Majorana Zero Modes and Topological Nature in Bi2Ta3S6-family Superconductors

Abstract

In this work, we report that Bi2Ta3S6-family superconductors exhibit nontrivial band topology. They possess a natural quantum-well structure consisting of alternating stacks of TaS2 and honeycomb Bi layers, which contribute superconducting and topological properties, respectively. Symmetry-based indicators $(\mathbb{Z}_4;\mathbb{Z}_{2}\mathbb{Z}_{2}\mathbb{Z}_{2})=(2;000)$ reveal that the topological nature arises entirely from the Bi layers, which belong to a quantum spin Hall phase characterized by a $p_x-p_y$ model on a honeycomb lattice. The topological zigzag (ZZ) and armchair (AC) edge states are obtained. Using VASP2KP, the in-plane $g$ factors of these topological edge states are computed from the ab initio calculations: $g_{x/y}^{\mathrm{ZZ}}=2.07/1.60$ and $g_{x/y}^{\mathrm{AC}}=0.50/0.06$. The strong anisotropy of the edge-state $g$ factors allows us to explore Majorana zero modes in the Bi monolayer on a superconductor, which can be obtained by exfoliation or molecular beam epitaxy. The relaxed structures of the Bi2Ta3Se6, Bi2Nb3S6 and Bi2Nb3Se6 are obtained. Their superconducting transition temperature $T_c$ are estimated based on the electron-phonon coupling and the McMillan formula. Furthermore, using the experimental superconducting gap $Δ$ and the computed $g$ factors, we obtain the phase diagram, which shows that the in-plane field $B_y>2.62\mathrm{ T}$ can generate corner Majorana zero modes in the Bi monolayer of the superconductor Bi2Ta3S6. A similar paradigm also applies to the Bi2Ta3S6 bulk with the emergence of Majorana hinge states. These natural quantum-well superconductors therefore offer ideal platforms for exploring topological superconductivity and Majorana zero modes.

Figures (5)

• Figure 1: Crystal structure and electronic band structure of Bi$_{2}$Ta$_3$S$_6$. (a, b) Top and side views of the crystal structure. The formation of S–Bi–S bonds align the S atoms in the two TaS$_2$ layers. (c) Electronic band structure without SOC. The size of the red (blue) circles indicate the Bi-$p_{x,y}$ ($p_z$) orbital components. (d) Electronic band structure with SOC, where a nontrivial gap opens at K (H) of approximately 1.0 eV.
• Figure 2: (a, b) Electronic bands (black lines) of the Bi-$p_{x,y}$ honeycomb model Eq. (\ref{['Bi-model']}) without and with SOC, which fit very well with the Bi-$p_{x,y}$ projections (red dots) from the DFT calculations. (c, d) Electronic bands of the ZZ-edge nanoribbon and the AC-edge nanoribbon with SOC, obtained from the Bi-$p_{x,y}$ honeycomb model. (e, f) DFT calculations of the ZZ-edge projections and AC-edge projections of the Bi layers in the Bi$_2$Ta$_3$S$_6$ slabs with SOC. They agree well with those of our model (highlighted by the blue dashed boxes).
• Figure 3: (a) Phonon properties and EPC for Bi$_2$Ta$_3$S$_6$. Left panel: phonon dispersion obtained from DFPT calculations including SOC. The sizes of the fatted bands are proportional to the mode- and momentum-resolved EPC strength $\lambda_{q\nu}$. Middle panel: total and atom-resolved phonon density of states (DOS). The low-frequency phonon modes are dominated by Ta and Bi atoms, while the high-frequency modes are dominated by S atoms. Right panel: the Eliashberg spectral function $\alpha^2F(\omega)$ (grey-shaded area) and the cumulative coupling constant $\lambda(\omega)$ (red curve). Most of the EPC strength are contributed by the Ta phonon modes. The effective screened Coulomb repulsion $\mu^* = 0.09$. (b) Atomic vibration patterns of the four lowest-frequency phonon modes along $\Gamma$-A (near the A point).
• Figure 4: (a) Evolution of the Zeeman gap on the ZZ edge as a function of applied in-plane magnetic fields $B_x$ and $B_y$, obtained from DFT calculations and the VASP2KP package. (b) Evolution of the Zeeman gap on the AC edge. The effective $g$ factors of the ZZ (AC) edge can be extracted from the slopes: $g_x^\mathrm{ZZ}=2.07$ and $g_y^\mathrm{ZZ}=1.60$ ($g_x^\mathrm{AC}=0.50$ and $g_y^\mathrm{AC}=0.06$).
• Figure 5: (a) A rectangle geometry of the honeycomb Bi monolayer with effective $g$ factors of the ZZ and AC edges obtained from DFT calculations. (b) Schematic plot of corner MZMs realized in Bi$_2$Ta$_3$S$_6$ superconductor. (c) Phase diagram for a $y$-directed magnetic field $B_y$ versus superconducting pairing $\Delta$, with $\alpha=1.85$ and $\mu=-106.7$ meV (which sets the ZZ-edge Dirac point at $E_F$). The orange region denotes the TSC phase that hosts corner MZMs. (d) Probability densities of the four MZMs (colored red) in a square geometry, which are localized at the four corners between the ZZ edges ($x$-directed) and the AC edges ($y$-directed). Inset: energy levels showing four MZMs. The parameters with $\Delta=0.123$ meV and $B_y=4.24$ T reside in the TSC phase and host corner MZMs.