  The recently discovered magnetism of two-dimensional (2D) van der Waals
crystals have attracted a lot of attention. Among these materials is CrI$_3$ -
a magnetic semiconductor exhibiting transitions between antiferromagnetic and
ferromagnetic orderings under the influence of an applied magnetic field. Here,
using first-principles methods based on density functional theory, we explore
spin-dependent transport in tunnel junctions formed of fcc Cu (111) electrodes
and a CrI$_3$ tunnel barrier. We find about 100% spin polarization of the
tunneling current for a ferromagnetically-ordered four-monolayer CrI$_3$ and
tunneling magnetoresistance of about 3,000% associated with a change of
magnetic ordering in CrI$_3$. This behavior is understood in terms of the spin
and wave-vector dependent evanescent states in CrI$_3$ which control the
tunneling conductance. We find a sizable charge transfer from Cu to CrI$_3$
which adds new features to the mechanism of spin-filtering in CrI$_3$-based
tunnel junctions. Our results elucidate the mechanisms of spin filtering in
CrI3 tunnel junctions and provide important insights for the design of
magnetoresistive devices based on 2D magnetic crystals.
