Stabilization of Majorana Modes in Magnetic Vortices in the Superconducting Phase of Topological Insulators using Topologically Trivial Bands

It has been shown that doped topological insulators, up to a certain level of doping, still preserve some topological signatures of the insulating phase such as axionic electromagnetic response and the presence of a Majorana mode in the vortices of a superconducting phase. Multiple topological insulators such as HgTe, ScPtBi, and other ternary Heusler compounds have been identified and generically feature the presence of a topologically trivial band between the two topological bands. In this Letter we show that the presence of such a trivial band can stabilize the topological signature over a much wider range of doping. Specifically, we calculate the structure of vortex modes in the superconducting phase of doped topological insulators, a model that captures the features of HgTe and the ternary Heusler compounds. We show that, due to the hybridization with the trivial band, Majorana modes are preserved over a large, extended doping range for $p$ doping. In addition to presenting a viable system where much less fine-tuning is required to observe the Majorana modes, our analysis opens a route to study other topological features of doped compounds that cannot be modeled using the simple ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ Dirac model.

Figure 1

Band structure and orbital content of the bands of HgTe [indicated by colors: blue (black) $|{\Gamma }_8,\pm 1/2⟩$, red (dark gray) $|{\Gamma }_6,\pm 1/2⟩$, green (light gray) $|{\Gamma }_8,\pm 3/2⟩$] for $\lambda =0$ (a) and $\lambda =1$ (d). Insets show zoomed-out energy spectrum. Corresponding Berry phase on the Fermi surfaces at $k_z=0$ as a function of chemical potential for $\lambda =0$ (b) and $\lambda =1$ (e). The Berry phases are calculated by a numerical line integral around each independent Fermi surface. The dashed green and solid green lines in (e) represent the two different HH Fermi surfaces. Inset shows zoomed-out Berry phase so that the $\pi$ Berry phase of the ${\Gamma }_6$ band can be seen. (c) Exact diagaonlization results for the lowest energy vortex core states for $\lambda =0$, 1 dashed and solid lines, respectively, where dots indicate critical points. This was calculated for $k_z=0$ for a lattice size of $180\times 180$ with HgTe parameters adapted from Ref. 17and $\Sigma =0.1$, $\Delta =0.3$. (f) Valence band ${\mu }_c^{-}$ as a function of $\lambda$.