Coexistence of multiple phases in the half-Heusler materials Co(Ti,Zr)(Sn,Sb)

We study coexisting phases of Co(Ti,Zr)(Sn,Sb) with the help of three coupled computational methods, which can capture the relevant physics at different length scales, namely, density functional theory, Monte Carlo simulations, and a mean field model. Our model material consists of three sublattices, one filled with Co, one with a Ti/Zr alloy, and one with a Sb/Sn alloy. The system forms phases in terms of spatial regions with specific Ti/Zr and Sb/Sn ratios. Depending on the overall stoichiometry and temperature, different spatial arrangements of coexisting phases are found. We found that the phase separation is controlled by two very different mechanisms. One is the mismatch of the lattices created by the smaller (larger) atomic radius of Ti (Zr); the second mechanism is given by the bond covalency whose strength is regulated by the change in the Sb/Sn valence electron number, which at the same time triggers magnetism. The simulations reveal up to three coexisting bulk phases, an optional interface phase, and the possible formation of magnetic domains. The occurrence of complex domain structures is of great importance for the reduction of the lattice thermal conductivity in this class of thermoelectric half-Heusler materials.