Native point defects in mono and bilayer phosphorene

We study the stability and electronic properties of intrinsic point defects, vacancy and self-interstitial, in mono and bilayer phosphorene. We calculate the formation energies, quasiparticle defect states, and charge transition levels (CTLs) of these defects using ab initio density functional theory (DFT) and $GW$ approximation to the electron self-energy. Using the $\mathrm{DFT}+GW$ two paths formalism for studying interstitial in monolayer phosphorene, we show that with the inclusion of electrostatic corrections CTLs can be calculated reliably. Our calculations show that all the native point defects have low formation energies 0.9–1.6 eV in neutral state. Furthermore, we find that vacancy in phosphorene behaves as an acceptorlike defect which can explain the $p$-type conductivity in phosphorene. On the other hand, interstitial can show both acceptor- and donorlike behavior.

Figure 1

(a) shows the variation of electrostatic energy of bilayer phosphorene with different supercell sizes. $\alpha$ is the scaling factor. Blue and red curves represent the variation with scaling $\alpha \times \alpha \times \alpha$ and $\alpha \times \alpha \times 1.5\alpha$, respectively. In (b) the formation energies of negatively charged interstitial in bilayer phosphorene without and with correction for three different cell sizes are shown by blue circles and red diamonds and extrapolated to infinite limit. The formation energy of neutral interstitial are shown by green squares.

Figure 2

The figure shows calculation of CTLs following different paths. The straight green and red arrows represent the quasiparticle energies calculated at the equilibrium structures of the defect with charge state $q$ and $q^{\prime }$, respectively. The curved arrows account for relaxation energies.

Figure 3

(a) Top view and (b) side view of interstitial defect in monolayer phosphorene with defect wave function. The isosurface is plotted at the charge density value of $3.4\times {10}^{-3}$ e/Å${}^3$. The interstitial atom is shown in yellow color and the neighbor atoms are in red. (c) Spin polarized DOS of the interstitial in three charged states. VBM is set to zero and is shown in a black dotted line. Fermi level is marked in a maroon dashed line. Red and blue are for two spin states. (d) CTLs calculated within DFT and $\mathrm{DFT}+GW$ formalism. The dashed and solid lines represent the formation energies calculated with DFT and $\mathrm{DFT}+GW$, respectively, and red, black, and blue are used for $+1$, 0, and $-1$ charge states, respectively. Formation energy of neutral defect remains the same. The VBM and CBM are also marked in the figure. We have set the vacuum to zero in the plot.

Figure 4

(a) and (b) show the top and side view of MV-(55|66) in $7\times 5$ supercell and the defect wave function. (c) and (d) show MV-(5|9) and the wave function localized at defect from the top and side view, respectively. The isosurface of wave functions is plotted at $3.4\times {10}^{-3}$ e/Å${}^3$. The neighbor to the vacancy is shown in yellow and the other neighbor atoms are shown in red. (e) shows the variation of formation energy while the structure changes from MV-(55|66) to MV-(5|9). The reaction coordinate is the distance between two images in hyper-surface. (f) Spin-polarized DOS of vacancy in all three charge states. The red and blue lines denote two spin states. The VBM of the corresponding cells are set to zero and shown in black dotted lines. The maroon dashed lines are Fermi levels of the corresponding systems. (g) depicts the variation of formation energies with Fermi energy both within DFT and $\mathrm{DFT}+GW$. The black line is for vacancy in neutral state and negatively charged vacancy is represented by the blue solid ($\mathrm{DFT}+GW$) and dashed (DFT) lines.

Figure 5

(a) and (b) are the top and side view of interstitial defect with the localized wave function in bilayer phosphorene. (c) Spin-polarized DOS of interstitial in bilayer phosphorene in positive, neutral, and negative charged states. VBM are set to zero and the Fermi level are shown in maroon dashed line. Red and blue lines represent two spin states. (d) CTLs calculated within both DFT and $\mathrm{DFT}+GW$. Solid lines denote the formation energies within $\mathrm{DFT}+GW$ and the dashed lines are used for DFT. Red, black, and blue are for $+1$, 0, and $-1$ charge states. Vacuum is set to zero.

Figure 6

(a) and (b) show the top and side view of MV-(5|9) in bilayer phosphorene. The defect wave function is plotted at isosurface value of $3.4\times {10}^{-3}$ e/Å${}^3$. (c) and (d) show the band diagrams of the defect structure relaxed with PBE and HSE, respectively. Red and blue lines denote two spin states. (e) is the formation energy plot of vacancy in neutral and negative state.

Figure 7

The figure shows calculation of CTL starting from an intermediate structure. The brown arrow represents the quasiparticle energy calculated at ${\vec{\mathrm{R}}}_{\mathrm{I}}$ and the curved blue arrows are the relaxation energy.