Optical absorption properties of few-layer phosphorene

We investigate the optical absorption and transmission of few-layer phosphorene in the framework of ab initio density functional simulations and many-body perturbation theory at the level of random-phase approximation. In bilayer phosphorene, the optical transition of the valence band to the conduction band appears along the armchair direction at about 0.72 eV, while it is absent along the zigzag direction. This phenomenon is consistent with experimental observations. The angle-resolved optical absorption in few-layer phosphorene shows that it is transparent when illuminated by near-grazing incidence of light. Also, there is a general trend of an increase in the absorption by increasing the number of layers. Our results show that the bilayer phosphorene exhibits greater absorbance compared to that of bilayer graphene in the ultraviolet region. Moreover, the maximal peak in the calculated absorption of bilayer ${\mathrm{MoS}}_2$ is in the visible region, while bilayer graphene and phosphorene are transparent. In addition, the collective electronic excitations of few-layer phosphorene are explored. An optical mode (in-phase mode) that follows a low-energy $\sqrt{q}$ dependence for all structures and another which is a damped acoustic mode (out-of-phase mode) with linear dispersion for multilayer phosphorene are obtained. The anisotropy of the band structure of few-layer phosphorene along the armchair and zigzag directions is manifested in the collective plasmon excitations.

Figure 1

Three stacking structures of bilayer phosphorene. (a) Top view of AA, AB, and AC stacking. (b) Side view of AA, AB, and AC stacking. Our numerical results of the ground-state energy show that the AB-stacked structure is energetically the most preferred for bilayer phosphorene.

Figure 2

The calculated band structures for (a) monolayer, (b) bilayer, and (c) trilayer phosphorene structures along the $X-\mathrm{\Gamma }-Y$ direction. The band gap for monolayer phosphorene is 0.98 eV, and it decreases to 0.6 and 0.52 eV for bilayer and trilayer phosphorene, respectively. The nature of the band gap remains direct for trilayer to single-layer black phosphorous. Isofrequency contour plots of the energies around the conduction band (CB) minimum and the valence band (VB) maximum in the $k$ space are illustrated for the (d) monolayer, (e) bilayer, and (f) trilayer phosphorene for $E_{\mathrm{F}}=0.11$ eV and $E_{\mathrm{F}}=-0.11$ eV, with a step of 0.02 eV. The shape of the Fermi surfaces in the systems is almost an elliptic shape, especially at low charge density.

Figure 3

(a) The real and (b) imaginary parts of the dielectric tensor of bilayer phosphorene vs $\omega$ for ${\varepsilon }_{xx}$ (blue line) and ${\varepsilon }_{yy}$ (red line). Note that the interband transitions are included in the dielectric tensor. The imaginary part of the dielectric function illustrates the excitations in the system. Note that the peaks in the imaginary part of the dielectric functions correspond to the drops in the real part of the dielectric functions. The inset shows $\text{Im}{\varepsilon }_{yy}\left(\omega \right)$ around $\omega =0$ for different values of $\eta$. $\text{Im}{\varepsilon }_{yy}(\omega =0)$ tends to zero by decreasing the value of $\eta$.

Figure 4

Optical (a) absorption and (b) reflection of pristine bilayer phosphorene of the $s$-polarized (along the armchair direction) and $p$-polarized (along the zigzag direction) normal incidence ($\theta =0$). Noticeably, the optical transition of the valence band to the conduction band appears along the armchair direction at about 0.72 eV, while it is zero up to 2.5 eV along the zigzag direction.

Figure 5

(a) Angle-resolved absorption and (b) transmission of bilayer phosphorene with incident angle $\theta =n\mathrm{\Delta }\theta$ for $\mathrm{\Delta }\theta ={10}^{\circ }$ and $n=0,1,\cdots ,8$. The optical absorbance monotonically decreases as the incident angle of light increases; however, the transmission increases. Notice that phosphorene is transparent when it is illuminated by a near-grazing-incidence light.

Figure 6

Optical absorption spectra of few-layer phosphorene along the (a) armchair and (b) zigzag directions. Light absorptivity improves by increasing the number of the layers in few-layer phosphorene. The inset shows the low-energy part of optical absorption. It shows that the absorption spectrum is redshifted with increasing the number of layers along the armchair direction, while it changes slightly with the addition of phosphorene layers along the zigzag direction.

Figure 7

The real part of the optical conductivity of few-layer phosphorene (in units of $e^2/4\hslash$) along the armchair direction. Notice that since ${\mathrm{\Pi }}^0(q=0,\omega )=\mathrm{\Pi }(q=0,\omega )$, the absorption becomes proportional to the optical conductivity in phosphorene structures.

Figure 8

The optical absorption of pristine bilayer phosphorene, graphene, and molybdenum disulfide along the armchair direction for normal incidence ($\theta =0$). Here, $\eta =30$ meV. and the value of the peak of the optical absorption depends on the damping constant $\eta$ used in the calculation. It can be noticed that the bilayer phosphorene exhibits greater absorbance compared to the optical absorption of bilayer graphene in the ultraviolet region.

Figure 9

(a) The optical and (b) acoustic plasmon modes of few-layer phosphorene for $E_{\mathrm{F}}=0.09$ eV. The diamonds, circles, and stars refer to plasmon modes of trilayer, bilayer, and monolayer phosphorene, respectively. Owing to the anisotropic band structure of few-layer phosphorene along the zigzag and armchair directions, we have an anisotropic plasmon mode in two directions. Notice that ${\omega }_p$ (monolayer) $\ge {\omega }_p$ (bilayer) $\ge {\omega }_p$(trilayer) for optical plasmon modes.