Transport Properties of Disordered Superconducting Thin Films. and Superconducting Islands

Transcription

1 Transport Properties of Disordered Superconducting Thin Films and Superconducting Islands A proposal for a PhD thesis submitted by Yonatan Dubi Physics department, Ben Guiron University

2 Contents Abstract 1 1. Introduction : Experiment and existing theory Experiment Existing theory 3 2. A phenomenological model for resistance of a 2D-superconducting sheet Introduction of the model Calculation Preliminary results Open questions and future study A Microscopic theory of transport through disordered superconductors Formation of superconducting islands Model Formation of superconducting islands preliminary results Magnetic field dependence preliminary results Coulomb interactions Tunneling through a single SC island Ultra-small SC islands in a disordered metal Tunneling current through a large SC island Tunneling conductance of a disordered superconductor Summary and future prospects Summary Future prospect 25 A. Conductance of a classical lattice 26 B. Ultra-small SC islands in a disordered metal 27 The low island concentration limit 27 Calculation of conductance 29 References 3

3 1 Abstract Understanding the physics describing the interplay between superconductivity and disorder is a long-standing issue of both experimental and theoretical physics. The common wisdom suggests that weak disorder has no dramatic effect on superconductivity. Strong disorder, however, may have a profound effect on the electronic properties of the system. Since the superconducting (SC) and the insulating states are two extremes in terms of conduction, one does not expect a direct (tunable) transition between them. It is a peculiarity of physics in two dimensions (2D) that such a transition is possible and was indeed observed in a large number of two-dimensional systems [1 3]. The aim of this work is to establish a microscopically well-based theory of the resistance of 2D disordered SC films which accounts for the detailed results observed in experiments. Primarily, we would like to explain both the existence of the SC-insulator transition (SIT) and the non-monotonic magneto-resistance (MR) observed in experiments [4, 5]. The central idea pursued in this work is that the physics of these systems is governed by the formation of superconducting islands, resulting from disorder-induced fluctuations in the amplitude of the SC order parameter. The emergence of these SC islands raises many interesting questions regarding transport properties of the system, and in this work we shall attempt to answer some of them. The main question raised is whether the formation of these SC islands accounts for the SIT and for the MR on the insulating side of the transition. Other issues raised are the influence of large and small islands on transport properties, and the details of the formation of the islands. We propose to address the above issues within several different approaches. In order to account for the SIT and positive-to-negative crossover of the MR we suggest a phenomenological model in which transport of electrons can take place either through the normal part of the sample or through the SC islands, via the Andreev channel. Due to Coulomb blockade, transport through the islands is thermally activated with the charging energy of the islands. At strong magnetic fields the islands are small and thus have a large charging energy, becoming inaccessible to itinerant electrons. As the island density grows with lowering the magnetic field, the area of the sample available for electron transport shrinks, leading to an exponentially enhanced resistance. As the island size becomes even larger, with decreasing magnetic field, their charging energy decreases, while at the same time, the resistance of the non-sc areas rises. Thus, at some value of the magnetic field the resistance of the SC and the normal areas become comparable, leading to a peak of the MR. For lower magnetic fields electrons prefer to tunnel into the SC islands and the resistance diminishes. At even lower magnetic field the islands percolate and the system becomes SC. This is numerically verified using variable-range hopping model calculations which include SC areas and the Coulomb blockade. Different aspects of transport properties of systems with SC islands are proposed to be treated in analytical methods, using perturbation theory and the Keldysh technique. For a system with small islands perturbation theory in the islands size (multiplied by the Fermi wave-vector) may be applied, leading to peculiar behavior of the system conductivity. Especially, we find that the conductance is reduced due to the existence of small SC islands. In the limit of large islands we use the Keldysh technique to calculate the tunneling conductance through the island. This technique may also be applied to other models suggested in this work. For the study of SC island formation and their behavior under the influence of magnetic field we propose the generalization of a scheme used by Ghosal et.al. [6], in which the Bugoliubov-de-Gennes (BdG) equations are numerically solved self-consistently on a disordered lattice. Beside showing the formation of SC islands due to disorder and their annihilation with magnetic field, this model may in the future allow us to incorporate Coulomb interactions and to give a microscopic description of the system, including its conductivity (using the Kubo formula or the Landauer

4 2 approach). Novel aspects The problem of the SIT and problems related to transport properties of normal-sc islands hybrid systems are regarded as major issues in mesoscopic physics, becoming ever more interesting as the experimental possibilities of fabricating, manipulating and measuring small SC islands increase. Thus, a better understanding of these systems is important, both from the theoretical and the experimental sides. In spite of its importance and of the plethora of experimental data, the topics presented in this research have not been extensively studied from the theoretical viewpoint. The influence of SC islands on transport properties is unknown, and there is no theoretical model for the non-monotonic MR in thin SC films. In this proposal we discuss several new ideas. These include the inclusion of Coulomb blockade on SC islands to describe the MR in thin films, the calculation of conductance of systems with small and large SC islands and studying the effect of magnetic fields and interaction on the formation of SC islands in disordered systems. Other ideas which are not discussed and are given at the summary as future directions. They include the study of finite size and interaction effects on the transport through large SC islands, the effect of interaction on SC island formation in disordered superconductors, etc. This research proposal is organized as follows: In section 1 an introduction to the relevant experimental findings is given, along with a presentation of existing theory. In section 2 we present our basic model for MR in thin SC films and numerical studies based on it, showing that a phenomenological description of the experiment is indeed feasible. In section 3 we discuss the various models for a microscopic description and obtain a microscopic theory describing the formation of SC islands. We then present a theory for calculating the conductance of systems with SC islands, first for transport through a single island and then for a disordered SC. We then summarize and conclude with future prospects.

5 3 1. INTRODUCTION : EXPERIMENT AND EXISTING THEORY In this section we briefly review the current experimental situation regarding the superconductor-insulator transition and the effects of magnetic field on the insulating side of the transition, and discuss the existing theory. We put special emphasis on open questions and unexplained experimental features Experiment The superconductor-to-insulator transition (SIT) is a phenomenon occurring in thin films of type-ii superconductors (SC) [1] such as Bi, Pb and Al (e.g. [2]), TiN [3] or amorphous Indium-Oxide (e.g. [4, 5, 7]). These films are amorphous, meaning that they are disordered but not granular, and that the disorder is spatially homogeneous. The experimental details vary from one experiment to the other, but the general description of the experiments is roughly the same. The films are SC below a critical temperature T c in the sense that as T the resistance becomes immeasurably low. As an external parameter is varied (e.g. perpendicular magnetic field, film thickness etc.) a transition occurs from a SC to an insulator, where the resistance seems to diverge as T. The value of the resistance at the transition seems to be non-universal, and varies from sample to sample. We are mainly interested in a set of experiments [4] where the transition is tuned by a magnetic field (B), and as B is further increased the resistance keeps rising, until at some field B max there is a peak in the resistance. Surprisingly, a further increase of B results in a decreasing resistance, which eventually seems to saturate at a value somewhat larger than the normal sheet resistance. Some experimental results are shown in Fig. 1. A theory intended for explaining these experiments should account for several pronounced experimental features. The first one is obviously the magnetoresistance (MR) line-shape, including the SIT itself, the peak in the MR and the saturation of the MR to values higher than the normal sheet resistance. The second one is the fact that the MR can rise up by several orders of magnitude compared with its value at the SIT, and that difference between the peak resistance and the resistance at the SIT gets higher with increasing disorder. The third feature is the temperaturedependence of the resistance (in zero field), showing a reentrant behavior near T c for strongly disordered samples (see Fig. 2(a)). Another feature is the temperature-dependence of the resistance in strong B, above the transition. Experiments show that there is a range of temperatures where activation behavior is observed, ρ exp(t 1 /T ), with T 1 having a non-monotonic B dependence, shown in Fig. 2(b). The theory should thus account for both this behavior of T 1 and for the breakdown of activated behavior at low temperatures. An additional feature is the dependence of the critical field B c on disorder, showing a decrease in B c as disorder is increased, alongside with an increase of the resistance at the transition point. The final item in this list is the critical behavior of the SIT, namely the critical exponent of the transition, which is experimentally found to be ν 1.3 [5] Existing theory The existing theory in this field is still incomplete. While some models account for the SIT with fair agreement with experiment, they are limited only to regions close to the transition and are unable to explain features such as

6 4 a) R [kω] b) R [kω] H [T] H [T] R [kω] H [T] mk mk 5 mk ρ (1 3 Ω) 15 (a) Ma1f 1 5 B c (b) Na1c.24 K K c) R [kω] H [T] H [T] 75 mk H [T] ρ (Ω) B c 5 1. K 1 B (T).7 K 15 FIG. 1: Experimental results taken from [5] (left column (a)-(c)) and [4] (right column). Both experiments use amorphous Indium Oxide films. The upper insets of the right column and upper figure in the left column show the SIT, where below some critical field B c the resistance decreases with increasing temperatures, and this behavior is reversed for fields larger than B c. In the left column the different graphs (a)-(c) refer to increasing level of disorder. Notice the lower inset of (a), in which the magnetoresistance seems to saturate at very large magnetic fields. In the left column a sample with very large aspect ratio was measured, giving rise to magnetoresistance which varies by several orders of magnitude. Further details are given in the references. the resistance peak structure. The reason is that all models (so-called dirty-boson models) are based on the strongcoupling limit of the BCS theory where all electrons are bound in Cooper-pairs, and hence only bosonic degrees of freedom are relevant. Clearly this assumption breaks down at strong magnetic fields, where unpaired electrons start to participate in transport. Here we give a brief example for such a theory (following [8]). Other examples are found in the literature [9]. The starting point is the lattice-gas Bose-Hubbard Hamiltonian, H = U i,j ˆn iˆn j t i (a i a i+1 + a i a i+1 ) i (h + δh i )ˆn i, (1.1) where a i, a i, ˆn i are creation, annihilation and number operators for Cooper-pairs at the i-th lattice site, respectively. Taking the limit of infinite on-site repulsion excludes double boson occupancy, and a finite repulsive interaction U for nearest neighbors is considered. The hopping term t is restricted for nearest neighbors, and randomness is introduced via an on-site random potential h + δh i. The fact that the boson occupancy is either one or zero allows for a mapping

7 5 8 Strong K Intermediate 1.5 R [kω] K 3.35 K Weak T [K] (a) T I (K) B c 1 4 ρ (Ω) T -1 (K -1 ) 1 B (T) (b) 15 2 FIG. 2: (a) Zero-field temperature dependence of the resistance for samples with different degrees of disorder. Notice the peak in the resistance for samples with strong disorder, so-called reentrant behavior (taken from [5]). Similar behavior has been observed in other experiments, e.g. [7]. (b) Activation energy T 1 as a function of magnetic field, taken from [4]. Similar behavior has been observed in, e.g., [5]. Inset: Activated behavior of the resistance, valid only at high temperatures. to a spin 1 2 system, a i Ŝ i 1 2 (σx i iσ y i ) ˆn i 1 2 Ŝz i 1 2 (1 σz i ), (1.2) leading to the Hamiltonian H = J z i Ŝ a i Ŝa i+1 J xy 2 (Ŝ+ i i Ŝ i+1 + Ŝ i Ŝ + i+1 ) i h i Ŝ z i, (1.3) where J z = 4U, J xy = 2t, h i = h + δh i. Superconductivity is thus related to the ordering of pseudo-spins in the x y plain, a Ŝ. The problem is therefore reduced to calculating the statistical average of the spin operators. This is done by calculating the Green s function and averaging over the disorder. In the calculation higher-order correlations were decoupled using [Ŝ+ i Ŝj z, Ŝz k ] Ŝz [Ŝ+ i, Ŝz k ] (where [, ] denotes the commutator and denotes thermal average), the low-temperature limit was taken and averaging over disorder was done. The result is given in terms of the parameters η = h J xy, κ = Jz J xy, ζ = h J xy, S x (ζ2 + η 2 ) 2 4ζ 2 (1 κ) 2 (1.4) The two phases of this model, i.e. the superconducting phase in which S x = and the insulating phase in which S x =, are separated by a critical strength of randomness η c = 2ζ(1 κ) ζ 2. (1.5) This example demonstrates the pros and cons common to all dirty-boson models. While simple assumptions easily

8 6 lead to the SIT itself, one cannot use the model to move further away from the transition, since the system is only described by the bosonic pairs. Even within the range of applicability of the dirty-boson models some of the experimental details are unaccounted for. For example, different models predict a critical exponent ν 1 while most experiments indicate that it is closer to 1.3. Another prediction is the existence of a universal value of the resistance at the transition point, R c = h/4e 2 which was not observed in some experiments. 2. A PHENOMENOLOGICAL MODEL FOR RESISTANCE OF A 2D-SUPERCONDUCTING SHEET 2.1. Introduction of the model We propose a model for the MR for the entire range of magnetic fields. The model is based on three assumptions. The first one is that the disorder induces the creation of superconducting islands (SCI) due to fluctuations in the amplitude of the SC order parameter. This concept has already been used to interpret the results of some experiments [4, 1] and was corroborated by numerical simulations [6]. The second assumption is that as the magnetic field is increased the concentration and size of these SCI is decreased (it is probable that their strength, i.e. the SC order parameter amplitude also diminishes with magnetic field, but this is of no importance to the present model). While this is a major assumption of our model, we are unaware of any theoretical nor experimental work regarding the relation between SCI and magnetic field. The third assumption is that the SCI have some charging energy, and thus, an electron has to overcome the Coulomb blockade in order to enter into an island (as a Cooper pair via the Andreev channel). In principal, the charging energy is inversely proportional to the island area, and thus increases with increasing magnetic field. In order to analyze the mechanism by which the MR can be negative, consider such a system in the strong magnetic field regime, B >> B max (Fig. 3(a)). Due to the strong magnetic field the SCI are small and have a large charging energy. There are two types of trajectories available for electron transport, trajectories which follow normal areas of the sample ( normal paths, solid line in Fig. 3(a)) and those in which an electron enters and leaves a SCI ( island paths, dashed line in Fig. 3(a)). The resistance of the normal paths has some value (which may depend e.g., on length, temperature, etc.) and is assumed to be unaffected by magnetic field. Due to Coulomb blockade, transport through the island paths is thermally activated, and its resistance is of the form R exp(e c /T ), where E c is the charging energy of the island. If the charging energy is large then transport along the normal paths will be favored. As the magnetic field is decreased (but still in the regime B > B max, Fig. 3.(b)), more SCI are created and their size increases, but they are still small enough such that transport along normal paths is favorable. Thus, some of the paths which were normal in stronger fields (bottom solid line in Fig. 3(a)) now become island paths and hence unavailable for electron transport (bottom dashed line in Fig. 3(b)). As a result, the effective phase space for the transport electrons diminishes, resulting in a negative MR. At a certain magnetic field B = B max (Fig. 3(c)) the SCI are large enough so that their charging energy is small and the resistance through them is comparable to the resistance through normal paths of the same length. This point corresponds to the MR peak, since as the magnetic field is further decreased (Fig. 3(d)) the SCI are so large that transport through them is always preferable over transport through normal paths. Enlargement of the SCI will thus result in a decrease in the resistance, as expected. At the critical field B c some of the SCI percolate through the system, resulting in an insulator-to-superconductor transition, with the (classical) percolation critical exponent ν = 4/3. It should be noted that at low enough temperatures one should

9 7 observe the effect of Josephson coupling between different SC islands. If the temperature is lower than the Josephson coupling between two neighboring islands, then a proximity effect will allow transport of Cooper-pairs between the islands, resulting in an effective increase of the island size (in case the Josephson coupling is stronger than the island charging energy, which is always the case near the transition where the islands are large). Thus, the effective size of the islands should be temperature dependent (at least for low enough temperatures) and as a result the transition point should be smeared over a range of magnetic fields, as indeed is observed in some experiments [7]. (a) (b) (c) (d) FIG. 3: Representation of the model :(a) At strong magnetic fields, B >> B max, The system is composed of small superconducting islands (SCI) with large charging energy. In this regime transport through normal paths (solid lines) is always preferable than transport through the SCI (see text). (b) As the magnetic field is decreased, but still B > B max, more SCI appear, resulting in a decrease in available trajectories for transport ( bottom solid line in (a) and bottom dashed line in (b)) and hence negative magnetoresistance. (c) At a certain field B max the resistance of normal paths and paths that include SCI is comparable, resulting in a peak in the resistance. (d) For even lower fields, B c < B < B max, transport through SCI is always favoured, resulting in positive magnetoresistance Calculation The mechanism introduced in the previous section can be made quantitative in the following scheme: we model the system by a lattice (taken to be square for computational convenience). Each site can be either normal (with probability p) or SC (with probability 1 p), representing normal and SC regions of the sample respectively. The parameter p is thus related to the applied magnetic field, p = p(b), but the relation between them is a matter of further investigation. In order to model the disorder, the conductance between every two normal sites is given by the formula [11, 12] G ij exp( 2αr ij ( ɛ i + ɛ j + ɛ i ɛ j )/2kT ), (2.1)

10 8 where α is the inverse localization length, r ij is the distance between the sites i and j, ɛ i is the energy of the ith site (measured from the chemical potential and taken from some energy distribution) and T is the temperature. Since on the SC side the resistance vanishes with decreasing temperature but is finite (though small) for finite temperatures, the conductance between two SC sites is taken to be very large with comparison to Eq. (2.1), but still not infinite and temperature dependent, in such a way that it diverges as T. The calculation was conducted with several functional forms (power law dependence, exponential dependence etc.) and no qualitative difference between them was found. The conductance between a normal site and a SC site (a local N-S junction) is taken to be G NS exp( E c /kt ), (2.2) where E C is the charging energy of the island. In the following calculation the charging energies of the islands were taken to be a constant, independent on island size. Even in this approximation the above reasoning is valid, as a reduction in the magnetic fields (at large fields, B > B max ) leads to an increase in the number of local N-S junctions and thus to an increase in the number of paths which are characterized by a conductance given by (2.2). In this approxiamtion negative MR will presist as long as trajectories which traverse around the islands will have better conductance then (2.2), and since this conductance dependes on trajectory length, this will only happen as long as the islands are small enough, and at a certain magnetic field (or probability p) this trend will reverse and the MR will become positive [13]. The lattice is connected to electrodes (see Fig. 4) and the conductance of the sample is then calculated numerically using Kirchoff s laws (see appendix A). x x x x x x x x x x x FIG. 4: The lattice model is composed of regular sites and superconducting sites, which are connected in clusters (blue islands). The resistance between two normal sites (small circles) is of the VRH type, exponential in distance and energy difference between every two sites (not necessarily nearest neighbors). The resistance between two SC sites (filled squares) is very small (and grows smaller with decreasing temperature, see text) for neighboring sites (wiggly lines) and is infinite (no link) between non-neighboring sites. The resistance between neighboring normal and SC sites (thick lines) is much larger then the resistance between two normal sites, and is exponential in the charging energy of the SC island Preliminary results Calculations were performed on a square lattice (the lattice constant is taken to be a = 1). The numerical parameters were taken to be α =.4 and E c = 2. Temperatures range from.14 to.34 in steps of.8 (the Bolzman constant is taken to be k = 1). On-site energies were taken from a uniform distribution [W/2, W/2] with W = 4. The conductance between SC sites was (rather arbitrarily) taken to be of the form G SS = 35 exp ( ).5 T. In Fig. 5 the numerical results are presented and compared with the experimental result. A good qualitative agreement is clearly

11 observed. Following the experimental work, we took our numerical data and fitted it to an activation-like temperature 9 4 Log R 2-2 Log R p p FIG. 5: Numerical results for the resistance as a function of SC island concentration on a semi-log scale. The results are in qualitative agreement with the experimental results of Sambandamurthy et.al [4] (right column of Fig.1). Inset: the SIT. dependence, R exp ( ) T1, (2.3) T for different values of p. From this fit the value of T 1 was extracted, as shown in Fig. 6(a), again in good qualitative agreement with theory (Fig. 2(b)). The origin of such a field dependence of the activation energy is that the dominant energy scale varies, from being that of hopping conductance in strong fields, to that of island charging energies in intermediate and weak fields. One should note that as the probability is closer to p = 1 we found that the fit to ( ( activated behavior is worse, and in fact a crossover to a variable-range-hopping (VRH) behavior [14], R exp T1 ) x ) T with x = 1/3 (see Fig. 6(b)) was found. We claim that the change in temperature dependence, also observed in experiment (see inset of Fig.2(b)), is responsible for the orders-of-magnitude change in the resistance. The reason is that if the resistance would be activated through the entire range of magnetic field, the only change in the resistance (as a function of temperature) would come from changes in T 1 of Eq.(2.3), a change which does not suffice to trigger such a change in the resistance. We note that T 1 is somewhat smaller than the charging energy E c, but linearly dependent on it, the coefficient of proportionality depending on model parameters (α, W etc.). For instance, at the MR peak, where an activation dependence was clearly observed, we found T 1 /E c.75. This is due to the fact that the resistance at the peak is determined not only by the resistance of the N-S links (which is activated with E c ), but also by the hopping resistance between normal sites (which constitute about half of the sample at this value of p). This resistance is also activated, but the activation energy is given by the average over energies in (2.1), which is typically much smaller than E c. Thus, we claim that the interpretation of [4] that the fact that T 1 is of the same order as the SC energy gap points on a common origin for the SC and insulating behavior might not be correct. The effect of increasing disorder can also be demonstrated within this model, although it is somewhat incomplete. The reason for that is that the only place in the present model where the strength of disorder enters is in the energy distribution. Thus, we do not expect that increasing disorder within the model would change neither the critical point (which is determined by the percolation transition point) nor the value of the resistance at the resistance peak (which is determined by the value of E c ). In order to qualitatively describe the results of Fig. 1 (left column), we conjecture that the strength of disorder has an effect on the charging energy, since the stronger the disorder is the smaller the

12 1 T T(k) B (T) (a) p 6 log R T -1 p (b) VRH exponent FIG. 6: (a) Activation energy T 1 as a function of probability. This figure should be qualitatively compared with the experimental data (inset). (b) The VRH exponent obtained from fitting the numerical data to a VRH dependence. The results show a crossover from an x = 1 activation-like behavior to a x = 1/3 VRH behavior. Inset : An Arhenius plot of the experimental data for magnetic fields of 6 (red),7,...,11 (purple) Tesla. While for magnetic fields an activation behavior can be fitted, it is clear that for larger magnetic fields the resistance in not activated, but can be better fitted with a VRH formula. size of the SC islands generated and thus the charging energy becomes higher. We thus repeated the calculation, now varying both the disorder strength W and the charging energy E c. The results of the calculation or for disorder values W = 5, 3, 1 and E c = 7, 3, 1 respectively are shown in Fig. 7. Indeed, the experimental situation in which the resistance maximum increases with increasing disorder is obtained. The second effect of disorder, i.e. the shift in R(p) R(p) R(p) 4 2 (a) (b) (c) p FIG. 7: Effect of increasing disorder from weak (a) to strong (c) disorder, along with an increase in the charging energy:resistance as a function of concentration p for different temperatures. Notice the increase in the vertical scale. the critical field, should be determined by a disorder-dependent change in the functional dependence p(b), which we intend to study in the future.

13 Open questions and future study Some detailes of the model need further study. For instance, the dependence of the activation energy T 1 on model parameters should be examined. As another example, the influence of a distribution of charging energies (corresponding to different island sizes) on the results should be examined. Preliminary study shows that taking the charging energy to be either p-dependent (with a linear dependence) or to be taken from a uniform distribution but with no consideration in the island size, leads to qualitatively similar results to those presented above. The next step would be to adjust the charging energy to the cluster size. We conclude this section by pointing out that the above model was applied in the linear response regime and at this point does not allow investigation of non-linear phenomenae, which were observed in this system at very low temperatures and large bias voltage [15]. These authors observe that for fields B c < B < B max there is an abrupt change of the differential conductance from effectively zero at bias voltage V < V c to finite differential conductance for V > V c. For fields larger the B max the differential conductance rises smoothly with V for V > V c. While they attribute this behavior to the emergence of a collective insulating state, we believe that this phenomena can be explained with an expanded version of the model. A step towards such a theory is described in section A MICROSCOPIC THEORY OF TRANSPORT THROUGH DISORDERED SUPERCONDUCTORS Although the results deduced from the model presented in the previous section are in good agreement with experiment, a microscopic theory for transport through disordered SC is required, for example in order to include nonlinear effects, Josephson coupling effects, etc. In this section we start from a microscopic description and explore possible routes for using this description for examining various aspects of the system Formation of superconducting islands 1. Model The first goal of a microscopic theory is to justify one of the main assumptions of the model of the previous section, that of the formation of SC islands in a disordered SC and their reduction with disorder and magnetic field. A model presenting the formation of SC islands has already been introduced by Ghosal et.al [6], where the system was considered at finite temperatures but with no magnetic field. In what follows we introduce the model (including magnetic fields) at zero temperature [2]. The system is described by the discrete Hamiltonian H = H + H int, where H = <ij>,σ(t ij c iσ c jσ + H.c.) + (V i µ)n iσ i,σ = U n i n i (3.1) i H int where c iσ (c iσ) is the creation (destruction) operator for an electron with spin σ on a site r i of a square lattice with lattice spacing a = 1, U is the pairing interaction, n iσ = c iσ c iσ, and µ is the chemical potential. The random potential V i is an independent random variable at each site r i, uniformly distributed over [ V, V ], and V thus controls

14 the strength of the disorder. The nearest neighbor hopping integral t ij is magnetic field dependent, and in the Landau gauge (the magnetic field is taken to be perpendicular to the sample plane) can be written as [21] 12 t ij = t exp( iφy i /φ ) (3.2) if r ij is in the x-direction, and t ij = t otherwise. Here φ = c/e is the flux quanta, φ is the flux per plaquette and y i is the y-direction coordinate (in units of lattice spacing). We now transform the Hamiltonian (3.1) into the mean-field BdG Hamiltonian by defining the local pairing amplitude (r i ) = U c i c i (3.3) and decoupling the interaction term in 3.1. The results is the effective Hamiltonian H eff = <ij>,σ(t ij c iσ c jσ + H.c.) + (V i µ i )n iσ i + [ (r i )c i c i + (r i )c i c i ] (3.4) i where µ i = µ + U n i /2 incorporates the site-dependent Hartree shift in the presence of disorder. Here n i = σ n i,σ. H eff can be diagonalized with a Bugoliubov transformation c i = n [γ n u n (r i ) γ n v n(r i )] c i = n [γ n u n (r i ) + γ n v n(r i )], (3.5) where γ (γ ) are the quasiparticle (QP) operators, and the us and vs are the quasielectron and quasihole amplitudes. Demanding that the transformation (3.5) diagonalize the Hamiltonian (3.4) leads to the so-called BdG equations [16] ˆξ ˆ ˆ ˆξ u n(r i ) v n (r i ) = E n u n(r i ), (3.6) v n (r i ) from which the coefficients u n (r i ) and v n (r i ) are obtained. In (3.6) ˆξun (r i ) = ˆδ t i,i+δ u n (r i + ˆδ) + (V i µ i )u n (r i ) where ˆδ = ±ˆx, ±ŷ, and ˆ u n (r i ) = (r i )u n (r i ) and similarly for v n (r i ), and the energies are the QP excitation energies E n. Using the transformation (3.5), we can write the pairing amplitude (r i ) and the occupation n i in terms of the QP amplitudes, (r i ) = U n u n (r i )v n(r i ) n i = 2 n v n (r i ) 2. (3.7) We now solve the self-consistent BdG equations (3.6) numerically on a finite lattice. Starting from some initial values of (r i ) (we have tried different initial guesses to ensure that the initial condition do not alter the final result) we obtain the eigenenergies E n and the QP amplitudes u n (i), v n (i). Next we determine i and n i (and hence the local

15 chemical potential µ i ), using the obtained solutions. The new values are then used to again solve the BdG equations. This procedure is repeated until both the pairing amplitude and the occupation cease to change from one iteration to another. At each iteration the chemical potential is altered in such a way that the average density n = 1 N i n i remains constant Formation of superconducting islands preliminary results First we demonstrate the evolution of the spatial fluctuations of the pairing amplitude as a function of disorder at zero field. The results are shown here for interaction strength U /t = 3 (the results are qualitatively the same for other values of interaction), averaged over 5 realizations of disorder on a lattice. In Fig. 8(a) the distribution of the pairing amplitude is plotted for increasing disorder W/t =.2,.6,..., 6 (orange line-w/t =.2, red line-w/t = 6). At low disorder the distribution is strongly peaked around a finite value.83t, which corresponds to the value obtained from BCS theory for a uniform order parameter, = 2t exp( 1/N()U). As disorder is increased the distribution is both broadened and shifted towards a peak at =. Notice that the distribution has a tail of large values of even for strong disorder, meaning that there are regions in the sample where the pairing amplitude is finite. This is indeed seen in Fig. 9, where the spatial fluctuations of the pairing amplitude are plotted in real space in a color-density plot (red-large amplitude, blue-small amplitude). The formation of SC islands is clearly observed. In Fig. 8(b) we plot the distribution of local density (the total density remaining fixed, n 1.35). As the disorder is increased, the distribution changes from being peaked around the mean value (meaning small fluctuations in density) to a bimodal distribution, with peaks at n = and n = 2. Another remarkable result of this model comes from G( ) (a) (b) n G(n) FIG. 8: (a) Distribution of the pairing amplitude for different values of disorder, W/t =.2 (orange),.6, 1,..., 6 (purple). The distribution flows from being strongly peaked at /t.83 to being peaked around = with a long tail at larger values. (b) Distribution of the local occupation for different values of disorder, W/t =.2 (orange),.6, 1,..., 6 (red), flowing from a peaked distribution to a bimodal distribution. investigating the density of states (DOS), plotted in Fig. 1 for different values of disorder (for a discrete model the DOS is g(ω) = n δ(ω E n), and the δ-functions are broadened to Lorentzians, with a width of the order of the mean level spacing). As in [6], we find that the energy gap persists even in the strong disorder regime (inset of Fig. 1). This is explained by noticing that regions where the SC correlations are destroyed are those where the disorder strongly fluctuates. Thus, this will happen at regions of high peaks or deep valleys of disorder. The deeps are doubly occupied by a Cooper-pair, and the peaks are unoccupied, and hence an energy gap is formed.

16 14 W= W =.6 W = 1.2 W = 1.8 W = 2.4 W = 3. W = 3.6 W = 4.2 W = 4.8 FIG. 9: Color-density plot of the spatial distribution of the pairing amplitude for increasing value of disorder W (red-large amplitude, green-small amplitude). As the disorder amplitude increases the pairing amplitude starts to fluctuate, resulting in regions of large amplitude (SC islands) surrounded by normal regions (vanishingly small pairing amplitude). g(ω) E gap W ω FIG. 1: The DOS for different values of disorder, W/t =.2 (orange),.6, 1,..., 6 (purple). The regular shape of the superconducting DOS is smeared with disorder, but the energy gap remains finite even for strong disorder. Inset: The energy gap as a function of disorder.

17 15 3. Magnetic field dependence preliminary results We now repeat the above calculation but for finite magnetic field. We choose the value of disorder to be constant, W/t = 1. In Fig. 11 we plot the distribution of pairing amplitude (a) and the distribution of occupation (b) for different values of magnetic field (or flux per plaquette) α = φ/φ =.5,.1,...,.25. The pairing amplitude follows the expected flow, from being peaked around some finite BCS value.8t (the distribution is somewhat smeared, but bear in mind that this is with finite disorder), to being peaked around =, marking the destruction of SC correlations with magnetic field. This is similar to the effect of increasing disorder. The distribution of local occupation, however, shows different behavior. While as the magnetic field starts to increase a flow towards a bimodal distribution is observed, this flow suddenly stops at a certain magnetic field, and remains fairly constant, with peaks at n.4 and n 1.6. We plan to study this puzzling effect in more detail in the future. In Fig. 12 we plot the G( ) (a) (b) G(n) n FIG. 11: (a) Distribution of the pairing amplitude for different values of magnetic field, α = (orange),.2,.4,...,.24 (green). The distribution flows from being peaked at.83t to being peaked around = with a long tail at larger values. similar to the effect of increasing disorder. (b) Distribution of the local occupation for different values of magnetic field, α = (orange),.2,.4,...,.24 (green). While a flow towards a bimodal distribution is observed, at some value of magnetic field the flow seems to stop. spatial distribution of the pairing amplitude as a function of magnetic field, given in terms of the flux per plaquette. As in the case of increasing disorder, the existence of the SC islands and the reduction of their size with increasing field is evident. The DOS for different values of magnetic field is given in Fig. 13, the inset showing the energy gap as a function of field. As expected, we find that the energy gap is decreased with increasing field, with no sign of saturation. This is just the manifestation of the known result that magnetic fields destroyed SC correlations. In order to further study the dependence of island size on magnetic field, it would be usefull to calculate the correlation functions of the order parameter on different locations on the lattice. This may lead us a step toward a detailed understanding of island behavior in magnetic fields. 4. Coulomb interactions An important aspect of the phenomenological model is the effect of Coulomb blockade and the appearance of another energy scale, the charging energy of the islands, in the system. While the model presented above shows that SC islands are formed, one must include the Coulomb repulsion between electrons in order to obtain the Coulomb

18 16 Α Α.3 Α.6 Α.9 Α.12 Α.15 Α.18 Α.21 Α.24 FIG. 12: Spatial distribution of the pairing amplitude on a color density plot (red-large amplitude, green-small amplitude) for different values of flux per plaquette α = φ/φ. The reduction of SC island size with magnetic field is evident Egap.3 g(ω) α ω FIG. 13: DOS for different values of magnetic field, α =.5 (orange),.3,...,.23 (purple). The vanishing of the energy gap and smearing of the SC DOS can be observed. Inset : Energy gap as a function of field. blockade. This may be done by inserting another term to the Hamiltonian (3.1) of the form H C = g i,j n i n j, g > (3.8) describing the Coulomb repulsion between electrons occupying neighboring sites. This Hamiltonian can then be treated in a Hartree-Fock approximation. The Hartree term is obtained by contracting the interaction term according to n i n j n i n j, leading to the mean-field interaction Hamiltonian H C = g ij n i n j. The Fock term is

19 17 given from the contraction n i n j = σσ c iσ c iσc jσ c jσ σσ c iσ c jσ c jσ c iσ, leading to a mean-field interaction correction to the hopping elements. Work in this direction is currently in progress Tunneling through a single SC island A starting point for calculating the resistance of a system with many SC islands would be to consider first the results of a single island. We show here two approaches, corresponding to the two extreme cases of very small islands and very large islands. For the former, the conductance can be evaluated in a Drude-like approximation. For the latter one may use the Keldysh scheme (to be described in the following) for calculating the current. 1. Ultra-small SC islands in a disordered metal The idea that SC correlations, when limited to small regions in space (i.e. SC islands), may cause a reduction in the conductance is somewhat counterintuitive. While a qualitative argument for such a mechanism was presented in Sec. 2, in this section we show that this mechanism may arise from treating ultra-small SC islands in a dirty metal, even when the charging energy of the islands is not taken into account. While this is motivated by the large magnetic-field limit of the SIT, it is an interesting question by itself, as such transport measurements of a 2D diffusive metal with small SC grains may be feasible. The first aim is to determine the single-particle properties near a SC island. We start with a Hamiltonian of a diffusive metal with a single SC island, H = d 2 rψ σ(r)( µ + V (r))ψ σ (r) + H (3.9) σ where V (r) is a random potential and H = a/2 a/2 a/2 dx dy( Ψ (x)ψ (x) H.c.) a/2, (3.1) is the reduced BCS Hamiltonian (so-called the Bogoliubov-de Gennes (BdG) Hamiltonian [16]), with the difference that the pairing occurs at a finite region in space, i.e. the SC island (taken to be at the origin). When a (or a L for a finite sample, not in the thermodynamic limit) we obtain the usual BdG Hamiltonian, in which only opposite wave-vectors are coupled via the pairing potential. Here we take the limit where a is smaller then any other length-scale in the system, specifically smaller then the Fermi wave-vector k F, i.e. k F a << 1. We expand H with a as a small parameter and obtain H a 2 Ψ ()Ψ () + H.c ! a4 2 ( Ψ (x)ψ (x) + H.c. ) (x,y)=(,), (3.11) and since 2 ψ (x)ψ (x) (x,y)=(,) kf 2 Ψ ()Ψ (), The second term in Eq.(3.11) is smaller than the first term by a factor of (k f a) 2 and can be neglected, and we are left with the first term in Eq.(3.11) which can be treated as a local perturbation. Notice that we do not assume to be small, as it might not be. Since the Anderson condition [19] for a small island to be superconducting is > δ, where δ is the island mean level spacing, and since δ a 2,

20 18 in order for these islands to still be SC might be fairly large. Single particle properties We can now solve the Dyson equation for the single particle Green s function in real space, given by G(x, x, ω) = g(x, x, ω) + 2 a 4 g(x,, ω)g(,, ω)g(, x, ω), (3.12) depicted graphically in Fig. 14. = + = + FIG. 14: The Dyson equation for the single particle Green s function. Scattering into the island results in the creation of a freely propagating hole, finally annihilated again at the island. Scattering in and out of the SC island is depicted as a square in the second line. Using the free particle Green s function (with no disorder ) in two dimensions g(x, x, ω) = iπωj ( 2mω x x ), (3.13) where Ω in the density of states (DOS) in two dimensions and J is the zeroth order Bessel function of the first kind, the Dyson equation can be solved, yielding the result for the Green s function G(x, x, ω) = g(x, x π 3 Ω 3 2 a 4, ω) + i 1 + π 2 Ω 2 2 a 4 J ( 2mω x )J ( 2mω x ). (3.14) This leads to the local DOS ρ(x) 1 ( π IG(x, x, ω) = Ω 1 π2 Ω 2 2 a π 2 Ω 2 2 a 4 J ( ) 2mω x ) 2. (3.15) Since Ω 2 a 4 δ 2 I, where δ 2 I is the island level spacing, then α = π 2 Ω 2 2 a 4 ( ) 2, δ which may be large, and hence α/(1 + α) may be close to 1. We can therefore write (3.15) in the following form ρ(x) = Ω[1 where L is the system linear size and δ is the level spacing at the Fermi energy. δ α ( ) 2 ω 1 + α J x ], (3.16) δ L This shows that ρ(x) changes on the length scale of x ω L. Since ω ɛ F, the condition that a is the smallest length scale in the system determines the validity of the above argument, a/l δ/ɛ F. The single particle DOS is plotted in Fig. 15 for the parameters α =.5 and ω/δ = N 1 1 (estimated for a:ino samples of [5]) and is in qualitative agreement with the experimentally observed DOS near a SC island [17]. The DOS shows an opening of a minigap at the position of the island and at small energies, followed by Friedel oscillations, eventually converging to the normal two-dimensional

21 19 DOS. These parameters yield an estimate for island size, a < 1nm, in which this approximation is valid. We note that although the BdG Hamiltonian was expanded in the point-island limit, the Green s function was calculated to infinite order. This is an important point since, as was previously marked, although a is small, the factor a 2 might not be. 1 ρ(x)/ω.9.8 α.5 ε F /δ=ν x/l FIG. 15: Single particle DOS for a sample with N 1 1 electrons and α =.5 (evaluated at the Fermi energy). The DOS shows a minigap at the position of the SC island. The next step is to calculate the conductance. Conductance For this aim we calculate the single particle Green s function for a randomly distributed realization of N superconducting point-islands. In the approximation where successive scattering events only occur from the same island (corresponding to the low island concentration limit, see appendix B) the generalization of the Green s function given in Eq.(3.14) for many islands leads to G(x, x ) = g(x, x γ ) i g( x R i, )g(, R i x ) (3.17) 1 + πωγ where γ = πω 2 a 4, R i is the location of the ith point island and the explicit dependence on ω was omitted. In momentum space the Green s function is G kk and averaging over island positions gives i γ = g k δ kk i e i(k k )R i g k g k, (3.18) 1 + πωγ i nγ G kk = g k (1 i 1 + πωγ g k)δ kk, (3.19) where n is the concentration of SC islands. This can be used to calculate the conductivity in linear response (see appendix B) in the Drude approxiamtion, and we obtain σ(ω) = σ (ω) + 2n ee 2 Plotted in Fig. 16 (for arbitrary parameter values). m nγτ 2 ω 2 τ πωγ (ω 2 τ 2 + 1) 2, (3.2) As seen in the Fig. 16, the correction leads to an inhomogeneous frequency dependence of the conductivity. Taking

22 2 1.1 σ(ω) σ ω FIG. 16: The correction to the Drude frequency-dependent conductivity arising from scattering from SC islands. The Drude DC conductivity (without SC islands) is marked with a dashed line. the limit ω gives the correction to the Drude conductivity σ SCI = σ 2nγτ 1 + πωγ. (3.21) This correction has a negative sign and hence reduces the conductance. We can understand this reduction in the following way. As an electron is scattered into the SC island, a Cooper-pair is formed and a hole is generated. This hole travels freely in the system until it scatters back to the island, the pair is broken, the hole is annihilated and the electron is again free. During that time, the electron (bound in the pair) is unavailable for transport and hence the carrier density (or the DOS) is reduced. We note here that in contrast to the case of normal impurities where the diffuson term does not contribute significantly to the conductance, in the case of superconducting impurities the diffuson term diverges (see appendix B), which may lead to a dramatic effect on the conductance. This issue is a matter of further study. Another correction to the Drude approximation would be the inclusion of inter-island scattering events. Such processes will give rise to Josephson physics, and may have an influence on the conductance of the system. It is indeed one of the directions we intend to follow with this model. We conclude this section with a remark. While we indeed show that small SC islands may cause a decrease in the conductance, the applicability of this model to the experiments described in Sec. 1 is questionable. We have not included many important aspects of the system (e.g. magnetic field, Coulomb repulsion, inter-island coupling) which may change the above results. Even for granular systems this model is still lacking, for example in the absence of a normal-sc domain wall potential barrier. Nevertheless, this model may serve as a starting point for the investigation of more elaborated systems. Another direction one might take is including Coulomb interaction in the islands, and treating them perturbatively. 2. Tunneling current through a large SC island In the case of a large SC island, we employ the so-called tunneling method, in which we interpret the transport through the islands as resulting from tunneling from states in the normal parts of the sample to states in the SC