Erratum: Superheating fields of superconductors: Asymptotic analysis and numerical results [Phys. Rev. B53, 5650 (1996)]

Effect of impurities on the superheating field of type-II superconductors

“…The effect of finite κ increases H s (T ) since the condition v s (0) = v c is no longer sufficient to cause the instability of the Meissner state in the surface layer of thickness ξ where the superfluid velocities v s (x) ≃ (1 − x/λ)v s (0) decreases below v c because of the London screening. The GL calculations [7][8][9]11 have shown that the finite-κ effects increase H s (κ) ≃ 1 + 0.7κ −1/2 H s (∞), giving a correction ≃ 7 − 16% as compared to H s (∞) at κ → ∞ for κ = 20 − 100. These effects also make H s (κ) dependent on α since κ(α) increases as α increases approaching κ ∼ ακ 0 in the dirty limit.…”

“…To study the stability of the Meissner state with respect to small perturbations, we consider a planar type-II superconductor occupying the region x > 0 with magnetic field H applied along the z-axis. For a large-κ superconductor, the previous calculations based on the GL [7][8][9]11 and Eilenberger 16 theories have shown that the instability at H = H s is driven by coupled fluctuations of δΨ(x, y) and δJ(x, y) that rapidly oscillate on the scale ∼ ξκ 1/4 parallel to the surface, and decay on a longer length ∼ ξκ 1/2 perpendicular to the surface. Taking these inhomogeneous unstable modes into account is essential for the calculation of small corrections ∼ H c κ −1/2 to H s .…”

“…In this case H s is reached as the current density at the surface becomes of the order of the depairing current density, which corresponds to the wavelength 2π/q ≃ ξ 30 . Slow decrease of the Meissner screening currents over the London penetration depth increases H s which is now limited by perturbations δJ(x, y) and δf (ω n , θ, x, y) oscillating on the scale 2π/k ∼ (ξ 3 λ) 1/4 along the x−axis and decaying over ∼ (ξλ) 1/2 along the y−axis [7][8][9]11 . Thus, Eqs.…”

“…It was shown that as the applied field H reaches H s , the Meissner screening current density at the surface becomes of the order of J c ≃ cH c /4πλ, which makes the superconducting state unstable. The value of H s in the limits of large and small GL parame-ter κ is given by [6][7][8][9][10][11]…”

“…( 1) reflects the suppression of the order parameter by the Meissner currents in the local limit κ ≫ 1, while the enhancement of H s by the factor κ −1/2 for κ ≪ 1 results from the proximity effect reduction of the Meissner pairbreaking localized in a narrow surface layer of thickness λ ≪ ξ, where λ is the London penetration depth and ξ is the coherence length. The extensive calculations of H s based on the GL equations [6][7][8][9][10][11] have shown that, for κ ≫ 1, the Meissner state becomes absolutely unstable with respect to small 2D perturbations of current and order parameter with the wavelength ∼ (ξ 3 λ) 1/4 along the surface and decaying over the length ∼ √ λξ perpendicular to the surface. Such perturbations describe the initial stage of penetration of vortex rows with the period ∼ (φ 0 /H s ) 1/2 corresponding to the equilibrium vortex lattice at the field H = H s .…”