Moiré dispersion of edge states in spin chains on superconductors

Our calculations of ferromagnetic spin chains on $s$-wave superconductors show that the energy oscillations of edge states with the chain's length are due to a moiré pattern emerging from the electronic interactions in the superconductor and the discreteness of the spin-chain lattice. By modifying the spin lattice, the moiré dispersion of edge states can be controlled. In particular, we can engineer nondispersive edge states that remain at fixed energy regardless of the size distribution of the spin chains. This is an important step in the study of edge states of spin chains that can be fabricated with a certain size dispersion.

Figure 1

Topological phase diagram and evolution of edge states with number of atoms. Inset: Winding number calculated for infinite FM chains as a function of the magnetic coupling $J$ and the Fermi vector $k_F$. The green and magenta areas correspond to the topological phases $w=1$ and $w=-1$, respectively, and the white area is the trivial region. (a)–(d) Projected density of states (PDOS) as a function of the number of magnetic atoms in the spin chain. The $J$ and $k_F$ parameters are indicated by the position of the corresponding shape. (a) $J=2.4$ eV, $k_F=0.6a_0^{-1}$, red square. (b) $J=2.7\mathrm{eV}$, $k_F=0.24a_0^{-1}$, orange star. (c) $J=4.5$ eV, $k_F=0.4a_0^{-1}$, yellow pentagon. (d) $J=4.5$ eV, $k_F=0.75a_0^{-1}$, pink diamond. Other parameters are $\mathrm{\Delta }=0.75$ meV, $N_0=0.037$/eV, ${\alpha }_R=3.0$ eV Å, $K=5.5$ eV, and $a_c=3.36$ Å. The correlation length is the BCS one, and for (d) it is $\xi =4852$ Å.

Figure 2

Projected density of states (PDOS) on the edge atom of a spin chain as a function of the number of atoms in the chain for an (a) even and (b) odd number of atoms, respectively. When (a) and (b) are combined, we recover Fig. 1. (c) Dimer PDOS as the distance between the two atoms is varied in a continuous way. Inset: Zoom-in area between distances 25–$27.5a_c$. The PDOS from (c) is plotted at selected distances in (d) and (e) corresponding to integer multiples of $a_c$, even and odd integers, respectively. This clearly shows that the observed oscillations are a consequence of the discrete sampling of the continuous PDOS in (c) and they match the periodicity of the oscillations in (a) and (b), revealing the moiré nature of the latter [the parameters are the same as in Fig. 1 and the above (a) and (b)].

Figure 3

PDOS on the edge atom of spin chains for slightly different interimpurity distances $a_c$. In the right panels, $a_c=3.30$ Å, the period of the resulting oscillations is $\sim 107$ atoms, shown for (a) even and (c) odd numbers of atoms. In (b) and (d), $a_c=3.322$ Å, we do not observe any evolution of the edge states as we vary the number of atoms. These plots can be compared with Figs. 2 and 2(b), $a_c=3.36$ Å, where we find the period of the oscillations to be 60 atoms. The parameters other than $a_c$ are the same as in Fig. 1.

Figure 4

PDOS obtained on the edge atom of Mn chains of increasing length on Nb (110). (a) The atomic chains are oriented along the $\left[1\overline{1}0\right]$ direction, resulting in a distance between impurities of $a_c=4.67$ Å. (b) The chains are oriented along the $[\overline{\frac{1}{2}},\frac{1}{2},\frac{5}{2}]$ direction, where the distance between impurities is $a_c=8.56$ Å. The parameters of the calculation are $k_F=0.1886a_0^{-1}$, such that $2\pi /k_F=17.6$ Å roughly twice $a_c$, $J=0.125$ eV, $\mathrm{\Delta }=1.5$ meV, $N_0=1.0{\mathrm{eV}}^{-1}$, ${\alpha }_R=1.0$ eV Å, $K=0.063$ eV, and $a=3.294$ Å. The BCS correlation length is $\xi =576$ Å. The insets show schemes of the crystal surface and the chain orientations.