Topological Dirac Semimetal Phase in Bismuth Based Anode Materials for Sodium-Ion Batteries
Wei-Chi Chiu, Bahadur Singh, Sougata Mardanya, Johannes Nokelainen, Amit Agarwal, Hsin Lin, Christopher Lane, Katariina Pussi, Bernardo Barbiellini, Arun Bansil
TTopological Dirac Semimetal Phase in Bismuth Based AnodeMaterials for Sodium-Ion Batteries
Wei-Chi Chiu, ∗ Bahadur Singh, † and Arun Bansil Department of Physics, Northeastern University,Boston, Massachusetts 02115, USA
Sougata Mardanya and Amit Agarwal
Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
Johannes Nokelainen, Katariina Pussi, and Bernardo Barbiellini
Department of Physics, School of Engineering Science,Lappeenranta University of Technology, FI-53851 Lappeenranta, Finland
Hsin Lin
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
Christopher Lane
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA andCenter for Integrated Nanotechnologies,Los Alamos National Laboratory, Los Alamos, NM 87545, USA a r X i v : . [ c ond - m a t . m t r l - s c i ] J un bstract Bismuth has recently attracted interest in connection with Na-ion battery anodes due to its highvolumetric capacity. It reacts with Na to form Na Bi which is a prototypical Dirac semimetalwith a nontrivial electronic structure. Density-functional-theory based first-principles calculationsare playing a key role in understanding the fascinating electronic structure of Na Bi and othertopological materials. In particular, the strongly-constrained-and-appropriately-normed (SCAN)meta-generalized-gradient-approximation (meta-GGA) has shown significant improvement over thewidely used generalized-gradient-approximation (GGA) scheme in capturing energetic, structural,and electronic properties of many classes of materials. Here, we discuss the electronic structureof Na Bi within the SCAN framework and show that the resulting Fermi velocities and s -bandshift around the Γ point are in better agreement with experiments than the corresponding GGApredictions. SCAN yields a purely spin-orbit-coupling (SOC) driven Dirac semimetal state inNa Bi in contrast with the earlier GGA results. Our analysis reveals the presence of a topologicalphase transition from the Dirac semimetal to a trivial band insulator phase in Na Bi x Sb − x alloysas the strength of the SOC varies with Sb content, and gives insight into the role of the SOC inmodulating conduction properties of Na Bi. . INTRODUCTION Since lithium is a nonrenewable resource , its widespread use in Li-ion batteries can beexpected to lead to increasing costs of batteries in the coming years. This has motivatedextensive research on Na-ion batteries as an alternative to Li based batteries. However,since Na ions have a larger size and greater weight compared to Li ions, they diffuse withgreater difficulty through common electrode materials. It is important therefore to developelectrode materials with a high reversible capacity and good conducting properties . Thiseffort can benefit from first-principles computations within the framework of the densityfunctional theory (DFT) .The generalized gradient approximation (GGA) has been extensively used for identifyingmany classes of topological materials and their novel applications including electrodesfor Li-ion batteries . The presence of symmetry-enforced Dirac states in a topological ma-terial can provide robust carriers for high electronic conductivity, which is an importantfactor for improved battery performance. Despite its success in predicting the first topo-logical insulator , the Dirac semimetal , and the Weyl semimetal , the GGA suffers fromfundamental shortcomings in describing the structural and electronic properties of materials.In this connection, recent advances in constructing new classes of exchange-correlation func-tionals show that the strongly-constrained-and-appropriately-normed (SCAN) meta-GGAfunctional 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-GGAcan satisfy. SCAN has been shown to yield improved modeling of metal surfaces , 2Datomically thin-films beyond graphene , the noncollinear antiferromagnetic ground state ofmanganese , magnetic states of copper oxide superconductors , and cathode materialsfor Li ion batteries , among others materials.Among the various anode materials, antimony was recently shown to be a good candidatefor sodium-ion batteries with strong sodium cyclability and a high theoretical capacityof 660 mAh/g corresponding to the Na Sb phase. However, DFT calculations predict thatNa Sb is prone to being an insulator . Na Bi, on the other hand, is a three-dimensional(3D) nontrivial Dirac semimetal , and a number of recent studies discuss Na Bi as an anodematerial in sodium-ion batteries . Huang et al. identify a variety of different phasesof Na Bi such as NaBi, c-Na Bi (cubic), and h-Na Bi (hexagonal) as being involved in the3odiation process.Notably, h-Na Bi (henceforth Na Bi, for simplicity) is a 3D analog of graphene. It hostssymmetry-protected, four-fold degenerate band-touching points or nodes in its bulk energyspectrum around which the energy dispersion is linear in all momentum space directions .The aforementioned nodes or Dirac points are robust in the sense that they cannot be re-moved without breaking space-group symmetries. Na Bi is a prototypical band inversionDirac semimetal which was predicted theoretically before its Dirac semimetal character wasverified experimentally . The Dirac points in this material are protected by C rota-tional symmetry and lie along the hexagonal z -axis. GGA-based studies indicate that theband inversion in Na Bi is driven essentially by crystal-field effects and does not require thepresence of spin-orbit-coupling (SOC) effects, although the SOC is responsible for openingup gaps in the energy spectrum everywhere except at the Dirac points. However, the GGAhas well-known shortcomings in predicting sizes of bandgaps and crystal-field splittings, pro-viding motivation for investigating the topological structure of Na Bi using more advanceddensity functionals.Here, we revisit the topological electronic structure of Na Bi using the more accurateSCAN meta-GGA exchange-correlation functional and find that Na Bi is a trivial bandinsulator in the absence of the SOC in contrast to the GGA results. In our case, inclusion ofthe SOC drives the system into the topological Dirac semimetal state. We show that SCANyields the band energetics and Fermi velocities in substantial agreement with the availableexperimental results on Na Bi. We also discuss a topological phase transition from the Diracsemimetal to a trivial band insulator phase in Na Bi x Sb − x alloys with varying Sb content.Our results establish that Na Bi is an SOC-driven topological semimetal like the commontopological insulators such as Bi (Se, Te) . Since bismuth nano-sheets have been shownto display structural stability and good conduction properties after sodiation/desodiationcycles in sodium-ion batteries , our results further indicate the promise of Na Bi as ananode material.The remainder of this paper is organized as follows. In Section II, we describe com-putational details and discuss the crystal structure of Na Bi. The SCAN-based topologicalelectronic properties are discussed in Section III. In Section IV, we consider the bulk and sur-face electronic properties of Na Bi x Sb − x alloy. Finally, we present brief concluding remarksin Section V. 4 I. METHODOLOGY AND CRYSTAL STRUCTURE
Electronic structure calculations were carried out within the DFT framework with theprojector augmented wave (PAW) method using the Vienna ab initio Simulation Pack-age (VASP) . We used the GGA and SCAN meta-GGA energy functionals with thePerdew–Burke–Ernzerhof (PBE) parametrization to include exchange-correlation effectsin computations. An energy cut-off of 400 eV was used for the plane-wave-basis set anda Γ-centered 17 × × k -mesh was employed to sample the bulk Brillouin zone (BZ).SOC effects were included self-consistently. The topological analysis was performed by em-ploying a real-space tight-binding model Hamiltonian, which was obtained by using theVASP2WANNIER90 interface . Bi p and Na s and p states were included in generat-ing Wannier functions. The surface electronic structure was calculated using the iterativeGreens function method as implemented in the WannierTools package .Figure 1a shows the hexagonal crystal structure of Na Bi with lattice parameters a =5 .
448 ˚A and c = 9 .
655 ˚A and space group D h ( P /mmc , No. 194). It has a layeredcrystal structure where Na(1) and Bi atoms in the Wyckoff positions 2 b [ ± (0 , , )] and2 c [ ± ( , , )] form a shared honeycomb structure. The Na(2) atoms with Wyckoff position4 f [ ± ( , , u ) and ± ( , , + u ); u = 0 . z -axis. Here, Na(1) and Na(2) represent two nonequivalentNa atoms in the unit cell. The bulk and surface BZs are shown in Figure 1b. HK Γ L Bi Na(2) Na(1)(a) (b) [001][010] A M FIG. 1. ( a ) crystal structure of Na Bi visualized using the VESTA package. Na and Bi atomsare shown as blue and orange spheres. The two nonequivalent Na atoms are marked as Na(1)and Na(2); ( b ) bulk and [001] (orange) and [010] (green) surface projected Brillouin zones. Therelevant high-symmetry points are marked. II. SOC-DRIVEN TOPOLOGICAL DIRAC SEMIMETAL
The bulk electronic structure of Na Bi without SOC obtained with GGA is shown inFigure 2a. It unveils a band inversion semimetal state in which the doubly-degenerate Bi p xy and singly-degenerate Bi p z states cross along the Γ − A direction and form two triply-degenerate points. At Γ, Na s states are inverted and located 0 . p xy states.Figure 2c presents the band structure with SOC. The nodal states now become gappedeverywhere except at two discrete points along the Γ − A line ( C rotational axis). Thestrength of band inversion is enhanced such that Na s states are lowered to 0 . p xy states.Figure 2b shows the energy bands without SOC calculated using SCAN. An insulatingground state with a band gap of 90 meV is seen in sharp contrast to the GGA results. Whenthe SOC is included, SCAN yields an inverted band structure and the Dirac semimetal statewith a pair of Dirac points (Figure 2d), although the band inversion strength is reducedand the Na s states are lifted to lie 0.45 eV below the Bi p xy states. The shift of the Na s band can be verified via angle-resolved photoemission spectroscopy (ARPES) experiments.Since the band inversion in Na Bi is driven by SOC, our SCAN-based results suggest thatit should be possible to realize a topological phase transition from normal insulator to Diracsemimetal by modulating the strength of the SOC. In this connection, Figure 3c illustrateshow the GGA and SCAN based band structures can be expected to evolve in Na Bi as afunction of the strength of the SOC.We turn now to discuss details of the Dirac points and the related Fermi velocities andcompare our theoretical predictions with the corresponding experimental results. We ob-tained the Fermi velocity ν = ( ν x , ν y , ν z ), where ν x , ν y , and ν z are the velocities along the k x , k y , and k z directions in the bottom half of the Dirac cone, respectively, via a linear fit to theband structures over the momentum range of − . − to 0.2 ˚A − . The resulting values are: ν = (2 . , . , . · ˚A for GGA and ν = (2 . , . , . · ˚A for SCAN.The Fermi velocity obtained from ARPES measurements is ν = (2 . , . , .
6) eV · ˚A. Locations of the Dirac points k d are (0 , , ± . πc ), (0 , , ± . πc ), from GGA and SCAN,respectively, while the corresponding experimental value is (0 , , ± . πc ) with δk z = ± . πc . The SCAN-based Fermi velocity is seen to be in better agreement with ex-periment. The locations of the Dirac points from SCAN and GGA are both within the6 i p GGA+SOCGGA SCAN+SOCSCAN Bi p (a) (d)(c) (b) Na s Na s E n e r gy ( e V ) E n e r gy ( e V ) E n e r gy ( e V ) E n e r gy ( e V ) FIG. 2. (Bulk band structure of Na Bi obtained (without spin-orbit-coupling (SOC)) using ( a )GGA and ( b ) SCAN meta-GGA. The Na s and Bi p states are shown as blue and red markers; ( c )and ( d ) are same as (a) and (b) except that the SOC is included in the computations. experimental resolution.As we already pointed out, a topological phase transition in Na Bi could be realizedby tuning SOC. We demonstrate this by adding a scaling factor λ to the SOC term inthe Hamiltonian as H soc = λ (cid:16) µ B (cid:126) em e c r ∂U∂r L · S (cid:17) . The resulting energy dispersion obtainedself-consistently along the A − Γ − A direction is presented in Figure 3a for a series of λ values. An insulator state is realized at λ = 0. At λ = 0 .
17, the bandgap vanishes as the Na s -derived conduction band crosses the Bi p xy -derived valence band at the Γ point, and thesystem reaches a topological critical point. As λ is increased further, two Dirac points startto emerge along the A − Γ − A line at the Fermi level. At λ = 0 .
5, the Na- s and Bi- p z bandscross in the valence band continuum region to form a second critical point. Finally, at λ = 1,the Na- s band shifts down to 0.45 eV below the Fermi level, while the system preserves apair of Dirac states at the Fermi level. Evolution of energies of the Na- s and Bi- p states atΓ as a function of λ is shown in Figure 3b. The Na- s band evolves linearly as λ increases7rom 0 to 1, but the Bi p xy states are seen to be almost independent of λ . In this way, theSOC realizes the band inversion by breaking the degeneracy of the doubly-degenerate Bi p xy states and shifting up the Bi p z state. Dirac Semimetal
NormalInsulator λ = 0 λ = 0.17λ = 0.1 λ = 0.5 λ = 1 (a)(b)
SCANGGA
No SOC SOC (c) Na s Bi p xy Bi p z k z k z k z k z Critical pointNormal insulator Dirac semimetalDirac semimetalNodal line semimetal k z E n e r gy ( e V ) E n e r gy ( e V ) λ FIG. 3. ( a ) energy bands in Na Bi along the A − Γ − A symmetry line as the strength λ of theSOC is varied from 0 to 1. Red, blue, and green dots mark the Na- s , Bi- p xy , and Bi- p z derivedlevels, respectively, at Γ; ( b ) energies of the Na s (red), Bi p xy (blue), and Bi p z (green) levels atΓ as a function of λ . Orange shading marks the Dirac semimetal region; ( c ) a schematic of howthe 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 Na Bi evolveswithin GGA and SCAN as the strength λ of the SOC is varied from 0 to 1. GGA yields anodal-line semimetal at λ = 0, which evolves into a Dirac semimetal with increasing λ , sothat 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, thetopology 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 thereforeonly be achieved through an additional controlling parameter (other than the SOC) suchas lattice strain along the c-axis . In contrast, the SOC provides sufficient control within8CAN to realize a topological phase transition. IV. TOPOLOGICAL PROPERTIES OF NA BI x SB − x Modulation of the SOC strength could be realized experimentally in Na Bi by formingNa Bi x Sb − x solid solutions where the SOC will weaken as the Bi atoms are replaced by thelighter Sb atoms. Along this line, we consider the end-compound Na Sb in the Na Bi struc-ture, and find Na Sb to be a trivial insulator with SCAN-based optimized lattice parametersto be: a = 5 .
355 ˚A and c = 9 .
496 ˚A. Na Sb hosts a bandgap of 0.74 eV and the electronicstates around the Fermi level are derived from Na- s and Sb- p orbitals. Notably, SCAN givesa bandgap, which is larger than the GGA value of ∼ . . The SCAN bandgap is inbetter agreement with the experimental value of ∼ . .We have investigated the electronic structure of Na Bi x Sb − x alloys within the virtual-crystal-approximation (VCA), which is a reasonable description for alloys in which thedopant and host atoms have similar chemical compositions . Figure 4a shows the bulkband structure of Na Bi x Sb − x alloys for various values of x . The band structure in thevicinity of the Fermi level is seen to evolve with x along the lines discussed above in con-nection with the evolution of the band structure in Na Bi with varying SOC strength. At x = 0, there is a clear band gap between the Na- s and Bi/Sb- p xy levels. At x = 0 . x , a Dirac semimetal state with a pair of Dirac points on the A − Γ − A symmetry line emerges and the Na- s derived levels move down to the Fermi level.The energy variation of various levels at Γ as a function of x is illustrated in Figure 4b.The SCAN-based surface band structure for the [010] surface for various values of x ispresented in Figure 4d. There is no surface state connecting the valence and conductionbands at x = 0 as expected from the bulk band structure of Na Sb. At x = 0 . x , the critical point splits into two bulkDirac cones along the A − Γ − A symmetry line, and the nontrivial surface states connectingthe Dirac cones emerge and can be seen for 0 . < x < x . Although we have discussed the electronic structure of the Na Bi x Sb − x alloys using9 ighLowHigh Low
Low k y (1/Å) k y (1/Å) k y (1/Å) E n e r gy ( e V ) Dirac Semimetal
Normal Insulator(a)(b) x = 0 x = 0.5 x = 0.618 x = 0.76 x = 1 (c) E n e r gy ( e V ) x - k z ( / Å ) (d) E n e r gy ( e V ) Na s Bi p xy Bi p z x = 0.618 x = 0.76 x = 1x = 0 x = 0.618 x = 0.76 x = 1 x FIG. 4. Bulk band structure of Na Bi x Sb − x alloys along the A − Γ − A symmetry line in theBZ for various x values. Red, blue, and green markers identify Na- s , Bi/Sb- p xy , and Bi/Sb- p z derived levels, respectively; ( b ) evolution of the Na- and Bi/Sb-derived levels at Γ as a function of x . Shaded region marks the Dirac semimetal phase; ( c ) topological double-Fermi-arcs at x = 0 . x = 0 .
76 (middle), and x = 1(right); ( d ) surface band structure for the [010] surface ofNa Bi x Sb − x alloys at x = 0, x = 0 . x = 0 .
76, and x = 1. the relatively simpler VCA scheme, a more sophisticated treatment of disorder effects usingthe coherent-potential-approximation (CPA) will be interesting . V. CONCLUSION
We discuss the topological electronic structure of Na Bi using the recently developedSCAN meta-GGA functional within the first-principles DFT framework. Our SCAN-based10and structure and Fermi velocities of the Dirac states are in better accordance with thecorresponding experimental results compared to the earlier GGA-based results. Nature ofthe Dirac states in Na Bi is examined in depth by exploring effects of the strength of theSOC on the topology of the band structure. The SCAN-based Dirac semimetal state inNa Bi is shown to be driven by SOC effects in contrast to the GGA where the topologicalphase appears even in the absence of the SOC. We also consider Na Bi x Sb − x alloys andshow that a topological phase transition can be realized from the Dirac semimetal state toa trivial band insulator by varying the Bi/Sb concentration. Our analysis indicates that thetopological state of pristine Na Bi is very robust compared to other anode materials such asNa Sb and MoS , which can undergo metal-insulator transitions. The presence of Diracbands with high carrier velocities and the associated conductivity provide a favorable factorfor battery performance. ACKNOWLEDGMENTS
The work at Northeastern University was supported by the U.S. Department of Energy(DOE), Office of Science, Basic Energy Sciences Grant No. DE-FG02-07ER46352, andbenefited from Northeastern Universitys Advanced Scientific Computation Center and theNational Energy Research Scientific Computing Center through DOE Grant No. DE-AC02-05CH11231. J.N. is supported by the Finnish Cultural Foundation. H. L. acknowledgesAcademia Sinica (Taiwan) for the support under Innovative Materials and Analysis Tech-nology Exploration (AS-iMATE-107-11). ∗ [email protected] † [email protected] E. A. Olivetti, G. Ceder, G. G. Gaustad, and X. Fu, Joule , 229 (2017). P. K. Nayak, L. Yang, W. Brehm, and P. Adelhelm, Angewandte Chemie International Edition , 102 (2018). S. Mukherjee, S. Bin Mujib, D. Soares, and G. Singh, Materials , 1952 (2019). W. Luo, F. Shen, C. Bommier, H. Zhu, X. Ji, and L. Hu, Accounts of Chemical Research,Accounts of Chemical Research , 231 (2016). Q. He, B. Yu, Z. Li, and Y. 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