Elastic, mechanical, and thermodynamic properties of Bi-Sb binaries: Effect of spin-orbit coupling

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 ${\mathrm{Bi}}_x{\mathrm{Sb}}_{1-x}$ ($0<x<1$) 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 band structure. 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 ${\mathrm{Bi}}_x{\mathrm{Sb}}_{1-x}$. The effect of SOC on the 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 auxetic behavior due to the hingelike geometric structure of bonds. The Debye temperature and the magnitude of the elastic wave velocities monotonically increase with increasing Sb concentration. However, anomalies were observed at very low Sb 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.

Figure 1

The crystal structure of Bi-Sb binaries located on the Bi-Sb phase diagram [12]. Bi atoms are shown in purple, while Sb atoms are shown in green. Each crystal structure is shown from two different lattice orientations. The cutoff length for bonds was defined as 3.10 Å in Sb-rich compositions and 3.20 Å in Bi-rich compositions.

Figure 2

Mechanical properties of Bi-Sb binaries calculated with and without inclusion of SOC: (a) bulk modulus $B$ (in GPa), (b) shear modulus $G$ (in GPA), (c) Young's modulus $E$ (in GPA), (d) Poisson's ratio $\nu$, and (e) $B/G$ ratio. The green dotted line in (e) shows the boundary ($B/G=1.7$) below (above) which the material behaves as brittle (ductile). Experimental data at room temperature (black triangles) is from Ref. [26], theoretical data (green diamonds) is from Ref. [50], and experimental data at 4.2 K (purple circles) is from Ref. [27].

Figure 3

Top, middle, and bottom panels represent the calculated Poisson's ratio of ${\mathrm{Bi}}_1{\mathrm{Sb}}_7, {\mathrm{Bi}}_7{\mathrm{Sb}}_1$, and ${\mathrm{Bi}}_9{\mathrm{Sb}}_1$ binaries, respectively. All plots were generated using the elate software [61]. Green (red) corresponds to the positive (negative) values of $\nu$ (see text for details).

Figure 4

(a) Distribution of electron localization function (turquoise) in monoclinic ${\mathrm{Bi}}_9{\mathrm{Sb}}_1$ plotted at isosurface value $\eta =0.25$. Purple: Bi atoms; yellow: Sb atoms. The hinge structure or bowtie structure of the Bi-Bi bonds can be noticed in the selected region. (b) Illustration of the negative Poisson's effect under compression (left) and expansion (right) on auxetic materials. The dotted square represents the deformed shape of the original structure (solid lines) and the arrows represent the direction of strain.

Figure 5

(a) Elastic wave velocities and (b) Debye temperature (${\mathrm{\Theta }}_D$) of Bi-Sb binaries calculated with SOC.

Figure 6

$C\left(T\right)/T^3$ vs temperature $T$ data for Bi-Sb binaries.