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Vortex Dynamics and Pinning in CaKFe_{4}As_{4} Single Crystals from DC Magnetization Relaxation and AC Susceptibility

^{1}

^{2}

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## Abstract

**:**

_{4}As

_{4}(where Ae is an alkali-earth metal and A is an alkali metal) has high critical current density, a very high upper critical field, and a low anisotropy, and has recently received much interest for the possibility of high magnetic field applications at the liquid hydrogen temperature. We have performed DC magnetization relaxation and frequency-dependent AC susceptibility measurements on high-quality single crystals of CaKFe

_{4}As

_{4}with the aim of determining the pinning potential U*. The temperature dependence of U* displays a clear crossover between elastic creep and plastic creep. At temperatures around 27–28 K, U* has a very high value, up to 1200 K, resulting in an infinitesimally small probability of thermally activated flux jumps. From the dependence of the normalized pinning potential on irreversible magnetization, we have determined the creep exponents in the two creep regimes, which are in complete agreement with theoretical models. The estimation of the pinning potential from multifrequency AC susceptibility measurements was possible only near the critical temperature due to equipment limitations, and the resulting value is very close to the one that resulted from the magnetization relaxation data. Magnetic hysteresis loops revealed a second magnetization peak and very high values of the critical current density.

## 1. Introduction

_{2}As

_{2}“parent compound”, where Ae is an alkali-earth metal (Ca, Sr, Ba), became one of the most popular materials for fundamental studies and possible applications. The first reason for this was the possibility of growing quite large single crystals, which allowed a comprehensive investigation of their physical properties. It was revealed that they have a rather high critical temperature T

_{c}up to 38 K [1], very high upper critical fields µ

_{0}H

_{c2}(>70 T) [2,3] and low anisotropies γ (<2) [3], which make them strong candidates for high-field applications. Superconductivity in a non-superconducting “parent” compound AeFe

_{2}As

_{2}is induced by alkali metal A substitution at Ae sites (A = Na, K, Rb, Cs). (Ae

_{1−x}A

_{x})Fe

_{2}As

_{2}has the same crystal structure symmetry as the parent compound, I4/mmm, because Ae and A randomly occupy equivalent sites. Actually, (Ae

_{1−x}A

_{x})Fe

_{2}As

_{2}can be viewed as solid solutions between AeFe

_{2}As

_{2}and AFe

_{2}As

_{2}compounds with the same crystallographic structure. However, if the difference in the ionic radius of Ae and A is too large, Ae and A cannot occupy equivalent sites in the unit cell, and a new “family” of IBSs appears [4], with a different structure: CaAFe

_{4}As

_{4}(A = K, Rb, Cs) and SrAFe

_{4}As

_{4}(A = Rb, Cs), noted AeA1144, in which there is an alternate stacking of the Ae and A layers across the Fe

_{2}As

_{2}layer, changing the symmetry to P/4 mmm. The compounds are superconducting materials with critical temperatures T

_{c}between 31 and 36 K. The most studied material in this new family is CaKFe

_{4}As

_{4}(CaK1144). Considering potential high-field applications, the investigations of the types and strength of various effective pinning centers and of the critical current density and its dependence on temperature and field are of great importance. Planar defects, which are intergrowth of CaFe

_{2}As

_{2}layers, with separation between them of about 50 nm, were revealed by high-resolution transmission electron microscopy (HRTEM). Due to their high density, it was suggested that those intergrowths are the strongest pinning centers for vortices along the ab-plane (when the applied field is parallel to the ab-plane) and they are responsible for the enhancement of J

_{c//ab}at around 1 T [5]. More recently, HRTEM, combined with electron energy loss spectroscopy (EELS), showed that these planar defects consist of a network consisting of one or two layers of KFe

_{2}As

_{2}spread over the periodically ordered KFe

_{2}As

_{2}, and of CaFe

_{2}As

_{2}monolayers. In the same study, it was suggested that the regions with dark contrast are distributions of local strain due to the substitution of Ca by K in a unit cell [6]. In the above-mentioned studies on the excellent properties of CaKFe

_{4}As

_{4}(CaK1144), the strength of the pinning centers was determined as a bulk pinning force from DC magnetic hysteresis loops. Here we investigated the mean pinning potential of the pinning centers in CaK1144 single crystals from magnetization relaxation measurements and frequency-dependent AC susceptibility studies.

## 2. Results

#### 2.1. Magnetic Memory Effects

#### 2.2. Magnetization Relaxation

_{0}H

_{DC}= 1 T, plotted in a double-logarithmic scale, which are clearly straight lines.

_{c}/p)[(J

_{c0}/J)

^{p}− 1] = Tln(t/t

_{0}),

_{c}(T) is a characteristic pinning energy, J

_{c0}(T) is the creep-free critical current density, and t

_{0}is the macroscopic time scale for creep (~10

^{−6}–1 s), which varies weakly with J(t). The theoretical models stated that the vortex creep exponent p is positive in the case of a collective (elastic) vortex creep regime, is negative for plastic creep regime, and depends mainly on the magnetic field H and the ratio J/J

_{c0}. It should be noted that Equation (1) describes both elastic and plastic creep, while in a large number of publications, when dealing with (collective) elastic creep, creep exponent is expressed by μ, notation p being used only for the plastic creep. A hypothetical fit of the magnetization relaxation curves m

_{irr}(t) ∝ J(t) using Equation (1) implies four parameters, clearly too many for acceptable conclusions, and usually results in values of the creep exponent p that are not consistent with any accepted model. The analysis is much more simplified in the cases in which, for a reasonably long measurement time, the double-logarithmic plots of m

_{irr}(t) are straight lines, as is evident in Figure 2. In such cases, one can introduce and calculate a normalized relaxation rate S = −dln(|m

_{irr}|)/dln(t) = −dln(J)/dln(t) and a normalized pinning potential U* = T/S [11]. In these conditions, for constant H and T, and if the overall J relaxation is not very high, which is the case of high-performance superconductors, p and t

_{0}can be considered constant, and the dependence of the normalized pinning potential on the current density is given by

_{c}(J

_{c0}/J)

^{p}= U

_{c}(|m

_{0}|/|m

_{irr}|)

^{p}.

_{c}+ pTln(t/t

_{0}). For a moderate, window time for measurement and averaging t

_{w}, we can consider that ln(t/t

_{0})~ln(t

_{w}/t

_{0})~constant, and the normalized pinning potential is given by

_{c}+ pTln(t

_{w}/t

_{0}),

_{c}for elastic pinning is lower than the pinning energy for plastic pinning, when the plastic pinning structure can accommodate the vortices. Assuming that ln(U

_{c}) and ln(J

_{c0}) depend only weakly on temperature, which is the case for a relatively large T interval at fixed H

_{DC}, the creep exponent p can be determined from the results of the magnetization relaxation measurements using Equation (2). For samples with strong pinning, as in our case, the data in the double-logarithmic plot (Figure 2) are very well described by a straight line with the slope being the normalized relaxation rate S = −dln(|m|)/dln(t). If the time of the measurement t

_{w}is not too large, but is large in comparison with the macroscopic time for flux creep (~10

^{−6}–1 s), to stress the independence of the normalized relaxation rate on time, the normalized relation rate is often noted in the literature as S* = −Δln(|m|)/Δln(t). Using these considerations, we assumed that in the timeframe of our measurements (t

_{1}= 100 s < t < t

_{w}= 2700 s), the slope is constant in time. In Figure 2, it is obvious that the experimental results are straight lines in the double-logarithmic scale. It can also be seen that for a fixed temperature, the time dependence of |m| ∝ J is very small in comparison with the change due to the variation of temperature, which is important for the determination of creep parameter p. From the measurements showed in Figure 2, we have determined for the DC field of 1 T the temperature dependence of the normalized relaxation rate S = −dln(|m|)/dln(t), as shown in Figure 3, and the normalized pinning potential, U* = T/S, as shown in the inset of Figure 3. Similar qualitative results were also obtained for other DC magnetic fields and reported elsewhere [12].

_{w}, the decrease in |m| with time is small compared with the decrease in |m| with increasing temperature, which is the case for our sample (see Figure 2). With this approximation, we will consider a J

_{av}(T,H) ∝ |m

_{av}| (T,H) averaged over our measurement time. Averaging is made in the double-logarithmic scale, where the dependence is linear, as can be seen in Figure 2.

_{irr}|) + const], the dependence of ln(U*) on ln(1/|m

_{av}|) = ln(1/J

_{av}) taken from the experimental data for each temperature, shown in Figure 4, allows the distinction between elastic creep and plastic creep, and the determination of the creep exponents.

_{av}|) dependence is linear, represent the vortex creep exponent p, which is positive for elastic creep and negative for plastic creep.

#### 2.3. Frequency-Dependent AC Susceptibility

_{ac}, for many frequencies of the AC field, at fixed temperatures T and fixed DC fields μ

_{0}H

_{DC}. At a fixed T, for several μ

_{0}H

_{DC}, χ″(h

_{ac}) dependence may show a peak at h

_{ac}= h*. The field h* actually represents the AC field of full penetration of the excitation in the center of the sample, which, in the critical state model, can be correlated with the critical current density of the material [13]. The positions of the above-mentioned peak depend on the frequency of the AC field excitation due to the different timescale for the vortices to leave the pinning centers. In the case of our CaK1144 single crystal, due to the very high critical current density, the experimental window in which the method is applicable was very close to the critical temperature. In fact, we could obtain useful results only at 35 K very close to the critical temperature, and for four applied DC fields, 0.75, 1, 1.25 and 1.5 T. The results are shown in Figure 5, the frequencies of the AC excitation fields being indicated in the figure. The selected measurement frequencies were equally distributed, in a logarithmic scale, between a minimum and a maximum one, hence the rather unusual values. It should be noted that the same frequency corresponds to the same symbol and color of the experimental data in any of the four panels of Figure 5.

_{ac}) dependence as

_{c}= h*/αc

_{c}in A/cm

^{2}. By using Equation (4), from the h* values taken from the curves in Figure 5, we estimated the critical current densities at respective frequencies at 35 K and in the four μ

_{0}H

_{DC}mentioned above. Figure 6 presents the dependence of the critical current density J

_{c}, shown in logarithmic scale, on ln(f

_{0}/f), where f

_{0}is a macroscopic attempt frequency of about 10

^{6}Hz. The shape of the experimental curves in Figure 6 indicates which model of pinning is suitable for our sample [15]: a downward curvature indicates an Anderson–Kim model of pinning with a linear dependence of the pinning potential on the probing current, an upward curvature indicates a collective pinning model with a power-law dependence of the pinning potential on the current, and a straight line in the double-logarithmic plot, as is the case of our measurements, is consistent with a logarithmic dependence of the pinning potential on the probing current density.

_{0}= k

_{B}T(1 + 1/b). From the slopes in Figure 6, we estimated the resulted values of U

_{0}in K (k

_{B}= 1), for each μ

_{0}H

_{DC}: 260 ± 5 K (0.8 T), 210 ± 5 K (1 T),170 ± 5 K (1.25 T), and, respectively, 150 ± 5 K (1.5 T).

#### 2.4. Magnetic Hysteresis Loops and Critical Current Density

_{c}. For this purpose, we have performed magnetization hysteresis measurements to determine the field dependence of the irreversible magnetization, at various temperatures. Such measurements are shown in Figure 7, in which, for clarity, we present the results in three panels, with different scales for the magnetic moment m(H

_{DC}, T), for the increasing and, respectively, decreasing applied magnetic field.

_{irr}), magnetization is zero for both increasing (H)↑ and decreasing (H)↓ applied field. Also, a small Second Magnetization Peak (SMP) can be seen, which is a nonmonotonic variation of Δm(H) = m(H)↓ − m(H)↑. At an intermediate temperature, SMP is also very well seen, with a quite flat dependence of Δm(H) on the temperature. At low temperatures (bottom panel), SMP disappears, and Δm(H) has the normal decrease with increasing temperature and field.

_{c1}), from the Δm(H) data we can estimate the critical current density as a function of temperature and field, using the modified Bean critical state model [13]. With the dimensions of the rectangle (the face of the sample perpendicular to the magnetic field) l (length) and w (width), with l > w, and the sample dimension in the field direction d (thickness), the critical current density, in A/cm

^{2}, is given by

_{c}(H, T), as shown in the 3-dimensional plot in Figure 8.

_{c}is of the order of 10

^{5}A/cm

^{2}even at the highest field of our measurements, which is quite important for future applications.

## 3. Discussion

_{B}= 1), which, for 1 T at about 27–28 K (U* ≈ 1200 K), is exp (−1200/27.5) = exp (−43.64) ≈ 10

^{−19}, an infinitesimally low value. For the lowest values of U* of 200 K at 4 K, liquid He temperature, the flux creep probability is exp (−200/4) = exp (−25) ≈ 10

^{−11}. With these values, it is obvious that in CaK1144, at temperatures of practical interest, the dissipation due to thermally activated flux jump (flux creep) is negligible. Even at higher temperatures, in the plastic creep regime such probabilities are small, e.g., for U* ≈ 200 K at T ≈ 32 K, we have exp (−200/32) = exp (−6.25) ≈ 2 × 10

^{−3}.

_{ac}provided by the equipment is 16 Oe, with this method, we had to probe a region in which critical current density is small enough to ensure that the corresponding h* (see Equation (4)) will not exceed 16 Oe. We could obtain the desired maxima in χ″(h

_{ac}) only very close to the critical temperature in the plastic creep regime. For all four DC fields in which we obtained at least 4 curves, at various frequencies, that display a clear maximum, the frequency-dependent critical current is linear in a double-logarithmic plot, consistent with a logarithmic dependence of the pinning potential on the probing current as proposed by Zeldov et al., [19]. From the slopes of those dependences, we determined the values of the pinning potential.

_{ac}). Extrapolating the U*(T) dependence from the magnetization relaxation in the insert of Figure 3 towards T = 35 K, one can estimate U*(35 K, 1 T) to be between 200 and 250 K, and a closer look at Figure 6b will show a value from AC susceptibility at the same temperature and DC field, U

_{0}(35 K, 1 T) ≈ 210 K, which is quite unexpected if one takes into account the much different timescale of the two types of measurements, 10

^{3}s and 10

^{−3}s, respectively. The probable explanation of this fact is that, very close to T

_{c}, the viscosity of the plastic vortex matter is very small (practically becomes a vortex liquid), and the timescale of the low-viscosity plastic creep is less important.

## 4. Materials and Methods

_{c}and above the field for the first full vortex penetration (in increasing H), or at any H in a decreasing field, the irreversible magnetic moment is considered to be the measured moment m. For the AC susceptibility measurements, we have used a commercial Quantum Design Physical Property Measurement System (PPMS) with the possibility of DC fields up to 14 T, in the ACMS set-up with frequencies up to 10 kHz and AC field amplitudes up to 16 Oe. In all the measurements presented in this paper, both DC and AC applied fields were perpendicular to the largest plane a–b of the single crystal, i.e., parallel to the c-axis.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

_{4}As

_{4}single crystals.

## Conflicts of Interest

## References

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**Figure 1.**Temperature dependence of the in-phase (

**left**) and out-of-phase (

**right**) third-harmonic susceptibility response in an applied DC field μ

_{0}H

_{DC}= 5 T, measured with an AC field having the amplitude h

_{AC}= 10 Oe, frequency f = 447 Hz, in the three cooling procedures (ZFC, FC, FCW).

**Figure 2.**Time dependence of the modulus of magnetic moment |m| of the sample after zero-field-cooling in a DC applied field μ

_{0}H

_{DC}= 1 T, at various temperatures, in double-logarithmic scale. Due to the rearrangement of the magnetic lines, the initial data for t < 100 s were excluded from the analysis.

**Figure 3.**Dependence of the normalized relaxation rate S = −dln(|m|)/dln(t) on the temperature, for the l DC magnetic field of 1 T. Insert: Temperature dependence of the normalized pinning potential.

**Figure 4.**Double logarithmic plot of the normalized pinning potential U* as a function of the inverse of irreversible magnetization, averaged logarithmically over t

_{w}, for the DC field of 1 T.

**Figure 5.**The dependence of the out-of-phase susceptibility response as a function of the amplitude of the AC excitation field, at the temperature of 35 K, applied DC fields of 0.75, 1, 1.25 and 1.5 T. The figure only shows the measurements for which the χ″(h

_{ac}) dependence exhibits a peak and if they are not noisy, frequencies being indicated in the panels.

**Figure 6.**Double-logarithmic plot of the frequency dependence of the critical current density, at 35 K, in the four DC fields of 0.75, 1, 1.25 and 1.5 T.

**Figure 7.**Magnetic hysteresis loops for increasing (

**lower branches**) and decreasing (

**higher branches**) applied magnetic field.

**Figure 8.**Three-dimensional plot of the field and temperature dependence of the critical current density.

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## Share and Cite

**MDPI and ACS Style**

Ionescu, A.M.; Ivan, I.; Miclea, C.F.; Crisan, D.N.; Galluzzi, A.; Polichetti, M.; Crisan, A.
Vortex Dynamics and Pinning in CaKFe_{4}As_{4} Single Crystals from DC Magnetization Relaxation and AC Susceptibility. *Condens. Matter* **2023**, *8*, 93.
https://doi.org/10.3390/condmat8040093

**AMA Style**

Ionescu AM, Ivan I, Miclea CF, Crisan DN, Galluzzi A, Polichetti M, Crisan A.
Vortex Dynamics and Pinning in CaKFe_{4}As_{4} Single Crystals from DC Magnetization Relaxation and AC Susceptibility. *Condensed Matter*. 2023; 8(4):93.
https://doi.org/10.3390/condmat8040093

**Chicago/Turabian Style**

Ionescu, Alina M., Ion Ivan, Corneliu F. Miclea, Daniel N. Crisan, Armando Galluzzi, Massimiliano Polichetti, and Adrian Crisan.
2023. "Vortex Dynamics and Pinning in CaKFe_{4}As_{4} Single Crystals from DC Magnetization Relaxation and AC Susceptibility" *Condensed Matter* 8, no. 4: 93.
https://doi.org/10.3390/condmat8040093