Origin of the Dzyaloshinskii-Moriya Interactions in the Intermetallic RMn2Si2 ( R= La, Ce, Yb and Y) Materials: A DFT Study

The Dzyaloshinskii-Moriya interactions ( D M ) are investigated using first-principles calculations by means of the WIENNCM code, an implementation of the FP-LAPW method. The intermetallic RMn 2 Si 2 (R= La, Ce, Yb, and Y) materials exhibit a large spin-orbit effect after the density of states; they found a strong hybridization between Mn-Si and Mn-R atoms. Also, show a large noncollinear magnetic configuration depending on the R atoms. By using ab-initio calculations, the RKKY effect is observed in the RMn 2 Si 2 materials, which shows explicitly the existence of the giant magnetoresistance (GMR) in these materials. Explicitly, the mechanisms responsible for the magnetoelectric coupling are due to relatively the effect of the presence of the Dzyaloshinskii-Moriya term.


1-Introduction
Recently, the ternary intermetallic RT2X2 compounds (R is rare earth, T is transition metal and X is Si or Ge) have attracted significant attention because of their remarkable properties and potential application as high-temperature thermoelectric [1]. On the other hand, these materials have been reexamined by neutron diffraction experiments as they show a positive giant magnetoresistance [2]. Several studies indicate that in the majority of these compounds, the Mn metal (except Mn) carries no magnetic moment. R atoms usually order antiferromagnetically at low temperatures. Magnetic studies found that RMn2Si2 compounds are either ferromagnetic or antiferromagnetic [3,4]. Nevertheless, the various studies devoted to RMn2Si2 (R = La, Ce, Yb, Y, Pr, Nd, Sm, Gd) [2,[5][6][7][8] found some discrepancies.
Moreover, these materials show a rich variety of magnetic behaviors depending on R (Ba, Ca, La, Rare Earths) [9]. This magnetic diversity should give rise to different transport properties and magnetoresistance (GMR) coefficients. In addition, the noncollinear magnetism in LaMn2Si2 compound has been studied in detail in Ref. [2]. Furthermore, these materials crystallize in the ThCr2Si2 structure, whereas, the structural and magnetic properties of BaMn2P2 and BaMn2As2 [10,11] are similar to those of the archetypal magnetoelectric material chromium sesquioxide Cr2O3 [12,13]. Magnetoelectric has a wide interest in several applications as multifunctional devices such as magnetic-electric transducers, actuators, sensors, and spintronic devices [14,15]. There are two principal types of magnetoelectric materials: magnetostrictive-piezoelectric composites and single-phase crystals and polycrystals [16]. In the next year, another two interesting multiferroic materials were discovered: orthorhombic TbMn2O5 [17] and hexagonal HoMnO3 [18] is similar to the TbMnO3 compound, this demonstrated a strong magnetoelectric coupling, giving a switchable polarization upon magnetic field, although its ferroelectricity is poor. Moreover, the origin of magnetoelectric materials based on the spin-orbit coupling can be explained by two possible mechanisms, Dzyaloshinskii-Moriya (DM) interaction [19,20] and spin-dependent metalligand hybridization [19].
The present paper is to study the physical properties of RMn2Si2(R= La, Ce, Yb, and Y) materials by using a non-collinear first-principles calculation. Firstly, we found an important RKKY coupling in the intermetallic RMn2Si2 materials; this effect is related to the noncollinear magnetism, which also is the sign of GMR in this compound. Secondly, we calculated the Dzyaloshinskii-Moriya interaction (DM) in these types of materials; we found this effect is related to the spin-orbit coupling, also the relationship between magnetoelectric coupling and DM interaction was presented. For this reason, the intermetallic RMn2Si2 materials could be suggested as a good candidate for magnetoelectric applications.

2-Computational Methods
The intermetallic RMn2Si2 (R= La, Ce, Yb, and Y) compounds is studied using the full potential linearized augmented plane wave FP-LAPW method implemented in the WIENNCM package [21] with the noncollinear magnetism. While the exchange and correlation potentials have been treated by the Generalized Gradient Approximation parameterized by Perdew, Burke, and Erzehof (GGA-PBE) [22]. All the calculations take into account spin-orbit coupling with a 300 k-point in the irreducible wedge of the Brillouin zone.
The RMn2Si2 compounds crystallize with the tetragonal ThCr2Si2-type crystal structure in the following space group (I4/mmm) with the lattice parameter (see Table 1  The Hamiltonian of the system (see, Fig. 1) is described by Heisenberg model: where, − is the exchange coupling interactions between the first nearest neighbor. is the RKKY interaction between the first nearest neighbor's intra-plans.

3.2-Non-collinear calculation
The magnetic moment value of Mn in each compound for the different angles is illustrated in

3.2.2-Dzyaloshinskii-Moriya interaction
The antisymmetric Hamiltonian is expressed by = * ( ⃗⃗⃗ × ⃗⃗⃗ ), in Heiseinberg model ⃗⃗ = ( , , ). [28]: where , and are calculated coefficients of by choosing the configuration that take into consideration one direction and cancel the two other components.
The has been calculated for the RMn2Si2 (R=La, Y, Ce, and Yb) compounds using ab initio calculations. The results of are presented in Table 3. From these results the RMn2Si2 (R=La, Y, Ce and Yb) compound exhibit a strong except YMn2Si2 give a small ( = 0.36meV, =0.0meV and =0.65meV). This difference was related to the spinorbit effect in each compound, it's normal to find a in this type of compounds due to the noncollinear inter-layers coupling existing between Mn-Mn atoms.  compounds, this interaction allows coupling between magnetism and ferroelectricity [15,16].
This interaction resembles the form of antisymmetric superexchange interaction due to relativistic spin-orbit coupling, being proportional to the vector product of spins, whereas, the D M interaction favors non-collinear spin ordering.These types of materials are an excellent model system for studying magnetoelectric coupling.

4-Conclusions
In summary, the electronic and magnetic properties of the ternary intermetallic RMn2Si2 (R= La, Ce, Yb, and Y) compounds are calculated. Using the first-principles calculations, we demonstrate that these materials exhibit noncollinear magnetism in agreement with experimental results. By varying the distance between Mn-Mn atoms, we indicate that the presence of RKKY interactions. Also, the Dzyaloshinskii-Moriya interactions in these materials are discussed. These materials could be suggested as a good candidate for magnetoelectric applications.

Figure 2
Total and partial density of states for RMn2Si2 materials: a) LaMn2Si2 b) YMn2Si2, c) CeMn2Si2, and d)