Topological Superconductivity on the Surface of Fe-Based Superconductors

As one of the simplest systems for realizing Majorana fermions, the topological superconductor plays an important role in both condensed matter physics and quantum computations. Based on ab initio calculations and the analysis of an effective 8-band model with superconducting pairing, we demonstrate that the three-dimensional extended $s$-wave Fe-based superconductors such as ${\mathrm{Fe}}_{1+\mathrm{y}}{\mathrm{Se}}_{0.5}{\mathrm{Te}}_{0.5}$ have a metallic topologically nontrivial band structure, and exhibit a normal-topological-normal superconductivity phase transition on the (001) surface by tuning the bulk carrier doping level. In the topological superconductivity (TSC) phase, a Majorana zero mode is trapped at the end of a magnetic vortex line. We further show that the surface TSC phase only exists up to a certain bulk pairing gap, and there is a normal-topological phase transition driven by the temperature, which has not been discussed before. These results pave an effective way to realize the TSC and Majorana fermions in a large class of superconductors.

Figure 1

(a) Side view of the crystal structure of FST. (b) Top view of FST, where two Fe sites are marked by the brown circles, while two Se (Te) sites are marked by the light blue circles, with $\pm z_0$ labeling their height from the Fe plane. (c) The first BZ of FST with symmetry $P4/nmm$ and six types of inequivalent TRIPs. The light blue square shows the 2D BZ of the projected (001) surface, in which the high-symmetry $k$ points $\overline{\mathrm{\Gamma }}$, $\overline{X}$, and $\overline{M}$ are labeled. (d)–(f) The low energy dispersions of FST with SOC around the $\mathrm{\Gamma }$ and $Z$ points. Red lines are the band structures calculated by the DFT calculations, while the blue dashes are the results from our effective model fitting. (g) The surface states calculated by the effective Hamiltonian and the parameters listed in Table I in the Supplemental Material [39]. In all figures, 0 eV corresponds to the Fermi level of the stoichiometric FST.

Figure 2

(a)–(d) The schematic evolution of the surface MZMs in a vortex line. As the chemical potential is tuned from the trivial regime (a) towards the surface TSC regime, two Majorana zero modes arise [(b)] and then become more and more localized at the ends of the vortex line [(c) and (d)]. (e) The energy spectrum at the $\mathrm{\Gamma }$ point of a vortex line along the $z$ direction as a function of the chemical potential $\mu$. The energy gap closes at ${\mu }_1=31$ and ${\mu }_2=62\mathrm{meV}$, respectively, showing the (001) surface is a TSC in the range $\mu \in [{\mu }_1,{\mu }_2]$. (f) The energy spectrum at the $Z$ point of the vortex line as a function of the chemical potential $\mu$. There is no gap closing at the $Z$ point, so the phase transitions are solely determined by the gap closing at the $\mathrm{\Gamma }$ point. In all figures, chemical potential $\mu =0\mathrm{eV}$ corresponds to the Fermi level of the stoichiometric FST.

Figure 3

Low energy dispersions of the vortex line for the TSC phase [(a) and (d)], the transition point [(b) and (e)] and the NSC phase [(c) and (f)], respectively. Here $\mu =50$ and $\mu =70\mathrm{meV}$ are chosen to represent the TSC and NSC phase, while the results of the transition point are calculated at $\mu =62\mathrm{meV}$. (d),(e), and (f) are the enlargement of (a),(b), and (c) to show the gap clearly.

Figure 4

TSC phase space vs bulk superconducting gap. The red and blue lines are the upper and lower phase boundaries of the TSC phase, respectively. The gray dash at $\mu =35\mathrm{meV}$ indicates a TSC at 0 K ($\mathrm{\Delta }={\mathrm{\Delta }}_{\exp }$) to NSC ($\mathrm{\Delta }\sim 0$) phase transition with temperature increasing. The inset shows an evolution of the TSC region (red line minus blue line) with respect to the bulk pairing gap.