Vortices pinning in supraconductors - Phenomenological description of the linear response of a pinned vortex lattice

In this article we report on the investigation of the dynamics of vortices based on the high frequency linear response. We present a serie of measurem ents of the complex penetration depth in the mixed state in a variety of sa mples, including conventional materials (Nb, V, PbIn), the non-conventional heavy fermion UPt3, and the high-T-c cuprate YBaCuO. We have explored a la rge frequency range (1 kHz-10 MHz) so as to cover the cross-over from the q uasi-static response, which is dominated by elastic interactions between vo rtices and sample defects, to the high-frequency regime, which is governed by viscous damping due to vortex friction against the host crystal. For a quantitative description of the frequency spectrum we start from a ph enomenological theory which makes a rigorous distinction between vortex lin es, along a vortex field omega, and magnetic field lines B. It predicts a s econd electrodynamical mode, which is linked to the vortex line tension and has a rather short range. We show that, in the limit of small vortex oscil lations, amplitude and phase of the linear response are governed by an addi tional boundary condition for the vortex lattice at the sample surface; it takes the form of a slipping condition with a characteristic length that de pends on the surface roughness. The frequency spectrum deduced from this me chanism is clearly different from the Campbell spectrum, which is the commo n signature of all bulk pinning mechanisms. Our results on samples of PbIn, Nb, V, and YBaCuO entirely confirm our mode l, including some non-intuitive size effects which appear at low frequency when the sample becomes transparent to the Aux flow mode. However, our meas urements in the B and C phases of UPt3 reveal and important contribution of the bulk to the vortex pinning.