# Large magneto-optical effect and magnetic anisotropy energy in two-
dimensional metallic ferromagnet Fe3GeTe2

Ming-Chun Jiang  Department of Physics and Center for Theoretical Physics,
National Taiwan University, Taipei 10617, Taiwan  Guang-Yu Guo
<EMAIL_ADDRESS>Department of Physics and Center for Theoretical
Physics, National Taiwan University, Taipei 10617, Taiwan  Physics Division,
National Center for Theoretical Sciences, Taipei 10617, Taiwan

###### Abstract

The recent discovery of long-range magnetic orders in atomically thin
semiconductors Cr2Ge2Te6 and CrI3 as well as metal Fe3GeTe2 has opened up
exciting opportunities for fundamental physics of two-dimensional (2D)
magnetism and also for technological applications based on 2D magnetic
materials. Due to their unique metallic nature, atomically thin Fe3GeTe2 films
exhibit fascinating properties such as Fermi level-tunable Curie temperature,
magnetic anisotropy energy and anomalous Hall effect, thus offering valuable
applications for, e.g., voltage-controlled 2D spintronics at room temperature.
Nevertheless, to exploit these 2D metallic magnets, the mechanisms that
control their physical properties should be well understood. In this paper,
based on systematic first principle density functional theory calculations, we
study two relativity-induced properties, namely, magnetic anisotropy energy
(MAE) and magneto-optical (MO) effects, of multilayers [monolayer, bilayer,
trilayer, fourlayer and quintlayer] and bulk Fe3GeTe2 and also their
connections with the underlying electronic structures of the materials.
Firstly, all the considered Fe3GeTe2 structures are found to prefer the out-
of-plane magnetization and have gigantic MAEs of $\sim$3.0 meV/f.u., being
about 20 and 6 times larger than 2D ferromagnetic semiconductors Cr2Ge2Te6 and
CrI3, respectively. This gigantic perpendicular anisotropy results from the
large magnetocrystalline anisotropy energy (MCE) which is ten times larger
than the competing magnetic dipolar anisotropy energy (MDE) which always
favors an in-plane magnetization. The giant MCEs are attributed to the large
Te $p_{x,y}$ orbital density of states near the Fermi level and also to the
topological nodal point just below the Fermi level at the K points in the
Brillouin zone. Secondly, 2D and bulk Fe3GeTe2 also exhibit strong MO effects
with their Kerr and Faraday rotation angles being comparable or even larger
than that of best-known MO materials such as PtMnSb, Y3Fe5O12 and Bi3Fe5O12.
The features in the Kerr and Faraday rotation spectra are almost thickness-
independent although the Kerr rotation angles increase monotonically with film
thickness. The strong MO Kerr and Faraday effects are found to result from the
large MO conductivity (or strong magnetic circular dichroism) in these
ferromagnetic materials. In particular, the calculated MO conductivity spectra
are one order of magnitude larger than that of Y3Fe5O12. The calculated MO
conductivity spectra are analysed in terms of the symmetry of the band states
and dipole-allowed optical transitions at high symmetry $\Gamma$, K and K′
points, which further reveal that atomically thin Fe3GeTe2 films with odd
layer-number would exhibit anomalous ferrovalley Hall effect. All these
interesting findings thus suggest that 2D and bulk ferromagnetic Fe3GeTe2 are
promising materials for high density MO and spintronic nanodevices.

##  I INTRODUCTION

Recent material realization of intrinsic magnetism in atomically thin films of
semiconducting Cr2Ge2Te6 [1] and CrI3 [2] as well as metallic Fe3GeTe2 [3, 4]
has created numerous fascinating opportunities for two-dimensional (2D)
magnetism. Fundamentally, the famous Mermin-Wagner theorem [5] dictates that
thermal fluctuation prohibits any long-range magnetic order in isotropic 2D
systems at any finite temperature. The discovery of the long-range
ferromagnetic order in monolayers (MLs) CrI3 [2] and Fe3GeTe2 [3, 4] thus
demonstrates that theories of 2D magnetism need to go beyond the isotropic
Heisenberg model. Magnetism at 2D limit is not only highly desirable for the
fundamental physics but also for the technological applications ranging from
magnetic memories to sensing, to spintronics to novel functionalities based on
2D materials. Among the magnetic 2D materials, few-layer Fe3GeTe2 structures
are unique since they are apparently the only ferromagnetic metal that retains
the long-range magnetic order down to the 2D limit [7, 6], and thus attract
particularly strong attention. For example, owing to its metallic nature,
Curie temperature ($T_{C}$) of trilayer (TL) Fe3GeTe2 was raised from 100 K to
300 K by ionic gating [4], thus offering application potential for voltage-
controlled 2D spintronics at room temperature. Furthermore, it was
demonstrated theoretically that the magnetocrystalline anisotropy energy (MCE)
of ML Fe3GeTe2 is large and also tunable by either tensile strain [8] or
electric gating [9]. Strong layer-dependent anoamalous Hall effect in
atomically thin Fe3GeTe2 was also predicted recently [10]. Interestingly,
unlike 2D semiconductors CrGeTe3 and CrI3 which all have a centrosymmetric
crystalline structure, few-layer Fe3GeTe2 with odd number of MLs have the
broken inversion symmetry (see Table I below). Consequently, atomically thin
Fe3GeTe2 with odd layer-number are expected to exhibit novel properties such
as anomalous valley Hall effect [11] and magnetically tunable second-order
nonlinear optical responses (e.g., second-harmonic generation and bulk
photovoltaic effect) [12].

In this work, we concentrate on two relativity-induced properties of 2D
Fe3GeTe2, namely, magnetic anisotropy energy (MAE) and magneto-optical (MO)
effects. MAE is the energy needed to rotate the magnetization direction from
the easy axis to the hard axis. MAE plays a crucial role in suppressing
thermal fluctuation and thus stabilizes long-range magnetic orders in 2D
systems. It is also an important factor that characterizes a magnetic material
from the application view point. In particular, a thin film with a large
perpendicular magnetic anisotropy may find applications in high density
magnetic data storage. It has two contributions, namely, the
magnetocrystalline anisotropy energy (MCE) and the magnetic dipolar anisotropy
energy (MDE). MCE is due to the difference between the relativistic band
structures for two different magnetic orientations. On the other hand, MDE
originates from the classical magnetic dipole-dipole interaction in the
magnetic solid [13, 14]. Interestingly, MDE is determined solely by the
crystalline structure and geometric shape of the sample [13, 14]. In a layered
material, the MDE always prefers an in-plane magnetization while MCE could
favor either an in-plane or the out-of-plane magnetization [15]. Although the
MAE of bulk and ML Fe3GeTe2 has been studied both experimentally and
theoretically [8, 9, 16], the MAE of atomically thin Fe3GeTe2 films of few-
layer thickness has not been reported yet.

MO effects are manifestations of the interplay between magnetism and light in
magnetic solids [17, 18]. When a linearly polarized light beam hits a magnetic
material, the principal axis of the reflected and transmitted light rotates
with respect to the polarization direction of the incident light. The former
is called the magneto-optical Kerr effect (MOKE) and the latter is known as
the magneto-optical Faraday effect (MOFE). MO effects originate the
simultaneous presence of the relativistic spin-orbit coupling (SOC) and
spontaneous magnetization in the magnetic solid. The spontaneous magnetization
and the SOC result in energy band splitting and thus lead to different
refractive indexes for the right- and left-circularly polarized light. As will
be discussed in the next section, this magnetic circular dichroism (MCD) gives
rise to the MOKE and MOFE. MOKE has been widely applied to study the magnetic
properties of thin films and surfaces [18]. Indeed, in Fe3GeTe2 and other 2D
ferromagnets, long range magnetic orders were discovered by the MOKE and MCD
experiments [4, 3, 2, 1]. Large MOKE and MOFE effects can also be exploited
for fabricating high density MO data storage and MO sensors. [19, 20] On the
other hand, it is known that 3$d$ transition metal alloys that contain heavy
elements such as FePt would have large MO effects due to the large SOC
strength on the heavy element atoms [21, 22]. Since Fe3GeTe2 contains heavy Te
atoms, bulk and 2D Fe3GeTe2 are expected to have large MO effects. However,
there has been no theoretical study on the optical and MO properties of bulk
and few-layer Fe3GeTe2.

In this work, therefore, we carry out a systematic first principles density
functional theory (DFT) study on the magnetic, electronic, optical and MO
properties of monolayer (ML), bilayer (BL), trilayer (TL), fourlayer (FL) and
quintlayer (QL) as well as bulk Fe3GeTe2. Indeed, we find that all the
considered Fe3GeTe2 structures would exhibit strong MOKE and MOFE especially
in the infrared and visible frequency range. Furthermore, we also find that
all the structures have a large MAE, being comparable to that of FePt which
has the largest MAE among the ferromagnetic transition metals and their
alloys. These findings suggest that bulk and few-layer Fe3GeTe2 are promising
ferromagnetic materials for nanoscale MO and spintronic devices.

The rest of this paper is organized as follows. In the next section, a brief
description of the crystalline structure of the considered Fe3GeTe2 structure
as well as the used theoretical methods and computational details is given. In
Sec. III, the calculated magnetic properties and electronic band structures as
well as the optical and MO properties are presented. The possible origins of
the large MAEs are discussed in terms of orbital-decomposed density of states
(DOSs) and also the topological features of the band structure near the Fermi
level. Also, the peaks in the calculated MO conductivity spectra are analyzed
in terms of the symmetry of the energy bands and main interband optical
transitions at high symmetry $\Gamma$ and K points in the Brillouin zone (BZ).
Finally, the conclusions drawn from this work are summarized in section IV.

Figure 1: Structure and Brillouin zone (BZ) of bulk and 2D Fe3GeTe2. (a) Side
view of the hexagonal bulk structure with two Fe3GeTe2 monolayers per unit
cell and (b) the corresponding hexagonal BZ. (c) Top view of one Fe3GeTe2
monolayer whose side view is indicated by the red dashed rectangle in (a). The
$\Gamma$-K-M plane in (b) can be regarded as the corresponding 2D BZ. Note
that in (c), the FeII triangular lattice sits right underneath the Te lattice
and thus cannot be seen here.

##  II STRUCTURES AND METHODS

Bulk Fe3GeTe2 forms a layered hexagonal structure with space group P63/mmc
(No. 194) and point group $D_{6h}$, and its experimental lattice constants are
$a=3.991$ Å and $c=16.33$ Å [23]. Each unit cell contains two weak interacting
Fe3GeTe2 MLs [see Fig. 1(a)]. Each Fe3GeTe2 ML consists of five 2D triangular
atomic layers where two Te lattices sandwich three ABA-stacked Fe lattices
[see Fig. 1(a)] with the Fe and Ge triangular lattices in the central layer
forming a honeycomb lattice [see Fig. 1(c)]. The Fe atoms sit on two
inequivalent sites, namely, the FeI site with $C_{3v}$ site symmetry and the
FeII site with $D_{3h}$ site symmetry [see Fig. 1(a)].

The $ab$ $initio$ electronic structure and structural optimization
calculations are based on the density functional theory with the generalized
gradient approximation (GGA) [24] to exchange-correlation interaction. To
accurately describe the structural properties of layered Fe3GeTe2 structures,
we have included the DFT-D2 vdW correction of Grimme [25] in the present
calculations. The present calculations are performed using the accurate
projector-augmented wave (PAW) method [26] implemented in the Vienna $ab$
$initio$ simulation package (VASP) [27, 28]. The few-layer Fe3GeTe2 structures
are modelled utilizing the slab-superlattice approach with the separations
between the neighboring slabs being about 15 Å. A large plane wave cut-off
energy of 400 eV is used. For the Brillouin zone integrations using the
tetrahedron method [29], $\Gamma$-centered $k$-meshes of 16 $\times$ 16
$\times$ 4 and 16 $\times$ 16 $\times$ 1 are used for bulk and few-layer
Fe3GeTe2, respectively. Since the experimental structural parameters for few-
layer Fe3GeTe2 are unavailable, we have determined theoretically the lattice
constants and atomic positions of both bulk and few-layer Fe3GeTe2. The
calculated lattice constants are listed in Table I, together with the
experimental lattice constants of bulk Fe3GeTe2. Note that the calculated
lattice constants of bulk Fe3GeTe2 agree well with the corresponding
experimental values (within 0.5 %) (see Table I), suggesting that the
structural properties of few-layer Fe3GeTe2 should be well described by the
GGA functional [24] plus the DFT-D2 vdW correction [25].

Table 1: Theoretical lattice constants and crystallographic point group of
bulk and few-layer Fe3GeTe2. Experimental lattice constants of bulk Fe3GeTe2
[23] are also listed in brackets for comparison. Here $d$ is the effective
thickness of few-layer Fe3GeTe2

| Bulk | ML | BL | TL | FL | QL
---|---|---|---|---|---|---
Point Group | D6h | D3h | D3d | D3h | D3d | D3h
$a$ (Å) | 3.995 (3.991a) | 4.010 | 4.010 | 4.010 | 4.010 | 4.010
$c$ (Å) | 16.73 (16.33a) |  |  |  |  |
$d$ (Å) |  | 8.365 | 16.73 | 25.10 | 33.46 | 41.83

aReference [23] (X-ray diffraction experiment at 293K).

To determine the MCE, we first perform the total energy calculations for an
in-plane magnetization (e.g., along the $x$-axis) and the out-of-plane
magnetization (along the $z$-axis). Then, the MCE is given by the total energy
difference $\Delta E_{c}=E^{[100]}-E^{[001]}$. Denser $k$-point meshes of 24
$\times$ 24 $\times$ 6 and 32 $\times$ 32 $\times$ 1 are used for bulk and
few-layer Fe3GeTe2, respectively. Thus-obtained MCEs are converged within 3 %
with respect to the $k$-point meshes used.

Similarly, the MDE is given by the difference in the magnetic dipole
interaction energy ($E_{d}$) between the magnetization along the $x$-axis and
along the $z$-axis. For a ferromagnetic system, in atomic Rydberg units, [13,
14]

$E_{d}=\sum_{qq^{\prime}}\frac{2m_{q}m_{q^{\prime}}}{c^{2}}M_{qq^{\prime}}$
(1)

where the speed of light $c=274.072$ and the so-called magnetic dipolar
Madelung constant

$M_{qq^{\prime}}=\sum_{\bm{R}}\frac{1}{\bm{|}\bm{R}+\bm{q}+\bm{q^{\prime}}\bm{|}^{3}}\left\\{1-3\frac{[(\bm{R}+\bm{q}+\bm{q^{\prime}})\cdot\hat{m}_{q}]^{2}}{\bm{|}\bm{R}+\bm{q}+\bm{q^{\prime}}\bm{|}^{2}}\right\\}$
(2)

where $\bm{R}$ are the lattice vectors, $\bm{q}$ are the atomic position
vectors in the unit cell, and $m_{q}$ is the atomic magnetic moment (in units
of $\mu_{B}$) on site $q$. In a 2D system, all $\bm{R}$ and $\bm{q}$ are in-
plane, and hence the second term in Eq. (2) would vanish for the out-of-plane
magnetization. Thus, the $E_{d}$ for an in-plane magnetization is always lower
than that for the out-of-plane magnetization. Consequently, the MDE always
prefers an in-plane magnetization rather than the out-of-plane magnetization
in 2D magnetic systems [13, 15, 30]. This is a purely geometrical effect and
thus MDE is also called the magnetic shape anisotropy energy.

For a solid with at least threefold rotational symmetry and a magnetization
along rotational $z$-axis, the non-zero components for the optical
conductivity are $\sigma_{xx}$, $\sigma_{zz}$, and $\sigma_{xy}$. We calculate
these three independent elements using the linear-response Kubo formula [31,
32, 33]. Thus, the absorptive parts of these elements are given by

$\sigma^{1}_{aa}(\omega)=\frac{\pi e^{2}}{\hbar\omega
m^{2}}\sum_{i,j}\int_{BZ}\frac{d\bm{k}}{(2\pi)^{3}}\left|p^{a}_{ij}\right|^{2}\delta(\epsilon_{\bm{k}j}-\epsilon_{\bm{k}i}-\hbar\omega),$
(3) $\sigma^{2}_{xy}(\omega)=\frac{\pi e^{2}}{\hbar\omega
m^{2}}\sum_{i,j}\int_{BZ}\frac{d\bm{k}}{(2\pi)^{3}}Im\left[p^{x}_{ij}p^{y}_{ij}\right]\delta(\epsilon_{\bm{k}j}-\epsilon_{\bm{k}i}-\hbar\omega),$
(4)

where $\hbar\omega$ is the photon energy and $\epsilon_{\bm{k}i}$ is the $i$th
band energy at point $\bm{k}$. Summations $i$ and $j$ are over the occupied
and unoccupied bands, respectively. Dipole matrix elements
$p^{a}_{ij}=\left\langle\bm{k}j\left|\hat{p}_{a}\right|\bm{k}i\right\rangle$,
where $\hat{p}_{a}$ denotes the Cartesian component $a$ of the dipole
operator, are obtained from the relativistic band structure within PAW
formalism [34]. The integration is over the whole Brillouin zone using the
linear tetrahedron method [35]. The dispersive part of the optical
conductivity elements can be obtained from its corresponding absorptive parts
using the Kramers-Kronig relations,

$\sigma^{2}_{aa}(\omega)=-\frac{2\omega}{\pi}P\int^{\infty}_{0}\frac{\sigma^{1}_{aa}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}}d\omega^{\prime},$ (5)
$\sigma^{1}_{xy}(\omega)=\frac{2}{\pi}P\int^{\infty}_{0}\frac{\omega^{\prime}\sigma^{2}_{xy}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}}d\omega^{\prime},$ (6)

where $P$ denotes the principal value.

For a bulk magnetic material, the complex polar Kerr rotation angle is given
by [21, 36]

$\theta_{K}+i\epsilon_{K}=\frac{-\sigma_{xy}}{\sigma_{xx}\sqrt{1+i(4\pi/\omega)\sigma_{xx}}}.$
(7)

However, for a magnetic thin film on a nonmagnetic substrate, the complex
polar Kerr rotation is given by [37, 38]

$\theta_{K}+i\epsilon_{K}=\frac{-2(Z_{0}d\sigma_{xy})}{(n_{s}+Z_{0}d\sigma_{xx})^{2}-1}$
(8)

where $n_{s}$ is the refractive index of the substrate, $Z_{0}$ is the
impedance of free space, and $d$ stands for the thickness of the magnetic
layer. Since few-layer Fe3GeTe2 were usually deposited on a SiO2/Si substrate
[4, 3], the refractive index of bulk SiO2 ($n_{s}=1.5$) is used here.
Similarly, the complex Faraday rotation angle for a thin film can be written
as [39]

$\theta_{F}+i\epsilon_{F}=\frac{\omega d}{2c}(n_{+}-n_{-}),$ (9)

where $n_{+}$ and $n_{-}$ represent the refractive indices for left- and
right-polarized lights, respectively. The refractive indices are related to
the optical conductivity and the dielectric function via expression
$n_{\pm}=\sqrt{\epsilon_{\pm}}=\sqrt{1+\frac{4\pi
i}{\omega}\sigma_{\pm}}=\sqrt{1+\frac{4\pi i}{\omega}(\sigma_{xx}\pm
i\sigma_{xy})}\approx 1+\frac{4\pi
i}{\omega}(\sigma_{xx}\pm\frac{i}{2}\sigma_{xy})$. As a result,

$\theta_{F}+i\epsilon_{F}\approx-\frac{2\pi d}{c}\sigma_{xy}.$ (10)

Here the real parts of the optical conductivity $\sigma_{\pm}$ can be written
as

$\sigma^{1}_{\pm}(\omega)=\frac{\pi e^{2}}{\hbar\omega
m^{2}}\sum_{i,j}\int_{BZ}\frac{d\bm{k}}{(2\pi)^{3}}\left|\Pi^{\pm}_{ij}\right|^{2}\delta(\epsilon_{\bm{k}j}-\epsilon_{\bm{k}i}-\hbar\omega),$
(11)

where
$\Pi^{\pm}_{ij}=\left\langle\bm{k}j\left|\frac{1}{\sqrt{2}}(\hat{p}_{x}\pm
i\hat{p}_{y})\right|\bm{k}i\right\rangle$. Clearly,
$\sigma_{xy}=\frac{1}{2i}(\sigma_{+}-\sigma_{-})$ and therefore $\sigma_{xy}$
would be nonzero only if $\sigma_{+}$ and $\sigma_{-}$ are different. In other
words, magnetic circular dichroism is the fundamental cause of the nonzero
$\sigma_{xy}$ and hence the MO effect.

Table 2: Calculated total spin magnetic moment ($m^{s}_{t}$), atomic
(averaged) spin magnetic moment ($m^{s}_{FeI}$, $m^{s}_{FeII}$, $m^{s}_{Ge}$,
$m^{s}_{Te}$), and orbital magnetic moment ( $m^{o}_{FeI}$, $m^{o}_{FeII}$,
$m^{o}_{Ge}$, $m^{o}_{Te}$), as well as density of states at the Fermi level
[$D(E_{F})$], magnetocrystalline anisotropy ($\Delta E_{c}$), magnetic dipolar
anisotropy energy ($\Delta E_{d}$) and total magnetic anisotropy energy
($\Delta E_{ma}=\Delta E_{c}+\Delta E_{d}$) of bulk and few-layer Fe3GeTe2.
Positive $\Delta E$ values mean that the out-of-plane magnetization is
favored. The spin and orbital moments for two magnetization directions
(m$\parallel$[001] and m$\parallel$[100]) are listed. The related experimental
and theoretical values are listed for comparison.

System |  | $D(E_{F})$ | $m^{s}_{t}$ | $m^{s}_{FeI}(m^{o}_{FeI})$ |  $m^{s}_{FeII}$ ($m^{o}_{FeII}$) |  $m^{s}_{Ge}$ ($m^{o}_{Ge}$) |  $m^{s}_{Te}$ ($m^{o}_{Te}$) |  $\Delta E_{c}$ ($\Delta E_{d}$) | $\Delta E_{ma}$
---|---|---|---|---|---|---|---|---|---
|  | (1/eV/f.u.) | ($\mu_{B}/$f.u.) | ($\mu_{B}$/atom) | ($\mu_{B}$/atom) | ($\mu_{B}$/atom) | ($\mu_{B}$/atom) | (meV/f.u.) | (meV/f.u.)
bulk |  m$\parallel$[001] | 3.25 | 6.29 | 2.41 (0.075) | 1.53 (0.030) | -0.10 (0.001) | -0.04 (-0.016) | 3.37 (-0.086) | 3.28
|  |  | 4.95a | 2.18b (0.10c) | 1.54b (0.10c) | - | - |  $\sim$3.4d | -
|  m$\parallel$[100] |  | 6.29 | 2.41 (0.084) | 1.54 (0.050) | -0.10 (0.002) | -0.04 (-0.006) | - | -
ML |  m$\parallel$[001] | 3.24 | 6.27 | 2.44 (0.076) | 1.47 (0.033) | -0.10 (0.001) | -0.04 (-0.018) | 3.00 (-0.32) | 2.68
|  |  | - | 1.72e | 1.01e | - | - | 2.76e(-0.11e) | 2.0f, 2.7g
|  m$\parallel$[100] |  | 6.28 | 2.43 (0.085) | 1.49 (0.049) | -0.10 (0.003) | -0.04 (-0.007) | - | -
BL |  m$\parallel$[001] | 3.26 | 6.31 | 2.43 (0.076) | 1.54 (0.033) | -0.10 (0.001) | -0.04 (-0.017) | 3.02 (-0.33) | 2.69
|  m$\parallel$[100] |  | 6.31 | 2.42 (0.085) | 1.54 (0.052) | -0.10 (0.003) | -0.04 (-0.007) | - | -
TL |  m$\parallel$[001] | 3.34 | 6.31 | 2.43 (0.076) | 1.54 (0.032) | -0.10 (0.001) | -0.04 (-0.017) | 3.20 (-0.33) | 2.87
|  m$\parallel$[100] |  | 6.32 | 2.42 (0.085) | 1.55 (0.050) | -0.10 (0.003) | -0.04 (-0.006) | - | -
FL |  m$\parallel$[001] | 3.34 | 6.31 | 2.43 (0.076) | 1.54 (0.032) | -0.10 (0.001) | -0.04 (-0.017) | 3.22 (-0.32) | 2.90
|  m$\parallel$[100] |  | 6.32 | 2.43 (0.084) | 1.55 (0.051) | -0.10 (0.003) | -0.04 (-0.007) | - | -
QL |  m$\parallel$[001] | 3.15 | 6.32 | 2.43 (0.076) | 1.54 (0.032) | -0.10 (0.001) | -0.04 (-0.017) | 3.30 (-0.32) | 2.98
|  m$\parallel$[100] |  | 6.32 | 2.43 (0.084) | 1.55 (0.051) | -0.10 (0.002) | -0.04 (-0.006) | - | -

aReference[40] (SQUID experiment); bReference [41] (Neutron scattering
experiment at 4K); cReference[42] (XMCD experiment: averaged orbital moment);
dReference[16](GGA); eReference [8](LDA);fReference [4](RMCD
experiment);gReference [9](LDA)

##  III RESULTS AND DISCUSSION

###  III.1 Magnetic moments and magnetic anisotropy energy

Since both bulk and few-layer Fe3GeTe2 have been experimentally found to be
ferromagnetic [3, 4, 23, 40], we consider only the ferromagnetic configuration
in this paper. The calculated spin and orbital magnetic moments of the
considered Fe3GeTe2 structures are listed in Table II, together with related
previous experimental and theoretical results. First of all, Table II shows
that calculated magnetic moments in few-layer Fe3GeTe2 hardly depend on their
thickness (i.e., the number of MLs) and also they are very close to that in
bulk Fe3GeTe2. This can be expected from the fact that the interlayer
interaction is weak in the considered Fe3GeTe2 structures. As mentioned above,
in all the considered systems there are two inequivalent Fe sites (FeI and
FeII) with different site symmetries (see Fig. 1). Interestingly, Table II
shows that in all the considered systems, FeI and FeII have rather different
spin magnetic moments, and the difference is nearly as large as 0.9 $\mu_{B}$.
For example, the spin magnetic moments of FeI and FeII in bulk Fe3GeTe2 are
2.41 and 1.53 $\mu_{B}$, respectively. In the ionic picture, the valence state
of Fe3GeTe2 could be written as (FeII2+)(FeI3+)2Ge4-(Te2-)2 [4] and thus we
would expect FeI ($d^{5}$) and FeII ($d^{6}$) to have the spin moments of 5.0
and 4.0 $\mu_{B}$, respectively. The fact that calculated spin moments are
significantly smaller than these values, supports the notion that these
Fe3GeTe2 structures are itinerant ferromagnets [3, 4, 8].

We note that these theoretical spin magnetic moments are in good agreement
with the neutron scattering data [41] (Table II). This indicates the validity
of the GGA functional used here for describing the magnetic properties of the
considered Fe3GeTe2 structures. We can expect that the total magnetic moment
in these Fe3GeTe2 structures comes mostly from the Fe atoms, and this would
result in a total magnetic moment of 6.35 $\mu_{B}$/f.u. The small difference
between this value and the calculated total spin moment is due to the small
spin moments of Te and Ge which are antiparallel to that of Fe (see Table II).
We also notice that the calculated total magnetic moment is significantly
larger than that from the SQUID magnetization measurement (4.95
$\mu_{B}$/f.u.) [40]. This may indicate that the samples used [40] were
nonstoichiometric or contained defects and Fe vacancies. Due to the well-known
crystal-field quenching, the calculated orbital magnetic moments of the Fe
atoms (e.g., 0.076 $\mu_{B}$/Fe for FeI and 0.033 $\mu_{B}$/Fe for II in bulk
Fe3GeTe2) are much smaller than the spin magnetic moments. The averaged
calculated orbital moment of the Fe atoms is $\sim$0.06 $\mu_{B}$/Fe, being
smaller but in the same order of magnitude as that (0.10 $\mu_{B}$/Fe) from
the XMCD measurement [42]. It is known that the GGA and LDA functionals would
give rise to too small orbital moments by up to 40 % (see, e.g., [44, 43] and
references therein). Nonetheless, these discrepancies between the experiment
and theory can be largely removed by including the so-called orbital
polarization correction in the electronic structure calculations (see, e.g.,
[44, 43] and references therein). Interestingly, Te orbital magnetic moments
are comparable to Te spin magnetic moments and furthermore depend strongly on
the magnetization direction (Table II). This is because the SOC in Te atoms is
much stronger than in Fe and Ge atoms.

We present the calculated magnetic anisotropy energies ($\Delta E$) in Table
II. By definition ($\Delta E=E^{[100]}-E^{[001]}$), a positive $\Delta E$
value indicates an out-of-plane magnetization easy axis [i.e., the
perpendicular magnetic anisotropy (PMA)]. Strikingly, Table II shows that all
the considered Fe3GeTe2s structures have a very large PMA, being $\sim$3.0
meV/f.u. or $\sim$1.0 meV/Fe. These $\Delta E_{ma}$ values not only are
several times larger than that of 2D ferromagnetic semiconductors Cr2Ge2Te6
($\sim$0.1 meV/f.u.) [15] and CrI3 ($\sim$0.5 meV/f.u.) [30] but also are
comparable to that of heavy metal magnetic alloys such as FePt (2.75 meV/Fe)
[45], which is known to have the largest MAE among the transition metal
alloys. This large PMA would strongly suppress the magnetic fluctuations in
these 2D materials and thus result in a higher ferromagnetic ordering
temperature ($T_{c}$) than 2D ferromagnetic semiconductors CrI3 [2] and
Cr2Ge2Te6 [1]. We note that the calculated MAE (2.9 meV/f.u.) of ML Fe3GeTe2
is in good agreement with the RMCD experimental data (2.0 meV/f.u.) [4]. The
largeness of the PMA plus its electric tunability [4] suggests that 2D
ferromagnetic metallic Fe3GeTe2 would have promising applications in high-
density magnetic data storage and other spintronic devices.

Table II indicates that the MAE per f.u. increases slightly as one moves from
ML to BL and then TL. However, the MAE per f.u. remains unchanged as the film
thickness is further increased, e.g., to that of FL and QL (Table II). As
mentioned before, the MAE consists of two competing contributions, namely, the
MCE, which prefers PMA, and MDE, which always favors an in-plane
magnetization. The MDE in 2D Fe3GeTe2 is in the same magnitude as that in 2D
CrI3 and Cr2Ge2Te6. However, the magnitude of the MCE in 2D Fe3GeTe2 is nearly
ten times larger than that of the MDE. This results in a gigantic MAE in 2D
Fe3GeTe2. Note that these values are nearly three orders of magnitude larger
than that of ferromagnetic Fe and Ni ( 4 $\mu$eV/f.u.) [46]. Bulk Fe3GeTe2 has
an even larger MAE of 3.28 meV/f.u., simply because its MDE is much smaller
than that in 2D Fe3GeTe2 (Table II). We notice that only one previous study,
based on the local density approximation (LDA), on the MDE has been reported.
[8] Nevertheless, the calculated MDE for ML Fe3GeTe2 in Ref. [8] is nearly
three times smaller than the present calculations (Table II). This is mainly
because the LDA spin magnetic moments reported in [8] are significantly
smaller than the present GGA calculations (see Table II).

Table II also shows that while the spin magnetic moments are generally
independent of the magnetization orientation, the orbital magnetic moments
change significantly as the magnetization is rotated, e.g., from [001] to
[100]. In particular, the Te orbital moment in ML Fe3GeTe2 gets reduced by
nearly 60 % when the magnetization is rotated from [001] to [100], while that
of FeI increases by about 12 % (Table II). This may suggest some correlation
between the MCE and the anisotropy in the orbital magnetization. Indeed, a
previous perturbative theory analysis showed that in elemental transition
metal MLs, the easy axis is along the direction in which the orbital moment is
largest [47]. This was supported by the ab initio calculations for Fe
monolayers imbedded in noble metals [13]. Interestingly, the orbital magnetic
moments on heavy element Te atoms (which have the largest SOC) in all the
considered Fe3GeTe2 structures with the perpendicular magnetization are much
larger than that for an in-plane magnetization, thus indicating a possible
connection with the strong PMA in these systems. Nevertheless, Table II shows
that the Fe orbital moments in these Fe3GeTe2 structures are larger for an in-
plane magnetization than for the perpendicular magnetization, indicating that
the perturbative theory analysis for transition metal monolayers [47] may not
be wholely applicable to the present ternary compounds.

Figure 2: Band structures of ML (left column), BL (middle column) and bulk
(right column) Fe3GeTe2. Spin-polarized scalar-relativistic band structures
(upper row), relativistic band structures with the magnetization along
$z$-axis (middle row) and with an in-plane magnetization along $x$-axis. In
(d-i), the dotted ellipse circles the topological node [gapless: (d,f,g,i) and
gapped: (e,h)]. The Fermi level is at 0 eV.

###  III.2 Electronic structure

To understand the calculated magnetic and optical properties of the Fe3GeTe2
materials, we present the calculated electronic band structures. The band
structures of ML, BL and bulk Fe3GeTe2 are displayed in Fig. 2, and that of
TL, FL and QL Fe3GeTe2 in Fig. S1 in the supplementary material (SM) [48].
Since the band structures of bulk and ML Fe3GeTe2 have already been reported
in several previous papers [4, 8] and are also rather similar to that of TL,
FL and QL Fe3GeTe2 due to the weak interlayer interaction, here we only
summarize the salient features of the calculated band structures.

First of all, all the considered Fe3GeTe2 structures are metallic with
multiple Fermi surface pockets (Fig. 2 and Fig. S1 in the SM [48]) and hence a
large density of states (DOS) at the Fermi level ($E_{F}$) (Table II). In all
the considered systems, there are many hole Fermi surface pockets centered at
the $\Gamma$ point and many electron Fermi surface pockets centered at the K
point in the BZ (Figs. 2 and Fig. S1 in the SM). Secondly, because bulk
Fe3GeTe2 contains two MLs per unit cell and also the interlayer interaction is
weak, the band structure of BL Fe3GeTe2 [Figs. 2(d), 2(e) and 2(f)] are almost
identical to that of bulk Fe3GeTe2 [Figs. 2(g), 2(h) and 2(i)]. The band
structure of ML Fe3GeTe2 is very similar to that of BL Fe3GeTe2 (Fig. 2)
except that the number of the bands is only half of that for the BL.
Similarly, the band structures of TL, FL and QL Fe3GeTe2 are overall nearly
the same as that of the BL except with the increased number of bands (Fig. S1
in the SM).

Figure 3: Scalar-relativistic site-, orbital-, and spin-projected DOS of bulk
Fe3GeTe2. The Fermi level is at 0 eV. Figure 4: Scalar-relativistic site-,
orbital-, and spin-projected DOS of monolayer Fe3GeTe2. The Fermi level is at
0 eV.

Next, we present the total as well as site-, orbital-, and spin-projected DOS
spectra of bulk and 2D Fe3GeTe2 in Figs. 3 and 4 as well as Figs. S2 and S3 in
the SM [48]. Figure 3 shows that for bulk Fe3GeTe2, the lower valence bands
ranging from -5.0 to -2.2 eV result mainly from the hybridization among Fe
$d$, Ge $p$, and Te $p$-orbitals. The upper valence bands and lower conduction
bands (from -2.2 to 1.0 eV) are dominated by Fe $d$-orbitals together with
minor contributions of Te $p$-orbitals around the Fermi level. Furthermore,
Figs. 3(b) and 3(c) show that spin-splitting of Fe $d$ bands is large, being
more than 1 eV and thus indicating strong intra-atomic exchange interaction in
bulk Fe3GeTe2. Interestingly, the local DOS spectra for the FeI and FeII sites
are rather different, being consistent with the rather different magnetic
moments on these sites (Table II). These differences are caused by their
different local environments (coordination number and site symmetry). For
example, the main spin-down FeI $d_{z^{2}}$ DOS peak is very sharp and is
located at -0.8 eV [Fig. 3(b)] while the main spin-down FeII $d_{z^{2}}$ DOS
peak is rather broad and is centered at 0.4 eV [Fig. 3(b)]. This is because
there is no ligand atom lying above or below FeI [Figs. 1(a) and 1(c)] and
thus FeI $d_{z^{2}}$ orbitals form a rather localized narrow band. In
contrast, there is one Te atom sitting right above (or right below) the FeII
atom [Figs. 1(a)]. Thus, FeII $d_{z^{2}}$\- and Te $p_{z}$-orbitals hybridize
and pushes FeII $d_{z^{2}}$-orbital dominated antibonding band above the Fermi
level [Fig. 3(c)].

Table 3: The SOC matrix elements $\langle m\sigma\left|\vec{L}\cdot\vec{S}\right|m^{\prime}\sigma^{\prime}\rangle$ in the $p$-orbital basis. Here $\eta$ is the polar angle and $\phi$ is the azimuthal angle. For the perpendicular magnetization, $\eta=\phi=0$ while for the in-plane magnetization along the $x$-axis, $\eta=\pi/2$ and $\phi=0$.  | $|p_{x};\uparrow\rangle$ | $|p_{y};\uparrow\rangle$ | $|p_{z};\uparrow\rangle$
---|---|---|---
$|p_{x};\uparrow\rangle$ | 0 | $-\frac{i}{2}\cos\eta$ | $\frac{i}{2}\sin\eta\sin\phi$
$|p_{y};\uparrow\rangle$ | $\frac{i}{2}\cos\eta$ | 0 | $-\frac{i}{2}\sin\eta\cos\phi$
$|p_{z};\uparrow\rangle$ | $-\frac{i}{2}\sin\eta\sin\phi$ | $\frac{i}{2}\sin\eta\cos\phi$ | 0
$|p_{x};\downarrow\rangle$ | 0 | $\frac{i}{2}\sin\eta$ | $-\frac{1}{2}(\cos\phi-i\cos\eta\sin\phi)$
$|p_{y};\downarrow\rangle$ | $-\frac{i}{2}\sin\eta$ | 0 | $-\frac{1}{2}(\sin\phi+i\cos\eta\cos\phi)$
$|p_{z};\downarrow\rangle$ | $\frac{1}{2}(\cos\phi-i\cos\eta\sin\phi)$ | $\frac{1}{2}(\sin\phi+i\cos\eta\cos\phi)$ | 0

Let us now discuss the possible origin of the large MCE in the Fe3GeTe2
systems in terms of the Fe $d$-orbital decomposed DOS spectra in, e.g., Fig.
3. Accoring to perturbation theory analysis, the occupied and empty $d$ states
near the Fermi level are coupled by the SOC and thus make most important
contributions to the MCE [55]. Moreover, for the same spin channel, the SOC
matrix elements $\langle d_{xy}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle$ and
$\langle d_{yz}\left|H_{SO}\right|d_{xz}\rangle$ prefer the out-of-plane
anisotropy, while $\langle d_{yz}\left|H_{SO}\right|d_{xy}\rangle$, $\langle
d_{yz}\left|H_{SO}\right|d_{z^{2}}\rangle$, and $\langle
d_{yz}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle$ favour an in-plane anisotropy
[56]. The ratios of these matrix elements are $\langle
d_{xy}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{xz}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{xy}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{z^{2}}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle^{2}$ = 4 : 1 : 1 : 3 : 1. [56]
Figure 3(b) shows that FeI $d_{yz,xz}$ and $d_{xy,x^{2}-y^{2}}$ orbitals
dominate FeI $d$-orbital decomposed DOS spectra in spin-up and spin-down
channels, respectively. Consequently, the SOC matrix elements $\langle
d_{xy}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle$ and $\langle
d_{yz}\left|H_{SO}\right|d_{xz}\rangle$ would make dominating contributions to
the MCE, thereby giving rise to the large PMA in bulk Fe3GeTe2. For the FeII
sites, the situation is more complicated [Fig. 3(c)]. In addition to the
prominent $d_{yz,xz}$ orbitals, there are the pronounced $d_{z^{2}}$ orbital
in both spin channels and also the prominent spin-down $d_{xy,x^{2}-y^{2}}$
orbitals. Consequently, there are competing contributions to the MCE from the
SOC matrix elements of $\langle
d_{xy}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle$ and $\langle
d_{yz}\left|H_{SO}\right|d_{xz}\rangle$, which prefers the PMA, and $\langle
d_{yz}\left|H_{SO}\right|d_{z^{2}}\rangle$, which favors an in-plane
magnetization. However, since the ratios of these matrix elements are $\langle
d_{xy}\left|H_{SO}\right|d_{x^{2}-y^{2}}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{xz}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{xy}\rangle^{2}$ : $\langle
d_{yz}\left|H_{SO}\right|d_{z^{2}}\rangle^{2}$ = 4 : 1 : 1 : 3, we believe
that FeII atoms would make positive contributions to the PMA although they may
be much smaller than FeI atoms.

Figures 3 and 4 show that there are pronounced occupied and unoccupied
$p$-orbital DOSs from heavy Te atoms in the vicinity of $E_{F}$. To examine
the possible contribution of these $p$-orbital states to the MCE, following
[56] we derive the SOC matrix elements in the basis of $p$ orbitals, which are
listed in Table III. Among the nonzero SOC elements, we find that $\langle
p_{x};\uparrow\left|H_{SO}\right|p_{y};\uparrow\rangle$ prefers the out-of-
plane magnetization but $\langle
p_{y}(p_{x});\uparrow\left|H_{SO}\right|p_{z};\uparrow\rangle$ favors an in-
plane magnetization. Furthermore, $\langle
p_{x};\downarrow\left|H_{SO}\right|p_{y};\uparrow\rangle$ prefers an in-plane
magnetization while $\langle
p_{x}(p_{y});\downarrow\left|H_{SO}\right|p_{z};\uparrow\rangle$ has no
preference. Unlike that for $d$ orbitals [56], the expectation values of all
nonzero elements $\langle
p;\uparrow\downarrow\left|H_{SO}\right|p;\uparrow\downarrow\rangle^{2}$ are
identical. Interestingly, Fig. 3 and 4 show that the Te $p_{x}$ and $p_{y}$
make a prominent contribution to the spin-up DOSs near the Fermi level. Since
$\langle p_{x};\uparrow\left|H_{SO}\right|p_{y};\uparrow\rangle$ prefers the
perpendicular anisotropy and also Te atoms have a much stronger SOC than Fe
atoms, the large MCE found in bulk and 2D Fe3GeTe2 may result mainly from the
presence of of pronounced Te $p_{x,y}$-orbital DOSs near the Fermi level.

The site-, orbital- and spin-projected DOS spectra of ML Fe3GeTe2 are shown in
Fig. 4. All the features of the DOS spectra are similar to that of bulk
Fe3GeTe2 (Fig. 3). One pronounced difference is the appearance of FeII
$d_{z^{2}}$ orbital dominated peaks near -3.0 eV in the lower valence bands.
This can be attributed to the lack of the interlayer coupling because there is
no Fe3GeTe2 ML above and below it. The absence of the hybridization among FeII
$d_{z^{2}}$ orbitals from neighboring MLs (via Te $p_{z}$ orbitals) makes FeII
$d_{z^{2}}$ orbitals localized and the corresponding band narrow. On the
contrary, the peaks of FeI $d_{z^{2}}$ orbitals in bulk and ML Fe3GeTe2 are
nearly the same because no ligand atom sits along the $c$ direction. The
site-, orbital- and spin-projected DOSs of BL and TL Fe3GeTe2 are displayed in
Figs. S2 and S3 in the SM [48], where the spectral features fall in between
that of ML and bulk Fe3GeTe2.

Finally, we notice a topological nodal point just below $E_{F}$ at the K point
in the scalar-relativistic band structure of bulk Fe3GeTe2 [Fig. 2(g)]. This
nodal point extends along the K-H line, and thus forms a topological nodal
line. [49] Interestingly, for the perpendicular magnetization, this nodal line
becomes split by a large band gap of $\sim$60 meV when the SOC is switched-on
[Fig. 2(h)]. This results in the upper band being pushed upwards nearly above
$E_{F}$ at the K point and the lower band moving downwards [Fig. 2(h)].
Furthermore, as reported before [49], these SOC-split bands have very large
Berry curvatures with opposite signs near the nodal points, thus leading to
the large observed anomalous Hall effect in bulk Fe3GeTe2. [49] Here we want
to emphasize that this gap-opening at the nodal point also lowers the total
band energy. On the other hand, these nodal points remain gapless when the
magnetization is in-plane and hence the total band energy would remain much
unchanged. Clearly, this would give rise to a significant contribution to the
large MCE in bulk Fe3GeTe2 (see Table II). We notice that all 2D Fe3GeTe2
structures except ML Fe3GeTe2 have such a nodal point at the K point [see Fig.
2 and Fig. S1 in the SM]. Therefore, we may conclude that to some extent, the
large MCE found in bulk and 2D Fe3GeTe2 also originate from the gap-opening of
the topological nodal point at the K point in these structures. We also notice
that in ML Fe3GeTe2, although there is no such a nodal point, the top valence
band at the K point is lowered by $\sim$19 meV when the magnetization is out-
of-plane [Fig. 2(b)] but goes up by 9 meV when the magnetization becomes in-
plane. Clearly, such band movements due to the presence of the SOC and the
change of magnetization direction would also contribute significantly to the
large MCE in ML Fe3GeTe2.

###  III.3 Optical and magneto-optical conductivity

Figure 5: Real (a)[(e)] diagonal and (b)[(f)] off-diagonal, imaginary (c)[(g)]
diagonal components and (d)[(h)] off- diagonal components of the optical
conductivity tensor of bulk [ML] Fe3GeTe2 in the ferromagnetic state with the
out-of-plane magnetization. All the spectra have been convoluted with a
Lorentzian of 0.1 eV to simulate the finite electron lifetime effects.

We present the calculated optical and MO conductivity tensors for bulk and ML
Fe3GeTe2 in Fig. 5, and also for BL and TL as well as for FL and QL Fe3GeTe2
in Figs. S4 and S4 in the SM [48], respectively. First of all, we can see that
all spectra are rather similar. This similarity is due to the weak interlayer
interaction in these materials. Such behaviour has also been reported in other
2D magnetic materials such as few-layer CrI3 [30] and Cr2Ge2Te6 [15].
Therefore, below we will analyze only the main features in the optical and MO
conductivity spectra of ML and bulk Fe3GeTe2 in detail as representatives of
these materials.

Figures 5(a) and 5(e) show that the diagonal elements of the optical
conductivity $\sigma_{xx}$ and $\sigma_{zz}$ differ significantly. Since the
$\sigma_{xx}$ ($\sigma_{zz}$) is the linear optical response of the materials
to an in-plane electric field polarization $E\perp c$ (out-of-plane electric
field polarization $E||c$), such difference indicates a large optical
anisotropy. This can be expected for 2D or quasi-2D materials [15, 30].
Specifically, the absorptive part of the diagonal element ($\sigma^{1}_{xx}$)
for bulk Fe3GeTe2 (Fig. 5(a)) is larger than $\sigma^{1}_{zz}$ in the low-
energy region of 0.0-3.0 eV, but becomes smaller than $\sigma^{1}_{zz}$in the
high-energy region of 3.0-6.0 eV. Compared to that of bulk, ML Fe3GeTe2 has a
similar difference between $\sigma^{1}_{xx}$ and $\sigma^{1}_{zz}$.
Nevertheless, as expected, such difference gets slightly enhanced, because ML
Fe3GeTe2 is a truly 2D material. We notice that this optical anisotropy can be
further understood in terms of the calculated orbital-projected DOS spectra.
Note that the $p_{x,y}$ and $d_{xy,x^{2}-y^{2}}$ ($p_{z}$ and $d_{z^{2}}$)
states can only be excited by $E\perp c$ ($E||c$) polarized light while the
$d_{xz,yz}$ states can be excited by both. Figures 3 and 4 indicate that the
upper valence bands stem mainly from Fe $d$ as well as Ge and Te $p$ orbitals.
In the energy region between -3.0 and 0.0 eV, the overall weight of Fe
$d_{xy,x^{2}-y^{2}}$ orbitals (excited by $E\perp c$) is slightly larger than
that of Fe $d_{z^{2}}$ orbital (excited by $E||c$), thus leading to the
slightly higher peaks of $\sigma^{1}_{xx}$ in the photon energy region below
3.0 eV [Figs. 5(a) and 5(e)]. Further, broad peaks of Te $p_{x,y}$ DOS in the
energy range from -1.0 to 1.0 eV [Figs. 3 and 4(d)] also contribute to the
higher peaks of $\sigma^{1}_{xx}$ below 3.0 eV. On the other hand, significant
FeII $d_{z^{2}}$ and Te $p_{z}$ states appear at lower valance bands from -5.0
to -3.0 eV. This would explain the increase of $\sigma^{1}_{zz}$ ($E||c$)
above 4.4 eV. Interestingly, among all considered structures, ML Fe3GeTe2 has
a particularly sharp Te $p_{z}$ peak [see Figs. 4(c) and 4(d)] due to the lack
of interlayer coupling, as described above. This results in the largest
optical anisotropy in ML Fe3GeTe2.

The calculated real ($\sigma^{1}_{xy}$) and imaginary ($\sigma^{2}_{xy}$)
parts of the off-diagonal optical conductivity elements are displayed in Figs.
5(b) and 5(b) (d) for bulk Fe3GeTe2 and in Figs. 5(f) and 5(h) for ML
Fe3GeTe2. First of all, we note that in the DC-limit ($\omega\rightarrow 0$),
$\sigma^{1}_{xy}(0)$ is actually the anomalous Hall conductivity (AHC). The
AHCs calcuated this way are 233 S/cm, 312 S/cm, 241 S/cm and 287 S/cm for
bulk, ML, BL, and TL Fe3GeTe2, respectively. The AHC value for bulk Fe3GeTe2
agrees quite well with the experimental AHC value of $\sim$360 S/cm [57] and
also previous theoretical one of $\sim$287 S/cm [58]. Secondly, as expected,
the overall features in the $\sigma_{xy}$ spectra for bulk and ML Fe3GeTe2 are
similar. For example, for both structures, the $\sigma_{xy}(\omega)$ spectra
oscillates significantly with several high peaks. Prominent peaks occur mostly
between 0.0 and 4.4 eV and beyond 4.4 eV, the magnitude of
$\sigma_{xy}(\omega)$ get significantly reduced, indicating weak magnetic
circular dichroism. Specifically, $\sigma^{1}_{xy}$ for ML Fe3GeTe2 has
pronounced positive peaks at 0.4 eV and 3.8 eV as well as well as a negative
peak at 2.8 eV [Fig. 5(f)]. On the other hand, $\sigma^{2}_{xy}$ for ML
Fe3GeTe2 has large positive peaks at 0.6 eV and 2.6 eV as well as a large
negative peak at 3.6 eV [Fig. 5(h)]. In comparison, the $\sigma_{xy}(\omega)$
spectra of bulk Fe3GeTe2 have peak positions and shapes being rather similar
to that of ML but with significantly reduced magnitudes [Figs. 5(b) and 5(d)].

Figure 6: Relativistic site- and orbital-projected band structures of bulk
ferromagnetic Fe3GeTe2. The Fermi level is at 0 eV. The main interband
transitions at the $\Gamma$ and K point, as well as the corresponding peaks in
the $\sigma^{2}_{xy}$ spectrum in Fig. 5(d) are indicated by red and blue
arrows, respectively. Also, the optical transitions in the vicinity of the
$\Gamma$ point are indicated by the magenta arrows, namely, A1 is at
(0.04,0.04,0) $2\pi/a$ and A2 is at (0.03,0.03,0) $2\pi/a$.  Figure 7:
Relativistic site- and orbital-projected band structures of ferromagnetic
Fe3GeTe2 ML. The Fermi level is at 0 eV. The main interband transitions at the
$\Gamma$, K and K′ point, as well as the corresponding peaks in the
$\sigma^{2}_{xy}$ spectrum in Fig. 5(h) are indicated by red and blue arrows,
respectively. Also, the optical transitions in the vicinity of the $\Gamma$
point are indicated by the magenta arrows, namely, A1 is at (0.08,-0.04,0)
$2\pi/a$ and A2 is at (0.07,-0.035,0)$2\pi/a$.

Equations (3), (4) and (11) indicate that the absorptive parts of the optical
conductivity elements ($\sigma^{1}_{xx}$, $\sigma^{1}_{zz}$,
$\sigma^{2}_{xy}$, $\sigma^{1}_{\pm}$) are directly related to the dipole-
allowed interband transitions. This allows us to further understand the origin
of the prominent peaks in the optical spectra in terms of the band state
symmetries and dipole selection rules of the considered materials here. To
this end, we perform a symmetry analysis on the band states and dipole
selection rules at high symmetry $\Gamma$ and K points, as described in the
supplementary note 1 in the SM [48]. The derived dipole selection rules are
listed in Tables S2 and S3, and the calculated optical transition matrix
elements are given in Tables S4 and S5. As mentioned before, few-layer
Fe3GeTe2 with an odd number of MLs have the broken inversion symmetry and thus
the K′ and K points are not equivalent. Therefore, for ML Fe3GeTe2, we include
the K′ point in our symmetry analysis as well. Figures 6 and 7 depict the
orbital-projected relativistic band structures of bulk and ML Fe3GeTe2,
respectively. We label the main dipole-allowed optical transitions on high
symmetry points $\Gamma$ and K of bulk Fe3GeTe2 in Fig. 6 and also on high
symmetry points $\Gamma$, K and K′ for ML Fe3GeTe2 in Fig. 7. Based on the
calculated transition matrix elements (Tables S4 and S5) and the derived
selection rules (Tables S2 and S3), we assign the peaks in the absorptive part
of the magnetic-optical conductivity $\sigma^{2}_{xy}$ [see Figs. 5(d) and
5(h)] to the main optical transitions near the high symmetry points in Figs. 6
and 7, as indicated by red, blue and magenta symbols and arrows.

We can see from Figs. 6 and 7 that both bulk and ML Fe3GeTe2 have major
optical transitions in the energy range from -4.0 eV to 2.0 eV. Interestingly,
we find that in this energy range, significant optical transitions would occur
only from Fe $d$ orbital dominated states which hybridize with Te $p$ and Ge
$p$ orbitals. This results from a direct impact of the strong SOC of heavy
elements on the MO effects [59]. Thus, we could also link the main transitions
to the calculated DOS spectra presented in the previous section. For example,
peak P9 of $\sigma^{2}_{xy}$ of bulk Fe3GeTe2 [Fig. 5(d)] originates mainly
from an optical transition from the valence states at -3.0 eV to the
conduction states at $\sim$ 2.0 eV [Fig. 6], which coincides with the Te peak
A to the Ge peak B in Figs. 3(d) and 3(e), respectively. Similarly, peak P10
of bulk Fe3GeTe2 [Fig. 5(d)] originates mainly from an optical transition from
-4.0 eV to $\sim$ 0.8 eV [Fig. 6], which coincides with the Ge peak C to the
FeI and FeII peak D from Figs. 3(b) and 3(e). Such connection can also be seen
in ML Fe3GeTe2 since the DOS spectra from the two structures are similar.
Furthermore, we show that the considerable hybridization of Fe $d$ orbitals
with $p$ orbitals of heavy elements in the vicinity of the $\Gamma$ point in
the energy range from around -0.2 eV to $\sim$0.2 eV also contribute to the
first prominent peak in all the considered Fe3GeTe2 structures. We indicate
two of such transitions as a representative by magenta arrows in Figs. 6 and
7.

Importantly, the main optical transitions would reveal not only the effect of
orbital hybridizations but also the impact of crystalline symmetry and the
SOC. First let us look into the effect of SOC-lifted degeneracies. Figures 6
and 7 show that pairs of right- and left-circular dipole-allowed transitions
appear, such as peaks P7 and N7 in Fig. 6 for bulk Fe3GeTe2 and also peaks P6
and N6 in Fig. 7 for ML Fe3GeTe2. As explained in the supplementary note 1
[48], lifted degeneracies would lead to different irreducible representations
(irreps) and thus difference in the optical transitions due to left- and
right-circularly polarized light. From Table S4 and S5 we could directly
observe the magnetic circular dichroism by sign and value differences of the
transition matrix elements. Second, we examine the effect of the crystalline
symmetry. Figure 7 indicates that optical transitions of ML Fe3GeTe2 sometimes
come in pairs at the K and K′ point, e.g., peaks N1 and P1 as well as peaks N5
and P5. This results from the inversion symmetry breaking which leads to an
interchange of irreps between the K and K′ point. With the inclusion of the
SOC and the spontaneous exchange field due to the intrinsic magnetization, ML
Fe3GeTe2 could be a possible ferrovalley material [11]. From Table S5 in the
SM [48], we can see the transition energy ($\Delta E_{ij}$) to be different
for transitions P1 and N1. However, in the absence of ferromagnetism, the
dipole-allowed transitions would still be of opposite chirality but with the
same $\Delta E_{ij}$ due to the broken inversion symmetry. The large
difference of $\sim$0.13 eV in $\Delta E_{ij}$ is thus an indication of
possible strong ferrovalley effect in few-layer Fe3GeTe2 containing odd number
of MLs. Nevertheless, the $\sigma^{2}_{xy}$ spectra from different few-layer
structures (Fig. 5 as well as Figs. S4 and S5) are similar. This suggests that
although the broken inversion symmetry in the odd-layer structures affects the
transition matrix elements at the K and K′ point, it does not change the
magneto-optical spectrum significantly. Finally, it should be emphasized that
the crucial factor for the large MO conductivity in these structures is the
orbital hybridization between the magnetic (Fe) atoms which introduce
magnetization and heavy elements (Te) which bring about strong SOC. [22]

###  III.4 Magneto-optical Kerr and Faraday effects

Figure 8: Kerr rotation ($\theta_{K}$) and ellipticity ($\epsilon_{K}$)
spectra for (a) bulk, (b) ML, (c) BL, (d) TL, (e) FL and (f) QL Fe3GeTe2 in
the ferromagnetic state with the out-of-plane magnetization. In (a), the green
diamond denotes the experimental $\theta_{K}$ value [3]. Figure 9: Faraday
rotation ($\theta_{K}$) and ellipticity ($\epsilon_{K}$) spectra for (a) bulk,
(b) ML, (c) BL, (d) TL, (e) FL and (f) QL Fe3GeTe2 in the ferromagnetic state
with the out-of-plane magnetization.

Finally, we plot the calculated MOKE and MOFE spectra as a function of photon
energy in Fig. 8 and 9, respectively. Figures 8 and 9 show that for all the
few-layer structures of Fe3GeTe2, the patterns of the MOKE and MOFE spectra
look similar to that of bulk Fe3GeTe2 [Fig. 8(a) and Fig. 9(a)], especially
the Kerr and Faraday rotation spectra of BL, TL, FL and QL Fe3GeTe2. As for
the optical conductivity spectra, this similarity of the MO spectra among all
the considered structures is due to the weak interlayer coupling in these
Fe3GeTe2 systems.

Figure 8 also shows that bulk and few-layer Fe3GeTe2 all exhibit large
negative Kerr rotations above 1.0 ∘ in the low energy region below $\sim$0.4
eV. In particular, the Kerr angles of FL and QL Fe3GeTe2 are larger than 3.0 ∘
at $\sim$0.3 eV. As photon energy further increases, the negative Kerr
rotations decrease monotonically and become positive between 0.6 eV and 1.0
eV. Then they all peak at around 1.0 eV and stay positive until photon energy
of around 2.0 eV. The positive peaks near 1.0 eV are rather prominent and can
reach 1.0 ∘ in TL, FL and QL Fe3GeTe2 (Fig. 8). The Kerr rotation spectra then
have a negative peak of -1.0 ∘ again at around 2.5 eV and become positive
again as photon energy increases beyond $\sim$2.6 eV. They all have a positive
pronounced peak at around 3.0 eV except bulk and ML Fe3GeTe2 which have the
positive peak at 3.2 eV and 2.8 eV, respectively. Interestingly, the Kerr
ellipticity spectra of all the structures have two large negative peaks at
around 0.5 eV and 2.6 eV, respectively. MOKE experiments were carried out on a
340-nm thick sample using 633 nm HeNe laser (photon energy 1.96 eV). We note
that the measured Kerr angle of $\sim$0.33∘ [3] is compared well with our
calculated Kerr angle of bulk Fe3GeTe2 [see Fig. 8(a)].

To investigate the potential applications of the Fe3GeTe2 systems in, e.g., MO
devices, we now compare the magnitudes of Kerr rotation angles to several
popular MO materials. Let us start with 3$d$ transition metals and their
alloys. We notice that bcc Fe metal has a Kerr rotation of -0.5∘ at 1.0 eV,
hcp Co metal has a Kerr rotation of -0.42∘ at 5.0 eV, and fcc Ni metal has a
Kerr rotation of -0.25∘ at 4.0 eV [18, 21, 32]. Clearly, the calculated Kerr
rotation angles of few-layer and also bulk Fe3GeTe2 are generally larger than
that of elemental 3$d$ transition metals. In particular, the magnitudes of
both the Kerr rotation angle and ellipticity of all the considered Fe3GeTe2
structures are larger than 1.0∘ at around 2.5 eV. They are also comparable or
even larger than the MOKE in 3$d$ transition metal alloys with heavy elements
such as Pt and Bi which have the strong SOC. [22] For example, famous MO
transition metal alloys such as FePt, CoPt and PtMnSb have Kerr rotation
angles ranging from 0.4∘ to 0.5∘ [21, 60]. Nevertheless, thin film MnBi has a
record-high Kerr rotation of 2.3∘ at 1.84 eV [63], which is comparable to or
smaller than that of TL, FL and QL Fe3GeTe2 in the infra-red frequency range
(within 1.0 eV) (see Fig. 8).

For further comparison, let us examine famous magnetic semiconductors with
good MO properties. Ferromagnetic semiconductors Y3Fe5O12 and Bi3Fe5O12, which
have been widely used in spintronic research, have Kerr rotation angle of
-0.12∘ at 4.8 eV and -1.21∘ at 2.4 eV, respectively [61, 59]. For diluted
magnetic semiconductors Ga1-xMnxAs, a Kerr rotation of $\sim$0.4∘ near 1.8 eV
was reported [62]. Excitingly, layered ferromagnetic semiconductors CrI3 and
Cr2Ge2Te6 were recently thinned down to just one or two monolayer with the
ferromagnetic order retained at low temperatures [2, 1]. Moreover, TL
Cr2Ge2Te6 and TL CrI3 were predicted to have large Kerr rotation of 0.7∘ near
2.8 eV and $\sim$1.0∘ at 1.3 eV, respectively [15, 30]. Overall, in comparison
to these famous MO semiconductors, the MOKE in bulk and ML Fe3GeTe2 are
comparable and in particular, TL, FL and QL Fe3GeTe2 exhibit stronger MOKE.

Remarkably, from Fig. 9, we can observe large positive (negative) peaks of
value 82 (-97) deg/$\mu m$, 142 (-156) deg/$\mu m$, 109 (-111) deg/$\mu m$,
and 95 (-111) deg/$\mu m$ at $\sim$2.55 (3.0) eV for bulk, ML, BL, and TL
Fe3GeTe2, respectively. For comparison, MnBi thin films are known to possess
the largest Faraday rotation angle of $\sim$80 deg/$\mu m$ at 1.77 eV [39,
63]. The widely used semiconductor Y3Fe5O12 possesses a Faraday rotation of
7.2 deg/$\mu m$ at 3.9 eV [59]. With the substitution of Y by heavy element
bismuth, Bi3Fe5O12 can show a large Faraday rotation angle of 51.2 deg/$\mu m$
at 3.7 eV [59]. We notice that recently discovered 2D ferromagnetic
semiconductors Cr2Ge2Te6 and CrI3 have Faraday rotation angles of $\sim$120
deg/$\mu m$ [15] and of at most $\sim$ 108 deg/$\mu m$ [30, 38], respectively.
Clearly, bulk and 2D Fe3GeTe2 reported here have relatively large Faraday
rotation angles. The outstanding MO properties of the considered Fe3GeTe2
systems suggest their promising applications for nanoscale MO sensors and high
density MO data-storage devices.

Finally, we notice that MO Faraday rotation spectra of bulk and 2D Fe3GeTe2
(Fig. 9) are similar to their MO Kerr rotation spectra (Fig. 8). This may be
expected since Eqs. (7), (8) and (10) indicate that all these MO spectra are
more or less proportional to the MO conductivity ($\sigma_{xy}$). Indeed, the
large MOKE and MOFE in bulk and few-layer Fe3GeTe2 stem from their large MO
conductivity (i.e., strong magnetic circular dichroism). For example, overall,
the MO conductivity of bulk and 2D Fe3GeTe2 (Fig. 5 as well as Figs. S4 and S5
in the SM [48]) is about 20 times larger than Y3Fe5O12 and also 2 times larger
than Bi3Fe5O12 [59]. They are also around 2 times larger than bulk and 2D CrI3
[30].

##  IV CONCLUSIONS

Summarizing, we have investigated two relativity-induced properties, namely,
magnetic anisotropy energy (MAE) and magneto-optical (MO) effects, of ML, BL,
TL, FL and QL as well as bulk Fe3GeTe2, and also their connections with the
underlying electronic structures of the materials, based on systematic first
principle DFT calculations. First of all, we find that all the considered
Fe3GeTe2 structures prefer the out-of-plane magnetization and have gigantic
MAEs of $\sim$3.0 meV/f.u., being about 20 and 6 times larger than 2D
ferromagnetic semiconductors Cr2Ge2Te6 and CrI3, respectively. These MAE
values are also comparable to that of FePt, which is known to have the largest
MAE among the magnetic transition metals and their alloys. This gigantic
perpendicular anisotropy results from the large relativity-induced MCE of
$\sim$3.32 meV/f.u., which is ten times larger than the competing classical
MDE of $\sim$0.3 meV/f.u., which favors an in-plane magnetization (Table II).
The giant MCEs can be attributed to the presence of significant Te $p_{x,y}$
orbital DOS in the vicinity the Fermi level and also to some extent, to the
topological nodal point just below the Fermi level at the K points in the
Brillouin zone in these materials. This strong PMA thus stabilizes the long-
range ferromagnetic order in 2D Fe3GeTe2 at temperatures higher than that in
2D Cr2Ge2Te6 and CrI3.

Secondly, we also find that 2D and bulk Fe3GeTe2 exhibit strong MO effects
with the calculated Kerr and Faraday rotation angles being comparable or even
larger than that of best-known bulk MO materials such as PtMnSb, Y3Fe5O12 and
Bi3Fe5O12. In particular, all the Fe3GeTe2 structure are predicted to have
large Kerr rotation angles of $\sim$1.0∘ at $\sim$3.0 eV and FL and QL
Fe3GeTe2 have Kerr rotation angles as large as $\sim$3.0∘ at $\sim$0.25 eV
(Fig. 8). Furthermore, ML Fe3GeTe2 has a Faraday rotation angle of -156
deg/$\mu$m, which is three times larger than that of famous MO oxide
Bi3Fe5O12. The shape and position of the features in the Kerr and Faraday
rotation spectra are almost thickness-independent though the Kerr rotation
angles increase slightly with the film thickness. The strong MO Kerr and
Faraday effects are found to result from the large MO conductivity (i.e.,
strong magnetic circular dichroism) in these ferromagnetic materials. In
particular, the calculated MO conductivity spectra are one order of magnitude
larger than that of Y3Fe5O12. The calculated MO conductivity spectra are
analysed in terms of the symmetry of the band states and dipole-allowed
optical transitions at high symmetry $\Gamma$, K and K′ points, which also
reveal that atomically thin Fe3GeTe2 films with odd layer-number would exhibit
anomalous ferrovalley Hall effect. We notice that our calculated MAE values
for bulk and ML Fe3GeTe2 agree well with the corresponding experimental values
and our predicted Kerr angle of bulk Fe3GeTe2 at 1.96 eV is in good agreement
with the measured one. Also, the DC-limit of the real part of the calculated
MO conductivity of bulk Fe3GeTe2 agrees well with the measured AHC. ALL these
interesting findings therefore indicate that 2D and bulk ferromagnetic
Fe3GeTe2 may find valuable applications for high density MO and spintronic
nanodevices.

## ACKNOWLEDGMENTS

The authors acknowledge the support from the Ministry of Science and
Technology and the National Center for Theoretical Sciences (NCTS) of The
R.O.C. The authors are also grateful to the National Center for High-
performance Computing (NCHC) for the computing time.

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