We present the first quantitative experimental evidence for the underscreened Kondo effect, an incomplete compensation of a quantized magnetic moment by conduction electrons, as originally proposed by Noziegraveres and Blandin. The device consists of an even charge spin S=1 molecular quantum dot, obtained by electromigration of C-60 molecules into gold nanogaps and operated in a dilution fridge. The persistence of logarithmic singularities in the low temperature conductance is demonstrated by a comparison to the fully screened configuration obtained in odd charge spin S=1/2 Coulomb diamonds. We also discover an extreme sensitivity of the underscreened Kondo resonance to the magnetic field that we confirm on the basis of numerical renormalization group calculations

Balestro, F.

Costi, T. A.

Florens, S.

Roch, N.

Wernsdorfer, W.

arXiv

International audienceWe present the first quantitative experimental evidence for the underscreened Kondo effect, an incomplete compensation of a quantized magnetic moment by conduction electrons, as originally proposed by Nozières and Blandin. The device consists of an even charge spin S = 1 molecular quantum dot, obtained by electromigration of C60 molecules into gold nanogaps and operated in a dilution fridge. The persistence of logarithmic singularities in the low temperature conductance is demonstrated by a comparison to the fully screened configuration obtained in odd charge spin S = 1/2 Coulomb diamonds. We also discover an extreme sensitivity of the underscreened Kondo resonance to the magnetic field that we confirm on the basis of numerical renormalization group calculations

Roch, Nicolas

Florens, Serge

Costi, Theo A.

Wernsdorfer, Wolfgang

Balestro, Franck

Hal - Université Grenoble Alpes

Observation of the Underscreened Kondo Effect in a Molecular Transistor

Juelich Shared Electronic Resources

Observation of the Underscreened Kondo Effect in a Molecular TransistorNicolas Roch,1 Serge Florens,1 Theo A. Costi,2 Wolfgang Wernsdorfer,1 and Franck Balestro11Institut Ne´el, associe´ a` l’UJF, CNRS, BP 166, 38042 Grenoble Cedex 9, France2Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany(Received 5 May 2009; published 3 November 2009)We present the first quantitative experimental evidence for the underscreened Kondo effect, anincomplete compensation of a quantized magnetic moment by conduction electrons, as originallyproposed by Nozie`res and Blandin. The device consists of an even charge spin S ¼ 1 molecular quantumdot, obtained by electromigration of C60 molecules into gold nanogaps and operated in a dilution fridge.The persistence of logarithmic singularities in the low temperature conductance is demonstrated by acomparison to the fully screened configuration obtained in odd charge spin S ¼ 1=2 Coulomb diamonds.We also discover an extreme sensitivity of the underscreened Kondo resonance to the magnetic field thatwe confirm on the basis of numerical renormalization group calculations.DOI: 10.1103/PhysRevLett.103.197202 PACS numbers: 75.20.Hr, 71.10.Hf, 73.21.La, 73.63.bWhen a magnetic impurity is inserted in a piece ofmetal, its magnetic moment can be completely screenedby the conduction electrons, owing to their quantized spin1=2. This general phenomenon, the Kondo effect, has beenthoroughly studied in diluted magnetic alloys [1] and hasattracted considerable attention in the more recent quan-tum dot systems [2]. Clearly, impurities carrying a spin Sgreater that 1=2 need to bind several electronic orbitals inorder to fully quench their magnetism, and nature seems toconspire in always providing enough screening channelsfor that situation to occur in general [3]. Therefore, thepossibility that screening may happen to be incomplete, asinitially proposed on theoretical grounds by Nozie`res andBlandin [4], has remained elusive for almost 30 yearsdespite the great experimental control that one can achievewith artificial quantum dot setups. The observation of theunderscreened Kondo effect, in addition to its overscreenedcounterpart [5], is also especially appealing since it con-stitutes one of the simplest cases where standard Fermi-liquid theory is violated [6,7].We demonstrate in this Letter that molecular quantumdots obtained through electromigration [8] are perfectcandidates for achieving underscreened Kondo impurities.Indeed, point contact tunneling (single mode) and impor-tant left or right asymmetry of the transport electrodesensure a large window of energies where a single screeningchannel is active. In addition, the Kondo phenomenon inmolecules can set in already at several kelvins [9–12]thanks to relatively important charging energies, allowinga complete study of Kondo crossovers on a sufficient rangeof temperatures. Both conditions of single channel andlarge Kondo temperature are difficult to meet altogetherin other quantum dot devices, where Kondo effects asso-ciated with higher spin states have been previously foundbut not yet investigated in detail [13–17]. We report here onthe first observation of the anomalous logarithmic behaviorin the temperature and bias voltage dependent conduc-tance in a spin S ¼ 1 quantum dot below the Kondo scale,as previously predicted for underscreened impurities[4,6,7,18], and we successfully confront our results withquantitative numerical renormalization group (NRG) cal-culations. More strikingly, the experimental data demon-strate that underscreened impurities are extremely sensitiveto the application of a magnetic field. In that case, and incontrast to fully screened moments, the Kondo resonance issplit by a Zeeman energy much smaller than the Kondotemperature, reflecting the high degree of polarizability ofa partially screened spin. This surprising finding is con-firmed by new NRG calculations of the local density ofstates that we performed on the single-channel spin S ¼ 1Kondo model in a magnetic field.Our recent work [19] has shown that C60 quantum dotswith two electrons on the molecule can be gate tuned via aquantum phase transition between a molecular singlet anda spin S ¼ 1 Kondo state. Interestingly, the central condi-tion for this phase transition to occur, vindicated by theseexperimental findings [19], is the presence of a singlescreening channel in the accessible temperature range[20,21]. Since screening channels result from the overlapbetween the wave functions of the conduction electronsand those of the magnetic impurity orbitals [4], one under-stands that orbital quantum numbers must not be preservedin order for the underscreened Kondo effect to take place[see Fig. 1(a) discussing the situation of nsc channelscoupled to an impurity spin S]. We can describe ourmolecular transistor [19] by two orbital levels (1, þ1)coupled to two metallic leads (L, R) [see Fig. 1(b)], withthe tunneling matrix [20]:t ¼ tL;þ1 tR;þ1tL;1 tR;1 : (1)A screening channel  ¼ 1; 2 is associated to each eigen-value t of this matrix, from which antiferromagneticKondo couplings J ¼ 4jtj2=Eadd result between the lo-PRL 103, 197202 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending6 NOVEMBER 20090031-9007=09=103(19)=197202(4) 197202-1  2009 The American Physical Societycalized orbitals and the conduction electrons (Eadd is ameasure of the addition energy on the quantum dot). Onethen obtains two Kondo temperatures:kBTK ¼ffiffiffiffiffiffiffiffiffiDJpe1=ðJÞ; (2)whereD is the bandwidth of the metal and  its the densityof states. While it seems physically sound that at vanishingtemperature a spin S ¼ 1 will be fully screened [20],moderate left or right asymmetries in the tunneling ampli-tudes (1) can lead to vastly different Kondo scales becauseof the exponential behavior in (2), as pointed out in [18].This situation naturally occurs with electromigration, asthe molecule tends to stay preferably closer to one of theelectrodes so that, e.g., tL;1  tR;1. Figure 1 shows in-deed that our conductance maxima, here shown for twodifferent devices, are much lower than the quantum value2e2=h, with e the electron charge and h Planck’s constant.It is then possible to have a large range of temperaturesover which underscreened behavior prevails. A furthercrucial condition for the realization of the underscreenedKondo effect is the formation of a spin S ¼ 1 unit itself.This relies on a strong Hund’s rule that makes the separa-tion between singlet and triplet states much larger than thetemperature. These high-energy singlet excitations [22] areclearly spectroscopically resolved in all of our measure-ments through cotunneling lines [see Figs. 1(d) and 1(e)].Sample A in Fig. 1(c) shows an odd charge Coulombdiamond with a pronounced Kondo ridge for gate voltagesbetween Vg ¼ 0:8 V and 1.35 V (parity was argued in[19]), so that the next diamond seen up to Vg ¼ 2:3 Vhas an even number of electrons. Figure 1(d) representsthe conductance map of sample B, where a single chargedegeneracy point is observed, with Kondo ridges on bothsides. We estimate a low limit for the addition energyEadd  200 meV. Figure 1(e) focuses inside the even dia-mond of sample A, where the singlet to triplet quantumphase transition occurs [19]. We can thus assess that ourexperiment was carried out in the temperature rangeTK1  T  TK2. Similar singlet excitations are visibleon the left-hand side of sample B [see Fig. 1(d)], thusallowing us to label this region even.We now present the study of our two different devices[see Figs. 1(c) and 1(d)], both showing fully screened andunderscreened Kondo anomalies. In order to maximize theKondo temperature, gate voltages were chosen away fromthe center of the Coulomb diamonds while staying out ofthe mixed valence regime. In the fully screened Kondoeffect, the conductance versus lowering the temperaturehas a logarithmic increase above the Kondo temperatureand then saturates in a quadratic, Fermi-liquid-like fashion.On the other hand, the underscreened Kondo effect isexpected to show two distinct logarithmic behaviors[4,6,7,18] above and below the highest Kondo temperatureTK2. Indeed, partially screened moments are known to actas ferromagnetic Kondo impurities [4], leading to a slowlogarithmic scattering of the conduction electrons at lowenergy. This is best seen by plotting the derivative ofconductance with respect to temperature as a function ofinverse temperature, as suggested in [18]. Our data in theinset of Fig. 2(a), taken for the even diamond of sample A,clearly display two distinct 1=T regimes. Another way todiscriminate both Kondo effects is to perform a fit to NRGresults [3,23] of single-channel spin S ¼ 1=2 and S ¼ 1Kondo models. The obvious qualitative differences,namely, the apparent divergence in dGðTÞ=dT at low tem-perature T for spin S ¼ 1 and the presence of a maximumfor spin S ¼ 1=2, in addition to the quantitative agreementwith the NRG predictions, are strong indications of tworemarkably different Kondo states [see Figs. 2(a) and 2(b)].This comparison to theory allows us to extract the associ-ated Kondo temperatures TK and the conductance ampli-tude G0 for each curve, using the scaling formGðTÞ ¼ G0fNRGS ðT=TKÞ þGbg; (3)with S the impurity spin and Gbg a constant conductancebackground [24]. We stopped our measurements aroundTbase ¼ 100 mK in order to ensure that saturation effectsare not provoked by an uncontrolled electronic tempera-ture. We also limited the ac excitation of the lock-indetection to stay in the limit eVac  kBT. With thesewords of caution, we could not observe any saturation ofFIG. 1 (color online). (a) Summary of the different types ofKondo effects according to impurity spin S and number ofscreening channels nsc. (b) Tunneling model of our singlemolecule transistor: two orbital levels couple to two leads.(c) Conductance map of sample A recorded at T ¼ 35 mK.Dashed lines are positioned at the gate voltages where moredetailed studies where performed. (d) Conductance map forsample B. (e) Zoom inside the dotted rectangle defined in theeven diamond of sample A. The white arrows represent thesinglet-triplet splitting in both samples.PRL 103, 197202 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending6 NOVEMBER 2009197202-2the conductance over two decades of temperature in theeven diamond, contrary to what is witnessed in the fullyscreened case. Since the Kondo effect displays a universalbehavior with a single scale TK, this must be reflected bydifferent universal curves between screened and un-derscreened situations, as evidenced by Fig. 2(c). Thebias voltage dependence of the nonlinear conductance atthe base temperature was also examined [25,26]. Althoughthis study is more difficult to perform accurately, tworelatively different curves are nevertheless obtained withslower voltage behavior for the spin S ¼ 1 molecules [seeFig. 2(d)]. We also note that a measure of TK can beroughly deduced from the half width at half maximum(HWHM) of the nonlinear conductance peak; this methodwas used to have an alternative determination of TK foreach Kondo ridge in samples A and B after extraction of abackground contribution (see Table I). The higher errorbars for the bias method are due to the absence of reliablefinite voltage predictions from theory; thus, TK dependssensitively on the estimated background. Both methods,relying on thermal or voltage smearing of the Kondo peak,nicely converge to comparable TK values for both types ofKondo effects.The evolution of the Kondo peak as a function ofmagnetic field, as seen in the bias spectrum of Figs. 3(a)and 3(b), displays even more dramatic effects. Previousexperimental studies [17,27,28] and theoretical calcula-tions [29] on fully screened impurities have shown aZeeman splitting of the Kondo resonance at a critical fieldgBBc  0:5kBTK. Indeed, as seen in Table I, this estimateagrees with the TK determined by the temperature and biasdependence for the odd charge diamonds of both samples.In contrast, for the spin S ¼ 1 quantum dot, the Kondoresonance separates into Zeeman states at a much lowermagnetic field, gBBc  kBTK, so that the Kondo tem-perature naı¨vely determined from 2gBBc is in disagree-ment with the previous estimates (see Table I). Thisobservation can be understood as the combination of tworelated effects. First, a partially screened Kondo impurity isexpected to be highly polarizable in a small magnetic field,which constitutes a relevant perturbation to the free spinfixed point in the renormalization group sense. Second, thelogarithmic cusp of the Kondo peak at low bias makes thesplitting more pronounced than for the Lorentzian shapetypical of a Fermi liquid in the fully screened case. To putthis surprising observation on firmer ground, we haveperformed NRG calculations for the local density of statesin a magnetic field for the single-channel Kondo modelswith spins S ¼ 1=2 and S ¼ 1. This numerical solutionallows us to confirm that the underscreened Kondo reso-nance starts to unveil its magnetic excitations for Zeemanenergies as low as kBTK=16 [see Figs. 3(d) and 3(e)]. Afurther interesting check on our analysis lies on the ex-tracted Zeeman splitting as a function of magnetic field[Fig. 3(c)]. For odd (even) diamonds, we find that thedispersion of the magnetic states, rescaled to the corre-sponding Kondo temperature, clearly shows the presence(absence) of a threshold. Also remarkable is how these datafor our two different samples quite naturally fall on top ofeach other, another signature of Kondo universality.In conclusion, we have given comprehensive experimen-tal evidence for the occurrence of the Nozie`res and Blandinunderscreened Kondo effect in even charge molecularquantum dots. Our analysis, based on the stark differenceswith respect to regular transport properties of fullyscreened impurities, was strengthened by NRG calcula-tions. An unexpected magnetic field sensitivity of partiallyTABLE I. Kondo temperatures for each sample in the even andodd diamonds, determined by using the methods of temperature,bias, or magnetic field, as described in the text. Sample B wasnot stable enough to perform a detailed temperature study.Sample/Method Temperature Bias Magnetic FieldA (odd) 4:4 0:3 K 5:5 1:3 K 4:8 0:3 KA (even) 1:1 0:1 K 1:9 0:5 K 0:6 0:4 KB (odd) 4:4 0:8 K 5:4 0:3 KB (even) 1:9 0:3 K 0:2 0:1 KdG/dT(a.u)dG/dT(a.u)eVb/kBTK1/T (K-1)a bc d0.140.070.00(G-G’bg )/G’ o1.00.50.00.01 0.1 1 100.080.060.040.020.000.1 1 101010.11/T (K-1)0.1 1 101E-30.010.1 Even OddNRG S=1NRG S=1/2NRG S=1NRG S=1/2NRG S=1NRG S=1/2EvenOdd(G-Gbg )/Go1.00.5T/TK0.1 1 10EvenOddFIG. 2 (color online). (a) Derivative of the conductance versustemperature recorded at gate Vg ¼ 2:17 V and bias Vb ¼ 0 V inthe even Coulomb diamond of sample A (same data on a logscale in the inset, showing two distinct 1=T behaviors). The bestfit is obtained for an underscreened Kondo model (solid line) andgives G0 ¼ 0:14 (in units of 2e2=h) and TK ¼ 1:1 K.(b) Derivative of the conductance versus temperature recordedat Vg ¼ 1:2 V and Vb ¼ 0 V in the odd Coulomb diamond. Thebest fit is obtained for a fully screened Kondo model (dashedline) and gives G0 ¼ 0:34 and TK ¼ 4:4 K. (c) Conductancedata of (a) and (b) compared to the relevant NRG curves. Thebackground was adjusted to follow the convention GðTKÞ ¼G0=2, giving Gbg ¼ 0:055 in the fully screened case and Gbg ¼0:022 in the underscreened case. (d) Differential conductance atthe base temperature rescaled in universal form, with G00 ¼GðTbaseÞ and G0bg such that the conductance is G00=2 at eVb ¼kBTK.PRL 103, 197202 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending6 NOVEMBER 2009197202-3screened Kondo impurities was also discovered that wecould confirm theoretically. This work illustrates the strik-ing versatility of molecular electronics for the investigationof fundamental aspects in quantum magnetism.We thank E. Eyraud and D. Lepoittevin for their tech-nical contributions, E. Bonet, T. Crozes, and T. Fournier forlithography development, and C. Winkelmann, C. Thirion,M. Desmukh, and L. Calvet for useful discussions.Samples were fabricated in the NANOFAB facility of theNe´el Institute. This work is partially financed by ANR-PNANO Projects MolSpintronics No. ANR-06-NANO-27,MolNanoSpin No. ANR-08-NANO-002, ERC AdvancedGrant MolNanoSpin No. 226558, and STEP MolSpinQIP.[1] A. C. Hewson, The Kondo Problem to Heavy Fermions(Cambridge University Press, Cambridge, 1993).[2] D. Goldhaber-Gordon et al., Nature (London) 391, 156(1998).[3] T.A. Costi et al., Phys. Rev. Lett. 102, 056802 (2009).[4] P. Nozie`res and A. Blandin, J. Phys. (Paris) 41, 193(1980).[5] R.M. Potok, I. G. Rau, H. Shtrikman, Y. Oreg, and D.Goldhaber-Gordon, Nature (London) 446, 167 (2007).[6] P. Coleman and C. Pe´pin, Phys. Rev. B 68, 220405 (2003).[7] P. Mehta, N. Andrei, P. Coleman, L. Borda, and G. Zarand,Phys. Rev. B 72, 014430 (2005).[8] H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P.Alivisatos, and P. L. McEuen, Nature (London) 407, 57(2000).[9] J. Park et al., Nature (London) 417, 722 (2002).[10] W. J. Liang, M. P. Shores, M. Bockrath, J. R. Long, andH. Park, Nature (London) 417, 725 (2002).[11] L. H. Yu and D. Natelson, Nano Lett. 4, 79 (2004).[12] J. J. Parks et al., Phys. Rev. Lett. 99, 026601 (2007).[13] J. Schmid, J. Weis, K. Eberl, and K.V. Klitzing, Phys. Rev.Lett. 84, 5824 (2000).[14] W.G. van der Wiel et al., Phys. Rev. Lett. 88, 126803(2002).[15] A. Kogan, G. Granger, M.A. Kastner, D. Goldhaber-Gordon, and H. Shtrikman, Phys. Rev. B 67, 113309(2003).[16] G. Granger, M.A. Kastner, I. Radu, M. P. Hanson, andA. C. Gossard, Phys. Rev. B 72, 165309 (2005).[17] C. H. L. Quay et al., Phys. Rev. B 76, 245311 (2007).[18] A. Posazhennikova and P. Coleman, Phys. Rev. Lett. 94,036802 (2005).[19] N. Roch, S. Florens, V. Bouchiat, W. Wernsdorfer, andF. Balestro, Nature (London) 453, 633 (2008).[20] M. Pustilnik, L. I. Glazman, and W. Hofstetter, Phys.Rev. B 68, 161303 (2003).[21] P. Roura-Bas and A.A. Aligia, Phys. Rev. B 80, 035308(2009).[22] N. Roch, S. Florens, V. Bouchiat, W. Wernsdorfer, andF. Balestro, J. Low Temp. Phys. 153, 350 (2008).[23] F. Mallet et al., Phys. Rev. Lett. 97, 226804 (2006).[24] See EPAPS Document No. E-PRLTAO-103-001945 for adescription of temperature, bias, and magnetic field scal-ing. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.[25] M. Grobis et al., Phys. Rev. Lett. 100, 246601 (2008).[26] G. D. Scott, Z. K. Keane, J.W. Ciszek, J.M. Tour, andD. Natelson, Phys. Rev. B 79, 165413 (2009).[27] S.M. Cronenwett, T. H. Oosterkamp, and L. P.Kouwenhoven, Science 281, 540 (1998).[28] A. Kogan et al., Phys. Rev. Lett. 93, 166602 (2004).[29] T.A. Costi, Phys. Rev. Lett. 85, 1504 (2000).FIG. 3 (color online). (a), (b) Differential conductance versusbias voltage and magnetic field for sample B in the odd and evendiamonds (lines are linear fits to the B dependence of theZeeman peaks of sample B). (c) Positions of the Zeeman peaksof both samples and odd (even) diamonds as extracted fromabove. Linear fits provide the critical field Bc by extrapolation tozero bias. Both bias voltage and magnetic field were renormal-ized by TK deduced from the HWHM at B ¼ 0 T for each case.(d), (e) NRG calculations of the energy E dependent equilibriumlocal density of states AðE;B; T ¼ 0Þ at zero temperature, nor-malized to its E ¼ B ¼ 0 value, for the spin S ¼ 1=2 and S ¼ 1single-channel Kondo models and magnetic field valuesgBB=kBTK ¼ 0, 1=8, 1=4, 1=2, and 1 (top to bottom).PRL 103, 197202 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending6 NOVEMBER 2009197202-4

Observation of the underscreened Kondo effect in a molecular transistor

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Observation of the underscreened Kondo effect in a molecular transistor

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