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Journal of Siberian Federal University. Mathematics & Physics 2018, 11(2), 219-221
УДК 537.9
Analysis of Superconductor Magnetization Hysteresis
Denis M. Gokhfeld*
Kirensky Institute of Physics Federal Research Center KSC SB RAS Akademgorodok, 50/38, Krasnoyarsk, 660036
Russia
Received 05.12.2016, received in revised form 10.12.2017, accepted 20.03.2018 The critical state model and the extended critical state model are described to analyse a magnetization hysteresis and to find superconductor parameters. We discuss how geometric sizes and form influence on magnetization hysteresis, critical current and trapped flux.
Keywords: pinning, Abrikosov vortices, critical state, critical current density, trapped flux. DOI: 10.17516/1997-1397-2018-11-2-219-221.
An essential criterion of superconductivity in studied samples is sharp drop of the resistance at some temperature. Other essential criterion is a great diamagnetic response during the magnetization measurements. Temperature and magnetic field dependencies of the magnetization give the detailed characterization of the superconducting phase in single-crystal and polycrystalline samples.
The critical state model [1] and its many modifications describe entering and trapping of magnetic flux in a Il-type superconductor. The extended critical state model [2-4] accounts contributions to the magnetization loop from the surface and the internal volume of a superconducting sample. The extended model allows to parametrize the magnetization hysteresis with asymmetry relative to the M = 0 axis.
1. Determination of superconducting parameters from magnetic measurements
The critical current density jc is most important parameter for superconductor applications. The diamagnetic response is determined by jc. The critical state model [1] results in the Bean formula jc = 3AM/2R, such that one can estimate jc from the magnetic measurement. In the Bean formula AM is the hysteresis width, R is the radius of the current circulation.
Main stages of magnetization hysteresis analysis are given below.
1. The content of a superconducting phase should be estimated firstly. The virgin magnetization changes as M(H) = -xH for H < Hcl, here x is the content of a superconducting phase. The demagnetization factor can distort the value of x. Additional diamagnetic [5] / paramagnetic [6, 7] phases in a sample tilt the magnetization hysteresis clockwise / anticlockwise. Coexistence of superconducting and ferromagnetic phases in a sample gives a composite hysteresis [8]. For hysteresis of the superconducting phase the dependence of M on H approaches to 0 as H increases to the upper critical field Hc2.
2. Some special fields are remarkable on a hysteresis and can be estimated directly (Fig. 1). The lower critical field Hcl is the point in which the virgin M(H) dependence begins to deflect from the linear line. The virgin M(H) dependence becomes to coincide with the envelop magnetization loop at H = Hp, the full penetration field. The M(H) dependence becomes reversible at H higher than the irreversibility field Hirr. The superconductivity and the corresponding
* [email protected] © Siberian Federal University. All rights reserved
diamagnetic response disappear at H higher than the upper critical field Hc2. The trapped magnetic flux Btr in a sample equals to n0AM at H = 0.
3. The depth of the magnetic field penetration Ao can be estimated from the reversible part of magnetization hysteresis. The London model [9] results in formula for the equilibrium magnetization M = —<0 / (32n2 A0)ln(nHc2/H), where <0 is the magnetic flux quantum, n is a constant about 1.
4. The hysteresis width is defined by the product of jc and R such that the current circulation scale should be found for estimations of jc from magnetic measurements. There is a problem here because the size of current circulation R can differ from the sample radius. It may be the effective grain radius for polycrystalline samples or a size of grain clusters. The hysteresis asymmetry relative to the M = 0 axis allows to estimate R because this asymmetry depends on R but it is independent of jc. The asymmetry is defined by the ratio of ls/R, where ls is the depth of the surface layer with equilibrium magnetization [2, 3]. Abrikosov vortices are not pinned in this surface layer. The value of ls is not larger than A0. Noticeable asymmetry of hysteresis is observed for ls/R > 0.1 such that one can estimate R < 10A for asymmetric magnetization hysteresis. For analyzed polycrystalline superconductors the magnetization loops are good fitted by the extended critical state model with R equal to the average grain radius obtained from SEM.
5. The jc(H) dependencies are easy plotted from the magnetization loops by using the Bean formula. These jc values obtained are the critical current density averaged on the cross-section perpendicular to the external magnetic field H. Then the field dependence of the pinning force Fp(H) = ¡i0Hjc(H) is plotted that allows to find the maximal pinning field and Hirr also. If the second peak (fishtail, peak effect) is noticeable on a magnetization hysteresis and on field dependencies of jc and Fp than the peak position and its temperature evolution contain information about the vortex lattice transition or the phase separation in superconductor [5].
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i ] Hm J """
H/H
Fig. 1. Characteristic fields on magnetization hysteresis of superconductor. The Hc2 field is more larger than a maximal external field
As the extended critical state model describes, the averaged critical current density jc depends on the size and the form of the sample. Due to the surface layer the averaged critical current jc = jcb(1 — ls (H)/R)n, here jcb is the critical current density of a macroscopic sample with sizes more larger than A0, n is the parameter defined by the sample form, n = 3 for the long cylindric sample and n = 2 for the long plate parallel to H (Fig. 1).
Review article [3] has some additional references on application of the extended critical state model to analysis of magnetization loops of different superconductors.
Conclusion
Magnetization hysteresis analysis gives some parameters of superconductors: the lower Hc1 and upper Hc2 critical fields, the full penetration field Hp, the irreversibility field Hirr, the trapped flux Btr, the equilibrium layer depth ls, the critical current density jc. When sizes of a sample/grains are comparable with the magnetic field penetration depth Ao, the magnetization hysteresis is asymmetric and the extended critical state model should be applied to describe this. The averaged critical current density decreases with the sample/grain sizes due to equilibrium magnetization of the surface layer. Also the sample/grain sizes and their form influence on the magnetization hysteresis form, particularly on the pinning force maximum field.
The work is supported by Russian Foundation for Basic Research (grant 16-42-240445).
References
[1] C.P.Bean, Magnetization of hard superconductors, Phys. Rev. Lett., 8(1962), 250.
[2] D.M.Gokhfeld et al., Magnetization asymmetry of type-II superconductors in high magnetic fields, J. Appl. Phys., 109(2011), 033904.
[3] D.M.Gokhfeld, An extended critical state model: Asymmetric magnetization loops and field dependence of the critical current of superconductors, Phys. Solid State, 56 (2014), 2298.
[4] D.M.Gokhfeld, Critical current density and trapped field in HTS with asymmetric magnetization loops, J. Phys.: Conference Series, 695(2016), 012008.
[5] D.A.Balaev et al., Increase in the Magnetization Loop Width in the Ba0.6K0.4BiÜ3 Superconductor: Possible Manifestation of Phase Separation, J. Exp. Theor. Phys., 118(2014), 104.
[6] E.Altin, D.M.Gokhfeld, F.Kurt, M.E.Yakinci, Physical, electrical, transport and magnetic properties of Nd(Ba,Nd)2.iCu3Or system, J. Mater. Sci.: Mater. Electron., 24(2013), 5075.
[7] E.Altin et al., Vortex pinning and magnetic peak effect in Eu(Eu,Ba)2.125Cu3OK, J. Mater. Sci.: Mater. Electron., 25(2014), 1466.
[8] E.Altin et al., Hysteresis loops of MgB2 + Co composite tapes, J. Mater. Sci.: Mater. Electron., 24(2013), 1341.
[9] Z.Hao, J.R.Clem, Phys. Rev. Lett., 67(1991), 2371.
Анализ петель намагниченности сверхпроводников
Денис М. Гохфельд
Институт физики им. Л. В. Киренского СО РАН Академгородок, 50/38, Красноярск, 660036
Россия
Описано использование модели критического состояния и расширенной модели критического состояния для определения параметров сверхпроводников из измеренных петель намагниченности. Обсуждается влияние геометрических размеров и формы образцов на вид петель намагниченности, критический ток и замороженное магнитное поле.
Ключевые слова: пиннинг, вихри Абрикосова, критическое состояние, плотность критического тока, захваченное поле.
