Tuning topological superconductivity in helical Shiba chains by supercurrent

Recent experimental investigations of arrays of magnetic atoms deposited on top of a superconductor have opened a new chapter in the search for topological superconductivity. We generalize the microscopic model derived by Pientka et al. [Phys. Rev. B 88, 155420 (2013)] to accommodate the effects of finite supercurrent in the host material. Previously it was discovered that helical chains with nonplanar textures are plagued by a gapless phase. We show that by employing supercurrent it is possible to tune the chain from the gapless phase to the topological gapped phase. It is also possible to tune the chain between the trivial and the topological gapped phase, the size of which may be dramatically increased due to supercurrent. For planar textures supercurrent mainly contributes to proliferation of the gapless phase. Our predictions, which could be probed in scanning tunneling microscope experiments, are encouraging for the observation and manipulation of Majorana states.

Figure 1

(a) The studied system consists of a helical arrangement of magnetic atoms on a superconductor. Supercurrent can be employed to modify the topological state of the chain. The signatures of the topological phase can be observed by tunneling from an STM tip to the MBS localized at the end. (b) Minimum value of $E_{+}\left(k\right)$. Different phases are separated by the condition $min{E}_{+}\left(k\right)=0$. The labels stand for normal (N), topological (T), and gapless (G). The parameters are $\theta =2\pi /5,k_{h}a=\pi /10,{\varepsilon }_{\phi }=\left|\Delta \right|/3,\xi =50a$. The inset shows the phase diagram for vanishing supercurrent ${\varepsilon }_{\phi }=0$. In addition to adding gapless regions, finite supercurrent also pushes some gapless regions to the topologically nontrivial gapped phase indicated by the red arrows. (c) Same as (b), but with inverted supercurrent ${\varepsilon }_{\phi }=-{\varepsilon }_{\phi }$.

Figure 2

(a) Same quantities as in Fig. 1 but for $\theta =\pi /5,k_{h}a=\pi /3,{\varepsilon }_{\phi }=\left|\Delta \right|/3,\xi =50a$. The inset shows the phase diagram for vanishing supercurrent ${\varepsilon }_{\phi }=0$. Supercurrent significantly increases the size of the topological region. (b) Same as (a), but $k_{h}a=\frac{\pi }{2}$. Supercurrent opens up large topologically nontrivial regions that are completely absent when the supercurrent vanishes.

Figure 3

(a) Same quantities as in Fig. 2 but for a planar helix $\theta =\frac{\pi }{2},k_{h}a=\pi /10,{\varepsilon }_{\phi }=\left|\Delta \right|/3,\xi =50a$. Supercurrent cannot deform the phase boundaries between the gapped phases; it can only add gapless regions on top of the ${\varepsilon }_{\phi }=0$ phase diagram (in the inset). (b) Phase diagram for a finite-size chain of 50 atoms with the same parameters as in (a). Colors indicate the ratio of the lowest-lying and the first excited energy level $|E_0/E_1|$. Small values (dark blue) indicate the topological phase with Majorana end states and large values (white) signals the trivial gapped state. In a finite-size system oscillations (blue stripes) indicate the gapless phase.

Figure 4

(a) LDOS at the end of the chain of 100 atoms. The system is gapless in the absence of supercurrent and enters the topologically nontrivial gapped phase signaled by a ZBP and the opening of a minigap (white arrow). The parameters are $\theta =2\pi /5,k_{h}a=\pi /10,k_{F}a=4.8\pi ,{\varepsilon }_0=0.05\left|\Delta \right|,\xi =50a$. (b) Same as (a) but for a system that is in the trivial gapped phase in the absence of supercurrent. The parameters are $\theta =\pi /5,k_{h}a=\pi /2,k_{F}a=4\pi ,{\varepsilon }_0=0.015\left|\Delta \right|,\xi =50a$.