Tight Binding Model of Quantum Conductance of Cumulenic ...paveldyachkov.ru/images/dyza.pdfTight Binding Model of Quantum Conductance of Cumulenic and Polyynic Carbynes P. N. D’yachkov,*,†

Tight Binding Model of Quantum Conductance of Cumulenic andPolyynic CarbynesP. N. D’yachkov,*,† V. A. Zaluev,† E. Yu. Kocherga,‡ and N. R. Sadykov‡

†Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences, Leninskii pr. 31, 119991 Moscow,Russia‡Snezhinsk Physical-Technical Institute, National Nuclear Research University MIFI, 456776 Snezhinsk, Russia

*S Supporting Information

ABSTRACT: The linear carbon chains called the carbynes are possibly the most simple carbon molecular systems of interest asthe potential materials for nanoelectronics. In a metallic cumulenic form of carbyne, the C atoms form the double bonds (...CC...), but the single and triple bonds alternate in the semiconducting polyynic structure (...−CC−CC−...). In thiswork, the ballistic electrical conductance of cumulenic and polyynic carbon chains C40, C20, and C10 is calculated using the π-electron tight-binding methods. The transmission function and dependences of the current on the length of the carbon chain, thetype of carbon bonds, material properties of the electrodes, bias voltage, and temperature are obtained. For the calculations oftransmission function, we use the Schrodinger equation that meets the specified value of the energy E, the wave functions beingthe superposition of incident and reflected waves in the region of cathode, and describe the transmitted wave in the anode. Forsmall bias voltages, the problem of calculating the transmission function in the cumulenic and polyynic chains is solvedanalytically by applying the difference schemes approach which permits us to pass from the tight-binding algebraic equations tothe differential equations and to take advantage of their solutions. In the case of large voltages, we apply an iterative technique.The transmission functions τ(E,V) of the cumulenic and polyynic chains of similar composition differ dramatically. The energyregions of very high and low transparency of carbynes for electrons are obtained, and an oscillatory character of the energydependence of transmission functions is pointed out. Each electronic level of the molecule corresponds to a peak in the energydependence of transmission function with τ ≈ 1. The peaks are responsible for the resonant electron transfer between theelectrodes. The voltage-dependent variations of the τ are initially quite weak, but at the higher voltages their effect is a drasticreduction of the carbyne transparency. One important feature of the current−voltage I−V characteristics is that the currentinitially increases with growth of the bias voltage, reaches a peak, and then drops to give rise to the negative conductance. On thetypical I−V curves of the polyynic chains, there are both the peak and local minimum (valley) at higher voltages. Moreover, thereare some oscillations of voltage dependences of conductance due to the discrete nature of the C chain electron energy levels. Theπ electron model results are compared with ab initio data on conductance of similar chains having only several C atoms. Thepresence of the negative resistance regions and valley in the curve I−V indicate the possibilities of design of the resonancetunneling devices based on the C chains.

■ INTRODUCTION

The past decade has been characterized by the creation ofnanoelectronic devices based on the carbon compounds.Particularly, the transistors, diodes, electromechanical, andelectron emission devices are already implemented, theoperating elements of which are the carbon nanotubes,1−4

graphene nanoparticles,5 fullerenes,6,7 or the small carbon-

containing molecules.8−10 The monatomic chains are possibly

the most simple carbon molecular systems that may be of

interest as potential materials for nanoelectronics. The linear

Received: April 19, 2013Revised: July 15, 2013Published: July 22, 2013

Article

pubs.acs.org/JPCC

© 2013 American Chemical Society 16306 dx.doi.org/10.1021/jp4038864 | J. Phys. Chem. C 2013, 117, 16306−16315

pubs.acs.org/JPCC

carbon chains, called the carbynes, have been discussed in theliterature for more than 50 years;11−19 particularly, it has beensuggested that carbynes exist in the interstellar dust andmeteorites.20,21 However, their very existence in the form of asufficiently long polymer is sometimes questioned.17−19,22,23 Todate, the existence of stable long linear carbon chains is reliablyestablished.19,22,23 For example, a carbon chain with a length of20 nm containing more than 100 atoms enclosed in a shell ofmultiwalled carbon nanotubes was found in ref 24. The carbonchains linking the graphene sheet have recently beenobserved.25 In recent years, there has been considerableprogress in the synthesis of compounds containing the longcarbon chain and two bulky end groups required to prevent theformation of covalent bonds between the neighbor chains andthe reactions of chain ends.26−28 Thus, the compounds (i-Pr)3Si−C−Si(i-Pr)3 containing carbon chains of 8 to 16 atomswith end triisopropylsilyl groups were isolated in the solidphase. Their structure is determined by methods ofcrystallography,26 which show that the carbon chains have theexpected linear geometry; the maximum deviation of the bondangles C−C−C of 180° is equal to 3.1° only and is observed forthe longest carbon chain with n = 16. In 2010, the record chainwith the 44 C atoms was detected in the compound Tr−C44−Tr, where Tr = tris(3,5-di-t-butylphenyl)methyl moiety.27 Inthis compound, the presence of linear carbyne chains wasconfirmed by 13C NMR spectroscopy and X-ray analysis.According to the valence of the carbon atom, a formation of thetwo forms is possible for the linear carbon chains. In acumulenic form of carbyne, the C atoms form the double bonds(...CC...), but the single and triple bonds alternate inthe polyynic structure (...−CC−CC−...). The experimen-tally observed structure of carbon chains reveals alternatingbonds with the lengths of 1.205 ± 0.009 and 1.360 ± 0.004 Å.26

The short carbon cumulenic type chains are observed inorganic molecules with terminal phenyl groups.29 Themolecular structure of a tetraphenylbutatriene with threecumulenic double bonds is determined by X-ray diffractioncrystallographic analysis, and it was found that the length of thecenter CC bond is equal to 1.26 Å.30 Moreover, the presenceof cumulenic type carbyne was noted for the solid amorphouscarbon phases at high pressures, but formation of the branchedstructures with covalent bonds between neighboring chainsdestroying the perfect carbyne chain structure was suggested inthis case.31 The vertically oriented carbyne films on the SiO2substrates can be prepared, and a field-effect transistor withinterchain charge transfer using the 30 Å films was designedrecently.32−34

The conductance of the short cumulenic carbynes havingonly several C atoms was studied in refs 35−43. Particularly,the conductance of straight wires of between three and sevencarbon atoms in contact with metal electrodes metal−C−metalwas calculated in refs 35 and 36, the metal electrodes beingdescribed using a semi-infinite uniform background model44

and the carbon atomic cores using a pseudopotential. It isshown that the conductance varies in an oscillatory manner asthe number of carbon atoms is increased; a localized charge atthe metal−wire contacts generates Schottky-like barriers, and abending of the wire reduces its conductance.35,36 There is areport on a first-principles analysis of the conductance andcurrent−voltage curves of the cumulenic 4−7 carbon atoms incontact with two Al electrodes represented by a slab of Alatoms.37 The low-bias current−voltage characteristic ispredicted to be linear, but a negative differential resistance is

observed at higher bias due to a shift of conduction channelsrelative to the states of the electrodes by the external biaspotential.37 A scattering-state approach is used to study thetransport properties of short cumulenic carbon chainscovalently connecting metallic carbon nanotube leads at thefinite bias.38 In this system, the negative differential resistancewas also found for both even and odd cumulenic chainjunctions. The ab initio results for the electron transportproperties of the cumulenic C8 and C9 chains bridging thezigzag edges of graphene electrodes were presented anddemonstrated a spin-polarization of the transmission due to aspin-splitting of the graphene flake bands.41 The conductanceof polyynic carbyne was theoretically studied in ref 42 only, inwhich the current−voltage curves of the short thiolate-cappedpolyynes in contact with gold electrodes such as Au−S−hexayne−S−Au are simulated, and polyynes appear as nearlyperfect molecular wires. Finally, a tight binding method for thePeierls instability and electronic transport in strongly coupledelectron−phonon systems is applied to a model of polyynechains biased through metallic Au leads and predicted that thechains should manifest an insulator−metal transition driven bythe nonequilibrium charging.43

The aim of this work is to calculate, using the tight-bindingmethods, the electrical conductance of polyynic and cumuleniccarbon chains containing several tens of the C atoms. Theoutline of this paper is as follows. In Section II, we give a briefintroduction to a ballistic mechanism of electron transport innanomaterials, and we estimate the values of the π-electronHamiltonian matrix elements and present ideas of calculatingthe transmission function τ(E,V) of the linear chains using atheory of difference schemes and in terms of the iterativetechnique. The τ(E,V) dependences, conductance, and currentof carbynes as a functions of their structure and length, of theFermi energy level of the electrode material, of the bias voltage,and of temperature are presented in Section III. This section ofthe paper ends with a comparison of this model with thealready published results of similar, although shorter, chains.The paper concludes with Section IV. The details of thecomputational procedures are given in the SupportingInformation.

■ METHOD OF CALCULATIONCarbyne in Electrical Circuit. Let us consider the

monatomic carbon chains in contact with the two electrodesconnected to a constant current source. For this system, wehave to describe the dependences of the current on the lengthof the carbon chain, the type of carbon bonds, applied voltageV, temperature T, and material properties of the electrodes. Asis known, the electron transport in nanomaterials is determinedby a ballistic mechanism according to which the electronstunnel through the nanosized object without scattering bylattice vibrations, impurities, or defects that limit the electricalconductivity of bulk materials.45,46 The ballistic transport ofcharge in the nanomaterials is due to the fact that conventionallength of the nanowire is typically much smaller than theelectron mean free path.To calculate current I, we use the well-known Landauer

formula47,48 for the ballistic transport based on a concept of theelectrodes as the macroscopic reservoirs of electrons and anindependent particle model. According to the Landauerformula, the temperature dependence of the current isdetermined by the Fermi distributions fM(E,T) of the electronsat the macroscopic electrodes M, and the transmission function

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4038864 | J. Phys. Chem. C 2013, 117, 16306−1631516307

τ(E,V) depends on the electronic properties of nanomaterialsand defines the transparency of the nanosized material forelectrons. In this work, the latter is calculated using quantumchemistry methods.Linear Combination of Π-Orbitals. The carbon chains

are the π-conjugated systems. Therefore, the calculations oftransmission functions can be most simply carried out withinthe framework of π-electron approximation of a linearcombination of atomic orbitals (LCAO) approach. Thistechnique is well-known as the simple Huckel method inmolecular quantum chemistry49−53 and was widely used in thetheory of electron structure of the fullerenes, nanotubes, andgraphene starting from the early papers.54−58 In the following,45

we model the macroscopic electrodes as the infinite unlimitedon one side monatomic chains with the pairs of degenerate πAOs on each atom M

− − − − − − − −

− − − −− − −

+ + +

... M M M C C ... C C

M M M ...N N

N N N

2 1 0 1 2 1

1 2 3

The electronic structure of these chains will again beevaluated in terms of the simple Huckel method. For transitionmetals, the pairs of basis functions are the dxz and dyz AOs.In this work, the matrix elements for the π-orbitals of the C

atoms were taken from the vertical ionization potentials ofphotoelectron spectra of the benzene molecule59,60 andcorrected49 taking into account the C−C bond differences ofthe carbyne and benzene. For the polyynic structure, thehopping integrals are t1 = −2.94 eV and t3 = −3.86 eV forcharacteristic single and triple bonds, respectively. In the case ofcumulenic carbyne, t2 = −3.40 eV. A Coulomb integralεC =−6.4 eV in all cases; it is characteristic of the atom and does notdepend on the C−C bond lengths.In nanoelectronics, one typically uses the noble metals like

gold or platinum as the electrodes materials. For such metals,the numerical value of the integrals εM and tM can be estimatedas follows. Note that the ionization potential of the metal atomsis less by about 20% than that of the C atom (Pt 8.9 eV, Au 9.2eV, C 11.2 eV). This suggests that the levels of metals shift inthe high-energy region relative to the levels of carbon andpoints out that εM > εC. Without specifying the electrodematerials, we take εM = 0.8εC as the Coulomb integral values ofmetals. For the hopping integrals of the metals, we assume tM =2tC,0; if |tM| ≤ |tC,0|, that is, the metal band widths are less thanor about the width of the carbyne bands, this may limit theconductivity of the carbon chain [see eqs 14 and 16 in theSupporting Information]. Finally, to account for the interactionbetween the terminal carbon and metal atoms, the hoppingintegral tM−C was taken as the arithmetic mean tM−C = 0.5(tM +tC).Transmission Function. For the calculations of trans-

mission function, one can use the stationary Schrodingerequation

+ |Ψ⟩ = |Ψ⟩H U z E[ ( )]0 (1)

Here, H0 is the Hamiltonian of the system at zero voltage V = 0.The potential energy U(z) = −|eW|z is conveniently expressedin terms of a constant electric field W = V/l in the spacebetween the electrodes, which is determined by the appliedvoltage V, electron charge e, and the distance l between theboundary atoms of the left and right electrodes

=| | ≤ −−| | − ≤ ≤−| | ≥

⎧⎨⎪

⎩⎪U z

e V z l

e Vz l l z l

e V z l

( )

/2 for /2

/ for /2 /2

/2 for /2 (2)

Our challenge is to find the solutions to this equation that meetthe specified value of the energy E, the superpositions ofincident and reflected waves in the region of electrode L, |ΨL⟩ =|Ψinc⟩ + |Ψref⟩, and describe the transmitted wave in the regionof electrode R, |ΨR⟩ = |Ψtrn⟩. Substituting the expansion in theatomic basis |Ψ⟩ = Σnψn|n⟩ in eq 1, we get a system of algebraicequations for the expansion coefficients ψn

ψ ε ψ ψ+ + − + =− − + +t U E t( ) 0n n n n n n n n n, 1 1 , 1 1 (3)

where Un = U(zn). The solutions of this system for the atoms ofthe electrodes can easily be found because the electrodes areinfinite chains, and one can use the Bloch’s theorem. Forcathodes, the expansion coefficients in the atomic basis set areas follows: ψn = ake

inkdM + bke−inkdM. Here the first term on the

right-hand side determines the amplitude of the incident wave,and the second one corresponds to the amplitude of thereflected wave in the region of the atom n. dM is a M−M bondlength, and k and K are the wave numbers in the regions of thecathode and anode. For the anode, the expansion coefficients ofthe transmitted wave in atomic basis are ψn = cKe

inKdM. Now, theproblem of calculating the reflection rk and transmission τkcoefficients is reduced to the determination of the relationshipbk/ak (or ck/ak): rk = (bk/ak)

2 and τk = 1 − (bk/ak)2.

In some cases, the problem of calculating the transmissioncoefficient can be solved analytically by applying the approachused in the theory of difference schemes.61 In this case, it ispossible to pass from the algebraic equations of the LCAOmethod to the differential equations and to take advantage oftheir solutions. Indeed, we assume that the difference eq 3represents an approximation of some differential equation,where the positions of the atoms in the carbon chain act as thenodes of the difference scheme. In the case of cumulenic andpolyynic forms of carbynes, the difference grids with constantand variable pitches are implemented. The calculations aregreatly simplified for the sufficiently long carbon chains, lowbias voltages, and the Fermi energy region of carbynes.To calculate the ratio bk/ak and hence the transmission

function for the given values of energy and voltage, we havealternatively used here an iterative technique45 applicable to thewider voltage and energy regions as compared to the finitedifference method. In the iteractive technique, one starts fromBloch’s solution for the ψN+1 and ψN+2 coefficients for the firstand second atoms of the anode. Substituting the ψN+1 and ψN+2values in the secular eq 3, one can now calculate the values ofψN for the rightmost of the carbon atoms in the chain. Next,using the values of ψN+1 and ψN one can determine the value ofψN−1 for the next C atom of the carbon chain. This procedurecan be continued, and the ψn values can be found for the Catoms of carbyne and the rightmost cathode atom.

■ RESULTS OF CALCULATIONTransmission Function. Figure 1 shows the transmission

function τ(E,V) for the energy region εC − 3 eV ≤ E ≤ εC + 3eV and bias voltages up to the 2 V obtained in the terms of thefinite difference approach. The calculation results indicate avery high transparency of cumulenic carbyne in this region. Forexample, for V = 0 and E = εC, the transmission coefficients τ =1.0 for all the cumulenic chains studied. There are some



oscillations in the energy dependence of τ(E,V) as one goes tothe high- and low-energy regions relative to the εC. Theoscillation frequency increases with the carbon chainelongation. The presence of electric voltage is accompaniedby a decrease of the transmission function over this range ofenergies, the reductions of τ at low energy with respect to theεC being stronger than in the region E > εC. For V = 2 V and E= εC, τ = 0.8 independently on the cumulene length.For cumulenic forms of the C40, C20, and C10 carbynes,

Figure 2 shows the electron energy levels of molecules and theenergy and voltage dependencies of transmission functioncalculated using the iterative procedure. One can see that eachelectronic level of the molecule corresponds to a peak in theenergy dependence of transmission coefficient with τ ≈ 1. Thepeaks are responsible for the resonant electron transfer betweenthe electrodes. The iterative calculations show that theoscillations of the τ(E,V) are enhanced by the transition tothe high- and low-energy regions relative to the E ≈ εC;

particularly, near the lower occupied and higher vacantelectronic levels of cumulenes, the τ(E,V) functions oscillatebetween the τ ≈ 1 and τ ≈ 0.1. A comparison of Figures 1 and2 shows that there is quantitative agreement with the resultsobtained using the finite difference method at the energies E ≈εC and voltages up to 2 V.A comparison of Figures 1−4 shows that the transmission

functions of the cumulenic and polyynic chains of similarcomposition differ dramatically. In the region of E ≈ εC, there isthe maximum transparency of cumulenic carbyne with τ ≈ 1(Figures 1 and 2), but the polyynic carbyne is virtually nottransparent for electrons (Figures 3 and 4). The reason for thesharp decrease of the τ near the E = εC is that there are noelectronic levels in the vicinity of εC of the polyynic chains. Forexample, the εC is located exactly in the center of the band gapof the infinite semiconducting polyynic chain, but there areconducting states at the E = εC in the analogous metalliccumulenic carbyne. For the energies |E − εC| > (t3 − t1), thetransmission function of polyynic carbyne grows up reachingthe values of τ ≈ 1 and oscillates similarly to the τ(E,V) of thecumulenic chains.In the iterative calculations of the polyynic carbynes (Figure

4), the τ(εC,0) = 0.00011, 0.026, and 0.33 for the C40, C20, and

Figure 1. Transmission coefficients of cumulenic carbynes calculatedusing the finite difference method for the energy region εC − 3 eV ≤ E≤ εC + 3 eV and voltages up to 2 V.

Figure 2. Transmission function of finite cumulenic carbynescalculated using the iterative procedure in comparison with MOs ofisolated carbon chains.



C10, respectively. The voltage-dependent variations of the τinitially are quite weak, but at the high voltages they lead to thesignificant decrease in the peak height of functions τ(E,V)resulting in the drastic reduction of transparency of the materialfor the electrons.Current and Conductance. Using the τ(E,V) depend-

ences, we can determine the conductance and current ofcarbynes as the functions of the Fermi energy level EF

M of theelectrodes material, of the bias voltage V, and temperature T.For T = 0 K, the Fermi distribution of the electrode electrons isa step function, and the Landauer formula immediately leads tothe following expression for the quantum conductance acrossthe carbyne G(T = 0,V = 0) = 2G0τ(EF

M,0), where the G0 = 2e2/h = (12.906 kOhm)−1 is a conductance quantum; the factors oftwo are due to the double orbital degeneracy of the carbyne πMOs and the two possible directions of electron spin. In thecase of the zero temperature and voltage, if the Fermi levelposition EF

M of the electrode material coincides with theposition of maximum in the curve τ(E,0) with τ = 1, theconductance of the carbon chain reaches the value of 2G0, thatis, the theoretical limit of conductance of the molecular wirewith double orbital degeneracy.If one knows the transmission function τ(E,V) and specifies

the position of the Fermi level EFM of electrodes relative to the

electronic levels of carbyne (e.g., with respect to the εC), onecan calculate the current−voltage (I−V) characteristics of thecircuit with monatomic carbon wire using the Landauer

equation. For T = 0 K, Figures 5 and 6 show the calculateddependences of the current of the cumulenic and polyynicchains on the bias voltage V and the position δE = EF

M − εC ofthe electrode Fermi level EF

M relative to the Fermi level ofcarbynes εC. In these calculations, we applied the transmissionfunctions found using the iterative procedure (Figures 2 and 4),which allows us to define these functions in a wide range ofenergies and voltages. One can see that in all the cumulenicchains the current initially increases and then decreases withincreasing V. The growth of the current at voltages up to 8 Vreflects an increase in the number of conducting states in theregion between the EF

M + V/2 and EFM − V/2, and a further

decline of current is due to the rapid decrease of the τ(E,V) forV > 8 V. In all the cumulenic chains, the higher current valuesare observed if EF

M = εC, that is, at the coincidence of the Fermienergies of the electrode material and carbynes. The up ordown shifts of the EF

M relative to εC reduce the current. Allother conditions being equal, a shortening of cumulenic chainleads to a small increase in the current, but the maximumtunneling currents are observed at δE = 0 and V ≈ 8 V and areequal to about 1 mA almost regardless of the cumulenic chainlength. One important feature of these I−V characteristics isthat the current initially increases with growth of the bias

Figure 3. Transmission function for polyynic chains calculated usingthe finite difference method.

Figure 4. Transmission function of finite polyynic carbynes calculatedusing the iterative procedure in comparison with MOs of isolatedcarbon chains.



Figure 5. Zero-temperature bias voltage dependences of current of thecumulenic C40, C20, and C10 carbynes in contact with electrodes fordifferent relative positions of Fermi levels of carbyne and electrodematerial δE = EF

M − εC. For example, the line marked as +3 eVcorresponds to δE = 3 eV.

Figure 6. Zero-temperature bias voltage dependences of current of thepolyynic C40, C20, and C10 carbynes in contact with electrodes fordifferent relative positions of Fermi levels of carbyne and electrodematerial δE = EF

M − εC. For example, the line marked as +3 eVcorresponds to δE = 3 eV.



voltage, reaches a peak, and then drops to give rise to thenegative conductance (dI/dV) < 0 clearly seen in Figure 7.Moreover, there are some oscillations of voltage dependencesof conductance due to the discrete nature of the C chainelectron energy levels.

A comparison of Figures 1−6 shows that the more complextransmission function of the polyynic carbynes as compared tothe cumulenic ones complicates the current−voltage character-istics of polyynes. On the typical bias voltage dependences ofcurrent and conductance curve of the polyynic chains (Figures5−8), there are both the peak and local minima (valley) athigher voltages. For example, in the case of the polyynic C40chain, the maximum current is observed when the Fermi levelof the electrode material shifts by 3−4 eV up or down on theenergy axis relative to the εC. The peak current is observed at avoltage of 4 V, while near 8 V there is a local minimum of thecurrent, which is replaced by an increase in current at highervoltages. A peak to valley ratio is equal to 3 in this case.Particularly, the change of the conductivity sign and presence ofvalleys (Figure 8) point to the possibilities of design of theresonance tunneling devices based on the carbon chains.As expected, the increase in temperature is accompanied by

some smoothing of the peaks in the curves of the conductance.Figure 9 illustrates this effect on the example of C40 chains.Short Carbon Chains. In the absence of experimental data

on the electric properties of carbon chains, it is useful to

compare the π electron model with the already published abinitio results on conductance of similar although shorter chainshaving only several C atoms. Theoretically, several groups havemade predictions on the conductance of carbon atomic wires

Figure 7. Bias voltage dependences of conductance of the cumulenicchains.

Figure 8. Bias voltage dependences of conductance of the polyynicchains.

Figure 9. Temperature dependences of conductance of cumulenic(above) and polyynic (below) C40 carbynes.



connected between the different model electrodes. Some of theresults are reproduced regardless of the choice of the model ofthe electrodes and can be considered as the actual character-istics of the nanowire. Such properties will be the focus of ourcomparison.First of all in papers 35 and 36 the conductance of straight

cumulenic wires CN with 3 ≤ N ≤ 7 in contact with metalelectrodes was calculated, the metal electrodes being describedusing a semi-infinite uniform background (jellium) model andthe carbon atomic cores in the terms of a pseudopotential. Forzero temperature and bias voltage, it was shown in these worksthat the conductance of cumulenic chains varies in anoscillatory manner as the number N of carbon atoms increasesif the Fermi levels of the carbon chain and electrodes coincide,the odd-numbered chains having a lower resistance than theeven-numbered chains. Using a density functional theory first-principles scattering-state approach, such odd−even propertieswere later discovered in work with cumulenic 4−7 carbonatoms and the two Al electrodes represented by a slab of Alatoms37 and in the studies of the transport properties of carbonchains covalently connecting metallic carbon nanotube leads38

and graphene electrodes.39

Figure 10 shows that this oscillatory character of theconductance is also observed in the π electron model with

infinite monatomic model electrodes and can be explained interms of the π electron level scheme of the free chains. Really,there is one nonbonding level located exactly at the Fermi levelof cumulenes in the case of odd-numbered chains (Figure 11)that provides the higher value of transmission coefficient for theenergy E = εC in this case. Note that inclusion of bias voltageshifts the nonbonding level relative to the εC and destroys theoscillatory dependence of the conductance.Next, in the work 37, a detailed first-principles analysis of the

I−V characteristics of the Al−CN−Al with 4 ≤ N ≤ 7 systemwas reported. The low-bias current−voltage characteristic ispredicted to be linear, but a negative differential resistance isobserved at higher bias due to a shift of conduction channelsrelative to the states of the electrodes by the external biaspotential.37 The similar I−V characteristics with negativedifferential resistance at finite bias were reported for botheven and odd short cumulenic chains covalently connectingmetallic carbon nanotube leads.38 The regions of the linear I−Vcharacteristics and of the negative differential resistance areclearly seen in Figures 12 and 13 too. Note that the negativedifferential resistance was observed in carbon nanotube

junctions just before breaking and hypothesized to arise fromthe formation of monatomic carbon wires in the junction.40

Finally, all theorists conclude that linear cumulenic atomiccarbon wires should have good conductance. The conductancedoes not decrease with growth of chain length, and its low-biasvalues fall in the approximate range of G0 to 2G0. Obviously,some fine features of transport properties should depend on thedetailed structure of the electrode.In summary, we can conclude that there is resonable

agreement between the π-electron model and the first-principles electronic conductance calculations.

Figure 10. Conductance of the atomic wires in units of theconductance quantum as a function of the number of carbon atomsin the cumulenic wire.

Figure 11. πMOs and transmission coefficients of cumulenic carbynesC3−C9 calculated using the iterative technique.



■ CONCLUSION

In this work, the ballistic electrical conductance of cumulenicand polyynic carbon chains C40, C20, and C10 is calculated usingthe π-electron tight-binding methods. For small bias voltages,the problem of calculating the transmission function in thecumulenic and polyynic chains is solved analytically by applyingthe difference schemes approach. In the case of large voltages,the iterative technique is a more effective approach. It is shownthat the transmission functions τ(E,V) of the cumulenic andpolyynic chains differ dramatically. The energy regions of veryhigh and low transparency of carbynes for electrons areobtained, and oscillatory character of the energy dependence ofτ(E,V) is pointed out. Each electronic level of the moleculecorresponds to the resonant electron transfer. The voltage-dependent variations of the τ(E,V) are initially quite weak, butat the higher voltages their effect is a drastic reduction of thecarbyne transparency. The current initially increases withgrowth of the bias voltage, reaches a peak, and then drops togive rise to the regions of negative differential resistance.

■ ASSOCIATED CONTENT

*S Supporting InformationThe details of the computational procedures. This material isavailable free of charge via the Internet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected]. Fax: +74959541279.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The work was performed in frames of Russian Scientific andResearch Program ″Scientific and pedagogic personal forinnovations in Russia in 2009-2013 years″. The research waspartially supported by the Russian Basic Research Foundation(grant 11-03-00691) and EU under Project No. FP7-247007CACOMEL.

■ REFERENCES(1) Tans, S. J.; M.Verschueren, A. R.; Dekker, C. Room TemperatureTransistor Based on a Single Carbon Nanotubes. Nature 1998, 393,49−51.(2) Postma, H. W. C.; Teepen, T. Z.; Yao, M. G.; Dekker, C. CarbonNanotube Single-Electron Transistors at Room Temperature. Science2001, 293, 76−79.(3) Kreupl, F. Electronics: Carbon Nanotubes Finally Deliver. Nature2012, 484, 321−322.(4) Franklin, A. D.; Chen, Z. Length Scaling of Carbon NanotubeTransistors. Nat. Nanotechnol. 2010, 5, 858−862.(5) Schwierz, F. Graphene Transistors. Nat. Nanotechnol. 2010, 5,487−496.(6) Kitamura, M.; Kuzumoto, Y.; Kamura, M.; Aomori, S.; Arakawa,Y. High-Performance Fullerene C60 Thin-Film Transistors Operatingat Low Voltages. Appl. Phys. Lett. 2007, 91, 183514.(7) Ullah, M.; Singh, T.; Sitter, H.; Sariciftci, N. Meyer-Neldel Rulein Fullerene Field-Effect Transistors. Appl. Phys. A: Mater. Sci. Process.2009, 97, 521−526.(8) Kushmerick, J. Nanotechnology: Molecular Transistors Scruti-nized. Nature 2009, 462, 994−995.(9) Hwang, J.; Pototschnig, M.; Lettow, R.; Zumofen, G.; Renn, A.;Gotzinger, S.; Sandoghdar, V. A Single-Molecule Optical Transistor.Nature 2009, 460, 76−80.(10) Song, H.; Kim, Y.; Jang, Y.; Jeong, H.; Reed, M.; Lee, T.Observation of Molecular Orbital Gating. Nature 2009, 462, 1039−1043.(11) Sladkov, A.; Kudryavtsev, Y. Diamond, Graphite, Carbyne theAllotropic Forms of the Carbon. Priroda 1969, 58, 37−44.(12) Kudryavtsev, Y. P.; Evsyukov, S. E.; Guseva, M. B.; Babaev, V.G.; Khvostov, V. V. Carbyne − the Third Allotropic Form of Carbon.Izv. Akad. Nauk., Ser. Khim. 1993, 3, 399−413.(13) Korshak, V. V.; Kudryavtsev, Y. P.; Khvostov, V. V.; Guseva, M.B.; Babaev, V. G.; Rylova, O. Y. Electronic Structure of CarbynesStudied by Auger and Electron Energy Loss Spectroscopy. Carbon1987, 23, 735−738.(14) Kudryavtsev, Y.; Baytinger, E.; Kugeev, F.; Korshak, Y.;Evsyukov, S. Electronic Structure of Carbyne Studied by X-RayPhotoelectron Spectroscopy and X-Ray Emission Spectroscopy. J.Electron Spectrosc. Relat. Phenom. 1990, 50, 295−307.(15) Kudryavtsev, Y.; Heimann, R.; Evsyukov, S. Carbynes: Advancesin the Field of Linear Carbon Chain Compounds. J. Mater. Sci. 1996,31, 5557−5571.(16) Whittaker, A. Carbon: a New View of its High-TemperatureBehavior. Science 1978, 200, 763−764.(17) Smith, P.; Buseck, P. Carbyne Forms of Carbon: Do They Exist?Science 1982, 216, 984−986.(18) Heimann, R.; Kleiman, J.; Salansky, N. A Unified StructuralApproach to Linear Carbon Polytypes. Nature 1984, 306, 164−167.(19) Whittaker, A. Carbyne Forms of Carbon: Evidences for TheirExistence. Science 1985, 229, 486−487.(20) Hayatsu, R.; Scott, R. G.; Studier, M. H.; Lewis, R. S.; Anders, E.Carbynes in Meteorites: Detection, Low-Temperature Origin, andImplications for Interstellar Molecules. Science 1980, 209, 1515−1518.(21) Kroto, H. W.; Heath, J. R.; Obrien, S. C.; Curl, R. F.; Smalley, R.E. Long Carbon Chain Molecules in Circumstellar Shells. Astrophys. J.1987, 314, 352−355.(22) Smith, P.; Buseck, P. Carbyne Forms Carbon: Evidence of TheirExistence. Science 1985, 229, 486−487.

Figure 12. I−V curves for cumulenic carbynes C3−C9 for equilibriumconditions.

Figure 13. Conductance of cumulenic carbynes C3−C9 for equilibriumconditions.



http://pubs.acs.org

mailto:[email protected]

(23) Li, S.-Y.; Zhou, H.-H.; Gu, J.-L.; Zhu, J. Does Carbyne ReallyExist? − Carbynes in Expanded Graphite. Carbon 2000, 38, 929−941.(24) Zhao, X.; Ando, Y.; Liu, Y.; Jinno, M.; Suzuki, T. CarbonNanowire Made of a Long Linear Carbon Chain Inserted Inside aMultiwalled Carbon Nanotube. Phys. Rev. Lett. 2003, 90, 187401.(25) Jin, C. H.; Lan, H. P.; Peng, L. M.; Suenaga, K.; Iijima, S.Deriving Carbon Atomic Chains from Graphene. Phys. Rev. Lett. 2009,102, 205501.(26) Eisler, S.; Slepkov, A. D.; Elliott, E.; Luu, T.; McDonald, R.;Hegmann, F.; Tykwinski, R. Polyynes as a Model for Carbyne:Synthesis, Physical Properties, and Nonlinear Optical Response. J. Am.Chem. Soc. 2005, 127, 2666−2676.(27) Chalifoux, W.; Tykwinski, R. Synthesis of Polyynes to Model thesp-Carbon Allotrope Carbyne. Nature Chem. 2010, 2, 967−971.(28) Haley, M. M. On the Road to Carbyne. Nature Chem. 2010, 2,912−913.(29) Hino, S.; Okada, Y.; Iwasaki, K.; Kijima, M.; Shirakawa, H.Electronic Structures of Cumulene Type Carbyne Model Compounds:a Typical Example of One-Dimensional Quantum Well. Chem. Phys.Lett. 2003, 372, 59−65.(30) Berkovitch-Yellin, Z.; Leiserowitz, L. Electron DensityDistribution in Cumulenes: an X-Ray Study of Tetraphenylbutatrieneat 20oC and −160oC. Acta Crystallogr. 1977, 33, 3657−3669.(31) Varfolomeeva, T. D.; Popova, S. V.; Lyapin, A. G.; Brazhkin, V.V.; Lyapin, S. G.; Kudryavtsev, Y. P.; Evsyukov, S. E. StructuralTransformations of the Cumulene Form of Amorphous Carbyne atHigh Pressure. JETP Lett. 1997, 66, 255−260.(32) Kudryavtsev, Y. P.; Guseva, M. B.; Babaev, V. G. OrientedCarbyne Layers. Carbon 1992, 30, 213−221.(33) Prazdnikov, Y. E.; Bozhko, A. D.; Novikov, N. D. EnergeticBarrier Reduction at the Carbyne Film Interface. J. Russ. Laser Res.2005, 26, 55−65.(34) Prazdnikov, Y. E. Prospects of Carbyne Applications inMicroelectronics. J. Mod. Phys. 2011, 2, 845−848.(35) Lang, N. D.; Avouris, P. Oscillatory Conductance of Carbon-Atom Wires. Phys. Rev. Lett. 1998, 81, 3515−3518.(36) Lang, N. D.; Avouris, P. Carbon-Atom Wires: Charge-TransferDoping, Voltage Drop, and the Effect of Distortions. Phys. Rev. Lett.2000, 84, 358−361.(37) Larade, B.; Taylor, J.; Mehrez, H.; Guo, H. Carbon-Atom Wires:Charge-Transfer Doping, Voltage Drop, and the Effect of Distortions.Phys. Rev. B 2001, 64, 075420.(38) Khoo, K. H.; Neaton, J. B.; Son, Y. W.; Cohen, M. L.; Louie, S.G. Negative Differential Resistance in Carbon Atomic Wire-CarbonNanotube Junctions. Nano Lett. 2008, 8, 2900−2905.(39) Shen, L.; Zeng, M.; Yang, S.-W.; Zhang, C.; Wang, X.; Feng, Y.Electron Transport Properties of Atomic Carbon Nanowires betweenGraphene Electrodes. J. Am. Chem. Soc. 2010, 132, 11481−11486.(40) Yuzvinsky, T. D.; Mickelson, W.; Aloni, S.; Begtrup, G. E.; Kis,A.; Zettl, A. Shrinking a Carbon Nanotube. Nano Lett. 2006, 6, 2718−2722.(41) Furst, J. A.; Brandbyge, M.; Jauho, A.-P. Atomic Carbon Chainsas Spin-Transmitters: An Ab Initio Transport Study. EPL 2010, 91,37002.(42) Crljen, Z.; Baranovic, G. Unusual Conductance of Polyyne-Based Molecular Wires. Phys. Rev. Lett. 2007, 98, 116801.(43) Magna, A. L.; Deretzis, I.; Privitera, V. Insulator-MetalTransition in Biased Finite Polyyne Systems. Eur. Phys. J. B 2009,70, 313−316.(44) Lang, N. D. Resistance of Atomic Wires. Phys. Rev. B 1995, 52,5335−5342.(45) Chen, A.-B. In Handbook of Theoretical and ComputationalNanotechnology; Rieth, M., Schommers, W., Eds.; American ScientificPublishers: California, 2006; Vol. 1, Chap. Basic Theory of ElectronTunneling and Ballistic Transport in Nanostructures, pp 311−377.(46) Nazarov, Y. V.; Blanter, Y. M. Quantum Transport: Introductionto Nanoscience; Cambridge University Press: New York, 2009; p 591.(47) Landauer, R. Electrical Resistance of Disordered One-Dimen-sional Lattices. Phil. Mag. 1972, 21, 863−867.

(48) Buttiker, M. Scattering Theory of Current and Intensity NoiseCorrelations in Conductors and Wave Guides. Phys. Rev. B 1992, 46,12485−12507.(49) Streitwieser, A. Molecular Orbital Theory for Organic Chemists;Wiley: New York, 1961; p 489.(50) Huckel, E. Theory of Free Radicals of Organic Chemistry.Trans. Faraday Soc. 1934, 30, 40−52.(51) Yates, K. Hu ckel Molecular Orbital Theory; Academic Press Inc.:New York, 1978.(52) Dewar, M. J. S. The Molecular Orbital Theory of OrganicChemistry; McGraw-Hill: NY, 1969; p 484.(53) Dewar, M. J. S.; Dougherty, R. C. The PMO Theory of OrganicChemistry; Springer: New York, 1975.(54) Bochvar, D.; Gal’pern, E. Hypothetical Systems: Carbododeca-hedron, s-Icosahedron, and Carbo-s-Icosahedron. Dokl. Akad. NaukSSSR 1973, 209, 610−612.(55) Saito, R.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S.Electronic Structure of Chiral Graphene Tubules. Appl. Phys. Lett.1992, 60, 2204−2206.(56) Dunlap, B. I. Connecting Carbon Tubules. Phys. Rev. B 1992,46, 1933−1936.(57) Hamada, N.; Sawada, S. I.; Oshiyama, A. New One-DimensionalConductors: Graphitic Microtubules. Phys. Rev. Lett. 1992, 68, 1579−1581.(58) Slonczewski, J. C.; Weiss, P. R. Band Structure of Graphite. Phys.Rev. 1952, 109, 272−279.(59) Fung, K. H.; Henke, W. E.; Hays, T. R.; Selzle, H. L.; Schlag, E.W. Ionization Potential of the Benzene-Argon Complex in a Jet. J.Phys. Chem. 1981, 85, 3560−3563.(60) von Niessen, W.; Cederbaum, L. S.; Kraemer, W. P. TheElectronic Structure of Molecules by a Many-Body Approach. I.Ionization Potentials and One-Electron Properties of Benzene. J.Chem. Phys. 1976, 65, 1378−1386.(61) Samarskii, A. A. The Theory of Difference Schemes; CRC Press:Boca Raton, 2001.



Documents

Tight Binding Model of Quantum Conductance of Cumulenic ...paveldyachkov.ru/images/dyza.pdfTight Binding Model of Quantum Conductance of Cumulenic and Polyynic Carbynes P. N. D’yachkov,*,†