Complex exchange mechanism driven ferromagnetism in half-metallic Heusler Co$_{2}$TiGe: Evidence from critical behavior

We have investigated the critical phenomenon associated with the magnetic phase transition in the half-metallic full-Heusler Co$_2$TiGe. The compound undergoes a continuous ferromagnetic to paramagnetic phase transition at the Curie temperature $T_{C}$=371.5 K. The analysis of magnetization isotherms in the vicinity of $T_{c}$, following modified Arrott plot method, Kouvel-Fisher technique, and critical isotherm plot, yields the asymptotic critical exponents $\beta$=0.495, $\gamma$=1.324, and $\delta$=3.67. The self-consistency and reliability of the obtained exponents are further verified by the Widom scaling relation and scaling equation of states. The mean-field-like value of the critical exponent $\beta$ suggests long-range nature of the exchange interactions, whereas the values of the critical exponents $\gamma$ and $\delta$, imply sizeable critical spin fluctuations. The half-metallic itinerant character of Co$_{2}$TiGe in the presence of magnetic inhomogeneity may result in such a strong deviation from the three-dimensional Heisenberg values ($\beta$=0.369, $\gamma$=1.38 and $\delta$=4.8) of the critical exponents towards the mean field values ($\beta$=0.5, $\gamma$=1 and $\delta$=3). The results suggest complex nature of exchange couplings that stabilize the long-range ferromagnetic ordering in the system and are consistent with the earlier theoretical studies on the exchange mechanism in Co$_2$TiGe.