# Enhanced Spin-polarization via Partial Ge1-dimerization as the Driving Force
of the 2$\times$2$\times$2 CDW in FeGe

Yilin Wang<EMAIL_ADDRESS>School of Future Technology, University of
Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

A $2\times 2\times 2$ charge density wave (CDW) was recently observed deep
inside the antiferromagnetic phase of a Kagome metal FeGe. A key question is
whether the CDW in FeGe is driven by its electronic correlation and magnetism.
Here, we address this problem using density functional theory and its
combination with $U$ as well as dynamical mean-field theory. Our calculations
show that large dimerization ($\sim$1.3 Å) of Ge1-sites along $c$-axis will
enhance electronic correlation of the Fe-$3d$ orbitals and, as a result, it
enhances the spin-polarization and saves more magnetic exchange energies. We
find that the balance between magnetic energy saving and structural energy
cost via partially dimerizing Ge1-sites in an enlarged superstructure, could
induce a new local minimum in total energies. The response to the large
partial Ge1-dimerization will induce additional small modulations ($<$0.05 Å)
of other sites in the Kagome and honeycomb layers, which further reduces the
total energy and leads to a stable $2\times 2\times 2$ CDW ground state in
FeGe. Our results are in good agreement with the existing experiments and
reveal a novel CDW mechanism driven by the interplay of structure, electronic
correlation and magnetism.

Introduction.—Charge density wave (CDW), the static modulations of small
amplitude in the electron density distribution accompanied by a periodic
lattice distortion Grüner (1988, 2019), is one of the most important phases of
matter in condensed matter physics. Common driving force for a CDW includes
Fermi surface nesting Peierls (2001) and electron-phonon couplings McMillan
(1977); Varma and Simons (1983); Johannes and Mazin (2008); Zhu  _et al._
(2015). More exotic mechanism driven by strong electron-electron interactions
is also suggested, for example, in copper oxides Tranquada  _et al._ (1995);
Hoffman  _et al._ (2002); Kivelson  _et al._ (2003); Wise  _et al._ (2008);
Reznik (2012); Blackburn  _et al._ (2013); Comin  _et al._ (2014); Le Tacon
_et al._ (2014); Fujita  _et al._ (2014) and nickel oxides Tranquada  _et al._
(1994); Lee and Cheong (1997). Recently, an exotic chiral $2\times 2\times 2$
CDW that breaks time-reversal symmetry was observed Jiang  _et al._ (2021);
Mielke  _et al._ (2022) in a Kagome metal AV3Sb5 (A=K, Rb, Cs) Ortiz  _et al._
(2019, 2020); Yang  _et al._ (2020), which was suggested to be triggered by
van Hove singularities (VHSs) Feng  _et al._ (2021); Kiesel  _et al._ (2013);
Wang  _et al._ (2013); Tan  _et al._ (2021); Li  _et al._ (2021); Denner  _et
al._ (2021); Lin and Nandkishore (2021); Park  _et al._ (2021); Christensen
_et al._ (2021); Zhao  _et al._ (2021); Jiang  _et al._ (2022); Nie  _et al._
(2022). Since Kagome metals can simultaneously feature geometry frustration
Syôzi (1951); Sachdev (1992); Norman (2016), flat-bands-induced electronic
correlation Tang  _et al._ (2011); Yin  _et al._ (2018, 2020a); Liu  _et al._
(2020); Kang  _et al._ (2020); Wang (2023) and magnetism Tasaki (1992); Lin
_et al._ (2018); Yin  _et al._ (2019, 2020b); Zhang  _et al._ (2021); Huang
_et al._ (2022), non-trivial topology Ortiz  _et al._ (2020); Ye  _et al._
(2018); Liang  _et al._ (2021); Zhou and Wang (2022), VHSs Wang  _et al._
(2013); Kiesel  _et al._ (2013) as well as strong electron-phonon interactions
Ye  _et al._ (2022); Xie  _et al._ (2022); Liu  _et al._ (2022); Zhong  _et
al._ (2022), it has become an ideal platform for exploring CDWs driven by
various mechanisms.

Figure 1: (a) 1$\times$1$\times$2 ideal Kagome structure of FeGe. There are
two types of Ge sites. Ge1 (dark blue) is at the center of the hexagon of the
Fe Kagome layer. Ge2 (grey) forms a honeycomb layer in between two Kagome
layers. (b) All the Ge1 atoms deviate from the Kagome layers and form dimers
along the c-axis. The strength of Ge1 dimerization is defined as
$d=d_{\text{Ge1-Ge1}}^{0}-d_{\text{Ge1-Ge1}}$, where $d_{\text{Ge1-Ge1}}^{0}$
and $d_{\text{Ge1-Ge1}}$ are the bond lengths of Ge1-Ge1 before and after
dimerization, respectively. (c) A 2$\times$2$\times$2 superstructure of FeGe
with space group P6/mmm. It consists of a large dimerization ($|d|\sim 1.3$ Å)
of 1/4 of the Ge1-sites along c-axis and small distortion ($<0.05$ Å,
indicated by short arrows) of other sites.

Very recently, another $2\times 2\times 2$ CDW was observed around
$T_{\text{CDW}}=100$ K, deep inside the A-type antiferromagnetic (AFM) phase
($T_{\text{N}}\sim 410$ K) of a magnetic Kagome metal FeGe Teng  _et al._
(2022); Yin  _et al._ (2022). Both neutron Teng  _et al._ (2022) and x-ray
Miao  _et al._ (2022) scattering experiments indicate a first-order nature of
this CDW transition. Possible anomalous Hall effect Teng  _et al._ (2022) and
topological edge modes Yin  _et al._ (2022) were observed in its CDW phase,
which are similar to those observed in AV3Sb5 Yang  _et al._ (2020); Yu  _et
al._ (2021). However, there are more differences between the CDWs in FeGe and
AV3Sb5. (i) The ordered magnetic moments were found to be enhanced in the CDW
phase of FeGe Teng  _et al._ (2022). (ii) A sharp superlattice peak,
$Q=(0,0,2.5)$, that probes lattice distortion along $c$-axis, was observed in
FeGe by the x-ray scattering experiment with the same onset temperature as CDW
Miao  _et al._ (2022), which is absent in AV3Sb5. (iii) First-principle
calculations find only little phonon softening around the three $L$-points,
(0.5, 0, 0.5), (0, 0.5, 0.5), (-0.5, 0.5, 0.5), but the phonon frequencies
never become negative in FeGe in the parameter regime that produces the
correct ordered magnetic moment Miao  _et al._ (2022). Such behavior in phonon
spectrum is very different from AV3Sb5 Tan  _et al._ (2021); Ye  _et al._
(2022), while it shows some similarities to the electronic-correlation-driven
CDWs in cooper and nickel oxides. These indicate a different origin of the CDW
in FeGe, in sharp contrast to AV3Sb5. A natural question thus arises: whether
the CDW in FeGe is driven by its electronic correlation and magnetism? Despite
several investigations by experiment Teng  _et al._ (2022); Yin  _et al._
(2022); Teng  _et al._ (2023); Miao  _et al._ (2022) and theoretical
calculations Setty  _et al._ (2022); Shao  _et al._ (2022); Miao  _et al._
(2022); Zhou  _et al._ (2022); Wu  _et al._ (2023); Ma  _et al._ (2023), the
driving force of the CDW in FeGe still remains an open question.

In a previous work Miao  _et al._ (2022), guided by the soft phonon mode at
$L$-points and the first-order nature of the CDW transition in FeGe, we have
obtained a stable $2\times 2\times 2$ superstructure of FeGe with lower energy
than its ideal structure, via structural relaxation using density functional
theory (DFT). A key finding is that there is a large dimerization ($\sim$1.3
Å) of 1/4 of Ge1-sites along the $c$-axis in the $2\times 2\times 2$
superstructure. Motivated by this observation, in this work, we carry out more
calculations to identify the driving force for the CDW in FeGe, using DFT and
its combination with $U$ (DFT+$U$) as well as dynamical mean-field theory
(DFT+DMFT) Lichtenstein  _et al._ (2001); Kotliar  _et al._ (2006). Our
calculations show that large dimerization of Ge1-sites along $c$-axis will
enhance electronic correlation of the Fe-$3d$ orbitals and, as a result, it
enhances the spin-polarization and saves more magnetic exchange energies. We
find that the balance between magnetic energy saving and structural energy
cost via partially dimerizing Ge1-sites in an enlarged superstructure, could
induce a new local minimum in total energies. The response to the large
partial Ge1-dimerization will induce additional small modulations ($<$0.05 Å)
of other sites in the Kagome and honeycomb layers, which further reduces the
total energy and leads to a stable $2\times 2\times 2$ CDW ground state in
FeGe. Our results thus reveal a novel CDW mechanism driven by the interplay of
structure, electronic correlation and magnetism, which is in good agreement
with the existing experiments.

Methods.—As shown in Fig. 1(a), hexagonal FeGe is consisting of a Kagome layer
of Fe3Ge and a honeycomb layer of Ge2. There are two non-equivalent Ge-sites:
Ge1 (blue) in the center of the hexagon of the Fe Kagome structure and Ge2
(grey) in the honeycomb layer. The DFT calculations are performed using the
VASP package Kresse and Furthmüller (1996); Blöchl (1994), with exchange-
correlation functional of generalized gradient approximation (GGA) Perdew  _et
al._ (1996). Although FeGe is a strongly correlated magnet, the DFT
calculations without Hubbard $U$ correction have already correctly produced
the ordered magnetic moments of its AFM phase (around 1.5 $\mu_{B}$/Fe),
observed by neutron scattering experiment Teng  _et al._ (2022). Therefore,
DFT calculations are applicable to the AFM phase of FeGe. The experimental
lattice parameters of FeGe, $a=4.985$ Å and $c=4.048$ Å Meier  _et al._
(2020); Teng  _et al._ (2022), are used in calculations.

To better capture the strong electronic correlations of FeGe, we also perform
fully charge self-consistent single-site DFT+DMFT calculations for its
paramagnetic (PM) and AFM phases, using the EDMFTF code developed by Haule et
al. Haule  _et al._ (2010); Haule and Birol (2015) based on the WIEN2K package
Blaha  _et al._ (2020). For AFM calculation, a non-magnetic calculation is
performed in the DFT part, while the AFM spin-polarization is considered in
the DMFT part by breaking the spin degeneracy of the local self-energy. The
Hubbard $U$ and Hund’s coupling $J_{H}$ are chosen to be 3.9 eV and 0.85 eV,
respectively, to reproduce the ordered magnetic moments observed
experimentally. More computational details are presented in the Supplementary
Materials sup .

Figure 2: Enhanced spin-polarization via partial Ge1-dimerization calculated
by DFT. (a) The difference of total energy between the distorted and ideal
Kagome structure, $\Delta E=E_{\text{tot}}(d)-E_{\text{tot}}(d=0)$, as
functions of the strength of Ge1-dimerization $d$. Dashed and solid curves are
for non-magnetic and anti-ferromagnetic states, respectively. The black arrow
labels a local energy minimum around $d=1.3$ Å. (b) The magnetic exchange
energy, $\Delta E_{\text{AFM}}-\Delta E_{\text{NM}}$, and the average of
ordered magnetic moments per Fe in the AFM state as functions of $d$, are
shown in the left and right $y$-axis, respectively. (a) and (b) are for the
1$\times$1$\times$2 structure with large dimerization of all Ge1-sites.
(c)-(h) are in analogous to (a) and (b), but for enlarged superstructures and
two different cases labeled by “S1” and “S2” (see the main text). (c) and (d)
are for the $\sqrt{3}\times\sqrt{3}\times 2$ superstructure (P6/mmm) with
large dimerization of $1/3$ of Ge1-sites. (e) and (f) are for the $2\times
2\times 2$ superstructure (P6/mmm) with large dimerization of $1/4$ of
Ge1-sites. (g) and (h) are for the $\sqrt{5}\times\sqrt{5}\times 2$
superstructure (Cmmm) with large dimerization of $1/5$ of Ge1-sites.

Results.—Fig. 2 show the results calculated by DFT. Fig. 2(a) shows the total
energy difference of the $1\times 1\times 2$ superstructure of FeGe between
the one with large dimerization of all the Ge1-sites along $c$-axis and the
one without dimerization, $\Delta E=E_{\text{tot}}(d)-E_{\text{tot}}(d=0)$, as
functions of the Ge1-dimerization strength $d$ (see Fig. 1). $\Delta E$ for
the non-magnetic (NM) state keeps increasing with $|d|$, indicating that it
has to pay for energies for structure distortions. However, the increasing
rate slows down when the dimerization strength $d$ is around 1.0 Å$\sim$1.3 Å.
Although $\Delta E$ also increases with $|d|$ for AFM state, its magnitude is
much smaller than that of NM state, indicating that the spin-polarization are
enhanced and more magnetic exchange energies are saved by Ge1-dimerization.
This is shown in Fig. 2(b), where the magnetic exchange energies, $\Delta
E_{\text{AFM}}-\Delta E_{\text{NM}}$, are negative and their magnitude keep
increasing with $|d|$ (left $y$-axis), and the ordered magnetic moments also
increase with $|d|$ (right $y$-axis). This is a main finding of the present
work. The competition between magnetic energy saving from enhanced spin-
polarization and structural energy cost from Ge1-dimerization may induce a new
local minimum in the total energies of the AFM state, as illustrated by the
black arrow around $d=1.3$ Å in Fig. 2(a). It may further become a global
energy minimum if the magnetic energy wins, and drive a first-order phase
transition. This has not achieved in the $1\times 1\times 2$ superstructure
because it costs too much structural energies to dimerize all the Ge1-sites.
Therefore, a straightforward strategy to balance the magnetic energy saving
and structural energy cost is to enlarge the structure along the $a$ and $b$
crystal axis, but dimerize partial Ge1-sites.

Along this line, we construct three superstructures:
$\sqrt{3}\times\sqrt{3}\times 2$ with large dimerization of 1/3 of Ge1-sites
[Fig. 2(c) and (d)], $2\times 2\times 2$ with large dimerization of 1/4 of
Ge1-sites [Fig. 2(e) and (f)], and $\sqrt{5}\times\sqrt{5}\times 2$ with large
dimerization of 1/5 of Ge1-sites [Fig. 2(g) and (h)]. We then perform
calculations for two different cases. (S1) The superstructure with only fixed
large dimerization of partial Ge1-sites but other sites are not relaxed; (S2)
The superstructure with fixed large dimerization of partial Ge1-sites and all
the other sites are relaxed until the force of each of those atoms are less
than 1 meV/Å.

As shown by the solid blue curves in Fig. 2(c), (e) and (g), well-defined
local energy minimums have already formed around $d=1.3$ Å in all the three
superstructures with only large partial Ge1-dimerization (S1). The energy
minimums are only 2$\sim$3 meV/atom higher than that of the non-distorted
structures ($d=0$). The magnetic exchange energies and the ordered magnetic
moments shown in Fig. 2(d), (f) and (h) follow the same behaviors as the
$1\times 1\times 2$ superstructure. Large dimerization of partial Ge1-sites
will exert extra potential on other sites, such that they will slightly relax
their positions to further reduce the total energy, which are shown by the
solid orange curves in Fig. 2(c), (e) and (g). The energy minimums around
$d=1.3$ Å of all the three superstructures get much closer to that of the
ideal structure, but it only becomes a global minimum in the $2\times 2\times
2$ superstructure, consistent with the experimentally observed $2\times
2\times 2$ CDW in FeGe. This is because the 1/3 Ge1-dimerization in the
$\sqrt{3}\times\sqrt{3}\times 2$ superstructure pays for more structural
distortion energy and the 1/5 Ge1-dimerization in the
$\sqrt{5}\times\sqrt{5}\times 2$ superstructure cannot save enough magnetic
energy, comparing to the 1/4 Ge1-dimerization in the $2\times 2\times 2$
superstructure. Therefore, we can conclude that the form of $2\times 2\times
2$ CDW superstructure in FeGe results from a subtle balance between magnetic
exchange energy saving and structure energy cost via large structural
dimerization of 1/4 of Ge1-sites.

The distortions of other sites are very small ($<$0.05 Å). The arrows in Fig.
1(c) illustrates such distortions in the $2\times 2\times 2$ superstructure.
The atoms mainly move out-of-plane along $c$-axis in the Kagome layers, and
move in-plane in the honeycomb layers to form a Kekulé-type distortion Herrera
and Naumis (2020); Shao  _et al._ (2022). We note that the magnetic exchange
energies and ordered magnetic moments are not further enhanced by those
additional distortions [green curves in Fig. 2(d), (f) and (h)], indicating
that the driving force for the enhanced spin-polarization is primarily from
the large dimerization of Ge1-sites. The small $2\times 2$ charge modulations
in Kagome and honeycomb layers which are observed by the STM experiments Yin
_et al._ (2022); Teng  _et al._ (2022); Chen  _et al._ (2023), are induced as
a consequence to respond to the extra potential induced by the large
dimerization of partial Ge1-sites.

In order to understand how Ge1-dimerization will enhance the spin-
polarization, we compare the electronic correlations of the paramagnetic state
of the $1\times 1\times 2$ superstructure with and without Ge1-dimerization by
DFT+DMFT calculations. The calculated mass-enhancement
$m^{\star}/m^{\text{DFT}}$ of Fe-$3d$ orbitals at $T=290$ K, $U=3.9$ eV and
$J_{H}=0.85$ eV are tabulated in Table. 1. The mass-enhancement, in
particular, of $d_{x^{2}-y^{2}}$, $d_{xz}$ and $d_{yz}$ orbitals increase
substantially from the ideal Kagome structure to the structure with a
Ge1-dimerization of $d=1.3$ Å. It indicates that the height of Ge1 atoms
relative to the Fe Kagome plane will affect the electronic correlations of
Fe-$3d$ orbitals. This is similar to the iron-based superconductors, where the
electronic correlations of Fe-$3d$ orbitals are found to be very sensitive to
the heights of As or Se sites Yin  _et al._ (2011); Ieki  _et al._ (2014);
Haule and Pascut (2016). Stronger electronic correlations will enhance the
spin-polarization of the AFM state of FeGe. Analogous to Fig. 2, we also
perform DFT+DMFT calculations for the $2\times 2\times 2$ superstructure,
which are shown in Fig. 3. Similar profiles of energy and ordered magnetic
moments as functions of $d$ are produced, and a global energy minimum is also
found around $d=1.3$ Å, which validates the findings from the simple DFT
calculations.

Table 1: The mass enhancement, $m^{*}/m^{\text{DFT}}=1/Z$, of Fe-$3d$ orbitals from DFT+DMFT calculations of the paramagnetic phase of the $1\times 1\times 2$ superstructure with ($d=1.3$ Å) and without ($d=0$ Å) Ge1-dimerization, at $T=290$ K, $U=3.9$ eV and $J_{H}=$ 0.85 eV. | $d_{3z^{2}-r^{2}}$ | $d_{x^{2}-y^{2}}$ | $d_{xz}$ | $d_{yz}$ | $d_{xy}$
---|---|---|---|---|---
$d=0$ Å | 2.146 | 3.232 | 3.342 | 3.239 | 2.055
$d=1.3$ Å | $2.287$ | $3.971$ | $4.171$ | $3.642$ | $2.136$

The subtle competition between magnetic and structural energy indicates that
the $2\times 2\times 2$ CDW in FeGe is sensitive to the variation of its
magnetism and crystal structure. Thus, on the one hand, increasing Hubbard $U$
in DFT+$U$ calculations is expected to further save the magnetic energy and
stabilize the CDW state. The DFT+$U$ results at $U=1$ eV for the $2\times
2\times 2$ superstructure are shown in Fig. 4(a) and (b). The ordered magnetic
moment is enhanced to about 2 $\mu_{B}/\text{Fe}$, much larger than the
experimental value, and the energy of the CDW ground state is further reduced.
We note that the energy minimum around $d=1.3$ Å becomes a global energy
minimum even in the superstructure with only large dimerization of 1/4 of
Ge1-sites (solid blue curve), which further confirms that the large
dimerization of partial Ge1-sites is the driving force. We also find that the
$\sqrt{3}\times\sqrt{3}\times 2$ superstructure becomes the ground state
instead of the $2\times 2\times 2$ superstructure at $U=1$ eV. This is also
expected within our theory, since the magnetic energy saving becomes even
greater in the superstructure with more dimerized Ge1-sites at larger $U$,
such that the $\sqrt{3}\times\sqrt{3}\times 2$ superstructure saves the most
energies among the superstructures at $U=1$ eV. This result further supports
our conclusion that the CDW in FeGe results from a subtle competition between
magnetic energy saving and structural energy cost.

On the other hand, elongating the crystal $c$-axis of FeGe may cause it to pay
for more structural distortion energy when dimerizing Ge1-sites. We show this
in Fig. 4(c) and (d) by increasing $c$ to 4.4 Å. Indeed, the $2\times 2\times
2$ CDW state is not favored anymore.

Figure 3: Analogous to Fig. 2, but calculated by DFT+DMFT for paramagnetic
(PM) and AFM phases of the $2\times 2\times 2$ superstructure, at $T=290$ K,
$U=3.9$ eV and $J_{H}=0.85$ eV. Here, $E$ stands for total free energy. The
orange star in (a) and (b) stands for the total free energy and ordered
magnetic moments, respectively, of the fully relaxed $2\times 2\times 2$
superstructure.

Conclusion and Discussion.—To summarize, by DFT, DFT+$U$ and DFT+DMFT
calculations, we show that the driving force of the $2\times 2\times 2$ CDW in
FeGe is an enhanced spin-polarization via large dimerization of partial
Ge1-sites in an enlarged superstructure. We reveal that the enhancement of
spin-polarization is due to stronger electronic correlations of Fe-$3d$
orbitals caused by large dimerization of Ge1-sites. The small $2\times 2$
charge modulations in Kagome and honeycomb layers observed by STM Yin  _et
al._ (2022); Teng  _et al._ (2022); Chen  _et al._ (2023), are thus induced as
a consequence to respond the large dimerization of partial Ge1-sites.

Our results thus indicate a first-order CDW transition in FeGe, which is
consistent with both neutron Teng  _et al._ (2022) and x-ray Miao  _et al._
(2022) scattering experiments. The large Ge1-dimerization along the $c$-axis
is consistent with the observation that an additional sharp super-lattice peak
$Q=(0,0,2.5)$ was found in the elastic x-ray scattering experiment with the
same onset temperature as the CDW transition Miao  _et al._ (2022), since this
peak probes the structural distortion along $c$-axis. The enhancement of spin-
polarization in the CDW phase is consistent with the observation from neutron
scattering experiment Teng  _et al._ (2022).

Similar to AV3Sb5, there are also VHSs at $M$ point near the Fermi level in
the electronic structure of FeGe Teng  _et al._ (2022, 2023); Miao  _et al._
(2022), so VHSs may also play roles in driving the CDW in FeGe. However, Wan
et al. Wu  _et al._ (2023) have computed the nesting functions of FeGe and
found that the maximum of nesting function is at $K$ point instead of $M$
point. This excludes the possibility of a VHSs-induced CDW in FeGe.

We note that the Kagome FeSn, the sister compound of FeGe, shows similar
electronic structures Lin  _et al._ (2020) and antiferromagnetic orders Sales
_et al._ (2019) to FeGe, but no CDW was observed experimentally. We also
perform the same calculations for FeSn and find a similar local energy minimum
induced by large Sn1-dimerization (see Fig. S1 in sup ), but it is far from
becoming a global energy minimum in the regime of reasonable Hubbard $U$ that
produces the correct ordered magnetic moments ($\sim$ 1.85 $\mu_{B}$/Fe) Sales
_et al._ (2019). It may originate from that Sn has a much larger atomic radius
and, as a result, much larger lattice parameters of FeSn Meier  _et al._
(2020), such that it has to pay for much more structural energies than FeGe by
dimerizing the Sn1-sites.

Figure 4: Analogous to Fig. 2 for $2\times 2\times 2$ superstructure of FeGe.
(a) and (b) are calculated by DFT+$U$ at $U=1$ eV with the experimental
lattice parameters. (c) and (d) are calculated by DFT, but with a larger
lattice parameter of $c=4.4$ Å.

Acknowledgement.—We thank Hu Miao, Zhida Song and Yajun Yan for very helpful
discussion. This project was supported by the National Natural Science
Foundation of China (No. 12174365). All the calculations were preformed on
TianHe-1(A), the National Supercomputer Center in Tianjin, China.

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