Eliashberg Equations Research Articles

Prominent Cooper pairing away from the Fermi level and its spectroscopic signature in twisted bilayer graphene

We investigate phonon-mediated Cooper pairing in flat electronic band systems by solving the full-bandwidth multiband Eliashberg equations for superconductivity in magic angle twisted bilayer graphene using a realistic tight-binding model. We find that Cooper pairing away from the Fermi level contributes decisively to superconductivity by enhancing the critical temperature and ensures a robust finite superfluid density. We show that this pairing yields particle-hole asymmetric superconducting domes in the temperature-gating phase diagram and gives rise to distinct spectroscopic signatures in the superconducting state. We predict several such features in tunneling and angle resolved photoemission spectra for future experiments.

Strong-coupling superconductivity in LiB2C2 trilayer films

Coupling between $\sigma$-bonding electrons and phonons is generally very strong. To metallize $\sigma$-electrons provides a promising route to hunt for new high-T$_c$ superconductors. Based on this picture and first-principles density functional calculation with Wannier interpolation for electronic structure and lattice dynamics, we predict that trilayer film LiB$_2$C$_2$ is a good candidate to realize this kind of high-T$_c$ superconductivity. By solving the anisotropic Eliashberg equations, we find that free-standing trilayer LiB$_2$C$_2$ is a phonon-mediated superconductor with T$_c$ exceeding the liquid-nitrogen temperature at ambient pressure. The transition temperature can be further raised to 125 K by applying a biaxial tensile strain.

Superconductivity from Coulomb repulsion in three-dimensional quadratic band touching Luttinger semimetals

We study superconductivity driven by screened Coulomb repulsion in three-dimensional Luttinger semimetals with a quadratic band touching and strong spin-orbit coupling. In these semimetals, the Cooper pairs are formed by spin-3/2 fermions with non-trivial wavefunctions. We numerically solve the linear Eliashberg equation to obtain the critical temperature of a singlet s-wave gap function as a function of doping, with account of spin-orbit and self-energy corrections. In order to understand the underlying mechanism of superconductivity, we compute the sensitivity of the critical temperature to changes in the dielectric function $\epsilon(i\Omega_n,q)$. We relate our results to the plasmon and Kohn-Luttinger mechanisms. Finally, we discuss the validity of our approach and compare our results to the litterature. We find good agreement with some bismuth-based half-Heuslers, such as YPtBi, and speculate on superconductivity in the iridate Pr2Ir2O7.

The unreasonable effectiveness of Eliashberg theory for pairing of non-Fermi liquids

The paradigmatic Migdal–Eliashberg theory of the electron–phonon problem is central to the understanding of superconductivity in conventional metals. This powerful framework is justified by the smallness of the Debye frequency relative to the Fermi energy, and allows an enormous simplification of the full many-body problem. However, superconductivity is found also in many families of strongly-correlated materials, in which there is no a priori justification for the applicability of Eliashberg theory. In these systems, superconductivity emerges out of an anomalous metallic state, calling for a new theoretical framework to describe pairing out of a non-Fermi liquid. In this article, we review two model systems in which such behavior is found: a Fermi sea coupled to gapless bosonic fluctuations, and a system of fermions with local, strongly frustrated interactions. In both models, there is a well-defined limit in which the Eliashberg equations are asymptotically exact even in the strongly coupled regime. These models thus provide tractable examples of how superconductivity can emerge in the absence of coherent electronic quasiparticles; they also demonstrate the surprisingly wide applicability of the Eliashberg formalism, well beyond the conventional regime for which it was originally designed.

Eliashberg equations for an electron–phonon version of the Sachdev–Ye–Kitaev model: Pair breaking in non-Fermi liquid superconductors

We present a theory that is a non-Fermi-liquid counterpart of the Abrikosov–Gor’kov pair-breaking theory due to paramagnetic impurities in superconductors. To this end we analyze a model of interacting electrons and phonons that is a natural generalization of the Sachdev–Ye–Kitaev-model. In the limit of large numbers of degrees of freedom, the Eliashberg equations of superconductivity become exact and emerge as saddle-point equations of a field theory with fluctuating pairing fields. In its normal state the model is governed by non-Fermi liquid behavior, characterized by universal exponents. At low temperatures a superconducting state emerges from the critical normal state. We study the role of pair-breaking on Tc, where we allow for disorder that breaks time-reversal symmetry. For small Bogoliubov quasi-particle weight, relevant for systems with strongly incoherent normal state, Tc drops rapidly as function of the pair breaking strength and reaches a small but finite value before it vanishes at a critical pair-breaking strength via an essential singularity. The latter signals a breakdown of the emergent conformal symmetry of the non-Fermi liquid normal state.

Eliashberg theory in the weak-coupling limit: Results on the real frequency axis

We formulate and solve the Eliashberg equations on the imaginary frequency axis at temperatures below $T_c$ in the weak-coupling limit. We find an excellent scaling at all temperatures, for a given coupling strength, and the normalized order parameter exhibits a BCS-like temperature dependence. The hybrid real-imaginary axis equations are also solved to obtain numerically exact analytic continuations from the imaginary frequency axis to the real frequency axis. This provides a determination of the gap edge, which, in the weak-coupling limit, is identical to the order parameter from the imaginary axis. The analytical result for the zero-temperature gap edge deviates from the BCS result by a factor of $1/\sqrt{e}$, which was also obtained for the transition temperature $T_c$. We show that the normalized gap function on both the real and imaginary frequency axes, for an electron-phonon Einstein spectrum ($\delta$-function) of a given strength, is a universal function of frequency, independent of temperature. The $1/\sqrt{e}$ correction is a result of this non-trivial frequency dependence in the gap function. This modification, in the gap edge and in $T_{c}$, serves to preserve various dimensionless ratios to their BCS values.

Theoretical Explanation of Electric Field‐Induced Superconductive Critical Temperature Shifts in Indium Thin Films

Herein, ab initio density functional theory computations and the self‐consistent solution of one‐band s‐wave generalized Eliashberg equations with proximity effect are combined to explain 60 years old experimental data on how a static electric field can affect the superconductive critical temperature of indium thin films. Although electronic densities of states, Fermi energy shifts, and Eliashberg spectral functions can be computed ab initio, the only parameter in the theory that cannot be determined a priori is the thickness of the surface layer affected by the electric field. However, it is shown that in the weak electrostatic field limit, Thomas–Fermi approximation is valid and, therefore, no free parameters are left, as this perturbed layer is known to have a thickness of the order of the Thomas–Fermi screening length. Moreover, it is shown that the theoretical model can reproduce experimental data, even when the magnitudes of the induced charge densities are so small to be usually neglected.

Superconductivity of LaH10 and LaH16 polyhydrides

We explore high-pressure phase stability and superconductivity of lanthanum hydrides $LaH_m$ (m=4-11,16). We predict stability of a hitherto unreported polyhydride $P6/mmm$-$LaH_{16}$ at pressures above 150 GPa; at 200 GPa its predicted superconducting $T_C$ is 156 K, critical field $\mu_0$$H_C$(0) ~ 35 T and superconducting gap is up to 35 meV. We revisit superconductivity of the recently discovered $LaH_{10}$ and find its $T_C$ to be up to 259 K (170 GPa) from solving the Eliashberg equation and 271 K from solving the gap equation in SCDFT which also allowed us to compute the Coulomb pseudopotential $\mu^*$ for $LaH_{10}$ and $LaH_{16}$. Presence of several polymorph modifications of LaH10 may explain the variety in the experimentally measured $T_C$ values for $LaH_{10}$ [1,2]

Solvable Strong-Coupling Quantum-Dot Model with a Non-Fermi-Liquid Pairing Transition

We show that a random interacting model exhibits solvable non-Fermi-liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between M quantum dots each hosting N flavors of fermions and N^{2} bosons that self-tune to criticality at low energies. The diagrammatic expansion is controlled by 1/MN, and the results become exact in a large-M, large-N limit. We find that pairing only develops within a region of the (M,N) plane-even though the pairing interaction is strongly attractive, the incoherence of the fermions can spoil the forming of Cooper pairs, rendering the system a non-Fermi liquid down to zero temperature. By solving the Eliashberg equation and the renormalization group equation, we show that the transition into the pairing phase exhibits Kosterlitz-Thouless quantum-critical behavior.

Strongly Enhanced Superconductivity Due to Finite Energy Spin Fluctuations Induced by an Incipient Band: A FLEX Study on the Bilayer Hubbard Model with Vertical and Diagonal Interlayer Hoppings

We study the spin-fluctuation-mediated $s\pm$-wave superconductivity in the bilayer Hubbard model with vertical and diagonal interlayer hoppings. As in the two-leg ladder model with diagonal hoppings, studied previously by the present authors, superconductivity is strongly enhanced when one of the bands lies just below (or touches) the Fermi level, that is, when the band is incipient. The strong enhancement of superconductivity is because large weight of the spin fluctuations lies in an appropriate energy range, whereas the low energy, pair-breaking spin fluctuations are suppressed. The optimized eigenvalue of the linearized Eliashberg equation, a measure for the strength of superconductivity, is not strongly affected by the bare width of the incipient band, but the parameter regime where superconductivity is optimized is wide when the incipient band is narrow, and in this sense, the coexistence of narrow and wide bands is favorable for superconductivity.

Comment on “Superconductivity at low density near a ferroelectric quantum critical point: Doped SrTiO3 ”

W\"olfle and Balatsky Phys. Rev. B 98, 104505 (2018) have proposed a microscopic pairing mechanism for doped SrTiO$_3$ (STO) based on the ${\textit gradient}$ coupling of electronic density to the soft TO phonon mode. Since this coupling to TO phonons is usually weak, this conclusion is surprising, especially for a low density superconductor such as STO, where the density of states is small. A crucial step in the argument made by W\"olfle and Balatsky is that the displacement vector of the TO mode is not strictly perpendicular to the momentum vector, making a deformation coupling possible. We show that they have made a mistake in computing the eigenvector and have grossly overestimated this lack of orthogonality. When corrected, the coupling is negligible. We also use transport data to put upper bounds on the coupling constant which are much smaller than the estimate by W\"olfle and Balatsky. Finally, we also object to their use of the Eliashberg equation when the phonon frequency is larger than the Fermi energy.

Superconducting Symmetries of Sr_{2}RuO_{4} from First-Principles Electronic Structure.

Although correlated electronic-structure calculations explain very well the normal state of Sr_{2}RuO_{4}, its superconducting symmetry is still unknown. Here we construct the spin and charge fluctuation pairing interactions based on its correlated normal state. Correlations significantly reduce ferromagnetic in favor of antiferromagnetic fluctuations and increase interorbital pairing. From the normal-state Eliashberg equations, we find spin-singlet d-wave pairing close to magnetic instabilities. Away from these instabilities, where charge fluctuations increase, we find two time-reversal symmetry-breaking spin triplets: an odd-frequency s wave, and a doubly degenerate interorbital pairing between d_{xy} and (d_{yz},d_{xz}).

Superconductive critical temperature of Pb/Ag heterostructures

Recent experimental data (H. Nam et al., Phys. Rev. B100, 094512 (2019)) of critical temperature and gaps measured on superconductor/normal metal heterostructure (Pb/Ag), epitaxially grown, shown a interesting not usual behaviour. The critical temperature decreases strongly but, despite the large differences in the lattice constants and electronic densities of states in the separate components, this heterostructure shows a spatially constant superconducting gap. In the paper it is demonstrated that the proximity Eliashberg equations, whit no free parameters, cannot explain the dependence of critical temperature from the rate dPb/dAg. However it is sufficient to assume that the density of states at the Fermi level of silver is equal to that of lead in a layer adjacent to the separation interface, presumably of a thickness less than the coherence length of the lead, to perfectly explain the decrease in the critical temperature and the gap value in function of the rate dPb/dAg always without free parameters.

Possible Superconductivity Induced by a Large Spin–Orbit Coupling in Carrier Doped Iridium Oxide Insulators: A Weak Coupling Approach

We study possible superconductivity in a carrier-doped iridium oxide insulator Sr$_{2}$IrO$_{4}$ based on an effective $t_{2g}$ three-orbital Hubbard model on the square lattice with a large spin-orbit coupling (SOC). Numerically solving the linearized Eliashberg equation for the superconducting (SC) gap function with the random phase approximation, we systematically examine both singlet and triplet SC gap functions with possible pairing symmetries and the parameter dependence of the superconductivity. For the realistic SOC $\lambda$ and Hund's coupling $J/U$ relevant to Sr$_{2}$IrO$_{4}$, namely, for a large $\lambda$ and small $J/U$ region, we find that the intra-band antiferromagnetic (AF) pseudospin $\bm{j}_{\text{eff}} = -\bm{l}+\bm{s}$ fluctuations favor a $d_{x^{2}-y^{2}}$-wave pseudospin $j_{\text{eff}}=1/2$ singlet pairing in the electron-doping. We also find that the $d_{x^{2}-y^{2}}$-wave pairing is more stabilized with increasing the SOC and decreasing the Hund's coupling. Furthermore, we show for a small $\lambda$ and large $J/U$ region that an $s_{\pm}$-wave singlet pairing is favored in the hole-doped region. The origin of the $s_{\pm}$-wave pairing is due to the inter-band pair scattering arising from the intra-orbital AF spin $\bm s$ fluctuations. Although the possibility of a pseudospin triplet pairing is considered, we find it always unfavorable for all parameters studied here. The experimental consequences for other strongly correlated materials with a large SOC are also discussed.

Nearly degenerate px+ipy and dx2−y2 pairing symmetry in the heavy fermion superconductor YbRh2Si2

Recent discovery of superconductivity in YbRh$_2$Si$_2$ has raised particular interest in its pairing mechanism and gap symmetry. Here we propose a phenomenological theory of its superconductivity and investigate possible gap structures by solving the multiband Eliashberg equations combining realistic Fermi surfaces from first-principles calculations and a quantum critical form of magnetic pairing interactions. The resulting gap symmetry shows sensitive dependence on the in-plane propagation wave vector of the quantum critical fluctuations, suggesting that superconductivity in YbRh$_2$Si$_2$ is located on the border of $(p_x+ip_y)$ and $d_{x^2-y^2}$-wave solutions. This leads to two candidate phase diagrams: one has only a spin-triplet $(p_x+ip_y)$-wave superconducting phase; the other contains multiple phases with a spin-singlet $d_{x^2-y^2}$-wave state at zero field and a field-induced spin-triplet $(p_x+ip_y)$-wave state. In addition, the electron pairing is found to be dominated by the `jungle-gym' Fermi surface rather than the `doughnut'-like one, in contrast to previous thought. This requests a more elaborate and renewed understanding of the electronic properties of YbRh$_2$Si$_2$.

Hydrogen-Induced High-Temperature Superconductivity in Two-Dimensional Materials: The Example of Hydrogenated Monolayer MgB_{2}.

Hydrogen-based compounds under ultrahigh pressure, such as the polyhydrides H_{3}S and LaH_{10}, superconduct through the conventional electron-phonon coupling mechanism to attain the record critical temperatures known to date. Here we exploit the intrinsic advantages of hydrogen to strongly enhance phonon-mediated superconductivity in a completely different system, namely, a two-dimensional material with hydrogen adatoms. We find that van Hove singularities in the electronic structure, originating from atomiclike hydrogen states, lead to a strong increase of the electronic density of states at the Fermi level, and thus of the electron-phonon coupling. Additionally, the emergence of high-frequency hydrogen-related phonon modes in this system boosts the electron-phonon coupling further. As a concrete example, we demonstrate the effect of hydrogen adatoms on the superconducting properties of monolayer MgB_{2}, by solving the fully anisotropic Eliashberg equations, in conjunction with a first-principles description of the electronic and vibrational states, and their coupling. We show that hydrogenation leads to a high critical temperature of 67K, which can be boosted to over 100K by biaxial tensile strain.

Specific heat and pairing of Dirac composite fermions in the half-filled Landau level

A recent proposal argues that an alternate description of the half-filled Landau level is a theory of massless Dirac fermions. We examine the possibility of pairing of these Dirac fermions by numerically solving the coupled Eliashberg equations unlike our previous calculation (Wang and Chakravarty, 2016). In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian states. However, in our approximation scheme, pairing is not possible in the putative particle–hole symmetric state for ℓ=0 angular momentum. The specific heat at low temperatures of a system of massless Dirac fermions interacting with a transverse gauge field, expected to be relevant for the half-filled Landau level, is calculated. Using the Luttinger formula, it is found to be ∝TlnT in the leading low temperature limit, due to the exchange of transverse gauge bosons. The result agrees with the corresponding one in the nonrelativistic composite fermion theory of Halperin, Lee and Read of the half-filled Landau level.

Characterization of the superconducting phase in tellurium hydride at high pressure

At present, hydrogen-based compounds constitute one of the most promising classes of materials for applications as phonon-mediated high-temperature superconductors. Herein, the behavior of the superconducting phase in tellurium hydride (HTe) at high pressure (p = 300 GPa) is analyzed in detail, by using the isotropic Migdal–Eliashberg equations. The chosen pressure conditions are considered here as a case study which corresponds to the highest critical temperature value [Formula: see text] in the analyzed material, as determined within recent density functional theory simulations. It is found that the Migdal–Eliashberg formalism, which constitutes a strong-coupling generalization of the Bardeen–Cooper–Schrieffer (BCS) theory, predicts that the critical temperature value ([Formula: see text] K) is higher than previous estimates of the McMillan formula. Further investigations show that the characteristic dimensionless ratios for the thermodynamic critical field, the specific heat for the superconducting state, and the superconducting band gap exceed the limits of the BCS theory. In this context, also the effective electron mass is not equal to the bare electron mass as provided by the BCS theory. On the basis of these findings it is predicted that the strong-coupling and retardation effects play pivotal role in the superconducting phase of HTe at 300 GPa, in agreement with similar theoretical estimates for the sibling hydrogen and hydrogen-based compounds. Hence, it is suggested that the superconducting state in HTe cannot be properly described within the mean-field picture of the BCS theory.

Strong electron-boson coupling in the iron-based superconductor BaFe1.9Pt0.1As2 revealed by infrared spectroscopy

Understanding the formation of Cooper pairs in iron-based superconductors is one of the most important topics in condensed matter physics. In conventional superconductors, the electron-phonon interaction leads to the formation of Cooper pairs. In conventional strong-coupling superconductors like lead (Pb), the features due to electron-phonon interaction are evident in the infrared absorption spectra. Here we investigate the infrared absorption spectra of the iron arsenide superconductor BaFe1.9Pt0.1As2. We find that this superconductor has fully gapped (nodeless) Fermi surfaces, and we observe the strong-coupling electron-boson interaction features in the infrared absorption spectra. Through modeling with the Eliashberg function based on Eliashberg theory, we obtain a good quantitative description of the energy gaps and the strong-coupling features. The full Eliashberg equations are solved to check the self-consistency of the electron-boson coupling spectrum, the largest energy gap, and the transition temperature (Tc). Our experimental data and analysis provide compelling evidence that superconductivity in BaFe1.9Pt0.1As2 is induced by the coupling of electrons to a low-energy bosonic mode that does not originate solely from phonons.

Superconductivity near a nematic quantum critical point: Interplay between hot and lukewarm regions

We present a strong coupling dynamical theory of the superconducting transition in a metal near a QCP towards $Q = 0$ nematic order. We use a fermion-boson model, in which we treat the ratio of effective boson-fermion coupling and the Fermi energy as a small parameter $\lambda$. We solve, both analytically and numerically, the linearized Eliashberg equation. Our solution takes into account both strong fluctuations at small momentum transfers $\sim \lambda k_F$ and weaker fluctuations at large momentum transfers. The strong fluctuations determine $T_c$, which is of order $\lambda^2 E_F$ for both s- and d- wave pairing. The weaker fluctuations determine the angular structure of the superconducting order parameter $F(\theta_k)$ along the Fermi surface, separating between hot and lukewarm regions. In the hot regions $F(\theta_k)$ is largest and approximately constant. Beyond the hot region, whose width is $\theta_h\sim\lambda^{1/3}$, $F(\theta_k)$ drops by a factor $\lambda^{4/3}$. The s- and d- wave states are not degenerate but the relative difference $(T_c^s-T_c^d)/T_c^s\sim\lambda^2$ is small.