Kondo effect in carbon nanotube quantum dots with spin-orbit coupling

a r X i v :0807.3595v 1 [c o n d -m a t .s t r -e l ] 23 J u l 2008

Kondo e?ect in carbon nanotube quantum dots with spin-orbit coupling

Tie-Feng Fang,1Wei Zuo,1and Hong-Gang Luo 2,3,4

1

Institute of Modern Physics,Chinese Academy of Sciences,Lanzhou 730000,China 2

Center for Interdisciplinary Studies,Lanzhou University,Lanzhou 730000,China

3

Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education,Lanzhou University,Lanzhou 730000,China

4

Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100080,China

Motivated by recent experimental observation of spin-orbit coupling in carbon nanotube quantum dots [F.Kuemmeth et al.,Nature (London)452,448(2008)],we investigate in detail its in?uence on the Kondo e?ect.The spin-orbit coupling intrinsically lifts out the fourfold degeneracy of a single electron in the dot,thereby breaking the SU (4)symmetry and splitting the Kondo resonance even at zero magnetic ?eld.When the ?eld is applied,the Kondo resonance further splits and exhibits ?ne multipeak structures resulting from the interplay of spin-orbit coupling and Zeeman e?ect.Microscopic cotunneling process for each peak can be uniquely identi?ed.Experimental observability of these ?ne structures is brie?y discussed.Finally,a purely orbital Kondo e?ect in the two-electron regime is also obtained.

PACS numbers:73.23.-b,73.63.Fg,72.15.Qm,71.70.Ej

Introduction.—The Kondo e?ect is one of the most fas-cinating and extensively studied subjects in condensed matter physics [1].It describes a correlated many-body state formed by the exchange interaction between an im-purity spin and conduction electrons.Since the ?rst ex-perimental observation of the Kondo e?ect in semicon-ductor quantum dots (QDs)in 1998[2],after 10years of theoretical predictions [3],various aspects of this many-body e?ect have been explored in virtue of the tun-ability of relevant parameters in the QDs.The Kondo physics was further enriched by fabricating carbon nan-otube (CNT)QDs [4,5,6,7,8,9,10]where additional orbital degree of freedom originating from two electronic subbands can play a role of pseudospin.In CNT QDs,spin-orbit coupling is widely believed to be weak and two sets of spin-degenerate orbits are expected to yield a fourfold-degenerate energy spectrum which possesses an SU (4)symmetry.Consecutive ?lling of these orbits forms four-electron shells [5,6,7,8,9,10,11].In each shell,the SU (4)Kondo e?ect was observed in the valleys with one,two,and three electrons [6,7,8,9].Theoreti-cally,the SU (4)Kondo e?ect in CNT QDs has also been extensively studied [12,13,14,15]

However,recent theories [16,17]suggest that spin-orbit interaction can be signi?cant in CNTs due to their curvature and cylindrical topology.More recently,trans-port spectroscopy measurements on ultra-clean CNT QDs by Kuemmeth et al.[18]demonstrate that the spin and orbital motion of electrons are strongly coupled,thereby breaking the SU (4)symmetry of electronic states in such systems.This motivates us to reconsider the Kondo e?ect in CNT QDs by explicitly taking account of the spin-orbit coupling since this symmetry-breaking perturbation at the ?xed point must break the SU (4)Kondo e?ect studied previously.An important conse-quence of this coupling is that even at zero magnetic

?eld the Kondo e?ect manifests as split resonant peaks in the di?erential conductance.At ?nite ?elds,these peaks further split into much complicated subpeaks,re-?ecting the entangled interplay of spin and orbital de-grees of freedom.Concerning all microscopic cotunneling events involving spin or/and orbit ?ip,these ?ne multi-peak structures can be uniquely identi?ed.Moreover,the spin-orbit coupling also determines the ?lling order in the two-electron (2e )ground state [18],where we ?nd a purely orbital Kondo e?ect di?erent from that observed by Jarillo-Herrero et al.[6].

Model Hamiltonian and QD Green’s function.—We model a CNT QD coupled to a source and a drain leads by the Anderson Hamiltonian H =H d +H T +H L ,where

H d =

m

εm d ?m d m +

U

H L describes the two leads with a half bandwidth D. Electronic transport through the CNT QD is deter-mined by the dot retarded Green’s function,which is G m(ε)≡ d m|d?m =G0m(ε)(1+U m′ ?n m′d m|d?m ), where G0m(ε)= ε?εm?Σ0(ε) ?1withΣ0(ε)= k,α|Vα|2π dε′

[G0m(ε)]?1+ ′m′[Σ0(ε)A m′m?B m′m],(4) where the prime in the summation means m′=m and

1

?n m′ =?

dε′f0(ε′)[G m′(ε′)]?

π

dε′f0(ε′)1+ Σ0(ε′)G m′(ε′) ?

π

ΓLΓR

h

-20-1001020

0.5

1.0

(b)

V / T

(4)

K

d I /d V (e

2

/h )

d I /d V (

e 2

/h )

V / T

(4)

K

FIG.2:Fine structures of the Kondo resonance with spin-orbit coupling.(a)dI/dV versus V with various parallel magnetic ?elds,B/B 0=0(bottom),1/3,2/3,1,4/3,5/3(top).The red,green,and blue dashed lines are guides for the traces of peak pairs marked by the numbers 1,2,and 3,respectively.(b)dI/dV versus V with various perpendicu-lar magnetic ?elds,B/T (4)

K =0(bottom),0.1,0.2,0.4,0.7,1(top).(c)Schematic representation of three transition pro-cesses producing the peak pairs 1,2,and 3in (a),respectively.In (a)and (b),the curves corresponding to B =0are o?set for clarity.The parameters used are εd =?10Γ,μ=10,and

?SO =10T (4)

K .

[see Fig.1(a)].Though these ?ne multipeak structures appear to be complicated,the inherent physics and the B -evolution of each peak can be clari?ed by identifying all many-body cotunnelings that add up coherently to screen both spin and orbital degrees of freedom.

In the one-electron regime,the Kondo e?ect arises from the coherent superposition of six transition pro-cesses:two spin-?ip intraorbital transitions |σ,λ ?|?σ,λ ,two spin-?ip interorbital transitions |σ,λ ?|?σ,?λ ,and two spin-conserved interorbital transi-tions |σ,λ ?|σ,?λ .Each transition needs an energy of ?i (i =1,···,6)dependent on the spin-orbit coupling and the magnetic ?eld,and develops a pair of Kondo peaks at V =±?i .From energy di?erences between the initial and ?nal states,one readily obtains all six transition energies as ?1=|ε++?ε?+|=|?SO +2B |,?2=|ε+??ε??|=|?SO ?2B |,?3=|ε++?ε??|=|2(μcos θ+1)B |,?4=|ε+??ε?+|=|2(μcos θ?1)B |,?5=|ε++?ε+?|=|?SO +2μB cos θ|,and ?6=

-20-10010201.0

1.2

1.4

1.6

1.8-20-1001020

1.0

1.5

2.0

2.5

(a)(b)

V /T (4)K d I /d V (e

2

/h )

V /T

(4)K

FIG.3:SU (4)Kondo splitting without spin-orbit coupling.(a)dI/dV versus V with di?erent parallel magnetic ?elds,

B/T (4)

K =0(bottom),0.05,0.1,0.2,0.4,0.6(top).(b)dI/dV versus V with di?erent perpendicular magnetic ?elds,

B/T (4)

K =0(bottom),0.5,1,2,4,6(top).The curves corre-sponding to B =0are o?set for clarity.The parameters used are εd =?10Γ,μ=10,and ?SO =0.

|ε?+?ε??|=|?SO ?2μB cos θ|.Both the spin-orbit coupling and the magnetic ?eld are symmetry-breaking perturbations at the SU (4)Kondo ?xed point.For ?SO =B =0,the ?xed point can be reached and,all the cotunneling processes are elastic and constitute a highly symmetric SU (4)Kondo resonance at V =0(see the curve corresponding to B =0in Fig.3).On the other hand,one can uniquely identify the six transition pro-cesses from the split Kondo peaks as long as all ?i are di?erent from each other and hence the six peak pairs are well resolved,which requires ?SO =0,B =0,and cos θ=0.This is exactly the case of Fig.2(a).Thus,the six peak pairs (twelve peaks)in Fig.2(a)can be unam-biguously attributed to the six transitions,respectively.As an example,we trace the B -evolutions of three peak pairs marked by the numbers 1,2,and 3and identify the inherent transition processes as |?,+ ?|?,? ,|+,+ ?|?,+ ,and |+,+ ?|?,? ,respectively,which are schematically shown in Fig.2(c).We also note that such an unique identi?cation is unavailable in Silicon QDs [21]where no more than nine peaks are visible.When the ?eld is applied perpendicularly to the CNT,the orbital Zeeman e?ect is absent and only the spin Zeeman e?ect is involved.As shown in Fig.2(b),while the central peak splits into two subpeaks corresponding to the spin-?ip interorbital transitions,the two side peaks split into three subpeaks among which the two split-o?ones result from the spin-?ip intraorbital transitions and the rested one corresponds to the interorbital transitions without spin-?ip.In this case,some peaks are still not resolved because ?3=?4and ?5=?6.

It is useful to comment on the experimental observabil-ity of these ?ne multipeak structures.On the one hand,in our scheme the Kondo temperature,so the width of the Kondo resonance,is underestimated and relatively,

3

the splitting of the Kondo peaks is more evident.On the other hand,the decoherence [25]neglected in the present study has an e?ect to smear out the split peaks,especially those with high energy.These two features will make the observation a bit di?cult.However,in a ultra-clean CNT QD system and using high resolved spectroscopy measurement,it is still possible to observe part or all of these ?ne structures.

For comparison,we plot in Fig.3the multiple splitting of the SU (4)Kondo resonance by neglecting the spin-orbit coupling.On increasing the ?eld,the SU (4)Kondo peak at the zero bias splits in a simple way following the Zeeman e?ect,which produces characteristic structures in agreement with the results in the literature [6,8,12]but quite di?erent from that discussed above.

-20-1001020

1

2

d I /d V (

e 2

/h )

V / T

(2)

K

B=B

FIG.4:Purely orbital Kondo e?ect for the magnetic ?elds

near the degeneracy point,(B ?B 0)/T (2)

K =?0.6(bottom),?0.4,?0.2,0,0.2,0.4,0.6(top).Curves are o?set for clarity.The parameters used are ε′d =?3Γand μ=10.

The spin-orbit coupling further determines the ?lling order in the many-body ground states [18].In a parallel magnetic ?eld,the 2e ground state is |+,+ |?,? and |+,+ |?,+ for 0B 0,respectively [18].It is noted that there is always one electron occupy-ing the state |+,+ ,and the spin of another electron is ?xed (σ=?)and its orbital degree of freedom can ?uc-tuate to give rise to a purely orbital Kondo e?ect near the degenerate point B =B 0.In this case,the e?ective single-particle energy is ε?λ=ε′d +(1?λμ)(B ?B 0),

where ε′

d is a constant for th

e 2e ground state.Fig.4presents the resulting dI/dV as a function o

f V with dif-ferent magnetic ?elds.For B =B 0,there is a pronounced zero-bias peak which represents an SU (2)orbital Kondo e?ect and provides a conductin

g channel only for σ=?electrons.The peak splits due to the ?eld applied away from the degeneracy point.This orbital Kondo e?ect comes from the same shell and dwells in the 2e valley,therefore being distinct from the one recently observed in CNT QDs without spin-orbit coupling [6].

Conclusion.—We have studied the Kondo e?ect in a

CNT QD with spin-orbit coupling.It is shown that the Kondo e?ect manifests as rich ?ne multipeak structures in the di?erential conductance when a magnetic ?eld is applied.These ?ne structures are quite di?erent from the SU (4)Kondo e?ect studied previously and might be ob-servable in future experiments.In such a system,a purely orbital Kondo e?ect develops in the 2e ground state due to the particular multielectron ?lling order.Our results indicate that the spin-orbit coupling signi?cantly changes the low-energy Kondo physics in CNT QDs.

Support from the NSFC (10575119),the Major State Basic Research Developing Program (2007CB815004),and the Program for NCET of China is acknowledged.

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