The dark side of benzene: interference vs. interaction
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec The dark side of benzene: interference vs. interaction
Dan Bohr
Institut f¨ur Theorie der Kondensierten Materie, Karlsruher Institut f¨ur Technologie, 76131 Karlsruhe, Germany
Peter Schmitteckert
Institut f¨ur Nanotechnologie, Karlsruher Institut f¨ur Technologie, 76344 Eggenstein-Leopoldshafen, Germany andDFG Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany (Dated: November 15, 2018)We present the study of the linear conductance vs. applied gate voltage for an interacting sixsite ring structure, which is threaded by a flux of π and coupled to a left and a right lead. Thisring structure is designed to have a vanishing conductance for all gate voltages and temperaturesprovided interactions are ignored. Therefore this system is an ideal testbed to study the interplayof interaction and interference. First we find a Kondo type resonance for rather large hoppingparameter. Second, we find additional resonance peaks which can be explained by a populationblocking mechanism. To this end we have to extend the Kubo approach within the Density MatrixRenormalization Group method to handle degenerate states. I. INTRODUCTION
The continuing miniaturization trend in solid statephysics has lead to the discovery of significant new effectsand interesting properties of new devices. Two effectsare playing major roles: Interaction and interference. In-terference is the essential component in Aharonov-Bohmrings, whereas interaction is the essential componentin Coulomb blockade experiments and Kondo physics.Combining these two effects has turned out to be aformidable theoretical task.
The appearance of a Fanoanti resonance has been suggested as a possibility for aninterference based transistor and discussed in more de-tail in . The role of interference effects for transportproperties even for acyclic molecules has been discussedin . Going beyond perturbative approaches the interplayof interaction and interference in the transport proper-ties was studied by Roura-Bas et. al . There the authorsfound a crossover between a SU(2) and SU(4)Andersonmodel mediated by an external level splitting.In this work we employ the density matrix renormal-ization group (DMRG) method to investigate an ex-ample of a structure where both interference and inter-action play a strong role, a spinless tight-binding modelof a ring-structure. Threading half a quantum of fluxthrough the ring, and disregarding interactions the con-ductance of this specific ring is analytically zero for allgate voltages as each single-particle level is two-fold de-generate and displays perfect destructive interference, seeappendix.Including interaction this picture changes qualita-tively: A very broad resonance appears as the strengthof the interaction is increased. Introducing an asymme-try in the ring gradually destroys this resonance, and a“normal” Lorentzian line shape at a Coulomb blockadeposition appears, where the single particle levels of the isolated ring are half occupied.An exponentially decaying line shape combined with aconductance plateau indicates that the conductance peakat small gate voltages is not described by a simple singleparticle picture and is similar to conductance plateaus inKondo physics. Breaking the degeneracy by perturbing asingle hopping-matrix element on the ring destroys thisconductance peak. In contrast to usual Kondo physicsnew conductance peaks appear which we explain by alevel/interference blocking mechanism. In contrast to themodel studied in the Kondo effect discussed in work isstabilized by the charge density wave like ordering due tothe umklapp scattering in the presence of the underlyinglattice. II. MODEL
In this paper we consider a model consisting of a(hexagonal) ring of sites, through which is threaded halfa flux quantum. The ring is then coupled symmetricallyto leads in a two-terminal setup, as indicated in Fig. 1.The flux through the ring is modeled using Peierls substi-tution as a phase on the hopping-matrix elements on thering, such that the relative phase of the hopping elementson the ring changes by π going once around the ring. Thedistribution of this phase over the ring is insignificant asit can always be gauged into a single bond. In this workwe choose to modify a single hopping element only, de-noted t BC , as indicated in red in Fig. 1. Including aparticle-hole symmetric nearest-neighbor density-densityinteraction on the ring the Hamiltonian of the model is H = H Ring + H lead + H Contact , (1) H Ring = M S − X i =1 (cid:16) t i d † i d i +1 + t ∗ i d † i +1 d i (cid:17) + t BC d † M S d + t ∗ BC d † d M S + M S − X i =1 U i (cid:16) n i − (cid:17)(cid:16) n i +1 − (cid:17) + U M S (cid:16) n M S − (cid:17)(cid:16) n − (cid:17) + M S X i =1 V gate n i (2) H lead = M X i =1 X α = L,R (cid:16) t i,α c † i,α c i +1 ,α + t ∗ i,α c † i +1 ,α c i,α (cid:17) + X k X α = L,R (cid:16) ǫ k,α n k,α + t k,α c † M,α c k,α + t ∗ k,α c † k,α c M,α (cid:17) (3) H Contact = t L d † c ,L + t ∗ L c † ,L d + t R d † M S / c ,R + t ∗ R c † ,R d M S / , (4)where n ℓ = d † ℓ d ℓ is the local density operator of site ℓ ,and n k,α = c † k,α c k,α is the density operator for momen-tum level k in lead α . M S = 6 denotes the number ofsites in the ring, M the number of real-space sites inthe leads, and k labels the momentum-space sites of theleads. In this work we use the values t i = t Dot on thedot and give values of t BC in units of t Dot . Mostly weconsider the case t BC = − t Dot corresponding to half aflux-quantum through the ring, Φ = Φ . Further, we usethe values t L = t R = t ′ = 0 . t Dot for the coupling to theleads. Additionally we use a combination of a logarithmicdiscretization to cover a large energy-scale of the band,and switch to a linear discretization for the low-energysector close to the Fermi edge , for details of discretiza-tion issues we refer to . We model each lead by a real- FIG. 1: (Color on-line) Illustration of the finite size setup usedin the DMRG calculations. The sites in the ring are shownin blue, and the implementation of the flux, t BC , is shown asa red hopping between sites 1 and 6 on the ring. Each lead(shown in green) is described by a real-space part coupledto a momentum-space part. The real-space part of the leadsensures a proper representation of local physics, whereas themomentum-space part ensures that the low-energy spectrumis properly represented. Note that the outermost real-spacesite in each lead is coupled to all momentum-space sites ofthat lead. space tight-binding chain coupled to a momentum-spacepart , an advantageous setup for representing all rele-vant energy scales of a given problem. Within this setupwe evaluate the Kubo formula for conductance, explicitly the two correlators g J i J j = e h h ψ | J i πη ( H − E ) (cid:0) ( H − E ) + η (cid:1) J j | ψ i , (5) g J i N = − e h h ψ | J i πiη ( H − E ) + η N | ψ i , (6)where J ℓ = i ( t ∗ ℓ c † ℓ − c ℓ − t ℓ c † ℓ c ℓ − ) denotes the currentoperator on site ℓ and N = ( N L − N R ) is the rigid shiftof the levels in the leads corresponding to the appliedvoltage perturbation. Note that the Hamiltonian H inEqs. (5,6) contains all interactions and couplings but notthe voltage perturbation, and | ψ i is the ground state ofthis Hamiltonian. A. Degenerate ground-state
In the case where the ground-state | ψ i is (near-) de-generate the evaluation of the Kubo formula sketchedabove breaks down. Rather than a single ground-statecorrelator at zero temperature, a finite temperature mul-tiple ‘ground-state’ average must be used, since in generalit cannot be decided numerically whether a finite gap isa physical gap, or caused by numerical inaccuracies. Theevaluation of the Kubo-formula in this case is thus g = 1 Z N X n =1 e − β ( E n − E ) g n , (7)where g n is the conductance of the n ’th (near-) degener-ate ground-state level calculated using Eqs. (5,6), β is theinverse temperature, E n is the energy of the n ’th level,and Z = P Nn =1 e − β ( E n − E ) is the partition sum. Herewe set β of the order of the inverse level spacing of theleads, thus averaging over low lying states having an ex-citation gap smaller than the finite size resolution of theleads. In addition N is chosen sufficiently large to coverall relevant (near-) degenerate states. III. RESULTS
It was shown in that a continuum model of an 1Dring of non-interacting spinless fermion threated by a flux π shows antiresonances at flux φ = π leading to new phe-nomena in the high temperature limit when interactionis turned on. In the non-interacting limit the transportproperties of the benzene-like ring-structure sketched inFig. 1 can be calculated exactly and in the infinite leadlimit the conductance is identically zero for all gate volt-ages, for all couplings t ′ and at all temperatures due toa perfect interference between the two paths through thering. Note that on a lattice this property does not holdfor a ring consisting of four or eight sites. Using DMRGto evaluate the Kubo formula for conductance we calcu-late the conductance for different values of the strengthof the interaction. We typically use 400 −
800 statesper block in the DMRG procedure, and the momentum-space part of the lead is described by 40 logarithmicallydiscretized levels to cover the broad energy-range of theband, and additionally 10 linearly discretized levels closeto the Fermi-edge to ensure a good discretization here.The conductances obtained are shown in Fig. 2. Inthe non-interacting limit we do indeed find a vanishingconductance with only minor finite-size deviations, origi-nating from the finite size of the lead used in the DMRGsetup.Increasing the strength of the interaction, U , the val-ues of the conductance also increase, and eventually aresonance is formed at zero gate-potential. For the inter-action strength U = 2 the ‘resonant’ value of the conduc-tance is thus found to be g ≈ .
75. The shape of this res-onance i.e. the exponential decay of the resonance withthe gate voltage, differs significantly from the Lorentzianshape usually found in simple resonant systems , in-dicating that a more complicated mechanism is at play.Increasing the strength of the interaction further, U =2 .
25, a broad ‘plateau’ in the conductance is formedaround zero gate-potential. The plateau is significantlywider than a Lorentzian of the corresponding height, andresembles somewhat a split Kondo resonance, the split-ting introduced by the hopping to the leads that alsoallows for transport. To show the similarity to the singleimpurity Anderson model (SIAM) we label in Fig. 3 theleft (right) lead as up (down) electrons. Setting t Dot = 0no mixing of ‘up’ and ‘down’ states occurs, and stronginteraction forbids adding/removing an additional parti-cle. In order to have transport it is necessary to switchon the hybridization between the dot sites, which actslike a magnetic field suppressing the proposed geometricKondo effect. In our case, due to the added flux, the de-generacy of the single-particle levels is not lifted leavingroom for Kondo physics. Nevertheless, the hybridizationof the dot provides a mixing of the up and down statesproposed in Fig. 3, which is necessary to enable transportthrough the ring. However, by increasing U a charge den-sity wave (CDW) ordering again becomes preferred whenthe interaction reestablishes two well separated states. It g ( V g a t e ) V gate t BC =-1.0U=0.0U=1.0 U=2.0U=2.25 U=3.0U=4.0 U=5.0 FIG. 2: (Color on-line) Conductance versus gate-potentialfor the ring-structure varying the nearest-neighbor interac-tion strength U . In the non-interacting limit the conductanceis zero to the precision of our finite-size setup, whereas thevalues for moderate interaction strengths approaches the uni-tary limit. Lines are added to the DMRG data as guides tothe eye.FIG. 3: (Color on-line) Conjectured Kondo setup. is interesting to note, that the effect is most dominant forinteraction values close to where a phase transition to aCDW ordered state appears in the thermodynamic limitat U c = 2 t .In order to test this idea we have performed simi-lar calculations on slightly asymmetric rings, reducingthe magnitude of a single hopping slightly from unity,breaking the degeneracy between the single particle lev-els, and obtained the conductances shown in Fig. 4. Asthe figure shows for the interaction strength U = 2 . U = 2 U/ t ′ = t ′ / √ T K , atwhich the Kondo resonance of the SIAM is destroyed isgiven by T K = D exp( − /J ( U )), see , where D = 2 t is the band cut-off, and J = t ′ (1 / | ǫ d | + 1 / | ǫ d + U | ) isthe effective Kondo coupling. Setting ˜ t ′ = 0 . t/ √ ǫ d = − ˜ U / U = 2 . t a Kondo temperature of T K ≈ . · − t ,which is in reasonable agreement with our numerical re- g ( V g a t e ) N D o t V gate U=2.25t BC =-0.98t BC =-0.99t BC =-0.999t BC =-0.9999t BC =-1.0 N Dot , t BC =-0.98N Dot , t BC =-0.99N Dot , t BC =-0.999N Dot , t BC =-0.9999N Dot , t BC =-1.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2 2.2 2.4 2.6 2.8 3 g ( V g a t e ) N D o t V gate U=1.0
FIG. 4: (Color on-line) An asymmetry of the hopping in thering rapidly destroys the geometric Kondo-effect. This figureshows the effect on the conductance and the occupation ofthe ring when modifying a single hopping by 0.01-2%. Whenthe geometric Kondo-effect is destroyed a normal resonantstructure pattern is rediscovered, and the resonance movestowards the normal position where the occupation is half-integer. Lines are added to the DMRG data as guides to theeye. sults.Also plotted in Fig. 4 is the total density of the ring forthe various parameter choices. Remarkably the electron-density of the ring remains virtually unchanged whenthe asymmetry is varied, although the conductance ofthe ring changes significantly. This clearly demonstratesthat the observed effect is an interference-effect. Further-more, increasing the asymmetry of the ring and therebydestroying the interference effect, the resonance is pushedtowards the usual location for resonant structures, wherethe particle number on the structure is half-integer, andat the same time the line-shape becomes increasinglyLorentzian, although an asymmetry with a long tail per-sists.In the case U = 1 .
0, where no geometric Kondo-effectis present, reducing the symmetry in the ring also re-sults in changes. From the symmetric case, t BC = − t Dot ,where only a very small resonance is found, to the mostasymmetric case considered here, t BC = − . t Dot , aclear resonance develops.We explain this new resonance as the result of a level n j V gate Level blocking in interacting nanostructureU=1.0, t BC =-0.98, j=3U=1.0, t BC =-0.98, j=4U=2.25, t BC =-0.98, j=1U=2.25, t BC =-0.98, j=2 U=2.25, t BC =-0.98, j=3U=2.25, t BC =-0.98, j=4U=2.25, t BC =-0.98, j=5U=2.25, t BC =-0.98, j=6 FIG. 5: (Color on-line) Occupation n j = h ˜ c + j ˜ c j i of the j -th single-particle levels of the non-interacting, isolated ringinduced by the coupling to the leads and the interaction onthe ring. blocking mechanism, similar to : When applying a gatevoltage the upper level of the two levels close to the Fermisurface of the leads is pushed out of resonance first. Thesecond level remains occupied since emptying the levelwould cost interaction energy due to the particle-holesymmetric interaction. Therefore adding or removing aparticle costs interaction energy and the lower level re-mains occupied although it is pushed above the Fermilevel. This is observed in Fig. 5, where the single-particleoccupations are plotted for the two cases considered. For U = 1 . j remains occupied, whereas the level j is emptied much faster. Since the levels now have dif-ferent occupations the perfect destructive interference isdestroyed and a conductance peak appears at the posi-tion, where the lower level is on resonance. By furtherincreasing the gate voltage we finally empty the lowerlevel as well. Interestingly, for U = 2 .
25 the filling ofthe upper level increases during this intermediate regime.Finally, for large V gate both levels share the same occupa-tion again and destructive interference is reestablished.A note is in order about the calculations for small V gate :In a region around V gate = 0 the combined lead and ringsystem is effectively degenerate, and thus the degener-ate method is applied in this parameter range. However,even using this expression the evaluation of the conduc-tance for small gate-potentials remains difficult due tonumerically difficult resolvent equations, and the resultsobtained there are not expected to be accurate. IV. DISCUSSION
At the fundamental level the results shown in this workclearly demonstrate that the simple picture of individualelectrons passing through the transport region one afterthe other gives a significantly different result than whenincluding even moderate electron-electron interactions.Rather a complicated many-body interference effect isformed, and we have proposed/conjectured a Kondo-effect as the explanation for the remarkable line-shapeobserved. The proposed Kondo-effect lies in the geo-metrical degree of freedom, and hence differs from thestandard Kondo-effect in the spin-degree of freedom. In-troducing an asymmetry in the ring clearly destroys theeffect, in a manner similar to the effect of a magnetic fieldon the standard Kondo-effect. Evidence for the proposedKondo effect is given by the disappearance of the conduc-tance peak through a weak effective magnetic field, whilethe peak is robust against a small gate voltage. Whileat first sight our model seems to be impractical for ex-perimental realizations as one cannot thread a benzenering with half a flux quanta, such a model can actuallyappear as effective models in molecular electronics . Appendix
The solution of the non-interacting system can mosteasily be calculated by scattering theory. There onesearches for scattering states of the form e i kx + r e − i kx in the left lead, and τ e i kx in the right lead, and Ψ n , n = 0 , , . . . , H − E )Ψ = 0. In theleads the solution of the tight binding leads requires E ( k ) = − t cos( k ). Solving the linear set of equationsfor flux φ = π and an incoming wave at the Fermi sur-face of k = π/ τ = 0 for allgate voltage V gate . Therefore the linear conductance G = | τ | = 0 vanishes for arbitrary t L and t R . In con-trast, setting the flux to zero one obtains a finite con- ductance G = t t t ( t + t t ) e h for the linear conductanceat zero gate voltage.In order to estimate the Kondo temperature in thestrongly interacting case we defineˆ N ↑ = ˆ n + ˆ n + ˆ n (8)ˆ N ↓ = ˆ n + ˆ n + ˆ n . (9)If we then then look at the interaction term generated byˆ U alt . = − e U (cid:16) ˆ N ↑ − ˆ N ↓ (cid:17) = e U ˆ N ↑ ˆ N ↓ + e U (cid:16) ˆ N ↑ − ˆ N ↓ (cid:17) (10)we find that it matches our original interaction ˆ U up tothe addition of terms with distance d=3, i.e. 0–3, 1–4,2–5, at least as long as we are close to the half filleddot. In the case of a charge density wave like orderingthese terms leads to the same contribution as the nearestneighbour interaction and therefore have to reduce e U to2 U/ Acknowledgments
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