Determining the vibrational entropy change in the giant magnetocaloric material LaFe11.6Si1.4 by nuclear resonant inelastic x-ray scattering

Magnetocaloric ${\mathrm{LaFe}}_{13\ensuremath{-}x}{\mathrm{Si}}_{x}$-based compounds belong to the outstanding materials with potential for efficient solid-state refrigeration. We have performed temperature-dependent $^{57}\mathrm{Fe}$ nuclear resonant inelastic x-ray scattering measurements (in a field ${\ensuremath{\mu}}_{0}H$ of $\ensuremath{\sim}0.7$ T) of the vibrational (phonon) density of states, VDOS, in ${\mathrm{LaFe}}_{11.6}{\mathrm{Si}}_{1.4}$ across the metamagnetic isostructural first-order phase transition at ${T}_{C}\ensuremath{\sim}192$ K from the low-temperature ferromagnetic (FM) to the high-temperature paramagnetic (PM) phase, in order to determine the change in thermodynamic properties of the Fe lattice at ${T}_{C}$. The experimental results are compared with density-functional-theory-based first-principles calculations using the fixed-spin moment approach. Our combined experimental and theoretical results reveal distinct and abrupt changes in the VDOS of the Fe sublattice across ${T}_{C}$, occurring within a small temperature interval of $\mathrm{\ensuremath{\Delta}}T\ensuremath{\le}12$ K around ${T}_{C}$. This indicates that strong magnetoelastic coupling (at the atomic scale) is present up to ${T}_{C}$, leading to a pronounced lattice softening (phonon redshift) in the PM phase. These changes originate from the itinerant electron magnetism associated with Fe and are correlated with distinct modifications in the Fe-partial electronic density of states $D({E}_{F}$) at the Fermi energy ${E}_{F}$. From the experimental VDOS we can infer an abrupt increase (jump) in the Fe-partial vibrational entropy $\mathrm{\ensuremath{\Delta}}{S}_{\mathrm{vib}}$ of $+6.9\ifmmode\pm\else\textpm\fi{}2.6$ J/(kg K) and in the vibrational specific heat $\mathrm{\ensuremath{\Delta}}{C}_{\mathrm{vib}}$ of $+2.7\ifmmode\pm\else\textpm\fi{}1.6$ J/(kg K) upon heating. The increase in magnitude of the vibrational entropy $|\mathrm{\ensuremath{\Delta}}{S}_{\mathrm{vib}}|=6.9$ J/(kg K) of the Fe sublattice at ${T}_{C}$ upon heating is substantial, if compared with the magnitude of the isothermal entropy change $|\mathrm{\ensuremath{\Delta}}{S}_{\mathrm{iso}}|$ of 14.2 J/(kg K) in a field change $\mathrm{\ensuremath{\Delta}}B$ from 0 to 1 T, as obtained from isothermal magnetization measurements on our sample and using the Maxwell relation. We demonstrate that $\mathrm{\ensuremath{\Delta}}{S}_{\mathrm{vib}}$ obtained by nuclear resonant inelastic x-ray scattering is a sizable quantity and contributes directly and cooperatively to the total entropy change $\mathrm{\ensuremath{\Delta}}{S}_{\mathrm{iso}}$ at the phase transition of ${\mathrm{LaFe}}_{13\ensuremath{-}x}{\mathrm{Si}}_{x}$.