Hydrostatic pressure induced three-dimensional Dirac semimetal in black phosphorus

We present the first-principles studies on the hydrostatic pressure effect of the electronic properties of black phosphorus. We show that the energy bands crossover around the critical pressure $P_c=1.23$ GPa; with increasing pressure, the band reversal occurs at the $Z$ point and evolves into 4 twofold-degenerate Dirac cones around the $Z$ point, suggesting that pressured black phosphorus is a 3D Dirac semimetal. With further increasing pressure the Dirac cones in the $\mathrm{\Gamma }-Z$ line move toward the $\mathrm{\Gamma }$ point and evolve into two hole-type Fermi pockets, and those in the $Z-M$ lines move toward the $M$ point and evolve into two tiny electron-type Fermi pockets, and a band above the $Z-M$ line sinks below ${\mathrm{E}}_F$ and contributes four electron-type pockets. A clear Lifshitz transition occurs at $P_c$ from semiconductor to 3D Dirac semimetal. Such a 3D Dirac semimetal is protected by the nonsymmorphic space symmetry of bulk black phosphorus. These suggest the bright perspective of black phosphorus for optoelectronic and electronic devices due to its easy modulation by pressure.

Figure 1

Crystal structure of bulk black phosphorus (a) and the first Brillouin zone and some high symmetric points of bulk black phosphorus (b).

Figure 2

Lattice constants of bulk black phosphorus under various pressures. The experimental data from Iwasaki et al. [24] and Cartz et al. [9, 25] are also shown as open triangles and squares for comparison.

Figure 3

Evolution of energy band structures of bulk black phosphorus with increasing pressures from 0 GPa (a), 0.5 GPa (b), 1.23 GPa, the critical pressure for band crossover (c), 2.0 GPa (d). The first three figures and the last one are the mBJ and PBE results, respectively.

Figure 4

Bands structure with orbital projection of bulk black phosphorus at 2.0 GPa (a) and schematic diagram of $3p_z$ orbital distribution contributed to the Dirac cones (b).

Figure 5

Sketched distribution of four Dirac cones in the first Brillouin zone of bulk black phosphorus under the pressure of 2.0 GPa. The two Dirac cones lie on the upper and lower surfaces of the Brillouin zone are identical.

Figure 6

Density of states (DOS) of bulk black phosphorus under the pressure of $P_c=1.23$ GPa (blue solid line) and 3.00 GPa (red dashed line).

Figure 7

Fermi surface (a) and its projection (b) of bulk black phosphorus under the pressure of 2.0 GPa. The projection on the plane (0, $k_y, k_z$) is shown. The two pockets in the $\mathrm{\Gamma }-Z$ line are hole type, and the others are electron type.

Figure 8

Calculated Fermi surfaces for bulk black phosphorus from 1.23 GPa (a), 1.5 GPa (b), 3.0 GPa (c), and 4.0 GPa (d).

Figure 9

Dependence of the average Fermi velocity of bulk black phosphorus under the pressures from 1.5 GPa up to 4.0 GPa by the PBE (a) and the mBJ potentials (b), respectively.

Figure 10

Calculated phonon dispersion of bulk black phosphorus under pressures of $P=0$, 1, 2, 3, and 4 GPa.