Formation and function of vacancies in Si/Ge Clathrates: The importance of broken symmetries

One promising material class for improved thermoelectrics are the clathrates,~i.e.,~semiconducting host lattices encapsulating guest atom. Even in simple clathrates, such as, Si$_{46}$ and Ge$_{46}$, the introduction of guests can result in important but not yet understood effects: In Si hosts, the addition of K~(or~Ba) results in defect-free K$_8$Si$_{46}$~(Ba$_8$Si$_{46}$) phases. In spite of their structural and electronic similitude, Ge hosts behave fundamentally different upon filling, where, the spontaneously formed framework vacancies completely~(or partially) balance the electron donated by K~(or Ba) guests leading to K$_8$Ge$_{44}$~(or Ba$_8$Ge$_{43}$) clathrates. In this work, we use density-functional theory, carefully validating the exchange correlation functional, to compute the formation energies of vacancies and vacancy complexes in Si- and Ge-hosts as function of the filling of guests. By taking into account of the structural disorder, geometric relaxations, and vibrational entropies, we verify the experimentally found vacancy concentration and the thermodynamic stabilities of these compounds. We can trace back the contrasting behaviour of Si/Ge clathrates upon filling to a curious, charged vacancy induced break in symmetry that occurs in Si but not in Ge hosts.