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Electronic structures of the tetradymites, Bi$_2$Te$_3$, Bi$_2$Te$_2$Se andBi$_2$Se$_3$, containing various dopants and vacancies, are studied using thefirst principles calculations methods. We focus on the possibility of formationof the resonant levels (RL), confirming the formation of the RL by Sn inBi$_2$Te$_3$, and predicting similar behavior of Sn in Bi$_2$Te$_2$Se andBi$_2$Se$_3$. Vacancies, which are likely present on the chalcogen atoms in thereal samples of Bi$_2$Te$_2$Se and Bi$_2$Se$_3$, are also studied and theircharged donor and resonant behavior is discussed. Doping of thevacancy-containing materials with regular acceptors, like Ca or Mg, is shown tocompensate the donor effect of vacancies, and $n-p$ crossover, while increasingthe dopant concentration, is observed. We verify, that RL on Sn is notdisturbed by the chalcogen vacancies in Bi$_2$Te$_2$Se and Bi$_2$Se$_3$, andfor the Sn-doped materials with Se or Te vacancies, double-doping, instead ofheavy doping with Sn, is suggested as an effective way of reaching the resonantlevel. This should help to avoid the smearing of the RL, which was a possiblereason for an earlier unsuccessful experimental observation of the influence ofthe RL on thermoelectric properties of Sn doped Bi$_2$Te$_2$Se. Finally weshow, that Al and Ga are possible new resonant impurities in the tetradymites,hoping that it will stimulate further experimental studies.

Resonant levels, vacancies and doping in Bi2Te3, Bi2Te2Se and Bi2Se3 tetradymites