arX

iv:1

105.

2938

v6 [

cond

-mat

.mes

-hal

l] 9

Jul

201

2submitted to: The European Physical Journal B

—————————————————————————————————————

Electronic conductance via atomic wires: a phase field matching theory approach

D. Szczesniak1,2 and A. Khater1

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

2. Institute of Physics, Jan D lugosz University in Czestochowa,Al. Armii Krajowej 13/15, 42200 Czestochowa, Poland∗

(Dated: November 16, 2018)

A model is presented for the quantum transport of electrons, across finite atomic wire nanojunc-tions between electric leads, at zero bias limit. In order to derive the appropriate transmission andreflection spectra, familiar in the Landauer-Buttiker formalism, we develop the algebraic phase fieldmatching theory (PFMT). In particular, we apply our model calculations to determine the electronicconductance for freely suspended monatomic linear sodium wires (MLNaW) between leads of thesame element, and for the diatomic copper-cobalt wires (DLCuCoW) between copper leads on aCu(111) substrate. Calculations for the MLNaW system confirm the correctness and functionalityof our PFMT approach. We present novel transmission spectra for this system, and show thatits transport properties exhibit the conductance oscillations for the odd- and even-number wiresin agreement with previously reported first-principle results. The numerical calculations for theDLCuCoW wire nanojunctions are motivated by the stability of these systems at low temperatures.Our results for the transmission spectra yield for this system, at its Fermi energy, a monotonicexponential decay of the conductance with increasing wire length of the Cu-Co pairs. This is acumulative effect which is discussed in detail in the present work, and may prove useful for appli-cations in nanocircuits. Furthermore, our PFMT formalism can be considered as a compact andefficient tool for the study of the electronic quantum transport for a wide range of nanomaterial wiresystems. It provides a trade-off in computational efficiency and predictive capability as comparedto slower first-principle based methods, and has the potential to treat the conductance propertiesof more complex molecular nanojunctions.

PACS number(s): 73.63.Nm, 73.63.-b, 03.65.Fd, 31.15.xf

I. INTRODUCTION

Current technological needs motivate the intensive re-search of the properties of materials at the nanoscale.One of the most important areas in this respect at presentconcerns nanoelectronics [1]. The great interest in thisdomain arises from the potential reduction of the sizeof the circuit components, maintaining their quality andfunctionality, and aiming at greater efficiency and storagecharacteristics for physical devices [1], [2].Among the various physical components which con-

stitute the nanoelectronic devices, the nanojunctionstherein are of crucial importance and will become in-creasingly so in the future [2], [3]. Such low-dimensionalelements, which can act e.g. as switches or gates [2],are considered to be key circuit components connectingthe larger nanostructures. At present they are experi-mentally prepared and investigated by means of severaltechniques, such as the mechanically controllable breakjunction methods [1], [4], feedback-stabilized break junc-tion technique [5], atomic force microscopy [1], scanningtunneling microscopy [1], [6], as well as transmission elec-tron microscopy [7], [8].

∗Electronic address: d.szczesniak@ajd.czest.pl

Atomic wires represent a particular class of such nano-junctions in nanoelectronic circuits. The conductanceproperties of atomic wires are the most interesting fea-tures of these systems, they depend on the materials usedto fabricate them and on their structural properties. Forexample, the conductance of the monatomic wire nano-junctions does not stay constant or decrease when thelength of the wire increases, rather it oscillates as a func-tion of the number of the wire atoms. The periodicity ofthe oscillations may vary depending on the type of theatomic wire. It has been shown that, in the monovalentwire nanojunctions composed of alkali (Na and Cs) ornoble metals (Ag, Au and Cu), the conductance oscilla-tions exhibit the two-atom periodicity [9], [10], [11], [12].In contrast the Al wire nanojunctions present the oscil-lations with a period of four or six atoms, depending onthe considered interatomic distances between consecutivesites [13], [14]. As well as in Al wires, the conductance os-cillations have been observed in other multivalence wirenanojuctions, as platinum and iridium [15], carbon [16],[17], and silicon systems [18], [19].

Understanding the electronic transport properties ofsuch nanojunctions is hence of particular importance.The electrons which contribute to the transport via nano-junctions present characteristic wavelengths comparableto the size of these components, leading to quantum co-herent effects. The properties of the nanoelectronic de-vice and its functionality may be greatly affected or evenbuilt on such nanojunction quantum effects [20], and can-not be described in the framework of the classical regime

2

[21]. In particular, the transport characteristics are usu-ally described employing the scattering theory for elec-tronic excitations in nanostructures [22]. The relation-ship between quantum scattering and the electronic con-ductance is initially provided by the work of Landauer[23], together with its generalization to the case of mul-tiple scattering as developed by Buttiker [24]. The de-termination of the scattering transmission and reflectionprobabilities for the electrons as quantum particles innanostructures is consequently a key concern.

The scattering probabilities can be determined thanksto a number of theoretical methods which have a commonobjective in the framework of the Landauer-Buttiker for-malism, and can be employed by integrating in their ap-proach, the density functional theory (DFT) [25], [26] orthe tight-binding model (TB) [27], [28]. The most pop-ular methods are the matrix Green’s function methodby Ando [29] or the non-equilibrium Green’s functionformalism (NEGF), also known as the Keldysh formal-ism [30]. However, there has been a debate whetherthe Ando’s formalism is complete [31], and as has beenpointed out, the combination of the Keldysh formalismand DFT is still very often computationally demanding[21], [27], motivating intensive research for solutions tothis difficulty. An interesting discussion of different typesof the optimization techniques may be found in [25], [26],[32].

In this paper we present a compact and efficient theo-retical method to calculate the electronic quantum con-ductance, in the framework of the Landauer-Buttiker for-malism, for the class of nanojunctions made up of finiteatomic wires of conducting atoms. Our model calcula-tions are based on the method of the phase field match-ing theory (PFMT), which was originally and intensivelydeveloped for the scattering and transport of phonon andmagnon excitations in nanostructures [33]. While at firstsight, the formulation of the PFMT may be expectedto be formally similar for different types of excitations,i.e. replacing here the dynamical equation for phonons ormagnons by the Schrodinger equation, there are howeversignificant technical dissimilarities which play an impor-tant role. For example, we can refer to the fact that in theelectron scattering problem one has to consider allowedsymmetries of the atomic orbitals, which is not the casefor phonon and magnon excitations.

To fully benefit from the PFMT method, our modeldynamics are described in the framework of the linearcombination of atomic orbitals in the Slater-Koster tight-binding approximation [34]. The PFMT method, pre-sented in this work, can be classified as a finite differenceapproximation technique. In comparison, an early dis-cussion of this method can be found in [35], [36], using abasic analytical formulation for the elementary problemof the electronic transport across a single chemical defectin a model wire nanojunction.

The combination of PFMT and tight-binding modelleads to transparent matrix structures which are numer-ically solvable in a direct manner, in contrast with the

iterative matrix algebra characteristic of the NEGF cal-culations which require special numerical optimizationmethods [32]. Furthermore, integrated tight-binding ap-proximation allows us to discretize the entire model cal-culation in the real-space, and equally important, tomake time economies in the numerical computations.Here we note that the tight-binding theory has been usedsuccessfully in previous work as a basis for electronictransport calculations, for various low-dimensional sys-tems, such as polymer nanojunctions [45], carbon nan-otubes [46], and the presently popular graphene systems[47]. This reinforces our choice of the electronic dynamicsmodel.As an implementation of the PFMT method we con-

sider in the present work two types of atomic wire nano-junction systems at zero bias limit. In the first step ourmodel calculations are applied to various lengths of thefreely suspended monoatomic linear sodium wires (ML-NaW), as nanojunctions between one-dimensional semi-infinite atomic leads of the same material. These calcu-lations are motivated by the fact that the MLNaW nano-junctions, which have been investigated previously usingother methods [11], [12], [26], [37], [38], are known asgood benchmark systems for electronic conductance cal-culations [21], [39]. This initial analysis is hence carriedout in order to demonstrate the correctness and function-ality of the PFMT method by comparing with previousresults.Further, our PFMT method is applied to the diatomic

linear copper-cobalt wires (DLCuCoW) which are hith-erto untreated. The particular interest in these systemsstems from the fact that they are mechanically and ther-modynamically stable at low temperatures, supported asthey are on atomically flat Cu(111) surfaces [40], [41],[42]. Our results for this nanojunctions supplement in anatural way recent experimental investigations on finiteCu-Co wires [40], by providing a theoretical descriptionof the electronic transport properties for these systems.The present paper is organized as follows. In section II

we give an insight into the general and essential featuresof the PFMT approach. Section III presents the applica-tions of our model calculations to MLNaW and DLCu-CoW nanojunctions. The numerical analysis yields boththe transmission and the reflection probabilities for elec-trons, as well as the total electronic conductance for theconsidered systems at zero bias limit. The conclusionsare presented in section IV.

II. THEORETICAL MODEL AND METHOD

The general formulation of the PFMT method is pre-sented next. It is convenient to develop it initially, andto rewrite it afterwards into the form which is directlyapplied for our model calculations for the MLNaW andDLCuCoW nanojunctions. This allows us to introducethe PFMT technique in the most detailed and appro-priate manner. Furthermore, the general presentation

3

clarifies the mathematical approach which is applied inthe present work for the system of wire nanojunctions.It equally illustrates how the method may be applied ingeneral, treating general molecular forms for the nano-junction when conjugated between one-, two- or three-dimensional leads. The general formulation can also in-corporate the interactions between atomic sites beyondnearest-neighbors.Let us first refer to the Landauer-Buttiker approach for

the analysis of the electronic transport via nanostructures[23], [24], and divide an arbitrary nanojunction systeminto three main parts [33], [43], namely the left and rightleads, and the nanojunction domain itself, as is in Fig.1.To model the electronic properties of this system, the

starting point is to write the general form of the tight-binding secular equation, for the total M number ofatoms per unit cell, and the total L number of atomicorbitals per site, as follows

E

. . .. . .

. . . Sn−1,n−1 S†

n,n−1

Sn,n−1 Sn,n S†n+1,n

Sn+1,n Sn+1,n+1

. . .

. . .. . .

...c(rn−1,k)

c(rn,k)

c(rn+1,k)

...

=

. . .. . .

. . . Hn−1,n−1 H†n,n−1

Hn,n−1 Hn,n H†

n+1,n

Hn+1,n Hn+1,n+1

. . .. . .

. . .

...c(rn−1,k)

c(rn,k)

c(rn+1,k)

...

.(1)

Note that we assume spin degeneracy in Eq.(1). In whatfollows, the M by M overlap sub-matrix Sn,n′ for a given

n-th unit cell, incorporates the sl,l′

m,m′ overlap integrals

between the l and l′ atomic orbitals centered at the sitescorresponding to the real space vectors rm and rm′ , re-spectively. The unit cell may contain more than one in-equivalent site, denoted here by m. Furthermore, thecorresponding diagonal Hamitonian block matrix Hn,n′

for the n-th unit cell, is composed of binding energy pa-

rameters εlm and the interaction integrals hl,l′

m,m′ . Thenagain, the off-diagonal Hn,n′ sub-matrix contains only

hl,l′

m,m′ terms which describe interactions between differ-

ent unit cells. Finally, the c(rn,k) denotes vector of thec(rl − rm − rn,k) wave function coefficients which corre-spond to the l atomic orbital at the lattice site m in then-th unit cell. k is the appropriate wave vector of theBloch like wave function of the electronic excitation.In order to calculate the electronic transport properties

in the framework of the PFMT method for the nanoscalesystem presented in Fig.1, we have to analyze the elec-tronic scattering at the embedded nanostructure. To thatend, we consider the nanojunction irreducible boundarydomain, and the interactions of its electronic states with

matching

domain

matching

domain

irreducible region

H0,1H-1,0H-2,-1H-3,-2 HN-2,N-1 H N-1,N HN,N+1 HN+1,N+2

H0

,0

H1

,1

H-1

,-1

H-2

,-2

H-3

,-3

HN

-2,N

-2

HN

-1,N

-1

HN

,N

HN

+1

,N+

1

HN

+2

,N+

2

FIG. 1: Schematic planar projection in the direction of prop-agation of an arbitrary three-dimensional Landauer-Buttiker-type system. The unit cells of the nanojunction and two semi-infinite leads are denoted by red and blue colors, respectively[43]. The corresponding Hamiltonian on- and off-diagonalsub-matrices are also depicted. Note that in order to keepthe figure simple we show the Hamiltonian sub-matrices forinteractions between different unit cells only for n < n′.

the electronic excitations incident from the leads to theleft and right of the domain. The wave function coeffi-cients of consecutive unit cells in the leads are defined inthe sense of the Bloch-Flouqet theorem by the followingphase relation

c(rn,k) = zc(rn−1,k), (2)

where z = exp(±ikrn) denotes the generalized Blochphase factor. In our notation z = exp(+ikrn) describesthe electronic wave function propagating to the right, andz = exp(−ikrn) to the left. We should emphasize that inthe case of multi-channel scattering processes one has toconsider not only propagating but also evanescent waves[33]. The evanescent solutions can be obtained using dif-ferent procedures, however an elegant method presentedpreviously for phonon and magnon problems [33] shall bealso applied to the electronic problem.For an electron incident along the leads at a given en-

ergy E and wave vector k, where E = Eγ(k) denotesthe dispersion curves for the γ = 1, 2,..., propagatingwaves in the Brillouin zone, the scattering at the bound-ary yields coherent reflected and transmitted fields. Tocalculate the Landauer-Buttiker electronic conductancevia a given nanojunction, we write the linearized equa-tions of motion for the consecutive atomic sites within theirreducible nanojunction domain, on the basis of Eq.(1).This procedure generates a set of N + 2M equations ofmotion with N + 4M unknown variables, namely thewave function coefficients, extending to outside the do-main and inside the leads. Since the number of unknownvariables is always greater than the number of equations,such a set of equations cannot be solved directly.To resolve this problem, we use the phase field match-

ing method [33], [35], [36]. This implies that, for unit cellsin the leads distant from the nanojunction boundary, the

4

coefficients may be expressed in terms of an appropriatesuperposition of the eigenstates of the perfect leads at thesame energy and wave vector. Hence, the wave functioncoefficients for the region outside the irreducible nano-junction domain may be expressed as

c(rn,k) = cγ(k)z−n

+

Γ∑

γ′

cγ′(k)znrγ,γ′ for n 6 −1, (3)

c(rn,k) =

Γ∑

γ′

cγ′(k)zntγ,γ′ for n > N. (4)

In equation (3) and (4), the rγ,γ′ and tγ,γ′ factors de-note the reflection and transmission coefficients for theincident electronic wave function corresponding to theγ channel, which is reflected or transmitted into the γ′

channel. Here, by channel we understand one of the totalnumber Γ of the allowed eigenstate solutions for the gen-eralized Bloch phase factor in Eq.(2), which determinethe dispersion branches and the evanescent waves for theelectronic structure of the lead system. The cγ(k) termdenotes hence the lead Hamiltonian eigenvector whichcorresponds to the γ electronic eigenstate, and serves asa weighting factor in equations (3) and (4). For a com-plete description of the scattering processes it is essentialto know the evanescent as well as the propagating elec-tronic eigenstates [33].Such formulation of the problem is similar to those

presented in [29] and [49]. However, in contradiction tothe mentioned techniques, we use equations (3) and (4) inorder to directly rewrite the equations of motion for theregion outside the nanojunction domain [33], [35], [36].In this manner we create an irreducible vector of the wavefunction coefficients for the nanojunction domain and itsmatching regions in the input and output leads, in thefollowing way

c(k) = [z(k) × cnano(k)] + I(k), (5)

where z(k) is the N + 4M by N + 2M matrix composedof the phase factors which are mapped onto the atoms inthe matching region outside the irreducible nanojunctiondomain. The vector cnano(k) incorporates the vector ofwave function coefficients on the nanojunction domain,and the reflection and transmission coefficients for thewave function in the matching region in the input andoutput leads. The number of reflection and transmissioncoefficients is equal to the number of scattering chan-nels in an appropriately constructed Hilbert space. Thevector I(k) decouples terms which describe the incomingwave. On the basis of Eq.(5), we reduce next the resul-tant N + 2M set of equations for the irreducible region,to derive the following square matrix of inhomogenouslinearized equations

[ES−M(z,k)]× cnano(k) = −I(k). (6)

S, the overlap matrix, and M, composed of the Hamilto-nian elements and appropriate phase factors, are N+2Mby N + 2M square matrices, whereas I(k) is the I(k)vector supplemented by the corresponding Hamiltonianterms. We observe that, contrary to the Green’s functionmethods, Eq.(6) allows us to solve the scattering problemdirectly for real energies.Moreover, Eq.(6) yields directly the tγ,γ′ = tγ,γ′(E)

and rγ,γ′ = rγ,γ′(E) scattering amplitudes, and the wavefunction coefficients in the irreducible boundary domain,as a function of the electronic energy E or wave vec-tor k. Here E = Eγ(k) denotes the dispersion relationfor the propagating waves of the electrons in the firstBrillouin zone of the leads. It is these E = Eγ(k) elec-trons, incident from the perfect leads, which scatter atthe nanojunction.In what follows, the tγ,γ′ = tγ,γ′(E) and rγ,γ′ =

rγ,γ′(E) amplitudes yield the required Landauer-Buttikertotal transmission and reflection probabilities on thenanojunction, T (E) and R(E)

R(E) =∑

γ,γ′

vγ′

vγ|rγ,γ′(E)|

2, (7)

T (E) =∑

γ,γ′

vγ′

vγ|tγ,γ′(E)|

2, (8)

summing over all incident, reflected, and transmittedwaves. To ensure the unitarity character of the scatter-ing matrix, the amplitudes are normalized with respectto their electronic group velocities, vγ and vγ′ , for elec-trons incident in channel γ and reflected or transmittedin channel γ′. The unitarity condition, T (E)+R(E) = 1,corresponds to the necessary conservation of energy.Finally, assuming the zero bias limit and spin degen-

eracy, the overall conductance is then simply written as

G(N,EF ) = G0T (EF ), (9)

where G0 = 2e2/h is the conductance quantum. Theelectronic conductance is given at the Fermi energy sincethe electrons at this level give the only significant contri-bution to the overall conductance from the Fermi-Diracdistribution.

III. NUMERICAL RESULTS AND DISCUSSION

In this section, we present the dynamic equations forthe MLNaW and DLCuCoW nanojunctions on the ba-sis of the theoretical discussion presented in section II.The schematic representations of the freely suspendedMLNaW and the supported DLCuCoW nanojunctionsbetween one-dimensional atomic electric leads are pre-sented in Fig.2(A) and Fig.2(B), respectively. For bothsystems each of their atoms is characterized by a one va-lence s-type electronic state, and the interactions withadjacent atoms have a nearest-neighbor range, which is

5

usually proposed for the Na atoms [44]. As for the con-sidered Cu-Co systems, it has also been demonstrated,in view of previous experimental results [40], that theirelectronic dynamics can be described within the s-likemonovalence approximation, suggesting that one chan-nel is sufficient to model the electronic transmission.For the numerical calculations of the electronic trans-

port across MLNaW and DLCuCoW nanojuntions pre-sented in this section, the overall Hamiltonian of thewire nanojunctions connected to the semi-infinite one-dimensional leads can be written on the basis of Eq.(1)in the following reduced form

H =∑

n

εn +∑

n

hn,n′(δn′,n+1 + δn′,n−1), (10)

where we neglect the summation over different electronicstates per site, and index m is simply replaced by n dueto the fact that only one atom per unit cell is assumedin our calculations.In the case of the MLNaW systems (see Fig.2(A)),

the Hamitonian elements are assumed to be consistentwith the Harrison theory convention [48]. In the pres-ence of local charge neutrality, all the binding energieswhich characterize the atoms on the MLNaW nanojuc-tions are equal to each other, whereas the correspondinghn,n′ parameters are distance dependent and defined as

hn,n′ ≡ ηℏ2

med2, (11)

where η = −1.32 is the dimensionless Harrison’s co-efficient for the s-type orbitals, me denotes the elec-tron vacuum mass, and d depicts the nearest-neighborinteratomic distance. For the MLNaW nanojunctions,Eq.(11) allows us to model the effective nearest-neighborinteractions by considering different interatomic spac-ings. On the other hand, the tight-binding parametersfor DLCuCoW nanojunctions (see Fig.2(B)) are assumedafter [40], to be distance independent.In particular, for the MLNaW, and DLCuCoW sys-

tems Eq.(6) can be reduced to the following explicit form

h−1,−2 −h−1,0 0 · · · 0

−h0,−1z1 E − ε0

. . ....

0. . .

. . .. . . 0

.... . . E − εN−1 −hN−1,NzN−1

0 · · · 0 −hN,N−1 hN,N+1

×

rc(0, k)

...c[(N − 1)d, k]

t

= −

h−1,−2

−h0,−1z−1

0...0

. (12)

The total transmission and reflection probabilities, forthe left and right leads, which are identical atomic wirespresenting equal group velocities dE/dk for the electrons,

h1,2

irreducible region

matching

domain

matching

domain

right leadDLCuCoW nanojunction

ε0

left lead

ε1 ε2 εN-2 εN-1 εNεN-3ε-1ε-2ε-3 εN+1 εN+2

h0,1h-1,0h-2,-1 hN-3,N-2 hN-2,N-1 hN-1,N hN,N+1

A

B

left leadMLNaW nanojunction

irreducible region

matching

domain

matching

domain

right lead

dLLdLLdLWdWWdWWdWWdWWdLWdLLdLL

ε-3 ε-2 ε-1 ε0 ε1 ε2 εN-3 εN-2 εN-1 εN εN+1 εN+2

hN,N+1hN-1,NhN-2,N-1hN-3,N-2h1,2h0,1h-1,0h-2,-1 hN+1,N+2h-3,-2

hN+1,N+2h-3,-2

FIG. 2: Schematic representations of the: (A) N-atomic lin-ear nanojunction made of sodium atoms (light green color)between sodium leads (dark green color); (B) N-atomic lin-ear nanojunction made of pairs of copper (orange color) andcobalt (blue color) atoms between copper leads. The cor-responding binding energies εn for sites n, and the nearest-neighbor couplings hn,n′ between sites n and n′ are depicted.Note that in order to keep the figures simple we show thenearest-neighbor coupling terms only for n < n′. Addition-ally, in (A) three different interatomic spacings are considered:dWW for the nanojunction wire, dLL in the leads, and dWL atthe contact. The detailed description of the electronic andstructural properties is presented in section III.

are related to the single channel amplitudes t(E) andr(E) in the following way

T (E) = |t(E)|2 and R(E) = |r(E)|2. (13)

At this point we would like to draw attention to the factthat the PFMT formulation of the scattering problempresented in equations (3-6), allows us to introduce addi-tional simplifications to the resultant equations. In par-ticular, the boundary terms of matrix [EI −M(z, k)] inEq.(12), known as a self-energies in the Green’s func-tion formalism, are real and energy independent in thePFMTmethod. This simplification can be also applied inthe general case of Eq.(6) when the interactions betweenneighbor atoms in the leads are symmetric.Note, furthermore, that since the s-state electrons of

the Na or Cu one-dimensional semi-infinite leads popu-late up to half of the available band states, one can showby symmetry that EF is equal to the binding energy ofthe lead atoms.

6

A. Monatomic Sodium wire nanojunctions

The PFMT model calculation is next applied nu-merically for the MLNaW wire nanojunctions betweensodium leads as in Fig.2(A). Our numerical results aredetermined for wires of various lengths, in particular theyare presented typically here for N ∈ [1, 6].

As stated in section II, binding energies of Na atomsare constant and equal -4.96 eV, while the distance de-pendent coupling terms are modeled using Eq.(11). Theinteratomic distance on the Na leads dLL = 3.64A is con-sidered to correspond to the bulk Na nearest-neighbordistance. However, d of Eq.(11) on the MLNaW nano-junction takes on a value of dWW = 3.39A which is theaverage interatomic distance for the finite Na wires [12].In contrast, the lead-nanojunction spacing is consideredfor two numerical values, dWL which is equal to 3.99Aand 3.29 A. These numbers correspond to a choice rangeof approximately 20% weaker (WC) and stronger (SC)lead-nanojunction couplings, respectively, compared tothe intrinsic lead coupling. This is a physically reason-able range to test. Taking into account the above valuesof the interatomic distances, we note that the N=1 caseis equivalent to the situation of the infinite chain withthe two bond defects, however this is not true for theremaining wire lengths due to the fact that dLL 6= dWW.In Fig.3, the overall transmission T (E) and reflection

R(E) probabilities are presented as a function of the inci-dent electronic energy E, for various wire lengths N , andfor the two different lead-chain coupling values SC andWC defined above. For each sub-figure of Fig.3, T (E)and R(E) are calculated independently, and the unitar-ity requirement for the scattering processes is checked bythe sum T (E) + R(E). Note that EF is set as a zeroenergy reference in these figures.

The positions of the transmission spectral resonancesvary as a function of the lead-nanojunction couplings, asexpected and as may be seen from Fig.3. Due to the mir-ror symmetry and assumed local charge neutrality in theconsidered MLNaW systems, all energy levels, excludingthe central peak for odd N , are distributed symmetri-cally with respect to the referentialEF , for both even andodd N . Analyzing the transmission spectra, the numberof the resonances is always equal to the number of theatoms in the considered MLNaW wire up to N = 3. Thisequality is also conserved for N = 4 for the strong SCcouplings, but reduced to N − 2 for the weak WC cou-plings. The N − 2 states are also observed for the fiveodd- and the six even-numbered wires for both SC andWC cases. The displacement of the resonances to outsidethe transmission spectra for any N is directly related tothe lead-nanojunction couplings for the considered sys-tem. This displacement happens when the resonancegoes to a localized state outside the leads monovalents-type conduction band. This effect is equally observedfor resonances and localized vibration states on atomicnanojunctions [33]. Further, when the MLNaW wire in-creases in length this displacement is quicker for the WC

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

TSC

(E) TWC

(E) R

SC(E) R

WC(E)

SSC

(E) SWC

(E)

Pro

babi

lity

A

N=1

Pro

babi

lity

TSC

(E) TWC

(E) R

SC(E) R

WC(E)

SSC

(E) SWC

(E)

B

N=2

TSC

(E) TWC

(E) R

SC(E) R

WC(E)

SSC

(E) SWC

(E)

Pro

babi

lity

C

N=3

Pro

babi

lity

TSC

(E) TWC

(E) R

SC(E) R

WC(E)

SSC

(E) SWC

(E)

D

N=4

TSC

(E) TWC

(E) R

SC(E) R

WC(E)

SSC

(E) SWC

(E)

Pro

babi

lity

E

N=5

Pro

babi

lity

TSC

(E) TWC

(E) R

SC(E) R

WC(E)

SSC

(E) SWC

(E)

E-EF [eV]

F

N=6

FIG. 3: The overall transmission T (E) and reflection R(E)probabilities as a function of energy for the MLNaW wirenanojunctions composed of N atoms, with strong (SC) orweak (WC) lead-wire couplings. Subfigures (A), (C) and (E)correspond to the odd-number N , (B), (D) and (F) to theeven-number. EF is set as a zero energy reference, and theunitarity condition is represented by the sum S(E).

7

connected wires than for the SC ones.

As mentioned previously, the only significant contri-bution to the overall conductance comes from electronicstates in the neighborhood of the Fermi level EF . Itis hence noticeable that for the odd-number wires, oneof the resonant maxima appears systematically at EF ,which yields as a consequence a conductance maximumfor the corresponding systems. In contrast, the even-numbered wires do not present any maxima at EF , whichoutcome results in successive minima for the conductanceof these systems.

The total conductance G(N,EF ) for the consideredMLNaW wires is presented in Fig.4, as a function ofthe wire length N , reflecting the maxima and minimaof the transmission spectra at EF , and exhibiting theusual conductance oscillations, here with a two-atom pe-riod. This situation relates directly to the interaction ofthe free electrons in the lead band with the electronicstates of the nanojunction wire. In particular, since theσ state for a sodium atom can take two electrons by thePauli principle, one notes that there are half-empty statesavailable for the odd-number wires while non availablefor the even-number wires. Since the Fermi energy EF

is equal to the binding energy of the Na atoms, the odd-number wires present a greater local density of statesLDOS accessible at the Fermi level D(EF ), comparedto that for the even-number wires. This ensures that thehopping mobility is unimpeded for the odd-number wires,whereas this is not the case for the even-number wires.The corresponding electronic transport is consequentlya maximum at ∼ G0 for the odd-number MLNaW wirenanojunctions, for both the weak WC and the strongSC couplings. The electronic conductance decreases forthe even-number wires, and this decrease varies with thelead-nanojunction couplings, it is ∼ 0.065G0 for SC and∼ 0.221G0 for WC, respectively. Furthermore, one ob-serves that the conductance of the even-number MLNaWwire nanojunctions increases slightly but monotonicallywith the increase of the wire length. This situation maybe explained qualitatively by arguing that the electricleads have a bigger influence on the LDOS at the Fermilevel for the short wires than for the longer ones. An in-teresting discussion on the problem of how the differenttypes of leads may influence the overall electronic con-ductance can be found in [12].

In summary, the electronic conductance determined inthis subsection presents hitherto unknown detailed trans-mission spectra, and confirms the well known even-oddconductance oscillation effect [50], which is previously re-ported for the MLNaW nanojunctions [9], as well as forother wire systems [11], [13], [16]. We also note that ourresults are in agreement with those obtained on the basisof the first-principle calculations in other model calcula-tions [11], [26], [38]. This clearly confirms the validity ofthe PFMT approach. The minor quantitative differencesbetween our results for the conductance and those pre-viously reported, for the same N -number MLNaW wiresystems, appear mainly due to the assumed structures of

1 2 3 4 5 6

0.7

0.8

0.9

1.0

Odd Strong coupling Even Weak coupling

Number of atoms

G(E

F) [G

0]

0.778 0.779 0.781

0.9360.9350.934

FIG. 4: The total electronic conductance G(N,EF ) at theFermi level as a function of the number of atoms on the ML-NaW wire nanojunction, in units of G0.

the electric leads.

B. Diatomic Copper-Cobalt wire nanojunctions

The second considered system in this work concernsthe supported DLCuCoW wire nanojunctions presentedin section II. Finite - Cu - Co - wire systems have beenexperimentally prepared previously as grouped adatomson the flat Cu(111) surface by Lagoute et al. [40], withthe use of the low-temperature scanning tunneling mi-croscope technique. Initially, these systems have beendeveloped as magnetic/nonmagnetic finite atomic wiresfor the investigation of their potential magnetic proper-ties. Subjects concerning their spin dynamics [51], theirstorage potential [52], and the observation of the Kondoeffect in their structure [53], have been indeed addressedrecently. However, their electronic transport propertieshave not been considered previously, despite their intrin-sic interest under a DLCuCoW configuration, and theirmechanical and thermodynamical stability at low tem-peratures. Note in this respect the increasing interestin the stability of such supported wire systems [54] forarbitrary ranges of temperature.As it was stated in section II, the electron dynam-

ics of the supported DLCuCoW nanojunctions can bediscussed within the s-like single band effective tight-binding model proposed in [40]. This possibility arisesfrom the existence of unoccupied confined quantumstates formed along the Cu-Co wire nanojunctions bylinking the orbitals of spz character. Such confined stateshave been observed in other supported wire systems likethe monatomic Cu [41] and Au [55], the diatomic Au-Pdnanojunction systems [56], and Fe nano-islands [57].In this context the possibility of using semiconductor

surfaces which present a band gap is equally interestingas a support for the atomic wires. These surfaces may

8

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 20.0

0.2

0.4

0.6

0.8

1.0

-2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

234567891011121314151617

T(E) R(E) S(E)

Pro

babi

lity

NCuCo

=2

A

T(E) R(E) S(E)

NCuCo

=7

B

E-EF [eV]

T(E) R(E) S(E)

Pro

babi

lity

NCuCo

=12

C

E-EF [eV]

T(E) R(E) S(E)

NCuCo

=17

D

Tran

smis

sion

pro

babi

lity

E-EF [eV]

Num

ber o

f Cu-

Co

pairs

E

FIG. 5: (A) - (D): The selected spectra of the transmission T (E) and reflection R(E) probabilities for respectively 2, 7, 12and 17 copper-cobalt atomic pairs in the DLCuCoW scattering region. The unitarity condition is represented by the sumS(E). (E): Surface plot of the transmission probabilities as a function of energy and number of the Cu-Co atomic pairs in theDLCuCoW scattering region. For all subfigures the EF is set at zero of energy.

electronically decouple from the supported wire orbitalsin certain cases, allowing one to study the electronics ofstable one-dimensional wire systems [54].

From our point of view the diatomic nanojunctionwires may be used to control the value of the overallelectronic conductance in low-dimensional nanoelectroniccircuits. In particular, the foreign cobalt atoms injecteddirectly into the monatomic copper wire, act as control-lable chemical defects which modify the initial systemand its electronic properties as a consequence. Further-more, the substrate supported configurations of the lead- DLCuCoW wire nanojunctions - lead, suggest a greatermechanical and thermodynamic stability than for thefreely suspended wires, which is one of the most impor-tant features required for future nanoelectronic devices.

Our model calculations for the electronic quantum con-ductance of the DLCuCoW systems are presented forwires composed of different numbers of periodically ar-ranged Cu-Co atomic pairs, as in Fig.2(B). The lengthof the DLCuCoW wire may characterized by the numberNCuCo of such pairs, and the results in this subsectionare hence presented in terms of this latter variable for agiven DLCuCoW nanojunction.

In the case of the DLCuCoW nanojunctions, we as-sume the Hamiltonian elements of Eq.(10) to have thevalues proposed in [40]. Following these studies, the bind-ing energy 3.31 eV is considered to be the same for allthe copper atoms, whereas for cobalt it is 2.96 eV. Thecoupling terms between copper nearest-neighbors on theleads, and between copper and cobalt nearest-neighborson the wire nanojunction, are close to each other withvalues of −0.95 eV and −0.94 eV, respectively.

It is important to note that the NCuCo = 1 nanojunc-

tion wire is a very particular case since it has quite adifferent symmetry from all other lengths and is open todifferent structural definitions; we have calculated the to-tal conductance for this particular case to be 0.9658G0,but will not use it in our discussion of the properties ofthe conductance G(NCuCo, EF ) for a general wire length.

In Fig.5(A)-(D) we present the detailed transmissionT (E) and reflection R(E) spectra of the DLCuCoWnanojunction, for the selected values of NCuCo = 2, 7,12, and 17. The resonance maxima and minima can beobserved as for the MLNaW wires, where EF is again setas a zero energy reference. However, the transmissionand reflection spectra are not symmetric with respectto this reference, due the slight difference between thecobalt and copper binding energies. The most impor-tant conclusion concerns the value of the transmissionprobabilities close to EF . The transmission spectra inFig.5(A)-(D) show strong resonant minima for both theeven- and the odd-number wires. In particular, we ob-serve that the value of the transmission at the Fermi leveldecreases when the length of the effective nanojunctionincreases. This trend is summarized in Fig.5(E) for theensemble of the transmission spectra for NCuCo valuesranging from 2 to 17. At the upper limit the conduc-tance goes to G(NCuCo = 17, EF ) ∼ 10−1G0 which issufficient for the purpose of the present calculation.

The overall electronic conductance G(NCuCo, EF ) as afunction of the DLCuCoW wire length in terms of Cu-Coatomic pairs, is presented in Fig.6, along with a fittingexponential function

G(NCuCo, EF ) = G(2, EF )exp[−α(NCuCo − 2)]

for NCuCo ≥ 2. (14)

9

Fig.6 resumes the transmission spectra at the Fermi en-ergy, it shows the strong monotonic decay of the elec-tronic conductance with the increase of the number ofCu-Co pairs of the nanojunction. In the correspondingEq.(14), G(2, EF ) = 0.8760G0 is the conductance of awire made up of two Cu-Co pairs, and α = 0.1630 denotesthe decay constant. Eq.(14) allows one hence to read thenumerical values of the conductance G(NCuCo, EF ) of theDLCuCoW wires for NCuCo ≥ 2 using a simple formulawhich is useful in the analysis of potential applications ofcorresponding nanocircuits. This behavior is fundamen-tally different from the typical conductance oscillationsobserved for odd- and even-number monoatomic wires.

0 2 4 6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

Number of Cu-Co pairs

G(E

F) [G

0]

PFMT results G(N

CuCo, E

F)

FIG. 6: The total electronic conductance G(NCuCo, EF ) inunits of G0, read at the Fermi level EF as a function of thenumber of Cu-Co pairs on the DLCuCoW nanojunction wire.The PFMT results are represented by open circles, and thefitting function by the red curve.

To understand the decrease of the conductance inFig.6, one has to take a closer look at the band struc-ture of the infinite -Cu-Co- diatomic chain which is theinfinite limit of the DLCuCoW nanojunction. We havecalculated the band structure for such an infinite di-atomic wire as in Fig.7. This presents a band gap of∆=0.35 eV and renders the system theoretically insulat-ing. This band gap, which corresponds directly to thedifference of the binding energies for cobalt and copperatoms, presents a differential ∼ 10−1 with respect to theCu binding energy. It is an effective potential barrierfor electrons between successive Cu and Co sites, andresults in a small but cumulative decrease of the conduc-tance as the DLCuCoW wire nanojunction increases inlength, when comparing to the theoretical maximum ofG0 for the pure atomic Cu wire. This direct interpre-tation is confirmed by the exponential form of Eq.(14),where exp[−α(N − 2)] = exp(−α)N−2, expressing themonotonic decay of the electronic conductance with theincreasing length of the DLCuCoW nanojunction.

Z

1

2

3

4

5

Co

Ene

rgy

[eV

]

=0.35 eV

Cu

FIG. 7: Band structure of the infinite variant of the DL-CuCoW wire nanojunction over the first Brillouin zone. Theband gap (∆=0.35 eV) is marked by two dotted lines, betweenthe copper (εCu) and the cobalt (εCo) binding energies. TheFermi energy for the half-filled bands is represented by thedashed line.

IV. SUMMARY

In this work we present a general method based onthe algebraic phase field matching theory (PFMT), tocalculate the electronic quantum transport across nano-junctions of arbitrary form between material leads of n-dimensions, n = 1, 2 or 3. This PFMT method is ap-plied in particular to calculate the quantum electronicconductance across wire nanojunctions held between one-dimensional electric leads. The presented PFMT formal-ism provides a compact and efficient approach for theanalysis of the electronic quantum transport for a widerange of wire nanojunctions [58].

The model calculations are carried out for nanojunc-tions made up of sodium atomic wires (MLNaW) betweenleads of the same element. Our calculated results for theelectronic quantum transport are shown to be a functionof the physical parameters of the wires, and exhibit thewell known oscillation behavior for even- and odd-numberwire lenghts. They are also in agreement with the first-principle results [11], [26], [38]. This clearly confirms thevalidity of the presented PFMT model.

Our model calculations are also carried out for the di-atomic copper-cobalt wires (DLCuCoW) held betweencopper leads. This system is selected for its mechani-cal and thermodynamical stability at low temperatures,which is one of the important features required for futurenanoelectronic devices. In contrast to the MLNaW sys-tem, the electronic quantum transport of the DLCuCoWwire nanojunction, at the Fermi energy, exhibits expo-nential decay with increasing wire length. This behavioris explained notably in terms of the band gap of the infi-nite variant of the DLCuCoW nanojunction wire. A re-lation, depicting analytically the numerical values of the

10

conductance G(NCuCo, EF ) of the DLCuCoW wires forany length NCuCo ≥ 2, is given and may help to simplifythe analysis of potential applications for correspondingnanocircuits.The presented PFMT approach, a transparent and

time-saving method, should be especially interesting forthe treatment of complex systems presenting multichan-nel conductance, and for the treatment of the elec-tronic quantum transport of nanojunctions which can ex-hibit electron-electron and electron-phonon interactionsat high temperatures or due to structural disorder.

V. ACKNOWLEDGMENTS

D. Szczesniak would like to note that this work hasbeen financed by the Polish National Science Center

(grant DEC- 2011/01/N/ST3/04492). He also acknowl-edges support from the French Ministry of Foreign Af-fairs, and the University of Maine 3MPL GraduateSchool. The authors would like to thank Pr Z. Bak forhis kindness and support throughout this work, and PrW.A. Harrison, Dr R. Szczesniak, and Dr J. Lagoute foruseful discussions.

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