A peak in the critical current for quantum critical superconductors

Abstract

Generally, studies of the critical current I_c are necessary if superconductors are to be of practical use, because I_c sets the current limit below which there is a zero-resistance state. Here, we report a peak in the pressure dependence of the zero-field I_c, I_c(0), at a hidden quantum critical point (QCP), where a continuous antiferromagnetic transition temperature is suppressed by pressure toward 0 K in CeRhIn₅ and 4.4% Sn-doped CeRhIn₅. The I_c(0)s of these Ce-based compounds under pressure exhibit a universal temperature dependence, underlining that the peak in zero-field I_c(P) is determined predominantly by critical fluctuations associated with the hidden QCP. The dc conductivity σ_dc is a minimum at the QCP, showing anti-correlation with I_c(0). These discoveries demonstrate that a quantum critical point hidden inside the superconducting phase in strongly correlated materials can be exposed by the zero-field I_c, therefore providing a direct link between a QCP and unconventional superconductivity.

Introduction

Unconventional superconductivity (SC) often is observed in close proximity to a magnetically ordered phase, where the SC transition temperature T_c forms a dome against a non-thermal control parameter, such as the external pressure, chemical substitution, or magnetic field^1,2,3,4,5,6. At an optimal value of the tuning parameter, where T_c is the highest, normal state properties do not follow predictions for Landau–Fermi liquids: the electrical resistivity (ρ) does not exhibit a T² dependence, and the electronic specific heat coefficient (γ  = C/T) does not saturate, but rather diverges with decreasing temperature^{1, 2, 7}. These non-Fermi liquid (NFL) behaviors arise from incoherent critical fluctuations associated with a quantum critical point (QCP) hidden inside the SC dome of heavy fermion compounds and some Fe-based superconductors, such as BaFe₂(As_1−xP_x)₂^{1, 2, 4, 6, 8}. Because the zero-temperature quantum phase transition is typically not accessible without destroying SC, the role of critical magnetic fluctuations on properties of unconventional superconductors has yet to be explored in depth.

The critical current (I_c), which limits the current capacity of a zero-resistance state, is characteristically taken to depend on the strength of vortex pinning, which, in turn, is determined by the geometry and distribution of microstructural defects^9,10,11. Because application of pressure should not lead to the creation of different or additional defects or to a substantial change in sample dimensions, I_c in relation to T_c should be at most weakly pressure-dependent. A substantial variation in I_c(P) or I_c/T_c(P), then, logically, should be attributed to intrinsic changes in the superconducting state itself. For example, the zero-field critical current density J_c (equal to I_c/A, where A is the sample cross sectional area perpendicular to current) of the hole-doped high-T_c cuprate superconductor Y_0.8Ca_0.2Ba₂Cu₃O_y has a sharp peak that is centered on a critical hole-doping where the pseudogap boundary line projects to zero temperature, and that is attributed in model calculations to changes in the superfluid density^{12, 13}. These results indicate that I_c measurements may provide an opportunity to explore the relationship between unconventional SC and any QCP that is hidden beneath the SC dome.

Here we report a peak in the zero-field critical current, I_c(0), at a critical pressure P_c in pure CeRhIn₅ (Rh115) and 4.4% Sn-doped CeRhIn₅ (SnRh115), where their respective antiferromagnetic boundary T_N(P) extrapolates to T = 0 K inside a dome of pressure-induced SC. The temperature dependence of I_c(0)s for pure Rh115 and SnRh115 under pressure is similar to that of superconducting CeCoIn₅, which is close to quantum criticality at ambient pressure. Normalized values of I_c(T, P) follow a common universal curve for each material, suggesting an intrinsic, fundamental connection to quantum criticality. Supporting this conclusion, the magnetic field dependence of the flux-pinning force (F_p = I_c × μ₀H), normalized to its maximum value, also forms a pressure-invariant universal curve for each compound. As will be discussed, these discoveries demonstrate that the pressure evolution of zero-field I_c is determined mainly by quantum critical fluctuations, and that the peak in I_c is a direct link to the hidden QCP.

Results

Temperature–pressure phase diagrams

Figure 1a and b presents a contour plot of the zero-field I_c(P, T) in the SC phase and the in-plane resistivity ρ_ab(P, T) in the normal state for pure CeRhIn₅ (Rh115) and 4.4% Sn-doped CeRhSn_0.22In_4.78 (SnRh115) single crystals. The dependence on pressure of the in-plane resistivity and current–voltage curves upon which Fig. 1 is based is displayed in Supplementary Figs. 1 and 2, respectively. The quantum critical region veiled by the superconducting phase is fully exposed by the pressure dependence of zero-field I_c(T). A sharp peak in the value of I_c(T) is clearly observed for pressures around the QCP at P_c, where a large enhancement in the resistivity is accompanied by strong quantum fluctuations^3,4,14. In addition, I_c(P) abruptly increases at pressures around P_c*, the critical pressure where coexisting phases of magnetism and SC evolve into a single SC state. In undoped Rh115, large differences between T_cs measured by heat capacity (C) and resistivity (ρ) at pressures below P_c* are ascribed to textured SC originating from an incommensurate long-range magnetic order^15,16,17,18.

Fig. 1

Temperature–pressure phase diagrams of CeRhIn₅ and CeRhSn_0.22In_4.78 single crystals. In the superconducting state below T_c(P), false colors denote the magnitude of the zero-field critical current I_c(P, T). At temperatures above T_c(P), false colors reflect the magnitude of the in-plane resistivity ρ_ab(P, T). a CeRhIn₅ (Rh115) and b CeRhSn_0.22In_4.78 (SnRh115). For both materials, ρ_ab(P, T) is enhanced around the quantum critical point P_c due to pronounced incoherent inelastic scattering. Similarly, the zero-field I_c(P, T) is largest at P_c, where the QCP is expected, as indicated by the arrow. In both a and b the vertical hashed rectangle is at P_c*, the pressure that separates a phase of coexisting superconductivity and magnetism from a purely SC phase for P > P_c*. Open squares in both a and b represent the antiferromagnetic transition temperature (T_N). SC transition temperature (T_c) of Rh115 is evaluated from specific heat (T_c,C) and resistivity (T_c,ρ) measurements, and T_c of SnRh115 is determined as T_c onset (T_c,on) and 50% (T_c,50%) of the normal state resistivity value at T_c,on. AFM, SC, and NFL stand for antiferromagnetic, superconducting, and non-Fermi liquid regions, respectively

Temperature dependences of the zero-field critical current

The antiferromagnetic transition temperature (T_N ~ 3.8 K) in pure Rh115 is suppressed by Sn doping, which induces a shift of its extrapolated T = 0 K antiferromagnetic transition, and pressure-induced superconductivity emanates from the tuned QCP^4,5 (see Supplementary Fig. 3). Figure 2a and b shows the temperature dependence of the zero-field critical current, I_c(0), for Rh115 and SnRh115 at several pressures, respectively. Here, I_c is determined by using the voltage criterion of 0.1 μV (see Supplementary Fig. 2). Analysis of the flux-pinning force, F_p = I_c × μ₀H, shows that the normalized flux-pinning force follows a power-law dependence on magnetic field, f_p(h) ∝ h^p(1−h)^q, and is peaked around h_peak ≈ 0.6, which is characteristic of type-II superconductors with weak pinning (see Supplementary Fig. 4). Here, the normalized pinning force is f_p = F_p/F_p,max and the reduced field is h = H/H_irr, where F_p,max is the maximum flux-pinning force and H_irr is the irreversible field. The dependence on temperature of the critical current has been widely explained by I_c(t)/I_c(0) = (1−t²)^α(1 + t²)^β for type-II superconductors, such as high-T_c cuprates, Fe-based superconductors, and MgB₂^19,20,21,22, where t = T/T_c is the reduced temperature. When T_c variations surrounding defects are important (δT_c-pinning), α = 7/6 and β = 5/6, but α = 5/2 and β = −1/2 for δl-pinning that arises from spatial variations in the charge-carrier mean free path (l) near a lattice defect^{10, 19, 23}. These functional forms are shown by the dotted and dashed lines for δT_c-pinning to δl-pinning in Fig. 2c, respectively. A crossover of the mechanism from δT_c-pinning to δl-pinning has been often reported by introducing additional defects via chemical substitution or heavy ion irradiation, indicating that δT_c-pinning is preferred in clean crystals^19,20,21.

Fig. 2

Temperature dependences of the zero-field critical current for Ce-based heavy fermion materials under pressure. a Temperature dependence of the zero-field critical current, I_c(0), for CeRhIn₅ at various pressures. b Zero-field I_c for CeRhSn_0.22In_4.78 at various pressures. c Reduced temperature (t = T/T_c,0) dependence of I_c, I_c(t), normalized by its value at t = 0.4 for Rh115 and SnRh115 at representative pressures and for CeCoIn₅ at ambient pressure. The normalized values of I_c(P, t) for all crystals can be described by a single curve, I_c(t) ∝ (1 − t²)^5/6(1 + t²)^2/3, indicating universal behavior of I_c(t) with respect to pressure in the CeMIn₅ (M = Co, Rh) materials. Dotted and dashed curves are for δT_c-pinning and δl-pinning, respectively, as discussed in the text

The values of I_c(t) for SnRh115 at pressures around P_c are fitted together with those for Rh115 and CeCoIn₅ in Fig. 2c. The temperature dependence of I_c(0) for CeCoIn₅ at ambient pressure is measured to compare it with that of CeRhIn₅, because Rh115 is believed to have a SC pairing mechanism similar to that in CeCoIn₅. The values of I_c(0) for all samples can be expressed well by one curve with the relation I_c(t) ∝ (1−t²)^5/6(1 + t²)^2/3, which is distinct from that for I_c(t) controlled by either δT_c-pinning or δl-pinning. This universal curve underscores that the origin of the zero-field I_c is the same for each compound, and that it does not change under pressure for these Ce-based quantum critical materials. The fact that external pressure does not create new defects inside the crystals suggests that the pressure evolution of I_c should be related to the pressure dependence of the SC coupling strength.

Discussion

Figure 3a presents the pressure dependences of I_c(0) and T_c,0 for SnRh115, which are similar to each other. However, their relative fractional variations in I_c and T_c, γ_I ≡ I_c,0(P)/I_c,0(P_c) × 100 and γ_T ≡ T_c,0(P)/T_c,0(P_c) × 100, where I_c,0(P_c) is I_c extrapolated to zero temperature at P_c and T_c,0(P_c) is the SC transition temperature at P_c, are much different, as shown in Fig. 3b: at 1.0 GPa, the critical current is 45% of the maximum value, and T_c,0 is 67% of its maximal value. The stronger pressure dependence of I_c relative to that of T_c,0 is clearly visible in ratio I_c,0/T_c,0 for SnRh115, as presented in Fig. 3c, d. An abrupt enhancement in I_c,0/T_c,0 is observed at P_c*, and the peak in the pressure dependence of I_c,0/T_c,0 is achieved at P_c. The dc conductivity (=σ_dc) at T_c onset is shown as a function of pressure in the right ordinate of Fig. 3c, where a minimum value appears near P_c (see Supplementary Fig. 5). The anti-correlation between I_c,0/T_c,0 and σ_dc in these Ce-based quantum critical compounds may be related with the presence of the hidden QCP at P_c, because the associated critical quantum fluctuations not only act as the SC pairing glue, but also strongly enhance incoherent electron scattering, thus leading to a minimum in σ_dc at P_c^{24, 25}. Homes’ scaling relation^26,27,28 states that the superfluid density n_s is proportional to σ_dcT_c in many correlated superconductors and, consequently, that the ratio n_s/T_c should be proportional to σ_dc. The fact that σ_dc is the minimum at P_c, where I_c,0/T_c,0 is the maximum in these Ce-based compounds, suggests a violation of Homes’ scaling if the strength of the condensate n_s is proportional to the critical current I_c,0. Pressure-dependent optical conductivity and/or penetration depth experiments that directly measure n_s will be important to provide a stringent test for the validity of Homes’ law in quantum critical superconductors.

Fig. 3

Pressure evolution of the zero-field critical current in Sn-doped CeRhIn₅. a Pressure dependences of T_c,0 and I_c,0 for SnRh115, where I_c,0 is the value of I_c obtained from an extrapolation of data in Fig.  2b zero Kelvin. b Fractional variations in I_c,0 and T_c,0 for SnRh115 under pressure. The fractions are defined as γ_I (at 0 K) ≡ I_c,0(P)/I_c,0(P_c) × 100 and γ_T ≡ T_c,0(P)/T_c,0(P_c) × 100, where I_c,0(P_c) is I_c extrapolated to zero temperature at P_c and T_c,0(P_c) is the superconducting transition temperature at P_c. Values of γ_I are plotted as a function of pressure for measured or estimated I_c(T, P) at 0 K (squares) and 0.3 K (circles). c The ratio between the critical current and SC transition temperature, I_c,0/T_c,0, plotted together with the dc conductivity at T_c onset, σ_dc, as a function of the pressure difference P−P_c, where P_c = 1.35 GPa is the QCP. d A contour plot of I_c,0/T_c,0 displayed in the temperature (T) and pressure (P−P_c) plane. The ratio I_c,0/T_c,0 forms a dome centered around the quantum critical point P_c and its values decrease with distance from P_c

Our study demonstrates that the critical current, a fundamental superconducting parameter, is a powerful tool for investigating the presence of a hidden QCP inside the superconducting dome without destroying the superconducting phase. The dependence on temperature of the zero-field I_c for both pure Rh115 and Sn-doped Rh115 exhibits the same functional form under pressure, underscoring that the peak at P_c in the pressure dependence of I_c arises from an enhanced fluctuations around the hidden QCP. Even though these results are specific to the Ce115 heavy-fermion materials, the prediction of similar results for the hole-doping dependence of the critical current density J_c(x) in high-T_c cuprates²⁹ suggests a universal behavior of J_c among unconventional superconductors. These discoveries should stimulate more theoretical and experimental effort to understand the intimate link between quantum criticality and the origin of unconventional superconductivity in various families of correlated electronic systems.

Methods

Measurement outline

CeRhIn₅, Sn-doped CeRhIn₅, and CeCoIn₅ single crystals were synthesized by the indium (In) self-flux method^30,31,32. Pressure was generated in a hybrid clamp-type pressure cell with Daphne 7373 as the pressure-transmitting medium, and the pressure was determined by monitoring the shift in the value of T_c for lead (Pb). Measurements of current–voltage (I–V) characteristics under pressure were performed in a Heliox VL system (Oxford Instruments) with a vector magnet (y = 5 T and z = 9 T, American Magnetics Inc.) and in a Physical Property Measurement System (PPMS 9 T, Quantum Design), where the current was provided by a Keithley 6221 unit and the voltage was measured with a Keithley 2182A nanovoltmeter.

Measurement details

Measurements of I–V characteristics were performed in a pulsed mode to minimize Joule heating developed at Ohmic contacts to the samples and copper (Cu) wires between the pressure cell and the connector. The duration of the pulsed current was 10–11 ms, and the repetition rate was one pulse every 2 s, which was sufficient to eliminate Joule heat in the samples^{33, 34}. A standard four-probe method was used to determine I−V, and good Ohmic contact to samples was achieved by using silver epoxy. The critical current was based on a 10⁻⁷ V criterion³⁵, which was averaged over three measurements. The dimensions of the measured crystals were 920×330×20, 650×200×22, and 1100×200×47 μm³ for CeRhIn₅ (Rh115), CeRhSn_0.22In_4.78 (SnRh115), and CeCoIn₅, respectively. The magnetic-field dependence of the critical current was measured at several pressures and the flux-pinning force (F_p) was estimated from the relation F_p = I_c × μ₀H ^36,37,38

Data availability

The data sets generated and/or analyzed in this study are available from the corresponding author on reasonable request.