Minimal model for the magnetic phase diagram of CeTi$_{1-x}$Sc$_{x}$Ge, GdFe$_{1-x}$Co$_{x}$Si, and related materials

We present a theoretical analysis of the magnetic phase diagram of CeTi$_{1-x}$Sc$_{x}$Ge and GdFe$_{1-x}$Co$_{x}$Si as a function of the temperature and the Sc and Co concentration $x$, respectively. CeScGe and GdCoSi, as many other RTX (R=rare earth, T=transition metal, X=p-block element) compounds, present a tetragonal crystal structure where bilayers of R are separated by layers of T and X. While GdFeSi and CeTi$_{0.75}$Sc$_{0.25}$Ge are ferromagnetic, CeScGe and GdCoSi order antiferromagnetically with the R 4f magnetic moments on the same bilayer aligned ferromagnetically and magnetic moments in nearest neighbouring bilayers aligned antiferromagnetically. The antiferromagnetic transition temperature $T_N$ decreases with decreasing concentration $x$ in both compounds and for low enough values of $x$ the compounds show a ferromagnetic behavior. Based on these observations we construct a simplified model Hamiltonian that we solve numerically for the specific heat and the magnetization. We find a good qualitative agreement between the model and the experimental data. Our results show that the main magnetic effect of the Sc $to$ Ti and Co $to$ Fe substitution in these compounds is consistent with a change in the sign of the exchange coupling between magnetic moments in neighbouring bilayers. We expect a similar phenomenology for other magnetic RTX compounds with the same type of crystal structure.