# Tuning the magnetic interactions in van der Waals Fe3GeTe2 heterostructures:
A comparative study of ab initio methods

Dongzhe Li<EMAIL_ADDRESS>CEMES, Université de Toulouse, CNRS, 29 rue
Jeanne Marvig, F-31055 Toulouse, France  Soumyajyoti Haldar  Institute of
Theoretical Physics and Astrophysics, University of Kiel, Leibnizstrasse 15,
24098 Kiel, Germany  Tim Drevelow  Institute of Theoretical Physics and
Astrophysics, University of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany
Stefan Heinze  Institute of Theoretical Physics and Astrophysics, University
of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany  Kiel Nano, Surface, and
Interface Science (KiNSIS), University of Kiel, Germany

###### Abstract

We investigate the impact of mechanical strain, stacking order, and external
electric fields on the magnetic interactions of a two-dimensional (2D) van der
Waals (vdW) heterostructure in which a 2D ferromagnetic metallic Fe3GeTe2
monolayer is deposited on germanene. Three distinct computational approaches
based on ab initio methods are used, and a careful comparison is given: (i)
The Green’s function method, (ii) the generalized Bloch theorem, and (iii) the
supercell approach. First, the shell-resolved exchange constants are
calculated for the three Fe atoms within the unit cell of the freestanding
Fe3GeTe2 monolayer. We find that the results between methods (i) and (ii) are
in good qualitative agreement and also with previously reported values. An
electric field of ${\cal E}=\pm 0.5$ V/Å applied perpendicular to the
Fe3GeTe2/germanene heterostructure leads to significant changes of the
exchange constants. We show that the Dzyaloshinskii-Moriya interaction (DMI)
in Fe3GeTe2/germanene is mainly dominated by the nearest neighbors, resulting
in a good quantitative agreement between methods (i) and (ii). Furthermore, we
demonstrate that the DMI is highly tunable by strain, stacking, and electric
field, leading to a large DMI comparable to that of ferromagnetic/heavy metal
(FM/HM) interfaces, which have been recognized as prototypical multilayer
systems to host isolated skyrmions. The geometrical change and hybridization
effect explain the origin of the high tunability of the DMI at the interface.
The electric-field driven DMI obtained by method (iii) is in qualitative
agreement with the more accurate ab initio method used in approach (ii).
However, the field-effect on the DMI is overestimated by method (iii) by about
50%. This discrepancy is attributed to the different implementations of the
electric field and basis sets used in the ab initio methods applied in (ii)
and (iii). The magnetocrystalline anisotropy energy can also be drastically
changed by the application of compressive or tensile strain in the
Fe3GeTe2/germanene heterostructure. The application of an electric field, in
contrast, leads only to relatively small changes of the MAE for electric
fields of up to 1 V/Å.

##  I Introduction

Magnetic skyrmions Bogdanov and Yablonskii (1989) – topologically protected
chiral spin structures with particle-like properties – have attracted
tremendous attention due to their potential application in next-generation
spintronics devices such as racetrack memories Fert  _et al._ (2013), logic
gates Luo  _et al._ (2018), artificial synapses for neuromorphic computing
Song  _et al._ (2020), and qubits for quantum computing Psaroudaki and
Panagopoulos (2021). The formation of magnetic skyrmions is due to the
competition between the Heisenberg exchange and Dzyaloshinskii-Moriya
interaction (DMI) Moriya (1960); Dzialoshinskii (1957); Fert and Levy (1980);
Bode  _et al._ (2007) together with the magnetocrystalline anisotropy energy
(MAE). In particular, the DMI plays an essential role in stabilizing skyrmions
since it favors canted spin configurations with a unique rotational sense. The
DMI results from spin-orbit coupling and is only non-zero for systems with
broken inversion symmetry. Magnetic skyrmion lattices were discovered in
experiments on bulk chiral magnets Mühlbauer  _et al._ (2009); Yu  _et al._
(2010) and subsequently in epitaxial ultrathin films Heinze  _et al._ (2011).
Isolated magnetic skyrmions were observed in ultrathin transition-metal films
at low temperatures Romming  _et al._ (2013); Hervé  _et al._ (2018); Meyer
_et al._ (2019) and at room temperature in magnetic multilayers Moreau-
Luchaire  _et al._ (2016); Boulle  _et al._ (2016); Woo  _et al._ (2016);
Soumyanarayanan  _et al._ (2017), in ferrimagnets Caretta  _et al._ (2018),
and in synthetic antiferromagnets Legrand  _et al._ (2020).

Recently, long-range magnetism was reported in two-dimensional (2D) materials
Gong  _et al._ (2017); Huang  _et al._ (2017); Deng  _et al._ (2018). This
provides a promising alternative avenue for exploring topological spin
structures in atomically thin layers. Several recent experiments reported the
observation of skyrmions in 2D vdW heterostructures, such as at an
Fe3GeTe2/WTe2 interface Wu  _et al._ (2020), in a Fe3GeTe2/Co/Pd multilayer
Yang  _et al._ (2020), and at an Cr2Ge2Te6/Fe3GeTe2 interface Wu  _et al._
(2022). Moreover, magnetic domain walls were observed in the Fe3GeTe2 surface
Yang  _et al._ (2022). The origin of skyrmions in these systems was attributed
to the interfacial DMI. A comprehensive material survey has been done by ab
initio calculations to explore the DMI in 2D magnets. The family of monolayer
Janus vdW magnets has been predicted to possess large enough DMI to allow
stable skyrmions Liang  _et al._ (2020); Yuan  _et al._ (2020); Xu  _et al._
(2020); Cui  _et al._ (2020); Jiang  _et al._ (2021a); Shen  _et al._ (2022).
Néel-type magnetic skyrmions were also observed in Fe3GeTe2 crystals, and
attributed to the DMI due to oxidized interfaces Park  _et al._ (2021). In
addition, it has been proposed that skyrmions can be stabilized in 2D vdW
multiferroic heterostructures Sun  _et al._ (2020), Moiré of vdW 2D magnets
Tong  _et al._ (2018), and even in centrosymmetric materials Amoroso  _et al._
(2020) induced by exchange frustration. For most 2D magnets, the DMI is absent
due to inversion symmetry. It is possible to break the inversion symmetry by
designing various 2D vdW heterostuctures and applying an electric field, or
strain Gong and Zhang (2019); Jiang  _et al._ (2021b). This indicates the
possibility of tuning DMI via external stimuli in 2D vdW heterostuctures.

From the theoretical point of view, the calculation of the DMI at the ab
initio level is, in principle, relatively straightforward, nevertheless,
complications can arise in practice. Several approaches have been introduced
based on different first-principles methods and used by numerous groups to
calculate the DMI for various material classes Bode  _et al._ (2007); Ferriani
_et al._ (2008); Heide  _et al._ (2009); Yanes  _et al._ (2013); Kashid  _et
al._ (2014); Zimmermann  _et al._ (2014); Dupé  _et al._ (2014); Simon  _et
al._ (2014); Yang  _et al._ (2015a); Vida  _et al._ (2016); Yamamoto  _et al._
(2017); Simon  _et al._ (2018); Jadaun  _et al._ (2020); Liang  _et al._
(2020). Unfortunately, mostly without a sufficient cross-check between them.
This invites a detailed benchmark study to validate different approaches,
however, so far this has only been performed for the ultrathin film system of
a Co monolayer on Pt(111) Zimmermann  _et al._ (2019). Such comparative
studies are crucial to understanding the origin of skyrmion stability,
particularly for the newly discovered 2D magnets.

Here, using ab initio calculations, we compare systematically three current
state-of-the-art approaches to extract magnetic interaction parameters in
Fe3GeTe2 (FGT) based heterostructures, namely (i) by the Green’s function
method Liechtenstein  _et al._ (1987), (ii) by using the generalized Bloch
theorem (gBT) Sandratskii (1986); Kurz  _et al._ (2004); Heide  _et al._
(2009); Zimmermann  _et al._ (2019), or (iii) by using the supercell approach
Yang  _et al._ (2015a). First we study the shell-resolved exchange interaction
between the Fe atoms of the different layers in a freestanding FGT monolayer.
We find that the approaches (i) and (ii) are in good qualitative agreement. We
then focus on the structural and magnetic properties of the 2D vdW
heterostructure of an FGT monolayer deposited on germanene under strain,
stacking, and electric field. We find that a small compressive strain
($\gamma$) of about 3% can significantly enhance the DMI in FGT
heterostructures by more than 400% compared to the value without strain. The
variation of DMI is mainly due to the geometrical change of the FGT monolayer.
Such a large DMI is comparable to that in state-of-the-art FM/HM
heterostructures, which have been demonstrated as prototypical multilayer
systems to host individual skyrmions even at room temperature. Furthermore,
the DMI can be substantially modified via different stacking geometry due to
the hybridization effect at the interface.

Upon applying an electric field the strength of the DMI varies almost linearly
and can even change sign when a strong electric field (${\cal E}>$ 1 V/Å) is
applied. The exchange constants are also considerably modified due to an
electric field while the effect on the magnetocrystalline anisotropy energy
(MAE) is small. However, the MAE is dramatically reduced to 25% of its
original value at a compressive strain of $\gamma=-3\%$. In connection with
the exchange frustration in FGT/germanene these large changes of DMI and MAE
open the possibility of zero-field magnetic skyrmions Li  _et al._ (2022). For
the DMI in FGT/Ge, the three theoretical approaches are in good qualitative
agreement, and we also discuss quantitative comparison in detail.

The paper is organized as follows. In Sec. II, we describe the three
theoretical approaches used to obtain the relevant spin-spin interaction
parameters by mapping the ab initio DFT calculations onto an extended
Heisenberg model. In Sec. III, we examine the Heisenberg exchange for free-
standing FGT followed by the DMI and MAE for FGT heterostuctures. Different
theoretical approaches are carefully benchmarked. We further investigate the
effects of biaxial strain, stacking configuration as well electric field on
the magnetic interactions in FGT heterostuctures. Finally, we summarize our
main conclusions in Sec. IV.

##  II Methods and computational details

In order to describe the magnetic properties of FGT heterostructures, we use
the extended Heisenberg model for the spins of Fe atoms in the hexagonal
structure:

$\begin{split}H&=-\sum_{ij}J_{ij}(\mathbf{m}_{i}\cdot\mathbf{m}_{j})-\sum_{ij}\mathbf{D}_{ij}\cdot(\mathbf{m}_{i}{\times}\mathbf{m}_{j})\\\
&-\sum_{i}K_{i}(m_{i}^{z})^{2}\end{split}$ (1)

where $\mathbf{m}_{i}$ and $\mathbf{m}_{j}$ are normalized magnetic moments at
positions $\mathbf{R}_{i}$ and $\mathbf{R}_{i}$ respectively. $M$ denotes the
size of the magnetic moment at every site. The three magnetic interaction
terms correspond to the Heisenberg isotropic exchange, the DMI, and the MAE,
respectively, and they are characterized by the parameters $J_{ij}$,
$\mathbf{D}_{ij}$, and $K_{i}$ in the related terms. Note, that by using Eq.
(1) it is assumed that the magnetic moments are constant.

During the past decade, in order to obtain the parameters very accurately in
Eq. (1), several approaches based on density functional theory (DFT) have been
developed, which is frequently named ab initio atomistic spin model. In this
work, we apply three different approaches for the calculation of magnetic
interactions in FGT vdW heterostructures: (i) The Green’s function method
Liechtenstein  _et al._ (1987); Szilva  _et al._ (2013) (also known as the
Liechtenstein formula) employing infinitesimal rotations; (ii) The generalized
Bloch theorem (gBT) Sandratskii (1986) which allows calculating the total
energy of spin-spirals of any wave vector $\mathbf{q}$ in magnetic
nanostructures Kurz  _et al._ (2004); (iii) The supercell approach Yang  _et
al._ (2015a) which is straightforward but computationally heavy due to the
comparison of total energies in a supercell geometry. We performed DFT
calculations using two community ab initio codes which differ in their choice
of basis set: The QuantumATK (QATK) package Smidstrup  _et al._ (2019) uses an
expansion of electronic states in a linear combination of atomic orbitals
(LCAO) while the FLEUR code fle  is based on the full-potential linearized
augmented plane wave (FLAPW) formalism. The former is computationally very
efficient, while the latter ranks amongst the most accurate implementations of
DFT. In the following, we denote the three different approaches as LCAO-Green,
FLAPW-gBT, and LCAO-supercell for simplicity.

###  II.1 The Green’s function method: LCAO-Green

In the Green’s function method, we treat the local rigid spin rotation as a
perturbation. The total energy variation due to the two-spin interaction
between sites $i$ and $j$ is

$\delta
E_{ij}=-\frac{2}{\pi}\int_{-\infty}^{E_{\text{F}}}dE\imaginary\Tr[\delta\mathbf{H}_{ii}\mathbf{G}_{ij}\delta\mathbf{H}_{jj}\mathbf{G}_{ji}]$
(2)

where $\mathbf{H}$ and $\mathbf{G}$ are real-space Hamiltonian and Green’s
function, which are both matrices in the orbital space and can be determined
from the solution of the electronic structure problem.

Then, we define another central quantity for the Green function method, namely
the $\mathbf{A}$ matrix which has a $4{\times}4$ size as follows

$\mathbf{A}_{ij}^{\mu\nu}=-\frac{1}{4\pi}\int_{-\infty}^{E_{\text{F}}}dE\Tr[\Delta_{ii}\mathbf{G}_{ij}^{\mu}\Delta_{jj}\mathbf{G}_{ji}^{\nu}]$
(3)

where indices $\mu$ and $\nu$ run over 0, $x$, $y$, or $z$ and
$\Delta_{ii}=(\mathbf{H}_{ii}^{\uparrow}-\mathbf{H}_{ii}^{\downarrow})/2$ is
the on-site difference between the spin-up and -down part of the Hamiltonian
matrix. Then, the Heisenberg exchange and the DMI can be expressed by using
only the $\mathbf{A}$ matrix as follows

$J_{ij}^{\text{iso}}=2\imaginary(\mathbf{A}_{ij}^{00}-\mathbf{A}_{ij}^{xx}-\mathbf{A}_{ij}^{yy}-\mathbf{A}_{ij}^{zz})$
(4)

$\mathbf{D}_{ij}^{\mu}=2\real(\mathbf{A}_{ij}^{0\mu}-\mathbf{A}_{ij}^{\mu 0})$
(5)

If we neglect SOC, we arrive at the original Liechtenstein-Katsnelson-
Antropov-Gubanov (LKAG) formula Liechtenstein  _et al._ (1987) which was
proposed in 1987,

$J_{ij}^{\text{iso}}=-\frac{1}{4\pi}\int_{-\infty}^{E_{\text{F}}}dE\imaginary\Tr[\Delta_{ii}\mathbf{G}_{ij}^{\uparrow}\Delta_{jj}\mathbf{G}_{ji}^{\downarrow}]$
(6)

where $\mathbf{G}_{ij}$ becomes hermitian.

Our Green’s function calculations were performed using QATK Smidstrup  _et
al._ (2019) in two steps: (i) We performed LCAO-DFT calculations with SOC in
order to construct the tight-binding like Hamiltonian matrix $\mathbf{H}_{ij}$
and the overlap matrix $\mathbf{S}_{ij}$. (ii) The magnetic exchange
parameters were evaluated as described above by Eqs. (2-6). For LCAO-DFT
calculations on FGT monolayers and on FGT/Ge heterostructures, the energy
cutoff for the density grid sampling was set to 150 Hartree, and a
$28{\times}28$ $\mathbf{k}$-point mesh was adopted for the Brillouin zone (BZ)
integration. For magnetic exchange calculations, we used a much denser
$\mathbf{k}$-point mesh of $48{\times}48$, 60 circle contour points, and
13${}^{\text{th}}$ nearest neighbors (NN) in order to obtain accurate
numerical integration. Using these parameters, we extracted $J_{ij}$ and
$\mathbf{D}_{ij}$ parameters with an accuracy of 0.01 meV. Note that this
approach, i.e., infinitesimal spin rotations, fits well to magnetic skyrmions
in which we often have large noncollinear spin structures.

###  II.2 The generalized Bloch theorem: FLAPW-gBT

The second approach employs the FLAPW method as implemented in the FLEUR code
fle  and is based on the generalized Bloch theorem (gBT) Sandratskii (1986);
Kurz  _et al._ (2004). It allows considering spin-spirals of any wave vector
$\mathbf{q}$ for systems without SOC. We first self-consistently compute
within the scalar-relativistic approximation the energy dispersion,
$E_{\text{ss}}(\mathbf{q})$, of homogeneous flat spin spirals Kurz  _et al._
(2004) which are characterized by a wave vector $\mathbf{q}$ and an angle
$\phi=\mathbf{q}\cdot\mathbf{R}$ between adjacent magnetic moments separated
by lattice vector $\mathbf{R}$.

As a second step, the DMI is computed within first-order perturbation theory
on the self-consistent spin spiral state Heide  _et al._ (2009); Zimmermann
_et al._ (2014, 2019). The energy variation
$\delta\epsilon_{\mathbf{k},\nu}(\mathbf{q})$ of these states due to the SOC
Hamiltonian can be written as

$\delta\epsilon_{\mathbf{k},\nu}(\mathbf{q})=\bra{\Psi_{\mathbf{k},\nu}(\mathbf{q})}H_{\text{SOC}}\ket{\Psi_{\mathbf{k},\nu}(\mathbf{q})},$
(7)

where $\ket{\Psi_{\mathbf{k},\nu}(\mathbf{q})}$ are the self-consistent
solutions in the scalar-relativistic approximation, $\mathbf{k}$ is the Bloch
vector, and $\nu$ is the band index. By integration over the Brillouin zone
and summation over all occupied bands $\nu$ this gives the total energy
contribution for spin spirals due to SOC denoted as $E_{\rm DMI}(\mathbf{q})$.

We map the energy dispersion in the scalar-relativistic approximation,
$E_{\text{SS}}(\mathbf{q})$, and the energy contribution to spin spirals due
to DMI, $E_{\text{DMI}}(\mathbf{q})$, to the atomistic spin model, Eq. (1), in
order to extract the exchange constants, $J_{ij}$, and the magnitudes of the
$\mathbf{D}_{ij}$, respectively. One of the key advantages of the gBT approach
is that even incommensurate spin spirals and those with a large $\mathbf{q}$
can be treated very efficiently in the chemical unit cell, i.e., without the
need for large supercells.

We used a cutoff parameter for the FLAPW basis functions of $k_{\text{max}}$ =
4.0 a.u.-1, and we included basis functions including spherical harmonics up
to $l_{\text{max}}$ = 8. The muffin tin radii used for Fe, Ge, and Te are 2.10
a.u., 2.10 a.u., and 2.63 a.u., respectively. Moreover, we treated 3$s$, 3$p$
and 4$d$ states by local orbitals for Fe and Te, respectively. To extract the
$J_{ij}$ and $\mathbf{D}_{ij}$ parameters, we converge the total energy of
flat spin-spiral states (without SOC and with one-shot SOC) using a
$33{\times}33$ $\mathbf{k}$-point mesh. For conical spin spiral calculations,
we increased the $\mathbf{k}$-point mesh up to $49{\times}49$ since the energy
dispersion amplitude is much smaller than for the flat ones. We model the
effect of a uniform electric field by placing a charged sheet in the vacuum
region of FGT/Ge (see Fig. 1), i.e. using the same methodology and sign
convention of the electric field as in Ref. Paul and Heinze, 2022. We maintain
the charge neutrality of the whole system by adding or removing the same
amount of opposite charge to or from the interface. Finally, we computed the
MAE using the force theorem using a denser $\mathbf{k}$-mesh of 64 ${\times}$
64.

###  II.3 The supercell approach: LCAO-supercell

In this approach, the DMI within the nearest-neighbor approximation for a
specific system is calculated from the energy difference between a clockwise
(CW) and a counterclockwise (CCW) 90∘ spin spiral calculated within a
supercell. $d_{\mathbin{\\!/\mkern-5.0mu/\\!}}$ can be obtained by the
following formula Yang  _et al._ (2015a):

$d_{\mathbin{\\!/\mkern-5.0mu/\\!}}=(E_{\text{CCW}}-E_{\text{CW}})/8\sqrt{3}$
(8)

The corresponding micromagnetic DMI coefficient, $D$,

$D=\frac{3\sqrt{2}d_{\mathbin{\\!/\mkern-5.0mu/\\!}}}{N_{\text{F}}a^{2}}$ (9)

where $a$ and $N_{\text{F}}$ are the lattice constant and number of
ferromagnetic layers, respectively.

Note, that by varying the supercell size one can go beyond nearest-neighbor
DMI approximation. However, the computational effort is much larger than using
the spin spiral approach sketched above.

We used the QATK code Smidstrup  _et al._ (2019) and a $4{\times}1$ supercell
where first neighbor spins rotated as 90∘. A $7{\times}28$ $\mathbf{k}$-point
mesh was adopted for the BZ integration. We included the effect of SOC self-
consistently. Although the supercell approach is straightforward and SOC can
be treated self-consistently, it is only limited to very large wave vectors.

Figure 1:  (a) Sketch of the studied 2D magnetic vdW heterostructure of an
Fe3GeTe2 monolayer (ML) on germanene. Three nonequivalent Fe atoms are
indicated as red spheres: Fe1 (bottom, interface atom), Fe2 (middle), and Fe3
(top). Te and Ge atoms are given by grey and green spheres, respectively. (b)
The electric field is created by a charged sheet located at 5.3 Å above the
FGT. The direction of electric field lines for ${\cal E}<0$ is shown.

##  III Results and discussions

Throughout the paper, we use the following conventions. We use a minus sign in
the Heisenberg exchange terms (cf. Eq. (1)), meaning that $J_{ij}>0$
($J_{ij}<0$) favors ferromagnetic (antiferromagnetic) alignment between
moments $\mathbf{m}_{i}$ and $\mathbf{m}_{j}$. For the DMI constants, a
positive (negative) sign denotes a preferred CW (CCW) rotational sense. In
addition, without specification, we give in our work the in-plane component of
the DMI since the out-of-plane component has shown to be negligible forming
skyrmions in 2D magnets Liang  _et al._ (2020); Du  _et al._ (2022). All
effective quantities, such as the Heisenberg exchange, the DMI, and the MAE,
are measured in meV/unit cell. Note that there are 3 Fe atoms per unit cell of
FGT (Fig. 1).

###  III.1 Geometric properties

We consider FGT heterostructures in which an FGT monolayer is deposited on
germanene (denoted as FGT/Ge in the following). As shown in Fig. 1(a), FGT
adopts the space group (194) P63/mmc and can be seen from the perspective of
the atomistic spin model as a stack of three Fe hexagonal layers in hcp
stacking. In the following, the top, center, and bottom (interface) atoms are
denoted as Fe3, Fe2, and Fe1, respectively.

We have used the QATK code with plane-wave basis sets for the atomic
relaxation. We employed the generalized gradient approximation (GGA),
obtaining a relaxed lattice constant of 4.00 Å for the Fe3GeTe2 monolayer,
which is in good agreement with experimental data ranging between about 3.991
$\sim$ 4.03 Å Deiseroth  _et al._ (2006); Chen  _et al._ (2013). Then,
germanene is matched at the interface with FGT with a lattice mismatch smaller
than 1%. The structures were fully relaxed until the energy and the forces on
each atom were less than 10-8 Ry and 10-4 Ry/Bohr, respectively. We also took
into account van der Waals interactions using semi-empirical dispersion
corrections as formulated by Grimme Grimme  _et al._ (2010). We used the local
density approximation (LDA) for the magnetic calculations with LCAO basis
sets. We did not take into account Hubbard $U$ correction since LDA yields a
magnetic moment of 1.76 $\mu_{\text{B}}$/Fe that compares well with
experiments, as previously pointed out in Ref. Zhuang  _et al._ , 2016; Deng
_et al._ , 2018. The lowest-energy stacking configuration is the one where the
Te atom is right above the center of the hexagonal ring of germanene with an
optimized vdW gap of about 2.86 Å (see Fig. 1b), which agrees well with
previous results He  _et al._ (2021a).

###  III.2 Free-standing FGT monolayer: Heisenberg exchange

Let us first consider the Heisenberg pair-wise exchange in the free-standing
FGT monolayer (Fig. 2). Here, due to the lack of broken inversion symmetry,
Fe3 and Fe1 are equivalent. Therefore, we end up with four different spin
pairs, $\left\\{\text{Fe3-Fe3 (equivalently Fe1-Fe1), Fe2-Fe2}\right\\}$ and
$\left\\{\text{Fe2-Fe1 (equivalently Fe3-Fe2), Fe3-Fe1}\right\\}$, sorted by
intra- and inter-layer exchange interactions. We show in Fig. 2 the Heisenberg
exchange constants ($J_{ij}$) with respect to distance ($R_{ij}$) computed by
the Green’s function method. All $J_{ij}$ decrease quickly with distance, some
showing an oscillatory character. The intralayer interactions (Fig. 2(a,b))
favor antiferromagnetic (AFM) coupling with moderate strength of the nearest-
neighbor exchange constants $J^{1}_{\text{Fe3-Fe3}}=-5.43$ meV and
$J^{1}_{\text{Fe2-Fe2}}=-2.26$ meV while the interlayer exchange (Fig. 2(c,d))
favor much stronger ferromagnetic (FM) coupling with
$J^{1}_{\text{Fe3-Fe1}}=73.16$ meV and $J^{1}_{\text{Fe2-Fe1}}=18.08$ meV. In
particular, the competition between FM and AFM pairs yields geometric
frustration in a triangular sublattice, which can help to stabilize
noncollinear spin structures Leonov and Mostovoy (2015); von Malottki  _et
al._ (2017); Zhang  _et al._ (2020); Rijal  _et al._ (2021). Clearly, the most
significant exchange coupling originates from the interaction between Fe3 and
Fe1 atoms, which are on top of each other (cf. Fig. 1). It drops quickly to a
small negative value, i.e., AFM coupling, at the second nearest distance and
goes up again to a small positive value at the third nearest distance (Fig.
2(d)). Similar results have been reported for FGT bulk Zhu  _et al._ (2021).

Figure 2:  Calculated Heisenberg exchange constants as a function of distance
$R_{ij}$ between sites $i$ and $j$ for a free-standing FGT monolayer obtained
within the LCAO-Green approach. (a,b) Intralayer exchange interactions for
Fe3-Fe3 (Fe1-Fe1) and Fe2-Fe2 spin pairs, respectively. (c,d) Interlayer
exchange interactions for Fe3-Fe2 (Fe2-Fe1) and Fe3-Fe1 spin pairs,
respectively. Note, that $J_{ij}>0$ and $J_{ij}<0$ correspond to favored
ferromagnetic and antiferromagnetic alignment. The distances are given in Å.

In order to quantitatively compare the results presented above with different
DFT approaches, we have calculated the exchange constants also by the FLAPW-
gBT method. In Fig. 3, we present the energy dispersions $E(\mathbf{q})$ of
ﬂat homogeneous spin spirals (per unit cell) for a free-standing FGT
monolayer. The energy dispersions are calculated in scalar-relativistic
approximation, i.e. neglecting SOC, along the high symmetry directions
$\overline{\Gamma\text{M}}$ and $\overline{\Gamma\text{KM}}$ of the two-
dimensional (2D) hexagonal Brillouin zone (BZ). The high-symmetry points
represent special states: the $\overline{\Gamma}$ point corresponds to the FM
state, the $\overline{\text{M}}$ point to the row-wise AFM state, the
$\overline{\Gamma\text{M/2}}$ to the 90∘ spin spiral, and the
$\overline{\text{K}}$ point to the Néel-state with 120∘ between adjacent spins
(Fig. 3(a)).

Figure 3:  Energy dispersions of spin spirals without SOC for free-standing
FGT using the FLAPW-gBT approach. (a) Sketches of the 2D hexagonal Brillouin
zone and of the FM state ($\overline{\Gamma}$ point), the row-wise AFM state
($\overline{\text{M}}$ point), the 90∘ spin spiral (at
$\overline{\Gamma{\text{M}}}$/2), and the Néel-state with 120∘ between
adjacent spins ($\overline{\text{K}}$ point). (b,c) Intralayer spin spirals
for Fe3-Fe3 and Fe2-Fe2 spin pairs, respectively. (c,d) Interlayer spin
spirals for Fe3-Fe1 and Fe3-Fe2 spin pairs, respectively. All energies are
given relative to the energy of the FM state, $E_{\rm FM}$. Black points are
from FLAPW-gBT calculations and the dispersion curve is obtained by fitting
the FLAPW-gBT data to the Heisenberg model (up to seventh NN). Insets show the
spin spiral configurations in the three Fe layers of FGT used to determine the
given Fe-Fe interactions. Note, that to determine the Fe2-Fe2 exchange we used
conical spin spirals in the Fe2 layer (see inset of (c)) and re-scaled the
energy dispersion (see text for details).

We focus first on intralayer exchange interactions. To extract only the
Fe3-Fe3 exchange parameter, we rotate only Fe3 spins by fixing the Fe2 and Fe1
layers to the FM state (inset of Fig. 3(b)). The lowest energy for spin
spirals in the Fe3 layer is at the $\overline{\text{M}}$ point (Fig. 3(b)),
indicating an AFM Fe3-Fe3 coupling in a good agreement with corresponding
LCAO-Green calculations (cf. Fig. 2(a)). The FM state ($\overline{\Gamma}$
point) is about 83 meV/Fe atom higher in energy.

In the case of the Fe2-Fe2 pair, the direct calculation of flat spin spiral
curves propagating only in the Fe2 layer is technically unfeasible due to the
complete quenching of the magnetic moment on the Fe2 atom due to symmetry (for
details, see Appendix B). This shows that the magnetic moment of Fe2 is only
induced by the magnetic moments of Fe1 and Fe3 in the ferromagnetic state.
Therefore, we used conical spin spirals with a small cone angle of
$\theta=10^{\circ}$, i.e. close to the FM state, and transformed the obtained
energy dispersion back to a flat spin spiral:
$E_{\text{flat}}(\mathbf{q})=E_{\text{coned}}(\mathbf{q})/\sin^{2}{\theta}$.
As shown in Fig. 3(c), the energy dispersion $E_{\text{flat}}(\mathbf{q})$ for
the Fe2 layer becomes qualitatively different compared to the Fe3 layer: The
ground state is found to be at the $\overline{\rm K}$ point (Néel state). The
energy difference between FM and Néel state is much smaller than in the case
of Fe3-Fe3, resulting in a weaker AFM Fe2-Fe2 coupling (see Table 3 in
Appendix A).

To calculate the interlayer exchange interaction, i.e. between Fe3 and Fe2
atoms, we rotate the spins of the Fe3 and Fe2 layers simultaneously by fixing
the Fe1 atom to the FM state (Fig. 3(d)). Then, we remove the intralayer
contribution by calculating
$E(\mathbf{q})=E_{\text{Fe3-Fe2}}(\mathbf{q})-E_{\text{Fe3-Fe3}}(\mathbf{q})-E_{\text{Fe2-Fe2}}(\mathbf{q})$
(not shown). The FM state at the $\overline{\Gamma}$ point turns out to be the
state of lowest energy, and the dispersion $E(\mathbf{q})$ rises quickly for
spin spirals with increasing $\mathbf{q}$, resulting in a strong FM coupling
which is consistent with the LCAO-Green result (see Fig. 2(c)).

Table 1: Comparison of the calculated Heisenberg exchange constants (in meV)
for freestanding FGT using the LCAO-Green, the FLAPW-gBT, and the VASP-ML
approach (see Ref. Xu  _et al._ , 2022). The NN intralayer (Fe3-Fe3, Fe2-Fe2)
and interlayer exchange (Fe3-Fe1, Fe3-Fe2) constants are presented. The spin
moments are the averaged magnetic moment which are given in $\mu_{\text{B}}$.

|  Fe3-Fe3 |  Fe2-Fe2 |  Fe3-Fe1 |  Fe3-Fe2 |  $M_{s}$/Fe
---|---|---|---|---|---
LCAO-Green | $-5.43$ | $-2.26$ | 73.16 | 18.08 | 1.79
FLAPW-gBT | $-10.21$ | $-2.02$ | 84.37 | 16.50 | 1.76
VASP-ML Xu  _et al._ (2022) | $-13.4$ | $-5.1$ | 74.1 | 39.4 | 1.66

In contrast, the energy dispersion of spin spirals propagating only in the Fe1
and Fe3 layers looks very different (Fig. 3(e)). The energy minimum is located
at the $\overline{\rm M}$ point, and two maxima are observed in the
$\overline{\Gamma{\rm M}}$ and in the $\overline{\Gamma{\rm K}}$ directions,
indicating an AFM coupling. At first glance, this seems like a qualitative
difference to the result of a strong ferromagnetic Fe3-Fe1 coupling observed
in the LCAO-Green calculation (Fig. 2(d)). However, this is due to the fact
that the Fe3 and Fe1 atoms are on top of each other, and are not
distinguishable by spin spirals which propagate in-plane with respect to the
Fe layers. As a result, if we fit the spin spiral curve in Fig. 3(d), we
obtain the shell-resolved exchange constants between the Fe3 and Fe1 atoms,
i.e. $J_{2}$, $J_{3}$, $J_{4}$, etc., except for the nearest-neighbor term,
$J_{1}$, which arises from direct Fe3-Fe1 coupling. To include $J_{1}$
explicitly, we have performed an additional calculation with an AFM coupling
between the Fe3 and Fe1 atoms. After that, by comparing the total energies
between FM and AFM, we can obtain $J_{1}$ by counting the number of nearest-
neighbor atoms, $E^{\text{Fe3-Fe1}}_{\rm AFM}-E^{\text{Fe3-Fe1}}_{\rm
FM}=2J_{1}+12(J_{2}+J_{3}+J_{4}+J_{6}+J_{7})+24J_{5}$. The obtained value for
the nearest-neighbor Fe3-Fe1 coupling is $J_{1}=83.64$ meV, showing a strong
FM as expected. All values for the calculated magnetic interactions up to the
seventh NN are given in Appendix A.

Our main results on the exchange constants obtained by the two computational
approaches are summarized in Table 1. For comparison, the results from a
machine learning approach (denoted as VASP-ML) calculated by Xu et al. Xu  _et
al._ (2022) are also included. For simplicity, we compare for all Fe-Fe pairs
only the NN exchange constant. LCAO-Green, FLAPW-gBT, and VASP-ML data are in
excellent qualitative agreement. Quantitatively, LCAO-Green yields exchange
parameters very close to the corresponding FLAPW-gBT calculation. In
particular, the Fe2-Fe2 pair agrees surprisingly well ($-2.26$ as compared to
$-2.02$ meV), which indicates that the conical spin spirals used in FLAPW-gBT
are very close to the Green’s function method (FM state).

However, the energies of Xu et al. Xu  _et al._ (2022) have non-negligible
discrepancy. In particular, the Fe2-Fe2 and Fe3-Fe2 pairs are about 50% higher
than our FLAPW-gBT results. Reasons for this quantitative discrepancy may be
the different lattice constants and different relaxations of atomic positions.
Moreover, such a discrepancy might be related to the rather large higher-order
exchange interactions (HOI) which have been obtained for the monolayer of FGT
in Ref. Xu  _et al._ , 2022. Note, that the energy dispersions of spin spirals
include implicitly contributions from HOI which are effectively mapped to our
calculated exchange constants as discussed in detail in Ref. Paul  _et al._ ,
2020.

Figure 4:  (a) Microscopic and (b) micromagnetic DMI coefficients in a
monolayer FGT on germanene as a function of biaxial strain, $\gamma$, obtained
by the LCAO-supercell (black dots) and FLAPW-gBT (red squares) approach. The
inset in (a) shows sketches of the hexagonal Fe layers, the DMI vectors, and
the sign convention.

###  III.3 FGT/Ge: Dzyaloshinskii–Moriya interaction

We now turn our discussion to the DMI, which is plays a central role for the
emergence of non-collinear spin structures such as magnetic skyrmions. The DMI
originates from SOC, and it only exists in materials lacking inversion
symmetry. According to Moriya’s symmetry rules Moriya (1960), since the FGT
has a (001) mirror plane, $\mathbf{D}_{ij}$ for each pair of NN Fe atoms is
perpendicular to their bonds. Therefore, $\mathbf{D}_{ij}$ can be expressed as

$\mathbf{D}_{ij}=d_{\mathbin{\\!/\mkern-5.0mu/\\!}}~{}(\hat{\mathbf{u}}_{ij}{\times}\hat{\mathbf{z}})+d_{\perp}\hat{\mathbf{z}}$
(10)

where $\hat{\mathbf{u}}_{ij}$ being the unit vector between sites $i$ and $j$
and $\hat{\mathbf{z}}$ indicating normal to the plane.

The in-plane component of the DMI constant in the NN approximation,
$d_{\mathbin{\\!/\mkern-5.0mu/\\!}}$, can be directly obtained from the
supercell approach Yang  _et al._ (2015b) (see Eq. (8)). The perpendicular
component, $d_{\perp}$, does not play a significant role in 2D magnets, as
reported in Ref. Liang  _et al._ , 2020; Du  _et al._ , 2022, and is thus
neglected. In the following, without specification, our calculated DMI refers
to the in-plane DMI component, and
$d_{\mathbin{\\!/\mkern-5.0mu/\\!}}>0~{}(d_{\mathbin{\\!/\mkern-5.0mu/\\!}}<0)$
denotes a CCW (CW) rotational sense. For free-standing FGT, the DMI involving
either Fe1 or Fe3 have opposite signs (i.e., chirality) because of the (001)
mirror plane. Upon incorporating germanene in FGT, the inversion symmetry
breaking at the FGT/Ge interface gives rise to an emergent DMI. Since the DMI
is a key ingredient for the formation of skyrmions, methods for efficiently
controlling and manipulating the DMI are essential for designing novel
functional spintronics devices. We will show in the following three ways for
tuning the DMI in FGT/Ge.

Figure 5:  The DMI parameters $D_{ij}$ between different pairs of Fe atoms as
a function of distance for FGT/Ge at $\gamma=0\%$ (black) and $\gamma=-3\%$
(red), calculated by LCAO-Green. (a-c) intralayer and (d-f) interlayer DMI
constants are presented. The positive and negative signs denote CCW and CW
chirality, respectively.

####  III.3.1 Strain

Strain engineering is one of the most commonly used methods to tune the
properties of 2D layers. We study the strain-dependent DMI in FGT/Ge. The in-
plane biaxial strain is defined as

$\gamma=(a-a_{0})/a_{0}$ (11)

where $a$ and $a_{0}$ are the strained and unstrained lattice constants of the
FGT, respectively. For compressive biaxial strain ($\gamma>0$), the in-plane
lattice tends to increase while for tensile biaxial strain ($\gamma<0$) the
in-plane lattice exhibits a decreasing trend.

The calculated microscopic $d_{\mathbin{\\!/\mkern-5.0mu/\\!}}$ and
micromagnetic $D$ of DMI for FGT/Ge are shown in Fig. 4. The DMI is evaluated
quantitatively by calculating the self-consistent total energy of cycloidal
$90^{\circ}$ spin spirals with opposite rotational sense, namely using the
supercell approach (i.e., LCAO-supercell). Both compressive and tensile
strains are considered ranging from $-$3% to 6.25%. Note that FGT has been
demonstrated to be stable under such strain via DFT phonon spectrum
calculations Hu  _et al._ (2020). We emphasize that the value of $a_{0}=4.0$ Å
used in this work was evaluated by the GGA and is slightly larger than the
lattice constant calculated by LDA, $a_{0}=3.91$ Å. If we use the latter as a
reference, the largest compressive strain used in this work becomes less than
$-$1%. At equilibrium ($\gamma=0\%$), we find a moderate DMI of about
$d_{\mathbin{\\!/\mkern-5.0mu/\\!}}=-0.25$ meV, favoring a CW spin rotation.
Its corresponding micromagnetic DMI coefficient is about $|D|$ = 0.36 mJ/m2.
This value is comparable to that of the two FGT heterostructures: FGT/In2Se3
Huang  _et al._ (2022) ($\sim$0.28 mJ/m2) and FGT/Cr2Ge2Te6 Wu  _et al._
(2022) ($\sim$0.31 mJ/m2), which were demonstrated recently as promising vdW
heterostructures to stabilize skyrmions.

Figure 6:  (a) Stacking-dependent DMI constant in NN approximation,
$d_{\parallel}$, in FGT/Ge heterostructures obtained by LCAO-supercell. The
three types of stacking considered are displayed on the right (see text for
details). (b) Strain dependence of $\Delta{\rm z}_{1}-\Delta{\rm z}_{2}$
calculated from the Fe1-Fe2 vertical distance, $\Delta{\rm z}_{1}$, and the
Fe2-Fe3 vertical distance, $\Delta{\rm z}_{2}$. (c) Strain dependence of the
van der Waals gap $\Delta z_{\rm gap}$ (see sketch of stacking 1).

Even more interestingly, we find a significant increase in the DMI when a
small compressive strain is applied (Fig. 4). The extracted $|D|$ is about 1.5
mJ/m2 at $\gamma=-3\%$, which is comparable to the value previously estimated
in FM/HM thin-film systems Yang  _et al._ (2015a); Woo  _et al._ (2016). Such
a large DMI indicates that the FGT/Ge heterostructure can be a promising
interface for stable isolated skyrmions Li  _et al._ (2022). A more detailed
analysis shows that the enhancement of the DMI in FGT/Ge is mainly medicated
by the geometrical change due to the buckled structure of germanene as shown
in Sec. III.3.2.

The results presented above are reproduced by FLAPW-gBT if we calculate the
effective DMI from spin spiral calculations including SOC (red squares in Fig.
4). Here, the effective DMI constant is obtained from fits in the region of
low $|\mathbf{q}|$ around the FM state ($\overline{\Gamma}$ point) where the
energy contribution due to SOC varies linearly with $|\mathbf{q}|$.
Interestingly, the DMI predicted by the LCAO-supercell method is in excellent
quantitative agreement with the one calculated by FLAPW-gBT, showing that the
reported results are robust against different approaches used. This indicates
a possibility of strain control of magnetic skyrmions in these systems. In
contrast, a tensile strain ($\gamma>0$) has almost no effect on the DMI (Fig.
4).

To gain further insight into the local decomposition of the DMI, we show in
Fig. 5 the calculated DMI using the LCAO-Green approach for six possible Fe
pairs in FGT/Ge as a function of distance with and without strain. Evidently,
the external compressive strain ($\gamma=-3$ %) has a significant effect on
the DMI of the FGT monolayer, particularly for pairs connected to the
interface Fe1 atom. When the strain is applied, a strong enhancement of the
DMI favoring a CW (i.e., positive sign) rotational sense is clearly observed
for Fe1-Fe1 and Fe2-Fe1 pairs (Fig. 5c,f), indicating a strong modification of
electronic and magnetic properties at the interface. This is also reflected by
the projected density of states in FGT/Ge (see Fig. 11 in Appendix C).

Quantitatively, the nearest-neighbor DMI changes from 0.88 (0.72) meV to 2.46
(2.84) meV for the Fe1-Fe1 (Fe2-Fe1) pair. A similar behavior is also observed
for the second-nearest neighbor DMI in Fe3-Fe1 (Fig. 5d). The DMI changes from
0.1 meV to $-$0.80 meV, accompanying a sign change. On the other hand, Fe3-Fe3
and Fe3-Fe2 (Figs. 5(a,e) are much less affected by the strain. Moreover, due
to symmetry, the DMI in Fe2-Fe2 (Fig. 5(b)) and the nearest neighbor Fe3-Fe1
(on top of each other) are almost quenched. These results are also in
agreement with those obtained by the TB2J code He  _et al._ (2021b) which uses
a similar methodology.

|  $J_{1}$ |  $J_{2}$ |  $J_{3}$ |  $J_{4}$ |  $J_{5}$ |  $J_{6}$ |  $J_{7}$ |  $J_{8}$ |  $D_{1}$ |  $D_{2}$ |  $D_{3}$ |  $D_{4}$ |  $D_{5}$ |  $D_{6}$ |  $D_{7}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$+0.5$ V/Å | 23.73 | $0.16$ | $-1.61$ | 0.48 | 1.12 | $-0.28$ | 0.00 | $-0.15$ | 0.31 | 0.09 | $-0.01$ | $-0.01$ | $-0.01$ | 0.00 | 0.00
0.0 V/Å | 22.87 | $-0.21$ | $-1.78$ | 0.43 | 1.25 | $-0.31$ | 0.00 | $-0.15$ | 0.23 | 0.10 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
$-0.5$ V/Å | 22.35 | $-0.62$ | $-1.90$ | 0.38 | 1.30 | $-0.37$ | 0.06 | $-0.17$ | 0.14 | 0.08 | $-0.01$ | $-0.01$ | $-0.01$ | 0.00 | 0.00

Table 2: Shell-resolved Heisenberg exchange constants ($J_{i}$) and DMI
constants ($D_{i}$) obtained by fitting the energy contribution to spin
spirals without and with SOC from DFT calculations using the FLAPW method as
presented in Fig. 7 and Fig. 8 for three electric field values. A positive
(negative) sign of $D_{i}$ denotes a preference of CW (CCW) rotating cycloidal
spin spirals. All values are given in meV/unit cell. Figure 7:  Energy
dispersions of spin spirals propagating in all Fe layers (see inset) of FGT/Ge
in scalar-relativistic approximation, i.e. without SOC, calculated via the
FLAPW method along two high symmetry directions
($\overline{\Gamma{\text{K}}{\text{M}}}$ and $\overline{\Gamma\text{M}}$) at
electric fields of ${\cal E}=+0.5$ V/Å (blue), ${\cal E}=0.0$ V/Å (black), and
${\cal E}=-0.5$ V/Å (red). The filled circles represent DFT total energies and
the solid lines are fits to the Heisenberg model (see Table 2 for exchange
constants).

####  III.3.2 Stacking

To study the effect of stacking order on the DMI of the FGT/Ge
heterostructure, three fully optimized representative stacking geometries are
considered here (see sketches in Fig. 6): (i) stacking 1 (the most favorable
stacking geometry), in which the Te atom is right above the center of the
hexagonal ring of germanene, (ii) stacking 2, in which the Ge atom of FGT is
located above the center of the hexagonal ring of germanene, and (iii)
stacking 3, in which the Fe1 and Fe3 atoms of FGT are placed directly above
the center of the hexagonal ring of germanene.

Fig. 6(a) shows the variation of the DMI strength
$d_{\mathbin{\\!/\mkern-5.0mu/\\!}}$ with respect to strain for these three
stacking geometries. We observe a similar behavior for stackings 1 and 2 for
the range of $\gamma$ between about $-$2% to 6%, namely, a clear enhancement
of DMI at compressive strain ($\gamma<0$) and a nearly constant value for a
tensile strain ($\gamma>0$). Interestingly, when $\gamma<-2\%$, the DMI
increases for stacking 1 while it decreases for stacking 2.

To understand the origin of the variation of the DMI, we plot in Fig. 6(b-c)
the geometrical change by strain. As a negative strain is applied, $\Delta{\rm
z}_{1}-\Delta{\rm z}_{2}$ increases rapidly for both stackings 1 and 2, where
$\Delta{\rm z}_{1}$ and $\Delta{\rm z}_{2}$ denote the vertical distances
between Fe1-Fe2 and Fe2-Fe3. This leads to an increase in the degree of
structural asymmetry, resulting in an enhancement of the DMI. On the other
hand, the vdW gap induced by hybridization is decreased by strain and is more
pronounced for stacking 2 than for stacking 1. As analyzed in detail in Ref.
Li  _et al._ (2022), the hybridization effect for FGT/Ge decreases the DMI.
This explains why the DMI for stacking 2 is smaller than for stacking 1 for
$\gamma<-2\%$.

For stacking 3, we observe the most pronounced geometrical changes, the
variation of $\Delta{\rm z}_{1}-\Delta{\rm z}_{2}$ and $\Delta_{\text{gap}}$
are almost two times and three times larger than for stacking 1, respectively.
As a result, even at $\gamma=0\%$ the DMI for stacking 3 is about $-0.63$ meV,
which is 2.5 times higher than the corresponding DMI in stacking 1. However,
only a slight increase of the DMI up to about $-0.78$ meV is observed with
compressive strain. Herein, the increase of
$d_{\mathbin{\\!/\mkern-5.0mu/\\!}}$ is mainly due to the interplay between
the increase of $\Delta{\rm z}_{1}-\Delta{\rm z}_{2}$ and the decrease of
$\Delta_{\text{gap}}$, which have an opposite effect on the DMI. By applying a
tensile strain to the system, the DMI decreases nearly linearly for stacking
3.

Figure 8:  SOC induced energy contribution $E_{\text{DMI}}(\mathbf{q})$ to the
dispersion of flat cycloidal spin spirals propagating in all Fe layers of
FGT/Ge calculated via the FLAPW method along the high symmetry directions
($\overline{\Gamma{\text{K}}{\text{M}}}$ and $\overline{\Gamma\text{M}}$) at
${\cal E}=+0.5$ V/Å (blue), ${\cal E}=0.0$ V/Å (black), and ${\cal E}=-0.5$
V/Å (red). The filled circles represent DFT data and the solid lines are fits
to the DMI term of the atomistic spin model. The inset shows the DMI constant
in NN approximation for the FGT/Ge heterostructure as a function of the
perpendicular electric field, calculated by the LCAO-supercell (black) and the
value of the NN DMI constant obtained in the FLAPW-gBT (red) approach (for all
values see Table 2).

####  III.3.3 Electric field

The FLAPW method in film geometry as implemented in the FLEUR code is used to
include the effect of an external electric field as described in Ref. Weinert
_et al._ , 2009. An out-of-plane electric field is defined by adding a charge
plate in the vacuum and adding the same amount of opposite charge to FGT/Ge
(cf. Fig. 1) to maintain charge neutrality Weinert  _et al._ (2009); Oba  _et
al._ (2015); Paul and Heinze (2022). We chose electric field values of ${\cal
E}=\pm 0.5$ V/Å, which can be applied in scanning tunneling microscopy
experiments Hsu  _et al._ (2017) and allow to write and delete isolated
magnetic skyrmions in ultrathin transition-metal films as demonstrated
experimentally Hsu  _et al._ (2017) and based on atomistic spin simulations
with DFT parameters Paul and Heinze (2022); Goerzen  _et al._ (2022). Note,
that we use the same sign convention for the electric field as in Refs. Paul
and Heinze, 2022; Goerzen  _et al._ , 2022.

We show in Fig. 7 the energy dispersion E($\mathbf{q}$) of homogeneous flat
spin spirals propagating in all Fe layers of FGT/Ge in the scalar-relativistic
approximation, i.e. without SOC. Upon including a negative electric field with
a strength of ${\cal E}=-0.5$ V/Å, the energy rises more slowly at the
$\overline{\Gamma}$ point (FM state) than for ${\cal E}=0$ and the energy
difference with respect to the AFM state ($\overline{\rm M}$ point) and Néel
state ($\overline{\rm K}$ point) decreases. For a positive electric field of
${\cal E}=+0.5$ V/Å, we observe the opposite effect, i.e. a faster rise and
larger energy differences at the high symmetry points. This field effect is
similar to that observed for Fe monolayers on transition-metal surfaces Paul
and Heinze (2022).

By fitting the energy dispersions with and without an applied electric field,
we obtain the field effect on the exchange constants (see Table 2). Note, that
we effectively treat the FGT monolayer in this spin model of the exchange as a
single layer here, i.e. without considering the exchange interactions between
different Fe pairs. We find a nearly linear decrease of the nearest-neighbor
exchange constant, $J_{1}$, by about 6% upon applying an electric field of 1
V/Å. The exchange constants beyond nearest neighbors are also significantly
influenced by the electric field which shows that the exchange frustration
increases for a positive electric field. The effect of the electric field on
the exchange interaction can be explained based on its spin-dependent
screening at the surface as shown previously for ultrathin $3d$ transition-
metal films Oba  _et al._ (2015); Paul and Heinze (2022).

Fig. 8 shows the SOC induced DMI contributions to the energy dispersion of
cycloidal spin spirals in FGT/Ge, $E_{\text{DMI}}({\mathbf{q}})$, under
electric field. When an electric field is applied,
$E_{\text{DMI}}({\mathbf{q}})$ displays the same trend as in zero field, i.e.,
it favors cycloidal spirals with a CW rotational sense, but an electric-field
induced modification in $E_{\text{DMI}}({\mathbf{q}})$ is clearly seen. The
out-of-plane electric field breaks the inversion symmetry and leads to a
drastic change of the nearest-neighbor DMI constant, $D_{1}$, which increases
and decreases by approximately 50% for ${\cal E}=+0.5$ V/Å and ${\cal E}=-0.5$
V/Å, respectively (see Table 2). Additionally, we find that $D_{2}$ remains
almost the same.

In general, when an electric field is applied from negative to positive, the
DMI favors less and less a CW rotational sense. This is also reproduced by the
LCAO-supercell approach (see inset of Fig. 8) in which we even find a change
of sign of the DMI constant at ${\cal E}<-0.37$ V/Å. However, quantitatively,
we note that LCAO-supercell overestimates the DMI compared to FLAPW-gBT under
an electric field. We attribute this quantitative discrepancy to the different
implementations of the electric field and basis sets used in the FLAPW-gBT and
LCAO-supercell methods.

Note, that the electric-field effect on the exchange interaction is opposite
to that on the DMI with respect to the formation of non-collinear spin states.
For ${\cal E}<0$, the energy dispersion without SOC (Fig. 7) rises less
quickly and the exchange frustration increases which is favorable for the
emergence of spin structures such as skyrmions. However, the energy
contribution due to DMI (Fig. 8) shows a shallower energy minimum for ${\cal
E}<0$ and the nearest neighbor DMI constant drops accordingly (cf. Table 2).
This shows that both the effect on the exchange and on the DMI needs to be
taken into account in order to predict electric-field assisted formation of
non-collinear spin structures in line with previous theoretical studies Paul
and Heinze (2022); Goerzen  _et al._ (2022). In contrast, the electric-field
effect on the magnetocrystalline anisotropy energy is small (see next
section).

Figure 9:  (a) Strain and (b) electric-field dependent MAE in the FGT/Ge
heterostructure obtained by the LCAO-SCF (black dotted) and the FLAPW-FT (red
square) approach.

###  III.4 FGT/Ge: Magnetocrystalline anisotropy energy

We define the MAE as the difference in total energies between a configuration
in which the magnetization in the ferromagnetic state is in-plane
($\parallel$) and out-of-plane ($\perp$) with respect to the FGT monolayer

$\text{MAE}=E_{\perp}-E_{\parallel}$ (12)

In Fig. 9, we plot the MAE in the FGT/Ge heterostructure as a function of
strain and electric field. We calculate the MAE using two approaches: (i) The
MAE is defined as the energy difference between self-consistently converged
total energies including SOC, namely LCAO-SCF and (ii) the MAE is taken as the
band energy difference (also known as force theorem) obtained after a one-step
diagonalization of the full Hamiltonian including SOC, starting from the well
converged self-consistent scalar relativistic (without SOC) density/potential,
namely FLAPW-FT.

We find that FGT/Ge has a strong perpendicular MAE of more than 4.62 meV
(about 1.5 meV/Fe). Interestingly, when a mechanical strain is applied, the
MAE can be significantly reduced to 2.08 meV at $\gamma=-3\%$ and 1.44 meV at
$\gamma=6.1\%$, respectively. In general, the two DFT approaches yield a good
qualitative agreement concerning the change of the MAE with strain. From the
local decomposition of MAE evaluated by the grand-canonical formulation Li
_et al._ (2013), we find that Fe1 and Fe3 favor favor out-of-plane anisotropy
while the Fe2 layer favors an in-plane direction of the moments. In contrast,
we find that an applied electric field has a much smaller effect on the MAE of
FGT/Ge (Fig. 9(b)).

##  IV Conclusion

In summary, Fe3GeTe2/germanene has been investigated as a representative 2D
vdW magnetic heterostructure using three current state-of-the-art approaches
to map the ab initio DFT calculations to an atomistic spin model: (i) The
Green’s function approach performing infinitesimal rotations, (ii) the spin
spiral method employing the generalized Bloch theorem for various $\mathbf{q}$
vectors, and (iii) the supercell approach based on the chirality-dependent
total energy difference. We obtain good qualitative agreement for the
Heisenberg exchange and Dzyaloshinskii-Moriya interaction in FGT/Ge using
these three different approaches. We obtain almost quantitative agreement for
the DMI between methods (ii) and (iii), indicating that the nearest-neighbor
approximation is valid in the FGT/Ge heterostructure. Furthermore, we have
studied the electronic and magnetic ground states of the FGT/Ge
heterostructure under biaxial mechanical strain, stacking, and a perpendicular
electric field. We have shown that the strength of the DMI can be
significantly modified via strain and stacking order, tracing its origin to
the geometrical change and hybridization effect. In particular, when a small
compressive strain is applied, the DMI is strongly enhanced while the MAE, in
contrast, is significantly decreased, which allow the possibility of nanoscale
skyrmions at a low magnetic field Li  _et al._ (2022). On the other hand, an
electric field changes the DMI and the exchange constants almost linearly. If
we apply a large enough electric field, we also expect a change of the sign of
the DMI, i.e. a reversed rotational sense of the favored non-collinear spin
structures. The electric field effect on the magnetocrystalline anisotropy is
small for the FGT/Ge heterostructure.

## Acknowledgments

This study has been (partially) supported through the grant NanoX no.
ANR-17-EURE-0009 in the framework of the “Programme des Investissements
d’Avenir”. S. Ha, T. D. and S. He. gratefully acknowledge financial support
from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through SPP2137 “Skyrmionics” (project no. 462602351). This work was performed
using HPC resources from CALMIP (Grant 2022-[P21008]). D. Li thanks Felix
Nickel for valuable discussions.

## Appendix A: Fitted Heisenberg exchange parameters for free-standing FGT

In Table 3, we present the fitted parameters up to the seventh nearest
neighbors for free-standing FGT. Note that, in the main text, we only
benchmarked the NN exchange parameter.

Figure 10:  Magnetic moments of Fe1 (Fe3) and Fe2 atom in a freestanding FGT
monolayer as a function of the $\mathbf{q}$-vector for flat ($\theta=\pi/2$)
and conical (with a small cone angle of $\theta=\pi/20$) spin spirals
calculated via the FLAPW method. Table 3: Shell-resolved Heisenberg exchange
constants, $J_{i}$, for $i$-th neighbors for the different Fe-Fe pairs in FGT
fitted from the spin spiral DFT calculations via the FLAPW method presented in
Fig. 2. All values are given in meV.

|  $J_{1}$ |  $J_{2}$ |  $J_{3}$ |  $J_{4}$ |  $J_{5}$ |  $J_{6}$ |  $J_{7}$
---|---|---|---|---|---|---|---
Fe3-Fe3 (Fe1-Fe1) |  $-$10.42 |  $-$0.30 | 1.28 | 0.19 |  $-$0.28 | 0.03 | 0.04
Fe2-Fe2 |  $-$2.02 | 0.50 |  $-$0.53 |  $-$0.20 |  $-$0.01 | 0.03 |  $-$0.38
Fe3-Fe2 (Fe2-Fe1) | 16.50 | 5.42 |  $-$1.23 |  $-$0.82 | 1.01 | 1.11 | 0.50
Fe3-Fe1 | 83.47 |  $-$6.40 | 4.13 | 6.86 |  $-$1.08 | 1.09 | 1.25

## Appendix B: Quenching of magnetic moment

Spin spirals can be characterized by their reciprocal spin spiral vector
$\mathbf{q}$, which determines the propagation direction of the spiral. For a
rotation axis along the $z$ direction, their magnetization is defined as

$\mathbf{m}_{i}=\begin{pmatrix}\cos(\mathbf{q}\cdot\mathbf{R}_{i})\sin\theta\\\
\sin(\mathbf{q}\cdot\mathbf{R}_{i})\sin\theta\\\ \cos\theta\end{pmatrix}$ (13)

where $\mathbf{R}_{i}$ is the position of site $i$ and $\theta$ is the cone
angle. For the special value $\theta=\pi/2$, we obtain flat spin spirals.

Figure 11:  Calculated spin-averaged PDOS on (a) Fe1, (b) Fe2, and (c) Fe3
atoms in FGT/Ge bilayer in equilibrium, i.e. $\gamma=0\%$ (black), and under a
compressive strain of $\gamma=-3\%$ (red) obtain by the LCAO-Green approach.
The Fermi energy is set to zero.

We show in Fig. 10 the variation of magnetic moments along $\mathbf{q}$
vectors for (a) flat and (b) conical spin spirals for Fe2-Fe2 spin spirals in
a freestanding FGT monolayer, where we rotate only the spins of the Fe2 atoms
by fixing other two layers to the FM state. For flat spin spirals (Fig.
10(a)), only the Fe3 and Fe1 layers possess a magnetic moment, while the
moments of the Fe2 atom vanish due to symmetry. On the other hand, we obtain
the expected magnetic moments (close to the FM state) for all three atoms.

## Appendix C: The PDOS in FGT/Ge without and with strain

To further show that the mechanism of strong enhancement of DMI originates
mainly from the Fe1 atom related pairs (see Fig. 5), the PDOS of Fe atoms in
FGT/Ge under a strain of $\gamma=-$3% and 0% are analyzed in Fig. 11. The
largest difference between the two cases originates from the Fe1 atom, which
is located at the interface.

## Appendix D: Comparison of FLEUR and QATK

We plot in Fig. 12 the total DOS for FGT/Ge calculated by the FLEUR code
(left) and by the QATK code (right) at different electric fields. We note that
the effect of the electric field on the PDOS is much more pronounced in QATK
than in FLEUR, leading to the quantitative difference in the variation of the
DMI with electric field observed in Fig. 8 (inset).

Figure 12:  Total DOS of FGT/Ge obtained via (a) the FLEUR code and (b) QATK
at electric fields of ${\cal E}=+0.5$ V/Å (blue), ${\cal E}=0.0$ V/Å (black),
and ${\cal E}=-0.5$ V/Å (red).

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11institutetext: Computer Science, Jacobs University Bremen, Germany 11email:
<EMAIL_ADDRESS>22institutetext: Mathematics, Kazan State University, Russia, 22email:
<EMAIL_ADDRESS>
# Publishing Math Lecture Notes as Linked Data

Catalin David 11 Michael Kohlhase 11 Christoph Lange 11 Florian Rabe 11 Nikita
Zhiltsov and Vyacheslav Zholudev 2211

###### Abstract

We mark up a corpus of LaTeX lecture notes semantically and expose them as
Linked Data in XHTML+MathML+RDFa. Our application makes the resulting
documents interactively browsable for students. Our ontology helps to answer
queries from students and lecturers, and paves the path towards an integration
of our corpus with external sites.

##  1 Application: Computer Science Lecture Notes

Over the last seven years, the second author has accumulated a large corpus of
teaching materials, comprising more than 2,000 slides, about 1,000 homework
problems, and hundreds of pages of course notes, all written in LaTeX. The
material covers a general first-year introduction to computer science,
graduate lectures on logics, and research talks on mathematical knowledge
management. This situation is typical for educators and researchers and
represents the state of the art in mathematics, physics, computer science, and
engineering: LaTeX has proven suitable for writing high-quality lecture notes
and publishing them as PDF. However, in our educational setting, we would like
to benefit from the much larger degree of interactivity that screen reading
and e-books support. For example, while reading notes students want to
directly look up the meaning of a symbol (e. g. $\vDash$) in a formula, or
examples for a difficult concept (e. g. structural induction). They may want
to select advanced material for self-study from the whole body of lecture
notes, based on the topics covered in the lecture. They want to use a search
engine to find related material in other universities’ online course notes, on
mathematical web sites, or Wikipedia. Lecturers want to query their repository
for document parts reusable in an upcoming lecture, given the prerequisites
students are expected to meet and the material that has already been covered.
In a course for a special audience, e. g. mathematics for physicists, they
want to draw examples from that domain even though they are less familiar with
it. They also want to locate didactic gaps, such as concepts without examples,
or unjustified proof steps. These services require semantic annotations in the
lecture notes that are understandable for external search engines. Plain LaTeX
is barely usable for anything _beyond_ on-screen reading and printing. Even
simple semantic annotations are uncommon, rare exceptions are the `\title`
command making its meaning explicit or `\frac{a}{b}` focusing on functional
structure instead of visual layout. This is especially problematic for symbols
in formulæ, which are often overloaded with multiple definitions or
presentable in different notations. $\binom{n}{k}$ can be a vector or a
binomial coefficient, and a French or Russian would write the latter as
$\mathcal{C}^{k}_{n}$. Therefore, we have developed a semantic representation
of mathematical knowledge in LaTeX and a presentation process that preserves
these semantic structures as Linked Data in the output, exposing them to
mashups for interactive exploration, as well as semantic searching and
querying. These are based on an ontology for mathematical knowledge so that
mathematical content can be linked across different repositories.

##  2 Research Background and Related Work

LaTeX’s importance in scientific authoring and its extensibility by macros
have led to semantic extensions enabling modern publishing workflows. SALT
(semantically annotated LaTeX [8]) marks up rhetorical structures and fine-
grained citations in scientific documents. Its markup is not sufficiently
fine-grained for formulæ, and its vocabulary is limited to rhetorics and
citations and not extensible. Our own sTeX offers macros for introducing new
mathematical symbols and using arbitrary metadata vocabularies. Some math
e-learning systems, such as ActiveMath [1] or MathDox [17], use semantic
representations of formulæ and higher-level structures, e. g. proof steps or
course module dependencies, in the standard XML languages OpenMath [5] and
OMDoc [11]. They utilize semantic structures but do not _publish_ them in a
standard representation like RDF, which would promote general-purpose queries
beyond the built-in services and integration with other systems on the web.
The Linking Open Data movement promotes best practices for publishing data on
the web [9], as standalone RDF or embedded into HTML documents as RDFa [2].
Applications include Sindice, an engine that crawls and indexes Linked Data
[19], and the Sparks $O_{3}$ Browser, a mashup that utilizes RDFa annotations
in HTML for interactive browsing [20]. Our interactive documents work
similarly but additionally support annotations in MathML formulæ. MathML has
pioneered embedded annotations long before RDFa, albeit with a more limited
scope. Its _parallel markup_ interlinks the rendered appearance and the
semantic structure of mathematical expressions; the meaning of mathematical
symbols is usually defined in lightweight ontologies called OpenMath content
dictionaries [4]. HELM (Hypertext Electronic Library of Mathematics [3])
pioneered representing structures of mathematical knowledge in RDF, e. g. what
mathematical theory introduces a symbol, what of its properties have been
declared or asserted, and how the latter are proved. The HELM ontology has not
gained wide acceptance, though. At the time of its development, there was no
RDFa-like way of embedding RDF into web documents.

##  3 Architecture and Demo

Our architecture publishes semantically enriched LaTeX lecture notes as
XHTML+MathML+RDFa Linked Data. We kept LaTeX as an input language, as it is
familiar to authors and well supported by editors, and as high-quality PDF can
be obtained from it. With sTeX (semantically enhanced TeX), we have introduced
LaTeX macros for marking up the semantic structure of formulæ and documents
[12] and manually annotated our complete corpus using the sTeX plugin for the
Emacs editor. One can, e. g., declare a symbol union, formally define it, and
make its semantic representation `\union{A,B,C}` expand to `A\cup B\cup C` for
human-readable rendering. There are environments for mathematical statements
and theories, e. g. `\begin{example}[for=union]`. LaTeXML transforms this into
a semantically equivalent intermediate XML representation, using the standard
XML languages OpenMath for formulæ [5] and OMDoc for higher-level structures
[11]. Finally, our JOMDoc rendering library [10] generates human-readable
output from this XML – an output that still contains the full semantic
structure as annotations. A custom Java implementation renders formulæ as
parallel markup of Presentation MathML annotated with OpenMath111A proposal
for fully representing formulæ in RDF [16] has not gained wide acceptance.
RDF-based reasoners are often limited to decidable first order logic subsets,
which is insufficient for mathematical applications, and XML has a
straightforward notion of order (e. g. of the arguments of an operator or of a
set constructor).; rendering higher-level structures as XHTML+RDFa [2] is
implemented in XSLT. RDF is extracted from XML by our Krextor XML$\to$RDF
library [15], which generates URIs for all mathematical objects in a document.
It uses our OMDoc ontology (cf. [14]) as a vocabulary for representing all
mathematical structures (e. g. “$d$ is a definition, $e$ is an example for
$d$”) plus full text, inspired by HELM and designed as a more expressive
counterpart of the OMDoc XML schema.

The whole transformation process is integrated into our versioned XML database
TNTBase [22]; see http://kwarc.info/LinkedLectures. TNTBase has a Subversion-
compatible interface making it suitable as a lecture notes repository. The
TeX$\to$XML and XML$\to$RDF transformations are automatically triggered by a
hook upon committing a new revision of an sTeX lecture module. If the
generated OMDoc+OpenMath is not schema-valid, the commit is rejected. On the
other hand, it follows Linked Data best practices and, depending on the MIME
type an HTTP client requests, serves a document as OMDoc, as RDF (only a
structural outline, not the full text and formulæ), or as XHTML+MathML+RDFa.
The latter contains JavaScript code from our JOBAD library for interactive
documents [13, 7], which operationalizes the annotations – Linked Data and
other – in the rendered documents. JOBAD’s definition lookup determines the
OpenMath annotation of the Presentation MathML symbol the user clicked on,
from that obtains the URI of the symbol, and then requests XHTML from that URI
(resulting in the symbol’s declaration and definition), which is then
displayed in a popup. The RDFa annotations are used for making parts of a
document (e. g. steps of a structured proof) foldable, and for displaying the
local neighborhood in the RDF graph (e. g. related examples) in popups; this
is implemented using the rdfQuery library [18], relying on the Linked Data
structure in the latter case. Further third-party services can be integrated
in a mashup style; we have demonstrated this for a unit conversion service
[13, 7]. Besides enabling JOBAD’s services, we have implemented machinery to
load the extracted RDF into a triple store and query it using SPARQL. We also
provide a widget for formulating queries without knowing SPARQL and the OMDoc
ontology. It allows to ask some non-trivial queries, e. g. “find examples for
all concepts from graph theory (about which I’m planning a lecture), assuming
as prerequisites the concepts from formal languages (and their
prerequisites)”. This would yield the parse tree of a context-free language as
an example for the concept “tree” – as operating systems were not among the
prerequisites.

Our demo shows the complete pipeline in action:  (i) annotating a document
with our sTeX Emacs mode,  (ii) committing it to TNTBase,  (iii) automatic
translation to OMDoc, schema validation, and RDF extraction, (iv) loading the
extracted RDF data into a triple store,  (v) retrieving the document in
different representations,  (vi) browsing the XHTML+MathML+RDFa rendering,
(vii) interacting with the Linked Data in it,  (viii) and querying a triple
store . Additionally, we will demonstrate the generation of PDF from the sTeX
sources.

##  4 Conclusion and Outlook

Our architecture makes legacy LaTeX lecture notes available as Linked Data. We
expose these data to external clients but have also implemented services for
interactively exploring the XHTML+MathML+RDFa presentation of our data. We are
also working on preserving some of the semantics in the PDF output, as SALT
does. Evaluation of our enriched lecture notes by the student end users is
planned for the next semester. To the best of our knowledge, we are the first
provider of RDF-based Linked Data in the domain of mathematics and among the
first to operationalize the Linked Data structures of formula markup. Having
successfully transformed more than 300,000 normal, non-semantic LaTeX
documents from arxiv.org to XHTML+Presentation MathML [21] and working on
machinery for automatically annotating them using natural language processing,
we will soon be able to expose even more mathematical knowledge as Linked Open
Data; however, due to the inherent complexity of mathematical knowledge, with
a less formal semantics than manually annotated documents. Our lecture notes
are self-contained so far, but we are now starting to reap the benefits of
Linked Data by linking them to other data sets, e. g. DBpedia [6], whose
mathematical knowledge does not have a semantics as strong as ours, but which
provides abundant informal background knowledge, e. g. about the originators
of mathematical theories. On the other hand, hardly any well-known
mathematical site (e. g. planetmath.org and mathworld.wolfram.com) currently
exposes machine-understandable metadata. We promote our technology, starting
with lightweight RDFa annotation using the OMDoc ontology, as a migration path
towards their integration into a true mathematical Semantic Web.

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