---
title: Kagome-type IAMX Family
url: https://www.emergentmind.com/topics/kagome-type-iamx-family
type: topic
---

# Kagome-type IAMX Family

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 [2411.10635] [2503.17124] [2402.12700] [1902.10309] [1810.09201]. 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 [2205.04582]. In the high-throughput $AB_3C_5$ literature, the same label is applied to the kagome-type IA–M–X family derived from the AV$_3$Sb$_5$ prototype [2503.17124]. In the stacked kagome–honeycomb semimetal literature, IAMX denotes compounds with IA = alkali metal element, $M$ = rare earth metal element, and $X$ = carbon group element in space group $P\bar{6}2m$ [2402.12700]. Ti(III) fluorides such as Rb$_2$NaTi$_3$F$_{12}$, Cs$_2$NaTi$_3$F$_{12}$, and Cs$_2$KTi$_3$F$_{12}$ are also described as having IAMX stoichiometry, while atacamite-derived kagome hydroxides are discussed as kagome-type IAMX compounds in a broader chemical sense [1902.10309] [1810.09201]. 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 |
|---|---|---|
| $AM_6X_6$ kagome metals | HfFe$_6$Ge$_6$-type, $P6/mmm$ | density waves, rare-earth magnetism |
| $AB_3C_5$ kagome metals | AV$_3$Sb$_5$-type, $P6/mmm$ | CDW, superconductivity, topological metal behavior |
| Stacked kagome–honeycomb semimetals | Fe$_2$P-type, $P\bar{6}2m$ | nodal lines, Weyl points, mirror Chern numbers |
| Ti(III) fluorides | $P2_1/m$ | $S=1/2$ frustration, $1/3$ plateau |
| Atacamite-derived hydroxides | $P\bar{3}m1$ or $P2_1$ | 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 $z=\tfrac14$ and $\tfrac34$ planes of HfFe$_6$Ge$_6$-type compounds, three-site kagome nets in AV$_3$Sb$_5$-type layers, twisted kagome–honeycomb stacks in $P\bar{6}2m$, buckled distorted kagome planes in Ti fluorides, and Cu-based kagome sheets in atacamites [2411.10635] [2503.17124] [2402.12700] [1902.10309] [1810.09201].

## 2. Structural archetypes and stability frontiers

In the $AM_6X_6$ kagome metals, the pristine space group is $P6/mmm$ (No. 191), with typical lattice parameters $a \simeq 5.3$–$6.2$ Å and $c \simeq 8.5$–$11.5$ Å. For the HfFe$_6$Ge$_6$-type structure exemplified by LnNb$_6$Sn$_6$, the rare-earth $A$ site occupies $1a$ $(0,0,0)$, the transition-metal site occupies $6i$ $(1/3,2/3,1/4)$ and forms perfect 2D kagome nets in the $z=\tfrac14$ and $\tfrac34$ planes, and the main-group element occupies two sites, $X_1:2c(0,0,1/4)$ and $X_2:4f(1/3,2/3,z_2)$. For LuNb$_6$Sn$_6$ at 100 K, single-crystal refinement gives $a = 5.5231(2)$ Å, $c = 9.3164(4)$ Å, Lu–Nb $\simeq 3.55$ Å, Nb–Sn $\simeq 3.00$ Å, and in-plane Sn$_1$–Sn$_1$ $\simeq 3.29$ Å [2411.10635].

Ortiz et al. evaluated the known $AM_6X_6$ chemistries and constructed a stability diagram for the $\approx 125$ member family. Five major structural variants were identified: ordered HfFe$_6$Ge$_6$-type, “SmMn$_6$Sn$_6$”-type disorder, “Y$_{0.5}$Co$_3$Ge$_3$” half-filled/half-vacant kagome layers, various $2\times2\times1$ supercells or stacking variants, and orthorhombic distortions of the hexagonal prototype. Chemical and steric control is expressed through the $A$-site Shannon radius $r_A$ and the rigid kagome-network scale set by the $M$–$X$ bond lengths and unit-cell volume $V_{\rm cell}$. The empirical frontier is written as
\[
r_A+r_{X_1}\lesssim \frac12 a-\Delta
\]
with $\Delta \approx 0.3$ Å, or equivalently
\[
V_{\rm cell}>V_0+\alpha\,(r_c-r_A),
\]
with $\alpha \approx 50$ Å$^3$/Å [2411.10635].

The AV$_3$Sb$_5$-type $AB_3C_5$ kagome metals also adopt $P6/mmm$ (No. 191), with a hexagonal primitive cell, $a \approx 5$–$6$ Å and $c \approx 7$–$10$ Å. In the high-throughput study of kagome compounds in this family, the A site lies at Wyckoff $1a$, the kagome-forming M site at Wyckoff $3f$, and the X sites at Wyckoff $2d$ and $6i$. KV$_3$Sb$_5$, RbV$_3$Sb$_5$, and CsV$_3$Sb$_5$ 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
\[
E_{\rm form}(AB_3C_5)=E_{\rm total}(AB_3C_5)-\sum_i n_i\mu_i,
\]
with competing phases from the Alexandria convex-hull dataset [2503.17124].

A complementary high-throughput first-principles study proposed 24 dynamically stable novel kagome metals derived from the AV$_3$Sb$_5$ prototype. Twenty-two retain $P6/mmm$, while CsRu$_3$Ge$_5$ and RbCr$_3$Te$_5$ distort to $P\bar{6}2m$ via a twisted-triangle kagome net, and the lattice parameters of the 24 compounds span roughly $a = 5.0$–$7.0$ Å and $c = 8.5$–$10.0$ Å [2202.05588].

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

Density-wave behavior is a central theme in the hexagonal kagome metals. ScV$_6$Sn$_6$ is a hexagonal HfFe$_6$Ge$_6$-type compound that undergoes a first-order phase transition at $92$ K. Single-crystal X-ray and neutron diffraction revealed a charge density wave modulation with wave vector
\[
\mathbf{q}=\bigl(\tfrac13,\tfrac13,\tfrac13\bigr),
\]
which triples the in-plane cell and the repeat along $c$. Superlattice reflections appear at $(H+\tfrac13,K+\tfrac13,L+\tfrac13)$ below $T_{\rm CDW}$, the neutron $(\tfrac13,\tfrac13,\tfrac{10}{3})$ peak shows a sharp onset with hysteresis, $\chi_m$ drops by $\sim20\%$, $\rho_{ab}$ drops by $\sim35\%$, and $c$ increases by $\sim0.04\%$ while $a$ is nearly unchanged. The dominant atomic displacements involve Sc and Sn$_1$ moving along $c$ by up to $\Delta z\cdot c \simeq 0.16$ Å, while the V sites shift by only $\simeq 0.004$–$0.006$ Å in plane [2205.04582].

LuNb$_6$Sn$_6$ exhibits an analogous but not identical instability. It undergoes a first-order DW-like transition at $T_{\rm DW}=68$ K with ordering wave vector
\[
\mathbf{k}=\bigl(\tfrac13,\tfrac13,\tfrac13\bigr),
\]
corresponding to a $\sqrt{3}\times\sqrt{3}\times 3$ supercell. X-ray scattering shows diffuse scattering on planes $L=3n+0.5$ above $T_{\rm DW}$ and weak superlattice reflections with $I_{\rm peak}/I_{\rm main}\sim10^{-3}$ below $T_{\rm DW}$. The thermodynamic and transport signatures include a sharp drop in $\chi(T)$ for both $H\perp c$ and $H\parallel c$, a latent-heat feature and thermal hysteresis in $C_p(T)$, and a $20$–$30\%$ drop in $\rho_{ab}(T)$ through $T_{\rm DW}$. The structural analysis supports the “rattling mode” DW model proposed for ScV$_6$Sn$_6$, with the chain-displacement order parameter described by
\[
F=\sum_i\Bigl[\tfrac12\,a(T)\,\eta_i^2+\tfrac14\,b\,\eta_i^4\Bigr]+\sum_{i\ne j}V_{ij}\eta_i\eta_j,
\]
where $a(T)=\alpha(T-T_0)$ and $V_{ij}>0$ is a strain-mediated repulsive coupling on the kagome network [2411.10635].

The AV$_3$Sb$_5$-type family provides a broader instability context. In the high-throughput $AB_3C_5$ survey, the AV$_3$Sb$_5$ compounds have a known CDW transition $T^*\approx80$–$100$ K and superconducting $T_c\approx0.3$–$3$ K, while phonon dispersions show that 35 of 36 hull compounds have imaginary modes in pristine $P6/mmm$, indicating a tendency to undergo $2\times2$ Star-of-David or Inverse-Star distortions. INb$_3$Bi$_5$ is the only dynamically stable pristine kagome, with $\lambda=0.28$, $\omega_{\log}=141$ K, and $T_c\approx0.03$ K from the Allen–Dynes formula with $\mu^*=0.1$ [2503.17124].

Among the 24 dynamically stable kagome metals proposed in the AV$_3$Sb$_5$ prototype study, fourteen exhibit $\lambda\ge0.35$ and $0.3$ K $\le T_c\le 5.0$ K. KZr$_3$Pb$_5$ has the highest reported $T_c$, with $\lambda\approx0.91$ and $T_c=5.03$ K. NaZr$_3$As$_5$ shows pronounced imaginary phonon modes at the $M$ and $L$ points, and symmetry analysis yields two candidate CDW phases: CDW I in space group Ibam with $\Delta E\simeq-11.3$ meV per 72-atom cell and Zr displacement $\simeq0.12$ Å, and CDW II in $P6/mcc$ with $\Delta E\simeq-25.1$ meV and Zr displacement $\simeq0.15$ Å [2202.05588].

## 4. Electronic topology, surface states, and quantum confinement

The AV$_3$Sb$_5$-type high-throughput studies identify a recurring electronic template. All stable kagomes in the $AB_3C_5$ survey exhibit Dirac points at the $K$ point, Van Hove singularities at $M$ typically $100$–$300$ meV below $E_F$, and flat bands. Groups 12, 13, and Ce-based compounds have a nearly flat band just above or below $E_F$ along $\Gamma$–$A$–$L$, 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 $\sim200$ meV above $E_F$ in PmBe$_3$Au$_5$, a Van Hove singularity $\sim250$ meV below $E_F$ and a flat band crossing $E_F$ along $k_z$ in KPd$_3$Hg$_5$, and multiple Dirac points within $\pm100$ meV of $E_F$ in BaTi$_3$Bi$_5$ [2503.17124].

Angle-resolved photoemission on RV$_6$Sn$_6$ ($R=$ 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 $K$ with $E_D \approx E_F-0.15$ eV in GdV$_6$Sn$_6$ and $-0.18$ eV in HoV$_6$Sn$_6$, a saddle point at $M$ near $-0.40$ to $-0.42$ eV, and a nearly dispersionless flat band near $-1.00$ to $-1.05$ eV. Comparison of DFT and ARPES slopes gives a modest bandwidth renormalization factor $Z^{-1}\simeq1.1$–$1.2$, and the larger layer spacing relative to FeSn is associated with more two-dimensional character [2112.15007].

A distinct topological lineage is provided by the stacked kagome–honeycomb IAMX compounds in space group $P\bar{6}2m$ (No. 189). Starting from a minimal five-band tight-binding model with one $p_z$ orbital on the honeycomb sublattice and three $d_{z^2}$ orbitals on the kagome sublattice, the non-SOC Hamiltonian was written as
\[
H_{\rm nSOC}
=\sum_{i\in\alpha,\beta}\epsilon_i c_i^\dagger c_i
+\sum_{\langle i,j\rangle\in\alpha}t_1 c_i^\dagger c_j
+\sum_{\langle i,j\rangle\in\beta}t_2 c_i^\dagger c_j
+\cdots,
\]
and high-throughput screening of the ICSD yielded $\sim400$ IA–M–X compounds, reduced by DFT screening to 298 “ideal” topological semimetals, all crystallizing in $P\bar{6}2m$. 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 $k_z=0$ mirror gaps into six pairs of Weyl points, the mirror Chern numbers are ${\rm MCN}[M_{001}]=2$ and ${\rm MCN}[M_{1\bar10}]=1$, and six Fermi arcs appear on the (001) surface [2402.12700].

The SOC-driven extension of this program adds a phase diagram with three regimes: nodal-ring semimetal for $\lambda<\lambda_{c1}$, strong topological insulator for $\lambda_{c1}<\lambda<\lambda_{c2}$, and Weyl semimetal for $\lambda>\lambda_{c2}$. In the model analysis, one finds
\[
\lambda_c^\pm=\frac{1}{\eta}\sqrt{-\frac{M_0}{M_1}},
\]
and a typical parameter set gives $\lambda_{c1}\approx0.2$ eV and $\lambda_{c2}\approx0.35$ 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 $\sim0.12$ eV and $\mathbb{Z}_2=1$ [2508.17454].

Quantum confinement adds a further layer of kagome-surface physics in CsV$_3$Sb$_5$. 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 $\sim0.5$ Å, creating a potential well of width $\sim5$ Å and depth $\Delta V\simeq1$ eV. The effective-mass quantization is written as
\[
E_n=E_0+\frac{\hbar^2\pi^2 n^2}{2m^*d^2},
\]
with a fitted $m^*\simeq0.7m_e$ and $E_1-E_0\simeq0.25$ eV. ARPES resolves $k_z$-independent quantum-well subbands and a split Dirac cone at $K$, while slab projections show that the observed spectra are almost entirely contributed by the top two layers [2109.12778].

## 5. Magnetic and frustrated-spin manifestations

The LnNb$_6$Sn$_6$ series connects kagome-metal density-wave physics to rare-earth magnetism. Within the HfFe$_6$Ge$_6$-type members, Nb is nonmagnetic and RKKY-mediated coupling on the 2D $A$ plane dominates. Heat-capacity anomalies confirm bulk ordering, small Schottky upturns appear below 1 K in many compounds, and metamagnetic transitions in $M(H)$ correlate with spin reorientation within the AB-stacked kagome layers [2411.10635].

| Ln | $T_N$ (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 $c$-axis modulation |
| Er | $\lesssim 0.15$ | likely $T_N\sim0.15$ K; Schottky confound |
| Tm | none $>0.4$ | paramagnet above 0.1 K |

The Ti(III) fluorides Rb$_2$NaTi$_3$F$_{12}$, Cs$_2$NaTi$_3$F$_{12}$, and Cs$_2$KTi$_3$F$_{12}$ represent a different IAMX-related kagome magnetism. All crystallize in monoclinic $P2_1/m$, the Ti$^{3+}$ ions form a slightly distorted, buckled kagome plane, and DFT plus energy mapping yields four inequivalent in-plane exchanges $J_A$, $J_B$, $J_C$, and $J_D$. The sequence Rb$_2$Na $\rightarrow$ Cs$_2$Na $\rightarrow$ Cs$_2$K tunes the system from a nearly pure one-dimensional $\Delta$-chain to a fully two-dimensional frustrated kagome antiferromagnet. Exact diagonalization reproduces a pronounced plateau at $M/M_s=1/3$ in all three compounds, with plateau windows of $\sim20$–$60$ T for Rb$_2$NaTi$_3$F$_{12}$, $\sim15$–$40$ T for Cs$_2$NaTi$_3$F$_{12}$, and $\sim30$–$80$ T for Cs$_2$KTi$_3$F$_{12}$ [1902.10309].

Metallic kagome antiferromagnetism in the $P\bar{6}2m$ 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,
\[
H=E_0+\sum_{\langle i,j\rangle}J_{ij}\,\mathbf{s}_i\!\cdot\!\mathbf{s}_j
+L_1\sum_{\triangle_1}\Bigl[(\mathbf{s}_1\!\cdot\!\mathbf{s}_2)(\mathbf{s}_2\!\cdot\!\mathbf{s}_3)+\cdots\Bigr],
\]
with $J_2=+0.89$ meV, $J_1=-0.09$ meV, $J_0'=+0.19$ meV, and $L_1=+0.12$ meV. The resulting ground states are dominated by nearly isolated antiferromagnetic triangles adopting $120^\circ$ order with positive or negative vector chirality [2211.04646].

The atacamite-derived compounds EuCu$_3$(OH)$_6$Cl$_3$, Zn$_x$Cu$_{4-x}$(OH)$_6$(NO$_3$)$_2$, and haydeeite MgCu$_3$(OH)$_6$Cl$_2$ show yet another kagome-magnetic regime. EuCu$_3$(OH)$_6$Cl$_3$ is a frustrated antiferromagnet with $\Theta_W\approx-400$ K, transitions at $T_{N1}=14.7$ K and $T_{N2}=8.7$ K, and frustration parameter $f\sim30$. Zn$_x$Cu$_{4-x}$(OH)$_6$(NO$_3$)$_2$ is essentially unfrustrated, with $T_N\approx7.5$ K and $f\ll1$. Haydeeite is a ferromagnet with $T_C=4.31$ K, easy axis in the kagome plane, and saturation moment $M_s\approx1.0\,\mu_B$/Cu [1810.09201].

## 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 $AM_6X_6$ family, the stability diagram emphasizes sterics: the $A$-site Shannon radius governs packing of the $A$–$X_1$–$X_1$–$A$ chains, the $M$–$X$ bond lengths fix the size of the rigid kagome network, and bond-modulated instabilities emerge at small $r_A$ while disorder and supercells appear at large $r_A$. Within LnNb$_6$Sn$_6$, the Nb$_6$Sn$_6$ sublattice is almost rigid, with Nb–Sn, Sn–Sn, and Nb–Nb bonds changing by $<2\%$ across the Ln series, whereas the Sn$_1$–Sn$_1$ bond compresses by up to $4\%$ and $U_{33}({\rm Ln}_1)$ and $U_{33}({\rm Sn}_1)$ diverge for $r_A<0.86$ Å. The proposed interpretation is that under-filled voids for small $A$ produce large “rattling” amplitudes, soft low-energy phonons, and density-wave instability [2411.10635].

In the AV$_3$Sb$_5$-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 $+3/+4$ oxidation states. They also emphasize that heavy-$p$-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 [2503.17124]. The dynamically stable 24-compound set further indicates that expanded chemical flexibility from Li$\rightarrow$Cs, 3$d\rightarrow$5$d$ B-site choice, and Ge$\rightarrow$Bi C-site choice provides routes to stronger EPC, higher $T_c$, new CDW distortions, and nontrivial $\mathbb{Z}_2$ topology [2202.05588].

The stacked kagome–honeycomb semimetals admit explicit tight-binding design rules. In the $P\bar{6}2m$ family, stronger intralayer hopping favors nodal rings on vertical mirrors, larger interlayer hopping favors nodal chains over isolated rings, and the normalized ratios
\[
\eta_1=\frac{|t_2|}{|t_1|},\qquad \eta_z=\frac{t_z}{|t_1|}
\]
sort the nodal-line configurations: $\eta_1<0.8$ and $\eta_z<0.3$ give rings at $K$, $0.8\lesssim\eta_1\lesssim1.2$ and $\eta_z\lesssim0.25$ give rings at $\Gamma$, $\eta_1>1.2$ and $\eta_z\gtrsim0.25$ give nodal chains, and further increase of intralayer coupling yields nested rings [2402.12700]. The SOC-driven continuation of this framework proposes chemical substitution $X={\rm C}\rightarrow{\rm Ge}\rightarrow{\rm Pb}$ as a route to traverse nodal-ring semimetal $\rightarrow$ Weyl semimetal $\rightarrow$ strong topological insulator [2508.17454].

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 [1902.10309]. In the atacamite family, changing the $M$ ion and the anion $X$ 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 [1810.09201].

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.

Source: https://www.emergentmind.com/topics/kagome-type-iamx-family