Discovery of a long-ranged charge order with 1/4 Ge1-dimerization in an antiferromagnetic Kagome metal

Exotic quantum states arise from the interplay of various degrees of freedom such as charge, spin, orbital, and lattice. Recently, a short-ranged charge order (CO) was discovered deep inside the antiferromagnetic phase of Kagome magnet FeGe, exhibiting close relationships with magnetism. Despite extensive investigations, the CO mechanism remains controversial, mainly because the short-ranged behavior hinders precise identification of CO superstructure. Here, combining multiple experimental techniques, we report the observation of a long-ranged CO in high-quality FeGe samples, which is accompanied with a first-order structural transition. With these high-quality samples, the distorted 2 × 2 × 2 CO superstructure is characterized by a strong dimerization along the c-axis of 1/4 of Ge1-sites in Fe3Ge layers, and in response to that, the 2 × 2 in-plane charge modulations are induced. Moreover, we show that the previously reported short-ranged CO might be related to large occupational disorders at Ge1-site, which upsets the equilibrium of the CO state and the ideal 1 × 1 × 1 structure with very close energies, inducing nanoscale coexistence of these two phases. Our study provides important clues for further understanding the CO properties in FeGe and helps to identify the CO mechanism.

vacancy or a substitutional defect at Ge2-site.One thing should be mentioned is that such C3-symmetric defect is rarely observed on the Fe3Ge layer in our STM study.
3) Type 7 of defect on Fe3Ge layer is C2-symmetric and has three orientations, which is more likely an Fe-site defect, such as an Fe vacancy or a substitutional defect.Such type of defect is also rarely observed in our STM study and was only found in a small Fe3Ge region of sample #2.4) Type 8 of defect is observed on the Ge2 layer, it is C6-symmetric with the defect center located at the Ge1-site, which might also be the Ge1-site defect seen on the Ge2 layer.5) Type 9 of defect on the Ge2 layer is C3-symmetric and has two orientations, which should be the Ge2-site defect.One thing should be mentioned is that the abovementioned defect types on Fe3Ge layer and Ge2 layer may overlap, such as types 1-5 and type 8, type 6 and type 9, since STM measurements may include information of both the topmost layer and the layer underneath.
To simplify, we just count the total densities of Ge1-site, Ge2-site and Fe-site defects in multiple sample regions of samples #1 and #2, the results are shown in Supplementary Fig. 5.The total density of Ge1-site defects is reduced by approximately a third to a half in sample #2, from ~ 1.7% of the atomic proportion of Ge1-site in sample #1 to ~ 0.8% in sample #2.Fe-site defects are rarely observed and will not be discussed in detail here.The density of Ge2-site defect is as low as less than ~ 0.01 % of the atomic proportion of Ge2-site and is almost unchanged in both samples, which should have little influence on CO.Supplementary Figure 4 | Different Types of defects observed in the Fe3Ge and Ge2 layers.a, Crystal structure of the 1 × 1 × 1 phase of FeGe.b-d, Sketches of crystallographic symmetries of defects located at different atomic sites.Defect locations are indicated by the black crosses, and their possible influence on LDOS is marked by the green bars.e-h, Typical topographic images for a selected Fe3Ge region measured under different bias voltages.By comparing these images, we counted and classified all the defects into five types and marked them out by the dashed circles with different colors.i-m, Detailed topographic images of five types of Ge1-site defects, all are C6symmetric.n,o, Detailed topographic images of the other two types of defects that are occasionally observed in Fe3Ge layer.The defects of types 6 and 7 are C3-and C2-symmetric, respectively, which might be the Ge2-site and Fe-site defects judging from their crystallographic symmetries.p,q, Detailed topographic images of the two types of defects observed on Ge2 layer.The defects of types 8 and 9 are C6-and C3-symmetric, respectively, which might be the Ge1-site and Ge2-site defects judging from their crystallographic symmetries.

Supplementary Figure 5 | Statistical defect density in several sample regions of samples #1 and #2.
Every red or blue hollow circle represents a sample region in samples #1 and #2, respectively.The density of Ge1-site defects (types 1-5) is reduced in sample #2, while the density of Ge2-site defect (type 9) is very low for both samples.The total density of all defects is reduced by approximately a third to a half in sample #2.

Supplementary Note 4. Detailed SCXRD results for samples #1 and #2
SCXRD measurements were carried out on both samples #1 and #2 at 300 K and 85 K, respectively.The diffraction patterns of these two samples are similar at 300 K, but vary obviously at 85 K, as displayed in Fig. 3, Supplementary Fig. 6 and Supplementary Fig. 7.At 85 K, new spots are observed for sample #2 and signal a structural modulation with a wave vector of (-0.5, 0.5, 0.5), while they are absent in sample #1.
By using Olex2 software with Shlex program 4,5 , the corresponding crystal structures are solved and refined.Considering that the change of physical properties of FeGe by annealing is reversible 6,7 and the defect density observed in STM study is less than 2% as discussed above, we first adopt the full FeGe stoichiometry for simplicity, and the refined results are shown in Supplementary Table 1-6 and Supplementary Fig. 8.For both samples at 300 K, high quality refinements are achieved with the space group P6/mmm.The refined atomic coordinates are listed in Supplementary Table 2 and Table 5, and the corresponding crystal structure is sketched in Fig. 3c and Supplementary Fig. 8a-d, which is consistent with the previous report 8 .The same space group and atomic coordinates can also describe the diffraction patterns of sample #1 at 85 K (Supplementary Table 3).However, for the diffraction patterns of sample #2 at 85 K, a higher quality refinement is achieved with the space group P-6m2.The P-6m2 unit cell presented in Supplementary Table 4 is eight times larger than the primitive unit cell at 300 K, with the lattice doubling along all three lattice directions.The related atomic coordinates are listed in Supplementary Table 6, and the corresponding crystal structure is sketched in Fig. 3d and Supplementary Fig. 8e-h.The largest distortion occurs at the Ge1c sites in the Kagome layers, where the Ge1c atoms in the adjacent two layers dimerize and deviate from the Kagome layer by ~ 0.7 Å.Other atoms in the Kagome layer also undergo small distortions (< 5 pm), and are out-of-phase between adjacent Kagome layers.The other Ge1 atoms mainly distort along the c-axis, the Fe atoms are distorted both within the plane and along the c-axis.The distortion of Ge2 layer is simpler, showing a deformed Kekulé-type distortion and is out-of-phase between adjacent Ge2 layers.
Furthermore, we consider the effect of defects on the refinement.As shown in Supplementary Figure 9a, when scrutinizing the SCXRD data collected at 300 K, we find that there is moderate residual electron density around Ge1b-site (Q peaks indicated by the black arrow), indicating obvious occupational disorders at Ge1-site.By adding such occupational disorders of Ge1-site, the refinements of the SCXRD data for samples #1 and #2 at 300 K are moderately improved, as listed in Supplementary Table 7.The related atomic coordinates are listed in Supplementary Table 8 and Table 9 for samples #1 and #2, respectively, and the corresponding crystal structure is sketched in Supplementary Fig. New spots appear at 85 K, signaling that the lattice doubles along all three lattice directions.

Supplementary Figure 7 |
Figure9a, when scrutinizing the SCXRD data collected at 300 K, we find that there is moderate residual electron density around Ge1b-site (Q peaks indicated by the black arrow), indicating obvious occupational disorders at Ge1-site.By adding such occupational disorders of Ge1-site, the refinements of the SCXRD data for samples #1 and #2 at 300 K are moderately improved, as listed in Supplementary Table7.The related atomic coordinates are listed in Supplementary Table8and Table9for samples #1 and #2, respectively, and the corresponding crystal structure is sketched in Supplementary Fig.9b.The obtained proportion of Ge1b-site to the total Ge1-site and their averaged distance from the Kagome plane are ~ 3% and ~ 0.973 Å for sample #1 and ~ 2% and ~ 0.865 Å for sample #2, respectively.Other types of defects, such as Ge2-site defects, don't show up in SCXRD data and the artificial introduction of a small amount of them has little effect on the refinements.