Microscopic simulation of superconductor/topological insulator proximity structures

We present microscopic, self-consistent calculations of the superconducting order parameter and pairing correlations near the interface of an $s$-wave superconductor and a three-dimensional topological insulator with spin-orbit coupling. We discuss the suppression of the order parameter by the topological insulator and show that the equal-time pair correlation functions in the triplet channel, induced by spin-flip scattering at the interface, are of $p_x\pm i{p}_y$ symmetry. We verify that the spectrum at subgap energies is well described by the Fu-Kane model. The subgap modes are viewed as interface states with spectral weight penetrating well into the superconductor. We extract the phenomenological parameters of the Fu-Kane model from microscopic calculations, and find they are strongly renormalized from the bulk material parameters. This is consistent with previous results of Stanescu et al. for a lattice model using perturbation theory in the tunneling limit.

Figure 1

Schematic (not to scale) band diagrams in a superconductor-topological insulator (S-TI) proximity structure. $E_f$ is the Fermi energy of the metal described by $H_M$ measured from the band crossing point. $\mu$ is the chemical potential of TI measured from the band-gap center. The superconducting gap is much smaller than the band gap of TI.

Figure 2

The superconducting order parameter $\Delta (z)$ near a S-TI interface at $z=d=0.95L$. The superconductor occupies $0<z<d$, and the topological insulator occupies $d<z<L$. $L=300$ nm, $\mu =0$, the bulk gap ${\Delta }_0=0.6$ meV.

Figure 3

The order parameter $\Delta (z)$ near a S-TI interface at $z=d=0.9L$. $L=160$ nm, $\mu =0$, ${\Delta }_0~2.4$ meV.

Figure 4

The order parameter profile for two different chemical potentials of the topological insulator, $\mu =-0.1$ eV and $\mu =0.2$ eV. $L=160$ nm, ${\Delta }_0~5.2$ meV.

Figure 5

The lowest few energy levels ${\epsilon }_n(k_{\parallel })$. $\mu =0$, $L=160$ nm, and the bulk superconducting gap ${\Delta }_0~$5.2 meV. A well-defined interface mode is clearly visible at subgap energies. Solid lines show a fit to the Fu-Kane model, with ${\Delta }_s=1.8$ meV, $v_s=2.7$ eV Å, and ${\mu }_s=7.5$ meV.

Figure 6

The dispersion of the lowest-energy level for different $\mu$ (in eV). Other parameters are the same as in Fig. 5, $L=160$ nm and ${\Delta }_0~$5.2 meV. The Fu-Kane model well describes the lowest-energy mode. As $\mu$ is increased, ${\Delta }_s$ and $v_s$ stay roughly the same, while ${\mu }_s$ scales linearly with $\mu$.

Figure 7

The spectral function $N(k_{\parallel },z,\omega )$ of the lowest-energy level, $\omega =E_{\min }$, shown in Fig. 6. The interface is at $z=0.9L$, $L=160$ nm. The spectral function oscillates rapidly with $z$, so only its envelope is plotted.

Figure 8

The local density of states $N(E,z)$ at $z=0.8d$ and $z=0.85d$ (the interface is at $z=0.9d$). $\mu =0$, $L=160$ nm, and ${\Delta }_0~5.2$ meV. The subgap states are due to the interface mode. A level broadening $~$$0.01{\Delta }_0$ is used.

Figure 9

The lowest-energy level of an S-TI structure with $L=160$ nm, $d=0.9L$, ${\Delta }_0=2.4$ meV. $\mu$ is the chemical potential of the TI and measured in eV.

Figure 10

The imaginary part of triplet pair correlation function $F_{\uparrow \uparrow }(k_{\parallel },z)$. The S-TI interface is at $d=0.9L$. $\mu =0$, $L=160$ nm, ${\Delta }_0=5.2$ meV.

Figure 11

The imaginary part of $F_{\uparrow \uparrow }(k_{\parallel },z)$. $\mu =0$, $L=300$ nm, $d=0.95L$, ${\Delta }_0=0.6$ meV. As a comparison, the data points show the singlet pair correlation function $F_{\uparrow \downarrow }(k_{\parallel },z=0.9L)/3$.