# Effect of Co Substitution on Ferrimagnetic Heusler compound Mn3Ga

Quynh Anh T. Nguyen1 Thi H. Ho1 Myung-Hwa Jung2 Sonny H. Rhim1
<EMAIL_ADDRESS>1Department of Physics and Energy Harvest Storage Research
Center, University of Ulsan, Ulsan 44610, Republic of Korea
2Department of Physics, Sogang University, Seoul, 04107, Republic of Korea

###### Abstract

Effect of Co substitution on Mn3Ga is investigated using first-principles
study for structural and magnetic properties. Without Co, ferrimagnetic
Heusler compound Mn3Ga is in tetragonal phase. With Co substitution, depending
on Co concentration ($x$) Mn3Ga prefers tetragonal (cubic) phase when $x\leq
0.5$ ($x\geq 0.5$). Ferrimagnetism is robust regardless of $x$ in both phases.
While magnetic moments of two Mn do not vary significantly with $x$, Co
magnetic moment in two phases exhibit different behaviors, leading to distinct
features in total magnetic moment (${\rm M_{tot}}$). When $x\leq 0.5$, in
tetragonal phase, Co magnetic moment is vanishingly small, resulting in a
decrease of ${\rm M_{tot}}$ with $x$. In contrast, when $x\geq 0.5$, in cubic
phase, Co magnetic moment is roughly 1$\mu_{B}$, which is responsible for an
increase of ${\rm M_{tot}}$. Electronic structure is analyzed with partial
density of states for various $x$. To elucidate the counterintuitively small
Co moment, the magnetic exchange interaction is investigated where exchange
coefficient between Co and Mn is much smaller in $x\leq 0.5$ case than $x\geq
0.5$ one.

##  I Introduction

For decades, Heusler compounds with plethora of material classes have drawn
continuously renewed attentions [1, 2]. Rich physics has been platforms for
various research. Numerous functionalities and properties, such as
superconductivity, magnetism, multiferroicity, half-metallicity, topological
insulators, and so forth [4, 3, 5, 6], have attracted for applications in many
areas. In particular, in the context of rapid progress in magnetic devices and
spintronics, tetragonal Heusler compounds have been intensively and
extensively explored. Properties of tetragonal Heusler compounds,
perpendicular magnetic anisotropy (PMA) and low saturation magnetization
($M_{S}$) [7, 8, 9], are regarded advantageous in applications with low
switching current ($I_{S}$) [10, 11, 12].

Mn3Ga belonging to Heusler family in tetragonal structure, possesses
aforementioned magnetic properties: strong PMA around 0.89 MJ/m-3, high Curie
temperature ($T_{C}$) of 740 K [13, 14], low $M_{S}$ around 200 emu/cm3, and
approximately 58% of spin polarization [15]. Low $M_{S}$ stems from the
intrinsic ferrimagnetism of antiparallel alignment of two Mn moments with
unequal magnitudes. Efforts have been devoted to enhance the magnetic
properties of Mn3Ga. One is to replace Mn by other transition metal (TM) [16]
where V-shape magnetization with respect to TM content ($x$) is quite
distinct. Upon substitution of nonmagnetic TM, ${\rm M_{tot}}$ decreases with
$x$ as nonmagnetic TM replaces one Mn, moment is reduced but the antiparallel
moment with the other Mn is retained. ${\rm M_{tot}}=0$ is achieved for
certain $x$ above which the survived antiparallel moment dominates hence ${\rm
M_{tot}}$ increases.

The V-shape of magnetization is also prominent with magnetic TM substitution
such as Fe, Co, and Ni [17, 18]. However, ${\rm M_{tot}}$ nearly vanishes
above $x=0.5$ and is not compensated. Interestingly, Co substitution
accompanies a structural transition from tetragonal to cubic while the
substitution by Fe and Ni retains the tetragonal structure. Moreover, the
magnetic behavior of magnetic TM in two structural phases is different.
Magnetic moments in tetragonal phase are smaller than the bulk counterpart but
those in cubic phase are comparable to the bulk ones. While the substitution
of Fe results in hard magnet and that of Ni exhibits the shape memory
phenomena, the property of Co substitution depends on the Co content. It has
been reported earlier that Co-poor ($0<x<0.5)$ in tetragonal alloy is well
suited for spin-transfer torque (STT) whereas Co-rich $(0.5<x<1)$ favors cubic
phase with half-metallicity. As STT and half-metallicity are favorable
components in spintronic application, further study would shed light on
uncovering Mn3-xCoxGa.

In this paper, employing ab initio calculations, the effect of Co substitution
on Mn3-xCoxGa is investigated. Magnetic property of Mn3-xCoxGa is illustrated
along with total as well as atomic resolved magnetic moments. Analysis follows
to reveal the electronic structure of Mn3-xCoxGa. To elucidate the contrasting
behavior of Co magnetism in two phases, tetragonal and cubic, the exchange
coefficients are discussed in the framework of the magnetic force theorem
using the Heisenberg model.

##  II Structure and computational Methods

Figure 1: Crystal structure of (a) tetragonal Mn3Ga and (b) cubic Mn2CoGa.
Blue, red, green and grey spheres denote Mn(I), Mn(II), Co and Ga atoms,
respectively.

Tetragonal ($x=0$) and cubic phases ($x=1$) are presented in Fig. 1, which
correspond to $D0_{22}$ and $L_{21}$ structure, respectively. Both tetragonal
and cubic phase have two inequivalent Mn atoms distinguished by Mn(I) and
Mn(II). Space group of tetragonal phase is $I4/mmm$ (No. 139) and that of
cubic phase is ${Fm\bar{3}m}$ (No. 225). Wyckoff positions of both phases are
listed in Table 1.

Table 1:  Wyckoff positions of constituent atoms of tetragonal and cubic phases, whose space groups are $I4/mmm~{}(No.139)$ and $Fm\bar{3}m~{}(No.25)$, respectively. Co takes either $2c$ or $2d$ site.  | Tetragonal | Cubic
---|---|---
Mn(I) | $2b$ | $4a$
Mn(II) | $4d$ | $4c$
Ga | $2a$ | $4b$
Co | 2c/2d | $4d$

First-principles calculations are performed using Vienna Ab initio Simulation
Package (VASP) [19, 20] with projector augmented-wave basis [21]. The
exchange-correlation potential is treated using generalized gradient
approximation (GGA) with Perdew, Burke, and Ernzerhof (PBE) parametrization
[22, 23]. Cutoff energy of 500 eV is employed for wave function expansion. For
Brillouin zone integration, ${15\times 15\times 7}$ and ${15\times 15\times
11}$ k meshes are used for tetragonal and cubic phases, respectively, where
$\sqrt{2}/2\times\sqrt{2}/2\times 1$ unit cell is adopted for the primitive
cell.

From lattice constant optimization, we obtain $a=3.78$ Å ($c/a=1.88$) for
tetragonal phase and $a$ = 4.12 Å for cubic phase, respectively, which are
used for further calculations The exchange coefficients are calculated using
the Heisenberg model in the framework of the magnetic force theorem (MFT)
[24]. To do so, additional self-consistent calculations are performed using
OpenMX [25], where atomic basis sets for Mn, Ga, and Co are ${s3p2d1}$,
${s2p2d1}$, and H-$s3p2d1$, respectively with energy cutoff of 300 Ry and
cutoff radii of 10.0$\AA$ (Mn and Co) and 9.0$\AA$ (Ga), respectively. Then,
$Jx$ package is employed [26, 27, 28] to extract the exchange coefficients
($J$) in the framework of MFT.

##  III Results and Discussion

###  III.1 Magnetic Moments

Figure 2: (a) Magnetic moment (Mtot) per unit cell with respect to $x$, where
white and grey area are tetragonal and cubic phases, respectively. Right axis
is in emu/cm3 converted from $\mu_{B}$ on the left axis. (b) Atom-resolved
magnetic moment: blue, red, and green circles denote Mn(I), Mn(II), and Co,
respectively. Open (solid) circles represent unstable (stable) phase.

Mn3Ga is either tetragonal or cubic depending on Co concentration $x$. When
$x<0.5$, tetragonal phase with $c/a=1.88$ is favored whereas when $x>0.5$
cubic phase is preferred. As such, $x=0.5$ is a phase boundary of two phaes
[29].

Total magnetic moment (${\rm M_{tot}}$) as function of $x$ is plotted in Fig.
2, where each moment of individual Co, Mn(I), and Mn(II) is also shown. For
all $x$, Mn3-xCoxGa is ferrimagnetic: Moments of Mn(I) and Mn(II) are
antiparallel with different magnitudes. Without Co, i.e. $x=0$, Mn(I) and
Mn(II) have moments of 2.87 and -2.32 $\mu_{B}$, respectively, exhibiting
indeed ferrimagnetism. Total magnetic moment (${\rm M_{tot}}$) is 1.72
$\mu_{B}$ per formula unit, consistent with previous works [30, 14], which
corresponds to magnetization of $314$ emu/cm3. As shown in Fig. 2(a), ${\rm
M_{tot}}$ decreases (increases) with $x$ in tetragonal (cubic) phase. Notably,
at the phase boundary when $x=0.5$, ${\rm M_{tot}}$ of cubic phase is larger
by 0.4 $\mu_{B}$ than tetragonal phase. This difference is due to different Co
magnetism in both phases. Acknowledging ferromagnetism of Co, Co moment in
tetragonal phase is quite small [31], which is against to the conventional
wisdom. This is discussed later in Sec.III.2 and III.3.

Magnetic moments of individual atoms are shown in Fig. 2(b) [32]. For all $x$,
Mn(I) and Mn(II) have almost constant moments with 2.90 and -2.32 $\mu_{B}$,
respectively. However, Co moment in tetragonal phase nearly vanishes while in
cubic phase is around 1.0 $\mu_{B}$. Consequently, ${\rm M_{tot}}$ decreases
with $x$ in tetragonal phase: ${\rm M_{tot}}=1.72$ $\mu_{B}$ for $x=0$ drops
to 0.72 $\mu_{B}$ for $x=0.5$. On the other hand, in cubic phase, more Co
substitution results in increase of ${\rm M_{tot}}$ from 1.00 to 2.00
$\mu_{B}$. The vanishingly small Co magnetic moment in tetragonal phase is
also reported in previous work which has been also reported in other
theoretical work [33]. As mentioned earlier, this counterintuitive feature
will be visited in III.3.

Figure 3:  Partial density of states (PDOS) of $d$ orbitals. Left and right
panels for tetragonal and cubic phases, respectively. Tetragonal phase with
(a) $x=0$ and (b) $x=0.5$. Cubic phase with (c) $x=1.0$ and (d) $x=0.5$. Lower
panel is comparison between tetragonal and cubic phases with $x=0.5$. Blue,
red, and green lines denote $d$ orbitals of Mn(I), Mn(II), and Co,
respectively. Peaks are emphasized with triangles. Fermi energy ($E_{F}$) is
set to zero.

###  III.2 Electronic Structure

Partial density of states (PDOS) is presented in Fig. 3. Left (right) upper
panel is tetragonal (cubic) phase for $x=0$ ($x=1$). Lower panel is for
$x=0.5$ of tetragonal and cubic phases, respectively. Important peaks are
labelled with triangle marks: 1, 2, and $C$ stand for Mn(I), Mn(II), and Co,
respectively; \+ (-) for the majority (minority) spin state, respectively.
Different peaks in the same spin state of particular atom are distinguished by
primes. PDOS’s of $x=1/3$ and $2/3$ are shown in Fig. S3, where the vertical
line serves to manifest the overall trend upon $x$. The shift of $E_{F}$ by Co
substitution is evidence as Co has two more electrons than Mn[34].

In tetragonal phase with $x=0$ [Fig. 3(a)], Mn(I) and Mn(II) exhibit distinct
features. Mn(I) has two main peaks: $1$\+ ($1$-) in occupied (empty) states
around -2.7 eV (+1.0 eV) in the majority (minority) spin state. On the other
hand, Mn(II) has rather broader peak with smaller height than Mn(I): 2+ around
+1.3 eV in the majority spin state and 2-, 2′-, and 2′′\- around -2.5, -1.2
eV, and +1.0 eV in the minority spin state, respectively.

For $x=0.5$ in tetragonal phase [Fig. 3(b)], Mn(I) and Mn(II) PDOS do not
change much with respect to $x=0$ but shifted by +0.20 eV due to Co. [See
vertical line in Fig. S3]. On the other hand, several peaks emerge in Co PDOS.
As Co replaces Mn(II), most Co peaks overlap with Mn(II): $C$\- coincides with
2′\- and small hump just above $E_{F}$ overlap with 2′+. Nonetheless, the
overlap with Mn(I) in the majority spin state around -2.8 eV (marked with
double triangles) is evident. Broad $C$\+ appear between -1.4 and 0.4 eV owing
to hybridization with Ga $p$ state [See Fig. S2]. In particular, there are
small humps around +0.2 and +0.6 eV, where the former, just mentioned earlier,
overlaps with 2′\+ but the latter has no such feature. As the sum of occupied
PDOS of two spin states is almost equal, Co magnetic moment nearly vanishes,
which is prominent in tetragonal phase.

Now we switch to cubic phase with $x=1$ [Fig. 3(c)]. Compared to $x=0$ of
tetragonal phase, 1+ changes little but 1- shifts by +0.3 eV. However, changes
in Mn(II) and Co are apparent. In the minority spin state, 2- shifts by 0.2 eV
and 2′\- is replaced by $C$-. Overlaps between 2′′\- (not labeled) and
$C^{\prime}$\- are prominent around $E_{F}$+0.8 eV. On the other hand, in the
majority spin state, $C$\+ appears almost around the same place as $C$-.
Overlaps between $C^{\prime}$\+ and 2+ around $E_{F}$+0.6 eV are much closer
to $E_{F}$ than $x=0$. Notably, 2′\+ is occupied in contrast to $x=0$.

Cubic phase with $x=0.5$ is shown in Fig. 3(d). Here, we discuss changes with
respect to $x=0.5$ of tetragonal phase and $x=1$ of cubic phase. Hence in this
paragraphs $x=0.5$ ($x=1.0$) refers to that in tetragonal (cubic) phase. Mn(I)
is little affected: 1+ and 1- change negligibly with respect to $x=1$ whereas
1- shifts by +0.3 eV with respect to $x=0.5$. On the other hand, Mn(II) is
quite affected with negative (positive) shifts with respect to $x=0.5$
($x=1$). More specifically, compared to $x=1$, peaks of 2+ and 2′\+ shift by
+0.30 and +0.20 eV, respectively; that of 2- by +0.35 eV. Compared to $x=0.5$,
shifts of 2+ and 2′\+ are -0.60 and -0.40 eV, respectively; 2- by -0.50 eV. In
particular, 2′\+ accompanies occupation change. For Co PDOS, C+ and C- shift
by -0.4 and -0.1 eV, respectively, with respect to $x=1$, and 0 and -0.1 eV
with respect to $x=0.5$. Positive shift of Mn(II) with respect to $x=1$ is
well expected, as mentioned earlier, as Co has more electrons than Mn.

Figure 4: Calculated exchange coefficients ($J$) for nearest neighbor
interactions for various $x$ ($0\leq x\leq$ 1). $J_{0}$, $J_{1}$, and $J_{2}$,
represent the exchange coefficients of Mn(I)-Mn(II), Co-Mn(I), Co-Mn(II) in
black, blue, and red, respectively. Tetragonal and cubic regions are divided
by white and grey regions. Solid and dashed lines for tetragonal and cubic
phases, respectively.

###  III.3 Magnetic Exchange Interaction

So far PDOS analysis is presented. While Co magnetic moment in cubic phase
($x\geq 0.5$) is around 1 $\mu_{B}$, that in tetragonal phase $x\leq 0.5$
nearly vanishes, which is against the conventional wisdom. In order to clarify
this puzzling Co magnetism, the exchange coefficients, $J$, are discussed,
employing the Heisenberg model,
$\hat{H}=-\sum_{ij}{J_{ij}{{S_{i}}}\cdot{S_{i}}}$, where $J_{ij}$, the
exchange coefficients, with positive (negative) sign implies parallel
(antiparallel) magnetic coupling. As briefly mentioned in II, $J^{\prime}$s
are calculated in the framework of MFT [24, 26, 28, 25]. Fig.4 shows
calculations of the exchange coefficients for three cases, Mn(I)-Mn(II), Co-
Mn(I), and Co-Mn(II), denoted as $J_{0}$, $J_{1}$, and $J_{2}$, respectively.

$J_{0}$, the exchange coefficients for two Mn types, is negative for all
cases. This indicates the antiparallel magnetism between two Mn sites
confirming the ferrimagnetism. Moreover, $|J_{0}|$ is larger than $|J_{1}|$ or
$|J_{2}|$ roughly by an order of magnitude. In tetragonal phase ($x\leq 0.5$),
$J_{0}=-9.89$ meV for $x=0$ whose magnitude increases with $x$ ranging 18 – 21
meV. In cubic phase ($x\geq 0.5$), $J_{0}=-14.9$ meV for $x=0.5$, whose
magnitude also increases with $x$ up to 20 meV. From this, addition of Co
introduces stronger tendency of the antiparallel magnetism between Mn(I) and
Mn(II).

The exchange coefficients between Co-Mn(I) and Co-Mn(II), namely $J_{1}$ and
$J_{2}$, exhibit opposite sign behavior where magnitudes in cubic phase are
slightly larger. In contrast to $J_{0}$, both $|J_{1}|$ and $|J_{2}|$ are
little influenced by Co. In tetragonal phase, $|J_{1}|$ and $|J_{2}|$ are
approximately 1 and 5 meV, respectively, while in cubic phase$|J_{1}|$ and
$|J_{2}|$ are approximately 5 and 7 meV, respectively. The opposite signs of
$J_{1}$ and $J_{2}$ is due to the antiparallel magnetism between Mn(I) and
Mn(II). From $|J_{2}|>|J_{1}|$, the antiparallel coupling between Co-Mn(II) is
stronger than Co-Mn(I). Moreover, $J_{1}$ and $J_{2}$ change signs between
tetragonal and cubic phases, henceforth Co shows the opposite sign of magnetic
moment in both phases. From the analysis of the exchange coefficients, Mn(I)
and Mn(II) exhibit rather rigid magnetic moments. Moreover, the exchange
coefficients with Co in tetragonal phase are weaker than cubic phase. As a
result, Co moments in tetragonal phase is smaller than cubic one.

##  IV Conclusion

In summary, we have investigated the effect of Co substitution on Mn3Ga for
structural and magnetic properties. The ferrimagnetic feature is robust
regardless of Co concentration ($x$). When $x\leq 0.5$, tetragonal phase is
favored over cubic and vice versa, where the magnetic behavior of each phase
is different. In tetragonal phase, ${\rm M_{tot}}$ decreases with $x$ while in
cubic phase ${\rm M_{tot}}$ increases with $x$. Electronic structure upon Co
substitution is investigated with PDOS, whose peak structures are analyzed.
Except Co, magnetic moments of individual atom change little with $x$. Co
magnetic moment in tetragonal phase nearly vanishes in contrast to 1$\mu_{B}$
of cubic phase. To get more insight on this contrasting Co moment in two
phases, the exchange coefficients are estimated using the Heisenberg model.
Indeed, the antiparallel magnetism, more precisely, the ferrimagnetism between
Mn(I)-Mn(II) with different magnitudes is well manifested. On the other hand,
the exchange coefficients of Co-Mn(I) and Co-Mn(II) exhibit opposite signs in
tetragonal and cubic phase.

###### Acknowledgements.

We are grateful to Soon Cheol Hong for fruitful discussions and comments. This
work was supported by the National Research Foundation (NRF) of Korea
(2019R1I1A3A01059880, 2020R1A2C3008044).

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