VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999

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Electronic Correlations in Transport through Coupled Quantum Dots

Antoine Georges1 and Yigal Meir21Laboratoire de Physique Théorique de l’Ecole Normale Supérieure,* 24 rue Lhomond, 75231 Paris Cedex 05, Franc

2Department of Physics, Ben-Gurion University, Beer Sheva 84105, Israel(Received 23 November 1998)

The conductance through two quantum dots in series is studied using general qualitative argumentsand quantitative slave-boson mean-field theory. It is demonstrated that measurements of theconductance can explore the phase diagram of the two-impurity Anderson model. Competition betweenthe Kondo effect and the interdot magnetic exchange leads to a two-plateau structure in the conductanceas a function to the gate voltage and a two or three peak structure in the conductance versus interdottunneling. [S0031-9007(99)09017-1]

PACS numbers: 73.40.Gk, 72.15.Qm, 73.23.Hk

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The recent observation of the Kondo effect in transport through a quantum dot [1–3] opened a new path fthe investigations of strongly correlated electrons. Haing confirmed earlier theoretical predictions [4,5] thatquantum dot behaves as a magnetic impurity, theseperiments also serve as a critical quantitative test for eisting theories. In particular, unlike magnetic impuritiein metals which have physical properties determined bthe host metal and the impurity atom, the correspondiparameters in the quantum dot case can be varied contiously, enabling, for example, a crossover from the Kondto the mixed-valence and the empty dot regimes in thsame sample [1,2].

The behavior of a lattice of magnetic impuritiessuch as a heavy-fermion system, is characterized bycompetition between the Kondo effect and the magnecorrelations between the impurities. An important stetowards the understanding of this problem was takenJones and collaborators [6], who studied the two-impuriproblem. Their work demonstrated that this competitioleads to a second-order phase transition when partichole symmetry applies. When this symmetry is brokethis transition is replaced by a crossover [7–9]. In view othe extensive experimental research on transport throutwo dots in series [10,11], it is thus natural to try anunderstand how this phase transition is manifested in tdouble-dot system, both because such systems may himportant applications (such as a quantum-dot laser [12and because such a tunable system may reveal detainformation on the corresponding phase diagram.

Transport through a double-dot system (see insetFig. 1) has already received much theoretical attentioin particular in the high temperature, Coulomb blockaderegime [12,13]. In experiments the Coulomb charginenergy and the excitation energy are much larger thtemperature. Accordingly, only a single state on eacdot is important, and double occupancy of each dot cbe ignored. Denoting the energies of these statese1 ; e0 1 De ande2 ; e0 2 De, respectively, and thetunneling amplitude between the dots byt, the isolated

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double-dot system can contain zero, one, or two electrondepending on the chemical potential:N ­ 0 for m ,

e2, N ­ 1 for e2 , m , e1, andN ­ 2 for m . e1,with e6 ­ e0 6

pDe2 1 t2. In the presence of a finite

antiferromagnetic spin exchangeJ between the dots, onestill has the three possibilities withe1 replaced bye1 2

3Jy4 [14]. In the Coulomb blockade regime there will betwo peaks in the conductance versus chemical potentiat the degeneracy pointsm ­ e6. Alternatively, startingfrom theN ­ 2 regime fort ­ 0, e1 will increase withincreasingt, until it crosses the chemical potential and the

FIG. 1. Plot of the conductance vs the tunneling between thdots, t, obtained by slave-boson mean-field theory. Becausof the Kondo effect in the two-electron regimesN ­ 2d theconductance has a peak att ­ G. As t increases beyondt1,the Kondo effect is quenched, until the ground state containssingle electronsN ­ 1d, leading to a different Kondo state andan enhanced conductance. For finite antiferromagnetic couplinJ (finite a ­ G2yUT 0

K ), the conductance peak is pushed tosmaller values oft and becomes narrower, as the singletformation destroys the Kondo state. In addition, a seconmaximum in the conductance in theN ­ 2 regime emerges.Inset: The double-dot system discussed in this paper.

© 1999 The American Physical Society

VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999

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ground state will have a single electron in the double-dsystem. Again we expect a peak in the conductancet ­ t1 corresponding toe1st1d ­ m.

At low temperatures, in addition to a renormalizatioof the above energy scales,e6 ! e6, and t1 ! t1, theKondo effect starts to play a significant role in the tranport. The relevance of the Kondo effect in the doubldot system has been studied in [15,16]. Here, we focon the competition between the Kondo effect and anferromagnetic exchange, and present detailed predictifor the conductance. (Recently Andreiet al. [17] have in-vestigated a very different realization of this competitiowhich applies to coupledmetallic islands,close to pointsof charge degeneracy—i.e., near the Coulomb-blockapeaks [18].) We find a rich phase diagram leading to iteresting features in the conductance as a function of gvoltage and intradot tunneling. As the corresponding eergy and temperature scales are experimentally accessthese predictions are relevant to transport experimentsdouble-dot systems.

To simplify notations, we assume in the followinge1 ­ e2 ­ e0 and VL ­ VR ­ V , whereVL sVRd is thecoupling to the left (right) lead. Then the eigenstatesthe double-dot system are the even and odd states.even-odd symmetry is broken anyway by the tunnelint, one can show that the above assumptions have lieffect on the underlying physics. Following Ref. [19], thzero-temperature conductanceper spin channelthroughthe system can be expressed in terms of the retardGreen functions for the even and odd statesG ret

e,osvdas g ­ e2

h G2jG rete sv ­ 0d 2 G ret

o sv ­ 0dj2 where G ;prV 2 and r is the density of states in the leads athe Fermi energy. Defining the correspondingscatteringphase shifts,da ; p 1 argG ret

a smd (such thatda is inthe0 p range), the conductance formula simplifies to

g ­e2

hsin2sde 2 dod . (1)

The Friedel sum rule relates the total chargeq per spinchannel,on the double-dot system, to these phase shifq ­ sde 1 dodyp. There is, however, no individualrelation betweenda and the occupation of the corresponding state.

For m . e1 (or alternativelyt , t1), there areN ­ 2electrons in the systemsq ­ 1d, and states withN ­ 0 orN ­ 1 are high-energy states that can be eliminated frothe Hilbert space. The effective low-energy Hamiltoniainvolves only spin degrees of freedom on the dots. It cbe cast into the form of a model oftwo Kondo impurities,with Kondo couplings to the even and odd combinationof the conduction electrons in the leads, an interimprity magnetic exchangeJ ~ t2yU, and a potential scat-tering term in the leadsVe,o such thatVe 2 Vo ~ tV 2yse2

0 2 t2d.Previous studies of the two-impurity Kondo problem

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(NRG)] [6,8] have already yielded information on howthe phase shiftsde, do behave as a function of thecouplings atT ­ 0 in this regime. Let us start with thecaseJ ­ 0. For t ­ 0 there is an even-odd symmetryand each channel has its own Kondo effect, leadingde ­ do ­ py2, and naturally a zero conductance. AtyG is increased, the differencede 2 do increases toreach the valuepy4, at which point the conductance takeits maximum possible value:e2yh per spin channel. AstyG is further increased, the Kondo effect is gradualovercome by potential scattering and one reachesdo . 0,de . p in the largetyG limit (but still with t ø t1). Aslave-boson mean-field theory (SBMFT) presented belo(cf. also Ref. [16]) yields in this regimesJ ­ 0, q . 1dd ; de 2 do ­ 2 tan21tyG, leading to

g ­e2

h4t2G2

st2 1 G2d2 , (2)

which reaches its maximum value att ­ G (solid curvein Fig. 1). The Kondo temperature in this regime iof order TK ­ c1T0

Kec2tyG, where thec’s are weaklydependent ontyG andT0

K ; We2pje0jyG is the single-dotKondo temperature. (The SBMFT yieldsc1 ­ cosdy2and c2 ­ dy2.) The crucial content of that formula isthat the coupled-dot Kondo temperature can bemuchlarger than the single-dot Kondo temperaturefor smallJ and largetyG. This has important consequences for thobservability of the effects described in this paper.

Let us now consider the effect of a finiteJ. For t ­ 0,the effective two-impurity Kondo model has particlehole symmetry, and it is known from the work of [6that a phase transitionexists at a critical value of thecoupling JcyT0

K . 2.2. For J , Jc, the spin of eachdot undergoes a Kondo effect with the leads andde ­do ­ py2. For J . Jc, the two spins are locked intoa singlet state and the Kondo effect does not appyielding do ­ 0, de ­ p. The phase-shift differencedjumps discontinuously fromd ­ 0 for J , Jc to d ­ p

for J . Jc. (The conductance is, of course, zero foall J since t ­ 0). Turning on a small value oftyG isknown to be a “relevant perturbation” on this critical poin(with dimension1

2 , identical to that ofJ 2 Jc [7,8]) andtherefore smears the transition into a rapid crossover frod ­ 0 to d ­ p. For J close to Jc, this smearing isdescribed by a crossover scaling function:

d

p­ f

√sJ 2 JcdyT0

K

tyG

!, (3)

with fsx ! 2`d ­ 0 andfsx ! 1`d ­ 1. As a result,the conductance has a very sharp maximum astyG

is increased for a fixed value ofJ close to Jc. ForJ significantly larger thanJc, the conductance remainsvery small with only a shallow maximum astyG isincreased. For intermediate values oftyG and JyT0

K , aquantitative calculation ofd is needed in order to obtainthe conductance, using, e.g., NRG [6,8,20] or SBMF

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VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999

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However, much can be said on semiquantitative levby using existing knowledge on the two-impurity Kondoproblem. The phase shiftd is an increasing functionof J, which starts at the value given above Eq. (2and increases until it saturates atd ­ p at a scaleJp.From the above estimate ofTK the ratioJpyT0

K increasesexponentially withtyG. These considerations, and theknowledge of the crossover aroundJc [Eq. (3)], leadto a qualitative contour plot of the conductance in thsJyT0

K , tyGd parameter space, through theN . 2 regime,as displayed in Fig. 2.

In practice, the exchangeJ is not an independentparameter, but is a function of the interdot tunneling,J ,t2yU. The contour plot above must thus be intersected ba curveJyT0

K ­ astyGd2, with a ; G2yUT0K , in order to

follow the dependence of the conductance as a functiontyG. SinceT0

K is a very sensitive function of the energyscales (such ase0 and G), the control parametera canbe varied continuously over many orders of magnitudallowing an experimental investigation of most of thephase diagram. Thus, as a function oft, the maximumconductancee2yh is reached fort . G with a peak widthDt ~ G for small a, while the peak is pushed downto much lower transmissiont . Gy

pa and becomes

very narrowDt . Gya for large a. In addition, as thesaturation scaleJp increases exponentially witht, onemay expect, for an intermediatea (middle broken curvein Fig. 2), an additional peak in the conductance versustin theN ­ 2 regime. These results are indeed confirmeby the SBMFT calculation (see Fig. 1).

As t is further increasedst . t1d, the equilibriumcharge decreases toN ­ 1 sq ­ 1y2d. In this regime the

FIG. 2. Schematic contour map of the conductance in thN . 2 regime. Thicker lines denote higher conductance, ththickest one corresponding tog ­ e2yh. The broken lines arethree physical contours (for different values ofa ; G2yUT 0

K )along whichJ , t2yU.

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effective Hamiltonian is that of asingle-impurityKondoproblem in the even parity sector [15], leading to unitarscatteringde . py2. In the odd parity sector, we havean almost empty resonant level withd0 . 0 [note thatsde 1 dodyp ­ q . 1y2 consistent with Friedel sumrule]. Throughout this regime, we therefore expect thzero-temperature conductance to be maximumg ­ e2yhand essentially independent oft. In this regime, theinterdot exchangeJ plays little role.

Similar interesting behavior is expected as a functioof gate voltage that controls the depth of the level energe0 with respect to the chemical potential (see Fig. 3). Foa very deep level the Kondo temperature is exponentiasmall, and thusJyT

s0dK is large and quenches the Kondo

effect. Ase0 increases, the Kondo temperature increasand we enter thesN ­ 2d Kondo state, and a finiteconductance. This conductance remains constant (at ztemperature) at a valuesmaller thane2yh, determined bythe value oft, until e0 crosses the Fermi energy and a newsN ­ 1d Kondo state is formed. There the conductancis given by its maximum value,e2yh per spin. Ase0 isfurther increased the double-dot system becomes emand the conductance drops to zero.

To substantiate these semiquantitative argumentshave performed a quantitative calculation of the phashifts and the conductance using a slave-boson mefield approximation. This method becomes exact as tnumber of spin degrees of freedom goes to infinity, anhas been previously used in order to study the twimpurity Anderson model in Ref. [7]. It was recentlyapplied in the present context in Ref. [16] but only in

FIG. 3. Plot of the conductance versus the level energy,obtained from SBMFT (fort ­ 2 and UyG ­ 104). Theconductance rises from a very small value (the singlet regimJ ¿ TK ), to a t-dependent value (N ­ 2 Kondo regime,J ø TK ), and then tog ­ e2yh (N ­ 1 Kondo regime) beforedropping to zero for an empty dot.

VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999

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the caseJ ­ 0. We have solved numerically the fullset of equations including the tunnelingt and exchangeJ, but we quote here only the simplified version of thequations [7] that hold in theq ­ 1 regime (one electronin the double-dot system per spin state, correspondingN ­ 2). For small enough values ofJyT0

K , the phase-shift differenced is given by the solution of

2p

dedty2G

√sin

d

22

tG

cosd

2

!­

J

T0K

. (4)

As J is increased beyond a critical couplingJSBc , d

reaches the valuep: This is either a smooth transition fort . 1yp or a first-order jump fort , 1yp (determinedby free-energy considerations). The existence of a phatransition even for nonzero values oftyG is an artifactof the SBMFT approximation:JSB

c should actually beinterpreted as an estimate of the saturation scaleJp

discussed above. This spurious transition does not affequalitatively the behavior of the conductance, excewhen it becomes very small: there a strictly zero valuof g can be found (as evident on Fig. 1), whereas threal system would have only a very small but finiteg.The SBMFT also provides a quantitative estimate of thKondo scale for the coupled-dot system in theq . 1regime, as mentioned after Eq. (2).

In conclusion, we have demonstrated that measurments of the conductance through a double-dot system cexplore the phase diagram of the two-impurity Andersomodel. By changing the control parametera ­ G2yUT0

K(which depends sensitively on the gate voltage), one cmake various cuts through the phase diagram (Fig. 2leading to nontrivial features in the conductance vs gavoltage and interdot tunneling (Figs. 1 and 3). As threlevant temperature scale can be much higher thansingle-dot Kondo temperature we believe that these prdictions could be tested experimentally.

This work was supported in part by a grant fromthe French-Israeli Scientific and Technical CooperatioProgram (Arc en Ciel-Keshet-1998). Work at BGU wafurther supported by the The Israel Science FoundatioCenters of Excellence Program.

*Unité propre du CNRS (UP 701) associée à l’ENS etl’Université Paris-Sud.

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[2] S. M. Cronenwettet al., Science281, 540 (1998).[3] For reviews on transport through quantum dots see, e.

M. A. Kastner, Comments Condens. Matter Phys.17, 349(1996); R. Ashoori, Nature (London)379, 413 (1996).

[4] L. I. Glazman and M. E. Raikh, Pis’ma Zh. Eksp. TeorFiz. 47, 378 (1998) [JETP Lett.47, 452 (1988)]; T. K. Ngand P. A. Lee, Phys. Rev. Lett.61, 1768 (1988).

e

to

se

ctptee

e

e-ann

an),

teethee-

nsn-

à

g.,

.

[5] S. Hershfield, J. H. Davies, and J. W. Wilkins, Phys. RevLett. 67, 3720 (1991); Y. Meir, N. S. Wingreen, and P. A.Lee, Phys. Rev. Lett.70, 2601 (1993); N. S. Wingreen andY. Meir, Phys. Rev. B49, 11 040 (1994).

[6] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. RevLett. 61, 125 (1988); B. A. Jones and C. M. Varma, PhysRev. B 40, 324 (1989); B. A. Jones, Physica (Amsterdam171B, 53 (1991).

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[11] For recent experimental studies of a double-dot systemsee, e.g., T. H. Oosterkamet al., Phys. Rev. Lett.80, 4951(1998); R. H. Blicket al., Phys. Rev. Lett.81, 689 (1998).

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[14] However, for J . 8p

De2 1 t2y3 there will be a directjump from N ­ 0 to N ­ 2. Since in practiceJ stemsfrom superexchange between the dots,J , t2yU, thelatter case may not be experimentally relevant, asrequirest , U.

[15] T. Ivanov, Europhys. Lett.40, 183 (1997); T. Pohjolaet al., ibid.40, 189 (1997).

[16] T. Aono, M. Eto, and K. Kawamura, J. Phys. Soc. Jpn.67,1860 (1998).

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[18] K. Matveev, Sov. Phys. JETP72, 892 (1991).[19] Y. Meir and N. Wingreen, Phys. Rev. Lett.68, 2512

(1992).[20] W. Izumida, O. Sakai, and Y. Shimizu, cond-mat/

9805067.

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