Asymmetric doping effects on the quantum critical compound CeRhIn₅

Abstract

Chemical substitution in correlated electron systems often produces asymmetric doping effects, where different dopants induce qualitatively distinct changes in electronic or magnetic properties, profoundly influencing phase diagrams and quantum criticality. Here, we investigate the chemical substitution effects of 5% Hg on CeRhIn₅ by systematic transport measurements under pressure. In contrast to dilute hole-doped (0.45% Hg) CeRhIn₅, which exhibits superconductivity near a Kondo-breakdown type antiferromagnetic (AFM) quantum critical point (QCP), or electron-doped (4.4% Sn) CeRhIn₅, which shows superconductivity near a spin-density-wave (SDW) type QCP, the heavily hole-doped (5% Hg) compound shows no superconductivity and evolves through two distinct AFM phases before entering a Fermi-liquid regime, with the AFM QCP likely of SDW type. The contrasting responses to electron and hole substitutions arise from their distinct influences on the local electronic structure. Electron doping induces homogeneous effects, whereas hole doping nucleates localized magnetic droplets that grow with increasing concentration, stabilizing new AFM order and weakening Kondo coupling. These results demonstrate how asymmetric doping reshapes the interplay between magnetism, superconductivity, and quantum criticality, providing mechanistic insights into doping-tuned quantum phase evolution and criticality in heavy-fermion and other correlated electron systems.

Introduction

Disorder induced by chemical substitution is generally undesirable because it obscures the intrinsic material properties and complicates the interpretation of the experimental results. Nevertheless, chemical substitution remains a powerful tool for tuning the electronic, magnetic, and structural properties of correlated electron systems. In high-temperature superconductors, for example, unconventional superconductivity often emerges near a magnetic quantum critical point (QCP), where the magnetic transition temperature is suppressed to zero^1,2,3,4,5. Chemical substitution is widely employed to access the QCP and probe the underlying mechanisms of exotic ground states^2,3,4,5. Importantly, asymmetric doping effects have been observed across diverse correlated electron systems, including cuprates^6,7,8, pnictides^9,10,11, and heavy fermion compounds^{12,13,14,15,16}, where different types of dopants induce qualitatively distinct changes in electronic or magnetic properties, thereby reshaping phase diagrams and tuning quantum criticality. Despite their ubiquity, further systematic studies are essential to elucidate the microscopic origin of the asymmetric doping effect, especially regarding their interplay with quantum criticality.

CeRhIn₅ is a prototypical quantum critical heavy fermion material that belongs to the CeTIn₅ (T = Co, Rh, Ir) family. It forms a tetragonal HoCoGa₅ structure with the space group P4/mmm and exhibits long-range antiferromagnetic (AFM) order with an incommensurate magnetic structure below T_N = 3.8 K at ambient pressure^17,18. The contrasting response to electron (Sn) and hole (Hg, Cd, or Zn) substitutions has been reported in studies of chemical substitutions for the In sites of CeRh(In_1-xM_x)₅ (M represents the doping substitutions)^{12,13,14,15,16}. In the case of Sn substitution, the AFM transition monotonically decreases to zero temperature at a critical concentration of 7%¹². Under pressure, Sn doping rigidly shifts the temperature–pressure (T–P) phase diagram of CeRhIn₅ to lower pressures^19,20,21, and the quantum critical signatures remain near the pressure-induced AFM QCP^20,21. However, the nature of quantum criticality is tuned from the Kondo-breakdown type in pure CeRhIn₅ to the spin-density-wave (SDW) type in Sn-doped CeRhIn₅²¹. In contrast to the monotonic suppression of the AFM transition by Sn doping, two distinct AFM ground states emerge with increasing hole concentrations, accompanied by a change in the magnetic structure^13,14,15,16. The high-pressure studies on 0.45% Hg-doped CeRhIn₅ revealed that, despite the suppression of the superconducting transition temperature, the slight Hg doping neither shifts the quantum critical pressure of pure CeRhIn₅ nor affects the characteristics of quantum criticality²².

To further explore the non-trivial hole doping effects on the magnetic phase diagram and quantum criticality, we systematically investigated the transport properties of 5% Hg-doped CeRhIn₅ under pressure, which is located on the higher Hg-doping side with a commensurate magnetic structure^13,15. Unlike pure, 4.4% Sn-doped, and 0.45% Hg-doped CeRhIn₅, which exhibit pressure-induced superconductivity near the AFM QCP, 5% Hg-doped CeRhIn₅ undergoes two distinct AFM ground states without pressure-induced superconductivity. The asymmetric responses to the electron and hole chemical substitutions can be attributed to their distinct influences on the electronic structure around the dopants.

Experimental method

Single crystals of CeRh(In_1-xHg_x)₅ were synthesized using a standard In flux method^{12,13,14,15,16}. The high-quality starting materials, Ce (rod, 99.9%), Rh (powder, 99.99%), In (ingot, 99.9995%), and Hg (liquid, 99.999%), were sealed in a quartz tube under vacuum with a molar ratio of Ce: Rh: In: Hg = 1: 1: F(1-x): F(x), where F = 20. Most of the excess In flux was removed by centrifugation at 600 °C, and the residual flux on the crystal surface was further eliminated by polishing with sandpaper before measurements. The Hg concentration was determined by energy dispersive X-ray spectroscopy measurements. The actual concentration is used throughout this paper. The high-pressure transport measurements were performed using the Van der Pauw method²³ in a diamond-anvil cell made of Be-Cu alloy, with NaCl powder as the pressure medium to ensure a quasi-hydrostatic pressure environment. The pressure inside the cell was determined using the ruby fluorescence method²⁴. A HelioxVL system that generates a maximum magnetic field of 12 T was used for the transport measurements in the temperature range of 0.3–20 K. A commercial superconducting quantum interference device magnetometer (MPMS, Quantum Design) was used for the susceptibility measurements in the temperature range of 1.8 to 305 K.

Results and discussion

The AFM transition was confirmed by the susceptibility measurements at ambient pressure (Supplementary Fig. S1), which clearly show that T_N increases from 3.8 K in CeRhIn₅ to 5.1 K upon 5% Hg doping. To track how this AFM transition evolves under pressure, we performed low-temperature resistance measurements up to 24.1 GPa (Fig. 1, see also Supplementary Fig. S2). Figure 1a displays the in-plane resistance and its first derivative in the vicinity of the AFM transition temperature at a representative pressure of 0.3 GPa. The AFM transition can be identified from the inflection point in R(T) or the peak in dR/dT. With increasing pressure, T_N initially shifts to higher temperatures, reaching a maximum of 10.4 K at 5.0 GPa, and then decreases to 6.1 K at 7.3 GPa, as indicated by the gray dashed lines in Fig. 1b, c. In this pressure range, the resistance anomaly associated with the AFM transition gradually weakens. As pressure increases to 8.0 GPa, however, a sharper resistive drop appears at 2.5 K. This evolution is more clearly resolved in dR/dT curves (Fig. 2a, b), which broaden progressively up to 7.3 GPa and then sharpen at 8.0 GPa. As we will demonstrate, the anomaly above 8.0 GPa signals a distinct magnetic transition, which is denoted as T_M. With further compression, T_M exhibits a dome-shaped dependence, reaching a maximum of 3.6 K at 9.5 GPa (Figs. 1d and 2c), and is eventually suppressed at 12.8 GPa, where the system enters a Fermi-liquid (FL) state evidenced by a T² dependence of resistance (Fig. 1f).

Fig. 1: Electrical resistance for 5% Hg-doped CeRhIn₅ under pressure.

a Temperature dependence of resistance (black solid symbols, left y-axis) and its first derivative with respect to temperature dR/dT (red line, right y-axis) at 0.3 GPa. b–e Temperature dependence of resistance at various pressures. The gray dashed lines in (b, c) indicate the pressure-induced evolution of the AFM transition T_N. The violet arrow in (c) represents the appearance of the magnetic transition T_M at 8.0 GPa, and its evolution with pressure is indicated by the violet dashed line in (d). f Low-temperature FL behavior above 12 GPa. The red lines are the least-squares fit of R = R₀ + AT². The orange arrows denote the FL temperature T_FL.

Fig. 2: Temperature dependence of the first derivative of resistance dR/dT for 5% Hg-doped CeRhIn₅ at various pressures.

a 0.3–2.5 GPa, b 2.9–8.0 GPa, c 8.3–12.8 GPa, and d 14.0–24.1 GPa. The red and violet arrows indicate the magnetic transitions T_N and T_M below and above 8.0 GPa, respectively.

The magnetic field response of the magnetic state exhibits a strong pressure dependence (Fig. 3, see also Supplementary Fig. S3). At P = 0.5 GPa, the AFM transition at T_N shifts to lower temperatures with increasing magnetic field (Fig. 3a, b). Interestingly, when the magnetic field reaches 10 T, two distinct resistive anomalies emerge (inset of Fig. 3b): the higher-temperature anomaly corresponds to the original AFM transition at T_N, while the lower-temperature anomaly represents a field-induced magnetic transition, denoted as T_N2. With further increasing magnetic field, T_N2 is enhanced, consistent with the stabilization of a new field-induced phase. This picture is further corroborated by the magnetoresistance (MR) data (Fig. 3c), which display a pronounced peak characteristic of a metamagnetic transition^25,26,27,28. The critical field of the metamagnetic feature increases with temperature, in agreement with the phase boundary extracted from the resistance measurements. The obtained temperature–magnetic field (T–B) phase diagram for 5% Hg-doped CeRhIn₅ at P = 0.5 GPa (Fig. 3d) is analogous to that of 4.0% Hg-doped CeRhIn₅¹³, suggesting the presence of a field-induced tricritical point separating the two AFM phases (AFM1 and AFM3) from the paramagnetic state. By comparison, at P = 9.1 GPa, a qualitatively similar response is observed, namely, the suppression of the transition at T_M and the emergence of a field-induced magnetic transition at T_M2 indicated by the metamagnetic feature in MR (Fig. 3e–g). However, the response is substantially weaker, with the MR amplitude nearly an order of magnitude smaller and T_M suppressed by only 12% under 10 T, in contrast to the 41% reduction of T_N under the same condition. As summarized in Fig. 3h, the resulting high-pressure T–B phase diagram shares the same overall topology as the low-pressure case, featuring two AFM regimes (AFM2 and AFM4) under field, but with a much more robust phase boundary associated with T_M. Taken together, the two T–B diagrams reveal a common underlying mechanism of field-induced phase reconstruction, while the contrasting field sensitivities underscore the distinct nature of the ordered states in the low- and high-pressure regimes.

Fig. 3: Field effects on different magnetic phases.

Evolution of the magnetic phase with an applied magnetic field for 5% Hg-doped CeRhIn₅ at a–d 0.5 GPa and e–h 9.1 GPa. a, e Temperature dependence of the in-plane resistance under magnetic fields applied along the c-axis. b, f First derivative of resistance dR/dT as a function of temperature at various magnetic fields. The inset of b shows the magnified view of dR/dT at 10 T. T_N and T_N2 denote two anomalies in resistance. c, g Field dependence of the magnetoresistance (MR) at low temperatures. d, h Magnetic field–temperature phase diagram for 5% Hg-doped CeRhIn₅. The red and navy symbols indicate the magnetic transitions identified by the anomalies in R(T), while violet symbols correspond to transitions determined from MR(B). AFM1, AFM2, AFM3, and AFM4 represent the four possible different magnetic phases.

Figure 4a displays the T–P phase diagram of 5% Hg-doped CeRhIn₅, which is markedly different from those of the pure, 4.4% Sn-doped, and 0.45% Hg-doped CeRhIn₅^{18,19,20,21,22,29,30}. In the latter systems, the applied pressure suppresses the AFM transition and induces a superconducting dome. By contrast, in 5% Hg-doped CeRhIn₅, the AFM transition at T_N appears to be suppressed to zero temperature at a lower critical pressure P_c1 ~ 8 GPa, where a distinct magnetic phase emerges at T_M ~ 2.5 K. This phase is subsequently suppressed at a higher critical pressure P_c2 ~ 12 GPa, beyond which the system enters a FL state. Thus, unlike the single critical pressure point in pure, 4.4% Sn-doped, and 0.45% Hg-doped CeRhIn₅, the 5% Hg-doped system exhibits two critical pressure points, P_c1 and P_c2, which divide the ground state into three regimes: AFM1 below P_c1, AFM2 between P_c1 and P_c2, and a paramagnetic FL state above P_c2. Furthermore, the isothermal cuts of the low-temperature resistance indicated by the color contour plot reveal a clear phase boundary near P_c1.

Fig. 4: Phase diagram and the low-temperature resistance analysis.

a Phase diagram of 5% Hg-doped CeRhIn₅ under pressure. Color represents the in-plane resistance. The violet and magenta symbols indicate the magnetic transitions T_N and T_M, respectively, which were determined from resistance inflection points. The red symbols denote the Fermi-liquid (FL) temperature, below which the resistance follows a FL behavior as R = R₀ + AT². b Pressure dependence of the residual resistance (violet diamonds, left y-axis) and the A coefficient (red balls, right y-axis) obtained from the fits of R = R₀ + AT². P_c1 is the lower critical pressure where T_N extrapolates to zero temperature, while P_c2 is the higher critical pressure at which T_M is suppressed, and the system enters a FL state. The dashed lines are guides to the eyes. AFM1 and AFM2 correspond to two different magnetic phases. FL represents the Fermi-liquid regime. The squares and circles represent the transition temperatures obtained from different runs. c–f Non-Fermi-liquid behavior in the vicinity of two critical pressures. The red solid lines are least-squares fits to the form R = R₀ + A^*Tⁿ.

To further investigate the potential quantum critical behavior in the vicinity of P_c1 and P_c2, the low-temperature resistance was analyzed using the FL form R = R₀ + AT² (Supplementary Fig. S4). The resulting pressure dependence of the residual resistance R₀ and A coefficient is summarized in Fig.4b. The residual resistance begins to decrease sharply as pressure crosses P_c1, and then continues to decrease smoothly through P_c2 without the characteristic divergence typically expected from the enhanced scattering near a QCP. In contrast, the A coefficient exhibits peaks at both P_c1 and P_c2, reflecting the enhancement of effective mass³¹. Moreover, the generalized power-law fits of the form R = R₀ + A^*Tⁿ were performed in the vicinity of both critical pressures. To minimize the magnetic contributions, the fits were restricted to the paramagnetic regime above the AFM transitions. As shown in Fig. 4c–f, the resistance deviates from FL behavior (n = 2), with the exponent n < 2, consistent with the non-Fermi-liquid behavior typically associated with the quantum fluctuations. These anomalies provide strong evidence for abundant critical fluctuations at both P_c1 and P_c2. The phase boundaries and quantum critical behavior observed in 5% Hg-doped CeRhIn₅ are reminiscent of those reported in CeRh_0.58Ir_0.42In₅³², where two distinct types of quantum criticality have been identified, with P_c1 corresponding to a Kondo-breakdown QCP and P_c2 associated with a SDW QCP. The non-Fermi-liquid behavior and enhancement of effective mass near P_c2 in 5% Hg-doped CeRhIn₅ are consistent with the expectations for an SDW QCP. However, additional measurements, especially those that are sensitive to low-temperature spin excitations and Fermi-surface evolution, would be crucial to clarify the quantum criticality near the two critical pressure points.

Doping effects on phase boundaries and quantum criticality have been extensively studied in the isostructural compound CeCoIn₅, which is located in the vicinity of a QCP at ambient pressure and exhibits the highest superconducting transition temperature among rare-earth-based heavy fermion materials^33,34. When Sn is substituted for In, the superconductivity and the AFM fluctuations originating from the underlying QCP are gradually suppressed with increasing Sn concentration, without inducing additional phase transitions^35,36. In contrast, Cd substitution for In in CeCo(In_1-xCd_x)₅ initially suppresses the superconductivity at doping concentrations below x = 0.75% and induces a long-range AFM order above x = 1.2%, leading to the microscopic coexistence of AFM order and superconductivity in the intermediate doping range¹⁴. Although the application of pressure reverses the evolution of the ground state with Cd substitution and generates a phase diagram that closely resembles that of CeRhIn₅ under pressure¹⁴, the conventional signatures of quantum criticality are absent in pressurized CeCo(In_1-xCd_x)₅³⁷. The disparate consequences of electron and hole substitutions are attributed to the qualitative differences in the local electronic environments around the dopants^{37,38,39,40,41}. Sn substitution modifies the electronic state homogeneously, whereas Cd substitution creates a heterogeneous electronic state, in which local magnetic droplets are nucleated surrounding the dopants, while the majority of electronic states remain unchanged. As the Cd concentration increases, the local magnetic droplets overlap and eventually form a long-range AFM order beyond x = 0.75% in CeCo(In_1-xCd_x)₅ (Fig. 5a). When the pressure is applied, the size of the magnetic droplets decreases, and thus the corresponding AFM transition is suppressed. However, the magnetic droplets persist even at the extrapolated QCP, inducing a heterogeneous electronic state in pressurized 1% Cd-doped CeCoIn₅, which prevents the appearance of signatures for quantum critical behavior³⁷.

Fig. 5: Schematic illustration of the ground states in different doping cases.

Evolution of ground states in a 1% Cd-doped CeCoIn₅, b 0.45% Hg-doped CeRhIn₅, and c 5% Hg-doped CeRhIn₅. Left panels show the ground states at ambient pressure, while right panels depict those near the extrapolated AFM QCP where the AFM order is completely suppressed with increasing pressure (gray arrows). Green-shaded regions represent the spatial extent of magnetic droplets nucleated around nonmagnetic dopants (red circles). AFM, SC, and FL denote antiferromagnetic, superconducting, and Fermi-liquid states, respectively. Hole doping nucleates localized magnetic droplets, which grow with increasing doping concentration and stabilize new AFM order. Under pressure, the magnetic droplets decrease in size but persist at the extrapolated AFM QCP. In 1% Cd-doped CeCoIn₅, the overlap between Cd-induced magnetic droplets yields an AFM + SC state at ambient pressure, which transforms into a SC state as pressure suppresses the AFM order. In 0.45% Hg-doped CeRhIn₅, the correlation length between Hg-induced magnetic droplets is insufficient to stabilize additional order, so the AFM order maintains the same character as in pure CeRhIn₅ and is gradually suppressed by pressure, giving rise to pressure-induced SC. In 5% Hg-doped CeRhIn₅, a new AFM state is formed due to the enhanced magnetic droplet correlations, which is distinct from that in the pure compound and evolves differently under pressure. With the suppression of AFM order, the system enters a FL state without superconductivity.

The similarities between Hg- and Cd-doping effects in CeTIn₅ suggest that the response of CeRhIn₅ to Hg substitution can be understood in terms of local doping effects^13,14,15. At low Hg concentrations (Fig. 5b), the correlation length between magnetic droplets is insufficient to form an additional long-range magnetic order, so the long-range AFM order primarily originates from the unaffected regions and maintains the same character as in pure CeRhIn₅. With increasing pressure, the magnetic droplets are expected to decrease in size but persist even when the long-range AFM order is completely suppressed. The number of magnetic droplets, however, is likely too small to significantly influence the phase boundaries or quantum criticality in the high-pressure regime. Therefore, the phase diagram and quantum critical behavior of 0.45% Hg-doped CeRhIn₅ are comparable to those of pure CeRhIn₅²². As the Hg concentration increases to 5% (Fig. 5c), the average spacing between Hg dopants is reduced to approximately three in-plane lattice constants, implying that a substantial fraction of the crystal volume is magnetically perturbed. The magnetic correlation length is thus well developed, potentially comparable to the incommensurate AFM order in pure CeRhIn₅. Consequently, the heavy Hg doping stabilizes an AFM order that is distinct from that of the pure compound and evolves differently under pressure. Analogous to the case of 1% Cd-doped CeCoIn₅³⁷, the magnetic droplet size in 5% Hg-doped CeRhIn₅ is expected to decrease with increasing pressure, eventually becoming smaller than the average impurity spacing above the critical pressure P_c2, where long-range magnetic order is suppressed. In this presumed quantum critical regime, the residual magnetic droplets locally freeze the magnetic quantum fluctuation, thereby suppressing the canonical QCP signatures that arise from the divergence of magnetic fluctuations. Nevertheless, the magnetic fluctuations remain largely intact in regions sufficiently distant from impurity sites³⁷. In 1% Cd-doped CeCoIn₅, the spectral weight of these unfrozen fluctuations is sufficient to support the formation of unconventional superconductivity³⁷. In contrast, in 5% Hg-doped CeRhIn₅, the higher impurity density significantly reduces these fluctuation-unfrozen regions, and the residual quantum fluctuations may therefore be too weak to stabilize a superconducting state.

In addition to quantum fluctuations, the doping itself can strongly influence the stability of unconventional superconductivity. For 0.45% Hg-doped CeRhIn₅, the AFM transition occurs at T_N ~ 3.4 K, and the maximum of superconducting transition emerges below T_c ~ 2.0 K near the AFM QCP at P_c ~ 2.3 GPa²². In contrast, when the Hg concentration is increased to 5%, T_N is enhanced to 5.1 K, yet superconductivity is completely suppressed down to the lowest measured temperature of 0.3 K. The disappearance of superconductivity at the high doping level raises the important question about the role of Hg substitution. The random distribution of dopants increases the impurity scattering, which can be further enhanced by quantum critical fluctuations⁴². Such impurity scattering leads to pair breaking and thereby suppresses the superconductivity. According to Abrikosov–Gorkov (AG) theory for a d-wave superconductor with unitary scattering from nonmagnetic impurities, the initial suppression rate of T_c should be dT_c/d(1/τ) = –π/4, where 1/τ = (ne²Δρ)/m^* = Δρ/μ₀λ² is the impurity scattering rate, μ₀ is the magnetic permeability of free space, and λ is the superconducting penetration depth⁴³. Experimentally, however, the maximum suppression rate in 0.45% Hg-doped CeRhIn₅ is only dT_c/d(1/τ) = –0.07, which is significantly smaller than the AG prediction. Similar remarkable robustness of superconductivity against nonmagnetic impurities has been reported in CeCoIn₅⁴⁴ and Sn-doped CeRhIn₅^20,21, indicating that randomness alone cannot account for the disappearance of superconductivity. Beyond impurity scattering, the magnetic moments introduced by hole doping weaken the Kondo screening^13,16, which further contributes to the suppression of superconductivity in 5% Hg-doped CeRhIn₅.

Conclusion

We have constructed a comprehensive T–P phase diagram for 5% Hg-doped CeRhIn₅ to investigate the hole-doping effects on magnetism, superconductivity, and quantum criticality. Under pressure, the system exhibits two distinct AFM ground states without superconductivity. Analysis of low-temperature resistance identifies two critical pressures separating three ground states, with the higher AFM QCP likely of SDW type, in contrast to the Kondo-breakdown type in the pure system. By comparison, 0.45% Hg-doping does not alter the phase boundaries or quantum criticality, which remains Kondo-breakdown type, whereas 4.4% Sn-doping shifts the phase diagram to lower pressures and induces SDW-type quantum criticality. These contrasting behaviors highlight the asymmetric responses to electron and hole doping: electron substitution modifies the system homogeneously, while hole doping nucleates localized magnetic droplets, which grow with concentration and stabilize new AFM order. In the heavily hole-doped compound, impurity-induced magnetic droplets persist and locally freeze the magnetic quantum fluctuations near the presumed QCP, thereby suppressing the canonical quantum critical signatures and giving rise to a spatially heterogeneous electronic state. These findings highlight the importance of disentangling local versus homogeneous doping effects for understanding quantum phase evolution and criticality across strongly correlated systems.