Coulomb problem in iron-based superconductors

Elio J. König and Piers Coleman

Phys. Rev. B 99, 144522 – Published 26 April 2019

Abstract

We discuss the role of strong Coulomb interactions in iron-based superconductors (FeSCs). The presumed $s^{\pm}$ character of these superconductors means that the condensate is not symmetry protected against Coulomb repulsion. Remarkably, the transition temperatures and the excitation gap are quite robust across the large family of iron-based superconductors, despite drastic changes in Fermi-surface geometry. The Coulomb problem is to understand how these superconductors avoid the strong on-site Coulomb interaction of the iron atoms, while maintaining a robust transition temperature. Within the dominant space of $t_{2 g}$ orbitals, on-site repulsion in the FeSCs forces two linearly independent components of the condensate to vanish. This raises the possibility that iron-based superconductors might adapt their condensate to the Coulomb constraints by rotating the pairing state within the large manifold of entangled extended $s$ -wave gap functions with different orbital and momentum space structure. We examine this “orbital and k-space flexibility” (OKF) mechanism using both Landau theory and microscopic calculations within a multiorbital $t − J$ model. Based on our results, we conclude that OKF necessitates a large condensate degeneracy. One interesting possibility raised by our results is that a resolution to the Coulomb problem in FeSCs might require a reconsideration of triplet pairing.

Received 16 November 2018
Revised 1 April 2019

DOI:https://doi.org/10.1103/PhysRevB.99.144522

Physics Subject Headings (PhySH)

Electron-mediated pairing Multiband superconductivity Superconductivity

High-temperature superconductors Iron-based superconductors Strongly correlated systems

Landau-Ginzburg theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Elio J. König¹ and Piers Coleman^1,2

¹Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
²Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, England, United Kingdom

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Vol. 99, Iss. 14 — 1 April 2019

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Images

Figure 1
Schematic illustrating orbital and k-space flexibility. Here, the large space of superconducting order parameters is represented as a three-dimensional space. Each direction has an associated “bare” $T_{c} (\vec{Δ})$ which is determined in the fictitious case without on-site repulsion and which is represented as the distance from the origin. It varies continuously and thus forms a surface, e.g., a sphere or an ellipsoid. The orientation with maximal $T_{c}$ determines the pairing state. On-site Coulomb repulsion constrains the order-parameter space to a submanifold [see Eq. (5)]. The latter is material dependent and represented by a plane. Upper panels: When the bare $T_{c}$ is the same for all pairing directions, the orientation adapts to Coulomb repulsion without any cost in $T_{c}$ (orbital and k-space flexibility). Lower panels (realistic situation): The bare $T_{c}$ is direction dependent; orbital and k-space flexibility fails to explain comparable $T_{c}$ for different materials.
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Figure 2
Diagrammatic representation of the matrix pair susceptibility $χ_{Γ Γ^{'}}$ (a) and the coefficients $β_{Γ_{1} Γ_{2} Γ_{3} Γ_{4}}$ (b) which enter the Landau free energy [Eq. (12)].
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Figure 3
Graphical representation of several superconducting states based on the tight-binding model proposed in [52]. The dispersion relation along $Γ − X − M − Γ$ is colored red (blue) for positive (negative) sign of the gap function. Each panel also contains an inset where the Fermi surfaces are plotted with the same color coding. (a) Orbital antiphase, $Δ (k) = Δ λ_{8}$ (in the present tight-binding model, this state has nodes). (b) Conventional $s_{\pm}, Δ (k) = Δ c_{x} c_{y} λ_{0}$ . (c) A mixed state $Δ (k) = Δ [- 0.01 c_{x} c_{y} λ_{0} - 0.99 s_{x} s_{y} λ_{1} + 0.11 (c_{x} - c_{y}) λ_{3}]$ with strong orbital entanglement. (d) A mixed state $Δ (k) = Δ [- 0.97 c_{x} c_{y} λ_{0} - 0.03 s_{x} s_{y} λ_{1} + 0.24 (c_{x} - c_{y}) λ_{3}]$ , which is mainly $s_{\pm}$ . Panels (c) and (d) occur at $x = - 0.9$ and 0.7, respectively, in the microscopic calculation of Fig. 4. Note that orbital entanglement naturally explains the sign change within the two inner Fermi surfaces of hole doped FeSC as proposed in [53].
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Figure 4
Microscopic investigation of orbital flexibility based on Eqs. (12) and (30). The parameter $x$ is introduced in the density of states $ρ_{e} = (1 + x) / (20 J) [ρ_{h} = (1 - x) / (10 J)]$ of electron pockets (hole pockets) in order to smoothly interpolate between systems with different types of carriers. (a) Evolution of $T_{c}$ as a function of $x$ ; the solid, dashed, and dot-dashed curves represent $T_{c}$ for the flexible, $Δ_{4}$ , and $Δ_{3}$ state, respectively, all of which are normalized to the $T_{c}$ of the flexible state at $x = 0$ . (b) Evolution of the direction of the flexible state in the space of $Δ_{3, 4, 5}$ . For this plot we chose a moderate $\bar{U} / J = 3$ . For $\bar{U} / J = 10$ the graphs of both panels are qualitatively similar, however the local maximum near $x \approx 0.7$ drops down to $ln (T_{c} / T_{c}^{(x = 0)}) \approx - 20$ .
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Figure 5
Representation of the three $t_{2 g}$ orbitals in an iron plane. The sites of the lattice correspond to iron positions. Chalcogen and pnictogen atoms reside in alternating positions above and below a plaquette, respectively (indicated by a circle with a plus or minus sign, respectively). The sign of the lobes of the orbitals is represented by two different colors: light green and darker red.
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Figure 6
Fermi surfaces for a three-band model based on [52]. The black curves correspond to the solution of the full matrix Hamiltonian; colored dashed lines correspond to approximate solutions using the projection on the two orbitals indicated by each of the Fermi surfaces. This picture also includes the vectors ${\hat{n}}^{α, β, γ}$ which rotate in different orbital spaces for different Fermi surfaces.
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