- Letter
- Open Access
Phase diagram of for out-of-plane magnetic field
Phys. Rev. B 112, L060501 – Published 8 August, 2025
DOI: https://doi.org/10.1103/gt54-tvc2
Abstract
The heavy-fermion superconductor (SC) ( = 0.35 K) shows two SC phases SC1 and SC2 when a magnetic field is applied parallel to the axis of the tetragonal unit cell. All experiments to date have indicated that the change in SC order parameter detected at is due to strong Rashba spin-orbit coupling at the Ce sites caused by the locally noncentrosymmetric environments of the otherwise globally centrosymmetric crystalline structure. Another phase (phase I) exists in this material below = 0.54 K. In a previous specific heat study [K. Semeniuk et al., Phys. Rev. B 107, L220504 (2023)], we have shown that phase I persists up to a field , larger than . From thermodynamic arguments, we expected the phase-I boundary line to cross phase SC2 at a tetracritical point. However, we could not find any signature of the phase-I line inside the SC2 phase and speculated that this was because the line is almost perpendicular to the axis and therefore invisible to -dependent measurements. This would imply a weak competition between the two order parameters. Here, we report magnetic-field-dependent measurements of the magnetostriction and ac susceptibility on high-quality single crystals. We see clear evidence of the singularity at inside the SC2 phase and confirm our previous prediction. Furthermore, we observe the transition across the line in -dependent specific heat measurements, which show that the line is not perpendicular to the field axis but has a positive slope. Our findings support recent muon spin resonance results which suggest coexistence of phase I with SC.
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References (46)
- A. J. Leggett, A theoretical description of the new phases of liquid , Rev. Mod. Phys. 47, 331 (1975).
- R. A. Fisher, S. Kim, B. F. Woodfield, N. E. Phillips, L. Taillefer, K. Hasselbach, J. Flouquet, A. L. Giorgi, and J. L. Smith, Specific heat of : Evidence for unconventional superconductivity, Phys. Rev. Lett. 62, 1411 (1989).
- G. Bruls, D. Weber, B. Wolf, P. Thalmeier, B. Lüthi, A. de Visser, and A. Menovsky, Strain–order-parameter coupling and phase diagrams in superconducting , Phys. Rev. Lett. 65, 2294 (1990).
- S. Adenwalla, S. W. Lin, Q. Z. Ran, Z. Zhao, J. B. Ketterson, J. A. Sauls, L. Taillefer, D. G. Hinks, M. Levy, and B. K. Sarma, Phase diagram of from ultrasonic velocity measurements, Phys. Rev. Lett. 65, 2298 (1990).
- H. R. Ott, H. Rudigier, Z. Fisk, and J. L. Smith, Phase transition in the superconducting state of ( = 0–0.06), Phys. Rev. B 31, 1651 (1985).
- S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Nearly ferromagnetic spin-triplet superconductivity, Science 365, 684 (2019).
- D. Braithwaite, M. Vališka, G. Knebel, G. Lapertot, J. P. Brison, A. Pourret, M. E. Zhitomirsky, J. Flouquet, F. Honda, and D. Aoki, Multiple superconducting phases in a nearly ferromagnetic system, Commun. Phys. 2, 147 (2019).
- D. Aoki, F. Honda, G. Knebel, D. Braithwaite, A. Nakamura, D. Li, Y. Homma, Y. Shimizu, Y. J. Sato, J.-P. Brison et al., Multiple superconducting phases and unusual enhancement of the upper critical field in , J. Phys. Soc. Jpn. 89, 053705 (2020).
- L. Taillefer, Scattering and pairing in cuprate superconductors, Annu. Rev. Condens. Matter Phys. 1, 51 (2010).
- 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. Yoshida, M. Sigrist, and Y. Yanase, Pair-density wave states through spin-orbit coupling in multilayer superconductors, Phys. Rev. B 86, 134514 (2012).
- M. Sigrist, D. F. Agterberg, M. H. Fischer, J. Goryo, F. Loder, S.-H. Rhim, D. Maruyama, Y. Yanase, T. Yoshida, and S. J. Youn, Superconductors with staggered non-centrosymmetricity, J. Phys. Soc. Jpn. 83, 061014 (2014).
- J. F. Landaeta, P. Khanenko, D. C. Cavanagh, C. Geibel, S. Khim, S. Mishra, I. Sheikin, P. M. R. Brydon, D. F. Agterberg, M. Brando et al., Field-angle dependence reveals odd-parity superconductivity in , Phys. Rev. X 12, 031001 (2022).
- P. Khanenko, D. Hafner, K. Semeniuk, J. Banda, T. Lühmann, F. Bärtl, T. Kotte, J. Wosnitza, G. Zwicknagl, C. Geibel et al., Origin of the non-Fermi-liquid behavior in , Phys. Rev. B 111, 045162 (2025).
- D. Möckli, Y. Yanase, and M. Sigrist, Orbitally limited pair-density-wave phase of multilayer superconductors, Phys. Rev. B 97, 144508 (2018).
- E. G. Schertenleib, M. H. Fischer, and M. Sigrist, Unusual phase diagram of : The role of staggered noncentrosymmetricity, Phys. Rev. Res. 3, 023179 (2021).
- A. Skurativska, M. Sigrist, and M. H. Fischer, Spin response and topology of a staggered-Rashba superconductor, Phys. Rev. Res. 3, 033133 (2021).
- K. Nogaki, A. Daido, J. Ishizuka, and Y. Yanase, Topological crystalline superconductivity in locally noncentrosymmetric , Phys. Rev. Res. 3, L032071 (2021).
- A. Ptok, K. J. Kapcia, P. T. Jochym, J. Łażewski, A. M. Oleś, and P. Piekarz, Electronic and dynamical properties of : Role of layers and expected orbital order, Phys. Rev. B 104, L041109 (2021).
- D. Möckli and A. Ramires, Two scenarios for superconductivity in , Phys. Rev. Res. 3, 023204 (2021).
- D. C. Cavanagh, T. Shishidou, M. Weinert, P. M. R. Brydon, and D. F. Agterberg, Nonsymmorphic symmetry and field-driven odd-parity pairing in , Phys. Rev. B 105, L020505 (2022).
- K. Nogaki and Y. Yanase, Even-odd parity transition in strongly correlated locally noncentrosymmetric superconductors: Application to , Phys. Rev. B 106, L100504 (2022).
- J. Ishizuka, K. Nogaki, M. Sigrist, and Y. Yanase, Correlation-induced Fermi surface evolution and topological crystalline superconductivity in , Phys. Rev. B 110, L140505 (2024).
- B. K. Nally and P. M. R. Brydon, Phase diagram of strongly-coupled Rashba systems, New J. Phys. 26, 093015 (2024).
- C. Lee, D. F. Agterberg, and P. M. R. Brydon, Unified picture of superconductivity and magnetism in , Phys. Rev. Lett. 135, 026003 (2025).
- D. Hafner, P. Khanenko, E.-O. Eljaouhari, R. Küchler, J. Banda, N. Bannor, T. Lühmann, J. F. Landaeta, S. Mishra, I. Sheikin et al., Possible quadrupole density wave in the superconducting Kondo lattice , Phys. Rev. X 12, 011023 (2022).
- D. S. Christovam, M. Ferreira-Carvalho, A. Marino, M. Sundermann, D. Takegami, A. Melendez-Sans, K. D. Tsuei, Z. Hu, S. Rößler, M. Valvidares et al., Spectroscopic evidence of Kondo-induced quasiquartet in , Phys. Rev. Lett. 132, 046401 (2024).
- T. Takimoto and P. Thalmeier, Theory of induced quadrupolar order in tetragonal , Phys. Rev. B 77, 045105 (2008).
- M. Kibune, S. Kitagawa, K. Kinjo, S. Ogata, M. Manago, T. Taniguchi, K. Ishida, M. Brando, E. Hassinger, H. Rosner et al., Observation of antiferromagnetic order as odd-parity multipoles inside the superconducting phase in , Phys. Rev. Lett. 128, 057002 (2022).
- S. Ogata, S. Kitagawa, K. Kinjo, K. Ishida, M. Brando, E. Hassinger, C. Geibel, and S. Khim, Parity transition of spin-singlet superconductivity using sublattice degrees of freedom, Phys. Rev. Lett. 130, 166001 (2023).
- S. Khim, O. Stockert, M. Brando, C. Geibel, C. Baines, T. J. Hicken, H. Luetkens, D. Das, T. Shiroka, Z. Guguchia et al., Coexistence of local magnetism and superconductivity in the heavy-fermion compound revealed by studies, Phys. Rev. B 111, 115134 (2025).
- T. Chen, H. Siddiquee, Q. Xu, Z. Rehfuss, S. Gao, C. Lygouras, J. Drouin, V. Morano, K. E. Avers, C. J. Schmitt et al., Quasi-two-dimensional antiferromagnetic spin fluctuations in the spin-triplet superconductor candidate , Phys. Rev. Lett. 133, 266505 (2024).
- G. Chajewski and D. Kaczorowski, Discovery of magnetic phase transitions in heavy-fermion superconductor , Phys. Rev. Lett. 132, 076504 (2024).
- B. Schmidt and P. Thalmeier, Anisotropic magnetic and quadrupolar phase diagram of , Phys. Rev. B 110, 075154 (2024).
- K. Semeniuk, D. Hafner, P. Khanenko, T. Lühmann, J. Banda, J. F. Landaeta, C. Geibel, S. Khim, E. Hassinger, and M. Brando, Decoupling multiphase superconductivity from normal state ordering in , Phys. Rev. B 107, L220504 (2023).
- See Supplemental Material at https://link.aps.org/supplemental/10.1103/gt54-tvc2 for a comparison of the specific heat capacity of samples of various quality, single-crystal x-ray diffraction information, a full set of the thermal expansion data, a magnified view of the transitions at and , and SC phase diagrams of samples of various quality
- G. Chajewski, D. Szymański, M. Daszkiewicz, and D. Kaczorowski, Horizontal flux growth as an efficient preparation method of single crystals, Mater. Horiz. 11, 855 (2024).
- A. Minamide and Y. Yanase, Superconducting meron phase in locally noncentrosymmetric superconductors, Phys. Rev. Lett. 134, 026002 (2025).
- A. Schilling, R. A. Fisher, N. E. Phillips, U. Welp, W. K. Kwok, and G. W. Crabtree, Anisotropic latent heat of vortex-lattice melting in untwinned , Phys. Rev. Lett. 78, 4833 (1997).
- H. Wilhelm, T. Lühmann, T. Rus, and F. Steglich, A compensated heat-pulse calorimeter for low temperatures, Rev. Sci. Instrum. 75, 2700 (2004).
- S. Ogata, S. Kitagawa, K. Kinjo, K. Ishida, M. Brando, E. Hassinger, C. Geibel, and S. Khim, Appearance of -axis magnetic moment in odd-parity antiferromagnetic state in revealed by , Phys. Rev. B 110, 214509 (2024).
- J. Juraszek, G. Chajewski, D. Kaczorowski, M. Konczykowski, D. F. Agterberg, and T. Cichorek, Nodeless superconducting state in the presence of zero-field staggered magnetization in , arXiv:2502.14423.
- Y. Imry, On the statistical mechanics of coupled order parameters, J. Phys. C 8, 567 (1975).
- R. M. Fernandes and J. Schmalian, Competing order and nature of the pairing state in the iron pnictides, Phys. Rev. B 82, 014521 (2010).
- F. Jakubczyk, J. M. Link, and C. Timm, Composite superconducting orders and magnetism in , arXiv:2506.08097.
- P. Khanenko, J. F. Landaeta, S. Ruet, T. Lühmann, K. Semeniuk, M. Pelly, A. W. Rost, G. Chajewski, D. Kaczorowski, C. Geibel et al., Data for “The phase diagram of for out-of-plane magnetic field”, Phys. Rev. B Letter (2025), https://doi.org/10.17617/3.HMLCQN, Edmond, V1.