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Elastic constants and observed ferromagnetism in inverse Heusler alloy Ti2CoAs using kjpaw pseudopotentials: A first-principles approach
Accepted Manuscript Elastic constants and observed ferromagnetism in inverse Heusler alloy Ti2CoAs using kjpaw pseudopotentials: A first-principles approach O.E. Osafile, P.O. Adebambo, G.A. Adebayo PII:
S0925-8388(17)32070-4
DOI:
10.1016/j.jallcom.2017.05.338
Reference:
JALCOM 42151
To appear in:
Journal of Alloys and Compounds
Received Date: 25 February 2017 Revised Date:
9 May 2017
Accepted Date: 29 May 2017
Please cite this article as: O.E. Osafile, P.O. Adebambo, G.A. Adebayo, Elastic constants and observed ferromagnetism in inverse Heusler alloy Ti2CoAs using kjpaw pseudopotentials: A first-principles approach, Journal of Alloys and Compounds (2017), doi: 10.1016/j.jallcom.2017.05.338. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Elastic constants and observed Ferromagnetism in inverse Heusler Alloy Ti2CoAs using kjpaw Pseudopotentials: A First-Principles Approach O. E. Osafile1, P.O. Adebambo2* and G.A. Adebayo3* 1 Federal University of Petroleum Resources, Effurun, Nigeria 2 Department of Physical and Computer Sciences, McPherson University, Abeokuta, Nigeria 3 Department of Physics, Federal University of Agriculture, PMB 2240, Abeokuta, Nigeria
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Abstract We performed density functional calculations of the structural parameters, spin-polarized projected density of state and elastic properties of X2CoAs (X = Mn, Ti) inverse Heulser alloys, within the Generalised Gradient Approximation (GGA). At first, we determined the stable phases of the alloys and other parameters such as lattice parameters, bulk modulus, its pressure derivatives, as well as the three independent elastic constants. The elastic constants show that the alloys are stable and are brittle in nature as revealed by the Cauchy stress parameters (C12 – C44). The results also show Mn2CoAs inverse Heusler alloy is better plasticity material compared to Ti2CoAs inverse Heusler alloy. The spin polarised projected density of state of the Heusler alloys was obtained and reported. The present calculations predicted both alloys to be Ferromagnetic in nature with Mn2CoAs having better magnetic property than Ti2CoAs.
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INTRODUCTION The Heusler alloys are compounds that have become highly researched both theoretically and experimentally in recent times because of their promising multifunction [1-3]. The possible applications include spin electronics, magnetic shape memory (MSM), optoelectronics among others. Ever since the advent of the Heusler compound named after Von Heusler in 1903 [4, 5], different Heusler structural variants or phases have been discovered and studied. These morphs include the full Heusler alloy (X2YZ) which crystallises in the L21 structure with four FCC sublattices [6-8]. Another identified variant results when one of the sublattices could be left vacant, precisely the X = ½, ½, ½ sub-lattice; this is known as the half Heusler phase with an XYZ configuration which crystallises in the C1b structure. Other variants include the quaternary Heusler (XX’YZ) [9] and the inverse Heusler (XYXZ). In this case, X occupies (0, 0, 0) and ( ½, ½, ½ ); these different variants experience enormous diversities both in physical and magnetic properties. In the alloys, X and Y denote the transition metals, and Z is an s-p element such as Al, Sb, In, Ga, As, etc. Numerous studies have shown that heat treatment and the structural composition (stoichiometrically) of these variants are important factors that determine the magnetic properties of the system under study, despite the fact that no single set of parameters can adequately describe the entire Heusler family [10]. The inverse Heusler compounds have been studied in recent times by some researchers both experimentally and theoretically with Mn occupying the X state in the full Heusler alloy [1113]. In these alloys, the valence of the X-transition metal atom is smaller than that of the Ytransition metal atom. Hence it crystallises in the XA or Xα structure, where the sequence is of the form X-X-Y-Z, and the prototype is Hg2TiCu [14]. The XA structure has been observed to be more energetically favourable than the L21 structure. Unlike the conventional full-Heusler alloys, where both direct and indirect exchange interactions are present, the properties of the inverse Heusler alloys have more of short-range interactions [15, 16]. In 2009, Kubota et al. grew the Mn2VAl alloy using soft X-ray magnetic circular dichroism and found it to be a ferrimagnetic half-metal [11]. Also, using first-principles calculation [17],
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Galanakis et al. doped Mn2V(Al, Si) Heusler alloys. They observed the half-metallic state of the systems makes it suitable materials for spintronic applications since their zero magnetisation leads to lower stray fields and thus small energy losses. Mayasuki et al. performed NMR and magnetisation measurement of Mn2VAl and observed found it to be a simple ferromagnet [18]. Singh et al. in 2013 studied the disorder dependent half-metallicity in Mn2CoSi, inverse Heusler alloy, via ab-initio calculations and surmised a 100% spin polarisation for this compound [19]. Some of these researchers have pointed to the possibility of the attractive ferromagnetic half-metallic behaviour in these inverse Heuslers and have associated this with a Slater-Pauling behaviour of the total spin magnetic moment [20]. Typically, Slater-Pauling combines coherent-growth on semiconductors with high Curie temperatures [21].
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In this work, we study the X2CoAs inverse Heusler alloy with X = Mn, Ti. Both compounds obey the slater-Pauling rule of Mt = Zt – 24 for Mn2CoAs and Mt = Zt – 18 for Ti2CoAs. Slater and Pauling showed in their pioneering papers [22, 23] that on adding one valence electron to a binary magnetic alloy, the electron occupies the minority spin band and reduces the total spin magnetic moment by about 1µB. This behavioural pattern has been found to exist in half-metallic Heusler alloys [20]. These Slater-Pauling rules connect the electronic properties of the half-metallic ferromagnets (HMF) alloy (appearance of half-metallic behaviour) to the magnetic properties (total spin magnetic moments) [20]. Information abounds in the literature on the band structure, and density of states of Mn2CoAs [23, 24], while the lattice constants and magnetic moment of the 3d and s-p elements Ti2CoAl are also available [25]. However, such information is lacking on Ti2CoAs alloy. The present calculations further investigate the elastic properties of these inverse Heusler alloys to buttress the possibility of their usefulness in photovoltaic, spintronics and some other applications.
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METHODS AND COMPUTATIONAL DETAILS Calculations were performed using quantum espresso package [26] with the projector augmented wave (PAW) based pseudopotential [Ref]. The Perdew-Burke-Ernzerhof (PBE) version of the generalised gradient approximation (GGA) to the exchange-correlation function of the density functional theory (DFT) was adopted [27]. The calculations are without spin-orbit coupling, though all other relativistic effects are included making the calculation semi-relativistic. In this approach, the Kohn-Sham equation is: 1 2 (1) − 2 ∇ + Vnuc ( r ) + VH [ n( r ) ] + Vxc [ n( r ) ] Ψ i ( r ) = Ei Ψ i ( r ) Equation (1) is solved self-consistently using a plane wave basis set for a fixed nuclear (ionic) positions. Where the first term in Eqn. (1) is the kinetic energy of a non-interacting electrons system, the second term is the Hartree-like energy term, which accounts for the classical Coulomb interaction of a spatial charge distribution (r) and the third term represents the exchange-correlation energy [28, 29]. The only unknown quantity is the exchangecorrelation energy functional and, in principle, the quality of the solution of the full manybody problem is dependent on the quality of the approximation. The FCC structure of these Heusler alloys are periodic, so the plane wave is calculated using: 1 Ψ k (r ) = ΣCk .G ei ( k +G ).r (2) Ω And can be represented as a grid in the k-space. The Brillouin zone (sampled over the high symmetry points) aids proper convergence of the infinite numbers of plane waves in principle. To truncate the plane wave expansion at some value of |k + G|, we use a converged
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h2 K + G
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≤ Ecut 2m (3) The convergence test of the high symmetry k-points on k-point sampling is achieved using: 1 P = ∑ P ( k )W k N k k ∈ BZ (4)
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The Monkhorst-Pack [30] k-point mesh is used in this work to take advantage of an equally spaced mesh in the reciprocal lattice by Fast Fourier Transform. Computational time was saved using the weighted k-points from only the irreducible Brillouin zone (IBZ). The basis set valence configuration of the X-element Mn is 3s2, 4s2, 3p6 and 3d5 while that of Ti is 3s2, 4s2, 3p6, 4p2 and 3d2. For the Y-Co atom, it is 4s2, 3p6 and 3d7 while the Z-As atom is 4s2 and 4p3. In addition to reported properties in the literature, we investigated the elastic properties of the systems. The elastic properties describe how well the system withstands deformation when exposed to external forces. The optical properties can be computed (real and imaginary parts) using the approach in [26]; this can be achieved using norm-conserving pseudopotentials.
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The theories of the first principle methods are well established [28, 29] as applied to several systems. The calculations were done with a starting magnetisation of 0.5 μB in both compounds with the Magnetisation assigned to Mn and Ti, the essence of this is to provide information on the magnetic moment of the alloys. We found the stable magnetic state of the alloys to be Ferromagnetic. The above achieved with the elements Mn2 and Ti2 occupying X position at sublattices (0, 0, 0) and (½, ½, ½) and Co and As occupies sub-lattices (¼, ¼, ¼) and (¾, ¾, ¾) respectively. To proceed with the calculations, we determined the equilibrium lattice parameter, bulk modulus and the pressure derivatives of the bulk modulus of the alloys by the standard procedure for computing the total energy for different lattice constant and fitted these to Birch-Murnaghan’s equation of states [31-33]. The self-consistent field calculations were performed with plane wave’s energy cut-off of 45 Ry as well as 10 X 10 X 10 k-points mesh. The above then allows establishing a relationship between the volume of a body and the pressure to which that body is subjected. After selecting some high-symmetry kpoints, the band structure was calculated using the number of Kohn-Sham states [34]. A much denser uniform k-points grids coupled with the tetrahedron method allow the calculations of the density of states. Core electrons are treated explicitly by employing the Troullier-Martins scalar route [35]. However, for the elastic properties of the system, a normconserving pseudopotential was used while the conditions "no inversion and no symmetry" was set as true to enable the utilisation of a uniform grid of k-points. For all calculations except the density of state, the occupation smearing and a Marzari-Vanderbilt [36, 37] degauss of 0.05 was adopted. Atomically, to view the entire major and minor spin bands, a spin polarisation was implemented. RESULTS AND DISCUSSION To determine the stable state of inverse the Heusler alloys Mn2CoAs and Ti2CoAs, we performed the structural optimisation of these alloys in Nonmagnetic (NM), Ferromagnetic (FM) and Antiferromagnetic (AFM) states. The results of the lattice parameters and the corresponding total energy values of each state are plotted and presented in Figure 1. From the Figures, the optimised lattice parameters of FM states are 10.887 and 11.629 a.u. in Mn2CoAs and Ti2CoAs respectively. The energy values of each lattice parameters were further fitted into Murnaghan's equation of state [31] to obtain other structural parameters.
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Along with the equilibrium lattice constants presented on these two inverse Heusler alloys, structural parameters such as bulk modulus, pressure derivatives are also shown in Table 1 as well. As shown in Table 1, the present results on Mn2CoAs inverse Heusler alloys are in close agreements with the previous work of Beeri and his co-workers [24]. However, we could not lay our hands on any existing results on structural parameters of AFM states of Mn2CoAs inverse Heusler alloys. In all the calculations, the lattice parameters used are between 9.5 to 11.50 a.u. However, in the AFM state of Mn2CoAs inverse Heusler alloys, there is a deviation in the stability of the compounds in the region 10.60 to 11.00 a.u. Beyond the area mentioned above, the computations failed to converge as revealed in Figure 1a.
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On the other hand, there is no structural parameter for Ti2CoAs inverse Heusler alloys in the literature, and this makes it difficult to compare the present set of results with other work. Contrary to the discovery in Mn2CoAs, there is convergence in calculated total energy between 9.5 and 11.50 a.u. in the three states as presented in Figure 1b. Meanwhile, from the results (Figure 1b), it was shown clearly that the most stable state of Ti2CoAs Heusler compounds is in the FM state. Every other calculation on these inverse Heusler alloys is done based on FM state. It is also important to mention here that the FM state of Mn2CoAs has been reported earlier [24].
Fig. 1b
Figure 1: Calculated total energy versus lattice constants in NM, FM and AFM states Mn2CoAs and Ti2CoAs in inverse Heusler alloys (here the green line denotes the NM state, the red line denotes the FM state while the blue line represents the AFM state) Table I: Equilibrium lattice constants "a" (a.u.), Bulk Modulus B (GPa), and pressure derivatives of Bulk modulus B' of Mn2CoAs and Ti2CoAs inverse Heusler alloys. Alloys State a (a.u.) B B' a a Mn2CoAs NM 10.735 10.601 225.3 194.07 4.32 4.14a FM 10.887 10.733a 137.5 221.91a 5.50 6.23a AFM 10.930 106.0 6.28 NM 11.609 148.8 4.04 Ti2CoAs FM 11.629 142.2 4.19 AFM 11.614 130.4 5.17 a [39]
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The spin-polarized projected density of state for these two inverse Heusler alloys are plotted and presented in Figures 2a and Figure 2b. Here, we shifted the Fermi energy to zero, and the Spin-up and Spin-down sub-bands are represented by red and green line respectively. Figure 2a is similar to the previous work of [38], the discrepancy observed in the two curves can be attributed to different pseudopotential used in the two calculations. Between -10 eV and 4 eV, the two Figures look similar except for the fact that the positions of the peaks and the minima differ between -2 and 0 eV in the two cases. This energy range is due to the Mn- and Ti-3d electrons; whereas, the region between 0 to 4 eV, this is attributed to As-orbital. Also, between the energy range of -8 eV to 2 eV, there are three pronounced peaks which are due to the mixing of the orbitals Mn-4s, Ti-3s, Ti-3d, and Co-3d.
Fig. 2a
Fig. 2b
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Figure 2: The spin-polarized projected of density state of Mn2CoAs and Ti2CoAs inverse Heusler alloys respectively.
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Despite the availability of some basic electronic properties of Mn2CoAs inverse Heulser alloys, there is no information on the Elastic constant of the alloys. Sequel to this, we calculated the elastic constants of these Heulser alloys and presented along with other parameters. The three independent Elastic constants for the cubic system which are C11, C12 and C44 were first calculated and shown in Table II. From here, we subjected these values to the conditions of stability in the cubic system, i.e. C11 + 2C12 > 0 and C44 > 0. We compared our results with these circumstances, and we concluded that Mn2CoAs and Ti2CoAs inverse Heusler alloys are stable in nature. After which we determined the Bulk, Young and shear moduli using the values of Elastic constants obtained from these alloys as presented. We analysed these results using Cauchy stress parameters (C12 – C44) to predict the nature of bonding as well as the brittleness and ductility nature of these alloys [38]. Here it was reported that negative value for these two inverse Heusler alloys accounted for covalent bonding as well as the brittleness nature of these materials. One of the critical parameters that reveal information on the technological application of these materials is Poisson's ratio. The higher the value, it means the material possess better plasticity trait. As a result, from the present results, Mn2CoAs inverse Heulser alloy has a better plasticity when compared to Ti2CoAs inverse Heulser alloys.
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Table II: Calculated elastic constants C11, C12, C44, Bulk modulus B, Young modulus E, shear modulus G as well as Poisson's ratio "n" of Mn2CoAs and Ti2CoAs inverse Heusler alloys, all parameters are in the unit of GPa except Poisson ratio which is unitless. C11 C12 C44 B E G n Mn2CoAs 20.91 66.74 69.58 51.59 -61.25 -40.82 6.50 Ti2CoAs 116.67 194.79 218.94 168.75 156.74 58.26 0.34
Conclusion
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We have reported the density-functional calculations of the lattice constants of X2CoAs (X = Mn, Ti) inverse Heusler alloys in NM, FM and AFM states. Also reported are the corresponding bulk modulus and its pressure derivatives. From our results obtained in the minimization of energies against the lattice parameters, we concluded the FM state is the most stable state for the two inverse Heusler alloys. As a result, we calculated the spinpolarized projected density of state, elastic constants in the ferromagnetic state for these inverse Heusler alloys. From the analysis of elastic constants calculations, we compared the plasticity properties of these two inverse Heusler alloys and concluded that Mn2CoAs Heulser alloy has a better plasticity trait. Acknowledgment
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The authors are grateful to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for access to its clusters.
*Author to whom all correspondences should be sent: P. O. Adebambo* [email protected] G.A. Adebayo* [email protected]
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Highlights
We reported that ferromagnetic state is the stable state of Mn2CoAs and Ti2CoAs inverse Heusler alloys. The spin-polarized projected density of state of Mn2CoAs and Ti2CoAs revealed three pronounced peaks at -8 eV to 2 eV.
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The elastic properties revealed that Mn2CoAs Heulser alloy has a better plasticity trait than Ti2CoAs Heulser alloy.
S0925-8388(17)32070-4
DOI:
10.1016/j.jallcom.2017.05.338
Reference:
JALCOM 42151
To appear in:
Journal of Alloys and Compounds
Received Date: 25 February 2017 Revised Date:
9 May 2017
Accepted Date: 29 May 2017
Please cite this article as: O.E. Osafile, P.O. Adebambo, G.A. Adebayo, Elastic constants and observed ferromagnetism in inverse Heusler alloy Ti2CoAs using kjpaw pseudopotentials: A first-principles approach, Journal of Alloys and Compounds (2017), doi: 10.1016/j.jallcom.2017.05.338. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Elastic constants and observed Ferromagnetism in inverse Heusler Alloy Ti2CoAs using kjpaw Pseudopotentials: A First-Principles Approach O. E. Osafile1, P.O. Adebambo2* and G.A. Adebayo3* 1 Federal University of Petroleum Resources, Effurun, Nigeria 2 Department of Physical and Computer Sciences, McPherson University, Abeokuta, Nigeria 3 Department of Physics, Federal University of Agriculture, PMB 2240, Abeokuta, Nigeria
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Abstract We performed density functional calculations of the structural parameters, spin-polarized projected density of state and elastic properties of X2CoAs (X = Mn, Ti) inverse Heulser alloys, within the Generalised Gradient Approximation (GGA). At first, we determined the stable phases of the alloys and other parameters such as lattice parameters, bulk modulus, its pressure derivatives, as well as the three independent elastic constants. The elastic constants show that the alloys are stable and are brittle in nature as revealed by the Cauchy stress parameters (C12 – C44). The results also show Mn2CoAs inverse Heusler alloy is better plasticity material compared to Ti2CoAs inverse Heusler alloy. The spin polarised projected density of state of the Heusler alloys was obtained and reported. The present calculations predicted both alloys to be Ferromagnetic in nature with Mn2CoAs having better magnetic property than Ti2CoAs.
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INTRODUCTION The Heusler alloys are compounds that have become highly researched both theoretically and experimentally in recent times because of their promising multifunction [1-3]. The possible applications include spin electronics, magnetic shape memory (MSM), optoelectronics among others. Ever since the advent of the Heusler compound named after Von Heusler in 1903 [4, 5], different Heusler structural variants or phases have been discovered and studied. These morphs include the full Heusler alloy (X2YZ) which crystallises in the L21 structure with four FCC sublattices [6-8]. Another identified variant results when one of the sublattices could be left vacant, precisely the X = ½, ½, ½ sub-lattice; this is known as the half Heusler phase with an XYZ configuration which crystallises in the C1b structure. Other variants include the quaternary Heusler (XX’YZ) [9] and the inverse Heusler (XYXZ). In this case, X occupies (0, 0, 0) and ( ½, ½, ½ ); these different variants experience enormous diversities both in physical and magnetic properties. In the alloys, X and Y denote the transition metals, and Z is an s-p element such as Al, Sb, In, Ga, As, etc. Numerous studies have shown that heat treatment and the structural composition (stoichiometrically) of these variants are important factors that determine the magnetic properties of the system under study, despite the fact that no single set of parameters can adequately describe the entire Heusler family [10]. The inverse Heusler compounds have been studied in recent times by some researchers both experimentally and theoretically with Mn occupying the X state in the full Heusler alloy [1113]. In these alloys, the valence of the X-transition metal atom is smaller than that of the Ytransition metal atom. Hence it crystallises in the XA or Xα structure, where the sequence is of the form X-X-Y-Z, and the prototype is Hg2TiCu [14]. The XA structure has been observed to be more energetically favourable than the L21 structure. Unlike the conventional full-Heusler alloys, where both direct and indirect exchange interactions are present, the properties of the inverse Heusler alloys have more of short-range interactions [15, 16]. In 2009, Kubota et al. grew the Mn2VAl alloy using soft X-ray magnetic circular dichroism and found it to be a ferrimagnetic half-metal [11]. Also, using first-principles calculation [17],
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Galanakis et al. doped Mn2V(Al, Si) Heusler alloys. They observed the half-metallic state of the systems makes it suitable materials for spintronic applications since their zero magnetisation leads to lower stray fields and thus small energy losses. Mayasuki et al. performed NMR and magnetisation measurement of Mn2VAl and observed found it to be a simple ferromagnet [18]. Singh et al. in 2013 studied the disorder dependent half-metallicity in Mn2CoSi, inverse Heusler alloy, via ab-initio calculations and surmised a 100% spin polarisation for this compound [19]. Some of these researchers have pointed to the possibility of the attractive ferromagnetic half-metallic behaviour in these inverse Heuslers and have associated this with a Slater-Pauling behaviour of the total spin magnetic moment [20]. Typically, Slater-Pauling combines coherent-growth on semiconductors with high Curie temperatures [21].
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In this work, we study the X2CoAs inverse Heusler alloy with X = Mn, Ti. Both compounds obey the slater-Pauling rule of Mt = Zt – 24 for Mn2CoAs and Mt = Zt – 18 for Ti2CoAs. Slater and Pauling showed in their pioneering papers [22, 23] that on adding one valence electron to a binary magnetic alloy, the electron occupies the minority spin band and reduces the total spin magnetic moment by about 1µB. This behavioural pattern has been found to exist in half-metallic Heusler alloys [20]. These Slater-Pauling rules connect the electronic properties of the half-metallic ferromagnets (HMF) alloy (appearance of half-metallic behaviour) to the magnetic properties (total spin magnetic moments) [20]. Information abounds in the literature on the band structure, and density of states of Mn2CoAs [23, 24], while the lattice constants and magnetic moment of the 3d and s-p elements Ti2CoAl are also available [25]. However, such information is lacking on Ti2CoAs alloy. The present calculations further investigate the elastic properties of these inverse Heusler alloys to buttress the possibility of their usefulness in photovoltaic, spintronics and some other applications.
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METHODS AND COMPUTATIONAL DETAILS Calculations were performed using quantum espresso package [26] with the projector augmented wave (PAW) based pseudopotential [Ref]. The Perdew-Burke-Ernzerhof (PBE) version of the generalised gradient approximation (GGA) to the exchange-correlation function of the density functional theory (DFT) was adopted [27]. The calculations are without spin-orbit coupling, though all other relativistic effects are included making the calculation semi-relativistic. In this approach, the Kohn-Sham equation is: 1 2 (1) − 2 ∇ + Vnuc ( r ) + VH [ n( r ) ] + Vxc [ n( r ) ] Ψ i ( r ) = Ei Ψ i ( r ) Equation (1) is solved self-consistently using a plane wave basis set for a fixed nuclear (ionic) positions. Where the first term in Eqn. (1) is the kinetic energy of a non-interacting electrons system, the second term is the Hartree-like energy term, which accounts for the classical Coulomb interaction of a spatial charge distribution (r) and the third term represents the exchange-correlation energy [28, 29]. The only unknown quantity is the exchangecorrelation energy functional and, in principle, the quality of the solution of the full manybody problem is dependent on the quality of the approximation. The FCC structure of these Heusler alloys are periodic, so the plane wave is calculated using: 1 Ψ k (r ) = ΣCk .G ei ( k +G ).r (2) Ω And can be represented as a grid in the k-space. The Brillouin zone (sampled over the high symmetry points) aids proper convergence of the infinite numbers of plane waves in principle. To truncate the plane wave expansion at some value of |k + G|, we use a converged
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h2 K + G
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≤ Ecut 2m (3) The convergence test of the high symmetry k-points on k-point sampling is achieved using: 1 P = ∑ P ( k )W k N k k ∈ BZ (4)
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The Monkhorst-Pack [30] k-point mesh is used in this work to take advantage of an equally spaced mesh in the reciprocal lattice by Fast Fourier Transform. Computational time was saved using the weighted k-points from only the irreducible Brillouin zone (IBZ). The basis set valence configuration of the X-element Mn is 3s2, 4s2, 3p6 and 3d5 while that of Ti is 3s2, 4s2, 3p6, 4p2 and 3d2. For the Y-Co atom, it is 4s2, 3p6 and 3d7 while the Z-As atom is 4s2 and 4p3. In addition to reported properties in the literature, we investigated the elastic properties of the systems. The elastic properties describe how well the system withstands deformation when exposed to external forces. The optical properties can be computed (real and imaginary parts) using the approach in [26]; this can be achieved using norm-conserving pseudopotentials.
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The theories of the first principle methods are well established [28, 29] as applied to several systems. The calculations were done with a starting magnetisation of 0.5 μB in both compounds with the Magnetisation assigned to Mn and Ti, the essence of this is to provide information on the magnetic moment of the alloys. We found the stable magnetic state of the alloys to be Ferromagnetic. The above achieved with the elements Mn2 and Ti2 occupying X position at sublattices (0, 0, 0) and (½, ½, ½) and Co and As occupies sub-lattices (¼, ¼, ¼) and (¾, ¾, ¾) respectively. To proceed with the calculations, we determined the equilibrium lattice parameter, bulk modulus and the pressure derivatives of the bulk modulus of the alloys by the standard procedure for computing the total energy for different lattice constant and fitted these to Birch-Murnaghan’s equation of states [31-33]. The self-consistent field calculations were performed with plane wave’s energy cut-off of 45 Ry as well as 10 X 10 X 10 k-points mesh. The above then allows establishing a relationship between the volume of a body and the pressure to which that body is subjected. After selecting some high-symmetry kpoints, the band structure was calculated using the number of Kohn-Sham states [34]. A much denser uniform k-points grids coupled with the tetrahedron method allow the calculations of the density of states. Core electrons are treated explicitly by employing the Troullier-Martins scalar route [35]. However, for the elastic properties of the system, a normconserving pseudopotential was used while the conditions "no inversion and no symmetry" was set as true to enable the utilisation of a uniform grid of k-points. For all calculations except the density of state, the occupation smearing and a Marzari-Vanderbilt [36, 37] degauss of 0.05 was adopted. Atomically, to view the entire major and minor spin bands, a spin polarisation was implemented. RESULTS AND DISCUSSION To determine the stable state of inverse the Heusler alloys Mn2CoAs and Ti2CoAs, we performed the structural optimisation of these alloys in Nonmagnetic (NM), Ferromagnetic (FM) and Antiferromagnetic (AFM) states. The results of the lattice parameters and the corresponding total energy values of each state are plotted and presented in Figure 1. From the Figures, the optimised lattice parameters of FM states are 10.887 and 11.629 a.u. in Mn2CoAs and Ti2CoAs respectively. The energy values of each lattice parameters were further fitted into Murnaghan's equation of state [31] to obtain other structural parameters.
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Along with the equilibrium lattice constants presented on these two inverse Heusler alloys, structural parameters such as bulk modulus, pressure derivatives are also shown in Table 1 as well. As shown in Table 1, the present results on Mn2CoAs inverse Heusler alloys are in close agreements with the previous work of Beeri and his co-workers [24]. However, we could not lay our hands on any existing results on structural parameters of AFM states of Mn2CoAs inverse Heusler alloys. In all the calculations, the lattice parameters used are between 9.5 to 11.50 a.u. However, in the AFM state of Mn2CoAs inverse Heusler alloys, there is a deviation in the stability of the compounds in the region 10.60 to 11.00 a.u. Beyond the area mentioned above, the computations failed to converge as revealed in Figure 1a.
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On the other hand, there is no structural parameter for Ti2CoAs inverse Heusler alloys in the literature, and this makes it difficult to compare the present set of results with other work. Contrary to the discovery in Mn2CoAs, there is convergence in calculated total energy between 9.5 and 11.50 a.u. in the three states as presented in Figure 1b. Meanwhile, from the results (Figure 1b), it was shown clearly that the most stable state of Ti2CoAs Heusler compounds is in the FM state. Every other calculation on these inverse Heusler alloys is done based on FM state. It is also important to mention here that the FM state of Mn2CoAs has been reported earlier [24].
Fig. 1b
Figure 1: Calculated total energy versus lattice constants in NM, FM and AFM states Mn2CoAs and Ti2CoAs in inverse Heusler alloys (here the green line denotes the NM state, the red line denotes the FM state while the blue line represents the AFM state) Table I: Equilibrium lattice constants "a" (a.u.), Bulk Modulus B (GPa), and pressure derivatives of Bulk modulus B' of Mn2CoAs and Ti2CoAs inverse Heusler alloys. Alloys State a (a.u.) B B' a a Mn2CoAs NM 10.735 10.601 225.3 194.07 4.32 4.14a FM 10.887 10.733a 137.5 221.91a 5.50 6.23a AFM 10.930 106.0 6.28 NM 11.609 148.8 4.04 Ti2CoAs FM 11.629 142.2 4.19 AFM 11.614 130.4 5.17 a [39]
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The spin-polarized projected density of state for these two inverse Heusler alloys are plotted and presented in Figures 2a and Figure 2b. Here, we shifted the Fermi energy to zero, and the Spin-up and Spin-down sub-bands are represented by red and green line respectively. Figure 2a is similar to the previous work of [38], the discrepancy observed in the two curves can be attributed to different pseudopotential used in the two calculations. Between -10 eV and 4 eV, the two Figures look similar except for the fact that the positions of the peaks and the minima differ between -2 and 0 eV in the two cases. This energy range is due to the Mn- and Ti-3d electrons; whereas, the region between 0 to 4 eV, this is attributed to As-orbital. Also, between the energy range of -8 eV to 2 eV, there are three pronounced peaks which are due to the mixing of the orbitals Mn-4s, Ti-3s, Ti-3d, and Co-3d.
Fig. 2a
Fig. 2b
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Figure 2: The spin-polarized projected of density state of Mn2CoAs and Ti2CoAs inverse Heusler alloys respectively.
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Despite the availability of some basic electronic properties of Mn2CoAs inverse Heulser alloys, there is no information on the Elastic constant of the alloys. Sequel to this, we calculated the elastic constants of these Heulser alloys and presented along with other parameters. The three independent Elastic constants for the cubic system which are C11, C12 and C44 were first calculated and shown in Table II. From here, we subjected these values to the conditions of stability in the cubic system, i.e. C11 + 2C12 > 0 and C44 > 0. We compared our results with these circumstances, and we concluded that Mn2CoAs and Ti2CoAs inverse Heusler alloys are stable in nature. After which we determined the Bulk, Young and shear moduli using the values of Elastic constants obtained from these alloys as presented. We analysed these results using Cauchy stress parameters (C12 – C44) to predict the nature of bonding as well as the brittleness and ductility nature of these alloys [38]. Here it was reported that negative value for these two inverse Heusler alloys accounted for covalent bonding as well as the brittleness nature of these materials. One of the critical parameters that reveal information on the technological application of these materials is Poisson's ratio. The higher the value, it means the material possess better plasticity trait. As a result, from the present results, Mn2CoAs inverse Heulser alloy has a better plasticity when compared to Ti2CoAs inverse Heulser alloys.
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Table II: Calculated elastic constants C11, C12, C44, Bulk modulus B, Young modulus E, shear modulus G as well as Poisson's ratio "n" of Mn2CoAs and Ti2CoAs inverse Heusler alloys, all parameters are in the unit of GPa except Poisson ratio which is unitless. C11 C12 C44 B E G n Mn2CoAs 20.91 66.74 69.58 51.59 -61.25 -40.82 6.50 Ti2CoAs 116.67 194.79 218.94 168.75 156.74 58.26 0.34
Conclusion
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We have reported the density-functional calculations of the lattice constants of X2CoAs (X = Mn, Ti) inverse Heusler alloys in NM, FM and AFM states. Also reported are the corresponding bulk modulus and its pressure derivatives. From our results obtained in the minimization of energies against the lattice parameters, we concluded the FM state is the most stable state for the two inverse Heusler alloys. As a result, we calculated the spinpolarized projected density of state, elastic constants in the ferromagnetic state for these inverse Heusler alloys. From the analysis of elastic constants calculations, we compared the plasticity properties of these two inverse Heusler alloys and concluded that Mn2CoAs Heulser alloy has a better plasticity trait. Acknowledgment
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The authors are grateful to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for access to its clusters.
*Author to whom all correspondences should be sent: P. O. Adebambo* [email protected] G.A. Adebayo* [email protected]
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Highlights
We reported that ferromagnetic state is the stable state of Mn2CoAs and Ti2CoAs inverse Heusler alloys. The spin-polarized projected density of state of Mn2CoAs and Ti2CoAs revealed three pronounced peaks at -8 eV to 2 eV.
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The elastic properties revealed that Mn2CoAs Heulser alloy has a better plasticity trait than Ti2CoAs Heulser alloy.