Band filling effect on polaron localization in La$_{1-x}($Ca$_y$Sr$_{1-y}$)$_{x}$MnO$_3$ manganites

We report on an a $mu$SR and $^{55}$Mn NMR investigation of the magnetic order parameter as a function of temperature in the optimally doped La$_{5/8}$(Ca$_y$Sr$_{1-y}$)$_{3/8}$MnO$_3$ and in the underdoped La$_{1-x}$Sr$_{x}$MnO$_3$ and La$_{1-x}$Ca$_{x}$MnO$_3$ metallic manganite families. The study is aimed at unraveling the effect of lattice distortions, implicitly controlled by the Ca-Sr isoelectronic substitution, from that of hole doping $x$ on the Curie temperature $T_c$ and the order of the magnetic transition. At optimal doping, the transitions are second order at all $y$ values, including the $y=1$ (La$_{5/8}$Ca$_{3/8}$MnO$_3$) end member. In contrast, they are first order in the underdoped samples, which show a finite (truncated) order parameter at the Curie point, including La$_{0.75}$Sr$_{0.25}$MnO$_3$ whose $T_c$ is much higher than that of La$_{5/8}$Ca$_{3/8}$MnO$_3$. The order parameter curves, on the other hand, exhibit a very minor dependence on $x$, if truncation is excepted. This suggests that the effective exchange interaction between Mn ions is essentially governed by local distortions, in agreement with the original double-exchange model, while truncation is primarily, if not entirely, an effect of under- or overdoping. A phase diagram, separating in the $x-y$ plane polaron-driven first order transitions from regular second order transitions governed by critical fluctuations, is proposed for the La$_{1-x}($Ca$_y$Sr$_{1-y}$)$_{x}$MnO$_3$ system.