Excess Kondo resonance induced by Andreev-normal co-tunneling

We report on a novel Kondo phenomenon of interacting quantum dots coupled asymmetrically to a normal and a superconducting lead. The effects of intradot Coulomb interaction and Andreev tunneling give rise to Andreev bound resonances. As a result, a new type of co-tunneling process which we term Andreev-normal co-tunneling, is predicted. At low temperatures, coherent superposition of these co-tunneling processes induces a Kondo effect in which Cooper pairs directly participate formation of a spin singlet, leading to four Kondo resonance peaks in the local density of states, and enhancing the tunneling current. 72.15Qm, 73.40Gk, 72.15Nj Typeset using REVTEX 1 The Kondo effect is a prototypical many-body correlation effect involving interactions between a localized spin and free electrons. [1] Its recent observation in semiconductor quantum dots (QD) [2–4] has generated a great deal of theoretical and experimental interests and provided rich understanding to many-body phenomena at the mesoscopic scale. For a QD coupled to two normal (N) leads, the physical origin of the Kondo effect is now understood [1,2,4]. Consider a single spin degenerate level ǫd of the QD such that ǫd < μN < ǫd + U , where μN is the chemical potential of the leads and U the on-site e-e interaction energy. An electron of either spin up or spin down which occupies ǫd cannot tunnel out of the QD because ǫd < μN ; as well, an electron outside the QD cannot tunnel into it unless the on-site Coulomb energy U is overcome. Therefore, the first-order tunneling process is Coulomb blockaded. However, due to Heisenberg uncertainty, the virtual higher-order co-tunneling events can still take place [1,2,4] by which the electron inside QD tunnels out followed by an electron with opposite spin tunneling into the QD, on a time scale ∼ h̄/|μN − ǫd|. As a consequence, the local spin is flipped. At low temperatures, the coherent superposition of all possible co-tunneling events gives rise to the Kondo effect in which the time-averaged spin in the QD is zero due to frequent spin flips: the whole system, QD plus leads, forms a spin singlet, and a very narrow Kondo peak located at μN arises in the local density of states (LDOS). When one of the leads is a superconductor (S), another transport process—Andreev tunneling, will occur in the normal-superconductor interface in which an incident electron from the normal side is reflected as a hole while a Cooper pair is created in the superconductor. Andreev tunneling is very important [5] because it determines transport properties of many mesoscopic superconducting-normal hybrid devices. It is therefore not surprising that the Kondo effect in N-QD-S hybrid systems has attracted considerable attention [6–10]. So far, the focus on Kondo effects in N-QD-S devices has been on enhancement or reduction of conductance as compared to that of N-QD-N systems [6,7,9]; and the emergence of subKondo peaks in LDOS at ±∆ [6,7] where ∆ is the superconductor gap energy. In these studies Andreev tunneling precesses, in essence, happen alone while the Cooper pairs do not 2 participate the formation of spin-singlet. However, since the Kondo effect in a QD results from co-tunneling processes, for a N-QD-S hybrid system, it is very natural to ask: are there co-tunneling processes consisting of one virtual Andreev tunneling and one virtual normal electron tunneling? If there are, can coherent superpositions of these Andreev-normal cotunneling give rise to a Kondo effect? What are the consequences and characteristics of the Kondo effect induced this way? It is the purpose of this letter to report our theoretical investigation on these issues. In contrast to previous work [6–10], we emphasize the possibility of virtual Andreev tunneling directly participating the co-tunneling process so that the Cooper pairs directly participate formation of the spin-singlet: these physical processes give rise to Kondo effect in the first place. Our results predicts a new co-tunneling process formed by an Andreev tunneling and a normal tunneling, and the superposition of this type processes induce four Kondo peaks in the LDOS. We consider the standard model Hamiltonian [11] of a N-QD-S system H = HN +HS + HQD + HT where HN = ∑ kσ ǫNka † kσakσ and HS = ∑ kσ ǫSkb † kσbkσ + ∑ k (∆bk↓b−k↑ +H.c.) describe the normal lead and the superconducting lead, respectively, in here we have set μS = 0. HQD = ∑ σ ǫdd † σdσ + Ud † ↑d↑d † ↓d↓ models the QD with a single level having spin index σ =↑, ↓ and intradot e-e Coulomb interaction U ; HT = ∑ kσ[VNa † kσdσ +VSb † kσdσ +H.c] denotes the tunneling part of the Hamiltonian. The current from the normal lead flowing into the QD is calculated by the standard Keldysh nonequilibrium Green’s function theory, as (h̄ = 1) [11]: I = −4eIm ∫ dǫ 2π ΓN { fN (ǫ)G (ǫ) + 1 2 G(ǫ) }