Trapped flux in pure and Mn-substituted CaKFe4As4 and MgB2 superconducting single crystals

Measurements of temperature dependent magnetization associated with trapped magnetic flux in single crystals of CaKFe4As4, CaK(Fe0.983Mn0.017)4As4 and MgB2 using zero-field-cooled and field-cooled protocols are presented. The results allow for the determination of the values of superconducting transition temperature, lower critical field and self-field critical current density. These are compared with the literature data. Possible experimental concerns are briefly outlined. Our results, on these known superconductors at ambient pressure, are qualitatively similar to those recently measured on superhydrides at megabar pressures (Minkov et al 2023 Nat. Phys. https://doi.org/10.1038/s41567-023-02089-1) and, as such, hopefully serve as a baseline for the interpretation of high pressure, trapped flux measurements.


Introduction
The phenomenon of trapped flux in superconductors has been observed and studied for many decades [1]. Although this phenomenon is observed both in type I and type II superconductors, it is significantly more pronounced in the latter, where it can be understood by considering Bean's critical state model [2,3] and th pinning of vortices. With the discovery of high T c superconductivity in cuprates and MgB 2 , a large body of work on trapped flux in superconductors has focused on potential applications [4,5]. On the other hand, the importance of * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. this phenomenon for basic science, as one of the experimental proofs of superconductivity, was also well appreciated, [6] and the measurements of trapped field were used for evaluation of the lower critical field, H c1 [7][8][9].
This topic received renewed interest with the discovery of superconductivity in superhydrides under pressure of over a megabar [10][11][12]. Experimental realization of static pressures in this range requires the use of a diamond anvil cell that contains an extremely small amount of the sample. This makes measurements of physical properties very challenging and the results open to founded and unfounded criticism. In the discussion surrounding superconductivity in hydrides, it was suggested that the measurement of trapped magnetic flux '...should establish definitively that these materials are indeed high temperature superconductors' [13]. Indeed, such measurements were performed and confirmed high temperature superconductivity in H 3 S and LaH 10 [14], even if facing subsequent comments [15] and revised expectations from the authors of [13]. Moreover, the trapped flux measurements were shown to have a great advantage for a small superconducting sample enclosed in a bulky pressure cell, given that the large addenda signal from the cell is minimized by use of zero applied field measurements.
Despite the existing general understanding of trapped flux in superconductors, it has become apparent that having 'baseline' examples of such measurements on known superconductors (single crystals at ambient pressure) would be of value. In this study, we have chosen single crystals of iron-based superconductors with fairly large pinning, pure CaKFe 4 As 4 and Mn-substituted CaK(Fe 0.983 Mn 0.017 ) 4 As 4 . The superconducting properties of both materials are fairly well studied [16,17] and their moderately high superconducting transition temperatures make them convenient samples for such measurements. We also perform similar measurements on a single crystal of MgB 2 , another well-studied material that, in contrast, has a fairly small pinning relative to CaKFe 4 As 4 materials. We attempt to address not only the superconducting properties of these compounds but also discuss some limitations associated with the measurements.

Experimental details
Single crystals of CaKFe 4 As 4 and CaK(Fe 0.983 Mn 0.017 ) 4 As 4 with sharp superconducting transitions were grown using high-temperature solution growth [16][17][18]. The Mn substitution level x was determined by performing energy dispersive x-ray spectroscopy. MgB 2 single crystals [19][20][21] were grown from a solution with excess Mg under high pressure and high temperature in boron nitride crucibles (see supporting information in [20] for more details), similarly to the procedure described in [22].
All samples were thin plates with the c-axis perpendicular to the plates and approximately of cuboid shape with minor irregularities. The width and the length of the samples were measured using an optical microscope, whereas the thickness was calculated from these two dimensions, the mass and the density inferred from the unit cell parameters as determined by x-ray diffraction. The estimated dimensions of the samples (a × b × c) were 0.198 × 0.086 × 0.0032 cm 3 , 0.165 × 0.104 × 0.0045 cm 3 and 0.068 × 0.038 × 0.0036 cm 3 for the CaKFe 4 As 4 , CaK(Fe 0.983 Mn 0.017 ) 4 As 4 and MgB 2 samples, respectively. The magnetic field in the measurements was applied along the c-axis (H∥c) and along the longest, inplane edge, designated as the a-edge (H∥ab).
Measurements were performed in a Quantum Design MPMS3 SQUID magnetometer in the dc mode with a halfcylindrical quartz sample holder. Crystals were glued by a small amount of GE-7031 varnish directly to the holder for H∥ab measurements, a small L-shaped adapter made out of 0.05 mm thick copper foil was used for H∥c. For the latter measurements the accuracy of alignment was ∼5 • , inspection of the adapter and the samples after the measurements showed no effects of magnetic torque. In this work, all magnetic measurements were performed at ambient pressure (more precisely, at ∼5-10 Torr of helium exchange gas in the sample chamber of the magnetometer).
The following protocols were used in our trapped flux measurements. Zero-field cooled (ZFC): the sample was cooled down from above T c in H = 0; after the temperature was stabilized at the target base temperature, the magnetic field was increased to the target field, H M ; after a 1 min dwell time in the stable field, the magnetic field was decreased to H = 0; then, after another 1 min dwell time, M(T) measurements on warming started. Field-cooled (FC): the sample was cooled down from above T c in the target field; after the temperature was stabilized at the target base temperature, the magnetic field was decreased to H = 0; then after 1 min dwell time, M(T) measurements start. The linear mode was used to change the magnetic field. For the trapped field measurements in the ZFC protocol with H M ⩽ 100 Oe, as well as for H = 25 Oe ZFC M(T) measurements, the sample was cooled in H = 0 after; above T c , the demagnetization procedure for the superconducting magnet of MPMS3 was performed.
The demagnetization procedure used in this work (see appendix A.1 for details) results in negative remnant magnetic field of the order of, or less than 2 Oe. The protocols used for measurements of the magnetization associated with trapped flux result in larger remnant fields, up to ∼25 Oe, which depend on the values of the target fields H M . This is also discussed in detail in the appendix A.1.
For other, auxiliary, magnetic measurements the dc mode of the MPMS3 magnetometer was used as well.

CaKFe 4 As 4
3.1.1. H∥c. Figure 1 presents the low field, low temperature, temperature dependent ZFC and FC susceptibility, χ = M/H of CaKFe 4 As 4 for H∥c. The superconducting transition, with T c ≈ 34.7 K, is clearly seen in the data. Significant differences in the diamagnetic signal between ZFC and FC data point to a fairly large pinning. Assuming the ideal magnetic field screening χ ideal = −1/4π in the ZFC mode, we can evaluate the demagnetization factor N associated with the shape of the sample from χ measured = χ ideal × 1 1−N as 1 1−N ≈ 14.3. This value is not far from that obtained for rectangular cuboid (H∥c), N cuboid ≈ 4ab/[4ab + 3c(a + b)], using the dimensions above and equation (22) from [23]. Table A1 in the appendix A.2 contains values of demagnetization factor for all three samples used in this work, evaluated from the geometry of the sample and under the assumption of ideal magnetic field screening. These latter values will be used in the text.
Temperature dependent trapped flux magnetization data measured in H = 0 following ZFC and FC protocols are shown in figure 2. The samples were cooled to 5 K. The measurements were done for a set of different target fields. The data in figure 2 are qualitatively consistent with Bean's model [3]. For the FC protocol, initially the trapped flux magnetization increases until, at H ⩾ H * , it becomes target field independent. For the ZFC protocol initially the magnetic field does not penetrate the sample and no trapped flux magnetization is observed. At H ⩾ H p magnetic flux penetrates the sample, the vortices become pinned and trapped flux magnetization is observed. The trapped field magnetization increases with increase in the target field, until, at H ≈ 2H * + H p it becomes field-independent. The H p field corresponds to the lower critical field H c1 with the correction for demagnetization factor H c1 = H p × 1 1−N . The temperature dependence of the saturated value (corresponding to high target fields) of the trapped field magnetization is the same for both protocols and reflects the temperature dependence of the critical current.
Using the data from figure 2, we plot the trapped field magnetization at 5 K (in both FC and ZFC protocols) as a function of the target field H M in figure 3. (Note: see appendix A.3 for a short comment on an artifact due to a remnant field in the superconducting magnet of the magnetometer) M trap (H) saturates at about 3.5 kOe for the FC protocol and slightly below or about 7.5 kOe for the ZFC, a result that is consistent with expectations from the Bean's model. At low fields, M trap (H) approaches zero approximately linearly for the FC, whereas the ZFC data goes to zero at a finite, positive field H p > 0, and approaches H p with some upward curvature. We can infer a value for H p via two methods. To estimate H p we use a phenomenological expression M trap = A(H − H p ) α (A, H p and α are fitting parameters) to fit five to nine low field M trap > 0 points. The result of is H p = 45 ± 5 Oe, which translates into H c1 = 640 ± 70 Oe after the demagnetization factor correction, with the values of α between 1.5 and 1.8. Alternatively, we can take H p as the first point deviating from M trap = 0, resulting in H p = 50 ± 12 Oe, or H c1 = 715 ± 170 Oe. We can proceed further and analyze temperature-dependent critical current J c . Traditionally, this is done by measuring magnetization loops and using the Bean model to extract the values of J c (H, T) [2,3,24] (see appendix A.4). However, we can use the temperature dependent trapped field magnetization M trap (T) to evaluate self-field J c (T). For H∥c this will correspond to the in-plane critical current J ab c . For the cuboidshaped sample J c = 2 × 20 Mtrap w(1−w/(3l)) , where w and l are the width and length of the sample, respectively. The resulting data are shown in figure 4 together with the data evaluated from the magnetization loops in appendix A.4. Both sets of data show very similar temperature dependence, with slight (∼12%) difference in the absolute values. This difference is due to some

H∥ab.
We obtained and analyzed a similar set of data for H∥ab from the same crystal: low field, low temperature, temperature dependent ZFC and FC susceptibility (figure 5). The data show the same value of T c . Using the same arguments as in the case of H∥c above, we estimate 1 1−N ≈ 1.06 for this orientation. Temperature dependent trapped flux magnetization data measured in H = 0 following ZFC and FC protocols are shown in figure 6. Qualitatively, the data are similar to those for H∥c, with a measurably higher value of H * : H * ab ∼ 10 kOe vs H * c ∼ 3.5 kOe. It is also noteworthy that, whereas for the ZFC protocol, the M trap (T) appear to saturate above ∼2H * + H p . For the FC protocol, M trap (T) appears to have a maximum at ∼H * ≈ 10 kOe and then slightly decreases. This is seen more clearly in figure 7. The reason for this decrease is not clear at this point. Figure 7 presents trapped field magnetization at 5 K (in both FC and ZFC protocols) as a function of the target field H M . For this orientation H p = 66 ± 10 Oe (with the values of exponent α ranging from 1.2 to 1.4), which translates into H c1 = 70 ± 10 Oe. Alternatively, from the first deviation, we obtain H p = 50 ± 12 Oe, or H c1 = 53 ± 13 Oe.
The critical current density evaluated from the temperature dependent trapped field magnetization in comparison with the 5 K value obtained from the magnetization loop is presented in figure 8. It should be mentioned that, in contrast to the H∥c case above, the H∥ab measured magnetization has contributions from two, potentially different, critical current densities, J c2 and J c3 (see appendix A.6 and [24][25][26] for schematics of the Bean model for this field configuration and in-depth discussions). According to [25,26] in CaKFe 4 As 4 the value of J c2 is about an order of magnitude larger than J c3 , so in this material, the use of a simple Bean model for H∥ab yields values very close to J c2 . The difference between the values of J c2 at 5 K obtained from the magnetization loop and from the trapped field magnetization is ∼17%, a bit larger than for H∥c.

CaK(Fe 0.983 Mn 0.017 ) 4 As 4
We repeated a similar set of measurements on a CaK(Fe 0.983 Mn 0.017 ) 4 As 4 single crystal. Figure 9 presents the low temperature, low field susceptibility of CaK(Fe 0.983 Mn 0.017 ) 4 As 4 for two orientations of the magnetic field, (a) H∥c and (b) H∥ab. Significant differences between ZFC and FC data suggest large pinning. Superconducting transition is detected at T c ≈ 18.7 K, consistent with [17]. The values of 1 1−N are estimated as ≈ 13.6 and ≈ 1.11 for H∥c and H∥ab respectively.
The temperature dependent trapped flux magnetization data measured in H = 0 following ZFC and FC protocols are shown in figures 10 and 11 for H∥c and H∥ab respectively. Qualitatively, these data are very similar to those for pure CaKFe 4 As 4 shown above. Based on the data in figures 10 and 11 we can plot the trapped field magnetization at 1.8 K (in both FC and ZFC protocols) as a function of the target field H M for both orientations of the applied magnetic field in figure 12.  Using the same arguments and fits as for pure CaKFe 4 As 4 , we can estimate the values of H p = 9.3 ± 0.3 Oe for H∥c and 13.5 ± 10 Oe for H∥ab (the values of α were 1.9-2.0 and 1.6-1.7 respectively). Using the values of 1 1−N for two magnetic field directions evaluated from the low field M/H(T) data, we obtain the estimates of H c1 (1.8K) = 130 ± 10 Oe and 15 ± 11 Oe for H∥c and H∥ab respectively. Alternatively, from the first deviation, we obtain H c1 (1.8K) = 205 ± 70 Oe and 28 ± 6 Oe for H∥c and H∥ab respectively.
We analyze the self-field temperature-dependent critical current J c using anisotropic M trap (T) data for H M for which saturation is already achieved (20 kOe in this case). The results are presented in figure 13. As was the case for pure CaKFe 4 As 4 , for H∥ab we assume that J c2 is significantly larger than J c3 and that for this field direction the J c2 is (approximately) evaluated.

MgB 2
Below we extend these measurements to a system with much lower pinning: a MgB 2 single crystal. Figure 14 presents temperature-dependent, low field ZFC and FC magnetic susceptibility M/H. The superconducting transition temperature T c ≈ 37.7 K, is consistent with other single crystal data [20,22]. Note that, in contrast to CaKFe 4 As 4 and CaK(Fe 0.983 Mn 0.017 ) 4 As 4 crystals discussed earlier, the pinning in MgB 2 crystal is rather small, resulting in only ≈8% difference in ZFC and FC data for H∥ab ( figure 14(b)). As above, from the low field ZFC M/H data we can estimate 1 1−N values for H∥c (≈ 14.1) and H∥ab (≈ 1.01).
Temperature dependent trapped flux magnetization data for MgB 2 measured in H = 0 following ZFC and FC protocols are shown in figures 15 and 16 for H∥c and H∥ab respectively. Qualitatively, the behavior appears to be similar to that obtained in CaKFe 4 As 4 and CaK(Fe 0.983 Mn 0.017 ) 4 As 4 , except for a shoulder observed for high H M values at temperatures close to T c for FC and ZFC protocols in both orientations of the magnetic field. In our opinion, unfortunately, this shoulder is an artifact associated with a remnant magnetic field (negative) in the superconducting magnet of the magnetometer. Although the H = 0 conditions are reported by the instrument, the sample is experiencing a finite H ̸ = 0 field due to vortices pinned in the windings of the superconducting magnet. This remnant field depends on the history of the measurements and is typically larger if the magnet was going to higher field. A detailed discussion is presented in the appendix A.1. The recipes to mitigate this issue [27] are not readily applicable to trapped field measurement protocols. This artifact is much more pronounced in MgB 2 measurements due to significantly lower pinning and consequently a more sizable ZFC, low field, low temperature, signal (figure 14). Since this shoulder will ultimately affect M trap vs H M analysis, this additional contribution is subtracted, as briefly described in appendix A.7.
We plot the trapped field magnetization at 1.8 K (in both FC and ZFC protocols), with the corrections to the contribution from the remnant field of the magnet, as a function of the target field H M for both orientations of the applied magnetic field in figure 17. From the measurements using the ZFC protocol we estimate H p = 0.28 ± 0.02 kOe for H∥c (values of α range   from 1.8 to 2.6 for different selections of points. For H∥ab we did not perform fits but rather took the midpoint between the two highest H M apparent M trap = 0 points, resulting in H p = 1.05 ± 0.05 kOe. Using the estimates of the demagnetization factor for two orientations, we obtain the values of H c1 at 1.8 K as 3.9 ± 0.3 kOe and 1.06 ± 0.05 kOe for H∥c and H∥ab respectively. We use the data of figure 16 to estimate the self-field critical current density of MgB 2 crystal (figure 18). We present only J ab c (T) results (from H∥c data) as we do not know the ratio of J c2 and J c3 in H∥ab data (see discussion above). As expected, the critical current density for MgB 2 single crystal is significantly smaller than in CaKFe 4 As 4 and CaK(Fe 0.983 Mn 0.017 ) 4 As 4 crystals. It is noteworthy that the evaluation of J ab c from the magnetization loop and from the trapped field magnetization measurements show they are much closer to each other than for CaKFe 4 As 4 and CaK(Fe 0.983 Mn 0.017 ) 4 As 4 . One of the reasons could be that flux creep in MgB 2 is measurably smaller (see appendix A.8). . Note that the apparent noise in ZCF points between 1.05 and 2 kOe is probably due to imperfect subtraction of the contribution due to the magnet remnant field.

Discussion and summary
We start with an obvious, but necessary observation; by comparing the low field susceptibility data (figure 1 for example) and the trapped field magnetization data M trap (T) obtained using the ZFC and FC protocols with different values of the target fields H M (figure 2 for example), we see that indeed, trapped field magnetization offers a very accurate value of T c , whereas the measured signal is significantly higher. This increased signal, combined with a nominal zero field measurement, makes trapped flux measurements exceptionally attractive when very small samples are combined with very large addenda (i.e. diamond anvil pressure cells). A comparison of H c1 and J c values obtained in this work with the available literature data, where more traditional methods were used, is presented in table 1. Our data are within the range of the available literature data.
Normalized data for the temperature-dependent critical current density for all three materials are presented in figure 19. Several observations can be made: the temperature dependencies of the critical current densities for the same sample are reasonably close; whereas J c (T) for CaK(Fe 0.983 Mn 0.017 ) 4 As 4 and MgB 2 show rather simple functional behavior with a negative curvature; the observed behavior for CaKFe 4 As 4 is more complex, possibly indicating different pinning regimes at different temperatures. We can compare these results with two models, T c fluctuation induced pinning (δT c model) and mean free path fluctuation induced pinning (δl model) [37]. None of these models appear to work well for our data, although qualitatively the δl model might account for the   [36] features in J c (T) for CaK(Fe 0.983 Mn 0.017 ) 4 As 4 and MgB 2 , whereas a crossover between δT c and δl models could be observed in CaKFe 4 As 4 on cooling.
To summarize, in agreement with the statements in [14], trapped flux measurements are indeed a very powerful experimental approach to evaluate basic superconducting properties, in particular for the case of small samples or samples contained in a pressure cell or non-superconducting matrix, since the measurements are performed in nominally zero applied field, thus almost canceling the background signal associated with the pressure cell or the matrix.
Caution should be taken in consideration of the demagnetization factor associated with the sample for a particular orientation of the magnetic field, flux creep and remnant magnetic field in the superconducting magnet of the magnetometer. The latter issue is more significant in samples with weak and moderate pinning, as in this case it affects not only low field trapped flux magnetization measurements in the FC and ZFC protocols, but the high field measurements as well, causing a constant field magnetization contribution to the data. In this work, we suggest a procedure that allows for the mitigation of this issue. We show that flux creep can induce a moderate error in the evaluation of the critical current density from the temperature dependence of the trapped flux magnetization in comparison to the nonrelaxation J c value obtained from the analysis of the magnetization loops. This being the case, evaluating J c from the temperature dependence of the saturated trapped flux magnetization would still be a more accurate method if the signal of 'normal' magnetization loops is contaminated with a large background from a pressure cell (or any large signal sample holder). The magnetic flux penetration and so the exact functional behavior of M trap (H M ) in ZFC protocol, close to the first penetration field, even for uniform finite samples is shape-dependent [38] and could be evaluated numerically (this is beyond the scope of this work). Although, having in mind other experimental uncertainties, using some phenomenologically 'reasonable' criterion for the flux penetration field H p might be appropriate.
Last but not least, in many aspects our results for these known, single crystalline, superconductors that were well shaped and measured at ambient pressure are similar to those for superhydrides at megabar pressures [14] and as such hopefully serve as a baseline and help to clear the doubts in the interpretation of the measurements in [14].

Data availability statement
The data cannot be made publicly available upon publication because the cost of preparing, depositing and hosting the data would be prohibitive within the terms of this research project. The data that support the findings of this study are available upon reasonable request from the authors.

Acknowledgments
This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The Ames National Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. Useful conversations with Vladimir Kogan, Nestor Haberkorn, Ruslan Prozorov and Juan Schmidt are appreciated. This work stems from collaboration with V S Minkov, V Ksenofontov, E F Talantsev and M I Eremets on superhydrides [14]. P C C acknowledges Alonso Quijano for his single minded focus.

A.1. Demagnetization procedure and remnant field in superconducting magnet
The demagnetization procedure used in this work comprised the following commands in MPMS3 MultiVu: This procedure results in a negative remnant field of ⩽2 Oe. This can be evaluated, for example, by the following measurements that use common, broadly available materials. One can measure a temperature dependent magnetization of elemental Pb near the superconducting transition temperature as a function of applied small magnetic fields changing monotonically from some negative to positive value (see figure A1). The value of the applied field at the maximum of the measured T c values (as well as M = 0 in superconducting state) would then correspond to the field needed to offset the remnant field in the magnet. Alternatively one can use any sample without magnetic hysteresis, with linear-at-low-fields M(H) and reasonably large signal. The Pd standard supplied with the magnetometer is suitable for this. The measurements of field dependent magnetization from −30 Oe to 30 Oe at T = 10 K are shown in figure A2(a). The demagnetization procedure was performed before each of the runs. Again, the value of the applied field corresponding to M = 0 is the field required to offset the remnant field. The remnant field values after the demagnetization procedure obtained in the measurements above are −1.5 Oe when Pb was used, and −0.6, −1.9, −1.1 Oe in the Pd runs. The remnant field after the demagnetization procedure is reported for the particular command sequence listed above. Readers are advised to construct their own 'demagnetization' command sequences resulting in remnant magnetization fields suitable for their purpose.
Since we cannot use the Oscillate mode in the trapped flux measurements (see e.g. [39]) the nominal 'H = 0' measurements conditions in this case would be contaminated by the remnant field in the superconducting magnet [27]. We can evaluate the remnant field for different values of magnetic field H M reached before it was decreased to zero in the Linear mode by measuring the magnetization of the Pd sample after the following procedure: the magnetic field was increased to the target field, H M ; after a 1 min dwell time in a stable field, the magnetic field was decreased to H = 0; then, after another 1 min dwell time the magnetization measurement was performed. The results are shown in the inset to figure A2(b). Using the data in figure A2(a), we can evaluate the dependence of the remnant field in the magnet from the target field H M value ( figure A2(b)). H rem is ∼ 5 Oe for H M = 1 kOe, becomes negative at and below H M = 3 kOe and saturates to H rem ∼ −25 Oe for H M ⩾ 20 kOe. Several scattered points between 45 kOe and 70 kOe probably reflect some unstable measurements and can be ignored.
The data in figure A2(b) provide the overall guidance on the remnant fields expected in MPMS3 superconducting magnets when the applied field is decreased to zero from H M of 1 kOe and higher. These data also show that whereas the data of three runs are consistent, there exist some scattering of the values of the remnant field even when exactly the same measurements protocol was used. Despite the smaller absolute values, in the relative terms, the scattering is still large below ∼1 kOe. To better quantitatively address the effect of remnant field on the trapped flux magnetization measurements, the procedure described in the appendix A.3 below was used.
It should be noted that the remnant field in a superconducting magnet depends on many factors, including the particular magnet design, field history, rates of the changes in field, target field approach (Linear, No Overshoot, Oscillate), dwell time after the field is declared stable by the magnetometer software, among others. The data above should serve as just a baseline for a 70 kOe MPMS3 magnet, with the understanding that curious readers will need to perform their own evaluation of the specific unit associated with the text that they are intrigued by.

A.2. Evaluation of demagnetization factors
Demagnetization factor values for the samples in this work were calculated using two different approaches: (a) assuming ideal magnetic field screening −1/4π in superconducting state in low field, low temperature ZFC magnetic susceptibility measurements; and (b) assuming the samples are ideal uniform cuboids and using equation (22) from [23]. The   shown in figure A3. It is of note that this procedure is relevant only for low fields not far from H p , as the saturated trapped field magnetization is several orders of magnitude higher than that in small fields (compare figures 2 and A3(a)). Panels (b) and (c) of the figure A3 show this procedure in more detail for 50 Oe and 75 Oe data. In these two panels, scaled ZFC H ≈ 1 Oe M(T) data with a proper sign were used, not a constant offset. Indeed the procedure and the resultant 'corrected data' appear to be reasonable. For both datasets the remnant field scaling contribution corresponds to ∼ 0.1 Oe remnant fields, but with different signs. The small sharp 'spikes' in the corrected data are the results of subtraction and are due to the field-dependence of the measured width of the superconducting transition that manifests itself more in this H∥c geometry.  Field-dependent critical current density obtained from the magnetization loops using the Bean's model [2,3,24]. extracted from the magnetization data using the Bean's model [2,3,24] as J c = 20∆M w(1−w/(3l)) , where δM is the difference in magnetization between the top and the bottom branches of the hysteresis loop, w and l are the width and length of the sample respectively. For H∥c, the in-plane critical current J ab c is measured. The obtained values of the critical current are consistent with the published results [25,26,29,40]. Figure A5 shows the time dependent magnetization of CaKFe 4 As 4 after the sample was cooled in zero field from above T c to 5 K, at which temperature a magnetic field of 20 kOe was applied along the c-axis and then removed to H = 0. The relaxation is close to exponential. In the first 10 min. the magnetization decreases by ∼5%. These data suggest that flux creep might have some effect on the measurements of temperature dependence of trapped flux magnetization (since such measurement takes ∼2 h) and therefore has some effect on the evaluation of the temperature dependence of the self-field critical current density. Figure A6 shows the critical current densities in a rectangular sample for magnetic fields applied (a) parallel to the c-axis and (b) in the ab plane. Further, detailed discussion is provided in [24][25][26]40]. A.7. Additional effects of remnant field in superconducting magnet in a low pinning sample Figure A7 shows ZFC temperature dependent trapped flux magnetization in MgB 2 for two, intermediate and high, values of H M . To address the artifact (the shoulder near T c ) in the data when evaluating trapped flux magnetization at 1.8 K, or more broadly, temperature dependent trapped flux magnetization, we assume, as we did for low field data in the appendix A.3, that the measured M trap (T) is a sum of two contributions: M trap (T) measured in 'real' H = 0 and the M(T) contribution associated with the fact that the measurements were performed in a remnant field of the superconducting magnet. The low field data in figure 14 show that except for a very small temperature range near T c , this second contribution is expected to be temperature independent. Then, subtracting from the data a constant, defined as an intercept of linear extrapolation of M(T) just below the shoulder with the vertical line at T c (see figure A7), would result in a reasonable representation of the data in 'real' H = 0.   Using the evaluation of remnant fields in the magnet in figure A2(b) we obtain H rem ≈ −9 Oe and ≈−26 Oe for H M = 3.5 kOe and 70 kOe respectively. These values correspond to ∼0.7 emu cm −3 and ∼1.9 emu cm −3 (respectively) contributions to the measured M trap (T) (this evaluation is done using the data in figure 14). The consistency with the ∼0.9 emu cm −3 and ∼2.0 emu cm −3 values in figure A7 is noteworthy.

A.6. Sketch of critical current densities in rectangular samples
Additional measurements were performed (figure A8) to experimentally confirm the hypothesis of the effect of the remnant magnetic field in the magnet on trapped field magnetization measurements. Comparison of the measurements in nominal H = 0 after H M = 70 and −70 kOe confirm that the remnant field has the sign opposite to the sign of H M . Moreover, measurements in small applied fields instead of nominal H = 0 enable a decrease in the size of the shoulder near T c to zero by careful choice of this small offset field (∼30 Oe in the measurements in figure A8). A close to parallel shift of the curves measured in different small offset fields justify the procedure of evaluation of the trapped field described above and in figure A7. Figure A9 shows the time dependent normalized magnetization of CaKFe 4 As 4 at T = 5 K and CaK(Fe 0.983 Mn 0.017 ) 4 As 4 and MgB 2 at T = 1.8 K after the ZFC protocol with H M = 20 kOe. Although any detailed analysis is beyond the scope of this work, these data clearly show smaller flux creep in MgB 2 , which is consistent with the analysis in [41], since the Ginzburg parameter of MgB 2 is several orders of magnitude smaller than that of CaKFe 4 As 4 [16,42].