Tunable Orbital Pseudospin and Multi-level Kondo Effect in Carbon Nanotubes

Transcription

1 Tunable Orbital Pseudospin and Multi-level Kondo Effect in Carbon Nanotubes Pablo Jarillo-Herrero, Jing Kong, Herre S.J. van der Zant, Cees Dekker, Leo P. Kouwenhoven, Silvano De Franceschi Kavli Institute of Nanoscience, Delft University of Technology, PO ox 546, 6 GA Delft, The Netherlands Present address: Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts , USA Present address: Laboratorio Nazionale TASC-INFM, I-34 Trieste, Italy Abstract. We report the observation of a multi-level Kondo effect resulting from orbital degeneracy in carbon nanotube quantum dots. The orbital degree of freedom plays the role of a pseudospin whose energy splitting is controlled by an applied magnetic field, and it is an order of magnitude larger than the Zeeman splitting. At zero field and low temperature, spin and orbital states form a 4-fold degenerate shell leading to a highly symmetric Kondo state and a correspondingly high Kondo temperature. At finite magnetic field, a four-fold splitting of the Kondo resonance is observed due to lifting both spin and orbital degeneracy. Keywords: carbon nanotube, quantum dot, Kondo, orbital, spin. PACS: 73..-f, 73..Dj, 73.3.Hk, Fg The Kondo effect, namely the interaction of a magnetic impurity with the conduction electrons of a nonmagnetic host metal, is one of the basic topics in condensed matter physics []. Nanotechnology provides new experimental tools to nail down the deepest aspects of this fundamental phenomenon. Controllable singleimpurity Kondo systems have been realized in various nanostructures, including quantum dots (QDs) formed in two-dimensional electron gases [-4], carbon nanotubes [5,6] and individual molecules [7,8]. Experiments performed until now demonstrate that the electronic properties of such nanostructures can be strongly affected by Kondo correlations. The simplest Kondo system consists of a localized electron (spin S=/) coupled to a Fermi sea via a Heisenberg-like exchange interaction, J S σ, where σ is the spin of a conduction electron interacting with the impurity site, and J the antiferromagnetic coupling (J > ) [9]. elow a characteristic temperature T K, the so-called Kondo temperature, a many-body singlet state forms between the impurity spin and the surrounding conduction electrons. This bound state gives rise to a resonant level at the Fermi energy. On average, a single conduction electron occupies this level and fully screens the impurity spin (Fig. A). In the presence of both spin and orbital degeneracy, the exchange interaction can be written as J Σ α S α σ α, where α = 4N -, N being the number of degenerate orbitals []. The larger degeneracy effectively yields a higher exchange coupling and a higher Kondo temperature (T K ~ exp( /NJ)). 48 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

2 Also the symmetry is enhanced such that spin and charge degrees of freedom are fully entangled. The state of the impurity is represented by a N-component hyper-spin and the Kondo Hamiltonian is invariant under special unitary transformations SU(N) in the hyper-spin space. In the case of double orbital degeneracy (N=) the symmetry group is SU(4), a well-known symmetry in nuclear physics, where the spin and orbital roles are played by the nucleon spin and isospin. In the SU(4) Kondo effect, the screening of the local magnetic moment requires, on average, three conduction electrons (Fig. A). Despite the favourably higher T K, the observation of SU(4) Kondo effect in QDs is challenging. The difficulty lies in the absence of orbital degeneracy in conventional semiconductor QDs, due to symmetry imperfections and level repulsion. Recently, elaborated device geometries [-3] have been proposed to obtain orbital degeneracy and hence SU(4) Kondo. In this work we show that this condition can be met in carbon nanotubes owing to their unique band structure. The electronic structure of carbon nanotubes (CNTs) can be derived from the twodimensional band structure of graphene. The continuity of the electron wave function around the nanotube circumference imposes the quantization of the wave-vector component perpendicular to the nanotube axis, k. This leads to a set of onedimensional subbands in the longitudinal direction [4]. Due to symmetry, for a given subband at k = k o there is a second degenerate subband at k = -k o. Figure C shows in black solid lines the schematic D band structure of a gapped CNT near the energy band gap. oth valence and conduction bands have two degenerate subbands, labelled as + and. It was predicted [5] that the orbital degeneracy is lifted by a magnetic field,, parallel to the nanotube axis (Fig.C). Experimental evidence for this effect has been reported only recently [6-8]. In the case of finite-length nanotubes, a discrete energy spectrum is expected from size quantization. Assuming negligible inter-subband mixing [9], the level spectrum of a CNT QD can be described as two sets of spin-degenerate levels, E + (n) and E (n) with n =,,3, (see Fig. C). At =, there is a four-fold degenerate shell for every n (Fig. D). It is this degeneracy that gives rise to multi-level Kondo effects, as we will discuss below. The four-fold shell filling emerges in a measurement of the linear conductance, G, versus gate voltage, V G (Fig. A), for a QD device fabricated from a metallic nanotube with a small band gap [7,] (Fig. ). The thick red trace, taken at T =8 K, shows Coulomb blockade oscillations corresponding to the filling of the valence band of the nanotube. Going from left to right, electrons are consecutively added to the last three electronic shells, n=3, and, respectively. The shell structure emerges clearly from the V G -spacing between the Coulomb peaks. The addition of an electron to a higher shell requires an extra energy cost corresponding to the energy spacing, n,n+, between shells (we define n,n+ E + (n+) E + (n) = E (n+) E (n) ). This translates into a larger width of the Coulomb valley associated with a full shell []. We estimate, ~3 mev, and,3 ~5 mev. The Coulomb charging energy for adding an electron to the same shell is U~5 mev []. The first group of four Coulomb peaks on the left-hand side of Fig. A (n=3) are strongly overlapped due to a large tunnel coupling to the leads. The coupling decreases with V G and becomes very small near the band gap, which lies just beyond the right-hand side of the V G -range shown. The Coulomb peaks associated with the last two electrons (n=) are not visible. 483 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

3 A Fermi Lead I QD Ordinary Kondo - SU() Fermi Lead SWNT Gate QD Two-level Kondo - SU(4) V G V C E (k ) (+) E () + E () + E (3) + k Orbital splitting E () E () E (3) ( ) D E = +, > +, >, >, > +, > +, >, >, > FIGURE. (A) Representation of the bound singlet state for an ordinary spin-/ impurity (SU() symmetry) and for a spin-/ impurity with both spin and (two-fold) orbital degeneracy (SU(4) symmetry). () Device scheme. Carbon nanotubes were grown by chemical vapor deposition on p-type Si substrates with a 5-nm-thick oxide. Individual nanotubes were located by atomic force microscopy and contacted with Ti/Au electrodes defined by e-beam lithography. The highly-doped Si substrate was used as a back-gate. A magnetic field was applied parallel to the substrate at an angle ϕ with respect to the nanotube axis. (C) and structure of a CNT near its energy gap. lack lines represent the onedimensional energy dispersion relation, E(k ), at = (k is the wave vector along the nanotube axis). The valence (conduction) band has two degenerate maxima (minima). Size quantization in a finitelength nanotube results in a set of discrete levels with both spin and orbital degeneracy. The degeneracy is lifted by a magnetic field parallel to the nanotube. The D subbands (and the corresponding levels) at finite are represented by red and blue dotted (solid) lines. Only the orbital splitting of the energy levels is shown in this figure. (D) Energy schematics of a CNT QD. At = the ground state is four-fold degenerate. Spin and/or orbital states can flip by one-step cotunnelling processes (dotted lines). (E) At finite, both orbital and spin degeneracy are simultaneously lifted, suppressing the Kondo effect. The shell structure breaks up at finite magnetic field,. The -dependence of the orbital levels appears as a large shift in the position of the Coulomb peaks. This is shown in Fig., where G is plotted versus (V G, ) for the same V G -range as in Fig. 484 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

4 A. The two groups of four Coulomb peaks exhibit similar evolution in. The first (last) two Coulomb peaks in each group shift towards lower (higher) V G, in full agreement with the situation depicted in the right inset to Fig. A. This is a clear demonstration of the -dependence of the two-fold degenerate nanotube band structure, which was not observed before. For each shell, the orbital magnetic moment, µ orb, can be extracted from the shift, V G (n), in the position of the corresponding Coulomb peaks. Neglecting the Zeeman splitting, we use the relation α V G (n) = µ orb (n)cosϕ, where is the change in, ϕ is the angle between the nanotube and the field, and α is a capacitance ratio extracted from non-linear measurements. The values obtained are.9,.8 and.88 mev/t, for n=, and 3, respectively. These values are an order of magnitude larger than the electron spin magnetic moment (½gµ =.58 mev/t for g=), and in good agreement with an estimate of µ orb based on the nanotube diameter [3]. Having acquired the necessary understanding of the energy level spectrum and its -dependence, we now focus on the Kondo effect. At ~, the conductance is strongly enhanced in certain regions of V G (Fig. ). The enhancement is due to Kondo correlations, as clearly demonstrated by the T-dependence (see below). The Kondo effect appears in the regions where the shells are partially filled, and it is absent in the case of full shells for which the ground state is a spin singlet. Starting from an empty shell (region, Fig. ), electrons are added by increasing V G (,, 3 and 4 electrons for regions I, II, III and IV, respectively). The first electron can enter any of the two degenerate orbital states and with either spin up or down. This four-fold degeneracy results in a multi-level Kondo effect [4], with an enhanced Kondo temperature associated to SU(4) symmetry, as we will substantiate below. An analogous situation occurs for three electrons since this is the same as one hole [5]. In both cases the total spin in the shell is S=/. For two electrons in the shell, there are two possible spin configurations. At = the ground state is a spin triplet (S=) due to orbital degeneracy [9] and Hund s rule. y lifting the orbital degeneracy with a magnetic field, the system undergoes a triplet-to-singlet transition when the orbital splitting overcomes the exchange energy, E exc. We estimate E exc =.5 mev for n= []. At ~.35 T, the triplet-to-singlet transition leads to a Kondo effect (Fig. ), in agreement with previous experiments [6]. In the n=3 shell, the Kondo temperature is much higher than the exchange energy, and there is Kondo effect over the entire region from to 3 electrons. The shape of G(V G ) is remarkably similar to predictions for a multi-level Kondo effect in quantum dots [7,8]. We now turn to the T-dependence of the linear conductance (Fig. A). Starting from T =8 K (thick red trace), G increases by lowering T in the regions corresponding to partially filled shells and decreases for full shells. In the centre of valleys I and III, G exhibits a characteristic logarithmic T-dependence with a saturation around e /h at low T, indicating a fully-developed Kondo effect (see Fig A, top inset, second from left). Similar T-dependences are observed for the n= and n=3 shell. From fits of G(T), taken at the V G values indicated by arrows in Fig. A, we find T K = 6.5, 7.5, and 6 K, for n=, and 3, respectively. These Kondo temperatures are an order of magnitude higher than those previously reported for nanotube QDs [5,6] and comparable to those reported for single-molecule devices [7,8]. Such high T K values, and the fact that G exceeds e /h (the one-channel conductance limit), are signatures of a non- 485 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

5 conventional Kondo effect. The bottom inset in Fig. A shows the normalized conductance, G/G, versus normalized temperature, T/T K, for different shells and for both one and two electrons in the shell. The observed scaling reflects the universal character of the Kondo effect. The low-temperature behaviour is fully determined by a single energy scale, T K, independent of the spin and orbital configuration responsible for the Kondo effect. G (e /h) A (T) 3G (e/h) T (K). G/G O.5 T K =6K T K =7.7K T K =6.5K. T/T K G (e /h) T (K) G (e /h).7.5 T (K) I II III IV V G (V) 3 4 S= S=/ S= I II III IV S= S=/ S=,3, +,3 3 n = 3 n = n = FIGURE. (A) Linear conductance, G, vs gate voltage, V G, for different temperatures, T, between.34 K (black trace) and 8 K (red trace). The four-fold shell filling emerges more clearly at high T. At low T, the Kondo effect leads to an increase in conductance. Top-left insets: T-dependence for three values of V G in shells,, and 3, respectively (see arrows). The solid lines in the insets are fits to the empirical function G(T)=G /(+( /s -)(T/T K ) ) s [3], where G, and T K are fitting parameters, while s is fixed at.. ottom-left inset: scaling behaviour for the three T-dependences shown in the upper insets. Topright inset: qualitative energy spectrum of the CNT QD as a function of magnetic field, (Zeeman splitting and charging energy are neglected). () G vs on a color scale at T=.34 K (G increases from blue to red). The high-g regions in between Coulomb peaks are due to the Kondo effect. Within each shell, the first two Coulomb peaks move towards lower V G with increasing and the second two Coulomb peaks move towards higher V G, reflecting the opposite magnetic moments of the two orbital states. The small diagrams indicate the spin and orbital filling of electrons for the n= shell. +,, +, E In the non-linear regime the Kondo effect manifests itself as a resonance in the differential conductance, di/dv, at source-drain bias V~. This is clearly seen in Fig. 3A, where di/dv(v,v G ) is shown for the second shell (n=). Two large Coulomb diamonds separated by three small ones can be identified. These correspond to the regions to IV in Fig. A. In regions I and III, there is a Kondo peak in di/dv at zero bias. In the centre of the Kondo ridges, the full-width at half maximum (FWHM) of these peaks is.3 mev. This width gives T K ~8K (FWHM~k T K ), in agreement with 486 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

6 A 4 V (mv) II I III IV VG (V) di/dv C Orbital splitting.5 V. G (e /h) Zeeman splitting 3 (T). di/dv (e /h) D ev/ktk gµ/ktk µorb/ktk FIGURE 3. (A) Color-scale representation of the differential conductance, di/dv, versus V and VG for the second shell at T~.34K and = (di/dv increases from blue to red). The regions labelled, I, II, III and IV, correspond to,,, 3 and 4 electrons, in the n= shell. () Three-dimensional plot of di/dv versus V and in the center of Coulomb valley I. V ranges from -.5 to.5 mv, ranges from -3 to 3T. The Kondo peak at (V,) = (,) splits in 4 peaks at finite, due to the lifting of both orbital and spin degeneracies. (C) -dependence of the zero-bias conductance. G decreases on a ~T scale, i.e. much faster than expected from Zeeman splitting. So the orbital splitting sets the scale for conductance suppression. (D) G is plotted vs normalized Zeeman energy, gµ/ktk (black trace), and vs normalized orbital splitting, µorb/ktk (blue trace). TK = 7.5 K as deduced from a fit of G(T). (Note that G()=.5G() when µorb/ktk~). To compare the suppression due to magnetic field with that due to bias voltage, we also show a measurement of di/dv vs normalized bias voltage, ev/ktk (red trace). The blue and red traces fall almost on top of each other. the previous estimate based on the fits of G vs T. In region II, we observe two peaks at finite bias, reflecting the singlet-triplet splitting of the Kondo resonance [6]. At finite, the Kondo resonance is expected to split. In an ordinary spin-/ Kondo system, this follows from lifting spin degeneracy. Two split peaks are observed in di/dv(v), separated by twice the Zeeman energy [-4]. Here we show that a fundamentally different splitting is observed in the presence of orbital degeneracy. A four-fold peak splitting is found in region I as shown in Fig. 3. The large zero-bias resonance opens up in four peaks that move linearly with and become progressively smaller. The two inner peaks can be ascribed to Zeeman splitting and the two outer 487 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

7 peaks to orbital splitting. The smaller splitting corresponds to inelastic cotunneling from +, > to +, >, while the larger splitting reflects inelastic cotunneling from orbital + to orbital (Fig. E). In the latter case, inter-orbital cotunneling processes can occur either with or without spin flip. The corresponding substructure, however, cannot be resolved due to the broadening of the outer peaks. The slope dv/d of a conductance peak directly yields the value of the magnetic moment associated with the splitting. We obtain a spin magnetic moment µ spin =½ dv/d spin =.6 mev/t ~µ from the inner peaks [9], and an orbital magnetic moment µ orb =½ dv/d orb /cosϕ =.8 mev/t ~3 µ from the outer peaks. The importance of the orbital degree of freedom appears in an explicit measurement of the -dependence of G (Fig. 3C). If the Kondo effect was determined by spin only (this could be the case if one of the orbitals is weakly coupled), G should decrease due to Zeeman splitting on a field scale ~k T K /gµ [3]. Instead we find that G is suppressed on a much smaller scale, ~k T K /µ orb, being determined by the orbital splitting (see Fig. 3D). From the fact that the orbital splitting determines the Kondo resonance we come to the following important conclusion: the observed zerobias peak largely originates from the presence of an orbital degeneracy. Degenerate orbital states, theoretically described by a pseudospin, can thus lead to a Kondo effect, analogous to degenerate spin states. At =, however, we have both spin and pseudospin degeneracy. It is known from theory that in this case the Kondo ground state obeys SU(4) symmetry at low temperature (T << T K ) [-3]. This SU(4) Kondo effect is substantially different from the Kondo effect for an impurity with S=3/ (SU() symmetry), as well as from the Kondo effect for two independent parallel spin-/ systems (SU() SU() symmetry). According to calculations reported in Ref. 3, the observation of a fourfold splitting constitutes direct experimental evidence of SU(4) symmetry. An SU(4) Kondo effect was recently reported also in vertical semiconductor quantum dots with no evidence, however, for such four-fold splitting [3]. We have shown that the spin and orbital degeneracy are simultaneously lifted by an applied magnetic field. Due to the large µ orb, the orbital splitting can exceed the level spacing between shells for fields above a few Tesla. This offers the opportunity to recover orbital degeneracy at high fields, while having at the same time a large spin splitting. Eventually, this removal of spin degeneracy leads to a purely orbital Kondo effect [33]. We finally remark that the strong orbital Kondo effect discussed in this report has been measured in all devices (four devices obtained from three different fabrication runs) where Kondo effect and four-fold shell filling were observed. ACKNOWLEDGMENTS We thank G. Zaránd, and R. Aguado for helpful discussions. We acknowledge financial support from the Japanese Solution Oriented Research for Science and Technology and from the Dutch Organization for Fundamental Research, FOM. 488 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see

8 REFERENCES. rown, M. P., and Austin, K., The New Physique, Publisher City: Publisher Name, 997, pp rown, M. P., and Austin, K., Appl. Phys. Letters 65, (994). 3. Wang, R.T., "Title of Chapter," in Classic Physiques, edited by R.. Hamil, Publisher City: Publisher Name, 997, pp Smith, C. D., and Jones, E. F., "Load-Cycling in Cubic Press" in Shock Compression in Condensed Matter-997, edited by S. C. Schmidt et al., AIP Conference Proceedings 49, New York: American Institute of Physics, 998, pp Kondo, J., Prog. Theor. Phys. 3, 37 (964).. Goldhaber-Gordon, D., et al., Nature 39, 56 (998). 3. Cronenwett, S. M., Oosterkamp, T. H., Kouwenhoven, L. P., Science 8, 54 (998). 4. Schmid, J., Weis, J., Eberl, K., von Klitzing, K., Physica 56-58, 8 (998). 5. Nygard, J., Cobden, D. H., Lindelof, P. E., Nature 48, 34 (). 6. uitelaar, M. R., achtold, A., Nussbaumer, T., Iqbal, M., Schonenberger, C., Phys. Rev. Lett. 88, 568 (). 7. Park et al., J., Nature 47, 7 (). 8. Liang, W., Shores, M. P, ockrath, M, Long, J. R., Park, H., Nature 47, 75 (). 9. Hewson, A. C., The Kondo Problem to Heavy Fermions, Cambridge, Univ. Press Cambridge, We assume that J is independent of spin and orbital quantum numbers. More generally, the exchange coupling can be written as Z a p J a p S a p-a a p. In the strong coupling limit, which occurs at low T, this hamiltonian can be replaced by one where J a p=j, provided that the orbital splitting is smaller than T K (,).. orda, L., Zarand, G., Hofstetter, W., Halperin,.I., von Delft, J., Phys. Rev. Lett. 9, 66 (3).. Zarand, G., rataas, A., Golhaber-Gordon, D., Solid State Comm. 6, 463 (3). 3. Choi, M.-S., et al., e-print available at 4. Dresselhaus, M. S., Dresselhaus, G. & Eklund, P. C. Science offullerenes and Carbon Nanotubes, San Diego, Academic Press, Ajiki, H., Ando, T. J., Phys. Soc. Jpn 6, 55 (993). 6. Minot, E., Yaish, Y., Sazonova, V., McEuen, P. L., Nature 48, 536 (4). 7. Zaric, S., etal, Science 34, 9 (4). 8. Coskun, U. C., Wei, T-C., Vishveshwara, S., Goldbart, P. M., ezryadin, A., Science 34, 3 (4). 9. A subband mixing may result in level repulsion and an orbital splitting. We observe no such splitting and conclude that this is much smaller than T K. The condition for orbital degeneracy is thus fulfilled at the level required for the observation of SU(4) Kondo ().. This band gap can be due to different perturbations, such as curvature or strain. The measured value of the band gap is ~3meV.. Liang, W., ockrath, M., Park, H., Phys. Rev. Lett. 88, 68 ().. The value of U is extracted from the separation between the Coulomb peaks at high temperature. The accuracy in the values of U, A ; and A ;3 is determined by the accuracy in the a-factor, of order % due to large T broadening and the residual Kondo effect at T~8K. We estimate a % error bar in E exc and ^orb. 3. From the value for n=l, we extract a nanorube diameter D = 4ju orb /ev F = 4.5 nm, in agreement with the measured diameter of 4.±.5 nm as determined by atomic force microscopy. For this device 9=37. The decreasing value of p, orb from n=l to n= is in qualitative agreement with predictions (6). For n=3, the presence of Kondo effect almost throughout the entire -range makes the determination less accurate. 4. Inoshita, T., Shimizu, A., Kuramoto, Y., Sakaki, H., Phys. Rev. 48, 475 (993). 5. In fact the situation for and 3 electrons in n= is not completely symmetric, since A ; < A r3 and also the gate coupling varies with V G. This may be related to the observed differences in G vs in regions I and III (for both shells and 3). 6. Sasaki, S. et al., Nature 45, 764 (). 7. Izumida, W., Sakai, O., Shimizu, Y., J. Phys. Soc. Jpn 67, 444 (998). 8. Levy-Yeyati, A., Flores, F., Martin-Rodero, A., Phys. Rev. Lett. 83, 6 (999). 9. The linear fit yields the correct g-factor, but with an offset of-3 uev. See also Kogan, A., et al., Phys. Rev. Lett. 93, 666 (4). 3. Costi, T. A., Phys. Rev. Lett. 85, 54 (). 3. Golhaber-Gordon, D., et al., Phys. Rev. Lett. 8, 55 (998). 3. Sasaki, S., et al., Phys. Rev. Lett. 93, 75 (4). 33. Jarillo-Herrero, P., et al., Nature 434, 484 (5). 489 Downloaded Sep 7 to Redistribution subject to AIP license or copyright, see