Campbell penetration in the critical state of type-II superconductors

Abstract

The penetration of an $ac$ magnetic signal into a type II superconductor residing in the Shubnikov phase depends on the pinning properties of Abrikosov vortices. Within a phenomenological theory, the so-called Campbell penetration depth $\lambda_{\rm \scriptscriptstyle C}$ is determined by the curvature $\alpha$ at the bottom of the effective pinning potential. Preparing the sample into a Bean critical state, this curvature vanishes and the Campbell length formally diverges. We make use of the microscopic expression for the pinning force density derived within strong pinning theory and show how flux penetration on top of a critical state proceeds in a regular way.Comment: 8 pages, 6 figure

Strong pinning theory of thermal vortex creep in type-II superconductors

“…Two local minima at rp and r f appear for x− < x < x+ and are separated by the local maximum at rus defining the barrier U dp (x) for depinning (the barrier to escape the pin) and the barrier Up(x) for pinning (or jumping into the pin). Bottom: Minimizing epin(x; r) with respect to the tip position r at fixed asymptotic position x corresponds to solving the self-consistency equationC(r − x) = fp(r), see (15), here done graphically. Multiple solutions ri(x), (i = p, f, us), show up if the slopē C is smaller than the maximum slope of fp(r), what corresponds to a Labusch parameter κ > 1.…”

“…Unfortunately, no closed expressions for the branches e i pin (x) can be given since the equilibrium equation (15) fixing r(x) is in general not solvable analytically. Progress can be made in the limits of marginally strong pinning with κ − 1 1 or for very strong pinning κ 1.…”

Campbell penetration depth in low carrier density superconductor YPtBi

“…This means that the amplitude of the AC field excitation, H AC , is not large enough to displace the vortex out of the pinning potential well, and it only perturbs the vortex position within the validity of Hooke's law. In this case, the penetration depth is described by the Campbell penetration depth λ C that determines the attenuation range of the AC perturbation from the sample surface to the interior, B AC (x) ∝ μH AC e −x/λ C in a semi-infinite superconductor [21][22][23][24][25][26][27]. Here μ is magnetic permeability, and x is the distance from the surface.…”

Thermal hysteresis of the Campbell response as a probe for bulk pinning landscape spectroscopy

“…From a microscopic treatment of the pinning problem within the strong-pinning formalism 17 , it is known that the overall restoring force on the vortex lattice results from proper averaging of pinning forces acting on individual vortices. Conceptually the same applies to the harmonic response of pinned vortices in the Campbell regime [21][22][23] , where a small displacement U of the vortex system results in a linear restoring force F pin (U ) = −α sp U , with α sp = 2n p r ⊥ ∆f pin /a 2 0 (3) an (averaged) strong-pinning curvature 23 . This spring constant scales linearly with the density of defects n p , the (transverse) trapping radius r ⊥ , and the jump in the (longitudinal) force profile, denoted by ∆f pin .…”