# Prediction of Superconductivity in Clathrate Er Hydrides under High Pressure

^{1}

^{2}

^{*}

## Abstract

**:**

_{n}(n = 4~6) under pressures of 50~300 GPa. Two novel compounds, ErH

_{4}and ErH

_{6}, are identified as thermodynamically and dynamically stable above 50 GPa; ErH

_{4}and ErH

_{6}can stabilize in clathrate structures with the I4/mmm and Im$\overline{3}$m space groups, respectively. An analysis of the electronic density of states (DOS) suggests the metallic nature of the two phases. Then, the superconducting critical temperature (T

_{c}) is estimated using the Allen–Dynes modified McMillan equation; the results are 130.9~181.2 K for Im$\overline{3}$m-ErH

_{6}at 100~300 GPa, and 74.4~79.8 K for I4/mmm-ErH

_{4}at 150~300 GPa. The resultant high T

_{c}superconductivity in this system can be traced back to the combination of high density of states at the Fermi level and strong electron–phonon interactions.

## 1. Introduction

_{c}superconductors has, in turn, transformed to some other systems such as hydrogen-dominant compounds [2]. According to Ashcroft’s prediction [2], hydrogen-rich hydrides can be metallized at much lower pressures compared to elemental hydrogen due to chemical “pre-compression” effects, and therefore, may be potential superconductors with high T

_{c}. This finding promoted the subsequent upsurge in the study of hydrides and which has eventually led to the discovery of a plethora of superconducting hydrides.

_{F}). However, by compressing mixtures of AMH and H

_{2}, alkali metal subhydrides (AMH

_{x}, x > 1) have been predicted to become stable (or metastable) under high pressures [8,9]. Among these alkali metal subhydrides, LiH

_{6}(with R3m symmetry), LiH

_{8}(with I422 symmetry), and KH

_{6}(with C2/c symmetry) are superconductors with T

_{c}values of 38~82 K at 150~300 GPa, 31~37 K at 100~200 GPa, and 46~70 K at 230~300 GPa, respectively [10,11]. These calculations suggest that alkali metal hydrides with high DOS values at E

_{F}are more likely to have high T

_{c}superconductivity.

_{F}, BeH

_{2}and MgH

_{2}have been predicted to have smaller T

_{c}values of 32~44 K at 250 GPa [12] and 16~23 K at 180 GPa [13], respectively. However, by increasing the H content, CaH

_{6}[14] and MgH

_{6}[15], calculations have shown they can be stable above 150 GPa. Interestingly, both of the hydrides crystalize in sodalite-like clathrate structures (in a space group of Im$\overline{3}$m) with hydrogenic frameworks. The estimated T

_{c}values exhibit highs of 235 K for CaH

_{6}at 150 [14] GPa and 263~271 K at 300~400 GPa for MgH

_{6}[15]. The ultrahigh T

_{c}values in these systems arise from a substantial DOS value at E

_{F}, contributed primarily from hydrogen atoms which comprise the hydrogenic lattices. Therefore, it has been proposed that systems with extended hydrogenic lattices are more likely to become superconductors at higher temperatures. A recent experimental investigation successfully synthesized CaH

_{6}and verified the superconductivity with a measured T

_{c}of 215 K at 172 GPa [16].

_{6}, extensive theoretical and experimental investigations have been carried out to search for hydrides with extended hydrogenic lattices. The hydrides with d-block elements are typical examples. A large number of scandium, yttrium, and lanthanum hydrides have been predicted to be stable under high pressures, and some of them have had high calculated T

_{c}values. For example, ScH

_{6}was predicted to stabilize in a structure isotypic with CaH

_{6}at pressures above 265 GPa [17]. ScH

_{x}(x = 7~10 and 12) was predicted to be synthesized by compressing mixtures of ScH

_{3}and H

_{2}above 150 GPa, forming the Cmcm, Immm, P6

_{3}/mmc, Cmcm, and C2/c structures, respectively [18,19]. The calculated T

_{c}values of these systems were reported to be 120~169 K at pressures above 250 GPa [19].

_{6}, theoretical calculations have shown that H clathrate structures could also be formed in rare earth (RE) hydrides such as REH

_{6}, REH

_{9}, and REH

_{10}, where RE = Y, La, Ce, and Pr [4]. These compounds are peculiar, since the H atoms in their structures are weakly covalently bonded with each other, forming unusual H cages with stoichiometries of H

_{24}, H

_{29}, and H

_{32}. The estimated T

_{c}values of these systems are striking [3,4,20]. YH

_{10}and LaH

_{10}(both within the Fm$\overline{3}$m symmetry) have been predicted to be potential room temperature superconductors with maximum T

_{c}values of 303~326 K at 250~400 GPa and 274~286 K at 210~250 GPa, respectively [3,4]. Remarkably, some of these materials, consistent with the theoretically predicted structures, have been successfully synthesized in recent experiments [21]. For example, YH

_{6}and YH

_{9}were measured to have T

_{c}values of 220 K at 166 GPa [22] and 243 K at 201 GPa [23], respectively. LaH

_{10}was experimentally synthesized to exhibit a T

_{c}value of 250 K at 170 GPa [6]. Synthesized LaH

_{9}and CeH

_{9}alloys exhibited T

_{c}values of 148~178 K in the pressure range of 97~172 GPa [24]. At much lower pressures, the T

_{c}values of CeH

_{9}(with the Fm$\overline{3}$m structure) and CeH

_{10}(with the P6

_{3}/mmc structure) were measured to be 57 K at 88 GPa and 115 K at 95 GPa [25,26], respectively. However, the theoretically predicted YH

_{10}and its highest T

_{c}> 300 K has not yet been observed in experiments [23].

_{2}phase with a P2

_{1}/m structure are the only two phases that have been identified to exhibit significant electron–phonon coupling (EPC). The maximum T

_{c}values have been estimated to be 10.6 K for ZrH at 120 GPa [27] and 12.8 K for HfH

_{2}at 260 GPa [28]. In the group V metal hydrides, only the I4/mmm phase of NbH

_{4}has been calculated to have a T

_{c}of 47 K at 300 GPa [29]. The relatively higher T

_{c}in this phase can be traced back to the larger EPC parameter and average logarithmic frequency.

_{2}S is thermodynamically and dynamically stable, up to 200 GPa [30]. The calculated T

_{c}was 80 K in the Cmca-H

_{2}S [31]. H

_{3}S has been predicted to stabilize in the R3m and Im$\overline{3}$m phases at pressures of 111 and 180 GPa, respectively [31]. The two structures both have a particularly high EPC, which leads to high T

_{c}values of 155~166 K at 130 GPa and 191~204 K at 200 GPa [31]. Recent experimental investigations have confirmed the ultrahigh T

_{c}superconductivity in the Im$\overline{3}$m phase H

_{3}S under high pressure [31]. In addition, high T

_{c}superconductivity has also been predicted in other sulfides. For example, in the H-S-P system (H

_{3}S

_{0.925}P

_{0.075}), the maximum T

_{c}was theoretically estimated to be 280 K at 250 GPa [32]. In the C-S-H system, a T

_{c}of 260 K at 133 GPa was reported in a recent experiment with a modulated AC susceptibility technique adapted for a diamond anvil cell [33]. This experiment revealed a Pnma structure of the material which was responsible for the near room-temperature superconductivity of carbonaceous sulfur hydride at 107–133 GPa [33].

_{3}Se was predicted to stabilize in the Im$\overline{3}$m phase above 200 GPa [34], which is isostructural to H

_{3}S. Due to a smaller EPC parameter compared to H

_{3}S, DFT calculations yielded a lower T

_{c}value of 130 K for H

_{3}Se at 200 GPa [34]. In the Te-H system, the structures and stoichiometries were somehow different from those of sulfur and selenium. DFT calculations showed that P6/mmm-H

_{4}Te, C2/m-H

_{5}Te

_{2}, and P6

_{3}/mmc-HTe were the most stable phases at 200 GPa [35]. The stoichiometry of H

_{3}Te stabilized in a C2/m structure above 300 GPa. The largest T

_{c}in Te-H systems was obtained in the H

_{4}Te phase, of which the calculated value was 104 K at 170 GPa [35]. Furthermore, superconductivity has also been reported in some other hydrides. For example, an experimental study indicated that PH

_{3}exhibited superconductivity with a maximum measured T

_{c}of 100 K at 207 GPa [36]. DFT calculations estimated a T

_{c}of 150 K for AsH

_{8}at 350 GPa [37].

_{c}superconductivity approaching or exceeding room temperature. The reason why they have such a high T

_{c}can be traced back to the combination of a high density of states near E

_{F}and strong electron–phonon interactions.

_{2}with a CaF

_{2}-type structure and insulating ErH

_{3}with a HoD

_{3}-type structure [38]. Recent DFT calculations have demonstrated that ErH

_{2}could be a superconductor with a T

_{c}of ~80 K at 14.5 GPa [39]. If sufficient pressure is applied to ErH

_{3}, it can also be forced to be transformed into metallic states. However, no superconductivity has yet been observed for ErH

_{3}in previous theoretical and experimental studies at pressures up to 140 GPa [40,41,42]. In this paper, we perform systematic structure searches by increasing the H concentration in Er-H systems, with the aim of finding clathrate Er hydrides. Encouragingly, we do predict two novel stable clathrate Er hydrides with the stoichiometries of ErH

_{4}and ErH

_{6}in the pressure range of 50~300 GPa. It is found that both of the two clathrate hydrides can achieve high-T

_{c}superconductivity.

## 2. Methods and Computational Details

_{n}(n = 4~6) were performed using the swarm-intelligence-based CALYPSO method and software [43,44], which has been successfully applied to predict the structures of many systems [45,46,47,48]. The structure optimizations and electronic property calculations were performed following the plane-wave basis projector-augmented-wave (PAW) method [49] in the framework of density functional theory as implemented in the Vienna ab initio simulation package (VASP) code [50]. In these calculations, the generalized gradient approximation in the Perdew–Burke–Ernzerhof [51] form is applied for the exchange-correlation functionals. The configurations of 5s

^{2}5p

^{6}6s

^{2}4f

^{12}and 1s

^{1}of Er and H, respectively, are treated as valence electrons for the PAW pseudopotentials. The cutoff energy is set to 650 eV, and the Monkhorst–Pack grid with a maximum spacing of 0.03 Å

^{−1}is individually adjusted in reciprocal space to the size of each computational cell. The density functional perturbation theory, as implemented in the Quantum ESPRESSO (QE) package [52], is used to examine the dynamical stability and to calculate the electron–phonon coupling (EPC) parameters of these compounds. The Troullier–Martins norm-conserving scheme is used to generate the pseudopotentials for H and Er, wherein the Er 4f electrons are treated as valence electrons. The Brillouin zone is sampled using a 4 × 4 × 4 q-point grid in the calculations of the EPC parameters. A denser 24 × 24 × 24 k-point mesh is used to accurately calculate the electron–phonon interaction matrix. The kinetic energy cutoffs are set to 70 Ry for the plane wave functions and 600 Ry for the charge density. The formation enthalpies (ΔH) of ErH

_{n}are calculated by using: ΔH(ErH

_{n}) = H(ErH

_{n}) − H(ErH

_{3}) − (0.5n − 1.5)H(H

_{2}), where H is the enthalpy of the most stable structure of certain compositions at the given pressure. For ErH

_{3}, the Fm$\overline{3}$m and P6

_{3}/mmc structures are considered [39], and for elemental H

_{2}, the P6

_{3}/m, C2/c, and Cmca-12 phases are used [53].

## 3. Results and Discussion

#### 3.1. Crystal Structures and Stabilities

_{c}benefit from the clathrate structures. To explore the superconductivity of the Er-H system, first, we search stable structures for ErH

_{n}(n = 4~6) at 50~300 GPa. The formation enthalpies (with respect to decomposition into ErH

_{3}and H

_{2}) versus composition for the energy favorable structures of each compound are displayed in Figure 1, from which the thermodynamical stability can be obtained. The results show that no compounds are stable at 50 GPa, since all of the formation enthalpies are positive. When the pressure increases to 100 GPa, the formation enthalpies become negative. Meanwhile, ErH

_{4}and ErH

_{6}, which locate on the convex hull, become stable. As pressure further increases up to 200 and 300 GPa, ErH

_{4}and ErH

_{6}maintain their stabilities.

_{4}and ErH

_{6}crystalize in clathrate structures under high pressure. The pressure dependence of the enthalpy difference for the energy low-lying structures are plotted in Figure 2. In the case of ErH

_{6}, the clathrate structure stabilizes above 88 GPa and adopts an Im$\overline{3}$m symmetry which is similar to that of XH

_{6}(X = Tb, Ca, Mg, Y, and Sc) [12,54]. As shown in Figure 3a, the Im$\overline{3}$m-ErH

_{6}forms an H

_{24}cage structure with the Er atoms locating at the cage center. The H

_{24}cage consists of six squares and eight hexagons, and the H-H bond lengths in the both the squares and the hexagons are 1.17, 1.19, 1.21, 1.25, and 1.29 Å at 100, 150, 200, 250, and 300 GPa, respectively. These bond values are smaller than those in TbH

_{6}and CaH

_{6}, and larger than those in ScH

_{6}and MgH

_{6}at the same conditions [12,54]. The nearest H-Er distances are 2.04 Å and 1.85 Å at 100, 300 GPa, respectively. The crystal structure parameters of Im$\overline{3}$m-ErH

_{6}and the other predicted stable phases are presented in Table 1.

_{24}cage. For example, the value of ELF is 0.59 at 100 GPa. It is worth noting that systems with weak covalent bonds are expected to reach high T

_{c}superconductivity. From this point of view, it is necessary to investigate the superconductivity of Im$\overline{3}$m-ErH

_{6}.

_{6}is stable under pressures of 88~300 GPa. When the pressure decreases to below 88 GPa, it transforms to the P$\overline{1}$ phase. The H atoms in this phase form two types of H

_{2}units, as displayed in Figure 3d. The H-H bond lengths of the H

_{2}units are 0.97 and 1.44 Å at 80 GPa, respectively. The corresponding ELF (Figure 3e) values are 0.96 and 0.55, respectively, indicating different strengths of covalent bonding.

_{6}at different pressures are plotted in Figure 4. It is shown that there is no imaginary frequency in the phonon spectra of Im$\overline{3}$m-ErH

_{6}at 100 and 300 GPa, indicating that they are dynamically stable (Figure 4a). A prominent feature of the Im$\overline{3}$m phase is the drastic softening of the low-lying optical mode around the H point and along the N-Γ and P-N directions when the pressure decreases from 300 to 100 GPa. At 86 GPa, strong phonon softening leads to the emergence of imaginary frequencies at the H point and along the P-N direction, which is responsible for the structural phase transition from Im$\overline{3}$m to P$\overline{1}$. The dynamical stability of P$\overline{1}$-ErH

_{6}is also revealed by the phonon spectra, as presented in Figure 4b.

_{4}, the clathrate structure is the only thermodynamically stable phase above 50 GPa. This phase shares the same crystal structure of I4/mmm with TbH

_{4}[45]. As presented in Figure 3c, the H atoms in this phase form a cage-like structure of H

_{18}, containing eight quadrilaterals and four hexagons. There are two types of H-H bonds in I4/mmm-ErH

_{4}, and the bond lengths are 1.36 and 1.39 Å at 300 GPa, and 1.58 and 1.23 Å at 100 GPa, respectively. The corresponding values of ELF (Figure 3f) are 0.52 and 0.45 at 300 GPa, and 0.41 and 0.73 at 100 GPa, respectively. This result indicates that the H-H covalent bonds present opposite responses of compression, i.e., the former is strengthened by pressure and the latter is weakened. The dynamical stability of this phase can also be demonstrated by the phonon dispersions presented in Figure 4c.

#### 3.2. Electronic Characteristics and Superconductivity

_{n}, we calculate the electronic density of states (DOS). Our results indicate that all the stable Er-H phases predicted here exhibit metallic nature in the corresponding stable pressure range. It is worth noting that even crystallized in different structures, the DOS distributions of Im$\overline{3}$m-ErH

_{6}, P$\overline{1}$-ErH

_{6,}and I4/mmm-ErH

_{4}bear strong resemblance to each other. As shown in Figure 5, the projected DOS on different orbitals of Er and H atoms all cross at the Fermi level. The electronic states around the Fermi level are mainly contributed from Er-f, Er-d, and H-s orbits, while the contribution of Er-s and Er-p are relatively scanty. Another overall feature of the DOS results is the broadening of Er-f electronic states, indicating that the 4f electrons are delocalized and the electronic correlation effect is weakened by pressure.

^{2}F(ω)], the EPC parameter (λ), and the logarithmic average of phonon frequency (ω

_{log}) are then calculated. As shown in Figure 6, for Im$\overline{3}$m-ErH

_{6}, the EPC parameters are calculated to be 3.89 and 1.60 at 100 and 300 GPa, respectively. The corresponding values of ω

_{log}are 857.4 K and 1427.8 K, respectively. The vibrations of H atoms which dominate the medium- and high-frequency region in the phonon DOS contribute 84% and 80% to the total EPC at 100 and 300 GPa, respectively. The strong EPC of this phase at low pressure is attributed to the strong phonon softening (Figure 4). For P$\overline{1}$-ErH

_{6}, the absence of H cage structure leads to a much weaker EPC of λ = 0.48 at 80 GPa. However, the ω

_{log}still has a considerable value of 944.8 K at this pressure, attributing to the high content of H. Different from the Im$\overline{3}$m phase, the contribution of H-atom vibrations located in the medium frequency (15~46 THz) contribute mainly (75%) to the total EPC in the P$\overline{1}$-ErH

_{6}. The contribution of high-frequency vibrations is only 3%.

_{4}, the electron–phonon interaction is also considerable. The calculated values of λ are 0.96 and 0.85 at 150 and 300 GPa, respectively, and the corresponding ω

_{log}values are 1231.7 and 1379.9 K, respectively. Similar to Im$\overline{3}$m-ErH

_{6}, the vibrations of H atoms both in the medium- and high-frequency region contribute dominantly to the total EPC.

_{c}) values can be estimated in terms of the Allen–Dynes modified McMillan equation [56]. The obtained T

_{c}of Im$\overline{3}$m-ErH

_{6}reaches high values of 130.9 K and 181.2 K at 100 GPa and 300 GPa, respectively. However, the P$\overline{1}$-ErH

_{6}has a much lower T

_{c}of 9.3 K at 80 GPa, because of the absence of H cage structure. For I4/mmm-ErH

_{4}, the EPC is relatively weak and the resultant T

_{c}values are 74.4 K and 79.8 K at 150 GPa and 300 GPa, respectively. This result indicates that the cage structure of H atoms obviously leads to higher superconducting critical temperatures, which is consistent with the perspectives of Zurek et al [57].

## 4. Conclusions

_{4}and ErH

_{6}, using the CALYPSO crystal structure prediction method combined with first-principles calculations. ErH

_{4}was predicted to crystalize in a clathrate structure in the whole pressure range considered here. This phase has a space group of I4/mmm, wherein the H atoms bond to each other and form H

_{18}cages. ErH

_{6}was also predicted to stabilize in a clathrate structure with the space group of Im$\overline{3}$m at pressures above 88 GPa. In this phase, the H atoms form a H

_{24}cage sublattice. As pressure decreases to below 88 GPa, the Im$\overline{3}$m phase transforms to a low symmetrical phase with a space group of P$\overline{1}$, attributed to the strong phonon softening. All of the predicted phases exhibit good metallic nature at their stable pressure ranges. The estimated maximum value of T

_{c}in the Er-H system is obtained in the Im$\overline{3}$m phase of ErH

_{6}, which reaches a high value of 181.2 K at 300 GPa. Our results enrich the family of rare earth hydrides and may help with the further discovery of room temperature superconductors in metal hydrides.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Enthalpies of ErH

_{n}formation (n = 4~6) at 50~300 GPa. The solid symbols connected by solid lines on the convex hull denote the stable phases against any type of decomposition. The open symbols located above the convex hull represent unstable or metastable phases. The colorful dotted and black dashed lines are for the eye-guide only.

**Figure 2.**Pressure dependence of the enthalpy difference for the energy low-lying structures of: (

**a**) ErH

_{6}; (

**b**) ErH

_{4}.

**Figure 3.**Crystal structures and bonding properties: (

**a**) Structure of Im$\overline{3}$m-ErH

_{6}; (

**b**) ELF of Im$\overline{3}$m-ErH

_{6}at 100 GPa; (

**c**) structure of I4/mmm-ErH

_{4}; (

**d**) structure of P$\overline{1}$ -ErH

_{6}; (

**e**) ELF of P$\overline{1}$ -ErH

_{6}at 80 GPa; (

**f**) ELF of I4/mmm-ErH

_{4}at 100 GPa. The isosurface value of ELF is set to 0.5. Small and large spheres denote H and Er atoms, respectively.

**Figure 4.**Phonon dispersions of: (

**a**) Im$\overline{3}$m-ErH

_{6}at 100 GPa (dotted line) and 300 GPa (solid line); (

**b**) P$\overline{1}$ -ErH

_{6}at 80 GPa; (

**c**) I4/mmm-ErH

_{4}at 150 (dotted line) and 300 GPa (solid line), respectively.

**Figure 5.**Projected density of states (DOS) of: (

**a**) Im$\overline{3}$m-ErH

_{6}at 100 GPa; (

**b**) P$\overline{1}$ -ErH

_{6}at 80 GPa; (

**c**) I4/mmm-ErH

_{4}at 100 GPa. The Fermi energy is set at 0 eV.

**Figure 6.**Eliashberg electron–phonon spectral function α

^{2}F(ω) (solid line), the integral EPC parameter λ (dashed line) as a function of frequency, and the corresponding phonon DOS (dash area) of (

**a**) Im$\overline{3}$m-ErH

_{6}at 300 GPa; (

**b**) Im$\overline{3}$ m-ErH

_{6}at 100 GPa; (

**c**) P$\overline{1}$ -ErH

_{6}at 80 GPa; (

**d**) I4/mmm-ErH

_{4}at 150 GPa.

Phase | Pressure (GPa) | Lattice Parameters | Coordinates |
---|---|---|---|

Im$\overline{3}$m-ErH_{6} | 100 | a = 3.16 Å α = 109.47 b = 3.16 Å β = 109.47 c = 3.16 Å γ = 109.47 | H1 0.25000 0.50000 −0.25000 Er1 0.00000 0.00000 0.00000 |

P$\overline{1}$-ErH_{6} | 80 | a = 3.15 Å α = 89.99 b = 3.67 Å β = 100.40 c = 4.71 Å γ = 89.99 | H1 0.82939 0.51335 0.88270 H2 0.78475 0.54530 0.67573 H3 0.36777 0.74998 0.87671 H5 0.72069 0.24999 0.43394 H6 0.82940 0.98659 0.88268 H7 0.78476 0.95469 0.67571 Er1 0.29653 0.24999 0.73683 |

I4/mmm-ErH_{4} | 100 | a = 3.31 Å α = 129.59 b = 3.31 Å β = 129.59 c = 3.31 Å γ = 74.05 | H1 1.38370 1.38370 1.00000 H2 0.75000 1.25000 0.50000 Er1 0.00000 0.00000 0.00000 |

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## Share and Cite

**MDPI and ACS Style**

Yan, X.Z.; Zhang, Z.L.; Chen, Y.M.; Kuang, F.G.
Prediction of Superconductivity in Clathrate Er Hydrides under High Pressure. *Crystals* **2023**, *13*, 792.
https://doi.org/10.3390/cryst13050792

**AMA Style**

Yan XZ, Zhang ZL, Chen YM, Kuang FG.
Prediction of Superconductivity in Clathrate Er Hydrides under High Pressure. *Crystals*. 2023; 13(5):792.
https://doi.org/10.3390/cryst13050792

**Chicago/Turabian Style**

Yan, Xiao Z., Zhao L. Zhang, Yang M. Chen, and Fang G. Kuang.
2023. "Prediction of Superconductivity in Clathrate Er Hydrides under High Pressure" *Crystals* 13, no. 5: 792.
https://doi.org/10.3390/cryst13050792