Altermagnetism, Kagome Flat Band, and Weyl Fermion States in Magnetically Intercalated Transition Metal Dichalcogenides

TL;DR

Problem addressed: to realize altermagnetism together with Weyl topology and kagome flat-band physics within a single material family. Approach: comprehensive first-principles (DFT) study of XY$_4$Z$_8$ intercalated TMDs, identifying a geometric control parameter $u/v$ that governs magnetic ground states and spin splitting with SOC. Contributions: (i) AM ground states in several compounds, (ii) SOC converts altermagnetic nodal lines into Weyl points with surface Fermi arcs, and (iii) an effective kagome lattice emerges yielding flat bands near $E_F$. Significance: offers a general design principle for engineering correlated topological phases and a versatile platform for spintronic/topological devices.

Abstract

Altermagnetic (AM) compounds have recently emerged as a promising platform for realizing unconventional quantum phases, enabled by their unique spin-split band structure at zero net magnetization. Here, we present a first-principles investigation of magnetically intercalated transition metal dichalcogenides (TMDs) of the form XY$_4$Z$_8$ (X $=$ Mn, Fe, Co, Ni, Cr, or V; Y $=$ Nb or Ta; and Z $=$ Se or S), identifying a subset of new versatile AM candidates. Our results establish a direct correlation between interatomic geometry, quantified by the ratio of interlayer to intralayer spacing, and the selection of magnetic ground states. Systems with A-type antiferromagnetic order exhibit momentum-dependent spin splitting consistent with AM behavior. Crucially, the combination of the AM spin-splitting and the spin-orbit coupling leads to the emergence of Weyl nodes together with the corresponding topological Fermi arc surface states. Moreover, we identify flat bands near the Fermi level that originate from the intercalant-induced formation of an effective kagome-like sublattice in the TMD layer. These results collectively establish magnetically intercalated TMDs as a promising platform for engineering altermagnetism, flat bands, and Weyl fermions within a single material family, facilitating the development of topological and spintronic applications.

Figures (5)

• Figure 1: Crystal structure and magnetic configurations in intercalated transition metal dichalcogenides (TMDs). (a) Side view and (b) top view of the crystal structure, illustrating the atomic arrangement, where X represents intercalating transition metals (V, Cr, Mn, Fe, Co, Ni), Y represents host transition metals (Nb, Ta), and Z represents chalcogens (Se, S). The interlayer spacing ($u$) and intralayer distance ($v$) are indicated. (c) Schematic illustration of magnetic configurations, depicting antiferromagnetic/altermagnetic (AFM/AM) and ferromagnetic (FM) alignments of spins in the layered structure. Arrows indicate spin directions. (d) 3D Brillouin zone for the hexagonal lattice highlights symmetry points and paths used in band structure calculations. Spin-degenerate nodal planes are protected by the [$C_{6z}$] and [$M_z$] symmetries.
• Figure 2: DFT-calculated spin-resolved electronic band structures of altermagnetic intercalated TMDs with SOC: (a) CoNb$_4$Se$_8$, (b) FeNb$_4$Se$_8$, (c) CoTa$_4$Se$_8$, and (d) FeNb$_4$S$_8$. For each compound, the band structures are shown on the (left) nodal plane ($k_z = 0$) along the high-symmetry path $\Gamma$-M-K-$\Gamma$, and the (right) off-nodal plane ($k_z = \pi/2c$) along $\Gamma^{\prime}$-M$^{\prime}$-K$^{\prime}$-$\Gamma^{\prime}$. Red and blue lines represent spin-up and spin-down channels, respectively. Pronounced spin splitting is observed along the off-nodal $\Gamma'$–M$^{\prime}$ line.
• Figure 3: Fermi surface crossections of altermagnetic intercalated TMDs with (top panels) and without (bottom panels) SOC. Constant-energy contours are shown at the nodal plane ($k_{z}=0$, left) and the off-nodal plane ($k_{z}=\pi/2c$, right) for (a,c) CoNb$_4$Se$_8$ and (b,d) FeNb$_4$Se$_8$. Panels (a,b) display the spin-resolved Fermi surfaces in the absence of SOC, where red and blue contours correspond to spin-up and spin-down channels, respectively. Panels (c,d) show the corresponding spectra with SOC included, for which spin is no longer a good quantum number, and all contours are plotted in black. Nevertheless, the residual altermagnetic spin texture survives as momentum-dependent band splittings that reconstruct the Fermi surface topology across the off-nodal plane along $k_{z}$.
• Figure 4: Weyl points (WPs) and topological surface states in CoNb$_4$Se$_8$. (a) Distribution of WPs in the bulk Brillouin zone (BZ) and their projections onto the (001) surface BZs. Bulk WPs with chirality $\chi = \pm 1$ are indicated in dashed red and blue. In cases where two same-chirality WPs are projected onto the same point in the surface Brillouin zone, the projections appear to have effective topological charges $\chi = \pm 2$. (b) Top left: zoom of the surface spectrum along the K-path with the SOC-induced Weyl crossings. Top right: band structure showing band crossings along the same path with WPs located near $E = 0.17$ eV. Bottom: band dispersions along the $k_x$, $k_y$, and $k_z$ directions. (c) Calculated surface spectrum of the (001) surface with SOC. The inset highlights the surface Fermi arcs (SFAs), where overlapping projections of two bulk Weyl nodes give rise to double Fermi arcs (black arrows). (e) Evolution of spin-resolved hybrid Wannier charge centers around bulk WPs, with topological charges $\chi = +1$ and $\chi = -1$, demonstrating opposite chirality for isolated bulk WPs and nontrivial Berry curvature.
• Figure 5: (a) Orbital-projected band structure of CoNb$_4$Se$_8$ highlighting contributions from Co, Nb, and Se. (b) and (c) Orbital resolved Nb contributions obtained by comparing the site directly beneath the Co intercalant (left) with the kagome-like Nb sites (right). (d) Top view of the pristine hexagonal Nb network (left) can be mapped onto an effective kagome lattice (right) by removing the Nb site directly below each Co atom (Site 1). The remaining Nb sites (Sites 2, 3, and 4) define the kagome sublattice. Numbered Nb sites denote sites within the reduced lattice, while the dashed box corresponds to the unit cell chosen for the TB construction. Primitive lattice vectors $t_{1}$, $t_{2}$ and sublattice vectors $a_{1}$, $a_{2}$, $a_{3}$ define the geometry and connectivity of the model. (e) Energy spectra of the tight-binding model in Eq. \ref{['tb-h-kagome']} for three Nb lattices: hexagonal (top), kagome-like (middle), and ideal kagome (bottom). Nb' (red) denotes the site directly beneath Co, and Nb (blue) denotes the kagome sites. In the top panel (hexagonal), all hoppings and onsite potentials are equal. In the middle panel (kagome-like), we set $t'/t=-0.4$ and $\epsilon_{\mathrm{Nb}'} / t=-2.2$ (all other onsite terms zero). In the bottom panel (ideal kagome), we take $t'/t=0$ and $\epsilon_{\mathrm{Nb}'} / t=-100$.