- Open Access
Unconventional magnetization in the multiphase superconductor
Phys. Rev. B 113, 024502 – Published 6 January, 2026
DOI: https://doi.org/10.1103/kcd9-9175
Abstract
Unconventional superconductors have specific signatures in their magnetic properties, such as intrinsic magnetization at interfaces and around defects and fractional and multiquanta vortices, with much attention focused on heavy fermion and high- superconductors. Here, we report the observation of highly anomalous magnetization in β-, a layered superconductor previously shown to exhibit a magnetic-field-induced transition from -wave to nodal -wave superconductivity at a transition field . This transition is driven by the coupling between spin-polarized electronic bands and the in-plane magnetic field. In the unconventional phase (above ) we observe three striking features: strictly linear and nonhysteretic dc magnetization, a sharp drop in the ac susceptibility when the field is applied parallel to the plane, and a pronounced anisotropy in the magnetic response between parallel and perpendicular field orientations. We show that these features are directly correlated with the expected transition to the nodal -wave state and propose that the unusual magnetization behavior can be explained by a transition from a conventional vortex lattice in the low-field -wave phase to a domain structure corresponding to spatial phase separation into superconducting (-wave) and normal domains above . Our work identifies experimental signatures of unconventional multiphase superconductivity, offering an insight into the magnetic-field response of nodal states.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (66)
- N. T. Huy, D. E. de Nijs, Y. K. Huang, and A. de Visser, Unusual upper critical field of the ferromagnetic superconductor UCoGe, Phys. Rev. Lett. 100, 077002 (2008).
- S. de la Barrera, M. R. Sinko, D. P. Gopalan, N. Sivadas, K. L. Seyler, K. Watanabe, T. Taniguchi, A. W. Tsen, X. Xu, D. Xiao, and B. M. Hunt, Tuning Ising superconductivity with layer and spin–orbit coupling in two-dimensional transition-metal dichalcogenides, Nat. Commun. 9, 1427 (2018).
- S. Ran, I-Lin Liu, Y. S. Eo, D. J. Campbell, P. M. Neves, W. T. Fuhrman, S. R. Saha, C. Eckberg, H. Kim, D. Graf, F. Balakirev, J. Singleton, J. Paglione, and N. P. Butch, Extreme magnetic field-boosted superconductivity, Nat. Phys. 15, 1250 (2019).
- Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Pauli-limit violation and re-entrant superconductivity in moiré graphene, Nature (London) 595, 526 (2021).
- Y. Iguchi, R. A. Shi, K. Kihou, C.-H. Lee, M. Barkman, A. L. Benfenati, V. Grinenko, E. Babaev, and K. A. Moler, Superconducting vortices carrying a temperature-dependent fraction of the flux quantum, Science 380, 1244 (2023).
- J. Jang, D. G. Ferguson, V. Vakaryuk, R. Budakian, S. B. Chung, P. M. Goldbart, and Y. Maeno, Observation of half-height magnetization steps in , Science 331, 186 (2011).
- R. Joynt and L. Taillefer, The superconducting phases of , Rev. Mod. Phys. 74, 235 (2002).
- S. Khim, J. F. Landaeta, J. Banda, N. Bannor, M. Brando, P. M. R. Brydon, D. Hafner, R. Küchler, R. Cardoso-Gil, U. Stockert, et al., Field-induced transition within the superconducting state of , Science 373, 1012 (2021).
- T. M. Riseman, P. G. Kealey, E. M. Forgan, A. P. Mackenzie, L. M. Galvin, A. W. Tyler, S. L. Lee, C. Ager, D. McK. Paul, C. M. Aegerter, et al., Observation of a square flux-line lattice in the unconventional superconductor , Nature (London) 396, 242 (1998).
- G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori, H. Nakamura, and M. Sigrist, Time-reversal symmetry-breaking superconductivity in , Nature (London) 394, 558 (1998).
- M. Sakano, K. Okawa, M. Kanou, H. Sanjo, T. Okuda, T. Sasagawa, and K. Ishizaka, Topologically protected surface states in a centrosymmetric superconductor β-, Nat. Commun. 6, 8595 (2015).
- K. Iwaya, Y. Kohsaka, K. Okawa, T. Machida, M. S. Bahramy, T. Hanaguri, and T. Sasagawa, Full-gap superconductivity in spin-polarised surface states of topological semimetal β-, Nat. Commun. 8, 976 (2017).
- X. Zhang, Q. Liu, J.-W. Luo, A. J. Freeman, and A. Zunger, Hidden spin polarization in inversion-symmetric bulk crystals, Nat. Phys. 10, 387 (2014).
- T. Xu, B. T. Wang, M. Wang, Q. Jiang, X. P. Shen, B. Gao, M. Ye., and S. Qiao, Nonhelical spin texture in the normal states of the centrosymmetric superconductor , Phys. Rev. B 100, 161109(R) (2019).
- L. Powell, W. Kuang, G. Hawkins-Pottier, R. Jalil, J. Birkbeck, Z. Jiang, M. Kim, Y. Zou, S. Komrakova, S. Haigh, et al., Multiphase superconductivity in , Nat. Commun. 16, 291 (2025).
- E. Herrera, I. Guillamón, J. A. Galvis, A. Correa, A. Fente, R. F. Luccas, F. J. Mompean, M. García-Hernández, S. Vieira, J. P. Brison, and H. Suderow, Magnetic field dependence of the density of states in the multiband superconductor , Phys. Rev. B 92, 054507 (2015).
- J. B. Llorens, L. Embon, A. Correa, J. D. González, E. Herrera, I. Guillamón, R. F. Luccas, J. Azpeitia, F. J. Mompeán, M. Garcia-Hernández, et al., Observation of a gel of quantum vortices in a superconductor at very low magnetic fields, Phys. Rev. Res. 2, 013329 (2020).
- M. Soda, N. Kagamida, E. Campillo, E. M. Forgan, E. Blackburn, S. Mühlbauer, R. Cubitt, H. Suderow, I. Guillamón, E. Herrera, et al., Penetration depth and coherence length in the superconductor β-, J. Phys. Soc. Jpn. 91, 034706 (2022).
- J. Chen, G. Pang, H. Su, Y. Chen, and H. Yuan, Nodeless superconductivity in , Phys. Rev. B 101, 054514 (2020).
- J. Kačmarčík, Z. Pribulová, T. Samuely, P. Szabó, V. Cambel, J. Šoltýs, E. Herrera, H. Suderow, A. Correa-Orellana, et al., Single-gap superconductivity in , Phys. Rev. B 93, 144502 (2016).
- M. Soda, N. Kagamida, S. Mühlbauer, E. M. Forgan, E. Campillo, M. Kriener, H. Yoshizawa, and H. Kawano-Furukawa, Field dependence of superfluid density in β-, J. Phys. Soc. Jpn. 90, 104710 (2021).
- I. R. Shein and A. L. Ivanovskii, Electronic band structure and Fermi surface of tetragonal low-temperature superconductor as predicted from first principles, J. Supercond. Nov. Magn. 26, 1 (2013).
- L. Lyard, P. Samuely, P. Szabo, T. Klein, C. Marcenat, L. Paulius, K. H. P. Kim, C. U. Jung, H.-S. Lee, B. Kang, et al., Anisotropy of the upper critical field and critical current in single crystal , Phys. Rev. B 66, 180502(R) (2002).
- J. Shiogai, S. Kimura, S. Awaji, T. Nojima, and A. Tsukazaki, Anisotropy of the upper critical field and its thickness dependence in superconducting FeSe electric-double-layer transistors, Phys. Rev. B 97, 174520 (2018).
- J. Singleton and C. Mielke, Quasi-two-dimensional organic superconductors: A review, Contemp. Phys. 43, 63 (2002).
- D. Roditchev, F. Giubileo, F. Bobba, R. Lamy, E.-M. Choi, H.-J. Kim, W. N. Kang, S. Miraglia, J. Marcus, W. Sacks, et al., Two-gap interplay in : A tunneling spectroscopy study, Physica C 408–410, 768 (2004).
- A. Hamill, B. Heischmidt, E. Sohn, D. Shaffer, K.-T. Tsai, X. Zhang, X. Xi, A. Suslov, H. Berger, L. Forró, et al., Two-fold symmetric superconductivity in few-layer , Nat. Phys. 17, 949 (2021).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/kcd9-9175 for the details of crystal growth and characterization; details of dc magnetization and susceptibility measurements, including demagnetization correction; amplitude dependence of ac susceptibility in perpendicular fields; critical current and resistance measurements; and typical magnetization for conventional type-I superconductors, which also contains Refs. [21, 42, 65, 66].
- J. D. Livingston, Magnetic properties of superconducting lead-base alloys, Phys. Rev. 129, 1943 (1963).
- C. P. Bean, Magnetization of high-field superconductors, Rev. Mod. Phys. 36, 31 (1964).
- M. Tinkham, Introduction to Superconductivity (Dover, New York, 2004).
- R. A. Klemm, Layered Superconductors (Oxford University, New York, 2012).
- W. Kuang, G. Lopez-Polin, H. Lee, F. Guinea, G. Whitehead, I. Timokhin, A. I. Berdyugin, R. Krishna Kumar, O. V. Yazyev, N. Walet, A. Principi, A. K. Geim, and I. V. Grigorieva, Magnetization signature of topological surface states in a non-symmorphic superconductor, Adv. Mater. 33, 2103257 (2021).
- A. F. Khoder, The superconducting transition and the behavior of the ac susceptibility, Phys. Lett. A 94, 378 (1983).
- L. J. M. van de Klundert, E. A. Gijsbertse, and L. C. van der Marel, On the AC susceptibility of metals in the normal or superconducting state, Physica 69, 159 (1973).
- J. G. Park, Persistent currents induced in the superconducting surface sheath, Adv. Phys. 18, 103 (1969).
- H. J. Fink and L. J. Barnes, Critical state of the superconducting surface sheath, Phys. Rev. Lett. 15, 792 (1965).
- L. J. Barnes and H. J. Fink, Critical currents in the superconducting surface sheath, Phys. Rev. 149, 186 (1966).
- F. Gömöry, Characterization of high-temperature superconductors by ac susceptibility measurements, Supercond. Sci. Technol. 10, 523 (1997).
- P. P. J. Van Engelen, G. J. C. Bots, and B. S. Blaisse, Amplitude effects of the alternating field susceptibility of superconducting tantalum, Phys. Lett. 19, 465 (1965).
- M. Strongin, D. G. Schweitzer, A. Paskin, and P. P. Craig, Magnetic-field penetration and breakdown of surface superconductivity, Phys. Rev. 136, A926 (1964).
- N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Temperature and purity dependence of the superconducting critical field, . III. Electron spin and spin-orbit effects, Phys. Rev. 147, 295 (1966).
- G. Blatter, M. V. Feigel'man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Vortices in high-temperature superconductors, Rev. Mod. Phys. 66, 1125 (1994).
- E. Zeldov, D. Majer, M. Konczykowski, V. B. Geshkenbein, V. M. Vinokur, and H. Shtrikman, Thermodynamic observation of first-order vortex-lattice melting transition in , Nature (London) 375, 373 (1995).
- P. L. Gammel, L. F. Schneemeyer, J. V. Wasczak, and D. J. Bishop, Evidence from mechanical measurements for flux-lattice melting in single-crystal and , Phys. Rev. Lett. 61, 1666 (1988).
- D. E. Farrell, J. P. Rice, and D. M. Ginsberg, Experimental evidence for flux-lattice melting, Phys. Rev. Lett. 67, 1165 (1991).
- M. Sigrist and K. Ueda, Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 (1991).
- F. Kidwingira, J. D. Strand, D. J. Van Harlingen, and Y. Maeno, Dynamical superconducting order parameter domains in , Science 314, 1267 (2006).
- S. A. Kivelson, G. Aeppli, and V. J. Emery, Thermodynamics of the interplay between magnetism and high-temperature superconductivity, Proc. Natl. Acad. Sci. USA 98, 11903 (2001).
- S. Holm-Dahlin, J. Larsen, H. Jacobsen, A. T. Rømer, A.-E. Tutueanu, M. Ahmad, J.-C. Grivel, R. Scheuermann, M. v. Zimmermann, M. Boehm, et al., Field-induced electronic phase separation in the high-temperature superconductor , Phys. Rev. B 109, 174517 (2024).
- K. E. Avers, W. J. Gannon, S. J. Kuhn, W. P. Halperin, J. A. Sauls, L. DeBeer-Schmitt, C. D. Dewhurst, J. Gavilano, G. Nagy, U. Gasser, and M. R. Eskildsen, Broken time-reversal symmetry in the topological superconductor , Nat. Phys. 16, 531 (2020).
- D. C. Peets, E. Cheng, T. Ying, M. Kriener, X. Shen, S. Li, and D. Feng, Type-I superconductivity in , Phys. Rev. B 99, 144519 (2019).
- E. Herrera, I. Guillamón, J. A. Galvis, A. Correa, A. Fente, S. Vieira, H. Suderow, A. Y. Martynovich, and V. G. Kogan, Subsurface bending and reorientation of tilted vortex lattices in bulk isotropic superconductors due to Coulomb-like repulsion at the surface, Phys. Rev. B 96, 184502 (2017).
- B. Rosenstein, I. Shapiro, B. Y. Shapiro, and G. Bel, Vector vortices in -wave superconductors with arbitrary parameter, Phys. Rev. B 67, 224507 (2003).
- J. Garaud, E. Babaev, T. A. Bojesen, and A. Sudbø, Lattices of double-quanta vortices and chirality inversion in superconductors, Phys. Rev. B 94, 104509 (2016).
- P. Das, Y. Suzuki, M. Tachiki, and K. Kadowaki, Spin-triplet vortex state in the topological superconductor , Phys. Rev. B 83, 220513(R) (2011).
- S. Ooi, M. Tachiki, T. Konomi, T. Kubo, A. Kikuchi, S. Arisawa, H. Ito, and K. Umemori, Observation of intermediate mixed state in high-purity cavity-grade Nb by magneto-optical imaging, Phys. Rev. B 104, 064504 (2021).
- C. P. Bean and J. D. Livingston, Surface barrier in type-II superconductors, Phys. Rev. Lett. 12, 14 (1964).
- M. Konczykowski, L. I. Burlachkov, Y. Yeshurun, and F. Holtzberg, Evidence for surface barriers and their effect on irreversibility and lower-critical-field measurements in Y-Ba-Cu-O crystals, Phys. Rev. B 43, 13707 (1991).
- E. Zeldov, A. I. Larkin, V. B. Geshkenbein, M. Konczykowski, D. Majer, B. Khaykovich, V. M. Vinokur, and H. Shtrikman, Geometrical barriers in high-temperature superconductors, Phys. Rev. Lett. 73, 1428 (1994).
- E. H. Brandt, Superconductors in realistic geometries: Geometric edge barrier versus pinning, Physica C 332, 99 (2000).
- H. Q. Yuan, J. Singleton, F. F. Balakirev, S. A. Baily, G. F. Chen, J. L. Luo, and N. L. Wang, Nearly isotropic superconductivity in , Nature (London) 457, 565 (2009).
- S. A. Baily, Y. Kohama, H. Hiramatsu, B. Maiorov, F. F. Balakirev, M. Hirano, and H. Hosono, Pseudoisotropic upper critical field in cobalt-doped epitaxial films, Phys. Rev. Lett. 102, 117004 (2009).
- Y. T. Chan, P. L. Alireza, K. Y. Yip, Q. Niu, K. T. Lai, and S. K. Goh, Nearly isotropic superconductivity in the layered Weyl semimetal at 98.5 kbar, Phys. Rev. B 96, 180504(R) (2017).
- E. H. Brandt, Properties of the ideal Ginzburg-Landau vortex lattice, Phys. Rev. B 68, 054506 (2003).
- C. P. Poole, Jr., R. J. Creswick, H. A. Farach, and R. Prozorov, Superconductivity, ed. (Elsevier, New York, 2007), pp. 124–133.