Kondo blockade due to quantum interference in single-molecule junctions

Molecular electronics offers unique scientific and technological possibilities, resulting from both the nanometre scale of the devices and their reproducible chemical complexity. Two fundamental yet different effects, with no classical analogue, have been demonstrated experimentally in single-molecule junctions: quantum interference due to competing electron transport pathways, and the Kondo effect due to entanglement from strong electronic interactions. Here we unify these phenomena, showing that transport through a spin-degenerate molecule can be either enhanced or blocked by Kondo correlations, depending on molecular structure, contacting geometry and applied gate voltages. An exact framework is developed, in terms of which the quantum interference properties of interacting molecular junctions can be systematically studied and understood. We prove that an exact Kondo-mediated conductance node results from destructive interference in exchange-cotunneling. Nonstandard temperature dependences and gate-tunable conductance peaks/nodes are demonstrated for prototypical molecular junctions, illustrating the intricate interplay of quantum effects beyond the single-orbital paradigm.

The Kondo temperature associated with the single-orbital Anderson model follows from perturbative scaling as, 1 T K ∼ (ΓU ) 1/2 exp[π ( + U )/ΓU ] (2) with Γ = Γ s + Γ d and Γ α = πρ 0 t 2 α . In the presence of the gate described by H g , one has → − eV g , yielding the standard quadratic dependence on applied gate, ln T K /D ∼ ( − eV g )( − eV g + U ). However, note that this behaviour is not necessarily expected in the case of real single-molecule junctions, since nontrivial gate dependences arise in the generic multi-orbital case (as shown explicitly in subsection 'Gate-tunable QI in Kondoactive molecules' for the isoprene junction).
Zero-bias conductance through a single Anderson orbital can be obtained from the Meir- where f denotes the Fermi function and t ee (ω, T ) = −πρ 0 Im T ee (ω, T ) is the spectrum of the even channel T-matrix. For a single Anderson impurity, T ee (ω, T ) = (t 2 s + t 2 d )G imp (ω, T ), with G imp (ω, T ) = d σ ; d † σ the full retarded impurity Green's function. The simple form of Supplementary Equation 3 applies in the case of proportionate couplings, 8 automatically fulfilled in the single-orbital case. The source/drain asymmetry factor is given by which is maximal, G 0 = 1, for equal couplings t s = t d .
The universal form of t ee (ω, T ) in the Kondo regime gives the universal temperaturedependence of conductance 9 often successfully fit to experimental data. 10,11 Importantly, at particle-hole symmetry U = −2 , the Friedel sum rule 1 pins the T-matrix to t ee (0, 0) = 1 is the local odd-channel conduction electron operator. Since the odd channel is strictly decoupled from the rest of the system (impurity and even channel), K(t, T ) factorizes as, Some lengthy algebra then yields the following result, In Eq. (23), this gives the ac conductance in terms of the retarded impurity Green function

Supplementary Note 2: Generalized Kondo resonance and derivation of Eq. (5)
We now consider the generalized 2CK model describing off-resonant single-molecule junctions, Eq. (16), focusing on the particle-hole (ph) symmetric case with W = 0. Quantum interference effects give rise to such a potential scattering node, and can in principle be realized in any given system by tuning gate voltages. Generically, the junction is still conducting due 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-temperature limit G(T T K ) G(0) can be obtained analytically, as shown below.
Combining Eq. (27)and Eq. (28), and noting thatΓ α = 1/(πρ 0 ) and G (0) (0) = −iπρ 0 , we have, In the even/odd orbital basis (Eq. (4) of subsection 'Emergent decoupling'), T sd (ω, T ) = , in terms of the T-matrices of the even/odd channels. Importantly, as shown in subsection 'Emergent decoupling', the odd channel decouples asymptotically. The molecule undergoes a Kondo effect with the even channel since J e > J o ; this cuts off the RG flow with the odd channel and effectively disconnects it. This is an emergent low-energy phenomenon, not a property of the bare system. In all cases, therefore, we have T oo (0, 0) = 0, meaning that, Furthermore, at ph symmetry, iπρ 0 T ee (0, 0) = t ee (0, 0) = 1 due to the Kondo effect, and so 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 frustrated quantum critical point. 13,14 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. However, δ = 1 2 (J e − J o ) could still be small, especially near a quantum interference node in J sd . When δ 2 = J 2 sd + J 2 − < T K , 2CK quantum critical fluctuations control the junction conductance. Interestingly, exact analytic results can be obtained in this special regime, including the non-linear conductance away from thermal equilibrium. Importantly, however, the 2CK critical point has an emergent spin isotropy. 14 This means that 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, one can 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 same low-energy crossover must occur in the physical spin-isotropic model as at the Toulouse point. Therefore the Toulouse limit solution can be used for the low-temperature crossover -provided there is good scale separation T * T K (where T * ∼ δ 2 is the Fermi liquid crossover scale 14 generated by relevant perturbations J sd and/or J − ).
Following Supplementary Reference 19, we obtain Eq. (7) of subsection 'Kondo resonance'. The precise quantitative agreement between the predicted G(T ) at linear response and 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 through the molecular junction is mediated only by W sd (for simplicity, we again take the phsymmetric case W ss = W dd = 0). In general, J ss = J dd , meaning that a Kondo effect will develop with the more strongly coupled lead. For concreteness, we take now J ss > J dd .
The drain lead therefore decouples for T T K . As shown below, this produces an exact node in the total conductance G(T = 0) = 0. The perturbative result for the conductance G(T T K ) ∼ W 2 sd , valid at high temperatures, is quenched at low temperatures due to interactions and the Kondo effect.
This depletion of the lead electron density at the molecule is due to the Kondo effect, and arises only for T T K . The conductance through the molecule is therefore blocked because there are no available source lead states to facilitate transport. Formally, this is proved from the optical theorem, assuming that the low-energy physics can be understood in terms of a renormalized non-interacting system: the conductance (at T = 0) is then related to the Alternatively, we can use the Kubo formula, Eq. (23)-Eq. (5)to obtain an expression for the conductance without resorting to Fermi liquid theory. In this regime with J sd = 0 but W sd = 0, we have iΩ = W sd σ (c † sσ c dσ − H.c.). Since at T = 0 the drain channel is decoupled, K(t, T ) factorizes as in Supplementary Equation 6. Following the same steps as in Sec. S4, we finally obtain G(0) in terms of the retarded electron Green's function G ss (0, 0), The total conductance at T = 0 therefore exactly vanishes since t ss (0, 0) = 1. Note also that the high-temperature perturbative result is precisely recovered if one sets t ss = 0.
One might wonder whether the suppression of conductance due to quantum interference By contrast, there is no direct source-drain conductance pathway in single-molecule junction devices -cotunneling proceeds only throughs the molecule, and J sd = 0 arises due to intrinsic quantum interference (i.e., a characteristic property of the isolated molecule and its contacting geometry). The Kondo blockade is an exact conductance node of the strongly interacting system at T = 0, which arises because the molecule is asymptotically bound to only one of the two leads -its Kondo cloud is impenetrable at low energies to cotunneling embodied by W sd . At higher temperatures, the conductance G(T ) ∼ W 2 sd is 'blind' to the quantum interference node in J sd ; the Kondo blockade arises entirely from the interplay between intrinsic molecular quantum interference and Kondo physics. The Kondo blockade lineshape 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 Kondo model, conductance can also be suppressed. 22 However, this arises because two spin-1 2 quantum impurities are successively screened by a single conduction electron channel in a two-stage process. This mechanism is not related to the Kondo blockade, which involves a net spin-1 2 molecule and single-stage Kondo screening by two conduction electron channels.

Supplementary Note 5: Cotunneling amplitudes
The benzyl radical in Figure 5  (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. (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 = 6.17t d . Note that J sd has two nodes at finite gate voltage.