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Nonunitary triplet superconductivity tuned by field-controlled magnetization: URhGe, UCoGe, and UTe2

Kazushige Machida
Phys. Rev. B 104, 014514 – Published 20 July 2021
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  • INTRODUCTION
  • THEORETICAL FRAMEWORK
  • PROTOTYPES OF PHASE TRANSITIONS
  • MAGNETIZATION CURVES
  • APPLICATION TO EXPERIMENTS ON THREE…
  • PAIRING SYMMETRY
  • SUMMARY AND CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We report on a theoretical study on ferromagnetic superconductors URhGe and UCoGe and identify the pairing state as a nonunitary spin-triplet one with time-reversal symmetry broken, analogous to the superfluid He3−A phase. A recently found superconductor UTe2 with almost ferromagnetism is analyzed by the same manner. Through investigating their peculiar upper critical field Hc2 shapes, it is shown that the pairing symmetry realized in all three compounds can be tuned by their magnetization curves under applied fields. This leads to the reentrant Hc2 in URhGe, an S-shaped in UCoGe and an L-shaped Hc2 in UTe2 observed for the magnetic hard b-axis direction. The identification with double chiral form d(k)=(b̂+iĉ)(kb+ikc) in UTe2 naturally enables us to understand (1) multiple phases with A1, A2, and A0 phases observed under pressure, (2) the enhanced reentrant Hc2 for the off-axis direction associated with first-order metamagnetic transition, and (3) Weyl point nodes oriented along the magnetic easy a axis. All three compounds are found to be topologically rich solid-state materials worth further investigation.

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    • Received 19 January 2021
    • Revised 31 May 2021
    • Accepted 6 July 2021

    DOI:https://doi.org/10.1103/PhysRevB.104.014514

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Spin-triplet pairingSuperconducting phase transitionSuperconductivity
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Kazushige Machida

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

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    Vol. 104, Iss. 1 — 1 July 2021

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    • Figure 1
      Figure 1

      Generic phase diagram in the T and Mx planes. Tc1 (Tc2) for the A1 (A2) phase increases (decreases) linearly in Mx. The third phase A0 decreases quadratically in Mx away from the degenerate point at Mx=0.

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    • Figure 2
      Figure 2

      (a) Prototype phase diagram in the T and Hb planes where Hb is parallel to the magnetic hard b axis and the moment Ma points to the easy a axis. The two transition lines of Tc1 and Tc2 (red curves) are separated by 2Ma at Hb=0. Initially, Tc1 decreases and Tc2 increases as Hb increases toward the degeneracy point at TR. There the projection of the FM moment Ma vanishes. Up to this field the SOC is assumed to lock the d vector, thus, Tc1 and Tc2 follow Ma(Hb). Approximately above HR by rotating the d-vector direction to follow the magnetization Mb(Hb) (the green lines) instead of Ma(Hb), Hc2(1) and Hc2(2) turn around their directions. (b) Under the perpendicular field Hb the spontaneous moment Ma rotates toward the b direction. The projection Mb(Hb) of Ma on the b axis increases. (c) The rotation field HR is indicated as the red dot where Mb(Hb)=Ma(H=0).

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    • Figure 3
      Figure 3

      Two types (a) and (b) of the phase diagram for H∥b with the b axis (hard axis). (a) The same as in Fig. 2. (b) At HCR defined by Mb(Hb)=Ma(Hb), Hc2(1) turns around by rotating the d vector to follow Mb starting from Tc0. (c) H∥c with the c axis (another hard axis). (d) H∥a with the a axis (easy axis). The green lines are the respective magnetization curves and the red curves are Hc2(1) and Hc2(2).

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    • Figure 4
      Figure 4

      Hc2 changes due to the competition between the orbital depairing and Tc(M). (a) Tc(M) decreases as a function of the applied field H. The orbital depairing is added up to further depress Hc2 than Hc2WHH. (b) Tc(M) increases as a function of the applied field H, competing with the orbital depairing. The resulting Hc2 is enhanced compared with Hc2WHH. (c) Tc(M) increases strongly as a function of the applied field H. Hc2 has a positive initial slope and keeps growing until hitting the absolute upper limit Hc2AUL. Then, Hc2 follows this boundary.

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    • Figure 5
      Figure 5

      Schematic typical phase diagrams for H parallel to the b axis with A1, A2, and A phases, whose structure depends on the position of Tc0 and the separation of Tc1 and Tc2. The A phase is a mixture of A1 and A2 phases. The absolute upper limit Hc2AUL is indicated as the gray region. (a) The reentrant SC situated at high fields such as in URhGe. (b) S-shape Hc2 with the double transitions from the A1 to the A phase such as in UCoGe. (c) L-shape Hc2 where the high field phase is the A2 phase such as in UTe2.

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    • Figure 6
      Figure 6

      The ferromagnetic spontaneous moment Mc(Hb) rotation indicated by the green arrow under the field H∥b in URhGe. At Hb=HR, it completely orients along the b-axis direction via a first-order transition where Mb(Hb) shows a jump of the magnetization. HCR is defined by the field Mc(Hb)=Mb(Hb). The rotation angle α from the c axis is measured by neutron experiment [39]. The magnetization curves Mb(Hb) and Mc(Hc) are from [64].

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    • Figure 7
      Figure 7

      (a) The magnetization component of Mb(H) in URhGe under the field direction tilted from the b axis toward the c axis by θ, estimated from the experimental data of M(H) [69]. The inset shows the coordinates and the projections of Ma and Mb onto the ϕ axis. (b) The magnetization M(H) in URhGe under the field direction tilted from the b axis toward the a axis by ϕ estimated from the experimental data (dots) of M(H) [64], including magnetization curves for three a, b, and c directions for reference.

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    • Figure 8
      Figure 8

      (a) The magnetization curves for three a, b, and c axes in UCoGe. Here the crossing points HCRb and HCRa at which each curve surpasses the spontaneous moment Mc(H=0)=0.06μB. (b) The magnetization curves of Mb(H) for the field directions tilted from the b axis toward the c axis by the angle θ (degrees) in UTe2. θ=23.7∘ corresponds to H∥(011) direction measured by [72]. Those are estimated by the method explained in the main text. The inset shows the magnetization curves for three a, b, and c axes in UTe2. HR is the first-order transition for the moment rotation from the a axis to the b axis.

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    • Figure 9
      Figure 9

      The phase diagram for the Hb(T) versus T(K) plane. Mc(Hb) is estimated from the neutron scattering data in Ref. [39] and Mb(Hb) comes from the magnetization curve measured in Ref. [64]. The red dots for Hc2 are the experimental data points in Ref. [5]. The red continuous line indicates Hc2 which starts at Tc1 and is suppressed by the orbital depairing effect. It reappears again by following the formula Tc1(Hb)=Tc0+κMb(Hb) near HR=11T. Hc2′orb and Hc2′(M) are the slopes due to the orbital depairing and Tc1(Mb), respectively.

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    • Figure 10
      Figure 10

      Reentrant SC (Ref. [71]) for various θ values measured from the b axis (θ=0) toward the c axis in URhGe. As θ increases (0.79∘, 1.65∘, 3.64∘, and 5.64∘), the magnetization curves (far left scale) starting at Tc0 grow slowly, pushing up the RSC regions to higher fields. The magnetization data are from Fig. 7 for θ≠0 and Ref. [64] for θ=0.

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    • Figure 11
      Figure 11

      Detailed RSC structures (Ref. [71]) in T−H plane (left scale) are displayed. The triangle areas in each θ are RSC. RSC moves right as θ increases. The magnetization curve data (right scale) corrected as explained in Fig. 7 are originally from Ref. [69].

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    • Figure 12
      Figure 12

      RSC phase diagram in the T−H plane for various fields rotated from the b axis toward the a axis by the angle ϕ. This is constructed by using the magnetization data (right scale) shown in Fig. 7. When the magnetization hits the real axis T>0, RSC appears in high field regions. The lower field Hc2 is common for all ϕ.

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    • Figure 13
      Figure 13

      Phase boundary of the reentrance SC (RSC) as a function of the angle ϕ measured from the b axis to the a axis constructed from Fig. 12. The blue (green) line indicates the upper (lower) boundary of the RSC. The brown line is the magnetization rotation field HR(ϕ). The dots are experimental data points by Ref. [71]. The triangles denote the lower field Hc2 which is almost independent of ϕ.

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    • Figure 14
      Figure 14

      Phase diagram for H∥b taken under hydrostatic pressure (Ref. [74]) and uniaxial pressure along the b axis (Ref. [75]) on URhGe. All data at the rotation field HR line up along the Hc2AUL boundary, evidencing the existence of Hc2AUL.

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    • Figure 15
      Figure 15

      Multiple phase diagram consisting of the A1 and A2 phases under uniaxial pressure σ=1.0GPa in URhGe. The data points of Hc2∥b are taken from Ref. [75]. Two transitions at Tc1 and Tc2 separated by 2Mc are identified. HR is the moment rotation field found experimentally [75]. The green line indicates the magnetization curve of Mb starting at Tc0.

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    • Figure 16
      Figure 16

      Phase diagrams (H∥b) under uniaxial pressure, including the ambient pressure (a) in Fig. 9 and σ=1.0GPa (e) in Fig. 15. The data are from Ref. [75]. Continuous and systematic evolution of the multiple phase diagrams with guide lines are seen. (a) σ=0GPa, (b) σ=0.2GPa, (c) σ=0.6GPa, (d) σ=0.8GPa, (e) σ=1.0GPa, and (f) σ=1.2GPa.

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    • Figure 17
      Figure 17

      (a) The resulting Tc1, Tc0, and Tc2 obtained from the analysis in Fig. 16 are displayed. The linear changes of those characteristic temperatures Tc1, Tc0, and Tc2 are found, corresponding to the linear decrease in Mc. The second transition Tc2 begins appearing above σ>0.8GPa where the double transitions are expected at H=0. (b) The resulting Mc change as a function of uniaxial pressure σ. The observed Curie temperatures (Ref. [75]) are also shown. It is consistent with the obtained decreasing tendency of Mc as σ increases.

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    • Figure 18
      Figure 18

      The S-shaped phase diagram for UCoGe in H∥b. Hc2b starts at Tc1 and is initially depressed by the orbital depairing. At around the crossing field HCR it turns toward higher T due to the d-vector rotation to follow Mb(Hb) denoted by the green line, forming the S shape. At further high fields after hitting Hc2AUL, Hc2b follows it. The experimental data points come from [40] and the point at T=0 and 10 T from [62].

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    • Figure 19
      Figure 19

      (a) Weakened S shape Hc2a for H∥a in UCoGe because HCR moves up compared to Hc2b case shown in Fig. 18. The data are from Ref. [40]. (b) Hc2c for H∥c in UCoGe. The data [62] clearly show the anomaly around 0.3 T, indicating the multiple phases identified as A1, A2, and A3. The magnetization curve of Mc(Hc) is displayed as the green dots, showing the weak rise in this scale. Both Hc2c starting at Tc1 and Tc1 are thus dominated by the orbital depairing without help of the magnetization. The four points denoted by the red triangles are read off from the thermal-conductivity anomalies [76].

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    • Figure 20
      Figure 20

      Hc2(ϕ) for ϕ=0∘, 3.2∘, 6.8∘, and 11.4∘ from the b axis toward the a axis in UCoGe. The data are from Ref. [40]. As ϕ increases, Mc grows slowly as a function of H (the counterclockwise rotation of the Mc curves), pushing up HCR to higher fields. This results in the decrease of Hc2(ϕ) because the orbital suppression becomes dominant. The enhanced Hc2 becomes diminished as ϕ increases.

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    • Figure 21
      Figure 21

      The L-shaped Hc2b observed in [12] is shown (red dots). Hc2b starting at Tc1 follows the orbital suppression plus the Ma depression by Hb toward HCR. When it approaches the strong increasing Mb(Hb), the d vector rotates and follows Mb(Hb) to grow. This forms the upper part of the L shape. In further high fields Hc2b reaches HR=32T and disappears there by hitting Hc2AUL. The green curve denotes the magnetization curve Mb(Hb) shown in the inset of Fig. 8 [14].

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    • Figure 22
      Figure 22

      Hc2(ϕ) for ϕ=0∘, 2∘, 4∘, 5.2∘, and 6.3∘ from the b axis toward the a axis. Mb(H) grows slowly with increasing ϕ. Hc2(ϕ) curves bent over. Before hitting HR(ϕ) which ultimately limits it, Hc2(ϕ) turns around with the negative slope because they reach their own Hc2AUL(ϕ). Mb(H) for each ϕ is estimated by Eq. (25). The data (dots) are from [12]. The inset shows Hc2AUL(ϕ) estimated by extrapolating the straight lines toward higher fields beyond HR(ϕ).

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    • Figure 23
      Figure 23

      Hc2(θ) for various θ, which are measured from the b axis toward the c axis. The magnetization curves of Mb(H) starting at Tc2 and Tc0 evaluated before [see Fig. 8] lead to the reentrant SC for θ=35∘ in addition to the low Hc2. For the lower angle of θ=12∘ the two separate SC are formed. Here the θ=0∘ case (H∥b) is shown for reference. It is seen that the magnetization curves only around θ∼35∘ allow RSC to appear.

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    • Figure 24
      Figure 24

      Hc2 and the associated internal phase transition lines under hydrostatic pressure P in UTe2. (a) P=0.4GPa and H∥b. (b) P=0.4GPa and H∥a. (c) P=1.0GPa and H∥b. (d) P=0.7GPa and H∥a. The data denoted by the red dots are from Ref. [28]. Tc1 and Tc2 at H=0 are split by the magnetization Ma which decreases under the applied field Hb as shown in (a) and (c). This decrease of Ma(Hb) is compensated by growing of the magnetization Mb(Hb) as denoted by the green lines there.

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    • Figure 25
      Figure 25

      T−P phase diagram in UTe2 with three transition temperatures Tc1, Tc2, and Tc3 corresponding to the A1, A2, and A0 phases, respectively. At the degenerate point of Pcr=0.2GPa all three phases converge. The lines for Tc1 and Tc2 as a function of P indicate that the underlying symmetry-breaking field Ma changes linearly with P, leading to the globally quadratic variation of Tc3 from the degenerate point. The red (dark blue) round dots are from the experiment [28] ([13]) except for the three red triangle points at P=0.40, 0.54, and 0.70 GPa for Tc3, which are inferred from Fig. 24.

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