Kagome-type IAMX Family

• The kagome-type IAMX family comprises layered compounds with embedded kagome sublattices that drive unique electronic phases such as density waves, superconductivity, and frustrated magnetism.
• High-throughput DFT, ARPES, and model Hamiltonians systematically reveal distinct structural archetypes and instabilities, including CDW transitions and Weyl point formation.
• Chemical tuning—via A-site radius, ligand substitution, and interlayer coupling—enables precise control over magnetic frustration, quantum confinement, and topological phases across various prototypes.

In the arXiv literature, the expression “kagome-type IAMX family” does not denote a single stoichiometric series. It is used for several materials classes in which a kagome sublattice is embedded in layered or stacked solids, including the hexagonal $AM_6X_6$ and $AB_3C_5$ kagome metals, the $P\bar{6}2m$ stacked kagome–honeycomb semimetals, Ti(III) fluorides that may be written in IAMX stoichiometry, and atacamite-derived hydroxides (Ortiz et al., 2024, Silva et al., 21 Mar 2025, Zhou et al., 2024, Jeschke et al., 2019, Puphal et al., 2018). Across these studies, the kagome motif is associated with density waves, superconductivity, topological semimetallic and insulating phases, quantum confinement at surfaces, and a wide range of frustrated or metallic magnetic ground states.

1. Nomenclature and materials scope

The literature uses “IAMX” in more than one way. In the ScV$_6$Sn$_6$ work, the hexagonal “166” compounds adopt the generic stoichiometry $I\,A\,M_6\,X_6$, abbreviated as IAMX, with $I$ on the $1a$ site, $M$ forming the two kagome nets per cell, and $X$ occupying both triangular-lattice and prismatic sites (Arachchige et al., 2022). In the high-throughput $AB_3C_5$0 literature, the same label is applied to the kagome-type IA–M–X family derived from the AV$AB_3C_5$1Sb$AB_3C_5$2 prototype (Silva et al., 21 Mar 2025). In the stacked kagome–honeycomb semimetal literature, IAMX denotes compounds with IA = alkali metal element, $AB_3C_5$3 = rare earth metal element, and $AB_3C_5$4 = carbon group element in space group $AB_3C_5$5 (Zhou et al., 2024). Ti(III) fluorides such as Rb$AB_3C_5$6NaTi$AB_3C_5$7F$AB_3C_5$8, Cs$AB_3C_5$9NaTi$P\bar{6}2m$0F$P\bar{6}2m$1, and Cs$P\bar{6}2m$2KTi$P\bar{6}2m$3F$P\bar{6}2m$4 are also described as having IAMX stoichiometry, while atacamite-derived kagome hydroxides are discussed as kagome-type IAMX compounds in a broader chemical sense (Jeschke et al., 2019, Puphal et al., 2018). This suggests that “IAMX” functions in practice as a family label for kagome-centered chemistry rather than a single formal composition rule.

Materials class	Prototype / space group	Representative phenomena
$P\bar{6}2m$5 kagome metals	HfFe$P\bar{6}2m$6Ge$P\bar{6}2m$7-type, $P\bar{6}2m$8	density waves, rare-earth magnetism
$P\bar{6}2m$9 kagome metals	AV$_6$0Sb$_6$1-type, $_6$2	CDW, superconductivity, topological metal behavior
Stacked kagome–honeycomb semimetals	Fe$_6$3P-type, $_6$4	nodal lines, Weyl points, mirror Chern numbers
Ti(III) fluorides	$_6$5	$_6$6 frustration, $_6$7 plateau
Atacamite-derived hydroxides	$_6$8 or $_6$9	AFM, FM, highly frustrated kagome magnetism

A common misconception is that all kagome-type IAMX materials share one crystallographic archetype. The data instead show several distinct structural lineages, with the kagome geometry realized in different ways: perfect 2D kagome nets in the $_6$0 and $_6$1 planes of HfFe$_6$2Ge$_6$3-type compounds, three-site kagome nets in AV$_6$4Sb$_6$5-type layers, twisted kagome–honeycomb stacks in $_6$6, buckled distorted kagome planes in Ti fluorides, and Cu-based kagome sheets in atacamites (Ortiz et al., 2024, Silva et al., 21 Mar 2025, Zhou et al., 2024, Jeschke et al., 2019, Puphal et al., 2018).

2. Structural archetypes and stability frontiers

In the $_6$7 kagome metals, the pristine space group is $_6$8 (No. 191), with typical lattice parameters $_6$9–$I\,A\,M_6\,X_6$0 Å and $I\,A\,M_6\,X_6$1–$I\,A\,M_6\,X_6$2 Å. For the HfFe$I\,A\,M_6\,X_6$3Ge$I\,A\,M_6\,X_6$4-type structure exemplified by LnNb$I\,A\,M_6\,X_6$5Sn$I\,A\,M_6\,X_6$6, the rare-earth $I\,A\,M_6\,X_6$7 site occupies $I\,A\,M_6\,X_6$8 $I\,A\,M_6\,X_6$9, the transition-metal site occupies $I$0 $I$1 and forms perfect 2D kagome nets in the $I$2 and $I$3 planes, and the main-group element occupies two sites, $I$4 and $I$5. For LuNb$I$6Sn$I$7 at 100 K, single-crystal refinement gives $I$8 Å, $I$9 Å, Lu–Nb $1a$0 Å, Nb–Sn $1a$1 Å, and in-plane Sn$1a$2–Sn$1a$3 $1a$4 Å (Ortiz et al., 2024).

Ortiz et al. evaluated the known $1a$5 chemistries and constructed a stability diagram for the $1a$6 member family. Five major structural variants were identified: ordered HfFe$1a$7Ge$1a$8-type, “SmMn$1a$9Sn$M$0”-type disorder, “Y$M$1Co$M$2Ge$M$3” half-filled/half-vacant kagome layers, various $M$4 supercells or stacking variants, and orthorhombic distortions of the hexagonal prototype. Chemical and steric control is expressed through the $M$5-site Shannon radius $M$6 and the rigid kagome-network scale set by the $M$7–$M$8 bond lengths and unit-cell volume $M$9. The empirical frontier is written as

$X$0

with $X$1 Å, or equivalently

$X$2

with $X$3 Å$X$4/Å (Ortiz et al., 2024).

The AV$X$5Sb$X$6-type $X$7 kagome metals also adopt $X$8 (No. 191), with a hexagonal primitive cell, $X$9–$AB_3C_5$00 Å and $AB_3C_5$01–$AB_3C_5$02 Å. In the high-throughput study of kagome compounds in this family, the A site lies at Wyckoff $AB_3C_5$03, the kagome-forming M site at Wyckoff $AB_3C_5$04, and the X sites at Wyckoff $AB_3C_5$05 and $AB_3C_5$06. KV$AB_3C_5$07Sb$AB_3C_5$08, RbV$AB_3C_5$09Sb$AB_3C_5$10, and CsV$AB_3C_5$11Sb$AB_3C_5$12 lie exactly on the convex hull, and the screening identified 36 additional hull compounds together with 269 compounds within 50 meV/atom and 1,386 within 100 meV/atom. The formation energy and hull distance were defined as

$AB_3C_5$13

with competing phases from the Alexandria convex-hull dataset (Silva et al., 21 Mar 2025).

A complementary high-throughput first-principles study proposed 24 dynamically stable novel kagome metals derived from the AV$AB_3C_5$14Sb$AB_3C_5$15 prototype. Twenty-two retain $AB_3C_5$16, while CsRu$AB_3C_5$17Ge$AB_3C_5$18 and RbCr$AB_3C_5$19Te$AB_3C_5$20 distort to $AB_3C_5$21 via a twisted-triangle kagome net, and the lattice parameters of the 24 compounds span roughly $AB_3C_5$22–$AB_3C_5$23 Å and $AB_3C_5$24–$AB_3C_5$25 Å (Yi et al., 2022).

3. Density-wave, lattice-instability, and superconducting regimes

Density-wave behavior is a central theme in the hexagonal kagome metals. ScV$AB_3C_5$26Sn$AB_3C_5$27 is a hexagonal HfFe$AB_3C_5$28Ge$AB_3C_5$29-type compound that undergoes a first-order phase transition at $AB_3C_5$30 K. Single-crystal X-ray and neutron diffraction revealed a charge density wave modulation with wave vector

$AB_3C_5$31

which triples the in-plane cell and the repeat along $AB_3C_5$32. Superlattice reflections appear at $AB_3C_5$33 below $AB_3C_5$34, the neutron $AB_3C_5$35 peak shows a sharp onset with hysteresis, $AB_3C_5$36 drops by $AB_3C_5$37, $AB_3C_5$38 drops by $AB_3C_5$39, and $AB_3C_5$40 increases by $AB_3C_5$41 while $AB_3C_5$42 is nearly unchanged. The dominant atomic displacements involve Sc and Sn$AB_3C_5$43 moving along $AB_3C_5$44 by up to $AB_3C_5$45 Å, while the V sites shift by only $AB_3C_5$46–$AB_3C_5$47 Å in plane (Arachchige et al., 2022).

LuNb$AB_3C_5$48Sn$AB_3C_5$49 exhibits an analogous but not identical instability. It undergoes a first-order DW-like transition at $AB_3C_5$50 K with ordering wave vector

$AB_3C_5$51

corresponding to a $AB_3C_5$52 supercell. X-ray scattering shows diffuse scattering on planes $AB_3C_5$53 above $AB_3C_5$54 and weak superlattice reflections with $AB_3C_5$55 below $AB_3C_5$56. The thermodynamic and transport signatures include a sharp drop in $AB_3C_5$57 for both $AB_3C_5$58 and $AB_3C_5$59, a latent-heat feature and thermal hysteresis in $AB_3C_5$60, and a $AB_3C_5$61–$AB_3C_5$62 drop in $AB_3C_5$63 through $AB_3C_5$64. The structural analysis supports the “rattling mode” DW model proposed for ScV$AB_3C_5$65Sn$AB_3C_5$66, with the chain-displacement order parameter described by

$AB_3C_5$67

where $AB_3C_5$68 and $AB_3C_5$69 is a strain-mediated repulsive coupling on the kagome network (Ortiz et al., 2024).

The AV$AB_3C_5$70Sb$AB_3C_5$71-type family provides a broader instability context. In the high-throughput $AB_3C_5$72 survey, the AV$AB_3C_5$73Sb$AB_3C_5$74 compounds have a known CDW transition $AB_3C_5$75–$AB_3C_5$76 K and superconducting $AB_3C_5$77–$AB_3C_5$78 K, while phonon dispersions show that 35 of 36 hull compounds have imaginary modes in pristine $AB_3C_5$79, indicating a tendency to undergo $AB_3C_5$80 Star-of-David or Inverse-Star distortions. INb$AB_3C_5$81Bi$AB_3C_5$82 is the only dynamically stable pristine kagome, with $AB_3C_5$83, $AB_3C_5$84 K, and $AB_3C_5$85 K from the Allen–Dynes formula with $AB_3C_5$86 (Silva et al., 21 Mar 2025).

Among the 24 dynamically stable kagome metals proposed in the AV$AB_3C_5$87Sb$AB_3C_5$88 prototype study, fourteen exhibit $AB_3C_5$89 and $AB_3C_5$90 K $AB_3C_5$91 K. KZr$AB_3C_5$92Pb$AB_3C_5$93 has the highest reported $AB_3C_5$94, with $AB_3C_5$95 and $AB_3C_5$96 K. NaZr$AB_3C_5$97As$AB_3C_5$98 shows pronounced imaginary phonon modes at the $AB_3C_5$99 and $P\bar{6}2m$00 points, and symmetry analysis yields two candidate CDW phases: CDW I in space group Ibam with $P\bar{6}2m$01 meV per 72-atom cell and Zr displacement $P\bar{6}2m$02 Å, and CDW II in $P\bar{6}2m$03 with $P\bar{6}2m$04 meV and Zr displacement $P\bar{6}2m$05 Å (Yi et al., 2022).

4. Electronic topology, surface states, and quantum confinement

The AV$P\bar{6}2m$06Sb$P\bar{6}2m$07-type high-throughput studies identify a recurring electronic template. All stable kagomes in the $P\bar{6}2m$08 survey exhibit Dirac points at the $P\bar{6}2m$09 point, Van Hove singularities at $P\bar{6}2m$10 typically $P\bar{6}2m$11–$P\bar{6}2m$12 meV below $P\bar{6}2m$13, and flat bands. Groups 12, 13, and Ce-based compounds have a nearly flat band just above or below $P\bar{6}2m$14 along $P\bar{6}2m$15–$P\bar{6}2m$16–$P\bar{6}2m$17, while group 15 compounds show more extended flat-band manifolds across the Brillouin zone. The study also reports representative band placements, including a Dirac cone $P\bar{6}2m$18 meV above $P\bar{6}2m$19 in PmBe$P\bar{6}2m$20Au$P\bar{6}2m$21, a Van Hove singularity $P\bar{6}2m$22 meV below $P\bar{6}2m$23 and a flat band crossing $P\bar{6}2m$24 along $P\bar{6}2m$25 in KPd$P\bar{6}2m$26Hg$P\bar{6}2m$27, and multiple Dirac points within $P\bar{6}2m$28 meV of $P\bar{6}2m$29 in BaTi$P\bar{6}2m$30Bi$P\bar{6}2m$31 (Silva et al., 21 Mar 2025).

Angle-resolved photoemission on RV$P\bar{6}2m$32Sn$P\bar{6}2m$33 ($P\bar{6}2m$34 Gd, Ho) directly resolves the canonical kagome spectral features in two-dimensional surface states of the vanadium kagome layer. The surface-state band structure shows a Dirac cone at $P\bar{6}2m$35 with $P\bar{6}2m$36 eV in GdV$P\bar{6}2m$37Sn$P\bar{6}2m$38 and $P\bar{6}2m$39 eV in HoV$P\bar{6}2m$40Sn$P\bar{6}2m$41, a saddle point at $P\bar{6}2m$42 near $P\bar{6}2m$43 to $P\bar{6}2m$44 eV, and a nearly dispersionless flat band near $P\bar{6}2m$45 to $P\bar{6}2m$46 eV. Comparison of DFT and ARPES slopes gives a modest bandwidth renormalization factor $P\bar{6}2m$47–$P\bar{6}2m$48, and the larger layer spacing relative to FeSn is associated with more two-dimensional character (Peng et al., 2021).

A distinct topological lineage is provided by the stacked kagome–honeycomb IAMX compounds in space group $P\bar{6}2m$49 (No. 189). Starting from a minimal five-band tight-binding model with one $P\bar{6}2m$50 orbital on the honeycomb sublattice and three $P\bar{6}2m$51 orbitals on the kagome sublattice, the non-SOC Hamiltonian was written as

$P\bar{6}2m$52

and high-throughput screening of the ICSD yielded $P\bar{6}2m$53 IA–M–X compounds, reduced by DFT screening to 298 “ideal” topological semimetals, all crystallizing in $P\bar{6}2m$54. Without SOC, four dominant nodal-line configurations were identified; with SOC, each nodal ring gaps except for discrete Weyl points. For LiNdGe, the ring on the $P\bar{6}2m$55 mirror gaps into six pairs of Weyl points, the mirror Chern numbers are $P\bar{6}2m$56 and $P\bar{6}2m$57, and six Fermi arcs appear on the (001) surface (Zhou et al., 2024).

The SOC-driven extension of this program adds a phase diagram with three regimes: nodal-ring semimetal for $P\bar{6}2m$58, strong topological insulator for $P\bar{6}2m$59, and Weyl semimetal for $P\bar{6}2m$60. In the model analysis, one finds

$P\bar{6}2m$61

and a typical parameter set gives $P\bar{6}2m$62 eV and $P\bar{6}2m$63 eV. First-principles calculations place LiYC in the nodal-ring semimetal regime, LiNdGe in the Weyl semimetal window, and KLaPb in the strong TI phase with a bulk gap $P\bar{6}2m$64 eV and $P\bar{6}2m$65 (Wu et al., 24 Aug 2025).

Quantum confinement adds a further layer of kagome-surface physics in CsV$P\bar{6}2m$66Sb$P\bar{6}2m$67. ARPES and slab DFT show that surface relaxation on the polar (0001) surface enlarges the spacing between the topmost kagome layer and the second layer by $P\bar{6}2m$68 Å, creating a potential well of width $P\bar{6}2m$69 Å and depth $P\bar{6}2m$70 eV. The effective-mass quantization is written as

$P\bar{6}2m$71

with a fitted $P\bar{6}2m$72 and $P\bar{6}2m$73 eV. ARPES resolves $P\bar{6}2m$74-independent quantum-well subbands and a split Dirac cone at $P\bar{6}2m$75, while slab projections show that the observed spectra are almost entirely contributed by the top two layers (Cai et al., 2021).

5. Magnetic and frustrated-spin manifestations

The LnNb$P\bar{6}2m$76Sn$P\bar{6}2m$77 series connects kagome-metal density-wave physics to rare-earth magnetism. Within the HfFe$P\bar{6}2m$78Ge$P\bar{6}2m$79-type members, Nb is nonmagnetic and RKKY-mediated coupling on the 2D $P\bar{6}2m$80 plane dominates. Heat-capacity anomalies confirm bulk ordering, small Schottky upturns appear below 1 K in many compounds, and metamagnetic transitions in $P\bar{6}2m$81 correlate with spin reorientation within the AB-stacked kagome layers (Ortiz et al., 2024).

Ln	$P\bar{6}2m$82 (K)	Key feature
Gd	7.6	A-type AFM; metamagnetic fields 0.5, 1.2 T
Tb	8.2	multi-step spin-flop/flip; 0.2, 0.8, 3.5 T
Dy	5.8	similar A-type with anisotropy; 0.3, 1.0 T
Ho	2.1	incipient AFM, possible $P\bar{6}2m$83-axis modulation
Er	$P\bar{6}2m$84	likely $P\bar{6}2m$85 K; Schottky confound
Tm	none $P\bar{6}2m$86	paramagnet above 0.1 K

The Ti(III) fluorides Rb$P\bar{6}2m$87NaTi$P\bar{6}2m$88F$P\bar{6}2m$89, Cs$P\bar{6}2m$90NaTi$P\bar{6}2m$91F$P\bar{6}2m$92, and Cs$P\bar{6}2m$93KTi$P\bar{6}2m$94F$P\bar{6}2m$95 represent a different IAMX-related kagome magnetism. All crystallize in monoclinic $P\bar{6}2m$96, the Ti$P\bar{6}2m$97 ions form a slightly distorted, buckled kagome plane, and DFT plus energy mapping yields four inequivalent in-plane exchanges $P\bar{6}2m$98, $P\bar{6}2m$99, $_6$00, and $_6$01. The sequence Rb$_6$02Na $_6$03 Cs$_6$04Na $_6$05 Cs$_6$06K tunes the system from a nearly pure one-dimensional $_6$07-chain to a fully two-dimensional frustrated kagome antiferromagnet. Exact diagonalization reproduces a pronounced plateau at $_6$08 in all three compounds, with plateau windows of $_6$09–$_6$10 T for Rb$_6$11NaTi$_6$12F$_6$13, $_6$14–$_6$15 T for Cs$_6$16NaTi$_6$17F$_6$18, and $_6$19–$_6$20 T for Cs$_6$21KTi$_6$22F$_6$23 (Jeschke et al., 2019).

Metallic kagome antiferromagnetism in the $_6$24 setting is represented by CrRhAs. Noncollinear spin-density-functional calculations led to a Hamiltonian dominated by an antiferromagnetic second-nearest-neighbor in-plane coupling and an important ring-exchange term,

$_6$25

with $_6$26 meV, $_6$27 meV, $_6$28 meV, and $_6$29 meV. The resulting ground states are dominated by nearly isolated antiferromagnetic triangles adopting $_6$30 order with positive or negative vector chirality (Huang et al., 2022).

The atacamite-derived compounds EuCu$_6$31(OH)$_6$32Cl$_6$33, Zn$_6$34Cu$_6$35(OH)$_6$36(NO$_6$37)$_6$38, and haydeeite MgCu$_6$39(OH)$_6$40Cl$_6$41 show yet another kagome-magnetic regime. EuCu$_6$42(OH)$_6$43Cl$_6$44 is a frustrated antiferromagnet with $_6$45 K, transitions at $_6$46 K and $_6$47 K, and frustration parameter $_6$48. Zn$_6$49Cu$_6$50(OH)$_6$51(NO$_6$52)$_6$53 is essentially unfrustrated, with $_6$54 K and $_6$55. Haydeeite is a ferromagnet with $_6$56 K, easy axis in the kagome plane, and saturation moment $_6$57/Cu (Puphal et al., 2018).

6. Chemical control and design principles

A major result of the IAMX-related literature is that kagome physics can be steered by chemically transparent control parameters. In the $_6$58 family, the stability diagram emphasizes sterics: the $_6$59-site Shannon radius governs packing of the $_6$60–$_6$61–$_6$62–$_6$63 chains, the $_6$64–$_6$65 bond lengths fix the size of the rigid kagome network, and bond-modulated instabilities emerge at small $_6$66 while disorder and supercells appear at large $_6$67. Within LnNb$_6$68Sn$_6$69, the Nb$_6$70Sn$_6$71 sublattice is almost rigid, with Nb–Sn, Sn–Sn, and Nb–Nb bonds changing by $_6$72 across the Ln series, whereas the Sn$_6$73–Sn$_6$74 bond compresses by up to $_6$75 and $_6$76 and $_6$77 diverge for $_6$78 Å. The proposed interpretation is that under-filled voids for small $_6$79 produce large “rattling” amplitudes, soft low-energy phonons, and density-wave instability (Ortiz et al., 2024).

In the AV$_6$80Sb$_6$81-derived kagome metals, three independent chemical knobs recur: A, M, and X. The high-throughput surveys identify stable substitutions beyond Sb/Bi, including Au, Hg, Tl, and Ce on the C site, with unexpected Ce-based stability attributed to Ce’s variable $_6$82 oxidation states. They also emphasize that heavy-$_6$83-block C elements require small B-site metals for hull stability, and that many compounds place Dirac points, Van Hove singularities, or flat bands close to the Fermi level (Silva et al., 21 Mar 2025). The dynamically stable 24-compound set further indicates that expanded chemical flexibility from Li$_6$84Cs, 3%%%%375$I\,A\,M_6\,X_6$376%%%% B-site choice, and Ge$_6$87Bi C-site choice provides routes to stronger EPC, higher $_6$88, new CDW distortions, and nontrivial $_6$89 topology (Yi et al., 2022).

The stacked kagome–honeycomb semimetals admit explicit tight-binding design rules. In the $_6$90 family, stronger intralayer hopping favors nodal rings on vertical mirrors, larger interlayer hopping favors nodal chains over isolated rings, and the normalized ratios

$_6$91

sort the nodal-line configurations: $_6$92 and $_6$93 give rings at $_6$94, $_6$95 and $_6$96 give rings at $_6$97, $_6$98 and $_6$99 give nodal chains, and further increase of intralayer coupling yields nested rings (Zhou et al., 2024). The SOC-driven continuation of this framework proposes chemical substitution $_6$00 as a route to traverse nodal-ring semimetal $_6$01 Weyl semimetal $_6$02 strong topological insulator (Wu et al., 24 Aug 2025).

Other IAMX-related families encode analogous tuning principles. In the Ti(III) fluorides, monovalent spacer substitution continuously changes the exchange hierarchy and hence the effective dimensionality of the frustrated network (Jeschke et al., 2019). In the atacamite family, changing the $_6$03 ion and the anion $_6$04 modifies charge balance, buckling, interplane separation, and Cu–O–Cu superexchange angles, thereby moving between long-range ordered antiferromagnets, a ferromagnet, and a highly frustrated antiferromagnet close to the spin-liquid regime (Puphal et al., 2018).

Taken together, these results define the kagome-type IAMX family not as a single compound class but as a broad research program organized around kagome geometry, chemical tunability, and competing electronic, structural, and magnetic instabilities. The strongest unifying pattern is methodological: stability diagrams, high-throughput DFT, ARPES, diffraction, and model Hamiltonians are repeatedly used to connect microscopic chemical control to density waves, superconductivity, topology, and frustration across multiple crystallographic realizations of kagome matter.