Unusual electric polarization behavior in elemental quasi-two-dimensional allotropes of selenium

We investigate tunable electric polarization and electronic structure of quasi-two-dimensional (quasi-2D) allotropes of selenium, which are formed from their constituent one-dimensional (1D) structures through an interchain interaction facilitated by the multivalence nature of Se. Our ab initio calculations reveal that different quasi-2D Se allotropes display different types of electric polarization, including ferroelectric (FE) polarization normal to the chain direction in $\alpha$ and $\delta$ allotropes, noncollinear ferrielectric (FiE) polarization along the chain axis in $\tau$-Se, and anti-ferroelectric (AFE) polarization in $\eta$-Se. The magnitude and direction of the polarization can be changed by a previously unexplored rotation of the constituent chains. In that case, an in-plane polarization direction may change to out-of-plane in $\alpha$-Se and $\delta$-Se, flip its direction, and even disappear in $\tau$-Se. Also, the band gap may be reduced and changed from indirect to direct by rotating the constituent chains about their axes in these quasi-2D Se allotropes.

Figure 1

Atomic structure of 1D allotropes of Se, namely (a) $a$-helix, (b) $b$-chain, (c) $c$-helix, and (d) $d$-chain. (e) Definition of the bond angle $\gamma$ and the dihedral angle $\psi$ used to characterize the structures. Also shown is the atomic structure of the corresponding quasi-2D allotropes (f) $\alpha$-Se, (g) $\delta$-Se, (h) $\tau$-Se, and (i) $\eta$-Se. Red arrows indicate the local electric polarization. The unit cells of the quasi-2D structures are highlighted by the transparent green areas in (f), (g), (h), and (i).

Figure 2

Phonon spectra of quasi-2D (a) $\alpha$-Se, (b) $\delta$-Se, (c) $\tau$-Se, and (d) $\eta$-Se.

Figure 3

Charge redistribution caused by connecting isolated 1D Se chains to quasi-2D (a) $\alpha$-Se, (b) $\delta$-Se, (c) $\eta$-Se, and (d) $\tau$-Se assemblies. Calling the optimum geometry of the quasi-2D states the $A$ structure, we display the difference charge density $\mathrm{\Delta }\rho =\rho$(A)$-\rho$(isolated-1D) as isosurfaces bounding regions of electron excess at $+2.5\times {10}^{-3}\text{e}$/Å${}^3$, shown in red, and electron deficiency at $-2.5\times {10}^{-3}\text{e}$/Å${}^3$, shown in blue.

Figure 4

Axial chain rotation process about the angle $\theta$ in quasi-2D Se allotropes. Shown is the transition from the initial state $A$, across the rotational barrier state $B_r$, to the closest energetically degenerate state $A^{\prime }$ with the opposite polarization direction. [(a)–(c)] Morphology and polarization and [(d)–(f)] energy changes in $\alpha$, $\delta$, and $\tau$ allotropes of Se. The electric dipole moment is shown by the red arrow, with $⨀$ representing the out-of-plane and $⨂$ the into-the-plane directions.

Figure 5

Phonon spectra of the transition state structure $B_r$, which is unstable with respect to the rotation of constituent chains in (a) $\alpha$-Se, (b) $\delta$-Se, and (c) $\tau$-Se. The lowest branch of the phonon spectra with imaginary frequency is highlighted by the red line. Corresponding modes, initiating structural change, are indicated by the red arrows in the bottom panels.

Figure 6

Effect of the axial rotation angle $\theta$ of the constituent 1D structures on the electronic band structure $E\left(\mathbf{k}\right)$ in quasi-2D Se allotropes. $E\left(\mathbf{k}\right)$ in $\alpha$-Se is shown for (a) $\theta =0^{\circ }$ and (d) $\theta ={30}^{\circ }$, in $\delta$-Se for (b) $\theta =0^{\circ }$ and (d) $\theta ={90}^{\circ }$, in $\tau$-Se for (c) $\theta =0^{\circ }$ and (d) $\theta =10.{27}^{\circ }$. PBE-D2 results without spin-orbit coupling (SOC) are presented by the solid black lines and those with SOC by the dashed red lines.

Figure 7

Net polarization $P$ and deformation ${\sigma }_y$ along the ${\vec{a}}_2$ direction as a function of uniaxial strain ${\sigma }_x$ along the ${\vec{a}}_1$ direction in quasi-2D (a) $\alpha$-Se (b) $\delta$-Se, and (c) $\tau$-Se. The effect of ${\sigma }_x$ on $P$ is shown by the blue dotted lines, and corresponding changes in ${\sigma }_y$ are shown by the red dashed line. (d) Energy change $\mathrm{\Delta }E$ caused by changing the lattice constant $a$ in $\tau$-Se and $\alpha$-Se along the ${\vec{a}}_1$ direction. Results for supercells of $\alpha$-Se, with $a=4a_1(\alpha -\mathrm{Se})$, are shown by the red dashed line. Results for supercells of $\tau$-Se, with $a=3a_1(\tau -\mathrm{Se})$, are shown by the blue dotted line.

Figure 8

Effect of the uniaxial strain ${\sigma }_x$ along the chain direction on the electronic band structure $E\left(k\right)$ in (a) $\alpha$-Se (b) $\delta$-Se, and (c) $\tau$-Se. Lowest-energy transitions across the fundamental band gap are indicated by the red dashed lines and arrows.

Figure 9

Highly symmetric paraelectric phase $B_S$ of quasi-2D allotropes of Se placed in the $x\text{−}y$ plane. The structure of (a) $\alpha$-Se, (b) $\delta$-Se, (c) $\tau$-Se, and (d) $\eta$-Se is shown in the side and the end-on view. The unit cells are highlighted by the transparent green areas. (e)–(h) display the charge redistribution $\mathrm{\Delta }\rho =\rho \left(A\right)-\rho (B_S$) between the polarized phase of $\alpha$-Se, $\delta$-Se, $\tau$-Se, and $\eta$-Se and their highly symmetric PE phase with no local dipoles, which we call the $B_S$ state in the text. Isosurfaces of $\mathrm{\Delta }\rho$ are represented by bounding regions of electron excess at $+7\times {10}^{-2}\text{e}$/Å${}^3$ (red) and electron deficiency at $-7\times {10}^{-2}\text{e}$/Å${}^3$ (blue). The electric polarization is thus pointed from the red to the blue region.

Figure 10

Phonon spectra of the paraelectric phase $B_s$ of (a) $\alpha$-Se, (b) $\delta$-Se, (c) $\tau$-Se, and (d) $\eta$-Se. The vibration mode of the lowest branch of the phonon spectra which is highlighted by red line is represent by the red arrow in the right panel. $⨀$ and $⨂$ in (c) representing the out-of-plane and the into-the-plane displacements of Se atoms.

Figure 11

Structural changes causing a flip of the polarization direction in quasi-2D allotropes of Se. Atomic displacement, indicated by the red arrows, starts from the initial state $A$, crosses the paraelectric state $B_S$, and is completed in the final state $A^{\prime }$ of (a) $\alpha$-Se, (b) $\delta$-Se, and (c) $\tau$-Se. Corresponding energy changes $\mathrm{\Delta }E$ in (d) $\alpha$-Se, (e) $\delta$-Se, and (f) $\tau$-Se.

Figure 12

Representative snapshot of a 2 ps long MD simulation of $\alpha$-Se at $T=300$ K. The full video is presented in Ref. [37].

Figure 13

Representative snapshot of a 2 ps long MD simulation of $\eta$-Se at $T=300$ K. The full video is presented in the SM [37].

Figure 14

Representative snapshot of a 2 ps long MD simulation of $\tau$-Se at $T=200$ K. The full video is presented in the SM [37].