Odd-frequency superconducting pairing in topological insulators

We discuss the appearance of odd-frequency spin-triplet $s$-wave superconductivity, first proposed by Berezinskii [J. Exp. Theor. Phys. 20, 287 (1974)], on the surface of a topological insulator proximity coupled to a conventional spin-singlet $s$-wave superconductor. Using both analytical and numerical methods, we show that this disorder robust odd-frequency state is present whenever there is an in-surface gradient in the proximity induced gap, including superconductor-normal state junctions. The time-independent order parameter for the odd-frequency superconductor is proportional to the in-surface gap gradient. The induced odd-frequency component does not produce any low-energy states.

Figure 1

(a) SN junction along the edge of a 2D TI. (b) Even-frequency pairing $|F_s\left(0\right)|$, (c) odd-frequency pairing $|F_t^0\left(1\right)|$, and (d) $|F_t^0\left(6\right)|$ in a 2D TI SN junction with ${\mu }_{\mathrm{S}}=0.3$, ${\mu }_{\mathrm{N}}=0$, and $U=2.2$, giving ${\Delta }_{\mathrm{S}}=0.16$. (e) Maximum height $h_F$ (black, left axis) and average width $w_F$ (red, right axis) as function of $\tau$ for $|F_t^0|$ (crosses) and $|{\partial }_x{F}_s|$ (line). (f) Magnitude of odd-frequency order parameter ${\partial }_{\tau }{F}_t^0\left|{}_0\right.$ (crosses) and $\frac{\partial \Delta }{\partial x}$ scaled by a factor of 0.4 (line) for ${\Delta }_{\mathrm{S}}=0.16$ (black) and ${\Delta }_{\mathrm{S}}=0.30$ (red).

Figure 2

(a) Magnitude (height) of the odd-frequency pairing $F_t^0$ (crosses) and ${\partial }_x{F}_s$ (line) as function of $\tau$ for phase windings $k=\pi /50$ (black) and $\pi /10$ (red) along the TI edge. (Inset) Schematic showing a supercurrent $I$ along the edge of a 2D TI. (b) Magnitude of the odd-frequency order parameter ${\partial }_{\tau }{F}_t^0{|}_0$ (crosses) and $\frac{\partial \Delta }{\partial x}$ (line) as function of $k$. Inset shows their ratio. Here $U=2.5$, $\mu =0.3$, giving ${\Delta }_{\mathrm{S}}=0.30$.

Figure 3

(a) Odd-frequency pairing $|F_t^1\left(1\right)|$ in the SN junction in Fig. 1 with ${\lambda }_R=2\lambda$. (b) $|F_t^1|$ along the TI edge and (c) at the SN interface into the TI for $\tau =0.5$ (dashed black), 2 (black), 8 (dashed red), and 16 (red). (d) Spin components $|F_t^{+1}\left(1\right)|$ and (e) $|F_t^{-1}\left(1\right)|$ of $F_t^1\left(1\right)=(F_t^{+1}-F_t^{-1})/2$.