Inhomogeneous Superconductivity in Comb-Shaped Josephson Junction Networks

We show that some of the Josephson couplings of junctions arranged to form an inhomogeneous network undergo a non-perturbative renormalization provided that the network's connectivity is pertinently chosen. As a result, the zero-voltage Josephson critical currents $I_c$ turn out to be enhanced along directions selected by the network's topology. This renormalization effect is possible only on graphs whose adjacency matrix admits an hidden spectrum (i.e. a set of localized states disappearing in the thermodynamic limit). We provide a theoretical and experimental study of this effect by comparing the superconducting behavior of a comb-shaped Josephson junction network and a linear chain made with the same junctions: we show that the Josephson critical currents of the junctions located on the comb's backbone are bigger than the ones of the junctions located on the chain. Our theoretical analysis, based on a discrete version of the Bogoliubov-de Gennes equation, leads to results which are in good quantitative agreement with experimental results.

We show that some of the Josephson couplings of junctions arranged to form an inhomogeneous network undergo a non-perturbative renormalization provided that the network's connectivity is pertinently chosen. As a result, the zero-voltage Josephson critical currents Ic turn out to be enhanced along directions selected by the network's topology. This renormalization effect is possible only on graphs whose adjacency matrix admits an hidden spectrum (i.e. a set of localized states disappearing in the thermodynamic limit). We provide a theoretical and experimental study of this effect by comparing the superconducting behavior of a comb-shaped Josephson junction network and a linear chain made with the same junctions: we show that the Josephson critical currents of the junctions located on the comb's backbone are bigger than the ones of the junctions located on the chain. Our theoretical analysis, based on a discrete version of the Bogoliubov-de Gennes equation, leads to results which are in good quantitative agreement with experimental results. It is a common belief that Josephson Junction Networks (JJN) may be regarded as the prototype of a complex physical system with a variety of interesting physical behaviors, adjustable acting only on a few external parameters and, by means of the modern fabrication technologies, also on the building topology and geometry of the array [1]. It is now possible to build arrays with very small junctions [2] to detect effects due to single electrons in a range of temperatures related to another relevant energy scale: the charging energy. The possibility of controlling experimentally the competition between Josephson and charging energies makes JJNs useful devices to investigate quantum phase transitions [3], or to model the physical properties of some real materials like granular superconductors [4]. Many of the results valid for JJNs are shared by cold atoms in optical lattices [5] since, in these systems, bosonic Josephson junctions and arrays may be rather easily realized [6]; in addition, Josephson networks and devices pave a very promising avenue to the quantum engineering of states relevant for quantum computing [7].
Inhomogeneous superconducting networks have been studied [8] mainly to provide a better understanding of the properties of well controlled disordered granular superconductors [9]. The appealing perspective to realize devices for the manipulation of quantum information recently stimulated the analysis of inhomogeneous planar JJNs with non conventional connectivity [10], engineered to sustain a topologically ordered ground state [11]. Transport measurements on superconducting wire networks evidenced -in a pure system with non-dispersive eigenstates-interesting anomalies of the network critical current induced by the interplay between the network's geometry and topology and an externally applied magnetic field [12]; more recently, the theoretical analysis of rhombi chains has evidenced the exciting possibility of being able to detect 4e superconductivity through measurements of the supercurrent in presence of a pertinent external magnetic field [13].
In this Letter we show that, even in absence of an externally applied magnetic field, a JJN fabricated on a pertinent graph [14] may support anomalous behaviors of the Josephson critical currents, which are induced by a non-perturbative renormalization of some of the Josephson couplings of the array. Our analysis clearly evidences that this renormalization is only attainable for the class of graphs, whose adjacency matrix supports an hidden spectrum [14,15]; thus, our findings are not generic to any inhomogeneous JJN. For instance, in absence of an external magnetic field, the networks analyzed in [8][9][10]12,13] should not give rise to any of the anomalous behaviors of I c discussed in this paper.
In the following, we provide a theoretical and experimental study of the behavior of the Josephson critical currents measurable in a comb-shaped JJN made of N b grains located at the vertices of a "comb" graph and linked by Josephson junctions [see Fig.1]. We compare our results with those obtained for a linear Josephson junction chain fabricated with the same junctions. Since one may regard the backbone of a comb graph as a decorated chain, it appears natural to compare its superconducting properties with those of a linear chain since the latter is the simplest network with euclidean dimension one. The result of this comparison shows that the Josephson critical currents of the junctions located on the comb's backbone are sensibly bigger than the ones of the junctions located on the chain.
Another way to look at a comb-shaped JJN is to regard it as a linear chain immersed in an environment mimicked by the addition of the fingers [16]. As in many Josephson devices one should then expect that the nominal value of the Josephson energy E J of the junctions in the array gets renormalized by the interaction with the environment. This situation is often analyzed using either the Caldeira-Leggett [17] or the electromagnetic environment [18] models. In these approaches one usually assumes that the effective boundary conditions for the quantum fluctuations of the environment modes do not depend on the Josephson couplings or on the network's topology: while this assumption is perfectly legitimate for weak environmental fluctuations, better care should be used if these fluctuations are strong as it may well happen for one dimensional JJNs. A simple paradigmatic example of a non perturbative renormalization of Josephson couplings is given by the simple inhomogeneous onedimensional array analyzed in [19,20], where the source of inhomogeneity is given by putting on a site of the linear chain a test junction with a different nominal value of the Josephson coupling E J . In the sequel we show that, for a comb-shaped JJN, the Josephson couplings on the backbone get renormalized. Our explicit computation shows that this renormalization is indeed non perturbative since the peculiar connectivity of a comb modifies the spectrum of quantum modes living on linear chains by the (obviously non-perturbative) addition of an infinite set of localized states, which disappear in the thermodynamic limit (the hidden spectrum): adding the fingers to a backbone chain is, in fact, a topological operation since it amounts to a non trivial change of boundary conditions for the Josephson linear chain. In a different context, the interplay between an hidden spectrum and a change in boundary conditions has been recently used in [21].
We use the lattice Bogoliubov-de Gennes (LBdG) equations [22] to compare the properties exhibited by Josephson linear chains and comb-shaped Josephson networks. Using the eigenfunctions of the LBdG equations, a self-consistent computation yields for both systems the gap function, the chemical potential and the quasi particle spectrum. We show that, for a linear chain, the superconducting gap and critical temperature satisfy to the well-known BCS equations and that, on the backbone of a comb JJN, the BCS equations are satisfied with a renormalized value of the Josephson energy. Then, we compute the zero-voltage Josephson critical currents I c on the comb's backbone and compare our results for I c with the outcomes of experimental measurements: our computation not only confirms with good accuracy the experimental results of [23], but is also in good agreement with new data obtained at temperatures closer to the critical temperature for the onset of superconductivity in N b grains. The new data are shown in Fig.2.
To obtain a discrete version of the BdG equations suitable to describe the JJNs fabricated in [23], we make the ansatz that the eigenfunctions of the continuous BdG equations [22] may be written in a tight binding form as u α ( r) = i u α (i)φ i ( r) and v α ( r) = i v α (i)φ i ( r); i labels the position of a superconducting island while the contribution of the electronic states participating to superconductivity in a given island is effectively described by a field φ i ( r), whose specific form depends on the geometry of the islands and on the fabrication parameters of the connecting junctions. The LBdG equations then read where u α and v α satisfy to i |u α (i)| 2 + |v α (i)| 2 = 1.
The matrix ǫ ij is defined by ǫ ij = −tA ij + U (i)δ ij − µδ ij , with A ij being the adjacency matrix of the network [24] )I c is the nominal value of the Josephson energy of all the junctions in the network while I c is the unrenormalized zero-voltage Josephson critical current of each junction. U 0 ( r) mimics the effects of the barrier between the superconducting islands. Self-consistency requires ∆(i) =Ṽ α u α (i)v * α (i) tanh β 2 ∈ α and U (i) = −Ṽ α |u α (i)| 2 f α + |v α (i)| 2 (1 − f α ) , wherẽ V ≡ Vφ 2 ( r = r i ) is assumed to be independent on i. Topology is encoded in the term −tA ij appearing in the definition of the matrix ǫ ij , while the specific values of t andṼ depend -as a result of our ansatz on the form of the eigenfunctions of the BdG equations-only on the φ i ( r). To justify the assumptions involved in the derivation of Eqs.(1)-(2), we observe that, for the JJNs described in [23], capacitive (inter islands and with a ground) effects are negligible, that the total number of electrons on the island N is much larger than the number of electrons tunneling through the Josephson junction and that all islands contain approximately the same N (N (i) ≡ N ). Furthermore, the islands are big enough to support the same superconducting gap of the N b bulk material. As a result one may require φ i ( r) to be position-independent on each island except for a small region near the junction and to be the same on each island with a normalization given by d rφ i ( r)φ i ( r) = N (i) ≡ N and d rφ i ( r)φ j ( r) ≈ 0 for i = j. In our derivations we put N ≡ 1.
For a linear array, the LBdG may be readily solved leading to an uniform potential U (i) ≡ U c and an uniform pair potential ∆(i) ≡ ∆ c . From the eigenvalue equation The BCS-like behaviour is obtained when E k = e k , which happens since U c = 0 and µ = E F . When ∆ c /E J ≪ 1, for T = 0, one gets ∆ c (T = 0) = 8E J e −2πEJ /Ṽ , while, for T = T c (i.e., ∆ c (T = T c ) = 0), one obtains k B T c = CE J e −2πEJ /Ṽ , with C = 4.54. It is comforting that the assumptions on which our approach is based lead, for the chain, to results having the same functional form of the well-known BCS formulas for the gap at T = 0 (i.e., ∆(T = 0) = 2hω D e −1/n(0)VBCS ) and the BCS critical temperature (i.e., k B T c = 1.14hω D e −1/n(0)VBCS ), provided that n(0)V BCS ≪ 1 [22]: in addition, one gets also ∆ c (T = 0)/k B T c = 8/C ≈ 1.76.
Measurements on a chain made with N b grains yield T c ≈ 8.8K and ∆ c (T = 0) ≈ 1.4meV ≈ k B · 15.9K; furthermore, in the experimental setup described in [23] it is I c ≈ 18µA. The parameters E J andṼ, determined from the BCS equation yielding the chain's critical temperature, are then given by E J ≈ k B · 430K and V/E J = 1.185. In Fig.2 we plot for several temperatures the measured I c (circles) and the critical currents obtained inserting ∆ c (T ) in the well-known Ambegaokar-Baratoff expression [25] for the zero-voltage Josephson current (lower solid curve): the agreement is excellent.
For a comb network with N × N islands (see Fig.1), one finds a solution of the LBdG equations (1)- (2) where both the Hartree-Fock potential U (i) and the gap function ∆(i) are position dependent. We denote the islands by (x, y), x labeling the finger and |y| the distance from the backbone, expressed in lattice units. The eigenvalue equation −E J j A ij ψ α (j) = e α ψ α (i), admits [24], in addition to a set of delocalized states with energies ranging from −2E J to 2E J , a localized ground-state ψ 0 = (C 0 / √ N )e −y/ξ , corresponding to the eigenvalue e 0 = −2 √ 2E J (C 2 0 = 1/ √ 2 and ξ given by sinh (1/ξ) = 1) and an hidden spectrum made of other eigenstates localized around the backbone [24]. For a crude analytical estimate, one may require that, away from the backbone, the fingers may be regarded as a linear chain with uniform potentials (i.e., ∆(i) = ∆ c and U (i) = U c ). To get then coupled equations for ∆ b , ∆ c , U b , and U c , one writes the LBdG equations (1)-(2) on a backbone's grain i. We set u α (i) = U α ψ α (i) and v α (i) = V α ψ α (i), with U 2 α + V 2 α = 1. The self-consistency equation for U implies that, at T = 0, U b ≈ U c −Ṽ C 2 0 2 ; upon requiringμ ≈ U b one immediately sees that, due to the localized modes in the fermionic spectrum, the chemical potential on the comb's backbone is smaller than the one measured on the chain.
By substituting the wavefunctions of the eigenstates of the hidden spectrum [24] in Eqs.(1)-(2) and usingμ ≈ U b one gets where ∈ k = ∆ 2 b + 4E 2 J (1 + cos 2 k). The hidden spectrum eigenstates contribute to the gap function ∆ b through the second term in the rhs of Eq.(3): without them, ∆ b equals ∆ c . When Furthermore, at low temperatures, ∆ b (T )/∆ c (T ) ≈ ∆ b (T = 0)/∆ c (T = 0). Using the parameters E J andṼ obtained from the measurements carried on the JJ chain, for a JJ comb one gets K ≈ 1.13. Upon requiring that, as for the linear chain, the T = 0 backbone's gap function has a BCS like functional form, i.e. ∆ b (T = 0) = 8Ē J e −2πĒJ /V , withĒ J andV the renormalized Josephson energy and the renormalized interaction term, one is able to estimate the renormalization of the Josephson coupling within the LBdG approach. Namely, one has, which embodies the effects of the hidden spectrum on the Josephson couplings. In Fig.2 we plot, as a function of the normalized temperature, the values of I c measured with the methods described in [23] (squares) and the values of I c obtained from the Ambegaokar-Baratoff formula using both the renormalized coupling given by Eq.(4) and the gap function along the backbone for the comb-like JJN studied in [23] (solid curve): the agreement between theory and experiments is very good at low temperatures, while the theory gives a slight overestimate at higher temperature. We showed that a non perturbative (i.e. induced by the states of the hidden spectrum) renormalization of some of the Josephson couplings of a comb-shaped JJN is responsible for the observed enhancement of I c of the Josephson junctions located along the comb's backbone. The key assumption in our derivation is that the eigenfunctions of the BdG equations may be written in a tight binding form; once this assumption is made, one is able to derive Eqs. (1)-(2) and to account for all the dependence on the electronic states into the definition of the parameters E J andṼ, which, in this paper, we determined from the measurements carried on the linear chain. Our approach yields a value of the renormalized Josephson coupling of the junctions located on the comb's backbone in excellent agreement with the experimental results (see fig.2). We expect that similar phenomena happen for the class [15] of JJNs fabricated on graphs whose adjacency matrix supports an hidden spectrum.
We are much grateful to L.B. Ioffe and M. Rasetti for asking very useful questions at several stages of our work. We are indebted to C. Granata, V. Corato and F.P. Mancini, for their helpful assistance. We thank M. Cirillo for discussions. Our research has been partially supported by the MIUR Project Josephson Networks for Quantum Coherence and Information (grant No.2004027555).