Elastic behavior under pressure of semiconducting SmS

Abstract

From ultrasonic wave velocity measurements the hydrostatic pressure dependences of the second-order elastic constants of the semiconducting form of SmS have been obtained up to 6\ifmmode\times\else\texttimes\fi{}${10}^{8}$ Pa---just below the pressure (6.5\ifmmode\times\else\texttimes\fi{}${10}^{8}$ Pa) at which the isostructural first-order phase transition takes place. The major objectives have been to assess how the elastic behavior and the interatomic forces in SmS alter as the phase transition is approached. The results are compared and contrasted with those of Tm${\mathrm{Se}}_{0.32}$${\mathrm{Te}}_{0.68}$, which also undergoes an isostructural transition but of closely second-order character. An isostructural volume collapse is associated with the identical irreducible representation ${\ensuremath{\eta}}_{0}(={\ensuremath{\eta}}_{11}+{\ensuremath{\eta}}_{22}+{\ensuremath{\eta}}_{33})$ and consequently bulk modulus instability. As SmS approaches the transition from the semiconducting side, its bulk modulus does decrease under pressure above about 3\ifmmode\times\else\texttimes\fi{}${10}^{8}$ Pa but not to a great extent. This behavior is in contrast with that of Tm${\mathrm{Se}}_{0.32}$${\mathrm{Te}}_{0.68}$ whose bulk modulus decreases continuously under pressure to a small value at the transition. The elastic constant ${C}_{12}$ reduces rapidly with pressure above about 3\ifmmode\times\else\texttimes\fi{}${10}^{8}$ Pa but, unlike that in Tm${\mathrm{Se}}_{0.32}$${\mathrm{Te}}_{0.68}$, does not go through zero before the transition pressure is reached. ${C}_{11}$ softens somewhat at high pressure, but ${C}_{44}$ and $\frac{1}{2}({C}_{11}\ensuremath{-}{C}_{12})$ show pressure dependences typical of rocksalt-structure crystals. The third-order elastic constants at atmospheric pressure have also been measured. Much of the largest third-order elastic constant is ${C}_{111}$, for which the nearest-neighbor repulsive forces are largely responsible. The anisotropy of the acoustic-mode Gr\"uneisen $\ensuremath{\gamma}$ parameters in the long-wavelength limit is explicable in terms of the strong influence of the nearest-neighbor repulsive forces on the third-order elastic constant ${C}_{111}$. At a pressure of 6\ifmmode\times\else\texttimes\fi{}${10}^{8}$ Pa (just below ${P}_{t}$), $\frac{\ensuremath{\partial}{C}_{11}}{\ensuremath{\partial}P}=\ensuremath{-}6.3$, $\frac{\ensuremath{\partial}{C}_{12}}{\ensuremath{\partial}P}=\ensuremath{-}22.8$, and $\frac{\ensuremath{\partial}B}{\ensuremath{\partial}P}=\ensuremath{-}17.3$---anomalous pressure dependences which arise (i) from enhanced interatomic binding due to an increase in the electron density in the delocalized $d$ band as the gap between the highest $f$ level and the bottom of the $d$ band is reduced by application of pressure, and (ii) a reduction of the nearest-neighbor repulsive force at high pressure. This decrease in interatomic repulsive forces as ${P}_{t}$ is neared is consistent with the breathing-mode model of the isostructural collapse. To provide an assessment of the elastic behavior under pressure in an alloy in the intermediate-valence state, the effects of the pressure on ${C}_{11}$ and ${C}_{44}$ for the metallic alloy ${\mathrm{Sm}}_{0.576}$${\mathrm{Y}}_{0.424}$S have also been measured; $\frac{\ensuremath{\partial}{C}_{11}}{\ensuremath{\partial}P}(=+37.5)$ is large because the effect of pressure is to drive the crystal away from the transition, thereby stiffening the longitudinal mode.