Shandite Structure: Rhombohedral Kagome Framework

• Shandite structure is a rhombohedral kagome-layered framework in which transition-metal atoms form interlinked kagome sheets with inequivalent cation sites.
• The framework features an ABC stacking pattern and distinct in-plane versus interlayer roles, contributing to its magnetic anisotropy and topological behavior.
• Chemical flexibility and targeted substitutions in shandites enable tuning of electronic saddle points and structural distortions, impacting stability and emergent phenomena.

Shandite structure denotes a rhombohedral kagome-layered structural family most commonly written as $A_3M_2X_2$, $T_3M_2X_2$, or $M_3A_2Ch_2$, in which transition-metal atoms form kagome or Kagome-like sheets that are stacked along the crystallographic $c$ axis and linked by inequivalent post-transition-metal and chalcogen sites. In the literature considered here, the parent symmetry is typically the hexagonal setting of $R\bar{3}m$ (No. 166), and the defining architectural features are layered kagome-bearing metal sheets, distinct in-plane and interlayer cation roles, and rhombohedral ABC stacking rather than simple single-layer repetition (Aziz et al., 2016, Kassem et al., 2017).

1. Family definition and crystallographic symmetry

Shandites are described in several closely related notational schemes. One paper introduces the family as chalcogenides of general formula $A_3M_2X_2$, with $A=$ Ni, Co, Rh, Pd and $M=$ Pb, In, Sn, Tl, while another uses $T_3M_2X_2$ with $T=$ Ni, Co, Rh, Pd; $T_3M_2X_2$0 Sn, In, Pb; $T_3M_2X_2$1 S, Se. A broader structural survey of Rh- and Pd-based members uses $T_3M_2X_2$2 for the same underlying framework. Despite the change in symbols, these descriptions converge on a common structural type: rhombohedral $T_3M_2X_2$3 shandites built from transition-metal kagome layers embedded in a three-dimensionally stacked lattice (Aziz et al., 2016, Sharma et al., 4 Mar 2025, Buiarelli et al., 31 Jul 2025).

The symmetry is routinely expressed in the hexagonal setting of $T_3M_2X_2$4, but the lattice is equally described by a primitive rhombohedral cell. In a structural family analysis, the conventional hexagonal cell contains three symmetry-equivalent kagome layers stacked in an $T_3M_2X_2$5 sequence, whereas the primitive rhombohedral cell contains one kagome layer. The same study emphasizes that the parent $T_3M_2X_2$6 structure has point group $T_3M_2X_2$7, and that the threefold roto-inversion symmetry permits perfect, non-distorted kagome layers in the parent phase (Buiarelli et al., 31 Jul 2025).

This combination of layered motif and rhombohedral connectivity is central to the identity of the structure. A recurring misconception is to treat shandites as merely two-dimensional kagome sheets. The crystallographic evidence instead supports a layered but fully three-dimensional framework, in which the kagome layers are structurally privileged but not isolated.

2. Local architecture, inequivalent sites, and coordination topology

At the local level, shandites are defined by a split cation topology. In Ni$T_3M_2X_2$8Sn$T_3M_2X_2$9S$M_3A_2Ch_2$0, the structure is described as consisting of sheets of metal atoms, both $M_3A_2Ch_2$1 and one $M_3A_2Ch_2$2 site, in a Kagome-like hexagonal network capped above and below by $M_3A_2Ch_2$3 atoms and stacked in ABC sequence; a second $M_3A_2Ch_2$4 site lies between the Kagome sheets with trigonal anti-prismatic coordination to the chalcogen atoms. This two-site $M_3A_2Ch_2$5-sublattice is therefore a structural feature of the framework itself rather than a material-specific anomaly (Aziz et al., 2016).

In Co$M_3A_2Ch_2$6Sn$M_3A_2Ch_2$7S$M_3A_2Ch_2$8, the local coordination is described more explicitly. One Co atom is coordinated by four Sn atoms on the Sn2 site and two S atoms, producing a tetragonal bipyramid, equivalently a distorted octahedron. These Co-centered CoSn$M_3A_2Ch_2$9S$c$0 units face-share within the $c$1 plane and corner-share along $c$2. The slabs of CoSn$c$3S$c$4 octahedra then stack in hexagonal A–B–C sequence, while Sn1 atoms lie between the slabs and interconnect them. In that description, Sn2 belongs directly to the Co coordination shell, whereas Sn1 acts as an interlayer connector (Yan et al., 2018).

Two explicit crystallographic realizations reported in the literature are summarized below.

System	Explicit site data	Source
Co$c$5Sn$c$6S$c$7	Co: 9e $c$8; S: 6c $c$9; Sn: 3a $R\bar{3}m$0, 3b $R\bar{3}m$1	(Nagpal et al., 2022)
$R\bar{3}m$2Pb$R\bar{3}m$3Ch$R\bar{3}m$4 in $R\bar{3}m$5	$R\bar{3}m$6: 9e $R\bar{3}m$7; Pb: 3a $R\bar{3}m$8, 3b $R\bar{3}m$9; $A_3M_2X_2$0: 6c $A_3M_2X_2$1	(Basak et al., 2023)

These explicit site assignments show that the shandite lattice accommodates two inequivalent heavy-element positions even when the transition-metal kagome net remains unchanged. A plausible implication is that much of the chemical tunability of shandites follows from selective occupation of these non-equivalent in-plane and interlayer sites.

3. Kagome planes, rhombohedral stacking, and structural anisotropy

In Co$A_3M_2X_2$2Sn$A_3M_2X_2$3S$A_3M_2X_2$4, the layered motif is stated directly: the compound consists of Co–Sn metallic layers stacked along the $A_3M_2X_2$5-axis in the hexagonal setting of $A_3M_2X_2$6, separated by Sn–S blocks, and in each layer the Co atoms are arranged in a two-dimensional kagomé sublattice. The structurally relevant directional distinction is therefore between the kagome-bearing $A_3M_2X_2$7 plane and the stacking direction $A_3M_2X_2$8, a distinction that the same work ties to strong magnetic anisotropy and an easy axis along $A_3M_2X_2$9 (Kassem et al., 2017).

A transport study on micro-ribbons restates the same framework in slightly different language: Co$A=$0Sn$A=$1S$A=$2 crystallizes in the shandite crystal structure featuring metallic Co–Sn layers stacked along the $A=$3-axis in an ABC fashion, with Co atoms building a corner-sharing kagome lattice in the $A=$4 plane. The associated morphology is highly anisotropic: single crystals grow as thin hexagonal flakes with the $A=$5-direction as the surface normal, and focused-ion-beam micro-ribbons were prepared with [001] out of plane and [110], [$A=$6] in plane (Geishendorf et al., 2018).

Thin-flake work adds another structural rendering of the same motif. There, Co$A=$7Sn$A=$8S$A=$9 is described as a quasi-two-dimensional layered crystal in which magnetic Co atoms form a 2D kagome lattice, an Sn atom sits at the center of the hexagon, the resulting $M=$0 unit is sandwiched by two S layers, and these units are connected by hexagonal Sn layers. That work also states explicitly that Co$M=$1Sn$M=$2S$M=$3 has relatively strong interlayer coupling unlike van der Waals 2D crystals, which corrects any over-identification of shandites with weakly bonded exfoliable layered solids (Tanaka et al., 2020).

Taken together, these descriptions fix the structural meaning of the kagome motif in shandites. The kagome net is the in-plane organizing principle, but the framework remains rhombohedrally stacked, chemically differentiated, and interlayer-coupled.

4. Thin films, monolayers, and termination-dependent shandite derivatives

The shandite framework persists in thin-film form. In Co$M=$4Sn$M=$5S$M=$6-based films, the parent structure is described as rhombohedral $M=$7 with alternating $M=$8 and $M=$9 planes stacked along $T_3M_2X_2$0, and with the Co atoms forming a kagome lattice in the $T_3M_2X_2$1 plane. Undoped and substituted films grown on $T_3M_2X_2$2 are strongly $T_3M_2X_2$3-textured, clear Laue fringes appear around the $T_3M_2X_2$4 reflection, in-plane $T_3M_2X_2$5 scans of $T_3M_2X_2$6 show twinned domains separated by $T_3M_2X_2$7, and the crystal structure is maintained throughout the Ni- and In-substituted series because Co$T_3M_2X_2$8Sn$T_3M_2X_2$9S$T=$0, $T=$1, and $T=$2 are isostructural (Lau et al., 2022).

Ab initio studies of atomically thin Co$T=$3Sn$T=$4S$T=$5 extend the same framework to slabs containing one, two, or three Co kagome layers. These films retain nearly $T=$6 symmetry and fall into two structurally distinct classes: Sn-end films of composition $T=$7 and S-end films of composition $T=$8. Sn-end films remain close to the bulk in-plane lattice constant, whereas S-end films show stronger in-plane contraction, especially in the monolayer limit; most bond angles remain near bulk values except for surface $T=$9, which is notably modified (Nakazawa et al., 2022).

A separate monolayer study uses a one-kagome-layer slab derived from the bulk shandite lattice and optimized before symmetrization back to $T_3M_2X_2$00. It distinguishes a Sn-end monolayer with Sn atoms at both surfaces from an S-end monolayer with S atoms at both surfaces, while retaining the Co kagome core. The resulting two-dimensional structures preserve threefold rotational symmetry around $T_3M_2X_2$01, a twofold rotation around the $T_3M_2X_2$02 axis, and mirror symmetry perpendicular to the $T_3M_2X_2$03 axis (Nakazawa et al., 2024).

These reduced-dimensional realizations demonstrate that “shandite structure” remains meaningful below the bulk limit, but only if surface termination is treated as a structural variable rather than a secondary detail.

5. Magnetic and topological consequences of the shandite framework

The layered shandite framework is repeatedly used to interpret magnetic dimensionality. In Co$T_3M_2X_2$04Sn$T_3M_2X_2$05S$T_3M_2X_2$06, critical-exponent analysis yields a second-order ferromagnetic transition with exponents inconsistent with short-range three-dimensional models; instead, the exchange was described by $T_3M_2X_2$07 with $T_3M_2X_2$08, and the authors stated that the material is best described by $T_3M_2X_2$09, $T_3M_2X_2$10, consistent with the layered structure and the easy axis along $T_3M_2X_2$11 (Yan et al., 2018).

Effective-model work resolves the structural origin of the low-energy electronic sector in more detail. Co$T_3M_2X_2$12Sn$T_3M_2X_2$13S$T_3M_2X_2$14 is described there as a primitive rhombohedral unit cell containing three Co atoms, two Sn atoms, and two S atoms, with Co forming a kagome layer, Sn2 occupying the centers of the kagome hexagons, Sn1 forming a separate triangular layer between kagome layers, and two triangular layers of S also present. The minimal model keeps one Co $T_3M_2X_2$15 orbital and one interlayer Sn1 $T_3M_2X_2$16 orbital near the Fermi level, while the hexagon-centered Sn2 site is treated as the main source of effective spin–orbit coupling (Ozawa et al., 2019).

First-principles work on hole-doped Co$T_3M_2X_2$17In$T_3M_2X_2$18Sn$T_3M_2X_2$19S$T_3M_2X_2$20 uses the same structural logic. The parent shandite has two inequivalent Sn Wyckoff positions, one inside the Co kagome layer and one inter-layer; In substitution is modeled specifically on the inter-layer Sn site. Within that preserved average shandite framework, mirror-symmetry-protected nodal lines appear without spin–orbit coupling, and spin–orbit coupling gaps the nodal lines while leaving Weyl nodes in the parent compound (Yanagi et al., 2020).

A broader survey of Co-based shandites then sharpens the structure–topology relation: the nodal lines of the Co-based shandite structure originate from interlayer coupling between Co atoms in different kagome layers, whereas the number and type of Weyl points are mainly governed by the interaction between Co and the metal atoms occupying the non-Co sites. In ordered Co$T_3M_2X_2$21SnPbS$T_3M_2X_2$22, simply exchanging which element occupies the in-plane site and which occupies the interlayer site switches the phase between a three-dimensional quantum anomalous Hall metal and a magnetic Weyl semimetal (Luo et al., 2022).

Recent work extends the same structural interpretation to magnetic textures. In Co$T_3M_2X_2$23Sn$T_3M_2X_2$24S$T_3M_2X_2$25Se$T_3M_2X_2$26, a spin-chiral interaction was identified as arising from the kagome lattice topology itself, while the symmetry-allowed alternating-layer Dzyaloshinskii–Moriya interaction was found to be negligible in the centrosymmetric end members because the allowed layer-resolved DMI vectors sum to zero (Le et al., 14 Jan 2026).

6. Compositional flexibility, distortions, and stability limits

The shandite framework is chemically flexible but not infinitely so. In Ni$T_3M_2X_2$27In$T_3M_2X_2$28S$T_3M_2X_2$29Se$T_3M_2X_2$30, powder XRD and Rietveld analysis show that the whole series remains a rhombohedral $T_3M_2X_2$31 shandite, while Se substitution increases the lattice constants systematically, especially the $T_3M_2X_2$32-axis parameter. The reported sequence from Ni$T_3M_2X_2$33In$T_3M_2X_2$34S$T_3M_2X_2$35 to Ni$T_3M_2X_2$36In$T_3M_2X_2$37Se$T_3M_2X_2$38 therefore exemplifies an isostructural chalcogen-substitution series rather than a symmetry-breaking transformation (Sharma et al., 4 Mar 2025).

Other substitutions do destabilize the parent phase. In the $T_3M_2X_2$39Pb$T_3M_2X_2$40Ch$T_3M_2X_2$41 series, $T_3M_2X_2$42, $T_3M_2X_2$43, and $T_3M_2X_2$44 are dynamically stable in $T_3M_2X_2$45, but $T_3M_2X_2$46 is unstable in the ideal shandite phase and relaxes to $T_3M_2X_2$47. The driving mode is a soft phonon that distorts the kagome sublattice, doubles the periodicity along the rhombohedral axis, and shifts Pt by about $T_3M_2X_2$48 Å relative to the ideal kagome positions (Basak et al., 2023).

A large first-principles study of Rh- and Pd-based shandites places this behavior in a broader context. Among twenty hypothetical or candidate compounds treated in the ideal $T_3M_2X_2$49 structure, nineteen were found dynamically stable at ambient conditions and only Rh$T_3M_2X_2$50Sn$T_3M_2X_2$51Se$T_3M_2X_2$52 showed an ambient instability. More significantly, in compounds such as Rh$T_3M_2X_2$53Tl$T_3M_2X_2$54S$T_3M_2X_2$55 and Pd$T_3M_2X_2$56Sn$T_3M_2X_2$57Se$T_3M_2X_2$58, hole doping or hydrostatic pressure can drive F- or L-point phonon softening when kagome-derived electronic saddle points are moved close to the Fermi level; increasing the electronic smearing temperature restores stability in several of these cases, suggesting a charge-density-wave-like electronically driven structural instability rather than a purely steric one (Buiarelli et al., 31 Jul 2025).

Co-based shandites show a related stability pattern. A global structural search over $T_3M_2X_2$59 compositions found the shandite structure to be the ground state for Co–Sn–S, Co–Pb–S, Co–Pb–Se, and ordered Co$T_3M_2X_2$60SnPbS$T_3M_2X_2$61, metastable for several Ge- and Se-containing variants, and unstable for Co–Ge–Se and the Te-based systems. In that survey, the empirical structure tolerance factor $T_3M_2X_2$62 was introduced, and large $T_3M_2X_2$63, roughly $T_3M_2X_2$64, favored the shandite structure (Luo et al., 2022).

This body of evidence supports a restrained general conclusion. Shandite structure is a robust rhombohedral kagome-layer framework, but its stability is not merely geometric: it is highly sensitive to which elements occupy the inequivalent metal sites, how chalcogen substitution modifies the rhombohedral metric, and whether electronic saddle points at symmetry-selected momenta are brought close enough to the Fermi level to drive soft-mode distortions.