High-temperature Majorana corner modes in a $d+i{d}^{\prime }$ superconductor heterostructure: Application to twisted bilayer cuprate superconductors

The realization of Majorana corner modes generally requires unconventional superconducting pairing or $s$-wave pairing. However, the bulk nodes in unconventional superconductors and the low $T_c$ of $s$-wave superconductors are not conducive to the experimental observation of Majorana corner modes. Here, we show the emergence of a Majorana corner mode at each corner of a two-dimensional topological insulator in proximity to a $d+i{d}^{\prime }$ pairing superconductor, such as heavily doped graphene or especially a twisted bilayer of a cuprate superconductor, e.g., ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$, which has recently been proposed as a fully gapped chiral $d_{x^2-y^2}+i{d}_{xy}$ superconductor with $T_c$ close to its native 90 K, and an in-plane magnetic field. By numerical calculation and intuitive edge theory, we find that the interplay of the proximity-induced pairing and Zeeman field can introduce opposite Dirac masses on adjacent edges of the topological insulator, which creates one zero-energy Majorana mode at each corner. Our scheme offers a feasible route to achieve and explore Majorana corner modes in a high-temperature platform without bulk superconductor nodes.

Figure 1

Schematic diagram of the proposed setup. A heterostructure composed of a 2D topological insulator deposited on a high-$T_c$ fully gapped $d+i{d}^{\prime }$ pairing superconductor such as a twisted bilayer of cuprate superconductor monolayers and subject to an in-plane Zeeman field. The sphere at each corner represents one zero-energy Majorana corner mode.

Figure 2

Quasiparticle bands with edge spectra (red lines) and bulk spectra (light blue lines) for an open boundary condition along the $y$ direction for (a) $h_x=0$, (b) $h_x=0.75$, and (c) $h_x=1$. The gap for the edge spectrum closes at the critical Zeeman field $h_x=0.75$. (d) Quasiparticle bands with the edge spectrum (red lines) and bulk spectra (light blue lines) for open boundary conditions along the $x$ direction with the critical Zeeman field $h_x=0.75$. (e) Eigenvalues of the real-space TB Hamiltonian with $h_x=1$ for a $30\times 30$ square size sample. (f) The density plot displays the corner localized probability distribution of the four zero-energy MCMs in (e). I, II, III, and IV label the four edges. Common parameters are $m_0=1$, $t_x=t_y={\lambda }_x={\lambda }_y=2$, ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0.5$, $\mu =0.1$.

Figure 3

(a) Phase diagram vs $h_x$ and ${\mathrm{\Delta }}_1$. (b) Line-scan real-space spectra along the blue dashed line in (a). $\mu =0.1$. (c) Phase diagram as a function of the azimuth of the in-plane Zeeman field. Parameters are $h_{\parallel }=m_0$, $h_x=h_{\parallel }cos\phi$, $h_y=h_{\parallel }sin\phi$, $\mu =0$. (d) Energy spectrum of the real-space TB Hamiltonian at $\phi =0.4\pi$, $\mu =0.1$ for a $30\times 30$ square size sample. The density plot exhibits the corner localized probability distribution of the four zero-energy MCMs. Common parameters are $m_0=1$, ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0.5$, $t_x=t_y={\lambda }_x={\lambda }_y=2$.