Disorder-induced subgap states and Majorana zero-energy edge modes in two-dimensional topological insulator-superconductor hybrid structures

Contrary to the widespread belief that Majorana zero-energy modes, existing as bound edge states in two-dimensional topological insulator (TI) -superconductor (SC) hybrid structures, are unaffected by nonmagnetic static disorder by virtue of time-reversal symmetry, we show that such a protection against disorder does not exist in realistic multichannel TI/SC/ferromagnetic insulator (FI) sandwich structures of experimental relevance since the time-reversal symmetry is explicitly broken locally at the SC/FI interface where the end Majorana mode (MM) resides. We find that although the MM itself and the bulk topological superconducting phase inside the TI are indeed universally protected against disorder, extra fermionic subgap states are generically introduced at the TI edge due to the presence of the FI/SC interface as long as multiple edge channels are occupied. We discuss the important implications of our finding for the detection and manipulation of the edge MM in TI/SC/FI experimental systems of current interest.

Figure 1

(a) Schematic picture of the devices: a TI in contact with SCs and an FI. The position of MM is marked with a red dot. The normal metal N acts as an external lead and the FI acts as a tunnel barrier. The MM is then detected as a zero-bias conductance peak. (b) At the interface between the metallic SC and the narrow-gap semiconductor (i.e., the TI), in general band-bending effect at the interface leads to extra edge channels (hashed region).

Figure 2

(a) LDOS at the end of a SM nanowire, where the dashed red line shows the result for a clean wire while the solid blue line is for a disordered wire. The parameters used for the BdG Hamiltonian $H=(\frac{p^2}{2m}+\alpha p{\sigma }_y-\mu ){\tau }_z+B{\sigma }_z+\Delta {\tau }_x$ are $\mu =0$, $B=2\Delta$, and $m{\alpha }^2=2\Delta$. (b) LDOS deep in the SC region of the TI edge, with disorder in SC region. (c) LDOS at the SC/FI interface on a clean TI. (d) LDOS at the SC/FI interface on a TI edge with disorder in SC region.

Figure 3

(a),(b) Band structures of respectively a five-channel and three-channel TI edge. Here the parameters of Eq. (1) are chosen as $\mu =\mathrm{diag}\left(2,-2,1,-1,0\right)\Delta$, ${\tilde{\mu }}^x={\tilde{\mu }}^y=0$, ${\tilde{\mu }}_{\alpha \gtrless \beta }^z=\pm 2\Delta$, and ${\mathbf{p}}_{\alpha }=\left(+,-,+,-,+\right)\widehat{z}$. The Fermi level in the FI region ($E_F=0$) is shown in dashed line, while that in the SC region ($E_F=3\Delta$) is shown in dotted line. (c),(d) Their corresponding LDOS at the SC/FI interface, without disorder. (e),(f) Their corresponding LDOS at the SC/FI interface, with a single realization of nonmagnetic disorder.

Figure 4

(a),(b) LDOS at the SC/FI interface, averaged over 5000 realizations of nonmagnetic disorder, for a five-channel and three-channel TI edge, respectively. The three curves show the results for different strengths of disorder, and the MMs have been removed for clarity. (c),(d) Their corresponding averaged LDOS deep in the SC region.