High-Entropy Kagome Metal (Gd,Tb,Dy,Ho,Er)Mn6Sn6

• (Gd,Tb,Dy,Ho,Er)Mn6Sn6 is a high-entropy kagome metal with a hexagonal HfFe6Ge6-type structure featuring an undistorted Mn kagome net and randomly occupied rare-earth sites.
• It exhibits multiple spin reorientation transitions—from in-plane to out-of-plane and canted ferrimagnetic states—driven by competing anisotropies and exchange interactions.
• Robust Dirac crossings, strictly linear magnetoresistance, and intrinsic anomalous Hall effects indicate preserved topological kagome electronic features despite configurational disorder.

$$(\mathrm{Gd},\mathrm{Tb},\mathrm{Dy},\mathrm{Ho},\mathrm{Er})\mathrm{Mn}_6\mathrm{Sn}_6$$ denotes a high-entropy member of the $R\mathrm{Mn}_6\mathrm{Sn}_6$ kagome-metal family, realized experimentally as $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$. It retains the hexagonal HfFe$_6$Ge$_6$-type framework with space group $P6/mmm$, in which Mn forms an undistorted kagome net and the rare-earth $1a$ site is randomly occupied by five lanthanides. Within the broader $R\mathrm{Mn}_6\mathrm{Sn}_6$ series, this structural motif is associated with ferrimagnetism, Dirac crossings, flat bands, and large intrinsic anomalous Hall responses; in the high-entropy variant, the same backbone coexists with multiple spin-reorientation transitions, metamagnetic features, and strictly linear non-saturating magnetoresistance (Liu et al., 30 Jul 2025).

1. Crystal chemistry and high-entropy realization

Single crystals of the high-entropy compound were grown by Sn-flux using elemental Gd (99 %), Tb (99.9 %), Dy (99.8 %), Ho (99.9 %), Er (99.9 %), Mn (99.99 %) and Sn (99.9999 %), with nominal ratio $$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$. The charge was heated to $1100\,^\circ\mathrm{C}$, held for 48 h, cooled from $R\mathrm{Mn}_6\mathrm{Sn}_6$0 to $R\mathrm{Mn}_6\mathrm{Sn}_6$1 at $R\mathrm{Mn}_6\mathrm{Sn}_6$2, and the excess flux was removed at $R\mathrm{Mn}_6\mathrm{Sn}_6$3. The resulting crystals are plate-like, with typical size $R\mathrm{Mn}_6\mathrm{Sn}_6$4 (Liu et al., 30 Jul 2025).

Single-crystal X-ray diffraction shows that the structure is the same $R\mathrm{Mn}_6\mathrm{Sn}_6$5 phase as in the parent $R\mathrm{Mn}_6\mathrm{Sn}_6$6 compounds. The $R\mathrm{Mn}_6\mathrm{Sn}_6$7 site at $R\mathrm{Mn}_6\mathrm{Sn}_6$8, $R\mathrm{Mn}_6\mathrm{Sn}_6$9, is randomly occupied by $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$0; Mn occupies $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$1 at $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$2 with occupancy $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$3; and Sn occupies $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$4 and $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$5 positions. At 123 K the refined lattice constants are $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$6 Å and $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$7 Å, while at 300 K they are $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$8 Å and $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$9 Å. No symmetry-lowering or volume anomaly is seen between 123 K and 300 K. The selected bond lengths at 300 K are Mn–Mn $_6$0 Å, Mn–Sn $_6$1 Å, and $_6$2–Sn $_6$3 Å, and EDX mapping confirms a homogeneous distribution of all five $_6$4 elements (Liu et al., 30 Jul 2025).

As in the parent Tb, Dy, and Ho compounds, the structural building block along $_6$5 is $_6$6Sn$_6$7–Mn–Sn$_6$8–Sn$_6$9–Sn$_6$0–Mn–$_6$1Sn$_6$2, with $_6$3 atoms at the centers of graphene-like Sn$_6$4 hexagons and Mn atoms forming an undistorted kagome net in the $_6$5-plane (Gao et al., 2021). This continuity of the crystallographic scaffold is central to interpreting the high-entropy phase as a disorder-modified member of the same kagome-metal series rather than as a distinct structural class.

2. Magnetic phase sequence and anisotropy competition

Magnetization measurements in $_6$6 T show that the high-entropy compound is paramagnetic above 450 K. A sharp increase in $_6$7 at $_6$8 K marks the onset of in-plane ferrimagnetic order, denoted FIM-$_6$9. On cooling below $P6/mmm$0 K, the easy axis reorients from $P6/mmm$1 into $P6/mmm$2, giving a uniaxial FIM-$P6/mmm$3 state. A second spin reorientation occurs at $P6/mmm$4 K, evidenced by ZFC/FC splitting; below this temperature the low-$P6/mmm$5 state is described as a canted-FIM texture (Liu et al., 30 Jul 2025).

Temperature range	Magnetic regime
$P6/mmm$6 K	PM
$P6/mmm$7	FIM-$P6/mmm$8
$P6/mmm$9	FIM-$1a$0
$1a$1 K	canted-FIM

Curie–Weiss analysis above 500 K gives

$1a$2

with $1a$3, corresponding to $1a$4, and $1a$5 K. The positive $1a$6 is consistent with dominant ferromagnetic Mn–Mn coupling. Arrott plots of $1a$7 versus $1a$8 at $1a$9 K yield straight lines through the origin, indicating a second-order transition at $R\mathrm{Mn}_6\mathrm{Sn}_6$0 (Liu et al., 30 Jul 2025).

Field-dependent magnetization resolves the anisotropy competition in more detail. At 350 K, $R\mathrm{Mn}_6\mathrm{Sn}_6$1 is linear, consistent with the paramagnetic state. In the 250–400 K range with $R\mathrm{Mn}_6\mathrm{Sn}_6$2, rapid saturation by $R\mathrm{Mn}_6\mathrm{Sn}_6$3 T is consistent with FIM-$R\mathrm{Mn}_6\mathrm{Sn}_6$4. In the 100–200 K range with $R\mathrm{Mn}_6\mathrm{Sn}_6$5, the easy axis gradually rotates into $R\mathrm{Mn}_6\mathrm{Sn}_6$6 under $R\mathrm{Mn}_6\mathrm{Sn}_6$7 T. For $R\mathrm{Mn}_6\mathrm{Sn}_6$8–$R\mathrm{Mn}_6\mathrm{Sn}_6$9 K and field perpendicular to the easy axis, a metamagnetic jump appears at $$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$0 T, followed by a slow approach to saturation above 3 T, indicative of field-induced spin reorientation and possible chiral states. Below 50 K, pronounced hysteresis for both field orientations, together with growing remanent moment and coercivity, reflects the low-$$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$1 canted-FIM texture (Liu et al., 30 Jul 2025).

3. Exchange hierarchy in the parent series

The microscopic magnetic framework of the series is most fully resolved for TbMn$$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$2Sn$$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$3, which is modeled by a minimal local-moment Heisenberg Hamiltonian

$$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$4

The intralayer term is a ferromagnetic Mn–Mn exchange within each kagome layer,

$$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$5

with $$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$6 and $$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$7 spin-1 on each Mn. Interlayer couplings comprise Mn–Mn terms $$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$8, $$R:\mathrm{Gd}:\mathrm{Tb}:\mathrm{Dy}:\mathrm{Ho}:\mathrm{Er}:\mathrm{Mn}:\mathrm{Sn}=0.2:0.2:0.2:0.2:0.2:6:20$$9, $1100\,^\circ\mathrm{C}$0, and an antiferromagnetic Mn–Tb exchange $1100\,^\circ\mathrm{C}$1, with the Tb moment $1100\,^\circ\mathrm{C}$2. The anisotropy term is

$1100\,^\circ\mathrm{C}$3

with $1100\,^\circ\mathrm{C}$4 and $1100\,^\circ\mathrm{C}$5. Dzyaloshinskii–Moriya terms were found unnecessary to capture the main INS features (Riberolles et al., 2021).

INS establishes a very large Mn kagome magnon bandwidth, with the top of the Mn magnon band approaching $1100\,^\circ\mathrm{C}$6 meV. The low-energy sector contains several distinct scales: a $1100\,^\circ\mathrm{C}$7-point gap $1100\,^\circ\mathrm{C}$8 meV, a bilayer splitting $1100\,^\circ\mathrm{C}$9 meV, and an Mn–Tb ferrimagnetic exchange-field shift $R\mathrm{Mn}_6\mathrm{Sn}_6$00 meV. A global SpinW fit to $R\mathrm{Mn}_6\mathrm{Sn}_6$01 data points gives

$R\mathrm{Mn}_6\mathrm{Sn}_6$02

$R\mathrm{Mn}_6\mathrm{Sn}_6$03

The dominant intralayer ferromagnetism is therefore accompanied by weaker, competing interlayer Mn–Mn couplings and a substantial antiferromagnetic Mn–Tb interaction (Riberolles et al., 2021).

DFT generalizes this picture across the $R\mathrm{Mn}_6\mathrm{Sn}_6$04 family. For $R\mathrm{Mn}_6\mathrm{Sn}_6$05–Lu, $R\mathrm{Mn}_6\mathrm{Sn}_6$06 is always ferromagnetic and largest, whereas $R\mathrm{Mn}_6\mathrm{Sn}_6$07 and $R\mathrm{Mn}_6\mathrm{Sn}_6$08 are weaker and can change sign. $R\mathrm{Mn}_6\mathrm{Sn}_6$09 scales roughly with $R\mathrm{Mn}_6\mathrm{Sn}_6$10-4$R\mathrm{Mn}_6\mathrm{Sn}_6$11 overlap and is largest for Gd, then Tb, Dy, and so on. In TbMn$R\mathrm{Mn}_6\mathrm{Sn}_6$12Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$13, the ferrimagnetic state is strongly stabilized: using the fitted parameters, the classical energies are $R\mathrm{Mn}_6\mathrm{Sn}_6$14 meV/f.u. and $R\mathrm{Mn}_6\mathrm{Sn}_6$15 meV/f.u., so the Mn–Tb coupling overwhelmingly locks the layers into a rigid three-dimensional ferrimagnetic network. More broadly, the weaker $R\mathrm{Mn}_6\mathrm{Sn}_6$16 and $R\mathrm{Mn}_6\mathrm{Sn}_6$17 are the same competing interactions that drive helical magnetism in YMn$R\mathrm{Mn}_6\mathrm{Sn}_6$18Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$19; for $R\mathrm{Mn}_6\mathrm{Sn}_6$20Gd–Ho, Mn–$R\mathrm{Mn}_6\mathrm{Sn}_6$21 antiferromagnetic coupling and $R\mathrm{Mn}_6\mathrm{Sn}_6$22 single-ion anisotropy lock in a collinear ferrimagnet, whereas in ErMn$R\mathrm{Mn}_6\mathrm{Sn}_6$23Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$24 the Mn–Er coupling becomes so weak that Mn and Er sublattices order separately (Riberolles et al., 2021).

This suggests that the high-entropy compound superposes a robust Mn kagome exchange network, characterized in the parent series by $R\mathrm{Mn}_6\mathrm{Sn}_6$25 meV and a transferable high-$R\mathrm{Mn}_6\mathrm{Sn}_6$26 magnon bandwidth $R\mathrm{Mn}_6\mathrm{Sn}_6$27 meV, with a distribution of $R\mathrm{Mn}_6\mathrm{Sn}_6$28-dependent anisotropies and Mn–$R\mathrm{Mn}_6\mathrm{Sn}_6$29 exchange fields. A plausible implication is that the multiple reorientation transitions in $R\mathrm{Mn}_6\mathrm{Sn}_6$30 are a collective manifestation of this exchange hierarchy under configurational disorder.

4. Magnetotransport and Hall response

The high-entropy compound is metallic throughout the measured range. Its longitudinal resistivity has residual-resistivity ratio

$R\mathrm{Mn}_6\mathrm{Sn}_6$31

The derivative $R\mathrm{Mn}_6\mathrm{Sn}_6$32 shows a clear peak at $R\mathrm{Mn}_6\mathrm{Sn}_6$33 K, in register with a spin-reorientation feature. Magnetoresistance,

$R\mathrm{Mn}_6\mathrm{Sn}_6$34

remains strictly linear and non-saturating up to the highest fields reported. At 2 K and $R\mathrm{Mn}_6\mathrm{Sn}_6$35 T, $R\mathrm{Mn}_6\mathrm{Sn}_6$36. In pulsed fields at 4 K up to 35 T, two distinct linear MR slopes appear, with a kink around 20 T but no saturation even at 35 T. Over 0–20 T, the linear fit is

$R\mathrm{Mn}_6\mathrm{Sn}_6$37

with $R\mathrm{Mn}_6\mathrm{Sn}_6$38 and $R\mathrm{Mn}_6\mathrm{Sn}_6$39 (Liu et al., 30 Jul 2025).

The Hall response is decomposed as

$R\mathrm{Mn}_6\mathrm{Sn}_6$40

The ordinary Hall coefficient is $R\mathrm{Mn}_6\mathrm{Sn}_6$41, indicating an electronlike ordinary Hall channel. Conductivities are obtained from

$R\mathrm{Mn}_6\mathrm{Sn}_6$42

and the Tian–Ye–Jin scaling

$R\mathrm{Mn}_6\mathrm{Sn}_6$43

gives $R\mathrm{Mn}_6\mathrm{Sn}_6$44 in total, or

$R\mathrm{Mn}_6\mathrm{Sn}_6$45

per Mn kagome layer. The weak dependence of $R\mathrm{Mn}_6\mathrm{Sn}_6$46 on $R\mathrm{Mn}_6\mathrm{Sn}_6$47 indicates an intrinsic Berry-curvature origin (Liu et al., 30 Jul 2025).

The parent Tb, Dy, and Ho compounds provide a reference for this interpretation. In those materials, the low-temperature MR for $R\mathrm{Mn}_6\mathrm{Sn}_6$48 is positive and approximately $R\mathrm{Mn}_6\mathrm{Sn}_6$49-like below $R\mathrm{Mn}_6\mathrm{Sn}_6$50 K, while at higher temperature it becomes negative and is attributed to suppression of spin-disorder scattering. Their anomalous Hall response follows

$R\mathrm{Mn}_6\mathrm{Sn}_6$51

with $R\mathrm{Mn}_6\mathrm{Sn}_6$52 and $R\mathrm{Mn}_6\mathrm{Sn}_6$53 for Tb, $R\mathrm{Mn}_6\mathrm{Sn}_6$54 for Dy, and $R\mathrm{Mn}_6\mathrm{Sn}_6$55 for Ho, implying negligible skew scattering and dominant intrinsic Karplus–Luttinger physics. The anomalous Hall conductivities at 7 T and 100–300 K are about $R\mathrm{Mn}_6\mathrm{Sn}_6$56 for TbMn$R\mathrm{Mn}_6\mathrm{Sn}_6$57Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$58, $R\mathrm{Mn}_6\mathrm{Sn}_6$59 for DyMn$R\mathrm{Mn}_6\mathrm{Sn}_6$60Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$61, and $R\mathrm{Mn}_6\mathrm{Sn}_6$62 for HoMn$R\mathrm{Mn}_6\mathrm{Sn}_6$63Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$64 (Gao et al., 2021).

5. Kagome electronic structure and topological character

The electronic structure of the parent series is defined by the Mn kagome lattice. High-resolution ARPES and $R\mathrm{Mn}_6\mathrm{Sn}_6$65 calculations on $R\mathrm{Mn}_6\mathrm{Sn}_6$66 ($R\mathrm{Mn}_6\mathrm{Sn}_6$67 Dy, Tb, Gd, Y) show a Dirac-like crossing at $R\mathrm{Mn}_6\mathrm{Sn}_6$68 located within $R\mathrm{Mn}_6\mathrm{Sn}_6$69 meV of $R\mathrm{Mn}_6\mathrm{Sn}_6$70, a Fermi velocity $R\mathrm{Mn}_6\mathrm{Sn}_6$71, and two weakly dispersive flat bands at approximately $R\mathrm{Mn}_6\mathrm{Sn}_6$72 eV and $R\mathrm{Mn}_6\mathrm{Sn}_6$73 eV. Across Gd, Tb, and Dy, the Dirac-cone position, gap, Fermi velocity, and flat-band energies vary by less than 20 meV, indicating that the Mn-derived kagome bands are unusually robust against changes in $R\mathrm{Mn}_6\mathrm{Sn}_6$74 configuration and magnetic ground state (Gu et al., 2022).

A minimal tight-binding description of these states is the standard nearest- and next-nearest-neighbor kagome model,

$R\mathrm{Mn}_6\mathrm{Sn}_6$75

In the parent Tb and Ho compounds, first-principles bands show linear crossings at the $R\mathrm{Mn}_6\mathrm{Sn}_6$76 points, and spin–orbit coupling opens gaps $R\mathrm{Mn}_6\mathrm{Sn}_6$77 meV for Tb and $R\mathrm{Mn}_6\mathrm{Sn}_6$78 meV for Ho. The corresponding intrinsic anomalous Hall conductivity is expressed as the Brillouin-zone integral of Berry curvature,

$R\mathrm{Mn}_6\mathrm{Sn}_6$79

with the sharp Berry-curvature contribution associated with SOC-gapped Dirac crossings near $R\mathrm{Mn}_6\mathrm{Sn}_6$80 (Gao et al., 2021).

For the high-entropy compound, the reported topological-electronic interpretation is explicitly inherited from this parent-series kagome physics. The relevant features are Dirac crossings at $R\mathrm{Mn}_6\mathrm{Sn}_6$81, a nearly flat band around $R\mathrm{Mn}_6\mathrm{Sn}_6$82 eV, and Berry-curvature hot spots generated by partially gapped kagome bands. High-entropy alloying introduces a random $R\mathrm{Mn}_6\mathrm{Sn}_6$83-site potential that broadens the kagome bands but does not destroy the Dirac-point signatures, as inferred from the robust intrinsic anomalous Hall effect; disorder-induced scattering is argued to suppress extrinsic AHE channels, thereby sharpening the relative role of band-structure Berry curvature (Liu et al., 30 Jul 2025).

A common misconception is that configurational disorder on the rare-earth site should necessarily erase kagome-band topology. The reported behavior does not support that conclusion. Instead, the preserved $R\mathrm{Mn}_6\mathrm{Sn}_6$84 framework, intrinsic-like Hall scaling, and parent-series ARPES robustness together indicate that the Mn kagome electronic structure remains operative in the high-entropy alloy. This suggests that the topological response is controlled primarily by the Mn-derived band manifold rather than by fine details of any single rare-earth ion.

6. Position within the $R\mathrm{Mn}_6\mathrm{Sn}_6$85 landscape

Within the parent $R\mathrm{Mn}_6\mathrm{Sn}_6$86 family, rare-earth substitution is already a magnetic tuning parameter. The series shares a very strong intralayer Mn exchange and dominant ferromagnetic interlayer $R\mathrm{Mn}_6\mathrm{Sn}_6$87, while weaker $R\mathrm{Mn}_6\mathrm{Sn}_6$88 and $R\mathrm{Mn}_6\mathrm{Sn}_6$89 generate magnetic competition. In nonmagnetic-$R\mathrm{Mn}_6\mathrm{Sn}_6$90 members such as YMn$R\mathrm{Mn}_6\mathrm{Sn}_6$91Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$92, those competing interlayer terms can drive helical or spiral phases; in moment-bearing members, Mn–$R\mathrm{Mn}_6\mathrm{Sn}_6$93 antiferromagnetic coupling and $R\mathrm{Mn}_6\mathrm{Sn}_6$94-ion anisotropy stabilize ferrimagnetism, but the strength of Mn–$R\mathrm{Mn}_6\mathrm{Sn}_6$95 binding weakens from Gd to Tb to Dy and beyond (Riberolles et al., 2021).

The high-entropy phase occupies an intermediate but distinct position in this landscape. Its $R\mathrm{Mn}_6\mathrm{Sn}_6$96 K lies between GdMn$R\mathrm{Mn}_6\mathrm{Sn}_6$97Sn$R\mathrm{Mn}_6\mathrm{Sn}_6$98 at 350 K and TbMn$R\mathrm{Mn}_6\mathrm{Sn}_6$99Sn$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$00 at 450 K. Its two spin-reorientation temperatures, 210 K and 80 K, are described as new collective effects of competing $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$01 anisotropies, and no single-$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$02 parent shows the same cascade. Its magnetoresistance at 9 T, $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$03, is more strongly linear than in any single-$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$04 compound, for which $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$05 at 9 T is quoted as a reference scale. The sharp MR kink at 20 T, absent in most parents, is interpreted as likely reflecting a high-field modification of Dy–Mn/Er–Mn coupling carried over from ErMn$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$06Sn$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$07. Its anomalous Hall conductivity per kagome layer, $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$08, lies near the weighted average of parent values $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$09–$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$10, which is taken as evidence that topological bands survive high-entropy disorder (Liu et al., 30 Jul 2025).

The resulting physical picture is that alloying five rare-earth elements tunes the balance between in-plane and out-of-plane anisotropy and modifies Mn–$$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$11 exchange without destroying the Mn kagome backbone. In the language of the parent-series studies, rare-earth substitution functions as a magnetic knob that can tune between three-dimensional ferrimagnets, helical instabilities, and spin-reoriented phases, each with distinct transport and topological signatures (Riberolles et al., 2021). In the high-entropy material, this tuning is not achieved by moving from one end member to another, but by embedding several competing $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$12-sector tendencies within the same $$(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16})\mathrm{Mn}_6\mathrm{Sn}_6$$13 kagome lattice.