Kondo blockade due to quantum interference in single-molecule junctions
Andrew K. Mitchell, Kim G. L. Pedersen, Per Hedegaard, Jens Paaske
KKondo blockade due to quantum interference in single-molecule junctions
Andrew K. Mitchell,
1, 2, ∗ Kim G. L. Pedersen, Per Hedeg˚ard, and Jens Paaske School of Physics, University College Dublin, Dublin 4, Ireland Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CE Utrecht, Netherlands Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen University, 52074 Aachen, Germany Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen, Denmark
Molecular electronics offers unique scientific and technological possibilities, result-ing from both the nanometer scale of the devices and their reproducible chemicalcomplexity. Two fundamental yet different effects, with no classical analogue, havebeen demonstrated experimentally in single-molecule junctions: quantum interferencedue to competing electron transport pathways, and the Kondo effect due to entangle-ment from strong electronic interactions. Here we unify these phenomena, showingthat transport through a spin-degenerate molecule can be either enhanced or blockedby Kondo correlations, depending on molecular structure, contacting geometry, andapplied gate voltages. An exact framework is developed, in terms of which the quan-tum interference properties of interacting molecular junctions can be systematicallystudied and understood. We prove that an exact Kondo-mediated conductance noderesults from destructive interference in exchange-cotunneling. Nonstandard temper-ature dependences and gate-tunable conductance peaks/nodes are demonstrated forprototypical molecular junctions, illustrating the intricate interplay of quantum effectsbeyond the single-orbital paradigm.
Perhaps the most important feature of nanoscale devicesbuilt from single molecules is the potential to exploit ex-otic quantum mechanical effects that have no classicalanalogue. A prominent example is quantum interfer-ence (QI), which has already been demonstrated in anumber of different molecular devices.
QI manifestsas strong variations in the conductance with changes inmolecular conformation, contacting or conjugation path-ways, or simply by tuning the back-gate voltage in athree-terminal setup. Another famous quantum phe-nomenon, relevant to single molecule junctions with aspin-degenerate ground state, is the Kondo effect, which gives rise to a dramatic conductance enhancementbelow a characteristic Kondo temperature, T K . Strongelectronic interactions in the molecule cause it to bindstrongly to a large Kondo cloud of conduction elec-trons when contacted to source and drain leads. A hall-mark of the Kondo effect is the proliferation of spin flipsas electrons tunnel coherently through the molecule, ul-timately screening its spin by formation of a many-bodysinglet. In this article, we uncover the intricate in-terplay of these two quantum effects, finding that thecombined effect of QI and Kondo physics has highlynon-trivial consequences for conductance through single-molecule junctions, and can even lead to an entirely newphenomenon – the
Kondo blockade .The Kondo effect is also routinely observed in semicon-ductor and nanotube quantum dot devices, whichare regarded as lead-coupled artificial atoms and assuch are often well described in terms of a single ac-tive interacting quantum orbital, tunnel-coupled to asingle channel of conduction electrons comprising bothsource and drain leads. This Anderson impurity model FIG. 1.
Interplay between quantum interference andelectronic interactions in single molecule junctions. ( a ) Enhanced Kondo resonant conductance; ( b ) Kondo block-ade, where conductance precisely vanishes. Tuning between a and b by applying a back-gate voltage allows efficient ma-nipulation of the tunneling current. (AIM) is by now rather well understood, and a quanti-tative description of the Kondo peak in single quantumdots can be achieved within linear response using non-perturbative methods such as the numerical renormaliza-tion group (NRG). In particular, the conductance isa universal function of T /T K , meaning that data for dif-ferent systems collapse to the same curve when rescaled a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y in terms of their respective Kondo temperatures. In-deed, even for multi-orbital molecular junctions, experi-mental conductance lineshapes have in some cases beensuccessfully fit to the theoretical form of the AIM, sug-gesting that an effective single-orbital, single-channel de-scription is valid at low temperatures.
However, somemolecular junctions apparently manifest nonuni-versal behavior and unconventional gate voltage depen-dences of conductance and T K , hinting at new physicsbeyond the standard single-orbital paradigm.The breakdown of the AIM is well-known in the con-text of coupled quantum dot devices, which can beviewed as simple artificial molecular junctions due totheir multi-orbital structure and the coupling to distinctsource and drain channels. Already the extension totwo or three orbital systems has lead to the discoveryof striking phenomena such as the ferromagnetic Kondoeffect corresponding to a sign change of the exchangecoupling, and multistage or frustrated screen-ing.In the following, we argue that a similar kind ofmulti-channel, multi-orbital Kondo physics accounts forthe behavior of real molecular junctions, and can beunderstood as a many-body QI effect characteristic ofthe orbital complexity and strong electronic correlationsin molecules. On entirely general grounds, we constructan effective model describing off-resonant conductancethrough single-molecule junctions with a spin-degenerateground state, taking into account both interactions lead-ing to Kondo physics, and orbital structure leading toQI. The physics of this generalized two-channel Kondo(2CK) model (including both potential scattering andexchange-cotunneling) is discussed in relation to thelocal density of states and observable conductance.We demonstrate how renormalized Kondo resonantconductance evolves into a novel Kondo blockade regimeof suppressed conductance due to QI (see Fig. 1). As anillustration, we consider two simple relevant molecularexamples, whose properties can be tuned between theselimits using gate voltages to provide functionality as anefficient QI-effect transistor. Results
Models and mappings
The Hamiltonian describing single-molecule junctionscan be decomposed as, H = H mol + H g + H leads + H hyb . (1)Here H mol describes the isolated molecule, and containsall information about its electronic structure and chem-istry. The first-principles characterization of moleculesis itself a formidable problem when electron-electron in-teractions are taken into account. In practice however,the relevant molecular degrees of freedom associated withelectronic transport are often effectively decoupled. Thisis the case for many conjugated organic molecules, wherethe extended π system can be treated separately interms of an extended Hubbard model. Reduced multi- orbital models have also been formulated using ab initio methods.
The leads are modeled as non-interacting conductionelectrons with H leads = P ασk (cid:15) k c † ασk c ασk where c † ασk cre-ates an electron in lead α =s, d (source, drain) with mo-mentum (or other orbital quantum number) k , and spin σ = ↑ , ↓ . The dispersion (cid:15) k corresponds approximatelyto a flat density of states ρ ( E ) = ρ θ ( D − | E | ), inside aband of width 2 D .The molecule is coupled to the leads via H hyb = P ασ ( t α d † i α σ c ασ + H.c.), where c ασ = t − α P k t αk c ασk isthe localized orbital in lead α at the junction, and d i α σ is a specific frontier orbital i α of the molecule, deter-mined by the contacting geometry. The molecule-leadhybridization is local, and specified by Γ α = πρ | t α | .The number of electrons on the molecule, N = h P iσ d † iσ d iσ i , is controlled by a gate voltage, incorpo-rated in the model by H g = − eV g P iσ d † iσ d iσ whichshifts the energy of all molecular orbitals. Deep in-side a Coulomb diamond, a substantial charging en-ergy, E C , must be overcome to either add or removeelectrons from the molecule. Provided Γ s , d (cid:28) E C theHamiltonian (1) can therefore be projected onto the sub-space with a fixed number of electrons on the molecule.In general this requires full diagonalization of the iso-lated H mol in the many-particle basis. Charging a neu-tral/spinless molecule by applying a back-gate voltage toadd or remove an electron typically yields a net spin- state. For odd-integer N , we therefore assume that themolecule hosts a spin- degree of freedom, S . Higher-spin molecules can arise, but are not considered here(the generalization is straightforward). To second-orderin the molecule-lead coupling H hyb , we then obtain an effective model of generalized 2CK form, H = H leads + H ex , where H ex = X αα σσ (cid:0) J αα S · τ σσ + W αα δ σσ (cid:1) c † ασ c α σ . (2)Here τ denotes the vector of Pauli matrices. The form of H ex is guaranteed by spin-rotation invariance and Her-miticity. Further details of the 2CK mapping are pro-vided in Methods . The cotunneling amplitudes form ma-trices in source-drain space, J = (cid:20) J ss J sd J sd J dd (cid:21) , W = (cid:20) W ss W sd W sd W dd (cid:21) , (3)and are referred to as respectively exchange , and potentialscattering terms. These 2CK parameters depend on thespecifics of molecular structure and contacting geometryin a complicated way, and must be derived from first-principles calculations for the isolated molecule. Thisgeneralized 2CK model hosts a rich range of physics;the non-Fermi liquid critical point is merely a singlepoint in its parameter space. Furthermore, any conduct-ing molecular junction must have a Fermi liquid groundstate, as demonstrated below.Any off-resonant molecule hosting a net spin- is de-scribed by the above generalized 2CK model at low decoupled "Kondo Cloud" Fermi liquideven/odd basis Increasingeven-oddfrustration source/drain basis even chainodd chain
FIG. 2.
Decoupling and frustration due to the Kondo effect in molecular junctions.
Left panels:
Imaginary part ofthe T-matrix, characterizing the effective energy-dependent exchange, in the physical basis of source (blue) and drain (red) leadsat T = 0 for the effective 2CK model. t αα ( ω,
0) is related to the renormalized density of states in lead α at the junction. Wetake a representative molecule-lead coupling J + = 0 . D with small but finite source/drain coupling asymmetry J − = 10 − D ,and consider the effect of reducing the exchange-cotunneling J sd from top to bottom. Physically, this could be achieved bygate-tuning in the vicinity of a QI node. The frustration of Kondo screening is always eventually relieved on the lowest energyscales, below T FL ∼ min( T eK , T ∗ ), because δ ≥ J − is always finite in any realistic setting. Center panels:
Corresponding T-matrix in the even/odd (blue/red) lead basis.
Right panels:
Real-space competition between even/odd (blue/red) conductionelectron channels, illustrated for the case where the leads are 1D quantum wires. The Kondo cloud (yellow) corresponds tothe spatial region of high molecule-lead entanglement. For small δ . − D one has a ‘Kondo frustration cloud’ embodyingincipient overscreening. temperatures T (cid:28) E C . The physics is robust due tothe large charging energy deep in a Coulomb diamond(charge fluctuations only dominate at the very edge of aCoulomb diamond ). In fact, the physics of the effectivemodel can be regarded as exact in the renormalizationgroup (RG) sense, despite the perturbative derivation ofEq. (2). Corrections to H ex obtained in higher-order per-turbation theory are formally RG irrelevant, and can besafely neglected because they get smaller and asymptoti-cally vanish on decreasing the temperature. They cannotaffect the underlying physics; only the emergent energyscales can be modified (this effect is also small, since thecorrections are suppressed by E C ).Experimentally relevant physical observables such asconductance can therefore be accurately extracted fromthe solution of the effective 2CK model (see Methods ).This requires sophisticated many-body techniques suchas NRG, which theoretically “attach” the source and drain leads non-perturbatively.
All microscopic de-tails of a real molecular junction are encoded in the2CK parameters J and W , which serve as input for theNRG calculations. In particular, destructive QI pro-duces nodes (zeros) in these parameters. Furthermore,QI nodes can be simply accessed by tuning the back-gatevoltage V g , as was shown recently in Ref. 45 for the caseof conjugated organic molecules.Importantly, two different types of QI can arise inmolecular junctions due to the electronic interactions.The QI can either be of standard potential scatteringtype (zeros in elements of W ) or of exchange type (zerosin elements of J ). Potential scattering QI is analogousto that observed in non-interacting systems describedby molecular orbitals. For interacting systems such asmolecular junctions (which typically have large chargingenergies ), potential scattering QI can similarly be un-derstood in terms of extended Feynman-Dyson (FD) or-bitals, which are the generalization of molecular orbitalsin the many-particle basis. Information on the real-spacecharacter of these orbitals, and how QI relates to molec-ular structure, can be extracted from the 2CK mapping.By contrast, exchange QI has no single-particle analogue,and cannot arise in non-interacting systems. Indeed, in-teractions are a basic requirement for the molecule tohost a spin- via Coulomb blockade. The spin wavefunc-tion is again characterized by the FD orbitals; dependingon the molecule in question, the spin can be delocalizedover the entire molecule.In the following we uncover the effect of this QIon Kondo physics, highlighting two distinct scenariosfor the resulting conductance – Kondo resonance andKondo blockade. We then go on to show that thisphysics is indeed realized in simple examples of molecu-lar junctions, and can be manipulated with gate voltages. Emergent decoupling
The generalized 2CK model can be simplified by diag-onalizing the exchange term in Eq. (2) via the unitarytransformation c α σ = U α α ψ ασ such that U † JU = (cid:20) J e J o (cid:21) , J e / o = J + ± δ (4)where J ± = ( J ss ± J dd ) and δ = J − + J . Notethat W is not generally diagonalized by this transforma-tion. The ‘odd’ channel decouples ( J o = 0) if and only if J = J ss J dd , as is the case when starting with a single-orbital Anderson model (see Supplementary Note 1). Bycontrast, real multi-orbital molecules couple to both evenand odd channels (electronic propagation through the en-tire molecule yields J (cid:28) J ss J dd when off resonance).However, electronic interactions play a key role here:the exchange couplings become renormalized as the tem-perature is reduced. A simple perturbative RG treatmenthints at flow toward a two-channel strong-coupling state,since both J e and J o initially grow. But the true low-temperature physics is much more complex, as seen inFig. 2 from the imaginary part of the scattering T-matrix t αα ( ω, T ) = − πρ ImT αα ( ω, T ) obtained by NRG for thegeneralized 2CK model and plotted as a function of ex-citation energy ω at T = 0 (see Methods ). The moleculespin is ultimately always Kondo-screened by conductionelectrons in the more strongly coupled even channel since J e > J o for any finite δ . Indeed, any real molecular junc-tion will inevitably have some degree of asymmetry in thesource/drain coupling J − , so that δ ≥ J − is always finitein practice. At particle-hole (ph) symmetry, the Friedelsum rule then guarantees that t ee (0 ,
0) = 1, charac-teristic of the Kondo effect. On the other hand, Kondocorrelations with the less strongly coupled odd channelare cut off on the scale of T K , and therefore t oo (0 ,
0) = 0(consistent with the optical theorem). These analyticpredictions are verified by NRG results in the center pan-els of Fig. 2.In all cases the odd channel decouples on the low-est energy/temperature scales, and the problem becomes effectively single-channel. This is an emergent phe-nomenon driven by interactions, not a property of thebare model. Despite the emergent decoupling of theodd channel, the Kondo effect always involves conduc-tion electrons in both source and drain leads for any fi-nite J sd . From the transformation defined in Eq. (4),the T-matrix in the physical basis can be expressed as t αα ( ω, T ) = | U α, e | t ee ( ω, T ) + | U α, o | t oo ( ω, T ), such that t αα (0 ,
0) = | U α, e | at ph symmetry – see left panels ofFig. 2.Although the physics at T = 0 is effectively single-channel, the full temperature dependence is highly non-trivial due to the competing involvement of the odd chan-nel (only for the oversimplified single-orbital AIM is theodd channel strictly decoupled for all T ). The universalphysics of the AIM is lost for δ = ( J e − J o ) = J + (orequivalently J = J ss J dd ): conductance lineshapes nolonger exhibit scaling collapse in terms of T /T K . Indeed,Kondo screening by the even channel occurs on the scale T eK ∼ D exp[ − /ρ J e ], and hence depends on δ . TheKondo temperature itself can therefore acquire an un-conventional gate voltage dependence, beyond the AIMparadigm.For even smaller δ , the Kondo effect occurs as atwo-step process, with even and odd channels compet-ing to screen the molecule spin. Since in this case J e ≈ J o , a frustration of Kondo screening sets in onthe scale T eK ≈ T oK ≡ T . The incipient frustra-tion for T ∼ T results in only partial screening(the molecule is overscreened, producing non-Fermi liq-uid signatures ). The frustration is relieved onthe much smaller scale T ∗ ∼ D ( ρ δ ) . The even chan-nel eventually ‘wins’ for T (cid:28) T ∗ and fully Kondo-screensthe molecule spin, while the odd channel decouples. Thisdramatic breakdown of the single-orbital AIM paradigmis shown in Fig. 2, with the degree of even/odd frustrationincreasing from top to bottom. In practice, such frustra-tion arises in a nearly symmetrical junction (small J − ),tuning in the vicinity of a QI node in J sd such that theperturbation strength δ is reduced. The first signaturesof frustration appear in conductance when T ∗ . T eK .Only when J − = J sd = 0, such that δ = 0, does thefrustration persist down to T = 0; we do not considerthis unrealistic scenario in the present work.In real-space, the entanglement between the moleculeand the leads is characterized by the Kondo cloud –a large spatial region of extent ξ K ∼ ~ v F /k B T eK pene-trating both source and drain leads ( v F is the Fermi ve-locity). In the right panels of Fig. 2 we illustrate thisfor the case where the leads are 1D quantum wires; thereal-space physics is then directly related to the T-matrixplotted in the left panels, as shown in Ref. 17. Note thatif the source/drain leads are 1D quantum wires, thenthe even/odd leads are also 1D quantum wires as de-picted. For small δ (lower panels) we have instead aKondo frustration cloud. The frustration is only relievedat longer length scales ξ ∗ ∼ ~ v F /k B T ∗ , beyond which theodd channel decouples. a b c FIG. 3.
Kondo resonant conductance near a potential scattering quantum interference node. ( a ) Zero-temperaturelinear conductance G (0) as a function of derived 2CK parameters J ss , J dd and J sd at W ss = W dd = W sd = 0. ( b ) Limitinguniversal conductance curves G ( T /T K ) and G ( T /T ∗ ) in the single-channel regime (large δ , red line) and the frustrated two-channel regime (small δ , black line and points), respectively. ( c ) Exact non-equilibrium conductance G ( T, V sd ) as a function ofbias voltage V sd at various temperatures in the frustrated regime of small δ , from Eq. (7). Conductance
The current through a molecular junction is mediatedby the cross terms coupling source and drain leads; theexchange and potential scattering terms J sd and W sd constitute two distinct conductance mechanisms. Athigh temperatures, the overall conductance can be un-derstood from a simple leading-order perturbative treat-ment using Fermi’s golden rule and is simply additive, G/G ∼ (2 πρ ) [ W + 3 J ], with G = 2 e h − . How-ever, at lower temperatures, electronic interactions leadto strong renormalization effects and rather surprisingKondo physics. Non-perturbative methods such as NRGmust therefore be used to calculate the full temperature-dependence of conductance, as described in Methods .Note that conductance through a molecular junction can-not be obtained simply from the T-matrix (except at T = 0).The QI aspect of the problem is entirely encoded inthe effective 2CK parameters, providing an enormousconceptual simplification. In particular, we identifytwo limiting QI scenarios relevant for conductance: W sd = 0 or J sd = 0. Exact analytic results, supportedby NRG, show that the Kondo effect survives QI in thecase of W sd = 0 to give enhanced conductance at lowtemperatures (Figs. 1a & 3), while a Kondo-mediatedQI node in the total conductance is found for J sd = 0,a Kondo blockade (Figs. 1b & 4). We demonstrateexplicitly that this remarkable interplay between QI andthe Kondo effect arises in two simple conjugated organicmolecules upon tuning gate voltages in Fig. 5. Kondo resonance
First we focus on conductance mediated exclusively bythe exchange cotunneling term J sd , tuning to a poten-tial scattering QI node W = . Even though the bare J sd is typically small, it gets renormalized by the Kondo effect and becomes large at low temperatures. TheKondo effect therefore involves both source and drainleads (Fig. 2), leading to Kondo-enhanced conductance.As shown in Supplementary Note 2, the fact that theodd channel decouples asymptotically implies the follow-ing exact result for the linear conductance, G ( T = 0) = 4 G p t ss (0 , t dd (0 , G J J + ( J ss − J dd ) . (5)Note that any finite interlead coupling J sd yields uni-tarity conductance G = G at T = 0 in the symmet-ric case J ss = J dd . The analytic result is confirmed byNRG in Fig. 3(a), and further holds for all T (cid:28) T K , T ∗ .Eq. (5) is an exact generalization of the standard single-orbital AIM result, G (0) /G = 4 J ss J dd / ( J ss + J dd ) ≡ s Γ d / (Γ s + Γ d ) , and reduces to it when J = J ss J dd .The full temperature-dependence of conductance canalso be studied with NRG. In all cases, we find Fermi-liquid behavior G ( T ) − G (0) ∼ ( T /T FL ) at the low-est temperatures T (cid:28) T FL , with T FL = min( T eK , T ∗ )(although T FL itself may have a nontrivial gate depen-dence). At large δ ∼ J + ( T ∗ (cid:29) T eK ), the behavior of thesingle-channel AIM is essentially recovered for theentire crossover [see red line, Fig. 3(b)]. However, theuniversality of the AIM is lost for smaller δ due to thecompeting involvement of the odd screening channel. Infact, for T ∗ (cid:28) T eK , appreciable conductance only sets inaround T ∼ T ∗ (rather than T eK ), and the entire conduc-tance crossover becomes a universal function of T /T ∗ –different in form from that of the AIM [see black line,Fig. 3(b)]. The formation of the Kondo state is reflectedin conductance by the following limiting behavior, G ( T ) T (cid:29) T FL ∼ ( ln − | T /T eK | : T ∗ (cid:29) T eK , ( T /T ∗ ) − : T ∗ (cid:28) T eK . (6)Furthermore, the abelian bosonization methods ofRefs. 53–55 can be applied to single-molecule junctionsin the limit T ∗ (cid:28) T eK to obtain an exact analytic expres-sion for the full conductance crossover (see Supplemen-tary Note 3), G ( T, V sd ) /G = T ∗ πT Re ψ (cid:18)
12 + T ∗ πT + i eV sd πk B T (cid:19) , (7)where ψ is the trigamma function. Remarkably, thisresult also holds away from thermal equilibrium , atfinite bias V sd (cid:28) T eK . Within linear response, Eq. (7)is confirmed explicitly by comparison to NRG datain Fig. 3(b), while Fig. 3(c) shows the nonequilibriumpredictions. The condition T ∗ (cid:28) T eK pertains to nearlysymmetric junctions, tuned near a QI node in J sd .Eq. (7) should be regarded as a limiting scenario:conductance lineshapes for real single-molecule junctionswill typically interpolate between the red and black linesof Fig. 3(b). Kondo blockade
At a QI node in the exchange-cotunneling J sd = 0, con-ductance through a single molecule junction is medi-ated solely by W sd . In this case, the molecule spin isfully Kondo screened by either the source or drain lead(whichever is more strongly coupled). Only in the spe-cial but unrealistic case J ss = J dd and J sd = 0 does thefrustration persist down to T = 0. For concreteness wenow assume ph symmetry W ss = W dd = 0, and J ss > J dd such that the even conduction electron channel is simplythe source lead. The drain lead therefore decouples onthe scale of T sK . As shown in Supplementary Note 4, onecan then prove that, G ( T = 0) = G (2 πρ W sd ) [1 − t ss (0 , , (8)where t ss ( ω, T ) is the T-matrix of the source lead. TheKondo effect with the source lead, characterized by t ss (0 ,
0) = 1, therefore exactly blocks current flowingfrom source to drain. This is an emergent effect of in-teractions – at high temperatures T (cid:29) T K when t ss ≈ G pert /G ≈ (2 πρ W sd ) .The zero-temperature conductance node arising for J sd = 0 can be understood physically as a depletion ofthe local source-lead density of states at the junction, dueto the Kondo effect. Conductance vanishes because lo-cally, no source-lead states are available from which elec-trons can tunnel into the drain lead. From a real-spaceperspective, one can think of the Kondo cloud in thesource lead as being impenetrable to electronic tunnel-ing at low energies. The effect of this Kondo blockade isdemonstrated in Fig. 4, where full NRG calculations for
FIG. 4.
Kondo blockaded conductance near a quan-tum interference node in the exchange cotunneling.
NRG results for the conductance G ( T ) as a function ofrescaled temperature T /T K at J sd = 0 for various W sd , show-ing in all cases an overall conductance node G (0) = 0. Plottedfor J ss = 0 . D , J dd = 0 . D and W ss = W dd = 0. Dottedlines show the high-temperature perturbative expectation. the conductance are shown for J sd = 0. The conductancecrossover as a function of temperature is entirely charac-teristic of the Kondo effect; G ( T ) /G pert is a universalfunction of T /T K . At low temperatures, a node in J sd thus implies an overall conductance node, even though W sd remains finite.The Kondo blockade will be most cleanly observed inreal single-molecule junctions that have strong molecule-lead hybridization and do not have a nearby Kondo res-onance. In addition to the large Kondo temperature, theperturbative cotunneling conductance observed at hightemperatures T (cid:29) T K is also larger in this case, therebyincreasing the contrast of the blockade on lowering thetemperature.We emphasize that the Kondo blockade is unrelatedto the Fano effect, which arises due to QI in thehybridization rather than intrinsic QI in the interactingmolecule itself (see Supplementary Note 4). Unlikethe Kondo blockade, the Fano effect is essentially asingle-channel phenomenon that does not necessitateinteractions, and different (asymmetric) lineshapesresult. Gate-tunable QI in Kondo-active molecules
In real molecular junctions, the two conductance mecha-nisms discussed separately above (due to finite exchange J sd and potential scattering W sd ), are typically both op-erative. Their mutual effect can be complicated due torenormalization from cross-terms proportional to W sd J sd .However, as the gate voltage V g is tuned, both Kondoresonant and Kondo blockade regimes are often accessi- e fcba d FIG. 5.
Gate-tunable Kondo resonance and Kondo blockade in simple conjugated organic molecular junctions.
Single-molecule junctions based on a benzyl ( a,b,c ) and an isoprene-like molecule ( d,e,f ), with all carbons sp -hybridized, weremapped to an effective 2CK model and linear conductance was calculated with NRG. ( a,d ) Conductance G ( T ) as a functionof rescaled temperature T /T K for various gate voltages eV g . ( a ) shows Kondo blockade G (0) = 0 at eV g = 0 and Kondoenhanced conductance for | V g | >
0. ( d ) shows Kondo blockade at eV g = 2 .
625 eV and perfect (unitarity) Kondo resonanceat eV g = 2 . b,e ) G (0) as a function of gate voltage eV g at T = 0; ( c,f ) Corresponding Kondo temperatures. Note thesensitive gate dependence of G (0) in ( e ), and the corresponding unconventional non-monotonic gate dependence of the Kondotemperature in ( f ). ble due to QI nodes in either J sd or W sd . In practice,we observe that overall conductance nodes can also beshifted away from the nodes in J sd by marginal poten-tial scattering W ss and W dd (not considered above). Wespeculate that the conductance nodes are topological andcannot be removed by potential scattering – only shiftedto a different gate voltage. Precisely at the node, the low-temperature physics is universal and therefore commonto all such off-resonant spin- molecules.To demonstrate the gate-tunable interplay between QIand the Kondo effect in single molecule junctions, wenow consider two simple conjugated organic molecules asexamples. Following Ref. 45, exact diagonalization of thePariser-Parr-Pople (PPP) model for the sp -hybridized π system of the molecule allows the effective 2CK modelparameters to be extracted as a function of applied gatevoltage (Supplementary Note 5). The 2CK model is thensolved using NRG, and the conductance is calculatednumerically-exactly as a function of temperature. Thesesteps are described in detail in Methods .Fig. 5 shows the conductance G ( T ) for junctionsspanned by respectively a benzyl, (a) and an isoprene-like molecule (d), as a function of rescaled temperature T /T K at different gate voltages. Both systems exhibitKondo resonant and Kondo blockade physics. In panel (a), a pronounced Kondo blockade appears near V g = 0,corresponding to the midpoint of the Coulomb diamond.Finite conductance at higher temperatures due to cotun-neling W sd is blocked at low temperatures by the Kondoeffect. On increasing the gate voltage, we find numeri-cally that G (0) ∼ eV , with conductance enhancementdue to renormalized J sd [Fig. 5(b)]. The overall conduc-tance in this case remains rather small for all eV g ana-lyzed. We also note that the Kondo temperature variesas ln T K /D ∼ eV , see Fig. 5(c). This gate evolution of T K could be considered as conventional from the single-orbital AIM perspective, but the conductance itself isblockaded rather than enhanced by Kondo correlations.However, richer physics can be accessed in junction(d). The crossovers of G ( T ) show perfect Kondo reso-nant conductance at finite eV g = 2 . G (0) = 2 e h − . But increasing the gatevoltage slightly to V g = 2 .
625 eV yields almost perfectKondo blockade, with G (0) ’ G (0) as a function of gate voltage at T = 0(and in practice for all T (cid:28) T K ), which exhibits non-trivial behavior due to the interplay between QI and theKondo effect. The rapid switching between Kondo reso-nant and Kondo blockade conductance with applied gatevoltage might make such systems candidates for QI-effecttransistors, or other technological applications.Finally, in panel (f), we show that the Kondo temper-ature also displays an unconventional gate-dependence,with T K increasing as one moves in towards eV g =0, analogous to the effect observed experimentally inRef. 28. The Kondo temperature remains finite for all eV g , but takes its minimum value at the Kondo reso-nance peak. In practice, the Kondo temperature can varywidely from system to system because it depends sensi-tively (exponentially) on the molecule-lead hybridization.However, Kondo temperatures up to around 30K arecommonly observed in real single-molecule junctions. We did not attempt an ab initio calculation of theabsolute Kondo temperatures, but note that the effec-tive bandwidth cutoff D in the effective 2CK modelis essentially set by the large charging energy of themolecule. For the PPP models used for the conjugatedhydrocarbons in Fig. 5, this in turn is set by the onsiteCoulomb repulsion, taken to be 11eV within the stan-dard Ohno parametrization. With this identification,we have T K ∼
10K for the specific example shown inpanel (a) at the Kondo blockade, and 0 .
1K in (d). Weemphasize that the Kondo blockade arises on similar tem-perature scales to that of the standard Kondo effect inmolecules, and therefore signatures should generally beobservable at experimental base temperatures on gatetuning to a QI node.The stark difference in transport properties of the twomolecular junctions shown in Fig. 5 is due to differencesin their QI characteristics – specifically the number andposition of QI nodes in the effective 2CK parameters (seeSupplementary Note 5). In turn, this is related to theunderlying molecular structure and contacting geometry,as explored for these alternant hydrocarbons in Ref. 45.Both molecules exhibit a Kondo blockade due to a nodein J sd , but this arises at eV g = 0 for the benzyl radi-cal in (a-c), whereas there are two nodes at finite ± eV g for the isoprene-like molecule in (d-f). In general, J sd has an odd(even) number of nodes as a function of gatein odd-membered alternant molecules if the source anddrain electrodes are connected to sites of the moleculeon different sublattices(the same sublattice) of the bi-partite π system. The strong Kondo resonance aris-ing at eV g = 2 . J ss ≈ J dd (see Eq. (5)).By contrast, there is no such symmetry for the benzylmolecule and J dd happens to dominate.Although we have exemplified the gate-tunable inter-play between QI and Kondo effect with these conjugatedhydrocarbon moieties, we emphasize that a Kondoblockade should be found in any off-resonant spin- molecule with intramolecular interactions and sufficientorbital complexity to produce a QI node in the exchangecotunneling. Discussion
Transport through spinful Coulomb-blockaded single-molecule junctions requires a description beyond thestandard single-orbital Anderson paradigm. The relevantmodel is instead a generalized two-channel Kondo model,to which real molecular junctions can be exactly mapped.Experimental data for individual molecular junctions canbe understood within this framework, avoiding the needfor a statistical interpretation.Quantum interference can be classified as being of ei-ther exchange or potential scattering type. Althoughthese distinct conductance mechanisms are simply addi-tive at high energies where standard perturbation theoryholds, the low-temperature behavior is much richer dueto electron-electron interactions which drive the Kondoeffect. We show that the Kondo effect survives a quan-tum interference node in the potential scattering to giveenhanced conductance, while a novel Kondo blockadearises in the case of an exchange cotunneling node, en-tirely blocking the current through the junction. Thisrich physics is tunable by applying a back-gate voltage,as demonstrated explicitly for two simple conjugated or-ganic molecules, opening up the possibility of efficientKondo-mediated quantum interference effect transistors.The theoretical framework we present can be usedto systematically study candidate molecules and helpoptimize the type and location of anchor groups forparticular applications. Quantum chemistry techniquescould be used to accomplish the Kondo model mappingfor larger molecules. The effect of vibrations anddissipation (relevant at higher energies ) could also betaken into account within generalized Anderson-Holsteinor Bose-Fermi Kondo models. Methods
Schrieffer-Wolff transformation
We derive the effective Kondo model describing off-resonant single-molecule junctions by projecting outhigh-energy molecular charge fluctuations from the fulllead-coupled system. This is equivalent to a two-channel generalization of the standard Schrieffer-Wolfftransformation. That is, projecting onto the subspaceof Hilbert space where the number, N , of electrons onthe molecule is fixed. The effective Hamiltonian in thissubspace has the form H eff ( E ) = P [ H leads + H mol + H g + H hyb Q ( E − QHQ ) − QH hyb ] P, (9)where P is a projection operator onto the N -electron sub-space of the molecule, while Q = I − P projects onto theorthogonal complement. These subspaces are connectedby H hyb , and the resolvent operator ( E − QHQ ) − de-termines the propagation of excited states at energy E .Note that P H leads P = H leads since P acts only on themolecular degrees of freedom, and P ( H mol + H g ) P ismerely a constant and dropped in the following. For theisolated molecule, H mol (cid:12)(cid:12) Ψ Nn (cid:11) = E Nn (cid:12)(cid:12) Ψ Nn (cid:11) , where (cid:12)(cid:12) Ψ Nn (cid:11) denotes the n ’th N -electron many-body eigenstate withenergy E Nn , and E N is the ground state energy. (cid:12)(cid:12) Ψ Nn (cid:11) spans the entire molecule and generally has weight onall atomic/molecular basis orbitals in H mol . So far thetreatment is exact.To second order in H hyb , Eq. (9) reduces to the effec-tive Hamiltonian H eff = H leads + P H hyb Q ( E N − Q ( H mol + H g ) Q ) − QH hyb P. (10)Virtual processes involving a given excited state (cid:12)(cid:12) Ψ N ± n (cid:11) contribute to Eq. (10) with weight controlledby the energy denominator (cid:10) Ψ N ± n (cid:12)(cid:12) E N − Q ( H mol + H g ) Q (cid:12)(cid:12) Ψ N ± n (cid:11) = E N − E N ± n ± eV g , which must be neg-ative to ensure stability of the N -electron ground state(including the electrostatic shift from the backgate de-scribed by H g ). The perturbative expansion in H hyb iscontrolled by a large energy denominator, and thereforeuse of Eq. (10) is justified deep inside the N -electronCoulomb diamond. Inserting the tunneling Hamiltonian, H hyb = P ασk ( t αk d † i α σ c ασk + H.c.), one arrives at theeffective Hamiltonian H eff = H leads + H ex , with H ex = X α k σ αkσ t α k t ∗ αk c † αkσ c α k σ X m ,m (cid:12)(cid:12) Ψ Nm (cid:11) A α ασ σ,m m (cid:10) Ψ Nm (cid:12)(cid:12) = X α σ ασ t α t ∗ α c † ασ c α σ X m ,m (cid:12)(cid:12) Ψ Nm (cid:11) A α ασ σ,m m (cid:10) Ψ Nm (cid:12)(cid:12) , (11)where the last line follows from the definition of lo-cal lead-electron operators c ασ = t − α P k t αk c αkσ , with t α = P k | t αk | . Here m and m label (degenerate) molec-ular ground states with energy E N . For odd N , themolecule often carries a net spin- , and so m and m aresimply the projections S z = ± . The spin density neednot be spatially localized. We now focus on this stan-dard case, although the generalization to arbitrary spinis straightforward when the molecular ground state for agiven N is more than 2-fold degenerate.The cotunneling amplitudes can be decomposed as, A α ασ σ,m m = h α ασ σ,m m + p α ασ σ,m m (12)where the contributions from hole and particle propaga-tion are given respectively by, h α ασ σ,m m ( V g ) = X n (cid:10) Ψ Nm (cid:12)(cid:12) d † i α σ (cid:12)(cid:12) Ψ N − n (cid:11) (cid:10) Ψ N − n (cid:12)(cid:12) d i α σ (cid:12)(cid:12) Ψ Nm (cid:11) eV g − E N + E N − n − i + , (13) p α ασ σ,m m ( V g ) = X n (cid:10) Ψ Nm (cid:12)(cid:12) d i α σ (cid:12)(cid:12) Ψ N +1 n (cid:11) (cid:10) Ψ N +1 n (cid:12)(cid:12) d † i α σ (cid:12)(cid:12) Ψ Nm (cid:11) eV g + E N − E N +1 n + i + . (14)The matrix elements in the numerators (referred to asFeynman-Dyson orbitals) constitute a correlated gener- alization of molecular orbitals, and are computed in themany-particle molecular eigenstate basis.Since the total Hamiltonian must preserve its origi-nal spin-rotational invariance, the cotunneling amplitudemust take the form A α ασ σ,m m = J αα τ σσ · τ mm + W αα δ σσ δ mm . (15)This leads to the desired effective 2CK model [Eq. (2) ofsubsection ‘Models and mappings’]: H = H leads + X α σ ασ (cid:0) J α α S · τ σ σ + W α α δ σ σ (cid:1) c † α σ c ασ . (16)The 2CK model parameters themselves are obtained fromtraces with Pauli matrices, J αα = t α t ∗ α X σσ ,mm τ iσ σ A α ασ σ,m m τ im m for i = x, y, z, (17) W αα = t α t ∗ α X σσ ,mm A α ασσ,mm (18)which, by spin-rotation invariance, further simplify to J αα = 2 t α t ∗ α A α α ↑↓ , ↓↑ = 2 t α t ∗ α X m A α α ↑↑ ,mm τ zmm (19) W αα = 4 t α t ∗ α A α α ↑↑ , ↑↑ − A α α ↑↓ , ↓↑ = 2 t α t ∗ α X m A α α ↑↑ ,mm (20)such that in practice only two matrix elements areneeded to obtain the exchange couplings J αα and thepotential scattering amplitudes W αα .Equations (13) and (14) therefore encode all theproperties of the single-molecule junction inside an N -electron Coulomb diamond. The exchange and potentialscattering terms in the 2CK model are determined by theamplitudes A which, from Eq. (12), have contributionsfrom both particle ( p ) and hole ( h ) processes (that is,processes involving virtual states with N + 1 or N − J αα and W αα – quantuminterference nodes arise if and only if molecular states areconnected by particle and hole processes with equal butopposite amplitudes. As discussed in Reference 45, theappearance of such nodes can be understood in terms ofthe properties of the underlying Feynman-Dyson orbitals.We emphasize that the effective 2CK model is totallygeneral, applying for any molecule with a two-fold spin-degenerate ground state, at temperatures less than themolecule charging energy so that charge fluctuations onthe molecule are frozen (typically the charging energy is0large when deep inside a Coulomb diamond, and there-fore the molecule is off-resonant). The 2CK model pa-rameters can be obtained purely from a knowledge ofthe isolated molecule, and can therefore be calculated inpractice using a number of established techniques (exactdiagonalization, configuration interaction etc). The low-temperature properties of the resulting 2CK model arehowever deeply nontrivial, requiring sophisticated many-body methods to “attach the leads” and account fornonperturbative renormalization effects. In the presentwork, we do this second step using the numerical renor-malization group. An advantage of the effective theory is that it can alsobe analyzed exactly on an abstract level (independentlyof any specific realization). This allows us to identifyall the possible scenarios that could in principle arise inmolecular junctions. The basic physics is arguably obfus-cated rather than clarified by the complexity of a full mi-croscopic description: a brute-force method (even if thatwere possible) may not yield new conceptual understand-ing or provide general predictions beyond a case-by-casebasis.2CK parameters for the molecules presented in Fig. 5were obtained following Ref. 45; see SupplementaryFigures 1 and 2.
Exact diagonalization of H mol In this work, we model the isolated molecule by asemi-empirical Pariser-Parr-Pople Hamiltonian forthe molecular π -system:ˆ H mol = X h i,j i X σ = ↑ / ↓ (cid:16) t ij d † iσ d jσ + H.c. (cid:17) + X i U ( n i ↑ − )( n i ↓ − )+ 12 X i = j V ij ( n i − n j − . (21)The operator d † iσ creates an electron with spin σ on the p z -orbital | i i , n iσ = d † iσ d iσ and n i = n i ↑ + n i ↓ . TheCoulomb interaction is given by the Ohno parametriza-tion V ij = U/ ( p | ~r ij | U / . | ~r ij | isthe real-space distance between two p z -orbitals | i i and | j i measured in ˚Angstr¨om. For sp hybridised carbon,the nearest neighbor overlap, t ij , is t ≈ − . U ≈ .
26 eV. For suitably small molecules, Eq. (21) can be solvedusing exact diagonalization (exploiting overall conservedcharge and spin) to provide the many-particle eigenstates (cid:12)(cid:12) Ψ Nn (cid:11) and eigenenergies E Nn . For larger molecules, ap-proximate methods can also be used, provided interac-tions are accounted for on some level. Any molecule canbe addressed within our framework, provided the eigen-states and eigenenergies of the isolated molecule can bedetermined.Importantly, the perturbed two-channel Kondo modelderived in the previous section remains the generic Hamiltonian of interest to describe such off-resonantjunctions. The diagonalization of Eq. (21) is requiredonly to obtain the parameters J and W , which are thenused in subsequent numerical renormalization group cal-culations to treat the coupling to source and drain leads.Note that the calculation of physical quantities such asconductance at lower temperatures necessitates an ex-plicit and nonperturbative treatment of the leads, andcannot be achieved with single-particle methods or exactdiagonalization alone.However, once the generic physics of the underlying2CK model is understood (a key goal of this paper),the transport properties and quantum interferenceeffects of specific molecular junctions can already berationalized and predicted from their 2CK parameters.The suitability of candidate molecules and the positionsof anchor groups can therefore be efficiently assessed,opening up the possibility of rational device design. Calculation of conductance
The key experimental quantity of interest for single-molecule junction devices is the differential conductance G ( T, V sd ) = d h I sd i /dV sd . In this section we recap thegeneric framework for exact calculations of the linear con-ductance G ( T ) ≡ G ( T, V sd →
0) through a molecule,taking fully into account renormalization effects due toelectronic interactions. We then describe how the nu-merical renormalization group (NRG) can be used toaccurately obtain G ( T ) for a given system described bythe effective model, Eq. (16).To simulate the experimental protocol, we add a time-dependent bias term to the Hamiltonian, H = H + H ( t ), with H ( t ) = eV sd ωt ) (cid:16) ˆ N s − ˆ N d (cid:17) , (22)where ˆ N α = P k,σ c † αkσ c αkσ is the total number operatorfor lead α . We focus on the serial ac and dc conduc-tance at t = 0, after the system has reached an oscillat-ing steady state. Within linear response, taking the limit V sd →
0, the exact serial ac conductance follows from theKubo formula, G ac ( ω, T ) = e h × (cid:20) − π ~ Im K ( ω, T ) ~ ω (cid:21) , (23)where K ( ω, T ) is the Fourier transform of the retardedcurrent-current correlator, K ( t, T ) = i θ ( t ) D [ ˆΩ( t ) , ˆΩ(0)] E T , (24)where ˆΩ = ( ˙ N s − ˙ N d ) and ˙ N α = ddt ˆ N α . The dc conduc-tance is then simply, G ( T ) = lim ω → G ac ( ω, T ) . (25)In practice, NRG is used to obtain K ( ω, T ) numer-ically. The full density matrix NRG method, estab-lished on the complete Anders-Schiller basis, provides1essentially exact access to such dynamical correlationfunctions at any temperature T and energy scale ω . Weuse ˙ N α = i [ ˆ H, ˆ N α ] to find an expression for the currentoperator amenable for treatment with NRG: i ˆΩ = (cid:16) J sd S · s sd + W sd X σ c † s σ c d σ (cid:17) − H.c. , (26)where s αβ = P σσ c † ασ σ σσ c βσ .Since the ground state of any conducting molecularjunction must be a Fermi liquid, the system can beviewed as a renormalized non-interacting system at T =0. The zero-bias dc conductance at T = 0 can thereforealso be obtained from a Landauer-B¨uttiker treatment, G ( T = 0) = 2 e h × s ˜Γ d |G sd ( ω = 0 , T = 0) | , (27)in terms of the full retarded electronic Green’s function G αβ ( ω, T ) FT ↔ − iθ ( t ) h{ c ασ ( t ) , c † βσ (0) }i T , which must becalculated non-perturbatively in the presence of the inter-acting molecule. Here ˜Γ α = 1 / ( πρ ). Note that Eq. (27)applies to Fermi liquid systems only at T = 0 and in thedc limit. The full temperature dependence of G ( T ) mustbe obtained from the Kubo formula.In practice, G sd ( ω, T ) is obtained from the T-matrixequation, which describes electronic scattering in theleads due to the molecule, G αβ ( ω, T ) = G (0) ( ω ) δ αβ + h G (0) ( ω ) i × [ W αβ + T αβ ( ω, T )] , (28)where G (0) ( ω ) is the free retarded lead electron Green’sfunction when the molecule is disconnected, such thatIm G (0) ( ω ) = − πρ ( ω ), and ρ (0) = ρ .Within NRG, the T-matrix can be calculated directly as the retarded correlator T αβ ( ω, T ) FT ↔ − iθ ( t ) h{ a α ( t ) , a † β (0) }i T . For the presentproblem, the composite operators, a α = X γ h W αγ c γ ↑ + J αγ ( c γ ↑ S z + c γ ↓ S − ) i , (29)follow from Eq. (16) using equations of motion meth-ods. In subsection ‘Emergent decoupling’ we also presentNRG results for the spectrum of the T-matrix, definedas t αβ ( ω, T ) = − πρ Im T αβ ( ω, T ) . (30)Note that the simple Landauer form of the Meir-Wingreen formula, which relates the conductancethrough an interacting region to a generalized transmis-sion function, applies only in the special case of propor-tionate couplings. In single-molecule junctions, the var-ious molecular degrees of freedom couple differently tosource and drain leads (which are spatially separated),and therefore this standard form of the Meir-Wingreenformula cannot be used, and one has to resort to usingfull Keldysh Green’s functions (or the methods described above for linear response). The exception is when themolecule is a single orbital – this artificial limit is con-sidered in Supplementary Note 1.For the NRG calculations, even/odd conduction elec-tron baths were discretized logarithmically using Λ = 2,and N s = 15000 states were retained at each step of theiterative procedure. Total charge and spin projectionquantum numbers were exploited to block-diagonalizethe NRG Hamiltonians, and the results of N z = 2calculations were averaged. 2CK model parametersfor the molecules presented in Fig. 5 are discussed inSupplementary Note 5. Acknowledgements
We thank Eran Sela and Martin Galpin for fruitful dis-cussions. A.K.M. acknowledges funding from the D-ITP consortium, a program of the Netherlands Organ-isation for Scientific Research (NWO) that is funded bythe Dutch Ministry of Education, Culture and Science(OCW). The Center for Quantum Devices is funded bythe Danish National Research Foundation. We are grate-ful for use of HPC resources at the University of Cologne.
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O ´Ujs´aghy, J Kroha, L Szunyogh, and A Zawadowski,“Theory of the fano resonance in the STM tunneling den-sity of states due to a single Kondo impurity,” PhysicalReview Letters , 2557 (2000). K Ohno, “Some remarks on the Pariser-Parr-Poplemethod,” Theoretica Chimica Acta , 219–227 (1964). A Weichselbaum and J von Delft, “Sum-rule conserv-ing spectral functions from the numerical renormalizationgroup,” Physical Review Letters , 076402 (2007). R Pariser and RG Parr, “A semi-empirical theory of theelectronic spectra and electronic structure of complex un-saturated molecules. ii,” The Journal of Chemical Physics , 767–776 (1953). JA Pople, “Electron interaction in unsaturated hydrocar-bons,” Transactions of the Faraday Society , 1375–1385(1953). ZG Soos and S Ramasesha, “Valence-bond theory of lin-ear Hubbard and Pariser-Parr-Pople models,” Physical Re-view B , 5410 (1984). W Izumida, O Sakai, and Y Shimizu, “Many body ef-fects on electron tunneling through quantum dots in anAharonov-Bohm circuit,” Journal of the Physical Societyof Japan , 717–726 (1997). FB Anders and A Schiller, “Real-time dynamics inquantum-impurity systems: A time-dependent numericalrenormalization-group approach,” Physical Review Letters , 196801 (2005). A Oguri and AC Hewson, “NRG approach to the trans-port through a finite Hubbard chain connected to reser-voirs,” Journal of the Physical Society of Japan , 988–996 (2005). Y Meir and NS Wingreen, “Landauer formula for the cur-rent through an interacting electron region,” Physical Re-view Letters , 2512 (1992). upplementary Information Supplementary Note 1: Single-orbital Anderson model limit
Single molecule junctions that exhibit a zero-bias conductance peak attributed to theKondo effect are typically modeled using the single-orbital Anderson model, H mol → H AIM = X σ (cid:15) d † σ d σ + U d †↑ d ↑ d †↓ d ↓ , (1)and where H hyb = P ασ ( t α d † σ c ασ +H.c.). This highly simplified model, which entirely neglectsthe orbital/spatial structure of the molecule, yields a very particular form of the effectiveKondo model, Eq. (16), upon Schrieffer-Wolff transformation. For any U and (cid:15) , one obtains J = J ss J dd and W = W ss W dd , meaning that in the even/odd orbital basis (cf. Eq. (4)of subsection ‘Emergent decoupling’), a standard single-channel Kondo model results. Theodd channel is strictly decoupled ( J o = 0) on the level of the bare Hamiltonian, and nomulti-channel effects can manifest. This has significant consequences for the physics of asingle-orbital model – for example the Kondo temperatures and conductance lineshapes musttake a particular form.The Kondo temperature associated with the single-orbital Anderson model follows fromperturbative scaling as, T K ∼ (Γ U ) / exp[ π(cid:15) ( (cid:15) + U ) / Γ U ] (2)with Γ = Γ s + Γ d and Γ α = πρ t α . In the presence of the gate described by H g , onehas (cid:15) → (cid:15) − eV g , yielding the standard quadratic dependence on applied gate, ln T K /D ∼ ( (cid:15) − eV g )( (cid:15) − eV g + U ). However, note that this behaviour is not necessarily expected inthe case of real single-molecule junctions, since nontrivial gate dependences arise in thegeneric multi-orbital case (as shown explicitly in subsection ‘Gate-tunable QI in Kondo-active molecules’ for the isoprene junction).Zero-bias conductance through a single Anderson orbital can be obtained from the Meir-Wingreen formula, G ( T ) = 2 e h G Z dω (cid:18) − ∂f∂ω (cid:19) t ee ( ω, T ) , (3)where f denotes the Fermi function and t ee ( ω, T ) = − πρ Im T ee ( ω, T ) is the spectrum ofthe even channel T-matrix. For a single Anderson impurity, T ee ( ω, T ) = ( t + t ) G imp ( ω, T ),1 a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y ith G imp ( ω, T ) = hh d σ ; d † σ ii the full retarded impurity Green’s function. The simple formof Supplementary Equation 3 applies in the case of proportionate couplings, automaticallyfulfilled in the single-orbital case. The source/drain asymmetry factor is given by G = 4Γ s Γ d (Γ s + Γ d ) ≡ t t ( t + t ) ≡ J ss J dd ( J ss + J dd ) , (4)which is maximal, G = 1, for equal couplings t s = t d .The universal form of t ee ( ω, T ) in the Kondo regime gives the universal temperature-dependence of conductance often successfully fit to experimental data. Importantly, atparticle-hole symmetry U = − (cid:15) , the Friedel sum rule pins the T-matrix to t ee (0 ,
0) = 1– meaning that G (0) = (2 e h − ) G – characteristic of the Kondo resonance. Note that inthis case, the potential scattering terms vanish exactly. This is in fact a consequence ofdestructive quantum interference between particle and hole processes; conductance is dueto the Kondo exchange-cotunneling term alone.It is instructive to derive the Meir-Wingreen formula Supplementary Equation 3 from thegeneral Kubo formula Eq. (23), since the subsequent generalization to the two-channel caseprovides novel results of relevance to single-molecule junctions. The key step here relies onthe fact that the odd conduction electron channel decouples in the single-orbital Andersonmodel. First, note that the correlator K ( ω, T ) = hh ˙ N s − ˙ N d ; ˙ N s − ˙ N d ii can be written as K ( ω, T ) = t + t ) hh t ˙ N s − t ˙ N d ; t ˙ N s − t ˙ N d ii by current conservation. Using ˙ N α = i [ ˆ H, ˆ N α ]with Supplementary Equation 1 we then obtain, K ( ω, T ) = − (cid:16) t s t d t + t (cid:17) X σ,σ hh f † o σ d σ − d † σ f o σ ; f † o σ d σ − d † σ f o σ ii , (5)where f o σ = ( t + t ) − / [ t d c s σ − t s c d σ ] is the local odd-channel conduction electron operator.Since the odd channel is strictly decoupled from the rest of the system (impurity and evenchannel), K ( t, T ) factorizes as, K ( t, T ) = − iθ ( t ) (cid:18) t s t d t + t (cid:19) X σ,σ h h d σ ( t ) d † σ (0) i sys × h f † o σ ( t ) f o σ (0) i odd + h d † σ ( t ) d σ (0) i sys × h f o σ ( t ) f † o σ (0) i odd −h d σ (0) d † σ ( t ) i sys × h f † o σ (0) f o σ ( t ) i odd + h d † σ (0) d σ ( t ) i sys × h f o σ (0) f † o σ ( t ) i odd i (6)2ome lengthy algebra then yields the following result,Im K ( ω, T ) = (cid:18) t t t + t (cid:19) Z ∞−∞ dω Im G imp ( ω , T ) × [ ρ ( ω − ω ) f ( ω − ω ) − ρ ( ω + ω ) f ( ω + ω )] . (7)In Eq. (23), this gives the ac conductance in terms of the retarded impurity Green function G imp ( ω, T ), consistent with the findings of Refs. 7 and 12. Taking the limit ω → K ( t, T ) cannot be factorized as in Sup-plementary Equation 6. In general, both even and odd channels remain coupled to themolecule, and the total conductance then also involves a contribution from the odd channel,requiring to go beyond Supplementary Equation 3. Supplementary Note 2: Generalized Kondo resonance and derivation of Eq. (5)
We now consider the generalized 2CK model describing off-resonant single-molecule junc-tions, Eq. (16), focusing on the particle-hole (ph) symmetric case with W = . Quantuminterference effects give rise to such a potential scattering node, and can in principle be real-ized in any given system by tuning gate voltages. Generically, the junction is still conductingdue to the exchange-cotunneling term, since J sd = 0. The full conductance lineshape G ( T ),must be obtained numerically from NRG. However, the conductance in the low-temperaturelimit G ( T (cid:28) T K ) ’ G (0) can be obtained analytically, as shown below.Combining Eq. (27)and Eq. (28), and noting that ˜Γ α = 1 / ( πρ ) and G (0) (0) = − iπρ , wehave, G (0) = 2 e h × | πρ T sd (0 , | . (8)In the even/odd orbital basis (Eq. (4) of subsection ‘Emergent decoupling’), T sd ( ω, T ) = U se U ∗ de T ee ( ω, T ) + U so U ∗ do T oo ( ω, T ), in terms of the T-matrices of the even/odd channels. Im-portantly, as shown in subsection ‘Emergent decoupling’, the odd channel decouples asymp-totically. The molecule undergoes a Kondo effect with the even channel since J e > J o ; this3uts off the RG flow with the odd channel and effectively disconnects it. This is an emergentlow-energy phenomenon, not a property of the bare system. In all cases, therefore, we have T oo (0 ,
0) = 0, meaning that, G (0) = 2 e h × | πρ U se U ∗ de T ee (0 , | . (9)Furthermore, at ph symmetry, iπρ T ee (0 ,
0) = t ee (0 ,
0) = 1 due to the Kondo effect, and so G (0) = 2 e h × | U se U ∗ de | = 2 e h × J J + ( J ss − J dd ) . (10)Supplementary Equation 10 generalizes the Meir-Wingreen result for the single-orbital An-derson model, Supplementary Equation 4, to the generic multi-orbital (molecular junction)case, and reduces to it when J = J ss J dd . Supplementary Note 3: Non-equilibrium conductance and derivation of Eq. (7)
In the special case J e = J o , the effective 2CK model Eq. (16), is precisely at a frustratedquantum critical point. In practice, J e = J o requires both J sd = 0 and J ss = J dd ,and is therefore not expected to be relevant to real molecular junction systems. How-ever, δ = ( J e − J o ) could still be small, especially near a quantum interference node in J sd . When δ = J + J − < T K , 2CK quantum critical fluctuations control the junctionconductance. Interestingly, exact analytic results can be obtained in this special regime,including the non-linear conductance away from thermal equilibrium. Similar calculationshave been performed for the two-impurity Kondo model in Supplementary Reference 15 andfor charge-Kondo quantum dot devices in Supplementary Reference 16.The source of the exact results is the Emery-Kivelson solution of the regular 2CK modelat the Toulouse point. By using bosonization methods, Supplementary Reference 17 showedthat a spin-anisotropic generalization of the 2CK model drastically simplifies to a Majoranaresonant level at a special point in parameter space – the Toulouse point (analogous to asimilar spin-anisotropic point for the single-channel Kondo problem ). In SupplementaryReference 19 Schiller and Hershfield then obtained the nonequilibrium serial conductance atthis Toulouse point, exploiting the fact that the effective Majorana resonant level model isnon-interacting and exactly solvable. 4f course, physical systems are not near the Toulouse point, and therefore conductancelineshapes are in general different from those obtained using the Emery-Kivelson solution.Importantly, however, the 2CK critical point has an emergent spin isotropy. This meansthat properties of the critical point are independent of any spin-anisotropy in the bare model.Exploiting the RG principle that the flow from high to low energies has no memory, onecan argue that subsequent low-energy crossovers (due to perturbations to the critical point δ = 0), are also independent of spin-anisotropy in the bare model. In particular, the samelow-energy crossover must occur in the physical spin-isotropic model as at the Toulousepoint. Therefore the Toulouse limit solution can be used for the low-temperature crossover– provided there is good scale separation T ∗ (cid:28) T K (where T ∗ ∼ δ is the Fermi liquidcrossover scale generated by relevant perturbations J sd and/or J − ).Following Supplementary Reference 19, we obtain Eq. (7) of subsection ‘Kondo reso-nance’. The precise quantitative agreement between the predicted G ( T ) at linear responseand NRG results (see Figure 3b) validate the above lines of argumentation. Supplementary Note 4: Kondo blockade and derivation of Eq. (8)
We now consider the case of a quantum interference node J sd = 0. Conductance throughthe molecular junction is mediated only by W sd (for simplicity, we again take the ph-symmetric case W ss = W dd = 0). In general, J ss = J dd , meaning that a Kondo effectwill develop with the more strongly coupled lead. For concreteness, we take now J ss > J dd .The drain lead therefore decouples for T (cid:28) T K . As shown below, this produces an exactnode in the total conductance G ( T = 0) = 0. The perturbative result for the conductance G ( T (cid:29) T K ) ∼ W , valid at high temperatures, is quenched at low temperatures due tointeractions and the Kondo effect.First, note that the local retarded electron Green’s function of the more strongly coupledlead vanishes at T = ω = 0 due to the Kondo effect, G ss (0 ,
0) = 0. This follows from theT-matrix equation, Eq. (28), using G (0) (0) = − iπρ and iπρ T ss (0 ,
0) = t ss (0 ,
0) = 1, G ss (0 ,
0) = − iπρ [1 − iπρ T ss (0 , . (11)This depletion of the lead electron density at the molecule is due to the Kondo effect, andarises only for T (cid:28) T K . The conductance through the molecule is therefore blocked because5here are no available source lead states to facilitate transport. Formally, this is proved fromthe optical theorem, assuming that the low-energy physics can be understood in terms ofa renormalized non-interacting system: the conductance (at T = 0) is then related to thetotal reflectance G (0) = (2 e h − )[1 − | r | ], where r = 1 − s G ss (0 , J sd = 0but W sd = 0, we have i ˆΩ = W sd P σ ( c † s σ c d σ − H.c.). Since at T = 0 the drain channel isdecoupled, K ( t, T ) factorizes as in Supplementary Equation 6. Following the same steps asin Sec. S4, we finally obtain G (0) in terms of the retarded electron Green’s function G ss (0 , G (0) = 2 e h × iπρ W G ss (0 ,
0) = 2 e h × (2 πρ W sd ) [1 − t ss (0 , . (12)The total conductance at T = 0 therefore exactly vanishes since t ss (0 ,
0) = 1. Note also thatthe high-temperature perturbative result is precisely recovered if one sets t ss = 0.One might wonder whether the suppression of conductance due to quantum interferencein Kondo-active molecular junctions is related to the Fano effect observed e.g. in STMexperiments of single magnetic impurities on metallic surfaces. In fact the mechanisms arevery different – the Kondo blockade arising here is a novel phenomenon. The Fano effectarises simply because electronic tunneling in an STM experiment can take two pathways –either into a magnetic impurity, or directly into the host metal. The quantum interference iscompletely on the level of the effective hybridization and is a non-interacting effect. Thereis no intrinsic quantum interference on the interacting impurity. In the Fano effect, theeffective Kondo model must always have J sd >
0, and the Kondo effect therefore involvesconduction electrons in both the host metal and the STM tip. Furthermore, the problemcan always be cast in terms of a single effective channel with asymmetric density of states,yielding asymmetric lineshapes. Note that one also obtains Fano-like lineshapes for trivialresonant level defects with no interactions.By contrast, there is no direct source-drain conductance pathway in single-molecule junc-tion devices – cotunneling proceeds only throughs the molecule, and J sd = 0 arises due tointrinsic quantum interference (i.e., a characteristic property of the isolated molecule andits contacting geometry). The Kondo blockade is an exact conductance node of the strongly6nteracting system at T = 0, which arises because the molecule is asymptotically bound toonly one of the two leads – its Kondo cloud is impenetrable at low energies to cotunnelingembodied by W sd . At higher temperatures, the conductance G ( T ) ∼ W is ‘blind’ to thequantum interference node in J sd ; the Kondo blockade arises entirely from the interplaybetween intrinsic molecular quantum interference and Kondo physics. The Kondo blockadelineshape is particle-hole symmetric and a universal function of T /T K .In the context of double quantum dots realizing a side-coupled two-impurity Kondomodel, conductance can also be suppressed. However, this arises because two spin- quantum impurities are successively screened by a single conduction electron channel in atwo-stage process. This mechanism is not related to the Kondo blockade, which involvesa net spin- molecule and single-stage Kondo screening by two conduction electron channels. Supplementary Note 5: Cotunneling amplitudes
The benzyl radical in Figure 5(a) of subsection ‘Gate-tunable QI in Kondo-activemolecules’ comprises 7 carbons in a planar arrangement, all sp hybridized [formally ( λ -methyl)-2 λ , λ -benzene], while the isoprene-like molecule in Figure 5(d) involves 5 sp hy-bridized carbons [formally 2-( λ -methyl)-4 λ -buta-1,3-diene]. The extended π system of eachis described by the Pariser-Parr-Pople model, using the standard Ohno parametrization. As input to the NRG calculations, we computed the effective 2CK model parameters fromEq. (19) and Eq. (20) as a function of gate voltage eV g . These are shown in SupplementaryFigures 1 and 2. 7 -4 -3 -2 -1 -3 -2 -1 0 1 23-0.4-0.200.20.4 a b -3 -2 -1 0 1 23-0.4-0.200.20.4 c d -3 -2 -1 0 1 2310 -4 -3 -2 -1 Supplementary Figure Effective 2CK parameters for the benzyl molecule. ( a,b ) Dimensionless exchange coupling J αα / ( t α t α ); ( c,d ) dimensionless potential scattering W αα / ( t α t α ). A linear(logarithmic) scale is used for panels a & c ( b & d ). Computed for t s = t d .Note that J sd has a single node at eV g = 0. -4 -3 -2 -1 -3 -2 -1 0 1 23-0.2-0.100.10.2 -3 -2 -1 0 1 2310 -4 -3 -2 -1 -3 -2 -1 0 1 23-0.2-0.100.10.2 a bc d Supplementary Figure Effective 2CK parameters for the isoprene-like molecule. ( a,b ) Dimensionless exchange coupling J αα / ( t α t α ); ( c,d ) dimensionless potential scattering W αα / ( t α t α ). A linear(logarithmic) scale is used for panels a & c ( b & d ). Computed for t s = 6 . t d . Note that J sd has two nodes at finite gate voltage. upplementary References AC Hewson, “The Kondo problem to heavy fermions,” Cambridge University Press (1997). KGL Pedersen, M Strange, M Leijnse, P Hedeg˚ard, GC Solomon, and J Paaske, “Quantuminterference in off-resonant transport through single molecules,” Physical Review B , 125413(2014). R Bulla, TA Costi, and T Pruschke, “Numerical renormalization group method for quantumimpurity systems,” Reviews of Modern Physics , 395 (2008). K Ohno, “Some remarks on the Pariser-Parr-Pople method,” Theoretica Chimica Acta , 219–227 (1964). A Weichselbaum and J von Delft, “Sum-rule conserving spectral functions from the numericalrenormalization group,” Physical Review Letters , 076402 (2007). A Oguri and AC Hewson, “NRG approach to the transport through a finite Hubbard chainconnected to reservoirs,” Journal of the Physical Society of Japan , 988–996 (2005). AI T´oth, L Borda, J von Delft, and G Zar´and, “Dynamical conductance in the two-channelKondo regime of a double dot system,” Physical Review B , 155318 (2007). Y Meir and NS Wingreen, “Landauer formula for the current through an interacting electronregion,” Physical Review Letters , 2512 (1992). TA Costi, AC Hewson, and V Zlatic, “Transport coefficients of the Anderson model via thenumerical renormalization group,” Journal of Physics: Condensed Matter , 2519 (1994). D Goldhaber-Gordon, J G¨ores, MA Kastner, H Shtrikman, D Mahalu, and U Meirav, “Fromthe Kondo regime to the mixed-valence regime in a single-electron transistor,” Physical ReviewLetters , 5225 (1998). GD Scott and D Natelson, “Kondo resonances in molecular devices,” ACS Nano , 3560–3579(2010). M Sindel, W Hofstetter, J Von Delft, and M Kindermann, “Frequency-dependent transportthrough a quantum dot in the Kondo regime,” Physical Review Letters , 196602 (2005). Ph Nozieres and A Blandin, “Kondo effect in real metals,” Journal de Physique , 193–211(1980). I Affleck and AWW Ludwig, “Exact conformal-field-theory results on the multichannel Kondoeffect: Single-fermion greens function, self-energy, and resistivity,” Physical Review B , 7297 E Sela and I Affleck, “Nonequilibrium transport through double quantum dots: Exact resultsnear a quantum critical point,” Physical Review Letters , 047201 (2009). AK Mitchell, LA Landau, L Fritz, and E Sela, “Universality and scaling in a charge two-channelKondo device,” Physical Review Letters , 157202 (2016). VJ Emery and S Kivelson, “Mapping of the two-channel Kondo problem to a resonant-levelmodel,” Physical Review B , 10812 (1992). G Toulouse, “Infinite-U Anderson Hamiltonian for dilute alloys,” Physical Review B , 270(1970). A Schiller and S Hershfield, “Exactly solvable nonequilibrium Kondo problem,” Physical ReviewB , 12896 (1995). O ´Ujs´aghy, J Kroha, L Szunyogh, and A Zawadowski, “Theory of the fano resonance in theSTM tunneling density of states due to a single Kondo impurity,” Physical Review Letters ,2557 (2000). A Schiller and S Hershfield, “Theory of scanning tunneling spectroscopy of a magnetic adatomon a metallic surface,” Physical Review B , 9036 (2000). R ˇZitko, “Fano-Kondo effect in side-coupled double quantum dots at finite temperatures andthe importance of two-stage Kondo screening,” Physical Review B , 115316 (2010)., 115316 (2010).