  Antiferromagnetism (AF) in AB'-stacked centrosymmetric bilayer (BL) CrI$_3$
breaks both spatial inversion ($P$) and time-reversal ($T$) symmetries but
maintains the combined $PT$ symmetry, thus inducing novel second-order
nonlinear optical (NLO) responses such as second-harmonic generation (SHG),
linear electric-optic effect (LEO) and bulk photovoltaic effect (BPVE). In this
work, we calculate AF-induced NLO responses of the BL CrI$_3$ based on the
density functional theory with the generalized gradient approximation (GGA)
plus onsite Coulomb correlation (U), i.e., the GGA+U method. Interestingly, we
find that the magnetic SHG, LEO and photocurrent in the AF BL CrI$_3$ are huge,
being comparable or even larger than that of the well-known nonmagnetic
noncentrosymmetric semiconductors. For example, the calculated SHG coefficients
are in the same order of magnitude as that of MoS$_2$ monolayer (ML), the most
promising 2D material for NLO devices. The calculated LEO coefficients are
almost three times larger than that of MoS$_2$ ML. The calculated NLO
photocurrent in the CrI$_3$ BL is among the largest values predicted so far for
the BPVE materials. On the other hand, unlike nonmagnetic semiconductors, the
NLO responses in the AF BL CrI$_3$ are nonreciprocal and also switchable by
rotating magnetization direction. Therefore, our interesting findings indicate
that the AF BL CrI$_3$ will not only provide a valuable platform for exploring
new physics of low-dimensional magnetism but also have promising applications
in magnetic NLO and LEO devices such as frequency conversion, electro-optical
switches, and light signal modulators as well as high energy conversion
efficiency photovoltaic solar cells.
