Significant effect of stacking on the electronic and optical properties of few-layer black phosphorus

The effect of the number of stacking layers and the type of stacking on the electronic and optical properties of bilayer and trilayer black phosphorus are investigated by using first-principles calculations within the framework of density functional theory. We find that inclusion of many-body effects (i.e., electron-electron and electron-hole interactions) modifies strongly both the electronic and optical properties of black phosphorus. While trilayer black phosphorus with a particular stacking type is found to be a metal by using semilocal functionals, it is predicted to have an electronic band gap of 0.82 eV when many-body effects are taken into account within the $G_0{W}_0$ scheme. Though different stacking types result in similar energetics, the size of the band gap and the optical response of bilayer and trilayer phosphorene are very sensitive to the number of layers and the stacking type. Regardless of the number of layers and the type of stacking, bilayer and trilayer black phosphorus are direct band gap semiconductors whose band gaps vary within a range of 0.3 eV. Stacking arrangements that are different from the ground state structure in both bilayer and trilayer black phosphorus exhibit significant modified valence bands along the zigzag direction and result in larger hole effective masses. The optical gap of bilayer (trilayer) black phosphorus varies by 0.4 (0.6) eV when changing the stacking type. The calculated binding energy of the bound exciton hardly changes with the type of stacking and is found to be 0.44 (0.30) eV for bilayer (trilayer) phosphorous.

Figure 1

Side and top views of AA, AB, and AC stacked structures for bilayer black phosphorus. $d_{\mathrm{int}}^1$ denotes the interlayer distance. The red rectangles in the bottom figures denote the unit cell. We also show the structural definition of the zigzag and armchair directions of black phosphorus.

Figure 2

Side and top views of ABA, AAB, and ACA stacked structures for trilayer black phosphorus. $d_{\mathrm{int}}^1$ and $d_{\mathrm{int}}^2$ denote the interlayer distances. The red rectangles in the bottom figures denote the unit cell.

Figure 3

Band structures for different stacking configurations of bilayer (top figures) and trilayer (bottom figures) phosphorene computed with HSE06 along the $Y\text{−}\mathrm{\Gamma }\text{−}X$ direction. The band gap values are also given for each structure. The Fermi level is set at 0 eV.

Figure 4

Direction-dependent hole and electron effective masses at the $\mathrm{\Gamma }$ point for the different number of layers and different types of stacking. $x$ $\left(y\right)$ axis denotes the zigzag (armchair) direction of black phosphorus. Each data point represents the end point of a vector whose amplitude corresponds to the effective mass in units of $m_0$ and the direction of this vector corresponds to the direction in $k$ space along which the mass is calculated.

Figure 5

Side and top views of decomposed charge densities corresponding to the VBM and CBM for both bilayer (top figures) and trilayer (bottom figures) black phosphorus at the $\mathrm{\Gamma }$ point.

Figure 6

$G_0{W}_0+\text{BSE}$ absorption spectra for bilayer (top figures) and trilayer (bottom figures) black phosphorus with different stacking types. Blue vertical dashed line mark the electronic band gap calculated at the level of $G_0{W}_0$. Green vertical lines represent the relative oscillator strengths for the different optical transitions.