Odd-frequency superconductivity in dilute magnetic superconductors

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Odd-frequency superconductivity in dilute magnetic superconductors

Flávio L. N. Santos, Vivien Perrin, François Jamet, Marcello Civelli, Pascal Simon, Maria C. O. Aguiar, Eduardo Miranda, and Marcelo J. Rozenberg

Phys. Rev. Research 2, 033229 – Published 10 August 2020

Abstract

We show that dilute magnetic impurities in a conventional superconductor give origin to an odd-frequency component of superconductivity, manifesting itself in Yu-Shiba-Rusinov bands forming within the bulk superconducting gap. Our results are obtained in a general model solved within the dynamical mean field theory. By exploiting a disorder analysis and the limit to a single impurity, we are able to provide general expressions that can be used to extract explicitly the odd-frequency superconducting function from scanning tunneling measurements.

Received 25 May 2020
Accepted 14 July 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033229

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Impurities in superconductors Odd-frequency superconductivity

Dynamical mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Flávio L. N. Santos ^1,2, Vivien Perrin ², François Jamet ², Marcello Civelli ², Pascal Simon², Maria C. O. Aguiar ^1,2, Eduardo Miranda ³, and Marcelo J. Rozenberg ²

¹Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, Minas Gerais 30123-970, Brazil
²Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
³Gleb Wataghin Institute of Physics, The University of Campinas (Unicamp), 13083-859 Campinas, SP, Brazil

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Vol. 2, Iss. 3 — August - October 2020

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Figure 1
Dilute magnetic superconductors. Magnetic impurities sites are embedded in a superconducting lattice. The impurity-site magnetic moment $\vec{S}$ (represented by the blue arrow) interacts with electrons (in orange) via a magnetic coupling $J$ , as described by the Hamiltonian in Eq. (1).
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Figure 2
Numerical results obtained using the DMFT equation (3) with $t = 1, μ = - 0.05, δ_{μ} = - 0.5, Δ = - 0.1, J = - 0.65, x = 0.01$ , and broadening $η = 10^{- 3}$ . (a) Full spectrum for $N_{↑}$ and $N_{↓}$ . The inset shows these functions at low energies $| ω | < | Δ |$ . Also shown are the (b) real and (c) imaginary components of the superconducting function $F_{m} (ω)$ as well as its even- $ω$ and odd- $ω$ components, for low energy.
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Figure 3
Plots of Shiba bands centered at $E_{0} > 0$ using the same parameters as in Fig. 2, but with $η = 10^{- 4}$ and making a scaling with $x$ . The horizontal axes are multiplied by $x^{- 1 / 2}$ and the vertical ones by $x^{1 / 2}$ . (a) Plot of $N_{↑}$ using $x = 0.0001$ , 0.001, 0.005, 0.01, 0.015, 0.02, 0.025, and 0.03. (b) Plot of $Im [F_{m}^{o} (ω)]$ using $x = 0.0001$ , 0.001, 0.005, and 0.01. The YSR bands overlap for $x > 0.01$ and $Im [F_{m}^{o} (ω)]$ becomes the superposition of two semicircles. For this reason curves with higher values of $x$ are omitted in (b). It is important to consider a small broadening $η$ in order to cause the lines to overlap completely.
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Figure 4
Plot of $Im [F_{m}^{o} (ω)]$ obtained using the same parameters as in Fig. 2 and $η = 10^{- 3}$ , but with different values of $x, x = 0.01$ , where the two Shiba bands do not overlap, and $x = 0.03$ , where the two bands do overlap, giving rise to a resonance around $ω = 0$ . Solid blue lines were obtained using the DMFT equation (3), while dashed red lines correspond to Eq. (8).
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Figure 5
Numerical results for nonmagnetic sites obtained using the DMFT equation (3) with $t = 1, μ = - 0.05, δ_{μ} = - 0.5, Δ = - 0.1, J = - 0.65, x = 0.01$ , and broadening $η = 10^{- 3}$ . (a) Full spectrum for $N_{n m ↑}$ and $N_{n m ↓}$ . The inset shows these functions at low energies $| ω | < | Δ |$ . Also shown are the (b) real and (c) imaginary components of the superconducting function $F_{n m} (ω)$ as well as its even- $ω$ and odd- $ω$ components, at low energies.
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Figure 6
Plots of the density of states $N_{↑} (ω) + N_{↓} (ω)$ , using $t = 1, μ = - 0.05, δ_{μ} = - 0.5, Δ = - 0.1, J = - 0.65, x = 0.01$ , and $η = 10^{- 3}$ . Solid blue lines give the solution of the DMFT equation (3), as presented in Fig. 2. In (a) the dashed red line corresponds to Eq. (B5), showing that it is exact for the whole spectrum. In (b) the dashed red line corresponds to the approximate Eq. (10), showing that the latter is a good approximation for the former for a broad range of energies.
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