Statistical model for the formation of the Ge1-xSnx alloy

The electronic structures of most semiconductor alloys are smooth functions of their composition. Binary alloys of group IV semiconductors are usually easy to prepare at any concentration, but this is not the case for the Ge1-xSnx alloy. Homogeneous alloys as required for nano- and optoelectronics device applications have proved difficult to form for x above a temperature-dependent critical concentration, above which Sn exhibits the tendency to segregate in the metallic cubic β phase, spoiling the semiconducting properties. The underlying mechanism for this segregation and critical concentration was not known. Through previous accurate ab initio local defect calculations we estimated the scale of energies involved in the immediate environment around a large number of Sn defects in Ge, the relaxed configurations of the defects, and the pressure directly related to the elastic field caused by the defects. This detailed information allowed us to build a simple statistical model including the defects most relevant at low x, namely substitutional α-Sn and non-substitutional β-Sn (in which a single atom occupies the centre of a Ge divacancy). Our model enables us to determine at which concentration β defects, which exhibit a tendency to segregate, can be formed in thermal equilibrium. These results coincide remarkably well with experimental findings, concerning the critical concentration above which the homogeneous alloys cannot be formed at room temperature. Our model also predicts the observed fact that at lower temperature the critical concentration increases.