Eliashberg Equations Research Articles

Phonon-mediated superconductivity in orthorhombic XS (X = Nb, Ta and W)

Abstract The unique three-dimensional (3D) orthorhombic NbS ($o$-NbS) synthesized in 1969 has recently been experimentally confirmed to be a superconductor [Ruan B B $et$ $al.$ 2023 Phys. Rev. B 108 174517]. However, there is currently no theoretical research on its superconducting mechanism. In this work, we investigate superconducting properties of $o$-NbS from first-principles calculations. Based on Eliashberg equation, it is found that the superconductivity is mainly originated from the coupling between the electrons of Nb-$4d$ orbitals and the vibrations of Nb atoms in the low-frequency region and the vibrations of S atoms in the high-frequency region. A superconducting transition temperature ($T_{c}$) of 10.7 K is obtained, which is close to the experimental value and higher than most transition-metal chalcogenides (TMCs). The calculated thermodynamic properties in the superconducting state, such as specific heat, energy gap, isotope coefficient, etc. also indicate that $o$-NbS is a conventional phonon-mediated superconductor. These results are consistent with recent experimental reports and provide a good understanding on superconducting mechanism of $o$-NbS. Furthermore, the TMCs of $o$-TaS and $o$-WS are also investigated, which belong to the same and neighboring groups as Nb, and find that $o$-$X$S ($X$ = Ta and W) are also phonon-mediated superconductors with $T_{c}$ of 8.9 K and 7.2 K, respectively.

Designing multicomponent hydrides with potential high Tc superconductivity

While hydrogen-rich materials have been demonstrated to exhibit high Tc superconductivity at high pressures, there is an ongoing search for ternary, quaternary, and more chemically complex hydrides that achieve such high critical temperatures at much lower pressures. First-principles searches are impeded by the computational complexity of solving the Eliashberg equations for large, complex crystal structures. Here, we adopt a simplified approach using electronic indicators previously established to be correlated with superconductivity in hydrides. This is used to study complex hydride structures, which are predicted to exhibit promisingly high critical temperatures for superconductivity. In particular, we propose three classes of hydrides inspired by the Fm[Formula: see text]m RH[Formula: see text] structures that exhibit strong hydrogen network connectivity, as defined through the electron localization function. The first class [RH[Formula: see text]X[Formula: see text]Y] is based on a Pm[Formula: see text]m structure showing moderately high Tc, where the Tc estimate from electronic properties is compared with direct Eliashberg calculations and found to be surprisingly accurate. The second class of structures [(RH[Formula: see text])[Formula: see text]X[Formula: see text]YZ] improves on this with promisingly high density of states with dominant hydrogen character at the Fermi energy, typically enhancing Tc. The third class [(R[Formula: see text]H[Formula: see text])(R[Formula: see text]H[Formula: see text])X[Formula: see text]YZ] improves the strong hydrogen network connectivity by introducing anisotropy in the hydrogen network through a specific doping pattern. These design principles and associated model structures provide flexibility to optimize both Tc and the structural stability of complex hydrides.

Two-Gap Superconductivity and the Decisive Role of Rare-Earth d Electrons in Infinite-Layer Nickelates.

We present a theoretical prediction of a phonon-mediated two-gap superconductivity in infinite-layer nickelates Nd_{1-x}Sr_{x}NiO_{2} by performing abinitio GW and GW perturbation theory calculations. Electron GW self-energy effects significantly alter the characters of the two-band Fermi surface and enhance the electron-phonon coupling, compared with results based on density functional theory. Solutions of the fully k-dependent anisotropic Eliashberg equations yield two dominant s-wave superconducting gaps-a large gap on a band of rare-earth Nd d and interstitial orbital characters and a small gap on a band of transition-metal Ni d character. Increasing hole doping induces a non-rigid-band response in the electronic structure, leading to a rapid drop of the superconducting T_{c} in the overdoped regime in agreement with experiments.

Superconductivity in Monolayer Carbon Allotropes with High Thermal Stability.

Intrinsic superconductivity is rarely discovered in sp2-hybridized monolayer carbon allotropes. Here we design a carbon monolayer configured of pentagon, heptagon, and hexagon rings with p2 plane group symmetry. Full-sp2 hybridization is proposed to favor thermal metastability on a low Gibbs free energy. The extremely small thermal expansion coefficient is predicted to the turn negative value to positive with elevating temperature. Carbon polygon structures remain intact at a high thermal temperature of 3,000 K. The high specific surface area is found to approach 2,700 m2/g, with O2-adsorption being advantageous over pristine graphene. We reveal electronic Fermi surfaces mediated by phonon modes of carbon out-of-plane vibrations. By calculating the Eliashberg equation, we evaluate intrinsic superconductivity with a large electron-phonon coupling coefficient. The superconducting transition temperature is estimated to reach 20 K under a high logarithmic average frequency. These first-principles calculations shall stimulate experimentalists' interest in exploring low-dimensional carbon superconductors with gas sensitivity.

Eliashberg Theory for Dynamical Screening in Bilayer Exciton Condensation.

We study the effect of dynamical screening of interactions on the transition temperatures (T_{c}) of exciton condensation in a symmetric bilayer of quadratically dispersing electrons and holes by solving the linearized Eliashberg equations for the anomalous interlayer Green's functions. We find that T_{c} is finite for the range of density and layer separations studied, decaying exponentially with interlayer separation. T_{c} is suppressed well below that predicted by a Hartree Fock mean field theory with unscreened Coulomb interaction, but is above the estimates from the statically screened Coulomb interaction. Furthermore, using a diagrammatic framework, we show that the system is always an exciton condensate at zero temperature but T_{c} is exponentially small for large interlayer separation.

Doping dependence and multichannel mediators of superconductivity: calculations for a cuprate model

We study two aspects of the superconductivity in a cuprate model system, its doping dependence and the influence of competing pairing mediators. We first include electron–phonon interactions beyond Migdal’s approximation and solve self-consistently, as a function of doping and for an isotropic electron–phonon coupling, the full-bandwidth, anisotropic vertex-corrected Eliashberg equations under a non-interacting state approximation for the vertex correction. Our results show that such pairing interaction supports the experimentally observed dx2−y2 -wave symmetry of the superconducting gap, but only in a narrow doping interval of the hole-doped system. Depending on the coupling strength, we obtain realistic values for the gap magnitude and superconducting critical temperature Tc close to optimal doping, rendering the electron–phonon mechanism an important candidate for mediating superconductivity in this model system. Second, for a doping near optimal hole doping, we study multichannel superconductivity, by including both vertex-corrected electron–phonon interaction and spin and charge fluctuations as pairing mechanisms. We find that both mechanisms cooperate to support an unconventional d-wave symmetry of the order parameter, yet the electron–phonon interaction is mainly responsible for the Cooper pairing and high critical temperature Tc . Spin fluctuations are found to have a suppressing effect on the gap magnitude and critical temperature due to their repulsive interaction at small coupling wave vectors.

Anomalous superconductivity in Li/F modified two-dimensional molybdenene

Anomalous superconductivity in Li/F modified two-dimensional molybdenene

Van Hove singularity driven enhancement of superconductivity in two-dimensional tungsten monofluoride (WF)

Superconductivity in two-dimensional materials has gained significant attention in the last few years. In this work, we report phonon-mediated superconductivity investigations in monolayer Tungsten monofluoride (WF) by solving anisotropic Migdal Eliashberg equations as implemented in EPW. By employing first-principles calculations, our examination of phonon dispersion spectra suggests that WF is dynamically stable. Our results show that WF has weak electron–phonon coupling (EPC) strength (λ) of 0.49 with superconducting transition temperature (T c ) of 2.6 K. A saddle point is observed at 0.11 eV below the Fermi level (E F ) of WF, which corresponds to the Van Hove singularity (VHS). On shifting the Fermi level to the VHS by hole doping (3.7 × 1014 cm−2), the EPC strength increases to 0.93, which leads to an increase in the T c to 11 K. However, the superconducting transition temperature of both pristine and doped WF increases to approximately 7.2 K and 17.2 K, respectively, by applying the Full Bandwidth (FBW) anisotropic Migdal–Eliashberg equations. Our results provide a platform for the experimental realization of superconductivity in WF and enhancement of the superconducting transition temperature by adjusting the position of E F to the VHS.

Prediction of High‐Temperature Superconductivity in Monolayer h‐AlH2 at Ambient Pressure

Although hydrides such as are experimentally confirmed to possess high superconducting critical temperature () of 250–260 K under 170–200 GPa, it is still a tough challenge to be applied. It is highly anticipated to find hydride superconductors with relatively high at low or ambient pressure. Reducing the dimensionality of materials can induce unexpected properties that are distinct from their bulk counterparts, and whether it can modulate the superconducting properties deserves further investigation. Herein, a new 2D monolayer aluminum hydride h‐ is theoretically predicted under ambient pressure based on the first‐principles calculations. Since the electronic structures of h‐ reveal the metallicity, the electron–phonon coupling (EPC) and possible phonon‐mediated superconductivity are investigated. Based on the isotropic Eliashberg equation, the calculated EPC constant λ of h‐ is 1.16, and the is up to 42.6 K. The EPC mainly originates from the coupling between electrons of Al‐s,,, and H‐s orbitals and the in‐plane vibration modes of H atoms. Especially, the can be enhanced to 63.7 K by applying 3% biaxial tensile strain. Thus, the predicted h‐ provides a new platform for finding hydride superconductors in low‐dimensional materials at ambient pressure.

Superconducting two-dimensional penta materials

Successful experimental synthesis and theoretical prediction of two-dimensional (2D) penta materials with various properties have enriched the family of 2D materials. In this work, based on first-principles calculations, we systematically study the electronic structures, phonon dispersions, and electron-phonon coupling (EPC) of four metallic penta materials ‘XY2’ (NC2, PC2, BN2, and BP2). Three of them display weak phonon-mediated superconductivity with different low superconducting transition temperatures (Tc). The highest Tc among them is 12 K for BP2. Through applying biaxial tensile strain, the λ and Tc of PC2 (BN2) can be enhanced up to 1.64 (2.68) and 49.77 (52.29) K, respectively. By solving the anisotropic Eliashberg equations, single-gap superconducting nature is predicted for PC2 and BN2. The high sensitive of superconductivity in these systems clearly demonstrate that they are potential candidates for applications in using as superconducting nanodevices. Our work will stimulate further searching of high-Tc superconductivity in low-dimensional materials.

Influence of Coulomb interactions on Einstein phonons in a three-dimensional electron gas and superconductivity

This study investigates the influence of Coulomb interactions on Einstein phonons in a three-dimensional electron gas (3DEG) and their impact on superconducting properties. By considering Coulomb effects in all contributions to the phonon self-energies, we explore the renormalization of phonons and its implications for superconductivity. Numerical solutions of the isotropic Eliashberg equations are used to determine the gap functions, providing valuable insights into the behavior of renormalized phonons. Our analysis reveals that while the frequencies of renormalized phonons are lower compared to non-interacting Einstein phonons, their inclusion leads to higher superconducting critical temperatures and maximum gap functions. However, as the strength of Coulomb interactions increases, a decrease in the superconducting critical temperature and gap is observed. These findings highlight the significant role of Coulomb interactions in determining the behavior of phonons and the emergence of superconductivity in three-dimensional electron gases.

Electron-phonon coupling from GW perturbation theory: Practical workflow combining BerkeleyGW, ABINIT, and EPW

We present a workflow of practical calculations of electron-phonon (e-ph) coupling with many-electron correlation effects included using the GW perturbation theory (GWPT). This workflow combines BerkeleyGW, ABINIT, and EPW software packages to enable accurate e-ph calculations at the GW self-energy level, going beyond standard calculations based on density functional theory (DFT) and density-functional perturbation theory (DFPT). This workflow begins with DFT and DFPT calculations (ABINIT) as starting point, followed by GW and GWPT calculations (BerkeleyGW) for the quasiparticle band structures and e-ph matrix elements on coarse electron k- and phonon q-grids, which are then interpolated to finer grids through Wannier interpolation (EPW) for computations of various e-ph coupling determined physical quantities such as the electron self-energies or solutions of anisotropic Eliashberg equations, among others. A gauge-recovering symmetry unfolding technique is developed to reduce the computational cost of GWPT (as well as DFPT) while fulfilling the gauge consistency requirement for Wannier interpolation.

Superconductivity in two-dimensional MB4 (M = V, Nb, and Ta)

The study of two-dimensional (2D) superconductors is one of the most prominent areas in recent years and has remained a long-standing scientific challenge. Since the introduction of the new member of 2D material family consisting of transition-metal elements with Borides (MBenes), the study of various properties of 2D metal borides has attracted widespread interest. In this work, we systematically investigate the phonon-mediated superconductivity in 2D MB4 (M = V, Nb, and Ta), by using the first-principles calculations. Starting from the dynamical stabilities of these structures, we perform a detailed analysis of the electron–phonon coupling and superconductivity of AB-stacked MBenes by solving the anisotropic Eliashberg equation. NbB4 has the largest electron–phonon coupling and the highest superconducting transition temperature T c of 35.4 K. Our study broadens the 2D superconducting boride family, which is of great significance for the study of 2D superconductivity.

Cubic C20: An intrinsic superconducting carbon allotrope

Finding intrinsic carbon superconductors is an interesting topic. Based on density-functional first-principles calculations, we first study the phonon-mediated superconductivity in a cubic metallic carbon allotrope, namely sc-C20, which has been synthesized in an experiment. Electron–phonon coupling is accurately computed with the Wannier interpolation method. By solving the Eliashberg equations, we predict that sc-C20 is an intrinsic carbon superconductor, without introducing any guest atoms or doping, whose transition temperature is determined to be about 24 K. Our findings enrich the family of carbon-based superconductors.

The link between s and d components of electron boson coupling constants in one band d wave Eliashberg theory for high Tc superconductors

The phenomenology of overdoped high Tc uperconductors can be described by a one band d wave Eliashberg theory where the mechanism of superconducting coupling is mediated by antiferromagnetic spin fluctuations and whose characteristic energy Ω0 scales with Tc according to the empirical law Ω0 = 5.8 kBTc. This model presents universal characteristics that are independent of the critical temperature such as the link between the s and d components of electron boson coupling constants and the invariance of the ratio 2∆/kBTc. This situation arises from the particular structure of Eliashberg's equations which, despite being non-linear equations, present solutions with these simple properties.

УРАВНЕНИЕ ЭЛИАШБЕРГА, ЗАПИСАННОЕ НА МНИМОЙ ОСИ ДЛЯ ИЗОТРОПНОЙ СРЕДЫ: АЛГОРИТМ РЕШЕНИЯ И ПРИМЕРЫ РАСЧЕТОВ

At present, the task of searching for compounds with a high superconducting transition temperature ТС is a very relevant scientific direction. Usually, the calculation of ТС is performed by numerically solving the system of Eliashberg equations. This system is a system of integral equations, which, by discretizing the integrals included in it, is reduced to a system of transcendental non-linear equations, suitable for a numerical solution. In this paper, we present a set of programs for solving this system written in various forms on the imaginary axis for an isotropic medium. In addition, formulas for the Jacobian of this system are presented for the first time, the use of which in numerical calculations increases the speed and stability of calculations. As an example of the developed methods applications, the results of calculations of ТС and thermodynamic properties for the metallic hydrogen I41/amd phase under a pressure of 500 GPa are given. Also, the results of calculations of ТС for some metallic hydrides superconducting at high pressures, are given.

Prediction of two stable freestanding β12 borophanes: Structure, electronic properties, and superconductivity

Borophene has several configurations and shows many unique properties, such as the Dirac point, the negative Poisson's ratio, inherent metallicity, but it is oxidized easily. Recently, ordered chemical modulation of ${\ensuremath{\beta}}_{12}$ borophene grown on the Ag(111) substrate by hydrogenation was experimentally reported, and three configurations of ${\ensuremath{\beta}}_{12}$ borophane were proposed by scanning tunneling microscopy and spectroscopy [Science 371, 1143 (2021)]. In this paper, using first-principles calculations, we study seven configurations of hydrogen atoms adsorbed on freestanding ${\ensuremath{\beta}}_{12}$ borophene not only including the three adsorption structures in above literature, but also other four possible configurations with one or two hydrogen atoms adsorbed on it per unit cell. Among the seven configurations, we predict two stable ${\ensuremath{\beta}}_{12}$ borophanes. They possess the dynamic stability and different anisotropic metallicity from pristine ${\ensuremath{\beta}}_{12}$ borophene and show superconductivity. Based on the Eliashberg equation, the calculated electron-phonon coupling strength are about 0.82 and 0.52, and their superconducting transition temperature ${T}_{\mathrm{c}}$ are 20.51 and 5.90 K, respectively. Thus, these borophanes provide new platforms for two-dimensional superconductivity and antioxidation of borophene.

Prediction of π-electrons mediated high-temperature superconductivity in monolayer LiC12

Prediction and synthesis of two-dimensional high transition temperature (T C) superconductors is an area of extensive research. Based on calculations of the electronic structures and lattice dynamics, we predict that graphene-like layered monolayer LiC12 is a π-electrons mediated Bardeen–Cooper–Schrieffer-type superconductor. Monolayer LiC12 is theoretically stable and expected to be synthesized experimentally. From the band structures and the phonon dispersion spectrum, it is found that the saddle point of π-bonding bands induces large density of states at the Fermi energy level. There is strongly coupled between the vibration mode in the in-plane direction of the lithium atoms and the π-electrons of carbon atoms, which induces the high-T C superconductivity in LiC12. The T C can reach to 41 K under an applied 10% biaxial tensile strain based on the anisotropic Eliashberg equation. Our results show that monolayer LiC12 is a good candidate as π-electrons mediated electron-phonon coupling high-T C superconductor.

Theoretical Prediction of Superconductivity in Boron Kagome Monolayer: MB3 (M = Be, Ca, Sr) and the Hydrogenated CaB3

Using first-principles calculations, we predict a new type of two-dimensional (2D) boride MB3 (M = Be, Ca, Sr), constituted by boron kagome monolayer and the metal atoms adsorbed above the center of the boron hexagons. The band structures show that the three MB3 compounds are metallic, thus the possible phonon-mediated superconductivity is explored. Based on the Eliashberg equation, for BeB3, CaB3, and SrB3, the calculated electron–phonon coupling constants λ are 0.46, 1.09, and 1.33, and the corresponding superconducting transition temperatures T c are 3.2, 22.4, and 20.9 K, respectively. To explore superconductivity with higher transition temperature, hydrogenation and charge doping are further considered. The hydrogenated CaB3, i.e., HCaB3, is stable, with the enhanced λ of 1.39 and a higher T c of 39.3 K. Moreover, with further hole doping at the concentration of 5.8 × 1011 hole/cm2, the T c of HCaB3 can be further increased to 44.2 K, exceeding the McMillan limit. The predicted MB3 and HCaB3 provide new platforms for investigating 2D superconductivity in boron kagome lattice since superconductivity based on monolayer boron kagome lattice has not been studied before.

Ternary superconducting hydrides stabilized via Th and Ce elements at mild pressures

The discovery of covalent H3S and clathrate structure LaH10 with excellent superconducting critical temperatures at high pressures has facilitated a multitude of research on compressed hydrides. However, their superconducting pressures are too high (generally above 150 GPa), thereby hindering their application. In addition, making room-temperature superconductivity close to ambient pressure in hydrogen-based superconductors is challenging. In this work, we calculated the chemically “pre-compressed” Be-H by heavy metals Th and Ce to stabilize the superconducting phase near ambient pressure. An unprecedented ThBeH8 (CeBeH8) with a “fluorite-type” structure was predicted to be thermodynamically stable above 69 GPa (76 GPa), yielding a Tc of 113 K (28 K) decompressed to 7 GPa (13 GPa) by solving the anisotropic Migdal–Eliashberg equations. Be-H vibrations play a vital role in electron–phonon coupling and structural stability of these ternary hydrides. Our results will guide further experiments toward synthesizing ternary hydride superconductors at mild pressures.