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Percolation of Superconductivity

In case of superconductors whose electrons attract each other only if they are near certain centers, the question arises 'How many such centers are needed to make the ground state superconducting?' We shall examine it in the context of a random U Hubbard model. In short we study the case where U_i is -|U| and 0 with probability c and 1-c respectively on a lattice whose sites are labelled i using the Gorkov decoupling and the Coherent Potential Approximation (CPA). We argue that for this model there is a critical concentration c_0 below which the system is not a superconductor.


I. INTRODUCTION
In many attempts to construct a viable model for High Temperature Superconductors the notion of negative -U centers is invoked [1][2][3][4] . In this connection there is a simple, natural question that arises: How many such centers are needed to make a superconductor. In this contribution we shall argue that under certain circumstances there is a critical concentration c 0 below which there is no superconducting order. Moreover, we developed a strategy for investigating the factors which determine c 0 .
In order to deal with a well posed problem we shall study a Random -U Hubbard Model defined by the Hamiltonian where the coupling constant The question we shall ask is: Is there a finite concentration c 0 such that for c < c 0 the cofigurationally averaged, superconducting long range order parameter χ vanishes even at zero temperature?
As is natural we define χ by the relation where the local pairing amplitude is given by Following the conventional notation the bracket < ... > denotes a thermodynamic average and Θ, for an arbitrary operatorΘ, implies the average ofΘ over all configurations U i , such that the fraction of negative U sites is c, with equal weight. A sample will be said to be superconducting if χ = 0 This implies that χ i = 0 on a finite fraction of all sites. Namely, if χ i = 0 only on a finite number of sites χ will go to zero as N → ∞ and the system will be regarded as not superconducting.
To make progress we calculate χ i within the Hartree -Fock -Gorkov (HFG) Decoupling scheme for the Greens functions and the averaging over the U-configurations is accomplished with the help of the Coherent Potential Approximation (CPA) 5 . In short, at the risk of missing some important feature of the problem, like localization of electrons, we develop a mean field theory for the phenomena described by H in Eq. (1) This approach may be justified by noting that little is known about the problem at hand systematically 1-3 and hence as a preliminary study a mean field theory is called for.
Note that the simplest approximations to the problem would be to set U i at each site equal to its average value U = cU. In some contexts this is called the Virtual Crystal Approximation 7,5 . Since, as is well known 3 , any amount of attraction leads to superconductivity, U = cU implies superconductivity for all non zero concentrations with the transition temperature T C decreasing albeit non-analytically, with c. Thus before setting out the details of the above theory it is worthwhile to pause, briefly, to consider a number of fairly general arguments which suggest that the above conclusion is premature and that there is a critical concentration c 0 of negative U -centers for superconductivity.
i. Classical Percolation Theory for a mixture of two metals with resistivities ρ 1 and ρ 2 have been studied in the Effective Medium Approximation 6-8 . For ρ 1 = ρ 0 and ρ 2 = 0 ( Fig.   1), namely in the case where metal 2 is a superconductor, it yields an effective resistivity given by Taking δt = τ we conclude that for δE = W c (d−1)/d > U the pair will propagate for δE = W c (d−1)/d < U the pair will not propagate. Thus for c < c 0 , where a system of negative U centers will not be superconducting. Presumably, the Cooper pairs will be localized. On the other hand for c > c 0 it will be a superconductor.
iii. Localization of Cooper pairs by local charge and order -parameter -phase fluctuation is the third argument which we wish to recall briefly. It was explored in the present context by Doniach and Inui 1 . In a Ginzburg Landau theory on a lattice the relative phases of the local order parameters ψ i = |ψ i |e iΘ i are determined by the quadratic term in the free energy function F ({ψ i }). This may be written in the form of Josephson , where the precise relationship of the coefficients to various parameters of the theory need not concern us here. To describe charge fluctuations associated with Cooper pairs arriving and leaving a site a charging energy needs to be added to F . Because, the local potential is related to the phase by Josepson voltage relation V i =h 2eΘ i this has the form of a kinetic energy term. Finally, to recover a microscopic description the local phases are treated as quantum mechanical variables by the Hamiltonian This is a much studied Hamiltonian in connection with granular superconductivity. In particular it was investigated by Gosset and Györffy 10 in the Hartree -approximation. In short they factorized the wave -function as shown below and found the following self-consistent equation for the individual site wave function φ 0 (Θ) where The amplitude ρ, determined by solving the above equation numerically is shown in Fig In concluding this introduction we note that the specific task we shall undertake is a contribution to the general problem of treating disorder and electron-electron interactions simultaneously For a comprehensive discussion of the relevant issues in this field the reader is referred to the relatively recent review article by Belitz and Kirkpatrick 9 .

II. THE COHERENT POTENTIAL APPROXIMATION FOR THE RANDOM-U HUBBARD MODEL
The physics described by this simple model appears to be exceedingly rich. For instance, one might expect that, under some circumstances, the Cooper pairs are subject to Anderson localization 11 and hence they form a random set of Andreev scatterers for the quasi-particles 12 . Such system of scattering centers may then Anderson localize the quasiparticles themselves and turn the system into an insulator below the critical concentration c 0 for superconductivity. However very little systematic fully microscopic work has been done on the problem and hence, as a preliminary exercise, a mean field theoretic treatment is called for even at the risk of failing to capture some of its important features. In any case, as we shall show, even such limited description turns out to be of physical interest.
Formally, the task is to find the Greens function where the creation and annihilation operators c iσ (τ ) and c + iσ (τ ) evolve in complex time τ according the random-U Hamiltonian H in Eq. (1), T is the τ -ordering operator, brackets < ... > denote here the usual equilibrium thermal averages corresponding to H, and average the result with respect to all arrangement of the U-centers each denoted by {U i }. In short we wish to find where the probability distribution is assumed to be of form Note that the local order parameter defined by Eq. (3) is given by and hence the knowledge of the averaged one particle Greens functions matrix is sufficient to address the question whether or not there is superconducting long range order at a given concentration c.
As we have indicated above we shall now proceed to a mean field approximation to the above problem. This consists of two steps. Firstly, we make use of the Hartree-Fock-Gorkov decoupling scheme to find the following 'mean-field' equation of motion where Secondly, we find average of the solution to Eq. (13), namely G(i, j; ω n ; {u i }), over all U-configurations using the Coherent Potential Approximation (CPA). The justification for this second step is that the CPA is well known to be a reliable mean-field theory of disorder for wave propagation in a medium described by independent random variables 6-8, 13 .
To implement the CPA we rewrite Eq. (13) in the Dyson form where The CPA recipe for G(i, j; ω n ) is to set it equal to the coherent Greens function (15) for the case where the random potential V l is replaced by the energy dependent, complex coherent potential Σ(ω n ), the same on every site. To determine the coherent potential (self-energy) we study, in turn a U i = −|U| impurity in the coherent lattice. On the impurity site at i we find where Then, the usual CPA condition which determines the self-energy Σ(ω n ) is given by Similar equations have been used to describe random superconductors by Lustfeld 14 and more recently by Litak et al 15 . The principle difference between our present concerns and that of these earlier authors is that we are focusing on the randomness of the interaction parameter U i and not on the random site energies ǫ i as was their aim. To put it on other way we are studying a problem analogues to that of 'spin-glass' 4 rather than that of dirty superconductors.
Manipulating the CP A equations yield the following gap equation: In what follows we present results of solving the above equations numerically for various interesting regimes. Of particular interest is the large U limit. As U i change its values form 0 to −|U| there exists Mott-Hubbard metal-insulator transition for large enough interaction |U|. An other interesting feature of the problem at hand is that fluctuations of pairing potentials ∆ i , which changes randomly from 0 to ∆ U , invalidate the Anderson theorem [17][18][19] and hence states appear in the gap 20 .

III. ORDER PARAMETER FLUCTUATIONS
At first we have calculated T C and χ for zero temperature (T = 0) by means of VCA where, as we mentioned in the introduction effective interaction between electrons U ef f = cU. Figs. 3a and 3b show the critical temperature T As is clear from Eq. (14), U i fluctuating between 0 and −|U| has two distinct direct consequences. On the one hand it causes the Hartree potential 1 2 U i n i to fluctuate. On the other it gives rise to a fluctuating pairing parameter -∆ i . As it turns out these two effects have very different influence on the solutions to Eqs. (18,19,20,21). Therefore, we examine them separately. As disorder was treated by CPA, at first we made calculations after neglecting Hartee potential 1 2 U i n i , in Eq. (18) and studied the case of order parameter fluctuation on their own. This means that we took the impurity potential in Eq. (18) to be In Figures 4a, b,  Fermi energy in these plots: ǫ F = µ = 0. Thus changing c from 0 to 1 system changes from normal metal (Fig. 6a) to a superconductor (Fig. 6c) through an insulator (Fig. 6b). Remarkably, for a low concentration of negative centers (Fig. 6a)  2) Interestingly, that transition from superconducting to normal phase is accompanied by a large value of local charge occupation n U (Fig. 7c) and large jump of a chemical potential µ (from one subband to another) near c 0 (Fig. 7d). It appears that for c below c 0 n U ≈ 2. Namely, every U site is doubly occupied with a pair of electrons (Fig. 7c). Because there are no empty spare U sites in the system these pairs cannot move. That is to say, they are localized on the U sites. is not large enough to create a band splitting effect but the tendency with χ → 0 is still observable as concentration c is tending to some finite c 0 > 0 (c → c 0 ). Here c 0 is less that in former case of larger interaction |U| = 2W (Fig. 7). The occupation of negative centers is larger than n but clearly less than 2 electrons per site. Clearly, in the random interaction case this is not true and this kind of disorder leads to pair breaking.
Thus on account for the large fluctuations of pairing potential ∆ i in our system,due to disorder, we observe a qualitative change in quasiparticle density of states shown in Fig 9a. These fluctuations lead also to complicated gap equations where T C is determined not only by G C but also by Σ, G U (Eq. 21).