Abstract
Compressed ternary alloy superhydrides are currently considered to be the most promising competitors for high-temperature superconducting materials. Here, the stable stoichiometries in the Y–Hf–H ternary system under pressure are comprehensively explored in theory and four fresh phases are predicted: Pmna-YHfH6 and P4/mmm-YHfH7 at 200 GPa, P4/mmm-YHfH8 at 300 GPa and P-6m2-YHfH18 at 400 GPa. The four Y–Hf–H ternary phases are thermodynamically and dynamically stable at corresponding pressure. In addition, structural features, bonding characteristics, electronic properties, and superconductivity of the four ternary Y–Hf–H phases are systematically calculated and discussed. As the hydrogen content and the density of states of H atoms at the Fermi level increase, the superconducting transition temperatures (Tc) of Y–Hf–H system are significantly enhanced. The P-6m2-YHfH18 with high hydrogen content exhibits a high calculated Tc value of 130 K at 400 GPa.
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Introduction
In recent years, the searches for superconducting materials have been a hotspot in the field of condensed matter physics, and realizing room-temperature superconductivity at atmospheric pressure is a long-held dream pursued by researchers. Hydrogen ranks first in the periodic table and has the smallest atomic mass in nature, thus exhibiting a substantially high Debye temperature1. According to the Bardeen–Cooper–Schrieffer (BCS) theory, metallic hydrogen has been a prospective candidate for a room-temperature superconductor1, which inspired extensive experimental and theoretical studies on the metallization of hydrogen under high pressure2,3,4,5. However, the metallization of hydrogen requires extreme pressure over 500 GPa experimentally. In 2004, Ashcroft innovatively predicted that the lattice of hydrogen can be chemically “precompressed” to achieve metallization at lower pressures by alloying hydrogen with other elements6. This idea greatly reduces the pressure conditions required by room-temperature superconductors, which set off an upsurge in the investigation for room-temperature superconductivity of hydrogen-rich materials.
So far, the theoretical and experimental explorations for binary hydrides of almost all elements in the periodic table (except for some radioactive and magnetic elements) have been basically completed and considerable results have been achieved7,8,9,10,11,12,13,14. Especially, the Y–H and Hf–H binary systems stand out due to their unusual hydrogen configuration structures and significantly high superconducting transition temperatures under pressures. For hafnium hydrogen compounds, the dense superhydrides HfH10 with a planar “pentagraphenelike” sublattice can exhibit a predicted Tc up to 234 K at 250 GPa15. HfH9 with H12 tube and HfH3 framework has a predicted Tc of 110 K at 200 GPa16. A superconducting critical temperature with Tc ~ 83 K was observed in HfH14 at 243 GPa in the experiment17. For binary yttrium hydrides, the calculated Tc of YH6 with H24 cage-like clathrate structure achieves 251–264 K at 120 GPa18 and 290 K at 300 GPa19. Based on the calculation, YH9 with H29-cage structure has a Tc of 276 K at 150 GPa20. YH10 with H32-cage structure possesses an Tc value of 303 K at 400 GPa20 in theoretical aspect. Among them, the high-temperature superconductivity of YH6 and YH9 materials have been confirmed in subsequent high-pressure experiments21,22,23.
The general rules of hydrogen-rich superconductors are summarized based on binary hydrides and applied to ternary hydrides with more abundant hydrogen configurations by the diverse chemical compositions24,25,26,27,28,29. The explorations on ternary hydrogen-rich materials can be mainly divided into two aspects. (i) Add another element to the binary hydride for structural modulation to form a new ternary hydride with better superconductivity. For example, the designed Li2MgH16 with remarkably high predicted Tc of 473 K at 250 GPa via doping lithium into the parent MgH1630. (ii) Mix metal binary hydrides proportionally in the form of solid solutions to obtain alloy-type ternary hydrides, in which the metal elements have similar electronegativity and atomic radius. Theoretical predictions show that ternary hydrides such as (Y, Sr)H1131, YCaH1232, and YZrH1833 through alloying hydrogen-rich binary compounds are estimated to exhibit good superconductivity under high pressures, and the experimentally observed Tc values of (La, Y)H1034, (La, Y)H6, and (Y, Ce)H935 at various pressures are 253 K, 226 K, and 140 K, respectively. Obviously, ternary hydrides have made significant breakthroughs in the discovery of room-temperature superconductors both theoretically and experimentally.
It is widely known that Y and Hf elements are located diagonally in the periodic table with close electronegativity, which indicates prominent high-temperature superconductivity potentially exists in Y–Hf–H system. In our work, we perform an extensive calculation to investigate the ternary Y–Hf–H system under high pressure and analyze their thermodynamical stability and dynamical stability. Indeed, several new stoichiometries in the ternary Y–Hf–H system have been predicted under pressure. The structural features, Bader charges, band structures, density of states (DOS), electron–phonon coupling (EPC) and superconducting transition temperatures of the new ternary Y–Hf–H phases are systematically studied and they are found to possess the unusual structural features and superconductivity.
Results and discussion
Ternary phase diagrams and phase stability
In the first step, in order to determine the stable low-energy compounds of Y–Hf–H ternary system under high pressure, we thoroughly explored the potential stoichiometries of YxHfyHz (x = 1–3, y = 1–3, z = 2–24) at pressures of 200, 300 and 400 GPa, respectively, and constructed the ternary phase diagrams of Y–Hf–H system under selected pressures (see Fig. 1). Its corresponding stable elemental solids (Y, Hf and H) located at three vertexes of triangle and binary hydrides (Y–H and Hf–H systems) on three sides have been systematically investigated experimentally and theoretically. The elemental solids and binary compounds information referenced in the present structural predictions are from previous works15,16,17,18,19,20,21,22,23,36,37,38,39. From Fig. 1a, YHfH6 and YHfH7 are both thermodynamic stable stoichiometries, which are exactly located on the convex hull of 200 GPa. Except for directly compressing Y, Hf, and H, the two red lines exhibit others potential synthetic routes for YHfH6 (YH3 + HfH3 → YHfH6) and YHfH7 (YH4 + HfH3 → YHfH7) based on the “triangle straight-line method” (TSLM)40,41. As the pressure increased, YHfH6 gradually becomes thermodynamic unstable while YHfH7 can maintain energy stability up to 300 GPa, and a new stable component YHfH8 has emerged on the convex hull. Similarly, YHfH8 can be synthesized through the route YH4 + HfH4 → YHfH8 shown in the red line of Fig. 1b. Upon further compression, YHfH18 compound with the highest hydrogen content, as the only composition falling into the convex hull, can be stable energetically superior to other stoichiometries at 400 GPa (see Fig. 1c). The above analysis demonstrates that YHfH6, YHfH7, YHfH8 and YHfH18 are thermodynamically stable under high pressures. We also provide their potential synthetic routes, which will stimulate the experimental synthesis. Subsequently, we focus on the structural characteristics, electronic properties, and superconductivity of YHfH6 and YHfH7 at 200 GPa, YHfH8 at 300 GPa, and YHfH18 at 400 GPa, respectively.
Convex hulls of Y–Hf–H system at (a) 200 GPa, (b) 300 GPa, and (c) 400 GPa. Red and blue dots represent thermodynamically stable and unstable structures, respectively.
Structural features, bonding features, and electronic properties
The crystal structure diagrams of YHfH6 and YHfH7 at 200 GPa, YHfH8 at 300 GPa and YHfH18 at 400 GPa are depicted in Fig. 2 and homologous detailed structural parameters are summarized in Table S1 of Supplementary Information (SI). In addition, to better understand the interaction information, we also depict their electronic localization functions (ELF) in Fig. 3. The predicted YHfH6 is found to crystallize in the orthorhombic Pmna phase with the lattice parameters of a = 10.170 Å, b = 8.108 Å, and c = 8.209 Å at 200 GPa. This phase is observed in ternary hydrides for the first time. There are three inequivalent H atoms are located at the Wyckoff sites 4h (0.000, 0.360, 1.386), 4h (0.000, 0.124, 1.123) and 4g (0.250, 0.239, 0.750), respectively, and these H atoms form one-dimensional linear H4 chains with two types H–H separations of 1.626 and 1.669 Å, as shown in Fig. 2a. Its weak covalence is accurately described by the ELFs in Fig. 3a,b, while the absence of electron localization between metal atoms and H atom reveals the purely ionic of neighboring Hf–H and Y–H connections. For YHfH7 at 200 GPa and YHfH8 at 300 GPa, both are stable with tetragonal P4/mmm phase, which adopts the same symmetry as previously reported YZrH833, YPrH842, YBaH843 and YMgH844 hydrides. In this structure, Hf atoms occupy the center position of the tetragonal lattice while Y atoms are located at eight vertices, and all H atoms form H-clathrate structures around metal Hf atoms. The Hf-centered H14 cage in P4/mmm-YHfH7 contains eight twisted quadrilateral and four square “H4” units with only two types H–H distances of 1.51 and 1.78 Å (see Fig. 2b). The hydrogen sublattice of P4/mmm-YHfH8 is very similar to YHfH7, which can be considered to insert an H atom into each long edges of YHfH7 lattice, forming an H18 cage with three types H–H separations of 1.29, 1.40 and 1.42 Å (see Fig. 2c). It is noteworthy that this H atoms insertion facilitates the four squares “H4” of P4/mmm-YHfH7 to evolve into four hexagon “H6” units in P4/mmm-YHfH8. The shortest H–H distance 1.29 Å in “H6” units are greatly shortened compared to that in “H4” (1.78 Å), which makes it closer to the H–H distance (0.98 Å) in metallic monatomic H at 500 GPa45. The decrease in distance between H atoms enhances their mutual interactions, which not only promotes the stability of the hydrogen cage skeleton, but also plays a decisive role in the superconductivity of the system. From Fig. 3c–f, the H atoms are weakly covalently bonded to one another in the cages, which are evident in view of the charge localizations between the nearest-neighbor H atoms. In the case of YHfH18 at 400 GPa, it is stable in hexagonal structure with P-6m2 symmetry, which has the lattice parameters of a = b = 3.176 Å and c = 4.770 Å (see Fig. 2d). It is found that all H atoms form semi-cages with the H–H connections around 1.03–1.46 Å in this phase. The weak covalence between H–H connection is also demonstrated by the ELF diagrams plotted in Fig. 3h,i, as the ELF values between metal atoms and the nearest H atom are smaller than 0.5. In addition, the shortest H–H bond length in P-6m2-YHfH18 phase is closest to the atomic H under 500 GPa45 and famous ultra-high Tc hydride Li2MgH16 (1.02 Å, at 300 GPa)30, HfH10 (1.14 Å, at 200 GPa)15, YH10 (1.08 Å, at 200 GPa)20 and LaH10 (1.11 Å, at 200 GPa)46, etc.
Crystal structures of (a) Pmna-YHfH6 at 200 GPa, (b) P4/mmm-YHfH7 at 200 GPa, (c) P4/mmm-YHfH8 at 300 GPa and (d) P-6m2-YHfH18 at 400 GPa. The blue, green and pink spheres represent Hf, Y and H atoms, respectively.
Electronic localization functions (ELF) of (a,b) Pmna-YHfH6 at 200 GPa, (c,d) P4/mmm-YHfH7 at 200 GPa, (e,f) P4/mmm-YHfH8 at 300 GPa and (h,i) P-6m2-YHfH18 at 400 GPa. The blue, green and pink spheres represent Hf, Y and H atoms, respectively.
The Bader charge analysis47 was conducted to further gain insight into the chemical bonding features between Y, Hf and H atoms in Pmna-YHfH6, P4/mmm-YHfH7, P4/mmm-YHfH8 and P-6m2-YHfH18, as tabulated in Table 1. As the electronegativity of Y (1.2) and Hf (1.3) are much smaller than H (2.2), a large amount of charge should transfer from Y and Hf atoms to H atoms based on the periodic law48. The calculated results show that each Hf atom approximately loses 1.21, 1.25, 1.26 and 1.26 e electrons in YHfH6, YHfH7, YHfH8 and YHfH18, respectively, and one Y atom loses 1.23, 1.32, 1.27 and 1.32 e charges, respectively. For H atoms, per H gains charges of 0.41, 0.37, 0.32 and 0.14 e, respectively. We intuitively observe that the number of electrons accepted by H atoms gradually decreases in these four structures with the rise of pressure. The obtained electrons for H–H unit will reside in the H–H antibonding orbitals, which can elongate the H–H bond length in the high-pressure hydrides49,50. Consequently, the H–H bonds in four phases present a gradually decreasing trend as well. The discussion of Bader charge is consistent with the above ELF analysis, further confirming the strong ionic natures between metal atoms and H atoms in the four stable configurations.
The electronic band structures and DOS of Pmna-YHfH6, P4/mmm-YHfH7, P4/mmm-YHfH8 and P-6m2-YHfH18 at high pressure were calculated to reveal their electronic properties, as exhibited in Fig. 4. As a result, these four structures possess metallic properties due to the overlapping of the conduction and valence bands, and a large total density of states (TDOS) at the Fermi level also confirms this conclusion. Furthermore, it is found that the electronic density of states at Fermi level is obviously dominated by metal Y and Hf atoms in YHfHn (n = 6, 7 and 8) phases, so the metallic properties of the above three phases are contributed by metal atoms. Due to their less hydrogen content, the H atoms provide low total H-DOS with 0.21, 0.11 and 0.16 (states/eV)/f.u., respectively, and H-DOS merely account for 11%, 15% and 23% of the total density of states, respectively. Interestingly, the electronic band structures of YHfH7 and YHfH8 are quite similar, which is attributed to their similar crystal structures. As for P-6m2-YHfH18 phase with the highest H content at 400 GPa, more electrons participate in the formation of Cooper pairs and there is a steep large electron pocket passing through the Fermi level at Γ point, which increases the TDOS at Fermi level. In this structure, the total H-DOS at Fermi level significantly enhances to 0.67 (states/eV)/f.u., which is comparable to HfH916, YH618, YCa2H1851, YCaH2051, CaHfH1852, YLuH1253 and Rb2MgH1854, and the proportion of H-DOS in TDOS increases to 47% as well. In fact, the H-derived DOS at Fermi level is crucial for considerably improving the superconductivity of P-6m2-YHfH18 in Y–Hf–H ternary system and this will be demonstrated in the subsequent discussions.
Electronic band structures and density of states (DOS) of (a) Pmna-YHfH6 at 200 GPa, (b) P4/mmm-YHfH7 at 200 GPa, (c) P4/mmm-YHfH8 at 300 GPa and (d) P-6m2-YHfH18 at 400 GPa.
Dynamical properties and electron–phonon coupling
The computed phonon dispersion curves, projected phonon density of states (PHDOS), Eliashberg spectral function α2F(ω) and the integral EPC parameters λ(ω) of YHfH6, YHfH7, YHfH8 and YHfH18 are plotted in Fig. 5. The phonon dispersion curves can effectively confirm their dynamic stability because of the absence of any imaginary phonon frequencies throughout the whole Brillouin zone. In the PHDOS, the low-frequency phonon bands are mainly originated from the vibrations of Hf and Y atoms, while the middle-frequency and high-frequency phonon modes regions are associated with H atoms due to the Hf and Y atomic masses are much heavier than H atoms.
The phonon dispersion curves, the projected phonon density of states and Eliashberg spectral function α2F(ω) together with the electron–phonon integral λ(ω) of (a) Pmna-YHfH6 at 200 GPa, (b) P4/mmm-YHfH7 at 200 GPa, (c) P4/mmm-YHfH8 at 300 GPa and (d) P-6m2-YHfH18 at 400 GPa.
Based on the discussion of electronic properties of four stable phases, their excellent metallic properties under pressures prompt us to explore their potential superconductivity. Therefore, the electron–phonon coupling parameters λ, the logarithmic average phonon frequency ωlog and electron density of states at Fermi level N(Ef) were calculated in Table 2. Moreover, based on the Eliashberg spectral function α2F(ω) and the integral EPC parameters λ(ω) of Pmna-YHfH6 at 200 GPa, P4/mmm-YHfH7 at 200 GPa, P4/mmm-YHfH8 at 300 GPa and P-6m2-YHfH18 at 400 GPa in Fig. 5, the superconducting critical temperature values are evaluated through solving the Allen–Dynes modified McMillan equation55:
For the hydrides with λ < 1.5, this equation can estimate highly accurate Tc values. Among them, the Coulomb pseudopotential constants μ* are set to typical 0.10 and 0.13, respectively. It is found that the Tc values slightly decrease when μ* = 0.13.
In YHfH6 and YHfH7 phases at 200 GPa, their calculated EPC integral coefficients λ are close (~ 0.38), which results in their similar predicted Tc values around 3 K (see Table 2). Although YHfH6 has a large electronic density of states at the Fermi level [N(Ef) = 11.983 states/spin/Ry/Unit Cell], even the largest among the four stable structures, this large N(Ef) value is mainly provided by metal Hf and Y atoms, which suppresses the superconductivity of YHfH6. From Fig. 5, the low-frequency heavy Hf and Y atomic vibrations below 6 THz contribute 51% of the total λ, while the high-frequency region above 20 THz associated with light H atom vibration accounts for 49% in YHfH6. As for YHfH7 structure, its phonon modes can be divided into three main regions. The low frequency phonon modes below 10 THz provided by metal atoms account for 39%, while H atoms drive 10% and 51% in the middle (23–27 THz) and upper (> 40 THz) parts, respectively. In this situation, the contribution of the H-phonon band begins to increase. For YHfH8 at 300 GPa, the dominance of H atom in electron–phonon coupling is further clarified and the contribution of the H-phonon band enhances to 68%. In addition, the significant difference in superconductivity between P4/mmm-YHfH7 and P4/mmm-YHfH8 with highly similar structures can be further attributed to the fact that the H18 cage of YHfH8 bonds more tightly than the H14 cage in YHfH7. Based on these reasons, the Tc.
value of P4/mmm-YHfH8 has been effectively enhanced to 47 K. Notably, in YHfH18 phases at 400 GPa, metal atoms motions in the lowest part of the phonon bands (< 10 THz) only occupy 18% of the total λ, while all other vibrational modes up to 82% (> 25 THz) are dominated by H atoms. Obviously, YHfH18 possess the highest predicted Tc value 130 K among the four stable structures, which is close to the theoretical predictions of other H-rich materials such as YZrH18 (156 K)33 and LiP2H14 (169 K)56, but higher than the predicted results of MgH12 (60 K)57, YCH12 (112 K)58, YSH6 (91 K)59, Ca2B2H13 (89 K)60 and Mg(ScH4)3 (10 K)61. This result highlights that the high-frequency H atom vibration exhibits a significant contribution to the electron–phonon coupling in YHfH18 structure, which is coincide with the analysis result of large H-DOS proportion at the Fermi level. Thus, we have certificated that H-derived DOS at the Fermi level can effectively improve the Tc value of Y–Hf–H ternary hydrides under high pressure.
Conclusions
In conclusion, we conducted comprehensive structural searches of the Y–Hf–H ternary system under high pressure by applying unbiased CALYPSO method and first-principles calculations to reveal the phase stability. As a result, we have predicted four thermodynamically stable ternary hydrogen-rich compounds at selected pressures, namely Pmna-YHfH6, P4/mmm-YHfH7, P4/mmm-YHfH8 and P-6m2-YHfH18. YHfH6 contains linear H4 chains, while YHfH7 and YHfH8 adopt clathrate structures of H14 and H18 cages, respectively. Additionally, YHfH18 forms intriguing H semi-cages lattices. In these four stable structures, ionic bonds are formed between the metal and H atoms, and the two nearest H atoms are connected by weak covalent bonds. According to their metallic properties, the calculated superconducting transition temperatures of YHfH8 and YHfH18 are 47 K at 300 GPa and 130 K at 400 GPa, respectively. We believe that the density of states at the Fermi level dominated by H atoms are closely associated with the high-temperature superconductivity of the Y–Hf–H system.
Computational methods
The extensive variable-composition structure searches of Y–Hf–H ternary system under high pressures were implemented via swarm intelligence-based CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) code62,63 including 1–2 formula units (f.u.) at 200, 300 and 400 GPa. A similar method has successfully predicted the high-pressure structures of various systems64,65,66,67,68,69 and provides effective guidance for experimental synthesis. We have performed the structural relaxations, energetic and electronic structures in the VASP (Vienna ab initio simulation package) code70 through density functional theory (DFT) within the Perdew–Burke–Ernzerhof (PBE) exchange correlation function of generalized gradient approximation (GGA)71,72. The Projector-Augmented Wave (PAW)73 method with valence electron configurations 4s24p64d15s2 for Y, 6s25d2 for Hf and 1s1 for H was adopted with 800 eV cutoff energy for the plane-wave basis expansion of the electronic wave functions and appropriate Monkhorst–Pack scheme with a k-point grid of 2π × 0.03 Å−1 for sampling in the Brillouin zone to ensure that all the enthalpy calculations were well converged to less than 1 meV/atom74. The descriptions of electronic properties and bonding characteristics have been conducted through Bader charge analysis47 and ELF method75,76. Additionally, the phonon and electron–phonon coupling matrix elements were calculated within density functional perturbation theory (DFPT) as implemented in the QUANTUM ESPRESSO package77 with a kinetic cutoff energy of 80 Ry. In the first Brillouin zone, we have selected k-point meshes of 12 × 12 × 12, 20 × 20 × 12, 16 × 16 × 16 and 20 × 20 × 16 for Pmna-YHfH6, P4/mmm-YHfH7, P4/mmm-YHfH8 and P-6m2-YHfH18, respectively, and q-point meshes of 3 × 3 × 3, 5 × 5 × 3, 4 × 4 × 4 and 5 × 5 × 4 for Pmna-YHfH6, P4/mmm-YHfH7, P4/mmm-YHfH8 and P-6m2-YHfH18, respectively.
Data availability
The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11804031, 11904297 and 11964001), the Fundamental Research Funds for the Central Universities (No. SWU-KT22049) and the Chongqing Talent Plan for Young Top Notch Talents (202005007).
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Y.Q.W., Y.Y.J. and C.Z.Z. conceived the idea. Y.Q.W., F.L.Y., J.Q.Z., S.L. and S.B.C. performed the calculations. Y.Q.W., Y.Y.J. and C.Z.Z. wrote the manuscript and all authors contributed to revisions.
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Wang, Y., Jin, Y., Yang, F. et al. Prediction of potential high-temperature superconductivity in ternary Y–Hf–H compounds under high pressure. Sci Rep 14, 17670 (2024). https://doi.org/10.1038/s41598-024-68697-w
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DOI: https://doi.org/10.1038/s41598-024-68697-w
