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2592
ISSN 0036-0244, Russian Journal of Physical Chemistry A, 2021, Vol. 95, No. 13, pp. 2592–2599. © Pleiades Publishing, Ltd., 2021.
First-Principle Investigations of (Ti1–xVx)2FeGa Аlloys. A Study
on Structural, Мagnetic, Еlectronic, and Еlastic Рroperties
O. Örneka,*, A. İyigörb, A. S. Meriça, M. Çanlıc, M. Özdurand, and N. Arıkane,**
a Department of Materials and Metallurgical Engineering, Kırşehir Ahi Evran University, Kırşehir, 40200 Turkey
b Department of Machinery and Metal Technology, Kırşehir Ahi Evran University, Kırşehir, 40100 Turkey
c Department of Chemistry and Chemical Processing Technologies, Kırşehir Ahi Evran University, Kırşehir, 40500 Turkey
d Department of Physics, Faculty of Arts and Sciences, Kırşehir Ahi Evran University, Kırşehir, 40200 Turkey
e Department of Medical Services and Techniques, Osmaniye Korkut Ata Universit, Osmaniye, 80010 Turkey
* e-mail: osmanornek@ahievran.edu.tr
** e-mail: nihatarikan@osmaniye.edu.tr
Received August 28, 2020; revised December 18, 2020; accepted December 26, 2020
Abstract—The structural, magnetic, electronic and elastic properties of ternary and quaternary
(Ti1‒xVx)2FeGa alloys with inverse-Heusler (XA) structure were investigated at x = 0, 0.25, 0.50, 0.75, and 1.
The crystal structures of (Ti1–xVx)2FeGa compounds are cubic (space group: F3m) with Hg2CuTi prototype
for x = 0 and 1. At x = 0.5 the structure is also cubic (space group: F3m) with LiMgPdSn protype, while it
is tetragonal (space group: Pm2) at x = 0.25 and 0.75. Calculated optimized lattice parameters (aand c),
bulk modulus (B), and elastic constants (Cij) are consistent with the available data in the literature. Total and
partial magnetic moments of (Ti1–xVx)2FeGa alloys were obtained. An increase in the total magnetic
moment values were observed upon addition of V to the Ti2FeGa alloy. From spin polarized band calcula-
tions, Ti2FeGa, (Ti0.75V0.25)2FeGa, TiVFeGa, and V2FeGa have a minority-spin energy gap of 0.65, 0.38,
0.83, and 0.64 eV, respectively, and they are guessed as half-metallic ferromagnets. According to the results
of second-order elastic constants, these compounds met the Born mechanical stability criteria. In addition,
according to Pugh criteria, it was found that they have a ductile structure and show anisotropic behavior.
Keyword: ab initio calculations, band calculations, mechanical properties, Heusler phase
DOI: 10.1134/S003602442113015X
INTRODUCTION
Heusler alloys are widely studied since their dis-
covery in 1903 [1]. Spin electrons on the alloys allow
them to be used for especially in conducting magne-
tism, ferromagnetic shape memory [1, 2]. In the basis
of Heusler alloys, non-ferromagnetic materials turn
out to be ferromagnetic alloys with high Curie tem-
perature which have capacity of being used in spin-
tronics industry. There are three types of phases
according to their positioning called full-Heusler
alloys (X2YZ) with L21 (no. 225) crystal structure,
inverse-Heusler alloys (X2YZ) with XA (no. 216) crys-
tal structure and half-Heusler alloys (XYZ) with C1b
(no. 216) crystal structure. The full Heusler com-
pounds are ternary intermetallics with a 2 : 1 : 1 stoi-
chiometry (X2YZ) in which X and Y are transition
metals and Z is the main group element.
Bacon and Plant [3] created possible scenarios for
both X2YZ and XYZ chemical ordering. Heusler alloys
consist of four face-centered cubic sublattices, which
can be characterized by the Wyckoff coordinates A (0,
0, 0), B (0.25, 0.25, 0.25), C (0.5, 0.5, 0.5), and D
(0.75, 0.75, 0.75). Since most of the Heusler alloys are
half metallic, they are categorized as crystallizing in
L21 structure. Their space group can be either Fm m
(full Heusler) or F3m (inverse Heusler) [2, 4, 5].
Inverse Heusler materials can be obtained in structure
XA (X2YZ) by replacing one of the X atoms of one of
the full Heusler materials in the structure of L21 with
the Y or Z atom and have F3m space group; so, it is
necessary to investigate the effect of site occupation on
the half-metallicity of Heusler alloys. Wei et al. [6]
found Ti2-based compounds as insensitive to the lat-
tice distortion with fully spin polarization. Liping et al.
[7] focused on searching structural, electronic, and
magnetic properties of Ti2FeGa. The bonds between
Ti(A)–Ti(B) coupling and Fe atoms come from elec-
tron hybridization of the d states which lies on the
band gaps. According to Drief et al. [8], the gap energy
4
4
4
3
4
4
STRUCTURE OF MATTER
AND QUANTUM CHEMISTRY
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
FIRST-PRINCIPLE INVESTIGATIONS 2593
equal to 0.60 eV for Ti2FeGa, in the spin-down state
and divulge metallic intersections at the Fermi level
for the spin-up state. Zhang et al. [1] found Ti2FeGa is
most likely to form Cu2MnAl-type structure. Besides,
Goraus and Czerniewski [9] characterize Ti2FeGa as a
material with ferromagnetic ordering within the 30–
50 K range, with the latter showing some spin glass
type behavior, and metallic type density of states.
Although some of the structural features of
Ti2FeGa alloy has been studied in terms of their theo-
retical aspects, the vanadium doped properties of this
alloy have not been investigated at sufficient levels.
Unique band structures can be ob served in most of the
vanadium based inverse-Heusler compounds. In these
compounds, there is a higher opportunity to get half-
metallic ferromagnetism (HMFs). None of the previ-
ous studies has investigated quadratic elastic constants
and their respective elastic properties. One of the main
problems in estimating the properties of materials is
finding an accurate ab-initio method for simulation
modeling of materials. Simulation allows discovering
some properties that have not yet been experimentally
investigated or are difficult to measure experimentally.
In this study, the Ti2FeGa inverse-Heusler alloy,
which has a limited number of studies in the literature,
was added with various ratios of V atoms to get
(Ti1‒xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1) alloys.
Structural, electronic, magnetic and mechanical
properties of these alloys were determined in Density
Functional Theory (DFT) by using GGY-PBE pseu-
dopotentials. There was no prior study in literature
about the alloys formed by selecting x = 0.25, 0.5, 0.75
ratios. Therefore, this study will provide an insight into
structural, magnetic, electronic and mechanical prop-
erties of these alloys.
CALCULATION METHODOLOGY
The MedeA-VASP software package [10, 11] was
used for DFT calculations with the projected aug-
mented-wave (PAW) potentials. The generalized gra-
dient approximation (GGA) of Perdew–Burke–Ern-
zerhof (PBE) [12] was performed for the exchange–
correlation energy function. The stable structures that
can be formed when V atom is added instead of Ti
atom by using MedeA: Substitutional Search module
are determined as Pm2 tetragonal structure for
(Ti0.75V0.25)2FeG, and cubic structure (F3m) for
Ti2FeGa, TiVFeGa, and V2FeGa. A plane-wave basis
set with a 500 eV energy cut-off was employed for the
spin-polarized calculation of structural, electronic,
and elastic properties. The energy convergence criteria
were arranged to value 10−10 eV using the Normal
(blocked Davidson) algorithm and reciprocal space
projection operators. The k-points created with
gamma-points were performed on 8 × 8 × 8 in the
total energy calculation for the F3m cubic structures
4
4
4
and on 12 × 12 × 12 in the state density calculation,
and on 8 × 8 × 6 in the total energy calculation for the
Pm
2 tetragonal structure and on 10 × 10 × 7 in the
state density calculation. The Fermi distribution func-
tion with a smearing parameter of 0.2 eV was employed
to integrate the bands at Fermi level [13]. Elastic con-
stant was calculated via stress-strain technique [14].
RESULTS AND DISCUSSION
Ti2FeGa (x = 1) and V2FeGa (x = 0) alloys crystal-
lize in XA or Hg2CuTi prototype structure (space
group: F3m). The TiVFeGa (x = 0.5) alloy crystal-
lizes LiMgPdSn in the cubic structure (F3m space
group) as a prototype. Alloys of (Ti0.75V0.25)2FeG
(x= 0.25) and (Ti0.25V0.75)2FeG (x = 0.75) composi-
tion crystallize in the tetragonal phase. All three
phases were presented in Fig. 1. In the first step,
(Ti1‒xVx)2FeGa alloy’s lattice parameters (a and c),
the bulk module (B) and formation energy were
calculated (Table 1) and compared to available data.
The calculated lattice constants and formation
energies of the cubic alloys in the F3m space group
and tetragonal alloys in the Pm2 space group are
shown in Table 1. Computed lattice constants of
Ti2FeGa and V2FeGa inverse Heusler alloys were
compared with other theoretical studies in the litera-
ture. According to this, the lattice constant value
obtained for the Ti2FeGa alloy was determined with a
difference of 0.02% from a GAST-PW91 using the
CASTEP package program [1] and from the study
GGY-PBE using the VASP program package, and
with a difference of 1.3% from GGY-WC using the
WIEN2k program package calculated by Drief et al.
[8]. In addition, the lattice constant of the Ti2FeGa
alloy was experimentally obtained with a difference of
1.2% in the experimental and theoretical study of
Goraus and Czerniewski. The lattice constant value of
the V2FeGe alloy differs by 0.18% from the value
obtained by Ma et al. [15] using the VASP codes
according to the GGY-PBE method, and with a dif-
ference of 0.17% from the value obtained by Zhang
et al. [16] with the CASTEP program package accord-
ing to the GGY method. There has been no compari-
son made for the TiVFeGa (Ti0.75V0.25)2FeGa and
(Ti0.25V0.75)2FeGa alloys, since no studies have been
found in the literature.
In Table 1, considering the lattice constants of the
alloys, for the cubic structure at x = 0, 0.5, 1, with
increase in vanadium content, the lattice constant val-
ues decreases. For the structures with tetragonal sym-
metry (x = 0.25 and 0.75), it is clearly seen that both
the a and c lattice constant and the c/a ratio decrease
upon addition of vanadium. This is caused by the fact
that the atomic diameter of vanadium is lower than
that of titanium.
4
4
4
Δ
()
f
H
4
4
2594
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
ÖRNEK et al.
The formation energy is a significant parameter
which gives deep insight about whether studied com-
pound is structurally stable and experimentally syn-
thesizable or not [17]. The formation energy of
Ti2FeGa can be obtained from the following equation:
where is the equilibrium total energy, ,
, and are the energies per atom. As the com-
puted formation energy value is negative, these com-
pounds are thermodynamically stable and may be syn-
thesized experimentally.
Considering the spin additive, the calculations
were made according to spin polarizations for
(Ti1‒xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1) alloys.
Slater–Pauling behavior (SPB) containing informa-
tion on total magnetic moments can be applied for
Heusler alloys. Since the 9 valence bands of the alloys
in the Heusler structure (x = 0, 0.5, 1) examined in
this study are full, the total magnetic moment accord-
ing to SPB can be found by the rule
of [16, 17]. is the sum of the
valence electrons in each atom forming the alloy or the
sum of the number of electrons in the spin orbits as
. When total magnetic moments are
calculated according to SPB, it is 1 μB for Ti2FeGa
inverse-Heusler alloy, 2 μB for TiVFeGa four-Heusler
alloy, and as 3 μB for V2FeGa inverse-Heusler alloy. It
is clearly seen in Table 2 that the total magnetic
moments belonging to these Heusler alloys (x = 0, 0.5,
1) obtained because of calculations are in agreement
with the values in the literature and the values obtained
from the SPB. However, it may not always be true to say
that (Ti0.75V0.25)2FeGa and (Ti0.25V0.75)2FeGa alloys in
the tetragonal structure are compatible with SPD.
Therefore, no comparison was made for these alloys in
tetragonal structure by applying the SPD rule. Never-
theless, there is an increase observed in the total mag-
netic moment values with increasing contribution of
the V atom to the Ti2FeGa alloy. Since the total mag-
netic moments of (Ti1‒xVx)2FeGa (x = 0, 0.25, 0.5,
0.75, 1) alloys have non-zero values, spin orientations,
electronic band curves, and state intensities are anti-
symmetrical. The greatest contribution to the total
magnetic moments of these alloys comes from d orbit-
als of Ti, V, and Fe atoms which form the alloys when
the partial state density curves are examined.
Many of the physical properties of solids are
directly or indirectly related to the electronic band
structure. In the electronic band structure of half-
metal alloys, one spin-oriented electrons are conduc-
tive, while the other spin-oriented electrons act as
insulators or semiconductors. Electronic band struc-
tures of the alloys of (Ti1–xVx)2FeGa (x = 0, 0.25, 0.5,
0.75, 1) are given in Fig. 2. Examination of these elec-
tronic band curves showed that there is no forbidden
energy range in the Fermi energy level and the bands
Δ= − + +
2
Ti FeGa Ti Fe Ga
[2],
f
HE EE E
2
Ti FeGa
E
Ti
E
Fe
E
Ga
E
=−
B
(18)
μ
tt
MZ
t
Z
=↑+↓
()
t
ZN N
overlap the Fermi energy level in the spin up orienta-
tion of the inverse and quadruple Heusler alloys with
x= 0, 0.5, 1. Therefore, it can be said that these alloys
show metallic properties in spin up direction. In spin
down orientation, Ti2FeGa, (Ti0.75V0.25)2FeGa,
TiVFeGa, and V2FeGa alloys have a band gap around
the Fermi level, and these alloys show a semiconduc-
tor feature in the spin down state. In this case, these
four alloys exhibit half-metallic behavior. Also these
alloys have an indirect band gap between valence and
conductivity bands. The band gap values of these
alloys are presented in Table 2. It was observed that the
inverse-Heusler alloy Ti2FeGa has the band gap values
compatible to the data found in the available literature
[18, 19]. However, there is no data in the literature to
compare a band gap value for other alloys with. In
addition, the (Ti0.25V0.75)2FeGa tetragonal alloy has a
small band gap just below the Fermi energy level in the
spin downward direction. This is because the tetrago-
nal alloy exhibits only a half-metallic behavior.
Fig. 1. Structures (a, b) for x = 0 and 1 the space group
F3m inverse-Heusler alloys, (c, d) tetragonal and belongs
to the space group Pm2, (e) the quadruple Heusler struc-
ture in the F3m space group.
(a) x = 0 (b) x = 1
(c) x = 0.25 (d) x = 0.75
(e) x = 0.50
F43m
P4m2
F43m
ab
c
4
4
4
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
FIRST-PRINCIPLE INVESTIGATIONS 2595
The total state intensity (TDOS) and partial state
intensity (PDOS) was calculated to characterize the
electronic band structures of the (Ti1– xVx)2FeGa (x =
0, 0.25, 0.5, 0.75, 1) alloys in more detail. Figure 3
shows the TDOS and PDOS diagrams calculated for
different (Ti1– xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1)
alloys.
Considering the spin up bands of the five alloys,
whose total and partial state densities are examined,
the highest contribution to the Fermi level was the
electrons from the d-states of the alloys. In the alloys
of (Ti1–xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1), while
x= 0, 0.5, 1 contributed the inverse and quaternary
Heusler alloys had dominant Ti-d orbital electrons,
increase in the addition rate of the V atom showed
similar pattern in the contribution of Fe-d orbital
(states). When x = 0.25 or 0.75, the contribution of the
Fe-d orbital decreases in contrast to the cubic phase
with the increase of the V atom in tetragonal alloys. In
addition, these findings were compatible with the
studies in the literature considering the electronic
properties [1, 6–8, 15, 16].
Elastic constants (Cij) give information about the
strength of a material and its resistance to external
forces. Elastic properties of solids are closely related to
many physical properties such as melting point, heat
capacity, thermal expansion coefficient, and Debye
temperature. The materials with the cubic structure
Table 1. Calculated structural parameters (a, c in Å; V in Å3) and formation enthalpy , eV/atom)
Materials Ref. acc/aV
Ti2FeGa This study 6.101 — — 227.119 –0.372
Exam. [9] 6.177 — — —
GGY-PBE [13] 6.122 — — —
GGY-PBE [6] 6.120 — — —
GGY-PBE [7] 6.100 — — —
GGY-WC [8] 6.0226 — — —
PW [7] 6.177 — — —
GGY-PW91 [1] 6.100 — — —
(Ti0.75V0.25)2FeGa This study 4.246 6.150 1.448 110.857 –0.360
TiVFeGa This study 5.995 — — 215.464 –0.277
(Ti0.25V0.75)2FeGa This study 4.210 5.951 1.413 105.478 –0.268
V2FeGa This study 5.921 — — 207.656 –0.282
GGY-PBE [13] 5.910 — — —
CASTEP-GGY [14] 5.911 — — —
Δ
f
()
H
Δ
f
H
Table 2. (Ti1–xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1) alloys total and partial magnetic moments (Mi; μB) and forbidden energy
ranges in spin-down direction (Eg, eV)
Materials Ref. MTMa-Ti(V) Mb-V(Ti) MFe MGa Eg
Ti2FeGa This study 1.0074 0.998 0.754 –0.941 0.000 0.65
GGY-PBE [13] 1 1.005 0.757 –0.966 0.009 0.57
GGY-PBE [6] 1 1.22 0.93 –1.09 –0.06 0.56
GGY-PBE [2] —————0.56
GGY-PBE [7] 1.002 1.123 0.904 –1.196 0.008 0.59
GGY-WC [8] 1.0016 0.7886 0.5026 — 0.00537 0.64
GGY-PW91 [1] 0.98 1.00 0.76 –0.86 0.08 0.56
(Ti0.75V0.25)2FeGa This study 1.5024 0.560 0.983 –0.440 –0.002 0.38
TiVFeGa This study 1.8025 0.829 0.558 0.244 –0.008 0.83
(Ti0.25V0.75)2FeGa This study 1.9901 0.337 0.764 0.725 –0.005 0
V2FeGa This study 2.8365 1.698 –0.226 1.182 –0.010 0.64
GGY-PBE [13] 2.8487 1.803 –0.231 1.200 –0.001 —
CASTEP-GGY [14] 2.86 2.34 –0.50 1.18 –0.16 —
2596
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
ÖRNEK et al.
have three elastic constants, such as the second order
C11, C12 , and C44, independent of each other. Tetrago-
nal materials have six elastic constants, such as C11,
C12, C13, C33, C44, and C66, which are independent of
each other. Calculated Cij elastic constants for
(Ti1‒xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1) alloys are
summarized in Table 3. Born mechanical stability cri-
teria for elastic constants are expressed for cubic crys-
tals as following equation:
(1)
and for crystals in tetragonal structure [20, 21]:
(2)
Considering values of C11, C12, C13, C33, C44, and C66 in
Table 3, all of these alloys met the mechanical stability
conditions. Therefore, both cubic inverse and quater-
nary Heusler alloys and tetragonal alloys can be con-
sidered mechanically stable in crystal symmetries. At
the same time, when the elastic constants values of
these alloys are examined, the C11 value increases with
the increasing contribution of the V atom for all the
alloys. In C12 value, it can be said that cubic alloys
increase among themselves and tetragonal alloys
among themselves in proportion with V contribution.
In addition, the V2FeGa alloy is the highest in C11 , C12,
−> + > >
11 12 11 12 44
0, ( 2 ) 0, 0
CC C C C
><+
>>
2
11 12 13 33 11 12
44 66
,2 ( ),
0, 0.
CC CCCC
CC
and C44 values in the elastic constants among these
alloys. Since there are no experimental or theoretical
data about elastic constants for these alloys in the lit-
erature, no comparison could be done. Thus, elastic
constants and related mechanical properties of these
alloys are first time presented in this study.
The calculated elastic constants of the
(Ti1‒xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1) alloys were
used for finding bulk module (B), shear modulus (G),
Young modulus (E), Poisson ratio (σ), and anisotropy
factor (A) and presented in Table 4. Bulk module val-
ues of these alloys were between C11 and C12 elastic
constant values as expected. The bulk modulus value
of the Ti2FeGa inverse-Heusler alloy was compatible
with a difference of 0.7% from Wei et al. [6] and with
a slightly higher difference from other studies. Besides,
the bulk modulus value increased in direct proportion
to the V fraction in the Ti2FeGa alloy. As an expres-
sion of compressibility, the shear module, G, the stud-
ied alloys were found to be more compressible since
these alloys had a value below 100 GPa. It can also be
said that the inverse-Heusler Ti2FeGa alloy with the
lowest shear modulus value is more compressible than
the other alloys. In order to obtain information about
the fragile or ductile nature of materials, Pugh’s crite-
rion was used. According to this criterion, if the B/G
ratio is less than 1.75, the material is fragile, and if the
Fig. 2. Spin-polarized electronic band curves of the alloy (Ti1–xVx)2FeGa (the Fermi level is set to zero energy).
8
10
6
4
2
0
2Ti2FeGa
L
Energy, eV
XWKW*
8
10
6
4
2
0
2V2FeGa
L
Energy, eV
XWKW*
8
10
6
4
2
0
2(Ti0.75V0.25)2FeGa
M
Energy, eV
XZ ZAR** MX Z ZAR**
8
10
6
4
2
0
2TiVFeGa
L
Energy, eV
XWKW*
8
10
6
4
2
0
2(Ti0.25V0.75)2FeGa
Energy, eV
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
FIRST-PRINCIPLE INVESTIGATIONS 2597
ratio is higher, it has a ductile nature. Accordingly,
since the B/G ratio of these alloys is greater than 1.75,
they are all ductile. For Heusler alloys with cubic crys-
tal structure, the ductility decreases with increase in V
fraction. For the alloys with x = 0.25, 0.75 having
tetragonal crystal structure, the ductility increases
with increase in V content (Table 4).
Considering the Young’s module, which is an
expression of stiffness, in contrast to the B/G ratio, the
hardness increases with increase in V content for the
Fig. 3. Total and partial state intensities of (Ti1–xVx)2FeGa alloys.
0.8
0.4
0
0.4
0.8 810 642
0.6
0.3
0
0.3
0.6
8
4
0
Toplam
Ti d
Ti p
V p
Ga p
Ga s
Fe s
V s
Ti s
V d
Fe d
x = 0.25
4
8
0
DOS (states/eV CELL)
Energy, eV
2
0.4
0.2
0
0.2
0.4 810 642
0.4
0.2
0
0.2
0.4
6
3
0
Toplam
Ti d
Ti p
Ga p
Ga s
Fe s
Ti s
Fe d
x = 0
3
6
0
DOS (states/eV CELL)
Energy, eV 2
0.8
0.4
0
0.4
0.8 810 642
0.6
0.3
0
0.3
0.6
8
4
0
Toplam
Ti d
Ti p
V p
Ga p
Ga s
Fe s
V s
Ti s
V d
Fe d
x = 0.25
4
8
0
DOS (states/eV CELL)
Energy, eV
2
0.4
0.2
0
0.2
0.4 810 642
0.4
0.2
0
0.2
0.4
6
3
0
Toplam
Ti d
V p
Ga p
Ga s
Fe s
V s
Fe d
x = 1
3
6
0
DOS (states/eV CELL)
Energy, eV 2
0.4
0.2
0
0.2
0.4 810 642
0.2
0.1
0
0.1
0.2
4
2
0
Toplam
Ti d
Ti p
V p
Ga p
Ga s
Fe s
V s
Ti s
V d
Fe d
x = 0.50
2
4
0
DOS (states/eV CELL)
Energy, eV
2
Table 3. (Ti1–xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1) alloys in second order elastic constants (Cij, GPa)
Materials C11 C12 C13 C33 C44 C66
Ti2Fe Ga 137.3 4 121. 0 6 — — 75.98 —
(Ti0.75V0.25)2FeG a 23 6.5 3 8 0.7 7 12 2.6 8 181.7 9 92 .77 37.41
TiVFeGa 202.57 138.74 — — 55.89 —
(Ti0.25V0.75)2FeGa 245.21 94.25 139.11 211.46 73.54 38.81
V2FeGa 253.68 130.24 — — 91.23 —
2598
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
ÖRNEK et al.
cubic structures, and the hardness decreases with
increase in V fraction for the tetragonal structures.
The calculated Poisson’s ratio contains general infor-
mation about atomic bonding. The value of the Pois-
son’s ratio is close to 0.1 in covalent materials and
close to 0.25 in ionic materials [22, 23]. The calculated
Poisson’s ratios of these alloys are between 0.30 and
0.38. Since this values are closer to 0.25, they appear to
exhibit ionic character.
Another parameter obtained within the scope of
mechanical properties is anisotropy factor. Anisotropy
factor is equal to 1 for isotropic materials. Thus, the
two- and three-dimensional dependences of the
Young module were calculated for all materials using
the ELATE program [24] and presented in Fig. 2. The
plots for isotropic materials have spherical shape. The
amount of deviation in Fig. 2 indicates the degree of
anisotropy. As can be clearly seen from all planes in
Fig. 4, all alloys are anisotropic.
Table 4. Bulk modulus (B, GPa), shear modulus (G), B/G, Young modulus (E), Poisson ratio (σ), and anisotropy factor
(A) of the studied alloys
Materials Ref. B, GPa G, GPa B/GE, GPa σA
Ti2FeGa This study 126.49 33.19 3.81 91.57 0.38 9.32
GGY-PBE [6] 125.60
GGY-PBE [7] 172.63
GGY-WC [8] 152.74
(Ti0.75V0.25)2FeGa This study 145.14 60.22 2.41 158.70 0.32 2.82
TiVFeGa This study 160.01 44.64 3.58 122.53 0.37 1.75
(Ti0.25V0.75)2FeGa This study 160.69 55.78 2.88 149.99 0.34 2.14
V2FeGa This study 171.39 78.00 2.19 203.18 0.30 1.48
Fig. 4.Three-dimensional (3D) and two-dimensional (2D) directional dependences of the Young’s modulus of (Ti1–xVx)2FeGa
alloys.
Ti
2
FeGa
3D
2D
100
50
50
100
100
50
0
50
100
0
100
50
0
50
100
50 0
Young's modulus in (xy) plane
25
50
mo
pn
0+
25
50
50 50 0
Young's modulus in (xy) plane
25
50
mo
pn
0+
25
50
50 50 0
Young's modulus in (xy) plane
25
50
mo
pn
0+
25
50
50
N
yx
(Ti
0.75
V
0.25
)
2
FeGa
3D
2D
100
150
50
50
100
150
150
50
100
0
50
100
150
0
100
150
50
0
50
100
150
250 0
Young's modulus in (xy) plane
mo
pn
0+
250
250 0
Young's modulus in (xy) plane
mo
pn
0+
250
250 0
Young's modulus in (xy) plane
mo
pn
0+
250
z
yx
TiVFeGa
3D
2D
100
50
50
100
100
50
0
50
100
0
100
50
0
50
100
0
Young's modulus in (xy) plane
50
100
mo
pn
0+
50
100
0
Young's modulus in (xy) plane
50
100
mo
pn
0+
50
100
0
Young's modulus in (xy) plane
50
100
mo
pn
0+
50
100
z
yx
(Ti
0.25
V
0.75
)
2
FeGa
3D
2D
100
50
50
100
150
100
50
0
50
100
150
0
100
150
50
0
50
100
150
250 0
Young's modulus in (xy) plane
250
mo
pn
0+
250
250 250 0
Young's modulus in (xy) plane
250
mo
pn
0+
250
250 250 0
Young's modulus in (xy) plane
250
mo
pn
0+
250
250
z
yx
V
2
FeGa
3D
2D
100
150
50
50
100
150
100
150
50
0
50
100
150
0
100
150
50
0
50
150
100
250 0
Young's modulus in (xy) plane
250
mo
pn
0+
250
250 250 0
Young's modulus in (xy) plane
250
mo
pn
0+
250
250 250 0
Young's modulus in (xy) plane
250
mo
pn
0+
250
250
z
yx
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A Vol. 95 No. 13 2021
FIRST-PRINCIPLE INVESTIGATIONS 2599
CONCLUSIONS
The structural, magnetic, electronic, and elastic
properties of (Ti1–xVx)2FeGa (x = 0, 0.25, 0.5, 0.75, 1)
alloys were calculated using the generalized gradient
approach with the Density Functional Theory. Con-
sidering lattice constants of the alloys among them-
selves, x = 0, 0.5, 1 in the cubic structure, the lattice
constant values decrease with increase in V content.
For structures at x = 0.25 and 0.75 with tetragonal
symmetry, both a and c lattice constants and the c/a
ratio decrease with increase in V content. This is
caused by the fact that the atomic diameter of V is
lower than that of Ti. The bulk modulus value of the
Ti2FeGa inverse-Heusler alloy is in good agreement
with other theoretical studies. There was no available
data to compare bulk modulus values calculated for
other alloys. Total and partial spin magnetic moments
of all the alloys were calculated and found that the cal-
culated total and partial spin magnetic moments of all
the cubic alloys were in good agreement with other
theoretical calculations and Slater–Pauling rule. No
comparison was made for the alloys with the tetrago-
nal structure by applying the Slater–Pauling rule.
With increase in V content, there is an increase in the
total magnetic moment values.
Spin-polarized electronic structure calculations of
Ti2FeGa, (Ti0.75V0.25)2FeGa, TiVFeGa, and V2FeGa
alloys have shown that these alloys exhibit half-metal-
lic behavior. Tetragonal (Ti0.25V0.75)2FeGa alloy has a
small band gap just below the Fermi energy level in the
spin downward direction. It can be said that this
tetragonal alloy exhibits only a half-metallic behavior.
Second-order elastic constants and related elastic
properties were calculated and evaluated for the first
time. When Born stability conditions are taken into
consideration, it is seen that all these alloys obey
mechanical stability conditions. Equilibrium lattice
constants, elastic constants and values obtained from
bulk module (B), shear module (G), and Young mod-
ule (E) confirm that the alloys are elastic and structur-
ally stable. The studied alloys have a ductile nature, as
it follows from the calculated B/G ratio, anisotropy,
Pugh ratio, Poisson ratio, and Cauchy pressure. For
Heusler alloys with a cubic crystal structure, with
increasing V contribution, the ductility decreases, and
for the alloys with tetragonal crystal structure, the
ductility increases with increasing V contribution.
From the Poisson’s ratio calculated for these alloys, it
was found that all the alloys have ionic character. All
these alloys were determined to be anisotropic.
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