Detection of second-order topological superconductors by Josephson junctions

We study Josephson junctions based on second-order topological superconductors (SOTSs) which can be realized in quantum spin Hall insulators with a large inverted gap in proximity to unconventional superconductors. We find that tuning the chemical potential in the superconductor strongly modifies the induced pairing of the helical edge states, resulting in topological phase transitions. In a corresponding Josephson junction, a $0-\pi$ transition is realized by tuning the chemical potentials in the superconducting leads. This striking feature is stable in junctions with respect to different sizes, doping the normal region, and the presence of disorder. Our transport results can serve as novel experimental signatures of SOTSs. Moreover, the $0-\pi$ transition constitutes a fully electric way to create or annihilate Majorana bound states in the junction without any magnetic manipulation.

Figure 1

(a) Energy spectra of the model (1) in a ribbon along the $x$ direction for chemical potential $\mu =0$ (black), $0.44m_0$ (blue), and $0.52m_0$ (red), respectively. The thick curves close to zero energy are edge bands. The pairing gap ${\mathrm{\Delta }}_{\text{eff}}^x$ vanishes at around $\mu =0.44m_0$, as pointed by the blue arrows. (b) $\left|{\mathrm{\Delta }}_{\text{eff}}^x\right|$ as a function of $\mu$. The blue circles are numerical results from tight-binding calculation while the red curve is the plot of Eq. (5). Other parameters are $m_{x\left(y\right)}=2.5,v_{x\left(y\right)}=1,{\mathrm{\Delta }}_0=0,{\mathrm{\Delta }}_2=0.05$, and 200 lattice layers in the $y$ direction. The units for energy and wave number are $m_0$ and $a^{-1}$, respectively.

Figure 2

(a) Schematic for the SNS setup. (b) Current-phase relations for $L=a$ and ${\mu }_R=0.1m_0, 0.3m_0,0.436m_0,0.5m_0$, and $0.57m_0$, respectively. (b) Critical current $J_c$ as a function of ${\mu }_R$ for $L=a$ (red), $20a$ (green), and $50a$ (blue), respectively. The inset displays the results in the presence of disorder of strength $V_{\text{dis}}=m_0$ for $W=39a$ and $L=20a$ (green), $W=29a$ and $L=40a$ (purple), respectively. The error bars are 500 times enlarged for visibility. For all solid curves, ${\mu }_L={\mu }_N=0.1m_0, W=39a, k_{B}T={10}^{-3}{\mathrm{\Delta }}_L$, and other parameters are the same as those in Fig. 1. The circled dots in (c) are the same as the solid curves but for ${\mu }_N=-0.3m_0$.

Figure 3

Andreev bound states for (a), (b) ${\mu }_R=0.1m_0$ and (c), (d) ${\mu }_R=0.52m_0$, respectively. (a) and (c) are for short junctions with $L=a$, while (b) and (d) for long junctions with $L=50a$ and $150a$, respectively. For all plots, $W=39a$ and other parameters are the same as those in Fig. 1.