Eliashberg Formalism Research Articles

Scalability of non-adiabatic effects in lithium-decorated graphene superconductor

The analysis is conducted to unveil how the non-adiabatic effects scale within the superconducting phase of lithium-decorated graphene . Based on the Eliashberg formalism it is shown that the non-adiabatic effects notably reduce essential superconducting parameters in and arise as a significant oppressor of the discussed phase. Moreover, non-adiabaticity is found to scale with the strength of superconductivity, proportionally to the phonon energy scale and inversely with respect to the electron-phonon coupling. These findings are partially in contrast to other theoretical studies and show that superconductivity in is more peculiar than previously anticipated. In this context, the guidelines for enhancing superconducting phase in and sibling materials are also proposed.

Superconducting Energy Gap in Hole-Doped Graphene Beyond the Migdal's Theory

In this work we analyze impact of non-adiabatic effects on the superconducting energy gap in the hole-doped graphene. By using the Eliashberg formalism beyond the Migdal's theorem, we present that the non-adiabatic effects strongly influence the superconducting energy gap in the exemplary boron-doped graphene. In particular, the non-adiabatic effects, as represented by the first order vertex corrections to the electron-phonon interaction, supplement Coulomb depairing correlations and suppress the superconducting state. In summary, the obtained results confirm previous studies on superconductivity in two-dimensional materials and show that the corresponding superconducting phase may be notably affected by the non-adiabatic effects.

The carrier mobility and superconducting properties of monolayer oxygen-terminated functionalized MXene Ti2CO2.

In this study, the carrier mobility of monolayer Ti2CO2 was evaluated by employing the Boltzmann transport equation and superconducting transition temperature (Tc) of Ti2CO2 was determined by utilizing the Migdal and Eliashberg formalism in the first-principles framework. In contrast to previous studies, the results reveal that optical phonons in monolayer Ti2CO2 have dominant roles in scattering processes, which significantly reduce the mobility of carriers. Alongside the rigid band model, the jellium model is implemented to investigate the screening effects on electron-phonon interactions. Based on the jellium model and full-band electron-phonon calculations, the predicted maximum electron mobility at room temperature is 38 cm2 V-1 s-1 in which 80% of the total scattering rate originates from the intra-valley transitions within the M-valleys, indicating the crucial role of the long wavelength phonon wavevectors in scattering processes. On the other hand, for the p-type material, a maximum room temperature mobility of about 285 cm2 V-1 s-1 is calculated, which can be explained by a relatively small effective mass and tiny scattering phase space. Moreover, a maximum Tc of 39 (10) K is obtained for the n-type monolayer Ti2CO2 based on the rigid (jellium) model. Outcomes indicate that the important peaks of α2F(ω) are mainly caused by the optical phonons. The remarkable couplings between the electron states and phonons are related to the non-zero slope of (near the Brillouin zone center) the longitudinal optical branch denoted by Eu caused by the displacements of oxygen and carbon atoms at intermediate and high energy ranges of phonon dispersion, respectively.

Breakdown of Adiabatic Superconductivity in Ca-Doped h-BN Monolayer

In the present paper, we report the breakdown of the adiabatic picture of superconductivity in a calcium-doped hexagonal boron nitride (Ca-h-BN) monolayer and discuss its implications for the selected properties of this phase. In particular, it is shown that the shallow conduction band of the Ca-h-BN superconductor potentially cause a violation of the adiabatic Migdal’s theorem. As a result, the pivotal parameters that describe the superconducting state in Ca-h-BN are found to be notably influenced by the non-adiabatic effects. This finding is described here within the vertex-corrected Eliashberg formalism that predicts a strong reduction of the order parameter, superconducting transition temperature and superconducting gap in comparison to the estimates obtained in the framework of the adiabatic theory. The observed trends are in agreement with the recent results on superconductivity in hexagonal monolayers and confirm that the non-adiabatic effects have to be taken into account during the design of such future low-dimensional superconductors.

Superconducting properties of doped blue phosphorene: effects of non-adiabatic approach

We study the effects of Kohn anomalies on the superconducting properties in electron- and hole-doped cases of monolayer blue phosphorene (BLP), considering both adiabatic and non-adiabatic phonon dispersions using first-principles calculations. We show that the topology of the Fermi surface is crucial for the formation of Kohn anomalies of doped BLP. By using the anisotropic Eliashberg formalism, we further carefully consider the temperature dependence of the non-adiabatic phonon dispersions. In cases of low hole densities, strong electron–phonon coupling (EPC) leads to a maximum critical temperature of K for superconductivity. In electron-doped regimes, on the other hand, a maximum superconducting critical temperature of K is reached at a doping level that includes a Lifshitz transition point. Furthermore, our results indicate that the most prominent component of EPC arises from the coupling between an in-plane (out-of-plane) deformation and in-plane (out-of-plane) electronic states of the electron (hole) type doping.

Quantum critical Eliashberg theory, the Sachdev-Ye-Kitaev superconductor and their holographic duals

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 a gravity theory with emergent space-time from a quantum many-body Hamiltonian—the Yukawa Sachdev-Ye-Kitaev model—where the Eliashberg formalism is exact. Exploiting the power of holography, we then determine the dynamic pairing susceptibility of the model. Our holographic map comes with the potential to use quantum gravity corrections to go beyond the Eliashberg regime.

Fermi surface topology and anisotropic superconducting gap in electron-doped hydride compounds at high pressure

The recent theoretical discovery of the predicted high-temperature superconductivity (superconducting transition temperature ${T}_{\mathrm{c}}\ensuremath{\sim}351\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ at 300 GPa) in clathrate ${\mathrm{Li}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ is an important advance toward room-temperature superconductors. Here we use first-principle approaches to identify a new ternary hydride of clathrate structure of $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Rb}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ by Rb-doped $\mathrm{Mg}{\mathrm{H}}_{16}$ at 300 GPa. We first verified the thermal and dynamic stability for the $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Rb}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ through the formation enthalpy and phonon calculations, respectively. Next, we showcased that $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Rb}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ has two Fermi surface (FS) sheets, which are both dominated by the H2 $s$ orbital. Using the Eliashberg formalism, the ${T}_{\mathrm{c}}$ of $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Rb}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ was calculated to be 130 K at 300 GPa. Furthermore, the calculations of the electronic, phonon, and superconducting properties reveal that $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Li}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ has four FS sheets with mixed H1 $s$ and H2 $s$ orbital character and reaches the ${T}_{\mathrm{c}}$ of 352 K at 300 GPa. Compared with $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Rb}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$, $Fd\overline{3}m\text{\ensuremath{-}}{\mathrm{Li}}_{2}\mathrm{Mg}{\mathrm{H}}_{16}$ produces high-frequency phonon softening, leading to the enhancement of ${T}_{\mathrm{c}}$. Moreover, it is demonstrated that Rb substitution for Li produces the reduction of contribution of H orbitals to the FS sheets and the decrease in FS sheets in number, which tend to decrease the electron-phonon coupling strength and ${T}_{\mathrm{c}}$. Our observed FS sheets and their associated superconducting gap will provide key insights into the understanding of the superconductivity mechanism in these and related systems.

High-temperature multigap superconductivity in two-dimensional metal borides

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

Superconductivity in the calcium-decorated hexagonal boron nitride monolayer

In the present paper, we study the pivotal thermodynamic properties of the novel two-dimensional calcium-decorated hexagonal boron nitride superconductor. The analysis is motivated by the ongoing and growing interests in the two-dimensional superconductors and the fact that the discussed material is expected to exhibit relatively high critical temperature, which is well above the temperature of the liquid helium. The presented investigations are performed within the state-of-art Eliashberg formalism, according to the potential phonon-mediated strong-coupling character of the considered superconducting state. In particular, we calculate the thermodynamic properties that allow us to quantitatively determine values of the characteristic dimensionless parameters i.e. the zero-temperature energy gap to the critical temperature, the ratio for the specific heat, as well as the ratio corresponding to the critical magnetic field. The obtained result show that the discussed superconducting state is strongly influenced by the strong electron–phonon interactions as well as the retardation effects and cannot be adequately described within the Bardeen–Cooper–Schrieffer theory. This is to say, the analyzed material is yet another example of the two-dimensional superconducting crystal that may be of potential importance for the development of superconducting low-dimensional devices.

Electronic structure, electron-phonon coupling, and superconductivity in noncentrosymmetric ThCoC2 from ab initio calculations

Superconductors without inversion symmetry in their crystal structure are known to exhibit unconventional properties. Recently, based on the measured temperature dependence of the magnetic field penetration depth, superconductivity in noncentrosymmetric $\mathrm{ThCoC_2}$ was proposed to be a nodal $d$-wave and mediated by the spin fluctuations. Moreover, a non-BCS behavior of the temperature dependence of the electronic specific heat and the magnetic upper critical field were reported. In this work, the electronic structure, phonons and electron-phonon coupling are studied in $\mathrm{ThCoC_2}$ on the basis of \textit{ab initio} computations. The effect of the spin-orbit coupling on the electronic structure and electron-phonon interaction is analyzed, and a large splitting of the electronic band structure is found. The calculated electron-phonon coupling constant $\lambda = 0.59$ remains in decent agreement with the experimental estimates, suggesting that the electron-phonon interaction is strong enough to explain superconductivity with $T_c \simeq 2.5$~K. Nevertheless, we show that the conventional isotropic Eliashberg formalism is unable to describe the thermodynamic properties of the superconducting state, as calculated temperature dependence of the electronic specific heat and magnetic penetration depth deviate from experiments, which is likely driven by the strong spin-orbit coupling and inversion symmetry breaking. In addition, to shed more light on the pairing mechanism, we propose to measure the carbon isotope effect, as our calculations based on the electron-phonon coupling predict the observation of the isotope effect with an exponent $\alpha \simeq 0.15$.

Non-adiabatic superconductivity in the electron-doped graphene

In the present study, we investigate the impact of the non-adiabatic effects on the superconducting state in the electron-doped graphene. In particular, by using the Eliashberg formalism we analyze the case scenario of the nitrogen-doped graphene, showing that the non-adiabatic effects complement electron-electron interaction and notably reduce (up to ∼ 40%) pivotal thermodynamic properties, such as: the critical temperature, the superconducting gap and their characteristic ratio. Interestingly, the influence of the non-adiabatic effects is found to rise together with the increase of the depairing Coulomb interaction. These observations are elucidated based on the structure of the vertex corrections to the electron-phonon interaction. As a result, we draw a direction for the future research in the field of the two-dimensional non-adiabatic superconductivity.

Thermodynamic Properties of the Superconducting State in Metallic Hydrogen: Electronic Correlations, Non-conventional Electron-Phonon Couplings and the Anharmonic Effects

Thermodynamical properties of the superconducting state in metallic hydrogen were determined on the basis of the model of two compressed hydrogen planes. We took into account both the on-site and the inter-site electronic correlations (U and K), as well as the relevant non-conventional electron-phonon coupling functions (gU and gK). We proved, within the Eliashberg formalism, that the maximum value of the critical temperature of transition into the superconducting state is about 200 K for the harmonic approximation, and about 84 K for the Morse anharmonic approximation. Omission of the electronic correlations results in a considerable overstatement of the TC value. On the other hand, the TC value is remarkably understated if the non-conventional interactions are disregarded. Other thermodynamic quantities, such as the order parameter, the jump in the specific heat value, the thermodynamic critical field, and the upper critical field, take the values for which the non-dimensional ratios RΔ, RC, RH and RH2 do not differ substantially from the predictions of the BCS theory.

Apical oxygen vibrations dominant role in d-wave cuprate superconductivity and its interplay with spin fluctuations

Microscopic theory of a high T c cuprate Bi 2 Sr 2 CaCu 2 O 8+x based on main pairing channel of electrons in CuO planes due to 40mev lateral vibrations of the apical oxygen atoms in adjacent the SrO ionic insulator layer is proposed. The separation between the vibrating charged atoms and the 2D electron gas creates the forward scattering peak leading in turn to the d -wave pairing within Eliashberg formalism. The phonon mode naturally explain the kink in dispersion relation observed by ARPES and the and effect of the O 16 → O 18 isotope substitution in the normal state. To describe the pseudogap physics a single band fourfold symmetric Hubbard model, with the hopping parameters and the on site repulsion e U ∼ 6t. It described the Mott insulator at low doping, while at higher dopping the pseudogap physics (still strongly correlated) can be be approximated by the symmetrized mean field model and with renormalized U incorporating screening. The location of the transition line T *between the locally antiferromagnetic pseudogap and the paramagnetic overdoped phases and susceptibility (describing spin fluctuations coupling to 2DEG) are also obtained within this approximation. The superconducting d - wave gap mainly due to the phonon channel but is assisted by the spin fluctuations (15%–20%). The dependence of the gap and T c on doping and effect of the isotope substitution are obtained and is consistent with experiments.

Stability and superconductivity of Ca-intercalated bilayer blue phosphorene.

Superconductivity attracts much attention in two-dimensional (2D) compounds due to their potential application in nano-superconducting devices. Inspired by a recent experiment reporting the superconducting state in twisted bilayer graphene, here, based on the first-principles density-functional theory complemented by the Eliashberg formalism, we have verified the stability and predicted superconductivity in Ca-intercalated bilayer blue phosphorene. The electron and phonon properties and electron-phonon coupling show that AA- and AA'-stacking orders of the phosphorene bilayer are dynamically stable and exhibit conventional phonon-mediated superconductivity with superconducting transition temperatures (Tc) of 11.63 K and 11.74 K, respectively. Furthermore, we study the temperature-dependence of the superconducting energy gap (Δ(T)) and specific heat difference (ΔC(T)). According to our calculations, we found that the dimensionless parameters relative to the Δ(0) and the ΔC(Tc) differ slightly from the values predicted by the Bardeen-Cooper-Schrieffer (BCS) theory. We expect that our findings will broaden the knowledge of 2D superconducting materials and may stimulate more efforts in this field.

Single-layer polymeric tetraoxa[8]circulene modified by s-block metals: toward stable spin qubits and novel superconductors.

Tunable electronic properties of low-dimensional materials have been the object of extensive research, as such properties are highly desirable in order to provide flexibility in the design and optimization of functional devices. In this study, we account for the fact that such properties can be tuned by embedding diverse metal atoms and theoretically study a series of new organometallic porous sheets based on two-dimensional tetraoxa[8]circulene (TOC) polymers doped with alkali or alkaline-earth metals. The results reveal that the metal-decorated sheets change their electronic structure from semiconducting to metallic behaviour due to n-doping. Complete active space self-consistent field (CASSCF) calculations reveal a unique open-shell singlet ground state in the TOC-Ca complex, which is formed by two closed-shell species. Moreover, Ca becomes a doublet state, which is promising for magnetic quantum bit applications due to the long spin coherence time. Ca-doped TOC also demonstrates a high density of states in the vicinity of the Fermi level and induced superconductivity. Using the ab initio Eliashberg formalism, we find that the TOC-Ca polymers are phonon-mediated superconductors with a critical temperature TC = 14.5 K, which is within the range of typical carbon based superconducting materials. Therefore, combining the proved superconductivity and the long spin lifetime in doublet Ca, such materials could be an ideal platform for the realization of quantum bits.

Nonadiabatic superconductivity in a Li-intercalated hexagonal boron nitride bilayer.

When considering a Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the vertex corrections of electron–phonon interaction cannot be omitted. This is evidenced by the very high value of the ratio λωD/εF ≈ 0.46, where λ is the electron–phonon coupling constant, ωD is the Debye frequency, and εF represents the Fermi energy. Due to nonadiabatic effects, the phonon–induced superconducting state in Li-hBN is characterized by much lower values of the critical temperature (TLOVCC ∈ {19.1, 15.5, 11.8} K, for μ* ∈ {0.1, 0.14, 0.2}, respectively) than would result from calculations not taking this effect into account (TMEC∈ {31.9, 26.9, 21} K). From the technological point of view, the low value of TC limits the possible applications of Li-hBN. The calculations were carried out under the classic Migdal–Eliashberg formalism (ME) and the Eliashberg theory with lowest-order vertex corrections (LOVC). We show that the vertex corrections of higher order (λ3) lower the value of TLOVCC by a few percent.

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.

First-principles exploration of superconductivity in MXenes.

MXenes are an emerging class of two-dimensional materials, which in their thinnest limit consist of a monolayer of carbon or nitrogen (X) sandwiched between two transition metal (M) layers. We have systematically searched for superconductivity among MXenes for a range of transition metal elements, based on a full first-principles characterization in combination with the Eliashberg formalism. Thus, we identified six superconducting MXenes: three carbides (Mo2C, W2C and Sc2C) and three nitrides (Mo2N, W2N and Ta2N). The highest critical temperature of ∼16 K is found in Mo2N, for which a successful synthesis method has been established [Urbankowski et al., Nanoscale, 2017, 9, 17722-17730]. Moreover, W2N presents a novel case of competing superconducting and charge density wave phases.

Effect of hydrostatic pressure on superconductivity of MgB2

The effect of hydrostatic pressure on superconductivity of hexagonal bulk MgB2 was investigated using the first-principles calculations. The calculations were performed based on the density functional theory employing the very recently developed EPW code incorporated in Quantum ESPRESSO computational package. The phonon density of states (PhDOS), isotropic Eliashberg function α2F(ω), the superconducting gap along real and imaginary frequency axes, the electron-phonon coupling constant λ, and the superconducting critical temperature Tc were computed within the general framework of Eliashberg formalism. The results show that hydrostatic pressure enhances superconductivity of the material. At equilibrium, where the hydrostatic pressure of the system is almost zero, the calculated Tc value is 35.67 K. It is relatively nearer to the experimental value of 39 K. The slight underestimation is attributed to the LDA-DFT approximation employed. This value is enhanced to around 95 K at hydrostatic pressure of 6500 kbar and then starts to decrease slowly. We emphasize that our approach to lattice strain engineering, with the target of enhancing Tc as demonstrated in this work, can be applied to various strain-engineering problems in other functional materials.

Characteristics of the s–Wave Symmetry Superconducting State in the BaGe3 Compound

Thermodynamic properties of the s–wave symmetry superconducting phase in three selected structures of the BaGe 3 compound ( P 6 3 / m m c , A m m 2 , and I 4 / m m m ) were discussed in the context of DFT results obtained for the Eliashberg function. This compound may enable the implementation of systems for quantum information processing. Calculations were carried out within the Eliashberg formalism due to the fact that the electron–phonon coupling constant falls within the range λ ∈ 0.73 , 0.86 . The value of the Coulomb pseudopotential was assumed to be 0.122 , in accordance with the experimental results. The value of the Coulomb pseudopotential was assumed to be 0.122 , in accordance with the experimental results. The existence of the superconducting state of three different critical temperature values, namely, 4.0 K, 4.5 K and 5.5 K, depending on the considered structure, was stated. We determined the differences in free energy ( Δ F ) and specific heat ( Δ C ) between the normal and the superconducting states, as well as the thermodynamic critical field ( H c ) as a function of temperature. A drop in the H c value to zero at the temperature of 4.0 K was observed for the P 6 3 / m m c structure, which is in good accordance with the experimental data. Further, the values of the dimensionless thermodynamic parameters of the superconducting state were estimated as: R Δ = 2 Δ ( 0 ) / k B T c ∈ { 3.68 , 3.8 , 3.8 } , R C = Δ C ( T c ) / C N ( T c ) ∈ { 1.55 , 1.71 , 1.75 } , and R H = T c C N ( T c ) / H c 2 ( 0 ) ∈ { 0.168 , 0.16 , 0.158 } , which are slightly different from the predictions of the Bardeen–Cooper–Schrieffer theory ( [ R Δ ] B C S = 3.53 , [ R C ] B C S = 1.43 , and [ R H ] B C S = 0.168 ). This is caused by the occurrence of small retardation and strong coupling effects.