⇦Chinese Physics Letters, 2023, Vol. 40, No. 11, Article code 117103Express Letter Electron-Correlation-Induced Charge Density Wave in FeGe Lin Wu (武琳)†, Yating Hu (胡雅婷)†, Dongze Fan (樊东泽), Di Wang (王棣)*, and Xiangang Wan (万贤纲)* Affiliations 1National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China 2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Received 29 September 2023; accepted manuscript online 18 October 2023; published online 23 October 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: diwang0214@nju.edu.cn; xgwan@nju.edu.cn Citation Text: Wu L, Hu Y T, Fan D Z et al. 2023 Chin. Phys. Lett. 40 117103    Abstract As the first magnetic kagome material to exhibit the charge density wave (CDW) order, FeGe has attracted much attention in recent research. Similar to $A$V$_{3}$Sb$_{5}$ ($A$ = K, Cs, Rb), FeGe exhibits the CDW pattern with an in-plane 2$\times$2 structure and the existence of van Hove singularities near the Fermi level. However, sharply different from $A$V$_{3}$Sb$_{5}$ which has phonon instability at $M$ point, all the theoretically calculated phonon frequencies in FeGe remain positive. Based on first-principles calculations, we surprisingly find that the maximum of nesting function is at $K$ point instead of $M$ point. Two Fermi pockets with Fe-$d_{xz}$ and Fe-$d_{x^{2}-y^{2}}$/$d_{xy}$ orbital characters have large contribution to the Fermi nesting, which evolve significantly with $k_{z}$, indicating the highly three-dimensional (3D) feature of FeGe in contrast to $A$V$_{3}$Sb$_{5}$. Considering the effect of local Coulomb interaction, we reveal that the instability at $K$ point is significantly suppressed due to the sublattice interference mechanism. Meanwhile, the wave functions nested by vector $M$ have many ingredients located at the same Fe site, thus the instability at $M$ point is enhanced. This indicates that the electron correlation, rather than electron-phonon interaction, plays a key role in the CDW transition at $M$ point.
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 DOI:10.1088/0256-307X/40/11/117103 © 2023 Chinese Physics Society Article Text Kagome lattice materials[1-4] exhibit unique electronic structure signatures owing to their unconventional geometric characteristics, embracing the flat bands induced by destructive interference of nearest-neighbor hopping, a pair of van Hove singularities (vHSs) at $M$ point, and Dirac cone dispersion at $K$ point.[5-12] A wide array of exotic physical phenomena in the kagome lattice arises from different degrees of electron filling. When large density of states (DOS) from the kagome flat bands is located near the Fermi level, strong electron correlations can induce magnetic order.[5-7] Meanwhile, when vHSs are located near the Fermi level, interaction between the saddle points and lattice instability could induce charge density wave (CDW) order with a 2$\times$2 structure in $xy$ plane.[8-12] Therefore, kagome lattice materials serve an essential platform for studying non-trival topological physics,[13-17] CDW order,[18-38] superconductivity,[39-47] fractional quantum Hall effect,[48-50] and quantum anomalous Hall effect (QAHE).[51-56] As a well-known family of non-magnetic layered kagome compounds, $A$V$_{3}$Sb$_{5}$ ($A$ = K, Cs, Rb)[18] were reported to host CDW,[19-34,57] superconductivity,[39-47] giant QAHE,[55,56] and chiral flux phase.[35-37] In this system, vHSs are located near the Fermi level and the phonon spectrum exhibits two imaginary phonon frequencies locating at the Brillouin zone (BZ) boundary [$M$ ($\frac{1}{2},0,0$) and $L$ ($\frac{1}{2},0,\frac{1}{2}$) points],[29,30] which induce an in-plane 2$\times$2 CDW state identified in experiment.[21-26] On the other hand, there are many kagome magnetic materials such as FeSn,[58-62] Fe$_{3}$Sn$_{2}$,[63,64] Mn$_{3}$Sn,[14] Mn$_{3}$Ge,[50,54] and Co$_{3}$Sn$_{2}$S$_{2}$.[15-17] However, it may be due to the large energy separation between flat bands and vHSs that CDW order and magnetic order have not been generally observed simultaneously in one material.[65,66] Very recently, the discovery of CDW order coexists with magnetic order in the Kagome material FeGe,[65,67-73] reported from a joint experimental study of angle-resolved photoemission spectroscopy (ARPES),[65,67] neutron and x-ray scattering,[67-69] scanning tunneling microscopy[67,68,70,71] and photoemission,[68] which offers a fascinating platform to investigate interplay between the CDW and magnetism. The correlation-induced renormalization and the topological features in FeGe have also been explored.[72] FeGe exhibits a sequence of phase transitions: (1) An A-type antiferromagnetism (AFM) phase appears below the high magnetic transition temperature up to $T_{\rm c}\sim 410$ K.[74] (2) A CDW phase[67-70] takes place at $T_{\rm CDW}\sim 100$ K and the system becomes stable in 2 $\times $2$\times $2 superstructure and induces an emergent anomalous Hall effect (AHE) possibly associated with a chiral flux phase as previously reported in metal $A$V$_{3}$Sb$_{5}$.[21-26,35,36,55] (3) The magnetic moment weakly cants along the $c$ direction, giving a $c$-axis double cone AFM structure below a lower transition temperature $T_{\rm canting}$ around 60 K.[75-80] Subsequent reports reveal that the vHSs in FeGe, which are originally far from Fermi surface in a nonmagnetic system, are brought to the vicinity of Fermi level due to spin exchange splitting of about 1.8 eV caused by magnetic order.[65-67] Though FeGe exhibits the existence of vHSs at the $M$ point near the Fermi level similar to $A$V$_{3}$Sb$_{5}$, it is noteworthy that the imaginary phonon around the $M$ point present in $A$V$_{3}$Sb$_{5}$[29,32] is not observed in the theoretically calculated phonon spectrum of FeGe.[65,69,70] Therefore, a comprehensive study of the band structures, Fermi surfaces, nesting function, and the mechanism of CDW transition in kagome FeGe is an emergency issue. We address it in this work based on first principles study. In the present work, we perform a detailed analysis of the electronic structure and an investigation on the nesting function of A-type AFM kagome materials of FeGe by employing first-principles calculations. We analyze the variation of the energy band structure and the Fermi surface for different $k_{x}$–$k_{y}$ planes as $k_{z}$ changes from 0 to 0.5, and find that the electronic structures of FeGe have strong 3D feature. The positions of vHSs evidently shift as $k_{z}$ varies, and only in a small $k_{z}$ range the vHSs are in proximity to Fermi level ($\pm 0.1$ eV), distinct from the quasi-two-dimensional structural characteristics of $A$V$_{3}$Sb$_{5}$.[30,32] Our numerical results show that the maximum of nesting function is at the $K$ point instead of the $M$ point. Whereafter, we find that two pockets have large contribution to the nesting function, which are respectively contributed from the $d_{xz}$ orbital and a combination of $d_{x^{2}-y^{2}}/d_{xy}$ orbitals. To understand the conflict between the nesting function at the $K$ point and CDW transition at $M$ point, we consider the effect of local Coulomb interaction.[81] We find that the Fermi surface nesting at the $K$ point is ineffective due to different sublattice characters of the band structures, similar with the sublattice mechanism in superconductors.[11,82] On the other hand, the CDW instability at the $M$ point could be enhanced by the local Coulomb interaction, since the wave functions nested by vector $M$ are mainly distributed from the same Fe site. It implies that the strong 3D Fermi surfaces and local electron correlation play indispensable parts in the CDW transition in FeGe. We carried out first-principles density functional theory (DFT) calculations by employing the full-potential all-electron code Wien2k.[83] The local spin density approximation (LSDA)[84] was used as the exchange-correlation functional in our calculations for the A-type AFM state. We chose a fine $k$-mesh of 200 $\times $200$\times $100 in the irreducible BZ to ensure that the nesting function calculated from the eigenvalues is robust to the number of $K$ points. Since our conclusions are quite different from the typical kagome materials such as $A$V$_{3}$Sb$_{5}$,[29,30,32] we have also used the Vienna ab initio Simulation Package (VASP)[85,86] with the projector augmented wave[87,88] method and Quantum-ESPRESSO package (QE)[89,90] with the norm-conserving pseudopotentials[91] to confirm our electronic structure calculations. The results of the above methods are well consistent, and in this study we present the calculated results of electronic structure from Wien2k. Meanwhile, the electron-phonon coupling (EPC) calculations are performed using QE, and the detailed computational parameters are presented in the Supplemental Materials (SMs). We start by performing LSDA calculation for FeGe based on the experimental crystal structure (see the SMs) and A-type AFM ground state.[80] The band structure and DOS (see the SMs) are basically similar in many aspects to the isostructure FeSn.[58-62] Our calculations reproduce the results of previous studies[65] quite well. The magnetic moment of the Fe ions is estimated to be 1.55$\mu _{\scriptscriptstyle{\rm B}}$, which is close to the previous experimental value of about 1.72$\mu _{\scriptscriptstyle{\rm B}}$.[76,78] In addition, similar to $A$V$_{3}$Sb$_{5}$,[30,32] we find a pair of vHSs close to Fermi surface on both the $k_{z}=0$ and 0.5 high symmetric planes as shown in Fig. S2 of the SMs. In order to analyze the orbital components of the energy bands, it is important to take the proper local coordinate axis. Since the angles between the orientations of Fe$_{\rm A}$, Fe$_{\scriptscriptstyle{\rm B}}$, and Fe$_{\rm C}$ (see the SMs) to the same nearest neighbor Ge$^{\rm out}$ atom are 120$^{\circ}$ in the global coordinate, the $x/y$ components of Fe$_{\rm A/B/C}-d$ orbitals are not equivalent. The suitable local coordinate of Fe$_{\rm A}$, Fe$_{\scriptscriptstyle{\rm B}}$, and Fe$_{\rm C}$ we have chosen are shown in Fig. S1(c) of the SMs, any two of which could be transformed into each other by the $C_{3}$ symmetry operation. The local $x$-axis always points to the nearest-neighbor Ge$^{\rm in}$ atom and the local $z$-axis direction is the same as the one in the global coordinate. In this local coordinate, the five Fe-3$d$ orbitals are clearly divided into higher $d_{yz}$ and $d_{x^{2}-y^{2}}$ parts and lower $d_{z^{2}}$, $d_{xz}$, and $d_{xy}$ parts, consistent with the $e_{\rm g}$–$t_{\rm 2g}$ relationship in the ortho-octahedral crystal field. The selection of such a local coordinate is also helpful for analysis of the Fermi pockets as shown in the following.
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Fig. 1. The orbital-projected electronic band structure of FeGe near Fermi level. The Fe-3$d$ orbital projection is performed with the local coordinate. The orbital characters are labeled by different colors.
We present the orbital components of the energy bands in Fig. 1 using the above local coordinate. As mentioned above, the kagome lattice with nearest-neighbor hopping will present the typical three-band structure including flat band, vHS, and Dirac cone.[5,6,8-11] Along the high symmetry directions $\varGamma$–$M$–$K$–$\varGamma$ lying in $k_{z}=0$ plane, there are two different kagome structures near the Fermi level. One consists of Fe-$d_{xz}$ orbitals, which are located between $-2.0$ and 0.5 eV, and contains the vHS1 located at $-0.26$ eV below $E_{\rm F}$, as marked in red in Fig. 1. The other kagome structure containing vHS2 consists of a combination of $d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals shown in purple and blue, respectively. The vHS2 is located above the Fermi energy (0.07 eV). Three Fe-$d_{yz}$ orbitals (labeled in orange in Fig. 1) also form a kagome three-band structure. However, the interaction between the Fe atoms and the $p_{z}$ orbitals of the Ge$^{\rm out}$ leads to two bands shifting downward away from $E_{\rm F}$. Along the $A$–$L$–$H$–$A$ path in the $k_{z}=0.5$ plane there are three kagome structures. Due to the interaction with the $p$ orbitals of Ge$^{\rm out}$, the three-band kagome structure composed of Fe-$d_{z^{2}}$ orbitals is located below and away from the Fermi energy level in the $k_{z}=0$ plane (from $-2.5$ to $-1.0$ eV), while in the $k_{z}=0.5$ plane the energy bands rise and cross the Fermi level. Two vHSs at the $L$ point close to the Fermi energy level are contributed by Fe-$d_{x^{2}-y^{2}}/d_{xy}$ and Fe-$d_{z^{2}}$ orbitals, marked as vHS2 and vHS3, respectively, in Fig. 1 (the analysis below confirms that the vHS2 in $k_{z}=0$ plane slowly turns into the vHS2 in $k_{z}=0.5$ plane with $k_{z}$ shifts from 0 to 0.5). It is worth mentioning that vHSs near $E_{\rm F}$ have different orbital characters in $k_{z} =0$ and $k_{z}=0.5$ planes, indicating the important role of $k_{z}$ in electronic structures. Furthermore, using group theory,[92,93] we obtain the irreducible representations (irreps) of the little group for each vHS based on the wave functions from DFT calculations. At the $M$ point, our calculations identify the irrep of vHS1 as B$_{\rm 3g}$, while vHS2 corresponds to the irrep A$_{\rm g}$. On the other hand, at $L$ point, the irreps of the vHS2 and VHS3 are both A$_{u}$. The irreps of the vHSs closer to $E_{\rm F}$ are different in $k_{z}=0$ and $k_{z}=0.5$ planes, also indicating the strong 3D feature in FeGe, which will be carefully discussed in the following.
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Fig. 2. The nesting function $\xi (q_{x},q_{y},0)$ of FeGe. We neglect the peak at $q=0$ to better present the nesting function.
To understand the CDW instability, we calculate the Fermi surface nesting function $\xi (\boldsymbol{q})$.[94] The 2 $\times $2$\times $2 supercell structure of CDW phase corresponding to the high-temperature non-magnetic pristine phase is suggested by experimental results.[67-70] As mentioned above, the CDW transition in FeGe takes place at $T_{\rm CDW}$ around 100 K, below the magnetic transition temperature 410 K. Since the A-type AFM structure has already enlarged unit cells doubled along the $z$-direction, we focus the in-plane $\boldsymbol{q}$ vector in the following, and obtain the nesting function $\xi (q_{x},q_{y},0)$, as illustrated in Fig. 2. It can be seen that the maximum values of nesting function are located at the $K$ point instead of the $M$ point, which is different from the results of the tight-binding model of kagome lattice[8-11] and $A$V$_{3}$Sb$_{5}$.[30,32] For FeGe, an overall renormalization factor of 1.6 is relatively smaller than the Fe-based superconductors,[95-100] and it provides a reasonable match between the band dispersion of the DFT calculation and the experimental measurements.[65] In addition, in order to consider the effect of the renormalization of the band dispersion on the Fermi surface nesting, we calculate the nesting function $\xi (q_{x},q_{y},0)$ after Fermi energy shift from $E_{\rm F}-0.1$ eV to $E_{\rm F}+0.1$ eV, and the conclusions mentioned above remain unchanged (see the SMs). The difference in nesting function between FeGe and $A$V$_{3}$Sb$_{5}$ motivated a careful analysis of the band structure, Fermi surface, and the nesting function of FeGe. As shown in Fig. 3(a), for $k_{z}=0$ plane, the Fermi surface pocket formed by the $d_{xz}$-dominant energy band (labeled as $\beta $) presents a hexagon parallel to the BZ (hereafter called the h-hexagon), and the maximum of nesting function is located at $K$ point [bottom panel of Fig. 3(a)]. Differently, the Fermi surface structure in $A$V$_{3}$Sb$_{5}$[30,32] is v-hexagon, i.e., a hexagon with a difference of 60$^{\circ}$ rotation from the BZ edge direction, where the peak of nesting function is located at $M$ point. It is worth mentioning that in Fig. 2, besides the maximum of total nesting function located at the $K$ point, there are also peaks along the $\varGamma$–$M$ direction. Therefore, we carefully analyze the $k_{z}$ momentum-dependent evolution of band structure, Fermi surface and the nesting function, and show the results in Figs. 3(a)–3(f). As $k_{z}$ increases from 0 to 0.2, vHS1 gradually approaches the Fermi surface, and the Fermi surface pocket $\beta $ gradually changes from h-hexagon to v-hexagon as shown in Figs. 3(a)–3(c). The $d_{x^{2}-y^{2}}/d_{xy}$ bands form the pocket $\alpha $ and contain the vHS2 downshift as $k_{z}$ increases, and the shape of the pocket $\alpha $ changes from a v-hexagonal shape [see Fig. 3(c)] to a circle [see Fig. 3(d)], and then gradually to an h-hexagonal shape [see Figs. 3(e) and 3(f)] as $k_{z}$ increases. For $k_{z}=0.5$ plane, the Fermi surface evolved to contain h-hexagonal-shaped pockets, which has similar features with $k_{z} =0$ plane to some extent, as suggested in Ref. [70]. As can be seen from Fig. 3, although the shape of the Fermi surface changes with $k_{z}$, pockets $\alpha $ and $\beta $ both retain their hexagonal-like shape for a wide range of $k_{z}$ values, suggesting a large contribution to the Fermi surface nesting. For the $k_{z}= 0$ and 0.1 planes, the maxima of nesting function are located at or near the $K$ point, as shown in Fig. 3(a) and 3(b); while in the $k_{z}=0.2$ plane, the nesting function shows an enhancement along the $\varGamma$–$M$ direction in the momentum space, as shown in Fig. 3(c), which originates from the nesting of v-hexagonal pockets $\alpha $ and $\beta $ in Fig. 3(c). As $k_{z}$ continues to shift to around 0.4, the shape of the pockets $\alpha $ and $\beta $ changes again from a v-hexagon to an h-hexagon, with the maximum of nesting function gradually moving away from the $\varGamma$–$M$ direction to the $\varGamma$–$K$ direction. For $k_{z}=0.5$ plane, the h-hexagonal pockets around $\varGamma$ with similar area in the BZ plane lead to the maxima of the nesting function near the $\varGamma$ point along the $\varGamma$–$K$ direction, as shown in Fig. 3(f). The difference between the Fermi surface and nesting functions of $A$V$_{3}$Sb$_{5}$ and that of FeGe originates from their different crystal structures. In $A$V$_{3}$Sb$_{5}$ ($A$ = K, Cs, Rb), due to the presence of the $A$ layers, the distance between the interlayer V atoms is at least 9.308 Å. In FeGe, on the other hand, the minimal distance between nearest-neighboring interlayer Fe atoms is 4.041 Å, since in FeGe only Fe–Ge$^{\rm in}$ and Ge$^{\rm out}$ layers are alternately arranged. Thus, the hopping parameters of Fe-3$d$ orbitals in the $z$-direction are significantly larger than those of V-$d$ orbitals, and the bands near the Fermi surface in FeGe, which are mainly contributed by Fe-3$d$ orbitals, have strong 3D features.
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Fig. 3. (a)–(f) The orbital-projected electronic band structures (top panel) and 2D Fermi surface (middle panel), and the nesting function (bottom panel) of FeGe along the $DT(0,0,k_{z})$–$U(\frac{1}{2},0,k_{z})$–$P(\frac{1}{3},\frac{1}{3},k_{z})$–$DT(0,0,k_{z})$ path in the $k_{z}=0$, 0.1, 0.2, 0.3, 0.4, and 0.5 planes. For the orbital-projected band structures, the circle size shows the relative portion of each orbital. For the 2D Fermi surface, the pockets $\alpha$, $\beta$, $\gamma$, and $\delta$ represent the pockets dominated by a combination of Fe-$d_{xy}$ and $d_{x^{2}-y^{2}}$, Fe-$d_{xz}$, Fe-$d_{yz}$, and Fe-$d_{z^{2}}$ orbital characters, respectively.
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Fig. 4. The EPC distribution of FeGe in $q_{z} = 0$ plane.
In tight-binding models of kagome lattice[8-11] and $A$V$_{3}$Sb$_{5}$,[29,30,32] the vHSs near the Fermi level could induce the peak of nesting at $M$ point. Meanwhile, in $A$V$_{3}$Sb$_{5}$, the phonon instability[29,32] and the CDW transition[21-26] at $M$ point driven by Fermi surface nesting are suggested. In FeGe, it is also suggested that vHSs play an important role in the CDW phenomenon.[65] However, our calculations show that the Fermi surface in FeGe has strong 3D feature, and the vHSs are not always near the Fermi energy level. For example, vHS1 is close to the Fermi level ($\pm 0.1$ eV) only in the small range of $k_{z}=0.183$–0.267, which is 16.8% of the BZ. Meanwhile, the fraction of $k_{z}$ when vHS2 and vHS3 are close to the Fermi level are 53.4% and 13.4%, respectively. Our numerical results show that the ratios of nesting function contributed by vHS1, vHS2, and vHS3 around the Fermi level to the total nesting function are 11.02$\%$, 20.44$\%$, and 7.53$\%$, respectively. This means that unlike $A$V$_{3}$Sb$_{5}$,[29,30,32] vHSs near the Fermi surface are not the most important factors of the nesting function in FeGe. In FeGe, the maximum of nesting function is at the $K$ point, which does not correspond to the observed CDW wave vector at $M$ point. Meanwhile, the theoretically calculated phonon spectrum remains positive.[65,69,70] The nesting function originates from the imaginary part of the bare electronic susceptibility, which cannot reflect the susceptibility enhancement by the interaction, indicating that the interaction plays an important role in CDW transition.[81,101] Generally, the electron-phonon interaction is weakly momentum-dependent except for the singular behavior in non-traditional superconductivity.[102] In order to verify the $k$-dependence of FeGe's EPC, we calculate the distribution of the EPC constant ${\lambda}_{q} $ in $q_{z}=0$ plane of the BZ, as depicted in Fig. 4. The small EPC areas are located around $\varGamma $ point, while the difference between the maximal and minimal values of EPC is not significant. As shown in Fig. 4, the EPC values at $M$ point (${\lambda}_{M} = 0.68$) and $K$ point (${\lambda}_{K} = 0.70$) are similar, and the distribution is homogeneous around the boundary of BZ, which is different from CsV$_{3}$Sb$_{5}$.[33] It shows that the conflict between the nesting function at $K$ point and the CDW transition at $M$ point does not originate from electron-phonon interaction. Therefore, we next analyze the local electron–electron correlation interaction, which is of substantial importance in 3$d$ electron systems.[103]
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Fig. 5. The 2D Fermi surface in $k_{x}$–$k_{y}$ plane with (a) $k_{z}=0.267$ and (b) $k_{z}=0.433$. The sublattice characters of the Fermi level states including three different Fe$_{\rm A}$, Fe$_{\rm B}$, and Fe$_{\rm C}$ sites are marked in red, green, and blue, respectively. The red arrows indicate the nesting vectors $q_{K}=(1/3,1/3,0)$ and $q_{M}=(1/2,0,0)$.
Based on DFT calculations, we obtain the wave functions $\psi _{n,\boldsymbol{k}+\boldsymbol{q}}(\boldsymbol{r})$ and $\psi _{n,\boldsymbol{k}}(\boldsymbol{r})$, which are electronic Bloch states at the Fermi level connected by the vector $\boldsymbol{q}$. We expand the distribution of $\psi _{n,\boldsymbol{k}+\boldsymbol{q}}(\boldsymbol{r})$ and $\psi _{n,\boldsymbol{k}}(\boldsymbol{r})$ to the basis set of atomic orbitals in real space. We find that when $\boldsymbol{q}= \boldsymbol{q}_{K}(1/3,1/3,0)$, the wave functions $\psi _{n,\boldsymbol{k}+\boldsymbol{q}}(\boldsymbol{r})$ and $\psi _{n,\boldsymbol{k}}(\boldsymbol{r})$ exhibit unequal predominant sublattice occupancy. Meanwhile, when $\boldsymbol{q}= \boldsymbol{q}_{M}(1/2,0,0)$, the wave functions $\psi _{n,\boldsymbol{k}+\boldsymbol{q}}(\boldsymbol{r})$ and $\psi_{n,\boldsymbol{k}}(\boldsymbol{r})$ are mainly distributed from the same Fe site. We take a representative $k_{z}=0.267$ plane as an example to show the above-mentioned results for the nesting vector $\boldsymbol{q}_{K}$ in Fig. 5(a), with the characters of three sublattice Fe$_{\rm A}$, Fe$_{\scriptscriptstyle{\rm B}}$, and Fe$_{\rm C}$ indicated in red, green, and blue, respectively. The Fermi surface contours of $\beta $ pockets coincide when shifted along the nesting vector $\boldsymbol{q}_{K}(1/3,1/3,0)$, as shown by red arrows in Fig. 5(a), resulting in the peak of nesting function at $K$ point. However, the nesting vector $\boldsymbol{q}_{K}$ connects Fermi surface points with mainly different sublattice occupancy, as shown in Fig. 5(a). It is worth mentioning that, due to the locality of Coulomb correlation, the electron-electron correlation interaction is diagonal in the index of atomic sites. It means that the susceptibility is suppressed regardless of the peak of nesting function at $K$ point.[81] Meanwhile, we demonstrate the results of the nesting vector $\boldsymbol{q}_{M}$ by $k_{z}=0.433$ plane in Fig. 5(b). There are nested Fermi surfaces along $\boldsymbol{q}_{M}(1/2,0,0)$ connecting the opposite edges of Fermi pocket $\alpha $, as shown in red arrow of Fig. 5(b). It can be seen that the wave functions $\psi _{n,\boldsymbol{k}+\boldsymbol{q}}(\boldsymbol{r})$ and $\psi_{n,\boldsymbol{k}}(\boldsymbol{r})$ connected by the vector $\boldsymbol{q}_{M}$ are dominated by the same sublattice occupancy as mentioned above, leading to enhanced susceptibility at the $M$ point. Therefore, similar to the sublattice mechanism in superconductors,[11,82] the CDW instability at $M$ point is considered to be derived by the local electron correlation. Since the electronic instability can significantly affect phonons,[81,101] it may explain the experimentally observed phonon anomalies.[65,69] It is worth mentioning that $A$V$_{3}$Sb$_{5}$ was classified as a Z$_{2}$ topological metal.[40,41] Additionally, a giant AHE has also been observed in $A$V$_{3}$Sb$_{5}$,[55,56] and the onset of the AHE was found to coincide with the CDW order.[56] The concept of a chiral flux phase was introduced to explain phenomena related to the time-reversal symmetry-breaking CDW.[35,36] Similarly, in the case of FeGe,[67-70] the CDW enhances the AFM ordered moment and gives rise to an emergent AHE, possibly associated with a chiral flux phase as mentioned above. The Z$_{2}$ topological features in FeGe have also been explored in Ref. [72]. It is worth noting that the reasonable Heisenberg interactions and magnetic anisotropy cannot explain the low-temperature double cone magnetic structure observed in FeGe, and the net contribution of Dzyaloshinskii–Moriya interactions to the low-temperature magnetic structure prohibits the inversion and mirror symmetry.[65] Based on the experimental 2$\times$2$\times$2 supercell and group theoretical analysis, four potential CDW phases are proposed.[65] Among them, the two CDW structures with the space group $P6_{3}22$[65] hold the potential to support the chiral flux phase.[35,36] Nevertheless, the intricate interplay between CDW and topology in FeGe warrants further investigation. In summary, based on DFT calculations, we have comprehensively investigated the Fermi surface nesting and the microscopic origin of the CDW order in the kagome magnetic metal FeGe. Our results indicate that the energy bands and Fermi surfaces of FeGe vary significantly with $k_{z}$, and the maximum of nesting function is at the $K$ point instead of the CDW vector at $M$ point. We find that the susceptibility at the $K$ point is significantly suppressed due to the sublattice interference mechanism.[11,82] On the other hand, the CDW instability at the $M$ point is enhanced, which indicates that electron correlation plays an indispensable part in the CDW transition. Acknowledgement. We are grateful to Dawei Shen and Yan Zhang for helpful discussion about ARPES. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12188101, 11834006, 12004170, and 12334007), the National Key R&D Program of China (Grant No. 2022YFA1403601), the Natural Science Foundation of Jiangsu Province (Grant No. BK20200326), and the excellent programme in Nanjing University. This work was also supported by the New Cornerstone Science Foundation through the XPLORER PRIZE. 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