⇦Chinese Physics Letters, 2021, Vol. 38, No. 10, Article code 107401Express Letter High $T_{\rm c}$ Superconductivity in Heavy Rare Earth Hydrides Hao Song (宋昊)1, Zihan Zhang (张子涵)1, Tian Cui (崔田)2,1*, Chris J. Pickard3,4, Vladimir Z. Kresin5, and Defang Duan (段德芳)1* Affiliations 1State Key Laboratory of Superhard Materials, College of Physics, Jilin University, Changchun 130012, China 2Institute of High Pressure Physics, School of Physical Science and Technology, Ningbo University, Ningbo 315211, China 3Department of Materials Science & Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom 4Advanced Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba, Sendai, 980-8577, Japan 5Lawrence Berkeley Laboratory, University of California at Berkeley, Berkeley, CA 94720, USA Received 20 August 2021; accepted 2 September 2021; published online 8 September 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 12122405, 51632002, and 11974133), the Program for Changjiang Scholars and Innovative Research Team in Universities (Grant No. IRT_15R23). C.J.P. acknowledges financial support from the Engineering and Physical Sciences Research Council (Grant No. EP/P022596/1).
*Corresponding authors. Email: cuitian@nbu.edu.cn; duandf@jlu.edu.cn Citation Text: Song H, Zhang Z H, Cui T, Pickard C J, and Kresin V Z et al. 2021 Chin. Phys. Lett. 38 107401    Abstract Sulfur and lanthanum hydrides under compression display superconducting states with high observed critical temperatures. It has been recently demonstrated that carbonaceous sulfur hydride displays room temperature superconductivity. However, this phenomenon has been observed only at very high pressure. Here, we theoretically search for superconductors with very high critical temperatures, but at much lower pressures. We describe two of such sodalite-type clathrate hydrides, YbH$_{6}$ and LuH$_{6}$. These hydrides are metastable and are predicted to superconduct with $T_{\rm c} \sim 145$ K at 70 GPa and $T_{\rm c} \sim 273$ K at 100 GPa, respectively. This striking result is a consequence of the strong interrelationship between the $f$ states present at the Fermi level, structural stability, and the final $T_{\rm c}$ value. For example, TmH$_{6}$, with unfilled 4$f$ orbitals, is stable at 50 GPa, but has a relatively low value of $T_{\rm c}$ of 25 K. The YbH$_{6}$ and LuH$_{6}$ compounds, with their filled $f$-shells, exhibit prominent phonon “softening”, which leads to a strong electron-phonon coupling, and as a result, an increase in $T_{\rm c}$. DOI:10.1088/0256-307X/38/10/107401 © 2021 Chinese Physics Society Article Text It has been predicted for a long time that metallic monoatomic hydrogen under high pressure enters a superconducting state, with a high value of critical temperature $T_{\rm c}$.[1] However, very high pressures are required to observe this phenomenon; and it has yet to be realized.[2–7] A promising route to room temperature superconductivity is to study hydrogen-rich compounds (or hydrides). Because of chemical precompression effects, they metallize at lower pressures,[8] see reviews.[9–11] The presence of two ions, light (e.g., H) and heavy, in the unit cell, leads to an appearance of optical high frequency phonon modes (vibrations of H ions) with a large phonon density of states (DOS) across the whole range of phonon momenta. This increases the magnitude of the electron-phonon interaction. An additional contribution due to the heavy ions (acoustic modes) further increases the electron-phonon interaction. Among such hydrides are H$_{3}$S[12,13] and LaH$_{10} $,[14,15] for which high $T_{\rm c}$ values were predicted and later confirmed experimentally.[16–20] These materials display values of $T_{\rm c}$ exceeding 200 K at pressures above 150 GPa. Recently, it has been reported that the compound of hydrogen, carbon and sulfur displays the transition at 288 K,[21] that is, at room temperature. However, because the high pressures require ($267 \pm 10$ GPa), the observation and corresponding analysis are rather difficult, so are practical or technological applications. The observation of high temperature superconductivity in the hydrides requires pressures as high as 150 GPa. The next frontier is observation of room-temperature superconductivity at significantly lower pressures (with a clear final goal of reaching ambient pressure). In this Letter, we present a step in this direction. Indeed, it will be demonstrated that high values of $T_{\rm c}$ are expected to be observed in YbH$_{6}$ with $T_{\rm c} \sim 145$ K at 70 GPa and, most strikingly, in LuH$_{6}$ with $T_{\rm c} \sim 273$ K at 100 GPa. Given the large number of superconducting hydrides,[9–11,22] now it is time to develop general concepts and define major factors that favor high values of $T_{\rm c}$. Tm, Yb, and Lu occupy special positions among rare earth (RE) elements. Yb has filled 4$f$ orbital with the configuration 4$f^{14}$ and does not display ferromagnetic or antiferromagnetic order. Its electronegativity is 1.1, similar to those of Ca and Y. It exhibits a mixed valence and is a superconductor at high pressure.[23] Adjacent to ytterbium, thulium has 13$e$ filled 4$f$ orbitals with 4$f^{13}$. Lutetium has filled 4$f$ orbital and an extra 5$d$ electron with 4$f^{14}5d^{1}$, and it is the hardest and most dense metal among the RE elements. The $f$ electrons in the hydrides containing these elements are expected to affect superconductivity and structural stability. We will discuss the corresponding role of the $f$ states in these superconducting hydrides in detail below. As is well-known, on-site Coulomb interactions are particularly significant for localized $f$ electrons. We first evaluate the equation of state (EOS) for YbH$_{2}$. Then we compare it with the experimental EOS to assess the reliability of our DFT calculations (see Fig. S1). For the high-pressure phase $P6_{3}/mmc$ of YbH$_{2}$, the ultra-soft pseudo-potentials at the DFT level provides a good agreement between the theory and experiment (the detailed discussion is in the Supplementary Materials). Furthermore, we construct the high-pressure phase diagram of the Yb–H system with both DFT and DFT+$U$ ($U=5$ eV), through extensive structure searches via the ab initio random structure searching method as implemented in the AIRSS code.[24,25] The results are presented in Fig. S2. One can see that the convex hull is essentially unchanged between DFT+$U$ and DFT. In addition, we calculate $T_{\rm c}$ of Yb at DFT level and find that $T_{\rm c}$'s are 1.6 K at 86 GPa and 4.6 K at 180 GPa. There is a good agreement between our calculation and the experimental measurements[23] (1.4 K at 86 GPa and 4.6 K at 179 GPa). Therefore, we consider that the DFT level calculations for the Yb–H system above 50 GPa are acceptable. Combining the convex hull (Fig. S2), enthalpy differences (Fig. S3) and phonon dispersion (Figs. S4 and S5) of the Yb–H system under high pressure, we construct the pressure-composition phase diagram in Fig. 1(a). Note that eight stoichiometries YbH, YbH$_{2}$, Yb$_{2}$H$_{5}$, YbH$_{3}$, YbH$_{4}$, YbH$_{6}$, YbH$_{8}$ and YbH$_{12}$ are stable over different pressure ranges. The crystal structures of our predicted stable Yb hydrides are shown in Fig. S6; the lattice parameters are provided in Table S1. For YbH, the $Fm\bar{3}m$ phase, with hydrogens occupying octahedral interstices, transforms into a $Cmcm$ phase at 170 GPa. YbH$_{2}$ adopts the space group $P6_{3}/mmc$ at low pressure, which is consistent with the experimental measurements.[26] We find that it transforms to a $P6/mmm$ phase at 233 GPa. $R\bar{3}m$-Yb$_{2}$H$_{5}$ is stable between 145 and 250 GPa. YbH$_{3}$ favors an $Fm \bar{3}m$ structure below 145 GPa, the hydrogen atoms occupy both octahedral and tetrahedral interstices. YbH$_{4}$ maintains the space group $I4/mmm$ isotypic with CaH$_{4} $[27] and YH$_{4}$[28] in the pressure range of 70–300 GPa, which contains monatomic H and molecular H$_{2}$. YbH$_{8}$ with $P6_{3}/mmc$ symmetry is stable above 240 GPa; as determined by the phonon dispersion (see Fig. S5). The high hydrogen content YbH$_{12}$, with $R\bar{3}$ symmetry, is stable in the pressure range of 70–240 GPa. For the case of YbH$_{6}$, it has two phases of $C2/m$ and $Im \bar{3}m$, and the phase transition occurs at 158 GPa. The $Im \bar{3}m$ phase contains clathrate H$_{24}$ cage and consists of eight H$_{6}$ hexagons and six H$_{4}$ squares. It is isostructural with $Im \bar{3}m$ CaH$_{6} $[27] and YH$_{6}$,[28] which have been synthesized at high pressure to exhibit high $T_{\rm c}$ of 215 K and 224 K, respectively.[29,30] Electron localization functions (ELF) of cubic YbH$_{6}$ show weak covalent interaction between the H atoms, with an ELF of 0.58 (see Fig. 1(c) and Fig. S7). The high pressure phase diagram and crystal structure of the Yb–H system are very similar to the Ca–H and Y–H systems. Therefore, they could have similar properties, such as high-temperature superconductivity.
cpl-38-10-107401-fig1.png
Fig. 1. (a) Pressure-composition phase diagram of the Yb–H system. [(b), (c)] The crystal structures and electron localization functions (ELFs) of $Im \bar{3}m$-YbH$_{6}$. The large cyan balls and small pink balls represent ytterbium and hydrogen, respectively.
Structure searches for the other two heavy rare earth (RE = Tm, Lu) hydrides adjacent to Yb were also performed using AIRSS.[24,25] The resulting convex hull diagrams at different pressures are shown in Fig. S8. We also find that cubic TmH$_{6}$ and LuH$_{6}$ are located on the convex hull at 200 and 400 GPa, respectively, indicating that they are thermodynamically stable at these pressures. Combining enthalpy differences and phonon dispersions of the Tm–H and Lu–H systems under high pressures, we find that the cubic TmH$_{6}$ and LuH$_{6}$ are stable above 130 GPa and 370 GPa, respectively. The lattice parameters of predicted stable Tm–H and Lu–H compounds are provided in Table S2. We calculate the electronic properties of $Im \bar{3}m$-YbH$_{6}$, as shown in Fig. 2(a). Remarkably, the 4$f$ orbitals associated with the Yb atom form a set of localized and almost non-dispersive bands that appear about 1 eV below the Fermi level. To gain an insight into the role of the $f$ electrons in the heavy rare earth (RE = Tm, Yb, Lu) sodalite-like hydrides, we calculate the band structure and density of electronic states of TmH$_{6}$ and LuH$_{6}$ at 100 GPa. Compared with YbH$_{6}$ and LuH$_{6}$, TmH$_{6}$ has unfilled 4$f$ orbitals. We can see that the $f$ electrons of TmH$_{6}$ dominate the Fermi level [see Fig. 2(b)]. For LuH$_{6}$, the fully filled 4$f$ orbitals and one extra electron in the 5$d$ orbitals leads to the 4$f$ electrons moving downwards in the band structure [see Fig. 2(c)]. In addition, we investigate magnetic properties for TmH$_{6}$, YbH$_{6}$ and LuH$_{6}$ at the pressure range of 50 to 150 GPa in Fig. 2(d). The results show that YbH$_{6}$ and LuH$_{6}$ are nonmagnetic, while TmH$_{6}$ is magnetic, and the strength of magnetism decreases with increasing pressure [see Fig. 2(d)]. The Mulliken population scheme[31] provides a qualitative analysis on atomic charge, bond population, charge transfer. At 100 GPa, electrons of YbH$_{6}$ transfer from 4$f$ orbitals of Yb to 1$s$ orbital of H by 0.34$|e|$ ($e$ being the elementary charge). In YbH$_{6}$, the 4$f$ orbitals of Yb thus remain nearly fully filled (e.g., 13.66 out of 14$|e|$), exhibiting no local magnetism. Electrons also transfer from Yb 6$s$ and 5$p$ orbitals to Yb 5$d$ orbitals and H $s$ orbitals. In summary, Yb atoms play the role of electron donors and each of them loses approximately 1.0$|e|$; each H uniformly gains $\frac{1}{6}|e|$. The electron transfer in TmH$_{6}$ is very similar to that in YbH$_{6}$. Tm transfers about 1.0$|e|$ to H atoms, but unfilled 4$f$ orbitals have less electrons of 12.86$|e|$ resulting in magnetic moment of $\sim$$1.2 \mu_{\scriptscriptstyle {\rm B}}$ at 100 GPa. For LuH$_{6}$, there is almost no electron transfer out of the 4$f$ orbitals, but as compared to YbH$_{6}$ and TmH$_{6}$, three electrons transfer from Lu 6$s$ and 5$d$ orbitals to hydrogen atoms.
cpl-38-10-107401-fig2.png
Fig. 2. Band structure and projected density of electronic states (PDOS) of (a) YbH$_{6,}$ (b) TmH$_{6}$ and (c) LuH$_{6}$ at 100 GPa. (d) Magnetic moments of XH$_{6}$ (X = Tm, Yb, Lu) compounds at high pressure.
Generally, the high $T_{\rm c}$ superconducting state of the hydrides is created by strong electron-phonon coupling (EPC) to high frequency optical phonons (see, e.g., Refs. [12,14,32,33]). Firstly, we calculate the phonon spectrum for $Im \bar{3}m$-YbH$_{6}$ at 70 GPa and 200 GPa, for TmH$_{6}$ at 50 GPa and for LuH$_{6}$ at 100 GPa (Fig. 3). The absence of any imaginary frequency indicates their dynamical stability. It is notable that YbH$_{6}$ is dynamically stable down to relatively low pressures (70 GPa). The phonon modes are separated into two parts: the acoustic phonon modes (low frequencies $ < 220$ cm$^{-1}$) dominated by vibrations of Yb atoms and the optical phonon modes (high frequencies) from the vibrations of H atoms. For YbH$_{6}$, the EPC parameter $\lambda$ is 2.2 at 70 GPa, and the optical phonon modes contribute to 90% of the total $\lambda$. As shown in Fig. 3(a), a striking feature of the phonon spectrum for YbH$_{6}$ is the presence of soft phonon modes along $\varGamma$–$H$ and $H$–$N$ directions. However, with increasing pressure from 70 to 100 GPa, phonon branches rapidly harden and the EPC parameter $\lambda$ substantially decreases from 2.2 to 1.3, which lead to a large reduction of $T_{\rm c}$ from 145 to 116 K (see Fig. S9). We can see that the phonon softening at 70 GPa does boost the EPC strength. The phonon dispersion and EPC parameter $\lambda$ of TmH$_{6}$ and LuH$_{6}$ are also calculated [see Figs. 3(c) and 3(d)]. It is shown that the EPC parameter $\lambda$ of TmH$_{6}$ is 0.72, optical phonon modes contribute to 84% of the total $\lambda$; as for the critical temperature $T_{\rm c}$, it is only 25 K. In sharp contrast, LuH$_{6}$ is dynamically stable at 100 GPa, and its EPC parameter $\lambda$ is 3.60, optical phonon modes contribute to 85% of the total $\lambda$, yield a high $T_{\rm c} = 273$ K. Moreover, several phonon branches show significant phonon softening and it is clearly seen that $\lambda_{{\rm q },{\rm v}}$ is enhanced in these regions of phonon softening. The critical temperatures of the remaining predicted stable Lu–H compounds are calculated, for example, LuH$_{5}$ is superconducting with $T_{\rm c} \sim 89$ K at 150 GPa, see detail in Table S4. In each case, the EPC parameter $\lambda$ decreases with increasing pressure. The values of $T_{\rm c}$ for TmH$_{6}$, YbH$_{6}$ and LuH$_{6}$, mentioned above, are estimated with use of self-consistent iterations (IA) for solution of the Eliashberg equations, as shown in Fig. S11. We also calculate the critical temperature using the Allen–Dynes-modified McMillan equation (Mc-A-D) and the Gor'kov–Kresin equation (G-K). The obtained values of $T_{\rm c}$ appear to be slightly lower than those obtained with IA, see Table S3.
cpl-38-10-107401-fig3.png
Fig. 3. Phonon dispersion, phonon density of state, spectral function $\alpha^{2}F(\omega)$ and integral EPC $\lambda$ of $Im \bar{3}m$-YbH$_{6}$ (a) at 70 GPa and (b) 200 GPa, TmH$_{6}$ (c) at 50 GPa, LuH$_{6}$ (d) at 100 GPa. Red circles show mode-resolved electron-phonon coupling constants $\lambda_{{\rm q },{\rm v}}$ and the radius of the circles are proportional to the EPC strength.
The value of the isotope coefficient for the H$\to$D substitutions directly reflects to the interplay between the optical and acoustic phonon modes. We calculate the isotope coefficient for cubic XH$_{6}$ hydrides (X = Yb, Lu, Tm, Ca, Sc, Y) using the two-coupling constants method[34] [see Eq. (S10)]. One can see that the isotope coefficients for cubic lanthanide hydrides are large and close to the optimum value ($\alpha_{{\max}}=0.5$), see Fig. 4. Such large values reflect the dominant contribution of the optical modes (see Table S3). As for CaH$_{6}$ and ScH$_{6}$, the values of isotope coefficients are also large, while smaller than YbH$_{6}$ and LuH$_{6}$. They are comparable with those for H$_{3}$S, see Refs. [34,35]. The critical temperature of XD$_{6}$ ($T_{\rm c}^{\rm D}$) was estimated according to the equation of $T_{\rm c}/T_{\rm c}^{\rm D} = (M_{\rm D}/M_{\rm H})^{\alpha}$, where $T_{\rm c}$ is obtained from the G-K equation (see Table S5). We now focus on the impact of the $f$ states upon the superconducting properties. It is noted that highly localized $f$ electrons in RE hydrides affect superconductivity adversely.[15,36] For example, according to experimental observations at high pressure, LaH$_{10}$ and CeH$_{9}$ have high $T_{\rm c}$ of 250–160 K and 117 K, respectively, while $T_{\rm c}$'s of PrH$_{9} $[37] and NdH$_{9} $[38] are found to be as low as 10 K. It is interesting that the value of the critical temperature for RE hydrides gradually decreases with an increase in filling of the $f$-shell and superconductivity disappears for the compound EuH$_{9}$ with half-filled $f$-shells (Eu-$f^{7}$).[39]
cpl-38-10-107401-fig4.png
Fig. 4. Superconductivity critical temperature $T_{\rm c}$ and EPC parameter $\lambda$ of various clathrate hydrides at high pressures. The pressure values are 150, 120, 50, 70 and 100 GPa for CaH$_{6}$, YH$_{6}$, TmH$_{6}$, YbH$_{6}$ and LuH$_{6}$, respectively.
As mentioned above, we study various compositions of XH$_{n}$ (X = Tm, Yb, Lu). The most interesting hydrides at high pressure are the TmH$_{6}$, YbH$_{6}$, and LuH$_{6}$ compounds. They are dynamically stable at 50 GPa, 70 GPa and 100 GPa with $T_{\rm c}$ of 25 K, 145 K and 273 K (ice point temperature), respectively. We uncover a non-trivial and non-monotonic dependence of $T_{\rm c}$ on the degree of the $f$-shell filling. Beyond filling half of the $f$-shell, the $T_{\rm c}$ of heavy RE hydrides gradually increases upon further filling. It reaches a maximum in the last RE hydrides LuH$_{6}$, which is comparable with LaH$_{10}$. Importantly, the pressure providing the dynamical stability for LuH$_{6}$ is much lower than that for LaH$_{10}$ (171 GPa for experimental observation and 210 GPa for predictions in the harmonic approximation). As a result, one can expect that the superconducting state of LuH$_{6}$ with values of $T_{\rm c}$ that are close to room temperature can be observed at much lower pressure. To gain more insight into the correlation between the presence of the $f$ states, stability, and the value of $T_{\rm c}$, we compute the phonon dispersion and EPC in an extreme case of freezing the $f$ manifold, through the Yb pseudopotential (see Fig. S10). The frozen-$f$ calculations show that the cubic structure becomes dynamically stable just above 100 GPa; as for the EPC, the parameter $\lambda$ is approximately equal to 5.5; then $T_{\rm c} = 262$ K with $\mu^{\ast} =0.1$. An increase in pressure leads to sharp decrease in the value of the EPC parameter $\lambda$ ($\lambda = 2.2$ at 200 GPa), but $\lambda$ is still more than two times larger compared to its value in the presence of $f$ electrons. One can conclude that the presence of the $f$ electrons in valence state leads to suppression of the phonon “softening”. The dynamical stability of structure is enhanced, but the strength of the EPC is reduced. Therefore, the presence of the $f$ electronic states at the Fermi surface increases the dynamical stability but reduces the superconducting $T_{\rm c}$. The hydride LuH$_{6}$, with filled 4$f$ orbitals and 1$e$ filled 5$d$ valence orbitals, is a remarkable compound with the highest $T_{\rm c}$ among other cubic XH$_{6}$ (see Fig. 4); its $T_{\rm c}$ value is comparable with that of the LaH$_{10}$ ($T_{\rm c} = 250$–260 K, La also has 1$e$ filled 5$d$ orbitals). Moving from the light to heavy RE, one can observe that $T_{\rm c}$ initially decreases and then gradually increases upon further filling of the $f$-shells above its half filling. Finally, it reaches a maximum $T_{\rm c}$ for the last RE hydride, LuH$_{6}$. It has been a long-term goal to obtain high-temperature superconductors in hydrides at pressures below 100 GPa. Here, we report a prediction of the realization of this dream in YbH$_{6}$ and LuH$_{6}$ compounds. As mentioned above, multi-megabar pressures are currently required to force the hydrides into the superconducting states to achieve high $T_{\rm c}$. However, there should be a compromise and balance between the value of $T_{\rm c}$ and the required pressure. One can propose the figure of merit $S$ (see Eq. (8) in Ref. [40]), which makes the compromise explicit. Lower values of both $S$(H$_{3}$S) and $S$(LaH$_{10}$) equal to 1.3 reflect the very high pressures required to achieve the superconducting states. As for our results, $S$(YbH$_{6}$) = 1.8 and $S$(LuH$_{6}$) = 2.5; both values exceed those for $S$(H$_{3}$S) and $S$(LaH$_{10}$). When we plot these $S$ values and pressures in figure of merit $S$ (see Fig. S12 and Fig. 10 in Ref. [40]), we find that LuH$_{6}$ sits somewhere between the iron based superconductors and the cuprates. In summary, we have investigated the heavy RE hydrides, in particular those with the sodalite hydrogen cage structure, at high pressure. The temperature of the transition into the superconducting state correlates strongly with the presence of $f$ states at the Fermi level. For example, the TmH$_{6}$ compound is stable at low pressure (50 GPa), whereas its $T_{\rm c}$ is relatively low ($T_{\rm c}=25$ K). One predicts much higher $T_{\rm c}$ for YbH$_{6}$ ($T_{\rm c}= 145$ K at 70 GPa). The optimal case corresponds to LuH$_{6}$. Remarkably, the critical temperature for this compound is predicted to reach $T_{\rm c} = 273$ K at $P = 100$ GPa. Such a pressure is much below those required for other high $T_{\rm c}$ hydrides. One may speculate that cubic clathrate ternary hydrides doped with heavy RE elements Yb and Lu could be synthesized at much lower pressure, exhibiting high temperature superconducting states. Acknowledgement. We thank Professor Hanyu Liu and Professor Hongjian Zhao for interesting and stimulating discussions. Parts of the calculations were performed in the High Performance Computing Center of Jilin University and on TianHe-1(A) at the National Supercomputer Center in Tianjin. References Metallic Hydrogen: A High-Temperature Superconductor?Ultrahigh-pressure transitions in solid hydrogenOptical studies of solid hydrogen to 320 GPa and evidence for black hydrogenHigh-pressure Raman study of solid hydrogen up to 300 GPaEvidence for a new phase of dense hydrogen above 325 gigapascalsObservation of the Wigner-Huntington transition to metallic hydrogenStructure and Metallicity of Phase V of HydrogenHydrogen Dominant Metallic Alloys: High Temperature Superconductors?Structure and superconductivity of hydrides at high pressuresAb Initio Approach and Its Impact on SuperconductivityPressure-induced metallization of dense (H2S)2H2 with high-Tc superconductivityPressure-induced decomposition of solid hydrogen sulfidePotential high- T c superconducting lanthanum and yttrium hydrides at high pressureHydrogen Clathrate Structures in Rare Earth Hydrides at High Pressures: Possible Route to Room-Temperature SuperconductivityEvidence for Superconductivity above 260 K in Lanthanum Superhydride at Megabar PressuresSuperconductivity at 250 K in lanthanum hydride under high pressuresCrystal structure of the superconducting phase of sulfur hydrideConventional superconductivity at 203 kelvin at high pressures in the sulfur hydride systemSuperconductivity of Lanthanum Superhydride Investigated Using the Standard Four-Probe Configuration under High PressuresRoom-temperature superconductivity in a carbonaceous sulfur hydrideMetal-Element-Incorporation Induced Superconducting Hydrogen Clathrate Structure at High PressurePressure-Induced Superconductivity in Elemental Ytterbium MetalAb initio random structure searchingHigh-Pressure Phases of SilaneA New High-Pressure Phase and the Equation of State of YbH 2Superconductive sodalite-like clathrate calcium hydride at high pressuresPressure-stabilized superconductive yttrium hydridesExperimental observation of superconductivity at 215 K in calcium superhydride under high pressuresAnomalous High‐Temperature Superconductivity in YH 6Population analysis of plane-wave electronic structure calculations of bulk materialsHigh- T c state of lanthanum hydridesHydrogen Pentagraphenelike Structure Stabilized by Hafnium: A High-Temperature Conventional SuperconductorColloquium : High pressure and road to room temperature superconductivityCardioprotective effects of Cu(II)ATSM in human vascular smooth muscle cells and cardiomyocytes mediated by Nrf2 and DJ-1On Distribution of Superconductivity in Metal HydridesSuperconducting praseodymium superhydridesHigh-Pressure Synthesis of Magnetic Neodymium PolyhydridesNovel Strongly Correlated Europium SuperhydridesSuperconducting Hydrides Under Pressure
[1] Ashcroft N W 1968 Phys. Rev. Lett. 21 1748
[2] Mao H K and Hemley R J 1994 Rev. Mod. Phys. 66 671
[3] Loubeyre P, Occelli F, and LeToullec R 2002 Nature 416 613
[4] Huang X, Li F, Huang Y, Wu G, Li X, Zhou Q, Liu B, and Cui T 2016 Chin. Phys. B 25 037401
[5] Dalladay-Simpson P, Howie R T, and Gregoryanz E 2016 Nature 529 63
[6] Dias R P and Silvera I F 2017 Science 355 715
[7] Monserrat B, Drummond N D, Dalladay-Simpson P, Howie R T, López R P, Gregoryanz E, Pickard C J, and Needs R J 2018 Phys. Rev. Lett. 120 255701
[8] Ashcroft N W 2004 Phys. Rev. Lett. 92 187002
[9] Duan D F, Liu Y X, Ma Y B, Shao Z, Liu B B, and Cui T 2017 Natl. Sci. Rev. 4 121
[10] Duan D F, Yu H Y, Xie H, and Cui T 2019 J. Supercond. Novel Magn. 32 53
[11]Bi T, Zarifi N, Terpstra T, and Zurek E 2019 Reference Module in Chemistry, Molecular Sciences and Chemical Engineering (Amsterdam: Elsevier) pp 1–36
[12] Duan D F, Liu Y, Tian F, Li D, Huang X, Zhao Z, Yu H, Liu B, Tian W, and Cui T 2015 Sci. Rep. 4 6968
[13] Duan D F, Huang X, Tian F, Li D, Yu H, Liu Y, Ma Y, Liu B, and Cui T 2015 Phys. Rev. B 91 180502(R)
[14] Liu H, Naumov I I, Hoffmann R, Ashcroft N W, and Hemley R J 2017 Proc. Natl. Acad. Sci. USA 114 6990
[15] Peng F, Sun Y, Pickard C J, Needs R J, Wu Q, and Ma Y 2017 Phys. Rev. Lett. 119 107001
[16] Somayazulu M, Ahart M, Mishra A K, Geballe Z M, Baldini M, Meng Y, Struzhkin V V, and Hemley R J 2019 Phys. Rev. Lett. 122 027001
[17] Drozdov A P, Kong P P, Minkov V S, Besedin S P, Kuzovnikov M A, Mozaffari S, Balicas L, Balakirev F F, Graf D E, Prakapenka V B, Greenberg E, Knyazev D A, Tkacz M, and Eremets M I 2019 Nature 569 528
[18] Einaga M, Sakata M, Ishikawa T, Shimizu K, Eremets M I, Drozdov A P, Troyan I A, Hirao N, and Ohishi Y 2016 Nat. Phys. 12 835
[19] Drozdov A P, Eremets M I, Troyan I A, Ksenofontov V, and Shylin S I 2015 Nature 525 73
[20] Hong F, Yang L, Shan P, Yang P, Liu Z, Sun J, Yin Y, Yu X, Cheng J, and Zhao Z 2020 Chin. Phys. Lett. 37 107401
[21] Snider E, Dasenbrock-Gammon N, McBride R, Debessai M, Vindana H, Vencatasamy K, Lawler K V, Salamat A, and Dias R P 2020 Nature 586 373
[22] Ma J, Kuang J, Cui W, Chen J, Gao K, Hao J, Shi J, and Li Y 2021 Chin. Phys. Lett. 38 027401
[23] Song J, Fabbris G, Bi W, Haskel D, and Schilling J S 2018 Phys. Rev. Lett. 121 037004
[24] Pickard C J and Needs R J 2011 J. Phys.: Condens. Matter 23 053201
[25] Pickard C J and Needs R J 2006 Phys. Rev. Lett. 97 045504
[26] Olsen J S, Buras B, Gerward L, Johansson B, Lebech B, Skriver H L, and Steenstrup S 1984 Phys. Scr. 29 503
[27] Wang H, John S T, Tanaka K, Iitaka T, and Ma Y 2012 Proc. Natl. Acad. Sci. USA 109 6463
[28] Li Y, Hao J, Liu H, Tse J S, Wang Y, and Ma Y 2015 Sci. Rep. 5 9948
[29] Ma L, Wang K, Xie Y, Yang X, Wang Y, Zhou M, Liu H, Liu G, Wang H, and Ma Y 2021 arXiv:2103.16282 [cond-mat.supr-con]
[30] Troyan I A, Semenok D V, Kvashnin A G, Sadakov A V, Sobolevskiy O A, Pudalov V M, Ivanova A G, Prakapenka V B, Greenberg E, Gavriliuk A G et al. 2021 Adv. Mater. 33 2006832
[31] Segall M D, Shah R, Pickard C J, and Payne M C 1996 Phys. Rev. B 54 16317
[32] Song H, Duan D, Cui T, and Kresin V Z 2020 Phys. Rev. B 102 014510
[33] Xie H, Yao Y, Feng X, Duan D, Song H, Zhang Z, Jiang S, Redfern S A T, Kresin V Z, Pickard C J, and Cui T 2020 Phys. Rev. Lett. 125 217001
[34] Gor'kov L P and Kresin V Z 2018 Rev. Mod. Phys. 90 011001
[35] Gor'kov L P and Kresin V Z 2016 Sci. Rep. 6 7
[36] Semenok D V, Kruglov I A, Savkin I A, Kvashnin A G, and Oganov A R 2020 Curr. Opin. Solid State Mater. Sci. 24 100808
[37] Zhou D, Semenok D, Duan D, Xie H, Huang X, Chen W, Li X, Liu B, Oganov A R, and Cui T 2020 Sci. Adv. 6 eaax6849
[38] Zhou D, Semenok D V, Xie H, Huang X L, Duan D F, Aperis A, Oppeneer P M, Galasso M, Kartsev A I, Kvashnin A G, Oganov A R, and Cui T 2020 J. Am. Chem. Soc. 142 2803
[39] Semenok D V, Zhou D, Kvashnin A G, Huang X, Galasso M, Kruglov I A, Ivanova A G, Gavriliuk A G, Chen W, Tkachenko N V, Boldyrev A I, Troyan I, Oganov A R, and Cui T 2021 J. Phys. Chem. Lett. 12 32
[40] Pickard C J, Errea I, and Eremets M I 2020 Annu. Rev. Condens. Matter Phys. 11 57