Current Correlations in a Quantum Dot Ring: A Role of Quantum Interference
CCurrent Correlations in a Quantum Dot Ring: A Role of Quantum Interference
Bogdan R. Bu(cid:32)lka and Jakub (cid:32)Luczak
Institute of Molecular Physics, Polish Academy of Sciences,ul. M. Smoluchowskiego 17, 60-179 Pozna´n, Poland (Dated: May 29, 2019)We present studies of the electron transport and circular currents induced by the bias voltage andthe magnetic flux threading a ring of three quantum dots coupled with two electrodes. Quantuminterference of electron waves passing through the states with opposite chirality plays a relevant rolein transport, where one can observe Fano resonance with destructive interference. The quantuminterference effect is quantitatively described by local bond currents and their correlation functions.Fluctuations of the transport current are characterized by the Lesovik formula for the shot noise,which is a composition of the bond current correlation functions. In the presence of circular currents,the cross-correlation of the bond currents can be very large, but it is negative and compensates forthe large positive auto-correlation functions.
PACS numbers: 72.10.-d, 73.23.-b, 73.63.-b, 73.21.La
I. INTRODUCTION
In 1985, Webb et al. presented their pioneeringexperiment, showing Aharonov-Bohm oscillations in ananoscopic metallic ring and a role of quantum inter-ference (QI) in electron transport. Later, Ji et al. demonstrated the electronic analogue of the opticalMach–Zehnder interferometer (MZI), which was basedon closed-geometry transport through single edge statesin the quantum Hall regime. Theoretical studies pre-dicted coherent transport through single molecules witha ring structure, where, due to their small size, one couldshow constructive or destructive quantum interference ef-fects at room temperatures. From 2011, these predictionshave been experimentally verified, using mechanicallycontrollable break junction (MCBJ) and scanning tunnel-ing microscope break junction (STM-BJ) techniques in various molecular systems: Single phenyl, polycyclicaromatic, and conjugated heterocyclic blocks, as well ashydrocarbons (for a recent review on QI in molecularjunctions, see and the references therein).Our interest is in the internal local currents and theircorrelations in a ring geometry to see a role of quantuminterference. An interesting aspect is the formation of aquantum vortex flow driven by a net current from thesource to the drain electrode, which has been studied inmany molecular systems (see also ). It has alsobeen shown that, under some conditions, a circular ther-moelectric current can exceed the transport current .In particular, our studies focus on the role of the stateswith opposite chirality in the ring and on the QI effectand the circular current. Correlations of the electroncurrents (shot noise) through edge states in the Mach–Zehnder interferometer have been extensively studied byButtiker et al. (see also and the references therein).However, in a metallic (or molecular) ring, the situationis different than in the MZI, as multiple reflections arerelevant to the formation of the circular current. Ourstudies will show that the transition from laminar to vor-tex flow is manifested in the shot noise of local currents. In particular, it will be seen in a cross-correlation func-tion for the currents in different branches of the ring,which becomes negative and large in the presence of thecircular current.The paper is organized as follows. In the next chap-ter, Section II, we will present the model of three quan-tum dots in a ring geometry, which is the simplestmodel showing all aspects of QI and current correlations.The model includes a magnetic flux threading the ring,which changes interference conditions as well as inducinga persistent current. The net transport current and thelocal bond currents, as well as the persistent current (andtheir conductances), are derived analytically, by means ofthe non-equilibrium Keldysh Green function technique.It will be shown that the correlation function for the nettransport current can be expressed as a composition ofthe correlation functions for the local currents inside thering. We will, also, show all shot noise components; inparticular, the one for the net transport current (givenby Lesovik’s formula ). The next chapters, Sections III–V, present the analyses of the results for the case Φ = 0(without the magnetic flux), for the case with the per-sistent current only (without the source-drain bias V ),and for the general case (for V (cid:54) = 0, Φ (cid:54) = 0) showing theinterplay between the bond currents and the persistentcurrent. Finally, in Section VI, the main results of thepaper are summarized. II. CALCULATIONS OF CURRENTS ANDTHEIR CORRELATIONS IN TRIANGULARQUANTUM DOT SYSTEMA. Model
The considered system of three quantum dots (QDs) inan triangular arrangement is presented in Figure 1. Thissystem is described by the Hamiltonian H tot = H QD + H el + H QD − el , (1) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y which consists of parts corresponding to the electrons inthe triangular QD system, in the electrodes, and in thecoupling between the sub-systems, respectively. The firstpart is given by H QD = (cid:88) i ∈ QD ε i c † i c i + (cid:88) i,j ∈ QD (cid:16) ˜ t ij c † i c j + h.c. (cid:17) , (2)where the first term describes the single-level energy, ε i , at the i -th QD and the second term correspondsto electron hopping between the QDs. Here, the hop-ping parameters ˜ t = t e ıφ/ = ˜ t ∗ , ˜ t = t e ıφ/ =˜ t ∗ , and ˜ t = t e ıφ/ = ˜ t ∗ include the phase shift φ = 2 π Φ / ( hc/e ), due to presence of the magnetic fluxΦ; where hc/e denotes the one-electron flux quantum.The spin of electrons is irrelevant in our studies and soit is omitted. We consider transport in an open systemwith the left (L) and right (R) electrodes as reservoirs ofelectrons, each in thermal equilibrium with a given chem-ical potential µ α and temperature T α . The correspondingHamiltonian is H el = (cid:88) k,α ∈ L,R ε k,α c † kα c kα , (3)where ε k,α denotes an electron spectrum. The couplingbetween the 3QD system and the electrodes is given by H QD − el = (cid:88) k ( t L c † kL c + t R c † kR c + h.c. ) , (4)with tunneling from the electrodes given by the hoppingparameters t L and t R , respectively. The model omitsCoulomb interactions and, therefore, one can derive alltransport characteristics analytically. (cid:0) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) FIG. 1: Model of the triangular system of three quantumdots (3QDs) threaded by the magnetic flux Φ and attachedto the left (L) and the right (R) electrodes.
B. Calculation of Currents
We consider a steady-state current, with the net trans-port current through the 3QD system, I tr = I + I ,expressed as a sum of the bond currents through the up-per and the lower branches I ij = eı (cid:126) (˜ t ij (cid:104) c † i c j (cid:105) − ˜ t ji (cid:104) c † j c i (cid:105) ) . (5)We use the non-equilibrium Green function technique(NEGF), which is described in many textbooks (e.g., see ). To determine the currents, one calculates thelesser Green functions, G Here, we consider a single-particle interference ef-fect which takes place in a Mach–Zehnder or Michel-son interferometer, but not in a Hanbury Brown andTwiss situation with a two-particle interference effect.The current fluctuations are described by the operator∆ ˆ I ij ( t ) − (cid:104) ˆ I ij ( t ) (cid:105) , and the current–current correlationfunction is defined as S ij,nm ( t, t (cid:48) ) ≡ (cid:104) (cid:104) ˆ I ij ( t ) ˆ I nm ( t (cid:48) ) + ˆ I nm ( t (cid:48) ) ˆ I ij ( t ) (cid:105)− (cid:104) ˆ I ij ( t (cid:105)(cid:104) ˆ I nm ( t (cid:48) ) (cid:105) (cid:105) . (21)We consider the steady currents, for which the correlationfunctions can be represented, in the frequency domain,by their spectral density S ij,nm ( ω ) ≡ (cid:90) ∞−∞ dτ e ıωτ S ij,nm ( τ ) . (22)In this work, we shall restrict ourselves to studying thecurrent correlations at the zero-frequency limit ω = 0.As the net transport current is ˆ I tr = ˆ I + ˆ I , its currentcorrelation function can be expressed as a composition ofthe correlation functions for the bond currents S tr,tr = S , + S , + 2 S , . (23)The correlation functions S ij,in can be derived by meansof Wick’s theorem and are expressed as S ij,in = e π (cid:126) { t ij t in ( (cid:104) c † i c j (cid:105)(cid:104) c i c † n (cid:105) + (cid:104) c † i c n (cid:105)(cid:104) c i c † j (cid:105) )) − t ij t ni ( (cid:104) c † i c i (cid:105)(cid:104) c n c † j (cid:105) + (cid:104) c † n c j (cid:105)(cid:104) c i c † i (cid:105) ))+ t ji t ni ( (cid:104) c † j c i (cid:105)(cid:104) c n c † i (cid:105) + (cid:104) c † n c i (cid:105)(cid:104) c j c † i (cid:105) )) − t ji t in ( (cid:104) c † i c i (cid:105)(cid:104) c j c † n (cid:105) + (cid:104) c † j c n (cid:105)(cid:104) c i c † i (cid:105) )) } . (24)Once again, we use the NEGF method. As the lesserGreen functions, G Φ = 0 Let us analyse the bond currents in detail; first inthe absence of the magnetic flux, Φ = 0, and for a lin-ear response limit V → 0. Using the derivations fromthe previous section, one can easily calculate the bondconductances and current correlation functions. The re-sults are presented in Figure 2 for an equilateral trian-gle 3QD system (with all inter-dot hopping parameters t = t = t = − 1, which is taken as unity in ourfurther calculations) and for various values of the en-ergy level ε at the 3rd QD. The central column cor-responds to the case ε = ε = ε = 0, when theeigenenergies are given by E k = 2 t cos k , for the wave-vector k = 0 and the degenerated state for k = ± π/ T = 1 at E = − T = 0 at E = 1,where the Fano resonance takes place, with destructiveinterference of two electron waves. At low E < ≤ G , G ≤ S sh , (for the currents in both branches) is positive (seethe red curve in the bottom panel in Figure 2). Notethat, at the lowest resonant level, all correlation func-tions S sh , = S sh , = S sh , = 0, which means that thecurrents in both branches are uncorrelated.For E > 0, the conductances G and G can be nega-tive and exceed unity (with their maximal absolute valuesinversely proportional to the coupling Γ α ). This mani-fests a circular current driven by injected electronic wavesto the 3QD system, which can not reach the drain elec-trode; therefore, they are reflected backwards to the otherbranch of the ring. The circular current can be charac-terized by the conductance (see also ) G dr ≡ (cid:26) G for G < , −G for G < . (33)where the superscript “ dr ” marks the contribution to thecircular current driven by the bias voltage, in order todistinguish it from the persistent current induced by theflux (which will be analysed later). There is some am-biguity in definition of the circular current. Our defi-nition (33) is similar to the one given by the conditionsign[ G ] = − sign[ G ] for the vortex flow, used by Jayan-navar and Deo and Stefanucci et al. (see –whichrefers to ).For the considered case in Figure 2b, with ε = 0,the circular current is driven counter-clockwise for 0 F > ε = ∓ T = 1, where two of them areshifted to the left/right for ε = ∓ 2; however, the stateat E = 1 is unaffected. There is still mirror symmetry,for which one gets three eigenstates, where two of themare linear compositions of all local states, but the oneat E = 1 has the eigenvector 1 / √ c † − c † ) | (cid:105) , which isseparated for the 3rd QD. Therefore, the bond currentsare composed of the currents through all three eigen-states, and their contribution depends on E . From theseplots, one can see that the circular current is driven,for E > ε , when the cross-correlation S sh , becomesnegative. The direction of the current depends on the po-sition of the eigenlevels and their current contributions.For ε = − 2, the current circulates clockwise, whereasits direction is counter-clockwise for ε = 2.Here, we assumed a flat band approximation (FBA) forthe electronic structure in the electrodes (i.e., the Greenfunctions g r,aα = ∓ ıπρ , where ρ denotes the density ofstates). Appendix A presents analytical results for thecurrents and shot noise in the fully-symmetric 3QD sys-tem coupled to a semi-infinite chain of atoms. The re-sults are qualitatively similar. However, the FBA is moreconvenient for the analysis than the system coupled toatomic chains; in particular, for the cases with ε = ∓ − V → T = 0. However,the conductances exhibit sharp resonant characteristicsin the energy scale ∆ E ∝ Γ α and, therefore, one can ex-pect that these features will be smoothed out with an in-crease of voltage bias and temperature. Figure 3 presentsthe Fano factor F = S tr,tr / eI tr , which is the ratio of thecurrent correlation function to the net transport current,which was calculated numerically from Equations (17)and (26). At E = − 2, one can observe the evolutionfrom the coherent regime, from F = 0 to F = 1 / eV (cid:29) Γ α or k B T (cid:29) Γ α . Quantuminterference plays a crucial role at E = 1, leading to theFano resonance for which the transmission T = 0 and F = 1 in the low voltage/temperature regime. An in- (cid:0)(cid:1) (cid:0)(cid:2) (cid:0)(cid:3) (cid:4) (cid:3) (cid:2) (cid:1)(cid:5)(cid:0)(cid:1) (cid:0)(cid:2) (cid:0)(cid:3) (cid:4) (cid:3) (cid:2) (cid:1)(cid:5)(cid:4)(cid:4)(cid:6)(cid:2)(cid:4)(cid:6)(cid:7) (cid:8) (cid:9)(cid:10)(cid:11)(cid:9)(cid:10) (cid:12)(cid:11)(cid:12) (cid:8) (cid:13)(cid:14)(cid:11)(cid:13)(cid:14) (cid:12)(cid:11)(cid:12) (cid:8) (cid:13)(cid:15)(cid:11)(cid:13)(cid:15) (cid:12)(cid:11)(cid:12) (cid:16)(cid:8) (cid:13)(cid:14)(cid:11)(cid:13)(cid:15) (cid:0)(cid:1) (cid:0)(cid:2) (cid:0)(cid:3) (cid:4) (cid:3) (cid:2) (cid:1)(cid:5)(cid:0)(cid:2)(cid:0)(cid:3)(cid:4)(cid:3)(cid:2)(cid:1) (cid:17) (cid:11)(cid:12) (cid:18) (cid:13)(cid:14) (cid:12) (cid:11)(cid:12) (cid:18) (cid:13)(cid:15)(cid:19)(cid:20)(cid:19)(cid:20)(cid:19)(cid:20)(cid:19)(cid:20) (cid:21) (cid:22) (cid:23)(cid:24)(cid:25) (cid:21) (cid:22) (cid:23)(cid:26) (cid:21) (cid:22) (cid:23)(cid:25)(cid:27)(cid:28) (cid:29)(cid:28) (cid:30)(cid:28)(cid:31)(cid:28) (cid:28)!(cid:28) FIG. 2: (Top) Transmission and dimensionless bond conductances: T —black, G —blue, and G —green. (Bottom) Di-mensionless spectral function of the shot noise: S shtr,tr —black, S sh , —blue, S sh , —green, and −S sh , —red; calculated as afunction of the electron energy E for the equilateral triangle system of 3QDs (with the inter-dot hopping t = t = t = − V → 0. The dot levels are ε = ε = 0 and ε = − 2, 0, 2,for left, center, and right columns, respectively. The coupling with the electrodes is taken to be Γ L = Γ R = 0 . 25. Note thatthe cross-correlation function S sh , (red) is plotted negatively to show the zero crossing more clearly. crease of the voltage/temperature results only in a smallreduction of the Fano factor. IV. PERSISTENT CURRENT AND ITS NOISE:THE CASE V = 0 The persistent current and its noise has been stud-ied in many papers (e.g., by B¨uttiker et al. , Se-menov and Zaikin , Moskalates , and, more recently,by Komnik and Langhanke ) using full counting statis-tics (FCS), as well as in 1D Hubbard rings by exact diag-onalization by Saha and Maiti (see, also, the book byImry ).Here, we briefly present the results for the persistentcurrent and shot noise in the triangle of 3QDs. Noticethat, in the considered case, the phase coherence lengthof electrons is assumed to be larger than the ring cir-cumference, L φ (cid:29) L . The circular current is given byEquation (18), which shows that all electrons, up to thechemical potential in the electrodes, are driven by themagnetic flux Φ. Figure 4 exhibits the plots of I φ , de-rived from Equation (18), for different couplings with theelectrodes. In the weak coupling limit, where Γ → I φ = e (cid:88) k v k f k = − e (cid:126) (cid:88) k t sin( k + φ/ f k , (34) (cid:0)(cid:0)(cid:1)(cid:2)(cid:0)(cid:1)(cid:3)(cid:0)(cid:1)(cid:4)(cid:0)(cid:1)(cid:5)(cid:6)(cid:7) (cid:8)(cid:3) (cid:8)(cid:9) (cid:8)(cid:2) (cid:8)(cid:6) (cid:0) (cid:6) (cid:2) (cid:9) (cid:3)(cid:10) (cid:7) (cid:11) (cid:12) (cid:13)(cid:14)(cid:15)(cid:14)(cid:14)(cid:16)(cid:14)(cid:15)(cid:16)(cid:14)(cid:15)(cid:17)(cid:14)(cid:15)(cid:18)(cid:14)(cid:15)(cid:19) (cid:0)(cid:0)(cid:1)(cid:2)(cid:0)(cid:1)(cid:3)(cid:0)(cid:1)(cid:4)(cid:0)(cid:1)(cid:5)(cid:6)(cid:7) (cid:20)(cid:21)(cid:14)(cid:15)(cid:14)(cid:16)(cid:14)(cid:15)(cid:22)(cid:16)(cid:15)(cid:14)(cid:16)(cid:15)(cid:22)(cid:17)(cid:15)(cid:14) (cid:23)(cid:24)(cid:25)(cid:24) FIG. 3: Fano factor as a function of the Fermi energy E F forthe equilateral triangle 3QDs system ( t = t = t = − ε = ε = ε = 0) (a) for various bias voltages eV =0 . 01, 0.5, 1.0, 1.5, and 2.0, at T = 0; and (b) for varioustemperatures k B T = 0 . V → L = Γ R = 0 . µ L = E F − eV / µ R = E F + eV / where f k = 1 / (exp[( E F − E k ) /k B T ] + 1) is the Fermidistribution for the electron with wave-vector k , energy E k = 2 t cos( k + φ/ v k = (1 / (cid:126) ) ∂E k /∂k =( − t/ (cid:126) ) sin( k + φ/ φ = 2 π Φ / ( hc/e ) is thephase shift due to the magnetic flux Φ. The sum runsover k = 2 πn/ ( N a ) for n = 0 , ± 1, where N = 3 and a = 1 is the distance between the sites in the triangle.The current correlator is derived from Equation (24) S φ,φ = e (cid:126) (cid:88) k t sin ( k + φ/ f k (1 − f k ) . (35)This result says that fluctuations of the persistent cur-rent could occur when the number of electrons in the ringfluctuates (i.e., an electron state moves through the Fermilevel and I φ jumps). We show, below, that the couplingwith the electrodes (as a dissipative environment) resultsin current fluctuations , as well. (cid:0)(cid:1)(cid:0)(cid:2)(cid:3)(cid:2)(cid:1)(cid:4) (cid:5) (cid:0)(cid:3)(cid:6)(cid:7) (cid:0)(cid:3)(cid:6)(cid:1)(cid:7) (cid:3) (cid:3)(cid:6)(cid:1)(cid:7) (cid:3)(cid:6)(cid:7)(cid:5)(cid:8)(cid:9)(cid:10)(cid:0)(cid:11)(cid:0)(cid:1)(cid:0)(cid:2)(cid:3)(cid:2)(cid:1)(cid:11)(cid:4) (cid:5) (cid:12) (cid:13) (cid:14)(cid:15)(cid:16)(cid:17)(cid:14)(cid:18)(cid:16)(cid:19)(cid:17)(cid:15)(cid:16)(cid:17)(cid:20)(cid:21)(cid:22)(cid:21) (cid:23)(cid:24)(cid:25)(cid:26)(cid:25)(cid:27)(cid:23)(cid:24)(cid:25)(cid:26)(cid:28)(cid:29) FIG. 4: Persistent current I φ versus the flux φ threading theequilateral triangle system of 3QDs ( t = t = t = − ε = ε = ε = 0). The coupling is taken as Γ L = Γ R =Γ = 0 . 01 and 0.25; the Fermi energies are E F = − − T = 0. At the limit, V → 0, the integrand function of the noise S ij,in , Equation (28), is proportional to f ( E )(1 − f ( E )),which becomes the Dirac delta for T → S φ,φ = S , + S , − S , , where the components are S ij,in = S shij,in + G Lij G Lin + G Rij G Rin (see Equations (7)–(10) and (29)–(31)). Figure 5 presents the correlation function S φ,φ andits various components for the Fermi energy E F = − . L = Γ R = 1 when fluctua-tions are large. Notice that the fluctuations of the bondcurrents S , and S , (the blue and green curves, re-spectively) are different, although the average currentsare equal. The cross-correlation function S , is posi-tive at φ = 0, but it becomes negative for larger φ , due (cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:7) (cid:9)(cid:0)(cid:10)(cid:11) (cid:9)(cid:0)(cid:10)(cid:1)(cid:11) (cid:0) (cid:0)(cid:10)(cid:1)(cid:11) (cid:0)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) (cid:16) (cid:12)(cid:12) (cid:17)(cid:16) (cid:18)(cid:19)(cid:20)(cid:18)(cid:21) (cid:16) (cid:18)(cid:21)(cid:20)(cid:18)(cid:21) (cid:16) (cid:18)(cid:19)(cid:20)(cid:18)(cid:19) (cid:16) (cid:18)(cid:19)(cid:20)(cid:18)(cid:19) (cid:16) (cid:18)(cid:19)(cid:20)(cid:18)(cid:19) (cid:22) (cid:23)(cid:24)(cid:25)(cid:23)(cid:24)(cid:26)(cid:26)(cid:27)(cid:27) (cid:22) (cid:28)(cid:28) (cid:29)(cid:22) (cid:23)(cid:24)(cid:25)(cid:23)(cid:30) (cid:22) (cid:23)(cid:30)(cid:25)(cid:23)(cid:30) (cid:22) (cid:23)(cid:24)(cid:25)(cid:23)(cid:24) (cid:22) (cid:23)(cid:24)(cid:25)(cid:23)(cid:24) FIG. 5: Flux dependence of spectral function of the per-sistent current correlator S φ,φ (black) and its components: S , (blue), S , (green), - S , (red), and S LL , = ( G L ) (blue-dashed), S RR , = ( G R ) , (blue-dotted), respectively.We assume strong coupling: Γ L = Γ R = 1 . E F = − . T = 0. to the quantum interference between electron waves pass-ing through different states (as described in the previoussection).Figure 5 also shows ( G L ) (blue-dashed curve) and( G L ) (blue-dotted curve), which correspond to the lo-cal fluctuations of the injected/ejected currents to/fromthe upper branch on the left and right junctions, respec-tively (see Equation (28)). The magnetic flux breaks thesymmetry, inducing the persistent current and, therefore,the local conductances G L and G R are asymmetric. V. CORRELATION OF PERSISTENT ANDTRANSPORT CURRENTS, Φ (cid:54) = 0 AND V (cid:54) = 0 In this section, we analyze the currents and their cor-relations in the general case, derived from Equations (6),(14), (18), and (28), in the presence of voltage bias andmagnetic flux. The results for the conductances and thespectral functions of the shot noise are presented in Fig-ure 6. The magnetic flux splits the degenerated levels at E = 1 and destroys the Fano resonance. Figure 6a showsthat there is no destructive interference for a small flux φ = 2 π/ 16, and the transmission is T = 1 for all reso-nances. One can observe the driven circular current for E > 0, with negative G and G , but their amplitudesare much lower than in the absence of the flux (comparewith Figure 2b for φ = 0). For a larger flux, φ = 2 π/ G , G ≥ E = 0, the electronic waves passonly through the lower branch of the ring, and the upperbranch is blocked (with G = 1 and G = 0, respec-tively).The lower panel of Figure 6 presents the spectral func-tions of the shot noise. According the Lesovik formula, S shtr,tr = 0 at the resonant states (as T = 1). This seemsto be similar to the case φ = 0 presented in the lowerpanel in Figure 2. However, there is a great difference (cid:0)(cid:1) (cid:0)(cid:2) (cid:0)(cid:3) (cid:4) (cid:3) (cid:2) (cid:1)(cid:5)(cid:4)(cid:4)(cid:6)(cid:2)(cid:4)(cid:6)(cid:7) (cid:8) (cid:9)(cid:10)(cid:11)(cid:9)(cid:10)(cid:12) (cid:11)(cid:12) (cid:8) (cid:13)(cid:14)(cid:11)(cid:13)(cid:14)(cid:12) (cid:11)(cid:12) (cid:8) (cid:13)(cid:15)(cid:11)(cid:13)(cid:15)(cid:12) (cid:11)(cid:12) (cid:16) (cid:8) (cid:13)(cid:14)(cid:11)(cid:13)(cid:15) (cid:0)(cid:1) (cid:0)(cid:2) (cid:0)(cid:3) (cid:4) (cid:3) (cid:2) (cid:1)(cid:5)(cid:0)(cid:4)(cid:6)(cid:17)(cid:4)(cid:4)(cid:6)(cid:17)(cid:3)(cid:3)(cid:6)(cid:17) (cid:18) (cid:11)(cid:12) (cid:19) (cid:13)(cid:14) (cid:12) (cid:11)(cid:12) (cid:19) (cid:13)(cid:15)(cid:20)(cid:21)(cid:20)(cid:21)(cid:20)(cid:21)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) (cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:29)(cid:30)(cid:31) (cid:31)!(cid:31)"(cid:31) FIG. 6: (Top) Energy dependence of driven conductance G (blue), G (green) and transmission T (black). (Bottom)Shot noise S shtr,tr (black) with the components: S sh , (blue), S sh , (green), and −S sh , (red) for the considered triangular3QD system threaded by the flux φ = 2 π/ 16 (left) or φ =2 π/ L = Γ R = 0 . 25, and T = 0.Note that we plot − S sh , . in the components of the shot noise S shij,in , indicating thedifferent nature of transport through these states and therole of quantum interference. Let us focus on the lowestresonant state, at E = − 2, in Figure 6c, and comparewith that in Figure 2e, in the absence of the flux. In theformer case, the currents in both branches were uncorre-lated, and S sh , = S sh , = S sh , = 0. In the presenceof the flux, quantum interference becomes relevant, whichis seen in the shot noise (Figure 6c). Now, the currents inboth branches are correlated; S sh , is negative close toresonance and fully compensates for the positive contri-butions S sh , and S sh , at resonance. For φ = 2 π/ φ = 2 π/ 16 and for various bias voltages. Com-pared with the results in Figure 3 for φ = 0, one cansee how a small flux can destroy quantum interferenceand change electron transport. It is particularly seenclose to E = − 1, where the states with opposite chiralityare located. In the case φ = 0, one can observe the Fanoresonance with a perfect destructive interference, T = 0and F = 1. With an increase of the flux φ , the Fanodip disappears, the two states are split, and transmissionreaches its maximum value T = 1; the Fano factor F = 0when the splitting ∆ E > Γ α . A similar effect was seenin the case of Figure 2, where a change of the positionof the local level ε removed the state degeneracy anddestroyed the Fano resonance. (cid:0)(cid:0)(cid:1)(cid:2)(cid:0)(cid:1)(cid:3)(cid:0)(cid:1)(cid:4)(cid:0)(cid:1)(cid:5)(cid:6)(cid:7) (cid:8)(cid:3) (cid:8)(cid:9) (cid:8)(cid:2) (cid:8)(cid:6) (cid:0) (cid:6) (cid:2) (cid:9) (cid:3)(cid:10) (cid:7) (cid:11)(cid:12)(cid:13)(cid:14)(cid:13)(cid:15)(cid:13)(cid:14)(cid:16)(cid:15)(cid:14)(cid:13)(cid:15)(cid:14)(cid:16)(cid:17)(cid:14)(cid:13) FIG. 7: Fano factor as a function of E F for the considered3QD system threaded by the flux φ = 2 π/ 16 and for variousbias voltages eV = 0 . 01, 0.5, 1.0, 1.5, and 2.0. The coupling isΓ L = Γ R = 0 . 25, the chemical potentials are µ L = E F − eV / µ R = E F + eV / 2, and T = 0. For the strong coupling Γ L = Γ R = 1, the intensity ofthe transport current is comparable to the persistent cur-rent and, therefore, one can expect a significant drivencircular current. Figure 8 presents the flux dependenceof the total circular current I c and its driven compo-nent I dr , as well as the transport current I tr , for variousvoltages. For the considered case E F = 0 . 9, the drivencurrent circulates counter-clockwise and deforms the fluxdependence of the circular currents, which become asym-metric. (cid:0)(cid:1)(cid:0)(cid:2)(cid:3)(cid:4)(cid:2)(cid:2)(cid:3)(cid:4)(cid:1)(cid:1)(cid:3)(cid:4)(cid:5) (cid:6) (cid:7)(cid:8)(cid:7)(cid:5) (cid:9)(cid:10) (cid:7)(cid:8)(cid:7)(cid:5) (cid:11)(cid:10) (cid:0)(cid:2)(cid:3)(cid:4) (cid:0)(cid:2)(cid:3)(cid:12)(cid:4) (cid:2) (cid:2)(cid:3)(cid:12)(cid:4) (cid:2)(cid:3)(cid:4)(cid:13)(cid:14)(cid:15)(cid:16) (cid:17)(cid:18)(cid:19)(cid:20)(cid:19)(cid:21)(cid:19)(cid:20)(cid:22)(cid:21)(cid:20)(cid:19)(cid:21)(cid:20)(cid:22) FIG. 8: Circular current I c = I dr + I φ (solid curves), itsdriven component I dr (dashed curves), and the net transportcurrent I tr (dotted curves) versus φ for various bias voltages: eV = 0 . 01, 0.5, 1.0, and 1.5. We assume a strong couplingΓ L = Γ R = 1, the chemical potentials are µ L = E F − eV / µ R = E F + eV / E F = 0 . 9, and T = 0. VI. SUMMARY We considered the influence of quantum interferenceon electron transport and current correlations in a ringof three quantum dots threaded by a magnetic flux.We assumed non-interacting electrons and calculated thebond conductances, the local currents, and the currentcorrelation functions—in particular, the shot noise—bymeans of the non-equilibrium Keldysh Green functiontechnique, taking into account multiple reflections of theelectron wave inside the ring. As we considered elasticscatterings, for which Kirchhoff’s current law is fulfilled,the transmission T = G + G is a sum of the localbond conductances and the shot noise for the transportcurrent is a composition of the local current correlationfunctions, S shtr,tr = S sh , + S sh , + 2 S sh , = T (1 − T ),which gives the Lesovik formula.In the system, having triangular symmetry, the eigen-states E = − E ± = 1 (with the wavevector k = 0and k = ± π/ 3) play a different role in the transport,which is seen in the bond conductances and the shot noisecomponents. An electron wave injected with energy closeto E is perfectly split into both branches of the ring andthe current cross-correlation function S sh , is positive.At the resonance E , the transmission T = 1 and all cor-relation functions S sh , = S sh , = S sh , = 0, whichmeans that the bond currents are uncorrelated. Themagnetic flux changes quantum interference conditionsand correlates the bond currents; the cross-correlation S sh , becomes negative at the resonance and fully com-pensates the positive auto-correlation components S sh , and S sh , (with S shtr,tr = 0).Quantum interference plays a crucial role in transportthrough the degenerate states at E ± = 1, where one canobserve Fano resonance with destructive interference. Inthis region, the circular current I dr can be driven bythe bias voltage. The bond conductances have an oppo-site sign, their maximal value is inversely proportionalto the coupling, Γ α , with the electrodes, and they canbe larger than unity. The direction of I dr depends onthe bias voltage and the position of the Fermi energy E F , with respect to the degenerate state E ± . The auto-correlation functions S sh , , S sh , are large (inverselyproportional to Γ α ) close to the resonance. The cross-correlator S sh , is negative in the presence of the drivencircular current. Our calculations show that a smallmagnetic flux, φ = 2 π/ 16, can destroy the Fano reso-nance, and two resonance peaks (with T = 1) appear.The driven component, I dr , is reduced with an increaseof φ , and it disappears at φ = 2 π/ 4. However, quan-tum interference still plays a role; the bond currents arestrongly correlated (with large S sh , and S sh , and neg-ative S sh , ). For a large coupling, the driven part I dr can be large and can profoundly modify the total circularcurrent I c = I dr + I φ .We also performed calculations of the bond currentsand their correlations for rings with a various number ofsites; in particular, for the benzene ring in para-, metha-, and ortho-connection with the electrodes. The resultsare qualitatively similar to those presented above forthe 3QD ring: Quantum interference of the travellingwaves with the eigenstates of opposite chirality leads tothe driven circular currents, accompanied by large cur-rent fluctuations with a negative cross-correlation com-ponent. To observe this effect, the two conducting branches should be asymmetric; in particular, in the ben-zene ring, the driven circular current appears for themetha- and ortho-connections, but is absent in the para-connection, where both conducting branches are equiva-lent (see also ).An open problem is including interactions betweenelectrons into the calculations of the coherent transportand shot noise. Coulomb interactions can be taken intoaccount in the sequential regime , or by using the real-time diagrammatic technique ; however, in practice,one includes only first- and second-order diagrams withrespect to the tunnel coupling and the role of QI is dimin-ished. In principle, one can treat QI on an equal footingwith electron interactions in the framework of quantumfield theory , as was done for the Anderson single im-purity model, by means of full counting statistics (FCS),where the average current and all its moments were cal-culated . However, this is a formidable task, even forthe simple 3QD model. Acknowledgments The research was financed by National Science Centre,Poland project number 2016/21/B/ST3/02160. Appendix A: Coupling to atomic chain electrodes:Analytic results The results for the conductances and shot noise maybe simplified when we take all hopping integrals equalto t , the same position of the site levels ε = 0, and thesymmetric coupling t L = t R = t , with the electrodes as asemi-infinite atomic chain. In this case, the Green func-tions in the electrodes are g r = e ık /t and g a = e − ık /t and the electron spectrum is E k = 2 t cos k . From Equa-tions (7)–(10), one can calculate the dimensionless bondconductances as G L = 2 sin k [sin k + sin 3 k − sin(2 k + φ )] /A , (A1) G R = 2 sin k [sin k + sin 3 k − sin(2 k − φ )] /A , (A2) G L = 2 sin k [sin k − sin(2 k − φ )] /A , (A3) G R = 2 sin k [sin k − sin(2 k + φ )] /A , (A4)where the denominator A = 4 + cos 2 φ − φ (3 cos k − cos 3 k ) − cos 4 k. (A5)It is seen an asymmetry with respect to the direction ofthe magnetic flux (to φ ) for the conductances G Lij and G Rij from the left and the right electrode. The transmission, T , is expressed as T ≡ G L + G L = G R + G R = G + G = 2 sin k [1 − k (cos φ − cos k )] /A , (A6)where the driven part of the bond conductances are cal-culated using Equations (15) and (16) G = 4 sin k cos k (2 cos k − cos φ ) /A , (A7) G = 2 sin k (1 − φ cos k ) /A , (A8) G Lφ = G Rφ = 2 sin φ sin k cos 2 k/A , (A9)and, from Equations (19) and (20), the part induced bythe flux is G Lφ = G Rφ = 2 sin φ sin k cos 2 k/A . 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