Coexisting Kondo and Weyl-like transport in kagome heavy-fermion metal, Yb$_y$Co$_3$G$e_{3-x}$Sn$_x$

@numerusform contribute predominantly to Kondo and Weyl-like transport, despite hybridization, suggesting a route towards selectively accessing correlated and topological responses within one bulk material without external Fermi-level tuning.

Introduction

Combining strong electronic correlations (e.g. heavy-fermions) with topological conduction (e.g. massless quasiparticles in Dirac and Weyl materials) in a single material is rare, but offers a route to materials whose electronic responses can be accessed selectively. In heavy-fermion compounds, hybridization between localized $4f$ states and conduction electrons produces quasiparticles with strongly enhanced effective masses and reorganizes the low-energy electronic structure.[1, 2] At higher temperatures, incoherent scattering from localized moments gives the logarithmic resistivity increase characteristic of the single-impurity Kondo effect.[3] On cooling, coherent scattering across a lattice of these localized moments can produce a resistivity maximum followed by a decrease called the Kondo lattice effect, with the ground state governed by the competition between Kondo screening and intersite magnetic interactions.[4] Diluting the magnetic sublattice shifts the coherence scale down in temperature, as shown in $\mathrm{Ce}_{x}\mathrm{La}_{1-x}\mathrm{Cu}_{6}$, $\mathrm{Lu}_{1-x}\mathrm{Yb}_{x}\mathrm{Rh}_{2}\mathrm{Si}_{2}$ and related systems.[5, 6, 7, 8, 9, 10]

Weyl systems provide a complementary form of electronic behavior, where linear band crossings make them solid-state analogs of relativistic chiral fermions.[11] Parallel electric and magnetic fields can generate an imbalance between Weyl nodes, producing an additional negative longitudinal magnetoresistance proportional to $B^{2}$[11, 12, 13] In Weyl–Kondo semimetals that break inversion symmetry, the Kondo lattice effect itself generates Weyl states within strongly renormalized quasiparticle bands,[14] and in the leading candidate, the nonmagnetic $\mathrm{Ce}_{3}\mathrm{Bi}_{4}\mathrm{Pd}_{3}$, a giant spontaneous Hall effect is attributed to such nodes pinned near the Fermi level.[15, 16] A different evolution occurs in YbPtBi: conduction-band Weyl states produce a chiral-anomaly contribution at elevated temperatures, but this contribution becomes negligible on cooling as Kondo coupling renormalizes the bands and reduces their effective velocity.[17] These examples raise a complementary possibility: Kondo and Weyl-like contributions could remain distinguishable in transport if they originate predominantly from different electronic states. In particular, a magnetic field or the exchange field of polarized moments can generate Weyl nodes from host-band crossings without requiring Kondo hybridization to create them.[18, 19]

Rare-earth kagome metals offer a natural setting in which to explore this possibility. In the $\mathrm{AM}_{6}\mathrm{X}_{6}$ family, a transition-metal kagome network is interleaved with a rare-earth sublattice,[20] bringing electronic states associated with kagome geometry together with localized $4f$ moments. The resulting band structures can contain dispersive Dirac bands, flat bands and Van Hove singularities. Recent work has established a triangular Kondo lattice in $\mathrm{YbV}_{6}\mathrm{Sn}_{6}$ and revealed hybridization between its kagome-derived bands and Yb $4f$ states.[21, 22] In $\mathrm{YbCr}_{6}\mathrm{Ge}_{6}$ spectroscopy and calculations identify coexisting kagome and Kondo-derived flat bands and a Dirac–Kondo semimetallic state.[23] These findings establish the capacity of kagome hosts to combine correlated and topological electronic structures, but whether Dirac or Weyl-like bands can remain visible as Kondo coherence develops remains an open question.

$\mathrm{Yb}_{y}\mathrm{Co}_{3}\mathrm{Ge}_{3-x}\mathrm{Sn}_{x}$ provides a compositionally tunable platform for addressing it. A dilute member of the $\mathrm{AM}_{6}\mathrm{X}_{6}$ family, $\mathrm{Yb}_{y}\mathrm{Co}_{3}\mathrm{Ge}_{3-x}\mathrm{Sn}_{x}$ contains a Co kagome network and a partially occupied, structurally disordered Yb sublattice. Previous measurements found a low-temperature resistivity upturn associated with Kondo scattering,[24, 25, 26] while pressure induced a maximum suggestive of Kondo lattice coherence.[26] Calculations for the ordered structural counterpart $\mathrm{YbCo}_{6}\mathrm{Ge}_{6}$ place a flat band and a spin–orbit-gapped Dirac crossing near the Fermi energy,[27] providing an electronic setting in which correlated and potentially topological responses could coexist. Structural work on related compounds also establishes flexibility in rare-earth occupancy and Ge/Sn substitution.[28] Exploring Yb$_y$Co$_3$Ge$_{3-x}$Sn$_x$ therefore allows the evolution of Kondo transport to be examined within a host containing multiple electronic states near the Fermi level.

Here we show that Yb$_y$Co$_3$Ge$_{3-x}$Sn$_x$ combines heavy-fermion behavior with an additional longitudinal transport contribution consistent with a Weyl chiral anomaly. Low-temperature transport, heat capacity and magnetization establish Yb-associated Kondo scattering and emergence of composition-dependent lattice coherence, with extrapolated Sommerfeld coefficients of approximately 2 J mol$^{-1}_{\mathrm{Yb}}$ K$^{-2}$. X-ray photoemission spectroscopy (XPS) supports predominantly trivalent Yb, while Y and Tb analogues lack the corresponding transport signatures, pointing at Yb as the necessary component. For current along the c-axis, transverse magnetoresistance follows single-ion $j = 1/2$ Kondo scaling,[29] whereas longitudinal magnetoresistance remains unsaturated to 32 T and is described by an additional positive conductivity proportional to $B^2$. This contribution remains resolved below the coherence temperature, contrasting with the suppression of the longitudinal chiral-anomaly signal in YbPtBi.[17] Although longitudinal magnetoresistance alone does not uniquely establish a chiral anomaly,[12, 30] the combined evidence supports a scenario in which Kondo and Weyl-like conduction arise from distinct electronic states. Their simultaneous visibility without external Fermi-level tuning suggests a route towards selectively accessing correlated and topological responses within a single bulk material.

Results

Growing Yb crystals of different fillings

Single crystals grow from a Sn flux (Methods) as thin hexagonal needles along the c-axis. A higher Yb loading of the melt gives a higher refined Yb occupancy, as reported earlier,[31] and our crystals span Yb = 0.19 to Yb = 0.48 (Supplementary Tables S1 and S2). They adopt the disordered Yb$_{0.5}$Co$_3$Ge$_3$ structure type,[24] space group $P6/mmm$ (Figure 1a), locally a hybrid of Y$_{0.5}$Co$_3$Ge$_3$-type and CoSn-type cells (Figure 1b and Supplementary Fig. S1). Viewed down the c-axis (Figure 1c), the Co kagome and Ge honeycomb lattices leave one channel per unit cell along the c-axis that holds the Yb site. Each channel holds a random sequence of filled and empty Yb sites, so the Yb sublattice is dilute and aperiodic by construction rather than by doping.

Heavy fermion behavior

We first investigate the influence of Yb filling on transport behavior of the crystals. Figure 1d shows the resistance of three Yb fillings and of Tb and Y analogues. All three Yb crystals show an upturn below $T \approx25$ K, matching the logarithmic Kondo upturn reported earlier.[24, 25, 26] On further cooling, however, a maximum develops and the resistance drops steeply. This maximum is the coherence peak of a Kondo lattice, which prior work on this compound family only observed under applied pressure.[26] It moves monotonically down in temperature as the Yb filling decreases, from $T = 4.4$ K to $T = 1.0$ K for Yb = 0.48, 0.19 respectively, and up in temperature with field (inset of Figure 1d and Supplementary Fig. S2), which neither analog shows.

To test for heavy quasiparticles, we measured the specific heat of Yb$_{0.48}$ and Yb$_{0.36}$ crystals in fields perpendicular to the c-axis (Figure 1e). At zero field $C_p/T$ per mole of Yb keeps rising on cooling down to the lowest temperature and reaches $C_p/T \approx600$ mJ mol$^{-1}_{\mathrm{Yb}}$ K$^{-2}$ at 2 K in both crystals, roughly two orders of magnitude above the 9 mJ mol$^{-1}$ K$^{-2}$ per formula unit reported for the Y analog.[28] In field the rise turns into a broad maximum that moves up in temperature, the Schottky anomaly of a doublet split by the field. Fitting all fields at once to the Schotte-Schotte resonant-level model[32] (Supplementary § 2.1) gives the Sommerfeld coefficient $\gamma(0)$, the zero-field $C/T$ extrapolated to $T \to0$. The model needs no crystal-field term, because the $4f$ entropy reaches only $S_{4f} \approx R \ln2$ at $T = 20$ K (Supplementary Fig. S3). X-ray photoemission spectroscopy (XPS) supports predominantly trivalent Yb (Supplementary Fig. S4), consistent with the local-moment response inferred from magnetization (Supplementary Fig. S8). We obtain $\gamma(0) = 1.8(2)$ J mol$_{\mathrm{Yb}}^{-1}$ K$^{-2}$ for Yb$_{0.48}$ and 2.02(5) J mol$_{\mathrm{Yb}}^{-1}$ K$^{-2}$ for Yb$_{0.36}$, well above the 0.4 J mol$^{-1}$ K$^{-2}$ commonly taken as the threshold for a heavy-fermion system.[33]

Single-ion Kondo scaling of magnetoresistance and magnetization

To test whether the field response of the Kondo state is that of a single Yb doublet, we modeled the measured magnetoresistance isotherms $\Delta\rho/\rho_0 = (\rho(B) - \rho_0)/\rho_0$ of Yb$_{0.48}$Co$_3$Ge$_{2.95}$Sn$_{0.05}$ between 0.35 and 25 K, with $\rho_0$ the zero-field resistivity. We used the Schlottmann model for $j = 1/2$, which gives the resistivity as a universal function of $B/B^*(T)$, with a single field scale $B^*$ set by the Kondo temperature. All isotherms of one orientation are fitted at once, with one amplitude per isotherm (Supplementary § 4.3).

For $B \perp c$ the isotherms collapse onto the Schlottmann curve over three decades of reduced field (Figure 2a), with a root-mean-square residual of 2 % of the amplitude, and the fit gives a field scale of $B_0 = 10.1(1)$ T and Kondo temperature of $T^* = 2.16(7)$ K. Only at low field and below 0.72 K does a small overshoot appear, where a model of isolated Kondo ions is expected to stop working. The magnetization of a crystal from the same batch collapses as well (inset of Figure 2a), with $B_0 = 7(4)$ T and $T^* = 2.4(26)$ K, consistent with transport. The large fit errors come from the 5 T limit of the magnetometer (Supplementary § 4.3). The doublet Kondo scale near $T \approx2.2$ K is thus half the coherence peak of $T \approx4.4$ K (Figure 1d), yet the isotherms between the two temperatures still collapse, so the onset of lattice coherence does not destroy the single-ion scaling of the field response immediately.

For $B \parallel c$ (Figure 2b) the same fit gives $B_0 = 5.0(6)$ T and $T^* = 2.4(3)$ K. The zero-temperature field scale is 2.0(2) times larger for $B \perp c$, while $T^*$ agrees within its error, the pattern of a single anisotropic doublet with $g_{\parallel}/g_{\perp} \approx2.0$. The collapse is worse, with a residual of 4 % of the amplitude against 2 % and systematic structure at low and high reduced field (Supplementary § 4.3). The magnetization along the $c$-axis (inset of Figure 2b) does not support the simple anisotropic doublet picture. Its Schlottmann collapse returns $B_0 \to0$, with a field scale $B^*_M \propto T$, like a Curie paramagnet. At 2 K it gives $B^*_M = 1.9$ T compared to the magnetoresistance value of $B^* = 6.5$ T, differing by a factor of more than 3. The low-temperature Curie $g$ factors, 4.24 to 4.59 along the $c$-axis and about 1.3 in the plane, show the same anisotropy (Supplementary Figs. S8 and S9).

Single-ion Kondo scaling thus describes the perpendicular magnetoresistance well, but for $B \parallel c$ only marginally, up to the field that polarizes the doublet, and with a field scale 3 times bigger than the magnetization. Therefore, trying to model the longitudinal magnetoresistance with only Kondo-like behavior is not enough, which leaves an additional negative contribution.

Non-saturating negative longitudinal magnetoresistance

In $\mathrm{Yb}_{0.48}\mathrm{Co}_{3}\mathrm{Ge}_{2.95}\mathrm{Sn}_{0.05}$ the $B \parallel c$ magnetoresistance, with the field parallel to the current $B \parallel c \parallel I$ (Figure 3), reaches $-21\%$ at 32 T and 0.44 K without saturating (Figure 3a). Figure 3b shows the same measurement on $\mathrm{Yb}_{0.37}\mathrm{Co}_{3}\mathrm{Ge}_{2.74}\mathrm{Sn}_{0.26}$, which has a lower filling. In both crystals the magnetoresistance is negative at every temperature, and at low temperature a few tesla remove most of the zero-field Kondo resistivity. At high field, however, it does not saturate, which is unexpected for a Kondo system.

The Schlottmann resistivity alone, $\rho_{\mathrm{Kondo,SM}}$, fitted to the 0.44 K isotherm of $\mathrm{Yb}_{0.48}$ (Figure 3c), levels off above about 20 T while the resistivity data keeps falling (Supplementary Section 5.2, Figure 3d). We therefore fit each $B \parallel c$ isotherm with $\rho_{\mathrm{Kondo,SM}}$, with its own amplitude $A$ and field scale $B^{*}$, plus an extra conduction channel which decreases resistivity as $\rho(B^{2})$ (Supplementary Section 6.1). In $\mathrm{Yb}_{0.48}\mathrm{Co}_{3}\mathrm{Ge}_{2.95}\mathrm{Sn}_{0.05}$ the fit resolves the channel at every temperature from 0.44 to 10 K, and it shrinks as the temperature rises. In $\mathrm{Yb}_{0.37}\mathrm{Co}_{3}\mathrm{Ge}_{2.74}\mathrm{Sn}_{0.26}$ the trend is the same, but above 4 K no channel can be resolved. Figure 3c and d test whether another negative magnetoresistance could account for the whole field dependence on its own, with two common models held to their theory. Three-dimensional weak localization[34] ($\rho_{\mathrm{WL}}$) yields $-0.02\%$ at 32 T.

about a thousand times less than the measured $-21$ % (Supplementary Section 6.2). Spin-disorder scattering by the Yb moments[35] ($\rho_{\mathrm{Kondo,YS}}$), $\Delta\rho/\rho_{0} \propto-M^{2}$ with its field scale fixed by a fit up to 5 T (Supplementary Section 6.3 and Supplementary Table S3), saturates with the moment and misses the high-field data. Fitted freely to every isotherm of Figure 3a, neither model gives physical values (Supplementary Fig. S10), with similar results for Yb$_{0.37}$Co$_3$Ge$_{2.74}$Sn$_{0.26}$ (Supplementary Section 6.4, Supplementary Table S4 and Supplementary Fig. S11).

Of the forms tested, only $\rho_{\mathrm{Kondo,SM}}-\rho(B^{2})$ describes the parallel magnetoresistance over the whole field range. The extra channel grows with field without saturating and is resolved up to 10 K in Yb$_{0.48}$Co$_3$Ge$_{2.95}$Sn$_{0.05}$ and down to 0.44 K, deep into the Kondo lattice regime. Its origin will be examined in the discussion.

Electronic structure

Measurements of Yb$_{0.48}$Co$_3$Ge$_{2.95}$Sn$_{0.05}$ using angle-resolved photoemission spectroscopy (ARPES) reveal a combination of dispersive and flat electronic bands (Figure 4a). In particular, linear-like electronic bands that extend towards the Fermi level are observed, more clearly resolved in the second-derivative data (Figure 4b). Additionally, multiple flat bands are also observed at higher binding energies.

To model and connect the ARPES spectra (Figure 4a,Figure 4b) to the Co kagome network, we first consider pure CoGe in its primitive CoSn-type cell, Co$_3$Ge$_3$ (DFT+SOC; Figure 4c,Figure 4d). Both the Co $d_{xz}/d_{yz}$ and the Co $d_{z^{2}}$ orbitals form band structures reminiscent of the prototypical single-orbital kagome lattice: two Dirac-like bands touching at K together with a nearly flat band (for more details see Supplementary Section 6.9). The two sets, however, live in different $k_z$ planes. The $d_{xz}/d_{yz}$ kagome bands appear at $k_z=0$ (Figure 4c), whereas the $d_{z^{2}}$ kagome bands appear at $k_z=\pi/c$ (Figure 4d). In Yb$_{0.5}$Co$_3$Ge$_3$ the c axis is doubled, which folds both planes onto $k_z=0$. With Yb randomly distributed, we therefore expect ARPES to probe a superposition of the two, which it seems to do (see Supplementary Section 6.9). Importantly, at the intermediate plane $k_z=\pi/2c$, where the mirror symmetry of the kagome plane no longer separates the two orbital sets, CoGe hosts a pair of Dirac-like bands of hybridized $d_{xz}/d_{yz}$-$d_{z^{2}}$ character whose Dirac point lies at the Fermi level, opened by only $\approx50$ meV with spin-orbit coupling.

Including Yb completes this picture. Starting from a magnetically polarized Yb configuration, we can stabilize a magnetic solution with a single Yb $4f$ band pinned to the Fermi level. In the $Yb_{0.5}Co_{3}Ge_{3}$ cell, the $k_z = \pi/2c$ plane of CoGe becomes the zone boundary, so the hybridized Dirac bands appear around H, where this flat $4f$ band crosses and hybridizes with them (Figure 4e). Thus, already at the mean-field DFT+$U$+SOC level, an explicit treatment of the Yb $4f$ shell yields both ingredients – heavy-fermion-like flat $f$ bands and Dirac/Weyl-like dispersions – at the H point of $Yb_{0.5}Co_{3}Ge_{3}$.

Discussion

Our measurements show that $Yb_yCo_{3}Ge_{3-x}Sn_x$ is a heavy-fermion Kondo lattice, with Sommerfeld coefficients well above the usual heavy-fermion threshold. For $B \perp c$ the magnetoresistance and the heat capacity fit a single $j = 1/2$ Kondo doublet, with the magnetoresistance collapsing onto one Schlottmann curve. The longitudinal ($B \parallel c \parallel I$) Schlottmann field scale of 5.0 T is half the perpendicular one ($B \perp c \parallel I$), but the magnetization along $c$ does not share it, and the resistivity does not saturate up to 32 T. Only the combined $\rho_{\mathrm{Kondo,SM}}-\rho(B^2)$ describes it over the whole field range. Given that both orientations were measured on the same crystal with the same contacts (Supplementary Section 6.5 and Supplementary Figs. S12 and S13), the anisotropy must originate from the Yb single-ion physics, from the measurement geometry, or from the band structure of the Co-host.

The Yb single-ion physics does not describe it. Below 10 K the Schlottmann model leaves structure in its residuals (Figure 2b), misses the high-field data which keep falling, and does not fit with Yosida’s spin-disorder model as well (Figure 3c). Weak localization, the usual negative magnetoresistance of a disordered metal, can also be excluded for this reason (Figure 3c). Current jetting, the geometric alternative, is unlikely (Supplementary Section 6.6).

An additional negative magnetoresistance component that appears only for $B \parallel I$, keeps growing at high field, and persists above the Kondo scale as well down to 0.44 K, has the form expected from the chiral anomaly of Weyl fermions.[12, 13] The anomaly adds a negative magnetoresistance that grows as $B^2$ for $B \parallel I$,[13] the same form as the extra channel in our fits. Our electronic-structure results provide a basis for this interpretation: ARPES resolves flat and dispersive bands, and calculations identify kagome-derived dispersive states together with Yb-derived flat states near the Fermi level (Figure 4). These results complement earlier calculations identifying near-Fermi-level kagome features in the related ordered structure. Magnetic or exchange fields can generate Weyl nodes from suitable host-band crossings, motivating this interpretation.

Combined with the Kondo lattice, this would place $Yb_yCo_{3}Ge_{3-x}Sn_x$ alongside $YbPtBi$[17] and $Ce_{3}Bi_{4}Pd_{3}$[15, 16] as a third candidate heavy-fermion Weyl system, although the heavy-fermion Weyl candidates differ in how the Kondo effect and the Weyl fermions relate.

In a Weyl-Kondo semimetal the Kondo effect forms the Weyl fermions,[14] and in $YbPtBi$ it suppresses their signal at low temperature.[17] Our data suggest a third relation. In our fits the Kondo part keeps the Schlottmann form and the extra channel only adds its own conductance in parallel, without any coupling, so the two sets of carriers appear to interact only weakly. One cause of this could be the apparently very weak crystal electric field the Yb atoms experience.

In summary, low-temperature and high-field magnetotransport, thermodynamic and magnetic measurements, identify $Yb_yCo_{3}Ge_{3-x}Sn_x$ as a kagome heavy-fermion metal with simultaneously distinguishable Kondo and Weyl-like transport responses. Yb-associated Kondo scattering develops into lattice coherence, while longitudinal transport requires an additional negative magnetoresistance proportional to $B^2$, consistent with a chiral-anomaly contribution.[12] This contribution remains resolved below the coherence temperature and shows no saturation within the measured field range up to 32 T. The combined transport and electronic-structure literature supports a picture in which the correlated and Weyl-like responses arise predominantly from distinct electronic states.

This picture expands the possible relationship between strong correlations and electronic topology beyond Weyl states formed within Kondo-renormalized bands.[14, 15, 16] Rather than becoming indistinguishable as correlations develop, the two responses remain accessible within the same bulk material without external Fermi-level tuning. The kagome host therefore offers a setting for investigating how different electronic states retain distinct transport responses in the presence of a Kondo lattice.[22, 23] Establishing the conditions that preserve this distinction could provide a basis for selectively controlling correlated and topological contributions to conduction.

Methods

Crystal growth and structure

Single crystals of $\mathrm{Yb}_{y}\mathrm{Co}_{3}\mathrm{Ge}_{3-x}\mathrm{Sn}_{x}$ were grown using the flux growth method.[36] The sample $\mathrm{Yb}_{0.48}\mathrm{Co}_{3}\mathrm{Ge}_{2.95}\mathrm{Sn}_{0.05}$ was grown by combining Yb (99.9 % ingots), Co (99.998 % powder), Ge (99.999 % pieces) and Sn (99.99 % shots) into an alumina Canfield crucible[37] in a ratio of 3:2:7:52, respectively, and sealed under 1/3 atm of Ar gas. The crucible was then heated to 1175 °C from 300 °C at a rate of 100 °C/h in a programmable furnace. The mixture dwelled here for 24 h and then cooled to 815 °C at a rate of 3 °C/h. The crucible was subsequently removed from the furnace, inverted and centrifuged at 1500 rpm to separate the single crystals from the flux. Residual Sn flux on the surface was etched using a 1:1 ratio of H$_2$O and HCl by volume. The reaction yields thin hexagonal needles, which can be connected to the amount of disorder across the $ab$ plane promoting crystallization mainly along the $c$ axis. Larger crystals were obtained by lowering the amount of Sn using the ratio 3:2:7:20. To tune the Yb occupancy of $\mathrm{Yb}_{y}\mathrm{Co}_{3}\mathrm{Ge}_{3-x}\mathrm{Sn}_{x}$, an Yb-deficient stoichiometric ratio was employed. The $\mathrm{Yb}_{0.36}$ and $\mathrm{Yb}_{0.37}$ samples were grown by combining the elements in a ratio of 1.2:6:6:25 and 1:6:6:25, respectively. The $\mathrm{Yb}_{0.19}$ sample was grown by lowering the amount of Sn flux using the ratio 1:6:6:10. We hypothesized that rather than the Sn acting purely as the flux, lowering the relative stoichiometry allows Sn to behave more as a reactant. This explains the increased Sn incorporation on the CoSn site, inducing more occupancy of that structure type relative to the $\mathrm{Y}_{0.5}\mathrm{Co}_{3}\mathrm{Ge}_{3}$ structure type. The $\mathrm{Y}_{0.22}$ sample was grown using Y (99.9 % pieces), Co, Ge and Sn in a ratio of 1:6:6:20, respectively, sealed and heated to 1175 °C. The reaction dwelled here for 6 h before being cooled to 500 °C at a rate of 5 °C/h.[28] The $\mathrm{Tb}_{0.34}$ sample was grown using Tb (99.9 % ingots), Co, Ge and Sn in a ratio of 1.2:6:6:20, respectively, sealed and heated to 1100 °C. The reaction dwelled here for 24 h before being cooled to 980 °C at a rate of 50 °C/h. The reaction underwent an additional cooling step to 940 °C at a rate of 3 °C/h and dwelled here for 72 h.[38] Both reactions were inverted and centrifuged at 1500 rpm and etched with 1:1 H$_2$O:HCl to remove residual Sn flux.

The transport and magnetization of each filling were measured on different crystals of one growth batch, which have the same Yb content within 0.01. The heat capacity was measured on crystals of other batches with the same filling as the transport crystals. The structure, the Yb occupancy, the split Ge site and the Sn content of each batch were refined from single-crystal X-ray diffraction at room temperature.

Transport

The resistance was measured with a four-probe lock-in technique, with the current of 100 to 200 $\mu$A along the needle axis $c$. The contact spacings were measured through a microscope. On $\mathrm{Yb}_{0.48}\mathrm{Co}_{3}\mathrm{Ge}_{2.95}\mathrm{Sn}_{0.05}$ the voltage contacts V56 used in the main text are 230 $\mu\mathrm{m}$ apart, and the second pair V12 is 1100 $\mu\mathrm{m}$ apart. Below 4 K the temperature during a field sweep was held by a constant pressure over the pumped ${}^{4}\mathrm{He}$ or ${}^{3}\mathrm{He}$ bath, and above 4 K by a fixed heater output. Fields up to 32 T were applied along c ($B \parallel c \parallel I$) and perpendicular to it ($B \perp c$), on the same contacts. Each isotherm is symmetrized in field and binned as described in Supplementary Section 5.1.

Magnetization and heat capacity

The magnetization of $\mathrm{Yb}_{0.48}$ (0.17 mg) and $\mathrm{Yb}_{0.37}$ (0.34 mg) crystals was measured in a Quantum Design MPMS-XL magnetometer in DC mode between 1.8 and 300 K in fields up to 5 T, after zero-field cooling, for both field orientations, with $c$ aligned to the field by eye to better than a degree. The heat capacity of $\mathrm{Yb}_{0.48}$ (16.2 mg) and $\mathrm{Yb}_{0.36}$ (5.9 mg) was measured in a Quantum Design PPMS from 2.2 K in fields from 0 to 9 T perpendicular to $c$, and at zero field up to 200 and 120 K.

Angle-resolved photoemission

ARPES measurements were performed at the Bloch beamline (MAX IV Laboratory). Data was taken at 18 K, with a photon energy of $h\nu= 188$ eV using linear horizontally polarized light. Samples were mounted on a copper plate using silver epoxy and cleaved in situ prior to measurements. ARPES data was analyzed using WaveMetrics Igor Pro 9 software.

Density-functional calculations

Density-functional calculations were performed with the Vienna Ab initio Simulation Package (VASP 6.4.3)[39, 40] using the projector-augmented-wave (PAW) method[41, 42] and the generalized-gradient approximation of Perdew, Burke and Ernzerhof (PBE)[43]. The PAW datasets treat Yb $5s^{2}5p^{6}4f^{7}6s5d$ (24 valence electrons, $4f$ in the valence), Co $3s^{2}3p^{6}3d^{7}4s$ (17 electrons) and Ge $3d^{10}4s^{2}4p^{2}$ (14 electrons) as valence states, with a plane-wave cutoff of 550 eV. Spin-orbit coupling is included self-consistently in a noncollinear formalism.[44, 45] The local Coulomb interaction in the Yb $4f$ shell is treated within the rotationally invariant DFT+$U$ scheme[46] with $U = 6$ eV and $J = 1$ eV; no $U$ is applied to Co $3d$. $\mathrm{Yb}_{0.5}\mathrm{Co}_{3}\mathrm{Ge}_{3}$ is modelled in the hexagonal HfFe$_{6}$Ge$_{6}$-type structure (space group $P6/mmm$, 13 atoms per unit cell) with $a = 5.075$ Å and $c = 7.804$ Å. Brillouin-zone integrations use a $\Gamma$-centred $8 \times8 \times6$ $k$-point mesh and Gaussian smearing of 0.05 eV, and total energies are converged to $10^{-8}$ eV. Because density-mixing schemes did not converge once the Yb $4f$ states approach the Fermi level, the self-consistency cycle uses direct band-by-band minimization. All input files were generated with the Atomic Simulation Environment.[47]

Fitting

All fits are least-squares fits with a soft-$\ell_{1}$ loss, and the errors of jointly fitted parameters are jack-knife errors over the isotherms or fields. The fit functions are given in Supplementary Sections 3.1, 5.3 and 6.

Acknowledgements

We acknowledge the support of HFML-FELIX, a member of the European Magnetic Field Laboratory (EMFL). We acknowledge the MAX IV Laboratory for beamtime on the Bloch beamline under proposal 20250269. Research conducted at MAX IV, a Swedish national user facility, is supported by Vetenskapsrådet (Swedish Research Council, VR) under contract 2018-07152, Vinnova (Swedish Governmental Agency for Innovation Systems) under contract 2018-04969 and Formas under contract 2019-02496. M.P.D., Z.C., P.M., D.P., A.G.-Č., M.R., S.B., M.E.K., S.W. and M.N.A. acknowledge the research program “Materials for the Quantum Age” (QuMat) for financial support. This program (registration number 024.005.006) is part of the Gravitation program financed by the Dutch Ministry of Education, Culture and Science (OCW). T.M.K. and M.N.A. acknowledge support from the NWO Talent Programme VIDI financed by the NWO VI.Vidi.223.089. M.N.A. acknowledges the Kavli Institute Innovation Award 2023. M.P.D., T.M.K., V.N.V., and M.N.A. acknowledge the Kavli Institute of Nanoscience Delft. R.M. and J.Y.C. acknowledge NSF DMR-2505304 and Welch Foundation AA-2056-20240404 for partial support of this project. V.N.V. acknowledges financial support from the HFML-FELIX Incentive fund. X.L. acknowledges the Guangzhou Elite Project scholarship (S. J. [?] No. 1). A.G.-Č. acknowledges the financial support of the Zernike Institute for Advanced Materials. S.G. and M.R. acknowledge support from the Vidi ENW research programme of the Dutch Research Council (NWO) [Grant DOI: 10.61686/YDRHT18202] with File No. VI.Vidi.233.077.

Data availability

The data supporting the findings of this study are openly available in 4TU.ResearchData at https://doi.org/10.4121/8807a9d6-8e8d-4484-8e0d-6cd972e61e0e

Code availability

The code supporting the findings and creating the figures of this study are openly available in 4TU.ResearchData at https://doi.org/10.4121/4ca95cec-f0b2-41e5-81af-3f6b462cd3de

Use of AI tools

Claude (Claude Opus 5.5, Anthropic) and Codex (GPT-6.1-sol, OpenAI) were used for language editing and refinement of the manuscript text, for refinement of the code used to generate figures, and for refinement of analysis code. All AI-assisted text and code were reviewed, tested and verified by the authors, and all figures were checked against the underlying data. The study design, analyses, interpretations and conclusions are the authors’ own, and the authors take full responsibility for the content of this Article.