Electronic structure of bulk and multilayer black phosphorus under strain: a minimal model study

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The tight-binding method is an important theoretical tool to investigate the physics of black phosphorus. Recently, a fifteen-parameter tight-binding model was proposed to precisely describe the band features. The large number of parameters is an obstacle for understanding the physics, and a minimal model for black phosphorus that grasps the main physics with satisfied precision is quite valuable. We use a tight banding model with only three parameters, that can reproduce the correct energy gap for monolayer, few-layer and bulk black phosphorus, to study the band structure black phosphorus under strain. The energy gap can be decreased by tensile strain normal to the phosphorus layers or in-plane compressive strain. At a certain critical strain, the gap of the phosphorus (either bulk, multilayer, or monolayer) is closed, and the dispersion is linear in the armchair direction and is parabolic in the other two principle directions. The simplicity of our model allows us to analytically investigate the energy gap and the critical strain as functions of the layer number of multilayer black phosphorus. When the strain exceeds the critical value, the valence and conduction bands evolve into two tents with their ridges touching each other.