Enhancing superconductivity by disorder

Maria N. Gastiasoro and Brian M. Andersen

Phys. Rev. B 98, 184510 – Published 19 November 2018

Abstract

We study two mechanisms for enhancing the superconducting mean-field transition temperature $T_{c}$ by nonmagnetic disorder in both conventional (sign-preserving gaps) and unconventional (sign-changing gaps) superconductors (SCs). In the first scenario, relevant to multiband systems of both conventional and unconventional SCs, we demonstrate how favorable density-of-states enhancements driven by resonant states in off-Fermi-level bands lead to significant enhancements of $T_{c}$ in the condensate formed by the near-Fermi-level bands. The second scenario focuses on systems close to localization where random disorder-generated local density-of-states modulations cause a boosted $T_{c}$ even for conventional single-band SCs. We analyze the basic physics of both mechanisms within simplified models, and we discuss the relevance to existing materials.

Received 7 December 2017
Revised 11 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.184510

Physics Subject Headings (PhySH)

Coherence length Impurities in superconductors Multiband superconductivity Superconducting gap Superconducting phase transition

High-temperature superconductors

Methods in superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Maria N. Gastiasoro^* and Brian M. Andersen

Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark

^*Present address: School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA.

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Vol. 98, Iss. 18 — 1 November 2018

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Images

Figure 1
Superconducting critical temperature $T_{c} / T_{c}^{0}$ in the presence of nonmagnetic disorder in (a) an unconventional multiband $s_{\pm}$ SC vs disorder concentration, and (b) a one-band $s$ -wave SC with $15 %$ disorder vs impurity strength $V$ . The black curves show the results when disallowing spatial modulations of density and SC OP consistent with AG theory. The red curves show the self-consistent cases with spatial modulations of both quantities. In (b) $p = 2 %$ (dotted), $p = 5 %$ (line-dotted), and $p = 10 %$ (solid); see the text.
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Figure 2
(a) Real-space map of the total electron density in the presence of 1.3% disorder consisting of repulsive impurities, $V = 0.725$ eV. Total LDOS in the normal state $N (r, ω = 0) / N^{0} (ω = 0)$ (b), self-consistent SC fields $Δ_{μ} (r) / Δ_{μ}^{0}$ at $T / T_{c}^{0} = 1.5$ for $d_{3 z^{2} - r^{2}}$ (c) and $d_{x z}$ (d). The superscript 0 denotes parameters of the disorder-free system. $Δ_{μ}^{0}$ refers to the $T = 0$ gap value of orbital $μ$ given by (0.78,0.78,1.31,0.063,0.055) meV in the basis of ( $d_{x z}, d_{y z}, d_{x y}, d_{x^{2} - y^{2}}, d_{3 z^{2} - r^{2}}$ ). (e,f) Entropy $S$ (e) and specific heat $C$ (f) vs $T$ comparing the disordered case (colored curves) with the homogeneous system (black curves).
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Figure 3
(a) Band structure for the two-band toy model along $k = (k_{x}, π / 2)$ . (b) $N_{b} (r_{0}, ω)$ for a resonant potential ( $V_{b} = 1 / Re [g_{b}^{0} (ω = 0)], V_{a} = 0$ ) at the impurity site for different $γ$ . Note that here we set $V_{a} = 0$ to most clearly demonstrate the origin of the $T_{c}$ enhancement. For the results in Fig. 2 we used a more realistic orbital-independent potential as stated above. The finite width at $γ = 0$ arises from an imposed broadening $η = T$ . (c,d) Self-consistent induced fields $Δ_{μ} (r_{0}) - Δ_{μ}^{0}$ in units of $t$ for bands $b$ (c) and $a$ (d) as a function of $V_{b}$ and $γ$ for on-site pairing $Γ = 1.93 t$ and $k_{B} T = 0.1 t$ . The dotted lines in (c,d) show the $γ$ above which $Δ_{μ}^{0} = 0$ in the disorder-free system at this $T$ .
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Figure 4
(a)–(l) Real-space maps of $Δ (r) / Δ^{0}$ vs $T$ for a $15 %$ disordered system with varying impurity strength $V = 1.5 t$ (a)–(d), $V = 3.3 t$ (e)–(h), $V = 5 t$ (i)–(l) for a conventional $s$ -wave SC in a one-band model. (m) Spatially averaged SC OP $〈 Δ 〉 / Δ^{0}$ vs $T$ for varying disorder strength $V$ , and for Anderson disorder with $V_{A} \in [- 5, 5] t$ (blue curve). The clean case is shown by the black curve. (n) Real-space map of $Δ (r) / Δ^{0}$ at $T / T_{c}^{0} = 1.93$ for the case of Anderson disorder. The red dots indicate sites with large $T_{c} (r) / T_{c}^{0} \propto exp {- 1 / [| U | N (r)]} / exp [- 1 / (| U | N^{0})] > 0.4$ , displaying the clear correlation between the local LDOS enhancements and the increased SC.
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