A new double-layered kagome antiferromagnet ScFe$_6$Ge$_4$

ScFe$_6$Ge$_4$ with the LiFe$_6$Ge$_4$-type structure (space group $R{\bar{3}}m$), which has a double-layered kagome lattice (18$h$ site) of Fe crystallographically equivalent to that of a well-known topological ferromagnet Fe$_3$Sn$_2$, is newly found to be antiferromagnetic (AFM) with a high N\'eel temperature of $T_{\rm{N}} \approx 650$ K, in contrast to the ferromagnetic (FM) ground state previously proposed in a literature. $^{45}$Sc nuclear magnetic resonance experiment revealed the absence of a hyperfine field at the Sc site, providing microscopic evidence for the AFM state and indicating AFM coupling between the bilayer kagome blocks. The stability of the AFM structure under the assumption of FM intra-bilayer coupling is verified by DFT calculations.

Recently, a number of non-trivial quantum phases have been discovered in magnetic materials consisting of the kagome lattice [1].One of the intensively studied materials is the ferromagnetic (FM) Fe3Sn2 with a double-layered kagome lattice of Fe [2], which is a well-known topological ferromagnet (see, e.g., [3]).The space group of Fe3Sn2 is R3 __ and Fe occupies a site with Wykoff symbol 18h [2].
The Fe site is characterized by two adjacent breathing kagome layers (bilayer block).As a family of materials with the Fe sublattice crystallographically equivalent to that in Fe3Sn2, RFe6Ge4 compounds (R = Li, Mg, Zr, Sc) with the LiFe6Ge4-type structure are known [4] (compare Figs. 1(a) and (b)).
In RFe6Ge4, the Fe bilayer blocks and RGe8 hexagonal bipyramidal layers (R is the center of the Ge dodecahedron) are alternately stacked in the c-direction (Fig. 1(b)) [4].The Sn8 bipyramidal layer in Fe3Sn2 is replaced by the RGe8 layer in In RFe6Ge4.The vertices of the Sn8 bipyramids in Fe3Sn2 do not penetrate the Fe kagome plane, while those of the ScGe8 in RFe6Ge4 slightly penetrate the kagome plane (see Figs. 1(a) and (b)).On the other hand, RFe6Ge6 of the HfFe6Ge6-type (P6/ ) has Fe monolayers (non-breathing kagome, 6i site) stacked in phase, with the RGe8 bipyramidal layers and the Ge honeycomb layers alternating between the Fe kagome layers (Fig. 1(c)) [10,11].RFe6Ge6 is antiferromagnetic (AFM) when R is a nonmagnetic element [12][13][14][15], while Fe3Sn2 is FM [16].The magnetic anisotropy of both Fe3Sn2 and RFe6Ge6 is basically uniaxial with the c-axis as the easy axis: a canted component appears at low temperatures, but both show collinear structures just below the Curie and Néel temperatures (TC and TN), respectively [12,[15][16][17].The Fe moment of Fe3Sn2 is ~2.0 µB [16], while that of RFe6Ge6 is 1.5-2.1 µB [13,15], depending on R.These three crystal structures have in common that they are composed of the Fe kagome lattice, but the spacing and phase between the Fe layers are different, resulting in a diversity of magnetic coupling between the layers.
The only experimental study of the magnetic properties of the RFe6Ge4 series is for ScFe6Ge4 [18].
From magnetization measurements of polycrystalline ScFe6Ge4, the authors claimed that ScFe6Ge4 is FM with TC = 491 K and a spontaneous magnetization Ms of ~0.5 µB per formula unit (at 3 K), i.e.In this study, we newly found that ScFe6Ge4 is not FM but AFM with TN ≈ 650 K. 45 Sc nuclear magnetic resonance (NMR) shows that the internal field at the Sc site Hhf(Sc) is canceled in its magnetically ordered state, which microscopically proves that ScFe6Ge4 is not FM and indicates that the magnetic coupling between the Fe bilayer blocks is AFM.We also show by DFT calculations that the AFM state is more stable than the FM state.
We have synthesized ScFe6Ge4 from Sc (Johnson Matthey, 99.9% purity), Fe (Johnson Matthey, 99.95%), and Ge (Rare Metallic, 99.999%) by arc melting in an argon atmosphere.Powder X-ray diffraction measurements at RT showed that the sample has the LiFe6Ge4-type structure with lattice and natural abundance ratio 100%.DFT calculations were performed using the Vienna ab initio simulation package (VASP) [19][20][21][22].We used the projector augmented wave (PAW) pseudopotentials [23,24] with the generalized gradient approximation (GGA) scheme following the Perdew, Burke and Ernzerhof (PBE) functional [25].The conjugate gradient algorithm was used to relax the atoms.The Methfessele-Paxton scheme [26] was used for both geometry relaxation and total energy calculations.
All atoms were relaxed until the forces on the atoms were less than of 10 −2 eV/Å and the energy difference between two successive electronic steps was less than 10 −7 eV.The unit cell was doubled along the c-axis and the k-point mesh was set to 35 × 35 × 5 when spin-orbit coupling was not considered and 17 × 17 × 3 when it was considered.
The inset of Fig. 2 shows examples of isothermal magnetization curves (5 and 300 K).They vary linearly through the origin, indicating no Ms at the temperatures, contrary to what was reported in [18].
In our experiments, when the sample was synthesized with excess Fe, a small Ms was observed.This is most likely due to FM impurities based on the hexagonal Laves phase ScFe2 with TC = 542 K [27].
The temperature dependence of the susceptibility χ measured in a temperature range up to 700 K and a field of 10 kOe is shown in Fig. 2. χ is small at low temperatures, decreases slightly with increasing temperature up to ~400 K, increases at higher temperatures, reaches a maximum at ~650 K, and decreases at higher temperatures.Another small hump was observed at ~570 K (denoted by T * ).Except for the hump at T * , this behavior is typical for AFM materials, suggesting that ScFe6Ge4 is AFM below The Mössbauer spectrum of our sample at RT (not shown) was the same as reported in [18] (see Fig. 7 in [18]); the spectrum was practically a single sextet component with |Hhf(Fe)| ≈ 191 kOe.
Assuming a standard hyperfine coupling for the 3d transition metal elements, µFe is estimated to be ~1-2 µB.The observed |Hhf(Fe)| is nearly identical to the RT value for Fe3Sn2 (199-200 kOe) [28,29], suggesting that the µFe of ScFe6Ge4 is comparable to that of Fe3Sn2; the reported Ms in Fe3Sn2 is ~2 µB/Fe [16].
It is also interesting to note that the TN of ScFe6Ge4 (≈ 650 K) is almost the same as the TC of Fe3Sn2, indicating that the magnetic interaction strength in ScFe6Ge4 and Fe3Sn2 is almost the same and differs only in sign.The sign difference can be attributed to the different medium driving the interbilayer interaction: ScGe8 in ScFe6Ge4 and Sn8 in Fe3Sn2 (see Figs. 1(a) and (b)).Note also that the TN of ScFe6Ge6 (≈ 500 K) in the absence of the bilayer blocks is significantly lower than that of ScFe6Ge4 (≈ 650 K), indicating that the intra-bilayer interaction is stronger than the inter-monolayer interaction.In ScFe6Ge4, the in-plane Fe-Fe bond length is 2.48 Å or 2.60 Å, and the inter-plane Fe-Fe bond length in the bilayer block is 2.97 Å, suggesting that the Fe-3d electrons hybridize directly in the bilayer block.On the other hand, the bilayer blocks are separated by more than 4 Å, suggesting an indirect magnetic coupling between the bilayer blocks.Considering that the same bilayer block is common to both FM and AFM materials, it is reasonable to assume that the magnetic coupling within the bilayer block is FM both in-plane and inter-plane.
To investigate the field dependence of the susceptibility humps observed at high temperatures, the temperature dependence of M/H (M is the magnetization) was measured at several different fields H (inset of Fig. 3).Two humps corresponding to T * and TN were observed in the range of measured fields.
TN changed little with field, while T * decreased slightly with increasing field.Considering that T * tentatively corresponds to a certain phase transition, a magnetic phase diagram is shown in Fig. 3.
There are two distinct AFM phases.The low-temperature phase appears to be slightly destabilized by the application of the field.The change in the spin structure of Fe3Sn2 pointed out by Fenner etmal.[17] is suggestive; Fe3Sn2 undergoes a transition from the paramagnetic phase to a c-axis collinear FM state at TC, but at 520 K the moments begin to tilt from the c-axis to the c-plane, i.e., a non-trivial magnetic state with a non-collinear component begins to appear below the temperature close to T * of ScFe6Ge4.
It is very likely that the same phenomenon occurs in ScFe6Ge4 as in Fe3Sn2.
The inset of Fig. 4 shows an example of a 45 Sc-NMR field sweep spectrum at 4. These results provide microscopic and direct evidence that ScFe6Ge4 is not FM but AFM.Furthermore, one should consider a magnetic structure in which Hhf(Sc) is canceled out.
Based on the magnetic structures reported for Fe3Sn2 and ScFe6Ge6 with similar stacking in their crystal structures [2,10,11], we consider possible magnetic structures under the constraint Hhf(Sc) = 0. Fe3Sn2 is FM (TC ≈ 650 K) [16] and ScFe6Ge6 is AFM (TN ≈ 500 K) [12].In both compounds, the in-plane Fe-Fe bonds are FM.For simplicity, only the magnetic couplings between the Fe kagome layers are considered here, i.e.only up/down degrees of freedom in the moment direction are considered.In considering the model for ScFe6Ge4, it is reasonable to impose the following constraints.
(i) The in-plane Fe-Fe magnetic coupling is FM.(ii) The interplane coupling in the bilayer block is FM.(iii) The coupling between the bilayer blocks is AFM.Assumption (i) was made for commonality with Fe3Sn2 and ScFe6Ge6.The reason for assumption (ii) is discussed above.Assumption (iii) is based on the fact that Hhf(Sc) = 0. Since the hexagonal Fe atoms are symmetrically coordinated above and below the ScGe8 bipyramid, if the Fe moments of the kagome layers have AFM coupling with respect to the c-plane containing the Sc atoms, the Hhf(Sc) (both isotropic and anisotropic) cancel at the Sc site in terms of symmetry.This is consistent with the fact that in ScFe6Ge6, the coupling between the Fe layers sandwiching the ScGe8 bipyramids is AFM.The magnetic structure satisfying (i)-(iii) is uniquely determined and shown schematically in Fig. 1(b); the stacking of in-plane FM kagome layers is -UU-DD-UU-DD-UU-DD-(the magnetic unit cell is twice the size of the crystal unit cell, hereafter denoted -UU-DD-), where U and D denote the FM layers with up and down moments, respectively, and the dash (-) corresponds to the ScGe8 bipyramidal layer.Note that the discussion here does not exclude the existence of a noncollinear component or the possibility of non-uniformity in µFe.
To verify the validity of the proposed AFM structure, we calculated the total energies of the spin-polarized states by DFT.First, we compared the total energies of the FM and AFM states without considering the spin-orbit coupling (SOC).In addition to the model proposed above, we also calculated the total energies for the -UD-DU-UD-DU-UD-DU-and -UD-UD-UD-UD-UD-UDsequences.The results, listed in Table 1 as the difference from the value for the FM state, show that the AFM configurations are generally more stable than the FM one, and the -UU-DD-sequence proposed above is the most stable.Furthermore, we have calculated the total energies of the FM state and the proposed state by considering the SOC.The result is almost the same as that without the SOC (see Table 1).In this case, the stable solution is a structure with the magnetic moment parallel to the

constants a = 5 .
071 Å and c = 20.053Å, which are in good agreement with the literature values a = 5.066 Å and c = 20.013Å[4], a = 5.079 Å and c = 20.009Å[18].Magnetization was measured using a Quantum Design SQUID magnetometer MPMS in the temperature range 5-700 K and magnetic field range 0-70 kOe.45 Sc NMR experiments were performed by the conventional spin-echo method using a Thamway PROT-II spectrometer.The sample was powdered and fixed with paraffin to ensure random orientation of the particles.The parameters of the 45 Sc nucleus were nuclear spin I = 7/2, nuclear gyromagnetic ratio γ = 1.0343MHz/kOe, nuclear quadrupole moment Q = -0.220× 10 -24 cm 2 ,

3 __;
2 K.A sharp spectrum with satellites was observed near the field corresponding to ν/γ, where ν is the experimental frequency.The resonance field, corresponding to the position of the center line of the spectrum, was measured at different frequencies and plotted in Fig.4.Extrapolation of the data yields a straight line passing through the origin with a slope of 1.033(1) MHz/kOe, almost equal to γ of 45 Sc, thus assigning the observed signal to the 45 Sc resonance.We can also see that Hhf(Sc) is practically negligible.An electric field gradient with asymmetry parameter η = 0 is expected at the Sc site (3a) with site symmetry the satellites observed around the center line are due to quadrupole effects.The inset of Fig.4also shows the simulated result of the powder pattern assuming a small negative isotropic hyperfine shift Kiso = -0.3% and a quadrupole frequency νQ = 0.76 MHz.The agreement between experiment and calculation is good, proving that Hhf(Sc) ≈ 0. This is in contrast to |Hhf(Fe)| ≈ 191 kOe.
c-axis.The lattice constants of the AFM state obtained by structural optimization are a = 5.0524 Å and c = 20.0216Å, which are close to the experimental values of a = 5.071 Å and c = 20.053Å at RT.The calculated µFe is 2.0 µB.The experimental µFe is currently unknown for ScFe6Ge4, but it agrees with the Mössbauer result.Our results are inconsistent with those in[18], but the reason is not clear since the assumed AFM structures is not explicitly stated there.ScFe6Ge4 with a double-layered kagome lattice of Fe is shown for the first time to be an antiferromagnet with TN = 650 K.The absence of the internal field at the Sc site provides microscopic evidence for the AFM state.The analogy with the magnetic structures of ScFe6Ge6 and Fe3Sn2, which contain the same substructures as ScFe6Ge4, and Hhf(Sc) ≈ 0, suggests FM coupling within the bilayer block both in-plane and inter-plane, and AFM coupling between the blocks.The validity of the structure was confirmed by DFT calculations.Since the Fe sublattice in ScFe6Ge4 is crystallographically equivalent to that in Fe3Sn2, ScFe6Ge4 is of interest as an AFM alternative to Fe3Sn2 (FM at TC = 650 K), a known topological magnetic material.We must await single crystal experiments to obtain an accurate phase diagram and neutron diffraction experiments to know the evolution of the microscopic magnetism.

Figure 2 . 13 Figure 3 .
Figure 2. Temperature dependence of magnetic susceptibility measured at a magnetic field of 10 kOe for ScFe6Ge4.The inset shows examples of isothermal magnetization curves.

Figure 4 .
Figure 4. Frequency dependence of the resonance field of 45 Sc NMR for ScFe6Ge4 at 4.2 K.The inset shows an example of the field-sweep spectrum (measurement frequency 30.6 MHz) (closed circles) and powder patterns calculated assuming Kiso = -0.3% and νQ = 0.76 MHz (solid curve: bare intensity, dashed curve: Gaussian function with full width at half maximum of 150 Oe was convoluted).

Table 1 .
The total energy difference of the AFM states measured with respect to the FM state, as estimated by DFT calculations (eV/unit cell).