Electronic structure and vibrational entropies of fcc Au-Fe alloys

Phonon density of states (DOS) curves were measured on alloys of face-centered-cubic (fcc) Au-Fe using nuclear resonant inelastic x-ray scattering (NRIXS) and inelastic neutron scattering (INS). The NRIXS and INS results were combined to obtain the total phonon DOS and the partial phonon DOS curves of Au and Fe atoms from which vibrational entropies were calculated. The main effect on the vibrational entropy of alloying comes from a stiffening of the Au partial phonon DOS with Fe concentration. Force constants were calculated from first principles for several compositions and show a local stiffening of Au-Au bonds close to Fe atoms. The calculated phonon DOS curves reproduce the experimental trend. The stiffening is attributed to two main effects comparable in magnitude: (i) an increase in electron density in the free-electron-like states and (ii) stronger $sd$ hybridization.

Figure 1

${}^{57}$Fe partial phonon DOS curves, $g^{\mathrm{Fe}}\left(E\right)$, for the ${\mathrm{Au}}_{1-x_{\mathrm{Fe}}}{\mathrm{Fe}}_{x_{\mathrm{Fe}}}$ alloys at 300 K from NRIXS measurements. The Fe content $x_{\mathrm{Fe}}$ for each curve is indicated in the figure.

Figure 2

Experimental (top) and calculated (bottom panel) phonon DOS curves. (Top) Left axis is the neutron-weighted phonon DOS curves for pure Au and Au${}_{0.97}$Fe${}_{0.03}$ from INS measurements and the right axis is the ${}^{57}$Fe partial phonon densities of states for Au${}_{0.97}$${}^{57}$Fe${}_{0.03}$ from NRIXS measurements. Analogous curves were calculated for pure Au and an SQS of Au${}_{30}$Fe${}_2$, but the motions of Au atoms that are 1nn to a Fe atom and those that are 2nn are plotted separately.

Figure 3

Neutron-weighted (dashed curves without markers) and neutron-weight-corrected (open diamonds) phonon DOS curves, along with the concentration weighted Fe (solid squares) and Au (solid circles) partial phonon DOS curves for Au${}_{0.80}$Fe${}_{0.20}$ (top) and Au${}_{0.50}$Fe${}_{0.50}$ (bottom).

Figure 4

Calculated electronic DOS curves for $d$ electrons in the SQS of paramagnetic Au${}_{30}$Fe${}_2$ at the Fe and Au sites and electronic DOS of pure Au. The vertical line marks the Fermi energy.

Figure 5

Configurational entropy of mixing ($\Delta S_{\mathrm{config}}$) and excess vibrational entropy of mixing ($\Delta S_{\mathrm{vib}}$) for ${\mathrm{Au}}_{1-x_{\mathrm{Fe}}}{\mathrm{Fe}}_{x_{\mathrm{Fe}}}$ alloys with respect to ideal mixing of fcc Au and fcc Fe. Excess vibrational entropy contributions from Au and Fe are $\Delta S_{\mathrm{vib}}^{\mathrm{Au}}$ and $\Delta S_{\mathrm{vib}}^{\mathrm{Fe}}$.