We consider a lateral double-dot system in the Coulomb blockade regime with a
single spin-1/2 on each dot, mutually coupled by an anti-ferromagnetic exchange
interaction. Each of the two dots is contacted by two leads. We demonstrate
that the voltage across one of the dots will have a profound influence on the
current passing through the other dot. Using Poor Man's scaling, we find that
the Kondo-effect can lead to a strong enhancement of this {\it
transconductance}.Comment: updated to published versio

A. A. Abrikosov

J. Paaske

P. Wölfle

V. Koerting

English

arXiv

Körting, V.

Wölfle, P.

Paaske, J.

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Transconductance of a Double Quantum Dot System in the Kondo Regime

Koerting, V.

Wofle, P.

Paaske, Jens

Copenhagen University Research Information System

u n i ve r s i t y  o f  co pe n h ag e n  Københavns UniversitetTransconductance of a double quantum dot system in the kondo regimeKoerting, V.; Wofle, P.; Paaske, JensPublished in:Physical Review LettersDOI:10.1103/PhysRevLett.99.036807Publication date:2007Document versionEarly version, also known as pre-printCitation for published version (APA):Koerting, V., Wofle, P., & Paaske, J. (2007). Transconductance of a double quantum dot system in the kondoregime. Physical Review Letters, 99(3), 036807. https://doi.org/10.1103/PhysRevLett.99.036807Download date: 02. Feb. 2020Transconductance of a Double Quantum Dot System in the Kondo RegimeV. Koerting* and P. Wo¨lfleInstitut fu¨r Theorie der Kondensierten Materie, Universita¨t Karlsruhe, D-76128 Karlsruhe, GermanyJ. PaaskeThe Niels Bohr Institute and Nano-Science Center, University of Copenhagen, DK-2100 Copenhagen, Denmark(Received 21 December 2006; published 20 July 2007)We consider a lateral double-dot system in the Coulomb blockade regime with a single spin-1=2 oneach dot, mutually coupled by an antiferromagnetic exchange interaction. Each of the two dots iscontacted by two leads. We demonstrate that the voltage across one of the dots will have a profoundinfluence on the current passing through the other dot. Using poor man’s scaling, we find that the Kondoeffect can lead to a strong enhancement of this transconductance.DOI: 10.1103/PhysRevLett.99.036807 PACS numbers: 73.63.Kv, 72.10.Fk, 75.30.HxNanostructured electron systems offer the unique possi-bility to prepare and probe highly nonequilibrium station-ary states. Energy dissipation and therefore heating,characteristic of any nonequilibrium state, take place inthe bulk reservoirs connected to the nanostructure, where itis absorbed causing a negligible change of state. The natureof such a nonequilibrium state may be conveniently probedby transport measurements rather than thermodynamic oroptical experiments (cf., e.g., Refs. [1,2]). In an ordinaryquantum dot setup the probing field, e.g., the applied biasvoltage, is also responsible for driving the system out ofequilibrium. However, by separating the driving field andthe probing field many more possibilities for both studyingnonequilibrium states and exploiting their properties arise.In the simplest realization of such a system one must havetwo parts of the system strongly coupled by some interac-tion and each contacted by two leads. Applying a suffi-ciently high bias across one part will then influence thecurrent through the other part. This coupling is mediatedby the change in nonequilibrium occupation numbers onthe nanostructure caused by the driving field, which in turngoverns the current response to the probing field.In this Letter we investigate the properties of a system ofthis type (cf. Fig. 1), that is two quantum dots in theCoulomb blockade regime with odd electron occupation,giving rise to spins SL;R  1=2, coupled by an exchangeinteraction K. Each dot is contacted by two leads, modeledby exchange-tunneling amplitudes JL;R  t2=EC, in termsof lead-dot tunneling amplitudes t and charging energy EC.This setup is very similar to the double-dot devices studied,for example, in Refs. [3–5], where the central region waseither a quantum dot, a quantum wire, or a tunnelingbarrier, which could then support an exchange interactionbetween the two spins. The following discussion does notrely on the physical nature of the exchange coupling. Inpractice, it can be due to a simple superexchange mecha-nism [6] or a RKKY interaction, as is most likely the casein Refs. [3,4], or even due to the so-called Kondo-doubletinteraction suggested recently [7].The theory of the two-impurity Kondo model per se iswell established [8], but with the experimental progress inquantum dot systems, issues like the quantum criticalproperties remain an active field of research [9–11]. Theinfluence of the exchange interaction K and an appliedmagnetic field on the nonlinear conductance dIL=dVL wasstudied in Refs. [12,13], and it was demonstrated how thenature of the ground state (Kondo screened or local singlet)can be extracted from the temperature dependence of thelinear conductance.In the present setup with two sets of biased leads, we arefacing an entirely new problem in which correlations andnonequilibrium effects must be treated side by side. Wecalculate the transconductance dIL=dVR and demonstrate atransistor effect in which a current through the left dot IL asa response to a bias on the left dot VL is switched on onlywhen the bias over the right dot VR exceeds a thresholdgiven by K > 0. This threshold effect rests on the fact thatthe exchange-tunneling current from lead 1 to lead 2, IL, iszero as long as the spins are locked in a singlet. A suffi-ciently large bias across the right dot will, however, lead toa partial occupation of the triplet state, which does allowfor a finite current IL. As we shall demonstrate, the non-equilibrium polarization due to VR, and thereby the trans-lead 1lead 2lead 3lead 4KL RFIG. 1 (color online). Double quantum dot system: TwoKondo impurities denoted L, R each connected to two leads 1,2 and 3, 4, respectively, and coupled to each other by a spinexchange interaction K.PRL 99, 036807 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 20070031-9007=07=99(3)=036807(4) 036807-1 © 2007 The American Physical Societyconductance peak at VR  K, will be enhanced by theKondo effect. In the case of ferromagnetic coupling, thetriplet ground state gives rise to a spin-1 Kondo effect[12,13]. In this case, a bias voltage over the right quantumdot will influence the corresponding zero-bias conductancepeak and singlet satellite peaks in dIL=dVL, partly bydecoherence and partly by a redistribution of the singletor triplet occupations. However, these effects are lessdramatic and we therefore restrict our attention to theantiferromagnetic case, K > 0, which exhibits the afore-mentioned threshold effect.Model.—We model the double dot by two spin- 12 opera-tors SL;SR mutually coupled by exchange interaction Kand each coupled to a pair of leads (couplings JL;R > 0) assketched in Fig. 1. The Hamiltonian reads H  Xknk ncynkcnk  KSL  SR Xn;m1;2JnmL SL  snm Xn;m3;4JnmR SR  snm; (1)where n;m  1; 2; 3; 4 label the leads, assumed to be inequilibrium at chemical potentials n. cynk creates elec-trons in lead n of momentum k and spin  and the spinoperator snm  Pk;k0;;0cynk ~;0=2cmk00 denotes, re-spectively, the conduction electron spin (n  m) and theexchange-tunneling operator (n  m). We do not allow forcharge transfer between the two dots. Experimentally thecharge transfer may be suppressed by a suitable arrange-ment of tunneling barriers (cf. [9]).For K  0, the two spins will undergo separate screen-ing by the conduction electron spins in each set of leads asthe temperature is lowered beneath the two, generallydifferent, Kondo temperatures TK;L and TK;R. However, atsufficiently large bias voltages eVL  TK;L and/or eVR TK;R, where eVL; eVR are the chemical potential differ-ences of leads n  1; 2 or leads n  3; 4, respectively,these Kondo effects are partially suppressed [14,15]. Afinite antiferromagnetic exchange coupling K, on the otherhand, will correlate the spins SL;SR and, depending on therelative magnitude of K and the Kondo temperatures, thetwo spins will either lock into a singlet or be screened bytheir respective lead electrons [8]. We will be interested inthe regime K  maxfTK;L; TK;Rg, when the Kondo effectcannot develop fully. In this case it is necessary to treat theexchange coupling K exactly, whereas it is possible to treatthe Kondo effect perturbatively.The spin states of the isolated double dot are the singlet(S  0) and triplet (S  1) states of the two coupledspin-1=2. They form the appropriate basis states in thisregime even when the coupling to the leads is turnedon. For convenience, we employ a pseudoboson (bondoperator [16]) representation of the four impurity states,with creation operators fbyg  sy; ty; ty0 ; ty, i.e.,  2fs;; 0;g. The energy eigenvalues of the four statesare !s   34K, !0  !;  14K  !t, and thereforeKSL  SR  P!byb, subjected to the constraint Q sys ty0 t0  tyt  tyt  1. The constraint may beexactly imposed by adding a term Q to the Hamilton-ian and taking the limit  ! 1 [17]. In terms of thepseudobosons, the spin-1=2 operators are given bySzL;R  12 	syt0 	 ty0 s tyt  tyt and SL;R12p SxL;R iSyL;R SL;Ry  12	syt tys tyt0 ty0 t[16], or in compact notation SL;R  12P;0byTL;R;0b0 ,where TL;R;0 is a vector of three 4 4 matricesTx; Ty; Tz. Notice that this exchange-tunneling model con-tains no operators of the form sys, which immediatelyimplies that the current IL is zero unless transitions to thetriplet states become energetically allowed.It is known that away from the particle-hole symmetricpoint a finite potential scattering term will give rise to acurrent below the threshold. Nevertheless, we shall assumethat a set of individual gates on each dot can be adjusted soas to make the system (nearly) particle-hole symmetric andhence neglect potential scattering altogether. In the setupby Sasaki et al. [4], such tuning was made possible by twoindividual plunger gates.Nonequilibrium perturbation theory.—In equilibrium,i.e., for vanishing bias voltages, the states ji of the doubledot are thermally occupied, with occupation numbers n hbybi  exp!=T=P0 exp!0=T. In a stationarynonequilibrium situation with sufficiently large currentsthrough the nanostructure, the occupation numbers aredetermined by the currents, i.e., the couplings and thevoltages, rather than temperature [18]. A further compli-cation is that nondiagonal parts of the density matrixn0  hbyb0 i may be nonzero due to the mixing of statesby the coupling to the leads. Confining our considerationsto zero magnetic field this will not be the case and, fur-thermore, the triplet states will be equally occupied, i.e.,n  nt, for   ; 0;. It is useful to define the polar-ization p  ns  nt, and we note that hSL  SRi   34p.The nonequilibrium occupation numbers are determinedfrom the steady-state quantum Boltzmann equation,P00n0  0n  0, with transition rates givenby the golden rule expression 0  4Xn;mTmn0  Tnm0Z 11d!gnm0 !gmn0!!0 Fnm0 !; (2)where (n;m) takes the values (1; 2) or (3; 4) and Tnm  TLor TR, !0  ! !0 , and Fnm0 !  f!n1f!m !0 , f! being the Fermi function. Thedimensionless exchange couplings, gnm  FJnm (assum-ing same density of states F in all leads), will only acquirefrequency dependence under renormalization as we shalldiscuss later. The Boltzmann equation is readily solvedtogether with the constraintPn  1, and using thethreefold degeneracy of the triplet, one finds that p st  ts=st  3ts.PRL 99, 036807 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007036807-2The current through the left dot is obtained as ILV  22ehX0nTL;0  TL;0Z 11d!g210 ! g120!!0F120!  F210!; (3)and a corresponding expression holds for IRV, withleads 1; 2 replaced by 3; 4.From Eq. (3) it is clear that VR only enters via theoccupation numbers, and using the unrenormalized cou-plings the transconductance is given by dILdVR 62ehKg2cp2FVL  eVL ddVR FVR; (4)where F	VKeVnBKeV	KeVnBKeV,in terms of the Bose function nB"  1=e"=T  1, andg2c  g12g342=Pm;ngmn2. dIL=dVR is an odd function ofVR which diverges like T=K  eVR for eVR ! K. As weshall see below, this divergence is removed once the broad-ening of the excited triplet states is taken into account, andone is left with a pronounced peak in the transconductance.Assuming that gmn  gLR for m; n 2 f1; 2g; f3; 4g, onehas g2c  g2L  g2R 1 which shows that the transcon-ductance peak is expected to be largest in a symmetricsetup where gL  gR.Renormalized perturbation theory.—Since the higherorder terms in the perturbation series are plagued by loga-rithmic singularities characteristic of the Kondo effect, weapply poor man’s scaling to sum up the leading logarithmsin this series. As demonstrated earlier [15,19], this is con-veniently done by keeping track of the generated frequencydependence of the couplings. Within this approach therenormalized coupling functions are calculated from thefollowing set of renormalization group (RG) equationsdescribing the flow of couplings under a reduction of thebandwidth D: @gnmtt !@ lnD  12Xlfglmst lgnlts l  K!lK gnlst l  Kglmts l!lK 2glmtt lgnltt l!lg; (5) @gnmts !@ lnD Xlfglmts lgnltt l!l gnlts l  Kglmtt l!lKg; (6)with gnmst !  gmnts ! K. Here !  D2 !2 2 is the step function and  is a Korringa-like decoher-ence rate due to particle-hole excitations in the leads. Ingeneral,  involves both self-energy and vertex correctionsand may depend on the intermediate state for which itdescribes the broadening (cf. [20]). Since, however, appears only under the logarithm we will neglect thesedetails here and simply use the maximum   maxf0 gwith 0 defined in Eq. (2). The renormalized couplingsand  are calculated self-consistently, then we solve for theoccupation numbers and calculate the current using therenormalized coupling functions in Eq. (3). The perturba-tive RG approach is valid for K  TK, where TK D0e1=2gD0 in terms of bare coupling gD0 and thehalf-bandwidth D0.Renormalized conductances.—In Fig. 2 we plot thepolarization p as a function of eVR=K at VL  0, forseveral values of K=TK and at temperature T=TK  0:1.As eVR approaches K the triplet states become partiallypopulated and eventually p tends to zero as 1=VR. Since weuse T  K, temperature is practically zero and the broad-ening is governed by K and VR. In the inset of Fig. 2 weplot the derivative Kdp=dVR vs eVR=K, showing a pro-nounced peak which is increasingly smeared as the ratioK=TK is made smaller and the Kondo correlations becomemore pronounced.From the polarization p and the renormalized couplings,we can now determine the current from Eq. (3). As men-tioned above, electron transport via exchange tunnelingalways involves excitation out of the singlet ground state.This requires an energy of the order of K and thereforeleads to a threshold behavior near eVL  K, as sketchedalso in Ref. [12]. Alternatively, the singlet-triplet gap canbe overcome by a finite VR, which will lead to a finitedIL=dVL even for eVL < K. This is clearly seen in Fig. 3.When VR becomes larger than K, a zero-bias peak is seento develop, the magnitude of which is determined by theratio K=T2K. In the limit of zero probing voltage, we findthat limVL!0dILdVL 62e2hnt4 ln Kp =TK2 ; (7)where both nt and  depend on the driving-voltage VR. Theinset of Fig. 3 shows the increase of the linear conductancewith VR, as calculated, respectively, by perturbative RGand second order perturbation theory (PT). The initial0 1 2eVR/K0.00.50.751p0.10.150.20 1 200.250.5- K dp/dVR1/ln(K/TK)FIG. 2 (color online). Polarization p  ns  nt versus voltageeVR=K for different values of K=TK at T=TK  0:1 and voltageVL  0. Inset shows the derivative of the curves in the mainpanel.PRL 99, 036807 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007036807-3rise with VR is set by the triplet occupation number nt 1 p=4. The amplitude is logarithmically enhanced dueto the Kondo effect and falls off very slowly ( / VR) forVR  K.Figure 4 shows the transconductance versus eVR=K forVL ! 0. Enhancing the Kondo correlations by loweringK=TK is seen to amplify the transconductance but also tosmear the threshold. By scaling with K=eVL we eliminatethe 1=K dependence seen in Eq. (4) as well as most of thedependence on the probing field for eVL < K (cf. inset ofFig. 4).Conclusion.—We studied the mutual influence of spa-tially separated currents flowing through two quantum dotscoupled only by spin exchange. The strong interrelationbetween the currents and nonequilibrium occupation num-bers for the singlet and triplet spin states correlates the twocurrents, and we showed that this can be observed as amarked peak in the transconductance. Since the spin ex-change interaction requires well-defined spins on the quan-tum dots, the coupling to the leads should be relativelysmall, but opening up towards the Kondo regime ourperturbative RG calculations demonstrate that weakKondo correlations (TK  K) will in fact enhance thetransconductance peak. This pronounced signal in thetransconductance provides a new experimental means ofprobing the exchange coupling between two quantum dotspins. Moreover, as evident from Eq. (7), this mode ofoperation provides direct experimental evidence for thevoltage dependence of the nonequilibrium distributionfunction nt.We acknowledge discussions with Chung-Hou Chung,Lars Fritz, and Matthias Vojta. This work has been sup-ported by the DFG-Center for Functional Nanostructures(CFN) at the University of Karlsruhe under project B2.9(V. K. and P. W.), the Institute for Nanotechnology,Research Center Karlsruhe (P. W.), and the DanishAgency for Science, Technology and Innovation (J. P.).*Author to whom correspondence should be addressed.koerting@tkm.physik.uni-karlsruhe.de[1] I. Aleiner, P. Brouwer, and L. Glazman, Phys. Rep. 358,309 (2002).[2] W. G. van der Wiel et al., Rev. Mod. Phys. 75, 1 (2002).[3] N. J. Craig et al., Science 304, 565 (2004).[4] S. Sasaki et al., Phys. Rev. B 73, 161303(R) (2006).[5] H. Jeong, A. M. Chang, and M. R. Melloch, Science 293,2221 (2001).[6] V. N. Golovach and D. Loss, Europhys. Lett. 62, 83(2003).[7] J. Simonin, Phys. Rev. Lett. 97, 266804 (2006).[8] C. Jayaprakash et al., Phys. Rev. Lett. 47, 737 (1981);B. A. Jones and C. M. Varma, Phys. Rev. Lett. 58, 843(1987); Phys. Rev. B 40, 324 (1989).[9] G. Zarand et al., Phys. Rev. Lett. 97, 166802 (2006).[10] C.-H. Chung and W. Hofstetter, arXiv:cond-mat/0607772.[11] L. Zhu and C. M. Varma, arXiv:cond-mat/0607426.[12] M. G. Vavilov and L. I. Glazman, Phys. Rev. Lett. 94,086805 (2005).[13] P. Simon, R. Lo´pez, and Y. Oreg, Phys. Rev. Lett. 94,086602 (2005).[14] A. Kaminski, Yu. V. Nazarov, and L. I. Glazman, Phys.Rev. B 62, 8154 (2000).[15] A. Rosch, J. Kroha, and P. Wo¨lfle, Phys. Rev. Lett. 87,156802 (2001).[16] S. Sachdev and R. N. Bhatt, Phys. Rev. B 41, 9323(1990).[17] A. A. Abrikosov, Physics (N.Y.) 2, 5 (1965).[18] O. Parcollet and C. Hooley, Phys. Rev. B 66, 085315(2002); J. Paaske, A. Rosch, and P. Wo¨lfle, Phys. Rev. B69, 155330 (2004).[19] A. Rosch, J. Paaske, J. Kroha, and P. Wo¨lfle, Phys.Rev. Lett. 90, 076804 (2003); J. Phys. Soc. Jpn. 74, 118(2005).[20] J. Paaske et al., Phys. Rev. B 70, 155301 (2004); NaturePhys. 2, 460 (2006).0 0.5 1 1.5 2eVR/K00.10.20.3(K/eVL)  dIL/dVR0.20.150.12nd PT0 0.5 1 1.5 200.10.20.30.050.10.51.01.51/ln(K/TK)eVL/KFIG. 4 (color online). Transconductance dIL=dVR in units ofe2=h multiplied by (K=eVL) versus eVR=K for fixed T=TK 0:1, probing voltage eVL=K  0:05, and different values ofK=TK. Inset: Transconductance for 1= lnK=TK  0:2 and dif-ferent eVL=K.0000000000 000000000066666666 66 66 66666666-2 -1 0 1 2eVL/K00.20.40.60.8dIL/dVLK6.06 64.02.01.51.20.90.60.30.00 00 2 4 6eVR/K00.20.4eVR/RGPTeVR/KFIG. 3 (color online). Differential conductance dIL=dVL inunits of e2=h for fixed T=TK  0:1 and 1= lnK=TK  0:2versus eVL=K at different values of eVR=K. Inset: dIL=dVL atVL ! 0 versus eVR=K. Solid circles: perturbative RG; opencircles: second order PT.PRL 99, 036807 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007036807-4

Transconductance of a double quantum dot system in the kondo regime

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Transconductance of a double quantum dot system in the Kondo regime

Transconductance of a double quantum dot system in the Kondo regime

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