High T_c superconductivity in layered hydrides XH₁₅ (X = Ca, Sr, Y, La) under high pressures

Abstract

The theoretical predictions and experimental synthesis of H₃S and LaH₁₀ superconductors with record high superconducting transition temperatures (T_c) have promoted the hydrogen-based superconducors to be a research hotspot in the field of solid-state physics. Here, we predict an unprecedented layered structure CaH₁₅, with high T_c of 189 K at 200 GPa using ab initio calculations. As concerns the novel structure, one layer is made of a hydrogen nonagon, the other layer consists of a Ca atom and six H₂ molecular units surrounding the Ca atom. This layered structure was also found in SrH₁₅, YH₁₅, and LaH₁₅ at high pressures, each materials exhibit high T_c especially YH₁₅ can reach above 200 K at 220 GPa. It represents the second class of layered superhydrides with high value of T_c after pentagraphene like HfH₁₀.

Graphical abstract

Keywords

high pressure / hydrides / superconductivity / ab initio calculation

1 Introduction

Pressure is one of the important thermodynamic parameters that modulate the structures and properties of materials. Theory predicts that insulator molecular phase will become metallic atomic phase under enough pressures, including H₂ [1], N₂ [2], O₂ [3, 4], and F₂ [5]. Based on Bardeen−Cooper−Schrieffer (BCS) theory, metallic hydrogen with the lightest quality is predicted to be a potential room temperature superconductor [6-8]. However, utill now, there is no direct evidence in experiments to realize the metallic hydrogen at high pressures [9]. It is generally believed that at least 500 GPa may be required to achieve metallic hydrogen [10]. In order to reduce the metallization pressure of hydrogen, Ashcroft proposed the “chemical precompression” that in hydrogen-rich compounds, heavy atoms can interact with hydrogen providing chemical pressure [11]. In addition, hydrogen can provide high phonon frequencies and strong electron−phonon coupling in hydrogen-rich compounds, which make them possible to become high temperature superconductors. In the past decades, extensive researches have been conducted in field of hydrogen-based superconductors under high pressures both on theoretical prediction and experimental synthesis [12-18].

Two of the best known theories predicted hydrogen-based superconductors are H₃S [19, 20] and LaH₁₀ [21, 22] which have been successfully synthesized [23-27]. H₃S has a three-dimensional S−H covalent bonds network, in which the S atoms form a bcc lattice and each S atom contacts with six H atoms, and its superconducting transition temperature (T_c) reaches 203 K at 155 GPa. LaH₁₀ represents a class of three-dimensional H clathrate-like structures [21-29], in which metal atoms fill the clathrate cavities, providing electrons to stabilize the hydrogen cage and its T_c reaches 250−260 K at 170 GPa. Recently, Xieet al. [30] predicted a hexagonal HfH₁₀ in which the H atoms arranged in clusters to form a planar “pentagraphenelike” sublattice. It is a new high temperature superconducting hydride besides covalent H₃S and cagelike LaH₁₀, with high T_c of 234 K at 250 GPa.

It is reported that there are a variety of hydrogen structural motifs in alkaline earth metal hydrides ranging from monatomic H, linear and bent H₃ molecular units to spiral polymer chains, even H₂₄ and H₂₉ cages [31-38]. The C2/m-Ca₂H₅ [31] and I4/mmm-CaH₄ [32] with H atoms and H₂ molecular units have been synthesized at 22 GPa and 116 GPa, respectively. The well known predicted Im\overline{3}m-CaH₆ with H₂₄ cage [33] have been confirmed by experiments recently [34] with a maximum T_c of 215 K at 172 GPa. The C2/m-CaH₉ [35] with a distorted H₂₉ cage was predicted to have a high T_c of 266 K at 300 GPa. The CaH₁₂ with the highest H content in Ca−H system possesses three phases of R\overline{3}, C2/m and C2/c above 50 GPa [33, 35, 36], containing H₂ molecular units which are disadvantageous for high temperature superconductivity. For the Sr−H system, SrH₆ contain linear and bent H₃ molecular units transform into spiral polymer H chains at 250 GPa, and SrH₁₀ posseses graphene like H-layer at high pressures [37]. Recently, experiment reported semimetallic BaH₁₂ [38] contains H₂, H₃ molecular units and detached H₁₂ chains with T_c of 20 K at 140 GPa. The diversity of hydrogen motifs in alkaline earth metal hydrides under high pressure led us to explore more hydrides with complex hydrogen motifs and high T_c.

Here, we performed extensive crystal structure searches on the CaH_n (n = 10−20) with high hydrogen content under high pressures. We found an interesting CaH₁₅ that adopts two stacked layers, the top layer is a nonagon made of hydrogen atoms, the bottom layer consists of a Ca atom arrayed in six H₂ molecular units. The estimated T_c is about 171−189 K at 200 GPa for CaH₁₅. A new class of superhydrides, XH₁₅ (X = Sr, Y, and La), which are isostructural to CaH₁₅, have also been predicted to be high temperature superconductors, especially, YH₁₅ has high T_c of 192−208 K at 220 GPa.

2 Computational details

We used an ab initio random structure searching (AIRSS) approach [39] and the density functional theory (DFT) method as implemetend in the cambridge serial total energy package (CASTEP) [40] to predict the stable or mestable structures in CaH_n (n = 10−20), and XH₁₅ (X = Sr, Y, La) between 100 and 300 GPa. In the process of structure searching, a plane-wave cut-off energy of 350 eV and a Brillouin zone sampling grid spacing of 2π × 0.07 Å⁻¹ were selected.

Structural relaxations, electronic properties, and total energies were determined in the framework of DFT with Perdew−Burke−Ernzerhof parametrization of the generalized gradient approximation as implemented in the viennaab initio simulation package (VASP) [41]. The valence electrons 3s²3p⁶4s² for Ca, 4s²4p⁶5s² for Sr, 4s²4p⁶4d¹5s² for Y, 5s²5p⁶5d¹6s² for La and 1s¹ for H were employed with the energy cutoff of 1000 eV. The Brillouin zone was sampled with a k-point mesh of 2π × 0.03 Å⁻¹ to make the enthalpy calculations well converged to less than 1 meV/atom.

Phonon dispersion and electron−phonon coupling (EPC) were carried out using density functional perturbation theory as implemented in the Quantum-ESPRESSO [42] package. Ultrasoft pseudopotentials were used with a kinetic energy cutoff of 80 Ry. The k-point and q-point meshes in the first Brillouin zone are 20×20×28 and 5×5×7 for XH₁₅ (X = Ca, Sr, Y, La). The superconducting transition temperatures of these structures are estimated through the self-consistent solution of the Eliashberg equation [43, 44].

3 Results and discussion

Since hydrides with high hydrogen content are favorable for high-temperature superconductivity, we began with random structure searching for Ca−H system from 100 GPa to 300 GPa mainly focused on the stoichiometries CaH_n with n = 10−20. We compared the formation enthalpy per atom of the stable phases with CaH₂ and H₂ at selected pressures, as shown in Fig.1. The stable structures of element H₂ [45] and binary hydrides CaH_n (n = 2, 4, 6, 9) come from the previous studies [33, 35]. The convex hull reflects the stability of each stoichiometries. The compounds located on the hull are considered to be thermodynamically stable shown in solid dots, while the compounds above the convex hull are metastable shown in hollow dots. It can be found that CaH₄ and CaH₉ are thermodynamically stable in the pressure range of 100−300 GPa, CaH₆ falls on the hull above 140 GPa, which are all in line with the previous results [33, 35]. With regard to CaH₁₂, we reproduced three phases of R\overline{3}, C2/m and C2/c under different pressures and calculated their enthalpies as a function of pressure, as shown in the Fig. S1 of supplemental materials (SM). It is found that the R\overline{3} phase transforms to the C2/m phase at 90 GPa, then to the C2/c phase above 180 GPa. The CaH₁₂ lies on the convex hull at 100−200 GPa but leaves the convex hull at 300 GPa indicating it is thermodynamically stable below 300 GPa. In addition, we found a new hydride CaH₁₄ with R\overline{3} symmetry which lies on the convex hull at 140 GPa. This structure contains H₂ molecular units as shown in Fig. S2 of SM, and the lattice parameters are listed in Table S1 of SM. We will not discuss the properties of CaH₁₂ and CaH₁₄ in detail because they would not have good superconductivity based on the previous summarized four criteria for high T_c hydrides [30]. It is noteworthy that an unique structure P\overline{6}2m-CaH₁₅ falls on the convex hull at pressures above 140 GPa indicating that the structure is thermodynamically stable. In addition, there is no phase transitions throughout the pressure range we studied, and the structural parameters are listed in Table S2 of SM.

We study the geometry of \mathrm{P}\overline{6}2m-CaH₁₅ in more detail. This layered structure are stacked in ABAB fashion in which the A layer consists of a nonagon made of hydrogen atoms, the B layer contains six H₂ molecules ring and a Ca atom located in the centre of the ring, as shown in Fig.2(a) and (b). The calculated electronic localization function (ELF) maps and Bader charges show its chemical bonding more clear. For the A layer, nonagons [Fig.2(c)] are actually surrounded by three curved H₃ units. The ELF values surrounding H₃ units equal 0.8 indicating that there are covalent bonding among these H atoms. High pressure strengthens the interaction between the H₃ units, and makes three H₃ units concatenate into a nine-membered ring. And it can be seen from Fig.2(d), in B layer, the ELF value surround the neighboring H₂ molecular units is 0.5, indicating the electrons can flow in the intermolecular region of H₂. In addition, the ELF toward the neighboring H−Ca connections show none electron localization, suggesting Ca−H bonding is mainly ionic. Bader charge analysis shows that charge transfer is about 1 |e| from Ca to H proving its ionic character, as listed in Table S2 of SM.

We investigate the H−H bonding character in \mathrm{P}\overline{6}2m-CaH₁₅ by calculating the crystal orbital Hamiltonian population (COHP), which counts the population of wavefunctions on two atomic orbitals of a pair of selected atoms. Fig.3(a) and (b) show the top view of the hydrogen nonagon layer and H₂ molecular units layer, in which the intramolecular/intermolecular H−H distances represent by d1/d2 for hydrogen nonagon ring and d3/d4 for H₂ molecualr ring. The H−H distances are 0.877 Å (d1) and 1.051 Å (d2) for hydrogen nonagon ring, 0.832 Å (d3) and 1.14 Å (d4) for H₂ molecualr ring at 300 GPa. The COHP of \mathrm{P}\overline{6}2m-CaH₁₅ is shown in Fig.3(c), the left sides with negative values represent bonding states, the right sides are antibonding states. The integrated COHP (ICOHP) up to the Fermi level represent the bonding interactions between H atoms, larger negative values represent stronger H−H bonds. The ICOHP values of those short H−H distances like 0.877, 1.051, and 0.832 Å are −3.64, −2.21, and −4.21 eV, respectively, indicating strong bonding interactions among these H atoms. These intramolecular H−H distances are longer than that of isolated H₂ molecules (0.74 Å) which is attributed to the extra electrons occupying the antibonding orbitals of hydrogen molecules, weakening the H−H bonding and increasing the bond length.

The elements Ca and Sr have similar properties: the Pauling electronegativity is 1 for Ca and 0.95 for Sr, atomic radius is 1.97 Å for Ca and 2.15 Å for Sr, and they have the same valence electron configuration. We further searched the structures of SrH₁₅ between 100 and 300 GPa, and as expected we obtained the same P\overline{6}2m structure as CaH₁₅ which settles on the convex hull above 110 GPa, as shown in Fig. S3 of SM. In addition, the elements Y and La are diagonally adjacent with Ca and Sr in the periodic table, respectively, which may have similar structures and properties. We performed the random structure searching for YH₁₅ and LaH₁₅ at high pressures and found the same structure with CaH₁₅. Both YH₁₅ and LaH₁₅ are metastable which are close to the convex hull (20−33 meV/atom above tie line) at 300 GPa, as shown in Fig. S4 of SM. We note that the P\overline{6}2m structure was first predicted in ErH₁₅ by Dmitrii et al. [36], but it was not discussed in depth. The main focus of this work being on distribution of superconductivity in metal hydrides.

To further explore the electronic properties of XH₁₅ (X = Ca, Sr, Y, La), we calculated the electronic band structures (Fig. S5 of SM) and partial electronic density of states (Fig.4) of P\overline{6}2m-XH₁₅ at 300 GPa. These four compounds have similar band structures, and they are good metal with highly dispersive bands crossing the Fermi level, especially, two bands along Γ−A−H direction are very steep (Fig. S5 of SM). From the electronic density of states inFig.4, we can see that the electronic states at the Fermi level are mainly contributed by H s-orbitals, which is favorable to electron phonon coupling and superconductivity. Noteworthy, the H p-orbitals also play an important role for the density of electronic states at Fermi surface. We consider that H₃ molecular units interact with each other to form a nonagon and the interaction between H₃ units is enhanced under higher pressure which requires the participation of H p-orbitals [37]. The total DOS of XH₁₅ (X = Ca, Sr, Y, La) at the Fermi level are calculated to be 0.58, 0.51, 0.85 and 0.43 states/(eV·f.u.) at 300 GPa, respectively. It is worth noting that for YH₁₅, besides H s-orbitals, d-orbitals also contribute a lot to the electronic density of states at the Fermi level. So YH₁₅ has the largest total DOS at the Fermi level among these four compounds.

The calculated phonon band structures, partial phonon density of states (PHDOS), eliashberg spectral functions {\alpha }^{2}F\left(\omega \right), and integral EPC parameters λ of XH₁₅ (X = Ca, Sr, Y, La) at 300 GPa are shown in Fig.5. There are absent of imaginary frequency in the entire Brillouin zone for all structures, which demonstrates their dynamical stability. Due to the discrepancy of atomic mass, these PHDOS divide the contribution of λ into three parts: the low-frequency vibrational modes below 500 cm⁻¹ mainly correspond to the vibrations of metal atoms X (X = Ca, Sr, Y, La) with a contribution below 30% of the total λ; the middle parts range from 500 cm⁻¹ to 2200 cm⁻¹ mostly come from the interaction of metal atoms and H₂ or H₃ units with a contribution about 60%−70% of the totalλ; the high frequency above 2200 cm⁻¹ are primarily dominated by the modes corresponding to the symmetric H₂ and H₃ stretch vibration contributed about 10%−30% to the totalλ. Hydorgen atoms play a dominating role in eletron−phonon coupling, for CaH₁₅ and SrH₁₅, there are distinct anomalies (dips) in the phonon dispersion curves mainly concentrated in the middle parts of vibration modes, especially the fifth and sixth vibrational modes at the A point. For YH₁₅, there is a strong dip in phonon branch 5 at the H point. For LaH₁₅, there is no distinct phonon dip (phonon softening) along the high symmetry line. In addition, we projected the EPC λ on phonon dispersion curves and the sizes of circles are proportional to the strength of λ, as shown in Fig.5. It is clearly seen that the λ is particularly enhanced in these phonon dip regions of CaH₁₅, SrH₁₅, and YH₁₅ which is favorable to produce a high superconducting transition temperature.

We show some crucial parameters involved with the superconductivity of XH₁₅ (X = Ca, Sr, Y, La) compounds at different pressures including the EPC parameter λ, the logarithmic average phonon frequency ω_log, and the superconducting transition temperature T_c (Fig.6). It can be seen the total λ of CaH₁₅ and SrH₁₅ at 300 GPa are calculated as being 1.26 and 1.10, YH₁₅ has the largest λ of 1.42, while LaH₁₅ has the smallest λ less than 1. The YH₁₅ has the smallest ω_log of 1469 K, while its λ is the highest among these four compounds, which due to the large DOS at the Fermi level and the strong phonon dips at H poinit, as shown in Fig.4 and Fig.5. As the case of LaH₁₅, it renders to produce a relative high ω_log but small λ, due to the hard phonon modes and the small DOS at the Fermi level. The Coulomb pseudopotential μ* is often taking from 0.1 to 0.13 for hydrogen-dominant metallic alloys. Using μ* values of 0.1 and 0.13, the ranges of T_c values for the compouds CaH₁₅, SrH₁₅, YH₁₅, LaH₁₅ are estimated to be 168−187 K, 122−139 K, 177−196 K and 85−101 K at 300 GPa, respectively, via the numerical solution of Eliashberg equation. It can be seen thatT_c decreases in the order of YH₁₅ > CaH ₁₅ > SrH ₁₅ >LaH ₁₅, consistent with the decreasing trend of λ. Thus, the high T_c in these hydrides is mainly attributed to the EPC interaction. With decreasing pressures, the electron−phonon coupling parameterλ of XH₁₅ (X = Ca, Sr, Y, La) raise, while logarithmic average phonon frequency ω_log decrease. But superconducting transition temperature T_cs of these four compounds are somewhat different. In case of CaH₁₅, the λ increases to 1.51, the ω_log decreases to 1338 K, and the T_c reaches maximum of 189 K with μ* = 0.1 (177 K with μ* = 0.13) at 200 GPa which is only 2 K higher than T_c at 300 GPa. The estimated T_c of YH₁₅ increases with decreasing pressure at the rate of −0.15 K/GPa (d T_c/dP) and reaches maximum of 208 K with μ* = 0.1 (192 K with μ* = 0.13) at 220 GPa, exceeding the T_c (203 K) of H₃S. The pressure dependence T_c of LaH₁₅ is −0.12 K/GPa which is similar with that of YH ₁₅, and the maximum T_c reaches 113 K with μ* = 0.1 at 200 GPa (100 K with μ* = 0.13). On the contrary, the T_c of SrH₁₅ decreases with decreasing pressure and reduces to 125 K with μ* = 0.1 at 200 GPa (111 K with μ* = 0.13). Although the T_cs of XH₁₅ (X = Ca, Sr, Y, La) dependence on pressure are different, they do not change much. This effect is the consequence of a complicated balance between the λ and ω_log at different pressures.

4 Conclusions

We have used ab initio random structure searching approach to explore the crystal structures of CaH_n (n = 10−20) from 100 to 300 GPa and found a unique layer structure with hydrogen nonagon, P\overline{6}2m-CaH₁₅, which was calculated to be thermodynamical stable above 140 GPa. The same hydrogen nonagon structure was also found in SrH₁₅, YH₁₅ and LaH₁₅ under high pressures. The SrH₁₅ is thermodynamical stable above 110 GPa, while YH₁₅ and LaH₁₅ are metastable in our studied pressure range. Electron−phonon coupling calculations show that the XH₁₅ (X = Ca, Sr, Y, La) are potentially high temperature superconductors with estimated maximum T_c of 189 K at 200 GPa, 139 K at 300 GPa, 208 K at 220 GPa and 113 K at 200 GPa, respectively. The YH₁₅ has the highest T_c among these four compounds due to the largest λ which is attributed to the strong phonon dips at H point and large DOS at the Fermi level. Further calculations show that the superconducting critical temperature of XH₁₅ (X = Ca, Sr, Y, La) has only a weak dependence on pressure due to the banlance of the λ and ω_log at different pressures. Our work will stimulate the future discovery of more layered class of high temperature superconductors in compressed hydrides.

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