One-dimensional transport in hybrid metal-semiconductor nanotube systems
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec One-dimensional transport in hybrid metal-semiconductor nanotube systems
M. F. Gelin
Department of Chemistry, Technische Universit¨at M¨unchen, D-85747 Garching, Germany
I. V. Bondarev
Department of Mathematics and Physics, North Carolina Central University, Durham, NC 27707, USA
We develop an electron transport theory for the hybrid system of a semiconducting carbon nano-tube that encapsulates a one-atom-thick metallic wire. The theory predicts Fano resonances inelectron transport through the system, whereby the interaction of electrons on the wire with nano-tube plasmon generated near-fields blocks some of the wire transmission channels to open up the newcoherent plasmon-mediated channel in the nanotube forbidden gap outside the wire transmissionband. Such a channel makes the entire hybrid system transparent in the energy domain whereneither wire, nor nanotube is indivudually transparent. This effect can be used to manipulate bythe electron charge transfer in hybrid nanodevices built on metal-semiconductor nanotube systems.
PACS numbers:
I. INTRODUCTION
Over the last decade, electron transport studies inquasi-one-dimensional (1D) nanostructures have resultedin important discoveries, such as conductance quantiza-tion and oscillatory length dependence, molecular rectifi-cation, negative differential resistance, hysteresis behav-ior, etc.[1–4] At present, peculiar mechanisms of electrontransport are fairly well understood for pristine atomicwires (AWs) [1], carbon nanotubes (CNs) and some CN-based components [4, 5]. Carbon nanotubes — graphenesheets rolled-up into cylinders of one to a few nanometersin diameter and up to hundreds of microns in length [6]— have been successfully integrated into miniaturizedelectronic, electromechanical, chemical devices and intonanocomposite materials [7, 8], and have found a varietyof applications in optoelectronics [9–17].Enormous potential of carbon nanotubes as buildingblocks for designing optoelectronic nanodevices stemsfrom their extraordinary thermomechanical stabilitycombined with unique physical properties that originatefrom quasi-one-dimensionality to give rise to a pecu-liar quasi-1D band structure featuring intrinsic, spatiallyconfined, collective electronic excitations such as exci-tons and plasmons [18–23]. Due to the circumferentialquantization of the longitudinal electron motion, real ax-ial (along the CN axis) optical conductivities of singlewall CNs consist of series of peaks E , E , ... , repre-senting the first, second, etc. excitons, respectively [seeFig. 1 (a)]. Imaginary conductivities are linked with thereal ones by the Kramers-Kronig relation, and so real inverse conductivities show the resonances P , P , ... next to their excitonic counterparts. These are weaklydispersive, low-energy ( ∼ − semiconducting CNs. In these systems, at low bias voltages not exceed-ing the CN fundamental band-gap, the CN itself doesnot have any intrinsic open channels to conduct electrons.Hence, there is no electron exchange between the CN andthe metallic AW encapsulated into it, with the transportbeing totally dominated by the AW alone, while at thesame time being affected by local quasi-static fields ofnanotube’s collective interband plasmon excitations.As an example, Fig. 1 shows same scale energy depen-dences for characteristic parameters to represent semi-conducting CNs, their near-field interaction with encap-sulated atomic types species, and one-dimensional (1D)metallic atomic wires, respectively. Panel (a) shows thedynamical axial conductivities for the semiconducting(11,0) and (16,0) CNs. Shown are the real conductivities,the imaginary conductivities, and the real inverse con-ductivities of relevance to the electron energy-loss spec-troscopy response function used in studies of collectiveplasmon excitations in solids [20–23, 77]. Peaks of thereal conductivities and those of the real inverse conduc-tivities represent excitons ( E , E ) and interband plas-mons ( P ), respectively [22–28]. Panel (b) shows localdensities of photonic states (DOS) for the non-radiativespontaneous decay (relative to vacuum) of a two-leveldipole emitter placed on the symmetry axis (inset) insidethe (11,0) and (16,0) CNs. Panel (c) shows the electrontransmission band for the free 1D metallic AW of 100sodium atoms with energy counted from the bottom ofthe fundamental band gap of the (11,0) and (16,0) CNs(top and bottom, respectively). All AW transport chan-nels are seen to be inside of the CN forbidden gap. How-ever, AWs encapsulated into CNs will experience near-field EM coupling to the CN interband plasmon modesrepresented by the large DOS resonances in panel (b).Interband plasmons are standing charge density wavesdue to the periodic opposite-phase displacements of theelectron shells with respect to the ion cores in the neigh-boring elementary cells on the CN surface. Such peri-odic displacements induce local coherent oscillating elec-tric fields of zero mean, but nonzero mean-square magni-tude, concentrated near the surface across the diameterthroughout the length of the CN [24, 25]. The electricdipole interaction of atoms on the wire with these near-fields will largely affect the AW transmission propertiesas well as the properties of the entire hybrid system.The coupling strength of an individual atom to theplasmon-induced near-fields inside the nanotube can be FIG. 1: (Color online) (a) Dimensionless (normalized by e / π ~ ) surface axial conductivities σ zz for the semiconduct-ing (11,0) and (16,0) CNs. Peaks of Re σ zz and Re(1 /σ zz )represent excitons ( E , E ) and interband plasmons ( P ),respectively. (b) Photonic DOS for non-radiative spontaneousdecay with CN plasmon excitation for a two-level dipole emit-ter on the symmetry axis (inset) of the (11,0) and (16,0) CN.(c) Electron transmission band for the free AW of 100 sodiumatoms with energy counted from the bottom of the fundamen-tal band gap E g of the (11,0) and (16,0) CNs (top and bottom,respectively); also shown are the AW Fermi energies E F andCN first interband plasmon energies E p . Conductivities areobtained using the ( k · p )-scheme by Ando [18]. DOS func-tions are calculated as described by Bondarev and Lambinin Refs. [29, 30]. Transmission is plotted within the nearest-neighbor hopping model as discussed by Mujica et al. and byGelin and Kosov in Refs. [89–91]. See Sec. V for more details. estimated as follows. Modeled by a two-level system, anatom (ion) interacts with the CN medium-assisted fieldsvia an electric dipole transition d z = h u | ˆ d z | l i between itslower | l i and upper | u i states [29, 30], with the CN sym-metry axis set to be the z -quantization axis [Fig. 1(b),inset]. Transverse dipole orientations can be neglected inview of the strong transverse depolarization in individualCNs [21, 78–80]. The atom–CN electric dipole couplingconstant is then given by ~ g = (2 πd z ~ ˜ ω A / ˜ V ) / [32, 82],where ˜ ω A is the effective transition frequency in reso-nance with a local CN-medium-assisted field mode. Theeffective mode volume ˜ V ∼ πR cn (˜ λ A /
2) for the CN of ra-dius R cn . Evaluating d z ∼ er A ∼ e ( e / ~ ˜ ω A ), where r A isthe linear size of the atom (estimate valid for quantumsystems with Coulomb interaction [81]), and introducingthe fine structure constant α = e / ~ c = 1 / ~ g = (2 α /π ) / ( ~ c/R cn ), to give ~ g ∼ . ∼ ~ γ c (˜ ω A ) = 6 π ~ c / ˜ ω A ξ (˜ ω A ) ˜ V [function ξ rep-resents the local photonic DOS at the atomic location,also called the Purcell factor [82]; shown in Fig. 1(b)],one has the ratio g/γ c ∼ ≫ | l i and | u i transition statesare here a subset of the one-dimensional electronic bandstates, and so are not as clearly defined as those for anindividual atom, to most likely result in a weaker non-resonance CN–AW coupling strength.Here we study the inter-play between the intrinsic 1Dconductance of metallic AWs and CN mediated near-fieldeffects for semiconducting single wall CNs that encapsu-late atomic wires of finite length. We use matrix Green’sfunctions formalism to develop an electron transfer the-ory for such a complex hybrid quasi-1D CN system. Thetheory predicts Fano resonances in electron transmissionthrough the system. That is the AW–CN near-field inter-action blocks some of the pristine AW transmission bandchannels to open up new coherent channels in the CN for-bidden gap outside the pristine AW transmission band.This makes the entire hybrid system transparent in theenergy domain where neither of the individual pristineconstituents, neither AW nor CN, is transparent. Theeffect can be used to control electron charge transfer insemiconducting CN based devices for nanoscale energyconversion, separation and storage [83–86].The paper is structured as follows. Section II formu-lates the theoretical model for CN-mediated AW trans-mission. Section III presents analytical expressions de-rived for the transmission coefficient. The expressions ob-tained are analyzed qualitatively in Section IV, and thennumerically in Section V. Section VI discusses the model approach and the approximations used. A brief summaryof the work is given in Section VII. Appendix A derivesmathematical expressions presented in Section III.
II. THE MODEL
This Section formulates the model for the plasmon-mediated electron transport through the hybrid metalAW encapsulating single wall semiconductor CN system.The detailed physical justification for the model and itstheoretical interpretation are discussed in Section VI.
A. The Hamiltonian of the hybrid CN–AW system
The quantum system under consideration is a metallicAW encapsulated into a single-wall semiconducting CN.The system is assumed to be attached to two electrodes(leads) kept at a certain bias voltage. The quantum me-chanical observable of interest is the steady-state electroncurrent through the CN–AW system.We write the total Hamiltonian of the entire system asthe sum of the Hamiltonians for the AW, the CN, andtheir interaction as followsˆ H = ˆ H AW + ˆ H CN + ˆ H int . (1)To describe the AW of N atoms (lattice sites) in length,we adopt the standard second quantized tight-bindingmodel Hamiltonian with the nearest neighbor electronhopping rate V and the electron on-site energy E ,ˆ H AW = E N X k =1 B † k B k + V N − X k =1 (cid:16) B † k B k +1 + B † k +1 B k (cid:17) . (2)Here, the operators B † k and B k create and annihilate, re-spectively, the electronic excitations on site k of the AW(see, e.g., Refs. [89–91]). They obey the Pauli commuta-tion rules [ B k , B † n ] = δ kn (1 − B † k B n ).We assume the bias voltage not to exceed the funda-mental bandgap E g [ = E (11) g ] of the semiconducting CNas it shows in Fig. 1 (a)–(c). Then, regardless of whetherthe AW transmission band is narrower [as in Fig. 1 (c)]or broader than E g , the electrons are transported in be-tween the leads at energies E F . E < E g inside the for-bidden gap of the CN. As this takes place, the CN doesdisturb the transport due to the quasi-static near-fieldsof weakly dispersive, low-energy ( ∼ − H CN = X n Z ∞ dω ~ ω ˆ f † ( n , ω ) ˆ f ( n , ω ) . (3)In this equation, the scalar bosonic field operatorsˆ f † ( n , ω ) and ˆ f ( n , ω ) create and annihilate, respectively,the surface EM excitation of the frequency ω at an ar-bitrary point n = R n = { R CN , ϕ n , z n } that representsthe position of a carbon atom [nanotube lattice site —Fig. 1 (b), inset] on the surface of the CN of radius R CN ,[ ˆ f ( n , ω ) , ˆ f † ( n ′ , ω ′ )] = δ nn ′ δ ( ω − ω ′ ). Summation is takenover all the carbon atoms on the entire CN surface. Sinceonly one plasmon resonance is located inside E g , Eq. (3)can further be simplified to take the formˆ H CN = E p ˆ f † ˆ f . (4)Here, the operators ˆ f † and ˆ f ([ ˆ f , ˆ f † ] = 1) create and anni-hilate collective interband plasmon excitations of energy E p [Fig. 1 (a),(b)] that are delocalized all over the CNsurface in accordance with the correspondence relation X n ˆ f † ( n , ω ) ˆ f ( n , ω ) = ˆ f † ˆ f δ ( ω − E p / ~ ) . (5)The CN–AW interaction can then be written in the formˆ H int = N X k =1 µ k (cid:16) B k ˆ f † + B † k ˆ f (cid:17) , (6)where µ k is the AW–CN dipole coupling constant for site k of the AW. We assume it to be the same for all of theAW sites, that is µ k = µ . ~ g = s πd z ~ ˜ ω A ˜ V ≈ r α π ~ cR cn (7)in what follows, as discussed in the introduction above.The Hamiltonian in Eqs. (1)–(7) belongs to the well-known family of Fano-Anderson Hamiltonians [87, 88].However, it describes a physical picture opposite to thatnormally refereed to as the Fano–Anderson model. Thelatter deals with a bound (e.g., localized on a defect) elec-tron state inside (or outside) of the continuum of scat-tering (band, or free) electron states [87]. In our case,the band electron states represented by the AW Hamil-tonian (2), interact with the CN collective interband plas-mon excitations described by the Hamiltonian (4). Theseare standing charge density waves due to the periodicopposite-phase displacements of the electron shells withrespect to the ion cores in the neighboring elementarycells on the CN surface [24, 25]. They are extended co-herently all over the entire surface of the CN as describedby Eqs. (3) and (5). The main feature of the standardFano–Anderson model is still there though, offering twodifferent electron transmission paths. They are: (i) thedirect transfer through the AW, and (ii) the transfermediated by quasi-static near-fields due to the CN col-lective interband plasmon excitations. Thus, the modelwe present here is a non-trivial extension of the Fano-Anderson model to cover coherently delocalized electronstates, such as collective plasmon excitations, in additionto localized (defect-type) states studied originally. B. The matrix representation of the operators
Using the relevant single-quantum Hilbert space basisof N +1 basis vectors as follows n B † k | i o k =1 , ··· ,N , ˆ f † | i , with | i being the vacuum state of the entire system, onecan convert the total Hamiltonian (1) into the matrixrepresentation as follows H = E V . . . µV E V . . . µ V E . . . µ ... ... ... . . . ... ... ...0 0 0 . . . E V µ . . . V E µµ µ µ . . . µ µ E p . (8)Here, rows and columns 1 , . . . , N refer to the AW, enu-merating its sites, and the ( N +1)-st row/column refersto the CN interband plasmon mode. Similar matrix rep-resentations can be written for any other operator of rel-evance to the problem.Matrix (8) represents the tight-binding Hamiltonianfor the entire hybrid CN–AW system. Its diagonal matrixelements H , . . . , H NN are the site energies of the AWsites that incoming electrons hop through; its off-diagonalelements specify the rates at which the hopping occursthrough the AW, and its eigen energies determine theresonances of the electron transmission through the CN–AW system. The CN–AW coupling term in the ( N +1)-st row/column modifies resonance transmission energiesand hopping pathways. To explore the role of this latteringredient of the model is the goal of this work. C. The transmission coefficient
To calculate the transmission coefficient through themetal AW encapsulated into a semiconducting CN, wefollow the Landauer formalism [1–4], in which the pres-ence of the leads is accounted for by the electron self-energy operator Σ ( E ) = Λ ( E ) − i ∆ ( E ) , (9)Here, the real and imaginary part represents the shift andbroadening, respectively, of the terminal electron energylevel E due to the coupling between the wire and theleads. This operator enters the Green’s function of oursystem as follows [89–92] G ( E ) = [ E − H − Σ ( E )] − . (10)With the left and right leads attached to the 1-st and N -th AW site, respectively, the transmission probabilityis proportional to the Green’s function matrix element G N , according to the general scattering matrix formal-ism as applied to atomic and molecular wires [89]. Then,the electron transmission coefficient T ( E ) is given by T ( E ) = 4∆ ( E ) | G N ( E ) | . (11)Here, ∆ is the imaginary part of the self-energy functionΣ( E ) defined in terms of the only two non-zero matrixelements of the self-energy operator (9) as followsΣ( E ) = Λ( E ) − i ∆( E ) ≡ Σ ( E ) ≡ Σ NN ( E ) , (12)with the left and right leads assumed to be identical.The transmission coefficient (11) is the probability foran electron to be transferred between the leads at a con-stant energy E , that is 0 ≤ T ≤
1. The conductance ofthe CN–AW system in the linear regime, whereby volt-ages are small and temperatures are low, is given by theLandauer formula (in units of e / π ~ )g = T ( ǫ F ) , (13)where ǫ F is the Fermi energy of the leads [89]. This quan-tity only depends on the electronic structure of the wireand the leads, and does not depend on the field applied. III. THE TRANSMISSION OF ELECTRONS:EXACT ANALYTICAL SOLUTION
To evaluate the matrix element G N of the Green’sfunction (10) in Eq. (11), we introduce the matrix h ≡ E − H − Σ ( E ) , and follow the rules of calculating the matrix elements ofits inverse. One obtains G N ( E ) = h N det( h ) , (14)with h N representing the N h .Next, we use the result of the matrix Green’s functionpartitioning technique developed in Ref. [89], wherebydet( h ) = Σ D N − + 2 Σ D N − + D N (15)with Σ given by Eq. (12), and h N = ( − N +1 S N − , (16)with D N and S N − being the determinant and the N H − E , respectively (both determinedby the AW lattice site number N ; subscripts to indicatethat the latter is a polynomial of degree one less than theformer). The matrix H is given by Eq. (8).Next, if we make a standard assumption that the iden-tical leads are made of a broadband metal with the half-filled conduction band, as discussed in Ref. [90], then thereal part Λ of the self-energy function (12) cancels out.The imaginary part ∆ takes the energy independent form ∆ = V S /γ , with V S ≡ V = V N representing the lead–wireterminal chemisorption coupling constant and γ beingthe lead metal half-bandwidth. In view of Eqs. (14)–(16),the transmission coefficient (11) then takes the form T ( E ) = 4∆ (cid:12)(cid:12)(cid:12)(cid:12) S N − ∆ D N − + 2 i ∆ D N − − D N (cid:12)(cid:12)(cid:12)(cid:12) . (17)Here, the quantities D N and S N are functions of the AWlattice site number N , which are given by the analyticalexpressions as follows (see Appendix A for the derivation) D N = ε p d N + (18) µ ε + 2 V (cid:26) − N d N + 2 Vε + 2 V (cid:2) ( − N V N − V d N − − d N (cid:3)(cid:27) ,S N = ε p V N + µ ε + 2 V (cid:2) − ( N + 1) V N + ( − N d N (cid:3) , (19)where ε = E − E, ε p = E p − E , (20)and d N = λ N +11 − λ N +12 λ − λ (21)with λ , = ε ± p ε − V . (22) IV. QUALITATIVE ANALYSIS
The transmission coefficient given by Eqs. (17)–(22) isthe key quantity to describe the electron transfer throughthe atomic wire encapsulated into a carbon nanotube.There are two parameters to control the AW–CN cou-pling there. They are the atom-plasmon coupling con-stant µ and the plasmon energy detuning ε p in Eqs. (7)and (20), respectively. From Eqs. (18) and (19), we seethat it is the ratio µ /ε p = µ / ( E p − E ) that determinesthe T ( E ) energy dependence. [The conductance (13) isdetermined by µ / ( E p − ǫ F ), accordingly.] In view of this,increasing µ affects T ( E ) the same way as decreasing ε p ,and vice versa. Therefore, we restrict ourselves to theanalysis of the T ( E ) behavior versus µ in what follows. A. Pristine AWs
The pristine AW case follows from the general equa-tion (17) if one substitutes µ = 0 there. Then, we recoverthe known quantum wire transmission formula [89, 91] T ( E ) = 4∆ (cid:12)(cid:12)(cid:12)(cid:12) V N − ∆ d N − + 2 i ∆ d N − − d N (cid:12)(cid:12)(cid:12)(cid:12) . (23)This shows that there are two possible, qualitatively dif-ferent electron transport regimes there for pristine AWs,depending on whether | ε / V | <
1, or | ε / V | > | ε / V | <
1, the roots λ , of Eq. (22)are complex, yielding d N in Eq. (21) of the form d N = sin[( N + 1) φ ]sin φ V N (24)with φ given by the roots of the equation cos φ = ε / V .In this case Eq. (23) takes the form T ( E ) = (cid:12)(cid:12)(cid:12)(cid:12) ξ sin φ sin[( N +1) φ ] + 2 iξ sin( N φ ) − ξ sin[( N − φ ] (cid:12)(cid:12)(cid:12)(cid:12) with ξ = ∆ /V = ( V S /γ )( V S /V ) ≪ V S < γ and V S . V to be fulfilled. This is the resonance tun-neling regime, in which the energies of the transmissionmaxima are approximately given by the roots of the equa-tion sin[( N + 1) φ ] = 0 (corresponding to the minima ofthe denominator) as follows φ maxk = πkN + 1 , k = 1 , , · · · , N, (25)to result in the resonance transmission band of precisely N energy channels for the AW of N atoms in length.They are E maxk = E − V cos φ maxk = E − V cos πkN + 1 , (26) T maxk = T ( E maxk ) = 11 + ξ cos φ maxk . These transmission maxima channels interchange withtransmission minima given approximately by the rootsof the equations sin[( N + 1) φ ] = ± φ mink = π ( k + 1 / N + 1 , k = 1 , · · · , N − , (27)yielding E mink = E − V cos φ mink = E − V cos π ( k + 1 / N + 1 , (28) T mink = T ( E mink ) = 4 ξ sin φ mink (cid:2) − ξ cos 2 φ mink (cid:3) + 4 ξ cos φ mink . The magnitude of T mink is seen to increase with ∆ (aslong as ξ = ∆ /V < T mink = 4 ξ / (1 + ξ ) for E mink in the center of the trans-mission band ( E mink = E , whereby cos φ mink = 0), to give T mink ≈ (2∆ /V ) for ∆ ≪ V (weak AW-lead coupling) and T mink ≈ ∼ V (strong AW-lead coupling). In the case where | ε / V | >
1, the roots λ , of Eq. (22)are real. Approximating them with their respective lead-ing terms of the power series expansions in | V /ε | < λ , ≈ ( ε ± | ε | ) /
2. Then, d N in Eq. (21) is esti-mated to go asymptotically as d N ≈ ε N = ( E − E ) N . The transmission coefficient (23) takes a non-resonantform then that scales with N exponentially, T ( E ) ≈ ξ (cid:16) ε V (cid:17) − N = 4 ξ (cid:18) E − EV (cid:19) − N , (29)showing a fast exponential decrease as N increases. For | ε / V | &
1, on the other hand, λ , = ( ε ± | ε | ǫ ) / ǫ = p − (2 V /ε ) now being a small positive parameter,to result in the leading term d N ≈ ( N + 1) (cid:16) ε (cid:17) N = ( N + 1) (cid:18) E − E (cid:19) N of the power series expansion in ǫ . In this regime, thetransmission coefficient (23) is an energy independentconstant decreasing with N as follows T ( E ) ≈ ξ ( N + 1) . This can also be obtained using Eq. (24) for | ε / V | . B. Coupled CN–AW system
Non-zero µ changes drastically the electron transportthrough the coupled CN–AW system. Intuitively, onewould expect additional transmission resonances (Fano-like [88]) to appear in the transmission coefficient (11).In this section we analyze Eq. (17) qualitatively to showthat this is indeed the case. This analysis is continued inSections V and VI to discuss the numerical results.Dividing the numerator and denominator of Eq. (17)by V N , one obtains T ( E ) = (cid:12)(cid:12)(cid:12)(cid:12) ξ ρ N − ξ δ N − + 2 iξ δ N − − δ N (cid:12)(cid:12)(cid:12)(cid:12) , (30)where ρ N = S N /V N and δ N = D N /V N . With ξ ≪ δ N = 0, to minimize the denominator. If µ = 0,this becomes d N = 0, according to Eq. (18), to bring usback to Eqs. (26) and (29) for | ε / V | < | ε / V | > µ = 0 and | ε / V | >
1, we see from Eq. (18) that there exists onemore possibility to make δ N close to zero. This is where ε p ( ε + 2 V ) = N µ , to result in two additional energylevels as follows E , = 12 (cid:20) E + 2 V + E p ± q ( E + 2 V − E p ) + 4 N µ (cid:21) . (31)As the top and bottom edges of the pristine AW tun-neling band are given by E = E ± V [see Eq. (26)], the E (higher energy) level falls into the domain E > E +2 V (or ε / V < −
1) of the exponentially small transmissionof the pristine AW, thereby opening an extra transmis-sion channel in this opaque area. At fixed µ = 0, raisingin energy with N , this channel stays within the CN for-bidden gap as long as N µ < ( E g − E p )( E g − E − V ),crossing into the CN conduction band when the inequal-ity changes its sign. The E (lower energy) level falls intothe resonance tunneling band E − V < E < E + 2 V (or,equivalently, | ε / V | <
1) of the pristine AW and remainsthere as long as the inequality
N µ < V ( E p − E + 2 V )holds true, lowering in energy with N . For large enough N this inequality changes the sign, while the channelgoes into the exponentially small transmission domain E < E − V (or ε / V >
1) of the pristine AW.Inside the pristine AW transmission band, close to thecenter of the band where | ε / V | ≪
1, in Eq. (24) one hassin φ = sin[arccos( ε / V )] ≈ sin( π/
2) = 1 to within termsof the second order in | ε / V | . Then, Eq. (24) becomes d N V N ≈ sin (cid:20) ( N +1) π (cid:21) = cos (cid:18) N π (cid:19) = i N − N . (32)Using this in Eqs. (18) and (19) to evaluate ρ N − , δ N , δ N − , and δ N − in Eq. (30), one can simplify this equa-tion to the form T ( E ) ≈ ξ (cid:2) α N ( E )+ µ sin( N π/ (cid:3) [(1 + ξ ) q cos( N π/ η ) + (1 − ξ )( − N µ ] + 4 ξ [ q sin( N π/ η ) − ( − N µ ] , (33)where α N ( E ) = ε p ( ε + 2 V ) − N µ = ( E − E )( E − E ) , (34) η = arccos( α N +1 /q ), and q = q α N +1 + µ . If µ = 0, thenEqs. (32)–(34) bring us back to Eqs. (26) and (28) [with φ max ( min ) k = π/
2] for odd and even N , respectively. Fornon-zero µ the factor in the brackets in the numerator ofEq. (33) becomes either α N ( E ) if N is even, or α N ∓ ( E )if N is odd of the form 4 n ± n = 1 , , , ... being posi-tive integers. Then, in view of Eq. (34) and the fact thatthe denominator of Eq. (33) is always non-zero for µ = 0,the transmission coefficient T ( E = E ) = 0 both in theformer and in the latter case, for N and N ∓ N is fixed. Thus, the E energy level in thepristine AW transmission band blocks the transmissionentirely, resulting in the Fano resonance , in full accordwith the total resonant reflection effect of the standardFano-Anderson model for a bound state within the con-tinuum of scattering states [87, 88]. The Fano resonancewidth Γ can be estimated from the focal parameter of theparabola one has in the numerator of Eq. (33) by setting α N,N ± ≈ α N ≈ ( E − E )( E − E ) for not too small N inthe neighborhood of E according to Eq. (34), whereas α N +1 ≈ α N ≈ E , there should be E ≈ E ,and then E ≈ E p + 2 V by Vieta’s theorem, to result inΓ ≈ µ κ ( ξ, N ) | E − V − E p | , (35)where κ ( ξ, N ) = (cid:20) cos (cid:18) N π (cid:19) − ( − N (cid:21) + 14 ξ (cid:20) (1 + ξ ) sin (cid:18) N π (cid:19) − (1 − ξ )( − N (cid:21) . We see that the Fano resonance width is directly pro-portional to the square of the AW–CN coupling strengthand varies strongly with N , while also being dependenton the relative position of the CN plasmon resonance en-ergy and the pristine AW transmission band center. For E p ≈ E + 2 V , as it shows in Fig. 1 (c) in particular,Eq. (35) results in Γ ∼ µ /V in full accord with the stan-dard Fano-Anderson model [88].Outside of the pristine AW transmission band, in thedomain of the exponentially small transmission where | ε / V | >
1, Eq. (30) can be simplified by approximatingthe functions ρ N − , δ N , δ N − , and δ N − with their re-spective leading terms in | ε / V | , while keeping in mindthat ξ ≪
1. This brings one to the following expression T ( E ) ≈ ξ µ α N ( E )( ε / V ) + [ α N ( E )+ µ ] ξ (36)to allow for evaluating various asymptotic regimes of theelectron transfer through the additional plasmon-assistedtransmission channels. One can see, in particular, thatwhen E = E , , whereby α N = 0, Eq. (36) yields theperfect transmission T ( E , ) = 1. In the vicinity of the(more interesting) higher energy resonance transmissionchannel E ≈ E (and similar for E ≈ E ), Eq. (34) can bewritten as α N ≈ ( E − E )( E − E ) ∼ ( E − E )2 µ √ N ,while | ε | ≈ | E − E | ∼ µ √ N , for N large enough as isseen from Eq. (31). This brings the transmission coeffi-cient (36) to the form T ( E ) ≈ (∆ /N ) ( E − E ) + (∆ /N ) . (37)We see the plasmon-assisted transmission energy channelto have the Lorentzian lineshape of the half-width-at-half-maximum ∆ /N , that is proportional to the AW–leadcoupling and inversely proportional to the AW length.For energies far from E , resonances outside of thepristine AW transmission band, the function α N inEq. (34) is non-zero, allowing for two possible plasmon-mediated electron transmission regimes. If the AW is nottoo long, one can approximate α N ≈ ε p ( ε + 2 V ) inEq. (34). Then Eq. (36) takes the form T ( E ) ≈ (cid:20) µ ( E − E p )( E − E )( E − E − V ) (cid:21) , (38)in which | ( E − E ) / V | > E = E p . In this regime,the transmission coefficient shows no wire length depen-dence. For long enough AWs, one has α N ≈ − N µ , whichbeing substituted into Eq. (36), results in T ( E ) ≈ (cid:18) E − E (cid:19) N . (39)This algebraic ( ∼ N − ) transmission length dependenceis much slower than the exponential transmission lengthdependence of Eq. (29) for pristine AWs. It comes fromthe slow plasmon-mediated transmission channel narrow-ing ∼ N − in Eq. (37). The inverse quadratic length de-pendence in Eq. (39) contrasts with the inverse linear length dependence of the phonon-mediated transmissiontypical of quasi-1D molecular wire systems [93]. V. NUMERICAL ANALYSIS
We assume that the leads and the AW are made of thesame metal, and that the AW incapsulating carbon nano-tube is end-bonded into the leads [94, 95]. (Other possi-bilities for CN–lead contacts can be found in Ref. [96].)The equilibrium band lineup inside the hybrid AW–CNsystem is then determined by the self-consistent chargeredistribution through the entire metal-(CN–AW)-metaljunction [95], to position the Fermi level E F of the systemat equilibrium in the middle of the CN forbidden gap theway it occurs for unpinned semiconducting CNs [94]. Weanalyze two representative semiconducting CNs to showtwo possibilities for relative arrangement of the CN inter-band plasmon resonance with respect to the encapsulatedAW transmission band. They are the (11,0) CN and the(16,0) CN. With energy counted from the bottom of theCN fundamental bandgap E (11) g = E g , by summing upthe first bright exciton excitation and binding energies,1 .
21 and 0 .
76 eV, as reported by Ma et al. [97] and Ca-paz et al. [98], respectively, one arrives at E g = 1 .
97 eVfor the (11,0) CN. This makes E = E F = 0 .
985 eV forthe AW on-site energy and the equilibrium Fermi en-ergy of the complex hybrid system of the (11,0) CN en-capsulating the AW of the same metal as that of leads.For the (16,0) CN, we evaluate E g = 1 .
47 eV numer-ically using the ( k · p )-method by Ando [18], to give E = E F = 0 .
735 eV for the AW encapsulating hybrid(16,0) CN system. For the AW, we use Na metal param-eters, with the electron effective mass m ∗ = 1 . m [99] ( m is the free electron mass) and the lattice constant a = 4 .
225 ˚A [100]. This yields the nearest neighbor elec-tron hopping rate V = ~ / m ∗ a = 0 .
21 eV. We choose theAW–lead coupling to range within ∆ ∼ . − . µ varies broadly in accordance with Eq. (7) as discussedin the introduction above.Figure 1 (a), (b), (c) shows for the (11,0) and (16,0)CNs the low-energy real, imaginary and real inverse con-ductivities, and the local photonic DOS resonances orig-inating from them, to scale with the finite-length (100atoms) sodium AW transmission bands. To calculate thegraphs in (a), we used the ( k · p )-method by Ando withthe exciton relaxation time 100 fs for both CNs (con-sistent with earlier estimates [102, 103]). Many-particleCoulomb correlations are included in these calculationsby solving the Bethe-Salpeter equation in the momen-tum space within the screened Hartree-Fock approxima-tion as described in Ref. [18]. Real conductivities consistof series of peaks ( E , E , ... ) representing the 1st, 2nd,etc., excitons. As imaginary conductivities are linkedwith real ones by the Kramers-Kronig relation, the realinverse conductivities show the interband plasmon peaks P , P , ... right next to E , E , ... (first observed inRef. [20]; see Refs. [22–28] for more details). The graphsin (b) show the local photonic DOS functions for theexcited state non-radiative spontaneous decay of a two-level dipole emitter placed on the symmetry axis (inset)of the (11,0) and (16,0) CN. Details of these calculationsand similar graphs for a variety of other geometry con-figurations can be found in Refs. [27–34]. Comparing (a)and (b), we see the sharp single-peak DOS resonancesto come from the interband plasmons of respective CNs.These are responsible for the AW–CN near-field couplingin hybrid CN systems. The coupling is due to the virtual(vacuum-type) EM energy exchange between the AW andthe CN to create and annihilate plasmons on the CNsurface as described by the interaction Hamiltonian (6).Comparison with (c), which shows transmission bandsfor the 100 Na atoms chain calculated per Eqs. (23) and(24) with ∆ = 0 .
05 eV to scale E g for the (11,0) CN (top)and (16,0) CN (bottom), indicates that the 1st interbandplasmon energy E p ∼ E F + 2 V and can be located bothoutside ( E p & E F + 2 V ) and inside ( E p . E F + 2 V ) of thefree AW electron transmission band.These two possibilities are simulated and presented inFigure 2 (a) and (b). Here, we show the transmission asgiven by Eqs. (17)–(22) for the AW of 10 sodium atomsin length inside the (11,0) CN [(a), E p & E F + 2 V ] andinside the (16,0) CN [(b), E p . E F + 2 V ] under the AW–lead coupling ∆ = 0 .
05 eV with the AW–CN coupling µ varied from 0 up to 0.3 eV over the energy range to coverthe entire free AW transmission band [cf. Fig. 1 (c)]. Atzero µ , in accordance with Eqs. (26) and (28), we seethe free AW transmission band of 10 resonance electrontransfer channels T maxk =1 , ≈ T mink =1 , that are controlled by the magni-tude of ∆ as discussed following Eq. (28). As µ departs FIG. 2: (Color online) Transmission versus energy and AW-CN coupling strength as given by Eqs. (17)–(22) for the AW oflength N = 10 inside the (11,0) CN [ E p & E F + 2 V , panel (a)]and inside the (16,0) CN [ E p . E F + 2 V , panel (b)]. AW-leadcoupling constant ∆ = 0 .
05 eV. [Cf. Fig. 1 (c)]. from zero to increase, the near-field AW–CN interactionis seen to block some of the electron transfer channelsin the AW transmission band, while opening up extra(plasmon-induced) electron transfer channels in the CNforbidden gap outside of the AW transmission band. De-pending on whether E p is outside or inside of the AWtransmission band, the higher energy plasmon-inducedtransfer channel in the CN forbidden gap either showsup gradually [panel (a)], or splits off from the top of thefree AW transmission band [panel (b)]. The exact ener-gies of the emerging and blocked transmission channelsare given by E , in Eq. (31), and their behavior is inagreement with that discussed in the previous section.Figure 3 (a), (b), (c), and (d) illustrates in detail theFano resonance effect discussed in the previous section.We see the transmission versus energy calculated accord-ing to Eqs. (17)–(22) for the sodium AW of varied length N = 100, 101, 102, and 103 inside the (11,0) CN underthe AW–CN coupling µ = 0 .
045 eV and the AW–lead cou-pling ∆ = 0 . E g = 1 .
97 eV) and the first interband plas-
FIG. 3: (Color online) Transmission versus energy as givenby Eqs. (17)–(22) for the AW of varying length N = 100 − µ = 0 .
045 eV and ∆ = 0 . mon energy ( E p = 1 .
50 eV) for the (11,0) CN (cf. Fig. 1).Red thick dashed lines show the approximate transmis-sion curves given by Eq. (33) valid in the neighborhood ofthe AW transmission band center (hence the choice of µ and ∆ in this calculation). Green lines are the parabolasof Eq. (34). They are seen to intersect the abscise axis attwo points, E = E , given by Eq. (31). At E = E insidethe AW band, the transmission drops down to zero inview of the fact that this coupled AW–CN state (which0 FIG. 4: (Color online) Log-scaled transmission versus the AWlength as given by Eqs. (17)–(22) for the AW inside the (11,0)CN at E = 1 .
93 eV (CN forbidden gap outside of the AWtransmission band, cf. Fig. 3). AW–CN coupling µ = 0 .
045 eVin (a) and 0.015 eV in (b), AW–lead coupling ∆ = 0 . can also be interpreted as one of the two branches torepresent the ”dressed” states of the mixed CN plasmonand AW electron excitations [32, 33]) is not a well definedeigen state of the entire hybrid system. An electron canoccupy this state just temporarily, not permanently, asthere is always a high probability for it to leave for oneof the many band states that are available in this energydomain. That is why this coupled AW–CN state behavesas a scattering resonance to reflect an incident electronflux at E = E , thereby blocking transmission at this en-ergy. Another coupled AW–CN state, the second branchof the mixed CN plasmon and AW electron excitations,is isolated at E = E in the CN forbidden gap outside theAW band. This is a well defined eigen state of the entirehybrid system that opens up a new plasmon-mediatedresonance transmission channel.Figure 4 shows the transmission as a function of theAW length, calculated from Eqs. (17)–(22) for the AWinside the (11,0) CN. The energy is fixed at E = 1 .
93 eV,that is inside the CN forbidden gap but outside the pris-tine AW transmission band, and is close to the plasmon-mediated resonance transmission channel E in Fig. 3.Red lines indicate the approximations given by Eq. (37)in the neighborhood of E (middle line), and by Eqs. (38)and (39) away from E in the short AW (left line) andlong AW (right line) limits. In (a) the same coupling pa-rameters as in Fig. 3 are used, while in (b) the AW–CN FIG. 5: (Color online) Transmission versus the AW lengthas given by Eqs. (17)–(22) for the AW inside the (11,0) CNat E = E F = 0 .
985 eV [cf. Fig. 2 (a)]. Panel (a): µ = 0 eV,∆ = 0 .
05 eV. Panel (b): µ = 0 .
15 eV, ∆ = 0 .
05 eV. Panel (c): µ = 0 .
15 eV, ∆ = 0 . coupling is reduced by a factor of three. We see thatthe resonance plasmon-mediated transmission dependsstrongly on the AW–CN coupling strength µ , and canbe achieved both at shorter and at longer AW length forstronger and weaker coupling, respectively. The energy ofthe plasmon-mediated transmission channels is controlledby the product N µ as given by Eq. (31). Therefore, toreach the resonance transmission regime of Eq. (37) atfixed energy E with µ reduced by a factor of three, onehas to increase N by a factor of nine. That is exactly1what we see comparing (a) and (b) in Fig. 4. This pecu-liarity is the key to practical applications of the plasmon-mediated coherent resonance transmission phenomenon.It is interesting to see how the AW–CN coupling affectsthe transmission at the Fermi level energy E = E F . Forpristine monoatomic wires of finite length it is known,in particular, that depending on the valence and inter-atomic spacing their conductance shows both odd-evenatom number oscillations and more complicated featuressuch as four-atom and six-atom period oscillations (seeRefs. [104–108] and refs. therein for details]. Figure 5shows the transmission coefficient versus the AW lengthcalculated from Eqs. (17)–(22) for the AW inside the(11,0) CN at E = E F = 0 .
985 eV. The graphs for the AW–(16,0) CN system look similar, and so are not shown here.In (a) and (b) the AW–CN coupling µ = 0 and 0.15 eV,respectively, while ∆ = 0 .
05 eV as in Fig. 2 (a). In (c)the AW–lead coupling is increased by a factor of two,∆ = 0 . µ = 0 .
15 eV is the same as in (b). Theodd-even atom number oscillations of the pristine AWin panel (a) come from the oscillatory behavior of d N inEq. (24) at ε = 0, whereby d N is zero or non-zero to yieldmaximal or minimal transmission in Eq. (23) for odd oreven N , respectively. In panels (b) and (c), where theAW–CN coupling is non-zero, the distinct behavior canbe understood from Eq. (33), which is the exact repre-sentation of Eq. (30) for E = E = E F . With ξ ≪
1, thisequation is seen to have maxima when q cos (cid:18) N π η (cid:19) + ( − N µ = 0 , (40)where η = arccos( α N +1 /q ) and q = q α N +1 + µ with α N = 2 V ( E p − E F ) − N µ . The case where α N +1 = 0 wasdiscussed in the previous section. The transmission co-efficient experiences the Fano resonance at E = E F then.This can be clearly seen in panels (b) and (c) for N = 10[and also in Fig. 2 (a)] for the parameters chosen. Fornon-zero α N +1 such that α N +1 ≪ µ , that is for N givenby the inequality | N + 1 − V ( E p − E F ) /µ | ≪
1, one has q ≈ µ and η ≈ π/
2. Then Eq. (40) fulfils for all inte-ger n such that N = 4 n + 3, yielding four-atom periodictransmission maxima one can see at low and moderate N in panels (b) and (c). As N increases and becomeslarge enough, one necessarily obtains α N +1 ≈ α N ≫ µ ,to yield q ≈ | α N | ≈ N µ and η ≈ π . As this takes place,Eq. (40) takes the form cos( N π/
2) = ( − N /N ≈ µ dependence in Eq. (33) cancels, resulting in odd-even atom number transmission oscillations that are onlydependent on the AW–lead coupling ∆ as one can seefrom the graphs in panels (b) and (c). VI. DISCUSSION
In this study, we consider coherent electron transportthrough the one-atom-thick, finite-length metallic wireencapsulated in a semiconducting CN whose forbidden gap is broader than the conduction band of the wire.We ignore incoherent electron scattering processes suchas those that are typical and normally studied for com-plex molecular junction systems, including vibronic cou-pling [93, 109], coupling to defects [110], particularitiesof the coupling to the leads [101, 111], etc. [112] Forour hybrid AW–CN system, incoherent processes like thisalso include electron exchange between the wire and thenanotube, which we do not expect to be significant dueto rather strong ionization potentials of atomic metals ∼ ∼ . − . <
2) [113], that is insufficient to pullelectrons of metal over to carbon. Overall, incoherent ef-fects can be quite generally accounted for in our model byintroducing a phenomenological finite plasmon lifetime.This redefines E p to E p − i ∆ E p with the imaginary partrepresenting the half-width (inverse lifetime) of the plas-mon resonance, which is equivalent to the replacement δ ( ω − E p / ~ ) −→ π ∆ E p / ~ ( ω − E p / ~ ) + (∆ E p / ~ ) in Eq. (5) above. Substitution of thus modified Eq. (5)into the CN Hamiltonian (4) does not make any change toit provided that ∆ E p ≪ E p , in which case the plasmonresonance is sharp and plasmons are well-defined long-lived excitations of the nanotube, whereby Z ∞ dω ~ ω ∆ E p / ( π ~ )( ω − E p / ~ ) + (∆ E p / ~ ) ≈ E p Z ∞ dω ∆ E p / ( π ~ )( ω − E p / ~ ) + (∆ E p / ~ ) = E p π (cid:20) arctan (cid:18) E p ∆ E p (cid:19) + π (cid:21) ≈ E p , thus leaving our results unchanged up to terms of thefirst non-vanishing order in ∆ E p /E p .In our approach, the AW is treated within the single-hopping-parameter (or single-band) tight-binding model.Such a model is realistic for the 1D chains of atoms withhalf-filled outermost s -shells. These include monovalentalkali metals and transition metals with filled d - (and f -)shells such as copper, silver, and gold. Other transitionmetals would feature multi-band (multi-channel [107])conductance due to their under-filled d - (and f -) shells.However, the AW–CN near-field interaction (6) is univer-sal in its nature, and is hardly sensitive to conductancepeculiarities for them to be able to affect our results.Our main result is the prediction of the sharp Fanoresonances in electron transmission through hybrid quasi-1D nanostructures of semiconducting CNs that encapsu-late metal AWs. The resonances are due to the AW–CNnear-field interaction in Eqs. (6) and (7). The interac-tion couples AW electron and CN plasmon excitationsto form two branches of the mixed (”dressed” [32, 33])states to represent the eigen states of the entire hybridsystem. The quantity that controls the coupling is µ N E p and E F arepositioned relative to each other in the CN forbidden gap,a significant AW–CN coupling strength can be achievedin structures of varied length even though the single-atomcoupling constant µ is small. If, for certain µ and N ,a coupled AW–CN state falls into the AW transmissionband ( E in Figs. 2 and 3), then, being surrounded byother band states in its vicinity, it ceases to be the well-defined eigen state; it turns into the scattering resonanceto reflect an incident electron flux into the neighboringband states [87, 88], thereby blocking the transmission atthis energy. If it happens that a coupled AW–CN stateis isolated outside of the AW band inside of the CN for-bidden gap ( E in Figs. 2 and 3), then it remains a well-defined eigen state of the entire hybrid system to openup a new plasmon-mediated coherent transmission chan-nel in the energy domain where neither of the individualpristine constituents, neither AW nor CN, is transparent.Such a resonance coherent electron transport can be quiteefficient even though the actual coupling constant µ , theAW length N , and the transmission energy are out oftheir resonance values since the out-of-resonance trans-mission coefficient falls down with N relatively slowly, ∼ /N , as one can see from Eq. (39) shown in Fig. 4.The features described of the Fano resonances we pre-dict are quite generic. They originate from the similaritybetween our model Hamiltonian (1)–(7) and the generalFano-Anderson model for a bound quantum state insideor outside of the continuum of scattering states [87]. TheFano resonances can also manifest themselves in thosemetal-nanotube combinations where the AW transmis-sion band happens to be broader than the CN forbid-den gap. They may affect electron transport in the CNconduction band as well as hole transport in the CN va-lence band, since E enters the CN conduction band and E enters the CN valence band at large µ N (Fig. 2).The result will be transmission reduction for some of thechannels inside a band of states and/or an extra plasmon-mediated coherent transmission resonance in the energydomain where no band states available.In our model, the single-atom AW–CN coupling con-stant µ in Eq. (7) is considered to be site independent.In reality, the structure of the AWs encapsulated in thenanotube can be quite different from that of pristine AWsdue to factors such as atom clustering [71], dimeriza-tion [75], multiple atomic chains formation [54], as well asa variety of random spontaneous deformations of atomicchains inside the CN. In all these and other related cases,our model coupling constant µ should be considered asthe effective mean interaction constant. Local deviationsfrom the mean value due to the factors mentioned willdefinitely cause the inhomogeneous broadening of theFano resonances we predict — both inside of the AWtransmission band to increase the Γ estimate in Eq. (35),in particular, and outside of the AW band to broaden theplasmon-mediated coherent transmission channel in theCN forbidden gap (see Figs. 2 and 3).Overall, by selectively controlling the AW length N in the process of sample fabrication [107], one might beable, in principle, to manipulate by the electron trans-port regimes as it shows in Figs. 4 and 5 and is com-mented above — both inside and outside of the CN for-bidden gap, both to reduce and to enhance the trans-mission of the hybrid AW–CN system. Controlling theAW length can also be supplemented with other exter-nal means, such as the AW transmission band tune-upthrough chemical or electrostatic gate control [114, 115],electrostatic doping to adjust the CN forbidden gap [116],and the quantum confined Stark effect to tune the CNplasmon energy [24], thus allowing for flexible transportoptimization in hybrid metal-semiconductor CN systemsin ways desired for practical applications. VII. CONCLUSIONS
We study coherent electron transport through the one-atom-thick, finite-length metallic wire encapsulated intoa semiconducting carbon nanotube with the forbiddengap broader than the AW conduction band. We use ma-trix Green’s functions formalism to develop the electrontransfer theory for such a hybrid metal-semiconductorsystem. Our goal is to understand the inter-play be-tween the intrinsic 1D conductance of the atomic wireand nanotube mediated near-field effects.The theory we developed predicts the Fano resonancesin electron transmission through the system. That is theAW–CN near-field interaction blocks some of the pristineAW transmission band channels to open up new coher-ent channels in the CN forbidden gap outside the AWtransmission band. This makes the entire hybrid systemtransparent in the energy domain where neither AW norCN is individually transparent. These generic features ofthe Fano resonances we predict may also manifest them-selves in those metal-nanotube combinations where theAW transmission band is broader than the CN forbid-den gap. They may affect both electron transport in theCN conduction band and hole transport in the CN va-lence band to block some of the transmission channelsinside and/or to provide extra plasmon-mediated coher-ent transmission channels outside of bands of states. Thiseffect can be used to control and optimize charge trans-fer in hybrid metal-semiconductor CN based devices fornanoscale energy conversion, separation, and storage.
VIII. ACKNOWLEDGMENTS
Appendix A: Derivation of equations (18) and (19)
Using the matrix H in Eq. (8), one can derive the recur-sion relations for the quantities D N and S N to determinethe transmission coefficient formulas (17) and (30). Recursion relation for D N .Expanding the determinant D N = det( H − E ) along thefirst row, one has the set of recursion relations as follows D N = ε D N − − V A N + ( − N µ F N , (A1) A N = V D N − − ( − N µ F N − , (A2) F N = V F N − − ( − N µ d N − , (A3) d N = ε d N − − V d N − , (A4)where A N , F N , and d N are the determinants of the N × N matrixes A = V V . . . µ ε V . . . µ V ε . . . µ ... ... ... . . . ... ... ...0 0 0 . . . ε V µ . . . V ε µµ µ µ . . . µ µ ε p , F = V ε V . . . V ε V . . .
V ε . . . . . . V ε V . . . V ε µ µ . . . µ µ µ µ , and d = ε V . . . V ε V . . .
V ε . . . . . . ε V
00 0 0 . . . V ε V . . . V ε , respectively. Using Eqs. (A2) and (A3) to eliminate A N in Eq. (A1) results in D N +2 − ε D N +1 + V D N = 2( − N µ V F N +1 − µ d N +1 , (A5)This is the recursion relation for D N . It should be solvedtogether with recursion relations (A3) and (A4) underthe initial conditions as follows D = ε p , D = ε ε p − µ ; F = 0 , F = µ ; d = 1 , d = ε . (A6) Recursion relation for S N .According to Eqs. (14)–(16), quantity S N − of interestis the N H − E . This is given bythe determinant of the N × N matrix S = V . . . µε V . . . µV ε V . . . µ ... . . . . . . . . . ... ... ...0 . . . V ε V µ . . . V ε V µµ . . . µ µ µ µ ε p . Expanding S N − = det( S ) along the first row, one has S N − = V S N − − ( − N µ B N − , (A7) B N − = µ d N − − V B N − , (A8)where B N − is the determinant of the N × N matrix B = ε V . . . V ε V . . .
V ε . . . . . . ε V
00 0 0 . . . V ε Vµ µ µ . . . µ µ µ . Combining Eqs. (A7) and (A8), one arrives at the recur-sion relation as follows S N +2 − V S N +1 + V S N = ( − N µ d N , (A9)to be solved under the initial conditions S = ε p , S = V ε p − µ ; d = ε , d = ε − V . (A10)The d N initial condition is now one element downshifted[cf. Eq. (A6)] to reflect the fact of the dimensionalityreduction in Eq. (A9) compared to Eq. (A5). Solving recursion relations (A5) and (A9) .Recursion relations (A5) and (A9) are a convenientset of recursion formulas for numerical evaluation of thetransmission coefficient in Eq. (17). They do allow for ex-act solution, and so they will be solved here analytically.According to Ref. [117], the solution to the second orderconstant coefficient inhomogeneous recursive relation y N +2 + a y N +1 + b y N = f N (A11)( a and b are constant coefficients, f N is a known function)is given by the expression as follows y N = y ζ N − − y b ζ N − + N − X k =0 f k ζ N − k − . (A12)4Here ζ N = λ N +11 − λ N +12 λ − λ (A13)with λ , being the roots of the characteristic equation λ + a λ + b = 0. For λ = λ , Eq. (A13) takes the form ζ N = ( N + 1) λ N . (A14)Starting with Eq. (A4) and bringing it to the standardform (A11), one has d N +2 − ε d N +1 + V d N = 0 . This is to be solved with initial conditions (A6) and (A10)for recursion relations (A5) and (A9), respectively. UsingEq. (A11) with f N = 0, Eq. (A12) and Eq. (A13), oneobtains Eq. (21) under initial conditions (A6), and d N = λ N +21 − λ N +22 λ − λ (A15)under initial conditions (A10), where λ , are the rootsof the characteristic equation λ − ε λ + V = 0. Theyare given by Eq. (22), and are subject to Vieta’s formulaswhereby λ + λ = ε and λ λ = V .Similarly, bringing Eq. (A3) to the form (A11), one has F N +2 − V F N +1 = − ( − N µ d N +1 , which should be solved under initial conditions (A6).Then, Eq. (A12) with f N = − ( − N µ d N +1 , where d N is given by Eq. (21), results in F N = µ V N − − µ N − ≥ X k =0 ( − k λ k +21 − λ k +22 λ − λ V N − k − . Here, the second term can be found by summing up twogeometric series with common ratios − λ /V and − λ /V ,respectively, to result in the final expression as follows F N = µε + 2 V (cid:2) V N − ( − N ( d N + V d N − ) (cid:3) . (A16)With F N determined by Eqs. (A16) the right hand sideof Eq. (A5) becomes q N = µ ε + 2 V (cid:8) V (cid:2) d N + ( − N V N (cid:3) − ε d N +1 (cid:9) , which can be further rewritten as q N = µ ε + 2 V (cid:2) − N V N +2 − λ N +21 − λ N +22 (cid:3) (A17)using Eq. (A4) followed by Eq. (21) to express d N +2 and d N in terms of λ , . With f N = q N of Eq. (A17) and ζ N = d N , Eqs. (A11)–(A13) under initial conditions (A6)result in the solution to Eq. (A5) as follows D N = (cid:0) ε ε p − µ (cid:1) d N − − ε p V d N − + N − ≥ X k =0 q k d N − k − . (A18) Here, the first two terms can be written as ε p d N − µ d N − in view of Eq. (A4). The third term can be evaluated bysumming up the geometric series in the same way as itwas done to derive Eq. (A16). There are three contribu-tions to the total sum that originate from the three termsin Eq. (A17). Using Eqs. (21) and (A4) as well as thefact that λ + λ = ε and λ λ = V , one has N − ≥ X k =0 ( − k V k +2 d N − k − = V d N − + Vε + 2 V (cid:2) ( − N V N − V d N − − d N (cid:3) and N − ≥ X k =0 λ k +21 d N − k − + N − ≥ X k =0 λ k +22 d N − k − = N d N − ε d N − , to result in N − ≥ X k =0 q k d N − k − = µ d N − + µ ε + 2 V (cid:26) − N d N + 2 Vε + 2 V (cid:2) ( − N V N − V d N − − d N (cid:3)(cid:27) after elementary algebraic simplifications. Substitutingthis into the right hand side of Eq. (A18), one finallyarrives at Eq. (18).Equation (A9) must be solved with d N of Eq. (A15)consistent with the initial conditions (A10) as opposedto Eq. (A5) where d N on the right is given by Eq. (21).Following Eqs. (A11) and (A12) with ζ N = ( N + 1) V N ,one then has the solution of the form S N = ε p V N − µ N V N − (A19)+ µ N − ≥ X k =0 ( − k λ k +21 − λ k +22 λ − λ ( N − k − V N − k − . Here, the last term can be written as ∂∂V N − ≥ X k =0 ( − k λ k +21 − λ k +22 λ − λ V N − k − , whereupon summing up the geometric series and differ-entiation followed by the algebraic simplifications subjectto λ + λ = ε and λ λ = V , result in the expression asfollows λ − λ ε + 2 V " ( N − V N + ε N V N − + ( − N λ N +11 − λ N +11 λ − λ . Substituting this into the right hand side of Eq. (A19),after simplifications one finally arrives at Eq. (19).5 [1] N.Agrait, A.L.Yeyati, J.M.van Ruitenbeek, Phys. Rep. , 81 (2003).[2] S.Datta, Nanotechn. , S433 (2004).[3] A.Nitzan and M.Ratner, Science , 1384 (2003).[4] N.A.Zimbovskaya and M.R.Pederson, Phys. Rep. ,1 (2011).[5] J.-C.Charlier, X.Blase, and S.Roche, Rev. Mod. Phys. , 677 (2007).[6] R.Saito, G.Dresselhaus, and M.S.Dresselhaus, Scienceof Fullerens and Carbon Nanotubes (Imperial College,London, 1998).[7] M.Dresselhaus, G.Dresselhaus, and Ph.Avouris (eds.),
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