  Using first principles calculations, we systematically study the elastic
stiffness constants, mechanical properties, elastic wave velocities, Debye
temperature, melting temperature, and specific heat of several
thermodynamically stable crystal structures of Bi$_{x}$Sb$_{1-x}$ ($0 < x < 1$)
binaries, which are of great interest due to their numerous inherent rich
properties, such as thermoelectricity, thermomagnetic cooling, strong
spin-orbit coupling (SOC) effects, and topological features in the electronic
bandstructure. We analyze the bulk modulus ($B$), Young's modulus ($E$), shear
modulus ($G$), $B/G$ ratio, and Poisson's ratio ($\nu$) as a function of the Bi
concentration in Bi$_{x}$Sb$_{1-x}$. The effect of SOC on above mentioned
properties is further investigated. In general, we observe that the SOC effects
cause elastic softening in most of the studied structures. Three monoclinic
structures of Bi-Sb binaries are found to exhibit significantly large
auxeticity. The Debye temperature and the magnitude of the elastic wave
velocities monotonically decrease with increasing Bi-concentration. We also
discuss the specific heat capacity versus temperature data for all studied
binaries. Our theoretical results are in excellent agreement with the existing
experimental and theoretical data. The comprehensive understanding of the
material properties such as hardness, mechanical strength, melting temperature,
propagation of the elastic waves, auxeticity, and heat capacity is vital for
practical applications of the studied binaries.
