Supplementary Material: Stepping Stone Mechanism: Carrier-Free Long-Range Magnetism Mediated by Magnetized Cation States in Quintuple Layer

The long-range magnetism observed in group-V tellurides quintuple layers is the only working example of carrierfree dilute magnetic semiconductors(DMS),whereas the physical mechanism is unclear,except the speculation on the band topology enhanced van Vleck paramagnetism.Based on DFT calculations,we find a stable longrange ferromagnetic order in a single quintuple layer of Cr-doped Bi_(2)Te_(3) or Sb_(2)Te_(3),with the dopant separation more than 9?.This configuration is the global energy minimum among all configurations.Different from the conventional super exchange theory,the magnetism is facilitated by the lone pair derived anti-bonding states near the cations.Such anti-bonding states work as stepping stones merged in the electron sea and conduct magnetism.Further,spin orbit coupling induced band inversion is found to be insignificant in the magnetism.Therefore,our findings directly dismiss the common misbelief that band topology is the only factor that enhances the magnetism.We further demonstrate that removal of the lone pair derived states destroys the long-range magnetism.This novel mechanism sheds light on the fundamental understanding of long-range magnetism and may lead to discoveries of new classes of DMS.