Miguel J. Espitia R¹, John H. Díaz F¹, César Ortega López²*

1: Grupo GEFEM, Universidad Distrital Francisco José de Caldas, Bogotá Colombia.

2: Grupo Avanzado de Materiales y Sistemas Complejos GAMASCO, Departamento de Física, Universidad de Córdoba, Montería Colombia.

*e-mail: cortegal@sinu.unicordoba.edu.co

ABSTRACT

First principles calculations based on density functional theory were employed to investigate the structural, electronic, and magnetic properties of the Sc_xCr_1-xN (x =0.0, 0.25, 0.50, 0.75, and 1.0) compound, in the wurtzite structure. We applied the full-potential linearized augmented plane wave (FP-LAPW) method as implemented in the Wien2k code. The analysis of the structural properties revealed that the lattice constant increases linearly with the increment of the Sc atoms in the structure, in accord with Vegard's law, whereas the bulk modulus decreases. The studies of the electronic density showed that the ternary compounds of Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) exhibit a half-metallic behavior with magnetic spin polarization of 100% and a magnetic moment of 3 μ_β/Cr-atom. The ferromagnetic behavior comes from a double exchange mechanism and hybridization of the Cr-d and N-p states that cross the Fermi level. These compounds are good candidates for applications in spintronics.

Keywords: DFT, structural properties, half-metallic ferromagnetism.

Estudio teórico de las propiedades estructurales y electrónicas de los compuestos Sc_xCr_1-xN

RESUMEN

Palabras Claves: DFT, propiedades estructurales, ferromagnetismo semimetálico.

Recibido: 07-09-2014; Revisado: 14-10-2014

Aceptado: 07-11-2014; Publicado: 18-11-2014

1. INTRODUCTION

Spintronics is a new branch of microelectronics in which essentially the spin of the charge carriers is used [1]. The expected advantages of spintronic devices over conventional electronic ones are nonvolatility, increased data processing speed, increased transistor density, and decreased power consumption [2]. The study of half-metallic ferromagnetism in semiconductors doped with transition metals is a topic of much current research, because of the promising applications of these materials in the field of spintronics [1, 3, 4] and spin injectors [5]. In recent years, the indirect gap semiconductor ScN [6] has begun to be studied for possible applications in diluted magnetic semiconductors, spintronic devices, and spin injectors. Al-Brithen et al. [7] grew Sc_1-xMn_xN via the molecular beam expitaxial technique, while A. Alsaad et al. [8] carried out studies of the structural and magnetic properties of the diluted magnetic Mn_xSc_1-xN semiconductors using density functional theory (DFT). Additionally, Birch J. et al. [9] grew ScN/CrN multilayers epitaxially via the magnetron sputtering technique.

In the present paper, a theoretical study of the structural, electronic, and magnetic properties of Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) was carried out, due to its potential applications in spintronic devices and spin injectors.

2. COMPUTATIONAL METHOD

The calculations were performed within the framework of density functional theory (DFT), and the full-potential linearized augmented plane wave (FP-LAPW) method was used as implemented in the Wien2k software package [10]. The exchange and correlation effects of the electrons were dealt with using the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [11]. In the LAPW method, the cell is divided into two types of regions, namely spheres centered at the atomic nuclear sites and an interstitial region between non-overlapping areas. Within the atomic spheres, wave functions are replaced by atomic functions; whereas in the interstitial region, the function is expanded in the form of plane waves. The charge density and potential expand to form spherical harmonics up to lmax = 10 inside the atomic spheres, and the wave function in the interstitial region expands in the form of plane waves with a cutoff parameter of R_MT K_max = 8, where R_MT is the smallest radius of the atomic level within the unit cell and K_max is the magnitude of the largest k vector of the reciprocal lattice. To ensure convergence in the integration of the first Brillouin zone, 1600 points were used, which corresponds to 140 k-points at the irreducible part of the first Brillouin. The integrals over the Brillouin zone were solved using the special approximation of k points provided by the Monkhorst-Pack method. Selfconsistency was achieved by requiring that the convergence of the total energy be less than 10^-4 Ry. To achieve expansion of the potential in the interstitial region, it was considered that G_max = 12. The corresponding muffin-tin radii were 1.6 bohr for N, 1.85 bohr for Sc, and 1.95 bohr for Cr. Calculations were performed considering the spin polarization caused by the presence of Cr in the compounds. To calculate the lattice constant, the minimum volume, the bulk modulus, and the cohesive energy of the two structures studied, calculations were fit to the Murnaghan equation of state [12] Eq (1)

In order to study the relative stability of Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds in concentration of x ScN molecules and 1-x CrN molecules, the energy of formation was calculated. For the ternary compound, the energy of formation is defined as the difference between the total energy of the ternary phase Sc_xCr_1-xN and the total energy of the binary compounds in their ground state, GaN wurtzite and CrN zincblende, and , respectively. Therefore, the energy of formation is given by Eq. (1) [13, 14]

3. RESULTS AND DISCUSSION

3.1 Structural properties

To determine the structural properties in the ground state, such as the lattice constant (a₀), the bulk modulus (B₀), and the total energy (E₀) of ScN and CrN binary compounds and allowed ternary Sc_xCr_{1- x}N (x = 0.25, 0.50, and 0.75) compounds in the wurtzite structure, the variation of total energies and equilibrium volumes were fit to Murnaghan’s equation of state, Equation (1). Additionally, in order to investigate the magnetic phase most favorable of compounds (x = 0.25, 0.50 and 0.75), the total energies of different compounds as functions of volume have been calculated in both ferromagnetic phase (FM) and antiferromagnetic phase (AFM) states. Figure 1 shows the energyvolume curves of Sc_xCr_1-xN compounds for x = 0.25, 0.50 and 0.75 in the FM and AFM states.

The calculated total energy differences between the FM and AFM states are about -0.025, -0.042 and - 0.063 eV for x = 0.25, 0.5 and 0.75, respectively. For all cases, FM states are more favorable in energy than AFM states, so the FM phase is a stable ground state for Sc_xCr_1-xN. Our results and various theoretical data are listed in Table 1.

The ground state structural properties (a₀, B₀, E₀) for the wurtzite structure of the binary compounds ScN and CrN are very close to the theoretical results [15- 17]. For the allowed ternary Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds, structures are obtained by replacing Cr atoms with Sc atoms in the supercell. For x = 0.25 and x = 0.50, a Cr atom is replaced by a Sc atom in a supercell of 8 and 4 atoms, respectively. For x = 0.75, three Cr atoms are replaced by three Sc atoms in the supercell of 8 atoms. The figure 2 shows the crystal structure of the allowed ternary Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compound obtained after structural relaxation. In all cases, the space group obtained is the same, the tetragonal structure P3m1 (N° 156). As we can see in Table 1, the values of the bulk modulus of the allowed ternary Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds are larger, and therefore they are quite rigid, making them good candidates for possible applications in devices that must operate at high temperatures and high power, and in hard coatings.

Figure 3 shows the variation of the lattice constant as a function of the concentration of Sc. It can be observed that the lattice constant equilibrium increases with increasing concentration (x) of Sc atoms. This can be attributed to the fact that the atomic radius of Sc r_Sc = 1.62 Å is much larger than the atomic radius of Cr r_Cr = 1.27 Å. On the other hand, Figure 4 shows that the modulus decreases with an increasing amount (x) of the concentration of Sc atoms in all compounds. A similar result obtained was by López W. et al. [17] in their study of the structural properties of the Sc_1-xIn_xN compound in the wurtzite structure.

Figure 3 shows that the lattice constant increases linearly with the concentration of Sc atoms. Conventionally, the lattice constant a(x) of the ternary Sc_xCr_1-xN compound is described by Vegard’s law [18], which expresses the lattice constant of each ternary compound Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) as a linear combination of the lattice constants of the two binary compounds, as follows: a(x) = xa_ScN + (1-x)a_CrN. However, there is a small upward deviation of the linear behavior of the lattice constant of the allowed Sc_xCr_1-xN compounds with respect to Vegard's law. We calculated the value of the deviation λa, with the equation a(x) = xa_ScN +(1-x)a_CrN + λ_ax(1-x) [19]. We found a small value, λ_a = - 0.0405 Å, for the lattice constant deviation of allowed Sc_xCr_1-xN compounds in the wurtzite structure. The small value of the deviation can be attributed to the fact that the lattice constants of ScN and CrN compounds are similar. The lattice constant of ScN is 11.3% larger than the lattice constant of CrN. In this case, the ternary compound Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) satisfies Vegard’s law because interactions between inoes (Sc, Cr and N) which form the ternary compounds are weak.

Table 2 shows the energy formation values of the allowed ternary Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds, which we calculated using Equation 2. We can see in Table 1 that the energy of binary compounds in their ground state is negative, whereas, according to the results of Table 2, the value of the energy of formation of each ternary compound is positive; therefore, the Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds are metastable.

This implies that a ternary compound can’t grow under equilibrium conditions, so to grow them it is necessary to supply power to the system [12, 13]. The result that we found for the energy of formation is very important, because these values will improve the growth conditions and will therefore produce Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds of excellent quality.

According to Table 2, the Sc_0.25Cr_0.75N compound has the smallest value by far of the energy of formation, and therefore it is the most energetically stable.

3.2 Electronic properties

The total spin-polarized density of states (TDOS) and partial spin-polarized density of states (PDOS) for Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) are shown in Figure 5. It is clear that Sc_xCr_1-xN (x = 0.25) does not exhibit half-metallicity (i.e., valence and conduction bands cross the Fermi level). However, the TDOS for Sc_xCr_1-xN (x = 0.50 and 0.75) are half-metallic and ferromagnetic, since in the valence band near the Fermi level the majority spin (spinup) is metallic, and the minority spin (spin-down) is semiconducting. The Sc_0.50Cr_0.50N and Sc_0.75Cr_0.25N compounds exhibit 100% polarization of the conduction carriers in the ground state, which is required in spin injection [20]. This suggests that it can be used efficiently for injection of spin polarized charge carriers. Figure 5 shows that above the valence band the density of the up spin and below the conduction band the minority spins are mainly dominated by the hybrid state Cr-3d, and N- 2p makes a smaller contribution. Additionally, based on crystal field theory, when a Cr atom replaces an atom of Sc, the tetrahedral crystal field formed by the N ion at the cationic site splits the five-fold degenerate 3d states of the Cr ion into three degenerate states of high energy, t_2g (d_xy, d_xz, and d_yz) and two degenerate states of low energy e_g (d_z and d_x²−y²) [21]. The electron configuration of the Cr atom in the Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds can be attributed to the Cr³⁺ [22]. The figure 6 shows the Cr-3d level split in minority and majority spin due to the exchange and show the splitting into doubly degenerate e_g and triply degenerate t_2g levels, when put Cr³⁺ atom in a tetrahedral environment, in this case there is not contribution of spin down, according to the density of states figure 5. We explain better the configuration of Cr³⁺ atom, we know that Cr atoms have six valence electrons {[Ar]3d⁵4s¹}. When Cr occupies the site of Sc, it donates three electrons. Of the remaining three electrons, two electrons occupy the doubly degenerate e_g states and one electron and two holes the triply degenerate t_2g state. Therefore, the major spin states of Cr-3d are not filled, because although the doubly degenerate state is completely filled, the triply degenerate state is only one-third full. As a result, the three electrons produce a total magnetic moment of 3 μ_β/atom-Cr. Therefore the compounds exhibit a half-metallic ferromagnetic behavior.

Additionally, according to the theory by Jhi et al. [23], the hybridization of the metallic Cr-3d states and the nonmetallic N- 2p electrons that cross the Fermi level results in a strong covalent bond, which is responsible for the high rigidity of the ternary compounds.

Figure 7 shows the band structures of the Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) ternary compounds. For x = 0.25 and 0.50 (Sc_0.50Cr_0.50N and Sc_0.75Cr_0.25N), Figures 4a and 4c confirm the half-metallic nature of both compounds. For x = 0.75 (Sc_0.25Cr_0.75N), Figure 5c shows the metallic behavior of the two spin channels. However, we can see that the minority spin has a gap while the majority spin has a metallic character, because strong hybridization appears between Cr-3d(t_2g) and N-2p, dominating both the DOS and the band structure at the Fermi level in the three compounds. Additionally, increasing value of the concentration of Cr atoms results in the broadening of the partially filled t_2g band, so DOS weight transfers from the Fermi level to lower energies, thereby leading to a gain in energy [24, 25]. Energy gain due to band broadening is known as Zener’s double exchange, and is responsible for stabilizing the ferromagnetism [26].

4. CONCLUSIONS

We reported theoretical investigations of the structural and electronic properties of Sc_xCr_1-xN compounds, with concentrations x =0.0, 0.25, 0.50, 0.75, and 1.0 of Sc atoms, using first principles calculations with the full-potential linearized augmented plane wave (FP-LAPW) method within density functional theory and the generalized gradient approximation (GGA). We found that the variation of the lattice constant of Sc_xCr_1-xN as a function of the concentration x Sc atoms is in agreement with Vegard’s law, with a very small deviation, where the sizes of the lattices constant (ScN and CrN) of binary compounds are the determining factor for this deviation. Furthermore, we found that the bulk modulus diminishes with an increase of Sc atoms. However, the values of the bulk modulus of ternary Sc_xCr_1-xN (x = 0.25, 0.50 and 0.75) compounds are high, and therefore the compounds are rigid, making them good candidates for possible applications in devices that must operate at high temperatures and high power, and in hard coatings.

Additionally, we found that Sc_xCr_1-xN (x = 0.25, 0.50, and 0.75) compounds exhibit a half-metallic behavior with a magnetic moment of 3 μ_β/–Cr-atom. The ferromagnetic behavior in the ground state comes from a double exchange mechanism and the hybridization of the Cr-3d and N-2p orbitals that cross the Femi level. These compounds are good candidates for applications in spintronics.

The authors thank the Research Center of the University of Cordoba CUIC for its financial support.

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