Self-consistent mapping: Effect of local environment on formation of magnetic moment in \boldsymbol{\alpha\mathit{-FeSi_{2}}}.

Self-consistent mapping: Effect of local environment on formation of magnetic moment in .

Abstract

The Hohenberg-Kohn theorem establishes a basis for mapping of the exact energy functional to a model one provided that their charge densities coincide. We suggest here to use a mapping in a similar spirit: the parameters of the formulated multiorbital model should be determined from the requirement that the self-consistent charge and spin densities found from the ab initio and model calculations have to be as close to each other as possible. The analysis of the model allows for detailed understanding of the role played by different parameters of the model in the physics of interest. After finding the areas of interest in the phase diagram of the model we return to the ab initio calculations and check if the effects discovered are confirmed or not. Because of the last controlling step we call this approach as hybrid self-consistent mapping approach (HSCMA). As an example of the approach we present the study of the effect of silicon atoms substitution by the iron atoms and vice versa on the magnetic properties in the iron silicide . The DFT+GGA calculations are mapped to the model with intraatomic Coulomb and exchange interactions, hoppings to nearest and next nearest atoms and exchange of the delocalized electrons between iron atoms; the magnetic moments on atoms and charge densities of the material are found self-consistently within the Hartree-Fock approximation.

We find that while the stoichiometric is nonmagnetic, the substitutions generate different magnetic structures. For example, the substitution of three atoms by the atoms results in the ferrimagnetic structure whereas the substitution of four atoms by atoms gives rise to either the nonmagnetic or the ferromagnetic state depending on the type of local enviroment of the substitutional atoms. Besides, contrary to the commonly accepted statement that the destruction of the magnetic moment is controlled only by the number of nearest neighbors, we find that actually it is controlled by the next-nearest-neighbors’ hopping parameter. This finding led us to the counterintuitive conclusion: an increase of Si concentration in ordered alloys may lead to a ferromagnetism. This conclusion is confirmed by the calculation within GGA-to-DFT.

pacs:
71.20.Be, 71.20.Eh, 71.20.Gj, 75.20.Hr, 75.47.Np, 71.20.-b, 75.10.Lp, 71.15.Dx, 71.70.Ch, 71.45.Gm
1

I INtroduction

The method of mapping of first-principle density functional theory (DFT) calculations to the effective Heisenberg model for theoretical study of the magnetic properties of solids was developed in the series of works (1). The role played by the electronic subsystem in this approach is reduced to formation of the lowest-order pairwise effective exchange interaction of classical spins. In order to have an opportunity to use the well-developed many-body perturbation theory and to obtain the physical picture of the formation of the magnetic and, especially, the non-magnetic properties of the matter by the electronic subsystem one either have to use Hedin’s GW approximation(2) or a detailed model which includes all atoms, their key orbitals and symmetry of the lattice in question, hopping parameters and Coulomb interactions. The GW approximation (even without the vertex corrections) is extremely time and computer-resources consuming. For this reason the route with more simple model Hamiltonians seems to be more efficient for highlighting the physics.

The construction of the hopping parameters for certain symmetries has been described by Slater and Koster(3). Then, the phase diagram for the chosen model in the multidimensional space of these hopping parameters, Coulomb and exchange matrix elements can be constructed in a proper approximation. However, one point in this multi-parameter space corresponds to each real material, which can be described by such a model. A change of the external conditions for the material, like applying a pressure, temperature, or placing a film of the materaial on some substrate, will move this point from the initial position only slightly. Therefore, in order to be able to predict the behavior of real material, we have to know the material-in-question coordinates in the parameter space with good accuracy. Unfortunately, a unique receipt how to find the position of the material in the model parameter space does not exist. Here we suggest the following way to resolve this difficulty. Since the DFT-based calculations usually give a reasonably good description of metals and produce corresponding self-consistent spin- and charge densities, we can speculate in the DFT spirit: we use the requirement that the system with a model Hamiltonian has the spin- and charge densities as close as possible (ideally, the same) to the one obtained within the first-principle calculations for finding the parameters of the model. Then, having obtained some prediction within the model calculations we return to the first-principle ones in order to check the validity of the model prediction. This is an essence of the suggested here hybrid self-consistent mapping approach (HSCMA). It may seem that such approach should work only within the validity domain of the chosen approximation for the DFT (in our case GGA-to-DFT). However, the largest contribution to the energy of the system and formation of the local charge density comes from the Hartree part of interaction, which is treated in DFT pretty well. That is why we want to start at least from the point in the model parameter space which provides close to . Further the model with these parameters can be used for description of the phenomena beyond reach of DFT. Then the question arises why we want that the self-consistent model and DFT charge densities (magnetic moments) have to be close to each other? The matter is that we want to know the bare parameters of the model in order to be able to use the diagrammatic methods for dressing them and to avoid double counting. The most difficult question here is to find an approximation for the model calculation which would correspond to the one used in DFT. It clear, however, that the constraints in the accuracy of both methods allow to require a rough correspondence of charge densities. Indeed, on the one hand, in the DFT we use the exchange-correlation potential with restricted and, often, unknown validity domain (moreover, it is known that the calculations within the same approximation, but different packages, produce non-coinciding results). On the other hand, the model has to contain much smaller number of the interactions and hopping parameters (otherwise we will come to GW type of description at least). In spite of this uncertainty at the initial step this approach is attractive because it does not contain any fitting parameters, which should be taken from experiment. From this point of view it should be considered as first-principle approach. We have chosen to treat the model within the Hartree-Fock approximation.

The HSCMA is applied here for the analysis of the magnetic properties of -based ordered alloys.

The growth of Fe-silicides on silicon has been widely studied in recent years because, depending on their phase, crystal structure and composition, they can be semiconducting, metallic and/or ferromagnetic, and hence offer a large variety of potential applications when integrated into silicon-based devices (4)-(6). To this day, several Fe–silicide structures have been reported. At the Fe-rich side of the binary phase diagram, metallic as well as ferromagnetic and ( structure) (7); (8)have already been established as key materials for spintronics (8); (9). The Si-rich side of the phase diagram contains several variants of a disilicide stoichiometric compound, such as the high-temperature tetragonal metallic phase (11), with applications as an electrode or an interconnect material (12); (13), and the orthorhombic semiconducting phase (14), which due to its direct band gap is an interesting candidate for thermoelectric, photovoltaic and optoelectronic devices (15). While room-temperature stable -phase is well-studied, tetragonal -phase do not attract great interest until recently. This is due to this phase is metastable and exist only at temperatures above (11). However, the iron silicides, which do not exist in bulk, can be stabilized as films. In Refs. (12),(16)-(19) a successful fabrication of thin films was reported. Also, while the magnetic order is not observed in bulk stoichiometric disilicide , ferromagnetism was found (19) in the metastable phase , which was stabilized in epitaxial-film grown on the silicon substrate. The authors of Refs. (13), (20) reported that the magnetic moments on Fe atoms (13) and (20) in nanoislands and nano-stripes on Si (111) substrate arise. These experimental achievements have good perspective for the integration of the -based magnetic devices into silicon technology, and, therefore, demands for the detailed understanding of the physics of the magnetic moment formation in these compounds.

Traditionally, the appearance of the magnetic structure in alloys is related to the increase of the concentration of atoms. So, the unusual ferromagnetism in epitaxial-film form authors (19) explain by the appearance of substitutional Fe atoms on Si sublattice. According to the ab initio calculation in the framework of Coherent Potential Approximation (CPA) performed in (19) the ferromagnetism in thin films appears with the substitution of small percent of silicon atoms by the iron atoms. Particularly, when the concentration of substitution atoms reaches , these atoms acquire magnetic moment . Similar explanation of anomalously high total magnetic moment was suggested also by the authors of Refs. (13), (20). The decrease of the magnetic moments of Fe atoms with the increase of the Si concentration was observed experimentally in the iron silicides and discussed in the framework of the phenomenological local environment models (21)-(23). It was noticed, that the changing of magnetic moment of Fe atom in iron silicides , rather depends on the number of Si atoms in the nearest local environment of iron and not on the concentration of Si atoms. In our work (24) the mechanism of magnetic moment formation in is analysed in the framework of the multiorbital model, where it is shown that the neighboring Fe atoms along crystallographic axes as well as Si atoms in the first coordination sphere play the crucial role in the destruction of the Fe magnetic moments. Namely, the increase of the number of such neighbors leads to the decrease of the Fe magnetic moment. Iron atoms in have only silicon atoms as the nearest neighbors and from the traditional point of view (21)-(23) it is naturally to assume that the absence of the magnetism in this silicide is caused by the nearest silicon environment. However, the specific feature of the structure is the presence of the alternating and planes, which are perpendicular to the tetragonal axis of the cell (Fig.1a). In such plane Fe atoms are surrounded only by atoms arranged along the crystallographic axes. Our analysis (24) prompts that such mutual arrangement of atoms should results in the magnetic moment destruction. The target of this work is to investigate the influence of local environment on the formation of the magnetic moments on iron atoms in the silicide , its ordered - rich solid solutions with substitutional atoms and Si - rich one with substitutional Si atoms, . Particularly, we will address the question about the role, played by second neighbors of Fe ions in the physics of magnetic moment formation.

The paper is organized as follow. In Sec.II we provide the details of ab initio and model calculations. The results of the ab initio calculations of and its Fe-rich alloys are given in Sec. IIIA. The results of the model calculations of and its Fe-rich alloys and the dependence of magnetic moments on the hopping matrix elements are presented in Sec. IIIB. The results of the ab initio investigation of Si-rich alloys of are described in Sec. IIIC. Sec.IV contains the summary of the obtained results and conclusions.

Ii Hscma: The Hybrid ab initio and Model Calculation Method

In this work we combine the ab initio calculations with the model one. We use the following scheme. First we perform the calculation of electronic and magnetic properties of the compound of interest within the framework of DFT-GGA for different way of silicon atoms substitution by iron atoms taking into account the relaxation of atomic positions. Then we perform mapping the DFT-GGA results to the multiorbital model, suggested in Ref.(24). The guiding argument for the formulation of the model are: the model should 1) contain as little as possible parameters; 2) contain the specific information about the compound in question, i.e., contain proper number of orbitals and electrons, and to posess the symmetry of the corresponding crystal structure, and 3) contain main interactions, reflecting our understanding of the underlying physics. At last, we perfom the mapping following the DFT ideology: we find the parameters of the model from fitting the its self-consistent charge density to the one, obtained in the ab initio calculations. The latter step distinguishes our approach from other ones (1); (25); (26). Here we briefly outline the model Hamiltonian, the details of model calculation are described in (24). We include into the Hamiltonian of our model set of interactions between the d-electrons of Fe (5d-orbitals per spin) following Kanamori (27). The structure contains neighboring ions, for this reason the interatomic direct d-d-exchange and d-d-hopping are included too. The p-electrons (3p-orbitals per spin) are modeled by atomic levels and interatomic hoppings. Both subsystems are connected by -hoppings. Thus, the Hamiltonian of the model is:

(1)

where

and the Kanamori’s part of the Hamiltonian

(2)

Here and are the creation (annihilation) operators of -electrons on Si- and d-electrons on Fe -ions; is complex lattice index, (site, basis); labels the orbitals; is spin projection index; are the Pauli matrices; and are the intraatomic Kanamori parameters; is the parameter of the intersite exchange between nearest Fe atoms. At last, are hopping integrals between , and atoms, correspondingly. The dependences of hopping integrals of were obtained from the Slater and Koster atomic orbital scheme (3) in the two-center approximation using basic set consisting of five 3d orbitals for each spin on each Fe and three 3p orbital for each spin on each Si. In this two-centre approximation the hopping integrals depend on the distance between the two atoms, where are the unit vectors along cubic axis and l, m, n are direction cosines. Then, within the two-center approximation, the hopping integrals are expressed in terms of Slater – Koster parameters , and for hopping, , for and , for hoppings ( specifies the components of the angular momentum relative to the direction ). Their -dependence are given by the functions and , where . The expressions for hopping integrals can be obtained in Table I from (3). For example, , etc. The number of points in the Brillouin zone was taken 1000. Monkhorst-Pack scheme (28) was used for generation of the k-mesh. The model is solved within the Hartree-Fock approximation (HFA). The band structure arises due to hopping parameters, which connect nearest neighbors (NN) and next NN (NNN) sites. The calculations were performed for three initial states: ferromagnetic (FM), antiferromagnetic (AFM) and paramagnetic (PM) states. After achieving self-consistency the state with minimal total energy was chosen. The last step was done with the help of the Galitsky-Migdal formula for total energy ((10) in (24)), which we adopted for our model.

All ab initio calculations presented in this paper have been performed using the Vienna ab initio simulation package (VASP) (29) with projector augmented wave (PAW) pseudopotentials (30). The valence electron configurations are taken for atoms and for Si atoms. The calculations is based on the density functional theory where the exchange-correlation functional is chosen within the Perdew-Burke-Ernzerhoff (PBE) parametrization (31) and the generalized gradient approximation (GGA) has been used. Throughout all calculations, the plane-wave cutoff energy is 500eV, and Gauss broadening with smearing 0.05eV is used. The Brillouin-zone integration is performed on the grid Monkhorst-Pack (28) special points . The optimized lattice parameters and atom’s coordinates were obtained by minimizing the full energy.

Iii Results and Discussion

iii.1 Ab initio calculations

Stoichiometric compound has tetragonal space symmetry group with one formula unit per cell. The structure is shown in Fig.1a. The compound is nonmagnetic metal with lattice parameters from our ab initio calculations Å, Å that are in a good agreement with experimental values (32). The structure of consists of alternating planes of iron and silicon atoms , which are perpendicular to the tetragonal axis of the cell. Iron atoms are surrounded by 8 silicon atoms (Å) located in the corners of slightly distorted in [001] direction cube, the next nearest neighbors (NNN) of iron atoms are Fe atoms arranged along crystallographic axes x and y, forming the iron plane (Å). The full density of states of was calculated in the works (33)-(35) and in our recent work (36), thus, in the present paper we give only partial spin-projected density (pDOS) of Fe d - electron states in Fig.1b. As seen, both and electrons are delocalized in a wide energy range and a magnetism is absent.

Figure 1: (Color online) Left panel: the structure of ; atoms are shown by blue balls, atoms – by grey balls. Right panel: partial density of electronic states (pDOS) of atoms; black line shows states, red line shows states. Zero on the energy axis is chosen at the Fermi energy.

However, as was mentioned in Introduction, several recent studies (13); (19); (20) discovered that a ferromagnetic state arises in the films of . The explanation of the emergence of the magnetic structure, suggested in these works, is within the commonly accepted opinion, that the magnetism arises due to an increase of Fe concentration in the material. The used theoretical approaches, CPA in the Ref. (19) and phenomenological local environment models in Refs (13); (20), take into account, however, only a part of the local environment effects because full account of them is beyond the reach of the standard CPA methods by construction, whereas the local environment models (21); (22) take into account the nearest enviroment only. In Ref.(24) we found that the next nearest environment (NNN) plays a crucial role in the magnetic moment formation. This motivates us to include the NNN local environment effects into study of the magnetic properties of - rich ordered alloys both in the framework of DFT calculations and subsequent analysis in the suggested multiorbital model too. The different local environment of iron atoms was set by the different spatial arrangement and number of substitutional Fe atoms in the ordered alloys . In this part of paper we presented the results of our ab initio calculations of some of ordered alloys . We used for the calculations the supercell , where a and c – the lattice parameters of stoichiometric .

The ordered alloys considered in the present work are shown in Table 1. Alloys A and B contain one and three substitutional atoms at the sites in the -planes, correspondingly. In the last three alloys C, D, E four atoms were replaced by atoms in different ways: in the plane perpendicular to c axis (С), in the plane parallel to c axis (D) и chess-mate replacement (E). The lattice parameters and calculated magnetic moments on the host iron atoms in sublattice of () and on the substitutional iron atoms ( and ) obtained after full optimization of geometry are given in Table 1. The geometry optimization results in the elongation of all cells along c axis and to the compression in the (ab) plane which are most pronounced for the C and E alloys.

Table 1: (Color online) The structures of some of ordered alloys, the optimized lattice parameters and the calculated magnetic moments; the colors encode: atoms by blue, host atoms by grey, the substitutional and atoms by black and green, correspondingly.

The substitution of one atom by iron (A) results in the appearance of the large magnetic moments (2.7) on the substitutional atom. Alhough the alloy A is ordered the obtained result is coincides with the result obtained in CPA (19) for a random alloy. The value of the magnetic moment and pDOS on the substitutional Fe atom are in a good agreement with the ones from Ref.(19). The general feature of both DOS is the sharp peak at the energy eV, which originates from the minority state of -electrons (Fig. 2).

A further increase of substitutional concentration leads to the non-trivial results that clearly illustrate the dependence of magnetic moments on the local environment. As seen from the Table 1, the substitution of three atoms by ones (alloy B) results in the appearence of the ferrimagnetic state: the substitutional and atoms become inequivalent: they acquire large magnetic moments, which are not equal to each other, and directed into opposite directions. The absolute values of magnetic moments are close to the ones in the alloy A. The alloy В presents only one of possible ways to order three substitutional Fe atoms in the supercell. Other nonequivalent ordering of the substitutional atoms are shown in Table 2. Our ab initio calculations show that the type of the magnetic structure, ferrimagnetic or ferromagnetic, is determined by the spatial arrangement of substitutional Fe atoms. Indeed, the first two alloys in Table 2 are ferrimagnetic, and the last three are ferromagnetic. The same dependence of the iron magnetic moments on the spatial arrangement (and hence on the the local environment) arises for the alloys with four substitutional Fe atoms on Si sites (C, D, E in Table 1). The alloy С and are non-magnetic, while the magnetic moments in the alloys D and E appear on the substitutional and on the host atoms. The pDOSes of substitutional in alloys B, D, E are similar to ones in alloy A, pDOS of in alloy B atom is mirror-symmetric to pDOS of . Notice that the states form peak in pDOS of substitutional atom when the latter has magnetic moment while the pDOS of d-electrons in the non-magnetic alloy C is similar to the one for the Fe atom in (Fig. 1b): and electron states are delocalized in the wide energy range.

Thus, our ab initio calculations confirm only part of the conclusions, derived from the local environment models (21); (22): the ferromagnetism arises with an increase of the Fe concentration indeed, but the types of the magnetic structure of the ordered alloys, which we obtain, are essentially different even at the same concentration of substitutional Fe atoms (Table 1 and Table 2): the magnetic moments on Fe atoms are determined by the composition and the configuration of its local environment. These findings motivate us to investigate the role played by the different local environment on the magnetic moments formation in alloys more carefully in the framework of the multiorbital model suggested in (24) and briefly outlined in Sec. II. As was pointed out in (24) the crucial role in the magnetic structure formation in iron silicides is played by both nearest and next-nearest local environment. Both are taken into account in a model calculations.

0.2 2.7
Figure 2: (Color online) pDOS for host (left) anf substitutional (right) in the alloy A. Black line shows states, red line shows states. Zero on the energy axis is the Fermi energy .
Table 2: (Color online) The ordered alloys with three substitutional atom at the sites. atoms are shown by blue balls, host by grey, substitutional and atoms are shown by black and green balls, correspondingly.

iii.2 The model calculations

In this subsection we describe the results of model calculations for the stochiometric and its ordered Fe-rich alloys (B, C and D in Table 1). In all model calculations we have used the following parameters (see Sec.II): Hubbard , i.e. all other parameters are given in units of U; . In the general case there are five hopping parameters: (Fe-Fe) and (Fe-Si) between the nearest neighbors (NN); (Fe-Fe), (Fe-Si) between next-nearest neighbors (NNN), and for Si –Si hoppings. The relation for NN and for NNN was kept in all model calculations; for this reason further everywhere we will use . The values for these hopping parameters are found from the requirement that after achieving self-consistency in both the model and the ab initio calculations (GGA), the d-DOS and magnetic moments on Fe atoms have to be as close to each other as possible. The best fit of the model magnetic moments and DOS to the ab initio ones can be achieved only when the hopping integrals are positive for the NN and negative for NNN. Along all model calculations we used equlibrium lattice parameter, obtained from the ab initio calculation (see Table 1). We also take into account that the values of hopping integrals should correlate with the distance between neighbors in all ordered alloys and in . The values of hopping parameters which provide the best fit are shown in the Table 3.

B C D
d t d t d t d t
Fe-Si (NN) 2.36 1.0 2.38 0.95 2.37 1.0 2.39 0.95
Fe-Fe (NN) - - 2.40 0.9 2.44 0.85 2.43 0.85
Fe-Si (NNN) - - 2.62 -0.55 2.24 -0.8 2.56 -0.45
Fe-Fe (NNN) 2.70 -0.65 2.60 -0.70 2.53 -0.75 2.56 -0.72
2.78 -0.60
Si-Si (NN) 2.34 2.0 2.39 2.0 2.53 1.5 2.41 2.0
Si-Si (NNN) 2.80 1.0 2.61 1.5 - - 2.78 1.0
Table 3: The distances (Å) between nearest neighbors (NN) and next-nearest neighbors (NNN) and the values of hopping integrals , which provide the best fit of the model charge densities to GGA-DFT ones.

We begin with the stoichiometric (Fig. 1a). It has the tetragonal lattice with the space group . Each of atom in the has only atoms in the nearest local environment and only Fe atoms as the second neighbors, therefore, there are three hoppings integrals: between NN (), between NNN () and between (). These parameters were used for fitting the model d-DOS and the magnetic moments on atoms to the ab initio ones. The values of , and parameters which provide the best fitting are shown in the Table 3. The model and GGA Fe - population numbers for stoichiometric and corresponding partial DOS of Fe d-electrons are compared in Table 4. The accuracy of the statement that the model reflects the properties of real compounds and qualitatively the features of ab initio pDOS at this set of parameters is seen from the Table 4.

Figure 3: (Color online) Top panel: Nearest and next-nearest neighbors of with corresponding hopping integrals (Fe and Si atoms are shown by grey and blue balls correspondingly) and the -map of magnetic moments; the blue lines show the values of hopping integrals and from Table 4. Bottom panel: Model pDOS for hopping integral (left) and (right). Hopping integral =1.0
VASP Model
0.77 0.76 0.67 0.66
0.72 0.71 0.70 0.68
0.72 0.71 0.79 0.68
0.58 0.55 0.63 0.61
0.67 0.63 0.70 0.60
Table 4: The comparison of orbital population numbers (, ), magnetic moments () and the number of electrons () for in the model with GGA-DFT ones. The ab intio (blue lines) and the model (black lines) pDOS of Fe d-electrons (left: -electrons, right: -electrons) in are compared in the figure under the Table..

In order to understand the effect of NN and NNN neighbors in the local environment on the magnetic moment (MM) formation we calculaled the dependence of the MMs on the hopping integrals (NN Fe - Si) and (NNN ). The map of the magnetic moment dependences on the hopping integrals and is shown in the top panel of Fig. 3. As seen the crucial role in the MM formation is played by hoppings between NNN Fe - Fe (). Indeed, with the experimentally existing nonmagnetic state is stable, a decrease of leads to the transition into ferromagnetic state. Furthermore, the boundaries between region with magnetic states and non magnetic ones are very sharp (Fig. 3, top panel): the MM decreases till zero very fast as a function of hopping between iron atoms. The hopping between NN Fe - Si () has effect only on the magnitude of the MM in the ferromagnetic region. The mechanism of ferromagnetism destruction with hopping is clearly seen from the bottom panel of Fig. 3. Switching off the hopping between NNN Fe – Fe () makes the -bands atom-like with the slight smearing. An increase of the hopping leads to a delocalization of these atom-like d-bands and destruction the magnetism. Hence, an increase of the distance between NNN (or, a decrease the hopping integral ) would results in the transition from nonmagnetic phase to magnetic one. This conclusion from the analysis of the model is confirmed by the ab initio calculation: the increase of the lattice parameters a and b of (or the distance NNN ) by 7% (2.9Å, 5.13 Å) causes formation of MMs on the Fe atoms. Thus, it is rather the hopping integral between the NNN atoms, not the NN hopping, determines the existence of magnetic or nonmagnetic state in , because the NN of Fe atom consist of Si atoms in the both cases.

Fe-rich alloys

To emphasize the importance of the NNN in the MM formation on iron atoms we consider the alloys C and D from Table 1. These alloys reveal essentially different magnetic behavior at the same concentration but different spatial arrangements of the substitutional atoms.

Figure 4: (Color online) Alloy C and D. Top panel: NN and NNN environment of iron atoms. atoms are shown by blue balls, grey and black balls stand for and substitutional atoms, correspondingly. Middle panel: Dependence of the MMs on hopping integrals and (hopping integrals and are switch on). Bottom panel: Dependence of MMs on hopping integrals and (hopping integrals and are switch off). Blue lines show the values of hopping integrals and for alloy C and and for alloy D (Table 3); these values provide the best fitting to the ab initio charge density.

As it follows from ab initio calculation, the ordered alloy D reveals ferromagnetism, whereas the alloy С remains nonmagnetic (Table 1). These ordered alloys have two nonequivalent atoms: is the host iron atom in the iron sublattice of and is the substitutional Fe atoms in the Si sublattice. Different spatial arragenment of the substitutional Fe atoms results in the different environment of the host and sustitutional Fe atoms in alloys D and C. These environments are shown in Fig. 4 (top panel). There is the important difference in the NNN environment of and in C and D alloys. In the С alloy both and atoms have four atoms along crystallographic axes a and b as NNN at the same distances Å (Table 3). The host in the alloy D also has four NNN atoms, but at different distances: two neighbors along axis a with the distance Å and two ones along axis b with Å. These inequal distances arise due to the different symmetry of crystal lattices: the C lattice is tetragonal with the space group , while the D one is orthorhombic with space group . Thus, the distortions of the underlying tetragonal lattice of arising in these alloys are different. Notice, that the distances between the NN and the NN in both alloys are the same. Besides, the atom has only two NNN atoms at the distance Å in the alloy D. This distance is larger than the corresponding one in the alloy C. Therefore, we are forced to introduce in the alloy D two hopping integrals for the short and long distances between NNN Fe -Fe, while only one hopping integral is required for the description of the alloy C. The values of hopping integrals providing the best fitting to the ab initio calculation are given in Table 3. The Hartree-Fock self-consistent MMs generated by the model at these values of hopping parameters are in alloy C and in alloy D.

Let us compare the dependences of the Fe magnetic moments on the NN hopping integral and at fixed values of NNN hopping integral and , shown at the middle panel of Fig.4. The range of the magnetic moments existence on both and atoms in the alloy D is restricted by the values of . The magnetic state with moments close to ab initio values () is on the narrow boundary between ferro- and paramagnetic phases. In the alloy С the nonmagnetic state is stable in all range of the hoppings between NN and . Namely the circumstance that the magnetic moments are close to the instability line make them very sensitive to changes of the NNN hoppings. Indeed, which of solutions, magnetic or non-magnetic, will arise, is controlled by the value of hopping integral : leads to formation of the paramagnetic state in the alloy С, whereas a decrease of in the alloy D, gives birth to a ferromagnetic state in the alloy D. The increase of compared to occurs due to shorter distance between Fe atoms in the NNN environment in alloy С (Table 3). Moreover, a decrease of results in appearance of magnetic moments on both Fe atoms in the alloy C; at the map of magnetic moments in the alloy C becomes similar to one for the alloy D (Fig. 4, bottom panel). At first glance one could expect that the formation of the Fe-Si bond should destroy the moment on the Fe atom. However, the magnetic moments on atoms happen to be much less sensitive to the hopping parameter between NN Fe and Si atoms. Indeed, all the -maps for Fe moments, calculated within this model, are elongated along the axis .

The physics of the destruction of the magnetic moments on Fe atoms can be interpreted from the point of view of the -band formation. The Fig. 5 illustrates this for the alloy C via the evolution of the d-electron pDOS and corresponding magnetic-moment maps with the increasing of only hopping integral at all other hopping integrals kept fixed. As seen, at first steps of increase of a gradual smearing of initially (at ) atom-like levels and a slight change of the map of magnetic moments occurs. Then, similar to the case of , at the abrupt destruction of the magnetic moments arises and the difference between the minority and majority spin states in pDOS disappears.

Figure 5: (Color online) The alloy C: Model pDOS (left panel) and the map of the magnetic moments (right panel) for the different values of hopping integral . The blue lines at the last map show the values of hopping integrals and (Table 3), which provide the best fitting to the ab initio charge density.

Let us now discuss the origin of the unusual ferrimagnetic state in the type of the alloy B (Table 1), which contains three substitutional atoms on the sites. The ordered alloy B has the tetragonal lattice with space group . There are three non-equivalent atoms in the unit cell: is the host iron sublattice of , and are the non-equivalent substitutional atoms in the sublattice. In accordance with ab initio calculations, the absolute values of magnetic moments on and atoms are close to each other, but have opposite directions: . The model MMs obtained for the values of hopping integrals from Table 3 are , , and . The specific feature of -pDOS in the alloy B is that the -pDOS is mirror-symmetric to the pDOS of atoms. This feature arises in both first-principle and model calculation. The comparison of pDOSes for substitutional atoms is shown in Fig. 6. As in previous cases we built the -maps of MMs for three non-equivalent Fe atoms (Fig. 7). The bright illustration of the importance of NNN interactions is that in spite of the fact that the NN local environment of substitutional Fe atoms is the same (Fig. 7, first column), they have completely different maps of magnetic moments. There is a wide range (at ) of the negative MMs in the map for atom (Fig. 7, bottom panel, middle column) with the sharp boundary between positive and negative values of MMs, whereas in the same region of - map the MM remains positive. These distinctions occur due to different number of Fe atoms in the NNN environment. Indeed, switching off the hoppings between NNN neighbors ( and ) changes the behavior of magnetic moments on atom: the region with the negative moment disappears and the maps for and atoms became almost identical (cf. middle and bottom panel of Fig. 7, last column). This numeric experiment explicitely shows that the role played by the NNN local environment is critically essential for the emergence of the atoms with the opposite MMs and, correspondingly, for the development of the ferrimagnetic state.

Figure 6: (Color online) The comparison of the ab initio (blue lines) and the model (black lines) pDOS of d-electrons (left panel) and d-electrons (right panel) in the alloy B. Top: -electrons, bottom: -electrons
Figure 7: (Color online) The alloy B. Top panel, left: NN and NNN environment of atom; color encodings: Si are blue, are grey, substitutional and are black and green balls correspondingly;Top panel, center: the dependence of MMs on hopping integrals and (hopping iontegrals and are swich on); Top panel, right: the dependence of MM on atom on the hopping integrals and (hopping integrals and are swith off). Middle and bottom panels: the same for and atoms, correspondingly. Blue lines on the maps show the values of hopping integrals and (Table 3), which provide the best fitting to the ab initio charge density.

  alloy C alloy B

Figure 8: (Color online) The dependence of MMs on the host and the substitutional and atoms on hopping in , the alloys C and B (from left to right). The solid lines are the dependences, obtained in model calculations, the dots show the MMs from ab initio calculations at the distance R between NNN Fe-Fe, according to (3). The scale for the distances (in Å) is given on the tops of the figures.

Thus, our analysis of the Hartree-Fock solutions of the multiorbital models of iron silicides and supporting them first-principle calculations allow to conclude that the decisive role in the destruction/formation of the iron magnetic moments is played by the NNN local environment or, more specifically, by the number of neighbors and the by spacing between them. The results of our calculations show that the previous statement (21)-(23), that the destruction of magnetic moments in the iron silicides is caused by the increase of Si atoms in the NN environment is inaccurate. The obtained in our calculations strong influence of NNN couples is caused by the peculiarity of the crystall structure, where the the iron atoms form planes. Since NNN are arranged along crystallographic axes, the strong -bonds between Fe atoms are formed. So in the alloy C and which contains the iron (001) planes with the shorter distance between than in the alloy D, these d-bonds result in the delocalization of the electrons and a decrease of the MMs up till their destruction (Fig.5). At the same time in the alloys C and D atoms have the same number of atoms in the NN environment and this does not prohibit them to have different MMs. It is very instructive to have a look from this point of view at the MM formation in the alloy A where the substitutional Fe atoms have maximal MMs compared to the ones in all other alloys considered here. The NNN environment of the substitutional atom in the alloy A (Table 1, first column) consists of only Si atoms; the hoppings between which are responsible for the destruction of moment are absent. This facts lead to the formation of a large value of MM on this iron atom.

In order to demonstrate the decisive role of the -hopping integral between NNN on the formation of the MM on Fe atoms we calculate the dependence of MM on this hopping for and the alloys C and B. This dependence is shown at Fig. 8. As seen, the increase of in the alloy C and in causes a destruction of the Fe MMs, whereas in the alloy B the abrupt flip of the magnetic moment is occurred with an increase . The model results are confirmed by the ab initio calculations. Obviously, the hopping integral changes its value with an increase of the spacing between NNN . Since the integral of the hopping matrix element contains an overlap of the wave functions, we assume that it depends on the distance R between the ions exponentially,

(3)

where and (Å). Taking the values and from Table 3 we have found the parameter ). Using Eq.(3) we obtained the distances R between NNN corresponding to the model parameters . Then the values of the MMs for the lattice parameters? corresponding to these distances have been calculated within GGA-to-DFT. These values are shown in Fig. 8 by dots. Remarkably, although the only hopping was changed with the distance R in the model calculations (the values of the other hopping parameters were kept fixed according to Table 3) we obtained the good agreement between the model and the ab initio magnetic moments. This again proves the significance of the NNN couplings for the MM formation.

.

iii.3 Ab initio calculation of the Si-rich alloys.

Our model calculations lead to the conclusion that the local MM formation is controlled either by a decrease of the number of couples in layers or by an increase of the distance between atoms in pairs. Moreover, we can state that the increase of the cell’s magnetic moment with increase of in -rich alloys is associated namely with the appearance of high-spin species in the layers, which are surrounded mainly by the atoms. However, these conditions can be fulfilled also by an increase of the concentration. To make sure that this unexpected conclusion derived from the model is correct we carried out the ab initio GGA calculation of magnetic moments for the -rich ordered alloys . The alloy’s structures must satisfy the conditions listed above. By adding atoms into the iron planes we can decrease the number of the couples. Besides, the substitutional atoms increase the spacing between the atoms.

Figure 9: (Color online) Different environments for ions leads to the formation of local MMs when a part of ions are replaced by ions (the host atoms are not shown). The optimized lattice parameters and the calculated MMs of the Fe atoms are given under the structures.

Fig. 9 displays three different variants of substitution of Fe atoms in the Fe planes by the Si atoms. All calculations were carried out for the supercells of , containing six iron atoms and two additional atoms. After full optimization of the supercells all considered alloys become magnetic, but the magnitude of the magnetic moment per supercell depends on the particular arrangement of substitutional Si atoms: (Fig. 9a), (Fig. 9b), and (Fig. 9c). The emergence of local MM on different atoms in the first two alloys (Fig. 9a,b) corresponds to the expectations, derived from the model. Indeed, since the number of iron NNN surrounded atom in the first alloy (Fig. 9a) is decreased by two, the local magnetic moment on the atom arises. Similar local MM appears on the and the atoms in the second alloy (Fig. 9b) due to an increase of the distance between NNN till 2.8Å. The third alloy (Fig. 9c), however, presents an example where, it seems, the model is oversimplified: the GGA calculation produces zero moment on the atom without Fe atoms in NNN surrounding, while according to our model the biggest local magnetic moment have to arise on the in this case. We assume that the term responsible for it and which is missed in our model is the crystal electric field (CEF), created by the Si surrounding. The in the third alloy (Fig. 9c) sits in the most symmetrical local surrounding by Si atoms, where the CEF splitting has to be stronger than in the first two cases (Fig. 9a, b).

The statement that the magnetic moments in alloys can arise due to an increase of the concentration allows us to suggest an alternative explanation of the ferromagnetism in the(111) film on Si(001) substrate, successfully stabilized by the authors of (19). The authors of Ref.(19) explain the ferromagnetizm of the film by the small concentration (about 3%) of the additional substitional Fe atoms. The calculation in (19)were performed in the framework of CPA, which is not able to take into account the local-environment effects. We assume that the observed moment arises not due to an increase of the Fe concentration as stated in the work (19), but due to an increase of concentration that arises due to a diffusion of the atoms from the substrate. For example, the lattice parameters in the considered here -rich alloys are such that the (111) elementary-cells sizes of the are very close to the : for the first (Fig. 9a), second (Fig. 9b) and third (Fig.9c) alloys, correspondingly. This corresponds to the mismutch about ( -1.5% )-( -2.5% ). Such a low mismutch presents an opportunity to stabilize the epitaxial films of the structure with similar arrangments of atoms. The magnetic moment . arises for all types of the substitutions shown in Fig. 9, which is consistent with the observed in Ref.(19) values.

Iv Conclusions

.Today it is recognized, that the large, if not the decisive, role in the mechanism of the magnetic structure formation in different compounds is played by the local environment of the magnetic species. However, most of ab initio codes, based on DFT, are complicated and usually represents a ‘‘black box’’, impedes the physical interpretation of the results. In particularly, it is difficult to extract the contributions from different local environment of an atom. Effects of local environment are especially important in alloys, in which the slight difference in the local environment can result in significantly different magnetic structures. The substitutionally disordered systems such as metallic alloys play an increasingly important role in technological applications and, hence, a lot of efforts are invested into a theoretical understanding of their properties. Although CPA is nowadays the most successful ab initio theory for the calculations of disordered alloys, the standard formulation of it neglects the effects of NNN environment. Along with the development of the ab initio methods (as non local CPA(37)), the understanding of the certain property of the specific compound can be reached in the framework of the suitable models with the parameters obtained from the ab initio calculations for a given compound. Namely such combination of the ab initio calculations with the multiorbital model one for the iron silicides was used in our work. The feature which distiguish our model approach from other ones is that the parameters of the model are determined from the fitting its self-consistent charge density to the one, obtained by ab initio calculations. This allows to study the effects of NN and NNN local environment of the atoms on the MM formation.

The presented study of the effect of silicon-atoms’ substitution by the iron atoms and vice versa on the magnetic properties in the iron silicide within the suggested multiorbital model has shown that while the stoichiometric material is nonmagnetic, the appearance of substitutional iron atoms in the may result in different magnetic structures, either ferromagnatic or ferrimagnetic. Which particular structure emerges is determined by the number and the spatial arrangement of the substitutional iron atoms. The latter statement is strongly supported by the fact that different magnetic structures can appear at the same concentration of substitutional Fe atoms. Besides, as follows from the Hartree-Fock model calculations, the MMs formation is essentially determined not by the NN Si atoms but by the NNN environment, particularly, by the atoms along the crystallgraphic axes: the MMs on iron atoms are very sensitive to the values of NNN Fe - Fe hopping parameters . We demonstrated it by a comparison of the maps of moments dependence on the hopping parameters with and without taking into account NNN ones. It is important that the nonmagnetic states in the stoichiometric arise at NNN only. The model with NN hoppings only, even if all NN to Fe atoms are Si atoms, does not have the solutions with zero moments on Fe. This allows to suggest that the magnetism in the nonmagnetic can be induced by a negative pressure.

The various magnetic structures (ferro-, ferri- or nonmagnetic) in Fe-rich alloys also controlled by the NNN Fe - Fe hopping parameters . The different ways of Si atoms substitution by Fe atoms result in the diverse local distortions of the underlying lattice and, in turn, to quite different hopping parameters and magnetic properties. It is most clearly demostrated by the magnetic behavior of several alloys with the same concentration of substitutional atoms, e.g., alloys C and D considered in this work (Sec.IIB). The comparison of the magnetic-moments maps reveals that different values of NNN hopping parameters lead to the diverse magnetic behaivour: a nonmagnetic one in the alloy C and a ferromagnetic one in the alloy D. Notice, despite of the different lattice distortion, the spacing between NN (as well as number of NN Si atoms) is the same in the both cases, hence, the local environment models which do not take into account the NNN hoppings, cannot explain this distinction. Unlike the local environment models (21); (22) we observe a weak dependence of the Fe magnetic moment on the hopping between NN and atoms: all the -maps for moments, calculated within this model, are elongated along the axis . One more characteristic feature of these maps is the presence of sharp boundaries between magnetic states region and nonmagnetic one as a function of NN hopping integral . In general, our conclusion about the decisive role of NNN local environment in the magnetic moment formation contradicts to the conclusions of earlier (much less detailed) models of local environment, where a decrease of the moment on atoms was ascribed to the increase of number of in NN sphere. According to our calculations the main role in formation of local magnetic moment is played by decreasing of the number of pairs along the crystallographic axes and/or increasing of the distance between them. This conclusion is especially interesting since most of models do not take the NNN hoppings into account.

The unexpected and somewhat counter-intuitive conclusion, produced by the model calculations, is that not only an increase of the Fe concentration can lead to the emergence of local magnetic moment on Fe atoms, but also of the metalloid concentration. Indeed, the number of the pairs can be reduced by replacing of the Fe atoms in iron planes by atoms. Moreover the distances between Fe atoms in these planes are increased due to the distortion of the underlying lattice. So, the conditions leading to the emergence of magnetism are met. The ab initio calculation of the ordered -rich alloys confirms this conclusion. Hence we can explain the ferromagnetism in the film, obtained by the authors of Ref.(19), in a more realistic way. In our opinion, the observed in Ref.(19) moment results from the increase of concentration due to a diffusion of the atoms from the substrate, but not due to an increase of the concentration.

Based on the presented analysis, we can formulate the conditions promoting the appearence of a magnetism in the iron silicides. The key parameters responsible for the magnetism are the hoppings between atoms and , which are the most sensitive parameters to different types of pressure. The latter can be done by either by fitting the lattice parameter of the substrate for film (chemical pressure), or by a sustitution of or atoms. As was pointed out in Ref.(20), the best orientation relationships, that stabilize the epitaxial are , or . Such planes contain additional atoms in -rich alloys from Fig. 9 and the sizes of corresponding unit cells are very close to the -substrate one. Small mismatch has place for the all mutual orientations of film and substrate and presents an opportunity to stabilize the epitaxial films of the structure. Moreover the possibility of tuning the hopping parameter in iron silicides has the large tecnological interest, because it gives an opportutity to control the appeance of different magnetic configurations in the cause of fabrication of new alloys or nanostructures with the prospective magnetic properties. At last, the existence of the region with sharp transition from ferro- to paramagnetic or from ferro- to ferrimagnetic state strongly improves the perspectives of the practical applications of iron silicide films and, hopefully, will stimulate technologists to find a way to make the films near the instability line with desirable characteristics.

Acknowledgements.
This work was supported by the Russian Foundation for Basic Research, projects No 14-02-00186, 17-02-00161 and by the joint Krasnoyarsk regional scientific foundation and Russian Foundation for Basic Research, projects No 16-42-242036, 16-42-243035. The authors would like to thank AS Shinkorenko for the technical support.

Footnotes

  1. thanks: Corresponding author

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