PHYSICAL REVIEW B 85, 165401 (2012)

Kondo temperature and screening extension in a double quantum dot system

L. C. Ribeiro,1 E. Vernek,2 G. B. Martins,3,* and E. V. Anda4

1Centro Federal de Educacao Tecnologica Celso Suckow da Fonseca (CEFET-RJ/UnED-NI), RJ 26041-271, Brazil2Instituto de Fısica, Universidade Federal de Uberlandia, Uberlandia 38400-902, MG, Brazil

3Department of Physics, Oakland University, Rochester, Michigan 48309, USA4Departamento de Fısica, Pontifıcia Universidade Catolica do Rio de Janeiro, 22453-900, Brazil(Received 4 February 2012; revised manuscript received 15 March 2012; published 2 April 2012)

In this work we use the slave-boson mean-field approximation at finite U to study the effects of spin-spincorrelations in the transport properties of two quantum dots coupled in series to metallic leads. Different quantumregimes of this system are studied in a wide range of the parameter space. The main aspects related to theinterplay between the half-filling Kondo effect and the antiferromagnetic correlation between the quantumdots are reviewed. Slave-boson results for conductance, local density of states in the quantum dots, and therenormalized energy parameters are presented. As a different approach to the Kondo physics in a double-dotsystem, the Kondo cloud extension inside the metallic leads is calculated and its dependence with the interdotcoupling is analyzed. In addition, the cloud extension permits the calculation of the Kondo temperature of thedouble quantum dot. This result is very similar to the corresponding critical temperature Tc, as a function of theparameters of the system, as obtained by using the finite-temperature extension of the slave-boson mean-fieldapproximation.

DOI: 10.1103/PhysRevB.85.165401 PACS number(s): 73.23.Hk, 72.15.Qm, 73.63.Kv

I. INTRODUCTION

The Kondo regime in strongly correlated mesoscopicstructures has been extensively studied since its observationin a single quantum dot (QD) connected to metallic leads.1

The increasing interest in investigating the underlying physicsof these structures is motivated by potential technologicalapplications, such as the design of single-electron transistordevices based on the Coulomb blockade effect, as observedin single QDs.2,3 A system composed of two tunnel-coupledQDs connected in series with metallic leads [henceforth calleda double QD (DQD)] is particularly interesting because it is thesimplest geometry in which the interplay of two energy scalesdetermines its physical properties: (i) the Kondo correlationbetween the spin of each QD and the spins of the conductionelectrons, and (ii) the antiferromagnetic correlation betweenthe spins of the two QDs. Each of these two regimes prevailsin different regions of the parameter space and competes inthe crossover region. As discussed below, these regimes aremanifested very clearly in charge-transport measurements.

Several experimental and theoretical works have appearedin the last decade studying DQDs. The continued interest inDQDs stems from the early recognition4,5 of a possible non-Fermi liquid (NFL) quantum critical point (QCP) separating aFermi liquid (FL) local singlet antiferromagnetic phase from aKondo-screened FL-Kondo phase in the two-impurity Kondomodel (TIKM).6 Subsequent numerical renormalization group(NRG)7 calculations on the two-impurity Anderson model(TIAM) detailed the properties of this NFL QCP,8 but alreadypointed out that the interimpurity hopping suppresses thecritical transition.9 The great flexibility of mesoscopic systems,especially semiconducting lateral QDs, where a continuoustuning of the relevant physical parameters is possible, led to aseries of detailed theoretical studies of DQDs as a prototype forthe TIAM and TIKM. Several papers used different flavors ofthe slave-boson formalism to analyze either the TIAM10–15

or the TIKM.16,17 Detailed studies have also been done

using NRG,8,18 the embedded cluster approximation (ECA),19

and the noncrossing approximation.20 The results obtainedconfirmed that when the even-odd parity symmetry is broken,the critical transition is replaced by a crossover. In addition, inthe TIAM, it was found that, as the interdot hopping increases,a coherent superposition of the many-body Kondo states ofeach QD (forming bonding and antibonding combinations)results in a splitting of the Kondo resonance, which leadsto a splitting of the zero-bias anomaly in the differentialconductance.11,12,19 These many-body molecular states shouldnot be confused with the single-particle molecular states(separated by an energy equal to twice the interdot hopping). Inreality, the coherent single-particle molecular states were thefirst to be observed by Blick et al.21 through careful analysisof the charging diagram of a DQD. Concurrently, by using acombination of charge transport and microwave spectroscopy,Oosterkamp et al.22 probed the formation of single-particlemolecular states (which they called “covalent bonds”) in aDQD by varying the tunneling coupling between the QDs,showing the possibility of controlling the quantum coherencein single-electron devices. Subsequently, Qin et al.23 showedthat these coherent molecular states are robust even whencoupled to acoustic phonons created in the system. The firstobservation of a coherent Kondo effect in a DQD was reportedby Jeong et al..24 Indeed, the splitting of the Kondo resonance(into bonding and antibonding many-body states) was thenclearly observed, as had been theoretically predicted.11,12,19

However, no single Kondo peak has been experimentallyobserved to date, probably due to the small values of interdottunnel coupling required, which leads to a very small overallconductance at half filling.

In addition, several recent researchers have been trying todetermine under what experimental conditions it would bepossible to observe manifestations of the NFL QCP mentionedabove. On the theoretical side, Affleck et al.25 have usedbosonization to determine the energy scale below which

165401-11098-0121/2012/85(16)/165401(10) ©2012 American Physical Society

RIBEIRO, VERNEK, MARTINS, AND ANDA PHYSICAL REVIEW B 85, 165401 (2012)

Lt

L Rt

Rαβt

α βL R

FIG. 1. (Color online) The figure sketches an artificial moleculeconsisting of two QDs with intradot Coulomb interaction U , andtunnel coupled to each other through the interdot matrix element tαβ .In addition each QD is tunnel coupled to its adjacent metallic leadthrough the matrix element tL(R). The leads L and R act as chargereservoirs, being in thermodynamic equilibrium with the DQD.

the RG flows to a FL QCP (away from the NFL QCP, asdiscussed above; see also Ref. 26). More importantly, an exactfunctional form for the conductance in the crossover regionfrom NFL to FL was derived,25 opening the doors for itspossible experimental observation. Note that a very interestingexperiment involving carefully approaching a cobalt atom ina scanning tunneling microsope (STM) tip to another cobaltatom laying on a gold surface has nonetheless failed to observethe NFL QCP physics.27 Also, Lee et al. have found throughNRG a conduction-band mediated superexchange JI , whichcompetes with the direct superexchange term JU = 4t2/U anddominates for large values of U (intradot repulsion; t is theinterdot hopping matrix element). These two terms (related tointerlead charge transfer) are known to destabilize the NFLQCP by breaking parity. Indeed, as mentioned above,6 thisQCP is closely associated with the TCK fixed point, whichis believed to have been observed in only one system.28 In arecent paper, Jayatilaka et al.,6 using NRG, carefully analyzethe experimental possibility of observing this QCP in a DQD.

In this work, we study the properties of a DQD (see Fig. 1)using the slave-boson mean-field approximation (SBMFA)at finite U . The main property we are interested in is thebehavior of the so-called Kondo cloud in a DQD. The Kondocloud can be understood as the spatial region occupied by theconduction electrons that are collectively involved in screeningthe impurity spin.29,30 In the case of the DQD analyzed here(Fig. 1), at half filling, when the tunnel coupling betweenthem is weak (as compared to the coupling of each QDto its adjacent lead), each QD will form its correspondingKondo cloud with the conduction spins of the lead to whichit is directly connected. However, as the interdot couplingincreases, the system is driven through the crossover regionbetween the Kondo regime and the antiferromagnetic “molec-ular” regime. When passing through this region, the Kondocloud associated with each dot should “shrink” and disappearaccordingly.

In a one-dimensional (1D) system, the Kondo cloud is char-acterized by its length, which can be estimated by assumingthat the mean free path of the many-body quasiparticle isrelated to a time scale associated with the Kondo temperatureTK . If one assumes that the electrons scattered by the singleimpurity propagate with the Fermi velocity vF , the Kondocloud length ξ is estimated to be29

ξ ≈ hvF

kBTK

, (1)

where h and kB are Planck and Boltzmann constants, re-spectively. Therefore, in the present case, it is important todetermine whether for the DQD the Kondo cloud length alsoscales with the inverse of TK , as shown in Eq. (1).

From a theoretical point of view, the Kondo cloud has beenstudied through different approaches.31 The most commonone is based on the study of the dependence of the spin-spincorrelation with the distance between the impurity and the con-duction electrons.29 Using a variational approach, it has beensuggested that ξ does not play a significant role in the physicsof a system of impurities in two and three dimensions.32

However, the Kondo cloud is an important concept to analyzethe conductance properties of one-dimensional systems, suchas the one studied in this paper.

Very recently, the authors studied the behavior of thiscorrelation for arbitrary distances from a single impurity.31

This study has been done by analyzing the effect of the Kondoresonance on the local density of states (LDOS) away fromthe impurity. Using this approach, it was possible to show thatξ behaves according to Eq. (1). As a consequence, it is inprinciple possible to determine TK by studying the length ofthe associated Kondo cloud.

To be more precise, the main objective of this paper isto answer the question: How do the Kondo temperature andthe Kondo cloud depend on the parameters of the DQD? Tothis end, we employ the finite-U SBMFA, which requiresa more involved numerical calculation than the infinite-USBMFA,11,16 and therefore eliminates the misconceptions cre-ated by the artificial introduction of an extra antiferromagneticinterdot interaction J in the Hamiltonian. We also study thesystem at finite temperature and estimate TK by associating itwith the temperature above which the width of the Kondo peakin the LDOS vanishes. In this limit, within this approximation,the QDs decouple from the rest of the system. We then comparethe Kondo temperature obtained from the length of the Kondocloud with the one obtained from the slave-boson criterion justdescribed above. The results agree very well in the region ofparameter space where the system is in the Kondo regime.

The paper is organized as follows: In Sec. II we present themodel for the DQD and describe the finite-U SBMFA, whilein Sec. III we present SBMFA results at half filling. Section IVis dedicated to the study of the Kondo cloud inside the leads,presenting a way of calculating its extension and from it theKondo temperature. In Sec. V we compare the results obtainedfor TK from the calculation of the Kondo cloud extension withthose obtained by calculating the width of the Kondo peak atfinite temperature. In Sec. VI we present our conclusions, andrelegate to the appendixes some more technical results.

II. MODEL AND SLAVE-BOSON MEAN-FIELDAPPROACH

The system of two quantum dots presented in the Fig. 1is described by an Anderson Hamiltonian, which can beseparated into three parts, H = H0 + Ht + Hleads, where

H0 =∑i=α,β

σ

εic†iσ ciσ +

∑i=α,β

Uc†i↑c

†i↓ci↓ci↑ (2)

describes the isolated QDs, in which c†iσ (ciσ ) creates (anni-

hilates) an electron with energy εi and spin σ in the ith QD

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(i = α,β), and the second term corresponds to the Coulombinteraction U in each QD. Then

Ht =∑

σ

(tLc†−1σ cασ + tRc

†1σ cβσ + H.c.)

+∑

σ

tαβ(c†ασ cβσ + H.c.) (3)

describes the connection of each QD with its adjacent lead(first term) and the tunnel coupling between the QDs (secondterm). Finally,

Hleads = t

∞∑i=1σ

(c†iσ ci+1σ + c†−iσ c−i−1σ + H.c.) (4)

describes the leads, modeled as two semi-infinite tight-bindingchains of noninteracting sites connected through the hoppingterm t . For simplicity, we take the applied gate potential tobe equal in both QDs, i.e., εα = εβ = Vg , and we considersymmetric coupling to the leads, tL = tR = t ′. Hereafter, wechoose t = 1 as our energy unit and set h = kB = 1.

Within the slave-boson formalism, the physics underlyingthe Kondo regime is brought into the model by the introductionof the auxiliary bosons ei , piσ , and diσ , which project theHilbert space onto space sectors with zero, one, and twoelectrons, respectively. To accommodate the new bosonic fieldwith these operators, the Hilbert space is naturally enlarged,and the single-electron operator ciσ (c†iσ ) is replaced by aquasielectron operator Zciσ (Z†c†iσ ), where the Z operatorin the mean field approximation becomes just a real number,33

Ziσ = [1 − 〈di〉2 − 〈piσ 〉2]−1/2(〈ei〉〈piσ 〉 + 〈piσ 〉〈di〉)× [1 − 〈ei〉2 − 〈piσ 〉2]−1/2, (5)

where i denotes the ith QD and 〈ei〉, 〈piσ 〉, and 〈diσ 〉 the meanvalues of the slave-boson operators. The full Hilbert spacehas to be restricted to the physically meaningful sector byimposing the constraints

〈ei〉2 +∑

σ

〈piσ 〉2 + 〈di〉2 − 1 = 0 (6)

and

niσ − 〈piσ 〉2 − 〈di〉2 = 0 (7)

via Lagrange multipliers λi1 and λi

2σ . Considering the hy-bridization of the fermion operators in H0 and Ht , and alsointroducing the Lagrange multipliers λi

1 and λi2σ , we can write

the effective Hamiltonian as

Heff =∑i=α,β

σ

εiniσ +σ∑

i=L(j=α),R(j=β)

tiZ[c†iσ cjσ + H.c.]

+∑i=α,β

Ui〈di〉2 +∑

σ

tαβZ2[c†ασ cβσ + H.c.]

+∑i=α,β

λi1

[〈ei〉2 +

∑σ

〈piσ 〉2 + 〈di〉2 − 1

]

−∑i=α,β

σ

λi2σ

[〈piσ 〉2 + 〈di〉2] + Hleads, (8)

where εi = εi + λi2σ is the renormalized quasifermion en-

ergy. It is important to notice that the replacement of thesingle-fermion operator, in the mean-field approximation, isequivalent to a renormalization of the connections t ′ and tαβ

by multiplicative parameters Z and Z2, respectively. With theeffective Hamiltonian Heff we can obtain the free energy of thesystem, which is minimized with respect to each componentof the set of parameters

γ = {eα,eβ,pασ ,pβσ ,dα,dβ,λα

1 ,λβ

1 ,λα2σ ,λ

β

2σ

}. (9)

The minimization of the free energy provides a set of nonlinearequations that has to be solved in a self-consistent way. SeeAppendix B for details.

III. DIFFERENT MANY-BODY REGIMES

In this section, we present SBMFA numerical results forthe low-temperature physics of the DQD shown in Fig. 1.These results were obtained in the two different regions of theparameter space, as described in the Introduction, and allow theanalysis of transport properties in the two well characterizedquantum states of the DQD, as well as in the crossover regionbetween them. In addition, some results for the molecularKondo regime, which is characterized by an effective Coulombinteraction Ueff , will be presented as an appendix.

A. Atomic Kondo regime at half filling

To start the discussion concerning the different regimes ofthe system and the crossover between them, we present inFigs. 2–4 the renormalized energy level εα(β) and the localdensity of states (LDOS) of the QDs, the conductance G,and the renormalization parameter Z2, respectively. All thesequantities [except for the LDOS in Fig. 2(b)] are calculated as afunction of the gate potential Vg for different values of tαβ , andwith EF = 0 in both reservoirs. If the gate potential satisfiesVg ≈ −U/2 (i.e., the DQD is occupied by two electrons),then, for small values of tαβ , the system is expected to be in theKondo regime resulting from the singlet state created by theantiferromagnetic correlation of each QD with the conductionelectron spins of the corresponding lead. The formation of thisKondo state is characterized by the plateau structure aroundVg = −U/2, at the Fermi level, observed in the results inFig. 2(a) for tαβ = 0.025, for example. This plateau is relatedto the opening of a conducting channel (at ω = 0), whichallows charge transport through the DQD. Associated with thisconduction channel there is a Kondo resonance at the Fermilevel, clearly seen in Fig. 2(b) (black squares, for tαβ = 0.025),which shows results for the QDs’ local densities of states closeto the Fermi level for Vg = −U/2. Note that the Kondo peakis split (and suppressed) for the smallest value of tαβ = 0.025in Fig. 2(b), and that this splitting becomes more pronouncedas tαβ increases.34 Consequently, the conductance (shown inFig. 3) is also suppressed from its maximum value G = G0

at Vg = −U/2. As mentioned in Sec. II, another importantcharacteristic of the Kondo regime, in the finite-U slave-bosonapproach, is the renormalization of the couplings through theparameter Z (between both QDs and between each QD and itsadjacent lead), as shown in Fig. 4. Note that for tαβ = 0.025(black squares), this renormalization is the strongest. This

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t =0.05

t =0.075

t =0.125

t =0.25

t =0.45

~

(b)

ω

DO

S Q

D α

(β)

FIG. 2. (Color online) (a) behavior of the renormalized localenergy state εα(β) of the QDs as a function of the gate potential Vg .The parameters used are U = 0.5, t ′ = 0.15, Fermi energy Ef = 0,and different magnitudes of tαβ (see legend). As tαβ increases, theplateau observed at εα(β) = 0, characteristic of the Kondo regime, isgradually suppressed. Note that for the largest value of tαβ = 0.45one has now two plateaus located at εα(β) ≈ ±tαβ , which (as notedin Appendix A) are associated with the molecular Kondo regimesat quarter filling. (b) QDs LDOS for Vg = −U/2, and the same tαβ

values as in panel (a). For the smallest value of tαβ = 0.025 (blacksquares) a splitting of the Kondo resonance is already observed. Notethat for tαβ = 0.125 (green inverted triangles), the separation betweenthe peaks is already close to 2tαβ , which is the separation expected forsingle-particle molecular orbitals, as mentioned in the Introduction.

result shows a large reduction of the parameter Z, reachingZ ≈ √

0.05 ≈ 0.22, strongly suppressing the renormalizedhoppings t ′ and tαβ , by factors Z and Z2, respectively [seeEq. (8)]. This is an indication of the formation of a Kondo state,where spin fluctuations are enhanced and charge fluctuationsin the QDs are suppressed, which is reflected in the decreaseof the effective hoppings connecting the QDs to each otherand to the leads. We should remark that the split Kondo peak(black squares) in Fig. 2(b) is associated with the many-bodycoherent states that had been theoretically predicted11,12,19 andexperimentally observed.24

B. Molecular regime at half filling

With the gradual increase in the magnitude of the con-nection tαβ , an antiferromagnetic interaction J = tαβ

2/U

develops between the spins localized in each QD, competingwith the Kondo state and, for large values of tαβ , it isresponsible for the total suppression of the Kondo state at halffilling. From Fig. 2(a) we note the progressive destruction,as tαβ increases, of the plateau in the εα(β)vsVg curves,

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0.5

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0.7

0.8

0.9

1.0

Vg /U

Co

nd

uct

ance

(2e

2/h

)

FIG. 3. (Color online) Conductance as a function of Vg fortαβ = 0.025 (black squares), tαβ = 0.05 (red circles), tαβ = 0.075(blue triangles), tαβ = 0.125 (green inverted triangles), tαβ = 0.25(magenta diamonds), and tαβ = 0.45 (dark yellow left triangles). Theother parameters are t ′ = 0.15 and U = 0.5.

around half filling. Simultaneously, it is possible to observethe development of two other plateaus, for Vg/U > 0 andVg/U < −1, that correspond to the emergence of one- andthree-electron Kondo regimes, respectively, which will bediscussed in Appendix A. In Fig. 2(b) (where Vg = −U/2,i.e., the particle-hole symmetric point), the transition to themolecular regime, accompanied by the destruction of theKondo resonance (black squares), and the formation of themolecular antiferromagnetic state (blue triangles) is reflectedin the LDOS ρα(β) of the QDs. Note in Fig. 2(b) that, fortαβ = 0.125 (green inverted triangles), the separation betweenthe peaks is already ≈2tαβ , indicating that these peaks areassociated with the coherent single-particle molecular states,as experimentally observed in Refs. 21–23. The transition to

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Z2

Vg/U

tαβ

=0.025

tαβ

=0.05

tαβ

=0.075

tαβ

=0.125

tαβ

=0.25

tαβ

=0.45

Vg/U

Z2

FIG. 4. (Color online) Renormalization parameter Z2 as a func-tion of Vg for the same tαβ , U , and t ′ values as in Fig. 3. The insetshows the small renormalization of Z characteristic of the molecularKondo regime.

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KONDO TEMPERATURE AND SCREENING EXTENSION IN . . . PHYSICAL REVIEW B 85, 165401 (2012)

the molecular regime can also be observed in Fig. 4, showingthat the renormalization of the dot-hopping connections isreduced, as the renormalization parameter rapidly approachesthe value Z≈1.0 as tαβ increases. This effect is reflected inthe conductance of the system as presented in Fig. 3, which isstrongly suppressed at half filling for larger values of tαβ . Theplateaus outside the half-filling regime shown in Fig. 2(a), forthe largest values of tαβ correspond to the conductances of theDQD at the molecular one- and three-Kondo regime that, asmentioned, are discussed in Appendix A.

IV. KONDO CLOUD

A. Cloud extension function

In this section we introduce a new perspective to the Kondoproblem in a strongly coupled DQD at half filling, based onthe analysis of the extension of the Kondo cloud inside themetallic leads. To analyze the extension of the Kondo cloudwe use the method developed in Ref. 31, where the authorsanalyze the propagation into the leads (away from the QD)of the perturbation in the LDOS, introduced by the Kondoresonance at the QD. To study this propagation, a functionF (N ) was defined, which quantifies the perturbation producedby the presence of the Kondo cloud in the N th site of the lead(counted from the border of the semi-infinite chain). In thecurrent work we consider a similar expression,

F (N ) =∫ +∞

−∞

[ρk

N (ω) − ρnkN (ω)

]ρα(β)(ω)dω, (10)

to quantify this perturbation. Instead of using a Gaussianfunction with width TK to convolute the density of states andeliminate the Friedel oscillations,30 allowing the study of theregion near the Fermi level, we use the LDOS ρα(β) of the QD.Note that, as ρα = ρβ , we will omit the subscripts α and β

from the rest of the equations in this subsection. The use of thisdefinition for the convolution function (ρ) has the advantageof directly incorporating into F (N ) the physical informationassociated with the Kondo ground state. In this expression,ρk

N and ρnkN represent the LDOS calculated in the N th site

inside the lead (L or R) with the system in and away from theKondo regime, respectively. The last condition is enforced bydisconnecting the DQD from the leads, i.e., by calculating theLDOS for t ′ = 0. It is well known that, within the SBMFA, thisis equivalent to taking the system to a temperature T > TK .10,16

The LDOS appearing in the integrand of F (N ) is pro-portional to the imaginary part of the Green’s function GNN

defined at the N th site inside the semi-infinite metallic leads.Considering the left lead, for instance, we can write

GNN = gNN + gN1t′GαN. (11)

In this expression gNN is defined as the Green’s function at siteN of the left lead when t ′ = 0 and gN1 satisfies the equation

gN1 = tN−1gNL , (12)

where

gL = ω − √ω2 − 4t2

2t2(13)

is the Green’s function defined for the left lead. For GαN(Nα)

we have

GαN = GNα = gN1t′Gαα, (14)

where Gαα is the dressed function at the QD α. SubstitutingGαN into Eq. (11) yields

GNN = gNN + t ′2(gN1)2Gαα. (15)

We note that the Kondo physics is introduced into GNN throughthe term proportional to t ′2. The imaginary part of this functionis proportional to the LDOS ρk

N at the N th site. The non-Kondosolution is obtained by eliminating the effects resulting fromthe presence of the QD by considering t ′ = 0 in the expressionof GNN . So, ρnk

N is defined as ρnkN = −(1/π )Im gNN .

Considering the analytical expression for ρkN and ρnk

N , weobtain

ρkN − ρnk

N = − 1

πIm[t ′2(gN1)2Gαα]. (16)

Substituting gN1 into Eq. (16) we obtain

ρkN − ρnk

N = − 1

πIm

[t ′2g2N

L t2N−2Gαα

]. (17)

This expression, when substituted into Eq. (10), results in

F (N ) = − 1

π

∫ +∞

−∞Im

[t ′2g2N

L t2N−2Gαα(ω)]ρ(ω)dω, (18)

where all the effects of strong correlation present at siteN are contained in the Green’s function of the QDs.Note that when tαβ = 0 one should obtain the results fromprevious calculations for a single-QD Kondo cloud.31 Forfinite tαβ , the competition between Kondo and antiferro-magnetism is contained in Gαα(ω), and therefore should bereflected in F (N ).

B. LDOS

The LDOS ρkN and ρnk

N , calculated at the site N = 50 of theleads for the system with tαβ = 0, are shown in Fig. 5. As inthe rest of this paper, the results presented in this figure wereobtained for Vg = −U/2, U = 0.5, and t ′ = 0.2. The LDOSρnk

N was calculated for an isolated lead (t ′ = 0). The Kondoresonance at the QD is shown in the blue trangles of Fig. 5. Onthe other hand, the LDOS ρk

N=50 (red circles) shows a smallpeak at the Fermi level. The presence of this peak is a signof the Kondo resonance “propagating” through the sites ofthe leads and reflects the existence of the Kondo cloud, withan extension ξ , dependent on the Kondo temperature TK . Inour context, the extension of the Kondo cloud is obtained fromF (N ) (see next section for more details). It is important to notehere that the propagated Kondo peak along the leads appearsas a resonance or an antiresonance depending on whether thesite N is even or odd, respectively.31

Figure 6 shows the behavior of the LDOS for differentvalues of tαβ , illustrating how the competition between theKondo regime and the antiferromagnetic correlation manifestsitself in the LDOS at an arbitrary site (N = 50). The magentadashed curve is the LDOS for a lead with no connection to thedots, which, as a reference, corresponds to the expected resultobtained for a non-Kondo regime. The black inverted trianglescorrespond to tαβ = 0 (t ′ = 0.2) and show the characteristic

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ρΝΚ

Ν(ω )

ρΚ

Ν( ω )

ρα(β)

( ω )

FIG. 5. (Color online) The figure shows the effect produced bythe presence of the impurity α(β) in the LDOS calculated at the siteN = 50 [inside the lead L(R)]. The black squares show the LDOS(at site N = 50) for the isolated lead; the red circles show the LDOSwhen each lead is connected to its respective QD, for tαβ = 0; andthe blue triangles show the LDOS for the impurity.

Kondo resonance propagated to N = 50. It is clear from thefigure that an increase of the tunnel coupling tαβ drives thesystem from a Kondo regime to a non-Kondo ground state, asthe LDOS at (and around) the Fermi level is clearly suppressed,approaching the value for the disconnected DQD (magentadashed curve).

-0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.120.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

N=50E

Fermi=0.00

t=1.00U=0.50t'=0.20V

g=-U/2

ω

DO

S

tαβ=0.09

tαβ=0.06

tαβ=0.03

tα β=0.00

t'=0.0

FIG. 6. (Color online) The figure shows the effect produced bythe antiferromagnetic interaction J = 4t2

αβ/U between the QDs in theLDOS at site N = 50 inside the leads. The black inverted trianglesshow the LDOS calculated at N = 50 for tαβ = 0. The red triangles,blue circles, and cyan inverted triangles are obtained for tαβ = 0.03,tαβ = 0.06, and tαβ = 0.09, respectively. The magenta dashed curveshows the LDOS for the isolated leads (t ′ = 0). The other parametersare Vg = −U/2, U = 0.5, and t ′ = 0.2.

V. KONDO TEMPERATURE

A. Kondo temperature from the cloudextension function

Figure 7 shows the logarithmic dependence of the functionF (N ) for various interdot couplings tαβ . Differently from theresult obtained for a system of one QD connected to a metalliclead,31 this function presents now an oscillatory behavior witha frequency associated with the interdot coupling. However,similarly to the single-QD case, the extension ξ of the Kondocloud can still be obtained, in this case by the exponential decayof its envelop function. The physical information contained inthis function is extracted from the straight lines, tangent tothe logarithm of the F (N ) function. Specifically, the extensionof the Kondo cloud, as a function of the interdot connection,can be obtained from the slopes of these lines, which areproportional to 1/ξ [see Eq. (23) in Ref. 31].

In Fig. 7 we show the function ln F (N ) for tαβ = 0(black squares), tαβ = 0.04 (blue triangles), and tαβ = 0.07(red circles), as well as their respective tangent lines. Forintermediate values of tαβ we present only the tangents toln F (N ). As mentioned above, the slopes of these straightlines are proportional to 1/ξ and allow us to obtain theKondo temperature through the expression ξ = �

TK, proposed

in Ref. 31. The values obtained for the Kondo temperatureTK are presented in the red circles in Fig. 9, as functionof tαβ (we label it as T KC

K , i.e., the Kondo temperatureobtained through the extension of the Kondo cloud). We note

that, for4t2

αβ

U< TK , which implies tαβ < 0.07, TK presents an

exponential behavior in accordance with results obtained byAono and Eto11 using a slave-boson formalism with an infiniteHubbard U .

0 100 200 300 400 500 600 700 800 900 1000-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

lnF

(N)

EFermi

=0.00

t=1.00U=0.50t'=0.20V

g=-U/2

N

tαβ

=0.00

tαβ

=0.04

tαβ

=0.07

FIG. 7. (Color online) Natural logarithm of the cloud extensionfunction F vs. the distance N (from the border of the semi-infinitechain) for tαβ = 0.0 (black squares), tαβ = 0.04 (blue triangles), andtαβ = 0.07 (red circles). The straight lines are tangents to ln F andcorrespond to tαβ = 0.01 (purple line) , tαβ = 0.02 (black line), tαβ =0.03 (red line),..., tαβ = 0.12 (green line). The other parameters areVg = −U/2, U = 0.5, and t ′ = 0.2.

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KONDO TEMPERATURE AND SCREENING EXTENSION IN . . . PHYSICAL REVIEW B 85, 165401 (2012)

B. Kondo temperature in the finite-temperatureSBMFA

In this section we determine the Kondo temperature TK ofthe DQD by extending the slave-boson formalism for finitetemperature. In this formalism, the impurity is decoupledfrom the rest of the system for a critical temperature Tc.Although phase transitions are typical artifacts of mean-fieldsolutions, the temperature for which this decoupling occurscan be considered as a reasonable approximation to the Kondotemperature, which physically corresponds to a crossoverbetween two regimes. The decoupling occurs through theparameter Z, which renormalizes the coupling of the QDsto the leads, t ′ = Zt ′, and ˜tαβ = Z2tαβ , which is vanishinglysmall when T ≈ TK . The parameter Z as a function oftemperature is studied in Fig. 8 for various values of tαβ .We see from this result that the parameter Z2 vanishesrapidly when the temperature approaches the characteristicvalue Tc.

The values of Tc obtained for the different values oftαβ are represented by the black squares in Fig. 9 (labeledT Cut

K ), and they agree with the values obtained for T KCK from

the function ln F (N ) associated with the extension of theKondo cloud in the metallic leads. We observe a qualitativeand semiquantitative agreement between these two ways ofcalculating the Kondo temperature, giving support to theinterpretations we are proposing. We note that for increasingtαβ > 0.07 the values of Tc become increasingly differentfrom the TK values obtained from the extension of the Kondocloud. These are precisely the values for which tαβ

2/U > TK ,and the discrepancy starts at tαβ � 0.07, where the DQDenters a crossover region that extends up to values of tαβ

for which 4t2αβ/U > 2T

(0)K , where the DQD enters into an

antiferromagnetic phase (T (0)K is the Kondo temperature of

each QD when tαβ = 0.0), according to previous works (seeRef. 11 and references therein).

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Z2

T

FIG. 8. (Color online) Renormalization parameter Z2 as a func-tion of temperature T for values of tαβ ranging from 0.00 (blackleftmost curve) to 0.13 (purple rightmost curve). The other parametersare t ′ = 0.2, U = 0.5, Vg = −U/2, and Ef = 0.0.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.006

0.012

0.018

0.024

tαβ

TC

ut

K

t'=0.2U=0.5

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070.006

0.012

0.018

0.024

tαβ

TK

CK

t α β

TK

TKC

K

TCut

K

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

FIG. 9. (Color online) Kondo temperature T KCK and T Cut

K as afunction of tαβ . The red circles correspond to TK obtained from theextension ξ of the Kondo cloud, while the black squares correspond toTK as obtained through the critical temperature Tc for which the QD’sare decoupled (see text), with Z → 0. The insets show the exponentialfits obtained for each TK in the region where the system is expectedto be in the Kondo regime. The values for the other parameters areVg = −U/2, U = 0.5, and t ′ = 0.2.

VI. CONCLUSIONS

In this paper, the interplay between the antiferromagneticinteraction and the Kondo effect for a DQD at half fillinghas been studied in detail. In particular, we have analyzedthe dependence of the extension of the Kondo cloud on theinterdot tunnel coupling (tαβ) when the system is drivenfrom the half-filling Kondo regime to the antiferromagneticmolecular ground state, as tαβ increases. The extension ofthe Kondo cloud and its associated TK were obtained byanalyzing the propagation inside the leads of the Kondoresonance “originating” at the dots. In the Kondo regime,the TK results obtained through the Kondo cloud extensionwere almost identical to the “Kondo transition temperature”Tc obtained from the finite-temperature extension of theMFSBA. Although we know that the decoupling of the DQDfrom the leads (occurring at Tc) is an artifact of the MFSBA,our results confirm that the Tc can be reliably taken to be≈ TK in a DQD, very much in the same way as in the case ofone impurity. Note that the dependence of Tk obtained by bothapproaches agrees with previous results by Aono and Eto.11

The study of the Kondo regime in the DQD was done withinthe context of a finite-U treatment. As U is finite, the Hamilto-nian incorporates the correlation between the QD’s spins. Thisallows us to avoid introducing an artificial antiferromagneticinteraction between the QDs, normally incorporated in U = ∞approaches (as done, for example, in Ref. 16), which providesmore confidence to our results. Its important to note that thestudy developed in this paper opens a set of conceptual ideasthat can be useful to understand the effects of the Ruderman-Kittel-Kasuya-Yosida interaction between two QD’s locatedat an arbitrary distance between themselves, as well as thetransport associated with them. Note that a very recent work35

employs similar ideas, related to the use of the Kondo cloudextension, to study the two-impurity Kondo model.

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RIBEIRO, VERNEK, MARTINS, AND ANDA PHYSICAL REVIEW B 85, 165401 (2012)

APPENDIX A: MOLECULAR KONDO REGIMEAT QUARTER FILLING

Given the importance in a DQD, as mentioned in theIntroduction, of the concept of molecular states (or molecularorbitals), the authors will briefly describe the properties ofthe molecular orbitals in a DQD, as done previously forrelated systems.36 For recent work by some of the authors,see Refs. 37,38. A system of two or more coupled QDs is saidto be in the molecular regime when the charge transport occursthrough orbitals that involve linear combinations of levelsin each QD (which could be called “atomic” orbitals). Themolecular Kondo effect occurs when an unpaired spin residingin a molecular orbital is screened by conduction electrons.38 Itis clear that, in the molecular Kondo regime, the charge at eachQD in a multi-QD system does not correspond to an integervalue. As a consequence, the noninteger QD charge groundstate of a multi-QD structure cannot be used as a criterion toidentify the system as being in a fluctuating valence regime,as it could be in a molecular Kondo ground state, involvingmore than one QD. The determination of the real nature ofthe ground state of these systems requires a careful analysis,mainly for parameter values where the system is in a transitionbetween the molecular and the atomic regimes.

In the case here analyzed for a DQD, by adjusting Vg tohave one (or three) total number of electrons in the DQD, andconsidering relatively large values of tαβ , the quarter-filledmolecular Kondo regime is accessed. In the context of thefinite-U slave-boson formalism, this regime is characterizedby two plateau structures in εα(β) [as observed at half fillingin Fig. 2(a)] at εα(β) = ±tαβ . The formation of these plateausis associated with two new resonances which allow chargetransport through the DQD, as can be seen by the wellseparated double peaks in the conductance in Fig. 3 (darkyellow left triangles and magenta diamonds, for tαβ = 0.45and 0.25, respectively).

Figure 4 shows that, as the system is driven out of theKondo regime at Vg = −U/2 by increasing tαβ , the parameterZ → 1.0, thus eliminating the renormalization of the hoppingmatrix elements [see Eq. (8)]. As an important characteristicof this result, we note two small suppressions of Z2 for thedark yellow left triangles and magenta diamonds in the one-and three-electron regions of Vg . This region is amplified in theinset of Fig. 4 and reflects a small renormalization provided bythe Z parameter, characteristic of the molecular Kondo regime.

To investigate in more detail the effect of the differentregimes in the transport properties of the DQD, in Fig. 10we present the conductance as a function of tαβ for differentvalues of Vg . These results are fitted by the analytic expression

f (tαβ) = 4�2t2αβ(

�2 + t2αβ − ε2

i

)2 − 4ε2i �

2, (A1)

where � = πt2L(R)ρ(EF ) is the coupling constant and i =

α(β). This expression is an extension of the one obtained byGeorges and Meir16 and is valid also away from the particle-hole symmetric point, Vg = −U/2. Exactly at Vg = −U/2,the DQD has one electron in each QD and the conductanceis at a maximum when tαβ = �. For higher values of tαβ ,as Vg moves away from −U/2, the DQD approaches the

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

tαβ

Co

nd

uct

ance

(2e2

/h)

FIG. 10. (Color online) Conductance as a function of tαβ forVg/U = −0.5 (black squares), Vg/U = −0.14 (red circles), Vg/U =0.0 (blue triangles), Vg/U = 0.08 (cyan inverted triangles), Vg/U =0.22 (magenta diamonds), Vg/U = 0.6 (gold left-triangles), andVg/U = 1.2 (light blue right-triangles). The other parameters aret ′ = 0.15 and U = 0.5.

quarter-filling molecular Kondo regime. This occurs when thecharge of the system is nearly one (nα + nβ ≈ 1) or threeelectrons (nα + nβ ≈ 3). This regime in the conductance isreflected in Fig. 3 by the two peaks that occur near the regionof one- and three-electron occupation. Figure 10 shows theconductance as function of tαβ for various values of Vg thatcorrespond to the positions of the conductance peaks in Fig. 3(for Vg� − U/2). From the results obtained for Vg > 0 onecan identify two crossover regions for each Vg as tαβ increases:one corresponding to the transition from an empty-dot situationto a molecular Kondo regime with an increasing conductanceregion, and the other, for values of tαβ above the maximumof the conductance, corresponding to the antiferromagneticmolecular regime where the conductance decreases and theDQD’s occupancy increases.

To better understand the physics underlying the strongcoupled QDs in the transition region between the differentregimes in the parameter space defined by tαβ and Vg , we usethe exact solution obtained in Appendix D when the QDs aredisconnected from their leads. We obtain that E

(1e)0 = ε0 − tαβ

and E(2e)0 = 2ε0 + U−

√U 2+16t2

αβ

2 are the lowest energies in theone- and two-electron sectors, respectively. Considering theseexpressions in a situation in which the charge occupationis increasing, we conclude that the inclusion of an extraelectron into the DQD requires an extra energy given by,

E(2e)0 − 2E

(1e)0 = 2tαβ + U−

√U 2+16t2

αβ

2 . This extra energy canbe identified with an effective Coulomb interaction

Ueff = 2tαβ +U −

√U 2 + 16t2

αβ

2, (A2)

which, for small values of U/tαβ , is ≈U/2 + O(U 2/16tαβ ).The addition of a second electron in the DQD is achieved

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KONDO TEMPERATURE AND SCREENING EXTENSION IN . . . PHYSICAL REVIEW B 85, 165401 (2012)

by adjusting the gate potential Vg . In fact, for Vg ≈ tαβ −U/2, the DQD is basically double occupied, characterizing atransition from a molecular Kondo regime (at quarter filling)to an antiferromagnetic state, or to a two-impurity half-fillingKondo regime, depending upon the ratio TK/J being less orgreater than unity, respectively.

APPENDIX B: THE MINIMIZATION OF THE FREEENERGY IN THE SBMFA

The free energy of the DQD is given by

F (γl) = −kBT ln

[ ∑i

e−βEi (γl )

], (B1)

where β = 1/kBT , and γl denotes the Lagrange multipliers.Differentiating F with respect to γl we obtain

∂F

∂γl

= βT kB

∑i∂Ei (γl )∂(γl )

e−βEi (γl )∑i e

−βEi (γl )

=⟨∂Ei(γl)

∂γl

⟩≈ ∂〈Ei(γl)〉

∂γl

, (B2)

where γl denotes all the components of the γ set definedin Eq. (9). In addition, we have adopted an approximationassuming that the mean value of the derivative of the internalenergy with respect to the components γl is approximatelyequivalent to the derivative of the mean value of this energywith respect to these components.39

APPENDIX C: CONDUCTANCE CALCULATION

The conductance is obtained from the Green’s functionsof the system. For the QD’s α(β) we have the local Green’sfunctions

Gσαα(ββ) = gα(β)σ

1 − t2αβ gασ gβσ

, (C1)

obtained through a diagrammatic expansion which incorpo-rates in the QD α(β) the physics underlying the completesystem, including the electron reservoirs. The function gα(β)σ

that appears in this expression describes the subsystemcomposed of the QD α(β) connected to the L(R) reservoir.The calculation of this function results in

gα(β)σ = gα(β)σ

1 − t ′2gα(β)σ gL(R)σ, (C2)

where gα(β)σ is the single-particle Green’s function asso-

ciated with the QD α(β), while gL(R)σ = ω−√ω2−4t2

2t2 is theL(R) reservoir’s Green’s function projected onto its nearestQD α(β).

The nonlocal Green’s functions of the system are given by

Gσαβ(βα) = gα(β)σ tαβ gβ(α)σ

1 − t2αβgα(β)σ gβ(α)σ

(C3)

and

GσLα(Rβ) = gL(R)σ t ′gα(β)σ

1 − t2αβgασ gβσ

, (C4)

which are associated with charge transport between the QDsand from the L(R) reservoir to the QD α(β), respectively.

These propagators can be used to obtain the conductanceG of the system. Using Gσ

αβ , for example, we obtain fromthe Keldysh formalism40 the following expression for theconductance:

G = 2e2

hπ2 t2Lt2

R

∫ +∞

−∞fL(ω)fR(ω)

∣∣Gσαβ(ω)

∣∣2 ∂fL(R)

∂ωdω,

(C5)

where e is the electron charge, h is Planck’s constant, andfL(R)(ω) the Fermi distribution function associated with theL(R) reservoir.

APPENDIX D: EXACT SOLUTION OF ISOLATED DQDAT HALF FILLING

In order to better understand the characteristics of theantiferromagnetic and the Kondo molecular regime accessedwhen the DQD is at half or quarter filling, respectively, wecalculate the exact solution when the DQD is decoupled fromthe metallic leads. The exact solution obtained in this appendixallows the identification of the regions in the parameter spacecorresponding to different regimes and the crossovers betweenthem.

The Hamiltonian for the decoupled DQD is given by

H =∑i=α,β

σ

εiniσ +∑i=α,β

Uiniσ niσ

+∑

σ

tαβ(c†ασ cβσ + c†βσ cασ ), (D1)

where εi and Ui are respectively the energy of the local stateand the Coulomb interaction in the ith QD, tαβ is the hoppingterm and σ are the spin projections of the electrons in the QDs.The DQD may have an electron occupancy of N = 1,2,3,4.Thus, considering the system with two electrons, N = 2, wedefine the basis

|ϕ1〉 = |↑↓,0〉, (D2)

|ϕ2〉 = |0,↑↓〉, (D3)

|ϕ3〉 = 1√2

[|↑,↓〉 − |↓,↑〉], (D4)

|ϕ4〉 = 1√2

[|↑,↓〉 + |↓,↑〉], (D5)

|ϕ5〉 = |↑,↑〉, (D6)

|ϕ6〉 = |↓,↓〉, (D7)

where |ϕ1〉, |ϕ2〉, and |ϕ3〉 are states with total spin ST = 0,while |ϕ4〉, |ϕ5〉, and |ϕ6〉 are states with ST = 1. Writtenin this basis, the Hamiltonian can be separated into blocks,which correspond to the projections ST = 0 and ST = 1 ofthe total spin. The block matrix that corresponds to a spinprojection ST = 1 is already diagonal and its eigenvalues areE4 = E5 = E6 = 2ε0 (for simplicity we consider εα = εβ =ε0). These energies are associated with states with ST = 1 andSz = 1,0, − 1.

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RIBEIRO, VERNEK, MARTINS, AND ANDA PHYSICAL REVIEW B 85, 165401 (2012)

The block matrix corresponding to ST = 0 is given by

HAF(2e) =

⎛⎜⎜⎝

2ε0 + U 0 −√2tαβ

0 2ε0 + U√

2tαβ

−√2tαβ

√2tαβ 2ε0

⎞⎟⎟⎠ ,

with the following eigenvalues:

E1 = 2ε0 + U, (D8)

E2 = 12

(4ε0 + U +

√U 2 + 16t2

αβ

), (D9)

E3 = 12

(4ε0 + U −

√U 2 + 16t2

αβ

). (D10)

These results allow (as shown in Appendix A) a betterunderstanding of the processes underlying the transitionbetween the different regimes of the DQD.

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