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Conductance, on-site, and intersite charge fluctuations and spin correlations in the system of two side-coupled quantum dots are calculated using Wilson’s numerical renormalization group (NRG) technique. We also show the spectral density calculated using the density-matrix NRG, which for some parameter ranges remedies inconsistencies of the convent...
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... this work we study a double quantum dot (DQD) in a side-coupled configuration ( Fig. 1), connected to a single conduction-electron channel. Systems of this type were studied previously using non-crossing approximation 14 , embedding technique 15 and slave- boson mean field theory 13,16 ...
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... Kondo temperature rapidly drops with increasing κ (Fig. 9b). Keeping the temperature con- stant and increasing κ, the Kondo plateau in the region −U/2 < δ a < U/2 will therefore quickly evolve into two peaks separated by a Coulomb blockade valley and for very large κ it will not conduct at all. This prediction can also be verified using NRG (Fig. 10). ...
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... effective temperature T to be finite, i.e. T ∼ 10 −9 D, since calculations at much lower temperatures would be experimentally irrelevant. In this case one naively expects to obtain essentially iden- tical conductance as in the single-dot case. As δ decreases below δ ∼ U/2, G/G 0 indeed follows result obtained for the single-dot case as shown in Fig. 11a. In the case of DQD, however, a sharp Fano resonance appears at δ = U/2. This resonance coincides with the sudden jump in S, n, as well as with the spike in ∆n 2 , as shown in Figs 11b,c, and d, respectively. Fano resonance is a con- sequence of a sudden charging of the nearly decoupled dot a, as its level ǫ crosses the chemical ...
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... the case of DQD, however, a sharp Fano resonance appears at δ = U/2. This resonance coincides with the sudden jump in S, n, as well as with the spike in ∆n 2 , as shown in Figs 11b,c, and d, respectively. Fano resonance is a con- sequence of a sudden charging of the nearly decoupled dot a, as its level ǫ crosses the chemical potential of the leads, i.e at ǫ = 0. ...
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... resonance is a con- sequence of a sudden charging of the nearly decoupled dot a, as its level ǫ crosses the chemical potential of the leads, i.e at ǫ = 0. Meanwhile, the electron density on the dot d remains a smooth function of δ, as seen from n d in Fig 11c. With increasing t d , the width of the reso- nance increases, as shown for t d /D = 0.01 in Fig. 11f. ...
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... is a con- sequence of a sudden charging of the nearly decoupled dot a, as its level ǫ crosses the chemical potential of the leads, i.e at ǫ = 0. Meanwhile, the electron density on the dot d remains a smooth function of δ, as seen from n d in Fig 11c. With increasing t d , the width of the reso- nance increases, as shown for t d /D = 0.01 in Fig. 11f. For t d > ∼ 0.1, the resonance merges with the Kondo plateau and disappears (see Fig. ...
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... Results of the NRG calculation are compared to the prediction from Eq. 15 in the inset of Fig. 11g. We see that general features are adequately described, but there are subtle differences due to non-perturbative electron correlation effects. Numerically calculated Fano reso- nance is wider than the semi-analytical prediction and G/G 0 does not drop to zero. In particular, from Eq. 15 it follows that G = 0 at δ = U/2 (ǫ = 0) and G = ...
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... now return to the description of the results pre- sented in Fig. 11a in the regime where δ < U/2. As δ further decreases, the system enters a regime of the two- stage Kondo effect 17 . This region is defined by J eff < T K (see also Fig. 11h), where T K is the Kondo temperature, approximately given by the single quantum dot Kondo temperature, Eq. 8 ...
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... now return to the description of the results pre- sented in Fig. 11a in the regime where δ < U/2. As δ further decreases, the system enters a regime of the two- stage Kondo effect 17 . This region is defined by J eff < T K (see also Fig. 11h), where T K is the Kondo temperature, approximately given by the single quantum dot Kondo temperature, Eq. 8 ...
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... the lower Kondo temperature, corresponding to the gap in the spectral density ρ d (ω) at ω = 0 and α is of the order of one 17 . Note that NRG values of the gap in ρ d (ω) (open circles and squares), calculated at T ≪ T 0 K , follow analytical results for T 0 K (δ) when J eff < T K , see Fig. 11h, while in the opposite regime, i.e. for J eff > T K , they approach J eff ...
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... shown in Fig. 11a for 0.3D < ∼ δ < U/2, G/G 0 cal- culated at T = 10 −9 D follows results obtained in the sin- gle quantum dot case and approaches value 1. The spin quantum number S in Fig. 11b reaches the value S ∼ 0.8, consistent with the result obtained for a system of two decoupled spin-1/2 particles, wherê S 2 = 3/2. This re- sult is also in ...
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... shown in Fig. 11a for 0.3D < ∼ δ < U/2, G/G 0 cal- culated at T = 10 −9 D follows results obtained in the sin- gle quantum dot case and approaches value 1. The spin quantum number S in Fig. 11b reaches the value S ∼ 0.8, consistent with the result obtained for a system of two decoupled spin-1/2 particles, wherê S 2 = 3/2. This re- sult is also in agreement with n ∼ 2 and the small value of the spin-spin correlation function S a · S d , presented in Fig. 11c and 11e ...
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... sin- gle quantum dot case and approaches value 1. The spin quantum number S in Fig. 11b reaches the value S ∼ 0.8, consistent with the result obtained for a system of two decoupled spin-1/2 particles, wherê S 2 = 3/2. This re- sult is also in agreement with n ∼ 2 and the small value of the spin-spin correlation function S a · S d , presented in Fig. 11c and 11e ...
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... of δ, G/G 0 suddenly drops to zero at δ < ∼ 0.3D. This sudden drop is approximately given by T < ∼ T 0 K (δ), see Figs. 11a and h. At this point the Kondo hole opens in ρ d (ω) at ω = 0, which in turn leads to a drop in the conductivity. The position of this sudden drop in terms of δ is rather insensitive to the chosen T , as apparent from Fig. 11h. Below δ < ∼ 0.25D, which corresponds to the condi- tion J eff ∼ T K (δ), also presented in Fig. 11h, the sys- tem crosses over from the two stage Kondo regime to a regime where spins on DQD form a singlet. In this case S decreases and S a · S d shows strong antiferromagnetic correlations, Figs. 11b and e. The lowest energy scale in ...
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... < ∼ T 0 K (δ), see Figs. 11a and h. At this point the Kondo hole opens in ρ d (ω) at ω = 0, which in turn leads to a drop in the conductivity. The position of this sudden drop in terms of δ is rather insensitive to the chosen T , as apparent from Fig. 11h. Below δ < ∼ 0.25D, which corresponds to the condi- tion J eff ∼ T K (δ), also presented in Fig. 11h, the sys- tem crosses over from the two stage Kondo regime to a regime where spins on DQD form a singlet. In this case S decreases and S a · S d shows strong antiferromagnetic correlations, Figs. 11b and e. The lowest energy scale in the system is J eff , which is supported by the obser- vation that the size of the gap in ρ d (ω) (open ...
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... insensitive to the chosen T , as apparent from Fig. 11h. Below δ < ∼ 0.25D, which corresponds to the condi- tion J eff ∼ T K (δ), also presented in Fig. 11h, the sys- tem crosses over from the two stage Kondo regime to a regime where spins on DQD form a singlet. In this case S decreases and S a · S d shows strong antiferromagnetic correlations, Figs. 11b and e. The lowest energy scale in the system is J eff , which is supported by the obser- vation that the size of the gap in ρ d (ω) (open circles in Figs. 11h) is approximately given by J eff . The main dif- ference between t d /D = 0.001 and t d /D = 0.01 comes from different values of J eff = 4t 2 d /U . Since in the lat- ter case J ...
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... the sys- tem crosses over from the two stage Kondo regime to a regime where spins on DQD form a singlet. In this case S decreases and S a · S d shows strong antiferromagnetic correlations, Figs. 11b and e. The lowest energy scale in the system is J eff , which is supported by the obser- vation that the size of the gap in ρ d (ω) (open circles in Figs. 11h) is approximately given by J eff . The main dif- ference between t d /D = 0.001 and t d /D = 0.01 comes from different values of J eff = 4t 2 d /U . Since in the lat- ter case J eff is larger, the system enters the AFM singlet regime at much larger values of δ, as can be seen from comparison of Figs. 11g and f. Consequently, the regime ...
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... of the gap in ρ d (ω) (open circles in Figs. 11h) is approximately given by J eff . The main dif- ference between t d /D = 0.001 and t d /D = 0.01 comes from different values of J eff = 4t 2 d /U . Since in the lat- ter case J eff is larger, the system enters the AFM singlet regime at much larger values of δ, as can be seen from comparison of Figs. 11g and f. Consequently, the regime of enhanced conductance ...
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... at T ∼ T K from around 0.5 to 0.25 and at T ∼ T 0 K from around 0.25 to 0 17 . We now inves- tigate how this behavior changes when κ = 0 by compar- ing temperature dependent susceptibilities calculated for a range of parameters κ. We find that the two stage Kondo effect is robust against κ = 0 perturbation. In fact, it survives until κ = U/2, see Fig. 12. For κ < U/2 the main effect of κ is to reduce the lower Kondo temperature T 0 K . For κ > U/2, dot a becomes irrelevant at low temperatures and dot d is in the usual Kondo ...
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... These systems typically involve two or more QDs in the central region. Among them, the bimolecular transistor (BMT) involving double QDs has generated significant attention [58][59][60][61] www.nature.com/scientificreports/ has a simple structure, and also for it enhances the transport and thermo-electrical properties in both the Kondo regime 62,63 and Coulomb blockade regime 64 . ...
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... When the DQD levels are far below the chemical potential with infinitely-strong Coulomb interaction, the DQDs are half-occupied, and then they enter the Kondo regime. In such a case, the SBMFA is feasible to describe the AR processes influenced by the Kondo physics [73,74]. In Fig. 4, we present the curves of the ET, LAR, and CAR probabilities in the cases of = 0 , , and 3 2 . ...
In this paper, we study the quantum transport properties in a double-quantum-dot circuit with two laterally-coupled superconductors. It is clearly found that with the adjustment of the phase difference between the superconductors, the electron tunneling and the local and crossed Andreev reflections display different characteristics. This phenomenon can be observed in the linear and nonlinear regimes, even in the presence of Coulomb interaction terms. Moreover, appropriate superconducting phase differences can efficiently enhance the crossed Andreev reflection, accompanied by the suppression of the local Andreev reflection and electron tunneling processes. Therefore, this setup can be used to be a promising device to realize the Cooper-pair splitting.
... We avoid considering Kondo regime [39][40][41] and Aharonov-Bohm oscillations 28,[42][43][44] since they were previously extensively studied. The effects were already scrutinised for a variety of geometries containing quantum dots such as side-coupled quantum dots 29,30,45 , the cases when a quantum dot is localised in each of the tunneling channels [46][47][48] , a chain of quantum dots localised in one of the channels 32,49 and a double-dot geometry [50][51][52] . ...
In multi-channel tunneling systems quantum interference effects modify tunneling conductance spectra due to Fano effect. We investigated the impact of Hubbard type Coulomb interaction on tunneling conductance spectra for the system formed by several interacting impurity atoms or quantum dots localised between the contact leads. It was shown that the Fano shape of tunneling conductance spectra strongly changes in the presence of on-site Coulomb interaction between localised electrons in the intermediate system. The main effect which determines the shape of the tunneling peaks could be not Fano interference but mostly nonequilibrium dependence of the occupation numbers on bias voltage.
... The connection between one quantum dot (QD 1 ) and metallic electrodes leads to the widening of a dot level. When the second dot (QD 2 ) is side coupled to the first one, there is a possibility to obtain the Fano-like asymmetric line shapes in the linear conductance [3][4][5][6][7]14] which are obtained as a result of the interference between discrete QD 2 level with a broad band of QD 1 . Additionally, for interacting dots, one obtains a two-stage Kondo effect [2,3,14,29,30]. ...
... When the second dot (QD 2 ) is side coupled to the first one, there is a possibility to obtain the Fano-like asymmetric line shapes in the linear conductance [3][4][5][6][7]14] which are obtained as a result of the interference between discrete QD 2 level with a broad band of QD 1 . Additionally, for interacting dots, one obtains a two-stage Kondo effect [2,3,14,29,30]. The Fano destructive interference partially suppresses the Kondo resonance. ...
We study the sub-gap spectrum and the transport properties of a double quantum dot coupled to metallic and superconducting leads. The coupling of both quantum dots to the superconducting lead induces a non-local pairing in both quantum dots by the Andreev reflection processes. Additionally, we obtain two channels of Cooper pair tunneling into a superconducting lead. In such a system, the direct tunneling process (by one of two dots) or the crossed tunneling process (by both quantum dots at the same time) is possible. We consider the dependence of the Andreev transmittance on an inter-dot tunneling amplitude and the coupling between a quantum dot and the superconducting lead. We also consider the occurrence of interferometric Fano-type line shapes in the linear Andreev conductance spectra.
... The connection between one quantum dot (QD 1 ) and metallic electrodes leads to the widening of a dot level. When the second dot (QD 2 ) is side coupled to the first one, there is a possibility to obtain the Fano-like asymmetric line shapes in the linear conductance [3][4][5][6][7]14] which are obtained as a result of the interference between discrete QD 2 level with a broad band of QD 1 . Additionally, for interacting dots, one obtains a two-stage Kondo effect [2,3,14,29,30]. ...
... When the second dot (QD 2 ) is side coupled to the first one, there is a possibility to obtain the Fano-like asymmetric line shapes in the linear conductance [3][4][5][6][7]14] which are obtained as a result of the interference between discrete QD 2 level with a broad band of QD 1 . Additionally, for interacting dots, one obtains a two-stage Kondo effect [2,3,14,29,30]. The Fano destructive interference partially suppresses the Kondo resonance. ...
We study the sub-gap spectrum and the transport properties of a double quantum dot coupled to metallic and superconducting leads. The coupling of both quantum dots to the superconducting lead induces a non-local pairing in both quantum dots by the Andreev reflection processes. Additionally, we obtain two channels of Cooper pair tunneling into a superconducting lead. In such a system, the direct tunneling process (by one of two dots) or the crossed tunneling process (by both quantum dots at the same time) is possible. We consider the dependence of the Andreev transmittance on an inter-dot tunneling amplitude and the coupling between a quantum dot and the superconducting lead. We also consider the occurrence of interferometric Fano-type line shapes in the linear Andreev conductance spectra.
... In general DQDs, there are two basic kinds of interdot interactions. One is the tunneling type that can be described with the transfer coupling parameter, This type is related to the twochannel Kondo effects, [10,11] two-stage Kondo effects, [9,[12][13][14][15][16][17][18] and others. The other interaction type is the capacitive type, and can be described by interdot Coulomb repulsion. ...
We report capacitive coupling induced Kondo–Fano (K–F) interference in a double quantum dot (DQD) by systematically investigating its low-temperature properties on the basis of hierarchical equations of motion evaluations. We show that the interdot capacitive coupling U 12 splits the singly-occupied (S-O) state in quantum dot 1 (QD1) into three quasi-particle substates: the unshifted S-O 0 substate, and elevated S-O 1 and S-O 2 . As U 12 increases, S-O 2 and S-O 1 successively cross through the Kondo resonance state at the Fermi level ( ω = 0), resulting in the so-called Kondo-I (KI), K–F, and Kondo-II (KII) regimes. While both the KI and KII regimes have the conventional Kondo resonance properties, remarkable Kondo–Fano interference features are shown in the K–F regime. In the view of scattering, we propose that the phase shift η ( ω ) is suitable for analysis of the Kondo–Fano interference. We present a general approach for calculating η ( ω ) and applying it to the DQD in the K–F regime where the two maxima of η ( ω = 0) characterize the interferences between the Kondo resonance state and S-O 2 and S-O 1 substates, respectively.
... The double dot is assumed to form a T-shaped geometry with external contacts, where only one of the dots is directly coupled to the leads, while the second dot is attached indirectly through the first quantum dot. This system, in the absence of Majorana wire, is known to exhibit a nonmonotonic dependence of conductance with lowering temperature due to the two-stage Kondo effect [39][40][41][42][43][44]. Moreover, the interplay between the Fano and Kondo effects in such systems was shown to give rise to interesting spin-resolved transport behavior [45][46][47]. ...
... When the temperature is decreased further, such that T T * , the spin on the second quantum dot becomes screened by the Fermi liquid formed by the first dot strongly coupled to the leads. The second-stage Kondo temperature T * can be evaluated from [41,42,47] ...
The transport behavior of a double quantum dot side-attached to a topological superconducting wire hosting Majorana zero-energy modes is studied theoretically in the strong correlation regime. It is shown that Majorana modes can leak to the whole nanostructure, giving rise to a subtle interplay between the two-stage Kondo screening and the half-fermionic nature of Majorana quasiparticles. In particular, coupling to the topological wire is found to reduce the effective exchange interaction between the two quantum dots in the absence of normal leads. Interestingly, it also results in an enhancement of the second-stage Kondo temperature when the normal leads are attached. Moreover, it is shown that the second stage of the Kondo effect can become significantly modified in one of the spin channels due to the interference with the Majorana zero-energy mode, yielding low-temperature conductance equal to G=G0/4, where G0=2e2/h, instead of G=0 in the absence of the topological superconducting wire. We also identify a nontrivial spin-charge SU(2) symmetry present in the system at a particular point in parameter space, despite lack of the spin and charge conservation. Finally, we discuss the consequences of a finite overlap between two Majorana modes, as relevant for short Majorana wires.
... The two quantum dots are coupled through the hopping matrix elements t. Both dots are characterized by a single orbital level of energy εj and Coulomb correlations Uj. [39][40][41][42][43][44]. Moreover, the interplay between the Fano and Kondo effects in such system was shown to give rise to interesting spin-resolved transport behavior [45][46][47]. ...
... The spin-resolved spectral function of the quantum dot directly coupled to the normal leads calculated for different values of the coupling to Majorana wire V M and hopping between the quantum dots t is shown in Fig. 2. In the case of V M = 0 one observes the conventional twostage Kondo effect [41,43], see the first row of Fig. 2. The spectral function first increases with lowering the energy ω, which happens for ω T K , but then starts to decrease once ω T * , where T K (T * ) is the first-stage (second-stage) Kondo temperature. Because T * depends exponentially on the effective exchange interaction J eff between the two quantum dots generated by the hopping t [41,42,47], ...
The transport behavior of a double quantum dot side-attached to a topological superconducting wire hosting Majorana zero-energy modes is studied theoretically in the strong correlation regime. It is shown that the Majorana zero energy mode can leak through the dot directly attached to topological superconductor to the side attached dot, giving rise to a subtle interplay between the two-stage Kondo screening and the half-fermionic nature of Majorana quasiparticles. In particular, the coupling to the topological wire is found to reduce the effective exchange interaction between the two quantum dots in the absence of normal leads. Interestingly, it also results in an enhancement of the second-stage Kondo temperature when the normal leads are attached. Moreover, it is shown that the second stage of the Kondo effect can become significantly modified in one of the spin channels due to the interference with the Majorana zero-energy mode, yielding the low-temperature conductance equal to $G=G_0/4$, where $G_0 = 2e^2/h$, instead of $G=0$ in the absence of the topological superconducting wire. Finally, in the case of a short wire, a finite overlap between the wave functions of the Majorana quasiparticles localized at the ends of the wire suppresses the quantum interference and a regular two-stage Kondo effect is restored.
... Local exchange between the spin of a quantum dot (QD) and the spin of conduction band electrons gives rise to the Kondo effect [2,5]. Direct exchange arriving with an additional side-coupled QD may destroy it or lead to the two-stage Kondo screening [4,[6][7][8][9][10]. In a geometry where the two QDs contact the same lead, conduction band electrons mediate the RKKY exchange [11][12][13]. ...
We report on a novel exchange mechanism, mediating the Kondo screening, in a correlated double quantum dot structure in the proximity of superconductor, stemming from nonlocal Andreev reflection processes. We estimate its strength perturbatively and corroborate analytical predictions with accurate numerical renormalization group calculations using an effective model for the superconductor-proximized nanostructure. We demonstrate that superconductor-induced exchange leads to a two-stage Kondo screening. We determine the dependence of the Kondo temperature on the coupling to superconductor and predict a characteristic modification of conventional low-temperature transport behavior, which can be used to experimentally distinguish this phenomenon from other Kondo effects.
... This trend is very similar to the phase transition in Fig. 2(a). In fact, at the case of ε = − U 2 , in the Kondo regime where U π 1, the approximate value of the Kondo temperature can be given by Haldane's expression formula [30,31]: ...
... In the former case, the entropy flow is ln 2, manifested as the partial Kondo screening, whereas in the latter case the entropy flow of the system is 0 and the QD spin is thoroughly screened by the Kondo effect. We can see that when the QD levels increase from ε = −20 to ε = −17 , the system gets closer to the position of electron-hole symmetry, and the Kondo temperature in the system is reduced [31], so the current gradually decreases. As a result, the supercurrent magnitude is suppressed gradually in region B. When the system arrives at the case of T K < T F < , it is located at the point of the electron-hole symmetry, the system's entropy flow fixed point is ln 3, the system presents the transition phase π , like the analysis in Fig. 4. For the result in region C, Fig. 5(b) shows that the average electron occupation increases with the decrease of the QD levels, and N σ = 2.0 and Nσ = 0. ...
With the help of the numerical renormalization group method, we theoretically investigate the Josephson phase transition in a parallel junction with one quantum dot embedded in each arm. It is found that in the cases of appropriate dot levels and symmetrical dot-superconductor couplings, the Josephson phase transition will be suppressed. This is manifested as the fact that with the enhancement of the electron correlation, the supercurrent only arrives at its π′ phase but cannot enter its π phase. Moreover, when the dot levels are detuned, one π′-phase island appears in the phase diagram. Such a result is attributed to the interplay among the Kondo effect, RKKY effect, and the Cooper pairing correlation. We believe that this work can be helpful in understanding the special roles of the correlation mechanisms of the parallel junction in driving the Josephson phase transition.
