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Interplay of the Kondo effect with the induced pairing in electronic and caloric properties of T-shaped double quantum dots

Interplay of the Kondo effect with the induced pairing in electronic and caloric properties of... Interplay of the Kondo e ect with the induced pairing in electronic and caloric properties of T-shaped double quantum dots 1, 2, 2 Krzysztof P. W ojcik and Ireneusz Weymann Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan,  Poland Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan,  Poland (Dated: July 2, 2018) We examine the in uence of the superconducting proximity e ect on the transport properties of a T-shaped double quantum dot strongly coupled to two normal, nonmagnetic or ferromagnetic leads. We show that the two-stage Kondo screening may be suppressed or enhanced by the presence of pairing correlations, depending on the speci c geometric arrangement of the device. We explain our results by invoking e ective decrease of Coulomb interactions by proximity e ect and nd qualitatively correct description in many cases, although spin- ltering e ect stemming from spin- dependent Fano-Kondo interference occurs to be surprisingly fragile to the presence of induced superconducting pairing correlations. The results are obtained within the numerical renormalization group framework in the limit of large superconducting gap, which allows for a reliable examination of the low-temperature sub-gap properties of the considered system. Nevertheless, nite temperature e ects are also taken into account. I. INTRODUCTION other side, Yu-Shiba-Rusinov-like bound states [33{35] are formed, which have already been explored experimen- tally with the Andreev bias spectroscopy [36]. The quan- The two-stage Kondo e ect in double quantum dots tum phase transition is present only in the absence of the (DQDs) or double magnetic impurities has been stud- normal leads and gets smeared to a crossover otherwise. ied for over a decade, both theoretically [1{10] and ex- Nevertheless, even in the latter case, around the critical perimentally [11, 12], in various contexts. In particular, value of the quantum-dot|SC coupling, the BCS-like ex- its relation to the Fano-like interference [13{15] was pre- pectation value hd d i for spin- quantum dot annihila- " # cisely established [6] and the spin-dependent variant of tion operators d becomes nonzero [38]. this e ect for DQDs in an external magnetic eld [16] In this paper we show that the interplay of supercon- or coupled to ferromagnetic leads [17] was proposed as a ducting proximity e ect and correlations giving rise to method for obtaining electrically tunable spin-polarized the Kondo e ect is even more interesting, if a single quan- currents. Moreover, the Andreev transport properties tum dot is substituted by a T-shaped DQD; see Fig. 1. of T-shaped DQDs coupled to superconducting (SC) In this geometry one quantum dot (QD1 in Fig. 1) is em- and normal leads have also been considered [18{23]. In bedded between two normal (ferromagnetic) leads and such hybrid systems, for low temperatures and voltages coupled to the second quantum dot (QD2). We consider smaller than the superconducting energy gap, transport two possible scenarios, in which the superconductor is occurs due to Andreev re ection processes [24{26]. How- coupled to either the rst or the second quantum dot. ever, while most studies dealt with transport between Then, depending on which quantum dot is proximized, normal and superconducting electrodes, the normal elec- and what is the strength of the coupling to SC lead, dif- tronic and caloric transport through T-shaped DQDs ferent interesting e ects take place, as described in the coupled to two normal (ferromagnetic) leads and prox- following. They include, among others, an enhancement imized by the third, superconducting electrode, has been and a destruction of any of the two screening stages of hardly examined so far. Therefore, in this article we per- the Kondo phenomenon. form a detailed and accurate analysis of such a case. Since one of the most experimentally accessible phys- To begin with, it is instructive to notice that similar ical quantities of such a system is its conductance, we studies of a single quantum dot case unveiled an intrigu- base our discussion on the dependence of conductance ing interplay between the Kondo physics [27, 28] and the on model parameters, gate voltages and temperature. pairing induced by the superconducting contact [29{32]. This allows us to thoroughly examine the in uence of in- A hallmark of this interplay is a quantum phase transi- duced superconducting pairing on the two stages of the tion between the Kondo-screened singlet and the BCS- Kondo e ect. Moreover, further information is gained like singlet states, as the ratio of the Kondo temperature from the analysis of the Seebeck coecient, the so-called to the superconducting energy gap is varied [31, 32]. At thermopower, whose sign and magnitude change between the Kondo side of the transition, the Kondo temperature di erent Kondo states [10, 39{44], while temperature de- was found to be enhanced with increasing the coupling pendence allows for recognizing metallic and hopping-like strength to the superconducting lead [37, 38]. At the transport regimes [44, 45]. We note that the subgap transport through hybrid double quantum dot systems is currently undergoing an [email protected] extensive exploration. This has been stimulated by im- arXiv:1804.08270v2 [cond-mat.mes-hall] 29 Jun 2018 2 pressive experiments demonstrating controllable splitting of Cooper pairs in DQDs with both dots attached to a superconductor [46]. This has also provided a great mo- tivation to many researchers to analyze hybrid DQDs in terms of their Cooper pair splitting eciency. This is an undoubtedly interesting direction, however, here we focus on completely di erent geometry, with only one quantum dot directly coupled to the SC lead. While being less useful as a Cooper pair splitter, this system exhibits very interesting strongly-correlated physics. Si- multaneously, recent rapid experimental advances in the eld [47{54] give hope for a possibility of fabricating the device considered here. From this point of view, our re- sults are expected to stimulate further research in hybrid T-shaped DQD as well as to be of assistance in under- standing future experimental observations. Figure 1. Schemes of possible realizations of the considered system. The rst quantum dot (QD1) is coupled to the left It is interesting to note that the interplay between the and right normal (ferromagnetic) leads with coupling strength Kondo correlations and superconductivity has been also , where r = L=R for the left/right lead. The two dots are considered in the case of the Anderson model with at- coupled via hopping matrix elements t. The superconducting tractive on-site Coulomb interactions [55]. In such a case electrode can be attached either to (a) the rst (i = 1) or to the charge Kondo e ect may occur, manifesting itself in (b) the second (i = 2) quantum dot, with the corresponding the electronic [56], caloritronic [57] and spin-caloritronic coupling strength . Si [58] properties. Moreover, intensive theoretical and ex- perimental investigations have clearly shown that in Tl- doped PbTe the negative-U centers induce superconduc- II. MODEL tivity in the otherwise normal host, while the charge Kondo e ect takes place in the system [59{63]. The In the present paper we consider the T-shaped dou- charge Kondo e ect is, however, not present in our sys- ble quantum dot (DQD) coupled to two metallic (in tem. Instead of attractive-U center in uence on the nor- general ferromagnetic) leads, and proximized by one su- mal host, we examine the in uence of BCS superconduc- perconducting electrode. We analyze two possible re- tor on a double quantum dot structure. Furthermore, re- alizations of such system, in which the SC lead is at- cent experiments have also demonstrated the possibility tached either to the rst [Fig. 1(a)] or to the second of fabricating quantum dots with attractive Coulomb in- [Fig. 1(b)] quantum dot. In both cases, the Hamilto- teractions, which persist both below and above the criti- nian of the system can be written in the general form cal temperature for the superconducting transition in the H = H +H +H +H +H +H , where the sub- DQD L R T S TS leads [64, 65]. This gives rise to an interesting interplay sequent parts describe the isolated DQD, left and right between the electrostatic attraction and pairing, which leads, tunneling between DQD and these leads, supercon- leads to suppression of the super-current through the de- ductor, and nally the tunneling between SC and DQD, vice in the crossover region between the weak-coupling respectively. and strong-coupling unitary transmission regimes [66]. We assume that the normal leads contain quasi-free Moreover, unlike the spin Kondo e ect, its charge coun- P electrons, H = " c c , with r 2 fR; Lg r rk rk terpart may become enhanced under nonequilibrium spin k rk and c denoting the annihilation operator correspond- rk bias [67]. Although in this paper we focus on the repul- ing to electron in lead r possessing pseudo-momentum sive U case, our work shall contribute to the general un- k and spin . H has a form of spin-preserving lo- derstanding of the interplay between Kondo correlations cal hopping between QD1 and the electrodes, H = with the superconducting proximity e ect. v d c , where d annihilates spin- electron rk rk i rk 1 at QDi. Assuming the wide-band situation, for the hy- bridization function between QD1 and lead r we take a The paper has the following structure. In Sec. II a de- constant within the energy cut-o D around the Fermi tailed description of the model is provided. Section III level, =  jv j , where  is the (spin-resolved) r r kr r brie y summarizes the role of the magnitude of Coulomb normalized density of states in the lead r at the Fermi interactions for further reference. The results for the case level. With these approximations ferromagnetism of nor- of QD1 (QD2) coupled to the SC lead are then presented mal leads can be taken into account through the spin in Sec. IV (Sec. V), respectively, and the paper is sum- dependence of = (1 + p ), where p is the spin r r r r marized in Sec. VI. polarization in the lead r, provided their magnetization is 3 parallel. We also assume symmetric couplings, = =2, spin-dependent conductance allows for determining the and p = p = p. linear-response current spin polarization through P = L R In the present paper we focus on the low-temperature (G G )=G, with the total conductance G = G + G . " # " # physics. Therefore, having written H in the BCS form, In NRG calculations at least 2048 states per iteration P P y y y were kept, the discretization parameter  = 2 was used, H = " c c + ( c c + h:c:), we S Sk Sk k k Sk k Sk" Sk# while the quantities of interest were calculated directly assume isotropic pairing amplitude,  =  > 0, and from discrete data [74]. integrate out the single-electron states of the supercon- ductor lying outside the energy gap 2jj, to nally take While neglecting the presence of the states of the su- the limit of jj ! 1 [29, 30]. In this way we obtain an perconductor lying outside the gap is one of the strongest e ective Hamiltonian H = H + H + H + H , limitations of the presented model, one needs to keep in e SDQD L R T with SC-proximized DQD part mind that at low temperatures these states contribute X X quite weakly to the physics of the real systems. Moreover, H = " n + U n n + U (n 1)(n 1) SDQD i i i i" i# 1 2 the device is coupled to another continuum, namely to i i normal leads. Therefore, one can expect that the e ects y y y of the presence of gapped continuous part of the spectrum +t (d d + h:c:) (d d + h:c:); (1) 2 Si 1 i" i# of SC lead are only quantitative and rather weak at low temperatures. Nevertheless, detailed study of a single where " is the energy level of QDi, U denotes the respec- i i quantum dot coupled to superconductor [75] show the tive Coulomb interaction strength, U measures inter-dot sign change of the order parameter at the singlet-doublet Coulomb interactions, t is the inter-dot hopping matrix transition point, which is necessarily not captured in our element and describes the coupling to the supercon- Si model for the quantum dot directly coupled to the SC ductor of QDi (i = 1 or 2, depending on geometry). The electrode. operator n = n + n , while n = d d . Henceforth, i i" i# i i we use the detuning  = " +U =2 from the particle-hole i i i symmetry point of each dot to specify the energy lev- III. THE ROLE OF COULOMB INTERACTIONS els of QDs. The coupling is related to the hopping Si matrix element v between QDi and SC, and the nor- Si One of the most intuitive consequence of the presence malized density of states of SC in the normal state,  , of a pairing potential induced by SC proximity is an e ec- through =  jv j , and it is assumed to be energy- Si S Si tive reduction of the corresponding Coulomb repulsion. independent, similarly to the normal leads case. The neg- To be able to analyze the range of validity of this picture, ative sign in front of corresponds to the choice of real Si rst we summarize the e ects related to the on-dot and and positive  in the bulk superconductor Hamiltonian. inter-dot capacitative correlations, U and U , for further The second quantum dot, QD2, is by assumption coupled i reference. Therefore, in this section we consider the sys- to the normal leads only indirectly, through QD1; com- tem in the absence of SC lead. pare Fig. 1. Through an even-odd change of basis of the leads states [68], the model at equilibrium can be exactly mapped onto an e ective single-band system, possessing an e ective coupling and a spin polarization p. A. In uence on Kondo screening Then, the model is solved with the aid of the numerical renormalization group (NRG) technique [69, 70]. We use The essence of the Kondo e ect is the screening of the complete basis set [71, 72] to construct the full density the local moment by the conduction band electrons [27]. matrix of the system [73]. Once the energy spectrum of Since Coulomb interactions are inevitable for the forma- the discretized Hamiltonian is known, the spin-resolved tion of such a moment, they are clearly necessary for the ret: transmission coecient T (!) = Imhhd ; d ii (!) Kondo physics to occur. However, it should be also noted is calculated from the imaginary part of the Fourier trans- that for U & 4=, which is the most common situation, form of the retarded QD1 Green's function. The trans- the Kondo temperature T is a decreasing function of U port coecients, such as the linear-response conductance due to its exponential dependence on =U [76]. in spin-channel , G , and the thermopower, S, can be In T-shaped DQDs the Kondo e ect develops in two calculated from T (!) using the standard linear-response stages [4]. When the temperature is lowered, rst, the expressions magnetic moment of QD1 is screened by the conduc- tion electrons of the leads at the Kondo temperature T . G = L ; (2) Then, for T  T , the resulting Fermi liquid serves as a band of the half-width  T for the second quantum dot 1 L + L K 1" 1# S = ; (3) (QD2), the magnetic moment of which is screened at the eT L + L 0" 0# second stage of the Kondo e ect, with the corresponding with L = ! [@f (!)=@!]T (!)d!, f (!) de- Kondo temperature [4] n T  T noting the Fermi-Dirac distribution function, e (minus) the electron charge, and h the Planck constant. The T  T exp(T =J ); (4) K K 4 2 (a) t = Γ/4 netic (p = 0) and ferromagnetic (p = 0:5) metallic leads. In Fig. 2(a) one can see that the second-stage Kondo temperature T is indeed increased by nite U . In fact, 1.5 the e ective exchange coupling J increases by a factor 0 1 (1 U =U ) for nite capacitive coupling between the dots [7]. However, qualitative features remain the same. 1 p = 0 At PHS, with lowering the temperature, the conductance p = 0.5 U = 0 rst increases at T and almost reaches 2e =h. Then, it U = 0.1U 0.5 decreases to 0 for temperatures below T . This behav- δ = 0 ior is observed for both ferromagnetic and nonmagnetic δ = 0.001U δ = U/6 leads, although only at the PHS point. There, the role (b) of the leads' spin polarization p is reduced to a change in T [77] and, thus, the following change in T , cf. Eq. (4). 0 δ = 0 0.8 A small detuning from the PHS point results in only quantitative changes for p = 0, yet it completely changes −2 0.6 t = 2Γ the situation for nite p. As clearly visible in Fig. 2(a), T /U t = Γ G(T ) does not drop to 0 at low temperatures for nite  . −4 1 t = Γ/2 0.4 t = Γ/3 However, the residual conductance is quite small even for t = Γ/4 relatively large detunings in the case of p = 0, while for −6 0.2 0 0.5 1 1.5 2 nite p, the conductance remains large at low T . This U /U 0 is caused by the exchange eld induced by the ferromag- netic leads [9, 78]. This exchange eld strongly depends -0.2 −10 −8 −6 −4 −2 0 on the position of the quantum dot levels and vanishes 10 10 10 10 10 10 T/U precisely at the PHS point [9, 78]. Once the exchange eld becomes larger than T (which is in fact very small), Figure 2. (a) The conductance G and (b) the Seebeck coe- the second stage of the Kondo e ect is blocked and the cient S as functions of temperature T for di erent detunings conventional (i.e. single-stage) Kondo e ect is restored. , U = U = U = D=10, = U=5 and t = =4. Here D 1 1 2 On the other hand, for large detunings [compare the is the band halfwidth used as energy unit. Solid lines corre- curve for  = U=6 in Fig. 2(a)], the exchange eld is spond to nite spin polarization of the leads, p = 0:5, while comparable to T and also the conventional Kondo ef- dashed lines were used for p = 0. Thick (bright thin) lines fect becomes blocked. indicate the presence (absence) of inter-dot Coulomb inter- In the inset of Fig. 2 the dependence of T on U is action U = U=10. The inset shows the dependence of the presented for a few values of t and p = 0 (dashed lines) second-stage Kondo temperature T on U for the particle- hole symmetric case and di erent t. as well as p = 0:5 (solid lines). It was extracted from G(T ) dependences calculated for di erent U . As re- ported earlier by Ferreira and co-workers for the case of nonmagnetic leads [7], the capacitative coupling between where J is an e ective antiferromagnetic exchange in- the dots tends to increase J and leads to exponential in- teraction between the two dots, J  t =U . Note that crease of T in the physically relevant regime of U < U . estimations of T or T , such as Eq. (4), possess rather This remains true also for ferromagnetic leads. Actually, an order-of-magnitude precision and for qualitative com- the presence of Coulomb correlations between the dots parison of Kondo temperatures in di erent systems more reduces the di erence between the cases of nite p and precise de nition is necessary. Here, we follow the con- p = 0, which is an interesting result at PHS point, where vention of de ning T as a temperature at which the the only in uence of p is the T (p) dependence. conductance increases to half of its maximal value as the K temperature is lowered, such that G(T ) = G =2, with Additional information about the relevant regimes can K max G being the global maximum of G(T ). Moreover, in max be extracted from the temperature dependence of the this paper by T we mean in fact the Kondo temper- thermopower S [10, 40]. However, to achieve nite val- ature in the case of t = 0. Furthermore, in a similar ues of the Seebeck coecient, one needs to tune the sys- fashion we can de ne T as the temperature below T K tem from the PHS point, where S = 0. Let us now at which G(T ) drops to G =2 again (this happens only inspect this in more detail for the line corresponding to max for t 6= 0). p = 0,  = U=6 and U = U=10 shown in Fig. 2(b). At The picture of the two-stage screening presented above high temperatures the system is in the hopping transport does not include the in uence of capacitive coupling be- regime [44, 45], characterized by S  T . Negative sign tween the two dots, U , which will be discussed now. of S is caused by the fact that positive frequencies host Figure 2 demonstrates how nite values of U in uence more spectral weight. Then, with decreasing the temper- the Kondo physics in the considered nanostructure, de- ature, S exhibits rst a local minimum and then, while pending on detuning of QD1  from the particle-hole cooling the system further, its sign changes twice, before symmetry (PHS) point,  = 0, in the case of nonmag- another minimum occurs. The narrow region of posi- S [k /e] B G [e /h] 5 tive thermopower corresponds to the Coulomb blockade regime, which is hardly present due to relatively strong coupling = U=5 used in Fig. 2. The second minimum −1 in S is a consequence of asymmetric Kondo peak near the Fermi level. Despite the fact that T depends on p [77], −2 the position of the minimum related to the Kondo e ect is practically independent of p. Moreover, it also hardly −3 0 U = 0 depends on U ; cf. Fig. 2(b). This is not the case for U = 0.5U 1 2 the position of the maximum in thermopower, which is U = U 1 2 −4 present at even lower temperatures and is related to the 10 (a) second stage of screening. One can also see that the max- (b) imum is completely absent for p = 0:5, which is due to the fact that for assumed parameters the exchange eld is larger than T and the second stage of screening is sup- 0.5 pressed; compare with Fig. 2(a). Furthermore, as far as the e ect of U is concerned, the shift of the maximum in S due to capacitive coupling can be visible and it results from the corresponding change in T , which can be seen in the temperature dependence of the conductance. -0.5 Finally, it is worth to note that the maximum of S at T  T is much more pronounced as compared to -1 the minimum at T  T . This is caused by the fact, 0.25 0.3 0.35 0.4 0.45 that good thermoelectric materials are characterized by δ /U 2 2 sharp and asymmetric features in the spectral density near ! = 0 [79, 80]. For the parameters considered in Figure 3. (a) The low-temperature conductance G and (b) its spin polarization P plotted as a function of QD2 detuning Fig. 2, the Kondo temperature T is quite large and the for ferromagnetic leads (p = 0:5),  = 0:1U , U = 0:1U , 1 2 2 Kondo peak in the spectral density is relatively broad. U = D=10 and di erent U , as indicated. For comparison, 2 1 On the contrary, T is indeed cryogenic, and the dip in the same curves calculated for the case of U = 0 are shown T (!) corresponding to the second stage of screening is using a light grey color. very sharp. ered values of U , the minimum in G( ) is present (note 1 2 B. In uence on Fano interference and its spin the logarithmic scale on vertical axis). The total con- dependence ductance does not drop to 0 due to the spin-splitting of the resonance condition, which can be recognized from The Fano e ect is a consequence of the quantum in- the plot of conductance spin polarization P in Fig. 3(b). terference between a resonant level and the continuum of The latter varies continuously between P = 1 (for states [13]. It is therefore also present in DQD systems corresponding to the antiresonance in the majority spin (even noninteracting) and manifests itself through an an- channel) and P = 1 (for antiresonance in the minority tiresonance in the conductance as a function of DQD en- channel). Qualitatively, this situation is hardly changed ergy levels [14]. Finite Coulomb correlations can modify by nite Coulomb interactions in QD1 U or the inter-dot the conditions for Fano interference and result in another capacitative coupling U . It can be seen that U slightly interesting phenomena. Primarily, the Fano physics is changes the position of the antiresonance and a ects its obtained only at the zero-temperature limit, which may width and depth. On the other hand, U only shifts the be experimentally irrelevant due to cryogenic scale of T minima, not a ecting their depth or spin-splitting signif- occurring in the system. At nite T , deviations from icantly, as can be seen from comparison with the U = 0 Fano anti-resonance curve can be expected and has al- case, which is plotted in Fig. 3 with bright lines. ready been measured [6, 15]. In fact, the antiresonance Basing on these observations, one could naively think itself may be seen as the consequence of the second stage that a weak coupling of SC lead to QD1, e ectively re- 2 2 of the Kondo e ect, which leads to the suppression of the sulting in a reduction of U to U = U 4 , should 1 1 1 S1 conductance at T  T [6]. Moreover, when U > 0, a only quantitatively in uence the Fano e ect and its spin spin-splitting of the conductance antiresonance occurs in dependence. As shall be shown in Sec. IV C, this conjec- T-shaped DQD coupled to ferromagnetic leads without ture is not true. applying an external magnetic eld [17]. Summing up this section, we have found that the pres- The Fano-like antiresonance is visible in Fig. 3(a), ence of capacitive correlations between the two quantum where the conductance is plotted against detuning of dots does not change qualitative features of the presented QD2 energy level for a few values of the Coulomb in- results. However, the quantitative changes can be rela- teraction strengths of QD1, U . Clearly, for all consid- tively strong, due to exponential dependence of T on U . G [e /h] P 6 2 (a) Γ = 0 to the single-quantum-dot case, for which it was shown S1 Γ = U/10 S1 that nite ( = 0) results in an enhancement S1 S2 Γ = U/4 S1 of the Kondo temperature [37, 38]. One can thus ex- Γ = U/2 S1 1.5 pect, through exponential dependence of T on T , cf. Γ = U S1 Eq. (4), that even a small increase in T should give rise to much larger changes in T . This can be clearly seen in Fig. 4(a), which presents the conductance plot- ted against T for a few representative values of . In- S1 0.5 deed, while increasing the strength of coupling to the su- perconductor results in a slight enhancement of T , the second-stage Kondo temperature exhibits a strong sup- pression with raising . Additionally, for < U=4, S1 S1 (b) 0 2 2 one nds G(T = 0)  =U , with  3. Moreover, S1 the local maximum in G(T ) is slightly lowered as in- S1 0.8 -0.02 creases. This can be understood by referring to the case 0.6 of a proximized quantum dot, where the low-temperature -0.05 value of the conductance was found to be suppressed due 0.4 to the coupling to superconductor [37, 38]. We also note that both the low-temperature conductance as well as the 0.2 −2 0 10 10 local maximum in G(T ) are rather independent of U , al- though for U = 0 the minimum is achieved at slightly lower T , due to smaller T , see Fig. 4(a). -0.1 −10 −8 −6 −4 −2 0 10 10 10 10 10 10 Figure 4(b) presents how nite value of coupling S1 T/U a ects the thermopower of the system. The most visible feature is that, unlike the conductance, the Seebeck coef- Figure 4. (a) The conductance G and (b) thermopower S as cient is very sensitive to the presence of SC correlations. a function of temperature T calculated for U = U = D=10, Already as small pairing potential as the one induced by U = U=10, and di erent values of coupling to superconduct- 0 = U=10 leads to the reduction of maximal value of S1 ing lead at  = 0:05U and p = 0. The case of U = 0 S1 1 S to less than a half of the value for = 0. One could S1 is shown with bright lines for comparison. The inset shows a claim that at low temperatures the thermopower is pro- close-up on the region of high T in (b), marked by the rect- portional to T and this reduction can be understood as angle in the main gure. a consequence of decrease of T . However, usually the lower T corresponds to the sharper dip in the spectral Therefore, to make the analysis more realistic, we assume density, which compensates for the decrease of T . In fact, when reducing the second-stage Kondo tempera- U = U=10 in our further analysis, which is a reasonable value for typical experimental setups [81], and discuss its ture T by decreasing the hopping between the dots t, the maximum in S remains almost constant for t < =2 in uence on the results whenever important. [10]. Moreover, according to Fig. 4, the decrease caused by neglecting U also does not lead to the suppression of IV. EFFECT OF PAIRING INDUCED IN THE S, despite the fact that the corresponding decrease of T FIRST QUANTUM DOT is practically identical to the one caused by = U=10, S1 cf. Fig. 4(a). One can conclude that the suppression of the thermopower by SC proximity e ect cannot be ex- In this section we describe the properties of T-shaped plained by the e ective reduction of the Coulomb interac- DQD, in which the rst quantum dot is proximized by tions and can be seen as a manifestation of the sensitivity the superconductor, see Fig. 1(a). In Sec. IV A we ana- of caloric properties against the pairing correlations. lyze how the superconductor proximity a ects the two- stage Kondo e ect in the considered system. Then, in The values of thermopower at higher temperatures are Sec. IV B, we examine the in uence of the inter-dot hop- much smaller than at T  T , as already explained in ping on the phase transition in QD1 [38]. The interplay Sec. III A. However, the zoom of S in this regime (see between the spin-dependent Fano interference and the the inset in Fig. 4) unveils further interesting properties. pairing induced by the SC lead is discussed in Sec. IV C. First of all, as can be intuitively understood through the e ective reduction of U , the positive peak of S(T ) cor- responding to the Coulomb blockade regime, is quickly A. In uence of pairing correlations on the suppressed with increasing . Furthermore, the nega- S1 two-stage Kondo e ect tive peak related to the Kondo regime is enhanced and for strong ultimately merges with the negative peak S1 The in uence of the superconductor proximity on the corresponding to the thermal accessibility of the Hub- two-stage Kondo e ect can be understood by resorting bard peaks, see the curve for = U . This behavior, S1 S [k /e] 2 G [e /h] 7 0.5 −9 (c) (a) T = 10 U t = 0 0.4 t = Γ/25 1.5 t = Γ/10 0.3 t = Γ/2 t = Γ t = 2Γ 0.2 t = 4Γ 0.5 0.1 0 0 (d) 2 0.05 0.2 (b) Γ = 0.75U S1 0.04 0.15 0.03 0.02 −9 −7 −5 −3 −1 0.1 10 10 10 10 10 0.01 T/U 0.05 -0.01 -0.02 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ /U Γ /U S1 S1 Figure 5. (a) The conductance G, (b) the Seebeck coecient S, and the expectation values (c) hd d i and (d) hd d i as 1" 1# 2" 2# functions of the coupling strength of the rst quantum dot to the SC lead for a few values of inter-dot hopping t, as S1 9 0 indicated. The other parameters are  = 0:05U , for U = U = U = D=10, T = 10 U , = U=20, p = 0 and U = U=10. 1 1 2 The case of U = 0 is shown for comparison with bright lines. The inset in (b) presents the temperature dependence of the conductance for = 0:75U and t = 4. Vertical lines correspond to the singlet-doublet transition point for device decoupled S1 from normal leads, corresponding to t = 0 and t  . clearly di erent compared to that for the second stage the crossover from becoming very wide. In this way we of screening, shows that the competition between the SC can make reference to the physics of quantum phase tran- correlations and good thermoelectric properties is not a sition, which only gets smeared due to nite . We also general rule. use cryogenic yet nite temperature, T = 10 U , instead of T = 0, because for small values of t the second-stage Kondo temperature T can be even smaller, which is ex- B. In uence of inter-dot hopping on the phase perimentally completely irrelevant. transition For t = 0, the conductance smoothly changes from almost G = 2e =h at = 0 due to the conventional S1 Kondo e ect (the value is slightly lower due to small de- For negligible inter-dot hopping t = 0, the system con- tuning from PHS) to G  0 for strong , where the sidered here is reduced to the case of a single quantum S1 Kondo resonance at the Fermi energy is destroyed by the dot proximized by the SC lead, which has been studied, pairing correlations. As far as the thermopower is con- e.g. in Ref. [38], in the context of the phase transition cerned, one could expect a negative peak at T  T . between the Kondo singlet and the singlet being a super- K However, at low temperatures S  T [40], as follows position of empty and doubly occupied states of the dot, from the Sommerfeld expansion, and for the considered where the expectation value hd d i becomes nonzero. 1" 1# very low temperature one gets S  0. The crossover of This transition is a sharp quantum phase transition in the the order parameter at QD1 in the case of t = 0 from limit ! 0 only, while in the presence of normal leads hd d i = 0, in the absence of SC lead, to the univer- it becomes a smooth crossover of the width  . In the 1" 1# sal limit hd d i = 1=2, for ! 1, can be seen in following section we analyze the e ect of nite hopping t 1" 1# S1 Fig. 5(c). Obviously at the decoupled QD2, hd d i = 0. between the two dots on this crossover. To achieve this, 2" 2# We note that the above discussed results are also valid for we analyze the dependence of conductance, Seebeck co- nite t, as long as the hopping is such small that T  T . ecient and the order parameters hd d i and hd d i 1" 1# 2" 2# Otherwise, the landscape changes signi cantly. as functions of the coupling to the SC lead, which are shown in Fig. 5. The coupling of QD1 to the normal For = U=20, as assumed in Fig. 5, nite value of hop- leads was reduced in comparison to Fig. 4, to prevent ping of the order of t = =25 = U=500 is already large S [k /e] G [e /h] G [2e /h] hd d i hd d i 2↑ 2↓ 1↑ 1↓ 8 (a) enough to result in almost full development of the second- stage of screening for = 0 at the considered tempera- S1 ture. However, nite increases T and consequently S1 K decreases T [compare Eq. (4) and Fig. 4], leading to the restoration of the conventional Kondo e ect (suppression of its second stage of screening) for some critical , see S1 0.5 the curves for t = =25 and t = =10 in Fig. 5(a). This critical value of corresponds to T ( = ) = T . S1 S1 S1 As explained in Sec. IV A, for T  T , one can expect a large, positive peak in S(T ). This condition is ful- lled around = and, therefore, the corresponding S1 S1 (b) peak of S( ) can be observed in Fig. 5(b). Again, for S1 t  U=40 the couplings & U=2 lead to the crossover S1 to the Shiba state and the suppression of the Kondo ef- 0.5 fect, with almost una ected hd d i( ) dependence 1" 1# S1 and very small values of hd d i. In this sense, the 2" 2# crossover is qualitatively una ected by the presence of Γ /U = 0 S1 QD2, provided t  . Γ /U = 0.1 S1 Finally, let us analyze what happens for stronger val- Γ /U = 0.5 -0.5 S1 ues of hopping t & . Then, for = 0, the local Γ /U = 0.75 S1 S1 Γ /U = 1 singlet inside the DQD is formed and the Kondo e ect S1 is completely suppressed [4]. The transport is governed -1 0 0.1 0.2 0.3 0.4 0.5 by the spectrum of H and the matrix elements of SDQD δ /U d between its eigenstates. When is increased, at 1 S1 the critical value of , the ground state of H be- S1 SDQD Figure 6. (a) The low-temperature conductance G and (b) comes a spin doublet. In the limit of small t this doublet its spin polarization P as a function of QD2 detuning  for corresponds to a single electron in QD2 and QD1 in the di erent couplings between QD1 and SC lead, and U = S1 1 superconducting singlet state. Therefore, the doublet is U = U = D=10, = U=5, t =  = U = U=10, p = 0:5. 2 1 practically decoupled from the leads and the Kondo e ect Bright lines indicate the results in the case of U = 0, for comparison. is suppressed. However, inter-dot hybridization restores the matrix element of d between the aforementioned doublet and the excited states. Then, the Kondo e ect is always present, although the corresponding Kondo tem- H is a spin singlet, hd d i > 0, i.e. the order pa- SDQD 2" 2# rameter in the second dot has the same sign as hd d i. perature T vary strongly with . In particular, when K S1 1" 1# the singlet-doublet splitting becomes very large, the rel- However, hd d i( ) changes sign at critical , cor- 2" 2# S1 S1 responding approximately to the singlet-doublet transi- evant Kondo scale is strongly suppressed. This is vis- ible in Fig. 5(a) for t = U=5. On the other hand, for tion in a DQD isolated from the normal leads. The crit- ical values for the transition are indicated in Fig. 5 by 0:75U , the Kondo e ect is restored, as seen also S1 in the inset, where the temperature dependence of con- vertical lines. The sign change of the pairing expecta- tion value may be understood by recalling the fact that ductance for such a case is plotted. Higher values of S1 this is in fact expected beyond the  ! 1 approxima- correspond to larger singlet-doublet splitting, hence the tion, i.e. when quasiparticle states in SC are also avail- drop of T below the temperature assumed for calcula- able [75]. Since QD2 is proximized by the continuum of tions in the gure. We note that a similar suppression of the Kondo e ect due to singlet-doublet splitting was also states formed by QD1 and the leads, exhibiting also pair- ing correlations, the sign change of its order parameter reported in the case of DQDs in a Cooper pair splitting geometry [83]. at the singlet-doublet transition is visible. The di erence between the zero of hd d i( ) and the value of 2" 2# S1 S1 It seems worth emphasizing that the restoration of the corresponding to the singlet-doublet transition is a con- Kondo e ect for large t does not have the nature of sup- sequence of renormalization of DQD levels due to nite pressing the second stage of the Kondo e ect. On the coupling to normal leads . contrary, it happens rather at QD2, while QD1 only me- diates the coupling to the leads. This resembles the sit- uation, when QD1 is very far from particle-hole symme- C. In uence of pairing correlations on the try, described in Ref. [8]. Interestingly, despite that the spin-dependent Fano e ect positive peak of S( ) is only diminished, but not com- S1 pletely suppressed in this regime, although it does no longer coincide with maximum of G( ) slope. More- From the discussion in previous sections, one can see S1 over, for strong t, the order parameter at QD2 becomes that in many cases the main e ect of the presence of a nonzero; see Fig. 5(d). As long as the ground state of weakly coupled superconducting lead is an e ective de- P G [e /h] 9 crease of the relevant Coulomb interaction. However, this (a) is not always the case, as argued in this section. As shown in Sec. III B, in the case of ferromagnetic leads 1.5 and U 6= 0, the spin-dependent Fano e ect is present ir- respective of the Coulomb interaction strength in the rst quantum dot, U . Nevertheless, even relatively small val- 1 1 Γ = 0 S2 ues of result in a practically complete suppression of S1 Γ = U/10 S2 the spin splitting of the minimum in conductance. This is Γ = U/4 S2 0.5 visible in Fig. 6, presenting the conductance and its spin Γ = U/2 S2 polarization as functions of  for U = U = U and for Γ = U 2 1 2 S2 a few representative values of . Although relatively S1 low values of coupling do not suppress the minimum S1 1 (b) 0.02 in G( ), see curve for = 0:1U in Fig. 6(a), the spin 2 S1 ltering e ect is completely suppressed, as presented in 0.8 Fig. 6(b). Note that such a suppression e ect was not obtained by altering only U in Sec. III B. Moreover, this 0.6 -0.02 e ect does not depend on U either, as can be seen by comparison to the case of U = 0 shown with bright lines 0.4 −3 0 10 10 in Fig. 6. The fragility of the spin-dependence of the Fano interference to the superconducting proximity ef- 0.2 fect is, therefore, a consequence of a nontrivial interplay between the pairing and the spin correlations. 0 In the case of stronger coupling , even more dra- S1 −10 −8 −6 −4 −2 0 10 10 10 10 10 10 matic changes can be expected. Indeed, the Fano anti- T/U resonance is completely removed for  0:5U ; see S1 Fig. 6(a). Moreover, the transition between the singlet Figure 7. (a) The conductance G and (b) the Seebeck coef- and doublet ground states of H can give rise to cient S as functions of the temperature T for di erent cou- SDQD plings between the second dot and the SC lead calculated the change of sign of the spin polarization, as observed S2 for  = 0:05U . The other parameters are U = U = U = 1 1 2 in Fig. 6(b); see for example curve for = 0:5U at S1 D=10, = U=5, t = =4, p = 0, and U = U=10. The case of 0:22U . Nevertheless, the suppression of conduc- U = 0 is shown with bright lines for comparison. The inset tance is not complete in any of spin channels and the shows the zoom into the large temperature region, marked by absolute value jPj does not exceed 25% in this regime. rectangle in the main plot. One can thus conclude that superconducting pairing cor- relations have a clearly detrimental e ect on the spin ltering properties of the considered device. second quantum dot destroy the Fano e ect completely. V. EFFECT OF PAIRING INDUCED IN THE A. In uence of pairing correlations on the SECOND QUANTUM DOT two-stage Kondo e ect In the preceding section the focal point of the discus- For weak coupling between the second quantum dot sion was the phase transition in QD1 and its in uence and the SC lead,  U , the qualitative understand- S2 on the Kondo physics of the system. Now, in turn, we ing of the proximity e ect can be founded on the idea of move to the analysis of transport properties of a di er- e ective reduction of U . Therefore, the Kondo temper- ent setup, which is shown in Fig. 1(b). Even though the ature for the rst stage of screening the spin in the rst physics for small pairing correlations is in such a case quantum dot, T , hardly depends on . Furthermore, K S2 quite similar to the case of system presented in Fig. 1(a), from Eq. (4) one immediately recognizes that T depends there appear signi cant di erences which are discussed on U through J , and grows with decreasing U . Thus, 2 2 in the following. for the device shown in Fig. 1(b), T increases with S2 In the present section, the analysis of the Kondo e ect in a way similar to T increasing with for the one K S1 is continued for the case of small particle-hole asymme- presented in Fig. 1(a). Note that this is opposite to what try, allowing for non-zero Seebeck coecient to occur. happens to T then. This is illustrated in Fig. 7(a) for The normal leads are assumed to be nonmagnetic. The a few representative values of . The corresponding S2 Fano-like interference e ects occur to be very similar as change in the Seebeck coecient peak position can be in the case of pairing present in QD1 and are not dis- observed in Fig. 7(b). cussed in detail. In particular, small values of lead The physics changes, in comparison to pairing induced S2 to the Fano anti-resonance with suppressed spin- ltering at QD1, for stronger interdot hopping t. Here, the change e ect, while strong pairing correlations induced in the of H ground state corresponds to the formation of a SDQD S [k /e] B G [e /h] 10 singlet in QD2, which suppresses the second stage of the Kondo e ect for above the critical value  U=2. S2 S2 (a) This is re ected in the perfect conductance and lack of 1.5 the thermopower peak at low temperatures for > S2 S2 (for t > 0 and > 0 the transition is in fact a quite sharp crossover, as explained in the following subsection). In- terestingly, an additional sign change of S(T ) occurs at T  T for close to this critical value, as illustrated K S2 0.5 in the inset of Fig. 7(b). This may be accounted for by the splitting of the Kondo peak by a residual dip correspond- ing to the second stage of screening. In fact, T increases 0.25 (b) with quite strongly and becomes only slightly smaller S2 than T for  0:4U . Then, the slope of the QD1 K S2 0.8 spectral function at ! = 0 changes and implies the sign change of S. Nevertheless, for even stronger , the sec- S2 0.6 ond stage of the Kondo e ect becomes nally suppressed. Interestingly, the width of the dip in QD1 spectral den- 0.4 sity corresponding to the second stage of screening (which 0.2 can be taken as a measure of T ) is in fact still nite and 0 0.1 0.2 0.3 0.4 even growing further, and only its depth vanishes, and so Γ /U S2 does the related positive peak of S(T ) together with the 0.5 0.04 (c) two corresponding sign changes. 0.4 B. Phase transition in the second quantum dot 0.3 t = 0 t = Γ/6 0.2 The largest di erence between the phase transition at t = Γ/5 t = Γ/4 QD1 and the one at QD2 induced by pairing correlations 0.1 0 Γ /U 1 t = Γ/2 S2 is associated with the fact that while QD1 is directly cou- t = Γ pled to the metallic leads, QD2 is coupled only through QD1. Therefore the e ective broadening of QD2 levels 0 0.2 0.4 0.6 0.8 1 is in the leading order proportional to  t =. To Γ /U 2 S2 explore the Kondo correlations one needs to consider rel- atively strong coupling , which leads to smearing of the Figure 8. (a) The conductance G, (b) Seebeck coecient S and the order parameter in (c) QD2 hd d i as function of transition at QD1. On the contrary, the transition at 2" 2# coupling between the second quantum dot and the SC lead QD2 is even sharper for strong . The e ect is even for di erent inter-dot hoppings t. The other parameters S2 more pronounced due to the fact that the interdot hop- are  = 0:05U , U = U = U = D=10, = U=5, p = 0, 1 1 1 2 ping t in experimental setups can be quite small. There- 9 0 0 T = 10 U , and U = 0:1U . The U = 0 case is shown with fore, the crossover is in fact quite sharp and the simi- bright lines for comparison. The inset in (b) shows the ther- larity to the quantum phase transition, which occurs at moelectric gure of merit ZT as a function of . The inset S2 t = 0 or = 0, is even more evident than in the case of in (c) presents the order parameter in QD1 (note di erent QD1. However, low values of t also imply indeed cryo- scale). genic Kondo temperatures for screening the second quan- tum dot spin, T , as follows from Eq. (4). This makes the system vulnerable to perturbations [82] and sets the e ect takes place and the conductance G = G does max ground for an interesting interplay between the Kondo ef- not depend on (G < 2e =h due to particle-hole S2 max fect and the superconducting pairing correlations in the asymmetry); cf. Fig. 8(a). Similarly, the Seebeck coe- vicinity of the crossover region. cient S  T  0, as shown in Fig. 8(b). The main results concerning the in uence of the inter- dot coupling on the phase transition at QD2 are summa- For nite hopping t, the second stage of the Kondo ef- rized in Fig. 8. Similarly to Fig. 5, a nite yet very small fect develops at energy scales corresponding to T . Nev- T = 10 U was assumed in calculations. For t = 0, there ertheless, at nite temperature only for suciently strong is a strict phase transition, with discontinuous change of t does T exceeds the actual T used in calculations. This the order parameter hd d i at = U=2, as shown in can be visible for t = =6 in Fig. 8(a). Moreover, due 2" 2# S2 Fig. 8(c). At the same time, there are no consequences to the increase of T with , the relevant critical value S2 of this fact for transport properties between the nor- of t, at which T = T , diminishes. Consequently, the mal leads, since QD2 remains completely decoupled from conductance is suppressed and a peak appears in S( ) S2 them. Therefore, the conventional, single-stage Kondo dependence; see Fig. 8(b). However, unlike in the case hd d i S [k /e] G [e /h] 2↑ 2↓ B hd d i 1↑ 1↓ ZT 11 of pairing induced in QD1 discussed in previous sections, oughly examined the sub-gap physics of both devices and the obtained values of S are larger and the thermoelectric showed that, depending on the superconductor position, eciency is enhanced. This is illustrated by the thermo- the second-stage Kondo temperature T may be either electric gure of merit reaching almost ZT = 0:25, as pre- enhanced or decreased by a small coupling to the su- sented in the inset to Fig. 8(b). This should be compared perconductor. In both cases there appears a doublet- to ZT  0:01 for parameters assumed in Fig. 5 (result singlet crossover around some critical value of the SC not shown in the gure). Further increase of the coupling pairing potential and the properties of the system change to SC lead induces a crossover to the conventional Kondo completely for strong pairing correlations. Depending regime. Its width is set up by the e ective coupling of on the device's geometry, the conventional Kondo e ect QD2 to the normal leads, , as can be deduced from may be strongly supported or completely suppressed in Fig. 8(c). Therefore, for strong , the conductance is this transport regime. Moreover, the crossover becomes S2 maximized and the thermopower strongly suppressed. very sharp for superconductor attached to side-coupled It is interesting to note that in the geometry consid- quantum dot at the regime of strong coupling to nor- ered in this section, QD2 and the normal leads do not mal leads. We explain these e ects as consequences of form a common continuous medium exhibiting pairing e ective decrease of the corresponding Coulomb interac- correlations, to which QD1 is coupled. For this reason, tion and basic properties of coupled Kondo impurities. the pairing amplitude induced in QD1 by the coupling Moreover, we show that the spin-dependent Fano-Kondo to QD2 is always of the same sign and is simply caused interference, which develops in the considered systems, by the hybridization of single-electron states, cf. inset turns out to be very vulnerable to the proximity e ect. in Fig. 8(c). Nevertheless, the order parameter hd d i The spin- ltering e ects present in T-shaped DQDs with 1" 1# exhibits a peak at = . ferromagnetic contacts can be suppressed by even small S2 S2 Finally, the large t regime corresponds to the transport values of the coupling to the superconductor. through molecular levels of DQD in the proximity of SC The presented results show that the superconductor lead. The location of the crossover is only slightly shifted proximity e ect provides additional means for the con- due to the renormalization of the energy levels, but its trol of the two-stage Kondo physics in T-shaped double width is increased signi cantly due to large . As can quantum dots. It enables to either strongly favor or com- be seen in the inset in Fig. 8(c), hd d i remains posi- 1" 1# pletely suppress each stage of the Kondo screening and tive, which is due to the reasons explained above. Con- obtain interesting electric or thermoelectric properties. sequently, the strong t case does not di er quantitatively Furthermore, the analysis of transport properties of hy- much from the case corresponding to weaker inter-dot brid T-shaped DQD systems gives additional insight into hopping, unless caloric properties are concerned. Then, the nature of the interplay between the Kondo correla- of course, smoothed crossover leads to a small slope of tions and the superconductivity, which exhibits a sur- the spectral function at ! = 0, and consequently reduced prising combination of increase of the Kondo tempera- thermopower. ture and suppression of the related spectral features. We hope that our analysis will foster further endeavor in this direction. VI. CONCLUSIONS In the present paper we have analyzed the transport ACKNOWLEDGMENTS properties of a T-shaped double quantum dot system proximized by the superconductor, considering two dis- tinct geometries. In the rst one, the quantum dot di- This work was supported by the National rectly coupled to the normal leads was connected to the Science Centre in Poland through project no. superconductor, while in the second geometry, the side 2015/19/N/ST3/01030. Discussions with B. Bu lka coupled quantum dot was proximized. We have thor- are acknowledged. [1] M. 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Interplay of the Kondo effect with the induced pairing in electronic and caloric properties of T-shaped double quantum dots

Wójcik, Krzysztof P.; Weymann, Ireneusz
Condensed Matter , Volume 2018 (1804) – Apr 23, 2018
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10.1103/PhysRevB.97.235449
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Abstract

Interplay of the Kondo e ect with the induced pairing in electronic and caloric properties of T-shaped double quantum dots 1, 2, 2 Krzysztof P. W ojcik and Ireneusz Weymann Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan,  Poland Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan,  Poland (Dated: July 2, 2018) We examine the in uence of the superconducting proximity e ect on the transport properties of a T-shaped double quantum dot strongly coupled to two normal, nonmagnetic or ferromagnetic leads. We show that the two-stage Kondo screening may be suppressed or enhanced by the presence of pairing correlations, depending on the speci c geometric arrangement of the device. We explain our results by invoking e ective decrease of Coulomb interactions by proximity e ect and nd qualitatively correct description in many cases, although spin- ltering e ect stemming from spin- dependent Fano-Kondo interference occurs to be surprisingly fragile to the presence of induced superconducting pairing correlations. The results are obtained within the numerical renormalization group framework in the limit of large superconducting gap, which allows for a reliable examination of the low-temperature sub-gap properties of the considered system. Nevertheless, nite temperature e ects are also taken into account. I. INTRODUCTION other side, Yu-Shiba-Rusinov-like bound states [33{35] are formed, which have already been explored experimen- tally with the Andreev bias spectroscopy [36]. The quan- The two-stage Kondo e ect in double quantum dots tum phase transition is present only in the absence of the (DQDs) or double magnetic impurities has been stud- normal leads and gets smeared to a crossover otherwise. ied for over a decade, both theoretically [1{10] and ex- Nevertheless, even in the latter case, around the critical perimentally [11, 12], in various contexts. In particular, value of the quantum-dot|SC coupling, the BCS-like ex- its relation to the Fano-like interference [13{15] was pre- pectation value hd d i for spin- quantum dot annihila- " # cisely established [6] and the spin-dependent variant of tion operators d becomes nonzero [38]. this e ect for DQDs in an external magnetic eld [16] In this paper we show that the interplay of supercon- or coupled to ferromagnetic leads [17] was proposed as a ducting proximity e ect and correlations giving rise to method for obtaining electrically tunable spin-polarized the Kondo e ect is even more interesting, if a single quan- currents. Moreover, the Andreev transport properties tum dot is substituted by a T-shaped DQD; see Fig. 1. of T-shaped DQDs coupled to superconducting (SC) In this geometry one quantum dot (QD1 in Fig. 1) is em- and normal leads have also been considered [18{23]. In bedded between two normal (ferromagnetic) leads and such hybrid systems, for low temperatures and voltages coupled to the second quantum dot (QD2). We consider smaller than the superconducting energy gap, transport two possible scenarios, in which the superconductor is occurs due to Andreev re ection processes [24{26]. How- coupled to either the rst or the second quantum dot. ever, while most studies dealt with transport between Then, depending on which quantum dot is proximized, normal and superconducting electrodes, the normal elec- and what is the strength of the coupling to SC lead, dif- tronic and caloric transport through T-shaped DQDs ferent interesting e ects take place, as described in the coupled to two normal (ferromagnetic) leads and prox- following. They include, among others, an enhancement imized by the third, superconducting electrode, has been and a destruction of any of the two screening stages of hardly examined so far. Therefore, in this article we per- the Kondo phenomenon. form a detailed and accurate analysis of such a case. Since one of the most experimentally accessible phys- To begin with, it is instructive to notice that similar ical quantities of such a system is its conductance, we studies of a single quantum dot case unveiled an intrigu- base our discussion on the dependence of conductance ing interplay between the Kondo physics [27, 28] and the on model parameters, gate voltages and temperature. pairing induced by the superconducting contact [29{32]. This allows us to thoroughly examine the in uence of in- A hallmark of this interplay is a quantum phase transi- duced superconducting pairing on the two stages of the tion between the Kondo-screened singlet and the BCS- Kondo e ect. Moreover, further information is gained like singlet states, as the ratio of the Kondo temperature from the analysis of the Seebeck coecient, the so-called to the superconducting energy gap is varied [31, 32]. At thermopower, whose sign and magnitude change between the Kondo side of the transition, the Kondo temperature di erent Kondo states [10, 39{44], while temperature de- was found to be enhanced with increasing the coupling pendence allows for recognizing metallic and hopping-like strength to the superconducting lead [37, 38]. At the transport regimes [44, 45]. We note that the subgap transport through hybrid double quantum dot systems is currently undergoing an [email protected] extensive exploration. This has been stimulated by im- arXiv:1804.08270v2 [cond-mat.mes-hall] 29 Jun 2018 2 pressive experiments demonstrating controllable splitting of Cooper pairs in DQDs with both dots attached to a superconductor [46]. This has also provided a great mo- tivation to many researchers to analyze hybrid DQDs in terms of their Cooper pair splitting eciency. This is an undoubtedly interesting direction, however, here we focus on completely di erent geometry, with only one quantum dot directly coupled to the SC lead. While being less useful as a Cooper pair splitter, this system exhibits very interesting strongly-correlated physics. Si- multaneously, recent rapid experimental advances in the eld [47{54] give hope for a possibility of fabricating the device considered here. From this point of view, our re- sults are expected to stimulate further research in hybrid T-shaped DQD as well as to be of assistance in under- standing future experimental observations. Figure 1. Schemes of possible realizations of the considered system. The rst quantum dot (QD1) is coupled to the left It is interesting to note that the interplay between the and right normal (ferromagnetic) leads with coupling strength Kondo correlations and superconductivity has been also , where r = L=R for the left/right lead. The two dots are considered in the case of the Anderson model with at- coupled via hopping matrix elements t. The superconducting tractive on-site Coulomb interactions [55]. In such a case electrode can be attached either to (a) the rst (i = 1) or to the charge Kondo e ect may occur, manifesting itself in (b) the second (i = 2) quantum dot, with the corresponding the electronic [56], caloritronic [57] and spin-caloritronic coupling strength . Si [58] properties. Moreover, intensive theoretical and ex- perimental investigations have clearly shown that in Tl- doped PbTe the negative-U centers induce superconduc- II. MODEL tivity in the otherwise normal host, while the charge Kondo e ect takes place in the system [59{63]. The In the present paper we consider the T-shaped dou- charge Kondo e ect is, however, not present in our sys- ble quantum dot (DQD) coupled to two metallic (in tem. Instead of attractive-U center in uence on the nor- general ferromagnetic) leads, and proximized by one su- mal host, we examine the in uence of BCS superconduc- perconducting electrode. We analyze two possible re- tor on a double quantum dot structure. Furthermore, re- alizations of such system, in which the SC lead is at- cent experiments have also demonstrated the possibility tached either to the rst [Fig. 1(a)] or to the second of fabricating quantum dots with attractive Coulomb in- [Fig. 1(b)] quantum dot. In both cases, the Hamilto- teractions, which persist both below and above the criti- nian of the system can be written in the general form cal temperature for the superconducting transition in the H = H +H +H +H +H +H , where the sub- DQD L R T S TS leads [64, 65]. This gives rise to an interesting interplay sequent parts describe the isolated DQD, left and right between the electrostatic attraction and pairing, which leads, tunneling between DQD and these leads, supercon- leads to suppression of the super-current through the de- ductor, and nally the tunneling between SC and DQD, vice in the crossover region between the weak-coupling respectively. and strong-coupling unitary transmission regimes [66]. We assume that the normal leads contain quasi-free Moreover, unlike the spin Kondo e ect, its charge coun- P electrons, H = " c c , with r 2 fR; Lg r rk rk terpart may become enhanced under nonequilibrium spin k rk and c denoting the annihilation operator correspond- rk bias [67]. Although in this paper we focus on the repul- ing to electron in lead r possessing pseudo-momentum sive U case, our work shall contribute to the general un- k and spin . H has a form of spin-preserving lo- derstanding of the interplay between Kondo correlations cal hopping between QD1 and the electrodes, H = with the superconducting proximity e ect. v d c , where d annihilates spin- electron rk rk i rk 1 at QDi. Assuming the wide-band situation, for the hy- bridization function between QD1 and lead r we take a The paper has the following structure. In Sec. II a de- constant within the energy cut-o D around the Fermi tailed description of the model is provided. Section III level, =  jv j , where  is the (spin-resolved) r r kr r brie y summarizes the role of the magnitude of Coulomb normalized density of states in the lead r at the Fermi interactions for further reference. The results for the case level. With these approximations ferromagnetism of nor- of QD1 (QD2) coupled to the SC lead are then presented mal leads can be taken into account through the spin in Sec. IV (Sec. V), respectively, and the paper is sum- dependence of = (1 + p ), where p is the spin r r r r marized in Sec. VI. polarization in the lead r, provided their magnetization is 3 parallel. We also assume symmetric couplings, = =2, spin-dependent conductance allows for determining the and p = p = p. linear-response current spin polarization through P = L R In the present paper we focus on the low-temperature (G G )=G, with the total conductance G = G + G . " # " # physics. Therefore, having written H in the BCS form, In NRG calculations at least 2048 states per iteration P P y y y were kept, the discretization parameter  = 2 was used, H = " c c + ( c c + h:c:), we S Sk Sk k k Sk k Sk" Sk# while the quantities of interest were calculated directly assume isotropic pairing amplitude,  =  > 0, and from discrete data [74]. integrate out the single-electron states of the supercon- ductor lying outside the energy gap 2jj, to nally take While neglecting the presence of the states of the su- the limit of jj ! 1 [29, 30]. In this way we obtain an perconductor lying outside the gap is one of the strongest e ective Hamiltonian H = H + H + H + H , limitations of the presented model, one needs to keep in e SDQD L R T with SC-proximized DQD part mind that at low temperatures these states contribute X X quite weakly to the physics of the real systems. Moreover, H = " n + U n n + U (n 1)(n 1) SDQD i i i i" i# 1 2 the device is coupled to another continuum, namely to i i normal leads. Therefore, one can expect that the e ects y y y of the presence of gapped continuous part of the spectrum +t (d d + h:c:) (d d + h:c:); (1) 2 Si 1 i" i# of SC lead are only quantitative and rather weak at low temperatures. Nevertheless, detailed study of a single where " is the energy level of QDi, U denotes the respec- i i quantum dot coupled to superconductor [75] show the tive Coulomb interaction strength, U measures inter-dot sign change of the order parameter at the singlet-doublet Coulomb interactions, t is the inter-dot hopping matrix transition point, which is necessarily not captured in our element and describes the coupling to the supercon- Si model for the quantum dot directly coupled to the SC ductor of QDi (i = 1 or 2, depending on geometry). The electrode. operator n = n + n , while n = d d . Henceforth, i i" i# i i we use the detuning  = " +U =2 from the particle-hole i i i symmetry point of each dot to specify the energy lev- III. THE ROLE OF COULOMB INTERACTIONS els of QDs. The coupling is related to the hopping Si matrix element v between QDi and SC, and the nor- Si One of the most intuitive consequence of the presence malized density of states of SC in the normal state,  , of a pairing potential induced by SC proximity is an e ec- through =  jv j , and it is assumed to be energy- Si S Si tive reduction of the corresponding Coulomb repulsion. independent, similarly to the normal leads case. The neg- To be able to analyze the range of validity of this picture, ative sign in front of corresponds to the choice of real Si rst we summarize the e ects related to the on-dot and and positive  in the bulk superconductor Hamiltonian. inter-dot capacitative correlations, U and U , for further The second quantum dot, QD2, is by assumption coupled i reference. Therefore, in this section we consider the sys- to the normal leads only indirectly, through QD1; com- tem in the absence of SC lead. pare Fig. 1. Through an even-odd change of basis of the leads states [68], the model at equilibrium can be exactly mapped onto an e ective single-band system, possessing an e ective coupling and a spin polarization p. A. In uence on Kondo screening Then, the model is solved with the aid of the numerical renormalization group (NRG) technique [69, 70]. We use The essence of the Kondo e ect is the screening of the complete basis set [71, 72] to construct the full density the local moment by the conduction band electrons [27]. matrix of the system [73]. Once the energy spectrum of Since Coulomb interactions are inevitable for the forma- the discretized Hamiltonian is known, the spin-resolved tion of such a moment, they are clearly necessary for the ret: transmission coecient T (!) = Imhhd ; d ii (!) Kondo physics to occur. However, it should be also noted is calculated from the imaginary part of the Fourier trans- that for U & 4=, which is the most common situation, form of the retarded QD1 Green's function. The trans- the Kondo temperature T is a decreasing function of U port coecients, such as the linear-response conductance due to its exponential dependence on =U [76]. in spin-channel , G , and the thermopower, S, can be In T-shaped DQDs the Kondo e ect develops in two calculated from T (!) using the standard linear-response stages [4]. When the temperature is lowered, rst, the expressions magnetic moment of QD1 is screened by the conduc- tion electrons of the leads at the Kondo temperature T . G = L ; (2) Then, for T  T , the resulting Fermi liquid serves as a band of the half-width  T for the second quantum dot 1 L + L K 1" 1# S = ; (3) (QD2), the magnetic moment of which is screened at the eT L + L 0" 0# second stage of the Kondo e ect, with the corresponding with L = ! [@f (!)=@!]T (!)d!, f (!) de- Kondo temperature [4] n T  T noting the Fermi-Dirac distribution function, e (minus) the electron charge, and h the Planck constant. The T  T exp(T =J ); (4) K K 4 2 (a) t = Γ/4 netic (p = 0) and ferromagnetic (p = 0:5) metallic leads. In Fig. 2(a) one can see that the second-stage Kondo temperature T is indeed increased by nite U . In fact, 1.5 the e ective exchange coupling J increases by a factor 0 1 (1 U =U ) for nite capacitive coupling between the dots [7]. However, qualitative features remain the same. 1 p = 0 At PHS, with lowering the temperature, the conductance p = 0.5 U = 0 rst increases at T and almost reaches 2e =h. Then, it U = 0.1U 0.5 decreases to 0 for temperatures below T . This behav- δ = 0 ior is observed for both ferromagnetic and nonmagnetic δ = 0.001U δ = U/6 leads, although only at the PHS point. There, the role (b) of the leads' spin polarization p is reduced to a change in T [77] and, thus, the following change in T , cf. Eq. (4). 0 δ = 0 0.8 A small detuning from the PHS point results in only quantitative changes for p = 0, yet it completely changes −2 0.6 t = 2Γ the situation for nite p. As clearly visible in Fig. 2(a), T /U t = Γ G(T ) does not drop to 0 at low temperatures for nite  . −4 1 t = Γ/2 0.4 t = Γ/3 However, the residual conductance is quite small even for t = Γ/4 relatively large detunings in the case of p = 0, while for −6 0.2 0 0.5 1 1.5 2 nite p, the conductance remains large at low T . This U /U 0 is caused by the exchange eld induced by the ferromag- netic leads [9, 78]. This exchange eld strongly depends -0.2 −10 −8 −6 −4 −2 0 on the position of the quantum dot levels and vanishes 10 10 10 10 10 10 T/U precisely at the PHS point [9, 78]. Once the exchange eld becomes larger than T (which is in fact very small), Figure 2. (a) The conductance G and (b) the Seebeck coe- the second stage of the Kondo e ect is blocked and the cient S as functions of temperature T for di erent detunings conventional (i.e. single-stage) Kondo e ect is restored. , U = U = U = D=10, = U=5 and t = =4. Here D 1 1 2 On the other hand, for large detunings [compare the is the band halfwidth used as energy unit. Solid lines corre- curve for  = U=6 in Fig. 2(a)], the exchange eld is spond to nite spin polarization of the leads, p = 0:5, while comparable to T and also the conventional Kondo ef- dashed lines were used for p = 0. Thick (bright thin) lines fect becomes blocked. indicate the presence (absence) of inter-dot Coulomb inter- In the inset of Fig. 2 the dependence of T on U is action U = U=10. The inset shows the dependence of the presented for a few values of t and p = 0 (dashed lines) second-stage Kondo temperature T on U for the particle- hole symmetric case and di erent t. as well as p = 0:5 (solid lines). It was extracted from G(T ) dependences calculated for di erent U . As re- ported earlier by Ferreira and co-workers for the case of nonmagnetic leads [7], the capacitative coupling between where J is an e ective antiferromagnetic exchange in- the dots tends to increase J and leads to exponential in- teraction between the two dots, J  t =U . Note that crease of T in the physically relevant regime of U < U . estimations of T or T , such as Eq. (4), possess rather This remains true also for ferromagnetic leads. Actually, an order-of-magnitude precision and for qualitative com- the presence of Coulomb correlations between the dots parison of Kondo temperatures in di erent systems more reduces the di erence between the cases of nite p and precise de nition is necessary. Here, we follow the con- p = 0, which is an interesting result at PHS point, where vention of de ning T as a temperature at which the the only in uence of p is the T (p) dependence. conductance increases to half of its maximal value as the K temperature is lowered, such that G(T ) = G =2, with Additional information about the relevant regimes can K max G being the global maximum of G(T ). Moreover, in max be extracted from the temperature dependence of the this paper by T we mean in fact the Kondo temper- thermopower S [10, 40]. However, to achieve nite val- ature in the case of t = 0. Furthermore, in a similar ues of the Seebeck coecient, one needs to tune the sys- fashion we can de ne T as the temperature below T K tem from the PHS point, where S = 0. Let us now at which G(T ) drops to G =2 again (this happens only inspect this in more detail for the line corresponding to max for t 6= 0). p = 0,  = U=6 and U = U=10 shown in Fig. 2(b). At The picture of the two-stage screening presented above high temperatures the system is in the hopping transport does not include the in uence of capacitive coupling be- regime [44, 45], characterized by S  T . Negative sign tween the two dots, U , which will be discussed now. of S is caused by the fact that positive frequencies host Figure 2 demonstrates how nite values of U in uence more spectral weight. Then, with decreasing the temper- the Kondo physics in the considered nanostructure, de- ature, S exhibits rst a local minimum and then, while pending on detuning of QD1  from the particle-hole cooling the system further, its sign changes twice, before symmetry (PHS) point,  = 0, in the case of nonmag- another minimum occurs. The narrow region of posi- S [k /e] B G [e /h] 5 tive thermopower corresponds to the Coulomb blockade regime, which is hardly present due to relatively strong coupling = U=5 used in Fig. 2. The second minimum −1 in S is a consequence of asymmetric Kondo peak near the Fermi level. Despite the fact that T depends on p [77], −2 the position of the minimum related to the Kondo e ect is practically independent of p. Moreover, it also hardly −3 0 U = 0 depends on U ; cf. Fig. 2(b). This is not the case for U = 0.5U 1 2 the position of the maximum in thermopower, which is U = U 1 2 −4 present at even lower temperatures and is related to the 10 (a) second stage of screening. One can also see that the max- (b) imum is completely absent for p = 0:5, which is due to the fact that for assumed parameters the exchange eld is larger than T and the second stage of screening is sup- 0.5 pressed; compare with Fig. 2(a). Furthermore, as far as the e ect of U is concerned, the shift of the maximum in S due to capacitive coupling can be visible and it results from the corresponding change in T , which can be seen in the temperature dependence of the conductance. -0.5 Finally, it is worth to note that the maximum of S at T  T is much more pronounced as compared to -1 the minimum at T  T . This is caused by the fact, 0.25 0.3 0.35 0.4 0.45 that good thermoelectric materials are characterized by δ /U 2 2 sharp and asymmetric features in the spectral density near ! = 0 [79, 80]. For the parameters considered in Figure 3. (a) The low-temperature conductance G and (b) its spin polarization P plotted as a function of QD2 detuning Fig. 2, the Kondo temperature T is quite large and the for ferromagnetic leads (p = 0:5),  = 0:1U , U = 0:1U , 1 2 2 Kondo peak in the spectral density is relatively broad. U = D=10 and di erent U , as indicated. For comparison, 2 1 On the contrary, T is indeed cryogenic, and the dip in the same curves calculated for the case of U = 0 are shown T (!) corresponding to the second stage of screening is using a light grey color. very sharp. ered values of U , the minimum in G( ) is present (note 1 2 B. In uence on Fano interference and its spin the logarithmic scale on vertical axis). The total con- dependence ductance does not drop to 0 due to the spin-splitting of the resonance condition, which can be recognized from The Fano e ect is a consequence of the quantum in- the plot of conductance spin polarization P in Fig. 3(b). terference between a resonant level and the continuum of The latter varies continuously between P = 1 (for states [13]. It is therefore also present in DQD systems corresponding to the antiresonance in the majority spin (even noninteracting) and manifests itself through an an- channel) and P = 1 (for antiresonance in the minority tiresonance in the conductance as a function of DQD en- channel). Qualitatively, this situation is hardly changed ergy levels [14]. Finite Coulomb correlations can modify by nite Coulomb interactions in QD1 U or the inter-dot the conditions for Fano interference and result in another capacitative coupling U . It can be seen that U slightly interesting phenomena. Primarily, the Fano physics is changes the position of the antiresonance and a ects its obtained only at the zero-temperature limit, which may width and depth. On the other hand, U only shifts the be experimentally irrelevant due to cryogenic scale of T minima, not a ecting their depth or spin-splitting signif- occurring in the system. At nite T , deviations from icantly, as can be seen from comparison with the U = 0 Fano anti-resonance curve can be expected and has al- case, which is plotted in Fig. 3 with bright lines. ready been measured [6, 15]. In fact, the antiresonance Basing on these observations, one could naively think itself may be seen as the consequence of the second stage that a weak coupling of SC lead to QD1, e ectively re- 2 2 of the Kondo e ect, which leads to the suppression of the sulting in a reduction of U to U = U 4 , should 1 1 1 S1 conductance at T  T [6]. Moreover, when U > 0, a only quantitatively in uence the Fano e ect and its spin spin-splitting of the conductance antiresonance occurs in dependence. As shall be shown in Sec. IV C, this conjec- T-shaped DQD coupled to ferromagnetic leads without ture is not true. applying an external magnetic eld [17]. Summing up this section, we have found that the pres- The Fano-like antiresonance is visible in Fig. 3(a), ence of capacitive correlations between the two quantum where the conductance is plotted against detuning of dots does not change qualitative features of the presented QD2 energy level for a few values of the Coulomb in- results. However, the quantitative changes can be rela- teraction strengths of QD1, U . Clearly, for all consid- tively strong, due to exponential dependence of T on U . G [e /h] P 6 2 (a) Γ = 0 to the single-quantum-dot case, for which it was shown S1 Γ = U/10 S1 that nite ( = 0) results in an enhancement S1 S2 Γ = U/4 S1 of the Kondo temperature [37, 38]. One can thus ex- Γ = U/2 S1 1.5 pect, through exponential dependence of T on T , cf. Γ = U S1 Eq. (4), that even a small increase in T should give rise to much larger changes in T . This can be clearly seen in Fig. 4(a), which presents the conductance plot- ted against T for a few representative values of . In- S1 0.5 deed, while increasing the strength of coupling to the su- perconductor results in a slight enhancement of T , the second-stage Kondo temperature exhibits a strong sup- pression with raising . Additionally, for < U=4, S1 S1 (b) 0 2 2 one nds G(T = 0)  =U , with  3. Moreover, S1 the local maximum in G(T ) is slightly lowered as in- S1 0.8 -0.02 creases. This can be understood by referring to the case 0.6 of a proximized quantum dot, where the low-temperature -0.05 value of the conductance was found to be suppressed due 0.4 to the coupling to superconductor [37, 38]. We also note that both the low-temperature conductance as well as the 0.2 −2 0 10 10 local maximum in G(T ) are rather independent of U , al- though for U = 0 the minimum is achieved at slightly lower T , due to smaller T , see Fig. 4(a). -0.1 −10 −8 −6 −4 −2 0 10 10 10 10 10 10 Figure 4(b) presents how nite value of coupling S1 T/U a ects the thermopower of the system. The most visible feature is that, unlike the conductance, the Seebeck coef- Figure 4. (a) The conductance G and (b) thermopower S as cient is very sensitive to the presence of SC correlations. a function of temperature T calculated for U = U = D=10, Already as small pairing potential as the one induced by U = U=10, and di erent values of coupling to superconduct- 0 = U=10 leads to the reduction of maximal value of S1 ing lead at  = 0:05U and p = 0. The case of U = 0 S1 1 S to less than a half of the value for = 0. One could S1 is shown with bright lines for comparison. The inset shows a claim that at low temperatures the thermopower is pro- close-up on the region of high T in (b), marked by the rect- portional to T and this reduction can be understood as angle in the main gure. a consequence of decrease of T . However, usually the lower T corresponds to the sharper dip in the spectral Therefore, to make the analysis more realistic, we assume density, which compensates for the decrease of T . In fact, when reducing the second-stage Kondo tempera- U = U=10 in our further analysis, which is a reasonable value for typical experimental setups [81], and discuss its ture T by decreasing the hopping between the dots t, the maximum in S remains almost constant for t < =2 in uence on the results whenever important. [10]. Moreover, according to Fig. 4, the decrease caused by neglecting U also does not lead to the suppression of IV. EFFECT OF PAIRING INDUCED IN THE S, despite the fact that the corresponding decrease of T FIRST QUANTUM DOT is practically identical to the one caused by = U=10, S1 cf. Fig. 4(a). One can conclude that the suppression of the thermopower by SC proximity e ect cannot be ex- In this section we describe the properties of T-shaped plained by the e ective reduction of the Coulomb interac- DQD, in which the rst quantum dot is proximized by tions and can be seen as a manifestation of the sensitivity the superconductor, see Fig. 1(a). In Sec. IV A we ana- of caloric properties against the pairing correlations. lyze how the superconductor proximity a ects the two- stage Kondo e ect in the considered system. Then, in The values of thermopower at higher temperatures are Sec. IV B, we examine the in uence of the inter-dot hop- much smaller than at T  T , as already explained in ping on the phase transition in QD1 [38]. The interplay Sec. III A. However, the zoom of S in this regime (see between the spin-dependent Fano interference and the the inset in Fig. 4) unveils further interesting properties. pairing induced by the SC lead is discussed in Sec. IV C. First of all, as can be intuitively understood through the e ective reduction of U , the positive peak of S(T ) cor- responding to the Coulomb blockade regime, is quickly A. In uence of pairing correlations on the suppressed with increasing . Furthermore, the nega- S1 two-stage Kondo e ect tive peak related to the Kondo regime is enhanced and for strong ultimately merges with the negative peak S1 The in uence of the superconductor proximity on the corresponding to the thermal accessibility of the Hub- two-stage Kondo e ect can be understood by resorting bard peaks, see the curve for = U . This behavior, S1 S [k /e] 2 G [e /h] 7 0.5 −9 (c) (a) T = 10 U t = 0 0.4 t = Γ/25 1.5 t = Γ/10 0.3 t = Γ/2 t = Γ t = 2Γ 0.2 t = 4Γ 0.5 0.1 0 0 (d) 2 0.05 0.2 (b) Γ = 0.75U S1 0.04 0.15 0.03 0.02 −9 −7 −5 −3 −1 0.1 10 10 10 10 10 0.01 T/U 0.05 -0.01 -0.02 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ /U Γ /U S1 S1 Figure 5. (a) The conductance G, (b) the Seebeck coecient S, and the expectation values (c) hd d i and (d) hd d i as 1" 1# 2" 2# functions of the coupling strength of the rst quantum dot to the SC lead for a few values of inter-dot hopping t, as S1 9 0 indicated. The other parameters are  = 0:05U , for U = U = U = D=10, T = 10 U , = U=20, p = 0 and U = U=10. 1 1 2 The case of U = 0 is shown for comparison with bright lines. The inset in (b) presents the temperature dependence of the conductance for = 0:75U and t = 4. Vertical lines correspond to the singlet-doublet transition point for device decoupled S1 from normal leads, corresponding to t = 0 and t  . clearly di erent compared to that for the second stage the crossover from becoming very wide. In this way we of screening, shows that the competition between the SC can make reference to the physics of quantum phase tran- correlations and good thermoelectric properties is not a sition, which only gets smeared due to nite . We also general rule. use cryogenic yet nite temperature, T = 10 U , instead of T = 0, because for small values of t the second-stage Kondo temperature T can be even smaller, which is ex- B. In uence of inter-dot hopping on the phase perimentally completely irrelevant. transition For t = 0, the conductance smoothly changes from almost G = 2e =h at = 0 due to the conventional S1 Kondo e ect (the value is slightly lower due to small de- For negligible inter-dot hopping t = 0, the system con- tuning from PHS) to G  0 for strong , where the sidered here is reduced to the case of a single quantum S1 Kondo resonance at the Fermi energy is destroyed by the dot proximized by the SC lead, which has been studied, pairing correlations. As far as the thermopower is con- e.g. in Ref. [38], in the context of the phase transition cerned, one could expect a negative peak at T  T . between the Kondo singlet and the singlet being a super- K However, at low temperatures S  T [40], as follows position of empty and doubly occupied states of the dot, from the Sommerfeld expansion, and for the considered where the expectation value hd d i becomes nonzero. 1" 1# very low temperature one gets S  0. The crossover of This transition is a sharp quantum phase transition in the the order parameter at QD1 in the case of t = 0 from limit ! 0 only, while in the presence of normal leads hd d i = 0, in the absence of SC lead, to the univer- it becomes a smooth crossover of the width  . In the 1" 1# sal limit hd d i = 1=2, for ! 1, can be seen in following section we analyze the e ect of nite hopping t 1" 1# S1 Fig. 5(c). Obviously at the decoupled QD2, hd d i = 0. between the two dots on this crossover. To achieve this, 2" 2# We note that the above discussed results are also valid for we analyze the dependence of conductance, Seebeck co- nite t, as long as the hopping is such small that T  T . ecient and the order parameters hd d i and hd d i 1" 1# 2" 2# Otherwise, the landscape changes signi cantly. as functions of the coupling to the SC lead, which are shown in Fig. 5. The coupling of QD1 to the normal For = U=20, as assumed in Fig. 5, nite value of hop- leads was reduced in comparison to Fig. 4, to prevent ping of the order of t = =25 = U=500 is already large S [k /e] G [e /h] G [2e /h] hd d i hd d i 2↑ 2↓ 1↑ 1↓ 8 (a) enough to result in almost full development of the second- stage of screening for = 0 at the considered tempera- S1 ture. However, nite increases T and consequently S1 K decreases T [compare Eq. (4) and Fig. 4], leading to the restoration of the conventional Kondo e ect (suppression of its second stage of screening) for some critical , see S1 0.5 the curves for t = =25 and t = =10 in Fig. 5(a). This critical value of corresponds to T ( = ) = T . S1 S1 S1 As explained in Sec. IV A, for T  T , one can expect a large, positive peak in S(T ). This condition is ful- lled around = and, therefore, the corresponding S1 S1 (b) peak of S( ) can be observed in Fig. 5(b). Again, for S1 t  U=40 the couplings & U=2 lead to the crossover S1 to the Shiba state and the suppression of the Kondo ef- 0.5 fect, with almost una ected hd d i( ) dependence 1" 1# S1 and very small values of hd d i. In this sense, the 2" 2# crossover is qualitatively una ected by the presence of Γ /U = 0 S1 QD2, provided t  . Γ /U = 0.1 S1 Finally, let us analyze what happens for stronger val- Γ /U = 0.5 -0.5 S1 ues of hopping t & . Then, for = 0, the local Γ /U = 0.75 S1 S1 Γ /U = 1 singlet inside the DQD is formed and the Kondo e ect S1 is completely suppressed [4]. The transport is governed -1 0 0.1 0.2 0.3 0.4 0.5 by the spectrum of H and the matrix elements of SDQD δ /U d between its eigenstates. When is increased, at 1 S1 the critical value of , the ground state of H be- S1 SDQD Figure 6. (a) The low-temperature conductance G and (b) comes a spin doublet. In the limit of small t this doublet its spin polarization P as a function of QD2 detuning  for corresponds to a single electron in QD2 and QD1 in the di erent couplings between QD1 and SC lead, and U = S1 1 superconducting singlet state. Therefore, the doublet is U = U = D=10, = U=5, t =  = U = U=10, p = 0:5. 2 1 practically decoupled from the leads and the Kondo e ect Bright lines indicate the results in the case of U = 0, for comparison. is suppressed. However, inter-dot hybridization restores the matrix element of d between the aforementioned doublet and the excited states. Then, the Kondo e ect is always present, although the corresponding Kondo tem- H is a spin singlet, hd d i > 0, i.e. the order pa- SDQD 2" 2# rameter in the second dot has the same sign as hd d i. perature T vary strongly with . In particular, when K S1 1" 1# the singlet-doublet splitting becomes very large, the rel- However, hd d i( ) changes sign at critical , cor- 2" 2# S1 S1 responding approximately to the singlet-doublet transi- evant Kondo scale is strongly suppressed. This is vis- ible in Fig. 5(a) for t = U=5. On the other hand, for tion in a DQD isolated from the normal leads. The crit- ical values for the transition are indicated in Fig. 5 by 0:75U , the Kondo e ect is restored, as seen also S1 in the inset, where the temperature dependence of con- vertical lines. The sign change of the pairing expecta- tion value may be understood by recalling the fact that ductance for such a case is plotted. Higher values of S1 this is in fact expected beyond the  ! 1 approxima- correspond to larger singlet-doublet splitting, hence the tion, i.e. when quasiparticle states in SC are also avail- drop of T below the temperature assumed for calcula- able [75]. Since QD2 is proximized by the continuum of tions in the gure. We note that a similar suppression of the Kondo e ect due to singlet-doublet splitting was also states formed by QD1 and the leads, exhibiting also pair- ing correlations, the sign change of its order parameter reported in the case of DQDs in a Cooper pair splitting geometry [83]. at the singlet-doublet transition is visible. The di erence between the zero of hd d i( ) and the value of 2" 2# S1 S1 It seems worth emphasizing that the restoration of the corresponding to the singlet-doublet transition is a con- Kondo e ect for large t does not have the nature of sup- sequence of renormalization of DQD levels due to nite pressing the second stage of the Kondo e ect. On the coupling to normal leads . contrary, it happens rather at QD2, while QD1 only me- diates the coupling to the leads. This resembles the sit- uation, when QD1 is very far from particle-hole symme- C. In uence of pairing correlations on the try, described in Ref. [8]. Interestingly, despite that the spin-dependent Fano e ect positive peak of S( ) is only diminished, but not com- S1 pletely suppressed in this regime, although it does no longer coincide with maximum of G( ) slope. More- From the discussion in previous sections, one can see S1 over, for strong t, the order parameter at QD2 becomes that in many cases the main e ect of the presence of a nonzero; see Fig. 5(d). As long as the ground state of weakly coupled superconducting lead is an e ective de- P G [e /h] 9 crease of the relevant Coulomb interaction. However, this (a) is not always the case, as argued in this section. As shown in Sec. III B, in the case of ferromagnetic leads 1.5 and U 6= 0, the spin-dependent Fano e ect is present ir- respective of the Coulomb interaction strength in the rst quantum dot, U . Nevertheless, even relatively small val- 1 1 Γ = 0 S2 ues of result in a practically complete suppression of S1 Γ = U/10 S2 the spin splitting of the minimum in conductance. This is Γ = U/4 S2 0.5 visible in Fig. 6, presenting the conductance and its spin Γ = U/2 S2 polarization as functions of  for U = U = U and for Γ = U 2 1 2 S2 a few representative values of . Although relatively S1 low values of coupling do not suppress the minimum S1 1 (b) 0.02 in G( ), see curve for = 0:1U in Fig. 6(a), the spin 2 S1 ltering e ect is completely suppressed, as presented in 0.8 Fig. 6(b). Note that such a suppression e ect was not obtained by altering only U in Sec. III B. Moreover, this 0.6 -0.02 e ect does not depend on U either, as can be seen by comparison to the case of U = 0 shown with bright lines 0.4 −3 0 10 10 in Fig. 6. The fragility of the spin-dependence of the Fano interference to the superconducting proximity ef- 0.2 fect is, therefore, a consequence of a nontrivial interplay between the pairing and the spin correlations. 0 In the case of stronger coupling , even more dra- S1 −10 −8 −6 −4 −2 0 10 10 10 10 10 10 matic changes can be expected. Indeed, the Fano anti- T/U resonance is completely removed for  0:5U ; see S1 Fig. 6(a). Moreover, the transition between the singlet Figure 7. (a) The conductance G and (b) the Seebeck coef- and doublet ground states of H can give rise to cient S as functions of the temperature T for di erent cou- SDQD plings between the second dot and the SC lead calculated the change of sign of the spin polarization, as observed S2 for  = 0:05U . The other parameters are U = U = U = 1 1 2 in Fig. 6(b); see for example curve for = 0:5U at S1 D=10, = U=5, t = =4, p = 0, and U = U=10. The case of 0:22U . Nevertheless, the suppression of conduc- U = 0 is shown with bright lines for comparison. The inset tance is not complete in any of spin channels and the shows the zoom into the large temperature region, marked by absolute value jPj does not exceed 25% in this regime. rectangle in the main plot. One can thus conclude that superconducting pairing cor- relations have a clearly detrimental e ect on the spin ltering properties of the considered device. second quantum dot destroy the Fano e ect completely. V. EFFECT OF PAIRING INDUCED IN THE A. In uence of pairing correlations on the SECOND QUANTUM DOT two-stage Kondo e ect In the preceding section the focal point of the discus- For weak coupling between the second quantum dot sion was the phase transition in QD1 and its in uence and the SC lead,  U , the qualitative understand- S2 on the Kondo physics of the system. Now, in turn, we ing of the proximity e ect can be founded on the idea of move to the analysis of transport properties of a di er- e ective reduction of U . Therefore, the Kondo temper- ent setup, which is shown in Fig. 1(b). Even though the ature for the rst stage of screening the spin in the rst physics for small pairing correlations is in such a case quantum dot, T , hardly depends on . Furthermore, K S2 quite similar to the case of system presented in Fig. 1(a), from Eq. (4) one immediately recognizes that T depends there appear signi cant di erences which are discussed on U through J , and grows with decreasing U . Thus, 2 2 in the following. for the device shown in Fig. 1(b), T increases with S2 In the present section, the analysis of the Kondo e ect in a way similar to T increasing with for the one K S1 is continued for the case of small particle-hole asymme- presented in Fig. 1(a). Note that this is opposite to what try, allowing for non-zero Seebeck coecient to occur. happens to T then. This is illustrated in Fig. 7(a) for The normal leads are assumed to be nonmagnetic. The a few representative values of . The corresponding S2 Fano-like interference e ects occur to be very similar as change in the Seebeck coecient peak position can be in the case of pairing present in QD1 and are not dis- observed in Fig. 7(b). cussed in detail. In particular, small values of lead The physics changes, in comparison to pairing induced S2 to the Fano anti-resonance with suppressed spin- ltering at QD1, for stronger interdot hopping t. Here, the change e ect, while strong pairing correlations induced in the of H ground state corresponds to the formation of a SDQD S [k /e] B G [e /h] 10 singlet in QD2, which suppresses the second stage of the Kondo e ect for above the critical value  U=2. S2 S2 (a) This is re ected in the perfect conductance and lack of 1.5 the thermopower peak at low temperatures for > S2 S2 (for t > 0 and > 0 the transition is in fact a quite sharp crossover, as explained in the following subsection). In- terestingly, an additional sign change of S(T ) occurs at T  T for close to this critical value, as illustrated K S2 0.5 in the inset of Fig. 7(b). This may be accounted for by the splitting of the Kondo peak by a residual dip correspond- ing to the second stage of screening. In fact, T increases 0.25 (b) with quite strongly and becomes only slightly smaller S2 than T for  0:4U . Then, the slope of the QD1 K S2 0.8 spectral function at ! = 0 changes and implies the sign change of S. Nevertheless, for even stronger , the sec- S2 0.6 ond stage of the Kondo e ect becomes nally suppressed. Interestingly, the width of the dip in QD1 spectral den- 0.4 sity corresponding to the second stage of screening (which 0.2 can be taken as a measure of T ) is in fact still nite and 0 0.1 0.2 0.3 0.4 even growing further, and only its depth vanishes, and so Γ /U S2 does the related positive peak of S(T ) together with the 0.5 0.04 (c) two corresponding sign changes. 0.4 B. Phase transition in the second quantum dot 0.3 t = 0 t = Γ/6 0.2 The largest di erence between the phase transition at t = Γ/5 t = Γ/4 QD1 and the one at QD2 induced by pairing correlations 0.1 0 Γ /U 1 t = Γ/2 S2 is associated with the fact that while QD1 is directly cou- t = Γ pled to the metallic leads, QD2 is coupled only through QD1. Therefore the e ective broadening of QD2 levels 0 0.2 0.4 0.6 0.8 1 is in the leading order proportional to  t =. To Γ /U 2 S2 explore the Kondo correlations one needs to consider rel- atively strong coupling , which leads to smearing of the Figure 8. (a) The conductance G, (b) Seebeck coecient S and the order parameter in (c) QD2 hd d i as function of transition at QD1. On the contrary, the transition at 2" 2# coupling between the second quantum dot and the SC lead QD2 is even sharper for strong . The e ect is even for di erent inter-dot hoppings t. The other parameters S2 more pronounced due to the fact that the interdot hop- are  = 0:05U , U = U = U = D=10, = U=5, p = 0, 1 1 1 2 ping t in experimental setups can be quite small. There- 9 0 0 T = 10 U , and U = 0:1U . The U = 0 case is shown with fore, the crossover is in fact quite sharp and the simi- bright lines for comparison. The inset in (b) shows the ther- larity to the quantum phase transition, which occurs at moelectric gure of merit ZT as a function of . The inset S2 t = 0 or = 0, is even more evident than in the case of in (c) presents the order parameter in QD1 (note di erent QD1. However, low values of t also imply indeed cryo- scale). genic Kondo temperatures for screening the second quan- tum dot spin, T , as follows from Eq. (4). This makes the system vulnerable to perturbations [82] and sets the e ect takes place and the conductance G = G does max ground for an interesting interplay between the Kondo ef- not depend on (G < 2e =h due to particle-hole S2 max fect and the superconducting pairing correlations in the asymmetry); cf. Fig. 8(a). Similarly, the Seebeck coe- vicinity of the crossover region. cient S  T  0, as shown in Fig. 8(b). The main results concerning the in uence of the inter- dot coupling on the phase transition at QD2 are summa- For nite hopping t, the second stage of the Kondo ef- rized in Fig. 8. Similarly to Fig. 5, a nite yet very small fect develops at energy scales corresponding to T . Nev- T = 10 U was assumed in calculations. For t = 0, there ertheless, at nite temperature only for suciently strong is a strict phase transition, with discontinuous change of t does T exceeds the actual T used in calculations. This the order parameter hd d i at = U=2, as shown in can be visible for t = =6 in Fig. 8(a). Moreover, due 2" 2# S2 Fig. 8(c). At the same time, there are no consequences to the increase of T with , the relevant critical value S2 of this fact for transport properties between the nor- of t, at which T = T , diminishes. Consequently, the mal leads, since QD2 remains completely decoupled from conductance is suppressed and a peak appears in S( ) S2 them. Therefore, the conventional, single-stage Kondo dependence; see Fig. 8(b). However, unlike in the case hd d i S [k /e] G [e /h] 2↑ 2↓ B hd d i 1↑ 1↓ ZT 11 of pairing induced in QD1 discussed in previous sections, oughly examined the sub-gap physics of both devices and the obtained values of S are larger and the thermoelectric showed that, depending on the superconductor position, eciency is enhanced. This is illustrated by the thermo- the second-stage Kondo temperature T may be either electric gure of merit reaching almost ZT = 0:25, as pre- enhanced or decreased by a small coupling to the su- sented in the inset to Fig. 8(b). This should be compared perconductor. In both cases there appears a doublet- to ZT  0:01 for parameters assumed in Fig. 5 (result singlet crossover around some critical value of the SC not shown in the gure). Further increase of the coupling pairing potential and the properties of the system change to SC lead induces a crossover to the conventional Kondo completely for strong pairing correlations. Depending regime. Its width is set up by the e ective coupling of on the device's geometry, the conventional Kondo e ect QD2 to the normal leads, , as can be deduced from may be strongly supported or completely suppressed in Fig. 8(c). Therefore, for strong , the conductance is this transport regime. Moreover, the crossover becomes S2 maximized and the thermopower strongly suppressed. very sharp for superconductor attached to side-coupled It is interesting to note that in the geometry consid- quantum dot at the regime of strong coupling to nor- ered in this section, QD2 and the normal leads do not mal leads. We explain these e ects as consequences of form a common continuous medium exhibiting pairing e ective decrease of the corresponding Coulomb interac- correlations, to which QD1 is coupled. For this reason, tion and basic properties of coupled Kondo impurities. the pairing amplitude induced in QD1 by the coupling Moreover, we show that the spin-dependent Fano-Kondo to QD2 is always of the same sign and is simply caused interference, which develops in the considered systems, by the hybridization of single-electron states, cf. inset turns out to be very vulnerable to the proximity e ect. in Fig. 8(c). Nevertheless, the order parameter hd d i The spin- ltering e ects present in T-shaped DQDs with 1" 1# exhibits a peak at = . ferromagnetic contacts can be suppressed by even small S2 S2 Finally, the large t regime corresponds to the transport values of the coupling to the superconductor. through molecular levels of DQD in the proximity of SC The presented results show that the superconductor lead. The location of the crossover is only slightly shifted proximity e ect provides additional means for the con- due to the renormalization of the energy levels, but its trol of the two-stage Kondo physics in T-shaped double width is increased signi cantly due to large . As can quantum dots. It enables to either strongly favor or com- be seen in the inset in Fig. 8(c), hd d i remains posi- 1" 1# pletely suppress each stage of the Kondo screening and tive, which is due to the reasons explained above. Con- obtain interesting electric or thermoelectric properties. sequently, the strong t case does not di er quantitatively Furthermore, the analysis of transport properties of hy- much from the case corresponding to weaker inter-dot brid T-shaped DQD systems gives additional insight into hopping, unless caloric properties are concerned. Then, the nature of the interplay between the Kondo correla- of course, smoothed crossover leads to a small slope of tions and the superconductivity, which exhibits a sur- the spectral function at ! = 0, and consequently reduced prising combination of increase of the Kondo tempera- thermopower. ture and suppression of the related spectral features. We hope that our analysis will foster further endeavor in this direction. VI. CONCLUSIONS In the present paper we have analyzed the transport ACKNOWLEDGMENTS properties of a T-shaped double quantum dot system proximized by the superconductor, considering two dis- tinct geometries. In the rst one, the quantum dot di- This work was supported by the National rectly coupled to the normal leads was connected to the Science Centre in Poland through project no. superconductor, while in the second geometry, the side 2015/19/N/ST3/01030. Discussions with B. Bu lka coupled quantum dot was proximized. We have thor- are acknowledged. [1] M. 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