The effect of pressure and alloying on half-metallicity of quaternary Heusler compounds CoMnYZ (Z = Al, Ga, and In)

The structural prototype of the quaternary Heusler compounds – LiMgPdSb – is denoted as Y [10]. There are three possible different types of atom arrangement in the quaternary Heusler compound XX′YZ: Y-type 1:X (0, 0, 0), X′ (0.25, 0.25, 0.25), Y (0.5, 0.5, 0.5), and Z(0.75, 0.75, 0.75); Y-type 2: X (0, 0, 0), X′ (0.5, 0.5, 0.5), Y(0.25, 0.25, 0.25), and Z (0.75, 0.75, 0.75); Y-type 3: X(0.5, 0.5, 0.5), X′ (0, 0, 0), Y (0.25, 0.25, 02.5), and Z (0.75, 0.75, 0.75). In first step, in order to obtain the correct atomic arrangement and the magnetic ground state corresponding to the true ground state of the quaternary Heusler compounds CoMnYZ (Z = Al, Ga, In), we performed the energy minimization as a function of lattice constant with respect to the three different possible site occupation for every non-magnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) configurations and the obtained curves are shown in Fig. 1 and Fig. 2. The calculated total energies within GGA as a function of volume are fitted to Murnaghan’s equation of state to obtain the ground-state properties [25]. In Table 1, there are presented our calculated equilibrium lattice constant a₀, along with the bulk modulus B₀, and the total energy Etot in their different structural and magnetic configurations. As seen in Fig. 1 and Fig. 2, the optimization of the cubic lattice parameters for all three possible configurations in their respective three different magnetic configurations reveal the lowest energy for Y-type 1 structure with a ferromagnetic ground state for all compounds. In the absence of the experimental data regarding the lattice constant a₀ and bulk modulus B₀, the parameters of the material of interest, and hence our results are predictions. Also, the highest calculated bulk moduli for CoMnYZ (Z = Al, Ga, In) in Y-type 1+FM configuration confirm the stability of this structure. In the same way, the formation energy E_f determines whether a compound can be experimentally synthesized or not. E_f is the change in energy when a material is formed from its constituent elements in their bulk states and can be calculated for CoMnYZ (Z = Al, Ga, In) compounds as EfCoMnYZ = EtotCoMnYZ + (ECobulk + EMnbulk + EYbulk + EZbulk) Z = Al,Ga,In $$\begin{array}{} E_f^{CoMnYZ}{\text{ = }} E_{tot}^{CoMnYZ}{\text{ + (}}E_{Co}^{bulk} {\text{ + }}E_{Mn}^{bulk}{\text{ + }}E_Y^{bulk}{\text{ + }}E_Z^{bulk}) \\ \qquad\qquad\qquad\quad {\text{ Z = Al,Ga,In}} \\ \end{array} $$ (1)

where EtotCoMnYZ$\rm E _{tot}^{CoMnYZ} $ are the first-principles calculated equilibrium total energies of studied compounds per formula unit, ECobulk,EMnbulk,EYbulkandEZbulk$\rm E_{Co}^{bulk},\;E_{Mn}^{bulk},\;E_{Y}^{bulk}\;and\;E_{Z}^{bulk}$ are the calculated equilibrium total energies of these atoms in their stable bulk phases. In Table 1, there are given the values of formation energy for all types of structures and all magnetic configurations. These values imply that the three compounds can be fab-ricated spontaneously in experiment due to their negative formation energy. Also, according to Table 1, the calculated E_f values confirm the structural stability of type1+FM structure for all CoMnYZ (Z = Al, Ga, In) compounds (high negative formation energy). Among them, CoMnYAl is most easily synthesized because of its lowest formation energy. Based on this, all the further calculations on electronic and magnetic properties of CoMnYZ (Z = Al, Ga, In) were performed on this structure only, i.e. in the type 1+FM structure.

The spin-polarized band structures of ferromagnetic CoMnYZ (Z = Al, Ga, In) at equilibrium lattice constants are shown in Fig. 3. The Fermi level is set as 0 eV. For these compounds, the minority-spin channel is metallic whereas in the majority-spin channel there is an energy gap around the Fermi level. Its value, listed in Table 2, is about 0.51 eV, 0.59 eV and 0.54 eV for CoMnYAl, CoMnYGa and CoMnYIn, respectively. Therefore, these compounds are HM ferromagnets. The half-metallic gap reported in Table 2, which is determined as the minimum between the lowest energy of majority (minority) spin conduction bands with respect to the Fermi level, and the absolute values of the highest energy of the majority (minority) spin valence bands, are 0.158 eV, 0.294 eV and 0.195 eV, for CoMnYAl, CoMnYGa and CoMnYIn, respectively. On the other hand, the large gaps in these compounds containing localized magnetic orbitals due to d-d hybridization between the transition metal atoms [26] is essential for the gap formation as the p–d hybridization previously discussed [1]. According to Slater-Pauling rule the calculated magnetic moment has to be an integer value for a compound to be a half metal ferromagnet. Indeed, our total magnetic moments per formula unit reported in Table 2 are found to be integer value 4.00 μ_B for all the compounds, and obey the Slater-Pauling behavior of HM ferromagnets with Heusler structure [27, 28], M_tot = (Z_tot − 18) rather than M_tot = (Z_tot − 24) here M_tot and Z_tot are the total magnetic moment per formula unit and the number of total valence electrons, respectively. Z_tot is 22 for all the CoM-nYAl, CoMnYGa and CoMnYIn compounds.

As seen in Table 2, the main contribution to the total magnetic moment is mainly due to transition element (Mn) with little contribution of other transition elements (Co, Y) and the magnetic moments of the Z (Si, Ge, Sn) atoms are quite small. In order to explain the magnetic properties of these compounds and analyse in detail the influence of exchange splitting of the TM-d states, we have calculated the spin-total (spin-TDOS) and partial density of states (spin-PDOS) for the CoMnYAl, CoMnYGa and CoMnYIn compounds presented in Fig. 4, Fig. 5 and Fig. 6, respectively. As it can be seen, the general structure total DOS are similar for our compounds. So, the Fig. 4, Fig. 5 and Fig. 6 confirm that the main contribution to the magnetic moment of the transition elements (Co, Mn, Y) is due to d states and that the magnetic moment of the Z atom is due to p states. It can be seen that there is a strong hybridization between Co, Mn, and Y atoms, which splits the d orbitals of these atoms into bonding e_g and t_2g orbitals (below the Fermi level), non-bonding e_u and t_1u orbitals (around the Fermi level), and anti-bonding e_g and t_2g orbitals (above the Fermi level). This hybridization puts the Fermi level in the minority band gap. In fact, d-d hybridization between transition metals takes part in the formation of minority band gap known as d-d band gap The d-d band gap is the origin of the HM band gap in the full-Heusler alloys with AlCu2Mn structure [29]. A weaker hybridization is also observed between d states of Co, Mn, and Y atoms with p states of Z atom, which determine the degree of occupation of the p-d orbital. This hybridization affects the width of energy gap.

To illustrate the origin of minority band gap, the orbital hybridization of CoMnYZ is qualitatively shown in Fig. 7 based on the classical molecular orbital approach. Only the hybridization of d states of Co, Mn and Y atoms has been considered since the sp-bands are located at deep energy and scarcely make contribution to the gap. In the CoMnYZ alloys, the hybridization between Co and Mn is firstly considered since Mn is the first-neighbor atom, as shown in Fig. 7a. The d-orbitals of Co and Mn split into a double degenerate d_z², d_x²−y² (noted in Fig. 7 by d₄, d₅, respectively) and a triple degenerate d_xy, d_yx, d_zx (noted in Fig. 7 by d₁, d₂, d₃, respectively) orbitals. The double degenerate orbitals d_z², d_x²−y² of Co can only couple to the double degenerate orbitals d_z², d_x²−y² of Mn forming a bonding orbital e_g and an anti-bonding orbital e_u. The triple degenerate orbitals d_xy, d_yx, d_zx of Co couple to those of Mn forming a bonding orbital t_2g and an anti-bonding orbital t_1u. Since the Co and Mn atoms are seated in the center of an octahedron formed by each other, the crystal field splitting of the bonding orbitals is E(e_g) > E(t_2g) and the anti-bonding orbitals is E(e_u) > E(t_1u) [3]. Finally, the hybridization of Co–Mn orbitals with the Cr d-orbitals is considered. As shown in Fig. 7b, the e_g (t_2g) orbitals hybridize with the d_z², d_x²−y² (d_xy, d_yx, d_zx) of Y atoms and form a lower energy bonding orbital e_g (t_2g) and a higher energy antibonding orbital e_g (t_2g). As Co and Mn atoms are in the center of a Y tetrahedron, the crystal field splitting of the bonding orbitals is E(e_g) < E(t_2g), and anti-bonding orbitals is E(e_g) < E(t_2g). Since t_1u, e_u cannot couple with any Y d-orbitals, the energy of E(e_u) > E(t_1u) is still remained. The complete hybridization orbitals of CoMnYZ are shown in Fig. 7c. The sp orbitals, the bonding orbitals e_g, and t_2g orbitals of CoMnYZ can jointly accommodate 18 valence electrons. The total number of valence electrons of CoMnYrZ (Z = Al, Ga, In) is 22, so there are still four valence electrons whose three electrons occupy the t_1u orbitals and one occupies the eu orbitals of majority state, exclusively localized at Co and Mn sites. Therefore, the Fermi level falls in the DOS of e_u of majority states and the gap is created between occupied (Co-Mn)-Y t_12g and unoccupied Co-Mn t_1u minority states. Thus, the minority electrons of CoMnYZ (Z = Al, Ga, In) present semiconductor character and the majority ones present metallic behavior, which causes these alloys to be half-metallic alloys. At the Fermi level, the spin-up DOS shows a metallic property, mainly due to Mn-d and this explains the large magnetic moment of Mn atom (Table 2).

Finally, the HM stability of CoMnYAl, CoMnYGa and CoMnYIn has been investigated under uniform strain and tetragonal distortion. Because HM materials are usually used in spintronic devices in the form of thin films or multilayers, the lattice constant will change when the thin films or multilayers are grown on appropriate substrates, and correspondingly, the half-metallicity may be destroyed. In order to study the effect of uniform strain (i.e. corresponding to hydrostatic pressure), we calculated the variation of valence band maximum (Max-VB) and conduction band minimum (Min-CB) as a function of lattice constant for CoMnYAl, CoMnYGa and CoMnYIn plotted in Fig. 8a. It was found that the half-metallicity is kept in the wide range of 6.24 Å to 6.96 Å, 6.12 Å to 6.87 Å and 6.14 Å to 6.94 Å for CoMnYAl, CoMnYGa and CoMnYIn, respectively. Among these compounds the CoMnYIn has a considerable region of half-metallicity which makes it stable against negative and positive pressures. Moreover, in the CoMnYIn, the Fermi level is located in the middle of the minority band gap and therefore, this compound is the most stable compound against the effects which destroy the half-metallicity (such as temperature or external stresses). However, a common behavior in all CoMnYrZ (Z = Al, Ga, In) compounds is observed. With increasing the lattice constant, the minority valence band edge smoothly increases and subsequently cuts the Fermi level, while with decreasing the lattice constant the minority conduction band edge smoothly decreases and the half-metallicity disappears. Galanakis et al. [30] showed that the contraction and expansion of the lattice constant mainly influence the delocalized p electrons and do not affect the well localized d electrons of the transition metals considerably.

The effect of a tetragonal distortion with the c/a ratio, keeping the unit-cell volume the same as the equilibrium unit-cell volume, has been studied. The variations of valence band maximum (Max-VB) and conduction band minimum (Min-CB) as a function of c/a ratio for the CoMnYZ (Z =Al, Ga, In) compounds are shown in Fig. 8b. It can be seen that the CoMnYAl, CoMnYGa and CoMnYIn compounds show the half-metallic characteristics when c/a is in the range of 0.88 to 1.28, 0.88 to 1.26 and 0.83 to 1.31, respectively. The HM character sustains for relatively larger values of tetragonal strain. The (Min-CB) and (Max-VB) of minority spin channel are approximately maximum at the equilibrium lattice constant and the absolute value of them decreases monotonically with both positive and negative tetragonal strain. In a word, the considerable HM gaps and the robust half-metallicities under uniform and tetragonal strains make CoMnYAl, CoMnYGa and CoMnYIn promising candidates for spintronic applications. As the equilibrium lattice constants of CoMnYAl, CoMnYGa and CoMnYIn compounds (6.4117 Å, 6.387 Å and 6.6122 Å, respectively) are close to that of zinc blende semiconductors such as InSb (6.48 Å) and CdTe (6.49 Å) [31], it is suggested to experimentally realize these HM Heusler alloys in the form of thin films on appropriate substrates and to use them as new candidates for applications in spintronic field.

The effect of the substitution of Z sp atoms on the structural, electronic and magnetic properties of CoMnYGa_1−xAl_x, CoMnYGa_1−xIn_x and CoMnYAl_1−xIn_x Heusler alloys has also been investigated. The calculated lattice constants a, semi-conducting (spin-down) band gap E_g, half-metallic gap Ehm, total moment qtot and Curie temperature T_c at different compositions x are listed in Table 3, Table 4 and Table 5. The equilibrium lattice constants versus concentration are shown in Fig. 9, where our calculated lattice constants a have been found to vary almost linearly following Vegard’s law [32] with little marginal downward bowing parameters equal to 0.0122 Å for CoMnYGa_1−xAl_x, alloy obtained by adjusting the values calculated by a polynomial function. However, this systematic linear increase in the lattice parameter along the all series of alloys according to Vegard’s law [32] suggests a good structural stability instead of forming a secondary phase. It has also been found that the values of the band gaps E_g are almost not affected by changing the concentration x (almost 0.5 eV). This behavior is consistent with previous works for other alloys such as Fe_3−xMn_xSi [33], Fe_3−xCr_xSi [34] and Fe₂Mn_1−xV_xSi_0.5Al_0.5[35]. However, a little deviation of the band gap E_g from linear variation has been noticed. Fig. 10 shows the variation of the band gap Eg and half-metallic gap EHM with concentration x in the CoMnYGa_1−xAl_x, CoMnYAl_1−xIn_x and CoMnYGa_1−xIn_x Heusler alloys. The second-order polynomial fitting of (E_g, E_HM) − x data gives the following equations for the three alloys : Eg(x) = 0.3371x2 − 0.4174x + 0.59(CoMnYGa1 − xAlx) $$\begin{array}{} {E_g}{\text{(}}x{\text{) = 0}}{\text{.3371}}{x^{\text{2}}}{\text{ - 0}}{\text{.4174}}x \\ \qquad\quad {\text{+ 0}}{\text{.59(CoMnYG}}{{\text{a}}_{{\text{1 - }}x}}{\text{A}}{{\text{l}}_x}{\text{)}} \\ \end{array} $$ (2)Eg(x) = 0.3652x2 − 0.3355x + 0.51(CoMnYAl1 − xInx) $$\begin{array}{} {E_g}{\text{(}}x{\text{) = 0}}{\text{.3652}}{x^{\text{2}}}{\text{ - 0}}{\text{.3355}}x \\ \qquad\quad{\text{ + 0}}{\text{.51(CoMnYA}}{{\text{l}}_{{\text{1 - }}x}}{\text{I}}{{\text{n}}_x}{\text{)}} \\ \end{array} $$ (3)Eg(x) = 0.1971x2 − 0.2473x + 0.59(CoMnYGa1 − xInx) $$\begin{array}{} {\text{ }}{E_g}{\text{(}}x{\text{) = 0}}{\text{.1971}}{x^{\text{2}}}{\text{ - 0}}{\text{.2473}}x \\ \qquad\quad {\text{ + 0}}{\text{.59(CoMnYG}}{{\text{a}}_{{\text{1 - }}x}}{\text{I}}{{\text{n}}_x}{\text{)}} \\ \end{array} $$ (4)

and EHM (x) = 0.3168x2 − 0.4529x + 0.294(CoMnYGa1−<!-- - -->xAlx) $$\begin{array}{} {E_{HM}}{\text{ (}}x{\text{) = 0}}{\text{.3168}}{x^{\text{2}}}{\text{ - 0}}{\text{.4529}}x \\ \qquad \qquad \quad {\text{ + 0}}{\text{.294(CoMnYG}}{{\text{a}}_{1 - x}}{\text{A}}{{\text{l}}_x}{\text{)}} & \\ \end{array} $$ (5)EHM(x) = 0.2986x2 - 0.2618x + 0.158(CoMnYAl1 - xInx) $$\begin{array}{} {E_{HM}} {\text{(}}x{\text{) = 0}}{\text{.2986}}{x^{\text{2}}}{\text{ - 0}}{\text{.2618}}x \\ \qquad \qquad {\text{ + 0}}{\text{.158(CoMnYA}}{{\text{l}}_{{\text{1 - }}x}}{\text{I}}{{\text{n}}_x}{\text{)}} \\ \end{array} $$ (6)EHM (x) = 0.1948x2 − 0.2936x + 0.294(CoMnYGa1 − xInx) $$\begin{array}{} {\text{ }}{E_{HM}}{\text{ (}}x{\text{) = 0}}{\text{.1948}}{x^{{\text{2 }}}}{\text{ - 0}}{\text{.2936}}x \\ \qquad\qquad\quad {\text{ + 0}}{\text{.294(CoMnYG}}{{\text{a}}_{{\text{1 - }}x}}{\text{I}}{{\text{n}}_x}{\text{)}} \\ \end{array} $$ (7)

From the calculated results, the disorder is of the same order for all the alloys for E_g and E_HM, respectively. The half metallicity in the parent compounds CoMnYAl, CoMnYGa and CoMnYIn is also retained in all the alloys with the change of the concentration x of the doped Z atoms. The total magnetic moment is found to be integer (4µB) for all alloys, in accordance with the Slater-Pauling rule. Curie temperature is another important aspect of application for spintronic material. Only with high Curie temperature can the magnetic materials be used in practice. Using the mean field approximation (MFA) [36], the Curie temperature (T_C) can be calculated as: Tc=2Δ<!-- ? -->E3kB $${T_c}{\text{ }} = {\text{ }}\frac{{{\text{2}}\Delta E}}{{{\text{3}}{k_B}}} $$ (8)

where ΔE is the total energy difference between the antiferromagnetic and ferromagnetic states (ΔE = E_AFM − E_FM) and k_B is the Boltzmann constant. The results are given in Table 3, Table 4 and Table 5, and shown in Fig. 11. The Curie temperature has been calculated to be 482 K, 616 K and 805 K for CoMnYAl, CoMnYGa and CoMnYIn, respectively. It was also found that the Curie temperature significantly changes with Z content in all the three alloys CoMnYAl_1−xGa_x, CoMnYGa_1−xIn_x and CoMnYAl_1−xIn_x. The Curie temperature of the half-metallic Mn₂VAl compound, estimated by using the mean field approximation, is 638 K and its value is in good agreement with the experimental value of the Curie temperature of 760 K [37]. Also the results show that the Curie temperature of our half-metallic ferromagnetic doped alloys CoMnYAl_1−xGa_x, CoMnYGa_1−xIn_x and CoMnYAl_1−xIn_x is higher than room temperature, so a wide range of scenarios can be acomodated. In addition, the Curie temperature increases almost proportionally with Z, with high valence-doped linearly. This behavior is consistent with previous theoretical works for alloys such as FeCoZrGe_1−xAs_x and FeCoZr_1−xNb_xGe [38] and recent experimental studies on Co2Ti_1−xFe_xGe [39] and Co₂FeGa_1−xSi_x[40] alloys.