Intrinsic first- and higher-order topological superconductivity in a doped topological insulator

We explore higher-order topological superconductivity in an artificial Dirac material with intrinsic spin-orbit coupling, which is a doped ${\mathbb{Z}}_2$ topological insulator in the normal state. A mechanism for superconductivity due to repulsive interactions, pseudospin pairing, has recently been shown to naturally result in higher-order topology in Dirac systems past a minimum chemical potential [T. Li et al., 2D Mater. 9, 015031 (2022)]. Here we apply this theory through microscopic modeling of a superlattice potential imposed on an inversion-symmetric hole-doped semiconductor heterostructure, known as hole-based semiconductor artificial graphene, and extend previous work to include the effects of spin-orbit coupling. We find that spin-orbit coupling enhances interaction effects, providing an experimental handle to increase the efficiency of the superconducting mechanism. We show that the phase diagram of these systems, as a function of chemical potential and interaction strength, contains three superconducting states: a first-order topological $p+ip$ state, a second-order topological spatially modulated $p+i\tau p$ state, and a second-order topological extended $s$-wave state $s_{\tau }$. We calculate the symmetry-based indicators for the $p+i\tau p$ and $s_{\tau }$ states, which prove these states possess second-order topology. Exact diagonalization results are presented which illustrate the interplay between the boundary physics and spin-orbit interaction. We argue that this class of systems offers an experimental platform to engineer and explore first- and higher-order topological superconducting states.

Figure 1

Schematic view of the honeycomb superlattice patterned on the heterostructure 2DHG: a patterned dielectric or gate is placed on a quantum well, e.g., GaSb-InAs-GaSb. Superlattice Brillouin zone: the reciprocal lattice vectors ${\mathit{G}}_i$ connect zone corners corresponding to ${\mathit{K}}_j$, and connect corners corresponding to ${\mathit{K}}_j^{\prime }$, the parity reflections of ${\mathit{K}}_j$.

Figure 2

Parameters of the effective Dirac Hamiltonian (4). (a) Dirac velocity $v$, in units of $E_0/K_0$. (b) Spin-orbit gap $\eta$, in units of $E_0$.

Figure 3

(a) Spin-independent matrix elements of $V_I$. Solid lines: blue, orange, green $=v_{00},v_{12},v_{33}$. (b) Spin-dependent matrix elements of $V_I$. Dashed lines: blue, orange, green, red and purple $=v_{44},v_{77},v_{56},v_{07},v_{47}$. (c) Spin-independent matrix elements of $V_{II}$. Solid lines: blue, orange $=u_{00},u_{12}$. (d) Spin-dependent matrix elements of $V_{II}$. Dashed lines: blue, orange $=u_{33},u_{03}$. In units of $2\pi e^2/\left({\epsilon }_r{K}_0\right)$.

Figure 4

Phase diagram. The three superconducting phases $d_{z0}^{\ell =0},d_{zz}^{\ell =\pm },d_{\ell =\pm }^{\alpha \pm }$, which correspond to $s$-wave intervalley ($s_{\tau }$), ($p+ip$)-wave intervalley ($p+ip$), and $p+i\tau p$ intravalley ($p+i\tau p$), as well as the magnetic phase. (a) Fixing the bare values $\{v_{00},v_{33},v_{44},v_{77},v_{56},v_{07},v_{47},u_{00},u_{33},u_{03}\}$ to those computed and shown in Fig. 3, while allowing for a variable ${\tilde{v}}_{12}$ and ${\tilde{u}}_{12}$, which are substituted into the interaction structure in place of $v_{12}$ and $u_{12}$. Here we show the variable values as ratio of the calculated values. (ai), (aii), (aiii) Show the same parameters but with increasing chemical potential $\mu /{\mu }_0=0.75,1,1.25$, respectively, with ${\mu }_0\equiv 0.025v{K}_0$, with $d/L=0.375, L=30\mathrm{nm}$. (b) Same as (a), but allowing for a variable ${\tilde{v}}_{33}$ and ${\tilde{u}}_{00}$. (bi), (bii), (biii) Show the same parameters but with increasing chemical potential $\mu /{\mu }_0=0.75,1,1.25$.

Figure 5

Exact diagonalization results for the second-order topological $s_{\tau }$ spin-triplet phase. (a), (b) The 1D dispersion of infinite superconducting ribbons with (a) armchair and (b) zigzag terminations. Edge modes propagating along opposite edges are shown in different colors. (c) Wave-function profile of the six zero-energy eigenstates on a flake geometry. These are the subgap states marked in red in the corresponding spectrum displaying 60 eigenstates around zero in (d). Here we use the parameters $\eta =0.2t, \mu =0.4t, {\mathrm{\Delta }}^{\prime }=0.033t$ corresponding to a bulk superconducting gap $\mathrm{\Delta }\approx 0.16t$.

Figure 6

Exact diagonalization results for the second-order topological $p+i\tau p$ intravalley spin-triplet phase. (a), (b), The 1D dispersion of infinite superconducing ribbons with (a) armchair and (b) zigzag terminations. Edge modes propagating along opposite edges are shown in different colors. (c) Wave-function profile of the six lowest absolute energy eigenstates on a flake geometry. These are the subgap states marked in red in the corresponding spectrum displaying 60 eigenenergies around zero in (d). Here we use the parameters $\eta =0.2t, \mu =0.4t, {\mathrm{\Delta }}^{\prime }=0.13t$, and $\varphi =\pi /2$ corresponding to a bulk superconducting gap $\mathrm{\Delta }\approx 0.16t$.

Figure 7

Dependence of the spectrum (in units of $t$) on the pair density wave order parameter $\varphi$ in the $p+i\tau p$ state, for ribbons of various width. The energy spectrum at $k=0$ of an infinite superconducting ribbon with armchair termination of width (a) 35, (b) 36, (c) 37 unit cells as a function of $\varphi$, for parameters $\eta =0.2t,\mu =0.4t$, and ${\mathrm{\Delta }}^{\prime }=0.13t$ corresponding to a bulk superconducting gap $\mathrm{\Delta }\approx 0.16t$.

Figure 8

The 1D dispersion of infinite superconducting ribbons with (a) armchair and (b) zigzag terminations in the $p+ip$ intervalley spin-triplet phase, with parameters $\eta =0.2t, \mu =0.4t$, and ${\mathrm{\Delta }}^{\prime }=0.2t$ corresponding to a bulk superconducting gap $\mathrm{\Delta }\approx 0.18t$. Edge modes propagating along opposite edges are shown in different colors.

Figure 9

(a) 2DHG spectrum, $\epsilon \left(p\right)/E_0$ for InAs. Solid blue line corresponds to the doubly degenerate spectrum that enters the computations of the effective Dirac Hamiltonian (4), the dashed line is the next highest subband, which is ignored in our approximations. (b) Probability densities $|a_{S_z}{\left(p\right)|}^2$ of each physical spin component $S_z$, presented in (A14).