Transport through side-coupled multilevel double quantum dots in the Kondo regime J. A. Andrade,1,2 Pablo S. Cornaglia,3,2 and A. A. Aligia3,2 1Centro At´omico Bariloche, CNEA, 8400 Bariloche, Argentina 2Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina 3Centro Ato´mico Bariloche and Instituto Balseiro, CNEA, 8400 Bariloche, Argentina Weanalyzethetransportpropertiesofadoublequantumdotdeviceintheside-coupledconfigu- ration. Asmallquantumdot(QD),havingasinglerelevantelectroniclevel,iscoupledtosourceand drainelectrodes. AlargerQD,whosemultilevelnatureisconsidered,istunnel-coupledtothesmall QD.AFermiliquidanalysisshowsthatthelowtemperatureconductanceofthedeviceisdetermined 4 by the total electronic occupation of the double QD. When the small dot is in the Kondo regime, 1 an even number of electrons in the large dot leads to a conductance that reaches the unitary limit, 0 while for an odd number of electrons a two stage Kondo effect is observed and the conductance is 2 strongly suppressed. The Kondo temperature of the second stage Kondo effect is strongly affected by the multilevel structure of the large QD. For increasing level spacing, a crossover from a large r a KondotemperatureregimetoasmallKondotemperatureregimeisobtainedwhenthelevelspacing M becomes of the order of the large Kondo temperature. 1 1 I. INTRODUCTION ] l Double quantum dots (DQDs) laterally defined in l a semiconductor heterostructures are highly tunable elec- h tronic devices whose energy level spacings and charging - energiescanbedeterminedbysettingthesizeandgeome- s e tryofeachdot,andtheircouplingbetunedusingmetallic m gates.1–6 The great tunability of the DQDs parameters, . as the interdot coupling, open the possibility of applica- t a tions in both classical and quantum computing,7–11 and m allowadetailedanalysisoftheinterplaybetweeninterfer- - ence and correlation phenomena.12–15 The properties of d these devices can be probed through transport measure- n mentsusingmetallicelectrodes. WhenoneQDhavingan o odd number of electrons and a single relevant electronic c FIG.1. (coloronline)Schematicrepresentationofthedouble [ levelistunnelcoupledtometallicsource-drainelectrodes, quantum dot device in the side-coupled configuration. QD d the electric conductance increases below a characteristic is coupled to left (L) and right (R) metallic electrodes. The 2 temperatureT whichsignalsthebuildupofKondocor- orbitals of QD D are coupled to QD d through a tunneling v K relations. The Kondo effect is associated to the screen- barrier. 3 6 ingoftheQDsmagneticmomentbytheFermiseaofthe 9 metallic electrodes. In the so-called side-coupled config- 7 uration (See Fig. 1), were only one of the QDs is cou- hybridization between the small QD and the leads is . pledtotheelectrodes,theinterdotcouplingcancompete smaller than the thermal energy, the Coulomb blockade 1 0 with the Kondo effect leading to a rich variety of corre- diamonds change their structure as a function of the in- 4 lated regimes. Namely, two-stage Kondo physics for a terdot tunneling coupling signaling a change in the un- 1 smallside-coupledQDhavingasinglerelevantelectronic derlying electronic structure associated to the multilevel v: level,13,15andatwo-channelKondoeffectforalargeside- nature of the QDs. i coupled QD.12 A device with an intermediate size of the In this article we analyze the transport properties of a X side-coupleddothasbeenpredictedtobearealizationof DQD device in the side-coupled configuration. We con- r the Kondo box problem.16–25 The properties of this type sider the multilevel nature of the side-coupled dot and a of devices have received considerable theoretical and ex- the intra- and inter-dot Coulomb interactions are prop- perimental attention. However, most theoretical studies erly taken into account. We obtain an exact relation be- have focused on simplified models considering a single tween the zero-temperature conductance and the charge interacting level on each dot,13,14,26–29 a continuum of in the DQD device (assuming energy independent lead- levels of the side-coupled dot,30,31 or no interactions on dot couplings) which generalizes Friedel’s sum rule for the large dot.32 thedouble-dotdevice. Weanalyzethedifferentstrongly- Recent experimental results for a DQD device in the correlated-electron regimes that occur at finite temper- side-coupled configuration,33 show a change in the level atures using Wilson’s numerical renormalization group structure of the DQD as a function of the interdot tun- (NRG)andcharacterizetheelectronictransportthrough neling coupling. In a spectroscopic regime, where the the device. 2 In the weak coupling regime (for temperatures higher the energy level splitting on QD D we include a single thanthecouplingtotheleads)weuseaperturbativeap- electron energy term: proach,startingfromtheexacteigenstatesforthesystem (cid:88) oftwodotsisolatedfromtheleads. Inthestrongcoupling He = (cid:15)˜Dαd†DασdDασ. (4) regime, we also use a slave-boson mean-field approxima- σ,α tion to help the interpretation of the NRG results. We Finally, showthattheusualeven-oddasymmetryintheCoulomb blockade valleys, due to the Kondo effect, can be com- (cid:88) (cid:88) (cid:104) (cid:105) H = V c† d +h.c. , (5) pletely altered in the large interdot hopping regime. V kν νkσ dσ The effect of increasing the interdot hopping on the ν=L,R k,σ conductance is very different for different occupations. In the strong-coupling regime, we show that there is a describes the coupling between QD d and the left (L) subtlecompetitionbetweenthelevelspacinginthelarge andright(R)electrodes, whicharemodeledbytwonon- quantum dot and the effective Kondo coupling between interacting Fermi gases: both dots, leading to a crossover at a very small energy (cid:88) scale related with a second-stage Kondo effect. Hel = (cid:15)kc†νkσcνkσ. (6) The rest of this article is organized as follows. In Sec. ν,k,σ IIwedescribethemodelandbasicformulasforthetrans- The conductance through the system is given by3,35,36 portcalculations. InSec. IIIwepresenttheconductance abnetdwteheenDthQeDeloecctcruopdaetsioanndintthheeDreQgDimfeoorfthweeaekx-pceoruimpleinng- G= e(cid:126)2ΓΓR+ΓΓL (cid:88)(cid:90) d(cid:15)(cid:20)−∂f∂((cid:15)(cid:15))(cid:21)Adσ((cid:15)). (7) tal parameters of Ref. [33]. In Sec. IV we present exact R L σ results for the zero temperature conductance. In Sec. V Here A ((cid:15)) is the spectral density of the small QD and we present numerical results for the conductance in the d we have assumed proportional (Γ ∝ Γ ) and energy Kondo regime. In Sec.VI we calculate the conductance L R independent dot-lead hybridization functions: andthemagneticsusceptibilityforamodelwithtwoand three quasidegenerate levels in the side-coupled QD. We Γ =2πρ (E )V∗ (E )V (E ). (8) L(R) L(R) F L(R) F L(R) F also analyze an effective model in the slave-boson mean- field approximation. where E = 0 is the Fermi energy of the electrodes, F ρ ((cid:15)) is the electronic density of states of the left L(R) (right) electrode, and V ((cid:15)) equals V for (cid:15)=(cid:15) . L(R) kL(R) k II. MODEL In the zero temperature limit: −∂f((cid:15)) →δ((cid:15)), and the ∂(cid:15) conductanceisproportionaltothespectraldensityatthe The double quantum dot device is described by the Fermi level (cid:80) A (0). As we show in Sec. IV assuming σ dσ following Hamiltonian a Fermi liquid ground state, A (0) can be written as a dσ function of the total electronic occupation of the DQD. H =HC +Ht+He+HV +Hel . (1) In the regime of weak lead-QD couplings or high tem- peraturesΓ=Γ +Γ (cid:28)k T,wecancalculatethecon- L R B Here H describes the electrostatic interaction in the C ductance through the system, to lowest order in Γ/k T, constant interaction approximation34 B replacing in Eq. (7) the exact spectral density A ((cid:15)) of dσ the isolated DQD: H = (cid:88) U(cid:96)(Nˆ −N )2 C (cid:96) (cid:96) 2 1 (cid:88) (cid:96)=d,D Adσ((cid:15))= Z (e−βEi +e−βEj)|(cid:104)Ψj|d†dσ|Ψi(cid:105)|2 +U (Nˆ −N )(Nˆ −N ), (2) i,j dD D D d d ×δ[(cid:15)−(E −E )], (9) j i where Nˆ = (cid:80) d† d is the occupation of the small d σ dσ dσ dot (QD d), Nˆ = (cid:80) d† d is the occupation of D σ,α Dασ Dασ the large dot (QD D), N = C V /U , C is the ca- (cid:96) g(cid:96) g(cid:96) (cid:96) g(cid:96) where |Ψ (cid:105) and E are the exact eigenfunctions and i i pacitance of dot (cid:96) with its corresponding gate electrode, eigenenergies of the DQD, and Z =(cid:80) e−βEi is the par- U is the charging energy and U is given by the QDs i (cid:96) dD tition function. Replacing this in Eq. (7) mutual capacitance.2,4–6 Ht =(cid:88)tαdD(cid:16)d†dσdDασ+h.c(cid:17), (3) G= e(cid:126)2kBΓT (cid:88)(Pi+Pj)f(Ei−Ej)f(Ej −Ei) i,j σ,α (cid:88) × |(cid:104)Ψ |d† |Ψ (cid:105)|2 (10) describes the tunneling coupling between the different j dσ i n orbitals on the side-coupled QD D and the smaller QD d having a single relevant electronic level. To describe where Pi =e−βEi/Z. 3 The spectral function of the DQD can be calculated non-perturbativelyasafunctionofthetemperatureusing Wilson’sNumericalRenormalizationGroup(NRG).37–39 This allows a calculation of the conductance in the full range of temperatures with a high precision. III. PERTURBATIVE RESULTS IN THE WEAK-COUPLING REGIME In this section we analyze the conductance through the DQD in a regime of weak coupling to the electrodes in which the thermal energy k T is larger than the hy- B bridization energy Γ. In this regime, Γ can be treated perturbativelyandtheelectrodesserveasaspectroscopic probe of the DQD in transport measurements. Conduc- tance maps are generated sweeping the gate voltages of FIG.2. (coloronline)Leftpanel: Conductancemapforadou- eachQD(parametrizedherebyN andN )whichmod- d D blequantumdotdeviceintheside-coupledconfiguration. The ify the total charge on the DQD and the distribution of interdot coupling is t = 0.01meV. Other parameters are dD charge between the QDs. For a DQD in the spectro- U =0.25meV, U =0.7meV, U =0.1meV, δ=0.02meV, D d dD scopic regime, a hexagonal structure is expected in the andk T =0.01meV.Rightpanel: Totalchargeofanisolated B conductance maps with conductance peaks occurring at DQD for the same parameters and gate voltages as the con- gate voltages such that there is a charge degeneracy on ductance map on the left panel. The lines indicate a change the DQD which allows charge fluctuations between the by one electron in the charge of the ground state of isolated DQDandtheleads.34 Fortheside-coupledconfiguration, DQD.TheDQDisemptyatthelowerleftcornerofthefigure andisoccupiedwitheightelectronsattheupperrightcorner. the conductance is only sensitive to charge fluctuations TheapproximateoccupationofeachQDisindicatedbetween on QD d which is connected to the electrodes. As it can parentheses as ((cid:104)Nˆ (cid:105),(cid:104)Nˆ (cid:105)). beinferredfromEq. (10),toobtainalargeconductance, d D atleasttwolevelsoftheDQDdifferingintheirchargeby oneelectronneedtobequasidegenerate(|E −E |(cid:46)k T i j B so that the product of Fermi functions is not exponen- oftheDQDwhichcoincidewiththoseofQDd. Asitcan tially suppressed). Furthermore, the electron must be be seen in the right panel of Fig. 2, these segments of fluctuating thermally in and out of QD d (in order for highconductanceareassociatedtoachangeinthecharge the matrix element |(cid:104)Ψ |d† |Ψ (cid:105)|2 to be sizable). of QD d. In this figure, the approximate occupation of j dσ i each QD at the different regions of the map is indicated InwhatfollowswefocusouranalysisonaDQDmodel between parenthesis ((cid:104)Nˆ (cid:105),(cid:104)Nˆ (cid:105)). The charging of the having three levels on the side-coupled QD with level d D large dot is accompanied by much weaker peaks in the energies(cid:15)˜ =−δ,(cid:15)˜ =0,and(cid:15)˜ =δ. Thissimplified D1 D2 D3 conductance due to a small mixing between the states of modelpresents,intheweak-couplingregime,manyofthe the two QDs. features observed in systems having a larger number of levels on the side-coupled QD (see Ref. [33]). To further For an intermediate interdot coupling (tdD ∼ δ, see simplify the discussion of the results, we consider tα to Fig. 3), the DQD states are in a molecular regime where dD be level independent and drop the level index α. the charge on each dot is not well defined as all states We calculate conductance maps of the DQD in the of the DQD involve a large orbital mixing between the spectroscopic regime using Ec. (10). The Hamiltonian QDs. This is reflected on the conductance map that of the isolated DQD is diagonalized to obtain its eigen- presents maxima at the charge degeneracy points of the vectors and eigenvalues for each value of N and N . In DQDwherethetotalchargeintheDQDchanges(asitis d D Figs. 2, 3, and 4 we present the results for a set of pa- indicated in the right panel of the figure). The QDs lose rametersobtainedfromtheexperimentofBainesetal.,33 theiridentityandbehaveasasingleQDwithaneffective and different values of the interdot tunnel coupling t . charging energy and coupling to the leads. dD In the regime of weak interdot coupling (t <δ), there For larger values of the interdot coupling t > δ the dD dD is little mixing of the states between QDs (except when system gradually enters a regime in which some of the thereisadegeneracyoftheenergylevelsofthetwoQDs) wavefunctionspresentastrongmixingbetweentheQDs andthechargeoneachQDisgenerallywelldefined. Due and other are mostly localized on one of the QD.33 For to the topology of the device, a peak in the conductance our simplified model with a single level on QD d and is expected at the lines of charge degeneracy in QD d, three levels on the side-coupled QD, one energy level is which is the one coupled to the electrodes. This is ob- associated to a bonding state between the level on QD d served on the left panel of Fig. 2 where segments of high andasymmetriccombinationoflevelsonQDD,andan- conductance are obtained at the charge degeneracy lines otherlevelisassociatedtothecorrespondingantibonding 4 FIG. 3. (color online) Left panel: Conductance map for FIG. 4. (color online) Left panel: Conductance map for t = 0.04meV. Right panel: the lines indicate a charge de- t =0.08meV.Rightpanel: thelinesindicatethechargede- dD dD generacyontheisolatedDQD,thechargeintheDQDchanges generacypointsoftheDQD,thechargeintheDQDchanges by one at each line and increases from zero at the lower left by one at each line and increases from zero at the lower left corner to eight at the upper right corner. Other parameters corner to eight at the upper right corner. The approximate are as in Fig. 2. occupation of each QD is indicated between parentheses on different regions of the map. Other parameters are as in Fig. 2. state. Thesestateshaveapproximately1/2oftheweight on each QD. The two remaining states of the DQD have measurestheasymmetryofthecouplingofthesmallQD most of their weight on the side-coupled QD. This par- to the left and right electrodes. This allows a direct es- ticularelectronicstructureoftheDQDisreflectedonthe timation of the zero-temperature conductance from the conductance of the device as it can be observed in Fig. charging diagrams presented on the right panels of Figs. 4. The highest conductance peaks are obtained when 2, 3, and 4. the bonding (ground state) and anti-bonding (highest energy) states change their occupation and much lower peaks are obtained when the two states mainly localized IV. EXACT RESULTS AT ZERO on QD D get charged. TEMPERATURE This peculiar structure of the wave functions in the regime of strong interdot tunnel-coupling can be easily Assuming that the ground state of the system is a understood in the non-interacting limit or the high ca- Fermi liquid, we show in Appendix A that the zero tem- pacitance limit where U = U = U , see Ref. [33]. D d dD perature spectral density of QD d is given at the Fermi It appears in the general case considering more levels on energy by each QD. In the latter case, some states of the DQD are strongly hybridized between the two QDs, being a 2 combination of several orbitals from each QD. The re- Adσ(0)= sin2(πNσ), (11) πΓ mainingstatesoftheDQDaremostlylocalizedonQDd or in QD D. The value of the hopping t at which the where dD crossover takes place depends on the level spacing δ and 1 (cid:90) 0 on the relative value of the intradot (Ud, UD) and inter- Nσ =− ImTr Gσ(ω)dω, (12) dot (UdD) Coulomb interactions. For UD = Ud = UdD π −∞ the interaction energy given by Eq. (2) depends only on isthetotalchargeperspinontheDQDandG (ω)isthe the total number of electrons on the DQD and is inde- σ localzerotemperatureGreen’sfunctionoftheDQD(see pendentofthestructureoftheelectronicwave-functions. Appendix A). Replacing Eq. (11) in Eq. (7) we obtain For U ,U > U , states strongly hybridized between D d dD the two QDs have a larger interaction energy than those (cid:88)e2 having a well defined number of electrons on each QD. G(T =0)=α sin2(πN ), (13) (cid:101) σ h In the next sections, we analyze the low temperature σ conductanceinthedifferenttunnelcouplingregimes. As we will show in the next section, the conductance maps where α(cid:101)= (Γ4LΓ+LΓΓRR)2. inthezerotemperaturelimitareonlydeterminedbythe For a symmetric coupling of the small QD to the left total occupation N of the DQD and a parameter that and right electrodes (Γ = Γ ) we have α = 1 and the L R (cid:101) 5 zero temperature conductance reaches the quantum of symmetric situation where Γ = Γ = Γ/2, (cid:15)˜ = 0, L R Dα conductancewhenthenumberofelectronsN intheDQD N = N = 1, and U = 0, so that the average d D dD isodd(N =N =N/2). Conversely,foranevennumber charge on each QD is 1. When both QDs are in the ↑ ↓ of electrons in the DQD the conductance vanishes. This Kondo regime (U ,U (cid:29) πΓ,t ) charge fluctuations d D dD important result indicates that the maximum amplitude on each QD can be eliminated using a Schrieffer-Wolff oftheconductanceisgivenonlybytheasymmetryofthe transformation53 and the low energy properties of the couplingtotheleftandrightleads,andtheremainingde- system can be described using a Kondo Hamiltonian: pendenceontheparameters,includingchargingenergies, level energies and tunnel couplings enters only through H =J S ·s +J S ·S +H (14) K K d 0 dD d D el the value of the occupation in the DQD. For fixed α, all (cid:101) sets of parameters that lead to a given occupation in the Here, S with (cid:96) = d,D are spin operators associated to (cid:96) DQD will lead to the same value for the conductance at theQDs,ands0 = 21(cid:80)s,s(cid:48)c†0sσs,s(cid:48)c0s(cid:48) istheelectronspin zero temperature. density on the orbital coupled to QD d. The coupling The conductance maps analyzed in the previous sec- constants, to leading order in H and H , are J = t V dD tion are expected to be completely modified in the zero 8t2 /(U +U ) and J =2Γ/U . dD d D K d temperature limit. Irrespectively of the charge distribu- For a sufficiently weak interdot coupling J the sys- dD tion inside the DQD, the conductance will be ∼ α(cid:101)2e2/h tem presents a two-stage Kondo effect. As the tem- in the valleys where the total number of electrons is an perature is reduced, the spin 1/2 of the QD coupled oddintegerandverysmallinthosewherethetotalnum- to the leads is Kondo screened at a temperature T ∼ K ber of electrons is even. In the Kondo regime, where the We−1/ρJK,whereρisthelead’slocaldensityofstatesat hybridization Γ is smaller than the local interactions Ud theFermilevelandW isahighenergycutoff. TheKondo andUD,theoccupationasafunctionofthegatevoltages effect on QD d generates a peak on its spectral density is well approximated by the occupation of the isolated of width ∼ k T , at the Fermi energy. This Kondo B K DQD away from the charge degeneracy points. At the peak can be associated to a renormalized Fermi liquid charge degeneracy points, the total charge in the DQD of quasiparticles having a density of states ∼1/k T .54 B K is N = k+1/2, where k is an integer and the conduc- Its emergence results in an increase of the conductance tance at T = 0 is e2/h. Based on the calculation of the through the device as the temperature is lowered. total occupation of the DQD as presented in the right ForJ <T thespinatQDD isscreenedatalower dD K panels of Figs. 2, 3, and 4, we expect the conductance temperature13 maps at T = 0 to consist of diagonal stripes of large caoltnedrnuacttainngcew(itdhelsimtriitpeeds obfylothweccohnadrugcetadnecgee.neracy lines) TK(cid:63) ∼TKe−πJTdDK. (15) In the next section we show that the temperatures at We can interpret this expression as the Kondo screening which the Fermi liquid behavior sets in and the above of the spin-1/2 in the side-coupled QD by the renormal- equationissatisfiedcan,insomecases,beextremelylow. ized quasiparticles associated to the Kondo effect in QD At intermediate temperatures, the conductance can be d. WhilethefirststageKondoeffectleadstoanincrease very different to the value expected in the high temper- intheconductanceofthedevice,thesecondstageKondo ature and the low temperature regimes. Depending on effect suppresses the conductance in agreement with Eq. the value of the gate voltages (i.e. at different regions of (13). This suppression of the conductance can be un- theconductancemaps),thesystemcanbeinthespectro- derstood as a Fano antiresonance, or interpreted as a scopic regime or the low temperature regime for a given blocking effect in the conductance due to the formation set of parameters. of a strong singlet between the spins on the two QDs at Equation (13) is valid for the interactions considered low energies. For J > T the spins in the QDs are dD K in the Hamiltonian of Eq.(1). Its important to point out alsolockedinasingletnotonlyatlowtemperatures,but that different interactions to the ones considered (e.g. a already for temperatures T > T . The conductance is K Hund rule coupling in one of the QD) may change the small for T < J and the Kondo screening does not dD nature of the ground state and the value of the zero- takes place. Shifting the gate voltage of QD d to mod- temperature conductance.40–51 ify its occupation to ∼ 0 or ∼ 2 results in a single stage KondoeffectwithaneffectiveKondocouplingthatleads to a high conductance regime at low temperatures.55 V. NUMERICAL RESULTS Whentheside-coupledQDineitheremptyordoubleoc- cupied and QD d is in the Kondo regime, the transport We now discuss the temperature dependence of the properties are dominated by the Kondo effect on QD d. conductance. We first review the main results for a For sufficiently large tunnel coupling t the properties dD case with a single relevant level on each QD. This prob- ofthesystemcanbebetterunderstoodconsideringbond- lem has been studied using a variety of techniques in- ing and antibonding states formed between the two QDs cluding the Numerical Renormalization Group13,14 and and including effective Coulomb interactions for the hy- FunctionalRenormalizationGroup.52 Wefirstconsidera bridized levels and effective couplings to the leads.13 6 1 energylevelsofQDD inordertoincreasethehybridiza- h] 0.8 tionwithQDd. Inthisregime,thewave-functionweight 2e/ 0.6 tdD on QD d associated to the spin-1/2 is reduced and the 2 Kondo coupling between QD d and the electrodes is re- [ 0.4 G ducedaccordingly,asitcanbereadilyshownpreforming 0.2 aSchrieffer-Wolfftransformation. Whent isincreased dD 0 further, the level structure of the DQD changes (see also 1 Ref. [33]). Two electrons occupy the lowest lying state /h] 0.8 oftheDQDwhichisabondingstatebetweentheorbital 22e 0.6 tdD on QD d and a linear combination of the orbitals on QD G[ 0.4 D. ThethirdelectronismainlylocalizedinQDD which 0.2 leads to a strongly suppressed Kondo coupling and an 0 exponentially suppressed Kondo temperature. For the 10−5 10−4 10−3 T/Γ 10−2 10−1 100 highest values of tdD considered, the Kondo tempera- tures reach values well below the experimentally accessi- ble range (see top panel in Fig. 5). FIG. 5. (color online) Conductance for different values of theinterdotcouplingt =0.01, 0.015, 0.02, 0.03, 0.04, 0.05, On the bottom panel of Fig. 5 we consider a situation dD and 0.09 meV. The other parameters are U = 0.25meV, with a single electron of QD d and 3 electrons on QD D. D Ud = 0.7meV, UdD = 0.1meV, δ = 0.02meV, and Γ = In this case, the Fermi liquid theory predicts a vanishing 0.2meV. a) Total occupation of tree electrons: Nd = 1 and conductance at zero temperature. The ground state of ND = 2. b) Total occupation of four electrons: Nd = 1 and the isolated DQD has total spin equal to zero. As in the N =3. D case of a single level in QD D, if the singlet-triplet gap of the DQD is smaller than the Kondo temperature for QD d, a two stage Kondo effect occurs as the tempera- As we show below, the main features observed in the ture is decreased. First the spin 1/2 on QD d is Kondo single level case for the temperature dependence of the screened by the electrons and holes of the electrodes and conductance, are also observed when multiple levels are thelowenergyexcitationsofthesystemcanbedescribed consideredintheside-coupledQD.Wecalculatethecon- by a local Fermi liquid of heavy quasiparticles in QD d. ductance through the system and the magnetic suscepti- A second stage Kondo effect is observed as the spin 1/2 bility using the full density matrix numerical renormal- on QD D is Kondo screened by the Kondo quasiparti- ization group (FDM-NRG)56,57. In all calculations we cles on QD d. The Kondo screening of the side-coupled use a logarithmic discretization parameter Λ = 10 and QD leads to a decrease in the conductance below a char- the z-trick58,59 averaging over four values of z = 0.25, acteristic T(cid:63) which depends strongly on the antiferro- K 0.5, 0.75, and1. Wekeepupto5000statesandstartthe magnetic coupling between the QDs [see Eq. (15)]. This truncation after 4 NRG iterations. behavior can be observed in the bottom panel of Fig. 5 where for the lowest values of t the conductance In Fig. 5 we present the conductance as a function dD presents a non-monotonous behavior. As the interdot of the temperature for the multilevel system analyzed in coupling is increased, the Kondo temperature of the sec- the previous section. On the top panel of Fig. 5 we ondstageKondoeffectincreasesexponentially. Forlarge consider a situation with a single electron on QD d and ∼ 2 electrons on QD D. For small t (cid:46) 0.02meV the enough values of tdD the antiferromagnetic coupling be- dD tween the QDs exceeds T , there is no Kondo effect and results are very similar to what is observed in a decou- K theconductancedecreasesmonotonously. Whilequalita- pled QD situation (t = 0): there is an increase in the dD tivelythebehavioroftheconductanceisthesameasthe conductance associated to the Kondo screening of the one observed for a single level on the side-coupled QD, spin-1/2 on QD d. The conductance reaches the quan- the value of the second stage Kondo temperature can be tum of conductance, as expected from the Fermi liquid strongly modified in the multilevel case. As we show in predictions. In this regime, the Kondo temperature is the next section, it can have a strong dependence on the only slightly reduced by the presence of the side-coupled level spacing δ on QD D. QD. As t is increased there is a decrease of the Kondo dD temperature which sets the temperature scale at which The results of Fig. 5 describe qualitatively the behav- the Fermi liquid behavior is recovered. This reduction of ior of the conductance away from the charge degeneracy the Kondo temperature can be understood recalling the lines in the valleys with an odd or an even number of analysis of the nature of the electronic wave functions in electrons in the DQD. The temperature scales for the the different interdot coupling regimes presented in Sec. different regimes, however, can vary strongly from one III. For t = 0 two electrons occupy the ground state valley to the other. This is illustrated in Figs. 6 and 7 dD of QD D forming a singlet and there is a single electron where the behavior of the conductance as a function of on QD d; a situation that remains essentially unaltered the temperature and N is presented for N = 1 and D d for t (cid:28) δ. For t (cid:38) δ, however, it becomes ener- N = 0.2, respectively. For N = 1, and a weak inter- dD dD d d getically favorable to have a sizable occupation in higher dot coupling t = 0.015meV, a clear even odd asym- dD 7 FIG. 6. (color online) Conductance maps for N = 1 and FIG. 7. (color online) Same as Fig. 6 for N =0.2. d d different values of t . Other parameters as in Fig. 5. An dD even-oddasymmetryisobservedinthelowtemperaturecon- ductance. metry can be observed in the low temperature conduc- the QDs and has therefore a large Kondo temperature. tanceasthechargeinQDDismodifiedbythegatevolt- The valleys for N = 2,4, however, have the magnetic age. A large (small) conductance is obtained for the val- D momentmainlylocalizedonQDDleadingtoaverysmall leys with an odd (even) total number of electrons in the Kondo temperature that for t =0.1meV is well below DQD. There is, however, an intermediate temperature dD the temperature range numerically explored (excepting regime T(cid:63) <T <T where this even-odd asymmetry is K K valuesofN veryclosetothechargedegeneracypoints). lost. The behavior of the conductance in this parameter D regime is qualitatively described by the Hamiltonian of Eq. (14). The behavior of the conductance for N = 0.2 is pre- d Fort =0.035meVtheDQDisinamolecularregime, sented in Fig. 7. In this case a large conductance is dD andthetwostageKondoeffectisnotobserved. Thesys- expected at zero temperature for odd values of N . For D tem behaves as a single QD that presents the Kondo ef- low interdot coupling t = 0.015meV [see Fig. 7a)]the dD fect for an odd number of electrons on the DQD, and Kondo temperature is, however, well below the range of no Kondo physics otherwise. In the Kondo regime a temperatures numerically explored. As t is increased dD Schrieffer-Wolff transformation can be performed to ob- the Kondo temperature increases and becomes higher tain the Kondo coupling that is determined by the effec- than the minimum temperature considered close to the tive charging energy of the DQD and the magnetic mo- charge degeneracy points [see Fig. 7 b)]. For larger val- ment of QD d in the ground-state wave function of the ues of t [see Figs. 7c) and d)] the Kondo temperature dD isolated DQD. For larger values of t the Kondo tem- for N ∼1 is strongly enhanced as the interdot bonding dD D perature for some of the valleys is strongly suppressed state is formed. In the valley with N (cid:39) 3, the Kondo D as the orbital on QD d strongly hybridizes with a com- temperaturereachesamaximuminthemolecularregime bination of orbitals on QD D and the remaining wave t ∼δandbecomesstronglysuppressedforlarget as dD dD functions are mainly localized on QD D. The odd valley the magnetic moment in the DQD becomes increasingly with N ∼0 is associated to the bonding state between localized on QD D. D 8 1 1 0.9 0.9 0.8 0.8 0.7 0.7 /h] 0.6 /h] 0.6 2e 0.5 2e 0.5 2 2 G[ 0.4 G[ 0.4 N =1,N =7 0.3 0.3 N =2,N =6 0.2 0.2 N =3,N =5 0.1 0.1 N =4 0 0 1 2 3 4 5 6 7 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 N T/TK FIG.8. (coloronline)Conductancevs. totalDQDoccupation FIG.9. (coloronline)Conductancefordifferentvaluesofthe for a three level side-coupled QD. NRG (solid dots) and the occupationintheDQD.Foranevenoccupation,theconduc- analytic result (lines) of Eq. (13) . The gate voltage in the tanceissuppressedforT (cid:46)T(cid:63) wherethesecondKondostage QD d is set such that its average occupation is one, while K sets in. Parameters are as in Fig. 8. the gate voltage in QD D is varied to change its occupation from zero to six electrons. Parameters are N = 1, U /Γ = d D U /Γ = 7.143, U = 0, t /Γ = 0.0693, δ/Γ = 0.001428, d dD dD wide range of model parameters. andΓ=0.07inunitsofthehalf-bandwidthoftheconduction Figure 9 presents the conductance as a function of the band of the electrodes. temperature for different values of the total DQD occu- pation and the parameters of Fig. 8. For an odd num- ber of electrons in the DQD, the occupation per spin is VI. KONDO EFFECT IN MULTILEVEL N = N/2 and Eq.(13) leads to an unitary conductance QUANTUM DOTS σ at zero temperature G(T = 0) = 2e2/h. The observed increaseintheconductancewhenthetemperatureislow- In this section we analyze the effect of having several eredisassociatedtothebuildupofKondocorrelationsas electronic levels on QD D on the transport properties of the spin in QD d is screened. The transport properties thedeviceandontheKondocorrelationsinthetwo-stage are essentially unaltered by the presence of QD D when Kondo regime. We consider a few levels on QD D and it is charged by an even number of electrons. For an calculate the conductance for the full range of values of eventotalnumberofelectronsintheDQDthetwostage the level spacing δ. Since we consider a finite number of Kondo effect is obtained, as in the single level case, and levels, the limit of small level spacing δ →0 describes fi- thezero-temperatureconductancevanishes. Thesecond- nitesizeQDwithadegenerategroundstate. Weanalyze stageKondotemperaturegenerallydependsonthenum- a DQD system in a parameter regime which is different ber of electrons on QD D, here is the same for total to the one analyzed in the previous sections. We set occupation in the DQD of N and 8−N electrons due U = U = U, U = 0, and t (cid:28) U ,U . Now the D d dD dD d D to the electron-hole symmetry for the set of parameters Coulomb interactions determine the charge distribution considered. in the DQD, in particular the bonding state obtained in We now consider the effect of the level spacing δ on the tdD >δ regime is unfavored by the absence of inter- the second-stage Kondo temperature T(cid:63). When there is dot Coulomb repulsion (U = 0). This can be easily K dD a single electron on QD D, there are two limiting cases seen calculating the expectation value of the interaction that can be readily solved. For δ → ∞ there is a single energy term [Eq. 2] for a DQD wave-function with a relevant level on QD D and the problem reduces to the single electron on each QD or the two electrons on the single level case. For δ = 0, we can perform a change of bonding state. basisonthedegenerategroundstateofQDD, suchthat We first analyze the validity of Eq. (13) for a sys- H couples the level on QD d to only one of the states t tem having 3 quasidegenerate levels (δ (cid:28) tdD) on the of the new basis. Namely, the symmetric combination of side-coupled QD calculating the conductance using the the original orbitals on QD D. NumericalRenormalizationGroup. Wecheckedthatthe renormalization procedure had converged to the low en- H =√n t (cid:88)(cid:16)d† d˜ +h.c.(cid:17) (16) t D dD dσ Dσ ergy fixed point to calculate the conductance and the σ occupation. The results are presented in Fig. 8. The √ gate voltage of QD d was kept fixed (N = 1) and the where d˜ = (cid:80) d / n , and n is the degener- d Dσ α Dασ D D occupation on QD D was swept from 0 to 6 by changing acy of the ground state. The problem reduces again it’s gate voltage. There is an excellent agreement with to the single level case but with a larger hopping am- √ the Fermi liquid predictions. The small discrepancy be- plitude t˜ = n t . Performing Schrieffer-Wolff dD D dD tween the numerical and analytical results is due to a transformation to the single level problem we obtain small loss of spectral weight in the numerical solutions. J (δ = 0) = n J (δ → ∞), leading to a much dD D dD Identical results to those in Fig. 8 were obtained for a largersecond-stageKondotemperatureintheδ =0case 9 1 1 100 0.8 δ 2G[2e/h] 000...0246 eloccupation 000...468 N1 111000−−−321KKT/T(δ=⋆⋆ 00..56 Lev 0.2 N2 10−40) 0.4 2µ 00..23 δ 010−3 10−2 10−1 100 101 102 103 104 10510−5 0.1 δ/TK⋆(δ=0) 0 10−8 10−6 10−4 10−2 100 102 FIG. 11. (color online) Slave-boson mean-field theory results T/T fortheleveloccupationandtheKondotemperatureasafunc- K tionoftheenergylevelspacingδ inQDD. Forδ largerthan FIG.10. (coloronline)Conductance(toppanel)andeffective TK(δ = 0) , it becomes energetically favorable to form the magneticmomentsquared(bottompanel)asafunctionofthe Kondo correlations with the lowest lying energy level. This temperature for different values of the energy level spacing reducestheexchangecouplingwiththeelectronbathandthe δ/T(cid:63)(δ = 0) = 0,0.004,0.4,3.6,7.2, and δ → ∞ on QD D. Kondo temperature. K Other parameters as in Fig 8. Only for δ >T(cid:63)(δ =0) there K is a sizable change in T(cid:63). K odd (3). The system presents a two stage-Kondo effect as it can be observed in the figure. The conductance in- [T(cid:63)(δ =0)(cid:29)T(cid:63)(δ →∞)]. creasesattemperaturesbelowthefirst-stageKondotem- K K We therefore expect to obtain a crossover from a large perature T which is essentially independent of δ. The K KondotemperaturetoasmallKondotemperatureasδis secondstage-KondotemperatureT(cid:63) decreasesasδ isin- K increasedfromzero. PerformingaSchrieffer-Wolfftrans- creased but only when δ exceeds T(cid:63)(δ = 0). Note that K formation in a system with two levels in QD D we have for δ → ∞, the second stage Kondo temperature falls fortheHamiltonianoftheisolatedDQDtoleadingorder below the minimum temperature numerically explored. in Ht and δ/U The behavior of µ2 as a function of the temperature shows a contribution of the three electrons on QD D (cid:88) HDd =J1Sd·SD1+J2Sd·SD2+δ d†D2σdD2σ and suggests that a single electron in an effective or- σ bitalcouplesantiferromagneticallytoQDdandisKondo (cid:34) (cid:35) screened60 while the remaining two electrons are doubly (cid:88) +J12 d†D2σdD1σSd·SD1+h.c. (17) occupied and effectively decoupled from the rest of the σ system. For T (cid:29)δ these electrons levels contribute with µ2 =1/3 to the magnetic moment squared of the DQD with D2 and with zero for T (cid:28) δ presenting a fast crossover be- tweenthesetworegimesasafunctionofthetemperature t2 in contrast to the slow Kondo screening. J =4 dD 1 U As it was mentioned in the previous section, in the (cid:18) δ2 (cid:19) second-stage Kondo regime, one can consider that the J2 =J1 1+ U2 (18) magnetic moment on QD D couples to a renormal- ized Fermi liquid of quasiparticles formed by the Kondo J +J J = 1 2 screening of the magnetic moment of QD d. In the next 12 2 sectionweassumethatthefirst-stageKondoeffectiswell developed and explore the crossover of T(cid:63) as a function where, S with(cid:96)=1,2arespinoperatorsassociatedto K D(cid:96) of δ using a slave-boson theory in the saddle point ap- the levels of QD D. proximation. As we show below, the crossover from a large Kondo temperature to a small Kondo temperature does not oc- cur at δ ∼ J , as it might be naively inferred from an 1 analysis of the isolated DQD, but at a much smaller en- A. Slave-boson mean-field theory ergy scale δ ∼T(cid:63)(δ =0). K In Fig. 10 we present the conductance and the mag- Our starting point is the Hamiltonian of Eq. (17) for netic moment squared µ2 of the DQD as a function of a system with two levels on QD D and a single electron the temperature for different values of the energy level oneachQD.WeassumethatthefirststageKondoeffect spacing in QD D. The parameters are the same as in is well developed an take the local density of states on Fig. 9 and the number of electrons in the large dot is QD d as that of a non-interacting band with a density of 10 states ρ ∼1/πk T and a bandwidth πk T . The integrals in Eq. (23), Eq. (24), and Eq. (25) eff B K B K For simplicity we consider δ (cid:28)U and take J =J = can be solved analytically at T = 0. The resulting 1 2 J =J. The resulting Hamiltonian is: equations were solved numerically for the level occupa- 12 (cid:34) (cid:35) tions in QD D and the Kondo temperature T(cid:63) = ∆(cid:101) = k Hml =J SD·s(cid:48)0+(cid:88)d†D2σdD1σSD1·s(cid:48)0 (19) πρeffJ(cid:101)2. The results are presented in Fig. 11. We ob- serve a slow crossover from a large Kondo temperature σ +δd†D2σdD2σ+He(cid:48)l, (20) TK(cid:63)(δ = 0) = T2Ke−π2TJK to a small Kondo temperature where SD = SD1 + SD2 and s(cid:48)0 = 21(cid:80)s,s(cid:48)f0†sσs,s(cid:48)f0s(cid:48) δTK(cid:63)(cid:38)(δT→K(cid:63)(δ∞=) =0)Tw2Khee−reπTJtKh.eTlehveelsloocwcucpraotsisoonvserNs1etasnidnNfo2r is the electron spin density of the quasi-particles of the begin to differentiate. For large δ (cid:29)T(cid:63)(δ =0) only the Fermi liquid where f0†s (f0s) creates (destroys) a quasi- lowest lying level is occupied. K particle on QD d, and Ifweassociateanenergygain∼T(cid:63) totheformationof K (cid:88) the Kondo effect, for small δ it is energetically favorable H(cid:48) = (cid:15)(cid:48)f† f . (21) el k νkσ νkσ to occupy partially the highest energy level on QD D ν,k,σ forming a symmetric combination of the two levels in QD D in order to increase the hybridization with QD d The biquadratic interactions between the pseud- and the Kondo temperature. The energy in this case is ofermionsgeneratedbythespin-spininteractionsarede- ∼ −T(cid:63)(δ = 0)+δ/2. For δ > 2T(cid:63)(δ = 0) it is however coupled, introducing two Bose fields B (conjugate to K K i the amplitude (cid:80) d† f with i = 1,2) and the con- energetically favorable to empty the highest energy level σ Diσ 0σ (N → 0) at the expense of having lower energy gain 2 straint on the occupation of QD D is enforced introduc- due to the Kondo effect ∼ −T(cid:63)(δ → ∞). We therefore ingtheLagrangemultiplierλ. Thefreeenergyexpressed K expect a crossover to occur between the two regimes for in terms of the Bose fields has a saddle point at which δ ∼T(cid:63)(δ =0). the latter condense, (cid:104)B (cid:105) = (cid:104)B†(cid:105) = b = (cid:104)f d† (cid:105). At K i i i 0σ Diσ the saddle point the effective Hamiltonian is VII. CONCLUSIONS (cid:88)(cid:16) (cid:17) H(cid:101) =J(cid:101) d†Dασf0σ+h.c +δd†D2σdD2σ (22) σ,α We analyzed the electronic transport through a DQD +(cid:88)λ(d† d −1)+H(cid:48) , device in the side-coupled configuration. A small quan- Dασ Dασ el tumdothavingasinglerelevantelectroniclevelistunnel- σ,α coupled to source and drain electrodes, a larger QD is coupled to the small QD but not directly to the elec- where J(cid:101)=J(b1+b2). Using the equations of motion of trodes (side-coupled QD). We considered multiple levels the operators d we obtain: Diσ on the side-coupled QD and Coulomb interactions be- tween electrons in the DQD. Assuming a Fermi liquid 1 (cid:90) TK (cid:104) (cid:105) ground state, we obtained an exact relation between the b1 =− dωf(ω)Im G(cid:101)01σ , (23) zero-temperature conductance and the electronic occu- π −TK pation of the DQD. Additional interaction terms in the systemHamiltonian,asaHund’srulecoupling,40–51 may 1 (cid:90) TK (cid:104) (cid:105) leadtonon-Fermiliquidgroundstatesandtoamodified b2 =− dωf(ω)Im G(cid:101)02σ , (24) zero-temperature conductance formula.42 π −TK We explored numerically the conductance through the systemformodelparametersappropriatetodescribetwo 2 (cid:88)(cid:90) ∞ (cid:104) (cid:105) experimentally relevant situations: i) a system with a Ni =− dωf(ω)Im G(cid:101)iiσ , (25) large level spacing on the side-coupled QD and ii) a sys- π σ −∞ tem with up to three quasidegenerate levels in the side- whereN =(cid:80) (cid:104)d† d (cid:105)isoccupancyofthedifferent coupled QD. In the latter case we considered that the i σ Dασ Dασ quasidegeneratelevelsweretheonlyrelevantfortheelec- levels on QD D, G(cid:101)iiσ and G(cid:101)0iσ are the Green’s function tronic transport, as it is expected if the remaining levels atthei-thlevelofQDD andthecorrelatorofthequasi- in the side-coupled QD have a much larger energy. In particleswitheachlevelofQDD respectively. Theseare theformercase,whichdescribestheparameterregimeof given by Ref. [33], only a few levels in the side-coupled QD need tobeconsideredtoobtainaqualitativedescriptioninthe −1 weak electrodes-QD coupling regime. i J(cid:101) J(cid:101) πρeff We analyzed the low-temperature conductance and G(cid:101)σ = J(cid:101) ω−λ 0 . (26) the Kondo correlations for the interdot tunnel-coupling J(cid:101) 0 ω−δ−λ crossover observed in Ref. [33]. Depending on the parity