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

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## Abstract

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## 1. Introduction

## 2. Methodology and Crystal Structure

## 3. SOC-Driven Topological Dirac Semimetal

## 4. Topological Properties of Na${}_{3}$Bi${}_{x}$Sb${}_{1-x}$

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) crystal structure of Na${}_{3}$Bi visualized using the VESTA [47] package. Na and Bi atoms are shown as blue and orange spheres. The two nonequivalent Na atoms are marked as Na(1) and Na(2); (

**b**) bulk and $\left[001\right]$ (orange) and $\left[010\right]$ (green) surface projected Brillouin zones. The relevant high-symmetry points are marked.

**Figure 2.**Bulk band structure of Na${}_{3}$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**,

**d**) are same as (

**a**,

**b**) except that the SOC is included in the computations.

**Figure 3.**(

**a**) energy bands in Na${}_{3}$Bi along the $A-\Gamma -A$ symmetry line as the strength $\lambda $ of the SOC is varied from 0 to 1. Red, blue, and green dots mark the Na-s, Bi-${p}_{xy}$, and Bi-${p}_{z}$ derived levels, respectively, at $\Gamma $; (

**b**) energies of the Na s (red), Bi ${p}_{xy}$ (blue), and Bi ${p}_{z}$ (green) levels at $\Gamma $ as a function of $\lambda $. Orange shading marks the Dirac semimetal region; (

**c**) a schematic of how the Dirac semimetal forms in GGA (top) and SCAN (bottom) as $\lambda $ is varied.

**Figure 4.**(

**a**) Bulk band structure of Na${}_{3}$Bi${}_{x}$Sb${}_{1-x}$ alloys along the $A-\Gamma -A$ symmetry line in the BZ 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 $\Gamma $ as a function of x. Shaded region marks the Dirac semimetal phase; (

**c**) topological double-Fermi-arcs at $x=0.618$ (left), $x=0.76$ (middle), and $x=1$ (right); (

**d**) surface band structure for the [010] surface of Na${}_{3}$Bi${}_{x}$Sb${}_{1-x}$ alloys at $x=0$, $x=0.618$, $x=0.76$, and $x=1$.

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**MDPI and ACS Style**

Chiu, W.-C.; Singh, B.; Mardanya, S.; Nokelainen, J.; Agarwal, A.; Lin, H.; Lane, C.; Pussi, K.; Barbiellini, B.; Bansil, A.
Topological Dirac Semimetal Phase in Bismuth Based Anode Materials for Sodium-Ion Batteries. *Condens. Matter* **2020**, *5*, 39.
https://doi.org/10.3390/condmat5020039

**AMA Style**

Chiu W-C, Singh B, Mardanya S, Nokelainen J, Agarwal A, Lin H, Lane C, Pussi K, Barbiellini B, Bansil A.
Topological Dirac Semimetal Phase in Bismuth Based Anode Materials for Sodium-Ion Batteries. *Condensed Matter*. 2020; 5(2):39.
https://doi.org/10.3390/condmat5020039

**Chicago/Turabian Style**

Chiu, Wei-Chi, Bahadur Singh, Sougata Mardanya, Johannes Nokelainen, Amit Agarwal, Hsin Lin, Christopher Lane, Katariina Pussi, Bernardo Barbiellini, and Arun Bansil.
2020. "Topological Dirac Semimetal Phase in Bismuth Based Anode Materials for Sodium-Ion Batteries" *Condensed Matter* 5, no. 2: 39.
https://doi.org/10.3390/condmat5020039