Highly anisotropic magnetism in the vanadium-based kagome metal TbV₆Sn₆

The crystal structure of TbV₆Sn₆, obtained from the 300 K refinement of single crystal X-ray diffraction data in space group $P6/mmm$ is shown in Fig. 1. The room temperature model (via a laboratory X-ray source) agrees with the refinement of 300 K synchrotron data, and low-temperature synchrotron data show no symmetry breaking down to 10 K. Tb and V-ions occupy the unique hexagonal crystallographic sites 1b and 6i respectively; whereas Sn occupy three different hexagonal crystallographic sites 2d(Sn1), 2e(Sn2) and 2c(Sn3). The structure consists of layered units of [V₃Sn1][TbSn2][V₃Sn1][Sn3] stacking along the crystallographic c-axis. Fig 1(b) presents a topside view of the crystal structure, and it highlights the kagome layer of V-atoms within the ab-plane. Each unit cell is made by a slab of V-Sn1-Sn3-Sn1-V atoms and a TbSn2 layer. In the TbSn2 layer, Tb atoms occupy the center of a stanene lattice of Sn2 atoms. The interstitial Tb atoms form a triangular lattice network as shown in Fig. 1(c).

The structural parameters at 300 K, 100 K, and 10 K are shown in Table 1. Due to the presence of interlayer diffuse scattering in the synchrotron X-ray data, the thermal parameters of V-sites exhibited unstable behavior while optimizing a structural model during refinement. To compensate, the thermal parameters of V-sites determined via laboratory X-ray diffraction refinement at room temperature were used and subsequently fixed while refining the synchrotron data. At 300 K, the refined nearest-neighbor (NN) V-V distance of 2.777(2) Å within a kagome plane is much smaller than the out-of-plane V-V distance of 4.633(3) Å along the c-axis. Magnetic ions of Tb occupy the corners of the unit cell, yielding a distance between NN Tb ions equal to the value of lattice parameter in all directions. Upon cooling, the unit cell, V-V, Tb-Tb distances all contract in a conventional fashion.

Figs. 2(a) and (b) display 300 K synchrotron X-ray scattering data highlighting select Bragg peaks in the ($H$, $K$, 0) and (0, $K$, $L$) scattering planes, respectively. Diffuse scattering is apparent along the $L$ direction, likely arising from interlayer structural disorder. Correlation lengths within the plane and between the planes do not change appreciably upon cooling to 10 K. Representative minimum correlation lengths $\xi_{min}$ were extracted from the $(-1,0,4)$ peak by fitting resolution-limited Gaussian lineshapes along the $H$ and $K$ directions. These fits render $\xi_{H/K}=900(20)$ Åin the ab-plane. Deconvolving the resolution via fits to a Voigt lineshape for cuts along the $L$-axis. These yielded an interplanar correlation length $\xi_{L}=460(10)$  Å.

Magnetization data collected as a function of temperature and magnetic field are summarized in Fig. 3. For each set of data, the magnetic field is applied either parallel or perpendicular to the $(0,0,1)$ axis, and the magnetization exhibits a strongly anisotropic behavior. At low temperature the magnetic susceptibility, $\chi$, along the $c$-axis is approximately two orders of magnitude weaker than its value within the ab-plane (Fig. 3(b)). Just above the ordering temperature at $T=5$ K, the anisotropy reaches a value of $\chi_{c}/\chi_{ab}=197$ under a 10 mT field. This demonstrates a strong easy-axis anisotropy in TbV₆Sn₆, similar to a number of other Tb-based compounds [35, 36, 25, 37]. The temperature-dependent magnetization reveals a magnetic transition at $T_{C}$ = 4.3(2) K, shown most clearly in the c-axis aligned data in Fig. 3(b).

Isothermal, field-dependent magnetization data collected below $T_{C}$ are plotted in Fig. 3(d). Along the easy-axis direction, the moment polarizes rapidly and reaches saturation near 1 T. The value of the saturated moment along the easy axis is nearly 9 $\mu_{B}$, as expected for the $4f^{8}$ electronic configuration of Tb³⁺ ions with $J=6$ and $g_{Lande}=1.5$. When the field is aligned along the ab-plane, the moment only weakly responds and fails to saturate up to 7 T. This demonstrates that TbV₆Sn₆ is an extreme easy-axis ferromagnet in which the value of g-tensor is nearly zero perpendicular to the moment easy-axis. The ferromagnetic state is rather soft, and the coercivity is $\approx 0.015$ T.

Apart from $T_{C}$, an additional broad magnetic anomaly appears in the temperature-dependent magnetization data near 60 K. The inset of Fig. 3(a) illustrates this broad feature in $\chi$, which appears only when the magnetic field is oriented within the ab-plane. This broad feature is consistent with the presence of a low-lying crystal field of the split $J=6$ ground state multiplet. Upon heating, thermal occupation of the first excited state drives an excitonic mixture of a multiplet with greater easy-plane character, and, above this temperature, the susceptibility becomes increasingly isotropic.

The inverse magnetic susceptibility, $\chi^{-1}$, is plotted in Fig. 3(c). $\chi^{-1}$ varies linearly with temperature above 100 K once the first low-lying crystal field multiplet is appreciably populated. An empirical Curie-Weiss analysis, $\chi^{-1}=(T-\Theta_{eff})/C$; $C$ = Curie constant, $\Theta_{eff}$ = an empirical Curie-Weiss temperature in the high temperature, thermally mixed state, was performed above 100 K and the fits are overplotted with the data in Fig. 3(c). This analysis yields $\Theta_{eff}=33.1(1)$ K parallel and $\Theta_{eff}=-40.4(2)$ K perpendicular to the moment easy-axis in the excitonic regime. Curie constants of 11.69(1) and 13.41(1) cm³ mol^-1 are fit with the field parallel and perpendicular to the c-axis respectively. These Curie constants translate to effective magnetic moments of 9.6 and 10.3 $\mu_{B}$/Tb along the respective directions, consistent with the $J$= 6 ($L$=3, $S$=3) moment of Tb³⁺ ions. Higher magnetic fields or a solution for the non-Kramers ground state multiplet wave functions will be required to quantitatively determine the precise g-tensor anisotropy.

Temperature dependent electrical resistivity $\rho_{xx}(T)$ data measured under magnetic fields of $\mu_{0}H$ = 0, 0.05 and 1 T are displayed in Fig. 4. $\rho_{xx}$ decreases smoothly upon cooling toward $T_{C}$, demonstrating a residual resistivity ratio $\rho_{xx}(300$ K$)/\rho_{xx}(2$ K$)\approx 18$. Zero-field data show a weak decrease near 3.5 K associated with Sn-superconductivity. This effect arises from small quantities of Sn-flux inclusions likely within the bulk of the crystal. This effect is removed by applying a field larger than the critical field ($\approx 300$ Oe) of Sn, and after doing so, TbV₆Sn₆ exhibits an inflection in the value of $\rho_{xx}(T)$ when cooling below $T_{C}$. The inset of Fig. 4 illustrates this small drop in resistivity at $T_{C}$, and when the ordered state is fully saturated ($\mu_{0}H$= 1 T), the inflection about $T_{C}$ becomes significantly broader.

The resistivity data in the paramagnetic regime below 100 K is well fit to the form $\rho_{xx}(T)$ = $\rho(0)+AT^{2}$. The fit is overplotted with the data in Fig. 4 and gives $\rho_{xx}(0)=1.884(3)$ $\mu\Omega$ cm and $A$ = 6.154(5)$\times$ 10^-4 $\mu\Omega$ cm K^-2. The values of $\rho_{xx}(0)$ and $A$ are close to the values of 2 $\mu\Omega$ and 3.5 $\times$ 10^-4 $\mu\Omega$ cm K^-2 for similar materials GdV₆Sn₆ and YV₆Sn₆ respectively [6, 32]. At high temperatures near 300 K, $\rho_{xx}(T)$ switches to follow a conventional, linear temperature dependence.

The field dependence of the transverse magnetoresistance $\rho_{xx}(H)$ collected at various temperatures and the resulting magnetoresistance ratio (MR) are plotted in Fig. 5. $\rho_{xx}(H)$ reveals a weak asymmetry with the reversal of the applied field direction due to the mixing of the transverse magnetoresistance with a strong Hall signal. The magnetoresistance is isolated from the mixed Hall signal by extracting the symmetric component of $\rho_{xx}(H)$ via averaging over the negative and positive field directions [38, 39]. The field dependence of resistance changes from negative and quadratic at low fields to linear and positive at high fields, and the resulting minima in the MR at low temperature form as the field polarizes the Tb moments and suppresses spin fluctuations. At 1.8 K, positive MR appears near 1 T, the field where TbV₆Sn₆ is driven into a fully spin polarized state. Nonsaturating, positive MR is observed up to the maximum applied field of 9 T, where the MR reaches $\approx 50$ % at 1.8 K. .

Hall resistance $\rho_{xy}(T,H)$ data were collected to characterize the carrier concentrations and effective mobilities of the conduction bands. Fig. 6 shows $\rho_{xy}(H)$ measured at various temperatures with the magnetic field applied parallel to the c-axis. At 200 K and above, $\rho_{xy}(H)$ shows a linear field dependence, indicating a dominant single-carrier conduction channel. As the temperature is lowered, $\rho_{xy}$ switches to a nonlinear form, consistent with a multi-carrier model. As a result, $\rho_{xy}(H)$ was fit with two different models: a two-band model at low ($T<100$ K) and a single band model at high ($T>200$ K).

Fits to $\rho_{xy}$ are shown by solid lines in Fig. 6(a). With the exception of the low-field data at 1.8 K (below $T_{C}$), the fits capture the data well. The deviation from the two-band model below $T_{C}$ is visible in Fig. 6(b), and likely represents the onset of the anomalous Hall component within $\rho_{xy}$. Attempting to extract the anomalous Hall signal $\Delta\rho_{xy}$ via subtracting the two-band model from the experimental data is plotted in Fig 6(c); however it instead reveals an oscillatory component within the Hall signal. This could correspond to Hall data reflecting quantum oscillations arising from a small Fermi pocket, though the oscillations are not readily resolved in magnetoresistance $\rho_{xx}$ data [40]. An alternative analysis is to instead use a single band treatment and propagate the 1.8 K, high-field Hall response backward linearly and extract an anomalous Hall signal from the difference. This is also plotted in Fig 6(c), and, while the difference provides a signal more consistent with a traditional anomalous Hall effect, it also neglects the multiband character of the low-$T$ transport. A similar analysis using this single band treatment with nonmagnetic YV₆Sn₆, for instance, resolves a qualitatively similar signal [32]. Future, detailed electronic structure calculations and lower temperature measurements will be important for determining the intrinsic magnitude of the anomalous hall effect (AHE) in TbV₆Sn₆.

Above $T_{C}$ and below 100 K, two-band fits capture $\rho_{xy}$ well and both electron-like and hole-like carriers contribute to the Hall signal. The resulting electron-type carrier density, $n_{e}$, and hole-type carrier density, $n_{h}$, are plotted in the upper panel of Fig. 6(d). The mobilities of each are plotted within the lower panel of Fig. 6(d), where each carrier type’s mobility becomes comparable at the lowest measured temperature. Above 200 K, $\rho_{xy}$ data are dominated by a single carrier type and a single band model showing electron-like carriers whose mobility increases quickly upon cooling.

Temperature-dependent heat capacity data, $C_{p}(T)$, collected between 1.8 K and 300 K at zero field are plotted in Fig. 7. $C_{p}(T)$ data exhibit a sharp anomaly near 4.1 K, coinciding with the onset of magnetic order below $T_{C}$. The low-temperature $C_{p}(T)$ collected away from short-range magnetic fluctuations obeys the Sommerfeld form $C_{p}$ = $\gamma T$ + $\beta T^{3}$, and the resulting fit is plotted in Fig. 7(b). The fit yields the coefficients $\gamma=72.1(20)$ mJ mol^-1 K^-2 and $\beta=1.28(1)$ mJ mol^-1 K^-4 translating to a $\Theta_{D}\approx 111$ K—though the $\gamma$ value extracted is purely phenomenological. The magnetic contribution to the heat capacity, $C\rm{{}_{mag}}$, through the ordering temperature can be isolated by subtracting the approximate phonon contribution, derived from measurements of YV₆Sn₆. This is overplot with TbV₆Sn₆ data in the inset of Fig. 7(a). Fig. 7 (d) shows the integration of the magnetic entropy $\Delta S$ determined from this data from 2 K to 9 K. The value approaches that expected for a non-Kramers doublet ground state with $S=Rln(2)$; however there remains unaccounted magnetic entropy below 2 K that is not included in this sum.

$C\rm{{}_{mag}}$ at higher temperatures is plotted in Fig. 7(c). A broad hump appears near 40 K, consistent with a Schottky anomaly expected from the thermal population of a low lying crystal field state. A little discrepancy to the energy scale of first exited crystal field state, compared to the energy scale observed in magnetization measurements, may be associated with the large size difference of the R-site ions and the systematic errors associated with the measurements. We believe that the subtraction of phonon contributions to the heat capacity is not perfect due to the large size difference of Y and Tb-ions and stronger contribution of Einstein heat capacity in YV₆Sn₆. A simple, two level model is used to fit the data to the form $C_{Schottky}=s\times R(\frac{\Delta}{T})^{2}\frac{e^{\Delta/T}}{[1+e^{\Delta/T}]^{2}}$ with $s$ being an overall scale factor and $\Delta$ being the energy gap between the two-level system. This form captures the asymmetric peak reasonably well on the high temperature side; however, upon cooling $C\rm{{}_{mag}}$ drops more rapidly than expected. This misfit at the low temperature portion of the Schottky peak likely arises from a slight oversubtraction of the phonon contribution using the scaled YV₆Sn₆ data. The parameterization predicts the first excited state near $\Delta=114$ K, though determining the full level scheme will require either inelastic neutron or Raman spectroscopy.