Single-walled carbon nanotube weak links: from Fabry-Pérot to Kondo regime
F. Wu, R. Danneau, P. Queipo, E. Kauppinen, T. Tsuneta, P. J. Hakonen
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Single-walled carbon nanotube weak links: from Fabry-P´erot to Kondo regime
F. Wu , R. Danneau , P. Queipo , E. Kauppinen , T. Tsuneta , and P. J. Hakonen Low Temperature Laboratory, Helsinki University of Technology, Espoo, Finland Center for New Materials, Helsinki University of Technology, Espoo, Finland (Dated: October 31, 2018)We have investigated proximity-induced supercurrents in single-walled carbon nanotubes in theKondo regime and compared them with supercurrents obtained on the same tube with Fabry-P´erotresonances. Our data display a wide distribution of Kondo temperatures T K = 1 - 14 K, and themeasured critical current I CM vs. T K displays two distinct branches; these branches, distinguishedby zero-bias splitting of the normal-state Kondo conductance peak, differ by an order of magnitudeat large values of T K . Evidence for renormalization of Andreev levels in Kondo regime is also found. An odd, unpaired electron in a strongly coupled quan-tum dot makes the dot to behave as a magnetic impurityscreened by delocalized electrons. Such a Kondo impu-rity creates a peak in the density of states at the Fermilevel, thereby leading to characteristic Kondo resonanceswith enhanced conductance around zero bias, which hasbeen observed in various quantum dot systems during therecent years [1, 2, 3]. By studying low-bias transport andmultiple Andreev reflections (MAR) in multi-walled car-bon nanotubes (MWNT) contacted by superconductingleads, it has been demonstrated that the Kondo reso-nances survive the superconductivity of the leads whenthe Kondo temperature T K exceeds the superconductinggap ∆ [4]; thus, intricate interplay between Kondo be-havior and superconductivity can be studied in nanotubequantum dots.In quantum dots made out of single walled carbonnanotubes (SWNT), contacts play a crucial role in theirtransport properties: in the highly transparent regime,Fabry-P´erot (FP) interference patterns in the differen-tial conductance can be observed [5], whereas in the lesstransparent case, Coulomb blockade peaks occur [6]. Inthe intermediate regime, zero bias conductance peaksalternate with Coulomb blockaded valleys, highlightingKondo resonances below Kondo temperature T K due toodd numbers of spin in the cotunnelling process betweenthe dot and the leads [3].Gate-controlled, proximity-induced supercurrent hasbeen reported both in SWNTs [7, 8, 9, 10, 11, 12] and inMWNTs [13, 14]. Reasonable agreement with resonantquantum dot weak link theories [15] has been reached inbest of the samples (see e.g. Ref. 9). In some of theexperiments, Kondo-restored supercurrents were found[11, 16] in otherwise Coulomb blockaded Josephson junc-tion case. In addition, when T K < ∆, π Josephson junc-tions have been observed [10, 11].Here we report a study of gate-tunable proximity-induced supercurrents of an individual SWNT. We com-pare supercurrents in Fabry-P´erot and Kondo regimes atthe same normal state conductance, and find smaller crit-ical currents in the Kondo regime up to T K ∼ T K but also the shape of theKondo resonance conductance peak affects the magni- tude of the supercurrent: resonances with zero-bias split-ting, which appear in about every second of our Kondopeaks, result in a smaller critical current than for theregular Kondo maxima.Our nanotube samples were made using surface CVDgrowth with Fe catalyst directly on oxidized, heavily-doped SiO /Si wafer. The electrically conducting sub-strate works as a back gate, separated from the sampleby 150 nm of SiO . A sample with L = 0 . µ m lengthand φ = 2 nm diameter was located using an atomicforce microscope and the contacts on the SWNT weremade using standard e-beam overlay lithography. Forthe contacts, 10 nm of Ti was first evaporated, followedby 70 nm of Al, in order to facilitate proximity-inducedsuperconductivity in Ti. Last, 5 nm of Ti was depositedto prevent the Al layer from oxidation. The width of thetwo contacts was 200 nm and the separation between thethem was 0.3 µ m.The measurement leads were filtered using an RC filterwith time constant of 10 µ s at 1 K, followed by twistedpairs with tight, grounded electrical shields for filteringbetween the still and the mixing chamber, while the finalsection was provided by a 0.7-m long Thermocoax cableon the sample holder. In the measurements, differentialconductance G d = dI/dV was recorded using standardlock-in techniques. Voltage bias was imposed via a room-temperature voltage divider. The normal state data wereobtained by applying a magnetic field of B ∼
70 mTperpendicular to the nanotube. The superconducting gapof the contact material was found to be ∆ g = 125 µ eV,and gate capacitance C g = 1 . e /h asa consequence [17]. A scan of differential conductance G d ( V ds , V g ) versus bias voltage V ds and gate voltage V g is shown in Fig. 1(a) at B ∼
70 mT. In the absenceof magnetic field, a gate-voltage-dependent supercurrent
FIG. 1: (Color online) Normal state differential conductance G d on the plane spanned by bias voltage V ds and gate voltage V g in (a) Fabry-P´erot regime, and (b) Kondo regime bothat T = 30 mK. Normal states were achieved in all the caseswith a magnetic field of B = 70 mT. Red and blue arrowsin (b) refer to two types of resonance peaks, which have onemagnitude difference in I CM with similar Kondo temperature T K . See text for more details. is observed in the SWNT. The measured critical super-current I CM varies periodically with the gate voltage V g ,reaching a maximum of 4.8 nA at zero bias normal stateconductance G N = G d | V ds =0 = 2 . e /h . The I CM R N product is V g -dependent and it changes in a similar fash-ion as I CM and the inverse of the normal state resistance G N . This result is similar to what has been observed ina superconducting SWNT in Fabry-P´erot regime [9].After a few thermal cycles, the transport of SWNTchanged from Fabry-P´erot into Kondo type of behavioras seen in Fig. 1(b). The G d map displays a series ofCoulomb blockade diamonds (even number of electrons)alternating with Kondo ridges, marked by the arrows(odd number of electrons). The Kondo ridges are ratherwide and the Kondo temperatures, which are deducedfrom the half width at half maximum of the resonant con-ductance peaks versus bias voltage G d ( V ds ) (see below),range over T K = 1 −
14 K. We find that both the criticalcurrent and zero-bias conductance are smaller comparedwith Fabry-P´erot regime, even in the Kondo resonanceswith the highest T K .The superconducting state IV curves in both Fabry-P´erot and Kondo regimes are shown in Fig. 2. As thesample is voltage biased, negative differential resistance(NDR) is observable in Fabry-P´erot regime. However, inKondo regime, NDR occurs only at small measured crit-ical current I CM and it disappears around the maximumof the Kondo resonance peak where I CM is large. We note that zero bias resistance and the IV curves evolvesmoothly with V g around the Kondo resonance withoutany sudden jumps, and that T K > ∆ g . We ascribe thedisappearance of NDR to the presence of large MAR-induced subgap current, which is stronger with respectto the supercurrent in the Kondo regime than in the FPcase.The nanotube together with superconducting leads canbe considered as a resonant level quantum dot, andthus the two-barrier Breit-Wigner model is applicableto model the behavior [15]. In our case, the measured I CM is nearly one order of magnitude smaller than thetheoretical prediction I = e ∆ g / ~ ≈
30 nA with oneresonant spin-degenerate level [18]. Taking into accountthe phase diffusion in an underdamped, voltage-biasedJosephson junction [19], the measured I CM ∝ E J ∝ I C . With Breit-Wigner model for wide resonance limit h Γ >> ∆ g and transmission probability α BW , we have I C = I [1 − (1 − α BW ) / ], so the I CM − G N relation canbe written as I CM = I M [1 − (1 − g n ) ] , (1)where I M denotes the maximum measurable critical cur-rent when the scaled conductance g n = G N / ( e /h ) → g n , and the prefactor depends on T K ; inour case it also applies approximately to the asymmetricFP conduction as one of the spin-degenerate transmis-sion channels is greatly suppressed [17]. The fit of Eq.(1) to our data is displayed in Fig. 2 (c), with I M = 5 . . T K = 14 K).As in the FP regime, the largest critical current overa Kondo resonance corresponds to the peak value ofthe normal state conductance. In addition, the magni-tude of I CM depends on the width of the resonance inbias voltage, i.e. on T K . We have fitted the conduc-tance peaks G d ( V ds ) with a Lorentzian function in orderto extract the Kondo temperature T K . The resulting I CM − T K correlation is plotted in Fig. 3 which dis-plays two branches, instead of a single-valued correlationas observed by Grove-Rasmussen et al. [16]. The upperand lower branches involve the resonance peaks markedin Fig. 1 by red and blue arrows, respectively. Due tothe problem of trapped charge fluctuating on the backgate, we have been forced to present only data on whichwe are sure of the identification between critical currentand normal state conductance. As shown in the insetof Fig. 3, in the data of the lower branch, there is asmall dip on the zero-bias conductance peak signifyingzero-field splitting of the Kondo resonances marked byblue arrows. The Lorentzian fitting on the split peaksis somewhat approximative, and the fitted T K remains abit smaller than from the true half width. Nevertheless, F-P Kondo I C M ( n A ) G N (e /h)(c) -40 -20 0 20 40-202 (b) I ( n A ) V ds ( V) I*5 -40 -20 0 20 40-404 I ( n A ) V ds ( V)(a) FIG. 2: (Color online) Superconducting I − V curves at a few gate voltage values in (a) Fabry-P´erot regime, and (b) Kondoregime. The circles with different colors show how the measured critical current I CM was determined [20]. I CM versus zero-biasnormal state conductance G N , measured for a resonance with T K = 14 K, is displayed in (c) where the black dots and redtriangles refer to Fabry-P´erot (several resonances) and Kondo data, respectively. Data in (a) were measured in the same cooldown as Fig. 1(a) at T = 60 mK; data in (b) were taken from another cool down after Fig. 1(b) at T = 60 mK with unchanged G N . The current of the smallest I CM curve in (b) has been amplified by a factor of 5 for clarity. Black and red solid lines in(c) are theoretical fits using Eq. (1). -5 0 5 12 G N ( e / h ) V sd (mV) I C M ( n A ) k B T K / FIG. 3: (Color online) Measured critical current I CM ver-sus scaled Kondo temperature k B T K / ∆ for Kondo resonancesmarked by red and blue arrows in Fig. 1. Peaks with zero-bias splitting are denoted by blue circles, while red dots referto non-split peaks in Fig. 1(b). The solid red and blue curvesare to guide the eyes. The inset shows two typical G N − V ds re-lations for the different kinds of conductance peaks and theirLorentzian fits. The curve for non-split peak has been shifteddownwards by 0.3 units for clarity. this uncertainty is insignificant on the scale of separationof the upper and lower branches in Fig. 3. Notice that k B T K ∼ E C > ∆ >> k B T is valid for all of the measuredresonance peaks, which indicates that the double branchstructure originates from the competition between Kondoeffect and the Coulomb blockade.Zero-field splitting seems to take place in our data inevery second Kondo resonance, as seen in the nearly al-ternating sequence of red and blue arrows in Fig. 1 (b).Zero-field Kondo-peak splitting has previously been re- ported in Ref. 21, where the the splitting originates frommagnetic impurity, which is different from our case as thesplitting should then be seen at every Kondo resonance.Using the standard fourfold shell-filling sequence, it ishard to explain our findings. Split Kondo ridges may beobservable when the dot is occupied by two electrons ( N = 2) [3, 22], and the energy scale of the splitting equals tothe gap between singlet ground state and triplet excitedstate. This, however, should be bordered from both sidesby standard spin-half Kondo peaks, a sequence that wecannot identify in our data. From the normal state biasmaps, the characteristic zero-bias splitting energy can beestimated as ∆ ZBS ∼ . g = 0 .
25 meV andthe typical singlet-triplet excitation energy as found inRef. 3. We conjecture that the observed zero-field split-ting is related to the SU(4) Kondo effect which is peculiarto carbon nanotubes [23, 24] and which has been shownto lead to a dip in the density of states at small energies[25]. Alternatively, zero-field splitting may be relatedwith the recent observation of non-negligible spin-orbitcoupling in SWNTs [26]. In any case, SU(4) Kondo canexplain the unusually high T K by the enhanced degener-acy of a multiple-level quantum dot [27].According to theory [28], the width of Andreev levelscan be substantially renormalized by the Kondo effect,which would also modify the IV curve. In order to lookfor the gap renormalization, we have extracted the excesscurrent I ex as a function of normal state transmission co-efficient α , which is displayed in Fig. 4, with α calculatedfrom α = G N / (2 e /h ), and I ex determined by the differ-ence of integration from G d − V ds curves in superconduct-ing/normal state like in Ref. 29. The relation between I ex and α in a quantum point contact [30, 31, 32] can be (b) I CM (nA) α = G N /(2e /h) theory I e x ( n A ) (a) FIG. 4: (Color online) Excess current I ex of one Kondo reso-nance with T K = 14 K at T = 90 mK versus (a) transmissioncoefficient α = G N / (2 e /h ) and (b) measured critical current I CM . The blue line in (a) is the theoretical curve from Eq.(2) with ˜∆ = 100 µ eV, and red line in (b) gives linear fit of I ex /I CM = 4 . written as I ex = I ex + I ex , where I ex = e ∆ h α (2 − α ) √ − α ln (cid:20) √ − α/ (2 − α )]1 − [2 √ − α/ (2 − α )] (cid:21) ,I ex = e ∆ h α (cid:20) − α + 2 − α − α ) / ln (cid:18) − √ − α √ − α (cid:19)(cid:21) . (2)The blue curve in Fig. 4 (a) illustrates Eq. 2 with∆ ≡ ˜∆ = 100 µ eV, indicating a gap renormalization of˜∆ / ∆ g = 0 .
8. We have also investigated the relation be-tween I ex and I CM at different gate voltages. The datais shown in Fig. 4 (b), which yields a linear relation with I ex /I CM = 4 . T K = 14 K; this arises because bothare proportional to α . MAR-induced current at largebias voltage gives I AR = 4 e ∆ /h [33]. By taking into ac-count that I M ∼ I , we get I AR /I CM ∼ π , which isclose to the measured I ex /I CM value.In summary, we have investigated experimentally theproximity-effect-induced supercurrents in SWNTs in theKondo regime and compared them with results in theFabry-Perot regime with equivalent conductance. In theKondo regime, two different types of resonances, eithersplit or non-split at zero-bias, were observed and thisbehavior reflected also in the magnitude of supercurrentthat displayed two branches vs. T K . The excess currentin Kondo regime was analyzed using MAR theory andrenormalization of Andreev levels by 80 % was obtained.We wish to acknowledge fruitful discussions with S.Andresen, J. C. Cuevas, T. Heikkil¨a, J. Voutilainen,A. D. Zaikin, K. Flensberg, G. Cuniberti, T. Kontos,L. Lechner, P.-E. Lindelof, C. Strunk and P. Virtanen.This work was supported by the Academy of Finlandand by the EU contract FP6-IST-021285-2. [1] D. Goldhaber-Gordon et al. , Nature , 156 (1998). [2] S. M. Cronenwett , T. H. Oosterkamp, and L. P. Kouwen-hoven, Science , 540 (1998).[3] J. Nyg˚ard, D. H. Cobden, and P. E. Lindelof, Nature , 342 (2000).[4] M. R. Buitelaar, T. Nussbaumer, and C. Sch¨onenberger,Phys. Rev. Lett. , 256801 (2002).[5] W. Liang et al. , Nature , 665 (2001).[6] S. J. Tans et al. , Nature , 474 (1997).[7] A. Yu. Kasumov et al. , Science , 1508 (1999); A. Yu.Kasumov et al. , Phys. Rev. B , 214521 (2003).[8] A. F. Morpurgo et al. , Science , 263 (1999).[9] P. Jarillo-Herrero, J. A. van Dam, and L. P. Kouwen-hoven, Nature ,953 (2006).[10] J.-P. Cleuziou et al. , Nature Nanotech. , 53 (2006).[11] H. I. Jørgensen et al. , Nano Lett. , 2441 (2007).[12] Y. Zhang, G. Liu, and C. N. Lau, Nano Res. , 145(2008).[13] T. Tsuneta, L. Lechner, and P. J. Hakonen, Phys. Rev.Lett. , 087002 (2007).[14] E. Pallecchi et al. , Appl. Phys. Lett. , 072501 (2008).[15] See, e.g. , C. W. J. Beenakker and H. van Houten, Single-electron Tunneling and Mesoscopic Devices (eds H. Kochand H. L¨ubbig) (reprinted in arXiv:cond-mat/0111505)175–179 (Springer, Berlin, 1992).[16] K. Grove-Rasmussen, H. I. Jørgensen, and P. E. Lindelof,New J. Phys. , 124 (2007).[17] F. Wu et al. , Phys. Rev. Lett. , 156803 (2007).[18] This value does not include the effect of finite length ofthe nanotube, which would suppress the maximum I abit. See, e.g. , A. V. Galaktionov and A. D. Zaikin, Phys.Rev. B , 184507 (2002).[19] G.-L. Ingold, H. Grabert, and U. Eberhardt, Phys. Rev.B , 395 (1994).[20] When NDR exists, we extract I CM from the local cur-rent maximum I CM p and minimum I CM n using I CM =( I CM p − I CM n ) /
2. Around the Kondo resonance peakswithout NDR, I CM is obtained using the averaged volt-ages, V CM p and V CM n , respectively, of the I CM p and I CM n peak positions at lower I CM , and taking I CM =( I ( V CM p ) − I ( V CM n ) / et al. , arXiv:cond-mat/0410467v2.[22] B. Babi´c, T. Kontos, and C. Sch¨onenberger, Phys. Rev.B , 235419 (2004).[23] P. Jarillo-Herrero et al. , Nature , 484 (2005).[24] A. Makarovski et al. , Phys. Rev. B , 241407(R) (2007).[25] J. S. Lim et al. , Phys. Rev. B , 205119 (2006).[26] F. Kuemmeth et al. , Nature , 448 (2008).[27] T. Inoshita et al. , Phys. Rev. B , 14725 (1993).[28] A. Levy Yeyati, A. Mart´ın-Rodero, and E. Vecino, Phys.Rev. Lett. , 266802 (2003).[29] H. I. Jørgensen et al. , Phys. Rev. Lett. , 207003 (2006).[30] D. Averin and A. Bardas, Phys. Rev. Lett. , 1831(1995).[31] E. N. Bratus, V. S. Shumeiko, and G. Wendin, Phys.Rev. Lett. , 2110 (1995).[32] J. C. Cuevas, A. Mart´ın-Rodero, and A. Levy Yeyati,Phys. Rev. B , 7366 (1996).[33] Y. Avishai, A. Golub and A. D. Zaikin, Phys. Rev. B67