Mixed-valence state in the dilute-impurity regime of La-substituted SmB$_6$

Homogeneous mixed-valence (MV) behaviour is one of the most intriguing phenomena of $f$-electron systems. Despite extensive efforts, a fundamental aspect which remains unsettled is the determination of the limiting cases for which MV emerges. Here we address this question for SmB$_6$, a prototypical MV system characterized by two nearly-degenerate Sm$^{2+}$ and Sm$^{3+}$ configurations. By combining angle resolved photoemission spectroscopy (ARPES) and x-ray absorption spectroscopy (XAS), we track the evolution of the mean Sm valence, $v_{Sm}$, in the Sm$_x$La$_{1-x}$B$_6$ series. Upon substitution of Sm ions with trivalent La, we observe a linear decrease of valence fluctuations to an almost complete suppression at $x$$\,$=$\,$0.2, with $v_{Sm}$$\,$$\sim$$\,$2; surprisingly, by further reducing $x$, a re-entrant increase of $v_{Sm}$ develops, approaching the value of $v_{imp}$$\,$$\sim$$\,$2.35 in the dilute-impurity limit. Such observation departs from a monotonic evolution of $v_{Sm}$ across the whole series, as well as from the expectation of its convergence to an integer value for $x$$\,$$\rightarrow$$\,$0. Our ARPES and XAS results, complemented by a phenomenological model, demonstrate an unconventional evolution of the MV character in the Sm$_x$La$_{1-x}$B$_6$ series, paving the way to further theoretical and experimental considerations on the concept of MV itself, and its influence on the macroscopic properties of rare-earth compounds in the dilute-to-intermediate impurity regime.