  Double-perovskite oxides that contain both 3d and 5d transition metal
elements have attracted growing interest as they provide a model system to
study the interplay of strong electron interaction and large spin-orbit
coupling (SOC). Here, we report on experimental and theoretical studies of the
magnetic and electronic properties of double-perovskites
(La$_{1-x}$Sr$_x$)$_2$CuIrO$_6$ ($x$ = 0.0, 0.1, 0.2, and 0.3). The undoped
La$_2$CuIrO$_6$ undergoes a magnetic phase transition from paramagnetism to
antiferromagnetism at T$_N$ $\sim$ 74 K and exhibits a weak ferromagnetic
behavior below $T_C$ $\sim$ 52 K. Two-dimensional magnetism that was observed
in many other Cu-based double-perovskites is absent in our samples, which may
be due to the existence of weak Cu-Ir exchange interaction. First-principle
density-functional theory (DFT) calculations show canted antiferromagnetic
(AFM) order in both Cu$^{2+}$ and Ir$^{4+}$ sublattices, which gives rise to
weak ferromagnetism. Electronic structure calculations suggest that
La$_2$CuIrO$_6$ is an SOC-driven Mott insulator with an energy gap of $\sim$
0.3 eV. Sr-doping decreases the magnetic ordering temperatures ($T_N$ and
$T_C$) and suppresses the electrical resistivity. The high temperatures
resistivity can be fitted using a variable-range-hopping model, consistent with
the existence of disorders in these double-pervoskite compounds.
