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Multiband effects on Fulde-Ferrell-Larkin-Ovchinnikov states of Pauli-limited superconductors

Masahiro Takahashi, Takeshi Mizushima, and Kazushige Machida
Phys. Rev. B 89, 064505 – Published 18 February 2014
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  • INTRODUCTION
  • FORMULATION
  • TWO-BAND SYSTEMS WITH A PASSIVE BAND…
  • TWO-BAND SYSTEMS WITH g22≠0
  • CONCLUSIONS
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    Multiband effects on Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states of a Pauli-limiting two-band superconductor are studied theoretically in the wide range of parameters, based on the Bogoliubov-de Gennes equation. First, we examine the phase diagrams of two-band systems with a passive band in which the intraband pairing interaction is absent and superconductivity is induced by a Cooper pair tunneling from an active band. The critical field of Bardeen-Cooper-Schrieffer to FFLO states becomes lower than the Lifshitz point with increasing the interband tunneling strength. We also study the thermodynamics of Pauli-limiting two-band superconductors with nonzero intraband pairing interactions. As a consequence of a competing effect between two bands, the FFLO phase is divided into two phases: Q1- and Q2-FFLO phases. In a particular case, the latter is further subdivided into a family of FFLO states with rational modulation lengths, where the spatial structure of the pair potential is approximately describable with sinusoidal functions with multiple modulation wave numbers. The resultant phase diagram is qualitatively different from that in a single-band superconductor and gives rise to a devil's staircase structure in the field dependence of physical quantities.

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    • Received 7 January 2014
    • Revised 31 January 2014

    DOI:https://doi.org/10.1103/PhysRevB.89.064505

    ©2014 American Physical Society

    Authors & Affiliations

    Masahiro Takahashi1,*, Takeshi Mizushima2,†, and Kazushige Machida2,‡

    • 1Department of Physics, Gakushuin University, Tokyo 171-8588, Japan
    • 2Department of Physics, Okayama University, Okayama 700-8530, Japan

    • *masahiro.takahashi@gakushuin.ac.jp
    • †mizushima@mp.okayama-u.ac.jp
    • ‡machida@mp.okayama-u.ac.jp

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    Vol. 89, Iss. 6 — 1 February 2014

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    Images

    • Figure 1
      Figure 1

      Phase diagram for the case of two-band systems with g22=0. (i) The solid (black) line corresponds to the phase boundaries in a single-band superconductor (g12=g22=0), (ii) the dashed (blue) line is the two-band case with a passive band Γ=(0.1,0.0,0.5), and (iii) the dash-dotted (red) line is the case of Γ=(0.8,0.0,0.5), where Δ0(2)/Δ0(1)≃0.10 (0.49) in the former (latter) case. The temperature of the Lifshitz point “L” is a fixed point TL/Tc=0.56..., which is independent of the interband coupling g12=g21. The inset shows the spatial profile of the pair potential in the top row and the magnetization in the bottom row at (T,H) marked as a filled (red) circle in the case (iii). The solid (red) lines and the dashed (blue) lines correspond to the quantities of the first and second band, respectively.

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    • Figure 2
      Figure 2

      Field dependence of the FFLO modulation number Q(H), based on the single-band analysis. The left (right) figure corresponds to the case R>1(<1).

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    • Figure 3
      Figure 3

      Phase diagrams for Γ=(0.1,0.6,2.0), where R=2 and Δ0(2)/Δ0(1)≃0.31. The FFLO phase is separated to two regions in the low temperature regime, Q1- and Q2-FFLO phases. In the high temperature regime, it is ambiguous whether the state is Q1- or Q2-FFLO. The inset shows the phase diagram for a single-band superconductor.

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    • Figure 4
      Figure 4

      Thermodynamic potential Ω as a function of the FFLO modulation wave number Q at T=0 for various magnetic fields, μBH/Δ0(1)=0.586 (i), 0.591 (ii), 0.597 (iii), 0.602 (iv), 0.726 (v), 0.753 (vi), and 0.780 (vii). The parameters are the same as those in Fig. 3. The inset shows the enlarged plot of Ω(Q) in the smaller Q region. The (red) filled circles show the global minima of Ω(Q).

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    • Figure 5
      Figure 5

      Spatial profiles of the gap for μBH/Δ0(1)=0.591 (a1), 0.753 (b1), and 0.753 (c1) at T=0. The corresponding magnetization densities are displayed in (a2), (b2), and (c2). (c1) and (c2) correspond to the Q1-FFLO state and the others are identified as the Q2-FFLO states. The parameters are the same as those in Fig. 3. The bottom panels show their Fourier components where δM(γ)=max[M(γ)]−Mave(γ).

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    • Figure 6
      Figure 6

      (a) FFLO wave number Q and (b) spatially averaged magnetization, M and M(γ), as a function of H at T=0. The other parameters are the same as those in Fig. 3. The (black) thin lines indicate the Q(γ)(H) curves obtained from the single-band analysis. The lines at lower Q correspond to the family of the Q2-FFLO, Q(2)/(2n2+1), with n2∈N. The symbol “×” (“+”) indicates the local minimum state whose thermodynamic potential is lower (higher) than that in the BCS state. The right panels show the detailed structures in the lower H region.

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    • Figure 7
      Figure 7

      Thermodynamic potential Ω as a function of the FFLO modulation wave number Q at μBH/Δ0(1)=0.74 for various temperatures, T/Tc=0 (i), 0.03 (ii), 0.06 (iii), and 0.09 (iv). Each plot is scaled by i×5×10−5(i=0,1,2,3) for visibility. The inset shows the calculated points in the phase diagram in Fig. 3.

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    • Figure 8
      Figure 8

      Energy spectra of the first band (a1) and the second band (a2). The (red) solid line (i) corresponds to Q1-FFLO states at μBH/Δ0(1)=0.753. The (blue) dash-dotted line (ii) and the (green) dotted line (iii) correspond to Q2-FFLO states with n2=0 at μBH/Δ0(1)=0.753 and 0.591, respectively. The density of states in (i)–(iii) are displayed in (b1)–(b3), respectively. The other parameters are the same as those in Fig. 3.

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    • Figure 9
      Figure 9

      Phase diagrams for Γ=(0.0625,0.5,0.2) in the main frame and for Γ=(0.1,0.6,0.5) in the inset. The former (latter) case corresponds to R=0.2 (0.5) and Δ0(2)/Δ0(1)≃0.60 (0.53). The FFLO phase is separated by the first-order transition to Q1- and Q2-FFLO states. The Q2-FFLO phase is divided into a family of the Q2-FFLO states through the first-order transitions.

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    • Figure 10
      Figure 10

      Thermodynamic potential Ω(Q) at T=0 for μBH/Δ0(1)=0.509 (i), 0.516 (ii), 0.525 (iii), 0.605 (iv), 0.625 (v), and 0.646 (vi). The set of parameters are Γ=(0.0625,0.5,0.2). The left panel is an enlarged plot of the low field region. The (red) filled circles denote the global minima of Ω(Q).

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    • Figure 11
      Figure 11

      Spatial profiles of the gap for μBH/Δ0(1)=0.514 (a1), 0.555 (b1), and 0.706 (c1) at T=0. The corresponding magnetization densities are displayed in (a2), (b2), and (c2). (c1) and (c2) correspond to the Q1-FFLO state. (a1) and (a2) illustrate the Q2-FFLO states with n2=2 and (b1) and (b2) correspond to the Q2-FFLO states with n2=1. The parameters are the same as those in Fig. 9. The bottom panels show their Fourier components where δM(γ)=max[M(γ)]−Mave(γ).

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    • Figure 12
      Figure 12

      (a) FFLO wave number Q and (b) spatially averaged magnetizations M and M(γ) as a function of H for Γ=(0.0625,0.5,0.2) and T=0. The (black) thin lines indicate the Q(H) curve obtained from the single-band analysis. We also depict the curves Q2(H)/(2n2+1), corresponding to the family of the Q2-FFLO. The symbol “×” (“+”) denotes the local minimum state whose thermodynamic potential is lower (higher) than that in the BCS state. The right panels illustrate the detailed structures in the lower H region.

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    • Figure 13
      Figure 13

      Energy spectra of the first band (a1) and the second band (a2) for Γ=(0.0625,0.5,0.2). The (red) line (i) corresponds to Q1-FFLO states at μBH/Δ0(1)=0.706. The (blue) dash-dotted line (ii) and (green) dotted line (iii) illustrate the Q2-FFLO states with n2=1 at μBH/Δ0(1)=0.555 and with n2=2 at 0.514, respectively. The corresponding density of states to the cases (i)–(iii) are displayed in (b1)–(b3).

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    • Figure 14
      Figure 14

      Grids used in the FE-DVR method.

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