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Use of a Sigmoid Function to Describe Second Peak in Magnetization Loops

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Abstract

Order-disorder transitions of a vortex lattice transfer type-II superconductors from a low critical current state to a high one. The similar transition between different current states can be caused by electromagnetic granularity. A sigmoid curve is proposed to describe the corresponding peak in a field dependence of the macroscopic critical density. Using the extended critical state model, analytic expressions are obtained for the field dependencies of the local critical current density, the depth of equilibrium surface region, and the macroscopic critical current density. The expressions are well fit to published data.

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Correspondence to Denis Gokhfeld.

Appendix

Appendix

The dependence of the local critical current density j c on the inner magnetic field B is described by a decreasing function j c (B). The Kim [32], a power [33], and an exponential [21] model are usually used. We support the following generalized j c (B) dependence [34]:

$$ j_{c}(B) = j_{c0}\frac{1-|B/B_{c2}|^{\alpha}}{1+|B/B_{0}|^{\alpha}}~~, $$
(9)

where α is positive dimensionless coefficient. This function gives better agreement with experimental dependencies in field range from 0 to H c2 than the earlier generalized dependence [35].

The simple phenomenological l s (H) dependence is written as

$$ l_{s}(H) = l_{s0} \left( 1 + |H|/H_{1}\right)~~, $$
(10)

where H 1 is the increasing rate. The magnetization loops becomes reversible in H higher than the irreversibility field H irr. So the l s (H) dependence increases from l s0 at H = 0 to R at H = H irr. Equation (10) can be rewritten as

$$ l_{s}(H,T) = l_{s0} + (R - l_{s0}) |H|/H_{\text{irr}}~~. $$
(11)

As distinct from H 1, the value of H irr depends on the size R.

Expressing J c (H) at H = 0 as J c0 = j c0(1 − l s0/R)n, one obtains the magnetic field dependence of the macroscopic critical current density:

$$ J_{c}(H)=J_{\text{c0}} \frac{1-\left|H/ H_{c2}\right|^{\alpha}}{1+\left| H/H_{0} \right|^{\alpha}} \left( 1-|H/H_{\text{irr}}|\right)^{n}~~. $$
(12)

A scaling of pinning force at different temperatures is resulted from this equation [34]. Equation (12) successfully describes J c (H) dependencies for most superconductors without the peak effect.

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Gokhfeld, D. Use of a Sigmoid Function to Describe Second Peak in Magnetization Loops. J Supercond Nov Magn 31, 1785–1789 (2018). https://doi.org/10.1007/s10948-017-4400-2

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  • DOI: https://doi.org/10.1007/s10948-017-4400-2

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