Phase Coexistence and Metal-Insulator Transition in Few-Layer Phosphorene: A Computational Study

Based on ab initio density functional calculations, we propose $\gamma \text{−}\mathrm{P}$ and $\delta \text{−}\mathrm{P}$ as two additional stable structural phases of layered phosphorus besides the layered $\alpha \text{−}\mathrm{P}$ (black) and $\beta \text{−}\mathrm{P}$ (blue) phosphorus allotropes. Monolayers of some of these allotropes have a wide band gap, whereas others, including $\gamma \text{−}\mathrm{P}$, show a metal-insulator transition caused by in-layer strain or changing the number of layers. An unforeseen benefit is the possibility to connect different structural phases at no energy cost. This becomes particularly valuable in assembling heterostructures with well-defined metallic and semiconducting regions in one contiguous layer.

Figure 1

Equilibrium structure of (a) an $\alpha \text{−}\mathrm{P}$ (black), (b) $\beta \text{−}\mathrm{P}$ (blue), (c) $\gamma \text{−}\mathrm{P}$, and (d) $\delta \text{−}\mathrm{P}$ monolayer in both top and side views. Atoms at the top and bottom of the nonplanar layers are distinguished by color and shading, and the Wigner-Seitz cells are shown by the shaded regions.

Figure 2

Electronic band structure and density of states (DOS) of (a) $\gamma \text{−}\mathrm{P}$ and (b) $\delta \text{−}\mathrm{P}$ monolayers. Results for bilayer and bulk systems are shown for comparison in the DOS plots only. Top and side views of the electron density ${\rho }_{vc}$ near the top of the valence and the bottom of the conduction bands of (c) $\gamma \text{−}\mathrm{P}$ and (d) $\delta \text{−}\mathrm{P}$. Only states in the energy range $E_F-0.4\mathrm{eV}<E<E_F+0.4\mathrm{eV}$ are considered, as indicated by the green shaded region in (a) and (b). ${\rho }_{vc}$ is represented at the isosurface value ${\rho }_{vc}=1.1\times 10^{-3}\mathrm{e}/{\AA }^3$ for $\gamma \text{−}\mathrm{P}$ and $\delta \text{−}\mathrm{P}$ and superposed with a ball-and-stick model of the structure.

Figure 3

(a) Dependence of the fundamental band gap $E_g$ on the slab thickness in $N$-layer slabs of $\alpha \text{−}\mathrm{P}$ (black), $\beta \text{−}\mathrm{P}$ (blue), $\gamma \text{−}\mathrm{P}$, and $\delta \text{−}\mathrm{P}$. Dependence of the fundamental band gap on in-layer strain is presented in (b) for $\gamma \text{−}\mathrm{P}$ and in (c) for $\delta \text{−}\mathrm{P}$. The strain direction is defined in Fig. 1. The shaded regions in (a) and (b) highlight conditions under which $\gamma \text{−}\mathrm{P}$ becomes metallic. Dashed vertical lines in (b) and (c) indicate a direct-to-indirect band-gap transition.

Figure 4

Energetically favorable in-layer connections between (a) $\alpha \text{−}\mathrm{P}$ and $\beta \text{−}\mathrm{P}$, (b) $\beta \text{−}\mathrm{P}$ and $\gamma \text{−}\mathrm{P}$, and (c) $\gamma \text{−}\mathrm{P}$ and $\delta \text{−}\mathrm{P}$, shown in perspective and side view. $l$ represents the edge length at the interface, and $\phi$ is the connection angle. The color scheme for the different allotropes is the same as in Fig. 1.