Eliashberg Equations Research Articles

First-principles study of superconductivity in α and β gallium

Elemental gallium can exist in several phases under ambient conditions. The stable $\alpha$ phase has a superconducting transition temperature, $T_c$, of 0.9~K. By contrast, the $T_c$ of the metastable $\beta$ phase is around 6~K. To understand the significant improvement in $T_c$ in the $\beta$ phase, we first calculate the electronic structure, phonon dispersion, and the electron-phonon coupling of gallium in the $\alpha$ and $\beta$ phase. Next, we solve the Eliashberg equations to obtain the superconducting gaps and the transition temperatures. Using these results, we relate the increased $T_c$ in the $\beta$ phase to structural differences between the phases that affect the electronic and phonon properties. The structure motif of the $\alpha$ phase is Ga$_2$ dimers, which form strong covalent bonds leading to bonding and antibonding states that suppress the density of states at the Fermi level. The $\beta$-Ga structure consists of arrays of Ga chains that favor strong coupling between the lattice vibrations and the electronic states near the Fermi level. The increased density of states and strong coupling to the phonons for the $\beta$-Ga chains compared to the $\alpha$ Ga$_2$ dimers enhance superconductivity in the $\beta$-Ga phase.

Absence of superconductivity in the repulsive Hubbard model on a square lattice in the regime of strong coupling

The possibility of the superconducting transition in the repulsive Hubbard model on a square lattice is studied using the ladder approximation of the strong coupling diagram technique. The t–U and t–t′–t″–U models are considered in the regime of strong correlations, for the on-site Coulomb repulsion U = 8t, in the range of temperatures 0.02t ≲ T ≲ 0.3t. To avoid the influence of the phase separation and size effects the calculations are performed in an infinite crystal, in the part of the phase diagram without inhomogeneities, for the electron concentration . Solutions of the Eliashberg equation for singlet and triplet pairing, which are transformed according to one-dimensional representations of the lattice point group D 4, are considered. For both models and all considered symmetries, eigenvalues of the Eliashberg equation are less than unity and demonstrate no upward trend with decreasing temperature. This result points to the absence of superconductivity in the Hubbard model in the strong-coupling regime. We discuss the reason for the small eigenvalue of the Eliashberg equation in the case of the singlet pairing.

Material Optimization of Potential High-Tc Superconducting Single-Layer Cuprates

We investigated the material parameters of several single-layer cuprates, including those with fluorinated buffer layers, with the aim of identifying possible high-temperature superconductors. To evaluate the material parameters, we use the Wannierization techniques and the constrained random phase approximation. The obtained single-band Hubbard models are studied using the fluctuation–exchange approximation, and the superconductivity is studied by solving the linearized Eliashberg equation. Comparison among several cuprates reveals unknown high-Tc superconductors. In, Ga, Al, and Cd compounds in particular show the potential to exhibit higher-Tc superconductivity than Hg1201.

Inverted V-shaped evolution of superconducting temperature in SrBC under pressure* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11974194 and 11974207) and K. C. Wong Magna Fund in Ningbo University.

Based on density functional first-principles calculations and anisotropic Eliashberg equations, we have investigated the electronic structure, lattice dynamics, and phonon-mediated superconductivity in newly synthesized layered compound SrBC under pressure. Different from LiBC and MgB2, our calculations surprisingly reveal that SrBC is isotropic in compressibility, due to the accumulation of substantial electrons in the interstitial region. We find that the Sr phonons strongly couple with B-2p z orbital and the interstitial states, giving rise to a two-gap superconductivity in SrBC, whose transition temperature shows an inverted V-shaped dependence on pressure. The maximal transition temperature is about 22 K at 50 GPa. On both sides of 50 GPa, the transition temperature exhibits quasi-linear variation with positive and negative slopes, respectively. Such a variation of transition temperature is infrequent among phonon-mediated superconductors. The competition between enhanced electron–phonon matrix element and hardened phonons plays an essential role in governing the behavior of the critical temperature.

Superconducting Transition Temperature for Very Strong Coupling in the Antiadiabatic Limit of Eliashberg Equations

It is shown that the famous Allen -- Dynes asymtotic limit for superconducting transition temperature in very strong coupling region $T_{c}>\frac{1}{2\pi}\sqrt{\lambda}\Omega_0$ (where $\lambda\gg 1$ - is Eliashberg - McMillan electron - phonon coupling constant and $\Omega_0$ - the characteristic frequency of phonons) in antiadiabatic limit of Eliashberg equations $\Omega_0/D\gg 1$ ($D\sim E_F$ is conduction band half-width and $E_F$ is Fermi energy) is replaced by $T_c>(2\pi^4)^{-1/3}(\lambda D\Omega_0^2)^{1/3}$, with the upper limit for $T_c$ given by $T_c<\frac{2}{\pi^2}\lambda D$.

High-temperature superconductivity in SrB3C3 and BaB3C3 predicted from first-principles anisotropic Migdal-Eliashberg theory

Very recently, carbon-boron clathrate $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ has been successfully synthesized, in which carbon and boron atoms form $s{p}^{3}$-bonded truncated octahedral cages. Interestingly, the $s{p}^{3}$-hybridized $\ensuremath{\sigma}$-bonding bands are partially occupied. This may drive $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ into a superconducting state, like boron-doped diamond. By means of density functional first-principles calculations and Wannier interpolation technique, we have investigated the electron-phonon coupling and phonon-mediated superconductivity in $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$. Our calculations reveal that there exists strong coupling between $s{p}^{3}$-hybridized $\ensuremath{\sigma}$-bonding bands and boron-associated ${E}_{g}$ phonon modes. Based on the Migdal-Eliashberg theory, we self-consistently solve the anisotropic Eliashberg equations. It is found that $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ is a single-gap superconductor, with superconducting transition temperature being 40 K. The anisotropic ratio of superconducting energy gap is computed to be 32.8%. Further replacing Sr with Ba, the transition temperature can be boosted to 43 K in $\mathrm{Ba}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ due to phonon softening. These findings suggest that $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ and $\mathrm{Ba}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ are phonon-mediated high-temperature anisotropic $s$-wave superconductors.

Cascade of replica bands in flat-band systems: Predictions for twisted bilayer graphene

We investigate the effect of electron-phonon interactions (EPI) in systems exhibiting one or more flat electron bands close to the Fermi level and a comparatively large phonon energy scale. After solving the self-consistent full-bandwidth Eliashberg equations, we compute Angular Resolved Photo Emission Spectroscopy and Scanning Tunneling Spectroscopy/Microscopy spectra. We obtain a sequence of quasiparticle replica bands in both the normal and superconducting states that originate from frequency dependent features of the electron mass renormalization function. We show that these replica bands can be used to extract the relevant phonon energy scale from experiments. Focusing in particular on twisted bilayer graphene, we predict replica-band formation which, when observed, will shed light on the role of EPI in this archetypal flat-band system.

Pressure-induced superconducting CS2H10 with an H3S framework

Crystal structures of two novel phases P3m1 and Cmc21 of metastable CS2H10 with CH4 molecules inserted into an H3S framework. The Tc values of P3m1 and Cmc21 calculated with the Eliashberg equation are 95 K at 50 GPa and 155 K at 150 GPa, respectively.

Evidence of Phonon‐Mediated Superconductivity in LaH10 at High Pressure

AbstractMotivated by the recent experimental discovery of the high‐critical‐temperature superconductor LaH10 (250–260 K at high pressures 170–190 GPa), the influence of H isotope substitution on the phonon spectra, electron–phonon interactions, and thermodynamic properties of superconducting LaHDx () at 250 GPa are studied. On the basis of first‐principles calculations, it is found that all investigated systems are dynamically stable in a clathrate structure with space group Fmm and exhibit high superconducting critical temperatures ranging from 169 to 234 K. The dominant role of hydrogen in enhancing is examined by numerically solving the Eliashberg equations. The estimated critical temperature, superconducting energy gap, specific heat, and thermodynamic critical field demonstrate that the underlying mechanism of superconductivity in lanthanum hydride is conventional electron–phonon coupling, which manifests itself in terms of isotope effect.

Eliashberg Theory of a Multiband Non-Phononic Spin Glass Superconductor

I solved the Eliashberg equations for a multiband non-phononic s± wave spin-glass superconductor, calculating the temperature dependence of the gaps and of superfluid density. Their behaviors were revealed to be unusual: showing non-monotonic temperature dependence and reentrant superconductivity. By considering particular input parameters values that could describe the iron pnictide EuFe2(As1−xPx)2, a rich and complex phase diagram arises, with two different ranges of temperature in which superconductivity appears.

Fully anisotropic superconductivity with few Helmholtz Fermi-surface harmonics

We present an alternative representation for the anisotropic Eliashberg equations of superconductivity, whose numerical solution yields an efficiency gain of several orders of magnitude with respect to the conventional representation in momentum space. Our method is a practical realization of a long-sought approach, whose essence is a linear transformation from regular ${\bf k}$ space to a set of orthonormal functions defined as the solutions of the Helmholtz equation on the Fermi surface. In this way, all the anisotropy of the problem can be described by a handful of coefficients with built-in symmetry. We perform benchmark calculations on the gap anisotropy of MgB$_2$, and reproduce previous results at a remarkably reduced computational cost. Furthermore, we apply our methodology to efficiently determine the transition temperature of the compressed YH$_6$ hydride, obtaining very good agreement with recent experimental measurements. The simplification introduced by our method enables the high-throughput exploration of superconducting materials without having to resort to the isotropic approximation, and opens up possibilities towards first principles calculations of more advanced theories of superconductivity.

The unbalanced phonon-induced superconducting state on a square lattice beyond the static boundary

The paper presents our verification of induction of the superconducting state on a square lattice by the linear electron–phonon interaction for values of the unbalance parameter (γ=λD∕λND) less than γC=0.42. Symbols λD and λND denote the values of the coupling constant in the diagonal and the non-diagonal channel of the self-energy. Calculations were carried out using the Eliashberg equations, in which the order parameter (Δkiωn) and the wave function renormalising factor (Zkiωn) depend explicitly on the Matsubara frequency (ωn) and the wave vector (k). The value of γC in the static boundary (Δkiωn→Δkiωn=1), equal to (0.93), is significantly greater than the obtained limit value of 0.42. Values of the thermodynamic functions of the superconducting state determined for our assumptions are significantly different from the values calculated in accordance with the BCS theory. The results were obtained for the electron–phonon interaction function explicitly dependent on the momentum transfer between electron states.

Multiple odd-parity superconducting phases in bilayer transition metal dichalcogenides

We study unconventional superconductivity in a two-dimensional locally noncentrosymmetric triangular lattice. The model is relevant to bilayer transition metal dichalcogenides with $2{\mathrm{H}}_{b}$ stacking structure, for example. The superconducting instability is analyzed by solving the linearized Eliashberg equation within the random phase approximation. We show that ferromagnetic fluctuations are dominant owing to the existence of disconnected Fermi pockets near van Hove singularity, and hence odd-parity spin-triplet superconductivity is favored. In the absence of the spin-orbit coupling, we find that odd-parity $f$-wave superconducting state is stabilized in a wide range of carrier density and interlayer coupling. Furthermore, we investigate impacts of the layer-dependent staggered Rashba and Zeeman spin-orbit coupling on the superconductivity. Multiple odd-parity superconducting phase diagrams are obtained as a function of the spin-orbit coupling and Coulomb interaction. Especially, a topological chiral $p$-wave pairing state is stabilized in the presence of a moderate Zeeman spin-orbit coupling. Our results shed light on a possibility of odd-parity superconductivity in various ferromagnetic van der Waals materials.

Superconducting mechanism for the cuprate Ba2CuO3+δ based on a multiorbital Lieb lattice model

For the recently discovered cuprate superconductor $\mathrm{Ba_{2}CuO_{3+\delta}}$, we propose a lattice structure which resembles the model considered by Lieb to represent the vastly oxygen-deficient material. We first investigate the stability of the Lieb-lattice structure, and then construct a multiorbital Hubbard model based on first-principles calculation. By applying the fluctuation-exchange approximation to the model and solving the linearized Eliashberg equation, we show that $s$-wave and $d$-wave pairings closely compete with each other, and, more interestingly, that the intra-orbital and inter-orbital pairings coexist. We further show that, if the energy of the $d_{3z^2-r^2}$ band is raised to make it "incipient" with the lower edge of the band close to the Fermi level within a realistic band filling regime, $s\pm$-wave superconductivity is strongly enhanced. We reveal an intriguing relation between the Lieb model and the two-orbital model for the usual K$_2$NiF$_4$ structure where a close competition between $s-$ and $d-$wave pairings is known to occur. The enhanced superconductivity in the present model is further shown to be related to an enhancement found previously in the bilayer Hubbard model with an incipient band.

How non-Fermi liquids cure their infrared divergences

Non-Fermi liquids in $d=2$ spatial dimensions can arise from coupling a Fermi surface to a gapless boson. At finite temperature, however, the perturbative quantum field theory description breaks down due to infrared divergences. These are caused by virtual static bosonic modes and affect both fermionic and bosonic correlators. We show how these divergences are resolved by self-consistent boson and fermion self-energies that resum an infinite class of diagrams and correct the standard Eliashberg equations. Extending a previous approach in $d=3\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we find a new ``thermal non-Fermi liquid'' regime that violates the scaling laws of the zero-temperature fixed point and dominates over a wide range of scales. We conclude that basic properties of quantum phase transitions and quantum-classical crossovers at finite temperature are modified in crucial ways in systems with soft bosonic fluctuations, and we begin a study of some of the phenomenological consequences.

Combining Eliashberg Theory with Density Functional Theory for the Accurate Prediction of Superconducting Transition Temperatures and Gap Functions.

We propose a practical alternative to Eliashberg equations for the abinitio calculation of superconducting transition temperatures and gap functions. Within the recent density functional theory for superconductors, we develop an exchange-correlation functional that retains the accuracy of Migdal's approximation to the many-body electron-phonon self-energy, while having a simple analytic form. Our functional is based on a parametrization of the Eliashberg self-energy for a superconductor with a single Einstein frequency, and enables density functional calculations of experimental excitation gaps. By merging electronic structure methods and Eliashberg theory, the present approach sets a new standard in quality and computational feasibility for the prediction of superconducting properties.

Full-bandwidth Eliashberg theory of superconductivity beyond Migdal's approximation

We solve the anisotropic, full-bandwidth and non-adiabatic Eliashberg equations for phonon-mediated superconductivity by fully including the first vertex correction in the electronic self-energy. The non-adiabatic equations are solved numerically here without further approximations, for a one-band model system. We compare the results to those that we obtain by adiabatic full-bandwidth, as well as Fermi-surface restricted Eliashberg-theory calculations. We find that non-adiabatic contributions to the superconducting gap can be positive, negative or negligible, depending on the dimensionality of the considered system, the degree of non-adiabaticity, and the coupling strength. We further examine non-adiabatic effects on the transition temperature and the electron-phonon coupling constant. Our treatment emphasizes the importance of overcoming previously employed approximations in estimating the impact of vertex corrections on superconductivity and opens a pathway to systematically study vertex correction effects in systems such as high-$T_c$, flat band and low-carrier density superconductors.

Eliashberg theory with the external pair potential

Based on BCS model with the external pair potential formulated in a work Grigorishin (2017) [1], analogous model with electron-phonon coupling and Coulomb coupling is proposed. The generalized Eliashberg equations in the regime of renormalization of the order parameter are obtained. High temperature asymptotics and effect of Coulomb pseudopotential on them are investigated: as in the BCS model the order parameter asymptotically tends to zero as temperature rises, but the accounting of the Coulomb pseudopotential leads to existence of critical temperature. The effective Ginzburg-Landau theory is formulated for such model.

The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal

Near a quantum-critical point, a metal reveals two competing tendencies: destruction of fermionic coherence and attraction in one or more pairing channels. We analyze the competition within Eliashberg theory for a class of quantum-critical models with an effective dynamical electron–electron interaction V(Ωm)∝1∕|Ωm|γ (the γ-model) for 0<γ<1. We argue that the two tendencies are comparable in strength, yet the one towards pairing is stronger, and the ground state is a superconductor. We show, however, that there exist two distinct regimes of system behavior below the onset temperature of the pairing Tp. In the range Tcross<T<Tp fermions remain incoherent and the spectral function A(k,ω) and the density of states N(ω) both display “gap filling” behavior in which, i.e., the position of the maximum in N(ω) is set by temperature rather than the pairing gap. At lower T<Tcross, fermions acquire coherence, and A(k,ω) and N(ω) display conventional ”gap closing” behavior, when the peak position in N(ω) scales with the gap and shifts to a smaller value as T increases. We argue that the existence of the two regimes comes about because of special behavior of fermions with frequencies ω=±πT along the Matsubara axis. Specifically, for these fermions, the component of the self-energy, which competes with the pairing, vanishes in the normal state. We further argue that the crossover at T∼Tcross comes about because Eliashberg equations allow an infinite number of topologically distinct solutions for the onset temperature of the pairing within the same gap symmetry. Only one solution, with the highest Tp, actually emerges, but other solutions are generated and modify the form of the gap function at T≤Tcross. Finally, we argue that the actual Tc is comparable to Tcross, while at Tcross<T<Tp phase fluctuations destroy superconducting long-range order, and the system displays a pseudogap behavior.

Superconducting transition temperatures of metallic liquids

We develop a nonperturbative approach for calculating the superconducting transition temperatures (Tc's) of liquids. The electron-electron scattering amplitude induced by electron-phonon coupling (EPC), from which an effective pairing interaction can be inferred, is related to the fluctuation of the T matrix of electron scattering induced by ions. By applying the relation, EPC parameters can be extracted from a path-integral molecular dynamics simulation. For determining Tc, the linearized Eliashberg equations are reestablished nonperturbatively. We apply the approach to estimate Tc's of metallic hydrogen liquids. It indicates that metallic hydrogen liquids in the pressure regime from 0.5 to 1.5TPa have Tc's well above their melting temperatures and therefore are superconducting liquids.1 MoreReceived 29 January 2019Revised 28 January 2020Accepted 30 January 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.013340Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectron-phonon couplingMethods in superconductivityPressure effectsSuperconducting phase transitionTransition temperaturePhysical SystemsHigh-temperature superconductorsLiquidsUnconventional superconductorsTechniquesDensity functional theoryEliashberg theoryGreen's function methodsCondensed Matter, Materials & Applied Physics