Research ArticleCONDENSED MATTER PHYSICS

Evidence for a pressure-induced antiferromagnetic quantum critical point in intermediate-valence UTe2

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Science Advances  14 Oct 2020:
Vol. 6, no. 42, eabc8709
DOI: 10.1126/sciadv.abc8709

Abstract

UTe2 is a recently discovered unconventional superconductor that has attracted much interest because of its potentially spin-triplet topological superconductivity. Our ac calorimetry, electrical resistivity, and x-ray absorption study of UTe2 under applied pressure reveals key insights on the superconducting and magnetic states surrounding pressure-induced quantum criticality at Pc1 = 1.3 GPa. First, our specific heat data at low pressures, combined with a phenomenological model, show that pressure alters the balance between two closely competing superconducting orders. Second, near 1.5 GPa, we detect two bulk transitions that trigger changes in the resistivity, which are consistent with antiferromagnetic order, rather than ferromagnetism. Third, the emergence of magnetism is accompanied by an increase in valence toward a U4+ (5f2) state, which indicates that UTe2 exhibits intermediate valence at ambient pressure. Our results suggest that antiferromagnetic fluctuations may play a more substantial role on the superconducting state of UTe2 than previously thought.

INTRODUCTION

Spin-triplet superconductors have recently attracted renewed interest because of their potential topological properties (1). UTe2 is a newly found superconductor that has been argued to host spin-triplet pairing due to a large Hc2 that violates the paramagnetic limit (2, 3). Nuclear magnetic resonance measurements revealed a very small change in the Knight shift in the superconducting state, which is consistent with the spin-triplet scenario (4). Under a magnetic field applied along the b axis, reentrant superconductivity was found, which abruptly changes to the normal state at a metamagnetic transition at 34.5 T (5). In addition, scanning tunneling microscopy experiments found evidence of in-gap states argued to be evidence of chiral superconductivity (6). Last, Kerr effect measurements revealed field-trainable time-reversal symmetry breaking in the superconducting state, which is consistent with topological (Weyl) superconductivity (7).

Applied pressure is a clean, symmetry-preserving tuning parameter that may shed light on the spin-triplet superconducting state of UTe2. Prior hydrostatic pressure work found evidence for two superconducting transitions above 0.3 GPa (8). How these transitions extrapolate to zero pressure remains a matter of contention as sample dependence leads to a disagreement as to whether two transitions can be observed at ambient pressure (7, 9). As pressure is increased, magnetic order emerges, and superconductivity is rapidly suppressed. Notably, no superconducting transition was found to occur in the magnetically ordered state. Conversely, another pressure study argued that, at the pressure where magnetic order emerges, there is a heterogeneous coexistence of magnetic and superconducting states (10). In this scenario, the superconducting regions do not percolate at zero field, and the low-temperature resistance is finite. Application of magnetic field suppresses the magnetic order and enhances superconductivity, which causes a zero-resistance state to reemerge under magnetic field. The discrepancy between these two results in the high-pressure region invites a close evaluation of the phase diagram. More recent pressure studies added two new pieces of information. First, above 0.5 GPa, there is field-reinforced superconductivity for magnetic fields applied along the a axis (11, 12). Second, there may be a link between the ambient pressure field-induced metamagnetic transition at 34.5 T and the magnetic state induced with pressure (13).

Here, we perform electrical transport, ac calorimetry, and x-ray absorption measurements in UTe2 under hydrostatic pressure. We find that the superconducting transition temperature is maximized near a putative antiferromagnetic quantum critical point occurring at a pressure of Pc1 = 1.3 GPa. Similar to prior works, we clearly observe two superconducting transitions that have an opposite pressure dependence. Our results, however, reveal a missing piece in the puzzle: The onset temperatures of the two superconducting states cross at very low pressures. Our phenomenological model shows that these closely lying order parameters compete at atmospheric pressure. Applied pressure favors one superconducting state over the other, but it likely preserves a low-temperature phase in which both orders coexist microscopically and break time-reversal symmetry in the low-pressure regime.

Notably, we also find clear thermodynamic evidence for two phase transitions consistent with antiferromagnetic order: Tm1 (magnetic ordering temperature) sets in near 1.45 GPa, whereas Tm2 sets in at a slightly higher pressure of 1.51 GPa. The electrical resistivity displays a clear upturn at Tm1, which is usually a signature of antiferromagnetic order rather than ferromagnetic order (14, 15). This region of the phase diagram most closely mirrors that of antiferromagnetic CeRhIn5 (16), instead of known ferromagnetic superconductors (17). Further, the emergence of magnetism is accompanied by an increase in valence toward a U4+ (5f2) state, which indicates that UTe2 exhibits intermediate valence at ambient pressure. The increase in valence is accompanied by a decrease in the Kondo coherence temperature, which is generally expected to drive the system from superconducting to magnetic (18). Our results provide evidence that one of the two nearly degenerate superconducting instabilities is strongly enhanced upon approaching an antiferromagnetic transition, which raises important questions about the proposed spin-triplet nature of the pairing state in UTe2.

RESULTS

Figure 1 summarizes our ac calorimetry and electrical resistivity data collected at representative pressures. Heat capacity at ambient pressure, shown in Fig. 1A, displays a peak near Tc2 (superconducting critical temperature) = 1.65 K, which is consistent with the offset from a zero-resistance state. This main peak is followed by a shoulder in heat capacity occurring at a slightly lower temperature, near Tc1 = 1.45 K. Although evidence for two transitions at zero pressure has not been uniformly reported in all prior publications, it has been recently observed in detailed heat capacity measurements (7). The presence of two peaks and time-reversal symmetry breaking was taken as evidence for a nonunitary two-component order parameter because of the orthorhombic crystal symmetry of UTe2. As shown in (7), the splitting between these two transitions is small and sample dependent, which explains why this feature may have been missed in earlier reports.

Fig. 1 Ac calorimetry and resistivity under pressure.

(A) Heat capacity at zero pressure showing two superconducting transitions. The sample measured at zero pressure is different from the sample measured under pressure. (B) Ac calorimetry up to 1.32 GPa. Tc1 and Tc2 cross between 0.1 and 0.34 GPa. (C) Ac calorimetry between 1.40 and 1.57 GPa. Magnetic order emerges at 1.45 GPa, splitting into two magnetic transitions at higher pressure. The low-temperature tail observed in ac calorimetry for pressures above 1.25 GPa is of unknown origin, but it is unrelated to any superconducting transition as indicated by its presence even at 1.57 GPa where the low-temperature resistance does not approach zero. The curves in (B) and (C) were offset for clarity. (D) ΔC/T versus pressure for each of the superconducting transitions. ψ1 corresponds with Tc1, and ψ2 corresponds with Tc2. Specific heat jumps were determined by subtracting a baseline from each of the transitions and using the resulting peak value and temperature. (E) A comparison between ac calorimetry and resistivity at 1.47 GPa. The superconducting peak and the offset of the zero-resistance state differ by 0.5 K, as shown by dashed lines. (F) A comparison between ac calorimetry and resistivity at 1.49 GPa. A current density of 16 mA/cm2 was used for the resistivity measurements in (E) and (F). A.U., arbitrary units.

At low pressures, these bulk transitions have opposite pressure dependence and cross at 0.2 GPa. As shown in Fig. 1B, the small shoulder at Tc1 at zero pressure moves to higher temperature as pressure is increased and also gains more entropy relative to the transition at Tc2, which is suppressed fairly linearly with pressure and loses entropy. Further, the zero-resistance state always occurs at the higher of these two transition temperatures. Because the transitions cross, our result indicates that both arise from superconductivity, in agreement with (8).

These observations are consistent with a scenario in which pressure tunes the balance between two closely competing superconducting instabilities. Notably, this situation bears close resemblance to UPt3, for which the degenerate two-component superconducting order parameter is believed to be split by a small symmetry-breaking field (19). Besides the symmetry of the gap function, the other main difference between the two cases is the opposite pressure dependencies of the transition temperatures, although this is unrelated to the symmetries of the systems in question and likely originates from variations in their microscopic details. To expand this analysis, we consider the Landau free-energy expansion of two superconducting order parameters that transform as two different one-dimensional irreducible representations of the orthorhombic point group D2h, ψ1 and ψ2. The fact that UTe2 is a spin-triplet superconductor implies that both ψ1 and ψ2 are odd under inversion. This, combined with the field-trained Kerr effect (7), imposes sufficient conditions on the transformation properties of ψ1 and ψ2 for us to write down a free energy coupling the two=α(TTc)ψ12+α(TTc)ψ22+β1ψ14+β2ψ242gψ12ψ22+ϵ(ψ22ψ12)For details on the construction, see section S1. Here, ϵ represents hydrostatic pressure. Experimentally, ϵ = 0 therefore corresponds to P = 0.2 GPa, in which case the two superconducting transitions are accidentally degenerate. Note that ϵ > 0 favors ψ1, whereas ϵ < 0 favors ψ2. Because previous experiments reported time-reversal symmetry breaking at ambient pressure (7), the Landau parameters are constrained to g1 > 0 and (g1g2/2)2 < β1β2. As shown in section S1, if the only effect of pressure is to change the transition temperatures of the two superconducting states, then the sum of the specific heat jumps divided by each Tc, ΔC1/Tc1 + ΔC2/Tc2 (as well as each term in the sum individually), should be constant under pressure. This would not be the case if pressure were to substantially change the quartic coefficients of the Landau expansion, which could, in turn, affect the time-reversal symmetry-breaking nature of the low-temperature state. We note that our ac calorimetry measurements do not provide quantitative values. Nonetheless, the jumps in specific heat at a given temperature can be compared as a function of pressure as we do not expect the extrinsic contribution from the pressure medium to change drastically in the pressure and temperature range being investigated. As shown in Fig. 1D, the sum of the specific heat jumps divided by Tc is nearly constant at low pressures (P < 0.7 GPa), consistent with the expectation of the model. This indicates that pressure is mainly affecting the transitions’ temperatures (i.e., the quadratic terms in the Landau free energy), suggesting that time-reversal symmetry breaking is likely to take place below the second transition temperature across this pressure range (see section S1 for more details of the calculation). As pressure is further increased, however, the sum of the specific heat jumps divided by Tc also increases. One possible reason for this behavior is the proximity to a magnetic boundary, which is predicted to promote an increase in the specific heat jump in f-electron superconductors (20). This is the first hint of the proximity of UTe2 to a pressure-induced magnetic state.

At higher pressures, UTe2 does, in fact, develop additional ordered states that likely stem from magnetism. Figure 1C shows that at 1.45 GPa, a new peak appears in heat capacity at Tm1 = 3.8 K. This temperature has been previously associated with the onset of magnetic order. Notably, the signature of bulk superconductivity is still clearly observed in heat capacity at Tc1 = 1.8 K. The observation of bulk superconductivity occurring below magnetic order is in contrast to prior reports (8, 1013). Compared to 1.40 GPa, however, the superconducting peak is notably broadened and has less entropy. Similarly, the resistance drop to zero becomes broader at this pressure, although zero resistance is still reached at a similar temperature to the feature in heat capacity. This bulk-like coexistence extends over only a minute pressure range of less than 0.04 GPa, which is probably why it has not been observed previously. As pressure is further increased, the difference in temperature between the superconducting transition obtained from calorimetry and resistivity begins to grow. This separation is evident in Fig. 1E in which there is more than a 0.5-K difference between the two transition temperatures at 1.47 GPa. This indicates that the coexistence of superconductivity and magnetism is likely at the macroscopic scale, with the superconducting volume fraction decreasing below the percolation threshold as pressure moves the system deeper inside the magnetically ordered phase.

At a pressure of 1.51 GPa, a clear second magnetic transition emerges at Tm2 = 3.0 K. The higher temperature magnetic transition temperature increases rapidly at a rate of dTm1/dP = 16 K/GPa, whereas the lower magnetic transition temperature shows minimal pressure dependence. At this pressure and beyond, no evidence for a bulk superconducting transition is found via ac calorimetry at any temperature above 70 mK. The resistivity still reaches zero resistance at 1.9 K, as shown in Fig. 1F. As pressure is increased further, the zero-resistance state is suppressed to zero temperature continuously, also in contrast with prior reports (10). Further, no evidence for the Tm2 transition is observed at 1.49 GPa in either resistivity or ac calorimetry. The extrapolation of Tm2 to pressures below 1.51 GPa remains an open issue.

To investigate the relationship between superconductivity and magnetism, we measured the critical current necessary to induce a finite resistance. We highlight the behavior at a pressure of 1.57 GPa, where Tc1 = 0.9 K as determined from critical current measurements. Figure 2A shows that at this pressure, the current density (Jc) below which there is a zero-resistance state is extremely low, reaching a maximum of 21 mA/cm2 at 0.1 K. The critical current density increases fairly linearly with decreasing temperature below about 0.6 K. Above 0.6 K, Jc saturates to a nearly constant value of just below 1.4 mA/cm2 until no detectable evidence of a zero-resistance state occurs near 0.9 K. The reason for this plateau is not understood. To highlight the role of the current density, the inset of Fig. 2A shows resistivity versus temperature plotted at different current densities. The difference between the curves sets in at temperatures as high as 1.5 K. Sample heating cannot explain the effect. The main source of sample heating is contact resistance to the sample, which is of the order of 1 ohm. Even at the highest current density, this results in heating of the order of only 100 pW, which is negligible at these temperatures.

Fig. 2 Electrical resistivity measurements.

(A) Critical current density versus temperature at 1.57 GPa. Inset shows resistivity versus temperature measured at different current densities. (B) Resistivity versus angle of applied field at 1.57 GPa as the field is rotated from parallel to [010] to perpendicular to [010]. Inset shows a comparison of resistivity versus temperature for 0 T and for 0.85 T applied either parallel or perpendicular to [010]. (C) Resistivity versus temperature at higher temperatures. Inset shows dρ/dT at 1.57 GPa. Current density was 160 mA/cm2. Circular markers indicate transition temperatures determined from ac calorimetry.

Our results not only provide an explanation for why prior reports did not observe a zero-resistance state in this pressure range (i.e., the measurement current was too high) but also enable us to further investigate the claim of a reentrant superconducting state with applied magnetic field in this pressure region (10). To this end, we performed field-dependent measurements at 1.57 GPa using a vector magnet and a large current density (J = 160 mA/cm2). Figure 2B shows the results of these normal-state measurements. Even at low fields (H = 0.85 T), the resistivity increases by about 30% for fields applied along the hard [010] axis as compared to fields in the (101) plane, which hints at the tendency to move away from a zero-resistance state for fields applied along the hard axis. At a lower current density (J = 16 mA/cm2), the resistivity of UTe2 becomes zero within experimental resolution at 0.85 T for fields in the (101) plane, whereas it is finite when the same field magnitude is applied along the b axis. This field-angle dependence is consistent with a recent report in which no reentrant superconductivity is found for fields applied along the b axis (13).

Although superconductivity is enhanced for fields in the (101) plane, we reiterate that at lower current densities, a zero-resistance state is obtained for all field directions. At zero pressure, Jc, as determined from susceptibility measurements, is near 10 kA/cm2 (21), a factor of nearly 500,000 times larger than the value of 21 mA/cm2 obtained here. Such a low critical current is inconsistent with bulk superconductivity, which is also supported by the lack of any feature in ac calorimetry. Instead, our results are fully consistent with filamentary superconductivity (22). One possibility is that superconductivity is percolating either on the surface or between magnetic domains, which has been observed previously when antiferromagnetic order and superconductivity coexist in the prototypical heavy-fermion superconductor CeRhIn5 (23).

Remarkably, the magnetically ordered states occurring above Pc2 = 1.4 GPa seem inconsistent with a simple ferromagnetic phase. Figure 2C shows resistivity versus temperature for several pressures near the emergence of magnetic order. Initially, a slight upward inflection is shown at Tm1 at 1.45 GPa. As pressure is increased, two magnetic transitions become clear in resistivity, Tm1 and Tm2, in agreement with heat capacity measurements. At 1.57 GPa, both Tm1 and Tm2 show an upward inflection in resistivity as the sample is cooled. At higher pressures, Tm2 continues to show an upward inflection, but Tm1 shows a broad downward feature. As will be discussed later, this temperature dependence suggests antiferromagnetic ordering.

Figure 3A shows the pressure-temperature phase diagram constructed from data shown in Figs. 1 and 2. The pressure at which the superconducting transition temperature is maximum coincides with the pressure at which the magnetic order extrapolates to zero temperature, which suggests a quantum critical point at Pc1 = 1.3 GPa. Below 1.4 GPa, however, there is no evidence for magnetic order within the superconducting state. This, once again, is quite similar to antiferromagnetic CeRhIn5, in which magnetic order does not occur at temperatures below the superconducting transition in zero applied field (16). Figure 3B shows the exponent extracted from taking the logarithmic derivative of ρ(T) = ρ0 + ATn after subtracting off ρ0. The residual resistivity term was determined by performing a power-law fit over a small temperature range just above Tc or Tm. The resulting plot shows a region of linear-in-temperature resistivity centered on the critical pressure of 1.3 GPa. As shown in Fig. 2C, at 1.32 GPa, the resistivity is linear from just above Tc up to 8 K. As temperature is increased, the resistivity becomes sublinear because of the Kondo coherence temperature being reduced to 40 K at these pressures (see fig. S2).

Fig. 3 Phase diagram and electrical resistivity exponent.

(A) Temperature versus pressure phase diagram for UTe2. Shaded area indicates the region where heat capacity and resistivity show different superconducting temperatures. The dotted blue line for pressures below Pc2 is a guide to the eye. Below 1.25 GPa, the higher temperature superconducting transition had the same transition temperature in both heat capacity and resistivity. There is some uncertainty in the transition temperature for Tc2 at 1.18 GPa due to the potential for sample heating. Tc1 at 1.57 GPa in resistivity was determined using a current density of 16 mA/cm2. (B) A plot of the exponent in ρ(T) = ρ0 + ATn by taking d[ln(ρ − ρ0)]/d[ln(T)] = n. Dashed lines are a guide to the eye for boundaries of T-linear behavior, indicating a putative quantum critical point at a pressure near 1.3 GPa.

We now turn to the uranium valence in UTe2 under pressure. Figure 4A displays the uranium L3 x-ray absorption near-edge spectroscopy (XANES) spectra of UTe2 at low pressure as well as UF4 (U4+ reference) and UCd11 (U3+ reference) at zero pressure. Contrary to 4f systems, it is often difficult to determine the absolute valence of uranium from its L3 edge (24). Nevertheless, the peak of the white line of UTe2 is substantially shifted to higher energies compared to UCd11 and slightly lower energies compared to UF4, which points to an intermediate-valence state at ambient pressure closer to 4+ (24). Figure 4B compares the uranium L3 XANES spectra of UTe2 at two representative pressures. At 2.5 GPa, a small positive shift in the resonance energy is observed as compared to the lowest pressure point at 0.3 GPa. The uranium absorption edge was modeled using an arctangent step function combined with a Gaussian peak (see fig. S3). The resulting pressure dependence of the Gaussian peak position can be expressed as a change in uranium valence by using the available 3+ and 4+ references.

Fig. 4 Uranium L3 x-ray absorption near-edge spectroscopy spectra of UTe2.

(A) Edge step–normalized x-ray absorption near-edge spectroscopy (XANES) data for UTe2 at 0.3 GPa and reference materials UCd11 and UF4 at ambient pressure. UF4 data were adapted from (24). (B) Edge step–normalized XANES data for UTe2 at the minimum and maximum pressures, showing a small shift toward 4+ at higher pressures. (C) Energy shift of UTe2 as a function of pressure. Right axis shows estimated valence shift by taking UCd11 and UF4 as U3+ and U4+ references, respectively. An apparent increase in valence starts at pressures higher than 1.25 GPa.

A small positive shift in the resonance energy is detected starting above 1.25 GPa, which is in the same pressure range as the onset of magnetic order, as shown in Fig. 4C. The fact that the shift is positive further suggests that the valence is not an integer (i.e., not fully 4+) at ambient pressure. By taking UCd11 and UF4 as U3+ and U4+ references, respectively, the energy shift at 2.5 GPa implies an apparent reduction of 0.10(4) electrons toward 5f 2 (U4+). We note, however, that this number is an upper limit because either a shift in the 6d orbitals without 5f participation or structural changes as a function of pressure may partially explain the observed behavior.

DISCUSSION

Above Pc2 = 1.4 GPa, two magnetic transitions emerge, the signatures of superconductivity in heat capacity and electrical transport occur at different temperatures, and the uranium valence starts to increase. These experimental observations warrant further discussion. First, it is hard to reconcile two magnetic transitions with ferromagnetic order at zero field, which suggests that the pressure-induced magnetic order is antiferromagnetic. Further evidence for antiferromagnetic ordering can be obtained by considering the temperature dependence of the resistivity at each of the magnetic transitions. For a second-order magnetic transition in a metal, the resistivity is expected to follow the Fisher-Langer behavior, meaning that dρ/dT is proportional to the heat capacity with a positive constant of proportionality (14). Thus, the resistivity should decrease as the temperature is lowered through the transition. At an antiferromagnetic transition, however, the resistivity may either show an upward or downward inflection depending on the ordering wave vector Q (15, 25). Therefore, there are two possible scenarios to interpret our transport results. The first is that the transitions are ferromagnetic, and the Fisher-Langer scaling behavior is violated. The second, which seems more natural, is that the pressure-independent upward inflection at Tm2 implies an antiferromagnetic order with constant Q. As shown in the inset of Fig. 2C, Tm1 also shows an upward inflection (i.e., a dip in dρ/dT) at a pressure of 1.57 GPa. When the pressure is further increased, this changes to a downward inflection, as seen at 1.66 GPa. This change in character is also consistent with antiferromagnetic order but with pressure-dependent Q (25). Further evidence for antiferromagnetism has been reported in (12), in which it was claimed that Tm2 is suppressed with applied magnetic field for all field directions. It was also argued that that field-temperature phase diagram under pressure is similar to other heavy fermion antiferromagnets.

Although scattering measurements are needed to confirm the character of the high-pressure phase, the likely proximity to antiferromagnetism under pressure invites further consideration of the ambient pressure magnetic fluctuations in UTe2. We note that magnetization and muon spin resonance measurements at ambient pressure exhibit scaling consistent with ferromagnetic fluctuations (2, 26). Nonetheless, a plot of the magnetic susceptibility times temperature (χT) versus temperature indicates the dominance of antiferromagnetic correlations at low temperatures (fig. S1).

A recent spectroscopic work on the UM2Si2 (M: Pd, Ni, Ru, and Fe) family provides a framework for considering the effect of the valence shift toward 4+ in antiferromagnetic uranium-based materials. There, it was argued that the effect of a higher U4+ character is to cause an overall decrease in the exchange interaction between f and conduction electrons (18). As a result, the large-moment antiferromagnetic member UPd2Si2 exhibits a higher 5f2 contribution compared to Pauli paramagnet UFe2Si2. This balance is epitomized in superconducting URu2Si2, which becomes antiferromagnetic under applied pressure. A smaller exchange interaction under pressure promotes a smaller Kondo scale, which is expected to drive the system from a superconducting to an antiferromagnetic ground state due to the competition between Kondo and Ruderman–Kittel–Kasuya–Yosida (RKKY) energy scales. We see clear evidence for a suppression of the Kondo coherence temperature in UTe2 as the pressure is increased (see fig. S2). Although counterintuitive, pressure plays the opposite role as compared to CeRhIn5, for which applied pressure increases the coherence temperature above 1 GPa and yields a change from an antiferromagnetic to a superconducting ground state (27). This is why the temperature-pressure phase diagram of UTe2 mirrors that of CeRhIn5.

Our results unearth a more complex interplay between superconductivity and magnetism in UTe2 than previously thought, which unveil important consequences for the nature of the pairing state. Generally, spin-triplet superconductivity is expected to arise out of ferromagnetic fluctuations (17). The extremely large critical field (2), the small change in the Knight shift below Tc (4), and the reentrant superconductivity (5) are indeed strong evidence for triplet pairing. Evidence for two antiferromagnetic transitions at high pressure, however, is in contradiction with a simple scenario in which ferromagnetic fluctuations solely drive the phase diagram of UTe2. A recent theoretical study predicts that, as the on-site Coulomb interaction increases, the ground state of UTe2 changes from ferromagnetic to frustrated antiferromagnetic (28). A particularly notable feature of the temperature-pressure phase diagram of UTe2 is the fact that one of the two nearly degenerate superconducting transition temperatures at ambient pressure is enhanced by a factor of 2 near a putative antiferromagnetic critical point at Pc1 = 1.3 GPa, whereas the other superconducting transition is suppressed. Such an observation would be more naturally explained if the two superconducting states at ambient pressure were a singlet and a triplet state, rather than two triplet states. Even if the magnetic order at high pressures were ferromagnetic, one would expect an enhancement of both transition temperatures if the underlying pairing states were both triplet. It would be informative to examine whether a mixed singlet-triplet state can explain the small Knight shift below Tc, although such state would seemingly be at odds with the observation of a trainable Kerr effect in UTe2 upon application of a c-axis magnetic field (7). Applying negative chemical pressure via doping may tune the system toward a ferromagnetic ground state and provides an intriguing path for future studies to uncover the nature of the pairing state of UTe2.

MATERIALS AND METHODS

UTe2 crystals were grown using a vapor transport technique with a ratio of 1 U:1.5 Te, as reported elsewhere (2). The crystallographic structure was verified by single-crystal diffraction at room temperature using Mo radiation in a commercial diffractometer, which resulted in lattice parameters a = 4.1647(2) Å, b = 6.1368(3) Å, and c = 13.9899(6) Å within the Immm (71) space group. Attempts to grow UTe2 with a more stoichiometric mix of uranium and tellurium resulted either in the growth of the tetragonal phase of UTe2 or a lower-quality orthorhombic phase. Zero-pressure heat capacity was measured using a Quantum Design PPMS with a 3He option. Two single crystals of UTe2 from the same growth were measured simultaneously in a piston-clamp pressure cell using Daphne oil 7373 as the pressure medium. Resistivity was measured on one crystal using a standard four-point technique with current along the [100] direction. Error bars in the inset of Fig. 2B are calculated from root mean square noise measurements reported by the manufacturer of the resistance bridge. The zero-resistance state was determined as the point where dρ/dT reached zero within experimental uncertainty. Ac calorimetry measurements were performed on the second crystal mounted in the same pressure cell (29). Transition temperatures from ac calorimetry were determined as the peak in C/T, except for Tc2 at 0.97 and 1.18 GPa where the transition was identified by an inflection in C/T. The pressure was calibrated using high-purity lead as a reference manometer. The lead transition remained sharp across the pressure region investigated, which indicates a hydrostatic environment. Pressure-dependent resistivity and ac-specific heat were measured in a combination of 3He cryostat and 4He cryostats, as well as in an adiabatic demagnetization refrigerator.

The UTe2 uranium L3 XANES was measured as a function of pressure at the 4-ID-D beamline of the Advanced Photon Source, Argonne National Laboratory. High pressure was generated using a CuBe diamond anvil cell fitted with a set of 600-μm regular and partially perforated anvils, the latter used to mitigate the diamond x-ray absorption. Data were collected in transmission geometry using a pair of photodiodes to monitor the x-ray intensity before and after the sample. The x-ray energy was calibrated by measuring the yttrium K edge of a reference foil. While the energy was not recalibrated during the high-pressure experiments, we expect that the U L3 edge shifted less than 0.3 eV during the experiment because of extrinsic factors, which is then an upper limit for any energy drift during the measurements. The pressure cell was cooled using a helium flow cryostat. The temperature was kept at 1.7(1) K during data collection, whereas it was raised to 15 K during pressurization. Double-stage helium gas membranes were used to control pressure in situ. A stainless steel gasket was preindented to about 70 μm, and a sample space of 300 μm in diameter was laser drilled (30). Si oil was used as pressure media, and a ruby sphere was used as a manometer. US2 and UCd11 were also measured as U4+ and U3+ references, respectively, although US2 was later found to be an unreliable 4+ reference due to hybridization (31). These samples were measured in fluorescence geometry at room temperature. Data normalization was performed using the Demeter software package (32). The uranium absorption edge was modeled using an arctangent step function combined with a Gaussian peak (fig. S1). This model was adjusted to the data by varying all parameters, except for the width of the step function (set to the uranium L3 core-hole lifetime of 7.43 eV) (33), and its position (set to the maximum of the XANES first derivative).

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/42/eabc8709/DC1

https://creativecommons.org/licenses/by-nc/4.0/

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REFERENCES AND NOTES

Acknowledgments: We would like to thank P. Orth, A. Gillman, and C. H. Booth for fruitful discussions. Funding: The experimental work at Los Alamos was performed under the auspices of the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Science and Engineering under project “Quantum Fluctuations in Narrow-Band Systems.” F.B.S. was supported by FAPESP under grant nos. 2016/11565-7 and 2018/20546-1. This research used resources of the Advanced Photon Source, a U.S. DOE Office of Science User Facility operated for the DOE Office of Science by the Argonne National Laboratory under contract no. DE-AC02-06CH11357. Theory work (M.H.C. and R.M.F.) was supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division, under award no. DE-SC0020045. Author contributions: F.B.S., P.F.S.R., and E.D.B. synthesized the samples. S.M.T., P.F.S.R., J.D.T., and T.A. prepared the samples and collected data. G.F. performed the XANES measurements. M.H.C. and R.M.F. developed the phenomenological model and assisted in the data interpretation. F.R. assisted with interpretation of experimental data. All authors contributed in the writing of the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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