Exciton in phosphorene: Strain, impurity, thickness, and heterostructure

Reduced electron screening in two dimensions plays a fundamental role in determining exciton properties, which dictates optoelectronic and photonic device performances. Considering the explicit electron-hole interaction within the $GW$ plus Bethe-Salpeter equation (BSE) formalism, we first study the excitonic properties of pristine phosphorene and investigate the effects of strain and impurity coverage. The calculations reveal strongly bound excitons in these systems with anisotropic spatial delocalization. Further, we present a simplified hydrogenic model with anisotropic exciton mass and effective electron screening as parameters, and the corresponding results are in excellent agreement with the present $GW$-BSE calculations. The simplified model is then used to investigate exciton renormalization in few-layer and heterostructure phosphorene. The changes in carrier effective mass along with increasing electron screening renormalize the exciton binding in these systems. We establish that the present model, in which the parameters are calculated within computationally less expensive first-principles calculations, can predict exciton properties with excellent accuracy for larger two-dimensional systems, for which the many-body $GW$-BSE calculations are impossible.

Figure 1

(a) The top and side views of single-layer phosphorene are shown, indicating the armchair and zigzag crystallographic directions. (b) The QP and optical gaps are calculated with varying vertical separation $L_z$ between the layers and extrapolated to $L_z\to \infty$. The results are in excellent agreement with the experimental results [17, 24, 30]. Absorption coefficient calculated without and with the electron-hole interaction for $L_z=30\AA$. $\mathrm{\Lambda }\left(\omega \right)$ shows strong absorption anisotropy between the light polarization along the (c) armchair and (d) zigzag directions. The vertical lines indicate the band edges corresponding to the first absorption peak. The energy difference between the QP and optical gaps indicates a strongly bound exciton.

Figure 2

(a) The effects of strain on the absorbance and dipole oscillator strength calculated including the electron-hole interaction within the $GW\text{−}\mathrm{BSE}$ formalism ($L_z=30\AA$). The vertical lines indicate the relative dipole oscillator strengths and are significantly affected by the applied uniaxial strain. (b) The corresponding exciton spectra for the pristine and uniaxially strained SLP. Strain-dependent dielectric screening influences the spectra. $E=0$ eV refers to the quasiparticle gap, and excitons generated from the transitions with the nonvanishing dipole oscillator strengths are shown. The relative oscillator strengths are shown in (a). Absorption coefficient calculated without and with the electron-hole interaction ($L_z=30\AA$) for the light polarization along (c) the armchair and (d) zigzag directions, shown for 1-ML O coverage. The absorption edges are calculated from the corresponding $E$ vs ${\left(E\mathrm{\Lambda }\right)}^{1/2}$ plot for this indirect gap semiconductor. The absorption anisotropy along the armchair and zigzag directions is greatly reduced due to monolayer impurity coverage. The insets show the side and top views of SLP with 1-ML O coverage.

Figure 3

(a) Calculated ${\chi }_{2\mathrm{D}}=(\varepsilon -1)L_{\mathrm{v}}/4\pi$ increases with layer thickness $N_{\ell }$, indicating an increase in electron screening. (b) The renormalization of band gap $E_g$ and exciton binding $E_{\mathrm{x}}$ in few-layer phosphorene shows a power-law dependence with the layer thickness, and both quantities converge very slowly to the corresponding bulk value. The variation in exciton binding is largely determined by the effective hole mass $m_h^{*}$ along the zigzag direction in addition to the change in effective electron screening.

Figure 4

(a) and (b) The top and side views of the $h$-BN/SLP/$h$-BN van der Waals encapsule. The structure is optimized using the PBE+D3-vdW functional with zero damping. The encapsulation induces a compressive (tensile) strain of 5.2% (1.3%) in the phosphorene layer along the armchair (zigzag) direction. (c) The valence band maxima and conduction band minima originate from the SLP and thus indicate a type-I band alignment. While the apparent band structure looks similar to the pristine SLP, the corresponding carrier effective masses are severely altered, which in turn modifies the exciton character.