Edge states and local electronic structure around an adsorbed impurity in a topological superconductor

Recently, topological superconducting states have attracted much interest. In this paper, we consider a topological superconductor with $Z_2$ topological mirror order [Y.-Y. Tai et al., Phys. Rev. B 91, 041111(R) (2015)] and $s_{\pm }$-wave superconducting pairing symmetry, within a two-orbital model originally designed for iron-based superconductivity [Y.-Y. Tai  et al., Europhys. Lett. 103, 67001 (2013)]. We predict the existence of gapless edge states. We also study the local electronic structure around an adsorbed interstitial magnetic impurity in the system, and find the existence of low-energy in-gap bound states even with a weak spin polarization on the impurity. We also discuss the relevance of our results to a recent scanning tunneling microscopy experiment on a Fe(Te,Se) compound with an adsorbed Fe impurity [J.-X. Yin et al., Nat. Phys. 11, 543 (2015)], for which our density functional calculations show the Fe impurity is spin polarized.

Figure 1

The band structures show the gapless edge states of the coexistence of topological and superconducting orders. (a) The folded band structure of a 2-site per unit cell BZ. (b)–(e) The band structure of a strip of width of 20 lattice constants with open and close boundary conditions under a nonsuperconducting and superconducting phase. We take the periodicity along the $y$ direction of the strip, where $2000k_y$ points are taken. The inset of the middle panel of (e) is the enlargement of the crossing point of the Fermi surface, where we can find that there are totally four degenerated points around $k_y=\pm \frac{\pi }{2}$ (the green inset) and the others around $k_y=0$ and $\pm \pi$ are not degenerated.

Figure 2

Cartoon picture for an interstitial impurity in a 2D lattice. (a) The top view of the parent lattice structure and the orientation of $d_{xz}$ and $d_{yz}$ orbitals. (b) The position of an interstitial atom located at the center of a plaquette of the 2D square lattice, together with the local hopping process among the $d_{xz}/d_{yz}$ orbitals, where $t_{\sigma ,\pi }$ stand for orbital lobes aligned parallel (perpendicular) to the hopping direction between two nearest-neighboring sites of the square lattice. We set $t_{\sigma }=a{t}_{\pi }=\xi$ for the parametrization.

Figure 3

LDOS and band dispersion from the model Hamiltonian. (a) The calculated LDOS for various values of spin exchange interaction $J$ on the impurity site (blue lines) and a site deep in the bulk (red lines) as a reference. (b) The highly folded band structure for a supercell (of size $N\times N)$ lattice with a single interstitial impurity sitting in the center of each supercell. (c) The LDOS on the interstitial impurity site (red solid lines) and a site deep in the bulk (black solid lines) for ${\lambda }_{\text{AOH}}=0.0$ (top panel) and ${\lambda }_{\text{AOH}}=0.15$ (bottom panel) at a fixed $J=0.8$. (d) The spatial dependence of LDOS at zero energy in the entire supercell of size $50\times 50$. The inset is the cut from site indices $(-24,0)$ to $(25,0)$ along a bond direction through one side of the plaquette in which the interstitial impurity sits [with site coordinate $(0.5,0.5)]$. Unless specified otherwise, the order parameter values are ${\lambda }_{\text{AOH}}=0.15$ and $\mathrm{\Delta }=0.02$.

Figure 4

Band structure from the model Hamiltonian. (a) ${\lambda }_{\text{AOH}}=0$ and (b) ${\lambda }_{\text{AOH}}=0.15$. The high-symmetry momentum points of $\mathrm{\Gamma },X^{\prime \prime }$, and $M^{\prime \prime }$ are defined according to the Brillouin zone of a 1-site per unit cell. The superconducting order parameter $\mathrm{\Delta }=0.02$ is taken.

Figure 5

DFT-based local density of states. (a) The local density of states on the IFI, one of its nearest-neighboring Fe sites, far away into pristine FeSe for a nonmagnetic (upper panel) and spin-polarized IFI (lower panel). (b) The $d$-orbital-resolved partial density of states on the nonmagnetic (upper panel) and spin-polarized (lower panel) IFI site. The positive/negative value of each lower panel describes the up/down spin component.

Figure 6

Setup of a two-layered with an adsorbed Fe impurity. (a) The side/top view of the real-space geometry. The labels 1, 2, and 3 indicate the IFI, NN, and far-away sites. (b) Schematic picture showing the DFT calculated magnetic distribution (black arrow). The actual magnitude of each of them is $2.89{\mu }_B$ (IFI site), $0.23{\mu }_B$ (NN site), and $0.019{\mu }_B$ (NNN site). In each panel the red/yellow balls represent Fe/Se atoms while the blue one stands for the IFI.