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Kondo effect in carbon nanotube quantum dot in a magnetic field D. Krychowski and S. Lipinski Institute of Molecular Physics, Polish Academy of Sciences Smoluchowskiego 17, 60-179 Poznan, Poland Theout-of-equilibriumelectrontransportofcarbonnanotubesemiconductingquantumdotplaced inamagneticfieldisstudiedintheKondoregimebymeansofthenon-equilibriumGreenfunctions. The equation of motion method is used. For parallel magnetic field the Kondo peak splits in four 9 peaks,followingthesimultaneoussplittingoftheorbitalandspinstates. Forperpendicularfieldori- 0 entationthetriplepeakstructureofdensityofstatesisobservedwiththecentralpeakcorresponding 0 toorbital Kondo effect and thesatellites reflecting the spin and spin-orbital fluctuations. 2 n INTRODUCTION a J 6 Carbon nanotubes (CNTs) have emerged as a viable H= ǫkαmσc+kαmσckαmσ 2 electronic material for molecular electronic devices be- kXαmσ ] cause they display a large variety of behavior depending + tα(c+kαmσdmσ +c.c.) ll on their intrinsic properties and onthe characteristicsof kXαmσ a h their electrical contacts [1]. These systems also form the + ǫmσd+mσdmσ + Unm+nm− - powerful tool for the study of fundamental many-body kXαmσ Xm s phenomena. An example is the observed Kondo effect e in semiconducting carbon nanotube quantum dots (CN- + U12n1σn−1σ′ (1) m Xσσ′ TQD) [2,3]. The long spin lifetimes, the relatively high . t Kondo temperature and the fact that this effect can be a m seen over a very wide range of gate voltage encompass- where m = ±1 numbers the orbitals, the leads chan- ing hundreds of Coulomb oscillations [4] make CNTQDs nels are labeled by (m,α), α = L,R. ǫmσ = ǫ0 + - d interesting candidates for spintronic applications. µorbmhcos(Θ) + gσµBh, Θ specifies the orientation of n Thepurposeofthepresentworkistodiscussmagnetic magnetic field h relative to the nanotube axis, µorb is o the orbital moment. The first term of (1) describes elec- c field dependence of the Kondo conductance of CNTQD. trons in the electrodes, the second describes tunneling [ Perpendicularfieldcouplesonlytospinandparallelfield to the leads, the third represents the dot and the last influences both spin and orbital magnetic moments. For v1 vanishing magnetic field and orbitally degenerate states two terms account for intra (U) and interorbital (U12) Coulombinteractions. CurrentflowingthroughCNTQD 1 the Kondo effect appears simultaneously in spin and or- can be expressed in terms of the Green functions [6]: 8 bitalsectorsresultinginSU(4)Fermiliquidgroundstate 9 withtotallyentangledspinandorbitaldegreesoffreedom 3 [5]. Magneticfieldbreaksthe spin-orbitalsymmetry and 1. in accordance to the experiment [2,3] our calculations ıe +∞ dω < 90 show the occurrence of the multi-peak structure of the Iα = 2~Z−∞ 2π Xmσ Γαmσ(ω)·Gmσ(ω)+ :0 dspiffine-roernbtiitaallcflouncdtuucattaionnces.reflecting the spin, orbital and +Γαmσ(ω)·fα(ω)· [G+mσ(ω)−G−mσ(ω)] (2) v i X where G<,G+ and G− are lesser, retarder and advanced r Greenfunctions,respectively,fα is the Fermifunctionof a MODEL α lead and tunneling rate Γαmσ = 2π|tα|2̺αmσ, where ̺αmσ is the density of states of the leads. The total current is given by I = (IL −IR)/2. The lesser Green The low energy band structure of semiconducting car- function G< is found using Ng ansatz [7], according to bon nanotubes is orbitally doubly degenerate at zero which the lesser self-energy Σ< is proportional to the magnetic field. This degeneracy has been interpreted in self-energy of the corresponding noninteracting system a semiclassical fashion as the degeneracy between clock- Σ<(ω)=A·Σ<(ω), and A can be found by the Keldysh 0 wise and counterclockwise propagating electrons along requirement Σ<−Σ> =Σ+−Σ−. The Green functions thenanotubecircumference[1]. Inthepresentconsidera- are found by the equation of motion method using tionswerestricttothesingleshellandthedotismodeled the self-consistent decoupling procedure proposed by by double orbitalAndersonHamiltonian with additional Lacroix [8]. interorbital interaction: 2 RESULTS AND DISCUSSION The first point of our numerical analysis is addressed 2 to the experiment of Jarillo-Herrero et al. [2], in which 2 T)1 h( the conductance of CNTQD for the almost parallel field 0 orientation was examined (Θ ≃ 21◦). The calculations T) -2V(m0V) 2 were performed with Coulomb interaction parameters ( h U = U12 = 40meV, inferred from the size of Coulomb 1 diamonds. The addition energy spectrum indicates that the level spacing of examined CNTQDs ∆ǫ ≃ 4.3meV [9], what corresponds to the length of NCT L∼400nm. The estimated Kondo temperature is TK ∼ 7.7K [2]. 0 Our discussion is based on the single shell model (1) -2 0 2 with the level placed in the centre of Coulomb val- V(mV) ley (ǫ0 = −20meV). Such an oversimplified approach, which gives only a first crude insight is justified since ∆ǫ/kBTK ∼ 6.5 is large and the higher levels do not FIG. 1: Calculated differential conductance dI/dV of CN- TQD versus bias voltage V and magnetic field h in the cen- play an important role [10]. To get the experimental tre of Coulomb valley for T = 0.34K. The parameters used value of the Kondo temperature one has to assume a are: U = U12 = 40meV, Γ = 3.2meV, ǫ0 = −20meV and value of coupling to the leads Γ = 3.2meV, which is µorb = 13µB. The angle between the nanotube axis and the slightly higher than the observed broadening of atomic fieldΘ=21◦. Colorscale: 0.1to1.5e2/h. Insetshowsthecor- or Coulomb lines for NCTs examined by Jarillo-Herrero responding(V,h)conductancemapobtainedfromthedataof etal.[2,9]. Thefactthatthesingleleveldescriptionofthe Jarillo- Herrero et al.[2]. multilevelsystemsunderestimatesKondotemperature is well known in literature [10,11]. Orbital moment is estimated from the average slope between the two Coulomb 2 peaks thatcorrespondto the addition ofthe electrons to 2 h) the same orbital state and reads µorb ∼ 13µB [2]. We 2G(e /1 focus on the regime, where the quantum dot is occupied by a single electron. Fig.1apresents the calculated gray- 0 scale plot of conductance versus magnetic field and bias h) -0.25 ΔV0 .(0V0) +0.25 voltagefor T =0.34K comparedwith the corresponding 2 / 1 G e experimental plot (inset). The central bright spot of di- ( T= 0.34 K G T= 8 K mension determined by TK is the region of spin-orbital Kondo effect. For vanishing bias and magnetic field the Kondo effect appears simultaneously in spin and orbital sectors resulting in a SU(4) Fermi liquid ground state. 0 The conductance reaches in the centre a value G = -0.25 0.00 +0.25 1.3 × e2/h. Magnetic field breaks the degeneracy and ΔV (V) four high intensity lines appear. A pair of inner lines G observedforsmallbiascorrespondstoorbitalconserving fluctuations and the outer lines reflect the orbital fluc- FIG. 2: Calculated linear conductance G = dI/dV|V→0 versus gate voltage at T = 0.34K and 8 K for the CNTQD tuations and simultaneous spin and orbital fluctuations. specified by parameters as in Fig. 1a. Inset shows the corre- Thelattertwoprocessesarenotresolvedfortheassumed sponding curvesobtained from thedata reported in [2]. values of Γ and temperature. Fig. 1b presents the linear conductance versus gate voltage at T = 8K and 0.34K. ∆VG = 0 corresponds shouldersinthedensityofstatesabovetheFermiedgeon to the centre of Coulomb valley. Our calculations rea- the scale much larger than the Kondo temperature [10] sonably well reproduce the shape of the dependence but and this might lead to additional weakly bias dependent underestimate all the values of conductance roughly by contributiontotheconductance. Adetaileddiscussionof a constant value 0.5×e2/h . The source of this discrep- the mentioned point will be given in the following paper ancy is not clear, but we suggest that a possible expla- [12]. nation is a neglectof higher orbitallevels in our descrip- Now let us turn to the discussion of the influence of tion. Apart from the earlier mentioned renormalization perpendicular field, which breaks only the spin degen- ofKondotemperaturetheyalsocancauseaformationof eracy. Our numerical analysis describes the results of 3 bital Kondo effects for each spin orientations seperately. 0.12 0.12 This results in the occurrence of the central peak. For h]0.08 2e / bias voltage V = ±2(µB/e)h there occur also the satel- dV[0.04 lites induced by tunneling processes which mix different dI/ spin channels. 0.08 ] 0.00 h -2 0 2 2e / V[mV] [ V Summarizing,the presentpaperprovidesasimplepic- d / 0.04 tureoftheinfluenceofmagneticfieldontheconductance I d ofcarbonnanotubeQDsintheKondoregime. Although the experiments under consideration concern multilevel dots our calculations show that the essence of the trans- 0.00 port properties can be inferred from the effective single shell spin-orbital Kondo physics. V[mV] FIG. 3: Differential conductance of CNTQD in perpendicular magnetic field for T = 1.2K calculated for the centre of Coulomb valley. The parameters used are: U = U12 = [1] M.S. Dresselhaus, G. Dresselhaus, Ph. Avouris, Carbon 15meV, Γ = 1.25meV, ǫ0 = −8meV with asymmetry of the Nanotubes, Springer-Verlag, Berlin (2000). leads ΓL/ΓR = 6. Inset shows the corresponding curves for [2] P. Jarillo-Herrero, J. Kong, H.S.J van der Zant, C. the same values of magnetic fields obtained from the data of Dekker, L.P. Kouwenhoven, S. De Franceschi, Nature Makarowski et al. [3]. 434, 484 (2005). [3] A. Makarovski, A. Zhukov,J. Liu, G. Finkelstein, Phys. Rev. B 75, 241407 (2007). Makarovskiet al. [3] for 600nm-long nanotube quantum [4] J. Nygard, D.H. Cobden, P.E. Lindelof, Nature 408,342 (2000). dot (∆ǫ ∼ 3meV). The Coulomb parameters estimated [5] M. Choi, R. Lopez, R. Aquado, Phys. Rev. Lett. 95, again from the size of Coulomb diamonds are taken as 0672041 ( 2005). U = U12 = 15meV and the orbital level is placed in the [6] H. Haug, A.P. Jauho, Quantum Kinetics in Transport centre of Coulomb valley ǫ0 = −8meV. Γ is taken as and Optics of Semiconductors, Springer-Verlag, Berlin, the width of orbital or Coulomb peaks. Γ = 1.25meV , Heidelberg, New York (1998) together with the above parameters,well reproduces the [7] T.K. Ng, Phys.Rev.Lett. 76, 487 (1996). experimental value of TK ∼ 13K. The calculated differ- [8] C. Lacroix, J. Phys. F11, 2389 (1998). [9] P. Jarillo-Herrero, S. Sapmaz, C.Dekker, L.P. Kouven- ential conductance for several values of magnetic field is hoven, H.S.J. van derZant, Nature429, 389 (2004). comparedwithexperimentonFig.2. AquasiSU(4)type [10] D. Boese, W. Hofstetter, H. Schoeller, Phys. Rev. B 66, behavior is still observed in the low field range, what 125315 (2002). reflects in a moderate change of conductance and a sin- [11] K.Yamada,K.Yosida,K.Hanzawa,Progr.Theor.Phys. gle peak structure. For higher magnetic fields the spin- 71, 450 (1984). orbital Kondo effect SU(4) is transformed to SU(2) or- [12] S. Lipinski, D. Krychowski: (to bepublished)