Effect of van der Waals interactions on the structural and elastic properties of black phosphorus

The structural and elastic properties of orthorhombic black phosphorus have been investigated using first-principles calculations based on density functional theory. The structural parameters have been calculated using the local density approximation (LDA), the generalized gradient approximation (GGA), and with several dispersion corrections to include van der Waals interactions. It is found that the dispersion corrections improve the lattice parameters over LDA and GGA in comparison with experimental results. The calculations reproduce well the experimental trends under pressure and show that van der Waals interactions are most important for the crystallographic $b$ axis in the sense that they have the largest effect on the bonding between the phosphorus layers. The elastic constants are calculated and are found to be in good agreement with experimental values. The calculated $C_{22}$ elastic constant is significantly larger than the $C_{11}$ and $C_{33}$ parameters, implying that black phosphorus is stiffer against strain along the $a$ axis than along the $b$ and $c$ axes. From the calculated elastic constants, the mechanical properties, such as bulk modulus, shear modulus, Young's modulus, and Poisson's ratio are obtained. The calculated Raman active optical phonon frequencies and their pressure variations are in excellent agreement with available experimental results.

Figure 1

Crystal structure of black phosphorus. The nearest and next-nearest interlayer atomic distances are shown ($d_1$ and $d_2$).

Figure 2

(a)–(c) Pressure dependence of lattice parameters $a,b$, and $c$ up to 5 GPa within LDA, GGA, and GGA-D, compared to experimental values (Ref. 55). Note the different scales on the vertical axes, in particular, the small scale in (a).

Figure 3

Comparison of the pressure dependence of the crystal volume as calculated within LDA, GGA, and GGA-D, compared to experimental volumes (Ref. 55).

Figure 4

Internal coordinates, (a) $u$ and (b) $v$ of black phosphorus up to 5 GPa as calculated within LDA, GGA, and GGA-D, compared to experimental values (Ref. 55).

Figure 5

Calculated (a) nearest-neighbor ($d_1$) and (b) next-nearest neighbor $(d_2$) interlayer atomic separations (cf. Fig. 1) of black phosphorus up to 5 GPa with and without dispersion correction and compared with experiments (Ref. 55).

Figure 6

Snapshots of Raman active modes for primitive black phosphorus. The listed frequencies are as calculated within GGA-Grimme at $P=0$.

Figure 7

Calculated (PBE-Grimme) Raman frequencies under pressure compared to experiments (Ref. 26).