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Ab initio study of the structural and optoelectronic properties of the HalfHeusler CoCrZ (Z= Al and Ga)
Article in Canadian Journal of Physics · January 2014
DOI: 10.1139/cjp-2013-0474
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Canadian Journal of Physics
Ab initio study of the structural and optoelectronic
properties of the Half-Heusler CoCrZ (Z= Al and Ga)
Canadian Journal of Physics
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Journal:
Manuscript ID:
Manuscript Type:
Date Submitted by the Author:
Article
17-Sep-2013
Murtaza, Ghulam; Islamia College Peshawar, Physics
Khenata, R.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Seddik, T.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Missoum, A.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
Al-Douri, Y.; Institute of Nono Electronic Engineering, University Malaysia
Perlis, 01000 Kangar, Perlis, Malaysia, Perlis
Abdiche, A.; Engineering Physics Laboratory, Tiaret University, 14000Tiaret- Algeria, Physics
Meradji, H.; Laboratoire LPR, Département de Physique, Faculté des
Sciences, Université Badji Mokhtar, Annaba, Algeria., Physics
Baltache, H.; Laboratoire de Physique Quantique et de Modélisation
Mathématique, Université de Mascara, 29000- Algeria, Physics
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Complete List of Authors:
cjp-2013-0474
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Keyword:
Half Heusler, ab initio calculation, APW+lo, Elastic constants, Bandstrucutre
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Canadian Journal of Physics
Ab initio study of the structural and optoelectronic properties of the Half-Heusler
CoCrZ (Z= Al and Ga)
a
a
A. Missoum , T. Seddik , G. Murtaza b, R. Khenata a, Yarub Al-Douric, A. Abdiche d, H.
Meradji e, H. Baltache a
a
Laboratoire de Physique Quantique et de Modélisation Mathématique, Université de
Mascara, 29000- Algeria
b
Modeling Laboratory, Department of Physics, Islamia College Peshawar, Pakistan
c
Institute of Nono Electronic Engineering, University Malaysia Perlis, 01000 Kangar, Perlis,
Malaysia
d
Engineering Physics Laboratory, Tiaret University, 14000- Tiaret- Algeria
e
Laboratoire LPR, Département de Physique, Faculté des Sciences, Université Badji
Mokhtar, Annaba, Algeria.
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Abstract
In order to study the structural, electronic and optical properties of the half-Heusler
CoCrZ (Z= Al and Ga), we have performed ab initio calculations using the full-potential with
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the mixed basis (APW + lo) method within the generalized gradient approximation (GGA).
The structural properties as well as the band structures, total and atomic projected densities of
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states are computed. From electronic band structures of the hypothetical CoCrGa compound
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we have found semi metallic behavior just like CoCrAl. We also studied the evolution of
electronic structure of CoCrAl under external hydrostatic pressure. It is found that the pseudo
gap around Fermi level increases continuously with increasing pressure, while the electronic
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density of states at the Fermi level does not change significantly. Furthermore, the optical
properties include the dielectric function and refractive index were evaluated and discussed
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under pressure up to 20 GPa as well as electrical conductivity and electron energy loss were
calculated for radiations up to 30 eV. By the same way, we have studied the magnetic
properties of CoCrAl for lattice expansion up to ɑ =1.1ɑ0 where a transition from
paramagnetic phase to half-metallic phase is expected.
PACS: 71.15.Mb, 71.15.Ap, 71.20.Be,
Corresponding author: Tel.: +92 321 6582416
Electronic address: [email protected] (G. Murtaza); [email protected] (R. Khenata)
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1. Introduction
Heusler compounds and their alloys have exceptional physical properties exhibiting
exotic useful features into many devices, as giant magneto resistance sensors (GMR), tunnel
junctions, spin valves (spintronic) [1, 2], narrow gap semiconductors, semi-metals with a
concentration of charge carriers which is adjustable [3, 4], magneto-resistive materials,
thermo-electric materials with a high degree of spin polarization, superconductors, semimetals
and topological insulators [5, 6]. For example, some Heusler materials are highly sought for
their large thermoelectric power because they have high electrical conductivity, a high
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Seebeck coefficient and low thermal conductivity. Therefore, they can be used as sources of
clean energy in order to solve the problem of CO2 emissions [7-9].
Experimentally, the crystal structure and magnetic properties of CoCrAl alloys have
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been explored and the lattice parameter was measured by Luo et al. [10] using the X-ray
powder diffraction. A small magnetic moment of 0.06 µB/unit cell was detected, which does
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not agree with the non-magnetic state predicted by the Slater-Pauling rule. A similar result has
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been found in full-Heusler alloy Fe2TiSn, which has 24 valence electrons and should be nonmagnetic. But due to the Fe–Ti disorder a Curie temperature and a spin moment of 0.26
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µB/unit cell at 5 K is observed [11]. Theoretically, Luo et al. [12] have calculated the
electronic and magnetic properties for the Half-Heusler compounds XCrAl (X = Fe, Co, Ni),
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using the FP-LAPW method with the local spin density approximation.
To our knowledge, it seems there is a lack of experimental and theoretical data reported
in literature on the structural, elastic and optical properties and their pressure behavior for the
interested compounds. Further, let us notice that CoCrGa remains hypothetical compound to
be studied in details. Hence, using the full potential augmented plane wave plus local orbital
method (FP-APW+lo) with the generalized gradient approximation (GGA) formalism for
exchange and correlation (Xc) effects, we have focused on the calculation of structural
properties, band structure and optical properties for the Half-Heusler CoCrZ (Z= Al and Ga)
compounds. This work is organized as follow: in section 2, we have indicated a brief review
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of the computational techniques used in this study. Section 3 presents and discusses the results
of structural, electronic and optical properties of the CoCrAl and CoCrGa compounds. At the
end, we have summarized the main conclusions of our work in section 4.
2. Computational method
These first-principles calculations were performed using the full-potential augmented
plane wave with the mixed basis (APW+lo) method [13, 14] within the density functional
theory (DFT) [15], implemented in the WIEN2K code [16]. The exchange-correlation effects
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are calculated by the generalized gradient approximation (GGA) [17]. To ensure the
convergence of energy eigenvalues, wave functions in the interstitial region were expanded in
plane waves with a cutoff Kmax = 8/RMT, where RMT is the smallest atomic muffin-tin sphere
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radius and Kmax determines the upper value of K vector magnitude in the plane wave
expansion. RMT's values are chosen to be 2.2, 2.2, 2.0 and 2.1 atomic units (a.u.) for Co, Cr,
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Al and Ga respectively. The valence wave functions inside the muffin-tin spheres are
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expanded up to lmax =10. The Fourier coefficients of charge density was expanded up to
Gmax=14(a. u.)-1. The convergence of self-consistent calculations was performed so that the
total energy of the system is stable with an accuracy of 10-5 Ry and a deviation of charge
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density less than 10-4 e. The integration over the Brillouin zone is performed up to 19 k-points
in the irreducible Brillouin zone, using the Monkhorst–Pack special k-points approach [18].
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3. Results and discussion
3.1. Ground states properties
Half-Heusler compounds crystallize in the C1b structure corresponding to space group
F-43m (No.:216), which consists of four fcc sub lattices and have the chemical formula XYZ,
where X, Y and Z atoms are located at (1/4, 1/4, 1/4), (0, 0, 0) and (1/2, 1/2, 1/2),
respectively. Generally, it is established that X is taken as a high valent transition metal atom,
Y as a low valent transition metal atom and Z as sp atom [19].
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The structural properties of the said Half-Heusler compounds are predicted by
optimizing the volume, i.e. minimizing the total energy of the unit cell with respect to the
variation in the unit cell volume. The variation of the total energy versus unit cell volume for
both nonmagnetic (nm) and spin polarized ferromagnetic (sp) phases for CoCrAl and CoCrGa
is displayed shown in Fig.1. We have found that the nonmagnetic (nm) is energetically
favored if we neglect the energy difference (-0.0015 eV) for CoCrAl compound. The
structural parameters like lattice constant, bulk modulus and the pressure derivative of bulk
modulus are determined by fitting the variation of the total energy versus volume of the
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nonmagnetic to the Murnaghan’s equation of state [20]. They are listed in Table 1. Our
computed lattice constant is in very good agreement with other theoretical results and slightly
underestimated compared to the experimental one. To the best of our knowledge no reported
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experimental or theoretical data on structural properties for CoCrAl are available.
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3.2 Elastic properties and their related constant
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Elastic constants of materials are essential for better understanding of their properties.
The elastic constants describe the response of a solid material to a small loading which causes
reversible deformations. Some basic mechanical properties can be derived from the elastic
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constants such as the bulk modulus, Young’s modulus, shear modulus, Poisson’s ratio, which
play an important part in determining the strength of the materials. From a fundamental view
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point, the elastic constants are related to various fundamental solid-state properties such as
interatomic potentials, equation of state, structural stability, phonon spectra and they are
linked thermodynamically to the specific heat, thermal expansion, Debye temperature,
melting point and Grüneisen parameter. In our case, these compounds have a cubic symmetry
hence only three independent elastic C11, C12 and C44 should be calculated. The elastic
constants C ij of the herein studied compounds are obtained by calculating the total energy as
a function of volume conserving strains following the Mehl method [21]. The calculated
elastic constants are listed in table 2. One can notice that the unidirectional elastic constant
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C11 is much higher than the C44 indicating that these compounds present weaker resistance to
pure shear deformation compared to resistance to unidirectional compression. We can notice
that the calculated elastic constants of CoCrAl are not very different from those of CoCrGa.
To the best of our knowledge no reported experimental or theoretical data on the elastic
constants for CoCrAl and CoCrGa compounds. The existence of a crystal in a stable or
metastable state requires that the following conditions between their elastic constants must be
full filled [22]:
;
;
. Our results for elastic constants in
Table 2 satisfy these stability conditions meaning that the herein studied compounds are
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elastically stable.
The bulk and shear moduli (B, G) of both compounds were calculated using the Voigt,
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Reuss and Hill approaches [23-25]. For the cubic system G is expressed as follows:
(12)
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(13)
(14)
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The Young’s modulus E and Poisson’s ratio ν for an isotropic material are given by
(15)
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(16)
Using the relations above the calculated shear modulus G, Young’s modulus E and Poisson
ratio ν for CoCrAl and CoCrGa are given in Table 2. Young's modulus is defined as the ratio
of uni-axial stress on the uni-axial deformation within the limits of Hook's law. If the value of
Young's modulus is high, the material is stiffer. The calculated values of Young's modulus
indicate that CoCrAl is stiffer than CoCrGa. Typical values of Poisson’s ratio are around to be
0.1 for covalent materials, 0.25 for ionic materials and around 0.3-0.45 for metals [26]. In the
present case, the value of Poisson’s ratio is 0.31 and 0.33 for CoCrAl and CoCrGa
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respectively, indicating that the metallic bonding contribution to the atomic bond is dominant.
This result is in agreement with experimental results [10] and several theoretical studies on
the nature of bonds in the Heusler compounds [27-29] including our electronic structure study
in following section. We also have calculated the anisotropy factors for these compounds by
classical relation: A = (2C44 + C12)/C11. The calculated anisotropy factors are 0.97 and 0.99
for CoCrAl and CoCrGa, respectively. From these values, we can conclude that these
compounds are substantially isotropic. This is further confirmed by the fact that G ≈ C44. Up
to this date, no experimental or theoretical structural properties are available for comparison
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with our theoretical results. In future, other experimental or theoretical work will be a good
test for our results.
The calculation of Young's modulus E, bulk modulus B and shear modulus G allowed
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us to calculate the Debye temperature, which is a fundamental and very important parameter
for determining physical properties such as the electrical conductivity versus temperature
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described by Bloch-Gruneisen formula, the specific heat, the melting temperature and the
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elastic constants. To evaluate the Debye temperature (
) from the elastic constants, we use
the standard methods defined by the following classical relations [26]:
is the average sound velocity, h is the Plank’s constant,
is the Boltzmann’s
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constant,
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where
(1)
is the atomic volume and n is the number of atoms per unit cell.
In polycrystalline materials, the average sound velocity is determined by classical
relation:
(2)
where
and
are the longitudinal and transversal elastic wave velocities respectively. In
cubic systems which are assumed isotropic materials, one can calculate from the Navier’
relations [30]:
(4)
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3.3. Electronic properties
We have performed calculations of band structures, along high symmetry directions (WL-Γ-X-W-K) of the Brillouin zone for CoCrZ (Z = Al, Ga) which are shown in Fig. 2. It is
clear that, close to the high symmetry points (Γ, X, W), the conduction band and valence band
across the Fermi level have a slight overlap, where their band structure shows a pocket of
electrons at the X point and a pocket of holes at the W point. CoCrAl and CoCrGa have a
small density of states of 0.4 and 0.15 state/eV.unit cell, respectively, at the Fermi energy in
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the spectra as shown in Fig. 3. This result shows that the compounds CoCrZ (Z = Al, Ga) are
semi metallic. They have a symmetrical DOS for each spin polarization, majority (up) and
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minority (down); therefore the magnetic moment is zero. Our half Heusler CoCrZ (Z = Al,
Ga) compounds, with 18 valence electrons (VEC) split equally into two spin polarization, so
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they are distributed on the nine bands of lower energy levels. The density of states (DOS) in
spin polarized calculations shows that CoCrZ (Z = Al, Ga) are paramagnetic and semi
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metallic, having a deep pseudo gap with approximately 0.4 eV of width and centered at the
Fermi level. As it is seen on total DOS curves, the total density of states decreases and
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increases abruptly with sharp peaks around Fermi level energy, within which they have small
density of states. The calculated total and partial DOS for CoCrZ (Z=Al,Ga) are displayed in
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Fig. 3, where we can see that they have the same shape for both compounds CoCrAl and
CoCrGa. We notice on Fig. 2 that, in the vicinity of the Fermi level (
), the dispersion of the
band structure of the two compounds CoCrZ (Z = Al, Ga) have the same shape, comparable to
Fe2VAl which is paramagnetic and semi metallic with a deep pseudo gap [31, 32]. Our results
agree well with those obtained by Luo et al [10, 12] and other general studies [33, 34]. Half
Heusler compounds with VEC = 18 are either semiconductors having a narrow gap or
semimetals (defined as materials with low number of electrons N(
) at the Fermi-level).
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In order to elucidate the different contributions from the different components in the
materials to the conductivity, the total and projected DOS of the atomic contributions are
plotted in Fig. 3. From the partial DOS curves; we can identify the angular momentum
character of the different structures. The structure which is in the range -8.1 (-9.5) to -4.7 eV
(-6.2) eV for CoCrAl (CoCrGa) is mainly due to the combination of electronic states Co, Cr
and Al (Ga).
The valence region for CoCrAl (CoCrGa), which extends from -4.1 (-5) eV at the Fermi
level EF=0, is mainly due to the combination of Co-3d and Cr-3d states. This admixture is the
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same for the conduction region which extends from the Fermi level EF = 0 to 4.1 (5) eV,
while atom Al (Ga) has a small contribution. On the other hand, for CoCrGa only, the valence
region shows a peak at the bottom, located at -14 eV (not represented in the range of Fig. 2).
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The states Co-3d and Cr-3d are globally dispersed around the Fermi level; hence it is clear
that Cr-3d and Co-3d states are strongly hybridized near the Fermi level. For CoCrGa, it is
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noticeable that Ga-3d has a contribution to the bottom of structure with a sharp peak. For the
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two compounds, we can conclude that their different structures are mainly dominated by Co3d and Cr-3d in the neighbor of Fermi level.
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As shown in Fig. 4 for CoCrAl compound, the width of the pseudo gap increases with
increasing pressure; however, up to 20 GPa it stays semi metallic, while the density of states
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at the Fermi level shows a weak dependence of pressure. Our spin polarized calculations are
shown in Fig. 5, predict that for a=1.1 =6 , CoCrAl is a half semimetal ferromagnet with
0.05
in majority state (up) and a small gap in minority state (down) 0.1 eV, just like the
superconducting topological semi metal YPtBi [35]. These observations indicate that a
transition from semimetal to narrow gap semiconductor is possible. As noted by Coey [36], a
lattice parameter expansion leads to an increase of
and sometimes a change of magnetic
structure. We conclude that CoCrAl is in a transition region between a ferromagnetic
semimetal and half metallic ferromagnet, similar to that observed for the composite halfHeusler FeVSb [37].
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3.4. Optical properties
The CoCrZ (Z = Al, Ga) compounds have a cubic symmetry; it is enough to compute
only one component of the dielectric tensor, which can completely determine the linear
optical properties. We denote the dielectric function by
the frequency,
, where ω is
is its imaginary part which is given by the relation [38]:
(6)
where the integral is taken over the first Brillouin zone. The momentum dipole elements
are the matrix elements for direct transitions between the valence
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state
and the conduction band state
, A is the potential vector defining the
electromagnetic field, and the energy ћ
is the corresponding transition
of the dielectric function can be deduced from the imaginary part
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energy. The real part
using the Kramers-Kronig relation:
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(7)
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where P is the principal value of the integral. Once the real and imaginary parts of the
dielectric function are determined, we can calculate important functions such as optical
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refractive index n( ) and electrical conductivity
, which we can find in literature of
physical optics [39]:
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with relations (9) and (10):
;
(8)
(9)
For the calculations of optical properties, we require a dense mesh of k-points,
uniformly distributed in the k-space. Thus, the Brillouin zone integration was performed up to
506 k-points in the irreducible part of the Brillouin zone. The difference is insignificant by
increasing the number of k-points beyond 506. In this work, we presented calculations with
only 506 k-points and broadening is taken to be 0.03 eV. Figure 7 shows the curves of real
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and imaginary part of dielectric function for a radiation spectrum up to 30 eV. As shown, the
optical spectra of CoCrAl and CoCrGa have some similarities. For each compound, the
dielectric function
has two structures as shown in Fig. 7. Optical transition occurs at
0.1 and 0.15 eV for CoCrAl and CoCrGa, respectively. Beyond these pseudogaps, curve
increases abruptly. The main peak in the spectra is located inside infrared range at 0.9 eV and
1.1 eV for CoCrAl and CoCrGa, respectively, followed by another structure located at 2.8 eV
and 3 eV, respectively. To interpret the optical spectrum, it is necessary to give several
transitions allocated to peaks which are present in the curve of reflection spectrum in Fig. 8,
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where many transitions appear at the associated energy. For both compounds, the main peak
for dielectric functions
is mainly due to the transitions within the Cr-3d bands.
is convoluted from
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The real dielectric function
conversion. The negative values of
by the Kramers–Kronig
are in the near infrared from 0.7 to 1.3 eV,
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indicating that the two crystals CoCrZ (Z = Al and Ga) have a Drude behavior. The high
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reflectivity for energies less than 1 eV characterizes a high conductivity in the infrared region,
with a narrow peak at 0.9 eV just after the pseudogap which can be considered as a threshold
(Fig. 8, 9).
loss function
shows a sharp drop while the energy
shows a main peak (Fig. 11) and the real part of the dielectric
(Fig.7) vanishes at
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function
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In successive Fig’s (7, 8, 11), the reflectivity R
, located at 24.5eV and at 24.0 eV for CoCrAl and
CoCrGa, respectively, corresponding to the energy of the incident photon, at which a material
becomes transparent for frequencies upper than the screened plasma frequency
. If we
assume that there is no photonic contribution, the static dielectric constants obtained at the
zero frequency limits
are 90.4 and 70 for CoCrAl and CoCrGa, respectively.
The dispersion curves of refractive index for CoCrAl and CoCrGa (Fig. 10) show that
both compounds have the same features. The refractive index shows a maximum value of 9.6
and 8.4 at 0.1 and 0.7 eV for CoCrAl and CoCrGa, respectively, followed by a secondary
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peak at 2.6 and 3 eV, respectively with Drude-like behavior. These results indicate two
possible interbands transitions in agreement with the calculated results.
Figure 6 shows the variations of the static refractive index as a function of the pressure
for these compounds. By increasing the pressure up to 20 GPa, there is a linear decrease of the
static refractive index. By linear fitting, we can determine the pressure derivative of the
refractive index n of these compounds and deducing the pressure coefficient
which has value
and
for CoCrAl and CoCrGa,
respectively. To our knowledge, there is no theoretical or experimental data available on the
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optical properties of these compounds, except for the experimental lattice parameter, the
magnetic and electronic properties of CoCrAl which were studied by Luo et al. [10,12].
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4. Conclusion
In this work, we have carried out a detailed investigation on the structural, elastic, electronic
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and optical properties of the Half-Heusler compounds CoCrZ (Z = Al, Ga) by using the APW
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+ lo method within the generalized gradient approximation (GGA). We summarize the most
important discussions:
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(i) We have determined the ground state properties, including lattice parameter, bulk modulus
and its pressure derivative. For CoCrAl, the available data are consistent with the calculated
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ground state and the related lattice parameter.
(ii) For the considered compounds CoCrZ (Z= Al, Ga), the elastic constants
, Young’s
modulus E, shear modulus G, Poisson’s ratio , sound velocity and the Debye temperature are
computed. The computed values of Poisson’s ratio indicate that these compounds have a
metallic-like bonding.
(iv) Calculated band structure and DOS show that these compounds are semimetal with deep
pseudogap close to the high symmetry points (Γ, X, W), with a width approximately equal to
0.2 eV for CoCrAl and to 0.1 eV for CoCrGa. As well, in both compounds, pseudogap width
increases with increasing pressure.
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(v) Calculations of the electronic band structure, for an expansion of the lattice parameter of
the compound up to
show that CoCrAl is in a transition region, between a semimetal
and a half semi metallic ferromagnet.
(vi) We have analyzed the spectral curves of the complex dielectric function, the reflectivity
and the optical conductivity to identify possible optical transitions. Also, we have determined
the pressure derivative of the refractive index n of these compounds and deducing their
pressure coefficient.
(ix) The high reflectivity of the studied compounds in low energy region, make them useful
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candidates for a coating to avoid solar heating.
To the best of our knowledge, there are only two partial studies of these compounds,
mentioned above. Therefore, our study on the structural and optical properties of these
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compounds have not been measured or calculated yet, so it is assumed to be the first
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theoretical prediction of these properties, which awaits experimental confirmation. Hopefully,
this work and its results can be considered as foresight study and stimulating matter for future
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Table Captions
Table 1: The calculated lattice constant ܽ଴ (in ), bulk modulus
derivative
and energy difference
(in GPa) and its pressure
between the spin-polarized (sp) and
nonmagnetic (nm) state at equilibrium lattice constant for CoCrZ (Z=Al, Ga) compounds.
CoCrAl
B0
Present
5.483
165.84
Expt. [13]
5.742
Other calc. [14]
5.52
Present
5.47
r
Fo
CoCrGa
a0
173.84
Re
C12
-0.0015
5.25
0.03
(in GPa), shear modulus G (in GPa), Young’s
modulus E (in GPa) and Poisson’s ratios
C11
4.76
-0.002
Table 2: Calculated elastic constants
volume.
∆E
C44
for CoCrAl and CoCrGa at the equilibrium
GR
GV
GH
E
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262.64 117.44 70.34
71.24
71.22
71.23
186.94
0.31
CoCrGa
268.35 126.58 70.40
70.65
70.65
70.65
186.66
0.32
, longitudinal, transverse and average sound
Table 3: Calculated density (
,
,
On
velocity (
ew
CoCrAl
in m/s) from the isotropic elastic moduli, and Debye temperature (
for CoCrAl and CoCrGa compounds.
ρ
vL
vT
vm
CoCrAl
5.56
6850.23
3580.66
4005.67
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498
CoCrGa
7.33
6002.11
3038.81
3406.43
425
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in K)
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Figure Captions
Fig.1: Total energy of studied compounds as a function of lattice volume of CoCrAl (a) and
CoCrGa (b) for both phases, non-magnetic (nm) and ferromagnetic (sp).
Fig.2: Calculated band structure of CoCrAl (a) and CoCrGa (b).
Fig.3: Total and partial densities of states of CoCrAl (a) and CoCrGa (b).
Fig.4: Evolution of total density of states (DOS) of CoCrAl under pressure
Fig.5: Total density of states (DOS) for the CoCrAl compound for two different values of the
lattice constant. Positive values of the DOS correspond to the majority (spin-up) electrons and
negative values to the minority (spin-down) electrons. The zero of the energy has been chosen
r
Fo
at Fermi energy.
Fig.6: Pressure dependence of static refractive index
for CoCrAl and CoCrGa.
Fig. 7: Calculated real and imaginary parts of the dielectric function CoCrAl (a) and CoCrGa
(b).
Re
Fig.8: Calculated reflectivity R(ω) spectra of CoCrAl and CoCrGa.
Fig.9: Calculated real part of photoconductivity spectra of CoCrAl and CoCrGa.
Fig.10: Calculated refractive index n(ω) spectra of CoCrAl and CoCrGa.
vi
Fig.11: Calculated electron energy loss L(ω) spectra of CoCrAl and CoCrGa.
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(a)
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(b)
Fig.1
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Fig. 2
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Fo
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Fig.3
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r
Fo
Fig. 4
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Fig.5
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Fo
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Fig.6
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r
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Fig. 7
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Fig.8
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Fig.9
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Fig.10
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Fig.11
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