Dynamical renormalization of electron-phonon coupling in conventional superconductors

Nina Girotto and Dino Novko

Phys. Rev. B 107, 064310 – Published 27 February 2023

Abstract

The adiabatic Born-Oppenheimer approximation is considered to be a robust approach that very rarely breaks down. Consequently, it is predominantly utilized to address various electron-phonon properties in condensed matter physics. By combining many-body perturbation and density functional theories we demonstrate the importance of dynamical (nonadiabatic) effects in estimating superconducting properties in various bulk and two-dimensional materials. Apart from the expected long-wavelength nonadiabatic effects, we found sizable nonadiabatic Kohn anomalies away from the Brillouin zone center for materials with strong intervalley electron-phonon scatterings. Compared to the adiabatic result, these dynamical phonon anomalies can significantly modify electron-phonon coupling strength $λ$ and superconducting transition temperature $T_{c}$ . Further, the dynamically induced modifications of $λ$ have a strong impact on transport properties, where probably the most interesting is the rescaling of the low-temperature and low-frequency regime of the scattering time $1 / τ$ from about $T^{3}$ to about $T^{2}$ , resembling the Fermi liquid result for electron-electron scattering. Our goal is to point out the potential implications of these nonadiabatic effects and reestablish their pivotal role in computational estimations of electron-phonon properties.

Received 27 December 2022
Revised 16 February 2023
Accepted 17 February 2023

DOI:https://doi.org/10.1103/PhysRevB.107.064310

Physics Subject Headings (PhySH)

Electron relaxation Electron-phonon coupling Superconductivity Transport phenomena

Superconductors

Density functional theory Many-body techniques Perturbation theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Nina Girotto and Dino Novko ^*

Centre for Advanced Laser Techniques, Institute of Physics, 10000 Zagreb, Croatia

^*dino.novko@gmail.com

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Issue

Vol. 107, Iss. 6 — 1 February 2023

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Images

Figure 1
(a) Diagrammatic representation of the electron self-energy $Σ = g^{2} G D_{0}$ , where $g$ are the screened EPC matrix elements (gray circles), $G$ is the electron Green's function (straight line), and $D_{0}$ is the bare phonon propagator (single wavy line). (b) The electron self-energy $Σ = g^{2} G D$ , where the bare phonon propagator $D_{0}$ is replaced with the dynamical (NA) phonon propagator $D$ (double wavy line). (c) Dressing of the phonons via the dynamical EPC, i.e., the Dyson equation for the phonon propagator $D = D_{0} + D_{0} \tilde{π} D$ , where $\tilde{π}$ is the NA phonon self-energy due to EPC.
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Figure 2
Structure, electronic bands along high-symmetry points of the Brillouin zone, and Fermi surface in the first Brillouin zone for (a) ${MgB}_{2}$ , (b) graphane (1L Gr-H), and (c) arsenene (1L As). Red/green color highlights the relevant electronic states at the Fermi level that are involved in electron-phonon scatterings and formation of the (adiabatic and dynamical) Kohn anomalies. Black arrows depict the corresponding electronic transitions.
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Figure 3
(a) Adiabatic (A; blue) and nonadiabatic (NA; red) phonon dispersions of ${MgB}_{2}$ , monolayer graphane (1L Gr-H), and monolayer arsenene (1L As). Notice how strong dynamical corrections to the phonon bands can occur away from the center of the Brillouin zone, as it is the case in ${MgB}_{2}$ and 1L As. (b) The corresponding phonon linewidths $γ_{q ν}$ coming from the EPC. The size of the dots and colors (defined within the color bars) represents the intensity of $γ_{q ν}$ . (c) Eliashberg functions $α^{2} F (ω)$ and cumulative EPC constant $λ (ω)$ obtained for ${MgB}_{2}$ , 1L Gr-H, and 1L As. Three different results are shown: adiabatic (A), where frequencies are obtained within the adiabatic DFPT and without momentum and branch-resolved phonon broadenings $γ_{q ν}$ (blue), nonadiabatic (NA), where frequencies are corrected with NA effects, while the corresponding NA phonon broadening due to EPC is not included (red), and full nonadiabatic ( $NA + τ$ ), where both NA frequency renormalization and NA phonon linewidth effects are taken into account (yellow).
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Figure 4
Total EPC strengths $λ$ , the first moments of the phonon spectrum $λ 〈 ω 〉$ , and superconducting transition temperatures $T_{c}$ obtained with three different approaches: adiabatic (A; within standard adiabatic DFPT methodology), nonadiabatic (NA; by including NA phonon frequency renormalizations), and full nonadiabatic ( $NA + τ$ ; by including both NA frequency renormalizations and NA phonon broadenings $γ_{q ν}$ ). The results are presented for various bulk and 2D superconducting materials. Bottom panels show the corresponding relative changes $(x_{NA / NA + τ} - x_{A}) / x_{A}$ for $T_{c}$ (left axis) and $λ$ (right axis). All relative changes are positive, except the $NA + τ$ results of $x = λ$ for ${MgB}_{2}$ and C (red diamond signs), and of $x = T_{c}$ for C (green-yellow circles).
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Figure 5
Frequency and temperature dependence of the optical (or electron-hole pair) scattering rate $1 / τ_{op} (ω, T)$ and the mass enhancement (or energy renormalization) parameter $λ_{op} (ω, T)$ for various bulk and 2D materials as obtained with adiabatic (A), nonadiabatic (NA), and full nonadiabatic ( $NA + τ$ ) methods. Note how the scaling of the low-frequency and low-temperature regime of $1 / τ_{op}$ is modified under the effect of NA phonon broadening.
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