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A theoretical analysis of elastic and optical properties of half Heusler MCoSb (M=Ti, Zr and Hf)

  • Ensemble 3-Centre of Excellence, Warsaw, Poland
  • Pachhunga University College Mizoram university

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Ab initio calculation of the Elastic and Optical properties of cubic half-Heusler compounds MCoSb (M = Ti, Zr and Hf) are reported using the FP-LAPW 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 non-metallic behavior of the MCoSb compounds, with optical band gap being around 1 eV.
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A theoretical analysis of elastic
and optical properties of half
Heusler MCoSb (M[Ti, Zr
and Hf)
Himanshu Joshi
, D. P. Rai
, Lalhriatpuia Hnamte
, Amel Laref
, R. K. Thapa
Condensed Matter Theory Research Group, Department of Physics, Mizoram University, Aizawl, Mizoram 796004,
Division of Computational Physics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh
City, Vietnam
Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Department of Physics, College of Science, King Saud University, Riyadh, Saudi Arabia
Condensed Matter Theory Research Centre, Butwal, Rupendehi, Nepal
Corresponding author.
E-mail address: (D.P. Rai).
Ab initio calculation of the Elastic and Optical properties of cubic half-Heusler
compounds MCoSb (M ¼Ti, Zr and Hf) are reported using the FP-LAPW
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 Debyes 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
reectivity at the infrared region of the photon energy. The imaginary part of the
dielectric function reveled the optically non-metallic behavior of the MCoSb
compounds, with optical band gap being around 1 eV.
11 November 2018
18 January 2019
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.
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Keywords: Materials science, Condensed matter physics
1. Introduction
Heusler compounds have made a tremendous contribution to the eld 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 half-metallicity, ferromagnetism,
superconductivity, Hall eect and thermoelectricity [1]. These compounds also
possess high Curie temperature along with high spin-polarization, 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 half-metallic in nature.
Most Heusler compounds with VEC of 19 or 22 are half-metallic ferromagnets
which is favorable for spintronic applications [3,4,5]. Heusler compounds which
are non-magnetic 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, Debyes temperature and the Youngs 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 dierent constraints. Therefore, we have made a detailed inves-
tigation on the elastic properties of half Heusler (HH) TiCoSb, ZrCoSb and HfCoSb
from rst 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 Debyes temperature of the three HH compounds MCoSb employing
Gibbs package [15,16] based upon quasiharmonic Debyes approximation for better
comparison purpose. All the three compounds were found to be mechanically stable
as they satisfy the Born-Huang stability criteria [17] given by C
>0, C
>0, C
>0 and C
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 non-availability. Optical parameters like the real and imag-
inary dielectric constant, refractive index, reectivity, extension coecient 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 intra-band 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 rst time to the best of our knowledge.
2. Calculation
The lattice constants were calculated by a volume optimization method based upon
Murnaghans equation of state [19] and were performed using the WIEN2k code
[20]. The code is based upon the Full Potential Linearized Augmented Plane
Wave (FP-LAPW) approach of the Density Functional Theory. The lattice constants
which were obtained and used in the calculation are respectively 5.8839
A, 6.0912
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 (PBE-GGA) [17] was used to dene the elec-
tron energy exchange and interactions. In order to fulll a good convergence crite-
rion, 10,000 optimized k-points were integrated in the rst Brillouin zone to generate
a2020 20 Monkhorst-Pack mesh, with energy convergence set to 10
Ry and
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charge to 10
. The energy cut-obetween the valence and the semi-core states
was set to 8.0 Ry and were treated ignoring the spin-orbit coupling i.e. the semi-
core states were treated semi-relativistically. The number of plane waves was set by
limiting R
¼7 in the interstitial region and the charge density expansion
was set to G
¼12. The mun-tin radii (R
)ofdierent 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
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
energy-strain method, as implemented in the code ElaStic [12]. The structure under
investigation has cubic symmetry, so there are three independent elastic constants:
and C
. From the elastic constants, dierent elastic properties were calcu-
lated using the Voigt, Reuss and Hill averaging scheme [26,27,28]. Voigts approx-
imation assumes uniform strain in the structure whereas Reuss approximation
assumes uniform stress. Elastic moduli under dierent 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)
3ðC11 þ2C12Þ;ð1Þ
The bulk modulus for a cubic structure is same for Voigt, Reuss and Hill averages.
Shear modulus is generally dened 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 xed by other opposite forces. In Voigt average, the shear modulus for a
cubical symmetry is given by Eq. (2)
GV¼C11 C12 þ3C44
Fig. 1. R
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-
The Youngs modulus and the Poissons ratio are calculated using Eqs. (5) and (6)
Replacing G by G
and G
in Eqs. (5) and (6), one can calculate the Voigt and the
Reuss average of Youngs modulus and Poissons ratio.
Debyes temperatures is calculated using Eq. (7), which is based upon Debyes
assumption that the temperature of highest normal mode of vibration can be esti-
mated from the average sound velocity [29].
here, his the Planks constant, kis the Boltzmanns constant, N
is the Avogadros
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
is the average
sound velocity. The average sound velocity is further expressed in terms of
compressional (n
) and shear (n
) sound velocities as given by Eq. (8) [30]
The expressions for n
and n
is given by Eqs. (9) and (10) respectively
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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 dierent crystallographic directions
and is calculated using Eq. (11)
C11 C12
The elastic results are presented under Results and Discussionsection that follows
later in the paper and are compared with the results obtained using Debyes 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 eld is given by its complex dielectric functionεðuÞ, which
is dened by Eq. (12) as
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)
the momentum matrix elements between the occupied and unoccupied states gives
the imaginary part of complex dielectric function
ε2ðuÞ¼ Ve2
nn*knpkn*j2fðknÞx1fkn*vðEkn Ekn*uÞ;
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
is the eigen
value corresponding to the crystal wave function jknj. The refractive index nðuÞand
the extinction coecient kðuÞis calculated corresponding to Eqs. (13) and (14) using
Eqs. (15) and (16) [32].
nðuÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kðuÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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The optical properties are studied using the GGA energy exchange correlation
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
satises the stability criteria of Born-Huang 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
value is higher than C
and C
, which indicates
that these compounds are hard to compress along the X-axes. 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 Murnaghans 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 Youngs 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 sti.
The fundamental parameter closely related to the melting point and specic heats in
solid is the Debyes temperature. High Debye temperature indicates stier crystal
orientation and such crystals are found to have high melting points. Debye temper-
ature (Q
) being the temperature required to activate all the phonon mode of a
Table 1. Calculated elastic constants, Bulk modulus (B), Shear modulus (G), Youngs modulus (Y) and
Poissons ratio (h)
Comp. C
(GPa) C
(GPa) C
(GPa) G
(GPa) G
(GPa) G
(GPa) B (GPa) Y (GPa) hRef.
TiCoSb 254.8 94.0 88.4 85.20 85.02 85.11 147.59 205.36
0.26 This work
[10], expt.
ZrCoSb 263.0 78.1 71.7 80.0 78.77 79.39 139.78 202.03
0.25 This work
[10], expt.
HfCoSb 257.2 88.4 78.6 80.87 80.78 80.83 144.68 204.3
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
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 Debyes 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 Debyes temperature of ZrCoSb is in excellent agreement
with the experimental values in reference 36, while TiCoSb and HfCoSb diers
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 dierence
can be attributed to the temperature inuence. 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 Debyes 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
), Shear (n
) and average (n
) elastic wave velocity
in m/s, density (r) in g/cc, Debyes temperature (Q
) in K and the shear
anisotropy (A) for MCoSb.
Comp. n
(m/s) n
(m/s) n
(m/s) r(g/cc)Q
(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
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 non-fragile 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 Debyes
Similarly, using the same approximation, we have also calculated the Debyes tem-
perature as a function of pressure and temperature using Eq. (17) [34].
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
The plot of qDis shown in Fig. 3. From the gure 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
dierent temperatures are listed in Table 3.
3.2. Optical properties
The investigation of optical properties is important in order to nd the optoelectronic
application of HH MCoSb. The optical properties are studied using the GGA energy
Fig. 3. Plot of Debyes temperature with respect to (a) Temperature and (b) Pressure for MCoSb (M ¼
Ti, Zr and Hf).
Table 3. Debyes temperature calculated at dierent 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, inter-band transition and intra-band transition. The inter-band transition is
further classied into direct inter-band transition and indirect inter-band transition.
Usually, intra-band transition is prominent only in metals and our compound of
investigation being a semiconductor, shows no intra-band transition. The indirect
inter-band 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 inter-band 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 dierent 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 inter-band 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 specic energy regions (3.578 eV-20.45 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 reection 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
and ε
) of dielectric function (b) Dispersion curves of refractive
index n (u) and extinction coecient k (u).
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Energy shift for the location of the peaks is observed in dierent 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 veries the reliability of the optical calculations. The peak values are
obtained at 2.95 eV, 2.734 eV and 3.06 eV respectively for dierent 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 coecient 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 nð0Þsatises the
zε1ð0Þ, which further veries 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
coecient 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.
Reectivity or reection coecient R(u) is a measure of the amount of electromag-
netic radiation reected from the incident medium (Fig. 5a) and is calculated using
Eq. (19) [32]
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
) for TiCoSb, ZrCoSb and HfCoSb.
Compound TiCoSb ZrCoSb HfCoSb
(eV) 1.040 1.073 1.137
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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 reectivity 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 reectivity 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 reection 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) Reectivity R (u) and (b) Electron energy loss functional L (u) vs Photon energy.
Table 5. Calculated static dielectric constant ε
(0), optical band gap (DE
static refractive index n(0), static reectivity R(0) and static loss function L(0) for
TiCoSb, ZrCoSb and HfCoSb.
Compound ε
(0) DE
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 half-Heusler MCoSb (M ¼
Ti, Zr and Hf) using rst principle methods. The Bulk modulus values calculated
from ElaStic code are very close to the values calculated from Murnaghans equation
of state. The Debyes 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 benecial to
numerous applications like optoelectronic, thin lm growth etc.
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/P-264). Amel Laref was supported by the Research Center of Fe-
male Scientic and Medical Colleges, Deanship of Scientifc Research, King Saud
Competing interest statement
The authors declare no conict of interest.
Additional information
No additional information is available for this paper.
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... Moreover, they have suggested Rh-based half-Heuslers with J o u r n a l P r e -p r o o f high power factor and reliable electronic transport properties. In the same context, Joshi et al [22] explored the optical properties of MCoSb (M = Ti, Zr, and F), and their investigations were based on the analysis of the optical properties using first-principles methods. They have found that the half-Heuslers phases consider promising candidates for optoelectronic applications. ...
In this work, we have explored half-Heusler RhVZ (Z = Si, Ge, Sn) with 18 valence electrons per unit cell within the Density Functional Theory (DFT). The structural, electronic, optical, thermoelectric, and phonon properties have been investigated. The structural properties such as lattice constant, bulk modulus, and pressure derivative of bulk modulus have been reported. All materials are semiconductors with a narrow bandgap as it is concluded from the electronic band structure, and the phonon dispersion approve the dynamical stability, meanwhile, the absorption coefficient is found to be around 104 cm−1 in the visible region indicating that these materials can be used in Photovoltaic applications. The semi-classical Boltzmann transport equation used to calculate the thermoelectric characteristics and all materials under investigation show high performance in the p-type region due to the valence band degeneracy. The thermoelectric analysis provided show that the power factor of the materials lies on 3.12 ×10−3, 3.26 × 10−3, and 6.52 ×10−3 W/msK2 for RhVSi, RhVGe, and RhVSn respectively, the figure of merit ZT reaches the maximum value of 0.8 at room temperature indicating that those materials are suitable for thermo-generation application at near room temperature. However, we found that RhVSn is more suitable for thermoelectric applications with its high ZT values more than RhVSi, and RhVGe compounds.
In this work, structural, electronic, magnetic, thermal and mechanical properties of Mn2ZrZ (Z= Ge and Si) under pressure upto 50 GPa is studied using state of the art density functional theory. In structural properties, under pressure ground state optimizations are performed to check the thermodynamic stability of studied alloys. Furthermore, enthalpy of formation and elastic stability criteria affirms the thermodynamic stability in studied alloys. Pugh ratio suggests that, Mn2ZrGe and Mn2ZrSi remain ductile and brittle in nature, respectively throughout pressure upto 50 GPa. Moreover, large elastic anisotropic response is observed for both alloys. In electronic properties density of states and band gaps are discussed in detail which affirms the ferromagnetic half metallic nature of alloys. Our computed results, such as optimized ground state lattice constant, band-gap and magnetic moment are consistent and have matched excellently with available literature at ambient conditions. In mechanical properties, Debye temperature factor, minimum thermal conductivity and melting temperature is observed to increase with pressure while, Grüneisen anharmonicity factor decreases. However, to date, there are no reports available in literature with under pressure results upto 50 GPa. Therefore, this work illustrates new findings of Mn2ZrZ under pressure for potential applications in thermal actuators and spintronic devices.
The present work is intended to provide worthful information about the structure and half-metallicity of Co-based Heusler alloy. The first principle density functional theory was used to analyze Co2TiN. First, the structure of an alloy was optimized through the consideration of Cu2MnAl and Hg2CuTi, two structure types along with different magnetic phases. The resulting stable structure Hg2CuTi showed the metallic nature of an alloy in both spin configurations with GGA potential. However, an implication of Hubbard parameter (U) considerably affected the electronic structure and therefore bands got shifted with 100% polarization at Fermi level which increases the efficiency of spintronics. Under the effect of U, spin-up configuration showed semiconducting nature with 0.21 eV that revealed the half-metallicity in Co2TiN full-Heusler alloy.
The well-known half-metallic properties along with the electronic structure of series Co2YZ (Y = Sc, Y; Z = P, As, Sb, Bi) were calculated by generalized gradient approximation of the density functional theory (DFT). The lattice parameter in the magnetic phase was optimized for both Cu2MnAl and Hg2CuTi structures to determine the very energetically stable structure. The magnetic properties, electronic structures and the density of states (DOS) were calculated and studied in the minority and majority spins to predict half metallicity. The hybridization of d-orbitals of neighbouring atoms in the lattice is considered to explain the half-metallic nature of these compounds.
The structural and electronic properties of a two non-oxide Transparent Conductors (TC’s), Ir-based half-Heusler XIrSb (X=Ti,Zr) are studied using DFT based on plane waves pseudo potential method. We examined two aspects: (i) the effect of the exchange–correlation (XC) approximation, namely: PBE, PBE+U and meta-GGA SCAN (The strongly constrained and appropriately normed); and (ii) the spin–orbit coupling (SOC) effects of the heavy metals on these compounds. We found that: (i) SCAN, similar to PBE+U, yields larger band gap compared to the PBE corresponding values for both compounds. SCAN gives a band gap about halfway between PBE and experiment or Hybrid-GGA. Similarly to PBE and PBE+U, SCAN shows the semiconducting behavior of the compounds with indirect band gap at the same locations in Brillouin Zone; (ii) spin–orbit coupling causes an important splitting in the valence band maximum (VBM) of order 0.44 eV and 0.58 eV for TiIrSb and ZrIrSb respectively leading to reduction of the band gap. Thus, the heavier is the X atom the larger is the SOC splitting. The VBM is dominated by Jeff=3/2 state. However, the splitting is almost negligible on the conduction band minimum (CBM). Therefore, SOC is crucial for predicting the band gap and reproducing correctly the electronic structure of these compounds.
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Ab initio study on the family of ternary copper chalcogenides Cu 3 TaX 4 (X = 6 S, Se, and Te) is performed to investigate the suitability of these compounds to applications 7 as photovoltaic absorber materials. The density functional theory based full potential 8 linearized augmented plane wave method (FP-LAPW method) is employed for 9 computational purposes. The electronic structure and optical properties are determined 10 including electron−electron interaction and spin−orbit coupling (SOC), within the 11 generalized gradient approximation plus Hubbard U (GGA+U) and GGA+U+SOC 12 approximation. The large optical band gaps of Cu 3 TaS 4 and Cu 3 TaSe 4 considered 13 ineffective for absorber materials, and also the hole effective mass has been modulated 14 through applied pressure. These materials show extreme resistance to external pressure, and 15 are found to be stable up to a pressure range of 10 GPa, investigated using phonon 16 dispersion calculations. The observed optical properties and the absorption coefficients 17 within the visible-light spectrum make these compounds promising materials for 18 photovoltaic applications. The calculated energy and optical band gaps are consistent 19 with the available literature and are compared with the experimental results where available.
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(Hf,Zr,Ti)Co(Sb,Sn) Solid solutions were prepared by mechanical-alloying followed by hot-press method as an attempt to reduce Hf concentration and therefore the material’s cost without negatively affecting the thermoelectric performance. To this end, two different methods were applied: (a) Hf substitution with its lighter and cheaper homologue Zr; and (b) fine tuning of carrier concentration by the substitution of Sb with Sn. The isoelectronic substitution of Hf with Zr was investigated in Hf0.6-xZrxTi0.4CoSb0.8Sn0.2 solid solutions and resulted in lower power factors and ZTs. However, the low thermal conductivity of Hf0.4Zr0.2Ti0.4CoSb0.8Sn0.2 contributed in achieving a relatively good ZT~0.67 at 970 K. The effect of charge carrier concentration was investigated by preparing Hf0.4Zr0.2Ti0.4CoSb1-ySny (y = 0.15–0.25) compounds. Hf0.4Zr0.2Ti0.4CoSb0.83Sn0.17 composition prepared by six hours milling reached the highest ZT of 0.77 at 960 K.
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 3ub. 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.
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This paper presents tables of key thermoelectric properties, which define thermoelectric conversion efficiency, for a wide range of inorganic materials. The 12 families of materials included in these tables are primarily selected on the basis of well established, internationally-recognised performance and their promise for current and future applications: Tellurides, Skutterudites, Half Heuslers, Zintls, Mg-Sb Antimonides, Clathrates, FeGa3–type materials, Actinides and Lanthanides, Oxides, Sulfides, Selenides, Silicides, Borides and Carbides. As thermoelectric properties vary with temperature, data are presented at room temperature to enable ready comparison, and also at a higher temperature appropriate to peak performance. An individual table of data and commentary are provided for each family of materials plus source references for all the data.
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The key to designing a half-Heusler begins from the understanding of atomic interactions within the compound. However, this pool of knowledge in half-Heusler compounds is briefly segregated in many papers for specific explanations. The nature of the chemical bonding has been systematically explored for the large transition-metal branch of the half-Heusler family using density-of-states, charge-density, charge transfer, electron-localization-function, and crystal-orbital-Hamilton-population plots. This review aims to simplify the study of a conventional 18-electron configuration half-Heusler by applying rules proposed by renowned scientists to explain concepts such as Zintl-Klemm, hybridization, and valence electron content (VEC). Atomic and molecular orbital diagrams illustrate the electron orbital transitions and provide clarity to the semiconducting behavior (VEC = 18) of half-Heusler. Eighteen-electron half-Heusler usually exhibits good thermoelectric properties owing to favorable electronic structures such as narrow bandgap (<1.1 eV), thermal stability, and robust mechanical properties. The insights derived from this review can be used to design high-performance half-Heusler thermoelectrics.
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We have studied the half-Heusler compound TbPdBi through resistivity, magnetization, Hall effect, and heat capacity measurements. A semimetal behavior is observed in its normal-state transport properties, which is characterized by a large negative magnetoresistance below 100 K. Notably, we find the coexistence of superconductivity and antiferromagnetism in this compound. The superconducting transition appears at 1.7 K, while the antiferromagnetic phase transition takes place at 5.5 K. The upper critical field Hc2 shows an unusual linear temperature dependence, implying unconventional superconductivity. Moreover, when the superconductivity is suppressed by magnetic field, its resistivity shows plateau behavior, a signature often seen in topological insulators/semimetals. These findings establish TbPdBi as a platform for the study of the interplay between superconductivity, magnetism, and nontrivial band topology.
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The M-doping (M = Zr, Hf) effects on the electronic structures and thermoelectric performance of TiCoSb were studied by first-principles calculations. The band structure analysis shows that substituting Ti with M does not change the band structures of these systems significantly. Most of the M-doped systems have a lower band gap value than that of TiCoSb; especially Ti 0.5 Zr 0.5 CoSb has the lowest energy band gap value of 0.971 eV. Besides, the amplitudes of the density of states in the region of the valence bands for M-doped systems show a similar but slightly higher value than Ti-CoSb. Those suggest that these compounds could have better thermoelectric performance than TiCoSb. The phonon dispersion relations show that the larger mass of Zr/Hf with respect to Ti lowers the optical modes and induces mixing with the acoustic branches. Our calculations offer a valuable insight on how to characterize complicated crystal structures of thermoelectric materials and optimize the material composition.
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The half-Heusler (HH) alloy, YNiBi, was synthesized through a reaction between Bi and the intermediate YNi phase. The thermoelectric properties of HH YNiBi were measured most thoroughly. A moderate power factor of 13.3 μWcm−1K−2 was achieved at 485 K, and rather low lattice thermal conductivity was identified, consistent with the theoretical expectation for YNiBi. A significant bipolar contribution to thermal conductivity was observed in YNiBi, which induces a relatively low thermoelectric dimensionless figure-of-merit zT in this material. Enhancement of zT for YNiBi could be realized through appropriate doping to suppress the bipolar effect.
Machine learning (ML) is increasingly becoming a helpful tool in the search for novel functional compounds. Here we use classification via random forests to predict the stability of half-Heusler (HH) compounds, using only experimentally reported compounds as a training set. Cross-validation yields an excellent agreement between the fraction of compounds classified as stable and the actual fraction of truly stable compounds in the ICSD. The ML model is then employed to screen 71,178 different 1:1:1 compositions, yielding 481 likely stable candidates. The predicted stability of HH compounds from three previous high throughput ab initio studies is critically analyzed from the perspective of the alternative ML approach. The incomplete consistency among the three separate ab initio studies and between them and the ML predictions suggests that additional factors beyond those considered by ab initio phase stability calculations might be determinant to the stability of the compounds. Such factors can include configurational entropies and quasiharmonic contributions.
In the present work, the elastic constants and derived properties of tetragonal and cubic Heusler compounds were calculated using the high accuracy of the full-potential linearized augmented plane wave (FPLAPW). To find the criteria required for an accurate calculation, the consequences of increasing the numbers of $k$-points and plane waves on the convergence of the calculated elastic constants were explored. Once accurate elastic constants were calculated, elastic anisotropies, sound velocities, Debye temperatures, malleability, and other measurable physical properties were determined for the studied systems. The elastic properties suggested metallic bonding with intermediate malleability, between brittle and ductile, for the studied Heusler compounds. To address the effect of off-stoichiometry on the mechanical properties, the virtual crystal approximation (VCA) was used to calculate the elastic constants. The results indicated that an extreme correlation exists between the anisotropy ratio and the stoichiometry of the Heusler compounds, especially in the case of Ni$_{2}$MnGa.
Structural, electronic, elastic, optical, and vibrational properties of ternary half-Heusler compounds HfXSb (X = Co, Rh, Ru) were studied with means of ab initio calculations based on the density functional theory. The calculated lattice constants were in good agreement with the available data. The electronic structure and corresponding density of states (DOS) were also calculated. Indirect band gaps were observed for HfCoSb and HfRhSb. Due to some valence bands crossing the Fermi level, HfRuSb has metallic character. In addition to the electronic structure, elastic and optical properties, phonon dispersion curves and phonon DOS were calculated. A detailed comparison was made between these three half-Heusler compounds.
The lattice thermal conductivity κ of various classes of crystalline solids is reviewed, with emphasis on materials with κ > 0.5Wcm−1K−1. A simple model for the magnitude of the lattice thermal conductivity at temperatures near the Debye temperature is presented and compared to experimental data on rocksalt, zincblende, diamond, and wurtzite structure compounds, graphite, silicon nitride and related materials, and icosahedral boron compounds. The thermal conductivity of wide-band-gap Group IV and Group III-V semiconductors is discussed, and the enhancement of lattice thermal conductivity by isotopic enrichment is considered.
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of 23.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.
This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, F[n(r)], independent of v(r), such that the expression E≡∫v(r)n(r)dr+F[n(r)] has as its minimum value the correct ground-state energy associated with v(r). The functional F[n(r)] is then discussed for two situations: (1) n(r)=n0+ñ(r), ñn0≪1, and (2) n(r)=ϕ(rr0) with ϕ arbitrary and r0→∞. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.