⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 107401Express Letter Coexistence of Zero-Dimensional Electride State and Superconductivity in AlH$_{2}$ Monolayer Qiuping Yang (杨秋萍), Xue Jiang (蒋雪)*, and Jijun Zhao (赵纪军)* Affiliations Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Ministry of Education), Dalian University of Technology, Dalian 116024, China Received 8 August 2023; accepted manuscript online 29 August 2023; published online 13 September 2023 *Corresponding authors. Email: jiangx@dlut.edu.cn; zhaojj@dlut.edu.cn Citation Text: Yang Q P, Jiang X, and Zhao J J 2023 Chin. Phys. Lett. 40 107401    Abstract Electrides, which confine “excess anionic electrons” in subnanometer-sized cavities of a lattice, are exotic ionic crystals. We propose a non-stoichiometric strategy to realize intrinsic two-dimensional (2D) superconducting electride. AlH$_{2}$ monolayer, which is structurally identical to 1H-MoS$_{2}$, possesses zero-dimensionally confined anionic electrons in the interstitial sites of Al triangles, corresponding to a chemical formula of [AlH$_{2}$]$^{+}e^{-}$. The interaction between interstitial anionic electrons (IAEs) and host cation lattice mainly accounts for stabilization of 1H-AlH$_{2}$ electride. Impressively, 1H-AlH$_{2}$ monolayer is an intrinsic Bardeen–Cooper–Schrieffer superconductor with $T_{\rm c}=38$ K, which is the direct consequence of strong coupling of the H-dominated high electronic states with Al acoustic branch vibrations and mid-frequency H-derived phonon softening modes caused by Kohn anomalies. Under tensile strain, IAEs transform into itinerant electrons, favoring the formation of stable Cooper pairs. Therefore, $T_{\rm c}$ reaches up to 53 K at a biaxial fracture strain of 5%. Our findings provide valuable insights into the correlation between non-stoichiometric electrides and superconducting microscopic mechanisms at the 2D limit.
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 DOI:10.1088/0256-307X/40/10/107401 © 2023 Chinese Physics Society Article Text Electrides, a type of unconventional stoichiometric compound, contain more electrons than what would be expected from the classical valence bond theory. The excess electrons are trapped in lattice interstices and act as nucleus-free anions.[1-3] Electrides are similar to metal halides with the $F$-center (electrons trapped in an anion vacancy), but differ significantly in that the anionic electrons in electrides are periodically distributed, whereas the $F$-center is a random point defect.[4] According to the dimensionality of anionic electron distribution, electrides are classified into zero-dimensional (0D) lattice cavities,[5,6] one-dimensional (1D) linked channels,[7,8] two-dimensional (2D) interlayers,[9,10] and three-dimensional (3D) configurations.[11,12] The physical properties of electrides are strongly related to the topology of interstitial electrons.[13] As the first synthesized crystalline organic electride, Cs$^{+}$(18C6)$_{2}\cdot e^{-}$ crystallizes from dimethyl ether-trimethylamine mixtures, but it has low thermal stability.[14] Later, inorganic 0D electride [Ca$_{24}$Al$_{28}$O$_{64}$]$^{4+}$(4$e^{-})$ (C12A7:$e^{-})$ was prepared by removing O$^{2-}$ from the center of clathrate Ca–Al–O cages in 12CaO$\cdot$7Al$_{2}$O$_{3}$ (C12A7),[15] which is thermally and chemically stable under ambient conditions, exhibiting unique properties of high electronic conductivity[16] and low work function.[17] Moreover, as a semiconductor with pronounced band gap, the synthesized Sr$_{5}$P$_{3}$ is an ideal 1D electride.[7] In dicalcium nitride (Ca$_{2}$N), a representative 2D electride, the interconnected anionic electrons are loosely confined to interlayers,[18] thereby bringing the research on inorganic electrides into nano era. Previous studies have shown that the presence of interstitial anionic electrons (IAEs) is associated with the formation of new electronic states near the Fermi level with appreciable electron-phonon interaction, which in turn, benefits superconductivity.[19] Indeed, superconducting electrides have been identified for 3D compounds from both experimental and theoretical aspects. For example, the superconductivity of C12A7:$e^{-}$,[16] Nb$_{5}$Ir$_{3}$,[19] and Zr$_{5}$Sb$_{3}$[20] has been experimentally confirmed, whereas their superconducting critical temperatures ($T_{\rm c})$ are too low ($T_{\rm c} < 10$ K). Intriguingly, it was theoretically predicted that some binary compounds under high pressure can greatly enhance $T_{\rm c}$, such as Li$_{5}$C ($T_{\rm c}=48.3$ K at 210 GPa),[21] Li$_{5}$N ($T_{\rm c}= 48.97$ K at 150 GPa),[22] and Li$_{6}$P ($T_{\rm c}= 39.3$ K at 270 GPa).[23] Very recently, Zhang et al. proposed that strong electron-phonon coupling (EPC) in Li$_{8}$Au electride leads to a breaking-record $T_{\rm c}$ of 73.1 K at 250 GPa.[24] However, such effects cannot be maintained at ambient pressure, which severely limits their applications in superconducting quantum interference devices,[25] single-electron superconductor quantum dot devices,[26,27] etc. In contrast, 2D electrides could be intrinsic superconductors without external pressure; however, they usually suffer from low $T_{\rm c}$, e.g., 0.9 K for Y$_{2}$C,[28] 3.4 K for MgONa,[28] 4.7 K for Ca$_{2}$N,[29] and 3.4 K for Ba$_{2}$N.[30] Therefore, it is highly desirable to explore 2D superconducting electrides with relatively high-$T_{\rm c}$ and further uncover the microscopic mechanisms of superconductivity in low-dimensional electrides. In this Letter, we establish a unified picture about the relationship between IAEs, dynamic stability of lattice, and superconductivity in low-dimensional systems. A stabilized AlH$_{2}$ monolayer with isostructural 1H-MoS$_{2}$ is proposed, in which 0D anionic electrons are confined in the interstices of Al triangles. Notably, the interaction between IAEs and the lattice of host cations plays an essential role in stabilizing this 2D electride. AlH$_{2}$ electride exhibits an intrinsic high-$T_{\rm c}$ of 38 K. Contrary to the reported superconducting electrides, strong coupling of H 1$s$ electrons with Al phononic vibrations and mid-frequency H-derived phonon softening is responsible for superconductivity, rather than IAEs. If the interaction between IAEs and the host cation lattice is diminished, phonon modes are significantly softened, which has a positive effect on the enhancement of $T_{\rm c}$. The synergistic effect of electron and lattice vibration results in a highest $T_{\rm c}$ of 53 K. In hydrogen-based superconductors, H-derived high-frequency vibrations give rise to strong EPC, which is necessary for an excellent phonon-mediated superconductor.[31-34] Aluminum is an Earth abundant element with attractive properties like electride state[35,36] and superconductivity.[37] Inspired by the previously reported 2D electrides, we consider a 2D compound of AlH$_{2}$ with atomic thickness. In such a non-stoichiometric ratio between Al and H, Al atoms are expected to provide excess electrons to form a superconducting electride. After comparing several typical 2D compound structures with a stoichiometry of 1 : 2 (see Fig. S1 in the Supplemental Material), the dynamically stable structure for AlH$_{2}$ is identical to 1H-MoS$_{2}$ with $P\bar{6}m2$ space group symmetry[38] (thereafter referred to as 1H-AlH$_{2})$ (see Fig. S2). To assess the stability of the 1H-AlH$_{2}$ monolayer, we calculated the cohesive energy using the following formula: \begin{align} E_{\rm coh} =\frac{1}{3}(E_{\rm Al} +2E_{\rm H}-E_{\rm AlH_{2}}),\notag \end{align} where $E_{\rm Al}$, $E_{\rm H}$, and $E_{\rm AlH_{2}}$ are the total energy of a single Al atom, a single H atom, and one unit cell of AlH$_{2}$, respectively. The cohesive energy is 2.40 eV/atom, slightly higher than that of germanene (3.24 eV/atom)[39] and phosphorene (3.30 eV/atom)[40] but comparable to that of 2D transition metal hydrides (e.g., 2.06 eV/atom for $P6/mmm$ PdH$_{3}$ and 2.44 eV/atom for $P6/mm$ Ag$_{2}$H),[41] indicating the feasibility of experimental synthesis of the 1H-AlH$_{2}$ monolayer. The unit cell of 1H-AlH$_{2}$ consists of one Al and two equivalent H atoms, forming an H–Al–H sandwich structure with three atomic layers [Fig. 1(a)]. The Al layer consists of triangles with Al–Al bond length of 2.72 Å, which is slightly shorter than that for $fcc$ crystal of Al (2.86 Å),[42] indicating stronger Al–Al interaction in the AlH$_{2}$ monolayer. Each Al atom is six-fold coordinated with the neighboring H atoms, forming an AlH$_{6}$ trigonal prism [Fig. 1(a)] with Al–H distance of 1.90 Å. For comparison, in bulk AlH$_{3}$ crystal with $Fm\bar{3}m$ space group,[43] each Al atom is also six-fold coordinated by H atoms, but forms an AlH$_{6}$ octahedron with shorter Al–H bond length of 1.70 Å. Such difference can be explained by the non-stoichiometric ratio and the increased coordination number of H atoms (from two to three) in 2D 1H-AlH$_{2}$ crystal. Electron localization function (ELF) analysis revealed that the Al–H bond is ionic [Fig. 1(c)], which is also supported by Bader charge analysis,[44] that is, each H atom gains about 0.90$e$ from Al atom and almost reaches a complete electronic shell of 1$s^{2}$. According to its formal charge, Al atom tends to lose three valence electrons in compounds. However, H$^{-}$ anion cannot accommodate any more electrons donated by the Al atom. Alternatively, the remaining electrons provided by Al occupy the interstices of lattice and are located in the center of Al triangles, resulting in 0D electride state [Fig. 1(b)]. To further verify the existence of IAEs, we constructed a hypothetical structure ([AlH$_{2}$]$^{+})$ by removing one electron per formula unit from 1H-AlH$_{2}$ monolayer. By comparing the ELF plot before and after, it can be confirmed that the excess electron is responsible for the 0D electride state (Fig. S3). Further analysis of charge density difference reveals electron depletion around Al as well as electron accumulation nearby H and interstices of Al triangles [Fig. 1(d)]. By substituting Al with Mg (one electron less than Al), the resulting 1H-MgH$_{2}$ monolayer shows absence of the interstitial electrons in ELF (Fig. S4) and is dynamically unstable (Fig. S5). All these results demonstrate that 1H-AlH$_{2}$ monolayer is a 0D electride in formula of [AlH$_{2}$]$^{+}e^{-}$, similar to [Y$_{2}$C]$^{1.8+}$1.8$e^{-}$,[45] [MgONa]$^{+}e^{-}$,[28] and [Ba$_{2}$N]$^{+}e^{-}$[30] electrides.
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Fig. 1. (a) Top and side views of 1H-AlH$_{2}$ monolayer. (b) Three-dimensional electron localization function (ELF) of 1H-AlH$_{2}$ monolayer. (c) Two-dimensional ELF maps in ($h~l~k = 0$, 0, 1) (top) and ($h~l~k=-1$, 0, 10.57) (bottom) planes. (d) Isosurface of charge density difference (isosurface value: 0.003 $e$/Å$^{3})$. The yellow and cyan areas represent electron accumulation and depletion, respectively.
We also investigated the key factors for stabilization of 1H-AlH$_{2}$ electride using homologous element substitution (Ga, In, Tl) for Al. Apart from 1H-AlH$_{2}$, no structure is dynamically stable (Fig. S6). As the atomic radius increases, the nucleus has less attraction for the valence electrons. As a result, the Coulombic attraction between IAEs and host cation lattice is weakened, and the IAEs are converted to itinerant electrons [see Fig. S7(a) for GaH$_{2}$ as a representative], leading to lattice instability in GaH$_{2}$, InH$_{2}$, and TlH$_{2}$ monolayers. Remarkably, the heavier Ga atom does not contribute to the low-frequency acoustic branches [Fig. S7(b)], in sharp contrast to the vibration modes of lighter Al atoms [Fig. S7(c)]. This clearly demonstrates that the interaction between IAEs and host cation lattice is the driving force for lattice stability, which will be further supported by the discussion of electronic properties.
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Fig. 2. (a) Projected density of states (PDOS) and (b) projected electronic band structures of 1H-AlH$_{2}$ monolayer calculated at the Perdew–Burke–Ernzerhof level.
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Fig. 3. (a) Infrared (IR) and Raman (R) active modes with symmetry representations at the $\varGamma $-point. (b) Eliashberg spectral function (blue area) and frequency-dependent EPC parameters $\lambda $ (red line) of 1H-AlH$_{2}$ monolayer. (c) Phonon dispersion curves and phonon density of states (PHDOS) for 1H-AlH$_{2}$ monolayer. The size of red circles in the phonon spectra is proportional to partial EPC parameter $\lambda_{q,v}$. (d) $T_{\rm c}$ values of the reported superconducting electrides.[16,19,20,28-30,46] (e) The density of states in the superconducting state (SDOS) at $T=20$ K. (f) Energy distribution of superconducting gaps $\varDelta_{nk}$ versus $T$ for 1H-AlH$_{2}$ monolayer.
The electronic band structures in Fig. S8 show that 1H-AlH$_{2}$ is metallic, with one band crossing the Fermi level, forming a hole-like band along $\varGamma$–$K$ and $M$–$\varGamma$ paths. The Fermi surface does not show any visible nesting feature (Fig. S9) that was observed in the high-$T_{\rm c}$ TeH$_{4}$ superconductor.[47] To better visualize the contribution of the interstitial electrons, we placed a pseudoatom with a Wigner–Seitz radius of 1.2 Å at each interstitial site of Al triangle to calculate the projected density of states (PDOS) and projected electronic band structures. The PDOS analysis shows that its metallicity comes from the contribution of Al atoms, H atoms, and IAEs, whereas the H 1$s$ electrons dominate the states near the Fermi level [Figs. 2(a) and 2(b)]. There is strong overlap between Al and H states below the Fermi level, consistent with the observation of strong Al–H ionic bond in Fig. 1(c). Meanwhile, the IAEs strongly couple with Al 3$p$ states near the Fermi level, which contribute to the stability of 1H-AlH$_{2}$ monolayer structure. In PDOS [Fig. 2(a)], there is a remarkably sharp peak near the Fermi level, i.e., the van Hove singularity (vHs) caused by the flat bands [Fig. 2(b) and Fig. S10]. Indeed, the connection between the near-Fermi-level van Hove singularity and superconductivity has already been demonstrated for doped-graphene[48] and Ti$_{2}$B$_{2}$H$_{4}$ monolayer.[49] Hence, we performed EPC calculations to exploit its potential superconductivity. According to the irreducible representation of wave vector groups, we employed factor group analysis to classify the lattice vibrations. For 1H-AlH$_{2}$ with three atoms per unit cell and $D_{\rm 3h}$ symmetry, there are nine phonon modes with the following symmetry at $\varGamma$: $\varGamma_{\rm acoustic}=E'+A_{2}''$; $\varGamma_{\rm optic}= E'+E''+A_{2}''+A_{1}'$.[50] Here, $E'$ ($v=1$, 2, 4, 5) and $A_{2}''$ ($v=3$, 8) modes are infrared (IR) active, whereas the Raman (R) active ones are $E' (v=1$, 2, 4, 5), $E'' (v=6$, 7) and $A_{1}'$ ($v=9$) [Figs. 3(a) and 3(c)]. The degenerate $E'$ ($v=4$, 5) mode and $E''$ ($v=6$, 7) mode correspond to conjugate and disconjugate vibrations of the in-plane H atoms, respectively [Fig. 3(a)]. Interestingly, the contribution of $E' (v=4$, 5) modes to EPC is significantly larger than $E'' (v=6$, 7) ones [Fig. 3(c)]. As shown in Fig. 3(c), phonon density of states (PHDOS) can be separated into two regions. The low-frequency acoustic branches mainly come from the vibration of Al atoms, whereas the vibration of H atoms dominates the optical branches. A striking feature is the presence of soft phonon patterns that can be classified as Kohn anomalies. The distinct Kohn anomalies occur along the $\varGamma$–$K$ and $M$–$\varGamma$ paths in mid-frequency optical branch [denoted modes I and II, respectively, see Fig. 3(c)] and contribute significantly to EPC. Further analysis reveals that the in-plane vibration of H atoms is responsible for the Kohn anomaly in mode I, while the out-of-plane vibration of H atoms dominates mode II [see inset of Fig. 3(c)].
cpl-40-10-107401-fig4.png
Fig. 4. (a) $T_{\rm c}$ and $\lambda$ versus biaxial strain for 1H-AlH$_{2}$ monolayer. (b) Evolution of electron density of states (DOS) under different biaxial strains. (c) Eliashberg spectral function $\alpha^{2}F(\omega)$ as a function of strain. (d) ELF under biaxial strains of 0, 3, and 6%, respectively. (e) Electronic band structures and (f) phonon dispersions under different biaxial tensile strains. In (f), the size of orange circles is proportional to partial EPC parameter $\lambda_{q,v}$. The red arrows represent an upward shift of Fermi level and the softening of the phonon dispersion curve in (e) and (f), respectively.
Combining Eliashberg spectral function $\alpha^{2}F(\omega)$ [Fig. 3(b)], acoustic and mid-frequency optical branches contribute 47.4% and 52.3% to the total EPC strength of $\lambda=1.01$, respectively. Thus, coupling of H 1$s$ electrons with low-frequency vibration of Al and mid-frequency vibration of H plays a crucial role in strong EPC, rather than IAEs, which is in sharp contrast to the superconducting mechanism of most known superconducting electrides.[23,24,51-53] Among them, IAEs play a dominant role in the superconducting transition. Such a strong EPC parameter is considered to be a favorable prerequisite for high-$T_{\rm c}$ superconductors. The Allen–Dynes-modified McMillan formula is used to estimate $T_{\rm c}$ for materials with EPC strength less than 1.5. Using a typical Coulomb pseudopotential parameter of $\mu^*=0.1$,[54-57] the resulting $T_{\rm c}$ is 38 K, which is rather close to the “McMillan limit” (39 K) and much higher than those of previously reported electrides in bulk and 2D phases [0.4–9.4 K, see Fig. 3(d)].[16,19,28-30,46] To double check this exciting finding, we evaluated $T_{\rm c}$ using Migdal–Eliashberg's theoretical framework,[58,59] and obtained a consistent $T_{\rm c}$ value of 40 K [Fig. 3(f)]. Moreover, 1H-AlH$_{2}$ is a single-gap superconductor [Fig. 3(e)] that is similar to previously reported 2D Ba$_{2}$N[30] electride. For comparison, multi-gap superconductivity is found in other 2D compounds, such as Janus MoSH monolayer ($T_{\rm c}=28.58$ K)[60] and MgB$_{2}$ monolayer ($T_{\rm c}=20$ K).[61,62] Although the interaction of IAEs with Al 3$p$ electrons plays a crucial role in lattice stability, it is not responsible for superconductivity. If such interaction is broken by change in latent lattice stability, IAEs may be converted into itinerant electrons that are conducive to the formation of stable Cooper pairs with enhanced $T_{\rm c}$. As strain engineering has also been demonstrated as an effective way to modulate $T_{\rm c}$ of both conventional and unconventional superconductors,[63-66] herein we regulate the electron gas via external strain. From the perspective of electronic effect, the Coulombic attraction between IAEs and host cation lattice would be weakened as the biaxial tensile strain increases. Consequently, the original IAEs gradually disappear [Fig. 4(d)] and itinerant electrons emerge. Meanwhile, Fermi level shifts upwards [Fig. 4(e)] and DOS at the Fermi level is boosted [Fig. 4(b)], thereby elevating the probability of electron coupling with lattice vibrations. From the perspective of phononic effect, the whole profile of Eliashberg phonon spectral function $\alpha^{2}F(\omega)$ is red-shifted [Fig. 4(c)]; thus the low-frequency part is crucial for superconductivity. Based on the phonon dispersion curves with $\lambda_{q,v}$ weights [Fig. 4(f)], the presence of itinerant electrons evidently enhances EPC parameter $\lambda $ [Fig. 4(a)]. In turn, $T_{\rm c}$ is significantly elevated up to 53 K under a biaxial strain of 5%. Accompanied with the strain-induced disappearance of IAEs, significant phonon softening occurs. When strain reaches 6%, imaginary frequencies appear in the phonon spectra [Fig. 4(f)]. On the contrary, $T_{\rm c}$ is suppressed by in-plane compressive strain (e.g., $T_{\rm c}=33$ K at a biaxial strain of 1%) (Fig. S11). These results further indicate that IAEs do not make major contribution to the superconductivity, but are merely hybridized with Al 3$p$ electrons to maintain lattice stability. This mechanism is distinctly different from that in previously reported crystalline electrides under high pressure, i.e., IAEs make the major contribution to superconductivity in Li$_{6}$P, Li$_{8}$Au, and Li$_{5}$C, etc.[23,24,51-53] In summary, 1H-AlH$_{2}$ monolayer with 0D electride state and superconductivity has been identified by first-principles calculations. The interaction between IAEs and host cation lattice plays a crucial role in the structural stability. The H 1$s$ electrons strongly couple with acoustic phononic vibration of Al and intermediate-frequency phononic vibration of H, giving rise to a Bardeen–Cooper–Schrieffer superconductor with the highest known $T_{\rm c}$ of 38 K for 2D electrides. The tensile strain converts IAEs into itinerant electrons, which promotes the formation of stable Cooper pairs. Before lattice instability, $T_{\rm c}$ reaches 53 K at a tensile strain of 5%. These results shed light on the correlation between 0D electride state and superconductivity in low-dimensional materials, and provide an effective strategy to achieve high-$T_{\rm c}$ superconductor. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12274050 and 91961204), and the Fundamental Research Funds for the Central Universities (Grant Nos. DUT22LAB104 and DUT22ZD103). References CHEMISTRY: Electrons as AnionsAnionic electrons in electridesInorganic Electride: Theoretical Study on Structural and Electronic PropertiesAdvances in Materials and Applications of Inorganic ElectridesPredicting the structure and stability of titanium oxide electridesTernary inorganic electrides with mixed bondingExploration of Stable Strontium Phosphide-Based Electrides: Theoretical Structure Prediction and Experimental ValidationElectron Confinement in Channel Spaces for One-Dimensional ElectrideHigh-Throughput ab Initio Screening for Two-Dimensional Electride MaterialsFirst-Principles Prediction of Thermodynamically Stable Two-Dimensional ElectridesStructural Diversity and Electron Confinement in Li4 N: Potential for 0-D, 2-D, and 3-D ElectridesComputer-Assisted Inverse Design of Inorganic ElectridesElectrides: From 1D Heisenberg Chains to 2D Pseudo-MetalsFirst electride crystal structureHigh-Density Electron Anions in a Nanoporous Single Crystal: [Ca24Al28O64]4+(4e-)Superconductivity in an Inorganic Electride 12CaO·7Al2 O3 :e-Field Emission of Electron Anions Clathrated in Subnanometer-Sized Cages in [Ca24Al28O64]4+(4e–)†Dicalcium nitride as a two-dimensional electride with an anionic electron layerElectride and superconductivity behaviors in Mn5Si3-type intermetallicsSuperconductivity in the Mn 5 Si 3 -type Zr 5 Sb 3 systemPredicted Superconductivity in the Electride Li5 CPredicted superconductivity and superionic state in the electride Li5 N under high pressurePredicted Pressure-Induced Superconducting Transition in Electride Li 6 P Superconductivity in Li 8 Au electrideNanoscale Superconducting Quantum Interference Devices Add Another DimensionHybrid superconductor–quantum dot devicesScanning gate microscopy measurements on a superconducting single-electron transistorTwo dimensional superconductors in electridesElectron-phonon interaction in a Ca 2 N monolayer: Intrinsic mobility of electreneSuperconductivity in monolayer Ba 2 N electride: First-principles studyConventional superconductivity at 203 kelvin at high pressures in the sulfur hydride systemPressure-induced metallization of dense (H2S)2H2 with high-Tc superconductivityPotential high- Tc superconducting lanthanum and yttrium hydrides at high pressureSuperconductivity at 250 K in lanthanum hydride under high pressuresAluminium at terapascal pressuresThe Bonding Electron Density in AluminumHigh-Pressure Effects on the Superconducting Transition Temperature of AluminumFunctionalization of Single-Layer MoS2 Honeycomb StructuresB2 P6 : A Two-Dimensional Anisotropic Janus Material with Potential in Photocatalytic Water Splitting and Metal-Ion BatteriesPhase Coexistence and Metal-Insulator Transition in Few-Layer Phosphorene: A Computational StudyEnhanced superconductivity in CuH 2 monolayersPreparation and properties of aluminum hydrideA fast and robust algorithm for Bader decomposition of charge densityTwo-Dimensional Transition-Metal Electride Y2 CBand structure, electron-phonon interaction and superconductivity of yttrium hypocarbideTellurium Hydrides at High Pressures: High-Temperature SuperconductorsTwo-gap superconductivity in heavily n -doped graphene: Ab initio Migdal-Eliashberg theoryHigh-temperature superconductivity in two-dimensional hydrogenated titanium diboride: Ti2B2H4Raman study and lattice dynamics of single molecular layers of MoS 2 Pressure Stabilized Lithium-Aluminum Compounds with Both Superconducting and Superionic BehaviorsPressure-induced superconductivity in Li-Te electridesPressure-stabilized superconducting electride Li 5 C Transition Temperature of Strong-Coupled SuperconductorsTransition temperature of strong-coupled superconductors reanalyzedProperties of boson-exchange superconductorsDensity-Functional Theory for SuperconductorsTwo-gap superconductivity in a Janus MoSH monolayerThe origin of the anomalous superconducting properties of MgB2Superconductivity and topological properties of MgB 2 -type diborides from first principlesTuning the band structure and superconductivity in single-layer FeSe by interface engineeringRecent Progress in Strain Engineering on Van der Waals 2D Materials: Tunable Electrical, Electrochemical, Magnetic, and Optical PropertiesFirst-Principles Calculations on the Effect of Doping and Biaxial Tensile Strain on Electron-Phonon Coupling in GrapheneEnhanced superconductivity by strain and carrier-doping in borophene: A first principles prediction
[1] Dye J L 2003 Science 301 607
[2] Dye J L 1993 Nature 365 10
[3] Li Z Y, Yang J L, Hou J G, and Zhu Q S 2003 J. Am. Chem. Soc. 125 6050
[4] Hosono H and Kitano M 2021 Chem. Rev. 121 3121
[5] Zhong X, Xu M L, Yang L L, Qu X, Yang L H, Zhang M, Liu H Y, and Ma Y M 2018 npj Comput. Mater. 4 70
[6] Wang J J, Zhu Q, Wang Z H, and Hosono H 2019 Phys. Rev. B 99 064104
[7] Wang J J, Hanzawa K, Hiramatsu H, Kim J, Umezawa N, Iwanaka K, Tada T, and Hosono H 2017 J. Am. Chem. Soc. 139 15668
[8] Zhang Y Q, Xiao Z W, Kamiya T, and Hosono H 2015 J. Phys. Chem. Lett. 6 4966
[9] Tada T, Takemoto S, Matsuishi S, and Hosono H 2014 Inorg. Chem. 53 10347
[10] Ming W M, Yoon M, Du M H, Lee K, and Kim S W 2016 J. Am. Chem. Soc. 138 15336
[11] Tsuji Y, Dasari P L V K, Elatresh S F, Hoffmann R, and Ashcroft N W 2016 J. Am. Chem. Soc. 138 14108
[12] Zhang Y W, Wang H, Wang Y C, Zhang L J, and Ma Y M 2017 Phys. Rev. X 7 011017
[13] Dye J L 1997 Inorg. Chem. 36 3816
[14] Dawes S B, Ward D L, Huang R H, and Dye J L 1986 J. Am. Chem. Soc. 108 3534
[15] Matsuishi S, Toda Y, Miyakawa M, Hayashi K, Kamiya T, Hirano M, Tanaka I, and Hosono H 2003 Science 301 626
[16] Miyakawa M, Kim S W, Hirano M, Kohama Y, Kawaji H, Atake T, Ikegami H, Kono K, and Hosono H 2007 J. Am. Chem. Soc. 129 7270
[17] Toda Y, Matsuishi S, Hayashi K, Ueda K, Kamiya T, Hirano M, and Hosono H 2004 Adv. Mater. 16 685
[18] Lee K, Kim S W, Toda Y, Matsuishi S, and Hosono H 2013 Nature 494 336
[19] Zhang Y Q, Wang B S, Xiao Z W, Lu Y F, Kamiya T, Uwatoko Y, Kageyama H, and Hosono H 2017 npj Quantum Mater. 2 45
[20] Lv B, Zhu X Y, Lorenz B, Wei F Y, Xue Y Y, Yin Z P, Kotliar G, and Chu C W 2013 Phys. Rev. B 88 134520
[21] Pereira Z S, Faccin G M, and da Silva E Z 2021 J. Phys. Chem. C 125 8899
[22] Wan Z Y, Zhang C, Yang T Y, Xu W J, and Zhang R Q 2022 New J. Phys. 24 113012
[23] Zhao Z Y, Zhang S T, Yu T, Xu H Y, Bergara A, and Yang G C 2019 Phys. Rev. Lett. 122 97002
[24] Zhang X H, Yao Y S, Ding S C, Bergara A, Li F, Liu Y, Zhou X F, and Yang G C 2023 Phys. Rev. B 107 L100501
[25] Gallop J and Hao L 2016 ACS Nano 10 8128
[26] de Franceschi S, Kouwenhoven L, Schönenberger C and Wernsdorfer W 2010 Nat. Nanotechnol. 5 703
[27] Huefner M, May C, Kičin S, Ensslin K, Ihn T, Hilke M, Suter K, de Rooij N F, and Staufer U 2009 Phys. Rev. B 79 134530
[28] Ge Y F, Guan S, and Liu Y 2017 New J. Phys. 19 123020
[29] Zeng X Z, Zhao S T, Li Z Y, and Yang J L 2018 Phys. Rev. B 98 155443
[30] Qiu X L, Zhang J F, Yang H C, Lu Z Y, and Liu K 2022 Phys. Rev. B 105 165101
[31] Drozdov A P, Eremets M I, Troyan I A, Ksenofontov V, and Shylin S I 2015 Nature 525 73
[32] Duan D F, Liu Y X, Tian F B, Li D, Huang X L, Zhao Z L, Yu H Y, Liu B B, Tian W J, and Cui T 2014 Sci. Rep. 4 6968
[33] Liu H Y, Naumov I I, Hoffmann R, Ashcroft N W, and Hemley R J 2017 Proc. Natl. Acad. Sci. USA 114 6990
[34] 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
[35] Pickard C J and Needs R J 2010 Nat. Mater. 9 624
[36] Nakashima P N H, Smith A E, Etheridge J, and Muddle B C 2011 Science 331 1583
[37] Gubser D U and Webb A W 1975 Phys. Rev. Lett. 35 104
[38] Ataca C and Ciraci S 2011 J. Phys. Chem. C 115 13303
[39] Sun M L and Schwingenschlögl U 2020 Chem. Mater. 32 4795
[40] Guan J, Zhu Z, and Tománek D 2014 Phys. Rev. Lett. 113 46804
[41] Yan X, Ding S C, Zhang X H, Bergara A, Liu Y, Wang Y, Zhou X F, and Yang G 2022 Phys. Rev. B 106 14514
[42]Kittel C 1976 Introduction to Solid State Physics (New York: Wiley)
[43] Brower F M, Matzek N E, Reigler P F, Rinn H W, Roberts C B, Schmidt D L, Snover J A, and Terada K 1976 J. Am. Chem. Soc. 98 2450
[44] Henkelman G, Arnaldsson A, and Jónsson H 2006 Comput. Mater. Sci. 36 354
[45] Zhang X, Xiao Z, Lei H, Toda Y, Matsuishi S, Kamiya T, Ueda S, and Hosono H 2014 Chem. Mater. 26 6638
[46] Dilmi S, Saib S, and Bouarissa N 2018 Curr. Appl. Phys. 18 1338
[47] Zhong X, Wang H, Zhang J, Liu H, Zhang S, Song H F, Yang G, Zhang L, and Ma Y 2016 Phys. Rev. Lett. 116 57002
[48] Margine E R and Giustino F 2014 Phys. Rev. B 90 14518
[49] Han Y L, Li Y P, Yang L, Liu H D, Jiao N, Wang B T, Lu H Y, and Zhang P 2023 Mater. Today Phys. 30 100954
[50] Jiménez Sandoval S, Yang D, Frindt R F, and Irwin J C 1991 Phys. Rev. B 44 3955
[51] Wang X M, Wang Y, Wang J J, Pan S N, Lu Q, Wang H T, Xing D, and Sun J 2022 Phys. Rev. Lett. 129 246403
[52] Zhang X H, Li F, Bergara A, and Yang G C 2021 Phys. Rev. B 104 134505
[53] Wang Q, Cui W, Gao K, Chen J, Gu T, Liu M, Hao J, Shi J, and Li Y 2022 Phys. Rev. B 106 54519
[54] McMillan W L 1968 Phys. Rev. 167 331
[55] Allen P B and Dynes R C 1975 Phys. Rev. B 12 905
[56] Carbotte J P 1990 Rev. Mod. Phys. 62 1027
[57] Oliveira L N, Gross E K U, and Kohn W 1988 Phys. Rev. Lett. 60 2430
[58]Migdal A B 1968 Sov. Phys.-JETP 35 996
[59]Eliashberg G M 1960 Sov. Phys.-JETP 38 966
[60] Liu P F, Zheng F P, Li J Y, Si J G, Wei L M, Zhang J R, and Wang B T 2022 Phys. Rev. B 105 245420
[61] Choi H J, Roundy D, Sun H, Cohen M L, and Louie S G 2002 Nature 418 758
[62] An Y P, Li J, Wang K, Wang G T, Gong S J, Ma C L, Wang T X, Jiao Z Y, Dong X, Xu G L, Wu R Q, and Liu W M 2021 Phys. Rev. B 104 134510
[63] Peng R, Xu H C, Tan S Y, Cao H Y, Xia M, Shen X P, Huang Z C, Wen C H P, Song Q, Zhang T, Xie B P, Gong X G, and Feng D L 2014 Nat. Commun. 5 5044
[64] Qi Y P, Sadi M A, Hu D, Zheng M, Wu Z P, Jiang Y C, and Chen Y P 2023 Adv. Mater. 35 2205714
[65] Si C, Liu Z, Duan W H, and Liu F 2013 Phys. Rev. Lett. 111 196802
[66] Xiao R C, Shao D F, Lu W J, Lv H Y, Li J Y, and Sun Y P 2016 Appl. Phys. Lett. 109 122604