An inhomogeneous superconducting state, not yet conclusively identified, was predicted by Fulde and Ferrell1 and Larkin and Ovchinnikov2 (FFLO) to arise in superconductors with strong Pauli limiting, a consequence of the electrons' Zeeman (spin) energy in a magnetic field. Radovan et al.3 propose that the observed cascades of steps in magnetization of the heavy fermion superconductor CeCoIn5, within the recently discovered second low-temperature state3,4, are due to transitions between Landau-level (LL) states with different m-quanta vortices, expected under certain conditions when the magnetic field is swept within the FFLO state5,6,7. The authors then conclude that the observed steps in magnetization constitute a proof that the low-temperature state in CeCoIn5 is indeed an FFLO state. We argue that this interpretation of the observed steps in magnetization cannot be supported on either quantitative or qualitative grounds.
The relative effectiveness of the orbital8 and paramagnetic9 (Pauli) limiting effects in suppressing superconductivity is reflected in the Maki parameter α=√(2Hc20/Hp. A large α indicates a small Pauli limiting field, HP (stronger Pauli limiting), and/or a large orbital limiting field, Hc20. Radovan et al.3 obtain α≈13 by using HP≈4 T, which assumes an electron g-factor of 2 for CeCoIn5. But the experimental value of the superconducting critical field is Hc2=12 T, three times the Pauli (upper) limit used by Radovan et al. Therefore, HP=4 T is unphysical.
HP can be estimated both theoretically and experimentally. It has been estimated by fitting the critical field Hc2 of CeCoIn5 to a model for a d-wave superconductor with an FFLO state10 to give an electron g-factor of 0.64 or HP=12.8 T. A very conservative lower bound of HP>10 T can be made simply by noting that Hc2=10 T at 1 K, where an FFLO state cannot influence Hc2.
Theoretically, the Maki parameter must exceed 9 for an m≠0 LL state to become a ground state5. HP=12.8 T gives α≈4.5, which is a third of the value derived by Radovan et al., and half the minimum required for observation of m≠0 LL superconducting states. The number of observed steps (tens) also seems to be orders of magnitude greater than that expected theoretically5 for a material with α close to that of CeCoIn5. Only one non-zero state (m=1) is expected5 for α≥9, and two such states (m=1,2) appear5 at α≥20, far greater than α≈4.5 for CeCoIn5.
In addition, the number of higher LL states is theoretically expected to increase as the applied field approaches the parallel orientation (Θ→0), with m→∞ at Θ=0 (refs 6, 7). This prediction opposes the trend shown by the results of Radovan et al. Thus, we believe that the observed steps in magne-tization cannot be due to the higher LL states induced by the FFLO effects in CeCoIn5.
The heat-capacity data shown in Fig. 2 of Radovan et al.3 are inconsistent with our data4, and miss a large narrow peak associated with the first-order nature of the superconducting phase transition below 1 K. A related point is that it is also incorrect to identify the order of the phase transition on the basis of the presence or absence of a temperature swing at a phase transition in the described magneto-caloric setup (inset to their Fig. 2). Such swings are expected both at first- and at second-order phase transitions11.
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Movshovich, R., Bianchi, A., Capan, C. et al. Magnetic enhancement of superconductivity. Nature 427, 802 (2004). https://doi.org/10.1038/427802a
Journal of Physics: Condensed Matter (2009)
Anisotropy of thermal conductivity and possible signature of the Fulde-Ferrell-Larkin-Ovchinnikov state inCeCoIn5
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