Electronic Tuning in URu2Si2 Through Ru to Pt Chemical Substitution

Studies that control the unit cell volume and electronic composition have been useful in revealing what factors lead to hidden order and superconductivity in the strongly correlated electron system URu₂Si₂. For example, isoelectronic tuning that increases the hybridization between the f and conduction electron states (i.e., applied pressure and Ru → Fe/Os chemical substitution) 1) converts hidden order into antiferromagnetism and 2) destroys the superconductivity. The impact of nonisoelectronic chemical substitution has been less clear, but several unifying trends have recently emerged for chemical substitution vectors that qualitatively add electrons (e.g., Ru → Rh/Ir and Si → P). This includes 1) the rapid destruction of hidden order and superconductivity, 2) composition regions where the underlying Kondo lattice is preserved but does not harbor an ordered state, and 3) the emergence of complex magnetism at large substitutions. In order to assess the limits of this perspective, we have investigated the series U(Ru_1−xPt_x)₂Si₂ for x ≲ 0.19, where the Ru and Pt d-shells differ substantially from each other. Magnetic susceptibility, electrical resistivity, and heat capacity measurements unexpectedly reveal a phase diagram with notable similarities to those of other electron doping series. This result reinforces the viewpoint that there is a quasi-universal affect that results from electron doping in this material, and we anticipate that an understanding of these trends will be useful to isolate what factors are foundational for hidden order and superconductivity.

1 Introduction

The heavy fermion superconductor (T_c = 1.4 K) URu₂Si₂ has attracted sustained interest since its discovery in 1985, owing mainly to the enigmatic hidden order (HO) phase that it enters through a second-order phase transition at T₀ = 17.5 K (Palstra et al., 1985; Maple et al., 1986; Schlabitz et al., 1986). Like many of the related UT₂X₂ (T = transition metal and X = Si, Ge) compounds (Żołnierek and Mulak, 1995), URu₂Si₂ crystallizes in the ubiquitous ThCr₂Si₂-type tetragonal structure and exhibits Kondo-lattice hybridization of the f electrons with the conduction electrons (Endstra et al., 1993). However, in contrast to these chemical/structural analogs, many of which exhibit more conventional low-temperature behavior (e.g., antiferromagnetism in URh₂Si₂ (Umarji et al., 1987) and Pauli paramagnetism in UFe₂Si₂ (Szytuka et al., 1988)), extensive investigations have not yet led to a definitive understanding of hidden order, why it is able to host superconductivity, and what factors are key in promoting its unique behavior.

A productive strategy for addressing these questions has been to perturb the electronic state and structure using tuning parameters such as applied pressure (McElfresh et al., 1987; Motoyama et al., 2003; Hassinger et al., 2008; Butch et al., 2010), magnetic fields (De Boer et al., 1986; Nieuwenhuys, 1987; Kim et al., 2003a; Kim et al., 2003b), and chemical substitution (Amitsuka et al., 1988; Dalichaouch et al., 1989; Dalichaouch et al., 1990a; Dalichaouch et al., 1990b; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Bauer et al., 2005; Jeffries et al., 2007; Yokoyama and Amitsuka, 2007; Butch and Maple, 2009; Butch and Maple, 2010; Kanchanavatee et al., 2011; Kanchanavatee et al., 2014; Das et al., 2015; Ran et al., 2016; Wilson et al., 2016). Applied pressure (McElfresh et al., 1987; Motoyama et al., 2003; Hassinger et al., 2008; Butch et al., 2010) and isoelectronic chemical substitution at the Ru site [Ru → Fe, Os (Kanchanavatee et al., 2011; Kanchanavatee et al., 2014; Das et al., 2015; Ran et al., 2016; Wilson et al., 2016)] produce similar results: as the hybridization strength increases, HO is first enhanced, and then it is converted to antiferromagnetism (LMAFM). Superconductivity is only observed within the HO region. Nonisoelectronic substitution at the Ru site is less straightforward: Ru → Mn, Tc, Re substitution (Dalichaouch et al., 1989; Dalichaouch et al., 1990a; Dalichaouch et al., 1990b; Bauer et al., 2005; Jeffries et al., 2007; Butch and Maple, 2009; Butch and Maple, 2010) (effectively removing electrons) suppresses HO and stabilizes complex ferromagnetism, while Ru → Rh, Ir (Amitsuka et al., 1988; Dalichaouch et al., 1990a; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007) (effectively adding electrons) suppresses HO, stabilizes a region with no long range magnetic order, and eventually gives rise to a new antiferromagnetically ordered state. More recently, studies have shown that Si → P substitution produces a similar phase diagram (Gallagher et al., 2016a; Gallagher et al., 2016b; Chappell et al., 2020). This raises the possibility that nonisoelectronic chemical substitution that effectively adds electrons, regardless of how it is accomplished, may result in a unified phase diagram. This is unexpected given the differences between 1) substitution at the Ru or Si sites, which are crystallographically distinct, and 2) differences between substitution of 3d, 4d, 5d, and s/p electrons, which have differing valence-electron wave functions and may impact hybridization in different ways.

In order to probe the possibility that some nonisoelectronic substitution series share related phase diagrams, we investigated the chemical substitution series $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_2$. Here, Ru → Pt substitution is expected to be distinct from earlier electron doping series because 1) Ru is a 4d element while Pt is a 5d element, 2) Ru → Rh and Si → P are adjacent periodic table column substitutions while Ru → Pt is not, and 3) distinct bonding occurs for UPt₂Si₂ (Ptasiewicz-Ba̧k et al., 1985; Lee et al., 2018; Lee et al., 2020), as evidenced by its formation in the CaBe₂Ge₂ structure. These differences might be expected to drive the appearance of a distinct T–x phase diagram, as suggested by studies of other families of materials where transition metal → Pt substitution has been shown to be a versatile tuning parameter. (Zhu et al., 2010; Luo et al., 2013). Instead of this, temperature-dependent magnetic susceptibility, electrical resistivity, heat capacity, and high-field magnetoresistivity measurements reveal striking similarities to earlier reported phase diagrams in which tuning effectively adds electrons (Amitsuka et al., 1988; Dalichaouch et al., 1990a; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007; Gallagher et al., 2016a; Gallagher et al., 2016b). In particular, the HO phase boundary is rapidly suppressed and abruptly collapses before x ≈ 0.03. Superconductivity is only observed for the parent compound, which may suggest that HO is rapidly converted to AFM by x ≈ 0.02. This is followed by a region (0.03 ≲ x ≲ 0.05) with a paramagnetic (PM) Kondo lattice with a heavy-Fermi-liquid ground state and no ordering is seen down to low temperatures. Finally, magnetic order with an antiferromagnetic character emerges and strengthens for 0.06 ≲ x ≲ 0.19, although it is distinct from what is seen for other tuning series (Amitsuka et al., 1988; Dalichaouch et al., 1990a; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007; Gallagher et al., 2016a; Gallagher et al., 2016b; Rahn et al., 2021). These results provide evidence that although Ru → Pt, Rh and Si → P chemical substitutions are each distinct in their own way, they nonetheless provide semiunified tuning in URu₂Si₂ that is likely connected to charge doping.

2 Experimental Methods

Polycrystalline samples of $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_2$ were synthesized by arc melting elements (99.99+% pure, lump form) in a 1:(2–2x):2x:2 M ratio of U:Ru:Pt:Si. The chemical composition and x values were determined using electron-dispersive spectroscopy (EDS) by averaging the measured values taken from multiple areas within specimens (see Supplementary Figure S1). For nominal concentrations x_nom ≤ 0.05, the measured x_meas ≈ x_nom and there is little variation across the samples. For 0.075 ≤ x_{nom} ≤ 0.12, x_meas < x_nom, but there is little variation across the samples. Finally, for x_nom = 0.15 there is substantial variation across the sample, suggesting that this is near the limit for uniform Pt substitution and above this value chemical separation may be present. Throughout the rest of the manuscript x = x_{meas, avg}, as defined in Supplementary Figure S1. The crystal structure (see Supplementary Figure S2) was verified by powder x-ray diffraction spectroscopy (pXRD) using a Scintag diffractometer with a copper source. The a and c lattice constants were extracted from the data using Rietveld refinement with WinPrep90 software.

Magnetization measurements were done using a Quantum Design Magnetic Property Measurement System in an applied magnetic field H = 5 kOe and at temperatures T = 1.8–300 K. The samples that were used for these measurements were crushed into powder in order to measure the polycrystalline average magnetic susceptibility χ_avg = (χ_c + 2χ_ab)/3. Electrical resistivity ρ(T) measurements were done using a standard four-wire resistance probe in a Quantum Design Physical Property Measurement System (PPMS) at temperatures T = 1.8–300 K using a He4 cryostat and also for T = 0.14–20 K with an adiabatic demagnetization refrigerator PPMS insert. Heat capacity measurements were conducted using the thermal relaxation method in a PPMS for T = 1.8–55 K. Magnetoresistivity measurements were carried out for semi-aligned large grain specimens using a resistive magnet at the NHMFL facility in Tallahassee FL at T ≈ 0.39 K and with the applied field H ≤ 41 T parallel to the crystallographic c axis.

3 Results

The powder x-ray diffraction patterns for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_2$ (Figure 1A; Supplementary Figure S2) show that for x ≲ 0.19 specimens crystallize in the ThCr₂Si₂ structure and do not exhibit impurity peaks, even for the concentration x ≈ 0.19 that has a large distribution in the Pt chemical composition determined by EDS. Figure 1A focuses on several peaks that are associated with principal crystallographic directions, where the (004) and (200) peaks noticeably shift in opposite directions with increasing x. This indicates that the substitution of Pt has distinct influences on the in plane and out of plane bonding, and Rietveld refinement of the data shows that the c and a values increase and decrease, respectively, with increasing x. Furthermore, the calculated unit cell volume V and lattice parameter ratio c/a (Figure 1C) remain nearly constant for all x. Taken together, these trends suggests that Ru → Pt substitution is analogous to a negative uniaxial pressure that expands the c axis and compresses the ab plane, although the influence of nonisoelectronic chemical substitution cannot be neglected (discussed below). The exception is that the lattice constant a deviates from the low-x trend for x ≈ 0.19, which may be related to chemical inhomogeneity that was observed in chemical analysis measurements.

FIGURE 1

FIGURE 1. (A) The powder x-ray diffraction intensity normalized to the maximum value I/I_max vs. scattering angle 2θ for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_{2}x\lesssim 0.19$ (patterns are vertically shifted for clarity) zoomed in on the peaks displaying strong shifting with increases x. Black vertical bars denote the peak centers corresponding to the (hkl) crystallographic planes for URu₂Si₂ that are listed near the bottom axis. (B) The c (left axis) and a (right axis) lattice constants vs. x determined by Rietveld refinement of the powder patterns. (C) Lattice constants ratio c/a (left axis) and unit cell volume V (right axis) vs. x.

Results for the polycrystalline average magnetic susceptibility χ_avg(T) = (χ_c + 2χ_ab)/3 are shown in Figure 2. The curve for x = 0 shows the expected Curie-Weiss behavior at elevated temperatures, a Kondo coherence maximum at $T_{\text{coh},{\chi }_{\text{avg}}}\approx 55\mathrm{K}$, and a kink at the HO transition at T₀ ≈ 17.4 K (Palstra et al., 1985; Maple et al., 1986; Schlabitz et al., 1986; Baumbach et al., 2014). The high temperature behavior is unchanged for all x, but there is a distinct evolution for low temperatures. In particular, 1) T₀ shifts to lower temperature for x ≈ 0.02, 2) it is removed for x ≈ 0.03–0.05 (suggesting that there is no ordered state over this x range), and 3) for x ≈ 0.06–0.19 there is a strong reduction in χ_avg(T) that is consistent with the onset of antiferromagnetic ordering at T_N. This feature is emphasized in the derivative of the data (Figure 2B), where T_N is defined as the midpoint of the upturn in dχ_avg/dT (downward-facing triangles) and the bars indicate the width in temperature of the transitions. This yields an ordering temperature T_N ≈ 42 K for x = 0.19 and, although it weakens with decreasing x, the ordering potentially extends as low as x ≈ 0.06 where T_N is slightly suppressed. The electrical transport and heat capacity measurements shown below support this perspective. Interestingly, hysteresis is also seen at large x for T < T_N and appears to strengthen with increasing x. This suggests that, while this ordered state has an antiferromagnetic character, it likely hosts additional complexity. In order to resolve this question, an investigation of the magnetic anisotropy for single-crystal specimens or neutron-scattering measurements will be necessary.

FIGURE 2

FIGURE 2. (A) Polycrystalline average magnetic susceptibility χ_avg = (χ_c + 2χ_ab)/3 vs. temperature T for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_{2}x\lesssim 0.19$. The temperature at the maximum in χ_avg, T_coh, is indicated by the vertical-dashed line. (B) Derivative of the polycrystalline average magnetic susceptibility with respect T, dχ_avg/dT, vs. T zoomed in on the phase transitions. Phase transition temperatures are indicated with colored arrows in (A) and downward-facing triangles in (B). Bars indicate the width in temperature of the transitions. Curves are vertically shifted by Δ for clarity.

Temperature-dependent electrical resistivity ρ/ρ_300K(T) results are shown in Figure 3. The expected behavior is seen for x = 0 (Maple et al., 1986; Schlabitz et al., 1986; Baumbach et al., 2014; Palstra et al., 1986), where ρ/ρ_300K(T) first increases with decreasing temperature and subsequently goes through a maximum near T_coh,ρ ≈ 75 K as the coherent Kondo lattice forms. Similar behavior persists for x ≲ 0.05, where there is little change in T_coh,ρ. As is seen in χ_avg(T), the HO transition (defined as the local minimum in d(ρ/ρ_300K)/dT, Figure 3B) shifts to lower T for x ≈ 0.02 and is not present for larger concentrations. Although x ≈ 0.02 exhibits ordering that connects to the hidden order phase boundary, measurements as low as 280 mK reveal no evidence for superconductivity (SC) (Figure 3C, inset). This is distinct from what is seen in other substitution series (e.g., Si → P (Gallagher et al., 2016a; Gallagher et al., 2016b) and Ru → Fe (Kanchanavatee et al., 2011)), where hidden order hosts SC even for chemically substituted specimens. The reason for the absence of SC is not clear, but it may suggest that, instead of enabling a smooth suppression of hidden order, Pt substitution stabilizes a magnetic phase that is similar to the antiferromagnetism that is seen at low x in the Ru → Rh series (Dalichaouch et al., 1990a; Amitsuka et al., 1988; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007) or the parasitic small moment antiferromagnetism that occurs due to defects in the parent compound (Niklowitz et al., 2010). Recent results also suggest that antiferromagnetism intervenes in the Si → P substitution series at the boundary of the hidden order phase (Chappell et al., 2020). Once the low-x ordering is suppressed, there is no evidence for an ordered state for x ≈ 0.03–0.05 at temperatures above 140 mK. In ρ/ρ_300K(T) at large x, the antiferromagnetic ordering appears as a subtle humplike feature that is followed by a reduction of the resistivity at low temperatures. These features are more evident in the derivative of the data (Figure 3B), and this characteristic shape is observed for x ≳ 0.06. The transition temperature, defined as the midpoint of the upturn in d(ρ/ρ_300K)/dT (star symbols and bars to indicate the temperature widths, Figure 3B), shifts to slightly higher temperature with increasing x and is in agreement with the values of T_N that are seen in χ_avg(T). Also noteworthy is that the shape of the anomaly in ρ/ρ_300K(T) evolves, becoming gradually more humplike at larger x. This could indicate that a change in the magnetic ordering is taking place. Over the large-x range, T_coh,ρ is difficult to define, because it occurs at temperatures that are similar to that of the magnetic ordering; however, it appears to roughly remain constant. Finally, we note that the residual resistivity increases strongly with increasing x. This may be due to disorder effects, but note that over similar x ranges in other substitution series (e.g., URu_2−xRh_xSi₂ (Dalichaouch et al., 1990a), URu_2−x(Fe,Os)_xSi₂ (Kanchanavatee et al., 2011; Kanchanavatee et al., 2014), and URu₂Si_2−xP_x (Gallagher et al., 2016a; Gallagher et al., 2016b)) this effect is much weaker. Therefore, we speculate that this could instead be associated with an intrinsic electronic effect.

FIGURE 3

FIGURE 3. (A) The electrical resistivity normalized to the room temperature value ρ/ρ_300K vs. log(T) for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_{2}x\lesssim 0.19$. (B) The derivative of ρ/ρ_300K with respect to T, d(ρ/ρ_300K)/dT, vs. T zoomed in on the HO and AFM phase transitions. (C) ρ/ρ_300K vs. T for x ≲ 0.05 and T ≳ 140 mK zoomed in on the SC and HO phase transitions. The curves are vertically offset by Δ for clarity. The phase transition temperatures are indicated with arrows in (A,C) and star symbols in (B), where the bars indicate the widths.

The heat capacity results are shown in Figure 4 and Supplementary Figure S4. The peak that is associated with the HO transition decreases for x ≈ 0.02 and is removed for x ≈ 0.03 (Figure 4A). Similar to the Si → P (Gallagher et al., 2016a; Gallagher et al., 2016b) and Ru → Rh (Amitsuka et al., 1988) substitution series, the height of the peak decreases and the low temperature value of C/T increases over this x region. In the region x ≈ 0.03–0.05, there is no evidence for an ordered state, but Fermi-liquid behavior is observed where the electronic component γ remains enhanced. The AFM transition for x ≈ 0.06 (Figure 4B) develops from a broad peak to a more pronounced peak for x ≈ 0.19 and is similar to the AFM transition seen in the phosphorus-substituted series (Gallagher et al., 2016a; Gallagher et al., 2016b). T_N is taken as the midpoint of the local minimum and maximum, and the bars indicate the temperature widths of the transitions. We also note that for specimens that exhibit antiferromagnetism, the low-temperature value of C/T is reduced from what is seen at lower x.

FIGURE 4

FIGURE 4. The heat capacity divided by temperature C/T vs. T for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_2$ (A) x = 0, x ≈ 0.02, x ≈ 0.03, and (B) x ≈ 0.05, x ≈ 0.06, x ≈ 0.08, x ≈ 0.09, x ≈ 0.19. The ordering temperatures T₀ and T_N are indicated by open squares and triangles, respectively. The dashed curve is for ThRu₂Si₂, a nonmagnetic analog of URu₂Si₂. (C,D) The 5f orbital magnetic entropy S_5f vs. T of the data in (A,B), respectively, that is calculated as described in the text. Plots of the 5f contribution to the heat capacity are shown in Supplementary Figure S4.

In Figures 4C,D, the magnetic entropy S_5f is shown, which is calculated by subtracting C/T of the nonmagnetic analog ThRu₂Si₂ (gray-dashed curve) and integrating from 1.8 − 55 K. As seen earlier for the Si → P series (Gallagher et al., 2016a; Gallagher et al., 2016b), the entropy associated with the HO transition is reduced for the chemically substituted sample. For samples with AFM, S_5f gradually increases overall, and the reduction following T_N is consistent with the loss of entropy due to the magnetic ordering. The behavior of S_5f(T) and its value at T_N is comparable to what is seen for the Si → P series at large x.

Finally, results for electrical resistivity measurements under applied magnetic fields are shown in Figure 5. The behavior for x = 0 is well known (De Boer et al., 1986; Nieuwenhuys, 1987; Kim et al., 2003a; Kim et al., 2003b), where a complex family of high-field ordered states was previously described. For x ≈ 0.02, the curve is reminiscent of what is seen for the parent compound: there is initially a strong increase and broad peak around 31 T that is abruptly followed by a cascade of transitions, and an eventual strong reduction in the resistivity near 38 T when the system enters the high-field polarized phase. The behavior for x ≈ 0.03 is distinct from what is seen in the HO x region. Here, the initial increase in ρ/ρ_300K is weak and there is an abrupt increase near 31 T that is followed by an abrupt decrease at 38 T. This square shape curve is reminiscent of what was previously seen in Si → P (Wartenbe et al., 2017; Huang et al., 2019) and Ru → Rh (Kuwahara et al., 2013) substitution series within the x range that hosts a paramagnetic Kondo lattice without long-range ordered ground states. Finally, for x ≈ 0.19, ρ/ρ_300K decreases weakly with H until it undergoes a more rapid reduction near 33 T, indicating the occurrence of a metamagnetic phase transition. This is similar to what was earlier seen in the antiferromagnetic x region for the Si → P series (Wartenbe et al., 2017; Huang et al., 2019), although the critical field is substantially reduced.

FIGURE 5

FIGURE 5. The magnetoresistivity normalized to the room temperature value ρ/ρ_300K vs. applied magnetic field μ₀H for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_{2}x=\mathrm{0,0.02},0.03,0.19$ (curves are vertically offset for clarity). The red arrow indicates the hidden order phase transition, the magenta arrows indicate the field-induced phase, and the blue arrow indicates the AFM transition.

4 Discussion

The phase diagram for U(Ru_1−xPt_x)₂Si₂ x ≲ 0.19, constructed from χ_avg(T), ρ/ρ_300K(T), and C/T(T) is shown in Figure 6. The HO transition is suppressed for x ≲ 0.02 and abruptly collapses before x ≈ 0.03 (red shaded region). Superconductivity is only observed for the parent compound, which may suggest that hidden order is rapidly converted to antiferromagnetism, similar to what is seen in the Ru → Rh substitution series (Dalichaouch et al., 1990a; Amitsuka et al., 1988; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007) and under applied pressure (McElfresh et al., 1987; Motoyama et al., 2003; Hassinger et al., 2008; Butch et al., 2010; Niklowitz et al., 2010). This is followed by a region (0.03 ≲ x ≲ 0.05) with a paramagnetic (PM) Kondo lattice with a heavy-Fermi-liquid ground state that is similar to the high-temperature behavior for x = 0, but no ordering is seen down to low temperatures (arrows). Finally, magnetic order with an antiferromagnetic character emerges and strengthens for 0.06 ≲ x ≲ 0.19 (blue shaded region). The data for x ≈ 0.06 are highlighted in gray because the phase transition is difficult to detect and is only distinguished in the heat capacity measurement. Also noteworthy is that there is an evolution in the large-x magnetic behavior, which is characterized by the appearance of hysteresis in χ_avg(T) and a change in the shape of the ρ/ρ_300K(T) curve. Finally, the Kondo coherence temperature T_coh in χ_avg(T) (dashed gray line in Figures 2A, 6) appears to remain constant, or slightly decreases with increasing x, suggesting that the Kondo lattice itself is nearly insensitive to doping. Taken together, these results reveal a strong similarity between the Ru → Pt phase diagram and those for the Ru → Rh (Amitsuka et al., 1988; Dalichaouch et al., 1990a; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007) and Si → P (Gallagher et al., 2016a; Gallagher et al., 2016b) substitution series, all of which feature an evolution from the HO to NO to AFM regions with increasing x and a robust Kondo-lattice energy scale that is somewhat insensitive to chemical substitution. Furthermore, the evolution of the field induced phases in the various regions resembles earlier results in other series. These similarities are unexpected, given that in each case 1) the lattice tuning is distinct and 2) the type of electrons that are being added differ substantially from each other. By clarifying the reasons for this similarity, it may be possible to further understand the factors that lead to the stability of hidden order in this distinct region of the electronic phase space. One attempt to do this was recently seen for the Si → P series, where tight-binding Hartree-Fock calculations show 1) that the radial probability distributions for the phosphorus ions are more tightly bound than that of the silicon and 2) that the energy difference between the orbitals decreases with increasing x (Chappell et al., 2020). The cumulative effect is that Si → P substitution decreases the hybridization strength, which correlates with the weakening of HO. At larger x, electrical charge tuning has been proposed to play an important role in determining the ground-state behavior. From this perspective, it might be proposed that Ru → Rh and Pt substitution also evolve along similar tuning axes, where decreasing hybridization strength and addition of charge carriers simultaneously tune the behavior. We note that this scenario is supported by earlier work studying trends in the UT₂X₂ (T = transition metal and X = Si,Ge) family, which suggest that the hybridization strength tends to decrease going from the Fe column towards the Cu column. Furthermore, all examples in the Co, Ni, and Cu columns exhibit magnetic ordering due to the uranium ions. (Endstra et al., 1993).

FIGURE 6

FIGURE 6. The transition temperature T vs. x phase diagram for $\mathrm{U}{({\mathrm{R}\mathrm{u}}_{1-x}{\mathrm{P}\mathrm{t}}_x)}_2{\mathrm{S}\mathrm{i}}_{2}x\lesssim 0.19$ constructed from measurements of polycrystalline average magnetic susceptibility χ_avg (down-facing triangles), electrical resistivity ρ/ρ_300K (stars), and heat capacity C (up-facing triangles). The coherence temperatures T_coh from χ_avg are the square symbols and the dashed gray line is a guide for the eyes. Inset: Energy gap Δ from fits to ρ/ρ_300K data as described in the text.

Finally, in order to gain further insight into the different regions of the phase diagram, fits to the low-temperature-normalized resistivity were done using the expression,

$\rho /{\rho }_{300\text{K}}\left(T\right)={\rho }_0/{\rho }_{300\text{K}}+A{T}^2+B\frac{T}{\mathrm{\Delta }}\left(1+\frac{T}{\mathrm{\Delta }}\right)e^{-\mathrm{\Delta }/T}(1)$

where ρ₀/ρ_300K is the normalized residual resistivity, AT² is the Fermi-liquid term, and the final term represents electron-magnon scattering due to spin excitations with an energy gap Δ (Figure 7) (McElfresh et al., 1987; Hassinger et al., 2008; Chappell et al., 2020; Palstra et al., 1986; Jeffries et al., 2008; Motoyama et al., 2008). It is already established that this expression merely provides a phenomenological description of the data and does not suggest that the hidden ordering is due to simple magnetism. We nonetheless use it in order to make a direct comparison to earlier studies, where Δ is associated with an energy gap that opens over the Fermi surface; e.g., as seen in measurements such as the heat capacity (Maple et al., 1986), inelastic neutron scattering (Wiebe et al., 2007), and angle-resolved photo-emission spectroscopy (ARPES) (Santander-Syro et al., 2009; Yoshida et al., 2010; Meng et al., 2013; Bareille et al., 2014). The best fits to the data were determined by varying the maximum temperature of the fit range T_max, and evaluating the maximum of the goodness of the fits R. For concentrations exhibiting HO (x ≲ 0.02) or AFM (0.06 ≲ x ≲ 0.19) the maximum in R is greater than 0.999. For concentrations with no ordering (0.03 ≲ x ≲ 0.05) fits to the data using Eq. 1 yielded Δ = 0 (Figure 7B). Results for the energy gap Δ are shown in the inset of Figure 6, where Δ decreases with increasing x in the HO region, mirroring the HO phase boundary. In the AFM x-region, Δ grows with increasing x and saturates near 35 K. These trends are similar to what was earlier seen for Si → P, where Δ decreases from 70 to 40 K in the HO region (Gallagher et al., 2016a; Gallagher et al., 2016b; Chappell et al., 2020). However, the absolute value of Δ differs from what was seen for the Si → P series, where Δ saturates at a larger value near 100 K. This provides some evidence that this type of magnetic ordering is distinct from that seen in other tuning series.

FIGURE 7

FIGURE 7. ρ/ρ_300K vs. T for U(Ru_1−xPt_x)₂Si₂ x ≈ (A) 0.02, (B) 0.03, and (C) 0.09. The dashed and dotted lines are the best fits to the data as described in the text. Insets: Goodness of fit R vs. the maximum temperature of the fit range T_max.

5 Conclusion

We have introduced the phase diagram for U(Ru_1−xPt_x)₂Si₂ x ≲ 0.19 constructed from χ(T), ρ(T), and C(T) measurements. The HO transition is suppressed and abruptly collapses before x ≈ 0.03. Superconductivity is only observed for the parent compound, which may suggest that HO is rapidly converted to magnetic order by x ≈ 0.02. For 0.03 ≲ x ≲ 0.05, no ordering is observed for T ≳ 280 mK and there is a PM Kondo lattice with a heavy-Fermi-liquid ground state. For 0.06 ≲ x ≲ 0.19, magnetic order with an antiferromagnetic character and hysteresis for T < T_N emerges and strengthens with increasing x. The similarity of this phase diagram with those of the Si → P (Gallagher et al., 2016a; Gallagher et al., 2016b) and Ru → Rh (Amitsuka et al., 1988; Dalichaouch et al., 1990a; Miyako et al., 1991; Kawarazaki et al., 1994; Yokoyama et al., 2004; Yokoyama and Amitsuka, 2007) chemical substitution series is noteworthy. To explain this, we suggest that their phase diagrams are controlled by a semi-universal combination of weakening hybridization strength and electron substitution. In order to fully understand this series, it will be useful to investigate single-crystal specimens of this series in order to determine the anisotropy in ρ(T) and χ(T), and to carry out more detailed studies of the Fermi surface and magnetic ordering in order to make comparisons to the P and Rh doped compounds.

Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author Contributions

RB designed the project, supervised its completion, contributed to data analysis and wrote the manuscript. GC synthesized the specimens, carried out measurements, contributed to data analysis and wrote the manuscript. WN contributed to synthesis and experiments. DG contributed to measurements in high magnetic fields. All co-authors reviewed the data analysis and manuscript.

Funding

This work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1644779 and the State of Florida. Synthesis of crystalline materials and their characterization were supported by the Center for Actinide Science and Technology (CAST), an Energy Frontier Research Center (EFRC) funded by the US. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award No. DE-SC0016568.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/femat.2022.861448/full#supplementary-material

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