Hyperbolic Plasmon in Black Phosphorus¶

Optical Conductivity of bilayer black phosphorus¶

Imaginary Part	Real Part

The imaginary part of the optical conductivity of bilayer black phosphorus in units of \(\sigma_0 = \pi e^2/2h\), showing a hyperbolic region (shaded) where \(\text{Im}(\sigma_{xx}) \cdot \text{Im}(\sigma_{yy}) < 0\) starting at \(\omega =2.8 eV\).
\(\text{Im}(\sigma_{xx})\) is negative throughout the spectrum (solid red line), \(\text{Im}(\sigma_{yy})\) goes from negative to positive around (solid blue line) \(\omega_h\)= 2.8 eV, which results in a hyperbolic region starting at that frequency.

Under compressive strain, the band gap of bilayer black phosphorus becomes smaller. As a result, the optical peaks shift to lower frequencies and \(\omega_h\) goes down.
For increasing bias in the bilayer case [Figs. 4(e) and 4(f)], the hyperbolic onset frequency \(\omega_h\) first goes up, because the bands corresponding to the excitation causing the second peak move away from one another. Then it goes down as the band gap closes, and a new peak appears in between the two existing peaks, because breaking the mirror symmetry in the z direction allows for new hybrid transitions.