Robustness of topological superconductivity in proximity-coupled topological insulator nanoribbons

We study the low-energy physics of topological insulator (TI) nanoribbons proximity-coupled to $s$-wave superconductors (SCs) by explicitly incorporating the proximity effects that emerge at the TI-SC interface. We construct a low-energy effective theory that incorporates the proximity effect through an interface contribution containing both normal and anomalous terms and an energy-renormalization matrix. We show that the strength of the proximity-induced gap is determined by the transparency of the interface and the amplitude of the low-energy TI states at the interface. Consequently, the induced gap is strongly band-dependent and collapses for bands containing states with low amplitude at the interface. We find that states with energies within the bulk TI gap have surface-type character and, in the presence of proximity-induced or applied bias potentials, have most of their weight near either the top or the bottom surface of the nanoribbon. As a result, single-interface TI-SC structures are susceptible to experiencing a collapse of the induced gap whenever the chemical potential is far enough from the value corresponding to the bulk TI Dirac point and crosses weakly coupled bands. We also find that changing the chemical potential in single-interface structures using gate potentials may be ineffective, as it does not result in a significant increase of the induced gap. On the other hand, we find that symmetric structures, such as a TI nanowire sandwiched between two superconductors, are capable of realizing the full potential of TI-based structures to harbor robust topological superconducting phases.

Figure 1

Comparison between the low-energy spectrum of the 3D continuum model given by Eq. (2) [orange (light gray) lines] and that of the tight-binding model (3) (black dots). The parameters of the lattice model were optimized, starting from the values given by Eq. (4), to capture the correct dispersion along the $\Gamma \text{−}Z$ direction in the Brillouin zone, which controls the intersubband spacing in the spectrum of a TI thin film (see Fig. 2). The relevant parameters, corresponding to ${\mathrm{Bi}}_2$${\mathrm{Se}}_3$, are $({\epsilon }_{+},{\epsilon }_{-},t_{1+},t_{1-},t_{2+},t_{2-},{\alpha }_1,{\alpha }_2)$ $=$ $\left(17.6,-3.26,-0.11,0.06,-2.88,0.53,0.04,0.26\right)$. The inset shows the structure of the rhombohedral lattice (of lattice constants $a=4.138$ Å and $c=28.64$ Å) used in the construction of the tight-binding model. The triangular sublattices correspond to ${\mathrm{Bi}}_2$${\mathrm{Se}}_3$ quintuple layers and are staked in the $z$ direction.

Figure 2

Spectrum of a TI slab with $N_z$ quintuple layers staked along the $z$ direction. The Dirac-cone-like feature (top panel) corresponds to the topologically protected gapless surface states. Applying a magnetic field parallel to the slab (middle and bottom panels) removes the double degeneracy of the surface states and shifts the corresponding Dirac cones. The magnitude of this change depends on the thickness of the slab (i.e., on $N_z)$, as it is dominated by the orbital effect (see the main text). The effective $g$ factors corresponding to the Zeeman in Eq. (6) are [63] $(g_{+z},g_{-z},g_{+\parallel },g_{-\parallel })=(-25.4,-4.12,4.1,4.8)$.

Figure 3

Low-energy spectrum of a TI nanoribbon with rectangular cross section $L_x\times L_z=103\times 9$ nm (left) and a schematic representation of typical proximity-coupled nanostructures studied in this work (right). All the bands shown in the left panel are doubly degenerate. The bands with energies within the bulk gap result from confining the TI surface states (see Fig. 2) along the $x$ direction. Notice the small gap near zero energy. The cross-sectional views in the right panel show the main components of a TI-based hybrid structure, including back/top gates (structures c and d) for controlling the chemical potential. Note that interface-induced bias potentials generated by the proximity-coupled bulk SC or the substrate [59] (not shown) are always present in nonsymmetric structures (e.g., structure a), even in the absence of applied gate potentials.

Figure 4

Left: Low-energy nanoribbon spectrum in the presence of an external magnetic field parallel to the wire. The total flux through the ribbon is $\Phi =0.583{\Phi }_0$, which realizes the $k=0$ degeneracy condition for the lowest energy bands (see the main text). Note that for this value of $\Phi$, the higher-energy bands are not exactly degenerate at $k=0$. Right: Low-energy spectrum in the presence of a bias potential with $V\left(i\right)$ given by Eq. (20) and $V_{\max }=0.05$ eV. The spatial profiles of the states marked by letters (A–E) are shown in Figs. 5 (left panel) and 6 (right).

Figure 5

Typical transverse profiles $|{\psi }_n{(x,z)|}^2$ for low-energy states in TI nanoribbons. Yellow (light gray) regions correspond to the maxima of the wave functions. Panel A shows a bulk-type state with energy $E_{\mathrm{A}}=0.24$ eV, while panel E corresponds to a valence bulk-type band with $E_{\mathrm{E}}=-0.22$ eV (see Fig. 4). The surface-type bands with energies within the bulk gap (see Fig. 4) are characterized by wave functions with maxima near the boundaries of the system (panels B, C, and D).

Figure 6

The effect of a bias potential on the spatial distribution of low-energy wave functions. While the bulk-type states (A and E) are only slightly distorted by the applied potential, the surface-type states become practically localized near the top (panel B) or the bottom (panels C and D) surfaces (see Fig. 4). Proximity-coupling the TI ribbon to a bulk SC placed on the top (see Fig. 3, structure a) will result in a large proximity-induced gap in band B, but vanishingly small gaps in bands C and D.

Figure 7

Proximity-induced quasiparticle gap characterizing the low-energy spectrum of the TI nanoribbon–superconductor hybrid structure (b) from Fig. 3. The bulk pair potential is ${\Delta }_0=1.5$ meV, ${\varphi }_{\text{SC}}=0$, the TI-SC coupling strength is $\gamma =4{\Delta }_0$, and $\xi =0.5$. Top: Induced gap for $B=0$ and ${\mu }_{\text{TI}}=0.046$ eV. Bottom: Nonzero magnetic field corresponding to $\Phi =0.8{\Phi }_0$ and chemical potential ${\mu }_{\text{TI}}=-0.086$ eV. Note that, in the absence of time-reversal symmetry $\left(B\ne 0\right)$, the spectrum has particle-hole symmetry, $E(-k)=-E\left(k\right)$, but $E(-k)\ne E\left(k\right)$. As a result, the SC state corresponding to the lower panel is gapless.

Figure 8

Phase diagram for a SC–TI nanoribbon–SC structure (see Fig. 3) as a function of the magnetic flux and the chemical potential. The green (light gray) regions correspond to a topological superconducting phase, while for parameters within the white areas the system is topologically trivial. The phase boundaries have a characteristic zigzag shape for ${\mu }_{\text{TI}}$ within the bulk TI gap (note that the region $0.1<{\mu }_{\text{TI}}<0.2$ is not shown) and breaks down when $\mu$ reaches the bulk-type bands $({\mu }_{\text{TI}}>0.22$ eV). The evolution of the quasiparticle gap along the cuts corresponding to the black lines is discussed in Sec. 3c. Parameters: $L_x\times L_z=60\times 9.5$ nm, $\gamma =8{\Delta }_0,\xi =0.5$, and ${\varphi }_{\text{SC}}=0$.

Figure 9

Phase boundaries for proximity-coupled nanoribbons of thickness $L_x=40$ nm (black line), 60 nm (dots), and 80 nm [orange (light gray) line]. All other parameters are the same as in Fig. 8. The chemical potential difference $\Delta {\mu }_{\text{TI}}$ between two maximum width values is controlled by the interband spacing. The minimum width (of the topological phase) increases with $L_x$, and the “center line” of the topological phase (half-distance between the phase boundaries at any given value of ${\mu }_{\text{TI}})$ shifts toward $\Phi =0.5{\Phi }_0$.

Figure 10

Top: Phase boundaries in the vicinity of a minimum width of the topological region for $\gamma =4{\Delta }_0$ (black) and $\gamma =8{\Delta }_0$ [orange (light gray)]. Bottom: Dependence of the minimum width ${\Lambda }_{\Phi }$ on the strength of the TI-SC coupling.

Figure 11

Top: Dependence of the minimum width on the ratio $\xi ={\tilde{t}}_{-}/{\tilde{t}}_{+}$ of the interface hopping matrix elements for the antisymmetric and symmetric molecular orbitals (see Sec. 2c). Bottom: Dependence of the minimum width on the phase difference between the bulk SCs for $\gamma =8{\Delta }_0$ (blue) and $\gamma =4{\Delta }_0$ [orange (light gray)]. Notice the oscillatory behavior and the nearly zero values of ${\Lambda }_{\Phi }$ for ${\varphi }_{\text{SC}}\approx (0.8+2n)\pi$.

Figure 12

Quasiparticle gap as a function of the chemical potential corresponding to the vertical cut through the topological phase shown in Fig. 8. The system, corresponding to the SC-TI-SC structure (b) in Fig. 3, is characterized by the following parameters: $L_x\times L_z=60\times 9.5$ nm, $\gamma =4{\Delta }_0,\xi =0.5,{\varphi }_{\text{SC}}=0$.

Figure 13

Quasiparticle gap as a function of the magnetic flux for different values of the chemical potential corresponding to the horizontal cuts in Fig. 8. The model parameters are the same as in Fig. 12. The shaded regions correspond to the topological superconducting phase. The quasiparticle gap has a characteristic V-type shape in the vicinity of a topological quantum phase transition and vanishes at the transition. Note that for some values of the chemical potential (e.g., $\mu =-0.086$ eV), the vanishing of the gap is not associated with a phase transition.

Figure 14

Quasiparticle gap as a function of the chemical potential for a TI-SC system with one interface—structure (a) in Fig. 3—and bias potential $V_{\max }=0$ (thin line and dots), $V_{\max }=0.03$ eV [orange (light gray) line], and $V_{\max }=0.06$ eV (black). Unlike the case of a symmetric SC-TI-SC structure (see Fig 12), the range of chemical potential corresponding to an induced gap larger than the “reference” value $(0.25$ meV, shaded area) is rather small and corresponds to low band occupancy (single-band occupancy for $V_{\max }=0.03$ eV). Outside this range, the gap practically collapses with increasing bias potential.

Figure 15

Cross-section profile of the applied gate potential corresponding to hybrid structure (c) from Fig. 3 (panel A) and structure (d) (panel C). The yellow (light gray) color represents the gate potential $V_{\mathrm{gate}}$, while the dark region corresponds to the value of the potential at the TI-SC interface, $V\left(i_0\right)=0$. Panels B and D show the real-space structure of some representative low-energy states supported by the TI ribbon in the presence of the gate potential. For the single-gate structure (panel A), some states are localized near the TI-SC interface, while others (see panel B) are away from the interface. For the two-gate structure (panel C), all states have some weight near the two TI-SC interfaces (panel D).

Figure 16

Dependence of the induced quasiparticle gap on the applied gate potential for a single-gate structure with a potential profile like that in Fig. 15, panel A (double thin line), and a double-gate structure with potential profile given in Fig. 15, panel C [full blue (dark gray) line]. The chemical potential is fixed at the value ${\mu }_{\text{TI}}=0.22$ eV, close to the bulk TI gap edge. Applying a gate potential shifts the spectrum up in energy, but it also distorts it. For the single-gate structure, at least one weakly coupled band remains occupied and limits the size of the induced gap.