Unfolding the physics of URu2Si2 through silicon to phosphorus substitution

The heavy fermion intermetallic compound URu2Si2 exhibits a hidden-order phase below the temperature of 17.5 K, which supports both anomalous metallic behavior and unconventional superconductivity. While these individual phenomena have been investigated in detail, it remains unclear how they are related to each other and to what extent uranium f-electron valence fluctuations influence each one. Here we use ligand site substituted URu2Si2-xPx to establish their evolution under electronic tuning. We find that while hidden order is monotonically suppressed and destroyed for x≤0.035, the superconducting strength evolves non-monotonically with a maximum near x≈0.01 and that superconductivity is destroyed near x≈0.028. This behavior reveals that hidden order depends strongly on tuning outside of the U f-electron shells. It also suggests that while hidden order provides an environment for superconductivity and anomalous metallic behavior, it's fluctuations may not be solely responsible for their progression.

M aterials that defy straightforward description in terms of either localized or itinerant electron behavior are a longstanding challenge to understanding novel electronic matter [1][2][3][4] . The intermetallic URu 2 Si 2 is a classic example, where the itinerant electrons exhibit a giant magnetic anisotropy that is normally a characteristic of localized electrons 5,6 . URu 2 Si 2 further displays an unknown broken symmetry state ('hidden-order') and unconventional superconductivity for temperatures below T 0 ¼ 17.5 K and T c ¼ 1.4 K, respectively [7][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 A 2g channel have been identified by electronic Raman spectroscopy as a signature of the hidden order 11 , while elastoresistance 12 , resonant ultrasound 13 (Ramshaw, B.J., Private communication.) and spectroscopic measurements 14 suggest the presence of fluctuations in the B 2g , B 1g and E g channels, respectively. These studies follow high resolution X-ray diffraction 15 , torque magnetometry 16 and polar Kerr effect measurements 17 , which provide further insight into hidden order. It is also noteworthy that URu 2 Si 2 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 pairing [18][19][20][21] . In contrast, the superconductivity in URu 2 Si 2 is fully contained inside the ordered phase, as revealed by numerous tuning studies [22][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 URu 2 Si 2 À x P x , 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 URu 2 Si 2 enabled these experiments, which were previously inaccessible due to metallurgical challenges associated with the high-vapor pressure of phosphorus 34 . We report electrical transport and thermodynamic measurements, which reveal that while hidden order is destroyed for xr0.035, the superconducting strength evolves nonmonotonically with a maximum near xE0.01 and that superconductivity disappears near xE0.028. 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 felectrons on the Ru and U sites, making it an ideal and long desired tool for unraveling the physics of URu 2 Si 2 .

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 w(T)). The Kondo lattice behavior of the electrical resistance (that is, nonmonotonic temperature dependence below room temperature with a resistive peak near 80 K) for TZT 0 is unaffected by phosphorus substitution for xr0.035 (Fig. 1a), suggesting that the strength of the hybridization between the f-and conduction electron states does not change much in the range 0oxo0.035. At lower temperatures, the hidden order transition temperature T 0 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 T 0 as extracted from R, C/T, and w, indicating that disorder effects are negligible. We find no evidence for hidden order in x ¼ 0.035 for T420 mK, showing that there is a quantum phase transition from hidden order to a paramagnetic correlated electron metal between 0.028oxo0.035. 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 xE0.25 (nearly a factor of 10 larger x) where correlated electron antiferromagnetism appears (See Supplementary Fig. 2) (Results for the larger x URu 2 Si 2-x P x substitution series will be reported separately). Unlike for other tuning strategies [22][23][24][25][26][27][28][29][30][31][32][33] , magnetism is distant from hidden order in this phase diagram.
The resistive superconducting transition temperature T c,r (Fig. 1c) initially weakly increases with x and subsequently vanishes, with no evidence for bulk superconductivity above 20 mK for x40.02. While the value of T c,C extracted from heat  capacity is in close agreement with T c,r for xr0.01, these values separate for x ¼ 0.02, where T c,r 4T c,C . We note that a similar discrepancy between T c,r and T c,C is seen for high quality single crystal specimens of the correlated electron superconductor CeIrIn 5 and may be an intrinsic feature of the unconventional superconducting state 35 . From both r and C/T, we find that for x ¼ 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 ¼ 0.035 for T420 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 À 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 (RRREr 300 K /r 0 ) for the specimens reported here is RRR ¼ 10 (see Supplementary Fig. 3), which is comparable with the typical values for parent URu 2 Si 2 , where T 0 and T c 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 studies 26,27 .
Having established the T À 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 C 5f /T at T c (DC 5f /T c ) (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, C 5f refers to the heat capacity following subtraction of the nonmagnetic ThRu 2 Si 2 lattice term, as described in the Supplementary section ( Supplementary Fig. 5). The non-monotonic behavior is reflected in the behavior of S 5f,Tc (x) (Fig. 2c), which goes through a maximum near x ¼ 0.01. In contrast, for the hidden order DC/T 0 and S 5f,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 z ¼ DC 5f /gT c (x), (Fig. 2b) which, for conventional superconductors, is a numeric constant z BCS ¼ 1.43. By using C 5f /T at T c for the value of the normal state g we find that z(x) evolves non-monotonically through a maximum value of 1.2 at x ¼ 0.01 (Fig. 2b), where the x ¼ 0 value is near 0.7 as previously reported 18,19 . 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 H c2 (Fig. 3d) is highly anisotropic at all x and follows H c2 y ð Þ / 1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 2 c cos 2 y þ g 2 a sin 2 y p dependence (where y is measured from the c-axis), suggesting that the upper critical field is Pauli limited 5 . While there is little x dependence in g c (x), the a-axis g-factor g a (x) significantly decreases before the superconductivity is destroyed near xE0.028 (Fig. 3d). We note that the actual value of the g-factor in the a-direction may differ significantly from the fitted g a (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. Magnetoresistance measurements further highlight the nonmonotonic evolution with x of the superconductivity and the underlying metallic state. Figure 3e demonstrates Kohler scaling for Ho9 T applied parallel to the c-axis at all dopings 36 , suggesting that the magnetotransport is controlled by the same (temperature dependent) relaxation time as the zero field resistivity. At each composition in the range 0oxo0.028 the normalized magnetoresistance is described by a distinct f x (h) (where h ¼ H/r(0, T)), which itself evolves with doping. Notably, the function f x (h) evolves non-monotonically with x with a maximum near x ¼ 0.006 (Fig. 2d). The maximum in the value of

Discussion
Although much of the recent excitement surrounding URu 2 Si 2 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 welldeveloped 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, URu 2 Si 2 has also benefited from such studies. For example, pressure drives a first order phase transition from hidden order into antiferromagnetism near P c ¼ 0.5 GPa, with a simultaneous evolution of the Fermi surface [22][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 À x phase diagrams that closely resemble the T À P phase diagram [26][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 destroyed [29][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 URu 2 Si 2 in clean single crystals at ambient pressure. In this context, ligand site substitution in URu 2 Si 2 is an obvious target for investigation.
While in many theoretical scenarios for hidden order in URu 2 Si 2 the U-5f electrons are treated as having mostly fixed valence in a particular atomic crystal field state 6,11 , 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 physics 37 (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 URu 2 Si 2 .
The stark contrast in the evolution of the hidden order and superconductivity in URu 2 Si 2-x P x (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 nonmonotonic evolution of the normal state electrical transport, which is also common in correlated electron systems [18][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 (A 2g ) 11 , elastoresistance (B 2g ) 12 , resonant ultrasound (B 1g ) 13 (Ramshaw, B.J., Private communication.) and spectroscopic measurements (E g ) 14 . These studies should be extended into ligand site substituted URu 2 Si 2 .
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 (UT 2 Si 2-x P x , T ¼ transition metal), which can now be synthesized using molten metal flux growth (We have already successfully synthesized several transition metal analogues UT 2 Si 2 (T ¼ 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 URu 2 Si 2 À x P x were grown from elements with purities 499.9% in a molten in flux, as previously reported 34 . The reaction ampoules were prepared by loading the elements into a 5 cm 3  crucible was then loaded into an alumina tube spanning the bore of a hightemperature 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 URu 2 Si 2-x P x 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 mKoTo20 K. Magnetization M(T, H) measurements were carried out for mosaics of single crystals for temperatures T ¼ 1.8-350 K under an applied magnetic field of H ¼ 5 kOe applied parallel to the c-axis using a Quantum Design Magnetic Property Measurement System. Magnetic susceptibility w 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 mKoTo300 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 Ho18 T and T ¼ 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 ¼ 50 mK.