Transport and optical properties of single- and bilayer black phosphorus with defects

We study the electronic and optical properties of single- and bilayer black phosphorus with short- and long-range defects by using the tight-binding propagation method. Both types of defect states are localized and induce a strong scattering of conduction states, reducing significantly the charge carrier mobility. In contrast to properties of pristine samples, the anisotropy of defect-induced optical excitations is suppressed due to the isotropic nature of the defects. We also investigate the Landau level spectrum and magneto-optical conductivity and find that the discrete Landau levels are sublinearly dependent on the magnetic field and energy level index, even at low defect concentrations.

Figure 1

(top left) Schematic representation of the atomic structure and hopping parameters of the TB model of BP. (top right) Top view of the atomic structure; the red dot represents a point defect, and the blue dot represents a long-range defect at the center of the projected honeycomb lattice on the surface. (bottom left) Three-dimensional contour plot of the lowest valence and conduction bands. (bottom right) Lowest valence and conduction bands along armchair (solid lines) and zigzag (dashed lines) directions.

Figure 2

(top) Electron and hole velocities in single-layer BP as a function of wave vector along armchair (red solid line) and zigzag (black dashed line) directions. (bottom) Effective mass along armchair ($Y$) and zigzag ($X$) directions. The wave vector $k$ is in units of $1/a$, where $a\approx 2.216$ Å is the atomic distance between two nearest neighbors.

Figure 3

Density of states of single-layer (left columns) and bilayer (right columns) with point defects (upper rows) or electron-hole puddles (bottom rows). The value $n_x\left(n_c\right)=0.1\%$ corresponds to defect a concentration of $2.98\left(1.49\right)\times {10}^{12}$ per ${\mathrm{cm}}^2$.

Figure 4

Transport properties of single- and bilayer with defects. (top) dc conductivity as a function of doping energy. (bottom) carrier mobility as a function of carrier density.

Figure 5

Optical conductivity as (a)–(d) a function of energy and optical absorption and (e)–(l) as a function of polarized angle for suspended single- and bilayer BP with defects. The energies of the photon for single-layer are (e), (f) $1.4$ eV and (i), (j) $1.78$ eV, and for bilayer are (j), (k) $0.98$ eV and (h), (l) $1.36$ eV. The black lines are the results for pristine samples, and the colored lines are the results for disordered samples with different concentrations of defects. Throughout this work, we fix the temperature to $T=300$ K and the chemical potential to ${\mu }_F=0$ for the optical conductivity and normalize the conductivity to ${\sigma }_0=\pi e^2/2h$, the universal optical conductivity of single-layer graphene in the visible-light region.

Figure 6

The energy dependence of the anisotropic effective masses of single-layer BP in the TB model.

Figure 7

Landau level spectrum of single-layer and bilayer BP in high magnetic field. The red dashed lines are the lowest twenty LLs calculated from Eq. (10).

Figure 8

DOS and magneto-optical spectrum of disordered single-layer and bilayer BP with perpendicular magnetic field $B=50\text{T}$.