We analyze the phase transitions of an interacting electronic system weakly
coupled to free-electron leads by considering its zero-bias conductance. This
is expressed in terms of two effective impurity models for the cases with and
without spin degeneracy. We demonstrate using the half-filled ionic Hubbard
ring that the weight of the first conductance peak as a function of external
flux or of the difference in gate voltages between even and odd sites allows
one to identify the topological charge transition between a correlated
insulator and a band insulator.Comment: 4 pages, 5 figures, to appear in Phys. Rev. Let

A. A. Aligia

A. P. Kampf

A. P. Alivisatos

B. Normand

K. Hallberg

L. P. Kouwenhoven

N. Nagaosa

T. Egami

W. Izumida

English

arXiv

The phase transitions of an interacting electronic system weakly coupled to free-electron which results by zero-bias conditions, were investigated. The phase transitions were expressed in terms of two effective impurity models for cases with and without spin degeneracy. The weight of first conductance peak as a function of external flux or of difference in gate voltages between even and odd sites allows to identify topological charge transition was also demonstrated using half-filled ionic Hubbard ring. The results show that conductance is expressed in terms of spectral density of an EAM, which illustrates breaking of the Kondo effect by an applied field.Fil: Aligia, Armando Ángel. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; ArgentinaFil: Hallberg, Karen Astrid. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Area de Investigación y Aplicaciones No Nucleares. Gerencia de Física (Centro Atómico Bariloche); ArgentinaFil: Normand, B.. University Of Fribourg; SuizaFil: Kampf, A. P.. Universität Augsburg; Alemani

Aligia, Armando Ángel

Hallberg, Karen Astrid

Normand, B.

Kampf, A. P.

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:cond-mat/0406355v1  [cond-mat.str-el]  16 Jun 2004Detection of topological transitions by transport through molecules and nanodevicesA. A. Aligia1, K. Hallberg,1 B. Normand,2 and A. P. Kampf31Centro Atómico Bariloche and Instituto Balseiro,Comisión Nacional de Energía Atómica, 8400 Bariloche, Argentina2Département de Physique, Université de Fribourg, CH-1700 Fribourg, Switzerland3Institut für Physik, Theoretische Physik III, Elektronische Korrelationen und Magnetismus,Universität Augsburg, 86135 Augsburg, GermanyWe analyze the phase transitions of an interacting electronic system weakly coupled to free-electron leads by considering its zero-bias conductance. This is expressed in terms of two effectiveimpurity models for the cases with and without spin degeneracy. We demonstrate using the half-filled ionic Hubbard ring that the weight of the first conductance peak as a function of external fluxor of the difference in gate voltages between even and odd sites allows one to identify the topologicalcharge transition between a correlated insulator and a band insulator.PACS numbers: 73.23.-b, 81.07.Nb, 81.07.Ta, 73.40.QvProgress in nanotechnology has made it possible toperform transport experiments on systems as small assingle molecules [1]. Metallic [2] or semiconducting[3, 4, 5] quantum dots (QDs) can now be assembled intoartificial molecules [4] or solids [2]. QD molecules of dif-ferent materials and sizes are now being investigated anda wide range of new QD systems is expected to be synthe-sized in the near future [3, 4]. The transport propertiesof a finite chain of 15 QDs were first measured over 10years ago [5], and the metal-insulator transition has beenstudied experimentally in a hexagonal lattice of Ag QDs[2]. These advances open the route for new approachesto investigate novel phenomena and theoretical conceptsin interacting electron systems.Here we focus on one such phenomenon, the topologicalphase transition associated with a parity change of theground state. The ionic Hubbard model (IHM) with al-ternating diagonal energy ± 12∆ has received much recentattention [6, 7, 8, 9, 10]. It was proposed to describe theneutral-ionic transition in organic charge-transfer salts[11], and later applied to model ferroelectricity in per-ovskites [6, 12]. At half filling and in the atomic limit(t → 0), the ground state is a band insulator (BI) forU < ∆, but is a Mott insulator (MI) for U > ∆. Inone dimension, with non-zero hopping t, a spontaneouslydimerized insulator (SDI) phase appears between BI andMI. With increasing U , one finds first a charge transi-tion at U = Uc from BI to SDI, followed by the closingof the spin gap at Us > Uc, where the transition to theMI occurs. Although in finite systems conventional or-der parameters such as charge and spin structure factorsvary continuously at a phase transition, in a system ofL sites one may define charge and spin topological num-bers which change discontinuously at Uc(L) and Us(L)[8]. The charge Berry phase has a step at Uc(L), where aparity change of the ground state occurs for periodic (an-tiperiodic) boundary conditions if L = 4m (L = 4m+ 2,m integer) [6, 8].Here we show that this charge transition can be de-RtU0 nφt1ttt2-3 3-1-2                                                                           FIG. 1: Interacting electron system on a flux-threaded ring,connected by weak links (t′) to two conducting leads.tected using the total intensity or width w of the firstpeak in a zero-bias conductance measurement performedon a flux-threaded ring. The importance of the ring ge-ometry is that the topological transition is absent in anopen system [9]. In one of the two parity sectors, which isselected using an applied gate voltage, w → 0 for appliedflux φ → 0 if L = 4m (φ → φ0/2, where φ0 = hc/e isthe flux quantum, if L = 4m + 2). Thus the transition,which is observed by varying parameters such as ∆, mayalso be studied in ring-shaped molecules where it is notpossible to attain a significant threading flux.The configuration for the proposed experiment is il-lustrated in Fig. 1. The interacting electron system onthe ring is connected to conducting leads through twosites, labeled 0 and n, by weak links with hopping t′,and either two materials or two different gate voltagesat alternating sites may be used to model the IHM. Itis essential to distinguish between two cases dependingon the spin degeneracy of the isolated interacting systemfor odd particle number. We thus perform two differentcalculations of the conductance. The first assumes thatspin degeneracy is lifted by a Zeeman interaction, butallows orbital degeneracy of the states, which is impor-tant when including interference effects arising in a ringgeometry [13]. In the second, where spin degeneracy isretained, we map the problem into an effective Andersonmodel (EAM) and express the conductance as a functionof the spectral density of the latter.If the ground state |g〉 of the interacting system is non-2degenerate, for small t′ the links can be eliminated by acanonical transformation, leading to an effective, non-interacting Hamiltonian for the two leads in which thetwo “impurity” sites i connected to the links (−1 and 1in Fig. 1) have an energy shift ∆ǫi(ω), and are connectedby an effective hopping teff(ω),Heff =∑kL,σǫkLc†kLσckLσ +∆ǫ−1c†−1σc−1σ +∆ǫ1c†1σc1σ+∑kR,σǫkRc†kRσckRσ +∑σ(teffc†1σc−1σ +H.c.). (1)L (R) refers to the lead containing the site −1 (+1), andthe impurity parameters may be expressed in terms of theGreen functions for the isolated ring gij(ω) = 〈〈ciσ; c†jσ〉〉,∆ǫ−1(ω) = t′2g00(ω), ∆ǫ1(ω) = t′2gnn(ω),teff(ω) = t′2gn0(ω). (2)The transmittance of Heff may be calculated in dif-ferent ways. Without affecting the essential results, weassume identical leads and, for their states without sites(−1, 1), we consider two models for the density of statesρ(ω) and the hybridization Vj(ω) with sites (−1, 1). I)ρ(ω) and Vj(ω) constant: for j = ±1 and teff = 0,geff0jj (ω) = 1/(ω−∆ǫj+iΓ) with Γ = πρV2. II) Leads de-scribed by a tight-binding model with nearest-neighborhopping t, where geff0jj (ω) = 1/(ω/2−∆ǫj + iΓ(ω)) withΓ(ω) = πρ(ω)V 2(ω) =√t2 − ω2/4. By introducing aninteger m = 1 or 2 for cases I and II,T (ω, Vg) = (3)4Γ2(ω)|teff |2|( ωm −∆ǫ−1 + iΓ(ω))(ωm −∆ǫ1 + iΓ(ω))− |teff |2|2.Vg is a gate voltage which changes the on-site energy ofall sites of the ring by −eVg. In case II this equation isexact (∀ t′) in the non-interacting system, and generalizesa previous result [13]. Eq. (3) also generalizes previousapproaches in which only one intermediate state of thering is included [14, 15], and is appropriate for the studyof interference effects [13]. Spin degeneracy limits its va-lidity to magnetic fields B for which the Kondo effect isdestroyed, as discussed below. For sufficiently large Zee-man energy gµBB, the zero-bias conductance at steadystate is given by G = G0T (µ, Vg)/2, where G0 = 2e2/hand µ is the chemical potential of the leads. The re-sults of Ref. [16] suggest that Eq. (3) remains valid, withG = G0T (µ, Vg), also in the absence of Zeeman splittingin an intermediate temperature range T1 < T < 0.1w,where T1 is very small. Henceforth we set µ = 0, corre-sponding to half-filled leads.The transmittance as a function of gate voltage is verysmall except at the poles of gn0. If the ground state|g〉 has N particles for µ = 0, the ground-state energyEg(N) satisfies Eg(N)− eVg < Eg(N − 1) and Eg(N) +-10 -8 -6 -4 -2 0 2 4 6 8 100.00.20.40.60.81.0eVg/w G/G0         gµBB/w  0  10-4  10-3  10-2  0.1  1FIG. 2: Conductance as a function of gate voltage for theEAM at several magnetic fields. Circles correspond to theconductance of a resonant level for the majority spin only.eVg < Eg(N + 1). As Vg is increased, a pole in gn0(Vg)is reached, and for larger Vg the ground state has N + 1particles. Similarly, if Vg is lowered, a pole is reachedwhen z = eVg + Eg(N − 1)− Eg(N) = 0.Assuming that the state of lowest energy for N−1 par-ticles, |g(N−1)〉, is not degenerate, gij may be expressedas a Laurent expansion around z = 0,gij(z) =aiājz+ βij + γijz + . . . , (4)where βjj , γjj , . . . are real coefficients andaj = 〈g(N − 1)|cjσ|g(N)〉. (5)αj = |aj|2 is the spectral weight for a local photoemissionprocess. Substituting these expressions in Eqs. (2) and(3), retaining terms to lowest nontrivial order in t′/Γ(0),and using that T (z) ∼ O(1) for z . (t′)2/Γ(0), yieldsT (z) ∼=11 + (w/z)24α0αn(α0 + αn)2+O((t′/Γ(0))2), (6)where the half-width at half maximum peak height isw = (α0 + αn)(t′)2/Γ(0). (7)Using G = G0T (µ, Vg), this expression coincides at suffi-ciently low temperatures with Eq. (7) of Ref. [15].If sites 0 and n are equivalent by symmetry, then w =2(t′)2α0/Γ(0) and the integrated weight I =∫dz T (z) =πw are both proportional to (t′)2 and to α0. If Vg is in-creased instead of decreased, the same result is obtainedwith cjσ → c†jσ and N − 1 → N + 1 in Eq. (5). Thusa single transport measurement gives simultaneous spec-tral information related to photoemission and to inversephotoemission. We stress that this information is ob-tained with much finer energetic resolution (µeV) thanthat available by direct spectroscopic techniques (meV).We turn next to a calculation of the conductance forsmall or zero B, where either |g(N)〉 or |g(N ± 1)〉 isspin-degenerate. This degeneracy can be taken into ac-count accurately if w is small compared to the separation3between groups of spin-degenerate energy levels, becausethe problem becomes equivalent to an EAM. For simplic-ity we take Vg such that a nondegenerate state |g(N)〉 isclose in energy to the spin-degenerate state |g(N − 1)〉,but extension to other cases is straightforward. Neglect-ing other states, the EAM for any interacting system be-tween the leads isHA = −t∑σ,i6=0,1c†iσci−1σ −∑σd†σ(t′−1c−1σ + t′1c1σ)+H.c.+εd∑σndσ + UAnd↑nd↓ − gµBB(nd↑ − nd↓), (8)with ndσ = d†σdσ, t′−1 = t′ā0, t′1 = t′ān, εd = Eg(N) −Eg(N −1)−eVg, and infinite UA (justified because UA =Eg(N) + Eg(N − 2)− 2Eg(N − 1) ≫ w).The conductance is given byG =2e2h2πw|t′1t′−1|2[|t′1|2 + |t′−1|2]∑σρdσ(µ), (9)where ρdσ(ω) is the spectral density of the effective dσelectrons [17]. The conductance of the EAM computedas a function of gate voltage for several values of B usingslave bosons in the mean-field approximation (SBMFA)[19] is shown in Fig. 2. For B = 0, G increases abruptlyfrom 0 to G0 when z ∼ 0, and remains nearly perfect forhigher Vg due to pinning of the Kondo peak in ρdσ(ω)at the Fermi level. However, because the Kondo energyscale TK decreases exponentially with Vg, even for verysmall B the plateau becomes a broad peak, terminatedwhen TK < gµBB, beyond which G falls to zero. Theabruptness of the fall is an artifact of the SBMFA, butthe total width of the feature is well described. For largerB the peak width decreases, tending to w to recover theprevious result: for gµBB & w, G = G0T (z)/2, withT (z) given by Eq. (6).We now apply these results to the topological chargetransition in the IHM, which is defined byHIHM = −tR∑i,σ(c†i+1σciσeiΦ/L +H.c.)− 12∆∑i(−1)ini−gµBB∑i(ni↑ − ni↓) + U∑ini↑ni↓, (10)where niσ = c†iσciσ, ni =∑σ niσ, and N =∑i ni. Wecalculate T (Vg) by exact diagonalization for the isolatedring with L sites and N = L electrons, with leads at-tached to two sites of the same energy (−∆/2) in thepresence of a magnetic flux φ (Fig. 1), with Φ = 2πφ/φ0.The Green functions gij(Vg) are obtained numericallyand substituted in Eqs. (2) and (3). We choose L = 8,but similar results are obtained for any L = 4n (L =4n + 2 with Φ → Φ + π). The qualitative features areindependent of the lead position n, with the exception ofn = L/2 where the transmittance at Φ = π vanishes for0.00.20.40.60.81.02 G/G0 Φ=0.4π Φ=0.3π Φ=0.2π Φ=0.1π Φ=0.04π    2 G/G0 U=5 ∆=-3-0.7 -0.6 -0.5 -0.4 -0.3 -0.20.00.20.40.60.81.0  U=4 ∆=-3eVg+gµBBFIG. 3: Transmittance as a function of gate voltage for an 8-site IHM ring with leads at sites 0 and 2 for tR = 1, t′ = 0.2,∆ = −3, and values of U below and above Uc(L = 8) = 4.352.symmetry reasons. Because HIHM is invariant under si-multaneous particle-hole transformation and sign changeof ∆, we restrict our analysis to eVg < 0. We set tR = 1as the unit of energy unless otherwise stated.The topological transition is present at any value oftR. As a basis for our study we consider a realistic ringstructure containing 8 QDs with only their lowest levelssingly occupied, the centers separated by 200 nm [5], andthe parameters U,∆ ∼ 1 meV and tR ∼ U/4 (intermedi-ate coupling strength). We first assume the presence ofa strong magnetic field (B ∼1 T for a QD array), in theplane of the ring in order not to alter the threading flux,which destroys the Kondo effect. Fig. 3 shows the firstpeak in G(Vg) at ∆ = −3 and for two values of U . ForL = 8 and ∆ = ±3, the topological transition occurs atUc = 4.352. The peak width is parameter-dependent: forU = 5, w is approximately constant with decreasing flux,whereas for U = 4, w decreases and the peak disappearsat Φ = 0. If U < Uc, the ground state |g(L)〉 for Φ = 0 iseven under reflection through any site (corresponding toa BI), while for U > Uc |g(L)〉 is odd (MI or SDI) [8]. For∆ = 0, the lowest-energy hole enters the system with theFermi wave vector, ±π/2, leaving an orbitally degenerateground state with L− 1 particles, |g(L− 1)〉. For ∆ 6= 0,this degeneracy is broken and |g(L− 1)〉 is odd under re-flection through any even site if ∆ is negative [18]. As aconsequence, for U < Uc the matrix element a0 [Eq. (5)]vanishes by symmetry, whence G is negligible at Φ = 0.The flux acts as a symmetry-breaking field, which allowsone to follow the first peak in a continuous manner untilit disappears as Φ → 0.To demonstrate that these essential features are not af-fected by the presence of a Kondo resonance we considerthe 8-site ring with parameters (Fig. 4) similar to one ex-perimental realization [5] and B normal to the ring planeso that the threading flux and Zeeman splitting have the4-4 -3 -2 -1 0 1 20.00.20.40.60.8   Φ=0.04π Φ=0.1π Φ=0.2π Φ=0.3πG/G0Vg (µV)FIG. 4: Transmittance as a function of Vg for a ring of 8 QDsdescribed by the IHM. The parameters chosen are U = 1meV, ∆ = −3U/4, t = U/4, t′/t = 0.2, and gµB = 0.025meV/T. The curves have been shifted in Vg (cf. lower panelFig. 3) such that their maxima coincide.same origin. Fig. 4 shows the conductance in this regime,calculated using the EAM in the SBMFA. The disappear-ance of the feature with decreasing flux remains clear.The topological transition may now be characterizedusing α0. As shown in Fig. 5 for ∆ = 3, α0 as a func-tion of U (cf. peak widths in Fig. 3 [18]) vanishes dis-continuously at Uc for Φ = 0, indicating unambiguouslythe charge transition. The analogous result obtained forfixed U by varying ∆ is of direct experimental interest,because ∆ can be controlled by a difference in gate volt-age applied between even and odd sites. A finite fluxacts as a parity-breaking perturbation and smooths thetransition. This is the situation for artificial arrays ofQDs, in which perfect structural symmetry is difficult toattain. We note that curves for all flux values cross ap-proximately at the same point: transport measurementsunder different applied fields can therefore help to locatethe transition, even in the absence of perfect reflectionsymmetry. However, for a small molecule, which by defi-nition exhibits the Φ → 0 limit, the only source of asym-metry is the leads.In summary, we have considered the transport proper-ties of a ring-shaped interacting system connected weaklyto conducting leads. The conductance is expressed interms of the spectral density of an EAM, which illustratesthe breaking of the Kondo effect by an applied field. Out-side the Kondo regime we have derived a transmittanceformula including nontrivial interference effects. Theconductance peaks may be characterized by their totalweight, offering spectral information with far higher res-olution than conventional spectroscopic measurements.As an application of this result, we have demonstratedthat the conductance can be used to detect the chargetransition of molecules or quantum dot rings described bythe IHM. The method is relevant in the general contextof systems presenting phase transitions which involve asymmetry change of the ground state.A.A.A. and K.H. are fellows of CONICET, and ac-knowledge the support of the Fundación Antorchas,4.0 4.2 4.4 4.6 4.8 5.00.00.20.40.60.81.01.21.41.61.8UUcLα0   ∆ = 3 Φ = 0 Φ = 0.01π Φ = 0.03π Φ = 0.1πFIG. 5: Matrix element α0 as a function of U for ∆ = 3(Uc = 4.352) over a range of values of the flux Φ.Project 14116-168, and PICT 03-12742 of ANPCyT.A.P.K. acknowledges support through SFB484 of theDeutsche Forschungsgemeinschaft and B.N. the supportof the Swiss National Science Foundation.[1] M.A. Reed et al., Science 278, 252 (1997); D. Porath etal., Nature (London) 403, 635 (2000); J. Reichert et al.,Phys. Rev. Lett. 88, 176804 (2002).[2] C.P. Collier et al., Science 277, 1978 (1997); G. Medeiros-Ribeiro et al., Phys. Rev. B 59, 1633 (1999).[3] A.P. Aliviasatos, Science 271, 933 (1996).[4] L. P. Kouwenhoven, Science 268, 1440 (1995).[5] L. P. Kouwenhoven et al., Phys. Rev. Lett. 65, 361(1990).[6] R. Resta and S. Sorella, Phys. Rev. Lett. 82, 370 (1999).[7] M. Fabrizio, A.O. Gogolin, and A.A. Nersesyan, Phys.Rev. Lett. 83, 2014 (1999).[8] M.E. Torio, A.A. Aligia, and H.A. Ceccatto, Phys. Rev.B 64, 121105 (R) (2001), and references therein.205109 (2003).[9] A.P. Kampf et al., J. Phys.: Condens. Matter 15, 5895(2003); Y.Z. Zhang, C.Q. Wu, and H.Q. Lin, Phys. Rev.B 67, 205109 (2003); C.D. Batista and A.A. Aligia, Phys.Rev. Lett. in press (LJ9218); A. A. Aligia, Phys. Rev. B69, 041101(R) (2004).[10] S.R. Manmana et al., cond-mat/0307741 (unpublished).[11] J. Hubbard and J. B. Torrance, Phys. Rev. Lett. 47, 1750(1981); N. Nagaosa and J. Takimoto, J. Phys. Soc. Jpn.55, 2357 (1986).[12] T. Egami, S. Ishihara, and M. Tachiki, Science 261, 1307(1994).[13] E.A. Jagla and C.A. Balseiro, Phys. Rev. Lett. 70, 639(1993).[14] G. Chen et al., Phys. Rev. B 50, 8035 (1994).[15] C.A. Stafford, Phys. Rev. Lett. 77, 2770 (1996).[16] W. Izumida, O. Sakai, and S. Suzuki, J. Phys. Soc. Jpn.70, 1045 (2001).[17] A.A. Aligia and C.R. Proetto, Phys. Rev. B 65, 165305(2002).[18] For reflection through odd sites, |g(L − 1)〉 is odd forpositive ∆. Under a sign change of ∆ the parities of evenand odd sites are interchanged, and with these the chargesector (BI or SDI, MI) in which α0 = 0 at Φ = 0.[19] K. Kang et al., Phys. Rev. B 63, 113304 (2001).

Physical Review Letters

Detection of topological transitions by transport through molecules and nanodevices

Crossref

Detection of Topological Transitions by Transport Through Molecules and Nanodevices

Aligia, A. A.

Hallberg, K.

Normand, Bruce

Kampf, Arno P. (Prof.)

OPUS - Augsburg University Publication Server

CONICET Digital

OPUS Augsburg

VOLUME 93, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R Sweek ending13 AUGUST 2004Detection of Topological Transitions by Transport Through Molecules and NanodevicesA. A. Aligia,1 K. Hallberg,1 B. Normand,2 and A. P. Kampf31Centro Atómico Bariloche and Instituto Balseiro, Comisión Nacional de Energı́a Atómica, 8400 Bariloche, Argentina2Département de Physique, Université de Fribourg, CH-1700 Fribourg, Switzerland3Institut für Physik, Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Universität Augsburg,86135 Augsburg, Germany(Received 3 December 2003; published 10 August 2004)076801-1We analyze the phase transitions of an interacting electronic system weakly coupled to free-electronleads by considering its zero-bias conductance. This is expressed in terms of two effective impuritymodels for the cases with and without spin degeneracy. Using the half-filled ionic Hubbard ring, wedemonstrate that the weight of the first conductance peak as a function of external flux or of thedifference in gate voltages between even and odd sites allows one to identify the topological chargetransition between a correlated insulator and a band insulator.DOI: 10.1103/PhysRevLett.93.076801 PACS numbers: 73.23.–b, 73.40.Qv, 81.07.Nb, 81.07.TaRtU0 nφt1ttt2-3 3-1-2FIG. 1. Interacting electron system on a flux-threaded ring,connected by weak links (t0) to two conducting leads.Progress in nanotechnology has made it possible toperform transport experiments on systems as small assingle molecules [1]. Metallic [2] or semiconducting [3–5] quantum dots (QDs) can now be assembled into arti-ficial molecules [4] or solids [2]. QD molecules of differ-ent materials and sizes are now being investigated and awide range of new QD systems is expected to be synthe-sized in the near future [3,4]. The transport properties ofa finite chain of 15 QDs were first measured over ten yearsago [5], and the metal-insulator transition has beenstudied experimentally in a hexagonal lattice of AgQDs [2]. These advances open the route for new ap-proaches to investigate novel phenomena and theoreticalconcepts in interacting electron systems.Here we focus on one such phenomenon: the topologi-cal phase transition associated with a parity change of theground state. The ionic Hubbard model (IHM) with alter-nating diagonal energy  12 has received much recentattention [6–10]. It was proposed to describe the neutral-ionic transition in organic charge-transfer salts [11], andlater applied to model ferroelectricity in perovskites[6,12]. At half filling and in the atomic limit (t! 0),the ground state is a band insulator (BI) forU <  but is aMott insulator (MI) for U >. In one dimension, withnonzero hopping t, a spontaneously dimerized insulator(SDI) phase appears between BI and MI. With increasingU, one finds first a charge transition at U  Uc from BI toSDI, followed by the closing of the spin gap at Us >Uc,where the transition to the MI occurs. Although in finitesystems conventional order parameters such as charge andspin structure factors vary continuously at a phase tran-sition, in a system of L sites one may define charge andspin topological numbers which change discontinuouslyat UcL andUsL [8]. The charge Berry phase has a stepatUcL, where a parity change of the ground state occursfor periodic (antiperiodic) boundary conditions if L 4m (L  4m 2, m integer) [6,8].0031-9007=04=93(7)=076801(4)$22.50 Here we show that this charge transition can be de-tected using the total intensity or widthw of the first peakin a zero-bias conductance measurement performed on aflux-threaded ring. The importance of the ring geometryis that the topological transition is absent in an opensystem [9]. In one of the two parity sectors, which isselected using an applied gate voltage, w! 0 for appliedflux ! 0 if L  4m (! 0=2, where 0  hc=e isthe flux quantum, if L  4m 2). Thus the transition,which is observed by varying parameters such as , mayalso be studied in ring-shaped molecules where it is notpossible to attain a significant threading flux.The configuration for the proposed experiment is illus-trated in Fig. 1. The interacting electron system on thering is connected to conducting leads through two sites,labeled 0 and n, by weak links with hopping t0, and eithertwo materials or two different gate voltages at alternatingsites may be used to model the IHM. It is essential todistinguish between the two cases depending on the spindegeneracy of the isolated interacting system for oddparticle number. We thus perform two different calcula-tions of the conductance. The first assumes that spindegeneracy is lifted by a Zeeman interaction but allowsorbital degeneracy of the states, which is important whenincluding interference effects arising in a ring geometry[13]. In the second, where spin degeneracy is retained, wemap the problem into an effective Anderson model2004 The American Physical Society 076801-1VOLUME 93, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R Sweek ending13 AUGUST 2004(EAM) and express the conductance as a function of thespectral density of the latter.If the ground state jgi of the interacting system isnondegenerate, for small t0 the links can be eliminatedby a canonical transformation, leading to an effective,noninteracting Hamiltonian for the two leads in whichthe two ‘‘impurity’’ sites i connected to the links (  1and 1 in Fig. 1) have an energy shift i!, and areconnected by an effective hopping teff!,Heff XkL;kLcykLckL  1cy1c1  1cy1c1XkR;kRcykRckR Xteffcy1c1  H:c:: (1)L (R) refers to the lead containing the site 1 (  1). Theimpurity parameters are given in terms of Green func-tions for the isolated ring gij!  hhci; cyjii,1!  t02g00!; 1!  t02gnn!;teff!  t02gn0!:(2)Without affecting the essential results, we assumeidentical leads and, for their states without sites 1; 1,we consider two models for the density of states ! andthe hybridization Vj! with sites 1; 1: (I) ! andVj! constant: for j  1 and teff  0, geff0jj ! 1=! j  i with   V2; (II) leads describedby a tight-binding model with nearest-neighbor hoppingt, where geff0jj !  1=!=2 j  i! with ! !V2! t2 !2=4p. By introducing an integerm  1 or 2 for cases I and II, the transmittanceT!;Vg becomesT 42jteff j2j!m 1  i!m 1  i  jteff j2j2: (3)Vg is a gate voltage, which changes the on-site energy ofall sites of the ring by eVg. In case II this equation isexact (8 t0) in the noninteracting system and generalizes aprevious result [13]. Equation (3) also generalizes pre-vious approaches in which only one intermediate state ofthe ring is included [14,15] and is appropriate for thestudy of interference effects [13]. Spin degeneracy limitsits validity to magnetic fields B for which the Kondoeffect is destroyed, as discussed below. For sufficientlylarge Zeeman energy gBB, the zero-bias conductance atsteady state is given by G  G0T;Vg=2, where G0 2e2=h and  is the chemical potential of the leads. Theresults of Ref. [16] suggest that Eq. (3) remains valid, withG  G0T;Vg, also in the absence of Zeeman splittingin an intermediate temperature range T1 < T < 0:1w,where T1 is very small. Henceforth we set   0, corre-sponding to half-filled leads.TVg is very small, except at the poles of gn0. If theground state jgi has N particles for   0, the ground-state energy EgN satisfies EgN  eVg <EgN  1076801-2and EgN  eVg <EgN  1. As Vg is increased, apole in gn0Vg is reached, and for larger Vg the groundstate has N  1 particles. Similarly, if Vg is lowered, apole is reached when z  eVg  EgN  1  EgN  0.Assuming that the state of lowest energy for N  1particles, jgN  1i, is not degenerate, gij may be ex-pressed as a Laurent expansion around z  0,gijz ai ajz %ij  &ijz    ; (4)where %jj, &jj, and . . . are real coefficients andaj  hgN  1jcjjgNi: (5)'j  jajj2 is the spectral weight for a local photoemis-sion process. Substituting these expressions in Eqs. (2)and (3), retaining terms to lowest nontrivial order int0=0, and using that Tz O1 for z & t02=0yieldsTz 11 w=z24'0'n'0  'n2 Ot0=02; (6)where the half-width at half maximum peak height isw  '0  'nt02=0: (7)Using G  G0T;Vg, this expression coincides at suf-ficiently low temperatures with Eq. (7) of Ref. [15].If sites 0 and n are equivalent by symmetry, then w 2t02'0=0 and the integrated weight I RdzTz w are both proportional to t02 and to '0. If Vg isincreased instead of decreased, the same result is ob-tained with cj ! cyj and N  1 ! N  1 in Eq. (5).Thus a single transport measurement gives simultaneousspectral information related to photoemission and to in-verse photoemission. We stress that this information isobtained with much finer energetic resolution (eV) thanthat available by direct spectroscopic techniques (meV).We turn next to a calculation of the conductance forsmall or zero B, where either jgNi or jgN  1i is spindegenerate. This degeneracy can be taken into accountaccurately if w is small compared to the separation be-tween groups of spin-degenerate energy levels, becausethe problem becomes equivalent to an EAM. We take Vgsuch that a nondegenerate state jgNi is close in energy tothe spin-degenerate state jgN  1i. Extension to othercases is straightforward. Neglecting other states, theEAM for any interacting system between the leads isHA  tX;i0;1cyici1 Xdyt01c1  t01c1  H:c:"dXnd UAnd"nd#  gBBnd"  nd#; (8)with nd  dyd, t01  t0 a0, t01  t0 an, "d  EgN EgN  1  eVg, and infinite UA ( justified becauseUA  EgN  EgN  2  2EgN  1  w).076801-2-10 -5 0 5 100.00.20.40.60.81.0eVg /w G/G0       gµBB/w  0  0.0001  0.001  0.01  0.1  1FIG. 2. Conductance as a function of gate voltage for theEAM at several magnetic fields. Circles correspond to theconductance of a resonant level for the majority spin only.FIG. 3. Transmittance as a function of gate voltage for aneight-site IHM ring with leads at sites 0 and 2 for tR  1, t0 0:2,   3, and values of U below and above UcL  8.-4 -3 -2 -1 0 1 20.00.20.40.60.8  Φ=0.04π Φ=0.1π Φ=0.2π Φ=0.3πG/G0Vg (µV)FIG. 4. Transmittance as a function of Vg for a ring of eightQDs described by the IHM. The parameters chosen are U 1 meV,   3U=4, t  U=4, t0=t  0:2, and gB 0:025 meV=T. The curves have been shifted in Vg (cf. lowerpanel Fig. 3) such that their maxima coincide.VOLUME 93, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R Sweek ending13 AUGUST 2004The conductance is given byG 2e2h2wjt01t01j2jt01j2  jt01j2Xd; (9)where d! is the spectral density of the effective delectrons [17]. The conductance of the EAM computed asa function of gate voltage for several values of B usingslave bosons in the mean-field approximation (SBMFA)[18] is shown in Fig. 2. For B  0, G increases abruptlyfrom 0 to G0 when z 0 and remains nearly perfect forhigher Vg due to pinning of the Kondo peak in d! atthe Fermi level. However, because the Kondo energy scaleTK decreases exponentially with Vg, even for very smallB the plateau becomes a broad peak, terminated whenTK < gBB, beyond which G falls to zero. The abrupt-ness of the fall is an artifact of the SBMFA, but the totalwidth of the feature is well described. For larger B thepeak width decreases, tending to w to recover the pre-vious result: for gBB * w, G  G0Tz=2, with Tzgiven by Eq. (6).We now apply these results to the topological chargetransition in the IHM, which is defined byHIHM  tRXi;cyi1ciei=L  H:c: 12Xi1inigBBXini"  ni# UXini"ni#; (10)where ni  cyici, ni Pni, and N Pini. We cal-culate TVg by exact diagonalization for the isolated ringwith L sites and N  L electrons, with leads attached totwo sites of the same energy (  =2) in the presence of amagnetic flux  (Fig. 1), with   2=0. The Greenfunctions gijVg are obtained numerically and substi-tuted in Eqs. (2) and (3). We choose L  8, but similarresults are obtained for any L  4n (L  4n 2 with !  ). The qualitative features are independent ofthe lead position n, with the exception of n  L=2 wherethe transmittance at    vanishes for symmetry rea-sons. Because HIHM is invariant under simultaneous076801-3particle-hole transformation and sign change of , werestrict our analysis to eVg < 0. We set tR  1 as the unitof energy unless otherwise stated.The topological transition is present at any value of tR.As a basis for our study we consider a realistic ringstructure containing eight QDs with only their lowestlevels singly occupied, the centers separated by 200 nm[5], and the parameters U; 1 meV and tR U=4(intermediate coupling strength). We first assume thepresence of a strong magnetic field (B 1 T for a QDarray), in the plane of the ring in order not to alter thethreading flux, which destroys the Kondo effect. Figure 3shows the first peak in GVg at   3 and for twovalues of U. For L  8 and   3, the topologicaltransition occurs at Uc  4:352. The peak width is pa-rameter dependent: for U  5, w is approximately con-stant with decreasing flux, whereas for U  4, wdecreases and the peak disappears at   0. If U <Uc,the ground state jgLi for   0 is even under reflectionthrough any site (corresponding to a BI), while for U >Uc, it is odd (MI or SDI) [8]. For   0, the lowest-076801-34.0 4.2 4.4 4.6 4.8 5.00.00.20.40.60.81.01.21.41.61.8UUcLα0 ∆ = 3 Φ = 0 Φ = 0.01π Φ = 0.03π Φ = 0.1πFIG. 5. Matrix element '0 as a function of U for   3(Uc  4:352) over a range of values of the flux .VOLUME 93, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R Sweek ending13 AUGUST 2004energy hole enters the system with the Fermi wave vector,=2, leaving an orbitally degenerate ground state withL 1 particles, jgL 1i. For   0, this degeneracy isbroken and jgL 1i is odd under reflection through anyeven site if  is negative [19]. As a consequence, for U <Uc the matrix element a0 [Eq. (5)] vanishes by symmetry,whence G is negligible at   0. The flux acts as asymmetry-breaking field, which allows one to followthe first peak in a continuous manner until it disappearsas  ! 0.To demonstrate that these essential features are notaffected by the presence of a Kondo resonance, we con-sider the eight-site ring with parameters (Fig. 4) similarto one experimental realization [5] and B normal to thering plane so that the threading flux and Zeeman splittinghave the same origin. Figure 4 shows the conductance inthis regime, calculated using the EAM in the SBMFA.The disappearance of the feature with decreasing fluxremains clear.The topological transition may now be characterizedusing '0. As shown in Fig. 5 for   3, '0 as a functionof U (cf. peak widths in Fig. 3 [19]) vanishes discontin-uously at Uc for   0, indicating unambiguously thecharge transition. The analogous result obtained for fixedU by varying  is of direct experimental interest, be-cause  can be controlled by a difference in gate voltageapplied between even and odd sites. A finite flux acts as aparity-breaking perturbation and smooths the transition.This is the situation for artificial arrays of QDs, in whichperfect structural symmetry is difficult to attain. We notethat curves for all flux values cross approximately at thesame point: transport measurements under different ap-plied fields can therefore help to locate the transition,even in the absence of perfect reflection symmetry. Fora small molecule, which by definition exhibits the  ! 0limit, the only source of asymmetry is the leads.In summary, we have considered the transport proper-ties of a ring-shaped interacting system connectedweakly to conducting leads. The conductance is expressedin terms of the spectral density of an EAM, whichillustrates the breaking of the Kondo effect by an applied076801-4field. Outside the Kondo regime we have derived a trans-mittance formula including nontrivial interference ef-fects. The conductance peaks may be characterized bytheir total weight, offering spectral information with farhigher resolution than conventional spectroscopic mea-surements. As an application of this result, we have dem-onstrated that the conductance can be used to detect thecharge transition of molecules or quantum dot rings de-scribed by the IHM. The method is relevant in the generalcontext of systems presenting phase transitions whichinvolve a symmetry change of the ground state.A. A. A. and K. H. acknowledge support of CONICETand PICT 03-12742 of ANPCyT. A. P. K. acknowledgessupport through SFB484 of the DeutscheForschungsgemeinschaft and B. N. acknowledges the sup-port of the Swiss National Science Foundation.[1] M. A. Reed et al., Science 278, 252 (1997); D. Porathet al., Nature (London) 403, 635 (2000); J. Reichertet al., Phys. Rev. Lett. 88, 176804 (2002).[2] C. P. Collier et al., Science 277, 1978 (1997); G.Medeiros-Ribeiro et al., Phys. Rev. B 59, 1633 (1999).[3] A. P. Alivisatos, Science 271, 933 (1996).[4] L. P. Kouwenhoven, Science 268, 1440 (1995).[5] L. P. Kouwenhoven et al., Phys. Rev. Lett. 65, 361 (1990).[6] R. Resta and S. Sorella, Phys. Rev. Lett. 82, 370 (1999).[7] M. Fabrizio, A. O. Gogolin, and A. A. Nersesyan, Phys.Rev. Lett. 83, 2014 (1999).[8] M. E. Torio, A. A. Aligia, and H. A. Ceccatto, Phys. Rev.B 64, 121105(R) (2001), and references therein.[9] A. P. Kampf et al., J. Phys. Condens. Matter 15, 5895(2003); Y. Z. Zhang, C .Q. Wu, and H. Q. Lin, Phys. Rev. B67, 205109 (2003); C. D. Batista and A. A. Aligia, Phys.Rev. Lett. 92, 246405 (2004); A. A. Aligia, Phys. Rev. B69, 041101(R) (2004).[10] S. R. Manmana et al., cond-mat/0307741 [Phys. Rev. B(to be published)].[11] J. Hubbard and J. B. Torrance, Phys. Rev. Lett. 47, 1750(1981); N. Nagaosa and J. Takimoto, J. Phys. Soc. Jpn. 55,2735 (1986).[12] T. Egami, S. Ishihara, and M. Tachiki, Science 261, 1307(1993).[13] E. A. Jagla and C. A. Balseiro, Phys. Rev. Lett. 70, 639(1993).[14] G. Chen et al., Phys. Rev. B 50, 8035 (1994).[15] C. A. Stafford, Phys. Rev. Lett. 77, 2770 (1996).[16] W. Izumida, O. Sakai, and S. Suzuki, J. Phys. Soc. Jpn.70, 1045 (2001).[17] A. A. Aligia and C. R. Proetto, Phys. Rev. B 65, 165305(2002).[18] K. Kang et al., Phys. Rev. B 63, 113304 (2001).[19] For reflection through odd sites, jgL 1i is odd forpositive . Under a sign change of  the parities of evenand odd sites are interchanged, and with these the chargesector (BI or SDI, MI) in which '0  0 at   0.076801-4

https://ri.conicet.gov.ar/bitstream/11336/69909/6/CONICET_Digital_Nro.accd21c0-68b5-4ba9-96bf-6e425d61d01d_s.pdf

Detection of topological transitions by transport through molecules and
  nanodevices

Detection of topological transitions by transport through molecules and nanodevices

Abstract

Similar works

Full text

Available Versions

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

Crossref

OPUS - Augsburg University Publication Server

CONICET Digital

OPUS Augsburg