SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Highly Accessed Nano Express

Quantum conductance of silicon-doped carbon wire nanojunctions

Dominik Szczȩśniak12*, Antoine Khater1, Zygmunt Ba̧k2, Radosław Szczȩśniak3 and Michel Abou Ghantous4

Author affiliations

1 Institute for Molecules and Materials UMR 6283, University of Maine, Ave. Olivier Messiaen, Le Mans, 72085, France

2 Institute of Physics, Jan Długosz University in Czȩstochowa, Al. Armii Krajowej 13/15, Czȩstochowa, 42200, Poland

3 Institute of Physics, Czȩstochowa University of Technology, Al. Armii Krajowej 19, Czȩstochowa, 42200, Poland

4 Department of Physics, Texas A&M University, Education City, PO Box 23874, Doha, Qatar

For all author emails, please log on.

Citation and License

Nanoscale Research Letters 2012, 7:616  doi:10.1186/1556-276X-7-616

The electronic version of this article is the complete one and can be found online at: http://www.nanoscalereslett.com/content/7/1/616


Received:28 July 2012
Accepted:11 October 2012
Published:7 November 2012

© 2012 Szczesniak et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Unknown quantum electronic conductance across nanojunctions made of silicon-doped carbon wires between carbon leads is investigated. This is done by an appropriate generalization of the phase field matching theory for the multi-scattering processes of electronic excitations at the nanojunction and the use of the tight-binding method. Our calculations of the electronic band structures for carbon, silicon, and diatomic silicon carbide are matched with the available corresponding density functional theory results to optimize the required tight-binding parameters. Silicon and carbon atoms are treated on the same footing by characterizing each with their corresponding orbitals. Several types of nanojunctions are analyzed to sample their behavior under different atomic configurations. We calculate for each nanojunction the individual contributions to the quantum conductance for the propagating σ, Π, and σelectron incidents from the carbon leads. The calculated results show a number of remarkable features, which include the influence of the ordered periodic configurations of silicon-carbon pairs and the suppression of quantum conductance due to minimum substitutional disorder and artificially organized symmetry on these nanojunctions. Our results also demonstrate that the phase field matching theory is an efficient tool to treat the quantum conductance of complex molecular nanojunctions.

Keywords:
Nanoelectronics; Quantum wires; Electronic transport; Finite-difference methods; 85.35.-p; 73.63.Nm; 31.15.xf

Background

Quantitative analysis of electronic quantum transport in nanostructures is essential for the development of nanoelectronic devices [1]. The monatomic linear carbon wire (MLCW) systems are expected in this context to have potentially interesting technological applications, in particular as connecting junction elements between larger device components [2]. In this respect, electronic quantum transport properties are the key features of such wire nanojunctions [3].

Carbon exists in nature under a wide range of allotropic forms as the two-dimensional graphene [4], the cage fullerenes [5], and the quasi one-dimensional carbon nanotubes [6]. These forms exhibit exceptional physical properties and can be considered as promising components for future nanodevices [7]. The discovery of MLCW, [8-14] turns the attention to another intriguing carbon allotropic form. In the experiment conducted recently by Jin et al. [14], MLCW was produced by directly removing carbon atoms row by row from the graphene sheets, leading to a relatively stable freestanding nanostructure.

At present, the available experimental data do not provide essential knowledge about the electronic properties of MLCW systems, and only theoretical studies shed some light on these properties. Furthermore, although the MLCW systems were investigated for a long time from the theoretical point of view [15-26], their interest was not highlighted until recently due to the open attention paid to other carbon allotropic forms. It has been shown in particular that from the structural point of view, MLCW can form either as cumulene wires (interatomic double bonds) or polyyne wires (alternating interatomic single and triple bonds) [14,17,19,27,28]. However, there is no straightforward answer as to which of these two structures is the favorable one; experimental studies do not give a satisfactory answer, and theoretical calculations yield provisions which depend on applied computational methods. Density functional theory (DFT) calculations predict double-bond structures [29,30], whereas ab initio Hartree-Fock (HF) results favor alternating bond systems [15-18,27]. This situation arises from the fact that DFT tends to underestimate bond alternation (second-order Jahn-Teller effect), while HF overestimates it [27].

More recently, first-principle calculations have indicated [31] that both structures are stable and present mechanical characteristics of a purely one-dimensional nanomaterial. Moreover, on the basis of the first-principle calculations [31-42], the cumulene MLCW wires are expected to be almost perfect conductors, even better than linear gold wires [29], while the corresponding polyyne wires are semiconducting [41]. It is also worth noting that the MLCW cumulene system may exhibit conductance oscillations with the even and odd numbers of the wire atoms [28,42].

In the present work, we consider in particular the problem of the electronic quantum transport across molecular nanojunctions made up of silicon-doped carbon wires, prepared in ordered or substitutionally disordered configurations as in the schematic representation of Figure 1, where the nanojunctions are between pure MLCW wire leads. This problem has not been considered previously and is still unsolved to our knowledge. The interest in the quantum transport of such nanojunctions arises from the fact that chemical defects or substitutional disorder may have a significant impact on their transport properties [43]. Chemical impurities doping the nanojunction may even allow the control of the transport for such nanostructures [44]. The properties of the nanoelectronic device and its functionality may hence be greatly affected or even built on such ordered and disordered configurations. The interest in silicon carbide, furthermore, stems from the fact that it is considered a good substrate material for the growth of graphene [45] and may produce interesting effects in its interactions with Si or C [46].

thumbnailFigure 1. Schematic representation of finite silicon-doped carbon wire nanojunction between two semi-infinite quasi one-dimensional carbon leads. The irreducible region and matching domains are distinguished (please see subsection ‘Phase field matching theory’ in the ‘Methods’ section for more details). The binding energies for a given atomic site and the coupling terms between neighbor atoms with corresponding interatomic distances are depicted. The n and nindices for the coupling parameters are dropped for simplicity.

The electrons which contribute to transport present characteristic wavelengths comparable to the size of molecular nanojunctions, leading to quantum coherent effects. The transport properties of a given nanojunction are then described in terms of the Landauer-Büttiker theory [47,48], which relates transmission scattering to quantum conductance. Several approaches have been developed in order to calculate the scattering transmission and reflection cross sections in nanostructures, where the most popular are based on first-principle calculations [49,50] and semiempirical methods using the non-equilibrium Green’s function formalism [51,52].

In the present work, we investigate the electronic scattering processes on the basis of phase field matching theory (PFMT) [53,54], originally developed for the scattering of phonons and magnons in nanostructures [55-59]. Our theoretical method is based on appropriate phase matching of the Bloch states of ideal leads to the local states in the scattering region. In this approach, the electronic properties of the system are described in the framework of the tight-binding formalism (TB) which is widely exploited for electronic transport calculations [54,60-63] and for simulating the STM images of nanostructures [64,65]. In particular, we employ the appropriate Slater-Koster [66] type Hamiltonian parameters calculated on the basis of the Harrison’s tight-binding theory (HTBT) [67]. The PFMT method, which is formally equivalent to the method of non-equilibrium Green’s functions [68], can be considered consequently as a transparent and efficient mathematical tool for the calculation of the electronic quantum transport properties for a wide range of molecular-sized nanojunction systems.

The present paper is organized in the following manner. In the ‘Methods’ section, we give the detailed discussion of theoretical PFMT formalism. Our numerical results, which incorporate propagating and evanescent electronic states, are presented per individual lead modes in the ‘Results and discussion’ section. Also presented are the total conductance spectra; they are compared with results based on first-principle calculations when available. Finally, the discussion and conclusions are given in the ‘Conclusions’ section. Appropriate appendices which supplement the theoretical model are also presented.

Methods

Theoretical model and propagating states

The schematic representation of the system under study with an arbitrary nanojunction region is presented in Figure 1. With reference to the Landauer-Büttiker theory for the analysis of the electronic scattering processes [47,48], this system is divided into three main parts, namely the finite silicon-doped carbon wire nanojunction region, made up of a given composition of carbon (black) and silicon (orange) atoms, and two other regions to the right and left of the nanojunction which are semi-infinite quasi one-dimensional carbon leads. Moreover, for the purpose of quantum conductance calculations, the so-called irreducible region and the matching domains are depicted (see the ‘Phase field matching theory’ subsection for more details). Figure 1 is used throughout the ‘Methods’ section as a graphical reference for analytical discussion.

The system presented in Figure 1 is described by the general tight-binding Hamiltonian block matrix:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M1">View MathML</a>

(1)

This is defined in general for a system of Nxinequivalent atoms per unit cell, where Nldenotes the number of basis orbitals per atomic site, assuming spin degeneracy. In Equation 1, Ei,j denotes on-diagonal matrices composed of both diagonal <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M2">View MathML</a> and off-diagonal <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M3">View MathML</a> elements for a selected unit cell. In contrast, the Hi,jmatrices contain only off-diagonal elements for interactions between different unit cells. The index αidentifies the atom type, C or Si, on the nth site in a unit cell. Each diagonal element is characterized by the lower index l for the angular momentum state. The off-diagonal elements <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M4">View MathML</a> describe the m-type bond, (m=σ,Π), between l and l nearest-neighbor states. The index βidentifies the types of interacting neighbors, C-C, Si-Si, or Si-C.

The <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M5">View MathML</a> elements are consistent with the Slater-Koster convention [66] and may be expressed in the framework of the HTBT [67] by the following:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M6">View MathML</a>

(2)

where <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M7">View MathML</a> values are the dimensionless Harrison coefficients; me, the electron mass in vacuum; and dβ, the interatomic distance for interacting neighbors. Explicit forms of the Ei,j and Hi,j matrices are given in Appendix Appendix 1. The tight-binding parameter schemes are illustrated in Figure 1; however, it is noteworthy that the n and nindices for coupling parameters are dropped for simplicity in this figure.

In our calculations, the single-particle electronic wave functions are expanded in the orthonormal basis of local atomic wave functions ϕl(r) as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M8">View MathML</a>

(3)

In Equation 3, k is the real wave vector; RN, the position vector of the selected unit cell; and RN, the position vector of the nth atom in the selected unit cell. For ideal leads, the wave function coefficients cl(rnRN,k) are characterized under the Bloch-Floquet theorem in consecutive unit cells by the following phase relation:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M9">View MathML</a>

(4)

where z is the phase factor

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M10">View MathML</a>

(5)

which corresponds here to waves propagating to the right (+) or to the left (−).

The electronic equations of motion for a leads unit cell, independent of N, may be expressed in a square matrix form, with an orthonormal minimal basis set of local wave functions as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M11">View MathML</a>

(6)

E stands for the electron eigenvalues, and I is the identity matrix, while the dynamical matrix Mdcontains the Hamiltonian matrix elements and the z phase factors; c(k,E) is the Nx×Nl size vector defined as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M12">View MathML</a>

(7)

Equation 6 gives the Nx×Nleigenvalues with corresponding eigenvectors which determine the electronic structure of the lead system, where l under the vector clcorresponds to Nl=4 orbitals s,px,py,pz. Note that the choice of an orthonormal minimal basis set of local wavefunctions may result initially in an inadequate description of the considered electronic eigenvalues. However, as can be seen later, the proper choice of the TB on-site energies and coupling terms allows us to to obtain agreement with the DFT results. This is a systematic procedure in our calculations.

Evanescent states

The complete description of electronic states on the ideal leads requires a full understanding of the propagating and evanescent electronic states on the leads. This arises because the silicon-doped nanojunction breaks the perfect periodicity of the infinite leads and forbids a formulation of the problem only in terms of the pure Bloch states as given in Equation 5. Depending on the complexity of a given electronic state, it follows that the evanescent waves may be defined by the phase factors for a purely imaginary wave vectors k=iκ such that

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M13">View MathML</a>

(8)

or for complex wave vectors k=κ1 + iκ2such that

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M14">View MathML</a>

(9)

The phase factors of Equations 8 and 9 correspond to pairs of hermitian evanescent and divergent solutions on the leads. Only the evanescent states are physically considered where spatial evanescence occurs to the right and left, away from the nanojunction localized states. It is important to note that the l-type evanescent state corresponds to energies beyond the propagating band structure for this state.

The functional behavior of z(E) for the propagating and evanescent states on the leads may be obtained by various techniques. An elegant method presented previously for phonon and magnon excitations [59] is adapted here for the electrons. It is described on the basis of Equations 4 and 6 by the generalized eigenvalue problem for z:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M15">View MathML</a>

(10)

Equation 10 gives the 2NxNleigenvalues as an ensemble of NxNlpairs of z and z−1. Only solutions with |z|=1 (propagating waves) and |z|<1 (evanescent waves) are retained as physical ones. In Equation 10, kis then replaced by the appropriate energy E variable. Furthermore, for systems with more than one atom per unit cell, the matrices HN,N−1 and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M16">View MathML</a> in this procedure are singular. In order to obtain the physical solutions, the eigenvalue problem of Equation 10 is reduced from the 2NxNl size problem to the appropriate 2Nlone, using the partitioning technique (please see Appendix Appendix 2).

Phase field matching theory

The scattering problem at the nanojunction is considered next. An electron incident along the leads has a given energy E and wave vector k, where E=Eγ(k) denotes the available dispersion curves for γ = 1, 2,.., γ propagating eigenmodes, where γcorresponds to the total number of allowed solutions for the eigenvalue problem of phase factors in Equation 10. In any given energy interval, however, these may be evanescent or propagating eigenmodes and together constitute a complete set of available channels necessary for the scattering analysis.

The irreducible domain of atomic sites for the scattering problem includes the nanojunction domain itself, (N∈[0,D−1]), and the atomic sites on the left and right leads which interact with the nanojunction, as in Figure 1. This constitutes a necessary and sufficient region for our considerations, i.e., any supplementary atoms from the leads included in the calculations do not change the final results. The scattering at the boundary yields then the coherent reflected and transmitted fields, and in order to calculate these, we establish the system of equations of motion for the atomic sites (N∈[−1,D]) of the irreducible nanojunction domain.

This procedure leads to the following general matrix equation:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M17">View MathML</a>

(11)

Mnano is a (D + 2)×(D + 4) matrix composed of the block matrices <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M18">View MathML</a>, and the state vector Vof dimension D + 4 is given as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M19">View MathML</a>

(12)

Since the number of unknown coefficients in Equation 11 is always greater than the number of equations, such a set of equations cannot be solved directly.

Assuming that the incoming electron wave propagates from left to right in the eigenmode γ over the interval of energies E=Eγ, the field coefficients on the left and right sides of the irreducible nanojunction domain may be written as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M20">View MathML</a>

(13)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M21">View MathML</a>

(14)

where γΓis an arbitrary channel into which the incident electron wave scatters, and cl(rn,zγ,Eγ) denotes the the eigenvector of the lead dynamical matrix of Equation 6 for the inequivalent site n at zγ and Eγ. The terms <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M22">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M23">View MathML</a> denote the scattering amplitudes for backscattering and transmission, respectively, from the γ into the γeigenmodes and constitute the basis of the Hilbert space which describes the reflection and transmission processes.

Equations 13 and 14 are next used to transform the (D + 2)×(D + 4) matrix of the system of equations of motion, Equation 11, into an inhomogeneous (D + 2)×(D + 2) matrix for the scattering problem. This procedure leads to the new form of the following vector:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M24">View MathML</a>

(15)

The rectangular sparse matrix in Equation 15 has the (D + 4)×(D + 2) size. The vectors <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M25">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M26">View MathML</a> are column vectors of the backscattering and transmission Hilbert basis.

Substituting Equation 15 into Equation 11 yields an inhomogeneous system of equations as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M27">View MathML</a>

(16)

In Equation 16, M is the matched(D + 2)×(D + 2) square matrix, and the vector of dimension (D + 2) which incorporates the <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M28">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M29">View MathML</a> elements, regroups the inhomogeneous terms of the incident wave. The explicit forms of the M matrix elements and and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M30">View MathML</a> vectors are presented in Appendix Appendix 3.

In practice, Equation 16 can be solved using standard numerical procedures, over the entire range of available electronic energies, yielding the coefficient cl for atomic sites on the nanojunction domain itself as well as the γreflection <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M31">View MathML</a> and the γtransmission <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M32">View MathML</a> coefficients.

The reflection and transmission coefficients give the reflection <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M33">View MathML</a> and transmission <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M34">View MathML</a> probabilities, respectively, by normalizing with respect to their group velocities vγ in order to obtain the unitarity of the scattering matrix as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M35">View MathML</a>

(17)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M36">View MathML</a>

(18)

where vγvγ(E) denotes the group velocity of the incident electron wave in the eigenmode γ. The group velocities are calculated by a straightforward procedure as in Appendix Appendix 4. For evanescent eigenmodes, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M37">View MathML</a>. Although the evanescent eigenmodes do not contribute to the electronic transport, they are required for the complete description of the scattering processes.

Furthermore, using Equations 17 and 18, the overall reflection probability, Rγ(E), for an electron incident in the γ eigenmode and the total electronic reflection probability, R(E), from all the eigenmodes may be expressed, respectively, as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M38">View MathML</a>

(19)

Similarly, for transmission probabilities, we may write the equivalent equations as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M39">View MathML</a>

(20)

The Tγ(E) and T(E) probabilities are very important for the electronic scattering processes since they correspond directly to the experimentally measurable observables. Likewise, the total transmission T(Eγ) allows to calculate the overall electronic conductance. In this work, we assume the zero-bias limit and write the total conductance in the following way:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M40">View MathML</a>

(21)

In Equation 21, G0 is the conductance quantum and equals 2e2/h. Due to the Fermi-Dirac distribution, G(EF) is calculated at the Fermi level of the perfect lead band structure since electrons only at this level give the important contribution to the electronic conductance. The Fermi energy can be determined using various methods where, in the present work, EF is calculated as the basis of the density of state calculations.

Results and discussion

The tight-binding model and basic electronic properties

In this section, we present the results of our model calculations for the electronic structure of carbon, silicon, and silicon carbide wires under study. Our results are validated by comparison with DFT calculations [29,69], which allow us to establish unambiguously our choice of the tight-binding parameters for these systems.

In principle, we can develop our model calculations for the nanojunctions and their leads using any adequate type of orbitals; even a single orbital suffices to calculate the electronic quantum transport for carbon nanojunctions [44]. However, this approximation is inadequate for silicon atoms. To treat both types of atoms on the same footing, we thus characterize the atoms by the electronic states 2s and 2p for carbon and by 3s and 3p for silicon. Such a scheme gives us four different orbitals, namely s, px, py, and pz, for both types of atoms.

In the present work, our TB parameters are effectively rescaled from the Harrison’s data in order to match our model calculations for the electronic structure with those given by the DFT. The utilized TB parameters are presented in Table 1 in comparison with the values given by Harrison. It is worthy to note that the values of the on-site Hamiltonian matrix elements <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M41">View MathML</a> are identical for states px, py, and pz. The off-diagonal distance-dependent <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M42">View MathML</a> elements are calculated on the basis of Equation 2. For symmetry considerations, these latter elements are positive or negative, also hs,p,σ = ηs,p,σ = 0 and hp,p,σ = ηp,p,σ = 0, for py and pz, and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M43">View MathML</a>[70]. Table 1 is supplemented for the reader by Figure 2 which gives the dependence of the hopping integrals with distance as calculated in the present paper (continuous curves), in comparison with the Harrison’s data (open symbols).

Table 1. Tight-binding parameters and Harrison’s dimensionless coefficients proposed in this work and compared with original values

thumbnailFigure 2. The nearest-neighbor tight-binding coupling parameters with the interatomic distance (A, B, C). The curves represent our calculated TB results in comparison with those calculated using the Harrison parameters (squares, triangles, circles).

Figure 2 clearly indicates the fact that qualitatively, both Harrison’s and our rescaled coupling parameters for silicon, carbon and diatomic silicon carbide wires, present the same functional behavior, confirming the desired conservation of their physical character. However, most of the rescaled coupling parameters have somehow smaller values than those initially proposed by Harrison; this trend can be also traced in Table 1 for the onsite parameters. This difference stems from the influence of the low-coordinated systems are considered here, whereas the initial Harrison values are given to match tetrahedral phases [67]. Another general observation can be made for the tight-binding parameters of the σ-type interactions (the hs,p,σand hp,p,σones), which present much closer values over the considered interatomic distance range than in the case of Harrison’s data.

Our calculated electronic band structures for silicon, carbon, and diatomic silicon carbide infinite wires (continuous curves) are presented in Figure 3 in comparison with the DFT results [29,69] as in the right-hand side of the figures. We note for the carbon and silicon structures that our TB parameters correctly reproduce the DFT results up to energies slightly above the Fermi level. Electronic branches in the regions of high energies are in qualitative agreement. In the case of the diatomic silicon carbide structure, some of the electronic states perfectly match the DFT results even for high-energy domains. The left-hand side of Figure 3 compares our results (continuous curves) with those from the older TB values given by Harrison (open symbols); as is seen, our TB parameters constitute the most optimal set for the electronic transport calculations since their corresponding electronic band structures conform to the appropriate energy ranges highlighted by the DFT results and, what is even more important, correctly reproduce the Fermi level.

thumbnailFigure 3. Electronic structures of carbon (A), silicon (B), and diatomic silicon carbide (C). These structures are for infinite linear atomic wires presented over the first Brillouin zone φ=kd∈[−ΠΠ. Our calculated results (continuous curves), represented by a color scheme (details in the text), are compared on the right-hand side with the first-principle results (closed circles, φ∈[0,Π) [29,69] and on the left-hand side with results calculated using Harrison TB parameters [67] (diamonds, φ∈[−Π,0]). Our calculated Fermi levels are given as the zero-reference energies, and the calculated electronic DOS in arbitrary units are presented in the right-hand column.

In Figure 3A,B for silicon and carbon, the red and blue colors correspond, respectively, to the σ and σ bands. These arise from the spxorbital hybrids where the lowest lying bands are always occupied by two electrons. Bands marked by the red color have the Π character and are degenerate. Their origin in the pyand pzorbitals allows them to hold up to four electrons. In Figure 3C for the diatomic silicon carbide, starting from the band structure minimum, consecutive bands have their origin in the following orbitals: carbon 3s (red band), silicon 3s (green band), carbon 3p (blue and black bands), and silicon 3p (orange and violet bands). The blue and orange colors for the silicon carbide electronic structure indicate two doubly degenerate Π-type bands.

The metallic or insulating character of the considered atomic wires, following the Fermi level, is appropriate only when the wires are infinite. It is well known that this character can change for the case of finite size wires with a limited number of atoms or due to the type and quality of the leads.

Numerical characteristics for the carbon leads

In general, the infinite carbon wires which are considered as the leads in our work, present electronic band structure characteristics which incorporate not only propagating (see Figure 3A), but also evanescent states. Both of these types of states, which are derivable from the generalized eigenvalue problem as presented in Equation 10, constitute a complete set over the allowed energies for the electrons incident along the leads, which can be further scattered at the considered nanojunction. This complete set of eigenstates is used as the basis for the numerical calculations of the quantum conductance presented in the ‘Transport properties’ subsection.

Figure 4A presents the three-dimensional representation of the solutions of Equation 10 as a set of generalized functionals z(E) for the σ, σ, and Πelectronic states of the carbon leads. As described by Equations 5, 8, and 9, the eigenstates in Figure 4A characterized by |z|=1 correspond to the propagating electronic waves described by the real wave vectors, whereas those by |z|<1 correspond to the evanescent and divergent eigenstates for the complex wave vectors. Furthermore, for convenience, the corresponding moduli of the complex z factors are presented in Figure 4B. Note that |z|=1 solutions may be grouped into pairs for the two directions of propagation linked by time-reversal symmetry. Due to the fact that each of these two solutions provides the same information, we consider waves propagating only from left to right. However, this is not true for the |z|<1 solutions which are always considered for both left and right as spatially evanescent. As can be seen in Figure 4, the generalized results for σ, σ, and Π states are represented by the same colors as the corresponding states in Figure 3A, following their propagating character for |z|=1, and further extended to the physically |z|<1 evanescent solutions.

thumbnailFigure 4. Three-dimensional representation of the functionals z(E) and the evolution of their absolute values for carbon leads. (A) Three-dimensional representation of the functionals z(E) on a complex plane and (B) the evolution of their absolute values as a function of energy for carbon leads. The color scheme here is the same as that for carbon in Figure 3A.

Figure 4 provides a more complete description for the electronic states of a given system compared to a typical band structure representation as in Figure 3, since both the propagating and evanescent states are shown. Such a general representation clearly indicates the importance of the evanescent eigenstates for a full description of the scattering problem presented in the ‘Transport properties’ subsection. The energies considered in our calculations correspond to the range within the band structure boundaries, marked by two vertical dotted lines in Figure 4B. As a consequence, not only the propagating states, but also the evanescent solutions are included in the quantum conductance calculations in the ‘Transport properties’ subsection.

Transport properties

In this subsection, the electronic transport properties of nanojunction systems composed of silicon-doped carbon wires between carbon leads are calculated using the PFMT method. Figure 5A presents a number of these systems where we indicate the irreducible domains by the shaded grey areas. Note that these systems are always composed of finite nanojunction regions of silicon and carbon atoms, coupled with two carbon semi-infinite leads. The first three systems of Figure 5 correspond to periodic diatomic silicon carbide nanojunctions composed of 1, 2, and 3 Si-C atomic pairs, respectively. The next system corresponds to a nanojunction with a substitutional disorder, composed of three carbon and three silicon atoms. The last is a symmetric nanojunction of five silicon atoms and only one carbon atom in the middle. Figure 5B presents the group velocities of electrons in the carbon leads.

thumbnailFigure 5. Schematic representation of the five nanojunction systems and group velocities for propagating band structure modes. (A) Schematic representation of the five nanojunction systems composed of silicon and carbon atoms between one-dimensional carbon leads considered in the present work. The irreducible domains are marked by the shaded grey areas, whereas for the other cases, only the irreducible domains are shown. (B) The group velocities for the propagating band structure modes on the carbon leads.

The calculated transmission and reflection scattering cross sections for each of the four available transport channels are presented in Figure 6. Each row of the figure corresponds to a nanojunction system (NS) as follows: Figure 6A,B,C for NS 1, Figure 6D,E,F for NS 2, Figure 6G,H,I for NS 3, Figure 6J,K,L for NS 4, and Figure 6M,N,O for NS 5. The red and green continuous curves represent the transmission and reflection spectra, respectively. The blue histograms correspond to the free electronic transport on the carbon leads, i.e., to the electronic transport on the perfect infinite quasi one-dimensional carbon wire over the different propagating states. These histograms constitute the reference to the unitarity condition which is used systematically as a check on the numerical results. The leads’ Fermi level is marked by a dashed line and set as a zero-energy reference. Under the zero-bias limit, the total conductance is calculated at this Fermi level.

thumbnailFigure 6. Transmission and reflection probabilities across five types of silicon-doped carbon wires between two semi-infinite one-dimensional carbon leads. The arrangement of the figure is as follows: (A, B, C) for case 1, (D, E, F) for case 2, (G, H, I) for case 3, (J, K, L) for case 4, and (M, N, O for case 5. The Fermi level is set at the zero-energy reference position.

In Figure 6, the transmission spectra present strong scattering resonances, showing an increasing complexity with the increasing size and configurational order of the nanojunctions. The valence σstate exhibits negligible transmission for all of the considered nanojunctions. The degenerate Π states and the σstate present in contrast the finite transmission spectra. However, it is only the Π states which cross the Fermi level, giving rise to electronic conductance across the nanojunction within the zero-bias limit.

In particular, the first three considered systems represent increasing lengths of the diatomic silicon carbide nanojunction with the increasing number of ordered Si-C atomic pairs. The transmission at the Fermi level for these systems is nonzero (see Figure 3C), which contrasts with the insulating character of the infinite silicon carbide wire. One can connect this finite transmission to the indirect bandgap (Δ) around the Fermi level for the diatomic silicon-carbide infinite wire (for more details, please see Figure 3C). This gap, Δ∼1.5 eV, is indeed related to the difference between the binding energies of the silicon and carbon atoms and corresponds to an effective potential barrier for the propagating Π-state electrons. As the wire length increases by adding Si-C atomic pairs, as for systems 1 to 3 of Figure 5B, the transmission decreases due to cumulative barrier effects. We note that a similar effect for the monovalent diatomic copper-cobalt wire nanojunctions has been observed in a previous work [54].

Furthermore, it is instructive to compare the scattering spectra for the degenerate Πstates, for nanojunction systems 3 and 4. These two systems contain identical numbers of silicon and carbon atoms; however, system 3 is an ordered configuration of Si-C pairs, whereas system 4 presents substitutional disorder of the atoms. It is seen that the disorder suppresses the conductance of the Π-state electrons at the Fermi level within the zero-bias limit. Another general observation can be made from the results for nanojunction system 5 which contains more silicon than carbon atoms. Despite the finite size of this system, which is comparable to system 4, and despite the structural symmetry of its atomic configuration, the electronic transmission is suppressed at the Fermi level within the zero-bias limit. This implies that one of the main observations of our paper is that structural symmetry on the nanojunction is not a guarantee for finite transmission in the case of the multivalence diatomic wire nanojunctions.

Figure 6 also shows that the transmission spectra for the σ state are close to unity over a significant range of energies from approximately 1 to 7 eV for all of the five nanojunction systems. This result may prove useful for the electronic conductance across silicon-doped carbon nanojunctions under finite bias voltages.

In Figure 7, we present the total electronic conductance G(E) as a function of energy E and in units of G0=2e2/h for the considered nanojunction systems of a given length as depicted in Figure 5 (red). Moreover, the perfect electronic conductance on the carbon leads (blue) is given in comparison and constitutes effectively the conductance of the infinite and perfect quasi one-dimensional carbon wire. In Figure 7, the Fermi level is indicated by the dashed line as a zero-reference energy, and G(E) is calculated from all the contributing eigenstates of Figure 6, including the two degenerate Π states.

thumbnailFigure 7. Total electronic conductance. Total electronic conductance G(E) (A, B, C, D, E) as a function of energy E in units of G0=2e2/h for silicon-doped carbon wires. See text for details.

We note that the conclusions given for the results presented in Figure 6 are also followed by the more general representation of the electronic transport depicted in Figure 7. Furthermore, the results presented in Figure 7 confirm that only the electrons incident from the leads in the Π states are responsible for the electronic conductance at the zero-bias limit, which is readable from the Fermi level position. However, for all considered systems, the conductance at the Fermi level is theoretically limited to the value of 2 G0, and the biggest conductance maxima close to the perfect infinite carbon wire value of 3 G0can be observed only in the energy interval from approximately 1 to 7 eV hence for energies above the Fermi level. Once again, this follows our previous observations for the transmission results for the Πstates concluded from Figure 6. Nonetheless, only on the basis of the results presented in Figure 7 can we note that due to the summation over all possible state contributions which constitute the G(E) spectra, not only the σ-state electrons, but also some of those in the degenerate Πstates contribute to the high conductance values in the cited energy intervals. This important observation proves that the σ- and Π-state electrons are of crucial importance for both the zero-bias quantum conductance of the silicon-doped carbon wires and the possible finite bias ones. This implies that the use of only a single orbital for the description of the carbon atoms will result in an inadequate description of the transport processes across low-coordinated systems containing these atoms.

Conclusions

In the present work, the unknown properties of the quantum electronic conductance for nanojunctions made of silicon-doped carbon wires between carbon leads are studied in depth. This is done using the phase field matching theory and the tight-binding method. The local basis for the electronic wave functions is assumed to be composed of four different atomic orbitals for silicon and carbon, namely the s, px, py, and pzstates.

In the first step, we calculate the electronic band structures for three nanomaterials, namely the one-dimensional infinite wires of silicon, carbon, and diatomic silicon carbide. This permits a matching comparison with the available corresponding DFT results, with the objective to select the optimal TB parameters for the three nanomaterials.

This optimal set of the tight-binding parameters is then used to calculate the electronic conductance across the silicon-doped carbon wire nanojunctions. Five different nanojunction cases are analyzed to sample their behavior under different atomic configurations. We show that despite the nonconducting character of the infinite silicon carbide wires, its finite implementation as nanojunctions exhibit a finite conductance. This outcome is explained by the energy difference between the binding energies of the silicon and carbon atoms, which correspond to an effective potential barrier for the degenerate Π-state electrons transmitted across the nanojunction under zero-bias field.

The conductance effects that may arise due to minimal substitutional disorder and to artificially organize symmetry considerations on the silicon carbide wire nanojunction are also investigated. By exchanging the positions of two silicon and carbon atoms on an initial nanojunction to generate a substitutional disorder, we show that the total quantum conductance is suppressed at the Fermi level. This is in sharp contrast with the finite and significant conductance for the initial atomically ordered nanojunction with periodic configurations of the silicon and carbon atoms. Also, the analysis of a silicon carbide nanojunction of a comparable size as the one above, presenting symmetry properties, shows that quantum conductance is suppressed at the Fermi level.

In summary, we note that the biggest maxima of the conductance spectra for the zero-bias limit can be observed for high energies for all of the considered systems. This conclusion reveals the fact that electrons incident from the leads in both σand Πstates are crucial for the considerations of the electronic transport properties of the silicon-doped carbon wire nanojunctions.

Appendix 1

Explicit forms of the Ei,jand Hi,jmatrices

The explicit forms of the submatrices of Equation 1 are given in the following manner:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M50">View MathML</a>

(22)

and

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M51">View MathML</a>

(23)

where

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M52">View MathML</a>

(24)

and

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M53">View MathML</a>

(25)

Equations 22 and 23 denote NxNlsquare matrices, where matrix (Equation 23) is upper triangular. In this manner, component matrices (Equations 24 and 25) are of the dimension Nl×Nl. Additionally, matrix <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M54">View MathML</a> always denotes diagonal matrix, while <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M55">View MathML</a> matrix is much more complex, with possible nonzero elements at every position. Please note that some of the <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M56">View MathML</a> elements can vanish due to symmetry conditions and simplify the notation of the <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M57">View MathML</a> matrix.

Appendix 2

Partitioning technique

The partitioning technique is a suitable method which allows to avoid the singularity problem of the HN,N−1 and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M58">View MathML</a> matrices and calculates only nontrivial solutions of Equation 10. Detailed discussion of the partitioning technique is presented in the work of Khomyakov and Brocks [71], and this section gives only our short remarks on this method.

Following studies from Khomyakov and Brocks [71], Equation 10 is partitioned into two parts of D1D2 and D2 sizes where

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M59">View MathML</a>

(26)

and

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M60">View MathML</a>

(27)

In Equation 27, parameter Nnstands for the order of nearest-neighbor interactions assumed in calculations, e.g., Nn=1 for the first nearest-neighbor interactions. On the basis of Equations 26 and 27, the reduced 2Nleigenvalue problem is written as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M61">View MathML</a>

(28)

At this point, we correct the misprint from the study of Khomyakov and Brocks [71] and write the submatrices of Equation 28 in the following form:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M62">View MathML</a>

(29)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M63">View MathML</a>

(30)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M64">View MathML</a>

(31)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M65">View MathML</a>

(32)

Please note that the reduced problem of Equation 28 gives 2Nl eigenvalues with 2Nl corresponding eigenvectors; this Nxtimes less than can be expected from a physical point of view. Nevertheless, those solutions can be easily separated into NxNl eigenvalues and NxNleigenvectors of a purely physical character.

Appendix 3

Explicit forms of the Mi,j, <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M66">View MathML</a>, and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M67">View MathML</a> components

The submatrices of the matched(D + 2)×(D + 2) square matrix Min Equation 16, for a given i and j indices, are given as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M68">View MathML</a>

(33)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M69">View MathML</a>

(34)

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M70">View MathML</a>

(35)

except for the submatrices which describe the boundary atoms of the system and those that are expressed in the following manner:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M71">View MathML</a>

(36)

Finally, the <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M72">View MathML</a> and <a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M73">View MathML</a> of Equation 16 vector components are written as follows:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M74">View MathML</a>

(37)

and

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M75">View MathML</a>

(38)

Appendix 4

Group velocities

As specified in the ‘Phase field matching theory’ subsection, the group velocities for individual states can be calculated on the basis of Equation 6 rewritten in the following manner:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M76">View MathML</a>

(39)

where v denotes the eigenvalues of Equation 39 which yields all required electron group velocities for each propagating state. Further, Vis the Nx×Nlsize matrix of the following form:

<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/616/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/616/mathml/M77">View MathML</a>

(40)

Finally, v(RN,k) stands for eigenvectors of the problem of Equation 39. We note that, usually, Equation 40 includes the constant part dβ/h, where h is the Planck constant. However, for the purpose of electronic conductance calculations within the PFMT approach, this term can be omitted due to the fact that only the ratios of the given group velocities are important (please see Equations 17 and 18).

Competing interests

The authors declare that they have no competing interests.

Authors, contributions

DS participated in the design of this study, in analytical calculations, and in writing the code for numerical calculations, carried out numerical calculations, drafted the manuscript, and participated in writing the final version of the manuscript. AK coordinated this study, participated in its design and analytical calculations, and in writing the final version of the manuscript. ZB participated in the design of this study, its coordination, and substantial critical revision of the final version of the manuscript. RS participated in writing the code for numerical calculations and substantial critical revision of the final version of the manuscript. MAG participated in the substantial critical revision of the final version of the manuscript. All authors read and approved the final manuscript.

Acknowledgements

D Szczȩśniak would like to thank the French Ministry of Foreign Affairs for his PhD scholarship grant CNOUS 2009-2374, to the Polish National Science Center for their research grant DEC-2011/01/N/ST3/04492, and to the Graduate School of Sciences at the University du Maine for their support.

References

  1. Agraït N, Levy-Yeyati A, van Ruitenbeek JM: Quantum properties of atomic-sized conductors.

    Phys Rep 2003, 377:81-279. Publisher Full Text OpenURL

  2. Nitzan A, Ratner M: Electron transport in molecular wire junctions.

    Science 2003, 300:1384-1389. PubMed Abstract | Publisher Full Text OpenURL

  3. Wan CC, Mozos JL, Taraschi G, Wang J, Guo H: Quantum transport through atomic wires.

    Appl Phys Lett 1997, 71:419-421. Publisher Full Text OpenURL

  4. Geim AK, Novoselov KS: The rise of graphene.

    Nature Mater 2007, 6:183-191. OpenURL

  5. Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE: C60: buckminsterfullerene.

    Nature 1985, 318:162-163. Publisher Full Text OpenURL

  6. Iijima S, Ichihashi T: Single-shell carbon nanotubes of 1-nm diameter.

    Nature 1993, 363:603-605. Publisher Full Text OpenURL

  7. Euen PL: Nanotechnology: carbon-based electronics.

    Nature 1998, 393:15-16. Publisher Full Text OpenURL

  8. Heath JR, Zhang Q, O’Brien SC, Curl RF, Kroto HW, Smalley RE: The formation of long carbon chain molecules during laser vaporization of graphite.

    J Am Chem Soc 1987, 109:359-363. Publisher Full Text OpenURL

  9. Lagow RJ, Kampa JJ, Wei HC, Battle SL, Genge JW, Laude DA, Harper CJ, Bau R, Stevens RC, Haw JF, Munson E: Synthesis of linear acetylenic carbon: the “sp” carbon allotrope.

    Science 1995, 267:362-367. PubMed Abstract | Publisher Full Text OpenURL

  10. Derycke V, Soukiassian P, Mayne A, Dujardin D, Gautier J: Carbon atomic chain formation on the β-SiC(100) surface by controlled sp→sp3 transformation.

    Phys Rev Lett 1998, 81:5868-5871. Publisher Full Text OpenURL

  11. Troiani HE, Miki-Yoshida M, Camacho-Bragado GA, Marques MAL, Rubio A, Ascencio JA, Jose-Yacaman M: Direct observation of the mechanical properties of single-walled carbon nanotubes and their junctions at the atomic level.

    Nano Lett 2003, 3:751-755. Publisher Full Text OpenURL

  12. Zhao X, Ando Y, Liu Y, Jinno M, Suzuki T: Carbon nanowire made of a long linear carbon chain inserted inside a multiwalled carbon nanotube.

    Phys Rev Lett 2003, 90:187401. PubMed Abstract | Publisher Full Text OpenURL

  13. Yuzvinsky TD, Mickelson W, Aloni S, Begtrup GE, Kis A, Zettl A: Shrinking a carbon nanotube.

    Nano Lett 2006, 6:2718-2722. PubMed Abstract | Publisher Full Text OpenURL

  14. Jin C, Lan H, Peng L, Suenaga K, Iijima S: Deriving carbon atomic chains from graphene.

    Phys Rev Lett 2009, 102:205501. PubMed Abstract | Publisher Full Text OpenURL

  15. Kértesz M, Koller J, Az̆man A: Ab initio Hartree-Fock crystal orbital studies. II. Energy bands of an infinite carbon chain.

    J Chem Phys 1978, 68:2779-2782. Publisher Full Text OpenURL

  16. Kértesz M, Koller J, Az̆man A: Different orbitals for different spins for solids: fully variational ab initio studies on hydrogen and carbon atomic chains, polyene, and poly(sulphur nitride).

    Phys Rev B 1979, 19:2034-2040. Publisher Full Text OpenURL

  17. Karpfen A: Ab initio studies on polymers. I. The linear infinite polyyne.

    J Phys C Solid State Phys 1979, 12:3227-3237. Publisher Full Text OpenURL

  18. Teramae M, Yamabe T, Imamura A: Ab initio effective core potential studies on polymers.

    Theor Chim Acta 1983, 64:1-12. OpenURL

  19. Springborg M: Self-consistent, first principles calculations of the electronic structures of a linear, infinite carbon chain.

    J Phys C 1986, 19:4473-4482. Publisher Full Text OpenURL

  20. Rice MJ, Phillpot SR, Bishop AR, Campbell DK: Solitons, polarons, and phonons in the infinite polyyne chain.

    Phys Rev B 1986, 34:4139-4149. Publisher Full Text OpenURL

  21. Springborg M, Dreschel SL, Málek J: Anharmonic model for polyyne.

    Phys Rev B 1990, 41:11954-11966. Publisher Full Text OpenURL

  22. Watts JD, Bartlett RJ: A theoretical study of linear carbon cluster monoanions, Cn and dianions, Cn2− (n=2−10).

    J Chem Phys 1992, 97:3445-3457. Publisher Full Text OpenURL

  23. Xu CH, Wang CZ, Chan CT, Ho KM: A transferable tight-binding potential for carbon.

    J Phys Condens Matter 1992, 4:6047-6054. Publisher Full Text OpenURL

  24. Lou L, Nordlander P: Carbon atomic chains in strong electric fields.

    Phys Rev B 1996, 54:16659-16662. Publisher Full Text OpenURL

  25. Jones RO, Seifert G: Density functional study of carbon clusters and their ions.

    Phys Rev Lett 1997, 79:443-446. Publisher Full Text OpenURL

  26. Fuentealba P: Static dipole polarizabilities of small neutral carbon clusters Cn (n ⩽ 8).

    Phys Rev A 1998, 58:4232-4234. Publisher Full Text OpenURL

  27. Abdurahman A, Shukla A, Dolg M: Ab initio many-body calculations of static dipole polarizabilities of linear carbon chains and chainlike boron clusters.

    Phys Rev B 2002, 65:115106. OpenURL

  28. Cahangirov S, Topsakal M, Ciraci S: Long-range interactions in carbon atomic chains.

    Phys Rev B 2010, 82:195444. OpenURL

  29. Tongay S, Ciraci S: Atomic strings of group IV, III-V, and II-VI elements.

    Appl Phys Lett 2004, 85:6179-6181. Publisher Full Text OpenURL

  30. Bylaska EJ, Weare JH, Kawai R: Development of bond-length alternation in very large carbon rings: LDA pseudopotential results.

    Phys Rev B 1998, 58:R7488—R7491. OpenURL

  31. Zhang Y, Su Y, Wang L, Kong ESW, Chen X, Zhang Y: A one-dimensional extremely covalent material: monatomic carbon linear chain.

    Nanoscale Res Lett 2011, 6:577. PubMed Abstract | BioMed Central Full Text | PubMed Central Full Text OpenURL

  32. Lang ND, Avouris P: Oscillatory conductance of carbon-atom wires.

    Phys Rev Lett 1998, 81:3515-3518. Publisher Full Text OpenURL

  33. Lang ND, Avouris P: Carbon-atom wires: charge-transfer doping, voltage drop, and the effect of distortions.

    Phys Rev Lett 2000, 84:358-361. PubMed Abstract | Publisher Full Text OpenURL

  34. Larade B, Taylor J, Mehrez H, Guo H: Conductance, I-V curves, and negative differential resistance of carbon atomic wires.

    Phys Rev B 2001, 64:075420. OpenURL

  35. Tongay S, Dag S, Durgun E, Senger RT, Ciraci S: Atomic and electronic structure of carbon strings.

    J Phys Cond Matter 2005, 17:3823-3836. Publisher Full Text OpenURL

  36. Senger RT, Tongay S, Durgun E, Ciraci S: Atomic chains of group-IV elements and III-V and II-VI binary compounds studied by a first-principles pseudopotential method.

    Phys Rev B 2005, 72:075419. OpenURL

  37. Baranović G, Z̆ Crljen: Unusual conductance of polyyne-based molecular wires.

    Phys Rev Lett 2007, 98:116801. PubMed Abstract | Publisher Full Text OpenURL

  38. Okano S, Tománek D: Effect of electron and hole doping on the structure of, C, Si, and S nanowires.

    Phys Rev B 2007, 75:195409. OpenURL

  39. Chen W, Andreev AV, Bertsch GF: Conductance of a single-atom carbon chain with graphene leads.

    Phys Rev B 2009, 80:085410. OpenURL

  40. Wang Y, Lin ZZ, Zhang W, Zhuang J, Ning XJ: Pulling long linear atomic chains from graphene: molecular dynamics simulations.

    Phys Rev B 2009, 80:233403. OpenURL

  41. Song B, Sanvito S, Fang H: Anomalous I-V curve for mono-atomic carbon chains.

    New J Phys 2010, 12:103017. Publisher Full Text OpenURL

  42. Zhang GP, Fang XW, Yao YX, Wang CZ, Ding ZJ, Ho KM: Electronic structure and transport of a carbon chain between graphene nanoribbon leads.

    J Phys Cond Matter 2011, 23:025302. Publisher Full Text OpenURL

  43. Ke Y, Xia K, Guo H: Disorder scattering in magnetic tunnel junctions: theory of nonequilibrium vertex correction.

    Phys Rev Lett 2008, 100:166805. PubMed Abstract | Publisher Full Text OpenURL

  44. Nozaki D, Pastawski HM, Cuniberti G: Controlling the conductance of molecular wires by defect engineering.

    New J Phys 2010, 12:063004. Publisher Full Text OpenURL

  45. Strupiński W, Grodecki K, Wysmołek A, Stȩpniewski R, Szkopek T, Gaskell PE, Grüneis A, Haberer D, BoŻek R, Krupka J, Baranowski JM: Graphene epitaxy by chemical vapor deposition on SiC.

    Nano Lett 2011, 11:1786-1791. PubMed Abstract | Publisher Full Text OpenURL

  46. Wang F, Shepperd K, Hicks J, Nevius MS, Tinkey H, Tejeda A, Taleb-Ibrahimi A, Bertran F, Fèvre PL, Torrance DB, First PN, de Heer WA, Zakharov AA, Conrad EH: Silicon intercalation into the graphene-SiC interface.

    Phys Rev B 2012, 85:165449. OpenURL

  47. Landauer R: Spatial variation of currents and fields due to localized scatterers in metallic conduction.

    IBM J Res Dev 1957, 1:223-231. OpenURL

  48. Büttiker M: Four-terminal phase-coherent conductance.

    Phys Rev Lett 1986, 57:1761-1764. PubMed Abstract | Publisher Full Text OpenURL

  49. Zwierzycki M, Xia K, Kelly PJ, Bauer GEW, Turek I: Spin injection through an Fe/InAs interface.

    Phys Rev B 2003, 67:092401. OpenURL

  50. Pauly F, Viljas JK, Huniar U, Häfner M, Wohlthat S, Bürkle M, Cuevas JC, Schön G: Cluster-based density-functional approach to quantum transport through molecular and atomic contacts.

    New J Phys 2008, 10:125019. Publisher Full Text OpenURL

  51. Caroli C, Combescot R, Nozières P, Saint-James D: Direct calculation of the tunneling currents.

    J Phys C 1971, 8:916-929. OpenURL

  52. Deretzis I, Magna AL: Coherent electron transport in quasi one-dimensional carbon-based systems.

    Eur Phys J B 2011, 81:15. Publisher Full Text OpenURL

  53. Khater A, Szczȩśniak D: A simple analytical model for electronic conductance in a one dimensional atomic chain across a defect.

    J Phys Conf Ser 2011, 289:012013. OpenURL

  54. Szczȩśniak D, Khater A: Electronic conductance via atomic wires: a phase field matching theory approach.

    Eur Phys J B 2012, 85:174. OpenURL

  55. Khater A, Bourahla B, Abou Ghantous M, Tigrine R, Chadli R: Magnons coherent transmission and heat transport at ultrathin insulating ferromagnetic nanojunctions.

    Eur Phys J B 2011, 82:53-61. Publisher Full Text OpenURL

  56. Khater A, Belhadi M, Abou Ghantous M: Phonons heat transport at an atomic well boundary in ultrathin solid films.

    Eur Phys J B 2011, 80:363-369. Publisher Full Text OpenURL

  57. Tigrine R, Khater A, Bourahla B, Abou Ghantous M, Rafli O: Magnon scattering by a symmetric atomic well in free standing very thin magnetic films.

    Eur Phys J B 2008, 62:59-64. Publisher Full Text OpenURL

  58. Virlouvet A, Khater A, Aouchiche H, Rafli O, Maschke K: Scattering of vibrational waves in perturbed two-dimensional multichannel asymmetric waveguides as on an isolated step.

    Phys Rev B 1999, 59:4933-4942. Publisher Full Text OpenURL

  59. Fellay A, Gagel F, Maschke K, Virlouvet A, Khater A: Scattering of vibrational waves in perturbed quasi-one-dimensional multichannel waveguides.

    Phys Rev B 1997, 55:1707-1717. Publisher Full Text OpenURL

  60. Mardaani M, Rabani H, Esmaeili A: An analytical study on electronic density of states and conductance of typical nanowires.

    Solid State Commun 2011, 151:928-932. Publisher Full Text OpenURL

  61. Rabani H, Mardaani M: Exact analytical results on electronic transport of conjugated polymer junctions: renormalization method.

    Solid State Commun 2012, 152:235-239. Publisher Full Text OpenURL

  62. Wu Y, Childs PA: Conductance of graphene nanoribbon junctions and the tight binding model.

    Nanoscale Res Lett 2011, 6:62. OpenURL

  63. Chen J, Yang L, Yang H, Dong J: Electronic and transport properties of a carbon-atom chain in the core of semiconducting carbon nanotubes.

    Phys Lett A 2003, 316:101-106. Publisher Full Text OpenURL

  64. Hands ID, Dunn JL, Bates CA: Visualization of static Jahn-Teller effects in the fullerene anion C60.

    Phys Rev B 2010, 82:155425. OpenURL

  65. Delga A, Lagoute J, Repain V, Chacon C, Girard Y, Marathe M, Narasimhan S, Rousset S: Electronic properties of Fe clusters on a Au(111) surface.

    Phys Rev B 2011, 84:035416. OpenURL

  66. Slater JC, Koster GF: Simplified LCAO method for the periodic potential problem.

    Phys Rev 1954, 94:1498-1524. Publisher Full Text OpenURL

  67. Harrison WA: Elementary Electronic Structure. Singapore: World Scientific; 2004. OpenURL

  68. Zhang L, Wang JS, Li B: Ballistic magnetothermal transport in a Heisenberg spin chain at low temperatures.

    Phys Rev B 2008, 78:144416. OpenURL

  69. Bekaroglu E, Topsakal M, Cahangirov S, Ciraci S: First-principles study of defects and adatoms in silicon carbide honeycomb structures.

    Phys Rev B 2010, 81:075433. OpenURL

  70. Kaxiras E: Atomic and Electronic Structure of Solid. New York: Cambridge University Press; 2003. OpenURL

  71. Khomyakov PA, Brocks G: Real-space finite-difference method for conductance calculations.

    Phys Rev B 2004, 70:195402. OpenURL