Ab initio study of phase stability in Fe-Pd binary alloys.

Tetragonal FePd is an interesting magnetic material [1]–[3] whose formation involves several ordered and disordered phases. Experiment shows that the A1 to $\mathrm {L}1 _{0}$ structural transformation in FePd is a complex transformation of the cascade type, which proceeds via intermediate low-symmetry phases, namely the disordered tetragonal phase A6 (I4/mmm), the modified $\mathrm {L}1 _{0}$ phase(P4/mmm), the hexagonal ordered phase Fe 2 Pd (P3m1), and, probably, the orthorhombic (Cmmm) and the cubic ordered $\mathrm {L}1 _{2}$ (Pm-3m) phases [4], [5]. We present ab initio calculations to understand the experimentally observed situation in FePd. The density-functional calculations employ the generalized gradient approximation (GGA) for exchange and correlations. The calculations are based on the projector augmented wave (PAW) method implemented in the Vienna Ab-Initio Simulation Package (VASP). For the electronic wave functions, an energy cutoff of 500 eV has been taken. We have performed the calculations for ordered, deformed $\mathrm {L}1 _{0}$, and two chemically disordered structures. The well known ordered structures for Fe-Pd system are FePd $(\mathrm {L}1 _{0})$ and FePd $_{3}(\mathrm {L}1 _{2})$ [4]–[5]. In addition, we have considered the experimentally observed, hexagonal Fe 2 Pd (CdI 2 -type), and orthorhombic Fe 3 Pd 5 (Pt 5 Ga 3 -type) structures where order is yet to be established. For simplicity, we have considered only ordered form of these two structures. Furthermore, we have considered deformationsof the $\mathrm {L}1 _{0}$ structure, namely pseudocubic structures $(a \quad = \quad b \quad = \quad c)$ and structures with $c/ a \lt 1$. For the chemically disordered structures, we took the face-centered cubic A1 structure and a tetragonally distorted 2 x 2 x 2 supercell of $\mathrm {L}1 _{2}$ where $c/ a \quad =0.974$. The disorder was created by substituting one Pd by one Fe in FePd 3 $(\mathrm {L}1 _{2})$ to form the 50–50 composition for FePd alloy. Figure 1 shows the unit cells for all calculations. We have optimized the experimentally obtained lattice parameters for our calculations. Table 1 shows the optimized lattice parameters and formation energy of all the ordered and disordered phases. The formation-energy calculations, which use ground-state density-functional theory $(T=0)$ confirms that two ordered phases, Fig. 1(a) and (b) and two chemically disordered phases, (e) and (f), are the stable phases. The other ordered phases, namely(c) and (d) and two deformed versions of $\mathrm {L}1 _{0}$ namely the pseudocubic and $\mathrm {L}1 _{0}$ with $c/a \lt 1$ are also likely to be formed in the multiphase Fe-Pd sample as they have very low positive formation energies. In conclusion, we have performed ab initio calculations of several ordered and disordered phases of FePd alloys and calculated formation energy and the magnetic moments of Fe and Pd atoms in the alloy systems. The calculated negative formation energies are consistent with the experimentally observed phases. This work was partially supported by the Indian-Russian collaborative project (RFBR No. 17–52-45097 and INT/RUS/RFBR/P-267).