Ferromagnetic helical nodal line and Kane-Mele spin-orbit coupling in kagome metal ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$

The two-dimensional kagome lattice hosts Dirac fermions at its Brillouin zone corners $K$ and $K^{\prime }$, analogous to the honeycomb lattice. In the density functional theory electronic structure of ferromagnetic kagome metal ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$, without spin-orbit coupling, we identify two energetically split helical nodal lines winding along $z$ in the vicinity of $K$ and $K^{\prime }$ resulting from the trigonal stacking of the kagome layers. We find that hopping across A-A stacking introduces a layer splitting in energy while that across A-B stacking controls the momentum space amplitude of the helical nodal lines. We identify the latter to be one order of magnitude weaker than the former owing to the underlying $d$-orbital degrees of freedom. The effect of spin-orbit coupling is found to resemble that of a Kane-Mele term, where the nodal lines can either be fully gapped to quasi-two-dimensional massive Dirac fermions, or remain gapless at discrete Weyl points depending on the ferromagnetic moment orientation. Aside from numerically establishing ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ as a model Dirac kagome metal by clarifying the roles played by interplane coupling, our results provide insights into materials design of topological phases from the lattice point of view, where paradigmatic low dimensional lattice models often find realizations in crystalline materials with three-dimensional stacking.

Figure 1

Nodal lines and three-dimensional stacking of the kagome lattice. (a) The two-dimensional (2D) kagome lattice and (b) associated Dirac fermions in the hexagonal Brillouin zone (BZ). Blue and red Dirac fermions at $K$ and $K^{\prime }$ possess opposite chiralities. (c) Schematic of a three-dimensional (3D), A-A stacked kagome lattice, and (d) the corresponding vertical nodal lines in the hexagonal prism BZ. (e) Schematic of an A-B-C stacking of the kagome lattice and the corresponding helical nodal lines are shown in (f) and (g) from both an isometric (f) and a top (g) perspective. In (c) and (e), the dashed lines represent the interplane hopping $t_1$ while the in-plane kagome bonds are characterized by an hopping integral $t_0$. In (f), we show the rhombohedral BZ along with a hexagonal prism extended from the 2D BZ illustrated in (b). (h) Surface spectra weight of the A-B-C kagome tight-binding model. A drumhead surface state (DSS) can be identified as the flat and bright intensity within the projection of the helical nodal line to the surface BZ.

Figure 2

Scalar-relativistic electronic structure and helical nodal lines in ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$. (a) Crystal structure of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ with iron atoms shown in red and tin atoms in gray. The primitive rhombohedral unit cell containing a kagome bilayer is outlined in gray. (b) Schematic of the rhombohedral BZ of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ with high-symmetry points labeled and high-symmetry directions highlighted in blue. The gray hexagonal prism is extended from the hexagonal BZ in the 2D limit. (c) Scalar-relativistic generalized gradient approximation (GGA) DFT electronic structure of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ where the majority spin is shown in red and minority in blue. (d)-(f) DFT energy-momentum dispersion of the upper helical nodal line within a region of $0.16\times 0.16{\AA }^{-2}$ close to $K$ at (d) $k_z=0$, (e) $k_z=0.317{\AA }^{-1}$, (f) $k_z=-0.317{\AA }^{-1}$ planes, respectively. The black sphere denotes the $K$ point of the 2D BZ. (g) The helical nodal lines around $K$ and $K$′ in ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$; the upper nodal line is shown in red and lower nodal line in blue. [(h) and (i)] Magnified top view of the helical nodal lines at $K$ (h) and $K$′ (i), respectively. The color gradient in (h) and (i) reflects the value of $k_z$. (j) Energy dispersion of both the upper and lower nodal lines along $k_z$. (k) Surface spectra weight of the lower nodal line inferred from the $\mathit{k}·\mathit{p}$ model (see Methods). (l) Schematic of two distinct momentum regions DSS1 (blue) and DSS2 (purple) that host different number of drumhead surface states. The hypotrochoid curve represents the projection of lower nodal line to the top surface.

Figure 3

Electronic structure of ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ with spin-orbit coupling. (a) The gap along the helical nodal line at each $k_z$ with ferromagnetic moment out-of-plane (red) and in-plane along [100] (blue) for the upper nodal line. (b) A similar analysis for the lower nodal line. (c) Weyl points originated from the lower helical nodal line with magnetic moment along [100]. The nodal line itself is shown in magenta. (d) The Berry curvature ${\mathrm{\Omega }}_z$ distribution with the ferromagnetic moments pointing out-of-plane along $K$′-$\mathrm{\Gamma }\text{−}K$ high-symmetry line in the band structure. (e), (f) Distribution of integrated Berry curvature (see text) up to the upper Dirac gap in the 3D BZ at $E=-0.1$ eV in a 3D view (e) and top view (f), respectively.

Figure 4

Layer splitting and interplane hopping in ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$. (a) Schematic of AA-BB-CC kagome lattice model with in-plane hopping $t_0$ (dark solid bonds), interplane hopping $t_{aa}$ (light solid bonds) and $t_{ab}$ (dashed bonds). [(b)–(e)] Double Dirac structure near $K$ (momentum line illustrated in (b) inset) at selected $t_{aa}$ and $t_{ab}$ ($t_0=1$): (b) $t_{aa}=0.2,t_{ab}=0.03$, (c) $t_{aa}=0.1,t_{ab}=0.03$, (d) $t_{aa}=0.2,t_{ab}=0.05$, and (e) $t_{aa}=0.2,t_{ab}=0.1$. The insets show the upper (red) and lower (blue) nodal lines. Here the momentum $k_x$ is expressed in the unit of $a^{-1}$, where $a$ is the in-plane hexagonal lattice constant. [(f) and (g)] Contour plots of the energy splitting $\mathrm{\Delta }E$ (f) and momentum displacement $\mathrm{\Delta }k$ (g) in the $t_{aa}\text{−}{t}_{ab}$ phase space. (h) Initial wave function on a single kagome layer used to project out double Dirac structure in AA-BB-CC model. The color illustrates the phase of wavefunction: 1 (red), $\omega =e^{i2\pi /3}$ (green) and $\overline{\omega }=e^{-i2\pi /3}$ (blue). The other partner state is obtained by the mirror $\mathcal{M}$ operation defined by the dashed grey line. (i) The $\mathit{k}·\mathit{p}$ 4-band model (red) compared to the full six-band AA-BB-CC model. (j) Initial basis set of a single Fe kagome layer for $\mathit{k}·\mathit{p}$ projection of the double Dirac cones in ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$. The rotated local coordinate frames for Fe sites are shown with the out-of-plane ${\widehat{z}}^{\prime }$ axis. The wave function for projection is the product of the phases in (h) and the local $d_{xy}$ orbitals. The partner state on the same layer can be obtained by mirror $\mathcal{M}$. (k) ${\mathrm{Fe}}_3{\mathrm{Sn}}_2$ band structure (blue) compared with the projected four-band $\mathit{k}·\mathit{p}$ expansion (red) near $K$ point.