Topological Dirac Semimetal Phase in Bismuth Based Anode Materials for Sodium-Ion Batteries

Bismuth has recently attracted interest in connection with Na-ion battery anodes due to its high volumetric capacity. It reacts with Na to form Na$_3$Bi which is a prototypical Dirac semimetal with a nontrivial electronic structure. Density-functional-theory based first-principles calculations are playing a key role in understanding the fascinating electronic structure of Na$_3$Bi and other topological materials. In particular, the strongly-constrained-and-appropriately-normed (SCAN) meta-generalized-gradient-approximation (meta-GGA) has shown significant improvement over the widely used generalized-gradient-approximation (GGA) scheme in capturing energetic, structural, and electronic properties of many classes of materials. Here, we discuss the electronic structure of Na$_3$Bi within the SCAN framework and show that the resulting Fermi velocities and {\it s}-band shift around the $\Gamma$ point are in better agreement with experiments than the corresponding GGA predictions. SCAN yields a purely spin-orbit-coupling (SOC) driven Dirac semimetal state in Na$_3$Bi in contrast with the earlier GGA results. Our analysis reveals the presence of a topological phase transition from the Dirac semimetal to a trivial band insulator phase in Na$_{3}$Bi$_{x}$Sb$_{1-x}$ alloys as the strength of the SOC varies with Sb content, and gives insight into the role of the SOC in modulating conduction properties of Na$_3$Bi.


I. INTRODUCTION
Since lithium is a nonrenewable resource 1 , its widespread use in Li-ion batteries can be expected to lead to increasing costs of batteries in the coming years. This has motivated extensive research on Na-ion batteries 2,3 as an alternative to Li based batteries. However, since Na ions have a larger size and greater weight compared to Li ions, they diffuse with greater difficulty through common electrode materials. It is important therefore to develop electrode materials with a high reversible capacity and good conducting properties 4 . This effort can benefit from first-principles computations within the framework of the density functional theory (DFT) 5 .
The generalized gradient approximation (GGA) has been extensively used for identifying many classes of topological materials [6][7][8][9] and their novel applications 10 including electrodes for Li-ion batteries 11 . The presence of symmetry-enforced Dirac states in a topological material can provide robust carriers for high electronic conductivity, which is an important factor for improved battery performance. Despite its success in predicting the first topological insulator 12 , the Dirac semimetal 13 , and the Weyl semimetal 14 , the GGA suffers from fundamental shortcomings in describing the structural and electronic properties of materials.
In this connection, recent advances in constructing new classes of exchange-correlation functionals show that the strongly-constrained-and-appropriately-normed (SCAN) 15 meta-GGA functional provides a systematic improvement over the GGA in diversely bonded materials. SCAN is the first meta-GGA that satisfies all of the 17 exact constraints that a meta-GGA can satisfy. SCAN has been shown to yield improved modeling of metal surfaces 16 , 2D atomically thin-films beyond graphene 17 , the noncollinear antiferromagnetic ground state of manganese 18 , magnetic states of copper oxide superconductors [19][20][21] , and cathode materials for Li ion batteries 22 , among others materials.
Among the various anode materials, antimony was recently shown to be a good candidate for sodium-ion batteries with strong sodium cyclability 23,24 and a high theoretical capacity of 660 mAh/g corresponding to the Na 3 Sb phase. However, DFT calculations predict that Na 3 Sb is prone to being an insulator 25 . 26 Na 3 Bi, on the other hand, is a three-dimensional (3D) nontrivial Dirac semimetal 27 , and a number of recent studies discuss Na 3 Bi as an anode material in sodium-ion batteries [28][29][30][31][32] . Huang et al. 32 identify a variety of different phases of Na 3 Bi such as NaBi, c-Na 3 Bi (cubic), and h-Na 3 Bi (hexagonal) as being involved in the sodiation process.
Notably, h-Na 3 Bi (henceforth Na 3 Bi, for simplicity) is a 3D analog of graphene. It hosts symmetry-protected, four-fold degenerate band-touching points or nodes in its bulk energy spectrum around which the energy dispersion is linear in all momentum space directions 33-37 .
The aforementioned nodes or Dirac points are robust in the sense that they cannot be removed without breaking space-group symmetries. Na 3 Bi is a prototypical band inversion Dirac semimetal which was predicted theoretically before its Dirac semimetal character was verified experimentally 13,[38][39][40][41][42] . The Dirac points in this material are protected by C 3 rotational symmetry and lie along the hexagonal z-axis. GGA-based studies indicate that the band inversion in Na 3 Bi is driven essentially by crystal-field effects and does not require the presence of spin-orbit-coupling (SOC) effects, although the SOC is responsible for opening up gaps in the energy spectrum everywhere except at the Dirac points. However, the GGA has well-known shortcomings in predicting sizes of bandgaps and crystal-field splittings, providing motivation for investigating the topological structure of Na 3 Bi using more advanced density functionals.
Here, we revisit the topological electronic structure of Na 3 Bi using the more accurate SCAN meta-GGA exchange-correlation functional and find that Na 3 Bi is a trivial band insulator in the absence of the SOC in contrast to the GGA results. In our case, inclusion of the SOC drives the system into the topological Dirac semimetal state. We show that SCAN yields the band energetics and Fermi velocities in substantial agreement with the available experimental results on Na 3 Bi. We also discuss a topological phase transition from the Dirac semimetal to a trivial band insulator phase in Na 3 Bi x Sb 1−x alloys with varying Sb content.
Our results establish that Na 3 Bi is an SOC-driven topological semimetal like the common topological insulators such as Bi 2 (Se, Te) 3 . Since bismuth nano-sheets have been shown to display structural stability and good conduction properties after sodiation/desodiation cycles in sodium-ion batteries 32 , our results further indicate the promise of Na 3 Bi as an anode material.
The remainder of this paper is organized as follows. In Section II, we describe computational details and discuss the crystal structure of Na 3 Bi. The SCAN-based topological electronic properties are discussed in Section III. In Section IV, we consider the bulk and surface electronic properties of Na 3 Bi x Sb 1−x alloy. Finally, we present brief concluding remarks in Section V.

II. METHODOLOGY AND CRYSTAL STRUCTURE
Electronic structure calculations were carried out within the DFT framework with the projector augmented wave (PAW) method using the Vienna ab initio Simulation Package (VASP) [7][8][9]43,44 . We used the GGA and SCAN meta-GGA energy functionals with the Perdew-Burke-Ernzerhof (PBE) parametrization 15 to include exchange-correlation effects in computations. An energy cut-off of 400 eV was used for the plane-wave-basis set and a Γ-centered 17 × 17 × 10 k-mesh was employed to sample the bulk Brillouin zone (BZ).
SOC effects were included self-consistently. The topological analysis was performed by employing a real-space tight-binding model Hamiltonian, which was obtained by using the VASP2WANNIER90 interface 45 . Bi p and Na s and p states were included in generating Wannier functions. The surface electronic structure was calculated using the iterative Greens function method as implemented in the WannierTools package 46 . Figure 1a shows the hexagonal crystal structure of Na 3 Bi with lattice parameters a = 5.448Å and c = 9.655Å and space group D 4 6h (P 6 3 /mmc, No. 194). It has a layered crystal structure where Na(1) and Bi atoms in the Wyckoff positions 2b [±(0, 0, 1 4 )] and 2c [±( 1 3 , 2 3 , 1 4 )] form a shared honeycomb structure. The Na(2) atoms with Wyckoff position 4f [±( 1 3 , 2 3 , u) and ±( 2 3 , 1 3 , 1 2 +u); u = 0.583] form a triangular lattice which is inserted between the honeycomb layers along the z-axis. Here, Na(1) and Na(2) represent two nonequivalent Na atoms in the unit cell. The bulk and surface BZs are shown in Figure 1b.

III. SOC-DRIVEN TOPOLOGICAL DIRAC SEMIMETAL
The bulk electronic structure of Na 3 Bi without SOC obtained with GGA is shown in   As we already pointed out, a topological phase transition in Na 3 Bi could be realized by tuning SOC. We demonstrate this by adding a scaling factor λ to the SOC term in the Hamiltonian as H soc = λ In Figure 3c, we illustrate schematically how the bulk electronic structure of Na 3 Bi 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 p xy and shifts the Bi p z 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 p xy 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 49 . In contrast, the SOC provides sufficient control within SCAN to realize a topological phase transition. the relatively simpler VCA scheme, a more sophisticated treatment of disorder effects using the coherent-potential-approximation (CPA) will be interesting 25,55 .

V. CONCLUSION
We discuss the topological electronic structure of Na 3 Bi using the recently developed