Campbell Response in Type-II Superconductors under Strong Pinning Conditions

Abstract

Measuring the ac magnetic response of a type II superconductor provides valuable information on the pinning landscape (pinscape) of the material. We use strong pinning theory to derive a microscopic expression for the Campbell length λC, the penetration depth of the ac signal. We show that λC is determined by the jump in the pinning force, in contrast to the critical current jc which involves the jump in pinning energy. We demonstrate that the Campbell lengths generically differ for zero-field-cooled and field-cooled samples and predict that hysteretic behavior can appear in the latter situation. We compare our findings with new experimental data and show the potential of this technique in providing information on the material's pinscape. 74.25.Op, 74.25.Wx, 74.25.Ha Technologically useful superconductors are of second type and acquire their desired transport and magnetic properties through vortex pinning, i.e., vortices [1] get immobilized by material defects. The characterization of the pinning landscape (or pinscape) is of great importance but presents quite a formidable task. Measurements of dc transport properties, either dynamically through the current-voltage characteristic [2] or statically through magnetization [3], are standard techniques used to gain information on the pinscape. Similarly, the ac magnetic response of superconducting samples [4] provides insight into the shape of pinning potentials. Unfortunately, the relation between the measured penetration depth of the ac signal, the so-called Campbell length λ C , and the parameters of the pinscape is only known on a phenomenological level. In this letter, we present a microscopic derivation of the Campbell length within the framework of strong pinning theory, thereby providing access to microscopic parameters of pinning defects and substantially enlarging the scope of applications of this measurement technique.Probing superconductors via their ac magnetic response goes back to the 60-ies and culminated in Campbell's work [4] which provided the first consistent explanation of the penetration phenomenon (see Refs.[5] for further developments): for small ac magnetic-field amplitudes h ac and frequencies ω, vortices oscillate reversibly within their pinning potentials (described as harmonic wells α x 2 /2), with the external signal h ac penetrating the sample on a distance λ C ∝ B/ √ α of order micrometers. Later work by Lowell [6] and Campbell [7] provided a more quantitative but still phenomenological understanding within a model pinscape. Here, we make use of the strong pinning scenario allowing us to perform a quantitative and microscopic analysis of the ac magnetic response. In particular, we find the dependence of the Campbell penetration depth λ C on the vortex state, e.g., the critical (Bean [3]) state with a linear vortex density gradient supporting the critical current density j c or a field-cooled state with a constant induction B, and predict the occurrence of new hysteretic effects. The comparison with recent experiments [8] confirms...

Tunnel diode resonator for precision magnetic susceptibility measurements in a mK temperature range and large DC magnetic fields

“…[21][22][23] For a superconductor in the mixed state, the measured quantity, λ m , is related to the Campbell penetration depth λ C by a relation λ 2 m = λ 2 L + λ 2 C . 22,24 However, for precision quantitative measurements, even laboratory scale magnetic fields strongly affect the TDR circuit with complicated response. We note here that while we focus on easily accessible laboratory fields, the TDR technique has been used at much higher fields in excess of 60 T. 8 The effective inductance L eff of the circuit is a combination of contributions from the inductor coil itself and the sample.…”

Metastability and hysteretic vortex pinning near the order-disorder transition in NbSe2 : Interplay between plastic and elastic energy barriers

“…For instance, hysteretic Campbell response in the presence of strong pinning centers has been recently demonstrated through a simple model. 48 A similar approach for VLs in weak pinning landscapes should help to better understand the connection between thermal history leading to metastable VL configurations and hysteretic Campbell responses.…”

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 .…”

“…[24][25][26][27]. Hysteresis in the response of ac-driven vortex systems have been observed in many superconductors, including traditional BCS materials [29][30][31] , high T c cuprates 44,45 , and other non-conventional superconductors 21 . In all those cases, a dependence of pinning on thermal and/or dynamical history has been reported and associated with different underlying vortex lattice configurations.…”