Exciton binding energies and luminescence of phosphorene under pressure

The optical response of phosphorene can be gradually changed by application of moderate uniaxial compression, as the material undergoes the transition into an indirect gap semiconductor and eventually into a semimetal. Strain tunes not only the gap between the valence band and conduction band local extrema but also the effective masses, and in consequence, the exciton anisotropy and binding strength. In this article, we consider from a theoretical point of view how the exciton stability and the resulting luminescence energy evolves under uniaxial strain. We find that the exciton binding energy can be as large as 0.87 eV in vacuum for 5% transverse strain, placing it amongst the highest for two-dimensional materials. Further, the large shift of the luminescence peak and its linear dependence on strain suggest that it can be used to probe directly the strain state of single layers.

Figure 1

Elastic properties of strained phosphorene. (a) Ball-and-stick representation of unstrained phosphorene from the side view (top) and the top view (bottom). (b) Schematic representation of the strained phosphorene in a diamond anvil cell. The green arrows indicate the direction in which the force was applied. (c) Ab initio stress-strain curve as a function of the uniaxial strain ${\varepsilon }_z$. Polynomial regression is shown with the solid blue line and the dashed tangent line at ${\varepsilon }_z=0$.

Figure 2

Electronic properties of strained phosphorene. Electronic band structure of strained phosphorene with (a) ${\varepsilon }_z=0.0$, (b) ${\varepsilon }_z=5.0\%$, and (c) ${\varepsilon }_z=22.0\%$. (d) Direct ($E_1$) and indirect band gaps ($E_2$ and $E_3$) as a function of strain ${\varepsilon }_z$. The band structures were obtained with the PBE functional ($E_1$, $E_2$, and $E_3$) and the HSE06 hybrid functional ($E_1^{\mathrm{HSE}}$).

Figure 3

2D electric susceptibility of strained phosphorene. ${\zeta }_{xx}^{2D}$ (blue), ${\zeta }_{yy}^{2D}$ (red), and geometric mean ${\zeta }^{2D}$ (dashed black line) as a function of strain: (a) from 0.0% to 5.0% and (b) from 20.0% to 24.0%.

Figure 4

(a) Electrons' and holes' effective masses as a function of the strain in the armchair and zigzag directions and reduced effective masses in the armchair direction. (b) Schematic representation of the band crossing between the conduction bands $C_1$ (dark green) and $C_2$ (light green) at ${\varepsilon }_z=5\%$ (left) and ${\varepsilon }_z=8\%$ (right).

Figure 5

Optical properties of strained phosphorene. (a) Energy diagrams establishing a parallel between the quasiparticle gap in the one-electron formalism (left) and the exciton binding and photon energies in the many-body formalism (right). (b) and (c) Exciton binding energies (EBE) and luminescence energy ($E_L$) as functions of strain ${\varepsilon }_z$.