Competition of electron-phonon mediated superconductivity and Stoner magnetism on a flat band

Risto Ojajärvi, Timo Hyart, Mihail A. Silaev, and Tero T. Heikkilä

Phys. Rev. B 98, 054515 – Published 22 August 2018

Abstract

The effective attractive interaction between electrons, mediated by electron-phonon coupling, is a well-established mechanism of conventional superconductivity. In metals exhibiting a Fermi surface, the critical temperature of superconductivity is exponentially smaller than the characteristic phonon energy. Therefore, such superconductors are found only at temperatures below a few kelvin. Systems with flat energy bands have been suggested to cure the problem and provide a route to room-temperature superconductivity, but previous studies are limited to only BCS models with an effective attractive interaction. Here we generalize Eliashberg's theory of strong-coupling superconductivity to systems with flat bands and relate the mean-field critical temperature to the microscopic parameters describing electron-phonon and electron-electron interaction. We also analyze the strong-coupling corrections to the BCS results and construct the phase diagram exhibiting superconductivity and magnetic phases on an equal footing. Our results are especially relevant for novel quantum materials where electronic dispersion and interaction strength are controllable.

Received 7 March 2018
Revised 21 June 2018

DOI:https://doi.org/10.1103/PhysRevB.98.054515

Physics Subject Headings (PhySH)

Electron-phonon coupling Magnetism Odd-frequency superconductivity Phase diagrams Superconductivity Surface states

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Risto Ojajärvi¹, Timo Hyart^1,2, Mihail A. Silaev¹, and Tero T. Heikkilä¹

¹Department of Physics and Nanoscience Center, University of Jyvaskyla, P.O. Box 35 (YFL), FI-40014 Jyvaskyla, Finland
²Institut für Theoretische Physik, Universität Leipzig, D-04103 Leipzig, Germany

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Issue

Vol. 98, Iss. 5 — 1 August 2018

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Images

Figure 1
Strong-coupling phase diagram for flat-band systems as a function of electron-phonon attraction $λ$ for electron-electron repulsion $u = 0.5 ω_{E}$ [Eq. (8)]. $T_{C}^{CE}$ is the temperature at which the $T_{C}$ 's of magnetic and superconducting order coincide. In the striped region these phases can form metastable domains inside the sample. This diagram is for $N \to \infty$ . For finite $N$ the overlap region between the phases is smaller.
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Figure 2
Quasiparticle dispersions $E (p)$ for different kinds of symmetry breakings with $N = 5$ . (a) In the noninteracting case, the spin bands are degenerate with $E (p) = \pm ɛ (p)$ . (b) For the ferromagnetic (FM) or the superconducting (SC) phase with a $θ = π$ phase shift between the sublattices, one quasiparticle band is shifted up and the other down in energy. In this case, no energy gap is opened. (c) For the antiferromagnetic (AFM) or the SC phase with $θ = 0$ an energy gap is opened and quasiparticle bands are doubly degenerate.
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Figure 3
Critical temperatures for superconducting and magnetic phases for $N \to \infty$ . (a) Superconductivity is suppressed when $λ ≲ u / ω_{E}$ . Above the critical point $λ_{C} (u)$ , $T_{C}^{sc}$ is linear in $λ$ . With increasing $λ$ , the electron-phonon renormalization increases and this limits the critical temperature. The dashed line is the approximation in Eq. (14). (b) Critical interaction strength for superconductivity as a function of $u$ . When $λ < λ_{C} (u)$ , superconductivity is suppressed. The dashed line is the instantaneous approximation. (c) Magnetism is suppressed when $u / ω_{E} ≲ λ$ . Above the critical point $u_{C} (λ)$ , $T_{C}^{m}$ is linear in $u$ . (d) Critical interaction strength for magnetism as a function of electron-phonon interaction. When $u < u_{C} (λ)$ , magnetism is suppressed. The dashed line is the instantaneous approximation. In this figure, we do not take into account the possible magnetic instability of the superconducting state, or vice versa.
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Figure 4
Effect of finite $N$ on critical temperature when $u = 0$ . For small $\tilde{λ}$ , the results coincide with the instantaneous approximation of Eq. (12) (shown with the dashed lines). For large $\tilde{λ}$ , the strong-coupling corrections limit the increase in $T_{C}^{sc}$ .
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Figure 5
Mean-field phase diagram for $N = \infty$ obtained by determining the line on which the critical temperatures for superconductivity and antiferromagnetism are equal. The thin dashed line shows the phase boundary $λ = u / ω_{E}$ in the case of instantaneous interactions. When the energy scales of interactions are small compared to $ω_{E}$ we recover the BCS results. The phase diagram for finite $N$ looks similar but the retardation effects are weaker, so that the deviation from the BCS approximation is smaller.
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Figure 6
Sketch of a domain wall between magnetic (red) and superconducting (blue) domains. At the domain wall a triplet component (purple) is induced.
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