⇦Chinese Physics Letters, 2023, Vol. 40, No. 9, Article code 097402 Theoretical Predictions on Superconducting Phase above Room Temperature in Lutetium-Beryllium Hydrides at High Pressures Bin Li (李斌)1*, Yeqian Yang (杨业迁)2, Yuxiang Fan (范雨香)1, Cong Zhu (朱聪)2, Shengli Liu (刘胜利)1, and Zhixiang Shi (施智祥)3 Affiliations 1School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China 2College of Electronic and Optical Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210023, China 3School of Physics, Southeast University, Nanjing 211189, China Received 15 May 2023; accepted manuscript online 8 August 2023; published online 24 August 2023 *Corresponding author. Email: libin@njupt.edu.cn Citation Text: Li B, Yang Y Q, Fan Y X et al. 2023 Chin. Phys. Lett. 40 097402    Abstract High-pressure structural search was performed on the hydrogen-rich compound LuBeH$_8$ at pressures up to 200 GPa. We found an $Fm\bar{3}m$ structure that exhibits stability and superconductivity above 100 GPa. Our phonon dispersion, electronic band structure, and superconductivity analyses in the 100–200 GPa pressure range reveal a strong electron–phonon coupling in LuBeH$_8$, while the superconducting critical temperature $T_{\rm c}$ shows a decreasing trend as the pressure increases, with $T_{\rm c}=255$ K at 200 GPa and maximal $T_{\rm c}=355$ K at 100 GPa. This study demonstrated the room-temperature superconductivity in $Fm\bar{3}m$-LuBeH$_8$, thus enriching the family of ternary hydrides. These findings provide valuable guidance for identifying new high-temperature superconducting hydrides.
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 DOI:10.1088/0256-307X/40/9/097402 © 2023 Chinese Physics Society Article Text The search for room-temperature superconducting materials is widely regarded as the “holy grail” of condensed matter physics.[1] According to Bardeen–Cooper–Schrieffer theory,[2] the superconducting critical temperature $T_{\rm c}$ is proportional to the Debye temperature, which is inversely proportional to its mass. Metallic hydrogen, being the lightest element, has a high Debye temperature and strong electron–phonon (e–ph) coupling,[3,4] which can lead to high-temperature superconductivity. However, due to the extremely high pressure required for metallic hydrogen synthesis, it is technically difficult to achieve metallic hydrogen. Therefore, researchers focused on metal hydrides instead. The metallization of hydrides can be achieved at lower pressures due to the “chemical pre-compression” effect of heavier elements.[5] The search for high-temperature superconductors in hydrogen-rich compounds has been performed since the theory of “chemical precompression” was proposed. Initially, researchers focused on natural binary hydrides, such as SiH$_4$ and AlH$_3$.[6,7] These studies were followed by the investigation of binary hydrides with new proportions, such as H$_3$S (maximum $T_{\rm c}$ is 203 K[8,9]), CaH$_6$ ($T_{\rm c}$ is 210 K at 170 GPa[10,11]), YH$_6$ ($T_{\rm c}$ is 220 K at 183 GPa[12,13]), and LaH$_{10}$ ($T_{\rm c}$ is 250–260 K at 170–200 GPa[14-16]). After exploring almost all binary hydrides, research shifted to ternary hydrides. Ternary hydrides greatly expand the variety of phases by providing more element ratios, leading to the discovery of higher superconducting transition temperatures. For example, CaYH$_{12}$ ($T_{\rm c} = 258$ K at 200 GPa[17]), Li$_2$MgH$_{16}$ ($T_{\rm c} = 473$ K at 250 GPa[18]), LaBH$_8$ ($T_{\rm c}$ = 126–156 K at 50–55 GPa[19,20]). The most prominent high-pressure high-$T_{\rm c}$ compounds are known as “superhydrides”. Superhydrides have enveloping cage-shaped hydrogen-based lattices, wrapped in positively charged metal atoms. The most prominent metal atoms are rare earth elements including lanthanum, yttrium, and cerium. However, the lutetium hydride has not received much attention.[21,22] Lutetium and lanthanum have similar electronegativity, and can dissociate hydrogen molecules into atoms, the $f$-shell filled with lutetium superhydride is expected to carry high $T_{\rm c}$. Up to a recent study, superconducting properties were observed around room temperature ($T_{\rm c} = 294$ K) in nitrogen-doped lutetium hydride at mild pressure of 10 kbar.[23] Unfortunately, despite the use of various methods, such as x-ray diffraction, elemental analysis, and Raman spectroscopy, its composition and structure have not been clarified. Furthermore, recent experimental and theoretical endeavors have reported the absence of near-ambient superconductivity in nitrogen-doped lutetium hydrides,[24-28] which is contrary to the original work by Dasenbrock.[23] The existence of superconductivity in nitrogen-lutetium hydrides is a topic of debate. In this Letter, we predict a new ternary room-temperature superconductor, LuBeH$_8$ (space group: $Fm\bar{3}m$), by searching the stable structures of lutetium-beryllium-hydrogen systems. We study its phonon dispersion, electronic band structure, e–ph couplings, and superconducting critical temperatures. Its high symmetry facilitates its good superconductivity. Through our calculations, we find that LuBeH$_8$ remains stable at 100 GPa, and the superconducting critical temperature is as high as 355 K, which is already far beyond the room temperature. We employed the in-house developed machine-learning-based crystal structure prediction package CRYSTREE[29,30] to search the stable crystal structure of the Lu–Be–H system at 100, 150 and 200 GPa. The crystal structure prediction software CRYSTREE utilizes extremal random forest regression to expedite the optimization process and to generate prospective structure candidates. The population size at each iteration is predefined as 60. The initial generation comprises randomly produced structures. Within subsequent generations, 60% of the structures are generated via the extreme random forest approach, while the remained 40% are randomly produced. The search concludes when either the enthalpy difference over the preceding 10 iterations falls below a threshold of $10^{-6}$ Ry or the maximum number of 50 iterations is reached. The results are verified by the graph theory assisted universal structure searcher MAGUS.[31] We then re-optimized the structures using the ab initio calculation of the Quantum Espresso (QE) package,[32] and calculated the phonon spectrum at different pressures using the density functional perturbation theory (DFPT).[33] The charge density and the wave function cutoff values are 600 Ry and 60 Ry, respectively. Electronic structure calculations were performed by using the method of full-potential linearization enhanced plane wave[34] with the Perdew–Burke–Ernzerhof functional. The pseudopotentials were selected from the standard solid-state pseudopotentials (SSSP) library.[35] VESTA was used to visualize the crystal structure.[36] Fermi surfaces were visualized using Fermisurfer.[37] A $4\times4\times4$ $q$-grid and a $12\times12\times12$ $k$-point grid were selected to calculate electron phonon coupling and integration in the Brillouin zone using the optimized tetrahedron method.[38] Dense $24\times24\times24$ grids are used to evaluate precise e–ph interaction matrices. Finally, $T_{\rm c}$ was calculated using the Allen–Dynes modified McMillan equation.[11]
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Fig. 1. The crystal structure of $Fm\bar{3}m$-LuBeH$_8$. The yellow, blue, and pink balls represent the lutetium, beryllium, and hydrogen atoms, respectively.
The crystal structure of LuBeH$_8$ is shown in Fig. 1. The atoms Lu, Be, and H occupy the 4$b$ (0.5, 0.5, 0.5), 4$a$ (0, 0, 0), and 32$f$ (0.655, 0.155, 0.655) Wyckoff positions in the crystal structure. The H atoms form a polyhedron surrounding the Lu atom. Be atoms are inserted between the polyhedra. $Fm\bar{3}m$-LuBeH$_8$ is structurally similar to sodium hydrides, such as LaH$_{10}$, where guest atoms such as La act as scaffolds and can apply mechanical pressure to the lattice,[11,22] a mechanism commonly referred to as chemical precompression. By linking this mechanism to LuBeH$_8$, the Be atoms occupy the sites between the next closest Lu atoms, effectively filling the remaining interspace in the whole structure. The denser Lu–Be scaffold is then formed, which firmly binds the highly symmetrical metal hydrogen lattice, allowing it to remain stable at lower pressures.
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Fig. 2. $Fm\bar{3}m$-LuBeH$_8$ phonon dispersion, phonon density of states (PHDOS), Eliashberg spectral function $\alpha^2F(\omega)$ and e–ph coupling $\lambda(\omega)$ at 100 GPa. The PHDOS projections on Lu, Be, and H are color-coded in red, green, and blue, respectively, to aid in visual interpretation.
The phonon properties of $Fm\bar{3}m$-LuBeH$_8$ were calculated using the DFPT scheme. The calculation determined that the lower pressure limit of LuBeH$_8$ is 100 GPa, above which no imaginary branches of the phonon spectrum exist. We show the phonon dispersion curve and phonon state density (PHDOS) of LuBeH$_8$ at 100 GPa in Fig. 2. It can be seen that there is no imaginary vibration in the entire Brillouin zone, indicating that the structure is dynamically stable at this pressure, and the vibration of phonons is mainly distributed in the middle and low frequencies within the entire frequency range. From the PHDOS diagram, it can be seen that the vibration in the low frequency region is mainly from the Lu atom, and there is a significant phonon peak located at 100 cm$^{-1}$, and the Be and H atoms in this range are barely vibrating. The vibration in the middle and high frequency range ($\geq$ 100 cm$^{-1}$) mainly comes from the H and Be atoms. We also show the integration of the Eliashberg function $\alpha^{2}F(\omega)$ and e–ph coupling $\lambda({\omega})$ in the panel on the far right. By integrating the Eliashberg function $\alpha^{2}F(\omega)$, we can get $\lambda=2\int \alpha^2F(\omega)\omega^{-1}d\omega$ and logarithmic mean phonon frequency $\omega_{\log}={\exp}[2\lambda^{-1}\int d\omega\alpha^{2}F(\omega)\omega^{-1}{\log}\omega]$. According to the coupling curve, it is not difficult to find that the coupling integral below 1200 cm$^{-1}$ accounts for most of the total coupling contribution.
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Fig. 3. $Fm\bar{3}m$-LuBeH$_8$ electronic band structure and partial density of states (DOS) at 100 GPa. The unit of DOS is eV$^{-1}\cdot$f.u.$^{-1}$, and elements are color-coded, with red, green, and blue representing Lu, Be, and H, respectively. The Fermi level serves as the zero point of the energy scale.
In Fig. 3, we show the electronic band structure of $Fm\bar{3}m$-LuBeH$_8$ at 100 GPa and the atomic projection density of states (DOS) in units of eV$^{-1}\cdot$f.u.$^{-1}$. It shows that the structure exhibits metallic behavior, as evidenced by more than one band crossing the Fermi level. Near the Fermi level, DOS is dominated by Lu and H atoms. From the electronic energy band, it can be seen that there is a Dirac-cone like band crossing at point $W$ at $\sim$ $-3.2$ eV. The corresponding DOS curve around $-6$ eV has a very high peak, and the Lu atom provides a very large density state due to the $f$ orbital contribution. Figure 4 shows the Fermi surfaces of $Fm\bar{3}m$-LuBeH$_8$, shadowed by the distributions of Fermi velocity, with changing color from blue to red to indicate an increase in Fermi velocity. At 100 GPa, the Fermi surface of $Fm\bar{3}m$-LuBeH$_8$ consists mainly of three parts [Figs. 4(b)–4(d)], six semi-oval hollow pockets are regularly distributed in the Brillouin area, each pocket is wrapped by a four-leaf clover-shaped sheet with four sharp corners, a large electron sphere around the $\varGamma$ point, and eight small dots regularly around this electron sphere.
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Fig. 4. The Fermi surfaces of $Fm\bar{3}m$-LuBeH$_8$ at 100 GPa, highlighting the Fermi velocity with a color gradient from blue to red to indicate relative velocity. The overall Fermi surface is shown in (a), and the three distinct parts of the surface are shown in (b)–(d). (e) The Brillouin zone with high symmetry points. where $g_1$, $g_2$, and $g_3$ are the coordinate axes in reciprocal space.
In order to estimate the superconducting critical temperature of LuBeH$_8$ under pressure and to find the maximum $T_{\rm c}$, we performed a linear response calculation for the e–ph coupling, based on the calculated Eliashberg spectral function $\alpha^{2}F(\omega)$, \begin{align} {\alpha ^2}F(\omega)=\frac{1}{{2\pi N(0)}}\sum\limits_{_{\scriptstyle Qv}}\frac{\gamma_{_{\scriptstyle Qv}}}{\omega _{Qv}}\delta (\omega - \omega_{_{\scriptstyle Qv}}), \tag {1} \end{align} where $N(0)$ is density of states at Fermi level, $\gamma_{_{\scriptstyle Qv}}$ is the e-ph linewidth, and $\omega_{_{\scriptstyle Qv}}$ is the phonon frequency at phonon branch $v$ and wavevector $Q$. The e–ph coupling constant $\lambda$ is obtained as \begin{align} \lambda = 2\int_0^\infty {\frac{\alpha ^2F(\omega)}{\omega}} d\omega. \tag {2} \end{align} In addition, the critical temperature is calculated by the Allen–Dynes modified McMillan formula[39] \begin{align} T_{\rm c} = f_1f_2\frac{\omega _{\log}}{1.2}\exp\Big[-\frac{1.04(1+\lambda)}{\lambda-{\mu ^*}(1 + 0.62\lambda)}\Big], \tag {3} \end{align} where $\lambda$ is the e–ph coupling intensity, $\omega_{\log}$ is the logarithmic mean phonon frequency, and the coulomb pseudopotential parameter $\mu^*$ is set to 0.1; $\omega_{\log}$ is defined as \begin{align} {\omega _{\log}}=\exp \Big[\frac{2}{\lambda }\int_0^\infty {\frac{{d\omega }}{\omega }{\alpha ^2}F(\omega)\ln \omega}\Big]. \tag {4} \end{align} The factors $f_1$ and $f_2$ depend on $\lambda$, $\mu^*$, $\omega_{\log}$, and mean square frequency ${\overline{\omega}_{2}}$, \begin{align} {f_1}{f_2}=\,&\sqrt[3]{1 + \Big(\frac{\lambda }{2.46(1 + 3.8\mu ^*)}\Big)^{3/2}}\notag\\ &\cdot\Big(1 - \frac{{\lambda ^2}(1 - {\overline{\omega}_{2}}/\omega _{\log })}{\lambda ^2 + 3.312(1 + 6.3\mu ^*)^2}\Big). \tag {5} \end{align} The detailed calculation results are listed in Table 1.
Table 1. The main superconductivity performance of $Fm\bar{3}m$-LuBeH$_8$ at different pressures from 100 to 200 GPa.
Structure Pressure (GPa) $T_{\rm c}$ (K) $\lambda$ $\omega_{\log}$ (K)
$Fm\bar{3}m$-LuBeH$_8$ 100 355 7.0 785
120 293 4.2 900
150 269 3.0 1100
180 274.8 2.5 1160
200 255 2.4 1265
The calculation results show that $Fm\bar{3}m$-LuBeH$_8$ exhibits metallic behavior while maintaining kinetic stability at 100 GPa, and its superconducting critical temperature is as high as 355 K, which is far beyond the temperature required for room temperature. Compared to the $Fm\bar{3}m$-LaBeH$_8$ structure proposed by Zhang et al. with a similar composition ($T_{\rm c} = 185$ K at 20 GPa)[40] and other AXH$_8$ compounds,[41-44] LuBeH$_8$ exhibits the highest $T_{\rm c}$ due to its larger e–ph coupling constant $\lambda$ and higher logarithmic mean phonon frequency $\omega_{\log}$. The superconducting quality merit $S = T\cdot[P^2+T^{2}_{\rm MgB_{2}}]^{-1/2}$ is commonly used to evaluate the significance of superconductors.[45] Our calculation indicates that the quality merit $S$ in LuBeH$_8$ is 3.30, which is higher than the quality merits of other hydrides such as LaH$_{10}$ and H$_3$S ($1 < S < 2$).[40,45] As the pressure increases, the overall superconducting critical temperature tends to decrease, which may be attributed to the hardening of phonon branches that impede e–ph coupling and superconductivity. In conclusion, we have found the superconducting phase $Fm\bar{3}m$-LuBeH$_8$ of a novel superhydride through first-principles calculations and crystal structure prediction. LuBeH$_8$ remains dynamically stable above 100 GPa while reaching a maximum superconducting critical temperature of 355 K. Excellent superconducting properties result from high structural symmetry and the efficient stacking of beryllium in the lattice, which allows stable mechanical pressure to be applied. $T_{\rm c}$ shows a decreasing tendency with increasing pressure. The increasing pressure will harden the phonon branches and inhibit the e–ph coupling of the structure as well as superconductivity. Our study demonstrates the potential applicability and utility of machine learning for crystal structure predictions. Additionally, our findings contribute to the search for room-temperature superconducting hydrides. Acknowledgements. This work was supported by the National Key R&D Program of China (Grant No. 2018YFA0704300), the National Natural Science Foundation of China (Grant No. U1932217), and the Start-Up Fund of Nanjing University of Posts and Telecommunications (Grant Nos. NY219087 and NY220038). Some of the calculations were performed on the supercomputer in the Big Data Computing Center (BDCC) of Southeast University. We thank Professor Haihu Wen for valuable discussion. 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