The role of the indirect tunneling processes and asymmetry in couplings in orbital Kondo transport through double quantum dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p The role of the indirect tunneling processes and asymmetry in couplings in orbitalKondo transport through double quantum dots
Piotr Trocha ∗ Department of Physics, Adam Mickiewicz University, 61-614 Pozna´n, Poland (Dated: November 7, 2018)System of two quantum dots attached to external electrodes is considered theoretically in orbitalKondo regime. In general, the double dot system is coupled via both Coulomb interaction anddirect hoping. Moreover, the indirect hopping processes between the dots (through the leads) arealso taken into account. To investigate system’s electronic properties we apply slave-boson meanfield (SBMF) technique. With help of the SBMF approach the local density of states for bothdots and the transmission (as well as linear and differencial conductance) is calculated. We showthat Dicke- and Fano-like line shape may emerge in transport characteristics of the double dotsystem. Moreover, we observed that these modified Kondo resonances are very susceptible to thechange of the indirect coupling’s strength. We have also shown that the Kondo temperature becomesuppressed with increasing asymmetry in the dot-lead couplings when there is no indirect coupling.Moreover, when the indirect coupling is turned on the Kondo temperature becomes suppressed.By allowing a relative sign of the nondiagonal elements of the coupling matrix with left and rightelectrode, we extend our investigations become more generic. Finally, we have also included the levelrenormalization effects due to indirect tunneling, which in most papers is not taken into account.
PACS numbers: 73.23.-b, 73.63.Kv 72.15.Qm, 85.35.Ds
I. INTRODUCTION
Originally, Kondo effect was discovered in non-magnetic metal containing magnetic impurities at lowtemperature. The effect comes from strong electron cor-relations and can be regarded as interactions of the impu-rity spin with cloud of the conduction electrons in metal.Scattering of the conduction electrons from the localizedmagnetic impurities leads to increase of the resistivity atlow temperature. In recent two decades one could no-tice revival of this effect as it was predicted and observedin transport through quantum dots.
Although, in thiscase Kondo resonance leads to increasing conductivity(not resistivity as in original Kondo effect) with decreas-ing temperature below so-called Kondo temperature T K ,the physical mechanism of the phenomena is common.Here, the role of the magnetic impurity plays a spin onthe dot.However, despite of regarding spin degree of freedomone may assume any two-valued quantum numbers, as forinstance, an orbital degree of freedom, to realize Kondoeffect. In the case of (at least) two discrete orbital levelscoupled to the external leads one may deal with orbitalKondo effect. To explain mechanism of creation of theKondo state let us introduce spinless electrons in a sys-tem of two single-level quantum dots coupled to externalleads as shown in Fig. 1. The dots’ energy levels are wellbelow the Fermi level of the leads. Due to large inter-dot Coulomb repulsion only one of them is occupied byan electron. On the other hand, removing an existingelectron on the double dot requires adding energy to thesystem. Thus, the system is in the deep Coulomb block-ade and the sequential tunneling events are prohibited.However, according to the Heisenberg uncertainty princi-ple, the higher-order processes, however, may appear on a very short time scale. Assuming that initially the upperdot was occupied, then it can tunnel onto the Fermi levelof the lead (left or right) and simultaneously another elec-tron from the Fermi level of the left or right electrode maytunnel to the lower dot. As a result, the charge exchangeoccured between the dots. A coherent superpostion ofsuch coherent events give rise to the sharp resonance inthe density of states at the Fermi level.Altough orbital Kondo effect has been investigatedin different geometries of the two orbital level systemboth experimentally and theoretically there is alittle comprehensive studies on the influence of the indi-rect coupling and/or asymmetry couplings on this phe-nomenon. In the case of spin Kondo effect most of re-searchers assume the maximal value of this coupling.
However, the maximal value of the indirect coupling inthe case of the orbital Kondo effect may leads to suppres-sion of Kondo resonance. Specifically, this effect is de-stroyed when the magnetic flux achieves 2 nπ ( n ∈ Z ) due to the formation of the bound state in the continuum(BIC) . Recently, Kubo et al. have investigated bothspin and orbital Kondo effect in double quantum dot re-garding non-maximal values of the indirect coupling .They have found that in the condition of intermediate in-direct coupling strength the differential conductance re-veals two kinds of peaks. Altough, we considered similarsystem, the role of this paper is to give more insight intothe connection between the role of the indirect couplingand various interference effects. Moreover, we assumedifferent asymmetry coupling of the dots to the externalleads. More specifically, in the system considered hereone dot is coupled to the leads with constant strength,whereas the coupling of the second dot to the leads canbe continuously tuned. Time reversal symmetry impliesthat the amplitude of the indirect coupling strength maychange sign . Thus, we also include this case in ourconsideration.In very recent experiment Tarucha et al. observedthat the period of the Aharonov-Bohm oscillations arehalved and the phase changes by half a period for theantibonding state from those of the bonding. They con-clude that these features can be related to the indirectinterdot coupling via the two electrodes.We also point out that similar double dot systemhas been investigated in Refs. [29,30] where the authorshave found a novel pair of correlation-induced resonancesas well as they studied the charging of a narrow QDlevel capacitively coupled to a broad one. However, thecorrelation-induced effects are not related to the Kondophysics.As we mentioned before various quantum interferenceeffects, which were previously reported in atomic physicsor quantum optics , were also discovered in elec-tronic transmission through QDs systems attached tothe leads . Here, we consider Dicke and Fano res-onances. In original Dicke effect one observes in spon-taneous emission spectra a strong and very narrow reso-nance which coexist with much broader line. This occurswhen the distance between the atoms is much smallerthan the wavelength of the emitted light (by individualatom). The former resonance, associated with a statewhich is weakly coupled to the electromagnetic field, iscalled subradiant mode, and the latter, strongly coupledto the electromagnetic field refers to superradiant mode.In the case of electronic transport in mesoscopic systems(for instance, quantum dots) this effect is due to indirectcoupling of QDs through the leads. Generally, indirectcoupling can lead to the formation of the bonding and an-tibonding states. As a result, a broad peak correspondingto the bonding state and a narrow one referring to theantibonding state emerge in the density of states .Let us now tell something about Fano effect. Inexperiment, Fano effect manifests itself as asymmetricline shape in emission spectra. It comes from quan-tum interference of waves resonantly transmitted througha discrete level and those transmitted nonresonantlythrough a continuum of states. The effect was ob-served in optics and also in electronic transport throughQDs systems . However, in this case the Fano phe-nomenon is due to the quantum interference of electronwaves transmitted coherently through the dot and thosetransmitted directly between the leads .It is convenient to associate resonant channel with dis-crete level and nonresonant channel with continuum ofstates. When the electron wave passes through resonantchannel its phase changes by π (within Γ), whereas thephase of electron waves in nonresonant channel changesvery slowly around the resonant level (Γ is the widthof the discrete level). (Of course, Fano effect occursonly when discrete level is embedded into a continuum.)Consequently, on the one side of the discrete level elec-tron waves through two channels interfere constructively,whereas on the other side they interfere destructively. As Γα L Γ R Γ L L R t QD1QD2 Γα R FIG. 1: Schematic picture of the double dot system. Theparameter α takes into account difference in the coupling ofthe two dots to external leads ( α ∈ h , i ). Tuning parameter α one can change geometry of the system from the parallelone for α = 1 to the T-shaped geometry for α = 0. a result, one observes asymmetric line in conductancearound the discrete level position.This effect can be also observed in system of two quan-tum dots embedded in two arms of AB ring or inthe so-called T geometry . Here, very narrow (broad)level, which is weakly (strongly) coupled to the leads,corresponds to the resonant (nonresonant) channel. Thenarrow level must appear within the broad one. As men-tioned before the difference in the coupling strengths ofthe bonding and antibonding states is due to indirect cou-pling. The phase shift of wave function in the broad levelis negligible when the energy changes within the narrowlevel and Fano resonance may appear.In this paper the orbital Kondo effect in electronictransport through two coupled quantum dots is consid-ered theoretically. Generally, the quantum dots may in-teract via both Coulomb repulsion and hopping term.To calculate local density of states (LDOS) for bothdots, transmission, and differential conductance we em-ploy slave-boson mean field approach. To show the for-mation of the bound state in the continuum as the indi-rect coupling strength approaches its maximal value wecalculate the Friedel phase. Due to emergence of the BICthe Fiedel phase, usually continuous, changes abruptlyat the energy corresponding to the BIC. Finally, to bemore familiar with experiment we show differential con-ductance.The paper is organized as follows. In Section 2 we de-scribe the model of a double-dot system which is takenunder considerations. We also present there the slave-boson mean-field technique used to calculate the basictransport characteristics. Numerical results on the or-bital Kondo problem are shown and discussed in Section3. Final conclusions are presented in Section 4. II. THEORETICAL DESCRIPTIONA. Model
To investigate various interference effects in Kondoregime we consider (spinless) Anderson Hamiltonian fordouble quantum dots coupled to external leads. Exper-imentally, such system may be realized by applying amagnetic field which lifts spin degeneracy of each dot.Generally, this Hamiltonian consists of three parts,ˆ H = ˆ H c + ˆ H DQD + ˆ H T , (1)where the first term, ˆ H c , describes nonmagnetic elec-trodes in the non-interacting quasi-particle approxima-tion, ˆ H c = ˆ H L + ˆ H R , with ˆ H β = P k ε k β c † k β c k β (for elec-trodes β = L , R). Here, c † k β ( c k β ) creates (annihilates)an electron with the wave vector k in the lead β , whereas ε k β denotes the corresponding single-particle energy.The next term of Hamiltonian (1) describes two cou-pled quantum dots,ˆ H DQD = X i ε i d † i d i + t ( d † d + h.c. ) + U n n , (2)where n i = d † i d i is the particle number operator, ε i is thediscrete energy level of the i -th dot ( i = 1 , t denotesthe inter-dot hopping parameter (assumed real), whereas U is the inter-dot Coulomb integral.The last term, H T , of Hamiltonian (1) describes elec-tron tunneling between the leads and dots, and takes theform ˆ H T = X k β X i =1 , ( V βi k c † k β d i + h . c . ) , (3)where V βi k are the relevant matrix elements.Finite widths of the discrete dots’ energy levels comefrom coupling to the external leads and may be expressedin the form Γ βii ( ε ) = 2 π | V βi k | ρ , where ρ denotes den-sity of states in the left and right lead ( ρ L = ρ R ≡ ρ ).Furthermore, we assume that Γ βii is constant within theelectron band, Γ βii ( ε ) = Γ βii = const for ε ∈ h− D, D i , andΓ βii ( ε ) = 0 otherwise. Here, 2 D denotes the electron bandwidth.For the system taken under considerations, the dot-lead couplings can be written in a matrix form, Γ β = (cid:18) Γ β Γ β Γ β Γ β (cid:19) , (4)where the off-diagonal matrix elements are assumed tobe Γ β = Γ β = q β q Γ β Γ β . The off-diagonal matrixelements of Γ β take into account various interference ef-fects resulting from indirect tunneling processes betweentwo quantum dots via the leads. These off-diagonal ma-trix elements may be significantly reduced in comparison to the diagonal matrix elements Γ βii or even totally sup-pressed due to complete destructive interference. To takeall those effects into account, the parameters q L and q R are introduced. Furthermore, we assume that q β are realnumbers and obey the condition | q β | ≤
1. When q β isnonzero, the processes in which an electron tunnels fromone dot to the β -lead and then (coherently) to the an-other dot are allowed. Introducing parameter α , whichtakes into account difference in the coupling of the twodots to external leads, the coupling matrix Eq. (5) canbe rewritten in the form, Γ β = Γ β (cid:18) α q β √ αq β √ α (cid:19) . (5)Tuning parameter α one can change geometry of the sys-tem from the parallel one for α = 1 to the T-shaped ge-ometry for α = 0. In the T-shaped geometry upper dotis disconnected from the leads. All intermediate valuesof the α refer to an intermediate geometry where each ofthe two dots is coupled to leads with different strength.We further assume symmetric coupling Γ L = Γ R ≡ Γ / B. Method
We perform our calculations in large interdot charg-ing energy limit, more specifically, when U → ∞ .Slave boson approach is one of the techniques whichallows investigate strongly correlated fermions in lowtemperatures . This method relies on introducing aux-iliary operators for the dots and replacing of the dots’creation and annihilation operators by f † i b , b † f i respec-tively. Here, slave-boson operator b † creates an emptystate, whereas pseudo-fermion operator f † i creates singlyoccupied state with an electron in the i-th dot. To elim-inate non-physical states, the following constraint has tobe imposed on the new quasi-particles, Q = X i f † i f i + b † b = 1 . (6)Above constraint prevents double occupancy of the dots;dots are empty or singly occupied.In the next step the Hamiltonian (1) of the system isreplaced by an effective Hamiltonian, expressed in termsof the auxiliary boson b and pseudo-fermion f i operatorsas, ˜ H = X k β ε k β c † k β c k β + X i ε i f † i f i + ( tf † bb † f + H . c . )+ X k β X i ( V βi k c † k β b † f i + H . c . ) + λ X i f † i f i + b † b − ! . (7)To avoid double occupancy of the DQD system the con-striction condition (Eq. (6)) has been incorporated inHamiltonian (7) by introducing the term with the La-grange multiplier λ .However, after such transformation our Hamiltonianis still rather complex and hard to solve. To get rid ofthis problem we apply mean field (MF) approximationin which the boson field b is replaced by a real and in-dependent of time c number, b ( t ) → h b ( t ) i ≡ ˜ b . Thisapproximation neglects fluctuations around the averagevalue h b ( t ) i of the slave boson operator, but is sufficientto describe correctly those leading to the Kondo effect.It also restricts our considerations to the low bias regime( eV ≪ | ε i | ).With the following definitions of the renormalized pa-rameters: ˜ t = t ˜ b , ˜ V βi k = V βi k ˜ b and ˜ ε i = ε i + λ , one canrewrite the effective MF Hamiltonian in the form,˜ H MF = X k β ε k β c † k β c k β + X i ˜ ε i f † i f i + (˜ tf † f + h . c . )+ X k β X i ( ˜ V βi k c † k β f i + h . c . ) + λ (cid:16) ˜ b − (cid:17) . (8)The unknown parameters ˜ b and λ have to be found self-consistently with the help of the following equations;˜ b − i X σ Z dε π hh f i | f † i ii <ε = 1 , (9) − i X i Z dε π ( ε − ˜ ε i ) hh f i | f † i ii <ε + λ ˜ b = 0 , (10)where hh f i | f † j ii <ε is the Fourier transform of the lesserGreen function defined as G In the following numerical calculations we assumeequal dot energy levels, ε i = ε (for i = 1 , 2) ( ε is mea-sured from the Fermi level of the leads in equilibrium, µ L = µ R = 0). Moreover, we set the bare level of thedots at ε = − D = 120Γ. All the energy quantities are expressed inthe units of Γ. The parameters q L and q R are assumedto be equal q L = q R = q if not stated otherwise. Tak-ing into account the above parameters, the Kondo tem-perature T K for the symmetric couplings ( α = 1) anddisregarding both direct and indirect couplings, ( t = 0, q = 0) is estimated to be T K ≈ − Γ. In this sec-tion we consider the situation when dots are not directlycoupled, which corresponds to the case with vanishinginterdot tunnel coupling parameter t = 0. On the otherside, one should remember that dots are still coupled(indirectly) through the leads what is reflected in finitevalues of the off-diagonal coupling matrix elements, i.e., q = 0. In this situation bonding and antibonding levels,created due to indirect couplings, coincide. In Fig. 2 weshow local density of states for QD1 and QD2 for largeoff-diagonal matrix couplings ( q = 0 . 99) and for differ-ent values of the asymmetry parameter α . Firstly, it isclearly presented that the broad and narrow Kondo peaksin the LDOS are superimposed at energy ε = 0 and theLDOS displays behavior typical for the Dicke effect. Byanalogy to the original Dicke phenomenon, one may as-sociate the narrow (broad) central peak in LDOS with asubradiant (superradiant) state. The subradiant (super-radiant) state corresponds to longlived (shortlived) state.For symmetric coupling α = 1 the LDOS for QD1 andQD2 are the same (see Fig. 2(a) but when asymmetryappears in couplings ( α = 1) this ceases to be true andthe LDOS for both dots have different line shape. As α drops down the widths of the Kondo peaks for both dotsalso diminish. With decreasing α the broad part of theLDOS for the quantum dot strongly coupled to the leads(QD2) becomes more pronounced, whereas the narrowone becomes narrower. The two peaks corresponding tosubradiant and superradiant state are well distinguish-able. As a result, the Dicke effect in LDOS for QD2 ismore distinct. The Dicke effect is also noticed in thetransmission shown in Fig. 3. This situation is analogous -2 -1 0 1 20.00.20.40.60.81.0 L D O S QD1/QD2 α =1a)-0.2 -0.1 0.0 0.1 0.20.00.20.40.60.81.0 α =0.6 L D O S QD1 QD2 b)-0.02 -0.01 0.00 0.01 0.020.00.20.40.60.81.0 L D O S ε /T K α =0.15 QD1 QD2 c) FIG. 2: Local density of states for the quantum dots QD1and QD2 obtained for indicated values of α and for q = 0 . t = 0. -2 -1 0 1 20.0 α =1 T ( ε ) ε /T K FIG. 3: The transmission probability calculated for indicatedvalues of α and for q = 0 . 99 and for t = 0. The transmissionprobability has well-defined Dicke line shape. to that reported in the case of the spin Kondo effect inparallel double dot system. The Dicke peak appearsin the transmission because the phases of the transmis-sion amplitudes for bonding and antibonding channelsare equal at zero energy. Thus, the two contributionsadds constructively leading to the maximum transmis-sion at ε = 0. However, now there is no dip structureat zero energy for q = 1 (originating from the completedestructive interference), which will be explained further. θ F / π b) -0.025 0.000 0.0250.51.0 T ( ε ) ε /T K q=0.999 q=0.99 -2 -1 0 1 20.00.51.0 T ( ε ) ε /T K c) L D O S q=0.999 q=0.99 q=0.9 q=0.75 q=0.5 q=0 a) FIG. 4: Local density of states for the both dots (a) Friedelphase (b) and the transmission (c) calculated for indicatedvalues of the q and for α = 1 and for t = 0. The well-defined Dicke line shape is only preserved for indirect couplingparameter q close to 1. Moreover, the effect is preserved for various values of theasymmetry parameter α . It is worth noting that the lin-ear conductance (see Eq. (14)) reaches unitary limit (twoquanta of e /h ). When α is reduced the effective Kondotemperature also decreases what can be seen looking atthe energy scale in Figure 2. The origin of this effectcomes from the fact that when α decreases one of thedots becomes detached from the electrodes. Then, therate of the higher order tunneling events (which leads tothe Kondo anomaly-see explanation in the Introduction)also diminishes and finally for α = 0, when one of the dotis totally disconnected from the leads, there is no possi-bility for such events and no Kondo effect is expected.As we mentioned in Section II A the off-diagonal ma-trix elements Γ β may be significantly reduced, so it isdesired to analyze this case. In Fig. (4) we plotted theLDOS, Friedel phase and the transmission for various val-ues of the parameter q which is directly related with am-plitude of the off-diagonal matrix elements. The Friedelphase is related to the LDOS by the following equation: dθ F /dε = πD ( ε ) with D ( ε ) being relevant density ofstates. One can notice that Dicke effect in LDOS can befound only when off-diagonal matrix elements are large,i.e., q close to 1. With decreasing q the Dicke line shapeis transformed in usual Lorenzian line. Similar behavioris observed in the transmission (see Fig. 4(c)). In insetof Fig. 4(c)) we show that the narrow part of the peakhas also well defined shape as the broad one. Moreover,when q is closer and closer to 1 then the narrow peakbecomes more and more narrower. This is reflected inabrupt (but continuous as long as q = 1 ) change by π of Friedel phase around ε = 0 for q close to 1. It is alsofound that the Dicke effect also disappears in transmis-sion when q is maximal ( q = 1). When q = 1 the trans-mission probability has Lorenzian shape and is describedby formula, T ( ε ) = 12 (1 + α ) ˜Γ (1 + α ) ˜Γ + (˜ ε − ε ) ! , (15)where ˜Γ = ˜ b Γ / 2. This equation clearly shows that noDicke effect should be expected for q = 1. This resultresembles that obtained for a noninteracting system .However, in the Kondo regime the situation is muchmore complex. To show this, let us first consider thecase with q = 1 and symmetric couplings, α = 1. It iswell known that for symmetric ( α = 1) noninteracting( U = 0) system as q tends to 1 one of the peaks be-comes progressively narrowed and finally for q = 1 a BICemerges . As a result the transmission reveals simplelorenzian lineshape. One may naively believe that similarsituation occurs in the Kondo regime. However, this cannot be true because when the indirect coupling strengthis equal to the dot-lead coupling (Γ = Γ = Γ ),the indirect tunneling processes completely destroy co-herent higher-order tunelling events leading to completesuppression of the Kondo resonance. One can also lookat this from the another point of view and explain thisas follows: As the antibonding state becomes a BIC, it istotally decoupled from the leads and there is no possibil-ity for an electron to exchange between the two molecularstates. Thus, no Kondo effect appears as the correspond-ing Kondo temperature is equal to zero. To support thesepredictions we calculated the Kondo temperature for ar-bitrary value of the parameter q . The Kondo tempera-ture for the symmetric case, α = 1, acquires the follow-ing form T K ≡ q ˜ ε + ˜Γ (1 − q ) . In the deep Kondoregime the renormalized parameter ˜ ε is equal to zero,which is shown in the Appendix, and thus the above for-mula clearly show vanishing of the corresponding Kondotemperature as q tends to 1. In Fig. 5 the Kondo tem- T K ( q ) q T K (q) Γ + Γ - FIG. 5: The Kondo temperature calculated as a function ofparameter q and renormalized width ˜Γ + (˜Γ − ) corespondingto the bonding (antybonding) level calculated for α = 1 andfor t = 0. -2 -1 0 1 20.0 T ( ε ) ε /T K α =1 α =0.9 α =0.75 α =0.5 α =0.25 FIG. 6: Graphical illustration of the Equation 16. The trans-mission probability calculated for indicated values of the α and for q = 0. perature is displayed as a function of parameter q . Forcomparison we also plot the q dependence of the renor-malized widths of the bonding and antibonding levels, ˜Γ b and ˜Γ a , respectively. In the case q L = q R = q , the widthsacquire the following form ˜Γ b = ˜ b Γ b , ˜Γ a = ˜ b Γ a withΓ b,a = (Γ + Γ ) ± q √ Γ Γ and Γ ii = Γ Lii + Γ Rii for i = 1 , 2. The characteristic widths for both distinct chan-nels behave in different way with varying the strength ofthe off-diagonal tunneling processes. One can notice thatthe renormalized width for the bonding channel changesnon-monotonically but rather slowly, whereas the one forthe antibonding channel drops monotonically to zero as q reaches maximum value. This behavior is consistentwith the predictions given in Ref. 55 with one exception.The vanishing of ˜Γ a when q is maximal is responsible forthe suppression of the Dicke effect. In contrast to Ref. 55the width ˜Γ b drops to zero for q = 1 due to vanishing ofthe boson field. We also emphasize that for q ∈ (0 , α dependances of the renormalized widths for bothchannels are different. For large values of the q the width˜Γ a changes little with α .However, the Kondo effect is also destroyed in more T K ( α ) α FIG. 7: The Kondo temperature calculated as a function ofthe asymmetry parameter α for q = 0 and for t = 0. general case, i.e., for q = 1 and arbitrary α . This can beunderstood when one notice that transmission (15) de-pends on renormalized parameter ˜ b which vanishes in thiscase. In the SBMF formalism this is manifested as lackof solutions of the self-consistent equations (9) and (10)(which then form a contradictory system of equations).On the other side, for q = 0 the following formuladescribes the transmission probability, T ( ε ) = 12 " ˜Γ ˜Γ + (˜ ε − ε ) + α α ˜Γ + (˜ ε − ε ) . (16)Figure 6 illustrates graphically Eq. (16) for indicated val-ues of the asymmetry parameter α . This plot clearlyshows, as explained earlier, that the width of the Kondopeak decreases as α is reduced. This case resembles thespin Kondo effect in single-level quantum dot coupled toferromagnetic leads . Then the asymmetry parameter α can be assigned with lead’s (pseudo)polarization in thefollowing way ˜ p = (1 − α ) / (1 + α ) . Increasing pseu-dopolarization (decreasing α ) the Kondo effect is sup-pressed which is in agreement with Refs. 56,57. In thepresence of asymmetry in couplings ( α < 1) the Kondotemperature should be defined by geometric mean as: T K ≡ p ˜Γ ˜Γ = ˜Γ √ α (remembering that ˜ ε → α dependence of the Kondotemperature for case of q = 0. In the present case, q = 0,electrons may travel only directly through dot QD1 orQD2 thus the reduction of the Kondo temperature withdecreasing α clearly reflects the suppression of Kondofluctuations in the DQD (due to reduction of the chargeexchange between the dots) as one of the dot becomesdetached from the leads and Kondo effect diminishes .Similar dependance on parameter q can be noticed inthe differential conductance which is displayed in Fig. 8.The differential conductance is symmetric in respect tothe zero bias point, thus we plot only its bias dependancefor nonnegative bias voltage. For q close to unity thedifferential conductance acquires Dicke line shape. Atzero bias the differential conductance approaches unitarylimit (2 e /h ) for all q < 1. For q close to 1 the differentialconductance drops very fast to the half of the zero biasvalue with increasing the bias voltage and next decreases G d i ff ( e / h ) eV/T K q=0.99 q=0.9 q=0.75 q=0.5 q=0 FIG. 8: Differential conductance as a function of bias voltagecalculated for indicated values of parameter q ( q L = q R = q )and for α = 1, t = 0. -4 -2 0 2 40.00.20.40.6 L D O S ε /T K -4 -2 0 2 40.00.51.0 T ( ε ) ε /T K q 1 0.99 0.9 0.7 0.5 0.25 0 FIG. 9: The transmission coefficient calculated for indicatedvalues of the parameter q ( q L = − q R = q ) and for α = 1. Theinset show corresponding LDOS which is independent on thevalue of q . slowly with further increasing of the bias voltage. Sucha sudden drop of the differential conductance is absentfor smaller values of the parameter q . This feature in thedifferential conductance is related to Dicke resonance.As we mentioned in Sec. I the off–diagonal elementsof the coupling matrix can have opposite signs. Specifi-cally, we examine the case q L = q and q R = − q . At thebeginning we assume symmetric couplings, i.e., α = 1.Then for maximal value of the parameter q ( q = 1)each DQD’s molecular state couples to different reser-voirs and as there is no connection between them, thecurrent doesn’t flow. This effect originates from thetotally destructive quantum interference which leads tovanishing of the transmission even if the LDOS of eachdots is finite. It is wort noting that both LDOS andthe corresponding Kondo temperature do not depend onthe parameter q , as the contributions from nondiago-nal processes, refereing to the left L and right R lead,cancel each others. This implies that the presented ef-fect originates fully from the quantum interference. Onecan notice that decreasing the value of the parameter q ,the transmission can be recovered as is shown in Fig. 9.Finally, for q = 0 the maximal value of the transmis-sion is restored. This is because as the value of theoff-diagonal matrix elements are decreased the destruc-tive interference becomes totally suppressed, and thus,for zero value of the nondiagonal couplings the zero biasconductance is fully restored. This can be shown moreformally by performing the transformation of the dotsoperators to the bonding-like ( d b = 1 / √ d + d )) andantibonding-like ( d a = 1 / √ d − d )) state. The corre-sponding couplings to the left lead acquires form ˜Γ Lb,a = (˜Γ L + ˜Γ L ) ± q q ˜Γ L ˜Γ L , whereas couplings to the rightelectrode is given by ˜Γ Rb,a = (˜Γ R + ˜Γ R ) ∓ q q ˜Γ R ˜Γ R .Now, it is clear that for α = 1 and q = 1 the bond-ing state is coupled to (decoupled from) left (right) lead,whereas antibonding state is coupled to (decoupled from)right (left) electrode. As the parameter q becomes lessthan 1, the decoupled states from given leads for q = 1start to bound with them. As a result the transmissiongrows with decreasing the value of the parameter q .Another interesting quantum interference effect can befound changing the asymmetry in couplings of two dotsto the leads. At the beginning we keep q equal to 1and change the parameter α in the interval h , i . Asbefore the transmission becomes recovered as the asym-metry increases [see Fig. 10]. However, in comparison tothe previous case, a new feature emerges in the trans-mission. More specifically, the dip structure appears inthe vicinity of the zero energy. At ε = 0 the transmis-sion drops to zero which results in vanishing of the linearconductance. Thus, the zero bias Kondo anomaly is to-tally suppressed for any value of the asymmetry param-eter α . To verify this effect experimentally it is desiredto measure the differential conductance as a function ofbias voltage. In Fig. 11 we displayed bias voltage depen-dence of the differential conductance for indicated valuesof the asymmetry parameter α . These results show thatapplying finite bias voltage, the differential conductance(which is zero at zero bias for all α ) is restored. Thedifferential conductance becomes the most pronouncedwhen one of the dot is almost decoupled from the leads,i.e., as α → 0, the differential conductance tend to uni-tary limit. However, with increasing asymmetry in thecouplings (decreasing α ) the range of finite values of thedifferential conductance shrinks due to decreasing of theKondo temperature. It is worth noting that the corre-sponding Friedel phase is a continuous function of energy– it does not suffer discontinuity at ε = 0 because no BICappears. However, the transmission at ε = 0 vanishes,thus the phase of transmission amplitude should be dis-continuous at ε = 0. -0.01 0.00 0.010.00.20.4 T ( ε ) ε /T K α =0.25 α =0.1 -1.0 -0.5 0.0 0.5 1.00.0000.0050.010 T ( ε ) α =0.9 α =0.75 α =0.5 FIG. 10: The transmission coefficient calculated for indicatedvalues of the parameter α and for q L = − q R = 1. α =0.25 α =0.1 G d i ff ( e / h ) eV/T K α =1 α =0.9 α =0.75 G d i ff ( e / h ) eV/T K FIG. 11: Differential conductance as a function of bias voltagecalculated for indicated values of the parameter α and for q L = − q R = 1, t = 0. B. Fano effect Here, we show the results obtained for nonzero interdothopping parameter ( t = 0). In present situation the in-terdot hopping term lifts the degeneracy and the bondingand the antibonding levels are split away. As a result, thedensity of states of the DQD system coupled to the leadsconsists of two Kondo peaks; a broad peak centered atthe bonding state and a narrow one corresponding to theantibonding state. This is clearly showed in Fig. 12(a))where the LDOS for both dots is plotted for maximal off-diagonal matrix elements ( q = 1). It is very interestingthat the narrow peaks in LDOS for QD1 and QD2 haveopposite symmetries. The LDOS for QD1 reaches zerofor negative energy, whereas the one for QD2 comes tozero for positive energy. Usually, when the asymmetryin coupling of two dots to the leads is reduced the widthof bonding (antibonding) state increases (decreases). Fi-nally, for α = 1 (full symmetric system) the antibondingstates is decoupled from the leads and acquires δ -Diracshape (it becomes a BIC), whereas the bonding state ac-quires the width of 2˜Γ. On the other side, when theasymmetry in coupling of two dots to the leads is in-creased then the width of bonding (antibonding) statedecreases (increases) and for α = 0 the two resonancesacquire the same width. However, in current problem thesituation is more complex because the level’s widths arerenormalized by factor ˜ b which has to be obtained self-consistently for each α . Now it is only true that relativewidth of the bonding state to the width of the antibond-ing state grows as α increases, and are the same for α = 0.However, due to widths renormalization both peaks inLDOS may have smaller widths for small α with com-parison to the width of the narrow peak for large α (butnot α ≈ 1) (for instance for α = 0 . 15 both resonances aremerged into narrow resonance for α = 0 . α onlyLDOS for QD2 may reach zero value at some energy.The above behavior of LDOS for α ∈ (0 , 1) results in Fanoantiresonance in the transmission probability. The cor-responding transmission probability calculated for q = 1and α = 0 . T ( ε ) is plotted for greater asymmetry incoupling of two dots to the leads ( α = 0 . α = 0 . α the widths of both peaks also decrease which can beseen looking at the energy scales for both cases. Thismeans that the Kondo temperature is also lowered as α decreases (as was shown in the previous section). For q = 1 the transmission probability has poles at, ε = ˜ ε − √ α α ˜ t. (17)It is worth to mention that at this energy the phase ofthe transmission amplitude suffers discontinuity. Thisis no more true for q < 1, because then the transmis-sion probability has complex poles. In turn, for q < q antiresonance behavior is less visible.Even a small shift in q change the transmission proba-bility within the antibonding level considerably, which is -0.04 -0.02 0.00 0.02 0.040.00.5 T ( ε ) ε /T K α =0.15 -2 -1 0 1 20.00.5 T ( ε ) ε /T K b) α =0.8 L D O S QD1 QD2 q=1a) FIG. 12: Local density of states for the both dots (a) and thetransmission probability (b) calculated for q = 1, α = 0 . t = 0 . 8. The existence of the Fano-Kondo effect is wellvisible. The inset shows the transmission probability obtainedfor q = 1 and for α = 0 . -2 -1 0 1 20.00.5 T ( ε ) ε /T K α =0.8q=0.99 FIG. 13: The transmission probability calculated for α = 0 . t = 0 . 8, and for q = 0 . 99. Even a small shift in q leads to theconsiderably changes in the transmission probability withinthe antibonding level considerably (compare with Fig. 12). showed in Fig. 13.At this point we should remark that the slave bosonmean field approach does not take into account the levelrenormalization arising due to coupling of dots to theleads. Such a renormalization should lead to splittingof the zero bias anomaly for specific cases. The levelsplitting can occurs due to asymmetry in coupling of thedots to the leads, i.e., for α < q close to 1 the bonding-like level will be onlyweakly renormalized, whereas the antibonding-like levelwill experience strong renormalization.Thus, the transmission described in Sec. III A shouldresembles that from Fig. 13 but with the broad peakpinned close to ε = 0 and the narrow maximum shiftedup in energy for relatively large value of parameter q .One can show this more formally including corrections inmolecular-like levels due to mentioned renormalization.To correct the drawback of the SBMF method onecan by hand introduce the mentioned renormalization ofthe levels as follows: ε b = ε + δε b and ε a = ε + δε a (with t = 0 for the sake of simplicity). Here, δε i arethe corrections due to indirect coupling of the dots lev-els. These corrections should be determined using rel-evant technique, as for instance earlier mentioned scal-ing procedure. However, as the SBMF technique failsfor nondegenerate states, one can not introduce by handthe renormalized levels into self-consistent equations ofthe form (9) and (10) derived within bonding and anti-bonding states basis. Thus, we give here only rough es-timation and some predictions implying from the levelsscaling. Such approach should deliver qualitatively goodinsight into Kondo peak splitting phenomenon, howeverto obtain quantitatively consistent results more reliabletechnique should be applied for the considered problem.For the sake of simplicity we analyze only the symmetriccase ( α = 1). Introducing the level renormalization thetransmission acquires the following form: T ( ε ) = ˜Γ (1 − q ) ( ε − ˜ ε a ) + ˜Γ (1 − q ) + ˜Γ (1 + q ) ( ε − ˜ ε b ) + ˜Γ (1 + q ) (18)with renormalized bonding ( b ) and antibonding ( a ) levelsof the following assumed form : ε i = ε + Γ ¯ i ∆ (19)Here, ∆ stands for some function which in general de-pends on the system’s parameters (like bandwidth, dotsenergy levels, couplings). For the sake of simplicity weassume ∆ to be a constant number and being maximumvalue of function ∆( q ) obtained from the scaling pro-cedure, i.e. when q = 1. Equation (18) clearly showsthat Kondo peak becomes split due to the level renor-malization originating from indirect tunneling betweenthe dots. In Fig. 14 we display expected lineshape inthe transmission for relatively large value of parameter q . The splitting in the transmission should decrease withdecreasing the value of parameter q and beyond a certainvalue of q the splitting ceases to be visible. For q = 0(and α = 1) no splitting occurs. -2 -1 0 1 20.00.51.0 T ( ε ) ε /T K q=0.99 FIG. 14: The transmission coefficient including the levelrenormalization calculated for α = 1 and for q L = q R = 0 . IV. SUMMARY AND CONCLUSIONS In this paper we have investigated electronic proper-ties of double quantum dots coupled to external leads.The dots have been coupled both via hopping term andCoulomb interaction. Moreover, we have considered alsothe effects of indirect tunneling between the dots throughthe leads. Employing the slave-boson mean field ap-proach the local density of states for both dots, theFriedel phase and the transmission in Kondo regime havebeen calculated. Moreover, to be more familiar with ex-periment we have calculated the corresponding differen-tial conductance.We have shown that for some set of parameters theDicke-and Fano-like resonances may appear in the con-sidered system. More specifically, it has been noticedthat for zero interdot hopping parameter t the LDOS ofeach QD consist of broad and narrow Kondo peaks thatare superimposed. As in original Dicke effect one may as-sociate narrow (broad) central peak in LDOS with a sub-radiant (superradiant) mode. Moreover, Dicke line shapehas been found in the transmission and in the differentialconductance. We have observed that this effect is verysensitive to the strength of the off-diagonal matrix ele-ments; with reducing the value of the off-diagonal matrixelements the Dicke line both in LDOS and the transmis-sion is transformed into usual Lorenzian line. It has beenfound that when the interdot tunneling is allowed thetransmission probability may reveal the antiresonancebehavior with a characteristic Fano line shape. Moreover,we have noticed that the line shape of the antiresonanceis also very susceptible to the change of the value of theoff-diagonal matrix elements.We have also calculated the Kondo temperature, andalso have shown that the latter becomes suppressed withincreasing asymmetry in the dot-lead couplings whenthere is no indirect coupling. Moreover, when the in-direct coupling is turned on, the characteristic widths forboth distinct channels behave in different way with vary-ing strength of the off-diagonal tunneling processes. Wefound also that the corresponding Kondo temperature istotally suppressed for maximal value of the indirect cou-1pling and no Kondo effect occurs. Moreover, we havealso included level renormalization effects due to indirectcoupling phenomenon, which leads to the splitting of theKondo peak. Acknowledgments The author thanks prof. J´ozef Barna´s for constructivecriticism and fruitful discussions. This work was sup-ported by Ministry of Science and Higher Education asa research project N N202 169536 in years 2009-2011. Appendix A: Proof of ˜ ε → in deep Kondo regime Here, we show that in deep Kondo regime the renor-malized parameter ˜ ε is equal to zero and, thus, theKondo temperature strictly correspond to to the renor-malized width ˜Γ of the Abricolov-Suhl resonance. Thiscan be shown analytically by integrating self-consistentequations for slave-boson parameters, ˜ b , λ , written inrepresentation of bonding and antibonding states.In the basis of the bonding and antibonding states theHamiltonian of the system becomes diagonal for ε = ε ≡ ε and acquires the following form:ˆ H = ˆ H c + X i = b,a ε i d † i d i + U n b n a + X k β X i = e,o ( V βi k c † k β d i + H . c . ) , (A1)with ε b = ε + t , ε a = ε − t . In further considerationswe assume t = 0, thus, ε b = ε a = ε .In the mean field slave boson representation the Hamil-tonian (A1) acquires the following form:˜ H MF = ˆ H c + X i = b,a ˜ ε f † i f i + λ (cid:16) ˜ b − (cid:17) + X k β X i = b,a ( ˜ V βi k c † k β f i + H . c . ) . (A2) The self-consistent equations determining the un-known parameters ˜ b and λ have the form of Eqs. (9)and (10) with lesser Green’s function hh f i | f † j ii <ε definedin the basis of the bonding and antibonding states. Thelesser bonding and antibonding Green’s function has thefollowing form: G Introducing the solution for ˜ ε into Eq. (A7) one finds,˜ b = 2 D Γ (1 − q ) q − (1 + q ) q +12 exp (cid:16) πε Γ (cid:17) . (A9) It is worth noting that the above equation does not de-termine the value of ˜ b for q = 1 when the SBMF methodfails. ∗ Electronic address: [email protected] J. Kondo, Prog. Theor. Phys. , 37 (1964). S. M. Cronenwett, T. H. Oosterkamp and L. P. Kouwen-hoven, Science , 540 (1998); S. Sasaki,S. De Franceschi,J. M. Elzerman, W. G. van der Wiel, M. Eto, S. Taruchaand L. P. Kouvenhoven, Nature (London) , 764 (2000). D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D.Abusch-Magder, U. Meirav and M. A. Kastner, Nature(London) , 156 (1998). L. I. Glazman and M. E. Raikh