We solve the Lippmann-Schwinger equation describing elastic scattering of
preformed pairs (e.g. bipolarons) off a short-range scattering center and find
the two-particle transmission through a thin potential barrier. While the pair
transmission is smaller than the single-electron transmission in the
strong-coupling limit, it is remarkably larger in the weak coupling limit. We
also calculate current-voltage characteristics of a molecule - barrier -
molecule junction. They show unusual temperature and voltage behavior which are
experimentally verifiable at low temperatures.Comment: 5 pages, 2 figure

Alexandrov, A. S.

Bratkovsky, A. M.

Kornilovitch, P. E.

English

arXiv

We solve the Lippmann-Schwinger equation describing elastic scattering of preformed pairs (e.g. bipolarons) off a short-range scattering center and find the two-particle transmission through a thin potential barrier. While the pair transmission is smaller than the single-electron transmission in the strong-coupling limit, it is remarkably larger in the weak coupling limit. We also calculate current-voltage characteristics of a molecule - barrier - molecule junction. They show unusual temperature and voltage behavior which are experimentally verifiable at low temperatures

A.S. Alexandrov (7170284)

A.M. Bratkovsky (7170647)

P.E. Kornilovitch (7170395)

Loughborough University Institutional Repository

arXiv:cond-mat/0109226 v1   13 Sep 2001Three-body scattering problem and two-electron tunneling in molecular wiresA.S. Alexandrov1,2, A.M. Bratkovsky1, and P.E. Kornilovitch11Hewlett-Packard Laboratories, 1501 Page Mill Road, MS 1L-12, Palo Alto, California 943042Department of Physics, Loughborough University, Loughborough LE11 3TU, United KingdomWe solve the Lippmann-Schwinger equation describing elastic scattering of preformed pairs (e.g.bipolarons) off a short-range scattering center and find the two-particle transmission through a thinpotential barrier. While the pair transmission is smaller than the single-electron transmission in thestrong-coupling limit, it is remarkably larger in the weak coupling limit. We also calculate current-voltage characteristics of a molecule - barrier - molecule junction. They show unusual temperatureand voltage behavior which are experimentally verifiable at low temperatures.PACS: 21.45.+v, 71.38.Mx, 72.10.Fk,73.63.Nm, 85.65.+hMolecular-scale electronics is currently a very activearea of research [1]. It is envisaged that linear conju-gated molecules would be used as the “transmission lines”in molecular circuitry [2,3] in addition to active molec-ular elements discussed in the literature [1,4]. When aso-called “molecular wire” is short, the dominant mech-anism of transport is most likely a resonant tunnelingthrough electronic molecular states (see [5,6] and refer-ences therein). With increasing size of the wires one hasto take into account strong interaction between carriersand vibronic excitations of the molecule, leading to self-trapping of electrons in polaronic states. The formationof polarons (and charged solitons) in polyacetylene (PA)was discussed theoretically in Refs. [7] and formation ofbipolarons (bound states of two polarons) in Ref. [8]. Po-larons in PA were detected optically in Ref. [9] and sincethen studied in great detail. There is also an exceed-ing amount of evidence of the polaron and bipolaronformation in conjugated polymers such as polypheny-lene, polypyrrole, polythiophene, polyphenylene sulfide[10], Cs-doped biphenyl [11], n-doped bithiophene [12],polyphenylenevinylene(PPV)-based light emitting diodes[13], and other molecular systems. In many cases thedoped polymers have bipolaron-like charge states toyield, in particular, the enhanced nonlinear optical prop-erties [14].Many experimental data provide evidence for hoppingtransport of (bi)polaronic carriers. However, at suffi-ciently low temperatures there should be a crossoverto the band motion of polarons, as suggested long ago[15,16], and bipolarons [17,18]. Indeed, due to recentextraordinary improvements in preparation of “plastic”molecular conductors, it became possible to measuretheir conductivity in a wide interval of temperatures andobserve the crossover in two-dimensional films of organicconjugated molecules [19]. In one-dimensional (1D) wiresthe band motion is expected to be strongly hindered byimperfections, and those imperfections are likely to beintentionally introduced in the system as functionalizingunits [20]. Moreover, the polarons in extended molecu-lar wires/units are expected to be bound into real spacebipolarons with lowering temperature. As it is known inthe context of oxide semiconductors the bipolaron forma-tion may strongly affect transport properties [17,21].In this Letter we study elastic scattering of carriersbound into real-space pairs in one-dimensional organicand other conductors. We present an exact analyticalsolution in the limit of slow pairs. We also find an un-usual temperature and voltage dependence of the tunnel-ing conductance which may be experimentally verified atlow temperatures.In mathematical terms, the scattering of pairs is athree-body problem with the mass of the third particletaken to infinity. Let Uˆ(x1 − x2) be an attractive po-tential between the two moving particles and Vˆ (x1, x2)the repulsive external potential representing the barrier.The starting point is the Lippmann-Schwinger equation[22,23] for the two-particle wave function Ψ(k1, k2) in mo-mentum representation, which explicitly takes into ac-count a boundary condition of the three-body scatteringproblem. It can be written asΨ = −iγGˆ(E + iγ)Φ, (1)where Gˆ(E+iγ) is the exact two-particle Green’s function(GF) in the external potential, Φ(k1, k2) is the wave func-tion of a free (Vˆ = 0) real-space pair in momentum repre-sentation, Φ(k1, k2) = 2πδ(q −Q)φ(k). Here q = k1 + k2is the center-of-mass momentum, k = (k1 − k2)/2 is therelative momentum, E = −ǫ + Q2/4 < 0 is the pair to-tal energy in the absence of the external potential, and ǫis its binding energy. The wave function φ(k) describesthe internal structure of the pair. Hereafter we chooseh¯ = kB = m1 = m2 = 1, γ = +0, and define Gˆ(E + iγ)ΦasGˆΦ ≡∫ ∫dk′1dk′2(2π)2G(k1, k2|k′1, k′2;E)Φ(k′1, k′2), (2)for any Gˆ and Φ. Using the two identities, Gˆ =Gˆ3 − Gˆ3Uˆ Gˆ and Gˆ = Gˆ12 − Gˆ12Vˆ Gˆ and the Lippmann-Schwinger equation one readily derives the equation forthe Fourier component T (k1, k2) of the product UˆΨ [23]1T (k1, k2) = (E − k2 − q2/4)Φ− Tˆ12∆Gˆ3T, (3)where ∆Gˆ3 = Gˆ3 − Gˆ0, Gˆ0 is the two-particle GF inthe absence of any interaction (Uˆ = Vˆ = 0), Gˆ3 is theGF of noninteracting particles in the external field Vˆ (forUˆ = 0), and Gˆ12 is GF of two interacting particles withno external field, Vˆ = 0. Here the scattering operatorTˆ12,defined by the relation Tˆ12Gˆ0 ≡ Uˆ Gˆ12, is expressedvia the particle-particle scattering t-matrix asT12(k1, k2|k′1, k′2;E) = 2πt(k, k′;E − q2/4)δ(q − q′). (4)The t-matrix satisfies the equationt(k, k′;E) = u(k − k′)−∫dp2πu(k − p)t(p, k′;E)p2 − E − iγ , (5)where u(p) is the Fourier component of the attractivepotential Uˆ(x1 − x2).In many (in)organic semiconductors, the long-rangeCoulomb repulsion is usually significantly reduced by thestrong Fro¨hlich interaction with optical phonons [24], sothat a net (attractive) potential between carriers is ashort-range one, Uˆ(x1 − x2) = −αδ(x1 − x2), α > 0.Then Eq.(5) is readily solved resulting in the momenta-independent t-matrixt(k, k′, E) = − α√−E√−E − α/2 , (6)which is valid for all energies provided that the squareroot is understood as its principal value. The bindingenergy is ǫ = α2/4, and the normalized ground state wavefunction is φ(k) = 2−1/2α3/2/(k2+ǫ). It is known that fora short-range inter-particle interaction T (k1, k2), Eq.(3),is proportional to the Fourier component of the center-of-mass wave function Ω(q), T (k1, k2) = −2−1/2α3/2Ω(q).Then the problem of elastic pair scattering is reduced to asingle integral equation for the center-of-mass scatteringamplitude Υ(q). Substituting Φ and Tˆ12 in Eq. (3) oneobtainsΩ(q) = 2πδ(q −Q)− Υ(q)q2/4−Q2/4− iγ , (7)where Υ(q) satisfiesΥ(q) =W (q,Q)−∫dq′2πW (q, q′)Υ(q′)q′2/4−Q2/4− iγ . (8)The effective center-of-mass scattering potentialW (q, q′)is determined using GF of two noninteracting particlesin the external potential (Uˆ = 0 but Vˆ 6= 0) asW (q, q′) = αχ(q)∫ ∫dk2dk′2(2π)2∆G3(q − k2, k2|q′ − k′2, k′2;E),(9)where χ(q) = E − q2/4 + (α/4)(q2 − 4E)1/2. In the fol-lowing we restrict our consideration to the scattering ofslow pairs with Q2 ≪ 4ǫ. This condition allows us to re-placeW (q, q′) withW (0, 0) ≡W in all equations becausethe characteristic momenta q, q′ ≃ Q are much smallerthan√−E. Then the solution of Eq. (8) is given byΥ(q) = WQ/(Q + 2iW ) so that the pair transmissionprobability isT2(Q) = 1−∣∣∣∣2Υ(−Q)Q∣∣∣∣2=Q2Q2 + 4W 2. (10)In general, G3, W , and T2 can be found only numerically.However, in many applications the scattering potential isalso a short-range one, Vˆ (x1, x2) = β[δ(x1) + δ(x2)], sothat the full Hamiltonian takes the formH = −12∂2∂x21− 12∂2∂x22− αδ(x1 − x2) + β [δ(x1) + δ(x2)] .(11)This three-body problem was considered before in [25,26]but no general analytical solution was found. Here wepresent the analytical solution in the limit of slow pairs.Consider the equation for the two-particle G3(k21/2 + k22/2− E)G3(k1, k2|k′1, k′2;E) +β∫dp12πG3(p1, k2|k′1, k′2;E) + β∫dp22πG3(k1, p2|k′1, k′2;E)= (2π)2δ(k1 − k′1)δ(k2 − k′2), (12)which has a formal solutionG3(k1, k2|k′1, k′2;E) = G0(k1, k2|k′1, k′2;E)−D(k2|k′1, k′2;E) +D(k1|k′2, k′1;E)k21/2 + k22/2− E. (13)Here G0(k1, k2|k′1, k′2;E) = (2π)2δ(k1 − k′1)δ(k2 −k′2)(k21/2 + k22/2 − E)−1, and D(k1|k′2, k′1;E) ≡(2π)−1β∫dk2G3(k1, k2|k′1, k′2;E) satisfies the integralequationD(k1|k′2, k′1;E)[1 +β(k21 − 2E)1/2]=2πβδ(k1 − k′1)k21/2 + k′22 /2− E− β∫dk22πD(k2|k′1, k′2;E)k21/2 + k22/2− E. (14)We are interested in W = α3(2π)−2∫ ∫dkdpD(k| −p, p;E)(k2 − E)−1. Integrating Eq. (14) with respectto k′2 = −k′1 ≡ −p one obtains for B(k;E) ≡(2π)−1∫dpD(k| − p, p;E) the following equation:B(k; E )[1 +β(k2 − 2E)1/2]+ β∫dk′2πB(k′;E)k2/2 + k′2/2− E =βk2 − E . (15)2It has the solutionB(k;E) =β(k2 − E) (1 + β/√−E) , (16)which is verified by direct substitution into Eq. (15). Fi-nally we obtainW = α3∫dk2πB(k;E)k2 − E =2αβα+ 2β. (17)This result together with Eq. (10) solves the problemof the elastic scattering of slow bound pairs for anystrength of the short-range attractive and scattering po-tentials. It is instructive to compare the pair trans-mission T2(Q), Eq. (10), with the single electron trans-mission T1(p) = p2/(p2 + β2) for equal kinetic energiesp2/2 = Q2/4 ≡ K. If the binding potential is strongcompared with the scattering potential (α≫ 2β) the pairtransmission is just the single-particle transmission of aparticle with a double mass and double barrier strength,T2(Q) = Q2/(Q2 + 16β2), in accordance with a naiveexpectation. In the general case the ratio isT2(Q)T1(p) =K + β2K + 4β2(1 + 2β/α)−2. (18)When the binding potential is weaker than the scatteringpotential (α≪ β) the ratio isT2(Q)T1(p) =(βα)2≫ 1. (19)Quite remarkably, a weak attraction between carriershelps the first transmitted particle to “pull” its partnerthrough a strong potential barrier.Another important difference between pair and single-electron tunnelling occurs due to their different statis-tics. While electrons are fermions, preformed pairs arebosons, so that their center-of-mass motion obeys theBose-Einstein statistics Hence, tunnelling conductanceshould be temperature dependent even at low temper-atures T as has been already established in the bipo-laron tunnelling to a normal metal with a decay of thebound state [27]. Here we calculate the current-voltagecharacteristics of a molecular junction (MBM), i.e. thecurrent through a thin potential barrier between twomolecular wires. For simplicity, we restrict our calcu-lations to the strong-coupling regime, α ≫ β, T 1/2. Inthis regime single carriers are frozen out, and the trans-mission is due to the pairs alone, which are scatteredoff a double-strength barrier, W ≈ 2β, Eq.(17), analo-gously to single particles with the double carrier mass.Then, in the presence of a voltage drop at the junction,2eV (for a pair), the conductance can be readily foundby matching the center-of-mass wave function and itsderivative on the left, Ωl and on the right side, Ωr ofthe δ-function barrier. In the coordinate representation0.0 0.2 0.4 0.6 0.8 1.0T/TF0.01.02.03.04.05.06.0G/G00.010.050.10.20.41.0FIG. 1. Zero-voltage conductance of MBM as a functionof temperature (in units of TF ) for different relative strengthof the barrier 4β2/TF . G0 = (2e2)/h.one has Ωl(X < 0) = eiQX +Re−iQX , and Ωr(X > 0) =CeiP+X with 1 +R = C, CP+ − (1−R)Q = 8iβ(1 +R),and P+ = (Q2 + 8eV )1/2. The transmission is given byT2(Q,P+) ≡ 1− |R|2 = 4QP+(Q+ P+)2 + 64β2, (20)for real P+, and is zero otherwise. Multiplying the trans-mission by eQ and integrating with the Bose-Einsteindistribution function f(Q) = [exp(Q2/4T − µ/T )− 1]−1yields the current asI(V ) = e∫∞0dQ2πQf(Q)[T2(Q,P+)− T2(Q,P−)], (21)where P− = (Q2 − 8eV )1/2 and µ is the chemicalpotential determined by the number of pairs n using∫∞−∞dQ2pi f(Q) = n. It is easy to calculate the integralsin the linear voltage classical limit, 2eV, TF ≪ T by ex-panding the transmission in powers of eV and replacingthe Bose-Einstein distribution with the Boltzmann one,f(Q) ≈ (2TF /πT )1/2 exp(−Q2/4T ) (TF ≡ π2n2/2 is theFermi temperature of single carriers). The result for theconductance, σ ≡ (dI/dV )V=0 isσ =2e2π√2TFπT[1 +4β2Te4β2/TEi(−4β2/T )], (22)where Ei(x) is the exponential integral function. Theconductance behaves as σ = e2piβ2√TF T2pi at T ≪ 4β2, andas σ = 2e2pi√2TFpiT at T ≫ 4β2. In the last case it has auniversal magnitude independent of the barrier strength.Apart from numerical coefficients, conductance of tightly30.0 0.2 0.4 0.6 0.8 1.02|e|V/TF0.00.20.40.60.81.0dI/dV, G0T = 0.5 TFT = 0.3 TFT = 0.1 TFFIG. 2. Differential conductance of MBM as a functionof voltage for different temperatures and 4β2/TF = 0.5. Thethin solid lines is the conductance of fermions at T = 0.1 TF .bound pairs is, of course, the same as conductance of sin-gle electrons in the classical limit. It is not the case, how-ever, in a degenerate system, when T ≤ TF . Numericalintegration of Eq.(21) at fixed density n reveals a temper-ature dependence in this limit, Fig. 1, in comparison withthe temperature independent conductance of fermionicnoninteracting carriers at low temperatures. This re-markable difference is entirely due to the bosonic natureof pairs. The conductance is proportional to the meanvelocity of carriers which in the case of bosons grows as√T (while it is temperature-independent for fermions).This explains the low-temperature behavior of the con-ductance. Interpair correlations may reduce the differ-ence in 1D wires. However, higher-dimension correctionsreadily restore it. There is also a breakdown of Ohm’slaw when 2eV ≥ T , as shown in Fig. 2 for low tempera-tures, again in contrast with the Fermi statistics, where anon-linearity appears only at eV ≥ TF ≫ T . We suggestthat the most appropriate materials for experimental ob-servation of the unusual current-voltage characteristics(Figs.1,2) are doped molecular semiconductors such asCs-doped biphenyl [11], where bipolarons were explicitelydetected by photoelectron and electron-energy-loss spec-troscopies, and single crystals of pentacene, tetracene,rubrene, quaterthiophene (α-4T), sexithiophene (α-8T),where the coherent (bi)polaron tunnelling has been re-cently observed below room temperature [19].In conclusion, we have solved the Lippmann-Schwingerequation in the effective mass approximation for singlecarriers and pairs in 1D conductors (molecular wires),which is valid for (bi)polarons if their size is largerthan the lattice constant. While mapping of this prob-lem onto discrete lattices is straightforward with a neg-ative Hubbard U model, the model itself can be ap-plied to bipolarons only in the extreme nonadiabaticlimit when the characteristic phonon frequency is largerthan the binding energy [17]. In this limit a discreteLippmann-Schwinger equation also has the analytical so-lution for slow pairs [28], which shows that the contin-uous model remains qualitatively correct even for lat-tice size (small) nonadiabatic bipolarons. In the oppo-site adiabatic regime bipolaron tunnelling is not a three-body problem because of the emission and absorptionof (virtual) phonons [17]. We have found the scatteringamplitude of elastic scattering of slow bipolarons, andconductance of the molecular junction (MBM) with pre-formed pairs. While the pair transmission is smaller thanthe single-electron transmission in the strong-couplingregime, it is surprisingly larger in the weak couplingregime. 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Three-body scattering problem and two-electron tunneling in molecular wires

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