nd - m at . m es - h al l ] 2 4 Ju l 1 99 8 Nucleation of Superconducting pairing states at mesoscopic scales at zero temperature

We find that spin polarized disordered Fermi liquids are unstable to the nucleation of superconducting pairing states at mesoscopic scales even in magnetic fields substantially higher than the critical one. We study the probability of finding superconducting pairing states at mesoscopic scales in this limit. We find that the distribution function depends only on the film conductance. The typical length scale at which pairing takes place is universal, and decreases when the magnetic field is increased. The number density of these states determines the strength of the random exchange interactions between mesoscopic pairing states. Suggested PACS index category: 05.20-y, 82.20-w Typeset using REVTEX 1 The stability of a superconducting state with an order parameter ∆ can be characterized in term of the generalized curvature in the following way, O(r, r′) = δ 2E({∆(r)}) δ∆(r)δ∆(r) . (1) The curvature determines the stability to the spatial variation of the modulas of the order parameter and the stability to the creation of supercurrents. Here E({∆(r)}) is the energy of a configuration {∆(r)}. The curvature evaluated at the ground state should be positive defined, i.e. det O(r, r) > 0. For a dirty superconductor where √ D/∆ ≫ l ≫ p F , the curvature at ∆(r) = ∆ has mesoscopic fluctuations, like other physical quantities. Here D is the diffusion constant and l is the mean free path, pF is the Fermi momentum. However, in the absence of an external magnetic field, the mesoscopic fluctuations are small. Thus the curvature is almost positive defined, and the conventional homogenous superconducting state is stable. When the magnetic field is applied parallel to the disordered thin superconducting film, the suppression of superconductivity is mainly due to Zeeman splitting of electron spin energy levels. In the strong spin-orbital scattering limit τso∆0 ≪ 1, close to the critical field H c , the average spin polarization energy Ep is nearly equal to the average condensation energy Ec. Mesoscopic fluctuations of spin polarization energy due to mesoscopic fluctuations of spin sucseptibility become comparable or larger than the energy difference Ec−Ep. Therefore, the amplitude of the mesoscopic fluctuations of the curvature evaluated at ∆ = ∆(H) could be comparable with the average. Here ∆(H) is the order parameter at given H , ∆0 is the order parameter for H = 0 and 1/τso is the spin-orbit scattering rate. At H c −H/H0 c ∼ g, where g = eν0Dd/(2πh̄) is the dimensionless film conductance, ν0 the density of states and d the film thickness, the curvature, averaged over the area of size ξ(H) = √ D∆0/∆(H), has a random sign, and the system is unstable with respect to the creation of normal regions or spontaneous creation of supercurrents. One of the consequences of the mechanism discussed above is the instability of the spin polarized disordered Fermi liquid well above the critical magnetic field. This is because 2 though the average curvature of the normal metal state (O(r, r) evaluated at ∆ = 0)is positive defined, i.e. Ep > Ec, its mesoscopic fluctuations have random signs because of the mesoscopic fluctuations of the spin polarization energy. In the regions where the spin polarization energy cost to form superconducting pairing state is much lower than the average energy cost, the fluctuations of the curvature are of large negative value comparable to its positive average such that the normal metal with ∆ = 0 becomes unstable. As a result, above the critical field H c , the superconducting pairing correlations are established at mesoscopic scales in the different regions in the normal metal and couple with each other via exchange interactions of random signs. In this paper, we study the probability to find regions where the superconducting pairing states are formed at mesoscopic scales at H > H c . At high magnetic fields in the strong spin-orbit scattering limit, the statistics of these pairing states can be studied with the help of the generalized Landau-Ginsburg equation,