The magnetism in the saw-tooth lattice of Mn in the olivine chalcogenides, Mn$_2$SiS$_{4-x}$Se$_x$ ($x$ = 1$textendash$4) is studied in detail by analyzing their magnetization, specific heat and thermal conductivity properties and complemented with density functional theory calculations. The air-stable chalcogenides are antiferromagnets and show a linear trend in the transition temperature, $T_N$ as a function of Se-content ($x$) which shows a decrease from $T_N approx$ 86~K for {mss} to 66~K for {msse}. Additional new magnetic anomalies are revealed at low temperatures for all the compositions. Magnetization irreversibilities are also observed as a function of $x$. The specific heat and the magnetic entropy indicate the presence of short-range spin fluctuations in Mn$_2$SiS$_{4-x}$Se$_x$. A spin-flop antiferromagnetic phase transition in the presence of applied magnetic field is present in Mn$_2$SiS$_{4-x}$Se$_x$, where the critical field for the spin flop increases from $x$ = 0 towards 4 in a non-linear fashion. Density functional theory calculations show that an overall antiferromagnetic structure with ferromagnetic coupling of the spins in the $ab$-plane minimizes the total energy. The band structures calculated for mss and msse reveal features near the band edges similar to those reported for Fe-based olivines suggested as thermoelectrics; however the experimentally determined thermal transport data do not support superior thermoelectric features. The transition from long-range magnetic order in mss to short-range order and spin fluctuations in msse is explained using the variation of the Mn-Mn distances in the triangle units that constitutes the saw-tooth lattice upon progressive replacement of sulphur with selenium.
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