Superconducting proximity effect in topological metals

Much interest in the superconducting proximity effect in three-dimensional (3D) topological insulators (TIs) has been driven by the potential to induce Majorana bound states at the interface. Most candidate materials for 3D TI, however, are bulk metals, with bulk states at the Fermi level coexisting with well-defined surface states exhibiting spin-momentum locking. In such topological metals, the proximity effect can differ qualitatively from that in TIs. By studying a model topological metal-superconductor (TM-SC) heterostructure within the Bogoliubov–de Gennes formalism, we show that the pair amplitude reaches the naked surface, unlike in a topological insulator-superconductor (TI-SC) heterostructure where it is confined to the interface. Furthermore, we predict vortex-bound-state spectra to contain a Majorana zero mode localized at the naked surface, separated from the bulk vortex-bound-state spectra by a finite energy gap in such a TM-SC heterostructure. These naked-surface-bound modes are amenable to experimental observation and manipulation, presenting advantages of TM-SC over TI-SC.

Figure 1

(a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$-SC heterostructure considered in this paper. (b) Dispersion of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ on a slab of finite thickness $L_{\mathrm{TI}}$. Each point is doubly degenerate, and the color scale indicates the minimum $z_{\min }={min}_{\Psi }{\langle z\rangle }_{\Psi }$ that can be obtained within the degenerate space $\Psi \in \mathrm{span}\{{\Psi }_1,{\Psi }_2\}$. The dotted horizontal lines indicate representative chemical potentials associated with TI, TM, and M regimes as defined in the text. We present schematics of corresponding Fermi surfaces next to each dotted line, where red filled circles represent the bulk states and the black circles the surface states. Each arrow points along the direction of the spin of the surface state on one of the surfaces, which is locked to the momentum.

Figure 2

The pair amplitudes in singlet and triplet channels as a function of the distance from the interface boundary ($z$) in three regimes: (a) TI, (b) M, and (c) TM, with chemical potentials ${\mu }_{\mathrm{TI}}=25\mathrm{meV}$, ${\mu }_{\mathrm{M}}=75\mathrm{meV}$, and ${\mu }_{\mathrm{TM}}=50\mathrm{meV}$, respectively. The parameters used in the calculation are $L_{\mathrm{TI}}=500\AA$, $L_{\mathrm{SC}}=250\AA$, $a=5\AA$, ${\Delta }_0=5\mathrm{meV}$, ${\mu }_{\mathrm{SC}}=300\mathrm{meV}$, and with $k$ points on a $100\times 100$ grid. (One quintuple layer is roughly 10 Å.)

Figure 3

Panels (a) and (b) show the $z$ dependence of the gap profile used to compute vortex-bound-state spectra for TI ($\mu =25\mathrm{meV}$) and TM ($\mu =50\mathrm{meV}$) regimes, respectively. Panels (c) and (d) show the spatial probability density profile ${\rho }_n(r,z)$, as defined in Eq. (8), of the lowest-lying vortex bound state in two regimes. ${\rho }_n(r,z)$ has been normalized such that the maximum value is unity. The parameters used in the calculation are $a=5\AA$, $R=3000\AA$, $L=500\AA$, ${\Delta }_0=5\mathrm{meV}$, $z_0=a/2$, ${\xi }_R=100\AA$, and ${\xi }_L=8\AA$ for TI and $\gamma =1/4$ for TM. The inset in each case shows the vortex bound-state spectrum, i.e., the energy $E_n$ of the $n\text{th}$ excitation.