Critical Current Density and Vortex Dynamics in Pristine and Irradiated KCa2Fe4As4F2

We report the critical current density (Jc) and vortex pinning properties in single crystals of a novel iron-based superconductor (IBS) KCa2Fe4As4F2 with large Jc in the pristine state, before and after introduction of artificial defects by swift-particle irradiation. The effects of 2.6 GeV U and 3 MeV proton irradiations in KCa2Fe4As4F2 single crystals on transition temperature Tc and Jc, including its dose dependence, are systematically studied. Jc~8 MA/cm2 under a self-field at 2 K in the pristine crystal is strongly enhanced up to 19.4 and 17.5 MA/cm2 by irradiation of 2.6 GeV U-ions and 3 MeV protons, respectively. Suppression of Tc and dose dependence of Jc in KCa2Fe4As4F2 is different from that in a representative IBS of (Ba,K)Fe2As2, which can be explained by considering the presence of embedded defects in pristine KCa2Fe4As4F2. The vortex dynamics in the pristine and proton irradiated KCa2Fe4As4F2 single crystals are also investigated from the analyses of the field dependence of Jc and the normalized magnetic relaxation rate. In addition to the contribution of embedded defects, weak collective pinning is considered for comprehensive analyses. Vortex dynamics in KCa2Fe4As4F2 is similar to those in (Ba,K)Fe2As2 to some extent, and different from that in anisotropic Li0.8Fe0.2OHFeSe. Large anisotropy, due to the presence of insulating blocking layers in KCa2Fe4As4F2, which leads to much lower irreversibility field (Hirr) compared with 122-type IBSs, strongly affect the vortex dynamics.


Introduction
Iron-based superconductors (IBSs) are attracting a great deal of attention due to their notable features, such as their relatively high transition temperature (T c ), high upper critical field (H c2 ), and large critical current density (J c ) [1]. In typical IBSs, such as 122-type compound (Ba,K)Fe 2 As 2 [2], electronic states are weakly anisotropic, although their crystal structures are characterized by the presence of two-dimensional FePn (Pn: As, P) or FeCh (Ch: S, Se, Te) layers. This is in contrast to quasi two-dimensional electronic states in cuprate superconductors with two-dimensional CuO 2 layers [3,4]. Recently, however, it was suggested that quasi two-dimensional electronic states emerge in novel IBSs, such as 12442-type compounds [5][6][7][8][9] or some IBSs consisting of FeSe layers sandwiched by thick insulating layers [10,11]. The 12442-type IBSs have double Fe 2 As 2 conducting layers between two neighboring Ca 2 F 2 insulating layers [5]. Due to the presence of insulating layers, anisotropic electronic states are realized as revealed by the highly anisotropic H c2 [12,13] and torque analyses [14]. KCa 2 Fe 4 As 4 F 2 is the first and a well-studied 12442-type IBS [8] and its anisotropic physical properties have been demonstrated. In KCa 2 Fe 4 As 4 F 2 , quasi two-dimensional electronic behavior, which is similar to that of cuprate superconductors, has been revealed by neutron spin resonance [15]. The quasi two-dimensional electronic state was also suggested from the large anisotropy of electrical resistivity, with ρ c /ρ ab > 100, and semiconductor-like ρ c [16]. Furthermore, a large anisotropy parameter, γ (= H c2 ab /H c2 c )~8, was also evaluated near T c [12,16]. Notable features of this highly anisotropic KCa 2 Fe 4 As 4 F 2 are moderate T c of~34 K, and larger J c , and H c2 compared with 122-type IBSs. Large H c2 //ab above 700 kOe at low temperatures has been evaluated [17], although the irreversibility field (H irr ), which separate zero and finite J c regions, is relatively low [12,16]. The in-plane J c evaluated from the measurement of irreversible magnetization at 2 K under the self-field is 8.2 MA/cm 2 , which is significantly larger compared with that of 122-type IBSs [16].
One of the most intriguing topics in KCa 2 Fe 4 As 4 F 2 is how the significantly high J c is enhanced by adding artificial pinning centers. It is well known that J c in superconductors, such as cuprates and IBSs, can be enhanced by introducing defects using swift particle irradiations, and significant effects on physical properties, such as remarkable enhancements of J c by irradiations of heavy ions and protons have been demonstrated [16,[18][19][20][21][22][23][24][25][26][27][28][29][30]. For KCa 2 Fe 4 As 4 F 2 , enhancement of J c up to 19 MA/cm 2 at 2 K under self-field is also demonstrated by introducing columnar defects in terms of heavy-ion irradiation, although the irradiation dose is limited [16]. It is also well known that introducing point defects by proton irradiation is effective at enhancing J c in cuprate superconductors [31,32] and IBSs [24,27]. A systematic study of the effect of columnar and point defects on J c in KCa 2 Fe 4 As 4 F 2 is demanded.
Furthermore, investigation of vortex pinning properties in KCa 2 Fe 4 As 4 F 2 is also required to understand its J c behavior. It is needless to say that vortex dynamics is one of the central issues related, not only to basic solid-state physics, but also to applications. Vortex dynamics in cuprates have been extensively studied in past decades [33,34]. When the c-axis coherence length, ξ c , is larger than the layer distance, d, such as in YBa 2 Cu 3 O 7 , the magnetic field generates well-connected quantized vortices via interlayer coupling, which creep collectively when the thermal energy is stronger than the pinning energy [33,34]. On the other hand, when the interlayer coupling is very weak, with ξ c < d, such as in Bi 2 Sr 2 CuCa 2 O 8 + y , quantized vortices can be considered as stacks of pancake vortices, which can creep individually in each layer [33,34]. These results suggest that anisotropic crystal structure and electronic structure strongly affect the behavior of vortices. Vortex dynamics in IBSs has also been studied by comparing them with those in cuprate superconductors. For analyses of vortex dynamics, magnetic relaxation in clean single crystals, where a critical state is realized in the whole sample, has been utilized since transport measurements are challenging due to the extremely large J c . Up to now, 122-type compounds, such as (Ba,K)Fe 2 As 2 or Ba(Fe,Co) 2 As 2 , and 11-type compounds, such as Fe(Te,Se) and FeSe, have mainly been studied [24,27,35,36]. In these weakly anisotropic materials with an anisotropy parameter 2-3, vortex pinning properties have been interpreted in either weak-collective pinning or strong pinning scenarios [1,2,24,27,35]. On the other hand, in one of IBSs, Li 0.8 Fe 0.2 OHFeSe, with a highly anisotropic crystal structure, breakdown of conventional creep theory has been pointed out [37]. To understand general characteristics of vortex pinning in anisotropic high temperature superconductors, including cuprates and IBSs, further investigation of vortex dynamics in anisotropic materials, such as KCa 2 Fe 4 As 4 F 2 , is demanded.
In this study, we investigated the effect of columnar defects and point defects created by irradiation of U ions and protons in KCa 2 Fe 4 As 4 F 2 single crystals. Changes in T c and J c were systematically studied as a function of the irradiation dose. We also investigated the vortex pinning properties and the vortex dynamics in KCa 2 Fe 4 As 4 F 2 before and after the introduction of point defects. A part of the data of KCa 2 Fe 4 As 4 F 2 were compared to those of (Ba,K)Fe 2 As 2 , which belong to 122-type IBSs and is a representative IBS, and those of Li 0.8 Fe 0.2 OHFeSe, which has anisotropic crystal structure similar to KCa 2 Fe 4 As 4 F 2 .

Experimental Methods
Single crystals of KCa 2 Fe 4 As 4 F 2 were grown using the self-flux method with KAs as the flux. Details of sample growth and the basic physical properties are reported in Reference [12]. Ba 0.6 K 0.4 Fe 2 As 2 single crystals, which were used as reference materials to be compared with KCa 2 Fe 4 As 4 F 2 , were also synthesized using the FeAs self-flux method [27,38]. The 2.6 GeV U ions were irradiated parallel to the c-axis at room temperature at RIKEN Ring Cyclotron in RI Beam Factory, operated by RIKEN Nishina Center and CNS, the University of Tokyo. The irradiation dose was evaluated using the doseequivalent magnetic field called the "matching field", at which all defects are occupied by single vortices, B Φ = nΦ 0 . Here, n is the areal density of CDs and Φ 0 is a flux quantum. The 3 MeV protons were irradiated parallel to the c-axis at room temperature at the National Institute of Radiological Sciences Heavy Ion Medical Accelerator in Chiba, Japan (NIRS-HIMAC). Cross-sectional observations of the single crystals were performed with a scanning transmission electron microscope (STEM) (JEOL, JEM-2100F, Akishima, Tokyo, Japan). The spatial resolution of the JEOL JEM-2100F microscope was 0.2 nm; however, we set it as 0.5 nm to add contrast to the STEM image. The specimens for STEM were prepared by digging and milling using a focused-ion beam (FIB), which is called the microsampling technique. Final milling using FIB was conducted at an acceleration voltage of 30 kV and with a very weak ion current of~10 pA without tilting the specimen. Magnetization was measured in a commercial superconducting quantum interference device magnetometer (MPMS-XL5, Quantum Design, San Diego, CA, United States) with an applied magnetic field parallel to the c-axis. In this system, temperature accuracy was ±1%, and magnetic field resolutions were 0.1 G up to 5 kOe and 1 G up to 50 kOe.

Experimental Results and Discussions
In this section, first, the effects of 2.6 GeV U and 3 MeV proton irradiations on T c and J c estimated from magnetization measurements in KCa 2 Fe 4 As 4 F 2 are discussed. STEM images of both the pristine and 2.6-GeV U-irradiated KCa 2 Fe 4 As 4 F 2 are also presented. Second, the vortex dynamics of the pristine and 3 MeV proton irradiated crystals are discussed based on magnetization measurements and analyses of magnetic relaxation.
3.1. Effects of 2.6 GeV U and 3 MeV Proton Irradiations on T c and J c in KCa 2 Fe 4 As 4 F 2 First, STEM analyses of the pristine and 2.6-GeV U ion-irradiated KCa 2 Fe 4 As 4 F 2 crystals with an electron beam injected along a-axis were performed. As shown in the STEM images in Figure 1a,b, some black lines parallel to the ab-plane were observed in both pristine and U-irradiated KCa 2 Fe 4 As 4 F 2 . Although the existence of these sparsely distributed defects cannot explain the significantly larger J c in pristine KCa 2 Fe 4 As 4 F 2 , they suggest the presence of atomic scale defects or chemical inhomogeneities, which may have contributed as pinning centers. In the 2.6-GeV U-irradiated KCa 2 Fe 4 As 4 F 2 , clear columnar defects with a diameter~5 nm along the c-axis were observed, which were similar to the case of Ba(Fe,Co) 2 As 2 and (Ba,K)Fe 2 As 2 [20,23,39].
Next, the effects of columnar defects on T c and J c in KCa 2 Fe 4 As 4 F 2 were investigated. Figure 2a shows zero-field-cooled magnetization measurements with a H//c-axis for 2.6-GeV U-irradiated KCa 2 Fe 4 As 4 F 2 . It is clear that the onset of superconductivity is systematically suppressed by increasing B Φ . Normalized T c estimated from the linear extrapolation from the low temperature is summarized in Figure 2b. The B Φ dependence of T c is also shown in the inset of Figure 2b. For comparison, the B Φ dependence of T c for (Ba,K)Fe 2 As 2 are also shown [26]. T c of both KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 are monotonically decreased with increasing B Φ , although B Φ dependence of T c in KCa 2 Fe 4 As 4 F 2 is stronger than that of (Ba,K)Fe 2 As 2 . The rate of the reduction of the normalized T c estimated from Figure 2b is 0.3%/T and 0.9%/T for (Ba,K)Fe 2 As 2 and KCa 2 Fe 4 As 4 F 2 , respectively.
If columnar defects only destroy local superconductivity, T c should not be affected by introducing such defects. The suppression of T c can be explained by considering the effect of point defects created by secondary electrons, which are generated via interaction with highly energetic U ions and the lattice [26]. It is known that magnetic impurities lead to suppression of T c in s-wave superconductors, as discussed by Abrikosov and Gor'kov [40], and point defects may work similar to magnetic impurities in anisotropic KCa 2 Fe 4 As 4 F 2 . Unfortunately, however, it is not easy to estimate the dose and energy of such secondary electrons in a given material. Next, the effects of columnar defects on Tc and Jc in KCa2Fe4As4F2 were investigated. Figure 2a shows zero-field-cooled magnetization measurements with a H//c-axis for 2.6-GeV U-irradiated KCa2Fe4As4F2. It is clear that the onset of superconductivity is systematically suppressed by increasing B. Normalized Tc estimated from the linear extrapolation from the low temperature is summarized in Figure 2b. The B dependence of Tc is also  in s-wave superconductors, as discussed by Abrikosov and Gor'kov [40], and point defects may work similar to magnetic impurities in anisotropic KCa2Fe4As4F2. Unfortunately, however, it is not easy to estimate the dose and energy of such secondary electrons in a given material.   [26]. The B Φ dependence of T c is also shown in the inset of (b).
Next, the in-plane J c for H//c-axis in KCa 2 Fe 4 As 4 F 2 is evaluated by measuring the irreversible magnetization using the extended Bean model [20,41,42],  Figure 2c shows the B Φ dependence of J c in KCa 2 Fe 4 As 4 F 2 irradiated with 2.6 GeV U ions at T = 2 K under the self-field. For comparison, similar data for a (Ba,K)Fe 2 As 2 single crystal are also shown [26]. The B Φ dependence of J c in KCa 2 Fe 4 As 4 F 2 shows similar trends to that of (Ba,K)Fe 2 As 2 , although the B Φ for the largest J c are quite different. In the low B Φ region, J c is enhanced in proportion to B Φ −1/2 . This can be understood by a simple estimation based on half-loop excitations of vortices in the matrix of discontinuous columnar defects [26]. By increasing B Φ further, J c takes a maximum at a certain B Φ , and it starts to decline above this value. The maximum J c of 19.4 MA/cm 2 is realized in KCa 2 Fe 4 As 4 F 2 with B Φ = 2 T, and that of 15 MA/cm 2 is realized in (Ba,K)Fe 2 As 2 with B Φ = 32 T. This value is more than twice as large as that of J c of the pristine sample, and larger than that of irradiated (Ba,K)Fe 2 As 2~1 5 MA/cm 2 , as shown in Figure 2c [26]. The J c in KCa 2 Fe 4 As 4 F 2 starts to decline above B Φ = 4 T, which is much earlier than that in (Ba,K)Fe 2 As 2 , as shown in Figure 2c. This early J c suppression at a smaller B Φ is consistent with stronger suppression of T c , which could be due to stronger generation of secondary electrons, as discussed above.
Following the effect of columnar defects, the effects of point defects on T c and J c in KCa 2 Fe 4 As 4 F 2 were investigated. Similar to the case of columnar defects, the onset of superconductivity was systematically suppressed by increasing dose, as shown in Figure 3a. Figure 3b shows the dose dependence of normalized T c in 3 MeV proton irradiated KCa 2 Fe 4 As 4 F 2 . The dose dependence of T c is also shown in the inset of Figure 3b. For comparison, the dose dependence of T c for (Ba,K)Fe 2 As 2 is also shown in Figure 3b and in its inset [43]. The T c of both KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 monotonically decrease with increasing proton dose. It should be noted that the rate of T c reduction in both KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 estimated from Figure 3b are 0.9 and 1.0% per 1 × 10 16 ions/cm 2 , respectively. These results suggest that the density of created point defects by proton irradiation and their effects on T c is similar in KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 . This is in contrast to the case of 2.6-GeV U-irradiated KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 , where the amount of point defects generated by secondary electrons may depend on the material. Figure 3c show the dose dependence of J c in KCa 2 Fe 4 As 4 F 2 irradiated with 3 MeV proton at T = 2 K under a self-field. For comparison, similar data for (Ba,K)Fe 2 As 2 measured at H = 1 kOe are also shown. J c in KCa 2 Fe 4 As 4 F 2 reaches its maximum of 17.5 MA/cm 2 at a dose of 3 × 10 16 ions/cm 2 . On the other hand, J c of (Ba,K)Fe 2 As 2 reaches its maximum of 14 MA/cm 2 at a dose of 5 × 10 16 ions/cm 2 . The doses for maximum J c are similar, although that of KCa 2 Fe 4 As 4 F 2 is a little smaller. Such a difference can be explained by the fact that the pristine KCa 2 Fe 4 As 4 F 2 contains embedded point defects, which are atomic-scale defects due to strains or chemical inhomogeneities, and are responsible for high J c by working as pinning centers [16]. As discussed above, suppression of T c and the dose dependence of J c in 2.6-GeV U-ion or 3 MeV protonirradiated KCa 2 Fe 4 As 4 F 2 are different from those in typical IBS of (Ba,K)Fe 2 As 2 , which can be explained by considering the embedded defects in pristine KCa 2 Fe 4 As 4 F 2 . It should be emphasized that maximum J c in KCa 2 Fe 4 As 4 F 2 is larger than that in 122 compounds, which have been extensively studied as raw materials for wires and tapes for future high-field applications [44][45][46]. Fabrications of wires and tapes of KCa 2 Fe 4 As 4 F 2 are demanded.  netic field. These features are similar to those in optimally-doped (Ba,K)Fe 2 As 2 [27] and Li 0.8 Fe 0.2 OHFeSe [37]. By contrast, a broad peaks in J c -H curves, sometimes referred to as fish-tail effects, are observed in other IBSs, such as Ba(Fe,Co) 2 As 2 [24] and Fe(Te,Se) [35]. One of the possible mechanisms of the fish-tail effect is weak collective pinning, which is attributed to the presence of dense atomic-scale defects [47,48]. The absence of a fish-tail effect in KCa 2 Fe 4 As 4 F 2 is possibly explained by the dominance of strong pinning due to the presence of embedded defects in pristine crystal. Furthermore, at low temperatures below 15 K and above~10 kOe, J c shows a power-law decay with the field, J c ∝ H −α , with α~1. The decay of J c proportional to H −1 is observed in YBa 2 Cu 3 O 7 with columnar defects, which is explained as follows [19]; for example, at H = 10 kOe, the distance between vortices is (Φ 0 /H) 1/2~5 0 nm. If the pinning centers are sparse and their average separation is larger than 50 nm, all these pinning centers are occupied by vortices above 10 kOe. Hence above this field, the pinning force F p will stay constant, in spite of the increase in H. So, the value of J c will decrease in proportional to H −1 , since F p = 1/c·J c H [49,50]. In other words, observation of α~1 suggests that there are sparse strong pinning centers in the pristine KCa 2 Fe 4 As 4 F 2 , which may also explain the exceptionally large J c at low temperatures in this system. It may also explain the slightly lower T c in KCa 2 Fe 4 As 4 F 2 compared with (Ba,K)Fe 2 As 2 with a similar hole number per Fe. On the other hand, at higher temperatures above 15 K, J c decreases faster with magnetic field, and no power-law field dependence is observed. Similar behavior is also observed in 3 MeV proton irradiated KCa 2 Fe 4 As 4 F 2 , where J c takes its maximum at a dose of 3 × 10 16 ions/cm 2 . As shown in Figure 4b, at low temperatures below 15 K and above~20 kOe, J c again shows the power-law decay, H −α with α~1. Above 15 K, J c decreases strongly with the magnetic field. This trend is very similar to that in the pristine sample, although the field dependence of J c is a little weaker. As shown in Figure 4c, in 3 MeV proton irradiated KCa 2 Fe 4 As 4 F 2 with a dose of 10 × 10 16 ions/cm 2 , the power-law field dependence of J c with smaller α (~1/2) is observed at low temperatures, although such behavior is absent at temperatures above 10 K. With increasing the proton dose, α at low temperatures changes from~1 to~1/2. In the case of sparse strong pinning, α = 1/2 or 5/9 is predicted [48,51]. So, the change of α from 1 to~1/2 may indicate the change of pinning, from very sparse strong pinning to sparse strong pinning. Vortices in the pristine KCa 2 Fe 4 As 4 F 2 are pinned by embedded defects. After the proton irradiation, the strong pinning of vortices by point defects introduced by proton irradiation becomes dominant. We have reported changes of α from~1/2 to~1/3 by creating point defects via proton irradiation in various IBSs [24,27,52,53]. The difference in the field dependence of J c between KCa 2 Fe 4 As 4 F 2 and other IBSs may be caused by the existence of embedded defects in the pristine KCa 2 Fe 4 As 4 F 2 . Another remarkable feature of M(H) is fast decay of J c at higher temperatures and higher magnetic fields. Such behavior of J c may be related to highly anisotropic properties in KCa 2 Fe 4 As 4 F 2 , which will be discussed below.
Vortex dynamics in the pristine and proton irradiated KCa 2 Fe 4 As 4 F 2 crystal were investigated by measuring the normalized magnetic relaxation rate, S = |dlnM(t)/dlnt|. The decay of magnetization with time, M(t), due to creep motion of vortices was traced for an hour, after the critical state was prepared. Figure 5a,b show the magnetic field dependence of S at T = 5, 10, and 15 K for H//c in the pristine and 3 MeV proton irradiated KCa 2 Fe 4 As 4 F 2 at a dose of 3 × 10 16 ions/cm 2 , respectively. The dimensions of the pristine and proton irradiated samples were 0.0178 × 0.0212 × 0.0008 and 0.0235 × 0.049 × 0.0013 cm 3 , respectively. There were three notable features in the behavior of S. First, characteristic dip features at low magnetic fields below 10 kOe could be identified in both the pristine and 3 MeV proton irradiated KCa 2 Fe 4 As 4 F 2 . The characteristic field for the suppression of S is roughly equal to the self-field H sf = J c ·t, as we discussed in Ref. [54], where t is the thickness of the sample, as indicated by arrows. H sf values for the pristine sample were evaluated as 4.7 kOe, 2.0 kOe, and 0.7 kOe at 5 K, 10 K, and 15 K, respectively, while those for the proton-irradiated sample were evaluated as 10.5 kOe, 5.3 kOe, and 2.2 kOe at 5 K, 10 K, and 15 K, respectively. Similar suppressions of S at low magnetic fields below the self-field have been observed in pristine Ba(Fe,Co) 2 As 2 and (Ba,K)Fe 2 As 2 [24,27]. Second, at high magnetic fields, S in the pristine KCa 2 Fe 4 As 4 F 2 increases rapidly above a temperature dependent characteristic field, although S in the proton-irradiated KCa 2 Fe 4 As 4 F 2 increases monotonically up to 40 kOe. Third, the values of S in KCa 2 Fe 4 As 4 F 2 are larger than those in (Ba,K)Fe 2 As 2 and Ba(Fe,Co) 2 As 2 with values of 0.02-0.03 [24,27]. Vortex dynamics in the pristine and proton irradiated KCa2Fe4As4F2 crystal were investigated by measuring the normalized magnetic relaxation rate, S = |dlnM(t)/dlnt|. The decay of magnetization with time, M(t), due to creep motion of vortices was traced for an hour, after the critical state was prepared. Figure 5a,  respectively. Similar suppressions of S at low magnetic fields below the self-field been observed in pristine Ba(Fe,Co)2As2 and (Ba,K)Fe2As2 [24,27]. Second, at high m netic fields, S in the pristine KCa2Fe4As4F2 increases rapidly above a temperature dep ent characteristic field, although S in the proton-irradiated KCa2Fe4As4F2 increases m tonically up to 40 kOe. Third, the values of S in KCa2Fe4As4F2 are larger than tho (Ba,K)Fe2As2 and Ba(Fe,Co)2As2 with values of 0.02-0.03 [24,27].  Now, we show that α, which determines the field dependence of J c (∝H −α ), can be overestimated in KCa 2 Fe 4 As 4 F 2 due to a large and field-dependent S(H). This is explained conceptually in Figure 6. When the magnetic field is applied and critical state is formed at t = 0, magnetic relaxation starts right after that. On the other hand, magnetization measurements for the evaluation of J c are done after some time delay t. If S(H) does not depend on a magnetic field, reduction of J c , expressed by black arrows, is constant at all fields. In the case of KCa 2 Fe 4 As 4 F 2 with a field-dependent S(H) (dS(H)/dH > 0), however, reduction of J c with time becomes larger at higher fields, as expressed by red arrows. Such an increased reduction of J c at high fields makes apparent α larger, which may explain the relatively large α in KCa 2 Fe 4 As 4 F 2 . t = 0, magnetic relaxation starts right after that. On the other hand, magnetization meas-urements for the evaluation of Jc are done after some time delay t. If S(H) does not depend on a magnetic field, reduction of Jc, expressed by black arrows, is constant at all fields. In the case of KCa2Fe4As4F2 with a field-dependent S(H) (dS(H)/dH > 0), however, reduction of Jc with time becomes larger at higher fields, as expressed by red arrows. Such an increased reduction of Jc at high fields makes apparent α larger, which may explain the relatively large α in KCa2Fe4As4F2. Figure 6. Schematic magnetic field dependence of Jc in a double logarithmic plot at different times after preparing the critical state at t = 0. When the normalized magnetic relaxation rate increases appreciably with the field, measured Jc values at high fields after some time delay (t, 10t, …) become smaller, resulting in stronger apparent magnetic field dependence of Jc, namely a larger α.
Next, we discuss the behavior of S(H) in low and high magnetic fields. According to the collective creep theory, the H dependence of S is determined by the variation of the glassy exponent μ, as observed in YBa2Cu3O7 [55]. In this theory, the glassy exponent () is related to the vortex-bundle size [33]. In a three-dimensional system, it is predicted as  = 1/7, 5/2, 7/9 for single-vortex, small-bundle, and large-bundle regimes, respectively [33,56]. The collective creep theory is characterized by the J dependence of U. Collective creep theory, which considers vortex elasticity, predicts an inverse power law form for the energy barrier: where Jc0 is the temperature-dependent critical current density in the absence of flux creep, U0 is the flux activation energy in the absence of flux creep, and U(J) is flux activation energy [57]. This formula can be applied when relaxation process goes on and J is reduced (J ≪ Jc0). On the other hand, at low temperatures and fields, the simple linear relationship proposed in the Anderson-Kim model [33], as shown below, is often accurate: This model neglects vortex elasticity and vortex-vortex interactions. This relation is often limited to the early stages of the relaxation process (J~Jc0). To express behavior of U(J) for a wide range of J, described by Equations (2) and (3), the interpolation formula is commonly used, as shown below [57]: Figure 6. Schematic magnetic field dependence of J c in a double logarithmic plot at different times after preparing the critical state at t = 0. When the normalized magnetic relaxation rate increases appreciably with the field, measured J c values at high fields after some time delay (t, 10t, . . . ) become smaller, resulting in stronger apparent magnetic field dependence of J c , namely a larger α.
Next, we discuss the behavior of S(H) in low and high magnetic fields. According to the collective creep theory, the H dependence of S is determined by the variation of the glassy exponent µ, as observed in YBa 2 Cu 3 O 7 [55]. In this theory, the glassy exponent (µ) is related to the vortex-bundle size [33]. In a three-dimensional system, it is predicted as µ = 1/7, 5/2, 7/9 for single-vortex, small-bundle, and large-bundle regimes, respectively [33,56]. The collective creep theory is characterized by the J dependence of U. Collective creep theory, which considers vortex elasticity, predicts an inverse power law form for the energy barrier: where J c0 is the temperature-dependent critical current density in the absence of flux creep, U 0 is the flux activation energy in the absence of flux creep, and U(J) is flux activation energy [57]. This formula can be applied when relaxation process goes on and J is reduced (J J c0 ). On the other hand, at low temperatures and fields, the simple linear relationship proposed in the Anderson-Kim model [33], as shown below, is often accurate: This model neglects vortex elasticity and vortex-vortex interactions. This relation is often limited to the early stages of the relaxation process (J~J c0 ). To express behavior of U(J) for a wide range of J, described by Equations (2) and (3), the interpolation formula is commonly used, as shown below [57]: On the other hand, from the Arrhenius relation, U can be also described as: where k B is the Boltzmann constant and t eff is the effective hopping attempt time [34]. By combining Equations (4) and (5), time and temperature dependent J(T,t) and S are given as follows [57]: From this formula, we can observe that S is enhanced in the single vortex regime, where µ takes a relatively small value of 1/7. On the other hand, the density of vortices is relatively sparse at fields below H sf . In such a low-field regime, vortices may behave independently. So, one of the possible origins of the peak in S(H) at low magnetic fields is the enhanced S(H) with a smaller µ in the single vortex regime. It should be noted that, in the case of proton irradiated (Ba,K)Fe 2 As 2 , peaks in S(H) disappear and only dip structures are observed, and α changes from 1/2 to 1/3 [27]. From these facts, enhancement of contribution of strong pinning by introducing point defects in (Ba,K)Fe 2 As 2 is suggested [27]. On the other hand, the degree of suppression of S in KCa 2 Fe 4 As 4 F 2 after proton irradiation is small and α is still larger than 1/2. Comparing the effect of proton irradiation on S(H) at low magnetic fields in these two materials, it is suggested that strong pinning nature of vortices in KCa 2 Fe 4 As 4 F 2 is even more dominant after proton irradiation compared with that in (Ba,K)Fe 2 As 2 . This is consistent with the change of α from 1 to 1/2, as discussed above. At high magnetic fields, S(H) gradually increases with increasing H. As discussed above, one of possible explanations for the increase in S(H) is gradual reduction of µ caused by the change in the vortex bundle size, from small bundles (µ = 5/2) to large bundles (µ = 7/9). Another possible origin of gradual increase in S(H) with increase H is the significantly low H irr in KCa 2 Fe 4 As 4 F 2 originated from highly anisotropic crystal structure and resulting large anisotropy [12,16]. S(H) is expected to diverge as H approaches H irr , and H irr is reduced with increasing temperature. So, the strong anisotropy in KCa 2 Fe 4 As 4 F 2 significantly affects the behaviors of S at high temperatures and high magnetic fields. It should be pointed out that this S(H) behavior affects the magnetic field dependence of J c . At high magnetic fields and high temperatures, a simple power-law dependence of J c on H is not observed and J c decrease rapidly with increasing H as shown in Figure 4a-c. This is related to the rapid increase of S(H) with increasing H, as discussed in Figure 6. After the proton irradiation, field dependences of both J c and S become moderate, as shown in Figure 4b,c and Figure 5b. Although the true H irr defined by the onset of nonlinearity in KCa 2 Fe 4 As 4 F 2 may not be affected by the introduction of point defects, as reported in YBa 2 Cu 3 O 7 [31], the enhancement of pinning force by proton irradiation explains weaker field dependence of J c and S. Figure 7a,b show the temperature dependence of S at H = 10, 20, and 30 kOe for H//c in the pristine and 3 MeV proton irradiated KCa 2 Fe 4 As 4 F 2 at a dose of 3 × 10 16 ions/cm 2 , respectively. In both the pristine and proton irradiated samples, S(T) shows a monotonic increase with increasing temperature without a plateau-like behavior, and the slope of S(T) increases with increasing magnetic field. The temperature dependence of S(T) becomes weaker only at 10 kOe between 10 K and 15 K. According to the collective creep theory, S(T) is proportional to temperature at low temperatures, while it shows a plateau at intermediate T ( U 0 /(k B ln(t/t eff )) with a value of S = 0.02-0.04 [58]. So, one may say that the behavior of S(T) at 10 kOe in the above-T range can be explained in the framework of collective creep theory. However, the value of S(T) between 10 K and 15 K is larger than the expected value. Plateau-like behaviors of S(T) have been observed in, not only YBa 2 Cu 3 O 7 [58], but also in 122-type compounds with reasonable values of S of 0.01-0.04 [24,27]. However, with an increasing magnetic field, the weak temperature dependence of S(T) between 10 K and 15 K disappears, and S(T) strongly increases at higher temperatures. Low H irr in anisotropic KCa 2 Fe 4 As 4 F 2 can explain the stronger temperature dependence of S(T) at high fields. On the other hand, such a divergent behavior of S(T) at high temperatures is suppressed by proton irradiation, as shown in Figure 7b. the expected value. Plateau-like behaviors of S(T) have been observed in, not only YBa2Cu3O7 [58], but also in 122-type compounds with reasonable values of S of 0.01-0.04 [24,27]. However, with an increasing magnetic field, the weak temperature dependence of S(T) between 10 K and 15 K disappears, and S(T) strongly increases at higher temperatures. Low Hirr in anisotropic KCa2Fe4As4F2 can explain the stronger temperature dependence of S(T) at high fields. On the other hand, such a divergent behavior of S(T) at high temperatures is suppressed by proton irradiation, as shown in Figure 7b. . It is important to determine the value of  in discussing vortex dynamics since  includes information on the size of the vortex bundle in the collective creep theory. To extract this value, it is convenient to evaluate inverse current density dependence of effective pinning energy, U * , which is defined as follows: From this equation and "interpolation formula", U * can be calculated as: It is important to determine the value of µ in discussing vortex dynamics since µ includes information on the size of the vortex bundle in the collective creep theory. To extract this value, it is convenient to evaluate inverse current density dependence of effective pinning energy, U * , which is defined as follows: From this equation and "interpolation formula", U * can be calculated as: Thus, the slope in the double logarithmic plot of U * vs. 1/J gives the value of µ, as shown in Figure 8a,b. For this analysis, a proper choice of H is important to determine a region with a unique µ [27]. However, since power-law field dependence of J c is broken down in KCa 2 Fe 4 As 4 F 2 , we chose a magnetic field of H = 20 kOe, where α is nearly constant below 15 K to avoid the effect of fast relaxation at high temperatures and at high magnetic fields. In this way, we evaluated µ = 0.70 and 0.22 for the pristine and proton irradiated KCa 2 Fe 4 As 4 F 2 , respectively. For comparison, we also plotted U * vs. 1/J curves for the pristine and proton irradiated (Ba,K)Fe 2 As 2 in Figure 9a,b, respectively, using the data of relaxation analyses described in Ref. [27]. We evaluated µ = 0.82 and 0.39 for the pristine and 3 MeV proton irradiated (5.6 × 10 16 ions/cm 2 ) (Ba,K)Fe 2 As 2 , respectively. Note that µ~1 in pristine crystal is often reported in YBa 2 Cu 3 O 7 [32] and IBSs [24,59]. After proton irradiation, however, µ becomes significantly smaller in both KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 . This trend has also been reported in Ba(Fe,Co) 2 As 2 [24]. Values of µ in various IBSs are summarized in Table 1. The values of µ in most of pristine IBSs are~1, except for Li 0.8 Fe 0.2 OHFeSe. In Li 0.8 Fe 0.2 OHFeSe, a very large value of µ~4.1 was reported, although the anisotropy parameter is comparable to that in KCa 2 Fe 4 As 4 F 2 [37]. It is discussed that the very large µ in Li 0.8 Fe 0.2 OHFeSe may indicate that vortices in this material are in the crossover regime between elastic Abrikosov vortices to stacks of pancake vortices. The fact that the µ value in KCa 2 Fe 4 As 4 F 2 was~1 may suggest that vortices in this material were more like Abrikosov vortices, similar to the case of (Ba,K)Fe 2 As 2 . On the other hand, negative slopes of U * versus 1/J are also observed at small J. This negative slope is often denoted as p in the plastic creep scenario with p~−0.5 and is confirmed experimentally [60]. Evaluated p = −0.25 and −0.69 for pristine KCa 2 Fe 4 As 4 F 2 and (Ba,K)Fe 2 As 2 are roughly consistent with this scenario. In the case of proton-irradiated KCa 2 Fe 4 As 4 F 2 , however, we were unable to access the negative slope region at a small J, while p = −0.26 was obtained in the proton-irradiated (Ba,K)Fe 2 As 2 . Reduction of |p| after introduction of point defects was also observed in Ba(Fe,Co) 2 As 2 [24]. Dose dependent measurements of vortex dynamics in the plastic region may shed light on the evolution of the plastic behavior.      From the above experimental results and discussion, we can form a conjecture that predominant pinning in the pristine KCa 2 Fe 4 As 4 F 2 is as follows. The large J c~8 MA/cm 2 in the pristine sample is attributable to the embedded strong pinning centers, while artificial defects introduced by the irradiation enhance J c to more than double. Vortex dynamics in KCa 2 Fe 4 As 4 F 2 is similar to those in weakly anisotropic (Ba,K)Fe 2 As 2 rather than Li 0.8 Fe 0.2 OHFeSe with similar anisotropy parameters. However, the effects of proton irradiation on magnetization, magnetic relaxation, and their behaviors at high temperatures and high magnetic fields are different from those in (Ba,K)Fe 2 As 2 . Low H irr in KCa 2 Fe 4 As 4 F 2 due to large anisotropy causes a breakdown of clear power-law dependence of J c on the magnetic field. Since the slope of U * − 1/J curve shows negative slope in a wide range of 1/J in the pristine sample, the effect of the plastic creep should be considered for the comprehensive understanding of the vortex dynamics in this system. The small value of H irr and the large value of S, as well as the rapid decay of J c -H at high temperatures and high magnetic fields, support this idea. We leave it as an open problem since the study on such fast dynamics is beyond the scope of this paper.

Summary
J c and vortex pinning properties in KCa 2 Fe 4 As 4 F 2 single crystals before and after introduction of artificial defects by ion irradiations are systematically studied. Columnar and point defects are introduced by 2.6-GeV U-ions and 3 MeV protons irradiations, respectively. The in-plane J c evaluated from the measurement of irreversible magnetization at 2 K under the self-field is 8 MA/cm 2 , which is the largest among all IBSs. J c under self-field at 2 K is strongly enhanced up to 19.4 or 17.5 MA/cm 2 by irradiation of 2.6 GeV U-ions or 3 MeV protons, respectively. In both cases, suppression of T c is observed. Quantitative differences of irradiation dose dependence of T c compared with those of (Ba,K)Fe 2 As 2 can be explained by considering the presence of embedded defects in the pristine KCa 2 Fe 4 As 4 F 2 . Vortex dynamics in the pristine and proton irradiated KCa 2 Fe 4 As 4 F 2 single crystals are also investigated from the analyses of field dependence of J c and the normalized magnetic relaxation rate. The values of α from~1 to~1/2 before and after proton irradiation, and the behavior of S suggest that the vortex system in the pristine and proton irradiated KCa 2 Fe 4 As 4 F 2 can be described by strong pinning. Vortex dynamics in KCa 2 Fe 4 As 4 F 2 is similar to that in weakly anisotropic (Ba,K)Fe 2 As 2 rather than Li 0.8 Fe 0.2 OHFeSe with similar anisotropy parameter. Low H irr in KCa 2 Fe 4 As 4 F 2 due to large anisotropy can explain some of unique behaviors of J c such as break down of clear power-law dependence on the magnetic field.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.