Predicted High-Temperature Superconductivity in Rare Earth Hydride ErH$_{2}$ at Moderate Pressure

Yiding Liu(刘一丁)$^{1,2}$, Qiang Fan(范强)$^{3}$, Jianhui Yang(杨建会)$^{1}$, Lili Wang(王丽丽)$^{4}$,\\ Weibin Zhang(张伟斌)$^{5}$, and Gang Yao(姚钢)$^{6,7\hyperlink{s}{*}}$

doi:10.1088/0256-307X/39/12/127403

⇦Chinese Physics Letters, 2022, Vol. 39, No. 12, Article code 127403⇨ Predicted High-Temperature Superconductivity in Rare Earth Hydride ErH$_{2}$ at Moderate Pressure Yiding Liu (刘一丁)1,2, Qiang Fan (范强)3, Jianhui Yang (杨建会)1, Lili Wang (王丽丽)4, Weibin Zhang (张伟斌)5, and Gang Yao (姚钢)6,7* Affiliations 1College of Mathematics and Physics, Leshan Normal University, Leshan 614004, China 2Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China 3School of New Energy Materials and Chemistry, Leshan Normal University, Leshan 614004, China 4Institute of Computer Application, China Academy of Engineering Physics, Mianyang 621900, China 5College of Physics and Electronics Information, Yunnan Key Laboratory of Optoelectronic Information Technology, Yunnan Normal University, Kunming 650500, China 6School of Physical Science and Technology, Southwest University, Chongqing 400715, China 7Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Received 18 September 2022; accepted manuscript online 23 November 2022; published online 4 December 2022 ^*Corresponding author. Email: yaogang@swu.edu.cn Citation Text: Liu Y D, Fan Q, Yang J H et al. 2022 Chin. Phys. Lett. 39 127403 Abstract Hydrides offer an opportunity to study high critical temperature (high-$T_{\rm c}$) superconductivity at experimentally achievable pressures. However, the pressure needed remains extremely high. Using density functional theory calculations, herein we demonstrate that a new rare earth hydride ErH$_{2}$ could be superconducting with $T_{\rm c} \sim 80$ K at 14.5 GPa, the lowest reported value for compressed hydrides to date. Intriguingly, due to Kondo destruction, superconductivity was prone to exist at 15 GPa. We also reveal an energy gap at 20 GPa on the background of normal metallic states. At 20 GPa, this compressed system could act as a host of superconductor judged from a sharp jump of spontaneous magnetic susceptibility with an evanescent spin density of state at Fermi level. Finally, electron pairing glue for ErH$_{2}$ at these three typical pressures was attributed to the antiferromagnetic spin fluctuation.

DOI:10.1088/0256-307X/39/12/127403 © 2022 Chinese Physics Society Article Text Hydrogen in certain hydrides can become metallic at a lower pressure than pure hydrogen, presumably because of additive effects from chemical precompression.[1] These hydrides may be well superconducting up to high critical transition temperature (high-$T_{\rm c}$). Several theoretical and experimental efforts were underway to better understand the high-$T_{\rm c}$ superconductivity of such hydrides. These compounds include GeH$_{4}$ (64 K at 220 GPa),[2] YH$_{3}$ (40 K at 17.7 GPa),[3] H$_{3}$S (203 K at 90 GPa),[4] VH$_{8}$ (71.4 K at 200 GPa),[5] LaH$_{10}$ (274–286 K at 210 GPa,[6] 260 K at 180–200 GPa,[7] and 250 K at 170 GPa[8]), YbH$_{6}$ (145 K at 70 GPa), TmH$_{6}$ (25 K at 50 GPa),[9] LuH$_{6}$ (273 K at 100 GPa),[9] and other room-temperature superconductors YH$_{10}$ (305–326 K at 250 GPa,[6] 303 K at 400 GPa[10]), Li$_{2}$MgH$_{16}$ (473 K at 250 GPa).[11] Note that in all the above-mentioned hydrides, the major glue for electron pairing is phonon. Among rare-earth metals, Er showed a powerful getter for hydrogen.[12] The stoichiometry erbium dihydride (ErH$_{2}$) and its nonstoichiometric composition ErH$_{2+x}$ ($-0.15 < x < 0.15$) both crystallize with fcc-CaF$_{2}$ type crystal structure (space group: $Fm\bar{3}m$), which has often been described as the $\beta$-phase.[13,14] It should be noted that the existence of ErH$_{3}$ has been predicated and confirmed before, however, no superconductivity was observed.[15,16] Hence, in the following discussion, we focus on ErH$_{2}$. The conventional unit cell of ErH$_{2}$ is displayed in Fig. 1, in which the H atoms occupied the tetrahedral sites of the Er fcc lattice. In the meantime, the $f$ electrons in lattice are often neither fully localized around their host nuclei nor fully itinerant. This localized versus itinerant duality has proposed the involvement of $f$ electrons as valence ones in the systems, that is, strongly correlated-electron materials, therefore promoting the emerging researches of heavy fermion superconductors.[17] On the other hand, it was argued that antiferromagnetic (AFM) spin fluctuation was another plausible pairing mechanism driven by Coulomb repulsion in this system, and could lead to high-$T_{\rm c}$ superconductivity.[18-21] The Hubbard model could be consequently applied to this system.[22-24] AFM spin fluctuation could be reflected with mutation from a large Fermi surface (FS) to a small one,[25] and a jump of spin DOS at Fermi level ($E_{\rm F}$) or spontaneous magnetic susceptibility ($\chi_{\rm s}$).[19,21] One central issue that naturally arises is whether a compressed system can be superconducting, and if there is another pairing mechanism for it. Despite the investigation discussed above for $f$ electrons in a crystal, a systematic study of possible pressure-induced superconductivity in ErH$_{2}$ is still missing. In this work, by using first-principles calculations, we reveal that the heavy fermion hydride ErH$_{2}$ is a superconductor with high-$T_{\rm c}$ at much lower pressure. In particular, traces for several accompanying physical behaviors, including FS nesting, Kondo effect, Kondo destruction, and energy gap on the background of normal metallic states, are reported.

Fig. 1. A unit cell of ErH$_{2}$ crystal is displayed. The larger green atoms represent the erbium fcc lattice. The eight smaller grey atoms represent hydrogen occupying all of the tetrahedral sites.

Our spin-polarized calculations based on the density functional theory (DFT) were performed using the Cambridge serial total energy package (CASTEP) code.[26] The exchange and correlation interactions were described by the generalized gradient approximation (GGA) of Perdew and Wang (PW91).[27] Norm-conserving pseudopotentials were used to describe the properties of the perfect crystal. The valence electron configurations for Er and H were 4$f^{11}5d^{1}5s^{2}5p^{6}6s^{2}$ and 1$s^{1}$, respectively. The energy cutoff was set to be 380 eV. For the geometry optimizations of the shape and size of the cell, the Brillouin zone integration was performed using a $10 \times 10 \times 10$ Monkhorst–Pack $k$-points mesh, and for electronic structure calculations including FSs and electron localization functions (ELFs), a dense mesh with $36 \times 36 \times 36$ $k$-points was used. All above parameter settings were verifiable to convergence tests. In addition, the Er-$f$ states were treated employing the GGA + $U$ approach[23] with $U = 6$ eV, as suggested in the relevant previous reports.[28] The phonon calculations were performed using finite displacements method.[29]

Fig. 2. Band structures of ErH$_{2}$ under ambient pressures ranging from 0 to 21.5 GPa: (a) 0 GPa, (b) 14.5 GPa, (c) 15 GPa, and (d) 0.5 GPa. At the rest of this pressure range except 20 GPa, band structures are similar with (d), $\alpha$ and $\beta$ denote spin-up and spin-down, respectively.

ErH$_{2}$ possesses AFM[30] that is subjected to have an intimate link with both FS nesting[31] and superconductivity.[19] Firstly, the FS topologies, electronic density of states (DOS) and band structures were calculated at ambient pressures ranging from 0 to 21.5 GPa. Figure 2 shows the band structures under 0, 14.5, 15 GPa and other pressures except 20 GPa. As shown in Fig. 3, there are three types in topology, electron-like type under 0 GPa, hole-like type at 14.5 and 15 GPa (large and small respectively), and open type at other pressures except for 20 GPa under which DOS and band structure revealed a gap, thus showed no FS around its $E_{\rm F}$. Compared with 0 GPa, at which the corrugated FS reflected the influence of phonon on the properties of electrons around $E_{\rm F}$, the other smooth FSs indicated that the interaction between phonon and electrons decreased considerably.[32] Figure 3(b) shows the topology of the hole-like-FS at 14.5 GPa. It did show nesting property in accordance with its AFM.[30] This consistency between FS nesting and AFM had also been confirmed in ErGa$_{3}$.[31] This FS at 14.5 GPa is apparently the largest one in the whole applied pressures. As shown in Fig. 4(c), in the conduction band region at this pressure, there is a sharp spin-down $f (\beta)$ peak in the range 4–5 eV just corresponding to the spin-up conduction $d$ ($\alpha$) electrons in this energy range. Moreover, the itinerancy of these $f$ electrons could be embodied with the localized $s$ electrons of H in the ELF at this pressure [Fig. 4(a)]. Compared with the ELFs under other pressures [Fig. 4(b)] where electrons localized around Er, the ELF at 14.5 GPa [Fig. 4(a)] showed a distinct localization of electrons around H. Judging from Fig. 4(c) in which two large H-derived sharp $s$-electron DOS peaks [spin-up ($\alpha$) and spin-down ($\beta$) respectively] were most adjacent to the $E_{\rm F}$, H atoms were essentially isolated like atomic H in this lattice. Hence, the result at 14.5 GPa could infer the Kondo effect.[17] The fully itinerant $f(\beta$) electrons like magnetons immersed in conduction $d(\alpha)$ electrons, entangled with them sufficiently and contributed to the Fermi volume.[17,33] Accordingly, the full formation of a large FS could correspond to heavy fermion superconductivity.[25] We show in Fig. 4(e) the variation of spin DOS at $E_{\rm F}$ and $\chi_{\rm s}$ along with the applied pressures. Combining with the explicit hopping of spin DOS at $E_{\rm F}$ at 14.5 GPa without abrupt jump of $\chi_{\rm s}$ and the largest FS with its nesting [Fig. 3(b)], superconductivity could be implied at 14.5 GPa being mediated by AFM spin fluctuation.[18-21]

Fig. 3. Types of FSs for ErH$_{2}$ at ambient pressures ranging from 0 to 21.5 GPa: (a) electron-like type under 0 GPa, [(b), (c)] hole-like type at 14.5 and 15 GPa, respectively, (d) open type at 0.5 GPa. At the rest of this pressure range except 20 GPa, the FSs are of open type similar with the one at 0.5 GPa.

Fermi liquid (FL) could easily exist in this state.[17] Furthermore, based on phonon calculation, the lattice specific heat at 14.5 GPa was also calculated out.[34] The specific heat $C$ in the form of $C/T$ dependence on $T^{2}$, as shown in Fig. 5 under ambient pressures, was obtained with the formula[35,36] \begin{align} C=\gamma_{1}T^{3}+\gamma_{2}T. \tag {1} \end{align} Here, the first term represents the lattice specific heat, the second term denotes the electrons contribution, $\gamma_{1}$ is interatomic force constant, $\gamma_{2}=\frac{\pi^{2}}{3}k_{\scriptscriptstyle{\rm B}}^{2}g(E_{\rm F})$ with $k_{\scriptscriptstyle{\rm B}}$ being Boltzmann's constant, and $g(E_{\rm F})$ is the DOS at $E_{\rm F}$. For the FL state, the electron specific heat depends on $T$ linearly, i.e., $\gamma_{2}T$. In other words, $T_{\rm c}$ could be obtained by locating the kink point of the $C/T$ versus $T^{2}$ plot for the superconducting transition.[32] With these regards, the $T_{\rm c}$ of ErH$_{2}$ at 14.5 GPa is found to be 78.9 K from the simulated $C(T)$ curve shown in Fig. 5. Note that this $T_{\rm c}$ value is around boiling point of liquid nitrogen. In contrast to the Kondo effect, superconductivity could also be driven by Kondo destruction transforming from a large FS to a small one in heavy fermion superconductors, such as CeCu$_{2}$Si$_{2}$.[25] The FS of ErH$_{2}$ at 15 GPa collapsed abruptly as shown in Fig. 3(c), which was identified as destruction of Kondo effect.[17,25,37] Hence, this quantum criticality at 15 GPa could also be prone to possess superconductivity. This phenomenon implied the fully localization of $f$ electrons around their host nuclei, i.e., Er, as the ELF shown in Fig. 4(b). The fact that the localized $f$ electrons do not contribute to the Fermi volume[17,33] was consistent with the small FS [Fig. 3(c)].

Fig. 4. (a) Schematic diagram of electron localization function (ELF) at 14.5 GPa. (b) The ELFs under other pressures of the whole applied pressures. The primitive cell of ErH$_{2}$ crystal is displayed. Green spheres represented Er, while grey spheres represented H. [(c), (d)] The DOS at 14.5 and 15 GPa, respectively. (e) The variation of spin DOS at $E_{\rm F}$ and the spontaneous magnetic susceptibility ($\chi_{\rm s}$) along with the applied pressures.

Fig. 5. $T^{2}$ dependence of the specific heat in the form of $C/T$ under ambient pressures for ErH$_{2}$.

Subsequently, with the Kondo destruction [Fig. 3(c)], the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction played a dominant role that led to the crossover feature from FL at 14.5 GPa to non-fermi-liquid (NFL) behavior at 15 GPa, where the indirect interaction among the localized $f$ electrons with magnetic moments was mediated mainly by the conduction $d$ electrons.[17,25] The DOS is shown in Fig. 4(d). Therefore, despite the superconductivity as predicted at 15 GPa, the dependence of electrons specific heat on $T$ is nonlinear,[35,36] so the $T_{\rm c}$ value at 15 GPa could not be obtained from the kink point of the $C/T$ versus $T^{2}$ curve in Fig. 5. This superconductivity could also be inferred with the medium of AFM spin fluctuation from the noticeable hopping of spin DOS at $E_{\rm F}$ between 14.5 and 15 GPa without steep variation of $\chi_{\rm s}$ in this pressure region [Fig. 4(e)] as well as this FS jump.[18-21] The accompanying behaviors, which could also to some extent support the affirmation of superconductivity at 15 GPa, are presented in the Supplemental Material.[38]

Fig. 6. [(a), (c)] The band structures of this bulk at 20 GPa with electronic spin-down and spin-up, respectively. The band structures of (110) surface built with $1 \times 2\times 1$ supercell of this bulk at 20 GPa are also plotted with red color in (a) and (c). The energy scales of this metallic surface state are from $-6$ to 6 eV. (b) The DOS of this bulk at 20 GPa. Here, $\alpha$ and $\beta$ denote spin-up and spin-down, respectively.

From the spin-polarized DOS and band structures of ErH$_{2}$ at 20 GPa [Figs. 6(a)–6(c)], an energy gap of 6.6 eV can be observed on the background of normal metallic states under the rest of applied pressures. The itinerant electrons were from $s$ electrons of H and $s$, $p$, $d$ ones of Er, whose all spins with up and down directions had almost the same weight and symmetric distribution. The fully localized electrons were from $f$ electrons of Er as shown in the ELF [Fig. 4(b)] and the DOS [Fig. 6(b)]. This manifested that itinerant AFM at 20 GPa vanished abruptly, leading to a localized AFM insulating state.[39] This transition results in a sharp jump in $\chi_{\rm s}$ and an evanescent spin DOS at $E_{\rm F}$ [Fig. 4(e)], which could also signify an AFM spin fluctuation.[18-21] Therefore, ErH$_{2}$ at this pressure could act as a host of superconductor owing to this pairing glue.[18-21] This implication is supported by the competition between spin density wave (SDW) and superconductivity.[17,39-41] The existence of SDW could be determined with Pauli paramagnetic susceptibility, $\chi_{\scriptscriptstyle{\rm P}}$(0), which was expressed as follows:[39] \begin{align} \chi_{\scriptscriptstyle{\rm P}}(0)=2\mu_{\scriptscriptstyle{\rm B}}^{2}\mu_{0}g(E_{\rm F}), \tag {2} \end{align} where $\mu_{\scriptscriptstyle{\rm B}}$ is Bohr magneton, $\mu_{0}$ is vacuum permeability and $g(E_{\rm F})$ is the DOS at $E_{\rm F}$. The SDW is suppressed from the zero DOS at $E_{\rm F}$ [Fig. 6(b)] and Eq. (2). This hypothesis could also be supported by the tiny and clear deviation of the $V/V_{0}$ versus pressures relation calculated directly with DFT at this pressure away from the monotonous trend of the curve fitted with $E$–$V$ data of this work (see Fig. S1 in the Supplemental Material[38]). The hybridization between localized $f$ moments and itinerant electrons is very weak at this pressure, implying the NFL behavior. Meanwhile, as shown in Figs. 6(a) and 6(c), the band structures of (110) surface of this bulk at 20 GPa in the range from $-6$ to 6 eV exhibiting metallic, come within the gap of this bulk. This metallic surface state with bulk insulating state emerged also in the archetype Kondo insulator, i.e., SmB$_{6}$.[37] Our results could postulate a possibility for topological superconductor which could host Majorana zero mode.[42,43] Note that the ErH$_{2}$ under 0, 14.5, 15 and 20 GPa are dynamically stable as no imaginary phonon mode is observed (see Fig. S3 in the Supplemental Material[38]). To summarize, we have revealed superconductivity that emerged in ErH$_{2}$ under moderate pressures 14.5, 15 and 20 GPa by using DFT calculations. It is proposed that ErH$_{2}$ at 14.5 GPa is a potential high-$T_{\rm c}$ superconductor with $T_{\rm c} \sim 80$ K. The gapped bulk state with metallic surface state at 20 GPa could be conducive to the exploration of topological superconductor, which will stimulate further experimental and theoretical studies for the feasibility of ErH$_{2}$ to host Majorana zero mode. Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant No. 12104294), and the Research Project of Leshan Normal University (Grant No. 801/204190415). This work was carried out at Shanxi Supercomputing Center of China, and the calculations were performed on TianHe-2. References Hydrogen Dominant Metallic Alloys: High Temperature Superconductors?Superconducting High Pressure Phase of GermanePredicted High-Temperature Superconducting State in the Hydrogen-Dense Transition-Metal Hydride

{YH}_{3}

at 40 K and 17.7 GPaConventional superconductivity at 203 kelvin at high pressures in the sulfur hydride systemSuperconductivity of Pressure-Stabilized Vanadium HydridesPotential high- T_c superconducting lanthanum and yttrium hydrides at high pressureEvidence for Superconductivity above 260 K in Lanthanum Superhydride at Megabar PressuresSuperconductivity at 250 K in lanthanum hydride under high pressuresHigh T_c Superconductivity in Heavy Rare Earth HydridesHydrogen Clathrate Structures in Rare Earth Hydrides at High Pressures: Possible Route to Room-Temperature SuperconductivityRoute to a Superconducting Phase above Room Temperature in Electron-Doped Hydride Compounds under High PressureAb initio study of intrinsic, H, and He point defects in hcp-ErFirst principles site occupation and migration of hydrogen, helium, and oxygen in β-phase erbium hydrideRare‐earth dihydride compounds: Lattice thermal expansion and investigation of the thermal dissociationHigh-pressure phase transition of MH₃ (M: Er, Ho)Pressure-induced metallization in Erbium trihydrideQuantum criticality in heavy-fermion metalsEmergent Loop-Nodal

s_{\pm}

-Wave Superconductivity in

{CeCu}_{2} {Si}_{2}

: Similarities to the Iron-Based SuperconductorsNormal-state spin dynamics in the iron-pnictide superconductors BaFe

_{2}

(As

_{1 - x}

_{x}

)

_{2}

and Ba(Fe

_{1 - x}

_{x}

)

_{2}

_{2}

probed with NMR measurementsFinding New Superconductors: The Spin-Fluctuation Gateway to High T_c and Possible Room Temperature SuperconductivityAntiferromagnetic spin fluctuation and superconductivityBand theory and Mott insulators: Hubbard U instead of Stoner IElectron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U studyLinear response approach to the calculation of the effective interaction parameters in the

LDA + U

methodQuantum phase transitions in heavy fermion metals and Kondo insulatorsFirst-principles simulation: ideas, illustrations and the CASTEP codeAtoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlationStructure and energetics of Er defects in

{LiNbO}_{3}

from first-principles and thermodynamic calculationsLattice dynamics of TiO2 rutile: influence of gradient corrections in density functional calculationsMagnetic and metal-semiconductor transitions in ordered and disordered ErH(D

)_{2 + x}

Fermi surface nesting and magnetic structure of

{ErGa}_{3}

Itinerant Versus Localized Heavy-Electron MagnetismPhonons and related crystal properties from density-functional perturbation theoryRecent Advances in Ce-Based Heavy-Fermion Superconductivity and Fermi Surface PropertiesMultiband Superconductivity with Unexpected Deficiency of Nodal Quasiparticles in

{CeCu}_{2} {Si}_{2}

Quantum phase transition and destruction of Kondo effect in pressurized SmB6Density functional study of FeS, FeSe, and FeTe: Electronic structure, magnetism, phonons, and superconductivitySuperconductivity at 41 K and Its Competition with Spin-Density-Wave Instability in Layered

{CeO}_{1 - x} F_{x} FeAs

Majorana Zero Mode Detected with Spin Selective Andreev Reflection in the Vortex of a Topological SuperconductorNearly ferromagnetic spin-triplet superconductivity

[1]	Ashcroft N W 2004 Phys. Rev. Lett. 92 187002
[2]	Gao G, Oganov A R, Bergara A et al. 2008 Phys. Rev. Lett. 101 107002
[3]	Kim D Y, Scheicher R H, and Ahuja R 2009 Phys. Rev. Lett. 103 077002
[4]	Drozdov A, Eremets M, Troyan I, Ksenofontov V, and Shylin S 2015 Nature 525 73
[5]	Li X F and Peng F 2017 Lnorg. Chem. 56 13759
[6]	Liu H Y, Naumov I I, Hoffmann R, Ashcroft N W, and Hemley R J 2017 Proc. Natl. Acad. Sci. USA 114 6990
[7]	Somayazulu M, Ahart M, Mishra A K et al. 2019 Phys. Rev. Lett. 122 027001
[8]	Drozdov A P, Kong P P, Minkov V S et al. 2019 Nature 569 528
[9]	Hao S, Zhang Z H, Cui T, Pickard C J, Kresin V Z, and Duan D F 2021 Chin. Phys. Lett. 38 107401
[10]	Peng F, Sun Y, Pickard C J et al. 2017 Phys. Rev. Lett. 119 107001
[11]	Sun Y, Lv J, Xie Y, Liu H, and Ma Y 2019 Phys. Rev. Lett. 123 097001
[12]	Yang L, Peng S M, Long X G et al. 2010 J. Appl. Phys. 107 054903
[13]	Wixom R R, Browning J F, Snow C S, Schultz P, and Jennison D R 2008 J. Appl. Phys. 103 123708
[14]	Bonnet J and Daou J 1977 J. Appl. Phys. 48 964
[15]	Hou P G, Tian F B, Li D et al. 2014 J. Chem. Phys. 141 054703
[16]	Kuzovnikov M A, Eremets M I, Drozdov A P, and Tkacz M 2017 Solid State Commun. 263 23
[17]	Gegenwart P, Si Q, and Steglich F 2008 Nat. Phys. 4 186
[18]	Ikeda H, Suzuki M T, and Arita R 2015 Phys. Rev. Lett. 114 147003
[19]	Nakai Y, Iye T, Kitagawa S et al. 2013 Phys. Rev. B 87 174507
[20]	Pines D 2013 J. Phys. Chem. B 117 13145
[21]	Tôru M and Kazuo U 2003 Rep. Prog. Phys. 66 1299
[22]	Anisimov V I, Zaanen J, and Andersen O K 1991 Phys. Rev. B 44 943
[23]	Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J, and Sutton A P 1998 Phys. Rev. B 57 1505
[24]	Cococcioni M and de Gironcoli S 2005 Phys. Rev. B 71 035105
[25]	Si Q M and Paschen S 2013 Phys. Status Solidi B 250 425
[26]	Segall M, Lindan Philip J D, Probert M J et al. 2002 J. Phys.: Condens. Matter 14 2717
[27]	Perdew J P, Chevary J A, Vosko S H et al. 1992 Phys. Rev. B 46 6671
[28]	Xu H X, Lee D, Sinnott S B et al. 2009 Phys. Rev. B 80 144104
[29]	Montanari B and Harrison N M 2002 Chem. Phys. Lett. 364 528
[30]	Vajda P and Daou J 1994 Phys. Rev. B 49 3275
[31]	Biasini M, Ferro G, Kontrym-Sznajd G, and Czopnik A 2002 Phys. Rev. B 66 075126
[32]	Kittel C 1996 Introduction to Solid State Physics (New Jersey: John Wiley & Sons Inc.)
[33]	Hoshino S and Kuramoto Y 2013 Phys. Rev. Lett. 111 026401
[34]	Baroni S, de Gironcoli S, Dal C A, and Giannozzi P 2001 Rev. Mod. Phys. 73 515
[35]	Settai R, Takeuchi T, and Ōnuki Y 2007 J. Phys. Soc. Jpn. 76 051003
[36]	Kittaka S, Aoki Y, Shimura Y et al. 2014 Phys. Rev. Lett. 112 067002
[37]	Zhou Y, Wu Q, Rosa Priscila F S et al. 2017 Sci. Bull. 62 1439
[38]	See Supplemental Material for supporting information of the hypotheses of superconductivity of ErH$_{2}$ at 15 and 20 GPa.
[39]	Feng D and Jin G J 2012 Condensed Matter Physics (Beijing: Higher Education Press) vol 1 pp 503–524 (in Chinese)
[40]	Subedi A, Zhang L, Singh D J, and Du M H 2008 Phys. Rev. B 78 134514
[41]	Chen G F, Li Z, Wu D et al. 2008 Phys. Rev. Lett. 100 247002
[42]	Sun H H, Zhang K W, Hu L H et al. 2016 Phys. Rev. Lett. 116 257003
[43]	Ran S, Eckberg C, Ding Q P et al. 2019 Science 365 684