Magnetic Vortex Phase Diagram for a Non-Optimized CaKFe₄As₄ Superconductor Presenting a Wide Vortex Liquid Region and an Ultra-High Upper Critical Field

Featured Application

This work concerns the study of the vortex phase diagram in CaKFe₄As₄, whose knowledge can be helpful for the understanding of the applicative ranges in field and temperature of these materials with not-optimized fabrication characteristics, as usually is found in superconducting wires and cables for power applications.

Abstract

To draw a complete vortex phase diagram for a CaKFe₄As₄ polycrystalline iron-based superconductor, different kinds of magnetic measurements have been performed focusing on the critical parameters of the sample. Firstly, magnetic moment versus field measurements m(H) were performed at low fields in order to evaluate the lower critical field H_c1. After that, by performing relaxation measurements m(t), a field crossover H_cross was detected in the framework of a strong pinning regime. The irreversibility field H_irr as a function of the temperature curve was then drawn by plotting the critical current densities J_c versus the field for temperatures near T_c. J_c(H) has demonstrated a second magnetization peak effect phenomenon, and the second peak field H_sp has been identified and plotted as a function of temperature, providing information about an elastic to plastic transition in the vortex lattice. Finally, the upper critical field H_c2 as a function of the temperature has been obtained. H_c1, H_cross, H_sp, H_irr, H_c2 have been fitted and used for drawing the complete vortex phase diagram of the sample. It can be helpful for the understanding of the applicative ranges in the field and temperature of the materials with not-optimized fabrication characteristics, as usually is found in superconducting wires and cables for power applications.

1. Introduction

The iron based superconductors (IBS) found out in 2008 [1] are probably the most intriguing material to investigate since the high-T_c superconductors (HTS) discovery by Bednorz and Muller in 1986 [2]. In fact, although the HTS critical temperature T_c is well above the liquid nitrogen temperature in many cases, materials such as YBCO or BSCCO have been not exploited completely for their use in power applications of superconductivity due to some problems: high values of anisotropy [3,4], superconductor-insulator-superconductor (SIS) grain boundary junction [5,6]etc. On the other hand, although the IBS have a lower T_c than HTS, they show lower values of anisotropy [7,8] and superconductor-normal-superconductor (SNS) grain boundary junction [9,10] together with high values of J_c and H_c2 [11,12]. Among all the IBS families, the recently discovered 1144 family has attracted the attention of researchers due to its analogy in the crystal structure with the well-known 122 IBS family [13]. In the 1144 family, the CaKFe₄As₄ presents interesting peculiarities such as relatively high T_c ≈ 35 K [14,15], large upper critical field [16,17], and very high J_c at high fields as well [17,18]. Moreover, in very recent years, CaKFe₄As₄ wires and tapes have been successfully fabricated by the powder-in-tube method [19,20] opening perspectives in terms of using this material for practical purposes. It is worth underlining that Refs. [19,20] have not performed a thorough magnetic analysis of the material nor the construction of a complete vortex phase diagram as instead shown in this work. The CaKFe4As4 can form during the fabrication process also spurious phases such as CaFe₂As₂ and KFe₂As₂ [14] and a large number of impurities. So, it is clear that the superconducting properties and critical parameters can be drastically enhanced if these phases and the grain boundaries formation are reduced by improving the fabrication process. In this framework, one of the most common problems in applications of superconducting materials on a large scale is the difficulty of not being able to optimize their superconducting characteristics in the entire volume of the application. In fact, the characteristics of such materials can differ particularly in the case of using well-optimized samples with perfect crystalline structure compared to when considering volumes of material much larger than a single crystal. In this context, the study of non-optimized samples becomes very useful to understand the field and temperature behavior of their superconducting critical parameters. In general, the first and third harmonic of the AC magnetic susceptibility is a valid approach to studying the vortex dynamics [21,22] but it often requires difficult analysis processes. In this context, it is worth underlining that an in-depth study of the temperature dependence of the modulus of the first and third harmonic of the AC magnetic susceptibility gives very accurate information about the vortex melting line [23,24]. The results are usually influenced by the values of the amplitude and frequency of the AC magnetic field together with the possibility to use a superimposed DC field. So, it reveals a very useful investigation when it is necessary to individuate the vortex melting line of the superconductor with well-fixed AC and DC field parameters. Alternatively, DC magnetization can be used. Its response as a function of the temperature, the magnetic field, and the time gives information about the critical parameters of the sample useful to draw a complete vortex phase diagram. In the present study, we report the superconducting properties of a non-optimized CaKFe₄As₄ superconductor by using DC magnetization measurements in magnetic fields up to 9 Tesla and temperatures in the range 5–35 K. In particular, the attention has been focused on drawing the vortex phase diagram of the sample. At this aim, different critical parameters such as H_c1, H_irr, etc. have been obtained. The results showed that the sample is perspective in terms of energy/high-power applications due to its very large H_c2 values and the strong pinning behavior. Nevertheless, the observed very wide vortex liquid region makes us understand that the efficiency of the pinning is low. Then, the sample must be optimized by improving, for instance, the fabrication process.

2. Materials and Methods

A disk-shaped pellet having a diameter equal to 3 mm and a thickness equal to 0.65 mm has been analyzed by means of DC magnetic measurements. The sample has been obtained by using a mechanochemically assisted synthesis route [25]. The fabrication details are reported in Ref. [15]. The sample has been characterized by measuring the magnetic moment as a function of the temperature m(T), the magnetic field m(H), and the time m(t). The measurements have been performed in perpendicular field configuration (field applied perpendicular to the disk surface) by using a QD-PPMS doted of a vibrating sample magnetometer. The residual entrapped field in the PPMS superconducting magnet [26] has been decreased below 1 × 10⁻⁴ T by oscillating the field around zero [27]. For the m(T) measurements, the sample was cooled down to 2.5 K without a magnetic field. After that, the field was turned on and the data was acquired by increasing the temperature (Zero Field Cooling) up to 300 K. For the m(H) measurements, after cooling the sample at the target temperature, the field has been increased to +9 T, decreased to −9 T, and finally increased again to +9 T. For the m(t) measurements, firstly the target temperature has been reached in zero fields. Then, a field equal to 8 T has been set and then decreased to the measurement field. After waiting for about 100 s for avoiding a metastable magnetic states [28], the data have been acquired for 10,800 s.

Table 1. Measurement parameters values.

Measurement	Minimum Value	Maximum Value	Rate
m(T)	2.5 K	300 K	0.5 K/min
m(H)	−9 T	+9 T	0.01 T/s
m(t)	100 s	10,800 s	1 datapoint/s

summarizes the measurement parameter values used in this work.

3. Results and Discussion

To draw the complete sample vortex phase diagram, the lower critical field H_c1 has been the first studied critical parameter. In particular, the first magnetization was measured versus the field at several temperatures. In Figure 1, the m(H) curves have been reported up to 0.1 T at 5 K, 10 K, 15 K, 20 K, and 25 K. It is visible the initial linear behavior of the magnetic moment due to the Meissner state and the reduction of the superconducting signal due to the temperature increase.

In the main panel of Figure 2, H_c1 has been determined for T = 10 K. The dashed red line and the red arrow individuate the linear behavior due to the Meissner state and the H_c1 value, respectively. Considering all the temperatures, the H_c1(T) behavior has been obtained and fitted with the power law [29]:

$${\mathrm{H}}_{\mathrm{c}1}\left(\mathrm{T}\right)={\mathrm{H}}_{\mathrm{c}1}\left(0\right){\left(1-\mathrm{T}/{\mathrm{T}}^{*}\right)}^{\mathrm{n}}$$

(1)

where H_c1(0) is the value of Hc1 at T = 0 K and T^* indicates when the phenomenon is undetectable. The H_c1(T) curve together with its fit is reported in the inset of Figure 2. From the fit procedure, H_c1(0) ≈ 120 Oe, T^* ≈ 27.7 K, n ≈ 0.67 have been obtained. The study of the H_c1(T) line is interesting since below it the sample is in the Meissner state while above it the vortices start to enter the sample causing dissipation processes. From our previous work [15], we already know that a strong pinning regime characterizes our sample. Now, we want to determine how the vortex interact with each other by means of magnetic relaxation measurements. At this aim, m(t) measurements have been made at several temperatures and fields. Figure 3 shows relaxation measurements at 2.5 K between 0.1 T and 1 T.

From Figure 3, the pinning energy versus field U(H) values have been extracted by means of U = −Tdln(t)/dln(|M|) [30,31]. In Figure 4, the U(H) for T = 2.5 K has been shown. It can be noted how the U(H) behavior changes for increasing fields. When the field is low, U(H) decrease is slow while the U(H) decrease is faster when the field is increased. These curve behaviors were fitted with $\mathrm{U}\left(\mathrm{H}\right)\propto {\mathrm{H}}^{-\mathsf{\alpha }}$ where α specifies the pinning regime of the sample.

Specifically, a single vortex pinning regime is obtained when α ≈ 0 [32] while a collective pinning regime is individuated by α > 0.5 [33]. From the fitting procedure, the field where the U(H) behavior changes was obtained, i.e., H_cross ≈ 0.53 T. When H < H_cross, α ≈ 0.09 indicates that the vortices do not interact with each other (single vortex regime). When H > H_cross, α ≈ 0.7 indicates a collective pinning behavior. The same fitting procedure can be repeated at different temperatures. The temperature dependence of H_cross as is shown in the inset of Figure 4 together with its fit with the power law

$${\mathrm{H}}_{\mathrm{cross}}\left(\mathrm{T}\right)={\mathrm{H}}_{\mathrm{cross}}\left(0\right){\left(1-\mathrm{T}/{\mathrm{T}}^{*}\right)}^{\mathrm{n}}$$

(2)

where H_cross(0) ≈ 0.55 T is the value of H_cross at T = 0 K, T^* ≈ 20.75 K indicates when the phenomenon is undetectable, and n ≈ 0.27 is the exponent. The study of H_cross(T) line can be interesting since below it the sample shows a field behavior of the pinning energy almost constant which could allow a better transport of current. In order to find the irreversibility line, the critical current density J_c versus H curves have been shown in the main panel of Figure 5. A second magnetization peak phenomenon, similar to others observed in literature for iron-based superconductors [34,35,36,37], has been detected. The second peak position H_sp, individuated by a black dashed line in the main panel of Figure 5, typically identifies the transition between the elastic and the plastic distortion of the vortex lattice in the framework of a strong pinning [38,39,40,41]. The H_sp values have been extracted at the different temperatures and plotted in the inset(a) of Figure 5. In particular, the H_sp(T) values have been fitted with the power law

$${\mathrm{H}}_{\mathrm{sp}}\left(\mathrm{T}\right)={\mathrm{H}}_{\mathrm{sp}}\left(0\right){\left(1-\mathrm{T}/{\mathrm{T}}^{*}\right)}^{\mathrm{n}}$$

(3)

where H_sp(0) ≈ 1.3 T is the value of H_sp at T = 0 K, T^* ≈ 30.4 K indicates when the phenomenon is undetectable, and n ≈ 0.14 is the exponent. The H_sp(T) line is very appealing for applications since for H approaching H_sp the J_c is increasing with the field while for H > H_sp it is decreasing. On the other hand, the black solid line in the main panel of Figure 5 indicates the irreversibility field H_irr values. Specifically, H_irr can be obtained with the criterion of J_c = 100 A/cm² [42]. In the inset(b) of Figure 5, H_irr versus temperature is shown together with the fit with the equation

$${\mathrm{H}}_{\mathrm{irr}}\left(\mathrm{T}\right)={\mathrm{H}}_{\mathrm{irr}}\left(0\right){\left(1-\mathrm{T}/{\mathrm{T}}^{*}\right)}^{\mathrm{n}}$$

(4)

where H_irr(0) ≈ 26 T is the value of H_irr at T = 0 K, T^* ≈ 31.1 K indicates when the phenomenon is undetectable, and n ≈ 0.86 is the exponent. The H_irr(T) line is very important since it individuates the vortex liquid region where no pinning forces act and high dissipations occur in the material. Naturally, this region must be avoided in the framework of power applications of superconductivity.

Finally, m(T) curves have been performed in ZFC conditions at different magnetic fields (see the main panel of Figure 6) to evaluate the H_c2 values. Particularly, they have been estimated by determining the onset of the magnetization drop at the different fields. H_c2(T) curve is shown in the inset of Figure 6 together with the fit with the power law

$${\mathrm{H}}_{\mathrm{c}2}\left(\mathrm{T}\right)={\mathrm{H}}_{\mathrm{c}2}\left(0\right){\left(1-\mathrm{T}/{\mathrm{T}}_{\mathrm{c}}\right)}^{\mathrm{n}}$$

(5)

where H_c2(0) ≈ 212 T is the value of H_c2 at T = 0 K, T_c = 35 K is the critical temperature of the sample at H ≈ 0 T [15], and n ≈ 0.9 is the exponent. The H_c2(T) line is important to determine the maximum appliable magnetic field to the material without its transition to the normal state.

It is worth to underline the huge H_c2(0) value which overcomes 200 T. Moreover, the n exponent values for H_irr and H_c2 are minor than 1 different from what happens in iron-based materials of different families [36,43,44,45,46,47]. All the features discussed so far can be summarized by constructing the H(T) vortex phase diagram. In the main panel of Figure 7, H_c1, H_irr, H_c2 are reported together with their fit. It is possible to note a very wide vortex liquid region where the vortices are unpinned [48,49,50]. In the framework of power applications of superconductivity, it is worth having the H_irr line near the H_c2 one to hinder the vortices movement inside the material and the consequent dissipation with depression of critical current density. From this point of view, the sample must be optimized even if its critical current density values are in agreement with the literature [17,25,51]. It is worth to underline that the strong pinning region is very narrow compared with the vortex liquid one as shown in the main panel of Figure 7. This suggests the presence of a few strong pinning centers and a bad connection among the grains of the polycrystalline sample. This usually causes a fast J_c decrease as a function of the field due to the efficiency lack of the pinning acting in the sample which leads to wide vortex liquid regions. In the inset of Figure 7, the strong pinning region has been magnified. Even with the zoom, the region below the H_c1 line is barely visible indicating a very weak Meissner state. Between the H_c1 and H_cross lines, a single vortex behavior has been found. On the other hand, collective pinning characterized by elastic deformations of the vortex lattice can be noted between H_cross and H_sp lines. Between H_sp and H_irr lines, the plastic deformations rule the vortex dynamics of the sample. It is interesting to highlight that this plastic deformations area is the widest in the framework of the strong pinning region. When plastic deformations act in a sample, the pinning energy as a function of the temperature decreases quickly [52,53,54]. This is coherent with the weak pinning strength found in the sample. Finally, the next fascinating challenge is to improve the sample fabrication process in order to drastically reduce the vortex liquid region by nearing the H_irr line to the H_c2 one. Reaching this target, a huge strong pinning region will be achieved allowing the sample to sustain high J_c in very high magnetic fields so becoming an interesting candidate for power applications of superconductivity.

4. Conclusions

A CaKFe₄As₄ polycrystalline sample has been analyzed by means of DC magnetic measurements performed as a function of the temperature, the magnetic field, and the time in order to determine its superconducting critical parameters. By fitting all the reported characteristic fields with the power law $\mathrm{H}\left(\mathrm{T}\right)=\mathrm{H}\left(0\right){\left(1-\mathrm{T}/{\mathrm{T}}^{*}\right)}^{\mathrm{n}}$, a rich vortex phase diagram has been drawn showing a very wide vortex liquid region. An opening for the upcoming researchers can be to reduce this zero-pinning region by optimizing the fabrication process of the sample so making it more suitable for power applications. A further opening for future studies can also be the doping of non-optimized samples by using different elements in order to understand the behavior of the critical parameters analyzed in this work. Anyway, the study of non-optimized superconductors becomes very useful to understand how the superconducting critical parameters behave in presence of a magnetic field since this kind of sample can have an analogous magnetic behavior of wires and cables used in the power applications of superconductivity.