Abstract
The electronic structure of antiferromagnetic is investigated. Anisotropy of exchange interactions between chrome sublattices is determined . The behavior of exchange integrals from the magnetic structure is analyzed.
1. Introduction
Intermetallic compounds of 3d metals with or possessing a tetragonal crystal structure of the type (space symmetry group -P4/nmm) have garnered a significant interest from the research community for their diverse magnetic structures. For example, is a ferrimagnet [1], and [2], [3], and [4] are antiferromagnets. The antiferromagnetic phase of is distinguished by two aspects: the small magnetic moments (in terms of the Bohr magneton) of the atoms ( = 0.4B, = 1.34B [4]) and its magnetic structure (Figure 1(b)). The peculiarity of the latter is that the effective molecular field between the and subsystems is compensated in the isotropic exchange approximation. This should lead to uncorrelated ordering of both subsystems, i.e., to the existence of two transition temperatures.
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Initially, experimental studies revealed only one critical temperature ( = 393 K) [4], which led to the conclusion that the anisotropic exchange interaction of – was significant [5]. However, later, in [6], a second critical temperature was detected (Figure 2, = 175 K) corresponding to the ordering of the subsystem. Thus, the anisotropic part of the – exchange interaction is not crucial and “works” only in the region of , inducing a small magnetic moment of (Figure 2). Thus, calculation and subsequent analysis of the electronic structure and interatomic exchange integrals in are of interest to us.
2. Crystal Structure and Electronic Structure of
The basic parameters of the crystal and magnetic structures were taken from the experiment [4, 6]. has a -type tetragonal crystal structure with the symmetry group -, Å, and Å. occupies the positions of the type, and and occupy positions of the type with parameters = 0.325 and = 0.725, respectively. We considered all ten colinear magnetic structures (Table 1), but the calculation was converged only for four of them (Table 2). The magnetic moment of a specific atom at position is given by [7]: , where and is a single-center exchange integral discussed in detail in the next section. Calculations of the electronic structure and exchange integrals in are performed by fully relativistic Korringa–Kohn–Rostoker method (SPR-KKR software package [8]). The atomic sphere approximation was used for the crystal potential. The exchange-correlation energy was calculated in the local density approximation without gradient corrections [9].
According to the results of calculations, the lowest energy has an antiferromagnetic structure AF3 (Table 2, Figure 1(b)), which agrees with the experimental data. At the same time, the calculation shows the instability of the ferromagnetic phase. Therefore, as a starting point, a ferrimagnetic structure of the type was considered (Figure 1(a)).
The electronic structure of nonmagnetic is shown in Figure 3, and its magnetic counterpart is depicted in Figure 4. The conduction band is located above and is formed mainly by the states of and states of that indicates strong hybridization in this compound. In general, the electronic structure is typical for pnictides of transition metals and agrees with the results of other authors [10]. The compound is a metal, and all features of the magnetism of transition metal compounds are inherent in it. Chromium d-electrons create magnetic and transport properties. They are delocalized and so are the magnetic moments of chromium atoms (Table 2).
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The magnetic moments of chromium in the ferrimagnetic and antiferromagnetic phases agree with the experimental data [4] and the results of other calculations by LAPW [10] and KKR [11] methods. The high value of the magnetic moment is related to the used approximation for potential of crystal lattice [11].
3. Interatomic Exchange Interactions and Critical Magnetization Temperature
The interatomic exchange integrals were calculated by the method [12], based on the calculation of the second derivative of the total energy functional from the deviations of the selected pair of spins from equilibrium. The effective classical Heisenberg Hamiltonian for metals/alloys is as follows:where the direction of the local magnetic moments on the i site is described by unit vectors . The Hamiltonian contains isotropic (), symmetric anisotropic () exchange coupling, and Dzyaloshinskii–Moriya interaction (). The Dzyaloshinskii–Moriya interaction is reported in Figure 5. For brevity, these are depicted for the AF3 configuration. The dependence of the exchange couplings from interatomic distance is shown in Figure 6 for the FIM and AF states. Now let us fix the position of an atom of interest as and consider the exchange parameter with nearby atoms (for convenience, we may omit the zero subscript when it causes no confusion). In the ferrimagnetic phase, the exchange interaction that ensures the bonding of and sublattices is meV in the first coordination sphere and rapidly decreases subsequently (Figure 6). The negative exchange interaction between the nearest atoms ( meV) ensures their “unusual” orientations in the AF structure (in a nonmagnetic crystal, the atoms are symmetrically identical).
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Among the exchange integrals that are important for describing the magnetic structure of (Figure 6, Table 3), only can be considered to be created by the double indirect exchange of atoms. For the other significant exchange integrals, the distance between the cations is smaller or comparable with the distances of the cation-anion [13]. Therefore, it is more appropriate to talk about the mechanism of the classical d-d exchange, which can be either ferromagnetic or antiferromagnetic (ferromagnetic iron and antiferromagnetic manganese or chromium). In other words, the very existence of FM and AFM only phases (lack of intermediate magnetic structure) is due to large value of the -band electronic filling. This differs substantially from the case of diluted magnetic semiconductors such as . In the latter case, the interaction of magnetic atoms via nonmagnetic As becomes the dominant mechanism due to the low concentration of manganese atoms [13].
The main topic of interest in is the interrelation between two chromium sublattices, which can be provided only by anisotropic exchange interaction [5, 6]. To estimate its value, the interatomic exchange integrals for the different AF states (Figure 6, Table 3) were calculated. In the case of the AF3 structure, for an arbitrary , in one half of the nearest atoms, the magnetic moments are parallel, while for the second one, they are antiparallel to the magnetic moment of the chosen atom. The observed difference in the exchange integrals along the X and Y axes meV and meV confirms the existence of anisotropy of the exchange interaction 6. However, its magnitude is insufficient to ensure simultaneous transition of sublattices to a magnetically ordered state (Figure 2). The anisotropy part of exchange interaction is much smaller: meV, meV.
We estimate the temperatures of the magnetic ordering of the chromium sublattices by the well-established expression formula: which is valid for the Heisenberg model with classical spins [12], where is the effective exchange interaction of the chosen atom with the entire crystal. The effective ordering temperatures are = 180 K and = 382 K versus experimentally observed = 175 K and = 393 K. The detailed analyses of orientation dependence of magnetic moments and interatomic exchange interaction (Tables 2 and 3) showed that magnetic moment of is induced by exchange interaction within the sublattice.
As was indicated in [14], estimate of the critical temperature on the basis of mean field approximation is inaccurate and gives too high or too low value, depending on the model used. However for estimated (experimental results, as shown in Figure 2), the critical temperature for and differs by no more than 3%. Such accuracy justifies the mean field approximations used for calculation of the exchange integrals as compared to the more involved Monte Carlo simulations. The Monte Carlo method improves convergence, provided the underlying code considers the following: (a) the changes in the electronic structure occurring when the magnetic moments of chromium atoms are reoriented; (b) the corresponding changes in exchange integrals; and (c) the effects of percolation in our case are not as important as in diluted magnetic semiconductors [14].
SPR-KKR calculation is based on the LDA. Self-interaction corrections for Kohn–Sham density functional theory were reported in literature [15, 16]. Their physical meanings, formulations, and applications to the critical temperature of the relevant phase transition have been discussed. In essence, the self-interaction corrections get rid of the self-interaction error, which is the sum of the Coulomb and exchange self-interactions that remain because of the use of an approximate exchange functional. The most frequently used self-interaction correction is the Perdew–Zunger correction. However, this correction leads to instabilities in the electronic state calculations of molecules. To avoid these instabilities, the authors provided several self-interaction corrections on the basis of the characteristic behaviors of self-interacting electrons, which have no two-electron interactions. These include the von Weizscker kinetic energy and long-range (far from the nucleus) asymptotic correction. Applications of self-interaction corrections have shown that the self-interaction error has a serious effect on the states of core electrons, but it has a smaller than expected effect on valence electrons, as is the case described in our manuscript. The distribution of self-interacting electrons indicates that they are near atomic nuclei rather than in chemical bonds and has a limited effect on the mean field theory. We believe that, at this point, we shall leave those effects to further discussions.
4. Conclusion
In the literature, there is a paper [17] devoted to theoretical analysis by the Berto method [18] of possible magnetic structures and the conditions for their realization in magnets with a tetragonal lattice of the type. Fruchart [17] indicates as the crucial parameter for the existence of the AF structure, the large value of the indirect exchange of ( in our notation). However, the applicability of their conclusions to the description of the AF structure in seems doubtful due to the isotropic exchange approximation used [17] to obtain them. Preliminary analysis of magnetic structures by the Berto method [18] showed that, for the realization of the experimentally observed AF structure in , the existence of anisotropy of the exchange interaction is a necessary condition.
Data Availability
The data used to support the findings of this study are included within this article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was funded by PSC-CUNY grant and Fordham Research Grant (FRG).