# Microscopic Insights for Beyond Room-Temperature Ferromagnetism in Ni doped
Two-Dimensional Fe5GeTe2

Sukanya Ghosh, Soheil Ershadrad and Biplab Sanyal  Department of Physics and
Astronomy, Uppsala University, Box-516, 75120, Uppsala, Sweden
<EMAIL_ADDRESS>
###### Abstract

Enhancement of Curie temperature ($T_{\mathrm{C}}$) of two-dimensional (2D)
magnets is immensely desirable for room temperature spintronic applications.
Fe5GeTe2 is an exceptional van der Waals metallic ferromagnet due to its
tunable physical properties and relatively higher $T_{\mathrm{C}}$ than other
2D magnets. Using density functional theory combined with dynamical mean field
theory and Monte Carlo simulations, we show that the $T_{\mathrm{C}}$ of
Fe5GeTe2 monolayer can increase well-above room temperature by substitutional
doping with Ni. It is found that two specific sublattices (Fe1 and Fe4) are
the first and second most energetically preferred occupation sites for Ni.
$T_{\mathrm{C}}$ of Fe5-δNiδGeTe2 increases up to $\sim$ 400 K at
$\delta\sim$20%. Exchange interactions between particular Fe5-Fe4 pairs play a
dominating role in tuning the transition temperature, influenced by doping-
induced structural distortions. Finally, we highlight the effect of dynamical
electron correlation in site-specific electronic structure and quasi-particle
mass of Fe-$d$ orbitals with varying Ni doping.

##  1 Introduction

Atomically thin, layered quasi-two-dimensional (2D) van der Waals (vdW)
crystals exhibit exceptional physical properties[1, 2]. Regarding 2D magnets,
however, according to the Mermin-Wagner theorem, an intrinsic long-range
magnetic order can not exist in the isotropic 2D limit because strong thermal
fluctuations prohibit continuous symmetries to break spontaneously[3]. The
presence of weak magnetic anisotropy is sufficient to open up a sizable gap in
the magnon spectra, causing long-range magnetic order to persist in materials
with dimension $D\leq 2$ at a finite temperature.

Among the newly discovered vdW magnetic materials[1, 4, 5, 6, 7, 8, 9, 10, 11,
12], metallic FenGeTe2($n=3,4,5)$[13, 14, 15, 16, 17] systems, commonly
referred to as FGT are quite special. Ferromagnetism close to room temperature
increases their demand for spintronic applications. This can be further
enhanced by applying pressure[18, 19], gating[20, 21], carrier doping[22, 23],
ion intercalation[24], etc. It has also been found that substitutional doping
of Fe5GeTe2 with cobalt increases the magnetic ordering temperature to $\sim$
360 K influencing the magnetic ground state, interlayer stacking and magnetic
textures[23, 25, 26, 27]. A recent study by Chen et al. has reported an
enhancement of ferromagnetism in bulk Fe5-δGeTe2 up to 478 K with Ni
doping[16]. The authors showed that Ni doping triggers structural
modifications, influences saturation magnetization, and affects
$T_{\mathrm{C}}$. However, a detailed description of microscopic mechanisms
determining the observed trend for $T_{\mathrm{C}}$ in Fe5-δNiδGeTe2 is still
lacking. Especially, the exact nature of magnetic interactions governing
$T_{\mathrm{C}}$ should be unveiled. How the tuning of exchange interactions
caused by structural modifications in Fe5-δNiδGeTe2 determines
$T_{\mathrm{C}}$, demands a thorough investigation.

In this Letter, performing first-principles calculations, we investigate the
rationals responsible for the enhancement of $T_{\mathrm{C}}$ well above room
temperature by substitutional doping with Ni in Fe5GeTe2 monolayer. We find
out which particular Fe sites are more prone to host Ni dopant. Our study
explains how the tuning of exchange interactions between certain Fe pairs,
caused by structural modifications, plays the dominating role to increase
$T_{\mathrm{C}}$ up to a certain doping. Reduction in $T_{\mathrm{C}}$ for
higher doping is caused by the replacement of magnetic Fe atom by nonmagnetic
Ni, however, the dominating exchange still remains ferromagnetic, as observed
in experiment[16], but in contrast with a recent DFT study[28].

A $\sqrt{3}\times\sqrt{3}$ cell of Fe5GeTe2 monolayer in UDU (Up-Down-Up)
configuration of Fe atoms is considered in this study, where two (Fe1U) and
one (Fe1D) Fe atoms are situated directly above and below Ge, respectively[13,
29, 30, 31]. We investigate magnetic and electronic properties of the
energetically favored configurations, determined by comparing the total
energies of Fe5-δNiδGeTe2 monolayers varying the position(s) of Ni dopant(s).

Figure 1:  (a) Percentage change of in-plane lattice parameter $a$ (red
squares), thickness $d$ (green circles) and formation energy $E_{\mathrm{f}}$
(blue triangles), respectively, for Fe5-δNiδGeTe2 monolayer with Ni doping
($\delta$). (b) Schematics showing formation energy ($E_{\mathrm{f}}$) of
Fe5-XNiXGeTe2 for different Ni occupation site. Purple area indicates the
tendency of a Fe sublattice to get substituted with Ni, numbers show
$E_{\mathrm{f}}$ for different Fe sites at $\delta=$6.7%. (c) Side view of
Fe5-δNiδGeTe2 at $\delta=60\%$. The height of histograms shows rumpling of
each Fe/Ni atom along $x$, $y$ and $z$ directions present at the unit cell of
$\delta=60\%$ wrt the undoped monolayer, horizontal dashed line shows the
average rumpling.

Our results show that the in-plane lattice parameter $a$ remains almost
unaltered compared to the undoped system till $\delta=27\%$, and starts to
reduce when $\delta>30\%$, see Fig. 1(a). However, the thickness $d$ of the
monolayer increases with doping up to $\delta=20\%$ but reduces for
$\delta\geq 40\%$ as shown in Fig. 1(a). Substitutional doping of Ni in
Fe5GeTe2 monolayer becomes energetically less favored with increase in
$\delta$. This is evident in Fig. 1(a) (blue triangles) where the formation
energy $E_{\mathrm{f}}$ per Ni dopant is observed to increase with $\delta$.
It is also worth noting that during the substitutional doping, the replacement
of Fe1 species with Ni is energetically more favored than other Fe sites.
Between $\delta=$ 6.7%(Fe1U) and 20%(Fe1U+Fe1U+Fe1D) only Fe1 sublattice gets
substituted by Ni. After Fe1, the next energetically favored occupation site
for Ni dopant is Fe4. There is a significant increase in $E_{\mathrm{f}}$
between $\delta=$20% and 27%, when one of the Fe4 atoms gets substituted
together with Fe1U and Fe1D species. The presence of Ni causes an excess of
electrons (Fig S1), which might cause the lowering of $E_{\mathrm{f}}$. For
$\delta\geq$ 33%, Fe atoms belonging to other Fe sublattices (Fe2, Fe5 and
Fe3) start to get substituted along with Fe1 and Fe4. Fig. 1(b) shows how
$E_{\mathrm{f}}$ varies when Ni substitutes different Fe sublattices at
$\delta=6.7\%$.

Structural distortion or rumpling (along $x$, $y$ and $z$ directions) in the
monolayer increases with Ni-doping (Fig. S1). Fig. 1(c) shows the side view
when $\delta=$ 60%. The height of the histograms in Fig. 1(d) shows the
difference in $x$, $y$ and $z$ coordinates between Fe or Ni atoms present in
60% Ni-doped and undoped systems for each (Ni/Fe) site of
$\sqrt{3}\times\sqrt{3}$ cell. $\delta=60\%$ causes significant rumpling of
the atoms present at the sites of Fe5 (along $z$) and Fe2 sublattices.
Negative value of average rumpling (dashed horizontal line) supports the
compression of cell parameters (both $a$ and $d$) with Ni doping, as we see in
Fig. 1(a). It should be noted that Ref. 16 also finds a reduction in layer
thickness with an increase in Ni doping. Apart from substitutional doping, Ni
can occupy any vacant site of $\sqrt{3}\times\sqrt{3}$ cell of Fe5-δGeTe2,
including vdW gap between different layers, such scenario can be present in
experiments performed at a finite temperature[16].

Figure 2: (a) Total magnetic moment (Mtot) of Fe5-δNiδGeTe2 monolayer as a
function of doping concentration $\delta$. Insets with red and green squares
show the average of total magnetic moment $M_{tot}^{avg}$ and variation of
magnetic anisotropy energy (MAE) as a function of $\delta$, respectively. (b)
Variation of average magnetic moment of Fe
(M${}_{\mathrm{Fe}}^{\mathrm{avg}}$, orange squares) and Ni
(M${}_{\mathrm{Ni}}^{\mathrm{avg}}$, cyan squares) atoms plotted with
$\delta$. Inset shows average orbital moment $M_{\mathrm{orb}}^{\mathrm{avg}}$
for Fe (orange) and Ni (cyan) for different $\delta$.  Figure 3: (a) Variation
of $\sum_{NN=5}{J_{ij}}$ with $\delta$ for each Fe sublattice.
$\sum_{NN=5}{J_{ij}}$ is sum of isotropic symmetric exchange interactions
$J_{ij}$ over the first five nearest neighbors. (b) Variation of
$T_{\mathrm{C}}$ with $\delta$. (c) $J_{ij}$ interactions between $i=$Fe5 and
$j=\sum$Fe4, $\sum$Fe3, $\sum$Fe1D and $\sum$Ni with $\delta$.

In our previous study, we showed that the inclusion 3of dynamical electron
correlation effect is an appropriate approach to determine the magnetic
moment, exchange interactions and $T_{\mathrm{C}}$ of the FenGeTe2 systems,
compared to the standard GGA and GGA+U methods[14]. Therefore, we perform
charge self-consistent dynamical mean-field theory (DFT+DMFT) calculations as
implemented in the FP-LMTO code RSPt [32, 33] to investigate the magnetic and
electronic properties of Fe5-δNiδGeTe2 monolayer. The computational details
are mentioned in SI.

Doping with Ni influences the magnetism of Fe5-δNiδGeTe2 monolayer. The total
spin moment $M_{\mathrm{tot}}$ reduces as a function of doping ($\delta$).
Fig. 2(a) shows that $M_{\mathrm{tot}}$ decreases from 10 $\mu_{B}$ to 0
$\mu_{B}$ from $\delta=0\%$ to 100%. These results are in good agreement with
the saturation magnetic moment of bulk Fe5-δNiδGeTe2. [29] It is interesting
to note that the direction of easy axis or magnetic anisotropy energy, MAE (=
$E_{||}-E_{\perp}$) oscillates between in-plane and out-of-plane directions
with $\delta$. However, for most of the doped systems, the easy axis lies in
the $xy$-plane, see green squares presented as an inset in Fig. 2(a), which is
in agreement with experiment[16]. The strength of MAE for Fe5-δNiδGeTe2
monolayers is much weaker than pristine Fe3GeTe2 and Fe4GeTe2 monolayers[14,
21]. The trend observed for MAE at lower $\delta$ values can be correlated
with the value of orbital moments obtained for different directions of spin
axis[34], see Table S2. It is worth noting that the switching of the easy axis
is observed in Fe5-δGeTe2 with Co doping and electrical gating[23, 35].
Experiments report switching of easy axis for bulk Fe5GeTe2 depending on the
Fe concentration[36, 29]. The average spin moment of Fe atoms first increases
with doping, becomes maximum at $\delta=33\%$, then reduces, see Fig. 2(b).
Similar to Ref. 16, our calculations also find that Ni dopants carry
negligible spin moment (cyan squares in Fig. 2(b)) and are not responsible for
the origin or tuning of ferromagnetism. The average orbital moment
$M_{\mathrm{orb}}^{\mathrm{avg}}$ of Fe remains between 0.05 and 0.04 till
$\delta$=80%, see inset of Fig. 2(b). The orbital moment of Ni falls in the
range of 0.01-0.02 and reduces for higher $\delta$.

Figure 4: Spectral function plots for $\delta=$ (a)0% and (b) 40%. The Dirac-
cone-type features present at the high-symmetry points K (a) and M (b) are
highlighted by the red circles. (c) and (d) DOS at $\delta=0\%$ and 40%,
respectively. Effective mass, $m*/m$ for (e) Fe-$d_{z}^{2}$ and (f) $d_{xy}$
for different Fe sublattices and Ni with $\delta$. These results are obtained
at $T=155$ K.

We calculate the isotropic symmetric ($J_{ij}$) and antisymmetric ($D_{ij}$)
exchange interactions present in Fe5-δNiδGeTe2 monolayer varying $\delta$.
Incorporating $J_{ij}$, $D_{ij}$ and MAE in the Heisenberg Spin-Hamiltonian
(Eq. S4), $T_{\mathrm{C}}$ is computed performing Monte Carlo simulations.
Fig. 3(a) shows there is a monotonic increase of $T_{\mathrm{C}}$ up to
$\delta=20\%$, then it reduces. Qualitative trend of $T_{\mathrm{C}}$ vs.
$\delta$ plotted in Fig. 3 agrees well with experimental reports on bulk
Fe5-δNiδGeTe2[16]. The critical $\delta$ value at which $T_{\mathrm{C}}$ of
the monolayer becomes maximum, is not exactly the same as observed in the
experiment. This slight discrepancy happens because in experiments, during the
doping process, Ni can be placed at any vacant position present in the bulk
without replacing Fe, enhancing ferromagnetism and hence $T_{\mathrm{C}}$.
However, the rational behind such trend of $T_{\mathrm{C}}$ in Fe5-δNiδGeTe2
is not addressed in the previous study.

Comparing the strength of different magnetic interactions we expect $J_{ij}$
couplings must play the dominating role to determine $T_{\mathrm{C}}$, in
agreement with our study on pristine FGT systems[14]. In order to investigate
the tuning of $J_{ij}$ with $\delta$, we plot the $J_{ij}$ values summed over
the first five nearest neighbors (NNs) for different $i$th Fe sublattices, see
Fig. 3(b). We consider NN up to 5, because the $J_{ij}$ interactions decay
significantly beyond that, see Figs.S4-S7. Most of the Fe sublattices show
dominating ferromagnetic (FM) $J_{ij}$ interactions while Fe1D and Fe1U show
antiferromagnetic (AFM) interactions, till $\delta=$13.4%.

The $\sum_{NN=5}{J_{ij}}$ term for Fe5, Fe4, Fe3 and Fe2 first increases with
$\delta$, becomes maximum for $\delta=$20%, then reduces for higher
concentration. $\sum_{NN=5}{J_{ij}}$ is plotted for each Fe species present in
the $\sqrt{3}\times\sqrt{3}$ cell. A monotonic increase of
$\sum_{NN=5}{J_{ij}}$ for each Fe sublattice (except Fe1U and Fe1D) is
observed up to $\delta=20\%$. This happens because till $\delta=20\%$, except
Fe1, the number of Fe sublattices present in the unit cell is 3. For
$\delta>$20% the Fe atoms belonging to Fe4 sublattice start to get substituted
in addition with Fe1U and Fe1D, as we see in Fig. 1(d). The replacement of
magnetic Fe causes sharp reduction in $\sum_{NN=5}{J_{ij}}$ of Fe4 for
$\delta>20\%$. After 40% all three Fe4 atoms get substituted with Ni, see red
circles in Fig. 3(a). The gradual replacement of different Fe sublattices with
Ni causes rapid lowering in $\sum_{NN=5}{J_{ij}}$ for $\delta\geq$ 33%. In
addition to possessing negligible magnetic moment (Fig. 2), Ni dopants have
negligible contribution to the $J_{ij}$ interactions. The magnitude of
$\sum_{NN=5}{J_{ij}}$ for Ni is $\sim$ 10 times smaller than the Fe
sublattices for $\delta\leq 20\%$. As the number of Ni atoms present in
Fe5-δNiδGeTe2 increases with $\delta$, $\sum_{NN=5}{J_{ij}}$ for Ni becomes
comparable with Fe for $\delta\geq 60\%$. For $i$=Ni, non-zero exchange
couplings exist when $j$=Fe, otherwise, interactions between Ni themselves is
rather weak. The violet symbols show the average variation of
$\sum_{NN=5}{J_{ij}}$ with Ni doping, which increases from 0% to 20% and then
reduces. The same investigation has been made for $D_{ij}$ interactions as
well. For a given $\delta$, the magnitude of $\sum_{NN=5}{D_{ij}}$ for any
$i$th Fe species is $\sim 10$ times smaller than $\sum_{NN=5}{J_{ij}}$ (Fig
S9).

Further analysis of $J_{ij}$ couplings among individual Fe pairs reveals that
the interactions between Fe5 and Fe4 play the dominating role. Fig 3(c) shows
$J_{ij}$ interactions when $i=$ Fe5 and $j$th species are considered to be the
nearest neighbors of Fe5, i.e., Fe4, Fe3, Fe1D and Ni. We find, among these
neighbors, exchange coupling between Fe5-$\sum$Fe4 plays dominating role to
control or tune the $T_{\mathrm{C}}$. The $J_{ij}$ interaction between Fe5 and
Fe4 increases with $\delta$ and then reduces for $\delta\geq 27\%$. The
increase of $J_{54}$ up to $\delta=20\%$ occurs due to the following reasons:
i) reduction of Fe5-Fe4 bond length with $\delta$ (Fig. S2), ii) among the
first five NNs of Fe5, three of them are Fe4 species with the strongest FM
coupling. Therefore, $J_{54}=(J_{54_{1}}+J_{54_{2}}+J_{54_{3}})$ for
$\delta\leq 20\%$. The reduction in $J_{54}$ for $\delta>20\%$ occurs due to
the gradual replacement of Fe4 with Ni, causing a decrease in the number of
Fe4 belonging to the first five NN of Fe5, see Fig. S3 for details. No
particular trend is observed for $J_{53}$ (green squares), up to
$\delta=60\%$. $J_{ij}$ exchange interactions between Fe5 and Fe1D is AFM
(brown squares), and replacement of Fe1D with Ni triggers FM exchange coupling
(cyan squares). Comparing different $J_{5j}$ interactions we see $J_{54}$
follows the similar trend as $\sum_{NN=5}{J_{5j}}$ (black squares) and average
of $\sum_{NN=5}{J_{ij}}$ (purple symbols) in Fig. 3(b). Therefore, $J_{54}$
has major influence on $T_{\mathrm{C}}$, especially for $\delta\leq 20\%$.
From Fig. 3(a) - (c) we can establish the fact that $T_{\mathrm{C}}$ not only
depends on the strength of $J_{ij}$ but also on the nearest neighbor (NN)
distance or effective coordination number.

Now we discuss how Ni doping modifies the electronic structure. Figs. 4(a) and
(b) show the spectral function $A(k,\omega)$ for $\delta=$0% and 40%. The main
differences observed in $A(k,\omega)$ at these two values of $\delta$ are:i)
at the high-symmetry point $K$, there is presence of Dirac-cone-type feature
for $\delta$=0%, which disappears at $\delta=40\%$, ii) at $M$, Dirac-cone is
observed for $\delta=40\%$ which is not present for the undoped monolayer. The
density of states (DOS) plots for $\delta=0\%$ and 40% are plotted in Figs.
4(c) and (d), respectively, (see Fig. S16 for details). The intensity of DOS
projected on Fe reduces with Ni doping. The Fe states have dominating
contribution close to $E_{\mathrm{F}}$, maximum intensity of Ni states arises
away from $E_{\mathrm{F}}$. Similar to other FGT systems, the admixture of
localized and itinerant electrons exists in Fe5GeTe2 as well[37, 14]. The
effective mass ($m^{*}/m)_{l\sigma}$ provides the quantitative measurements of
electronic correlation[38] (see Eq. S7). Both qualitative and quantitative
trends of $m^{*}/m$ with $\delta$ remain almost unaltered for different Fe-$d$
states, see Figs. 4(e)-(f) and Fig.S17. $m^{*}/m$ of Fe reduces between
$\delta=0\%$ and 20%, then increases for $\delta>20\%$, especially for Fe5 and
Fe2. ($m^{*}/m)_{l\sigma}$ of Fe5, Fe2 are larger than for Fe3, Fe4. This
implies the $d$ electrons belong to Fe5 and Fe2 sublattices are more
correlated than Fe3 and Fe4. Similar to Fe5-δGeTe2, site-dependence of Fe
sublattices is present in Fe3GeTe2 as well[39]. Interestingly, the effective
mass of Ni-$d$ is lower than Fe-$d$, signifying the Ni-$d$ states are less
correlated or more delocalized than Fe-$d$.

In summary, we investigate the effect of Ni doping on the structural,
electronic, and magnetic properties of $\sqrt{3}\times\sqrt{3}$ Fe5-δNiδGeTe2
monolayer using DFT+DMFT and Monte Carlo simulations. Our results show that
$T_{\mathrm{C}}$ of the monolayer increases up to $\sim$400 K by
substitutional doping with Ni. The variation in $T_{\mathrm{C}}$ of
Fe5-δNiδGeTe2 monolayer with doping is in good agreement with a recent
experimental report on bulk Fe5-δNiδGeTe2. Moreover, we identify the dominant
exchange interactions responsible for the observed trend in $T_{\mathrm{C}}$.
Our results show the structural modifications caused by the Ni dopant, thereby
modifying the nearest neighbor distances and effective coordination numbers,
which affect the dominating exchange couplings. Our study also shows how Ni-
doping influences site-dependent spectral features and effective masses
arising from electron correlation.

Acknowledgments

B.S. and S.G. acknowledge a postdoctoral grant from Carl Tryggers Stiftelse
(CTS 20:378). B. S. acknowledges financial support from Swedish Research
Council (grant no. 2022-04309). The computations were enabled in project SNIC
2022/3-30 by resources provided by the Swedish National Infrastructure for
Computing (SNIC) at NSC, PDC, and HPC2N partially funded by the Swedish
Research Council (Grant No. 2018-05973). B.S. acknowledges allocation of
supercomputing hours by PRACE DECI-17 project ‘Q2Dtopomat’ in Eagle
supercompter in Poland and EuroHPC resources in Karolina supercomputer in
Czech Republic and LUMI supercomputer in Finland.

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