Spontaneous Appearance of the Spin-Triplet Fulde-Ferrell-Larkin-Ovchinnikov Phase in a Two-Band Model: Possible Application to LaFeAsO 1−x F x
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The possibility of a spontaneous spin-triplet paired phase of the Fulde-Ferrell-Larkin-Ovchinnikov type is studied. As it is shown in a system with the dominant interband pairing and two distinct Fermi surface sheets, the Fermi wave-vector mismatch can be compensated by a nonzero center-of-mass momentum of the Cooper pairs. This idea is examined with the use of a model which describes the two hole-like bands in the iron-based superconductor. It is shown that for the proper range of model parameters, the minima of the free energy appear which correspond to a nonzero Cooper pair momentum. Different superconducting gap symmetries are analyzed, and the corresponding phase diagrams are shown.
KeywordsUnconventional superconductivity FFLO phase Iron pnictides
The so-called Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) phase has been proposed decades ago [1, 2] and attracted much attention over the years. This unconventional superconducting phase can be induced by the external magnetic field in systems with high Maki parameter  for the case of spin-singlet pairing. The Fermi wave-vector mismatch which appears in such conditions can be compensated by a nonzero center-of-mass momentum of the Cooper pairs. Experimental signs of the FFLO phase have been reported in the heavy fermion compound CeCoIn 5 [4, 5, 6], as well as in organic superconductors [7, 8, 9, 10, 11]. Also, an indirect evidence of a superfluid FFLO phase in a system of ultracold atomic gas trapped in an array of one-dimensional tubes has been reported .
It has been proposed by us recently (M. Zegrodnik and J. Spałek, 2014, A spontaneous paired state with nonzeroCooper-pair momentum: Possible application to iron pnictides, unpublished) that a paired phase with nonzero Cooper pair momentum can appear in the absence of an external magnetic field in systems with dominant interband pairing and two distinct Fermi surface sheets. The high value of the Maki parameter would not be required for the formation of such phase. However, the electronic structure of the system at hand should exhibit certain features to create favorable conditions for nonzero momentum pairing. To study this idea, we use the interband spin-triplet pairing mechanism [14, 15] suggested for iron pnictides in . It should be noted that with respect to the iron-based superconductors both spin-singlet [17, 18, 19, 20] and spin-triplet [16, 20, 21], gap symmetries have been considered. In this work, we use the tight binding model which reflects the two hole-like bands of the iron-based compound LaFeAsO 1−x F x . The stability of the proposed phase against both the normal and the homogeneous paired phases is analyzed. Different symmetries of the superconducting gap are considered, and their influence on the properties of the nonzero momentum pairing is studied. One should note that our approach could be applied to other multiband systems with either spin-singlet or spin-triplet types of pairing.
2 Theoretical Model
3 Results and Discussion
In our study, we consider the following phases: the normal phase (NS) with Δ Q =0, the homogeneous superconducting phase (SC-A) with Δ Q ≠0, Q≡0, and the inhomogeneous superconducting phase (FF-A) with Δ Q ≠0, Q≠0.
In the subsequent discussion, n expresses the number of electrons per one Fe ion; the wave-vectors are given in the units of 1/a, where a is the lattice parameter; and all the energies have been normalized to the bare bandwidth W, whereas T represents the reduced temperature T≡k B T/W.
One should note that our model with the interband pairing between the two Fermi surface sheets shown in Fig. 1a resembles the situation of one-band model with the spin-singlet pairing between the two spin subbands. The difference is that, here, the bottoms of the bands between which the pairing occurs coincide but the shape of the dispersion relations leads to Fermi wave-vector mismatch (c.f. Fig. 1a); whereas in the original idea of the FFLO phase, under the influence of the Zeeman term, the spin subbands are shifted as a whole. In our model, the mismatch can be tuned by changing n, because by increasing the Fermi level, one increases the distance between the Fermi sheets (c.f. Fig. 1).
We have analyzed the possibility of a new kind of superconducting phase with a spontaneous nonzero Cooper pair momentum. This phase can occur without the external a magnetic field in systems with the dominant interband pairing and two distinct Fermi surface sheets. The corresponding Fermi wave-vector mismatch which appears in such situation can be compensated by nonzero center-of-mass momentum of the Cooper pairs. In our study, we use as an example a tight binding model which describes the two hole-like bands of the iron-based superconductor LaFeAsO 1−x F x . The calculations have been carried out for different even-parity gap symmetries (s-wave, extended s-wave, and d-wave). We have shown that for proper values of the band filling and of the pairing strength, the free-energy minima appear which correspond to nonzero Cooper pair momentum. The direction of the Q vector depends on the selected gap symmetry (c.f. Figs. 2 and 6). For the case of the d-wave symmetry, the values of the pairing strength J 1 have to be very large (J 1>1) to obtain a paired solution in the considered model.
In our approach, we use the mean field (BCS) approximation which overestimates both the values of the order parameters and the critical temperature, so it would be interesting to analyze the considered problem with the inclusion of interelectronic correlations. The spin-triplet interband pairing induced by the combined effect of Hund’s rule and the correlations have been analyzed by us recently within the Gutzwiller approximation but without the possibility of a nonzero momentum pairing [23, 24]. Also, application of the proposed idea to other systems with the interband pairing seems reasonable. Namely, pairing between two species of particles with different (effective) masses could lead to a similar Fermi wave-vector mismatch as that considered above. Such an unconventional phase could be realized in systems of ultracold atomic gases in optical lattices. Spin-singlet pairing between particles with different effective masses has been theoretically investigated in [25, 26]. However, in these considerations, the so-called spin-dependent masses are induced by interelectronic correlations and appear in an external magnetic field. As a result, the appearance of the nonzero momentum pairing is both due to the energy shift of the spin subbands and the corresponding modification of the dispersion relations due to spin-dependent renormalization factors.
As we have mentioned, the pairing induced by Hund’s rule has an interband character. However, when it comes to other mechanisms, both inter- and intra-band components of the pairing can appear. The former can lead to the non-zero momentum of the Cooper pairs, whereas when the latter is strong, the homogeneous superconducting phase should be favored. It would be interesting to see to what extent the energy gain coming from the nonzero momentum pairing can survive in a model with both inter- and intra-band pairing. Another issue which would require further studies is the appearance of the degeneracy of the spin-triplet and spin-singlet pairings within our approach. This degeneracy should be broken by the spin-orbit coupling which has not been included by us at this stage of research. Moreover, the spin-orbit coupling would probably lead to a mixed ground state. These issues should be analyzed separately and are beyond the scope of this paper.
The authors are grateful to the Foundation for Polish Science (FNP) for the support within the project TEAM, as well as to the National Science Center (NCN) through the Grant MAESTRO, No. DEC-2012/04/A/ST3/00342.
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