Multiband superconductors: Disparity between band length scales

Tiago T. Saraiva, C. C. de Souza Silva, J. Albino Aguiar, and A. A. Shanenko

Phys. Rev. B 96, 134521 – Published 27 October 2017

Abstract

Multiple condensates in a superconducting material can interfere constructively or destructively and this leads to unconventional effects not inherent in single-band superconductors. Such effects can be pronounced when the spatial scales (healing lengths) of different band condensates deviate from each other. Here we show that, contrary to usual expectations, this deviation can be considerable even far beyond the regime of nearly decoupled bands, being affected by difference between the band Fermi velocities. Our study is performed within the extended Ginzburg-Landau (GL) formalism that goes to one order beyond the GL theory in the perturbation expansion of the microscopic equations over the proximity to $T_{c}$ . The formalism makes it possible to obtain closed analytical results for the profiles of the band condensates and for their healing lengths and, at the same time, captures the difference between the healing lengths which does not appear in the standard GL domain.

Received 30 January 2017

DOI:https://doi.org/10.1103/PhysRevB.96.134521

Physics Subject Headings (PhySH)

Multiband superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Tiago T. Saraiva, C. C. de Souza Silva, J. Albino Aguiar, and A. A. Shanenko

Departamento de Física, Universidade Federal de Pernambuco, Av. Jorn. Aníbal Fernandes, s/n, Cidade Universitária 50740-560, Recife, PE, Brazil

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Vol. 96, Iss. 13 — 1 October 2017

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Images

Figure 1
Sketch of our calculational setup: the plane $x = 0$ separates domains with zero $(x < 0)$ and nonzero $(x > 0)$ gaps so we have the Dirichlet boundary condition for the superconducting gap(s) at the interface $x = 0$ . The system remains symmetric under translations in $y$ and $z$ directions.
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Figure 2
EGL solution for $Δ (x, τ)$ given in units of its asymptotic value $Δ_{\infty} (τ)$ as a function of the relative distance from the interface $x / ξ_{GL}$ for $τ = 0$ (dashed curve) and $τ = 0.4$ (solid curve). The dotted horizontal line marks $δ = tanh (1 / \sqrt{2})$ that determines the condensate healing length according to Eq. (21).
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Figure 3
Condensate healing length versus $τ$ : the analytical result of Eq. (23) is given by the dotted line, whereas the full numerical solution of Eq. (21) is represented by the solid curve. The difference between the numerical and analytical results becomes significant for $τ > 0.3 - 0.4$ , which defines the validity domain of Eq. (23) as $τ ≲ τ^{*} = 0.3 - 0.4$ ; see the related discussion in the main text.
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Figure 4
(a) Deviation rate of the band healing lengths in units of the GL coherence length [see Eq. (53)] versus the ratio of the band Fermi velocities $β$ and the relative interband coupling $λ_{12} / λ_{11}$ as calculated at $χ = N_{2} (0) / N_{1} (0) = 1$ . (b) The same quantity as in panel (a) but now given as a function of $β$ for the relative interband couplings $0.2, 0.3, 0.4$ , and 0.5.
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Figure 5
(a) Deviation rate (53) versus $β = v_{F 2} / v_{F 1}$ and $λ_{12} / λ_{11}$ for the same intraband couplings as in Fig. 4 but the band DOS's ratio is now changed to $χ = 0.2$ . (b) The deviation rate as a function of $β$ for $λ_{12} / λ_{11} = 0.2, 0.3, 0.4$ , and 0.5; the other microscopic parameters are the same as in panel (a).
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Figure 6
(a) EGL solution for the band gap $Δ_{i} (x, τ)$ [in units of its asymptotic value $Δ_{i, \infty} (τ)]$ as a function of the distance from the interface (in units of $ξ_{GL}$ ) calculated at $τ = 0.07$ and $β = β^{(a - b)}$ ; the solid and dashed curves represent bands 1 and 2, respectively. (b) The same as in panel (a) but for $β = β^{(c)}$ . (c) The relative band healing length $ξ_{i} / ξ_{GL}$ versus $τ$ at $β = β^{(a - b)}$ : the analytical result of Eq. (52) is given by the dashed line, while the full numerical solution of Eq. (51) is represented by the solid curve. (d) The same as in (c) but now for $β = β^{(c)}$ . The dotted lines in panels (a) and (b) mark $δ = tanh (1 / \sqrt{2})$ that determines the band healing length according to Eq. (51). The validity domain of the $τ$ expansion can be estimated as $τ ≲ τ^{*} = 0.4 - 0.5$ for $β = β^{(a - b)}$ and $τ ≲ τ^{*} = 0.05 - 0.07$ for $β = β^{(c)}$ ; see the discussion in the main text.
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