--- title: High-Entropy Kagome Metal (Gd,Tb,Dy,Ho,Er)Mn6Sn6 url: https://www.emergentmind.com/topics/gd-tb-dy-ho-er-mn6sn6 type: topic --- # High-Entropy Kagome Metal (Gd,Tb,Dy,Ho,Er)Mn6Sn6 \((\mathrm{Gd},\mathrm{Tb},\mathrm{Dy},\mathrm{Ho},\mathrm{Er})\mathrm{Mn}_6\mathrm{Sn}_6\) 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 [2507.22809]. ## 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 \(1100\,^\circ\mathrm{C}\) to \(600\,^\circ\mathrm{C}\) at \(1\,^\circ\mathrm{C}/\mathrm{h}\), and the excess flux was removed at \(600\,^\circ\mathrm{C}\). The resulting crystals are plate-like, with typical size \(\sim 3\times 5\times 0.1~\mathrm{mm}^3\) [2507.22809]. Single-crystal X-ray diffraction shows that the structure is the same \(P6/mmm\) phase as in the parent \(R\mathrm{Mn}_6\mathrm{Sn}_6\) compounds. The \(R\) site at \(1a\), \((0,0,0)\), is randomly occupied by \(\mathrm{Gd}_{0.21}\mathrm{Tb}_{0.22}\mathrm{Dy}_{0.22}\mathrm{Ho}_{0.19}\mathrm{Er}_{0.16}\); Mn occupies \(6i\) at \((x\approx 0.495,0,\tfrac14)\) with occupancy \(\approx 1\); and Sn occupies \(2c\) and \(6i\) positions. At 123 K the refined lattice constants are \(a=5.508(1)\) Å and \(c=9.133(2)\) Å, while at 300 K they are \(a=5.513(1)\) Å and \(c=9.139(2)\) 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 \(\sim 2.84\) Å, Mn–Sn \(\sim 2.78\) Å, and \(R\)–Sn \(\sim 3.25\) Å, and EDX mapping confirms a homogeneous distribution of all five \(R\) elements [2507.22809]. As in the parent Tb, Dy, and Ho compounds, the structural building block along \(c\) is \(R\)Sn\(_1\)–Mn–Sn\(_2\)–Sn\(_3\)–Sn\(_4\)–Mn–\(R\)Sn\(_1\), with \(R\) atoms at the centers of graphene-like Sn\(_1\) hexagons and Mn atoms forming an undistorted kagome net in the \(ab\)-plane [2109.01866]. 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 \(H=0.1\) T show that the high-entropy compound is paramagnetic above 450 K. A sharp increase in \(M(T)\) at \(T_C=420\) K marks the onset of in-plane ferrimagnetic order, denoted FIM-\(ab\). On cooling below \(T_{\mathrm{SR}1}\simeq 210\) K, the easy axis reorients from \(ab\) into \(c\), giving a uniaxial FIM-\(c\) state. A second spin reorientation occurs at \(T_{\mathrm{SR}2}\simeq 80\) K, evidenced by ZFC/FC splitting; below this temperature the low-\(T\) state is described as a canted-FIM texture [2507.22809]. | Temperature range | Magnetic regime | |---|---| | \(T>420\) K | PM | | \(210~\mathrm{K}0\), with the Tb moment \(S=3\). The anisotropy term is \[ H_{\mathrm{aniso}}=K^M\sum_i (s_i^z)^2+K^T\sum_j (S_j^z)^2, \] with \(K^T<0\) and \(K^M\lesssim 0\). Dzyaloshinskii–Moriya terms were found unnecessary to capture the main INS features [2110.14713]. INS establishes a very large Mn kagome magnon bandwidth, with the top of the Mn magnon band approaching \(\sim 230\) meV. The low-energy sector contains several distinct scales: a \(\Gamma\)-point gap \(\Delta\simeq 6.5\) meV, a bilayer splitting \(\omega_B\simeq 47\) meV, and an Mn–Tb ferrimagnetic exchange-field shift \(\omega_F\simeq 8\) meV. A global SpinW fit to \(\sim 100\) data points gives \[ J=-28.8(2)\ \mathrm{meV},\quad J^{MT}=1.42(6)\ \mathrm{meV},\quad J_1^{MM}=-4.4(4)\ \mathrm{meV}, \] \[ J_2^{MM}=-19.2(2)\ \mathrm{meV},\quad J_3^{MM}=+1.8(2)\ \mathrm{meV},\quad K^M=-1.30(6)\ \mathrm{meV},\quad K^T=-1.70(12)\ \mathrm{meV}. \] The dominant intralayer ferromagnetism is therefore accompanied by weaker, competing interlayer Mn–Mn couplings and a substantial antiferromagnetic Mn–Tb interaction [2110.14713]. DFT generalizes this picture across the \(R\mathrm{Mn}_6\mathrm{Sn}_6\) family. For \(R=\mathrm{Y},\mathrm{Gd}\)–Lu, \(J_2^{MM}\) is always ferromagnetic and largest, whereas \(J_1^{MM}\) and \(J_3^{MM}\) are weaker and can change sign. \(J^{MT}\) scales roughly with \(R\)-4\(f\) overlap and is largest for Gd, then Tb, Dy, and so on. In TbMn\(_6\)Sn\(_6\), the ferrimagnetic state is strongly stabilized: using the fitted parameters, the classical energies are \(E_{\mathrm{ferri}}\simeq -220\) meV/f.u. and \(E_{\mathrm{UDDU}}\simeq -110\) meV/f.u., so the Mn–Tb coupling overwhelmingly locks the layers into a rigid three-dimensional ferrimagnetic network. More broadly, the weaker \(J_1^{MM}\) and \(J_3^{MM}\) are the same competing interactions that drive helical magnetism in YMn\(_6\)Sn\(_6\); for \(R=\)Gd–Ho, Mn–\(R\) antiferromagnetic coupling and \(R\) single-ion anisotropy lock in a collinear ferrimagnet, whereas in ErMn\(_6\)Sn\(_6\) the Mn–Er coupling becomes so weak that Mn and Er sublattices order separately [2110.14713]. This suggests that the high-entropy compound superposes a robust Mn kagome exchange network, characterized in the parent series by \(J\simeq -30\) meV and a transferable high-\(T\) magnon bandwidth \(\sim 230\) meV, with a distribution of \(R\)-dependent anisotropies and Mn–\(R\) exchange fields. A plausible implication is that the multiple reorientation transitions in \((\mathrm{Gd},\mathrm{Tb},\mathrm{Dy},\mathrm{Ho},\mathrm{Er})\mathrm{Mn}_6\mathrm{Sn}_6\) 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 \[ \mathrm{RRR}=\rho(300~\mathrm{K})/\rho(2~\mathrm{K})\simeq 70. \] The derivative \(d\rho/dT\) shows a clear peak at \(\simeq 205\) K, in register with a spin-reorientation feature. Magnetoresistance, \[ \mathrm{MR}\equiv \frac{\rho_{xx}(B)-\rho_{xx}(0)}{\rho_{xx}(0)}\times 100\%, \] remains strictly linear and non-saturating up to the highest fields reported. At 2 K and \(\mu_0H=9\) T, \(\mathrm{MR}\simeq 225\%\). 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 \[ \Delta \rho/\rho_0=a\cdot B+b, \] with \(a\simeq 0.25~\mathrm{T}^{-1}\) and \(b\simeq 0\) [2507.22809]. The Hall response is decomposed as \[ \rho_{xy}=R_0B+R_sM. \] The ordinary Hall coefficient is \(R_0\simeq +1.2\times 10^{-9}~\Omega\cdot \mathrm{m/T}\), indicating an electronlike ordinary Hall channel. Conductivities are obtained from \[ \sigma_{xx}=\frac{\rho_{xx}}{\rho_{xx}^2+\rho_{xy}^2},\qquad \sigma_{xy}=\frac{\rho_{xy}}{\rho_{xx}^2+\rho_{xy}^2}, \] and the Tian–Ye–Jin scaling \[ \rho_{xy}^{A}=\sigma_{\mathrm{int}}\rho_{xx}^2+\sigma_{\mathrm{sk}}\rho_{xx} \] gives \(\sigma_{\mathrm{int}}\simeq 66~\Omega^{-1}\mathrm{cm}^{-1}\) in total, or \[ \sigma_{xy}^{A}\simeq 0.17~e^2/h \] per Mn kagome layer. The weak dependence of \(\sigma_{xy}\) on \(\sigma_{xx}\) indicates an intrinsic Berry-curvature origin [2507.22809]. The parent Tb, Dy, and Ho compounds provide a reference for this interpretation. In those materials, the low-temperature MR for \(H\parallel c\) is positive and approximately \(H^2\)-like below \(\simeq 50\) K, while at higher temperature it becomes negative and is attributed to suppression of spin-disorder scattering. Their anomalous Hall response follows \[ \rho_{yx}(H)=R_0\mu_0H+R_s\mu_0M, \] with \(\rho_{yx}^{A}\propto \rho_{xx}^{\alpha}\) and \(\alpha\simeq 2.0\) for Tb, \(1.9\) for Dy, and \(1.96\) 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 \(250~\Omega^{-1}\mathrm{cm}^{-1}\) for TbMn\(_6\)Sn\(_6\), \(40~\Omega^{-1}\mathrm{cm}^{-1}\) for DyMn\(_6\)Sn\(_6\), and \(95~\Omega^{-1}\mathrm{cm}^{-1}\) for HoMn\(_6\)Sn\(_6\) [2109.01866]. ## 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 \(ab\ initio\) calculations on \(X\mathrm{Mn}_6\mathrm{Sn}_6\) (\(X=\) Dy, Tb, Gd, Y) show a Dirac-like crossing at \(K\) located within \(\pm 5\) meV of \(E_F\), a Fermi velocity \(v_F\approx (4.5\pm 0.5)\,\mathrm{eV\cdot \AA}\), and two weakly dispersive flat bands at approximately \(-0.42\) eV and \(-0.10\) 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 \(4f\) configuration and magnetic ground state [2203.10542]. A minimal tight-binding description of these states is the standard nearest- and next-nearest-neighbor kagome model, \[ H=\sum_{i,\sigma}\varepsilon_0 c^\dagger_{i\sigma}c_{i\sigma} -t\sum_{\langle i,j\rangle,\sigma}c^\dagger_{i\sigma}c_{j\sigma} -t'\sum_{\langle\!\langle i,j\rangle\!\rangle,\sigma}c^\dagger_{i\sigma}c_{j\sigma} +i\lambda_{\mathrm{SOC}}\sum_{\langle\!\langle i,j\rangle\!\rangle}\nu_{ij}c^\dagger_{i\alpha}\sigma^z_{\alpha\beta}c_{j\beta}. \] In the parent Tb and Ho compounds, first-principles bands show linear crossings at the \(K\) points, and spin–orbit coupling opens gaps \(\Delta_K\simeq 35\) meV for Tb and \(\Delta_K\simeq 15\) meV for Ho. The corresponding intrinsic anomalous Hall conductivity is expressed as the Brillouin-zone integral of Berry curvature, \[ \sigma_{xy}^{\mathrm{int}}=\frac{e^2}{\hbar}\int_{\mathrm{BZ}}\frac{d^3k}{(2\pi)^3}\,\Omega_z(k), \] with the sharp Berry-curvature contribution associated with SOC-gapped Dirac crossings near \(K/K'\) [2109.01866]. 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 \(K\), a nearly flat band around \(-0.3\) eV, and Berry-curvature hot spots generated by partially gapped kagome bands. High-entropy alloying introduces a random \(R\)-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 [2507.22809]. 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 \(P6/mmm\) 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 \(R\mathrm{Mn}_6\mathrm{Sn}_6\) landscape Within the parent \(R\mathrm{Mn}_6\mathrm{Sn}_6\) family, rare-earth substitution is already a magnetic tuning parameter. The series shares a very strong intralayer Mn exchange and dominant ferromagnetic interlayer \(J_2^{MM}\), while weaker \(J_1^{MM}\) and \(J_3^{MM}\) generate magnetic competition. In nonmagnetic-\(R\) members such as YMn\(_6\)Sn\(_6\), those competing interlayer terms can drive helical or spiral phases; in moment-bearing members, Mn–\(R\) antiferromagnetic coupling and \(R\)-ion anisotropy stabilize ferrimagnetism, but the strength of Mn–\(R\) binding weakens from Gd to Tb to Dy and beyond [2110.14713]. The high-entropy phase occupies an intermediate but distinct position in this landscape. Its \(T_C=420\) K lies between GdMn\(_6\)Sn\(_6\) at 350 K and TbMn\(_6\)Sn\(_6\) at 450 K. Its two spin-reorientation temperatures, 210 K and 80 K, are described as new collective effects of competing \(R\) anisotropies, and no single-\(R\) parent shows the same cascade. Its magnetoresistance at 9 T, \(225\%\), is more strongly linear than in any single-\(R\) compound, for which \(\sim 150\%\) 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\(_6\)Sn\(_6\). Its anomalous Hall conductivity per kagome layer, \(0.17~e^2/h\), lies near the weighted average of parent values \(0.04\)–\(0.27~e^2/h\), which is taken as evidence that topological bands survive high-entropy disorder [2507.22809]. 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–\(R\) 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 [2110.14713]. In the high-entropy material, this tuning is not achieved by moving from one end member to another, but by embedding several competing \(R\)-sector tendencies within the same \(P6/mmm\) kagome lattice. Source: https://www.emergentmind.com/topics/gd-tb-dy-ho-er-mn6sn6