Lattice dynamics and structural stability of ordered ${\text{Fe}}_3\text{Ni}$, ${\text{Fe}}_3\text{Pd}$ and ${\text{Fe}}_3\text{Pt}$ alloys using density functional theory

We investigate the binding surface along the Bain path and phonon dispersion relations for the cubic phase of the ferromagnetic binary alloys ${\text{Fe}}_{3}X\left(X=\text{Ni},\text{Pd},\text{Pt}\right)$ for $\text{L}{1}_2$ and $\text{D}{0}_{22}$ ordered phases from first principles by means of density functional theory. The phonon dispersion relations exhibit a softening of the transverse acoustic mode at the $M$ point in the $\text{L}{1}_2$ phase in accordance with experiments for ordered ${\text{Fe}}_3\text{Pt}$. This instability can be associated with a rotational movement of the Fe atoms around the Ni-group element in the neighboring layers and is accompanied by an extensive reconstruction of the Fermi surface. In addition, we find an incomplete softening in [111] direction which is strongest for ${\text{Fe}}_3\text{Ni}$. We conclude that besides the valence electron density also the specific Fe-content and the masses of the alloying partners should be considered as parameters for the design of Fe-based functional magnetic materials.

Figure 1

(Color online) Primitive cells of fcc $\text{L}{1}_2$ (a) and $\text{D}{0}_{22}$ (b) ordered alloys of $A_{3}B$ stoichiometry as used in our calculations. The lower two images show both structures represented in the same supercell which consists of two primitive cells stacked along the $c$ axis in the case of $\text{L}{2}_1$ (c). Exchanging the $A$ and $B$ components, marked by dark (blue) and bright (orange) spheres in one of the $a\text{−}b$-planes yields the $\text{D}{0}_{22}$ ordering (d). Note, that the symmetry of the $\text{D}{0}_{22}$ structure is tetragonal. After reduction of the $c/a$ ratio by a factor of $\sqrt{1/2}$, the cell transforms to the bcc-type $\text{D}{0}_3$ structure with cubic symmetry.

Figure 2

(Color online) The energy profile along the Bain path for $\text{L}{1}_2{\text{Fe}}_3\text{Pd}$ obtained with VASP and WIEN2K for PBE and PW91 exchange-correlation potentials. The calculations were carried out at a fixed atomic volume of $13.08{\text{Å}}^3$, corresponding to an fcc lattice constant of $a=3.74\text{Å}$. For better comparison, all results are given relative to the energy of the bcc state $\left(c/a=\sqrt{1/2}\right)$. The labeling fcc and bcc refers to coordination of the atoms regardless of the species. With the VASP code, pseudopotentials with different numbers of semicore electrons were tested; the reference calculations (WIEN2K) account for $\text{Fe}3s$ and $\text{Pd}4s$ as semicore electrons. Most curves coincide along the full Bain path within 2 meV/atom.

Figure 3

(Color online) Contour plots of the binding surfaces (energy as a function of atomic volume $v$ and tetragonal distortion $c/a$) of ferromagnetic $\text{L}{1}_2$ ordered ${\text{Fe}}_3\text{Ni}$ (top), ${\text{Fe}}_3\text{Pd}$ (center), and ${\text{Fe}}_3\text{Pt}$ (bottom). The energy distance between two contour lines is 2 meV/atom. The global minima are located at $c/a=0.74$ and $v=11.633{\text{Å}}^3/\text{atom}$ for ${\text{Fe}}_3\text{Ni}$, $c/a=0.78$ and $v=13.054{\text{Å}}^3/\text{atom}$ for ${\text{Fe}}_3\text{Pd}$, and $c/a=1.00$ and $v=13.082{\text{Å}}^3/\text{atom}$ for ${\text{Fe}}_3\text{Pt}$.

Figure 4

(Color online) Comparison of total energy and magnetic moment as a function of tetragonal distortion $c/a$ between $\text{L}{1}_2$-ordered and $\text{D}{0}_{22}$-ordered ferromagnetic ${\text{Fe}}_3\text{Ni}$ (left), ${\text{Fe}}_3\text{Pd}$ (center) and ${\text{Fe}}_3\text{Pt}$ (right). The calculations were performed at a constant atomic volume close to the respective equilibrium volume of the $\text{L}{1}_2$ structure.

Figure 5

(Color online) Calculated phonon dispersions (right panels) of $\text{L}{1}_2$ ordered ${\text{Fe}}_3\text{Ni}$ (top), ${\text{Fe}}_3\text{Pd}$ (center) and ${\text{Fe}}_3\text{Pt}$ (bottom) along the main symmetry directions and the corresponding vibrational density of states (left panels). Imaginary frequencies are omitted from the density of states. The polarization of the phonon branches according to the atomic contributions to the respective eigenvectors is shown in the intensity (color) coding.

Figure 6

(Color online) Calculated phonon dispersions (right panel) of $\text{D}{0}_3$ ordered ${\text{Fe}}_3\text{Ni}$ along the main symmetry directions and the corresponding vibrational density of states (left panel). Again, the polarization of the phonon branches according to the atomic contributions to the respective eigenvectors is shown in the intensity (color) coding.

Figure 7

(Color online) Schematic representation of the two $M$ point soft modes in the phonon dispersions of fcc ${\text{Fe}}_3\text{Ni}$, ${\text{Fe}}_3\text{Pd}$, and ${\text{Fe}}_3\text{Pt}$ shown in Fig. 5. The picture shows a projection of the repeated $\text{L}{1}_2$ unit cell along the [0 0 1] direction. The brighter (red) arrows correspond to the M2 mode, while the black arrows show vibrations of the M4 mode. The main feature here is that both vibrations are identical, but orthogonal to each other. Both vibrations match with a translation along one half of the face diagonal ([110] direction) as shown by the thick (blue) arrow, whereby the two vibrations remain the same but shifted in phase by $\pi$.

Figure 8

(Color online) Total energy as a function of the displacement amplitude according to the two modes (M2, diamonds, and M4, circles) which may be made responsible for the (complete) softening of the acoustical phonons at the $M$ point (cf. Fig. 7). Results are shown for ${\text{Fe}}_3\text{Ni}$, ${\text{Fe}}_3\text{Pd}$, and ${\text{Fe}}_3\text{Pt}$. The lines are polynomials fitted to the calculated data points. For all three alloys, displacements according to the M2 mode lead to a decrease in energy.

Figure 9

(Color online) Comparison of the total and partial, site resolved electronic density of states (DOS) of ${\text{Fe}}_3\text{Pt}$ with perfect $\text{L}{1}_2$ order (left) and M2 distorted ${\text{Fe}}_3\text{Pt}$ with an displacement amplitude $\delta =0.0795\text{Å}$ (center) and M4 distorted ${\text{Fe}}_3\text{Pt}$ with $\delta =0.0748\text{Å}$ (right). The majority spin DOS are denoted by positive values, while negative values refer to the minority spin channel. In the M2 case, a pseudogap opens at the Fermi level due to a redistribution of the Fe states. For the M4 distortion, a narrower gap opens up 0.25 eV below the Fermi level, while the DOS at the Fermi level remains largely unchanged (pseudogaps denoted by arrows).

Figure 10

(Color online) Plots of the isoenergy surface in reciprocal space of the minority spin electronic states the Fermi level (Fermi surface) of ${\text{Fe}}_3\text{Ni}$ (left) and ${\text{Fe}}_3\text{Pt}$ (right). The Fermi level is intersected by six bands (shown online in different colors). The features are largely similar and differ mainly only in size due to the varying atomic volumes. Both surfaces exhibit large nearly flat portions which may give rise to a Kohn anomaly.

Figure 11

(Color online) Fermi surfaces of the 13th (a) and 15th (b) minority spin band of ${\text{Fe}}_3\text{Pt}$ (with an additional semitransparent image of the 13th band in the latter case). These two bands exhibit flat parts which may be connected with a wave vector close to $\left(\frac{1}{2},\frac{1}{2},0\right)$ as indicated by the arrows. The vector can either be applied tangentially between the flattened, horizontal walls of the cone-shaped extensions at the zone boundary entirely within the surface of the 13th band as shown in (a) or between the vertical walls on the left and the respective sections of the cube shaped surface centered around the $R$ point in (b). The eigenvalues in reciprocal space of the of the 13th and 15th bands along two horizontal planes ($k_z=0.5$ and $k_z=0.43$) are shown in (c) as contour plots. The intersection of the complete Fermi surface (all bands) with these planes are marked by black lines. The arrows denote again parallel vectors $\left(\frac{1}{2},\frac{1}{2},0\right)$. The energies are given relative to the Fermi level; the difference between two contour lines is 50 meV.