Native point defects in few-layer phosphorene

Using hybrid density functional theory combined with a semiempirical van der Waals dispersion correction, we have investigated the structural and electronic properties of vacancies and self-interstitials in defective few-layer phosphorene. We find that both a vacancy and a self-interstitial defect are more stable in the outer layer than in the inner layer. The formation energy and transition energy of both a vacancy and a self-interstitial $P$ defect decrease with increasing film thickness, mainly due to the upward shift of the host valence band maximum in reference to the vacuum level. Consequently, both vacancies and self-interstitials could act as shallow acceptors, and this well explains the experimentally observed $p$-type conductivity in few-layer phosphorene. On the other hand, since these native point defects have moderate formation energies and are stable in negatively charged states, they could also serve as electron compensating centers in $n$-type few-layer phosphorene.

Figure 1

Top (a) and side (b) views of the unit cell of black phosphorus.

Figure 2

Six inequivalent interstitial configurations in bilayer phosphorene. The defect and its nearest neighbors are colored differently.

Figure 3

(a) Energy band structures (a) of monolayer phosphorene calculated using HSE06-$35\%$ and GW0 methods, and side views of charge density of (b) CBM and (c) VBM. The vacuum level is set to zero and the charge density isosurface levels are shown at 40% of their maximum values.

Figure 4

Energy band structures (a) of bilayer phosphorene calculated using HSE06-$30\%$ and GW0 methods, and side views of charge density of (b) CBM and (c) VBM. The vacuum level is set to zero and the charge density isosurface levels are shown at 40% of their maximum values.

Figure 5

Band alignments for few-layer phosphorene. The vacuum level is taken as zero energy reference.

Figure 6

Static dielectric tensors $\varepsilon$ as functions of the inverse of vacuum thickness for (a) monolayer, (b) bilayer and (c) quadrilayer phosphorene, respectively.

Figure 7

Total energies of monolayer phosphorene containing a vacancy, $V_{\mathrm{P}}^{\mathrm{out}}$, or a self-interstitial, ${\mathrm{P}}_i^{\mathrm{out}}$, in the charge states of (a) $1-$ or (b) $1+$, as a function of the inverse of vacuum thickness. The total energies of the slabs with a vacuum thickness of 20 Å are taken as zero.

Figure 8

(a) Formation energy of $V_{\mathrm{P}}$ and ${\mathrm{P}}_i$ as a function of electron chemical potential ${\mu }_e$ in monolayer phosphorene. (b) Local structures of $V_{\mathrm{P}}$ and ${\mathrm{P}}_i$. The defect and its nearest neighbors are colored differently.

Figure 9

Formation energies of $V_{\mathrm{P}}$ and ${\mathrm{P}}_i$ in (a) bilayer and (b) quadrilayer phosphorene as a function of electron chemical potential.

Figure 10

Local structure of negatively charged (a) $V_{\mathrm{P}}^{\mathrm{out}}$, (b) $V_{\mathrm{P}}^{\mathrm{in}}$, and (c) ${\mathrm{P}}_i^{\mathrm{out}}$ and ${\mathrm{P}}_i^{\mathrm{in}}$ in bilayer phosphorene. The point defects and their nearest neighbors are colored differently.

Figure 11

Transition levels of $V_{\mathrm{P}}$ and ${\mathrm{P}}_i$ in few-layer phosphorene, referenced to the vacuum level.

Figure 12

Formation energies of $V_{\mathrm{P}}$ and ${\mathrm{P}}_i$ as functions of electron chemical potential in monolayer phosphorene given by the HSE06-$\alpha$ method. The solid and dashed lines represent $\alpha =35\%$ and $\alpha =25\%$ results. The gray region represents the HSE06-$25\%$ band gap. (b) Transition energy referenced to the vacuum level.