Fig. 3: Defect calculations for Bi₂Te₃, Bi₂Se₃, and Sb₂Te₃.

a–c Formation energy of native defects in Bi₂Te₃, Bi₂Se₃, and Sb₂Te₃ with varying chemical potentials of anions. d–f Formation energy of native defects in Bi₂Te₃, Bi₂Se₃, and Sb₂Te₃ at the crosspoint denoted in a–c, where the formation energy of the two antisite defects are equal. The insets show the schematics of the Te/Se₍₁₎–Bi/Sb–Te/Se₍₂₎–Bi/Sb–Te/Se₍₁₎ quintuple layer structure. Possible agglomeration processes of antisite defects in Bi₂Te₃, from g isolated antisite defects, to h antisite defect pairs, then to i a defect line, and finally agglomerated to form j a defect plane. k Formation energy of structures in g–j relative to that containing isolated antisite defects, which is calculated as \({E}_{{form}}=\left(E+\left(n-1\right){E}_{{perf}}-n{E}_{{iso}}\right)/n\), where E, E_iso, and E_perf are the energy of the structure containing n defect pairs, the structure containing two isolated antisite defects, and the structure without defects, respectively.