Chiral superconductivity in thin films of doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

Recent experimental evidences point to rotation symmetry-breaking superconductivity in doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, where the relevant order parameter belongs to a two-component odd-parity representation $E_u$ of the crystal point group. The $E_u$ channel admits two possible phases, the nematic phase, that well explains the reported rotation symmetry breaking, and the chiral phase, that breaks time-reversal symmetry. In weakly anisotropic three-dimensional (3D) systems the nematic phase is the stable one. We study the stability of the nematic phase versus the chiral phase as a function of the anisotropy of the Fermi surface and the thickness of the sample and show that by increasing the 2D character of the Fermi surface or by reducing the number of layers in thin slabs the chiral phases is favored. For the extreme 2D limit composed by a single layer of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ the chiral phase is always the stable one and the system hosts two chiral Majorana modes flowing at the boundary of the system.

Figure 1

Variation of the coefficient ratio $b_2/b_1$ as a function of the anisotropy of the system parametrized by $v_z/v$ and the chemical potential $\mu$, keeping the parameter ${\alpha }_z$ fixed. For sufficiently small $v_z/v$ the coefficient $b_2$ becomes negative $\left(b_1>0\right)$ and the chiral phase becomes possible.

Figure 2

Variation of the coefficient $b_2$ as a function of the number of layers $N$ for different chemical potential. The parameter used in the tight-binding model are $t_0=0,t_1=-4.1\mathrm{eV},t_2=0.75\mathrm{eV},t_z=-0.6\mathrm{eV},\lambda =0.5\mathrm{eV},a=5.0\AA$, and $a_z=9.1\AA$.

Figure 3

Bands structure of a TI single layer with chiral superconductivity on a ribbon geometry. The parameters used are $t_0=-0.1,t_1=-5t_2,t_2=1.0,\lambda =0.5,\mathrm{\Delta }=0.1$, and $\mu =0.2$. Two Majorana edge states copropagate on each side of the ribbon (blue on one side, red on the other side).