Josephson effect through a multilevel dot near a singlet-triplet transition

Josephson effect through a multilevel dot near a singlet-triplet transition

Minchul Lee Department of Applied Physics, College of Applied Science, Kyung Hee University, Yongin 449-701, Korea    Thibaut Jonckheere Centre de Physique Théorique, UMR6207, Case 907, Luminy, 13288 Marseille Cedex 9, France    Thierry Martin Centre de Physique Théorique, UMR6207, Case 907, Luminy, 13288 Marseille Cedex 9, France Université de la Méditerranée, 13288 Marseille Cedex 9, France
Abstract

We investigate the Josephson effect through a two-level quantum dot with an exchange coupling between two dot electrons. We compute the superconducting phase relationship and construct the phase diagram in the superconducting gap–exchange coupling plane in the regime of the singlet-triplet transition driven by the exchange coupling. In our study two configurations for the dot-lead coupling are considered: one where effectively only one channel couples to the dot, and the other where the two dot orbitals have opposite parities. Perturbative analysis in the weak-coupling limit reveals that the system experiences transitions from 0 to (negative critical current) behavior, depending on the parity of the orbitals and the spin correlation between dot electrons. The strong coupling regime is tackled with the numerical renormalization group method, which first characterizes the Kondo correlations due to the dot-lead coupling and the exchange coupling in the absence of superconductivity. In the presence of superconductivity, many-body correlations such as two-stage Kondo effect compete with the superconductivity and the comparison between the gap and the relevant Kondo temperature scales allows to predict a rich variety of phase diagrams for the ground state of the system and for the Josephson current. Numerical calculations predicts that our system can exhibit Kondo-driven 0--0 or -0- double transitions and, more interestingly, that if proper conditions are met a Kondo-assisted -junction can arise, which is contrary to a common belief that the Kondo effect opens a resonant level and makes the 0-junction. Our predictions could be probed experimentally for a buckminster fullerene sandwiched between two superconductors.

pacs:
73.63.-b, 74.50.+r, 72.15.Qm, 73.63.Kv

I Introduction

The Josephson effectJosephson (); Tinkham96 () is one of the most celebrated manifestation of many body correlations in condensed matter physics: a Cooper pair currentBardeen57 () between two bulk superconductors separated by an intermediate region or arbitrary nature can flow even in the absence of an applied bias. Over the last few decades, the Josephson effect has become a very active field of theoreticalShiba69 (); Glazman89 (); Spivak91 (); Yeyati97 (); Shimizu98 (); Rozhkov99 (); ChoiMS00 (); Rozhkov01 (); Makhlin01 (); Zaikin04 (); Choi04 (); Siano04 (); Lee08 (); Karrasch08 () and experimentalKasumov99 (); Kasumov01 (); Scheer01 (); Ryazanov01 (); Kontos02 (); Buitelaar02 (); Kasumov05 (); vanDam06 (); Jarillo-Herrero06 (); Jorgensen06 (); Cleuziou06 (); Eichler07 (); Sand-Jespersen07 (); Winkelmann09 (); Eichler09 () investigation in the context of mesoscopic devices, devices which are small enough that electron transport occurs in a phase coherent manner. Because the tunneling of Cooper pairs through the junction is greatly affected by the physical properties of the segment between superconducting electrodes, the study of the Josephson current provides a new way to investigate the electronic properties of the medium. In early days, thin layers of insulators and metals were used to form the Josephson junction.Golubov04 () Advance in nanofabrication technology can now enable one to make the middle segment small enough to be considered as a quantum dot (QD), a zero dimensional entity bridging the two superconductors.Buitelaar02 (); Cleuziou06 (); Eichler07 (); Sand-Jespersen07 (); Eichler09 () Furthermore, even a real (or artificial) molecule can be inserted between two closely positioned superconducting leads to form a molecular Josephson junction (MJJ).Kasumov01 (); Kasumov05 (); Winkelmann09 () Quantum dots connected to normal metal leads are known to exhibit the Coulomb blockade phenomenon due to their large charging energies.Grabert92 () Interestingly, at low enough temperatures, when an odd number of electrons occupy the QD, one can reach the Kondo regime.Goldhaber-Gordon98 (); Cronenwett98 (); vanderWiel00 () Yet the leads can be chosen to be superconductors which lead to a competition between Kondo physics and superconductivity.Choi04 (); Siano04 (); Lee08 (); Karrasch08 (); Winkelmann09 (); Eichler09 () The purpose of the present work is precisely to study the Josephson effect through a multilevel quantum dot in this context, with applications to molecular spintronics.

Indeed, the QD-JJ have received a great theoretical and experimental attention because they can exhibit an interesting competition between two many-body correlations: the superconductivity and the Kondo effect. Due to its small size, the QD has a large Coulomb charging energy. The Kondo effect then emerges for such a small QD coupled strongly to the leads when the QD has a localized magnetic moment, that is, nonzero total spin of electrons in it. At temperatures below the so-called Kondo temperature ,Haldane78 () the conduction electrons in the leads screen the localized moment through multiple cotunneling spin-flip processes, forming a spin-singlet ground state, and induces a resonance level at the Fermi energy, which increases the linear conductance up to the unitary-limit value () that is otherwise completely suppressed due to the strong Coulomb repulsion. If the leads consist of -wave superconductors, the conduction electrons form spin-singlet Cooper pairs incapable of flipping the QD spin. It has been known,Shiba69 (); Glazman89 (); Spivak91 (); Rozhkov99 (); Rozhkov01 () and recently probed,Ryazanov01 (); Kontos02 (); vanDam06 () that in the weak dot-lead coupling limit the large Coulomb repulsion only allows the electrons in a Cooper pair to tunnel one by one via virtual processes in which the spin ordering of the pair is reversed, leading to a junction, and that the localized moment remains unscreened. In the opposite limit where the Kondo temperature exceeds the superconducting gap , however, the induced Kondo resonance level restores the 0 junction state of the supercurrent.Glazman89 (); Choi04 (); Siano04 (); Karrasch08 () As a result, one can drive a phase transition between spin singlet (0 junction) and doublet ( junction) states by changing the relative strengths of and .

Current issues about electronic transport through a QD or a molecule go beyond the spin-degenerate single-level model and take into account multi-level structures and/or possible magnetic interactions. For example, the theoretical prediction that the two-level quantum dots (TLQDs) with spin exchange interaction coupled to normal-metal leads can experience a quantum phase transition, specifically the singlet-triplet transition,Izumida01 (); Hofstetter02 () was recently confirmed by two independent experiments.Quay07 (); Roch08 () The transition was observed to accompany a drastic change in the transport mechanism, and it was also found that the spin exchange coupling between electrons could suppress the Kondo correlation completely or alter its physical nature by changing the screening mechanism. The influence of such a magnetic interaction on the Josephson current was also studied for a MJJ where the molecule is modeled by a single-level QD having spin exchange couplingElste05 () between spins of QD electron and a metal ion.Lee08 () It was predicted that the state of the supercurrent can be switched between 0 and junctions by tuning the magnetic interaction. On the other hand, theoretical calculationsShimizu98 (); Rozhkov01 () and experimentsvanDam06 () have shown that the Josephson junction made of a multi-level quantum dot in the weak-coupling limit can behave as a junction even when the dot is nonmagnetic without a localized spin and vice versa. The studies found out the significant roles of (1) the off-diagonal Cooper pair tunneling processShimizu98 () in which two electrons in the pair are transferred via different orbitals in the QD and (2) the parity of the QD orbital wave functionsRozhkov01 () that determine the relative sign of the dot-lead couplings.

In this paper we study the electronic transport through a Josephson junction having in it a TLQD with the spin exchange interaction between electrons in two orbital levels. Here we focus on the regime where the doubly-occupied QD experiences the singlet-triplet transition due to the spin exchange coupling that is tunable by the gate voltage. The physical properties of the ground state and the supercurrent-phase relation (SPR) through the junction are examined as the strengths of the superconductivity and the spin exchange coupling are varied. In order to study both of the weak- and strong-coupling limits we exploit the numerical renormalization group (NRG) method which can take into account the Coulomb interaction in a nonperturbative way. In additions, the physical understanding of the numerical outcome is supplemented by the analytical analysis such as fourth-order perturbation theory and scaling theory.

Our main findings are summarized as follows: (1) The sign of the supercurrent is determined by the competition between diagonal and off-diagonal tunneling processes whose strength and sign can be controlled by the parity of the orbital wave functions and the spin correlation present in the dot. (2) The origin and physical property of the TLQD-JJ can be explained in terms of the competition between the superconductivity and the Kondo correlation found from the normal-lead counterpart of the system. For example, the two-stage Kondo effect leads to 0--0 or -0- double transitions with the exchange coupling or the superconducting gap. (3) When the superconducting phase difference between two leads is maximal, the existing Kondo correlation is greatly affected. Interestingly, we observed that a Kondo-assisted -junction can arise if some conditions are met.

This paper is organized as follows: In Sec. II we describe the model Hamiltonian of the TLQD-JJ and specify the regimes that we are interested in. The weak-coupling limit is studied by using the fourth-order perturbation analysis in Sec. III. Section IV presents the results of the NRG calculations applied to the weak- and strong-coupling limits and the phase diagrams of the system with respect to the properties of the SPR. In Sec. V we summarize our study.

Ii Model

Figure 1: (color online) (LEFT) Sketch of the TLQD connected to two -wave superconducting leads. (RIGHT) Energy levels of two-electron states in the TLQD as functions of the exchange coupling . The shaded region is of our interest, where the singlet-triplet transition occurs in the ground state.

The TLQD connected to two single-channel -wave superconducting leads as shown in Fig. 1 is modeled by the two-impurity Anderson model: , where

(1)
(2)
(3)

Here () destroys an electron with energy () with respect to the fermi level and spin on lead (in orbital on the dot); and are occupation operators for the leads and the dot orbitals. The Coulomb energy of the strength is assumed to depend on the total number of electrons in the dot. The Hund’s rule in the dot results in the ferromagnetic exchange coupling denoted as between the electron spins , where are Pauli matrices. The left and right leads are assumed to have identical dispersion energy and superconducting gap , while a finite phase difference is applied between them. The energy-independent dot-lead tunneling amplitudes hybridize the electron states between the dot and the leads, which are well characterized by tunneling rates , where is the density of states of the leads at the Fermi energy.

Since we are interested in the regime of the singlet-triplet transition of an isolated dot, we focus on the parameter region in which the dot is doubly occupied. In addition, we consider the nondegenerate case with a finite splitting between two orbitals. Figure 1 displays the energy levels of two-electron states of the isolated dot as functions of : three singlet states, with and three triplet states , where the states are labeled as with the charge number , the spin , and the component of the spin . The singlet states,

(4a)
(4b)
(4c)

have the energies, , , , respectively, and the triplet states,

(5a)
(5b)
(5c)

are degenerate with the energy . Due to the finite splitting and the existence of the inter-orbital Coulomb interaction, the singlet-triplet transition is driven by the competition between the states and [see Fig. 1]. The bare singlet-triplet splitting is then defined by

(6)

The external gate voltage can tune the singlet-triplet splitting by affecting the level splitting ,Kogan03 (); Holm08 () the exchange coupling strength ,Quay07 (); Roch08 () or both of them. For simplicity, we assume that the gate-voltage dependency is implemented only through and that or are independent of . Our simplification can still capture the main physics of the system as long as the regime close to the singlet-triplet transition is concerned.

The configuration of the dot-lead coupling is another important source that can govern the physics of the system. First, the number of the effective channels coupled to the dot can be controlled.Hofstetter02 () If the condition,

(7)

is satisfied, the dot-lead coupling matrix has a zero eigenvalue, and one of the two channels can be completely decoupled from the dot under a proper unitary transformation, resulting in a one-channel problem. This reduction of the effective channels then affects the Kondo effect greatly, which will be discussed later. Secondly, the phase of the coupling coefficients has an influence on the interference and consequently on the electron transport through the dot.Rozhkov01 (); vanDam06 () Even though no magnetic field is applied in our system, the (real-valued) coupling coefficients can acquire an additional phase depending on the parity of the orbital wave functions on the dot.vanDam06 () Two distinctive cases can then be conceived: when two orbitals have the same parity and when they have the opposite parities. Taking into account the essential impacts of the dot-lead coupling and focusing on the consequent qualitative features of system states and electron transport, we consider two representative cases in this paper:

(8)

with . The case I deals with the effective one-channel problem with Eq. (7) satisfied, while the case II reflects the two-channel problem with the negative product of coupling coefficients. The effect of asymmetric coupling with respect to the orbitals is also examined by setting . Another kind of asymmetric junction such as that can happen frequently in realistic experimental setups like break junctionsRoch08 () is not considered in our study because this asymmetry is observed to make no qualitative impact on the Josephson current.

Finally, since we are interested in the low temperature behavior, we concentrate for the most part on the Kondo regime. The hybridizations are chosen to be far smaller than the particle or hole excitations with respect to the two-electron states in order to suppress the resonant tunneling. Specifically, throughout our study, we choose , , , and , where the half band width is taken as the unit of energy. Here we have also used the particle-hole symmetry condition .

Iii Weak Coupling Limit:

iii.1 Fourth-Order Perturbation Theory

First, we consider the weak coupling limit where the superconducting gap is much larger than the Kondo temperature , which will be defined in Sec. IV. In this case the supercurrent can be calculated via fourth-order perturbation theory in .Rozhkov01 (); Spivak91 (); ChoiMS00 () We apply degenerate perturbation theory that takes into account the singlet state and the triplet states simultaneously since they are almost degenerate close to the singlet-triplet transition point of isolated dot. Unlike the single-level quantum dot studiesRozhkov01 (); Spivak91 () where it is enough to collect only terms that depend on the phase difference , on the other hand, one must keep track of all the -independent terms in the TLQD study because they contribute to the renormalization of the singlet-triplet splitting,ChoiMS00 () and the transition point is shifted from its unnormalized position, . Due to the singlet nature of the Cooper pair, there exists no coupling between the singlet and the triplet states to any order of the perturbation, and the energy of each state is separately shifted: for with and . The energy shifts are given by

(9)

where we have defined . Figure 2 shows typical virtual hopping processes that contribute to each term in Eq. (9). The detailed expressions for the coefficients can be found in the Appendix.

Figure 2: (color online) Listing of typical virtual tunneling processes contributing to . The arrows indicate the direction of the charge transfers for processes contributing to (a) , (b) , (c) , (d) , (e) , and (f) .

The effective singlet-triplet splitting then becomes

(10)

with

(11)

where and consist of the terms that are proportional to and , respectively. We find that is mostly positive in the parameter regime of our interest, favoring the singlet formation. The singlet-triplet transition point when , which now becomes -dependent, is then shifted from its bare value to a more negative value. It should be noted that the second-order contribution to

(12)

is finite in contrast to the previous study of parallel double-dot systemChoiMS00 () where the leading contribution is found to be of the order of . The main difference comes from the characteristics of the singlet states in two systems. In the double-dot system studied by Choi et al., the two quantum dots, each of which is singly occupied, are identical and have no Coulomb interaction between them so the lowest-lying singlet state is , while it is in our system due to the existence of the finite splitting and the inter-orbital Coulomb interaction. The singlet state has the same charge distribution as the triplet states , so the second-order perturbation does not give rise to any additional splitting between two states [see Fig. 2 (a)]. On the other hand, having as the lowest-lying singlet states, our system can exhibit a rather huge renormalization of the singlet-triplet splitting that is of the order of . This second-order term is numerically found to increase as is decreased. This tendency is opposite to the expectation that the renormalization, which is due to the tunneling of Cooper pairs whose amplitude increases with , should be weakened as decreases: in other words, . This discrepancy should be resolved by the higher-order terms of the order of that are more involved as decreases: In fact, the fourth-order term is observed to become negative for smaller so that the renormalization is diminished. Owing to this opposite -dependencies of and the other higher-order terms, varies non-monotonically with , which in turns implies that the transition point also displays a non-monotonic dependency on : see Figs. 3 and 5.

The supercurrent can be calculated via the derivative of the energy with respect to the phase difference :

(13)

with

(14)

As a matter of fact, only the virtual processes in Figs. 2 (e) and (f) contribute to the Cooper pair tunneling. The -term [(e)] arises from the diagonal processes where both electrons in a Cooper pair travel through either the orbital 1 or 2, while the -term [(f)] from the off-diagonal processes with one electron traveling through the orbital 1 and the other traveling through the orbital 2. Depending on the order of the sequence of electron tunneling and the spin correlation of dot electrons, the coefficients can acquire a relative minus sign owing to Fermi statistics. For the singlet state, one can find that

(15)

The negative sign for is attributed to the processes with one electron traveling through a filled level (orbital 1) and the other electron through an empty level (orbital 2). It should be noted that it is necessary to take into account the dot electron correlation exactly in order to determine the supercurrent sign correctly. Not all the processes contributing to acquire the phase: For example, the processes with the intermediate state acquire no phase at all [see Eq. (76)], while their amplitudes are always smaller than those of the other processes, and finally is negative. For the triplet state,

(16)

because of the presence of local magnetic moments in both orbitals.Spivak91 () Apart from the sign, we have found numerically that the off-diagonal processes usually have larger amplitude than the diagonal ones:

(17)

Hence, when the product is comparable to in magnitude, the sign of the supercurrent dictates the sign of the off-diagonal term, or that of the product regardless of the spin state: the current exhibits the 0()-junction for the negative (positive) product. Otherwise, that is, if , the diagonal term prevails in determining the sign of the supercurrent so that the singlet (triplet) state features the 0()-junction behavior regardless of the sign of the product.

In the following sections, we identify the system state according to its ground-state spin and the sign of the supercurrent in the - plane. We use the labels and to denote the spin singlet and triplet state, respectively. Since the phase transition depends on the superconducting phase difference as well, the phase boundaries are located at three different values of : 0 (red line), (green line), and (blue line). Between and boundaries the system is in the intermediate state having a stable ground state and a meta-stable state. The intermediate states are tagged with a subscript that represents the meta-stable state spin. For example, the ground state in the state is mostly of the spin triplet, while it is of the spin singlet at and near , and the system experiences a phase transition from spin doublet to singlet as is varied from 0 to . The state identification is then supplemented by the SPR calculated from Eq. (14), classifying whether it is of either or junctions. Two states with same ground-state spin can be distinguished if their SPRs are different and the boundary between them will be colored in yellow line.

iii.2 Case I:

Figure 3: (color online) Phase diagrams in the - plane for the case I with [(a)], 0.6 [(b)], and 0.1 [(c)]. The phase boundaries are located when the ground-state spin is changed at (red line), (green line), and (blue line). The yellow line separates two states with same ground-state spin but different SPRs. Each phase is shaded in gray scale according to its SPR: lighter gray for 0 junction and darker gray for junction. Refer the detailed classification of the states to the text. The insets show the SPR for at the points indicated by the arrows. Here the value of is swept from to , and for the above diagrams are not valid.
Figure 4: (color online) Critical currents as functions of in units of for the case I with [(a)], 0.6 [(b)], and 0.1 [(c)] for various values of : (top), , , , and (bottom). The insets display the critical currents as functions of in units of for (in the spin triplet state) and (in the spin singlet state).

Figure 3 shows the phase diagrams in the - plane in the case I for various values of . The lower bound of is set to because the perturbation theory works only when . For , the Josephson coupling

(18)

is always negative because and , and the current exhibits the -junction behavior, no matter what values and have [see Fig. 3 (a)]. For , on the other hand, the contribution from the off-diagonal term becomes negligible since

(19)

and the sign of is governed solely by the term. The spin singlet state is then of the 0 junction since , and the singlet-triplet transition accompanies the 0- transition with the intermediate states as shown in Fig. 3 (c). Figure 3 (b) shows that for intermediate values of , both of the 0 and junctions can appear in the spin singlet state: the 0 and junctions take place in the regions with larger and smaller values of , respectively. The phase boundary separating two regions moves toward the smaller as is decreased.

Different strength of the Josephson coupling in the spin singlet and triplet states gives rise to a discontinuous change in the SPR in the intermediate states [see the insets in Fig. 3] and a rapid change in the critical current across the singlet-triplet transition as shown in Fig. 4. The numerical calculation of the supercurrent finds that the supercurrent is stronger in the spin triplet state than in the spin singlet state: . In the spin singlet state the diagonal and the off-diagonal processes make the opposite contributions (), resulting in a partial cancellation. This is not the case in the spin triplet state in which both processes contribute to the junction (). For small , on the other hand, such a cancellation does not make a significant role since the term becomes much smaller than the term, so the critical currents in both spin states become comparable as can be seen in Fig. 4 (c).

The critical current exhibits a non-monotonic dependence on : see the insets in Fig. 4. In two extreme limits, and , the supercurrent should vanish. The supercurrent, induced by the proximity effect that is proportional to , should vanish in the limit . In the opposite limit, the high energy cost of the quasiparticles created during the virtual processes suppresses the current. Consequently, the critical current has a maximum as a function of . For the intermediate values of when the 0- transition occur in the spin singlet state, the critical current can become zero at the transition [see the inset in Fig. 4 (b)].

iii.3 Case II:

Figure 5: (color online) Phase diagrams in the - plane for the case II with [(a)], 0.72 [(b)], and 0.1 [(c)]. Refer to Fig. 3 for the details.
Figure 6: (color online) Critical currents as functions of in units of for the case II with [(a)], 0.72 [(b)], and 0.1 [(c)] for various values of : (top), , , , and (bottom). Refer the guide for the insets to Fig. 5.

The phase diagram and the critical current in the case II are shown in Figs. 5 and 6, respectively. In this case one of the dot-lead tunneling amplitude changes its sign, making the product negative and accordingly reversing the sign of the off-diagonal contributions. It then switches the junction characteristics from to 0 junction for the case when the off-diagonal term prevail over the diagonal term: compare Fig. 3 (a) and Fig. 5 (a). However, for when the off-diagonal contributions are negligible, the negative product does not affect the supercurrent and the phase diagram so much: Fig. 3 (c) and Fig. 5 (c) are almost identical. As a result, for the intermediate values of , the additional 0- transition now takes place in the spin triplet state in contrast to the case I: compare Fig. 3 (b) and Fig. 5 (b). Another difference from the case I is that the critical current is now much larger in the spin singlet state than in the spin triplet state as long as is not so small: see Fig. 6. The same argument used in the case I applies as well: With the negative product, the term which is negative cuts down the positive contribution from the term, while in the spin singlet state both of two terms contributes to the 0 junction.

In addition to the properties of the supercurrent, the shape of the phase boundaries are also different from those in the case I. Figures 5 (a) and (b) show that for the moderate values of the phase boundaries are much shifted toward the spin triplet side, implying that the spin singlet state is being further favored. Furthermore, the transition point displays a monotonic dependence on and does not approach its bare value in the limit , which is contradictory to our expectation from the previous weak-coupling argument. The inclination to the spin singlet state is accounted for by looking at the term in that is proportional to [see the last term in Eq. (9)]. Numerical calculations observe and , which means that with the negative product the spin singlet state is much lowered than the spin triplet state [see the Appendix for expressions of and ]. This term is also observed to make the fourth-order splitting term positive, which is the cause of the monotonic behavior of . One may suspect that this favoring of the spin singlet state in the limit is the artifact of the fourth-order perturbation close to its limit of validity, . However, the non-perturbative NRG study in the following section finds that the system should be of the spin singlet state in the vanishing limit and that it is attributed to the complete screening of dot spins by the two-channel conduction electrons, which will be discussed in details in the next section. Considering that the Kondo effect which is responsible for the screening cannot be correctly captured by the perturbation theory, it is quite interesting that it still reflects correct asymptotic behaviors in the limit : the approaching of to its bare value in the case I and the precursor of the disappearance of the spin triplet state in the case II.

Iv Strong Coupling Limit:

In this section we extend our study to the strong-coupling limit by using the NRG method that is known to be suitable for the non-perturbative study of the low-temperature properties of the impurity system. Even though the standard NRG procedureWilson75 () can be directly applied to the original Hamiltonian, we have introduced a unitary transformation

(20)

which makes all the matrix elements of the Hamiltonian real in order to boost up the speed of the numerical computation. Here the unitary matrices are chosen to be

(21)

where and in the cases I and II, respectively. Under the unitary transformation, each part of the Hamiltonian is transformed into

(22)
(23)
(24)

respectively, where and . Here the transformed dot-lead coupling matrix is given by

(25)

The Wilson’s NRG techniqueWilson75 (); Krishnamurthy80 () consists of the logarithmic discretization of the conduction bands, the mapping onto a semi-infinite chain, and the iterative diagonalization of the properly truncated Hamiltonian. Following the standard NRG procedures extended to superconducting leads,Yoshioka00 () we evaluate various physical quantities from the recursion relation

(26)

for with the initial Hamiltonian given by

(27)

Here the fermion operators have been introduced as a result of the logarithmic discretization of the conduction bands and the accompanying tridiagonalization, is the logarithmic discretization parameter (we choose ), and

(28)
(29)

with . The original Hamiltonian is recovered by

(30)

It has been knownKrishnamurthy80 (); Campo05 () that the logarithmic discretization underestimates the coupling between the conduction-band electrons and the dot electrons. In order to avoid this problem, we multiply by a correction factor given byKrishnamurthy80 (); Campo05 ()

(31)

Within the NRG procedure, the spin of the ground state, the occupation , and the spin correlation can be directly calculated from the expectation values of the corresponding operators. The supercurrent can be also obtained by calculating the expectation value

(32)

where . In terms of the fermion operators , the current expectation value is expressed as

(33)

with the current matrix defined by

(34)

The Andreev levels are located from the subgap many-body excitations which are identified as the poles of the dot Green’s functions.

iv.1 Normal Leads:

In the presence of Coulomb interaction and spin exchange coupling, strong dot-lead coupling can induce nontrivial many-body correlations that may compete with and even suppress superconductivity. A promising candidate of such many-body correlations in the QD system is the Kondo effect. In order to identify nontrivial correlations in our system and to elaborate the analysis of the strongly-coupled Josephson junction, it is quite useful to investigate the normal-lead case with . The NRG procedure described above is then applied by setting and : The latter condition, though not being essential, is imposed in order to simplify the dot-lead coupling matrix [Eq. (25)]. The normal-lead version of our system has been well studied in the literature, so we briefly summarize the known theoretical analyses and present relevant numerical results in our parameter regime for comparison with the superconducting case.

iv.1.1 Case I: Single Channel

Figure 7: (color online) Energy-resolved transmission coefficient for a two-level QD coupled to normal leads (case I) with ferromagnetic [(a)] and antiferromagnetic [(b)] exchange coupling for various values of (as annotated).

In the case I only the lead-, that is, the symmetrized conduction-band channel is coupled to the dot, with the other channel completely detached: see Eq. (25). The two-level QD system attached to a single conduction channel has been well studied in the context of the quantum phase transition in a vicinity of singlet-triplet degenerate point.Hofstetter02 () In this case the system can be mapped onto an exchange-coupled Kondo model through a Schrieffer-Wolff transformation:Schrieffer66 ()

(35)

Here the Kondo spins and are fictitious QD spins defined on the basis of the spin singlet state and the spin triplet states . Both of the Kondo spins are coupled to the localized spin of the conduction channel associated to the symmetrized combination of left and right leads [see Eq. (21)]: Here we define the localized spins of conduction-band electron spins as

(36)

The effective spin couplings are, up to linear order in ,

(37a)
(37b)

Note that one has as long as .

The ground state of the Kondo Hamiltonian, Eq. (35) can be of the spin singlet or doublet depending on the strength of the effective exchange coupling and is known to undergo a phase transition at the critical coupling , which is of the Kosterlitz-Thouless-type.Vojta02 () The ferromagnetic side corresponds to an underscreened Kondo model where the conduction electrons screen one of the Kondo spins and the remaining spin then couples ferromagnetically to the conduction band and becomes asymptotically free at low energies.Cragg79 () The corresponding Kondo temperature decreases with increasing [see Fig. 7 (a)]. On the other hand, on the antiferromagnetic side , a two-stage Kondo effect takes place for small .Hofstetter02 (); SCQD (); vanderWiel02 () First, the Kondo effect leads to a screening of one of the Kondo spins, with the larger coupling (for example we assume ) which therefore defines the larger Kondo temperature . For temperatures lower than , the second spin is decoupled from the conduction band. At a much lower energy scale (denoted as ), the effective antiferromagnetic exchange coupling between and then induces the second screening due to the local Fermi liquid that is formed on the first spin. is then the Kondo temperature of the second spin screened by electrons of a bandwidth and density of states :Hofstetter02 (); SCQD ()

(38)

The second Kondo effect leads to a Fano resonance and makes a dip in the energy-resolved transmission coefficientHofstetter02 ()

(39)

where we have introduced the retarded QD Green’s functions and a coupling matrix with , , and . As shown in Fig. 7 (b), the dip becomes widened with increasing and eventually overrides the Kondo peak until at which the Kondo effect completely vanishes.

iv.1.2 Case II: Two Channels

Unless the zero-eigenvalue condition, Eq. (7) is satisfied, the dot is always coupled to both of the two conduction-band channels. The low-energy physics of the system is then governed by the two-channel two-impurity Kondo model with an exchange coupling. Similarly to the case I, the effective spin model can be derived via the Schrieffer-Wolff transformation:

(40)

The exchange coupling coefficients are given by

(41a)
(41b)
(41c)

while one obtains the same expression for as Eq. (37b). Here each of two Kondo spins is coupled to composite localized spins of conduction-band channels. The effective Hamiltonian, Eq. (40) is not convenient for further analysis since it contains cross terms ( and ) that do not conserve the channel degrees of freedom. We introduce a unitary transformations that diagonalizes the conduction-band spin operator in the channel basis that is coupled to for :

(42)

with for (upper sign) and (lower sign), respectively. In terms of rotated conduction-band spins , the spin exchange terms in the effective Hamiltonian read

(43)

where for denotes the index of the Kondo spin for which the coupled conduction-band spin operator is not diagonalized, and the coefficients are given by

(44a)
(44b)
(44c)

with and . The index is chosen between and such that either or is the largest among the coefficients. Now the scaling analysis is ready with Eq. (43). Suppose that is the largest one. Upon decreasing temperature, the Kondo spin is first screened by the conduction-band spin , which defines a Kondo temperature . Below this Kondo temperature, the spins and are energetically frozen so that the remaining degrees of freedom is approximately governed by the exchange coupling,

Figure 8: (color online) Scaled NRG eigenenergy flows with the iteration for a two-level QD coupled to normal leads (case II) with (a) and and (b) and .
(45)

The antiferromagnetic coupling will eventually screen out the remaining Kondo spin at a lower Kondo temperature since . Hence the system undergoes two-stage Kondo effectsPustilnik01 () as the temperature goes down: Two Kondo spins are screened out one by one since their couplings to relevant conduction-band degrees of freedom are different in magnitude. The ground state is of the spin singlet at low temperatures due to complete screening, while the partial screening in the intermediate temperature leaves the system in the spin doublet. We have confirmed this scaling analysis numerically by examining the RG flow of the scaled low-lying eigenenergies in the NRG procedure. Figure 8 (a) clearly shows that the flow is in the high-temperature regime attracted by an unstable fixed point and then goes to the stable fixed point in the lower temperature.

The spin exchange coupling can interrupt the Kondo correlation. In fact, we have observed that the two-stage Kondo effect ceases to happen if the exchange coupling is so antiferromagnetic that . In this regime, the Kondo spins are frozen to form a spin singlet by themselves before the conduction-band electrons screen them out. Hence, at zero temperature the system undergoes a transition between a Kondo state and an antiferromagnetic state as the exchange coupling is varied. In contrast to the case I, however, the transition does not involve any change in the spin state: The ground state in both states is of the spin singlet. Note that the two-stage Kondo effect arises in the ferromagnetic side in the two-channel case while the one in the single-channel case happens in the antiferromagnetic side.

It may be interesting to consider a special case when the two Kondo temperatures are equal to each other: . This can happen when so that , , and , giving rise to the exchange Hamiltonian:

(46)