Coexistence of Superconductivity, Magnetism and FFLO States

FFLO States

The BCS (Bardeen, Cooper and Schrieffer, 1957) theory successfully explains conventional superconductivity in terms of electrons with opposite spin and momentum that condense into pairs. A sufficiently large magnetic field destroys superconductivity by coupling to the orbital motion of the electrons. This orbital limit, i.e. the upper critical field Hc₂, separates the uniform superconducting state from the normal metallic state. For a superconductor in the paramagnetic limit the magnetic field couples predominantly to the spins of the electrons (Zeeman effect). Fulde & Ferrell, and independently, Larkin & Ovchinnikov (FFLO), have shown in 1964 that this coupling would give rise to a state different to the conventional BCS-state.

The magnetic field polarizes the spins and the Fermi surfaces of up- and down-spins acquire different Fermi momenta. Consequently, the superconducting order parameter develops nodes in real space, leading to alternating regions of superconductivity and spin-polarized normal state (magnetic walls). The periodicity is given by the inverse of the difference in the Fermi momenta. The FFLO state can only develop in very pure and strongly anisotropic single crystals. Until very recently the FFLO state was elusive to experimental observation. The FFLO state manifests itself as a wedge in the H-T phase diagram at low temperatures. Fig. 1 shows the phase diagram for CeCoIn₅, a strongly anisotropic heavy fermion compound and superconductor, as measured at the Magnet Lab by H.A. Radovan, N.A. Fortune, T.P. Murphy, S.T. Hannahs, E.C. Palm, S.W. Tozer and D. Hall (Nature 425, 51 (2003)), which provides the first solid evidence for the experimental realization of the FFLO state.

Coexistence of Superconductivity and Magnetism

The interplay of magnetism and superconductivity covers quite different aspects, e.g. conventional superconductors, organic superconductors, high T_c oxides and heavy fermion systems. In strongly correlated electron systems the superconductivity pairing mechanism, the symmetries of the order parameter and competing ground states with transitions from insulator or ordered magnetic phases to superconductivity are important facets to be considered.

Magnetism breaks the time reversal symmetry of the spin-singlet Cooper pairs (BCS) and is detrimental to conventional superconductivity. For instance, magnetic impurities lead to a reduction of the superconducting T_c (Abrikosov & Gor’kov, 1961) and the onset of a first order ferromagnetic transition destroys the superconductivity in the compounds ErRh₄B₄ and HoMo₆S₈. Exceptions for the coexistence of superconductivity and ferromagnetism are the compounds UGe₂ and URhGe, although it is believed that these are examples of spin-triplet pairing, rather than the conventional singlet pairing. On the other hand, antiferromagnetism can coexist with spin-singlet Cooper pairs, if the periodicity of the antiferromagnet is much shorter than the coherence length of the superconductor, as found in rare earth (RE) molybdenum sulfide (REMo₆S₈) compounds, RE rhodium boride alloys (RERh₄B₄), and RE borocarbides.

Heavy fermion systems are intermetallic compounds with 4f or 5f electrons with quasi-particles of very large effective mass (100m₀ – 1000m₀). A large number of heavy fermion superconductors has been discovered. Similarly to high T_c oxides, the superconductivity in heavy fermions is mediated by spin-fluctuations, rather than phonons. Hence, antiferromagnetism and superconductivity are intimately entangled and antiferromagnetic long-range order is detrimental to the pairing mechanism. For instance, coexistence of the two phases in a small region of the phase diagram has been found in CeCu₂(Si_1-xGe_x)₂ and CeRhIn₅, while for CeIn₃, CePd₂Si₂ and CeRh₂Si₂ a superconducting dome appears when the antiferromagnetic order is suppressed under high hydrostatic pressure.

The competition of superconductivity with nonmagnetic long-range ordered phases is also an exciting topic. URu₂Si₂ is a superconductor with T_c = 1.5 K and a second order transition at T₀ = 17.5 K from the paramagnetic into a phase of unknown nature, which has been called the “hidden order” (HO) state. A complex T-H phase diagram with many phases develops as a function of magnetic field, which does not change qualitatively under high hydrostatic pressure (see Fig. 2). Shubnikov-de Haas measurements performed at the Magnet Lab revealed an abrupt reconstruction of the Fermi surface at high fields within the HO phase. Rather than phase separation between HO and antiferromagnetism, the observations suggest adiabatic continuity between the two forms of order with field and pressure changing their relative weight (Y.-J. Jo, L. Balicas, C. Capan, K. Behnia, P. Lejay, J. Flouquet, J.A. Mydosh, and P. Schlottmann, Phys. Rev. Lett. 98, 166404 (2007)). Above T₀ neutron scattering experiments revealed well-correlated, itinerant-like spin-excitations up to at least 10 meV, emanating from incommensurate wave vectors. The large entropy change associated with the presence of an energy gap in the excitations explains the “missing entropy”, i.e. the reduction in the electronic specific heat through the transition (C.R. Wiebe, J.A. Janik, G.J. MacDougall, G.M. Luke, J.D. Garrett, H.D. Zhou,Y.-J. Jo, L. Balicas, Y. Qiu, J.R.D. Copley, Z. Yamani, and W.J.L. Buyers, Nature Physics 3, 96 (2007)).

Rather than preventing the formation of Cooper pairs, a strong magnetic field can as well induce superconductivity via the Jaccarino-Peter effect. This is the case in the quasi-two-dimensional organic insulator λ-(BETS)₂FeCl₄, which below a temperature of 4 K and with increasing (in-plane) magnetic field develops from antiferromagnetic insulator into a metal and between 18 T and 41 T into a superconductor (L. Balicas, J.S. Brooks, K. Storr, S. Uji, M. Tokumoto, H. Tanaka, H. Kobayashi, A. Kobayashi, V. Barzyskin, and L.P. Gor’kov, Phys. Rev. Lett. 87, 067002 (2001)). The Jaccarino-Peter mechanism postulates that the external magnetic field compensates the exchange field of the aligned Fe³⁺ spins, so that the effective field experienced by the ions is small and Cooper pairs can form. Close to the metal/superconductor boundaries the interlayer resistivity shows characteristic dip structures in the superconducting state, which have been interpreted as evidence for an oscillating order parameter in real space of the FFLO type (S. Uji, T. Terashima, M. Nishimura, Y. Takahide, T. Konoike, K. Enomoto, H. Cui, H. Kobayashi, A. Kobayashi, H. Tanaka, M. Tokumoto, E.S. Choi, T. Tokumoto, D. Graf, and J.S. Brooks, Phys. Rev. Lett. 97, 157001 (2006)).

Researchers

• Luis Balicas
• James S. Brooks
• Eun Sang Choi
• Lev P. Gor’kov
• Scott Hannah
• Timothy Murphy
• Eric Palm
• Pedro Schlottmann
• Stanley Tozer
• Christopher Wiebe
• Kun Yang

---

For more information, contact Pedro Schlottmann of the Condensed Matter Science / Theory group.