The surface impedance Z(f ) of conventional isotropic materials has been carefully measured for frequencies f ranging from 1 kHz to 3 MHz, allowing a detailed investigation of the depinning transition. Our results exhibit the irrelevance of classical ideas on the dynamics of vortex pinning. We propose a new picture, where the linear ac response is entirely governed by disordered boundary conditions of a rough surface, whereas in the bulk vortices respond freely. The universal law for Z(f ) thus predicted is in remarkable agreement with experiment, and tentatively applies to microwave data in YBaCuO films.PACS numbers: 74.60. Ge, 74.25.Nf. A perfect sample of a type-II superconductor in the vortex (or mixed) state would be transparent to an electromagnetic wave at very low frequencies. But defects are always present and strongly alter the quasistatic and low-frequency response; low frequencies here means Ω = 2πf ≪ Ω d , a so-called "depinning frequency" [1] depending on the material and vortex density. It is important for applications to know what kind of defects can pin vortices, how they hinder small vortex oscillations and thereby restrain the penetration of an ac ripple. In this respect, a study at low levels of excitation of both the resistive and inductive part of the surface impedance Z(Ω) = R − iX as a function of the frequency provides much information about the dynamics of pinning. It is generally accepted that bulk pinning centers limit the quasistatic skin effect to a pinning (or Campbell's) length λ C ∼ 1 − 100 µm, while dissipation is vanishingly small, as observed [1,2]. No model however has been able to account for variations of Z at higher frequencies. In particular, as the first increasing of R(f ) is stronger than expected, the understanding of dissipation remains a puzzling problem, including in high T c materials [3].Experiments are performed on a series of slabs of coldrolled polycrystalline PbIn and pure single-crystralline Nb. The slabs (xy) are immersed in a normal magnetic field B; unless specified their thickness 2d is much larger than the flux-flow penetration depth δ f (see below). At equilibrium, up to the upper critical field B c2 , a regular lattice of vortex lines parallel to z is formed, with the density n = B/ϕ 0 , where ϕ 0 is the flux quantum. Both faces of the slab, z = ±d, are then subjected to an ac magnetic field b 0 e −iΩt parallel to the length (xdirection) of the sample. Under such conditions, induced currents and electric fields, J(z) and e(z)e −iΩt , are along the y-direction, while vortices oscillate in the xz-planes. For low exciting fields (b 0 ∼ 1 µT), vortex displacements u(z) ∼ 1Å are very small compared with the vortex spacing a ≃ n − 1 2 (∼ 1000Å, for B ∼ 0.1 T) (Fig. 1a). The electric field e 0 at the surface z = d, e 0 = e(d) = −e(−d), is measured by means of a pick-up wound coil. The main experimental difficulty in such measurements is to ensure a precise calibration of the phase ϕ of e 0 (within better than 0.5 • at 100 kHz). Thus we get the surface impe...

We have measured the surface impedance of thick superconductors in the mixed state over a broad 2 kHz -20 MHz frequency range. The depinning cross-over is observed; but it is much broader than expected from classical theories of pinning. A striking result is the existence of size effects which invalidate the common interpretation of the low-frequency surface inductance in terms of a single penetration depth. Instead, a two-mode description of vortex dynamics, assuming free vortex flow in the bulk and surface pinning, accounts quantitatively for the spectrum of the complex apparent penetration depth.

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