Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus

By performing ab initio calculations for one- to four-layer black phosphorus within the $GW$ approximation, we obtain a significant difference in the band gap ($\sim$1.5 eV), which is in line with recent experimental data. The results are analyzed in terms of the constructed four-band tight-binding model, which gives accurate descriptions of the mono- and bilayer band structure near the band gap, and reveal an important role of the interlayer hoppings, which are largely responsible for the obtained gap difference.

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Figure 1

(a) Top view of the crystal structure of monolayer BP, and side views of the occupied orbitals, corresponding to (b) bonding orbitals and (c) lone pairs. The orbitals are given in terms of the maximally localized Wannier functions [25, 26] obtained in this work.

Figure 2

Band structures of bulk BP calculated by using (a) the DFT-GGA and (b) a more accurate $GW$ approach along the high-symmetry points of the Brillouin zone. The corresponding path is shown by a green line in (c). Zero energies in (a) and (b) correspond to the Fermi level (GGA) and center of the band gap ($GW$).

Figure 3

Band structures for $n$-layer BP calculated within the $GW$ approach for $n=1–4$. Zero energy corresponds to the center of the band gap. Blue circles show the band splitting near the gap.

Figure 4

Band structures calculated by using the tight-binding parametrization (see text for details) in comparison with the original $GW$ bands for (a) monolayer and (b) bilayer BP. Hopping parameters of the TB model are sketched in (c). (d) and (e) show the dependence of the monolayer TB model on the inlayer ($t_2^{\parallel }$) and nearest-neighbor interlayer ($t_{\mathrm{NN}}^{\perp }$) hopping parameters.