Temperature dependence of phonons in ${\mathrm{Pd}}_3\mathrm{Fe}$ through the Curie temperature

Iron phonon partial densities of states of ${{\mathrm{Pd}}_3}^{57}\mathrm{Fe}$ were measured from room temperature through the Curie transition at 500 K using nuclear resonant inelastic x-ray scattering. The experimental results were compared to ab initio spin-polarized calculations that model the finite-temperature thermodynamic properties of $L{1}_2$-ordered ${\mathrm{Pd}}_3\mathrm{Fe}$ with stochastically generated atomic displacements, coupled with magnetic special quasirandom structures of noncollinear magnetic moments. The scattering measurements and first-principles calculations show that the Fe partial vibrational entropy is close to what is predicted by the quasiharmonic approximation owing to a cancellation of effects. Anharmonicity and a magnon-phonon interaction approximately cancel a ferromagnetic optical phonon stiffening.

Figure 1

(a) Schematic of supercells with Fe atoms (dark blue) stochastically displaced from their ideal positions (light blue) in the 0 K ferromagnetic calculations, where the magnetic moments (red arrows) are aligned in the same direction. (b) Supercells with randomly oriented magnetic moments and stochastically displaced Fe atoms in the 800 K paramagnetic calculations. Each set of randomly oriented magnetic moments is a magnetic special quasirandom structure (SQS). Pd atoms are not shown for this illustration.

Figure 2

The magnetization curve of ${\mathrm{Pd}}_3\mathrm{Fe}$ obtained from an empirical fit of a magnetic shape function [54] to hyperfine magnetic fields obtained from the NFS spectra in this study (green) and Mössbauer data from a study by Longworth [19] (orange). The shaded region indicates the temperature range where ${\mathrm{Pd}}_3\mathrm{Fe}$ exhibits ferromagnetic order.

Figure 3

The ${}^{57}\mathrm{Fe}$ nuclear forward-scattering spectra from $L{1}_2$-ordered ${\mathrm{Pd}}_3\mathrm{Fe}$ at several temperatures. The fits (black curves) overlie experimental data (points). The spectra are displayed using a log scale and offset for clarity.

Figure 4

The normalized ${}^{57}\mathrm{Fe}$ PDOS extracted from NRIXS measurements at various temperatures. The spectra from measurements above 298 K are offset and compared with the 298 K PDOS (black curve). Error bars are from counting statistics.

Figure 5

Average energies of the Fe PDOS from NRIXS measurements (blue squares) plotted with the average Fe phonon energies from the Grüneisen parameter model (green line) and the QH DFT model (red line).

Figure 6

Total, Pd partial, and Fe partial phonon DOS curves of ${\mathrm{Pd}}_3\mathrm{Fe}$ calculated with the s-TDEP method from 0 to 800 K.

Figure 7

(a) NRIXS ${}^{57}\mathrm{Fe}$ PDOS curves compared at 298 and 786 K. (b) s-TDEP Fe PDOS curves compared at 300 and 800 K. Phonon difference spectra are shown for both NRIXS and s-TDEP.

Figure 8

(a) The Fe partial vibrational entropy from the NRIXS measurements compared with the entropy from the Grüneisen parameter model (QH ${\gamma }_{\mathrm{T}})$ and the QH DFT model. (b) The s-TDEP Fe partial vibrational entropy calculated for ${\mathrm{Pd}}_3\mathrm{Fe}$ with changing magnetic order (blue), ferromagnetic order (green), and the absence of phonon-phonon interactions (orange). The red line is the entropy from the QH DFT model. The insets in (a) and (b) show the nonharmonic contributions to the vibrational entropy.

Figure 9

Calculated phonon dispersions for the ferromagnetic and paramagnetic states at 800 K. The dispersions displayed do not include effects from phonon-phonon interactions. Displacement patterns are shown for two high-energy optical phonon modes that soften with decreasing magnetization. The orange and green spheres represent Fe and Pd atoms, respectively. The Fe partial phonon DOS curves of ${\mathrm{Pd}}_3\mathrm{Fe}$ calculated with the s-TDEP method for the ferromagnetic and paramagnetic states at 800 K are shown in the lower left.

Figure 10

${\mathrm{Pd}}_3\mathrm{Fe}$ spectral functions (logarithmic intensity scale) calculated with s-TDEP along the high-symmetry directions at 0, 300, and 800 K. Measurements of the 80 K phonon dispersion by inelastic neutron scattering [60] are shown on top of the 0 K spectral function.

Figure 11

${\mathrm{Pd}}_3\mathrm{Fe}$ phonon line shapes at the $X$ high-symmetry point at 800 K. The orange and green peaks are the optical modes that shift with changing magnetic order. The black dashed peak is the line shape of the optical mode after the Pd-Pd 1NN cubic force constant is set to zero.