# The Depairing Current Density of a Fe(Se,Te) Crystal Evaluated in Presence of Demagnetizing Factors

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{1−x}K

_{x}Fe

_{2}As

_{2}compound, it became clear that the importance of this new class of materials (which can present not-oxide superconductors) breaks the link with cuprates. This highlights the fact that, for fabricating high-temperature superconductors, oxygen does not necessary play a crucial role. The 11 family in the class of iron-based superconductors has been intensively investigated in recent years so that we may understand the basic mechanism governing its superconductivity [3]. These materials have a layered structure, like cuprates, and they show interesting peculiarities such as high values in the upper critical field, critical current density, and irreversibility field [4,5,6,7]. Moreover, they have lower anisotropy values than cuprates, with associated high pinning energy values [8,9,10,11]. Among these properties, there are also non-monotonic responses to magnetization with applied magnetic fields; this has led to particular phenomena that have been deeply studied in recent years [12,13,14,15,16]. These phenomena are strictly correlated with vortex dynamics [17,18,19,20,21]. On the other hand, the presence of vortices inside type-II superconductors can also be investigated by analyzing the lower critical field, ${H}_{c1}$, together with the London penetration depth, ${\lambda}_{L}$. ${H}_{c1}$ and ${\lambda}_{L}$ are useful parameters for gathering information regarding the bulk thermodynamic properties of a sample, and they have been investigated for iron-based compounds in the past [22,23,24]. In particular, compared with other physical quantities, penetration depth is a useful parameter to study the superconductivity of a compound intrinsically; this is because it is not sensitive to the aspects related to surface conditions. In particular, the study of the behavior of ${\lambda}_{L}^{-2}$, as a function of temperature, can provide information on the superconductivity typology characterizing the sample (e.g., single gap BCS theory, the two gap model, etc.) [25,26,27]. Moreover, the ${\lambda}_{L}$ values indicate the strength of the interactions between vortices, thus allowing us to deduce the magnitude of the effective pinning energy of the sample. ${\lambda}_{L}$, together with the coherence length, $\mathsf{\xi}$, represent the fingerprints of a superconductor, and this study becomes crucial when first characterizing a new superconductor. Moreover, by combining them, it is possible to obtain the de-pairing current density, ${J}_{dep}$, which fixes the upper limit of the presence of superconductivity inside a superconducting material. The de-pairing current density is of significant importance for understanding the existing limits for increasing ${J}_{c}$ [28,29], and since it directly provides data on the critical velocity of the superfluid, it is essential for the investigation of the superconducting mechanism and the symmetry of the superconducting gap [30]. Using this framework, by following the Ginzburg–Landau theory, the de-pairing current density, ${J}_{dep}$, depends on the characteristic critical parameters ${H}_{c1}$ and ${H}_{c2}$, and more specifically, the London penetration depth, ${\lambda}_{L}$, and the coherence length, $\mathsf{\xi}$ [30,31]. In this work, the influence of the demagnetization effects on the de-pairing current density, ${J}_{dep}$, has been analyzed by studying a Fe(Se,Te) iron-based superconductor. We started by measuring the first magnetization curve at different temperatures in order to obtain the lower critical field ${H}_{c1}$ values. We have noted that the demagnetization effects acting on the sample were significant, and they resulted in an underestimation of the real ${H}_{c1}$ values. From the ${H}_{c1}$ values, the London penetration depth, ${\lambda}_{L}$, as a function of the temperature, was obtained, and it was noted that it is not possible to fit the penetration depth with the typical exponential behavior that characterizes the s-wave superconductor. In this context, the plot of ${\lambda}_{L}^{-2}$, as a function of T, confirmed that our sample shows peculiarities which can be ascribed to a multigap superconducting behavior. Finally, after determining the coherence length, $\mathsf{\xi}$, from the upper critical field, ${H}_{c2}$, the ${J}_{dep}$ values were calculated as a function of the temperature, by considering the demagnetization effects and not considering them; this provided very high values in the framework of the iron-based superconductors.

## 2. Results and Discussion

^{2}is the magnetic flux quantum and k is the Ginzburg–Landau parameter. Using the Ginzburg–Landau theory, and by following the approach reported in Ref. [40], k can be calculated using the relation k = ${H}_{c2}$(0)/(2

^{1/2}${H}_{c}$(0)), where ${H}_{c}$(0) is the thermodynamic critical field. ${H}_{c}$(0) is calculated using ${H}_{c1}$, ${H}_{\mathrm{c}1}^{demag}$, and ${H}_{c2}$ values at a temperature of zero (i.e., ${H}_{c}\left(0\right)={\left({H}_{c1}\left(0\right)\times {H}_{c2}\left(0\right)\right)}^{1/2}$ ≈ 8 kOe and ${H}_{c}^{demag}\left(0\right)={\left({H}_{\mathrm{c}1}^{demag}(0)\times {H}_{c2}(0)\right)}^{1/2}$ ≈ 16.5 kOe); therefore, obtaining k ≈ 40 and k

^{demag}≈ 20 aligns with other Fe-chalcogenide superconductors [40,41,42,43]. For the calculation of ${\lambda}_{L}$, Equation (5), both ${H}_{c1}$ and ${H}_{\mathrm{c}1}^{demag}$, together with the k and k

^{demag}values, were used, and the results are reported in Figure 3. Both the ${\lambda}_{L}\left(T\right)$ curves were fitted using the following equation:

^{demag}values for the ${\lambda}_{L}$ calculation, respectively. At low temperatures, ${\lambda}_{L}$ does not show the typical exponential behavior expected for a fully gapped clean s-wave superconductor [44]. In general, a power law temperature dependence of ${\lambda}_{L}$ implies the presence of low-energy quasiparticle excitations [45].

^{−15}Tm

^{2}is the magnetic flux quantum.

_{dep}values, especially at low temperatures. The ${J}_{dep}$ values. obtained using Equation (9), are reported in Figure 6. It is worth noting the presence of two ${J}_{dep}$(T) curves. In particular, the black curve was obtained using ${\lambda}_{L}$ without the demagnetization correction, whereas the red curve was obtained by considering ${\lambda}_{L}^{demag}$ in conjunction with the demagnetization correction. Both sets of ${J}_{dep}$ values align with the highest values reported for the different iron-based families; this demonstrates the very good quality of this crystal [62,63,64,65,66]. It is evident that the ${J}_{dep}^{demag}$ values that take the demagnetizing factor into account are five times higher than the ${J}_{dep}$ values obtained without considering the demagnetizing factor. This helps us understand how important the demagnetizing factor is when estimating different important superconducting parameters such as ${H}_{c1}$, ${\lambda}_{L}$, and ${J}_{dep}$. It is important to note that the role of ${\lambda}_{L}$ is crucial to Equation (9) since it is squared, and therefore, a small ${\lambda}_{L}$ change generates a large ${J}_{dep}$ variation. In conclusion, in light of the fact that ${H}_{c1}$ and ${\lambda}_{L}$ can strongly depend on stoichiometry, the possibility of tuning it by modifying the fabrication process and parameters could be exploited to enhance the de-pairing current density.

## 3. Materials and Methods

_{0.5}Te

_{0.5}(nominal composition) crystal with the following dimensions: 3 × 3 × 0.2 mm

^{3}. The crystal was created using the Bridgman technique, and ${T}_{c}$ = 14.5 K. Details concerning the creation of the crystal are reported elsewhere [4]. A SEM-EDX analysis was performed on the sample, which showed the presence of twin boundaries and a slight deviation from the nominal composition in terms of stoichiometry (Fe

_{0.96}Te

_{0.59}Se

_{0.45}) [67]. This is probably due to micro inhomogeneity and the phase separation of magnetic premises, which is typical for crystal growth and synthesis in FeSeTe [68,69,70] and its basic compound FeSe [71,72,73]. The sample was characterized in a dc magnetic field that was applied perpendicularly to its largest face (H||c). In particular, the dc magnetic moment, as a function of the field, m(H), was measured using a Quantum Design PPMS-9T equipped with a VSM option. To avoid the effect on the sample response caused by the residual trapped field inside the PPMS dc magnet [74], this field was reduced below 1 × 10

^{−4}T [75]. Regarding the m(H) measurements, the sample was first cooled down to the measurement temperature in the zero field and thermally stabilized for at least 20 min. Then, the field was ramped with the fixed sweep rate value to +9 T, then it was reduced to −9 T, and finally, to +9 T again in order to acquire the complete hysteresis loop.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kamihara, Y.; Watanabe, T.; Hirano, M.; Hosono, H. Iron-based layered superconductor La[O
_{1−x}F_{x}]FeAs (x = 0.05–0.12) with T_{c}= 26 K. J. Am. Chem. Soc.**2008**, 130, 3296–3297. [Google Scholar] [CrossRef] - Bednorz, J.G.; Müller, K.A. Possible high T
_{c}superconductivity in the Ba-La-Cu-O system. Z. Phys. B Condens. Matter**1986**, 64, 189–193. [Google Scholar] [CrossRef] - Hsu, F.-C.; Luo, J.-Y.; Yeh, K.-W.; Chen, T.-K.; Huang, T.-W.; Wu, P.M.; Lee, Y.-C.; Huang, Y.-L.; Chu, Y.-Y.; Yan, D.-C.; et al. Superconductivity in the PbO-type structure α-FeSe. Proc. Natl. Acad. Sci. USA
**2008**, 105, 14262–14264. [Google Scholar] [CrossRef] [PubMed] - Galluzzi, A.; Buchkov, K.; Tomov, V.; Nazarova, E.; Leo, A.; Grimaldi, G.; Nigro, A.; Pace, S.; Polichetti, M. Evidence of pinning crossover and the role of twin boundaries in the peak effect in FeSeTe iron based superconductor. Supercond. Sci. Technol.
**2018**, 31, 015014. [Google Scholar] [CrossRef] - Galluzzi, A.; Buchkov, K.; Nazarova, E.; Tomov, V.; Grimaldi, G.; Leo, A.; Pace, S.; Polichetti, M. Transport properties and high upper critical field of a Fe(Se,Te) iron based superconductor. Eur. Phys. J. Spec. Top.
**2019**, 228, 725–731. [Google Scholar] [CrossRef] - Galluzzi, A.; Buchkov, K.; Nazarova, E.; Tomov, V.; Leo, A.; Grimaldi, G.; Pace, S.; Polichetti, M. Magnetic field sweep rate influence on the critical current capabilities of a Fe(Se,Te) crystal. J. Appl. Phys.
**2020**, 128, 073902. [Google Scholar] [CrossRef] - Hosono, H.; Yamamoto, A.; Hiramatsu, H.; Ma, Y. Recent advances in iron-based superconductors toward applications. Mater. Today
**2018**, 21, 278–302. [Google Scholar] [CrossRef] - Yuan, H.Q.; Singleton, J.; Balakirev, F.F.; Baily, S.A.; Chen, G.F.; Luo, J.L.; Wang, N.L. Nearly isotropic superconductivity in (Ba,K)Fe
_{2}As_{2}. Nature**2009**, 457, 565–568. [Google Scholar] [CrossRef] [PubMed] - Yamamoto, A.; Jaroszynski, J.; Tarantini, C.; Balicas, L.; Jiang, J.; Gurevich, A.; Larbalestier, D.C.; Jin, R.; Sefat, A.S.; McGuire, M.A.; et al. Small anisotropy, weak thermal fluctuations, and high field superconductivity in Co-doped iron pnictide Ba(Fe
_{1−x}Co_{x})_{2}As_{2}. Appl. Phys. Lett.**2009**, 94, 062511. [Google Scholar] [CrossRef] - Grimaldi, G.; Leo, A.; Martucciello, N.; Braccini, V.; Bellingeri, E.; Ferdeghini, C.; Galluzzi, A.; Polichetti, M.; Nigro, A.; Villegier, J.-C.; et al. Weak or Strong Anisotropy in Fe(Se,Te) Superconducting Thin Films Made of Layered Iron-Based Material? IEEE Trans. Appl. Supercond.
**2019**, 29, 1–4. [Google Scholar] [CrossRef] - Leo, A.; Braccini, V.; Bellingeri, E.; Ferdeghini, C.; Galluzzi, A.; Polichetti, M.; Nigro, A.; Pace, S.; Grimaldi, G. Anisotropy effects on the quenching current of Fe(Se,Te) Thin Films. IEEE Trans. Appl. Supercond.
**2018**, 28, 8234633. [Google Scholar] [CrossRef] - Eley, S.; Willa, R.; Chan, M.K.; Bauer, E.D.; Civale, L. Vortex phases and glassy dynamics in the highly anisotropic superconductor HgBa
_{2}CuO_{4+δ}. Sci. Rep.**2020**, 10, 10239. [Google Scholar] [CrossRef] - Polichetti, M.; Galluzzi, A.; Buchkov, K.; Tomov, V.; Nazarova, E.; Leo, A.; Grimaldi, G.; Pace, S. A precursor mechanism triggering the second magnetization peak phenomenon in superconducting materials. Sci. Rep.
**2021**, 11, 7247. [Google Scholar] [CrossRef] [PubMed] - Llovo, I.F.; Sónora, D.; Mosqueira, J.; Salem-Sugui, S.; Sundar, S.; Alvarenga, A.D.; Xie, T.; Liu, C.; Li, S.L.; Luo, H.Q. Vortex dynamics and second magnetization peak in the iron-pnictide superconductor Ca
_{0.82}La_{0.18}Fe_{0.96}Ni_{0.04}As_{2}. Supercond. Sci. Technol.**2021**, 34, 115010. [Google Scholar] [CrossRef] - Yi, X.; Xing, X.; Meng, Y.; Zhou, N.; Wang, C.; Sun, Y.; Shi, Z. Anomalous Second Magnetization Peak in 12442-Type RbCa
_{2}Fe_{4}As_{4}F_{2}Superconductors. Chin. Phys. Lett.**2023**, 40, 027401. [Google Scholar] [CrossRef] - Lopes, P.V.; Sundar, S.; Salem-Sugui, S.; Hong, W.; Luo, H.; Ghivelder, L. Second magnetization peak, anomalous field penetration, and Josephson vortices in KCa
_{2}Fe_{4}As_{4}F_{2}bilayer pnictide superconductor. Sci. Rep.**2022**, 12, 20359. [Google Scholar] [CrossRef] - Prozorov, R.; Ni, N.; Tanatar, M.A.; Kogan, V.G.; Gordon, R.T.; Martin, C.; Blomberg, E.C.; Prommapan, P.; Yan, J.Q.; Bud’ko, S.L.; et al. Vortex phase diagram of Ba(Fe
_{0.93}Co_{0.07})_{2}As_{2}single crystals. Phys. Rev. B**2008**, 78, 224506. [Google Scholar] [CrossRef] - Sun, Y.; Taen, T.; Tsuchiya, Y.; Pyon, S.; Shi, Z.; Tamegai, T. Magnetic relaxation and collective vortex creep in FeTe
_{0.6}Se_{0.4}single crystal. Europhys. Lett.**2013**, 103, 57013. [Google Scholar] [CrossRef] - Pramanik, A.K.; Harnagea, L.; Nacke, C.; Wolter, A.U.B.; Wurmehl, S.; Kataev, V.; Büchner, B. Fishtail effect and vortex dynamics in LiFeAs single crystals. Phys. Rev. B
**2011**, 83, 094502. [Google Scholar] [CrossRef] - Sundar, S.; Salem-Sugui, S.; Amorim, H.S.; Wen, H.H.; Yates, K.A.; Cohen, L.F.; Ghivelder, L. Plastic pinning replaces collective pinning as the second magnetization peak disappears in the pnictide superconductor Ba
_{0.75}K_{0.25}Fe_{2}As_{2}. Phys. Rev. B**2017**, 95, 134509. [Google Scholar] [CrossRef] - Taen, T.; Tsuchiya, Y.; Nakajima, Y.; Tamegai, T. Critical current densities and vortex dynamics in FeTe
_{x}Se_{1−x}single crystals. Phys. C Supercond. Appl.**2010**, 470, 1106–1108. [Google Scholar] [CrossRef] - Ren, C.; Wang, Z.S.; Luo, H.Q.; Yang, H.; Shan, L.; Wen, H.H. Evidence for two energy gaps in superconducting Ba
_{0.6}K_{0.4}Fe_{2}As_{2}single crystals and the breakdown of the uemura plot. Phys. Rev. Lett.**2008**, 101, 257006. [Google Scholar] [CrossRef] [PubMed] - Wang, X.L.; Dou, S.X.; Ren, Z.A.; Yi, W.; Li, Z.C.; Zhao, Z.X.; Lee, S.I. Unconventional superconductivity of NdFeAsO
_{0.82}F_{0.18}indicated by the low temperature dependence of the lower critical field H_{c1}. J. Phys. Condens. Matter**2009**, 21, 205701. [Google Scholar] [CrossRef] - Martin, C.; Gordon, R.T.; Tanatar, M.A.; Kim, H.; Ni, N.; Bud’Ko, S.L.; Canfield, P.C.; Luo, H.; Wen, H.H.; Wang, Z.; et al. Nonexponential London penetration depth of external magnetic fields in superconducting Ba
_{1−x}K_{x}Fe_{2}As_{2}single crystals. Phys. Rev. B**2009**, 80, 020501. [Google Scholar] [CrossRef] - Abdel-Hafiez, M.; Ge, J.; Vasiliev, A.N.; Chareev, D.A.; Van De Vondel, J.; Moshchalkov, V.V.; Silhanek, A.V. Temperature dependence of lower critical field H
_{c1}(T) shows nodeless superconductivity in FeSe. Phys. Rev. B**2013**, 88, 174512. [Google Scholar] [CrossRef] - Song, Y.J.; Ghim, J.S.; Yoon, J.H.; Lee, K.J.; Jung, M.H.; Ji, H.S.; Shim, J.H.; Bang, Y.; Kwon, Y.S. Small anisotropy of the lower critical field and the s±- wave two-gap feature in single-crystal LiFeAs. Europhys. Lett.
**2011**, 94, 57008. [Google Scholar] [CrossRef] - Prozorov, R.; Kogan, V.G. London penetration depth in iron-based superconductors. Rep. Prog. Phys.
**2011**, 74, 124505–124525. [Google Scholar] [CrossRef] - Rosenstein, B.; Li, D. Ginzburg-Landau theory of type II superconductors in magnetic field. Rev. Mod. Phys.
**2010**, 82, 109–168. [Google Scholar] [CrossRef] - Tahara, S.; Anlage, S.M.; Halbritter, J.; Eom, C.B.; Fork, D.K.; Geballe, T.H.; Beasley, M.R. Critical currents, pinning, and edge barriers in narrow YBa
_{2}Cu_{3}O_{7−δ}thin films. Phys. Rev. B**1990**, 41, 11203–11208. [Google Scholar] [CrossRef] [PubMed] - Tinkham, M. Introduction to Superconductivity; Dover Publications: Mineola, NY, USA, 2004; ISBN 0486134725. [Google Scholar]
- Arpaia, R.; Nawaz, S.; Lombardi, F.; Bauch, T. Improved nanopatterning for YBCO nanowires approaching the depairing current. IEEE Trans. Appl. Supercond.
**2013**, 23, 1101505. [Google Scholar] [CrossRef] - Wang, T.; Ma, Y.; Li, W.; Chu, J.; Wang, L.; Feng, J.; Xiao, H.; Li, Z.; Hu, T.; Liu, X.; et al. Two-gap superconductivity in CaFe
_{0.88}Co_{0.12}AsF revealed by temperature dependence of the lower critical field H_{c1}^{c}(T). npj Quantum Mater.**2019**, 4, 33. [Google Scholar] [CrossRef] - Musolino, N.; Bals, S.; Van Tendeloo, G.; Clayton, N.; Walker, E.; Flükiger, R. Modulation-free phase in heavily Pb-doped (Bi,Pb)2212 crystals. Phys. C Supercond. Its Appl.
**2003**, 399, 1–7. [Google Scholar] [CrossRef] - Galluzzi, A.; Leo, A.; Masi, A.; Varsano, F.; Nigro, A.; Grimaldi, G.; Polichetti, M. Magnetic Vortex Phase Diagram for a Non-Optimized CaKFe
_{4}As_{4}Superconductor Presenting a Wide Vortex Liquid Region and an Ultra-High Upper Critical Field. Appl. Sci.**2023**, 13, 884. [Google Scholar] [CrossRef] - Felner, I.; Kopelevich, Y. Magnetization measurement of a possible high-temperature superconducting state in amorphous carbon doped with sulfur. Phys. Rev. B
**2009**, 79, 233409. [Google Scholar] [CrossRef] - Yadav, C.S.; Paulose, P.L. Upper critical field, lower critical field and critical current density of FeTe
_{0.60}Se_{0.40}single crystals. New J. Phys.**2009**, 11, 103046. [Google Scholar] [CrossRef] - Stoner, E.C. XCVII. The demagnetizing factors for ellipsoids. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1945**, 36, 803–821. [Google Scholar] [CrossRef] - Prando, G.; Giraud, R.; Aswartham, S.; Vakaliuk, O.; Abdel-Hafiez, M.; Hess, C.; Wurmehl, S.; Wolter, A.U.B.; Büchner, B. Evidence for a vortex–glass transition in superconducting Ba(Fe
_{0.9}Co_{0.1})_{2}As_{2}. J. Phys. Condens. Matter**2013**, 25, 505701. [Google Scholar] [CrossRef] [PubMed] - Yeshurun, Y.; Malozemoff, A.P.; Shaulov, A. Magnetic relaxation in high-temperature superconductors. Rev. Mod. Phys.
**1996**, 68, 911–949. [Google Scholar] [CrossRef] - Maheshwari, P.K.; Gahtori, B.; Gupta, A.; Awana, V.P.S. Impact of Fe site Co substitution on superconductivity of Fe
_{1−x}Co_{x}Se_{0.5}Te_{0.5}(x = 0.0 to 0.10): A flux free single crystal study. AIP Adv.**2017**, 7, 15006. [Google Scholar] [CrossRef] - Murugesan, K.; Lingannan, G.; Ishigaki, K.; Uwatoko, Y.; Sekine, C.; Kawamura, Y.; Junichi, H.; Joseph, B.; Vajeeston, P.; Maheswari, P.K.; et al. Pressure Dependence of Superconducting Properties, Pinning Mechanism, and Crystal Structure of the Fe
_{0.99}Mn_{0.01}Se_{0.5}Te_{0.5}Superconductor. ACS Omega**2021**, 6, 30419–30431. [Google Scholar] [CrossRef] [PubMed] - Lei, H.; Hu, R.; Petrovic, C. Critical fields, thermally activated transport, and critical current density of β-FeSe single crystals. Phys. Rev. B
**2011**, 84, 014520. [Google Scholar] [CrossRef] - Dutta, P.; Pramanick, S.; Chatterjee, S. Effect of S-doping on the magnetic and electrical properties of FeSe superconductor. Phys. C Supercond. Appl.
**2022**, 602, 1354126. [Google Scholar] [CrossRef] - Fletcher, J.D.; Serafin, A.; Malone, L.; Analytis, J.G.; Chu, J.H.; Erickson, A.S.; Fisher, I.R.; Carrington, A. Evidence for a nodal-line superconducting state in LaFePO. Phys. Rev. Lett.
**2009**, 102, 147001. [Google Scholar] [CrossRef] [PubMed] - Takahashi, H.; Imai, Y.; Komiya, S.; Tsukada, I.; Maeda, A. Anomalous temperature dependence of the superfluid density caused by a dirty-to-clean crossover in superconducting FeSe
_{0.4}Te_{0.6}single crystals. Phys. Rev. B**2011**, 84, 132503. [Google Scholar] [CrossRef] - Bendele, M.; Weyeneth, S.; Puzniak, R.; Maisuradze, A.; Pomjakushina, E.; Conder, K.; Pomjakushin, V.; Luetkens, H.; Katrych, S.; Wisniewski, A.; et al. Anisotropic superconducting properties of single-crystalline FeSe
_{0.5}Te_{0.5}. Phys. Rev. B**2010**, 81, 224520. [Google Scholar] [CrossRef] - Milošević, M.V.; Perali, A. Emergent phenomena in multicomponent superconductivity: An introduction to the focus issue. Supercond. Sci. Technol.
**2015**, 28, 060201. [Google Scholar] [CrossRef] - Weyeneth, S.; Puzniak, R.; Mosele, U.; Zhigadlo, N.D.; Katrych, S.; Bukowski, Z.; Karpinski, J.; Kohout, S.; Roos, J.; Keller, H. Anisotropy of superconducting single crystal SmFeAsO
_{0.8}F_{0.2}studied by torque magnetometry. J. Supercond. Nov. Magn.**2009**, 22, 325–329. [Google Scholar] [CrossRef] - Gonnelli, R.S.; Daghero, D.; Tortello, M.; Ummarino, G.A.; Stepanov, V.A.; Kremer, R.K.; Kim, J.S.; Zhigadlo, N.D.; Karpinski, J. Point-contact Andreev-reflection spectroscopy in ReFeAsO
_{1−x}F_{x}(Re = La, Sm): Possible evidence for two nodeless gaps. Phys. C Supercond. Appl.**2009**, 469, 512–520. [Google Scholar] [CrossRef] - Szabó, P.; Pribulová, Z.; Pristáš, G.; Bud’ko, S.L.; Canfield, P.C.; Samuely, P. Evidence for two-gap superconductivity in Ba
_{0.55}K_{0.45}Fe_{2}As_{2}from directional point-contact Andreev-reflection spectroscopy. Phys. Rev. B**2009**, 79, 012503. [Google Scholar] [CrossRef] - Mu, G.; Luo, H.; Wang, Z.; Shan, L.; Ren, C.; Wen, H.H. Low temperature specific heat of the hole-doped Ba
_{0.6}K_{0.4}Fe_{2}As_{2}single crystals. Phys. Rev. B**2009**, 79, 174501. [Google Scholar] [CrossRef] - Bekaert, J.; Vercauteren, S.; Aperis, A.; Komendová, L.; Prozorov, R.; Partoens, B.; Milošević, M.V. Anisotropic type-I superconductivity and anomalous superfluid density in OsB
_{2}. Phys. Rev. B**2016**, 94, 144506. [Google Scholar] [CrossRef] - Klein, T.; Braithwaite, D.; Demuer, A.; Knafo, W.; Lapertot, G.; Marcenat, C.; Rodière, P.; Sheikin, I.; Strobel, P.; Sulpice, A.; et al. Thermodynamic phase diagram of Fe(Se
_{0.5}Te_{0.5}) single crystals in fields up to 28 tesla. Phys. Rev. B**2010**, 82, 184506. [Google Scholar] [CrossRef] - Diaconu, A.; Martin, C.; Hu, J.; Liu, T.; Qian, B.; Mao, Z.; Spinu, L. Possible nodal superconducting gap in Fe
_{1+y}(Te_{1−x}Se_{x}) single crystals from ultralow temperature penetration depth measurements. Phys. Rev. B**2013**, 88, 104502. [Google Scholar] [CrossRef] - Kumar, R.; Varma, G.D. Study of TAFF and vortex phase of Fe
_{x}Te_{0.6}0Se_{0.40}(0.970 ≤ x ≤ 1.030) single crystals. Phys. Scr.**2020**, 95, 045814. [Google Scholar] [CrossRef] - Poole, C.; Farach, H.; Creswick, R.; Prozorov, R. Superconductivity; Academic Press: Cambridge, MA, USA, 2007; ISBN 0080550487. [Google Scholar]
- Peri, A.; Mangel, I.; Keren, A. Superconducting Stiffness and Coherence Length of FeSe
_{0.5}Te_{0.5}Measured in a Zero-Applied Field. Condens. Matter**2023**, 8, 39. [Google Scholar] [CrossRef] - Bardeen, J. Critical fields and currents in superconductors. Rev. Mod. Phys.
**1962**, 34, 667–681. [Google Scholar] [CrossRef] - Maiorov, B.; Mele, P.; Baily, S.A.; Weigand, M.; Lin, S.Z.; Balakirev, F.F.; Matsumoto, K.; Nagayoshi, H.; Fujita, S.; Yoshida, Y.; et al. Inversion of the upper critical field anisotropy in FeTeS films. Supercond. Sci. Technol.
**2014**, 27, 044005. [Google Scholar] [CrossRef] - Her, J.L.; Kohama, Y.; Matsuda, Y.H.; Kindo, K.; Yang, W.H.; Chareev, D.A.; Mitrofanova, E.S.; Volkova, O.S.; Vasiliev, A.N.; Lin, J.Y. Anisotropy in the upper critical field of FeSe and FeSe
_{0.33}Te_{0.67}single crystals. Supercond. Sci. Technol.**2015**, 28, 045013. [Google Scholar] [CrossRef] - Sun, Y.; Pan, Y.; Zhou, N.; Xing, X.; Shi, Z.; Wang, J.; Zhu, Z.; Sugimoto, A.; Ekino, T.; Tamegai, T.; et al. Comparative study of superconducting and normal-state anisotropy in Fe
_{1+y}Te_{0.6}Se_{0.4}superconductors with controlled amounts of interstitial excess Fe. Phys. Rev. B**2021**, 103, 224506. [Google Scholar] [CrossRef] - Mishev, V.; Nakajima, M.; Eisaki, H.; Eisterer, M. Effects of introducing isotropic artificial defects on the superconducting properties of differently doped Ba-122 based single crystals. Sci. Rep.
**2016**, 6, 27783. [Google Scholar] [CrossRef] [PubMed] - Kondo, K.; Motoki, S.; Hatano, T.; Urata, T.; Iida, K.; Ikuta, H. NdFeAs(O,H) epitaxial thin films with high critical current density. Supercond. Sci. Technol.
**2020**, 33, 09LT01. [Google Scholar] [CrossRef] - Li, J.; Yuan, J.; Yuan, Y.H.; Ge, J.Y.; Li, M.Y.; Feng, H.L.; Pereira, P.J.; Ishii, A.; Hatano, T.; Silhanek, A.V.; et al. Direct observation of the depairing current density in single-crystalline Ba
_{0.5}K_{0.5}Fe_{2}As_{2}microbridge with nanoscale thickness. Appl. Phys. Lett.**2013**, 103, 62603. [Google Scholar] [CrossRef] - Bristow, M.; Knafo, W.; Reiss, P.; Meier, W.; Canfield, P.C.; Blundell, S.J.; Coldea, A.I. Competing pairing interactions responsible for the large upper critical field in a stoichiometric iron-based superconductor CaKFe
_{4}As_{4}. Phys. Rev. B**2020**, 101, 134502. [Google Scholar] [CrossRef] - Sun, Y.; Ohnuma, H.; Ayukawa, S.-Y.; Noji, T.; Koike, Y.; Tamegai, T.; Kitano, H. Achieving the depairing limit along the c axis in Fe
_{1+y}Te_{1−x}Se_{x}single crystals. Phys. Rev. B**2020**, 101, 134516. [Google Scholar] [CrossRef] - Galluzzi, A.; Buchkov, K.; Tomov, V.; Nazarova, E.; Kovacheva, D.; Leo, A.; Grimaldi, G.; Pace, S.; Polichetti, M. Mixed state properties of iron based Fe(Se,Te) superconductor fabricated by Bridgman and by self-flux methods. J. Appl. Phys.
**2018**, 123, 233904. [Google Scholar] [CrossRef] - Tsurkan, V.; Deisenhofer, J.; Günther, A.; Kant, C.; Klemm, M.; von Nidda, H.-A.; Schrettle, F.; Loidl, A. Physical properties of FeSe
_{0.5}Te_{0.5}single crystals grown under different conditions. Eur. Phys. J. B**2011**, 79, 289–299. [Google Scholar] [CrossRef] - Wittlin, A.; Aleshkevych, P.; Przybylińska, H.; Gawryluk, D.J.; Dłuzewski, P.; Berkowski, M.; Puźniak, R.; Gutowska, M.U.; Wiśniewski, A. Microstructural magnetic phases in superconducting FeTe
_{0.65}Se_{0.35}. Supercond. Sci. Technol.**2012**, 25, 065019. [Google Scholar] [CrossRef] - Sivakov, A.G.; Bondarenko, S.I.; Prokhvatilov, A.I.; Timofeev, V.P.; Pokhila, A.S.; Koverya, V.P.; Dudar, I.S.; Link, S.I.; Legchenkova, I.V.; Bludov, A.N.; et al. Microstructural and transport properties of superconducting FeTe
_{0.65}Se_{0.35}crystals. Supercond. Sci. Technol.**2017**, 30, 015018. [Google Scholar] [CrossRef] - McQueen, T.M.; Huang, Q.; Ksenofontov, V.; Felser, C.; Xu, Q.; Zandbergen, H.; Hor, Y.S.; Allred, J.; Williams, A.J.; Qu, D.; et al. Extreme sensitivity of superconductivity to stoichiometry in Fe
_{1+δ}Se. Phys. Rev. B**2009**, 79, 014522. [Google Scholar] [CrossRef] - Onar, K.; Yakinci, M.E. Solid state synthesis and characterization of bulk β-FeSe superconductors. J. Alloys Compd.
**2015**, 620, 210–216. [Google Scholar] [CrossRef] - Fiamozzi Zignani, C.; De Marzi, G.; Grimaldi, G.; Leo, A.; Guarino, A.; Vannozzi, A.; della Corte, A.; Pace, S. Fabrication and Physical Properties of Polycrystalline Iron-Chalcogenides Superconductors. IEEE Trans. Appl. Supercond.
**2017**, 27, 1–5. [Google Scholar] [CrossRef] - Galluzzi, A.; Nigro, A.; Fittipaldi, R.; Guarino, A.; Pace, S.; Polichetti, M. DC magnetic characterization and pinning analysis on Nd
_{1.85}Ce_{0.15}CuO_{4}cuprate superconductor. J. Magn. Magn. Mater.**2019**, 475, 125–129. [Google Scholar] [CrossRef] - Galluzzi, A.; Mancusi, D.; Cirillo, C.; Attanasio, C.; Pace, S.; Polichetti, M. Determination of the Transition Temperature of a Weak Ferromagnetic Thin Film by Means of an Evolution of the Method Based on the Arrott Plots. J. Supercond. Nov. Magn.
**2018**, 31, 1127–1132. [Google Scholar] [CrossRef]

**Figure 1.**The field dependence of the initial magnetic moment curve is plotted for different temperatures. The black dashed line provides the linear fit for the low field m(H) curves.

**Figure 2.**(

**a**) Comparison between the temperature dependence of the lower critical field ${H}_{c1}$ (black squares) (where the black solid line is the fit of the data using the following equation, ${H}_{c1}\left(T\right)={H}_{c1}\left(0\right){\left(1-\frac{T}{{T}_{c}}\right)}^{n}$) and the temperature dependence of the lower critical field after considering the demagnetization effects ${H}_{\mathrm{c}1}^{demag}$ (red circles) (where the red dashed line is the fit of the data using the following equation ${H}_{c1}^{demag}\left(T\right)={H}_{c1}^{demag}\left(0\right){\left(1-\frac{T}{{T}_{c}}\right)}^{n}$). When performing the fit, ${H}_{c1}$(0) = 143 Oe and n = 1.54 were obtained without considering the demagnetizing factor, and ${H}_{c1}^{demag}$(0) = 597 Oe and n = 1.54 were obtained by considering the demagnetizing factor. (

**b**) Ratio of the ${H}_{\mathrm{c}1}^{demag}\left(\mathrm{T}\right)$ and ${H}_{\mathrm{c}1}\left(\mathrm{T}\right)$ values. The red star indicates the ratio of ${H}_{\mathrm{c}1}^{demag}\left(0\right)/{H}_{c1}$(0).

**Figure 3.**Temperature dependence of the London penetration depth ${\mathsf{\lambda}}_{\mathrm{L}}$ obtained from ${H}_{c1}$ (black squares) and ${H}_{\mathrm{c}1}^{demag}$ (red circles). Both the curves were fitted with ${\lambda}_{L}\left(T\right)={\lambda}_{L}\left(0\right){\left(1-\frac{T}{{T}_{c}}\right)}^{n}$. When performing the fit, ${\lambda}_{L}$(0) = 208 nm and n = −0.72 were obtained without considering the demagnetizing factor, whereas ${\lambda}_{L}^{demag}$(0) = 92 nm and n = −0.72 were obtained by considering the demagnetizing factor.

**Figure 4.**Temperature dependence of the London penetration depth ${\lambda}_{L}^{-2}$, obtained from ${H}_{c1}$ (black squares) and ${H}_{\mathrm{c}1}^{demag}$ (red circles). The solid lines are a guide for the eyes.

**Figure 5.**Temperature dependence of the coherence length, $\mathsf{\xi}$ (black squares), together with its fit and the following equation $\mathsf{\xi}\left(T\right)=\mathsf{\xi}\left(0\right){\left(1-\frac{T}{{T}_{c}}\right)}^{n}$. After performing the fit, $\mathsf{\xi}\left(0\right)$ ≈ 3 nm and $n\approx -0.64$ were obtained.

**Figure 6.**Temperature dependence of the de-pairing current density, ${J}_{dep}$ with (red closed circles) and without (black closed squares) consideration of the demagnetizing factor. The solid lines are a guide for the eyes.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Galluzzi, A.; Buchkov, K.; Tomov, V.; Nazarova, E.; Leo, A.; Grimaldi, G.; Crisan, A.; Polichetti, M.
The Depairing Current Density of a Fe(Se,Te) Crystal Evaluated in Presence of Demagnetizing Factors. *Condens. Matter* **2023**, *8*, 91.
https://doi.org/10.3390/condmat8040091

**AMA Style**

Galluzzi A, Buchkov K, Tomov V, Nazarova E, Leo A, Grimaldi G, Crisan A, Polichetti M.
The Depairing Current Density of a Fe(Se,Te) Crystal Evaluated in Presence of Demagnetizing Factors. *Condensed Matter*. 2023; 8(4):91.
https://doi.org/10.3390/condmat8040091

**Chicago/Turabian Style**

Galluzzi, Armando, Krastyo Buchkov, Vihren Tomov, Elena Nazarova, Antonio Leo, Gaia Grimaldi, Adrian Crisan, and Massimiliano Polichetti.
2023. "The Depairing Current Density of a Fe(Se,Te) Crystal Evaluated in Presence of Demagnetizing Factors" *Condensed Matter* 8, no. 4: 91.
https://doi.org/10.3390/condmat8040091