A Quenched Disorder in the Quantum‐Critical Superconductor CeCoIn5

Abstract Emergent inhomogeneous electronic phases in metallic quantum systems are crucial for understanding high‐T c superconductivity and other novel quantum states. In particular, spin droplets introduced by nonmagnetic dopants in quantum‐critical superconductors (QCSs) can lead to a novel magnetic state in superconducting phases. However, the role of disorders caused by nonmagnetic dopants in quantum‐critical regimes and their precise relation with superconductivity remain unclear. Here, the systematic evolution of a strong correlation between superconductive intertwined electronic phases and antiferromagnetism in Cd‐doped CeCoIn5 is presented by measuring current–voltage characteristics under an external pressure. In the low‐pressure coexisting regime where antiferromagnetic (AFM) and superconducting (SC) orders coexist, the critical current (I c) is gradually suppressed by the increasing magnetic field, as in conventional type‐II superconductors. At pressures higher than the critical pressure where the AFM order disappears, I c remarkably shows a sudden spike near the irreversible magnetic field. In addition, at high pressures far from the critical pressure point, the peak effect is not suppressed, but remains robust over the whole superconducting region. These results indicate that magnetic islands are protected around dopant sites despite being suppressed by the increasingly correlated effects under pressure, providing a new perspective on the role of quenched disorders in QCSs.

Here, each external magnetic field was applied after zero-field cooling, and then current sweeps were performed sequentially with systematically increasing current values.The I-V curves are rigidly shifted downwards for comparison.The slight decrease in Ic in the 2nd to 6th current sweeps compared to I1st is believed to be due to the unstable vortex state driven by the first current sweep (1st).However, when the applied current is greater than I*, the voltage value exhibits an anomalous kink or dip.
Since the vortices can be strongly trapped by strong pinning sites, a current higher than I1st, i.e.Subsequently, the average distance (aavg) between Cd dopants, which is the same as with d at ambient pressure, was determined from the inverse cube root of the Cd concentration; for x = 0.7%, aavg = (7/2000) -1/3  6.59 unit cell [S6].Here, we assumed that the proportions of the Cd atom on In(1) and In(2) sites were equal.The change in the droplet size against applied pressure is calculated based on the pressure dependence of the spin-lattice relaxation rate 1/T1 for CdCo in Ref. S6: d  6.59, 6.01, 4.16, and 2.11 unit cells at 1 bar, 1.9 (P < Pc1), 6.1 (Pc1 < P < Pc2), and 12.4 kbar (Pc2 ≤ P), respectively.and CeCoIn5 has an in-plane penetration depth ab(t) = ab(0)/(1-t 2.17 ) -0.5 , the Hdip can be thought to be associated with the elastic interaction between vortices [S7, S8].

Figure S1 Current
Figure S1 Current-voltage characteristics at 1.8 K for 1% Cd-doped CeCoIn5 under various pressures.Magnetic field dependencies of current-voltage (I-V) characteristic curves at pressures of 1.9, 6.1, 10.0, 12.4, 15.0, and 21.1 kbar.Here, the I-V curves at 1.8 K for the first current sweep are shown as representative.The critical current (Ic) is determined using the 1 μV criterion, indicated by a horizontal red line.Black arrows represent the field variation over which the I-V curves were measured.

Figure S2
Figure S2 Peak effect in pressurized 1% Cd-doped CeCoIn5.Pressure evolution of Ic with respect to the magnetic field for selective temperatures obtained from the first run of the current sweep.An anomalous increase in Ic(H) appears for temperatures T > 1.0 K at 7.7 kbar and in all superconducting regions for P  10.0 kbar, representing the emergence of the peak effect after the breakup of coupling of magnetic droplets.

Figure S4 Current
Figure S4 Current-voltage curves of sequential current sweeps at fixed magnetic fields for 1% Cd-doped CeCoIn5.a) and b) I-V characteristics at a fixed magnetic field of 15 kOe and 22 kOe, respectively, at 0.5 K under the pressure of 21.1 kbar.Here, each external magnetic

I2nd,
is required to generate flux flow.

Figure S5
Figure S5 The first and second current sweeps for 1% Cd-doped CeCoIn5 at 21.1 kbar.I-V characteristic curves for the pressurized CdCo at 1.8 K and selective magnetic fields.The difference between I-V curves measured by the first and second current sweep gradually increases with increasing applied magnetic field but begins to be suppressed at the boundary of 13 kOe (= Hdip).Subsequently, the first and second sweep I-V curves merge at 20 kOe, corresponding to Hpeak.The disparate I-V curves for the first and second sweeps lead to different critical currents: i1st and i2nd.The third current sweep has the same I-V curve as that of the second sweep and is thus not shown here.Both the first and second sweeps were performed with increasing electrical current.

Figure S6
Figure S6 Magnetic field dependence of the critical current for 1% Cd-doped CeCoIn5 under various pressures.The critical currents corresponding to I1st and I2nd with respect to the magnetic field for selective temperatures are obtained from the first and second sweeps, respectively.The difference between the I1st and the I2nd at the same magnetic field is associated with the change in vortex configuration from the unpinned to the pinned state, and the local magnetic droplets formed by the applied pressure are believed to be the main source of flux pinning.

Figure S7
Figure S7 Simple cartoons for the pressure dependence of the droplet size in 1% Cd-doped CeCoIn5.The effective droplet size (d) at ambient pressure is estimated considering the critical doping concentration (x ~ 0.7%) for the emergence of long-range AFM in CeCo(In1-xCdx)5 [S2, S3].The unit cell of the parent CeCoIn5 is denoted by a small black mesh constituting a 60  60 unit cell, with red symbols indicating the Cd atoms doped in the In(2) positions.In addition, for the substitution of Cd into the positions of In, we only considered In(2) sites because the In(1) sites replaced by Cd atoms are unrelated to the static magnetic order [S4, S5].

Figure S8
Figure S8 Pressure dependence of the upper critical field for 1% Cd-doped CeCoIn5.The upper critical field (Hc2) for CdCo under pressure was estimated from the temperature and magnetic field dependences of in-plane resistance (Rab).a) and b) show representative data measured at 21.1 kbar for the R-T curve and magnetoresistance at various magnetic fields and temperatures, respectively.Hc2(T) is determined from the midpoint of the superconducting transition following the red arrows.c) Temperature dependences of Hc2 and d) reduced upper critical field (hc2) as a function of normalized temperature (t) for CdCo under various pressures.The Hc2 at zero Kelvin, Hc2(0), was determined from the empirical curve Hc2(T) = H(0)[1-(T/Tc) 2 ] 0.8 , as plotted in d) with a red line.e) and f) show Hc2 at zero Kelvin, Hc2(0), and superconducting coherence length (ξab) as a function of pressure, respectively.Here, ξab is estimated from the relation Hc2 = 0/2πξ 2 .

Figure S9
Figure S9 Temperature dependence of Hdip at P ≥ Pc2.a) Temperature dependences of Hdip (P ≥ Pc2), at which Ic(H) has a minimum value before the peak effect.b) The Hdip at various pressures shows the same temperature dependencies, Hdip(t)  (1-t 2 )  with  = 1.11 0.04, as denoted by the solid line.Since the energy of vortex-vortex repulsion is proportional to 1/ 2

Figure S10
Figure S10 Pressure dependences of flux-pinning force for 1% Cd-doped CeCoIn5 under various pressures.The magnetic field dependence of flux-pinning force (Fp) is obtained from the relation Fp(H) = Ic(H)  H.The Hpeak, at which Ic(H) exhibits an anomalous peak in the vicinity of Hc2 in the pressurized CdCo, is located near the Fp maximum point, indicating that the softening of the vortex lattice induces a rapid decrease in Ic(H) at H > Hpeak, accompanied by a disordered vortex phase [S9, S10].

Figure S11
Figure S11 Temperature dependence of the critical current at zero Tesla for 1% Cddoped CeCoIn5 under various pressures.The Ic, measured at 0 T, shows a significant improvement up to the critical pressure at which long-range AFM is assumed to disappear, though the increase in Tc is insignificant.Because the supercurrent carrying ability of superconductors is related to the superfluid density [S11], suppressing the antiferromagnetic area in CdCo through the application of pressure can enhance Ic(at 0 T) owing to an increase in superconducting volume fraction.