Mean Field Theory of the Kondo Effect in Quantum Dots with an Even Number of Electrons Mikio Eto1,2 and Yuli V. Nazarov1 1Department of Applied Physics/DIMES, Delft University of Technology, 1 Lorentzweg 1, 2628 CJ Delft, The Netherlands 0 2Faculty of Science and Technology, Keio University, 0 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan 2 (January 9, 2001) n We investigate the enhancement of the Kondo effect in quantum dots with an even number of a electrons, using a scaling method and a mean field theory. We evaluate the Kondo temperature J TK as a function of the energy difference between spin-singlet and triplet states in the dot, ∆, and 1 the Zeeman splitting, EZ. If the Zeeman splitting is small, EZ ≪TK, the competition between the 1 singlet and triplet states enhances the Kondo effect. TK reaches its maximum around ∆ = 0 and ] decreases with ∆ obeying a power law. If the Zeeman splitting is strong, EZ ≫ TK, the Kondo l effect originates from the degeneracy between the singlet state and one of the components of the l a triplet state at −∆ ∼ EZ. We show that TK exhibits another power-law dependence on EZ. The h mean field theory provides a unified picture to illustrate the crossover between these regimes. The - enhancement of the Kondo effect can be understood in terms of the overlap between the Kondo s e resonant states created around the Fermi level. These resonant states provide the unitary limit of m the conductanceG∼2e2/h. . t 73.23.Hk, 72.15.Qm, 85.30.Vw a m - d n I. INTRODUCTION o c The Kondo effect observed in semiconductor quantum dots has attracted a lot of interest.1–5 In a quantum dot, [ the number of electrons N is fixed by the Coulomb blockade to integer values and can be tuned by the gate voltage. 1 Usually the discrete spin-degenerate levels in the quantum dot are consecutively occupied, and the total spin is zero v or 1/2 for an even and odd number of electrons, respectively. The Kondo effect takes place only in the latter case. 2 The spin 1/2 in the dot is coupled to the Fermi sea in external leads through tunnel barriers, which results in the 5 formationofthe Kondoresonantstateatthe Fermilevel.6–8 The conductancethroughthe dotis enhancedto avalue 1 of the order of e2/h at low temperatures of T T (Kondo temperature).9–13 This is called unitary limit. When N 1 ≪ K is even, there is no localized spin and thus the Kondo effect is not relevant. 0 1 Recently Sasaki et al. has found a large Kondo effect in so-called “vertical” quantum dots with an even N.14 0 The spacing of discrete levels in such dots is comparable with the strength of electron-electron Coulomb interaction. / Hence the electronic states deviate from the simple picture mentioned above.15,16 If two electrons are put into nearly t a degeneratelevels,theexchangeinteractionfavorsaspintriplet(Hund’s rule).15 Thisstateischangedtoaspinsinglet m by applying a magnetic field which increases the level spacing. Hence the energy difference between the singlet and - triplet states, ∆, can be controlled experimentally by the magnetic field. The Kondo effect is significantly enhanced d around the degeneracy point between the triplet and singlet states, ∆ = 0. Tuning of the energy difference between n the spin states is hardly possible in traditional Kondo systems of dilute magnetic impurities in metal and thus this o c situation is quite unique to the quantum dot systems. : The Kondoeffect in multilevelquantum dots has been investigatedtheoretically by severalgroups.17–20 They have v shownthat the contributionfrommultilevels enhancesthe Kondoeffect. In ourpreviouspaper,21 we haveconsidered i X theexperimentalsituationbySasakiet al.inwhichthe spin-singletandtripletstatesarealmostdegenerate. Wehave r calculated the Kondo temperature TK as a function of ∆, using the poor man’s scaling method.22–24 We have shown a that T (∆) is maximal around ∆ = 0 and decreases with increasing ∆ obeying a power law, T (∆) 1/∆γ. The K K ∝ exponent γ is not universal but depends on a ratio of the initial coupling constants. Our results indicate that the Kondo effect is enhanced by the competition between singlet and triplet states, in agreement with the experimental findings.14 We have disregardedthe Zeeman splitting of the spin-triplet state, E M (M =0, 1 is z-componentof the total Z spin S =1), since this is a smallenergy scale in the experimentalsitua−tion, E T .1±4 Pustilnik et al. and Giuliano Z K et al. have studied another situation where the Zeeman effect is relevant, E ≪T .25,26 Usually the Zeeman effect Z K ≫ lifts off the degeneracy of the spin states and, as a result, breaks the Kondo effect. They have found that the Kondo effect can arise from extra degeneracy between one of the components of the spin-triplet state, SM = 11 , and a | i | i 1 singlet state, 00 , if the value of ∆ is tuned to fulfill that E = ∆. Their mechanism might explain some other Z experimental r|esuilts of the Kondo effect in quantum dots under hig−h magnetic fields.4,27 The purpose of the present paper is to construct a general theory for the enhancement of the Kondo effect in quantum dots with an even number of electrons, for various values of ∆ and E . We adopt the poor man’s scaling Z method along with the mean field theory. It is well known that the characteristic energy scale of the Kondo physics, theKondotemperatureT ,isdeterminedbyalltheenergiesfromT uptotheuppercutoff.7,8Bythescalingmethod, K K we can evaluate T (its exponentialpartat least)by taking allthe energies properly.22–24 When E is negligible, the K Z energies from ∆ to the upper cutoff would feel fourfold degeneracy of the dot states, 1M (M = 0, 1) and 00 , | i ± | i which enhances the Kondo temperature. With increasing ∆, T decreases by a power law.21 We extend our previous K calculations to the case of E = ∆ T which has been discussed by Pustilnik et al.25 and Giuliano et al.26 We Z K − ≫ take into account the energies not only from T to E , where only two degenerate states 11 and 00 are relevant, K Z | i | i but also from E to the upper cutoff, where the dot states seem fourfold degenerate. The latter energy region has Z been neglected in Refs. 25 and 26. In consequence we find a power law dependence of T on E again. K Z The meanfieldtheoryofthe Kondoeffectwaspioneeredby YoshimoriandSakurai28 andiscommonlyusedforthe Kondo lattice model.29 It is useful to capture main qualitative features of the Kondo effect; renormalizability at the scaleofT ,resonancesattheFermilevel,andresonanttransmission. Thesimplicityanduniversalityofthemeanfield K theory have driven us to apply it to the problem in question. Generally the Kondo effect gives rise to a many-body groundstate whichconsists of the dotstates SM =f† 0 andthe conductionelectrons Πc† 0 . The total spinof | i SM| i kσ| i this groundstateis less thanthe originalspin S localizedinthe dot. The binding energyis ofthe orderoftheKondo temperatureT . We takeinto accountthe spincouplingsbetweenthe dotstatesandconductionelectrons, f† c , K h SM kσi by the mean field, neglecting their fluctuations.30 These spin couplings give rise to resonant states around the Fermi level µ with the width of the order of T . The conduction electrons can be transported through the resonant levels, K which yields the unitary limit of the conductance G 2e2/h. For our study, the mean field calculations have the ∼ following advantages. (i) The enhancement of T by the competition between the singlet and triplet states can be K directlyunderstoodintermsoftheoverlapbetweentheirKondoresonantstates. (ii)ThepowerlawdependenceofT K on∆orE isobtained,whichis inaccordancewiththe calculatedresultsbythe scalingmethod. (iii)The meanfield Z calculations are applicable to the intermediate regions where two of T , ∆, and E , are of the same order. The poor K Z man’s scaling method hardly gives any results in these regions. Hence we canexamine the whole parameter regionof ∆andE by the meanfieldtheory. The disadvantageofthe meanfieldcalculationsis that they only givequalitative Z answers.30 Hence the mean field theory and scaling method are complementary to each other for understanding the Kondo effect. WeshalldiscusstherelationofourapproachtotherenormalizationgroupanalysisofthemultilevelKondoeffect.23,24 Ourmodeleffectivelyreducesto the onewithtwochannelsinthe leadsandspin-triplet(andsinglet)stateinthe dot. The ground state of this model would be believed to be a spin singlet, which corresponds to the full screening of the dot spin. The poor man’s scaling approach and our mean field theory, however, show a tendency to the formation of the underscreened Kondo ground state with spin 1/2. We should mention that the exact ground state can not be determined within the limits of the applicability of these approaches. Pustilnik and Glazman have recently proposed a different model for the “triplet-singlet Kondo effect.”31 In our notations, they set C = √2,C = 0 in Eq. (4) for 1 2 the singlet state. Their model can be directly mapped onto a special case of the two-impurity Kondo model,32 for whichthe groundstate is a spin singlet. We are concernedaboutthe caseof C C , andwe find that the difference 1 2 between C and C reduces as a result of the renormalization.21 This suggests t≈hat the case considered in Ref. 31 is 1 2 by no means a generic one. This paper is organized as follows. Our model is presented in the next section. In section III, we rederive T (∆) K when the Zeeman splitting is irrelevant, using the poor man’s scaling method, in a simpler form than our previous work.21 Then we extend our calculations to the case of E = ∆ T . The section IV is devoted to the mean field Z K − ≫ theory for the Kondo effect in quantum dots. First we explain this theory for the usual Kondo effect in a quantum dot with S = 1/2. Then we apply the mean field scheme to our model with an even number of electrons in the dot. The conclusions and discussion are given in the last section. II. MODEL We are interested in the competition between the spin-singlet and triplet states in a quantum dot. To model the situation, it is sufficient to consider two extra electrons in a quantum dot at the background of a singlet state of all otherN 2electrons,whichwewillregardasthevacuum 0 . Thesetwoextraelectronsoccupytwolevelsofdifferent orbitalsy−mmetry.33 The energiesofthe levels areε andε|.iPossibletwo-electronstates are(i) the spin-tripletstate, 1 2 (ii) the spin-singlet state of the same orbital symmetry as the triplet state, 1/√2(d† d† d† d† )0 , and (iii) two 1↑ 2↓ − 1↓ 2↑ | i 2 singlet states of different orbital symmetry, d† d† 0 , d† d† 0 . Among the singlet states, we only consider a state 1↑ 1↓| i 2↑ 2↓| i of the lowest energy, which belongs to the group (iii). Thus we restrict our attention to four states, SM : | i 11 =d† d† 0 (1) | i 1↑ 2↑| i 1 10 = (d† d† +d† d† )0 (2) | i √2 1↑ 2↓ 1↓ 2↑ | i 1 1 =d† d† 0 (3) | − i 1↓ 2↓| i 1 00 = (C d† d† C d† d† )0 , (4) | i √2 1 1↑ 1↓− 2 2↑ 2↓ | i where d† creates an electron with spin σ in level i. The coefficients in the singlet state, C , C (C 2+ C 2 = 2), iσ 1 2 | 1| | 2| are determined by the electron-electron interaction and one-electron level spacing δ =ε ε . We set C =C = 1. 2 1 1 2 This is the case for δ =0.34 Although C =C in general, the scaling analysis shows that−the Kondo temperature is 1 2 6 the same as that in the case of C =C =1, apart from a prefactor.21 The energies of the triplet state are given by 1 2 E =E E M (5) S=1,M S=1 Z − and the energy of the singlet state is denoted by E . We define ∆ by 00 ∆=E E . (6) 00 S=1 − The energy diagram for the spin states is indicated in Fig. 1(a). The dot is connected to two external leads L, R with free electrons being described by H = ε(i)c(i)† c(i) , leads k α,kσ α,kσ α=L,Rkσi X X where c(i)† (c(i) ) is the creation (annihilation) operator of an electron in lead α with momentum k, spin σ, and α,kσ α,kσ orbitalsymmetryi(=1,2). Thedensityofstatesν intheleadsremainsconstantintheenergybandof[ D,D]. The − tunneling between the dot and the leads is written as H = (V c(i)† d +H.c.). T α,i α,kσ iσ α=L,Rkσi X X We assume thatthe orbitalsymmetry is conservedinthe tunneling processes.33 To avoidthe complicationdue to the factthat there aretwo leadsα=L,R,weperforma unitary transformationfor electronmodes inthe leads alongthe lines ofRef.9; c(i) =(V∗ c(i) +V∗ c(i) )/V ,c¯(i) =( V c(i) +V c(i) )/V , with V = V 2+ V 2. The kσ L,i L,kσ R,i R,kσ i kσ − R,i L,kσ L,i R,kσ i i | L,i| | R,i| modes c¯k(iσ) are notcoupled to the quantum dotand shallbe disregardedhereafter. ThenHleadps andHT are rewritten as H = ε(i)c(i)†c(i), (7) leads k kσ kσ kσi X H = V (c(i)†d +H.c.). (8) T i kσ iσ kσi X We assume that the state of the dot with N electrons is stable, so that addition/extraction energies, E± ≡ E(N 1) E(N) µ where µ is the Fermi energy in the leads, are positive. We are interested in the case where ± − ∓ E± ∆, δ and also exceed the level broadening Γi = πνV2 and temperature T (Coulomb blockade region). In ≫ | | i this case we can integrate out the states with one or three extra electrons. This is equivalent to Schrieffer-Wolff transformationwhichisusedtoobtaintheconventionalKondomodel.7,8 Weobtainthe followingeffectivelow-energy Hamiltonian H =H +H +HS=1+HS=1↔0+H′ . (9) eff leads dot eff The Hamiltonian of the dot H reads dot H = E f† f , (10) dot SM SM SM S,M X 3 using pseudo-fermion operators f† (f ) which create (annihilate) the state SM . The condition of SM SM | i f† f =1 (11) SM SM SM X shouldbefulfilled. ThethirdtermHS=1 representsthespinflipprocessesamongthreecomponentsofthespin-triplet state. This resembles a conventional Kondo Hamiltonian for S =1 in terms of the spin operator Sˆ HS=1 = J(i) Sˆ c(i)†c(i)+Sˆ c(i)†c(i)+Sˆ (c(i)†c(i) c(i)†c(i)) + k′↓ k↑ − k′↑ k↓ z k′↑ k↑ − k′↓ k↓ Xkk′ iX=1,2 h i = J(i) √2(f† f +f† f )c(i)†c(i)+√2(f† f +f† f )c(i)†c(i) 11 10 10 1−1 k′↓ k↑ 10 11 1−1 10 k′↑ k↓ Xkk′ iX=1,2 h +(f† f f† f )(c(i)†c(i) c(i)†c(i)) . (12) 11 11− 1−1 1−1 k′↑ k↑ − k′↓ k↓ i The exchange coupling J(i) is accompanied by the scattering of conduction electrons of channel i. The fourth term HS=1↔0 in H describes the conversionbetween the spin-triplet and singlet states accompaniedby the interchannel eff scattering of conduction electrons HS=1↔0 = J˜ √2(f† f f† f )c(1)†c(2)+√2(f† f f† f )c(1)†c(2) 11 00− 00 1−1 k′↓ k↑ 00 11− 1−1 00 k′↑ k↓ Xkk′ h (f† f +f† f )(c(1)†c(2) c(1)†c(2))+(1 2) . (13) − 10 00 00 10 k′↑ k↑ − k′↓ k↓ ↔ i The coupling constants are given by V2 J(i) = i , (14) 2E c V V J˜= 1 2, (15) 2E c where 1/E = 1/E+ +1/E−. Note that J˜2 = J(1)J(2). The last term H′ represents the scattering processes of c eff conduction electrons without any change of the dot state and is not relevant for the current discussion. The spin-flip processes included in our model are shown in Fig. 1(b). III. SCALING CALCULATIONS In this section we calculate the Kondo temperature T using the poor man’s scaling technique.22–24 By changing K the energy scale (bandwidth of the conduction electrons) from D to D dD , we obtain the scaling equations using −| | the second-orderperturbationcalculationswithrespecttothe exchangecouplings,J(1),J(2),andJ˜. Withdecreasing D, the exchange couplings are renormalized. The Kondo temperature is determined as the energy scale at which the exchange couplings become so large that the perturbation breaks down. A. In the absence of Zeeman effect When the Zeeman splitting is small and irrelevant, E T , we obtain a closed form of the scaling equations for Z K J(1), J(2), andJ˜in two limits.21 (i) When the energyscale≪D is muchlargerthan the energy difference ∆, H can dot | | be safely disregarded in H . The scaling equations can be written as eff d J(1) J˜ J(1) J˜ 2 = 2ν . (16) dlnD J˜ J(2) − J˜ J(2) (cid:18) (cid:19) (cid:18) (cid:19) (ii) For D ∆, the ground state of the dot is a spin triplet and the singlet state can be disregarded. Then J(1) and ≪ J(2) evolve independently 4 d J(i) = 2νJ(i)2, (17) dlnD − whereas J˜does not change. In the case of ∆ T , the scaling equations (16) remain valid till the scaling ends. The matrix in Eq. (16) has K | | ≪ eigenvalues of J =(J(1)+J(2))/2 (J(1) J(2))2/4+J˜2 ± ± − =J(1)+J(2),0. q (18) The larger one, J , diverges upon decreasing the bandwidth D and determines T + K T (0)=D exp[ 1/2νJ ] K 0 + − =D exp[ 1/2ν(J(1)+J(2))]. (19) 0 − Here D is the initial bandwidth, which is given by √E+E−.35 0 When ∆>D , the scaling equations (17) work in the whole scaling region. This yields 0 T ( )=D exp[ 1/2νJ(1)] (20) K 0 ∞ − for J(1) J(2). This is the Kondo temperature for spin-triplet localized spins.36 ≥ In the intermediate region of T (0) ∆ D , the exchange couplings develop by Eq. (16) for D ∆. Around K 0 ≪ ≪ ≫ D =∆,J˜saturateswhileJ(1) andJ(2) continuetogrowwithdecreasingD,followingEq.(17)forD ∆. Wematch ≪ the solutions of these scaling equations at D ∆ and obtain a power law of T (∆) K ≃ T (∆)=T (0) (T (0)/∆)tan2θ, (21) K K K · with tanθ =J˜/[ (J(1) J(2))2/4+J˜2+(J(1) J(2))/2] − − q = J(2)/J(1) (22) q for J(1) J(2). Here (cosθ,sinθ)T is the eigenfunction of the matrix in Eq. (16) corresponding to J . θ 0 for + J(1) J≥(2) and θ =π/4 for J(1) =J(2). In general, 0<θ π/4 and thus 0<tan2θ 1. ∼ ≫ ≤ ≤ Finally, for ∆ < 0, all the coupling constants saturate and no Kondo effect is expected, provided ∆ T (0). K | | ≫ ThusT quicklydecreasestozeroat∆ T (0). TheKondotemperatureasafunctionof∆isschematicallyshown K K ∼− in Fig. 2(a). B. Case of EZ =−∆ When E = ∆, the energies of states 00 and 11 are degenerate. Then the Kondo effect is expected even Z when ∆ T (−0).25,26 In this subsection w|e evialuate|Ti in this special case of E = ∆ by the poor man’s scaling K K Z | |≫ − method. (i) For the energy scale of D ∆ = E , H can be disregarded in H . The exchange couplings, J(1), J(2), Z dot eff ≫ | | and J˜, evolve following Eq.(16). (ii) In another limit of D ∆ =E , only the states 00 and 11 are relevant. In Z ≪| | | i | i H , eff 1 1 H|00i,|11i = J(i)(f† f f† f )(c(i)†c(i) c(i)†c(i))+ J(i)(f† f +f† f )(c(i)†c(i) c(i)†c(i)) eff 2 s 11 11− 00 00 k′↑ k↑ − k′↓ k↓ 2 c 11 11 00 00 k′↑ k↑ − k′↓ k↓ Xkk′ iX=1,2h i + √2J˜ f† f c(1)†c(2)+f† f c(1)†c(2)+(1 2) . (23) 11 00 k′↓ k↑ 00 11 k′↑ k↓ ↔ Xkk′ h i J(i) =J(i) =J(i) initially. The scaling procedure yields s c 5 d J(i) = 4νJ˜2 dlnD s − (24) ( dlndDJ˜=−ν(Js(1)+Js(2))J˜, and J(i) do not change. These scaling equations are nearly equivalent to those of the anisotropic Kondo model with c S =1/2,22 as pointed out in Refs. 25,26. When ∆ = E > D , the scaling equations (24) remain valid in the whole scaling region. This yields the Kondo Z 0 | | temperature T ( )=D exp[ A(θ)/2ν(J(1)+J(2))] (25) K 0 ∞ − with 1 ln 1+λ (0<θ π/8) A(θ)= λ 1−λ ≤ (26) ( 2 tan(cid:16)−1λ(cid:17) (π/8<θ π/4), λ ≤ where λ = cos4θ . A(θ) decreases monotonically with increasing θ. A(θ) as θ 0. A(π/8) = 2 and A(π/4) = π/2.| When| J(1) +J(2) is fixed, T ( ) is the largest at J(1) = J(2) (θ→=∞π/4) an→d becomes smaller with decreasing Jp(2)/J(1)(=tan2θ). K ∞ In the intermediate region,T (0) ∆ =E D , we matchthe solutionsof Eqs.(16)and(24)at D ∆. We K Z 0 ≪| | ≪ ≃| | obtain a power law T (∆)=T (0) (T (0)/∆)A(θ)−1. (27) K K K · | | Figure 2(b) shows the behaviors of T (∆) in the case of E = ∆. K Z − IV. MEAN FIELD CALCULATIONS A. Kondo resonance for spin S =1/2 To illustrate the mean field theory for the Kondo effect in quantum dots, we begin with the usual case of S =1/2. We assume that one level (E ) in a quantum dot is occupied by an electron with spin either up or down (σ = , ). 0 ↑ ↓ The effective low-energy Hamiltonian is H = εkc†kσckσ + Eσfσ†fσ+J fσ†fσ′c†k′σ′ckσ, (28) kσ σ kk′ σ,σ′ X X XX with the constraint of f†f +f†f =1. (29) ↑ ↑ ↓ ↓ ForelectronsinleadsL,R,wehaveperformedaunitarytransformationofc =(V∗c +V∗c )/ V 2+ V 2 where V is the tunneling coupling to lead α.9 The last term in Eq. (28) rkeσpresenLtsLth,keσexcRhanRg,keσcoup|linLg| bet|weRe|n α p S =1/2 in the dot and conduction electrons (Appendix A). In the mean field theory, we introduce the order parameter 1 Ξ = ( f†c + f†c ) (30) h i √2 h ↑ k↑i h ↓ k↓i k X to describe the spin couplings between the dot states and conduction electrons. The mean field Hamiltonian reads H = ε c† c + E f†f (√2J Ξ c† f +H.c.)+2J Ξ 2+λ f†f 1 . (31) MF k kσ kσ σ σ σ− h i kσ σ |h i| σ σ− ! kσ σ k,σ σ X X X X The constraint, Eq. (29), is taken into account by the last term with a Lagrange multiplier λ. By minimizing the expectation value of H , Ξ is determined self-consistently (see Appendix A). MF h i 6 In the absence of the Zeeman effect, E = E = E . The mean field Hamiltonian, H , represents a resonant ↑ ↓ 0 MF tunneling through an “energy level,” E˜ = E +λ, with “tunneling coupling,” V˜ = √2J Ξ . V˜ provides a finite 0 0 − h i widthoftheresonance,∆˜ =πν V˜ 2,withν beingthedensityofstatesintheleads. Theconstraint,Eq.(29),requires 0 | | that the states for the pseudo-fermions are half-filled, that is, E˜ =µ. Hence the Kondo resonant state appears just 0 at the Fermi level µ, as indicated in the inset (A) in Fig. 3(a). The self-consistent calculations give us the resonant width 2 ∆˜ =πν √2J Ξ =D exp[ 1/2νJ]. (32) 0 0 h i − (cid:12) (cid:12) This is identical to the Kondo temperature T(cid:12). (cid:12) K(cid:12) (cid:12) In the presence of the Zeeman splitting, E = E E and E = E +E . Hence the resonant level is split for ↑ 0 Z ↓ 0 Z − spin-upanddownelectrons,E˜ =E +λ. The constraint,Eq.(29), yields E +λ=µ(see inset(B) inFig.3(a)). ↑/↓ ↑/↓ 0 The resonant width ∆˜ is determined as ∆˜2+E2 =∆˜2, (33) Z 0 where ∆˜ is given by Eq. (32). The Kondo temperature is evaluated by this width, T (E )=∆˜. T decreases with 0 K Z K increasing E and disappears at E =T (0), as shown in Fig. 3(a). Z Z K The conductance G through the dot is expressed, using Γ =πν V 2, as α α | | 2e2 4Γ Γ E 2 L R Z G= 1 . (34) h (ΓL+ΓR)2 " −(cid:18)TK(0)(cid:19) # This is the conductance in the unitary limit for E =0. Figure 3(b) presents the E dependence of the conductance. Z Z With increasing E , the splitting between the resonant levels for spin up and down becomes larger. In consequence Z the amplitude of the Kondo resonance decreases at µ, which reduces the conductance. B. Kondo resonance in the present model Now we apply the mean field theory to our model which has the spin-triplet and singlet states in a quantum dot. The spin states of the coupling to a conduction electron are (S = 1) (S = 1/2) = (S = 3/2) (S = 1/2) for the ⊗ ⊕ former, and (S =0) (S =1/2)=(S =1/2) for the latter (Appendix B). To represent the competition between the ⊗ triplet and single states, therefore, the order parameter should be a spinor of S =1/2. It is Ξ~ where h i cosϕ √2f† c(1)+f† c(1) /√3+sinϕf† c(2) ~Ξ= 11 k↑ 10 k↓ 00 k↓ (35) cosϕ (cid:16)√2f† c(1)+f† c(1)(cid:17) /√3 sinϕf† c(2)  Xk 1−1 k↓ 10 k↑ − 00 k↑  (cid:16) (cid:17)  for J(1) >J(2). A mode of the largest coupling is taken into account in this approximation. The Hamiltonian reads H =H +H J Ξ~† Ξ~ +Ξ~† Ξ~ Ξ~ 2 +λ f† f 1 , (36) MF lead dot− MF h i h i−|h i| SM SM − ! h i XSM where J =J(1)+ J(1)2+3J˜2, (37) MF p and tanϕ=√3J˜/J . (38) MF The last term in H considers the restriction of Eq. (11). The expectation value of H is minimized with respect MF MF to Ξ~ 2. The Kondo temperature can be estimated by | | T =πν J Ξ~ 2, (39) K MF | h i| 7 using Ξ~ determined by the self-consistent calculations (Appendix B). h i Firstletusconsiderthe caseinthe absenceofthe Zeemaneffect, E =E andE =E +∆. The resonant 1M S=1 00 S=1 level for the triplet state is threefold degenerate at E˜ =E +λ whereas the resonant level for the singlet state S=1 S=1 is at E˜ = E +λ. These levels are separated by the energy ∆. The Lagrange multiplier λ is determined to fulfill 0 00 Eq. (11). Figure 4(a) shows the calculated results of T as a function of ∆. Both of T and ∆ are in units of K K D exp( 1/νJ ). We find that (i) T (∆) reaches its maximum at ∆=0, (ii) for ∆ T (0), T (∆) obeys a power 0 MF K K K − ≫ law T (∆) ∆tan2ϕ =const., (40) K · and (iii) for ∆<0, T decreases rapidly with increasing ∆ and disappears at ∆=∆ T (0); K c K | | ∼− ∆ = D exp( 1/νJ )(1+tan2ϕ)(tan2ϕ)−sin2ϕ. (41) c 0 MF − − These features are in agreement with the results of the scaling calculations. The behaviors of T (∆) can be understood as follows. The inset of Fig. 4(a) schematically shows the Kondo K resonant states. The resonance of the triplet state is denoted by solid lines whereas that of the singlet state is by dottedlines. (A)When∆ T (0),thetripletresonanceappearsaroundµwhereasthesingletresonanceisfarabove K ≫ µ. (B) With a decrease in ∆, the two resonant states are more overlapped at µ, which raises T gradually. This K resultsinapowerlawofT (∆), Eq.(40). ThelargestoverlapyieldsthemaximumofT at∆=0. (C)When∆<0, K K the singlet and triplet resonances are located below and above µ, respectively, being sharper and farther from each other with increasing ∆. Finally the Kondo resonance disappears at ∆=∆ . c | | The conductance through the dot is given by 4Γ1Γ1 ∆˜2 ∆˜2 4Γ2Γ2 ∆˜2 G/(e2/h)= L R 11 + 10 + L R 00 , (42) (Γ1L+Γ1R)2 (ε−E˜11)2+∆˜211 (ε−E˜10)2+∆˜210! (Γ2L+Γ2R)2(ε−E˜00)2+∆˜200(cid:12)(cid:12)ε=µ (cid:12) (cid:12) where Γi =πν V 2. The resonantwidths are ∆˜ /∆˜ =2cos2ϕ/3,∆˜ /∆˜ =cos2ϕ/3, and∆˜ /∆˜ (cid:12)=sin2ϕ with α | α,i| 11 0 10 0 00 0 ∆˜ = πν J Ξ~ 2. The conductance G as a function of ∆ is shown in Fig. 4(b), in a symmetric case of Γi = Γi 0 | MFh i| L R (i = 1,2). G = 2e2/h for ∆ > 0 whereas G goes to zero suddenly for ∆ < 0. Around ∆ = 0, G is larger than the value in the unitary limit, 2e2/h, which is attributable to nonuniversal contribution from the multichannel nature of our model.21 In the presence of the Zeeman splitting, E = E E M, the resonant level of the triplet state is split into 1M S=1 Z − three. With increasing E , the Kondo effect is rapidly weaken except in the region of ∆ E . In Fig. 5(a), we Z Z ∼ − show the Kondo temperature T in E -∆ plane, in the case of ϕ = 0.15π. Figure 5(b) presents T as a function of K Z K ∆ for several values of E . When E is large enough, the Kondo effect takes place only when the resonant state of Z Z 11 is overlapped with that of 00 . Then T is the largest at ∆= E and decreases with ∆ being away from this K Z | i | i − value. At ∆= E , T obeys a power law Z K − T (∆) ∆1/(2+3tan2ϕ) =const., (43) K ·| | which is indicated by a broken line in Fig. 5(b). This is qualitatively in agreement with the calculated results by the scaling method. Figure 6 indicates the conductance G in E -∆ plane, when ϕ = 0.15π and Γi = Γi (i = 1,2). G takes the value Z L R of 2e2/h around E = 0 and ∆ > 0, and also along the line of E = ∆. (G > 2e2/h in the neighborhood of Z Z − E = ∆=0, as discussed above.) For sufficiently large E , our model is nearly equivalent to the anisotropic Kondo Z Z model with S = 1/2.25,26 Hence G = 2e2/h at ∆ = E and reduces to zero as ∆ deviates from this value, in the Z − same way as in Fig. 3(b) for the case of S =1/2. V. CONCLUSIONS AND DISCUSSION TheKondoeffectinquantumdotswithanevennumberofelectronshasbeeninvestigatedtheoretically. TheKondo temperature T has been calculated as a function of the energy difference ∆=E E and the Zeeman splitting K 00 S=1 − E , using the poor man’s scaling method and mean field theory. The scaling calculations have indicated that the Z competition between the spin-triplet and singlet states significantly enhances the Kondo effect. When the Zeeman effect is irrelevant, E T , T is maximal around ∆ = 0 and decreases with ∆ obeying a power law. In a case of Z K K ≪ ∆=E , the Kondoeffect takes place from the degeneracybetween two states, 00 and 11 . Evenin this case, the Z − | i | i 8 contribution from the other states of higher energy, 10 and 1 1 , plays an important role in the enhancement of | i | − i T . As a result, T is maximal around E =0 and depends on E by a power law again. K K Z Z The meanfieldtheory yieldsa clearcutview for the Kondoeffect inquantumdots. Consideringthe spincouplings between the dot states and conduction electrons as a mean field, f† c(i) , we find that the resonant states are h SM k,σi created around the Fermi level µ. The resonant width is given by the Kondo temperature T . The unitary limit of K the conductance, G 2e2/h, can be easily understood in terms of the tunneling through these resonant states. In ∼ our model, the overlap between the resonant states of S = 1 and S = 0 in the dot enhances the Kondo effect. The mean field calculations have led to a power law dependence of T on ∆ and on E , in accordance with the scaling K Z calculations. The mean field theory is not quantitatively accurate for the evaluation of T .30 (In the case of S =1/2, the exact K value of T is obtained accidentally.) In our model, the scaling calculations indicate that all the exchange couplings, K J(1), J(2), and J˜, are renormalized altogether following Eq. (16) when ∆ and E are much smaller than the energy Z | | scale D. In consequence two channels in the leads are coupled effectively for an increase in T . In the mean field K calculations,theinterchannelcouplingsaretakenintoaccountinEq.(37)onlypartly. Infact,conductionelectronsof channel 1 and 2 independently take partin the conductance, Eq. (42). By the perturbationcalculations with respect to the exchange couplings, we find that mixing terms between the channels appear in the logarithmic corrections to the conductance.21 We could improve the mean field calculations by adopting another form of the order parameter than Eq. (35). Our calculated results explain the experimental findings by Sasaki et al.:14 The Kondo effect is largely enhanced around ∆ = 0 when the Zeeman effect is irrelevant. The behavior of T in the presence of the Zeeman effect may K be observed experimentally under higher magnetic fields. In experiments the value of the Zeeman splitting can be controlled by applying a magnetic field parallel to the quantum dot. More generally, it is possible to control several parametersinsemiconductorquantumdots andto realizenewsituationswhichcannotbe reachedin traditionalsolid state context. The quantum dot systems, therefore, have the potential of tools to explore the Kondo physics further beyond the present theory. ACKNOWLEDGMENTS The authors are indebted to L. P. Kouwenhoven, S. De Franceschi, J. M. Elzerman, K. Maijala, S. Sasaki, W. G. van der Wiel, Y. Tokura, L. I. Glazman, M. Pustilnik, and G. E. W. Bauer for valuable discussions. The authors acknowledge financial support from the “Netherlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO). M. E. is also grateful for financial support from the Japan Society for the Promotion of Science for his stay at Delft University of Technology. APPENDIX A: MEAN FIELD CALCULATIONS FOR S =1/2 The original Hamiltonian for a quantum dot with one energy level reads H = ε c† c + (V c† d +H.c.)+H (A1) k α,kσ α,kσ α α,kσ σ dot α=L,R kσ α=L,R kσ X X X X with H = E d†d +Ud†d d†d . (A2) dot 0 σ σ ↑ ↑ ↓ ↓ σ X Forthestateofoneelectroninthedot,the additionandextractionenergiesaregivenbyE+ =E +U µandE− = 0 − µ E ,respectively. Theparameters,E andU,inEq.(A2)shouldbedeterminedtofittheseenergiestoexperimental 0 0 da−ta. For conduction electrons in leads L, R, we perform a unitary transformation, c = (V∗c +V∗c )/V, kσ L L,kσ R R,kσ c¯ = ( V c +V c )/V, with V = V 2+ V 2, along the lines of Ref. 9. We disregard the modes c¯ kσ R,i L,kσ L,i R,kσ L R kσ − | | | | which are uncoupled to the quantum dot. p We consider the Coulomb blockade region for one electron, where both E+ and E− are much larger than the level broadening Γ = πνV2 (ν being the density of states in the leads) and temperature. Integrating out the dot states with zero or two electrons by the Schrieffer-Wolff transformation,7,8 we obtain the effective low-energy Hamiltonian H = ε c† c + E f†f +J Sˆ c† c +Sˆ c† c +Sˆ (c† c c† c ) (A3) k kσ kσ σ σ σ + k′↓ k↑ − k′↑ k↓ z k′↑ k↑− k′↓ k↓ Xkσ Xσ Xkk′ h i 9 under a constraint of Eq. (29). In the second term we have included the Zeeman effect, E = E E . The ↑,↓ 0 Z third term represents the exchange coupling between the dot spin and conduction electrons with J = V±2/E where c 1/E =1/E++1/E−. By expressing the spin operator Sˆ as Sˆ =f†f , Sˆ =f†f , Sˆ =(f†f f†f )/2, one finds c + ↑ ↓ − ↓ ↑ z ↑ ↑− ↓ ↓ that Eq. (A3) is identical to Eq. (28). ThemeanfieldHamiltonian,Eq.(31),includes“energylevels”forpseudo-fermions,E˜ =E +λ,whicharecoupled σ σ to the leads with “tunneling amplitude,” V˜ = √2J Ξ . The Green function for the pseudo-fermions is − h i 1 G (ε)= , (A4) σ ε E˜ +i∆˜ σ − where ∆˜ =πν V˜ 2. This represents the resonant tunneling with the resonant width ∆˜. | | The expectation value of the Hamiltonian, Eq. (31), is written as ∆˜ E˜ ∆˜ ∆˜ E˜2+∆˜2 ∆˜ E = + σ tan−1 + ln σ λ+ , (A5) MF σ "−π π E˜σ 2π D02 #− πνJ X where D is the bandwidth in the leads.7 We set µ=0 in this appendix. The constraint of Eq. (29) is equivalent to 0 the condition ∂E 1 ∆˜ MF = tan−1 1=0. (A6) ∂λ π E˜ − σ σ X This yields E +λ=0. The minimization of E with respect to ∆˜ (or Ξ 2) determines ∆˜ 0 MF |h i| ∂E 1 E˜2 +∆˜2 1 MF = ln σ + =0. (A7) ∂∆˜ 2π D2 πνJ σ 0 X For E =0, we find Z ∆˜ =D exp[ 1/2νJ] ∆˜ . (A8) 0 0 − ≡ This is equal to the Kondo temperature, T . For E =0, Eq. (A7) yields K Z 6 ∆˜2+E2 =∆˜2. (A9) Z 0 Using the T-matrix, Tˆ, the conductance through the dot, G, is given by e2 G= (2πν)2 R,k′σ Tˆ L,kσ 2 h Xσ |h | | i| (cid:12)εk=εk′=µ = eh2(2πν)2(|V|LV|L2|+2|V|VRR|2|2)2 Xσ |hψ(cid:12)(cid:12)k′σ|Tˆ|ψkσi|2(cid:12)εk=εk′=µ (cid:12) e2 4Γ Γ ∆˜2 (cid:12) L R = (A10) h (ΓL+ΓR)2 Xσ (ε−E˜σ)2+∆˜2(cid:12)(cid:12)ε=µ (cid:12) (cid:12) whereΓ =πν V 2. This yieldsEq.(34)inthe text. Onthe second(cid:12)line inEq.(A10), ψ =c† 0 =(V L,kσ + α | α| | kσi kσ| i L| i V R,kσ )/V, and the T-matrix is evaluated in terms of the Green function, Eq. (A4), V˜ 2G (ε=ε ). R σ k | i | | APPENDIX B: MEAN FIELD CALCULATIONS IN THE PRESENT MODEL ForthespinstatesofthecouplingbetweenthespintripletS =1inthedotandaconductionelectron,weintroduce spinors of S =1/2 and 3/2. Using the Clebsch-Gordan coefficients, they are given by 10