Interplay of Magnetism and Topological Superconductivity in Bilayer Kagome Metals

The binary intermetallic materials, $M_3{\mathrm{Sn}}_2$ ($M=3d$ transition metal) present a new class of strongly correlated systems that naturally allows for the interplay of magnetism and metallicity. Using first principles calculations we confirm that bulk ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ is a ferromagnetic metal, and show that $M=\text{Ni}$ and Cu are paramagnetic metals with nontrivial band structures. Focusing on ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ to understand the effect of enhanced correlations in an experimentally relevant atomistically thin single kagome bilayer, our ab initio results show that dimensional confinement naturally exposes the flatness of band structure associated with the bilayer kagome geometry in a resultant ferromagnetic Chern metal. We use a multistage minimal modeling of the magnetic bands progressively closer to the Fermi energy. This effectively captures the physics of the Chern metal with a nonzero anomalous Hall response over a material relevant parameter regime along with a possible superconducting instability of the spin-polarized band resulting in a topological superconductor.

Figure 1

(a) Layered arrangement in bulk $M_3{\mathrm{Sn}}_2$. (b) Bilayer $M_6{\mathrm{Sn}}_6$ derived out of the bulk layered structure. (c) Stacking of two kagome layers within the bilayer, viewed along the out-of-plane direction.

Figure 2

$\mathrm{GGA}+U+\mathrm{SOC}$ band structure of bulk ferromagnetic ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ (a), paramagnetic ${\mathrm{Ni}}_3{\mathrm{Sn}}_2$ (b), and ${\mathrm{Cu}}_3{\mathrm{Sn}}_2$ (c). The nontrivial Weyl and Dirac-like crossings are encircled. In (a) the Fe $d$ orbital character is shown—red, $d_{xy}$; green, $d_{yz}$; cyan, $d_{xz}$; blue, $d_{3z^2-r^2}$; yellow, $d_{x^2-y^2}$. The corresponding non-spin-polarized projected DOS are shown for the Fe (d), Ni (e), and Cu (f) compounds.

Figure 3

$\mathrm{GGA}+\mathrm{SOC}+U$ band structure (a) and Fe $d$ projected spin-polarized DOS, with positive (negative) axis corresponding to DOS in up (down) spin channel (b) of bilayer Fe compound in the FM state. In (a) the Fe $d$ orbital characters are denoted with different colors—red, $d_{xy}$; green, $d_{yz}$; cyan, $d_{xz}$; blue, $d_{3z^2-r^2}$; and yellow, $d_{x^2-y^2}$. (b) Shows a comparison between the bilayer and bulk DOS, while the inset shows the van Hove features of the two-dimensional electronic structure of the bilayer. (c) Band structure of bilayer in nonmagnetic state. (d) The Fermi surface in the nonmagnetic (blue) and ferromagnetic states (red).

Figure 4

Panel (a) shows tight-binding bands obtained for a three orbital model with parameters tuned to reproduce qualitative agreement with the DFT results. Panel (b) shows the evolution of ${\sigma }_{xy}$ as a function of the breathing anisotropy parameter $r$ (see text). Panel (c) shows the superconducting order parameter in the reduced one orbital spin-polarized tight-binding model with nearest neighbor attractive interactions mediated by ferromagnetic fluctuations. The thickness of the bonds is proportional to the amplitude of the superconducting order parameter and the colors encode its phase. Panel (d) shows the topological superconductor and magnetic metal (with ${\sigma }_{xy}\ne 0$) as a function of $r$ for the one-orbital tight-binding model.