STRUCTURAL AND ELASTIC PROPERTIES OF FE-GE ALLOYS: AB INITIO STUDIES

In this paper, with the help of the density functional theory, the structural and elastic properties of A2, B2, D03, and L12 phases of Fe100–xGex alloys (12,5 ≤ x ≤ 28,125 at. %) have been studied. The electronic and full ionic relaxations were used for the investigation of crystal structures. The concentration dependencies of the atomic volumes, structural phase transition temperatures, tetragonal and rhombohedral shear moduli have been calculated. We show that the atomic volume curves correlate with the sequence of phase transitions observed experimentally: A2→B2→D03 (x ≤ 22 at. % of Ge content). The structural phase transition temperatures increase with the Ge concentration. The calculated tetragonal moduli for the D03, A2, and L12 structures decrease with the increasing of the Ge content, what agrees with the experimental results. The dependence of rhombohedral shear moduli as a function of Ge concentration does not change significantly with increasing Ge atoms. The C44 is increased for the D03 phase, while for A2, B2, and L12, it decreases.


Introduction
The discovery of large magnetostrictive strains in iron-gallium alloys in 1999 started the extensive study of rare-earth-free binary alloys based on α-Fe [1,2]. These alloys are promising materials for sensors and actuator applications. Among them, iron-gallium alloys are the most thoroughly investigated. The phase diagram of Fe-Ge alloys is very similar to Fe-Ga systems in the Fe-rich region [3]. Ga and Ge are p-elements that have a significant influence on electronic structures of binary compounds, which, in turn, determines their structural and magnetic properties. In both alloys, in the range of Ga(Ge) content up to 12 at. %, the phase diagram is characterized by the existence of the disordered α-phase (A2 structure). At these compositions, the values of magnetostriction (λ 100 ) for Fe-Ga and Fe-Ge are similar and positive [1]. The further increase of Ga content up to 19 at. % leads to the formation of mixing phase D0 3 +A2 [4]. The magnetostriction of Fe 81 Ga 19 reaches 340·10 -6 in slowly cooled samples. In the case of Fe 81.5 Ge 18.5 , B2 and D0 3 phases are observed [6], and λ 100 = -96·10 -6 [1]. In contrast to Fe-Ga alloys, the properties of Fe-Ge systems are not well investigated. Experimental studies of phase formation and transitions in alloys with Ge additives are presented in [5-13, etc.]. For Fe-Ge alloys in the phase region x ≤ 22 at. %, three types of the base-centered cubic (bcc) structures with different ordering (fully disordered A2, partially ordered B2, and ordered D0 3 ) exist [5,7,9,10,13]. In the concentration range of Ge content 22 ≤ x ≤ 28 at. %, low temperature face-centered cubic (fcc) L1 2 and high-temperature hexagonal D0 19 phases were also observed [5,6,8,[10][11][12]. The effect of the addition of Ge atoms on the elastic properties of Fe-Ge alloys is considered in [1,14]: with the increase of Ge atoms in Fe lattice the tetragonal elastic modulus decreases.
The magnetic moments and Curie temperatures of Fe 100-x Ge x alloys were investigated theoretically in [15][16][17][18]. With adding of Ge atoms the total magnetic moment and Curie temperature reduced. Cao et al. [19] with the help of a full-potential-linearized augmented plane wave method studied the magnetostriction as a function of Ge concentration. They found that λ 100 increased linearly with x up to 11 at. % and then decreased. In our recent work [15,18] based on the total energy calculation of Fe 100-x Ge x alloys with different structures, the phase diagram as a function of x was constructed. Nevertheless, the existing theoretical results are insufficient to understand the relation between phase transformations and magneto-elastic properties.
Therefore, this study aims to investigate the structural and elastic properties of cubic phases of Fe 100-x Ge x (12,5 ≤ x ≤ 28,125 at. %) alloys within different approaches to geometry optimization. The paper is organized as follows. Section 2 presents the details of ab initio calculations. Section 3 contains

Calculation results
The calculated equilibrium lattice parameters a 0 , total energies per atom E 0 , and formation energies E form for electronic and ionic relaxation are presented in Table. The formation energy can be defined as a difference between the total energy per atom of an alloy and total energies per atom of its components in their equilibrium bulk structures: is the total energy per atom of alloys components, x is the Ge content concentration. For A2, B2, D0 3 , and L1 2 cubic structures, the lattice parameter increases with Ge content. In the case of B2 and D0 3 phases, the lattice constant decreases for systems with an excess of Ge (x > 25 at. %). For the comparison, the experimentally obtained lattice constants are also included in Table. For both relaxations, the values of lattice parameters are in good agreement with each other and with experimental results. The difference between a 0 el and a 0 ion is less than 0,5 %, and between a 0 el and a 0 exp is approximately 1 %. The differences between the obtained total energy values are negligible, and the D0 3 structure is energetically favorable for all considered Ge concentrations. B2, D0 3 , and L1 2 structures are stable because their formation energies are negative (E form < 0). A2 phase is stable at Ge content x < 18 at. %. However, in the disordered A2 structure, the arrangement of atoms in the lattice has a significant effect on the ground state properties and formation energy, and we considered only one configuration. Fig. 1(a) shows the atomic volume V a as a function of Ge concentration in the range of 12,5 ≤ x ≤ 28,125 at. %. The available experimental values [6,9,10] for the A2 structure are also presented in Fig. 1(a). The closest to the experiment are A2 phase results obtained with electronic relaxation and Ge content of up to x = 21,875 at. %. In the range of x > 22 at. %, the experimental volume changes slightly, while the theoretical estimation continues to increase. The lowest and the largest V a are observed for the most stable phase D0 3 and A2 structure, respectively. The V a of the B2 structure is close to D0 3 . Under the transition from disordered to ordered state, the unit-cell parameters decrease slightly and, therefore, the atomic volume also decreases [25][26][27]. The obtained dependencies of V a on Ge content correspond to the sequence of phase transitions observed experimentally [5,13]: A2→B2→D0 3 (8≤ x ≤22 at. %). The fcc phase L1 2 in the range of 21,875 ≤ x ≤ 28,125 at. % has a minimum of V a in stoichiometric composition Fe 75 Ge 25 , which is in agreement with the experimental data [11]. The L1 2 phase is experimentally observed in the narrow Ge concentration range x ≈ 22÷25,7 at. % [5,11,28,29]. Here, we simulated a wider range of concentrations for the L1 2 phase, since the minimal concentration step in the 32-atoms supercell is 3,125 at. %.  [10]. 3 Data were taken from [7]. 4 Data were taken from [11].  Mechanics. Physics, 2020, vol. 12, no. 2, pp. 49-56 52 Fig. 1 (b) presents the calculated temperatures of structural phase transitions ph tr T as a function of Ge concentration and their comparison with the available experimental data [8,[11][12][13] 19 , and L1 2 phases [5,8,13]. The concentration dependencies of tetragonal C' and rhombohedral C 44 shear moduli are presented in Fig. 2 (a, b) together with the room-temperature experimental results. For both elastic moduli, the closest to the experimental values were calculation results for the D0 3 structure obtained by electronic relaxation. The increase of Ge concentration up to x = 25 at. % leads to a decrease in the tetragonal elastic modulus. This indicates a pronounced softening of the D0 3 structure. The rhombohedral shear modulus C 44 does not change significantly with x, only slightly decrease for structures with Ge excess (x > 25 at. %). The same concentration dependencies for both C' and C 44 were obtained theoretically for the D0 3 phase in the Fe-Ga system [24]. In the case of A2 and L1 2 structures, the tetragonal shear modulus decreases in the considered range of 12,5 ≤ x ≤ 28,125 at.%. For A2, B2, and L1 2 structures, the rhombohedral shear modulus C 44 has a trend similar to C'.

Conclusion
We have studied the structural and elastic properties of Fe-Ge alloys by using the first-principles methods. Crystal structure optimization was performed for phases A2, B2, D0 3 , and L1 2 of Fe 100-x Ge x (12,5 ≤ x ≤ 28,125 at. %). We considered two types of relaxations: electronic and full ionic. We showed that the lattice constants increase with Ge concentration in both approaches, and the difference between obtained lattice constant, total energy, and formation energy is negligible. The D0 3 structure is energetically favourable for all considered Ge concentrations. The dependence of atomic volume   a Vx on Ge content corresponds to the sequence of phase transitions observed experimentally (A2→B2→D0 3 ) in the range 8 ≤ x ≤ 22 at. %. We estimated the temperature of structural phase transitions ph tr T as a function of Ge concentration and found that the slope of the calculated curve for the A2 phase is steeper than for the experimental one. Moreover, we obtained the dependencies of tetragonal C' and rhombohedral C 44 shear