Influences of strains on the formation of the quasi-Dirac cone and the Landau levels in black phosphorus

A tunable tight-binding (TB) Hamiltonian, which can be tuned continuously by applying the strains, is proposed based on the two tight-binding Hamiltonian models used to describle the pristine and compressed black phosphorus (BP). This tunable TB Hamiltonian should be of fundamental significance in two aspects: i) A general tunable TB Hamiltonian making it possible to study the BP more conveniently and deeply; ii) An extension of the study of the strain effect from the monolayer phosphorene to the multilayer phosphorene and even the bulk BP in the presence of both the in-plane and out-of-plane strains. Furthermore, based on the tunable TB Hamiltonian we can find the strain-induced quasi-Dirac cone, and clearly disclose how the strains affect the formation of the quasi-Dirac cone. Then we investigate the Landau levels (LLs) of BP under different magnetic fields, uncovering the linear, square-root, or complex dependence of the LLs on the LL number or the magnetic field. Moreover, the double-step and triple-step transitions of the LLs are found to be induced by the strains for the BP in the in-plane and out-of-plane magnetic fields. This research successfully incorporates the out-of-plane strain effect into the tunable TB Hamiltonian, enabling the comprehensive investigation of the strain-correlated effect in BP.