Anisotropic elasticity drives negative thermal expansion in monocrystalline SnSe

Negative thermal expansion (NTE) materials have been at the center of attention for the past few decades as thermal expansion compensators in the fields of engineering, photonics, electronics, and medicine. Numerous crystalline materials exhibit NTE, wherein a combination of positive and negative linear thermal expansion coefficients results from their highly anisotropic elasticity. In this study, we selected SnSe, an anisotropic uniaxial NTE material as a model system where theoretical studies have linked its NTE along the $c$ direction to transverse phonons and to positive Grüneisen parameters along all crystallographic axes. However, the fundamental origin of NTE in SnSe have not been experimentally resolved. Here we performed temperature-dependent resonant ultrasound spectroscopy (between 295–773 K) on single-crystalline SnSe to experimentally measure all nine independent elastic constants ($C_{11}, C_{22}, C_{33}, C_{44}, C_{55}, C_{66}, C_{12}, C_{13}, C_{23}$). Our data revealed a high degree of anisotropy in the temperature-dependent elastic constants with shear anisotropic factors show a contrasting pattern with increasing temperature. From this data we also deduced its material compressibility and negative Poisson's ratios in the major crystallographic directions that could explain its colossal linear thermal expansion coefficient along the $c$ direction, reaching $\sim –12\times {10}^{-5}{\mathrm{K}}^{-1}$ at 773 K as reported in this study. Furthermore, we confirmed positive Grüneisen parameters along all the crystallographic axes and observe that SnSe behaves like a semicompressible parallelepiped with elastically coupled $a$ and $b$ axes, with the NTE being driven by the displacement of Sn atoms in the $c$ direction.

Figure 1

Sample crystal structure and experimental setup. (a) Crystal structures of the Pnma and Cmcm phases of SnSe below and above the phase transition $T_c\sim 810\mathrm{K}$. (b) Single crystalline SnSe bulk sample and (c) RUS experimental setup with (i) direct contact transducer system for measurements up to 800 K, and (ii) buffer rod transducer system for measurements up to 1200 K.

Figure 2

Elastic constants and moduli of SnSe. Temperature-dependent (a) extensional, (b) shear, and (c) off-diagonal elastic constants (the error bars represent the standard deviations of 20 different fitting routines), (d) axial bulk moduli (fitted with the empirical Varshni function [30] to elicit their temperature dependence, and (e) Young's modulus, bulk modulus, and shear modulus.

Figure 3

Elastic anisotropy in SnSe. Temperature-dependent (a) shear isotropic factors, (b) a schematic of crystallographic planes of SnSe as the crystal transitions from the Pnma to the Cmcm phase, (c)–(e) elastic compliance constants, and (d) Poisson's ratio.

Figure 4

Temperature-dependent crystal expansion. Temperature-dependent (a) thermal expansion coefficients obtained from the temperature-dependent XRD reported in a previous study [9], and (b) the axial Grüneisen parameters of SnSe from the thermal expansion coefficients and RUS data [see Eqs. (13, 14, 15)].