Second-Order Topological Superconductivity in $\pi$-Junction Rashba Layers

We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom $s$-wave superconductor with phase difference $\varphi$ close to $\pi$. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If $\varphi =\pi$, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference $\varphi$ deviates from $\pi$ by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference $\varphi$ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling $\varphi$ by magnetic flux, the other involves an additional layer of randomly oriented magnetic impurities responsible for the phase shift of $\pi$ in the proximity-induced superconducting pairing.

Figure 1

Schematics of the bilayer setup consisting of two 2DEG Rashba layers (blue) separated by a tunnel barrier (orange) and coupled to $s$-wave superconductors (green) that form a Josephson junction. The two Rashba SOI vectors are aligned in opposite directions due to the structural asymmetry. A magnetic flux ensures a phase difference in the proximity-induced superconductivity between two layers. Alternatively, the phase difference of $\pi$ can be induced by an additional layer of randomly oriented magnetic impurities placed between one of the SCs and the Rashba layer.

Figure 2

Topological phase diagram of the $\pi$-junction bilayer as a function of tunnel amplitude $\mathrm{\Gamma }$ and Zeeman energy ${\mathrm{\Delta }}_Z$. Topological phase transitions occur at $\mathrm{\Gamma }=|{\mathrm{\Delta }}_Z\pm \mathrm{\Delta }|$ (black lines). For ${\mathrm{\Delta }}_Z=0$ and $\mathrm{\Gamma }>\mathrm{\Delta }$, the system is in the HTSC phase (green line) hosting a Kramers pair of helical edge states. For ${\mathrm{\Delta }}_Z<\mathrm{\Delta }$ and $\mathrm{\Gamma }>|\mathrm{\Delta }+{\mathrm{\Delta }}_Z|$, the system is in the SOTSC phase (red) hosting at two of the corners each a MCS. For $|{\mathrm{\Delta }}_Z-\mathrm{\Delta }|<\mathrm{\Gamma }<|{\mathrm{\Delta }}_Z+\mathrm{\Delta }|$, the system is in the Weyl SC phase (blue) hosting flat zero-energy edge states. Otherwise, the system is in the trivial phase (white).

Figure 3

(a) Energy spectrum in the HTSC phase ($\mathrm{\Gamma }>\mathrm{\Delta }$) for semi-infinite geometry, obtained by exact diagonalization. The bulk states (green dots) are gapped, while both edges host a Kramers pair of dispersive helical edge states (orange dots). (b) Color plot of the probability density $|\psi (x,y){|}^2$ of a helical edge state in the finite geometry. The states are exponentially localized at the edge (yellow strips). Numerical parameters: $k_{\mathrm{so}}{L}_x=k_{\mathrm{so}}{L}_y\approx 860$, $\mu =0$, $\mathrm{\Delta }/E_{\mathrm{so}}\approx 0.2$, $\varphi =\pi$, and $\mathrm{\Gamma }/E_{\mathrm{so}}\approx 0.6$.

Figure 4

Color plots of probability density and spectrum (insets) of low-energy states in SOTSC phase. (a) If the field direction $\mathbf{n}$ has projection on both edges as well as $|\delta \varphi |<\min \{\delta {\varphi }_{c,\parallel },\delta {\varphi }_{c,\perp }\}$ (here, $\theta =\pi /3$ and $\delta \varphi =0$), the zero-energy MCSs (yellow dots in inset) are separated from bulk states (blue dots in inset) by the gap and are located at two opposite corners. For $0<\theta <\pi /2$, the sign of the gap ${\mathrm{\Delta }}_s$ on the $s=0$, 1 ($s=2$, 3) edge is positive (negative) and shown in blue (red): thus, the lower left (upper right) corner acts as a domain wall at which MCSs are localized. (b) If $|\delta \varphi |=\delta {\varphi }_{c,\parallel /\perp }$ (here, $\theta =0$, and $\delta \varphi =\delta {\varphi }_{c,\perp }=0$), the helical edge states propagating along the corresponding edge stays gapless. The other edges are still gapped. (c) A trivial square region in the center (of the size $L_x/2\times L_x/2$) leads to two additional MCSs localized at the inner corners. (d) If $\min \{\delta {\varphi }_{c,\parallel },\delta {\varphi }_{c,\perp }\}<|\delta \varphi |<\max \{\delta {\varphi }_{c,\parallel },\delta {\varphi }_{c,\perp }\}$ (here $\theta =\pi /3$ and $\delta \varphi \approx \pi /3$), only the sign of the gap on the $s=1$ edge is positive. As a result, the two MCSs are located now at two neighboring corners. Numerical parameters are the same as in Fig. 3, except $k_{\mathrm{so}}{L}_x=k_{\mathrm{so}}{L}_y=430$ and ${\mathrm{\Delta }}_Z/E_{\mathrm{so}}=0.15$.