Structural phase transition, mechanics, and thermodynamics of heavy fermion metal UPt₃ under pressure

Abstract

Crystal structures, electronic structures, mechanics, and thermodynamics of the heavy fermion superconductor UPt₃ under a pressure of up to 300 GPa have been investigated by a particle swarm optimization structure prediction method together with detailed first-principles calculations. A pressure-induced structural phase transition (P_T) is predicted at 155.9 GPa, where the hexagonal crystal structure with the space group P6₃/mmc transforms into an orthorhombic structure with the space group Cmmm. The molar volume of UPt₃ drops about 2.52% at 155.9 GPa, while the distance between the first-nearest neighbor of U atoms (d_U–U) decreases, implying a switch from the heavy electronic states to the weakly correlated electronic states. The metal nature is well retained upon the phase transition and upon further compression to 300 GPa. Phonon dispersions and elastic constants are used to confirm the dynamical and mechanical stability of both phases under different pressures. The bulk modulus B, shear modulus G, and Young's modulus E of the Cmmm are all higher than those of the P6₃/mmc, indicating enhanced mechanical properties of the Cmmm phase at the same pressure. The highest phonon vibration frequency increases with pressure, suggesting strengthened atom–atom interactions. Thermodynamic properties, evaluated using the quasi-harmonic approximation (QHA), reveal that the P6₃/mmc phase remains stable in the 0–155.9 GPa range, while the Cmmm phase emerges under higher pressures. Our results provide theoretical insights into the pressure-driven phase transition of UPt₃ and provide its detailed electronic, phononic, mechanical, and thermodynamic properties under external pressure.

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Introduction

Heavy fermion materials, as typical strongly correlated systems with a high specific heat coefficient, are commonly found in lanthanide or actinide intermetallic compounds containing f electrons.^1–4 Due to the f electrons exhibiting both itinerant and localized states, the unfilled 4f or 5f electron shells coherently hybridize with conduction band electrons to form heavy fermion states. These states give rise to a variety of quantum phenomena, including unconventional superconductivity,^3,5 multiple magnetic order,⁶ non-Fermi liquids,^7,8 and topological states.⁹ To date, a series of uranium-based heavy fermion superconductors have been extensively studied, such as UTe₂,¹⁰ UPt₃,¹¹ and UBe₁₃.¹² They exhibit the coexistence of spin-triplet superconductivity and magnetic order.

In UPt₃, two distinct superconducting states (A and B phases) were observed at zero field with superconducting transition temperatures (T_c) of around 0.5 K.^13,14 In the B phase, the broken time-reversal symmetry was observed by the polar Kerr effect, providing the evidence of complex, multicomponent nature for its superconducting order parameter.^15,16 After including the effect of magnetic field, UPt₃ transitions into a new superconducting state C phase.^17,18 Neutron diffraction experiments have confirmed the coexistence of superconductivity and antiferromagnetic order in both the B and C phases, where the Néel temperature (T_N) is 5 K.^19–21 Notably, all superconducting phases have been identified as topological superconducting states with different topological invariants and surface states.²² Recently, the angle-resolved photoemission spectroscopy studies have revealed significant quasiparticle renormalization in the electronic structure of UPt₃, which can be attributed to the strong correlation effects of the 5f electrons.²³ With its abundant array of quantum phenomena, UPt₃ serves as a paradigmatic system in condensed matter physics, offering a unique platform for exploring heavy-fermion superconductivity and the interplay of competing quantum orders.

Pressure is a crucial parameter to regulate crystal structures, which further influences the electronic and various physical properties of materials. In uranium-based compounds, pressure can induce quantum phase transitions, enhance or suppress superconductivity, modulate magnetic order, and drive other quantum phenomena. For instance, UO₂²⁴ and UO₃²⁵ were reported to undergo structural phase transitions under pressure. The volume collapse of UO₃ under high pressure leads to an insulator-to-metal electronic transition.²⁵ For UGe₂, superconductivity was observed in the pressure range of 1–1.6 GPa, with the highest T_c of approximately 0.7 K at 1.2 GPa.^26–28 A pressure-induced symmetry transition, from Immm to I4/mmm, suggests that less 5f electron participation in bonding due to the weakly correlated superconducting phase appears in the tetragonal structure of UTe₂.^29,30 Spin-triplet p-wave superconductivity in UBe₁₃ has been revealed through low-temperature high-pressure experiments, providing a direct case of triplet pairing in strongly correlated electron systems.¹² For UPt₃, no structural phase transitions have been observed up to 52 GPa, as confirmed by the X-ray diffraction experiments.³¹ However, when the pressure exceeds 0.3 GPa, the low-temperature superconducting A phase was suppressed. Under an external magnet field, the superconducting B phase transformed into the C phase,^32,33 with the C phase being the most stable state under uniaxial pressure.³⁴ The experimental results of Gouchi et al. demonstrated that when the pressure exceeds the critical threshold of 0.4 GPa, the Josephson effect in the B phase was suppressed, leading to significant changes in its superconducting properties.³⁵ Furthermore, neutron diffraction experiments revealed that pressure not only suppresses the superconductivity but also affects the antiferromagnetic order in UPt₃.³⁶

In the present study, we report the discovery of a novel orthorhombic Cmmm phase of UPt₃ above 155.9 GPa, obtained through the particle swarm optimization (PSO) structure prediction as well as detailed first-principles calculations. The structure and physical properties of P6₃/mmc and Cmmm-UPt₃ are calculated at different pressures. Compared with the P6₃/mmc phase, the volume of the Cmmm phase shrinks by about 2.52% and the U–U bond length shortened by 36% at the P_T point. The electronic density of states (DOSs) at the Fermi level, N(E_F), of Cmmm-UPt₃ is higher than that of P6₃/mmc-UPt₃, and the itinerancy of the U 5f electrons is released upon phase transition. Excellent dynamical and mechanical stabilities of both phases at different pressures are confirmed by calculating the phonon dispersions and elastic constants, respectively. The highest vibration frequency increases with pressure, indicating that the interactions among atoms are strengthened. The elastic moduli B, G, and E of Cmmm-UPt₃ are higher than those of P6₃/mmc-UPt₃ at P_T. The QHA is used to analyze the thermodynamic properties. Our results clearly illustrate the structural phase transition of an important heavy fermion metal and provide its detailed electronic, phononic, mechanic, and thermodynamic properties under a wide pressure range of 0–300 GPa.

Computational details

The structure predictions in our present work were performed by using the PSO technique as implemented in the CALYPSO code,^37,38 which has been proven to be effective and accurate in predicting the stable and metastable structures of a large variety of materials including actinide compounds.^39–42 An extensive search of stable UPt₃ crystal structures within 1–4 molecular formula units were carried out using CALYPSO between 0 and 300 GPa. As shown in Fig. 1, the candidate eight structures with the lowest enthalpy were picked to optimize with high accuracy to obtain the most stable structure of UPt₃. The structural relaxations and physical properties were determined using the Vienna ab initio simulation package (VASP)^43,44 with the projector augmented wave scheme (PAW).⁴⁵ The exchange–correlation function with the local density approximation (LDA) was performed to deal with the Kohn–Sham equations.⁴⁶ The valence electrons for U and Pt were 6s²6p⁶6d²5f²7s² and 5d⁹6s¹, respectively. The kinetic energy cutoff of 550 eV was set to ensure convergence of total energies and forces better than 1 × 10⁻⁶ eV and 0.001 eV Å⁻¹, respectively. The Monkhorst–Pack⁴⁷ and Γ-centered k-point meshes with a reciprocal space resolution of 2π × 0.03 Å⁻¹ were chosen for the orthorhombic and hexagonal crystals, respectively. Dudarev's type on-site Coulomb repulsion⁴⁸ among the localized U 5f electrons was considered. The LDA+U scheme has been verified to be effective in better describing the electronic structures and/or magnetic states of UPt₃⁴⁹ and other uranium-based compounds.^24,25 Here, the Hubbard effective parameter U–J, which can be labeled as one single parameter U for simplicity, of 2 eV is used.⁴⁹ Such value has been carefully checked and selected in our previous study of UPt₃ under ambient conditions.⁴⁹

Fig. 1 (a) Relative enthalpy and (b) volume curves as a function of pressure for UPt₃ from 0 to 300 GPa. The enthalpy of the P6₃/mmc phase is set as zero for reference. The experimental results of the P6₃/mmc UPt₃ at 0–50 GPa are presented for comparison.³¹

To characterize the lattice dynamic stability, the density functional perturbation theory (DFPT)⁵⁰ as implemented in the PHONOPY code⁵¹ was used to calculate the phonon dispersion curves. The 2 × 2 × 2 supercell and 3 × 3 × 4 (3 × 3 × 2) q-point meshes were utilized in calculations of phonon spectra for different structural phases. At the same time, the QHA⁵² was used to reveal the thermal properties and the pressure–temperature (P–T) diagram.

Results and discussion

1. Pressure-induced phase transition

In this work, a series of pressure points are set to predict the stable phases of UPt₃ in the pressure range of 0 to 300 GPa. After high-precision structural optimization, it is observed that only two phases exhibit the lowest enthalpy within a pressure range of 0–300 GPa: P6₃/mmc-UPt₃ and Cmmm-UPt₃ [see Fig. 1(a)]. The P6₃/mmc-UPt₃ is an experimentally observed phase at ambient pressure⁵³ and no other competitive structures are discovered during 0–155.9 GPa, which is consistent with the X-ray diffraction results (0–50 GPa) reported by Benedict et al.³¹ Note that we also check the effects of ferromagnetic and antiferromagnetic states to the relative enthalpy of different phases. As shown in Fig. S1 in the ESI,† the P_T of the nonmagnetic states are close to the results of the ferromagnetic and antiferromagnetic states. In addition, the Néel temperature (T_N) is only 5 K^19–21 for UPt₃. Thus, we only consider the nonmagnetic state in our following discussions. For comparison, Table 1 lists the lattice parameters, Wyckoff positions, energies, and enthalpies for two phases of UPt₃ at different pressures, including data from the previous experiment.⁵³ For the P6₃/mmc phase, the optimized lattice constant a(c) is 5.647 (4.932) Å, differing from the experimentally reported values⁵³ by 1.1 (1.3) % under normal pressure. Fig. S2(a) (ESI†) shows the dependence of the lattice constant on pressure. As pressure increases, the lattice constants of UPt₃ decrease in both phases. When the pressure is up to 155.9 GPa, the a and c of P6₃/mmc-UPt₃ are reduced by 8 and 11%, respectively, corresponding to the a/a₀ and c/c₀ ratio are 0.91 and 0.88. This suggests that the [001] direction of UPt₃ is more sensitive to pressure, demonstrating anisotropic compression behavior. From Table 1, the predicted orthorhombic phase Cmmm-UPt₃ (19.806 eV f.u.⁻¹) is nearly isenthalpic with the P6₃/mmc-UPt₃ (19.805 eV f.u.⁻¹) at 155.9 GPa. When P > 155.9 GPa, Cmmm-UPt₃ is enthalpically more favorable than the P6₃/mmc, indicating the phase transition from the P6₃/mmc phase to the Cmmm phase at 155.9 GPa. Here, the lattice constants a, b, and c for the high-pressure Cmmm phase are 7.691, 10.162, and 2.516 Å, respectively, (see in Table 1). As the pressure increases to 300 GPa, the lattice constants of Cmmm phases decrease by 3.6, 3.9, and 4.8% for a, b, and c, respectively. To determine whether the phase transition of UPt₃ is first-order or second-order, the volume–pressure curves for each phase are calculated and are shown in Fig. 1(b). The results show a decreasing behavior in volume upon compression. Notably, a 2.52% volume collapse occurs during the transition from the P6₃/mmc to the Cmmm phase at 155.9 GPa, suggesting a first-order. The equation of state (EOS) of UPt₃ are calculated and plotted in Fig. S2(b) (ESI†). For P6₃/mmc-UPt₃, the equilibrium volume (V₀) is 136.35 ų, and the bulk modulus (B) and pressure derivative of B(B′) are 250.79 GPa and 5.04, respectively. The B of P6₃/mmc-UPt₃ calculated via the EOS is consistent with the B value (247.36 GPa) obtained by the Voigt–Reuss–Hill (VRH) method.⁵⁴ The V₀, B, and B′ values of high-pressure phase Cmmm-UPt₃ are 135.23 ų, 254.54 GPa and 5.06, respectively. Meanwhile, the spin–orbital coupling (SOC) effect is considered to reveal the P_T pressure of UPt₃. As shown in Fig. S3 (ESI†), the P_T pressure is approximately 179 GPa with SOC, when P6₃/mmc-UPt₃ transformed into Cmmm-UPt₃. Due to the P_T pressure without the SOC effect is lower than that with the SOC effect and there are no experimental results to verify, and also since the inclusion of SOC is very expansive in calculations, SOC is not considered in the subsequent calculations.

Table 1 Calculated lattice parameters (Å, deg), Wyckoff positions, energy (eV f.u.⁻¹), and enthalpy (eV f.u.⁻¹) of UPt₃ phases at select pressures (GPa)

Space group	Pressure	Lattice parameters	Wyckoff positions	Energy	Enthalpy
P63/mmc	0	a = b = 5.647, c = 4.932	U 2c 0.333 0.667 0.250	−35.75
		α = β = 90°, γ = 120°	Pt 6h 0.833 1.667 0.250
		a = b = 5.712, c = 4.867^53
	155.9	a = b = 5.168, c = 4.363		−29.29	19.806
		α = β = 90°, γ = 120°
Cmmm	155.9	a = 7.691, b = 10.162, c = 2.516	U 4j −0.5 −0.115 0.5	−28.03	19.805
		α = β = γ = 90°	Pt1 4j −0.5 0.365 0.5
			Pt2 4e −0.25 0.25 0.0
			Pt3 4g −0.224 0.0 0.0
	300	a = 7.412, b = 9.760, c = 2.395		−19.93

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Different from P6₃/mmc-UPt₃, the Wyckoff positions of U and Pt in Cmmm-UPt₃ are: U 4j (−0.5, −0.115, 0.5), Pt 4j (−0.5, 0.365, 0.5), 4e (−0.25, 0.25, 0.0), and 4g (−0.224, 0.0, 0.0). As illustrated in Fig. 1(b), a 9-coordinated “single-capped cube” is featured in the high-pressure Cmmm phase and it originated from the 12-coordinated U–Pt tetra-decahedron in the P6₃/mmc phase. From Fig. S4 (ESI†), it can be found that the three bond angles between Pt–U–Pt are 60.80°, 59.20°, and 60.33° in P6₃/mmc. The minimum and maximum bond angles in the Cmmm phase are 54.89° and 101.97°, respectively. In the Cmmm phase, the nearest neighbor U–Pt1 distance is 2.538 Å, which is slightly shorter than the U–Pt distance (2.653 Å) in the P6₃/mmc UPt₃ phase. The other U–Pt2 and U–Pt3 distances in the Cmmm phase are 2.675 and 2.729 Å, respectively. For U compounds, the Hill limit (∼3.5 Å) represents the boundary between localization and delocalization of the 5f electrons.⁵⁵ In observing the behavior of P6₃/mmc-UPt₃ under pressure (0–155.9 GPa), the U–U bond length (d_U–U) decreases from 4.090 Å⁴⁹ to 3.696 Å, indicating that the 5f electrons become more itinerant under pressure. The d_U–U in the Cmmm is 2.349 Å, significantly shorter than that in the P6₃/mmc (3.696 Å) at 155.9 GPa, indicating a stronger interaction between U atoms in Cmmm-UPt₃. This high-to-low symmetry structural transition suggests more 5f electron participation in transferring.³⁰

2. Electronic structure

As is well known, crystal structural transition often leads to electron redistribution, which underlines the importance of studying the pressure-induced electronic transition in materials. To obtain further insight into the electronic properties of UPt₃ under different pressures, we calculate the total electronic density of states (TDOSs) as well as the projected density of states (PDOSs) of the P6₃/mmc- and Cmmm-UPt₃, as shown in Fig. 2. In P6₃/mmc, the Van Hove peaks associated with U-5f orbitals, located between −0.5 and 1 eV at 0 GPa, transform into broader energy levels (−1 to 2 eV) at 155.9 GPa. Meanwhile, in the Cmmm phase, the U-5f DOS peaks at −0.3 eV gradually disappear as the pressure increases from 155.9 GPa to 300 GPa. As seen in Fig. 3(a), the DOSs at the Fermi level, N(E_F), decrease as pressure increases both in the P6₃/mmc and Cmmm phases. The N(E_F) of the P6₃/mmc phase decreases from 5.78 to 3.22 states per eV f.u. as the pressure increases from 0 to 155.9 GPa. In the Cmmm phase, the N(E_F) further declines from 3.84 states per eV f.u. at 155.9 GPa to 3.08 states per eV f.u. at 300 GPa. The N(E_F) in the P6₃/mmc phase is lower than that in the Cmmm phase at 155.9 GPa, indicating that the metallicity of P6₃/mmc-UPt₃ is weaker than that of Cmmm-UPt₃. For both P6₃/mmc- and Cmmm-UPt₃, the contribution of Pt-5d orbitals to the Fermi level is slightly lower than that of U-5f orbitals. Although pressure diminishes the metallic characteristics of UPt₃, it does not induce a insulator-to-metal transition like that observed in UO₃.²⁵

Fig. 2 TDOSs and PDOSs for (a)–(d) P6₃/mmc and (e)–(h) Cmmm-UPt₃ at different pressures. (i) and (j) Electronic band structures, TDOSs, and PDOSs for UPt₃ at 155.9 GPa.

Fig. 3 (a) Total N(E_F) as well as N(E_F) of the U-5f and Pt-5d orbitals and (b) averaged Bader charges of U and Pt atoms in each UPt₃ phase.

It can be seen from Fig. 2 that the U-5f electrons occupy the conduction and several valence bands (−1 to 0 eV), while the lower valence band energy levels are localized mainly by the Pt-5d electrons. In the Cmmm phase, the contribution of both U-5f and Pt-5d states to the valence and conduction bands near the Fermi level suggests hybridization between U-5f and Pt-5d electrons. Compared to the localized 5f electrons in the P6₃/mmc phase, the 5f electrons in the Cmmm phase exhibit more itinerant behavior. Pressure and the structural transition suppress the localization of U-5f electrons, which may be associated with the reduction in the U–U distance.

The Bader charges of U and Pt atoms under different pressures are calculated and are presented in Fig. 3(b). The average valence state of U (Pt) for P6₃/mmc-UPt₃ decreased from +1.682 (−0.560) at 0 GPa to +1.414 (−0.471) at 155.9 GPa, while for the Cmmm phase, the U atom loses electrons, decreasing from 0.812 to 0.701 e over the pressure range of 155.9–300 GPa. The calculated partial charges of UPt₃ at 155.9 GPa are listed in Table 2. In the P6₃/mmc and Cmmm phases, each U atom loses 1.414 and 0.812 electrons, respectively, with these electrons being unevenly distributed around the Pt atoms, indicating an anisotropic bonding environment. Similar to U₂X (X = Nb and Zr)^39,56 and U–Te systems,^30,57 the differences in charge transfer of U atoms between the two phases indicate that the structural phase transition leads to a redistribution of the electrons. The electron localization function (ELF) is generally used to characterize the localized distribution of electrons. The ELF values are defined as 0, 0.5, and close to 1, corresponding to non-localized electrons, electron gas, and strong electrons, respectively.⁵⁸ In both the P6₃/mmc and Cmmm phases, the high ELF regions are 0.819 and 0.833 at 155.9 GPa, respectively, and are mainly concentrated around U atoms, as shown in Fig. S5 (ESI†). For P6₃/mmc-UPt₃, the localized electrons around the U atom form a “hexagonal ring”. Meanwhile, both U and Pt ions are immersed in an almost uniform electron sea with a normalized electron density around 0.4, indicating a typical metallic bond combination.

Table 2 Compared bond length (Å), band angles (deg °) and partial charges (eV atom⁻¹) for U and Pt of UPt₃ at 155.9 GPa

Space group	Bond	Bond length	Bond angels		Atom	Partial charge
P63/mmc	U–U	3.696			U	+1.414
	U–Pt1	2.653	Pt1–U–Pt2	60.33	Pt1	0.483
	U–Pt2	2.584	Pt2–U–Pt2	60.80/59.20	Pt2	0.449
Cmmm	U–U	2.341	Pt1–U–Pt2	59.19	U	+0.812
	U–Pt1	2.538	Pt2–U–Pt3	56.24	Pt1	−0.168
	U–Pt2	2.675	Pt2–U–Pt2	56.10/91.89	Pt2	−0.329
	U–Pt3	2.729	Pt3–U–Pt3	54.89/101.97	Pt3	−0.314

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3. Mechanical properties

In addition to the electronic properties, pressure-induced structural phase transitions can also induce changes in mechanical properties, including elasticity, Debye temperature, Hardness, etc.^24,25 The elastic constants C_ij of the material are essential for understanding its mechanical stability, ductility, plasticity, and anisotropy. The C_ij, elastic moduli (B, G, and E), sound velocities (v_t, v_l, and v_m), and Debye temperature (θ_D) for the two phases of UPt₃ are calculated at different pressures, as summarized in Fig. 4. Obviously, all the elastic constants of UPt₃ at different pressures are positive and also satisfy the corresponding stability criteria⁵⁹ of the hexagonal and orthorhombic crystals. Thus, both the P6₃/mmc and Cmmm phases are mechanically stable at corresponding pressures. Based on the modified criterion,⁶⁰ UPt₃ is mechanically unstable at 90 GPa, as indicated by the negative value (Fig. S6, ESI†). For high-pressure mechanically unstable materials α-quartz and AlO, the phonon spectrum exhibits negative frequencies, and the modified criteria are not satisfied.^61,62 However, UPt₃ is dynamically stable within the pressure range shown in Fig. 5 and Fig. S7 (ESI†), indicating that UPt₃ is metastable. Of course, the mechanical stability of UPt₃ under high pressure requires further theoretical and experimental works to verify. For P6₃/mmc, the C_ij (i, j ≠ 4, 6) exhibits an obviously increasing trend with pressure. The slopes of C₄₄ and C₆₆ related to shear deformation with increasing pressure are smaller than other C_ij (i, j ≠ 4, 6). In addition, the changes of B, G, and E with pressure are consistent with our previous study of Mott–Hubbard insulator UO₂²⁴ and metal Zr.⁶³ It is worth noting that, within the pressure range of 120–155.9 GPa, the C₃₃ and C₄₄ values of the P6₃/mmc phase exhibit slight oscillations. This phenomenon may suggest that UPt₃ could be in the transition stage of transforming into a new phase. As shown in Fig. S8 (ESI†), similar trends to those of C₄₄ can be observed in the pressure-dependence of G, E, v_t, v_l, v_m, and θ_D as well. These behaviors can be attributed to the interdependence [eqn (S1)–(S5) in ESI†] between C₄₄ and these parameters. The detailed elastic constants C_ij of UPt₃ at 155.9 GPa are presented in Table 3. The values of C₂₂ and C₃₃ for the Cmmm phase are 1165.1 and 1020.6 GPa, respectively, which are higher than those of the P6₃/mmc phase. This indicates that the b and c directions of the Cmmm phase exhibit greater resistance to compression at 155.9 GPa. However, the resistance along the a direction of the Cmmm is slightly weaker than that of the P6₃/mmc phase, as the C₁₁ value for the former is lower than that of the latter. Meanwhile, the C_ii (i = 1–3) is higher than the C_jj (j = 4–6), indicating greater resistance to the axial compression than shear deformation in both P6₃/mmc and Cmmm phases of UPt₃.

Fig. 4 Calculated (a) elastic constants, (b) elastic moduli, (c) sound velocities, and (d) Debye temperatures as a function of pressure for UPt₃. While the points stand for the calculated values, the solid curves are derived from the polynomial fitting of the calculation data. For comparison, the theoretical data of α-U,⁶⁴ UIr₃,⁶⁵ and UPd₃⁶⁶ are provided.

Fig. 5 Calculated phonon dispersions as well as PhDOSs for (a) P6₃/mmc-UPt₃ and (b) Cmmm-UPt₃ at 155.9 GPa.

Table 3 Calculated elastic constants C_ij (in GPa) of UPt₃ at 155.9 GPa

Space group	C 11	C 12	C 13	C 22	C 23	C 33	C 44	C 55	C 66
P63/mmc	1040.5	640.2	580.6			897.4	88.5		200.2
Cmmm	1022.7	663.6	608.1	1165.1	482.6	1020.6	31.9	244.4	318.2

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The B, G, and E are evaluated by the VRH approximations.⁵⁴ From Table 4, the polycrystalline moduli B, G, and E of the Cmmm phase are larger than those of the P6₃/mmc phase, indicating that the mechanical properties of the high-pressure Cmmm phase are superior at 155.9 GPa. For comparison, we provide the result of B, G, and E for α-U,⁶⁴ UIr₃,⁶⁵ and UPd₃⁶⁶ at different pressures in Fig. 4(b). As shown, the pressure-induced behaviors for these mechanical properties for these compounds are similar. With respect to α-U,⁶⁴ which shows brittle elasticity at ambient pressure as reflected by the low values of B/G, alloying with Pt, Ir, and Pd greatly enhances the bulk modulus B and change the brittle nature to ductile. Alloying with platinum, the shear modulus G and Young's modulus E are enhanced slightly with respect to α-U.⁶⁴ Compressing over around 40 GPa, the G and E for UPt₃ become lower than those of orthorhombic α-U.⁶⁴ The B/G and σ values for the P6₃/mmc (Cmmm) phase of UPt₃ at 155.9 GPa are 5.09 (4.63) and 0.41 (0.40), respectively, both exceeding the thresholds of the Pugh's criteria (1.75) and Poisson's ratio (0.26), respectively. The B/G and σ of UPt₃ are close to those of gold (5.39 and 0.45),⁶⁷ indicating that UPt₃ has the similar elasticity to that of pure gold. These results provide direct evidence that UPt₃ exhibits ductile properties in both the P6₃/mmc and Cmmm phases at 155.9 GPa. According to the results shown in Table 4, we conclude that both P6₃/mmc and Cmmm-UPt₃ exhibit strong anisotropy (A^U > 0), with the latter showing more pronounced anisotropy due to its lower structural symmetry.

Table 4 Calculated elastic moduli (B, G, and E, in GPa), B/G ratio, Poisson's ratio (σ), transverse v_t (m s⁻¹), longitudinal v_l (m s⁻¹), average v_m (m s⁻¹) sound velocities, and Debye temperature θ_D (K) of UPt₃ at 155.9 GPa

Space group	B	G	E	B/G	A ^U	σ	v l	v t	v m	θ D
P63/mmc	727.3	143.0	402.0	5.09	0.84	0.41	5820.82	2297.57	2603.64	333.0
Cmmm	743.7	160.5	449.3	4.63	5.26	0.40	5869.28	2402.92	2719.90	350.9

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At the same time, the sound velocities (v_t, v_l, and v_m) and Debye temperature θ_D are calculated using the elastic constants. As pressure increases, the v_t, v_l, v_m and θ_D of UPt₃ increase steadily. The growth rates of v_t, v_l, v_m and θ_D in the P6₃/mmc phase initially decrease and then fluctuate between 120 and 155.9 GPa, while v_t, v_l, v_m and θ_D increase almost monotonically during the pressure range of 155.9–300 GPa for the Cmmm phase. The variations of these parameters clearly indicate that pressure enhance atomic interactions and strengthens atomic bonding. The obtained values of v_t, v_l, v_m, and θ_D for the P6₃/mmc and Cmmm phases at 155.9 GPa are presented in Table 4. The θ_D of the Cmmm phase is higher than that of the P6₃/mmc phase, indicating that the phase transition from P6₃/mmc to Cmmm enhances the bonding ability of UPt₃. As shown in Fig. 4, the v_m and θ_D of UPt₃ in pressure range of 0–25 GPa are largely lower than those of UIr₃⁶⁵ and UPd₃,⁶⁶ which can be attributed to the heavier atomic mass of Pt compared to Ir and Pd and also can be reflected by low values of G for UPt₃.

4. Dynamic and thermodynamic properties

Here, the phonon vibrational properties of UPt₃ under different pressures are calculated to evaluate its dynamical stability. As shown in Fig. S7 (ESI†) and Fig. 5, the phonon spectra for both the P6₃/mmc and Cmmm phases of UPt₃ exhibit dynamic stability at all pressures, confirming their dynamical stability. The coupling between the optical and acoustic modes in the P6₃/mmc and Cmmm phases is clearly observable. With increasing pressure, the highest frequency of the optical branch increases from 22 meV to 39 meV in the P6₃/mmc phase (0–155.9 GPa) and from 42 meV to 53 meV in the Cmmm phases (155.9–300 GPa). These results clearly indicate that pressure can strengthen the interaction between cations and anions in UPt₃. By comparing the phonon dispersions of the P6₃/mmc and Cmmm phases at 155.9 GPa (Fig. 5), it is observed that the highest vibration frequency of the Cmmm phase is higher than that of the P6₃/mmc phase, indicating stronger bonding and greater lattice stiffness of the former one, which may contribute to its superior mechanical properties. Similar to the phononic results at ambient pressure,⁴⁹ both the U and Pt atomic vibrations contribute widely in the full energy range for both P6₃/mmc and Cmmm phases. The rigid dispersions of the acoustic branches clearly demonstrate their excellent mechanical properties. The flat phonon bands as well as the van Hove singularities of the bands contribute evident peaks in the phonon density of states (PhDOSs).

According to the group theory analysis of the P6₃/mmc and Cmmm space groups, the symmetry decomposition of the modes are as follows.

For P6₃/mmc, Γ_acoustic = A_2u ⊕ E_1u,

Γ_optical = A_1g ⊕ A_2g ⊕ A_2u ⊕ B_1u ⊕ B_2u ⊕ 2B_2g ⊕ 2E_1g ⊕ 6E_2g ⊕ 4E_1u ⊕ 2E_2u.

For Cmmm, Γ_acoustic = B_1u ⊕ B_2u ⊕ B_3u,

Γ_optical = 3B_1u ⊕ 4B_2u ⊕ 4B_3u ⊕ 3B_1g ⊕ B_2g ⊕ 2B_3g ⊕ A_u ⊕ 3A_g.

As listed in Table S2 (ESI†), the optical modes E_1u and A_2u are infrared (IR) active for the P6₃/mmc phase, whereas A_1g, E_1g and E_2g are Raman (R) active. A_2g, B_2g, B_1u, B_2u and E_2u mode vibrations are nonactive. In the case of Cmmm phase, it possesses the point group symmetry of D_2h. Only an A_u mode is nonactive with a corresponding frequency of 15.92 meV at the Γ-point. The B_iu (i = 1, 2, and 3) modes are IR active and can absorb or emit infrared light. At 155.9 GPa, additional information regarding the vibration modes at the Γ point for both the P6₃/mmc and Cmmm phases are provided in Table S2 (ESI†). This information will be useful for future Raman and other related experiments.

It is well established that the specific heat C_V and entropy S can be calculated from the total energy and PhDOSs. The temperature dependencies of C_V and S for the P6₃/mmc and Cmmm phases of UPt₃ under different pressures are evaluated and exhibited in Fig. 6. In the low-temperature region (T < 300 K), pressure significantly affects C_V. For instance, in the P6₃/mmc phase, the C_V values are 195.0 J mol⁻¹ K⁻¹ and 184.3 J mol⁻¹ K⁻¹ at 0 GPa and 155.9 GPa, respectively. However, for T > 300 K, the C_V values at different pressures approach the Dulong–Petit limit as the temperature increases. Meanwhile, pressure suppresses the increase of S with temperature. Compared to the P6₃/mmc phase, the Cmmm phase exhibits lower values of C_V and S, indicating more stable thermodynamic properties at 155.9 GPa. Based on the QHA, the temperature dependence of the bulk modulus B(T), specific heat capacity at constant pressure C_P, and thermal expansion coefficient α_V are calculated at 155.9 GPa, and the P–T phase diagram for the P6₃/mmc and Cmmm is plotted in Fig. 7.

Fig. 6 Temperature dependencies of (a) specific heat at constant volume and (b) entropy of UPt₃.

Fig. 7 Temperature dependencies of (a) bulk modulus B, (b) specific heat at constant pressure C_P, and (c) thermal expansion coefficients α_V for UPt₃ at 155.9 GPa. (d) P–T phase diagram of P6₃/mmc and Cmmm phases of UPt₃ computed using the QHA model.

The B(T) of the P6₃/mmc and Cmmm decreases by 3.63% and 2.74%, respectively, when the temperature increases to 1000 K [see Fig. 7(a)]. The B(T) of the Cmmm phase is less sensitive to temperature compared to the P6₃/mmc phase. At the same time, the C_P and α_V for Cmmm are slightly lower than those of the P6₃/mmc in the overall temperature range at 155.9 GPa. The P–T phase diagram [Fig. 7(d)] of the two phases of UPt₃ is revealed by comparing the free energies (Fig. S9, ESI†) of the two phases as a function of temperature. Note that the calculated values of the transition pressure are 158 and 160 GPa at T = 380 and 850 K, respectively. Different from the α-U,⁶⁴ UO₃²⁵ and U₂Nb,⁵⁶ UPt₃ exhibits a relatively well-defined P–T phase diagram, showing its strong stability under extreme conditions.

Conclusions

In conclusion, we have systematically investigated the high-pressure behavior of UPt₃ by employing the PSO method and first-principles calculations. We report the discovery of a novel orthorhombic Cmmm phase of UPt₃ that emerges above 155.9 GPa. This phase transition is accompanied by a 2.52% reduction in volume and a 36% shortening of the U–U bond length. Compared with the P6₃/mmc phase, the electronic properties of Cmmm-UPt₃ reveal an increased N(E_F) and enhanced itinerant properties of the U 5f orbitals. The phonon dispersion and elastic constants confirm the dynamical and mechanical stability of both phases, with the Cmmm phase exhibiting superior mechanical properties, including higher bulk modulus, shear modulus, and Young's modulus. With increasing pressure, many mechanical properties are enhanced. Meanwhile, the specific heat, entropy, thermal expansion coefficients, and P–T phase diagram of UPt₃ are calculated and analyzed. Our work provides a theoretical basis for an important uranium-based heavy fermion metal UPt₃ under extreme conditions. The quantum states of chiral superconductivity and topological state in our predicted high-pressure Cmmm phase deserve theoretical and experimental efforts in the future.

Data availability

All data included in this study are available upon request by contacting the corresponding author.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors gratefully acknowledge financial support from the NSAF (Grant No. U2330104), the National Natural Science Foundation of China (Grant No. 12074381 and 12204482) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2024A1515010484). The calculations were performed at the CSNS Scientific Computing Platform of the Institute of High Energy Physics of CAS and the GBA Sub-center of the National HEP Science Data Center.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5tc00379b

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