Critical Current Density and Vortex Dynamics in Pristine and Irradiated KCa₂Fe₄As₄F₂

Abstract

We report the critical current density (J_c) and vortex pinning properties in single crystals of a novel iron-based superconductor (IBS) KCa₂Fe₄As₄F₂ with large J_c 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 T_c and J_c, including its dose dependence, are systematically studied. J_c~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 T_c and dose dependence of J_c 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 J_c 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 Li_0.8Fe_0.2OHFeSe. Large anisotropy, due to the presence of insulating blocking layers in KCa₂Fe₄As₄F₂, which leads to much lower irreversibility field (H_irr) compared with 122-type IBSs, strongly affect the vortex dynamics.

1. 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)Fe₂As₂ [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 CuO₂ 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 Fe₂As₂ conducting layers between two neighboring Ca₂F₂ 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]. KCa₂Fe₄As₄F₂ is the first and a well-studied 12442-type IBS [8] and its anisotropic physical properties have been demonstrated. In KCa₂Fe₄As₄F₂, 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 KCa₂Fe₄As₄F₂ 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/cm², which is significantly larger compared with that of 122-type IBSs [16].

One of the most intriguing topics in KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂, enhancement of J_c up to 19 MA/cm² 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 KCa₂Fe₄As₄F₂ is demanded.

Furthermore, investigation of vortex pinning properties in KCa₂Fe₄As₄F₂ 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 YBa₂Cu₃O₇, 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 Bi₂Sr₂CuCa₂O_{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)Fe₂As₂ or Ba(Fe,Co)₂As₂, 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_0.8Fe_0.2OHFeSe, 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 KCa₂Fe₄As₄F₂, is demanded.

In this study, we investigated the effect of columnar defects and point defects created by irradiation of U ions and protons in KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ before and after the introduction of point defects. A part of the data of KCa₂Fe₄As₄F₂ were compared to those of (Ba,K)Fe₂As₂, which belong to 122-type IBSs and is a representative IBS, and those of Li_0.8Fe_0.2OHFeSe, which has anisotropic crystal structure similar to KCa₂Fe₄As₄F₂.

2. Experimental Methods

Single crystals of KCa₂Fe₄As₄F₂ 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_0.6K_0.4Fe₂As₂ single crystals, which were used as reference materials to be compared with KCa₂Fe₄As₄F₂, 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 dose-equivalent magnetic field called the “matching field”, at which all defects are occupied by single vortices, B_Φ = nΦ₀. Here, n is the areal density of CDs and Φ₀ 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.

3. 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 KCa₂Fe₄As₄F₂ are discussed. STEM images of both the pristine and 2.6-GeV U-irradiated KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂

First, STEM analyses of the pristine and 2.6-GeV U ion-irradiated KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂. Although the existence of these sparsely distributed defects cannot explain the significantly larger J_c in pristine KCa₂Fe₄As₄F₂, 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 KCa₂Fe₄As₄F₂, clear columnar defects with a diameter ~5 nm along the c-axis were observed, which were similar to the case of Ba(Fe,Co)₂As₂ and (Ba,K)Fe₂As₂ [20,23,39].

Next, the effects of columnar defects on T_c and J_c 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_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)Fe₂As₂ are also shown [26]. T_c of both KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂ are monotonically decreased with increasing B_Φ, although B_Φ dependence of T_c in KCa₂Fe₄As₄F₂ is stronger than that of (Ba,K)Fe₂As₂. 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)Fe₂As₂ and KCa₂Fe₄As₄F₂, 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 KCa₂Fe₄As₄F₂. Unfortunately, however, it is not easy to estimate the dose and energy of such secondary electrons in a given material.

Next, the in-plane J_c for H//c-axis in KCa₂Fe₄As₄F₂ is evaluated by measuring the irreversible magnetization using the extended Bean model [20,41,42],

J_c [A/cm²] = 20ΔM/a(1 − a/3b)

(1)

where ΔM [emu/cm³] is M_down − M_up. M_up and M_down are the magnetization when sweeping the field up and down, respectively, and a [cm] and b [cm] are the sample width and length (a < b). Figure 2c shows the B_Φ dependence of J_c in KCa₂Fe₄As₄F₂ irradiated with 2.6 GeV U ions at T = 2 K under the self-field. For comparison, similar data for a (Ba,K)Fe₂As₂ single crystal are also shown [26]. The B_Φ dependence of J_c in KCa₂Fe₄As₄F₂ shows similar trends to that of (Ba,K)Fe₂As₂, 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/cm² is realized in KCa₂Fe₄As₄F₂ with B_Φ = 2 T, and that of 15 MA/cm² is realized in (Ba,K)Fe₂As₂ 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)Fe₂As₂~15 MA/cm², as shown in Figure 2c [26]. The J_c in KCa₂Fe₄As₄F₂ starts to decline above B_Φ = 4 T, which is much earlier than that in (Ba,K)Fe₂As₂, 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 KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂. 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)Fe₂As₂ is also shown in Figure 3b and in its inset [43]. The T_c of both KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂ monotonically decrease with increasing proton dose. It should be noted that the rate of T_c reduction in both KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂ estimated from Figure 3b are 0.9 and 1.0% per 1 × 10¹⁶ ions/cm², respectively. These results suggest that the density of created point defects by proton irradiation and their effects on T_c is similar in KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂. This is in contrast to the case of 2.6-GeV U-irradiated KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂, 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 KCa₂Fe₄As₄F₂ irradiated with 3 MeV proton at T = 2 K under a self-field. For comparison, similar data for (Ba,K)Fe₂As₂ measured at H = 1 kOe are also shown. J_c in KCa₂Fe₄As₄F₂ reaches its maximum of 17.5 MA/cm² at a dose of 3 × 10¹⁶ ions/cm². On the other hand, J_c of (Ba,K)Fe₂As₂ reaches its maximum of 14 MA/cm² at a dose of 5 × 10¹⁶ ions/cm². The doses for maximum J_c are similar, although that of KCa₂Fe₄As₄F₂ is a little smaller. Such a difference can be explained by the fact that the pristine KCa₂Fe₄As₄F₂ 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 proton-irradiated KCa₂Fe₄As₄F₂ are different from those in typical IBS of (Ba,K)Fe₂As₂, which can be explained by considering the embedded defects in pristine KCa₂Fe₄As₄F₂. It should be emphasized that maximum J_c in KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ are demanded.

3.2. Vortex Dynamics in Pristine and Proton Irradiated KCa₂Fe₄As₄F₂

Figure 4a shows magnetic field dependence of J_c for H//c at various temperatures in the pristine KCa₂Fe₄As₄F₂. At all temperatures, J_c monotonically decreases with magnetic field. These features are similar to those in optimally-doped (Ba,K)Fe₂As₂ [27] and Li_0.8Fe_0.2OHFeSe [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)₂As₂ [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 KCa₂Fe₄As₄F₂ 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 YBa₂Cu₃O₇ with columnar defects, which is explained as follows [19]; for example, at H = 10 kOe, the distance between vortices is (Φ₀/H)^1/2~50 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_cH [49,50]. In other words, observation of α~1 suggests that there are sparse strong pinning centers in the pristine KCa₂Fe₄As₄F₂, 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 KCa₂Fe₄As₄F₂ compared with (Ba,K)Fe₂As₂ 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 KCa₂Fe₄As₄F₂, where J_c takes its maximum at a dose of 3 × 10¹⁶ ions/cm². 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 KCa₂Fe₄As₄F₂ with a dose of 10 × 10¹⁶ ions/cm², 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 KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ and other IBSs may be caused by the existence of embedded defects in the pristine KCa₂Fe₄As₄F₂. 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 KCa₂Fe₄As₄F₂, which will be discussed below.

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,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 KCa₂Fe₄As₄F₂ at a dose of 3 × 10¹⁶ ions/cm², respectively. The dimensions of the pristine and proton irradiated samples were 0.0178 × 0.0212 × 0.0008 and 0.0235 × 0.049 × 0.0013 cm³, 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 KCa₂Fe₄As₄F₂. 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)₂As₂ and (Ba,K)Fe₂As₂ [24,27]. Second, at high magnetic fields, S in the pristine KCa₂Fe₄As₄F₂ increases rapidly above a temperature dependent characteristic field, although S in the proton-irradiated KCa₂Fe₄As₄F₂ increases monotonically up to 40 kOe. Third, the values of S in KCa₂Fe₄As₄F₂ are larger than those in (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 KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂.

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 (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:

$$U\left(J\right)=U_0{\left(J_{c0}/J\right)}^{\mu },$$

(2)

where J_c0 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 $\ll$ 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:

$$U\left(J\right)=U_0\left(1-J_{c0}/J\right).$$

(3)

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]:

$$U\left(J\right)=\frac{U_0}{\mu }\left[{\left(J_{c0}/J\right)}^{\mu }-1\right].$$

(4)

On the other hand, from the Arrhenius relation, U can be also described as:

$$U=k_{\mathrm{B}}T\ln \left(t/t_{\mathrm{eff}}\right),$$

(5)

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]:

$$J\left(T,t\right)=\frac{J_{c0}}{{\left[1+\left(\mu k_{\mathrm{B}}T/U_0\right)\ln \left(t/t_{\mathrm{eff}}\right)\right]}^{1/\mu }}$$

(6)

$$S=\frac{k_{B}T}{U_0+\mu k_{B}T\ln \left(t/t_{\mathrm{eff}}\right)}.$$

(7)

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)Fe₂As₂, 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)Fe₂As₂ is suggested [27]. On the other hand, the degree of suppression of S in KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ is even more dominant after proton irradiation compared with that in (Ba,K)Fe₂As₂. 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 KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ may not be affected by the introduction of point defects, as reported in YBa₂Cu₃O₇ [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 KCa₂Fe₄As₄F₂ at a dose of 3 × 10¹⁶ ions/cm², 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 ($\gg$U₀/(k_Bln(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 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 H_irr 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 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:

U^* = k_BT/S.

(8)

From this equation and “interpolation formula”, U^* can be calculated as:

$$U^{*}=U_0+\mu k_{B}T\ln \left(t/t_{\mathrm{eff}}\right)=U_0{\left(J_{c0}/J\right)}^{\mu }.$$

(9)

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 KCa₂Fe₄As₄F₂, 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 KCa₂Fe₄As₄F₂, respectively. For comparison, we also plotted U^* vs. 1/J curves for the pristine and proton irradiated (Ba,K)Fe₂As₂ 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 × 10¹⁶ ions/cm²) (Ba,K)Fe₂As₂, respectively. Note that μ~1 in pristine crystal is often reported in YBa₂Cu₃O₇ [32] and IBSs [24,59]. After proton irradiation, however, μ becomes significantly smaller in both KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂. This trend has also been reported in Ba(Fe,Co)₂As₂ [24]. Values of μ in various IBSs are summarized in

Table 1. Glassy exponents for elastic creep μ in various IBS compounds, estimated from U* vs. 1/J plots.

	KCa2Fe4As4F2	(Ba,K)Fe2As2	Ba(Fe,Co)2As2	Li0.8Fe0.2OHFeSe	Fe(Te,Se)	FeSe
Pristine	0.70	0.82	1.09 [24]	4.1 [37]	1.34 [35]	0.71 [36]
Proton liubinirradiated	0.22	0.39	0.82 [24]	-	-	-

. The values of μ in most of pristine IBSs are ~1, except for Li_0.8Fe_0.2OHFeSe. In Li_0.8Fe_0.2OHFeSe, a very large value of μ~4.1 was reported, although the anisotropy parameter is comparable to that in KCa₂Fe₄As₄F₂ [37]. It is discussed that the very large μ in Li_0.8Fe_0.2OHFeSe 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 KCa₂Fe₄As₄F₂ was ~1 may suggest that vortices in this material were more like Abrikosov vortices, similar to the case of (Ba,K)Fe₂As₂. 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 KCa₂Fe₄As₄F₂ and (Ba,K)Fe₂As₂ are roughly consistent with this scenario. In the case of proton-irradiated KCa₂Fe₄As₄F₂, 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)Fe₂As₂. Reduction of |p| after introduction of point defects was also observed in Ba(Fe,Co)₂As₂ [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 KCa₂Fe₄As₄F₂ is as follows. The large J_c~8 MA/cm² 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 KCa₂Fe₄As₄F₂ is similar to those in weakly anisotropic (Ba,K)Fe₂As₂ rather than Li_0.8Fe_0.2OHFeSe 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)Fe₂As₂. Low H_irr in KCa₂Fe₄As₄F₂ 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.

4. Summary

J_c and vortex pinning properties in KCa₂Fe₄As₄F₂ 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/cm², 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/cm² 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)Fe₂As₂ can be explained by considering the presence of embedded defects in the pristine KCa₂Fe₄As₄F₂. Vortex dynamics in the pristine and proton irradiated KCa₂Fe₄As₄F₂ 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 KCa₂Fe₄As₄F₂ can be described by strong pinning. Vortex dynamics in KCa₂Fe₄As₄F₂ is similar to that in weakly anisotropic (Ba,K)Fe₂As₂ rather than Li_0.8Fe_0.2OHFeSe with similar anisotropy parameter. Low H_irr in KCa₂Fe₄As₄F₂ 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.