Prediction of an unusual trigonal phase of superconducting LaH$_{\bf 10}$ stable from 250 to 425 GPa pressure

Based on evolutionary crystal structure searches in combination with ab initio calculations, we predict an unusual structural phase of the superconducting LaH$_{10}$ that is stable from about 250 GPa to 425 GPa pressure. This new phase belongs to a trigonal $R\bar{3}m$ crystal lattice with an atypical cell angle, $\alpha_{rhom}$ $\sim$ 24.56$^{\circ}$. We find that the new structure contains three units of LaH$_{10}$ in its primitive cell, unlike the previously known trigonal phase, where primitive cell contains only one LaH$_{10}$ unit. In this phase, a 32-H atoms cage encapsulates La atoms, analogous to the lower pressure face centred cubic phase. However, the hydrogen cages of the trigonal phase consist of quadrilaterals and hexagons, in contrast to the cubic phase, that exhibits squares and regular hexagons. Surprisingly, the shortest H-H distance in the new phase is shorter than that of the lower pressure cubic phase and of atomic hydrogen metal. We find a structural phase transition from trigonal to hexagonal at 425 GPa, where the hexagonal crystal lattice coincides with earlier predictions. Solving the anisotropic Migdal-Eliashberg equations we obtain that the predicted trigonal phase (for standard values of the Coulomb pseudopotential) is expected to become superconducting at a critical temperature of about 175 K, which is less than $T_c \sim$250 K measured for cubic LaH$_{10}$.

Lanthanum superhydride has so far provided the highest transition temperature of the rare-earth-hydrides. Its underlying crystal structure is therefore a topic of concurrent theoretical investigations [5,[19][20][21][22]. Several groups have carried out crystal structure searches, especially in the 100−300 GPa pressure region. Initial crystal searches showed that LaH 10 adopts a stable face centred-cubic lattice, F m3m, above 210 GPa, while at lower pressures the cubic phase becomes dynamically unstable [5]. However, later a combined theoretical and experimental study * hpps@barc.gov.in † fabian.schrodi@physics.uu.se ‡ peter.oppeneer@physics.uu.se identified a low symmetry monoclinic structure C2/m as the most likely low pressure phase [19]. In this work it was noticed that, despite the overall monoclinic crystal symmetry, the lanthanum sublattice can be described by a trigonal R3m symmetry, which was confirmed by the accompanying x-ray diffraction measurements in decompression experiments [19]. Most recently, two additional crystal structures of monoclinic C2 and body-centred orthorhombic Immm symmetries were added to the list of possible low pressure structures by further crystal structure searches [20]. In addition, at high pressure (> 400 GPa) a hexagonal P 6 3 /mmc structure was recently predicted [22]. It was also noted that the inclusion of anharmonic nuclear quantum corrections reduces the low pressure, low symmetry C2 and Immm structures to the cubic F m3m structure and thus ruled out the existence of lower symmetry structures for LaH 10 [20]. The observed R3m lanthanum sublattice, however, yet awaits a satisfactory explanation. In this respect, it is pertinent to mention that former low-symmetry structures are associated with the face-centred cubic lattice through suitable deformations. The crystal structure of LaH 10 at higher pressures has not yet been fully understood and thus more studies, in this pressure region, offer an exciting possibility for the discovery of new crystal structures.
In this article, we study the structural behavior of LaH 10 superhydride by performing evolutionary crystal structure searches under pressure, especially, above 250 GPa. We predict a completely new phase that belongs to a trigonal R3m crystal symmetry. We analyze the stability of this phase and show that it has a lower enthalpy than the previously predicted face-centred cubic F m3m and hexagonal P 6 3 /mmc phases. By solving the anisotropic Migdal-Eliashberg equations with ab initio input [23] we analyze the superconducting properties and find that the superconducting critical temperature is reduced to ∼ 175 K, compared to T c = 250 K of the cubic phase [6,8] at lower pressures, for a realistic value of the Coulomb pseudopotential (µ = 0.1). We further find that, although the electron-phonon interaction is responsible for superconductivity, the ratio ∆/k B T c (with ∆ the superconducting gap) deviates from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) value, placing the lanthanum superhydrides in the strong-coupling regime.

II. METHODOLOGY
To start with, we performed crystal structure searches using the evolutionary algorithm as implemented in the USPEX code [24][25][26]. Over the years, this method has been established as a versatile tool for the predictions of novel stoichiometries and crystal structures of materials at high pressures [27][28][29][30][31][32]. We performed crystal structure searches in the pressure range of 100 − 500 GPa using crystal models of one to four formula units of LaH 10 . The first-generation crystal structures are always created randomly, while subsequent generations contain 20% random structures, and the remaining 80% of structures are created using heredity, softmutation, and transmutation operators. We computed the crystal structure optimizations, enthalpies and electronic structures within the framework of the density functional theory (DFT) while phonon dispersions and electron-phonon interactions (discussed below) are calculated within the framework of density functional perturbation theory (DFPT). All calculations employed the Perdew-Burke-Ernzerhof version of the exchange-correlation energy functional [33]. For structure optimizations and enthalpies calculations, we use the VASP code [34][35][36][37] and PAW potentials with 600 eV plane-wave kinetic energy cut-off, and Brillouin zone (BZ) grids of 2π × 0.01Å −1 interval.

A. Structural optimization and stability
We began with 100 GPa crystal structure searches and these searches readily reproduce previously known structures such as face-centred cubic F m3m, trigonal R3m, monoclinic C2/m and body-centred orthorhombic Immm [7,20]. Since the low pressure (< 300 GPa) phase diagram has been explored extensively in the past by many researchers, we turn our attention to the 300 − 500 GPa pressure region. Our crystal structure searches in this region produce two crystal structures, namely a trigonal structure R3m and a hexagonal structure P 6 3 /mmc. The primitive cell of our trigonal structure, shown in Fig. 1) consists of three formula units of LaH 10 , Here, the big grey spheres represent La atoms and small cyan spheres represent H atoms and coordinate system shows orientation of crystal lattice. These structural models are rendered using VESTA software [38].
unlike the previously known trigonal structure which consists of only one formula unit of LaH 10 [5,19,20]. We also notice that the new trigonal structure has an anomalously small cell angle, α rhom ∼ 24.56 • , unlike the earlier trigonal structure for which α rhom ∼ 60 • [5,20]. In this trigonal phase, the lanthanum atoms are surrounded by cages consisting of 32-H atoms, each of which is linked to six neighboring cages via cuboids of 8-H atoms. The 8-hexagonal faces of each 32-H atoms cage are shared by the surrounding 32-H atoms cages. Contrary to the low pressure cubic phase, the hydrogen cage in the trigonal structure is made of quadrilaterals and hexagons, as shown in Fig. 1.
Interestingly, the new trigonal phase has slightly lower enthalpy, < 2.0 meV/atom, than the face-centred cubic phase even for lower pressures, < 250 GPa, thus making these phases energetically indistinguishable in this pressure region. In addition, the earlier mentioned argument, that nuclear quantum corrections (zero-point vibrations) of hydrogen destabilize low-symmetry structures in favor of the cubic F m3m structure [20] can play a role here. However, the enthalpy difference between trigonal and cubic phases grows with increasing pressure, reaching to a value of ∼ 9.0 meV/atom at 400 GPa, as shown in Fig. 2. We also find that the new trigonal phase transforms to a hexagonal phase above 425 GPa. Our hexagonal phase belongs to the same crystal lattice (P 6 3 /mmc) as that of previous works [22]. In this study, we have not attempted to estimate the phonon contributions of the free energies due to the involvement of computationally expensive phonon calculations at several pressures for all candidate structures. We did not find any other lower enthalpy structure up to pressures of 500 GPa. Similar to the cubic F m3m phase the trigonal phase also has a 32-atoms hydrogen cage around the La atoms, see Fig. 1(b). However, square and hexagonal faces are now distorted, probably to accommodate the symmetry changes. Here, the square and regular hexagonal faces of the cubic H-cage deform into quadrilateral and irregular hexagonal faces [7]. As shown in Fig. 3, this leads to a splitting of the two H-H dis- tances of the face-centred cubic phase into many different H-H distances. Notably, the smallest H-H distances are smaller than those of the cubic phase as well as those of hydrogen metal at similar pressures [39].
We now turn our attention to the electronic and phonon properties of the new phase. These were calculated using DFPT as implemented in the Quantum Espresso package [40]. We use ultrasoft pseudopotentials with 50 and 500 Ry cut-off for plane-wave kinetic energy and charge density, respectively. We used a 24 × 24 × 24 Monkhorst-Pack [41] k-point grid for electronic properties and a 12 × 12 × 12 k-point mesh for phonons. Force constants, phonons, and electron-phonon couplings were calculated on a 4 × 4 × 4 q-point mesh. A test run with denser 36 × 36 × 36 k-point meshes does not show significant changes.
In Figure 4 we show the electronic band structure and density of states (DOS) at 350 GPa. The electronic band structure clearly shows that the trigonal phase is a good metal, similar to the lower pressure cubic phase. Many electronic bands cross the Fermi level along various directions of the BZ, see Fig. 4(a). The orbital projection of the DOS reveals that La-d and La-p states contribute most to the DOS at the Fermi level, while La-s contributions are insignificant and hence not shown in the plot. Similar DOS trends were also noticed for the cubic phase [5]. However, the trigonal phase has considerably smaller DOS values (N 0 ) at the Fermi level than the cubic phase. For example, at 350 GPa we compute N 0 = 0.60 states/eV and N 0 = 0.88 states/eV per LaH 10 unit for the trigonal and cubic phases, respectively.
The phonon dispersions of the trigonal phase are shown in Fig. 5(a) along high symmetry lines of the BZ, and the respective phonon density of states (PHDOS) in panel (b) of the same graph. It is pertinent to mention that the highest phonon frequency, ≈ 2500 cm −1 , of this phase lies between the highest frequencies of the cubic phase and of atomic hydrogen metal. These phonon frequencies are ≈ 2000 cm −1 at 300 GPa [5] and ≈ 2600 cm −1 at 400 GPa [39] for the cubic LaH 10 and hydrogen metal, respectively. Evidently, the La atoms contribute mainly to the low frequency part of the phonon spectrum, below 500 cm −1 , whereas H atoms contribute mainly to the high frequency part of the phonon spectrum, as can be recognized in Fig. 5(b).
In Fig. 6(a)  close to the archetypal value µ = 0.1 and plot the temperature dependence of ∆ in Fig. 6(b). As a measure of momentum anisotropy of the gap, we show here the range [min k ∆ k,m=0 , max k ∆ k,m=0 ] for each µ and T . As is directly apparent, each shaded area in Fig. 6(b) is very thin, reflecting a negligible degree of gap anisotropy, a property which goes in line with results for most hydride superconductors. The critical temperature for the typical value of µ = 0.1 [5,6,13,15] is approximately 175 K, which is a significant decrease compared to T c ≤ 250 K in the lower-pressure cubic phase of LaH 10 [5,8]. This change in T c can partially be understood in terms of the different values for the DOS at the Fermi energy, N 0 , in the trigonal and cubic phase, as was mentioned before. Additionally, the electron-phonon coupling strength λ = 1.69 as found here is smaller than the λ = 2.2 that has been estimated for the cubic phase at 250 GPa [5]. It deserves further to be mentioned that suitably doping of LaH 10 could provide an increase of the Fermi-energy DOS and hence enhance T c [46].
In Table I we list some characteristics of the superconducting state for the same values of the Coulomb pseudopotential. The limiting values ∆(T = 0) have been obtained via the fitting function ∆(T ) Re α − T β /γ. Remarkably, for all choices of µ we find the ratio ∆/k B T c in the strong-coupling regime, in contrast to the weak-coupling BCS value of 1.76. Thus, although the electron-phonon interaction is prevalent, lanthanum superhydride cannot be considered as a BCS superconductor.

IV. CONCLUSIONS
In summary, we have predicted a new trigonal phase of superconducting LaH 10 at pressures above 250 GPa. The new phase has an anomalously small cell angle and its primitive cell is made of three units of LaH 10 , unlike the previously known trigonal phase where only one unit of LaH 10 resides in the primitive cell. Analogous to the lower pressure cubic phase, this phase also has a 32-H atoms cage encapsulating the lanthanum atom and each cage is interconnected to the neighboring six such cages. However, the crystal symmetry reduction from cubic to trigonal results in significant distortions of the hydrogen cages around the La atoms. Interestingly, the smallest H-H distances in the trigonal phase are smaller than those of the cubic phase and those of atomic hydrogen metal. We have also found that the trigonal phase transforms to a hexagonal phase, where the crystal lattice is identical with that given in earlier predictions. Solving the anisotropic Migdal-Eliashberg equations on the basis of ab initio input we predict strong-coupling superconductivity with a high transition temperature, T c ≈ 175 K for the archetypal value µ = 0.1, at 350 GPa, a T c value which is thus somewhat lower than T c ≈ 250 K that was measured [8] and computed [5] for the cubic LaH 10 phase.