We analyze the validity of Eliashberg theory of phonon-mediated superconductivity in 2D systems in light of recent extensive Monte-Carlo studies of the Holstein model. Conventional wisdom says that Eliashberg theory is applicable as long as vertex corrections remain small. For small ratio of the phonon energy $Ω_0$ and the Fermi energy $E_F$, this condition is supposed to hold even when the dimensionless electron-phonon coupling $λ$ is larger than one, i.e., in the strong coupling regime. A comparison between various quantities computed in the Migdal approximation and those computed by Quantum Monte Carlo prove that this belief is wrong, and we identify analytically some of the ways in which this breakdown occurs for various "normal state" properties at $λ= λ_{cr}$, where $λ_{cr} = O(1)$. The breakdown occurs at temperatures high enough that neither superconducting nor charge-density wave correlations extend over any significant range of distances, so it cannot be associated with the onset of an instability toward any of the relevant ordered ground-states - rather it is associated with the local physics of classical bipolaron formation. Still, we show that certain properties, including the superconducting $T_c$ and the superconducting gap structure below $T_c$, can be accurately inferred from the strong-coupling limit of Eliashberg theory at $λ\leq λ_{cr}$.
We study the influence of the rattling phonons on superconductivity in $β$-pyrochlore KOs$_{2}$O$_{6}$ compound based on the strong-coupling Eliashberg approach. In particular, analyzing the specific heat data we find that the rattling phonon frequency changes discontinuously at the critical temperature of the first order phase transition. Solving the strong-coupling Eliashberg equations with temperature dependent $α^{2}F(ω)$, we investigate the consequence of this first order phase transition for the anomalous temperature dependence of the superconducting gap. We discuss our results in contex…
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…
We study the influence of inelastic rare-earth impurity scattering on electron-phonon mediated superconductivity and mass renormalization in (La$_{1-x}$Pr$_{x}$)Os$_{4}$Sb$_{12}$ compounds. Solving the strong coupling Eliashberg equations we find that the dominant quadrupolar component of the inelastic scattering on Pr impurities yields an enhancement of the superconducting transition temperature T$_c$ in LaOs$_{4}$Sb$_{12}$ and increases monotonically as a function of Pr concentration. The calculated results are in good agreement with the experimentally observed T$_c (x)$ dependence. Our anal…
High-Tc pairing mechanism absent of hole-pockets in heavily electron-doped FeSe is one of the key unsolved problems in Fe-based superconductors. Here, this problem is attacked by focusing on the higher-order many-body effects neglected in conventional Migdal-Eliashberg formalism. We uncover two significant many-body effects for high-Tc superconductivity: (i) Due to the "vertex correction", the dressed multiorbital Coulomb interaction acquires prominent orbital dependence for low-energy electrons. The dressed Coulomb interaction not only induces the orbital fluctuations, but also magnifies the …
We report the full complex dielectric function of high-purity $\textrm{Ba}_{0.68}\textrm{K}_{0.32}\textrm{Fe}_2\textrm{As}_2$ single crystals with $T_{\mathrm{c}}=38.5\ \textrm{K}$ determined by wide-band spectroscopic ellipsometry at temperatures $10\leq T\leq300\ \textrm{K}$. We discuss the microscopic origin of superconductivity-induced infrared optical anomalies in the framework of a multiband Eliashberg theory with two distinct superconducting gap energies $2Δ_{\mathrm{A}}\approx6\ k_{\mathrm{B}}T_{\mathrm{c}}$ and $2Δ_{\mathrm{B}}\approx2.2\ k_{\mathrm{B}}T_{\mathrm{c}}$. The observed un…
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 {\it a priori} justification for the applicability of Eliashberg theory. In these systems, superconductivity emerges out of an anomalous metallic state, calling …
We address the problem of superconductivity for non-Fermi liquids using two commonly adopted, yet apparently distinct methods: 1) the renormalization group (RG) and 2) Eliashberg theory. The extent to which both methods yield consistent solutions for the low energy behavior of quantum metals has remained unclear. We show that the perturbative RG beta function for the 4-Fermi coupling can be explicitly derived from the linearized Eliashberg equations, under the assumption that quantum corrections are approximately local across energy scales. We apply our analysis to the test case of phonon medi…
We discuss the compatibility of the most accurate experimental data with the ordinary Migdal-Eliashberg theory of superconductivity in the fullerene compound Rb_3C_60. By using different model phonon spectra we conclude that the experimental data can be fitted only by invoking an electron-phonon coupling of order λ\simeq 3. This exceedingly high value is unphysical and it is not consistent with the basic assumptions of the Migdal-Eliashberg theory. On the contrary, by relaxing the adiabatic hypothesis on which the Migdal-Eliashberg theory rests, the experimental data can be fitted by much more…
We present a first-principles approach to describe magnetic and superconducting systems and the phenomena of competition between these electronic effects. We develop a density functional theory: SpinSCDFT, by extending the Hohenberg-Kohn theorem and constructing the non-interacting Kohn- Sham system. An exchange-correlation functional for SpinSCDFT is derived from the Sham Schlüter connection between the SpinSCDFT Kohn-Sham and a self-energy in Eliashberg approximation. The reference Eliashberg equations for superconductors in the presence of magnetism are also derived and discussed.
Results are presented for the temperature and frequency dependence of the real and imaginary parts of the diagonal self energy for a d-wave superconductor. An Eliashberg analysis, which has been successful in recent fitting of superconductor-insulator-superconductor tunnel junction conductances for BiSrCaCuO (Bi-2212), is extended to finite temperatures. The effect of the temperature dependence of the 40 meV spin resonance mode, measured in inelastic neutron scattering (INS) in Bi-2212, on the finite temperature self energies is investigated.
Driving a conventional superconductor with an appropriately tuned classical electromagnetic field can lead to an enhancement of superconductivity via a redistribution of the quasiparticles into a more favorable non-equilibrium distribution -- a phenomenon known as the Eliashberg effect. Here we theoretically consider coupling a two-dimensional superconducting film to the quantized electromagnetic modes of a microwave resonator cavity. As in the classical Eliashberg case, we use a kinetic equation to study the effect of the fluctuating, dynamical electromagnetic field on the Bogoliubov quasipar…
We formulate an efficient scheme to perform Migdal-Eliashberg calculation considering the retardation effect from first principles. While the conventional approach requires a huge number of Matsubara frequencies, we show that the intermediate representation of the Green's function [H. Shinaoka et al., Phys. Rev. B 96, 035147 (2017)] dramatically reduces the numerical cost to solve the linearized gap equation. Without introducing any empirical parameter, we demonstrate that we can successfully reproduce the experimental superconducting transition temperature of elemental Nb ($\sim 10$ K) very a…
I solved the Eliashberg equations for multiband non-phononic $s\pm$ wave spin glass superconductor and I calculated the temperature dependence of the gaps and superfluid density that reveals unusual behavior as non monotonic temperature dependence and reentrant superconductivity. For particular values of input parameters the phase diagram is still more complex with two different ranges of temperature where the superconductivity appears.
Graphene is the only member of the carbon family from zero- to three-dimensional materials for which superconductivity has not been observed yet. At this time, it is not clear whether the quest for superconducting graphene is hindered by technical challenges, or else by the fluctuation of the order parameter in two dimensions. In this area, ab initio calculations are useful to guide experimental efforts by narrowing down the search space. In this spirit, we investigate from first principles the possibility of inducing superconductivity in doped graphene using the fully anisotropic Migdal-Elias…
We describe how the fermionic functional renormalization group (fRG) flow of a Cooper+forward scattering problem can be continued into the superconducting state. This allows us to reproduce from the fRG flow the fundamental equations of the Eliashberg theory for superconductivity at all temperatures including the symmetry-broken phase. We discuss possible extensions of this approach like the inclusion of vertex corrections.
Eliashberg's foundational theory of superconductivity is based on the application of Migdal's approximation, which states that vertex corrections to first order electron-phonon scattering are negligible if the ratio between phonon and electron energy scales is small. The resulting theory incorporates the first Feynman diagrams for electron and phonon self-energies. However, the latter is most commonly neglected in numerical analyses. Here we provide an extensive study of full-bandwidth Eliashberg theory in two and three dimensions, where we include the full back reaction of electrons onto the …
We extend the two leading methods for the \emph{ab initio} computational descrip tion of phonon-mediated superconductors, namely Eliashberg theory and density fu nctional theory for superconductors (SCDFT), to include plasmonic effects. Furth ermore, we introduce a hybrid formalism in which the Eliashberg approximation fo r the electron-phonon coupling is combined with the SCDFT treatment of the dynam ically screened Coulomb interaction. The methods have been tested on a set of we ll-known conventional superconductors by studying how the plasmon contribution a ffects the phononic mechanism in …
We present a novel method for embedding spin and charge fluctuations in an anisotropic, multi-band and full-bandwidth Eliashberg treatment of superconductivity. Our analytical framework, based on the random phase approximation, allows for a selfconsistent calculation of material specific characteristics in the interacting, and more specifically, the superconducting state. We apply this approach to bulk FeSe as representative for the iron-based superconductors and successfully solve for the superconducting transition temperature $T_c$, the gap symmetry and the gap magnitude. We obtain $T_c \app…
The influence of antiadiabatic phonons on the temperature of superconducting transition is considered within Eliashberg - McMillan approach in the model of discrete set of (optical) phonon frequencies. A general expression for superconducting transition temperature $T_c$ is proposed, which is valid in situation, when one (or several) of such phonons becomes antiadiabatic. We study the contribution of such phonons into the Coulomb pseudopotential $μ^{\star}$. It is shown, that antiadiabatic phonons do not contribute to Tolmachev's logarithm and its value is determined by partial contributions f…
The pairing mechanism and gap structure in Ba122 pnictides have been hotly discussed for long time as one of the central issues in Fe-based superconductors. Here, we attack this problem by taking account of the vertex corrections (VCs) for the Coulomb interaction $U$ ($U$-VCs), which are totally dropped in conventional Migdal-Eliashberg formalism. The $U$-VC in the charge susceptibility induces strong orbital fluctuations, and the $U$-VC also enlarges the orbital-fluctuation-mediated attractive interaction. By analyzing the effective multiorbital Hubbard model for Ba122 pnictides, we find that…
In this paper, through an exhaustive analysis within the Migdal-Eliashberg theory, we show the incompatibility of experimental data of Rb$_3$C$_{60}$ with the basic assumptions of the standard theory of superconductivity. For different models of the electron-phonon spectral function $α^2F(Ω)$ we solve numerically the Eliashberg equations to find which values of the electron-phonon coupling $λ$, of the logarithmic phonon frequency $Ω_{ln}$ and of the Coulomb pseudopotential $μ^*$ reproduce the experimental data of Rb$_3$C$_{60}$. We find that the solutions are essentially independent of the par…
We present a necessary condition for odd-frequency (odd-f) superconductivity (SC) to occur in a large class of materials described by Eliashberg theory. We use this condition to prove a no-go theorem ruling out the occurrence of odd-f SC in standard one-band superconductors with pairing interactions mediated by phonon exchange. We also present a corresponding no-go theorem for superconductors with interactions mediated by spin-fluctuations. Our results explain why odd-f SC is rare in conventional materials, and they open up the possibility for a search for materials with interactions designed …
Density-functional calculations of the phonon spectrum and electron-phonon coupling in MgB$_2$ are presented. The $E_{2g}$ phonons, which involve in-plane B displacements, couple strongly to the $p_{x,y}$ electronic bands. The isotropic electron-phonon coupling constant is calculated to be about 0.8. Allowing for different order parameters in different bands, the superconducting $λ$ in the clean limit is calculated to be significantly larger. The $E_{2g}$ phonons are strongly anharmonic, and the non-linear contribution to the coupling between the $E_{2g}$ modes and the p$_{x,y}$ bands is signi…
We examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO$_3$ developing a material's specific multiband, full bandwidth, and anisotropic Eliashberg theory for this system. Our self-consistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak-coupling constant λ$_m$ = 0.4, in mediating the high T$_c$, and explain other puzzling experimental observations, such as the s-wave symmetry and replica bands. We discover that the formation of replica bands has a T$_c$ decreasing effect that is ne…
We perform Eliashberg calculations for magnon-mediated superconductivity in a normal metal, where the electron-magnon interaction arises from interfacial coupling to antiferromagnetic insulators. In agreement with previous studies, we find $p$-wave pairing for large doping when the antiferromagnetic interfaces are uncompensated, and $d$-wave pairing close to half-filling when the antiferromagnetic interfaces are compensated. However, for the $p$-wave phase, we find a considerable reduction in the critical temperature compared to previous weak-coupling results, as the effective frequency cutoff…
The newly discovered iron pnictide superconductors apparently present an unusual case of interband-channel pairing superconductivity. Here we show that, in the limit where the pairing occurs within the interband channel, several surprising effects occur quite naturally and generally: different density-of-states on the two bands lead to several unusual properties, including a gap ratio which behaves inversely to the ratio of density-of-states; the weak-coupling limit of the Eliashberg and the BCS theory, commonly taken as equivalent, in fact predict qualitatively different dependence of the $Δ_…
Superconductivity is abundant near quantum-critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of…
Migdal-Eliashberg theory describes the properties of the normal and superconducting states of electron-phonon mediated superconductors based on a perturbative treatment of the electron-phonon interactions. It is necessary to include both electron and phonon self-energies self-consistently in Migdal-Eliashberg theory in order to match numerically exact results from determinantal quantum Monte Carlo in the adiabatic limit. In this work we provide a method to obtain the real-axis solutions of the Migdal-Eliashberg equations with electron and phonon self-energies calculated self-consistently. Our …
We study the normal-state and superconducting properties of NaFe$_{1-x}$Co$_x$As system by specific heat measurements. Both the normal-state Sommerfeld coefficient and superconducting condensation energy are strongly suppressed in the underdoped and heavily overdoped samples. The low-temperature electronic specific heat can be well fitted by either an one-gap or a two-gap BCS-type function for all the superconducting samples. The ratio $γ_NT_c^2/H_c^2(0)$ can nicely associate the neutron spin resonance as the bosons in the standard Eliashberg model. However, the value of $ΔC/T_cγ_N$ near optim…
We study the graphite intercalated compound CaC$_6$ by means of Eliashberg theory, focusing on the anisotropy properties. An analysis of the electron-phonon coupling is performed, and we define a minimal 6-band anisotropy structure. Comparing with Superconducting Density Functional Theory (SCDFT) the condition under which Eliashberg theory is able to reproduce the SCDFT gap structure is determined, and we discuss the role of Coulomb interactions. The Engelsberg-Schrieffer polaron structure is computed by solving the Eliashberg equation on the Matsubara axis and analytically continuing it to th…
Even- and odd-frequency superconductivity coexist due to broken time-reversal symmetry under magnetic field. In order to describe this mixing, we extend the linearized Eliashberg equation for the spin and charge fluctuation mechanism in strongly correlated electron systems. We apply this extended Eliashberg equation to the odd-frequency superconductivity on a quasi-one-dimensional isosceles triangular lattice under in-plane magnetic field and examine the effect of the even-frequency component.
We perform single- and multi-band Migdal-Eliashberg (ME) calculations with parameters exctracted from density functional theory (DFT) simulations to study superconductivity in the electric-field-induced 2-dimensional hole gas at the hydrogenated (111) diamond surface. We show that according to the Eliashberg theory it is possible to induce a high-T$_{\text{c}}$ superconducting phase when the system is field-effect doped to a surface hole concentration of $6\times10^{14}\,$cm$^{-2}$, where the Fermi level crosses three valence bands. Starting from the band-resolved electron-phonon spectral func…
We review recent progress in the theory of electromagnetic response of dirty superconductors subject to microwave radiation. The theory originally developed by Eliashberg in 1970 and soon after that elaborated in a number of publications addressed the effect of superconductivity enhancement in the vicinity of the transition temperature. This effect originates from nonequilibrium redistribution of quasiparticles and requires a minimal microwave frequency depending on the inelastic relaxation rate and temperature. In a recent series of papers we generalized the Eliashberg theory to arbitrary tem…
The standard Eliashberg - McMillan theory of superconductivity is essentially based on the adiabatic approximation. Here we present some simple estimates of electron - phonon interaction within Eliashberg - McMillan approach in non - adiabatic and even antiadiabatic situation, when characteristic phonon frequency $Ω_0$ becomes large enough, i.e. comparable or exceeding the Fermi energy $E_F$. We discuss the general definition of Eliashberg - McMillan (pairing) electron - phonon coupling constant $λ$, taking into account the finite value of phonon frequencies. We show that the mass renormalizat…
Recent work on an electron-phonon model in two-dimensions is reviewed and some new results are presented. We utilize both Quantum Monte Carlo simulations and Migdal-Eliashberg theory. The type of electron-ion coupling considered is on-site. Competing instabilities are charge density wave (CDW) and singlet superconductivity. When the electron band is half-filled, the charge density wave dominates. Away from half-filling CDW correlations actually suppress superconductivity.
The disclosure of spinon superconductivity and superconductivity mediated by spin-waves in hole-doped Bi2212 cuprate raises the question about the origin of the superconductivity in other cuprates and specially in an electron-doped NCCO cuprate.
The unconventional superconductivity in the noncentrosymmetric LaNiC$_2$, and its evolution with pressure, is analyzed basing on the {\it ab initio} computations and the full Eliashberg formalism. First principles calculations of the electronic structure, phonons and the electron-phonon coupling are reported in the pressure range 0-15 GPa. The thermodynamic properties of the superconducting state were determined numerically solving the Eliashberg equations. We found that already at $p=0$ GPa, the superconducting parameters deviate from the BCS-type, and a large value of the Coulomb pseudopoten…
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 (MAB phases). First, we identified the MBene layers with potential for superconductivity via isotropic Eliashberg calculations, considering a wide range of metal elements, with focus on alkaline earth and transiti…
In this paper, we calculate the superconducting critical temperature as a function of pressure, Tc(P ), using a method based on the functional derivative of the critical temperature with the Eliashberg function, dTc/dA2F. The coulomb electron-electron repulsion parameter, mu*(p) at each pressure is obtained in a consistent way by solving the linearized Migdal-Eliashberg equation. This method requires as the starting input only the knowledge of Tc(P ) at the starting pressure. It applies to superconductors for which the Migdal-Eliashberg equations hold. We study Al, a typical BCS weak coupling …
We determine the behavior of the critical temperature of magnetically mediated p-wave superconductivity near a ferromagnetic quantum critical point in three dimensions, distinguishing universal and non-universal aspects of the result. We find that the transition temperature is non-zero at the critical point, raising the possibility of superconductivity in the ferromagnetic phase.
We investigated the characteristics of the spin fluctuations mediated superconductivity employing the Eliashberg formalism. The effective interaction between electrons was modeled in terms of the spin susceptibility measured by the inelastic neutron scattering experiments on single crystal La2-xSrxCuO4 superconductors. The diagonal self-energy and off-diagonal self-energy were calculated by solving the coupled Eliashberg equation self-consistently for chosen spin susceptibility and tight-binding dispersion of electrons. The full momentum and frequency dependence of the self-energy is presented…
Superconducting Pr$_2$CuO$_x$, $x\simeq 4$ films with $T^\prime$ structure and a $T_c$ of 27 K have been investigated by millimeter-wave transmission and broadband (infrared-to-ultraviolet) reflectivity measurements in the normal and superconducting state. The results obtained by both experimental methods show a consistent picture of the superconducting condensate formation below $T_c$. An Eliashberg analysis of the data proves $d$-wave superconductivity and unitary-limit impurity scattering of the charge carriers below $T_{c}$. The derived electron-exchange boson interaction spectral function…
We review the evidence supporting valence-fluctuation mediated superconductivity in CeCu2Si2 and CeCu2Ge2, where Tc reaches 2.4 K at high pressure. In these systems the valence and magnetic critical points, at p_V and p_c respectively, are well separated. Characteristic signatures associated with both phenomena are distinct. In contrast, the valence and spin fluctuation regions appear much closer in most Ce based compounds. Concerning d-transition metals, superconductivity in pure iron emerges in the pressure window 15-30 GPa with the onset of Tc up to almost 3 K. All relevant observations poi…
Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. BCS theory is often viewed as the weak coupling limit of Eliashberg theory, in spite of a handful of papers that have pointed out that this is not so. Here we present very accurate numerical solutions in the weak coupling limit to complement the existing analytical results, and demonstrate more convincingly the validity of this limit by extending the analytical results to first order …
The ab initio anisotropic Migdal-Eliashberg formalism has been used to examine the pairing mechanism and the nature of the superconducting gap in the recently discovered lithium-decorated monolayer graphene superconductor. Our results provide evidence that the superconducting transition in Li-decorated monolayer graphene can be explained within a standard phonon-mediated mechanism. We predict a single anisotropic superconducting gap and a critical temperature Tc = 5.1-7.6 K, in very good agreement with the experimental results.
We consider two issues related to the condensation energy in superconductors described by the Eliashberg theory for various forms of the pairing interaction, associated either with phonon or electronic mechanisms of superconductivity. First, we derive a leading correction to the BCS formula for the condensation energy to first order in the coupling $λ$. Second, we show that at a given $λ$, the value of the condensation energy strongly depends on the functional form of the effective pairing interaction $Γ(ω)$.
In PrOs4Sb12, the lowest-lying singlet and triplet states in a Pr 4f^2 configuration hybridize with conduction electrons having local a_u and t_u point-group symmetries. It is shown that for an attractive triplet pairing interaction, the orbital degrees of freedom of the t_u component are important. In addition, the Th point-group symmetry characteristic of skutterudites plays an important role in stabilizing triplet superconductivity.
