Supercurrent as a probe for topological superconductivity in magnetic adatom chains

A magnetic adatom chain, proximity coupled to a conventional superconductor with spin-orbit coupling, exhibits locally an odd-parity, spin-triplet pairing amplitude. We show that the singlet-triplet junction, thus formed, leads to a net spin accumulation in the near vicinity of the chain. The accumulated spins are polarized along the direction of the local $\mathbf{d}$ vector for triplet pairing and generate an enhanced persistent current flowing around the chain. The spin polarization and the “supercurrent” reverse their directions beyond a critical exchange coupling strength at which the singlet superconducting order changes its sign on the chain. The current is strongly enhanced in the topological superconducting regime where Majorana bound states appear at the chain ends. The current and the spin profile offer alternative routes to characterize the topological superconducting state in adatom chains and islands.

Figure 1

(a) Variation of the BdG spectrum with impurity moment $JS$ for a chain of length $N_{\text{imp}}=50$, on a superconducting square lattice of size $61\times 61$. The yellow lines are the pair of YSR states for a single magnetic impurity in the same superconducting host. The insets on the left show the adatom configurations. The pair of red lines at zero energy represents the MBS within the range $J_{c1}S\le JS\le J_{c2}S$. (Right inset) Expanded view of the spectrum for the chain within the topological range. (b), (c), (d), and (e) show, respectively, the $z$ component of spin-expectation value $S_z$, singlet pairing gap ${\mathrm{\Delta }}_s$, and triplet pairing amplitudes ${\mathrm{\Delta }}_{\uparrow \uparrow }^{t1}, {\mathrm{\Delta }}_{\uparrow \uparrow }^{t3}$, averaged along the chain, at the adatom sites or bonds with varying $JS$ for a chain and a single impurity. The black square symbols in (b) show the total $S_z$, divided by $N_{\text{imp}}=15$, for a $21\times 21$ lattice. The vertical green lines in (d) and (e) denote the value of $JS$ at which $\mathrm{Re}({\mathrm{\Delta }}_{\uparrow \uparrow }^{t1,\text{Chain}}$) is maximum and $\mathrm{Im}({\mathrm{\Delta }}_{\uparrow \uparrow }^{t3,\text{Chain}}$) changes sign. Parameters taken: $t=1, \mu =0, \alpha =0.1$, and $U_s=3=U_t$.

Figure 2

The LDOS profile of an adatom chain of length $N_{\text{imp}}=50$, on a superconducting square lattice of size $61\times 61$, corresponding to the pair of lowest energy eigenvalues for exchange-coupling strength (a) $JS=1$ (nontopological phase), and $JS=3$ (topological phase). The sharp peaks at the ends of the chain in (b) represent the zero-energy localized MBS. Parameters are the same as in Fig. 1.

Figure 3

Total number of up- and down-spin electrons, $n_{\uparrow }$ (blue) and $n_{\downarrow }$ (red), vs exchange coupling $JS$ in the grand canonical ground state for (a) a single impurity and (b) an adatom chain. Calculations were performed on a $21\times 21$ lattice. Parameters are the same as in Fig. 1.

Figure 4

The spatial profile of (a) $z$ component of spin-expectation value $S_z$, (b) singlet $s$-wave superconducting order parameter ${\mathrm{\Delta }}_s$, (c) real part of triplet pairing gap for the bond along $+x$ direction, $\mathrm{Re}\left({\mathrm{\Delta }}_{\uparrow \uparrow }^{t,1}\right)$, and (d) imaginary part of triplet pairing gap for the bond along $+y$ direction, $\mathrm{Im}\left({\mathrm{\Delta }}_{\uparrow \uparrow }^{t,3}\right)$, calculated for $JS=3$ (topological superconducting phase). A square lattice of size $61\times 61$ was used for the calculation with impurity-chain length $N_{\text{imp}}=50$ and parameters of Fig. 1.

Figure 5

Variation of the triplet order parameter ${\mathrm{\Delta }}_t$ in natural logarithmic scale with the attractive interaction strength $U_t$ for impurity moment $JS=3$. Inset shows the variation of $ln({\mathrm{\Delta }}_t/U_t)$ with $ln\left(U_t\right)$. Other parameters are the same as in Fig. 4.

Figure 6

Profile of the accumulated spins near the adatom chain for (a) $JS=1(<J_{c1}S)$, (b) $JS=3(J_{c1}S\le JS\le J_{c2}S)$, and (c) $JS=5(>J_{c2}S)$, calculated on a $21\times 21$ lattice with $N_{\text{imp}}=15$. The red symbols mark the adatom sites. The length of the arrows is amplified for visualization purpose. (d) Variation of the spin-expectation value $S_y$, averaged over the sites just below the chain, with impurity moment $JS$ for finite $U_t$ and $U_t=0$ (solid symbols and dashed lines, respectively) in case of a single magnetic impurity (blue) and a chain (red). Parameters are the same as in Fig. 1.

Figure 7

The real-space profile of the local $\mathbf{d}$-vector of the triplet pairing amplitude for (a) $JS=1$ (trivial phase), (b) $JS=3$ (topological superconducting phase), and (c) $JS=5$ (trivial phase). Other parameters are the same as in Fig. 4.

Figure 8

Current flow around the adatom chain for (a) $JS=1(<J_{c1}S)$, (b) $JS=3(J_{c1}S\le JS\le J_{c2}S)$, and (c) $JS=5(>J_{c2}S)$, calculated on a $21\times 21$ lattice with $N_{\text{imp}}=15$. The red symbols mark the adatom sites. (d) Variation of the magnitude of the current ${\mathbf{j}}_{\mathbf{s}}$, normalized by the critical supercurrent for the host superconductor ${\mathbf{j}}_{\mathbf{sc}}$, with the impurity moment $JS$ for a single magnetic impurity (blue squares) and an adatom chain (red circles). The dashed line shows the variation of the current with $JS$ for $U_t=0$. Parameters are the same as in Fig. 1.