Nonmonotonic band gap evolution in bent phosphorene nanosheets

Nonmonotonic bending-induced changes of fundamental band gaps and quasiparticle energies are observed for realistic nanoscale phosphorene nanosheets. Calculations using stochastic many-body perturbation theory show that even slight curvature causes significant changes in the electronic properties. For small bending radii ($<4$ nm) the band gap changes from direct to indirect. The response of phosphorene to deformation is strongly anisotropic (different for zigzag vs armchair bending) due to an interplay of exchange and correlation effects. Overall, our results show that fundamental band gaps of phosphorene sheets can be manipulated by as much as 0.7 eV depending on the bending direction.

Figure 1

Phosphorene is characterized by puckered honeycomb lattice with ridges along the armchair ($x$) axis. Two nearest neighbor distances are denoted in the left panel ($d_1$ and $d_2$). Bending of PNS is illustrated on the right for bending radius $R=5$ nm; PNS bent along zigzag and armchair axes are on the top denoted as (a) and (b), respectively. Note that a plane separating the “inner” and “outer” phosphorus atoms is on the surface of a cylinder with radius $R$.

Figure 2

Orbital character for the band-edge states. Simplified overlaps of two nearest neighbors with interatomic distances $d_1$ and $d_2$ are depicted separately for clarity. Identical colors in the $p$-orbital lobes correspond to a bonding overlap; distinct colors depict an antibonding overlap. The rightmost column shows details of the orbital isosurfaces. The character of the HOMO state does not change with bending. ${\mathrm{LUMO}}_1$ is the lowest unoccupied state for radii $R\ge 4$ nm. ${\mathrm{LUMO}}_2$ (see text for reference) has either an antibonding $p_x$ character (for $R<4$ nm in zigzag bending) or mixed $p_x+p_z$ character (for $R<4$ nm in armchair bending).

Figure 3

Top: orbital isosurfaces for bending along the zigzag axis; the phase of the wave function is distinguished by its color. Bottom: QP energies in zigzag bending obtained by LDA (squares) and $\overline{\mathrm{\Delta }}G{W}_0$ (circles). The stochastic error of $\overline{\mathrm{\Delta }}G{W}_0$ for each point is smaller than the symbol size. The HOMO state is shown in red, ${\mathrm{LUMO}}_1$ (denoted ${\mathrm{L}}_1$) in blue, and ${\mathrm{LUMO}}_2$ (denoted ${\mathrm{L}}_2$) in black. For simplicity, we do not distinguish ${\mathrm{LUMO}}_1$ and ${\mathrm{LUMO}}_2$ in the DFT results, which always refer to the lowest unoccupied state. The fundamental band gap $E_g$ is shown for $R=2$ nm and the specific values are reproduced in Table 1. Note that ${\mathrm{LUMO}}_2$ is identified for $R=5$ as the fifth state above ${\mathrm{LUMO}}_1$ (for clarity we do not depict intermediate states). The solid lines are linear fits of the changes of the QP energies in Table 3; the dotted lines show the exchange contribution to the changes of the QP energies. These are described by Eq. (4) and the coefficients given in Table 2.

Figure 4

Top: orbital isosurfaces for bending along the armchair axis. Bottom: QP energies for PNS bent along the armchair direction obtained by LDA (squares) and $\overline{\mathrm{\Delta }}G{W}_0$ (circles). We only show the lowest energy LUMO in the DFT results.

Figure 5

Eigenvalue and fundamental band gaps ($E_g$) of phosphorene sheets bent along the armchair (black) and zigzag (red) directions. Bent structures for $R=2$ nm are illustrated in the insets: top for armchair bending and bottom for zigzag. Error bars show the stochastic errors. The lines are guides for the eye.