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Pauli paramagnetic effects on mixed-state properties in a strongly anisotropic superconductor: Application to Sr2RuO4

Yuujirou Amano, Masahiro Ishihara, Masanori Ichioka, Noriyuki Nakai, and Kazushige Machida
Phys. Rev. B 91, 144513 – Published 27 April 2015
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  • INTRODUCTION
  • QUASICLASSICAL THEORY INCLUDING PAULI…
  • SPATIAL STRUCTURES OF VORTICES
  • FLUX-LINE LATTICE FORM FACTORS
  • JUMPS AT FIRST-ORDER Hc2 TRANSITION
  • FIELD DISTRIBUTIONS
  • MAGNETIC TORQUE
  • DISCUSSIONS ON PHASE DIAGRAM AND…
  • CONCLUSION AND UNSOLVED PROBLEMS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study theoretically the mixed-state properties of a strong uniaxially anisotropic type-II superconductor with the Pauli paramagnetic effect, focusing on their behaviors when the magnetic field orientation is tilted from the conduction layer ab plane. On the basis of Eilenberger theory, we quantitatively estimate significant contributions of the Pauli paramagnetic effects on a variety of physical observables, including transverse and longitudinal components of the flux-line lattice form factors, magnetization curves, Sommerfeld coefficient, field distributions, and magnetic torques. We apply these studies to Sr2RuO4 and quantitatively explain several seemingly curious behaviors, including the Hc2 suppression for the ab-plane direction, the larger anisotropy ratio and intensity found by the spin-flip small-angle neutron scattering, and the first-order transition observed recently in magnetocaloric, specific-heat, and magnetization measurements in a coherent and consistent manner. Those lead us to conclude that Sr2RuO4 is either a spin-singlet or a spin-triplet pairing with the d-vector components in the ab plane.

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    • Received 26 March 2015
    • Revised 10 April 2015

    DOI:https://doi.org/10.1103/PhysRevB.91.144513

    ©2015 American Physical Society

    Authors & Affiliations

    Yuujirou Amano, Masahiro Ishihara, Masanori Ichioka*, Noriyuki Nakai, and Kazushige Machida†

    • Department of Physics, Okayama University, Okayama 700-8530, Japan

    • *ichioka@cc.okayama-u.ac.jp
    • †Present address: Department of Physics, Ritsumeikan University, Kusatsu 525-8577, Japan; machida@mp.okayama-u.ac.jp

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    Vol. 91, Iss. 14 — 1 April 2015

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    • Figure 1
      Figure 1

      (a) Unit vectors u1 and u2 of the vortex lattice. Circles indicate vortex centers. The gray region is a unit cell of our calculations. (b) |Δ(r)|. (c) Mpara(r). (d) Bz(r)−B¯. (e) Bx(r). (f) By(r). (b)–(f) The density plot within a unit cell, when θ=89∘ at B¯=1.5 and μ=0.04.

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    • Figure 2
      Figure 2

      (a) B¯ dependence of the pair potential when θ=89∘ (solid lines) and θ=90∘ (dashed lines). Spatial averaged values of |Δ(r)| are presented for μ=0 and μ=0.04. The latter (μ=0.04) exhibits first-order transitions for both 89∘ and 90∘.

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    • Figure 3
      Figure 3

      B¯ dependence of the FLL form factors. (a) |Fz(1,0)|2 and |Fz(1,1)|2 for μ=0, when θ=89∘ (solid lines) and 90∘ (dashed lines). (b) The same as (a), but for μ=0.04. (c) |Fx(1,0)|2 for μ=0 and 0.04 when θ=89∘. (d) |Fy(1,1)|2 for μ=0 and 0.04 when θ=89∘.

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    • Figure 4
      Figure 4

      B¯ dependence of the FLL form factors when θ=89∘. We plot renormalized values |Fy(1,1)/Hc2,ab|2 as a function of B¯/Hc2,ab for μ=0.04 and 0. The points + indicate experimental values [4] on Sr2RuO4. We also present |Fz(1,0)/Hc2,ab|2 and |Fz(1,1)/Hc2,ab|2. The vertical axis is a logarithmic scale.

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    • Figure 5
      Figure 5

      Field-orientation θ dependence of the transverse FLL form factor. (a) |Fy(1,1)|2 as a function of θ for μ=0 at B¯=1.5, 4.5, 9.0, and 13.5. (b) |Fy(1,1)|2 as a function of θ for μ=0.04 at B¯=1.5, 3.0, and 4.5. (c) |Fy(1,1)/Hc2,ab|2 in a logarithmic scale as a function of θ for μ=0.04 at B¯/Hc2,ab∼0.33 and 0.5 (B¯=3.0 and 4.5). The points + indicate experimental values [4] on Sr2RuO4 at 0.5 (T) and 0.7 (T). We also plot |Fy(1,1)/Hc2,ab|2 for μ=0 at B¯=9.0 (B¯/Hc2,ab=0.21). (d) θ dependence of the longitudinal FLL form factor |Fz(1,1)/Hc2,ab|2 for μ=0.04 at B¯/Hc2,ab=0.17, 0.33, and 0.50.

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    • Figure 6
      Figure 6

      B¯ dependence of the magnetization. (a) Mtotal(B¯)=Mdia(B¯) for μ=0, when θ=89∘ (solid line) and 90∘ (dashed line). Inset: Mtotal(B¯) is enlarged near Hc2. (b) Mtotal(B¯),Mpara(B¯), and Mdia(B¯) for μ=0.04 when θ=89∘ (blue solid lines) and 90∘ (red dashed lines). (c) Mdia(B¯) is focused near Hc2 for θ=89∘ and 90∘ to see the jumps of ΔMdia at Hc2. (d) The scaling behaviors of γ(B¯) and χspin(B¯) for θ=90∘. The jumps of Δγ and Δχspin relative to its normal values are seen at the first-order Hc2 transition.

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    • Figure 7
      Figure 7

      Topographic maps of (a) Bz(r)−B¯ and (b) Mpara(r) within one unit cell at B¯=1.5, 4.5, and 8.5 for μ=0.04 and θ=90∘. The field distribution (c) P(B) and (d) P(M) associated to (a) and (b), respectively.

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    • Figure 8
      Figure 8

      The θ dependences of the free energies for various fields B¯. (a) μ=0 and (b) μ=0.04. The corresponding torque curves τ(θ)=dF/dθ. (c) μ=0 and (d) μ=0.04.

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    • Figure 9
      Figure 9

      The scaling behaviors between |Fy(1,1)|2 and |τ(θ)| as a function of θ. (a) μ=0 and (b) μ=0.04. Magnitude of each quantity is scaled by the maximum value.

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    • Figure 10
      Figure 10

      (a) Comparison with the theoretical torque curves |τ(θ)| for B¯=1.5, 3.0, 4.5, 8.6 (lines) and experimental data [8] for B¯=0.2, 0.5, 0.7, 1.4 (T) (empty symbols). We have adjusted the maximum values of the torque curves and displayed those curves by changing the maximum values arbitrarily to be clearly seen. The highest-field theoretical data B¯=8.6 and experimental data 1.4 (T) clearly show the jumps associated with the first-order transition. (b) Maximum angles of the form factors (triangles) and the torque curves (circles) in the B¯ and angle θ plane. Theoretical results (filled symbols) are compared with the corresponding experimental data [4, 8] (empty symbols). In the scale of vertical axis, Hc2,ab=9.1 in the theoretical estimate is assigned to be 1.5 (T).

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    • Figure 11
      Figure 11

      (a) The θ dependences of the vortex lattice anisotropy ΓVL(θ). Open circles indicate the experimental data for B=0.5 (T) [4]. Other symbols are for ΓVL(θ) evaluated by the free-energy minimum at B¯=2, 4, 8, and 16 for μ=0. The line presents Γ(θ) of the effective mass model in Eq. (9) with Γ=60. (b) The θ dependences of Hc2(θ)/Hc2,c at T=0.1Tc for μ=0.04. (c) Enlarged figure of (b). The results of numerical calculations by Eilenberger theory are presented by square points. The experimental data [8] are shown by circles. There, the filled (empty) symbols indicate the first- (second-) order transition. The solid line shows Γ(θ) of the effective mass model with Γ=60. The dashed lines correspond to the theoretical curves calculated by Eq. (A3) with Γ=60, where μ=0.04 for fitting to numerical calculations and μ=0.0293 for fitting to experimental data. (d) The anisotropy ΓHc2(T)=Hc2,ab(T)/Hc2,c(T) as a function of T. The experimental data [67] (circles) and the numerical results by Eilenberger theory with μ=0.04 (squares) are shown. The three continuous lines are evaluated by Eqs. (A3) and (A4) with (1) Γ=180 and μ=0.0293, (2) Γ=60 and μ=0.0293, and (3) Γ=60 and μ=0.04.

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