Oct 21, 2005 - superconductor (SNS) junction below the Thouless gap. ... In a diffusive SNS structure with transparent SN in- .... that the operator H...

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arXiv:cond-mat/0510570v1 [cond-mat.mes-hall] 21 Oct 2005

Alessandro Silva Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway 08854, New Jersey, USA. (Dated: October 5, 2018) We study the tails of the density of states (DOS) in a diffusive superconductor-normal metalsuperconductor (SNS) junction below the Thouless gap. We show that long-wave fluctuations of the concentration of impurities in the normal layer lead to the formation of subgap quasiparticle states, and calculate the associated subgap DOS in all effective dimensionalities. We compare the resulting tails with those arising from mesoscopic gap fluctuations, and determine the dimensionless parameters controlling which contribution dominates the subgap DOS. We observe that the two contributions are formally related to each other by a dimensional reduction. PACS numbers: 74.45.+c; 74.40.+k; 74.81.-g.

I.

INTRODUCTION.

The properties of hybrid superconductor-normal metal structures (SN) continue to attract considerable attention both experimentally1 and theoretically2,3,4,5,6,7,8,9 , though the fundamental process governing the physics of such systems, Andreev reflection10 , has been discovered long ago. In fact, while it is well known that generically the proximity to a superconductor leads to a modification of the density of states in the normal metal, the nature and extent of this effect depends on the details the hybrid structure. In particular, it was recently pointed out2 that when a closed mesoscopic metallic region is contacted on one side to a superconductor, the resulting DOS turns out to depend on its shape. If integrable, the DOS is finite everywhere but at the Fermi level, where it vanishes as a power law. On the contrary, in a generic chaotic metallic region one expects the opening of a gap around the Fermi level, the Thouless gap3 . In analogy with the considerations above, a diffusive metallic region sandwiched between two bulk superconducting electrodes has been predicted to have a gapped density of states, the gap being at energies comparable to to the Thouless energy ET h = D/L2z , where D is the diffusion constant and Lz the width of the normal layer4,5,6,7 [see Fig.1]. In a diffusive SNS structure with transparent SN interfaces, the density of states in the normal part, averaged over its thickness, and at energies Epright above the (E − Eg )/∆30 , gap edge Eg ≃ 3.12ET h , is ν ∝ 1/πV 2 1/3 where ∆0 = (Eg δ ) , δ = 1/(ν0 V ), and V = Lx Ly Lz is the volume of the normal region. This dependence is reminiscent of the density of states at the edge of a Wigner semicircle in Random Matrix Theory [RMT], ∆0 being the effective level spacing right above the gap edge. Using this analogy, Vavilov et al.8 realized that the disorder averaged DOS should not display a real gap, but have exponentially small tails below the gap edge, analogous to the Tracy-Widom tails11 in RMT. A rigorous study in terms of a Supersymmetric Sigma Model description of the SNS structure has shown that this is in-

deed the case9 . However, in analogy to the theory of Lifshits tails12 in disordered conductors, the nature of the resulting subgap quasiparticle states depends additionally on the effective dimensionality d, determined by comparing the interface length scales Lx , Ly , with the typical length scale of a subgap quasiparticle state, L⊥ . In particular, if Lx ≫ L⊥ > Ly or Lx , Ly ≫ L⊥ the subgap quasiparticle states are localized either in the x direction or in the x − y plane along the interface, respectively. Correspondingly, the asymptotic tails of the DOS deviate from the universal RMT result, applicable only in the zero dimensional case [Lx , Ly < L⊥ ]. The analogy with RMT applies, within the appropriate symmetry class, to other physical situations, such as diffusive superconductors containing magnetic impurities8,13,14 , and superconductors with inhomogeneous coupling constants15 . In both cases, at mean field level the density of states has a square root singularity close to the gap edge16,17 . Correspondingly, accounting for mesoscopic RM-like fluctuation, the disorder averaged density of states has tails below the gap edge, with an asymptotics similar to the one calculated in Ref.[9] for SNS structures. On the other hand, in the case of diffusive superconductors containing magnetic impurities, it was shown18,19 that, in addition to mesoscopic fluctuations , subgap quasiparticle states can form as a result of classical fluctuations , i.e. long-wave fluctuations of the concentration of magnetic impurities associated to their Poissonian statistics. Similarly, also in superconductors with inhomogeneous coupling constant long-wave fluctuations of the coarse grained gap lead to the appearance of subgap quasiparticle states, and consequently to tails of the DOS17 . Interestingly, in both cases the tails originating from mesoscopic fluctuations and from classical ones are formally related by a dimensional reduction18 . In this paper, we close this set of analogies, studying the contribution to the subgap tails of the DOS in a diffusive SNS junction arising from long-wave fluctuations of the concentration of impurities in the normal layer. Combining the results of this analysis with those obtained by Ostrovsky, Skvortsov, and Feigel’man9 , who considered

2 the subgap tails originating from mesoscopic fluctuations, we provide a consistent picture of the physics of the subgap states. In particular, a quantitative comparison of the two contribution shows that mesoscopic fluctuations dominate in long and dirty junctions, while classical fluctuations dominate in wider and/or cleaner ones. In analogy with diffusive superconductors with magnetic impurities, and superconductors with inhomogeneous coupling constants, also in the present case the two contributions to the subgap tails, arising from mesoscopic and classical fluctuations, are related by a dimensional reduction. The rest of the paper is organized as follows: in Sec.II we present the details of the analysis of the subgap DOS arising from fluctuations of the concentration of impurities nimp in an SNS junction. In Sec.III, we compare the two contributions to the subgap DOS associated to mesoscopic and classical fluctuations. In Sec.IV, we present our conclusions.

II.

SUBGAP DOS ASSOCIATED TO FLUCTUATIONS OF nimp .

Let us start considering a diffusive metallic layer in between two superconducting bulk electrodes, a geometry represented schematically in Fig.1. Assuming kF l >> 1, where l is the mean free path, this system can be described in terms of the quasiclassical approximation. In particular, at mean field level [ i.e., neglecting both mesoscopic and classical fluctuations ], neglecting electronelectron interaction, and assuming the thickness of the metallic layer Lz >> l, one can describe the SNS structure by the Usadel equation20,21 D 2 ∇ θ + i E sin[θ] = 0, 2

(1)

where D = vF2 τ /3 is the diffusion constant, E is the energy measured from the Fermi level, assumed to be | E |≪ ∆, where ∆ is the gap in the bulk electrodes. The field θ is related to the quasiclassical Green’s functions and the anomalous Green’s function by the relations g(r, E) = cos[θ(r, E)], f (r, E) = i sin[θ(r, E)]. In addition, assuming the interfaces to be perfectly transparent, the proximity to the two superconducting regions can be described by the boundary conditions θ(z = ±Lz /2) = π/2. It is convenient to measure all lengths in units Lz , and set θ = π/2 + iΨ. Therefore, Eq.(1) becomes ∇2 Ψ + 2

E cosh[Ψ] = 0, ET h

(2)

where ET h = D/L2z is the Thouless energy. The boundary conditions for the field Ψ are simply Ψ(z = ±1/2) = 0. In terms of Ψ the DOS is ν = 2ν0 Im[sinh[Ψ]], where ν0 is the density of states of the normal metal at the Fermi

Lx

S

Ly

N

S

Lz FIG. 1: A schematic plot of an SNS junction: two bulk superconducting electrodes (S) connected to a diffusive metal (N) of thickness Lz . The interfaces have linear size Lx , Ly .

level. The DOS can be calculated looking for solutions of Eq.(2) uniform in the x − y plane4,5,6,9 . In particular, for E < Eg ≡ C2 ET h [C2 ≃ 3.122] all solutions of Eq.(2) are real, implying ν = 0. Therefore, one identifies Eg with the proximity induced gap within the normal metal layer. The mean field DOS right above Eg averaged over the z direction is found to be s E − Eg . (3) ν ≃ 3.72 ν0 Eg Let us proceed analyzing the tails of the DOS at energies E < Eg arising from fluctuations of the concentration of impurities, i.e. long-wave inhomogeneities in the x − y plane of 1/τ . We first consider an SNS structure such that the linear size of the SN interfaces is much larger than the thickness of the metallic layer [Lx , Ly ≫ Lz ]. In the framework of the Usadel description of the metallic layer [Eq.(2)] one can account for long-wave transversal fluctuations of the concentration of impurities by promoting ET h , or equivalently Eg = C2 ET h , to be a position dependent random variable, characterized by the statistics Eg (x) = Eg + δEg (x), hδEg (x)i = 0, hδEg (x)δEg (x′ )i =

Eg2 δ(x − x′ ), nd Ldz

(4) (5) (6)

where d is the effective dimensionality of the system, and nd the effective concentration of impurities. As shown below, d is determined by comparing the linear sizes of the interface Lx , Ly to the linear scale of the subgap states L⊥ ≃ Lz /((Eg − E)/Eg )1/4 . If Lx , Ly ≫ L⊥ the system is effectively two dimensional, and n2 = nimp Lz . On the other hand, if Lx < L⊥ ≪ Ly [or Ly < L⊥ ≪ Lx ], the system is effectively one dimensional, and n1 = nimp Lz Lx . Accounting for these fluctuations, the Usadel equation Eq.(2) becomes ∂z2 Ψ + ∇2x Ψ + 2C2 where δǫg = δEg /Eg .

E (1 − δǫg (x)) cosh[Ψ] = 0, Eg

(7)

3 Our purpose is to calculate the DOS averaged over fluctuations of δEg at energies E < Eg . For this sake, let us introduce δE = Eg − E, and δΨ(z, x) = Ψ(z, x) − Ψ0 (z), where Ψ0 is the solution of Eq.(2) at E = Eg . Expanding Eq.(7) and keeping the lowest order nonlinearity in δΨ one obtains (∂z2 + f0 (z))δΨ + ∇2x δΨ +

g0 (z) 2 δΨ = g0 (z)(δǫ − δǫg ),(8) 2

where δǫ = δE/Eg , g0 (z) = 2C2 cosh[Ψ0 (z)], and f0 (z) = 2C2 sinh[Ψ0 (z)]. In order to simplify further Eq.(8), it is useful to notice that the operator H = −∂z2 − f0 (z), diagonalized with zero boundary conditions at ±1/2, admits an eigenstate Φ0 with zero eigenvalue. Physically, Φ0 determines the shape of the mean field z-dependent DOS obtained from Eq.(2). Therefore, it is natural to set p (9) δΨ(z, x) ≃ A1 /A2 χ(x) Φ0 (z), R R with A1 = dz g0 Φ0 ≃ 7.18, and A2 = dz g20 Φ30 ≃ 2.74. Substituting Eq.(9) in Eq.(8), and projecting the resulting equation on Φ0 , one obtains ∇2 χ + χ2 = δǫ − δǫg (x)

(10)

(11)

with η ≡ (A1 A2 )1/4 /(nd Ldz ). Let us now split χ = −u + iv, and obtain the system −∇2 u + u2 − v 2 = δǫ − δǫg , 1 − ∇2 v + u v = 0. 2

(12) (13)

Interestingly, this set of equations is analogous to the equations obtained by Larkin and Ovchinikov in the context of the study of gap smearing in inhomogeneous superconductors17 , and to the equations obtained by the author and Ioffe in the context of the study of subgap tails in diffusive superconductors containing magnetic impurities18 . Let us now proceed with the calculation of the DOS. In the present notation, the DOS averaged over the thickness of the normal layer is given by ν(x, δǫ | δǫg (x)) ≃ 3.72 v(x, δǫ | δǫg (x)). ν0

where we used Eq.(12) to express δǫg in terms of u, v, and, with exponential accuracy, neglected the term v 2 in the action. In order to find the optimal fluctuation one has to find a nontrivial saddle point u0 of S, tending asymptotically √ to the solution of the homogeneous problem [u0 → δǫ], and subject to the constraint of having nontrivial solutions for v of Eq.(13). Since the normal metal layer is diffusive, and momentum scattering isotropic, it is natural to assume the optimal fluctuation to be spherically symmetric. The EulerLagrange equation associated to S is 1 (− ∆(d) + u) (∆(d) u − u2 + δǫ) = 0 2

(14)

We are interested in calculating the average density of states hνi/ν0 ≃ 3.72hvi at energies below the Thouless gap [δǫ > 0]. In this parameter range, the corresponding functional integral R R 2 D[δǫg ]v(x, δǫ | δǫg (x))e−1/(2η) dx(δǫg (x)) R ,(15) hvi ≃ R 2 D[δǫg ]e−1/(2η) dx(δǫg (x))

(17)

where ∆(d) ≡ ∂r2 +

where we rescaled the length by (A1 A2 )−1/4 , and hδǫg (x)δǫg (x′ )i = η δ(x − x′ ),

receives its most important contributions by exponentially rare instanton configurations of δǫg such that, at specific locations along the interfaces of the junction, δǫg (x) ≥ δǫ. The remaining task is to select among all these fluctuations the one that dominates the functional integral Eq.(15), i.e. the optimal fluctuation . The action associated to a configuration of δǫg is Z Z 1 2 dx(δǫg ) ≃ dx(∇2 u − u2 + δǫ)2 , (16) S= 2η

d−1 ∂r , r

(18)

is the radial part of the Laplacian in spherical coordinates. An obvious solution to Eq.(17) is obtained setting ∆(d) u − u2 + δǫ = 0.

(19)

This equation is equivalent to the homogeneous Usadel equation with uniform Eg , i.e. Eq.(10) with δǫg = 0. Though this equation has definitely nontrivial instanton solutions for u with the appropriate asymptotics, it is possible to show that the constraint of Eq.(13) is satisfied only by v = 0. This is physically obvious since Eq.(19) describes a uniform system where all long-wave fluctuations of 1/τ have been suppressed, and thus, within the present approximation scheme, the subgap DOS must vanish. However, it should be pointed out that, accounting for mesoscopic fluctuations, the instanton solutions of Eq.(19) describe the optimal fluctuation associated to mesoscopic gap fluctuations, as shown in Ref.[9]. Let us now look for the nontrivial saddle point. Equation (17) is equivalent to the system 1 (− ∆(d) + u)h = 0, 2 ∆(d) u − u2 + δǫ = h.

(20) (21)

which can be reduced to a single second order instanton equation setting h = (2∂r u)/r. With this substitution, Eq.(20) becomes the derivative of Eq.(21), which now reads ∆(d−2) u − u2 + δǫ = 0.

(22)

4 Notice that this equation is, upon reduction of the dimensionality by 2, identical in form to the one associated to mesoscopic fluctuations, Eq.(19). As we will see later, this reduction of dimensionality relates in a similar way the dependence of the action associated to classical and mesoscopic fluctuations on δǫ. It is now straightforward to see that the instanton solution u0 of this equation with the appropriate asymptotics describes indeed the optimal fluctuation, the constraint of Eq.(13) being automatically satisfied in virtue of Eq.(20), with v0 ∝ (2∂r u0 )/r. Moreover, the corresponding optimal fluctuation of δǫg is δǫg = 2∂r u0 /r. It is clear that the√instanton solutions of Eq.(22) must have the form u0 = δǫυ(r/λ), with λ = 1/(δǫ)1/4 . The corresponding equation for υ(r) is ∂r2 υ + (d − 3)/r∂r υ − υ 2 +1 = 0. The instanton solution of this equation can be easily found numerically, and the corresponding action S calculated. The result is Sd = ad nd Ldz δǫ

8−d 4

(23)

where the constants ad are a1 ≃ 0.88, and a2 ≃ 7.74. Within our approximation scheme, the density of states is hνi ∝ W exp[−S], where W is a prefactor due to gaussian fluctuations around the instanton saddle point. The calculation of W can be performed using the standard technique due to Zittarz and Langer, and is similar to those reported in Ref.[18,17]. To leading order in the saddle point approximation, the final result is q d(10−d)−12 hνi 8 ≃ βd nd Ldz δǫ e−Sd , (24) ν0

where β1 ≈ 0.1 and β2 ≈ 0.5. The result in Eq.(24) relies on a saddle point approximation, which is justified provided Sd ≫ 1. This translates into the condition 4 8−d 1 δǫ ≫ . (25) ad nd Ldz As mentioned before, the effective dimensionality, and therefore the asymptotic density of states, is determined by comparing the linear size of the optimal fluctuation, in dimensionfull units L⊥ ≃ Lz λ = Lz /δǫ1/4 , to the linear dimensions of the interfaces Lx , Ly . If Lx , Ly ≫ L⊥ the asymptotics is effectively two dimensional [d = 2], while for Ly ≫ L⊥ , Lx ≪ L⊥ the asymptotic DOS is effectively one dimensional [d = 1]. Since L⊥ increases as the energy gets closer to the average gap edge, it is clear that in any finite size system the applicable asymptotics might exhibit various crossovers, 2d → 1d → 0d, as δǫ → 0. In particular, the tails are zero dimensional when Lx , Ly < L⊥ , in which case the asymptotic form of the DOS is obtained by calculating the integral Z δǫ2 d(δǫg ) p hνi − 2ηg √ ≃ 3.72 δǫg − δǫ e 0 ν0 2πη0 1 (26) e−S0 , ≈ δǫ3/2

where η0 = 1/(nimp V ) [V 1/(2η0 )δǫ2 . III.

= Lx Ly Lz ] and S0 =

MESOSCOPIC VS. CLASSICAL FLUCTUATIONS.

In the previous section we have discussed the asymptotic density of states below the Thouless gap originating from classical fluctuations, i.e. inhomogeneities in the concentration of impurities or equivalently in 1/τ . As discussed in the introduction, this mechanism to generate subgap states is complementary to mesoscopic fluctuations of the gap edge. The tails associated to mesoscopic gap fluctuations have been calculated by Ostrovsky, Feigel’man and Skvortsov in Ref.[9]. To exponential accuracy, the subgap DOS associated to mesoscopic fluctuations is hνi/ν0 ∝ exp[−S˜d ], where 6−d S˜d ≃ a ˜d Gd (δǫ) 2 ,

(27)

where a ˜d is a constant [˜ a0 ≃ 1.9, a ˜1 ≃ 4.7, and a ˜2 ≃ 10], and Gd is the effective dimensionless conductance Lx Ly , Lz = 4πν0 DLx , = 4πν0 DLz .

G0 = 4πν0 D

(28)

G1 G2

(29) (30)

The scale of the optimal fluctuation associated to mesoscopic fluctuations is also L⊥ ≃ Lz /(δǫ)1/4 . Therefore, the effective dimensionality d is to be determined according to the criteria presented in the previous section. Before discussing the comparison of mesoscopic and classical fluctuations, let us first explain the rationale behind the separation these two contributions. Though it is clear that the only physical fluctuations in a real sample are associated to fluctuations in the positions of impurities, these fluctuations can affect the DOS in two ways: i)- depress the Thouless gap edge by increasing locally the scattering rate [classical fluctuations], or ii)take advantage of interference effects in the quasiparticle wave functions to generate quasiparticle states that couple inefficiently to the superconducting banks [mesoscopic fluctuations]. It makes sense to think of two types of effects separately if the actions associated to them are very different in magnitude [S˜ ≫ S or vice versa]. Obviously, in the crossover region, where S ≈ S˜ the separation of these two mechanisms is meaningless, because the system can take advantage of both at the same time. With this caveat, let us proceed in the comparison of these two contributions, starting with the zero dimensional case. Since the dimensionless conductance is G0 ≈ Eg /δ, where δ ≈ 1/(ν0 V ) is the level spacing, then the d = 0 action associated to mesoscopic fluctuations can be written as 3/2 δE ˜ S0 ≈ , (31) ∆0

5 where ∆0 = (Eg δ 2 )1/3 , where δ = 1/(ν0 V ) is the level spacing in the metallic layer. Physically, ∆0 can be interpreted as being the effective level spacing right above the gap edge. Indeed, from Eq.(3) one sees that s δE 1 ν≈ . (32) πV ∆30

to a zero dimensional action calculated within the typical volume of the optimal fluctuation. The latter is V⊥ = Lx L⊥ Lz for d = 1, and V⊥ = L2⊥ Lz in d = 2. For example, for d = 1 one can write

Therefore, the result of Eq.(31) indicates that tails originating from mesoscopic fluctuations of the gap edge are universal [in d = 0], in accordance to the conjecture formulated in Ref.[8] on the basis of Random Matrix Theory. In turn, in the zero dimensional case the action associated to classical fluctuations is 2 δE , (33) S0 ≈ δE0 p where δE0 = Eg / nimp V is the scale of typical fluctuations of the gap edge associated to fluctuations of the concentration of impurities. The dimensionless parameter controlling which which mechanism dominates is therefore

where δEef f

γ0 =

∆0 . δE0

(34)

Clearly, for γ0 ≫ 1 mesoscopic fluctuations dominate the subgap tails, while for γ0 ≪ 1 classical fluctuations give the largest contribution to the subgap DOS24 . Let us now write γ0 in terms of elementary length scales, one can estimate γ0

1 (Lz /l)7/6 1 p ≈ kF l kF2 σ (Lx Ly /l2 )1/6 ≈

1 (Lz /l)7/6 , kF l (Lx Ly /l2 )1/6

S1 ≈ nimp Lx L⊥ Lz (δǫ)2 2 δE ≈ , δEef f p = Eg / nimp V⊥ . Similarly, 2 δE , S˜1 ≈ ∆ef f

(36)

(37)

2 1/3 where ∆ef f = (Eg δef , δef f = 1/(ν0 V⊥ ) being the f) level spacing in the volume of the optimal fluctuation. In analogy to the zero dimensional case, one is therefore led to conclude that also in for one dimensional tails long and dirty junctions are dominated by mesoscopic fluctuations, while wider and/or cleaner junctions favor classical ones. This qualitative statement is indeed correct, but the proof is complicated by the energy dependence L⊥ . The appropriate way to proceed for d = 1, 2 is to write the actions associated to classical and mesoscopic fluctuations in compact form as 8−d Eg − E 4 S = , (38) δEd 6−d 4 E − E g (39) S˜ = ∆d

where δEd = Eg /(ad nd Ldz )4/(8−d) , and ∆d = Eg /(˜ ad Gd )4/(6−d) . Therefore, the dimensionless parameter that determines which contributions dominates the subgap DOS is (35)

where we used the fact that the scattering cross section of a single impurity σ is typically of the same order of λ2F . Within the assumptions of the theory, γ0 is the ratio of two large numbers, and therefore its precise value depends on the system parameters. However, from Eq.(35) we see that making the junction longer and longer, i.e. increasing Lz , tends to favor mesoscopic fluctuations. Intuitively, this is due to the fact that as Lz increases, the dimensionless conductance of the junction diminishes while the average number of impurities increases, therefore suppressing the associated fluctuations of the gap edge. At the same time, increasing the area of the junction, or making them cleaner, reverses the situation. In summary, mesoscopic fluctuations are favored in long and dirty junctions, while classical fluctuations are favored in wider and/or cleaner ones. Since in higher dimensionalities the linear scale of the optimal fluctuation associated to the two mechanism is identical [L⊥ = Lz /(δǫ)1/4 ], it is possible, and physically suggestive, to reduce the form of the actions in d = 1, 2

γd ≡

∆d . δEd

(40)

If γd ≫ 1, the subgap DOS is dominated by mesoscopic gap fluctuations, and the applicable result is Eq.(27). On the other hand, for γd ≪ 1 the DOS below the gap is determined by long-wave fluctuations of 1/τ [Eq.(24)]. Finally, estimating γd in terms of elementary length scales, one obtains 1 (Lz /l)8/7 , (kF l)16/35 (Lx /l)8/35 1 (Lz /l). ≈ (kF l)2/3

γ1 ≈

(41)

γ2

(42)

In analogy to Eq.(35), the fact that γd is proportional to a power of Lz /l implies that mesoscopic fluctuations are dominant in long junctions, while the inverse proportionality of γd on a power of kF l and of the linear size of the interface [in d = 0, 1] implies that wide interfaces and/or cleaner samples may favor the contribution arising from classical fluctuations.

6 IV.

CONCLUSIONS

In this paper, we discussed the effect of inhomogeneous fluctuations of the concentration of impurities, or equivalently of 1/τ , on the tails of the DOS below the Thouless gap in diffusive SNS junctions. We have shown that these classical fluctuations lead to the formation of subgap quasiparticle states and are complementary to mesoscopic fluctuations in determining the asymptotic DOS. Finding the dimensionless parameter that controls which mechanism gives the dominant contribution to the subgap tails, one finds that, qualitatively, mesoscopic fluctuations are favored in long and dirty junctions, while classical ones dominate in wider and/or cleaner ones. We have observed that, as for diffusive superconductors containing magnetic impurities, and for diffusive superconductors with an inhomogeneous coupling constant,

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the two contributions are formally related by a dimensional reduction by 2, both at the level of instanton equations determining the optimal fluctuation, and in the dependence of the DOS on the distance from the gap edge δǫ. As in other physical systems25 , it is natural to expect that supersymmetry is at the root of dimensional reduction also in this context. This fact could in principle be elucidated generalizing the Sigma Model describing mesoscopic fluctuations to include the physics associated to classical fluctuations.

V.

ACKNOWLEDGEMENTS.

I would like to thank E. Lebanon, A. Schiller, and especially L. B. Ioffe and M. M¨ uller for discussions. This work is supported by NSF grant DMR 0210575.

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