# Novel three-dimensional Fermi surface and electron-correlation-induced
charge density wave in FeGe

Lin Wu  Yating Hu  Di Wang<EMAIL_ADDRESS>Xiangang Wan
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and School
of Physics, Nanjing University, Nanjing 210093, China  Collaborative
Innovation Center of Advanced Microstructures, Nanjing University, Nanjing
210093, China

###### Abstract

As the first magnetic kagome material to exhibit the charge density wave (CDW)
order, FeGe has attracted much attention in recent studies. Similar to AV3Sb5
(A = K, Cs, Rb), FeGe exhibits the CDW pattern with an in-plane 2$\times$2
structure and the existence of van Hove singularities (vHSs) near the Fermi
level. However, sharply different from AV3Sb5 which has phonon instability at
$M$ point, all the theoretically calculated phonon frequencies in FeGe remain
positive. Here, we perform a comprehensive study of the band structures, Fermi
surfaces, nesting function and the mechanism of CDW transition through first-
principles calculations. Surprisingly, we 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
AV3Sb5. Meanwhile, the vHSs are close to the Fermi surface only in a small
$k_{z}$ range, and does not play a leading role in nesting function.
Considering the effect of local Coulomb interaction, we reveal that the Fermi
level eigenstates nested by vector $K$ are mainly distributed from unequal
sublattice occupancy, thus the instability at $K$ point is significantly
suppressed. 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.

Kagome lattice materials Sy$\widehat{o}$zi (1951); Yin  _et al._ (2022a); Chen
_et al._ (2022a); Nguyen and Li (2022) exhibit unique electronic structure
signatures owing to their unconventional geometric characteristics, embracing
the flat bands induced by destructive interference of nearest-neighbour
hopping, a pair of van Hove singularities (vHs) at $M$ point, and Dirac cone
dispersion at $K$ point Mielke (1991); Tanaka and Ueda (2003); Pollmann  _et
al._ (2008); Kiesel  _et al._ (2013); Park  _et al._ (2021); Yu and Li (2012);
Kiesel and Thomale (2012); Guo and Franz (2009). 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 are located near the Fermi level, strong electron correlations can
induce magnetic order Mielke (1991); Tanaka and Ueda (2003); Pollmann  _et
al._ (2008). 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$2 structure in xy-plane Kiesel
_et al._ (2013); Park  _et al._ (2021); Yu and Li (2012); Kiesel and Thomale
(2012); Guo and Franz (2009). Therefore kagome lattice materials serve an
essential platform for studying non-trival topological physics Yin  _et al._
(2020); Kuroda  _et al._ (2017); Morali  _et al._ (2019); Liu  _et al._
(2019); Yin  _et al._ (2019), CDW order Ortiz  _et al._ (2019); Subedi
(2022a); Miao  _et al._ (2021); Subedi (2022b); Jiang  _et al._ (2021); Lin
and Nandkishore (2021); Li  _et al._ (2021); Mielke  _et al._ (2022); Denner
_et al._ (2021); Nie  _et al._ (2022); Yu  _et al._ (2021a); Xiang  _et al._
(2021); Tan  _et al._ (2021); Wu  _et al._ (2022); Kang  _et al._ (2022); Si
_et al._ (2022); Luo  _et al._ (2022); Zheng  _et al._ (2022); Feng  _et al._
(2021a); Jiang  _et al._ (2022); Feng  _et al._ (2021b); Wang  _et al._
(2013), superconductivity Chen  _et al._ (2021); Ortiz  _et al._ (2020, 2021);
Zhao  _et al._ (2021); Jeong  _et al._ (2022), fractional quantum Hall effect
Tang  _et al._ (2011); R$\ddot{u}$egg and Fiete (2011); Brien  _et al._ (2010)
and quantum anomalous Hall effect (QAHE) Ohgushi  _et al._ (2000); Liu  _et
al._ (2018); Nayak  _et al._ (2016); Kiyohara  _et al._ (2016); Yang  _et al._
(2020); Yu  _et al._ (2021b).

As a well-known family of non-magnetic layered kagome compounds, AV3Sb5 (A =
K, Cs, Rb) Ortiz  _et al._ (2019) were reported to host CDW Subedi (2022a);
Miao  _et al._ (2021); Subedi (2022b); Jiang  _et al._ (2021); Lin and
Nandkishore (2021); Li  _et al._ (2021); Mielke  _et al._ (2022); Denner  _et
al._ (2021); Nie  _et al._ (2022); Yu  _et al._ (2021a); Xiang  _et al._
(2021); Tan  _et al._ (2021); Wu  _et al._ (2022); Kang  _et al._ (2022); Si
_et al._ (2022); Luo  _et al._ (2022); Zheng  _et al._ (2022),
superconductivity Chen  _et al._ (2021); Ortiz  _et al._ (2020, 2021); Zhao
_et al._ (2021); Jeong  _et al._ (2022), giant QAHE Yang  _et al._ (2020); Yu
_et al._ (2021b) and chiral flux phase Feng  _et al._ (2021a); Jiang  _et al._
(2022); Feng  _et al._ (2021b). In this system, vHSs are located near the
Fermi level and the phonon spectrum exhibit 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) Tan  _et al._ (2021); Si  _et al._
(2022), which induce an in-plane 2$\times$2 CDW state identified in experiment
Jiang  _et al._ (2021); Lin and Nandkishore (2021); Li  _et al._ (2021);
Mielke  _et al._ (2022); Denner  _et al._ (2021); Nie  _et al._ (2022). On the
other hand, there are many kagome magnetic materials such as FeSn Kang  _et
al._ (2020); Xie  _et al._ (2021); Sales  _et al._ (2019); Do  _et al._
(2022); Zhang  _et al._ (2022), Fe3Sn2 Ye  _et al._ (2018); Lin  _et al._
(2018), Mn3Sn Kuroda  _et al._ (2017), Mn3Ge Nayak  _et al._ (2016); Kiyohara
_et al._ (2016) and Co3Sn2S2 Morali  _et al._ (2019); Liu  _et al._ (2019);
Yin  _et al._ (2019). 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 Teng  _et al._ (2022a); Zhou
_et al._ (2022).

Very recently, the discovery of CDW order coexists with magnetic order in the
Kagome material FeGe was reported from a joint experimental study of angle-
resolved photoemission spectroscopy (ARPES) Teng  _et al._ (2022a, b), neutron
and x-ray scattering Teng  _et al._ (2022b); Yin  _et al._ (2022b); Miao  _et
al._ (2022), scanning tunneling microscopy (STM) Teng  _et al._ (2022b); Yin
_et al._ (2022b); Shao  _et al._ (2022); Chen  _et al._ (2022b) and
photoemssion Yin  _et al._ (2022b), which offers a fascinating platform to
investigate interplay between the CDW and magnetism. FeGe exhibits a sequence
of phase transitions: (1) an A-type antiferromagnetism (AFM) phase appears
below the high magnetic transition temperature up to $T_{c}\sim$ 410K Ohoyama
_et al._ (1963), (2) a CDW phase Teng  _et al._ (2022b); Yin  _et al._
(2022b); Miao  _et al._ (2022); Shao  _et al._ (2022) takes place at
$T_{CDW}\sim$ 100K and the system becomes stable in 2$\times$2$\times$2
superstructure as previously reported in metal AV3Sb5 Jiang  _et al._ (2021);
Lin and Nandkishore (2021); Li  _et al._ (2021); Mielke  _et al._ (2022);
Denner  _et al._ (2021); Nie  _et al._ (2022), and (3) the magnetic moment
weakly cants along the c direction, giving a c-axis double cone AFM structure
below a lower transition temperature $T_{canting}$ around 60K Beckman  _et
al._ (1972); H$\ddot{a}$ggstr$\ddot{a}$m  _et al._ (1975); G$\ddot{a}$fvert
_et al._ (1977); Forsyth  _et al._ (1978); Bernhard  _et al._ (1984, 1988).
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 1.8eV caused by magnetic order
Teng  _et al._ (2022a, b); Zhou  _et al._ (2022). Though FeGe exhibits the
existence of vHSs at the $M$ point near the Fermi level similar to AV3Sb5, it
is note-worthy that the imaginary phonon around the $M$ point present in
AV3Sb5 Tan  _et al._ (2021); Si  _et al._ (2022) is not observed in the
theoretically calculated phonon spectrum of FeGe Teng  _et al._ (2022a); Shao
_et al._ (2022); Miao  _et al._ (2022). 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, which we will address 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 FeGe by employing first-principles calculations. We analyse 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 AV3Sb5 Si  _et al._ (2022); Wu  _et
al._ (2022). 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 respectively
contribute form 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 Doniach and Sondheime (1998). 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 Kiesel and Thomale (2012); Wu  _et
al._ (2021). 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 Blaha  _et al._
(2020). The local spin density approximation (LSDA) Vosko  _et al._ (1980) 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 AV3Sb5 Tan  _et al._
(2021); Si  _et al._ (2022); Wu  _et al._ (2022), we have also used the Vienna
ab initio Simulation Package (VASP) Kresse and Hafner (1993); Kresse and
Furthm$\ddot{u}$ller (1996) with the projector augmented wave (PAW)
Bl$\ddot{u}$chl (1994); Kresse and Joubert (1999) method to confirm our
electronic structure calculations. The results of the two methods are well
consistent, and in this paper we present the calculated results from Wien2k.

We start by performing LSDA calculation for FeGe based on the experimental
crystal structure (see Supplemental Materials (SM)) and A-type AFM ground
state Bernhard  _et al._ (1988). The band structure and the density of states
(DOS) (see SM) are basically similar in many aspects to the isostructure FeSn
Kang  _et al._ (2020); Xie  _et al._ (2021); Sales  _et al._ (2019); Do  _et
al._ (2022); Zhang  _et al._ (2022). Our calculations reproduce the results of
previous studies Teng  _et al._ (2022a) quite well. The magnetic moment of the
Fe ions is estimated to be 1.55 $\mu_{B}$, which is close to the previous
experimental value of about 1.72 $\mu_{B}$H$\ddot{a}$ggstr$\ddot{a}$m  _et
al._ (1975); Forsyth  _et al._ (1978). Besides, similar to AV3Sb5 Si  _et al._
(2022); Wu  _et al._ (2022), we find a pair of vHSs close to Fermi surface on
both the $k_{z}$=0 and 0.5 high symmetric plane as shown in Fig. 6 of SM.

In order to analyse the orbital components of the energy bands, it is
important to take the proper local coordinate axis. Since the angle between
the orientations of FeA, FeB, and FeC (see SM) to the same nearest neighbour
Geout atom is 120∘ in the global coordinate, the x/y components of FeA/B/C-$d$
orbitals are not equivalent. The suitable local coordinate of FeA, FeB, and
FeC we have chosen are shown in Fig. 5(c) of SM, 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 Gein 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 $e_{g}$-$t_{2g}$ relationship in the ortho-
octahedral crystal field. The selection of such local coordinate is also
helpful to the analysis of the Fermi pockets as shown in the following.

Figure 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 labelled 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-
neighbour hopping will present the typical three-band structure including flat
band, vHS, and Dirac cone Mielke (1991); Tanaka and Ueda (2003); Kiesel  _et
al._ (2013); Park  _et al._ (2021); Yu and Li (2012); Kiesel and Thomale
(2012). 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 of the Kagome structures consists of Fe-$d_{xz}$ orbitals, which
are located between -2.0 and 0.5 eV, and contain the vHS1 located at -0.26 eV
below $E_{F}$, 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 Geout leads to two bands shifting
downward away from $E_{F}$. 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 Geout, 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 turn 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 $E_{F}$ 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 Bradley and Cracknell ; Tang and Wan (2021),
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 B3g, while vHS2 corresponds to
the irrep Ag. On the other hand, at $L$ point, the irrep of the vHS2 and VHS3
are both Au. The irreps of the vHSs closer to $E_{F}$ are different in
$k_{z}$=0 and $k_{z}$=0.5 planes, also indicating the strong 3D feature in
FeGe, which will be discussed carefully in the following.

Figure 2: The nesting function $\xi(q_{x},q_{y},0)$ of FeGe. We neglected the
peak at $q$=0 to better present the nesting function.

To understand the CDW instability, we calculate the Fermi surface nesting
function $\xi(\mathbf{q})$ Johannes and Mazin (2008). The 2$\times$2$\times$2
supercell structure of CDW phase for the non-magnetic pristine phase is
suggested by experimental results Shao  _et al._ (2022); Miao  _et al._
(2022); Teng  _et al._ (2022b); Yin  _et al._ (2022b). Since the A-type AFM
structure has already enlarged unit cells doubled along the z-direction, we
focus the in-plane $\mathbf{q}$ vector in the following, and obtain the
nesting function $\xi(q_{x},q_{y},0)$, as is 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 Kiesel  _et al._ (2013); Park  _et al._
(2021); Yu and Li (2012); Kiesel and Thomale (2012) and AV3Sb5 Si  _et al._
(2022); Wu  _et al._ (2022). This motivated a careful analysis of the band
structure, Fermi surface and the nesting function of FeGe.

Figure 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 $\alpha$, $\beta$, $\varGamma$,
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.

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 AV3Sb5 Si  _et al._ (2022); Wu
_et al._ (2022) is v-hexagon, i.e. a hexagon with a difference of 60∘ 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 analyse the $k_{z}$ momentum-
dependent evolution of band structure, Fermi surface and the nesting function,
and show the results in Fig. 3(a-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 Fig. 3(a-c). The
$d_{x^{2}-y^{2}}$/$d_{xy}$ bands forming the pocket $\alpha$ and containing
vHS2 shift down 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 (f)) as $k_{z}$
increases. For $k_{z}$=0.5 plane, the Fermi surface evolved to contain three
different h-hexagonal-shaped pockets. 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 and planes, the maximum of nesting function are
located at or near the $K$ point, as shown in Fig. 3(a) and (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 is
originate 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 to 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 AV3Sb5 and
that of FeGe origin from their different crystal structures. In AV3Sb5 (A = K,
Cs, Rb), due to the presence of the $A$ layers, the distance between the
interlayer V atoms is at least 9.308 $\mathring{A}$. In FeGe, on the other
hand, the minimum distance between nearest-neighboring interlayer Fe atoms is
4.041 $\mathring{A}$, since in FeGe only Fe-Gein and Geout 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.

In tight-binding models of kagome lattice Kiesel  _et al._ (2013); Park  _et
al._ (2021); Yu and Li (2012); Kiesel and Thomale (2012) and AV3Sb5 Tan  _et
al._ (2021); Si  _et al._ (2022); Wu  _et al._ (2022), the vHSs near the Fermi
level could induce the peak of nesting at $M$ point. Meanwhile, in AV3Sb5, the
phonon instability Tan  _et al._ (2021); Si  _et al._ (2022) and the CDW
transition Jiang  _et al._ (2021); Lin and Nandkishore (2021); Li  _et al._
(2021); Mielke  _et al._ (2022); Denner  _et al._ (2021); Nie  _et al._ (2022)
at $M$ point driven by Fermi surface nesting is suggested. However, in FeGe,
where the Fermi surface has strong 3D feature, 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 ratio 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. It means that unlike AV3Sb5 Tan  _et al._ (2021); Si
_et al._ (2022); Wu  _et al._ (2022), vHSs near the Fermi surface is not the
most important factors of the nesting function in FeGe.

Figure 4: 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 FeA, FeB, and FeC 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).

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, indicating the
interaction plays an important role in CDW transition Mahan (2000); Doniach
and Sondheime (1998). Meanwhile, the theoretically calculated phonon spectrum
remain positive Teng  _et al._ (2022a); Shao  _et al._ (2022); Miao  _et al._
(2022). The electron-phonon interaction is generally weakly momentum-dependent
except for the singular behaviour in non-traditional superconductivity Wan and
Savrasov (2014). Therefore, we believe 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 analyze the local electron-electron
correlation interaction, which is of substantial importance in 3$d$ electron
systems Imada  _et al._ (1998).

Based on DFT calculations, we obtain the wave functions
$\psi_{n,\mathbf{k}+\mathbf{q}}\left(\mathbf{r}\right)$ and
$\psi_{n,\mathbf{k}}\left(\mathbf{r}\right)$, which are electronic Bloch
states at the Fermi level connected by the vector $\mathbf{q}$. We expand the
distribution of $\psi_{n,\mathbf{k}+\mathbf{q}}\left(\mathbf{r}\right)$ and
$\psi_{n,\mathbf{k}}\left(\mathbf{r}\right)$ to the basis set of atomic
orbitals in real space. We find that when $\mathbf{q}=$
$\mathbf{q}_{K}(1/3,1/3,0)$, the wave functions
$\psi_{n,\mathbf{k}+\mathbf{q}}\left(\mathbf{r}\right)$ and
$\psi_{n,\mathbf{k}}\left(\mathbf{r}\right)$ are with unequal predominant
sublattice occupancy. Meanwhile, when $\mathbf{q}=$ $\mathbf{q}_{M}(1/2,0,0)$,
the wave functions $\psi_{n,\mathbf{k}+\mathbf{q}}\left(\mathbf{r}\right)$ and
$\psi_{n,\mathbf{k}}\left(\mathbf{r}\right)$ 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 $\mathbf{q}_{K}$ in
Fig. 4(a), with the characters of three sublattice FeA, FeB, and FeC indicated
in red, green and blue, respectively. The Fermi surface contours of $\beta$
pockets coincide when shifted along the nesting vector
$\mathbf{q}_{K}(1/3,1/3,0)$, as shown in red arrows of Fig. 4(a), resulting in
the peak of nesting function at $K$ point. However, the nesting vector
$\mathbf{q}_{K}$ connects Fermi surface points with mainly different
sublattice occupation, as shown in Fig. 4(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 Doniach and Sondheime (1998). Meanwhile, we demonstrate the results of
the nesting vector $\mathbf{q}_{M}$ by $k_{z}$=0.433 plane in Fig. 4(b). There
are nested Fermi surfaces along $\mathbf{q}_{M}(1/2,0,0)$ connecting the
opposite edges of Fermi pocket $\alpha$, as shown in red arrow of Fig. 4(b).
It can be seen that the wave functions
$\psi_{n,\mathbf{k}+\mathbf{q}}\left(\mathbf{r}\right)$ and
$\psi_{n,\mathbf{k}}\left(\mathbf{r}\right)$ connected by the vector
$\mathbf{q}_{M}$ are dominated by same sublattice occupancy as mentioned
above, leading to enhanced susceptibility at the $M$ point. Therefore, similar
with the sublattice mechanism in superconductors Kiesel and Thomale (2012); Wu
_et al._ (2021), the CDW instability at $M$ point is considered to be derived
by the local electron correlation. Since the electronic instability can
significantly affect phonons Mahan (2000); Doniach and Sondheime (1998), it
may explain the experimentally observed phonon anomalies Teng  _et al._
(2022a); Miao  _et al._ (2022).

In summary, based on DFT calculations, we 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 Kiesel and Thomale (2012); Wu
_et al._ (2021), on the other hand the CDW instability at the $M$ point is
enhanced, which indicates that the electron correlation plays an indispensable
part in the CDW transition.

##  I Acknowledgement

This work was supported by the NSFC (No. 12188101, 11834006, 12004170,
11790311, 51721001), Natural Science Foundation of Jiangsu Province, China
(Grant No. BK20200326), and the excellent programme in Nanjing University.
Xiangang Wan also acknowledges the support from the Tencent Foundation through
the XPLORER PRIZE.

##  II Supplemental Materials

Figure 5: (a)The crystal structure of FeGe. The light blue, dark blue, and
orange spheres represent Gein, Geout, and Fe atoms, respectively. The Fe atoms
located at three different sites are labeled as FeA, FeB, and FeC. (b) The
local structure of the FeGe6 octahedron. The octahedron includes two Gein
atoms and four Geout atoms. (c) Our local coordinate for each Fe atom. The
local x-axis always points to the nearest-neighbor Gein atom, and the z-axis
of each local coordinate is perpendicular to the paper surface.

###  II.1 Crystal Structure

Hexagonal FeGe is an intermetallic compound of the CoSn structure and
crystallize into the $P6/mmm$ (No. 191) space groupOhoyama  _et al._ (1963).
As shown in Fig. 5(a), there are two distinct types of Ge atoms in a unit
cell, labeled Gein (site 1a) and Geout (site 2d) respectively, depending on
whether they are on the same layer as Fe atoms. In the Fe-Gein plane, three Fe
atoms at different sites (noted as FeA, FeB, and FeC in Fig. 5(a)) form Kagome
lattices, and Gein atoms are located in the center of the hexagons. Geout
atoms compose honeycomb structures above and below the Fe-Gein plane.

The local structure of the FeGe6 octahedron is shown in Fig. 5(b). It can be
seen that each Fe atom is surrounded by six Ge atomic octahedrons, including
two Gein atoms and four Geout atoms. In the $O_{h}$ crystal field, ortho-
octahedral structure leads to $t_{2g}-e_{g}$ energy splitting. Here the
octahedron is distorted and induces further splitting of the five Fe-3$d$
orbitals.

Figure 6: Band structure (left panel) and DOS (right panel) of A-type AFM
FeGe. The Fermi level is aligned to 0eV. The DOS of Fe-3$d$ orbitals and
Ge-4$p$ orbitals are indicated by the solid orange and dashed blue lines,
respectively.

###  II.2 Band structure and density of states

We perform LSDA calculation for FeGe based on the experimental A-type AFM
ground state Bernhard  _et al._ (1988), and show the band structure and the
DOS in Fig. 6. As shown in Fig. 6, it is clear that the Fe-3$d$ orbitals
dominate the DOS around $E_{F}$ (-2 to 2 eV relative to $E_{F}$), while the
Ge-4$p$ orbitals are mainly located between -6.0 and -2.0 eV. In the AFM
phase, there is an upshift of the spin minority bands due to the exchange
splitting induced by the ordered moment. Two peaks located near 0.4 and -1.3
eV correspond to the flat bands of the spin minority and spin majority states,
respectively. The contribution of the Fe-3$d$ and Ge-4$p$ orbitals is
relatively close in the range from -6 to -4 eV, suggesting hybridization
between the Fe-3$d$ and Ge-4$p$ orbitals.

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