Superconducting praseodymium superhydrides

We have successfully realized the synthesis and superconductivity of praseodymium superhydrides above megabar pressures.

Abstract

Superhydrides have complex hydrogenic sublattices and are important prototypes for studying metallic hydrogen and high-temperature superconductors. Previous results for LaH₁₀ suggest that the Pr-H system may be especially worth studying because of the magnetism and valence-band f-electrons in the element Pr. Here, we successfully synthesized praseodymium superhydrides (PrH₉) in laser-heated diamond anvil cells. Synchrotron x-ray diffraction analysis demonstrated the presence of previously predicted F$\overline{4}$3m-PrH₉ and unexpected P6₃/mmc-PrH₉ phases. Experimental studies of electrical resistance in the PrH₉ sample showed the emergence of a possible superconducting transition (T_c) below 9 K and T_c dependent on the applied magnetic field. Theoretical calculations indicate that magnetic order and likely superconductivity coexist in a narrow range of pressures in the PrH₉ sample, which may contribute to its low superconducting temperature. Our results highlight the intimate connections between hydrogenic sublattices, density of states, magnetism, and superconductivity in Pr-based superhydrides.

INTRODUCTION

The idea that hydrogen-rich compounds may be high–critical temperature (T_c) superconductors can be traced back to 2004 (1), when chemical precompression of hydrogen by other elements was proposed as an effective way to reduce the metallization pressure of hydrogen. Recent experimental results of T_c exceeding 200 K in compressed H₃S (2–4) and 250 to 260 K in LaH₁₀ system (5–8) have indicated compressed hydrogen-rich compounds as potential room-temperature superconductors.

It is recognized that superconductivity in these hydrides owes its origin to electron-phonon coupling (EPC). Three parameters determine T_c: the characteristic phonon frequency, EPC, and Coulomb pseudopotential (9). Recent theoretical studies have covered almost all binary hydrides and found several metal superhydrides with extraordinary high-T_c superconductivity, such as CaH₆ (10), MgH₆ (11), YH_6–10 (12, 13), AcH_10–16 (14), and ThH_9–10 (15). Peng et al. (16) first studied all the candidate structures of rare earth superhydrides with H-rich cages at high pressure and proposed that only several hydrides could be superconductors with T_c > 77 K. At the same time, superhydrides with H₂ units are recognized to have relatively low critical temperature, e.g., LiH₆ (17), NaH₇ (18), Xe(H₂)₇ (19), and HI(H₂)₁₃ (20). The question is why some superhydrides are high-T_c superconductors, while others, with the same structure and stoichiometry, are not.

Continuing studies of lanthanide superhydrides, in this work, we studied high-pressure behavior of the Pr-H system above 100 GPa. Chesnut and Vohra (21) studied the crystal structure of metallic Pr and determined the phase sequence above megabar pressure. Pr can readily absorb hydrogen at high temperature and form hydrides: Face-centered cubic dihydride PrH₂ and hexagonal close-packed trihydride PrH₃ were found at ambient pressure. Subsequent filling of octahedral voids in the structure of dihydrides leads to nonstoichiometric PrH_2+x composition, which exhibits considerable variations of magnetic structures (22). Here, through high-pressure and high-temperature (HPHT) synthesis, two unexpected Pr superhydrides were obtained and studied. In particular, we investigated superconducting behavior of synthesized Pr superhydrides by electrical resistance measurements. Theoretical calculations are used to unravel the relationship among their magnetic properties, electronic band structures, phonon spectra, and superconductivity. Comparison with already detailed studies of La and Ce superhydrides allows us to elucidate the great influence of metal atoms on superconductivity of superhydrides.

RESULTS AND DISCUSSION

The stability and structures predicted by theoretical calculations

Before describing the experimental results, we have compared our theoretical findings with the previous ab initio study (16), which is different from ours in a number of aspects. These differences are crucial for understanding our experimental results and motivated us to further perform independent variable-composition searches for stable compounds in the Pr-H system at pressures of 50, 100, and 150 GPa using the Universal Structure Predictor: Evolutionary Xtallography (USPEX) (23–25) package and Ab Initio Random Structure Searching (AIRSS) (26) code (see fig. S1). The current theoretical results performed by Vienna Ab-initio Simulation Package (VASP) (27–29) are also checked by an independent code Cambridge serial total energy package (CASTEP) (30). The results of CASTEP can be found in fig. S1. These two codes give the same results in principle. The only difference is the symmetry of PrH₃ that CASTEP gives C2/m-PrH₃, while VASP gives Pm$\overline{3}$m-PrH₃ without magnetism and spin-orbit coupling (SOC) effects.

Results of the structure search exhibit large differences depending on including or excluding magnetism and SOC effects, which can be seen in Fig. 1 and fig. S1. However, previous calculations (16) did not include these effects. In agreement with previous results (16), our search gives Pm$\overline{3}$m as the most stable symmetry for monohydride PrH and Fm$\overline{3}$m for trihydride PrH₃, but important metastable phases P4/nmm-PrH₃ (~70 meV per atom above the convex hull) and Fm$\overline{3}$m-PrH were not reported. Previous work indicated that superhydride F$\overline{4}$3m-PrH₉ is stable between 100 and 200 GPa but did not report P6₃/mmc-PrH₉, which is about 19 meV per atom above the convex hull at 100 GPa. We also updated the convex hull and phase diagram of Pr-H system at 150 GPa.

Fig. 1. Calculated convex hulls for Pr-H system at various pressures.

Convex hulls for Pr-H system with the inclusion of SOC and magnetism at (A) 50, (B) 100, and (C) 150 GPa.

Synthesis of polyhydrides Fm$\overline{\mathbf{3}}$m-PrH₃ and P4/nmm-PrH_3-δ

To synthesize previously unknown hydrides, we carried out several experiments by directly compressing Pr and hydrogen in the Diamond Anvil Cells (DACs). The diamond used in this experiment was coated with 150-nm alumina film by magnetron sputtering. The metallic Pr sample was loaded and sealed with a little pressure in the argon-protected glove box. After loading hydrogen into the cell, the sealed pressure was about 10 GPa, and selected x-ray diffraction (XRD) patterns are shown at various pressures (see fig. S3C). Figures 2 and 3 summarize the data for PrH₃ and PrH₉, respectively. Before laser heating, the diffraction pattern at 30 GPa included peaks from Fm$\overline{3}$m-PrH₃ in Fig. 2A, the structure of which can be viewed as cubic close packing of Pr atoms with all octahedral and tetrahedral voids filled by H atoms (see Fig. 2C). After compression to 40 GPa, the sample was laser-heated to 1400 K. We found stronger signal from Fm$\overline{3}$m-PrH₃, while peaks from Fm$\overline{3}$m-PrH disappeared (see fig. S3C). Upon further compression, the diamonds broke.

Fig. 2. XRD patterns and crystal structures of PrH₃ at pressures.

(A) Refinement of the experimental XRD patterns obtained in Pr + H₂ cell by cold compression to 30 GPa. arb. units, means arbitrary units. (B) Refinement of the XRD pattern by Fm$\overline{3}$m-PrH₃ and P4/nmm-PrH₃ after laser heating at 43 GPa. Red line, experimental data; black line, model fit for the structure; green line, residues. R-factors for the refinement are R_p = 14.2% and R_wp = 24.5%. Crystal structures of (C) Fm$\overline{3}$m-PrH₃ and (D) P4/nmm-PrH₃ phases at 50 GPa.

Fig. 3. Refinement of the experimental XRD pattern, pressure-volume data, and crystal structure of PrH₉.

(A) Refinement of the XRD pattern by F$\overline{4}$3m-PrH₉ and P6₃/mmc-PrH₉. Red line, experimental data; black line, model fit for the structure; green line, residues. R-factors are as R_p = 12.4% and R_wp = 22.0%. (B) EoS of the synthesized Pr-H phases; theoretical results include magnetism and SOC effects. Inset: The distinction among PrH₈, PrH₁₀, and PrH₉ phases. Crystal structures of (C) F$\overline{4}$3m-PrH₉ with H₂₈ cages and (D) P6₃/mmc-PrH₉ with H₂₉ cages.

The experimental volumes of cubic PrH₃ are in good agreement with those predicted for Fm$\overline{3}$m-PrH₃ structure in the pressure range of 10 to 53 GPa (see Fig. 3B).The experimental equation of state (EoS) of this phase was fitted by the third-order Birch-Murnaghan EoS, which gave V₀ = 37.7 (3) ų, K₀ = 113 (2) GPa, and K₀′ = 3.0 (5). Fm$\overline{3}$m-PrH, proposed for explanation of the XRD pattern, is slightly nonstoichiometic from the EoS (Fig. 3C), and it seems more correct to define as Fm$\overline{3}$m-PrH_1+x, where x = 0.08 to 0.13.

It is well known that experimental studies of hydrides are greatly affected by the hydrogen permeability contributing to the failure of diamonds in the high-pressure experiments. To minimize this problem, we synthesized the new hydrides by replacing of pure hydrogen with ammonia borane (AB), which is an excellent source of hydrogen (released during decomposition of AB). Several experiments were performed according to the reaction: Pr + NH₃BH₃→PrH_x + c-BN through HPHT treatment (31–33). Figure 2B shows the diffraction pattern after laser heating at 43 GPa. The reaction products are dominated by Fm$\overline{3}$m-PrH₃ with a small quantity of tetragonal phase P4/nmm-PrH_3-δ (0.05≤δ≤0.15) with smaller unit cell volume. At 50 GPa in the P4/nmm-PrH₃ structure, each Pr atom is bonded to 9 H atoms with 2.09 Å ≤d(Pr-H)≤ 2.17 Å. Experimental cell parameters of found compounds are shown in table S3.

Synthesis of F$\overline{\mathbf{4}}$3m-PrH₉ and P6₃/mmc-PrH₉

To obtain higher hydrides of Pr, we conducted further experiments at pressures above 100 GPa. To overcome problems with hydrogen permeation, we also used NH₃BH₃ (AB) as the source of hydrogen, which proved to be effective for synthesis of superhydrides at megabar pressures (7, 34). The original sample containing Pr with AB was laser-heated to 1650 K at 115 GPa. Measurements after laser heating did not show any changes in pressure, and Raman signal of H₂ was detected at 4147 cm⁻¹, indicating the generation of hydrogen. Figure 3A shows the XRD pattern with the presence of two praseodymium superhydrides F$\overline{4}$3m-PrH₉ and P6₃/mmc-PrH₉. Experimental lattice parameters at 120 GPa are a = 4.967 (1) Å and V = 122.52 (9) ų for F$\overline{4}$3m-PrH₉ and a = 3.588 (1) Å, c = 5.458 (4) Å, and V = 60.84 (9) ų for P6₃/mmc-PrH₉. This sample was compressed to 130 GPa and then gradually decompressed down to the lowest pressure of 105 GPa to determine its experimental EoS (Fig. 3B and table S4). Both EoS of PrH₉ are very close to the calculated curve of Pr + 9H and correspond well with the calculated values. After decompression down to 53 GPa, the recorded XRD pattern demonstrates the presence of two lower hydride phases: Fm$\overline{3}$m-PrH₃ with experimental parameters of a = 4.832 (1) Å at 50 GPa and P4/nmm-PrH_3-δ with a = 2.801 (1) Å and c = 6.280 (2) Å at 50 GPa, which is consistent with the low pressure results.

Both structures have almost the same volume and energy on convex hull at studied pressure range (Fig. 1, B and C). The stability of F$\overline{4}$3m-PrH₉ was previously predicted (16), while its coexistence with metastable P6₃/mmc-PrH₉ is unexpected. According to our theoretical calculations, the enthalpy difference between P6₃/mmc-PrH₉ and F$\overline{4}$3m-PrH₉ is about 19 meV per atom, which is near the limit of density functional theory (DFT) accuracy. According to recent studies (35, 36), 20% of experimentally synthesized materials are metastable, some of which even have high positive formation enthalpy.

Properties of F$\overline{\mathbf{4}}$3m-PrH₉ and P6₃/mmc-PrH₉

We performed a series of experiments to investigate superconductivity of PrH₉ via measurements of electrical resistance R(T) in the range of 1.6 to 300 K at pressures from 100 up to 150 GPa (see Fig. 4). The XRD pattern of the prepared sample at 126 GPa, deposited with four electrodes, shows presence of both F$\overline{4}$3m-PrH₉ and P6₃/mmc-PrH₉ phases (Fig. 4C). Possible superconducting transitions were detected with the resistance drop below 9 K, so we proposed that the superconducting transition temperature is below 9 K, far below LaH₁₀ of the same group. The superconducting resistance drop R(T) is also dependent on the applied magnetic field, further proving that this is a superconducting transition (see Fig. 4D). Another run of experiments confirmed the existence of the pronounced superconducting resistance drop in PrH₉ below 9 K (see Fig. 4E and fig. S9). The complexity of the experiments prevented us from accurately determining the pressure dependence of superconducting T_c. We did not observe zero resistance of the superhydrides samples due to their complex geometries, and the samples were mixed phase. The same phenomenon of incompletely dropping to zero in resistance have also been reported in the measurement of superconducting resistance of boron (37) and iron (38) at high pressure.

Fig. 4. Electrical resistance measurements of PrH₉.

(A) The sample inside the diamond anvil cell connected with four electrodes before and after laser heating for sample 1. (B) The photos of sample 2 from different sides of cell after heating. (C) XRD pattern proves that cubic and hexagonal PrH₉ were synthesized in the sample at around 120 GPa from a mixture of Pr and AB. (D) Resistance steps of sample 1 at different magnetic fields. (E) Resistance steps of sample 2 at different pressures.

Further theoretical calculations were aimed at understanding why both F$\overline{4}$3m-PrH₉ and P6₃/mmc-PrH₉ have such low T_c. As shown in Fig. 3 (C and D), both structures have clathrate structures, which are also found in other rare earth hydrides. Calculations of the electron localization function reveal weak covalent H-H interactions. In F$\overline{4}$3m-PrH₉ structure, the nearest H-H distance is 1.135 Å at 120 GPa, which is a bit longer than the known shortest H-H distance in P6₃/mmc-CeH₉ (~1.1 Å) (39, 40) but shorter than in Fm$\overline{3}$m-LaH₁₀ (5). At the same time, P6₃/mmc-PrH₉ with Pr@H₂₉ cages has the nearest H-H distance of ~1.170 Å (at 120 GPa), which is longer than d_min(H-H) in atomic hydrogen and in CeH₉ at the same pressure (for details, see fig. S4) (16).

Calculations demonstrate that both PrH₉ structures are dynamically stable (fig. S6) and exhibit metallic properties (Fig. 5). However, only 6 to 9% of the total densities of electron states (DOS) at the Fermi level comes from the hydrogen atoms, the rest being due to f-electrons of Pr. Relatively high values of the density of states above 3 to 4 eV⁻¹ f.u.⁻¹ (per eV per formula unit) at or near (±1 eV) the Fermi level, caused by a series of Van Hove singularities make it impossible to use constant DOS approximation when calculating parameters of the superconducting state in PrH₉ (41). Low contribution of hydrogen to DOS is associated with weak EPC at 150 GPa, resulting in low superconducting T_c. EPC calculations for both PrH₉ with the selected pseudopotential (PP) give the estimated T_c of 0.8 K for cubic PrH₉ and 8.4 K for hexagonal PrH₉ at 120 GPa with μ* = 0.1, which is in good agreement with experiments (see figs. S10 to S12).

Fig. 5. Electronic properties of PrH₉.

Electron localization function of (A) F$\overline{4}$3m-PrH₉ and (B) P6₃/mmc-PrH₉. Calculated DOS and band structure in (C) F$\overline{4}$3m-PrH₉ and (D) P6₃/mmc-PrH₉ at 150 GPa. DOS (E_F) is mostly due to f-electrons of Pr and has very high value in both cases.

We summarized magnetic properties for all studied praseodymium hydrides at the pressure range of 0 to 150 GPa in Fig. 6. We find that all Pr-H compounds are magnetic: Fm$\overline{3}$m-PrH₃ and Fm$\overline{3}$m-PrH have strong magnetism and retain almost constant magnetic moments at high pressures, while tetragonal PrH₃ and both phases of PrH₉ lose magnetism under pressure. P6₃/mmc-PrH₉ loses magnetism at 120 GPa, while F$\overline{4}$3m-PrH₉ retains a very low magnetic moment. Magnetic order and likely superconductivity coexist in a very close range of pressures in praseodymium hydrides, which may have an effect on the low superconducting transition temperature T_c.

Fig. 6. Magnetism of Pr hydrides at pressures up to 150 GPa.

(A) Magnetic moments of Pr-H compounds at high pressure and (B) magnetic map of Pr-H system as a function of pressure.

CONCLUSIONS

Using in situ decomposition reaction of NH₃BH₃ under HPHT conditions previously used for synthesis of lanthanum superhydrides, we synthesized two novel metallic superhydrides F$\overline{4}$3m-PrH₉ and P6₃/mmc-PrH₉, two trihydrides Fm$\overline{3}$m-PrH₃ and P4/nmm-PrH_3-δ, and one monohydride Fm$\overline{3}$m-PrH_1+x in the pressure range of 0 to 130 GPa. For most synthesized phases, the EoS and unit cell parameters are in good agreement with our DFT calculations. Resistance measurements of praseodymium hydrides indicated possible superconducting transitions in both PrH₉ below 9 K, which is in agreement with theoretical calculations: 8.4 K for hexagonal and 0.8 K for cubic PrH₉ at 120 GPa. Magnetic order and likely superconductivity coexist in a very close range of pressures in praseodymium hydrides, which may have an effect on the low superconducting transition temperature. Present results on Pr superhydrides show that superconductivity declines along the La-Ce-Pr series, while magnetism becomes more and more pronounced. Metallic atoms are not just donors of the electrons to the “metallic hydrogen” sublattice but play a more profound role in determining superconducting T_c.

METHODS

Experimental method

The praseodymium powder samples were purchased from Alfa Aesar with a purity of 99.99%. Molybdenum electrodes were sputtered onto the surface of one diamond anvils in the van der Pauw four-probe geometry. A four-probe measurement scheme was essential to separate the sample signal from the parasitic resistance of the current leads. We prepared an isolated layer from cubic boron nitride (or a mixture of epoxy and CaF₂). We performed laser heating of three diamond anvil cells (100- and 150-μm culets) loaded with metallic Pr sample and ammonia borane in the argon-protected glove box. The diamonds used for electrical DACs had a culet with a diameter of 100 μm. Thickness of the tungsten gasket was 20 ± 2 μm. Heating was carried out by pulses of infrared laser with a wavelength of 1 μm (Nd:YAG), and temperature measurements were carried out by the MAR 345 detector. Pressure was measured by the edge position of diamond Raman signal (42). XRD patterns studied in diamond anvil cells samples were recorded on the BL15U1 synchrotron beamline (43) at Shanghai Synchrotron Research Facility (China) with the use of a focused (5 μm × 12 μm) monochromatic beam. Additional syntheses with electrodes were carried out at the 4W2 High-Pressure Station of Beijing Synchrotron Radiation Facility (China). The beam size was about 32 μm × 12 μm. Both facilities are with the incident x-ray beam (20 keV, 0.6199 Å) and a Mar165 charge-coupled device two-dimensional detector. The experimental XRD images were integrated and analyzed using the Dioptas software package (44). The full profile analysis of the diffraction patterns, as well as the calculation of the unit cell parameters, was performed in the Materials Studio (45) and Jana2006 program (46) by the Le Bail method (47).

Theoretical calculations

We have carried out variable-composition searches for stable compounds in the Pr-H system at pressures of 50, 100, and 150 GPa using the USPEX (23–25) package coupled with the VASP code (27–29)and AIRSS (26) code coupled with the CASTEP plane-wave code (30) and on the fly pseudopotentials (48). The first generation of USPEX search (120 structures) was created using a random symmetric generator, while all subsequent generations (100 structures) contained 20% random structures and 80% created using heredity, soft mutation, and transmutation operators.

We calculated the EoS for PrH, both PrH₃, and two PrH₉ phases. To calculate the EoS, we performed structure relaxations of phases at various pressures using DFT (49, 50) within the generalized gradient approximation (Perdew-Burke-Ernzerhof functional) (51, 52) and the projector augmented-wave method (53, 54) as implemented in the VASP code (27–29). Plane-wave kinetic energy cutoff was set to 1000 eV, and the Brillouin zone was sampled using Γ-centered k-points meshes with a resolution of 2π × 0.05 Å⁻¹. Obtained dependences of the unit cell volume on pressure were fitted by three-order Birch-Murnaghan equation (55) to determine the main parameters of the EoS, namely, V₀, K₀, and K′, where V₀ is equilibrium volume, K₀ is bulk modulus, and K′ is derivative of bulk modulus with respect to pressure using the EosFit7 software (56). We also calculated phonon densities of states for studied materials using finite displacement method [VASP (57) and Phonopy (58)].

Calculations of phonons, EPC, and superconducting T_c were carried out with Quantum ESPRESSO package (59) using density-functional perturbation theory (60), using plane-wave pseudopotential method and local density approximation exchange-correlation functional (61). Norm-conserving pseudopotentials for H (1s¹) and Pr (5s²5p⁶4f³6s²) were used with a kinetic energy cutoff of 90 Rybderg (Ry). In our ab initio calculations of the EPC parameter λ, the first Brillouin zone was sampled using a 6 × 6 × 6 q-points mesh with a denser 24 × 24 × 24 k-points mesh for F$\overline{4}$3m-PrH₉ and a 3 × 3 × 2 q-points mesh with a denser 15 × 15 × 10 k-points mesh for P6₃/mmc-PrH₉ (with Gaussian smearing and σ = 0.035 Ry, which approximates the zero-width limits in the calculation of λ). Critical temperature T_c was calculated from the Allen-Dynes–modified McMillan formula (62)$T_{\mathrm{c}}=\frac{{\mathrm{\omega }}_{\text{log}}}{1.2}\text{exp}[-\frac{1.04(1+\mathrm{\lambda })}{\mathrm{\lambda }-{\mathrm{\mu }}^{*}(1+0.62\mathrm{\lambda })}]$, with ${\mathrm{\omega }}_{\text{log}}=\text{exp}[\frac{2}{\mathrm{\lambda }}\int \text{ln}(\mathrm{\omega })\frac{{\mathrm{\alpha }}^{2}F(\mathrm{\omega })}{\mathrm{\omega }}d\mathrm{\omega }]$ and $\mathrm{\lambda }=2\int \frac{{\mathrm{\alpha }}^{2}F(\mathrm{\omega })}{\mathrm{\omega }}d\mathrm{\omega }$, where μ*, α²F(ω), and λ are Coulomb pseudopotential, the electron-phonon spectral function, and the EPC parameter, respectively.

SUPPLEMENTARY MATERAILS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/9/eaax6849/DC1

Table S1. Crystal structure of predicted Pr-H phases.

Table S2. Experimental parameters of DACs.

Table S3. Experimental cell parameters and volumes of lower praseodymium hydrides along with calculated cell volumes (V_DFT).

Table S4. Experimental cell parameters and volumes of two praseodymium superhydrides along with calculated cell volumes (V_DFT).

Table S5. EoS of metallic Pr from reference.

Table S6. Calculated EoS parameters of third Birch-Murnaghan equation for all studied Pr-H phases.

Fig. S1. Calculated convex hulls for Pr-H system at various pressures.

Fig. S2. Convex hulls without and with zero-point energy (ZPE) correction of found praseodymium hydrides at 120 GPa.

Fig. S3. Experimental XRD patterns dependence of pressure in the range of 0 to 130 GPa.

Fig. S4. Pressure dependence of the nearest H-H distances and nearest Pr-H distances from experimental cell parameters.

Fig. S5. Raman spectra of Z1 cell under decompression.

Fig. S6. Calculated phonon density of states and band structure for PrH₉.

Fig. S7. Calculated phonon density of states and band structure for PrH₈ and PrH₃.

Fig. S8. Electron density of states for PrH₃.

Fig. S9. Enlarged figure of electrical resistance measurements of PrH₉ in sample 2.

Fig. S10. Calculated superconducting parameters of F$\overline{4}$3m-PrH₉ at 120 GPa as a function of electronic smearing σ and the pseudopotential.

Fig. S11. Eliashberg spectral functions, the electron-phonon integral λ(ω), and critical transition temperature T_c(ω) calculated at 120 GPa for cubic PrH₉ with σ = 0.035 Ry.

Fig. S12. Calculated superconductivity of hexagonal PrH₉ by Eliashberg spectral functions at 120 GPa.

References (63–67)