Recently, the iridate double perovskite Sr$_2$YIrO$_6$ has attracted considerable attention due to the report of unexpected magnetism in this Ir$^{5+}$ (5d$^4$) material, in which according to the J$_{eff}$ model, a non-magnetic ground state is expected. However, in recent works on polycrystalline samples of the series Ba$_{2-x}$Sr$_x$YIrO$_6$ no indication of magnetic transitions have been found. We present a structural, magnetic and thermodynamic characterization of Sr$_2$YIrO$_6$ single crystals, with emphasis on the temperature and magnetic field dependence of the specific heat. Here, we demonstrate the clue role of single crystal X-ray diffraction on the structural characterization of the Sr$_2$YIrO$_6$ double perovskite crystals by reporting the detection of a $sqrt{2}a times sqrt{2}a times 1c$ supercell, where $a$, $b$ and $c$ are the unit cell dimensions of the reported monoclinic subcell. In agreement with the expected non-magnetic ground state of Ir$^{5+}$ (5d$^4$) in Sr$_2$YIrO$_6$, no magnetic transition is observed down to 430~mK. Moreover, our results suggest that the low temperature anomaly observed in the specific heat is not related to the onset of long-range magnetic order. Instead, it is identified as a Schottky anomaly caused by paramagnetic impurities present in the sample, of the order of $n sim 0.5(2)$ %. These impurities lead to non-negligible spin correlations, which nonetheless, are not associated with long-range magnetic ordering.
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