SU(4) and SU(2) Kondo Effects in Carbon Nanotube Quantum Dots

a r X i v :0709.1288v 1 [c o n d -m a t .m e s -h a l l ] 9 S e p 2007

SU(4)and SU(2)Kondo E?ects in Carbon Nanotube Quantum Dots

A.Makarovski,1A.Zhukov,1J.Liu,2and G.Finkelstein 1

Departments of 1Physics and 2Chemistry,Duke University,Durham,NC 27708

We study the SU(4)Kondo e?ect in carbon nanotube quantum dots,where doubly degenerate orbitals form 4-electron “shells”.The SU(4)Kondo behavior is investigated for one,two and three electrons in the topmost shell.While the Kondo state of two electrons is quenched by magnetic ?eld,in case of an odd number of electrons two types of SU(2)Kondo e?ect may https://www.sodocs.net/doc/6613607795.html,ly,the spin SU(2)state is realized in the magnetic ?eld parallel to the nanotube (inducing primarily orbital splitting).Application of the perpendicular ?eld (inducing Zeeman splitting)results in the orbital SU(2)Kondo e?ect.

PACS numbers:PACS numbers:73.23.Hk,73.23.-b,75.20.Hr,73.63.Fg

At low temperatures,a variety of nanoscale Coulomb blockade [1]systems with degenerate ground states ex-hibit the Kondo e?ect [2].This many-body phenomenon has now been observed in semiconductor quantum dots,molecules,carbon nanotubes,and magnetic addatoms on metallic surfaces (see Ref.[3]for a review).In high qual-ity nanotubes the quantum-mechanical orbitals originat-ing in two electronic subbands are doubly degenerate,forming 4-electron “shells”([4,5],see also Ref.[6]for additional references).In each shell,the Kondo behavior develops in the valleys with one,two,and three electrons [4,7,8].The Kondo e?ect with one electron in a shell is expected [9]to obey the SU(4)symmetry [10,11],as studied recently in Ref.[12](see also Ref.[13]).In this paper,we investigate the SU(4)Kondo e?ect in the 1,2,and 3-electron valleys in magnetic ?eld.

The nanotubes are grown on a Si/SiO 2substrate by Chemical Vapor Deposition using CO as a feedstock gas [14].This method was veri?ed to produce mostly single-wall nanotubes with diameters of about 2nm.Cr/Au electrodes separated by 200nm (Figures 1-3)or 600nm

(Figure 4)are deposited on top of the nanotubes.All the measurements are conducted at temperatures between 1.2K and 2K.We choose to work with several small-gap semiconducting nanotubes [15],which demonstrate high p-type conductance at negative gate voltages.At positive gate voltages,the middle section of the nanotube ?lls with electrons.The part of the nanotube adjacent to the electrodes stays p-type (“leads”).Therefore,a quantum dot is formed within a nanotube,de?ned by p-n and n-p junctions.As a result,Coulomb blockade sets in at low temperatures (Figure 1).

Figure 1shows conductance map of a 200nm -long nanotube quantum dot measured as a function of the source-drain bias V SD and gate voltage V gate .The “Coulomb diamonds”[1]demonstrate clear 4-electron shell ?lling.The p-n junction transparency grows with V gate ,resulting in the enhancement of the Kondo e?ect in each successive shell.The zero-bias Kondo ridge ap-pears in Coulomb diamonds with 1,2,and 3electrons (visible for V gate >10V).

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1 2 3FIG.1:Di?erential conductance map of a 200nm -long semiconducting nanotube quantum dot measured as a function of V gate and V SD (grayscale map:0.1to 2e 2/h ;T =2K,B =0).“Coulomb diamonds”demonstrate 4-electron periodicity.Six such 4-electron shells are visible.Contact transparency grows with V gate .For V gate >10V,the Kondo ridge is visible at V SD =0for 1,2and 3electrons in the topmost shell.Schematic:the quantum dot is formed within the semiconducting nanotube.

The ambipolar semiconductor nanotubes as studied here are uniquely suited for observation of the SU(4)

2

Kondo e?ect:the electrons are re?ected adiabatically from the p-n junctions at the ends of the quantum dot,resulting in little mode mixing.Therefore,the level mis-match between the two orbitals level in a shell is very small,as evidenced by observation of the Kondo ridge in the 2e valleys of many successive shells (Figure 1).It is also important for the observation of the SU(4)sym-metry that the “leads”to the dot are formed within the same nanotube ,and thus have the same orbital

symme-try,which should be conserved in tunneling processes [9].In nanotubes,parallel magnetic ?eld B couples to both the spin and orbital magnetic moment of electrons [15,16,17].The orbital magnetic moment μcorrespond-ing to the electron motion around the nanotube circum-ference is signi?cantly larger than the spin magnetic mo-ment μ0(we estimate μ≈7μ0,see below).Therefore,in magnetic ?eld the levels in a 4-electron shell should spilt into two doublets moving up or down in energy with B .Each doublet corresponds to an orbital with a clockwise or counterclockwise direction of rotation with respect to magnetic ?eld,occupied by two electrons (spin-up and spin-down,schematic in Figure 2a).Exactly this behav-ior is observed in Figure 2a,which shows conductance of a 200nm-long nanotube quantum dot as a function of the gate voltage and parallel magnetic ?eld.Three 4-electron shells are visible in zero ?eld,which are split into pairs of doublets in B .

Figure 2b demonstrates conduction maps measured as a function of V SD and V gate at B =3T.The Kondo ridges split at ?nite ?eld in several horizontal lines at ?nite V SD (indicated by triangles)visible inside the Coulomb diamonds.These lines mark inelastic cotun-neling thresholds;their appearance indicates that the ground state degeneracy is (partially)lifted.In the co-tunneling processes,electrons tunnel in and out of the nanotube through a virtual state,leaving behind an ex-citation [18].The energy of the excitation may be ex-tracted from the value of e |V SD |at which a cotunneling feature is observed.The enhancement of the cotunneling thresholds known as the out-of-equilibrium Kondo e?ect [19,20,21,22,23]results in the appearance of peaks in the di?erential conductance (indicated by triangles in Figure 2b).

The dependence of the co-tunneling features on mag-netic ?eld can be best traced in Figure 3,where we show the conductance measured as a function of V SD and B in the three valleys.(In each ?eld,the gate voltage is ad-justed to stay in the centers of the 1e,2e,or 3e valleys.)Let us consider the 2e valley ?rst.There are six di?erent low-energy states of the two electrons in the nanotube:three di?erent singlet states and one three-component triplet state (schematic in Figure 3b).The energy di?er-ences between these states,due to the orbital mismatch,the exchange interaction,and the excess Coulomb inter-action,are found to be very small [6].In the presence of the lifetime broadening Γ,the states shown in Figure 3b

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FIG.2:a)Di?erential conductance of a 200nm -long nan-otube quantum dot (similar to Figure 1)measured as a func-tion of V gate and B .At zero ?eld,three shells are visible.Within each shell,two lower (higher)single-electron traces move down (up)in magnetic ?eld.Each doublet corresponds to spin-up and spin-down electrons ?lling an orbital with a certain direction of rotation in magnetic ?eld.Scale of the colormap:0to 1.2e 2/h .b)Di?erential conductance map as a function V gate and V SD at B =3T.Two top shells of Figure 2a are shown.In the 2e valleys,the zero-bias Kondo ridge splits into horizontal cotunneling features at V SD ≈±2meV (indicated by red triangles),corresponding to an elec-tron excitation from the lower to the higher orbital.The 1e valleys demonstrate two Zeeman-split features at |V SD | 1meV,while the 3e valleys show a single feature close to zero bias (all indicated by green triangles).Grayscale:0.1to 1.0e 2/h .

become e?ectively degenerate and all participate in the formation of the Kondo resonance.The Kondo state ob-served here in the two-electron valley is expected [24,25]to obey the SU(4)symmetry.It should be di?erent from the two-electron singlet-triplet Kondo e?ect induced by level crossing in magnetic ?eld [26],where only four (and not six)degenerate states participate in the Kondo pro-

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[27].

As B is applied,the singlet state with two electrons occupying the lower orbital splits down from the rest (the lowest state in the schematic of Figure 3b).When the energy splitting becomes greater than the Kondo temper-ature (μB ～k B T K )at B ～1T,the zero-bias Kondo ridge disappears (Figure 3b).It is replaced by the inelas-tic cotunneling thresholds,which correspond to the exci-tation of the ground state and appear at eV SD =±2μB (one of the two electrons is moved from the lower to the higher energy orbital in the shell,see red arrows in the schematic of Figures 3b).These thresholds indeed evolve linearly with ?eld and can be used to estimate the elec-tron angular momentum as μ≈7 and the diameter of the nanotube as 2nm [12,16].

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FIG.3:Dependence of the a)1e,b)2e and c)3e Kondo features in perpendicular magnetic ?eld.Top:schematics of the di?erent states (rectangular boxes)and the allowed transitions induced between them by the tunneling processes (arrows).These states are degenerate at zero B ,but are split in the ?eld.The lower row represents the states with the lowest orbital energies.The transitions to the states with higher orbital energies are indicated by red arrows.The transitions within the lowest energy Zeeman doublets (for 1e and 3e)are shown by green arrows.Grayscale maps:conductance as a function of the V SD and B .The gate voltage was adjusted to stay in the center of a valley when the magnetic ?eld was stepped.Kondo zero-bias peak (visible at V SD =0,B =0)splits into four,two,and three features in the 1e,2e and 3e valleys respectively.The larger energy cotunneling peaks in all three images,marked by dashed lines at negative V SD (and the symmetric features at positive V SD ),correspond to the orbital splitting.In the 1e valley,the lower energy features marked by a dotted line at negative V SD (and the symmetric feature at positive V SD ),roughly correspond to the Zeeman splitting.In the 3e valley,the single peak close to zero bias marked by a dotted line survives to higher ?elds.Lower row:the same di?erential conductance data shown as a function of the V SD at di?erent B ranging from 0T to 9T in 0.25T increments (top to bottom).The curves are o?set by 0.05e 2/h per 0.25T starting from 9T.

4 In addition to the excitations at±2μB ,the1e and3e

valleys also demonstrate lower energy features:two reso-

nant cotunneling thresholds at eV SD≈±gμ0B in the1e

valley(Figure3a)and a zero-bias peak in the3e valley

(Figure3c).In both valleys,after the orbitals in the shell

are split by more than the SU(4)Kondo temperature

T SU(4) K ,there is one electron left unpaired on the lower or

the higher orbital,respectively.This electron can form

the SU(2)Kondo state as long as gμ0B k B T SU(2)

K .

Apparently this scenario is realized in the3e valley,re-sulting in the appearance of a single Kondo ridge close to zero bias(Figures2b and3c).If the SU(2)Kondo tem-perature is less than gμ0B /k B,the SU(2)Kondo state is not formed and cotunneling features at eV SD≈±gμ0B should appear,as seen in the1e valley.A

similar1e be-havior was recently observed and attributed[9,12]to the SU(4)Kondo e?ect.

We observe the di?erence between the1e and3e be-haviors in several samples and for several cool-downs.At least in two samples,we can exclude the possibility of the two orbitals in a shell having di?erentΓ,and hence di?er-

ent T SU(2)

K .The e?ect was also found in successive shells

(Figure2b),so a monotonic change of some parameter with V gate may be ruled out.Overall,while we cannot explain the e?ect,it appears to be generic.Another open question presented by Figure3a concerns the low energy cotunneling features.While at high?elds their positions eV SD are close to±gμ0B ,at low?elds they appear at energies signi?cantly exceeding±gμ0B .In particular, the peaks seem to?rst emerge from the background at

the energy of eV SD～k B T SU(4)

K (at B ≈1.5T).The or-

bital cotunneling features are already well-formed in this ?eld.We believe these observations call for theoretical interpretations.

We?nd that the odd-electron valleys also exhibit co-tunneling features whose energies decrease with?eld. These features are best visible in Figure3c,in which case the electron occupying the higher orbital of the partially ?lled shell can be excited to the lower orbital of the next, unoccupied shell.The energy of such an excitation is ??2μB ,where?is the splitting between the shells at zero?eld.This energy becomes lower in magnetic?eld, as the corresponding levels come closer.Extrapolating this energy to zero?eld,we?nd?≈10meV,consistent with other measurements.Similarly,in1e valley,an elec-tron can be excited from a lower,completely?lled shell, to the lower orbital of the partially?lled shell,which is occupied by one electron.This process also requires en-ergy of??2μB (the corresponding faint feature may be visible in Figure3a).In contrast,in a two-electron valley the inter-shell processes have energy of at least?,so that no extra low-energy co-tunneling features are observed. Finally,in Figure4we study the dependence of the Kondo conductance on magnetic?eld perpendicular to the nanotube axis,B⊥.In this orientation,the?eld pri-

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FIG.4:Conductance of a600nm-long nanotube quantum dot as a function of B⊥in the a)1e and b)2e valleys measured at di?erent magnetic?elds.B⊥ranges from0T to8T(top to bottom)in1T increments(the curves are not o?set).The vertical arrow indicates the position of the orbital SU(2)peak in the one electron valley.

marily couples to the electron spins[17].As a result, in the2e valley,the ground state becomes an electron triplet,with the two electrons occupying di?erent or-bitals.The Kondo peak is suppressed(Figure4b,B⊥～3T)and non-equilibrium Kondo(cotunneling)features appear at energies slightly above±gμ0B⊥,corresponding to the spin-?ip excitation of one of the electrons.In the 1e valley,on the other hand,the ground state remains de-generate in magnetic?eld:the electron can occupy one of two orbitals and we may expect to observe the orbital SU(2)Kondo e?ect[28].Indeed,the Kondo peak in Fig-ure4a splits three-ways:the two side peaks correspond to the spin-?ip processes,while the center peak corre-sponds to the orbital SU(2)Kondo e?ect.The orbital (pseudospin)Kondo e?ect was observed earlier in double quantum dots[29].

In conclusion,we study the transitions between the SU(4)and SU(2)Kondo in nanotube quantum dots in magnetic?eld.The two-electron Kondo e?ect is sup-pressed both by parallel and perpendicular magnetic ?elds,due to formation of a non-degenerate ground state. In parallel magnetic?eld,the odd-electron SU(4)Kondo may be completely suppressed,or turn into the SU(2) (spin)Kondo e?ect.In perpendicular magnetic?eld,the one-electron SU(4)Kondo e?ect is transformed to SU(2)

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orbital Kondo e?ect.

Acknowledgements:We thank H.Baranger,A.Chang, L.Glazman,K.Le Hur,E.Novais,K.Matveev,G.Mar-tins,M.Pustilnik,and D.Ullmo for valuable discussions. The work is supported by NSF DMR-0239748.

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