Impurity scattering and Friedel oscillations in monolayer black phosphorus

We study the impurity scattering effect in black phosphorene (BP) in this work. For a single impurity, we calculate the impurity-induced local density of states (LDOS) in momentum space numerically based on a tight-binding Hamiltonian. In real space, we calculate the LDOS and Friedel oscillation analytically. The LDOS shows strong anisotropy in BP. Many impurities in BP are investigated using the $T$-matrix approximation when the density is low. Midgap states appear in the band gap with peaks in the DOS. The peaks of midgap states are dependent on the impurity potential. For finite positive potential, the impurity tends to bind negative charge carriers and vice versa. The infinite-impurity-potential problem is related to chiral symmetry in BP.

Figure 1

Lattice structure of monolayer black phosphorene. (a) Red (blue) atoms represent the upper (lower) layer; there are four atoms in a unit cell, which are labeled A, B, C, and D. The unit-cell size along the $x$ direction is $a_x=4.43\AA$, and that along the $y$ direction is $a_y=3.27\AA$. (b) The first Brillouin zone (BZ) of BP. (c) Energy dispersion of monolayer BP along the $x$ (solid) and $y$ (dashed) directions. (d) Momentum transferred by impurity scattering at the energy contour.

Figure 2

FT-LDOS in BP with (a) $E=0.3$ eV, (b) $E=0.7$ eV, (c) $E=1.5$ eV, and (d) $E=2.5$ eV. The red dashed rectangle denotes the first BZ. In the calculation of FT-LDOS, we set $u=2$ eV. The largest part in the LDOS is shown by the elliptic contours.

Figure 3

DOS on lattices (a) A, (b) B, (c) C, and (d) D in the case of $u\to \infty$. The DOS of midgap states on sites A and C is very low. DOS on sites B and D shows peaks in the band gap which indicate midgap states. The impurity density $n$ is proportional to height of peaks.

Figure 4

DOS on sites (a) A, (b) B, (c) C, and (d) D with $t_4=0$ and $u\to \infty$. With $t_4=0$, BP preserves chiral symmetry, due to which vacancies have bound states exactly at $E=0$ eV

Figure 5

DOS on sites (a) A, (b) B, (c) C, and (d) D at $u=10$ eV. For finite $u$, DOS on sites A and C have finite amplitudes. The positions of the peaks move towards valence bands compared to $u\to \infty$.

Figure 6

DOS on sites (a) A, (b) B, (c) C, and (d) D at $u=-10$ eV. For finite $u$, the DOS on sites A and C has a finite amplitude. The positions of the peaks move towards conduction bands compared to $u\to \infty$.

Figure 7

DOS on site A for impurity potential $u\to \infty$ with the form of Eq. (31), which has amplitudes on all four sites in BP. The DOS on sites B, C, and D are the same as that on site A, which we have not shown here. This kind of impurity does not bind states in the band gap.

Figure 8

Self-consistent $T$-matrix calculation of impurity of the form in Eq. (5). We set $u=10$ eV as in Fig. 5. Self-consistent calculation takes the finite lifetime of carriers into account, so the width of the peaks is larger than results using the zeroth Green's function.