Abstract
The Kondo effect is a manybody phenomenon arising due to conduction electrons scattering off a localized spin^{1}. Coherent spinflip scattering off such a quantum impurity correlates the conduction electrons, and at low temperature this leads to a zerobias conductance anomaly^{2,3}. This has become a common signature in bias spectroscopy of singleelectron transistors, observed in GaAs quantum dots^{4,5,6,7,8,9} as well as in various singlemolecule transistors^{10,11,12,13,14,15}. Although the zerobias Kondo effect is well established, the extent to which Kondo correlations persist in nonequilibrium situations where inelastic processes induce decoherence remains uncertain. Here we report on a pronounced conductance peak observed at finite bias voltage in a carbonnanotube quantum dot in the spinsinglet ground state. We explain this finitebias conductance anomaly by a nonequilibrium Kondo effect involving excitations into a spintriplet state. Excellent agreement between calculated and measured nonlinear conductance is obtained, thus strongly supporting the correlated nature of this nonequilibrium resonance.
Main
For quantum dots accommodating an odd number of electrons, a suppression of charge fluctuations in the Coulombblockade regime leads to a local spin1/2 degree of freedom and, when temperature is lowered through a characteristic Kondo temperature T_{K}, the Kondo effect shows up as a zerobias peak in the differential conductance. In a dot with an even number of electrons, the two electrons residing in the highest occupied level may either form a singlet or promote one electron to the next level to form a triplet, depending on the relative magnitude of the level splitting δ and the ferromagnetic intradot exchange energy J. For J>δ, the triplet state prevails and gives rise to a zerobias Kondo peak^{7,8}, but when δ>J the singlet state is lower in energy and no Kondo effect is expected in the linear conductance. Nevertheless, spinflip tunnelling becomes viable when the applied bias is large enough to induce transitions from singlet to triplet states. Such interlead exchange–tunnelling may give rise to Kondo correlations and concomitant conductance peaks near V ∼±δ/e, where e is the elementary charge. However, because the tunnelling involves excited states with a rather limited lifetime, the extent to which the coherence of such inelastic spin flips, and hence the extent to which the Kondo effect is maintained, remains unanswered.
The possibility of finitebias conductance anomalies induced by the Kondo effect was discussed in 1966 by Appelbaum^{16}, and a detailed account of the subsequent development of this problem can be found in ref. 17. In the context of a doubledot system, a qualitative description of a finitebias Kondoeffect, treating decoherence and nonequilibrium effects in an incomplete way, is given in ref. 18. Finitebias conductance peaks have already been observed in carbon nanotubes^{11,14,19} as well as in GaAs quantum dots^{8,9,20}. However, owing to the lack of a quantitative theory for this nonequilibrium resonance, characterization of the phenomenon has not yet been possible. As pointed out in refs 17, 21, 22 a biasinduced population of the excited state (here the triplet), may change a simple finitebias cotunnelling step into a cusp in the nonlinear conductance. Therefore, to quantify the strength of correlations involved in such a finitebias conductance anomaly, a proper nonequilibrium treatment will be necessary. As we demonstrate below, the qualitative signature of Kondo correlations is a finitebias conductance peak which is markedly sharper than the magnitude of the threshold bias.
We have examined a quantum dot based on a singlewall carbon nanotube (see Fig. 1a). Electron transport measurements of the twoterminal differential conductance were carried out in a cryostat with a base electron temperature of T_{el}≈80 mK and a magnetic field perpendicular to the nanotube axis. The lowtemperature characteristics of the device are seen from the density plot in Fig. 1b, showing dI/dV as a function of source–drain voltage V and gate voltage V_{g}. The dominant blue regions of low conductance are caused by Coulomb blockade (CB), whereas the sloping white and red lines are edges of the CB diamonds, where the blockade is overcome by the finite source–drain bias. Moreover, white and red horizontal ridges of high conductance around zero bias are seen. These ridges occur in an alternating manner, for every second electron added to the nanotube dot. These are the Kondo resonances induced by the finite electron spin S=1/2 existing for an odd number N of electrons where an unpaired electron is localized on the tube. The zerobias resonances are absent for the other regions (with even N) where the groundstate spin is S=0. Over most of the measured gatevoltage range (not all shown), the diamond plot shows a clear fourelectron periodicity, consistent with the consecutive filling of two nondegenerate subbands, corresponding to the two different sublattices of the rolledup graphene sheet, within each shell^{19}. This shellfilling scheme is illustrated in Fig. 1c.
We observe inelastic cotunnelling features for all even N at eV ∼±Δ for a filled shell and eV ∼±δ for a halffilled shell. Reading off the addition, and excitation energies throughout the quartet near V_{g}=−4.90 V, shown in Fig. 1b, we can estimate the relevant energy scales within a constant interaction model^{19,23,24}. We deduce a charging energy E_{C}≈3.0 meV, a level spacing Δ≈4.6 meV, a subband mismatch δ≈1.5 meV, and rather weak intradot exchange, and intraorbital Coulomb energies J,dU<0.05δ. The fact that δ>J is consistent with a singlet ground state for a halffilled shell, involving only the lower subband (orbital), together with a triplet at excitation energy δ−J and another singlet at energy δ. It should also be noted that the regions with odd N and doublet ground state have an inelastic resonance at an energy close to δ. We ascribe these to (possibly Kondoenhanced) transitions exciting the valence electron from orbital 1 to orbital 2, but we shall not investigate these resonances in detail. Focusing on the halffilled shell, that is N=2, Fig. 2a shows the measured line shape, dI/dV versus V, at V_{g}=−4.90 V and for T_{el} ranging from 81 to 687 mK. The conductance is highly asymmetric in bias voltage and exhibits a pronounced peak near V ∼δ/e which increases markedly when lowering the temperature (see Fig. 2a, inset).
For N=2, we model the nanotube quantum dot by a twoorbital Anderson impurity occupied by two electrons coupled to the leads by means of four different tunnelling amplitudes t_{iα}, between orbitals i=1,2 and leads α=L,R. In terms of the conduction electron density of states, ν_{F}, which we assume to be equal for the two leads, the tunnelling induces a level broadening . In the Kondo regime, Γ_{i}≪E_{C}, charge fluctuations are strongly suppressed and the Anderson model becomes equivalent to a Kondo model incorporating secondorder superexchange (cotunnelling) processes, like that illustrated in Fig. 3a, in an effective exchange–tunnelling interaction (see Supplementary Information, Section S2). This effective model describes the possible transitions between the five lowestlying twoelectron states on the nanotube (see Fig. 3b). As J≪δ, we neglect J and assume that the four excited states are all degenerate with excitation energy δ. The next excited state is a singlet entirely within orbital 2 with excitation energy 2δ (see Supplementary Information, Section S2). Although this state is merely a factor of two higher in energy than the five lowestlying states, we shall neglect this and all higherlying states. As there are no spinflip transitions connecting the groundstate singlet to the excited singlet at energy 2δ, we expect this omission to have a minute influence on the conductance for V <2δ.
To deal with the logarithmic singularities arising in loworder perturbation theory (PT) within the Kondo model, it is convenient to use the socalled poor man’s scaling method^{25}. As we have demonstrated earlier^{26}, this method can also be generalized to deal with nondegenerate spin states, such as a spin1/2 in a magnetic field, as well as with nonequilibrium systems where the local spin degrees of freedom are out of thermal equilibrium with the conduction electrons in the leads. The present problem is complicated by the presence of the two finite energy scales, eV and δ, causing the renormalization of the couplings to involve more than just scattering near the conductionelectron Fermi surfaces. To accommodate for this fact, we must allow for different renormalization at different energy scales, and the poor man’s scaling method should therefore be generalized to produce a renormalization group (RG) flow of frequencydependent coupling functions. The details of this method are presented in refs 26,
These coupled nonlinear RG equations do not lend themselves to analytical solution but may be solved numerically with relatively little effort. From this solution, we obtain renormalized coupling functions with logarithmic peaks at certain frequencies determined by eV and δ (see, for example, Fig. 3c). As resonant spin flips only take place through the excited states at energy δ, Korringalike spin relaxation, through excitation of particle–hole pairs in the leads, partly degrades the coherence required for the Kondo effect^{28}. Therefore, even at zero temperature, the couplings remain weak and increase roughly as , with Γ being the effective (Vdependent) spinrelaxation rate. Using the parameters t_{αi} from the fit in Fig. 4, we calculate the Kondo temperature and the spinrelaxation rate and find that T_{K}≈2 mK and Γ≈250 mK (see Supplementary Information, Sections S3–S5). This in turn implies a small parameter 1/ln(δ/T_{K})≈0.1, which justifies our perturbative approach. Notice that the smallness of T_{K} corresponds roughly to a mere 30% reduction of the total hybridization to orbital 1 as an electron is added to the nanotube. This tiny T_{K} does not show up directly in the Vdependence of the nonlinear conductance, for example, in the width of the peak, which is characterized instead by the excitation energy δ and the spinrelaxation rate Γ.
Having determined the renormalized coupling functions, the electrical current is finally calculated from the golden rule expression: where μ_{L/R}=±eV/2 denotes the two different chemical potentials, ħ is Planck’s constant, n_{γ} are the nonequilibrium occupation numbers of the five impurity states and g_{R,σ′;L,σ}^{γ′;γ}(ω) is the renormalized exchange–tunnelling amplitude for transferring a conduction electron from left to right lead and changing its spin from σ to σ′, while switching the impurity state from γ to γ′. Figure 4 shows a comparison of this calculation to the data, obtained by fitting the four tunnelling amplitudes t_{iα} determining the unrenormalized exchange couplings. Note that the measured conductance at V =0 and at V ≫δ, together with the asymmetry between positive and negative bias, places strong constraints on these four amplitudes. The resulting fit is highly satisfactory, with only slight deviations at the highest voltages.
To quantify the importance of Kondo correlations, we also make a comparison with plain nonequilibrium cotunnelling. This unrenormalized secondorder tunnelling mechanism clearly underestimates the conductance peak at positive bias, and most severely so at the lowest temperature. Qualitatively, simple nonequilibrium cotunnelling gives rise to a cusp in the conductance which is roughly as wide as the magnitude of the threshold bias itself. As is evident from the data and analysis presented here, nonequilibrium Kondo correlations instead produce a conductance anomaly which is sharper than the magnitude of the threshold—that is a proper finitebias peak.
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Acknowledgements
We thank L. DiCarlo and W. F. Koehl for experimental contributions and D. H. Cobden and V. Körting for useful discussions. This research was supported by the Center for Functional Nanostructures of the DFG (J.P., P.W.), the European Commission through project FP6003673 CANEL of the IST Priority (J.P.), ARO/ARDA (DAAD190210039), NSFNIRT (EIA0210736) (N.M., C.M.M.) and the Danish Technical Research Council (J.N.).
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Affiliations
The Niels Bohr Institute & The NanoScience Center, University of Copenhagen, DK2100 Copenhagen, Denmark
 J. Paaske
 & J. Nygård
Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany
 A. Rosch
Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
 P. Wölfle
Department of Physics, University of Illinois at Urbana Champaign, Urbana, Illinois 618013080, USA
 N. Mason
Department of Physics, Harvard University, Cambridge, Massachusetts 0213, USA
 N. Mason
 & C. M. Marcus
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