Combined fast reversible liquidlike elastic deformation with topological phase transition in ${\mathrm{Na}}_3\mathrm{Bi}$

By means of first-principles calculations, we identified the structural phase transition of ${\mathrm{Na}}_3\mathrm{Bi}$ from the hexagonal ground state to the cubic $cF16$ phase above 0.8 GPa, in agreement with the experimental findings. Upon the releasing of pressure, the $\mathit{cF}16$ phase of ${\mathrm{Na}}_3\mathrm{Bi}$ is mechanically stable at ambient condition. The calculations revealed that the $cF16$ phase is topological semimetal (TS), similar to well-known HgTe, and it even exhibits an unusually low $C^{\prime }$ modulus (only about 1.9 GPa) and a huge anisotropy $A^u$ of as high as 11, which is the third-highest value among all known cubic crystals in their elastic behaviors. These facts render $\mathit{cF}16$-type ${\mathrm{Na}}_3\mathrm{Bi}$ very soft with a liquidlike elastic deformation in the (110)$\langle 1\overline{1}0\rangle$ slip system. Importantly, accompanying this deformation, ${\mathrm{Na}}_3\mathrm{Bi}$ shows a topological phase transition from a TS state at its strain-free cubic phase to a topological insulator (TI) at its distorted phase. Because the $C^{\prime }$ elastic deformation has almost no energy cost in a reversible and liquidlike soft manner, $\mathit{cF}16$-type ${\mathrm{Na}}_3\mathrm{Bi}$ would potentially provide a fast on/off switching method between TS and TI, which would be beneficial to quantum electronic devices for practical applications.

Figure 1

The DFT-derived pressure-dependent phase transitions of ${\mathrm{Na}}_3\mathrm{Bi}$. (a) $hP8$ hexagonal phase, experimentally claimed to be stable at the ground state under ambient pressure. (b) $hP24$ hexagonal phase, theoretically found to be stable at the ground state under ambient pressure. (c) $cF16$ cubic phase, more stable in energy above 0.8 GPa than both $hP8$ and $hP24$ phases. (d) $oC16$ orthorhombic phase, more stable above 118 GPa. (f)–(i) The derived phonon dispersions of $hP8$ (0 GPa), $hP24$ (0 GPa), $cF16$ (0 GPa), and $oC16$ (120 GPa), respectively. (e),(j) The pressure-dependent enthalpies of the $hP8, hP24$ and $cF16, oC16$ phases.

Figure 2

The liquidlike elastic behavior of ${\mathrm{Na}}_3\mathrm{Bi}$. (a) $C^{\prime }$ vs universal anisotropic ratio $A^u$, showing that ${\mathrm{Na}}_3\mathrm{Bi}$ has an extremely low $C^{\prime }$ and a highly large $A^u$ among a variety of cubic crystals. (b) Anisotropic ratio $A$ ($A^u$) vs Chung anisotropy $A^c$ for a variety of cubic crystals. (c) The DFT-derived deformation energy as a functional of strain according to the definition of elastic energies of bulk modulus, $C_{44}$, and $C^{\prime }$ at 0 GPa. (d) The pressure-dependent universal anisotropic ratio $A^u$ and $C^{\prime }$.

Figure 3

Huge elastic anisotropy of ${\mathrm{Na}}_3\mathrm{Bi}$. (a),(c) The stereographic projections of the Young's and shear moduli, respectively. (b),(d) The representation surfaces for the crystallographic-orientation-dependent Young's and shear moduli, respectively. (e)–(g) The electron localization function (ELF) isosurface maps for the (001), (110), and (111) planes, respectively.

Figure 4

Topological phase transition from TS to TI under the $C^{\prime }$ tetragonal deformation. Band structures for (a),(c) cubic $cF16$-type ${\mathrm{Na}}_3\mathrm{Bi}$ and for (b),(d) its distorted structure via the $C^{\prime }$ deformation with a 4% strain, from DFT (black solid lines), HSE (green dashed lines), and GW approximation (orange dotted lines). (a),(b) No SOC inclusion; (c),(d) with SOC inclusion. The solid red circles denote the $s$-like states.