Role of spin-orbit coupling in the alloying behavior of multilayer ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ solid solutions revealed by a first-principles cluster expansion

We employ a first-principles cluster-expansion method in combination with canonical Monte Carlo simulations to study the effect of spin-orbit coupling on the alloying behavior of multilayer ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$. Our simulations reveal that spin-orbit coupling plays an essential role in determining the configurational thermodynamics of Bi and Sb atoms. Without the presence of spin-orbit coupling, ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ is predicted to exhibit at low-temperature chemical ordering of Bi and Sb atoms, leading to formation of an ordered structure at $x\approx 0.5$. Interestingly, the spin-orbit-coupling effect intrinsically induced by the existence of Bi and Sb results in the disappearance of chemical ordering of the constituent elements within an immiscible region existing at $T$ < 370 K, and consequently ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ displays merely a tendency toward local segregation of Bi and Sb atoms, resulting in coexistence of Bi-rich and Sb-rich ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ solid solutions without the formation of any ordered structure of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ as predicted in the absence of spin-orbit coupling. These findings distinctly highlight an influence of spin-orbit coupling on the alloying behavior of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ and probably other alloys composed of heavy elements, where the spin-orbit-coupling effect is supposed to be robust.

Figure 1

(a) $A7$-type structure of bismuth (Bi), antimony (Sb), and disordered solid solutions of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$. (b) A structurally ordered compound of bismuth antimonide (BiSb). The purple spheres in (a) represent either Bi or Sb atoms, while the red and blue spheres in (b) explicitly denote, respectively, Bi and Sb atoms. The thin black lines in both (a) and (b) outline the conventional hexagonal unit cells of the materials with each containing six atoms, representing three buckled sheets with a vertical stacking sequence of $ABC$.

Figure 2

Ground-state diagram at $T$ = 0 K of multilayer ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ with (a) the absence of the effect of SOC and (b) the presence of the effect of SOC. Red crosses are the CE-predicted energies of mixing $\mathrm{\Delta }{E}_{\mathrm{mix}}$ of all generated $\sigma$. Open black circles are the DFT-calculated $\mathrm{\Delta }{E}_{\mathrm{mix}}$ of the selected $\sigma$, included in the final cluster expansions. Thick black lines, connecting two large filled black circles both in (a) and in (b), represent the DFT-derived ground-state lines of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$, while filled blue squares stand for the DFT-calculated $\mathrm{\Delta }{E}_{\mathrm{mix}}$ of the completely random solid solutions of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ modeled by the SQS method.

Figure 3

(a) $\mathrm{\Delta }{G}_{\mathrm{mix}}$ of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ at $T$ = 300, 320, 340, 360, 380, and 400 K, evaluated by taking into account the effect of SOC. (b) Isostructural phase diagram of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ in the presence of the effect of SOC (see the main text for description).

Figure 4

(a) Strength of effective cluster interactions (ECIs), obtained from the final expansion including the 187 input $\sigma$ of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ in the presence of the effect of SOC. (b) $\mathrm{\Delta }{G}_{\mathrm{mix}}$ of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ at $T$ = 300, 450, 600, 750, and 900 K under the influence of SOC, as obtained from canonical MC simulations (open circles) and mean-field (MF) approximation (shaded squares).

Figure 5

$a$ and $c$ lattice parameter of random solid solutions of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ as a function of composition $x$, calculated in this work (red triangles). The red dashed lines indicate the lattice parameters calculated according to the Vegard's law between Bi and Sb. Comparison is made with the experimental data, previously reported by Dismukes et al. [19] (shaded black circles).

Figure 6

Electronic density of states around the highest occupied state, indicated by the vertical dotted lines at 0 eV, of (a) ${\mathrm{Bi}}_{0.875}{\mathrm{Sb}}_{0.125}$ and (b) ${\mathrm{Bi}}_{0.5}{\mathrm{Sb}}_{0.5}$ disordered solid solutions.