Dependence of the optical conductivity on the uniaxial and biaxial strains in black phosphorene

By using the Kubo formula, the optical conductivity of strained black phosphorene was studied. The anisotropic band dispersion gives rise to an orientation dependent optical conductivity. The energy gap can be tuned by the uniaxial and biaxial strains which can be observed from the interband optical conductivity polarized along the armchair ($x$) direction. The preferential conducting direction is along the $x$ direction. The dependence of the intraband optical conductivity along the zigzag ($y$) direction on the Fermi energy and strain exhibits increasing or decreasing monotonously. However, along the $x$ direction this dependence is complicated which originates from the carriers' inverse-direction movements obtained by two types of the nearest phosphorus atom interactions. The modification of the biaxial strain on the energy structure and optical-absorption property is more effective. The imaginary part of the total optical conductivity ($\mathrm{Im}\sigma$) can be negative around the threshold of the interband optical transition by modifying the chemical potential. Away from this frequency region, $\mathrm{Im}\sigma$ exhibits positive value. It can be used in the application of the surface plasmon propagations in multilayer dielectric structures.

Figure 1

The band structure of single layer BP in the absence of strain along $\mathrm{\Gamma }-X$ (a), $\mathrm{\Gamma }-Y$ (b), $\mathrm{\Gamma }-S$ (c) directions. $\mathrm{\Gamma }, X, Y$, and $S$ are $(0,0),(\pi /a,0),(0,\pi /b)$, and $(\pi /a,\pi /b)$ points in a $(k_x,k_y)$ Brillouin zone. Red dashed lines represent the energy in the midpoint between the valence band (black dotted line) and the conduction band (blue solid line) at $\mathrm{\Gamma }$ point.

Figure 2

The energy gap at $\mathrm{\Gamma }$ point under biaxial strains, with strains applied along $(x,y)$ (a), $(x,z)$ (b), and $(y,z)$ (c) directions. The color bar represents the energy gap value. The unit is eV. The black solid lines represent the zero energy gap.

Figure 3

The energy difference between the conduction band $\left(E_{+}\right)$ and the valence band $\left(E_{-}\right)$ in the absence of strain (a), under uniaxial tensile strain ${\epsilon }_x=25\%$ (b), ${\epsilon }_y=15\%$ (c), and compressive strain ${\epsilon }_z=-10\%$ (d). The black dashed lines represent the maximum continuous equal energy line in a Brillouin zone. The color bar represents the energy difference value. The unit is eV.

Figure 4

The real (a),(c) and imaginary (b),(d) parts of the intraband optical conductivities along the armchair ${\sigma }_{xx}^{\mathrm{intra}}$ and zigzag ${\sigma }_{yy}^{\mathrm{intra}}$ directions as a function of the optical energy at different chemical potential ${\mu }_c$ in the absence of strain. $E_F=E_c\left(\mathrm{\Gamma }\right)+{\mu }_c$. Conductivities are normalized with respect to ${\sigma }_0=e^2/\hslash$.

Figure 5

The coefficients of the intraband optical conductivities along the armchair (${\mathrm{Coe}}_{xx}^{\mathrm{intra}}$) and zigzag (${\mathrm{Coe}}_{yy}^{\mathrm{intra}}$) directions with strain applied along the $x$ (a),(b) and $y$ (c),(d) directions at different chemical potential ${\mu }_c$.

Figure 6

The real (a),(c),(e) and imaginary (b),(d),(f) parts of the interband optical conductivities ${\sigma }_{xx}^{\mathrm{inter}}$ vs the optical energy at different uniaxial tensile and compressive strains, with strain applied along the $x$ (a),(b), $y$ (c),(d), and $z$ (e),(f) directions.

Figure 7

Same as Fig. 6 except for the interband optical conductivity ${\sigma }_{yy}^{\mathrm{inter}}$.

Figure 8

The real (a),(c) and imaginary (b),(d) parts of the interband optical conductivities ${\sigma }_{xx}^{\mathrm{inter}}$ (a),(b) and ${\sigma }_{yy}^{\mathrm{inter}}$ (c),(d) as a function of biaxial tensile and compressive strains along ($y,z$) directions.

Figure 9

The real (a),(c) and imaginary (b),(d) parts of the total optical conductivities along the armchair direction ($x$) (a),(b) and along the zigzag direction ($y$) (c),(d) vs the incident optical energy.