Superconductivity-induced ferromagnetism and Weyl superconductivity in Nb-doped ${\mathbf{Bi}}_2{\mathbf{Se}}_3$

A recent experiment on Nb-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ showed that zero field magnetization appears below the superconducting transition temperature. This gives evidence that the superconducting state breaks time-reversal symmetry spontaneously and is possibly in the chiral topological phase. This is in sharp contrast to the Cu-doped case which is possibly in the nematic phase and breaks rotational symmetry spontaneously. By deriving the free energy of the system from a microscopic model, we show that the magnetic moments of the Nb atoms can be polarized by the chiral Cooper pairs and enlarge the phase space of the chiral topological phase compared to the nematic phase. We further show that the chiral topological phase is a Weyl superconducting phase with bulk nodal points which are connected by surface Majorana arcs.

Figure 1

Phase diagrams of doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$; the blue region denotes the chiral phase while the green region denotes the nematic phase. (a) The case for $J{\mu }_d=0$ which is relevant to the Cu-doped case when ${\mu }_d=0$. If only the ${\mathrm{\Gamma }}_0$ terms in Eqs. (3) and (4) are finite as assumed in Ref. [29], the system is always in the nematic phase with ${\lambda }_a=-{\lambda }_b,{\lambda }_c=0$, as shown by the dashed line. (b) The case for finite $J{\mu }_d$ which is relevant to the Nb-doped case. The chiral phase regime is enlarged.

Figure 2

(a) The schematic top view (from the $z$ direction) of nodal points of the chiral superconductor in the Brillouin zone. The four nodes shown are located near the north pole, while their inversion partners (not shown) are the other four nodal points near the south pole. In the general case, the Majorana arcs connecting nodes with opposite chiralities emerge on the surface of superconductors as shown in (b) and (c). (b) Majorana arcs on the $xz$ plane. (c) Majorana arcs on the $yz$ plane. (d) The Majorana arcs on the $xz$ plane in the special case with ${\mathrm{\Gamma }}_0={\mathrm{\Gamma }}_{xy}={\mathrm{\Gamma }}_z=0$ and $\mathrm{\Gamma }=0.9$ in Eq. (3) that four nodal points merge into one in the north and south poles, respectively. The signs $\pm$ and numbers $\pm 2$ denote the chirality of nodes. In (b)–(d), the surface spectral function $A$ at zero energy is calculated using the Hamiltonian in Eq.(12), with the parameters $v=\sqrt{3}{v}_z=5\sqrt{3},m=3,\mu =4,JM=0.2,\mathrm{\Delta }=1$. For (b) and (c), ${\mathrm{\Gamma }}_0=0.9,\mathrm{\Gamma }=0.2,{\mathrm{\Gamma }}_{xy}=0.5,{\mathrm{\Gamma }}_z=0.1$.