Prediction of high-Tc superconductivity in ternary lanthanum borohydrides

PHYSICAL REVIEW B 104, 134501 (2021) Prediction of high-Tcsuperconductivity in ternary lanthanum borohydrides Xiaowei Liang,1Aitor Bergara,2,3,4Xudong Wei,1Xiaoxu Song,1Linyan Wang,1Rongxin Sun,1Hanyu Liu,5,6,* Russell J. Hemley,7Lin Wang,1Guoying Gao ,1,†and Yongjun Tian1 1Center for High Pressure Science, State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China 2Departmento de Física de la Materia Condensada, Universidad del País Vasco, UPV/EHU, 48080 Bilbao, Spain 3Donostia International Physics Center (DIPC), 20018 Donostia, Spain 4Centro de Física de Materiales CFM, Centro Mixto CSIC-UPV/EHU, 20018 Donostia, Spain 5International Center for Computational Method & Software and State Key Laboratory of Superhard Materials, College of Physics, Jilin University, Changchun 130012, China 6Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), International Center of Future Science, Jilin University, Changchun 130012, China 7Departments of Physics and Chemistry, University of Illinois Chicago, Chicago, Illinois 60607, USA (Received 19 June 2021; revised 19 September 2021; accepted 22 September 2021; published 4 October 2021) The study of superconductivity in compressed hydrides is of great interest due to measurements of high critical temperatures (Tc) in the vicinity of room temperature, beginning with the observations of LaH10 at 170–190 GPa. However, the pressures required for synthesis of these high-Tcsuperconducting hydrides currently remain extremely high. Here we show the investigation of crystal structures and superconductivity in the La-B-H system under pressure with particle-swarm intelligence structure-searches methods in combination with first-principles calculations. Structures with seven stoichiometries, LaBH, LaBH4,LaBH 6,LaBH 7,LaBH 8,La(BH) 3,and La(BH4)3were predicted to become stable under pressure. Remarkably, the hydrogen atoms in LaBH8were found to bond with B atoms in a manner that is similar to that in H3S. Lattice dynamics calculations indicate that LaBH7and LaBH8become dynamically stable at pressures as low as 109 and 48 GPa, respectively. Moreover, the two phases were predicted to be superconducting with a critical temperature Tcof 93 K and 156 K at 110 GPa and 55 GPa, respectively (μ∗=0.1). The present results provide guidance for future experiments targeting hydride superconductors with both low synthesis pressures and high Tc. DOI: 10.1103/PhysRevB.104.134501 I. INTRODUCTION Exploration of superconductivity in materials at ever increasing temperatures is a burgeoning research topic in condensed matter physics, chemistry, and materials science. Conventional electron-phonon coupling considerations point to compressed hydrogen-rich materials as excellent candidates for superconductors having high critical temperatures (Tc’s) due to the potential for formation of atomic hydrogen lattices in which the low mass leads to both high vibrational frequencies and strong electron-phonon coupling. As originally proposed by Ashcroft [1], this concept has inspired numerous studies (see Refs. [2–6] for reviews), specifically the recent progress on pressurized hydrides predicted and observed to have high Tc’s above 200 K in pursuit of superconductivity at, or even above, room temperature [7–22]. However, pressures in the megabar range (>100 GPa) are required to synthesize and stabilize the high-Tchydrides considered so far. For example, high-Tcsuperconductivity was established above 170 and 166 GPa for clathrate metal *[email protected] †[email protected] hydrides LaH10 and YH6[17,20], and near 155 and 267 GPa for p-block element hydrides H3S and C-S-H [14,22], respectively, where the pressures are those of the reported Tcmaxima. Given the very high pressures required to create these high critical temperatures, the pursuit of high-Tc superconductivity in hydrides that can persist in stable or metastable compounds at lower, and even ambient, pressure remains an important goal. The stability of binary hydrides having potential superconducting Tc’s above 100 K has been largely limited to pressures above 100 GPa [23]. For example, synthesis of superhydride UH7has been reported at a low pressure of 31 GPa, but its Tcis estimated to be 44 K [24,25]. The lowest pressures reported for stabilization of a superhydride include those of CeH9at 80 GPa [26] and BaH12 at 75 GPa [27] for atomic and molecular-based hydrogen structures, respectively. Predictions of lower pressure stability of hydrogen-rich binary hydrides include that of RbH12, which is calculated to be stable at 50 GPa with a Tcnear 115 K [23]. With the additional degrees of freedom made possible by expanding the chemical space available, ternary hydrides are receiving growing interest as means both to increase Tcand to enhance stability over a broader range of pressures. As such, a Tcof 287 K has been reported in a C-S-H mixture at about 267 GPa, while 2469-9950/2021/104(13)/134501(8) 134501-1 ©2021 American Physical Society XIAOWEI LIANG et al. PHYSICAL REVIEW B 104, 134501 (2021) the structure and composition for the high-Tcphase remain under study [22]. Theoretical calculations predict that hydride perovskite structures based on the above elements could be a route to stabilizing lower pressure hydride superconductors, for example, by sublattice replacement of SH3with CH4in H3S to produce structures of composition CSH7 with predicted dynamical stability, and therefore kinetic stability, at lower pressures than that of pure H3S[28,29]. Lower level CH4substitution in the material, either as stoichiometric compounds or doped structures, could enhance low-pressure stability, as well as significantly enhance Tcas recently predicted for the C-S-H superconductor [30]. These results further suggest that ternary hydride systems may be a useful venue for discovering high-Tcsuperconductors at low pressures. Metal hydrides having clathrate and related structures (e.g., MgH6,CaH 6,YH 6,LaH 10, and H3S) [7–10,12,13] have been predicted, and in several cases now observed, to have high-Tc values that are higher than those with higher H content, such as MgH12 and MgH16 [31]. This trend arises from the different forms of H atoms in the structures. In high-Tchydrides, H2 molecules accept electrons, inducing dissociation to form a monoatomic and metallic structure in which there is an increase in the H contribution to the electronic states around the Fermi level and increasing the critical temperature. Therefore, the presence of atomic H in the structure is an important feature of high-Tcsuperconductivity of hydrides. Theoretical studies indicate that adding Li and Ca to the B-H binary stabilizes phases that accommodate more atomic H atoms, leading to ternary hydrides with higher Tc’s relative to those found for B-H phases [32–36]. Given its larger ionic radius at ambient pressure, La can accommodate more atomic H (e.g., as a superhydride) compared to Li and Ca [12]. Therefore, the stable phases with higher H content might be obtained in the La-B-H system under pressure. Moreover, a recent experimental study reported evidence for superconductivity at and above room temperature in La-hydride samples upon thermal annealing [37]. Given that ammonia borane (NH3BH3) was used as a hydrogen source, the formation of La-H phases containing boron was suggested as giving rise to the high Tc[17,37]. In this paper, we examine theoretically high-pressure structures, stability, and superconducting properties of stoichiometric La-B-H phases, with a focus on lower pressure stability. Detailed study of phases with composition LaBHx (x =1–10) and La(BHx)3(x =1–5) reveals intriguing H-rich LaBH7(P¯ 3m1) and LaBH8(Fm¯ 3m) structures containing BH6and BH8units, respectively. Moreover, LaBH7 and LaBH8are dynamically stable at pressures as low as 109 and 48 GPa, with predicted Tc’s of 93 and 156 K at 110 and 55 GPa, respectively. Our results indicate that continued exploration of ternary hydrides in these and related chemical systems may be an effective route to realizing a high-temperature superconductivity at lower, or even ambient, pressure. II. COMPUTATIONAL DETAILS The structure searches of La-B and La-B-H system were performed under pressure using the particle swarm optimization technique implemented in the CALYPSO code [38,39]. The structural relaxations and electronic properties were calculated using density functional theory with the Perdew-Burke-Ernzerhof generalized gradient approximation as implemented in the VASP code [40,41]. The ion-electron interaction was described by projector-augmented-wave potentials, where 5s25p65d16s2,2s22p1, and 1s1configurations were treated as valence electrons for La, B, and H atoms, respectively [42]. Plane wave kinetic energy cutoff was set to 700 eV and corresponding Monkhorst-Pack (MP) k-point meshes for different structures were adopted to ensure that the enthalpy converges to 1 meV/atom. Phonon calculations were performed by using the supercell method or density functional perturbation theory (DFPT) with PHONOPY [43] and Quantum-ESPRESSO codes [44], respectively. Electronphonon coupling (EPC) calculations were carried out with the Quantum-ESPRESSO code using ultrasoft pseudopotentials for all atoms. We adopted a kinetic energy cutoff of 60 Ry. 7×7×5, and 9 ×9×9q-point meshes in the first Brillouin zones (BZ) were used for P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8, respectively. Correspondingly, we chose MP grids of 28 × 28 ×20 and 36 ×36 ×36 to ensure k-point sampling convergence. III. RESULTS AND DISCUSSION Before investigating the phase stability of ternary La-BH compounds under pressure, we first assessed information about the La-H, B-H, and La-B binaries. The high-pressure behavior of the B-H [34–36] and La-H [12,13,45–47] binaries has been well-studied in recent years, whereas information on the La-B system under pressure is lacking. We therefore performed structure-search calculations for LanBm(n =1, m=1–8; n =2, m =1) with system sizes containing up to 4 or 8 formula units (f.u.) per simulation cell at pressures of 0–300 GPa. To identify the stability of different stoichiometries, convex hulls were constructed by calculating the formation enthalpies for predicted LanBmstructures relative to the elemental La and B (Fig. S1 within the Supplemental Material, SM [48]). All the stoichiometries were found to have negative formation enthalpies within 0–300 GPa, showing that they are thermodynamically stable with respect to decomposition into La and B elements. The increase in formation enthalpy indicates that the stability of the phase decreases with increasing pressure; in addition, a stoichiometry located on the hull is stable with respect to other binary compounds, otherwise it is metastable. At 1 atm and 10 GPa, the experimentally observed LaB4and LaB6were predicted to be stable, which confirms the reliability of our method. At 50 GPa, with exception for LaB4and LaB6, a new stoichiometry LaB is also located on the convex hull. With increasing pressure to 100 GPa, we found that LaB, LaB4,LaB 5, and LaB8are stable, while LaB6was predicted to possibly decompose into LaB5+LaB8. At 200 GPa, only LaB and LaB8remain on the convex hull, and LaB8has the lowest formation enthalpy. At 300 GPa, LaB8remains stable, and LaB becomes metastable. The predicted stable structures of La-B compounds under pressure are shown in Fig. S2 [48]. In R¯ 3m-LaB, B atoms form graphene-like layers. With increasing B content, the B atoms are apt to form polyhedral configurations with decahedra in 134501-2 PREDICTION OF HIGH-TCSUPERCONDUCTIVITY IN … PHYSICAL REVIEW B 104, 134501 (2021) FIG. 1. Calculated convex hull of the La-B-H system is presented at 100, 150, 200, and 300 GPa, respectively. Thermodynamically stable and metastable stoichiometries are shown as purple circles and blue triangles, respectively. Cmmm-LaB4, octahedra in P4/mmm-LaB5and octadecahedra in R¯ 3m-LaB8, respectively. Calculated La-B-H ternary phase diagrams at 100– 300 GPa are presented in Fig. 1. All the ternary hydrides are stable relative to dissociation into elements at the pressures studied. Moreover, all the ternary hydrides LaBHxstudied have lower enthalpies than those of LaB and H2, suggesting that they are stable against decomposition into LaB and H2 (Fig. S3 within the SM [48]). We note that LaBH, LaBH4, LaBH6,La(BH) 3, and La(BH4)3fall on the 3D convex hull at 100 GPa, indicating that they are also stable phases with respect to decomposition into binary and other ternary phases. At 150 GPa, an additional composition, LaBH7, appears on the convex hull. With further increase in pressure to 200 GPa, the originally stable LaBH, LaBH6, and La(BH4)3are predicted to decompose into other compounds, and the higher H content LaBH8begins to become stable. At 300 GPa, the formation enthalpy of LaBH8is increasingly negative. LaBH7 becomes metastable phases with higher enthalpies relative to 1/4LaBH 4+3/4LaBH 8. To determine accurate stability pressures, we also plot the specific enthalpy curves of LaBH4, LaBH6,LaBH 7, and LaBH8relative to other compounds (Fig. S4 within the SM [48]). After including zero-point energy corrections (Fig. S5 within the SM [48]) [49], we found that LaBH4remains thermodynamically stable in a P21/mstructure within the entire pressure range studied. C2/c-LaBH6 is stable below 134 GPa. For LaBH7,theP¯ 3m1 structure is predicted to become stable against dissociation into other stoichiometries at pressures of 103–233 GPa. LaBH8is predicted to crystallize in the cubic Fm¯ 3mstructure, which is stable relative to LaBH7and H2above 161 GPa. More recently, Cataldo et al. [50] reported simulations of the structure, superconductivity, and stability of LaBH8. FIG. 2. The predicted crystal structures of ternary hydrides under pressure. (a) P6/mmm-LaBH, (b) Pmma-LaBH3,(c)P21/m-LaBH4, (d) C2/c-LaBH6,(e)P¯ 3m1-LaBH7, and (f) Fm¯ 3m-LaBH8. Magenta, blue and green balls represent La, B, and H atom, respectively. Although the results are largely consistent with our results, several differences should be noted. First, Cataldo et al. predicted that F¯ 43m-LaBH5and Fm¯ 3m-LaBH8are thermodynamically stable compositions at 100 and 110 GPa, respectively. On the other hand, we find that F¯ 43m-LaBH5and Fm¯ 3m-LaBH8are thermodynamically unstable with respect to decomposition into LaBH4+LaBH6and LaBH6+H2at 100 and 110 GPa, respectively. Relative enthalpy-pressure curves for LaBH5are shown in Fig. S6 within the SM [48]. Second, we found additional stoichiometries (LaBH, LaBH4, and LaBH6) to be stable at 100 GPa that were not mentioned in Ref. [50]. In addition, we explored systematically the crystal structures of different stoichiometries and their stability at different pressures. Stable pressures of the predicted La-B-H compounds were determined that will provide clear guidance for experiments, including specific composition and structure information. Pressure favors the formation of more H-rich compounds in the La-B-H system, e.g., with LaBH5, LaBH6,LaBH 7, and LaBH8, increasing stable under pressure. Finally, we point out interesting structural trends among the La-B-H system are associated with this increase in H content. Specifically, the H atoms gradually bond with the surrounding B atoms, changing from BH units in LaBH3and LaBH4to BH4in LaBH6,BH 6in LaBH7, and eventually BH8units in LaBH8. The predicted stable structures of the LaBHxsystem are shown in Fig. 2. LaBH adopts the hexagonal P6/mmm structure, in which B atoms form honeycomb sheets and B, La, and H atomic layers are alternately arranged. In LaBH3and LaBH4, zigzag B chains stretch along specific directions while B atoms are surrounded by H atoms to form covalent bonds. Since more H atoms are filled in C2/c-LaBH6, no bonds exist between B atoms and each B atom forms a BH4unit with the adjacent four H atoms. With the increasing H content of LaBH7, each B atom accommodates six H atom to form BH6 units. The BH6groups are located on the vertices and edges of the hexagonal structure, with the BH6units distributed on the edges connected to each other by H atoms. The Hricher LaBH8assumes a high-symmetry Fm¯ 3mstructure in which B atoms accommodate all the H atoms to form BH8 134501-3 XIAOWEI LIANG et al. PHYSICAL REVIEW B 104, 134501 (2021) covalent units that occupy the octahedral interstices of the face-centered cubic (fcc) lattice formed by La atoms. In H3S, the S atoms located on a body-centered cubic lattice with each S atom covalently bonded to the surrounding six H atoms. The H atoms in LaBH8are found to bond with B in a manner that issimilartothatofSinH 3S[51]. In addition, the atomic positions of La and eight H atoms in LaBH8are the same as those of La and eight of the H atoms in LaH10.Given the high Tcof H3S and LaH10, this similarity in bonding and high symmetry structure suggests interesting superconducting properties of LaBH8as well. The enthalpy curves and predicted stable structures of La(BH)3and La(BH4)3 are shown in Fig. S7 within the SM [48]. P¯ 1-La(BH)3is stable between 100–265 GPa and transforms to a Cm structure above 265 GPa. In P¯ 1-La(BH)3, B atoms bond to each other forming layered structures composed of 4and 10membered rings and H atoms locate between layers to bond with the surrounding B atoms. Cm La(BH)3is an open-framework structure formed by B and H atoms with La-filled channels along the crystallographic b axis. La(BH4)3is stable in the Cm structure between 100 and 150 GPa, where B atoms form chains and bond with surrounding H atoms. As a result, the remaining H atoms adopt the form of H2units. In addition, several metastable structures are also predicted as shown in Fig. S8 within the SM [48]. At 100 GPa, the F¯ 43mstructure was predicted to have the lowest enthalpy in LaBH5, although it is metastable relative to other stoichiometries. As the pressure decreases to 50 GPa, the structure tends to become stable. As shown in Fig. S9 within the SM [48], F¯ 43m-LaBH5is constructed by fcc lattices formed by La atoms with BH4units and H atoms located on the octahedral and tetrahedral interstices respectively. C2/c-LaBH6, which is stable at higher pressures, has a similar structure to F¯ 43m-LaBH5but a little distorted, with more individual H atoms in the lattice. In LaBH8, a low-symmetry P21/mstructure is thermodynamically more favorable than the Fm¯ 3mstructure below 154 GPa, where B atoms bond with seven H atoms forming BH7units. Under pressure, this structure will transform into more densely packed Fm¯ 3mstructure with BH8units. This low-pressure P21/mstructure can be viewed as the distorted Fm¯ 3mstructure. The La and B atoms always maintain NaCl-type structure in F¯ 43m-LaBH5,C2/cLaBH6,P21/m-LaBH8, and Fm¯ 3m-LaBH8, showing that pressure has less effect on the La-B bonding. Moreover, these H-rich compounds have common structural features of B-H units, and the number of H atoms bonded to B increases with increasing pressure. To investigate the thermodynamic stability mechanism of Fm¯ 3m-LaBH8, we calculated the internal energies (U) and the pressure-volume (P-V) contributions to the enthalpy relative to LaBH7+H2and P21/m-LaBH8(Fig. S10 within the SM [48]). The P-V term of Fm¯ 3m-LaBH8is always lower than those of LaBH7+H2and P21/m-LaBH8, while Uis opposite, showing that denser structure of cubic Fm¯ 3m-LaBH8is the key factor for its stability. Lattice dynamics calculations were carried out for the phases in the pressure ranges of their predicted thermodynamic stability. The lack of imaginary frequencies in the calculated phonon dispersion curves indicates that all structures are dynamically stable within the harmonic approximation (Figs. 5and S11 within the SM [48]). On the other FIG. 3. The Calculated ELF with isosurface value of 0.6 and ELF in the (1 1 0) plane for (a) P¯ 3m1-LaBH7and (b) Fm¯ 3m-LaBH8at 110 and 55 GPa, respectively. hand, phonon softening is evident for LaBH7and LaBH8, an effect that can enhance the EPC [12]. With decreasing pressure, these phonon modes further soften and eventually have imaginary frequencies. Figure S12 within the SM [48] shows the frequency of the softest mode as a function of pressure. In contrast to LaH10, however, LaBH7and LaBH8 maintain dynamical stability to pressures as low as 109.2 and 48.3 GPa, respectively, the latter being much lower than that predicted for other H-rich hydride superconductors. To understand the origin of relatively low-pressure stability of P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8, we explored the bonding of these structures by calculating the electron localization function (ELF) [52,53] and Bader charge transfer [54] among atoms. ELFs with an isosurface of 0.6 are shown in Figs. 3(a) and 3(b) for the two phases at 110 and 55 GPa, respectively. Electron density at the La atoms is due to their inner valence shells. Many electrons are clearly localized between B and H atoms and closer to the H atoms. The ELF slice in the (110) plane containing La, B, and H atoms for LaBH7and LaBH8[Fig. 3(c)] also shows that the ELF values between B and H atoms gradually increase toward H atoms, suggesting the polar covalent character of the B-H bond. For comparison, we calculated the ELF of the well-known hydrogen storage material LiBH4at 1atm (Fig. S13 within the SM [48]), which contains a strong BH4covalent unit. The ELFs around B and between B and H atoms in LaBH7and LaBH8are similar to thoseinLiBH 4. As shown in Fig. S9 within the SM [48], the bond lengths of the B-H bonds in La-B-H compounds are comparable to those of 1.23 Å in LiBH4at 1 atm. Moreover, similar results were also found in Im¯ 3m-H3S[7]. In LaBH7, atom H2 appears to form a covalent bond to H1 with an ELF value of 0.64 connecting BH6units on the edges. In both phases, the ELF values at the center of the shortest La-H and La-B are below 0.3, indicative of an ionic character between La and B-H units. To provide a deeper insight into the bonding, we calculated the Crystal Orbital Hamilton Population (COHP) and its 134501-4 PREDICTION OF HIGH-TCSUPERCONDUCTIVITY IN … PHYSICAL REVIEW B 104, 134501 (2021) integration ICOHP [55–57] projected onto B-H and H-H pairs in LaBH7and LaBH8at different pressures, and H-S bonds in H3S at 200 GPa for comparison. Positive and negative -COHP indicate bonding and antibonding interactions, respectively. The ICOHP up to the Fermi level can describe the atom pair interaction strength. Figure S14 within the SM [48] clearly shows there are strong B-H bonding states below the Fermi level in LaBH7and LaBH8, which are similar to the case of H-S in H3S. The H1-H2 interaction in LaBH7is weak and mainly distributed around 10 eV below the Fermi energy, while the interaction strength of H2-H3 in LaBH7and H-H in LaBH8are negligible, which suggests that the H-H bond will not play a major role in structural stability and properties. The -ICOHP at the Fermi level increases with pressure, which indicates the enhancement of the B-H interactions. At 200 GPa, the calculated -ICOHP between B and H atoms at Fermi level are 3.34 and 3.18 in LaBH7and LaBH8, values that are comparable to that of 3.50 between S and H in H3S, indicating the H atoms in LaBH7and LaBH8bond with B in a manner as H-S in H3S. Bader charge calculations show that electrons transfer from La and B to H atoms. In LaBH7, each La atom and B atom located on the vertex and edge of the lattice loses 1.44, 1.37, and 1.16 electrons, respectively. Correspondingly, each H atom in BH6at the vertex (H3) and edge (H1) accepts 0.45 and 0.39 electrons, respectively. The H atom (H2) that only bonds with a H atom gets 0.15 electrons. The existence of the H1-H2 covalent bond weakens the B-H1 bond connected to it. In LaBH8, each La and B atom transfers 1.47 and 1.04 electrons to eight H atoms, respectively. With increasing pressure, the number of electrons transferred by La and B atom decreases and increases, respectively. Fm¯ 3m-LaBH8possesses the same fcc lattices composed by La atoms as in Fm¯ 3m-LaH10.The difference is that B and H atoms form strong covalent bonds in LaBH8, while the H atoms in LaH10 are connected by weak covalent bonds forming a cage structure. At 200 GPa, the calculated -ICOHP for the B-H bond in LaBH8is 3.18, which is much higher than 0.17 for the H-H bond, revealing strong interactions between B and H atoms in LaBH8. The calculated -ICOHP for H-H pairs in LaH10 (Fig. S15 within the SM [48]) is 1.6, showing weak covalent bonds. By analyzing the eigenvectors of soft modes for LaBH8and LaH10, it is found that their structural stabilities are mainly associated with the vibrations of the B-H and H-H bond, respectively (Fig. S16 within the SM [48]). With decreasing pressure, the distance between atoms becomes longer and the interactions weaker, which eventually leads to a structural instability and phase transition (Fig. S17 within the SM [48]) [12,45]. The results indicated that the B-H bonds in LaBH8are much stronger than H-H interactions in LaH10, and the B and H atoms can maintain a bonding interaction over large pressure range. Therefore, the strong interactions between B and H atoms play an important role in determining its relatively low-pressure dynamical stability. We further investigated the electronic properties of the stable structures found in the La-B-H system. The calculated electronic density of states (DOS) for P6/mmm-LaBH, Pmma-LaBH3,P21/m-LaBH4and C2/c-LaBH6within their ranges of pressure stability are shown in Fig. S18 [48]. The electronic DOS at the Fermi level indicates that they are all FIG. 4. The calculated electronic band structure, density of states, and Fermi surfaces of P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8at 110 and 55 GPa, respectively. metallic. P6/mmm-LaBH, Pmma-LaBH3, and P21/m-LaBH4 all have relatively high DOS values at the Fermi level. However, this metallicity is mainly derived from the contribution of La and B atoms: there is a negligible H contribution to the DOS at Fermi level, which is unfavorable to superconductivity. In C2/c-LaBH6, the Fermi level falls at the valley of the electronic DOS, showing poor metallicity. We focused on the H-richer LaBH7and LaBH8compounds. Figure 4illustrates the calculated electronic band structures, DOS and Fermi surface of P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8at 110 and 55 GPa, respectively. They are the metallic phases with some bands crossing the Fermi level. In LaBH7, a flat band with more localized electronic states appears near the Fermi level at the point, which might enhance the electron-phonon interactions. Flat-steep band features are beneficial for superconductivity [58]. As such, the steep and flat bands are found for LaBH8along the Γ-Xand X-Wdirections near the Fermi level, respectively. According to the calculated ELFs, the distributions of ELF between H2 and H3 in LaBH7and H atoms in neighboring BH8units in LaBH8 can be seen as the picture of two tangent circles, respectively. The ELF values decrease gradually from the center of one atom to the middle point of the two atoms. Therefore, we roughly considered half of the distance between H2 and H3 in LaBH7and H atoms in neighboring BH8units in LaBH8as the H projected radius for the projected DOS calculations. The results show that the contribution of H atoms to the DOS at 134501-5 XIAOWEI LIANG et al. PHYSICAL REVIEW B 104, 134501 (2021) FIG. 5. Calculated phonon dispersion curves (red circle area proportional to associated EPC), projected phonon density of states (PDOS), the Eliashberg phonon spectral function α2F(ω)/ω and its integral λ(ω)of(a)P¯ 3m1-LaBH7at 110 GPa, (b) Fm¯ 3m-LaBH8at 200 GPa, and (c) Fm¯ 3m-LaBH8at 55 GPa. the Fermi level exceeds that of La and B atoms in LaBH7, and the metallicity are dominated by H atoms in LaBH8, which suggest that P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8maybehigh-Tc superconductors. Both the Fermi surfaces of P¯ 3m1-LaBH7 and Fm¯ 3m-LaBH8are composed of three sheets and one of them is an electron-like ellipsoid and polyhedron around the Γpoint, respectively. In LaBH7, one Fermi surface sheet is like a tube along the Γ-Adirection and the other is distributed in a large outer region of the Brillouin zone. In LaBH8, with the exception of a cross-like sheet, there are small electron and hole pockets at Land Xpoint, respectively. Given their promising electronic properties, we calculated the superconducting properties of LaBH7and LaBH8. We calculated their phonon spectra, projected phonon DOS, Eliashberg phonon spectral function α2F(ω)/ω and integral λ(ω) for the two phases at 110 and 200 GPa, respectively [Figs. 5(a) and 5(b)]. Similar to the hydrides studied previously, the projected phonon DOS can be separated into three regions. The La atom with the heaviest atomic mass dominates the low-frequency region, whereas the vibrations of the B and H atoms are associated with the midand high-frequency phonon branches, respectively. The spectral function α2F(ω)/ω for LaBH7is mainly distributed below 30 THz, especially between 8–15 THz [Fig. 5(a)], which results in an EPC constant λof 1.46 at 110 GPa. However, the value of the phonon DOS between 8–15 THz is negligible. Further analysis reveals a soft mode in this frequency range with a potentially large EPC contribution. The distribution of the EPC strength on the different phonon modes are also plotted with the spectra. The soft mode associated to H atoms below 20 THz around the Mpoint shows a quite large EPC. Similarly, for LaBH8the calculated EPC λis 0.72 at 200 GPa, and the contribution to λof the vibrations related to H atoms above 30 THz accounts for 83% of the total value. The soft mode near 30 THz at Γpoint makes an important contribution to the EPC. Previous studies of related superconducting hydrides indicate that the total EPC may be enhanced by further phonon softening induced by decompression toward the structural instability predicted by this harmonic approximation of the lattice dynamics [7,12]. Calculations for LaBH8indicate that λincreases to 1.97 and 2.29 near its predicted instability at 55 [Fig. 5(c)] and 50 GPa (Fig. S19 within the SM [48]), which are comparable with the value of 2.19 found for H3S at 200 GPa. As discussed above, our results show that the interaction between B and H atoms is stronger than that between H atoms in LaBH8. The contribution to the strong EPC at 55 GPa arising from soft modes (5–20 THz) is about 59%, which is mainly associated with the vibrations of B-H and H-H [Fig. 5(c)]. We adopted the Allen-Dynes modified McMillan equation to estimate the Tcof P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8at different pressures (Table I)[59]. For LaBH7, the calculated λ and phonon frequency logarithmic average ωlog is 1.46 and 837 K at 110 GPa, leading to a Tcof 93 K with μ∗=0.1. As pressure decreases from 200 to 100, 55, and 50 GPa, the calculated λfor LaBH8increases from 0.72 to 1.11, 1.97, and 2.29, whereas ωlog decreases from 1557 to 1189, 807, and 692 K. As a result of these two effects, the calculated Tc first increases from 58 to 115 K and then decreases to 108 K assuming μ∗=0.1, which follows the trend of λand with pressure, respectively. Since the λof LaBH8at 55 and 50 GPa are much greater than 1.5, the accuracy of Tcvalues was improved by considering the strong-coupling and the shape corrections (f1and f2). The estimated Tcvalues are 139 and 138 K with μ∗=0.1. By numerically solving the Eliashberg equation [60], the Tc’s increase a little, becoming 156 and 154 K, respectively. IV. CONCLUSIONS Density functional theory-based structure-search calculations have identified seven phases in the ternary La-B-H system at pressures of 100–300 GPa that are potential targets for experimental synthesis. Most significant are the predictions of stability of H-rich P¯ 3m1-LaBH7at 103–223 GPa and Fm¯ 3m-LaBH8above 161 GPa, with the latter calculated to be dynamically stable as low as 48 GPa. Structural trends among these phases are observed as the H content increases. In LaBH, the B atoms form graphene-like layers, whereas in LaBH3and LaBH4, the B atoms not only bond with each other to form zigzag chains, but bond with H atoms. In LaBH6 and LaBH7, there are no B-B bonds and B atoms are coordinated by Hs to form BH4and BH6units. LaBH8is stable in the high-symmetry Fm¯ 3mstructure, in which the B atoms accommodate all the H atoms to form BH8units. The La 134501-6 PREDICTION OF HIGH-TCSUPERCONDUCTIVITY IN … PHYSICAL REVIEW B 104, 134501 (2021) TABLE I. The calculated electron-phonon coupling parameter λ, phonon frequency logarithmic average ωlog and critical temperature Tc(μ∗ =0.1–0.13) from Allen-Dynes modified McMillan, McMillan with the strong-coupling and the shape corrections (f1and f2) and Eliashberg equations for P¯ 3m1-LaBH7and Fm¯ 3m-LaBH8. Tc(K) Tc(K) Tc(K) Pωlog McMillan McMillan (f1and f2) Eliashberg (GPa) λ(K) μ∗=0.1–0.13 μ∗=0.1–0.13 μ∗=0.1 LaBH7(P¯ 3m1) 110 1.46 837 93–85 LaBH8(Fm¯ 3m) 200 0.72 1557 58–45 LaBH8(Fm¯ 3m) 100 1.11 1189 96–84 LaBH8(Fm¯ 3m) 55 1.97 807 115–107 127–139 156 LaBH8(Fm¯ 3m) 50 2.29 692 108–102 126–138 154 atom acts as an electron donor in the structures to stabilize the higher H content B-H units. Moreover, EPC calculations show that LaBH7and LaBH8are potential superconductors. Softening of phonons dominated by H-atom vibrations in these structures makes a large contribution to superconductivity. The estimated Tcof LaBH7is 93 K at 110 GPa, whereas the Tcof LaBH8is calculated to be as high as 156 K at 55 GPa. The expanded range of dynamical stability to low pressures together with its predicted relatively high Tcmake Fm¯ 3m-LaBH8a promising candidate superconductor for lowpressure stabilization experiments. Similar results for LaBH8 were reported during the preparation and submission of this paper [50,61,62]. Additional chemical substitution of these phases could be used to enhance both Tc(e.g., by electron or hole doping) or structural stability at still lower pressures. Additional theoretical work could explore potential anharmonic and quantum effects on the stability and the calculated critical temperatures [45,46,63]. The present paper is thus expected to stimulate further research on ternary and more complex superconducting hydrides with high critical temperatures and expanded ranges of stability. 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[48] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.104.134501 for formation enthalpies per atom of LanBmand LaBHx(x =1–10) at 100–300 GPa, the predicted stable structures of La-B system under pressure, relative enthalpy-pressure curves for LaBH4,LaBH 6,LaBH 7, and LaBH8relative to the corresponding binary hydrides as a function of pressure without and with zero-point energy, the enthalpy curves of structures for LaBH5relative to other ternary La-B-H compounds as a function of pressure, the enthalpy curves and predicted stable structures of La(BH)3and La(BH4)3, crystal structures of several predicted metastable La-B-H compounds, crystal structures of F¯ 43m-LaBH5,C2/cLaBH6,P21/m-LaBH8,andFm¯ 3m-LaBH8, calculated internal energies U and PV components of the enthalpy of the Fm¯ 3m-LaBH8, phonon dispersion relations for the predicted stable structures at different pressures, the calculated frequency of the softest mode of LaBH7and LaBH8as a function of pressure, calculated ELF of Pnma LiBH4and Im¯ 3m-H3S, the calculated COHP and its integration ICOHP for B-H and HH pairs in LaBH7and LaBH8and H-S in H3Satdifferent pressures; the calculated COHP and its integration ICOHP for H-H pairs in LaH10, eigenvectors of the soft phonon modes in LaBH7,LaBH 8and LaH10, the calculated B-H, H-H distances as a function of pressure for LaBH8, LiBH4,andLaH 10, electronic density of states (DOS) of predicted stable LaBH, LaBH3,LaBH 4,andLaBH 6at their stable pressures, superconductivity of LaBH8at 50 GPa, detailed structural parameters of the predicted La-B compounds and La-B-H ternary hydrides. 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