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Anomalous proximity effect in d-wave superconductors

The anomalous proximity effect between a d-wave superconductor and a surface layer with small electronic mean free path is studied theoretically in the framework of the Eilenberger equations. The angular and spatial structure of the pair potential and the quasiclassical propagators in the interface region is calculated selfconsistently. The variation of the spatially-resolved quasiparticle density of states from the bulk to the surface is studied. It is shown that the isotropic gapless superconducting state is induced in the disordered layer.


I. INTRODUCTION
There is continuing experimental evidence that the behavior of high temperature superconductors (HTS) can be understood in terms of the d-wave pairing scenario, rather than in the conventional s-wave picture. On the other hand it is well known that the d-wave order parameter is strongly reduced by electron scattering at impurities and therefore can be formed only in clean materials. However, the condition of clean limit is not fulfilled in the vicinity of the grain boundaries or other HTS interfaces even if the material is clean in the bulk. There are at least two reasons for that. The first one is that quasiparticle reflection from realistic interfaces is diffusive, thus providing the isotropisation in the momentum space and the suppression of the d-wave component of the order parameter. The second one is the contamination of the material near interfaces as a result of fabrication process or electromigration in large scale application devices. As a result the formation of a thin disordered layer near HTS surfaces and interfaces is highly probable. An important question is whether or not superconducting correlations vanish in such a layer in the limit of small mean free path and what is the orbital structure of the superconducting state in the interface region.
Surface peculiarities in d-wave superconductors were extensively discussed in the framework of the theoretical models based on specular quasiparticle reflection from clean interfaces [1][2][3][4][5][6][7] . Zero-and finite-bias anomalies predicted in these papers were recently observed experimentally in Refs. 8,9 . In this paper we focus on the problem of the anomalous proximity effect between a d-wave superconductor and a thin disordered layer in the limit of strong disorder. It is shown that an isotropic order parameter is nucleated in such a layer even in the absence of the subdominant pairing interaction in the s-wave channel. The spatially-resolved quasiparticle density of states is calculated. It is shown that zero-and finite-energy peaks are present in the surface density of states in the d-wave region. Zero-energy peaks are fully smeared out in the disordered layer, which is in a peculiar gapless superconducting state.

II. PROXIMITY EFFECT AT THE INTERFACE
Two approaches to the study of surface roughness effects in unconventional superconductors were used previously. In the first one it is assumed that the interface consists of facets with random orientations compared to the crystallographic axes of the material 6 . According to the second approach, both sides of an ideal interface are coated by a so-called Ovchinnikov's thin disordered layer 5,10,11 . In the latter case the degree of disorder (or interface roughness) is measured by the ratio of the layer thickness d to the quasiparticle mean free path in the layer ℓ. Up to now both approaches were used to study the smearing of Andreev surface bound states by weak disorder. Here we will concentrate on the regime of strong disorder.
We consider the surface or a weakly transparent barrier in a d-wave superconductor oriented normal to the crystallographic ab plane. We assume that the crossover from the clean to the dirty limit takes place in a thin layer near the surface with mean free path ℓ and thickness d < √ ξ 0 ℓ, where ξ 0 is the coherence length of the bulk material.
To study the proximity effect at the interface we use the quasiclassical Eilenberger equations 12 with impurity scattering taken in the Born limit. For our purpose it is convenient to rewrite these equations in terms of functions Φ + = (f (r, θ) + f (r, θ + π))/2 and Φ − = f (r, θ) − f (r, θ + π) 2v cos θ dg dx ∆ ln Here g and f are respectively the normal and the anomalous quasiclassical propagators, ω = πT (2n + 1) are the Matsubara frequencies, v is the Fermi velocity, x is the coordinate in the direction of the surface normal, θ is the angle between the surface normal and quasiparticle trajectory, τ = ℓ/v and ... = (1/2π) 2π 0 (...)dθ. We assume that the Fermi surface has a cylindrical shape.
For pure d-wave interaction the coupling constant may be written in the form 13 where α is the misorientation angle between the crystallographic a axis and the surface normal. Then according to the self-consistency equation (4) angular and spatial dependencies of the pair potential are factorized ∆ = √ 2∆(x) cos(2(θ − α)), i.e. ∆ has pure d-wave angular structure everywhere in the interface region.
Far from the interface the bulk anomalous propagator also has the d-wave symmetry At the same time, as will be shown below, the angular structure of the propagator Φ + (x, θ) is essentially modified near the interface and an s-wave component of To proceed further we have to supplement equations (1)-(4) with the appropriate boundary conditions for the function Φ + (x) and its derivative dΦ + (x)/dx at the interface between the clean and the disordered regions of a d-wave superconductor (at x = 0). These conditions can be derived by integration of the Eilenberger equations (1)-(2) in a small region near the interface. In accordance with Ref. 14 , the first boundary condition is the continuity of Φ − at the interface and can be written in the form This condition manifests the current conservation across the interface. The second boundary condition depends on the backscattering properties of the interface. To account for such a backscattering we introduce a strongly disordered thin layer located near the interface at −δ ≤ x ≤ 0, which is characterized by the mean free path ℓ δ , where ℓ δ , δ ≪ d, ℓ. Assuming that all the boundaries are transparent, integrating Eq.(2) in the interval −δ ≤ x ≤ 0 and taking the limit δ → 0 we arrive at where D = 2δ/ℓ δ .
For D = 0 Eq. (7) is a direct consequence of the continuity of the Eilenberger functions along quasiclassical trajectories which is valid for transparent SNboundaries 14 . With increase of D the probability of quasiparticle penetration into the N -layer decreases as . It means that most quasiparticles are diffusively reflected back to the d-wave region at a length scale smaller than ℓ.
In the following we will consider the case of strong disorder ℓ ≪ d. Then in accordance with Refs. 14,15 it follows from the symmetry of the problem that the boundary condition at the totally reflecting free interface ( Since ℓ ≪ d and d < √ ξ 0 ℓ, the dirty limit condition ℓ ≪ ξ 0 is fulfilled in the disordered layer. It is straightforward to show from Eqs.(1), (4) that in this case the pair potential ∆ in the disordered layer vanishes due to impurity pair-breaking. Then the angle-averaged functions Φ + and g at −d ≤ x ≪ −ℓ obey the dirty limit Usadel equations 16 in the form which formally coincides with the one valued for a normal metal with T cn = 0. Since the scale of variation of Φ + and g in this regime is of the order of the dirty limit coherence length √ ξ 0 ℓ, the functions Φ + and g in a thin disordered layer d < √ ξ 0 ℓ are spatially-independent.
As a result the Eilenberger equations in the region −d ≤ x ≤ 0 are essentially simplified and have the solution where k = 1/ℓ |cos θ| .
Making use of the boundary conditions Eqs.(6), (7) at x = 0 and of Eqs.(9), (10), one can further reduce the problem to the solution of the Eilenberger equations (1)-(4) in the clean d-wave superconductor (x ≥ 0) with the condition (5) in the bulk (x ≫ ξ 0 ) and the following boundary condition at x = 0 Since the boundary condition (13) has a closed form. In the following we will limit ourselves to the situation when the disordered layer produces the most strong effect, namely when D = 0. The isotropic Usadel function Φ + (0) has to be determined selfconsistently by an iteration procedure.

III. RESULTS AND DISCUSSION
In the limit κ ≪ ξ 0 the pair potential ∆(x) is a smooth function of x at distances of the order of κ. Then the boundary value problem Eqs.(11)-(13) is essentially simplified and has the asymptotic solution where , Here prime denotes the derivative with respect to the coordinate x. As is seen from the solution (14), the anomalous Green's function Φ + at x = 0 is proportional to the sum of three terms with different angular symmetry. Two of them are the isotropic part Φ + (0) and the term with the d-wave symmetry Ψ(0), while the third term is proportional to the product ∆ ′ (0) |cos θ| cos(2(θ − α)). The latter term is a source for nucleation of a nonzero s-wave component of Φ + .
Note that according to Eq.(14) the solution Φ + at x = 0 would have pure d-wave symmetry in the approximation of a spatially independent pair potential ∆. The reason is that in this case the characteristic length κ(θ) of spatial variation of Φ + cancels out from the solution Eq.(14) for Φ + (x), since κ(θ) is present both in Eq. (11) and in the boundary condition Eq. (13). At the same time, in the selfconsistent approach a nonzero angleaveraged value Φ + (x) appears at the interface, since the above cancellation is incomplete in the presence of a pair potential gradient.
As suggested in 6 It is worth mentioning that since a subdominant swave pairing interaction is not included in the present model, the pair potential ∆ still has pure d-wave angular structure everywhere in the d-wave region. For the same reason there is no source for the phase shift between sand d-wave components of Φ + (x), thus the surface d + is state which breaks time-reversal symmetry should not occur in the case considered.
In the general case of arbitrary κ values the problem was solved numerically. The isotropic function Φ + (0) and the spatially dependent pair potential ∆(x) were calculated by iterating the equations (11), (12) making use of the boundary condition (13) and the selfconsistency equation (4). The results of numerical calculations shown in Figs.1-3 confirm the considerations presented above. Fig.1 shows the angular dependence of Φ + (x) far from the boundary (x ≫ ξ 0 ), as well as at x = ξ 0 and x = 0, for two different orientations α of the a axis with respect to the interface normal. In both cases far from the interface the angular distribution is typical for a d-wave superconductor. At x = ξ 0 the positive lobe (horizontal) is suppressed stronger than the negative one (vertical), since the characteristic length κ(θ) in the direction perpendicular to the interface is small compared to κ(θ) in the direction parallel to the interface. Hence at x ≈ ξ 0 negative lobes of Φ + (x) practically reach the local value Ψ(x), while positive ones still do not. This difference leads to the negative sign of the s-component Φ + (x) (see Fig.2). In the vicinity of the interface (x ≤ 0.3ξ 0 ) the situation is just the opposite. In accordance with solution (14), due to the angular dependence of κ(θ) ∝ |cos θ| the negative lobes are suppressed stronger than the positive ones, the function Φ + (x) changes sign to positive and reaches its maximum at x = 0.
Note that for quasiparticle trajectories parallel to the surface, θ = π/2, it follows from Eq.(14) that Φ + (0, π/2) = Ψ(0, π/2), while lim θ→±π/2 Φ + (0, θ) = Ψ(0, π/2) + η. This discontinuity is the manifestation of the simple fact that quasiparticles which propagate exactly parallel to the interface have information about the disordered region only via the local value of the pair potential, while for all other directions the direct interaction between both regions takes place. However, this discontinuity at θ = π/2 does not contribute to the result of the angular averaging of Φ + in the boundary condition Eq.(13). Fig.3 shows the spatial variations of ∆(x) for different values of the angle α. As follows from Eq. (14), the function Φ + (0, θ) near the interface has a contribution proportional to |cos θ| cos(2(θ − α)). This immediately leads to the result that the amplitude of the s-component induced into the disordered layer scales with misorientation angle α as Φ + (0, α = 0) cos 2α. At α = π/4 the superconducting correlations are not induced into the disordered layer, i.e. Φ + (0) = 0. Further increase of α leads to a sigh change of the s-component.
As is seen from Fig.3, these qualitative considerations are in a good agreement with the results of exact numerical calculations. In particular, for the d xy case (α = π/4) it follows that Φ + (0) = 0. At the same time, it is worth mentioning that the pair potential at the interface, ∆(0, α = π/4), is nonzero, in contrast to the case of a specular reflecting boundary when ∆ at α = π/4 vanishes. In the case considered of diffusive surface scattering there is no symmetry requirement for the vanishing of Φ + (0, α = π/4).
In the whole temperature range the amplitude of the s-wave component Φ + induced into the disordered layer is an order of magnitude smaller compared to the amplitude of the order parameter in the bulk superconductor (see Fig.2). That means that g(0) is close to unity for all temperatures. Thus, taking into account that g(0) is independent of the Matsubara frequencies and that ξΦ ′ + (0) ≈ Φ + (0), we obtain from the boundary condition Eq.(13) that at low temperature Φ + (0) ∝ ω for ω ≤ ∆. As soon as ω exceeds the value of ∆, the function Φ + (0) behaves as Φ + (0) ∝ ω −2 . The density of states N (ε) = N (0)Re g(0, ε = iω) , where g = 1 − Φ + 2 and N (0) is the normal state density of states. Therefore it follows from the property Φ + ∝ ω at small ω that the density of states N (ε = 0)/N 0 = 1, i.e. there is a gapless superconducting state in the disordered layer.
To demonstrate this behavior explicitly we have calculated the density of states by numerical integration of Eqs.(11)-(13) on the real energy axis making the substitution ω = −iε in these equations. The results of calcula-tions of the normalized low-temperature density of states N (ε)/N (0) = 1 at T = 0.1T c and α = 0 are presented in Fig.4. To take into account an inelastic scattering we have introduced the complex energy ε = ε + iγ with γ = 0.05T c .
As is seen from Fig.4, the density of states in the disordered layer is gapless and has a rather weak and broad peak at an energy below the maximum value of the bulk pair potential ∆ bulk . This peak is a signature of the Andreev bound states at finite energies. Note that there is no midgap (zero-energy) peak in the density of states in the disordered layer since there is no sign change of the order parameter in this region. At the same time the density of states at the surface of the d-wave region (at x = 0) exhibits a sharp zero-energy peak. In accordance with the known results for specular interfaces 2,5,6 this peak is due to the midgap states in the surface of the d-wave region .
An important difference between specular and rough interfaces is that in the former case the midgap states occur only at nonzero values of the misorientation angle α = 0, whereas in the latter case these states occur even for α = 0. This result has a simple physical interpretation. Due to the presence of the disordered surface layer incident and reflected quasiparticle trajectories are uncorrelated. Therefore for any α there is a finite probability for an incident quasiparticle to experience a sign reversal of the pair potential upon reflection. Then the averaging over incoming trajectories yields the midgap states and as a result the nonvanishing zero-energy peak in the surface density of states appears.

IV. CONCLUSIONS
In conclusion, the proximity effect between a d-wave superconductor and a disordered surface layer is studied theoretically in the regime of strong disorder. The boundary conditions for the Eilenberger equations are derived at the interface between the clean and the disordered regions. It is shown that superconducting correlations in the disordered layer do not vanish in the limit of small electronic mean free path and the isotropic superconducting state is induced in such a layer. The crossover from this state to the d-wave pairing state in the bulk is studied by solving the Eilenberger equations.
The quasiparticle density of states in the disordered layer is gapless and shows a broad peak at finite energies, while the density of states at the surface of the d-wave region exhibits a zero-bias peak due to sign reversal of the order parameter. This peak is fully smeared out in the disordered layer.
The above phenomena have important consequences for the description of rough HTS interfaces and Josephson junctions, in particular for SNS junctions with a normal metal interlayer. These effects will be discussed elsewhere.