Eliashberg Equations Research Articles

The encapsulation of atomically-thin MgB2-based superconductors in two-dimensional material bilayers

Two-dimensional (2D) superconductors are a class of materials with unique properties that can potentially exhibit novel superconductivity at a reduced dimensionality, as well as pave the way for miniaturizing superconducting devices at the atomic limit. However, such materials are highly sensitive to interaction with their substrates and the environment, and therefore it is necessary to find potential methods for chemically isolating them. We address this problem by examining the possibility of isolating the medium-temperature 2D MgB2-based superconductors. We examine two possible structures: the monolayer stoichiometric MgB2 and the B-caged Mg3B8 structure. By solving the anisotropic Eliashberg equations, we predict that the latter possesses K upon application of a small lateral (3% tensile) strain, with an electron–phonon coupling of 0.88. To investigate protecting the superconductivity of these two layers, we apply density functional theory calculations to examine their encapsulation in a graphene bilayer or a hexagonal boron nitride bilayer. We find that both have high potential as encapsulation systems to protect the superconductivity of the 2D MgB2-based superconductors, in particular B-caged (B–Mg) n systems.

Superconductivity in correlated multiorbital systems with spin-orbit coupling: Coexistence of even- and odd-frequency pairing, and the case of Sr2RuO4

The superconducting order parameter of strontium ruthenate is the center of a lasting puzzle calling for theoretical studies that include the seldom-considered effects of spin-orbit coupling and the frequency-dependence of the order parameters. Here we generalize the frequency-dependent theory of superconductivity mediated by spin and charge fluctuations to include spin-orbit coupling in multi-orbital systems and we characterize the superconducting states using the spin-parity-orbital-time $SPOT$ quantum numbers, group theory, and phase distributions in the complex plane. We derive a pseudospin formulation that maps the inter-pseudospin sector of the normal state Eliashberg equation to a pseudospin-diagonal one. Possible superconducting order parameters for strontium ruthenate are obtained starting from a realistic density-functional-theory normal state. We find that spin-orbit coupling leads to ubiquitous entanglement of spin and orbital quantum numbers, along with notable mixing between even- and odd-frequency correlations. We propose a phase diagram obtained from the temperature dependence of the leading and subleading symmetries in the pseudospin-orbital basis. An accidental degeneracy between leading inter-pseudospin symmetries in strontium ruthenate, B$_{1g}^+$ and A$_{2g}^-$, could resolve apparent experimental contradictions.

Superconducting Gap of Pressure Stabilized (Al0.5Zr0.5)H3 from Ab Initio Anisotropic Migdal-Eliashberg Theory.

Motivated by Matthias’ sixth rule for finding new superconducting materials in a cubic symmetry, we report the cluster expansion calculations, based on the density functional theory, of the superconducting properties of Al0.5Zr0.5H3. The Al0.5Zr0.5H3 structure is thermodynamically and dynamically stable up to at least 200 GPa. The structural properties suggest that the Al0.5Zr0.5H3 structure is a metallic. We calculate a superconducting transition temperature using the Allen–Dynes modified McMillan equation and anisotropic Migdal–Eliashberg equation. As result of this, the anisotropic Migdal–Eliashberg equation demonstrated that it exhibits superconductivity under high pressure with relatively high-Tc of 55.3 K at a pressure of 100 GPa among a family of simple cubic structures. Therefore, these findings suggest that superconductivity could be observed experimentally in Al0.5Zr0.5H3.

Enhanced superconductivity in CuH2 monolayers

Achieving superconductivity in hydrides at lower pressures is a long-standing scientific challenge. Here, we propose that reducing their effective dimensionality might be a possible route to achieve this goal. First-principles structural search calculations identify several two-dimensional transition metal hydrides (TMHs) with phonon-mediated superconductivity. Among them, the two ${\mathrm{CuH}}_{2}$ phases with $P\text{\ensuremath{-}}3m1$ and $P\text{\ensuremath{-}}6m2$ symmetries are predicted to have the highest ${T}_{\mathrm{c}}$ values up to 44.4 and 47.8 K, comparable to the well-known ${\mathrm{MgB}}_{2}$ superconductor. Their superconductivity mainly originates from the coupling of Cu $3{d}_{xz}/{d}_{yz}$ and $\mathrm{H}\text{\ensuremath{-}}1s$ electrons with in-plane H-vibrational phonons. Interestingly, both ${\mathrm{CuH}}_{2}$ structures show a unique superconducting energy gap by solving the anisotropic Eliashberg equations. Furthermore, the superconductivity of TMHs is closely related to the polarity of the TM-H bond. Our paper paves the way for finding hydride superconductors in low-dimensional materials.

Migdal-Eliashberg theory as a classical spin chain

We formulate the Migdal-Eliashberg theory of electron-phonon interactions in terms of classical spins by mapping the free energy to a Heisenberg spin chain in a Zeeman magnetic field. Spin components are energy-integrated normal and anomalous Green's functions and sites of the chain are fermionic Matsubara frequencies. The Zeeman field grows linearly with the spin coordinate and competes with ferromagnetic spin-spin interaction that falls off as the square of the inverse distance. The spin-chain representation makes a range of previously unknown properties plain to see. In particular, infinitely many new solutions of the Eliashberg equations both in the normal and superconducting states emerge at strong coupling. These saddle points of the free-energy functional correspond to spin flips. We argue that they are also fixed points of kinetic equations and play an essential role in far from equilibrium dynamics of strongly coupled superconductors. Up to an overall phase, the frequency-dependent gap function that minimizes the free energy must be non-negative. There are strong parallels between our Eliashberg spins and Anderson pseudospins, though the two sets of spins never coincide.

Van Hove singularity induced phonon-mediated superconductivity above 77 K in hole-doped SrB3C3

To design a phonon-mediated superconductor with transition temperature above the liquid-nitrogen temperature under ambient pressure, we align the Fermi level of compound ${\mathrm{SrB}}_{3}{\mathrm{C}}_{3}$ to a Van Hove singularity by introducing 0.4 hole/f.u., namely ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$. Based on density functional first-principles calculations and the Wannier interpolation technique, the electronic structure, lattice dynamics, and electron-phonon coupling (EPC) of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ are investigated. Both the enlarged density of states at the Fermi level and the softened phonons play important roles to enhance the EPC. By solving the anisotropic Eliashberg equations, we find that the transition temperature of ${\mathrm{Rb}}_{0.4}{\mathrm{Sr}}_{0.6}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ is 83 K, successfully exceeding the liquid-nitrogen temperature at ambient pressure condition. Moreover, our calculations reveal that there is a single-gap to two-gap superconductivity transition in comparison with ${\mathrm{SrB}}_{3}{\mathrm{C}}_{3}$.

Theoretical prediction of superconductivity in monolayer B3N

Using first-principles calculations, we predict the two-dimensional (2D) monolayer boron-nitrogen compound ${\mathrm{B}}_{3}\mathrm{N}$, which shows a honeycomb lattice similar to graphene. Its stability is proved by phonon spectra and ab initio molecular dynamics simulations. Since pristine ${\mathrm{B}}_{3}\mathrm{N}$ is a metal by band structure calculation, we investigate the electron-phonon coupling and possible phonon-mediated superconductivity. Based on the Eliashberg equation, the calculated electron-phonon coupling strength is about 0.66, and the superconducting transition temperature ${T}_{c}$ is 14.1 K, which is comparable to that of borophene. Furthermore, when 0.04 electron/cell doping and 10% biaxial tensile strain are applied, ${T}_{c}$ can be increased to 17.4 K. Thus, the predicted ${\mathrm{B}}_{3}\mathrm{N}$ provides a platform for 2D superconductivity.

Interorbital singlet pairing in Sr2RuO4 : A Hund's superconductor

We study the superconducting gap function of Sr$_2$RuO$_4$. By solving the linearized Eliashberg equation with a correlated pairing vertex extracted from a dynamical mean-field calculation we identify the dominant pairing channels. An analysis of the candidate gap functions in orbital and quasiparticle band basis reveals that an inter-orbital singlet pairing of even parity is in agreement with experimental observations. It reconciles in particular the occurrence of a two-component order parameter with the presence of line-nodes of quasiparticles along the c-axis in the superconducting phase. The strong angular dependence of the gap along the Fermi surface is in stark contrast to its quasi-locality when expressed in the orbital basis. We identify local inter-orbital spin correlations as the driving force for the pairing and thus reveal the continuation of Hund's physics into the superconducting phase.

Euclidean Q-Balls of Fluctuating SDW/CDW in the ‘Nested’ Hubbard Model of High-Tc Superconductors as the Origin of Pseudogap and Superconducting Behaviors

The origin of the pseudogap and superconducting behaviors in high-Tc superconductors is proposed, based on the picture of Euclidean Q-balls formation that carry Cooper/local-pair condensates inside their volumes. Euclidean Q-balls that describe bubbles of collective spin-/charge density fluctuations (SDW/CDW) oscillating in Matsubara time are found as a new self-consistent solution of the Eliashberg equations in the ‘nested’ repulsive Hubbard model of high-Tc superconductors. The Q-balls arise due to global invariance of the effective theory under the phase rotation of the Fourier amplitudes of SDW/CDW fluctuations, leading to conservation of the ‘Noether charge’ Q in Matsubara time. Due to self-consistently arising local minimum of their potential energy at finite amplitude of the density fluctuations, the Q-balls provide greater binding energy of fermions into local/Cooper pairs relative to the usual Frohlich mechanism of exchange with infinitesimal lattice/charge/spin quasiparticles. We show that around some temperature T* the Q-balls arise with a finite density of superconducting condensate inside them. The Q-balls expand their sizes to infinity at superconducting transition temperature Tc. The fermionic spectral gap inside the Q-balls arises in the vicinity of the ‘nested’ regions of the bare Fermi surface. Solutions are found analytically from the Eliashberg equations with the ‘nesting’ wave vectors connecting ‘hot spots’ in the Brillouin zone. The experimental ‘Uemura plot’ of the linear dependence of Tc on superconducting density ns in high-Tc superconducting compounds follows naturally from the proposed theory.

Revisiting homogeneous electron gas in pursuit of properly normed ab initio Eliashberg theory

We address an issue of how to accurately include the self energy effect of the screened electron-electron Coulomb interaction in the phonon-mediated superconductors from first principles. In the Eliashberg theory for superconductors, self energy is usually decomposed using the $2\times 2$ Pauli matrices in the electron-hole space. We examine how the diagonal ($\sigma_{0}$ and $\sigma_{3}$) components resulting in the quasiparticle correction to the normal state, $Z$ and $\chi$ terms, behave in the homogeneous electron gas in order to establish a norm of treating those components in real metallic systems. Within the $G_{0}W_{0}$ approximation, we point out that these components are non-analytic near the Fermi surface but their directional derivatives and resulting corrections to the quasiparticle velocity are nevertheless well defined. Combined calculations using the $G_{0}W_{0}$ approximation and Eliashberg equations show us that the effective mass and pairing strength strikingly depend on both $Z$ and $\chi$, in a different manner. The calculations without the numerically demanding $\chi$ term is thus shown to be incapable of describing the homogeneous electron gas limit. This result poses a challenge to accurate first-principles Eliashberg theory.

Carbonaceous sulfur hydride system: The strong-coupled room-temperature superconductor with a low value of Ginzburg–Landau parameter

The superconducting state in a carbonaceous sulfur hydride (C–S–H) system is probably characterized by the record-high critical temperature of 288 K (p≈267 GPa). We determined the properties of the C–S–H superconducting phase within the scope of both classical Eliashberg equations and the Eliashberg equations with vertex corrections. We took into account the scenarios pertinent to either the intermediate or the high value of an electron–phonon coupling constant (λ≈0.75 or λ≈3.3, respectively). The scenario for the intermediate value, however, cannot be actually realized due to the anomalously high value of the logarithmic phonon frequency (ωln/kB=7150 K) it would require. On the other hand, we found it possible to reproduce correctly the value of TC and other thermodynamic quantities in the case of strong coupling, with all the reservations discussed in the presented paper. The vertex corrections lower the order parameter values within the range from ≈50 K to ≈275 K. For the upper critical field HC2≈27 T, the Ginzburg–Landau parameter κ is of the order of 1.7. The strong-coupling scenario for the C–S–H system is also suggested by the high values of λ estimated for H3S (λ≈2.1, κ≈1.5), LaH10 (λ≈2.8–3.9, κ≈1.6), and YH6 (λ≈1.7, κ≈1.3) compounds. In the case of the C–S–H system, we also anticipate the presence of the antiferromagnetic state above the superconducting state like in the dense CS2 superconductor. For p≈174 GPa and TC≈180 K, the magnetic ordering transition occurs at TN≈213 K.

High-temperature multigap superconductivity in two-dimensional metal borides

Using first-principles calculations in combination with the Eliashberg formalism, we systematically investigated phonon-mediated superconductivity in two-dimensional (2D) metal-boride crystals, consisting of a boron honeycomb network doped by diverse metal elements. Such 2D metal-boride compounds, named MBenes, are chemically exfoliable from single-crystalline layered ternary borides. First we identified the MBene layers with potential for superconductivity via isotropic Eliashberg calculations, considering a wide range of metal elements, with a focus on alkaline earth and transition metals. Subsequently, we performed a detailed analysis of the prominent superconducting MBenes by solving the anisotropic Eliashberg equations. The obtained high critical temperatures (up to 72 K), as well as the rich multigap superconducting behavior, recommend these crystals for further use in multifunctional 2D heterostructures and superconducting device applications.

Machine learning of superconducting critical temperature from Eliashberg theory

The Eliashberg theory of superconductivity accounts for the fundamental physics of conventional superconductors, including the retardation of the interaction and the Coulomb pseudopotential, to predict the critical temperature Tc. McMillan, Allen, and Dynes derived approximate closed-form expressions for the critical temperature within this theory, which depends on the electron–phonon spectral function α2F(ω). Here we show that modern machine-learning techniques can substantially improve these formulae, accounting for more general shapes of the α2F function. Using symbolic regression and the SISSO framework, together with a database of artificially generated α2F functions and numerical solutions of the Eliashberg equations, we derive a formula for Tc that performs as well as Allen–Dynes for low-Tc superconductors and substantially better for higher-Tc ones. This corrects the systematic underestimation of Tc while reproducing the physical constraints originally outlined by Allen and Dynes. This equation should replace the Allen–Dynes formula for the prediction of higher-temperature superconductors.

Unconventional superconductivity mediated solely by isotropic electron-phonon interaction

Unconventional superconductivity is commonly linked to electronic pairing mechanisms, since it is believed that the conventional electron-phonon interaction (EPI) cannot cause sign-changing superconducting gap symmetries. Here, we show that this common understanding needs to be revised when one considers a more elaborate theory of electron-phonon superconductivity beyond standard approximations. We self-consistently solve the full-bandwidth, anisotropic Eliashberg equations including vertex corrections beyond Migdal's approximation assuming the usual isotropic EPI for cuprate, Fe-based, and heavy-fermion superconductors with nested Fermi surfaces. In the case of the high-${T}_{c}$ cuprates we find a $d$-wave order parameter, as well as a nematic state upon increased doping. For Fe-based superconductors, we obtain ${s}_{\ifmmode\pm\else\textpm\fi{}}$ gap symmetry, while for heavy-fermion ${\mathrm{CeCoIn}}_{5}$ we find unconventional $d$-wave pairing. These results provide a proof of concept that EPI cannot be excluded as a mediator of unconventional and of high-${T}_{c}$ superconductivity.

Metallic Hydrogen: A Liquid Superconductor?

Metallic hydrogen is expected to exhibit remarkable physics. Of particular interest in this work is the possibility of high-temperature superconductivity. Comparing calculations of the superconducting critical temperatures of the solid phase to melting temperatures over a range of pressures leads to an interesting question: Will the solid, in a superconducting state, melt to a liquid that remains a superconductor? In this work, the possibility of liquid superconductivity in metallic hydrogen is investigated. This is done by first-principles simulations, and using the results of these to solve the Eliashberg equations. These are carried out over the pressure (and temperature) conditions where molecular dissociation is expected to first occur in the solid phase. Over the pressure range $386.8(4)$--$783.7(4)$ GPa, $T_c$ increases from $308(6)$ to $372(2)$ K with a maximum uncertainty of $10$ K; it then decreases to $356(2)$ K at $883.7(3)$ GPa. Comparisons to the solid phase show that the critical temperature is not significantly changed between the two phases, though the physics behind their superconductivity is different. Careful comparisons of these values to recent results in the context of the hydrogen phase diagram show that they are higher than the melting temperatures and that the solid will melt to liquid atomic hydrogen. The results of this work (in this context) therefore suggest that liquid atomic hydrogen will indeed exist in a superconducting state. They also provide the pressure and temperature conditions over which to look for it.

Exploring multichannel superconductivity in ThFeAsN

We investigate theoretically the superconducting state of the undoped Fe-based superconductor ThFeAsN. Using input from $ab~initio$ calculations, we solve the Fermi-surface based, multichannel Eliashberg equations for Cooper-pair formation mediated by spin and charge fluctuations, and by the electron-phonon interaction (EPI). Our results reveal that spin fluctuations alone, when coupling only hole-like with electron-like energy bands, can account for a critical temperature $T_c$ up to $\sim7.5\,\mathrm{K}$ with an $s_{\pm}$-wave superconducting gap symmetry, which is a comparatively low $T_c$ with respect to the experimental value $T_c^{\mathrm{exp}}=30\,\mathrm{K}$. Other combinations of interaction kernels (spin, charge, electron-phonon) lead to a suppression of $T_c$ due to phase frustration of the superconducting gap. We qualitatively argue that the missing ingredient to explain the gap magnitude and $T_c$ in this material is the first-order correction to the EPI vertex. In the noninteracting state this correction adopts a form supporting the $s_{\pm}$ gap symmetry, in contrast to EPI within Migdal's approximation, i.e., EPI without vertex correction, and therefore it enhances tendencies arising from spin fluctuations.

Phonon-mediated high-temperature superconductivity in the ternary borohydride KB2H8 under pressure near 12 GPa

The discovery of high-temperature superconductivity in hydrogen-rich compounds has fueled the enthusiasm for finding materials with more promising superconducting properties among hydrides. However, the ultrahigh pressure needed to synthesize and maintain high-temperature hydrogen-rich superconductors hinders the experimental investigation of these materials. For practical applications, it is also highly desired to find more hydrogen-rich materials that superconduct at high temperatures but under relatively lower pressures. Based on first-principles density functional theory, we calculate the electronic and phonon band structures for a ternary borohydride formed by intercalating ${\mathrm{BH}}_{4}$ tetrahedrons into a fcc potassium lattice, ${\mathrm{KB}}_{2}{\mathrm{H}}_{8}$. Remarkably, we find that this material is dynamically stable and one of its $s{p}^{3}$-hybridized $\ensuremath{\sigma}$-bonding bands is metallized (i.e., partially filled) above a moderate high pressure. This metallized $\ensuremath{\sigma}$-bonding band couples strongly with phonons, giving rise to a strong superconducting pairing potential. By solving the anisotropic Eliashberg equations, we predict that the superconducting transition temperature of this compound is 134--146 K around 12 GPa.

Phonon-mediated superconductivity in bismuthates by nonadiabatic pairing

In the present paper, the impact of small Fermi energy on the selected parameters of the superconducting state in Ba$_{1-x}$K$_{x}$BiO$_{3}$ (BKBO) is studied at $x \in (0.3, 0.4, 0.5)$. This is done by employing the adiabatic and non-adiabatic Eliashberg equations in context of the available experimental data. It is found that the retardation, strong-coupling and the non-adiabatic effects notably influence superconducting phase in BKBO. In particular, the electron-electron interaction, approximated here by the Coulomb pseudopotential, is argued to be reduced by the non-adiabatic effects that supplement retardation and allow for the phonon-mediated superconductivity. These findings are reinforced by further analysis of the isotope effect showing reduction of the isotope coefficient with respect to the canonical Bardeen-Cooper-Schrieffer (BCS) level, as caused by the interplay of all effects mentioned above. Although physics behind the isotope effect appears to be complex, its resulting behavior comply with the scenario for the conventional superconductors. In summary, obtained results confirm recent theoretical and experimental studies that suggest phonon-mediated mechanism of superconductivity in BKBO. However, they also point out that this phase cannot be properly described with the BCS theory due to the existence of somewhat unusual effects.

Superconducting and normal properties of the low-dimensional phases of the metallic hydrogen

In the first part of the paper we consider properties of a two-dimensional low density (n = 1) electron system with strong onsite Hubbard attraction U >W (W is the bandwidth) in the presence of a strong random potential V uniformly distributed in the range from -V to +V. In the framework of the Bogoliubov-de Gennes approach we observed an appearance of inhomogeneous state of spatially separated Fermi-Bose mixture of Cooper pairs and unpaired electrons with the formation of bosonic droplets of different size in the unpaired fermionic matrix. Software package to solve the system of Eliashberg equations that describes the transition of a metal to a superconducting state is presented. We use a version of the Eliashberg system of equations written on the imaginary axis. The system is presented in two versions: without taking into account the corrections for the chemical potential of electrons and with taking them into account. The calculations were performed for the I41/amd phase of metallic hydrogen at a pressure of 500 GPa.

Strong-coupling theory of condensate-mediated superconductivity in two-dimensional materials

We develop a strong-coupling theory of Bose-Einstein condensate-mediated superconductivity in a hybrid system, which consists of a two-dimensional electron gas with either (i) parabolic spectrum or (ii) relativistic Dirac spectrum in the vicinity of a two-dimensional solid-state condensate of indirect excitons. The Eliashberg equations are derived and the expressions for the electron pairing self-energy due to the exchange interaction between electrons mediated by a single Bogoliubov excitation (a bogolon) and the bogolon pairs are found. Furthermore, we find the superconducting order parameter and estimate the critical temperature of the superconducting transition. The critical temperature reveals its linear dependence on the dimensionless coupling constant. It is shown, that the bogolon-pair-mediated interaction is the dominant mechanism of electron pairing in hybrid systems in both the weak and strong coupling regimes. We calculate the effective bogolon-electron interaction constant for both parabolic and linear electron dispersions and examine the dependence of the critical temperature of electron gas superconducting transition on exciton condensate density.