Topological crystalline superconductivity in Dirac semimetal phase of iron-based superconductors

Takuto Kawakami and Masatoshi Sato

Phys. Rev. B 100, 094520 – Published 13 September 2019

Abstract

In iron-based superconductors, band inversion of $d$ and $p$ orbitals yields Dirac semimetallic states. We theoretically investigate their topological properties in normal and superconducting phases based on the tight-binding model involving full symmetry of the materials. We demonstrate that a Cooper pair between electrons with $d$ and $p$ orbitals relevant to the band structure yields odd-parity superconductivity. Moreover, we present the typical surface states by solving the Bogoliubov–de Gennes equation and characterize them by topological invariants defined with crystal symmetry. It is found that there appear various types of Majorana fermions such as surface flat band, Majorana quartet, and Möbius twisted surface states. Our theoretical results show that iron-based superconductors are promising platforms to realize rich topological crystalline phases.

Received 1 July 2019

DOI:https://doi.org/10.1103/PhysRevB.100.094520

Physics Subject Headings (PhySH)

Majorana fermions Superconductivity Surface states Symmetry protected topological states Topological materials Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Takuto Kawakami^1,2,* and Masatoshi Sato^2,†

¹Department of Physics, Osaka University, Osaka 560-0043, Japan
²Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

^*t.kawakami@qp.phys.sci.osaka-u.ac.jp
^†msato@yukawa.kyoto-u.ac.jp

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Vol. 100, Iss. 9 — 1 September 2019

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Images

Figure 1
Lattice structure of the Fe(Se,Te) system. The white and blue spheres indicate the Fe and chalcogen (Se,Te) atoms, respectively, with sublattices 1 and 2 due to the buckling of the chalcogens.
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Figure 2
(a) Band dispersion obtained from the tight-binding model (3) along a typical high-symmetry cut in the Brillouin zone. The color code indicates the orbital property of the band (blue, $d$ orbital; red, $p_{z}$ orbital; green, mixed). The size of the gray circle indicates the weight of the $λ_{η_{1}} = - 1$ component (see also main text). The sign on the band indicates the inversion eigenvalue. (b) Bulk and surface Brillouin zones with the parity product of the two lower bands at the time-reversal symmetric momenta. The green sphere represents bulk Dirac points. The green spots on the surface Brillouin zone represent the projected Fermi surface. The red curves on the surface Brillouin zone represent the surface Fermi loop. The parameters are $t_{d} = - 1.5, t_{p} = 1, t_{d}^{'} = - 0.6, t_{p}^{'} = 1, t_{1}^{0} = 0.5, t_{2}^{0} = 1, t_{1}^{\pm} = t_{2}^{\pm} = 0.4 \mp 0.2, δ μ = 2 (t_{d} - t_{p})$ , and $μ = - 0.5$ .
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Figure 3
Schematics of the band structure and position of the Fermi level. In the case where the Fermi level is close to the Dirac points, odd-parity superconductivity is possible.
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Figure 4
(a) Fermi surface around the Dirac point, where the color code indicates the expectation value of the parity operator $P$ on it. (b)–(d) The normalized superconducting gap on the Fermi surface of the $B_{1 u}, B_{2 u}$ , and $E_{u}$ representations.
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Figure 5
Energy spectrum of the system with the (100) surface. (a)–(f) indicate the $B_{1 u}$ states. (a) shows the dispersion along a typical high-symmetry cut in the surface Brillouin zone. The bands in green and red indicate the bulk and surface states, respectively. (b)–(d) magnify the momentum regions around the surface Fermi loop. (e) shows the schematics of the surface zero-energy states on the surface Brillouin zone in the normal state. The projected bulk Fermi surface and the surface Fermi loop are depicted by the green oval and red curve, respectively. (f) is the surface zero-energy states in the $B_{1 u}$ superconducting state. In (f), the bulk Fermi surface and Fermi loop in the normal state (indicated by green and red dashed curves, respectively) are gapped at the generic point. At intersections with mirror invariant lines, the surface Fermi loop survives as Majorana fermions in the superconducting state. Around the $Γ$ point, they form a Majorana quartet [10]. (g)–(l) are the same as (a)–(f) but for the $B_{2 u}$ state. The thick red line in (l) shows the Majorana flat band. The parameters are the same as in Fig. 2.
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Figure 6
Energy spectrum of the system with the (110) surface in a manner similar to Fig. 5. (a)–(e) and (f)–(j) indicate the $B_{1 u}$ and $B_{2 u}$ states, respectively. In contrast to the (100) surface shown in Fig. 5, the $B_{1 u}$ state exhibits a flat band on the $Γ Z$ path, and $B_{2 u}$ has a Majorana quartet. In addition, the whole surface Fermi loop remains gapless for the $B_{1 u}$ superconductor, associated with the diagonal $C_{2}^{″}$ -odd property of the gap function. Both $B_{1 u}$ and $B_{2 u}$ states have Möbius twisted surface states (hourglass fermions), passing through the bulk spectrum, as shown by the dashed line in (a) and (f). They are typical for glide-protected time-reversal-invariant topological phases. (See text for more details.) The parameters are the same as in Figs. 2 and 5.
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