  The 3$d$-5$d$ based double perovskites offer an ideal playground to study the
interplay between electron correlation ($U$) and spin-orbit coupling (SOC)
effect, showing exotic physics. The Sr$_2$FeIrO$_6$ is an interesting member in
this family with ionic distribution of Fe$^{3+}$ (3$d^5$) and Ir$^{5+}$
(5$d^4$) where the later is believed to be nonmagnetic under the picture of
strong SOC. Here, we report detailed investigation of structural, magnetic and
electronic transport properties along with electronic structure calculations in
(Sr$_{1-x}$Ca$_x$)$_2$FeIrO$_6$ series with $x$ from 0 to 1. While the basic
interactions such as, $U$ and SOC are unlikely to be modified but a structural
modification is expected due to ionic size difference between Sr$^{2+}$ and
Ca$^{2+}$ which would influence other properties such as crystal field effect
and band widths. While a nonmonotonic changes in lattice parameters are
observed across the series, the spectroscopic investigations reveal that 3+/5+
charge state of Fe/Ir continue till end of the series. An analysis of magnetic
data suggests similar nonmonotonic evolution of magnetic parameters with
doping. Temperature dependent crystal structure as well as low temperature (5
K) magnetic structure have been determined from neutron powder diffraction
measurements. The whole series shows insulating behavior. The electronic
structure calculations show, SOC enhanced, a noncollinear antiferromagnetic and
Mott-type insulating state is the stable ground state for present series with a
substantial amount of orbital moment, but less than the spin magnetic moment,
at the Ir site and the magnetocrystalline anisotropy. The obtained results
imply that the Ca$^{2+}$ has large influence on the magnetic and transport
properties, further showing a large agreement between experimental results as
well as theoretical calculations.
