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A theoretical analysis of elastic
and optical properties of half
Heusler MCoSb (M[Ti, Zr
and Hf)
Himanshu Joshi
a
, D. P. Rai
b,c,∗
, Lalhriatpuia Hnamte
a
, Amel Laref
d
, R. K. Thapa
a,e
a
Condensed Matter Theory Research Group, Department of Physics, Mizoram University, Aizawl, Mizoram 796004,
India
b
Division of Computational Physics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh
City, Vietnam
c
Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
d
Department of Physics, College of Science, King Saud University, Riyadh, Saudi Arabia
e
Condensed Matter Theory Research Centre, Butwal, Rupendehi, Nepal
∗
Corresponding author.
Email address: dibya@tdt.edu.vn (D.P. Rai).
Abstract
Ab initio calculation of the Elastic and Optical properties of cubic halfHeusler
compounds MCoSb (M ¼Ti, Zr and Hf) are reported using the FPLAPW
approach of the Density Functional Theory. Generalized Gradient Approximation
was used as the exchange and correlation potential for investigating these
properties. It was found that the Bulk modulus decreases with the increase in
temperature and increases with the increase in pressure for all of the three
Heusler compounds under study. The Debye’s temperature along with
compressional, Shear and average elastic wave velocities has also been
calculated. The elastic results are compared with the available theoretical and
experimental works. The optical investigation of the compounds shows high
reﬂectivity at the infrared region of the photon energy. The imaginary part of the
dielectric function reveled the optically nonmetallic behavior of the MCoSb
compounds, with optical band gap being around 1 eV.
Received:
11 November 2018
Revised:
18 January 2019
Accepted:
21 January 2019
Cite as: Himanshu Joshi,
D. P. Rai,
Lalhriatpuia Hnamte,
Amel Laref,
R. K. Thapa.A theoretical
analysis of elastic and optical
properties of half Heusler
MCoSb (M¼Ti, Zr and Hf).
Heliyon 5 (2019) e01155.
doi: 10.1016/j.heliyon.2019.
e01155
https://doi.org/10.1016/j.heliyon.2019.e01155
24058440/Ó2019 Published by Elsevier Ltd. This is an open access article under the CC BYNCND license
(http://creativecommons.org/licenses/byncnd/4.0/).
Keywords: Materials science, Condensed matter physics
1. Introduction
Heusler compounds have made a tremendous contribution to the ﬁeld of material sci
ence since its discovery in 1903. It is rather surprising that the number of potential
applications Heusler compounds exhibit owing to their simple crystalline structure.
They are one of the most studied compounds for halfmetallicity, ferromagnetism,
superconductivity, Hall eﬀect and thermoelectricity [1]. These compounds also
possess high Curie temperature along with high spinpolarization, which is of great
importance in technological applications. The potentiality of a Heusler compound
for a particular type of application can be easily determined by counting their
valence electrons. Heusler compounds with a valence electron count (VEC) of 18
or 24 are narrow band semiconductors and are potential thermoelectric materials
[2]. VEC other than 18 or 24 makes these compounds halfmetallic in nature.
Most Heusler compounds with VEC of 19 or 22 are halfmetallic ferromagnets
which is favorable for spintronic applications [3,4,5]. Heusler compounds which
are nonmagnetic and have a VEC of 27 or 18 are found to be superconductors
[1,6]. Therefore, due to their vast technological applications new Heusler com
pounds are in continuous demand. Some new Heusler compounds which were theo
retically found to be stable violated the stability criterion when synthesized
experimentally, thus arising a serious concern about stability from theoretical
approach [7]. One of the method to ensure the stability of a structure through theo
retical calculations is to check its mechanical stability which according to Born [8] is
a necessary condition for thermodynamic and structural stability. Thus, the role of
elastic constants is important to determine the mechanical stability in order to further
verify the stability criterion and the order parameter of a structure [9]. Very few
experimental and theoretical works are available on the investigation of elastic prop
erties of these compounds. Sekimoto et al. (2005) [10] had experimentally investi
gated the sound velocities, Debye’s temperature and the Young’s modulus of the
materials. Unfortunately, no other experimental works on the elastic properties of
the compounds are reported on the available literature. Coban et al. (2016) [11]
had theoretically investigated the elastic constants of HfCoSb within DFT formula
tion. The elastic constants of other two compounds are not known yet. Most of the
works reported on these compounds are on their electronic and thermoelectric prop
erties. However, elastic properties being one of the fundamental properties, its
knowledge is essential as they provide information about the nature of bonding
forces and the mechanical strength of the system which is of great importance for
applications under diﬀerent constraints. Therefore, we have made a detailed inves
tigation on the elastic properties of half Heusler (HH) TiCoSb, ZrCoSb and HfCoSb
from ﬁrst principle method using the code ElaStic [12], based upon Density
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Functional Theory (DFT) [13,14]. Further, we have also calculated the Bulk
modulus and Debye’s temperature of the three HH compounds MCoSb employing
Gibbs package [15,16] based upon quasiharmonic Debye’s approximation for better
comparison purpose. All the three compounds were found to be mechanically stable
as they satisfy the BornHuang stability criteria [17] given by C
11
>0, C
44
>0, C
11

C
12
>0 and C
11
þ2C
12
>0.
Optical properties of the compounds are also reported in addition to the elastic prop
erties. These compounds still lack optical studies and experimental works are
encouraged due to their nonavailability. Optical parameters like the real and imag
inary dielectric constant, refractive index, reﬂectivity, extension coeﬃcient and the
energy loss functional were investigated as a function of photon energy. It was found
that the optical band gap in case of HfCoSb is higher than TiCoSb and ZrCoSb and
varies as HfCoSb >ZrCoSb >TiCoSb. All other static optical parameters calcu
lated varies in a reverse way with TiCoSb as highest and HfCoSb as lowest. HfCoSb
has the highest energy band gap among MCoSb (M ¼Ti, Zr and Hf) and is the
reason to also have highest optical gap. TiCoSb had the least band gap and also
has the least optical gap. Our calculated energy band gap values were 1.04 eV,
1.073 eV and 1.137 eV respectively for TiCoSb, ZrCoSb and HfCoSb from Gener
alized Gradient Approximation (GGA) [18] energy exchange functional. The com
pounds under investigation being semiconductors, their intraband transitions were
neglected in the study of optical properties.
In this work, we have mainly focused on the theoretical investigation of elastic and
optical properties of HH MCoSb. The investigated properties revel the fundamental
nature of a material from which other characteristics can be extracted and is thus given
importance in this paper. Elastic properties of Ti and Zr based MCoSb and the optical
properties of ZrCoSb are being reported for the ﬁrst time to the best of our knowledge.
2. Calculation
The lattice constants were calculated by a volume optimization method based upon
Murnaghan’s equation of state [19] and were performed using the WIEN2k code
[20]. The code is based upon the Full Potential Linearized Augmented Plane
Wave (FPLAPW) approach of the Density Functional Theory. The lattice constants
which were obtained and used in the calculation are respectively 5.8839
A, 6.0912
A
and 6.0574
A for TiCoSb, ZrCoSb and HfCoSb. They are found to be in close agree
ment with the experimental values [21,22,23,24,25]. PerdeweBurkeeErnzerhof
Generalized Gradient Approximation (PBEGGA) [17] was used to deﬁne the elec
tron energy exchange and interactions. In order to fulﬁll a good convergence crite
rion, 10,000 optimized kpoints were integrated in the ﬁrst Brillouin zone to generate
a2020 20 MonkhorstPack mesh, with energy convergence set to 10
5
Ry and
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charge to 10
3
e
. The energy cutoﬀbetween the valence and the semicore states
was set to 8.0 Ry and were treated ignoring the spinorbit coupling i.e. the semi
core states were treated semirelativistically. The number of plane waves was set by
limiting R
MT
K
max
¼7 in the interstitial region and the charge density expansion
was set to G
max
¼12. The muﬃntin radii (R
MT
)ofdiﬀerent atoms in the unit cell
was calculated by optimization method and the optimized values are Ti ¼1.81230;
Co ¼2.01608; Sb ¼2.28841 for TiCoSb, Zr ¼2.34603; Co ¼2.07325; Sb ¼
2.57325 for ZrCoSb and Hf ¼2.57664; Co ¼2.03498; Sb ¼2.31289 for HfCoSb.
The R
MT
optimization curve is shown in Fig. 1.
The elastic properties are calculated within the DFT framework using the Lagrangian
theory of elasticity, in which a solid is assumed to be an anisotropic and homoge
neous elastic medium. The second order elastic constants are calculated using the
energystrain method, as implemented in the code ElaStic [12]. The structure under
investigation has cubic symmetry, so there are three independent elastic constants:
C
11
,C
12
and C
44
. From the elastic constants, diﬀerent elastic properties were calcu
lated using the Voigt, Reuss and Hill averaging scheme [26,27,28]. Voigt’s approx
imation assumes uniform strain in the structure whereas Reuss approximation
assumes uniform stress. Elastic moduli under diﬀerent averaging scheme are
described here under as follows
Bulk modulus, which is the measure of resistance to compressibility was calculated
using the expression in Eq. (1)
B¼1
3ðC11 þ2C12Þ;ð1Þ
The bulk modulus for a cubic structure is same for Voigt, Reuss and Hill averages.
Shear modulus is generally deﬁned as the deformation that occurs in a solid when a
force is applied to any of the parallel face while the other face opposite to the parallel
face is kept ﬁxed by other opposite forces. In Voigt average, the shear modulus for a
cubical symmetry is given by Eq. (2)
GV¼C11 C12 þ3C44
5;ð2Þ
Fig. 1. R
MT
optimization of (a) TiCoSb, (b) ZrCoSb and (c) HfCoSb.
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The following expression shown in Eq. (3) gives the Reuss average
GR¼5ðC11 C12ÞC44
4C44 þ3ðC11 C12Þ;ð3Þ
The arithmetic mean of the Voigt and the Reuss average in Eq. (4) gives the Hill
shear modulus
GH¼GVþGR
2;ð4Þ
The Young’s modulus and the Poisson’s ratio are calculated using Eqs. (5) and (6)
Y¼9BG
3BþG;ð5Þ
h¼3B2G
2ð3BþGÞ;ð6Þ
Replacing G by G
V
and G
R
in Eqs. (5) and (6), one can calculate the Voigt and the
Reuss average of Young’s modulus and Poisson’s ratio.
Debye’s temperatures is calculated using Eq. (7), which is based upon Debye’s
assumption that the temperature of highest normal mode of vibration can be esti
mated from the average sound velocity [29].
QD¼h
k3n
4prNA
M1=3
ym;ð7Þ
here, his the Plank’s constant, kis the Boltzmann’s constant, N
A
is the Avogadro’s
number, nis the number of atoms per molecule or number of atoms per formula
unit, Mis the molar mass, ris the density of the unit cell and n
m
is the average
sound velocity. The average sound velocity is further expressed in terms of
compressional (n
l
) and shear (n
s
) sound velocities as given by Eq. (8) [30]
ym¼1
32
y3
s
þ1
y3
l1=3
;ð8Þ
The expressions for n
s
and n
l
is given by Eqs. (9) and (10) respectively
ys¼ﬃﬃﬃﬃ
G
r
s;ð9Þ
nl¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3Bþ4G
3r
s;ð10Þ
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Elastic properties also relate to another physical parameter known as the shear
anisotropy (A). It gives the nature of bonding in diﬀerent crystallographic directions
and is calculated using Eq. (11)
A¼2C44
C11 C12
;ð11Þ
The elastic results are presented under “Results and Discussion”section that follows
later in the paper and are compared with the results obtained using Debye’s quasi
harmonic approximation employing Gibbs package.
The optical parameters calculated are the response of the compound MCoSb when
electromagnetic radiation is introduced to them. The optical response of a material
to an external electric ﬁeld is given by its complex dielectric functionεðuÞ, which
is deﬁned by Eq. (12) as
εðuÞ¼ε1ðuÞþε2ðuÞ;ð12Þ
where, ε1ðuÞis real and ε2ðuÞthe imaginary part of the dielectric function εðuÞ.
The real and imaginary part of the dielectric function is calculated using the
KramerseKronig relation [31] given by Eq. (13)
ε1ðuÞ¼1þ2
pZ
N
0
ε2ðuÞu*du*
u*2u2;ð13Þ
the momentum matrix elements between the occupied and unoccupied states gives
the imaginary part of complex dielectric function
ε2ðuÞ¼ Ve2
2pm2u2xZd3kX
nn*knpkn*j2fðknÞx1fkn*vðEkn Ekn*uÞ;
ð14Þ
In Eq. (14),pis the momentum matrix element between nand n* states, jknjis the
crystal wave function while f(kn) is the Fermi distribution function, E
kn
is the eigen
value corresponding to the crystal wave function jknj. The refractive index nðuÞand
the extinction coeﬃcient kðuÞis calculated corresponding to Eqs. (13) and (14) using
Eqs. (15) and (16) [32].
nðuÞ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ε2
1ðuÞþε2
2ðuÞ
pþε1ðuÞ
2!1=2
;ð15Þ
kðuÞ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ε2
1ðuÞþε2
2ðuÞ
pε1ðuÞ
2!1=2
;ð16Þ
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The optical properties are studied using the GGA energy exchange correlation
functional.
3. Results and discussion
3.1. Elastic properties
The calculated elastic parameters are tabulated in Table 1. From the calculated
elastic constant values, one can see that all the three compounds under investigation
satisﬁes the stability criteria of BornHuang as discussed earlier. Therefore, it is
envisaged that the HH compounds MCoSb (M ¼Ti, Zr and Hf) are mechanically
stable. For all of MCoSb, the C
11
value is higher than C
12
and C
44
, which indicates
that these compounds are hard to compress along the Xaxes. The Bulk modulus
values calculated from ElaStic code using Eq. (1) are respectively 147.59 GPa,
139.78 GPa and 144.68 GPa for TiCoSb, ZrCoSb and HfCoSb, which are very close
to the values calculated from Murnaghan’s equation of state (146.914 GPa, 139.865
GPa and 145.117 GPa). The similarity between the two results estimates the accu
racy of elastic calculations. Our calculated Bulk modulus value of HfCoSb is higher
than the theoretical report of Coban et al. (2016) [11] obtained from equation of
state, which is 137.712 GPa. However, their Bulk modulus value obtained after
elastic investigation is close to the one that we have calculated. Unfortunately for
TiCoSb and ZrCoSb, no Bulk modulus results are available for comparison. Our
calculated value of Young’s modulus is 4.7% and 6.4% higher than the available
experimental data for TiCoSb and HfCoSb, whereas for ZrCoSb, it is 2.4% lower.
High value of Y in MCoSb shows that the covalent bonding component dominates
these compounds. Strong covalent bond indicates that the material is stiﬀ.
The fundamental parameter closely related to the melting point and speciﬁc heats in
solid is the Debye’s temperature. High Debye temperature indicates stiﬀer crystal
orientation and such crystals are found to have high melting points. Debye temper
ature (Q
D
) being the temperature required to activate all the phonon mode of a
Table 1. Calculated elastic constants, Bulk modulus (B), Shear modulus (G), Young’s modulus (Y) and
Poisson’s ratio (h)
Comp. C
II
(GPa) C
I2
(GPa) C
44
(GPa) G
V
(GPa) G
R
(GPa) G
H
(GPa) B (GPa) Y (GPa) hRef.
TiCoSb 254.8 94.0 88.4 85.20 85.02 85.11 147.59 205.36
196
0.26 This work
[10], expt.
ZrCoSb 263.0 78.1 71.7 80.0 78.77 79.39 139.78 202.03
207
0.25 This work
[10], expt.
HfCoSb 257.2 88.4 78.6 80.87 80.78 80.83 144.68 204.3
192
0.26 This work
[10], expt.
274.57 77.6 75.5 84.41 143.31 210.2 0.25 [11], theo.
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crystal, large Q
D
values indicate high energy is required to excite the phonons in a
crystal and such materials are highly favorable in thermoelectric power generation.
In the compound MCoSb, the Debye’s temperature varies as TiCoSb >ZrCoSb >
HfCoSb and is found to agree with the experimental results of Li et. al. [33] and Se
kimoto et. al. [10]. The Debye’s temperature of ZrCoSb is in excellent agreement
with the experimental values in reference 36, while TiCoSb and HfCoSb diﬀers
by 4.34% and 1.51% respectively. It is listed in Table 2. The calculated elastic
wave velocities are higher than the available experimental results. The diﬀerence
can be attributed to the temperature inﬂuence. The values that we report are observed
at 0 K, whereas the values reported in experiments are observed in the temperature
range of 300 Ke900 K. Further, in theoretical calculations we consider perfect sin
gle crystals where as in experiments, crystal imperfection is taken into consideration
which varies the two results. Our calculated values of A are 1.09, 0.76 and 0.93 for
MCoSb (M ¼Ti, Zr and Hf) respectively. The calculated A values are either greater
or less than 1 but not equal to. For any isotropic crystal, the value of A equals to 1.
Values less than or greater than 1 is a measure of shear anisotropy possessed by the
crystal. Thus HH MCoSb is purely anisotropic.
We have also calculated the unit cell density of MCoSb compounds and it was found
that HfCoSb has the highest density among the three compounds investigated. The
density is found to vary as HfCoSb >ZrCoSb >TiCoSb and the values are 10.7 g/
cc, 7.9 g/cc and 7.4 g/cc respectively.
The plot of Bulk modulus is important as it alone can reveal the temperature and
pressure characteristics of other moduli of elasticity and also that of elastic constants.
We have used the quasiharmonic Debye’s approximation to plot the variation of bulk
modulus with respect to temperature at constant pressure, see Fig. 2 (a) and that with
Table 2. The compressional (n
l
), Shear (n
s
) and average (n
m
) elastic wave velocity
in m/s, density (r) in g/cc, Debye’s temperature (Q
D
) in K and the shear
anisotropy (A) for MCoSb.
Comp. n
l
(m/s) n
s
(m/s) n
m
(m/s) r(g/cc)Q
D
(K) A Ref.
TiCoSb 5918.413 3379.224 3755.187 7.453 435.08 1.09 This work
5699 3237 ee417 e[10], expt.
5691 3230 ee416 e[33], expt.
ZrCoSb 5544.085 3379.224 3503.823 7.991 392.073 0.76 This work
5623 3192 ee399 e[10], expt.
5488 3134 ee392 e[33], expt.
HfCoSb 4866.790 2766.790 3075.741 10.734 346.152 0.93 This work
4743 2721 ee341 e[10], expt.
4703 2709 ee340 e[33], expt.
4841.62 2811.77 3119.98 e220.77 0.77 [11], theo.
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respect to pressure at constant temperature, see Fig. 2 (b). It is seen that the bulk
modulus decreases abruptly with the increase of temperature. This indicates that
the elastic constants C
ij
will also decrease with the application of temperature.
Thus, it can be predicted that all of the moduli of elasticity decreases with the in
crease of temperature at constant pressure. This abrupt decrease of Bleads to an
important property of HH compound MCoSb. That is, it shows the high temperature
working range of these compounds because at high temperatures, due to the decrease
of Band G, the G/B ratio will also decrease, making these compounds nonfragile at
higher temperature ranges, which is a prime condition for number of applications
like thermoelectricity, superconductivity etc. On the other hand, the pressure charac
teristics of Bulk modulus shows the disadvantages of the HH compound MCoSb at
high pressure ranges. At constant temperature (0 K), Bincreases linearly with in
crease of pressure, thereby indicating the abrupt increase of elastic constants and
thus the other moduli of elasticity. The boiling point and the melting point of the ma
terial also increases and thus gets cut out from number of applications like optical
applications, solder applications etc. Further the G/B ratio also increases and thus
the HH compound MCoSb becomes unfavorable for thermoelectric, superconducti
vity and other energy applications.
The value of B at 0 K, pressure remaining constant and that at 0 BPa, temperature
remaining constant is very close to the B values listed in Table 1. Thus, it acts as
an estimation of accuracy for results calculated from quasiharmonic Debye’s
approximation.
Similarly, using the same approximation, we have also calculated the Debye’s tem
perature as a function of pressure and temperature using Eq. (17) [34].
qD¼Z
kB6pV1=2n1=3fðsÞﬃﬃﬃﬃﬃ
B
M
r;ð17Þ
M is the molecular weight per formula unit and B is the bulk modulus which is
assumed to be equal to the static bulk modulus as expressed by Eq. (18)
Fig. 2. Plot of bulk modulus with respect to (a) Temperature and (b) Pressure for MCoSb (M ¼Ti, Zr
and Hf).
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B¼Bstatic ¼Vd2EðVÞ
dV2;ð18Þ
The plot of qDis shown in Fig. 3. From the ﬁgure it is seen that the qDvalue decreases
with the increase in temperature and increases with the increase in pressure. For Ti
CoSb and HfCoSb, where qDwas found to be higher than the experimental results
from elastic calculation, whereas quasiharmonic calculations shows excellent agree
ment with the experimental results at 700 K and 500 K. The values obtained at
diﬀerent temperatures are listed in Table 3.
3.2. Optical properties
The investigation of optical properties is important in order to ﬁnd the optoelectronic
application of HH MCoSb. The optical properties are studied using the GGA energy
Fig. 3. Plot of Debye’s temperature with respect to (a) Temperature and (b) Pressure for MCoSb (M ¼
Ti, Zr and Hf).
Table 3. Debye’s temperature calculated at diﬀerent temperatures from quasi
harmonic approximation.
Temperature (K) TiCoSb ZrCoSb HfCoSb
0 436.94 397.84 352.05
100 436.43 397.25 351.4
200 434.23 394.98 349.21
300 431.37 392.13 346.58
400 428.27 389.08 343.81
500 425.07 385.95 340.97
600 421.81 382.77 338.1
700 418.52 379.56 335.21
800 415.21 376.33 332.3
900 411.88 373.08 329.37
1000 408.53 369.82 326.44
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exchange correlation functional and the optical parameters are plotted in the energy
range of 0e60 eV. Fig. 4(a) gives the plot of dielectric function εðuÞwith respect to
photon energy. Generally, there are two types of contribution to the dielectric func
tion, interband transition and intraband transition. The interband transition is
further classiﬁed into direct interband transition and indirect interband transition.
Usually, intraband transition is prominent only in metals and our compound of
investigation being a semiconductor, shows no intraband transition. The indirect
interband transition is negligibly small because electron excitation by photon across
an indirect band gap is extremely rare due to low momentum of photons, when
compared to the direct interband transition, thus we have neglected it in our calcu
lation. The real part of the dielectric function shows sharp peaks at 1.78 eV, 1.92 eV
and 1.91 eV respectively for TiCoSb, ZrCoSb and HfCoSb in the visible region of
the spectrum. The obtained peaks are related with the nature of the functionals and
can change with diﬀerent functionals. After a peak value is obtained, the peaks re
duces gradually and tends towards minimum value which is obtained at 8.61 eV,
6.35 eV, 6.08 eV respectively for TiCoSb, ZrCoSb and HfCoSb. This is due to
the interband transition between the VCM and the CBM.
ε1ðuÞincreases from HfCoSb <ZrCoSb <TiCoSb, this is expected as the atomic
radius increases from Ti <Zr <Hf. The static dielectric function ε1ð0Þincreases
and attains the maximum value and then it declines until it attains a negative value
for speciﬁc energy regions (3.578 eV20.45 eV for TiCoSb; 3.932 eVe18.027 eV
for ZrCoSb; 5.156 eVe18.925 eV for HfCoSb). There after ε1ð0Þtends towards a
constant value. The negative values indicates the reﬂection of incident radiation
from the surface. Our calculated values of real part of static dielectric constants
are respectively 21.505, 18.987 and 18.403 for TiCoSb, ZrCoSb and HfCoSb.
Fig. 4. (a) Real and imaginary parts (ε
1
and ε
2
) of dielectric function (b) Dispersion curves of refractive
index n (u) and extinction coeﬃcient k (u).
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Energy shift for the location of the peaks is observed in diﬀerent MCoSb HH com
pounds but the general pattern of the ε1ðuÞcurves are similar to each other. The op
tical band gap of a compound is given by the zero frequency value of the imaginary
part of the dielectric function, ε2ðuÞ. The calculated optical band gaps are 0.775 eV,
1.102 eV and 1.156 eV respectively for TiCoSb, ZrCoSb and HfCoSb. The value of
this gap and the one calculated from band structure plots (Table 4) are close to each
other, which veriﬁes the reliability of the optical calculations. The peak values are
obtained at 2.95 eV, 2.734 eV and 3.06 eV respectively for diﬀerent components
of MCoSb. This peaks originates due to the transition from 3d,4dand 5dstates
of M and Co atom to the 5pstates of Sb atom. A red shift is observed between Ti
CoSb eHfCoSb and HfCoSb eZrCoSb as the peak values of ε2ðuÞhas shifted to
lower energy range between these compounds.
The refractive index nðuÞand the extinction coeﬃcient kðuÞis calculated using
Eqs. (15) and (16),itisshowninFig. 4 (b). The high peaks of the refractive in
dex is observed in the visible region of the spectrum. The static refractive index
obtained is 4.64, 4.36 and 4.29 for Ti, Zr and Hf components of MCoSb respec
tively and it increases from Hf to Ti. The static refractive index nð0Þsatisﬁes the
relationnð0Þ2
zε1ð0Þ, which further veriﬁes the accuracy of the calculation. It
can be seen that the trends of kðuÞis similar to that of ε2ðuÞ, it is because
kðuÞalso provides a measure of absorption of incident radiation. The extinction
coeﬃcient kðuÞbecomes very low in the energy range between 0.0136
eVe1.7007 eV and between 23.088 eVe27.875 eV (the energy region being
much wider for TiCoSb and HfCoSb). It indicates very low absorption of light
which is favorable for transparent properties of the material in this range of
the photon energy.
Reﬂectivity or reﬂection coeﬃcient R(u) is a measure of the amount of electromag
netic radiation reﬂected from the incident medium (Fig. 5a) and is calculated using
Eq. (19) [32]
RðuÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
εðuÞ
p1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
εðuÞ
pþ1
2
;ð19Þ
As an application of the external radiation incident on the material, some of the
valence electrons may undergo inelastic scattering, leading to loss of energy. The
energy lost by the electron can be calculated using Eq. (20)
Table 4. Calculated band gap (DE
G
) for TiCoSb, ZrCoSb and HfCoSb.
Compound TiCoSb ZrCoSb HfCoSb
DE
G
(eV) 1.040 1.073 1.137
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LðuÞ¼Im1
εðuÞ¼ε2ðuÞ
ε2
1ðuÞþε2
2ðuÞ;ð20Þ
The high peaks in the L(u) vs energy plots represents the plasma resonance behavior
(see Fig. 5b) and the frequencies at which these peaks originates are known as plasma
frequency. In these frequencies, ε1ðuÞ¼0 and dε1ðuÞ
du>0. At higher energy range, the
amplitude of L(u) gets larger as ε2ðuÞgets smaller. After plasma frequency is at
tained, the compound tends towards transparency asRðuÞ/0. The highest peaks
of reﬂectivity and loss function are positioned at 1.9184 eV, 2.7891 eV, 3.1157 eV
and 21.5106 eV, 20.6126 eV, 21.40 eV respectively for Ti, Zr and Hf components
of MCoSb. The static reﬂectivity Rð0Þand the static loss function Lð0Þare presented
in Table 5. In the energy range between 0.0136 eVe1.7007 eV and between 23.088
eVe27.875 eV, L(u) is very low indicating very less loss of electron energy due to
scattering. R(u) is also appreciably low indicating less reﬂection of the incident radi
ation in that energy range. Therefore, corresponding to ε2ðuÞ,kðuÞ,R(u) and L(u)
results, we report that the HH compound MCoSb shows transparent properties in the
energy range between 0.0136 eVe1.7007 eV and between 23.088 eVe27.875 eV.
In Fig. 6, the absorption and the conduction spectra of the compounds are shown.
From the conduction spectra, we see that the threshold point occurs very close to
0 eV, indicating the narrow energy band gaps in the compounds. Therefore, the com
pounds are characterized as narrow band gap semiconductors which is evident from
the band structure plots of the compounds reported earlier [11,35,36]. High peaks
are observed both in the infrared as well as in the visible region of both the
Fig. 5. Plot of (a) Reﬂectivity R (u) and (b) Electron energy loss functional L (u) vs Photon energy.
Table 5. Calculated static dielectric constant ε
1
(0), optical band gap (DE
OG
),
static refractive index n(0), static reﬂectivity R(0) and static loss function L(0) for
TiCoSb, ZrCoSb and HfCoSb.
Compound ε
1
(0) DE
OG
n(0) R(0) L(0)
TiCoSb 21.505 0.775 4.64 0.416 0.000821
ZrCoSb 18.987 1.102 4.36 0.392 0.000809
HfCoSb 18.403 1.156 4.29 0.387 0.000796
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conduction and absorption spectrum. Absorption spectra has peaks higher in the
visible region and may be due to the transition between widely separated energy
levels. In reference to the optical results obtained, it can be said that the compound
ZrCoSb fails to show any type of optical behavior in the energy range 23e27 eV.
The same holds for TiCoSb and HfCoSb, the energy range being much wider.
Unfortunately we could not compare our optical results due to lack of experimental as
well as theoretical results. We have summarized our optical results in Table 5.
HfCoSb results are in agreement with the optical results obtained by Coban et al [11].
4. Conclusion
We have presented the elastic and optical properties of halfHeusler MCoSb (M ¼
Ti, Zr and Hf) using ﬁrst principle methods. The Bulk modulus values calculated
from ElaStic code are very close to the values calculated from Murnaghan’s equation
of state. The Debye’s temperature are calculated within the DFT framework using
the Lagrangian theory of elasticity as well as by using the quasiharmonic approxi
mations. The values obtained from these two approaches are close to one another
and also to the available experimental data. The optical band gaps calculated from
the imaginary part of the dielectric function are found to be close to the energy
band gap of the materials. This revels that the compounds acts as semiconductors
to optical conduction as well as to electrical conduction. Further, the optical proper
ties of the material varies according to the photon energies and can be beneﬁcial to
numerous applications like optoelectronic, thin ﬁlm growth etc.
Declarations
Author contribution statement
Himanshu Joshi, Dibya P. Rai, Lalhriatpuia Hnamte, Amel Laref, R. K. Thapa:
Conceived and designed the analysis; Analyzed and interpreted the data; Contrib
uted analysis tools or data; Wrote the paper.
Fig. 6. (a) The conduction and (b) the absorption spectra of MCoSb.
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Funding statement
Himanshu Joshi and R. K. Thapa were supported by SERB Govt. of India (EMR/
2015/001407). D. P. Rai was supported by DST New Delhi India & RFBR (DST/
INT/RUS/RFBR/P264). Amel Laref was supported by the Research Center of Fe
male Scientiﬁc and Medical Colleges, Deanship of Scientifc Research, King Saud
University.
Competing interest statement
The authors declare no conﬂict of interest.
Additional information
No additional information is available for this paper.
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