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Condens. Matter 2020, 5, 39 5 of 9 (a) (b) Normal Insulator 0.17 λ = 0 λ = 0.1 λ = 0.17 λ = 0.5 λ = 1 Dirac semimetal SOC (c) GGA Nodal line semimetal Na s Bi pxy Bi pz Dirac Semimetal 0.475 λ kz Critical point kz No SOC Dirac semimetal kz kz kz Normal insulator SCAN Figure 3. (a) energy bands in Na3Bi along the A − Γ − A symmetry line as the strength λ of the SOC is varied from 0 to 1. Red, blue, and green dots mark the Na-s, Bi-pxy, and Bi-pz derived levels, respectively, at Γ; (b) energies of the Na s (red), Bi pxy (blue), and Bi pz (green) levels at Γ as a function of λ. Orange shading marks the Dirac semimetal region; (c) a schematic of how the Dirac semimetal forms in GGA (top) and SCAN (bottom) as λ is varied. In Figure 3c, we illustrate schematically how the bulk electronic structure of Na3Bi evolves within GGA and SCAN as the strength λ of the SOC is varied from 0 to 1. GGA yields a nodal-line semimetal at λ = 0, which evolves into a Dirac semimetal with increasing λ, so that the SOC is a secondary effect that breaks the degeneracy of Bi pxy and shifts the Bi pz level up to invert with Na s level. However, for the bands which form the Dirac points, the topology is dominated in the GGA by the crystal field which inverts the Bi pxy and the Na s levels. If the symmetry is preserved, a topological phase transition in GGA can therefore only be achieved through an additional controlling parameter (other than the SOC) such as lattice strain along the c-axis [48]. In contrast, the SOC provides sufficient control within SCAN to realize a topological phase transition. 4. Topological Properties of Na3BixSb1−x Modulation of the SOC strength could be realized experimentally in Na3Bi by forming Na3BixSb1−x solid solutions where the SOC will weaken as the Bi atoms are replaced by the lighter Sb atoms. Along this line, we consider the end-compound Na3Sb in the Na3Bi structure, and find Na3Sb to be a trivial insulator with SCAN-based optimized lattice parameters to be: a = 5.355Å and c = 9.496 Å. Na3Sb hosts a bandgap of 0.74 eV and the electronic states around the Fermi level are derived from Na-s and Sb-p orbitals. Notably, SCAN gives a bandgap, which is larger than the GGA value of ∼0.5 eV [25]. The SCAN bandgap is in better agreement with the experimental value of ∼1.1 eV observed through absorption and photoconductivity measurements [49]. We have investigated the electronic structure of Na3BixSb1−x alloys within the virtual-crystal-approximation (VCA), which is a reasonable description for alloys in which the dopant and host atoms have similar chemical compositions [50–53]. Figure 4a shows the bulk band structure of Na3BixSb1−x alloys for various values of x. The band structure in the vicinity of the Fermi level is seen to evolve with x along the lines discussed above in connection with the evolution of the band structure in Na3Bi with varying SOC strength. At x = 0, there is a clear band gap between the Energy(eV) Energy(eV)PDF Image | Topological Dirac Semimetal Phase in Bismuth Based Anode Materials for Sodium-Ion Batteries
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