Impurity-induced excitations in a topological two-dimensional ferromagnet/superconductor van der Waals moiré heterostructure

The emergence of a topological superconducting state in van der Waals heterostructures provides a new platform for exploring novel strategies to control topological superconductors. In particular, impurities in van der Waals heterostructures, generically featuring a moiré pattern, can potentially lead to the unique interplay between atomic and moiré length scales, a feature absent in generic topological superconductors. Here we address the impact of nonmagnetic impurities on a topological moiré superconductor, both in the weak and strong regime, considering both periodic arrays and single impurities in otherwise pristine infinite moiré systems. We demonstrate a fine interplay between impurity-induced modes and the moiré length, leading to radically different spectral and topological properties depending on the relative impurity location and moiré lengths. Our results highlight the key role of impurities in van der Waals heterostructures featuring moiré patterns, revealing the key interplay between length and energy scales in artificial moiré systems.

Figure 1

(a) Schematic view of an artificial ${\mathrm{CrBr}}_3/{\mathrm{NbSe}}_2$ moiré topological superconductor. Panels (b) and (c) show the real-space modulation of exchange coupling $J\left(\mathbf{r}\right)$ (b) and superconductivity $\mathrm{\Delta }\left(\mathbf{r}\right)$ (c) in the moiré unit cell. Panels (d), (e), and (f): Electronic band structure of a (d) uniform, (e) moiré, and (f) moiré with an impurity topological superconductor. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t, {\delta }_J=2J_0, {\delta }_{\mathrm{\Delta }}=1.4{\mathrm{\Delta }}_0$.

Figure 2

(a) Gap as a function of impurity location in the strong-impurity regime $w=2t$. Panels (b), (c), and (d) show the electronic structure for three impurity locations, showing that halfway between exchange maxima, the gap is minimal (b). In comparison, close to the exchange maxima, the gap remains nearly unchanged [(c), (d)]. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t, {\delta }_J=2J_0, {\delta }_{\mathrm{\Delta }}=1.4{\mathrm{\Delta }}_0$.

Figure 3

(a) Gap as a function of impurity location in the weak-impurity regime $w=t/2$. Panels (b), (c), and (d) show the electronic structure for three impurity locations, showing that halfway between exchange maxima, the gap is minimal (b). In contrast, close to the exchange maxima, the gap remains nearly unchanged [(c), (d)]. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t, {\delta }_J=2J_0, {\delta }_{\mathrm{\Delta }}=1.4{\mathrm{\Delta }}_0$.

Figure 4

Topological gap as a function of the location of an impurity in a moiré unit cell, for (a) $11\times 11$, (b) $13\times 13$, (c) $15\times 15$, (d) $17\times 17$ supercells. A direct correlation is observed between the location of the impurity in the unit cell and the moiré pattern, leading to drastic changes in the energy gap. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t, {\delta }_J=2J_0, {\delta }_{\mathrm{\Delta }}=1.4{\mathrm{\Delta }}_0$.

Figure 5

Normalized topological gap as a function of moiré amplitude [Eq. (15)] for weak [(a), (c)] and strong [(b), (d)] impurity, for a $9\times 9$ [(a), (b)] and a $11\times 11$ [(c), (d)] moiré. It is observed that in the absence of the moiré the location of the impurities does not impact the magnitude of the gap, whereas with moiré, the topological gap shows a strong dependence on the size and location of the impurity. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t$.

Figure 6

Density of states [(a)–(d)] and local density of states [(e)–(h)] at the energy of the in-gap state ${\epsilon }_V$ for a single impurity in an otherwise pristine system. Panels (a) and (e) correspond to the uniform case, whereas panels (b), (c), (d), (f), (g), and (h) to different locations of a single impurity [insets in (f), (g), (h)] for the moiré case. Both in the absence (a) and presence [(b), (c), (d)] of a moiré, a strong nonmagnetic impurity gives rise to an in-gap state. It is observed that in the presence of the moiré pattern, the in-gap state leads to a strong interference with moiré length, yielding a spatial dependence with respect to the location of the impurity [(f), (g), (h)]. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t, {\delta }_J=2J_0, {\delta }_{\mathrm{\Delta }}=1.4{\mathrm{\Delta }}_0$, and moiré $5\times 5$.

Figure 7

Moire topological superconducting states of a pristine system [(a), (b), (c), (d)] and of a defective system [(e), (f), (g), (h)]. Panels (a) and (e) show the electronic structure of a ribbon infinite in the $x$ direction and finite in the $y$ direction. Panels (b) and (f) show the momentum-resolved edge spectral function, and panels (c) and (g) the momentum-resolved bulk spectral function. Panels (d) and (h) show the local density of states at $\omega =0$, highlighting the emergence of topological zero modes at the top and bottom edges following the moiré pattern. The structure insets in (d) and (h) show a schematic of the boundary conditions used. We used $J_0=2{\mathrm{\Delta }}_0, \lambda =2{\mathrm{\Delta }}_0, \mu =3t, {\delta }_J=2J_0, {\delta }_{\mathrm{\Delta }}=1.4{\mathrm{\Delta }}_0$, and moiré $5\times 5$.