Phonon density of states of Sn in texturedSnOunder high pressure: Comparison of nuclear inelastic x-ray scattering spectra to a shell model

Abstract

The local phonon density of states (DOS) at the Sn site in tin monoxide $(\mathrm{SnO})$ is studied at pressures up to $8\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ with $^{119}\mathrm{Sn}$ nuclear resonant inelastic x-ray scattering (NRIXS) of synchrotron radiation at $23.88\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$. The preferred orientation (texture) of the $\mathrm{SnO}$ crystallites in the investigated samples is used to measure NRIXS spectra preferentially parallel and almost perpendicular to the $c$ axis of tetragonal $\mathrm{SnO}$. A subtraction method is applied to these NRIXS spectra to produce projected local Sn DOS spectra as seen parallel and perpendicular to the $c$ axis of $\mathrm{SnO}$. These experimentally obtained local Sn DOS spectra, both in the polycrystalline case as well as projected parallel and perpendicular to the $c$ axis, are compared with corresponding theoretical phonon DOS spectra, derived from dispersion relations calculated with a recently developed shell model. Comparison between the experimental projected Sn DOS spectra and the corresponding theoretical DOS spectra enables us to follow the pressure-induced shifts of several acoustic and optic phonon modes. While the principal spectral features of the experimental and theoretical phonon DOS agree well at energies above $10\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$, the pressure behavior of the low-energy part of the DOS is not well reproduced by the theoretical calculations. In fact, they exhibit, in contrast to the experimental data, a dramatic softening of two low-energy modes, their energies approaching zero around $2.5\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, clearly indicating the limitations of the applied shell model. These difficulties are obviously connected with the complex $\mathrm{Sn}\text{\ensuremath{-}}\mathrm{O}$ and $\mathrm{Sn}\text{\ensuremath{-}}\mathrm{Sn}$ bindings within and between the $\mathrm{Sn}\text{\ensuremath{-}}\mathrm{O}\text{\ensuremath{-}}\mathrm{Sn}$ layers in the litharge structure of $\mathrm{SnO}$. We derived from the experimental and theoretical DOS spectra a variety of elastic and thermodynamic parameters of the Sn sublattice, such as the Lamb-M\"ossbauer factor, the mean force constant, and Debye temperatures, as well as the vibrational contributions to the Helmholtz free energy, specific heat, entropy, and internal energy. We found, in part, good agreement between these values, for instance, for the Gr\"uneisen parameters for some selected phonon modes, especially for some optical modes studied recently by Raman spectroscopy. We discuss in detail a possible anisotropy in the elastic parameters resulting from the litharge-type structure of $\mathrm{SnO}$, for instance for the Lamb-M\"ossbauer factor, where we can compare with existing data from $^{119}\mathrm{Sn}$-M\"ossbauer spectroscopy.