Tuning the magnetic interactions in van der Waals Fe 3GeTe 2heterostructures A comparative study of ab initio methods Dongzhe Li1Soumyajyoti Haldar2Tim Drevelow2and Stefan Heinze2 3

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Tuning the magnetic interactions in van der Waals Fe3GeTe2heterostructures:

A comparative study of ab initio methods

Dongzhe Li,1, ∗Soumyajyoti Haldar,2Tim Drevelow,2and Stefan Heinze2, 3

1CEMES, Universit´

e de Toulouse, CNRS, 29 rue Jeanne Marvig, F-31055 Toulouse, France

2Institute of Theoretical Physics and Astrophysics, University of Kiel, Leibnizstrasse 15, 24098 Kiel, Germany

3Kiel Nano, Surface, and Interface Science (KiNSIS), University of Kiel, Germany

(Dated: February 14, 2023)

We investigate the impact of mechanical strain, stacking order, and external electric ﬁelds on the magnetic in-

teractions of a two-dimensional (2D) van der Waals (vdW) heterostructure in which a 2D ferromagnetic metallic

Fe3GeTe2monolayer 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 Fe3GeTe2monolayer. We ﬁnd that the results between meth-

ods (i) and (ii) are in good qualitative agreement and also with previously reported values. An electric ﬁeld of

E=±0.5V/ ˚

A 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 (MAE) 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/ ˚

I. Introduction

Magnetic skyrmions [1] – 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 mem-

ories [2], logic gates [3], artiﬁcial synapses for neuromor-

phic computing [4], and qubits for quantum computing [5].

The formation of magnetic skyrmions is due to the compe-

tition between the Heisenberg exchange and Dzyaloshinskii-

Moriya interaction (DMI) [6–9] together with the magne-

tocrystalline anisotropy energy (MAE). In particular, the DMI

plays an essential role in stabilizing skyrmions since it fa-

vors 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 chi-

ral magnets [10, 11] and subsequently in epitaxial ultrathin

ﬁlms [12]. Isolated magnetic skyrmions were observed in ul-

trathin transition-metal ﬁlms at low temperatures [13–15] and

at room temperature in magnetic multilayers [16–19], in fer-

rimagnets [20], and in synthetic antiferromagnets [21].

Recently, long-range magnetism was reported in two-

dimensional (2D) materials [22–24]. This provides a promis-

ing alternative avenue for exploring topological spin struc-

tures in atomically thin layers. Several recent experi-

∗dongzhe.li@cemes.fr

ments reported the observation of skyrmions in 2D vdW

heterostructures, such as at an Fe3GeTe2/WTe2interface

[25], in a Fe3GeTe2/Co/Pd multilayer [26], and at an

Cr2Ge2Te6/Fe3GeTe2interface [27]. Moreover, magnetic do-

main walls [28] and nonreciprocal magnons [29] were re-

ported in the Fe3GeTe2surface. The origin of skyrmions in

these systems was attributed to the interfacial DMI. A com-

prehensive material survey has been done by ab initio calcula-

tions to explore the DMI in 2D magnets. The family of mono-

layer Janus vdW magnets has been predicted to possess large

enough DMI to allow stable skyrmions [30–35]. N´

eel-type

magnetic skyrmions were also observed in Fe3GeTe2crystals,

and attributed to the DMI due to oxidized interfaces [36]. In

addition, it has been proposed that skyrmions can be stabi-

lized in 2D vdW multiferroic heterostructures [37], Moir´

e of

vdW 2D magnets [38], and even in centrosymmetric materi-

als [39] induced by exchange frustration. For most 2D mag-

nets, the DMI is absent due to inversion symmetry. It is possi-

ble to break the inversion symmetry by designing various 2D

vdW heterostuctures and applying an electric ﬁeld, or strain

[40, 41]. This indicates the possibility of tuning DMI via ex-

ternal 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 straightfor-

ward, nevertheless, complications can arise in practice. Sev-

eral approaches have been introduced based on different ﬁrst-

principles methods and used by numerous groups to calculate

the DMI for various material classes [9, 30, 42–53]. Unfortu-

nately, mostly without a sufﬁcient cross-check between them.

arXiv:2210.15351v2 [cond-mat.mtrl-sci] 13 Feb 2023

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) [54].

Such comparative studies are crucial to understanding the ori-

gin of skyrmion stability, particularly for the newly discovered

2D magnets.

Here, using ab initio calculations, we compare systemati-

cally three current state-of-the-art approaches to extract mag-

netic interaction parameters in Fe3GeTe2(FGT) based het-

erostructures, namely (i) by the Green’s function method [55],

(ii) by using the generalized Bloch theorem (gBT) [43, 54, 56,

57], or (iii) by using the supercell approach [49]. 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 qualita-

tive agreement. We then focus on the structural and magnetic

properties of the 2D vdW heterostructure of an FGT mono-

layer deposited on germanene under strain, stacking, and elec-

tric ﬁeld. We ﬁnd that a small compressive strain (γ) of about

3% can signiﬁcantly enhance the DMI in FGT heterostruc-

tures 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 (E>1 V/ ˚

A) 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 γ=−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 [58]. For the DMI in FGT/Ge,

the three theoretical approaches are in good qualitative agree-

ment, 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. Dif-

ferent theoretical approaches are carefully benchmarked. We

further investigate the effects of biaxial strain, stacking con-

ﬁguration as well electric ﬁeld on the magnetic interactions in

FGT heterostuctures. Finally, we summarize our main con-

clusions in Sec. IV.

II. Methods and computational details

In order to describe the magnetic properties of FGT het-

erostructures, we use the extended Heisenberg model for the

spins of Fe atoms in the hexagonal structure:

H=−X

Jij (mi·mj)−X

Dij ·(mi×mj)

−X

Ki(mz

i)2(1)

where miand mjare normalized magnetic moments at po-

sitions Riand Rirespectively. The three magnetic interaction

terms correspond to the Heisenberg isotropic exchange, the

DMI, and the MAE, respectively, and they are characterized

by the parameters Jij ,Dij , and Kiin 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 parame-

ters 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 calcula-

tion of magnetic interactions in FGT vdW heterostructures:

(i) The Green’s function method [55, 59] (also known as the

Liechtenstein formula) employing inﬁnitesimal rotations; (ii)

The generalized Bloch theorem (gBT) [56] which allows cal-

culating the total energy of spin-spirals of any wave vec-

tor qin magnetic nanostructures [57]; (iii) The supercell

approach [49] which is straightforward but computationally

heavy due to the comparison of total energies in a supercell

geometry. We performed DFT calculations using two com-

munity ab initio codes which differ in their choice of basis

set: The QuantumATK (QATK) package [60] uses an ex-

pansion of electronic states in a linear combination of atomic

orbitals (LCAO) while the FLEUR code [61] 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 implemen-

tations of DFT. In the following, we denote the three differ-

ent approaches as LCAO-Green, FLAPW-gBT, and LCAO-

supercell for simplicity. Apart from the methods presented

above, there are also other approaches widely used in the

community for calculations of spin-spin interactions, e.g., the

four-state method [62] and the machine learning approach

[63].

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

Throughout this paper, vectors are denoted with bold char-

acters while matrices are represented by bold plus single un-

derline (e.g., Gij ). Moreover, Lrepresent orbital index while

σ= (σx, σy, σz)is spin index.

The variation of total energy due to the spin interactions in

Eq. 1, we obtain the following variation with respect to the

miand mj:

δEij =−2Jij (δmi·δmj)

−2δmiJani

ij δmj

−2Dij ·(δmi×δmj)

(2)

where the ﬁrst term represents the isotropic exchange

(i.e., Heinsenberg), the second term is the the symmetric

anisotropic exchange, where Jani

ij is a 3×3symmetric tensor.

The last term corresponds to the DMI,

Green’s function method treats the local rigid spin rotation

as a perturbation. Using the force theorem, the total energy

variation due to the two-spin interaction between sites iand j

δEij =−2

πZEF

−∞

dE Im TrδHiiGij δHjj Gji(3)

where Hii =δei·σand Gij =G0

ij I+Gij ·σare real-space

Hamiltonian and Green’s function. Here, ei=miis a unit

orientation vector (normalized to 1).

Then, in Eq. 3, if we take trace in both orbital (L) and spin

space (σ), we end up with the following expression:

TrL,σ[δHiiGij δHjj Gji] = −2[G0

ij G0

ji −X

µ∈(x,y,z)

Gµ

ij Gµ

ji]δeiδej

−2X

µ,ν∈(x,y,z)

δeµ

i(Gµ

iiGν

ji +Gµ

ij Gν

ji)δeν

−2Dij ·(δei×δej)]

(4)

To simplify for the notation of Eq. 4, we deﬁne a central

quantity for the Green function method, namely the Amatrix,

which has a 4×4size as follows.

Aµν

ij =−1

4πZEF

−∞

dE TrL[Gµ

ij Gν

ji](5)

where indices µand νrun over 0, x,y, or z.

Finally, comparing Eq. 4 to Eq. 2, the Heisenberg exchange

and the DMI can be expressed by using only the Amatrix as

follows.

Jij = 2 Im(A00

ij −Axx

ij −Ayy

ij −Azz

ij )∆ii∆jj

4(6)

Jani

ij = 2 Im(Aµν

ij +Aνµ

ij )∆ii∆jj

4(7)

Dµ

ij = 2 Re(A0µ

ij −Aµ0

ij )∆ii∆jj

4(8)

where ∆ii = (H↑

ii −H↓

ii)is the on-site difference between

the spin-up and -down part of the Hamiltonian matrix.

If we neglect SOC, the DMI vanishes, Gx=Gy= 0,

G0= 1/2(G↑+G↓),Gz= 1/2(G↑−G↓), we arrive

at the original Liechtenstein-Katsnelson-Antropov-Gubanov

(LKAG) formula [55] which was proposed in 1987,

Jij =−1

4πZEF

−∞

dE Im Trh∆iiG↑

ij ∆jj G↓

jii(9)

where Gij becomes hermitian.

Note that the derivations above are general for orthogonal

and non-orthogonal basis sets. Please refer to Ref. [64] for a

detailed demonstration within the non-orthogonal basis set.

Our Green’s function calculations were performed using

QATK [60] in two steps: (i) We performed LCAO-DFT cal-

culations with SOC in order to construct the tight-binding like

Hamiltonian matrix Hij and the overlap matrix Sij . (ii) The

magnetic exchange parameters were evaluated as described

above by Eqs. (3-9). For LCAO-DFT calculations on FGT

monolayers and on FGT/Ge heterostructures, the energy cut-

off for the density grid sampling was set to 150 Hartree, and a

28×28 k-point mesh was adopted for the Brillouin zone (BZ)

integration. For magnetic exchange calculations, we used a

much denser k-point mesh of 48×48, 60 circle contour points,

and 13th nearest neighbors (NN) in order to obtain accurate

numerical integration. Using these parameters, we extracted

Jij and Dij 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 non-

collinear spin structures.

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

The second approach employs the FLAPW method as im-

plemented in the FLEUR code [61] and is based on the gen-

eralized Bloch theorem (gBT) [56, 57]. It allows consider-

ing spin-spirals of any wave vector qfor systems without

SOC. We ﬁrst self-consistently compute within the scalar-

relativistic approximation the energy dispersion, Ess(q), of

homogeneous ﬂat spin spirals [57] which are characterized by

a wave vector qand an angle φ=q·Rbetween adjacent

magnetic moments separated by lattice vector R.

As a second step, the DMI is computed within ﬁrst-order

perturbation theory on the self-consistent spin spiral state [43,

46, 54]. The energy variation δk,ν (q)of these states due to

the SOC Hamiltonian can be written as

δk,ν (q) = hΨk,ν (q)|HSOC |Ψk,ν (q)i,(10)

where |Ψk,ν (q)iare the self-consistent solutions in the scalar-

relativistic approximation, kis the Bloch vector, and νis the

band index. By integration over the Brillouin zone and sum-

mation over all occupied bands νthis gives the total energy

contribution for spin spirals due to SOC denoted as EDMI(q).

We map the energy dispersion in the scalar-relativistic ap-

proximation, ESS(q), and the energy contribution to spin

spirals due to DMI, EDMI(q), to the atomistic spin model,

Eq. (1), in order to extract the exchange constants, Jij , and

the magnitudes of the Dij , respectively. One of the key ad-

vantages of the gBT approach is that even incommensurate

spin spirals and those with a large qcan be treated very ef-

ﬁciently in the chemical unit cell, i.e., without the need for

large supercells.

We used a cutoff parameter for the FLAPW basis functions

of kmax = 4.0 a.u.−1, and we included basis functions includ-

ing spherical harmonics up to lmax = 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 3s, 3pand 4dstates by

local orbitals for Fe and Te, respectively. To extract the Jij

and Dij parameters, we converge the total energy of ﬂat spin-

spiral states (without SOC and with one-shot SOC) using a

33×33 k-point mesh. For conical spin spiral calculations,

we increased the k-point mesh up to 49×49 since the energy

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TuningthemagneticinteractionsinvanderWaalsFe3GeTe2heterostructures:AcomparativestudyofabinitiomethodsDongzheLi,1,SoumyajyotiHaldar,2TimDrevelow,2andStefanHeinze2,31CEMES,Universit´edeToulouse,CNRS,29rueJeanneMarvig,F-31055Toulouse,France2InstituteofTheoreticalPhysicsandAstrophysics,UniversityofKiel...

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Tuning the magnetic interactions in van der Waals Fe 3GeTe 2heterostructures A comparative study of ab initio methods Dongzhe Li1Soumyajyoti Haldar2Tim Drevelow2and Stefan Heinze2 3

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