Majorana bound states in topological insulators with hidden Dirac points

We address the issue of whether it is possible to generate Majorana bound states at the magnetic-superconducting interface in two-dimensional topological insulators with hidden Dirac points in the spectrum. In this case, the Dirac point of edge states is located at the energies of the bulk states such that two types of states are strongly hybridized. Here, we show that well-defined Majorana bound states can be obtained even in materials with a hidden Dirac point provided that the width of the magnetic strip is chosen to be comparable with the localization length of the edge states. The obtained topological phase diagram allows one to extract precisely the position of the Dirac point in the spectrum. In addition to standard zero-bias peak features caused by Majorana bound states in transport experiments, we propose to supplement future experiments with measurements of charge and spin polarization. In particular, we demonstrate that both observables flip their signs at the topological phase transition, thus providing an independent signature of the presence of topological superconductivity. All features remain stable against substantially strong disorder.

Figure 1

(a) The setup consisting of a 2D TI (green slab) proximity coupled to an $s$-wave SC (yellow slab) with a magnetic strip (red slab) placed on one of edges. (b) The energy spectrum obtained numerically in a slab geometry of the TI with PBC along the $y$- and OBC along the $x$-direction. The color code refers to contributions of the $E$ and $H$ orbitals. Black arrows in the zoomed inset indicate specific positions of the chemical potential $\mu$. Nonzero parameters are chosen to be $A=365\mathrm{meV}$, $B=-686\mathrm{meV}$, $D=-512\mathrm{meV}$, $M=-40\mathrm{meV}$, ${\mu }_{\text{Edge}}=-100\mathrm{meV}$, $L_{\text{Edge}}/a=10$, $\mu =50\mathrm{meV}$, and $W/a=100$, with $a=1$ nm.

Figure 2

Topological phase diagram as a function of the Zeeman field ${\mathrm{\Delta }}_{\text{Z}}$ and the chemical potential $\mu$ found numerically for the finite-size system ($n_W=75$ and $n_L=150$). The color indicates the energy of the lowest state $E_0$. In the topological phase (blue area), the system hosts MBSs, $E_0\approx 0$. In the trivial phase (green area), the lowest state corresponds to a gapped edge or bulk state. In particular, for small ${\mathrm{\Delta }}_{\text{Z}}$, $E_0\approx \mathrm{\Delta }$. The white blurred line is the phase boundary, found numerically from a gap closing condition at $k_y=0$ with PBC. The parameters are chosen to be the same as in Fig. 1 with $W_{\text{Z}}/a=L_{\text{Z}}/a=10$. The red stars mark three points in the parameter space, upon which we elaborate more in detail in Sec. 3. With this figure, we confirm that the MBSs can emerge also in the system with hidden DPs, in which bulk and edge states are strongly hybridized at the DP. Also in this case, the phase boundary has a parabolic shape [see Eq. (7)] and its extremum can be used to find the position of the DP.

Figure 3

(a) Probability density of the lowest energy state evaluated along the $y$ edge at $x=W$ for three different positions of the chemical potential $\mu$. Only for the chemical potential tuned into the topological phase, i.e., $\mu ={\mu }_{\mathrm{DPC}}$, do the lowest energy states correspond to zero-energy MBSs (blue solid line), localized at the two ends of the magnetic strip. Outside of the topological phase, the lowest energy state is at the finite energy $E_0>0$. For ${\mu }_{\mathrm{EC}}$ (${\mu }_{\mathrm{BC}}$) shown by the orange dashed line (dotted green lines), the state is localized under the magnetic strip (spread over the entire system). (b) The 20 lowest energy states calculated for the values of the chemical potential specified in panel (a). A pair of zero-energy states emerges for $\mu ={\mu }_{\mathrm{DPC}}$. The parameters are the same as in Fig. 2 and the value of the Zeeman field is fixed to ${\mathrm{\Delta }}_{\text{Z}}/\mathrm{\Delta }=7.5$.

Figure 4

The 20 lowest eigenenergies $E_n/\mathrm{\Delta }$ as a function of Zeeman field ${\mathrm{\Delta }}_Z/\mathrm{\Delta }$ (normalized by the gap $\mathrm{\Delta }$) obtained numerically at $\mu ={\mu }_{\text{DPC}}$. The color code denotes the charge polarization ${\langle Q\rangle }_n$ [see Eq. (9)]. The length and direction of the black arrows correspond to the spin polarization along the $x$-axis ${\langle S_x\rangle }_n$. For the low-energy edge states, both observables flip their sign across the topological phase transition. As expected, the bulk states remain unaffected by the local magnetic field. The parameters are the same as in Fig. 2, and the domain $\mathcal{D}$ is chosen to be twice as wide as the magnetic strip.

Figure 5

Spatially resolved spin (charge) polarization $\langle S_x(i,j)\rangle \left[\langle Q(i,j)\rangle \right]$ determined by taking contributions of the five lowest energy states below the chemical potential. The chemical potential $\mu$ is tuned to ${\mu }_{\text{DPC}}$. Panels (a) and (c) [(b) and (d)] correspond to the trivial (topological) phase at ${\mathrm{\Delta }}_{\text{Z}}/\mathrm{\Delta }=3.5$ (${\mathrm{\Delta }}_{\text{Z}}/\mathrm{\Delta }=7.5$). We find that during the topological phase transition, both quantities undergo a sign flip close to the edge, where the dominant contribution to the total polarization is localized. In the topological phase, $\langle S_x(i,j)\rangle$ and $\left[\langle Q(i,j)\rangle \right]$ are strongly suppressed at the ends of the magnetic strip where MBSs are localized. The cross-section plots, denoted by the black dashed lines, are presented in Appendix pp3. The remaining set of parameters is the same as in Fig. 2.

Figure 6

The spectrum of the 20 lowest-lying energy states in the topological phase (${\mathrm{\Delta }}_{\text{Z}}/\mathrm{\Delta }=7.5$) as a function of the disorder strength in the chemical potential ${\sigma }_{\mu }$. The color code denotes (a) the spin polarization along the $x$ axis and (b) the charge polarization. At disorder strength of several $\mathrm{\Delta }$, the bulk gap closes. Below this value, the polarization of bulk states is well-defined and can be used to distinguish between topological and trivial phases. All nonzero parameters are the same as in Fig. 2.

Figure 7

Spatially resolved spin (charge) polarization $\langle S_x(i,j)\rangle \left[\langle Q(i,j)\rangle \right]$ in the presence of disorder in the chemical potential is shown in panels (a,b) [(c,d)]. (a,c) In the weak disorder regime, the bulk gap is still open and the polarization patterns are similar to the ones in the clean case. (b,d) Such features disappear in the strong disorder regime where the bulk gap is closed. The set of parameters is the same as in Fig. 6.

Figure 8

Topological phase diagram as a function of ${\mathrm{\Delta }}_{\text{Z}}$ and the chemical potential $\mu$ obtained for different widths of the magnetic strip: (a) $W_{\text{Z}}/a=15$, (b) $W_{\text{Z}}/a=17$, (c) $W_{\text{Z}}/a=20$, and (d) $W_{\text{Z}}/a=30$. We find that the unique features associated with the topological phase at the system edge get smeared out as a result of the increased coupling between the edge and the bulk states as well as a result of the suppression of the superconductivity induced in the bulk states at the edge. The wider the magnetic strip, the stronger are these effects. We use the red lines to outline the region of the phase diagram, where the bulk effects dominate and the edge features of the topological phase get lost. The rest of the parameters are the same as in Fig. 2 as well as the color code.

Figure 9

The probability density of the four lowest energy states at the DP. For ${\mu }_{\text{Edge}}=0\mathrm{meV}$ (dashed blue), the DP is inside the TI bulk gap. The corresponding wave functions are clearly localized on the right and left edges and are clearly separated from bulk states. For ${\mu }_{\text{Edge}}=100\mathrm{meV}$ (solid orange), the DP is hidden. The wave functions have contributions from both bulk and edge states, which demonstrates strong hybridization between them. The parameters are the same as in Fig. 1.

Figure 10

Topological phase diagram of a system with a nonhidden DP (${\mu }_{\text{Edge}}=0$) as a function of the Zeeman field ${\mathrm{\Delta }}_{\text{Z}}$ and the chemical potential $\mu$ found numerically for the finite-size system. The color indicates the energy of the lowest state $E_0$. The white blurred line is the phase boundary, found analytically by using Eq. (3) of Ref. [35] for the $g$-factor. We note that in this regime the effective $g$-factor of the edge states is twice as small ($\alpha \approx 8$) as the one we found in Fig. 2 for the system with hidden DP, which results in higher values of the critical magnetic field needed to enter the topological phase. The remaining parameters are the same as in Fig. 2 of the main text.

Figure 11

The probability density for the two zero-energy modes from Fig. 3. The inset shows a zoom into the MBS at the upper right edge, close to the ends of the magnetic strip. The MBSs are localized at the end of the magnetic strip and extend the most along the edge in the $x$-direction. The nonzero parameters are the same as in Fig. 3.

Figure 12

Cross-section of the spin (left panels) and charge (right panels) polarizations for (a,b) a hidden DP and (c,d) a nonhidden DP. The Zeeman field varies from $0.25\mathrm{\Delta }$ to $10{\mathrm{\Delta }}_{\text{Z}}$ across the topological phase transition. The trivial phase is shown in blue, while the topological phase is shown in red. The sign-flip of the polarization is most pronounced in the spin polarization and observed most strongly directly under the magnetic strip and in the case of a nonhidden DP. The hybridization between bulk and edge states suppresses the spin-flip away from the strip. The black dashed lines mark the edge of the magnetic strip. The rest of the parameters are the same as in Fig. 5.