1 Introduction

It is well known that magnetic measurements in superconducting materials provide very useful information such as the magnitude, field dependence, temperature dependence and time dependence (flux creep) of the critical current density, activation energy, lower and upper critical fields, irreversibility line, AC loses, magnetostriction, and volume fractions of the grains for granular superconductors. In any type of magnetic measurement, one should note that the thermomagnetic history is crucial to identify any physical parameter using the comparison of experimental and theoretical data. The evolution of magnetization < M > , previously exposed to various magnetic field-temperature histories, has been studied by many researchers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Many researchers have considered the situation where the applied external magnetic field is reduced to zero [8, 20] to facilitate the study of the trapped flux or remanent magnetization. In previous studies, two procedures, named by Hcool and Hcycle, have been proposed to describe a state of critical remanent magnetization [16, 17]. In a Hcool procedure, the specimen cooled down to temperature T0 while under effect of applied external magnetic field (Ha), represented by Hcool. After this procedure, the sample’s temperature fixed at temperature (T0) and magnetic field applications has been removed. In the Hcycle procedure, the specimen is subjected to magnetic field application and removal process after zero field cooling to constant temperature (T0), where the maximum magnetic field was labeled as Hcycle. Clem and Hao [1] and Cave [2] have investigated the evolution of magnetization as a function of temperature theoretically for monolithic and isotropic structured and idealized slab shaped superconductors that magnetization process carried out with the various procedures just outlined. Reversible behavior of magnetization < M > versus temperature (T) for temperatures just below to the critical temperature (Tc) has been reported in Ref. [5]. To account reversible behavior of magnetization < M > as a function of temperature T for temperatures just below to the critical temperature Tc investigated in Ref. [5], Clem and Hao introduced a range of temperature defined as ΔT = Tc − Tirr. In the range of temperature ΔT, bulk pinning critical current density Jc has disappeared, but an equilibrium magnetic field screening Meissner current exists whose magnitude decreases as temperature T increases over the range from Tirr to Tc. The evolution of magnetization of type-II superconductors with time, in stationary magnetic field and at ambient temperature conditions, magnetized via various procedures outlined above, has been investigated in Ref. [21,22,23]. LeBlanc et al. [24] carried out an experiment to observe thermal disclosure of hidden magnetic moments in conventional and high temperature superconductors and supported their results with theoretical calculations by means of flux conservations principle [25, 26]. Rezeq et al. [27] has reported that including observations about trapped flux in niobium superconductor and detailed mathematical model to describe behavior of the sample studied. According to their study, after application of external magnetic field on granular superconductors, such as niobium superconductor, effect of return fields of magnetized grains on induced permanent macroscopic current must be considered in the analysis of granular superconductors. This situation reflects the most important feature of their study.

Nonlinear electric field E and current density Jn relation are often used in simulating of superconductors, where n is a parameter showing the dependence of E on J and is usually a large number [28,29,30]. This relationship can be imposed by a commercial software using finite element method to simulate physical situations that based on partial differential equations in superconducting structure. Therefore, a general numerical method is used to simulate the response of superconductors against time-varying external magnetic field by many researchers [31,32,33,34,35,36,37,38,39,40,41,42].

The primary objective of this work is examination of validity of E-J relationship for simulating results of remanent magnetization in lutetium doped YBCO high temperature superconductor. Our numerical computation reproduce successfully the features of remanent magnetization as a function of temperature, upon field cooling at selected fields and removing the fields at 20 K. It is worth to mention two notes regarding the novelty of this work.

First, we have chosen the target sample having the highest critical current density among the samples given in our previous communication [43] prepared by a modified MPMG procedure. A great number of papers have been published on adding elements in Y123 system for increasing the critical current density (Jc) values of YBCO samples [44, references therein]. Raychaudhuri et al. [45] reported that partial Lu substituted YBCO (Y0.75Lu0.25Ba2Cu3O7) sample shows much more Meissner effect than pure YBCO samples measured by DC magnetic susceptibility data at 20 K. It was also published that after the further annealing treatment, the sample doped with 0.5 wt% Lu oxide, prepared by top-seeding melt-texture-growth process, exhibits better critical current density than the undoped Y123 sample [46].

Second, remanent magnetization or trapped flux has been calculated by many workers for idealized geometry (infinite long slab or cylinder). For example, Xu [20] reported his calculations of remanent magnetization for hard superconductors in various critical-state models where he considered an infinite long slab of thickness 2d with an external field parallel to the surface of the sample. Celebi and Leblanc [16] have calculated remanent magnetization for idealized (i.e., infinite) cylindrical geometry, using Maxwell’s equation \(\nabla \times {\varvec{H}}\) = j together with the critical state assumption in order to account the experimental observations on flux trapping in sintered tubes of high-Tc superconductors. In order to be able to design and manufacture better superconducting wires and devices, it is necessary to develop a better understanding of the magnetic behavior such as flux trapping, ac loss mechanisms and of the internal distributions of magnetic field and current density. This requires good modeling capabilities, since high-Tc superconductors are characterized by a very non-linear current–voltage relationship. Finite-element (FE) simulations are a reliable technique to compute these quantities in details, not only in individual conductors but also in complex devices where the electromagnetic interaction between the different components is very strong. Hence, we have used E-J relationship for simulating results of remanent magnetization in lutetium doped YBCO high-Tc superconductor.

2 Experimental Procedure

The lutetium doped YBCO high-Tc superconducting bulk specimen investigated in this work was prepared by a modified MPMG procedure. In the modified MPMG preparation process, the final temperature at quenching stage was the liquid nitrogen temperature (77 K). Detailed information for sample production and magnetic characterization processes for six samples are given in Ref. [43]. The target sample studied in this work is the one having the highest critical current density, which is sintered in a tube furnace in air temperature of 940 °C and has the doping amount of x = 0.5 in Y1−xLuxBa2Ca3O7−δ. The X-ray diffraction (XRD) data were taken using Rigaku D/Max-IIIC diffractometer with Cu K_ radiation over the range of 20–60° with a scan speed of 0.2°/min at room temperature. The orthorhombic lattice parameters were calculated from observed peaks using the least square methods as a = 3.7943 Å, b = 3.9017 Å, and c = 11.6208 Å. These values are consistent with the ones reported in Mellekh et al. [47] and the ones reported for lutetium doped YBCO superconductors prepared by solid state reaction method [44]. We assessed our sample studied in this paper is mainly Y123 phase. We note that the detailed structural analysis is beyond the scope of this work since we have focused on magnetic measurements and simulated results based on H-Formulation.

DC magnetic measurements were conducted in Quantum Design Physical Property Measurement System (PPMS) by using its related component called vibrating sample magnetometer (VSM). The HTS bulk specimen is properly placed to VSM module to ensure that the applied field is parallel to the c-axis of the specimen. Three different magnetization measurements have been caried out to investigate how the magnetization of the specimen varies with temperature, namely, MZFC, MFC, and MREM. Superconducting transition temperatures were obtained by means of the AC Measurement System (ACMS) option of the Quantum Design PPMS. The transition temperature of our sample is 89.5 K which is consistent with Ref. [44], where the samples prepared with solid state reaction method.

For MZFC(T) measurements, the specimen is in a situation that there is no significant external magnetic field except for background magnetic field when the sample is above the critical temperature Tc. Under these circumstances, the sample is cooled down to temperature T0 = 20 K without magnetic field. In the literature, such processes are called zero field cooling (ZFC) process. Then, the sample is magnetized at this temperature by the application of selected external magnetic field, Ha = 40 kA/m. Then, the temperature of the specimen is increased to transition temperature, Tc, in the presence of selected applied magnetic field and MZFC(T) is measured during heating.

In the procedure to measure MFC(T), first the selected magnetic field is applied when the sample is above the transition temperature, Tc. When the specimen is cooled down to selected temperature, T0 in the presence of applied magnetic field Hcool, MFC(T) is recorded as a function of temperature. Finally, at temperature T0, this selected field is removed to induce flux trapping current circulating in the sample, the corresponding magnetization is called remanent magnetization, MREM. Hence, MREM(T), as a function of temperature, is measured during warming process from T0 to above Tc. These processes are repeated for some selected fields Hcool = 40, 80, 160, 320 kA/m, to map out the magnetic behavior of the specimen.

3 Numerical Computations

Numerical calculations to model the magnetic behavior of the Y0.5Lu0.5Ba2Ca3Oy superconductor has been achieved with commercial finite element method (FEM) using the software COMSOL Multiphysics 4.3b [48]. In many studies, for example Ref. [32,33,34,35,36,37], numerical calculations based on FEM have been used to investigate electromagnetic behavior of high temperature superconductors such as induced current density, magnetization, levitation and guidance forces, magnetic flux and magnetic field profiles, and AC loses. Maxwell equations are used as governing equations and nonlinear E-J relationship is used to reproduce superconducting behavior while transition takes place between normal and superconducting states. Most challenging part of modeling of superconducting materials is nonlinear E-J characteristics of them. To handle this challenging situation, some researchers in Ref. [34] have introduced a calculation technique based on edge element.

In the simulations, two different kinds of materials by means of electromagnetic characteristics are used as follows:

  1. (i)

    Superconducting material with highly nonlinear resistivity relation ρ(J) and B = µ0H

  2. (ii)

    Vacuum area that encircles superconducting material and defined as an insulator with the equation B = µ0H

For superconducting materials, resistivity-ρ is not linear, unlike traditional conductors, and it can be described as such a nonlinear power-law expression [39].

$$\rho = \frac{{E}_{c}}{{J}_{c}}{\left|\frac{J}{{J}_{c}}\right|}^{n-1}$$
(1)

The numerical calculations in the simulation have been carried out with a personal computer that has 2.1 GHz processor and 4 GB of RAM in duration of nearly half an hour. The speed of calculation depends on two main factors, first one originates from simulation such as meshing features, the number of time steps of the simulation and the second one originates from computer’s specifications such as computing capability of the processor in unit time and performance of graphics processing unit while rendering visual outputs. For simplicity and saving time, this work is limited to two-dimensional (2D) situation. In our numerical computations, typical temperature dependence of the critical current densities are selected as Jc(T) = 6.5 × 108(1 − T/Tc)2.5 A/m2, Jc(T) = 8.70 × 108(1 − T/Tc)2.5 A/m2, and the other parameters are chosen as Ec = 10−4 V/m, f = 5 Hz, and n = 25 to construct literature compatibility. Simulation geometry with coordinate system and meshed case of the system are given in Fig. 1. As can be seen from Fig. 1, simulation geometry has infinite length in z-direction which is perpendicular to the plane of the paper.

Fig. 1
figure 1

Sample geometry and its meshed case using in the simulations (the units of axis are in meters)

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The governing equations can be listed as follows:

$$\mu \frac{\partial \overrightarrow{H}}{\partial t}+\overrightarrow{\nabla }\times \overrightarrow{E} = 0$$
(2)
$$\overrightarrow{E} = \rho \overrightarrow{\nabla }\times \overrightarrow{H}$$
(3)

What we mean by two-dimensional (2D) approximation is that usage of modeling environment that consisting with the cross-sections of the superconductor and encircling material, supposing on infinite length on one direction, particularly in z-direction like ours. As a result of this assumption, the magnetic and electric fields have only two transverse components H = [Hx (x, y), Hy (x, y), 0] and one longitudinal component, E = [0, 0, Ez (x, y)], respectively. Equation 3 can be transformed into the scalar form:

$${E}_{z} = \rho \left({J}_{z}\right){J}_{z}$$
(4)
$${J}_{z} = \frac{\partial {H}_{y}}{\partial x}-\frac{\partial {H}_{x}}{\partial y}$$
(5)

The condition of zero divergence of the magnetic field

$$\frac{\partial {H}_{x}}{\partial x}+\frac{\partial {H}_{y}}{\partial y} = 0$$
(6)

can be considered as an initial condition.

Equation (2) can be rewritten in the x- and y-component as

$$\mu \frac{\partial {H}_{x}}{\partial t}+\frac{\partial E}{\partial y} = 0$$
(7)
$$\mu \frac{\partial {H}_{y}}{\partial t}-\frac{\partial E}{\partial x} = 0$$
(8)

Taking the x and y derivatives of Eqs. (7) and (8), respectively, and adding them, we obtain the following equation:

$$\mu \frac{\partial }{\partial t}\left(\frac{\partial {H}_{x}}{\partial x}+\frac{\partial {H}_{y}}{\partial y}\right) = \mu \frac{\partial }{\partial t}\left(\nabla .H\right) = 0$$
(9)

From which \(\nabla .H\) = constant = 0, i.e., the divergence value is fixed equal to the values at t = 0. In numerical solutions, this condition must be applied at each time step. A Dirichlet boundary condition is used at the outer boundary of the air region, which forms outer space as a function of time.

Using the equation below, we have calculated the magnetization along the y-axis:

$$M = \frac{\frac{1}{2}{\int}\!_{D}x{J}_{z}dS}{wh}$$
(10)

where w is the sample width and h is the sample height used in the simulation. Integration is evaluated over the domain D, i.e., the cross-section of the sample.

4 Results and Discussions

Figure 2(a) shows the zero-field cooled (ZFC), field cooled (FC), and remanent (REM) magnetization as a function of temperature for Lu0.5Y0.5Ba2Cu3Oy superconducting sample for the field of 40 kA/m. Measurements of the zero field cooled magnetization have been done for by first cooling the sample to a very low temperature in zero magnetic field, then applying a measuring magnetic field of 40 kA/m, and finally monitoring the magnetization with increasing temperature. The remanent magnetization was obtained by removing the field after field cooling process at selected magnetic field of 40 kA/m and measuring with increasing temperature. We note that the difference between field cooled magnetization (MFC) and zero field cooled magnetization (MZFC) shown by open circles almost overlaps with remanent magnetization (MREM) as reported by several researchers [1, 3, 8]. Figure 2(b) shows the variation of MFC as a function of temperature in enlarged scale. We note that the MFC data is saturated at about 60 K and negligibly small compared with MZFC which can be attributed to the fact that the sample has strong pinning force.

Fig. 2
figure 2

a The zero field cooled (ZFC), field cooled (FC), and remanent (REM) magnetization as a function of temperature for the sample studied. We note that the field cooled (FC) magnetization MFC is saturated at about 60 K as shown in b

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Figure 3 shows the remanent magnetization of the sample during the heating process after the removal of selected applied fields denoted Hcool given in the legend. After each field cooling process, the field is removed at 25 K, hence flux trapping current induced in the sample is maximum at this temperature among the temperatures above 25 K. As the sample is heated, remanent magnetization and corresponding bulk current decreased and finally reaches to zero at the critical temperature of 92 K. Therefore, thermomagnetic history was erased. The same procedure was repeated for different cooling fields (Hcool = 40, 80, 160, 320 kA/m). The remanent magnetization corresponding Hcool = 320 kA/m has the maximum values, the upper curve of MREM(T) shown in Fig. 3. MREM(T) for 160, 80, and 40 kA/m merge to the curve of MREM(T) for 320 kA/m at the temperature of about 65, 74 and 82 K, respectively. Note that lowest curve near zero level in Fig. 3 is monitored during the field cooling process, resulting in negligibly small field cooled magnetization, MFC (about 0.5 emu/cm3).

Fig. 3
figure 3

The remanent magnetization of the sample during the heating process after the removal of applied field denoted Hcool given in the legend

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Figure 4 shows the numerical calculations of magnetization based on the finite element method (FEM) described above to reproduce the experimental data shown in Fig. 2. In our calculations, we have used Ec = 10−4 V/m, n = 25, temperature exponent of 2.5, and critical current density (at 25 K) of 6.5 × 108 A/m2. The red horizontal line represents the field cooled magnetization MFC during cooling. The green circles are for remanent magnetization during heating after removal of Hcool = 40 kA/m. The black squares represent the zero-field cooled magnetization MZFC with increasing temperature, where the applied field of 40 kA/m is present upon zero field cooling.

Fig. 4
figure 4

The numerical calculations of magnetization based on the finite element method (FEM) described above to reproduce the experimental data shown in Fig. 2

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Figure 5 displays the numerical calculations of remanent magnetization for selected values of Hcool given in the legend to reproduce the experimental data shown in Fig. 3. In our calculations, we have used Ec = 10−4 V/m, n = 25, and temperature exponent of 2.5. For comparison, the computed results for critical current density (at 25 K) of 6.5 × 108 A/m2 and 8.7 × 108 A/m2 are shown by continuous lines and symbols, respectively. For example, the black circles and black lines are for remanent magnetization with increasing temperature, respectively, after removal of Hcool = 40 kA/m.

Fig. 5
figure 5

The numerical calculations of remanent magnetizations as a function of temperature by curves (Jc0 = 6.5 × 108 A/m2 at 25 K) and symbols (Jc0 = 8.7 × 108 A/m2 at 25 K) with different colors for selected values of Hcool given in the legend to reproduce the experimental data shown in Fig. 3

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We note that our numerical computations shown in Figs. 4 and 5 qualitatively reproduce the experimental data given in Figs. 2 and 3, respectively. The small discrepancies observed in high temperature range can be attributed to the granularity of the sample. In our numerical computations, we considered the sample as a monolithic specimen for the simplicity and convenience. In fact, the sample is granular superconductor. Therefore, the total magnetic moment MREM produced by the removal of Hcool is considered to arise from two contributions: (i) macroscopic flux retaining intergranular persistent currents induced to circulate through the matrix by reducing the applied magnetic field and concurrently (ii) localized flux retaining persistent currents induced to circulate within the individual grains.

In Fig. 6, we have presented both measured and computed remnant magnetization MREM as a function of selected cooling field, namely, Hcool at 25 K. The open squares correspond to the experimental remnant magnetization at 25 K after removal of Hcool. Black solid curve is for the numerical calculation of the remnant magnetization described above. For comparison, we also presented the corresponding calculations using Bean model for infinite cylinder geometry by red curve with triangles (see Ref. [16]). We note that numerical calculations give quite a good fit to the experimental data.

Fig. 6
figure 6

Measured and computed remnant magnetization MREM as a function of cooling field, Hcool. The open squares correspond to the experimental remnant magnetization at 25 K after removal of Hcool. Black solid curve is for numerical calculation for remnant magnetization. Red curve with triangles denotes the calculated remnant magnetization for infinite cylinder geometry based on Bean model

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Figure 7 shows the flux density profiles after field cooling at HFC = 320 kA/m, for (a) during the decrease of applied field from HFC = 320 kA/m and (b) during the heating process upon reaching zero field, across the width of the sample. In Fig. 7(a), blue, green, and red colors represent the flux density profile inside and outside of the sample corresponding to the applied fields of 213,333, 106,667, and 0 A/m, respectively. In Fig. 7(b), the flux density profiles are shown during heating process. As the temperature increases, the slope of the flux density profile (sometimes it is called B-profile) decreases. Hence, the flux retaining critical current flowing inside the sample decreases as the temperature increases consistent with Maxwell’s equation.

Fig. 7
figure 7

Flux density profiles after field cooling at HFC = 320 kA/m, for a during the decrease of applied field from HFC = 320 kA/m and b during the heating process upon reaching zero field, across the width of sample

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Figure 8 shows the distributions of current density over the cross-section of the sample for selected times a 0.004 s, b 0.008 s, c 0.012 s, and d 0.06 s, corresponding to applied fields of 213,333, 106,667, 0 A/m (where MREM = 202,406 A/m just before heating after removal of HFC = 320 kA/m), and 0 A/m (where temperature is increased so that MREM(T) decreased to 106,013 A/m), respectively. Removing the external field induces a continuous flux retaining current to flow through a depth adjacent to the surface, proportional to Hcool or Ha. The current density is directed along the axis perpendicular to the plot. The axis units are in meters and the color bar represents Jz/Jc, the current density normalized with the critical current density. We note that the current is flowing out of plane in the left side of the sample, represented by red color, and the current is flowing into the plane in the right side of the sample, represented by blue color. Lines in each figure correspond to magnitude-controlled streamline positioning of flux lines

Fig. 8
figure 8

The distribution of current density over the cross-section of the sample for selected time a 0.004 s, b 0.008 s, c 0.012 s, and d 0.06 s, corresponding to Ha fields of 213,333, 106,667, 0 A/m (MREM = 202,406 A/m just before heating after removal of HFC = 320 kA/m), and 0 A/m (where temperature is increased so that MREM(T) decreased to 106,013 A/m), respectively. The color bar represents the current density normalized with the critical current density Jz/Jc. Lines in each figure correspond to magnitude-controlled streamlines positioning of flux lines

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5 Conclusions

In this paper, we have explored the validity of the E-J approach for modeling the experimental results of remanent magnetization in a Y0.5Lu0.5Ba2Cu3Oy sample synthesized by a modified melt powder melt growth (MPMG) method. From our results, we could draw the conclusion that numerical computations based on H-formulation can reproduce quite well the experimental data for remanent magnetization as a function of temperature, which provides us an estimation of temperature dependence of the critical current density. The critical current density at 25 K was estimated to be a 6.50 × 108 A/m2 and temperature dependence is determined as (1 − T/Tc)2.5 from the best fit curve of MREM(T) for the sample studied. In numerical calculations, we have used Jc(H) = Jc0/(1 + H/HREF), where HREF is a constant characterizing the superconducting sample. We have confirmed by both experimental and numerical computation using H-formulation in the finite element method (FEM) that remanent magnetization MREM accurately equals to the difference of the field cooled magnetization MFC and zero field cooled magnetization MZFC in the low field and low temperature range. Furthermore, both flux density profile and current density profile have been obtained from the numerical calculations for various stages of the field applications and temperature values during heating process.