J. Nig. Soc. Phys. Sci. 3 (2022) 138–145

Journal of the
Nigerian Society

of Physical
Sciences

Half-metallic Characteristics of the Novel Half Heusler Alloys
XCrSb(X = T i, Zr, Hf )

B. E. Iyorzor∗, M. I. Babalola

Department of Physics, University of Benin, Benin City, Nigeria.

Abstract

Ab-initio calculations are performed to examine the structural, mechanical, electronic, magnetic and thermodynamic properties of the half-Heusler
ternary alloys XCrSb (X = H f , Ti, Zr). In this study, the spin-polarized density functional theory (DFT) method that is spin-polarized with
generalised gradient approximation (GGA) are used to perform ab-initio calculations to investigate the physical properties of a novel half-Heusler
ternary alloys XCrSb (X = H f , Ti, Zr). It was confirmed that the alloys are stable mechanically and exhibit ferromagnetic states (FM). The study
reveals that the alloys portray half-metallic character with narrow energy gaps. And it also shows that they have a total magnetic moment of
approximately 3µβ. From the formation energy calculation, it shows that the alloys can be synthesized experimentally. Also, it was observed that
they are mechanically stable. The heat capacities and Debye temperatures were also computed and they show high thermodynamic stability.

DOI:10.46481/jnsps.2021.297

Keywords: Half-Heusler Alloys, Half-metallic ferromagnet (HMF), Spin-polarization, Band Structure, Density of state, Quasi-Harmonic
Approximation (QHA)

Article History :
Received: 08 July 2021
Received in revised form: 22 August 2021
Accepted for publication: 24 August 2021
Published: 28 February 2022

c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: E. A. Emile

1. Introduction

The Heusler alloys have been studied extensively in the re-
cent past, and one of the major results is that they are among the
best half-metallic ferromagnets (HMF) to achieve 100% spin
polarization at room temperature. Density Functional Theory
(DFT) has proven to be a reliable tool for calculating elec-
tronic structures for many decades, this is due to its simple
and usefull approach in approximating the ground state func-
tionals of real many-body electrons. The theory is built on the
fact that the properties of many-body interacting system can

∗Corresponding author tel. no:
Email addresses: beniyorzor@uniben.edu (B. E. Iyorzor ),

michael.babalola@uniben.edu (M. I. Babalola )

be seen as a functional of the ground state electron density
[1]. From the works of Mehmood et al [2-3], it was gathered
that the half-Heusler alloys YZSb (Z = Cr, Mn) and RhCrZ
(Z = S i, Ge) exhibit half-metallic properties within the frame-
work (DFT) among others. Also, the electronic, magnetic and
optical properties were equally studied. The field of spintron-
ics is quite different from the conventional electronics in which
charges are responsible for transfer of information, whereas in
spintronics spins are responsible for the transfer of information.
Some of these enhancements and advantages are; increasing the
speed of the data processors, drop in the consumption of electric
power and the increase in integration density [4−6]. The study
of half Heusler (HH) alloys present interesting and different
magnetic occurances, and this have attracted researchers in re-

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Iyorzor & Babalola / J. Nig. Soc. Phys. Sci. 3 (2022) 138–145 139

cent time [7−15]. Ternary half-Heusler compounds are usually
denoted by the chemical formula XYZ, where X is rare-earth
metal, Y is transition metal and Z is the main group elements in
the periodic table. This area has attracted a lot of attention by
many researchers for about a decade now. Relatively, the HH
compounds form a big family of materials with several physi-
cal properties and applications especially in the areas of opto-
electric semiconductors and spintronics [16 − 18], thermoelec-
tiric semiconductors [19 − 21], piezoelectric semiconductors
[22] and topological insulators [23, 24]. Considering the half
metallic materials, the majority and minority spin-bands exhibit
different behaviours. The majority spin band portrays metallic
behaviour while the minority spin band displays a semiconduc-
tor character with a narrow band gap within the Fermi level.
And around the Fermi level, the gap results in a 100% spin-
polarization [25, 26]. Since the discovery of the half-metallicity
in the (HH) compounds, numerous researchers have done vigor-
ous studies determining the stability of the physical/mechanical,
electronic/phonon dispersion and as well as thermodynamic prop-
erties using First-Principles calculations. Amoung them are:
ZrMnAs [27], PtXBi (X = Fe, Co, Mn and Ni) [28], ZrNiPb
[29], and XVSb (X = Fe, Ni, Co) [30]. The works of Rogl and
coworkers [31] provides an extensive review on the mechani-
cal properties of many HH systems, like the XNiSn and XCoSb
where X = T i, Zr and Hf. They studied the fracture toughness,
hardness/elastic moduli and density-porosity of alloys empiri-
cally and theoretically. However, nothing has been mentioned
about the replacement of Co atom with Cr atom in the XCoSb
series. Considering the industrial uses of chromium, ranging
from hardening of steel, chromium plating, leather tanning, as
well as industrial catalysts to pigmentation. Therefore, it be-
came imperative in this present study, to predict the physical,
mechanical, electronic, magnetic and thermodynamic proper-
ties of the novel half-Heusler alloys using ab-initio calculations

2. Methodology

The ab-initio total energy calculations were executed em-
ploying the QUANTUM ESPRESSO (QE) as applied in the
works of Giannozi [32]. The Projected Augumented Wave pseudo
potential was used in the computation [33]. The exchange inter-
relationship between electrons, bonding and magnetic features
were treated with the GGA-PBE approach. The (HH) alloy
XYZ, has an fcc structure, in space group F − 43m No. 216
with its Structurbericht designation of C1 b as recorded in the
International Table of Crystallography. The crystal structures
for the three alloys are presented in Figure 1. We use the spin-
polarized density funtional theory (DFT). The energy cut-off
value and the k-points are essential flags (in the input-file) in
ensuring the accuracy of the calculations [34]. The k-points are
used for the structural optimization and static SCF calculations
for the alloys XCrSb (X = H f ,Ti and Zr). Convergence tests
for the energy cut-off and the k-points are carried out before
determining their appropriate values for the calculations. The
Monkhorst Pack format with 8 x 8 x 8 framework was used,
and the kinetic energy cut-off were fixed to 60Ry for HfCrSb

Figure 1. The crystal structure of XCrSb where X=Ti, Zr and Hf.

and ZrCrSb, 65Ry for TiCrSb alloys. The Pseudo-atomic con-
figurations used as valence electrons, in this study, are Hf 5d2

6s2 , Ti 3d2 4s2 , Zr4d2 5s2 , Cr 3d5 4s1 , and Sb 4d10 5s2 5p3

respectively. The (X = H f ,Ti and Zr), Sb and Cr occupy the
atomic positions 4a(0, 0, 0), 4b( 12 ,

1
2 ,

1
2 ) and 4c(

1
4 ,

1
4 ,

1
4 ) respec-

tively.

3. Results and Discussions

3.1. Structural and Mechanical Properties

Investigating the structural properties of the ground state
configuration of the XCrSb compounds (where X = T i, Zr and
Hf), firstly, the structural/geometry optimisation was carried out
by minimizing the total energy of the alloys with respect to the
variation of the lattice parameters. The lattice constant a, the
bulk modulus B and its pressure derivative B′ were obtained
when the energy-lattice parameter was fitted to the Birch Mur-
naghan equations of state. These results are presented in Table
1 and the obtained energy-lattice curves for the various alloys
are shown in Figure 2. The three HH alloys in different mag-
netic states were studied in order to ascertain the true nature
of the alloys. The results for the ferromagnetic (FM) states,
non-magnetic (NM) states as well as for the anti-ferromagnetic
(AFM) states were presented in Fig. 3. It is observed that the
(FM) states posses the lowest ground state energy for the three
HH alloys, hence they are all ferromagnets. It is the ferromag-
netic states of these alloys that have been used to compute the
other properties in this work. In solids, the elastic constants
and some structural features play vital roles in determining the
mechanical stability of the material [35]. For cubic phases, the
stability is measured by the following criteria C11 + 2C12 > 0,
C11 − C12 > 0, , and C11 > 0 [36]. The alloys are mechanically
stable since all the necessary conditions were satisfied, see Ta-
ble 2. The bulk modulus B, Young Modulus E and shear modu-
lus G are parameters used to quantify the mechanical properties
of solids. From Table 1, the results of the B and E show that
the deformation resistance decreases in trend from HfCrSb to
ZrCrSb to TiCrSb alloys, except for the shear modulus that has
a contrary behaviour. An expression to establish the plasticity
of a material is given by the BG ratio. The threshold value for
distinguishing between ductility and brittleness of materials is
about 1.75 [37]

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Iyorzor & Babalola / J. Nig. Soc. Phys. Sci. 3 (2022) 138–145 140

Figure 2. Relationship between total energies per unit cell and lattice parameters for HfCrSb, TiCrSb and ZrCrSb

From our results presented in Table 2, it is obvious that the
alloys are ductile in nature since the values of the B/G ratio are
more than 1.75. Although, there was no available experimental
or theoretical reports on the XCrSb series for comparison, but
our results for B, E, and the elastic properties are still within
appreciable range when compared with the XCoSb series in
literatures [31, 38]. It is observed that the values of the me-
chanical properties obtained for XCoSb HH alloys are higher
than that of XCrSb obtained in this work. This is due to the
fact that the bonding during the hybridization of the d-orbital
of Cr atom and the d-orbital of X(Ti, Zr and Hf) atom is weak
as shown in Figures 7 − 9. The present calculated results of
Zener anisotropy A, shows that the three alloys are anisotropic,
since the values are approaching 1, which is an indication of
high elastic anisotropy. These results were obtained using the
relation, from equation (1), as recorded in [39].

A =
2C44

C11 − C12
(1)

Another factor that can affect the stability of a material against
shear stress is the Poisson′s ratio µ. It reveals the nature of the
bonding forces in materials. The value range is of the order 0

¡ ν ¡0.5 [39]. From the results of ν in Table 2, it shows that
the three alloys are of good plasticity. The Cauchy relation is
another parameter which expresses the ductility and brittleness
of materials. When the value is positive, the material is consid-
ered ductile, otherwise it is brittle. The alloys under consider-
ation are ductile as reported in Table 2. The formation energy
∆ H of the alloys were investigated to verify whether they can
be synthesized experimentally. The calculation was done using
Equation 2:

∆H = Ect −
(

Ee1b
Na

+
Ee2b
Nb

+ ...

)
(2)

where Ect is the total energy of the alloy, E
e1
b and E

e2
b are the total

energies of the constituent elements respectively, and Na repre-
sents the number of atom per cell. From the calcultions, the for-
mation energies of XCrSb (X = H f , Ti, and Zr) are −1.49eV,
−1.10eV and −1.0eV respectively. These results confirm that
the alloys can be synthesized easily because the enthalpies have
negative values.

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Iyorzor & Babalola / J. Nig. Soc. Phys. Sci. 3 (2022) 138–145 141

Figure 3. The total energies per unit cell as a function of lattice constants for the Ferromagnetic (FM), Antiferromagnetic (AFM) and Non-magnetic (NM) states of
the alloys.

3.2. Electronic and Magnetic Properties

From the crystal structure of the (HH) compounds in Figure
1, the X(X = T i, Zr and Hf) atom forms a tetrahedral position
with the Cr atom as the cation at the center. The zincblende
sublattice [CrX]3+ formed hybridizes with the [S b]−3 ion. With
this configuration the Cr atom gains one extra electron added
to its initially six valence electrons thereby resulting to seven
electrons that occupy thed-orbital. During the filling of the
d-orbital, three electrons are left unpaired which constitue the
magnetic moment of the alloys. The spin magnetic moment
for each of the alloys is approximately 3µ as shown in Table 3.
This satisfies the spin magnetic moment predicted by the Slater-
Pauling rule [40]. The rule asserts that the total spin magnetic
moment per formula unit Mtot is given by Mtot = |Nv − 18| for
(HH) compounds, where Nv is the sum of electrons at the va-
lence. The sum of electrons at the valence for Ti, Zr and Hf are
4 each while that of Cr and Sb are 6 and 5 respectively. Hence
the total number of electrons at the valence for each compounds
is 15. The total and partial magnetic moments are presented in
Table 3.

In order to obtain quantitative results to reveal the half metal-

lic character from the electronic band structure calculations, the
spin polarization P was computed for the minority and majority
bands. From the three alloys, we have 0ev, 0.25ev for the up and
down DOS at the Fermi level respectively. Using Equation (3)
for spin polarization [41], we obtained 100% spin polarization
in the alloys.

P =
Dos

(
E f (up)

)
− Dos

(
E f (down)

)
Dos

(
E f (up)

)
+ Dos

(
E f (down)

) × 100% (3)
The electronic properties of the three (HH) alloys have been

computed in the form of band structures and are presented in
Figures 4 − 6. The (HH) alloys XCrSb (X = T i, Zr and Hf)
are found to posses the half-metallic properties since their band
structures with the majority spin channels have indirect band
gaps while their band structures for the minority spin channels
show metallic behaviour. The measured band gaps for TiCrSb,
ZrCrSb and HfCrSb are 0.31eV, 0.49eV and 0.42eV respec-
tively. The PDOS for the three compounds have also been com-
puted and are presented in Figures 7 − 9. The purpose of the
PDOS plot is to give insight into the nature of bonding between
the orbitals of the individual atoms. The PDOS of the three half

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Iyorzor & Babalola / J. Nig. Soc. Phys. Sci. 3 (2022) 138–145 142

Table 1. The Structural properties: the a (Å), B (GPa), and B′, energy of formation ∆H(eV), band gap near Fermi energy Eg(eV) and magnetic ground state Mg.

Alloy a B B´ ∆H Eg Mg
TiCrSb 6.197 84.96 4.43 -1.10 0.31 FM
ZrCrSb 6.383 95.80 4.65 -1.00 0.49 FM
HfCrSb 6.341 99.66 4.75 -1.49 0.42 FM

Table 2. The mechanical/elastic properties: the C11,C12and C44 (GPa), B / G ratio, E (Gpa), Poisson’s ratio υ, Zener anisotropy A, and the Cauchy relation.

Alloy TiCrSb ZrCrSb HfCrSb
C11 108.80 112.76 128.29
C12 73.02 69.87 85.33
C44 33.70 30.98 32.33
G 27.38 27.17 27.99

B/G 3.10 3.53 3.56
E 74.17 73.58 76.76
ν 0.35 0.35 0.37
A 1.88 1.44 1.51

C12 - C44 39.32 38.89 53.00

Table 3. The total and partial magnetic moments of the half-Heusler alloy XCrSb (X=Hf, Ti, Zr).

Alloy Mx(µβ) MCr (µβ) MS b(µβ) Mtot(µβ)
TiCrSb 0.2899 2.7133 -0.0025 3.0007
ZrCrSb 0.1835 2.8653 -0.0386 3.0102
HfCrSb 0.1900 2.7696 0.0001 2.9597

Table 4. The specific heat capacity Cv , Debye temperature ΘD , Zero Point energy Eo , and Debye sound velocity Vs.

Alloy Cv(J/Nmol.) ΘD(K) E0(kJ/Nmol.) Vs(m/s)
TiCrSb 72.33 253 7.11 2302
ZrCrSb 72.63 238 6.69 2232
HfCrSb 73.16 209 5.87 1946

Heusler alloys are some how similar this is due to the fact that
they have similar atoms inhabiting the Y and Z atomic sites and
the three different atoms inhabiting the X atomic site are in the
same group. From the minority-spin channels of the three HH
alloys, the t2g states of Cr atom dominate the region below the
Fermi energy while the eg states dominate in the region above
the Fermi energy. Since both states are occupied around the
Fermi energy, hence they show metallic character. The gaps
seen around the Fermi energy in the majority spin straits for
the three half-Heusler compounds are due to the crystal field
splitting of the d-orbital of Cr atom into t2g and eg. Due to the
exchange interactions between the Cr d-orbital and the X(Ti,
Zr and Hf) d-orbital, there is depletion of electrons in the ma-
jority spin states leading to unoccupied states. This results to
the development of half-metallic gap around the Fermi energy.
We also observed the bonding states between the Cr d-orbital
and the X(Ti, Zr and Hf) d-orbital just below the Fermi energy
and the anti-bonding states between the Cr d-orbital and the
X(Ti, Zr and Hf) d-orbital just above the Fermi energy. This ob-
servation is in line with observations by other researchers who
worked on half Heusler alloys. It is also important to investigate
the relationship between the magnetic moment and the lattice

constant in order to determine the range of lattice constants at
which the magnetic moment is preserved. From Figure 10, it is
observed that the magnetic moment of 3µβ is preserved within
the range of 11.2a.u to 14a.u for three HH alloys

3.3. Thermodynamic Properties
In Solid State Physics, the Debye temperature ΘD is one ba-

sic factor in describing phenomena associated with many phys-
ical properties, like melting points, lattice vibration, specific
heat, thermal expansion etc. The ΘD is used for delineating high
temperature regions from low temperature regions in solids.
Basically, the ΘD depends on the elastic constants. The ΘD
can be calculated from the average sound velocity, Vm by the
expression [42, 43]

ΘD =
h
kB

(
3n

4πVa

)1/2
Vm

(4)

Vm =
[
1
3

(
1

V 3t
+

2
V 3t

)]−1
3

(5)

Vt =
(

3B + 4G
3ρ

) 1
2

(6)

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Iyorzor & Babalola / J. Nig. Soc. Phys. Sci. 3 (2022) 138–145 143

Figure 4. Band structure of TiCrSb alloy

Figure 5. Band structure of ZrCrSb alloy

A =
2C44

C11 − C12
(7)

where B, G and ρ are the bulk modulus, shear modulus and
densities of the various alloys respectively. The calculated spe-
cific heat capacity, Debye temperatures, Zero Point energy and
the Debye sound velocity for the various alloys are presented in
Table 4. The Debye temperatures computed in this work are in
reasonable range when compared with other half Heuslers [31,
38].

The Debye temperature is related to the strenght of the co-
valent bonds in solids, that is, the higher the ∆D of a compound
indicates a stronger covalent bond which leads to high melting
point. Also, a strong covalent bond is associated with the hard-
ness of the material 44. The other thermodynamic properties
such as the enthalpy H, Free energy F , heat capacity at con-
stant volume Cv and the entropy S of the alloys are evaluated
by using the QHA. The results are shown in Figure 8. We can
see that the Cv of the alloys increases swiftly as the temperature
increases, such that at very high temperature it tends to reach
the Dulong-Petit terminal point. It was also observed that at
very low temperature, Cv is proportional to T 3. The entropy

Figure 6. Band structure of HfCrSb alloy

Figure 7. PDOS for TiCrSb alloy

Figure 8. PDOS for ZrCrSb alloy

and enthalpy graphs show that as the temperature increases the
values of the properties increased, while the reverse is observed
for that of Free energy.

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Iyorzor & Babalola / J. Nig. Soc. Phys. Sci. 3 (2022) 138–145 144

Figure 9. PDOS for HfCrSb alloy

Figure 10. Total magnetic moment per formular unit as a function of lattice
constants for the three alloys.

4. Conclusion

The physical properties of a novel (HH) compounds XCrSb
(where X = H f , Ti, and Zr) have been investigated using ab-
initio calculations. The ferromagnetic state is found to be more
stabled and favourable than the antiferromagnetic and non- mag-
netic states structurally in term of total energy. The alloys are
found to be stable from the mechanical stability conditions.
Also, the B/G ratio and the Cauchy-relation reveals that the al-
loys are ductile. The electronic band structure and the DOS cal-
culated reveal that the alloys exhibit half metallic ferromagnetic
property and having approximately 3µβ magnetic moment. The
total magnetic moment of these alloys is produced through re-
markable exchange splitting between the majority-spin states
and the minority-spin states of Cr orbital. From Table 3, the
Cr atom is found to be the major contributor. Furthermore,
the alloys satisfied the Slater-Pauling rule. The alloys are half-
metallic in nature with energy-gaps of 0.4196eV, 0.309eV, 0.4946eV
for HfCrSb, TiCrSb, and ZrCrSb respectively. Theoretically, it
is shown that these new alloys can be synthesized easily exper-
imentally because of their negative formation energies. Finally,

∆D and Cv computed for the three HH alloys are within accept-
able range when compared with other HH alloys in that series.
Finally, the half-metallicity, the stabilty in the ferromagnetic
phase and the 100% spin-polarization around the Fermi energy
make these alloys promising candidates for future spintronics
applications.

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