Introduction

Kagome-lattice materials have attracted tremendous attentions in recent years due to lattice geometry for geometrical frustration physics and correlated electron states1,2,3,4. Typical kagome-lattice electronic bands are featured by Dirac dispersions, saddle points, and flat bands exhibiting in recently discovered layered kagome-lattice d-transition metals5. The latter two features with natural electronic correlation effects are promising to induce correlation quantum phases, such as charge density waves (CDW) and superconductivity (SC). However, both correlated ordering phases are hard to realize in real kagome materials. For instance, paramagnet CoSn with textbook kagome bands exhibits high density-of-states (DOS) flat bands and saddle points near the Fermi energy (EF) and can form the giant fermion-boson interaction by coupling to pronounced phonons6,7. Not as expected, neither SC nor CDW driven by this fermion-boson interaction is found even through pressure tuning or chemical engineering8. While, in most magnetic kagome metals, such as Fe3Sn29, Mn3Sn10, Co3Sn2S211, and YMn6Sn612, they show correlated electronic states interplay with bands topology, but lacking SC and CDW phases. Latterly, the CDW state is observed in the antiferromagnetic ordered phase of kagome-lattice FeGe but without SC13. The CDW in FeGe arises from electron-correlations-driven magnetic order and instability, which is evidently different from the force of electron-phonon coupling (EPC), necessitating further detailed investigations. In this way, although the saddle points and flat bands with high-DOS in kagome materials are naturally accompanied by strong correlation effects, it’s still a great challenge to achieve SC by systematically modulating the correlation effect, DOS, and carrier concentration near EF. Additionally, kagome materials with both SC and CDW are expected to study not only the mechanism of the ordering phases but also the interplay between them.

Kagome metals AV3Sb5 (A = K, Rb, Cs) have offered a novel platform for exploring correlated superconducting states intertwined with charge ordering5,14,15,16,17,18,19,20. The superconducting transition temperatures (Tcs) of KV3Sb5 and RbV3Sb5 are around 1 K, while in CsV3Sb5 rises to 8 K at approximately 2 GPa14,21,22,23,24. The presence of CDW order is revealed by high-resolution scanning tunneling microscopy and single-crystal X-ray diffraction14,16,25. In these materials, the unique vanadium-kagome structure effectively regulates the interaction between charge and orbital degrees of freedom thus achieving diverse states of matter26,27,28. However, because d-electron states from kagome-lattice hybridization with s/p-states from adjacent layers dominate the low-energy exactions, these multiple bands make it more complicated to study the microscopic mechanisms and the relationship between CDW and SC29. There are widely divergent perspectives about the force of CDW and SC formations and the form of electron pairs19. For example, the EPC was observed in all the vanadium-derived bands in KV3Sb515. However, without acoustic phonon anomaly25 and the size of CDW gaps more than the theoretical value29 from EPC mode indicate that the EPC force may not be strong enough to support the form of CDW, and electronic-driven mechanism should be taken into considered seriously. The interplay of electron-phonon with electron-electron interactions may lead to enhanced transition temperatures which have long been discussed but remain elusive in kagome superconductors and cuprate as well17,30. It’s crucial to devise a simple kagome-lattice to elucidate the underlying mechanisms governing the formation of CDW and SC, and reveal their correlation mechanism.

Compressed hydrides under high-pressures have been widely recognized as one of the promising approaches for obtaining high-Tc due to the high Debye temperature of element hydrogen31,32,33. And its high-phonon frequencies coupling with large DOS at EF forming Cooper pairs mediated by EPC interactions are conducive to SC34,35. Inspired by the unique van Hove singularity (vHs) and flat-band inherent in kagome lattice, we are committed to artificially fabricating hydrogen-kagome metals to investigate the formation mechanism of CDW and SC, and unveil the uncharted charge ordering states.

In this work, we propose a series of prototypes of hydrogen-kagome metals AH3Li5 (A = C, Si, P), obtained from our designed methodology in conjunction with first-principles calculations. Notably outperforming vanadium-kagome superconductors, the representative PH3Li5 exhibits the highest Tc of 120.09 K under compression among known kagome-lattice materials. Being substituted element phosphorus by carbon to achieve single-hole doping, the dynamically stable pressure of derived CH3Li5 was down to 30 GPa while maintaining a Tc of 36.55 K, which can be potentially synthesized in large-volume presses equipped with more convenient in-situ measurement techniques. Moreover, CH3Li5 shows a dome-shaped SC upon decompression, and eventually transforms into a 1 × 1 × 2 CDW phase. This result demonstrates the competitive interactions between CDW and SC with pressure evolution. Importantly, the origin of CDW stems from the synergistic interplay between robust EPC and Fermi surface nesting (FSN), rather than their independent occurrences. Our study provides instructive insights into the exploration of hydrogen-kagome materials, highlighting their potential to serve as a novel platform for investigating the interplay between CDW and SC.

Results

Hydrogen-based kagome hydrides and superconductivity

The interstitial quasiatoms (ISQs) in crystalline cavities can function as a driving force in stabilizing structures36. Through utilizing the under-occupied H-s orbital to absorb ISQs, we insert hydrogen atoms into the Wyckoff positions of ISQs in parent electrides (Li)+5(X)m-ne- with special P6/mmm configurations to identify hydrogen-kagome metals37 (see Fig. S1 in Supporting Information). In Fig. 1a, a series of paradigmatic prototype of hydrogen-kagome superconductors AH3Li5 (A = C, Si, P) isostructural to CsV3Sb5 have been identified. The optimized P6/mmm-PH3Li5 at 250 GPa crystallizes into a layered structure of hydrogen-kagome sheets intercalated by Li atoms, as shown in Fig. 1b and Table S1. Meanwhile, the thermodynamic stability is demonstrated by the enthalpy difference curves, as depicted in Fig. S6. The H atoms form a standard kagome-lattice with additional P atoms site at the hydrogen-kagome lattice’s hexagonal center. Above and below the hydrogen-kagome layer, there are two honeycomb lattice planes formed by the out-of-plane Li atoms sitting above and below the centers of the H-triangles in kagome-lattice layer.

Fig. 1: The identification and electronic structures of kagome hydride.
figure 1

a The methodology for identifying hydrogen-kagome materials. b Pristine structure of PH3Li5 at 250 GPa and its atomic arrangement along the ab-plane. Panels (c, d) represent its band structure and projected DOS.

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The electronic band structure and projected DOS calculated in Fig. 1c, d and Fig. S2 reveal that the metallic character stems from bands overlapping at EF, where the Li-/P-p and H-s orbitals display sp-hybridization. Compared to Li-/P-p orbitals, significant H-sp-hybridized electrons contribute dominant component to electronic states at EF [N(EF)]. The band dispersion around the H point, above EF, exhibits a “hole pocket”, and a Dirac cone marked as DC1 has been observed. There exists a vHs at EF and two flat bands with small dispersion near EF. One band is located at approximately -1 eV along K to Γ direction, while the other arises at +1 eV along Γ to M direction attributed to the high localization of sp-hybridized electronic states that originated from atoms H and P. The emergence of vHs and “flat-steep” bands in close proximity to EF signifies a potential of strong EPC interaction, thereby contributing to the formation of Cooper pairs38. Moreover, we note that the nonlocal correlation effects have been proven to be significant for superconductivity in materials such as bismuthate (Ba, K)BiO3 and antimonate (Ba, K)SbO339,40,41,42. Herein, the higher degrees of orbital multiplicity characteristics of H-sp-hybridized orbitals relative to H-s orbital at EF, which are associated with the unique hydrogen-kagome bonding properties in hydrogen-kagome superconductors, also exhibit nonlocal correlation effects36,43. For the hydrogen-based superconductors, the EPC evaluated by density functional perturbation theory and generalized-gradient approximation method is shown to be strong enough to account for the superconductivity according to previous studies44,45,46,47,48.

To investigate the potential superconductivity, we calculated the phonon spectrum, phonon DOS, α2F(ω) and accumulated λ(ω) of PH3Li5 at 250 GPa in Fig. S3a. The distinguishable high-frequency vibrations above 55 THz are derived from H-stretching vibrations while the contribution is only about 7.5% to the total λ. The λ is mainly contributed by the mediate frequency modes ranged from 15 to 45 THz, giving rise to large peaks in the α2F(ω). This result can be reflected by the larger projected phonon linewidth, which play a significant contribution of 73.1% to EPC. However, the low-frequency translational vibrations below 15 THz accounts for 19.4% to the integral λ. Subsequently, we adopted the Allen-Dynes modified McMillan equation to assess Tc of PH3Li5 (see Fig. S3b). A sizable estimated Tc of 120.09 K has been predicted, and the corresponding λ and phonon frequency logarithmic average ωlog reach 1.37 and 1054 K. Boosting pressure from 200 to 300 GPa, the ωlog demonstrates a consistent rise with a pressure coefficient, dωlog/dP of 0.85 K/GPa. These analyses suggest that pressure-induced increase of Tc in PH3Li5 is primarily governed by the λ and N(EF).

Single hole doping and lattice dynamic stability

Aiming at lower the dynamic stabilization pressure, we then substitute phosphorus by carbon to achieve the single-hole doping in PH3Li549. Figure S4 demonstrates that hole doping effectively eliminates the imaginary phonon frequencies, and the derived hydrogen-kagome CH3Li5 remains dynamically stable down to 30 GPa. Combined with thermodynamic stability analysis in Fig. S6, CH3Li5 presents the latent capacity to be synthesized within attainable synthetic pressure range of large cavity compressors. In Fig. 2a and Table S2, the ωlog is determined to be 655 K and λ reaches 0.98 for CH3Li5 at 150 GPa, resulting in a Tc of 47.34 K. The integral λ(ω) is mainly contributed by two parts: (i) the low frequencies below 10 THz, characterized by soft phonon modes corresponding to the lowest acoustic branch along qA → qH and qM → qL → qH directions, contribute 36.95% to total λ. While further decompressing CH3Li5 to 30 GPa, the softening effect of acoustic branch along qA → qH → qK paths become more pronounced (see Fig. S4), and the Tc maintains 36.55 K. (ii) The EPC strength is originated from significant phonon linewidths above 10 THz, indicating a pronounced dependence of optical phonon frequencies on wave vector due to robust interatomic interactions36. These phonon vibration modes enhance the likelihood of satisfying the formation energy of Cooper pairs, which accounts for 63.45% of total λ36,50.

Fig. 2: Electron-phonon coupling related parameters of CH3Li5.
figure 2

a Phonon dispersion, projected phonon DOS, integral λ and α2F(ω) of CH3Li5 at 150 GPa with (b) its Tcs at different pressures. c Evolution curve of phonon linewidth and (d) nesting function ξq along typical (q) paths with pressures for CH3Li5.

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Moreover, the pressure dependency of Tc in CH3Li5 shows a dome-like shape (Fig. 2b): the hexagonal phase of CH3Li5 exhibits a maximum Tc of 57.18 K at 100 GPa, then decrease monotonically with decompression, and finally transform into a CDW phase (will be proved later). The evolution of the dome-like superconductivities can be demonstrated by examining the strength of linewidth, \({\gamma }_{{\bf{q}}v}=\pi {{\rm{\omega }}}_{{\boldsymbol{q}}v}\sum _{{mn}}\sum _{{\bf{k}}}{\left|{g}_{{mn}}^{v}\left({\bf{k}},{\bf{q}}\right)\right|}^{2}\delta ({\varepsilon }_{m{,}{\bf{k}}+{\bf{q}}}-{\varepsilon }_{f})\times \delta ({\varepsilon }_{n,{\bf{k}}}-{\varepsilon }_{f})\), which reflected the electron-phonon matrix element \({g}_{{mn}}^{\upsilon }\left({\bf{k}},{\bf{q}}\right)\) and FSN characteristics. As pressures decrease from 200 GPa to 100 GPa, the strength of γqυ shows an ascending tendency along the qA → qH → qK paths and a relatively small increasement along qM → qL → qH paths (Fig. 2c), resulting in a boosted Tc. Conversely, during the depressurization from 100 GPa to 30 GPa, a noticeably declined trend in phonon linewidth strength are observed along qM → qL → qH paths and a relatively slight decrease along qA → qH → qK, illustrating that these phonon vibrations corresponding to abovementioned special q-paths to scatter electrons played a crucial role on EPC.

To elucidate the deep-seated factors that affect EPC strength, we detected the N(EF), FSN strength and \({g}_{{mn}}^{\upsilon }\left({\bf{k}},{\bf{q}}\right)\) reflected by \(\left\langle {I}^{2}\right\rangle\) in Table S2. The FSN was calculated based on the nesting function: \(\xi {\bf{q}}=\frac{1}{N}\sum _{{\boldsymbol{k}},i,j}{\delta }({\varepsilon }_{{\boldsymbol{k}},i}-{{\rm{\varepsilon }}}_{F}){\times }{\delta }\left({\varepsilon }_{{\boldsymbol{k}}+{\boldsymbol{q}},j}-{\varepsilon }_{F}\right)\). The strengths of FSN increased with depressurization as plotted in Fig. 2d, which agree well with the evolution in N(EF). When decompressing from 200 to 100 GPa, Tc shows increasing trend similar to FSN strength but was in inversely proportional to the rapid declined \(\left\langle {I}^{2}\right\rangle\), illustrating that the expanded FSN enhances EPC thus dominating Tc. Simultaneously, the increased N(EF) promotes the likelihood of Cooper pairs being formed by phonon scattering. On the contrary, despite the continued increase in N(EF) and FSN strength, the strength of \(\left\langle {I}^{2}\right\rangle\) has rapidly declined, leading to a precipitous drop in Tc as pressure decreases from 100 to 30 GPa. This result indicates that the decreasing Tc stems from the reduction of \({g}_{{mn}}^{\upsilon }\left({\bf{k}},{\bf{q}}\right)\) due to the weakened atomic bonding strength.

We further revealed the effect of FSN on phonon vibrations. As pressure decreased from 200 to 30 GPa, Figs. S4, S5 expose that the soften mode along qΓ → qA route strengthened. The route of phonon softening matches well with the nesting directions along this route in FS calculation in Fig. 2d. Especially, the FSN along qΓ to qA comes from the cone-shaped (electron-type) and “tubular-shaped” (hole-type) FSs (Fig. S7), explaining soft phonon modes were caused by FSN. As proved in Table S2, during decompression from 200 to 100 GPa, the phonon-softening caused by FSN make a major contribution to EPC. In the decompression process from 100 to 30 GPa, the strength of FSN slightly enhanced but the EPC continues to decrease, confirming that the \({g}_{{mn}}^{\upsilon }\left({\bf{k}},{\bf{q}}\right)\) dominates the EPC.

Phonon instability and CDW phases

We noticed that the phonon spectrum of acoustic lattice waves of CH3Li5 approaching the A point and optical lattice wave at the Γ point softened continuously with depressurization, until an imaginary frequency emergent at 25 GPa in Fig. 3a. This dynamic instability is associated with the intrinsic CDW with modulations shown in Fig. 3c51. The instability at A and Γ points is related to a 1 × 1 × 2 supercell, where the displacement vectors possess an in-plane component. Detailed analysis of vibration mode reveals that the lowest imaginary phonon frequency at Γ point corresponds to the vibration of the central H-trimer layer, as indicated by the blue arrows in Fig. 3c. Such phonon softening regarding to point A leads to a periodic lattice distortion of H-trimers and Li-layers. The pristine phase went through P-62m structure and then transformed into the final CDW Cmcm phase, where the phonon spectrum in Fig. 3b confirms the dynamic stability of the CDW phase.

Fig. 3: Structural modulation of P6/mmm-CH3Li5 and the projected DOS of CDW phase.
figure 3

Modulation of Phonon dispersions of (a) P6/mmm-CH3Li5 and (b) CDW phase at 25 GPa. c The generating process of CDW phase. And the displacement vectors that associated with the instabilities of CDW order at Γ and A points are plotted as arrows. d Projected DOS of CDW phase.

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Subsequently, we calculated the projected DOS to investigate the electronic properties of CDW phase. As illustrated in Fig. 3d, this phase is metallic with a tiny Tc of 0.05 K (Table S2). The valence band is mainly consisted by C-p orbitals, which hybridize strongly with Li-p orbitals. A comparison between the hybrid feature of pristine P6/mmm structure and CDW phase of CH3Li5 reveals that the enhanced hybridization strength caused by Li−C interaction in CDW phase lowers the energy of the valence bands and raises the energy of conduction bands, which is more conducive to energy stability of CDW phase at lower pressures52.

The origin of CDW and its competition with superconductivity

Having established the intrinsic CDW phase of CH3Li5, we investigate the underlying mechanisms driving the phase transition to understand the origin of CDW order. Combining with band structure analyses in Fig. S8, we first examined the FSs of the pristine phase at 30 GPa and observed electron-type (cone-shaped, yellow) and hole-type (pot-shaped, magenta; and tubular network, cyan) FSs. These FSs along qΓ to qA and qΓ to qM paths are nested with each other corresponding to the obvious nesting peaks as shown in Fig. 2d. To separate the purely electronic effects from that of the EPC interaction and quantitatively estimate the strength of FSN, we calculate the electron susceptibility based on the Lindhard response function53,54: (i) \({\chi }^{{\prime} }\left({\bf{q}}\right)=\sum _{{\bf{k}}}\frac{f\left({\varepsilon }_{{\bf{k}}}\right)-f({\varepsilon }_{{\bf{k}}{\boldsymbol{+}}{\bf{q}}})}{{\varepsilon }_{{\bf{k}}}-{\varepsilon }_{{\bf{k}}{\boldsymbol{+}}{\bf{q}}}}\) and (ii) \(\mathop{\mathrm{lim}}\nolimits_{\omega \to 0}\frac{{\chi }^{{\prime} {\prime} }({\bf{q}},\omega )}{\omega }=\sum _{{\bf{k}}}\delta ({\varepsilon }_{{\bf{k}}}-{\varepsilon }_{F})\delta ({\varepsilon }_{{\bf{k}}{\boldsymbol{+}}{\bf{q}}}-{\varepsilon }_{F})\). The real part of electron susceptibility χ′(q) is defined as (i), and the low-frequency limit of the imaginary part χ′(q, ω → 0) is (ii)54. Figure 4a, b illustrates the momentum dependences of the imaginary part χ″(q, ω → 0) and real part χ′(q) at 30 GPa, which exhibit a larger value around the qCDW point along qΓ to qA direction55. Moreover, there is a dominant peak of χ′(q) close to qA point as pressure decreases in Fig. 4c, indicating that the FSN-driven Peierls-like electronic instability is an important driving force for CDW order56.

Fig. 4: Lindhard response function and electron-phonon coupling strength.
figure 4

a The imaginary and (b) real part of normalized Lindhard response function for the pristine phase in the xz plane at 30 GPa. c The real part of static bare susceptibility and (d) λq strength along high symmetry paths in the pristine structure at various pressures, which corresponds to three softened phonon modes becomes imaginary at 25 GPa, i. e. two lowest acoustic modes and one softened optical mode.

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We further calculated the EPC strength λq for the soft phonon modes in the pristine structure to reveal whether the instability arises from electron-phonon interactions. In Fig. 4d, the λq exhibits a conspicuous peak at the high-symmetry qCDW points and increase with depressurization from 100 to 35 GPa. This result suggests a gradual enhancement of EPC effects on CDW, further proving the tight connection between EPC and CDW. Accordingly, the depressurization-induced enhancement of phonon-modulated lattice distortion and CDW ordering stems from both the FSN and EPC interaction. Next, we illustrate the correlation mechanism of CDW and related SC. In the pressure range of 100 to 30 GPa, the application of depressurization suppresses the SC, which is subjected to the declined total λ in Table S2. However, this result is inversely proportional to the increasing tendency of CDW dominated by increased FSN and EPC interaction at qCDW point, which is reflected by the enhanced phonon softening. These results provide critical insights for underlying the collaborative and competitive interplay between CDW and SC in CH3Li5 kagome hydrides.

Discussion

Our discovery of hydrogen-kagome superconductors is crucial for observing new physical phenomena and understanding their fundamental origins, therefore, offering new playgrounds to study the intricate mechanisms governing the emergence of superconductivity and charge ordering in kagome materials. Our results demonstrate that the capacity to enhance superconductivity and the unusual competition between SC and CDW state in hydrogen-kagome compounds represent a characteristic of electronic states than was appreciated previously in vanadium-based kagome metals. In contrasted to divergent perspectives about the force of CDW and SC formations and the form of electron pairs in d-orbital kagome systems, our findings concern the role of electron-electron interactions in simply phonon-mediated hydrogen-based kagome superconductors, which are expected to be experimentally synthesized under high pressures. Taking hydrogen-based kagome PH3Li5 as an illustration, our first-principles calculations reveal that PH3Li5 at 250 GPa exhibits a high-Tc of 120.09 K, accompanied by perfect Dirac cones, a van Hove singularity and a flat band. Significantly, we uncover that the high-Tc is mainly attributed to the EPC that derived from the strong dependency of optical phonon frequencies on wave vector and acoustic phonon softening to scatter sp-hybridized H-electronic states. Completely substituting element phosphorus by carbon to realize single-hole doping, the dynamic stability of CH3Li5 was modulated to 30 GPa, which can be potentially synthesized in large-volume presses equipped with more convenient in-situ measurement techniques. Being attributable to the both FSN and EPC interactions, CH3Li5 undergoes a CDW phase transition at 25 GPa, and the lattice is reconstructed from hexagonal to orthorhombic system, accompanied by a sharp decline in superconductivity, which illustrates the compete interplay between CDW and superconducting state.

By combining the features of prominent electronic properties inherent in kagome lattice and the well-known high Debye temperature of hydrogen that dominated the high-Tc in hydrides, our proposed methodology of extracting the fingerprints of electride states from the lattices of parent electrides to identify hydrogen-based kagome superconductors has been proved to be an efficient strategy than the energy-based traditional searching method does. As a result, a series of hydrogen-based kagome superconductors identified in compressed hydride AH3Li5 (A = C, Si, P) family offer a paradigm for lightweight elements to form kagome materials. Our study provides a scientific method to identify high-Tc hydrogen-based kagome family, and offers a new avenue to fundamentally understand the mechanism of superconductivity and CDW in kagome materials.

Moreover, compared to the high-Tc reported in compressed hydrides with either high hydrogen content or under extremely high pressures, the unique hydrogen-kagome structure effectively regulates the interaction between charge and orbital degrees of freedom, thereby serving as an efficient route to high-Tc at low pressures. Subsequently, the experimental synthesis for high-Tc materials could be easier to achieve. Although the origin and correlation mechanism between CDW and superconductivity in hydrogen-based kagome have been pioneeringly demonstrated in our research, further research on improving superconductivity and revealing intricate charge ordering, as well as physical properties are urgently demanded. Based on our proposed hydrogen-based kagome model and mechanism demonstration, we believe the simple kagome materials composed of other elements will provide significant theoretical guidance for the macroscopic preparation of high-Tc superconductors and further unveil even more exotic physical properties.

Methods

Ab initio calculations

The density functional theory (DFT) in coordination with the Perdew-Burke-Ernzerhof parameter of the Generalized Gradient Approximation (GGA) as included in the Vienna ab initio simulation program (VASP) was employed to perform the structural optimization and property analysis57,58. The ion and electron interactions were treated with the projector-augmented wave (PAW) method57. The tested plane-wave basis set cutoff of 1000 eV, Monkhorst-Pack k-meshes spacing of 2π × 0.03 Å–1, and self-consistent field tolerance of 0.1 × 10-5 eV/atom were used. Dynamic stability and superconductivities were calculated based on DFT theory and the plane-wave pseudopotential method with Super-soft pseudo-potentials, as implemented in the QUANTUM ESPRESSO code59. Super-soft pseudo-potentials were used with a kinetic energy cut-off of 100 Ry. The charge density cut-off of the plane wave basis was chosen to be 1000 Ry. After convergence accuracy test, the k- and q-points meshes in the first Brillouin zone were 24 × 24 × 24 and 4 × 4 × 4 for AH3Li5 (A = C, N, Si, P, S).

Superconductivity calculations

The superconducting Tcs were estimated by the Allen-Dynes modified McMillan equation60,61. \({T}_{{\rm{c}}}=\frac{{\omega }_{\log }}{1.2}\exp \left[-\tfrac{1.04\left(1+{\rm{\lambda }}\right)}{{\rm{\lambda }}-{\mu }^{* }\left. (1+0.62{\rm{\lambda }}\right)}\right]\), where λ is the EPC parameter, ωlog is the logarithmic average frequency and μ* is the Coulomb pseudopotential parameter62. Referring to previous theoretical reports, the μ* can be best chosen of 0.1 ~ 0.1333. A degauss value of 0.025 was adopted to guarantee the EPC parameter and Tc values are converged. The EPC λ can be obtained by the integral: \({\rm{\lambda }}=2{\int }_{0}^{{\infty }}\frac{{\alpha }^{2}F(\omega )}{\omega }{\rm{d}}\omega\), and ωlog is defined as \({\omega }_{\log }=\exp \left(\frac{2}{{\rm{\lambda }}}{\int }_{0}^{{\infty }}\frac{{\rm{d}}\omega }{\omega }{\alpha }^{2}F(\omega )\mathrm{ln}\omega \right)\). Eliashberg spectral function uses the formula: \({\alpha }^{2}F\left(\omega \right)\,{=}\,\frac{1}{2\pi N({\varepsilon }_{F})}\sum _{{\bf{q}}v}\frac{{\gamma }_{{\bf{q}}v}}{{\omega }_{{\bf{q}}v}}\delta \left(\omega -{\omega }_{{\bf{q}},v}\right)\)63, and phonon linewidth is defined as: \({\gamma }_{{\bf{q}}v}=\pi {{\rm{\omega }}}_{{\bf{q}}v}\sum _{{mn}}\sum _{{\bf{k}}}{\left|{g}_{{mn}}^{v}\left({\bf{k}},{\bf{q}}\right)\right|}^{2}\delta ({\varepsilon }_{m{,}{\bf{k}}+{\bf{q}}}-{\varepsilon }_{f})\times \delta ({\varepsilon }_{n,{\bf{k}}}-{\varepsilon }_{f})\), where \(N\left({\varepsilon }_{F}\right)\) is the DOS at the EF, \({\omega }_{{\boldsymbol{q}}v}\) is the phonon frequency of mode υ and wave vector q and \({g}_{{mn}}^{v}\left({\bf{k}},{\bf{q}}\right)\) is the electron-phonon matrix element between two electronic states with momenta k and k + q at the EF and the \({\gamma }_{{\bf{q}}v}\) represents linewidth.