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Violation of the orbital depairing limit in a nonunitary state: High-field phase in the heavy fermion superconductor UTe2

Kazushige Machida
Phys. Rev. B 107, 224512 – Published 21 June 2023
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  • INTRODUCTION
  • THEORETICAL FRAMEWORK
  • UPPER CRITICAL FIELD
  • ANALYSIS OF Hc2(T) in UTe2
  • DISCUSSIONS AND PERSPECTIVES
  • CONCLUSION AND SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    A theoretical study is reported on the origin of an extremely high upper critical field ∼70 T observed in UTe2 with the transition temperature Tc=1.6–2 K, which exceeds the conventional orbital depairing limit set by the Fermi velocity and Tc for a superconductor (SC) in the clean limit. We investigate possible violation of the orbital limit in terms of a spin-triplet nonunitary state, which is effectively coupled to the underlying magnetization induced by an external field. This produces the reduced internal field by canceling it via magnetization. We formulate a theory within the Ginzburg-Landau framework to describe this orbital limit violation and analyze experimental data on the upper critical fields for various field orientations in UTe2. We show that the orbital limit violation for a spin-triplet SC, as well as the Pauli-Clogston limit violation for a spin-singlet SC, constitutes a complete and useful framework for examining the high field physics of superconductors in the clean limit.

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    • Received 12 January 2023
    • Revised 24 April 2023
    • Accepted 9 June 2023

    DOI:https://doi.org/10.1103/PhysRevB.107.224512

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Critical fieldSpin-triplet pairingSuperconductivity
    1. Physical Systems
    Heavy-fermion systems
    1. Techniques
    Landau-Ginzburg theory
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Kazushige Machida

    • Department of Physics, Ritsumeikan University, Kusatsu 525-8577, Japan

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    Vol. 107, Iss. 22 — 1 June 2023

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    • Figure 1
      Figure 1

      (a) Case of a magnetization curve M(H)=χH (top). Heff is reduced compared with the external field. The permitted region with light blue color bounded by α0Tc0 extends to a higher field (middle). Hc2 is enhanced compared with Hc2orb (bottom). (b) When the magnetization has the jump at the metamagnetic field Hm, Heff goes outside the allowed region at Hc2(1). However, it returns above Hm and HSC appears, separated from LSC. The extrapolated Tc for HSC is higher than Tc0 for LSC (dotted curve in the bottom panel). (c) The metamagnetic jump is smaller than the case in (b). LSC and HSC overlap to appear. The light blue regions in the middle panels in (a), (b), and (c) show the permitted region for Hc2.

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    • Figure 2
      Figure 2

      (a) Resulting H−T phase diagram with the A1 (LSC) and A2 (HSC) phases. The dashed lines are not realized. (b) Constructed Heff (green curve) at T=0 as a function of the external field H using the measured magnetization curve [38] of Mb(μB) (red curve). Hrot is the d-vector rotation field. Hm is the metamagnetic transition field. Hc2AUL is the absolute upper limit of Hc2.

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    • Figure 3
      Figure 3

      Phase diagram in the H−T plane for H∥b and the field orientations tilted by the angle φ measured from the b axis toward the a axis. A2 or HSC quickly shrinks as φ increases, whereas A1 or LSC remains almost unaffected. Hc2AUL(φ) becomes low as φ increases indicated by the left-hand side because the projection of Mb(φ) strongly decreases as postulated in the inset. The resulting upper critical field Hc2(φ) is shown.

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    • Figure 4
      Figure 4

      Magnetization curve Mb(H) (red curve) for H∥b axis obtained experimentaly [38, 39]. The other magnetization curves for various angles of θ are reconstructed by projecting Mb(H) onto the magnetic field direction. Heff(θ)=H−α0κM(θ) is constructed from M(θ) thus obtained. The light blue band at the center indicates the permitted region for Hc2. The intersection point between Heff(θ) and the light blue band yields the absolute upper limit Hc2AUL.

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    • Figure 5
      Figure 5

      H−T phase diagram for various θ, including the case H∥b axis for comparison. The permitted region of Heff is indicated on the left-hand side in light blue, which is the same as in Fig. 4. For θ=35∘, HSC (A2) is permitted for Hm<Heff<Hc2AUL. The Hc2 curve starts at Hc2AUL toward Tc2 at H=0. However, the HSC terminates abruptly at Hm, below which Heff is outside the permitted region. The permitted region at the low field is not available for the A2 because Tc2<0. It is used by LSC (A1), which is relatively unchanged with varying θ, including the case for H∥b axis. The inset shows the HSC (A2) and LSC (A1) as a function of θ.

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    • Figure 6
      Figure 6

      H−T phase diagram under hydrodynamic (P) and uniaxial (σ) pressure for URhGe. The extrapolated straight line to lower T defines Hc2AUL and Tc2(H=0) to higher T, respectively. The pressure dependences of Hc2AUL and Tc2(H=0) are shown in the inset, indicating the linear scaling for both quantities with the linear decrease of Mc. The dotted points are the experimental data [51, 52].

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    • Figure 7
      Figure 7

      H−T phase diagram [25] for H∥b axis in UCoGe. The extrapolated straight line to lower T and higher T defines Hc2AUL=24T and Tc2(H=0)=1.0 K, respectively. The red dots are the experimental data [53, 54].

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    • Figure 8
      Figure 8

      H−T phase diagrams [24] for H∥a axis where the dots are experimental data points [67] for (a) P=0.40, (b) 0.54, and (c) 0.40GPa. The labeling of various phases and transition temperatures are explained in the text of Appendix.

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