Introduction

Materials that defy straightforward description in terms of either localized or itinerant electron behavior are a longstanding challenge to understanding novel electronic matter1,2,3,4. The intermetallic URu2Si2 is a classic example, where the itinerant electrons exhibit a giant magnetic anisotropy that is normally a characteristic of localized electrons5,6. URu2Si2 further displays an unknown broken symmetry state (‘hidden-order’) and unconventional superconductivity for temperatures below T0=17.5 K and Tc=1.4 K, respectively7,8,9,10. The development of new experimental techniques and access to ultra-high-purity single crystal specimens has recently advanced our understanding of the ordered states in this compound. For instance, excitations in the A2g channel have been identified by electronic Raman spectroscopy as a signature of the hidden order11, while elastoresistance12, resonant ultrasound13 (Ramshaw, B.J., Private communication.) and spectroscopic measurements14 suggest the presence of fluctuations in the B2g, B1g and Eg channels, respectively. These studies follow high resolution X-ray diffraction15, torque magnetometry16 and polar Kerr effect measurements17, which provide further insight into hidden order. It is also noteworthy that URu2Si2 differs from most other unconventional superconductors, which are typically found near the zero temperature termination point of a line of phase transitions, a ‘quantum critical point,’ where strong fluctuations are believed to be favourable for the superconducting pairing18,19,20,21. In contrast, the superconductivity in URu2Si2 is fully contained inside the ordered phase, as revealed by numerous tuning studies22,23,24,25,26,27,28,29,30,31,32,33.

To explore the mechanisms of hidden order, anomalous metallic behavior and superconductivity in this material we synthesized high purity single crystal specimens of chemically substituted URu2Si2−xPx, where ligand site substitution is a ‘gentle’ way to tune the electronic state. Recent advances developing a molten metal flux growth technique to produce high quality single crystal specimens of URu2Si2 enabled these experiments, which were previously inaccessible due to metallurgical challenges associated with the high-vapor pressure of phosphorus34. We report electrical transport and thermodynamic measurements, which reveal that while hidden order is destroyed for x≤0.035, the superconducting strength evolves non-monotonically with a maximum near x≈0.01 and that superconductivity disappears near x≈0.028. While the rapid suppression of hidden order indicates the importance of itinerant electrons, the unexpected maximum in superconducting strength may suggest the presence of a critical point that is defined by the termination of a phase boundary other than that of the hidden order. Owing to the relatively small chemical difference between silicon and phosphorus, we propose that in this series the dominant tuning effect is simply to change the chemical potential. Extrinsic factors such as disorder, as well as some intrinsic factors including changes in the unit cell volume or bond angles, spin orbit coupling and ligand hybridization strength play a minor role. Phosphorus substitution further has the advantage that it does not directly affect either the local electron count or the balance of the spin–orbit and Coulomb interactions of the d- or f-electrons on the Ru and U sites, making it an ideal and long desired tool for unraveling the physics of URu2Si2.

Results

Electrical transport and heat capacity

In Fig. 1 we show normalized electrical resistance R and heat capacity C versus temperature T for several phosphorus concentrations x (see Supplementary Fig. 1 for magnetic susceptibility χ(T)). The Kondo lattice behavior of the electrical resistance (that is, non-monotonic temperature dependence below room temperature with a resistive peak near 80 K) for T≥T0 is unaffected by phosphorus substitution for x≤0.035 (Fig. 1a), suggesting that the strength of the hybridization between the f- and conduction electron states does not change much in the range 0<x<0.035. At lower temperatures, the hidden order transition temperature T0 and the size of the anomalies in R and C associated with it are monotonically suppressed with increasing x (Fig. 1b,d). There is an excellent agreement between the values of T0 as extracted from R, C/T, and χ, indicating that disorder effects are negligible. We find no evidence for hidden order in x=0.035 for T>20 mK, showing that there is a quantum phase transition from hidden order to a paramagnetic correlated electron metal between 0.028<x<0.035. However, the data does not preclude the possibility of a broad fluctuation regime around the quantum phase transition. The paramagnetic region subsequently extends up to x≈0.25 (nearly a factor of 10 larger x) where correlated electron antiferromagnetism appears (See Supplementary Fig. 2) (Results for the larger x URu2Si2-xPx substitution series will be reported separately). Unlike for other tuning strategies22,23,24,25,26,27,28,29,30,31,32,33, magnetism is distant from hidden order in this phase diagram.

Figure 1: Electrical transport and heat capacity for URu2Si2-xPx.
figure 1

(a) Electrical resistance normalized to the value at 300 K R/R300 K versus temperature T for phosphorus concentrations x=0 and 0.035. (b) Electrical resistance normalized to the value at 21 K R/R21 K versus T for 0≤x≤0.035. (c) R/R21 K versus T in the low temperature region, emphasizing the superconducting transitions. (d) The 5f contribution to the heat capacity C5f divided by T versus T for 0≤x≤0.028. (e) C5f/T versus T in the low T region, showing the bulk superconducting transitions. See Supplementary section for a description of the phonon background subtraction.

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The resistive superconducting transition temperature Tc,ρ (Fig. 1c) initially weakly increases with x and subsequently vanishes, with no evidence for bulk superconductivity above 20 mK for x>0.02. While the value of Tc,C extracted from heat capacity is in close agreement with Tc,ρ for x≤0.01, these values separate for x=0.02, where Tc,ρ>Tc,C. We note that a similar discrepancy between Tc,ρ and Tc,C is seen for high quality single crystal specimens of the correlated electron superconductor CeIrIn5 and may be an intrinsic feature of the unconventional superconducting state35. From both ρ and C/T, we find that for x=0.028 there is a transition into the hidden order state near 13.5 K, but no bulk superconductivity down to 20 mK. There is no evidence for superconductivity in x=0.035 for T>20 mK.

Silicon to phosphorus phase diagram

These results are summarized in Fig. 2a, where the superconducting region is enclosed by hidden order in the T−x phase space. Over this concentration range, the ground state is mainly tuned by electronic variation, as indicated by the comparably small changes in other intrinsic and extrinsic factors. The lowest residual resistivity ratio (RRR≈ρ300 K/ρ0) for the specimens reported here is RRR=10 (see Supplementary Fig. 3), which is comparable with the typical values for parent URu2Si2, where T0 and Tc depend weakly on RRR in the range 10–500 (ref. 34). The high crystal–chemical quality of these specimens is further highlighted by the observation of quantum oscillations in electrical transport measurements (see Supplementary Fig. 4), indicating that disorder effects are negligible. The unit cell volume and bond angles are also unchanged by phosphorus substitution (Fig. 2e), in contrast to some previous studies26,27.

Figure 2: Summary of physical quantities for URu2Si2-xPx.
figure 2

(a) Temperature T versus phosphorus concentration x phase diagram for URu2Si2-xPx constructed from heat capacity (circles), magnetic susceptibility (squares) and electrical resistance (diamonds). The T−x phase boundary T0(x) separates the paramagnetic heavy electron liquid phase from the hidden order phase. Tc(x) separates the hidden order and superconducting phases. The dotted lines are guides to the eye. (b) Left axis: The size of the discontinuity in the heat capacity divided by the superconducting transition temperature Tc and the electronic coefficient of the heat capacity γ, ΔC5f/γTc versus x. Right axis: The width of the superconducting phase transition ΔTc versus x. (c) Left axis: The electronic coefficient of the heat capacity γ versus x. Right axis: The 5f contribution to the entropy S5f at Tc versus x. (d) Left axis: The value of the Kohler scaled curve at H/R0=50 versus x. Right axis: The anisotropy of the upper critical field curves versus x. (e) The lattice constants, a(x) (left axis) and c(x) (right axis), obtained from single crystal X-ray diffraction measurements.

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Having established the T−x phase diagram (Fig. 2a), we now discuss the region beneath the hidden order phase boundary, where unexpectedly rich behavior occurs. As evidenced by the jump size in C5f/T at Tc (ΔC5f/Tc) (and the transition width), which evolve though a maximum (and a minimum) between 0.006 and 0.01, respectively (Figs 1e and 2b), phosphorus substitution non-monotonically enhances the thermodynamic signature of the superconductivity. Here, C5f refers to the heat capacity following subtraction of the nonmagnetic ThRu2Si2 lattice term, as described in the Supplementary section (Supplementary Fig. 5). The non-monotonic behavior is reflected in the behavior of S5f,Tc(x) (Fig. 2c), which goes through a maximum near x=0.01. In contrast, for the hidden order ΔC/T0 and S5f,T0 are monotonically suppressed with increasing x (see Supplementary Fig. 6). Further evidence for non-monotonic evolution of the superconductivity is provided by the doping evolution of the ratio ζ=ΔC5f/γTc(x), (Fig. 2b) which, for conventional superconductors, is a numeric constant ζBCS=1.43. By using C5f/T at Tc for the value of the normal state γ we find that ζ(x) evolves non-monotonically through a maximum value of 1.2 at x=0.01 (Fig. 2b), where the x=0 value is near 0.7 as previously reported18,19. This suggests that the superconducting coupling strength may evolve through a maximum. We note that there is no evidence that all electrons from phosphorus substitution contribute directly to the conduction bands that are important for superconductivity.

Electrical transport in magnetic fields

Magnetoresistance data (Fig. 3) show quantum oscillations, emphasizing the high quality of these specimens (see Supplementary Fig. 4). Similar to the parent compound, the upper critical field Hc2 (Fig. 3d) is highly anisotropic at all x and follows dependence (where θ is measured from the c-axis), suggesting that the upper critical field is Pauli limited5. While there is little x dependence in gc(x), the a-axis g-factor ga(x) significantly decreases before the superconductivity is destroyed near x≈0.028 (Fig. 3d). We note that the actual value of the g-factor in the a-direction may differ significantly from the fitted ga(x) because of the increased importance of diamagnetic effects as the field rotates into the ab-plane. It remains to be seen whether these trends are consistent with recent theoretical proposals such as ref. 6.

Figure 3: Electrical transport in magnetic fields for URu2Si2-xPx.
figure 3

(a) The x=0.006 electrical resistance normalized to the room temperature value R/R300 K versus magnetic field H for several different angles θ. The data were collected at the temperature T=20 mK. The electrical current was applied in the ab-plane and θ=0 is the configuration where, H is parallel (||) the crystallographic c-axis. (b) R/R300 K versus H for x=0.01 at T=20 mK for select θ. (c) R/R300 K versus H at T=20 mK for x=0.02 for select θ. (d) The upper critical field Hc2, defined as the extrapolated zero resistance intercept, for T=20 mK for 0≤x≤0.02. Data for x=0 is from ref. 38. (e) The Kohler scaled electrical resistivity versus the reduced field H/R0 for 0≤x≤0.028.

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Magnetoresistance measurements further highlight the non-monotonic evolution with x of the superconductivity and the underlying metallic state. Figure 3e demonstrates Kohler scaling for H<9 T applied parallel to the c-axis at all dopings36, suggesting that the magnetotransport is controlled by the same (temperature dependent) relaxation time as the zero field resistivity. At each composition in the range 0<x<0.028 the normalized magnetoresistance is described by a distinct fx(h) (where h=H/ρ(0, T)), which itself evolves with doping. Notably, the function fx(h) evolves non-monotonically with x with a maximum near x=0.006 (Fig. 2d). The maximum in the value of fx(h) nearly coincides with the maximum in the thermodynamic signatures of the superconductivity inside the hidden order phase.

Discussion

Although much of the recent excitement surrounding URu2Si2 has focused on the uranium electronic structure and the symmetry of the hidden-order phase, a more fundamental question is the degree to which the f-electrons can be treated as being localized and the role of valence fluctuations. The continuity of experimental information extracted from well-developed applied pressure (P) and chemical substitution (x) series has proven essential to disentangle such effects in other correlated systems including high temperature superconducting cuprates, pnictides and heavy fermion compounds. To some extent, URu2Si2 has also benefited from such studies. For example, pressure drives a first order phase transition from hidden order into antiferromagnetism near Pc=0.5 GPa, with a simultaneous evolution of the Fermi surface22,23,24,25, but the resulting insight is limited by the small number of pressure-cell compatible experimental probes. Ruthenium site substitution with Fe and Os produces T−x phase diagrams that closely resemble the T−P phase diagram26,27,28, but the information gained from these series is constrained by strong disorder. Moreover, ruthenium site substitution is particularly disruptive, as evidenced by the rather different phase diagrams resulting from Rh and Re substitution studies where the hidden order and superconductivity are rapidly destroyed29,30,31,32,33. To understand the complex interplay between different phenomena in this compound a more ‘gentle’ tuning scheme has long been desired, which could provide access to the physics of URu2Si2 in clean single crystals at ambient pressure. In this context, ligand site substitution in URu2Si2 is an obvious target for investigation.

While in many theoretical scenarios for hidden order in URu2Si2 the U-5f electrons are treated as having mostly fixed valence in a particular atomic crystal field state6,11, it is now believed that they have a dual character: that is, the dynamic nature of the U-5f valence electrons allows for fluctuations between different configurations. However, measurements of the pure compound so far give no insight into the role of these fluctuations in favouring hidden order and superconductivity. The rapid changes in the hidden order and superconductivity in our measurements confirm the importance of the itinerant electron states. This is further supported by the weak evolution of Kondo lattice physics37 (which tracks the hybridization strength between f- and conduction electrons) and strong evolution in the g-factor anisotropy (which is a marker for local moment character). Together these results point towards this series as a platform for unraveling the relationship between local and itinerant behavior in URu2Si2.

The stark contrast in the evolution of the hidden order and superconductivity in URu2Si2-xPx (monotonic versus non-monotonic) further suggests that hidden order, although necessary, may not be directly responsible for the superconducting pairing. Instead, the observation of superconductivity completely contained inside the hidden order region and non-monotonic evolution of thermodynamic and electrical transport properties may indicate the presence of an independent collapsing phase boundary within the hidden order state, as is ubiquitous in other unconventional superconductors. This scenario is reinforced by the observed non-monotonic evolution of the normal state electrical transport, which is also common in correlated electron systems18,19,20,21, where the strongest deviation from Fermi liquid behavior is seen near the critical point. Alternatively, the independent evolution of hidden order and superconductivity may suggest several competing order parameters in the hidden order phase, as evidenced by electronic Raman (A2g)11, elastoresistance (B2g)12, resonant ultrasound (B1g)13(Ramshaw, B.J., Private communication.) and spectroscopic measurements (Eg)14. These studies should be extended into ligand site substituted URu2Si2.

Finally, the existing theoretical landscape focuses on f-electron physics with no guidance regarding the specificity of the transition metal ion. It is especially puzzling that hidden order and superconductivity are only observed in the U–Ru duo. Examination of silicon site substituted transition metal analogues (UT2Si2-xPx, T=transition metal), which can now be synthesized using molten metal flux growth (We have already successfully synthesized several transition metal analogues UT2Si2 (T=Mn, Fe, Ir and Pt) using the same flux growth recipe.), may be particularly illuminating in addressing the universality of hidden order and superconductivity in this fascinating uranium compound.

Methods

Single crystal synthesis using molten indium flux

Single crystals of URu2Si2−xPx were grown from elements with purities >99.9% in a molten in flux, as previously reported34. The reaction ampoules were prepared by loading the elements into a 5 cm3 tantalum crucible in the ratio 1(U):2(Ru):2(Si):22(In). The crucible was then loaded into an alumina tube spanning the bore of a high-temperature horizontal tube furnace. Argon gas was passed through the tube and a zirconium getter was placed in a pot before the tantalum crucible in order to purify the argon at high temperatures. The crucible was heated to 500 °C at 50 °C/h, dwelled for 5 h, heated to 600 °C at 50 °C/h, dwelled for 5 h and heated to 1,450 °C at 70 °C/h. The dwells at intermediate temperature are intended to allow the phosphorus to completely dissolve into the indium flux without producing a dangerous high-vapor pressure. The crucible was then cycled between 1,450 and 1,400 °C at 100 °C/h 10 times. Finally, the furnace was turned off and quickly cooled to room temperature. The indium flux was subsequently removed using hydrochloric acid, to which the URu2Si2-xPx crystals are insensitive. This technique produced single-crystal platelets similar to the ones previously reported.

Bulk thermodynamic and electrical transport measurements

Heat capacity measurements were performed for mosaics of single crystals using the He3 option in a Quantum Design Physical Properties Measurement System for temperatures 400 mK<T<20 K. Magnetization M(T, H) measurements were carried out for mosaics of single crystals for temperatures T=1.8–350 K under an applied magnetic field of H=5 kOe applied parallel to the c-axis using a Quantum Design Magnetic Property Measurement System. Magnetic susceptibility χ is defined as the ratio M/H. Zero magnetic field electrical resistance R was measured using the He3 option in Quantum Design Physical Properties Measurement System for temperatures 400 mK<T<300 K. Several individual crystals were measured for each concentration, which revealed a high degree of batch uniformity. The angular dependence of the superconducting upper critical field was measured using the superconducting magnet (SCM-1) dilution refrigerator system at the National High Magnetic Field Laboratory for H<18 T and T=20 mK. Additional magnetoresistance measurements were performed at the National High Magnetic Field Laboratory, Tallahassee, up to magnetic fields of 35 T and at T=50 mK.

Additional information

How to cite this article: Gallagher, A. et al. Unfolding the physics of URu2Si2 through silicon to phosphorus substitution. Nat. Commun. 7:10712 doi: 10.1038/ncomms10712 (2016).