Topological surface states and Andreev bound states in superconducting iron pnictides

The nontrivial topology of the electronic structure of iron pnictides can lead to the appearance of surface states. We study such states in various strip geometries with a focus on the superconducting phase. In the presence of unconventional superconducting pairing with $s_{\pm }$-wave gap structure, the topological states are quite robust and partly remain in the superconducting gap. Furthermore, Andreev bound states appear, which coexist with the topological states for small superconducting gaps and merge with them for larger gap values. The bulk and surface dispersions are obtained from exact diagonalization for two-orbital and five-orbital models in strip geometries.

Figure 1

Sign structure of the superconducting gap function with $s_{\pm }$-wave structure. The red (blue) areas denote regions in the BZ where the sign of $cos{k}_{x}cos{k}_y$ is negative (positive). The dashed lines show the nodes of this function. In addition, the Fermi surfaces of (a) the two-orbital model and (b) the five-orbital model are plotted.

Figure 2

New unit cell and (11) edges: (a) Comparison of the old unit cell (dashed square) and the new unit cell (solid square). The new unit cell is rotated by ${45}^{\circ }$, and its sides are stretched by a factor of $\sqrt{2}$. $a$ is the lattice constant. (b) Strip with two (11) edges in the rotated coordinate system. Black (gray) dots belong to sublattice $A\left(B\right)$. The strip width is $W=N_X$. The new unit cell is depicted by a solid square.

Figure 3

Quasiparticle spectra in the superconducting phase for the two-orbital model. Bands of a (10) strip of width $W=40$ (red) are compared to the bulk bands projected onto the 1D BZ for the strip (blue). Only the low-energy part of the spectra is shown. (a) $\Delta =0.5$ for $s_{++}$-wave pairing. For comparison, the topological surface bands, modified according to $\xi \to \pm \sqrt{{\xi }^2+{\left|\Delta \right|}^2}$, are also plotted (dashed green lines). (b) $\Delta =0.5$ for $s_{\pm }$-wave pairing, (c) $\Delta =2.0$ for $s_{\pm }$-wave pairing, and (d) $\Delta =6.0$ for $s_{\pm }$-wave pairing. In (b)–(d), the dashed dark blue lines denote topological surface bands of the normal state.

Figure 4

Scattering processes in the two-orbital model for the (10) strip. The shaded regions indicate $k_y$ values for which surface states of topological origin exist in the normal, paramagnetic phase. The bold lines are exemplary lines with constant $k_y$ for which sign-changing scattering processes are possible. Relevant states are indicated by white circles. (a) Small $\Delta$: sign-changing processes can only occur outside of the shaded regions. Topological states and Andreev bound states are separated. (b) Larger $\Delta$: states in a broader region around the Fermi surfaces become relevant. Topological states merge with Andreev bound states.

Figure 5

Quasiparticle spectra in the superconducting phase for the five-orbital model. Bands of a (10) strip of width $W=40$ (red) are compared to the bulk bands projected onto the 1D BZ for the strip (blue). Only the low-energy part of the spectra is shown. (a) $\Delta =0.15(s_{++}$ pairing). For comparison, some of the topological surface bands modified according to $\xi \to \pm \sqrt{{\xi }^2+{\left|\Delta \right|}^2}$ are also plotted (dashed green lines). (b) $\Delta =0.15$ ($s_{\pm }$ pairing) and (c) $\Delta =0.4$ ($s_{\pm }$ pairing). In (b) and (c), the dashed dark blue lines denote topological surface bands of the normal state.

Figure 6

Energy bands in the paramagnetic phase for the two-orbital model. Bands of a (11) strip of width $W=40$ (red) are compared to the bulk bands projected onto the 1D BZ for the strip (blue). The black dash-dotted line denotes the Fermi energy at half filling.

Figure 7

Vector field $(cos\varphi (\mathbf{k}),sin\varphi (\mathbf{k}\left)\right)$ (green arrows) and exemplary path at constant $k_Y$ (black arrow) in the extended BZ. The shaded regions are the areas for which an effective 1D system with fixed $k_Y$ has a bulk gap. The gap is not necessarily at the Fermi energy.

Figure 8

Energy bands in the paramagnetic phase for the five-orbital model. Bands of a (11) strip of width $W=20$ (red) are compared to the bulk bands projected onto the 1D BZ for the strip (blue). The black dash-dotted line denotes the Fermi energy at a filling factor of $0.6$.

Figure 9

Quasiparticle spectra in the superconducting phase for the two-orbital model. Bands of a (11) strip of width $W=40$ (red) are compared to the bulk bands projected onto the 1D BZ for the strip (blue). (a) $\Delta =0.5$ ($s_{++}$ pairing), (b) $\Delta =0.1$ ($s_{\pm }$ pairing), (c) $\Delta =0.5$ ($s_{\pm }$ pairing), and (d) $\Delta =2.0$ ($s_{\pm }$ pairing). Only the low-energy part of the spectra is shown. Note that there were no surface states in the normal phase (see Fig. 6).

Figure 10

Scattering processes in the two-orbital model for the (11) strip. The extended BZ is illustrated along with the Fermi surfaces. The bold lines are exemplary lines with constant $k_Y$. Relevant states are drawn as white circles. (a) Small $\Delta$: sign-changing processes can never occur. (b) Larger $\Delta$: relevant states are also found in the vicinity of the Fermi surfaces, and sign-changing scattering processes become possible. Andreev bound states can emerge.

Figure 11

Quasiparticle spectra in the superconducting phase for the five-orbital model. Bands of a (11) strip of width $W=20$ (red) are compared to the bulk bands projected onto the 1D BZ for the strip (blue). Only the low-energy part of the spectra is shown. (a) $\Delta =0.15$ ($s_{++}$ pairing). For comparison, some of the topological surface bands modified according to $\xi \to \pm \sqrt{{\xi }^2+{\left|\Delta \right|}^2}$ are also plotted (dashed green lines). (b) $\Delta =0.15$ ($s_{\pm }$ pairing). The dashed dark blue lines denote topological surface bands of the normal state.