Quantized charge transport in chiral Majorana edge modes

Majorana fermions can be realized as quasiparticles in topological superconductors, with potential applications in topological quantum computing. Recently, lattices of magnetic adatoms deposited on the surface of $s$-wave superconductors—Shiba lattices—have been proposed as a new platform for topological superconductivity. These systems possess the great advantage that they are accessible via scanning-probe techniques and thus enable the local manipulation and detection of Majorana modes. Using a nonequilibrium Green's function technique we demonstrate that the topological Majorana edge modes of nanoscopic Shiba islands display universal electronic and transport properties. Most remarkably, these Majorana modes possess a quantized charge conductance that is proportional to the topological Chern number, $\mathcal{C}$, and carry a supercurrent whose chirality reflects the sign of $\mathcal{C}$. These results establish nanoscopic Shiba islands as promising components in future topology-based devices.

Figure 1

(a) Sketch of a Shiba island with STS tip. The LDOS is shown for $\mu =-4t$ ($\mathcal{C}=-1$) in (b),(d), for $\mu =-2t$ ($\mathcal{C}=0$) in (e), and for $\mu =0$ ($\mathcal{C}=+2$) in (c),(f). Panels (d)–(f) display the energy-resolved spectra at the location marked by arrows in (b),(c); equally spaced in-gap states are clearly visible in the topological phases in (d) and (f). Panels (b),(c) show the spatial weight distribution (color coded) of the low-energy in-gap states marked by arrows in (d),(f); the insets display the corresponding normal-state Fermi surfaces. Parameters here and below are $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$.

Figure 2

Differential conductance $G=dI/dV$ vs chemical potential $\mu$ for the finite Shiba lattice (red) and the Shiba island (black), measured in a low-energy window; see text. $G$ not only distinguishes between the different phases but is even quantized to a remarkable accuracy as $G=\frac{2e^2}{h}\left|\mathcal{C}\right|$. Here, $t_{\mathrm{tip}}=0.02t$. Inset: real-space Chern number normalized by coverage, $\tilde{\mathcal{C}}$, demonstrating the topological character of the finite-size system; see Appendix pp4 for details.

Figure 3

Induced superconducting triplet correlations $\langle c_{\mathbf{r},\sigma }^{†}{c}_{{\mathbf{r}}^{\prime },\sigma }^{†}\rangle$ for $\sigma =\uparrow ,\downarrow$. The upper row shows the sign of the real part and the lower row the sign of the imaginary part of the correlations — positive (negative) sign is show in blue (red). The parameters are $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$, and $\mu =-0.3t$ corresponding to the $\mathcal{C}=2$ phase. Shown are just quarters of the Shiba island with radius $R=15a_0$.

Figure 4

Spatial distribution of the supercurrents ($V=0$) carried by the lowest energy edge modes in the (a) $\mathcal{C}=-1(\mu =-4t)$ and (b) $\mathcal{C}=+2(\mu =0)$ phases. The direction of the current flow depends on $\mathrm{sgn}\left(\mathcal{C}\right)$ (see insets) with the currents flowing clockwise in (a) and counterclockwise in (b).

Figure 5

$\pi$-flux threaded through the center of the Shiba island shifts states $E=\pm \epsilon /2$ (black) to almost zero energy (purple), with the remaining splitting being due to the small island size; the LDOS was measured at the location indicated by the red arrow. The spatial LDOS profile corresponds to one of the almost zero-energy state (arrow) and reveals that spectral weight is bound to the $\pi$ flux as well as to the island edge, reflecting the presence of two Majorana zero modes for large islands.

Figure 6

Disordered Shiba island of magnetic adatoms with no rotational or mirror symmetries. (a) Energy-resolved LDOS in the $\mathcal{C}=-1$ phase $(\mu =-4t)$. (b) Spatial LDOS of the lowest-energy edge mode shown in (a). (c) Supercurrent carried by the lowest-energy edge mode shown in (a). (d) Differential conductance, $G$, and (e) modified Chern number $\tilde{\mathcal{C}}$, as a function of $\mu$. Parameters used: $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$.

Figure 7

(a)–(c) Spin-resolved LDOS at the edge of the Shiba island in the $\mathcal{C}=-1(\mu =-4t$), $\mathcal{C}=0(\mu =-2t$), and $\mathcal{C}=2(\mu =0$) phases, respectively, for a set of parameters $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.8,1.2,2.0)t$. Total LDOS for the lowest-energy topological edge modes in the topological (d) $\mathcal{C}=-1(\mu =-4t$) and (e) $\mathcal{C}=2(\mu =0$) phases. Supercurrents carried by the lowest-energy topological edge modes in the topological (f) $\mathcal{C}=-1(\mu =-4t$) and (g) $\mathcal{C}=2(\mu =0$) phases.

Figure 8

(a) Schematic picture of a finite Shiba lattice with STS tip. Differential conductance, $G=dI/dV$, and current-voltage dependence $I\left(V\right)$ for a finite Shiba lattice of size $N_x=N_y=41$ in the (b) $\mathcal{C}=2$ ($\mu =0$), (c) $\mathcal{C}=0$ ($\mu =-2t$), and (d) $\mathcal{C}=-1$ ($\mu =-4t$) phases. Here, $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$ and $t_{\mathrm{tip}}=0.2t$. The light blue shaded area indicates the range of $\mathrm{\Delta }E$.

Figure 9

Real-space Chern number (RSCN) as a function of $\mu$ for a fully covered $41\times 41$ Shiba lattice with periodic boundary conditions. Parameters used: $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$ leading to ${\mu }_c^{\left(1\right)}=-0.4t$ and ${\mu }_c^{\left(2\right)}=-3.6t$.

Figure 10

Real-space Chern number (RSCN) as a function of coverage, i.e., the ratio of sites covered by magnetic adatoms and the total number of sites, for $\mu =0$ corresponding to the $\mathcal{C}=2$ phase. The lattice is a $41\times 41$ superconductor with periodic boundary conditions where a stripe of magnetic adatoms (blue) or an island of adatoms (red) is deposited. In addition, a covered system is considered where an island of adatoms is missing, yielding a hole (green). Parameters used: $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.8,1.2,2.0)t$.

Figure 11

Real-space Chern number divided by coverage, $\tilde{\mathcal{C}}$, as a function of $\mu$ which reproduces the phase diagram of an infinitely large Shiba lattice [18]. Results correspond to a Shiba island with a diameter of 30 atoms on a $41\times 41$ superconducting square lattice. In addition to the results for the Shiba island (red), we also present $\tilde{\mathcal{C}}=\mathcal{C}$ for the fully covered system (coverage = 1) with periodic boundary conditions (black) and open boundary conditions (blue). Parameters used: $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$.

Figure 12

Cylinder spectra on a ribbon consisting of 100 unit cells. Top row: $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.2,0.3,0.5)t$. Bottom row: $(\alpha ,{\mathrm{\Delta }}_s,J)=(0.8,1.2,2.0)t$. Systems with Chern numbers $\mathcal{C}=-1,0$, and 2 correspond to $\mu /t=-4,-2$, and 0, respectively.