See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/260516872 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 CITATIONS READS 48 440 9 authors, including: Ghulam Murtaza Rabah Khenata Islamia College Peshawar University Mustapha Stambouli of Mascara 169 PUBLICATIONS 2,477 CITATIONS 652 PUBLICATIONS 10,165 CITATIONS SEE PROFILE Abdelmadjid Bouhemadou University Ferhat Abbas - Setif 1 SEE PROFILE Yarub Al-Douri 429 PUBLICATIONS 8,359 CITATIONS 297 PUBLICATIONS 6,370 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: First principle study of Quaternary chalchogenide to find optical and thermo-electric properties. View project Wien2k Papers View project All content following this page was uploaded by Ghulam Murtaza on 04 May 2014. 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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 r Fo 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 ew vi Re Complete List of Authors: cjp-2013-0474 On Keyword: Half Heusler, ab initio calculation, APW+lo, Elastic constants, Bandstrucutre ly http://mc06.manuscriptcentral.com/cjp-pubs Page 1 of 24 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. r Fo 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 Re 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 vi states are computed. From electronic band structures of the hypothetical CoCrGa compound ew 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 On 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 ly 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) http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 2 of 24 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 r Fo 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 Re 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 vi not agree with the non-magnetic state predicted by the Slater-Pauling rule. A similar result has ew 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 On 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), ly 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 http://mc06.manuscriptcentral.com/cjp-pubs Page 3 of 24 Canadian Journal of Physics 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 r Fo 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 Re 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, vi Al and Ga respectively. The valence wave functions inside the muffin-tin spheres are ew 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 On 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]. ly 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]. http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 4 of 24 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 r Fo 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 Re experimental or theoretical data on structural properties for CoCrAl are available. vi 3.2 Elastic properties and their related constant ew 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 On 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 ly 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 http://mc06.manuscriptcentral.com/cjp-pubs Page 5 of 24 Canadian Journal of Physics 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 r Fo elastically stable. The bulk and shear moduli (B, G) of both compounds were calculated using the Voigt, Re Reuss and Hill approaches [23-25]. For the cubic system G is expressed as follows: (12) vi ew (13) (14) On The Young’s modulus E and Poisson’s ratio ν for an isotropic material are given by (15) ly (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 http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 6 of 24 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 r Fo 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 Re 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 vi described by Bloch-Gruneisen formula, the specific heat, the melting temperature and the ew 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 ly constant, On 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) http://mc06.manuscriptcentral.com/cjp-pubs Page 7 of 24 Canadian Journal of Physics 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 r Fo 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 Re 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 vi 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 ew 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 On 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 ly 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). http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 8 of 24 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 r Fo 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). Re 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 vi noticeable that Ga-3d has a contribution to the bottom of structure with a sharp peak. For the ew two compounds, we can conclude that their different structures are mainly dominated by Co3d and Cr-3d in the neighbor of Fermi level. On 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 ly 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]. http://mc06.manuscriptcentral.com/cjp-pubs Page 9 of 24 Canadian Journal of Physics 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 r Fo 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 Re energy. The real part using the Kramers-Kronig relation: vi (7) ew 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 On refractive index n( ) and electrical conductivity , which we can find in literature of physical optics [39]: ly 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 http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 10 of 24 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, r Fo 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 Re 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, vi indicating that the two crystals CoCrZ (Z = Al and Ga) have a Drude behavior. The high ew 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 ly function On 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 http://mc06.manuscriptcentral.com/cjp-pubs Page 11 of 24 Canadian Journal of Physics 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 r Fo 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]. Re 4. Conclusion In this work, we have carried out a detailed investigation on the structural, elastic, electronic vi and optical properties of the Half-Heusler compounds CoCrZ (Z = Al, Ga) by using the APW ew + lo method within the generalized gradient approximation (GGA). We summarize the most important discussions: On (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 ly 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. http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 12 of 24 (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 r Fo 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 Re compounds have not been measured or calculated yet, so it is assumed to be the first vi 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 On References ew work on these materials. 1. K. A. Kilian, R. H. Victora, J. Appl. Phys. 87, 7064 (2000). ly 2. C. T. Tanaka, J. Nowak, J. S. Moodera, J. Appl. Phys. 86, 6239 (1999). 3. J. A. Caballero, Y. D. Park, J. R. Childress, J. Bass, W.-C.Chiang, A. C. Reilly, W. P. Pratt Jr., and F. Petroff, J. Vac. Sci. Technol. A 16, 1801 (1998). 4. C. Hordequin, J. P. Nozières, and J. Pierre. J. Magn. Magn.Mater.183, 225 (1998). 5. S.A.Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Moln·r, M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science. 294, 1488 (2001). 6. R. A. de-Groot, F. M. Mueller, P. G. V. Engen, K. H. J. 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Voigt, Lehrburch der Kristallphysik_Teubner, Leipzig, (1928). http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 14 of 24 26. M. H. Ledbetter, Materials at Low Temperatures, edited R. P. Reed and A. F. Clark American Society for Metals, OH, (1983). 27. Nanda B R K and Dasgupta S. J. Phys.: Condens. Matter 15, 7307 (2003). 28. Galanakis I, Dederichs P H and Papanikolaou N. Phys. Rev. B 66, 134428 (2002). 29. H. C. Kandpal, C. Felser, R. Seshadri, J. Phys. D: Appl. Phys. 39, 776 (2006). 30. E. Schreiber, O.L. Anderson, N. Soga, Elastic Constants and their Measurements, McGraw-Hill, New York, (1996). 31. E. Shreder, S. Streltsov, A. Svyazhin, A. Lukoyanov, V. Anisimov. Proceedings of the r Fo Third Moscow International Symposium on Magnetism, 220-223, (2005). 32. Ye Feng, J. Y. Rhee, T. A. Wiener, D. W. Lynch, B. E. Hubbard, A. J. Sievers, D.L.Schlagel, T. A. Lograsso, and L. L. Miller, Phys. Rev. B 63, 165109 (2001). Re 33. I. Galanakis, E. Sasioglu, K. Ozdogan Phys. Rev. B 77, 214417 (2008). 34. L. Offernes, P. Ravindran, A. Kjekshus, Journal of Alloys and Compounds 439, 37 vi (2007). ew 35. N. P. Butch, P. Syers, K. Kirshenbaum, A. P. Hope, J. Paglione, Phys. Rev. B 84, 220504 (2011). On 36. J. M. D. Coey Trinity College, Dublin. Magnetism and Magnetic Materials (2010). 37. Bo Kong, Bo Zhu, Yan Cheng, Lin Zhang, Qi-Xian Zeng, Xiao-Wei Sun, Physica B ly 406, 3003 (2011). 38. Ambrosch-Draxl C, Sofo JO. Comput Phys. Commun., 175,1,(2006). 39. M.Dressel, G.Gruner .Electrodynamics of Solids, Optical Properties of Electrons in Matter Cambridge, (2003). http://mc06.manuscriptcentral.com/cjp-pubs Page 15 of 24 Canadian Journal of Physics 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 vi 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 ly 498 CoCrGa 7.33 6002.11 3038.81 3406.43 425 http://mc06.manuscriptcentral.com/cjp-pubs in K) Canadian Journal of Physics Page 16 of 24 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. ew ly On http://mc06.manuscriptcentral.com/cjp-pubs Page 17 of 24 Canadian Journal of Physics r Fo (a) ew vi Re ly On (b) Fig.1 http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 18 of 24 r Fo ew vi Re ly On Fig. 2 http://mc06.manuscriptcentral.com/cjp-pubs Page 19 of 24 Canadian Journal of Physics r Fo ew vi Re ly On Fig.3 http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 20 of 24 r Fo Fig. 4 ew vi Re ly On Fig.5 http://mc06.manuscriptcentral.com/cjp-pubs Page 21 of 24 Canadian Journal of Physics r Fo vi Re Fig.6 ew ly On http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 22 of 24 r Fo ew vi Re ly On Fig. 7 http://mc06.manuscriptcentral.com/cjp-pubs Page 23 of 24 Canadian Journal of Physics r Fo ew vi Re Fig.8 ly On Fig.9 http://mc06.manuscriptcentral.com/cjp-pubs Canadian Journal of Physics Page 24 of 24 r Fo Re Fig.10 ew vi ly On Fig.11 http://mc06.manuscriptcentral.com/cjp-pubs View publication stats
