Aashish A. Clerk and Vinay Ambegaokar Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853 (Received 21 October 1999) Using a generalization of the noncrossing approximation which incorporates Andreev reflection, we study the properties of an infinite- U Anderson impurity coupled to two superconducting leads. In the regime where △ and T g are comparable, we find that the position of the subgap resonance in the impurity spectral function develops a strong anomalous phase dependence—its energy is a minimum when the phase difference between the superconductors is equal to π . Calculating the Josephson current through the impurity, we find that π-junction behavior is lost as the position of the bound state moves above the Fermi energy. What is the Josephson current through an interacting Anderson impurity? As this phenomenon involves the coher- ent transport of pairs of electrons through the impurity, one would expect that a strong on-site repulsion would greatly diminish the effect. Though this intuitive expectation is par- tially correct, there is a variety of nonintuitive behavior also associated with this system. If the on-site repulsion is large and the impurity is singly occupied, a lowest-order perturbative calculation in the impurity-superconductor coupling reveals that the sign of the Josephson coupling can become negative, meaning that the corresponding ground state of the system has a phase differ- ence of δ= π between the two superconductors. 1,2 An ap- pealing explanation for this behavior was provided by Spi- vak and Kivelson,² who showed that in the limit of single occupancy, it is impossible to transport a pair across the impurity while preserving its spin ordering, leading to an extra factor of ( − 1). More generally, π junction behavior is expected whenever spin-flip tunneling processes dominate. 3,4 Recent work using a Hartree-Fock type procedure indicates this behavior can survive beyond lowest-order perturbation theory. 5 Alternatively, in the limit of large impurity- superconductor coupling, the physics of the Kondo effect will also become significant. A resulting enhancement of the Josephson current through the impurity has been predicted in the regime where the superconducting gap △ is much smaller Significantly, no than the Kondo temperature T K -1 π-junction behavior is expected here, as the spin of the im- purity is completely screened by the Kondo effect. Given these prior results, it is natural to ask how large the ratio T k / △ must be to see the destruction of π -junction be- havior, and what the properties of the junction are in this crossover region; these questions are the motivation for the current paper. We examine the regime where △ is compa- rable in magnitude to T K , a regime in which the effects of both Kondo physics and superconductivity must be treated on an equal footing. Note this parameter range is also of interest as it is more appropriate to mesoscopic quantum dot systems, which would be one possible experimental realiza- tion of the model. We find, somewhat surprisingly, that the properties of our system can be understood in terms of an- other well-studied feature of magnetic impurities coupled to superconductors—the subgap bound state. Typically, the po- sition of this state is a function of the ratio T K /Δ. 6~9 We find now that this resonance also develops a pronounced depen- dence on the phase difference—as δ increases from zero, the energy of the bound state decreases . This phase dependence is anomalous in the sense that it indicates an energetic pref- erence away from δ = 0. Interpreting this subgap state as a current carrying Andreev bound state, 10 we are lead to the conclusion that the system is a π junction if the subgap bound state is located below the Fermi energy E F . This re- lation is confirmed by making an explicit calculation of the Josephson current through the impurity. Formalism . The model we study consists of an infinite-U Anderson impurity coupled to two superconducting leads, each having a different phase. Using a slave-boson represen- tation, we have
The c α,kσ operators here create band electrons, with σ de- noting spin and α = L,R labeling the left and right supercon- ducting leads. Δα = |Δ|exp(iφα) represents the pair potential in lead α , with the phase difference δ being defined as φ R − φ L . The Anderson impurity has bare energy ε d , and is represented in the usual manner using auxiliary fermion ( f ) and boson ( b ) operators; the U = ∞ constraint of single occu- pancy takes the form ∑ σ f σ + b i b = 1. The NS-NCA . We calculate the impurity spectral function (also called the impurity density of states) for our system using the NS-NCA, 11 an extension of the self-consistent non- crossing approximation (NCA) (Ref. 12) to superconducting systems. The NCA amounts to an infinite order resummation of perturbation theory, and has been shown to be quantita- tively reliable down to temperatures below T K . 12,13 The modification we employ self-consistently includes multiple- Andreev reflection processes in a manner which is formally exact to order 1/N, where N = 2 is the spin degeneracy of the impurity. As the success of the normal NCA is attributed to the fact that it too is exact to order 1/N (in the absence of superconductivity), the NS-NCA employed here can be viewed as a natural extension to systems with superconduc- tivity. The graphs determining the f -fermion and b slave bo- son propagators are given in Fig. 1. Note that the anomalous graphs appearing in Fig. 1 al- ways involve two Andreev reflections. As these Andreev re- flection events pick up the phase of the superconductor in which they occur, and each Andreev reflection is free to occur in either of the two superconductors, the phase differ- ence δ naturally enters the impurity self energies through an interference term. A previous study using the NCA to study the regime △≫ T κ (Ref. 14) neglected these diagrams, and thus omitted the only phase-dependent contribution to the impurity Green function which survives in the limit of a strong on-site repulsion. The Dyson equations pictured diagrammatically in Fig. 1 lead to a set of coupled integral equations for the f -fermion and slave boson propagators. Letting F(ω)＝[ω−εd - Σ(ω)]-1 and B(ω)=[ω−Π(ω)]−1 represent the f-fermion and slave boson retarded propagators, respectively, the equations read
In these equations, ρ(ε) is the electronic density of states, Г= πW 2 ρ(0) the bare tunneling rate, f the Fermi distribu- tion function, and τ(ε) is an effective electron-hole coher- ence parameter defined by
where u k,α and υ k,α are the usual BCS coherence factors. Results. We use a Gaussian density states for the band electrons having half width D . For the results shown here, we choose model parameters ε d = − 0.67D and Γ = 0.15D, which yields an approximate T K = 0.0005D.15 We have solved numerically via iteration the NS-NCA equations in equilibrium for various temperatures, gap sizes and phase differences. Within the NCA, the impurity spectral function A d (ω) can be directly related to the f -fermion and slave bo- son spectral functions.
where the auxiliary particle spectral functions are defined by A f = − (1/π)Im F, A b = − (1/π)Im B . Note that the equality in Eq. (6) reflects a neglect of vertex corrections which is consistent with the large- N nature of the NCA. The solid curve in Fig. 2 (with points indicated by circles) shows the position y (in units of Δ) of the subgap resonance in the impurity density of states as a function of ∆ / T κ at δ = 0 and at a constant ratio T/Δ = 0.5; this resonance corre- sponds to the subgap bound state. As has been observed in earlier studies, we find that as ∆ / T κ decreases, the position of the bound state increases, with the transition across the Fermi energy occurring when ∆ is on the order of T k . Note that as we are not using a model with particle-hole symme- try, we only have a single subgap state. A cogent interpretation of this behavior was provided in, 7,8 where the position of the subgap bound state was sug- gested to result from the competition between two possible ground states. For T K > Δ , the ground state of the system is the Kondo singlet, and the excited subgap state is a spin doublet. Recall that in the Kondo singlet ground state of the infinite-U Anderson model, there is a small probability of finding the impurity empty, despite the fact that the bare site energy ε d is well below E F ; this is not true for the doublet state, where the impurity is expected to be singly occupied. Thus, the fact that the bound state is located above E F in this limit (i.e., y>0) reflects the fact that one must add a particle at the impurity to the singlet ground state to reach the dou- blet state. The magnitude of y indicates the energy splitting between these two states,
where E doub is the energy of the doublet state and E k is that of the Kondo singlet state. Similarly, for △ > T K , the ground state is the fully-paired doublet favored by the superconduc- tivity; the bound state is located below E F as one must re- move a particle from the doublet ground state to reach the excited singlet state. An identical expression holds for y:
The crossing of E F by the subgap state is thus seen to indi- cate a substantial change in the nature of the ground state. While the behavior of the bound state at δ = 0 has re- ceived much attention, the effects of having a phase differ- ence has not, to our knowledge, been previously studied. The triangles in Fig. 2 indicate the bound state positions y for the same values of T K / ∆ as the solid curve, but now with a phase difference of δ= π between the superconductors. We find that for all values tested, y decreases as we increase δ to π —the subgap bound state has an anomalous phase depen- dence. This behavior is seen more explicitly in the inset of Fig. 2, where we plot the subgap resonance for various val- ues of δ. The phase dependence of the subgap state can also be interpreted in terms of a competition between the Kondo singlet and the fully paired doublet states. The Kondo state is again expected to be the ground state for T K > Δ , while the the doublet state will be the ground state in the opposite regime. Significant now, however, is the fact that the phase dependence of these two states is quite different. In the dou- blet state, the impurity can be viewed as a local moment and we thus expect a negative Josephson coupling and that E doub will be a minimum at δ= π . In the Kondo singlet state, the impurity spin has been screened and thus we do not expect any anomalous Josephson coupling— E K will be a maximum at δ= π . With these associations, it follows from Eqs. (7) and (8) that y is minimized at δ= π , meaning that the subgap state has an anomalous phase dependence. Note that for small values of Δ / T κ , it is possible to make the subgap state cross E F and thus change markedly the nature of the system ground state by only changing δ. At this point, it is natural to attempt to make a connection to the Josephson effect. Recall that for a noninteracting sys- tem, it is possible to discuss the Josephson current as being at least partially carried by Andreev bound states existing in the weak link between the two superconductors. 10 These states have a phase-dependent energy, and thus contribute to the current through the relation I = (2e/ ˆ ˆ h )dF/dδ , where F is the free energy of the junction. For the simple case where a noninteracting impurity couples the two superconductors, there are always two Andreev bound states with energies ± | ε( δ ) | ; as the lower-energy state − | ε( δ ) | has a ''normal'' phase dependence ( dE/d δ> 0 ), the Josephson current is al- ways positive. In the present case, matters are quite different due to the strong on-site interaction. We have only one apparent An- dreev bound state, and its phase dependence is anomalous (dE/dδ<0). We would thus naively expect that the Joseph- son current would both become negative and enhanced in magnitude as the bound state crosses below E F and becomes ''occupied.'' Of course, this picture is oversimplified, as there are other possible contributions to the Josephson current—due to interactions, the impurity contribution to the free energy F is not just a simple function of the impurity density of states. The conjecture made above can be tested by making an explicit calculation of the dc Josephson current through the impurity. Using a technique similar to that used in Ref. 16 to calculate the normal current through an interacting impurity, we arrive at the following exact expression for the Josephson current: 17
Here, g R , 12 (ω) indicates an anomalous BCS propagator and G β , 21 ( ω, δ ) is the anomalous impurity propagator; without loss of generality, we have chosen φ t + φ r = 0. We calculate the anomalous impurity propagator within the NCA using the lowest-order contributing graph,11,12 without making any further approximation. To calculate an approximate value of the 0 temperature Josephson current, we use Eq. (9) with the Fermi function taken at T = 0, but with all spectral functions calculated at T = 0.5Δ [the NCA is unreliable at T = 0 (Ref. 13)]. This is a reasonable procedure, as we expect no qualitative changes in the spectral functions as the temperature is further lowered; the subgap resonances will only be sharpened. Retaining a finite T Fermi function would smear the contribution of the subgap resonances, and thus obscure the behavior we are interested in. Plotted in Fig. 3 is the approximate 0 temperature phase derivative of the Josephson current ( dI j /d δ ) thus obtained as a function of Δ / T κ , taken at δ = 0. Comparison to Fig. 2 indicates that π-junction behavior is indeed lost when the bound state crosses above the Fermi energy, as expected from a noninteracting picture. Similar results hold at other values of δ−( d1 , /d δ ) changes sign at roughly the same value of Δ / T κ at which the bound state crosses the Fermi energy. It would thus appear that the subgap bound state is the entity responsible for carrying the reversed-sign Joseph- son current associated with π -junction behavior. Note that in some respects, our conclusion here resembles what was found in a study of the Josephson current through a noninteracting magnetic insulating layer; 18 there also, the transition to π-junction behavior is accompanied by an anomalous Andreev bound state crossing below the Fermi energy. We stress again that the present situation is quite different due to the strong on-site repulsion—there is no a priori reason guaranteeing that the general noninteracting theory relating I , to Andreev bound states should be appli- cable to the interacting system studied here. We find only a single subgap state in the impurity density of states, whereas in the noninteracting case bound states always occur in pairs, with one member of each having dE/dφ<0. Also of interest in the present system is the behavior of the superconducting phase of the impurity itself. We find that this phase undergoes a shift by π as the bound state passes through the Fermi energy. When the bound state is above E F , the impurity simply has the average phase of the two superconducting leads—at low energies, the signs of both the real and imaginary parts of the anomalous impurity Green function ( G R ; 21 ) are the same as those of the average anoma- lous Green functions of the superconductors. However, when the bound state moves below E F , we find that the anomalous impurity green function changes sign. We interpret this to mean that the impurity has acquired an additional phase of 1 L.I. Glazman and K.A. Matveev, Pis'ma Zh. Eksp. Teor. Fiz. 49, 570 (1989) [JETP Lett. 49, 659 (1989)]. ² B.I. Spivak and S.A. Kivelson, Phys. Rev. B 43, 3740 (1991). 3 I.O. Kulik, Zh. Eksp. Teor. Fiz. 49, 1241 (1965) [Sov. Phys. JETP 22, 841 (1966)]. 4 H. Shiba and T. Soda, Prog. Theor. Phys. 41, 25 (1969). 5 A.V. Rozhkov and D.P. Arovas, Phys. Rev. Lett. 82 , 2788 (1999). 6 J. Zittartz, A. Bringer, and E. Muller-Hartmann, Solid State Com- mun. 10 , 513 (1971). 7 T. Matsuura, Prog. Theor. Phys. 57, 1823 (1977). 8 M. Jarrell, D.S. Sivia, and B. Patton, Phys. Rev. B 42, 4804. (1990). 9 W. Chung and M. Jarrell, Phys. Rev. Lett. 77, 3621 (1996). 10 C.W.J. Beenakker, in Transport Phenomena in Mesoscopic Sys- tems , edited by H. Fukuyama and T. Ando (Springer-Verlag, π . This behavior is shown Fig. 4. It is clear from Eq. (9) that this additional factor of ( − 1) will cause a sign change in the Josephson current. A similar phase shift was found in earlier work involving a single superconductor, 9 though a transition was not observed. Conclusions . Using an extension of the NCA, we have studied the properties of an infinite-U Anderson impurity coupled to two superconductors. We find that the subgap bound state develops an anomalous phase dependence, and that π-junction behavior is lost when the bound state crosses above E F . A.C. acknowledges the support of the Olin Foundation. This work was supported in part by the NSF under Grant No. DMR-9805613. Berlin, 1992).