Magnetism and fermiology of kagome magnet YMn₆Sn₄Ge₂

Abstract

Kagome lattice magnets are an interesting class of materials as they can host topological properties in their magnetic and electronic structures. YMn₆Sn₆ is one such compound in which various exotic magnetic and electronic topological properties have been realized. Here, by means of a partial substitution of Sn with an isovalent and slightly smaller atom Ge, we demonstrate the sensitivity of such chemical substitution on the magnetic structure and its influence in the electronic properties. Magnetic structure of YMn₆Sn₄Ge₂ determined by neutron diffraction reveals an incommensurate staggered magnetic spiral with a slightly larger spiral pitch than in YMn₆Sn₆. This change in magnetic structure influences the Fermi surface enhancing the out-of-plane conductivity. Such a sensitivity to the partial chemical substitution provides a great potential for engineering the magnetic phases and associated electronic properties not only in YMn₆Sn₆, but also in the large family of 166 rare-earth kagome magnet.

Introduction

Materials exhibiting exotic magnetic textures in the presence of nontrivial electronic structures have shaped the direction of much research in recent years. Investigations into these materials have led to many predictions and experimental observations of quantum phenomena including the quantum anomalous Hall effect^1,2, chiral Majorana modes^3,4, and the topological magnetoelectric effect^5,6. These phenomena rely on the nature of the electronic structure in the presence of time-reversal symmetry breaking driven by the material’s magnetism^7,8,9. An increasingly popular class of materials to investigate the interplay of topological magnetic and electronic structures are the kagome-net magnets^{10,11,12,13,14,15}.

RT₆Sn₆ compounds, where R represents a rare-earth element, and T is a 3d transition metal element, are the recent addition to the class of kagome magnets^{16,17,18,19,20,21,22}. In these compounds, the T atoms are arranged in the kagome geometry, and the crystal structure provides a rich materials space to tune both the electronic and magnetic properties. Compounds with non-magnetic R atoms are simpler as magnetism comes only from the T sublattice. The most widely studied, and arguably the most interesting member of this family, is YMn₆Sn₆ where multiple exotic properties have been realized including a series of competing magnetic phases and the topological Hall effect^16,23,24, Dirac bands^25,26, a rarely observed magnetization-driven Lifshitz transition²⁷, a large anomalous transverse thermoelectric effect²⁸, and an emergent electromagnetic induction²⁹.

The magnetic structure drives most of these phenomena in YMn₆Sn₆ and the interesting magnetism in this compound comes from a parametric frustration facilitated by its crystal structure shown in Fig. 1a¹⁶. It consists of Mn kagome layers in the ab-plane that are separated alternately by two distinct blocks along the c-axis: a layer containing only Sn atoms (Sn-layer) and a layer containing both Y and Sn atoms (Y-Sn layer) as depicted in Fig. 1a. In the first approximation, the energy of this system can be described by the Hamiltonian¹⁶:

$$\begin{array}{rcl}{{{\mathcal{H}}}}&=&\mathop{\sum}\limits_{i,j}{J}_{n}{{{{\bf{n}}}}}_{i}\cdot {{{{\bf{n}}}}}_{j}+\mathop{\sum}\limits_{i,j}{J}_{p}{{{{\bf{n}}}}}_{i}\cdot {{{{\bf{n}}}}}_{j}\\ &+&\mathop{\sum}\limits_{i}{J}^{z}{n}_{i}^{z}\cdot {n}_{i+1}^{z}+K\mathop{\sum}\limits_{i}{({n}_{i}^{z})}^{2}+\mathop{\sum}\limits_{i}{{{{\bf{n}}}}}_{i}\cdot {{{\bf{H,}}}}\end{array}$$

(1)

where H is the external field and n is a unit vector along the local magnetization direction. The first sum runs over the three nearest neighbors along the c-axis, the second over the first nearest neighbors in the ab-plane, the next over all vertical bonds (i+1 denotes the nearest c-neighbor), and last two over each Mn ion. K is the single-ion anisotropy, and the Ising-type anisotropic exchange, J^z, is the only one allowed by symmetry for the vertical bonds. As a further simplification, in our previous paper¹⁶ we limited the latter term to nearly-ferromagnetic vertical bonds only.

Fig. 1: Crystal structure and magnetic phase diagram of YMn₆Sn₆.

a Sketch of crystal structure of YMn₆Sn₆ where J_i represents the magnetic exchange interaction in different Mn planes indicated by the red lines. b Sketch of magnetic phase diagram of YMn₆Sn₆ stabilized by in-plane magnetic field.

The in-plane interaction among the Mn atoms within a kagome layer is strongly ferromagnetic (J_p < 0). The spins are forced to lie in the ab-plane^16,30 due to the net effect of the single-ion term (easy-axis) and the Ising exchange (easy-plane and stronger). The most interesting part of the magnetic order is the helical magnetic spiral at T = 0 and H = 0 which comes from the competition between J₁, J₂, and J₃, and is therefore very sensitive to the stacking pattern of the Mn-layers along the c-axis. In YMn₆Sn₆, the exchange interactions through two inequivalent layers along the c-axis within a unit cell are opposite in sign. Specifically [Fig. 1a], the interaction across the Sn-layer is ferromagnetic (FM) (J₁ < 0) and that across the Sn-Y layer is antiferromagnetic (J₂ > 0). The exchange interaction between the like-Mn layers (i.e., the next-nearest neighbor interaction) in YMn₆Sn₆ is weak and ferromagnetic J₃ < 0. In the absence of J₃, a commensurate antiferromagnetic structure is expected²³. In fact, this is the case just below the Néel temperature (T_N) of 345 K. The presence of J₃ introduces parametric frustration and the system transitions into a unique staggered spiral phase below 333 K that persists down to the lowest measured temperatures¹⁶.

The application of an external magnetic field in the ab-plane stabilizes a series of competing magnetic phases. A sketch of the magnetic field (B)-temperature (T) phase diagram of YMn₆Sn₆ obtained in previous studies^16,23 is depicted in Fig. 1b. At low temperatures there are four main magnetic phases, namely distorted spiral (DS), transverse conical spiral (TCS), fan-like (FL), and forced ferromagnetic (FF). In addition, there are three smaller phases labelled “I”, which has a mixture of commensurate and incommensurate peaks, the canted antiferromagnetic phase (CAF), and the antiferromagnetic phase (AF)²³.

As these various magnetic structures in YMn₆Sn₆ are directly related to the crystal structure, mainly set by the spacing blocks between the Mn layers, replacement of Sn atoms by isoelectronic Ge provides a clean way to influence the magnetic properties by tuning these exchange interactions. In this work, we report the synthesis of YMn₆Sn₄Ge₂ and the study of its magnetic and magnetotransport properties by means of magnetic and transport measurements, and neutron diffraction experiments; and, by means of first principles calculations, we provide the microscopic insight into the influence of the magnetic structure on its electronic structure and thus the transport properties. We find that in YMn₆Sn₄Ge₂, Ge replaces Sn preferentially from one of the three sites [see structural details in “Crystal chemistry” and Table 1]. Directly related to this particular replacement, YMn₆Sn₄Ge₂ orders directly into the incommensurate spiral phase below T_N of 345 K with the spins lying in the ab-plane, and gives rise to a distinct magnetic and electronic structure. Magnetization and magnetotransport measurements show marked differences from that of the parent compound and indicate that new magnetic phases are stabilized. Counterintuitively, conductivity is enhanced, more significantly along the c-axis due to the change in fermiology introduced by the Ge doping. Our results provide an important insight into the doping study of not only YMn₆Sn₆, but all RMn₆Sn₆ compounds that are currently attracting significant attention for electronic and magnetic topological states.

Table 1 Crystallographic data of YMn₆Sn₆.

Results

Crystal chemistry

YMn₆Sn₆ crystallizes in the HfFe₆Sn₆-type structure in the hexagonal space group P6/mmm (#191). The crystallographic data taken from Ref. ¹⁶ are presented in Table 1. In this structure, there are three inequivalent Sn positions where Sn₁, Sn₂, and Sn₃ take Wyckoff positions 2e, 2d, and 2c, respectively. Previous doping studies have reported the dopant atoms entering into different Sn positions. For example, in YMn₆Sn_6−xGa_x, Ga enters into the 2c site³¹, while in YMn₆Sn_6−xIn_x, In takes the 2d site³². An analysis of calculated x-ray diffraction patterns of YMn₆Sn₄Ge₂ with Ge in each of the three positions, as depicted in Fig. 2a, shows that the intensity distribution in each of these cases is distinct, which makes it easier to identify the position taken by Ge in our sample. In Fig. 2a, we plot the experimental powder x-ray diffraction pattern of YMn₆Sn₄Ge₂ together with the calculated patterns in different possible structures. The intensity distribution of the Bragg peaks marked by asterisks clearly shows that Ge in this compound takes the 2c position. This analysis is further verified by the Rietveld refinement of the powder x-ray pattern as shown in Fig. 2b. The results of the refinement are presented in Table 2. The best refinement result was obtained for 93% of Ge and 7% of Sn at the 2c site. For simplicity, we use the formula YMn₆Sn₄Ge₂ throughout this article and the same is used in the analysis of magnetic measurements as it does not make a noticeable difference. The refinement shows that the YMn₆Sn₄Ge₂ lattice parameters a and c are smaller than in the parent compound, with the c (a) axis shrinking by 3.8% (2.5%). It is to be noted that it is the Sn in the Sn-Y layer that is replaced by Ge in YMn₆Sn₄Ge₂. This essentially influences J₂ and J₃, but leaves J₁ unaffected [see Fig. 1a].

Fig. 2: Structural characterization of YMn₆Sn₄Ge₂.

a Calculated powder x-ray diffraction patterns of YMn₆Sn₆ and YMn₆Sn₄Ge₂ with Ge in the three crystallographic sites 2c, 2d and 2e plotted together with the measured room temperature x-ray powder pattern of YMn₆Sn₄Ge₂. The same lattice parameters were used in the calculation for the three structures of YMn₆Sn₄Ge₂. Here, the x-ray powder patterns are shown between 5° ≤ 2θ ≤ 37°. Asterisks are used to emphasize some diffraction peaks. b Rietveld refinement of room temperature powder x-ray diffraction pattern of YMn₆Sn₄Ge₂. The model with Ge in the 2c position is used for the refinement. Inset shows an optical image of YMn₆Sn₄Ge₂ single crystals on a 2 mm × 2 mm grid (blue lines on the background). The cylindrical morphology of YMn₆Sn₄Ge₂ single crystals is markedly different than that of the parent compound which grow in a plate-like shape. The crystallographic c-axis in these crystals is along the length of the cylinder.

Table 2 Selected data from Rietveld refinement of powder x-ray diffraction collected on ground crystals of YMn₆Sn₄Ge₂, and from our DFT optimization.

This result is in agreement with our DFT calculations. In the latter, we first fully optimized the crystal structure with one Sn replaced by Ge in each of the three crystallographically inequivalent Sn sites. We found a very strong preference for the 2c position and the calculated total energy per formula was 0.35 eV (0.48 eV) lower than for the 2d (2e) position (see Supplementary Table 1). Next, we have fully optimized the structure of YMn₆Sn₄Ge₂ with Ge in the 2c position. The results are shown in Table 2 together with our experimentally obtained structure.

Magnetic properties

DC magnetic susceptibility (M/B, where M is the magnetic moment and B is the external magnetic field) of YMn₆Sn₄Ge₂ for B = 0.1 T, as a function of temperature (T) is depicted in Fig. 3a. The black (red) curve is the susceptibility measured with B parallel to the ab-plane (c-axis). Both of these curves peak at 345 K indicating that the magnetic ordering takes place below this temperature.

Fig. 3: Magnetic properties of YMn₆Sn₄Ge₂.

a Magnetic susceptibility (M/B) of YMn₆Sn₄Ge₂ measured with a magnetic field B = 0.1 T, parallel and perpendicular to the c-axis. Field-cooled (FC) protocol was used for these measurements. b, c Magnetization M of YMn₆Sn₄Ge₂ as a function of external magnetic field (B) applied along the c-axis, and in the ab-plane at selected temperatures. d Magnetic phase diagram of YMn₆Sn₄Ge₂ for a magnetic field applied in the ab-plane, constructed from a color contour plot of dM/dB data measured between 1.8 K and 360 K every 20 K. The white dashed lines are guides to the eye at phase boundaries. DS and FF are distorted spiral and forced ferromagnetic phases, respectively. U1 and U2 are two unknown magnetic phases. Inset shows the magnetic field derivative of magnetization at 1.8 K.

Magnetization (M vs B) data of YMn₆Sn₄Ge₂ for B∣∣ c-axis at some representative temperatures are presented in Fig. 3b. At 1.8 K, M increases gradually with increasing B, and saturates above 8 T, attaining the saturated moment of 12.2 μ_B per formula unit. This behavior of M remains the same in the entire temperature range below T_N with the exception that the saturation field and the saturated moment (as expected) decrease with increasing temperature. Above T_N (at 360 K), the M vs B curve shows a rather monotonic increase, which indicates the presence of spin correlations above T_N, expected in this class of materials, as described in Ref. ¹⁶.

Magnetization data measured with the magnetic field in the ab-plane are more interesting and are depicted in Fig. 3c. At 1.8 K, M first increases linearly with increasing B up to 2.2 T, where it shows a sharp jump representing a metamagnetic transition. It then increases with increasing B, but makes a cusp-like feature, followed by a linear increase before saturating above 6 T. The saturated moment at 1.8 K is 12.8 μ_B per formula unit. With increase in temperature, the metamagnetic transition remains in the entire temperature range below T_N. This transition, however, occurs at lower B as the temperature increases. The magnetic saturation field also decreases with increasing temperatures. The cusp-like feature, observed distinctly just above the metamagnetic transition at 1.8 K, gradually flattens above 100 K and is linear above 200 K.

Magnetization along the c-axis in YMn₆Sn₄Ge₂ has similar behavior as that in the parent compound, except that the saturation is attained at much smaller field (8 T vs 13 T at 1.8 K). However, the magnetization of YMn₆Sn₄Ge₂ in the ab-plane has marked differences [see Fig. 3c, d]. One similarity is that the metamagnetic transition at 1.8 K occurs at the same magnetic field of 2.2 T, but the behavior after the metamagnetic transition is quite different. While there is a cusp-like feature immediately after the metamagnetic transition in YMn₆Sn₄Ge₂, the parent compound has a linear M dependence and is in the TCS magnetic phase. The cusp-like feature in the parent compound appears only in the FL phase between 6.7 T and 9.8 T (at 1.8 K). The FL phase in YMn₆Sn₆ is short lived and disappears above 170 K.

In Fig. 3d, we show the magnetic phase diagram of YMn₆Sn₄Ge₂ constructed by plotting the field derivative of magnetic moment (dM/dB) as a function of magnetic field in the temperature interval from 1.8 K to 350 K. The phase below the metamagnetic transition is the distorted spiral (DS) as this phase is similar to the DS phase of the parent compound [based on the magnetic structure determined in zero external magnetic field described in “Neutron Diffraction: Magnetic structure in zero external magnetic field” section below]. The phase at magnetic saturation is the FF phase. We mark the intermediate phases as the unknown phases U1 and U2 because these phases are different from both the TCS and FL phase of the parent compound as discussed in more details in the following sections.

Magnetotransport properties

Electrical resistivity of YMn₆Sn₄Ge₂ measured with the electric current applied along the c-axis (ρ_c), and in the ab-plane (ρ_ab) is shown by solid red and blue lines, respectively, in Fig. 4a. In each direction, the resistivity decreases with decreasing temperature indicating the metallic behavior of the material in the entire temperature range from 400 to 1.8 K. Both ρ_c and ρ_ab show a marked kink at 345 K indicating the reduction of the spin scattering of the charge carriers with the onset of the magnetic ordering. It is consistent with the T_N determined from susceptibility measurements in Fig. 3a. Resistivity for YMn₆Sn₆ is shown for comparison by the dashed lines in Fig. 4a and is very sensitive to the spiral ordering, especially when measured with the current along the c-axis²⁷. The resistivity does not show any noticeable change when YMn₆Sn₆ orders into the commensurate antiferromagnetic phase at 345 K, but shows a remarkable change of slope at 333 K as it enters into the incommensurate spiral phase. The fact that a remarkable kink is observed in resistivity at 345 K in YMn₆Sn₄Ge₂ indicates that this compound directly orders into the incommensurate spiral state (which is verified by neutron diffraction, presented below). The c-axis resistivity, therefore, can provide important information about the nature of the magnetic ordering in this compound as it does in YMn₆Sn₆²⁷. Even more interesting is that the resistivity (in either direction) in YMn₆Sn₄Ge₂ is smaller than that in the parent compound, more significantly along the c-axis than in the ab-plane. It suggests that the c-axis conductivity is significantly enhanced by Ge doping of YMn₆Sn₆. This effect can be seen more clearly in Fig. 4b where the ratio of c-axis to ab-plane conductivity (σ_c/σ_ab, where σ = 1/ρ) for both YMn₆Sn₆ and YMn₆Sn₄Ge₂ are plotted between 1.8 K and 400 K. It can be seen that Ge doping, despite being isoelectronic doping, introduces a significant change in the fermiology.

Fig. 4: Electrical resistivity and conductivity of YMn₆Sn₄Ge₂.

a Electrical resistivity of YMn₆Sn₄Ge₂ (solid lines) and YMn₆Sn₆ (dashed lines) as a function of temperature measured with the current applied along the c-axis (ρ_c) and in the ab-plane (ρ_ab). The YMn₆Sn₆ data are adopted from Ref. ^27,42. b Ratio of the c-axis to ab-plane conductivity of the two compounds.

To understand the effect of the different magnetic phases on magnetotransport, we measured magnetoresistance (MR). The MR is defined by (ρ_B − ρ₀)/ρ₀ × 100% where ρ_B(ρ₀) is resistivity in finite (zero) B. Let us first look at the c-axis MR (MR_c) which is depicted in Fig. 5a. At 1.8 K, MR_c first increases slightly until it reaches the metamagnetic transition, where it jumps sharply by ≈42%. With further increase in B it shows a plateau until about 4.5 T where it decreases sharply and then flattens above 6.2 T where magnetic saturation is attained. On increasing temperature, the same MR behavior persists up to 200 K. At and above 240 K, the MR decreases with increasing magnetic field above the metamagnetic transition. The overall behavior of MR measured with current in the ab-plane (MR_ab) as depicted in Fig. 5b is quite similar to that of MR_c, with the exception of the magnitude of jump at the metamagnetic transition, especially at lower temperatures. The MR_ab jump at 1.8 K is very small compared to that of MR_c as can be seen in Fig. 5c. Here, the four magnetic phases inferred from the magnetization [red curve in Fig. 5c; also see Fig. 3d] are clearly observed in the MR measurements in both directions.

Fig. 5: Magnetoresistance and magnetization of YMn₆Sn₄Ge₂ and YMn₆Sn₆.

Magnetoresistance of YMn₆Sn₄Ge₂ at selected temperatures measured with current applied along the (a) c-axis, and (b) in the ab-plane. Magnetoresistance (black and blue lines, left axis) and Magnetization (red line, right axis) of (c) YMn₆Sn₄Ge₂, and (d) YMn₆Sn₆ at 1.8 K. The YMn₆Sn₆ data in (d) are adopted from Ref. ^27,42. In each MR measurement, the magnetic field was applied in the ab-plane. In (c, d) the black (blue) curve represents the MR measured with current applied along the c-axis, I_c (ab-plane, I_ab). The green dashed lines in (c, d) are a guide to eye for the indicated magnetic fields. It is to be noted that demagnetization considerations in these materials do not significantly affect the comparison between MR vs B and M vs B²⁷.

Comparison of MR and magnetization for an in-plane magnetic field at 1.8 K of YMn₆Sn₄Ge₂ and YMn₆Sn₆ can be seen in Fig. 5c, d, respectively. First, let us compare MR_c [black curves in Fig. 5c, d]. In YMn₆Sn₆, MR_c changes very little, if at all, at the metamagnetic transition and shows a sharp drop at around 4 T due to the Lifshitz transition²⁷. MR_c then sharply increases before entering into the FL phase, then decreases continuously, and flattens after attaining magnetic saturation (not shown here). MR_c of YMn₆Sn₄Ge₂, on the other hand, shows a significant jump at the metamagnetic transition, plateaus until about 4.5 T, above which it decreases rapidly, and then flattens after entering into the saturated state. This MR_c behavior provides crucial information about the possible magnetic phases stabilized by the magnetic field in YMn₆Sn₄Ge₂. In our previous study of YMn₆Sn₆²⁷, we found that MR_c is sensitive to the angle of rotation of spins between two consecutive planes. For example, this angle changes negligibly at the DS to TCS phase transition at the metamagnetic transition, but the change is larger at the TCS to FL phase transition. Thus, the MR_c shows negligible change during the former transition while it increases by about 10% during the latter. Now, looking at MR_c of YMn₆Sn₄Ge₂, the fact that it increases by about 42% at the metamagnetic transition suggests that the inter-layer angle change between DS and U1 is much larger than that between the DS and TCS of the parent compound. The sharp drop in MR_c at 4.7 T is consistent with a second magnetic phase transition to U2 inferred from the magnetization data that shows a sudden slope change at the same field. The forced ferromagnetic (FF) phase is attained above 6.2 T.

Neutron diffraction: magnetic structure in zero external magnetic field

The zero-field magnetic structure of YMn₆Sn₆ has a temperature dependent periodicity along the c-axis with wavevector k = (0, 0, k_z), where k_z ranges from commensurate at 0.5 reciprocal lattice units (r.l.u.) just below the onset of T_N down to 0.26 r.l.u. (incommensurate) at 4 K. YMn₆Sn₄Ge₂ is also incommensurate along the c-axis, however, there is no sign of the initial commensurate phase. The total change in pitch between the onset of magnetic order and the lowest temperature measured is much smaller than that in the parent compound ranging from k_z = 0.27 r.l.u. at T_N = 345 K to 0.23 r.l.u. at 4 K. Fig. 6 shows a summary of the results from the Taipan experiment, where the temperature evolution of the magnetic wavevector [Fig. 6c] and area [Fig. 6d] were tracked via L-scans across the (0, 0, 2) + k Bragg peak. At lower temperatures, a one-peak Gaussian fit is sufficient to analyze the data. However, at temperatures above ≈ 100 K, the FWHM of the Bragg peak monotonically increases with increasing temperature, with values larger than that of the calculated instrumental resolution. Above ≈ 320 K, it is clear that there are actually two magnetic Bragg peaks in close proximity to one another (similar to that in the parent compound). Fits comprised of two Gaussian peaks were therefore used to analyze the temperature dependent parameters. The FWHM of each Gaussian peak was fixed to be that of the calculated instrumental resolution for the (0, 0, 2) + k position (0.018 Å⁻¹). The peak with the smaller-q value is denoted by the wavevector, k₁, and the peak with the larger-q value is denoted by the wavevector, k₂. The fits for representative data at 15, 260, and 330 K can be seen in Fig. 6b. The neutron data show that the magnetic structure associated with the k₂ wavevector is dominant as compared to that belonging to k₁, although it does appear that the k₁ magnetic Bragg peak persists to the base temperature measured [fit results for the k₁ structure are shown in Supplementary Fig. 1c, d]. This contrasts with the parent compound, where the volume of the k₁ and k₂ magnetic domains is approximately equal throughout the same temperature range.

Fig. 6: Neutron diffraction of YMn₆Sn₄Ge₂.

L-scans spanning the magnetic Bragg peaks at (0, 0, 2) + k_i. a Scans between 15 K and 340 K, where scans are offset from one another along the intensity-axis by an amount proportional to the temperature. b Individual scans at 15 K, 260 K, and 330 K, offset along the Intensity-axis for clarity, with two-Gaussian fits to the data. The green solid line is the fit for the k₁ magnetic peak, and the blue solid line is the fit for k₂. The black and grey solid lines are for the linear background and the total fit, respectively. c The evolution of the k₂ wavevector and (d) the area of the (0, 0, 2) + k₂ magnetic peak with temperature. The solid orange line in (d) is a power law fit to the data.

The power law, $I={I}_{0}{(1-\frac{T}{{T}_{{{{\rm{N}}}}}})}^{2\beta }$, was fit to the integrated area data for the Bragg peak, (0, 0, 2) + k₂, shown as the solid line in Fig. 6d. The fit was performed using only data points near T_N, between 300 and 346 K, with I₀ = 14.0 ± 0.6, T_N = 345.95 K ± 0.02 K, and β = 0.323 ± 0.008. The β value extracted from the power law fit is $\approx \frac{1}{3}$, which is typical for various three-dimensional magnetic materials.

Data from the HB-3 triple-axis spectrometer were used to refine the nuclear [Supplementary Fig. 1a] and magnetic structure. Data were collected as θ − 2θ scans across Bragg peaks, and the integrated area of Gaussian fits to the peaks were corrected for the Lorentz factor to extract the observed structure factor, ${\left\vert F\right\vert }_{obs}^{2}$. These values were used to refine the magnetic structure using the program, FULLPROF³³, and the results are shown in Fig. 7. Irreducible representational analysis was used to model the structure and the details of the refinement can be found in the Supplementary Note 1. The refined structure is the staggered spiral where all moments within a Mn kagome plane are ferromagnetically ordered and constrained to the ab-plane. The moments between Mn planes (along the c-axis) are rotated relative to one another by an angle α within a unit cell (between Mn layers separated by the Sn-layer) and by an angle β between unit cells (i.e., between the Mn layers separated by the Sn-Y layer) [see Fig. 1a]. The sum α + β determines the pitch length, and hence wavevector, of the spiral. At 1.5 K, k = (0, 0, 0.23) and the refined value α = 7.9(7)°, leading to a value of β = 75.0(7)°. The total ordered moment per Mn atom is 2.19(3) μ_B. A smaller set of data was collected at 250 K, where the refinement yielded μ = 1.61(8) μ_B and α = 4(2)°, with an agreement R_F = 10.6 [refinement shown in Supplementary Fig. 1b]. The refined moment values at both 1.5 K and 250 K are in agreement with those from magnetization measurements [Fig. 3b, c].

Fig. 7: Magnetic structure of YMn₆Sn₄Ge₂ at 1.5 K.

a View of the magnetic structure showing the moment rotation along the c-axis. Only Mn atoms are shown and the two layers of Mn atoms within a unit cell are shaded black and grey. The nuclear unit cell boundaries are outlined in grey. b View of the magnetic structure in the ab-plane of one unit cell. The Mn atoms within a unit cell are nearly ferromagnetically aligned, however, a small angle α = 7.9(7)° separates them. c The results of the magnetic structure refinement at 1.5 K, where the agreement between the observed (${\left\vert F\right\vert }_{obs}^{2}$) and calculated (${\left\vert F\right\vert }_{calc}^{2}$) structure factor is R_F = 5.63.

Discussion

One marked difference between YMn₆Sn₄Ge₂ and YMn₆Sn₆ observed in neutron diffraction data at base temperature is in the spiral pitch, k_z, which is 0.23 (T = 1.5 K) in the former compound and 0.26 (T = 4 K) in the latter. As such, the refined values of the two turning angles, α and β, in YMn₆Sn₄Ge₂ are 7.9° and 75.0°. These angles are quite different in the parent compound where α ≈ − 5°, and β ≈ 99° (the Supplementary Table 3 contains a quick reference to these comparisons). This suggests that unlike the parent compound where J₁ < 0 and J₂ > 0, both of these exchange interactions are negative (i.e. J_1,2 < 0) in YMn₆Sn₄Ge₂. In fact, the ratio of the exchange interactions J₂/J₁ and J₃/J₁ can be calculated from α and β using the following relations^16,34:

$$\alpha =-{{{\rm{sign}}}}({J}_{1}{J}_{3}){\cos }^{-1}\left(\frac{{J}_{2}{J}_{3}}{{J}_{1}^{2}}-\frac{{J}_{3}}{{J}_{2}}-\frac{{J}_{2}}{4{J}_{3}}\right),$$

(2)

$$\beta ={\cos }^{-1}\left(\frac{{J}_{3}{J}_{1}}{{J}_{2}^{2}}-\frac{{J}_{1}}{4{J}_{3}}-\frac{{J}_{3}}{{J}_{1}}\right),$$

(3)

$$\alpha +\beta ={\cos }^{-1}\left(\frac{{J}_{1}{J}_{2}}{8{J}_{3}^{2}}-\frac{{J}_{2}}{2{J}_{1}}-\frac{{J}_{1}}{2{J}_{2}}\right).$$

(4)

Experimental values of α, and β yield J₂/J₁ = 0.142 and J₃/J₁ = −0.0692, confirming that J₁ and J₂ are both ferromagnetic in YMn₆Sn₄Ge₂. This means that replacing Sn by Ge in the 2c position reverses the sign of both J₂ and J₃. As one can anticipate, the effect of Ge substitution is the strongest regarding J₂, but the effect on J₃, while sign-changing, is small in terms of the absolute change. The change of sign of J₂ is a large and unexpected effect. Indeed, intercalating a magnetic ion such as Tb, in the same plane, leads to an effective ferromagnetic J₁ interaction, which is readily understood in terms of exchange coupling between Tb and Mn^20,35 (regardless of the sign of the Tb-Mn exchange coupling). Ge is nonmagnetic, so the sign change, on the first glance, seems rather mysterious.

A useful hint is provided by our resistivity measurements, which show that Ge substitution increases the conductivity along the z direction (where z∥c-axis here) [Fig. 4a], thus making the material more metallic. This is consistent with the DFT calculations, which show the Fermi surface with quasi-2D sheets (in the spin-majority channel), which is absent in the parent compound [Fig. 8a, b]^16,27. Direct calculations of the plasma frequency squared $({\omega }_{p}^{2})$, which determines the conductivity in the single scattering rate approximation, also support this assertion; the calculated ${({\omega }_{p}^{2})}_{z}$ increases by 50% upon Ge substitution as shown in Fig. 8c (the in-plane conductivity also increases, albeit less, but is irrelevant for this discussion)!. The calculated σ_c/σ_ab presented in Fig. 8c agrees well with the experimental data depicted in Fig. 4b. Furthermore, better metallicity translates into a kinetic energy advantage for a ferromagnetic coupling, which is exactly what we deduced from our neutron measurements.

Fig. 8: Calculated electronic and transport properties of YMn₆Sn₄Ge₂ and YMn₆Sn₆.

Calculated Fermi surface of ferromagnetic YMn₆Sn₄Ge₂ for (a) spin majority, and (b) spin minority bands (c) Calculated c-axis plasma frequency squared ${({\omega }_{p}^{2})}_{z}$ (left axis), and ratio of c-axis to ab-plane conductivity (right axis), for the Fermi surface depicted in (a, b) (blue lines), and those of YMn₆Sn₆ (red lines).

This provides insight into the role of Ge substitution at the 2c position for both the magnetic ordering and fermiology of YMn₆Sn₄Ge₂ that change both the magnetic and transport properties measured in zero or small magnetic fields. These changes in magnetic and electronic structures are certainly responsible for the observed marked difference in the magnetic phases and the magnetoresistance behavior at higher magnetic field. Magnetic field dependent neutron diffraction experiments are required to determine the magnetic structure stabilized at the higher magnetic fields. More importantly, it will also be intriguing to perform several other measurements such as ARPES, STM²⁶, thermoelectric effect²⁸, and electromagnetic induction²⁹ to understand the role of the altered magnetic and electronic structure in these various properties observed in the parent compound YMn₆Sn₆.

YMn₆Sn₄Ge₂ is a compound between YMn₆Sn₆ and YMn₆Ge₆. Both of the end compounds first order into the commensurate antiferromagnetic structure, with YMn₆Sn₆ (YMn₆Ge₆) having easy plane (easy axis) anisotropy in the entire temperature range. In contrast to both of the end compounds, YMn₆Sn₄Ge₂ orders directly into an incommensurate spiral state. In addition, and interestingly, the saturation magnetic field of YMn₆Sn₄Ge₂ (6 T) is smaller than that of both YMn₆Sn₆ (9.8 T) and YMn₆Ge₆ (>30 T), despite the fact that complete replacement of Sn by Ge leads to a threefold increase of the saturation field. This is an indication that partial replacement of Ge in the RMn₆Sn₆ compounds leads to markedly different magnetic and transport properties than one would expect form just a linear interpolation between the two compounds. Our study thus provides a new prospective into the study of the R166 stannates to tune the magnetic and magnetotransport properties.

In summary, we have performed systematic transport, magnetic, and neutron diffraction experiments in a wide temperature and magnetic field range on single crystals of YMn₆Sn₄Ge₂. The experimental studies have been supported by DFT calculations. Our magnetotransport measurements confirm that the magnetic structure of the parent compound YMn₆Sn₆ can be tuned through Ge doping. From neutron diffraction, we observed that this compound orders directly into an incommensurate staggered spiral magnetic structure that persists from the Néel temperature to 1.5 K. An in-plane external magnetic field stabilizes two magnetic phases that are markedly different from those in the parent compound. Both experimental and calculated results reveal that Ge substitution onto the 2c site alters the signs of the interplanar exchange interactions J₂ and J₃ influencing the pitch of the staggered spiral. This change in magnetic structure influences the Fermi surface making the material more conductive along the c-axis. Altogether, our detailed studies of the kagome magnet, YMn₆Sn₄Ge₂, provide important information that will be crucial in understanding the magnetotransport properties and the magnetic structures in the class of R166 kagome magnets that is currently attracting a great deal of interest for both magnetic and electronic topological states.

Methods

Crystal growth

Single crystals of YMn₆Sn₄Ge₂ were grown by the self-flux method using Sn as a flux. Y pieces (Alfa Aesar; 99.9%), Mn pieces (Alfa Aesar; 99.95%), Sn shots (Alfa Aesar; 99.999%), and Ge pieces (Alfa Aesar; 99.9999%) were loaded into a 2-ml aluminum oxide crucible in a molar ratio of 1:6:18:2. The crucible was then sealed in a fused silica ampule under vacuum. The sealed ampule was heated to 1150 °C over 10 h, kept at 1150 °C for 10 h, and then cooled to 650 °C at the rate of 5 °C/h. Once the furnace reached 650 °C, the tube was centrifuged to separate the crystals in the crucible from the molten flux. Several well-faceted hexagonal crystals [see the inset in Fig. 2b for an optical image of the crystals] up to 40 mg were obtained in the crucible.

Powder x-ray diffraction

The crystal structure was verified by Rietveld refinement³⁶ of a powder x-ray diffraction pattern collected on a pulverized single crystal using a Rigaku Miniflex diffractometer with a copper target. The Rietveld refinement was performed using the FULLPROF software³³. As it was clear from the comparison of powder pattern with simulations shown in Fig. 2a that Ge has replaced Sn from 2c site, only the occupancy of Ge and Sn in 2c site was refined.

Magnetic and magnetotransport measurements

DC magnetization, resistivity, and magnetoresistance measurements were performed in a Quantum Design Dynacool Physical Property Measurement System (PPMS) with a 9 T magnet. ACMS II option was used in the same PPMS for DC magnetization measurements. Single crystals of YMn₆Sn₄Ge₂ were trimmed to adequate dimensions for electrical transport measurements. Crystals were oriented with the [0,0,1] and [1,1,0] directions parallel to the applied field for the c-axis and ab-plane measurements. Resistivity and Hall measurements were performed using the 4-probe method. Pt wires of 25 μm were used for electrical contacts with contact resistances <30 Ohms. Contacts were affixed with Epotek H20E silver epoxy. An electric current of 2 mA was used for the electrical transport measurements. Contact misalignment in the magnetoresistance measurement was corrected by symmetrizing the measured data in positive and negative magnetic fields. The magnetic and magnetotransport data (with current along c-axis) presented here were measured on the same single crystal. First, magnetic properties were measured in both directions. Then the same crystal was polished to measure the magnetotransport measurements. The resistivity and magnetotransport data with current in the ab-plane were measured in a second crystal from the same growth batch after characterizing the magnetic properties.

Neutron diffraction

Neutron diffraction data were taken on the thermal triple-axis neutron spectrometer, TAIPAN³⁷, at the Australian Centre for Neutron Scattering. A PG(002) vertically focusing monochromator and a PG(002) flat analyzer were used at a wavelength of 2.345 Å. Söller slit collimators with full-width-at-half-maximum (FWHM) angular divergences of $3{0}^{{\prime} }$-$2{0}^{{\prime} }$-$2{0}^{{\prime} }$ were placed before the monochromator, before the sample, and after the sample. Data were taken between temperatures of 360 K and 4 K using a cryo-furnace with He exchange gas for temperatures below 290 K. The sample used was a 19.7 mg single crystal of YMn₆Sn₄Ge₂ oriented in the (H, H, L) scattering plane. The sample mosaic was determined to be 0.37°, reflective of a good quality crystal. Additional neutron diffraction data were taken on the thermal triple-axis neutron spectrometer, HB-3, at the High Flux Isotope Reactor at Oak Ridge National Laboratory. A PG(002) vertically focusing monochromator and a PG(002) flat analyzer were used at a wavelength of 2.359 Å. Söller slit collimators with FWHM angular divergences of $4{8}^{{\prime} }$-$2{0}^{{\prime} }$-$2{0}^{{\prime} }$-$7{0}^{{\prime} }$ were placed before the monochromator, before the sample, after the sample, and after the analyzer. The sample was the same single crystal used in the TAIPAN experiment, but the crystal was oriented in the (H, 0, L) scattering plane and placed in a cryostat (1.5–300 K).

Density functional theory calculations

For the density functional theory (DFT) calculations we first used a projector augmented wave basis as implemented in the Vienna ab initio simulation package (VASP)^38,39, for structure optimization, and then used an augmented plane wave code WIEN2K⁴⁰ for Fermi surface analysis. In all cases a generalized gradient approximation for the exchange and correlation functional⁴¹ was utilized. No LDA+U or other corrections beyond DFT were applied. Up to 11 × 11 × 6 k-point mesh (64 irreducible points) was used to structural optimization, and 48 × 48 × 25 for the Fermi surface and transport analyses.

Error bars displayed in plots and uncertainties listed throughout the manuscript represent plus and minus one standard deviation.