Migdal Eliashberg Research Articles

The superconducting properties of a Pb/MoTe2/Pb heterostructure: First-principles calculations within the anisotropic Migdal–Eliashberg theory** Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20141441).

The spin-polarized band structures of an ultrathin Pb/MoTe2/Pb heterostructure are calculated via first-principles density functional theory. The electron–phonon interaction and the superconducting properties of the ultrathin Pb/MoTe2/Pb heterostructure are studied by using the fully anisotropic Migdal–Eliashberg theory powered by Wannier–Fourier interpolation. Due to the complex Fermi surface in this low-dimensional system, the electron–phonon interaction and the superconducting gap display significant anisotropy. The temperature dependence of the superconducting gap can be fitted by solving numerically the Bardeen–Cooper–Schrieffer (BCS) gap equation with an adjustable parameter α, suggesting that phonon-mediated mechanism as its superconducting origin. Large Rashba spin-splitting and superconductivity coexist in this heterostructure, suggesting that this hybrid low-dimensional system will have some specific applications.

Superconducting properties of under- and over-doped BaxK1−xBiO3 perovskite oxide

In this study, we investigate the thermodynamic properties of the Ba[Formula: see text]K[Formula: see text]BiO3 (BKBO) superconductor in the under- (x = 0.5) and over-doped (x = 0.7) regime, within the framework of the Migdal–Eliashberg formalism. The analysis is conducted to verify that the electron–phonon pairing mechanism is responsible for the induction of the superconducting phase in the mentioned compound. In particular, we show that BKBO is characterized by the relatively high critical value of the Coulomb pseudopotential, which changes with doping level and does not follow the Morel–Anderson model. In what follows, the corresponding superconducting band gap size and related dimensionless ratio are estimated to increase with the doping, in agreement with the experimental predictions. Moreover, the effective mass of electrons is found to take on high values in the entire doping and temperature region. Finally, the characteristic dimensionless ratios for the superconducting band gap, the critical magnetic field and the specific heat for the superconducting state are predicted to exceed the limits set within the Bardeen–Cooper–Schrieffer theory, suggesting pivotal role of the strong-coupling and retardation effects in the analyzed compound. Presented results supplement our previous investigations and account for the strong-coupling phonon-mediated character of the superconducting phase in BKBO at any doping level.

Superconductivity in Li-intercalated bilayer arsenene and hole-doped monolayer arsenene: a first-principles prediction

Using first-principles calculations, we find Li-intercalated bilayer arsenene with AB stacking is dynamically stable, which is different from pristine bilayer with AA stacking. Electron–phonon coupling of the stable Li-intercalated bilayer arsenene are dominated by the low frequency vibrational modes (E″(1), (1), E′(1) and acoustic modes) and lead to an superconductivity with Tc = 8.68 K with isotropical Eliashberg function. Small biaxial tensile strain (2%) can improve Tc to 11.22 K due to the increase of DOS and phonon softening. By considering the fully anisotropic Migdal–Eliashberg theory, Tc are found to be enhanced by 50% and exhibits a single anisotropic gap nature. In addition, considering its nearly flat top valence band which is favorable for high temperature superconductivity, we also explore the superconducting properties of hole-doped monolayer arsenene under different strains. the unstrained monolayer arsenene superconducts at Tc = 0.22 K with 0.1 hole/cell doping. By applying 3% biaxial strain, Tc can be lifted up strikingly to 6.69 K due to a strong Fermi nesting of the nearly flat band. Then Tc decreases slowly with strain. Our findings provide another insight to realize 2D superconductivity and suggest that the strain is crucial to further enhance the transition temperature.

Breakdown of the Migdal-Eliashberg theory: A determinant quantum Monte Carlo study

The superconducting (SC) and charge-density-wave (CDW) susceptibilities of the two dimensional Holstein model are computed using determinant quantum Monte Carlo (DQMC), and compared with results computed using the Migdal-Eliashberg (ME) approach. We access temperatures as low as 25 times less than the Fermi energy, $E_F$, which are still above the SC transition. We find that the SC susceptibility at low $T$ agrees quantitatively with the ME theory up to a dimensionless electron-phonon coupling $\lambda_0 \approx 0.4$ but deviates dramatically for larger $\lambda_0$. We find that for large $\lambda_0$ and small phonon frequency $\omega_0 \ll E_F$ CDW ordering is favored and the preferred CDW ordering vector is uncorrelated with any obvious feature of the Fermi surface.

Superconducting critical temperature under pressure

The present record on the critical temperature of a superconductor is held by sulfur hydride (approx. 200 K) under very high pressure (approx. 56 GPa.). As a consequence, the dependence of the superconducting critical temperature on pressure became a subject of great interest and a high number of papers on of different aspects of this subject have been published in the scientific literature since. In this paper, we calculate the superconducting critical temperature as a function of pressure, Tc(P), by a simple method. Our method is based on the functional derivative of the critical temperature with the Eliashberg function, δTc(P)/δα2F(ω). We obtain the needed coulomb electron-electron repulsion parameter, μ*(P) at each pressure in a consistent way by fitting it to the corresponding Tc using the linearized Migdal–Eliashberg equation. This method requires as input the knowledge of Tc at the starting pressure only. It applies to superconductors for which the Migdal–Eliashberg equations hold. We study Al and β−Sn two weak-coupling low-Tc superconductors and Nb, the strong coupling element with the highest critical temperature. For Al, our results for Tc(P) show an excellent agreement with the calculations of Profeta et al. which are known to agree well with experiment. For β−Sn and Nb, we found a good agreement with the experimental measurements reported in several works. This method has also been applied successfully to PdH elsewhere. Our method is simple, computationally light and gives very accurate results.

Polaronic behavior in a weak-coupling superconductor

The nature of superconductivity in the dilute semiconductor SrTiO3 has remained an open question for more than 50 y. The extremely low carrier densities ([Formula: see text]-[Formula: see text] cm-3) at which superconductivity occurs suggest an unconventional origin of superconductivity outside of the adiabatic limit on which the Bardeen-Cooper-Schrieffer (BCS) and Migdal-Eliashberg (ME) theories are based. We take advantage of a newly developed method for engineering band alignments at oxide interfaces and access the electronic structure of Nb-doped SrTiO3, using high-resolution tunneling spectroscopy. We observe strong coupling to the highest-energy longitudinal optic (LO) phonon branch and estimate the doping evolution of the dimensionless electron-phonon interaction strength ([Formula: see text]). Upon cooling below the superconducting transition temperature ([Formula: see text]), we observe a single superconducting gap corresponding to the weak-coupling limit of BCS theory, indicating an order of magnitude smaller coupling ([Formula: see text]). These results suggest that despite the strong normal state interaction with electrons, the highest LO phonon does not provide a dominant contribution to pairing. They further demonstrate that SrTiO3 is an ideal system to probe superconductivity over a wide range of carrier density, adiabatic parameter, and electron-phonon coupling strength.

Nonempirical Calculation of Superconducting Transition Temperatures in Light-Element Superconductors.

Recent progress in the fully nonempirical calculation of the superconducting transition temperature (Tc ) is reviewed. Especially, this study focuses on three representative light-element high-Tc superconductors, i.e., elemental Li, sulfur hydrides, and alkali-doped fullerides. Here, it is discussed how crucial it is to develop the beyond Migdal-Eliashberg (ME) methods. For Li, a scheme of superconducting density functional theory for the plasmon mechanism is formulated and it is found that Tc is dramatically enhanced by considering the frequency dependence of the screened Coulomb interaction. For sulfur hydrides, it is essential to go beyond not only the static approximation for the screened Coulomb interaction, but also the constant density-of-states approximation for electrons, the harmonic approximation for phonons, and the Migdal approximation for the electron-phonon vertex, all of which have been employed in the standard ME calculation. It is also shown that the feedback effect in the self-consistent calculation of the self-energy and the zero point motion considerably affect the calculation of Tc . For alkali-doped fullerides, the interplay between electron-phonon coupling and electron correlations becomes more nontrivial. It has been demonstrated that the combination of density functional theory and dynamical mean field theory with the ab initio downfolding scheme for electron-phonon coupled systems works successfully. This study not only reproduces the experimental phase diagram but also obtains a unified view of the high-Tc superconductivity and the Mott-Hubbard transition in the fullerides. The results for these high-Tc superconductors will provide a firm ground for future materials design of new superconductors.

The electron–phonon interaction with forward scattering peak is dominant in high Tc superconductors of FeSe films on (TiO2)

The theory of the electron–phonon interaction (EPI) with strong forward scattering peak (FSP) in an extreme delta-peak limit (Kulić and Zeyher 1994 Phys. Rev. B 49 4395; Kulić 2000 Phys. Rep. 38 1–264; Kulić and Dolgov 2005 Phys. Status Solidi b 242 151; Danylenko et al 1999 Eur. Phys. J. B 9 201) is recently applied in (Lee et al 2014 Nature 515 245; Rademaker et al 2016 New J. Phys. 18 022001; Wang et al 2016 Supercond. Sci. Technol. 29 054009) for the explanation of high in a monolayer FeSe grown on (Lee et al 2014 Nature 515 245) and TiO2 (Rebec et al 2016 arXiv:1606.09358v1) substrates. The EPI is due to a long-range dipolar electric field created by high-energy oxygen vibrations ( meV) at the interface (Lee et al 2014 Nature 515 245; Rademaker et al 2016 New J. Phys. 18 022001; Wang et al 2016 Supercond. Sci. Technol. 29 054009). In leading order (with respect to ) the mean-field critical temperature ∼ and the gap are due to an interplay between the maximal EPI pairing potential and the FSP-width qc. For K one has meV in a satisfactory agreement with ARPES experiments. In leading order Tc0 is mass-independent and a very small oxygen isotope effect is expected in next to leading order. In clean systems Tc0 for s-wave and d-wave pairing is degenerate but both are affected by non-magnetic impurities, which are pair-weakening in the s-channel and pair-breaking in the d-channel. The self-energy and replica bands at T = 0 and at the Fermi surface are calculated and compared with experimental results at ( Rademaker et al 2016 New J. Phys. 18 022001; Wang et al 2016 Supercond. Sci. Technol. 29 054009). The EPI coupling constant is mass-dependent () and at makes the slope of the self-energy and the replica intensities mass-dependent. This result, overlooked in the literature, is contrary to the prediction of the standard Migdal–Eliashberg theory for EPI. The small oxygen isotope effect in and pronounced isotope effect in and ARPES spectra Ai of the replica bands in FeSe films on SrTiO3 and TiO2 is a smoking-gun experiment for validity of the EPI–FSP theory to these systems. The EPI–FSP theory predicts a large number of low-laying pairing states, thus causing internal pair fluctuations. The latter reduce Tc0 additionally, by creating a pseudogap state for . Possibilities to increase Tc0, by designing novel structures are discussed in the framework of the EPI–FSP theory.

EPW: Electron–phonon coupling, transport and superconducting properties using maximally localized Wannier functions

The EPW (Electron-Phonon coupling using Wannier functions) software is a Fortran90 code that uses density-functional perturbation theory and maximally localized Wannier functions for computing electron–phonon couplings and related properties in solids accurately and efficiently. The EPW v4 program can be used to compute electron and phonon self-energies, linewidths, electron–phonon scattering rates, electron–phonon coupling strengths, transport spectral functions, electronic velocities, resistivity, anisotropic superconducting gaps and spectral functions within the Migdal–Eliashberg theory. The code now supports spin–orbit coupling, time-reversal symmetry in non-centrosymmetric crystals, polar materials, and k and q-point parallelization. Considerable effort was dedicated to optimization and parallelization, achieving almost a ten times speedup with respect to previous releases. A computer test farm was implemented to ensure stability and portability of the code on the most popular compilers and architectures. Since April 2016, version 4 of the EPW code is fully integrated in and distributed with the Quantum ESPRESSO package, and can be downloaded through QE-forge at http://qe-forge.org/gf/project/q-e. Program summaryProgram title: EPWCatalogue identifier: AEHA_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHA_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public Licence 3No. of lines in distributed program, including test data, etc.: 1635099No. of bytes in distributed program, including test data, etc.: 22533187Distribution format: tar.gzProgramming language: Fortran 90, MPI.Computer: Non-specific.Operating system: Unix/Linux.RAM: Typically 2GB/coreClassification: 7.3, 7.8, 7.9.External routines: LAPACK, BLAS, MPI, FFTW, Quantum- ESPRESSO package [1]Does the new version supersede the previous version?: YesNature of problem:Calculation of electron and phonon self-energies, linewidths, electron–phonon scattering rates, electron–phonon coupling strengths, transport spectral functions, electronic velocities, resistivity, anisotropic superconducting gaps and spectral functions within the Migdal–Eliashberg theory.Solution method:The code relies on density-functional perturbation theory and maximally localized Wannier functions.Reasons for new version:New features (listed in the paper) and optimization of the code.Summary of revisions:Recent developments and new functionalities are described in Section 2 of the paper.Running time:Up to several hours on several tens of processors.

High pressure driven superconducting critical temperature tuning in Sb2Se3 topological insulator

In this letter, we are reporting the change of superconducting critical temperature in Sb2Se3 topological insulator under the influence of an external hydrostatic pressure based on first principles electronic structure calculations coupled with Migdal–Eliashberg model. Experimentally, it was shown previously that Sb2Se3 was undergoing through a transition to a superconducting phase when subjected to a compressive pressure. Our results show that the critical temperature increases up to 6.15 K under the pressure unto 40 GPa and, subsequently, drops down until 70 GPa. Throughout this pressure range, the system is preserving the initial Pnma symmetry without any structural transformation. Our results suggest that the possible relevant mechanism behind the superconductivity in Sb2Se3 is primarily the electron–phonon coupling.

Aspects of electron–phonon interactions with strong forward scattering in FeSe Thin Films on SrTiO3 substrates

Mono- and multilayer FeSe thin films grown on SrTiO3 and BiTiO3 substrates exhibit a greatly enhanced superconductivity over that found in bulk FeSe. A number of proposals have been advanced for the mechanism of this enhancement. One possibility is the introduction of a cross-interface electron–phonon (e–ph) interaction between the FeSe electrons and oxygen phonons in the substrates that is peaked in the forward scattering (small direction due to the two-dimensional nature of the interface system. Motivated by this, we explore the consequences of such an interaction on the superconducting state and electronic structure of a two-dimensional system using Migdal–Eliashberg (ME) theory. This interaction produces not only deviations from the expectations of conventional phonon-mediated pairing but also replica structures in the spectral function and density of states, as probed by angle-resolved photoemission spectroscopy, scanning tunneling microscopy/spectroscopy, and quasiparticle interference imaging. We also discuss the applicability of ME theory for a situation where the e–ph interaction is peaked at small momentum transfer and in the FeSe/STO system.

Enhanced superconductivity due to forward scattering in FeSe thin films on SrTiO3 substrates

We study the consequences of an electron–phonon (e–ph) interaction that is strongly peaked in the forward scattering () direction in a two-dimensional superconductor using Migdal–Eliashberg theory. We find that strong forward scattering results in an enhanced Tc that is linearly proportional to the strength of the dimensionless e–ph coupling constant in the weak coupling limit. This interaction also produces distinct replica bands in the single-particle spectral function, similar to those observed in recent angle-resolved photoemission experiments on FeSe monolayers on SrTiO3 and BaTiO3 substrates. By comparing our model to photoemission experiments, we infer an e–ph coupling strength that can provide a significant portion of the observed high Tc in these systems.

On the thermodynamic properties of the Rb3C60 superconductor

The thermodynamic properties of the superconducting state in Rb3C60 fulleride have been studied in the framework of the Migdal–Eliashberg approach. In the first step, the high physical value of the Coulomb pseudopotential has been predicted (μC★=0.33), which corresponds to the screened Coulomb potential UC=0.61eV. Next, the temperature dependence of the order parameter and the wave function renormalization factor has been calculated. On the basis of the obtained results, the ratio of the energy gap to the critical temperature (RΔ≡2Δ(0)/kBTC), the ratio of the specific heat jump to the normal state specific heat (RC≡ΔCTC/CNTC), and the parameter connected with the thermodynamic critical field (RH≡TCCNTC/HC2(0)) have been estimated. It has been shown that the above parameters significantly differ from the BCS predictions: RΔ=4.06,RC=2.15, and RH=0.145. Finally, the temperature dependence of the electron effective mass (me★) has been presented; me★ assumes maximum at the critical temperature: 2.27me, where me denotes the electron band mass. In the paper, the obtained theoretical results have been also compared with the experimental data.

Quantitative reliability study of the Migdal-Eliashberg theory for strong electron-phonon coupling in superconductors

We reassess the validity of the Migdal-Eliashberg (ME) theory for coupled electron-phonon systems for large couplings $\ensuremath{\lambda}$. Although model calculations have found that the ME theory breaks down for $\ensuremath{\lambda}\ensuremath{\sim}0.5$, it is routinely applied for $\ensuremath{\lambda}>1$ to strong-coupling superconductors. To resolve this discrepancy, it is important to distinguish between bare parameters, used as input in models, and effective parameters, derived from experiments. We show explicitly that ME gives accurate results for the critical temperature and the spectral gap for large effective $\ensuremath{\lambda}$. This provides quantitative theoretical support for the applicability of the ME theory to strong-coupling conventional superconductors.

Electron–phonon properties of pnictide superconductors

In this paper we discuss the normal and superconducting state properties of two pnictide superconductors, LaOFeAs and LaONiAs, using Migdal–Eliashberg theory and density functional perturbation theory. For pure LaOFeAs, the calculated electron–phonon coupling constant λ = 0.21 and logarithmic-averaged frequency ω ln = 206 K , give a maximum T c of 0.8 K, using the standard Migdal–Eliashberg theory. Inclusion of multi-band effects increases the T c only marginally. To reproduce the experimental T c , a 5–6 times larger coupling constant would be needed. Our results indicate that standard electron–phonon coupling is not sufficient to explain superconductivity in the whole family of Fe–As based superconductors. At the same time, the electron–phonon coupling in Ni–As based compounds is much stronger and its normal and superconducting state properties can be well described by standard Migdal–Eliashberg theory.

Ab initio theory of complex electronic ground state of superconductors: I. Nonadiabatic modification of the Born–Oppenheimer approximation

The ARPES of high- T c cuprates and theoretical results of low-Fermi energy band structure fluctuation for different groups of superconductors indicate that electron coupling to pertinent phonon modes drive system from adiabatic into anti-adiabatic state ( ω> E F ). At these circumstances, not only Migdal–Eliashberg approximation is not valid, but basic adiabatic Born–Oppenheimer approximation (BOA) does not hold. At these circumstances, electronic structure has to be studied as explicitly dependent on instantaneous nuclear coordinates Q as well as on instantaneous nuclear momenta P. In the present paper—part I, it has been shown that Q, P-dependent modification of the BOA for ground electronic state can be derived by sequence of canonical transformations of the basis functions. The effect of nuclear coordinates and momenta on electronic structure is presented in the form of corrections to zero-, one- and two-particle terms of clamped nuclear Hamiltonian. In the anti-adiabatic state, correction to electronic ground state energy (zero-particle term correction) is negative and system can be stabilized in the anti-adiabatic state at distorted geometry with respect to adiabatic equilibrium structure and gap in one-particle spectrum of quasi-continuum states at Fermi level can be opened. Stabilization effect is solely the consequence of nuclear dynamics ( P) that is crucial in anti-adiabatic state. It has been shown that nuclear dynamics also increases electron correlation until system at nuclear motion remains in a bound state. Corresponding corrections to electronic wave function are also specified. On the other hand, when system remains at vibration motion of nuclei in adiabatic state, the influence of nuclear dynamics ( P-dependence) is negligible. In this case, all basic effects are covered through nuclear coordinates ( Q-dependence) within the adiabatic BOA and standard results of solid-state (or molecular) physics are recovered.

Superconductivity, nonadiabaticity and strong correlation in the light of recent experiments

Recently a series of experiments in Scanning Tunneling Spectroscopy, angle-resolved photoemission spectroscopy (ARPES) and isotope effects have provided a great deal of information about the nature of the superconducting state, even thought many questions are still open. In this paper we present a discussion of the various theoretical frameworks in the light of these new experiments. Strong correlations started as the possible path to an entirely novel physics but, in many recent versions, they are reduced to the modest role of just raising the coupling in a BCS gap equation with phonon mediators. We argue instead that a conceptual generalization of the Migdal-Eliashberg (ME) theory is necessary and we discuss the consequences arising from the breakdown of the adiabatic assumption. The Nonadiabatic Theory of Superconductivity represents a complex generalization of the ME theory which is natural and, in many respects, unavoidable. In fact Migdal's theorem is strongly violated in these materials for a variety of reasons. The quantification of these effects is technically complicated and it is rather model dependent. In this framework strong correlations can have an important role in locating the system in a favourable range of parameters (forward scattering). We present a critical discussion of these developments in the light of the recent experiments and we also propose some crucial tests.

Breakdown of Migdal–Eliashberg Theory via Catastrophic Vertex Divergence at Low Phonon Frequency

We investigate the applicability of Migdal–Eliashberg (ME) theory by revisiting Migdal’s analysis within the dynamical mean-field theory framework. First, we compute spectral functions, the quasi-particle weight, the self energy, renormalised phonon frequency and resistivity curves of the half-filled Holstein model. We demonstrate how ME theory has a phase-transition-like instability at intermediate coupling, and how the Engelsberg–Schrieffer (ES) picture is complicated by low-energy excitations from higher order diagrams (demonstrating that ES theory is a very weak coupling approach). Through consideration of the lowest-order vertex correction, we analyse the applicability of ME theory close to this transition. We find a breakdown of the theory in the intermediate coupling adiabatic limit due to a divergence in the vertex function. The region of applicability is mapped out, and it is found that ME theory is only reliable in the weak coupling adiabatic limit, raising questions about the accuracy of recent analyses of cuprate superconductors which do not include vertex corrections.

Coupling to Phonons in the Migdal–Eliashberg Approach

The relevance of the strong correlations in the high critical temperature superconductors (HTSC) is well experimentally documented. However, if the properties of the normal and superconducting state in HTSC oxides are interpreted in terms of the standard Eliashberg the- ory, which holds in low temperature superconductor sys- tems, the Migdal-Eliashberg approach implies serious lim- itations in the reproduction of experimental spectroscopies whose contributions are inter-band and not intra-band. Re- cent angle-resolved photoemission spectroscopy (ARPES) measurements on HTSC oxides whose contributions are intra-band show a kink in the quasiparticle spectrum at char- acteristic phonon frequencies in the normal and supercon- ducting state. In contrast with our theoretical discussion, we include our theoretical results for the renormalized energy Ek as a function of the bare band energy ek obtained from ARPES in a Pb sample and in a Bi2Sr2CaCu2O8+δ opti- mally doped sample (Bi2212). This is clear evidence that electron-phonon is strong and involved in pairing.

Relevance of multiband Jahn-Teller effects on the electron-phonon interaction inA3C60

Assessing the effective relevance of multiband effects in fullerides is of fundamental importance in understanding the complex properties of these compounds. In this paper we investigate the role of the multiband effects on the electron-phonon (el-ph) properties of the ${t}_{1u}$ bands coupled with the Jahn-Teller intramolecular ${H}_{g}$ vibrational modes in ${\mathrm{C}}_{60}$ compounds. We show that, assuming perfect degeneracy of the electronic bands, vertex diagrams arising from the breakdown of the adiabatic hypothesis, are one order of magnitude smaller than the noncrossing terms usually retained in the Migdal-Eliashberg (ME) theory. These results permit to understand the robustness of the ME theory found by numerical calculations. The effects of the nondegeneracy of the ${t}_{1u}$ in realistic systems are also analyzed. Using a tight-binding model we show that the el-ph interaction is mainly dominated by intraband scattering within a single electronic band. Our results question the reliability of a degenerate band modeling and show the importance of these combined effects in the ${A}_{3}{\mathrm{C}}_{60}$ family.