Second-Order Topological Superconductors with Mixed Pairing

We show that a two-dimensional semiconductor with Rashba spin-orbit coupling could be driven into the second-order topological superconducting phase when a mixed-pairing state is introduced. The superconducting order we consider involves only even-parity components and meanwhile breaks time-reversal symmetry. As a result, each corner of a square-shaped Rashba semiconductor would host one single Majorana zero mode in the second-order nontrivial phase. Starting from edge physics, we are able to determine the phase boundaries accurately. A simple criterion for the second-order phase is further established, which concerns the relative position between Fermi surfaces and nodal points of the superconducting order parameter. In the end, we propose two setups that may bring this mixed-pairing state into the Rashba semiconductor, followed by a brief discussion on the experimental feasibility of the two platforms.

Figure 1

(a). Heterostructure of a RS and SC with mixed pairing (top view). Majorana zero modes (denoted by red solid ellipses) emerge at the four corners in the second-order topological superconducting phase. Two sets of coordinate systems connected by ${\mathcal{C}}_4$ rotation are shown, in both real space and reciprocal space. (b). Hybrid Josephson junction with FeSC ($s_{\pm }$ pairing), cuprate SC ($d_{x^2-y^2}$ pairing), and a single RS layer sandwiched between them. The junction interface is parallel to the $ab$ plane of the two SCs. (c). Phase diagram in absence of ${\mathrm{\Delta }}_{sd}$, with ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=1$. Phase I: first-order TSC; Phase II: nodal SC; Phase III: fully gapped trivial SC. For $s+id$ pairing, Phase I and II would be driven into the second-order phase. All parameters in this and the following figures are in the unit of $t$.

Figure 2

Edge states in the nodal phase. (a) Bulk nodes in BZ are denoted by stars, and those in the same color have the same topological charge. (b) Winding number for topologically distinct nodes and its evolution with chemical potential. Clearly, only in the nodal phase would the winding number take nonzero values. (c) Energy spectrum in the cylinder geometry. Red lines denote the four energy levels closest to zero and each of the lines is doubly degenerate. (d) Variations of the (scaled) wave function amplitude $|{\varphi }_j|$ with lattice site $j$ and wave vector $k_y$ for the energy levels denoted by red lines in (c). The parameters chosen are $N_x=200$, $A=2$, ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=1$, ${\mathrm{\Delta }}_2=0$. In (c) and (d), $\mu =5$.

Figure 3

Determination of the second-order phase from the bulk spectrum. (a)–(d) Fermi surfaces ${ϵ}_{\pm }=0$, nodal lines of ${\mathrm{\Delta }}_s$, ${\mathrm{\Delta }}_{sd}$ and ${{\mathrm{\Delta }}_{sd}}^{\prime }$ in the first quadrant of BZ are plotted for different chemical potential ($\mu$) and $s+d$ pairing form ($\delta$, $\eta$). Signs of the pair $[{\mathrm{\Delta }}_{sd}(\mathit{k}),{\mathrm{\Delta }}_{sd}^{\prime }(\mathit{k})]$ are indicated in corresponding areas. The system resides in second-order phases when the signs of ${\mathrm{\Delta }}_{sd}$ and ${\mathrm{\Delta }}_{sd}^{\prime }$ at ${\mathit{k}}^{\mathit{c}}$ (marked by stars in magenta) are opposite, or equivalently, when the nodal point ${\mathit{k}}^{\mathit{n}}$ (marked by magenta circle) of the pairing term lies between the two Fermi surfaces. Distributions of MZMs for an $80\times 80$ lattice are shown in the insets, as well as several low-lying energy levels. Clearly, MZMs (red points in the insets) are fourfold degenerate and separated from other energy levels with a finite gap. In all the figures, $A={\mathrm{\Delta }}_2=0.5$, ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=1$.