(a) Band structure of phosphorene layers (4 L) with applied external electric field of (a) 0.3 V/Å and (b) 0.45 V/Å without including SOC. The red and blue lines show the top valence and bottom conduction states, respectively. (c) Inversion energy Δ_inv and (d) Z₂ invariant of 2-, 3-, and 4-layer phosphorene as a function of electric field. The negative values of Δ_inv show non-trivial topological phases. It can be seen that the phase transition takes place for 3- and 4-layer phosphorene at a critical field of 0.55 V/Å and 0.3 V/Å, respectively.^[80]