule LDOS’s are also expected to behave in an analogous to their symmetriccounterparts way, their resistance may increase, decrease or remain unaffected.The qualitative aspects of LDOS redistribution reported in this paragraph should be evidenced within the fullrange of semiemprirical/first-principles descriptions of a biased molecular system with a proper treatment of electron-electron interactions, even if the level of accuracy is expected to vary from case to case. On the other hand though,the intrinsic shortcomings of each method should influence on the correctness of the presented results. In this work forexample, a tight-binding representation of the device and the contacts could result inappropriate for small diameterCNTs[17]or alkane dithiol molecules, and lacking of realism at the contact-device interface schematization[17]. IV. SPATIAL DISTRIBUTION OF NONEQUILIBRIUM CHARGES ON MOLECULAR DEVICES
The investigation of molecular conduction in terms of local density of states only implicitly copes with the actualprocess of transport, which is the movement of charges. Principal quantum transport theory teaches that when thetwo electrodes of a molecular junction are bias-driven out of equilibrium, the electrochemical potentials of the latterseparate and tend to pump in/pull out electrons from those molecular levels that lie within them[10]. In the idealcase where the energy levels of the device are perfectly correlated to the contacts we can expect no accumulation ofcharges on the device body. This picture changes when dealing with real molecular systems where, as seen previously,the spatially anisotropic bidirectional movement of the LDOS’s towards the energies of the electrochemical potentialscan increase the number of localized states on the device (which could also exist prior to the application of bias dueto symmetry/coupling factors). In this case we can expect a charge accumulation on the device body correspondingto the occupation of localized states that can condition the transport mechanism by reducing the easiness with whichnew charges can flow in/out of the device. This phenomenon can be visualized at figures 6 and 7, where the absolutevalue of the change of electronic density between equilibrium and non-equilibrium conditions along the device body(∆ ρ = ρ − ρ eq ) is plotted for the (8,0) and (5,5) CNTs for a 2V bias, accompanied by the respective self-consistentpotential profile. Both electronic densities and potential values (interrelated via equation 2) are calculated on theatomic sites, whereas in the case of the potential profile mean values have been assigned for atoms with an identicalprojection on the device axis (atoms that belong to the same CNT ring). There are two distinct CNT areas where thechange of electronic density has distinguishable characteristics, a) the area close to the metallic contacts and b) theinner part of the nanotube. In the first case, an initial screening can be observed for both types of CNTs (up to about5˚A distance from the CNT ends), which could be perceived as the adjustment of the infinite metallic propagationmodes to the finite modes of the molecular device. Such screening is quantitatively higher for the (5,5) tube and hasan steep impact on the shaping of the potential profile near the CNT edges, contrary to the (8,0) case, where thedeviation from the linear Laplace component of a neutral system is smaller (see fig. 6). On the other hand, regardingthe inner CNT body, a finite difference density distribution can be mainly observed for the semiconducting (8,0) tube,while minimal differences are evidenced for the (5,5) metallic one. This effect is directly related to the increase oflocalized states for the semiconducting CNT, which provokes an encapsulation of charges on the device body. V. DISCUSSION
In this article the influence of nonequilibrium charge transfer (NECT) on the conduction mechanism of moleculardevices (CNTs, alkane dithiols) has been analyzed on the basis of local density of states alterations provoked by theapplication of bias. The main characteristic of this process can be individualized on the interaction between themolecular LDOS and the electrochemical potentials of the contacts, where a tracking relationship has been clearlymanifested. Such process provokes a nonuniform reorganization of the density of states throughout the body ofthe molecular device, which, especially in geometrically symmetric devices with a conduction gap, can increase thenumber of localized states at energies close to the Fermi level. These states, when occupied by charges, can giverise to localized electronic wavefunctions that yield a non-zero difference between equilibrium and non-equilibriumelectronic densities at the inner parts of the molecule. On the other hand it has been demonstrated that the extentof such effect is smaller in molecular systems with a metallic character.It can be expected that the presence of the aforementioned processes is limited to the quantum description ofelectronic transport. Indeed, moving towards the semiclassical limit, the device can be thought of as the sum ofnumerous nanosize discretized units where quantum rules can be implemented. In this case, the potential profile thateach of this units perceives is flat and therefore no bidirectional LDOS modifications are expected, rather than aunified movement of all states towards the respective potential level. Therefore no alteration of the local electronicstructure should be observed with respect to the zero-bias case other than this shift.The concepts discussed above can reflect both practically and theoretically on the perception of transport inmolecular systems. From an engineering point of view, findings could be useful in the understanding of nanoscaledevice operation, since the dependence of the conduction gap formation from the applied source-drain bias magnitudeis something that should not be overlooked. From a theoretical point of view, the presence of NECT phenomena caninvoke a discussion both in a conceptual and in methodological level. Conceptually it becomes evident that molecularconduction issues cannot be solely treated in equilibrium terms. Methodologically it is confirmed that a self-consistentapproach to the computer-aided design of molecular systems is fundamental for the capturing of charging phenomena,either in or out of equilibrium.
Acknowledgments
This work has been partially supported by the Sicilian Region under the contract POR-Regione Sicilia-Misura 3.15. [1] G. Arena, I. Deretzis, G. Forte, F. Giannazzo, A. La Magna, G. Lombardo, V. Raineri, C. Sgarlata and G. Spoto, New J.Chem. , 756 (2007)[2] M. P. Anantram and F. L´eonard, Rep. Prog. Phys. , 507 (2006)[3] M. Hulman, H. Kuzmany, O. Dubay, G. Kresse, L. Li and Z. K. Tang, J. Chem. Phys. , 3384 (2003)[4] Z. Chen, J. Appenzeller, Y.-M. Lin, J. Sippel-Oakley, A. G. Rinzler, J. Tang, S. J. Wind, P. M. Solomon, and P. Avouris,Science , 1735 (2006) [5] S. Heinze, J. Tersoff, R. Martel, V. Derycke, J. Appenzeller, and Ph. Avouris, Phys. Rev. Lett. , 106801 (2002)[6] Z. Chen, J. Appenzeller, J. Knoch, Y.-M. Lin, and Ph. Avouris, Nano Lett. , 1497 (2005)[7] A. Javey, J. Guo, Q. Wang, M. Lundstrom, and H. Dai, Nature , 654 (2003)[8] D. Mann, A. Javey, J. Kong, Q. Wang, and H. Dai, Nano Lett. , 1541 (2003)[9] P. Pomorski, C. Roland, and H. Guo, Phys. Rev. B , 115408 (2004)[10] F. Zahid, M. Paulsson, and S. Datta, Electrical Conduction Through Molecules: Advanced Semiconductors and OrganicNano-Techniques , edited by H. Morkoc, Academic, New York (2003)[11] Y. Xue and M. A. Ratner, Phys Rev B , 205416 (2004)[12] I. Deretzis and A. La Magna, Appl. Phys. Lett. , 163111 (2007)[13] F. Zahid, M. Paulsson, E. Polizzi, A. W. Ghosh, L. Siddiqui, and S. Datta, J. Chem. Phys. , 064707 (2005)[14] R. Hoffmann, J. Chem. Phys , 1397 (1963)[15] D. Kienle, J. I. Cerda, and A. W. Ghosh, J. Appl. Phys. , 043714 (2006)[16] S. Datta, Electronic Transport in Mesoscopic Systems , edited by H. Ahmed, M. Pepper, and A. Broers, CambridgeUniversity Press, Cambridge (1995)[17] I. Deretzis and A. La Magna, Nanotechnology , 5063 (2006)[18] J. Hinze and H. H. Jaff´e, J. Am. Chem. Soc. , 540 (1962)[19] M. Brandbyge, J.-L. Mozos, P. Ordej´on, J. Taylor, and K. Stokbro, Phys. Rev. B , 165401 (2002)[20] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing ,Cambridge University Press, Cambridge (1992)[21] V. Eyert, J. Comp. Phys. , 271 (1996)[22] B. Shan and K. Cho, Phys Rev B , 233405 (2004)[23] J. J. Palacios, P. Tarakeshwar, and D. M. Kim, arXiv:0705.1328v1[24] F. Triozon, P. Lambin and S. Roche, Nanotechnology , 230 (2005)[25] N. Gonzalez-Lakunza, N. Lorente and A. Arnau, J. Phys. Chem. C , 12383 (2007)[26] For the Laplace term: U Laplace = − qV S at the source and U Laplace = − qV D at the drain. For the Image term: U Image = − U Poisson ( (cid:126)r ) at the position of the device contacts. T r a n s m i ss i on , T -3 -2 -1 0 1 2 3 Energy, E-E F [eV] (3,2)(3,3)(8,0)(5,5) FIG. 1: Transmission as a function of energy for a) a 4-unit-cell (3,2), b) a 15-unit-cell (3,3), c) an 8-unit-cell (8,0) and d) a12-unit-cell (5,5) carbon nanotube system, for 0V (shaded area) and 2V (black line) applied bias. The contact metal is Au(111)whereas zero energy refers to the charge neutrality level of each CNT.
Voltage [V] C u rr e n t [ µ A ] a) b) FIG. 2: Current-Voltage curves for a) an 8-unit-cell (8,0) and b) a 12-unit-cell (5,5) carbon nanotube system. Solid linesrepresent the current value based on self-consistent calculations, whereas dotted lines are obtained without self-consistency,based on the simple potential profile of references [10, 17], where the potential fully drops in the interface area between thedevice and the contacts, while throughout the device it remains flat.
FIG. 3: Geometry, transmission coefficient and ring LDOS evaluation as a function of energy for an 8-unit-cell (8,0) CNT for0V (shaded area) and 2V (black line) applied bias.FIG. 4: Geometry, transmission coefficient and ring LDOS evaluation as a function of energy for a 12-unit-cell (5,5) CNT for0V (shaded area) and 2V (black line) applied bias.